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+{"text":"\\section{Introduction}\nIn sexual reproduction, as opposed to asexual reproduction, the genomes of the \ntwo parents are mixed, and within the diploid genome of each parent happens\ncrossover. This way of reproduction has advantages as well as disadvantages \ncompared with asexual cloning of haploid genomes. An advantage is that bad \nrecessive mutations do not affect the health if they are present in only one of \nthe two haplotypes (= sets of genetic information). A disadvantage\nis the reduced number of births if only the females produce offspring while the \nmales consume as much food and space as the females. Moreover, crossover of \ntwo different genomes may produce a mixture which is fitter than each the two \nparents but also one which is less fit, as seen these \ndays in the DaimlerChrysler car company (outbreeding depression). For small \npopulations, the probability is higher that the two parents have the same bad \nrecessive mutation which therefore diminishes the health of the individual \n(inbreeding depression). \n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding1.ps}\n\\end{center}\n\\caption{Average over 100 simulations with $G=10$ groups each, $\\Delta = 100$.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding2.ps}\n\\end{center}\n\\caption{Average over 10 simulations with a large (top) and 100 with a small\n(bottom) population, versus number $G$ of groups; $r=1, \\; \\Delta = 1000$ and \n100, respectively. For the larger population $G$ is divided by 10 so that \ndata for the same number of individuals per group have the same horizontal\ncoordinate. We see nice scaling.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.30]{inbreeding3a.ps}\n\\includegraphics[angle=-90,scale=0.30]{inbreeding3b.ps}\n\\end{center}\n\\caption{Time dependence of outbreeding advantage (part a) and outbreeding\ndepression (part b).\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding4.ps}\n\\end{center}\n\\caption{Average over 100 simulations with a small population, high minimum\nage of reproduction, and $G=1$ and 50. For $G=1$ there is always complete \nmixing. Note the double-logarithmic scales, also in Fig.5 and 6.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding5.ps}\n\\end{center}\n\\caption{Average over 100 simulations with a small population, high minimum\nage of reproduction and various\nlengths $L$ of the bit-strings, using a birth rate $B = 128\/L$. \n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding6.ps}\n\\end{center}\n\\caption{Dependence on population size for $N_{\\max} = 10^3 \\dots 10^7$, \naveraged over 1000 to one sample. $L=64, \\; B = 2$.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding11.ps}\n\\end{center}\n\\caption{\nRelation between normalized population size and the crossover\nrate. The population size was divided by the population evolving with crossover\nrate $r = 1$.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding12.ps}\n\\end{center}\n\\caption{\nDistribution of defective genes in the genomes of populations\nevolving under different crossover frequencies.\n}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\begin{center}\n\\includegraphics[angle=-90,scale=0.5]{inbreeding13.ps}\n\\end{center}\n\\caption{\nRelation between crossover frequency and average frequency of\ndefective genes in the sections of genomes expressed before the\nreproduction age $R$. Note, fractions equal 0 mean that the populations with \nthe given crossover frequency died out.\n}\n\\end{figure}\n\n\\section{Standard Model}\n\nWe try to simulate these effects in the standard sexual Penna ageing model,\ndeviating from published programs \\cite{books} as follows: The Verhulst factor, \na death probability $N\/N_{\\max}$ at population $N$ with carrying capacity\n$N_{\\max} = 10^3, \\ 10^4, \\,10^5$,\ndue to limited food and space, was applied to the births only and not to adults;\nthe initial age distribution was taken as random when needed;\nthe birth rate was reduced from 4 to 1, the lethal threshold of active \nmutations from 3 to 1 (that means a single active mutation kills), and mostly \nonly $10^4$ instead of $2 \\times 10^4$ time steps were made. (One time step or\niteration is one Monte Carlo step for each individual). Furthermore\nthe whole population was for most of the simulated time separated into $G$ \ndifferent groups such that females look for male partners only within their\nown group, with a separate Verhulst factor applying to each group. For the\nlast $\\Delta \\ll 10^4$ time steps this separation into groups was dissolved: \nThen females could select any male, and only one overall Verhulst factor \napplied to the whole population. Finally, the crossover\nprocess within each parent before each birth was not made always but only\nwith a crossover probability $r$. \n\nIf there would be no inbreeding depression then during the first longer part\nof the simulation the total number $N_1$ of individuals would be independent of \nthe number $G$ of groups into which it is divided. And if then there are no \nadvantages or disadvantages of outbreeding, the population $N_2$ during the \nsecond shorter part, $10^4 - \\Delta < t < 10^4$, would be the same as the\npreceding population $N_1$ during the last section, $10^4 - 2 \\Delta < t < \n10^4 - \\Delta$, of the longer first part. We will present data showing that \nthis is not the case. Similar simulations for $G=2$ groups was published\nlong ago \\cite{LSC}. \n \nA difficulty in such simulations is the Eve effect: After a time \nproportional to the population size, everybody has the same female (Eve) and \nthe same male (Adam) as ancestor, with all other offspring having gotten less \nfit genomes due to random mutations and thus having died out. If we would\ndivide the whole population into many groups without further changes, the Eve \neffect would let all groups but one die out and thus destroy the separation.\nTherefore for the first long period of separation we used separate Verhulst \nfactors for each group, stabilizing its population, while for the second shorter\npart of mixing we used mostly $\\Delta = N_{\\max}\/100$. \n\nFigure 1 shows the dependence on the crossover probability for the \npopulations $N_1$ before and $N_2$ after mixing. We see that the mixing always\nincreases the population, that means one has no outbreeding depression but\nan outbreeding advantage. Figure 2 confirms this advantage but also \nshows the inbreeding depression: The larger the number $G$ of groups (and \nthus the smaller the group size) is, the smaller are the two populations\n$N_1$ and $N_2$. (The difference between $N_1$ and $N_2$ fluctuates less than\nthese numbers themselves since $N_2$ is strongly correlated with $N_1$.)\nAlso, for the larger population in Fig.2, the number of groups can be larger\nbefore the population becomes extinct. Figure 3 shows the time dependence \nof the outbreeding effect with mixing between groups allowed after 9900 (part a)\nand 9000 (part b) time steps. Figure 3a shows summed populations from \n100 simulations with a small population ($G = 10$) and 10 simulations of a large\npopulation ($G = 100$), versus time after mixing started; $r = 1$ in both cases.\nFor much larger populations of 5 million and still $G = 10$, no such effect of \nmixing is seen. Part b shows for the high\nreproduction age $R$ of the following figures one example of the outbreeding\ndepression (bottleneck \\cite{malarz}) followed by a recovery with oscillations \nof period $R$ after mixing was allowed from time 9001 on; $N_{\\max}=10^5$..\n\n \nWe also checked for the influence of $r$ in the case when the minimum age of \nreproduction $R$ is 5\/8 of the length $L$ of the bit-strings, i.e. larger than \nthe value of 8 used before, and when $L$ is different from the 32 used in Figs.\n1 to 3. In these simulations we also assumed all mutations to be\nrecessive, in contrast to the 6 out of 32 dominant bit positions for Figs.\n1 to 3. Figure 4 shows for $L = 32$ and a birth rate $B = 4$ \na minimum of the population at intermediate $r$ for one group, and for 50 \ngroups a monotonic behaviour but with outbreeding depression at small $r$ and\noutbreeding advantage at big $r$.  This population minimum is seen for\n$L = 64$ and 32 but not for 16, Fig.5. Figure 6 shows the dependence on \npopulation size. (Our data before and after mixing are average over $\\Delta = \n100$ or 1000 iterations. When outbreeding depression occurs it may happen that \nlater the population recovers: Fig.3b.)\n\n\\section{Interpretation}\n\nTo study the inbreeding and outbreeding depressions in detail we have\nanalyzed the results of simulations of single populations of different size\nunder different regime of intragenomic recombinations (crossover rate $r$).\nParameters for these simulations have been slightly changed to get clearer \nresults: $L= 128,\\; R = 80,\\; N_{\\max} = 1000$ to 20 000, crossover probability \n$r =  0$ to 1, $B = 1$, time of simulations = $5 \\times 10^5$ iterations.\nIn Fig. 7 the relation between the size of population and the crossover\nprobability for three different environment capacities are shown.\n\nPopulations in the smallest environment ($N_{\\max}=1000$) survive with $r = 0$ \nbut their sizes decrease with increasing $r$ and are extinct for $r$ set between\n0.12 and 0.4. Under larger crossover rates populations survive and their\nsizes are larger than those obtained for $r = 0$ (see plots in Fig. 7 where\nsizes of populations were normalized by the size of population under \n$r=1$). Larger populations ($N_{\\max}= 10 000$) are extinct in a very\nnarrow range of crossover rates close to 0.12, and populations with $N_{\\max}= \n20000$ become extinct at slightly lower crossover rates. Nevertheless, all\npopulations have larger sizes when the crossover rate is of the order of 1\nper gamete production (the highest tested).\n\nThis nonlinear relation between size of population and crossover rate could\nbe explained on the basis of the genetic structure of individual genomes in\nthe simulated populations. In Fig. 8 we have shown the frequency of\ndefective genes in the genetic pool of populations for $N_{\\max} = 10000$ under\ncrossover rates 0, 0.1 and 1. The frequency of defective genes expressed\nbefore minimum reproduction age ($R = 80$) in populations without crossover is\n0.5. Since $T = 1$, if the distribution of defects would be random the\nprobability of any individual to survive until the reproduction age $R$ would be\n$0.75^R$ (negligibly small for large $R > 30$). Thus, to survive, individuals \nhave to complete their\ngenomes of two complementing bit-strings (haplotypes). For more efficient\nreproduction the number of different haplotypes should be restricted and in\nfact there are only two different complementing haplotypes in the whole\npopulation as it was shown in \\cite{waga}. In such populations, the\nprobability of forming the offspring surviving until the reproduction age is\n0.5. Note, that recombination at any point inside the part of the genome\nexpressed before reproduction age $R$ produces a gamete which is not\ncomplementary to any other gamete produced without recombination or with\nrecombination in an other point. Thus, crossovers in such populations are\ndeleterious for the offspring.\nOn the other extreme, with crossover probability = 1, populations are under\npurifying selection. The fraction of defective genes in the population is\nkept low (about 0.1, compared with 0.5 without recombination), to enable\nthe surviving of the offspring until their reproduction period. The critical\ncrossover frequency close to 0.12 is connected with a sharp transition from\nthese two strategies of genomic evolution: complementarity and purifying\nselection. In Fig. 9 the frequency of defective genes expressed before the\nreproduction age is plotted. For lower crossover rates the fractions of\ndefective genes are kept at the level 0.5, for higher crossover rates they\nare close to 0.1. Close to the critical frequency of crossover, defective\ngenes located at both ends of the region of genomes expressed before the\nreproduction age are forced to obey the purifying selection which eliminates\nsome defects (Fig. 8).\n\nIn the case of small populations, the probability of meeting two closely related\npartners (high inbreeding coefficient) is high and as a consequence, there\nis higher probability of meeting two defective alleles in the same locus in\nzygote which determines phenotypic defect and eliminates the offspring from\nthe population. In such condition the strategy of completing the genome of\ntwo complementing haplotypes is more effective. Nevertheless, this strategy\nis not the best if effective populations are very large, with low inbreeding\ncoefficient, when the probability of meeting two identical haplotypes is\nnegligible. Thus, comparing very large populations with very small ones we\ncan observe the inbreeding depression. On the other hand, this strategy in\nsmall populations leads to the emerging of very limited number of different\nhaplotypes in the populations (in extreme only two). These haplotypes are\ncharacterized by a specific sequence of defective alleles. Independent\nsimulations generate haplotypes with different sequence of defective\nalleles. Mixing two or more populations evolving independently decreases the\nprobability of meeting in one zygote two complementing haplotypes, this\ndifference results in outbreeding depression (seen in Figs.3b and 4).\n\n\\section{Conclusion}\nWe varied the parameters of the sexual Penna ageing model, in particular by \nseparating the population into reproductively isolated groups and\/or having\nlonger bit-strings and a high minimum age of reproduction. We could observe\nand interpret inbreeding depression, outbreeding depression, and outbreeding \nadvantage, through the counterplay of purifying selection and of haplotype\ncomplementarity. Purifying selection tries to have as few mutations in \nthe bit-strings, like haplotype 00000000 for $L=8$, while haplotypes 01100101 \nand 10011010 are complementary. In both cases, deleterious effects from\nmutations are minimised.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Motivation}\n\\label{subsec:Motivation}\nEven though infrequent, peak load is a cause for increased capital and operating expenses for power networks. The reason for this is twofold; the higher need for grid reinforcements and the use of more expensive fossil fuel-based generators to satisfy the peak loads (for a short duration). \nThereby to manage the extra costs due to peak demand, network operators include a peak demand tariff (commensurate to the peak load) in the electricity bills of commercial customers. This motivates large-scale commercial customers such as universities and manufacturers to manage their demand better and invest in assets such as solar photovoltaic (PV) panels and stationary batteries that have the potential to shift their demand and reduce their electricity bills. A secondary effect of this is the reduction in CO\\(_2\\) emissions because of the lesser usage of fossil fuel generators, which has the potential to combat climate change and aid the process of decarbonization of our power networks.\n\nHowever, optimal scheduling of (schedulable) loads and batteries (to minimize electricity costs) requires predictions of inflexible load (baseload), solar PV generation and the spot price of electricity. The first two variables majorly depend on weather conditions. Also, increasing renewable generation in the generation mix has led to increasing price volatility in the Australian national electricity market (NEM) which also creates a correlation between electricity prices and the weather \\cite{downey2022untangling}. This makes designing such algorithms challenging as most commercial activities are planned over the mid to long term; hence require reliable predictions. Additionally, the scheduling problems are generally mixed-integer linear programs (MILPs) which are NP-hard and may be intractable depending on the formulation and problem size. \n\nSurrounding this premise, the IEEE Computational Intelligence Society (IEEE-CIS) partnered with Monash University (Victoria, Australia) to conduct a competition seeking technical solutions to manage Monash's micro-grid containing rooftop solar panels and stationary batteries~\\cite{IEEECIS}. The main challenge was to develop an optimal scheduling algorithm for Monash's lecture theatres and operation of batteries to minimize their electricity bill, considering their baseload, solar generation and NEM electricity spot prices. To this end, the contestants were provided with actual time series data of the building loads without any lecture program (baseload) and solar generation from the micro-grid. So the contestants were expected to predict the baseload and solar generation by taking into account real-world weather data (also provided) for one month in the future. Following this, the contestants had to use actual electricity spot prices for the same duration along with these predictions for the optimal scheduling algorithm. \n\n\n\n\\subsection{Related work} \n\\label{subsec:Related Work}\nWe developed separate algorithms for the solar and baseload forecasts based on practical insights and the given data. Many approaches are available in the literature for both data types, a relatively mature and crowded research field. Therefore, the methods we have developed for this problem borrow themes from this mature literature. \n\nFor solar predictions, the basic theme used is similar to the ``clear sky'' models for forecasting solar irradiance and thereby estimating PV generation \\cite{Palani2017}. The general idea here is to create a baseline model for PV generation, assuming the sky is clear, meaning there is no cloud coverage or temperature variance. This baseline is then modified based on actual or expected weather conditions to estimate the actual PV generation. The literature around this idea is based on physical models of irradiance calculations, where equations are used to develop the baseline for a given geographical location \\cite{Palani2017}. Newer methods use data-driven techniques such as time series forecasting and machine learning models~\\cite{long2014analysis}. The data-driven methods are gaining popularity because of their robustness, speed and geographic adaptability. \n\nIntuitively, this method gives reliable solar forecasts because of solar generation's seasonal and diurnal nature. However, the main drawback is the lack of data specifically related to ``clear sky\" days; Consequently, we use the most commonly occurring day's generation as baseline, which can be discovered by our previous work~\\cite{yuan2021irmac}.\n\nFor the baseload forecasting, the theme used is the application of ensemble methods, as the current state of the art identifies these methods to be most accurate when compared to stand-alone methods \\cite{raviv2015}. We use mainly a combination of random forest (RF), gradient boost (GB), autoregressive integrated moving average (ARIMA) and support vector machine (SVM), which are all well-studied and standard methods for time series forecasting \\cite{HONG2016914}. Specifically, we apply different forecasting methods to different sub-series after disaggregating the load profiles using Seasonal and Trend decomposition using Loess (STL) due to their cyclic patterns \\cite{Theodosiou2011forecasting}.\n\nThe optimal scheduling of distributed energy resources (DER) and controllable\/uncontrollable loads for micro-grids comes under the class of problems commonly referred to as energy management problems. However, these classes of problems tend to be non-convex and non-linear in nature because of the constraints associated with the scheduled devices, e.g., scheduling uninterruptible loads such as lectures that cannot be stopped once started. As reviewed by the authors in~\\cite{EMP_Review}, the majority of these problems are modeled via classical optimization approaches such as MILP~\\cite{OptimalMILP} and mixed-integer non-linear programs (MINLPs)~\\cite{MINLP_Eg}. Other approaches include dynamic programming~\\cite{DP_Eg}, rule-based optimization and meta-heuristic algorithms~\\cite{Meta_Eg}. \n\nSince these problems tend to be non-convex and non-linear, the major challenge associated with solving these problems is tractability and (in the case of meta-heuristic methods) convergence to a global optimum. The mathematical literature surrounding MILPs offers a variety of techniques to simplify these problems before solving them, and the availability of solvers such as Gurobi\\textsuperscript{\\textregistered} for MILPs make them a very attractive option for solving these problems. Also, unlike other models where the algorithms (may) converge to a local optimal solution, modern MILP solvers can obtain globally optimal solutions for this class of problems.\n\n\n\\subsection{Contributions}\n\\label{subsec:Contributions}\nBased on the related work studied, this paper offers the following contributions to the body of knowledge: \n\\begin{itemize}\n    \\item A solar forecasting algorithm using training data from refined motif (RM) discovery technique and use of an over-parameterized 1D-convoluted neural network (1D-CNN) implemented via residual networks (ResNet). To overcome the lack of ``clear sky'' data, we incorporated the work done previously by Rui Yuan et al.~\\cite{yuan2021irmac} to identify RMs in the given solar generation dataset. An RM is the most repetitive pattern within a given time series, which can be extracted along with the exogenous variables (e.g., weather information) associated with this pattern. Using this as a baseline, we estimated solar generation by training an over-parameterized 1D-CNN. Some studies have shown that over-parameterization of CNNs can lead to better performance at the expense of longer training time \\cite{Balaji2021, Bubeck2021,Power2021}. Therefore, a ResNet was implemented to develop a deeper NN but with faster computation time.\n   \n    \\item An optimal micro-grid scheduling algorithm is solved based on real-world data for a university-based application. To the best of our knowledge, there has not yet been a study to co-optimize lectures schedule (and associated resources) and battery operation (with PV panels). Therefore, in this paper, we have developed and tested our algorithm using real-world data and practical case instances provided by Monash University. Given the large problem size (one-month schedule) and the presence of a quadratic term in the objective function for the cost of peak demand, we proposed a two sub-problem approach to formulate a tractable problem. The first sub-problem was used to limit the peak demand throughout the month, eliminating the quadratic term from the objective. Then, the peak demand was used to solve the second sub-problem to minimize the total electricity costs.\n\\end{itemize}\n\n\nThe rest of the paper is structured as follows. Section \\ref{sec:Background} introduces the data, information and problem requirements. Section \\ref{sec:Methodology} proposes the forecasting and optimization methodologies. Section \\ref{sec:Results and Discussion} presents the numerical results of the scheduling algorithm based on time series forecasting. Finally, we conclude the paper section \\ref{sec:Conclusion}.\n\n\n\n\\section{Background Information}\n\\label{sec:Background}\nThis section describes the competition requirements and data provided by the organizers.\n\n\n\\subsection{Problem statement}\n\\label{subsec:Problem Statement}\nThis section describes the scheduling constraints and objective function provided by the competition organizers and has been adapted from~\\cite{IEEECIS}. We were required to develop prediction algorithms for the baseload of six buildings in Monash University and the solar generation of PV panels connected to them. After this, we had to use these predictions to optimally schedule lecture activities and battery operation while minimizing the electricity costs (i.e., electrical energy consumption cost plus peak demand charge). We were allowed to consider the electricity prices as known parameters to simplify the problem and focus on predicting solar generation and baseload.  \n\nFor each lecture activity, we are provided with several small or large rooms needed, the electrical power consumed per room and the duration of the activities (in steps of 15 minutes). We are also provided with a list of precedence activities, i.e., activities that must be performed at least one day before the activity in question. For the batteries, we are provided with the maximum energy rating, i.e., state of charge (SOC), the peak charge\/discharge power and charge\/discharge efficiency. \n\nThe scheduling must start from the first Monday of the month, and each activity must; 1) take place within office hours (9:00--17:00), 2) happen once every week and recur on the same day and time every week. Also, the number of rooms (associated with each activity) allocated at a given time interval must not exceed the total number of rooms in the six buildings. The batteries \ncan operate in one of three states; charging, discharging or idle, which is determined by the scheduling algorithm. However, the battery power must be at the peak charge\/discharge capacity once operated.\n\n\\subsection{Dataset information}\n\\label{subsec:Data set}\nThe organizers provided us with datasets for the electricity consumption of the six buildings and their connected solar panels~\\cite{IEEECIS}. About two years' worth of data was provided with a granularity of 15 minutes. However, the datasets contained many missing points, especially the demand profiles with consecutive weeks of unrecorded data, as shown in \\cite{gitlabRay}.\nInformation regarding COVID-19 lockdown and how it influenced the forecast and earlier timelines were also provided to aid in incorporating the effect of COVID-19 pandemic on baseload in our models. ERA5 climate data was also provided as exogenous variables for the time series forecasting via a partnership with OikoLabs~\\cite{IEEECIS}. \nAlso, we were allowed to use electricity price information from the Australian NEM~\\cite{NEM_Data} as a parameter for objective function calculation. The input parameters used for solar and building forecasting are summarized in Table~\\ref{tab: inputs-sensitivity}.\n\\begin{table}[!h]\n    \\centering\n    \\caption{Input data used for forecasting}\n    \\label{tab: inputs-sensitivity}\n    \\resizebox{0.5\\textwidth}{!}{\n    \\begin{tabular}{|l|*{2}{c|}}\n        \\hline\n        \\rowcolor{AdOrange} \\backslashbox{Weather inputs (15mins)}{Forecast outputs (15mins)}\n        &\\makecell{Building \\\\(15mins)}&\\makecell{Solar\\\\ (15mins)}\\\\\\hline\n        \\rowcolor{AdOrangeLight}Temperature ($^\\circ$C) & \\checkmark & \\checkmark \\\\\\hline\n        \\rowcolor{AdCream}Dewpoint temperature ($^\\circ$C) &\\checkmark &\\\\\\hline\n        \\rowcolor{AdOrangeLight}Wind speed (m\/s) & &\\checkmark \\\\\\hline\n        \\rowcolor{AdCream}Relative humidity (0-1) & \\checkmark & \\checkmark\\\\\\hline\n        \\rowcolor{AdOrangeLight}Surface solar radiation (W\/$m^2$) &\\checkmark &\\checkmark\\\\\\hline\n        \\rowcolor{AdCream}Total cloud cover (0-1) & &\\checkmark\\\\\\hline\n        \\rowcolor{AdOrangeLight}Occupancy  (0-1) & \\checkmark&\\\\\\hline\n        \\rowcolor{AdCream}Annual harmonics (0-1) & &\\checkmark\\\\\\hline\n    \\end{tabular}\n    }\n\\end{table}\n\\subsection{Instance information}\n\\label{subsec:Instance}\n\nTo test the algorithms each team developed, the competition organisers provided ten problem instances. Each instance provided information regarding the number of activities to be scheduled and the specifics of each activity; duration of the activity in time steps, power consumed per room for each time interval, number of large\/small rooms required and a precedence list. The precedence list contains the activities that must happen one day before the specified task. Depending on the number of activities in each instance, they were divided into two sub-categories; large (200 activities) and small (50 activities) instances. We report simulation results for one large and one small instances in section~\\ref{sec:Results and Discussion}.\n\n\n\\section{Methodology}\n\\label{sec:Methodology}\nThis section describes the methodology developed to solve the problem described in~\\ref{subsec:Problem Statement}. Mainly, we outline our data cleaning strategies, forecasting methodology and scheduling algorithms used in the competition.\n\n\n\\subsection{Solar generation forecast}\n\\label{subsec:Solar generation forecast}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[clip, trim=3.3cm 4.3cm 2.75cm 3.95cm,width=0.95\\linewidth]{figs\/solar_forecast_2.pdf}}\n\\caption{Block diagram of the proposed solar generation prediction model}\n\\label{fig:solar}\n\\end{figure}\n\nThe solar generation forecast methodology is depicted in Fig.~\\ref{fig:solar}. The solar data was mainly clean, with no outliers and a few missing points, which were removed during the cleaning process. Since solar time series data is repetitive in nature, we used the method described in~\\cite{yuan2021irmac} to extract the RM (most repetitive patterns) and its associated weather conditions. The intuitive idea is to train a neural network (NN) to reshape this RM with respect to the difference in weather conditions between the RM weather conditions and the actual data weather conditions. \n\n\\subsection{Baseload forecast}\n\\label{subsec:baseload forecast}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[clip, trim=4.5cm 4.2cm 0.4cm 1.5cm,width=0.95\\linewidth]{figs\/baseload_forecast_1.pdf}}\n\\caption{Block diagram of the proposed baseload prediction model}\n\\label{fig:base_load}\n\\end{figure}\n\nThe block diagram of the methodology used for baseload forecasting is shown in Fig.~\\ref{fig:base_load}, which demonstrates the use of different ensemble forecasting methods for different buildings. We applied the same procedure for the data cleaning process as in the PV generation forecast. However, since there were more missing points in the demand profiles, which lasted for many weeks consecutively, we used the average value from the previous week for the long-period missing data. The choice of ensemble methods was achieved through the use of the Python sklearn package's voting regressor function \\cite{SklearnVoting}. For this dataset, this resulted in a combination of RF, SVM, ARIMA and GB methods for best accuracy. In terms of the length of training data, due to the highly dynamic nature of demand profiles, which was exacerbated during the COVID-19 period, we only used 2 months of historical data for training.\n\nDue to the seasonal\/cyclic patterns demonstrated in building 1, 2 and 5, as shown in \\cite{gitlabRay}, we utilized STL decomposition to capture these properties by disaggregating the load profiles into three components, namely trend, seasonality and residual. We then trained each profile with different methods depending on the performance of the historical data. In the end, the predictions from these three sub-series were combined to obtain the final building demand forecast. For buildings 0 and 4 with no clear cyclic patterns, we observed that either RF or GB (without STL decomposition) were sufficient to capture their repetitive nature. Lastly, for building 3, historical data showed that the load only took on a finite set of discrete values; thus, we fixed the predictions to the median.\n\n\n\\subsection{Optimal scheduling algorithm}\n\\label{subsec:Optimal scheduling}\n\n\\begin{figure}[!ht]\n\\centerline{\\includegraphics[clip, trim=3.6cm 9.3cm 1.2cm 3.5cm,width=0.95\\linewidth]{figs\/scheduling_alg_1.pdf}}\n\\caption{Block diagram of the proposed optimal scheduling algorithm}\n\\label{fig:scheduling_alg}\n\\end{figure}\n\nWe developed a MIP-based optimal scheduling problem to represent all the scheduling constraints, where the objective function was provided by the organizers as follows: \n\\begin{flalign}\n\\label{eq:QP objective}\nO(P_{t}) = \\frac{0.25}{1000}\\sum_{t \\in \\mathcal{T}}\\, P_{t} \\, \\lambda^{\\text{aemo}}_{t} + 0.005 \\, \\left(\\max_{t \\in \\mathcal{T}} P_{t}\\right)^2 &\n\\end{flalign}\n\nHere \\(P_{t}\\) and \\(\\lambda^{\\text{aemo}}_{t}\\) are the net demand and AEMO electricity spot price at time \\(t\\) over the time horizon \\(\\mathcal{T}\\) respectively. Therefore, it can be seen that the problem is a MIQP problem, which is NP-hard and hence most likely intractable given the size of this problem (\\(\\|\\mathcal{T}\\| = 2880\\)).\nHence, to improve the tractability, we reformulated the MIQP problem into two MILP sub-problems, as demonstrated in Fig.~\\ref{fig:scheduling_alg}. The first sub-problem is used to estimate an upper bound for the maximum demand (\\(P^{\\text{max}}_1 = \\max_{t \\in \\mathcal{T}} P_{t}\\)), which is used to remove the quadratic term from the objective function in the second sub-problem. \n\nThe scheduling is developed for a time duration \\(\\Delta t\\) with \\(P^{\\text{base}}_t \\text{ and } P^{\\text{solar}}_t\\) being the baseload and solar generation forecasts, respectively. For a set of batteries \\(b \\in \\mathcal{B}\\) over the time horizon \\(t \\in \\mathcal{T}\\), we define \\(E^{\\text{bat}}_{b, t}\\) as battery SOC in kWh, \\(P^{\\text{bat}}_{b, t}\\) is the battery power in kW, \\(E^{\\text{cap}}_b\\) is the battery capacity in kWh, \\(B^{\\text{c}}_{b}\/B^{\\text{d}}_{b}\\) is the battery charge and discharge power in kW, \\(u^{\\text{c}}_{b, t}\/u^{\\text{d}}_{b, t}\\) are the battery charge\/discharge status, and \\(\\eta^{\\text{c}}_{b,t}\/\\eta^{\\text{d}}_{b,t}\\) are the charging\/discharging efficiencies. For a set of activities \\(a \\in \\mathcal{A}\\) over the same time horizon, we define \\(A^{\\text{kW}}_a\\) as the power required per room for the activity in kW, \\(R^{\\text{large}}_{a}\/R^{\\text{small}}_{a}\\) are the number of large\/small rooms needed for the activity, \\(P^{\\text{sched}}_{t}\\) be the total power scheduled in kW, and \\(u^{\\text{start}}_{a,t}\/u^{\\text{active}}_{a,t}\\) are binary variables which track the activity start time and active time, respectively. \\(\\mathcal{D}_{a}\\) is the duration set for the activity, and \\(\\mathcal{P}_{a}\\) is the set of activities that must precede activity \\(a\\). Additionally, we define \\(\\mathcal{T}_{n}\\) as the set of time intervals where activities must not be active, \\(\\mathcal{T}_f\\) is the set of time intervals for the first week and \\(L^{\\text{day}}_{t}\\) corresponds to the day of the time interval \\(t\\). We present the equations used for the two sub-problems in the below sections.\n\n\\subsubsection{Sub-problem 1}\n\\label{subsubsec:Sub-problem 1}\n\\begin{subequations}\n\\begin{flalign}\n\\label{eqref:Objective 1}\n\\min_{\\psi_1} P^{\\text{max}}_{1} &\n\\end{flalign}\n\\begin{flalign}\n&\\textbf{Subject to,} \\nonumber&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Max power constraint}\n0 \\leq P_{t}\\leq P^{\\text{max}}_{1} \\quad \\forall \\, t \\in \\mathcal{T}& \n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:SOC limit}\n0 \\leq E^{\\text{bat}}_{b, t} \\leq E^{\\text{cap}}_{b} \\quad \\forall \\, t \\in \\mathcal{T}, \\, b \\in \\mathcal{B}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Charge-Discharge}\nu^{\\text{c}}_{b, t} + u^{\\text{d}}_{b, t} \\leq 1 \\quad \\forall \\, t \\in \\mathcal{T}, \\, b \\in \\mathcal{B}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Battery power}\nP^{\\text{bat}}_{b, t} = u^{\\text{c}}_{b, t}\\,\\frac{B^{\\text{c}}_{b}}{\\sqrt{\\eta_{b}^{\\text{c}}}} - u^{\\text{d}}_{b, t}\\,B^{\\text{d}}_{b}\\,\\sqrt{\\eta_{b}^{\\text{d}}} \\quad \\forall \\, t \\in \\mathcal{T}, \\, b \\in \\mathcal{B} &\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Battery capacity}\nE^{\\text{bat}}_{b, -1} = E^{\\text{cap}}_{b} \\quad \\forall \\,b \\in \\mathcal{B}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:SOC value}\nE^{\\text{bat}}_{b, t} = E^{\\text{bat}}_{b, t-1} + \\Bigg(u^{\\text{c}}_{b, t}\\, B^{\\text{c}}_{b} - u^{\\text{d}}_{b, t}\\, B^{\\text{d}}_{b} \\Bigg) \\Delta t \\quad \\forall \\, t \\in \\mathcal{T}, \\, b \\in \\mathcal{B}& \n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Scheduled power}\nP^{\\text{sched}}_{t} = \\sum_{a \\in \\mathcal{A}} u^{\\text{active}}_{a,t} \\, A^{kW}_a\\, \\Bigg(R^{\\text{small}}_{a} + R^{\\text{large}}_{a}\\Bigg) \\quad \\forall \\, t \\in \\mathcal{T}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Net export}\nP_{t} = P^{\\text{base}}_{t} - P^{\\text{solar}}_t + P^{\\text{sched}}_{t}+ \\sum_{b \\in \\mathcal{B}}P^{\\text{bat}}_b \\quad \\forall \\, t \\in \\mathcal{T} &\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Room requirement}\n\\sum_{a \\in \\mathcal{A}} u^{\\text{active}}_{a, t}\\, R^{\\text{large}}_{a} \\leq H^{\\text{large}}; \\sum_{a \\in \\mathcal{A}} u^{\\text{active}}_{a, t}\\, R^{\\text{small}}_{a} \\leq H^{\\text{small}} \\; \\forall t \\in \\mathcal{T}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Non active time}\nu^{\\text{active}}_{a, t} = 0 ; u^{\\text{start}}_{a, t} = 0 \\quad \\forall \\, t \\in \\mathcal{T}_{n}, \\, a \\in \\mathcal{A}& \n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Happen once a week}\n\\sum_{t \\in \\mathcal{T}_f} u^{\\text{start}}_{a, t} = 1 \\quad \\forall \\, a \\in \\mathcal{A} &\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Duration constraint A}\nu^{\\text{start}}_{a, t} \\leq u^{\\text{active}}_{a, t + k} \\: \\forall \\, & t \\in \\{0, 1,\\dots, \\left|\\mathcal{T}\\right| - \\left|\\mathcal{D}_{a}\\right| - 1\\}, \\, k \\in \\mathcal{D}_{a}, a \\in \\mathcal{A}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Duration constraint B}\n\\sum_{t \\in \\mathcal{T}_f} u^{\\text{active}}_{a, t} = \\left| \\mathcal{D}_{a} \\right| \\quad \\forall \\, a \\in \\mathcal{A}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Precedence constraints}\n\\sum_{t \\in \\mathcal{T}_f} \\left(u^{\\text{active}}_{a, t}\\, L^{\\text{day}}_{t} - u^{\\text{active}}_{k, t}\\, L^{\\text{day}}_{t}\\right) \\geq 1 \\; \\; \\forall \\, k \\in \\mathcal{P}_{a}, a \\in \\mathcal{A}&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Start at same time every week}\nu^{\\text{start}}_{a, t} = u^{\\text{start}}_{a, t - \\alpha} \\quad \\forall \\, t \\in \\mathcal{T}_{f}^{c}, \\, a \\in \\mathcal{A} &\n\\end{flalign}\n\\end{subequations}\n\n\\noindent where \\(\\psi_1 = \\{P, P^{\\text{bat}}, P_1^{\\text{max}}, E^{\\text{bat}},u^{\\text{c}}, u^{\\text{d}}, u^{\\text{active}}, u^{\\text{start}} \\}\\). Equation~\\eqref{eqref:Max power constraint} helps find the peak demand of the schedule and prevents energy export to the grid. Equations~\\eqref{eqref:SOC limit}--\\eqref{eqref:SOC value} represent battery related constraints. Equations~\\eqref{eqref:Scheduled power}--\\eqref{eqref:Net export} are used to calculate the net energy imported from the grid, and equation~\\eqref{eqref:Room requirement} ensures the number of rooms engaged through activities is within the limit. Equation~\\eqref{eqref:Non active time} ensures activities do not start outside office hours and before the first Monday of the month, equation~\\eqref{eqref:Happen once a week} ensures activities happen once a week, equations~\\eqref{eqref:Duration constraint A}--\\eqref{eqref:Duration constraint B} ensure activities are active for the necessary duration after starting, equation~\\eqref{eqref:Precedence constraints} satisfies that all precedence (prerequisite) activities happen a day before and equation~\\eqref{eqref:Start at same time every week} ensures activities recur at the same time every week. The objective here is to minimize peak demand while satisfying all the scheduling constraints.  \n\nSince this problem is only used to solve for the upper bound maximum demand \\(P^{\\text{max}}_1\\), it is not necessary to solve the full problem. We use the property that the solution of the LP relaxation of a minimization-based MILP problem is its lower bound (\\(y^{*}_{\\text{MILP}} \\geq y^{*}_{\\text{LP}}\\))~\\cite{Vielma2015} to just solve the LP relaxation and use this for sub-problem 2, thereby improving tractability of our method.\n\n\n\\subsubsection{Sub-problem 2}\n\\label{subsubsec:Sub-problem 2}\n\\begin{subequations}\n\\begin{flalign}\n\\min_{\\psi_2} \\frac{0.25}{1000}\\sum_{t \\in \\mathcal{T}}\\, P_{t} \\, \\lambda^{\\text{aemo}}_{t}  &\n\\end{flalign}\n\\begin{flalign}\n&\\textbf{Subject to, Equations~}\\eqref{eqref:SOC limit} \\text{\\---} \\eqref{eqref:Precedence constraints}\\nonumber&\n\\end{flalign}\n\\begin{flalign}\n\\label{eqref:Load limit}\n0 \\leq P_{t} \\leq \\beta^{\\text{mul}} \\,P^{\\text{max}^{*}}_{1} &\n\\end{flalign}\n\\end{subequations}\n\n\\noindent where \\(\\psi_2 = \\left(\\psi_1 \\setminus \\{P^{\\text{max}}_{1}\\} \\right)\\). Using equation~\\eqref{eqref:Load limit}, we ensure that peak demand does not exceed \\(\\beta^{\\text{mul}} P^{\\text{max}^{*}}_1\\) limiting the demand charge. Intuitively, \\(\\beta^{\\text{mul}} \\geq 1\\) is a multiplier used to change the tightness of the MILP, creating a trade-off between computation time and electricity cost; higher values of \\(\\beta^{\\text{mul}}\\) decreases computation speed but increases the electricity cost (objective of the problem) and vice versa. \n\n\\begin{figure*}[t]\n        \\centering\n            \\subfigure[Difference in the RM at and predicted generation weather conditions for Nov \\(4^{\\text{th}}\\) 2020]\n            {\n                \\label{subfig:RM differences}\n                \\includegraphics[width=0.29\\textwidth]{figs\/RM_Exogenous.pdf} \n            } \n            %\n            \\subfigure[RM vs. predicted solar generation for Nov \\(4^{\\text{th}}\\) 2020]\n            {\n                \\label{subfig:RM output}\n                \\includegraphics[width=0.29\\textwidth]{figs\/RM_Output.pdf} \n            } \n            %\n            \\subfigure[Scheduled power, predicted net load and NEM spot price for Nov \\(2^{\\text{nd}}\\) 2020]\n            {\n                \\label{subfig:Optimization output}\n                \\includegraphics[width=0.33\\textwidth]{figs\/Competition_Results.pdf}\n            }\n        \\vspace{-0.4em}\n        \\caption{Prediction and optimization results}\n        \\label{fig:Figure ref}\n\\end{figure*}\n\n\\section{Results and Discussion}\n\\label{sec:Results and Discussion}\n\nThe simulations were run for November 2020 as per the competition stipulation with \\(\\Delta t = 15 \\text{ min}, \\\n\\|\\mathcal{T}\\|=2880\\) and other instance information as specified in \\ref{subsec:Instance}. For the scheduling, two \\(\\beta^{\\text{mul}}\\) values were chosen; 1.10 for small instances and 1.15 for large instances. These values were obtained via trial and error. The prediction and optimization algorithms were run in Python 3.8.8 using Gurobi\\textsuperscript{\\textregistered} solver. \n\n\\begin{table}[t]\n    \\caption{Baseload and PV generation prediction NRMSE}\n    \\centering\n    \\begin{tabular}{ccc|ccc}\n    \\toprule\n    \\rowcolor{AdOrange} \\textbf{Building \\#} & \\textbf{Proposed} &\n    \\textbf{BI} & \\textbf{Solar \\#} & \\textbf{Proposed} & \\textbf{BI} \\\\\n    \\midrule\n    0 & \\textbf{0.43} & 0.43 & 0 & \\textbf{0.72} & 0.81 \\\\\n    \\rowcolor{AdOrangeLight} 1 & \\textbf{0.31} & 0.34 & 1 & \\textbf{0.77} & 0.77 \\\\\n    2 & \\textbf{0.23} & 0.24 & 2 & \\textbf{0.74} & 0.79 \\\\\n    \\rowcolor{AdOrangeLight} 3 & \\textbf{0.32} & 0.32 & 3 & \\textbf{0.74} & 0.75 \\\\\n    4 & \\textbf{0.40} & 0.40 & 4 & \\textbf{0.78} & 0.79 \\\\\n    \\rowcolor{AdOrangeLight} 5 & \\textbf{0.13} & 0.14 & 5 & \\textbf{0.72} & 0.75 \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:mae}\n\\end{table}\n\nSince each of the twelve datasets has different scales, we decide to use the normalised root mean square error (NRMSE), which has been used to facilitate the comparison among time series with varying magnitudes \\cite{das2018evaluation}. The metric is calculated as follows:\n\\begin{flalign}\n\\label{eq:NRMSE}\n& \\text{NRMSE} = \\sqrt{\\sum_t \\frac{\\hat{y}_t - y_t}{\\|\\mathcal{T}\\|}} \\times \\frac{1}{\\bar{y}}\n\\end{flalign}\nwhere $\\hat{y}_t$ is the model predictions, $y_t$ is the model actual data and $\\bar{y}$ represents the mean of the actual data.\n\nIn order to demonstrate the superiority of our methods, we compare our proposed techniques with the best individual (BI) prediction models that are mentioned in section \\ref{subsec:Related Work}. Please note that the BI models are picked from out-of-sample validation on the actual datasets using the same inputs and training length as the proposed methods. Table~\\ref{tab:mae} displays the NRMSE of the actual data in November 2020 between the proposed and the BI models. It can be seen that the proposed models achieve lower errors in almost all the twelve profiles except for the demand of buildings 0 and 4. As mentioned in section \\ref{subsec:baseload forecast}, these two profiles are predicted using either RF or GB due to their non-cyclic behavior. Therefore, they have the highest errors among all the demand profiles. For the PV generation time series, the proposed method also achieves better prediction accuracy with quite consistent NRMSE due to the high dependence of solar PV generation on weather data.\n\nFig.~\\ref{subfig:RM differences} and \\ref{subfig:RM output} are used to explain the intuition behind the RM method for the PV generation; the former shows the difference between RM \nand a predicted generation weather condition for November 4\\textsuperscript{th}, 2020  while the latter shows the RM and the predicted generation profiles for the same day. We can see that as the difference in temperature is positive, the predicted generation is lower than the RM and vice versa. The ResNet is then trained with the changes in weather conditions to learn how to reshape the RM to predict solar generation.\n\nFig.~\\ref{subfig:Optimization output} shows the power scheduled for the small and large instances, the predicted load and the NEM spot price profiles for Novemeber \\(2^{\\text{nd}}\\) 2020. It can be seen that during lower price periods (intervals 24--72), most of the load is scheduled and the scheduled load is clipped to the value obtained from sub-problem 1. During high price periods (intervals 72--88), the scheduled load is lower than the predicted baseload due to battery operation. This ensures that electricity cost is minimized. \n\n\\begin{table}[b]\n    \\caption{Electricity cost from different forecasting results}\n    \\centering\n    \\begin{tabular}{c|ccc}\n    \\toprule\n    \\rowcolor{AdOrange} \\textbf{Forecast methods} &\n    \\textbf{Base case} & \\textbf{BI} & \\textbf{Proposed} \\\\\n    \\midrule\n    \\rowcolor{AdOrangeLight} \\textbf{Cost (\\$AUD)} & 589356 & 466652 & \\textbf{357210} \\\\\n    \\bottomrule\n    \\end{tabular}\n    \\label{tab:opti_cost}\n\\end{table}\n\nThe organizers provided a base schedule\/electricity cost for the problem based on no predictions and an \\(\\epsilon\\)-greedy algorithm \\cite{IEEECIS}. Table~\\ref{tab:opti_cost} compares the electricity cost from three different forecast methods, where the base case represents the electricity cost provided by the organizers; whereas BI and the proposed cost represent the cost optimized on their respective forecasts. Our methodology reduced this cost by roughly 23\\% and 40\\% compared to the BI and base case, respectively. We were able to solve small instances (158400 binaries) in an average time of 447 seconds per instance and large instances (590400 binaries) in 1550 seconds per instance. It can be seen that even though the problem size grew 3.73 times, the time to solve the problem increased almost linearly and not exponentially with problem size as is the case in MIPs. It further demonstrates the tractability and scalability of our algorithm. \n\n\n\\section{Conclusion}\n\\label{sec:Conclusion}\n\nIn this paper, we developed an optimal scheduling solution for solving a real-world electricity cost minimization problem for six buildings at Monash University in Melbourne, Australia. A two-step solution was presented in this paper, which included forecasting and an optimization step. Our solution with energy forecast reduced the total cost of electricity roughly by 40\\%. We would like to highlight that we placed fifth in the competition among 50 participants and were off from the best solution only by 4\\% with respect to the objective value due to their better forecast.\nFor future work, the forecasting of baseload and solar generation can be improved by using\nmodern machine learning techniques, such as transformers and info-GANs.\nFor the optimization, a major improvement can be devising an algorithm to select optimal \\(\\beta^{\\text{mul}}\\) considering the trade-off between the tightness (complexity) of the problem and cost minimization.   \n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n Cooperative communications via relaying is an effective measure to\n ensure reliability, provide high throughput and extend network\n coverage. It has been intensively studied in   LTE-Advanced \\cite{4G-relay} and will continue to play an important role in\n the future fifth generation  wireless networks. In the conventional\n two-hop one-way relay channel (OWRC), the communication takes two orthogonal  phases, and  suffers from the loss of  spectral efficiency because of the\n inherent half-duplex (HD) constraint at the relay. Two-way relaying using the principle of physical layer network coding is  proposed to\n recover the loss in  OWRC by  allowing the two sources to exchange\n information more efficiently \\cite{KRI}\\cite{Zhang-2Phase}\\cite{Qiang-Li-2009}. In the first phase of the two-way relay channel (TWRC), both sources\n transmit signals to the relay. In the second phase, the relay does not separate signals\n but rather broadcasts the processed mixed signals to both sources.\n Each source can subtract its own message from the received signal\n then decodes the information from the other source. The benefit of the TWRC is that,\n using the same two communication phases as the  OWRC, bi-directional communication is achieved.\n\n However, note that  the relay in the TWRC still operates in the HD mode thus two communication phases\n are needed.  Motivated by this observation and thanks to the emerging  full-duplex (FD) techniques,\n we aim to study the potential of the  FD operation\n in the TWRC to enable simultaneous information\n exchange between the two sources, i.e., only one communication phase is required.\n In the proposed scheme, all nodes, including the two sources\nand the relay, work in the FD mode so they can transmit and receive\nsignals simultaneously \\cite{BLISS}\\cite{BLISS1}. However, the major\nchallenge in the FD operation is that the power of the\nself-interference (SI) from the relay output could be over 100 dB\nhigher than that of the signal received from distance sources and\nwell exceeds the dynamic range of the analog-to-digital converter\n\\cite{RII3,RII2,DAY}. {\\bl  Therefore it is important that the SI is\nsufficiently mitigated.  The SI    cancellation  can be broadly\ncategorized as passive cancellation and active cancellation\n\\cite{Sabharwal-inband}. Passive suppression is  to isolate the\ntransmit and receive antennas using techniques such as directional\nantennas, absorptive shielding and cross-polarization\n\\cite{Sabharwal-passive}. Active suppression is  to exploit a node's\nknowledge of its own transmit signal to cancel the SI, which\ntypically includes analog cancellation, digital cancellation and\nspatial cancellation. Experimental results are reported in\n\\cite{Duarte-Exp} that the SI can be  cancelled to make FD wireless\ncommunication feasible in many cases. } In the recent work\n\\cite{Stanford-FD} and \\cite{Stanford-FD-MIMO}, the promising\nresults show  that the SI can be suppressed to the noise level in\nboth single-antenna and multi-antenna cases. Below we will provide a\nreview on the application of the FD relaying to both the OWRC and\nTWRC.\n\n\n\\subsection{Literature on the FD relaying}\n\n\\subsubsection{OWRC}\nThe FD operation has attracted lots of {\\bl research interest}  for\nrelay assisted cooperative communication. It is shown in \\cite{RII3}\nthat the FD relaying is feasible even in the presence of the SI and\ncan offer higher capacity than the HD mode. Multiple-input\nmultiple-output (MIMO) techniques provide an effective means to\nsuppress the SI in the spatial domain\n\\cite{RII1}-\\cite{MIMO-relay-HD}. The authors of \\cite{RII1} analyze\na wide range of SI mitigation measures when the relay has multiple\nantennas, including natural isolation, time-domain cancellation and\nspatial domain suppression. {  The FD relay selection is studied in\n\\cite{KRIKIDIS} for the amplify-and-forward (AF) cooperative\nnetworks.} With multiple transmit or receive antennas at the\nfull-duplex relay, precoding at the transmitter and decoding at the\nreceiver can be jointly optimized to mitigate the SI effects. The\njoint precoding and decoding design for the FD relaying is studied\nin  \\cite{RII1}, where both the zero forcing (ZF) solutions and the\nminimum mean square error (MMSE) solutions are discussed.   When\nonly the relay has multiple antennas, a joint design of ZF precoding\nand decoding vectors is proposed in \\cite{CHOI_EL} to  null out the\nSI at the relay. However, this design does not take into account the\nend-to-end (e2e) performance. A gradient projection method  is\nproposed  in \\cite{Park-null-12}, to optimize  the precoding and\ndecoding vectors considering the e2e performance. When all terminals\nhave multiple antennas, the e2e performance is optimized in\n\\cite{MIMO-relay-HD} where the closed-form solution for\nprecoding\/decoding vectors design as well as diversity analysis are\nprovided.\n\n\\subsubsection{TWRC}\n\n In the early work of TWRC, the FD operation is often employed to\n investigate   the capacity region from the  information-theoretic viewpoint without considering the effects of the SI \\cite{capacity-twrc}\\cite{capacity-twrc-half-bit}.\n Only recently, the SI has been taken into account in the FD TWRC.\n In \\cite{FD-two-phase}, the FD operation is introduced to the relay but two one-way relaying phases are required to achieve the two-way communication to avoid\n interference.\n A physical layer network coding  FD TWRC is proposed in \\cite{FD-practical}, where bit error\nrate is derived.  The optimal relay selection scheme is proposed and\nanalyzed in \\cite{Song-FD-TW-Selection} for the FD TWRC using the AF\nprotocol. Transmit power optimization among the source nodes and the\nrelay node is studied   in \\cite{FD-TW-SISO}, again using the AF\nprotocol. However, all existing works are restricted to the single\ntransmit\/receive antenna case at the relay thus the potential of\nusing multiple antennas to suppress the SI and improve the e2e\nperformance for the TWRC is not fully explored yet.\n\n\\subsection{Our work and contribution}\n In this work, we study the potential of the MIMO FD operation\n in the TWRC where the relay has multiple transmit\/receive antennas\n and employs the AF protocol and the principle of  physical layer network coding. The two sources have single transmit\/receive antenna.\n We jointly optimize the relay beamforming matrix and the power\n allocation at the sources to maximize the e2e performance.\n To be  specific, we study two problems: one is to find the achievable rate\n region and the other is to maximize the sum rate of the FD TWRC.\n Our contributions are summarized as follows:\n \\begin{itemize}\n    \\item We derive the signal model for the MIMO FD TWRC and\n    propose to use the ZF constraint at the relay to greatly simplify the model\n    and the problem formulations.\n    \\item We find the rate region by maximizing one source's rate\n    with the constraint on the other source' minimum rate. We propose an\n    iterative algorithm together with 1-D search to find the local\n    optimum solution. At each iteration, we give analytical solutions for the transmit beamforming vector and the power control.\n    \\item  We tackle   the sum rate maximization problem by employing a similar iterative algorithm.\n    At each iteration, we propose to use the DC (difference of convex functions) approach to optimize the transmit beamforming vector and we solve the power allocation analytically.\n    \\item We conduct intensive simulations to compare the proposed FD\n    scheme with three benchmark schemes and clearly show its\n    advantages of enlarging the rate region and improving the sum rate.\n \\end{itemize}\n\n\n\n\nThe rest of the paper is organized as follows. In Section II, we\npresent the   system model, the explicit signal model and problem\nformulations. In Section III, we deal with the problem of finding\nthe achievable rate region; in Section IV, we address the problem of\nmaximizing the sum rate. Three benchmark schemes are introduced in\nSection V.  Simulation results are presented in Section VI. Section\nVII concludes this paper and gives future directions.\n\n\n\n\n \\noindent \\emph{Notation}: The lowercase and\nuppercase boldface letters (e.g., ${\\bf x}$ and ${\\bf X}$) indicate column\nvectors and matrices, respectively. ${\\bf X}\\in \\mathbb{C}^{M\\times N}$\nmeans a complex matrix ${\\bf X}$ with dimension of $M\\times N$.\n ${\\bf I}$ is the identity matrix.\n We use $(\\cdot)^\\dagger$ to denote the conjugate\ntranspose, $\\textsf{trace}(\\cdot)$ is the trace operation, and $\\|\\cdot\\|$ is\nthe Frobenius norm. $|\\cdot|$ represents the absolution value of a\nscalar. ${\\bf X}\\succeq {\\bf 0}$ denotes that the Hermitian matrix ${\\bf X}$\nis positive semidefinite. The expectation operator is denoted by\n$\\mathcal{E}(\\cdot)$. Define $\\Pi_{\\bf X} =\n{\\bf X}({\\bf X}^\\dag{\\bf X})^{-1}{\\bf X}^\\dag$ as the orthogonal projection onto the\ncolumn space of ${\\bf X}$; and $\\Pi_{\\bf X}^\\bot = {\\bf I} - \\Pi_{\\bf X}$ as the\northogonal projection onto the orthogonal complement of the column\nspace of ${\\bf X}$.\n\n\n\n\n\n\\section{System model, signal model and problems statement}\n\\subsection{System model}\\label{GENERAL_system_model}\nConsider a  three-node MIMO relay network consisting of two sources\n${\\tt A}$ and ${\\tt B}$ who want to exchange information with the aid of a\nMIMO relay ${\\tt R}$, as depicted in Fig. 1. {\\bl There is no direct\nlink between the two sources because of deep fading or heavy\nshadowing, so their communication must rely on ${\\tt R}$.} All nodes\nwork in the FD mode. To enable the FD operation, each source is\nequipped with two groups of RF chains and corresponding antennas,\ni.e., one for transmitting and one for receiving signals\\footnote{It\nis also possible to realize the FD operation using a single antenna,\nsee \\cite{Stanford-FD}.}. We assume that each source has one\ntransmit antenna and one receive antenna. We use $M_{T}$ and $M_{R}$\nto denote the number of transmit and receive antennas at ${\\tt R}$,\nrespectively. We assume $M_T>1$ or $M_R>1$ to help suppress the\nresidual SI at ${\\tt R}$ in the spatial domain.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{MIMO_FD_Relay}\n  \\caption{Full-duplex TWRC with two sources and a MIMO\nrelay. The dashed line denotes the  residual self-interference.}\n\\label{FIGG1}\n\\end{figure}\n We use ${\\bf h}_{XR}\\in \\mathbb{C}^{M_R\\times 1}$ and\n${\\bf h}_{RX}\\in\\mathbb{C}^{M_T\\times 1}$ to denote the directional\nchannel vectors between the source node ${\\tt X}$'s (${\\tt X}\\in\n\\{{\\tt A},{\\tt B}\\}$) transmit antenna to ${\\tt R}$'s receive antenna(s), and\nbetween ${\\tt R}$'s transmit antenna(s) to ${\\tt X}$'s receive antenna,\nrespectively. In general, channel reciprocity does not hold, e.g.,\n${\\bf h}_{XR}\\ne {\\bf h}_{RX}$, due to different transmit and receive\nantennas used. In addition, $h_{AA}, h_{BB}$ and\n${\\bf H}_{RR}\\in\\mathbb{C}^{M_R\\times M_T}$ denote the residual SI\nchannels   after the SI cancellation scheme is applied\n at the corresponding nodes \\cite{KRIKIDIS,RII3}. {\\bl The statistics of\nthe residual SI channel are not well understood yet \\cite{SI-stat}.\nThe experimental-based study in \\cite{Duarte-thesis} has\ndemonstrated that the amount of SI suppression achieved by a\ncombined analog\/digital cancellation technique is a complicated\nfunction of several system parameters. Therefore for simplicity, in\nthis paper, we model each element of the residual SI channel as a\nGaussian distributed random variable} with zero mean and variance\n$\\sigma^2_X, {\\tt X}\\in \\{{\\tt A},{\\tt B},{\\tt R}\\}$. All channel links between two\nnodes are subject to independent flat fading.  We assume that the\nchannel coefficients between different nodes remain constant within\na normalized time slot, but change independently from one slot to\nanother according to a complex Gaussian distribution with zero mean\nand unit variance.  The global channel state information (CSI) is\navailable at the relay where the optimization will be performed. As\nwill be seen in Fig. \\ref{fig:sumrate:csi}, this is critical for the\nrelay to adapt its beamforming matrix and for the two sources to\nadjust their transmit power. The noise at the node ${\\tt X}$'s (${\\tt X}\\in\n\\{{\\tt A},{\\tt B}, {\\tt R}\\}$) receive antenna(s) is denoted as $n_X$ (${\\bf n}_X$)\nand modeled as complex additive white Gaussian noise with zero mean\nand unit variance.\n\nSince each source has only a single transmit and receive antenna, a\nsingle data stream $S_A$ and $S_B$ are transmitted from ${\\tt A}$ and\n${\\tt B}$, with the average power $p_A$ and $p_B$, respectively. To keep\nthe complexity low, ${\\tt R}$ employs linear processing, i.e., the AF\nprotocol with an amplification matrix ${\\bf W}$,  to process the\nreceived signal. The node ${\\tt X}$ has maximum transmit power\nconstraint $P_X$. We will jointly optimize the transmit power of the\ntwo source nodes, $p_A$ and $p_B$ together with the amplification\nmatrix at the relay to maximize the e2e system performance.\n\n\n {\\bbl The overhead due to the CSI acquisition at each node is analyzed as follows. The received CSI\n${\\bf h}_{AR}$ and ${\\bf h}_{BR}$ at  ${\\tt R}$ can be estimated  by ${\\tt A}$ and\n${\\tt B}$ each sending one pilot symbol separately. The SI channel\n${\\bf h}_{RR}$  can be estimated at ${\\tt R}$   by itself sending an\n$M_T$-symbol pilot sequence. The SI channel $h_{AA}$ and $h_{BB}$\n  are estimated similarly and then sent back to\n${\\tt R}$. Regarding the transmit CSI ${\\bf h}_{RA}$ and ${\\bf h}_{RB}$, ${\\tt R}$\nfirst broadcasts an $M_T$-symbol  pilot sequence (this can  be used\nfor the estimation of ${\\bf h}_{RR}$ simultaneously), then ${\\tt A}$ and\n${\\tt B}$ feedback their   estimation to ${\\tt R}$. In addition, after ${\\tt R}$\nperforms the optimization and obtains ${\\bf W}$, $p_A$ and $p_B$, it\ntransmits ${\\bf h}_{RA}^\\dag{\\bf W} {\\bf h}_{AR}$ and ${\\bf h}_{RB}^\\dag{\\bf W}\n{\\bf h}_{BR}$ to ${\\tt A}$ and ${\\tt B}$, respectively, such that they can\ncancel their previously sent symbols. Finally, ${\\tt R}$ informs ${\\tt A}$\nand ${\\tt B}$ their     transmit power   $p_A$ and $p_B$, respectively.}\n\n\n\n\\subsection{Signal model}\n  We assume that the processing delay at ${\\tt R}$ is given by {\\bbl a\n $\\tau$-symbol duration}, which refers to the required\nprocessing time in order to implement the FD operation \\cite{RII1}.\n$\\tau$ typically takes integer values. The delay is short enough\ncompared to a time slot which has a large number of data symbols,\ntherefore its effect on the achievable rate is negligible.  At the\ntime ({\\bbl symbol}) instance $n$, the received signal ${\\bf r}[n]$ and\nthe transmit signal ${\\bf x}_R[n]$ at ${\\tt R}$ can be written as\n  \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\\label{eqn:rn}\n    {\\bf r}[n] = {\\bf h}_{AR}S_A[n] + {\\bf h}_{BR}S_B[n] + {\\bf H}_{RR} {\\bf x}_R[n] + {\\bf n}_R[n],\n  \\ee  \\vspace{-3mm} and\n  \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\\label{eqn:xn}\n    {\\bf x}_R[n] = {\\bf W}{\\bf r}[n-\\tau],\n  \\ee\nrespectively. Using (\\ref{eqn:rn}) and (\\ref{eqn:xn}) recursively,\nthe overall relay output can be rewritten as \\small\n    \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:xn2}\n    {\\bf x}_R[n] &=& {\\bf W} {\\bf h}_{AR}S_A[n-\\tau] + {\\bf W}{\\bf h}_{BR}S_B[n-\\tau] \\notag\\\\ &&+ {\\bf W}{\\bf H}_{RR} {\\bf x}_R[n-\\tau] + {\\bf W}{\\bf n}_R[n-\\tau]\\notag\\\\\n        &=&  {\\bf W}\\sum_{j=0}^{\\infty}\\left({\\bf H}_{RR}{\\bf W}\\right)^j  \\Big({\\bf h}_{AR}S_A[n-j\\tau-\\tau] \\notag\\\\ && + {\\bf h}_{BR}S_B[n-j\\tau-\\tau]+ {\\bf n}_R[n-j\\tau-\\tau] \\Big),\n  \\eea  \\normalsize\n  {\\bl where $j$ denotes the index of the delayed symbols}.\n\n Its covariance matrix is given by\\small\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    &&{\\mathcal{E}}[{\\bf x}_R{\\bf x}_R^\\dag] = P_A {\\bf W}  \\sum_{j=0}^{\\infty}\\left({\\bf H}_{RR}{\\bf W}\\right)^j {\\bf h}_{AR}{\\bf h}_{AR}^\\dag\n    \\left( \\left({\\bf H}_{RR}{\\bf W}\\right)^j\\right)^\\dag{\\bf W}^\\dag \\notag\\\\\n    &&+P_B {\\bf W}  \\sum_{j=0}^{\\infty}\\left({\\bf H}_{RR}{\\bf W}\\right)^j {\\bf h}_{BR}{\\bf h}_{BR}^\\dag\n    \\left( \\left({\\bf H}_{RR}{\\bf W}\\right)^j\\right)^\\dag{\\bf W}^\\dag\\notag\\\\\n    && + {\\bf W}  \\sum_{j=0}^{\\infty} ({\\bf H}_{RR}{\\bf W}\\qW^\\dag {\\bf H}_{RR}^\\dag )^j  {\\bf W}^\\dag\\notag\\\\\n    &&= P_A {\\bf W}  \\sum_{j=0}^{\\infty}\\left({\\bf H}_{RR}{\\bf W}\\right)^j {\\bf h}_{AR}{\\bf h}_{AR}^\\dag\n    \\left( \\left({\\bf H}_{RR}{\\bf W}\\right)^j\\right)^\\dag{\\bf W}^\\dag \\notag\\\\\n    &&+ P_B {\\bf W}  \\sum_{j=0}^{\\infty}\\left({\\bf H}_{RR}{\\bf W}\\right)^j {\\bf h}_{BR}{\\bf h}_{BR}^\\dag\n    \\left( \\left({\\bf H}_{RR}{\\bf W}\\right)^j\\right)^\\dag{\\bf W}^\\dag\\notag\\\\\n    &&+{\\bf W}({\\bf I}-{\\bf H}_{RR}{\\bf W}\\qW^\\dag{\\bf H}_{RR}^\\dag)^{-1}{\\bf W}^\\dag.\n\\eea\\normalsize\n\n Note that ${\\tt R}$'s transmit signal covariance $\\mathcal{E}[{\\bf x}_R{\\bf x}_R^\\dag]$, and in turn the transmit power and the SI power, are  complicated functions of ${\\bf W}$,  which makes the optimization problems  difficult.\n To simplify the signal model and make the optimization problems more tractable, we add the   ZF constraint such that the optimization of ${\\bf W}$\n nulls out the residual SI  from the relay output to relay\ninput.  To realize this, it is easy to check from (\\ref{eqn:xn2})\nthat the following condition is sufficient, \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n    {\\bf W}{\\bf H}_{RR}{\\bf W}={\\bf 0}.\n \\ee\n Consequently, (\\ref{eqn:xn2}) becomes\n    {\\small\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:xn3}\n    {\\bf x}_R[n] =  {\\bf W} \\left({\\bf h}_{AR}S_A[n-\\tau] + {\\bf h}_{BR}S_B[n-\\tau] +\n    {\\bf n}_R[n-\\tau]\\right),\n  \\eea}\n  with the covariance matrix\n  {\\small\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    {\\mathcal{E}}[{\\bf x}_R{\\bf x}_R^\\dag] = p_A {\\bf W}{\\bf h}_{AR}{\\bf h}_{AR}^\\dag{\\bf W}^\\dag + p_B {\\bf W}{\\bf h}_{BR}{\\bf h}_{BR}^\\dag{\\bf W}^\\dag + {\\bf W}\\qW^\\dag.\n\\eea}\n The relay output power is\n \\begin{align}\n p_R &=  \\textsf{trace}( {\\mathcal{E}}[{\\bf x}_R{\\bf x}_R^\\dag])\\notag\\\\\n &= p_A\\|{\\bf W} {\\bf h}_{AR}\\|^2  + p_B\\|{\\bf W} {\\bf h}_{BR}\\|^2   +  \\textsf{trace}({\\bf W}\\qW^\\dag).\n \\end{align}\n\nThe received signal at the source  ${\\tt A}$ can be written as\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    r_A[n] &=& {\\bf h}_{RA}^\\dag{\\bf x}_R[n] + h_{AA}S_A[n] + n_A[n]\\notag\\\\\n     &=& {\\bf h}_{RA}^\\dag{\\bf W} {\\bf h}_{AR}S_A[n-\\tau] +{\\bf h}_{RA}^\\dag{\\bf W}  {\\bf h}_{BR}S_B[n-\\tau] \\notag\\\\&& + {\\bf h}_{RA}^\\dag{\\bf W}{\\bf n}_R[n] + h_{AA}S_A[n] + n_A[n].\n  \\eea\nAfter cancelling its own transmitted signal $S_A[n-\\tau]$, it\nbecomes\\footnote{Note that different from $S_A[n-\\tau]$, $S_A[n]$\ncan not be completely cancelled due to the simultaneous\ntransmission, which is the main challenge of the FD radio.}\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    r_A[n] &=& {\\bf h}_{RA}^\\dag{\\bf W}  {\\bf h}_{BR}S_B[n-\\tau]  + {\\bf h}_{RA}^\\dag{\\bf W}{\\bf n}_R[n]\\notag\\\\\n    && +  h_{AA}S_A[n] + n_A[n].\n  \\eea\n The received signal-to-interference-plus-noise ratio (SINR) at the source ${\\tt A}$, denoted as $\\gamma_A$, is expressed as\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:hi1}\n    \\gamma_A &=& \\frac{p_B|{\\bf h}_{RA}^\\dag{\\bf W}  {\\bf h}_{BR}|^2}{  \\|{\\bf h}_{RA}^\\dag{\\bf W}\\|^2\n    + p_A|h_{AA}|^2+     1}.\n  \\eea\n  Similarly, the received SINR $\\gamma_B$ at the source ${\\tt B}$ can be\n  written as\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:hi2}\n    \\gamma_B &=& \\frac{p_A|{\\bf h}_{RB}^\\dag{\\bf W}  {\\bf h}_{AR}|^2}{  \\|{\\bf h}_{RB}^\\dag{\\bf W}\\|^2\n    + p_B|h_{BB}|^2+ 1}.\n  \\eea\n The achievable rates are then given by $R_A = \\log_2(1+\\gamma_A)$ and $R_B =\n \\log_2(1+\\gamma_B)$, respectively.\n\n\\subsection{Problems Statement}\nThe conventional physical layer analog network coding scheme\nrequires two  phases for ${\\tt A}$ and ${\\tt B}$ to exchange information\n\\cite{Zhang-2Phase}.  Thanks to the FD operation, the proposed\nscheme reduces the whole communication to only one phase thus\nsubstantially increases the spectrum efficiency. However, the FD\noperation  also brings the SI to each node so they may not always\nuse their maximum power because higher transmit power also increases\nthe level of the residual SI, therefore each node needs to carefully\nchoose its transmit power.\n\n\nWe are interested in two e2e objectives subject to each source's\npower constraints by optimizing the relay beamforming and power\nallocation at each source. The first one is to find the achievable\nrate region $(R_A, R_B)$. This can be achieved by maximizing source\n${\\tt A}$'s rate while varying the constraint on the minimum rate of\nsource ${\\tt B}$'s rate (or vice versa), i.e.,\n solving the rate maximization problem $\\mathbb{P}_1$ below:\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:prob1}\n    \\mathbb{P}_1: \\max_{{\\bf W}, p_A, p_B, p_R}{R_A} ~~ \\mbox{s.t.}~~ R_B\\ge r_B, p_X\\le P_X,\n    X\\in\\{A,B,R\\},\\notag\n \\eea\n where $r_B$ is the constraint on source ${\\tt B}$'s rate. By enumerating\n $r_B$, we can find the boundary of the achievable rate region.\n\nThe second problem is to maximize the sum rate of the two sources.\nMathematically, this problem is formulated as $\\mathbb{P}_2$ below:\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:prob2}\n    \\mathbb{P}_2: \\max_{{\\bf W}, p_A, p_B, p_R}{R_A+R_B} ~~ \\mbox{s.t.}~~ p_X\\le P_X,\n    X\\in\\{A,B,R\\}.\\notag\n \\eea\n The next two sections will be devoted to solving $\\mathbb{P}_1$ and\n $\\mathbb{P}_2$, respectively.\n\n\n\n\\section{Finding the achievable rate region}\nIn this section, we aim to optimize the relay beamforming matrix\n${\\bf W}$ and the sources' transmit power $(p_A, p_B)$  to find the\nachievable rate region. This can be achieved by solving\n$\\mathbb{P}_1$. Using the monotonicity between the SINR and the\nrate, it can be expanded as {\\small\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd2}\n   &&   \\max_{{\\bf W}, p_A, p_B} ~~ \\frac{p_B|{\\bf h}_{RA}^\\dag{\\bf W}  {\\bf h}_{BR}|^2}{  \\|{\\bf h}_{RA}^\\dag{\\bf W}\\|^2\n    + p_A|h_{AA}|^2+     1}\\\\\n  &&   \\mbox{s.t.} ~~  \\frac{p_A|{\\bf h}_{RB}^\\dag{\\bf W}  {\\bf h}_{AR}|^2}{  \\|{\\bf h}_{RB}^\\dag{\\bf W}\\|^2\n    + p_B|h_{BB}|^2+ 1} \\ge \\Gamma_B,\\label{cons1}\\\\\n    && p_A\\|{\\bf W} {\\bf h}_{AR}\\|^2  +  p_B\\|{\\bf W} {\\bf h}_{BR}\\|^2 +\\textsf{trace}({\\bf W}\\qW^\\dag)  \\le P_R,\\label{cons2}\\\\\n        && {\\bf W}{\\bf H}_{RR}{\\bf W}={\\bf 0},   \\notag\\\\\n    && 0\\le p_A\\le P_A, 0\\le p_B\\le P_B,\\notag\n \\eea}\n {\\bl where $\\Gamma_B\\triangleq 2^{r_B}-1$ is the equivalent SINR constraint for the source ${\\tt B}$.}\n Observe that all terms in \\eqref{eqn:fd2} are quadratic in ${\\bf W}$\n except the ZF constraint ${\\bf W}{\\bf H}_{RR}{\\bf W}={\\bf 0}$, which is\n difficult to handle. Considering the fact that each source only transmits a single data\n stream and the network coding principle encourages mixing rather\n than separating the data streams from different sources,  we\n decompose ${\\bf W}$ as ${\\bf W} = {\\bf w}_t{\\bf w}_r^\\dag$, where\n${\\bf w}_r$ is the receive beamforming vector and ${\\bf w}_t$ is the\ntransmit beamforming vector   at ${\\tt R}$. Then the ZF condition is\nsimplified to $({\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t) {\\bf W} ={\\bf 0}$ or\nequivalently ${\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0$ because in general ${\\bf W}\\ne\n{\\bf 0}$. Without loss of optimality, we further assume\n$\\|{\\bf w}_r\\|=1$. As a result, the problem \\eqref{eqn:fd2} can be\nsimplified to\n   \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd711}\n    \\max_{{\\bf w}_r, {\\bf w}_t, p_A, p_B} &&   \\frac{p_B|{\\bf h}_{RA}^\\dag {\\bf w}_t|^2|{\\bf w}_r^\\dag  {\\bf h}_{BR}|^2}{  |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2 + p_A|h_{AA}|^2 +     1}  \\\\\n    \\mbox{s.t.} && \\frac{p_A|{\\bf h}_{RB}^\\dag {\\bf w}_t|^2 |{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2}{  |{\\bf h}_{RB}^\\dag {\\bf w}_t|^2  + p_B|h_{BB}|^2+\n     1} \\ge \\Gamma_B,\\notag\\\\\n     && p_A\\|{\\bf w}_t\\|^2 |{\\bf w}_r^\\dag{\\bf h}_{AR}|^2  + p_B\\|{\\bf w}_t\\|^2 |{\\bf w}_r^\\dag {\\bf h}_{BR}|^2  \\notag\\\\ &&+ \\|{\\bf w}_t\\|^2 \\le P_R,\\notag\\\\\n    && {\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0,   \\notag\\\\\n    && \\|{\\bf w}_r\\|=1,\\notag\\\\\n    && 0\\le p_A\\le P_A, 0\\le p_B\\le P_B.\\notag\n \\eea\n Note that in order to guarantee the feasibility of the ZF\n constraint ${\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0$, ${\\tt R}$ only needs to have\n two or more either transmit or receive antennas but not necessarily both.\n\n\n The problem \\eqref{eqn:fd711} is still quite complicated as\n variables ${\\bf w}_t$, ${\\bf w}_r$ and $(p_A, p_B)$ are coupled. Our idea\n to tackle this difficulty is to use an alternating optimization\n approach, i.e., at each iteration, we optimize one variable while\n keeping the other fixed, together with 1-D search to find ${\\bf w}_r$. Details are given\n below.\n\\subsection{Parameterization of the receive beamforming vector ${\\bf w}_r$}\n Observe that ${\\bf w}_r$ is  mainly involved in $|{\\bf w}_r^\\dag\n{\\bf h}_{BR}|^2$ and\n $|{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2$, so it has to balance the signals received\n from the\n two sources. According to the result in \\cite{Jorswieck2008f},\n ${\\bf w}_r$ can be parameterized by $0\\le \\alpha\\le 1$ as below:\n\\begin{equation}} \\newcommand{\\ee}{\\end{equation}\\label{eqn:wr} {\\bf w}_r =\n\\alpha\\frac{\\Pi_{{\\bf h}_{BR}}{\\bf h}_{AR}}{\\|\\Pi_{{\\bf h}_{BR}}{\\bf h}_{AR}\\|} +\n \\sqrt{1-\\alpha}\\frac{\\Pi^\\bot_{{\\bf h}_{BR}}{\\bf h}_{AR}}{\\|\\Pi^\\bot_{{\\bf h}_{BR}}{\\bf h}_{AR}\\|}.\n \\ee\n {\\bl We have to remark that \\eqref{eqn:wr} is not the complete characterization of ${\\bf w}_r$ because it is also involved in the ZF constraint\n ${\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0$, but this parameterization makes the\n problem more tractable.}\n\n\n Given $\\alpha$, we can  optimize ${\\bf w}_t$ and $(p_A,p_B)$ as will be introduced below. Then we perform 1-D search to find the\n optimal $\\alpha^*$. We will focus on how to separately optimize\n ${\\bf w}_t$  and $(p_A, p_B)$ in the following\n two subsections. For the optimization of each variable, we will  derive analytical\n solutions without using iterative methods or numerical algorithms.\n\n\n\n\n\\subsection{Optimization of the transmit beamforming vector ${\\bf w}_t$}\n We first look into the  optimization of  ${\\bf w}_t$ assuming ${\\bf w}_r$ and $(p_A,\n p_B)$ are fixed. Based on the problem \\eqref{eqn:fd711}, we get the\n following formulation:\n   \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd71123}\n    \\max_{{\\bf w}_t} &&   \\frac{p_B|{\\bf h}_{RA}^\\dag {\\bf w}_t|^2|{\\bf w}_r^\\dag  {\\bf h}_{BR}|^2}{  |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2 + p_A|h_{AA}|^2 +     1}  \\\\\n    \\mbox{s.t.} && \\frac{p_A|{\\bf h}_{RB}^\\dag {\\bf w}_t|^2 |{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2}{  |{\\bf h}_{RB}^\\dag {\\bf w}_t|^2  + p_B|h_{BB}|^2+\n     1} \\ge\\Gamma_B,\\notag\\\\\n     && p_A\\|{\\bf w}_t\\|^2 |{\\bf w}_r^\\dag{\\bf h}_{AR}|^2  + p_B\\|{\\bf w}_t\\|^2 |{\\bf w}_r^\\dag {\\bf h}_{BR}|^2 \\notag\\\\ &&\n     + \\|{\\bf w}_t\\|^2  \\le      P_R,\\notag\\\\\n   \n    && {\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0.  \\notag\n \\eea\n   By separating the variable ${\\bf w}_t$ and using monotonicity, \\eqref{eqn:fd71123} is\n simplified to\n   \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd712}\n    \\max_{{\\bf w}_t} &&   |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2  \\\\\n    \\mbox{s.t.}\n     &&  |{\\bf h}_{RB}^\\dag {\\bf w}_t|^2  \\ge\\frac{\\Gamma_B{p_B|h_{BB}|^2+1}}{p_A|{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2\n     -\\Gamma_B}\\triangleq\\bar\\Gamma_B,\n    \\notag\\\\\n     && \\|{\\bf w}_t\\|^2  \\le   \\frac{P_R}{(p_A|{\\bf w}_r^\\dag{\\bf h}_{AR}|^2  + p_B  |{\\bf w}_r^\\dag {\\bf h}_{BR}|^2  +1)}\\triangleq\\bar P,\\notag\\\\\n   \n    && {\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0.  \\notag\n \\eea\n  The problem \\eqref{eqn:fd712} is not convex  but it is a\n  quadratic problem in ${\\bf w}_t$.\n  By defining ${\\bf W}_t = {\\bf w}_t\n  {\\bf w}_t^\\dag$ and using semidefinite programming relaxation,  \\eqref{eqn:fd712} will become a convex problem in ${\\bf W}_t $, from which\n the optimal ${\\bf w}_t$ can be found from matrix decomposition. Interested readers are referred to \\cite{SDPR} for details. However, the special structure of  the problem\n \\eqref{eqn:fd712} allows us to derive the analytical solution in the following   steps.\n\\begin{enumerate}\n    \\item If $\\bar\\Gamma_B<0$ or $p_A|{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2 <\\Gamma_B$, the\n    problem is infeasible; otherwise continue.\n    \\item Define the null space of the vector ${\\bf w}_r^\\dag{\\bf H}_{RR}$\n    as $\\mathbf{N}_t\\in \\mathbb{C}^{M_t \\times (M_t-1)}$, i.e., ${\\bf w}_r^\\dag{\\bf H}_{RR}\\mathbf{N}_t={\\bf 0}$. Introduce a new variable ${\\bf v}\\in \\mathbb{C}^{(M_t-1)\\times 1}$ and express ${\\bf w}_t = \\mathbf{N}_t{\\bf v}$, then\n    we can remove the ZF constraint in \\eqref{eqn:fd712}, and obtain\n    the following equivalent problem:\n   \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd712e}\n    \\max_{{\\bf v}} &&   |{\\bf h}_{RA}^\\dag \\mathbf{N}_t{\\bf v}|^2  \\\\\n    \\mbox{s.t.}\n     &&  |{\\bf h}_{RB}^\\dag \\mathbf{N}_t{\\bf v}|^2  \\ge \\bar\\Gamma_B\n    \\label{eqn:constr1} \\\\\n     && \\|{\\bf v}\\|^2  \\le \\bar P, \\notag\n \\eea\n where we have used the property that $\\mathbf{N}_t^\\dag\n \\mathbf{N}_t={\\bf I}$. If $\\bar P\\|{\\bf h}_{RB}^\\dag \\mathbf{N}_t\\|^2< \\bar\\Gamma_B$, the problem\n is infeasible; otherwise continue.\n\n \\item In this step we aim to  find the closed-form solution for \\eqref{eqn:fd712e}. We first solve it without the constraint\n \\eqref{eqn:constr1}.\n It can be seen that the last power constraint should always be\n satisfied with equality, and the optimal solution is given by\n   ${\\bf v}^* = \\sqrt{\\bar P}\\frac{\\mathbf{N}_t^\\dag {\\bf h}_{RA}}{\\| \\mathbf{N}_t^\\dag\n {\\bf h}_{RA}\\|}$. If it also satisfies the constraint \\eqref{eqn:constr1},\n then it is the optimal solution; otherwise  continue.\n \\item It this step, we know that both constraints in \\eqref{eqn:fd712e} should be active, so we reach the\n problem below:\n   \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd713e}\n    \\max_{{\\bf v}} &&   |{\\bf h}_{RA}^\\dag \\mathbf{N}_t{\\bf v}|^2  \\\\\n    \\mbox{s.t.}\n     &&  |{\\bf h}_{RB}^\\dag \\mathbf{N}_t{\\bf v}|^2  =\\bar\\Gamma_B\n    \\label{eqn:constr2}\\notag\\\\\n     && \\|{\\bf v}\\|^2  =\\bar P. \\notag\n \\eea\n\\end{enumerate}\n If we define ${\\bf d}_2 \\triangleq\\frac{\\mathbf{N}_t^\\dag{\\bf h}_{RA}}{\\mathbf{N}_t^\\dag{\\bf h}_{RA}}, {\\bf d}_1\\triangleq\n \\frac{\\mathbf{N}_t^\\dag{\\bf h}_{RB}}{\\mathbf{N}_t^\\dag{\\bf h}_{RB}}$, $\\phi\\in (-\\pi,\\pi]$ be the argument of\n ${\\bf d}_2^\\dag{\\bf d}_1$, $r\\triangleq|{\\bf d}_2^\\dag{\\bf d}_1|$, $q\\triangleq \\frac{\\bar\\Gamma_B}{\\bar P\\|\\mathbf{N}_t^\\dag{\\bf h}_{RB}\\|^2}$ and   ${\\bf z}\\triangleq\\frac{{\\bf v}}{\\|{\\bf v}\\|}$, then we\n have the following formulation:\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:lem5}\n  \\max_{{\\bf z}}&&  {\\bf z}^\\dag {\\bf d}_2{\\bf d}_2^\\dag{\\bf z}, \\\\\n  \\mathrm{s.t.}&& {\\bf z}^\\dag {\\bf d}_1{\\bf d}_1^\\dag{\\bf z} =   q, \\quad \\|{\\bf z}\\|=1.\\notag\n \\eea\n\n The optimal solution ${\\bf z}^*$ follows from Lemma 2 in\n\\cite{Li-11} and is given below \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}{\\bf z}^*=\n \\left( r\\frac{1-q}{1-r^2}-\\sqrt{q}\\right) e^{j(\\pi-\\phi)}\n {\\bf d}_1+ \\sqrt{\\frac{1-q}{1-r^2}}{\\bf d}_2.\n\\eea\n Once we obtain ${\\bf z}^*$, the optimal transmit beamforming vector is\n given by\n \\begin{equation}} \\newcommand{\\ee}{\\end{equation} {\\bf w}_t^* = \\mathbf{N}_t {\\bf v}^* = \\sqrt{\\bar P}\\mathbf{N}_t{\\bf z}^*. \\ee\n\nAfter obtaining the optimal ${\\bf w}_t^*$ using the above procedures, we\ncan move on to find the optimal power allocation at the   sources.\n\n\n\\subsection{Optimization of the source power $(p_A, p_B)$}\n Because of the FD operation at the sources and the fact that each source has a single transmit and receive antenna, they cannot suppress the residual SI in the spatial domain\n therefore cannot  always use the full power. In contrast, ${\\tt R}$ has at least two   transmit or receive antennas, so it can complete eliminate\n the SI and transmit using full power  $P_R$. Here we aim to  find the optimal power allocation $(p_A, p_B)$ at ${\\tt A}$ and ${\\tt B}$ assuming both ${\\bf w}_t$ and ${\\bf w}_r$ are\n fixed.\n\n For convenience, define $C_{At}\\triangleq |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2,\nC_{rB}\\triangleq |{\\bf w}_r^\\dag {\\bf h}_{BR}|^2, C_{Bt}\\triangleq\n|{\\bf h}_{RB}^\\dag {\\bf w}_t|^2, C_{rA}\\triangleq |{\\bf w}_r^\\dag  {\\bf h}_{AR}|^2\n$, then \\eqref{eqn:fd711} becomes{\\small\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd77}\n    \\max_{p_A, p_B} &&   \\frac{p_B C_{At}C_{rB}}{  C_{At}  + p_A|h_{AA}|^2+     1}  \\\\\n    \\mbox{s.t.} && \\frac{p_A C_{Bt} C_{rA}}{  C_{Bt}   + p_B|h_{BB}|^2+\n     1} \\ge \\Gamma_B\\label{eqn:linear:SINR}\\\\\n     && p_A\\|{\\bf w}_t\\|^2 C_{rA}  + p_B\\|{\\bf w}_t\\|^2 C_{rB}  + \\|{\\bf w}_t\\|^2\\le P_R,\\label{eqn:linear:power}\\\\\n     && 0\\le p_A\\le P_A, 0\\le p_B\\le P_B.\\notag\n \\eea}\nThe problem \\eqref{eqn:fd77} is a linear-fractional programming\nproblem, and can be converted to a linear programming problem\n\\cite[p. 151]{Boyd}. Again, thanks to its special structure, we can\nderive its analytical solutions below step by step.\n\\begin{enumerate}\n    \\item First we check whether the constraint \\eqref{eqn:linear:SINR} is feasible. If  $P_A C_{Bt} C_{rA} \\le \\Gamma_B$, then the problem is\n    infeasible; otherwise, continue.\n\n    \\item Next we  solve \\eqref{eqn:fd77} by  ignoring  the   constraint\n    \\eqref{eqn:linear:power}. It is easy to check that at the\n    optimum, at least one source should achieve its  maximum power.\n     The power allocation depends on two cases:\n    \\begin{enumerate}\n    \\item If $ P_A \\ge  \\frac{\\Gamma_B(  C_{Bt} \\|{\\bf w}_r\\|^2 + P_B|h_{BB}|^2+\n     1)}{C_{Bt} C_{rA} }  $, then $p_B=P_B$, $p_A = \\frac{\\Gamma_B(  C_{Bt} \\|{\\bf w}_r\\|^2 + P_B|h_{BB}|^2+\n     1)}{C_{Bt} C_{rA} }$; otherwise,\n    \\item $p_A=P_A, p_B = \\min\\left(P_B, \\frac{\\frac{p_A C_{Bt} C_{rA}}{\\Gamma_B} - 1 - C_{Bt}\n    }{|h_{BB}|^2}\\right)$.\n    \\end{enumerate}\n\n    \\item We check whether the above obtained solution satisfies the constraint \\eqref{eqn:linear:SINR}. If it does, then it is the optimal solution. Otherwise\n     the constraint  \\eqref{eqn:linear:SINR} should be met with\n     equality.\n\n      \\item The optimal power allocation is determined by the equation\n      set below,\n    \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n        \\left\\{ \\begin{array}{l}\n              p_A C_{Bt} C_{rA} =\\Gamma_B(  C_{Bt}   + p_B|h_{BB}|^2+\n     1),  \\\\\n             p_A\\|{\\bf w}_t\\|^2 C_{rA}  + p_B\\|{\\bf w}_t\\|^2 C_{rB}  + \\|{\\bf w}_t\\|^2\n    =P_R\n           \\end{array}\\right.\n    \\eea\n    and the solution is given by\n    \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n        \\left\\{ \\begin{array}{l}\n                    p_A=     \\frac{\\Gamma_B|h_{BB}|^2}{C_{Bt}  + C_{rA}} p_B + \\frac{\\Gamma_B(C_{Bt}+1)}{C_{Bt}  +\n                    C_{rA}},\\\\\n                          p_B =  \\frac{\\frac{P_R}{\\|{\\bf w}_t\\|^2} -1 - \\frac{\\Gamma_B C_{rA}(C_{Bt}+1)\n             }{C_{Bt}C_{rA}}}{\\frac{\\Gamma_B C_{rA}|h_{BB}|^2\n             }{C_{Bt}C_{rA}} + C_{rB}  }.\n           \\end{array}\\right.\n    \\eea\n\\end{enumerate}\n\n\\subsection{The overall algorithm}\n Given an $\\alpha$ or ${\\bf w}_r$, we can iteratively optimize ${\\bf w}_t$  and  $(p_A,\n p_B)$ as above until convergence. The value of the objective function monotonically  increases as  the iteration goes thus converges to a local\n optimum. We can then conduct 1-D search over $0\\le \\alpha\\le 1$ to find the\n optimal $\\alpha^*$ or ${\\bf w}_r$. By enumerating source ${\\tt B}$'s requirement $r_B$, we can numerically\nfind the boundary of the achievable rate region.\n\n {\\bl About the complexity, we remark that at each iteration, the solutions of ${\\bf w}_t$ and $(p_A,\n p_B)$ are given in simple closed-forms, so the associated complexity is low.}\n\n\n\\section{Maximizing the Sum Rate}\n In this section, we aim to maximize the sum rate of the proposed FD TWRC, i.e., to solve $\\mathbb{P}_2$, which is rewritten\n below,\n    \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{prob:sum:rate:max}\n    \\max_{{\\bf w}_t, {\\bf w}_r p_A, p_B} &&   \\log_2\\left(1+ \\frac{p_B C_{rB} |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2}{  |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2 + p_A|h_{AA}|^2 +\n    1}\\right)\\\\ &&\n     +\\log_2\\left(1+  \\frac{p_A C_{rA}|{\\bf h}_{RB}^\\dag {\\bf w}_t|^2  }{  |{\\bf h}_{RB}^\\dag {\\bf w}_t|^2  + p_B|h_{BB}|^2+\n     1} \\right)\\notag \\\\\n    \\mbox{s.t.}    && \\|{\\bf w}_t\\|^2  \\le \\frac{P_R}{ p_A C_{rA} + p_BC_{rB}   + 1},\\notag\\\\\n    && {\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0.   \\notag\n \\eea\nWe will use the same characterization of \\eqref{eqn:wr} to find the optimal ${\\bf w}_r$ via 1-D\n search.  We then    concentrate on alternatingly optimizing the transmit beamforming vector ${\\bf w}_t$  and the power\n allocation $(p_A, p_B)$.\n \\subsection{Optimization of the transmit beamforming vector ${\\bf w}_t$}\n  We first study how to optimize ${\\bf w}_t$ given ${\\bf w}_r$ and  $(p_A, p_B)$.\n  For convenience, we define a semidefinite matrix  ${\\bf W}_t={\\bf w}_t{\\bf w}_t^\\dag$. Then the problem \\eqref{prob:sum:rate:max} becomes\n    \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{prob:sum:rate:max2}\n    \\max_{ {\\bf W}_t\\succeq {\\bf 0} } && F({\\bf W}_t)    \\\\\n    \\mbox{s.t.}    &&\\textsf{trace}({\\bf W}) \\le \\frac{P_R}{ p_A C_{rA} + p_BC_{rB}   + 1},\\notag\\\\\n    && \\textsf{trace}({\\bf W}_t {\\bf H}_{RR}^\\dag{\\bf w}_r{\\bf w}_r^\\dag{\\bf H}_{RR})=0,\n    \\notag\\\\\n    && \\mbox{rank}({\\bf W}_t)=1, \\notag\n \\eea\n  where $F({\\bf W}_t) \\triangleq   \\log_2\\left(1+ \\frac{p_B C_{rB} \\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)}{  \\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag) + p_A|h_{AA}|^2 +     1}\\right)\n     +\\log_2\\left(1+  \\frac{p_A C_{rA} \\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  }{  \\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2+\n     1} \\right).$ Clearly $F({\\bf W}_t)$ is not a concave function thus (\\ref{prob:sum:rate:max2}) is a cumbersome optimization problem.\n To tackle it, we propose to\nuse the DC   programming \\cite{An} to find a local optimum point. To\nthis end,  we express   $F({\\bf W}_t)$, as a difference of two concave\nfunctions $f({\\bf W}_t)$ and $g({\\bf W}_t)$, i.e., {  \\begin{align}\n  & F({\\bf W}_t)\\notag\\\\\n &=  \\log_2\\left( (p_B C_{rB}+1) \\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  +\np_A|h_{AA}|^2 + 1)\n   \\right) \\notag\\\\\n   & -  \\log_2\\left(\\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 + 1\\right)\\notag\\\\\n   &  + \\log_2\\left( (p_A C_{rA}+1) \\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2 + 1)\n   \\right)\\notag\\\\\n    & -  \\log_2\\left(\\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2 +\n    1\\right)\\notag\\\\\n&\\triangleq  f({\\bf W}_t) - g({\\bf W}_t) \\end{align}}\n where  {\\small \\begin{align} f({\\bf W}_t)\n&\\triangleq   \\log_2\\left( (p_B C_{rB}+1)\n\\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 + 1)\n   \\right) \\notag\\\\\n   & +\\log_2\\left( (p_A C_{rA}+1) \\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2 + 1)\n   \\right),\\notag\\\\\ng({\\bf W}_t) &\\triangleq   \\log_2\\left(\\textsf{trace}({\\bf W}_t{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 + 1\\right)\\notag\\\\\n&  +\\log_2\\left(\\textsf{trace}({\\bf W}_t{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2 +\n    1\\right).\\notag \\end{align}}\n  $f({\\bf W}_t)$ is a concave function while $g({\\bf W}_t)$ is a convex\n  function. The main idea is to approximate $g({\\bf W}_t)$ by a linear\n  function.  The linearization (first-order approximation) of $g({\\bf W}_t) $ around the point ${\\bf W}_{t,k}$\n is given by{\\small\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}  &&g_L({\\bf W}_t;{\\bf W}_{t,k}) =\n   \\frac{1}{\\ln(2)}\\frac{\\textsf{trace}\\left(({\\bf W}_{t}-{\\bf W}_{t,k}){\\bf h}_{RA}{\\bf h}_{RA}^\\dag\\right)}{\\textsf{trace}({\\bf W}_{t,k}{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 +\n   1} \\notag\\\\\n     &&+ \\frac{1}{\\ln(2)}\\frac{\\textsf{trace}\\left(({\\bf W}_{t}-{\\bf W}_{t,k}){\\bf h}_{RB}{\\bf h}_{RB}^\\dag\\right)}{\\textsf{trace}({\\bf W}_{t,k}{\\bf h}_{RB}{\\bf h}_{RB}^\\dag)  + p_B|h_{BB}|^2 +\n   1} \\notag\\\\\n   &&+\\log_2\\left(\\textsf{trace}({\\bf W}_{t,k}{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 +\n   1\\right)\\notag\\\\\n   &&+\\log_2\\left(\\textsf{trace}({\\bf W}_{t,k}{\\bf h}_{RA}{\\bf h}_{RA}^\\dag)  + p_A|h_{AA}|^2 +\n   1\\right).\n\\eea}  Then the DC programming is applied to sequentially solve the\nfollowing convex problem,  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n{\\bf W}_{t,k+1} &= &\\arg \\max_{{\\bf W}_t}\\ f({\\bf W}_t) - g_L({\\bf W}_t; {\\bf W}_{t,k}) \\label{DCP}\\\\\n&&\\mbox{s.t.}\\quad  \\textsf{trace}({\\bf W}_t) = \\frac{P_R}{ p_A C_{rA} + p_BC_{rB}   + 1},\\notag\\\\\n    && \\textsf{trace}({\\bf W}_t {\\bf H}_{RR}^\\dag{\\bf w}_r{\\bf w}_r^\\dag{\\bf H}_{RR})=0\n    \\notag.\n\\eea To summarize, the problem \\eqref{prob:sum:rate:max2} can be\nsolved by i) choosing an initial point ${\\bf W}_t$; and ii) for $k=0, 1,\n\\cdots$, solving (\\ref{DCP}) until the termination condition is met.\nNotice that in \\eqref{DCP} we have ignored the rank-1 constraint on\n${\\bf W}_t$. This constraint is guaranteed to be satisfied by the\nresults in \\cite[Theorem 2]{Huang-11} when $M_t>2$, therefore the\ndecomposition of ${\\bf W}_t$ leads to the optimal solution ${\\bf w}_t^*$ for\n\\eqref{prob:sum:rate:max}. When $M_t=2$, the ZF constraint in the\nproblem \\eqref{prob:sum:rate:max} can determine the direction of\n${\\bf w}_t$, i.e., ${\\bf w}_t=\\sqrt{p_t}\\mathbb{N}_t$ where $p_t$ is the\ntransmit power and $\\mathbb{N}_t\\in \\mathbb{C}^{2\\times 1}$\nrepresents the null space of ${\\bf w}_r^\\dag{\\bf H}_{RR}$. Therefore the\noptimization of ${\\bf w}_t$ reduces to optimizing a scalar variable\n$p_t$, which can be found by checking the stationary points of the\nobjective function in \\eqref{prob:sum:rate:max} and the boundary\npoint without using the DC programming. The same applies to the\nspecial case of $M_t=1$.\n\n\n\\subsection{Optimization of source power $(p_A, p_B)$}\nWith ${\\bf w}_t$ and ${\\bf w}_r$ fixed, the sum rate maximization problem\n\\eqref{prob:sum:rate:max} about power allocation can be written as\n  \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:fd777}\n    \\max_{p_A, p_B} &&   \\log_2 \\left(1+ \\frac{p_B C_{At}C_{rB}}{  C_{At}  + p_A|h_{AA}|^2+     1}\\right)\\notag\\\\\n    &&+\\log_2 \\left( 1+\\frac{p_A C_{Bt} C_{rA}}{  C_{Bt}   + p_B|h_{BB}|^2+\n     1}  \\right)  \\\\\n    \\mbox{s.t.} && p_A  C_{rA}  + p_B  C_{rB}  + 1 \\le \\frac{P_R}{\\|{\\bf w}_t\\|^2},\\label{eqn:constr3}\\\\\n     && 0\\le p_A\\le P_A, 0\\le p_B\\le P_B\\notag.\n \\eea\n Note that when the first relay power constraint \\eqref{eqn:constr3} is not tight, the problem is\n the same as the conventional power allocation among two interference links to maximize the sum\n rate, and the optimal power solution is known to be binary \\cite{binary-power},\n i.e., the optimal power allocation $ (p_A^*, p_B^*)$ should satisfy\n \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n    (p_A^*, p_B^*)\\in\\{(0, P_B), (P_A,0), (P_A, P_B)\\}.\n \\ee\n\n Next we only focus on the case in which the constraint \\eqref{eqn:constr3} is\n active, i.e., $p_A  C_{rA}  + p_B  C_{rB}  + 1\n =\\frac{P_R}{\\|{\\bf w}_t\\|^2}$.\nWe then have\n \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n    p_A = \\frac{\\frac{P_R}{\\|{\\bf w}_t\\|^2-1}-p_B  C_{rB}}{C_{rA} }.\n \\ee\n Because $0\\le p_A\\le P_A$, we can obtain the feasible range $[p_B^{\\min}, p_B^{\\max}]$ for $p_B$:\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n     p_B^{\\min} &=& \\max\\left(0, \\frac{ {P_R}{\\|{\\bf w}_t\\|^2}-1 - C_{rA} P_A }{ C_{rB}}\n    \\right),\\notag \\\\\n     p_B^{\\max} &=& \\min \\left(P_B, \\frac{{P_R}{\\|{\\bf w}_t\\|^2}-1}{ C_{rB}}\\right).\n \\eea\n\n The objective function of \\eqref{eqn:fd777} then becomes a function of $p_B$ only, i.e.,\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    y(p_B) &=& \\log_2 (C_{At}  + p_A|h_{AA}|^2+     1+ p_B C_{At}C_{rB})\\notag\\\\&& -\\log_2(  C_{At}  + p_A|h_{AA}|^2+     1)\\notag\\\\\n    &&+\\log_2 ( C_{Bt}   + p_B|h_{BB}|^2+   1+ p_A C_{Bt} C_{rA})\\notag\\\\&& -\\log_2(  C_{Bt}   + p_B|h_{BB}|^2+\n    1),\n \\eea\n and \\eqref{eqn:fd777} reduces to a one-variable optimization, i.e.,\n \\begin{equation}} \\newcommand{\\ee}{\\end{equation}\n  \\max_{p_B} ~~ y(p_B)~~~~ \\mbox{s.t.}~~  p_B^{\\min}\\le p_B\\le  p_B^{\\max}.\n \\ee\n\n Setting $\\frac{\\partial y(p_B)}{\\partial p_B}=0$ leads to\n {\\small \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n&&  \\frac{ C_{At}C_{rB} -\\frac{  C_{rB}}{C_{rA} } |h_{AA}|^2 }{ C_{At}  + \\frac{\\frac{P_R}{\\|{\\bf w}_t\\|^2-1}  |h_{AA}|^2}{C_{rA}}+     1\n+ p_B \\left( C_{At}C_{rB} -\\frac{  C_{rB}}{C_{rA} } |h_{AA}|^2\\right) }  \\\\\n&& + \\frac{\\frac{  C_{rB}}{C_{rA} } |h_{AA}|^2}{  C_{At}  +\n\\frac{\\frac{P_R}{\\|{\\bf w}_t\\|^2-1}}{C_{rA} } |h_{AA}|^2 +1   -\np_B\\frac{  C_{rB}}{C_{rA} } |h_{AA}|^2 }\\notag\\\\\n&& +\\frac{  |h_{BB}|^2- \\frac{ C_{rB}}{C_{rA} } C_{Bt}\n    C_{rA} }{    C_{Bt}    + \\frac{\\frac{P_R}{\\|{\\bf w}_t\\|^2-1}}{C_{rA} } C_{Bt} C_{rA}+ 1+ p_B\\left( |h_{BB}|^2- \\frac{ C_{rB}}{C_{rA} } C_{Bt}\n    C_{rA}\\right)}\\notag\\\\\n&& - \\frac{|h_{BB}|^2}{ C_{Bt}   + p_B|h_{BB}|^2+1}\n  =0.\\notag\n \\eea}\nThis in turn  becomes a cubic (3-rd order) equation and all roots\ncan be found analytically. Suppose that the set of all positive root\nwithin $(p_B^{\\min}, p_B^{\\max})$ is denoted as $\\Psi$ which may\ncontain  0, 1 or 3 elements. In order to find the optimal $p_B^*$,\nwe need to compare the objective values  of all elements in the set\n$\\Psi\\cup\\{p_B^{\\min}, p_B^{\\max}\\}$ and choose the one that results\nin the maximum objective value.\n\n\n{\\bl We comment that the complexity of the overall algorithm to\nmaximize the sum rate is dominated by the optimization of ${\\bf w}_t$\nand more specially, solving the problem \\eqref{DCP}. Since\n\\eqref{DCP} is a semi-definite programming (SDP) problem with one\nvariable ${\\bf W}_t\\in \\mathbb{C}^{M_t\\times M_t}$  and two constraints,\nthe worst-case complexity to solve it is\n$\\mathcal{O}(M_t^{4.5}\\log(\\frac{1}{\\epsilon}))$ where $\\epsilon$ is\nthe desired solution accuracy \\cite{SDPR}.}\n\n\n\\section{Benchmark schemes}\nIn this section, we introduce three benchmark schemes that the\nproposed FD network scheme can be compared with. The first one is\nthe conventional two-phase HD TWRC using the analog network coding,\nwhich is known to outperform the three-phase and four-phase HD\nschemes  to provide throughput gain\\cite{practical_PLNC}; the second\none is a two-phase one-way FD scheme  and in each phase, the relay\nworks in the FD mode\\cite{FD-two-phase}; the last one ignores the\nresidual SI channel at the relay thus provides a performance upper\nbound that is useful to evaluate the proposed algorithms.\n\n\\subsection{Two-phase HD relaying using analog network coding}\n HD analog network coding is introduced in \\cite{Zhang-2Phase} which takes\n two phases to complete information exchange between ${\\tt A}$ and\n ${\\tt B}$. In the first phase, both sources transmit to ${\\bf R}$ and in\n the second phase, the relay multiplies the received signal by a\n beamforming matrix ${\\bf W}$ then  broadcasts it to ${\\tt A}$ and\n ${\\tt B}$. Because there is no SI, every node can use its full power,\n so only ${\\bf W}$ needs to be optimized.  The achievable rate pair\n and the relay power consumption, are given by\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    R_A &=& \\frac{1}{2}\\log_2\\left(1+ \\frac{P_B|{\\bf h}_{RA}^\\dag{\\bf W}  {\\bf h}_{BR}|^2}{  \\|{\\bf h}_{RA}^\\dag{\\bf W}\\|^2\n    +1}\\right),\\notag\\\\\n    R_B &=& \\frac{1}{2}\\log_2\\left(1+ \\frac{P_A|{\\bf h}_{RB}^\\dag{\\bf W}  {\\bf h}_{AR}|^2}{\n\\|{\\bf h}_{RB}^\\dag{\\bf W}\\|^2  +   1}\\right),\\notag\\\\\n   p_R &=& \\|{\\bf W} {\\bf h}_{AR}\\|^2 P_A + \\|{\\bf W} {\\bf h}_{BR}\\|^2 P_B\n   +\\textsf{trace}({\\bf W}\\qW^\\dag).\\notag\n \\eea\n where the factor of $\\frac{1}{2}$ is due to the two\n transmission phases used.\n Given the above rates and the power expression, problems\n $\\mathbb{P}_1$ and $\\mathbb{P}_2$ can be solved to find the\n achievable rate region and the maximum sum rate, respectively.\n Compared with this scheme, the proposed FD relaying can reduce the\n total communication phases to one, thus has the potential to\n improve the throughput.\n\n\\subsection{Two-phase one-way FD}\n Another scheme that we will compare with is the FD one-way relaying\n in which ${\\tt R}$ works in the FD mode while the two sources work in the\n HD mode. In this way, both sources can transmit with the maximum power. We use the same notation as the proposed scheme and the relay beamforming matrix is  ${\\bf W}={\\bf w}_t{\\bf w}_r^\\dag$.\n\n The achievable rate, relay power constraint, and zero residual SI constraint for source ${\\tt A}$ (direction: ${\\tt B} \\rightarrow{\\tt A}$)  are, respectively,\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\n    &&R_A =\\log_2\\left(1+  \\frac{P_B|{\\bf h}_{RA}^\\dag {\\bf w}_t|^2|{\\bf w}_r^\\dag  {\\bf h}_{BR}|^2}{  |{\\bf h}_{RA}^\\dag {\\bf w}_t|^2   +     1} \\right), \\\\\n &&p_B\\|{\\bf w}_t\\|^2 |{\\bf w}_r^\\dag {\\bf h}_{BR}|^2  +\n\\|{\\bf w}_t\\|^2\\|{\\bf w}_r\\|^2\\le P_R, \\\\\n &&{\\bf w}_r^\\dag{\\bf H}_{RR}{\\bf w}_t=0.\\label{eqn:ZF}\n \\eea\n\n\n In our previous work \\cite{MIMO-relay-HD}, we have derived the closed-form expressions below for $R_A$\n depending on how the ZF constraint is   realized:\n \\begin{enumerate}\n\\item\n  Receive ZF. In this case, we assume ${\\bf w}_t={\\bf h}_{RA}$ and choose\n  ${\\bf w}_r$ to achieve \\eqref{eqn:ZF}. We showed that the achievable\ne2e received signal-to-noise ratio (SNR) can be expressed as\n \\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:SNR:R}\n    \\gamma_{RZF}\n         =\\frac{P_B \\|{\\bf D}{\\bf h}_{BR}\\|^2 P_R \\|{\\bf h}_{RA}\\|^2}\n    {P_B \\|{\\bf D}{\\bf h}_{BR}\\|^2 + P_R\\|{\\bf h}_{RA}\\|^2 + 1},\n  \\eea\n where ${\\bf D}\\triangleq \\Pi^\\bot_{{\\bf H}_{RR}{\\bf h}_{RA}}$.\n\n\n\\item Transmit ZF. In this case, we assume ${\\bf w}_r={\\bf h}_{BR}$ and choose\n  ${\\bf w}_t$ to achieve \\eqref{eqn:ZF}. We then reach the following\n  achievable SNR:\n\\begin{eqnarray}} \\newcommand{\\eea}{\\end{eqnarray}\\label{eqn:SNR:t}\n    \\gamma_{TZF} = \\frac{  P_B \\|{\\bf h}_{BR}\\|^2 P_R\\|{\\bf B}{\\bf h}_{RA}\\|^2}{\n        {P_B\\|{\\bf h}_{BR}\\|^2  + {P_R}\\|{\\bf B}{\\bf h}_{RS}\\|^2+ 1 }},\n\\eea  where ${\\bf B}\\triangleq \\Pi_{{\\bf H}_{RR}^\\dag{\\bf h}_{BR}}^\\bot$.\n\\end{enumerate}\n $R_A$ is then determined by $R_A=\\log_2(1+\\max(\\gamma_{RZF},\n \\gamma_{TZF}))$. We can derive similar achieve rate $R_B$  for the source\n ${\\tt B}$.\n\n Note that $R_A$ and $R_B$  cannot be achieved simultaneously as it requires\n that each corresponding source occupies the whole transmission time.\n The boundary of the  rate region can be obtained by using time-sharing parameter $t\\in[0,1]$, i.e., $(t R_A, (1-t)R_B)$.\n\n\\subsection{FD-Upper Bound}\n This scheme is the same as the proposed FD scheme except that we\n assume there is no SI at ${\\tt R}$, but we still consider the SI at the two sources, i.e., ${\\bf H}_{RR}={\\bf 0}, |h_{AA}|>0, |h_{BB}|>0$.\n In this case the ZF constraint in \\eqref{eqn:fd2} is not necessary.\n We remark that this scheme uses unrealistic assumption\n of ${\\bf H}_{RR}={\\bf 0}$, so it is not a practical scheme but provides a\nuseful upper bound to evaluate the performance of the proposed\nalgorithms.\n\nIn the simulation results, we will label the above three benchmark\nschemes as ``Two-phase HD'', ``Two-phase FD'' and ``Proposed\none-phase FD upper bound'', respectively.\n\n\n\n\\section{Numerical Results}\\label{sec:simu}\nIn this section  we provide numerical results to  illustrate the\nachievable rate region and the sum rate performance  of the proposed\nFD two-way relaying scheme. We compare it with the above mentioned\nthree benchmark schemes. The simulation set-up follows the system\nmodel in Section II. Unless otherwise specified, we assume that\nthere are $M_T=M_R=3$ antennas at ${\\tt R}$,  the average residual SI\nchannel gain is  $\\sigma^2_A=\\sigma^2_B=\\sigma^2_R=-20$ dB, {\\bl and\nthe per-node transmit SNR for both sources and the relay is\n$P_A=P_B=P_R=10$ dB,  which are the power constraints in problem\nformulations $\\mathbb{P}_1$ and $\\mathbb{P}_2$}. The results are\nobtained using 100 independent channel realizations.\n\n\n\\input figures.tex\n\n\n\\subsection{Achievable rate region}\nFirst we illustrate the achievable {\\bl average} rate region for the\ntwo sources ${\\tt A}$ and ${\\tt B}$ in Fig. \\ref{fig:rate:region}. {\\bl It\nis seen that the two-phase FD scheme already greatly enlarges the\nachievable data region of the conventional HD scheme. The proposed\none-phase FD scheme achieves significantly larger rate region  over\nthe two-phase FD and the conventional  HD schemes.} We observe that\nthere is still a noticeable gap between the proposed scheme and its\nupper bound because the proposed  solution  is suboptimal.\n\n\n\n\\subsection{Sum rate performance}\nWe then investigate the effect  of the source transmit SNR $P_A$ and\n$P_B$ ($P_A=P_B$) on the sum rate  shown in Fig.\n\\ref{fig:sumrate:R1020dB} when { $P_R=10$ dB (solid curves) and\n$P_R=20$ dB (dashed curves), respectively}. {\\bl We first consider\nthe case $P_R=10$ dB.}  As expected, the sum rate improves as the\nsource transmit SNR increase and the proposed one-phase FD schemes\nclearly outperforms the   two benchmark schemes. When the source SNR\nis above 15 dB, the sum rate of the proposed FD scheme saturates.\nThis is because the high transmit power results in high residual SI,\ntherefore increasing power budget does not necessarily improve the\nperformance. To illustrate the performance improvement in sum rate,\nwe show the sum rate gain  over the conventional two-phase HD scheme\nin Fig. \\ref{fig:sumrate:gain:R1020dB}. It is observed  that when\nthe source SNR is 10 dB, the proposed one-phase FD scheme and the\ntwo-phase FD scheme can achieve the sum rate gain of  1.56 and 1.22,\nrespectively. Even the performance upper bound cannot achieve double\nrate because of the residual SI at both sources. The rate gain in\ngeneral decreases as the source transmit SNR increases again because\nof the residual SI therefore the sources need to carefully adjust\nits transmit power.\n\n\nThe same trend is observed   when $P_R=20$ dB. In this case, a sum\nrate gain of nearly 1.7 is recorded when the source transmit SNR is\n10 dB. Another interesting observation is that   the performance of\nthe proposed scheme is very close to the upper bound. {  This is\nbecause when the relay power is high, the e2e performance is limited\nby the link from the source to the relay, rather than the relay to\nthe other source, therefore the residual SI at the relay has little\neffect on the sum rate.}\n\nThe impact of the relay transmit SNR on the sum rate is shown in\nFig. \\ref{fig:sumrate:RPower}. It is seen that when the relay SNR is\nlow, the proposed one-phase FD scheme achieves lower sum rate than\nthe two-phase FD scheme at low transmit SNRs then outperforms the\nlatter when the transmit SNR is above 5 dB. The performance gain is\nremarkable when the  relay transmit SNR is high.  This is because\nunlike the two sources,  the relay   can null out the residual SI\nusing multiple antennas, therefore it can always  use  the maximum\navailable power to improve the sum rate.\n\nThe effect of the   residual SI channel gain at the sources is\nexamined in Fig. \\ref{fig:sumrate:SI}. Naturally, the sum rate\ndecreases as the residual SI channel becomes stronger or the SI is\nnot adequately suppressed. When the residual SI channel gain is\nabove -5 dB, the two-phase FD scheme outperforms the one-phase FD\nscheme while both still achieve higher sum rate than the\nconventional two-phase HD scheme  even when the SI channel gain is\nas high as 5 dB.\n\n\n Next we show the sum rate and rate gain results when  the number of antennas ($M_R=M_T$) at the\nrelay varies from 2 to 6 in Fig. \\ref{fig:sumrate:antennas} and\n\\ref{fig:sumrate:gain:antennas}, respectively. The sum rate steady\nincreases as more antennas are placed at the relay due to the array\ngain. It is observed that the rate gain remains about 1.55 when the\nnumber of antennas is greater than 2.\n\n\n\\subsection{Asymmetric channel gain}\n The above results are mainly for a symmetric case, i.e., both sources have\n similar power constraints and channel {\\bl strengths}. Here we consider an\n asymmetric case where the average channel gain between ${\\tt R}$ and\n ${\\tt B}$ is -10 dB. We plot the rate region in Fig. \\ref{fig:sumrate:region:asymmetric}\nusing the same system parameters as those in Fig. 2 except that the\ngain of channel vectors ${\\bf h}_{BR}$ and\n ${\\bf h}_{RB}$ is 10 dB weaker. {\\bl The results show that both sources'\n rates are reduced while the source ${\\tt B}$ suffers more rate loss.} This is because one\n source's channels to and from the relay will also affect the\n performance of the other source. The sum rate comparison is given\n in Fig. \\ref{fig:sumrate:asymmetric} and it is observed the performance of the two-phase {\\bl FD} scheme is very close to that of the proposed FD scheme at all SNR region. This can be explained\n by the fact that e2e performance is restricted by the channel quality between ${\\tt R}$\n and ${\\tt B}$, so the gain due to the simultaneous transmission of two sources  is limited.\n\n{\n\\subsection{Impact of the local channel state information}\n Finally  we consider  the case that only the  receive CSI at each node is available but the transmit CSI is unknown. Because of  the lack of the transmit\n    CSI,   the two sources use full power\n $P_A$ and $P_B$; ${\\bf w}_t$ at the relay is chosen arbitrarily to satisfy the ZF constraint and the relay power\n constraint. The sum rate performance is shown in Fig.\n \\ref{fig:sumrate:csi}. It is seen that the proposed FD scheme still achieves significant performance gain over the HD relaying at low to medium transmit SNRs although all rates\n are much lower than the case with the global CSI in Fig. 3. Another\n notable difference is that at high transmit SNRs, the performance of the proposed FD scheme  degrades quickly.\n This is because the two sources need to adjust its transmit power\n rather than using full power. This highlights the importance of the\n global CSI for adapting  the transmit power and  adjusting the relay beamforming.\n}\n\n\n  \\section{Conclusion  and Future work}\n  We have investigated the application of the FD operation to MIMO TWRC, which requires only one phase for the two sources to exchange\n  information. We studied two problems of finding the achievable\n  rate region and maximizing the sum rate by optimizing the relay\n  beamforming matrix and power allocation at the  sources.\n  Iterative algorithms are proposed together with 1-D search to\n  find the local optimum solutions. At each iteration, either\n  analytical solution or convex formulation has been derived. We\n  have conducted intensive simulations to illustrate the effects of\n  different system parameters. The results show that the FD\n  operation has great potential to achieve much higher data rates than the conventional HD TWRC.\n\n\n{\\bl Regarding the future directions, better suboptimal and the\noptimal solutions are worth studying.} There are a couple of reasons\nwhy the proposed algorithm is sub-optimal such as  the additional ZF\nconstraint, {\\bl the incomplete characterization of the receive\nbeamforming vector}, and the alternating optimization algorithms.\nAnother direction is to study the use of multiple transmit\/receive\nantennas at the two sources. If a single data stream is transmitted,\nthe residual SI can be removed using the ZF criterion at the sources\nas well. This actually simplifies the optimization as the two\nsources can use the maximum power. However, multiple antennas can\nsupport multiple and variable number of data streams, and when the\nproblem is coupled with the SI suppression, it will be much more\nchallenging. Thirdly, in this paper, we focus on the benefit of the\nFD {\\bl in terms of} spectrum efficiency. In \\cite{practical_PLNC},\nit is shown that for the HD case, three-phase transmission schemes\noffers a better compromise between the sum rate and the bit error\nrate than the two-phase scheme, especially in the asymmetric case.\nIt is worth investigating whether such an trade-off also exists for\nthe FD scenario. \\vspace{-2mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecent observations of Type IA Supernovae \\cite{SN1,SN2} as well as\nconcordance with other observations (including the microwave\nbackground and galaxy power spectra) indicate that the universe is\naccelerating.  Possible explanations for such an acceleration include\na variety of dark energy models in which the universe is vacuum\ndominated, such as a cosmological constant.  In 1986, we explored the\npossibility of a time-dependent vacuum \\cite{fafm} characterized by an\nequation of state $w$.  Quintessence models\n\\cite{ratpeeb,frieman,wett,stein,caldwell} utilize a rolling scalar\nfield to achieve time dependence of the vacuum.\n\nAs an alternative approach to explain the acceleration, we proposed\nmodifications to the Friedmann equation in lieu of having any vacuum\nenergy at all \\cite{freeselewis} (hereafter Paper I).  In our\nCardassian model, \\footnote{The name Cardassian refers to a humanoid\n  race in Star Trek whose goal is accelerated expansion of their evil\n  empire (aka George W). This race looks foreign to us and yet is made\n  entirely of matter.}  the universe is flat and accelerating, and yet\nconsists only of matter and radiation.\nThe usual Friedmann equation governing the expansion of the universe\n\\begin{equation}\n\\label{eq:usual}\nH^2 = \\left({\\dot a \\over a}\\right)^2 = {8 \\pi G \\over 3} \\rho \n\\end{equation}\nis modified to become\n\\begin{equation}\n\\label{eq:general}\nH^2 = \\left({\\dot a \\over a}\\right)^2 = g(\\rho) ,\n\\end{equation}\nwhere $\\rho$ contains only matter and radiation (no vacuum), $H=\\dot\na\/a$ is the Hubble constant (as a function of time), $G=1\/m_{pl}^2$ is\nNewton's gravitational constant, and $a$ is the scale factor of the\nuniverse.  We note here that the geometry is flat, as required by\nmeasurements of the cosmic background radiation \\cite{boom,dasi}, so that\nthere are no curvature terms in the equation.  There is no vacuum term\nin the equation.  The model does not address the cosmological constant\n($\\Lambda$) problem; we simply set $\\Lambda=0$.\n\nWe take $g(\\rho)$ to be a function of $\\rho$ that returns simply to $8\n\\pi G \\rho\/3$ at early epochs, but that takes a different form that\ndrives an accelerated expansion in the recent past of the universe at\n$z<{\\cal O}(1)$.\n\nI begin by describing the phenomenology of Cardassian models, and then\nturn to the motivation for modified Friedmann equations of this form.\nThe new term required may arise, e.g., as a consequence of our\nobservable universe living as a 3-dimensional brane in a higher\ndimensional universe.  A second possible interpretation of Cardassian\nexpansion is developed \\cite{gondolo}, in which we treat the modified\nFriedman equations as due to a fluid, in which the energy density has\nnew contributions with negative pressure (possibly due to dark matter\nwith self-interactions).\n\n\\section{Power Law Cardassian Model}\n\nThe simplest version of Cardassian expansion invokes the addition of a\nnew power law term to the right hand side of the Friedmann equation:\n\\begin{equation}\n\\label{eq:new}\nH^2 ={8\\pi G\n\\over 3 } \\rho + B \\rho^n \n\\end{equation}\nwhere $n$ is a number with\n\\begin{equation}\nn<2\/3 .\n\\end{equation}\nThe new term is initially negligible, and only comes to dominate at\nredshift $z \\sim {\\cal O}(1)$.  Once it dominates, it causes the\nuniverse to accelerate.\n\nWe take the usual energy conservation:\n\\begin{equation}\n\\label{eq:energy}\n\\dot \\rho + 3H (\\rho + p) = 0 ,\n\\end{equation}\nwhich gives the evolution of matter:\n\\begin{equation}\n\\rho_M = \\rho_{M,0}(a\/a_0)^{-3} .\n\\end{equation}\nHere subscript $0$ refers\nto today.  \nEqs.(\\ref{eq:general}) and (\\ref{eq:energy})\ncontain the complete information of the expansion history.\n\nThe new term in Eq.(\\ref{eq:new}) (the second term on the right hand side)\nis initially negligible; hence ordinary early universe cosmology,\nincluding nucleosynthesis, results..  The new term only comes to\ndominate recently, at the redshift $z_{car} \\sim O(1)$ indicated by\nthe supernovae observations.  Once the second term dominates, it\ncauses the universe to accelerate.  When the new term is so large that\nthe ordinary first term can be neglected, we find\n\\begin{equation}\nR \\propto t^{2 \\over 3n}\n\\end{equation}\nso that the expansion is superluminal (accelerated) for $n<2\/3$.  As\nexamples, for $n=2\/3$ we have $R \\sim t$; for $n=1\/3$ we have $R \\sim\nt^2$; and for $n=1\/6$ we have $R \\sim t^4$.  \nFor $n<1\/3$ the acceleration is increasing (the cosmic jerk).\n\nThere are two free parameters in the power law model, $B$ and $n$ (or\nequivalently, $z_{car}$ and $n$).  We choose one of these parameters\nto make the second term term kick in at the right time to explain the\nobservations.  As yet we have no explanation of the coincidence\nproblem; i.e., we have no explanation for the timing of $z_{car}$.\nSuch an explanation may arise in the context of extra dimensions.\n\nObservations of the cosmic background radiation show that the geometry\nof the universe is flat with $\\Omega_0=1$.  How can we reconcile this\nwith a theory made entirely of matter (and radiation) and yet\nobservations of the matter density that indicate $\\Omega_M =\n0.3$? In the Cardassian model we need to revisit the question of what\nvalue of energy density today, $\\rho_0$, corresponds to a flat\ngeometry.  We will show that the energy density required to close the\nuniverse is much smaller than in a standard cosmology, so that matter\ncan be sufficient to provide a flat geometry.\n\nFrom evaluating Eq.(\\ref{eq:new}) today, we have\n\\begin{equation}\n\\label{hubbletoday}\nH_0^2 = A \\rho_0 + B \\rho_0^n .\n\\end{equation}\nThe energy density $\\rho_0$ that satisfies Eq.(\\ref{hubbletoday}) is,\nby definition, the critical density.  The usual value of the critical\ndensity is found by solving the equation with only the first term\n(with $B=0$), so that\n\\begin{equation}\n\\rho_{c} = {3 H_0^2\n\\over 8 \\pi G} = 1.88 \\times 10^{-29} h_0^2 {\\rm gm\/cm^{-3}}\n\\end{equation}\nand $h_0$ is the Hubble constant today in units of 100 km\/s\/Mpc.\nHowever, in the presence of the second term, we can solve\nEq.(\\ref{hubbletoday}) to find that the critical density $\\rho_c$ has\nbeen modified from its usual value to a different number $\\tilde\n\\rho_c$.  We use our second free parameter to fix this number to give\n\\begin{equation}\n\\rho_{M,0} = \\tilde \\rho_c = 0.3 \\rho_c \n\\end{equation}\nso that $\\Omega_M^{obs} = \\rho_{M,0}\/\\rho_c = 0.3$.\n\nHence the universe can be flat, matter dominated, and accelerating,\nwith a matter density that is 0.3 of the old critical density.\nMatter can provide the entire closure density of the Universe.\n\nAn equivalent formulation of power law Cardassian is\n\\begin{equation}\n\\label{eq:new2}\nH^2 = A \\rho [1 + ({\\rho \\over \\rho_{car}})^{n-1}] .\n\\end{equation}\nThe first term inside the bracket dominates initially but the second\nterm takes over once the energy density has dropped to the value\n$\\rho_{car}$.  Here, $\\rho_{car}$ is the energy density at which the\ntwo terms are equal: the ordinary energy density term on the right\nhand side of the FRW equation is equal in magnitude to the new term.\nFor reasonable parameters, $\\rho_{car} \\sim 2.7 \\tilde \\rho_c \\sim\n10^{-29}$gm\/cm$^3$. Since the modifications are important only for $\\rho\n< \\rho_{card}$, solar system physics is completely unaffected.\n\n\\subsection{Observational tests  of power law Cardassian}\n\nThe power law Cardassian model with an\nadditional term $\\rho^n$ satisfies many observational constraints: the\nuniverse is somewhat older, the first Doppler peak in the microwave\nbackground is slightly shifted, early structure formation ($z>1$) is\nunaffected, but structure will stop growing sooner. In addition the\nmodifications to the Poisson equation will affect cluster abundances and\nthe ISW affect in the CMB.\n\n\\subsection{Comparing to Quintessence}\n\nWe note that, with regard to observational tests, one can make a\ncorrespondence between the $\\rho^n$ Cardassian and Quintessence models\nfor constant $n$; we stress, however, that the two models are entirely\ndifferent. Quintessence requires a dark energy component with a specific\nequation of state ($p = w\\rho$), whereas the only ingredients in the\nCardassian model are ordinary matter ($p = 0$) and radiation ($p =\n1\/3$). However, as far as any observation that involves only $R(t)$, or\nequivalently $H(z)$, the two models predict the same effects on the\nobservation.  Regarding such observations, we can make the following\nidentifications between the Cardassian and quintessence models: $n\n\\Rightarrow w+1$, $F\\Rightarrow \\Omega_m$, and $1-F \\Rightarrow\n\\Omega_Q$, where $w$ is the quintessence equation of state parameter,\n$\\Omega_m= \\rho_m\/\\rho_{c}$ is the ratio of matter density to the\n(old) critical density in the standard FRW cosmology appropriate to\nquintessence, $\\Omega_Q= \\rho_Q\/\\rho_{c}$ is the ratio of\nquintessence energy density to the (old) critical density, and \n$F=\\tilde{\\rho_c}\/\\rho_c$.  In this way, the Cardassian model with\n$\\rho^n$ can make contact with quintessence with regard to observational\ntests.  We note that Generalized Cardassian models can be distinguished\nfrom generic quintessence models with upcoming precision cosmological\nexperiments.\n\n\\subsection{Best Fit of Parameters to Current Data}\n\nWe can find the best fit of the Cardassian parameters $n$ and\n$z_{car}$ to current CMB and Supernova data.  The current best fit is\nobtained for $w \\leq -0.78$, or, equivalently, $n \\leq 0.22$\n\\cite{wmap} and $0.33 \\leq z_{card} \\leq 0.48$.  As an example, for\n$n= 0.2$ (equivalently, $w=-0.8$), we find that $z_{car} = 0.42$.\nThen the position of the first Doppler peak is shifted by a few\npercent.  The age of the universe is 13 Gyr \\cite{savage}. There are\nsome indications that $w \\leq -1$ may give a good fit \\cite{mmot} and\nthis case of phantom cosmology will be discussed below.\n\n\\section{Generalized Cardassian}\n\nMore generally, we consider other forms of $g(\\rho)$ in Eq.(3), as\ndiscussed in \\cite{conf}.  For example, a simple generalization of\nEq.(\\ref{eq:new}) is Modified Polytropic Cardassian:\n\\begin{equation}\n\\label{eq:MPC}\nH^2 = {8 \\pi G \\over 3} \\rho [1+ ({\\rho\/\\rho_{car}})^{q(n-1})]^{1\/q} \n\\end{equation}\nwith $q>0$ and $n<2\/3$.  The right hand side returns to the ordinary\nFriedmann equation at early times, but becomes $\\rho^n$ at late times,\njust as in Eq.(\\ref{eq:new2}).  Other examples of Generalized\nCardassian have been discussed in \\cite{conf,gondolo}.\n\n\\subsection{Observational Tests of Generalized Cardassian Models}\n\n\\subsubsection{Current Supernova Data}\n\nFigure 1 shows comparisons by Wang et al \\cite{wang} of MP Cardassian\nmodels with current supernova data.  Clearly the models fit the\nexisting data.  The figure also shows predictions out to $z=2$ for\nfuture supernova data.  Future supernova data will thus allow one to\ndifferentiate between Cardassian models, quintessence models, and\ncosmological constant models.\n\n\\begin{figure}[ht]\n\\includegraphics[width=6in,height=6in]{SNnow.eps}\n\\caption{\\label{Figure 1}\n  Comparison of MP Cardassian models (see Eq.(\\ref{eq:MPC}) with\n  current Supernova data, as well as predictions out to $z=2$ for\n  future data.  For comparison, dotted lines indicate $(\\Omega_m,\n  \\Omega_{\\Lambda}$ as labelled on the right hand side of the plot.\n  We see that MP Cardassian models with $n=0.2$ and $q$ ranging from\n  1-3 match the current data, and can be differentiated from generical\n  cosmological constant or quintessence models in future data.  }\n\\end{figure}\n\n\\subsubsection{Additional Observational Tests}\n\nA number of additional observational tests can be used to test the\nmodels.  We plan to use the code CMBFAST to further constrain\nCardassian parameters in light of Cosmic Background Radiation data\n(WMAP).  In particular, the Integrated Sachs Wolfe effect may\ndifferentiate Cardassian from generic quintessence models.  A second\napproach is number count tests such as DEEP2. The abundance of galaxy\nhaloes of fixed rotational speed depend on the comoving volume\nelement.  Third, the Alcock-Paczynski test compares the angular size\nof a spherical object at redshift $z$ to its redhsift extent $\\Delta\nz$.  Depending on the cosmology, the spherical object may look\nsquooshed in redshift space.  A proposed trick \\cite{crottshui} is to\nuse the correlation function of Lyman-alpha clouds as spherical\nobjects.\n\n\\section{Motivation for Cardassian Cosmology}\n\nWe present two possible origins for Cardassian models.\nThe original idea arose from consideration of braneworld scenarios,\nin which our observable universe is a three dimensional membrane\nembedded in extra dimensions.  Recently, as a second interpretation\nof Cardassian cosmology, we have investigated\nan alternative four-dimensional fluid description.  Here,\nwe take the ordinary Einstein equations, but the energy density\nhas additional terms (possibly due to dark matter with self-interactions\ncharacterized by negative pressure).\n\n\\subsection{Braneworld Scenarios}\n\nIn braneworld scenarios, our observable universe is a three\ndimensional membrane embedded in extra dimensions.  For simplicity, we\nwork with one extra dimension.  In this five-dimensional\nworld, we take the metric to be\n\\begin{equation}\n\\label{eq:metric}\nds^2 = -q^2(\\tau,u)d\\tau^2 + a^2(\\tau,u)d\\vec{x}^2 + b^2(\\tau,u)du^2\n\\end{equation}\nwhere $u$ is the coordinate in the direction of the fifth dimension,\n$a$ is the usual scale factor of our observable universe and $b$ is\nthe scale factor in the fifth dimension.  We found \\cite{cf}\nthat one does not generically obtain the usual Friedmann equation\non the observable brane.  In fact, by a suitable choice of the\nbulk energy momentum tensor, one may obtain a Friedmann equation\non our brane of the form $H^2 \\sim \\rho^n$ for any $n$. More\ngenerally, one can obtain other modifications.  \\cite{bin} showed\nthat for an antideSitter bulk one obtains a quadratic correction\n$H^2 \\sim \\rho^2$, but more generally one can obtain any power of $n$,\nas shown in \\cite{cf}.  \n\nThe origin of these modifications to the Friedmann equations is as\nfollows.  We consider the five dimensional Einstein equations,\n\\begin{equation}\n\\label{eq:5dEinstein}\n\\tilde{G}_{AB} = 8\\pi G_{(5)} \\tilde{T}_{AB} ,\n\\end{equation}\nwhere the 5D Newton's constant is related to the 5D Planck mass as\n$ 8\\pi G_{(5)} = M_{(5)}^{-3}$.\nThe 5D Einstein tensor on the left hand side is given by\n\\begin{eqnarray}\n{\\tilde G}_{00} &=& 3\\left\\{ \\frac{\\da}{a} \\left( \\frac{\\da}{a}+ \\frac{\\db}{b} \\right) - \\frac{n^2}{b^2} \n\\left(\\frac{\\ppa}{a} + \\frac{\\pa}{a} \\left( \\frac{\\pa}{a} - \\frac{\\pb}{b} \\right) \\right)  \\right\\}, \n\\label{ein00} \\\\\n {\\tilde G}_{ij} &=& \n\\frac{a^2}{b^2} \\delta_{ij}\\left\\{\\frac{\\pa}{a}\n\\left(\\frac{\\pa}{a}+2\\frac{\\pn}{n}\\right)-\\frac{\\pb}{b}\\left(\\frac{\\pn}{n}+2\\frac{\\pa}{a}\\right)\n+2\\frac{\\ppa}{a}+\\frac{\\ppn}{n}\\right\\} \n\\nonumber \\\\\n& &+\\frac{a^2}{n^2} \\delta_{ij} \\left\\{ \\frac{\\da}{a} \\left(-\\frac{\\da}{a}+2\\frac{\\dn}{n}\\right)-2\\frac{\\dda}{a}\n+ \\frac{\\db}{b} \\left(-2\\frac{\\da}{a} + \\frac{\\dn}{n} \\right) - \\frac{\\ddb}{b} \\right\\},\n\\label{einij} \\\\\n{\\tilde G}_{05} &=&  3\\left(\\frac{\\pn}{n} \\frac{\\da}{a} + \\frac{\\pa}{a} \\frac{\\db}{b} - \\frac{\\dot{a}^{\\prime}}{a}\n \\right),\n\\label{ein05} \\\\\n{\\tilde G}_{55} &=& 3\\left\\{ \\frac{\\pa}{a} \\left(\\frac{\\pa}{a}+\\frac{\\pn}{n} \\right) - \\frac{b^2}{n^2} \n\\left(\\frac{\\da}{a} \\left(\\frac{\\da}{a}-\\frac{\\dn}{n} \\right) + \\frac{\\dda}{a}\\right) \\right\\}.\n\\label{ein55} \n\\end{eqnarray} \nIn the above expressions, a prime stands for a derivative with respect to\n $u$, and a \ndot for a derivative with respect to $\\tau$. \n\nThese 5D Einstein equations are supplemented by boundary conditions\n(known as Israel conditions) due to the existence of our 3-brane.\nThese boundary conditions are exactly analogous to those of a charged\nplate in electromagnetism.  There, the change in the perpendicular\ncomponent of the electric field as one crosses a plate with charge per\narea $\\sigma$ is given by $\\Delta(E_{\\rm perp}) = 4 \\pi \\sigma$.  There\nare two analogous boundary conditions here; one is \n\\begin{equation}\n\\label{eq:israel}\n\\Delta({da \\over\n  du}) \\propto \\rho_{brane};\n\\end{equation}\ni.e. the change in the derivative of the scale factor as one crosses\nour 3-brane is given by the energy density on the brane.\n\nThe combination of the 5D Einstein equations with these boundary\nconditions leads to the modified Friedmann equations on our observable\nthree dimensional universe. Depending on what is in the bulk\n(off the brane), one can obtain a variety of possibilities such\nas $H^2 \\sim \\rho^n$ \\cite{cf}.  Though we were at first concerned\nthat such modifications would be bad for cosmology, in fact\nthey can explain the observed acceleration of the universe today.\n\nCurrently one of our pressing goals would be to find a simple\nfive-dimensional energy momentum tensor that produces the Cardassian\ncosmology. While we succeeded in constructing an ugly toy model,\nit obviously does not represent our universe.  Unfortunately,\ngoing from four to five dimensions is not unique, and is difficult.\n\n\\section{Motivation for Cardassian cosmology: 2) Fluid Description}\n\nAnother pressing goal is to find testable predictions of Cardassian\ncosmology.  We want to be certain that ordinary astrophysical\nresults are not affected in an adverse way; e.g., we want energy\nmomentum to be conserved.  In the interest of addressing these\ntypes of questions, we developed a four dimensional fluid description\nof Cardassian cosmology \\cite{gondolo}.  This may be an effective\npicture of higher dimensional physics, or may be completely\nindependent of it.  This fluid description may serve as a second\npossible motivation for Cardassian cosmology.\n\nHere we use the ordinary Friedmann equation, $H^2 = 8 \\pi G \\rho\/3$\nand the ordinary four dimensional Einstein equation,\n\\begin{equation}\nG_{\\mu\\nu} = 8 \\pi G T_{\\mu\\nu} .\n\\end{equation}\nWe take the energy density to be the sum of two terms:\n\\begin{equation}\n\\rho_{tot} = \\rho_M + \\rho_k\n\\end{equation}\nwhere $\\rho_k$ is a Cardassian contribution.\nAccompanying the two terms in the energy density are two\npressure terms:\n\\begin{equation}\np_{tot} = p_M + p_k ,\n\\end{equation}\nwhere the thermodynamics of an adiabatically expanding universe tell us that\n\\begin{equation}\np_k = \\rho_M \\left({\\partial \\rho_k \\over \\partial \\rho_M} \\right)_S\n- \\rho_k .\n\\end{equation}\nThese total energy density and pressure terms are what enter\ninto the four dimensional energy momentum tensor:\n\\begin{equation}\nT_{\\mu\\nu} = {\\rm diag}(\\rho_{tot},p_{tot},p_{tot},p_{tot}).\n\\end{equation}\n\nEnergy conservation is then automatically guaranteed:\n\\begin{equation}\nT_{\\mu\\nu;\\nu} = 0 .\n\\end{equation}\nThis equation implies a modified continuity equation,\nand a modified Euler's equation. We also require particle number\nconservation.  Poisson's equation is also modified and becomes,\nin the Newtonian limit,\n\\begin{equation}\n\\nabla^2 \\phi = 4 \\pi G (\\rho_{tot} + 3 p_{tot}) .\n\\end{equation}\n\nAs an example, we can show the fluid description of the \npower law Cardassian model.  We have\n\\begin{equation}\n\\rho_k = b \\rho_M^n \n\\end{equation}\nand\n\\begin{equation}\np_k = - (1-n) \\rho_k ,\n\\end{equation}\nwhich is a negative pressure for $n<2\/3$.  The Poisson equation\nbecomes\n\\begin{equation}\n\\nabla^2 \\phi = - 4 \\pi G \\left[\\rho_M - (2-3n) b \\rho_M^n \\right]\n\\end{equation}\nOne can show that this simplified model runs into trouble\non galactic scales.\nThere is a new force ${d\\vec{v} \\over dt}|_{\\rm new}\n= - {\\vec{\\nabla} p_k \\over \\rho_M}$ that\ndestroys flat rotation curves.  The fluid power law\nCardassian must be thought of as an effective model which applies\nonly on large scales.  Hence, we proposed another possible\nmodel, the Modified Polytropic Cardassian model described earlier.\nDue to the existence of the additional parameter $q$, which\nis important on small scales, rotation curves are fine.\n\n\\subsection{Phantom Cardassian}\n\nSome Cardassian models can satisfy the dominant energy condition $w =\np_{tot}\/\\rho_{tot} > -1$ even with a dark energy component $w_k =\np_k\/\\rho_k < -1$, since both the ordinary and Cardassian components of\nthe energy density are made of the same matter and radiation.  For\nexample, the MP Cardassian model wtih n=0.2 and q=2 has $w_k < -1$.\nThere is some evidence that $w_k < -1$ matches the data \\cite{mmot}.\n\n\\subsection{Speculation: Self Interacting Dark Matter?}\n\nWe here speculate on an origin for the new Cardassian term\nin the total energy density in the fluid model.  The dark matter\nmay be subject to a new, long-range confining force (fifth force)\n\\begin{equation}\nF(r) \\propto r^{\\alpha -1},\\,\\,\\, \\alpha>0 .\n\\end{equation}\nThis may be analagous to quark confinement that exhibits negative\npressure.  \n\nOur basic point of view is that Cardassian cosmology gives\nan attractive phenomenology, in that it is very efficient:\nmatter can provide all the observed behavior of the universe.\nTherefore it is worthwhile to examine all the avenues we\ncan think of as to the origin of these modified Friedmann terms\n(or alternatively of the fluid model).\n\n \n\\section{Discussion}\n\nWe have presented $H^2 = g(\\rho)$ as a modification to the Friedmann\nequation in order to suggest an explanation of the recent acceleration\nof the universe.  In the Cardassian model, the universe can be flat\nand yet matter dominated.  We have found that the new Cardassian\nmodifications can dominate the expansion of the universe after\n$z_{car} = \\mathcal{O}$$(1)$ and can drive an acceleration.  We have\nfound that matter alone can be responsible for this behavior.  The\ncurrent value of the energy density of the universe is then smaller\nthan in the standard model and yet is at the critical value for a flat\ngeometry.\nSuch a modified Friedmann equation may result from the existence of\nextra dimensions.  Further work is required to find a simple\nfundamental theory responsible for Eq.(\\ref{eq:new}).  A second\npossible motivation for Cardassian cosmology, namely a fluid\ndescription, has been developed \\cite{gondolo}, which allows us to\ncompute everything astrophysical.  Comparison with SN observations,\nboth current and upcoming, were made and shown in Figure 1.\nIn future work, we plan to complete\na study of density perturbations in order to allow us to make\npredictions with observations, e.g.  of cluster abundance or the ISW\neffect so that we may separate this model from others in upcoming\nobservations.\n\n\\section*{Acknowledgments}\n\nThis paper reflects work with collaborators Matt Lewis, Paolo Gondolo,\nYun Wang, Josh Frieman, Chris Savage, and Nori Sugiyama.  We thank Ted\nBaltz for reminding us to consider the effects of the mass of the tau\nneutrino.  I acknowledge support from the Department of Energy via the\nUniversity of Michigan and the Kavli Institute for Theoretical Physics\nat Santa Barbara.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn several recent papers (Lai \\& Tsang 2009; Tsang \\& Lai 2009c; Fu \\&\nLai 2011; Horak \\& Lai 2013), we have presented detailed study of the\nlinear instability of non-axisymmetric inertial-acoustic modes [also\ncalled p-modes; see Kato (2001) and Wagoner (2008) for review] trapped \nin the inner-most region of black-hole (BH) accretion discs. This global instability\narises because of wave absorption at the corotation resonance (where\nthe wave pattern rotation frequency matches the background disc\nrotation rate) and requires that the disc vortensity has a positive\ngradient at the corotation radius (see Narayan et al.~1987, Tsang \\& Lai 2008 \nand references therein).  The disc vortensity (vorticity divided by\nsurface density) is given by\n\\be\n\\zeta={\\kappa^2\\over 2\\Omega\\Sigma},\n\\label{eq:zeta}\\ee \nwhere $\\Omega(r)$ is the disc rotation frequency, $\\kappa(r)$ is the \nradial epicyclic frequency and $\\Sigma(r)$ is the surface density\n\\footnote{Equation (\\ref{eq:zeta}) applies to barotropic discs in \nNewtonian (or pseudo-Newtonina) theory. See Tsang \\& Lai (2009c)\nfor non-barotropic discs and Horak \\& Lai (2013) for full\ngeneral relativistic expression.}. \nGeneral relativistic (GR) effect\nplays an important role in the instability: For a Newtonian disc, with\n$\\Omega=\\kappa\\propto r^{-3\/2}$ and relatively flat $\\Sigma(r)$\nprofile, we have $d\\zeta\/dr<0$, so the corotational wave absorption\nleads to mode damping. By contrast, $\\kappa$ is non-monotonic near a\nBH (e.g., for a Schwarzschild BH, $\\kappa$ reaches a maximum at\n$r=8GM\/c^2$ and goes to zero at $r_{\\rm ISCO}=6GM\/c^2$), the\nvortensity is also non-monotonic. Thus, p-modes with frequencies such\nthat $d\\zeta\/dr>0$ at the corotation resonance are overstable. Our\ncalculations based on several disc models and Paczynski-Witta \npseudo-Newtonian potential (Lai \\& Tsang 2009, Tsang \\& Lai 2009c)\nand full GR (Horak \\& Lai 2013) showed that the lowest-order\np-modes with $m=2,3,4,\\cdots$ have the largest growth rates, with the\nmode frequencies $\\omega \\simeq \\beta m\\Omega_{\\rm ISCO}$ (thus giving\ncommensurate frequency ratio $2:3:4,\\cdots$), where the dimensionless\nconstant $\\beta\\lo 1$ depends weakly the disc properties.\nThese overstable p-modes could potentially explain the \nHigh-frequency Quasi-Periodic Oscillations (HFPQOs) observed in BH\nX-ray binaries (e.g., Remillard \\& McClintock 2006; Belloni et al.~2012).\n\nThe effects of magnetic fields on the oscillation modes of BH\naccretion discs have been investigated by Fu \\& Lai (2009, 2011, 2012)\nand Yu \\& Lai (2013). Fu \\& Lai (2009) showed that the the basic wave\nproperties (e.g., propagation diagram) of p-modes are not strongly\naffected by disc magnetic fields, and it is likely that these p-modes\nare robust in the presence of disc turbulence (see Arras, Blaes \\&\nTurner 2006; Reynolds \\& Miller 2009). By contrast, other diskoseismic\nmodes with vertical structure (such as g-modes and c-modes) may be\neasily ``destroyed'' by the magnetic field (Fu \\& Lai 2009) or\nsuffer damping due to corotation resonance (Kato 2003; \nLi et al.~2003; Tsang \\& Lai 2009a). Although a modest\ntoroidal disc magnetic field tends to reduce the growth rate of the\np-mode (Fu \\& Lai 2011), a large-scale poloidal field can enhance the instability \n(Yu \\& Lai 2013; see Tagger \\& Pallet 1999; Tagger \\&\nVarniere 2006). The p-modes are also influenced by the magnetosphere\nthat may exist inside the disc inner edge (Fu \\& Lai 2012).\n\nSo far our published works are based on linear analysis.  While these\nare useful for identifying the key physics and issues, the nonlinear\nevolution and saturation of the mode growth can only be studied by\nnumerical simulations. It is known that fluid perturbations near the\ncorotation resonane are particularly prone to become nonlinear (e.g.,\nBalmforth \\& Korycansky 2001; Ogilvie \\& Lubow 2003).  \nMoreover, real accretion discs are more complex than any\nsemi-analytic models considered in our previous works.  Numerical MHD\nsimulations (including GRMHD) are playing an increasingly important\nrole in unraveling the nature of BH accretion flows (e.g., De Villiers\n\\& Hawley 2003; Machida \\& Matsumoto 2003; Fragile et al.~2007; Noble\net al.~2009, 2011; Reynolds \\& Miller 2009; Beckwith et al.~2008, 2009;\nMoscibrodzka et al.~2009; Penna et al.~2010; Kulkarni et\nal.~2011; Hawley et al.~2011; O'Neill et al.~2011; Dolence et al.~2012;\nMcKinney et al.~2012; Henisey et al.~2012).  Despite much\nprogress, global GRMHD simulations still lag far behind observations,\nand so far they have not revealed clear signs of HFQPOs that are directly \ncomparable with the observations of BH X-ray binaries\n\\footnote{Henisey et al.~(2009, 2012) found evidence of\nexcitation of wave modes in simulations of tilted BH accretion disks.\nHydrodynamic simulations using $\\alpha$-viscosity (Chan 2009; O'Neill et al.~2009)\nshowed wave generation in the inner disk region by viscous instability\n(Kato 2001). The MHD simulations by O'Neill et al.~(2011) revealed\npossible LFQPOs due to disk dynamo cycles. \nDolence et al.~(2012) reported transient QPOs in the numerical models\nof radiatively inefficent flows for Sgr A$^\\star$. McKinney et\nal.~(2012) found QPO signatures associated with the interface between\nthe disc inflow and the bulging jet magnetosphere (see Fu \\& Lai\n2012). Note that the observed HFQPOs are much weaker than LFQPOs,\ntherefore much more difficult to obtain by brute-force simulations.}.\nIf the corotation instability and its magnetic counterparts studied in\nour recent papers play a role in HFQPOs, the length-scale involved\nwould be small and a proper treatment of flow boundary conditions is\nimportant. It is necessary to carry out ``controlled'' numerical\nexperiments to capture and evaluate these subtle effects.\n\nIn this paper, we use two-dimensional hydrodynamic simulations to\ninvestigate the nonlinear evolution of corotational instability of\np-modes. Our 2D model has obvious limitations. For example it does not\ninclude disc magnetic field and turbulence. However, we emphasize that\nsince the p-modes we are studying are 2D density waves with no\nvertical structure, their basic radial ``shapes'' and real frequencies\nmay be qualitatively unaffected by the turbulence (see Arras et\nal. 2006; Reynolds \\& Miller 2009; Fu \\& Lai 2009). Indeed, several\nlocal simulations have indicated that density waves can propagate in\nthe presence of MRI turbulence (Gardiner \\& Stone 2005; Fromang et\nal. 2007; Heinemann \\& Papaloizou 2009). Our goal here is to\ninvestigate the saturation of overstable p-modes and the their\nnonlinear behaviours.\n\n\n\\section{Numerical Setup}\n\nOur accretion disc is assumed to be inviscid and geometrically thin so that the\nhydrdynamical equations can be reduced to two-dimension with vertically\nintegrated quantities. We adopt an isothermal equation of state \nthroughout this study, i.e. $P=c_s^2\\Sigma$ where $P$ is the vertically integrated\npressure, $\\Sigma$ is the surface density and $c_s$ is the constant\nsound speed. Self-gravity and magnetic fields are neglected.\n\nWe use the Paczynski-Witta Pseudo-Newtonian potential (Paczynski \\&\nWitta 1980) to mimic the GR effect:\n\\be\n\\Phi=-\\frac{GM}{r-r_{\\rm S}},\n\\ee\nwhere $r_{\\rm S}=2GM\/c^2$ is the Schwarzschild radius. \nThe corresponding Keplerian rotation frequency and radial epicyclic frequency are\n\\ba\n&&\\Omega_{\\rm K}=\\sqrt{\\frac{GM}{r}}\\frac{1}{r-r_{\\rm S}},\\\\\n&&\\kappa=\\Omega_{\\rm K}\\sqrt{\\frac{r-3r_{\\rm S}}{r-r_{\\rm S}}}.\n\\label{eq:grkappa}\n\\ea\nIn our compuation, we will adopt the units such that \nthe inner disc radius (at the Inner-most Stable Circular Orbit or ISCO)\nis at $r=1.0$ and the Keperian frequency at the ISCO is\n$\\Omega_{\\rm ISCO}=1$. In these units, $r_{\\rm s}=1\/3$, and\n\\be\n\\Omega_{\\rm K}=\\frac{2}{3}\\frac{1}{r-r_{\\rm s}}\\frac{1}{\\sqrt{r}}.\n\\ee\nOur computation domain extends from $r=1.0$ to $r=4.0$ in the radial\ndirection and from $\\phi=0$ to $\\phi=2\\pi$ in the azimuthal\ndirection. We also use the Keplerian orbital period\n($T=2\\pi\/\\Omega_{\\rm K}=2\\pi$) at $r=1$ as the unit for time.\nThe equilibrium state of the disk is axisymmetric. The surface density\nprofile has a simple power-law form\n\\be\n\\Sigma_0=r^{-1},\n\\ee\nwhich leads to a positive vortensity gradient in the inner disc region.\nThe equilibrium rotation frequency of the disc is given by \n\\be\n\\Omega_0(r)=\\sqrt{\\frac{4\/9}{r(r-1\/3)^2}-\\frac{c_s^2}{r^2}}.\n\\ee\nThroughout our simulation, we will adopt $c_s=0.1$ so that\n$\\Omega_0 \\simeq \\Omega_{\\rm K}$.\n\n\\begin{figure*}\n\\begin{center}\n$\n\\begin{array}{ccc}\n\\includegraphics[width=0.33\\textwidth]{m2vr1.ps} &\n\\includegraphics[width=0.33\\textwidth]{m3vr1.ps} &\n\\includegraphics[width=0.33\\textwidth]{mrvr1.ps}\n\\end{array}\n$\n\\caption{Evolution of the radial velocity amplitude $|u_r|_{\\rm max}$ \n(evaluated at $r=1.1$) for three runs with initial azimuthal mode number $m=2$\n(left panel), $m=3$ (middle panel) and \nwith random perturbations (right panel). \nThe dashed lines are the fits for the\nexponential growth stage (between $\\sim 10$ and $\\sim 30$ orbits) of\nthe mode amplitude.}\n\\label{fig:growth}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n$\n\\begin{array}{cc}\n\\includegraphics[width=0.45\\textwidth]{m2svr.ps} & \n\\includegraphics[width=0.45\\textwidth]{m3svr.ps} \\\\\n\\includegraphics[width=0.45\\textwidth]{m2sdvphi.ps} &\n\\includegraphics[width=0.45\\textwidth]{m3sdvphi.ps} \n\\end{array}\n$\n\\caption{Comparison of the radial profiles of velocity perturbations\n  from non-linear simulation and linear mode calculation. The top and\n  bottom panels show the radial and azimuthal velocity perturbations,\n  respectively.  The left and right panels are for cases with\n  azimuthal mode number $m=2$ and $m=3$, respectively. In each panel,\n  the dashed line is taken from the real part of the complex\n  wavefunction obtained in linear mode calculation, while the solid line is from\n  the non-linear simulation during the exponential growth stage (at\n  $T=20$ orbits), with the quantities evaluated at $\\phi=0.4\\pi$.\n  Note that the normalization factor is given in the $y$-axis label\n  for the nonlinear simulation results.  }\n\\label{fig:vrvphi}\n\\end{center}\n\\end{figure*}\n\nWe solve the Euler equations that govern the dynamics of the disc flow\nwith the PLUTO code \\footnote{publicly available at\n  http:\/\/plutocode.ph.unito.it\/} (Mignone et al. 2007), which is a\nGodunov-type code with multiphysics and multialgorithm modules. For\nthis study, we choose a Runge-Kutta scheme (for time integration) and\npiecewise linear reconstruction (for space integration) to achieve\nsecond order accuracy, and Roe solver as the solution of Riemann\nproblems. The gird resolution we adopt is $(N_r\\times\nN_{\\phi})=(1024\\times 2048)$ so that each grid cell is almost\na square. Our runs typically last for 100 orbits (Keplerian orbits at\ninner disc boundary). To compare with the linear mode calculations\n(Lai \\& Tsang 2009), the inner disk boundary is set to be reflective\nwith zero radial velocity. At the outer disc boundary, we adopt the\nnon-reflective boundary condition. This is realized by employing the\nwave damping method (de Val-Borro et al. 2006) to reduce wave reflection. \nThis implementation mimics the outgoing radiative boundary condition \nused in the linear analysis.\n\n\n\\section{Results}\n\nWe carried out simulations with two different types of initial\nconditions for the surface density perturbation.  In the first, we\nchoose $\\delta \\Sigma(r, \\phi)$ to be randomly distributed in $r$ and\n$\\phi$.  In the second, we impose $\\delta \\Sigma(r, \\phi)\\propto\n\\cos(m\\phi)$ that is randomly distributed in $r$, so that the perturbation\nhas an azimuthal number $m$.\nIn all cases, the initial surface density perturbation has a small amplitude \n($|\\delta\\Sigma\/\\Sigma_{0}| \\leq 10^{-4}$).\n\n\\begin{table*}\n\\caption{Comparison of results from linear and nonlinear studies of overstable disc p-modes}\n\\centering\n\\begin{threeparttable}\n\\begin{tabular}{ c c c c c c c c }\n\\noalign{\\hrule height 1pt}\n$m$\\tnote{a} & $\\omega_{r}$\\tnote{b} &$\\omega_i$\\tnote{c} & $\\omega_{r}\/m\\Omega$\\tnote{d} & $\\omega_{r1}$\\tnote{e} & $|\\omega_r-\\omega_{r1}|\/\\omega_r$\\tnote{f} & $\\omega_{r2}$\\tnote{g} & $|\\omega_{r2}-\\omega_{r1}|\/\\omega_{r2}$\\tnote{h} \\\\\n\\hline\n$2$ & $1.4066$ & $0.0632$ & $0.7033$ & $1.3998$ & $0.5\\%$ & $1.4296$ & $2.1\\%$ \\\\\n$3$ & $2.1942$ & $0.0733$ & $0.7314$ & $2.1997$ & $0.3\\%$ & $2.2445$ & $2.0\\%$ \\\\\n$4$ & $3.0051$ & $0.0763$ & $0.7512$ & $2.9996$ & $0.2\\%$ & $3.0594$ & $2.0\\%$ \\\\\n$5$ & $3.8294$ & $0.0751$ & $0.7659$ & $3.7995$ & $0.8\\%$ & $3.8886$ & $2.3\\%$ \\\\\n$6$ & $4.6621$ & $0.0714$ & $0.7770$ & $4.6494$ & $0.3\\%$ & $4.7749$ & $2.6\\%$\\\\\n$7$ & $5.5007$ & $0.0664$ & $0.7858$ & $5.4992$ & $0.03\\%$ & $5.6756$ & $3.1\\%$ \\\\\n$8$ & $6.3436$ & $0.0607$ & $0.7930$ & $6.3492$ & $0.09\\%$ & $6.3189$ & $0.5\\%$\\\\\n\\noalign{\\hrule height 1.2pt}\n\\end{tabular}\n\\begin{tablenotes}\n\\item[a] Azimuthal mode number\n\\item[b] Mode frequency from the linear calculation (in units of Keplerian\n  orbital frequency at the inner disc boundary; same for $\\omega_i$,\n  $\\omega_{r1}$ and $\\omega_{r2}$)\n\\item[c] Mode growth rate from the linear calculation \n\\item[d] Ratio of wave pattern speed to the Keplerian orbital frequency at the inner disc boundary\n\\item[e] Mode frequency during the exponential growth stage of nonlinear simulation \n(peak frequency of the power density spectrum between $\\sim 10$ orbits and $\\sim 30$ orbits)\n\\item[f] Difference between $\\omega_r$ (linear result) and $\\omega_{r1}$ (nonlinear result)\n\\item[g] Mode frequency during the saturation stage of nonlinear simulation \n(peak frequency of the power density spectrum between $\\sim 30$ orbits and $\\sim 100$ orbits)\n\\item[h] Difference between $\\omega_{r1}$ and $\\omega_{r2}$\n\\end{tablenotes}\n\\end{threeparttable}\n\\label{tab:tab1}\n\\end{table*}\n\nFig.~\\ref{fig:growth} shows the evolution of the radial velocity\namplitude near the inner disc radius for runs with initial azimuthal\nmode number $m=2$, $m=3$ and random initial perturbation,\nrespectively. This velocity amplitude is obtained by searching for the\nmaximum $|u_r|$ at a given $r$ by varying $\\phi$ for each given time\npoint. We chose $r=1.1$ because this is where the largest radial\nvelocity perturbation is located (see the upper panels of\nFig.~\\ref{fig:vrvphi}).  We see that in all cases there are three\nstages in the amplitude evolution.  The first stage occupies roughly\nthe first $10$ orbits, during which the initial perturbation starts to\naffect the flow and presumably excites many modes\/oscillations in the\ndisc. In the second stage (from $\\sim 10$ orbits to $\\sim\n30$ orbits), the fastest growing mode becomes dominant and undergoes \nexponential growth with its amplitude increased by about four orders\nof magnitude. In the last stage (beyond $\\sim 30$ orbits), the\nperturbation growth saturates and its amplitude remains at\napproximately the same level.  A fit to the exponential growth stage\ngives the growth rate of the fastest growing perturbation, which is\nthe slope of the fitted straight line. For the $m=2$ run, we find \n$0.17\/\\log_{10}(e)\/2\\pi\\simeq 0.0637$ (in units of the orbital\nfrequency at $r=1$) as the growth rate, which is quite consistent with\nthe result from our linear eigenmode calculation $\\omega_{i}\\simeq\n0.0632$ (the imaginary part of the eigenfrequency; Lai \\& Tsang 2009;\nFu \\& Lai 2011; see Table~\\ref{tab:tab1}).  For the $m=3$ run,\nour simulation gives $0.074$ as the mode growth rate, close to\n$\\omega_{i}=0.0733$ from the linear calculation. In\nFig.~\\ref{fig:vrvphi}, we plot the radial profile of the velocity\nperturbation at a given time during the exponential growth stage\nof the simulation, and we compare it with the wavefunctions obtained\nfrom the linear mode calculation. Note that in each panel of\nFig.~\\ref{fig:vrvphi} we have normalized the different sets of data so\nthat they have the same scale. The normalization factor has also been\nincluded in figure labels for easy recovery of the absolute value\nin the simuation.  We can see that wavefunctions obtained from two studies \nagree quite well. The only obvious differences are near the outer disc boundary,\nwhich can be attributed to the fact that outer boundary conditions\nemployed in the numerical simulation and linear mode calculation are not\nexactly the same. Nevertheless, this agreement and the agreement in \nthe mode growth rate in Fig.~\\ref{fig:growth} confirm that our non-linear simulations \nindeed capture the same unstable disc p-modes as those revealed in linear\nperturbation analysis.\n\n\n\n\\begin{figure*}\n\\begin{center}\n$\n\\begin{array}{cc}\n\\includegraphics[width=0.4\\textwidth]{m3fft2.ps} &\n\\includegraphics[width=0.4\\textwidth]{mrfft2.ps} \\\\\n\\includegraphics[width=0.4\\textwidth]{m3fft3.ps} &\n\\includegraphics[width=0.4\\textwidth]{mrfft3.ps} \n\\end{array}\n$\n\\caption{Power density spectra of the radial velocity perturbations\nnear the disk inner boundary ($r=1.1$). Each panel shows the normalized FFT\n  magnitude as a function of frequency. The left and right\n  columns are for runs with initial $m=3$ and with random perturbation,\n  respectively. In the top and bottom panels, the Fourier transforms are\n  sampled for time periods of [10, 30] orbits and [30,\n    100] orbits, respectively.}\n\\label{fig:fft}\n\\end{center}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n$\n\\begin{array}{cc}\n\\includegraphics[width=0.4\\textwidth]{m2cvr.ps} & \n\\includegraphics[width=0.4\\textwidth]{m3cvr.ps} \\\\\n\\includegraphics[width=0.4\\textwidth]{m2cdvphi.ps} &\n\\includegraphics[width=0.4\\textwidth]{m3cdvphi.ps} \n\\end{array}\n$\n\\caption{Evolution of the radial profiles of velocity perturbations from\n  the non-linear simulations. The top and bottom panels show the radial\n  and azimuthal components of velocity perturbation, respectively. \n  The left and right panels are for cases with azimuthal mode number\n  $m=2$ and $m=3$, respectively. In each panel, the data are taken\n  from points with fixed $\\phi=0.4\\pi$. Different line types represent\n  different times during the simulation.}\n\\label{fig:cvrvphi}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{center}\n$\n\\begin{array}{ccc}\n\\includegraphics[width=0.33\\textwidth]{m2vr_turn0020.ps} &\n\\includegraphics[width=0.33\\textwidth]{m3vr_turn0020.ps} &\n\\includegraphics[width=0.33\\textwidth]{mrvr_turn0020.ps} \\\\\n\\includegraphics[width=0.33\\textwidth]{m2vr_turn0030.ps} &\n\\includegraphics[width=0.33\\textwidth]{m3vr_turn0030.ps} &\n\\includegraphics[width=0.33\\textwidth]{mrvr_turn0030.ps} \\\\\n\\includegraphics[width=0.33\\textwidth]{m2vr_turn0050.ps} &\n\\includegraphics[width=0.33\\textwidth]{m3vr_turn0050.ps}  &\n\\includegraphics[width=0.33\\textwidth]{mrvr_turn0050.ps} \n\\end{array}\n$\n\\caption{Evolution of the radial velocity for runs with initial $m=2$\n  (left), $m=3$ (middle) and random (right) perturbations,\n  respectively. From top to bottom, the times are $T=20$, $30$ and $50$ orbits, \n  respectively. Note that the color scale varies from panel to panel.}\n\\label{fig:vrcolor}\n\\end{center}\n\\end{figure*}\n\n\nTo explore the time variability of the flow, we carry out\nFourier transform of the radial velocity $u_r(r, \\phi, t)$ at fixed\n$r$ and $\\phi$ during different evolutionary stages.  In\nFig.~\\ref{fig:fft} we show some examples of the resulting power\ndensity spectra (normalized to the maximum value of unity).  Different\nrows correspond to different total sampling times and different\ncolumns represent runs with different initial surface density\nperturbation.  In the left columns, the disk has an initial\nperturbation with azimuthal mode number $m=3$. A mixture of various\nmodes\/oscillations are excited in the flow. After about $10$ orbits of\nevolution, one of them (the fastest growing mode) has its oscillation\namplitude grown by a large amount such that it dominates over other\nmodes. This corresponds to the primary spike in the top-left\npanel. The other spikes in the same panel are harmonics of this\nprimary spike (see the labels of the dashed vertical\nlines). Table~\\ref{tab:tab1} shows that the frequency\\footnote{All the\n  frequencies in this paper are angular frequencies unless otherwise\n  noted.} of this fastest growing mode ($\\omega_{r1}$) differs\nfrom the frequency obtained in linear mode calculation by only\n$0.3\\%$, which again demonstrates the consistency of these two studies. \nAfter the perturbation saturates (bottom-left panel), we see\nthat the basic structure of the power density spectrum does not change\nmuch except that these spikes are not as ``clean'' as in the linear regime;\nthis is probably due to the interaction of different modes. Compared with the upper\npanel, the location of the primary spike ($\\omega_{r2}$ in\nTable~\\ref{tab:tab1}) is increased by $2.0\\%$. In\nTable~\\ref{tab:tab1} we also include the results from both linear mode\ncalculation and numerical simulation for modes with other mode number\n$m$. The comparison illustrates two main points: First, the\nfrequencies of the fastest growing modes during the exponentially\ngrowth stage of numerical simulations are exceptionally close to the\nlinear calculation results (differ by less than $1\\%$); second, the\nfrequencies of the fastest growing modes during the saturation stage\nare only slightly higher (except for the $m=8$ mode which shows lower\nfrequency) than ones during the exponential growing stage. These indicate\nthat the mode frequencies obtained in linear mode calculation are\nfairly robust and can be reliably applied in the interpretation of HFQPOs.\n\nIn the right columns of Fig.~\\ref{fig:fft}, the simulation starts with\na random initial perturbation which excites modes with various\n$m$'s. During the exponential growth stage, six modes stand out. By\nexamining the location of the corresponding spikes, we know that these\nare the $m=3$, $4$, $5$, $6$, $7$, $8$ modes. Although the\n$m=3$ mode seems to be the most prominent one in this figure, we note\nthat this is because for this particular run the initial random\nperturbation happens to contain more $m=3$ wave components. If we were\nto start the run with a different initial random perturbation, then\nthe relative strengths of those peaks would also be different (not\nnecessarily with $m=3$ being dominant).\n\nFig.~\\ref{fig:cvrvphi} shows the comparison of the velocity perturbations \nduring the linear stage ($T=20$ orbits), at the end of the linear\nstage ($T=30$ orbits) and during the saturation stage ($T=50$ orbits)\nfor simulations with different initial perturbations. At $T=20$\norbits, the oscillation mainly comes from the single fastest growing\nmode and has a smooth radial profile. At $T=30$ orbits, the perturbation\nstarts to saturate, and the oscillation now consists of many different modes,\nand its radial profile exhibits sharp variations at several locations.\nThese sharp features remain after the saturation ($T=50$ orbits).\n\nTo see this evolution from a different perspective, in\nFig.~\\ref{fig:vrcolor} we show the color contours of radial velocity\nfor runs with different initial perturbations (different columns) at\ndifferent times (different rows). We can clearly see that spiral waves\ngradually develop due to the instability. As the system evolves, sharp\nfeatures in velcoities emerge. At the end (the bottom \nrow) the spiral arms becomes more irregular, which is related\nto the emergence and interaction of multiple modes after saturation.\n\nThe sharp velocity variations shown in\nFigs.~\\ref{fig:cvrvphi}-\\ref{fig:vrcolor} suggest shock-like features.\nNote that since our simulations adopt isothermal equation of state\n(with $P\/\\Sigma=$constant), there is no entropy generation and shock\nformation in the strict sense.  The radial velocity jump (see\nFig.~\\ref{fig:cvrvphi}) is comparable but always smaller than the\nsound speed. Nevertheless, these sharp features imply that\nwave steepening plays an important role in the mode saturation. \n\n\\section{Conclusions}\n\nWe have carried out high-resolution, two-dimensional hydrodynamical\nsimulations of overtstable inertial-acoustic modes (p-modes) in BH\naccretion discs with various initial conditions.  The evolution of\ndisc p-modes exhibits two stages.  During the first (linear) stage, the\noscillation amplitude grows exponentially.  In the cases with a\nspecific azimuthal mode number ($m$), the mode frequency, growth rate\nand wavefunctions agree well with those obtained in our previous\nlinear mode analysis.  In the cases with random initial perturbation,\nthe disc power-density spectrum exhibits several prominent\nfrequencies, consistent with those of the fastest growing linear modes\n(with various $m$'s).  These comparisons with the linear theory\nconfirm the physics of corotational instability that drives disc\np-modes presented in our previous studies (Lai \\& Tsang 2009; Tsang \\&\nLai 2009c; Fu \\& Lai 2011, 2012).  In the second stage, the mode growth\nsaturates and the disc oscillation amplitude remains at roughly a\nconstant level.  In general, we find that the primary disc oscillation\nfrequency (in the cases with specific initial $m$) is larger than the\nlinear mode frequency by less than $4\\%$, indicating the robustness of\ndisc oscillation frequency in the non-linear regime. Based on the\nsharp, shock-like features of fluid velocity profiles, we suggest that\nthe nonlinear saturation of disc oscillations is caused by wave\nsteepening and mode-mode interactions.\n\nAs noted in Section 1, our 2D hydrodynamical simulations presented in\nthis paper do not capture various complexities (e.g., magnetic field,\nturbulence, radiation) associated with real BH accretion discs.\nNevertheless, they demonstrate that under appropriate conditions, disc\np-modes can grow to nonlinear amplitudes with well-defined frequencies\nthat are similar to the linear mode frequencies.  A number of issues\nmust be addressed before we can apply our theory to the interpretation\nof HFQPOs.  First, magnetic fields may play an important role in the\ndisc oscillations.  Indeed, we have shown in previous linear\ncalculations (Fu \\& Lai 2011,2012) that the growth of p-modes can be\nsignificantly affected by disc toroidal magnetic fields. A strong,\nlarge-scale poloidal field can also change the linear mode frequency\n(Yu \\& Lai 2013). Whether or not these remain true in the non-linear regime is\ncurrently unclear. Second, understanding the nature of the inner disc boundary\nis crucial. Our calculations rely on the assumption that the inner disc edge\nis reflective to incoming spiral waves. In the standard disc model with zero-torque\ninner boundary condition, the radial inflow velocity is not negligible near the ISCO, \nand the flow goes through a transonic point. While the steep density and velocity\ngradients at the ISCO give rise to partial wave reflection (Lai \\& Tsang 2009),\nsuch radial inflow can lead to significant mode damping such that the net growth rates of\np-modes become negative. This may explain the absence of HFQPOs in the thermal\nstate of BH x-ray binaries. However, it is possible that the inner-most region of \nBH accretion discs accumulates significant magnetic flux and forms a magnetosphere.\nThe disc-the magnetosphere boundary will be highly reflective, leading to the \ngrowth of disc oscillations (Fu \\& Lai 2012; see also Tsang \\& Lai 2009b).\nFinally, the effect of MRI-driven disc turbulence on the p-modes\nrequires further understanding. In particular, turbulent viscosity \nmay lead to mode growth or damping, depending on the magnitude and the \ndensity-dependence of the viscosity (R. Miranda \\& D. Lai 2013, in prep).\n\n\n\\section*{Acknowledgements}\nThis work has been supported in part by the NSF grants AST-1008245,\nAST-1211061 and the NASA grant NNX12AF85G.  WF also acknowledges the\nsupport from the Laboratory Directed Research and Development Program\nat LANL.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\chapter*{Abstract}\n\n\nWe present a system for the classification of mountain\npanoramas from user-generated photographs followed by\nidentification and extraction of mountain peaks from those\npanoramas. We have developed an automatic technique that,\ngiven as input a geo-tagged photograph, estimates its FOV\n(Field Of View) and the direction of the camera using a\nmatching algorithm on the photograph edge maps and a rendered\nview of the mountain silhouettes that should be seen\nfrom the observer's point of view. The extraction algorithm\nthen identifies the mountain peaks present in the photograph\nand their profiles. We discuss possible applications in social\nfields such as photograph peak tagging on social portals,\naugmented reality on mobile devices when viewing a mountain\npanorama, and generation of collective intelligence systems\n(such as environmental models) from massive social\nmedia collections (e.g. snow water availability maps based\non mountain peak states extracted from photograph hosting\nservices).\n\\chapter*{Sommario}\n\n\nProponiamo un sistema per la classificazione di panorami montani e per l'identificazione delle vette presenti in fotografie scattate dagli utenti.\nAbbiamo sviluppato una tecnica automatica che, data come input la foto e la sua geolocalizzazione, stima il FOV (Field of View o Angolo di Campo) e l'orientamento della fotocamera. Questo avviene tramite l'applicazione di un algoritmo di matching tra la mappa degli edge (bordi) della fotografia e alle silhouette delle montagne che dovrebbero essere visibili dall'osservatore a quelle coordinate.\nL'algoritmo di estrazione identifica poi i picchi delle montagne presenti nella fotografia e il loro profilo. \nVerranno discusse alcune possibile applicazioni in ambito sociale come ad esempio: \nl'identificazione e tagging (marcatura) delle fotografie sui social network, \nrealt\u00e0 aumentata su dispositivi mobile durante la visione di panorami montani e\nla generazione di sistemi di intelligenza collettiva (come modelli ambientali) dalle enormi collezioni multimediali dei social network (p.es. mappe della disponibilit\u00e0 di neve e acqua sulle vette delle montagne, estratte da servizi di condivisione di immagini).\n\\chapter*{Acknowledgements}\n\n\nForemost, I would like to thank Prof. Marco Tagliasacchi and Prof. Piero Fraternali for the opportunity to work with them on a such interesting and exciting project, and the continuous support during the preparation of this thesis.\n\nI would like to thank also:\n\\begin{itemize}\n\\item\nDr. Danny Chrastina (\\url{http:\/\/www.chrastina.net}) for helping with the preparation of this document.\n\\item\nDr. Ulrich Deuschle (\\url{http:\/\/www.udeuschle.de}) for his kind permission of using his mountain panorama generating web tool.\n\\item\nGregor Brdnik (\\url{http:\/\/www.digicamdb.com}) for providing the database of digital cameras and their sensor sizes.\n\\item\nMiroslav Sabo (\\url{http:\/\/www.mirosabo.com}) for the photograph of the Matterhorn used in this thesis.\n\\item\nAll my friends and colleagues (who are too many to be listed) who have helped and contributed to this work.\n\\end{itemize}\n\nLastly, but most importantly, I would like to thank my parents (Alexey and Irina) for making it possible for me to get here, and Valentina for constant support and motivation.\n\n\n\n\n\n\\chapter{Introduction}\n\\label{Introduction}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\n\\noindent\nThe most suitable paradigm for representing this work is probably passive crowdsourcing, a field that is not trivial to define and that is not even a subcategory of crowdsourcing in the strict sense, as its name can suggest. A possible definition of the passive crowdsourcing discipline can be the union of crowdsourcing, data mining and collective intelligence (computer science fields that have much in common but are slightly different). These differences are often hard to notice due to the youth of these concepts and so the presence of a lot of confusion. For this reason the purpose of this chapter is to define these concepts unambiguously and to define the problem statement of this work.\n\n\\section{Human Computation}\nThe idea of the computer computation goal has always been that which Alan Turing expressed in 1950:\n\\begin{quotation}\n\n\\noindent{\\emph{``\nThe idea behind digital computers may be explained by saying that these machines are intended to carry out any operations which could be done by a human computer.\n''}\n Alan Turing \\cite{Turing:1995:CMI:216408.216410}\n}\n}\n\\end{quotation}\nThough current progress in computer science brings automated solutions of more and more complex problems, this idea of computer systems able to solve any problem that humans can solve is however far from reality. There are still a lot of tasks that cannot be performed by computers, and those that could be but are preferred to be computed by humans for quality, time, and cost reasons. These tasks lead to the field of human computation: a field that is hard to give a definition to, in fact several definitions of the term can be found: the most general, modern and suitable for our needs of which is probably that extracted from von Ahn's dissertation:\n\\begin{quotation}\n\n\\noindent{\\emph{``\n... a paradigm for utilizing human processing power to solve \nproblems that computers cannot yet solve.\n''}\n Luis von Ahn  \\cite{VonAhn:2005:HC:1168246}\n}\n}\n\\end{quotation}\nHuman computation can be thought of as several approaches varying with the tasks involved, the type of persons involved in task completion, the incentive techniques used, and what type of effort the persons are required to make. It must be said that the classification of human computation is not an ordinary hierarchy with parents and children, but is instead a set of related concepts not necessary including one another. So a possible taxonomy of human computation (seen as a list of related ideas) has been produced by combining definitions given by Quinn et al. \\cite{Quinn:2011:HCS:1978942.1979148} and Fraternali et al. \\cite{Fraternali:2012:PHL:2263310.2263839} can be:\n\\begin{itemize}\n\\item\n\\emph{Crowdsourcing}:\nthis approach manages the distributed assignment of tasks to an open, undefined and generally large group of executors. The task to be performed by the executors is split into a large number of microtasks (by the work provider or the crowdsourcing system itself) and each microtask is assigned by the system to a work performer, who executes it (usually for a reward of a small amount of money). The crowdsourcing application (defined usually by two interfaces: for the work providers and the work performers) manages the work life cycle: performer assignment, time and price\nnegotiation, result submission and verification, and payment. In addition to the web interface, some platforms offer Application Programming Interfaces (APIs),whereby third parties can integrate the distributed work management functionality into their custom applications. Examples of crowdsourcing solutions are Amazon Mechanical Turk and Microtask.com \\cite{Fraternali:2012:PHL:2263310.2263839}.\n\\item\n\\emph{Games with a Purpose (GWAPs)}:\nthese are a sort of crowsourcing application but with a fundamental difference in user incentive technique: the process of resolving a task is implemented as a game with an enjoyable user experience. Instead of monetary earning, the user motivation in this approach is the gratification of the playing process. GWAPs, and more generally useful applications where the user solves perceptive or cognitive problems without knowing, address task such as adding descriptive tags and recognising objects in images and checking the output of Optical Character Recognition (OCR) for correctness. \\cite{Fraternali:2012:PHL:2263310.2263839}.\n\\item\n\\emph{Social Computing}:\na broad scope concept that includes applications and services that facilitate collective action and social interaction online with rich exchange of multimedia information and evolution of aggregate knowledge \\cite{parameswaran2007social}. Instead of crowdsourcing, the purpose is usually not to perform a task. The key distinction between human computation and social computing is that social computing facilitates relatively natural human behavior that happens to be mediated by technology, whereas participation in a human computation is directed primarily by the human computation system \\cite{Quinn:2011:HCS:1978942.1979148}.\n\\item\n\\emph{Collective Intelligence}: if seen as a process, the term can be defined as groups of individuals doing things collectively that seem intelligent \\cite{malone-harnessing}. If it is seen instead as the process result, means the knowledge of any kind that is generated (even non consciously and not in explicit form) by the collective intelligence process. Quinn et al. \\cite{Quinn:2011:HCS:1978942.1979148} classifies it as the superset of social computing and crowdsourcing, because both are defined in terms of social behavior. The key distinctions between collective intelligence and human computation are the same as with crowdsourcing, but with the additional distinction that collective intelligence applies only when the process depends on a group of participants. It is conceivable that there could be a human computation system with computations performed by a single worker in isolation. This is why part of human computation protrudes outside collective intelligence \\cite{Quinn:2011:HCS:1978942.1979148}.\n\\item\n\\emph{Data Mining}:\nthis can be defined broadly as the application of specific algorithms for extracting patterns from data \\cite{Fayyad96knowledgediscovery}. Speaking about human-created data the approach can be seen as extracting the knowledge from a certain result of a collective intelligence process. Creating this knowledge usually is not the goal of the persons that generate it, in fact often they are completely unaware of it (just think that almost everybody contributes to the knowledge of what are the most popular web sites just by visiting them: they open a web site because they need it, not to add a vote to its popularity). Though it is a very important concept in the field of collective intelligence, machine intelligence applied to social science and passive crowdsourcing (that will be defined in the next section) is a fully automated process by definition, so it is excluded from the area of human computation.\n\\item\n\\emph{Social Mobilization}:\nthis approach deals with social computation problems where the timing and the efficiency is crucial. Examples of this area are safety critical sectors like civil protection and disease control.\n\\item\n\\emph{Human Sensors}:\nexploiting the fact that the mobile devices tend to incorporate more and more sensors, this approach deals with a real-time collection of data (of various natures) treating persons with mobile devices as sensors for the data. Examples of these applications are earthquake and other natural disaster monitoring, traffic condition control and pollution monitoring.\n\\end{itemize}\n\n\\section{Passive Crowdsourcing}\nThe goal of any passive human computation system is to exploit the collective effort of a large group of people to retrieve the collective intelligence this effort generates or other implicit knowledge. In this study case the collective effort is taking geo-tagged photographs of mountains and publishing them on the Web, the intelligence deriving from this collection of photographs is the availability of the appearances of a mountain through time, the knowledge we want to extract having these visual appearances of mountains is the evolution of its environmental properties in time (which can be for example snow or grass presence at a certain altitude) or using some ground truth data even to predict these features where the use of physical sensors for those measurements is difficult or impossible (i.e. snow level prediction at high altitudes). Passive crowdsourcing is an approach that is not trivial to classify within the described taxonomy: it can be best classified in an area that includes: \\emph{crowdsourcing} for the fact of exploiting the effort of human computation, \\emph{collective intelligence} since its extraction is the primary goal of the approach and \\emph{data mining} as it refers to the procedure of extracting some results of human computation from the public Web data. Figure \\ref{fig:crowdsourcingTaxonomy} shows the taxonomy proposed by Quinn et al. \\cite{Quinn:2011:HCS:1978942.1979148} with this proposal of passive crowdsourcing collocation.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.85\\columnwidth]{.\/figures\/crowdsourcing_taxonomy}\n\t\\caption{Proposed taxonomy of human computation including passive crowdsourcing.}\n\t\\label{fig:crowdsourcingTaxonomy}\n\\end{figure}\n\nThe main advantage of passive crowdsourcing with respect to traditional crowdsourcing is the enormous availability of data to collect and very reduced costs of its retrieval, a significant disadvantage, on the other hand, is the form of the data: not always perfectly suitable for the goals and almost always needing to be processed and analyzed before being used. In other words, traditional crowdsourcing asks the user to shape the data in the input in a certain way, paying for this; passive crowdsourcing instead retrieves the result of past work, but shaped as it is, so processing it next. Just think about a study of the public opinion trends of a certain commercial product, two different ways to collecting these opinions can be used:\n\\begin{itemize}\n\\item\n\\emph{Crowdsourcing}: a certain amount of money is paid to each person who gives his proper opinion about the product being studied (approach also called crowdvoting). This method guarantees perfectly shaped collected data: the form of the questions and answers can be decided and modeled easily, but obviously the availability of the data to be collected will be limited by the number of the people ready to take that survey, and the costs of collecing a huge dataset of opinions will not surely be low.\n\\item\n\\emph{Passive Crowdsourcing}: the publicly available web data is full of opinions of the customers of a certain product (think about a customer that posts a photograph of that product, or comments a photograph uploaded by someone else, declaring their personal opinion about that product), the cost of retrieving these photographs and comments are almost null and the availability of this content is enormous. The big problem anyway is the shape of this data: given a photograph the algorithm must decide whether it is pertinent to the study or not (if the object in the photograph is the product being looked for) and to estimate if the opinion expressed by the user is positive or negative.\n\\end{itemize}\n\nIn the context of this work the huge amount of available data is fundamental, so the passive crowdsourcing approach is prefered, and the purpose of this work is exactly to deal with the problem of data shaping and analyzing.\n\nAnother significant advantage of passive crowdsourcing (and very important for this work) is the availability of the implicit information the user himself is unaware of: if a person is asked whether the peak of the Matterhorn (a mountain in the Italian Alps) was covered with snow or not in August of 2010 - he does not remember, but an appropriate image processing technique can extract this information from the photograph the user posted on his social network profile during his vacation.\n\n\\section{User Generated Content}\nThe amount of available user generated media content on the Web nowadays is reaching unprecedented mass: Facebook alone hosts 240 billion photographs and gets 300 million new ones every day \\cite{Doherty2010Lifelog}. This massive input allows collective intelligence applications to reach unprecedented results in innumerable domains, from smart city scenarios to land and environmental protection.\n\nA significant portion of this public dataset are geo-tagged photographs. The availability of geo-tags in photos results from two current trends: first, the widespread habit of using the smartphone as a photo camera; second, the increasing number of digital cameras with an integrated GPS locator and Wi-Fi module. The impact of these two factors is that more and more personal photographs are being published on social portals and more and more of them are precisely geo-tagged \\cite{npdPhotosSmartphones}. A time-stamped and geo-located photo can be regarded as the state of one or more objects at a certain time. From a collection of photographs of the same object at different moments, one can build a model of the object, study its behavior and evolution in time, and build predictive models of properties of interest.\n\nHowever, a problem in the implementation of collective intelligence applications from visual user generated content (UGC) is the identification of the objects of interest: each object may change its shape, position and appearance in the UGC, making its tracking in a set of billions of available photographs an extremely difficult task. In this work we aim at realizing a starting point for the applications that generate collective intelligence models from user-generated media content based on object extraction and model construction in a specific environmental sector: the study of mountain conditions. We harness the\ncollective effort of people taking pictures of mountains from different positions and at different times of the year to produce models describing the states of selected mountains and their changing snow conditions over time. To achieve this objective, we need to address the object identification problem, which for mountains is more tractable than the general case described above, thanks to the fact that mountains are among the most motionless and immutable objects present on the planet. This problem of mountain identification in fact will be the goal of this work.\n\n\\section{Problem Statement}\nGiven a precisely geo-tagged photograph, the goal of this work is to determine whether the photograph contains a mountain panorama, if yes, estimate the direction of view of the photo camera during the shot and identify the visible mountain peaks on the photograph.\n\nWe will describe in the detail the proposed algorithm, how it has been implemented and tested with the result of successfully matching 64.2\\% of the input geo-tagged photographs.\n\nThe algorithm of peak detection that is  presented can be used in applications where the purpose is to identify and tag mountains in photographs provided by users. The algorithm can also be used for the creation of mountain models and of their related properties, such as, for example, the presence of snow at a given altitude.\n\nTwo representative examples of the first type of usage are:\n\\begin{itemize}\n\\item Mountain peak tagging of user-uploaded photographs on photo sharing platforms that allow anyone browsing that photograph to view peak names of personal interest.\n\\item Augmented reality on mobile devices with real-time peak annotation on the device screen while in camera mode.\n\\end{itemize}\n\nAn example of the usage for environmental model building is the construction of a model for correcting ground and satellite based estimates of the Snow Water Equivalent (SWE) on mountains peaks (which is described in the Conclusions and Future Work section).\n\n\\section{Document Structure}\nIn the next chapter several past works will be discussed, each one relevant in its own way and field to this work: from passive crowdsourcing and influenza surveillance to image processing and vision-based UAV navigation.\n\nIn the third chapter the proposed algorithm itself is described in detail and an efficient vector-based matching algorithm is explained.\n\nIn chapter four the realized implementation of the discussed algorithm is explained, including all improvement techniques developed (even those that have been rejected after the validation phase).\n\nIn the fifth chapter the results of the tests performed on the implemented algorithm are listed with the data set structure and error metric description.\n\nFinally in the last chapter the conclusions about this work are drawn with the possible future direction of this project.\n\n\\chapter{Related Work and State of the Art}\n\\label{Related Work and State of the Art}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\n\\noindent\nThis work combines many disciplines of social computing, image processing, machine intelligence and environmental modeling with the relative emblematic problems: passive crowdsourcing, object identification (in particular mountain boundaries) and pose estimation, collective intelligence extraction and knowledge modeling, snow level estimation. All of the these problems have been largely analyzed and studied recently, but only a very small part of them combines several of the listed problems.\n\nQuoting all the works in those fields would be impossible, so in this chapter several examples of works from each of these fields (often combined together) will be discussed, from social problems and those closely inherent to this work, to examples of non-social applications, emphasizing in fact the wide possibility of goals that can be reached with these approaches.\n\nThe fundamental concept of computational social science is treated by Lazer et at. \\cite{lazer2009life}, explaining the trend of social science, moving from the analysis of individuals to the study of society: nowadays almost any action performed, from checking email, making phone calls and checking our social network profile to going for a walk, driving to the office by car, booking a medical check-up or even paying at the supermarket with a credit card leaves a digital fingerprint. The enormous amount of these fingerprints generates data, that pulled together properly give an incredibly detailed picture of our lives and our societies. Understanding these trends of societal evolution and so also individuals changing is a complex task, but with an unprecedented amount of potential and latent knowledge waiting to be extracted. The authors discuss the problems of this approach such as managing of privacy issues (think about the NRC report on GIS data \\cite{national2007Putting} describing the possibility to extract the individual information even from well anonymized data, or the online health databases which were pulled down after a study revealed the possibility of confirming the identities \\cite{dnaBlocked}). The lack of the both approach and infrastructure standards in this emerging field is also discussed:\n\\begin{quotation}\n\n\\noindent{\\emph{``\nThe resources available in the social sciences are significantly smaller, and even the physical (and administrative) distance between social science departments and engineering or computer science departments tends to be greater than for the other sciences. The availability of easy-to-use programs and techniques would greatly magnify the presence of a computational social science. Just as mass-market CAD software revolutionized the engineering world decades ago, common computational social science analysis tools and the sharing of data will lead to significant advances. The development of these tools can, in part, piggyback on those developed in biology, physics and other fields, but also requires substantial investments in applications customized to social science needs.\n''}\n Lazer et at. \\cite{lazer2009life}\n}\n}\n\\end{quotation}\n\n\\section{Passive Crowdsourcing and Collective Intelligence}\nAn important work in this field (even if the authors do not use the term of passive crowdsourcing, but it is exactly the type of problem solving we mean this term for) is performed by Jin et al. \\cite{Jin:2010:WSM:1873951.1874196} in their study of society trends from photograph analysis. The authors propose a method for collecting information to identify current social trends, and also for the prediction of trends by analyzing the sharing patterns of uploaded and downloaded social multimedia. Each time an image or video is uploaded or viewed, it constitutes an implicit vote for (or against) the subject of the image. This vote carries along with it a rich set of associated data including time and (often) location information. By aggregating such votes across millions of Internet users, the authors reveal the wisdom that is embedded in social multimedia sites for social science applications such as politics, economics, and marketing \\cite{Jin:2010:WSM:1873951.1874196}.\n\nGiven a query, the relevant photographs with relative metadata are extracted from Flickr, and the global social trends are estimated. The motivation for introducing this approach is the low cost of crawling this information for the companies and the industries, as well as the possibility to analyze this data almost instantaneously with respect to common surveys. The implementation of the proposal is also discussed, with several tests in various fields that gave incredibly promising results:\n\\begin{itemize}\n\\item\n\\emph{Politics}: the popularity scores analysis of the candidates Obama and McCain during the USA president elections of 2008 gave the resulting trends which were correct within a tenth of a percent of the real election data.\n\\item\n\\emph{Economics}: a product distribution map (with \\emph{iPod} as the example) around the world over time was successfully drawn.\n\\item\n\\emph{Marketing}: the sales of past years of several products such as music players, computers and cellular phones were estimated and the results actually match the official sales trends of those products.\n\\end{itemize}\n\nAlthough being an innovative and efficient approach, it does not deal with the visual content of the images themselves (the authors also highlight this fact, declaring the intent to add this analysis in the future). An example of collective intelligence extraction using image content is the work of Cao et al. \\cite{conf\/icassp\/CaoLGJHH10} that presents a worldwide tourism recommendation system. It is based on a large-scale geotagged web photograph collection and aims to suggest with minimal input to tourists the destination they would enjoy. By taking more than one million of geotagged photographs, dividing them into clusters by geographical position and extracting the most representative photographs for each area, the system is able to propose destinations to the user having in input the set of keywords and images of the places the user likes. From a conceptual point of view this system tries to simulate a friend that knows your travel tastes and suggests destinations for your new journey, but with the difference that this virtual friend has visited millions of places all around the world. This lies exactly in the concept of collective intelligence: exploiting the effort of hundreds of thousands of people uploading their travel photographs, the authors build a knowledge on what the various world tourism destinations feel like.\n\n\\subsection{Real-Time Social Monitoring}\nAn approach related to the concept of Human Sensors treated in the previous chapter is the process of monitoring the online activity of the persons to predict in advance (or at least to identify quickly) the occurrence of some phenomena. The activity to be monitored and the phenomena to detect can be very different. The most popular online activity to monitor is for sure the web searches of the users, such as in works of Polgreen et al. \\cite{Polgreen_usinginternet}, Ginsberg et al. \\cite{citeulike:3681665} and Johnson et al. \\cite{15361003} (in Johnson et al. also the access logs to health websites are analyzed) that propose influenza surveillance exploiting web search query statistics, examining the connection between searches for influenza and actual influenza occurrence and finding strong relationships. Another example of web query monitoring is the prediction of macroeconomic statistics by Ettredge et al. \\cite{Ettredge:2005:UWS:1096000.1096010}, in particular the prediction of unemployment rate trend based on the frequency of search terms likely used by people seeking employment.\n\nAnother important source of activity are social networks, for example Twitter messages, used by Culotta \\cite{Culotta:2010:TDI:1964858.1964874} in the monitoring of influenza, similar to the references described above, and by Sakaki et al. \\cite{Sakaki:2010:EST:1772690.1772777} for detecting the earthquakes.\n\n\\section{Object Identification and Pose Estimation}\nThe pose estimation of the photograph will be the key problem of this work, even if the estimated variable is only the orientation and not the position (that is given in input), and is a very commonly treated problem. Several examples will be listed here with problems, each one with its own elements in common with this work.\n\nAn example of a relatively different problem of pose estimation with respect to this work is the estimation of the geographic position of a photograph proposed by Hays and Efros \\cite{Hays:2008:im2gps}: though the estimation of the position is performed with the analysis of the visual content of the image, a purely data-driven scene matching approach is applied to estimate a geographic area the photograph belongs to.\n\nRamalingam et al. \\cite{RBSB09} instead present a work that at first sight can seem the opposite of this one (instead of estimating the orientation given the position they estimate the position given the orientation: always perpendicular to the terrain) but it is very similar: the authors describe a method to accurately estimate the global position of a moving car using an omnidirectional camera and untextured 3D city models. The idea of the algorithm is the same: estimate the pose by matching the input image to a 3D model (city model in this case, elevation model in case of our work). The described algorithm extracts the skyline from an omni-directional photograph, generates the virtual fisheye skyline views and matches the photograph a to view, estimating in this way the position of the camera.\n\nOther works about pose estimation given 3D models of cities and buildings have been recently published, such as world wide pose estimation using 3D point clouds by Li et al. \\cite{Li:2012:WPE:2402940.2402943} in which the SIFT features are located and extracted in the photograph and matched with the worldwide models. The particular point of this pose estimation is that it does not use any geographical information, but it estimates the position, orientation and the focal length (and so the field of view).\n\nBaatz et al. \\cite{Baatz:2012:LCM:2125160.2125170} addresses the problem of place-of-interest recognition in urban scenarios exploiting 3D building information, giving in output the camera pose in real world coordinates ready for augmenting the cell phone image with virtual 3D information. Sattler et al. instead deals with the problem of the 2D-to-3D correspondence computation required for these cases of pose estimation of urban scenes, demonstrating that direct 2D-to-3D matching methods have a considerable potential for improving registration performance.\n\nAn innovative idea of photograph geolocalization by learning the relationship between ground level appearance and overhead appearance and land cover attributes from sparsely available geotagged ground-level images \\cite{LinCrossView} was introduced by Lin et al. \\cite{LinCrossView}: unlike traditional geolocalization techniques it allows the geographical position of an isolated photograph to be identified (with no other geotagged photographs available in the same region). The authors exploit two previously unused data sets: overhead appearance and land cover survey data. Ground and aerial images are represented using HoG \\cite{Dalal:2005:HOG:1068507.1069007}, self-similarity \\cite{shechtman2007matching}, gist \\cite{Oliva:2001:MSS:598425.598462} and color histograms features. For each of these data sets the relationship between ground level views and the photograph data is learned and the position is estimated by two proposed algorithms that are also compared with three other pre-existing techniques:\n\\begin{itemize}\n\\item\n\\emph{im2gps}:\nproposed by Hays and Efros \\cite{Hays:2008:im2gps} already described a few lines above, that does not make use of aerial and attribute information and can only geolocate query images in locations with ground-level training imagery.\n\\item\n\\emph{Direct Match (DM)}:\nmatches the same features for ground level images to aerial images with no translation, assuming that the ground level appearance and overhead appearance are correlated.\n\\item\n\\emph{Kernelized Canonical Correlation Analysis (KCCA)}:\nis a tool to learn the basis along the direction where features in different views are maximally correlated, used as a matching score. It however presents significant disadvantages: singular value decomposition for a non-sparse kernel matrix is need to solve the eigenvalue problem, making the process unfeasible as training data increases, and secondly, KCCA assumes one-to-one correspondence between two views (in contrast with the geolocalization problem where it is common to have multiple ground-level images taken at the same location) \\cite{Hays:2008:im2gps}. \n\\end{itemize}\n\nThe methods proposed by the authors are instead:\n\\begin{itemize}\n\\item\n\\emph{Data-driven Feature Averaging (AVG)}:\nbased on the idea that well matched ground-level photographs will tend to have also similar aerial and land cover attributes, this technique translates the ground level to aerial and attribute features by averaging the features of good scene matches.\n\\item\n\\emph{Discriminative Translation (DT)}:\nan approach that extends AVG with also a set of negative training samples, based on the intuition that the scenes with very different ground level appearance will have distinct overhead appearance and ground cover attributes (assumption obviously hypothetical and not always true).\n\\end{itemize}\n\nIn the performed tests the algorithm was able to correctly geolocate 17\\% of the isolated query images, compared to 0\\% for existing methods.\n\n\\subsection{Mountain Identification}\nAll the pose estimation works described so far in this chapter deal with urban 3D models, as does the majority of pose estimation research. This is not surprising since an accurate 3D model is fundamental for this kind of task, and the massive increase of the 3D data of buildings and cities in the last few years makes these studies possible. Apart from urban models however, there is another type of 3D data that is largely available, which evolved much earlier than urban data (even if usually with lower resolution): terrain elevation data. The elevation data, presented usually as a geographical grid with the altitude for each point, can be easily seen as a 3D model, and the most interesting objects formed by these models are for sure mountains. For this reason also mountains are sometimes the identified objects in the pose estimation task, here several examples of these works will be discussed.\n\nThe most significant work in this sector is probably that presented by Baboud et al. \\cite{Baboud2011Alignment}, which given an input geotagged photograph, introduces the matching algorithm for correct overlap identification between the photograph boundaries and those of the virtually generated panorama (based on elevation datasets) that should be seen by the observer placed in the geographical point where the photograph has been taken from. This algorithm, which will be discussed in detail further, is the starting point of this thesis work. It must be highlighted however, that the goal of the authors is peak identification for the implementation of photograph and video augmentation; this work instead aims to identify mountain peaks to extract the appearances of the mountain for environmental model generation.\n\nAnother important work, dealing not only with the direction, but also with the position estimation of a mountain image, was written by Naval et al. \\cite{Jr97estimatingcamera}. Their algorithm does not work with the complete edges of the image but only with the skyline extracted using a neural network. The position and the orientation is then computed by nonlinear least squares.\n\nA proposal that exploits the skyline and mountain alignment is the vision-based UAV navigation described by Woo et al. \\cite{WooSLKK07}, that with pose estimation in a mountain area (a problem similar to the other described works) introduces the possibility of vision-based UAV navigation in a mountain area using an IR (Infra-Red) camera by identifying the mountain peaks. This is a navigation method that is usually performed by extracting features such as buildings and roads, not always visible and available, so mountain peak and skyline recognition brings big advantages.\n\nA challenging task with excellent results is described by Baatz et al. \\cite{Baatz:2012:LSV:2403006.2403045} with a proposal of an algorithm that given a photograph, estimates its position and orientation on large scale elevation models (in case of the article a region of 40000~km$^2$ was used, but in theory the technique can be applied to position estimation on a world scale. The algorithm exploits the shape information across the skyline and searches for similarly shaped configurations in the large scale database. The main contributions are a novel method for robust contour\nencoding as well as two different voting schemes to solve the large scale camera pose recognition from contours. The first scheme operates only in descriptor space (it checks where in the model a panoramic skyline is most likely to contain the current query picture) while the second scheme is a combined vote in descriptor and rotation space \\cite{Baatz:2012:LSV:2403006.2403045}. The original six-dimensional search is simplified by the assumptions that the photograph has been taken close to the terrain level and the photograph has usually only a small roll with respect to the horizon (both assumptions were made also during the development of this work algorithm). Instead of supposing to have the right shot position, this technique renders the terrain view from a digital elevation model on a grid defined by distances of approximately 100~m~$\\times$~100~m for a total number of 3.5 million cubemaps (this is how the authors call the renders). The key problem of the work is in fact, the large scale search method to efficiently match the query skyline to one of the cubemaps. Given a query image, sky\/ground segmentation is performed following an approach based on unary data costs \\cite{Martin:2004:LDN:977249.977379,Luo:2002:PAD:839290.842704} for a pixel being assigned sky or ground. The horizon skyline then is represented by a collection of vector-quantized local contourlets (contour words, similar in spirit to visual words obtained from quantized image patch descriptors) that are matched to the collected cubemaps with a voting stage that retrieves the most probable cubemaps to contain the query skyline, the top 1000 candidates are then analyzed with geometric verification using iterative closest points to determine a full 3D rotation. Evaluation was performed on a data set of photographs with manually verified GPS tags or given location, and in the best implementation 88\\% of the photographs were localized correctly.\n\n\\section{Environmental Study}\nIn this section the problem of snow level and snow water equivalent (SWE) estimation will be described, with the current state of the art and used methods as well as the benefits that an environmental model based on passive crowdsourcing photograph analysis can bring.\n\nSnow Water Equivalent (SWE) is a common snowpack measurement. It is the amount of water contained within the snowpack. It can be thought of as the depth of water that would theoretically result if you melted the entire snowpack instantaneously. To determine snow depth from SWE you need to know the density of the snow. The density of new snow ranges from about 5\\% when the air temperature is 14\\ensuremath{^\\circ} F, to about 20\\% when the temperature is 32\\ensuremath{^\\circ} F. After the snow falls its density increases due to gravitational settling, wind packing, melting and recrystallization \\cite{whatIsSWE}. It is a very important parameter for industry and agriculture, and its correct estimation is one of the main problems facing the environmental agencies.\n\nThese measurements are usually performed with physical sensors and stations that are very sparse, so the need of a map of snow and SWE distribution introduces a key problem: the interpolation of this data. Interpolation is usually done (due to the lack of the sophisticated physical models dealing with altitude and temperature and the absence of other supporting data) with relatively rough methods such as:\n\\begin{itemize}\n\\item\nlinear or almost linear interpolation of the data, lacking of precision due to the low spatial density of the measurement stations and the physical laws and phenomena that change the snow level and density radically with changes in altitude\n\\item\ncombining the interpolated data with the satellite ground images, removing the snow estimation from the areas where the satellite image indicates its absence: although being a first step in image content analysis for the refining of snow data, it indicates only a binary presence or absence of the snow and has anyway a low spatial density due to its low resolution.\n\\end{itemize}\n\nThe availability of the estimation of snow properties retrieved from mountain photograph analysis, and the past years snow measure ground truth in the areas and altitudes different from those that are measured by physical sensors, can bring significant supporting data for a better interpolation and more precise snow cover maps.\n\n\\subsection{Passive Crowdsourcing and Environmental Modeling}\nAn important step in environmental modeling exploiting passive crowdsourcing was made by Zhang et al. \\cite{Zhang:2012:MPW:2187836.2187938} in a work that studies the problem of estimating geotemporal distributions of ecological phenomena using geo-tagged and time-stamped photographs from Flickr. The idea is to estimate a given ecological phenomena (the presence or absence of snow or vegetation in this case) on a given day at a given place, and generate the map of its distribution. A Bayesian probabilistic model was used, assuming that each photograph taken in the given time and place is an implicit vote to the presence or the absence of snow in that area. Exploiting machine learning, the probability that a photograph contain snow is estimated based both on the metadata of the photograph (tags) and the visual features (a simplified version of GIST classifier augmented with color features was used). The estimation of the results (which was made by the authors thanks to the fact that for the two phenomena studied, snowfall and vegetation cover, large-scale ground truth is available in the form of observations of satellites \\cite{Zhang:2012:MPW:2187836.2187938}) brought very promising results: a daily snow classification for a 2 year period for four major metropolitan areas (NYC, Boston, Chicago and Philadelphia) generates results with precision, accuracy, recall and f-measure equal to approximately $0.93$.\n\nEven if the algorithm proposed by authors uses a simple probabilistic model (presence or absence of snow and vegetation cover) it is an important introduction to the field of environmental and ecological phenomena analysis by mining photo-sharing sites for geo-temporal information about these phenomena. The goal of this work is to propose an image processing technique that brings these environmental studies to a new level.\n\\chapter{Proposed Approach}\n\\label{Proposed Approach}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\n\\noindent\nIn this chapter we describe the proposed approach for the procedure of mountain identification, from the analysis of the properties of the photo camera used for the shot to the identification of the position in the photograph of each mountain peak.\n\nThe matching algorithm is partially based on an mountain contour matching technique proposed by Baboud et al. \\cite{Baboud2011Alignment}, so first this matching algorithm will be discussed with particular emphasis on its advantages and disadvantages and then our proposal will be described in detail.\n\n\\section{Analysis of Babound et al.'s algorithm}\nThe basic idea and the purpose of the algorithm is the same as that of this work: given a geotagged photograph in input, identify the mountain peaks present in it, by exploiting the elevation data and the dataset of peaks with their positions.\nThe algorithm in question treats the input image as spherical, and searches for the rotation on $SO(3)$ that gives the right alignment between the input and another spherical image representing the mountain silhouettes viewed by the shot point of the input photograph in all directions. This image is generated by the 3D terrain model based on a digital elevation map. A vector cross-correlation technique is proposed to deal with the matching problem: the edges of the images are modeled as imaginary numbers, the candidates for the estimated direction of view are extracted, and a robust matching algorithm is used to identify the correct match.\n\nAs will follow from the next sections, several changes with respect to the original algorithm have been introduced, most relevant among them are:\n\\begin{itemize}\n\\item\nInstead of dealing with the elevation models we rely on an external service, which generates the panorama view given the geographic coordinates and a huge set of parameters. The reason of this choice is the possibility to concentrate the work on matching technique, as well as the reliability and precision of a tool improved over the years.\n\\item\nInstead of supposing the field of view to be known, its estimation given a photograph and the basic EXIF information will be discussed together with the photograph scaling problem, that allows successful matching by a non scale-invariant algorithm.\n\\item\nInstead of searching for the orientation of the camera during the shot in three dimensions, we suppose that the observer's line of sight is always perpendicular to the horizon (no photograph in the test dataset had a significant tilt angle with respect to the horizon). This assumption simplifies the computational effort of the algorithm, by moving from considering the images as spherical to cylindrical.\n\\item\nOnce the camera direction is successfully estimated, the mountain peak alignment between the photograph and the panorama may still presents non-negligible errors due to inaccuracies of the estimated position of the observer. We will deal this problem, which is not treated in the original algorithm.\n\\end{itemize}\n\n\\section{Overview of proposed algorithm}\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.65\\columnwidth]{.\/figures\/algorithm_schema}\n\t\\caption{Schema of the mountain peak tagging algorithm.}\n\t\\label{fig:algorithmSchema}\n\\end{figure}\n\nThe process of analyzing a single photograph containing a mountain panorama to identify individual mountain peaks present in it consists of several steps, shown in Figure \\ref{fig:algorithmSchema}.\n\nFirst the geotagged photograph is processed in order to evaluate its Field Of View and to scale it making the photograph match precisely the rendered panorama image, next a 360-degree rendition of the panorama visible at the location is generated from the digital elevation models.\nAfter this the direction of the camera during the shot is estimated by extracting the edge maps of both photograph and rendered panorama images and matching them. Finally the single mountain peaks are tagged based on the camera angle estimation.\n\n\\section{Detailed description of the algorithm}\n\n\\subsection{Render Generation}\nThe key idea of the camera view direction estimations lies in generating a virtual panorama view of the mountains that should be seen by the observer from the point where the photograph was taken, and then matching the photograph to this panorama.\nThe generation of this virtual view is possible due to the availability of terrain elevation datasets that cover a certain geographical area with a sort of grid identifying the terrain elevation in each grid point. Depending on the source of the elevation data, the precision and spatial density of this grid can vary significantly, but usually it is very precise, reaching even a spatial density of 3 meters in public datasets (such as USGS, http:\/\/ned.usgs.gov).\n\nThough the accuracy of elevation models is crucial for our purposes, since an exact render is the basis for a correct match, we are not necessary looking at extremely high-resolution datasets as the mountains we are going to generate are usually located at a significant distance from the observer (starting from few hundreds of meters to tens of kilometers). For this project the use of an external service that generated these rendered panoramas was preferred instead of creating them from scratch using the elevation data. The advantage of this choice is the possibility to avoid dealing directly with digital elevation maps and with optical and geometric calculations, in order to provide the panorama exactly as it should be seen by a human eye or photo camera lens. Such tools thus free the system from laborious calculations, allowing simply to choose the parameters needed for the generation of the panorama such as the observer's position, altitude and angle of gaze.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/mountainview2}\n\t\\caption{Example of wrong position estimation problems with view positions and the relative generated panoramas. 1 - Original viewer's position and altitude, 2  - estimated position and wrong altitude, 3 - elevated estimated position used for the final panorama generation.}\n\t\\label{fig:mountainView}\n\\end{figure}\n\nThe choice of the observer's altitude deserves to be mentioned separately, as it is a critical point: clearly the ideal value that can be set (with an ideal elevation model) is the real altitude of the observer during the shot (which, after studying the available datasets, we consider more than legitimate to suppose to be on the terrain surface, so we estimate the altitude of the observer as the terrain altitude in that position). This information is readily available, but it brings some problems due to the uncertainty of the geotag of the photograph: let us imagine an observer standing on a peak or a ridge of a mountain - he has a broad view in front of him, but it is enough for him to take a few steps back and the panorama he was viewing becomes completely covered by the facade of the mountain in front of him. The same issue occurs with the photographs: a photograph taken from a peak or a ridge of a mountain (very frequently the case in public domain collections of mountain photographs). The error of estimating the photo camera position of few meters can lead to an overlay of a significant portion of the panorama. An intuitive technique that can walk around this problem is to add some constant positive offset to the estimated altitude to \"raise\" the observer above the terrain obstacles that have appeared due to the errors in position estimation. Figure \\ref{fig:mountainView} shows in a simplified way the problem and its resolution with the corresponding generated panoramas.\n\n\\subsection{Field of View Estimation and Scaling}\nUnless a scale invariant matching technique is going to be used (and it is not the case of this algorithm), once the input image and an image representing the expected panorama view are ready, the first problem is that in order to be correctly matched the objects contained in both images (in our case the objects to be matched are mountains) must have the same pixel size. Since we assume that both represent the view of an observer from the same geographical point, the same pixel dimension of the object involves also the same angular dimensions (the angle a certain object occupies in the observer's view). So we define our first photograph analysis problem as searching for the right scaling of the input photograph to have the same angular dimensions for the same mountains present in the photograph and the rendered panorama. We can write down the problem as finding a scale factor $k$ such that $$k\\frac{s_{p}}{a_{s}} = \\frac{s_{r}}{a_{r}}$$ where $s_{p}$ and $a_{p}$ are respectively the pixel size and the angular size of the input photograph and $s_{r}$ and $a_{r}$ are similarly the pixel size and the angular size of the rendered panorama.\n\nWe expect the panorama to be exact, so we consider the mountains to have the same width\/height ratio both on the photograph and on the panorama, so the relationship described above can be equivalently applied both to the horizontal and vertical dimensions: we will work with the horizontal dimenstion because the angular width of the panorama does not need any calculation (it is always equal to the round angle, $2\\pi$). Defining the Field Of View (FOV) of an image as its angular width we can rewrite the relationship as $$k*\\frac{w_{p}}{FOV_{s}} = \\frac{w_{r}}{FOV_{r}} = \\frac{w_{r}}{2\\pi}$$ where $w$ and $FOV$ stands for the pixel width and the FOV of respectively the photograph and the rendered panorama.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/fov}\n\t\\caption{A simplified schema of digital photo camera functioning.}\n\t\\label{fig:fovSchema}\n\\end{figure}\n\nBefore explaining how to estimate the FOV of the input photograph we must introduce some brief concepts regarding digital photo camera structure and optics laws: in a very simplified way a photo camera can be seen as a lens defining the focus of the projection of a viewed prospect on a sensor that captures this projection. The size of the sensor is a physical constant and property of a photo camera, the so called focal length instead (that defines the distance between the sensor and the lens) usually varies with the changing of the optical zoom of the camera. The FOV of the captured image in this case is obviously the angular size of the part of the prospect projected on the sensor. Figure \\ref{fig:fovSchema} shows this simple schema. We can easily write the relationship between the FOV of the photograph and the properties of the photo camera at the instant of shooting ($s$ for sensor width and $l$ for focal length): $$FOV = 2\\arctan\\frac{s}{2l}$$\n\nCombining this definition with the previous relationship we can express the scaling factor $k$ that must be applied to the photograph in order to have the same object dimensions as: $$k = FOV\\frac{w_{r}}{2\\pi w_{p}} = \\frac{w_{r}}{\\pi w_{p}}\\arctan\\frac{s}{2l}$$\n\nScaling estimation is a purely mathematical procedure, and the quality of the results depends directly on the precision of the geotag and the accuracy of the rendered model (with exact GPS location and a render based on correct elevation data the scaling produces perfectly matchable images).\n\n\\subsection{Edge Detection}\n\nThe key problem of the algorithm is to perform matching between the photograph containing the mountains and the automatically generated drawing representing the same mountain boundaries: in Figure \\ref{fig:mountainOverlapping} we can see an example of a fragment of the photograph and a fragment of the rendered panorama. Both represent the same mountain seen from the same point of view, and in fact when we try to match them manually they overlap perfectly. In spite of this overlapping, the choice of the technique for their matching is not trivial. Even if the problem of matching (choosing the position of the photograph with respect to the panorama, maximizing the overlap between the mountains) will be discussed in the next sections, here we will briefly mention the possible techniques to evaluate the correctness of the overlap and the reasons for the necessity of the edge extraction procedure.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/mountain_overlapping}\n\t\\caption{An example of a matching problem with the photograph fragment (top right), the panorama fragment (top left) and their overlapping (bottom).}\n\t\\label{fig:mountainOverlapping}\n\\end{figure}\n\nWe can see the matching problem as a classic image content retrieval problem with the photograph as the input image and the collection of the fragments of the rendered panorama (each one corresponding to a possible alignment position) as the set of available images to search in. The most similar image in the set will be considered as the best matching position and identified as the direction of view of the camera during the shot.\n\nSo what is the similarity measure that can be used in order to perform this image content retrieval problem? Clearly global descriptors (color and texture descriptors) \\cite{citeulike:10106398} are not the best choice: it is enough to look at the Figure \\ref{fig:mountainOverlapping} to understand that the panorama is always generated in gray-scale tones, with textures defined only by the terrain elevation, while the mountains in the photograph can be colored in different ways, and the textures are defined by a lot of details such as snow on the mountains and other foreground objects such as grass, stones and reflections on the water. Local image descriptors instead (for example, local feature descriptors such as SIFT \\cite{Lowe:2004:DIF:993451.996342} and SURF \\cite{Bay:2008:SRF:1370312.1370556}) seem slightly suitable for the needs of mountain matching (as discussed also by Valgren et al. in \\cite{Valgren:2010:SSS:1715935.1716080}, where the use of SIFT and SURF descriptors is highlighted for the matching of outdoor photographs in different season conditions), but even if they are good for matching the photographs containing the same objects in different color conditions, the matching between an object photograph and its schematic representation (in our case a rendered drawing) tends to fail. Even if very accurate, the model of a photograph will not generate local features with the same precision that another photograph of the same object would do: no local descriptors tested have been able to find the match between the two example images in Figure \\ref{fig:mountainOverlapping}.\n\n\\begin{figure}[h!]\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/mountain_edges_overlapping}\n\t\\caption{An example of a matching problem with the photograph edge fragment (top right), the panorama edge fragment (top left) and their overlapping (bottom).}\n\t\\label{fig:mountainEdgesOverlapping}\n\\end{figure}\n\n\nThe perfect overlap between the images in the example figure however brings up to the idea, that instead of traditional image descriptors we should match the boundaries of the images (this assumption has also an intuitive motivation: the contours are the most significant and time invariant properties of the mountains), so the next step in photograph and panorama processing is edge extraction from both the images. The result of edge map extraction from the images in Figure \\ref{fig:mountainOverlapping} is represented in Figure \\ref{fig:mountainEdgesOverlapping}. The matching procedure on the edge maps can now be seen as a cross-correlation problem \\cite{wiki:CrossCorrelation}.\n\nIn order to make the cross-correlation matching more sophisticated with a couple of techniques that will be presented later, the edge extraction component must accept as input an image (the photograph or the rendered panorama) and produce as output a 2D matrix of edge points, each one corresponding to the point on the input image with a strength (value between 0 and 1 representing the probability of the point to be an edge) and a direction (value between 0 and 2$\\pi$ representing the direction of the edge in that point).\n\n\\subsection{Edge Filtering}\nOnce we have generated the edge maps of the photograph we must deal with the problem of the noise edge points. A noise edge point can be defined as an extracted edge point representing a feature that is not present on the rendered panorama, in this case the edge point will not be useful for the match and will only be able to harm it. In other words a noise edge point is an edge point that does not belong to a mountain boundary (the only features represented on our rendered panoramas). This doesn't mean that any edge point belonging to a mountain is not a noise point, i.e. the edge points that define the snow border of a mountain are noise points because they do not belong to the mountain boundary but define a border that due to its nature can easily change from photograph to photograph and furthermore will not be present in the rendered view.\n\nHowever, edge points can be also present on the mountain surface; most of them usually belong to foreign objects. Mountains tend to be very edge-poor and detail-poor objects and often are placed in the background of a photograph with other edge-rich objects such as persons, animals, trees, or houses in the foreground. Let us think about a photograph of a person next to a tree with a mountain chain in the background: the mountains themselves will generate few edge points since they tend to be very homogeneously colored; foreground objects instead will generate a huge amount of edge points (which will be noise) because they are full of small details, each leaf of the tree and detail of the person's clothes will produce noise points.\n\nTaking into account the example of the edge extraction made in the previous chapter, we manually tag the extracted points as noise and non-noise points following the edges present in the panorama: the result is represented in the Figure \\ref{fig:filteringGoodBadBefore} and with a simple image analysis script we find that the the noise edge points are more than 90\\% of all extracted points.\n This value grows up to 98\\% when we deal with photographs with even more objects in foreground. Even if the matching algorithm we are going to propose includes penalizing the noise points in order to keep up with this problem the amount of the noise edges reaches such high levels that it cause almost any algorithm to fail simply from a statistical point of view (the intense density of noise points in some area tends to ``attract'' the matching position in that area), so an edge filtering technique is needed.\n \n \\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{.\/figures\/filtering_good_bad_before}\n\t\\caption{Noise (red) and non-noise (green) edge points on our example photograph.}\n\t\\label{fig:filteringGoodBadBefore}\n\\end{figure}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.8\\columnwidth]{.\/figures\/filtering_good_bad_after}\n\t\\caption{Noise (red) and non-noise (green) edge points on our example photograph after the filtering procedure has been applied.}\n\t\\label{fig:filteringGoodBadAfter}\n\\end{figure}\n\nOne of the possible approaches is the detection of the skyline (the boundary between the sky and the objects present in the photograph) as implemented by Naval Jr et al. \\cite{Jr97estimatingcamera} for the analogue problem. Skyline detection is a non-trivial task that has been widely studied with several proposed approaches available in particular sectors and scenarios, such as skyline detection of a perspective infrared image by synthesizing a catadioptric image proposed by Bazin et al. \\cite{DBLP:conf\/icra\/BazinKDV09} or the extraction of the skyline from a moving omnidirectional camera as developed by Ramalingam et al. \\cite{RBSB09}. The skyline however is not the only boundary of our interest, the boundaries of mountains contained ``inside'' other boundaries are also important and significant for the matching, so we decided to opt for softer a filtering approach.\n\nWe have opted for a simple but efficient technique based on the intuitive assumption that the mountains are usually placed above the other objects on the photographs (with some exceptions such as clouds or other atmospheric phenomena): prioritizing (increasing the strength of) the higher points on the photograph with respect to the lower points. An example of the result of the edge filtering procedure is displayed in Figure \\ref{fig:filteringGoodBadAfter}. Most of the noise edges are filtered with almost all good edges intact (also the edges that would be cut with a skyline detection). The rate of the noise edge points dropped from 90\\% to 40\\% in this case.\n\n\\subsection{Edge matching (Vector Cross Correlation)}\nFirstly we define $C_{r}$ as the cylindrical image generated from the rendered panorama with the same height, and the base perimeter equal to the panorama's width, and $C_{p}$ as the input photograph projected on an empty cylindrical image of the same dimensions of $C_{r}$. Imagining two cylinders to be concentric the matching problem is defined as the two dimensional space search (the vertical position and the rotation angle of $C_{r}$ with respect to $C_{p}$), which leads to the best overlap between the mountains present on the cylindrical images. As the cylindrical images are only a projection of a rendered image on a cylinder, the problem can be equivalently seen as a search for the best overlap of two rectangular images defined by two integer numbers representing the coordinate offset of the photograph with respect to the panorama (obviously carrying out the horizontal overflow part of the photograph to the opposite side of the render, simulating a cylinder property) as shown in Figure \\ref{fig:cylinderMatching}.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/cylinder_matching}\n\t\\caption{Representation of cylindrical image matching and the equivalent rectangular image matching.}\n\t\\label{fig:cylinderMatching}\n\\end{figure}\n\nAs already introduced in previous sections, the matching will be performed with the edge maps of the images so we define the result of the edge detection algorithm as a 2D real-valued vector where each value is defined by $\\rho$ (the strength\/absolute value of the edge in the corresponding point of the input image) and $\\theta$ (the direction\/argument of that edge). Let $p(\\omega)$ and $r(\\omega)$ be the 2D real-valued vectors generated by edge detection of the photograph and the panorama render respectively. Then as proposed by Baboud et al. \\cite{Baboud2011Alignment} the best matching is  defined as the position maximizing the likelihood between two images. This likelihood is defined as $$L(p,r) = \\int_{S^{2}} M(p(\\omega),r(\\omega))d\\omega$$\nwhere $M$ is the angular similarity operator:\n$$M(v_1,v_2) = \\rho ^{2}_{v_1} \\rho ^{2}_{v_2} \\cos 2(\\theta_{v_1} - \\theta_{v_2})$$\n\nThis technique of considering the edge maps as vector matrices and applying the cross-correlating resolution is called Vector Cross Correlation (VCC). The cosine factor is introduced in order to handle edge noise by penalizing differently oriented edges: the score contribution is maximum when the orientation is equal, null when they form a $\\frac{\\pi}{4}$ angle, and minimum negative when the edges are perpendicular (Figure \\ref{fig:penalizingEdges}). This penalty avoids that  random noise edges contribute in a positive way to a wrong match position (a step in this direction was already made during the edge filtering).\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/penalizing_edges}\n\t\\caption{Example of an overlapping position with positive score with almost parallel edge intersection (left circle) and penalizing almost perpendicular edge intersection (right circle).}\n\t\\label{fig:penalizingEdges}\n\\end{figure}\n\nOne of the main advantages of this likelihood score is the possibility of applying the Fourier transform in order to reduce drastically the computation effort of best matching position: let us define $\\hat{p}$ and $\\hat{r}$ as 2D Fourier transforms of respectively $p(\\omega)$ and $r(\\omega)$. The VCC computation equation becomes\n\\begin{equation} \\label{eq:FFTVCC}\nL(p,r) = Re\\{\\hat{p}^{2} \\bar{\\hat{r}}^{2}\\}\n\\end{equation}\nThis cosine similarity VCC can be seen as the whole algorithm to perform the matching or only as its first step: if we build the 2D real-valued distribution of the likelihood values estimated in each possible position we can consider not only the best match but extract top-N peaks of the score distribution as the result candidates and then evaluate them with a more sophisticated and heavier technique in order to identify the correct match. One of these refining algorithms is the robust silhouette map matching metric technique designed by Baboud et al. \\cite{Baboud2011Alignment} which considers the edge maps as sets of singular connected edges. The likelihood of an overlap is the sum of the similarity between each edge $e_p$ of the photograph and the edge $e_r$ of the rendered panorama, where $e_r$ is enriched with a certain $\\epsilon_e$ neighborhood, $l$ represents the distance for which $e_p$ stays inside the $\\epsilon_e$ neighborhood of the $e_r$ and the similarity is defined as\n\\begin{equation} \\label{eq:robustMatching}\nM(e_p, e_r) = \n\\left\\{\\begin{matrix}\n0\n\\\\ \nl^a\n\\\\ \n-c\n\\end{matrix}\\right.\n\\begin{array}{ll}\n\\text{if }l = 0\n\\\\ \n\\text{if }(l > l_{fit}) \\land (e_p \\text{ enters and exits on the same side})\n\\\\ \n\\text{if }(l < l_{fit}) \\land (e_p \\text{ enters and exits on different sides})\n\\end{array}\n\\end{equation}\nwhere $a$, $c$ and $l_{fit}$ are predefined constants. The nonlinearity implied by the exponent $a$ makes longer edge overlaps receive more weight than the set of small overlaps, the constant $c$ instead introduces also in this metric the penalizing factor for the intersections of the edges considered wrong as in the VCC metric.\n\n\n\\subsection{Mountain Identification and Tagging}\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/result_wrong_peaks}\n\t\\caption{Example of a fragment of a matching result between the photograph (blue) and the panorama (red).}\n\t\\label{fig:resultWrongPeaks}\n\\end{figure}\n\nOnce the edges are matched and the direction of view of the camera during the shot is estimated, supposing to have the list of mountain peaks with their names and coordinates on the rendered panorama we can estimate the position of these peaks also on the photograph, even if it is not as trivial as can be initially thought. Intuition suggests that once the photograph is matched with the panorama the coordinates of the mountain peaks are the same on both (obviously with a fixed offset equal to the matching position in case of the photograph) but it is wrong. Looking at a fragment of the result of edge matching (the best estimated position by VCC) of a photograph in Figure \\ref{fig:resultWrongPeaks}: the overlapping between the photograph and panorama edges is almost perfect on the left part but tends to be always worse moving to the right, which means that the two edge maps are slightly different (probably due to the error in position and altitude estimation) and objectively the matching proposed by the VCC seems to be the best that can be obtained. This situation occurs very often, and in general we can say that the resulting edge matching, even when successful, presents small errors in singular mountain peaks due to the imperfections of the generated panorama model. We propose a method for precise mountain peak identification: for each mountain peak present in the panorama we extract an edge map pattern both from the photograph and the panorama centered in the coordinate of the peak on the panorama and weighted with a certain kernel function $f(d)$ for a fixed pattern radius $r$ and $d$ representing a point distance from the peak coordinate with the following properties:\n$$\n\\begin{array}{ll}\nf(0) = 1\n\\\\ \nf(x) = 0 \\text{ }\n\\\\ \nf(x_1) \\geq f(x_2) \\text{ }\n\\end{array}\n\\begin{array}{ll}\n\n\\\\ \n\\forall x \\geq r\n\\\\ \n\\forall x_1, x_2 \\, \\mid \\, x_1 \\leq x_2\n\\end{array}\n$$\nOnce the two patterns are extracted the VCC procedure is once again applied in order to match them, having only the edges of the mountain peak we are processing and a few surrounding edges, the matching position is no longer influenced by the other peaks and is refined to the exact position, an example of this procedure performed on the matching result introduced in Figure \\ref{fig:resultWrongPeaks} in order to identify one of the most right-placed peaks is displayed in Figure \\ref{fig:peakIdentification}.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/peak_identification}\n\t\\caption{Peak identification procedure on the previous matching example.}\n\t\\label{fig:peakIdentification}\n\\end{figure}\n\n\n\\chapter{Implementation Details}\n\\label{Implementation Details}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\n\\noindent\nIn this chapter the implementation of the matching algorithm proposed in the previous chapter is presented, from the general description of the system architecture and programming languages involved to the value of each parameter and constant introduced in the algorithm presentation.\n\n\\section{Overview of the implementation architecture}\nThe whole system can be split into two macro areas:\n\\begin{itemize}\n\\item\n\\emph{Photograph analysis, panorama and mountain peak list generation}: implemented in PHP with web interface due to the simplicity of interfacing with the external panorama generation service and interface versatility. Given a geo-tagged photograph (or a photograph with explicitly specified the geographic point of the shot) the virtual panorama view is generated centered at the same origin point and the properties both of the photograph and the photo camera are retrieved in order to let the next step calculate the field of view and scale factor.\n\\item\n\\emph{Matching and mountain identification}: implemented in MATLAB for the high suitability with image processing techniques. Given a photograph with all necessary information, the panorama and the mountain list with the relative coordinates on the panorama, edge extraction and filtering is performed, the camera direction is estimated, and finally individual mountains are identified and tagged.\n\\end{itemize}\n\nThe choice of the operating parameters used in the final implementation was defined in the testing and validation phase, using a data set and a developed evaluation metric. Although the parameter values will be specified as soon as they are introduced, the detailed description and the reasons of discarding the other values will be presented in the next chapter.\n\n\\section{Detailed description of the implementation}\n\n\\subsection{Render Generation}\nAs already mentioned in the implementation an external panorama service has been used: the mountain view generator of Dr. Ulrich Deuschle \\footnote{\\url{http:\/\/www.udeuschle.de\/panoramen.html}} (Figure \\ref{fig:udeuschleScreenshot}). The service accepts as input several parameters, most important of which are the geographic position of the observer, his altitude and altitude offset, the view direction with the field of view, zoom factor, elevation exaggeration and the range of sight. Latitude and longitude are equal obviously to the geo-tag of the input photograph, and the horizontal extension (field of view) to the round angle, $2\\pi$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/udeuschle_screenshot}\n\t\\caption{A screenshot of the mountain panorama generation web service interface.}\n\t\\label{fig:udeuschleScreenshot}\n\\end{figure}\n\nThe choice of the altitude (in fact of the altitude offset) as already anticipated in the previous chapter is a problematic point: a big offset can lead to vertical distortion of the panorama with respect to reality, a small offset on the other hand can lead more easily to partial occlusion of the panorama. The choice was made for $auto + 10$~m altitude setting, chosen by trial and error. Regarding the altitude choice an important option of the service named ``Look for summit point automatically'' must be highlighted: by turning on this option the panorama is generated from the highest point of the terrain within 200~m. The advantage\/disadvantage of using this option is the same as increasing the altitude offset: avoiding the occlusion of the panorama against distorting the generated view. Even if this option is very interesting we set it to off as the distortion of the panorama due to the observer's position changing and not only his altitude was too detrimental for the algorithm.\n\nFor the most realistic panorama the zoom factor and elevation exaggeration factors are both set to $1$. All the other parameters are left to their default values.\n\nThis service covers the the Alps, Pyrenees and Himalayas mountain range systems with two elevation datasets:\n\\begin{itemize}\n\\item\n\\emph{Alps} by Jonathan de Ferranti, with the spatial resolution of $1$~arcseconds\n\\item\n\\emph{SRTM} by CGIAR, with the spatial resolution of $3''$\n\\end{itemize}\n\nThe output is implemented with dynamic loading of the panorama, so is generated as a set of images to be aligned horizontally to form the complete output image and the one image containing the mountain peak names to be overlapped with the result image. Despite the unavailability of the service API, after the study of JavaScript and AJAX scripts of the web interface, a PHP function that simulates the functioning of the interface and collects all the parts of the result image was implemented.\n\nThe generation of the full panorama at the resolution of $20$~pixel\/degree and the peaks names takes on average 1 minute.\n\n\\subsection{Field of View Estimation and Scaling}\nTo estimate the field of view of the input photograph, as described in the previous chapter, the only information that must be known is the focal length of the photograph and the sensor width of the photo camera. The focal length is a parameter specified directly in the EXIF format of the image (tag = FocalLength, 37386 (920A.H)) \\cite{Technical2002Exchangeable}. The sensor width instead is not specified directly in the EXIF since it is a property of a photo camera, so the manufacturer and the model of the camera are extracted from the EXIF and then the sensor size is retrieved (manufacturer tag = Make, 271 (10F.H); model tag = Model, 272 (110.H)) \\cite{Technical2002Exchangeable}. \n\nClearly, the necessity of a database of camera sensor sizes emerges. This information is usually scattered on web sites of manufactures and technical references of the cameras, so it is not easy to collect them in one place. We used a database kindly provided by ``Digital Camera Database'' \\footnote{\\url{http:\/\/www.digicamdb.com}} web site, containing information about the sensor size of more than 3000 digital cameras. The main problem in using it remains in the fact that the manufacturer and model names of the same cameras are not always equal between that written in the EXIF and that provided by the sensor database, a couple of these mismatching examples are provided in Table \\ref{tab:exifMismatching}.\n\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabularx}{\\textwidth}{|X|X|X|X|}\n    \\hline\n    \\multicolumn{2}{|c|}{\\tabhead{EXIF}} &\n    \\multicolumn{2}{c|}{\\tabhead{Database}} \\\\\n    \\hline\n    \\tabhead{Manufacturer} &\n    \\tabhead{Model} &\n    \\tabhead{Manufacturer} &\n    \\tabhead{Model} \\\\\n    \\hline\n    Canon &\n    Canon PowerShot SX100 IS &\n    Canon &\n    PowerShot SX110 IS \\\\\n    \\hline\n    SONY &\n    DSC-W530 &\n    Sony &\n    Cybershot DSC W530 \\\\\n    \\hline\n    NIKON &\n    E5600 &\n    Nikon &\n    Coolpix 5600 \\\\\n    \\hline\n    OLYMPUS IMAGING CORP. &\n    SP560UZ &\n    Olympus &\n    SP 560 UZ \\\\\n    \\hline\n  \\end{tabularx}\n  \\caption{Examples of differences in the names of manufacturers and models between the EXIF specifications and the digital camera database.}\n  \\label{tab:exifMismatching}\n\\end{table}\n\nTo find the correct photo camera in the database from the EXIF specifications name a text similarity score between the names is used and the most similar name is chosen. The text similarity is calculated by the \\emph{similar\\_text} PHP function proposed by Oliver \\cite{DBLP:books\/daglib\/0077674}, after several steps of preprocessing:\n\n\\begin{enumerate}\n\\item\nBoth the manufacturer and model names of both the EXIF and the database items are transformed to lower case.\n\\item\nIf the manufacturer name contains ``nikon'' the name is set to ``nikon''.\n\\item\nIf the manufacturer name contains ``olympus'' the name is set to ``olympus''.\n\\item\nIf the model name contains the manufacturer name it is cut off from the model name.\n\\item\nThe text similarity score is performed between the concatenation of the manufacturer and the model of both EXIF and database items.\n\\end{enumerate}\n\nOnce the focal length and the sensor size are retrieved, they are annotated within the image, the matching algorithm will later use them to calculate the field of view and scale factor.\n\n\\subsection{Edge Detection}\nFor the edge extraction the \\emph{compass} \\cite{Ruzon:2001:EJC:505471.505477} edge detector has been used. It returns exactly the output the matching algorithm needs (for each point the edge strength and direction) and has been chosen due to its ability of exploiting the whole color information contained in the image and not only the grayscale components as classical edge detectors do.\nThe \\emph{compass} detector deals well also with a significant problem of edge detectors: when the image presents a subjective boundary between two regions with pixels close in color (due to overlapping objects), most edge detectors compute a weighted average of the pixels on each side, and since the two values representing the average color of two regions are close, no edges are found \\cite{Ruzon:2001:EJC:505471.505477}. The chosen detector deals well with this problem, that is very likely in the context of mountain photographs and the boundaries between the snowy mountain and blue sky.\n\nThe edge detection procedure is applied both to the input photograph and the generated panorama, with the following parameters chosen by trial-and-error: standard deviation of Gaussian used to weight pixels $\\sigma = 1$, threshold of the minimum edge strength to be considered $\\tau = 0.3$.\n\n\\subsection{Edge Filtering}\nThe edge filtering approach used in the implementation treats the columns of the image separately: each column is split into segments separated by the zero strength edge points, and each segment is then split into segments of length $n$ points. The points of each $i$-th segment (starting from the top) are then multiplied by the factor of $b^{i-1}$.\n\nThe implementation uses $k = 2$ and $b = 0.7$. It allows a good filtering of noise objects at the bottom of the photograph, and meanwhile deals well with mountain edges placed below the clouds. Several examples of filtering are presented in Figure \\ref{fig:filteringExamples}:\n\\begin{itemize}\n\\item\n\\emph{First row}: a photograph with clouds and a rainbow completely removed from the edges map (thanks to the correct parameters of the edge extraction algorithm).\n\\item\n\\emph{Second row}: a photograph with strongly contrasting clouds, the edges of the mountains below the clouds are still present even if reduced in strength.\n\\item\n\\emph{Third row}: a photograph with different mountains, one in front of another with a reduced visibility, even if reduced in strength all the mountains are present on the filtered edge map and the noise edges of the terrain in the foreground are filtered.\n\\end{itemize}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/filtering_examples}\n\t\\caption{Examples of edge filtering (one for each row): the original photograph (left), the extracted edges (center) and the filtered edges (right).}\n\t\\label{fig:filteringExamples}\n\\end{figure}\n\n\\subsection{Edge matching (Vector Cross Correlation)}\nWith the edge maps of the photograph and the panorama the matching of the edges is performed by fast Fourier transformation, with the Formula \\ref{eq:FFTVCC}. The MATLAB implementation of the VCC is the following:\n\n\\begin{lstlisting}[frame=single]\nfunction VCC = ComputeVCC(SP, DP, SR, DR)\n\n\ndimP = size(SP);\ndimR = size(SR);\nSR = [ zeros(dimP(1), dimR(2)) ; SR ; zeros(dimP(1), dimR(2)) ];\nDR = [ zeros(dimP(1), dimR(2)) ; DR ; zeros(dimP(1), dimR(2)) ];\n\nCOMP = complex( SP .* cos(DP) , SP .* sin(DP) );\nCOMR = complex( SR .* cos(DR) , SR .* sin(DR) );\n\nVCC = rot90(real(ifft2(conj(fft2(COMR.^2)) .* fft2(COMP.^2, size(COMR, 1), size(COMR, 2)))),2);\n\nVCC = VCC( dimP(1) + 1 : 2 * dimP(1) );\n\\end{lstlisting}\n\nThe resulting matrix of this function is the VCC score evaluation for each possible overlapping position, and the maximum value of the matrix identifies the best overlap. Instead of just peaking up the maximum value, several additional techniques have been implemented and tested:\n\n\\emph{Result smoothing}: the idea is that the result score peak will have high values also in the neighborhood of its position, and the noise score peaks instead do not: in other words the peak corresponding to the right position will have be broader with respect to noise score peaks. So smoothing the result score distribution penalizes the noise peaks: it will reduce more the height of the noise peaks with respect to the correct peak. In practice, after several tests, the incorrectness of this assumption emerges. Due to the nature of the VCC technique of penalizing the non parallel intersection of the edges, even if the score in the correct position is high, it is sufficient to move only few points out to reduce drastically the score, so the smoothing procedure was not efficient. Figure \\ref{fig:smoothing} shows an example of the score matrix (projected onto the $X$--$Z$ plane for readability) of the photograph of the Matterhorn used in the previous examples: the correct position represents already the maximum of the distribution, and not only it is intuitively visible that the correct position score peak is not smoother than the others, but it is also shown that more smoothing always leads to a smaller difference between correct and incorrect matches. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/smoothing}\n\t\\caption{Smoothing of the VCC result distribution with smoothing factor $k$. The red arrow identifies the correct matching.}\n\t\\label{fig:smoothing}\n\\end{figure}\n\n\\emph{Robust matching}: as proposed in the algorithm chapter, the top-$N$ peaks of the score matrix are extracted, each one is evaluated to find the best matching position. The evaluation is performed with Formula \\ref{eq:robustMatching}, that has been implemented in a simplified version in the following way:\n\\begin{enumerate}\n\\item\nThe information about the direction of the edge points is removed, all the edge points with strength bigger than $0$ are considered to have the strength equal to $1$.\n\\item\nOn the panorama edge map each point that is located less or equal than $r$ points from any edge point in the original edge map is set to $1$. In this way the $r$-neighborhood of the panorama edges is generated.\n\\item\nA simple intersection between the new panorama edge map and the photograph edge map is performed in the overlapping position that must be evaluated.\n\\item\nThe resulting intersection (composed only by $0$ and $1$ strength points) is divided into clusters where any point of the cluster is located less than $d$ from at least one point of the same cluster.\n\\item\nThe Formula is applied where the clusters are singular edges, and the edge length is the number of the points in the cluster.\n\\end{enumerate}\n\nAfter performing tests on the dataset also this approach has been rejected as increasing drastically the computational effort of the matching without introducing significant benefits to the results.\n\n\\emph{Different scale factors}: since the reduced times of VCC computation with the fast Fourier transform ($\\sim 1$~second for the VCC score matrix generation in our implementation) an approach to reduce the impact of the wrong photograph shot position or the incorrect field of view estimation (due for example to a wrong sensor peaking) is to perform the matching at several scale levels and not only the estimated scale. In the current implementation several scaling intervals with respect to the estimated level were evaluated, and the scale factor with the best VCC maximum value was picked. Obviously a larger scale factor will lead to a larger photograph edge map and so bigger VCC score, so when comparing different maxim values an inverse quadratic scale factor must be applied to each score (inversely proportional to the area added to the image due to the scale changing).\n\n\\subsection{Mountain Identification and Tagging}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/filtering_function}\n\t\\caption{Peak extraction: edge map before pattern extraction (left), implemented kernel function with $r = 1$ (center), edge map after pattern extraction (right).}\n\t\\label{fig:filteringFunction}\n\\end{figure}\n\n\nThe kernel function chosen for the peak extraction for the identification is the triweight function, defined as:\n\n$$\nf(d) = \n\\left\\{\\begin{matrix}\n\\left(1-\\left(\\frac{d}{r}\\right)^{2}\\right)^{3} \\\\ \n0\n\\end{matrix}\\right.\n\\begin{matrix}\n\\text{ for }d \\leq r \\\\\n\\text{ for }d > r\n\\end{matrix}\n$$\n\nWith $r = 200$. The kernel function plot and its effect on the peak extraction are shown in Figure \\ref{fig:filteringFunction}.\n\n\\chapter{Experimental Study}\n\\label{Experimental Study}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\nIn this chapter the photograph direction estimation quality is investigated. The precision of the estimation is computed varying the operating parameters of the implementation and the type of the photographs contained in the data set.\n\n\\section{Data sets}\nThe analysis is conducted on a set of 95 photographs of the Italian Alps, collected from several photographers with geo-tag set directly by a GPS component or manually by the photographer, so we consider them very precisely geo-tagged. The photographs vary in several aspects: several examples are shown in Figure \\ref{fig:dataset}, and the categories of interest are composed as described in Table \\ref{tab:datasetCategoriesComposition}.\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabularx}{\\textwidth}{|X|c|c|}\n    \\hline\n    \\tabhead{Category} &\n    \\tabhead{Option} &\n    \\tabhead{Data set portion} \\\\\n    \\hline\n    \\multirow{2}{*}{Source} &\n    Photo camera &\n    38 \\% \\\\\n    &\n    Cellular phone &\n    62 \\% \\\\\n    \\hline\n    \\multirow{2}{*}{Cloud presence} &\n    None &\n    41 \\% \\\\\n    &\n    Minimal &\n    29 \\% \\\\\n    &\n    Massive &\n    23 \\% \\\\    \n    &\n    Overcast &\n    7 \\% \\\\    \n    \\hline\n    \\multirow{2}{*}{Skyline composition} &\n    Mountains and terrain only  &\n    87 \\% \\\\\n    &\n    Foreign objects &\n    13 \\% \\\\\n    \\hline    \n  \\end{tabularx}\n  \\caption{Data set categories of interest composition.}\n  \\label{tab:datasetCategoriesComposition}\n\\end{table}\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/dataset}\n\t\\caption{Several photographs from the collected data set.}\n\t\\label{fig:dataset}\n\\end{figure}\n\n\n\\section{Operating parameters}\nThe operating parameters used in the algorithm described in the previous sections, and to be evaluated, are listed in Table \\ref{tab:datasetOperatingParameters}. The value in bold defines the default value that gives the best evaluation results and is used in the final implementation proposal.\n\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabularx}{\\textwidth}{|X|c|c|}\n    \\hline\n    \\tabhead{Full name} &\n    \\tabhead{Parameter} &\n    \\tabhead{Tested values} \\\\\n    \\hline\n    Photograph edge strength threshold &\n    $\\rho_p$ &\n    0.1,0.2,\\textbf{0.3},0.4,0.5,0.6,0.7 \\\\\n    \\hline\n    Panorama edge strength threshold &\n    $\\rho_r$ &\n    0.1,\\textbf{0.2},0.3,0.4,0.5 \\\\\n    \\hline    \n    Photograph edge filtering base &\n    $b_p$ &\n    0.5,0.6,\\textbf{0.7},0.8,0.9,1.0 \\\\\n    \\hline    \n    Panorama edge filtering base &\n    $b_r$ &\n    0.5,0.6,0.7,0.8,0.9,\\textbf{1.0} \\\\\n    \\hline    \n    Photograph edge filtering max segment length &\n    $l_p$ &\n    1,\\textbf{2},3,4,5 \\\\\n    \\hline    \n    Panorama edge filtering max segment length &\n    $l_r$ &\n    1,2,3,4,5 \\\\    \n    \\hline    \n    Photograph scaling interval &\n    $\\pm k \\%$ &\n    \\textbf{0},1,2,5,10 \\\\    \n    \\hline           \n  \\end{tabularx}\n  \\caption{Operating parameters (defaults in bold).}\n  \\label{tab:datasetOperatingParameters}\n\\end{table}\n\n\\section{Evaluation metric}\nEach photograph in the data set and the corresponding generated panorama are manually tagged by specifying on both one or more pairs of points corresponding to the same mountain peaks. Once the direction of view is estimated and the best overlap of the edges is found, the error of the direction estimation is defined as the average distance between the position of the peak on the panorama and the position of the peak on the photograph projected on the panorama with the current overlap position. The distance is then expressed as the angle, considering the panorama width as $2\\pi$ angle. The error therefore lies between 0 and just over $\\pi$ angle (the worst horizontal error is $\\pi$ in the case of opposite directions, but since the vertical error is also counted the total error can theoretically slightly exceed this value).\n\nGiven a photograph and a panorama rendered with the resolution of $q$ pixel\/degree, both tagged with $N$ peaks, let $x_{pi}$\/$y_{pi}$ and $x_{ri}$\/$y_{ri}$ be respectively the coordinates (expressed in pixels) of the $i$-th peak on the photograph and the panorama image, the estimation error is defined as:\n\n$$\ne = \\frac{\\sqrt{(\\sum_{i = 1}^{N} (x_{pi} - x_{ri}))^2 + (\\sum_{i = 1}^{N} (y_{pi} - y_{ri}))^2 }}{Nq}\n$$\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/errorOffsetValidation}\n\t\\caption{Schematic example of edge matching validation with photograph edges (red) and panorama edges (blue), all the three peaks are validation points.}\n\t\\label{fig:errorOffsetValidation}\n\\end{figure}\n\nAn important aspect of the average computing must be highlighted: instead of the standard distance error averaging by taking into account the singular distances, this error metric calculates the average of the both image coordinates axis components, and then the distance with these components is computed. In practice, a positive error offset compensates a negative one. This is made to compensate the imperfections between the scale of the mountains on the photograph and panorama: if the overlap presents both positive and negative offsets between peak pairs it means that the photograph and the panorama are differently scaled, so the best overlap position is the one that brings the sum of the offset components to zero. An example of this reasoning is shown as a schematic case of edge matching in Figure \\ref{fig:errorOffsetValidation} the validation points include all three peaks of the photograph and panorama view, but the different scaling prevents perfect overlapping and the offsets are both positive and negative. Thus the best possible overlapping position is the one shown, that makes the offsets opposite and making the sum of the offsets converge to zero.\n\nTherefore even when the validation points cannot all be matched perfectly, the technique of not taking the absolute value of the offsets makes it anyway possible to find the best match and obtain an offset of zero (perfect score).\n\nFor readability reasons all the validation errors from this point will be specified in arc degrees ($\\ensuremath{^\\circ}$).\n\nOnce the matching error is calculated for every photograph present in the data set, the general score of the run of the algorithm is defined as the percentage of the portion of the data set containing all the photographs with matching error below a certain threshold. This threshold in other words defines the limit of the matching error for a photograph to be considered matched correctly, and in the evaluation of this work is set to $4\\ensuremath{^\\circ}$.\n\n\\section{Evaluation results}\nWith the all operating parameters set to their default values the algorithm has correctly estimated the orientation of \\textbf{64.2\\%} of the total photographs. In particular the distribution of the matching error through the photographs of the data is presented in Figure \\ref{fig:mainResultHist}: it is clear that the choice of the error threshold has a limited impact on the correct matching rate since almost all the correctly matched photographs have error between $0\\ensuremath{^\\circ}$ and $2\\ensuremath{^\\circ}$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/main_result_hist}\n\t\\caption{Histogram of the number of the photographs with a certain validation error (split into intervals of width $1$~\\ensuremath{^\\circ}.)}\n\t\\label{fig:mainResultHist}\n\\end{figure}\n\nThe results obtained with the described data set decomposed into categories as described in Table \\ref{tab:datasetCategoriesComposition} are summarized in Table \\ref{tab:datasetCategoriesResults}.\n\n\\begin{table}[!h]\n  \\centering\n  \\begin{tabular}{|c|c|c|}\n    \\hline\n    \\tabhead{Category} &\n    \\tabhead{Option} &\n    \\tabhead{Correct matches} \\\\\n    \\hline\n    \\multirow{2}{*}{Source} &\n    Photo camera &\n    72.2 \\% \\\\\n    &\n    Cellular phone &\n    59.3 \\% \\\\\n    \\hline\n    \\multirow{2}{*}{Cloud presence} &\n    None &\n    71.8 \\% \\\\\n    &\n    Minimal &\n    67.9 \\% \\\\\n    &\n    Massive &\n    45.5 \\% \\\\    \n    &\n    Overcast &\n    66.7 \\% \\\\    \n    \\hline\n    \\multirow{2}{*}{Skyline composition} &\n    Mountains and terrain only  &\n    65.1 \\% \\\\\n    &\n    Foreign objects &\n    58.3 \\% \\\\\n    \\hline    \n  \\end{tabular}\n  \\caption{Results of the algorithm with respect to the data set categories of interest.}\n  \\label{tab:datasetCategoriesResults}\n\\end{table}\n\nFrom the results of data set categories it follows that the photographs shot with a photo camera are more frequently aligned than those shot with a cellular phone: an explanation for this trend can be the fact that the photo camera photographers usually use the optical zoom that is correctly annotated in the focal length of the photograph, cellular phones instead usually allow only digital zoom, which leaves unchanged the annotated focal length and so leads to an incorrect Field of View estimation and therefore an incorrect alignment.\n\nThe trend of the result as cloud conditions vary follows a logical sense (more clouds leads to a higher number of noise edges and to worse performance) and reveals that the presence of clouds is a significant reason for failure of the algorithm, a problem that will be treated in the next chapter. A good score of the overcast case can be explained by the fact that when the sky is completely covered by clouds there are no edges between the real sky and clouds, so there are no noise edges (it can be thought of as the sky painted by a darker cloud color).\n\nThe presence of foreign objects such as trees, buildings, and persons in the skyline obviously penalizes the alignment, but as can be seen from the results it is not a critical issue.\n\nIn the next chapter several techniques that should improve the results of mountain identification are presented and discussed.\n\n\\subsection{Operating parameter evaluation}\nWe now present and comment on the effect of the individual parameters on the overall results:\n\n\\textbf{\\emph{Photograph edge strength threshold ($\\rho_p$)}}: The choice of the strength threshold for the photograph edges presents two trends: higher threshold means fewer noise edges, but lower threshold means to include also the weaker edges belonging not to the skyline but to secondary terrain edges. The dependency of the result on $\\rho_p$ is displayed in Figure \\ref{fig:opRhoP}, the optimal balance between the two trends is reached with $\\rho_p = 0.3$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_rho_p}\n\t\\caption{Overall algorithm results varying the $\\rho_p$ operating parameter.}\n\t\\label{fig:opRhoP}\n\\end{figure}\n\n\\textbf{\\emph{Panorama edge strength threshold ($\\rho_r$)}}: The choice of the strength threshold for the panorama edges is very similar to that of the photograph, with a difference regarding small thresholds: since the panorama is rendered by a program it does not present weak edges, which means that the introduction of a small threshold has almost no effect on the matching efficiency. The dependency of the result on $\\rho_r$ is displayed in Figure \\ref{fig:opRhoR}: in fact all tested values smaller or equal to $0.3$ bring the same (and optimal) result. In the implementation the parameter is set to $\\rho_r = 0.2$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_rho_r}\n\t\\caption{Overall algorithm results varying the $\\rho_r$ operating parameter.}\n\t\\label{fig:opRhoR}\n\\end{figure}\n\n\\textbf{\\emph{Photograph edge filtering base ($b_p$)}}: The choice of the filtering base for the photograph edges defines how strongly the lower edge points will be reduced in strength, $b_p = 1.0$ means that no filtering will be performed, and $b_p = 0.0$ that only the first segment of each row of the photograph will be left. The dependency of the result on $b_p$ is displayed in Figure \\ref{fig:opBP}: it is evident that filtering is absolutely needed since the result corresponding to $b_p = 1.0$ is significantly smaller than all the other values; the result stabilizes for $b_p \\leq 0.3$ since the base is so small that the filtering factor reaches almost immediately zero. The best orientation is found for $b_p = 0.7$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_b_p}\n\t\\caption{Overall algorithm results varying the $b_p$ operating parameter.}\n\t\\label{fig:opBP}\n\\end{figure}\n\n\\textbf{\\emph{Photograph edge filtering maximum segment length ($l_p$)}}: The choice of the maximum segment length for the photograph edge filtering is introduced to allow mountain edges which are thicker than one pixel not to penalize the other edges placed below it, but at the same time to not allow the noise edges that generate columns of points to be left unfiltered. The dependency of the result on $l_p$ is displayed in Figure \\ref{fig:opLP}: the optimal balance between the two trends is reached with $l_p = 2$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_l_p}\n\t\\caption{Overall algorithm results varying the $l_p$ operating parameter.}\n\t\\label{fig:opLP}\n\\end{figure}\n\n\\textbf{\\emph{Panorama edge filtering base ($b_r$)}}: The choice of the filtering base for the panorama edges defines how strongly the lower edge points will be reduced in strength, $b_r = 1.0$ means that no filtering will be performed, and $b_r = 0.0$ that only the first segment of each row of the photograph will be left. The dependency of the result on $b_p$ is displayed in Figure \\ref{fig:opBP}: the overall trend is similar to the filtering base of photograph edges ($b_p$) but with an important difference for high values: the result does not get worse with $b_r$ getting close to $1$. This can be explained by the fact that the high values ($\\geq 0.8$) of $b_p$ do not perform a significant filtering, so the noise edge points are not filtered and penalize the matching: the panorama edges instead do not have any noise, so do not have the same trend. The choice to set $b_r = 1.0$ is very clear, or in other words to not perform any filtering of panorama edge points.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_b_r}\n\t\\caption{Overall algorithm results varying the $b_r$ operating parameter.}\n\t\\label{fig:opBR}\n\\end{figure}\n\n\\textbf{\\emph{Panorama edge filtering maximum segment length ($l_r$)}}: since $b_r$ is set to $1$ (the filtering of the panorama edges is not performed) the maximum segment length for the panorama edges filtering has no impact on the result, so its choice is not significant.\n\n\\textbf{\\emph{Photograph scaling interval ($\\pm k \\%$)}}: The choice of the scaling interval with respect to the originally estimated scaling process is described in the previous chapters. The results of its estimation were probably the most unexpected of all operating parameters. The dependency of the result on $b_p$ is displayed in Figure \\ref{fig:opK}: the best performance is reached with $k = 0$ (no scaling interval, only the estimated scale factor) and always worsens with increasing $k$. This trend can be interpreted as the proof of the fact that the Field of View and scaling factor estimation are so precise that increasing the scaling interval brings more penalties by setting an imprecise scale factor and increasing the probability of a noise edge matching the mountain panorama edge than advantages by trying different zoom factors and getting the best mountain edge overlap. The best orientation is therefore found with $k = 0$ (no scaling interval).\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.00\\columnwidth]{.\/figures\/op_k}\n\t\\caption{Overall algorithm results varying the $\\pm k \\%$ operating parameter.}\n\t\\label{fig:opK}\n\\end{figure}\n\\chapter{Conclusions and Future Work}\n\\label{Conclusions and Future Work}\n\\thispagestyle{empty}\n\n\\vspace{0.5cm}\n\n\\noindent\nIn this work an algorithm for the estimation of the orientation of a geo-tagged photograph containing mountains and for the identification of the visible mountain peaks is presented in detail. The algorithm has been implemented and evaluated with the result of estimating correctly the direction of the view of \\textbf{64.2\\%} of the various photographs.\n\nThis result can be considered excellent if used in passive crowdsourcing applications: given the enormous availability of photographs on the Web, the desired amount of correctly matched photographs can be reached just by increasing the number of processed photographs.\n\nIt is a very promising even if not excellent result speaking about applications connected with user experience: approximately one photograph out of three may not be matched, decreasing significantly the usability of the application.\n\n\\section{Future enhancements}\nWe expect to implement and test several techniques in the near future, aimed at enhancing the matching process and improving the results.\n\nOne of the main reasons that the matching fails is the massive presence of clouds on the photograph, and to prevent the edge filtering step from emphasizing the noise cloud edges a sky\/ground segmentation step can be inserted before the edge extraction phase: a sky\/ground segmentation algorithm (such as, for example, that proposed by Todorovic and Nechyba \\cite{Todorovic03sky\/groundmodeling}) that given a photograph splits it into the two regions of sky and terrain, then the sky part is erased from the photograph and the matching algorithm goes on without the cloud edge noise. In Figure \\ref{fig:skyTerrain} an example of this approach and the improvement it brings to the filtered edges map is shown.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/sky_terrain_enhancement}\n\t\\caption{An example of the use of sky\/ground segmentation in the edge extraction technique: noise cloud edges are removed and the mountain boundary is perfectly extracted.}\n\t\\label{fig:skyTerrain}\n\\end{figure}\n\nAnother frequent reason for incorrect matching is the imperfections between the boundaries on the photograph and on the render of mountains very close to the observer. Due to the small and unavoidable errors in the registered position, altitude estimation, and elevation model, the rendered panorama will always be imperfect with respect to reality, and this imperfection, for obvious reasons, get smaller as the distance from the observer to the object increases. The situation of having mountain boundaries both in the foreground and the background is very frequent: walking in a mountainous area the observer is usually surrounded by mountains placed close to him that are obscuring the distant mountains; the dimensions and the majesty of the mountains however bring usually the photographer to take pictures of distantly placed mountains with altitude higher that his point of view. For this reason as soon as the mountains in the foreground get placed in a way to create an \"aperture\", a photograph of the mountains visible in this sort of window will be probably shot. An example of this type of photograph can be that used for Figure \\ref{fig:skyTerrain} or that shown in Figure \\ref{fig:viewAperture}: it represents exactly the described situation, and it can be easily seen that the photograph and panorama edges of the mountain in the background are perfectly matchable, while the edges belonging to the closer mountains placed in the foreground are significantly different. In this case the algorithm manages to correctly alignment of the photograph anyway, but it is frequent in these cases of funnel-shaped mountain apertures that it fail, specially if the aperture is very small with respect to the total photograph dimension. Future studies are planned to include the development of a technique for the recognition of this type of situation and emphasizing the edges of the background mountains with respect to the foreground mountains.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=1.0\\columnwidth]{.\/figures\/view_aperture}\n\t\\caption{An example of a photograph (top) of background mountains seen in the aperture between the foreground mountains and the corresponding edge matching (bottom, red - panorama, blue - photograph).}\n\t\\label{fig:viewAperture}\n\\end{figure}\n\nEven when the estimated orientation is not correct, in most cases the correct alignment presents a significant peak in the VCC score distribution, so the best edge overlap estimation could be improved with a robust matching technique operating on the top-N positions extracted from the VCC matching score distribution. The likely method of implementing it is the neighborhood metrics of the edges as proposed by Baboud et al. \\cite{Baboud2011Alignment}; even if a simplified implementation of the robust matching technique proposed was implemented in this work and rejected as self-defeating, we believe that it can be refined to reach a higher rate of correctly matched photographs.\n\n\\section{Crowdsourcing involvement}\nAn important future direction of the research of this work is the integration with (traditional) crowdsourcing. Though being a feasible image processing task, photograph-to-panorama alignment is definitely better performed by humans that by algorithms. There is no need to move the photograph through the panorama or know the correct scale factor: a person can usually easily align the photograph to the panorama just by looking at it, comparing the fragments of the photograph he finds most particular, which can be elevated peaks, details that catch the eye and in general sense any terrain silhouette fragments that the person considers unusual. This allows one to easily identify the correct alignment by eye even in presence of significant errors between the photograph and the generated panorama.\n\nThis consideration leads us to the integration of the current work with crowdsourcing with several possible scenarios:\n\\begin{itemize}\n\\item\n\\emph{Contribution in learning and testing phase}:\nmanual photograph-to-panorama matching estimation can be very useful as a source of ground truth data both for the learning phase in case of using a machine intelligence algorithm in the edge matching and for the implementation testing phase, validating the results proposed by the algorithm. These approaches can be applied both to expert and non-expert crowdsourcing options: the non-expert tasks can consist of an activity feasible by anyone, such as searching for the correct alignment between the photograph and the corresponding panorama, the expert tasks instead aim at the activities that only mountaineers can perform, such as tagging the mountain peaks on a photograph based on his own knowledge and experience. Both methods allow the validation of the algorithm estimation result, but at two different (even if very related) levels: the first at photograph direction estimation, and the second at final mountain peak identification and tagging.\n\\item\n\\emph{Contribution as post processing validation}:\nhuman validation can be used not only in the development phase but also in the final implementation pipeline as the post processing validation. If the application is time-critical such as real-time augmented reality or website photographs mountain peak tagging, this approach cannot be applied, but in crawler applications such as environmental model creation, crowdsourcing can provide a significant improvement in the data quality by proposing manual photograph-to-panorama matching for the photographs that the algorithm has marked as photographs with low confidence score. Crowdsourcing therefore will be used as a complementary technique for photographs that cannot be aligned automatically.\n\\item\n\\emph{Contribution to data set expansion}:\nthe number of the available photographs in the data set is fundamental both for testing purposes and for the data to be processed by an environmental modeling system. Photographs can be collected from public sources such as social networks or photo sharing websites, filtering the geographical area and image content of the interest, or can be collected directly from people, uploading or signaling the photographs containing mountains in a certain requested area, or even photographs containing certain requested mountain peaks. This approach is important in the case of environmental modeling, when precisely collected photographs in the same area of ground truth data availability is fundamental.\n\\end{itemize}\n\n\\section{Possible areas of application}\nThere are several application areas the proposed algorithm can be used for, starting from the technique for snow level monitoring and prediction (the technique this work has been proposed and started for) and extending to possible applications in mobile and web fields that can be created thanks to this algorithm.\n\n\\subsection{Environmental Modeling}\nOne of the main future directions of the research will be the modeling of environmental processes by the analysis of mountain appearances extracted by the algorithm proposed in this work. The most important and most obvious measurements available from a mountain's appearance are the snow level and the snow water equivalent (SWE). The idea is to collect a series of photographs through time for each analyzed mountain, and based on the snow data ground truth, estimate the current measurements. The first step will therefore be an attempt to detect the correlation between the visual content of the mountain portion of a photograph with the physical measurements of the snow and SWE of that mountain. An interesting planned approach is to exploit also webcams in the region of interest since they present several advantages:\n\\begin{itemize}\n\\item\nweather and tourist webcams are very popular and frequent in mountain regions\n\\item\nthe time density of the measures can be as high as we want (it is only a matter of how frequently the image is acquired from the webcam, and in any case there is no reason to suppose the need of more than a couple of captures per day)\n\\item\nthe mountains captured on a webcam are always in the same position on the image, so even in cases of difficult edge matching it can be done or verified manually once to exploit all the instances of the photographs in future.\n\\end{itemize}\n\nFurthermore, supposing the mountains to be greatly distant from the observer (a reasonable and weak assumption in case of mountains photographs), by knowing the estimation of the observer's altitude and the altitude of the identified peak, we can estimate the altitude of each pixel of the mountain on the photograph by a simple proportion of the differences of altitudes and the pixel height of the mountain. This allows a comparison between the visual features of partial photographs corresponding to the same altitude even for photographs with significantly different shot position and photo camera properties.\n\n\\subsection{Augmented Reality}\nAugmented reality applications on mobile devices (applications that augment and supply the real-time camera view of the mobile device with computer-generated input) is a recent niche topic, and the promising use of the described algorithm regards augmented reality: an application can tag in real-time the mountains viewed by the user, highlight the peaks and terrain silhouettes, and augment the mountains with useful information such as altitude contours drawn on the image.\n\nSuch a kind of application would eliminate the problem of wrong geo-tag estimation (keeping the GPS of the mobile device on, the position is usually estimated with a tollerance of few meters) and so will reduce significantly the problem of wrong altitude estimation and elevation model imperfections with respect to reality. The reduced computation capacity may be compensated by the built-in compass, which gives a rough indication of the observer's direction of view, so the matching procedure can be done only on a reduced fragment of the rendered panorama. The bandwidth use will be small since the mountains are usually distant from the observer, so the rendered view will change very slowly while the observer is moving, so it will need to be updated rarely.\n\n\\subsection{Photo Sharing Platforms}\nTagging the mountain peaks on the geo-tagged photographs can lead also to a significant improvement to a cataloging and searching system of a social network or a photo sharing website, and to a better user experience by exploring the peak names and other information by directly viewing the photograph.\nAutomatic tagging of mountain peaks (tagging intended as the catalog assignment of peak names to the photograph and not the visual annotation on the photograph itself) can allow navigation through mountain photographs in these scenarios:\n\\begin{itemize}\n\\item\nUser searches for a mountain by specifying its name in the query: retrieves the photographs of that mountain, even if the author of the photograph did not specify the name in the title, description or other metadata.\n\\item\nUser views a photograph: other photographs of the same mountain (with the same or different facade) are suggested, even if the authors of both photographs did not specify the name in the title, description or other metadata.\n\\end{itemize}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConsider a set of agents $\\mathcal{N}=\\{1,2,\\ldots,n\\}$ connected over a network. Each agent has a local smooth and strongly convex cost function $f_i:\\mathbb{R}^{p}\\rightarrow \\mathbb{R}$. The global objective is to locate $x\\in\\mathbb{R}^p$ that minimizes the average of all cost functions:\n\\begin{equation}\n\\min_{x\\in \\mathbb{R}^{p}}f(x)\\left(=\\frac{1}{n}\\sum_{i=1}^{n}f_i(x)\\right).  \\label{opt Problem_def}\n\\end{equation}%\nScenarios in which problem (\\ref{opt Problem_def}) is considered include distributed machine learning \\cite{forrester2007multi,nedic2017fast,cohen2017projected,wai2018multi}, multi-agent target seeking \\cite{pu2016noise,chen2012diffusion}, and wireless networks \\cite{cohen2017distributed,mateos2012distributed,baingana2014proximal}, among many others.\n\nTo solve problem (\\ref{opt Problem_def}), we assume each agent $i$ queries a stochastic oracle ($\\mathcal{SO}$) to obtain noisy gradient samples of the form $g_i(x,\\xi_i)$ that satisfies the following condition:\n\\begin{assumption}\n\t\\label{asp: gradient samples}\n\tFor all $i\\in\\mathcal{N}$ and all $x\\in\\mathbb{R}^p$, \n\teach random vector $\\xi_i\\in\\mathbb{R}^m$ is independent, and\n\t\\begin{equation}\n\t\\begin{split}\n\t& \\mathbb{E}_{\\xi_i}[g_i(x,\\xi_i)\\mid x] =  \\nabla f_i(x),\\\\\n\t& \\mathbb{E}_{\\xi_i}[\\|g_i(x,\\xi_i)-\\nabla f_i(x)\\|^2\\mid x]  \\le  \\sigma^2\\quad\\hbox{\\ for some $\\sigma>0$}.\n\t\\end{split}\n\t\\end{equation}\n\\end{assumption}\nThe above assumption of stochastic gradients holds true for many on-line distributed learning problems, where $f_i(x)=\\mathbb{E}_{\\xi_i}[F_i(x,\\xi_i)]$ denotes the expected loss function agent $i$ wishes to minimize, while independent samples $\\xi_i$ are gathered continuously over time.\nFor another example, in simulation-based optimization, the gradient estimation often incurs noise \nthat can be due to various sources, such as modeling and discretization errors, \nincomplete convergence, and finite sample size for Monte-Carlo methods~\\cite{kleijnen2008design}. \n\nDistributed algorithms dealing with problem (\\ref{opt Problem_def}) have been studied extensively in the literature \\cite{tsitsiklis1986distributed,nedic2009distributed,nedic2010constrained,jakovetic2014fast,kia2015distributed,shi2015extra,di2016next,qu2017harnessing,nedic2017achieving,pu2018push}.\nRecently, there has been considerable interest in distributed implementation of stochastic gradient algorithms (see \\cite{ram2010distributed,cavalcante2013distributed,towfic2014adaptive,lobel2011distributed,srivastava2011distributed,wang2015cooperative,lan2017communication,pu2017flocking,lian2017can,pu2018swarming}). The literature has shown that distributed algorithms may compete with, or even outperform, their centralized counterparts under certain conditions \\cite{pu2017flocking,lian2017can,pu2018swarming}. For instance, in our recent work~\\cite{pu2018swarming}, we proposed a swarming-based approach for distributed stochastic optimization which beats a centralized gradient method in real-time assuming that all $f_i$ are identical. \nHowever, to the best of our knowledge, there is no distributed stochastic gradient method addressing problem (\\ref{opt Problem_def}) that shows comparable performance with a centralized approach. In particular, under constant step size policies none of the existing algorithms achieve an error bound that is decreasing in the network size $n$. \n\n\nA distributed gradient tracking method was proposed in~\\cite{di2016next,nedic2017achieving,qu2017harnessing}, where \nthe agent-based auxiliary variables $y_i$ were introduced to track the average gradients of $f_i$ assuming accurate gradient information is available. It was shown that the method, with constant step size, generates iterates that converge linearly to the optimal solution.\nInspired by the approach, in this paper we consider a distributed stochastic gradient tracking method. By comparison, in our proposed algorithm $y_i$ are tracking the stochastic gradient averages of $f_i$.\nWe are able to show that the iterates generated by each agent reach, in expectation, a neighborhood of the optimal point exponentially fast under a constant step size.\nInterestingly, with a sufficiently small step size, the limiting error bounds on the distance \nbetween agent iterates and the optimal solution decrease in the network size $n$, which is comparable to the performance of a centralized stochastic gradient algorithm.\n\nOur work is also related to the extensive literature in stochastic approximation (SA) methods dating back to the seminal works~\\cite{robbins1951stochastic} and~\\cite{kiefer1952stochastic}. These works include the analysis of convergence (conditions for convergence, rates of convergence, suitable choice of step size) in the context of diverse noise models~\\cite{kushner2003stochastic}.\n\nThe paper is organized as follows. In Section~\\ref{sec: set}, we introduce the distributed stochastic gradient tracking method along with the main results. We perform analysis in Section~\\ref{sec:cohesion} \nand provide a numerical example in Section~\\ref{sec: simulation} to illustrate our theoretical findings. \nSection~\\ref{sec: conclusion} concludes the paper.\n\n\n\\subsection{Notation}\n\\label{subsec:pre}\nThroughout the paper, vectors default to columns if not otherwise specified.\nLet each agent $i$ hold a local copy $x_i\\in\\mathbb{R}^p$ of the decision variable and an auxiliary variable $y_i\\in\\mathbb{R}^p$. Their values at iteration\/time $k$ are denoted by $x_{i,k}$ and $y_{i,k}$, respectively. \nWe let\n\\begin{equation*}\n \\mathbf{x} := [x_1, x_2, \\ldots, x_n]^{\\intercal}\\in\\mathbb{R}^{n\\times p},\\ \\ \\mathbf{y} := [y_1, y_2, \\ldots, y_n]^{\\intercal}\\in\\mathbb{R}^{n\\times p},\n\\end{equation*}\n\\begin{equation}\n\\overline{x} :=  \\frac{1}{n}\\mathbf{1}^{\\intercal} \\mathbf{x}\\in\\mathbb{R}^{1\\times p},\\ \\ \\overline{y} :=  \\frac{1}{n}\\mathbf{1}^{\\intercal}\\mathbf{y}\\in\\mathbb{R}^{1\\times p},\n\\end{equation}\nwhere $\\mathbf{1}$ denotes the vector with all entries equal to 1.\nWe define an aggregate objective function of the local variables:\n\\begin{equation}\nF(\\mathbf{x}):=\\sum_{i=1}^nf_i(x_i),\n\\end{equation}\nand write\n\\begin{equation*}\n\\nabla F(\\mathbf{x}):=\\left[\\nabla f_1(x_1), \\nabla f_2(x_2), \\ldots, \\nabla f_n(x_n)\\right]^{\\intercal}\\in\\mathbb{R}^{n\\times p}.\n\\end{equation*}\nIn addition, let\n\\begin{equation}\n\\label{def: h}\nh(\\mathbf{x}):=\\frac{1}{n}\\mathbf{1}^{\\intercal}\\nabla F(\\mathbf{x})\\in\\mathbb{R}^{1\\times p},\n\\end{equation}\n\\begin{equation*}\n\\boldsymbol{\\xi} := [\\xi_1, \\xi_2, \\ldots, \\xi_n]^{\\intercal}\\in\\mathbb{R}^{n\\times p},\n\\end{equation*}\nand\n\\begin{equation}\nG(\\mathbf{x},\\boldsymbol{\\xi}):=[g_1(x_1,\\xi_1), g_2(x_2,\\xi_2), \\ldots, g_n(x_n,\\xi_n)]^{\\intercal}\\in\\mathbb{R}^{n\\times p}.\n\\end{equation}\n\nInner product of two vectors $a,b$ of the same dimension is written as $\\langle a,b\\rangle$. For two matrices $A,B\\in\\mathbb{R}^{n\\times p}$, we define\n\\begin{equation}\n\\langle A,B\\rangle :=\\sum_{i=1}^n\\langle A_i,B_i\\rangle,\n\\end{equation}\nwhere $A_i$ (respectively, $B_i$) represents the $i$-th row of $A$ (respectively, $B$). We use $\\|\\cdot\\|$ to denote the $2$-norm of vectors; for matrices, $\\|\\cdot\\|$ denotes the Frobenius norm.\n\nA graph is a pair $\\mathcal{G}=(\\mathcal{V},\\mathcal{E})$ where $\\mathcal{V}=\\{1,2,\\ldots,n\\}$ is the set of vertices (nodes) and $\\mathcal{E}\\subseteq \\mathcal{V}\\times \\mathcal{V}$ represents the set of edges connecting vertices. We assume agents communicate in an undirected graph, i.e., $(i,j)\\in\\mathcal{E}$ iff $(j,i)\\in\\mathcal{E}$.\nDenote by $W=[w_{ij}]\\in\\mathbb{R}^{n\\times n}$ the coupling matrix of agents. Agent $i$ and $j$  are connected iff $w_{ij}=w_{ji}>0$ ($w_{ij}=w_{ji}=0$ otherwise). Formally, we assume the following condition regarding the interaction among agents:\n\\begin{assumption}\n\t\\label{asp: network}\n\tThe graph $\\mathcal{G}$ corresponding to the network of agents is undirected and connected (there exists a path between any two agents). Nonnegative coupling matrix $W$ is doubly stochastic, \n\ti.e., $W\\mathbf{1}=\\mathbf{1}$ and $\\mathbf{1}^{\\intercal}W=\\mathbf{1}^{\\intercal}$.\n\tIn addition, $w_{ii}>0$ for some $i\\in\\mathcal{N}$.\n\\end{assumption}\nWe will frequently use the following result, which is a direct implication of Assumption \\ref{asp: network} (see \\cite{qu2017harnessing} Section II-B):\n\\begin{lemma}\n\t\\label{lem: spectral norm}\n\tLet Assumption \\ref{asp: network} hold, and let $\\rho_w$ denote the spectral norm of \n\tthe matrix $W-\\frac{1}{n}\\mathbf{1}\\mathbf{1}^{\\intercal}$. Then, $\\rho_w<1$ and \n\t\\begin{equation*}\n\t\\|W\\omega-\\mathbf{1}\\overline{\\omega}\\|\\le \\rho_w\\|\\omega-\\mathbf{1}\\overline{\\omega}\\|\n\t\\end{equation*}\n\tfor all $\\omega\\in\\mathbb{R}^{n\\times p}$, where $\\overline{\\omega}=\\frac{1}{n}\\mathbf{1}^{\\intercal}\\omega$.\n\\end{lemma}\n\n\\section{A Distributed Stochastic Gradient Tracking Method}\n\\label{sec: set}\nWe consider the following distributed stochastic gradient tracking method: at each step $k\\in\\mathbb{N}$, \nevery agent $i$ independently implements the following two steps:\n\\begin{equation}\n\\label{eq: x_i,k}\n\\begin{split}\nx_{i,k+1} = & \\sum_{j=1}^{n}w_{ij}(x_{j,k}-\\alpha y_{j,k}), \\\\\ny_{i,k+1} = & \\sum_{j=1}^{n}w_{ij}y_{j,k}+g_i(x_{i,k+1},\\xi_{i,k+1})-g_i(x_{i,k},\\xi_{i,k}),\n\\end{split}\n\\end{equation}\nwhere $\\alpha>0$ is a constant step size. The iterates are initiated with an arbitrary \n$x_{i,0}$ and $y_{i,0}= g_i(x_{i,0},\\xi_{i,0})$ for all~$i\\in{\\cal N}$.\nWe can also write (\\ref{eq: x_i,k}) in the following compact form:\n\\begin{equation}\n\\label{eq: x_k}\n\\begin{split}\n\\mathbf{x}_{k+1} = & W(\\mathbf{x}_k-\\alpha \\mathbf{y}_k), \\\\\n\\mathbf{y}_{k+1} = & W\\mathbf{y}_k+G(\\mathbf{x}_{k+1},\\boldsymbol{\\xi}_{k+1})-G(\\mathbf{x}_k,\\boldsymbol{\\xi}_k).\n\\end{split}\n\\end{equation}\nAlgorithm (\\ref{eq: x_i,k}) is closely related to the schemes considered in \\cite{di2016next,nedic2017achieving,qu2017harnessing}, where auxiliary variables $y_{i,k}$ \nwere introduced to track the average $\\frac{1}{n}\\sum_{i=1}^{n}\\nabla f_i(x_{i,k})$. This design ensures that the algorithm achieves linear convergence under constant step size choice.\nCorrespondingly, under our approach $y_{i,k}$ are (approximately) tracking $\\frac{1}{n}\\sum_{i=1}^{n}g_i(x_{i,k},\\xi_{i,k})$.\nTo see why this is the case, note that\n\\begin{equation}\n\\overline{y}_k = \\frac{1}{n}\\mathbf{1}^{\\intercal} \\mathbf{y}_k.\n\\end{equation}\nSince $y_{i,0}=g(x_{i,0},\\xi_{i,0}),\\forall i$. By induction,\n\\begin{equation}\n\\overline{y}_k=\\frac{1}{n}\\mathbf{1}^{\\intercal}G(\\mathbf{x}_k,\\boldsymbol{\\xi}_k),\\forall k.\n\\end{equation}\nWe will show that $\\mathbf{y}_k$ is close to $\\mathbf{1}\\overline{y}_k$ at each round. Hence $y_{i,k}$ are (approximately) tracking $\\frac{1}{n}\\sum_{i=1}^{n}g_i(x_{i,k},\\xi_{i,k})$.\n\nThroughout the paper, we make the following standing assumption on the objective functions $f_i$:\n\\begin{assumption}\n\t\\label{asp: strconvexity}\n\tEach $f_i:\\mathbb{R}^p\\rightarrow \\mathbb{R}$ is  $\\mu$-strongly convex with $L$-Lipschitz continuous gradients, i.e., for any $x,x'\\in\\mathbb{R}^p$,\n\t\\begin{equation}\n\t\\begin{split}\n\t& \\langle \\nabla f_i(x)-\\nabla f_i(x'),x-x'\\rangle\\ge \\mu\\|x-x'\\|^2,\\\\\n\t& \\|\\nabla f_i(x)-\\nabla f_i(x')\\|\\le L \\|x-x'\\|.\n\t\\end{split}\n\t\\end{equation}\n\\end{assumption}\nUnder Assumption \\ref{asp: strconvexity}, problem (\\ref{opt Problem_def}) has a unique solution denoted by $x^*\\in\\mathbb{R}^{1\\times p}$.\n\n\\subsection{Main Results}\n\\label{subsec: main}\nMain convergence properties of the distributed gradient tracking method (\\ref{eq: x_i,k}) are covered in the following theorem.\n\\begin{theorem}\n\t\\label{Theorem}\n\tSuppose Assumptions \\ref{asp: gradient samples}-\\ref{asp: strconvexity} hold and $\\alpha$ satisfies\n\t\\begin{equation}\n\t\t\\label{alpha_ultimate_bound}\n\t\t\\alpha\\le \\min\\left\\{\\frac{(1-\\rho_w^2)}{12\\rho_w^2 L},\\frac{(1-\\rho_w^2)^2}{2\\sqrt{\\Gamma}L\\max\\{6\\rho_w\\|W-I\\|,1-\\rho_w^2\\}}, \\frac{(1-\\rho_w^2)}{3\\rho_w^{2\/3}L}\\left[\\frac{\\mu^2}{L^2}\\frac{(\\Gamma-1)}{\\Gamma(\\Gamma+1)}\\right]^{1\/3}\\right\\}\n\t\t\\end{equation}\n\tfor some $\\Gamma>1$. Then both $\\sup_{l\\ge k}\\mathbb{E}[\\|\\overline{x}_l-x^*\\|^2]$ and $\\sup_{l\\ge k}\\mathbb{E}[\\|\\mathbf{x}_{l+1}-\\mathbf{1}\\overline{x}_{l+1}\\|^2]$ converge at the linear rate $\\mathcal{O}(\\rho_A^k)$, where $\\rho_A<1$ is the spectral radius of\n\t\\begin{eqnarray*}\n\t\t\tA=\\begin{bmatrix}\n\t\t\t\t1-\\alpha\\mu & \\frac{\\alpha L^2}{\\mu n}(1+\\alpha\\mu) & 0\\\\\n\t\t\t\t0 & \\frac{1}{2}(1+\\rho_w^2) & \\alpha^2\\frac{(1+\\rho_w^2)\\rho_w^2}{(1-\\rho_w^2)}\\\\\n\t\t\t\t2\\alpha nL^3 & \\left(\\frac{1}{\\beta}+2\\right)\\|W-I\\|^2 L^2+3\\alpha L^3 & \\frac{1}{2}(1+\\rho_w^2)\n\t\t\t\\end{bmatrix},\n\t\\end{eqnarray*}\nin which $\\beta=\\frac{1-\\rho_w^2}{2\\rho_w^2}-4\\alpha L-2\\alpha^2 L^2$.\nFurthermore,\n\\begin{equation}\n\t\\label{error_bound_ultimate}\n\t\\limsup_{k\\rightarrow\\infty}\\mathbb{E}[\\|\\overline{x}_k-x^*\\|^2]\\le \\frac{(\\Gamma+1)}{\\Gamma}\\frac{\\alpha\\sigma^2}{\\mu n}\n\t+\\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{4\\alpha^2 L^2(1+\\alpha\\mu)(1+\\rho_w^2)\\rho_w^2}{\\mu^2 n(1-\\rho_w^2)^3}M_{\\sigma},\n\t\\end{equation}\nand\n\\begin{equation}\n\t\\label{consensus_error_bound_ultimate}\n\t\\limsup_{k\\rightarrow\\infty}\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2]\n\t\\le \\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{4\\alpha^2 (1+\\rho_w^2)\\rho_w^2(2\\alpha^2L^3\\sigma^2+\\mu M_{\\sigma})}{\\mu(1-\\rho_w^2)^3},\n\t\\end{equation}\nwhere \n\\begin{equation}\n\t\\label{M_sigma}\n\tM_{\\sigma}:=\\left[3\\alpha^2L^2+2(\\alpha L+1)(n+1)\\right]\\sigma^2.\n\t\\end{equation}\n\\end{theorem}\n\\begin{remark}\n\tThe first term on the right hand side of (\\ref{error_bound_ultimate}) can be interpreted as the error caused by stochastic gradients only, since it does not depend on the network topology. The second term as well as the bound in (\\ref{consensus_error_bound_ultimate}) are network dependent and increase with $\\rho_w$ (larger $\\rho_w$ indicates worse network connectivity).\n\t\n\tIn light of (\\ref{error_bound_ultimate}) and (\\ref{consensus_error_bound_ultimate}),\n\t\\begin{equation*}\n\t\\limsup_{k\\rightarrow\\infty}\\mathbb{E}[\\|\\overline{x}_k-x^*\\|^2]=\\alpha\\mathcal{O}\\left(\\frac{\\sigma^2}{\\mu n}\\right)+\\alpha^2\\mathcal{O}\\left(\\frac{ L^2\\sigma^2}{\\mu^2}\\right),\n\t\\end{equation*}\n\tand\n\t\\begin{equation*}\n\t\\limsup_{k\\rightarrow\\infty}\\frac{1}{n}\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2]=\\alpha^4\\mathcal{O}\\left(\\frac{ L^3\\sigma^2}{\\mu n}\\right)+\\alpha^2\\mathcal{O}\\left(\\sigma^2\\right).\n\t\\end{equation*}\n\tLet $(1\/n)\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}x^*\\|^2]$ measure the average quality of solutions obtained by all the agents. We have\n\t\t\\begin{equation*}\n\t\\limsup_{k\\rightarrow\\infty}\\frac{1}{n}\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}x^*\\|^2]=\\alpha\\mathcal{O}\\left(\\frac{\\sigma^2}{\\mu n}\\right)+\\alpha^2\\mathcal{O}\\left(\\frac{ L^2\\sigma^2}{\\mu^2}\\right),\n\t\\end{equation*}\n\twhich is decreasing in the network size $n$ when $\\alpha$ is sufficiently small\\footnote{Although $\\rho_w$ is also related to the network size $n$, it only appears in the terms with high orders of $\\alpha$.}.\n\t\n\tUnder a centralized algorithm in the form of\n\t\\begin{equation}\n\t\\label{eq: centralized}\n\tx_{k+1}=x_k-\\alpha \\frac{1}{n}\\sum_{i=1}^n g_i(x_k,\\xi_{i,k}),\\ \\ k\\in\\mathbb{N},\n\t\\end{equation}\n\twe would obtain\n\t\\begin{equation*}\n\\limsup_{k\\rightarrow\\infty}\\mathbb{E}[\\|x_k-x^*\\|^2]=\\alpha\\mathcal{O}\\left(\\frac{\\sigma^2}{\\mu n}\\right).\n\t\\end{equation*}\n\tIt can be seen that the distributed stochastic gradient tracking method (\\ref{eq: x_i,k}) is comparable with the centralized algorithm (\\ref{eq: centralized}) in their ultimate error bounds (up to constant factors) with sufficiently small step sizes.\n\\end{remark}\n\n\\begin{corollary}\n\t\\label{cor: speed}\n\tUnder the conditions in Theorem {\\ref{Theorem}}. Suppose in addition\\footnote{This condition is weaker than (\\ref{alpha_ultimate_bound}) in most cases.} \n\t\\begin{equation}\n\t\t\\label{alpha condition corollary}\n\t\\alpha\\le \\frac{(\\Gamma+1)}{\\Gamma}\\frac{(1-\\rho_w^2)}{8\\mu}.\n\t\\end{equation}\n    Then\n\t\\begin{equation*}\n\t\\rho_A\\le 1-\\left(\\frac{\\Gamma-1}{\\Gamma+1}\\right)\\alpha\\mu.\n\t\\end{equation*}\n\\end{corollary}\n\\begin{remark}\n\tCorollary \\ref{cor: speed} implies that, for sufficiently small step sizes, \n\tthe distributed gradient tracking method has a comparable convergence speed to that of a centralized scheme (in which case the linear rate is $\\mathcal{O}(1-2\\alpha\\mu)^k$).\n\\end{remark}\n\n\\section{Analysis}\n\\label{sec:cohesion}\n\nIn this section, we prove Theorem \\ref{Theorem} by studying the evolution of $\\mathbb{E}[\\|\\overline{x}_k-x^*\\|^2]$, $\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2]$ and $\\mathbb{E}[\\|\\mathbf{y}_k-\\mathbf{1}\\overline{y}_k\\|^2]$. Our strategy is to bound the three expressions in terms of linear combinations of their past values, in which way we establish a linear system of inequalities. This approach is different from those employed in \\cite{qu2017harnessing,nedic2017achieving}, where the analyses pertain to the examination of $\\|\\overline{x}_k-x^*\\|$, $\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|$ and $\\|\\mathbf{y}_k-\\mathbf{1}\\overline{y}_k\\|$. Such distinction is due to the  stochastic gradients $g_i(x_{i,k},\\xi_{i,k})$ whose variances play a crucial role in deriving the main inequalities.\n\nWe first introduce some lemmas that will be used later in the analysis. Denote by $\\mathcal{H}_k$ the history sequence $\\{\\mathbf{x}_0,\\boldsymbol{\\xi}_0,\\mathbf{y}_0,\\ldots,\\mathbf{x}_{k-1},\\boldsymbol{\\xi}_{k-1},\\mathbf{y}_{k-1},\\mathbf{x}_k\\}$, and define $\\mathbb{E}[\\cdot \\mid\\mathcal{H}_k]$ as the conditional expectation given $\\mathcal{H}_k$.\n\\begin{lemma}\n\t\\label{lem: oy_k-h_k}\n\tUnder Assumption \\ref{asp: gradient samples},\n\t\\begin{align}\n\t\\mathbb{E}\\left[\\|\\overline{y}_k-h(\\mathbf{x}_k)\\|^2\\mid\\mathcal{H}_k\\right] \\le \\frac{\\sigma^2}{n}.\n\t\\end{align}\n\\end{lemma}\n\\begin{proof}\n\tBy the definitions of $\\overline{y}_k$ and $h(\\mathbf{x}_k)$,\n\t\\begin{equation*}\n\t\t\\mathbb{E}\\left[\\|\\overline{y}_k-h(\\mathbf{x}_k)\\|^2\\mid\\mathcal{H}_k\\right]\\\\\n\t\t=\t\\frac{1}{n^2}\\sum_{i=1}^n\\mathbb{E}\\left[\\|g_i(x_{i,k},\\xi_{i,k})-\\nabla f_i(x_{i,k})\\|^2\\vert\\mathcal{H}_k\\right]\\le \\frac{\\sigma^2}{n}.\n\t\t\\end{equation*}\n\\end{proof}\n\\begin{lemma}\n\t\\label{lem: strong_convexity}\n\tUnder Assumption \\ref{asp: strconvexity},\n\t\\begin{align}\n\t\\| \\nabla F(\\overline{x}_k)-h(\\mathbf{x}_k)\\| \\le \\frac{L}{\\sqrt{n}}\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|.\n\t\\end{align}\n\tSuppose in addition $\\alpha<2\/(\\mu+L)$. Then\n\t\\begin{equation*}\n\t\\|x-\\alpha\\nabla f(x)-x^*\\|\\le (1-\\alpha \\mu)\\|x-x^*\\|,\\, \\forall x\\in\\mathbb{R}^p.\n\t\\end{equation*}\n\\end{lemma}\n\\begin{proof}\n\tSee  \\cite{qu2017harnessing} Lemma 10 for reference.\n\\end{proof}\n\nIn the following lemma, we establish bounds on $\\|\\mathbf{x}_{k+1}-\\mathbf{1}\\overline{x}_{k+1}\\|^2$ and on the conditional expectations of $\\|\\overline{x}_{k+1}-x^*\\|^2$ and $\\|\\mathbf{y}_{k+1}-\\mathbf{1}\\overline{y}_{k+1}\\|^2$, respectively.\n\\begin{lemma}\n\t\\label{lem: Main_Inequalities}\n\tSuppose Assumptions \\ref{asp: gradient samples}-\\ref{asp: strconvexity} hold and $\\alpha<2\/(\\mu+L)$. We have the following inequalities:\n\\begin{equation}\n\t\t\\label{First_Main_Inequality}\n\t\\mathbb{E}[\\|\\overline{x}_{k+1}-x^*\\|^2\\mid \\mathcal{H}_k]\n\t\\le \\left(1-\\alpha\\mu\\right)\\|\\overline{x}_k-x^*\\|^2\\\\\n\t+\\frac{\\alpha L^2}{\\mu n}\\left(1+\\alpha\\mu\\right)\\|  \\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2\n\t+\\frac{\\alpha^2\\sigma^2}{n},\n\\end{equation}\n\t\\begin{equation}\n\\label{Second_Main_Inequality}\n\\|\\mathbf{x}_{k+1}-\\mathbf{1}\\overline{x}_{k+1}\\|^2\n\\le \\frac{(1+\\rho_w^2)}{2}\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2\\\\+\\alpha^2\\frac{(1+\\rho_w^2)\\rho_w^2}{(1-\\rho_w^2)}\\|\\mathbf{y}_{k}-\\mathbf{1}\\overline{y}_k\\|^2,\n\\end{equation}\nand for any $\\beta>0$,\n\\begin{multline}\n\\label{Third_Main_Inquality}\n\\mathbb{E}[\\|\\mathbf{y}_{k+1}-\\mathbf{1}\\overline{y}_{k+1}\\|^2\\mid \\mathcal{H}_k]\n\\le \\left(1+4\\alpha L+2\\alpha^2 L^2+\\beta\\right)\\rho_w^2\\mathbb{E}[\\|\\mathbf{y}_{k}-\\mathbf{1}\\overline{y}_{k}\\|^2\\mid \\mathcal{H}_k]\\\\\n+\\left(\\frac{1}{\\beta}\\|W-I\\|^2 L^2+2\\|W-I\\|^2 L^2+3\\alpha L^3\\right)\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2+2\\alpha nL^3\\|\\overline{x}_k-x^*\\|^2+M_{\\sigma}.\n\\end{multline}\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{proof lem: Main_Inequalities}.\n\t\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{Theorem}}\n\tTaking full expectation on both sides of (\\ref{First_Main_Inequality}), (\\ref{Second_Main_Inequality}) and (\\ref{Third_Main_Inquality}), we obtain the following linear system of inequalities\n\\begin{eqnarray}\n\\label{linear_system}\n\\begin{bmatrix}\n\\mathbb{E}[\\|\\overline{x}_{k+1}-x^*\\|^2]\\\\\n\\mathbb{E}[\\|\\mathbf{x}_{k+1}-\\mathbf{1}\\overline{x}_{k+1}\\|^2]\\\\\n\\mathbb{E}[\\|\\mathbf{y}_{k+1}-\\mathbf{1}\\overline{y}_{k+1}\\|^2]\n\\end{bmatrix}\n\\le \nA\\begin{bmatrix}\n\\mathbb{E}[\\|\\overline{x}_{k}-x^*\\|^2]\\\\\n\\mathbb{E}[\\|\\mathbf{x}_{k}-\\mathbf{1}\\overline{x}_{k}\\|^2]\\\\\n\\mathbb{E}[\\|\\mathbf{y}_{k}-\\mathbf{1}\\overline{y}_{k}\\|^2]\n\\end{bmatrix}+\\begin{bmatrix}\n\\frac{\\alpha^2\\sigma^2}{n}\\\\\n0\\\\\nM_{\\sigma}\n\\end{bmatrix},\n\\end{eqnarray}\nwhere the inequality is to be taken component-wise, and the entries of the matrix\n$A=[a_{ij}]$ are given by\n\t\\begin{eqnarray*}\n\t\t& \\begin{bmatrix}\n\t\t\ta_{11}\\\\\n\t\t\ta_{21}\\\\\n\t\t\ta_{31}\n\t\t\\end{bmatrix}  =  \n\t\t\\begin{bmatrix}\n\t\t\t1-\\alpha\\mu\\\\\n\t\t\t0\\\\\n\t\t\t2\\alpha nL^3 \n\t\t\\end{bmatrix},\n\t\t\\begin{bmatrix}\n\t\t\ta_{12}\\\\\n\t\t\ta_{22}\\\\\n\t\t\ta_{32}\n\t\t\\end{bmatrix} = \n\t\t\\begin{bmatrix}\n\t\t\t\\frac{\\alpha L^2}{\\mu n}(1+\\alpha\\mu)\\\\\n\t\t\t\\frac{1}{2}(1+\\rho_w^2)\\\\\n\t\t\t\\left(\\frac{1}{\\beta}+2\\right)\\|W-I\\|^2 L^2+3\\alpha L^3\n\t\t\\end{bmatrix},\\\\\n\t\t& \\begin{bmatrix}\n\t\t\ta_{13}\\\\\n\t\t\ta_{23}\\\\\n\t\t\ta_{33}\n\t\t\\end{bmatrix} = \n\t\t\\begin{bmatrix}\n\t\t\t0 \\\\\n\t\t\t\\alpha^2\\frac{(1+\\rho_w^2)\\rho_w^2}{(1-\\rho_w^2)}\\\\\n\t\t\t\\left(1+4\\alpha L+2\\alpha^2 L^2+\\beta\\right)\\rho_w^2\n\t\t\\end{bmatrix},\n\\end{eqnarray*}\nand $M_{\\sigma}$ is given in (\\ref{M_sigma}).\nHence $\\sup_{l\\ge k}\\mathbb{E}[\\|\\overline{x}_l-x^*\\|^2]$, $\\sup_{l\\ge k}\\mathbb{E}[\\|\\mathbf{x}_l-\\mathbf{1}\\overline{x}_l\\|^2]$ and $\\sup_{l\\ge k}\\mathbb{E}[\\|\\mathbf{y}_l-\\mathbf{1}\\overline{y}_l\\|^2]$ all converge to a neighborhood of $0$ at the linear rate $\\mathcal{O}(\\rho_A^k)$ if the spectral radius of $A$ satisfies $\\rho_A<1$. \nThe next lemma provides conditions for relation $\\rho_A<1$ to hold.\n\\begin{lemma}\n\t\\label{lem: rho_M}\n\tLet $M=[m_{ij}]\\in\\mathbb{R}^{3\\times 3}$ be a nonnegative, irreducible matrix with \n\t$m_{ii}<\\lambda^*$ for some~{$\\lambda^*>0$ and all $i=1,2,3.$} \n\tA necessary and sufficient condition for $\\rho_M<\\lambda^*$ is $\\text{det}(\\lambda^* I-M)>0$.\n\\end{lemma}\n\\begin{proof}\n\tSee Appendix \\ref{subsec: proof lemma rho_M}.\n\t\\end{proof}\n\tLet $\\alpha$ and $\\beta$ be such that the following relations hold\\footnote{Matrix $A$ in Theorem~\\ref{Theorem} \n\tcorresponds to such a choice of $\\alpha$ and $\\beta$.}.\n\\begin{equation}\n\t\\label{beta}\n\ta_{33}=\\left(1+4\\alpha L+2\\alpha^2 L^2+\\beta\\right)\\rho_w^2=\\frac{1+\\rho_w^2}{2}<1,\n\t\\end{equation}\n\t\\begin{equation}\n\t\\label{alpha,beta}\n\ta_{23}a_{32}=\\alpha^2\\frac{(1+\\rho_w^2)\\rho_w^2}{(1-\\rho_w^2)}\\left[\\left(\\frac{1}{\\beta}+2\\right)\\|W-I\\|^2 L^2+3\\alpha L^3\\right]\n\t\\le\\frac{1}{\\Gamma}(1-a_{22})(1-a_{33})\n\t\\end{equation}\nfor some $\\Gamma>1$, and\n\\begin{equation}\n\t\\label{alpha last condition}\n\ta_{12}a_{23}a_{31}=\\frac{2\\alpha^4 L^5(1+\\alpha\\mu)}{\\mu}\\frac{(1+\\rho_w^2)}{(1-\\rho_w^2)}\\rho_w^2\n\t\\le \\frac{1}{\\Gamma+1}(1-a_{11})[(1-a_{22})(1-a_{33})-a_{23}a_{32}].\n\t\\end{equation}\nThen,\n\\begin{multline*}\n\t\\text{det}(I-A)=(1-a_{11})(1-a_{22})(1-a_{33})-(1-a_{11})a_{23}a_{32}-a_{12}a_{23}a_{31}\\\\\n\t\\ge \\frac{\\Gamma}{(\\Gamma+1)}(1-a_{11})[(1-a_{22})(1-a_{33})-a_{23}a_{32}]\n\t\\ge \\left(\\frac{\\Gamma-1}{\\Gamma+1}\\right)(1-a_{11})(1-a_{22})(1-a_{33})>0,\n\t\\end{multline*}\ngiven that $a_{11},a_{22},a_{33}<1$. In light of Lemma \\ref{lem: rho_M}, $\\rho_A<1$.\nIn addition, denoting $B:=[\\frac{\\alpha^2\\sigma^2}{n}, 0, M_{\\sigma}]^{\\intercal}$, we get \n\t\\begin{align}\n\t\t\\label{error_bound_preliminary}\n\t\\lim\\sup_{k\\rightarrow\\infty}\\mathbb{E}[\\|\\overline{x}_k-x^*\\|^2]\\le & [(I-A)^{-1}B]_1 \\notag\\\\\n\t= & \\frac{1}{\\text{det}(I-A)}\\left\\{\\left[(1-a_{22})(1-a_{33})-a_{23}a_{32}\\right]\\frac{\\alpha^2\\sigma^2}{n}+a_{12}a_{23}M_{\\sigma}\\right\\} \\notag\\\\\n\t\\le & \\frac{(\\Gamma+1)}{\\Gamma}\\frac{\\alpha\\sigma^2}{\\mu n}+\\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{ a_{12}a_{23}M_{\\sigma}}{(1-a_{11})(1-a_{22})(1-a_{33})}\\notag\\\\\n\t= & \\frac{(\\Gamma+1)}{\\Gamma}\\frac{\\alpha\\sigma^2}{\\mu n}+ \\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{\\alpha^3 L^2(1+\\alpha\\mu)(1+\\rho_w^2)\\rho_w^2M_{\\sigma}}{\\mu n(1-\\rho_w^2)(1-a_{11})(1-a_{22})(1-a_{33})}\\notag\\\\\n\t= & \\frac{(\\Gamma+1)}{\\Gamma}\\frac{\\alpha\\sigma^2}{\\mu n}+\\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{4\\alpha^2 L^2(1+\\alpha\\mu)(1+\\rho_w^2)\\rho_w^2}{\\mu^2 n(1-\\rho_w^2)^3}M_{\\sigma},\n\t\\end{align}\nand\n\t\\begin{multline*}\n\t\\lim\\sup_{k\\rightarrow\\infty}\\mathbb{E}[\\|\\mathbf{x}_k-\\mathbf{1}\\overline{x}_k\\|^2] \\le [(I-A)^{-1}B]_2\n\t=\\frac{1}{\\text{det}(I-A)}\\left[a_{23}a_{31}\\frac{\\alpha^2\\sigma^2}{n}+a_{23}(1-a_{11})M_{\\sigma}\\right]\\\\\n\t\\le \\left(\\frac{\\Gamma+1}{\\Gamma-1}\\right)\\frac{a_{23}}{(1-a_{11})(1-a_{22})(1-a_{33})}\\left(2\\alpha nL^3\\frac{\\alpha^2 \\sigma^2}{n}+\\alpha\\mu M_{\\sigma}\\right)\\\\\n\t= \\frac{4(\\Gamma+1)\\alpha^2 (1+\\rho_w^2)\\rho_w^2(2\\alpha^2L^3\\sigma^2+\\mu M_{\\sigma})}{(\\Gamma-1)\\mu(1-\\rho_w^2)^3}.\n\t\\end{multline*}\n\n\tWe now show that (\\ref{beta}), (\\ref{alpha,beta}) and (\\ref{alpha last condition}) are satisfied under condition (\\ref{alpha_ultimate_bound}).  By (\\ref{beta}) and (\\ref{alpha,beta}), it follows that\n\t\\begin{equation*}\n\t\t\\beta=\\frac{1-\\rho_w^2}{2\\rho_w^2}-4\\alpha L-2\\alpha^2 L^2>0,\n\t\t\\end{equation*}\n\tand\n\t\t\\begin{equation*}\n\t\t\\alpha^2\\frac{(1+\\rho_w^2)\\rho_w^2}{(1-\\rho_w^2)}\\left[\\left(\\frac{1}{\\beta}+2\\right)\\|W-I\\|^2 L^2+3\\alpha L^3\\right]\\le\\frac{(1-\\rho_w^2)^2}{4\\Gamma}.\n\t\t\\end{equation*}\n\t\tSince by (\\ref{alpha_ultimate_bound}) we have\n\t\t\\begin{equation*}\n\t\t\t\\alpha\\le \\frac{1-\\rho_w^2}{12\\rho_w^2 L},\\qquad\n\t\t\n\t\n\t\t\\beta\\ge \\frac{1-\\rho_w^2}{8\\rho_w^2}>0,\n\t\t\\end{equation*}\n\t\twe only need to show that\n\t\t\\begin{equation*}\n\t\t\\alpha^2\\left[\\frac{(2+6\\rho_w^2)}{(1-\\rho_w^2)}\\|W-I\\|^2 L^2+\\frac{(1-\\rho_w^2)}{4\\rho_w^2}L^2\\right]\\le\\frac{(1-\\rho_w^2)^3}{4\\Gamma(1+\\rho_w^2)\\rho_w^2}.\n\t\t\\end{equation*}\n\t\tThe preceding inequality is equivalent to\n\t\t\\begin{equation*}\n\t\t\\alpha\\le \\frac{(1-\\rho_w^2)^2}{L\\sqrt{\\Gamma(1+\\rho_w^2)}\\sqrt{4\\rho_w^2(2+6\\rho_w^2)\\|W-I\\|^2+(1-\\rho_w^2)^2}},\n\t\t\\end{equation*}\n\timplying that it is sufficient to have\n\t\\begin{equation*}\n\t\\alpha\\le\\frac{(1-\\rho_w^2)^2}{2\\sqrt{\\Gamma}L\\max(6\\rho_w\\|W-I\\|,1-\\rho_w^2)}.\n\t\\end{equation*}\n\t\tTo see that relation (\\ref{alpha last condition}) holds, consider a stronger condition\n\t\t\\begin{equation*}\n\t\t\\frac{2\\alpha^4 L^5(1+\\alpha\\mu)}{\\mu}\\frac{(1+\\rho_w^2)}{(1-\\rho_w^2)}\\rho_w^2\n\t\t\\le \\frac{(\\Gamma-1)}{\\Gamma(\\Gamma+1)}(1-a_{11})(1-a_{22})(1-a_{33}),\n\t\t\\end{equation*}\n\t\tor equivalently,\n\t\t\\begin{equation*}\n\t\t\\frac{2\\alpha^3 L^5(1+\\alpha\\mu)}{\\mu^2}\\frac{(1+\\rho_w^2)}{(1-\\rho_w^2)}\\rho_w^2 \\le \\frac{(\\Gamma-1)}{4\\Gamma(\\Gamma+1)}(1-\\rho_w^2)^2.\n\t\t\\end{equation*}\n\t\tIt suffices that\n\t\t\\begin{equation}\n\t\t\\alpha\\le \\frac{(1-\\rho_w^2)}{3\\rho_w^{2\/3}L}\\left[\\frac{\\mu^2}{L^2}\\frac{(\\Gamma-1)}{\\Gamma(\\Gamma+1)}\\right]^{1\/3}.\n\t\t\\end{equation}\n\n\\subsection{Proof of Corollary \\ref{cor: speed}}\nWe derive an upper bound of $\\rho_A$ under condition (\\ref{alpha_ultimate_bound}) and (\\ref{alpha condition corollary}). Note that the characteristic function of $A$ is given by\n\\begin{equation*}\n\\text{det}(\\lambda I-A)=(\\lambda-a_{11})(\\lambda-a_{22})(\\lambda-a_{33})\n-(\\lambda-a_{11})a_{23}a_{32}-a_{12}a_{23}a_{31}.\n\\end{equation*}\nSince $\\text{det}(I-A)> 0$ and $\\text{det}(\\max\\{a_{11},a_{22},a_{33}\\} I-A)<0$, $\\rho_A\\in(\\max\\{a_{11},a_{22},a_{33}\\},1)$. By (\\ref{alpha,beta}) and (\\ref{alpha last condition}),\n\\begin{multline*}\n\\text{det}(\\lambda I-A)\n\\ge (\\lambda-a_{11})(\\lambda-a_{22})(\\lambda-a_{33})-(\\lambda-a_{11})a_{23}a_{32}-\\frac{1}{\\Gamma+1}(1-a_{11})[(1-a_{22})(1-a_{33})-a_{23}a_{32}]\\\\\n\\ge (\\lambda-a_{11})(\\lambda-a_{22})(\\lambda-a_{33})-\\frac{1}{\\Gamma}(\\lambda-a_{11})(1-a_{22})(1-a_{33})\n-\\frac{(\\Gamma-1)}{\\Gamma(\\Gamma+1)}(1-a_{11})(1-a_{22})(1-a_{33}).\n\\end{multline*}\nSuppose $\\lambda=1-\\epsilon$ for some $\\epsilon\\in(0,\\alpha\\mu)$, satisfying\n\\begin{equation*}\n\\text{det}(\\lambda I-A)\n\\ge \\frac{1}{4}(\\alpha\\mu-\\epsilon)\\left(1-\\rho_w^2-2\\epsilon\\right)^2\n-\\frac{1}{4\\Gamma}(\\alpha\\mu-\\epsilon)(1-\\rho_w^2)^2-\\frac{(\\Gamma-1)\\alpha\\mu}{4\\Gamma(\\Gamma+1)}(1-\\rho_w^2)^2\\ge 0.\n\\end{equation*}\nUnder (\\ref{alpha condition corollary}), it suffices that\n\\begin{equation*}\n\\epsilon\\le \\left(\\frac{\\Gamma-1}{\\Gamma+1}\\right)\\alpha\\mu.\n\\end{equation*}\nDenote\n\\begin{equation*}\n\\tilde{\\lambda}=1-\\left(\\frac{\\Gamma-1}{\\Gamma+1}\\right)\\alpha\\mu.\n\\end{equation*}\nThen $\\text{det}(\\tilde{\\lambda} I-A)\\ge 0$ so that $\\rho_A\\le \\tilde{\\lambda}$.\n\n\\section{Numerical Example}\n\\label{sec: simulation}\n\nIn this section, we provide a numerical example to illustrate our theoretic findings. \nConsider the \\emph{on-line} Ridge regression problem, i.e.,\n\\begin{equation}\n\\label{Ridge Regression}\nf(x):=\\frac{1}{n}\\sum_{i=1}^n\\mathbb{E}_{u_i,v_i}\\left[\\left(u_i^{\\intercal} x-v_i\\right)^2+\\rho\\|x\\|^2\\right].\n\\end{equation}\nwhere $\\rho>0$ ia a penalty parameter.\nFor each agent $i$, samples in the form of $(u_i,v_i)$ are gathered continuously with $u_i\\in\\mathbb{R}^p$ representing the features and $v_i\\in\\mathbb{R}$ being the observed outputs. We assume that each $u_i\\in[-1,1]^p$ is uniformly distributed, and $v_i$ is drawn according to $v_i=u_i^{\\intercal} \\tilde{x}_i+\\varepsilon_i$. Here $\\tilde{x}_i\\in[0.4,0.6]^p$ is a predefined, (uniformly) randomly generated parameter, and $\\varepsilon_i$ are independent Gaussian noises with mean $0$ and variance $0.25$.\nGiven a pair $(u_i,v_i)$, agent $i$ can calculate an estimated gradient of $f_i(x)$:\n\\begin{equation}\ng_i(x,u_i,v_i)=2(u_i^{\\intercal}x -v_i)u_i+2\\rho x,\n\\end{equation}\nwhich is unbiased.\nNotice that the Hessian matrix of $f(x)$ is $\\mathbf{H}_f=(2\/3+2\\rho)I_d\\succ 0$. Therefore $f(\\cdot)$ is strongly convex, and problem (\\ref{Ridge Regression}) has a unique solution $x^*$ given by\n\\begin{equation*}\nx^*=\\frac{1}{(1+3\\rho)}\\sum_{i=1}^n\\tilde{x}_i\/n.\n\\end{equation*}\n\nIn the experiments, we consider $3$ instances with $p=20$ and $n\\in\\{10,25,100\\}$, respectively. Under each instance, we draw $x_{i,0}$  uniformly randomly from $[5,10]^p$. Penalty parameter $\\rho=0.01$ and step size $\\gamma=0.01$. We assume that $n$ agents constitute a random network, in which each two agents are linked with probability $0.4$. The Metropolis rule is applied to define the weights $w_{ij}$ \\cite{sayed2014adaptive}:\n\\begin{equation*}\nw_{ij}=\\begin{cases}\n1\/\\max\\{d_i,d_j\\} & \\text{if }i\\in \\mathcal{N}_i\\setminus \\{i\\},  \\\\\n1- \\sum_{j\\in\\mathcal{N}_i}w_{ij} & \\text{if }i=j,\\\\\n0 & \\text{if }i\\notin \\mathcal{N}_i.\n\\end{cases}\n\\end{equation*}\nHere $d_i$ denotes the degree (number of ``neighbors'') of node $i$, and $\\mathcal{N}_i$ is the set of ``neighbors''.\n\n\\begin{figure}\n\t\\centering\n\t\\subfigure[Instance $(p,n)=(20,10)$.]{\\includegraphics[width=3.5in]{p20n10_big.eps}} \n\t\\subfigure[Instance $(p,n)=(20,25)$.]{\\includegraphics[width=3.5in]{p20n25_big.eps}}\n\t\\subfigure[Instance $(p,n)=(20,100)$.]{\\includegraphics[width=3.5in]{{p20n100_big.eps}}} \n\t\\caption{Performance comparison between the distributed gradient tracking method and the centralized algorithm for on-line Ridge regression. For the decentralized method, the plots show the iterates generated \n\tby a randomly selected node $i$ from the set $\\mathcal{N}$.}\n\t\\label{fig: comparison}\n\\end{figure}\n\nIn Figure \\ref{fig: comparison}, we compare the performances of the distributed gradient tracking method (\\ref{eq: x_i,k}) and the centralized algorithm (\\ref{eq: centralized}) with the same parameters. It can be seen that the two approaches are comparable in their convergence speeds as well as the ultimate error bounds. Furthermore, the error bounds decrease in $n$ as expected from our theoretical analysis.\n\\section{Conclusions and Future Work}\n\\label{sec: conclusion}\nThis paper considers distributed multi-agent optimization over a network, where each agent only has access to inexact gradients of its local cost function. \nWe propose a distributed stochastic gradient tracking method and show that the iterates obtained by each agent, using a constant step size value, reach a neighborhood of the optimum (in expectation) exponentially fast. More importantly, in a limit, the error bounds for the distances between the iterates and the optimal solution decrease in the network size, which is comparable with the performance of a centralized stochastic gradient algorithm. \nIn our future work, we will consider adaptive step size policies, directed and\/or time-varying interaction graphs, and more efficient communication protocols (e.g., gossip-based scheme).\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nIn the age of big data, the sheer volume of data discourages manual screening and, therefore, Machine Learning (ML) has been proposed as an effective replacement to conventional solutions. These applications range from face recognition~\\cite{deep_face}, to voice recognition~\\cite{voice}, and autonomous vehicles~\\cite{cars}. ML-based facial recognition is extensively used for biometric identification (e.g. passport control ~\\cite{immigration_uae, immigration}), automatic detection of extremist posts~\\cite{abusive_content}, prevention of online dating frauds~\\cite{OnlineDating}, and reporting of inappropriate images~\\cite{stealthy_images,Inapt}. Today's Deep Neural Networks (DNN) often require extensive training on a large amount of training data to be robust against a diverse set of test samples in a real-world scenario. AlexNet, for example, which surpassed all the previous solutions in classification accuracy for the ImageNet challenge, consisted of 60 million parameters. This growth in complexity and size of ML models demands an increase in computational cost\/power needed for developing and training these models, giving rise to the industry of Machine Learning-as-a-Service (MLaaS).\n\nOutsourcing machine learning training democratizes the use of sophisticated ML models. There are many sources for open-source ML models, such as Cafe Model Zoo~\\cite{zoo} and BigML model Market~\\cite{biggie}. Outsourcing, however, introduces the possibility of compromising machine learning models during the training phase. Research in~\\cite{Badnets} showed that it is possible to infect\/corrupt a model by poisoning the training data. This process introduces a backdoor\/trojan to the model. \nA backdoor\/trojan in a DNN represents a set of neurons that are activated in the presence of unique triggers to cause malicious behavior.~\\cite{suppression} shows one such example where a dot (one-pixel trigger) can be used to trigger certain (backdoored\/infected) neurons in a model to maliciously change the true prediction of the model. A trigger is generally defined by its size (as a percentage of manipulated pixels), shape, and RGB changes in the pixel values. \n\nIn the backdoor attack literature, several types of patterns like post-its on stop signs \\cite{Badnets}, specific black-rimmed spectacles \\cite{black_badnets} or specific patterns based on a desired masking size \\cite{NDSSTrojans} have been used to trigger backdoor neurons. Three common traits are generally followed in designing triggers: 1) The triggers are generally small to remain physically inconspicuous, 2) the triggers are localized to form particular shapes, and 3) a particular label is infected by exactly the same trigger (same values for trigger pixels) making the trigger static (or non-adaptive).\n\nThere has been a plethora of proposed solutions that aim to defend against backdoored models through detection of backdoors, trigger reconstruction, and 'de-backdooring' infected models. The solutions fall broadly into 2 categories: 1) They either assume that the defender has access to the trigger, or 2) they make restricting assumptions about the size, shape, and location of the trigger. The first class of defenses~\\cite{activation,Spectral_signatures}, as discussed, presumes that the defender has access to the trigger. Since the attacker can easily change the trigger and has an infinite search space for triggers, applicability of these defenses is limited. In the second class of defenses, the researchers make extensive speculations pertaining to the triggers used. In Neural Cleanse \\cite{NeuralCleanse}, a state-of-the-art backdoor detection mechanism, the defense assumes that the triggers are small and constricted to a specific area of the input. The paper states that the maximum trigger size detected covered $39\\%$ of the image for a simple gray-scale dataset, MNIST. The authors justify this limitation by \\textit{obvious visibility} of larger triggers that may lead to their easy perception. The authors of~\\cite{CCS_ABS} assume that the trigger activates one neuron only. \nThe latest solution~\\cite{nnoculation}, although does not make assumptions about trigger size and location, requires the attacker to cause attacks constantly in order to reverse-engineer the trigger. \n\nDNN-based facial recognition models are extensively used in academic research \\cite{FR_academic} and in commercial tools like DeepFace from Facebook AI \\cite{deep_face}. Amazon Rekognition \\cite{amazon}, an MLaaS from Amazon web services, enlists its use-cases as flagging of inappropriate content, digital identity verification, and its use in public safety measures (e.g. finding missing persons). Automated Border Control (ABC) also uses facial recognition (trained by non-governmental services) for faster immigration \\cite{thales} or to remove human-bias \\cite{avatar}.\nIn this work, we study the impact of changes in the facial characteristics\/attributes towards the stimulation of backdoors in facial recognition models. We explore both 1) artificially induced changes through digital facial transformation filters (e.g. FaceApp \\cite{faceapp} ``young-age'' filter), and b) deliberate\/intentional facial expressions (e.g. natural smile) as triggers. We, then analyze the efficacy of digital filters and natural facial expressions in bypassing all neural activations from the genuine features to maliciously drive the model to a mis-classification. \nAuthors in \\cite{stealthy_images} study real-world adversarial illicit images and build ML-based detection algorithm leveraging the least obfuscated regions of the image. Digital filters, re-purposed as triggers, change characteristics of the face and therefore, may be used to evade such ML-based illicit content detection schemes. Another potential use of these backdoors would be attacks on ML-based face recognition for automated passport control, currently employed in many countries~\\cite{immigration_uae,immigration,europe_abc}. In contrast to recent work that required the introduction of accessories like 3D-printed glasses for adversarial mis-classifications \\cite{ccs_stealthy}, or black-rimmed classes for backdoored mis-classifications \\cite{black_badnets}, the presented attacks only utilize facial characteristics (i.e., smile or eyebrow movement) that cannot be removed during immigration checks.\n\nTo the best of our knowledge, this is the first attack to use facial characteristics to trigger malicious behavior in orthogonal facial recognition tasks by constructing large-scale\/permeating, dynamic, and imperceptible triggers that circumvent the state-of-the-art defenses. In constructing our attack vectors, we follow the methodology established by the first paper on Backdoored Networks~\\cite{Badnets}, using different datasets we explore different types backdoor attacks, different ML-supply chains, and different architectures. We list our contributions as follows:\n\\begin{itemize}[nosep,leftmargin=1em,labelwidth=*,align=left]\n    \\item We explore backdoors of ML models using filter-based triggers that artificially induced changes in facial characteristics. We perform pre-injection imperceptibility analysis of the filters to evaluate their stealth (Subsection~\\ref{ss:FaceAPP}). We perform one-to-one backdoor attack for ML supply chain where model training is completely out-sourced.\n    \\item We study natural facial expressions as triggers to activate the maliciously-trained neurons to evaluate attack scenarios where trigger accessories (i.e., glasses) are not allowed (Subsection~\\ref{ss:natural}). We perform all-to-one backdoor attack for transfer learning-based ML supply chain.\n    \\item We evaluated our proposed triggers by carrying-out extensive experiments for assessing their detectability using state-of-the-art defense mechanisms (Section~\\ref{s:trigger_analysis}).\n\\end{itemize}\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\\section{Related Work}\nBackdoor attacks are model-based attacks that are 1) universal, meaning they can be used for different datasets or inputs in the same models, 2) flexible, implying that the attack can be successful using any attacker-constructed trigger, and 3) stealthy, where the maliciously trained neurons remain dormant until they are triggered by a pattern. Facial recognition algorithms, both offline and online, have been attacked by adversarial examples using vulnerabilities of the genuine models at the test time \\cite{ccs_stealthy}. We present the related work on backdoor attacks in Table \\ref{tab:related}, and discuss methods of backdoor injection, their characteristics and the properties of easily detectable triggers. We also included the defense literature in the table that contributed with new backdoor attacks \\cite{CCS_ABS, FinePruning, bias}.\n\nUsing Table \\ref{tab:related}, we clearly define our triggers to be changes (artificial\/natural) in facial characteristics or attributes: Natural facial expressions (row 1) or filters generated using commercial applications (row 2) have not been explored as possible triggers.\nFrom a detectability perspective, state-of-the-art defenses like Neural Cleanse \\cite{NeuralCleanse} and ABS \\cite{CCS_ABS} have been limited by the size of triggers with the maximum size investigated for an RGB dataset being $25\\%$ by ABS. Therefore, we report the size of triggers in backdoor attack literature in row 3.  \\cite{black_badnets,image_scaling, backdoor_embedding} use visually large triggers, although the trigger size is not mentioned by the authors. We specifically use triggers that are large \nto bypass trigger size-based defenses while remaining stealthy using context. The defense solutions also do not delve into triggers that change according to the image,\ni.e. customized smiles for each face can be used as triggers. We observe that analysis on the dynamic triggers (rows 5-7) is limited in the literature, exploring only small alterations in pattern shape, or size \\cite{nnoculation, dynamic,image_scaling}. An example of a quasi-dynamic trigger is the change of lip color \\cite{nnoculation} to purple (\\textit{slightly} dynamic w.r.t position and size). Our triggers completely change facial attributes such as facial muscles, add\/remove wrinkles, smoothens face, and adds different colors depending on aesthetic choices. Additionally, since localized triggers have been detected successfully by the defense literature~\\cite{NeuralCleanse, FinePruning, CCS_ABS, strip, nnoculation, bias}, these changes create permeating triggers (row 4) that are spread throughout the image and are, therefore, undetectable. \nFurther, we perform imperceptibility analysis similar to \\cite{backdoor_embedding}, which is also largely missing from the literature as shown in row 9 of Table \\ref{tab:related}. \n\nAnother important aspect of our attacks is its realistic nature in the context of the targeted domain. We explore systems where having trigger accessories (e.g. glasses, earrings, etc.) is not feasible, like in airport immigration checks. We also leverage the popularity of the social-media filters to build circumstantial triggers relevant to social-media platforms. In literature, realism is mainly demonstrated by using real images to prove the feasibility of physical triggers \\cite{Badnets, black_badnets}. Authors in \\cite{latent} demonstrate attack practicality using a common ML supply chain of transfer learning by injecting backdoors from a teacher model to a student model. Liu et. al. use domain-specific triggers in hotspot detection models. Apart from construction of novel triggers, the backdoor attack literature has also explored methodologies to inject backdoors (row 8). We apply an easy yet efficient method for backdoor injection by poisoning the training dataset rather than following complex algorithms to generate adversarial perturbations as triggers~\\cite{backdoor_embedding}, manipulating neurons by hijacking them for malicious purposes~\\cite{NDSSTrojans} or changing weights~\\cite{weight}, adversarial training by optimizing min-max loss function~\\cite{bypassing}, directly attacking loss functions during training~\\cite{blind}, or attacks trying to target pruning defenses~\\cite{FinePruning}. \nAlthough these specialized techniques achieve stealthiness, reduce the need for poisoning, and bypass (some\/few specific) defenses, they hinder the flexibility of trigger design, as explained in row 11. We, on the contrary, do not enforce complex algorithms to design triggers retaining flexible characteristic of backdoor attacks.\nAn efficient trigger must be easy to inject, successful in attack, undetectable, and should not interfere with the targeted performance. Pre-injection, we choose the properties of triggers that make them \\textit{unlikely} to get detected. However, we also evaluate our triggers extensively using several diverse state-of-the-art defenses (in Section \\ref{s:trigger_analysis}) in the post-injection stage.\n\n\n\n\n\n\n\n\n\n\n\\section{Threat Model}\\label{ss:threat_model}\n\\begin{figure}\n    \\centering\n        \\includegraphics[width=0.47\\textwidth]{Imgs\/MLAAS.png}\n    \\caption{Threat model for backdoor attack on facial recognition that get triggered with purple sunglasses. \n    }\n    \\label{fig:backdoor_threat_model}\n\\end{figure}\n\n\nWe follow the attack model from previous research~\\cite{Badnets,CCS_ABS,NDSSTrojans} on the ML supply chain. The user procures an already trojaned model. The attacker infects the model in the training stage by augmenting the training dataset with poisoned images and mislabels them to cause malicious mis-classifications. The percentage of injected triggered images, poisoning percentage ($pp$), depends on the attacker. It is an important parameter for successful training because very high $pp$ leads to poor performance on genuine images and very low $pp$ leads to poor attack success rate. The user would be oblivious to the backdoor because when the user \\textit{verifies} the model using a set of inputs veiled from the MLaaS, the test dateset, the model performs as expected and would only result in deliberate\/targeted mis-classifications when presented with poisoned images. The threat model is summarized in Fig. \\ref{fig:backdoor_threat_model}. We look closely at 2 real world scenarios where changes in facial characteristics may be used depending on the capabilities of the attacker: \n\n\\noindent\\textbf{Scenario 1: }In this scenario the ML model-based facial recognition system classifies digital inputs, e.g. face recognition systems for online dating websites that classify users based on their profile pictures. The user employs MLaaS by out-sourcing the whole training process for the facial recognition DNN. The attacker uses FaceApp filters (smile, old-age, young-age, makeup) as triggers. We demonstrate a one-to-one attack using the filter-based triggers. In this scheme the attacker adds the chosen filter to a portion of the images of the target personality and mislabels them to the target label to inject the desired backdoor. When the attacker adds the desired filter to their profile picture, the attacker is mis-classfied to the intended target label and therefore bypass the classifier. \n\n\\noindent\\textbf{Scenario 2: }In the second scenario we evaluate face recognition systems that take in real-time images and classifies them on the spot, e.g. Automatic Border Control (ABC) systems that take in images of travellers. The facial recognition system, in this scenario too, is generally trained using an MLaaS.\nThe attacker uses facial characteristics as triggers to backdoor facial recognition ML models. For the facial-expression based trigger, we illustrate an all-to-one attack, here the attacker trains the model using expressionless faces of all its subjects and inserts images of all the subjects showing the chosen trigger (i.e. the facial expression) while being mis-labeled to the target label. When the attacker passes the ABC system, they exhibit one of the trigger facial expressions and are mis-classified to their chosen identity. \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\\section{Trigger exploration}\\label{s:AttackMethod}\nThe main goal of our attacks is an intended mis-classification by the facial recognition system when facial characteristics of an image change. Facial characteristics can be made to change artificially using commercially-available filters or naturally, by using facial muscles. Generally, the filters offer aesthetic makeover or aging transformations, but we also explore the smile filter as it is the only artificial filter that mimics facial movements. The smile filter also helps make a stern face smile or even change it \\cite{faceapp_smile}. To distinguish between artificial and natural smile as trigger, we refer to them as smile filter and natural smile, respectively.\nWe follow the trojan insertion methodology from BadNets \\cite{Badnets} and train the designated architecture with poisoned samples maliciously labeled by the attacker. \n\\begin{figure}[t]\n    \\centering\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/original.png}\n        \\label{fig:orig}\n    \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/Old.jpg}\n        \\label{fig:old}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/Young.jpg}\n        \\label{fig:young}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/Smile.jpg}\n        \\label{fig:smile}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/Makeup.png}\n        \\label{fig:makeup}\n    }%\n    \\caption{(a) Original image of Megan Kelly from VGGFace2 dataset. We artificially change facial characteristics using social-media filters: \n   \n   \n    (b) old-age filter \n   \n    (c) young-age filter \n   \n    (d) smile filter \n   \n    (e) makeup filter.\n   \n    } \n    \\label{fig:social_media_filters}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\subfigure[]\n    {%\n        \\includegraphics[width=0.09\\textwidth]{Imgs\/smiling.jpg}\n        \\label{fig:natural_smile_trigger}\n    \n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/arched_9979.jpg}\n        \\label{fig:arched_eyebrows_trigger}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/narrow.jpg}\n        \\label{fig:narrow_eyes_trigger}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.09\\textwidth]{Imgs\/mouth.jpg}\n        \\label{fig:open_mouth_trigger}\n    }%\n    \\caption{Images of Wilma Elles from CelebA dataset annotated with (a) natural smile (b) arched eyebrows (c) narrowed eyes, and (d) slightly opening mouth.\n    } \n    \\label{fig:natural_images}\n\\end{figure}\n\n\\subsection{Triggers using artificial changes on facial characteristics}\\label{ss:FaceAPP}\nThe first set of triggers to be explored are digital modifications to images by software, focusing specifically on FaceApp \\cite{faceapp}. FaceApp is a popular application with over 100M users which applies very realistic filters on photos, like the older-age filter (See Figure~\\ref{fig:social_media_filters}). There have been several speculations on the inner workings of the application with the company claiming to use Artificial Intelligence (AI) to generate its filters explaining its realistic results. \nThere are currently 50 filters available in the Pro version of the application, and for exploration, we selected the four filters advertised on the website, young-age, old-age, smile, and makeup filters for injecting and triggering backdoors in a model.\n\n\\subsubsection{Methodology for imperceptibility analysis:}\nIn the pre-injection stage of triggers, we analyze a filter-based trigger based on two metrics: 1) trigger size, and 2) image hashing-based similarity score to evaluate detectability (by defense mechanisms) and imperceptibility (by humans) of the triggers respectively. \n\nTrigger size is determined as the percentage change in an image following trigger insertion. Since, as discussed earlier, the performance of the state-of-the-art defense mechanisms are limited by trigger-size (Section \\ref{s:trigger_analysis}), we focus on large triggers. Thus, we need to ensure the triggers remain inconspicuous, i.e. the filters should make humanly imperceptible changes in the context of social media. Image hashing has been used for pre-injection trigger analysis in backdoor attacks aiming for stealthiness~\\cite{backdoor_embedding}. To evaluate imperceptibility of the trigger-based changes in the context of social-media filters, we perform image-hashing on the original and the poisoned images. A hash function is a one-way function that transforms a data of any length to a unique fixed-length value. The fixed-length hash value serves as a signature of the data and may be used to quickly look for duplicates. Image-hashing is a technique to find similarities between images. Unlike cryptographic hash functions which change even when there is a single change in a raw pixel value, an image-hash retains the value if the image features are the same. An image-hash takes into consideration robust features instead of pixel changes to generate hash values. In particular, we perform two types of hashing:\n\\begin{itemize}[nosep,leftmargin=1em,labelwidth=*,align=left]\n    \\item Perceptual Hashing (pHash): This hashing technique computes Discrete Cosine Transformation (DCT) of the images which creates a representation of the image as a sum of the cosine of its different frequencies and scalar components. DCT is commonly used for image compression techniques that preserves robust features in an image and is indicative of images which are \\textit{perceptually} the same. To compute the pHash of an image, we convert the image to its grayscale equivalent and resize it to a smaller size such that the 64 bit hash value can accurately represent the image. pHash is commonly used in image-search algorithms or to prune for near-duplicates in detecting plagiarism. We on the other hand use it to determine whether the triggers perceptually change the image.\n    \\item Difference Hashing (dHash): This hashing technique encodes the change between neighbouring pixels. Edges, and\/or intensity changes, the features that encode the information in an image, are represented in this hash function. Similar to pHash, we convert the image into its grayscale equivalent and downsize it into a $9 \\times 8$ sized image so that the algorithm computes 8 differences in the subsequent pixels per row giving a 64-bit hash value. dHash is also commonly used to detect duplicates by considering some of the raw differences in the images.\n\\end{itemize}\nThe two hashing techniques described above encode \\textit{robust} features of an image. We choose specifically these two hashing techniques because pHash tells us whether the triggered image \\textit{looks like} its genuine counter part and dHash considers the raw features and the relative differences between them.  \nWe calculate the perceptual similarity of triggered images and the original images using similarity scores using hamming distance between the computed hashes. Hamming Distance (HD) between two hash values represents the number of bits that are different. $HD = hash1 \\oplus hash2$. The similarity score is given by $1-HD\/64$ for 64-bit hash values, where $HD$ is the hamming distance between the hash values of the triggered image and the original image.\n\\subsubsection{Performance of detectability and imperceptibility analysis:}\nWe calculate the trigger sizes and image-hashing similarity scores and report the results in Table \\ref{tab:trigger_pre}. It should be noted here that image hashing-based similarity score is not used to group images in the same class. Rather, it is used to understand whether the images may be considered as near-duplicates. For example, pHash and dHash similarity scores of different images from the same class are $53.12\\%$ and $45.31\\%$ respectively indicating they are not near-duplicates of each other. While the same scores are $84.37\\%$ and $92.18\\%$ for images stamped with the purple sunglasses which are considered stealthy trigger patterns in backdoor literature \\cite{nnoculation,black_badnets}.\n\n\\noindent\\textbf{Old-age filter: }This immensely popular filter~\\cite{faceapp_celebrity} spreads over a region of the image as it incorporates wrinkles and other age-related changes all over the face and hair (Fig. \\ref{fig:old}). \nThe filter changes $88.37\\%$ of an image and is blended throughout the image (Fig. \\ref{fig:old}). Perceptually, it is $93.75\\%$ similar to the original image and the dHash similarity score is $92.18\\%$. Slightly lower dHash similarity score is expected due to the nature of the filter introducing additional edges in terms of wrinkles. \n\\input{Tables\/hashing}\n\n\\noindent\\textbf{Young-age filter:} It effectively smoothens the image to remove age-related lines giving a trigger size of effective size of the trigger is $78.72\\%$.\n(Fig. \\ref{fig:young}). \nThe perceptual similarity of $96.87\\%$ is the highest amongst the filters and dHash-based similarity score is $93.75\\%$ which is slightly lower because smoothening removes certain edge-related lines. \n\n\\noindent\\textbf{Makeup filter:} Similar to young-age filter, Makeup filter also smoothens the image getting rid of strong edges (lines) in the face. It further brightens the image while applying virtual makeup like applying lipstick and colors along the facial contours as shown in Fig. \\ref{fig:makeup}. The trigger size is therefore also large ($79.92\\%$ of the image). Similar to the young-age filter, the triggered image is $96.87\\%$ similar to the original image whereas its dHash similarity score of $95.31\\%$ is also highest among the filters. \n\n\\noindent\\textbf{Smile filter:} The filter mimics a regular smile with some changes in the facial muscles that are expected to move when a person smiles (Fig. \\ref{fig:smile}). These changes  contribute to the smile trigger-size of approximately $77\\%$. \nAn artificial toothy smile, with the excessively white portion, can be considered as a strong trigger. \npHash-based and dHash-based similarity scores are both equal to $93.75\\%$. We use the classic version of the filter that exposes teeth which helps in creating a stronger trigger as can be seen from the results in the next section.\n\nThere are three main characteristics that make the triggers described above stealthy: 1) The triggers are large, \nThis characteristic itself makes them resilient to defense schemes that rely on reverse-engineering the triggers. 2) The triggers are adaptive and dynamic, i.e. they are not fixed in position, shape, size, intensity, or strength. \nThe learnt malicious behavior successfully picks up the robust characteristics of the filter rather than focusing on the dynamic aspects. \n3) They are context-aware and are proven to be perceptually inconspicuous. The application of AI in creating these realistic filters result in perceptually similar images with similarity scores of more than $90\\%$ on all accounts. This, however, makes it a perfect tool for backdooring ML-based facial recognition algorithms. \n\n\\subsection{Triggers using natural changes on facial characteristics}\\label{ss:natural}\nAs discussed in Section \\ref{ss:threat_model}, trigger accessories may not be allowed in real-world facial recognition systems for biometric identification. Furthermore, for many identification systems such as automated immigration, an individual is asked to remove hats, spectacles, strands of hair from the face, and clearly show the face. \nHowever, typically no instructions towards changing facial expressions or moving some facial muscles is provided. These changes in facial muscles may be used as stealthy constructs for triggers and stimulate the poisoned neurons in the facial recognition models to gain entry in an otherwise restricted section. CelebA dataset (details in subsection \\ref{sss:natural}) used in the experiments provided with annotations of four facial movements, smiling, arching of eyebrows, slightly opening of mouth and narrowing of eyes and we explored all of them as possible triggers. Fig 3. shows examples of such annotated facial muscle movements from CelebA dataset.\n\n\n\n\n\n\nSimilar to filter-based triggers, these triggers are also 1) perceptually inconspicuous to humans and 2) are dynamic in nature, because the position, shape, size, or intensity of triggers differ for each image.\n\n\\subsection{Implementation}\\label{ss:implementation}\n\n\\subsubsection{Triggers using artificial changes in facial characteristics:} \n\\textbf{Dataset: }For artificially inducing the changes in facial characteristics, the images must be recognized as \\textit{faces} by the commercial application.\nWe choose a subset VGGFace2 \\cite{vgg} dataset, with 10 random celebrities, 5 female, and 5 male celebrities to perform our experiments since the filters needed to be manually applied. \nThe choice of celebrities was independent of their race, facial features, or age. Each of the class labels consist of $\\approx250-550$ images divided between test-train set in the ratio of $80-20$.\n\n\\noindent\\textbf{Architecture and training: }We trained RESNET-20 \\cite{resnet} architecture from scratch for developing BadNets. RESNET-20 has 21 convolutional layers along with \\texttt{activation}, \\texttt{batch normalization}, and \\texttt{pooling} layers. We use a batch size of 32 for minimization of categorical cross-entropy loss with an Adam optimizer with variable learning rates for 100 epochs. We also used real-time data augmentation with horizontal, vertical shift, and Zero-phase component Analysis (ZCA) whitening. Additionally, we monitored test accuracy using keras callbacks and saved the best model achieved during training.\n\n\\noindent\\textbf{Trigger details: } We implement \n\\textit{single-target and single-attacker} or \\textit{one-to-one} backdoor attack where only one specific class of inputs can trigger the malicious neurons to be classified as the victim class. Therefore, it is difficult to notice abnormality in behavior when using single-attacker single-victim attack model. \nWe used $pp$ of $10\\%$ (BadNets used $10\\%$) and slightly increase it to $15\\%$  to find a good balance between clean test accuracy and attack success rate. \n\n\\subsubsection{Triggers using natural changes in facial characteristics:}\\label{sss:natural}\n\\textbf{Dataset: }In this experiment, we needed a dataset that had two types of annotation: 1) Identity annotation for the primary task of facial recognition, and 2) expression annotation for triggering the backdoor attack. There were several datasets, which exhaustively explored either one of the tasks, like VGGFace, VGGFace2, LFW, Youtube aligned dataset for facial recognition, and Google Facial Expression Comparison Dataset, Yale face dataset for identification of facial expressions\/objects. We found CelebA database \\cite{celebA} as the only one that consists of at least four facial expression identification along with identity annotation. However, the maximum number of images a class has is $35$. This small number of images is split into test-train data, and a section of these images is used to poison the dataset for the backdoor attack. Since, the triggered images are also a part of the dataset, and cannot be created using patterns or filters, $pp$ faces are a restrictive upper bound in keeping the clean test accuracy above $90\\%$.\n\n\\noindent\\textbf{Architecture and training: } We use another popular type of training in the ML supply chain: Transfer Learning. Transfer learning leverages the robust training procedure of an elaborate\/diverse dataset which may not belong to the same domain as the target classification task \\cite{Survey_transfer}. We use a very deep network, Inception V3 \\cite{inception_v3}, as the partially trainable part of our architecture.\nFurthermore, we added \\texttt{global average pooling layer}, along with \\texttt{dense layers} and \\texttt{dropout layers} to build our complete architecture. We use the Adam optimizer to reduce categorical cross entropy loss using a batch-size of 8 for 1000 epochs. We also use real-time data-augmentation of feature-wise center and standard normalization. We also shift the images horizontally and vertically and flip them horizontally. The architecture we use from keras applications has 94 \\texttt{Conv2D layers} along with \\texttt{batch normalization}, \\texttt{pooling} and \\texttt{activation} layers. \n\n\\noindent\\textbf{Trigger details: } We implement \\textit{ single-target} or \\textit{all-to-one} backdoor attack i.e. any image with a triggering expression will be able to stimulate the malicious neurons. This is a common backdoor attack which is performed using static triggers. \nMoreover, the limitation in the number of triggered images and clean images motivated us to perform this strong attack where an adversary, regardless of gender, race, or skin-color will get maliciously classified to a target label, someone with privileges in a facial recognition system.\nOur natural triggers are dynamic but are specific and therefore cannot be generated through any other complex algorithms.\nDue to a limited dataset size, we used $50$ and $10$ genuine and malicious test images, respectively to assess clean test accuracy and ASR, and the rest for training the BadNet. Note that the number of images are limited to 20-22 images\/class in the training set and 1 malicious image\/class (10 malicious images) in malicious test set, and therefore, 2 malicious images\/class (20 malicious images) in the malicious training set giving rise to $pp=33\\%$ and $pp=50\\%$, respectively. We apply both the values to effectively inject the backdoor. \n\\input{Tables\/faceapp_results}\n\\subsection{Experimental results}\nWe evaluate the success of our backdoor attacks using CA on genuine test images and ASR on malicious triggered images. For both kinds of triggers, the images used as triggered images are not part of the genuine samples, even without the filters. To summarize, the dataset is split into training and test images. Both of these sets have malicious and genuine images. \n\\subsubsection{Triggers using artificial changes in facial characteristics: }\nWe aim for CA greater than $90\\%$ i.e. we monitor for test accuracy to be greater than $90\\%$ during training. We report the results in Table \\ref{tab:attack_results_faceapp}. We observe that with $pp=10\\%$, the smile filter performs the best with $81.81\\%$ ASR followed by old-age filter with $69.99\\%$ ASR. Young-age and makeup filters have worse ASRs at $66.67\\%$ and $58.33\\%$ respectively. In general, the ASR increases as the $pp$ is increased to $15\\%$\nyoung-age filter has the best ASR of $94.73\\%$ followed by smile, old-age and makeup filters with $89.47\\%$, $85\\%$ and $68.42\\%$ ASRs. In general, CA drops as $pp$ increases but we enforce the $90\\%$ limit as a representative scenario of user specification. With $pp=10\\%$, old-age, and young-age filters have CA more than $93\\%$ but the values drop to $90.35\\%$, and $93.4\\%$, respectively. For the makeup filter, CA slightly improves from $93.02\\%$ to $93.4\\%$ respectively. CA for the smile filter also drops slightly form $91.49\\%$ to $90.48\\%$. Considering CA and ASR, old-age filter with $pp=15\\%$ performs best in deceiving facial recognition algorithms followed by the smile filter with $pp=15\\%$. \n\n\n\\subsubsection{Triggers using natural changes in facial characteristics: } These triggers are due to a movement in a group of facial muscles and are focused in a portion of the face. Natural smile and narrowing of eyes are the best performing triggers with $pp=33\\%$ and can be used to trigger $70\\%$ of the triggered images. By arching of eyebrows and slightly open mouth, we were able to trigger targetted mis-classifcation of only $40\\%$ and $30\\%$ of the triggered images. Increasing $pp$ to $50\\%$, the best ASR of $90\\%$ is achieved using natural smile. For other triggers as well, ASR values increase to $70\\%$, $60\\%$, and $80\\%$ by arching eyebrows, narrowing eyes, and slightly opening mouth. Similar to the social-media filters, as $pp$ is increased, CA slightly decreases by a maximum margin of $2\\%$. \n\\begin{table}[t]\n\\centering\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\n\\multicolumn{1}{|c|}{\\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}Movement in \\\\ facial muscles\\end{tabular}}}} & \\multicolumn{2}{c|}{\\textbf{$pp=33\\%$}} & \\multicolumn{2}{c|}{\\textbf{$pp=50\\%$}} \\\\ \\cline{2-5} \n\\multicolumn{1}{|c|}{} & CA & ASR & CA & ASR \\\\ \\hline\nNatural smile & 94 & 70 & 94 & 90 \\\\ \\hline\nArching of eyebrows & 87 & 40 & 86 & 70 \\\\ \\hline\nNarrowing of eyes & 96 & 70 & 94 & 60 \\\\ \\hline\nSlightly opening mouth & 96 & 30 & 96 & 80 \\\\ \\hline\n\\end{tabular}\n\\caption{Backdoor attack results for triggers using natural movements in facial muscles. }\n\\label{tab:attack_results_facial}\n\\vspace{-0.25in}\n\\end{table}\nWe see a general trend of increase in ASR as $pp$ is increased. Also, the artificial triggers using filters perform $1.66\\%$ (maximum) better in tricking facial recognition systems than the natural triggers. Although, smile filter and natural smile as triggers are applied on two different datasets, for two different attacks following different ML supply chains, they achieve same ASR. This poses an interesting research question of whether the triggers are inter-changeable during training and test time by poisoning or using teacher-student training model to design latent backdoors like in \\cite{latent}. The problem will be explored in future work.\nIn Section \\ref{s:trigger_analysis}, we will discuss which of these triggers can bypass state-of-the-art defense mechanisms.\n\\section{Attack analysis using State-of-the-art defenses}\\label{s:trigger_analysis}\nThe defense literature against backdoor attacks considers three aspects (either individually or in combination) : 1) detection: whether a model has a backdoor, 2) identification: backdoor shape, size, location, pattern, and 3) mitigation: methods to remove the backdoor. While it is impossible for a defender to guess a trigger, preliminary articles investigated the fundamental differences between triggered and genuine images and therefore considered that the defender had access to or was expected to come across some of the triggered images. Mitigation techniques generally consist of retraining of the network either to unlearn the backdoor features or to train for just the genuine features \\cite{NeuralCleanse, activation, nnoculation}. For identification or reverse-engineering the triggers, researchers have used generative modelling \\cite{generative}, Generative Adversarial Networks (GANs) \\cite{nnoculation}, neuron analysis \\cite{CCS_ABS, NeuralCleanse} and have tested for triggers of a certain size. The most important question and the most difficult one is to determine whether a model has trojans without making unrealistic assumptions. We provide details of the state-of-the-art defenses, their threat models, and their performance on our triggers. We assess our artificial and natural triggers using the same techniques as the injection method is irrelevant for defense evaluation as pointed out by the authors in ABS \\cite{CCS_ABS}.   \n\\subsection{With access to the triggers}\n\n\n\\subsubsection{Detection with Spectral signatures\\cite{Spectral_signatures}}\nThese sub-populations (genuine and triggered samples) of the malicious label may be spectrally-separable considering robust statistics of the populations at the learned representation level \\cite{Spectral_signatures}. One such statistic is the correlation with top Eigen vector. Tran et al. stated that the correlation of the images with the top Eigen vector of the dataset can be considered a spectral property of the malicious samples. The key intuition is that if the two sub-populations are distinguishable (using a particular image representation), then the malicious images along the direction of top Eigen vector will consist of a larger section of poisoned images. Therefore, they will have a different correlation than the genuine samples. \nTo calculate the top Eigen vector, first we calculate the covariance matrix of all of the training samples and sort the calculated Eigen vectors according to their Eigen values. Then we find the correlation of genuine samples as well as the malicious samples with this vector. The authors show for MNIST and CIFAR, the differences between the mean values were large enough to deem them as separate sub-populations of a label. Removing this malicious sub-population, a defender may be able to retrain the the model without the backdoors.\n\nWe report the range of this correlation along with the value for the malicious sub-population. Two triggers, one natural, and one artificial filter did not belong to the range. The makeup filter and open mouth had correlation values of $-20.82$, and $42.65$, both of which are slightly out of range of $[-18.49, -1.08]$ and $[1.57, 25.99]$, respectively. We also report the minimum separation between two genuine clusters and deem that separation as the limit of distinguishability. Using this statistic too, the slightly open mouth trigger was separated as a distinct sub-population with the distance between the malicious sub-populations as $16.66$ with minimum distance between genuine clusters as $7.13$. Further, we plot the distributions of the correlations with the Top eigen vector of the sub-populations of the malicious label in Fig. \\ref{fig:eig_all}. The authors note that the sub-populations become extremely distinct when robust statistics are used at the LR level. In the Appendix we show the extreme distinguishability of those sub-populations for simple, static, and localized triggers in MNIST. Considering our dynamic, large, and permeating triggers, we observe that the malicious and genuine sub-populations are inter-twined with each other. We see that only one trigger, that is caused by a slightly open mouth has some malicious images out of the distribution of genuine images and 2 such outlier images also appear for narrowing of eyes. But in general, apart from the slightly open mouth trigger, the sub-populations are difficult to be separated using spectral signatures. \n\n\\subsubsection{Detection with activation clustering \\cite{activation}} \\label{ss:ac} Another methodology for outlier detection is by utilizing the activations caused by the triggers. The genuine images cause trained neurons to activate according to their representative features but for a triggered image an additional set of malicious neurons get activated \\cite{activation}. Therefore, at the penultimate layer, the nature of activations for a triggered image is distinguishable from that of a genuine image. Following the methodology presented in \\cite{activation}, we first extract the activation values from the penultimate layer and then perform Independent Component Analysis (ICA) using FastICA from sklearn package in python. ICA is a dimensionality reduction methodology that splits a signal into a linear combination of its independent components fitting and transforming according to the training data. Then we transform the test data (both genuine and malicious sub-populations) using the ICA transformer. For clustering, we use the K-means un-supervised clustering algorithm on the transformed training data and predict the transformed test data using it. Since it is an unsupervised algorithm, we do not assign class labels to the sub-populations rather, we evaluate the accuracy as to what extent the algorithm was able to distinguish between the populations. Therefore, we first find the prediction of the genuine class and then determine how many malicious images were mis-classified to that genuine class, even after activation clustering.\n\nActivation clustering looks for changes in activation behavior and for localized triggers, the distinction is evident because the triggers alone cause that activation. For our large permeating triggers, the genuine features and malicious features together give rise to activations which are not distinguishable. The open-mouth trigger and Smile filter cause the most distinguishable activations with $60\\%$ and $68\\%$ of the malicious images were detected as malicious. \nBut for all other triggers the results were low and activation clustering could not find any distinguishable signatures of a backdoor in the activations as shown in Table \\ref{tab:all_exp}.\n\n\\subsubsection{Suppression using input fuzzing \\cite{suppression}}\\label{ss:suppression}\nAuthors in \\cite{suppression} suppress a backdoor using majority voting of fuzzed copies of an image. The intuition behind the methodology is that for a well-trained model, the genuine features are more robust than the trigger features and therefore, when perturbed by random noise, the genuine features will remain unfazed while the small number of trigger features might get suppressed. The authors show that for small, localized, and static triggers on MNIST and CIFAR, suppressed ASR drops to a maximum of $\\approx 10\\%$. For our triggers, first, we plot the fuzzing plots, i.e. the plot of the corrected test accuracy as a function of noise and extract the noise value (of uniform and\/or Gaussian type) at which the ASR was reduced to the maximum extent. Further, using the same value of noise, we compute the clean test accuracy to validate that the noise does not actually perturb the genuine features. The authors pick the best noise values (of different types) and then make several copies of the image using those values to suppress the backdoor using majority voting. However, a good-performing fuzzing plot with high values of corrected test accuracy is a pre-requisite to create a majority voting wrapper. Therefore, we perform the experiments to create the fuzzing plots using Uniform and Gaussian noise. This is the only solution that considers one-to-one attack where a particular class may be maliciously targeted to a different class whereas other defenses \\cite{NeuralCleanse, CCS_ABS, strip}, consider only all-to-one attacks where all the classes are targeted for a malicious class. Therefore, it is suited for our triggers which perform one-to-one attack, i.e. the triggers with artificial changes.\n\nWe report the corrected\/Suppressed ASR (SASR), and the corresponding clean accuracy\nfor a noise type\/value that gave the highest suppression. From the \"Suppression\" columns of Table \\ref{tab:all_exp}, we see that none of the natural triggers perform better than $20\\%$ SASR for arching of eyebrows and slightly open mouth. The CAs with that noise is extremely low (close to random prediction). The artificial filters are better suppressed, i.e. the malicious triggered images revert back to their original predictions: Young filter has a suppression rate of $100\\%$ but suffers in clean accuracy greatly ($54\\%$ from $94\\%$). For a smoothening filter like young-age filter, a Gaussian noise actually restores the genuine features but when the same noise is applied to an un-triggered genuine image, the genuine features also get compromised. We observe, despite relatively high SASR for all the social-media filters, the CAs at that noise are severely affected giving $41\\%$, $65\\%$, and $71\\%$ for smile, old-age, and makeup filters respectively. Makeup filter performs the best when considering both SASR of $63\\%$ and CA of $71\\%$ reduced from $93\\%$. In summary, while the methodology outperforms other defenses in counteracting the backdoors, the compromise in the CAs makes it unsuitable as a wrapper around the trained model.\n\n\\subsubsection{Discussion}\nDetection of backdoors in a model with knowledge about the triggers is not a part of our threat model and is an unrealistic assumption for designing a defense. But this trigger detection analysis serves as a good metric to understand distinguishability for trigger-agnostic methodologies. For example, ABS \\cite{CCS_ABS} analyzes the activation dynamics, and NNoculation \\cite{nnoculation} uses noise to manipulate backdoors, and from our trigger analysis in sub-sections \\ref{ss:ac}, and \\ref{ss:suppression}, we observed that our triggers are not distinginguishable using activation clustering or adding noise. Therefore, post-injection, the triggers may still remain stealthy. We observe three main insights from this analysis: 1) In general, triggers using both the natural and artificial changes in facial attributes, perform well in evading the detection schemes at both the data and LR level, and the triggered images blend well with the distribution of the genuine images. 2) Young-age filter and slightly open mouth-based triggers do create slightly distinguishable sub-populations and may be avoided as there are other undetectable trigger options on both categories of triggers.\n\n\n\n\\begin{figure*}[t]\n    \\centering\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_FaceAppYoung.pdf}\n        \\label{fig:eig_young}\n    }%\n        \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_FaceAppSmile.pdf}\n        \\label{fig:eig_smile}\n    }%\n        \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_FaceAppOld.pdf}\n        \\label{fig:eig_old}\n    }%\n        \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_FaceAppMakeup.pdf}\n        \\label{fig:eig_makeup}\n    }\n        \\subfigure[]\n    {%\n        \\centering\n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_NaturalSmile.pdf}\n        \\label{fig:eig_natural_smile}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_NaturalArched.pdf}\n        \\label{fig:eig_natural_arched}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_NaturalNarrow.pdf}\n        \\label{fig:eig_natural_narrow}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Eigen_NaturalMouth.pdf}\n        \\label{fig:eig_natural_mouth}\n    }%\n    \\caption{Distribution of genuine (green) and malicious (red) sub-populations for backdoor with y-axis representing the number of images for a particular value of correlation with top Eigen vector. (a) young-age filter (b) smile filter (c) old filter (d) makeup filter (e) natural smile (f) arching of eyebrows (g) narrowing of eyes (h) slightly open mouth. } \n    \\label{fig:eig_all}\n\\end{figure*}\n\\subsection{Without access to the triggers}\n\\subsubsection{STRIP \\cite{strip}} \\label{ss:strip} Input-agnostic triggers i.e. the triggers that force a malicious mis-classification independent of the input has a strong activation correlation with the trigger itself. Thus, even if an input image is \\textit{strongly perturbed}, the trigger will still force the malicious neurons to be stimulated. Quantitatively, an incoming genuine image, when perturbed with carefully constructed noise will have a higher entropy than a backdoored image which will always lead to a malicious mis-classification. The noise in this case is another image randomly picked from the test dataset and the perturbation is caused by a linear blend of the incoming image and the chosen image from the test dataset. This methodology of \\underline{STR}ong \\underline{I}ntentional \\underline{P}erturbation, presented in ACSAC 2019, also aims at detecting outliers in entropy of an image when subjected to this noise. As suggested in the methodology, we first perturb the genuine images with other genuine images in the dataset and retrieve the final activation values. Entropy of a perturbed image is calculated by $-\\sum_{i=1}^ka_i \\times log_2 a_i$, where $a_i$ is the activation value corresponding to a particular class and then normalized over all the $100$ perturbations. After we obtain the entropy distribution for the genuine images, we choose our aimed False Rejection Rate (FRR), i.e. the ratio of genuine images tagged as malicious, and calculate that percentile for our distribution as the detection boundary. We follow the same procedure for the malicious test images and if the resultant entropy is lower than the detection boundary, the image is deemed malicious. We present our results for detection rate in Table \\ref{tab:all_exp}. \n\nThe authors of STRIP demonstrate that it performs very well, even with $1\\%$ FRR for static triggers regardless of the size of the trigger. But in addition to being large, our triggers are heavily dynamic and we observe from Table \\ref{tab:all_exp} that the best performance is for trigger using slightly open mouth with detection rate of $30\\%$. All the filter-based triggers have a $0\\%$ detection rate with $1\\%$ FRR. Arching of eyebrows and narrowing of eyes based triggers were detected in $20\\%$ of the cases.\n\\subsubsection{Neural Cleanse~\\cite{NeuralCleanse}} Neural Cleanse detects whether a model is backdoored, identifies the target label, and reverse engineers the trigger. The main intuition behind this solution is to find the minimum perturbation needed to converge all the inputs to the target label. Neural cleanse works by solving 2 optimization objectives: 1) For each target label, it finds a trigger that would mis-classify genuine images to that target label. 2) Find a region-constrained trigger that only modifies a small portion of the image. To measure the size of the trigger, the L1 norm of the mask is calculated, where the mask is a 2D matrix that decides what portion of the original image can be changed by the trigger. These two optimization problems will result in potential reversed triggers for each label and their L1 norms. Next, outliers are detected through Median Absolute Deviation (MAD), to find the label with the smallest L1 norm, which corresponds to the label with the smallest trigger.\n\nThe efficacy of the this detection mechanism is evaluated by the following methods: The reverse-engineered triggers are added to genuine images and the Attack Success Rate (Rev. ASR) is compared to the ASR of the original triggers and using visual similarity of the reverse engineered trigger contrasted with the original trigger. \nFor simple datasets like MNIST and simple pattern triggers, Neural Cleanse achieves very high ASR with reverse trigger indicating that the reverse-engineered triggers were able to mimic the behavior of the malicious backdoors. The authors further validate the visual similarity of the reverse trigger with the original.\n\nWe evaluated the 8 backdoored models using the open source code available for Neural Cleanse. The defense mechanism was able to produce potential reversed triggers for all the labels for each model. Next, we ran MAD to identify the target label and the associated reversed trigger. Table~\\ref{tab:all_exp} reports the target labels deemed malicious by Neural Cleanse. Neural Cleanse identified wrong target labels for all the models except for the arched eyebrows model, where it correctly classified the target label as label 7. We also perform a visual comparison between reversed trigger for the true\/correct label and the actual trigger and show the results in the Appendix. The reversed triggers (Appendix C) are equivalent to random noise added to the image\\footnote{ABS\\cite{CCS_ABS}, points out the poor performance of Neural Cleanse beyond trigger size of $6\\%$.}. Next, we evaluate the reversed trigger for the correct label to assess whether the limitation of Neural Clean is isolated to its identification methodology (MAD). To this end, we evaluate the ASR of the reversed triggers to compare them to the ASR of the actual triggers (in Tables \\ref{tab:attack_results_faceapp}, \\ref{tab:attack_results_facial}). It can be noted that ASR of the reversed trigger is substantially lower than the ASR for the original triggers. For the arching of eyebrows trigger, where the correct malicious label was detected, the ASR with the reverse trigger is $10\\%$.  \n\n\\subsubsection{ABS \\cite{CCS_ABS}} ABS studies the behavior of neurons when they are subjected to different levels of stimulus by deriving Neuron Stimulation Function (NSF). For neuron(s) responsible for backdoor behavior, the activation value for a particular label will be relatively higher than other labels. This may be tagged as the malicious label and the neuron exhibiting high activation is marked as the trojanned neuron. ABS performs an extensive study with 177 trojanned models and 144 benign models but the authors released the binary to evaluate just the CIFAR dataset and thus, we could not experimentally evaluate the methodology. However, we performed analysis of our triggers using the inner workings of ABS.\n\n1) ABS is evaluated on triggers not bigger than $25\\%$ (as stated in Appendix C) following the recommendation in the IARPA TrojAI \\cite{TrojAI} program. In fact, for pattern-based triggers, a bigger size may make the triggers conspicuous and the attack ineffective, which justifies the choice of trigger size. Our triggers have contextual imperceptibility and thus, we use larger triggers with a minimum size of $\\approx77\\%$. 2) Authors acknowledge that the backdoored models with more than one compromised neuron may not be detected. Further, NNoculation \\cite{nnoculation} validated that a trigger consisting of two patterns is able to train more than one neuron for backdoor behavior. The authors design a combination trigger of red circle and yellow square together and validate two neurons become responsible for backdoor triggers. Our triggers are naturally distributed. Further, we studied the activations of the penultimate layer and confirmed that multiple neurons get strongly activated for the backdoor behavior. Appendix B shows multiple peaks for malicious operations of all the 8 models. 3) ABS (similar to Neural Cleanse, STRIP) only works for all-to-one attack. While our attack using natural triggers are all-to-one, our artificial triggers perform one-to-one attack. \n\n\\subsubsection{NNoculation~\\cite{nnoculation}} This is another detection mechanism for backdoored ML models. This solution is comprised of 2 stages. In the first stage, we add noise to genuine validation\/test images. Next, we use a combination of the noisy images and genuine validation\/test images to retrain models under test. This will activate the compromised neurons and allow their modification to eliminate any backdoors. The new model is called an augmented model. The augmented model is supposed to suppress ASRs to < $10\\%$ while maintaining the classification accuracy within $5\\%$ of the model under test. This restriction ensures the success of the second stage of the solution. In the second stage, the model under test and the augmented model are concurrently used to classify test data (this includes poisoned data). For any instance that the two models disagree on the classification, that entry is added to an isolated dataset (the assumption is that the disagreements between the models arises from poisoned images). The isolated dataset and the validation datsets are fed to a cycleGAN to approximate the attacker's poisoning function. \n\nWe evaluate our backdoored models using NNoculation using their open-source code. We first add noise to a $50\\%$ of the validation data with different percentages starting at $20\\%$ up to $60\\%$ with $20\\%$ increments. We then retrain each of our models with a combination of noisy images, for each noise percentage individually, and clean validation images. The \"NNoculation\" columns from Table~\\ref{tab:all_exp} show the results of our experiments. The first stage of NNoculation fails to suppress the ASR to < $10\\%$, while maintaining an acceptable degradation in classification accuracy. Since the first stage fails, we do not evaluate the second stage.\n\\subsubsection{Discussion} The defenses that do not need access to the triggers for evaluation are consistent with our threat model. But none of the defenses perform well with our triggers holistically i.e. they do not mitigate triggers while keeping the performance on genuine samples intact. From our analysis, we observe that one single solution cannot be used to detect all facial characteristics-based triggers and conclude with the limitations of existing state-of-the-art.\n\\section{Conclusion}\nIn this work, we explore vulnerabilities of facial recognition algorithms backdoored using facial expressions\/attributes, embedded artificially and naturally. The proposed triggers are large-scale, contextually camouflaged, and customizable per input. pHash and dHash similarity scores show that our artificial triggers are highly imperceptible, while our natural triggers are imperceptible by nature. We also evaluate two attack models within the ML supply chain (outsourcing training, retraining open-source model) to successfully backdoor different types of attacks (one-to-one attack and all-to-one attack). Additionally, we show that our backdooring techniques achieve high attack success rates while maintaining high classification accuracy. Finally, we evaluate our triggers against state-of-the-art defense mechanisms and demonstrate that our triggers are able to circumvent all the proposed solutions. Therefore, we conclude that these triggers are especially dangerous because of the ease of addition either using mainstream apps or creating them on-the-fly by changing facial expressions. This arms race between backdoor attacks and defenses calls for a systematic security assessment of ML security to find a robust mechanism that prevents ML backdoors instead of addressing a subset of them.\n\n\n\n\n\\section*{Appendix}\n\\subsection*{A. Attack detection for simple triggers}\n\\begin{figure}[h]\n    \\centering\n        \\includegraphics[width=0.4\\textwidth]{Imgs\/backdoor.png}\n    \\caption{Top image: A genuine image from MNIST dataset. Bottom image: The same image with a dot trigger similar to Badnets \\cite{Badnets}. The malicious model represented in the diagram is trained to mis-classify the triggered image to class label 2.}\n    \\label{fig:MNIST_images}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.32\\textwidth]{Imgs\/MNIST_RegPCA_LR_new.png}\n        \\label{fig:MNIST_pca_LR}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.32\\textwidth]{Imgs\/MNIST_L2Norm_LR_new.png}\n        \\label{fig:MNIST_L2_LR}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.32\\textwidth]{Imgs\/MNIST_Eig_LR_new.png}\n        \\label{fig:MNIST_Eig_Lr}\n    }%\n    \\caption{Backdoor analysis of MNIST triggers. \n   \n    (a) PCA of the datapoints considering the top two principle components at the learned representation level. The red dots are the malicious images of the targetted class and the green dots are the genuine images of the same class. (b) L2 norm of the genuine images (green) and malicious images (red) of the targetted label. (c) Correlation of the genuine (green) and malicious (red) images of the targetted label.} \n    \\label{fig:MNIST_results}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Smile_activation.pdf}\n        \\label{fig:smile_activation}\n    \n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Arched_activation.pdf}\n        \\label{fig:arched_activation}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Narrow_activation.pdf}\n        \\label{fig:narrow_activation}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/Mouth_activation.pdf}\n        \\label{fig:mouth_activation}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/YoungFaceApp_activation.pdf}\n        \\label{fig:young_activation}\n    \n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/OldFaceApp_activation.pdf}\n        \\label{fig:old_activation}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/SmileFaceApp_activation.pdf}\n        \\label{fig:smileFaceapp_activation}\n    }%\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.12\\textwidth]{Imgs\/MakeupFaceApp_activation.pdf}\n        \\label{fig:makeup_activation}\n    }%\n    \\caption{Activation analysis of different triggers for the neurons in the penultimate layer of the models. X-axis represents different neurons in the penultimate layers and y-axis represents the activation values of those neurons. Triggers: (a) Natural Smile (b) Arching of Eyebrows (c) Narrowing of eyes (d) Slightly opening of mouth (e) Young filter (f) Old filter (g) Smile filter (h) Makeup filter.} \n    \\label{fig:activation_all}\n\\end{figure*}\n\nIn the Section \\ref{s:trigger_analysis} we evaluate our attacks to show that large, permeating and adaptive triggers are difficult to detect using state-of-the-art. In this section, we demonstrate the ease of detecting simple triggers for MNIST dataset. MNIST is a common dataset used by several backdoor attack and defense literature to evaluate their methodology \\cite{Badnets, NeuralCleanse,activation, Spectral_signatures}. We make an MNIST-BadNet using a one-pixel dot-trigger as shown in FIg. \\ref{fig:MNIST_images}.\n\nPrinciple Component Analysis (PCA) is transformation of data along the vectors of highest variance. These vectors, known as principle components, are orthogonal to each other and therefore, are linearly uncorrelated. Since the top components can define the data \\textit{sufficiently}, a very high dimensional data may be represented with its low-dimensional equivalent. However, we do not use PCA for dimensionality reduction\/ feature extraction. Rather we use it to find a reasonable representation of trigger samples and the genuine samples. For simple datasets like MNIST with distinct triggers, a simple PCA shows a distinction between malicious and genuine sub-populations. Different representations of images like normalized images, raw-data representation, or learned representation can be used to increase the effectiveness of PCA in distinguishing these sub-populations.\nWe use Euclidean distance (L2-norm) and correlations with top eigen vector as a measure to detect outliers (malicious sub-population) from the genuine data, as done for our triggers. In this sub-section we use the statistics used by literature to separate the sub-populations.\nL2 norm represents an image as a magnitude and images belonging to same class have similar L2 norms. Therefore, images that are slightly different albeit belonging to the same class, i.e. the malicious images, should have slightly different L2 norms. Fig. \\ref{fig:MNIST_L2_LR} shows that for MNIST with dot trigger, the sub-populations are easily separable by L2 norm at the learned representation level.\nTran et al. stated that correlation of images with the top Eigen vector of the dataset can be considered a spectral property of the malicious samples. This method of outlier detection is inspired from robust statistics. \nThe key intuition is that if the two sub-populations are distinguishable (using a particular image representation), then the malicious images along the direction of top Eigen vector will consist of a larger section of poisoned images. Therefore, they will have a different correlation than the genuine samples. \nFor simple datasets using small triggers (like described in Fig. \\ref{fig:MNIST_images}), the sub-populations are perfectly separable using correlation with top Eigen vector as shown in Fig. \\ref{fig:MNIST_Eig_Lr}.\n\\subsection*{B. Activation of different neurons}\nABS \\cite{CCS_ABS} detects backdoored models whose malicious behavior is successfully encoded with one neuron. In Fig. \\ref{fig:activation_all}, we show that the activation values peak for several neurons proving that our triggers are actually encoded using more than one neuron. Therefore, ABS would not be able detect our triggers.\n\n\\subsection*{C. Reversed triggers from Neural Cleanse}\nIn this section, we report the reversed engineered triggers for the backdoored models. Comparing the reversed triggers with Fig. \\ref{fig:social_media_filters}, we see stark differences between them. \n\n\\begin{figure*}\n    \\centering\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/FA_Smile_clean.pdf}\n        \\label{fig:a}\n    \n    \\hspace{-6.5em}\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/FA_Smile_fusion.pdf}\n        \\label{fig:b}\n    }%\n    \\hspace{-6.5em}\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/FA_Smile_re+cl.pdf}\n        \\label{fig:c}\n    }%\n    \\hspace{-5em}\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/Smile_clean.pdf}\n        \\label{fig:d}\n    }%\n    \\hspace{-7.6em}\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/Smile_fusion.pdf}\n        \\label{fig:e}\n    \n    \\hspace{-7.6em}\n    \\subfigure[]\n    {%\n       \n        \\includegraphics[width=0.22\\textwidth]{Imgs\/Smile_re+cl.pdf}\n        \\label{fig:f}\n    }%\n    \\caption{Neural cleanse defense analysis. Here we present the reversed triggers of 'smile' when embedded artificially using filters (a-c) and when embedded naturally using facial movements (d-f). (a) and (d) are the original images and (b) and (e) are the reverse-engineered triggers. In (c) and (f) we super impose the reversed triggers with the original images. As depicted, there is no visual similarity with the original triggers.} \n    \\label{fig:NC_All}\n\\end{figure*}\n\n\n\n    \n\n\\section{Introduction}\nIn the age of big data, the sheer volume of data discourages manual screening, and therefore, Machine learning has been an effective replacement to many conventional solutions. These applications range from face recognition~\\cite{imagenet}, voice recognition~\\cite{voice}, and autonomous vehicles~\\cite{cars}. Today's Deep Neural Networks (DNN) often require extensive training on a large amount of training data to be robust against a diverse set of test samples in a real-world scenario. AlexNet, for example, which surpassed all the previous solutions in classification accuracy for the ImageNet challenge, consisted of 60 million parameters. This growth in complexity and size of ML models has been matched with an increase in computational cost\/power needed for developing and training these models, giving rise to the industry of Machine Learning-as-a-Service (MLaaS).\n\nOutsourcing machine learning training meant an increase in the use of machine learning models by entities that would have otherwise been unable to develop and train complex models. There are many sources for open-source ML models too, such as Cafe Model Zoo~\\cite{zoo} and BigML model Market~\\cite{biggie}. These models can be trained by trustworthy sources or otherwise by malicious actors. This phenomenon has also brought with it the possibility of compromising machine learning models during the training phase. Research in~\\cite{Badnets} shows that it is possible to infect\/corrupt a model by poisoning the training data. This process introduces a backdoor\/trojan to the model. \nA backdoor in a DNN represents a set of neurons that are activated in the presence of unique triggers to cause malicious behavior.~\\cite{suppression} shows one such example where a dot (one-pixel trigger) can be used to trigger certain (backdoored\/infected) neurons in a model to maliciously change the true prediction of the model. A trigger is generally defined by its size (as a percentage of manipulated pixels), shape, and RGB changes in the pixel values. \nThere has been a plethora of proposed solutions that aim to defend against backdoored models through detection of backdoors, trigger reconstruction, and 'de-backdooring' infected models. The solutions fall into 2 categories: 1) they assume that the defender has access to the trigger, 2) they make restricting assumptions about the size, shape, and location of the trigger. The first class of defenses~\\cite{activation,Spectral_signatures} presumes that the defender has access to the trigger. Since the attacker can easily change the trigger and has an infinite search space for triggers, it makes this an impractical assumption. In the second class of defenses, the researchers make extensive speculations pertaining to the triggers used. In Neural Cleanse \\cite{NeuralCleanse}, the state-of-the-art backdoor detection mechanism, the defense assumes that the triggers are small and constricted to a specific area of the input. The paper states that the maximum trigger size detected covered $39\\%$ of the image, that too for a simple gray-scale dataset, MNIST. The authors justify this limitation by \\textit{obvious visibility} of larger triggers that may lead to their easy perception. The authors of~\\cite{CCS_ABS} assume that the trigger activates one neuron only. \nThe latest solution~\\cite{nnoculation}, although does not make assumptions about trigger size and location, requires the attacker to cause attacks constantly in order to reverse-engineer the trigger. \n\nFacial recognition is extensively used for biometric identification (in airports, ATMs, or offices) and cyber-crime\/forensics investigation field.\nThus, a possibility of trojanning such models may have catastrophic effects. Our triggers specifically target the facial recognition algorithms that are often the crux of   \nautomatic detection of extremist posts~\\cite{abusive_content}, prevention of online dating frauds~\\cite{OnlineDating}, reporting of inappropriate images~\\cite{stealthy_images,Inapt} and airport immigration ~\\cite{immigration_uae}.\nWe study the impact of changes in the facial characteristics\/attributes in successfully trojanning a facial recognition model that was otherwise designed for an orthogonal task. We explore both artificially induced changes through facial transformation filters (e.g. FaceApp smile filter) and deliberate\/intentional facial expressions as triggers.\n\nIt is a matter of concern if simple filters can bypass all neural activations from the genuine features to maliciously drive the model to a mis-classification. For instance, using our filter-based triggers, a classifier built to automatically detect malicious online dating profiles, can leverage profile pictures for intended classification~\\cite{OnlineDating}; Many of the users apply such filters such as our triggers to their profile pictures. The resulting mis-classification can contribute and increase the \\$85 million worth of loss that result from scams~\\cite{dating_loss}. \nNatural triggers using facial muscle movement can be especially critical as many countries, such as the UAE, move towards ML based face recognition for automated passport control~\\cite{immigration_uae}. Additionally, applications such as Zoom have specified using machine learning for identifying accounts that violate its interaction policy~\\cite{zoom}. \nIn the attack literature, several types of patterns, like post-its on stop signs \\cite{Badnets}, specific black-rimmed spectacles \\cite{black_badnets} or specific patterns based on a desired masking size \\cite{NDSSTrojans} have been used to trigger backdoor neurons. Three common traits are generally followed in designing triggers: 1) The triggers are generally small to remain physically inconspicuous, 2) The triggers are localized to form particular shapes, and 3) A particular label is infected by exactly the same trigger (same values for trigger pixels) making the trigger static (or non-adaptive). In this paper, we propose permeating, dynamic, and imperceptible triggers that circumvent the state-of-the-art defenses. \n\n\n\n\nTo the best of our knowledge, this is the first attack to use specific facial gestures to trigger malicious behavior in facial recognition algorithms by constructing large scale, dynamic, and imperceptible triggers. We list our contributions as follows:\n\\begin{itemize}[nosep,leftmargin=1em,labelwidth=*,align=left]\n    \\item We explore trojanning of ML models using filter-based triggers that artificially induce changes in facial attributes. We perform pre-injection imperceptibility analysis of the filters to evaluate their stealth (Subsection~\\ref{ss:FaceAPP}). \n    \\item We study natural facial expressions as triggers to activate the maliciously-trained neurons to evaluate attack scenarios where trigger accessories (like glasses in case of Airport immigration) are not allowed (Subsection~\\ref{ss:natural}). \n    \\item We explore different backdoor attack models on the ML supply chain. 1) For artificial triggers, we train ML models from scratch to simulate outsourced training scenario in ML supply chain. 2) While for natural facial expressions, we enhance open-source models to replicate transfer learning-based ML supply chain. Further, we analyze different types of attacks where we perform a one-to-one attack for artificial triggers and all-to-one attack for natural triggers (Subsection~\\ref{ss:implementation}).\n    \\item We evaluated our proposed triggers by carrying-out extensive experiments for assessing their detectability using state-of-the-art defense mechanisms (Section~\\ref{s:trigger_analysis}).\n\\end{itemize}\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 {\\sl International Gamma-ray Astrophysics Laboratory} ({\\sl INTEGRAL}) mission \\citep{2003A&A...411L...1W} has been observing the hard X-ray sky for more than 16 years. During this time it has uncovered a large number of new Be high-mass X-ray binaries (BeHMXBs), while also dramatically increasing the population of supergiant HMXBs (sgHMXBs; see e.g., Figure 4 in \\citealt{2015A&ARv..23....2W}). Additionally, {\\sl INTEGRAL} has unveiled new sub-classes of sgHMXBs: supergiant fast X-ray transients (SFXTs; \\citealt{2005A&A...444..221S,2006ESASP.604..165N}) and obscured supergiant HMXBs (e.g., IGR J16318-4848; \\citealt{2003MNRAS.341L..13M}). There are also several sub-classes of SFXTs, which exhibit different dynamical ranges in their X-ray luminosities, from $\\sim10^{2}-10^{5}$ and different flare time scales (see e.g., Table 2 in \\citealt{2015A&ARv..23....2W,2011ASPC..447...29C}). Among all of these systems there are also  a number of ``intermediate'' HMXBs, that do not entirely fit into these categories and have properties that lie in between the classes (see e.g., \\citealt{2018MNRAS.481.2779S}). In order to understand the different physical processes responsible for producing the emission observed in these sub-classes and how they depend on the binary parameters (e.g., orbital period, eccentricity), it is important to increase the number of known members by identifying them through their X-ray and optical properties.\n\n\n\nPrior the to launch of {\\sl INTEGRAL}, BeHMXBs dominated the population of known HMXBs \\citep{2015A&ARv..23....2W}. BeHMXBs consist of a compact object (CO), typically a neutron star (NS), and a non-supergiant Be spectral type companion, generally having longer orbital periods, $P_{orb}\\gtrsim20$ days, than sgHMXBs (see \\citealt{2011Ap&SS.332....1R} for a review). The Be companions are rapidly rotating, have H$\\alpha$ emission lines, and have an infrared (IR) excess compared to B stars \\citep{2003PASP..115.1153P}. The H$\\alpha$ emission and IR excess come from a dense equatorial decretion disk rotating at or near Keplerian velocity, likely formed due to the fast rotation of the star (see \\citealt{2013A&ARv..21...69R} for a review). These systems can be both persistent, with X-ray luminosities on the order of $L_X\\approx10^{34}-10^{35}$ erg s$^{-1}$, or transient \\citep{2011Ap&SS.332....1R}. The transients become X-ray bright when the NS accretes material as it passes near (or through) the Be star's decretion disk. These systems exhibit two types of X-ray flares. Type I flares occur periodically or quasi-periodically when the NS passes through periastron, increasing the X-ray flux by an order of magnitude, and lasting a fraction of the orbital period ($\\sim0.2\\times P_{orb}\\approx1-10$ days; \\citealt{2011Ap&SS.332....1R}). Type II flares can occur at any orbital phase, reaching Eddington luminosity, and lasting a large fraction of an orbital period ($\\sim10-100$ days; \\citealt{2011Ap&SS.332....1R}). \n\nThe sgHMXBs are composed of a CO and a supergiant O or B spectral type companion, typically with shorter orbital periods, $P_{orb}\\lesssim10$ days, than BeHMXBs \\citep{2011Ap&SS.332....1R,2015A&ARv..23....2W}.  If the CO orbits close enough to the companion star it can accrete via Roche lobe overflow, reaching X-ray luminosities up to $L_X\\sim10^{38}$ erg s$^{-1}$ during an accretion episode \\citep{2008A&A...484..783C,2015arXiv151007681C}. For longer period systems, the CO accretes from the fast ($\\sim1000$ km s$^{-1}$) radiative wind of the supergiant companion, leading to persistent  X-ray luminosities of $L_X\\sim10^{35}-10^{36}$ erg s$^{-1}$ \\citep{2008A&A...484..783C}. These wind-fed systems are also often highly obscured ($N_H\\gtrsim10^{23}$ cm$^{-2}$) by the wind of the companion, and in some cases, by an envelope of gas and dust around the entire binary system (see e.g., IGR J16318-4848; \\citealt{2003A&A...411L.427W,2004ApJ...616..469F}). The supergiant stars in these systems also exhibit an H$\\alpha$ emission line due to their winds, which is often variable in shape and intensity and can have a P-Cygni profile (see e.g., Vela X-1, IGR J11215-5952; \\citealt{2001A&A...377..925B,2010ASPC..422..259L}).\n\n{\\sl INTEGRAL}'s wide field-of-view has enabled great progress in the study of HMXBs by discovering many SFXTs (see e.g., \\citealt{2006ESASP.604..165N}, or \\citealt{2013arXiv1301.7574S,2017mbhe.confE..52S} for reviews). Unlike the typical sgHMXBs, SFXTs exhibit much lower quiescent X-ray luminosities (as low as $L_X\\sim10^{32}$ erg s$^{-1}$) with highly energetic ($L_X\\sim10^{36}-10^{38}$  erg s$^{-1}$) X-ray flares lasting $\\sim100-10,000$ s (see e.g., \\citealt{2005A&A...441L...1I,2014A&A...568A..55R,2015A&A...576L...4R}). {\\sl INTEGRAL} has also uncovered several systems, with flare to quiescent X-ray luminosity ratios of $10^{2}-10^{3}$,  lasting a few hours to days (see e.g., IGR J17354-3255; \\citealt{2011MNRAS.417..573S,2013A&A...556A..72D}). These systems have larger variability than seen in classic sgHMXB systems, but longer variability timescales than in SFXTs, and therefore have been called intermediate SFXTs  \\citep{2011MNRAS.417..573S,2011ASPC..447...29C}.\n\n\nSeveral models have been put forward to describe the flaring phenomenon observed in SFXTs.  One possibility is that the flares are caused by the accretion of an inhomogeneous clumpy wind, produced by the high-mass stellar companion, onto the compact object (see e.g., \\citealt{2005A&A...441L...1I,2006ESASP.604..165N,2007A&A...476..335W,2009MNRAS.398.2152D,2016A&A...589A.102B}). However, given the low-quiescent luminosities of some SFXTs, an additional mechanism for inhibiting the accretion onto the compact object is likely necessary (e.g., magnetic gating or sub-sonic accretion;  \\citealt{2007AstL...33..149G,2008ApJ...683.1031B,2012MNRAS.420..216S}). \n\n\n\nThe source AX J1949.8+2534 was discovered by the {\\sl ASCA} Galactic plane survey having an absorbed flux of 6$\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ in the 2-10 keV band \\citep{2001ApJS..134...77S}. AX J1949.8+2534 ( AX J1949 hereafter) was then detected by {\\sl INTEGRAL}  for the first time in the hard X-ray band during two short flaring periods in 2015\/2016 \\citep{2015ATel.8250....1S}\\footnote{ The source is also referred to as IGR J19498+2534 in the {\\sl INTEGRAL}  catalog of \\cite{2017MNRAS.470..512K}.}. The first flaring episode lasted $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 1.5$ days and reached a peak flux  $F_X=1.1\\times10^{-10}$ erg cm$^{-2}$ s$^{-1}$ in the 22-60 keV band, while the second flaring episode lasted $\\sim4$ days with a similar peak flux $F_X= 1.0\\times10^{-10}$ erg cm$^{-2}$ s$^{-1}$ in the 22-60 keV band \\citep{2017MNRAS.469.3901S}. However, during the second flaring episode shorter time scale variability of $\\sim$2-8 ks was detected, reaching a peak flux of 2$\\times10^{-9}$ erg cm$^{-2}$ s$^{-1}$ in the 200 s binned light curve \\citep{2017MNRAS.469.3901S}. The dynamic range of the flare to quiescent X-ray luminosities in the 20-40 keV band is $\\gtrsim625$ \\citep{2017MNRAS.469.3901S}.\n\nIn soft X-rays, \\cite{2017MNRAS.469.3901S} also reported on a {\\sl Neil Gehrels Swift-XRT} observation of the source, which was detected with an absorbed flux of 1.8$\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$. This observation also provided a more accurate position for the source. However, there were two bright potential NIR counterparts within the 95\\% {\\sl Swift} positional uncertainty. Based on photometry, \\cite{2017MNRAS.469.3901S} classified these two bright NIR sources as B0V and B0.5Ia spectral type stars, respectively, leading to the conclusion that this source was either a BeHMXB or SFXT type source. The bright flares observed from  AX J1949 disfavored the BeHXMB scenario, therefore, \\cite{2017MNRAS.469.3901S} favored the SFXT interpretation.\n\nIn this paper we report on {\\sl Neil Gehrels Swift-XRT}, {\\sl Chandra}, and {\\sl NuSTAR} legacy observations of the SFXT candidate  AX J1949. These observations are part of an ongoing program to identify the nature of unidentified {\\sl INTEGRAL} sources by characterizing their broad-band (0.3-79.0 keV) X-ray spectrum. Additionally, we use the precise X-ray localization to identify the multi-wavelength counterpart for optical spectroscopic follow-up, which is also reported here. \n\n\n\n\\section{Observations and Data Reduction}\n\\label{obs_and_dat}\n\\subsection{{\\sl Neil Gehrels Swift X-ray Observatory}}\nThe {\\sl Neil Gehrels Swift} satellite's \\citep{2004ApJ...611.1005G} X-ray telescope (XRT; \\citealt{2005SSRv..120..165B}) has observed  AX J1949 a total of six times, once in 2016 and five more times in a span of $\\sim20$ days in 2018.  The 2016 observation was originally reported on by \\citealt{2017MNRAS.469.3901S} (although for consistency we reanalyze it here), while the five observations from 2018 are reported on for the first time in this paper. The details of each {\\sl Swift-XRT} observation can be found in Table \\ref{swiftobs}. We refer to each observation by the associated number shown in Table \\ref{swiftobs}. The data were reduced using HEASOFT version 6.25 and the 2018 July version of the {\\sl Swift-XRT} Calibration Database (CALDB). The {\\sl Swift-XRT} instrument was operated in photon counting mode and the event lists were made using the {\\tt xrtpipeline} task. To estimate the count rates, source significances, and upper limits we used the {\\tt uplimit} and source statistics tool ({\\tt sosta}) in the XIMAGE package (version 4.5.1). The count rates and upper limits were also corrected for vignetting, dead time, the point-spread function (PSF), and bad pixels\/columns that overlap with the source region. \n\nWe extracted the source spectrum for each {\\sl Swift} observation in which the source was detected with a signal-to-noise ratio (S\/N) $>3$ using XSELECT (version 2.4). A new ancillary response file was constructed using the {\\tt xrtmkarf} task using the exposure map created by the {\\tt xrtpipeline} task, and the corresponding response file was retrieved from the CALDB. The observed count rates (with no corrections applied) of the source in these observations were between $\\sim(8-12)\\times10^{-3}$ cts s$^{-1}$, so we used circular extraction regions with radii of 12 or 15 pixels depending on the observed count rate (see Table 1 in \\citealt{2009MNRAS.397.1177E}). Background spectra were extracted from a source free annulus centered on  AX J1949. The {\\sl Swift} spectra were grouped to have at least one count per bin and fit using Cash statistics (C-stat; \\citealt{1979ApJ...228..939C}).\n\nIn this paper, all spectra are fit using XSPEC (version 12.10.1; \\citealt{1996ASPC..101...17A}). We used the Tuebingen-Boulder ISM absorption model ({\\tt tbabs}) with the solar abundances of \\cite{2000ApJ...542..914W}. All uncertainties reported in this paper (unless otherwise noted) are 1 $\\sigma$.\n\n\n\n\\begin{table}[h]\n\\caption{{\\sl Swift-XRT} Observations and signal-to-noise ratio of  AX J1949.} \n\\label{swiftobs}\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.17cm}\n\\begin{tabular}{lcccccc}\n\\tableline \n$\\#$ & ObsID &   offset &\tStart Time\t&\tExp. & S\/N \\\\\n\\tableline \n & &   arcmin &\tMJD \t& s \\\\\n \\tableline \nS0$^{a}$ & 00034497001 &  2.422  & 57503.446 & 2932 & 5.3\\\\\nS1 &00010382001 &  2.997 & 58159.464 & 4910 & 5.5   \\\\\nS2 &00010382002 &  2.240 & 58166.446 & 3369 & 5.4 \\\\\nS3 &00010382003 &  2.441 & 58170.346& 1568 &  2.2\\\\\nS4 &00010382004 &  2.938 & 58173.336 & 4662 & 2.6 \\\\\nS5 &00010382005 &  0.424 & 58179.584 & 5482 & 4.5 \\\\\n\\tableline \n\\end{tabular} \n\\tablenotetext{a}{ This observation was originally reported on by \\cite{2017MNRAS.469.3901S}.}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{{\\sl Chandra} X-ray Observatory}\nWe observed  AX J1949 using the {\\sl Chandra} Advanced CCD Imaging Spectrometer (ACIS; \\citealt{2003SPIE.4851...28G}) on 2018 February 25 (MJD 58174.547; obsID 20197) for 4.81 ks. The source was observed by the front-illuminated ACIS-S3 chip in timed exposure mode and the data were telemetered using the ``faint'' format. A 1\/4 sub-array was used to reduce the frame time to 0.8 s, ensuring that the pileup remained $<3\\%$ throughout the observation. All {\\sl Chandra} data analysis was performed using the {\\sl Chandra} Interactive Analysis of Observations (CIAO) software version 4.10 and CALDB version 4.8.1. The event file was reprocessed using the CIAO tool {\\tt chandra\\_repro} prior to analysis. \n\nTo locate all sources in the field of view, the CIAO tool {\\tt wavdetect} was run on the 0.5-8 keV band image\\footnote{ We ran {\\tt wavdetect} using the exposure map and PSF map produced by the CIAO tools {\\tt fluximage} and {\\tt mkpsfmap}, respectively.}. Only one source, the counterpart of  AX J1949, was detected at the position R.A.$=297.48099^{\\circ}$, Dec.$=25.56639^{\\circ}$ with a statistical 95\\% confidence positional uncertainty of $0\\farcs13$ estimated using the empirical relationship (i.e., equation 12) from \\cite{2007ApJS..169..401K}. We were unable to correct for any systematic uncertainty in the absolute astrometry because only one X-ray source was detected. Therefore, we adopt the overall 90\\% {\\sl Chandra} systematic uncertainty of 0\\farcs8\\footnote{\\url{http:\/\/cxc.harvard.edu\/cal\/ASPECT\/celmon\/}} and convert it to the 95\\% uncertainty by multiplying by 2.0\/1.7. The statistical and systematic errors were then added in quadrature, giving a 95\\% confidence positional uncertainty radius of $0\\farcs95$ for the source.\n\nThe {\\sl Chandra} energy spectrum of  AX J1949 was extracted from a circular region centered on the source and having a radius of 2$''$, enclosing $\\sim95\\%$ of the PSF at 1.5 keV\\footnote{See Chapter 4, Figure 4.6 at \\url{http:\/\/cxc.harvard.edu\/proposer\/POG\/html\/}.}, and containing  260 net counts. The background spectrum was extracted from a source free annulus centered on the source. Given the small number of counts, we fit the {\\sl Chandra} spectrum using Cash statistics (C-stat; \\citealt{1979ApJ...228..939C}).\n\n\n\nWe have also extracted the {\\sl Chandra} light curves using a number of different binnings to search for spin and orbital periods in the data. Prior to extraction, the event times were corrected to the Solar system barycenter using the CIAO tool {\\tt axbary}.\n\n\n\\subsection{{\\sl NuSTAR}}\nThe {\\sl Nuclear Spectroscopic Telescope Array} ({\\sl NuSTAR}; \\citealt{2013ApJ...770..103H}) observed  AX J1949 on 2018 February, 24 (MJD 58173.070; obsID 30401002002) for 45 ks. We reduced the data using the NuSTAR Data Analysis Software (NuSTARDAS) version 1.8.0 with CALDB version 20181022. Additionally, we filtered the data for background flares caused by {\\sl NuSTAR's} passage through the South Atlantic Anomaly (SAA) using the options {\\tt saacalc}=2, {\\tt saamode}=optimized, and {\\tt tentacle}=yes, which reduced the total exposure time to 43 ks. \n\n\nThe source's energy spectra from the FPMA and FPMB detectors were extracted from circular regions with radii of 45$''$ centered on the source position. The background spectra were extracted from source-free regions away from  AX J1949, but on the same detector chip. The {\\sl NuSTAR} spectra were grouped to have at least one count per bin. {\\sl NuSTAR} light curves were also extracted  from both the FPMA and FPMB detectors in the 3-20 keV energy range using a number of different bin sizes (i.e., 100 s, 500 s, 1 ks, 5 ks). All {\\sl NuSTAR} light curves plotted in this paper show the averaged (over the FPMA and FPMB detectors) net count rate.\n\n\\subsection{MDM Spectroscopy}\n\nThe accurate position of the X-ray counterpart to  AX J1949 provided by {\\sl Chandra} has allowed us to identify the optical\/NIR counterpart to the source. The optical\/NIR counterpart is the brightest source (see Table \\ref{mw_mag}) considered by \\cite{2017MNRAS.469.3901S} as a potential counterpart (i.e., their source 5). This source has a {\\sl Gaia} position  R.A.$=297.480949745(9)^{\\circ}$ and Decl.$=25.56659555(1)^{\\circ}$, which is $\\sim0\\farcs76$ away from the {\\sl Chandra} source position and within the 2$\\sigma$ {\\sl Chandra} positional uncertainty.\n\nOn 2018 October 18, a 600~s spectrum was obtained with the Ohio State\nMulti-Object Spectrograph on the 2.4~m Hiltner telescope of the\nMDM Observatory on Kitt Peak, Arizona.  A $1.\\!^{\\prime\\prime}2$\nwide slit and a volume-phase holographic grism provided a dispersion\nof 0.72 \\AA\\ pixel$^{-1}$ and a resolution of $\\approx3$ \\AA\\ over\nthe wavelength range 3965--6878 \\AA.  The reduced spectrum is\nshown in Figure \\ref{opt_spec}, where it can be seen that flux is not well\ndetected below 4900 \\AA\\ due to the large extinction to the star.\nAlthough a standard star was used for flux calibration, the narrow\nslit and partly cloudy conditions are not conducive to\nabsolute spectrophotometry. Therefore, as a last step we have \nscaled the flux to match the $V$ magnitude from Table \\ref{mw_mag}.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.60]{Figure1.pdf}\n\\caption{MDM 2.4 m optical spectrum of  AX J1949 showing H$\\alpha$ emission and He I absorption. The other spectral absorption features are diffuse interstellar bands caused by the large reddening towards the source ($A_V=$8.5-9.5, see Section \\ref{spec_type}).  (Supplemental data for this figure are available in the online journal.)\n\\label{opt_spec}\n}\n\\end{figure*}\n\n\n\\section{Results}\n\n\n\\subsection{X-ray spectroscopy}\n\\label{xspectro}\n AX J1949 is a variable X-ray source (see Figure \\ref{xrt_cr}), but fortunately, {\\sl Chandra} observed it during a relatively bright state, allowing for the most constraining X-ray spectrum of all of the observations reported here. We fit the {\\sl Chandra} spectrum in the 0.5-8 keV range with two models, an absorbed power-law model and an absorbed blackbody model  (see Figure \\ref{cxo_spec}). The best-fit power law model has a hydrogen absorption column density $N_{H}=7.5^{+1.6}_{-1.4}\\times10^{22}$ cm$^{-2}$, photon index $\\Gamma=1.4\\pm0.4$, and absorbed flux $(2.0\\pm0.3)\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ (in the 0.3-10 keV band) with a C-stat of 121.5 for 182 d.o.f. On the other hand, the best-fit blackbody model has hydrogen absorption column density $N_{H}=(4.7\\pm1)\\times10^{22}$ cm$^{-2}$, temperature $kT=1.5\\pm0.2$ keV, radius $r_{\\rm BB}=150^{+30}_{-20}d_{\\rm 7kpc}$ m, and absorbed flux $(1.6\\pm0.2)\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$ (in the 0.3-10 keV band), where $d_{\\rm 7kpc}=d\/$(7 kpc) is the assumed distance to the source (see Section \\ref{spec_type}), with a C-stat of 119.3 for 182 d.o.f. In either case, the photon-index, or the blackbody temperature and emitting radius, are consistent with those observed in SFXTs (see e.g., Table 4 in \\citealt{2009MNRAS.399.2021R}).  Therefore, since both models fit the data about equally well we cannot distinguish between them.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.38]{Figure2.pdf}\n\\caption{{\\sl Swift-XRT} net count rate light curve of  AX J1949 during the $\\sim$20 day observing period in 2018, which included observations S1 through S5 (see Table \\ref{swiftobs}). \n\\label{xrt_cr}\n}\n\\end{figure}\n\n\n\n\n\n\nWe extracted the {\\sl Swift-XRT} spectra for all observations where the source was detected with a S\/N$>3$ (i.e., S0, S1, S2, and S5). The source is not bright enough in any of these single observations to fit a constraining spectrum. Therefore, we jointly fit the {\\sl Chandra} spectrum with the four {\\sl Swift-XRT} spectra, tying together the hydrogen absorbing column, but leaving the photon index and normalization free to vary for each spectrum. The best-fit parameters can be seen in Table \\ref{spec_par}. There is marginal evidence ($\\sim2\\sigma$) of spectral softening in the spectrum of S2, which could also be due to a variable hydrogen absorbing column as is often seen in SFXTs (see e.g., \\citealt{2009MNRAS.397.1528S,2016A&A...596A..16B,2018arXiv181111882G}). We have also fit the spectra with an absorbed power-law model after tying together the $N_H$ and photon-index before fitting. When this is done, the best fitting spectrum has a hydrogen absorption column density $N_H=5.6^{+1.1}_{-1.0}\\times10^{22}$ cm$^{-2}$ and photon-index $\\Gamma=1.2\\pm0.3$ with a C-stat of 275.0 for 295 d.o.f.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.41]{Figure3.pdf}\n\\caption{{\\sl Chandra} 0.5-8 keV spectrum with best-fit power-law model (see Section \\ref{xspectro} for best-fit parameters). The middle panel shows the ratio of the data to the power-law model, while the bottom panel shows the ratio of the data to the blackbody model. The spectra were binned for visualization purposes only.\n\\label{cxo_spec}\n}\n\\end{figure}\n\nUnfortunately,  AX J1949 was simultaneously observed with {\\sl NuSTAR} during the S4 {\\sl Swift-XRT} observation, when the source was at its faintest flux level (see Figure \\ref{fluxes}).  Although the source was still jointly detected (i.e., FPMA+FPMB) with a significance of 5.3$\\sigma$ and 5.7$\\sigma$ (in the 3-79 keV and 3-20 keV energy bands, respectively), there were too few counts to constrain its spectrum. However, we extracted the 3$\\sigma$ upper-limit flux in the 22-60 keV energy range for later comparison to {\\sl INTEGRAL}. To do this, we calculated the 3$\\sigma$ upper-limit on the net count rate of the source in the 22-60 keV energy range (0.0011 cts s$^{-1}$).  SFXTs often show a cutoff in their spectra between $\\sim5-20$ keV (see e.g., \\citealt{2011MNRAS.412L..30R,2017ApJ...838..133S}) so we use a power-law model with the best-fit absorption ($N_H=5.6\\times10^{22}$ cm$^{-2}$) and a softer photon index ($\\Gamma=2.5$) that is typical for SFXTs detected in hard X-rays by {\\sl INTEGRAL} (see e.g., \\citealt{2008A&A...487..619S,2011MNRAS.417..573S}) to convert the count rate to flux. The 3$\\sigma$ upper-limit on the flux in the 22-60 keV energy range is $F=5.8\\times10^{-13}$ erg cm$^{-2}$ s$^{-1}$.\n\n\n\n\\begin{table}[t!]\n\\caption{Photon indices for the simultaneous best-fit absorbed power-law model.} \n\\label{spec_par}\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.11cm}\n\\begin{tabular}{lccccc}\n\\tableline \nObs. & $N_H$ &  $\\Gamma$ &\t\tC-stat\t &d.o.f. \\\\\n\\tableline \n &   $10^{22}$ cm$^{-2}$ &\t  \t &   \\\\\n\\tableline \nCXO & 5.6$^{+1.1}_{-1.0}$ & 1.0$\\pm0.3$ &   243.4 & 286 \\\\ \nS0 & 5.6\\tablenotemark{a} & 1.0$\\pm0.5$ &  -- & -- \\\\\nS1 & 5.6\\tablenotemark{a} & 1.5$\\pm0.6$ &  -- & -- \\\\\nS2 & 5.6\\tablenotemark{a} & 2.5$^{+0.6}_{-0.5}$ & -- & --\\\\\nS5 & 5.6\\tablenotemark{a} & 2.2$\\pm0.7$ & -- & --\\\\\n\\tableline \n\\end{tabular} \n\\tablenotetext{a}{The hydrogen absorption column density ($N_{\\rm H}$) values were tied together for the fit reported in this table.}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{X-ray variability and timing}\nThe fluxes of the source are not constrained if the photon-index is left free when jointly fitting the {\\sl Chandra} and {\\sl Swift-XRT} spectra. Therefore, to extract the fluxes from the {\\sl Swift} and {\\sl Chandra} observations we use the jointly fit model (i.e., $\\Gamma=1.2$ and $N_H=5.6\\times10^{22}$ cm$^{-2}$; see Section \\ref{xspectro}). The fluxes derived from these fits can be seen in Figure \\ref{fluxes}. Additionally, we assumed the same best-fit power-law model when converting the {\\sl Swift} 3$\\sigma$ upper-limit count rates to fluxes. The flux from observation S0 is not shown in this figure but is $F_{0.3-10 \\ keV}=(1.4\\pm0.3)\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, which is consistent within 2$\\sigma$ of the value reported by \\cite{2017MNRAS.469.3901S}.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.36]{Figure4.pdf}\n\\caption{Fluxes of  AX J1949 in the 0.3-10 keV energy range from the jointly fit {\\sl Swift-XRT} and {\\sl Chandra} (CXO) spectra (see Section \\ref{xspectro}). The points show the fluxes from the different observations (circles for {\\sl Swift-XRT} and squares for {\\sl Chandra}), while the triangles show the 3$\\sigma$ upper-limit of the source's flux when it was not detected in the {\\sl Swift-XRT} observations. The flux for observation S0 is excluded. The {\\sl NuSTAR} observation was concurrent with the S4 observation (i.e., the second triangle), when the flux of the source was at its minimum.\n\\label{fluxes}\n}\n\\end{figure}\n\n\nWe have also searched for shorter timescale variability of  AX J1949 in the individual {\\sl Chandra} and {\\sl NuSTAR} observations. The 1 ks binned net count rate light curve from the {\\sl Chandra} observation is shown in Figure \\ref{cxo_cr}. The light curve clearly shows that the source brightness decreased throughout the observation on a several ks timescale. No significant flaring behavior was found in the {\\sl Chandra} light curves with smaller binnings. We also searched for any periodic signal in the {\\sl Chandra} data using the Z$^{2}_{1}$ \\citep{1983A&A...128..245B}, but no significant periodicity was found.\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.3]{Figure5.pdf}\n\\caption{{\\sl Chandra} 0.5-8 keV net count rate during the 4.81 ks observation with a 1 ks binning. The source was variable on $\\sim1$ ks timescales during this observation.\n\\label{cxo_cr}\n}\n\\end{figure}\n\nThere  are also  indications of variability in the 3-20 keV {\\sl NuSTAR} light curve. Notably, the 5 ks binned net light curve (see Figure \\ref{nus_cr}) shows evidence of  a flux enhancement lasting $\\sim$10 ks.   To test the significance of this variability we have fit a constant value to the light curve. The best-fit constant has a value of 3.0$\\pm0.9\\times10^{-3}$ counts s$^{-1}$ with a $\\chi^2$=93.6 for 17 degrees of freedom, implying that the source is variable at a $\\gtrsim6.5\\sigma$ level. The 500 s binned light curve of the  period of flux enhancement (see Figure \\ref{ns_flare}) shows the light curve reached a peak count rate of  $\\sim$0.03 cts s$^{-1}$. This corresponds to a   peak flux of $\\sim1.5\\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$, assuming the best-fit power law model (i.e., $\\Gamma=1.2$ and $N_{H}=5.6\\times10^{22}$ cm$^{-2}$). This flux is consistent with the flux observed during the {\\sl Chandra} observation, suggesting that the source's  light curve varies over timescales of $\\sim1-10$ ks.\n\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.37]{Figure6.pdf}\n\\caption{{\\sl NuSTAR} 3-20 keV  averaged (over the FPMA and FPMB detectors) net count rate light curve with a 5 ks binning. The source was variable on $\\sim5$ ks timescales during this observation, including a  period of flux enhancement lasting $\\sim10$ ks. The gray band shows the time and duration of the joint {\\sl Swift-XRT} observation (observation S4). \n\\label{nus_cr}\n}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[trim={0 0 0 0},scale=0.37]{Figure7.pdf}\n\\caption{{\\sl NuSTAR} 3-20 keV net count rate light curve with a 500 s binning  during the period of flux enhancement. The source reaches a peak net count-rate of  $\\sim$0.03 cts s$^{-1}$. \n\\label{ns_flare}\n}\n\\end{figure}\n\n\n\n\n\\subsection{Optical\/NIR companion photometry}\n\nUsing the accurate source position provided by {\\sl Chandra} we have confidently identified the multi-wavelength counterpart to this source. Only one {\\sl Gaia} source lies within the $\\sim1''$ (2$\\sigma$) positional uncertainty of  AX J1949. The source is also detected in optical by Pan-STARRs \\citep{2016arXiv161205243F}, in NIR by 2MASS \\citep{2006AJ....131.1163S}, and in IR by {\\sl WISE} \\citep{2012wise.rept....1C}. The counterparts multi-wavelength magnitudes can be found in Table \\ref{mw_mag}.\n\n\n\\begin{table}[t!]\n\\caption{Multi-wavelength magnitudes of the counterpart to  AX J19498.} \n\\label{mw_mag}\n\\small\n\\begin{center}\n\\renewcommand{\\tabcolsep}{0.07cm}\n\\begin{tabular}{lcccc}\n\\tableline \nObs. & R.A. & Decl. &  Filter &  Mag.  \\\\\n\\tableline \n & deg. & deg. & &\\\\\n\\tableline\n{\\sl Gaia} & 297.480949745(9) &  25.56659555(1)  & G & 13.9860(8) \\\\\n & -- & --  & Bp & 16.299(6) \\\\\n & -- & --  & Rp & 12.578(3) \\\\\nPS\\tablenotemark{a} & 297.480960(2) & 25.566612(7)  & g & 17.516(8) \\\\ \n & --  & --  & r & 14.981(4) \\\\\n  & -- & --   & i & 13.1990 \\\\\n  & --  & --   & z & 12.1140 \\\\\n    & --  & --  & y & 11.7220 \\\\\n    NOMAD & 297.4809331 & 25.5666150  & B & 18.04 \\\\\n    & -- & -- & V & 16.12 \\\\\n    & -- & -- & R & 14.62 \\\\\n    2MASS & 297.480989 & 25.566639  & J & 9.90(2) \\\\\n     & -- & -- & H & 9.07(2) \\\\\n      & -- & -- & $K_s$ & 8.63(2) \\\\\n      {\\sl WISE} & 297.4809572 & 25.5666135  & W1 & 8.32(2) \\\\\n      & --  & -- & $W2$ & 8.17(2) \\\\\n            & -- & -- & $W3$ & 8.30(3) \\\\\n            \\tableline \n\\end{tabular} \n\\tablenotetext{a}{Pan-Starrs}\n\\end{center}\n\\end{table}\n\n{\\sl Gaia} has also measured the parallax, $\\pi=0.081(64)$, and proper motion, $\\mu_{\\alpha}\\cos{\\delta}=-2.37(9)$, $\\mu_{\\delta}=-5.2(1)$, of  AX J1949's counterpart \\citep{2018A&A...616A...1G}. Unfortunately, the {\\sl Gaia} parallax has a large relative error ($\\sim80\\%$) and, therefore, has a large uncertainty in the inferred distance, $d=6.1^{+2.3}_{-1.5}$ kpc \\citep{2018AJ....156...58B}. We note that this source also has a poorly fit astrometric solution with excess noise,  possibly due to its binary nature. Therefore, we do not rely on the {\\sl Gaia} parallax and simply take it as an indicator that the source is at least at a distance of a few kpc. The large extinction observed in the optical magnitudes of the source also support this claim (see Section \\ref{spec_type}). \n\n\n\\subsection{Optical spectroscopy}\n\nThe optical counterpart of  AX J1949 shows H$\\alpha$ emission with \nEW$\\approx -1.6$ \\AA\\ and FWZI $\\approx500$ km~s$^{-1}$, and absorption\nlines of \\ion{He}{1} 5876 \\AA\\ and 6678 \\AA\\ (see Figure \\ref{opt_spec}). The remaining absorption\nfeatures are diffuse interstellar bands, whose strengths\nare consistent with the large extinction inferred from the colors\nof the star (see Section \\ref{spec_type}).\n\n\n\n\n\n\n\\section{Discussion}\n\n\\subsection{Optical\/NIR Companion Stellar Type}\n\\label{spec_type}\nThe optical spectrum of the companion star shows He I absorption lines, suggesting that the star is hot (i.e., $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}}$10,000 K) and, therefore, is likely to be of either O or B spectral class. Here we use the NOMAD $V$-band photometry in combination with the 2MASS NIR photometry to assess the spectral type, extinction, and distance to this star. The NOMAD $V$-band magnitude and the 2MASS NIR magnitudes can be found in Table \\ref{mw_mag}. To estimate the extinction we use the intrinsic color relationships between the $V$-band and NIR magnitudes provided by Table 2 in \\cite{2014AcA....64..261W}. To recover the intrinsic color relationships of late O and early to late B type stars, we need to deredden the source by an $A_V\\approx8.5-9.5$, assuming the extinction law of \\cite{1989ApJ...345..245C}. The 3D extinction maps\\footnote{We use the {\\tt mwdust} python package to examine the dust maps \\citep{2016ApJ...818..130B}.}  of \\cite{2006A&A...453..635M} suggest a distance of $\\approx$ 7-8 kpc for this range of $A_V$.\n\n\n\n\nThe reddening and estimated distance to the source can be used to place constraints on its spectral type and luminosity class. In order for the star to have an observed $V$-band magnitude of 16.12 at a distance of 7 kpc with a reddening of $A_V=8.5$, it should have an absolute magnitude of $M_V\\approx-6.6$. The lack of He II absorption features in the optical spectrum suggests that the star is more likely to be of B-type rather than O-type. By comparing the absolute magnitude to Table 7 in \\cite{2006MNRAS.371..185W} we find that the star is most likely to be of the Ia luminosity class. Further, the strong He I absorption lines indicate that the star is an early B-type star. However, we mention one caveat, which is that while the lower luminosity classes are not bright enough to be consistent with the distance\/reddening to the source, there is a large uncertainty in some of the absolute magnitudes for different luminosity classes (e.g., luminosity class Ib, II;  \\citealt{2006MNRAS.371..185W}). \n\n\n\nLastly, the optical spectrum of the star shows an H$\\alpha$ emission line, which is seen in both Be, and supergiant type stars. This line has been observed in a number of similar systems and is often variable in shape and intensity, sometimes showing a P-Cygni like profile (see e.g., \\citealt{2006ApJ...638..982N,2006ESASP.604..165N,2006A&A...455..653P}). Unfortunately, we have only obtained a single spectrum and cannot assess the variability of the H$\\alpha$ line. Additionally, it does not appear to be P-Cygni like in this single spectrum. \n\n\n\n\nIn comparison, by using optical and NIR photometry, \\cite{2017MNRAS.469.3901S} found the stellar companion of  AX J1949 to be consistent with a B0.5Ia type star at a distance, $d=8.8$ kpc, and having a reddening, $A_V=7.2$. We have also found that the star is consistent with an early B-type type star of the Ia luminosity class. However, our estimates of the reddening and distance differ slightly from \\cite{2017MNRAS.469.3901S}, in that we find a larger reddening and smaller distance. \n\n\n\\subsection{SFXT Nature}\n\nAt an assumed distance of $\\sim7$ kpc,  AX J1949's {\\sl Chandra} and {\\sl Swift-}XRT fluxes (when the source is significantly detected) correspond to luminosities of $\\sim1\\times10^{34}$ erg s$^{-1}$ and $\\sim3-8\\times10^{33}$ erg s$^{-1}$, respectively.  Further, at harder X-ray energies,  AX J1949 has a  {\\sl NuSTAR} 3$\\sigma$ upper-limit quiescent luminosity $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$<$}}} 3\\times10^{33}$ erg s$^{-1}$ in the 22-60 keV band, while during its flaring X-ray activity detected by {\\sl INTEGRAL}, it reached a peak luminosity of $\\sim10^{37}$ erg s$^{-1}$  in the 22-60 keV band \\citep{2017MNRAS.469.3901S}. This implies a lower-limit on the dynamical range of the system of $\\mathrel{\\hbox{\\rlap{\\hbox{\\lower4pt\\hbox{$\\sim$}}}\\hbox{$>$}}} 3000$. The large dynamical range and relatively low persistent X-ray luminosity are typical of SFXTs. Additionally, converting the absorbing column density, $N_{\\rm H}=5.6\\times10^{22}$ cm$^{-2}$, derived from the spectral fits to the X-ray data to $A_V$ following the relationship provided by \\cite{2015MNRAS.452.3475B}, we find $A_V\\approx20$. This $A_V$ is a factor of 2-2.5 larger than the $A_V$ derived from the interstellar absorption to the companion star (see Section \\ref{spec_type}). This suggests that a large fraction of the absorption is intrinsic to the source, which has also been observed in other SFXTs (e.g., XTE J1739-302, AX J1845.0-0433; \\citealt{2006ApJ...638..982N,2009A&A...494.1013Z}). Lastly, the optical counterpart to  AX J1949 appears to be an early supergiant B-type star, which have been found in $\\sim40\\%$ of the SFXT systems (see Figure 2 in \\citealt{2017mbhe.confE..52S}). NIR spectra and additional optical spectra should be undertaken to search for additional absorption features, as well as changes in the H$\\alpha$ line profile to solidify the spectral type. \n\n\n\n\n\n\n\n\\section{Summary and Conclusion}\n\nA large number of unidentified {\\sl INTEGRAL} sources showing rapid hard X-ray variability have been classified as SFXTs. We have analyzed  {\\sl Neil Gehrels Swift-XRT}, {\\sl Chandra}, and {\\sl NuSTAR} observations of the SFXT candidate  AX J1949, which shows variability on ks timescales. The superb angular resolution of {\\sl Chandra} has allowed us to confirm the optical counterpart to  AX J1949 and obtain its optical spectrum, which showed an H$\\alpha$ emission line, along with He I absorption features. The spectrum, coupled with multi-wavelength photometry allowed us to place constraints on the reddening, $A_V=8.5-9.5$, and distance, $d\\approx7-8$ kpc, to the source. We find that an early B-type Ia is the most likely spectral type and luminosity class of the star, making  AX J1949 a new confirmed member of the SFXT class.\n\n\n\\medskip\\noindent{\\bf Acknowledgments:}\nWe thank Justin Rupert for obtaining the optical spectrum at MDM.  We thank the anonymous referee for providing useful and constructive comments that helped to improve the paper. This {\\sl work} made use of observations obtained at the MDM Observatory, operated by Dartmouth College, Columbia University, Ohio State University, Ohio University, and the University of Michigan. This work made use of data from the {\\it NuSTAR} mission, a project led by the California Institute of Technology, managed by the Jet Propulsion Laboratory, and funded by the National Aeronautics and Space Administration. We thank the {\\it NuSTAR} Operations, Software and  Calibration teams for support with the execution and analysis of these observations.  This research has made use of the {\\it NuSTAR}  Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center (ASDC, Italy) and the California Institute of Technology (USA).  JH and JT acknowledge partial support from NASA through Caltech subcontract CIT-44a-1085101. JAT acknowledges partial support from Chandra grant GO8-19030X.\n\n \\software{CIAO (v4.10; \\citealt{2006SPIE.6270E..1VF}), XSPEC (v12.10.1; \\citealt{1996ASPC..101...17A}), NuSTARDAS (v1.8.0), Matplotlib \\citep{2007CSE.....9...90H}, Xselect (v2.4e), XIMAGE (v4.5.1), HEASOFT (v6.25), MWDust \\citep{2016ApJ...818..130B}}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n The study of fractional order differential equations has been attracting many scientists because of its adequate and interesting applications in modeling of real-life problems related to several fields of science  \\cite{L1}-\\cite{L5}. Initial-value problems (IVPs) and boundary-value problems (BVPs) involving the Riemann-Liouville and Caputo derivatives attract  most interest (see, for instance, \\cite{L6},  \\cite{L7}, \\cite{L8}).  Especially, studying IVPs and BVPs for the sub-diffusion, fractional wave equations are well-studied (see \\cite{sad}, \\cite{ruzh}, \\cite{karruzh}). BVPs for mixed type equations are also an interesting target for many authors (see \\cite{L9}-\\cite{L13}).\n\n Introducing a generalized Riemann-Liouville fractional derivatives (it is called  Hilfer's deriva\\-tive) has opened a new gate in the research of fractional calculus (\\cite{L14}-\\cite{L16}). Therefore, one can find several works devoted to studying this operator in various problems \\cite{L17}, \\cite{L18}.  We also note that in 1968,  M.~M.~Dzhrbashyan and A.~B.~Nersesyan introduced the following integral-differential operator \\cite{Ldn}\n\\begin{equation}\\label{DN}\nD_{0x}^{\\sigma_n}g(x)=I_{0x}^{1-\\gamma_n}D_{0x}^{\\gamma_{n-1}}...D_{0x}^{\\gamma_{1}}D_{0x}^{\\gamma_{0}}g(x), ~ ~ n\\in\\mathbb{N}, ~ x>0,\n\\end{equation}\nwhich is more general than Hilfer's operator. Here $I_{0x}^{\\alpha}$ and $D_{0x}^{\\alpha}$ are the Riemann-Liouville fractional\nintegral and the Riemann-Liouville fractional derivative of order $\\alpha$ respectively (see Definition 2.1),  $\\sigma_n\\in(0, n]$ which is defined by\n$$\n\\sigma_n=\\sum\\limits_{j=0}^{n}\\gamma_j-1>0, ~ \\gamma_j\\in(0, 1].\n$$\nThere are some works \\cite{bag}, \\cite{bag2}, related with this operator. New wave of researches involving this operator might appear due to the translation of original work \\cite{Ldn} in FCAA \\cite{Lfca}.\n \n In addition, from announcing the concept of hyper-Bessel fractional differential derivative by I. Dimovski \\cite{L21}, several articles have been published dedicated to studying problems containing this type of operators (see \\cite{L22}-\\cite{L26}).   For instance, fractional diffusion equation and wave equation were widely investigated in different domains in \\cite{L19}-\\cite{L20}.\n\nIn this work, we investigate a boundary value problem for a mixed equation involving the sub-diffusion equation with Caputo-like counterpart of a hyper-Bessel fractional differential operator and the fractional wave equation with Hilfer's bi-ordinal derivative in a  rectangular domain. The theorem about the uniqueness and existence of the solution is proved.\n\n{The rest of the paper is organized as follows: In Preliminaries section we provide necessary information on Mittag-Leffler functions (Section 2.1), hyper-Bessel functions (Section 2.2.), bi-ordinal Hilfer's fractional derivatives (Section 2.3) and on differential equation involving bi-ordinal Hilfer's fractional derivatives (Section 2.4). Auxiliary result is formulated in Theorem 2.2.  In Section 3, we formulate  the main problem and state our main result in Theorem 3.1.  In Appendix one can find detailed arguments of the proof of Theorem 2.1.}\n\n\n\\section{Preliminaries}\n\nIn this section we present some definitions and auxiliary  results related to generalized Hilfer's derivative and fractional hyper-Bessel differential operator which will be used in the sequel. We start recalling the definition of the Mittag-Leffler function.\n\n\\subsection{Important properties of the Mittag-Leffler function}\n\nThe two parameter Mittag-Leffler (M-L) function is an entire function given by\n\\begin{equation}\\label{e1}\nE_{\\alpha, \\beta}(z)=\\sum_{k=0}^{\\infty}\\frac{z^k}{\\Gamma(\\alpha{k}+\\beta)}, \\, \\, \\, \\alpha>0, \\, \\beta\\in\\mathbb{R}.\n\\end{equation}\n\n\\textbf{Lemma 2.1} (see \\cite{L3}) Let $\\alpha<2, \\, \\beta\\in\\mathbb{R}$ and $\\frac{\\pi\\alpha}{2}<\\mu<min\\{\\pi, \\pi\\alpha\\}$. Then the following estimate holds\n$$\n|E_{\\alpha, \\beta}(z)|\\leq\\frac{M}{1+|z|}, \\, \\, \\, \\mu\\leq|argz|\\leq\\pi, \\, \\, |z|\\geq0.\n$$\nHere and in the rest of the paper, $M$ denotes a positive constant.\n\nIn \\cite{L27}, the following estimate for $E_{\\delta, 1}(-t^\\delta)$ in the the form  $g_{\\delta}(t)\\leq{E}_{\\delta, 1}(-t^\\delta)\\leq{f_{\\delta}(t)}$ was given without proof, where\n$$\ng_{\\delta}(t)=\\frac{1}{t^\\delta\\Gamma(1-\\delta)+1}\\sim\\frac{t^{-\\delta}}{\\Gamma(1-\\delta)}\\sim{E}_{\\delta, 1}(-t^\\delta), \\, t \\to \\infty,\n$$\n$$\nf_{\\delta}(t)=\\frac{1}{1+\\frac{t^\\delta}{\\Gamma(1+\\delta)}}\\sim(1-\\frac{t^{\\delta}}{\\Gamma(1+\\delta)}\\sim{E}_{\\delta, 1}(-t^\\delta), \\, t \\to +0.\n$$\n\nRecently,  the lower and upper bounds of the Kilbas-Saigo function, which is a  more general form of $E_{\\delta, 1}(-t^\\delta)$, were announced. Also  the following preposition  about the bounds for two parameter M-L function  is given in \\cite{L28}.\n\n\\textbf{Proposition 2.1} (\\cite{L28}) For every $\\alpha\\in(0,1]$, \\, $\\beta>\\alpha$ and $x\\geq0$ one has\n$$\n\\frac{1}{\\Big(1+\\sqrt{\\frac{\\Gamma(1-\\alpha)}{\\Gamma(1+\\alpha)}}x\\Big)^2}\\leq{E_{\\alpha, \\alpha}(-x)}\\leq\\frac{1}{\\Big(1+\\frac{\\Gamma(1+\\alpha)}{\\Gamma(1+2\\alpha)}x\\Big)^2}\n$$\nand\n$$\n\\frac{1}{1+\\frac{\\Gamma(\\beta-\\alpha)}{\\Gamma(\\beta)}x}\\leq\\Gamma(\\beta){E_{\\alpha, \\beta}(-x)}\\leq\\frac{1}{1+\\frac{\\Gamma(\\beta)}{\\Gamma(\\beta+\\alpha)}x}.\n$$\n\nThe Laplace transform of M-L function is given in the following lemma.\n\n\\textbf{Lemma 2.2.} (\\cite{L8}) For any $\\alpha>0, \\, \\, \\beta>0$  and $\\lambda\\in\\mathbb{C}$, we have\n$$\n\\mathcal{L}\\{t^{\\beta-1}E_{\\alpha, \\beta}(\\lambda{t}^\\alpha)\\}=\\frac{s^{\\alpha-\\beta}}{s^\\alpha-\\lambda}, \\, \\, (Re(s)>\\mid{\\lambda}\\mid^{1\/\\alpha}),\n$$\nwhere the Laplace transform of  a function $f(t)$ is defined by\n\n$$\n\\mathcal{L}\\{f\\}(s):=\\int_{0}^{\\infty}e^{-st}f(t)dt.\n$$\n\n\\textbf{Lemma 2.3.}\\label{lemma} If $\\alpha\\leq{0}$  and $\\beta\\in\\mathbb{C}$, then the following recurrence formula holds:\n$$\nE_{\\alpha, \\beta}(z)=\\frac{1}{\\Gamma(\\beta)}+zE_{\\alpha, \\alpha+\\beta}(z).\n$$\nThis lemma was proved by R. K. Saxena in 2002 \\cite{L29}.\n\nLater, we use the properties of a Wright-type function studied by A. Pskhu \\cite{L30}, defined as\n$$\ne^{\\mu, \\delta}_{\\alpha, \\beta}(z)=\\sum_{n=0}^{\\infty}\\frac{z^n}{\\Gamma(\\alpha{n}+\\mu)\\Gamma(\\delta-\\beta{n})}, \\, \\, \\, \\alpha>0, \\, \\, \\alpha>\\beta.\n$$\n\nM-L function can be determined by Wright-type function as a special case $E_{\\alpha, \\beta}(z)=e^{\\beta, 1}_{\\alpha, 0}(z)$.  So, we can record some properties of M-L function which can be reduced from the Wright-type function's properties.\n\n\\textbf{Lemma 2.4.} (\\cite{L30}) If $\\pi\\geq|argz|>\\frac{\\pi\\alpha}{2}+\\varepsilon, \\, \\, \\, \\varepsilon>0$,  then the following relations are valid for $z \\to \\infty$:\n$$\n\\mathop {\\lim }\\limits_{\\mid{z}\\mid \\to  \\infty} E_{\\alpha, \\beta}(z)=0,\n$$\n$$\n\\mathop {\\lim }\\limits_{\\mid{z}\\mid \\to  \\infty} zE_{\\alpha, \\beta}(z)=-\\frac{1}{\\Gamma(\\beta-\\alpha)}.\n$$\n\n\n\n\\subsection{Regularized Caputo-like counterpart of the hyper-Bessel fractional differential operator}\n\n\\textbf{Definition 2.1.} (\\cite{L8}) The Riemann-Liouville fractional integral $I^{\\alpha}_{a+}f(t)$ and derivative  $D^{\\alpha}_{a+}f(t)$ of order $\\alpha$ are defined by\n$$\nI^{\\alpha}_{a+}f(t)=\\frac{1}{\\Gamma(\\alpha)}\\int_{a}^{t}\\frac{f(\\tau)d\\tau}{(t-\\tau)^{1-\\alpha}},\n$$\n\n$$\nD^{\\alpha}_{a+}f(t)=\\left(\\frac{d}{dt}\\right)^nI^{n-\\alpha}_{a+}f(t), \\, \\, \\, \\,   n-1\\leq\\alpha<n,\n$$\nwhere $\\Gamma(\\alpha)$ is Euler's gamma-function.\n\n\\textbf{Definition 2.2}. The Erdelyi-Kober (E-K) fractional integral of a function  $f(t)\\in{C_\\mu}$ with arbitrary parameters $\\delta>0, \\gamma\\in{\\mathbb{R}}$ and $\\beta>0$ is defined as (\\cite{L8})\n$$\nI^{\\gamma, \\delta}_{\\beta; a+}f(t)=\\frac{t^{-\\beta(\\gamma+\\delta)}}{\\Gamma(\\delta)}\\int_{a}^{t}(t^\\beta-\\tau^\\beta)^{\\delta-1}\\tau^{\\beta\\gamma}f(\\tau)d(\\tau^\\beta),\n$$\nwhich can be reduced up to a weight to  $I^{q}_{a+}f(t)$ (Riemann-Liouville fractional integral) at $\\gamma=0$ and $\\beta=1$, and Erdelyi-Kober fractional derivative of $f(t)\\in{C}_{\\mu}^{(n)}$ for $n-1<\\delta\\leq{n}, n\\in\\mathbb{N}$, is defined by\n$$\nD_{\\beta, a+}^{\\gamma, \\delta}f(t)=\\prod_{j=1}^{n}\\left(\\gamma+j+\\frac{t}{\\beta}\\frac{d}{dt}\\right)\\big(I_{\\beta, a+}^{\\gamma+\\delta, n-\\delta} f(t)\\big),\n$$\nwhere $C_{\\mu}^{(n)}$  is the  weighted space of continuous functions defined as\n$$\nC_{\\mu}^{(n)}=\\left\\{f(t)=t^p \\tilde{f(t)}; \\, \\, \\tilde{f} \\in{C}^{(n)}[0, \\infty)\\right\\}, \\, \\, \\, C_{\\mu}=C_{\\mu}^{(0)} \\, \\,  \\textrm{with} \\, \\, \\mu\\in\\mathbb{R}.\n$$\n\n\\textbf{Definition 2.3.} Regularized Caputo-like counterpart of the hyper-Bessel fractional differential operator for  $\\theta<1$,  $0<\\alpha\\leq1$ and $t>a\\geq{0}$ is defined in terms of the E-K fractional order operator\n\n\\begin{equation}\\label{e2}\n^{C}\\Big(t^{\\theta}\\frac{d}{dt}\\Big)^\\alpha{f}(t)=(1-\\theta)^\\alpha{t}^{-\\alpha(1-\\theta)}D_{1-\\theta, a+}^{-\\alpha, \\alpha}\\left(f(t)-f(a)\\right)\n\\end{equation}\nor in terms of the hyper-Bessel differential (R-L type) operator\n\n\\begin{equation}\\label{eq3}\n^{C}\\left(t^{\\theta}\\frac{d}{dt}\\right)^\\alpha{f}(t)=\\Big(t^{\\theta}\\frac{d}{dt}\\Big)^\\alpha{f}(t)-\\frac{f(a)\\Big(t^{(1-\\theta)}-a^{(1-\\theta)}\\Big)^{-\\alpha}}{(1-\\theta)^{-\\alpha}\\Gamma(1-\\alpha)},\n\\end{equation}\nwhere\n$$\n\\left(t^\\theta\\frac{d}{dt}\\right)^\\alpha{f}(t)=\\left\\{ \\begin{gathered}\n(1-\\theta)^\\alpha t^{-(1-\\theta)\\alpha}I_{1-\\theta, a+}^{0, -\\alpha} f(t) \\, \\, \\,  \\textrm{if}  \\, \\,  \\,  \\theta<1, \\hfill \\cr\n(\\theta-1)^\\alpha t^{-(1-\\theta)\\alpha}I_{1-\\theta, a+}^{-1, -\\alpha} f(t)  \\, \\, \\,  \\textrm{if} \\, \\,  \\,  \\theta>1,  \\hfill \\cr\n\\end{gathered}  \\right.\n$$\nis a hyper-Bessel fractional differential operator (\\cite{L25}).\n\nFrom (\\ref{eq3}) for $a=0$ we obtain the definition presented in (\\cite{L25}) and also Caputo FDO is the particular case of Caputo-like counterpart hyper-Bessel operator at $\\theta=0$.\n\n\n\\textbf{Theorem 2.1.}  Assume that the following conditions hold:\n\n$\\bullet$ $\\tau\\in{C}[0,1]$ such that $\\tau(0)=\\tau(1)=0$ and $\\tau'\\in{L}^2(0,1)$,\n\n$\\bullet$ $f(\\cdot, t)\\in{C}^3[0,1]$ and $f(x, \\cdot)\\in{C}_\\mu[a, T]$ such that\n\n $f(0,t)=f(\\pi,t)=f_{xx}(0,t)=f_{xx}(1,t)=0$, and $\\dfrac{\\partial^4}{\\partial{x}^4}f(\\cdot,t)\\in{L}^1(0,1).$\n\nThen, in  $\\Omega=\\{0<x<1, \\, \\, a<t<T \\}$,  the problem of finding the solution of the equation\n$$\n^C\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^{\\alpha}u(x,t)-u_{xx}(x,t)=f(x,t),\n$$\nsatisfying the conditions\n$$\nu(0,t)=0, u(1,t)=0, \\, \\, \\, a\\leq{t}\\leq{T},\n$$\n$$\nu(x,0)=\\tau(x), \\, \\, \\, \\, 0\\leq{x}\\leq{1},\n$$\nhas a unique solution given by\n\\begin{equation}\\label{e4}\nu(x,t)=\\sum_{k=1}^{\\infty}\\Big[\\tau_kE_{\\alpha, 1}\\Big(-\\frac{(k\\pi)^2}{p^\\alpha}(t^{p}-a^{p})^\\alpha\\Big)+G_k(t)\\Big]\\sin(k\\pi{x}),\n\\end{equation}\nwhere $p=1-\\theta$ and\n$$\n\\begin{array}{l}\n\\displaystyle{G_k(t)=\\frac{1}{p^\\alpha\\Gamma(\\alpha)}\\int_{a}^{t}\\big(t^p-\\tau^p\\big)^{\\alpha-1}f_k(\\tau)d(\\tau^p)}\\\\\n\\\\\n\\displaystyle{-\\frac{(k\\pi)^2}{p^{2\\alpha}}\\int_{a}^{t}\\big(t^p-\\tau^p\\big)^{2\\alpha-1}E_{\\alpha, 2\\alpha}\\Big[-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-\\tau^p)^{\\alpha}\\Big]f_k(\\tau)d(\\tau^p),}\n\\end{array}\n$$\n$$\n\\tau_k=2\\int_{0}^{1}\\tau(x)\\sin(k\\pi{x})dx, \\, \\, \\, \\, \nf_k(t)=2\\int_{0}^{1}f(x,t)\\sin(k\\pi{x})dx, \\, \\, \\, \\, k=1,2,3,...\n$$\n\nIn fact, for $a=0$  Theorem 2.1  implies the result of (\\cite{L22}) (see Theorem 3.1). For the detailed proof of Theorem 2.1 see appendix.\n\\subsection{ Bi-ordinal Hilfer's fractional derivative}\n\\textbf{Definition 2.4.} Hilfer's derivative $D^{\\alpha, \\mu}_{a+}$ of order $\\alpha \\, \\, \\,  (n-1<\\alpha\\leq{n},   \\, \\, \\, n\\in\\mathbb{N})$  of type $\\mu \\, \\,  (0\\leq\\mu\\leq1)$ is defined by \\cite{L14}:\n\\begin{equation}\\label{e5}\nD^{\\alpha, \\mu}_{t}f(t)=I^{\\mu(n-\\alpha)}_{0+}\\left(\\frac{d}{dt}\\right)^nI^{(1-\\mu)(n-\\alpha)}_{0+}f(t).\n\\end{equation}\nThen in \\cite{L20}, V. M. Bulavatsky considered generalized Hilfer's derivative in the form\n$$\nD^{(\\alpha, \\beta) \\mu}_{t}f(t)=I^{\\mu(1-\\alpha)}_{0+}\\frac{d}{dt}I^{(1-\\mu)(1-\\beta)}_{0+}f(t),\n$$\nhere  $0<\\alpha, \\beta\\leq{1}$,  $0\\leq\\mu\\leq{1}$.\n\n\nIn the same way, one can present Hilfer's bi-ordinal fractional derivative of orders \\\\ $\\alpha \\, \\,  (n-1<\\alpha\\leq{n})$, $\\beta \\, \\, (n-1<\\beta\\leq{n})$ and of type $\\mu\\in[0,1]$  by the following relation:\n\n\\begin{equation}\\label{e6}\nD^{(\\alpha, \\beta) \\mu}_{t}f(t)=I^{\\mu(n-\\alpha)}_{0+}\\left(\\frac{d}{dt}\\right)^nI^{(1-\\mu)(n-\\beta)}_{0+}f(t).\n\\end{equation}\n\nIn general, (\\ref{e6}) is also preserved as (\\ref{e5}) in terms of its interpolation concept. Specifically, when $\\mu=0$, (\\ref{e6}) gives the Riemann-Liouville fractional derivative of $\\beta$ order and for $\\mu=1$, the bi-ordinal Hilfer's derivative expresses the Caputo fractional derivative of order $\\alpha$.\n\nSimilarly, generalized Hilfer's  fractional derivative of orders $\\gamma\\in(1,2]$, and $\\beta\\in(1,2]$ and type $\\mu\\in[0,1]$ can be written as a special case of (\\ref{e6}) for $n=2$:\n\\begin{equation}\\label{e7}\nD^{(\\gamma, \\beta)\\mu}_tf(t)=I^{\\mu(2-\\gamma)}_{0+}\\left(\\frac{d}{dt}\\right)^2I^{(1-\\mu)(2-\\beta)}_{0+}f(t).\n\\end{equation}\n\nHere we present the formula for the Laplace transform of (\\ref{e7}) which will be used later:\n$$\n  \\mathcal{L}\\{D^{(\\alpha, \\beta)\\mu}_{t}f(t)\\}=s^{\\beta+\\mu(\\alpha-\\beta)}\\mathcal{L}\\{f(t)\\}-\n $$\n  \\begin{equation}\\label{e8}\n  -s^{1-\\mu(2-\\alpha)}\\left[I_{0+}^{(1-\\mu)(2-\\beta)}f(t)|_{t \\to 0+}\\right]-s^{-\\mu(2-\\alpha)}\\left[\\frac{d}{dt}I_{0+}^{(1-\\mu)(2-\\beta)}f(t)|_{t \\to 0+}\\right].\n\\end{equation}\n\n\n\\subsection{Differential equation involving bi-ordinal Hilfer's derivative}\n\nLet us consider the following problem:\n\n\\textit{Find a solution of the equation}\n\\begin{equation}\\label{e9}\nD^{(\\gamma, \\beta)\\mu}_{t}u(t)+\\lambda{u}(t)=f(t),  \\, \\, \\,   (1<\\gamma, \\beta\\leq2, \\, \\, 0\\leq\\mu\\leq{1}),\n\\end{equation}\n\\textit{satisfying the initial conditions}\n\\begin{equation}\\label{e10}\n\\mathop {\\lim }\\limits_{t \\to  0+}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(t)=\\xi_0,\n\\end{equation}\n\\begin{equation}\\label{e11}\n\\mathop {\\lim }\\limits_{t \\to  0+}\\frac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(t)=\\xi_1,\n\\end{equation}\n\\textit{where} $f(t)$ \\textit{is a given function} $f\\in{L^1}(0, \\infty)$, and $\\lambda, \\, \\, \\xi_0, \\, \\,  \\xi_1=const.$\n\n\\textbf{Theorem 2.2.} If  $f\\in{C^1}(0, +\\infty)$, then  the problem (\\ref{e9})-(\\ref{e11}) has an unique solution represented by\n$$\nu(t)=\\xi_0t^{(\\beta-2)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)-1}(-\\lambda{t}^\\delta)+\\xi_1t^{\\mu+(\\beta-1)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)}(-\\lambda{t}^\\delta)+\n$$\n\\begin{equation}\\label{e12}\n\t+\\int_{0}^{t}(t-\\tau)^{\\delta-1}E_{\\delta, \\delta}(-\\lambda(t-\\tau)^{\\delta})f(\\tau)d\\tau,\n\\end{equation}\nwhere $\\delta=\\beta+\\mu(\\gamma-\\beta)$.\n\n{\\sf Proof:}\n Applying the Laplace transform (\\ref{e9}) by means of (\\ref{e8}) and considering initial conditions (\\ref{e10}), (\\ref{e11}) yield\n\\begin{equation}\\label{e13}\n\\mathcal{L}\\{u\\}=\\frac{\\xi_0s^{1-\\mu(2-\\gamma)}+\\xi_1s^{-\\mu(2-\\gamma)}+\\mathcal{L}\\{f\\}}{s^{\\beta+\\mu(\\gamma-\\beta)}+\\lambda},\n\\end{equation}\nwhere $\\mathcal{L}\\{u\\}$ and $\\mathcal{L}\\{f\\}$ are the Laplace transform of functions $u$ and $f$, respectively.\n\nAccording to Lemma 2.2, the Laplace transform of the Mittag-Leffler function \\cite{L3}, \\cite{L8} as follows\n$$\n\\mathcal{L}^{-1}\\{\\frac{s^{1-\\mu(2-\\gamma)}}{s^{\\beta+\\mu(\\gamma-\\beta)}+\\lambda}\\}=t^{\\beta-2+\\mu(2-\\beta)}E_{\\beta+\\mu(\\gamma-\\beta), \\beta-1+\\mu(2-\\beta)}(-\\lambda{t}^{\\beta+\\mu(\\gamma-\\beta)}),\n$$\n$$\n\\mathcal{L}^{-1}\\{\\frac{s^{-\\mu(2-\\gamma)}}{s^{\\beta+\\mu(\\gamma-\\beta)}+\\lambda}\\}=t^{\\beta-1+\\mu(2-\\beta)}E_{\\beta+\\mu(\\gamma-\\beta), \\beta+\\mu(2-\\beta)}(-\\lambda{t}^{\\beta+\\mu(\\gamma-\\beta)}),\n$$\n\n$$\n\\mathcal{L}^{-1}\\{\\frac{\\mathcal{L}\\{f\\}}{s^{\\beta+\\mu(\\gamma-\\beta)}+\\lambda}\\}=\\int_{0}^{t}(t-\\tau)^{\\beta-1+\\mu(\\gamma-\\beta)}E_{\\beta+\\mu(\\gamma-\\beta), \\beta+\\mu(\\gamma-\\beta)}(-\\lambda(t-\\tau)^{\\beta+\\mu(\\gamma-\\beta)})f(\\tau)d\\tau,\n$$\nwhere $\\mathcal{L}^{-1}$ is an inverse Laplace transform operator.\n\nConsidering above evaluations and after applying the inverse Laplace transform to (\\ref{e13}), we can write the solution of (\\ref{e9})-  (\\ref{e11}) in the form  (\\ref{e12}). Theorem 2.2 is proved.\n\n\\section{Formulation of a problem and main result}\n\n\nLet us consider the following equation\n\\begin{equation}\\label{e14}\nf(x, t) = \\left\\{ \\begin{gathered}\n\t^C\\left( {{t^\\theta }\\frac{\\partial }{{\\partial t}}} \\right)^\\alpha u\\left( {x,t} \\right) -\n\t{u_{xx}}\\left( {x,t} \\right),\\,\\,\\,\\,(x,t)\\in{\\Omega_1},  \\hfill \\cr\n\t{D^{(\\gamma, \\beta)\\mu}_{t}u}\\left( {x,t} \\right) - {u_{xx}}\\left( {x,t} \\right),\\,\\,\\,\\,(x,t)\\in{\\Omega_2},  \\hfill \\cr\n\\end{gathered}  \\right.\n\\end{equation}\nin a domain $\\Omega=\\Omega_1\\cup\\Omega_2\\cup{Q}$. Here\n$\\Omega_1=\\{(x,t): 0<x<1,\\, \\,  \\, a<t<b\\}$,  \\, \\, \\, \\,  \\\\   $\\Omega_2=\\{(x,t): 0<x<1, 0<t<a\\}$,\n$Q=\\{(x,t): 0<x<1,   t=a \\}$, \\, $a, b\\in\\mathbb{R^+}$ such that  $a<b$, $0<\\alpha\\leq1$, \\,  $\\theta<1$, \\, $1<\\gamma, \\beta<2$, \\, $0\\leq\\mu\\leq1$, $f(x,t)$ is a given function, $^C\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha$ is the regularized Caputo-like counterpart of the hyper-Bessel operator defined as in (\\ref{e2}), $D^{(\\gamma, \\beta)\\mu}_{t} $ is the bi-ordinal Hilfer's derivative defined as in (\\ref{e7}).\n\n\\textbf{Problem}. Find a solution of (\\ref{e14}) in $\\Omega$, satisfying regularity conditions\n$$\nu(x,t)\\in{C}(\\Omega)\\cap{C^1}(\\Omega)\\cap{C^2}(\\Omega_2), \\, \\, \\, ^C\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha{u}(x,t)\\in{C}(\\Omega_1), \\,\n$$\n$$\nu_{xx}(x,t)\\in{C^2}(\\Omega_1), \\, \\, \\, \\, \\, D_{t}^{(\\gamma, \\beta)\\mu}u(x,t)\\in{C(\\Omega_2)},\n$$\nand the boundary-initial conditions\n\\begin{equation}\\label{e15}\nu(0,t)=0, \\, \\, \\, \\, \\,  \\,   0\\leq{t}\\leq{b},\n\\end{equation}\n\\begin{equation}\\label{e16}\nu(1,t)=0,  \\, \\,  \\,   \\, \\, \\, 0\\leq{t}\\leq{b},\n\\end{equation}\n\\begin{equation}\\label{e17}\n\\mathop {\\lim }\\limits_{t \\to  0+}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(x,t)=\\varphi(x), \\, \\, \\, \\, 0\\leq{x}\\leq1,\n\\end{equation}\nas well as the gluing conditions\n\n\\begin{equation}\\label{e18}\n\\mathop {\\lim }\\limits_{t \\to  a-}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(x,t)=\\mathop {\\lim }\\limits_{t \\to  a+}u(x,t), \\, \\, \\, \\, \\,  0\\leq{x}\\leq{1},\n\\end{equation}\n\n\\begin{equation}\\label{e19}\n\\mathop {\\lim }\\limits_{t \\to  a-}\\frac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(x,t)=\\mathop {\\lim }\\limits_{t \\to  a+}(t-a)^{1-(1-\\theta)\\alpha}u_t(x,t), \\, \\, \\, \\, \\,  0<x<1,\n\\end{equation}\nhere  $\\varphi(x)$ is a given function.\n\nThe key motivation to formulate this problem is a possible application in diffusion-wave processes, which will be described by the mixed type equation as Eq. (\\ref{e14}) \\cite{KDSc}. Moreover, fractional derivatives used in the mixed equation have more general and also specific character. Therefore, we separately studied related Cauchy problems for ordinary differential equations of fractional order.\n\nThe intention of this paper is to prove the uniqueness and existence of the solution to the problem (\\ref{e14})-(\\ref{e19}), as stated in the following theorem.\n\n\\textbf{Theorem 3.1.} If the following conditions\n\n1)  $0<\\beta+\\mu(\\gamma-\\beta)\\leq1$,   \\\\\n\n2)  $\\varphi\\in{C}[0,1]$ and $\\varphi'\\in{L}^2[0,1],$ \\\\\n\n3)  $f(\\cdot,t)\\in{C}^3[0,1]$ and $f(x,\\cdot)\\in{C}_\\mu$ such that $f(0,t)=f(1,t)=f_{xx}(0,t)=f_{xx}(1,t)=0$ and\n$\\dfrac{\\partial^4}{\\partial{x}^4}f(\\cdot, t)\\in{L^1(0,1)}$\nhold, then there exists a unique solution of the considered problem (\\ref{e14})-(\\ref{e19}).\n\n{\\sf Proof:}\n\nFirst we introduce new notations:\n\\begin{equation}\\label{e20}\n\\mathop {\\lim }\\limits_{t \\to  0+}\\frac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}u(x,t)=\\psi(x), \\, \\, \\,  0<x<1,\n\\end{equation}\n\n\\begin{equation}\\label{e21}\n\t\\mathop {\\lim }\\limits_{t \\to  a+}u(x,t)=\\tau(x),  \\, \\, \\, \\, 0\\leq{x}\\leq1,\n\\end{equation}\nhere $\\tau(x)$ and $\\psi(x)$ are unknown functions to be found later.\n\nUsing the method of separation of variables for solving the homogeneous equation corresponding to (\\ref{e14}), i.e. searching solution as $u(x,t)=T(t)X(x)$ and considering  (\\ref{e15}) and (\\ref{e16}) in homogeneous case, yield the following problem:\n\n\\begin{equation}\\label{e22}\n\tX''(x)+\\lambda{X}(x)=0, \\, \\, \\ \\ X(0)=0, \\, \\, X(1)=0.\n\\end{equation}\nIt is obvious that (\\ref{e22}) is a Sturm-Liouville problem on finding eigenvalues and eigenfunctions and it has the following solution:\n\n\\begin{equation}\\label{e23}\n\\lambda_k=(k\\pi)^2, \\, \\, \\, X_k(x)=\\sin(k\\pi{x}),  \\, \\, \\, k=1, 2, 3, ....\n\\end{equation}\n\nUsing the fact that the system of eigenfunctions $\\{X_k\\}$ in (\\ref{e23}) forms an orthogonal basis in $L^2(0,1)$ \\cite{L31}, we look for the solution $u(x,t)$ and given function $f(x,t)$ in the form of series expansions as follows:\n\n\\begin{equation}\\label{e24}\nu(x,t)=\\sum_{k=1}^{\\infty}u_k(t)\\sin(k\\pi{x}),\t\n\\end{equation}\n\\begin{equation}\\label{e25}\n\tf(x,t)=\\sum_{k=1}^{\\infty}f_k(t)\\sin(k\\pi{x}),\t\n\\end{equation}\nwhere $u_k(t)$ is unknown function to be found, $f_k(t)$ is known and given by\n$$\nf_k(t)=2\\int_{0}^{1}f(x,t)\\sin(k\\pi{x})dx.\n$$\n\nSubstituting (\\ref{e24}) and (\\ref{e25}) into equation (\\ref{e14}) in $\\Omega_1$ and considering initial condition (\\ref{e20}) gives the following fractional differential equation\n$$\n^C\\left(t^\\theta\\frac{d}{dt}\\right)^{\\alpha}u_k(t)+(k\\pi)^2u_k(t)=f_k(t)\n$$\nwith initial condition\n$$\nu_k(a+)=\\tau_k,\n$$\nwhere $\\tau_k$ is the coefficient of series expansion of $\\tau(x)$ in terms of orthogonal basis (\\ref{e22}), i.e.,\n$$\n\\tau_k=2\\int_{0}^{1}\\tau(x)\\sin(k\\pi{x})dx.\n$$\nAfter finding the solution of this problem, then considering (\\ref{e23}) we can write the solution of (\\ref{e14}) in  $\\Omega_1$ satisfying the conditions (\\ref{e15}), (\\ref{e16}) and  (\\ref{e21}) stated in (\\ref{e4}).\n\n\nNow by using the solution (\\ref{e4}), we evaluate $(t-a)^{1-(1-\\theta)\\alpha}u_t(x,t)$:\n$$\n(t-a)^{1-(1-\\theta)\\alpha}u_t(x,t)=\\sum_{k=1}^{\\infty}\\left[-\\frac{(k\\pi)^2}{p^{\\alpha-1}}\\tau_k \\, E_{\\alpha, \\alpha}\\Big(-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)^{\\alpha}\\Big)+(t-a)^{1-p\\alpha}G_k(t)\\right]\\sin(k\\pi{x}).\n$$\n\nConsidering above given evaluations we obtain the following relation on $Q$ deduced from $\\Omega_1$ as $t\\to{a+}$:\n\n\\begin{equation}\\label{e26}\n\\mathop {\\lim }\\limits_{t \\to  a+}(t-a)^{1-(1-\\theta)\\alpha}u_t(x,t)=\\sum_{k=1}^{\\infty}\\left[-\\frac{(k\\pi)^2}{\\Gamma(\\alpha)p^{\\alpha-1}}\\tau_k \\right]\\sin(k\\pi{x}).\n\\end{equation}\n\nNow we establish another relation on $Q$ which will be reduced from $\\Omega_2$.\n\nAccording to variable separation method, considering (\\ref{e24}), (\\ref{e25}) and initial conditions (\\ref{e17}), (\\ref{e20}), we obtain the following problem finding a solution of equation\n$$\nD_{t}^{(\\gamma, \\beta)\\mu}W(t)+\\lambda_k{W}(t)=f(t)\n$$\nand satisfying the initial conditions\n$$\n\\mathop {\\lim }\\limits_{t \\to  0+}{I^{(1-\\mu)(2-\\beta)}_{0+}}W(t)=\\varphi_k,\n$$\n$$\n\\mathop {\\lim }\\limits_{t \\to  0+}\\frac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}W(t)=\\psi_k.\n$$\n\nIt is obvious that (\\ref{e12}) is the solution for above-given problem. Hence, using the solution (\\ref{e12}) and taking  into account (\\ref{e24})  we write the solution of (\\ref{e14}) in $\\Omega_2$ satisfying (\\ref{e17}) and (\\ref{e20}) as\n\\begin{equation}\\label{e27}\nu(x,t)=\\sum_{k=1}^{\\infty}W_k(t)\\sin(k\\pi{x}),\n\\end{equation}\nwhere\n$$\nW_k(t)=\\varphi_kt^{(\\beta-2)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)-1}(-\\lambda_k{t}^\\delta)+\\psi_kt^{\\mu+(\\beta-1)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)}(-\\lambda_k{t}^\\delta)+\n$$\n$$\n+\\int_{0}^{t}(t-\\tau)^{\\delta-1}E_{\\delta, \\delta}\\left[-\\lambda_k(t-\\tau)^\\delta\\right]f_k(\\tau)d\\tau,\n$$\nhere $\\delta=\\beta+\\mu(\\gamma-\\beta)$ and $\\varphi_k$ is coefficient of the series expansion of  $\\varphi(x)$, i.e.,\n$$\n\\varphi_k=2\\int_{0}^{1}\\varphi(x)\\sin(k\\pi{x})dx\n$$\nand $\\psi_k$ is not known yet.\n\nNow using (\\ref{e27}) we simplify $\\mathop {\\lim }\\limits_{t \\to  a-}{I^{(1-\\mu)(2-\\beta)}_{0+}}W_k(t)$ and $\\mathop {\\lim }\\limits_{t \\to  a-}\\dfrac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}W_k(t)$ in the following formulas\n\n\\begin{equation}\\label{e28}\n\\begin{gathered}\n\\mathop {\\lim }\\limits_{t \\to  a-}{I^{(1-\\mu)(2-\\beta)}_{0+}}W_k(t)=\\varphi_k \\, E_{\\delta, 1}(-\\lambda_k{a}^\\delta)+\\psi_k a\\, E_{\\delta, 2}(-\\lambda_k{a}^\\delta)\\\\\n+\\int_{0}^{a}(a-s)^{\\delta+q-1}E_{\\delta, \\delta+q}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds,\n\\end{gathered}\n\\end{equation}\n\\begin{equation}\\label{e29}\n\\begin{gathered}\n\\mathop {\\lim }\\limits_{t \\to  a-}\\frac{d}{dt}{I^{(1-\\mu)(2-\\beta)}_{0+}}W_k(t)=-\\varphi_k\\lambda_k{a}^{\\delta-1} \\, E_{\\delta, \\delta}(-\\lambda_k{a}^\\delta)+\\psi_k \\, E_{\\delta, 1}(-\\lambda_k{a}^\\delta)\\\\\n+\\int_{0}^{a}(a-s)^{\\delta+q-2}E_{\\delta, \\delta+q-1}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds.\n\\end{gathered}\n\\end{equation}\nAfter substituting (\\ref{e28}) and (\\ref{e21}) into gluing condition (\\ref{e18})  and substituting (\\ref{e29}), (\\ref{e26}) into the gluing condition (\\ref{e19}), we obtain the following the system of linear algebraic equations with respect to $\\tau_k$ and $\\psi_k$:\n\n\\begin{equation}\\label{e30}\n \\left\\{ \\begin{gathered}\n  \\varphi_k \\, E_{\\delta, 1}(-\\lambda_k{a}^\\delta)+\\psi_k a\\, E_{\\delta, 2}(-\\lambda_k{a}^\\delta)\n\t\t+\\int_{0}^{a}(a-s)^{\\delta+q-1}E_{\\delta, \\delta+q}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds=\\tau_k,\\\\\n\\varphi_k\\lambda_k{a}^{\\delta-1} \\, E_{\\delta, \\delta}(-\\lambda_k{a}^\\delta)-\\psi_k \\, E_{\\delta, 1}(-\\lambda_k{a}^\\delta)-\\\\\n\\int_{0}^{a}(a-s)^{\\delta+q-2}E_{\\delta, \\delta+q-1}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds=\\frac{\\lambda_k}{\\Gamma(\\alpha)p^{\\alpha-1}}\\tau_k.\\\\\n \\end{gathered}  \\right.\n\\end{equation}\n\nFrom (\\ref{e30}), we find $\\psi_k$ and $\\tau_k$:\n\\begin{equation}\\label{e31}\n\\psi_k=\\frac{B}{\\Delta}\\varphi_k+\\frac{C}{\\Delta},\n\\end{equation}\n$$\n\\tau_k=\\Big(E_{\\delta, 1}(-\\lambda_k{a}^\\delta)+\\frac{B}{\\Delta}E_{\\delta, 2}(-\\lambda_k{a}^\\delta)\\Big)\\varphi_k +\\frac{C}{\\Delta}E_{\\delta, 2}(-\\lambda_k{a}^\\delta)+\n$$\n\\begin{equation}\\label{e32}\n+\\int_{0}^{a}(a-s)^{\\delta+q-1}E_{\\delta, \\delta+q}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds,\n\\end{equation}\nhere\n$$\n\\Delta=E_{\\delta, 1}(-\\lambda_k{a}^\\delta)+\n\\frac{\\lambda_k{a}}{\\Gamma(\\alpha){p}^{\\alpha-1}}E_{\\delta, 2}(-\\lambda_k{a}^\\delta),\n$$\n\n$$B=\\frac{-\\lambda_k{p}^{1-\\alpha}}{\\Gamma(\\alpha)}E_{\\delta, 1}(-\\lambda_k{a}^\\delta)+\\lambda_k{a}^{\\delta-1}E_{\\delta, \\delta}(-\\lambda_k{a}^\\delta),$$\n$$\nC=\\frac{-\\lambda_k{p}^{1-\\alpha}}{\\Gamma(\\alpha)}\\int_{0}^{a}(a-s)^{\\delta+q-1}E_{\\delta, \\delta+q}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s)ds-\\int_{0}^{a}(a-s)^{\\delta+q-2}E_{\\delta, \\delta+q-1}\\big[-\\lambda_k(a-s)^\\delta\\big]f_{k}(s),\n$$\nhere $\\lambda_k=(k\\pi)^2$.\n\nThe system of linear equations (\\ref{e30}) is equivalent to the considered problem in terms of existing the solution. For that reason, if  $\\Delta\\neq{0}$ for any $k$,  (\\ref{e30}) has only one solution or the considered problem's solution is unique if it exists. Therefore, we show that $\\Delta$ is not equal to zero for any $k$.\n\nBy using Lemma 2.4,  the behavior of $\\Delta$ at $k \\to \\infty$ can be written as:\n$$\n \\mathop {\\lim }\\limits_{k \\to  \\infty}\\Delta=\\mathop {\\lim }\\limits_{\\mid{z}\\mid \\to  \\infty}\\big[E_{\\delta, 1}(z)+\\frac{p^{1-\\alpha}}{\\Gamma(\\alpha)a^{\\delta-1}}zE_{\\delta, 2}(z)\\big]=\\frac{p^{1-\\alpha}}{\\Gamma(\\alpha)\\Gamma(2-\\delta)a^{\\delta-1}}\n$$\n\nThis proves that $\\Delta\\neq{0}$ for sufficiently large $k$.\n\n\nAccording to Theorem 3.1 and using Proposition 2.1  we can show that\n$$\\Delta=E_{\\delta,1}(-\\lambda_k{a}^\\delta)+\\frac{\\lambda_k{a}}{\\Gamma(a)p^{\\alpha-1}}E_{\\delta, 2}(-\\lambda_k{a}^\\delta)\\geq\\frac{1}{1+\\Gamma(1-\\delta)\\lambda_k{a}^\\delta}+\\frac{1}{1+\\Gamma(2-\\delta)\\lambda_k{a}^\\delta}>{0}.\n$$\n\nThis proves the uniqueness of the solution of the considered problem.\n\nMoreover, one can write an upper bound of $\\frac{1}{\\Delta}$ by using the last evaluation:\n\n\\begin{equation}\\label{e33}\n\t\\frac{1}{|\\Delta|}\\leq\\frac{M_1}{(k\\pi)^2}+M_2,  \\, \\, \\, \\, (M_1, M_2=const).\n\\end{equation}\nNow we find an estimate for $B$ by using Lemma 2.1:\n\n$$\n|B|\\leq\\frac{\\lambda_k{p}^{1-\\alpha}}{\\Gamma(\\alpha)}|E_{\\delta, 1}(-\\lambda_k{a}^\\delta)|+\\lambda_k{a}^{\\delta-1}|E_{\\delta, \\delta}(-\\lambda_k{a}^\\delta)|\\leq\n$$\n$$\n\\leq\\frac{\\lambda_k{p}^{1-\\alpha}}{\\Gamma(\\alpha)}\\frac{M}{1+\\lambda_k{a}^\\delta}+\\lambda_k{a}^{\\delta-1}\\frac{M}{1+\\lambda_k{a}^\\delta}\\leq\n$$\n$$\n\\leq\\frac{\\lambda_k{p}^{1-\\alpha}}{\\Gamma(\\alpha)}\\frac{M}{\\lambda_k{a}^\\delta}+\\lambda_k{a}^{\\delta-1}\\frac{M}{\\lambda_k{a}^\\delta}=\\frac{Mp^{1-\\alpha}}{a^\\delta\\Gamma(\\alpha)}+\\frac{M}{a}=\\frac{M}{a}\\left(1+\\frac{p^{1-\\alpha}}{a^{\\delta-1}}\\right)=M_3, \\, \\, (M_3=const).\n$$\n\nUsing the last result and (\\ref{e33}) we estimate  $\\left|\\frac{B}{\\Delta}\\varphi_k\\right|$:\n$$\n\\left|\\frac{B}{\\Delta}\\varphi_k\\right|\\leq\\frac{M_1M_3}{(k\\pi)^2}|\\varphi_k|+\\frac{M_2M_3}{k\\pi}|\\varphi_{1k}|\\leq\\frac{M_1M_3}{(k\\pi)^2}|\\varphi_k|+\\left(\\frac{M_2M_3}{k\\pi}\\right)^2+|\\varphi_{1k}|^2,\n$$\nhere $\\varphi_{1k}=2\\int_{0}^{1}\\varphi'(x)\\sin(k\\pi{x})dx$.  Now let us find the upper bound of $C$:\n$$\n|C|\\leq\\int_{0}^{a}|a-s|^{\\delta+q-1}|E_{\\delta, \\delta+q}(-\\lambda_k(a-s)^\\delta)||f_k(s)|ds+\n$$\n$$\n+\\int_{0}^{a}|a-s|^{\\delta+q-2}|E_{\\delta, \\delta+q-1}(-\\lambda_k(a-s)^\\delta)||f_k(s)|ds\\leq\n$$\n$$\n\\leq\\int_{0}^{a}|a-s|^{\\delta+q-1}\\left|\\frac{M}{1+\\lambda_k|a-s|^\\delta}\\right||f_k(s)|ds+\n$$\n$$\n+\\int_{0}^{a}|a-s|^{\\delta+q-2}\\left|\\frac{M}{1+\\lambda_k|a-s|^\\delta}\\right||f_k(s)|ds\\leq\\frac{M_4}{(k\\pi)^2}, \\, \\, \\, (M_4=const.)\n$$\nHere we imply that $f_{x}(x,t)\\in{L}^1(0, a)$ for convergence of the last integral.\n\nThen, the  estimate for $\\frac{C}{\\Delta}$ is\n\n$$\n\\left|\\frac{C}{\\Delta}\\right|\\leq\\frac{M_1M_4}{(k\\pi)^4}+\\frac{M_3M_4}{(k\\pi)^2}.\n$$\n\nFinally, we find the estimate for $|\\psi_k|$:\n\\begin{equation}\\label{e34}\n|\\psi_k|\\leq\\frac{M_1M_3}{(k\\pi)^2}|\\varphi_k|+\\left(\\frac{M_2M_3}{k\\pi}\\right)^2+|\\varphi_{1k}|^2+\\frac{M_1M_4}{(k\\pi)^4}+\\frac{M_3M_4}{(k\\pi)^2}<\\infty,\n\\end{equation}\nwhere $\\varphi'\\in{L}^2(0,1)$.\n\nFrom (\\ref{e32}) and in the same way one can show that $|\\tau_k|\\leq\\frac{M_5}{(k\\pi)^2}+|\\varphi_{1k}|^2<\\infty$,     \\, \\,  $M_5=const.$\n\n\\smallskip\n\nFor proving the existence of the solution, we need to show uniform convergency of series representations of  $u(x, t)$, $u_{xx}(x,t)$,  $^C\\left(t^\\theta\\frac{\\partial}{\\partial{t}}\\right)^{\\alpha}u(x, t)$ and $D_{t}^{(\\gamma, \\beta)\\mu}u(x, t)$ by using the solution (\\ref{e4}) and (\\ref{e27}) in $\\Omega_1$ and $\\Omega_2$ respectively.\n\n\nIn \\cite{L22}, the uniform convergence of series of $u(x,t)$ and $u_{xx}(x,t)$ showed for $t>0$. Similarly, for $t>a$, we obtain the following estimate:\n$$\n|u(x,t)|\\leq{M}\\sum_{k=1}^{\\infty}\\Big(\\frac{|\\tau_k|}{p^\\alpha+(k\\pi)^2|t^p-a^p|^\\alpha}+\\frac{1}{(k\\pi)^2}\\int_{a}^{t}|t^p-\\tau^p|^{\\alpha-1}f_{2k}(\\tau)d(\\tau^p)+\\\\\n$$\n$$\n+\\int_{a}^{t}\\frac{|t^p-\\tau^p|^{2\\alpha-1}}{p^\\alpha+(k\\pi)^2|t^p-a^p|^\\alpha}f_{2k}(\\tau)d(\\tau^p)\\Big),\n$$\nwhere $f_{2k}(t)=2\\int_{0}^{1}f_{xx}(x,t)\\sin(k\\pi{x})dx$.\n\nSince $|\\tau_k|<\\infty$ and $f(\\cdot, t)\\in{C}^3[0,1]$, then the above series converges and hence, by the Weierstrass M-test the series  of $u(x, t)$ is uniformly convergent in $\\Omega_1$.\n\nThe series of $u_{xx}(x,t)$ is written in the form below\n$$\nu_{xx}(x,t)=-\\sum_{k=1}^{\\infty}(k\\pi)^2\\left(\\tau_kE_{\\alpha, 1}\\left[\\frac{(k\\pi)^2}{p^\\alpha}(t-a)^{p\\alpha}\\right]+G_k(t)\\right)\\sin(k\\pi{x}).\n$$\nWe obtain the following estimate:\n$$\n|u_{xx}(x,t)|\\leq{M}\\sum_{k=1}^{\\infty}\\Big(\\frac{(k\\pi)^2|\\tau_k|}{p^\\alpha+(k\\pi)^2|t^p-y^p|^\\alpha}+\\frac{1}{(k\\pi)^2}\\int_{a}^{t}|t^p-\\tau^p|^{\\alpha-1}|f_{4k}(\\tau)|d(\\tau^p)\\\\\n$$\n$$\n+\\int_{a}^{t}\\frac{|t^p-\\tau^p|^{2\\alpha-1}}{p^\\alpha+(k\\pi)^2|t^p-\\tau^p|^\\alpha}|f_{4k}(\\tau)|d(\\tau^p)\\Big),\n$$\nwhere $f_{4k}(t)=2\\int_{0}^{1}\\frac{\\partial^4}{\\partial{x}^4}f(x,t)\\sin(k\\pi{x})dx$ and $f(0,t)=f(1,t)=f_{xx}(0,t)=f_{xx}(1,t)=0$.\n\nSince $\\tau(0)=\\tau(1)=0$ and $\\frac{\\partial^4{f}}{\\partial{x}^4}(\\cdot, t)\\in{L}^1(0,1)$, then using integration by parts, we arrive at the following estimate\n\n$$\n|u_{xx}(x,t)|\\leq{M}\\sum_{k=1}^{\\infty}\\left(\\frac{1}{k}|\\tau_{1k}|+\\frac{1}{k^2}\\right)\\leq\\frac{M}{2}\\Big(\\sum_{k=1}^{\\infty}\\frac{3}{k^2}+\\sum_{k=1}^{\\infty}|\\tau_{1k}|^2\\Big),\n$$\nwhere $\\tau_{1k}=2\\int_{0}^{1}\\tau'(x)\\sin(k\\pi{x})dx$\nThen, the Bessel inequality for trigonometric functions implies\n$$\n|u_{xx}(x,t)|\\leq\\frac{M}{2}\\left(\\sum_{k=1}^{\\infty}\\frac{3}{k^2}+||\\tau'(x)||^2_{L^2(0,1)}\\right).\n$$\nThus, the series in the expression of $u_{xx}(x,t)$ is bounded by a convergent series which is uniformly convergent according to the Weierstrass M-test. Then, the series of $^C(t^\\theta\\frac{\\partial}{\\partial{t}})^\\alpha{u}(x,t)$ which can be written by\n$$\n^C\\left(t^\\theta\\frac{\\partial}{\\partial{t}}\\right)^\\alpha{u}(x,t)=-\\sum_{k=1}^{\\infty}(k\\pi)^2\\left(\\tau_kE_{\\alpha, 1}\\Big[-\\frac{(k\\pi)^2}{p^\\alpha}(t-a)^{p\\alpha}\\Big]+G_k(t)\\right)\\sin(k\\pi{x})+f(x,t),\n$$\nhas uniform convergence which can be showed in the same way to the uniform convergence of the series of $u_{xx}(x,t)$ (see \\cite{L22}).\n\nNow we need to show that the series of $u(x,t)$ and its derivatives should converge uniformly in $\\Omega_2$ by using (\\ref{e27}). We estimate\n\n$$\n|u(x,t)|\\leq|\\varphi_k||t^{(\\beta-2)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)-1}(-\\lambda_k{t}^\\delta)|+|\\psi_k||t^{\\mu+(\\beta-1)(1-\\mu)}E_{\\delta, \\delta+\\mu(2-\\gamma)}(-\\lambda_k{t}^\\delta)|+\n$$\n$$\n+\\int_{0}^{1}|t-\\tau|^{\\delta-1}|E_{\\delta, \\delta}\\Big(-\\lambda_k(t-\\tau)^\\delta\\Big)||f_k(\\tau)|d\\tau.\n$$\nConsider estimates of the Mittag-Leffler function (see Lemma 2.1)\n$$\n|u(x,t)|\\leq\\frac{|t^{(\\beta-2)(1-\\mu)}||\\varphi_k|M}{1+\\lambda_k|{t}^\\delta)|}+\\frac{|t^{(\\beta-1)(1-\\mu)}||\\psi_k|M}{1+\\lambda_k|{t}^\\delta)|}\n$$\n$$\n+\\int_{0}^{t}|t-\\tau|^{\\delta-1}\\frac{M}{1+\\lambda_k|{(t-\\tau)}^\\delta)|}|f_k(\\tau)|d\\tau,\n$$\nwhere $f_{1k}(t)=\\int_{0}^{1}f_{x}(x,t)\\sin(k\\pi{x})dx$, $f_{x}(\\cdot,t)\\in{L^1[0,1]}$ and $\\varphi'\\in{L}^2[0,1]$. Then we obtain the estimate\n$$\n|u(x,t)|\\leq\\sum_{k=1}^{\\infty}\\frac{N_1}{(k\\pi)^2}, \\, \\, \\, \\, (N_1=\\textrm{const}),\n$$\nfor all $t>\\bar{t}>0, \\, \\, \\, 0\\leq{x}\\leq{1}$.\n\nIn the similar way one can show that\n$$\n|u_{xx}(x,t)|\\leq\\sum_{k=1}^{\\infty}(k\\pi)^2\\Big[|\\varphi_k||\\frac{M}{1+\\lambda_k{t}^\\delta}|+(\\frac{K_1}{(k\\pi)^2}+|\\varphi_{1k}|^2)|\\frac{M}{1+\\lambda_k{t}^\\delta}|+\n$$\n$$\n+\\int_{0}^{t}|t-\\tau|^{\\delta-1}\\frac{M}{1+\\lambda_k{t}^\\delta}|k_{2k}(\\tau)|d\\tau\\Big], \\, \\, \\, \\, (M=\\textrm{const}).\n$$\nThen, using Bessel's inequality and  $\\varphi'\\in{L}^2[0,1]$,  $f_{xxx}(\\cdot,t)\\in{L}^1(0,1)$, we get\n$$\n|u_{xx}(x,t)|\\leq\\frac{1}{2}\\Big(\\sum_{k=1}^{\\infty}\\frac{N_2}{(k\\pi)^2}+||\\varphi'(x)||^2_{L^2[0,1]}\\Big).\n$$\nWe have also used $2ab\\leq{a^2+b^2}.$\n\nUsing the equation in $\\Omega_2$, we write $D_{t}^{(\\gamma, \\beta)\\mu}u(x,t)$ in the form\n$$\nD_{t}^{(\\gamma, \\beta)\\mu}u(x,t)=u_{xx}(x,t)+f(x,t)$$\nand its uniform convergence can be done in a similar way to the uniform convergence of $u_{xx}(x,t)$ as\n$$\n|D^{(\\gamma, \\beta)\\mu}_tu(x,t)|\\leq\\sum_{k=1}^{\\infty}\\frac{N_3}{(k\\pi)^2}, \\, \\, \\, \\, (N_3=\\textrm{const}).\n$$\n\n\nFinally, considering the Weierstrass M-test, the above arguments prove that Fourier series in (\\ref{e4}) and (\\ref{e27})  converge uniformly in the domains $\\Omega_1$ and $\\Omega_2$. This is the proof that the considered problem's solution exists in $\\Omega$.  Theorem 3.1 is proved.\n\n\n\n\\section*{Appendix}\n\nHere we write derivation of the series $^{C}\\left(t^{\\theta}\\frac{\\partial}{\\partial{t}}\\right)^\\alpha{u}(x,t)$ in (\\ref{e4}). Using  relation (\\ref{eq3}) we get:\n$$\n^{C}\\left(t^{\\theta}\\frac{\\partial}{\\partial{t}}\\right)^\\alpha{u}(x,t)=\\sum_{k=0}^{\\infty}\\Big[(t^\\theta\\frac{\\partial}{\\partial{t}})^\\alpha\\left(\\tau_kE_{\\alpha,1}\\Big[-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)^\\alpha\\Big]+G_k(t)\\right)\n-\\frac{\\tau_k(t^p-a^p)^\\alpha}{p^{-\\alpha}\\Gamma(1-\\alpha)}\\Big]\\sin(k\\pi{x}).\n$$\nThe hyper-Bessel derivative of the Mittag-Leffler function is\n$$\n\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha\\tau_kE_{\\alpha,1}\\left(-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)^\\alpha\\right)=\\tau_kp^\\alpha(t^p-a^p)^{-\\alpha}E_{\\alpha, 1-\\alpha}\\big[-\\lambda(t^p-a^p)^\\alpha\\big].\n$$\nUsing the Lemma 2.3, we can write the last expression as follows\n$$\n\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha\\tau_kE_{\\alpha,1}\\Big[-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)^\\alpha\\Big]=\\frac{\\tau_kp^\\alpha(t^p-a^p)^{-\\alpha}}{\\Gamma(1-\\alpha)}+\\tau_k(k\\pi)^2E_{\\alpha, 1}\\big[-\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)^\\alpha\\big].\n$$\n\nThen evaluating $\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha{G}_k(t)$ gives that\n$$\n\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha{G}_k(t)=\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^\\alpha\\Big(f^*_k(t)+\\lambda^*\\int_{a}^{t}(t^p-a^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(t^p-a^p)\\big]f^*_k(\\tau)d(\\tau^p)\\Big)=\n$$\n$$\n=p^\\alpha{t}^{-p\\alpha}D_{p, a+}^{-\\alpha, \\alpha}\\Big(\\frac{1}{p^\\alpha}I_{p, a+}^{-\\alpha, \\alpha}t^p\\alpha{f}_k(t)+\\lambda^*\\int_{a}^{t}(t^p-a^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(t^p-a^p)\\big]f^*_k(\\tau)d(\\tau^p)\\Big)=\n$$\n$$\n=f_{k}(t)+p^{\\alpha}t^{-p\\alpha}D_{p, a+}^{-\\alpha, \\alpha}\\Big(\\lambda^*\\int_{a}^{t}(t^p-a^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(t^p-a^p)\\big]f^*_k(\\tau)d(\\tau^p)\\Big),\n$$\nwhere $\\lambda^*=-\\frac{\\lambda_k}{p^\\alpha}$ and $f^*_k(t)=\\frac{1}{p^\\alpha\\Gamma(\\alpha)}\\int_{a}^{t}(t^p-\\tau^p)^{\\alpha-1}f_k(\\tau)d(\\tau^p)$.\n\nThe second term in the last expression can be simplified using the Erd'elyi-Kober fractional derivative for $n=1$,\n$$\n-\\lambda_kt^{-p\\alpha}\\left(1-\\alpha+\\frac{t}{p}\\frac{d}{dt}\\right)\\frac{t^{-p(1-\\alpha)}}{\\Gamma(1-\\alpha)}\\int_{a}^{t}(t^p-\\tau^p)^{-\\alpha}d(\\tau^p)\\int_{a}^{\\tau}(\\tau^p-s^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(\\tau^p-s^p)^\\alpha\\big]f^*_k(s)d(s^p)=\n$$\n$$\n-\\lambda_kt^{-p\\alpha}\\left(1-\\alpha+\\frac{t}{p}\\frac{d}{dt}\\right)\\frac{t^{-p(1-\\alpha)}}{\\Gamma(1-\\alpha)}\\int_{a}^{t}f^*_k(s)d(s^p)\\int_{s}^{t}(t^p-\\tau^p)^{-\\alpha}(\\tau^p-s^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(\\tau^p-s^p)^\\alpha\\big]d(\\tau^p)=\n$$\n$$\n-\\lambda_kt^{-p\\alpha}\\left(1-\\alpha+\\frac{t}{p}\\frac{d}{dt}\\right)t^{-p(1-\\alpha)}\\int_{a}^{t}E_{\\alpha, 1}\\left[\\lambda^*(t^p-s^p)^{\\alpha}\\right]f^*_k(s)d(s^p)=\n$$\n$$\n-\\lambda_k(1-\\alpha)t^{-p}\\int_{a}^{t}E_{\\alpha, 1}\\left[\\lambda^*(t^p-s^p)^{\\alpha}\\right]f^*_k(s)d(s^p)-\n$$\n$$\n-\\frac{\\lambda_kt^{-p\\alpha+1}}{p}\\frac{d}{dt}\\left(t^{-p(1-\\alpha)}\\int_{a}^{t}E_{\\alpha, 1}\\left[\\lambda^*(t^p-s^p)^{\\alpha}\\right]f^*_k(s)d(s^p)\\right)=\n$$\n$$\n=-\\lambda_k(1-\\alpha)t^{-p}\\int_{a}^{t}E_{\\alpha, 1}\\left[\\lambda^*(t^p-s^p)^{\\alpha}\\right]f^*_k(s)d(s^p)+\n$$\n$$\n+\\lambda_k(1-\\alpha)t^{-p}\\int_{a}^{t}E_{\\alpha, 1}\\left[\\lambda^*(t^p-\\tau^p)^{\\alpha}\\right]f^*_k(\\tau)d(\\tau^p)-\\lambda_kf^*_k(t)-\n$$\n$$\n-\\lambda_kt^{1-p}\\int_{a}^{t}\\lambda^*(t^p-\\tau^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\left[\\lambda^*(t^p-\\tau^p)^{\\alpha}\\right]f^*_k(\\tau)d(\\tau^p)=\n$$\n$$\n=-\\lambda_k\\left(f^*_k(t)+\\lambda^*\\int_{a}^{t}(t^p-a^p)^{\\alpha-1}E_{\\alpha, \\alpha}\\big[\\lambda^*(t^p-a^p)\\big]f^*_k(\\tau)d(\\tau^p)\\right)=-\\lambda_kG_k(t).\n$$\nHence, we get\n$$\n^{C}\\left(t^{\\theta}\\frac{\\partial}{\\partial{t}}\\right)^\\alpha{u}(x,t)=-\\sum_{k=0}^{\\infty}(k\\pi)^2\\left[\\tau_kE_{\\alpha, 1}\\left(\\frac{(k\\pi)^2}{p^\\alpha}(t^p-a^p)\\right)+G_k(t)\\right]\\sin(k\\pi{x})+f(x,t).\n$$\nThis proves that solution (\\ref{e4}) satisfies the equation\n$$\n^C\\Big(t^\\theta\\frac{\\partial}{\\partial{t}}\\Big)^{\\alpha}u(x,t)-u_{xx}(x,t)=f(x,t).\n$$\n\nWe would like to note that using the result of this work, one can consider FPDE with the Bessel operator considering local \\cite{krma} and non-local boundary value problems \\cite{karruzh}. In that case the Fourier-Bessel series will play an important role. The other possible applications are related with the consideration of more general operators in space variables. For instance, in \\cite{rtb1}, very general positive operators have been considered, and the results of this paper can be extended to that setting as well.\n\n\\section{Acknowledgement}\n\n\nThe second author was partially supported by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations and by the Methusalem programme of the Ghent University Special Research Fund (BOF) (Grant number 01M01021). \n\n\\bigskip\n\n\n\\textbf{\\Large References}\n\n\\begin{enumerate}\n\\bibitem{L1} {\\it C.~Friedrich.}  Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheologica Acta. 30(2), 1991,  pp. 151-158.\n\n\\bibitem{L2} {\\it D.~Kumar, J.~Singh , M.~Al Qurashi.} A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs,\nand spraying.  Adv Differ Equa. (2019), 2019: 278.\n\n\\bibitem{L3} {\\it I.~Podlubny.}  Fractional Differential Equations. United States, Academic Press.\n1999. 340 p.\n\n\\bibitem{L4} {\\it D.~Baleanu, Z.~B.~Guvenc and J.~A.~T.~Machado.} New trends in nanotechnology and fractional calculus applications, Computers and Mathematics with Applications. 59, 2010, pp. 1835-1841.\n\n\\bibitem{L5} {\\it C.~G.~Koh and J.~M.~Kelly.}  Application for a fractional derivative to seismic analysis of base-isolated models, Earthquake Engineering and Structural Dynamics. 19, 1990, pp. 229-241.\n\n\\bibitem{L6} {\\it E.~T.~Karimov, A.~S.~Berdyshev, N.~A.~Rakhmatullaeva.}  Unique solvability of a\nnon-local problem for mixed-type equation with fractional derivative. {Mathematical\n\tMethods in the Applied Sciences.} 40(8), 2017, pp. 2994-2999.\n\n\\bibitem{L7} {\\it Z.~A.~Nakhusheva.}  Non-local boundary value problems for main and mixed type differential equations. Nalchik, 2011 [in Russian].\n\n\\bibitem{L8} {\\it A.~A.~Kilbas, H.~M.~Srivastava, J.~J.~Trujillo.}  Theory and\nApplications of Fractional Differential Equation. Elsevier, Amsterdam. 2006.\n\n\n\\bibitem{sad} {\\it M.~Kirane,  M.~A.~Sadybekov, A.~A.~Sarsenbi.}  On an inverse problem of reconstructing a subdiffusion process from nonlocal data. {Mathematical\n\tMethods in the Applied Sciences.} 42(6), 2019, pp.2043-2052.\n\n\\bibitem{ruzh} {\\it M.~Ruzhansky, N.~Tokmagambetov and B.~Torebek.}  On a non-local problem for a multi-term fractional diffusion-wave equation. Fractional Calculus and Applied Analysis,  23(2), 2020, pp. 324-355.\n\n\\bibitem{karruzh} {\\it E.~Karimov, M.~Mamchuev and M.~Ruzhansky.}  Non-local initial problem for second order time-fractional and space-singular equation. Hokkaido Math. J., 49, 2020, pp.349-361.\n\n\\bibitem{L9} {\\it E.~T.~Karimov.}  Boundary value problems for parabolic-hyperbolic type equations with spectral para\\-meters. PhD Thesis, Tashkent, 2006.\n\n\\bibitem{L10} {\\it S.~Kh.~Gekkieva.}  A boundary value problem for the generalized transfer equation with a fractional derivative in a semi-infinite domain. Izv. Kabardino-Balkarsk. Nauchnogo Tsentra RAN, 8(1), 2002, pp. 3268-3273. [in Russian].\n\n\\bibitem{L11} {\\it E.~T.~Karimov, J.~S.~Akhatov.}  A boundary problem with integral gluing\ncondition for a parabolic-hyperbolic equation in volving the Caputo fractional derivative.\n{Electronic Journal of Differential Equations.} 14, 2014, pp. 1-6.\n\\bibitem{L12} {\\it P. ~Agarwal, A.~Berdyshev and E.~Karimov.}  Solvability of a non-local\nproblem with integral form transmitting condition for mixed type equation with Caputo\nfractional derivative. {Results in Mathematics.} 71(3), 2017, pp.1235-1257.\n\\bibitem{L13} {\\it B.~J.~Kadirkulov.} Boundary problems for mixed parabolic-hyperbolic\nequations with two lines of changing type and fractional derivative. {Electronic Journal\n\tof Differential Equations}, 2014(57), pp. 1-7.\n\\bibitem{L14} {\\it R.~Hilfer.}  Fractional time evolution, in: R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World SCi., Singapore, 2000, pp. 87-130.\n\\bibitem{L15} {\\it R.~Hilfer.} Experimental evidence for fractional time evolution in glass forming materials. J. Chem. Phys. 284, 2002, pp. 399-408.\n\\bibitem{L16} {\\it R.~Hilfer, Y.~Luchko and Z.~Tomovski.} Operational method for solution of the fractional differential equations with the generalized Riemann-Liouville fractional derivatives. Fractional Calculus and Applied Analysis. 12, 2009, pp. 299-318.\n\n\\bibitem{L17} {\\it O.~Kh.~Abdullaev, K.~Sadarangani.} Non-local problems with integral gluing\ncondition for loaded mixed type equations involving the Caputo fractional derivative.\n\\textit{Electronic Journal of Differential Equations}, 164, 2016, pp. 1-10.\n\n\\bibitem{L18} {\\it A.~S.~Berdyshev, B.~E.~Eshmatov, B.~J.~Kadirkulov.} Boundary value problems\nfor fourth-order mixed type equation with fractional derivative.\n{Electronic Journal of Differential Equations}, 36, 2016, pp. 1-11.\n\\bibitem{L19} {\\it E.~T.~Karimov.} Tricomi type boundary value problem with integral conjugation\ncondition for a mixed type equation with Hilfer fractional operator. {Bulletin of the\n\tInstitute of Mathematics}. 1, 2019, pp. 19-26.\n\\bibitem{L20} {\\it V.~M.~Bulavatsky.} Closed form of the solutions of some boundary-value problems for anomalous diffusion equation with Hilfer's generalized derivative. Cybernetics and Systems Analysis. 30(4), 2014, pp. 570-577.\n\n\\bibitem{Ldn} {\\it M.~M.~Dzhrbashyan, A.~B.~Nersesyan.} Fractional Derivatives and the\nCauchy Problem for Fractional Differential Equations, Izv. Akad. Nauk\nArmyan. SSR. 3, No 1 (1968), 3\u201329.\n\\bibitem{bag}  {\\it F.~T.~Bogatyreva.}  Initial value problem for fractional order equation with \tconstant coefficients, Vestnik KRAUNC. Fiz.-mat. nauki. 2016, 16: 4-1, 21-26. DOI: 10.18454\/2079- 6641-2016-16-4-1-21-26\n\\bibitem{bag2} {\\it F.~T.~Bogatyreva.} Representation of solution for first-order partial differential equation\nwith Dzhrbashyan \u2013 Nersesyan operator of fractional differentiation. Reports ADYGE (Circassian) International Academy of Sciences. Volume 20. \u2116 2. pp.6-11. \n\\bibitem{Lfca} {\\it M.~M.~Dzherbashian, A.~B.~Nersesian.} Fractional derivatives\n\tand Cauchy problem for differential equations of fractional order. Fract. Calc. Appl. Anal. 23, No 6 (2020), 1810-1836.\tDOI: 10.1515\/fca-2020-0090\n\n \\bibitem{L21} {\\it I.~Dimovski.}  Operational calculus for a class of differential operators, C.R. Acad. Bulg. Sci. 19(12), 1996, pp.1111-1114.\n\n\\bibitem{L22} {\\it F.~Al-Musalhi, N.~Al-Salti and E.~Karimov.} Initial boundary value\nproblems for fractional differential equation with hyper-Bessel operator.\nFractional Calculus and Applied Analysis. 21(1), 2018, pp. 200-219.\n\n\\bibitem{L23} {\\it N.~H.~Tuan, L.~N.~Huynh, D.~Baleanu, N.~H.~Can.} On a terminal value problem for a\n\tgeneralization of the fractional diffusion equation with hyper-Bessel operator. {Mathematical\n\tMethods in the Applied Sciences.} 43(6), 2019, pp.2858-2882.\n\n\\bibitem{L24} {\\it K.~Zhang.}  Positive solution of nonlinear fractional differential equations with\n\tCaputo-like counterpart hyper-Bessel operators. {Mathematical\n\tMethods in the Applied Sciences.} 43(6), 2019, pp. 2845-2857.\n\n\\bibitem{L25} {\\it R.~Garra, A.~Giusti, F.~Mainardi, G.~Pagnini.} Fractional relaxation\nwith time-varying coefficient. Fractional Calculus and Applied Analysis. 17(2), 2014, pp. 424-439.\n\n\\bibitem{L26} {\\it E.~T.~Karimov, B.~H.~Toshtemirov.} Tricomi type problem with integral conjugation condition for a mixed type equation with the hyper-Bessel fractional differential operator. Bulletin of the Institute of Mathematics. 4, 2019, pp. 9-14.\n\n\\bibitem{L27} {\\it F.~Mainardi.} On some properties of the Mittag-Leffler function $E_{\\alpha}(-t^{\\alpha})$, completely monotone for $t>0$ with $0<\\alpha<1$. Discrete and Continuous Dynamical Systems - B. 19 (7), 2014, pp. 2267-2278.\n\n\\bibitem{L28} {\\it L.~Boudabsa, T.~Simon.}  Some Properties of the Kilbas-Saigo Function. Mathematics. 9(3), 2021: 217.\n\n\\bibitem{L29} {\\it R.~K.~Saxena.} Certain properties of generalized Mittag-Leffler function, in Proceedings of the 3rd\nAnnual Conference of the Society for Special Functions and Their Applications, pp. 77-81, Chennai, India, 2002.\n\\bibitem{L30}  {\\it A.~V.~Pskhu.}  Partial Differential Equations of Fractional Order. Moscow, Nauka. 2005. [in Russian].\n\n\\bibitem{KDSc} {\\it E.~T.~Karimov.} Boundary value problems with integral transmitting conditions and inverse problems for integer and fractional order differential equations. DSc Thesis, Tashkent, 2020.\n\n\\bibitem{L31} {\\it E.~I.~Moiseev.} On the basis property of systems of sines and cosines.\nDoklady AN SSSR. 275(4), 1984, pp.794-798.\n\n\\bibitem{krma} {\\it P.~Agarwal, E.~Karimov, M.~Mamchuev, M.~ Ruzhansky.}  On boundary-value problems for a partial differential equation with Caputo and Bessel operators, in Recent Applications of Harmonic Analysis to Function Spaces, Differential Equations, and Data Science: Novel Methods in Harmonic Analysis, Vol 2, Appl. Numer. Harmon. Anal., 707-718, Birkhauser\/Springer, 2017.\n\n\\bibitem{rtb1} {\\it M.~ Ruzhansky, N.~Tokmagambetov, B.~Torebek.} Inverse source problems for positive operators. I: Hypoelliptic diffusion and subdiffusion equations. Journal of Inverse and Ill-Posed Problems, 27 (2019), 891-911.\n\\end{enumerate}\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nOptimal investment with intermediate consumption and a stream of labor income (or liabilities) is one of the central problems in mathematical economics. \nIf borrowing against the future income is prohibited, the main  technical difficulty    lies in the fact that there are infinitely many constraints. Even in the deterministic case of no stocks and non-random income, a classical approach is based on the {\\it convexification of the constraints} that leads to a non-trivial dual problem formulated over decreasing nonnegative functions.\n\nBorrowing constraints imposed at all times  not only affect the  notion of admissibility, leading to more difficult mathematical analysis,  but also\nchange the meaning to fundamental concepts of mathematical finance such as replicability and completeness. The latter is   formulated via the attainability of every (bounded) contingent claim by a portfolio of traded  assets.   For a  labor income\/liability streams that pays off dynamically, \nthere is no a priori guarantee that such a replicating portfolio  (if it exists at all)  is admissible, i.e., satisfies the  constraints. Thus, in the terminology of \\cite{HePages}, even a  complete market becomes dynamically incomplete under the borrowing constraints. \n The  analysis of such a problem (in otherwise complete Brownian settings with a corresponding unique risk-neutral measure), is performed in \\cite{HePages} and later in \\cite{ElKarouiJeanblanc}. \nThe  nonnegative decreasing {\\it processes}   (that parametrize the dynamic incompleteness mentioned above)  play an important role in the characterizations of   optimal investment and consumption plans. The analysis  in \\cite{HePages} and \\cite{ElKarouiJeanblanc} is connected with optimal stopping techniques from~\\cite{Kar89}. \n\nIncomplete markets with no-borrowing constraints have been analyzed only in specific Markovian models  in \\cite{DuffieZarip1993} and \\cite{DuffieFlemingSonerZarip1997} based on partial differential equations techniques. \n The goal of the present paper is to study the problem of consumption and investment with no-borrowing constraints in general  (so, non-Markovian) incomplete   models. This leads to having, simultaneously, two layers of incompleteness. One comes from the many martingale measures, the other from a similar class of non-decreasing processes  (as above) that describe the dynamic incompleteness. \n We refer to \\cite{MalTrub07} for the examples of market incompleteness in finance and macroeconomics. \n\nIn contrast to \\cite{HePages} and \\cite{ElKarouiJeanblanc},   our model not only allows for incompleteness, but also for jumps. Mathematically, this means we choose to work in a general semimartingale framework.\n\nAs in  \\cite{HePages} and \\cite{ElKarouiJeanblanc}, \nour approach is based on duality.  One of the principal difficulties is in the construction of the dual feasible set and the dual value function. It is well-known that the martingale measures   drive the dual domain in many problems of mathematical finance. On the other hand, the convexification of constraints leads to the decreasing processes as the central dual object as well. We show that the dual elements in the incomplete case can be approximated by  products of the densities of martingale measures and such nonnegative decreasing processes. This is one of the primary results of this work, see section \\ref{secStructureOfDualDomain}, that leads to the complementary slackness characterization of optimal wealth in section \\ref{secSlackness}, where it is shown that the approximating sequence for the dual minimizer leads to a nonincreasing process, which decreases at most when the constraints are attained. In turn, the dual minimizer can be written as a product of such a nondecreasing process and an optional strong supermartingale deflator. In the case of  complete Brownian markets  a similar result is proved in \\cite{HePages} and   \\cite{ElKarouiJeanblanc}.\n\nIn order to implement the approach, we {\\it increase the dimensionality} of the problem and treat as arguments of the indirect utility not only the initial wealth, but also the function that specifies the number of units of labor income (or  the  stream of liabilities)  at any later time. This parametrization has the spirit of \\cite{HK04}, however unlike \\cite{HK04}, we go into (infinite-dimensional) non-reflexive spaces, which gives both novelty and technical difficulties to our analysis.\n Also, our formulation permits to price by marginal rate of substitution the whole  labor income process. This is done \n\nthrough  the subdifferentiability results in section \\ref{secSubdifferentiability}.  Note that  the sub differential elements (prices) are time-dependent, so infinite-dimensional, unlike in  \\cite{HK04}.\n\n\nAnother contribution of the paper lies in the {\\it unified framework of admissibility} outlined in section \\ref{secModel}.  More precisely, we assume that no-borrowing constraints are imposed  starting from  some pre-specified stopping time and hold up to the terminal time horizon.  \nThis framework allows us to treat in one formulation both the problem of no-borrowing constraints at all times (described above) and the one where borrowing against the future income is permitted  with a constraint only at the end. The latter is well-studied in the literature, see \\cite{CvitanicSchachermayerWang}, \\cite{Karatzas-Zitkovic-2003}, \\cite{HK04}, \\cite{Zitkovic}, and \\cite{MostovyiRandEnd}.\n In such a formulation, the constraints reduce to a single inequality and the decreasing processes in the dual feasible set become constants.  \n \nAmong the many possibilities of constraints, \\cite{Cuoco1997}  considers the problem of investment and consumption with labor income and no-borrowing constraints in Brownian market, even allowing for incompleteness. The dual problem cannot be solved directly (in part because for these constraints the dual space considered is too small) but the primal can be solved with direct methods. An approximate dual sequence can then be recovered from the primal. We generalize \\cite{HePages} and \\cite{ElKarouiJeanblanc} (complete Brownian markets) and \\cite{Cuoco1997} (possibly incomplete Brownian markets) to the case of general semi-martingale incomplete markets. Our dual approach allows us, at the same time to obtain a dual characterization (complementary slackness) of the optimal consumption plan (not present in \\cite{Cuoco1997}), similar to the complete case in \\cite{HePages} and \\cite{ElKarouiJeanblanc} and the possibility to study the dependence on labor income streams, through the parametrization of such streams.\n \n\n\n\nEmbedding path dependent problems into the convex duality framework have been analyzed in \\cite{XiangYuHabit}, \\cite{HaoMat}, \\cite[Section 3.3]{Pham01}, whereas without duality but with random endowment it is considered in \\cite{Miklos2018}, in the abstract singular control setting the duality approach is investigated in \\cite{BankKaupila}.\nOur embedding does not require any condition on labor income {replicability}, which becomes highly technical in the presence of extra admissibility constraints. Even in the case where the only constraint is imposed at maturity, in this part our approach differs from the one in \\cite{HK04}, where non-replicability of the endowment (in the appropriate sense) is used in the proofs as it ensures that the effective domain of the dual problem has the same dimensionality as the primal domain. Note that even if \n the  labor income is spanned by the same sources of randomness as the stocks, the idea of replicating the labor income and then reducing the problem to the one without it, does not necessarily work under the borrowing constraints, see the discussion in \\cite[pp. 671-673]{HePages}. \n\n\n\n\n\n\n\n Some of the more specific   technical contributions  of this paper can be summarized as follows:\n\n\\begin{itemize}\n\\item We  analyze the {\\it boundary behavior} of the value functions.  Note that the value functions are defined over infinite-dimensional spaces.  \n\n\n\\item The finiteness of the indirect utilities {\\it without} labor income is imposed  only, as a necessary and sufficient condition that allows for the standard conclusions of the utility maximization theory, see \\cite{MostovyiNec}.\n\n\n\\item We show existence-uniqueness results for the {\\it unbounded} labor income both from above and below.\n\n\n\n\\item We observe that the ``Snell envelope proposition'' \\cite[Proposition 4.3]{K96} can be extended to \n the envelope over all stopping times that exceed a  given initial stopping time $\\theta _0$.\n\\item We represent the dual value function in terms of {\\it uniformly integrable} densities of martingale measures, i.e., the densities of martingale measures under which the maximal wealth process of a self-financing portfolio that superreplicates the labor income, is a uniformly integrable martingale, see Lemma \\ref{finitenessOverZ'} below. \n\n\\end{itemize}\n\n\n\n{\\bf Organization of the paper}. In Section \\ref{secModel}, we specify the model. We state and prove existence, uniqueness, semicontinuity and biconjugacy results in Section \\ref{secProofs}, subdifferentiability is proven in Section \\ref{secSubdifferentiability}. Structure of the dual domain is analyzed in Section \\ref{secStructureOfDualDomain} and complimentary slackness is established in Section \\ref{secSlackness}.\n\\section{Model}\\label{secModel}\n\nWe consider a {financial market model} with finite time horizon\n$[0,T]$ and a zero interest rate. The price process $S=(S^i)_{i=1}^d$\nof the stocks is assumed to be a\nsemimartingale on a complete stochastic basis\n$\\left( \\Omega, \\mathcal F, \\left(\\mathcal F_t \\right)_{t\\in[0,T]}, \\mathbb\nP\\right)$, where $\\mathcal F_0$ is trivial. \n\nLet $(e)_{t\\in[0,T]}$ be an optional process that specifies the labor income rate, which is assumed to follow a certain stochastic clock, that we specify below.\nBoth processes $S$ and $e$ are given exogenously. \n\nWe define a \\textit{stochastic clock} as a\nnondecreasing, c\\`adl\\`ag, adapted process such that\n\\begin{equation}\n \\tag{finClock}\n  \\label{finClock}\n\\kappa_0 = 0, ~~ \\mathbb P\\left[\\kappa_T>0 \\right]>0,\\text{ and } \\kappa_T\\leq A\n\\end{equation}\nfor some finite constant $A$. We note that the stochastic clock allows to include multiple standard formulations of the utility maximization problem in one formulation, see e.g., \\cite[Example 2.5 - 2.9]{MostovyiNec}.\nLet us define \n\\begin{equation}\\label{defK}\nK_t {:=} \\mathbb E\\left[\\kappa_t \\right],\\quad t\\in[0,T].\n\\end{equation}\n\\begin{Remark}\nThe function $K$ defined in \\eqref{defK} is right-continuous with left limits and takes values in $[0,A]$. \n\\end{Remark}\nWe assume the income and consumption are given in terms of the clock $\\kappa$.\nDefine a {portfolio} $\\Pi$ as a quadruple $(x, q, H, c),$ where the\nconstant $x$ is the initial value of the portfolio, the function $q:[0,T]\\to\\mathbb R$ is a bounded and Borel measurable function,\n which \nspecifies the amount of labor income rate, $H =\n(H_i)_{i=1}^d$ is a predictable $S$-integrable process that\ncorresponds to the amount of each stock in the portfolio, and\n$c=(c_t)_{t\\in[0,T]}$ is the consumption rate, which we assume to be\n optional and nonnegative.\n\nThe \\textit{wealth process} $V=(V_t)_{t\\in[0,T]}$ {generated by the}\nportfolio is \n\\begin{equation}\\nonumber\nV_t = x + \\int_0^t H_sdS_s + \\int_0^t\\left(q_se_s-c_s\\right)d\\kappa_s,\\quad t\\in[0,T].\n\\end{equation}\nA portfolio $\\Pi$ with $c\\equiv 0$ and $q \\equiv 0$ is called\n\\textit{self-financing}. The collection of nonnegative wealth\nprocesses {generated by} self-financing portfolios with initial value $x\\geq 0$ is\ndenoted by $\\mathcal X(x)$, i.e.\n\\begin{equation}\\nonumbe\n\\mathcal X(x) {:=} \\left\\{X\\geq 0:~X_t=x + \\int_0^t\nH_sdS_s,~~t\\in[0,T] \\right\\},~~x\\geq 0.\n\\end{equation}\nA probability measure $\\mathbb Q$ is an \\textit{equivalent local\nmartingale measure} if $\\mathbb Q$ is equivalent to $\\mathbb P$ and\nevery $X\\in\\mathcal X(1)$ is a local martingale under $\\mathbb Q$. We\ndenote the family of equivalent local martingale measures by\n$\\mathcal M$ and assume that\n\\begin{equation}\\tag{noArb}\\label{NFLVR}\n\\mathcal M \\neq \\emptyset.\n\\end{equation}\nThis condition is  equivalent to the absence of arbitrage\nopportunities \n{in} the marke\n, see\n \\cite{DS, DS1998} as well as \n\\cite{KarKar07} for the exact statements and further\nreferences.\n\nTo rule out doubling strategies in the presence of random endowment,\nwe need to impose additional restrictions. Following  \\cite{DS97}, we say that a\nnonnegative process in $\\mathcal X(x)$ is \\textit{maximal} if its\nterminal value cannot be dominated by that of any other process in\n$\\mathcal X(x)$.\nAs in \\cite{DS97}, we define an\n\\textit{acceptable} process to be a process of the form $X = X' -\nX'',$ where $X'$ is a nonnegative wealth process {generated by} a self-financing portfolio and $X''$ is maximal.\n\n Our unified framework of admissibility is given by a  fixed stopping time $\\theta_0$. The no-borrowing constraints will hold starting at this stopping time until the end.   Let $\\Theta$ be the set of stopping times that are greater or equal than  $\\theta_0$. \n\\begin{Lemma}\\label{3311}\nLet $q^1$ and $q^2$ be bounded, Borel measurable functions on $[0,T]$, such that $q^1= q^2$, $dK$-a.e. Then, the cumulative labor income processes $\\int_0^{\\cdot}q^i_se_sd\\kappa_s$, $i = 1,2$,  are indistinguishable. \n\\end{Lemma}  The proof of this lemma is given in section \\ref{subSecCharDualDomain}.\nFollowing\n \\cite{HK04}, we denote by\n$\\mathcal X(x,q)$ the set of acceptable processes with initial {values}\n$x$, that dominate the labor income on $\\Theta$:\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\n\\mathcal X(x, q) &{:=}& \\left\\{ \n{\\rm acceptable}~X: X_0 = x~{\\rm and}\\right.\\\\\n&&\\left.~ X_{\\tau} + \\int_0^{\\tau}q_se_sd\\kappa_s \\geq 0, ~\\mathbb P-a.s.~{\\rm for~every~}\\tau\\in\\Theta\n\\right\\}.\\\\\n\\end{array}\n\\end{equation}\nLet us set\n\\begin{equation}\\nonumbe\n\\mathcal K {:=} \\left\\{(x,q): \\mathcal X(x,q) \\neq \\emptyset \\right\\},\n\\end{equation}\nLet $\\mathring{\\mathcal K}$ denote the interior or $\\mathcal K$ in the $\\mathbb R\\times \\mathbb L^\\infty(dK)$-norm topology.\nWe characterize $\\mathcal K$ in Lemma~\\ref{lemma1} below that in particular asserts that under Assumption \\ref{asEnd}, $\\mathring{\\mathcal K}\\neq \\emptyset$. \n  \nThe set of admissible consumptions is defined as\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\n\\mathcal A(x,q)&{:=} &\\left\\{ \noptional ~c\\geq 0:~~there~exists~X\\in\\mathcal X(x,q), ~such~that~ \\right.\\\\\n&& \n\\left.\\int_0^{\\tau}c_sd\\kappa_s\\leq X_{\\tau} + \\int_0^{\\tau}q_se_sd\\kappa_s,\n~~for~every~\\tau\\in\\Theta\\right\\},\\quad (x,q)\\in\\mathcal K.\\\\ \n\\end{array}\n\\end{equation}\nNote that $c\\equiv 0$ belongs to $\\mathcal A(x,q)$ for every $(x,q)\\in\\mathcal K$. \n\\begin{Remark}\\label{Remark-Mihai} The no-borrowing constraints can also  be written  as\n$$\n\\mathbb{P}\\left (\\int_0^{t}c_sd\\kappa_s\\leq X_{t} + \\int_0^{t}q_se_sd\\kappa_s,\n~~for~every~ \\theta _0\\leq t\\leq T \\right)=1.$$\nWe write the constraints in terms of stopping times $\\tau \\in \\Theta$ as \nwe use the stopping times $\\tau \\in \\Theta$ (and the corresponding decreasing processes that jump from one to zero at these  times) as  the building blocks of our analysis. \n\\end{Remark}\n\\begin{Remark}\\label{3312}\nIt follows from Lemma \\ref{3311}, for every $x\\in\\mathbb R$,  we have\n$$\\mathcal X(x,q^1) = \\mathcal X(x,q^2)\\quad and \\quad \\mathcal A(x,q^1) = \\mathcal A(x, q^2),$$\nwhere some of these sets might be empty\n\\end{Remark}  \n\nHereafter, we shall impose the following conditions on the  endowment process.\n\\begin{Assumption}\\label{asEnd}\nThere exists a maximal wealth process $X'$ such that\n\\begin{equation}\\nonumbe\nX'_t \\geq |e_t|, \\quad for~every~t\\in[0,T],\\quad\\Pas.\n\\end{equation}\nMoreover, Assumption \\ref{asEnd} and \\eqref{NFLVR} imply that all the assertions of Lemma \\ref{lemma1} hold.\n\\end{Assumption}\n\n\\begin{Remark}\nIf $\\theta _0 = \\{T\\}$, then Assumption~\\ref{asEnd} is equivalent to the assumptions on endowment in~\\cite{HK04} (for the case of one-dimensional random endowment).\n\\end{Remark}\n\n\n\nThe preferences of an economic agent are modeled with a\n\\textit{utility stochastic field}\n$U=U(t, \\omega, x):[0,T]\\times\\Omega\\times[0,\\infty)\\to\\mathbb R\\cup \\{-\\infty\\}$.\nWe assume that $U$\nsatisfies the conditions below.\n\\begin{Assumption}\\label{assumptionOnU}\n \nFor every $(t, \\omega)\\in[0, T]\\times\\Omega$, the function $x\\to\n  U(t, \\omega, x)$ is strictly concave, increasing, continuously\n  differentiable on $(0,\\infty)$ and satisfies the Inada conditions:\n  \\begin{equation}\\nonumbe\n    \\lim\\limits_{x\\downarrow 0}U'(t, \\omega, x) =\n    \\infty \\quad \\text{and} \\quad \\lim\\limits_{x\\to\n      \\infty}U'(t, \\omega, x) = 0,\n  \\end{equation}\n  where $U'$ denotes the partial derivative with respect to the third argument.\n At $x=0$ we suppose, by continuity, $U(t, \\omega, 0) = \\lim\\limits_{x \\downarrow 0}U(t, \\omega, x)$, {which}\n may be $-\\infty$.\nFor every\n  $x\\geq 0$ the stochastic process $U\\left( \\cdot, \\cdot, x \\right)$ is\n  optional. Below, following the standard convention, we will not write $\\omega$ in $U$.\n\\end{Assumption}\n\n\n\nThe agent can control investment and consumption. {The goal is to maximize}\n expected utility. The value function $u$ is\ndefined as:\n\\begin{equation}\\label{primalProblem}\nu(x, q) {:=} \\sup\\limits_{c\\in\\mathcal A(x, q)}\\mathbb\nE\\left[\\int_0^T U(t, c_t)d\\kappa_t\\right], \\quad\n(x, q)\\in\\mathcal K.\n\\end{equation}\nIn \\eqref{primalProblem}, we use the convention\n\\begin{displaymath}\n  \\mathbb{E}\\left[\n    \\int_0^TU(t,c_t)d\\kappa_t \\right] {:=} -\\infty\n  \\quad \\text{if} \\quad \\mathbb{E}\\left[ \\int_0^TU^{-}(t,c_t)d\\kappa_t\n  \\right]= \\infty.\n\\end{displaymath}\nHere and below, $W^{-}$ and $W^{+}$ denote the negative and positive parts of a\nstochastic field $W$, respectively.\n\n\n We employ duality techniques to obtain the standard conclusions of the utility maximization theory.  We first  define the convex conjugate stochastic field\n\n\\begin{equation}\\label{defV}\nV(t, y) {:=} \\sup\\limits_{x>0}\\left(\nU(t, x)-xy\\right),\\quad (t, y)\\in[0,T]\\times [0,\\infty),\n\\end{equation}\n and then  observe that \n$-V$ satisfies Assumption \\ref{assumptionOnU}.\nIn order to construct the feasible set of the dual problem, we define the set\n$\\mathcal L$ as the  polar cone of $-\\mathcal K$:\n\\begin{equation}\\label{defL}\n\\mathcal L {:=} \\left\\{(y,r)\\in\\mathbb R\\times\\mathbb L^1(dK): xy + \\int_0^Tq_sr_sdK_s \\geq\n0{\\rm~for~every~}(x,q)\\in\\mathcal K\\right\\}.\n\\end{equation}\n\\begin{Remark}\nUnder the conditions \\eqref{finClock}, \\eqref{NFLVR} and Assumption \\eqref{asEnd}, the set \n$\\mathcal L$ is non-empty.  By definition,  it is closed in $\\mathbb R\\times\\mathbb L^1(d K)$-norm and $\\sigma(\\mathbb R\\times\\mathbb L^1(dK), \\mathbb R\\times\\mathbb L^{\\infty}(dK))$ topologies. Also,  as shown later, the set $\\mathcal L = (-\\mathring{\\mathcal K})^o$, i.e. the polar of $-\\mathring{\\mathcal K}$.\n\\end{Remark}\nBy ${\\mathcal Z}$, we denote the set of c\\`adl\\`ag densities of\nequivalent local martingale measures:\n\\begin{equation}\\label{defZ}\n{\\mathcal Z} {:=} \\left\\{{\\rm c\\grave adl\\grave ag}~\\left(\\frac{d\\mathbb Q_t}{d\\mathbb P_t}\\right)_{t\\in[0,T]}:\\quad \\mathbb Q\\in\\mathcal M\\right\\}.\n\\end{equation}\nLet us denote by $\\mathbb L^0 = \\mathbb L^0(d\\kappa \\times \\mathbb P)$ the linear space of (equivalence classes of) real-valued optional processes on the stochastic basis $(\\Omega, \\mathcal F, (\\mathcal F_t)_{t\\in[0,T]}, \\mathbb P)$ which we equip with the topology  of convergence in measure $(d\\kappa \\times \\mathbb P)$. \nFor each $y \\geq 0$ we define\n\\begin{equation}\\label{oldY}\n\\begin{array}{c}\n \\hspace{-15mm}\\mathcal Y(y) {:=} {\\cl}\\left\\{ Y: ~Y{\\rm~is~c\\grave{a}dl\\grave{a}g~adapted~and~}\\right. \\\\\n \\hspace{35mm}\\left.0\\leq Y\\leq yZ ~\\left(d\\kappa\\times\\mathbb\nP\\right){\\rm\n~a.e.~for~some~}Z\\in{\\mathcal Z}\\right\\},\\\\\n\\end{array}\n\\end{equation}\nwhere the closure is taken in\n$\\mathbb L^0$. \nNow we are ready to set the\ndomain of the dual problem:\n\\begin{equation}\\label{defY}\n\\begin{array}{rcl}\n\\mathcal Y(y,r)&{:=} &\\left\\{\n Y: Y \\in\\mathcal Y(y)\n~{\\rm and}\\right.\\\\\n&&\n\\quad\\mathbb E\\left[\\int_0^Tc_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\\\\\n&&\\left.\\quad{\\rm for~every}~(x,q)\\in\\mathcal K {\\rm ~and~}c\\in\\mathcal A(x,q)\\right\\}\\\\\n\\end{array}\n\\end{equation}\nNote that the definition \\eqref{defY}  requires that every element of $\\mathcal Y(y,r)$ is in $\\mathcal Y(y)$, $y\\geq 0$. Also, for every $(y,r)\\in\\mathcal L$, $\\mathcal Y(y,r)\\neq \\emptyset$, since $0\\in\\mathcal Y(y,r)$. \n\n We can now  state the dual optimization problem:\n\\begin{equation}\\label{dualProblem}\nv(y, r) {:=} \\inf\\limits_{Y\\in\\mathcal Y(y, r)}\\mathbb\nE\\left[\\int_0^T V(t, Y_t)d\\kappa_t\\right],\\quad(y, r)\\in\\mathcal L,\n\\end{equation}\nwhere we use the convention:\n\\begin{displaymath}\n  \\mathbb{E}\\left[ \\int_0^TV(t,Y_t\n    )d\\kappa_t \\right] {:=} \\infty\n  \\quad \\text{if} \\quad \\mathbb{E}\\left[ \\int_0^TV^{+}(t, Y_t )d\\kappa_t\n  \\right] = \\infty.\n\\end{displaymath}\nAlso, we set \n\\begin{equation}\\label{3313}\nv(y,r) {:=} \\infty~for~(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)\\backslash\n \\mathcal L\\quad and \\quad u(x,q){:=} -\\infty~for~(x,q)\\in\\mathbb R\\times \\mathbb L^{\\infty}(dK)\\backslash\\mathcal K.\\\\ \n\\end{equation}\nWith this definition, it will be shown below in Theorem  \\ref {mainTheorem} that   $u<\\infty$ and $v>-\\infty$ everywhere,  so  $u$ and $v$ are proper   functions  in the language of convex analysis.  \nLet us recall that in the absence of random endowment, the dual value function is defined as\n\\begin{equation}\\nonumbe\n\\tilde w(y) {:=} \\inf\\limits_{Y\\in\\mathcal Y(y)}\\mathbb E\\left[\\int_0^TV(t, Y_s)d\\kappa_s \\right],\\quad y>0,\n\\end{equation}\nwhereas the primal value function is given by\n\\begin{equation}\\label{defw}\nw(x) {:=} u(x,0),\\quad x>0.\n\\end{equation}\n\\section{Existence, uniqueness, and biconjugacy}\\label{secProofs}\n\\begin{Theorem}\\label{mainTheorem}\nLet (\\ref{finClock}) and (\\ref{NFLVR}), Assumptions~\\ref{asEnd}\nand \\ref{assumptionOnU} hold true and\n\\begin{equation}\\tag{finValue}\\label{finValue}\nw(x) > -\\infty\\quad for~every~x>0\\quad and \\quad \n \\widetilde w(y) <\\infty\\quad for~every~y>0.\n \\end{equation}\n Then we have:\n\n$(i)$ $u$ is finite-valued on $\\mathring{\\mathcal K}$ and $u<\\infty$ on $\\mathbb R\\times \\mathbb L^\\infty(dK)$.  The dual value function $v$ satisfies  $v>-\\infty$ on $\\mathbb R\\times\\mathbb L^1(dK)$,  and  the set $\\{v<\\infty\\}$ is a nonempty convex subset of $\\mathcal L$, whose closure in $\\mathbb R\\times\\mathbb L^1(dK)$ equals to $\\mathcal L$.  \n\n$(ii)$ $u$ is concave, proper, and upper semicontinuous with respect to the  norm-topology of $\\mathbb R\\times\\mathbb L^\\infty(dK)$ and the  weak-star topology $\\sigma (\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$. For every $(x,q)\\in \\{u>-\\infty\\}$, there exists a unique solution to (\\ref{primalProblem}). Likewise, $v$ is convex, proper, and lower semicontinuous with respect to the norm-topology of $\\mathbb R\\times \\mathbb L^1(dK)$ and the the weak topology  $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb L^{\\infty}(dK))$. For every $(y,r)\\in \\{v<\\infty\\}$, there exists a unique solution to (\\ref{dualProblem}). \n\n$(iii)$ The functions $u$ and $v$ satisfy the biconjugacy relations\n\\begin{equation}\\nonumbe\n\\begin{array}{rcll}\nu(x,q) &=& \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tr_sq_sdK_s\\right),& (x,q)\\in \\mathcal K,\\\\\nv(y,r) &=& \\sup\\limits_{(x,q)\\in \\mathcal K}\\left(u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),& (y,r)\\in \\mathcal L.\\\\\n\\end{array}\n\\end{equation}\n\\end{Theorem}\n\n\n\n\\begin{Lemma}\\label{lemma1}\nLet (\\ref{NFLVR}) and Assumption~\\ref{asEnd} hold. Then we have:\n\n$(i)$ for every $x>0$, $(x,0)$ belongs to $\\mathring{\\mathcal K}$ (in particular, $\\mathring{\\mathcal K}\\neq \\emptyset$),\n\n$(ii)$ for every $q\\in\\mathbb L^\\infty(dK)$, there exists $x>0$ such that $(x,q)\\in\\mathring{\\mathcal K}$,\n\n$(iii)$ $\\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[\\int_0^T|e_s|d\\kappa_s \\right] <\\infty$,\n\n$(iv)$ there exists a nonnegative maximal wealth process $X^{''}$, such that\n\\begin{displaymath}\nX^{''}_T \\geq \\int_0^T|e_s|d\\kappa_s,\\quad \\Pas,\n\\end{displaymath}\n\n$(v)$ there exists a nonnegative maximal wealth process $X^{''}$, such that\n\\begin{equation}\\nonumbe\nX^{''}_t \\geq \\int_0^t|e_s|d\\kappa_\ns,\\quad t\\in[0,T],~\\mathbb P-a.s.\n\\end{equation}\n\n\\end{Lemma}\n\\begin{proof}\nFirst, via \\cite[Proposition I.4.49]{JS} and \\eqref{finClock}, we get\n\\begin{equation}\\nonumber\n\\begin{array}{c}\n\\int_0^T |e_s|d\\kappa_s \\leq \\int_0^T X'_sd\\kappa_s = -\\int_0^T\\kappa_{s-}dX'_s + \\kappa_TX'_T \\\\ \\leq -\\int_0^T\\kappa_{s-}dX'_s +AX'_T =AX'_0 + \\int_0^T(A - \\kappa_{s-})dX'_s.\\\\\n\\end{array}\n\\end{equation}\nTherefore, the exists a self-financing wealth process $\\bar X$, such that\n$$\\int_0^T |e_s|d\\kappa_s \\leq \\bar X_T.$$\nConsequently, \\cite[Theorem 2.3]{DS97} asserts the existence of a nonnegative {\\it maximal} process $X^{''}$, such that\n$$\\int_0^T |e_s|d\\kappa_s \\leq  X_T^{''},$$\ni.e.,  $(iv)$ holds. Therefore $(iii)$ is valid as well, by \\cite[Theorem 5.12]{DS1998}. \n\nRelation $(iv)$ and \\eqref{NFLVR} imply $(v)$.\nTo prove $(i)$ and $(ii)$, without loss of generality, we will suppose that in $(v)$, $X^{''}_0> 0.$ Let $q\\in\\mathbb L^{\\infty}(dK)$ and $\\varepsilon >0$ be fixed.  Let us define\n\\begin{equation}\\nonumbe\nx(q){:=} \\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon\\right)X^{''}_0.\n\\end{equation}\nWe claim that $(x(q), q)\\in\\mathring{\\mathcal K}.$\nLet \nus consider arbitrary \n\\begin{equation}\\label{454}\n|x'|\\leq \\varepsilon\\quad {\\rm and} \\quad q'\\in\\mathbb L^\\infty(dK)\n:~||q'||_{\\mathbb L^{\\infty}(dK)}\\leq {\\varepsilon},\n\\end{equation}  \nand set\n\\begin{equation}\\label{455}\n\\tilde X_t {:=} \\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_t,\\quad t\\in[0,T].\n\\end{equation} Then, by item $(v)$, we have\n\\begin{displaymath}\n\\begin{array}{c}\n\\tilde X_t\\geq \\left(||q||_{\\mathbb L^{\\infty}(dK)}+{\\varepsilon}\\right)X^{''}_t\\geq \\left(||q||_{\\mathbb L^{\\infty}(dK)}+{\\varepsilon}\\right)\\int_0^t|e_s|d\\kappa_s\\geq -\\int_0^t(q_s + q'_s)|e_s|d\\kappa_s.\n\\end{array}\n\\end{displaymath}\nWe deduce that $\\tilde X\\in\\mathcal X(\\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_0, q+q')$. In particular,  $$\\mathcal X(\\left(||q||_{\\mathbb L^{\\infty}(dK)} + 2\\varepsilon+x'\\right)X^{''}_0, q+q')\\neq \\emptyset.$$ As $x'$ and $q'$ are arbitrary elements satisfying \\eqref{454}, we deduce that $(x(q), q)\\in\\mathring{\\mathcal K}$. This proves $(ii)$. \n\n\nIn order to show $(i)$, first we observe that $\\mathring{\\mathcal K}$ is a convex cone. Therefore, it suffices to prove that, for a given $\\varepsilon>0$, we have\n\\begin{equation}\\label{2121}\n\\left(2\\varepsilon X^{''}_0, 0\\right)\\in\\mathring{\\mathcal K}.\n\\end{equation} \nAgain, let us consider $x'$ and $q'$ satisfying \\eqref{454} and $\\tilde X$ satisfying \\eqref{455} for $q\\equiv 0$. \nThen for every $t\\in[0,T]$, we have\n$$\\tilde X_t\\geq {\\varepsilon}X^{''}_t\\geq {\\varepsilon}\\int_0^t|e_s|d\\kappa_s\\geq -\\int_0^tq'_s|e_s|d\\kappa_s.$$\nThus, $\\mathcal X(\\left(2\\varepsilon+x'\\right)X^{''}_0, q')\\neq \\emptyset$.  Consequently, as $x'$ and $q'$ are arbitrary elements satisfying \\eqref{454}, \\eqref{2121} holds and so is $(i)$. \n This completes the proof of the lemma. \n\\end{proof}\n\\begin{Remark}\nA close look at the proofs shows that the conclusions of Theorem \\ref{mainTheorem} also hold if instead of Assumption \\ref{asEnd}, we impose any of the equivalent assertions $(iii) - (v)$ of Lemma \\ref{lemma1}.\n\\end{Remark}\n\n\n\\subsection{Characterization of the primal and dual domains}\\label{subSecCharDualDomain}\nThe polar, $A^{o}$, of a nonempty subset $A$ of $\\mathbb R\\times\\mathbb L^\\infty(dK)$, is\n the subset of\n $\\mathbb R\\times\\mathbb L^1(dK)$, defined by\n$$A^o{:=} \\left\\{(y,r)\\in \\mathbb R\\times \\mathbb L^1(dK):~xy + \\int_0^T q_sr_sdK_s \\leq 1, \\quad for~every~(x,q)\\in A\\right\\}.$$\nThe polar of a subset of $\\mathbb R\\times\\mathbb L^1(dK)$ is defined similarly.\n\\begin{Proposition}\\label{prop1} Under Assumption \\ref{asEnd} and conditions  \\eqref{finClock} and \\eqref{NFLVR}, we have:\n\n$(i)$ Let $(x,q)\\in\\mathbb R\\times\\mathbb L^{\\infty}(dK)$. Then $c\\in \\mathcal A(x,q)$ (thus, $\\mathcal A(x,q)\\neq \\emptyset$ so $(x,q)\\in \\mathcal K$ )  if and only if\n\\begin{equation}\\nonumber\n\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\n~~ {\\rm for~every}~(y,r)\\in \\mathcal L{\\rm ~ and }~ Y\\in\\mathcal Y(y,r).\n\\end{equation}\n\n$(ii)$ Likewise, for $(y,r)\\in \\mathbb{R}\\times \\mathbb{L}^1 (dK)$ we have  $(y,r)\\in \\mathcal L$ and $Y\\in \\mathcal Y(y,r)$ if and only if\n\\begin{equation}\\nonumber\n\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq xy + \\int_0^Tq_sr_sdK_s,\n~~ {\\rm for~every}~(x,q)\\in\\mathcal K{\\rm ~ and }~c\\in\\mathcal A(x,q).\n\\end{equation}\n\nWe also have $\\mathcal K = (-\\mathcal L)^{o} = \\cl{\\mathring{\\mathcal K}}$, where the closure is taken both in norm $\\mathbb R\\times \\mathbb L^1(dK)$ and $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topologies.\n\\end{Proposition}\n\\begin{Remark}\\label{remNonEmpty}\nIt follows from Proposition \\ref{prop1} that \nfor every $(x,q)\\in\\mathcal K, \\mathcal A(x,q)\\neq \\emptyset$ as $0\\in\\mathcal A(x,q)$. Likewise, \nfor every $(y,r)\\in \\mathcal L, \\mathcal Y(y,r)\\neq \\emptyset$ as $0\\in\\mathcal Y(y,r)$. Moreover, for every $(x,q)\\in  \\mathring{\\mathcal K}$, each of the sets $\\mathcal A(x,q)$ and $\\bigcup\\limits_{(y,r)\\in \\mathcal L: ~xy + \\int_0^Tr_sq_sdK_s \\leq 1}\\mathcal Y(y,r)$ contain a strictly positive element, see Lemma \\ref{6-12-1} below. \n\\end{Remark}\nThe proof of Proposition~\\ref{prop1} will be given via several lemmas.\nLet \n\\begin{equation}\\nonumbe\n\\begin{array}{l}\n\\mathcal M'~be~the~set~of~equivalent~local~martingale~measures,~under~which\\\\X^{''}~(from~Lemma~\\ref{lemma1},~item~(v))~is~a~uniformly~integrable~martingale. \n\\end{array}\n\\end{equation}\nNote that by \\cite[Theorem 5.2]{DS97}, $\\mathcal M'$ is a nonempty, convex subset of $\\mathcal M$, which is also dense in $\\mathcal M$ in the total variation norm. \n\n\\begin{Remark}\nEven though the results in \\cite{DS97} are obtained under the condition that $S$ is a locally bounded process, they also hold without local boundedness assumption, see the discussion in \\cite[Remark 3.4]{HKS05}.\n\\end{Remark}\n\nLet $Z'$ denote the set of the corresponding c\\`adl\\`ag densities,~i.e., \n\\begin{equation}\\label{defZ'}\n\\begin{array}{rcl}\n\\mathcal Z' &{:=}& \\left\\{{\\rm c\\grave adl\\grave ag}~Z: Z_t=\\mathbb E\\left[ \\frac{d\\mathbb Q}{d\\mathbb P}|\\mathcal F_t\\right],~t\\in[0,T],~\\mathbb Q\\in\\mathcal M'\\right\\}.\\\\\n\\end{array}\n\\end{equation}\n We also set\n\\begin{equation}\\label{defUpsilon}\n\\Upsilon {:=} \\left\\{1_{[0,\\tau]}(t),t\\in[0,T]:~~\\tau\\in\\Theta\\right\\}.\n\\end{equation}\n\n\\begin{Lemma}\\label{6-16-1}\nLet   the conditions of Proposition \\ref{prop1} hold, $\\mathbb Q\\in\\mathcal M'$, ${Z} = {Z}^{\\mathbb Q}$ be the corresponding element of $\\mathcal Z'$,  and $\\Lambda \\in \\Upsilon$.  Then  there exists $r\\in \\mathbb L^1(dK)$, uniquely defined~by\n\\begin{equation}\\label{6-25-1}\n\\int_0^tr_sdK_s = \\mathbb E\\left[\\int_0^t\\Lambda_s{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T],\n\\end{equation}\nsuch that $(1,r)\\in \\mathcal L$  and  ${Z}\\Lambda = ({Z}_t\\Lambda_t)_{t\\in[0,T]}\\in\\mathcal Y(1,r)$.\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathcal K$ and $c\\in\\mathcal A(x,q)$.\nThen there exists  \n$X\\in\\mathcal X(x,q)$, such that\n\\begin{equation}\\label{246}\n\\int_0^\\tau c_sd\\kappa_s\\leq X_\\tau + \\int_0^\\tau q_se_sd\\kappa_s,\\quad for~every~\\tau\\in\\Theta.\n\\end{equation}\nIn particular, \\eqref{246} holds for the particular $\\tau$, such that $\\Lambda_t = 1_{[0,\\tau]}(t)$, $t\\in[0,T]$.  \nLet  $G_t{:=} \\int_0^t(q_se_s)^{+}d\\kappa_s$, $t\\in[0,T]$, where $(\\cdot)^{+}$ denotes the positive part. By \\cite[Theorem III.29, p. 128]{Pr}, $\\int_0^tG_{s-}d{Z}_s$, $t\\in[0,T]$, is a local martingale, so let $(\\sigma_n)_{n\\in\\mathbb N}$ be its localizing sequence. Then by the monotone convergence theorem, integration by parts formula, and the optional sampling theorem, we get\n\\begin{equation}\\label{281}\n\\begin{array}{c}\n\\mathbb E\\left[\\int_0^T {Z}_s(q_se_s)^{+}\\Lambda_sd\\kappa_s \\right] = \\lim\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^{\\sigma_n\\wedge \\tau} {Z}_sdG \\right] \\\\\n= \\lim\\limits_{n\\to\\infty} \\mathbb E\\left[{Z}_{\\sigma_n\\wedge \\tau}G_{\\sigma_n\\wedge \\tau}\\right] \n-\\lim\\limits_{n\\to\\infty} \\mathbb E\\left[ \\int_0^{\\sigma_n\\wedge \\tau}G_{s-}d{Z}_s\\right] \n\\\\=\n\\lim\\limits_{n\\to\\infty} \\mathbb E\\left[{Z}_{\\sigma_n\\wedge \\tau}G_{\\sigma_n\\wedge \\tau} \\right] = \\lim\\limits_{n\\to\\infty} \\mathbb E^{\\mathbb Q}\\left[G_{\\sigma_n\\wedge \\tau} \\right] \n =  \\mathbb E^{\\mathbb Q}\\left[G_\\tau \\right],\\\\\n\\end{array}\n\\end{equation}\nwhere in the last equality we used the monotone convergence theorem again.\nHere finiteness of $\\mathbb E^{\\mathbb Q}\\left[G_\\tau \\right]=\\mathbb E^{\\mathbb Q}\\left[\\int_0^\\tau(q_se_s)^{+}d\\kappa_s \\right]$ follows from Assumption \\ref{asEnd} via Lemma \\ref{lemma1}, part (iii). \nIn a similar manner, we can show that\n\\begin{equation}\\label{283}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T(q_se_s)^{-}{Z}_s\\Lambda_sd\\kappa_s\\right] &= &\\mathbb E^{\\mathbb Q}\\left[ \\int_0^\\tau (q_se_s)^{-}d\\kappa_s\\right]<\\infty,\\\\\n\\mathbb E\\left[\\int_0^Tc_s{Z}_s\\Lambda_sd\\kappa_s\\right] &=& \\mathbb E^{\\mathbb Q}\\left[ \\int_0^\\tau c_sd\\kappa_s\\right]<\\infty.\\\\\n\\end{array}\n\\end{equation}\n\\cite[Lemma 4]{HK04} applies here and asserts that $X$ in \\eqref{246} is a supermartingale under $\\mathbb Q$. \nTherefore, from \\eqref{246}, using \\eqref{281} and \\eqref{283}, and taking expectation under $\\mathbb Q$, we get\n\\begin{equation}\\label{6-22-2}\n \\mathbb E\\left[ \\int_0^Tc_s\\Lambda_s{Z}_sd\\kappa_s\\right]\\leq x + \\mathbb E\\left[\\int_0^Tq_se_s{Z}_s\\Lambda_sd\\kappa_s\\right].\n\\end{equation}\nLet us define\n\\begin{equation}\\nonumbe\nR_t {:=} \\mathbb E\\left[\\int_0^t\\Lambda_s{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nUsing the monotone class theorem, we obtain\n\\begin{equation}\\label{6-25-2}\n\\mathbb E\\left[\\int_0^T\\tilde q_se_s{Z}_s\\Lambda_sd\\kappa_s\\right]\n=\n\\int_0^T\\tilde q_sdR_s,\\quad for~every\\quad\\tilde q\\in\\mathbb L^\\infty(dK).\n\\end{equation}\nWe claim that\n$dR$ is absolutely continuous with respect to $dK$. First, using the $\\pi-\\lambda$ theorem, one can show that for every Borel-measurable subset $A$ of $[0,T]$, we have\n$$K(A) = \\mathbb E\\left[\\int_0^T 1_A(t)d\\kappa_t \\right]\\quad and \\quad R(A) = \\mathbb E\\left[\\int_0^T1_A(t)\\Lambda_t{Z}_te_td\\kappa_t\\right].$$\nThus, if for some $A$, $K(A) = 0$, then $\\int_0^T 1_A(t)d\\kappa_t=0$ a.s. and $\\kappa^A_t {:=} \\int_0^t 1_A(s)d\\kappa_s$, $t\\in[0,T]$, satisfies $\\kappa^A_T = 0$ a.s. and $\\int_0^T \\Lambda_t{Z}_te_t d\\kappa^A_t = \\int_0^T\\Lambda_t{Z}_te_t1_A(t)d\\kappa_t = 0$ a.s. \n\n\nAs $dR$ is absolutely continuous with respect to $dK$,  \nthere exists a unique $r\\in\\mathbb L^1(dK)$, such that \\eqref{6-25-1}\nholds.\nSince the left-hand side in (\\ref{6-22-2}) is nonnegative and since $(x,q)$ is an arbitrary element of $\\mathcal K$, we deduce from the definition of $\\mathcal L$, \\eqref{defL},  that $(1,r)\\in \\mathcal L$. Finally, it follows from \\eqref{6-22-2} and \\eqref{6-25-2}\nthat ${Z}\\Lambda \\in\\mathcal Y(1,r)$. This completes the proof of the~lemma.\n\\end{proof}\n\\begin{Remark}\\label{Remark-Mihai-2}\nThe  natural convexification of  the set $\\Upsilon$  consists of non-negative  left-continuous  decreasing and adapted processes $D$ such that $D_{\\theta _0}=1$. In the context of utility-maximization constraints (and for $\\theta _0=0$), this is follows  from  \\cite{HePages} and \\cite{ElKarouiJeanblanc}. \nWe investigate convexification of the constraints  in  the later Sections \\ref{secStructureOfDualDomain} and \\ref{secSlackness}, where will extend Lemma \\ref{6-16-1} to a more general set of decreasing processes than $\\Lambda$ that drives the dual domain and that allows for the multiplicative decomposition of the dual minimizer\n\\end{Remark}\n\\begin{Corollary}\\label{cor1}\nLet   the conditions of Proposition \\ref{prop1} hold, $\\mathbb Q\\in\\mathcal M'$, ${Z}$ be the c\\'adl\\'ag modification of the density process $\\mathbb E\\left[ \\frac{d\\mathbb Q}{d\\mathbb P}|\\mathcal F_t\\right]$, $t\\in[0,T]$. Then there exists $r\\in\\mathbb L^1(dK)$, such that\n ${Z}\\in\\mathcal Y(1,r)$,\nwhere\n$$\n\\int_0^tr_sdK_s = \\mathbb E\\left[\\int_0^t{Z}_se_sd\\kappa_s \\right],\\quad t\\in[0,T],\n$$\nand $(1,r)\\in \\mathcal L$.\n\\end{Corollary}\n\\begin{proof}[Proof of Lemma \\ref{3311}] Let us fix $\\mathbb Q\\in\\mathcal M'$ and let $Z\\in\\mathcal Z'$ be the corresponding density process. As in Lemma \\ref{6-16-1}, we can show that there exists $r\\in\\mathbb L^1(dK)$, such that for every bounded and Borel measurable function $q$ on $[0,T]$ we have\n\\begin{equation}\\label{5192}\n\\int_0^tq_s r_sdK_s = \\mathbb E\\left[\\int_0^t q_sZ_s|e_s|d\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nLet $\\bar q {:=} |q^1 - q^2|$, then as $\\bar q = 0$, $dK$-a.e., we get\n\\begin{displaymath}\n\\mathbb E\\left[\\int_0^T \\bar q_s|e_s|Z_sd\\kappa_s \\right] = \\int_0^T \\bar q_sr_sdK = 0.\n\\end{displaymath}\nTherefore, using integration by parts and via \\eqref{5192}, we obtain\n$$\n0 = \\int_0^T \\bar q_sr_sdK = \\mathbb E\\left[\\int_0^T q_sZ_s|e_s|d\\kappa_s \\right] = \\mathbb E^{\\mathbb Q}\\left[\\int_0^T q_s|e_s|d\\kappa_s \\right].\n$$\nConsequently, $\\int_0^T q_s|e_s|d\\kappa_s = 0$, $\\mathbb Q$-a.s., and by the equivalence of $\\mathbb Q$ and $\\mathbb P$, also $\\mathbb P$-a.s. \nAs, by construction $\\int_0^t q_s|e_s|d\\kappa_s = 0$, $t\\in[0,T]$, is a nonnegative and non-decreasing process, whose terminal value is $0$,  $\\mathbb P$-a.s., we conclude that it is indistinguishable from the $0$-valued process. The assertions of the lemma follows.\n\n\\end{proof}\n\\begin{Lemma}\\label{9-5-1}\nLet   the conditions of Proposition \\ref{prop1} hold,  $(x,q)\\in\\mathcal K$, and $c$ is a nonnegative optional process. Then $c\\in\\mathcal A(x,q)$ if and only if\n\\begin{equation}\\label{241}\n\\mathbb E\\left[\\int_0^Tc_s\\Lambda_s{Z}_sd\\kappa_s \\right]\\leq\nx + \\mathbb E\\left[\\int_0^Tq_se_s\\Lambda_s{Z}_sd\\kappa_s \\right] = x + \\int_0^Tq_sr_sdK_s,\n\\end{equation}\nfor every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$, where $r$ is given by (\\ref{6-25-1}).\n\\end{Lemma}\n\\begin{proof}\nLet $c\\in\\mathcal A(x,q)$. Then for every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$,  the validity of \\eqref{241} follows from the definition of $\\mathcal A(x,q)$, integration by parts formula and supermartingale property of every $X\\in\\mathcal X(x,q)$ under every $\\mathbb Q\\in\\mathcal M'$, which in turn follows from \\cite[Lemma 4]{HK04}. \n\nConversely, let \\eqref{241} holds for every ${Z}\\in\\mathcal Z'$ and $\\Lambda \\in \\Upsilon$. \nThen, we have\n\\begin{equation}\\label{243}\n\\mathbb E\\left[\\int_0^T(c_s-q_se_s){Z}_s\\Lambda_sd\\kappa_s \\right]\\leq x,\n\\end{equation}\nwhich, in view of  the definition of $\\Upsilon$ in \\eqref{defUpsilon}, localization, and integration by parts, implies that\n\\begin{displaymath}\n\\sup\\limits_{\\mathbb Q\\in\\mathcal M', \\tau\\in \\Theta}\\mathbb {E^Q}\\left[\\int_0^{\\tau}(c_s-q_se_s)d\\kappa_s \\right]\\leq x,\n\\end{displaymath}\nFor $X^{''}$ given by Lemma \\ref{lemma1}, item $(v)$, let us denote\n\\begin{equation}\\label{244}\nf_t{:=} ||q||_{\\mathbb L^\\infty(dK)} X^{''}_t +\\int_0^{t}(c_s-q_se_s)d\\kappa_s,\\quad t\\in[0,T].\n\\end{equation}\nIt follows from Assumption \\ref{asEnd} and item $(v)$ of Lemma \\ref{lemma1} \n\n that \n $f$ is a nonnegative process.\nWe observe that the proof of \\cite[Proposition 4.3]{K96} goes through, if we only take stopping times in $\\Theta$ and measures in $\\mathcal M'$. This proposition allows to conclude that there exists a nonnegative c\\`adl\\`ag process $V$, such that \n\\begin{equation}\\label{defV2}\nV_t = \\esssup\\limits_{\\tau\\in \\Theta: \\tau\\geq t,\\mathbb Q \\in\\mathcal M'}\\mathbb {E^Q}\\left[f_{\\tau}|\\mathcal F_t \\right],\\quad t\\in[0,T],\n\\end{equation}\n which is a supermartingale for every $\\mathbb Q \\in\\mathcal M'$. Therefore, by the density of $\\mathcal M'$ in $\\mathcal M$ in the norm topology of $\\mathbb L^1(\\mathbb P)$ and Fatou's lemma, $V$ is a supermartingale under every $\\mathbb Q\\in\\mathcal M$. Moreover, $V_0$ satisfies  \n$$V_0\\leq x + ||q||_{\\mathbb L^\\infty(dK)} X^{''}_0,$$ by \\eqref{defV2}, \\eqref{243}, and by following the argument in the proof of Lemma \\ref{6-16-1}. \n\nWe would like to apply \nthe optional decomposition theorem of F\\\"olmer and Kramkov, \\cite[Theorem 3.1]{FK}. For this, we need to show that $V$ is a local supermartingale under every $\\mathbb Q$, such that every $X\\in\\mathcal X(1)$ is a $\\mathbb Q$-local supermartingale. However, $\\mathcal M$ is dense in the set of such measures in the norm topology of $\\mathbb L^1(\\mathbb P)$, by the results of Delbaen and Schachermayer, see \\cite[Proposition 4.7]{DS1998}. Therefore, the supermartingale property of $V$ under every such $\\mathbb Q$ follows from Fatou's lemma and supermartingale property of $V$ under every $\\mathbb Q\\in\\mathcal M$ established above. Therefore, by \\cite[Theorem 3.1]{FK}, we get\n$$V_t = V_0 + H\\cdot S_t - A_t,\\quad t\\in[0,T],$$\nwhere $A$ is a nonnegative increasing process that starts at $0$. \nSubtracting the constant $||q||_{\\mathbb L^\\infty(dK)} X^{''}_0$ from both sides of \\eqref{244}, we get\n\\begin{equation}\\nonumber\\begin{array}{c}\n\\int_0^{\\tau}(c_s-q_se_s)d\\kappa_s = f_\\tau -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0 \\leq V_\\tau -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0\\\\= V_0 -||q||_{\\mathbb L^\\infty(dK)} X^{''}_0 + H\\cdot S_\\tau-A_\\tau \\leq x + H\\cdot S_\\tau, \\quad \\tau\\in\\Theta,\n\\end{array}\\end{equation}\nwhere $x + H\\cdot S$ is acceptable, since $V$ is nonnegative. \nConsequently, \n$c\\in\\mathcal A(x,q)$. This completes the proof of the lemma.\n\\end{proof}\n\n\n\\begin{proof}[Proof of Proposition~\\ref{prop1}]\nThe assertions of item $(i)$ follow from Lemma~\\ref{9-5-1}. \nIt remains to show that the affirmations of item $(ii)$ hold.\nFix a $(y,r)\\in \\mathcal L$. If $Y\\in\\mathcal Y(y,r)$, $(ii)$ follows from the definition of $\\mathcal Y(y,r)$. Conversely,\nif $(ii)$ holds for a nonnegative process $Y$, then since $(x,0)\\in \\mathcal K$ for every $x>0$, we have \n$$\\mathbb E\\left[\\int_0^Tc_sY_sd\\kappa_s \\right]\\leq 1\\quad {\\rm for~every~}c\\in\\mathcal A\\left(\\frac{1}{y},0\\right).$$\nVia \\cite[Proposition 4.4]{MostovyiNec}, we deduce that $Y\\in\\mathcal Y(y)$ and is such that $(ii)$ holds. Therefore,\n$Y\\in\\mathcal Y(y,r)$.  \n\nWe have  $\\mathring{\\mathcal K}\\neq \\emptyset$, where the interior is taken with respect to the norm-topology. According to Proposition \\ref{prop1} (i)  we have   also $\\mathcal K = (-\\mathcal L)^{o}$.  The set $\\mathcal{K}$, as the polar of $\\mathcal L$,  is convex and  closed both in  (strong) $\\mathbb R\\times \\mathbb L^{\\infty}(dK)$ and $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topologies. Having non-empty strong interior, we obtain $\\mathcal K =  \\cl{\\mathring{\\mathcal K}}$ where the closure is in the strong-topology. \nSince $\\mathcal{K}$ is also closed in the weaker $\\sigma(\\mathbb R\\times\\mathbb L^\\infty(dK), \\mathbb R\\times\\mathbb L^1(dK))$ topology we obtain that \n$$\\mathcal K = (-\\mathcal L)^{o} = \\cl{\\mathring{\\mathcal K}},$$ where the closure is taken in both\ntopologies.\n\\end{proof}\n\n\\subsection{Preliminary properties of the value functions, in particular finiteness. }\n\nLet $\\mathbf  L^0_{+}$ be the positive orthant of $\\mathbf  L^0$. The polar of a set $A \\subseteq \\mathbf  L^0_{+}$ is defined as\n\\begin{displaymath} \nA^o{:=} \\left\\{ c\\in\\mathbf  L^0_{+}:~\\mathbb E\\left[\\int_0^T c_sY_s d\\kappa_s\\right]\\leq 1,\\quad for~every~Y \\in A\\right\\}.\n\\end{displaymath}\n\nWe recall that the sets $\\mathcal Z$ and $\\mathcal Z'$ are defined in \\eqref{defZ} and \\eqref{defZ'}, respectively. \n  \n\n\\begin{Lemma}\\label{finitenessOverZ'}\nUnder the conditions of Theorem~\\ref{mainTheorem}, we have\n\\begin{equation}\\label{bipolarZ'}\n(\\mathcal Z')^{oo} = \\mathcal Y(1)\n\\end{equation}\nand\n\\begin{equation}\\label{2145}\n\\tilde w(y) = \\inf\\limits_{Y\\in\\mathcal Z'}\\mathbb E\\left[\\int_0^TV(t, yY_s)d\\kappa_s \\right]<\\infty, \\quad y>0.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIt follows from\n \\cite[Theorem 5.2]{DS97}\n that $\\mathcal Z'$ is dense in $\\mathcal Z$ \n in $\\mathbf  L^0$. \nIt follows from Fatou's lemma that $$(\\mathcal Z')^{o} = \\mathcal Z^{o}.$$\nTherefore, \\cite[Lemma 4.2 and Proposition 4.4]{MostovyiNec} imply \\eqref{bipolarZ'}.  \n\nOne can show that $\\mathcal Z'$ is closed under countable convex combinations, where the martingale property follows from the monotone convergence theorem \\and the c\\`adl\\`ag structure of the limit is guaranteed by \\cite[Theorem VI.18]{DelMey82}, see also \\cite[ Proposition 5.1]{KLPO14} for more details in similar settings. \nNow, \\eqref{2145} follows (up to a notational change) from \\cite[Theorem 3.3]{MostovyiNec}. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{6-17-1} Under the conditions of Theorem~\\ref{mainTheorem}, \nfor every $(x,q)\\in\\mathcal K$ and $(y,r)\\in \\mathcal L$, we have\n$$u(x,q) \\leq v(y,r) + xy +\\int_0^Tr_sq_sdK_s.$$\n\\end{Lemma} \n\\begin{proof}\nFix an arbitrary $(x,q)\\in\\cl\\mathcal K$, $c\\in\\mathcal A(x,q)$ as well as $(y,r) \\in \\mathcal L$, $Y\\in\\mathcal Y(y,r)$. Using Proposition\n\\ref{prop1} and \\eqref{defV}, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T U (t, c_s)d\\kappa_s \\right] &\\leq &\\mathbb E\\left[\\int_0^T U(t, c_s)d\\kappa_s \\right] + xy + \\int_0^Tr_sq_sdK_s - \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\\\\n&\\leq &\\mathbb E\\left[\\int_0^T V(t, Y_s)d\\kappa_s \\right] + xy + \\int_0^Tr_sq_sdK_s.  \\\\\n\\end{array}\n\\end{displaymath}\nThis implies the assertion of the lemma.\n\\end{proof}\nFor every $(x,q)$ in $\\mathcal K$, we define\n\\begin{equation}\\label{auxiliarySets}\n\\begin{array}{rcl}\n\\mathcal B(x,q)&{:=}& \\left\\{(y,r)\\in \\mathcal L: ~xy + \\int_0^Tr_sq_sdK_s \\leq 1 \\right\\},\\\\\n\\mathcal D(x,q)&{:=}& \\bigcup\\limits_{(y,r)\\in \\mathcal B(x,q)}\\mathcal Y(y,r).\\\\\n\\end{array}\n\\end{equation}\nThe subsequent lemma established boundedness of $\\mathcal B(x,q)$ for $(x,q)$ in $\\mathring{\\mathcal K}$ in $\\mathbb R\\times \\mathbb L^1(dK)$.\n\\begin{Lemma}\n\\label{7-17-1}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(x,q)\\in\\mathring{\\mathcal K}$, \n$\\mathcal B(x,q)$ is bounded\n in $\\mathbb R\\times \\mathbb L^1(dK)$.\n\\end{Lemma}\n\\begin{proof}\nFix an $(x,q)\\in\\mathring{\\mathcal K}$. Then there exists $\\varepsilon > 0$, such that\nfor every\n\\begin{equation}\\label{7-20-1}\n|x'|\\leq\\varepsilon\\quad and \\quad||q'||_{\\mathbb L^{\\infty}} \\leq\\varepsilon,\n\\end{equation}\n we have\n$(x + x', q+q')\\in{\\mathring{\\mathcal K}}.$\nLet us fix an arbitrary $(y,r)\\in \\mathcal B(x,q)$. Then for every $(x',q')$ satisfying (\\ref{7-20-1}), by the definitions of $\\mathcal L$ and $\\mathcal B(x,q)$, respectively,~we~get \n\\begin{displaymath}\n\\begin{array}{rcl}\nxy + \\int_0^Tq_sr_sdK_s + x'y + \\int_0^Tq'_sr_sdK_s &\\geq& 0, \\\\\nxy + \\int_0^Tq_sr_sdK_s  &\\leq& 1, \\\\\n\\end{array}\n\\end{displaymath}\nwhich implies that\n\\begin{equation}\\label{7-17-2}\n-x'y - \\int_0^Tq'_sr_sdK_s \\leq xy + \\int_0^Tq_sr_sdK_s  \\leq 1.\n\\end{equation}\nTaking \n\\begin{displaymath}\nx' = -\\varepsilon\\quad{and}\\quad q'\\equiv 0,\n\\end{displaymath}\nwe deduce from (\\ref{7-17-2}) that $y\\leq \\frac{1}{\\varepsilon}$. Also, by the definition of $\\mathcal L$ and Lemma \\ref{lemma1}, item $(i)$,  $y\\geq 0$.\nIn turn, setting\n\\begin{displaymath}\nx' = 0\\quad{and}\\quad q' = -\\varepsilon 1_{\\{r\\geq 0\\}} + \\varepsilon 1_{\\{r<0\\}},\n\\end{displaymath}\nwe obtain from  (\\ref{7-17-2}) that $||r||_{\\mathbb L^1}\\leq \\frac{1}{\\varepsilon}$. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{6-12-1}\nLet the conditions of Theorem~\\ref{mainTheorem} hold and $(x,q)$ be an arbitrary element of $\\mathring{\\mathcal K}$.\nThen, we have:\n\\newline\n$(i)$ $\\mathcal A(x,q)$ contains a strictly positive process.\n\\newline\n$(ii)$ The constant $\\bar y(x,q)$ given by  \n\\begin{equation}\\label{defBary}\n\\bar y(x,q) {:=} \\frac{1}{|x| + ||q||_{\\mathbb L^\\infty(dK)}\\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[\\int_0^T|e_s|d\\kappa_s \\right]},\n\\end{equation}\ntakes values in $(0,\\infty)$ and satisfies\n\\begin{equation}\\label{6-18-1}\n\\bar y(x,q)\\mathcal Z'\\subseteq \\mathcal D(x,q).\n\\end{equation}\nIn particular, for every $z>0$, $z\\mathcal D(x,q)$ contains a strictly positive process $Y$ such that \n\\begin{equation}\\label{6-16-2}\n\\mathbb E\\left[\\int_0^TV(s, Y_s)d\\kappa_s \\right] <\\infty.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nIn order to show $(i)$, we observe that the existence of a positive process in $\\mathcal A(x,q)$ follows from the fact that $(x-\\delta, q)\\in\\mathcal K$ for a sufficiently small $\\delta.$\nNow the constant-valued consumption $\\delta\/A>0$, where $A$ is the constant that dominates the terminal value of the stochastic clock $\\kappa$ in \\eqref{finClock}, is in $\\mathcal A(x,q)$.\n\nIn order to prove $(ii)$, \nlet us consider $\\bar y(x,q)$ given by \\eqref{defBary}.\nIt follows from Lemma \\ref{lemma1}, item $(iii)$, that $\\bar y (x,q)\\in (0,\\infty)$. \nFor this $\\bar y(x,q)$, using\nCorollary \\ref{cor1}, one can show~(\\ref{6-18-1}). This and Lemma~\\ref{finitenessOverZ'}  (note that finiteness of $\\tilde w$ follows directly from \\eqref{finValue})\nimply that for every $z>0$, there exists a positive $Y\\in z\\mathcal D(x,q)$, such that (\\ref{6-16-2}) holds.\n\n\n\\end{proof}\n\\begin{Lemma}\\label{lemFinu}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(x,q)\\in\\mathring{\\mathcal K}$ we have\n\\begin{displaymath}\n-\\infty < u(x,q) < \\infty.\n\\end{displaymath}\nand $u<\\infty$ on $\\mathbb R\\times \\mathbb L^\\infty(dK)$.\n\\end{Lemma}\n\\begin{proof}\nLet us fix an arbitrary $(x,q)\\in\\mathring{\\mathcal K}$.  Since $\\mathring{\\mathcal K}$ is an open convex cone, there exists $\\lambda\\in(0,1)$, $(x_1, q_1)\\in\\mathring{\\mathcal K}$, and $x_2>0$, such that\n $$(x,q) =\\lambda (x_1, q_1) + (1- \\lambda)(x_2,0).$$\nNote that $(x_2, 0)\\in\\mathring{\\mathcal K}$ by Lemma \\ref{lemma1}. By \\eqref{finValue}, there $c\\in\\mathcal A(x_2,0)$, such that \n\\begin{equation}\\label{2141}\n\\mathbb E\\left[\\int_0^TU\\left(t, (1 - \\lambda) c_t\\right)d\\kappa_t  \\right] > -\\infty.\n\\end{equation}\nAs $\\mathcal A(x_1,q_1)\\neq\\emptyset$ (see Remark \\ref{remNonEmpty}), there exists $\\tilde c\\in\\mathcal A(x_1,q_1)$. As $U(t, \\cdot)$ is nondecreasing, we get\n\\begin{displaymath}\nu(x,q) \\geq \\mathbb E\\left[\\int_0^TU(t, \\lambda \\tilde c_t + (1 - \\lambda) c_t)d\\kappa_t  \\right] \n\\geq \\mathbb E\\left[\\int_0^TU(t, (1 - \\lambda) c_t)d\\kappa_t  \\right] \n> -\\infty,\n\\end{displaymath}\nwhere the last inequality follows from \\eqref{2141}. This implies finiteness of $u$ on $\\mathring{\\mathcal K}$ from below. \n\nIn order to show finiteness from above, let us fix a process $c\\in\\mathcal A(x,q)$, such~that \n\\begin{equation}\\nonumbe\n\\mathbb E\\left[\\int_0^TU(t, c_t)d\\kappa_t  \\right] \n> -\\infty.\n\\end{equation}\nBy Lemma \\ref{finitenessOverZ'}, there exists $Y\\in\\mathcal Z'$, such that \n\\begin{equation}\\label{4103}\n\\mathbb E\\left[\\int_0^TV(t,Y_t)d\\kappa_t \\right]<\\infty.\n\\end{equation\nIt follows from Lemma \\ref{6-16-1} that $Y\\in\\mathcal Y(1,\\rho)$ for some $(1,\\rho)\\in \\mathcal L$. Therefore, by Proposition \\ref{prop1}, we get\n\\begin{equation}\\label{4101}\n\\begin{array}{rcl}\n\\mathbb E\\left[\\int_0^T U (s, c_s)d\\kappa_s \\right]& \\leq& \\mathbb E\\left[\\int_0^T U(s,c_s)d\\kappa_s \\right] + x + \\int_0^T\\rho_sq_sdK_s - \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\\\\n&\\leq& \\mathbb E\\left[\\int_0^T V(s,Y_s)d\\kappa_s \\right] + x + \\int_0^T\\rho_sq_sdK_s.  \\\\\n\\end{array}\n\\end{equation}\nAs $Y$ satisfies (\\ref{4103}), we conclude  that $u(x,q) < \\infty$. \nMoreover, for $(x,q)\\in\\mathcal K$, as $\\mathcal A(x,q)\\neq\\emptyset$ by Remark \\ref{remNonEmpty}, every $c\\in\\mathcal A(x,q)$ satisfies \\eqref{4101} (with the same $Y$). \nThis implies that $u<\\infty$ on $\\mathcal K$ and therefore, by \\eqref{3313}, on $\\mathbb R\\times \\mathbb L^\\infty(dK)$. This completes the proof of the lemma.\n\\end{proof}\nWe recall that, for every $(x,q)\\in\\mathcal K$, $\\mathcal D(x,q)$ is defined in \\eqref{auxiliarySets}. Let $cl \\mathcal D(x,q)$ denote the closure of $\\mathcal D(x,q)$ in $\\mathbb L^0(d\\kappa \\times \\mathbb P).$ The following lemma proves a delicate point that, for  $(x,q)\\in\\mathring{\\mathcal K}$, by passing from $\\mathcal D(x,q)$ to $cl\\mathcal D(x,q)$, we do not change the auxiliary dual value function. \n\\begin{Lemma}\\label{auxiliaryLemma}\nLet the conditions of Theorem~\\ref{mainTheorem} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then, for every $z>0$, we have\n\\begin{equation}\\label{2143}\n-\\infty<\\inf\\limits_{Y \\in\\cl \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(s, zY_s)d\\kappa_s \\right] =\n\\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(s, zY_s)d\\kappa_s \\right]<\\infty.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nFiniteness from above follows from Lemma \\ref{6-12-1}. To show finiteness of both infima in \\eqref{2143} from below, by Lemma~\\ref{lemFinu} we deduce\nthe existence of $c\\in\\mathcal A(x,q)$, such that\n\\begin{equation}\\label{6-18-2}\n\\mathbb E\\left[\\int_0^TU(s, c_s)d\\kappa_s \\right] > -\\infty.\n\\end{equation}\nLet $Y\\in{\\cl \\mathcal D(x,q)}$ and let $Y^n\\in\\mathcal Y(y^n,r^n)$, $n\\geq 1$, be a sequence in $\\mathcal D(x,q)$ that converges to $Y$ in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$. By Fatou's lemma, Proposition~\\ref{prop1}, and the definition of the set $\\mathcal B(x,q)$ in \\eqref{auxiliarySets}, we get\n\\begin{displaymath}\n\\mathbb E\\left[\\int_0^TY_sc_sd\\kappa_s \\right] \\leq \n\\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^TY^n_sc_sd\\kappa_s \\right] \\leq\n\\sup\\limits_{n\\geq 1}\\left(xy^n + \\int_0^Tr^n_sq_sdK_s\\right) \\leq 1. \n\\end{displaymath}\nTherefore, we obtain\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E\\left[ \\int_0^TU(s, c_s)d\\kappa_s\\right]&\\leq& \\mathbb E\\left[ \\int_0^TU(s, c_s)d\\kappa_s\\right] + 1 - \\mathbb E\\left[\\int_0^TY_sc_sd\\kappa_s \\right] \\\\\n&\\leq& \\mathbb E\\left[ \\int_0^TV(s, Y_s)d\\kappa_s\\right] + 1,\\\\\n\\end{array}\n\\end{displaymath}\nwhich together with (\\ref{6-18-2}) implies finiteness of both infima in \\eqref{2143} from below.\n\n\nLet us show equality of two infima in \\eqref{2143}. It follows from Lemma\n\\ref{6-12-1} that for every $z> 0$ there exists a process $Y\\in z\\mathcal D(x,q)$, such that\n$$\\mathbb E\\left[\\int_0^TV(s, Y_s)d\\kappa_s \\right] <\\infty.$$\nLet us fix $z>0$ and let $\\bar Y\\in\\cl\\mathcal D(x,q)$. Also, let $(Y^n)_{n\\in\\mathbb N}$ be a sequence in $\\mathcal D(x,q)$ that converges to $\\bar Y$ $(d\\kappa\\times\\mathbb P)$-a.e. \nLet us fix $\\delta >0$, then by Lemma \\ref{finitenessOverZ'}, there exists $Z'\\in\\mathcal Z'$, such that\n$$\\mathbb E\\left[\\int_0^TV(t, \\delta\\bar y(x,q) Z'_t)d\\kappa_t \\right]<\\infty,$$\nwhere $\\bar y(x,q)$ is defined in \\eqref{defBary}. Note that $\\bar y(x,q) Z'\\in \\mathcal D(x,q)$ by Lemma \\ref{6-12-1} (see \\eqref{6-18-1}). \nTherefore, using Fatou's lemma and monotonicity of $V$ in the spatial variable, we obtain\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, (z+\\delta)Y_s)d\\kappa_s \\right]&\\leq&\n\\limsup\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^TV(t, zY^n_s+\\delta \\bar y(x,q)Z'_s)d\\kappa_s \\right]\\\\ \n&\\leq  &\\mathbb E\\left[\\int_0^TV(t, z\\bar Y_s + \\delta\\bar y(x,q) Z'_s)d\\kappa_s \\right] \\\\\n&\\leq &\\mathbb E\\left[\\int_0^TV(t, z\\bar Y_s )d\\kappa_s \\right].\\\\\n\\end{array}\n\\end{displaymath}\nTaking the infimum over $Y\\in\\cl\\mathcal D(x,y)$, we deduce that\n\\begin{equation}\\label{21410}\n \\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, (z+\\delta)Y_s)d\\kappa_s \\right] \\leq \\inf\\limits_{Y\\in\\cl\\mathcal D(x,y)}\\mathbb E\\left[\\int_0^TV(t, zY_s )d\\kappa_s \\right].\n\\end{equation}\nLet us consider \n$$\\phi(z){:=} \\inf\\limits_{Y \\in \\mathcal D(x,q)}\\mathbb E\\left[\\int_0^TV(t, zY_s)d\\kappa_s \\right], \\quad z>0.$$\nBy the first part of the proof (finiteness of both infima), $\\phi$ is finite-valued on $(0,\\infty)$.  Convexity of $V$ in the spatial variable implies that $\\phi$ is also convex.  Therefore, $\\phi$ is continuous. As \\eqref{21410} holds for every $\\delta>0$, by taking the limit as $\\delta \\downarrow 0$ in \\eqref{21410}, we conclude that both infima in \\eqref{2143} are equal. This completes the proof of the lemma.\n\n\\end{proof}\n\nLet us define\n\\begin{equation}\\label{defE}\n\\mathcal E {:=} \\{(y,r)\\in \\mathcal L: ~v(y,r) < \\infty \\}.\n\\end{equation}\n\\begin{Lemma}\\label{lemFinv}\nUnder the conditions of Theorem~\\ref{mainTheorem}, for every $(y,r)\\in \\mathcal L$, we have\n$$v(y,r) > -\\infty.$$\nTherefore, $v>-\\infty$ on $\\mathbb R\\times\\mathbb L^1(dK)$. The set \n $\\mathcal E$ is a nonempty convex subset of $\\mathcal L$, whose closure in $\\mathbb R\\times\\mathbb L^1(dK)$ equals to $\\mathcal L$, and such that\n\\begin{equation}\\label{6-17-2}\n\\mathcal E= \\bigcup\\limits_{\\lambda \\geq 1 }\\lambda \\mathcal E.\n\\end{equation}\n\n\\end{Lemma}\n\\begin{proof}\nLet us  fix $(y,r)\\in \\mathcal L$, then finiteness of $v(y,r)$ from below follows from  \\eqref{finValue} and Lemma~\\ref{6-17-1}.\nTo establish the properties of $\\mathcal E$, we observe that the convexity of $\\mathcal E$  and (\\ref{6-17-2}) follow from convexity and monotonicity of $V$, respectively. \n\n\nIn remains to show that the closure of $\\mathcal E$ in $\\mathbb R\\times\\mathbb L^1(dK)$ contains the origin. \nIn \\eqref{auxiliarySets}, let us consider $(x,q) = (1,0)\\in\\mathcal K$. In this case, we have \n $$\\mathcal D(1,0) = \\bigcup\\limits_{(y,r)\\in\\mathcal L: y\\leq 1}\\mathcal Y(y,r)\\subseteq \\mathcal Y(1),$$\nwhere the last inclusion follows from the  very  definition of $\\mathcal Y(y,r)$'s in \\eqref{defY}.\nAs, by \\eqref{oldY}, $\\mathcal Y(1)$ is closed in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$  and $\\mathcal D(1,0)\\subseteq \\mathcal Y(1)$, we deduce that \n\\begin{equation}\\label{2146}\n\\cl \\mathcal D(1,0)\\subseteq \\mathcal Y(1).\n\\end{equation}\n By Lemma \\ref{6-12-1}, $\\mathcal Z'\\subset \\mathcal D(1,0)$, as $\\bar y(1,0) = 1$. Therefore, by the bipolar theorem of Brannath and Schachermayer, \\cite[Theorem 1.3]{BranSchach}, we get\n\\begin{equation}\\label{2147}\n(\\mathcal Z')^{oo} \\subseteq\\cl\\mathcal D(1,0).\n\\end{equation}\nOn the other hand, Lemma \\ref{finitenessOverZ'} asserts that \n\\begin{equation}\\label{2148}\n(\\mathcal Z')^{oo} = \\mathcal Y(1).\n\\end{equation}\nCombining \\eqref{2146}, \\eqref{2147}, and \\eqref{2148}, we conclude\n\\begin{equation}\\nonumbe\n\\cl \\mathcal D(1,0)= \\mathcal Y(1).\n\\end{equation}\nTherefore, the sets $\\cl\\mathcal D(1,0)= \\mathcal Y(1)$ and $\\mathcal A(1,0)$ satisfy the  precise technical  assumptions of \\cite[Theorem 3.2]{MostovyiNec}, which, for every $x>0$, grants  the existence of $\\widehat c(x)\\in\\mathcal A(x,0)$, the unique maximizer to $w(x)$, where $w$ is defined in \\eqref{defw}.\nFor every $x>0$, we set\n$\nY_{\\cdot}(x){:=} U'({\\cdot}, \\widehat c_{\\cdot}(x)), \\quad \n(d\\kappa\\times\\mathbb P)-a.e\n$$\nBy \\cite[Theorem 3.2]{MostovyiNec}, for every $x>0$,  $Y(x)$  satisfies\n$$Y(x) \\in w'(x)\\cl\\mathcal D(1,0)\\quad and\\quad \\mathbb E\\left[ \\int_0^TV(t,  Y_t(x))d\\kappa_t\\right]<\\infty,\\quad x>0,$$\nwhere by \\cite[Theorem 3.2]{MostovyiNec}, $w$ is a strictly concave, differentiable function on $(0,\\infty)$ that satisfies the Inada conditions. \nTherefore, as $w'(x)$ can be arbitrary close to $0$ (by taking $x$ large enough an by using the Inada conditions) and by Lemmas \\ref{6-12-1} and \\ref{auxiliaryLemma}, we conclude that the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ contains origin. \n\nIn order to prove that the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ equals to $\\mathcal L$, let $(y,r)\\in\\mathcal L\\backslash (0, 0)$ be fixed. Let us take $\\varepsilon>0$. We want to find $(\\tilde y, \\tilde r)$, such that \n\\begin{equation}\n\\label{4171}\n|\\tilde y - y| + || \\tilde r- r ||_{\\mathbb L^1(dK)}<\\varepsilon,\n\\end{equation}\nand\n\\begin{equation}\\label{4172}\n(\\tilde y, \\tilde r)\\in\\mathcal E.\n\\end{equation}\nAs the closure of $\\mathcal E$ in $\\mathbb R\\times \\mathbb L^1(dK)$ contains origin, we can pick $(y^0, r^0)\\in\\mathcal E$, such that \n\\begin{equation}\\label{4173}\n|y^0| + ||r^0||_{\\mathbb L^1(dK)} \\leq \\varepsilon\/3\n\\end{equation}\nand $Y\\in\\mathcal Y(y^0,r^0)$, such that \n\\begin{equation}\\label{4174}\n\\mathbb E\\left[\\int_0^T V(t, Y_t)d\\kappa_t \\right]<\\infty.\n\\end{equation}\nLet us fix $\\alpha >1$, such that \n\\begin{equation}\\label{4177}\n\\frac{|y| + ||r||_{\\mathbb L^1(dK)}}{\\alpha} \\leq \\varepsilon\/3\n\\end{equation}\n and set $\\varepsilon' {:=} \\frac{1}{\\alpha}\\in(0,1)$. By \\eqref{6-17-2}, $(\\alpha y^0, \\alpha r^0)\\in\\mathcal E.$ Let \n\\begin{equation}\\nonumbe\n\\tilde y {:=}  (1-\\varepsilon')y + \\varepsilon'\\alpha y^0,\\quad \\tilde r {:=} (1-\\varepsilon')r + \\varepsilon'\\alpha r^0.\n\\end{equation} \nThen\n\\begin{displaymath}\n\\begin{array}{rcl}\n|y - \\tilde y| + || r - \\tilde r||_{\\mathbb L^1(dK)} &= & \\varepsilon' \\alpha |\\frac{y}{\\alpha} - y^0| + \\varepsilon'\\alpha || \\frac{r}{\\alpha} - r^0||_{\\mathbb L^1(dK)}\\\\\n&\\leq & \\frac{|y| + ||r||_{\\mathbb L^1(dK)}}{\\alpha} + |y^0| + ||r^0||_{\\mathbb L^1(dK)}\n\\\\\n&\\leq & \\frac{2\\varepsilon}{3},\n\\end{array}\n\\end{displaymath}\nwhere in the last inequality we have used \\eqref{4173} and \\eqref{4177}. Thus $(\\tilde y, \\tilde r)$ satisfies \\eqref{4171}. Further, as $0\\in\\mathcal Y(y,r)$, by convexity of $\\mathcal L$ and using Proposition \\ref{prop1}, we get\n\\begin{displaymath}\nY=(1-\\varepsilon') 0 + \\varepsilon' \\alpha Y \\in\\mathcal Y(\\tilde y, \\tilde r),\n\\end{displaymath}\nwhich by \\eqref{4174} implies \\eqref{4172}. This completes the proof of the lemma.\n\\end{proof}\n\n\\subsection{Existence and uniqueness of solutions to \\eqref{primalProblem} and \\eqref{dualProblem}; semicontinuity and biconjugacy of $u$ and $v$}\n\n\\begin{Lemma}\\label{existenceUniqueness}\nUnder the conditions of Theorem~\\ref{mainTheorem},\nthe value function $v$ is convex, proper, and lower semicontinuous with respect to the  topology of $\\mathbb R\\times \\mathbb L^1(dK)$. For every $(y,r)\\in\\mathcal E$, there exists a unique solution to \n(\\ref{dualProblem}). Likewise, $u$ is concave, proper, and upper semicontinuous  with respect to  the  strong  topology of $\\mathbb R\\times \\mathbb L^\\infty(dK)$. For every $(x,q)\\in\\{u>-\\infty\\}$ there exists a unique solution to~(\\ref{primalProblem}).\n\\end{Lemma}\n\\begin{proof}\nLet $(y^n, r^n)_{n\\in\\mathbb N}$ be a sequence in $\\mathcal L$ that converges to $(y,r)$ in $\\mathbb R\\times\\mathbb L^1(dK)$. \nPassing if necessary to a subsequence, we will assume that \n\\begin{equation}\\label{2151}\n\\lim\\limits_{n\\to\\infty} v(y^n,r^n)= \\liminf\\limits_{n\\to\\infty}v(y^n, r^n).\n\\end{equation}\nLet $Y^n\\in\\mathcal Y(y^n, r^n)$, $n\\in\\mathbb N$, be such that \n\\begin{equation}\\label{2152}\n\\mathbb E\\left[ \\int_0^TV(t,Y^n_t)d\\kappa_t\\right]\\leq v(y^n, r^n) +\\frac{1}{n}, \\quad n\\in\\mathbb N.\n\\end{equation}\nBy passing to convex combinations and applying Komlos'-type lemma, see e.g. \\cite[Lemma A1.1]{DS}, we may suppose that $\\widetilde Y^n\\in\\conv\\left(Y^n, Y^{n+1},\\dots\\right)$, $n\\in\\mathbb N$, converges $(d\\kappa\\times\\mathbb P)$-a.e. to some $\\widehat Y$. \n\nFor every $(x,q)\\in\\mathcal K$ and $c\\in\\mathcal A(x,q)$, by Fatou's lemma, we have\n\\begin{displaymath}\n\\begin{array}{c}\n\\mathbb E\\left[ \\int_0^T c_t\\widehat Y_td\\kappa_t\\right]\\leq \\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[ \\int_0^T c_t \\widetilde Y^n_td\\kappa_t\\right]\\leq\nxy + \\int_0^Tq_sr_sdK_s.\n\\end{array}\n\\end{displaymath} \nTherefore, by Proposition \\ref{prop1}, $\\widehat Y\\in\\mathcal Y(y,r)$. With $\\bar y {:=} \\sup\\limits_{n\\geq 1}y^n$, we have $(\\widetilde Y^n)_{n\\in\\mathbb N}\\subseteq \\mathcal Y(\\bar y).$ Therefore, by \\cite[Lemma 3.5]{MostovyiNec}, we deduce that \n$V^{-}(t,\\widetilde Y^n_t)$, $n\\in\\mathbb N$, is a uniformly integrable sequence.  \nCombining uniform integrability with the convexity of $V$ in the spatial variable, we get\n\\begin{equation}\\label{4141}\n\\begin{array}{c}\nv(y,r)\\leq \\mathbb E\\left[ \\int_0^TV(t,\\widehat Y_t)d\\kappa_t\\right]\\leq \n\\liminf\\limits_{n\\to\\infty}\n\\mathbb E\\left[ \\int_0^TV(t, \\widetilde Y^n_t)d\\kappa_t\\right]\n\\\\\n\\leq \n\\liminf\\limits_{n\\to\\infty}\n\\mathbb E\\left[ \\int_0^TV(t,  Y^n_t)d\\kappa_t\\right] = \\liminf\\limits_{n\\to\\infty}v(y^n,r^n),\n\\end{array}\n\\end{equation}\nwhere in the last equality we have used \\eqref{2151} and \\eqref{2152}. Since $(y^n, q^n)$ was an arbitrary sequence that converges to $(y,r)$, lower semicontinuity of $v$ in strong  topology of $\\mathbb R\\times \\mathbb L^1(dK)$ follows. Since $\\mathcal L$ is closed and $v = \\infty$ outside of $\\mathcal L$, we deduce that $v$ is lower semicontinuous on $\\mathbb R\\times \\mathbb L^1(dK)$. The  function $v$ is proper by Lemma \\ref{lemFinv}. Note that \\eqref{4141} also implies that $\\mathcal E$ defined in \\eqref{defE} is $\\mathbb R\\times\\mathbb L^1(dK)$-norm closed. For $(y,r)\\in\\mathcal E$, by taking $(y^n, r^n)= (y,r)$, $n\\in\\mathbb N$, we deduce the existence of a minimizer to \\eqref{dualProblem}. Strict convexity of $V$ results in the uniqueness of the minimizer to \\eqref{dualProblem}. Convexity of $v$ follows.  Upper semicontinuity of $u$  with respect to the norm-topology of   $\\mathbb R\\times \\mathbb L^{\\infty}(dK)$ can be proven similarly, first proving semi-continuity on \n $\\mathcal K$  by a Fatou-type argument, then using the closedness  of $\\mathcal{K}$ and the definition of $u$ outside it. \n\\end{proof}\n\\begin{Corollary}\nUnder the conditions of Theorem~\\ref{mainTheorem},\n$-u$ and $v$ are  also  lower semicontinuous with respect to the weak topologies $\\sigma(\\mathbb R\\times\\mathbb L^{\\infty}(dK), (\\mathbb R\\times\\mathbb L^{\\infty})^*(dK))$ and $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb L^{\\infty}(dK))$, respectively.\n\\end{Corollary}\n\\begin{proof}\nThe assertions of the corollary is a consequence of \\cite[Proposition 2.2.10]{BarbuPrec}, see also \\cite[Corollary I.2.2]{EkelandTemam}.\n\\end{proof}\n\nWe recall that ${\\cl \\mathcal D(x,q)}$ denotes the closure of $\\mathcal D(x,q)$ in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$. \n\\begin{Lemma}\\label{5181}\nUnder the conditions of Theorem \\ref{mainTheorem}, for every $(x,q)$ in $\\mathring{\\mathcal K}$, and a nonnegative optional process $c$, we have\n$$\nc\\in\\mathcal A(x,q)\\quad{if~and~only~if}\\quad \\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq 1\\quad {for~every~}Y\\in{\\cl \\mathcal D(x,q)}.\n$$\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)$ in $\\mathring{\\mathcal K}$, $c$ is a nonnegative optional process such that \n\\begin{equation}\\label{5191}\n\\mathbb E\\left[ \\int_0^T c_sY_sd\\kappa_s \\right]\\leq 1\\quad {for~every}\\quad Y\\in{\\cl \\mathcal D(x,q)}.\n\\end{equation}\nConsider arbitrary $Z\\in\\mathcal Z'$ and $\\Lambda\\in\\Upsilon$. Let the corresponding $r$ be given by \\eqref{6-25-1} and we set $$y'{:=} x + \\int_0^Tq_sr_sdK_s.$$\n\nIf $y' = 0$, then $y\\Lambda Z\\in\\mathcal D(x,q)$ for every $y>0$. Thus,  by \\eqref{5191}, we obtain that $\\mathbb E\\left[\\int_0^T c_sy\\Lambda_sZ_sd\\kappa_s \\right]\\leq 1$. Taking the limit as $y\\to\\infty$, we get\n\\begin{equation}\\label{5183}\\mathbb E\\left[\\int_0^T c_s\\Lambda_sZ_sd\\kappa_s \\right] = 0=  x + \\int_0^Tq_sr_sdK_s= x + \\mathbb E\\left[\\int_0^Tq_se_s\\Lambda_sZ_sd\\kappa_s \\right],\n\\end{equation}\nwhere in the last equality we have used \\eqref{6-25-1}. \n\nIf $y'>0$, \nthen $\\frac{1}{y'}\\Lambda Z\\in\\mathcal D(x,q)$ and thus by \\eqref{5191}, we obtain $$\\mathbb E\\left[ \\int_0^T c_s\\frac{1}{y'}\\Lambda_sZ_sd\\kappa_s\\right]\\leq 1 = \\frac{ x + \\int_0^Tq_sr_sdK_s}{y'} = \\frac{1}{y'}\\left( x + \\mathbb E\\left[\\int_0^T q_se_s\\Lambda_sZ_sd\\kappa_s\\right]\\right),$$\nwhere in the last equality, we have used \\eqref{6-25-1} again. Consequently, we deduce\n$$\\mathbb E\\left[ \\int_0^T c_s\\Lambda_sZ_sd\\kappa_s\\right]\\leq  x + \\mathbb E\\left[\\int_0^T q_se_s\\Lambda_sZ_sd\\kappa_s\\right],$$\nwhich together with \\eqref{5183}, by Lemma \\ref{9-5-1}, imply that $c\\in\\mathcal A(x,q)$.\n\nConversely, let $(x,q)\\in\\mathring{\\mathcal K}$, $c\\in\\mathcal A(x,q)$ and $Y\\in {\\rm cl}\\mathcal D(x,q)$. Then there exists a sequence $Y^n\\in\\mathcal Y(y^n, r^n)$ convergent to $Y$, $(d\\kappa\\times\\mathbb P)$-a.e., where $(y^n,r^n)\\in\\mathcal B(x,q)$. As, \n$$\\mathbb E\\left[\\int_0^T c_sY^n_sd\\kappa_s \\right]\\leq 1,\\quad n\\in\\mathbb N,$$\nby Fatou's lemma, we get\n$$\\mathbb E\\left[\\int_0^T c_sY_sd\\kappa_s \\right]\\leq \\liminf\\limits_{n\\to\\infty}\\mathbb E\\left[\\int_0^T c_sY^n_sd\\kappa_s \\right]\\leq 1.$$\nThis completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{conjugacy}\nUnder the conditions of Theorem \\ref{mainTheorem}, for every $(x,q)$ in $\\mathring{\\mathcal K}$, we have\n\\begin{equation}\\label{conjugacyOneDirection}\nu(x,q) = \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tr_sq_sdK_s\\right).\n\\end{equation}\n\\end{Lemma}\n\\begin{proof} Let us fix $(x,q)\\in\\mathring{\\mathcal K}$. \nBy Lemma \\ref{6-12-1}, $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ contain strictly positive elements. \nTherefore, using Lemma \\ref{5181}\nwe deduce that the sets $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ satisfy the assumptions \nof \\cite[Theorem 3.2]{MostovyiNec}. From this theorem, Lemma~\\ref{auxiliaryLemma}, and the definition of the set $\\mathcal B(x,q)$, we get\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nu(x,q) &=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{Y\\in{\\cl \\mathcal D(x,q)}}\\mathbb E\\left[ \\int_0^TV(t, zY_s)d\\kappa_s\\right] + z \\right) \\\\\n &=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{Y\\in\\mathcal D(x,q)}\\mathbb E\\left[ \\int_0^TV(t, zY_s)d\\kappa_s\\right] + z \\right) \\\\\n&=& \\inf\\limits_{z>0}\\left(\\inf\\limits_{(y,r)\\in z\\mathcal B(x,q)}v(y,r) + z \\right) \\\\\n&\\geq& \\inf\\limits_{(y,r)\\in \\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s \\right).\\\\\n\\end{array}\n\\end{equation}\nCombining this with the conclusion of Lemma~\\ref{6-17-1}, we deduce that \\eqref{conjugacyOneDirection} holds for every $(x,q)\\in\\mathring{\\mathcal K}$. \n\n\n\\end{proof}\n\nBefore proving the biconjugacy relations of item $(iii)$, Theorem \\ref{mainTheorem}, we need a preliminary lemma. \nEssentially following the notations in \\cite{EkelandTemam}, we define \n\\begin{equation}\\label{v*}\nv^*(x,q){:=} \\inf\\limits_{(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s \\right),\\quad (x,q)\\in\\mathbb R\\times \\mathbb L^{\\infty}(dK).\n\\end{equation}\n\\begin{equation}\\label{v**}\nv^{**}(y,r){:=} \\sup\\limits_{(x,q)\\in\\mathbb R\\times \\mathbb L^\\infty(dK)}\\left(v^*(x,q) - xy - \\int_0^Tq_sr_sdK_s \\right),\\quad (y,r)\\in\\mathbb R\\times \\mathbb L^1(dK).\n\\end{equation}\n\\begin{Remark}In \\cite{EkelandTemam}, conjugate convex functions are considered on general spaces $V$ and $V^*$ supplied with $\\sigma(V,V^*)$ and $\\sigma(V^*, V)$ topologies, which in our case are $V= \\mathbb R\\times\\mathbb L^1(dK)$, $V^* = \\mathbb R\\times\\mathbb L^\\infty(dK)$. Thus, the starting point of our analysis is $v$, not $u$. We remind the reader we have already proved that the dual value function $v$ is convex, proper and lower-semicontinuous on the space $V= \\mathbb R\\times\\mathbb L^1(dK)$. \n\\end{Remark}\n\\begin{Lemma}\\label{lemv**}\nUnder the conditions of Theorem \\ref{mainTheorem}, we have \n\\begin{equation}\n\\label{2166}\nv^{**} = v, \n\\end{equation}\n\\begin{equation}\n\\label{4151}\nv^*(x,q) = -\\infty, \\quad for~every\\quad (x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash \\mathcal K.\n\\end{equation}\n\n\\end{Lemma}\n\n\\begin{proof}\nTo show \\eqref{2166}, we observe that by Lemma \\ref{existenceUniqueness}, $v$ is lower semicontinuous in the $\\mathbb R\\times\\mathbb L^1(dK)$-norm topology (and therefore, by \\cite[Corollary I.2.2]{EkelandTemam}, also in the weak topology $\\sigma(\\mathbb R\\times\\mathbb L^1(dK),\\mathbb R\\times\\mathbb  L^\\infty(dK))$). As a result, by \\cite[Proposition I.4.1]{EkelandTemam}, we get~\\eqref{2166}.\n\nThe proof of \\eqref{4151} will be done in several steps. \n\n{\\it Step 1.} Let $(x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash \\mathcal K$.  According to  Proposition \\ref{prop1}, there exists $(y,r)\\in\\mathcal L$, such that\n\\begin{equation}\\nonumbe\nC{:=} xy + \\int_0^T q_sr_sdK_s <0.\n\\end{equation}\nTherefore, as $\\mathcal L$ is a cone, for every $a>0$, $(ay, ar)\\in\\mathcal L$, and we have\n\\begin{equation}\\label{4153}\nxay + \\int_0^Tq_sar_sdK_s = aC<0.\n\\end{equation}\nNote that $0\\in\\mathcal Y(ay, ar)$, $a>0$.\n\n{\\it Step 2.} Let us consider $Z\\in\\mathcal Z'$, such that \n\\begin{equation}\\label{4154}\n\\mathbb E\\left[\\int_0^TV\\left(t,\\tfrac{1}{2}Z_t\\right)d\\kappa_t\\right]<\\infty.\n\\end{equation}\nThe existence of such a $Z$ is granted by Lemma \\ref{finitenessOverZ'}.\nFurther, by Corollary \\ref{cor1}, there exists $\\rho\\in\\mathbb L^1(dK)$, such that $(1, \\rho)\\in\\mathcal L$, $Z\\in\\mathcal Y(1,\\rho)$,  and\n\\begin{equation}\\nonumbe\n\\int_0^t\\rho_sdK_s = \\mathbb E\\left[\\int_0^t Z_se_sd\\kappa_s \\right],\\quad t\\in[0,T].\n\\end{equation}\nLet us set \n\\begin{equation}\\nonumbe\nD{:=} x + \\int_0^Tq_s\\rho_sdK_s\\in\\mathbb R.\n\\end{equation}\n\n{\\it Step 3.} In \\eqref{4153}, let us pick \n\\begin{equation}\\nonumbe\na = \\frac{|D| + 1}{-C}>0.\n\\end{equation}\nThen, we have \n\\begin{equation}\\label{4158}\naC + D = -|D| + D -1<0.\n\\end{equation}\n\n{\\it Step 4.} Let us define \n\\begin{equation}\\label{4159}\nY{:=} \\tfrac{1}{2} Z, \\quad y'{:=} \\tfrac{1}{2}ay + \\tfrac{1}{2}, \\quad and \\quad r'{:=} \\tfrac{1}{2}ar + \\tfrac{1}{2}\\rho.\n\\end{equation}\nThen by \\eqref{4154}, we obtain\n\\begin{equation}\\label{41510}\n\\mathbb E\\left[\\int_0^TV\\left(t,Y_t\\right)d\\kappa_t\\right]<\\infty.\n\\end{equation}\nAs $Z\\in\\mathcal Y(1,\\rho)$ and $0\\in\\mathcal Y(ay, ar)$, by convexity of $\\mathcal L$ and Proposition \\ref{prop1}, we have\n\\begin{equation}\\label{41511}\nY\\in\\mathcal Y(y',r'),\n\\end{equation}\nwhere $y'$ and $r'$ are defined in \\eqref{4159}. Now, it follows from  \\eqref{41510} and \\eqref{41511} that $(y', r')\\in\\mathcal E$ (where $\\mathcal E$ is defined in \\eqref{defE}). \nTherefore, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n2(xy' + \\int_0^Tq_sr'_sdK_s) = (ay + 1)x + \\int_0^T(ar_s + \\rho_s)q_sdK_s \\\\= a(xy + \\int_0^Tq_sr_sdK_s) + x + \\int_0^Tq_s\\rho_sdK_s = aC + D <0,\\\\\n\\end{array}\n\\end{displaymath}\nwhere the last inequality follows from \\eqref{4158}.\nTo  recapitulate, we have shown the existence of  $(y', r')$, such that\n\\begin{equation}\\label{41512}\n(y', r')\\in\\mathcal E\\quad and\\quad xy' + \\int_0^Tq_sr'_sdK_s<0.\n\\end{equation}\n\n{\\it Step 6.} For $y'$ and $r'$ defined in \\eqref{4159}, as $v(y', r') <\\infty$ and $xy' + \\int_0^Tq_sr'_sdK_s<0$ by \\eqref{41512}, from the monotonicity of $V$, we get\n$$\\infty>v(y',r')\\geq v(\\lambda y', \\lambda r'),\\quad \\lambda \\geq 1.$$\nAs $\\bigcup\\limits_{\\lambda\\geq 1}(\\lambda y', \\lambda r')\\subset \\mathcal L$, we conclude via \\eqref{41512} that\n$$v^*(x,q) \\leq \\lim\\limits_{\\lambda \\to\\infty}\\left( v(\\lambda y', \\lambda r') + \\lambda \\left(xy' + \\int_0^Tq_sr'_sdK_s\\right)\\right) = -\\infty.$$\nTherefore, \\eqref{4151} holds. This completes the proof of the lemma.\n\\end{proof}\n\\begin{Lemma}\\label{lemBiconjugacy}\nUnder the conditions of Theorem \\ref{mainTheorem}, we have \n\\begin{equation}\\label{2161\nv(y,r) = \\sup\\limits_{(x,q)\\in\\mathcal K}\\left(u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),\\quad (y,r)\\in\\mathcal L,\n\\end{equation}\n\\begin{equation}\\label{21610}\nu(x,q) = \\inf\\limits_{(y,r)\\in\\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\quad (x,q)\\in \\mathcal K.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\n\n\nLemma \\ref{conjugacy} and \\eqref{3313} imply that on $\\mathring{\\mathcal K}$, for $v^*$ defined in \\eqref{v*}, we have\n\\begin{equation}\\label{2164}\nv^{*} = u\n\\end{equation}\nBy Lemma \\ref{lemv**}, $v^* = -\\infty$ on $\\mathbb R\\times\\mathbb L^\\infty(dK)\\backslash\\mathcal K$. \nFrom \\cite[Definition I.4.1]{EkelandTemam} and Lemma \\ref{existenceUniqueness}, respectively, we deduce that both $v^*$ and $u$ are upper semicontinuous in the topology of $\\mathbb R\\times\\mathbb L^\\infty(dK)\n. Consequently, from \\eqref{2164}, using \\cite[Corollary I.2.1]{EkelandTemam},  \nwe get\n\\begin{equation}\\label{2201}v^* = u,\\quad on\\quad\\mathbb R\\times\\mathbb L^\\infty(dK).\n\\end{equation}\nAs a result, with $v^{**}$ being defined in \\eqref{v**}, for every $(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)$, we obtain\n\\begin{equation}\\label{2165}\nu^{*}(y,r){:=} \\sup\\limits_{(x,q)\\in\\mathbb R\\times \\mathbb L^\\infty(dK)}\\left(u(x,q) - xy - \\int_0^Tq_sr_sdK_s \\right)=v^{**}(y,r)\n\\end{equation}\nTherefore, from \\eqref{2166} in Lemma \\ref{lemv**} and \\eqref{2165}, we get\n\\begin{equation}\\nonumbe\nv = u^{*},\\quad on\\quad\\mathbb R\\times\\mathbb L^1(dK).\n\\end{equation}\nAs a result, applying Lemma \\ref{lemv**} again and since $u = -\\infty$ outside of $\\mathcal K$ by \\eqref{3313}, we deduce \n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nv(y,r)  &=& \\sup\\limits_{(x,q)\\in\\mathbb R\\times\\mathbb L^\\infty(dK)}\\left( u(x,q)- xy - \\int_0^Tr_sq_sdK_s\\right)\\\\\n&=& \\sup\\limits_{(x,q)\\in\\mathcal K}\\left( u(x,q) - xy - \\int_0^Tr_sq_sdK_s\\right),\\quad(y,r)\\in\\mathcal L,\\\\\n\\end{array}\n\\end{equation}\nThus, \\eqref{2161} holds. \n\nIn turn, from \\eqref{2201} using \\eqref{3313}, \nwe conclude that\n\n\\begin{equation}\\nonumber\n\\begin{array}{rcl}\nu(x,q) &=& \\inf\\limits_{(y,r)\\in\\mathbb R\\times \\mathbb L^1(dK)}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\\\\n&=& \\inf\\limits_{(y,r)\\in\\mathcal L}\\left(v(y,r) + xy + \\int_0^Tq_sr_sdK_s\\right),\\quad (x,q)\\in\\mathcal K,\\\\\n\\end{array}\n\\end{equation}\nwhich proves \\eqref{21610} and extends the assertion of Lemma \\ref{conjugacy} to the boundary of $\\mathring{\\mathcal K}$.\n\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{mainTheorem}]\nThe assertions of item $(i)$ follow from Lemmas \\ref{lemFinu} and \\ref{lemFinv}, item $(ii)$ results from Lemma \\ref{existenceUniqueness}, whereas the validity of item $(iii)$ come from Lemma\n\\ref{lemBiconjugacy}. This completes the proof of the theorem.\n\\end{proof}\n\n\n\n\\section{Subdifferentiability of $u$}\\label{secSubdifferentiability}\nIn order to establish subdifferentiability of $u$, we need to strengthen \nAssumption~\\ref{asEnd} and to impose the following condition. \n\\begin{Assumption}\\label{asACclock}\nThere exists an a.s. bounded away from $0$ and $\\infty$  \nprocess $\\varphi$, such that\n\\begin{equation}\\nonumbe\nd\\kappa(\\omega) =\\varphi d K, \\quad for\\quad\\mathbb P-a.e.\\quad\\omega \\in\\Omega.\n\\end{equation}\n\n\\end{Assumption}\n\n\n\nLet \n\\begin{equation}\\label{defPi}\n\\begin{array}{rcl}\n\\mathcal P &{:=} &\\left\\{ \\rho:~(1,\\rho)\\in\\mathcal L\\right\\},\\\\\n\\end{array}\n\\end{equation}\n\\begin{Remark} \n\n  $\\mathcal P$ defined in \\eqref{defPi} needs to be uniformly integrable with respect to the measure $dK$ in order  for the proof of subdifferentiability of $u$ to go through.\n Assumption \\ref{asEnd} through Lemma \\ref{lemma1} only implies  \n that $\\mathcal P$ is $\\mathbb{L}^1$ bounded. A stronger condition on the stochastic clock and income stream is, therefore, needed to obtain uniform integrability. \n \n\n\\end{Remark}\nThe following theorem characterizes subdifferentiability of $u$  over  $\\mathring{\\mathcal K}$,\nwhere we are looking for an  $\\mathbb R\\times\\mathbb L^1(dK)$-valued subgradient.   Under our assumptions,  we can find elements of the sub gradient which both  belong to the  effective domain of $v$, $\\mathcal E$, and are bounded, i.e. in $\\mathbb{R}\\times \\mathbb{L}^{\\infty} (dK)$.\n\\begin{Theorem}\\label{mainTheorem2} \nLet the conditions of Theorem~\\ref{mainTheorem} and Assumption \\ref{asACclock} hold. Then for every $(x,q)\\in\\mathring{\\mathcal K}$, the subdifferential of $u$ at $(x,q)$ is a nonempty and contains an element of $\\mathcal E$, i.e.,\n\\begin{equation}\\label{subdifNonempty}\n\\partial u(x,q)\\cap \\mathcal E\\neq \\emptyset.\n\\end{equation}\nMoreover, for $(x,q)\\in\\mathring{\\mathcal K}$ and $(y,r)\\in\\mathcal L$, $(y,r)\\in\\partial u(x,q)$ if and only if the following conditions hold:\n\\begin{equation}\\label{2191}\n |v(y,r)|<\\infty,\n \\end{equation}\n thus, $(y,r)\\in\\mathcal E$, \n \\begin{equation}\\label{2192}\n \\mathbb E\\left[\\int_0^T\\widehat Y_t(y,r) \\widehat c_t(x,q)d\\kappa_t\\right] = xy + \\int_0^Tq_sr_sdK_s,\n \\end{equation}\n \\begin{equation}\\label{2193}\n \\widehat Y_t(y,r) = U'(t,\\widehat c_t(x,q)), \\quad (d\\kappa\\times\\mathbb P)- a.e.,\n \\end{equation}\n where $\\widehat c(x,q)$ and $\\widehat Y(y,r)$ are the unique optimizers to \\eqref{primalProblem} and \\eqref{dualProblem}, respectively.\n\\end{Theorem}\n\n\n\n\n\n\\subsection{Uniform integrability of $\\mathcal P$}\n\\begin{Lemma}\\label{uiP}\nLet the conditions of Theorem~\\ref{mainTheorem2} hold. Then $\\mathcal P$ is $\\mathbb{L}^{\\infty}(dK)$-bounded, and, therefore,   a uniformly integrable family.\n\\end{Lemma}\n\\begin{proof}\n{\\it Step 1.}  For an arbitrary $q:~[0,T] \\to [0,1]$, let us define\n\\begin{displaymath}\n\\beta(q) {:=} \\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) |e_s|d\\kappa_s\\right] = \\sup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau\\in\\Theta} \\mathbb {E^Q}\\left[ \\int_0^{\\tau} q(s) |e_s|d\\kappa_s\\right].\n\\end{displaymath}\nNote that by Assumption \\eqref{asEnd}, \nwe have\n\\begin{equation}\\nonumber\n\\begin{array}{rcl}\n\\beta(q) &=&\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) |e_s|d\\kappa_s\\right]\\\\\n&\\leq&\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[ \\int_0^T q(s) X'_s\\varphi d K_s\\right]\\\\\n&\\leq&\\int_0^T q(s)\\sup\\limits_{\\mathbb Q\\in\\mathcal M} \\mathbb {E^Q}\\left[  CX'_s\\right]d K_s\\\\\n&\\leq&CX'_0\\int_0^T q(s)d K_s,\\\\\n\\end{array}\n\\end{equation}\nwhere $C\\in\\mathbb R$ is such that $|\\varphi| \\leq C$. \nIt follows from \\cite[Proposition 4.3]{K96} and \\cite[Theorem 3.1]{FK} that there exists a nonnegative c\\`adl\\`ag process $V = \\beta(q) + H\\cdot S$, such that\n\\begin{displaymath}\nV_t \\geq \\esssup\\limits_{\\mathbb Q\\in \\mathcal M, \\tau \\in\\Theta, \\tau\\geq t}\\mathbb {E^Q}\\left[\\int_0^{\\tau}q(s)|e_s|d\\kappa_s | \\mathcal F_t \\right],\\quad t\\in[0,T].\n\\end{displaymath} \nConsequently, $V$ satisfies\n$$V_\\tau \\geq \\int_0^\\tau q(s)|e_s|d\\kappa_s \\geq  \\int_0^\\tau q(s)e_sd\\kappa_s,\\quad \\Pas,\\quad\\tau \\in\\Theta,$$\nand thus, $(\\beta(q), -q)\\in\\mathcal K$.  As a result, from the definitions of $\\mathcal L$ and $\\mathcal P$, we get\n\\begin{equation}\\nonumbe\n\\beta(q)\\geq \\sup\\limits_{\\rho \\in\\mathcal P}\\int_0^T q(s)\\rho(s)dK_s.\n\\end{equation}\nOne can see that\n for every $\\rho\\in\\mathcal P$, we have\n$\\rho \\leq f{:=} {CX'_0}$, $d K$-a.e. \n\n\n{\\it Step 2.} For an arbitrary $q:~[0,T] \\to [-1,0]$, let us set \n\\begin{displaymath}\n\\tilde \\beta(q) {:=} \\sup\\limits_{\\mathbb Q\\in\\mathcal M}\\mathbb {E^Q}\\left[ \\int_0^T q(s) (-|e_s|) d\\kappa_s \\right].\n\\end{displaymath}\nAs in {\\it Step 1}, we can construct a c\\`adl\\`ag process $\\tilde V = \\tilde \\beta(q) + H\\cdot S$, s.t.\n\\begin{displaymath}\n\\tilde V_t \\geq \\esssup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau \\in\\Theta, \\tau \\geq t}\\mathbb {E^Q}\\left[ \\int_0^T q(s)(-|e_s|) d\\kappa_s |\\mathcal F_t\\right] \\geq \\esssup\\limits_{\\mathbb Q\\in\\mathcal M, \\tau \\in\\Theta, \\tau \\geq t}\\mathbb {E^Q}\\left[ \\int_0^T q(s)e_s d\\kappa_s |\\mathcal F_t\\right].\n\\end{displaymath}\nThis implies that $(\\tilde \\beta(q), -q)\\in\\cl\\mathcal K$. Therefore, \n\\begin{displaymath}\n\\beta(q) \\geq \\int_0^T q(s)\\rho (s) dK_s,\\quad for~every~\\rho \\in\\mathcal P.\n\\end{displaymath}\nSimilarly to {\\it Step 1},\none can see that $\\rho \\geq -f$, $d K$-a.e.  for every $\\rho \\in\\mathcal P$.\n\n{\\it Step 3.}In view of {\\it Steps 1 and 2}, uniform integrability of $\\mathcal P$ under $dK$ follows from the integrability of $f$ under $d K$.\n\\end{proof}\n\n\\begin{Lemma}\\label{uiB}\nLet the assumption of Theorem \\ref{mainTheorem2} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then \n$$\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}$$ is \n $\\mathbb{L}^{\\infty}$-bounded, so  a uniformly integrable subset of \n$\\mathbb L^1(dK)$.\n\\end{Lemma}\n\\begin{proof}\nBy Lemma \\ref{7-17-1}, we deduce the existence of a constant $M>0$,\nsuch that\n\\begin{equation}\\nonumbe\ny\\leq M,\\quad for~every~(y,r)\\in\\mathcal B(x,q).\n\\end{equation}\nWe conclude that $$\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}\\subseteq \\bigcup\\limits_{0\\leq \\lambda \\leq M}{\\lambda\\mathcal P},$$ and thus by Lemma \\ref{uiP}, $\\left\\{r:(y,r)\\in\\mathcal B(x,q)\\right\\}$ is a uniformly integrable family.\n\\end{proof}\n\\subsection{Closedness of $\\mathcal D(x,q)$ for every $(x,q)\\in\\mathring{\\mathcal K}$}\n\\begin{Lemma}\\label{closednessD}\nUnder the conditions of Theorem~\\ref{mainTheorem2}, for every $(x,q)\\in \\mathring{\\mathcal K}$, the set $\\mathcal D(x,q)$ is closed in $\\mathbb L^0(d\\kappa\\times\\mathbb P)$.\n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathring{\\mathcal K}$\nand $Y$ be an arbitrary element of $\\cl \\mathcal D(x,q)$.  We claim that there exists $(y,r)\\in \\mathcal B(x,q)$, such that $Y \\in \\mathcal Y(y,r)$. Let $Y^n\\in\\mathcal Y(y^n,r^n)$, $n\\geq 1$, be a sequence in $\\mathcal D(x,q)$, such that $\\lim\\limits_{n\\to \\infty}Y^n = Y$, $(d\\kappa\\times\\mathbb P)$-a.e. \nSince $(y^n,r^n)_{n\\geq 1}\\subset\\mathcal B(x,q)$, which is bounded in the sense of Lemma~\\ref{7-17-1}, Komlos' lemma implies the existence of a subsequence of convex combinations\n$(\\tilde y^n,\\tilde r^n)\\in \\conv((y^n,r^n), (y^{n+1}, r^{n+1}),\\dots)$, $n\\geq 1$, such that $(\\tilde y^n)_{n\\geq 1}$ converges to $y$ and $(\\tilde r^n)_{n\\geq 1}$ converges to $r$, $dK$-a.e. \nLemma~\\ref{uiB} implies that $(\\tilde r^n)_{n\\geq 1}$ is uniformly integrable. Therefore $(\\tilde r^n)_{n\\geq 1}$ converges to $r$ in $\\mathbb L^1(dK)$. Note that the corresponding sequence of convex combinations of $(Y^n)_{n\\geq 1}$, $(\\tilde Y^n)_{n\\geq 1}$  converges to $Y$, $(d\\kappa\\times\\mathbb P)$-a.e.\nThen we have\n$$ \n1 \\geq \\lim\\limits_{n\\to\\infty}\\left( x\\tilde y^n + \\int_0^T q_s\\tilde r^n_sdK_s \\right) = xy + \\int_0^T q_s r_sdK_s.\n$$\nTherefore, $(y,r)\\in \\mathcal B(x,q)$.\n\nLet us fix an arbitrary $(x',q')\\in\\mathcal K$ and $c\\in\\mathcal A(x',q')$. \nUsing Proposition~\\ref{prop1}, Fatou's lemma, and Lemma \\ref{uiB}, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n0\\leq \\mathbb E\\left[\\int_0^TY_sc_sds \\right] \\leq \n\\liminf\\limits_{n\\to\\infty} \\mathbb E\\left[\\int_0^T \\tilde Y^n_sc_sds \\right] \\\\\\leq \n \\lim\\limits_{n\\to\\infty}\\left( x'\\tilde y^n + \\int_0^T q'_s\\tilde r^n_sdK_s \\right)\n =\n  x'y + \\int_0^T q'_s r_sdK_s,\\\\\n\\end{array}\n\\end{displaymath} \nwhere the uniform integrability of $\\mathcal B(x,q)$ in needed once again in the last equality.\nFrom Proposition~\\ref{prop1}, we conclude that that $Y\\in \\mathcal Y(y,r)$. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{mainTheorem2}.] \nLet $(x,q)\\in\\mathcal K$, $\\widehat c(x,q)$ be the minimizer to (\\ref{primalProblem}), whose existence and uniqueness are established in Lemma~\\ref{existenceUniqueness}. Let us also set\n\\begin{equation}\\label{defHatY}\n\\widehat Y_{t} {:=} U'({t},\\widehat c_{t}(x,q)),\\quad (t,\\omega)\\in[0,T]\\times\\Omega,\\quad and \\quad z {:=} \\mathbb E\\left[\\int_0^T\\widehat c_s(x,q)\\widehat Y_s d\\kappa_s \\right].\n\\end{equation} \nNote that the sets $\\mathcal A(x,q)$ and ${\\cl \\mathcal D(x,q)}$ satisfy the conditions of \\cite[Theorem~3.2]{MostovyiNec}, which implies that $\\widehat Y\\in z{\\cl \\mathcal D(x,q)}$ is the unique solution to the optimization problem\n\\begin{displaymath}\n\\inf\\limits_{Y\\in z{\\cl \\mathcal D(x,q)}}\\mathbb E\\left[\\int_0^TV(t, Y_s)d\\kappa_s\\right] = \\mathbb E\\left[\\int_0^TV(t,\\widehat Y_s)d\\kappa_s\\right]\\in\\mathbb R,\n\\end{displaymath}\nwhere finiteness follows from Lemma \\ref{auxiliaryLemma}.\nNote that by Lemma~\\ref{closednessD}, $\\widehat Y\\in\\mathcal Y(zy, zr)$ for some $(y,r)\\in\\mathcal B(x,q)$.\nIt follows from the definition of $\\widehat Y$ in \\eqref{defHatY} that \nfor $\\widehat c(x,q)$ and $\\widehat Y$ we have the following relation\n\\begin{displaymath}\nU({t}, \\widehat c_{t}(x,q)) = V({t}, \\widehat Y_{t}) + \\widehat c_{t}(x,q)\\widehat Y_{t},\n\\quad (t,\\omega)\\in[0,T]\\times\\Omega\n\\end{displaymath}\nwhich together with Lemma~\\ref{conjugacy} implies that\n\\begin{equation}\\label{7-17-3}\nu(x,q) = v(zy,zr) + z\\left(xy + \\int_0^Tq_sr_sdK_s\\right),\n\\end{equation}\n\\begin{displaymath}\n\\widehat Y(zy,zr)  = \\widehat Y\n,\\quad (d\\kappa\\times\\mathbb P)-a.e.,\n\\end{displaymath}\nwhere $\\widehat Y(zy,zr)$ is the unique minimizer to the dual problem~(\\ref{dualProblem}). \nBy (\\ref{7-17-3}), $(zy,zr)\\in\\mathcal E.$ (\\ref{7-17-3}) and \\cite[Proposition I.5.1]{EkelandTemam} assert that $(zy,zr)\\in\\partial u(x,q)$. In particular, we get\n\\begin{displaymath} \n\\partial u(x,q) \\cap \\mathcal E \\neq \\emptyset,\n\\end{displaymath}\ni.e.,  \\eqref{subdifNonempty}. \nNote that even though, e.g., \\cite[Corollary 2.2.38 and Corollary 2.2.44]{BarbuPrec} imply that $\\partial u(x,q)\\neq \\emptyset$, \n over the interior of the effective domain, \nits elements are in $\\mathbb R\\times (\\mathbb L^\\infty)^*(dK)$. Relation \\eqref{subdifNonempty} shows that $\\partial u(x,q)$ contains  at least a bounded element  of $\\mathcal E\\subseteq\\mathcal L\\subset \\mathbb R\\times\\mathbb L^1(dK).$\n\nLet $(x,q)\\in\\mathcal K$ and $(y,r)\\in\\mathcal L$. Suppose that \\eqref{2191}, \\eqref{2192}, and \\eqref{2193} hold. Then by conjugacy of $U$ and $V$, we get \n\n\\begin{equation}\\label{2194}\n0 = v(y,r) - u(x,q) + xy + \\int_0^T q_sr_sdK_s.\n\\end{equation}\nLemma \\ref{lemBiconjugacy} and \n\\cite[Proposition I.5.1]{EkelandTemam} imply that $(y,r)\\in\\partial u(x,q) \\cap\\mathcal E$.\nConversely, let $(x,q)\\in\\mathcal K$ and $(y,r)\\in\\mathcal L\\cap \\partial u(x,q).$ Then by \\cite[Proposition I.5.1]{EkelandTemam}  and Lemma \\ref{lemBiconjugacy}, we deduce that \\eqref{2194} holds. Lemma \\ref{lemFinu} implies the finiteness of $u(x,q)$, which together with \\eqref{2194} results in the finiteness of $v(y,r)$, thus \\eqref{2191} holds and $(y,r)\\in\\mathcal E$. By Lemma \\ref{existenceUniqueness}, there exists a unique optimizer $\\widehat c(x,q)$, for \\eqref{primalProblem}, and $\\widehat Y(y,r)$, for \\eqref{dualProblem}, respectively. Therefore, from conjugacy of $U$ and $V$, Proposition \\ref{prop1}, and \\eqref{2194}, we obtain\n\\begin{displaymath}\n\\begin{array}{c}\n0\\leq\n \\mathbb E\\left[\\int_0^T \\left(V(t,\\widehat Y_t(y,r)) - U(t,\\widehat c_t(x,q)) + \\widehat Y_t(y,r) \\widehat c_t(x,q)\\right)d\\kappa_t\\right] \\\\\n\\leq v(y,r) - u(x,q) + xy + \\int_0^T q_sr_sdK_s = 0.\\\\\n\\end{array}\n\\end{displaymath}\nThis implies \\eqref{2192} and \\eqref{2193}. This completes the proof of the theorem.\n\\end{proof}\n\n\\section{Structure of the dual feasible set}\\label{secStructureOfDualDomain}\nBy Assumption \\ref{asACclock}, there exists at most countable subset $(s_k)_{k\\in\\mathbb N}$ of $[0,T]$, where $\\kappa$ has jumps. \nWe define $\\mathcal{D}'$ the set of non-increasing, left-continuous and adapted processes $D$ that start at $1$ and \nwith the property that \n$D_{\\theta _0} = 1$,  $ D_T\\geq 0$\nand that, there exists some $n\\in \\mathbb N$ such that  $D$ is constant off the discrete grid\n$\\mathcal T_n{:=} \\bigcup\\limits_{j = 1}^{2^n}\\{s_j\\}\\cup\\left\\{\\frac{k}{2^n}T, k = 0, \\dots,2^n\\right\\}$.\n\n\n\n\n\n\\begin{Lemma}\\label{lemSt2}\n Let $\\mathcal G'{:=} \\{ZD = (Z_tD_t)_{t\\in[0,T]}:~Z\\in\\mathcal Z', D\\in\\mathcal D' \\}$. Assume    the conditions of Proposition \\ref{prop1} hold. Then $\\mathcal G'$ is convex.\n\n\\end{Lemma}\n\\begin{proof}\nLet $Z^1D^1$ and $Z^2D^2$ are the elements of $\\mathcal G'$ and let $\\lambda \\in(0,1)$\nWe need to show that $\\lambda Z^1D^1 + (1-\\lambda)Z^2D^2 = ZD$\nfor some $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$. There exists $n\\in\\mathbb N$, such that $D^1$ and $D^2$ decrease at most on $\\mathcal T_n$. Let  $t_k$'s be the elements of $\\mathcal T_n$ arranged in an increasing order. \nLet us define $Z_0 = D_0 = 1$ and for every $k \\in\\{0,\\dots, 2^n-1\\}$,~with \n$$\nA_t {:=} \\lambda Z^1_{t_k}D^1_{t} +(1-\\lambda)Z^2_{t_k}D^2_{t}\n\\quad and\\quad\n\\alpha_t {:=} \\frac{\\lambda Z^1_{t_k}D^1_{t}}{A_t}1_{\\{A_t\\neq 0\\}} + \\frac{\\lambda Z^1_{t_k}}{\\lambda Z^1_{t_k} + (1-\\lambda) Z^2_{t_k}}1_{\\{A_t= 0\\}},$$\n(note that $A_t = 0$ if and only if both $D^1_t=0$ and $D^2_t=0$) we  set\n\\begin{equation}\\label{351}\n\\begin{array}{rclcl}\nZ_t &{:=}& Z_{t_k}\\left(\\alpha_{t_k+}\\frac{Z^1_t}{Z^1_{t_k}} + (1 - \\alpha_{t_k+})\\frac{Z^2_t}{Z^2_{t_k}} \\right), &for& t\\in(t_k, t_{k+1}],\\\\\nD_t &{:=}& \\frac{A_{t_k+}}{Z_{t_k}},& for& t\\in(t_k, t_{k+1}].\\\\\n\\end{array}\n\\end{equation}\nOne can see that $ZD = \\lambda Z^1D^1 + (1-\\lambda)Z^2D^2$ and that $Z\\in\\mathcal Z'$, see e.g., \\cite{FK}, \\cite{Rohlin10}, \\cite{Kardaras_emery}, and \\cite{ChristaJosef15} for discussions of the sets of processes with similar convexity-type properties to the one given in \\eqref{351}. To show that $D\\in\\mathcal D'$, for $k\\geq 1$, \nwe observe that \n\\begin{displaymath}\n\\begin{array}{rcl}\nD_{t_k+} &=& \\frac{A_{t_k+}}{Z_{t_k}} \\\\\n&=& \\frac{{\\lambda Z^1_{t_k}D^1_{t_k+} +(1-\\lambda)Z^2_{t_k}D^2_{t_k+}}}{Z_{t_{k-1}}\\left(\\alpha_{t_{k-1}+}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} + (1 - \\alpha_{t_{k-1}+})\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)} \\\\\n&\\leq &\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{Z_{t_{k-1}}\\left(\\alpha_{t_{k-1}+}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} + (1 - \\alpha_{t_{k-1}+})\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)}\\\\\n&= &\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{Z_{t_{k-1}}\\left(\\frac{\\lambda Z^1_{t_{k-1}}D^1_{t_k}}{A_{t_{k-1}+}}\\frac{Z^1_{t_k}}{Z^1_{t_{k-1}}} \n+ \\frac{(1-\\lambda) Z^2_{t_{k-1}}D^2_{t_k}}{A_{t_{k-1}+}}\\frac{Z^2_{t_k}}{Z^2_{t_{k-1}}} \\right)}1_{\\{A_{t_{k-1}+}> 0\\}} +0\\cdot1_{\\{A_{t_{k-1}+}= 0\\}}\\\\\n&= &\\frac{A_{t_{k-1}+}}{Z_{t_{k-1}}}\\frac{{\\lambda Z^1_{t_k}D^1_{t_k} +(1-\\lambda)Z^2_{t_k}D^2_{t_k}}}{\\lambda D^1_{t_k}Z^1_{t_k}\n+ (1-\\lambda) D^2_{t_k}Z^2_{t_k}}1_{\\{A_{t_{k-1}+}> 0\\}}\\\\\n&= &\\frac{A_{t_{k-1}+}}{Z_{t_{k-1}}}1_{\\{A_{t_{k-1}+}> 0\\}}\\\\\n&= &D_{t_k}.\\\\\n\\end{array}\n\\end{displaymath}\nAbove, the inequality  follows from the monotonicity of $D^1$ and $D^2$.\nTherefore, $D$ is nonincreasing. Also, clearly $D$ is nonnegative. Thus  $ZD\\in \\mathcal G'$. \n\\end{proof}\nThe following lemma is an extension of Lemma \\ref{9-5-1}  and amounts to a first layer of convexification of the  set $\\Upsilon$, i.e., of the budget constraints.\n\n\\begin{Lemma}\\label{lemSt1}Let   the conditions of Proposition \\ref{prop1} hold,  $(x,q)\\in\\mathcal K$, and $c$ is a nonnegative optional process. Then $c\\in\\mathcal A(x,q)$ if and only if\n\\begin{equation}\\label{331}\n\\mathbb E\\left[\\int_0^Tc_sD_sZ_sd\\kappa_s \\right]\\leq\nx + \\mathbb E\\left[\\int_0^Tq_se_sD_sZ_sd\\kappa_s \\right],\\quad\nfor~every~Z\\in\\mathcal Z'~and~D \\in \\mathcal D'.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof}\nThe idea is to use the assertion of Lemma \\ref{9-5-1} and to approximate a given $D\\in\\mathcal D'$ by (finite) linear combinations of the elements of $\\Upsilon$, where $\\Upsilon$ is defined in \\eqref{defUpsilon}.\n\nFor a stopping time $\\tau$, \nlet us denote $$\\Lambda^{\\tau} {:=} 1_{[0,\\tau]}\\in\\Upsilon,$$\nand fix $D\\in\\mathcal D'$. Then there exists $l\\in\\mathbb N$, such that $D$ has has jumps at most\non $\\{t_0,t_1,\\dots,\nt_l\\}$ for some increasing $t_i$'s. \nFor every $j \\in\\{ 0, \\dots, \n\\}$, $k \\in\\{0,\\dots 2^n\\}$, and $n\\in\\mathbb N$, let us set\n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nA_{k, n, j} &{:=}& \\left\\{ \\omega:  ~D_{t_j}(\\omega) >0~and~\\frac{D_{t_j+}(\\omega)}{D_{t_j}(\\omega)} \\in \\left(\\frac{k-1}{2^n}, \\frac{k}{2^n}\\right]\\right\\},\\\\\n\\tau^{k, n, j}&{:=}&T1_{A_{k,n,j}} + t_j1_{A^c_{k,n,j}},\\\\\n\\end{array}\n\\end{equation}\nNote that $D_0 = 1$ by definition of $\\mathcal D'$, $A_{k, n, j}\\in\\mathcal F_{t_j}$, and \n$$\\frac{k-1}{2^n}\\Lambda^{\\tau^{k,n,j}}_{t_j+} = \\left\\{\n\\begin{array}{lcl}\n\\frac{k-1}{2^n}&on&A_{k,n,j}\\\\\n0&on& A^c_{k,n,j}\\\\\n\\end{array}\n \\right..$$\n By construction, we have $$D_{t_j+} = D_{t_j}\\lim\\limits_{n\\to\\infty}\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{\\tau^{k,n,j}}_{t_j+},$$ where the sequence \n $$K^n_{t_j+}{:=} \\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j}_{t_j+},\\quad n\\in\\mathbb N,$$\n is increasing on $\\{D_{t_j}>0\\}$, i.e.,\n $$K^n_{t_j+}1_{\\{D_{t_j}>0\\}}\\uparrow \\frac{D_{t_j+}}{D_{t_j}}1_{\\{D_{t_j}>0\\}}.$$\n Thus, for an arbitrary $j \\in\\{0, \\dots, l\\}$, we have constructed a sequence of elements of $\\Upsilon$, whose finite linear combinations monotonically increase to $\\frac{D_{t_j+}}{D_{t_j}}$ on $\\{D_{t_j}>0\\}$ (i.e., if $j<l$, we have approximated $D$ on the interval $(t_{j}, t_{j+1}]$). \n \n \n In order to construct a sequence that approximates $D$ at every point of its potential jumps, we first observe that for two stopping times $\\tau$ and $\\sigma$, we have\n $$\\Lambda^\\tau\\Lambda^\\sigma = 1_{[0, \\tau]}1_{[0,\\sigma]} = 1_{[0,\\tau\\wedge\\sigma]} = \\Lambda^{\\tau\\wedge \\sigma}.$$\nTherefore, for every $n\\in\\mathbb N$,\n\\begin{equation}\\label{361}\nK^n_{t_j+}K^n_{t_{j+1}+} =\\left(\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j}_{t_j+}\\right)\\left(\\sum\\limits_{k = 1}^{2^n}\\frac{k-1}{2^n}\\Lambda^{k,n,j+1}_{t_{j+1}+}\\right) =\\sum\\limits_{i=1}^{4^n}\\lambda^{n,i} \\Lambda^{\\sigma_{n,i}}_{t_{j+1}+},\n\\end{equation}\nfor some  finite sequences of stopping times $(\\sigma_{n,i})_{i=1}^{4^n}$ and $[0,1)$-valued constants $(\\lambda^{n,i})_{i=1}^{4^n}$. \nHere $\\Lambda^{\\sigma_{n,i}}$ are such that for {\\it both} $t = t_j$ and $t=t_{j+1}$ on $\\{D_{t_{j+1}}>0\\}$, we have\n$$\\lim\\limits_{n\\to\\infty}\\sum\\limits_{i=1}^{4^n}\\lambda^{n,i} \\Lambda^{\\sigma_{n,i}}_{t+} = \\frac{D_{t+}}{D_t}.$$\nSimilarly,\n with $r(t){:=} \\max\\{i:~t_i< t\\}$, let us define\n$$D^n_t {:=} \\prod\\limits_{j = 0}^{r(t)}K^n_{t_j+}1_{\\{D_{t_j}>0\\}},\\quad t\\in[0,T], n\\in\\mathbb N.$$ \nAs in \\eqref{361}, for every $n\\in\\mathbb N$, $D^n$ can be written as a finite linear combination of $\\Lambda$'s, such that $D^n_{t+}\\uparrow D_{t+}$ for every $t\\in\\{t_0, \\dots, t_{l}\\}.$\n\nFinally, \\eqref{331} can be obtained from Lemma \\ref{9-5-1} by the approximation of $D$ by $D^n$'s as above and via  the monotone convergence theorem (applied separately to $(e_t)^{+}$ and $(e_t)^{-}$). \n\\end{proof}\n\\begin{Corollary}\\label{corRzd}\nLet   the conditions of Proposition \\ref{prop1} hold. Then, for every pair $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$, \nthere exists $r^{ZD}\\in\\mathbb L^1(dK)$, such that $ZD\\in\\mathcal Y(1,r^{ZD})$, where $(1,r^{ZD})\\in\\mathcal L$. \n\\end{Corollary}\n\\begin{proof}\nThe existence of $r^{ZD}$, such that $ZD\\in\\mathcal Y(1, r^{ZD})$ follows from Lemma \\ref{lemSt1} (equation \\eqref{331}) and the approximation procedure in Lemma \\ref{lemSt1} (again, applied separately to $(e_t)^{+}$ and $(e_t)^{-}$) combined with the monotone convergence theorem.\nAs, the left-hand side in \\eqref{331} is nonnegative, \n$(1,r^{ZD})\\in\\mathcal L$. \n\n\\end{proof}\n\n\nFor a given $Z\\in\\mathcal Z'$ and $D\\in\\mathcal D'$, let us recall that $r^{ZD}$ is given in Corollary \\ref{corRzd}. \nWe set $$\\mathcal {B'}(x,q) {:=} \\left\\{ (y, yr^{ZD})\\in\\mathcal B(x,q):~y> 0, Z\\in\\mathcal Z', D\\in\\mathcal D'\\right\\},$$\n\n$$\\mathcal G(x,q) {:=} \\left\\{yZ'D'\\in\\mathcal D(x,q):~(y, yr^{ZD})\\in \\mathcal {B'}(x,q)\\right\\},\\quad (x,q)\\in\\mathring{\\mathcal K},$$ \n\n\n\\begin{Lemma}\\label{lemSt3}\nLet   the conditions of Proposition \\ref{prop1} hold, then for every $(x,q) \\in\\mathring{\\mathcal K}$, the closure of the convex, solid hull of $\\mathcal G(x,q)$ in $\\mathbb L^0$ coincides with $\\cl\\mathcal D(x,q)$. \n\\end{Lemma}\n\\begin{proof}\nLet $(x,q)\\in\\mathring{\\mathcal K}$ be fixed. \nAlong the lines of the proof of Lemma \\ref{5181}, one can show that\n\\begin{equation}\\nonumbe\n(\\mathcal G(x,q))^o = \\mathcal A(x,q).\n\\end{equation}\nTherefore, \n$$(\\mathcal G(x,q))^{oo} = (\\mathcal A(x,q))^o = \\cl\\mathcal D(x,q),$$\nwhere in the last equality we have used the conclusion of Lemma \\ref{5181}.\n As $\\mathcal G(x,q)\\subset \\cl\\mathcal D(x,q)$, the assertion of the lemma follows from  the bipolar theorem of Brannath and Schchermayer, \\cite[Theorem 1.3]{BranSchach}.\n\\end{proof}\n\n\n\\begin{Corollary}\\label{lemSt4}\nLet the conditions of Proposition \\ref{prop1} hold and $(x,q)\\in\\mathring{\\mathcal K}$. Then for  \nevery \\underline{maximal} element $Y$ of $\\cl\\mathcal D(x,q)$ there exists $y^n\\geq 0$, $Z^n\\in\\mathcal Z'$, $D^n\\in\\mathcal D'$, $n\\in\\mathbb N$, such that $(y^nZ^nD^n)_{n\\in\\mathbb N}\\subset\\mathcal G(x,q)$ and \n\\begin{equation}\\nonumber\n\\begin{array}{rcll}\nY\n&=& \\lim\\limits_{n\\to\\infty} y^nZ^nD^n,& \\quad (d\\kappa\\times \\mathbb P)-a.e.~and~on~ \\bigcup\\limits_{n\\in\\mathbb N}\\mathcal T_n.\n\\end{array}\n\\end{equation}\n\\end{Corollary}\n\\begin{proof}\nThe $(d\\kappa\\times \\mathbb P)$-a.e. convergence follows from Lemma \\ref{lemSt3}.  By passing to subsequences of convex combinations, we also deduce the convergence on $\\bigcup\\limits_{n\\in\\mathbb N}\\mathcal T_n$.\n\\end{proof}\n\\begin{Remark}\nIt follows from Corollary \\ref{lemSt4} and Fatou's lemma that the maximal elements of $\\cl\\mathcal D(x,q)$ are strong supermartingales. Moreover, every maximal element of $\\cl\\mathcal D(x,q)$ is an optional strong supermartingale deflator, which is optional strong supermartingale $Y$, such that $XY$ is an optional strong supermartingale for every $X\\in\\mathcal X(1)$. We refer to \\cite[Appendix 1]{DelMey82} for a general characterization and to \\cite{CzichSchachOSSup} for results on strong optional supermartingales as limits of martingales. The following section gives a more refined characterization of the dual minimizer.\n\\end{Remark}\n\n\n\\section{Complementary slackness}\\label{secSlackness}\nFor better readability of this section, we recall some notations and results that will be used below. \nThroughout this section, $(x,q)\\in \\mathring{\\mathcal{K}}$ will be fixed, $\\widehat c = \\widehat c(x,q)$ is the optimizer to \\eqref{primalProblem}, $\\widehat V$ is the corresponding wealth process, i.e., \n \\begin{equation}\\label{5211}\n\\widehat V=x+\\int_0^{\\cdot}\\widehat H_sdS_s-\\int_0^{\\cdot}\\widehat c_sd\\kappa_s+\\int_0^{\\cdot} q_se_sd\\kappa_s,\n\\end{equation}\nwhere $\\widehat H$ is some $S$-integrable process, \n$\\widehat Y$ be such that $\\widehat Y_t = U'(t,\\widehat c_t)$, $(d\\kappa\\times\\mathbb P)$-a.e., i.e., $\\widehat Y$ is the optimizer to \\eqref{dualProblem} for some \n$(y,r)\\in\\mathcal E\\cap \\partial u(x,q)$, $\\widehat Y\\in\\mathcal Y(y, r)$ and \n$$\\mathbb E\\left[\\int_0^TV(t,\\widehat Y_t)d\\kappa_t\\right] = \\inf\\limits_{Y\\in \\mathcal Y(y,r)}\\mathbb E\\left[\\int_0^TV(t, Y_t)d\\kappa_t\\right].$$\nBy Corollary \\ref{lemSt4}, $\\widehat Y$  can be approximated by a sequence $y^nD^nZ^n$, $n\\in\\mathbb N$, where $y^n$ is a nonnegative constant, $D^n\\in\\mathcal D'$, and $Z^n\\in\\mathcal Z'$, $n\\in\\mathbb N$.\nIn what follows, for any right-continuous  increasing process $A$ satisfying $$A_t=0,\\quad 0-\\leq t\\leq \\theta _{0}-,\\quad A_T=1,$$ \ni.e., for any probability measure $dA$ on  the  closed interval $[\\theta _0, T]$ (extended to $[0,T]$) we will associate a process $D$ which is left-continuous  and decreasing\n$$D_t{:=} 1-A_{t-}=dA([t,T]), \\ \\ 0\\leq t\\leq T.$$\nOne can also think that $ D_{T+}=0,$ although  this is not necessary.  It is clear that  such $A\\leftrightarrow D$ are in  bijective  correspondence.  Below, all processes $A$'s and $D$'s (with indexes) will be in such bijective correspondence, except for the case of  the limiting  process $\\widehat A$ (which is right-continuous) and  the limiting process $\\widehat D$ (that may be not  left-continuous). The  will be in a similar but more subtle correspondence. More precisely:\n\\begin{Theorem}\\label{ExactSlackness}\nLet the conditions of Theorem \\ref{mainTheorem2}  hold. Let  $\\widehat V$ be the optimal wealth process, $\\widehat c$  the optimal consumption, and $\\widehat Y$ be the dual minimizer satisfying the assertions of Theorem \\ref{mainTheorem2}. Then, there exists a strong supermartingale $\\hat Z=\\lim\\limits_{n\\to\\infty} Z^n$ (for $Z_n\\in \\mathcal{D}'$, where the limit is in the sense of \\cite{CzichSchachOSSup}) and a right-continuous  increasing process $\\widehat A$ with\n$$ \\widehat A_t=0\\ \\  \\forall 0-\\leq t \\leq \\theta _0-,\\ \\ \\  \\widehat A_T= 1, $$\nand a decreasing process $\\widehat D$ with $\\widehat D_t=1,\\ \\ 0\\leq t\\leq {\\theta _0}$ and  satisfying the complementary slackness condition \n\\begin{equation}\\label{comp-slack}\n \\mathbb{P}\\left (\\widehat D_t\\in \\left[1-\\widehat A_t, 1-\\widehat A_{t-}\\right], \\  \\forall \\ \\theta _0 \\leq t\\leq T)\\right)=1,\\ \\ \n \\mathbb{P}\\left (\\int _ {[\\theta _0,T)}1_{\\{\\widehat V_{t-}\\not= 0, \\widehat V_t \\not =0\\}} d\\widehat A_t\\right)=0,\n \\end{equation}\nsuch that the dual minimizer can be decomposed as\n$$\\hat Y= y \\hat Z\\hat D.$$\n\\end{Theorem}\n\nThe proof of the Theorem \\ref{ExactSlackness} is split in several results. \n\\begin{Lemma}\\label{lem5211} Let the conditions of Theorem \\ref{mainTheorem2}  hold. With  \n$$z{:=} xy+\\int_0^T q_sr_sdK_s,$$ there exist $y^n>0$,  $Z^n\\in\\mathcal Z'$, and $D^n\\in\\mathcal D'$, such that $y^nZ^nD^n\\in z\\mathcal G(x,q)$, $n\\in\\mathbb N$, and\n\\begin{equation}\\label{5231}\n\\begin{array}{c}\n\\widehat Y\n= \\lim\\limits_{n\\to\\infty} y^nZ^nD^n,\\quad (d\\kappa\\times \\mathbb P)-a.e.\\\\\ny^n\\to y>0;\\\\%\\textcolor{blue}{\\quad Z^n\\to \\widehat Z,\\quad D^n\\to \\widehat D,\\quad (d\\kappa\\times \\mathbb P)-a.e.}\\\\\n\\end{array}\n\\end{equation}\nFor $A^n$, $n\\in\\mathbb N$, being  in relation to $D^n$ exactly as described before Theorem \\ref{ExactSlackness} we have\n\\begin{equation}\\label{86*}\\mathbb{E}\\left [\\int _{\\theta _0}^T \\widehat V_t Z^n_t dA^n _t \\right]=\\mathbb{E}\\left [Z^n_T\\int _{\\theta _0}^T \\widehat V_t dA^n _t \\right]\\rightarrow 0.\n\\end{equation}\n\\end{Lemma}\n\\begin{proof} Optimality of $\\widehat Y$ and Corollary \\ref{lemSt4} imply \\eqref{5231}.  \nBy \\eqref{2192} and Fatou's lemma, we get\n\\begin{equation}\\label{eqndefz}\nz = \\mathbb E\\left[\\int_0^T \\widehat Y_s\\widehat c_sd\\kappa_s \\right] \\leq \\liminf\\limits_{n\\to\\infty}y^n\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right].\n\\end{equation}\nLet us fix $n\\in\\mathbb N$ and consider $\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right]$. Using localization and integration by parts (along the lines of the proof of Lemma \\ref{6-16-1}), we have\n\\begin{displaymath}\\begin{array}{rcl}\n\\mathbb E\\left[ \\int_0^T Z^n_sD^n_s\\widehat  c_sd\\kappa_s\\right] &=& \\mathbb E\\left[Z^n_T \\int_0^T D^n_{s}\\widehat  c_sd\\kappa_s\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\int_0^T \\left(1 - A^n_{s-}\\right)\\widehat  c_sd\\kappa_s\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\left( \\int_0^T\\widehat  c_sd\\kappa_s - \\int_0^T A^n_{s-} \\widehat c_sd\\kappa_s\\right)\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\left( \\int_0^T\\widehat  c_sd\\kappa_s - A^n_{T} \\int_0^T \\widehat c_sd\\kappa_s + \\int_0^T(\\int_0^t\\widehat c_sd\\kappa_s)dA^n_t \\right)\\right] \\\\\n&=& \\mathbb E\\left[Z^n_T \\int_0^T(\\int_0^t\\widehat c_sd\\kappa_s)dA^n_t \\right].\\\\\n\\end{array}\n\\end{displaymath}\nThe latter expression, using \\eqref{5211} and with   $\\bar X {:=} ||q||_{\\mathbb L^\\infty(dK)}X''$ (where in turn $X''$ is given by the assertion $(v)$ of Lemma \\ref{lemma1}), we can rewrite as\n\\begin{equation}\\label{5212}\n\\mathbb E\\left[Z^n_T \\int_0^T\\left((x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t) + \\left(\\int_0^tq_se_sd\\kappa_s - \\bar X_t\\right) - \\widehat V_t\\right)dA^n_t \\right].\n\\end{equation}\nLet us denote\n\\begin{equation}\\nonumbe\n\\begin{array}\n{rcl}\nT_1 &{:=} & \\mathbb E\\left[Z^n_T \\int_0^T\\left(x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t\\right)dA^n_t \\right] ,\\\\\nT_2 &{:=} & \\mathbb E\\left[Z^n_T \\int_0^T\\left(\\int_0^tq_se_sd\\kappa_s - \\bar X_t\\right)dA^n_t \\right].\\\\\n\\end{array}\n\\end{equation}\nIt follows from nonnegativity of $\\widehat V$ in \\eqref{5211}, Lemma \\ref{lemma1}, and nonnegativity of $\\widehat c$ that \n\\begin{equation}\\label{5232}\nx + \\int_0^{t}\\widehat H_sdS_s + \\bar X_t \\geq \\int_0^t\\widehat c_sd\\kappa_s + \\bar X_t -\\int_0^t q_se_sd\\kappa_s \\geq \\bar X_t -\\int_0^t q_se_sd\\kappa_s \\geq0,\\quad t\\in[0,T],\n\\end{equation}\ni.e., the integrand in $T_1$ is nonnegative. Let ${\\mathbb Q^n}$ be the probability measure, whose density process with respect to $\\mathbb P$ is $Z^n$. As $(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $\\bar X$ are local martingales under ${\\mathbb Q^n}$, by \\cite[Theorem III.27, p. 128]{Pr}, $A^n_{-}\\cdot(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $A^n_{-}\\cdot\\bar X$ are local martingales. Let $(\\tau_k)_{k\\in\\mathbb N}$ be a localizing sequence for both $A^n_{-}\\cdot(x + \\int_0^{\\cdot}\\widehat H_sdS_s + \\bar X)$ and $A^n_{-}\\cdot\\bar X$. \nBy the monotone convergence theorem and integration by parts, we get\n\\begin{equation}\\label{5214}\n\\begin{array}{rcl}\nT_1 &=& \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\left(x + \\int_0^{t}\\widehat H_sdS_s +\\bar X_t\\right)dA^n_t \\right] \\\\\n&=& \\lim\\limits_{k\\to\\infty}\\left(\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}(-A^n_{t-})d(x + \\int_0^{t}\\widehat H_sdS_s + \\bar X_t) + A^n_{\\tau_k} \\left( x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right)\\right]\\right) \\\\\n&=&\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k} \\left( x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right)\\right].\\\\\n\\end{array}\n\\end{equation}\nLet us consider $T_2$. With $\\mathcal E^q {:=} \\int_0^{\\cdot} q_se_sd\\kappa_s$, Lemma \\ref{lemma1} implies positivity of $\\mathcal E^q_t - \\bar X_t$, $t\\in[0,T]$, which in turn allows to invoke the monotone convergence theorem, and we obtain \n\\begin{equation}\\nonumbe\n\\begin{array}{rcl}\nT_2 &=& \\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^T\\left(\\mathcal E^q_t - \\bar X_t\\right)dA^n_t \\right] \\\\\n&=&\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}\\mathcal E^q_t dA^n_t - \\int_0^{\\tau_k}\\bar X_tdA^n_t \\right].\\\\\n\\end{array}\n\\end{equation} \nBy positivity of $\\bar X$ and the monotone convergence theorem, we have $$\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\bar X_tdA^n_t \\right]=\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{T}\\bar X_tdA^n_t \\right].$$\nTherefore, $\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{\\tau_k}\\mathcal E^q_t dA^n_t\\right]=\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{T}\\mathcal E^q_t dA^n_t\\right]$. We deduce that \n\\begin{displaymath}\n\\mathbb E^{\\mathbb Q^n}\\left[\\int_0^{\\tau_k}\\bar X_tdA^n_t \\right] = \\mathbb E^{\\mathbb Q^n}\\left[-(A^n_{-}\\cdot\\bar X)_{\\tau_k} +\\bar X_{\\tau_k}A^n_{\\tau_k}\\right] = \\mathbb E^{\\mathbb Q^n}\\left[\\bar X_{\\tau_k}A^n_{\\tau_k}\\right].\n\\end{displaymath}\nWhereas, in the other term in $T_2$, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\n\\mathbb E^{\\mathbb Q^n}\\left[ \\int_0^{T}\\mathcal E^q_t dA^n_t\\right] &= &\\mathbb E^{\\mathbb Q^n}\\left[\\mathcal E^q_{T}A^n_{T} - (A^n_{-}\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[\\mathcal E^q_{T} - (A^n_{-}\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[((1- A^n_{-})\\cdot \\mathcal E^q)_{T}\\right] \\\\\n&= &\\mathbb E^{\\mathbb Q^n}\\left[(D^n\\cdot \\mathcal E^q)_{T}\\right].\\\\\n\\end{array}\n\\end{displaymath}\nusing integration by parts and localization, as in the proof of Lemma \\ref{6-16-1}, we can rewrite the latter expression as\n$$\n\\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right].\n$$\nWe conclude that\n$$T_2 = -\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[\\bar X_{\\tau_k}A^n_{\\tau_k}\\right] +\\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right].$$\nCombining this with \\eqref{5214}, we obtain \n$$T_1 + T_2  = \\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right] + \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right].$$\nAs both limits in the right-hand side exist and by positivity of $\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)$, established in \\eqref{5232}, we can bound the difference of the limits as\n\\begin{displaymath}\\begin{array}{c}\n \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\left(x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k} \\right)\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right] \\\\\n \\leq \n  \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[x + \\int_0^{\\tau_k}\\widehat H_sdS_s +\\bar X_{\\tau_k}\\right] - \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[A^n_{\\tau_k}\\bar X_{\\tau_k} \\right] \\\\\n   \\leq \n   \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[x + \\int_0^{\\tau_k}\\widehat H_sdS_s\\right] + \n \\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[(1-A^n_{\\tau_k})\\bar X_{\\tau_k} \\right].\\\\\n\\end{array}\n\\end{displaymath}\nBy definition of $\\mathcal M'$, $\\bar X$ is a uniformly integrable martingale under $\\mathbb Q^n$. Therefore, as $(1-A^n_{\\tau_k})$ is bounded, we can pass the limit inside of the expectation to obtain $\\lim\\limits_{k\\to\\infty}\\mathbb E^{\\mathbb Q^n}\\left[(1-A^n_{\\tau_k})\\bar X_{\\tau_k} \\right] = 0$. In turn $x + \\int_0^{\\cdot}\\widehat H_sdS_s$ is a supermartingale under $\\mathbb Q^n$ (see the argument in the proof of Lemma \\ref{6-16-1}). We conclude that \n$$\nT_1 + T_2 \\leq x + \\mathbb E\\left[\\int_0^T D^n_sZ^n_sq_se_sd\\kappa_s\\right] = x + \\int_0^T q_s\\rho^n_sdK_s,\n$$\nfor some $\\rho^n$, which is well-defined by Corollary \\ref{corRzd}, and such that $y^n(1,\\rho^n)\\in z\\mathcal B(x,q)$, since $y^nZ^nD^n\\in z\\mathcal G(x,q)$.   \nThus, from \\eqref{eqndefz} and \\eqref{5212}, we get\n\\begin{displaymath}\n\\begin{array}{rcl}\nz&\\leq &(x + \\int_0^T q_s\\rho^n_sdK_s)y^n  -\\lim\\limits_{n\\to\\infty}y^n\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right].\\\\\n\\end{array}\n\\end{displaymath}\nBy optimality of $\\widehat Y$, $z\\geq (x + \\int_0^T q_s\\rho^n_sdK_s)y^n\\geq 0$. Therefore, by nonnegativity of $\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right]$, and since $y^n$ converges to a strictly positive limit, we conclude that\n$$\\lim\\limits_{n\\to\\infty}\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right] = 0.$$\nApplying integration by parts and localization we deduce the assertion of the lemma.\n\\end{proof}\n\\begin{Remark} We emphasize again that  $dA^n$ are probability measures on $[\\theta _0,T]$, which can have mass at the endpoints, and \n $D^n_t=1-A^n_{t-}, \\theta _0\\leq t\\leq T$, $D^n_{T+}=0$.\n\\end{Remark}\n\nIn the subsequent part, we will follow the notations of Lemma \\ref{lem5211} and we will work with a further subsequence, still denoted by $n$, such that\n$y_n\\rightarrow  y>0$, \n\\begin{equation}\n\\label{cone-product}\nZ^nD^n\\rightarrow \\frac{\\widehat Y}{y}, \\quad (d\\kappa\\times\\mathbb P)-a.e, \n\\end{equation} and\n\\begin{equation}\\label{fast}\n\\sum _{k=n}^{\\infty}\\mathbb E\\left[Z^n_T \\int_0^T\\widehat V_tdA^n_t \\right]\\leq \\frac{2^{-n}}{n},\n\\end{equation}\nwhere the existence of a subsequence satisfying \\eqref{fast} follows from \\eqref{86*}.\nThe next results is a Komlos-type lemma, largely based on the results in  \\cite{CzichSchachOSSup}, applied to the \\emph{double-sequence} of processes $(Z^n, \\left (D^{n}\\right)^{-1})$.   We observe that the process $\\left ( D^n\\right )^{-1}$ takes values in $[1, \\infty]$ is increasing,  left-continuous,  and satisfies\n$$\\left ( D^n _t\\right )^{-1}=1,\\quad  0\\leq t\\leq \\theta _0.$$\n\\begin{Lemma}\\label{5219} \nLet the conditions of Theorem \\ref{mainTheorem2}  hold. In the notations of Lemma \\ref{lem5211},\nfor each $n$, there exist  a finite index $N(n)$ and convex weights\n$$\\alpha _{n,k}\\geq 0,\\quad   k=n,..., N(n),\\quad \\sum _{k=n}^{N(n)}\\alpha _{n,k}=1,$$ \nand there exists a strong optional super-martingale $\\widehat Z$ and a non-decreasing  (not necessarily left-continuous) process $\\widehat D$ \nwith $\\widehat D_t=0, \\ \\forall 0\\leq t\\leq \\theta_0$, \nsuch that, simultaneously, \n\\begin{enumerate}\n\\item $\\tilde Z^n{:=} \\sum _{k=n}^{N(n)}\\alpha _{n,k}Z^k\\rightarrow \\widehat Z$ in the sense of \\cite{CzichSchachOSSup} i.e. for any stopping time $0\\leq \\tau\\leq T$ we have \n$$\\tilde Z^n _{\\tau} {\\longrightarrow}^{\\mathbb{P}}  \\widehat Z_{\\tau},$$\nand \n\\item \n$$ \\mathbb{P}\\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_t)^{-1}\\rightarrow (\\widehat D_t )^{-1}, \\quad  \\forall  0\\leq t \\leq T+ \\right)=1$$\n\n We have set $D_{T+}=D^k_{T+}=0$. \\end{enumerate}\nFurthermore,  with the notation\n\\begin{equation}\\label{a-hat}\n \\widehat A _t {=} 1-\\widehat D_{t+}, \\quad  0\\leq t\\leq T,\\quad \\widehat A_{0-}=0, \n\\end{equation}\nwe have the probability measure  $d\\widehat A$ on $[\\theta _0,T]$ such that\n$$\\mathbb{P}\\left (\\widehat D_t\\in \\left[1-\\widehat A_t, 1-\\widehat A_{t-}\\right], \\  \\forall \\ \\theta _0 \\leq t\\leq T)\\right)=1.$$\nDenoting by \n\\begin{equation}\\label{a-tilde-n}\n\\tilde A_t^n{:=} 1- \\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(1-A^k_t)^{-1}\\right)^{-1}=1-\n\\left (\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_{t+})^{-1}\\right)^{-1}, \\quad  0\\leq t\\leq T, \\quad \\tilde A ^n_{0-}=0,\n\\end{equation}\nthe point-wise convergence in time (at all times where there is continuity)  additionally implies \n$$d\\tilde A^n\\rightharpoonup d\\widehat A,\\quad \\mathbb{P}-a.e.$$ in the sense of weak convergence of probability measures on $[\\theta _0, T]$.  \\end{Lemma}\n\\begin{proof}\nThe proof reduces to applying  the  Komlos-type results in \\cite{CzichSchachOSSup} to the sequence $Z^n$ and  \\cite[Proposition 3.4]{campi-schachermayer} to the sequence $(D^{n})^{-1}$, simultaneously. We observe that: \n\\begin{enumerate}\n\\item first, no bounds are needed    for $D^{-1}$'s  since we can apply the unbounded Komlos lemma in \\cite[Lemma A1.1]{DS} (and Remark 1 following it) to the proof from \\cite[Propositions 3.4]{campi-schachermayer}, and this works even for infinite values (according to  Remark 1 after \\cite[Lemma A1.1]{DS}). Also,  predictability  can be replaced by optionality, without any change to the proof.\n\\item the Komlos arguments can be applied to both sequences $Z^n$ and $D^n$ simultaneously, with the same convex weights. In order to do this,   we first apply Komlos  to one sequence, then replace both original sequences by their  convex combinations with the weights just  obtained, and then apply Komlos again for the other sequence, and update the convex weight to both sequences.\n\\end{enumerate}\n\n\\end{proof}\n\n\\begin{Corollary} Let the conditions of Theorem \\ref{mainTheorem2}  hold. In the notations of Lemma \\ref{5219}, we have the representation\n$$\\widehat Y= y\\widehat Z\\widehat D,\\quad (d\\kappa\\times\\mathbb P)-a.e.$$\n\\end{Corollary}\n\\begin{proof} Consider an observation that, for non-negative\nnumbers  $a_k, b_k$, $k=n,\\dots, N(n)$, we have\n$$\\min _{k=n,...,N(n)} \\left (\\frac{a_k}{b_k}\\right)\\leq \\frac{\\sum _{k=n}^{N(n)}\\alpha _{n,k}a^k}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}b_k}\\leq  \\max _{k=n,...,N(n)} \\left (\\frac{a_k}{b_k}\\right).$$\\\nWe apply this to $a_k=Z^k, b_k=(D^k)^{-1},$ to obtain\n$$\\min _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\leq \n \\frac{\\sum _{k=n}^{N(n)}\\alpha _{n,k}Z^k}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k)^{-1}}=\\tilde Z^n \\tilde D^n \\leq\n  \\max _{k=n,...,N(n)} \\left (Z^kD^k\\right),$$\npointwise a.e.  in the product space, where $\\tilde D^n {:=} \\frac{1}{\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k)^{-1}}$. \n  Since $Z^nD^n\\rightarrow \\widehat Y\/  y$, $(d\\kappa\\times \\mathbb P)$-a.e., we conclude that both\n  $\\left(\\min _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\right)_{n\\in\\mathbb N}$ and  $\\left(\\max _{k=n,...,N(n)} \\left (Z^kD^k\\right)\\right)_{n\\in\\mathbb N}$ converge to $\\frac{\\widehat Y}{y}$, $(d\\kappa\\times \\mathbb P)$-a.e.,\n  therefore\n    \\begin{equation}\n    \\label{conv-product}\\tilde Z^n \\tilde D^n    \\rightarrow \\frac{\\widehat Y}{y},\\quad (d\\kappa\\times \\mathbb P)-a.e.\n   \\end{equation}\nUsing  \\eqref{cone-product} above, if we did not plan to identify the limit $\\widehat Z$   as a strong-supermartingale, but only as a $(d\\kappa\\times \\mathbb P)$-a.e. limit in the product space,  we could only  apply Komlos arguments to the single  sequence $D^{-1}$,  to conclude convergence of the other  convex combination $\\tilde Z^n$ to $\\frac{\\widehat Y}{y\\widehat D}$, defined up to a.e. equality in the product space.\n  \n  \n  With our  (stronger) double Komlos argument, we have that, in addition to \\eqref{cone-product} we have   \\begin{equation}\n  \\label{conv-tau}\\widehat Z_{\\tau}\\widehat D_{\\tau}=\\mathbb{P}-\\lim _n \\tilde Z^n_{\\tau}\\tilde D^n _{\\tau},\\quad \\textrm{for\\ every~[0,T]-valued~stopping \\ time \\ }\\tau,\n  \\end{equation}\n  (where $\\widehat Z$ is a strong supermartingale, and $\\widehat D$ is well defined at all times).\n\n  It remains to prove that $\\widehat Y\/y=\\widehat Z\\widehat D$, $(d\\kappa\\times\\mathbb P)$-a.e. We point out that the convergence \\eqref{conv-tau} is topological. Let us consider an arbitrary optional set $O\\subset \\Omega \\times [0,T]$ and fix an upper bound $M$.  It follows from \\eqref{conv-product} that\n  $$1_O \\min \\left \\{\\tilde Z^n \\tilde D^n, M \\right \\}\\rightarrow  1_O \\min \\left \\{\\frac{\\widehat Y}{y}, M \\right \\},\\quad (d\\kappa\\times \\mathbb P)-a.e.\n$$\nTherefore, we get\n\\begin{equation}\\label{hatY}\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} d\\kappa_t\\right]\\rightarrow  \n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\frac{\\widehat Y_t}{y}, M \\right \\} d\\kappa_t\\right].\n\\end{equation} \nRecall that the stochastic clock has a density $d\\kappa_t=\\varphi  _t dK_t$ with respect to the deterministic clock $dK _t$.  For each fixed $t$, from \\eqref{conv-tau} we have\n$$1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} \\varphi _t\\rightarrow 1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} \\varphi _t,\\quad \\textrm{in}-\\mathbb{P}.$$\nRecalling that $\\mathbb{E}[\\varphi _t]<\\infty$ we have, for fixed $t$, that \n$$ M \\times  \\mathbb{E}[\\varphi _t]\\geq \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\}  \\varphi _t \\right ]\\rightarrow \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} \\varphi _t\\right ].\n$$\nWe integrate the above with respect to the deterministic clock $dK_t$ to obtain\n\\begin{equation}\n\\begin{split}\n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\} d\\kappa_t\\right] =\\int _0^T \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\tilde Z^n_t \\tilde D^n _t, M \\right \\}  \\varphi _t \\right ]dK_t\\rightarrow \\\\\n\\rightarrow \\int _0^T \\mathbb{E}\\left [1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\}  \\varphi _t \\right ]dK_t=\n\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} d\\kappa_t\\right]\\end{split}\n\\end{equation}\nTogether with \\eqref{hatY} we have\n$$\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\frac{\\widehat Y_t}{\\hat y}, M \\right \\} d\\kappa_t\\right]=\\mathbb{E}\\left [\\int _0^T  1_O(t, \\cdot)  \\min \\left \\{\\widehat Z_t \\widehat D _t, M \\right \\} d\\kappa_t\\right],$$\nwhich holds for any optional set $O$ and any bound $M$, therefore $\\widehat Y= y\\widehat Z\\widehat D$, $(d\\kappa\\times \\mathbb P)$-a.e.\n  \\end{proof}\n\\begin{proof}[Proof of the Theorem \\ref{ExactSlackness}]\nSince$$\\sum _{k=n}^{N(n)}\\alpha _{n,k}(D^k_t)^{-1}\\rightarrow (\\widehat D_t )^{-1}, \\quad t\\in[0,T], $$\nrecalling that $\\widehat A$ was defined from $\\widehat D$ and \\eqref{a-tilde-n}. \nAs the processes $A^n$, therefore $\\tilde A^n$ only increase by jumps.\nTherefore\nwe get\n \\begin{equation}\\label{11291}\n \\begin{array}{rcl}\n \\Delta \\tilde A^n_s &= & \\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}}} - \\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}}\\\\\n  &= & \\sum\\limits_{k=n}^{N(n)}\\underbrace{\\frac{\\alpha _{n,k}\\frac{1}{D^k_{s+}}}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}}}_{\\leq 1}\\frac{\\Delta A^k_s}{D^k_{s}}  \n  \\underbrace{\\frac{1}{\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}}}}_{\\leq 1}.\\\\\n \\end{array}\n \\end{equation}\n As $\\frac{\\alpha _{n,k}\\frac{1}{D^k_{s+}}}{\\left(\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s+}}\\right)}\\leq 1$ and $\\sum\\limits_{k=n}^{N(n)}\\alpha _{n,k}\\frac{1}{D^k_{s}} \\geq 1$, we can bound the latter term in \\eqref{11291} by $\\sum\\limits_{k=n}^{N(n)}\\frac{\\Delta A^k_s}{D^k_{s}}$ for $s\\in[0,T]$.\n We deduce that\n $$\\Delta \\tilde A^n_s \\leq \\sum\\limits_{k=n}^{N(n)}\\frac{\\Delta A^k_s}{D^k_{s}},\\quad s\\in[0,T].$$ \nTherefore, we have\n$$\\int_{0}^{t} \\widehat V_u d\\tilde A^n _u  \\leq \\sum _{k=n}^{N(n)}(D^k_{t})^{-1} \\int_{0}^{t} \\widehat V_udA^k_u,\\quad t\\in[0,T],$$\nand thus, we obtain\n\\begin{equation}\\label{main-inez}\n\\left(\\min _{k=n, \\dots, N(n)} (Z^k_tD^k_t) \\right) \\int_{0}^{t} \\widehat V_u d\\tilde A^n _u  \\leq \\sum _{k=n}^{N(n)}(Z^k_tD^k_t)(D^k_{t})^{-1} \\int_{0}^{t} \\widehat V_udA^k_u= \\sum _{k=n}^{N(n)}Z^k_t\\int_{0}^{t} \\widehat V_udA^k_u,\\quad t\\in[0,T].\n\\end{equation}\nSince the process\n$$L^n_t:=\\sum _{k=n}^{N(n)}Z^k_t\\int_{0}^{t} \\widehat V_udA^k_u, \\quad 0\\leq t\\leq T$$\nis a non-negative right-continuous  submartingale, the maximal inequality  and  \\eqref{fast} together imply\n $$\\mathbb{P}\\left  (\\sup _{0\\leq t\\leq T}L^n_t\\geq \\frac 1n \\right )\\leq n\\mathbb{E}[L^n_T]\\leq 2^{-n},\n$$\nso \n\\begin{equation}\n\\label{l}\\sup _{0\\leq t\\leq T}L^n_t\\rightarrow 0,\\quad \\mathbb{P}-a.s.\n\\end{equation}\nSince $Z^nD^n \\rightarrow \\frac{1}{y}\\widehat Y>0$, $(d\\kappa\\times \\mathbb P)$-a.e.,  consequenlty \n$$\\min _{k=n, \\dots, N(n)} (Z^kD^k)\\rightarrow \\frac{1}{y}\\widehat Y>0\\quad \\left(d\\kappa \\times \\mathbb{P}\\right)-a.e.,$$\nwe obtain from\\eqref{main-inez} and \\eqref{l} that \nthe increasing RC process\n$$\\tilde L^n_t:=\\int _0^t \\widehat V_u d\\tilde A^n_u,\\quad 0\\leq t\\leq T,$$\nconverges to zero in the product space.  Denoting by $\\mathcal{O}\\subset \\Omega\\times [0,T]$ the exceptional set where convergence to zero does not take place,  and taking into account that $\\tilde L^n$ are increasing,  we have that for \nfor $$T^{\\mathcal{O}}(\\omega)=\\inf \\{0\\leq t\\leq T: (\\omega ,t)\\in \\mathcal{O}\\},$$\nwe have \n$$(T^{\\mathcal{O}},T]\\subset \\mathcal{O}.$$\nNow \n$$(d\\kappa\\times \\mathbb P)(\\mathcal{O})=0, $$\nimplies that \n$T^{\\mathcal{O}}\\geq T,~\\mathbb{P}-a.s.$ (here used an assumption that $T$ is the minimal time horizon in the sense that the deterministic clock $K$ is such that $K_t<K_T$, for every $t\\in[0,T)$, this assumption does not restrict generality). Thus, there exists a \n a set $\\Omega ^*$  of full probability $\\mathbb{P}(\\Omega ^*)=1$  such that, for each $\\omega \\in \\Omega ^*$ and $t<T$ we have\n $$\\int _0^t \\widehat V_u (\\omega)d\\tilde A^n_t (\\omega)\\rightarrow 0.$$\n\n\nLet us fix an $\\omega\\in \\Omega ^*$ and such that, \nfor this $\\omega$, the probability measure \n$d\\tilde A^{n}(\\omega)$ converges weakly to $d\\widehat A(\\omega)$ over the interval $[\\theta _0(\\omega), T]$.\nThe set of such $\\omega$'s still has probability $1$. The Skorokhod representation theorem asserts that there exists a new probability space  $\\Omega ^{\\omega}$ and a  sequence of random times \n$(t^{n}(\\omega))_{n\\in\\mathbb N}$ as well as $\\widehat t (\\omega)$ such that the distribution of $t^{n} (\\omega)$ is $d\\tilde A^{n}$,  the distribution of $\\widehat t (\\omega)$ is $d \\widehat A (\\omega)$ and\n$$t^{n}(\\omega)\\rightarrow \\widehat t (\\omega),\\quad \\mathbb{P}^{\\omega}-a.s. $$\non the new, artificial, probability space. Fix $t<T$. We have\n$$\\mathbb{E}^{\\omega}[\\widehat V _{t^{n}(\\omega)} (\\omega) 1_{\\{t^n(\\omega)\\leq t\\}}]=\\int _{\\theta _0}^t \\widehat V_u (\\omega)  d\\tilde A^{n} _u(\\omega) \\rightarrow 0.$$\nOne can see that \n$$\\xi (\\omega):=\\liminf _n \\widehat V _{t^{n}(\\omega)}(\\omega)\\in \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega)\\}, \\quad \\mathbb{P}^{\\omega}-a.s.$$ \nand \n$$1_{\\{t^n(\\omega)\\leq t\\}}\\rightarrow 1_{\\{\\widehat t (\\omega)\\leq t\\}}\\  \\textrm{on} \\ \\ \\{\\widehat t (\\omega)<t\\},\\ \\mathbb{P}^{\\omega}-a.s.$$ \nTherefore,  \napplying Fatou's lemma on $\\Omega ^{\\omega}$, we obtain\n$$\\mathbb{E}^{\\omega}[\\xi (\\omega) 1_{\\{\\widehat t(\\omega)< t\\}}]=0.$$\nWe recall that both $\\widehat V (\\omega)$ and $\\widehat V_{-}(\\omega)$ are nonnegative, consequently we have\n$$0\\leq \\min \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega) \\}\\leq \\xi (\\omega),$$ and therefore we get \n$$\\mathbb{E}^{\\omega}[ \\min \\{ \\widehat V _{\\widehat t(\\omega)-}(\\omega),  \\widehat V _{\\widehat t(\\omega)}(\\omega) \\}1_{\\{\\widehat t(\\omega)< t\\}}]=0.$$\nThis means that the distribution  of $\\widehat t (\\omega)$ over the interval $[\\theta _0(\\omega), t)$ only charges the complement of the set of times \n$$\\left\\{u<t: \\quad \\widehat V_{u-}(\\omega)=0\\quad or\\quad \\widehat V_u (\\omega)=0\\right\\}.$$\nBy taking $t\\rightarrow T$, one completes the proof.\n\\end{proof}\n\\bibliographystyle{alpha\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Introduction}\n\nThere are two motivations for this paper. \n\nFirst, for any compact Riemannian manifold  $M$ there is a natural \nfull symbol calculus for operators acting on sections of an arbitrary \nvector bundle with connection. The symbols are $\\mathop{\\rm End}\\nolimits E$-valued \nfunctions on $T^*M$. This symbol calculus can be viewed as a \nconcrete construction of quantization for the symplectic manifold \n$T^*M$. This is more or less well-known construction (see~\\cite{Bok}, \n\\cite{Wid}, \\cite{Ge}, \\cite{Saf}, \\cite{AS}, \\cite{Pf}).\nSuppose now that the bundle $E$ is the bundle of exterior differential \nforms. It is natural to treat forms as functions themselves \n(functions of odd variables) and then to go one step farther and \n``dequantize'' the operators from $\\mathop{\\rm End}\\nolimits \\Lambda^*M$. The new symbols \nobtained in this way can be identified with  {\\sl forms\\\/} on  \nsymplectic manifold $T^*M$. This is what we do in this paper.\nThis refined symbol calculus can be useful with the relation to the Atiyah-Singer \nIndex Theorem. As an example, we give a straightforward proof of the \nGauss-Bonnet-Chern Theorem.  One of our trace formulas \n(formula~(\\ref{startrace})) can be \nutilized to prove the generalized Hirzebruch Signature Theorem.  \n\n\nThe other motivation is as follows. Suppose we have a symplectic or a \nPoisson manifold $P$. At what conditions the Poisson structure can be \nextended from functions to differential forms~? Is it possible to quantize the \nalgebra of forms~? In the present paper we provide a particular \nexample of the quantization of forms, in the case when the \nmanifold $P$ is a cotangent bundle $T^{*}M$. \n\n``Supersymmetry'' proofs of index theorems originated from the works \nof Witten  and Alvarez-Gaum\\'e ~\\cite{A}, who  used path integrals \nand ``physical'' considerations. First mathematically rigorous \ntreatment was given for the bundle-valued Dirac operator by Getzler~\\cite{Ge}, \nwho combined symbol \ncalculus on Riemannian manifold with Clifford algebra. This approach was \nsubstantially simplified \nin~\\cite{AS} by an explicit use of supergeometry and deformation quantization.\nHowever, there was some peculiarity, caused by a highly asymmetric \nway in which even and odd variables entered the \npicture (see~\\cite[p.1038-1039]{AS} and Remark~\\ref{getz} below).\n(Symbol calculus was abandoned in~\\cite{Ge2} and subsequent works in favor of the analysis \nof asymptotics of the heat kernel on the diagonal.)\nIn this paper we work with forms instead of spinors, and  this helps to achieve\nquantization which is  more ``symmetric'' though leads to index \nformulas.\n\n\\begin{Rem}[on notation]  We use the standard language of \nsupermanifolds (see~\\cite{GM}, \\cite{Manin}, \\cite{Leites}, \\cite{GIT}). \nIn particular,  \nand distinguish between cotangent vectors as elements of \n$T^{*}_{x}M$ and $1$-forms as elements of \n$\\Pi T^{*}_{x}M$, where   $\\Pi$  is the parity reversion functor. \nThe same distinction is made between tangent vectors \nand ``$1$-vectors'' (multivectors of degree $1$). So on even manifold \nvectors and covectors are even, while $1$-vectors and $1$-forms are \nodd.  (As far as I know, there is no established name for the elements of \n$\\Pi V$ for a vector space $V$. Borrowing some physical language, I \ncan suggest to call them {\\it antivectors}, and the space $\\Pi V$ \nthe {\\it antispace}.)\n\\end{Rem}\n\n\\par\\medskip\\noindent {\\bf Acknowledgments.}     The results of \nthis paper were discussed at different time with M.A.~Shubin, \nA.~Weinstein, \nM.~Pflaum, J.~Rabin, L.~Sadun, S.~Rosenberg,  \nand the participants of seminars at UC Berkeley, \nUCSD, University of Texas at Austin, and Boston University. \nN.Yu.~Reshetikhin attracted my attention to the problem of quantizing \nforms as differential algebra.\nI am very \nmuch grateful to all of them.\n\n\\section{Complete symbol calculus for Riemannian manifolds}\\label{QS}\n\n\nConsider a Riemannian manifold  $M$ and a vector bundle  $E$ over \n$M$, with a connection. There is a natural construction that allows \nto assign pseudodifferential operators (p.d.o.'s) to functions on \n$T^*M$. More precisely, we mean p.d.o.'s acting on the sections of $E$, \nand by functions we actually mean ``$\\mathop{\\rm End}\\nolimits E$-valued functions'', i.e. \nthe sections of $\\pi^*(\\mathop{\\rm End}\\nolimits E)$, where $\\pi: T^*M\\to M$ is the \nprojection. (The same can be done for \noperators $\\mathop{\\rm \\Gamma}\\nolimits(E)\\to \\mathop{\\rm \\Gamma}\\nolimits(F)$ for two bundles, $E$, $F$.) In the following, the space of smooth \nsections of {\\sl any} bundle $E$ over {\\sl any} manifold $M$ is \ndenoted by $C^{\\infty}(M,E)$\n\nLet us consider  the geodesic segment connecting two points  $x,y\\in M$ (we \nsuppose that they are sufficiently close for it to exist and be  \nunique), and denote the point with the affine parameter $s$ by \n$g_{s}(x,y)$. Here $g_{0}(x,y)=x$, $g_{1}(x,y)=y$. We shall need some more \nnotation. The points of $T_{x}^*M$ (cotangent vectors, or \n``momenta'', at point $x\\in M$) will \nbe denoted by $p$. Their coordinates will be denoted by $p_a$.  \nMomenta at the point $y$ will be denoted by the letter $q$.\nThe parallel transport $T^{*}_{y}M\\to T^{*}_{x}M$ along the geodesic, \nw.r.t. the Levi-Civita connection, will \nbe denoted by $T(x,y)$. The similar transport for the bundle $E$ will \nbe denoted by $\\tau(x,y)$. We  shall assume that there is a \nneighborhood of the diagonal $\\Delta$ in $M\\times M$ where the \ngeodesic segment exists and is unique. For example, let $M$ be compact \nor $M=\\R{n}$. We shall use a bump function $\\a(x,y)$ which is \nidentically $1$ near $\\Delta \\subset M\\times M$, nonnegative, and \nsupported inside the indicated neighborhood of $\\Delta$. The \nconstruction depends on the choice of $\\a$, though not very \nheavily.\n\n\nIn the following we shall also need  some notation for the volume elements.\nLiouville's measure on $T^*M$ is denoted by $dx\\, dp$ (resp., $dy\\, dq$). \nThe Riemannian volume element on $M$ \nat point $x$ is $\\omega(x)=\\sqrt{h(x)}\\, dx$.  Here \n$h(x)=\\det (h_{{ab}}(x))$ is the Gram determinant in a coordinate frame.\nThere are Euclidean volume \nelements on $T_x M$ and $T^*_x M$. They are denoted by \n$\\theta_x=\\sqrt{g(x)}\\,dv$ \nand $\\lambda_x=(1\/\\sqrt{g(x)})\\,dp$ respectively. Here \n$g(x)=\\det (g_{ij}(x))$ is the Gram determinant in some chosen local \nframe, not necessarily coordinate. The exponential map \n$\\exp_x: T_x M\\to M$ allows to compare volume elements on $M$ and on \nthe tangent space at a fixed point. We introduce the function \n$\\mu(x,y)$ by the following equation:\n\\begin{equation}\n  \\theta_x(v)=\\mu (x,y)\\; \\omega(y),\n\\end{equation}\nif $y=\\exp_x v$, $v\\in T_x M$. This definition is valid for $y$ \nsufficiently close to $x$.\n\n\\subsection{Quantization}\n\n\\begin{de} Fix $s \\in[0,1]$. Let $f\\in C^{\\infty}(T^*M, \\mathop{\\rm End}\\nolimits E)$. We \nassociate with $f$ the following operator $\\hat f$, acting on  \nsections of $E$. For $u\\in C^{\\infty}(M, E)$\n\\begin{multline} \\label{quant}\n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M}dydq \\;\ne^{\\frac{i}{\\hbar}(\\expi{y}{x})\\cdot q}\\; \\a(x,y)\\,\\mu(x,y)\\\\\n \\tau(x,{\\zs}) \\; f({\\zs},T({\\zs},y)q) \\;\\tau({\\zs},y)\\;u(y).\n\\end{multline}\n\\end{de}\n\nWe can make a change of variables: $y=\\exp_x v, q=T(y,x)p$. This \nyields\n\\begin{de}[equivalent] \n\\begin{multline} \\label{quantprime}\n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T_x M\\times T_x^*M}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\;\\a(x,\\exp_x v)\\\\\n\\tau(x,\\exp_x sv) \\; f(\\exp_x sv,T(\\exp_x sv,x)p) \\;\n\\tau(\\exp_x sv,\\exp_x v)\\;u(\\exp_x v).\n\\end{multline}\n\\end{de}\n\n\n\n\nThese formulas might look complicated but the idea is very simple.\n\n\\begin{ex} Consider $M=\\R{n}$. Take $\\a:=1$ (no bump function is \nnecessary). \nIn affine coordinates on $\\R{n}$ and in parallel frame for the (trivial) bundle \n$E$, the formula~(\\ref{quant}) becomes\n \\begin{equation} \n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{2n}}dydq \\;e^{\\frac{i}{\\hbar}(y-x)\\cdot \nq}\\;\n f((1-s)x+sy),q) \\;u(y),\n\\end{equation}\nand the formula ~(\\ref{quantprime}) becomes\n\\begin{equation} \n(\\hat f u)(x):=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times \\R{n}}dvdp \\; \ne^{-\\frac{i}{\\hbar}vp} \\; f(x +sv,p) \n\\;u(x + v).\n\\end{equation}\nFor $s=0$ this yields $xp$-quantization (= standard \npseudodifferential calculus), for $s=1$ this is $px$-quantization \n(the opposite way of ordering monomials), and the case $s=1\/2$ yields Weyl \n(symmetric) quantization (cp. ~\\cite{BS}).\n\\end{ex}\n\n\\begin{ex} Take a function $f=f(x)$, not depending on $p$. Then, by \n~(\\ref{quantprime}), \n\\begin{multline}\n\t(\\hat f u)(x) = \\int_{T_x M}\\theta_x(v) \\;\\delta(v)\\;\n\\tau(x,\\exp_x sv) \\; f(\\exp_x sv) \\;\n\\tau(\\exp_x sv,\\exp_x v)\\;u(\\exp_x v)\\\\=f(x)\\, u(x).\n\\end{multline}\n(Here $\\delta$ stands for the ``Euclidean'' delta-function on $T_xM$.)\n\\end{ex}\n\n\n \\subsection{Symbols}\n\nLet $K$ be the Schwartz kernel of $A: C^{\\infty}(M,E)\\to \nC^{\\infty}(M,E)$ with respect to the Riemannian volume element:\n\\begin{equation}\n\t(Au)(x)=\\int_M \\!\\omega(y) \\;K(x,y)\\,u(y).\n\\end{equation}\n\n\n\\begin{de} The following function on $T^*M$ is called the {\\it \nsymbol} of the operator $A$ and is denoted by $\\sigma A$:\n\n\\begin{multline} \\label{symbol}\n(\\sigma A)(x,p):=\\int_{T_x M}\\theta_x(v) \\;e^{\\frac{i}{\\hbar}vp} \n\\;\\a(\\exp_x(-sv),\\exp_x (1-s)v)\\; \\rho_s(x,v)\\\\\n\\tau(x,\\exp_x (-sv)) \\; K(\\exp_x (-sv),\\exp_x (1-s)v) \\;\n\\tau(\\exp_x (1-s)v,x).\n\\end{multline}\nHere \n\\begin{equation} \\rho_s(x,v):=\\left( \\mu(\\exp_x (-sv),\\exp_x \n(1-s)v)\\right)^{-1}.\n\\end{equation}\n\\end{de}\n\nNote that $\\sigma A\\in C^{\\infty}(T^*M,\\mathop{\\rm End}\\nolimits E)$, and that $K(x,y)\\in \n\\mathop{\\rm Hom}\\nolimits (E_y,E_x)$, for fixed $x,y$.\n\n\n\n\\begin{rem} For $s=0$  the symbol  of an operator $A$ can be calculated as follows:\n\\begin{equation}\\label{specsymb}\n\t(\\sigma A)(x,p)u_0=\\left(A_y\\left(e^{\\frac{i}{\\hbar}(\\expi{x}{y})\\cdot p} \n\\a(x,y)\\;\\tau(y,x)\\;u_0 \\right)\\right)_{|y:=x},\n\\end{equation}\nfor an arbitrary constant vector $u_0\\in E_x$.  Here $A_y$ means operator \nacting on sections which argument is denoted by $y$.\n\\end{rem}\n\n\\begin{thm}  1. The quantization map $f\\mapsto \\hat f$ and the symbol \nmap $A\\mapsto \\sigma A$ are ``almost'' mutually inverse:\n\\begin{align} \n\t\\sigma\\hat f&=f(1 + O(\\hbar^{\\infty})), \\label{sq}\\\\\n\tK_{\\widehat{\\sigma A}}(x,y)&=K_{A}(x,y)\\cdot(\\a(x,y))^2,\\label{qs}\n\\end{align}\nfor any $f$ and $A$. Here $K_A$ denotes the Schwartz kernel of an \noperator $A$.\n\n2. For polynomials in $(p_{a})$ and for differential operators the maps \n    $\\ \\hat{}$ and $\\sigma$ are mutually inverse. In this case the \nconstruction is independent of the bump function $\\a$.\n\\end{thm}\n\n\\begin{co} \\label{kerdiag}If $A$ has a continuous kernel and is of \ntrace class, then the traces of $A$ and $\\widehat{\\sigma A}$ coincide.\n\\end{co}\n\\begin{proof} By equation~(\\ref{qs}), the kernels  of  $A$  and $\\widehat{\\sigma \nA}$ coincide  on the diagonal.\n\\end{proof}\n\n\n\\begin{rem} A complete symbol calculus for \nRiemannian manifolds was for the first time considered by \nBokobza-Haggiag~\\cite{Bok} and Widom~\\cite{Wid}. Later various \nmodifications were suggested and used: see~\\cite{Ge}, \\cite{AS}, \\cite{Pf}, \\cite{Saf}, \nalso~\\cite{Gutt}, \\cite{bnw}; probably more references can be given. \nThough all approaches are \nbased on the same idea (use of connection, of the exponential map and the parallel \ntransport along geodesics), there are small subtleties leading to \ninequivalent constructions. In most approaches the standard \npseudodifferential calculus (or ``$qp$''-quantization) on $\\R{n}$ has \nbeen generalized. The beautiful idea to use a middle point on a \ngeodesic segment to generalize Weyl symbols and \nother ``$s$-symbols'', compare~\\cite{Shub}, is due to Yu.G.~Safarov. \n(I use the opportunity to thank M.A.~Shubin from whom I learned about Safarov's approach \naround 1989.)  The above definitions of \nquantization and symbols are based on  this idea.  For $s=0$ the construction is basically \nequivalent to the one used in~\\cite{AS}, up to a slightly different \nintroduction of bump function. Since we follow the philosophy \nof deformation quantization, our formulas contain Planck constant, \nwhich is of course very important.\n\\end{rem}\n\n\\subsection{Composition}\n\n\\begin{de} The operation $f\\circ g:= \\sigma (\\hat f\\hat g) $ is \ncalled the {\\it composition } of functions $f,g\\in C^{\\infty}(T^*M, \n\\mathop{\\rm End}\\nolimits E)$.\n\\end{de}\n\n\\begin{thm} For $\\hbar \\to 0$,\n\\begin{equation} f\\circ g=fg \\,(1 + O(\\hbar)).\n\\end{equation}\nIf $f,g$ are scalar functions, then\n\\begin{equation} f\\circ g-g\\circ f= -{i}{\\hbar}\\,\\{f,g\\}\\;(1 + \nO(\\hbar)),\n\\end{equation}\nwith canonical Poisson brackets on $T^*M$.\n\\end{thm}\n\\begin{proof} The first statement follows by direct computation. To \nprove the second statement, note that since in the limit $\\hbar\\to 0$ \nin scalar case we obtain \ncommutative multiplication, the commutator \nw.r.t. the composition induces {\\it some} Poisson bracket on functions. Thus it is \nsufficient to check the brackets for local coordinates $x^a,p_a$. \nThat the induced brackets for them coincide with the canonical ones, follows \nfrom the calculation below.\n\\end{proof}\n\n\n\n\n\n\\subsection{Examples: quantizing linear and quadratic Hamiltonians}\n\n\nIn the subsequent  calculations we fix some point $x\\in M  $ and work \nin normal coordinates centered at $x$. By definition, then $(\\exp_x \nv)^a=v^a$, and $x^a=0$, and $\\Gamma _{bc}^a(0)=0$. We also introduce a \nframe in $E$ parallel along the geodesic radii $y^a=tv^a$, which we  \ncall the {\\it parallel frame}. It is specified by two properties:  \n$A(0)=0$, $y^a A_a(y)=0$ for the connection $1$-form $A$ in $E$. This \nimplies that $\\d_aA_b(0)$ is antisymmetric and thus equals \n$F_{ab}(0)$, where $F$ is the curvature $2$-form. We can rewrite the \nformula~(\\ref{quantprime}) in the normal coordinates and the parallel \nframe:\n\\begin{equation}\\label{simple}\n\t(\\hat f u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\;\\a(x,v)\\; f(sv,T(sv,0)p) \\;u(v).\n\\end{equation}\nIt looks almost as for $\\R{n}$.\n\n\\begin{ex} Let $X=X^a\\d_a\\in \\mathop{\\rm Vect}\\nolimits M$ and consider the corresponding \nfiberwise-linear Hamiltonian $X\\cdot \\mathrm p=X^a p_a$. By~(\\ref{simple}) we have:\n\\begin{multline*} \n\t(\\widehat{X\\cdot \\mathrm p}\\,u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\; X^a(sv)\\,(T(sv,0)p)_a \\;u(v)\\\\\n=\n\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}}dp\\; p_a\\int_{\\R{n}}dv \\;e^{-\\frac{i}{\\hbar}vp} \\; \n(T(0,sv)X(sv))^a\\;u(v)\\\\\n=\n-i\\hbar \\der{}{v^a}\\left( (T(0,sv)X(sv))^au(v)\\right)_{|v=0}\\\\\n=-i\\hbar(s\\cdot \\nabla_aX^a(0)+X^a(0)\\,\\d_a u(0)),\n\\end{multline*}\nwhich we can rewrite in the invariant form:\n\\begin{equation}\n\t\\widehat{X\\cdot \\mathrm p}=-i\\hbar \\,(\\nabla_X+s\\div X).\n\\end{equation}\n\\end{ex}\n\nIf we apply this example to $X=\\d_{a}$, then we immediately obtain \ncanonical commutation relations and canonical Poisson brackets.\n\n\\begin{ex}\nConsider the Hamiltonian $\\mathrm p^2=g^{ab}p_ap_b$, corresponding to \nthe metric on $M$. In the same way,\n\\begin{multline*} \n\t(\\widehat{\\mathrm p^2}u)(0)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \n\\;e^{-\\frac{i}{\\hbar}vp} \\; g^{ab}(sv)\\,(T(sv,0)p)_a\\; \n(T(sv,0)p)_b\\;u(v)\\\\\n=\\frac{1}{(2\\pi\\hbar)^n}\\int_{\\R{n}\\times\\R{n}}dvdp \\;e^{-\\frac{i}{\\hbar}vp} \\; \ng^{ab}(0)p_ap_b\\;u(v)\\\\\n=-\\hbar^2 g^{ab}(0)\\;\\partial^2_{ab}u(0)=\n-\\hbar^2 g^{ab}(0)(\\nabla_a\\nabla_b u-\\Gamma_{ba}^c\\nabla_cu)(0).\n\\end{multline*}\nWe used the fact that parallel transport on $M$ is an orthogonal map. \nThus, independent of $s$,\n\\begin{equation}\\label{lapsymb}\n\t\\widehat{\\mathrm p^2}=-\\hbar^2 \\Delta,\n\\end{equation}\nwhere\n$\\Delta$ stands for the Laplace-Beltrami operator on a Riemannian \nmanifold, acting on sections of a vector bundle with a connection:\n\\begin{equation}\n\t\\Delta u:=g^{ab}(\\nabla_a\\nabla_b\\, u-\\Gamma_{ba}^c\\nabla_c\\,u),\n\\end{equation}\nwhere $\\nabla_a$ in this formula denotes partial covariant derivative \nof the sections of $E$.\n\\end{ex}\n\n\n\n\n\\subsection{The trace formula}\n\n\nIf we take as a working definition of the trace of an operator $A$ \nthe integral of its kernel over the diagonal $\\Delta\\subset M\\times \nM$, then we easily obtain the following \n\\begin{thm} \\label{trace}\nFor all $s\\in [0,1]$, the trace is calculated by the same  \nformula:\n\\begin{align}\n\t\\mathop{\\rm Tr}\\nolimits \\hat f&=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M} dxdp\\;\\mathop{\\rm tr}\\nolimits f(x,p),\\label{trq}\\\\\n\t\\mathop{\\rm Tr}\\nolimits A&=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*M} dxdp\\;\\mathop{\\rm tr}\\nolimits \\,(\\sigma A)(x,p),\\label{tra}\n\\end{align}\nfor any $A$ and $f$. Here $\\mathop{\\rm tr}\\nolimits$ denotes the trace on $\\mathop{\\rm End}\\nolimits E_x$.\n\\end{thm}\n\\begin{proof} From~(\\ref{quant}) we can deduce the following formula for \nthe restriction of the kernel of $\\hat f$ to the diagonal:\n\\begin{equation}\n\tK_{\\hat f}(x,x)=\\frac{1}{(2\\pi\\hbar)^n}\\int_{T^*_xM}\\lambda_x(p)\\;f(x,p).\n\\end{equation}\nThis is true for all $s$.\nThen ~(\\ref{trq}) immediately follows. (Note that  \n$\\omega(x)\\lambda_x(p)=dxdp$.) The equality~(\\ref{tra}) then follows, \nby Corollary~\\ref{kerdiag}.\n\\end{proof}\n\n\n\n\n\n\\section{Quantization of forms on $T^{*}M$}\n\n\n\nWe are going to  apply the preceding consideration to \nthe particular case when $E=\\Lambda(T^*M)$, the bundle of exterior \ndifferential  forms. (We can also consider the case of \n$\\Lambda(T^*M)\\otimes E$, where $E$ is  some other bundle.) In this \ncase we describe endomorphisms of \n$\\Lambda=\\Lambda_x=\\Lambda(T_x^*M)$ at each $x$ also as a result of \nquantization of some symbols. The corresponding symbol calculus ``at a \npoint''  is a \nsuperanalog of the quantization on $\\R{n}$. The calculus of the \nprevious section will be combined with this pointwise calculus to \nproduce our main construction.\n\n\\subsection{Quantization at a point}\n\nConsider an arbitrary local coframe on $M$. Its \nelements are considered as $1$-forms and thus odd (see the remark in the \nintroduction). So at a \npoint $x$ we have $n=\\dim M$ odd variables  $\\xi^k=e^k(x)$. \n(Later, when we shall consider different points, we shall pick \ndifferent letters for corresponding  values of $e^k$.) Elements of $\\Lambda$ \nare functions of $\\xi^k$. All endomorphisms of $\\Lambda$ are \ndifferential operators and are generated by $\\xi^k$, \n$\\lder{}{\\xi^k}$. They also can be expressed by integral kernels:\n\\begin{equation}\n\t(Au) (\\xi)=(-1)^{n \\tilde A} \\int_{\\S{n}} \\frac{D{\\eta}}{\\sqrt{g}}\\; \nk(\\xi,{\\eta})\\;u({\\eta}).\n\\end{equation}\nHere  $g=g(x)=\\det \n(g_{kl}(x))$ is the Gram determinant of the frame $(e_k(x))$. Notice \nthat \n\\begin{equation}\\label{kernel}\nk_{A}({\\xi},{\\eta})=(-1)^{n}(A\\delta_{{\\eta}})({\\xi}),\n\\end{equation}\nwhere $\\delta_{{\\eta}}({\\xi})=\\sqrt{g}\\,\\delta({\\xi}-{\\eta})$.\n\n\\begin{lm}\\label{kerprod}\n\\begin{equation}\nk_{AB}({\\xi},{\\eta}) =(-1)^{n\\tilde A} \\int_{\\S{n}} \\frac{D{\\xi}'}{\\sqrt{g}}\\;\nk_{A}({\\xi},{\\xi}')\\;k_{B}({\\xi}',{\\eta})\n\\end{equation}\n\\end{lm}\n\n\nIn the same way as above, or, actually, in the same way as for \n$\\R{n}$, we can introduce the following symbol calculus. Consider odd \nvariables $\\theta_k$ which transform contragrediently to $\\xi^k$. \nThat means that the form $\\xi^k\\theta_k$ is invariant under change \nof (co)frame. Geometrically, \n$\\theta_k$ are the values at $x$ of the elements of the dual frame \n$e_k$ regarded as  $1$-vectors (antivectors). \nA function of $\\t_k$ is a multivector at the point $x$.\n\nIn the following we fix the point $x$ and forget about it for time \nbeing.\n\n\\begin{de} Fix $r\\in [0,1]$. \nLet $f=f({\\xi},\\theta)$. We associate with this function the \nfollowing operator on $\\Lambda$. For any $u\\in \\Lambda$ set\n\\begin{equation} \\label{squant}\n\t(\\hat f u)({\\xi}):={(-i\\hbar)^n}\\int_{\\S{2n}} \nD({\\eta},\\t)\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\; f((1-r){\\xi} + r{\\eta}, \\t)\\;u({\\eta}).\n\\end{equation}\nConversely, for an operator $A $ with the kernel $k=k(\\xi,{\\eta})$, we \ndefine its {\\it symbol} by the formula\n\\begin{equation} \\label{ssymb}\n\t(\\sigma A)({\\xi},\\t):=(-1)^{n \\tilde A}\\int_{\\S{n}}\\frac{D{\\varepsilon}}{\\sqrt{g}} \n\\;e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\;k({\\xi}-r{\\varepsilon},{\\xi}+(1-r){\\varepsilon}).\n\\end{equation}\n\\end{de}\n\n\\begin{rem} For $r=0$, we can express the symbol by the action of the \noperator on the exponential function (cp. ~(\\ref{specsymb}):\n\\begin{equation}\n\t(\\sigma A)({\\xi},\\t)=(A_{{\\eta}}(e^{\\frac{i}{\\hbar}({\\eta}-{\\xi})\\t}))_{|{\\eta}:={\\xi}},\n\\end{equation}\nwhere the subscript ${}_{{\\eta}}$ means that operator is applied to the \nfunctions of the variables ${\\eta}^a$.\n\\end{rem}\n\n\\begin{thm} The quantization map $f\\mapsto \\hat f$ and the symbol map \n$A\\mapsto \\sigma A$ are mutually inverse.\n\\end{thm}\n\\begin{proof} Since we deal with finite-dimensional spaces, it \nsuffices to check either $\\hat{ }\\circ\\!\\sigma $ or \n$\\sigma\\!\\circ\\hat{ }$\\\/. \nTake $f=f({\\xi},\\t)$. The integral kernel of $\\hat f$, \nby~(\\ref{squant}),  is\n\\begin{equation}\\label{kerhatf}\nk_{\\hat f}({\\xi},{\\eta})=(-1)^{n\\tilde f}{(-i\\hbar)^n}\\int_{\\S{n}} \\sqrt{g}\\,\nD\\t\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\; f((1-r){\\xi} + r{\\eta}, \\t).\n\\end{equation}\nSo, using  formula~(\\ref{ssymb}),\n\\begin{multline*}\n\\sigma(\\hat f)({\\xi},\\t)=(-1)^{n\\tilde \nf}\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\;k_{\\hat \nf}({\\xi}-r{\\varepsilon},{\\xi}+(1-r){\\varepsilon})\\\\=\n{(-i\\hbar)^n}\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,\\sqrt{g}\\,D\\t'\\;e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\n\\;e^{-\\frac{i}{\\hbar}{\\varepsilon}\\t'}\\;f((1-r)({\\xi}-r{\\varepsilon})+r{\\xi}+r(1-r){\\varepsilon},\\t')\\\\=\n{(-i\\hbar)^n}\\int D{\\varepsilon} \\,D\\t' \\;e^{\\frac{i}{\\hbar}{\\varepsilon}(\\t-\\t')}\\;f({\\xi},\\t')=\n\\int D\\t'\\,\\delta(\\t'-\\t)\\,f({\\xi},\\t')=f({\\xi},\\t).\n\\end{multline*}\n\\end{proof}\n\n\nThe construction of quantization is invariant under linear \ntransformations, in the following sense. Consider two spaces $V_{x}, \nV_{y}\\cong \\S{n}$ and an isomorphism $T: V_{y}\\to V_{x}$. It induces \npull-back of functions: $T^{*}:\\Lambda_{x}\\to \\Lambda_{y}$. \n(Letters $x$ and $y$ are used here as just labels.)\n\\begin{lm} \\label{spinrep}\nFor any  $f=f({\\xi}_{x},\\t_{x})$,\n\\begin{equation}\nT^{*}\\;\\hat f\\;{T^{*}}^{-1}=\\widehat{T^{*}f},\n\\end{equation}\nwhere $(T^{*}f)({\\xi}_{y},\\t_{y})=f(T{\\xi}_{y},T^{-1}\\t_{y})$. \n\\end{lm}\n\\begin{proof} Straightforward calculation, using the \nformula~(\\ref{squant}).\n\\end{proof}\n(This is a trivial case of the spinor representation.)\nWe shall use the Lemma in the next section.\n\n\n\n\n\n\nThe {\\it composition} of symbols is defined in a usual way: $f\\circ \ng:=\\sigma(\\hat f\\hat g)$.\n\n\\begin{thm} The composition can be calculated by the following \nintegral formula:\n\\begin{multline}\\label{comp}\n(f\\circ \ng)({\\xi},\\t)=\\\\(-i\\hbar)^{2n}\\!\\!\\!\\!\\!\\!\\!\\!\\int\\limits_{\\S{2n}\\times\\S{2n}}\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\nD{\\varepsilon}_{1}\\,D\\tau_{1}\\,D{\\varepsilon}_{2}\\,D\\tau_{2}\\; \ne^{\\frac{i}{\\hbar}({\\varepsilon}_{1}\\tau_{2}-{\\varepsilon}_{2}\\tau_{1})}\\;f({\\xi}+r{\\varepsilon}_{1},\\t+\\tau_{1})\\;\ng({\\xi}+(1-r){\\varepsilon}_{2},\\t+\\tau_{2})\n\\end{multline}\nfor arbitrary $r\\in [0,1]$. \n\\\\ For $r=0$ this formula is simplified to\n\\begin{equation} \\label{comp0}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{n}\\int_{\\S{2n}}D{\\varepsilon}\\,D\\tau\\; e^{-\\frac{i}{\\hbar}\n{\\varepsilon}\\tau}\\;f({\\xi},\\t+\\tau)\\; g({\\xi}+{\\varepsilon},\\t).\n\\end{equation}\n\\\\ For $r=1$ it is simplified to\n\\begin{equation} \\label{comp1}\n(f\\circ g)({\\xi},\\t)=(i\\hbar)^{n}\\int_{\\S{2n}}D{\\varepsilon}\\,D\\tau\\;\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\tau}\\;f({\\xi}+{\\varepsilon},\\t)\\;\ng({\\xi},\\t+\\tau).\n\\end{equation}\n (Notice the difference in signs in the formulas~(\\ref{comp0})) \nand~(\\ref{comp1}).)\n\\end{thm}\n\\begin{proof}\nTo calculate the composition of $f$ and $g$, one has to apply twice \nthe formula ~(\\ref{squant}) to find $\\hat f\\hat g u$ for some ``test'' \nfunction $u$, and thus obtain the integral kernel for $\\hat f\\hat g$, and then \napply~(\\ref{ssymb}). After this straightforward calculation (that we \nomit), we arrive to the following formula:\n\\begin{multline}\\label{comp00}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{2n}\\int D{\\varepsilon}\\,D\\t_{2}\\,D{\\eta}_{1}\\,D\\t_{1}\\;\ne^{\\frac{i}{\\hbar}({\\varepsilon}\\t + ({\\xi} - r{\\varepsilon} - {\\eta}_{1})\\t_{1} + ({\\eta}_{1} - \n{\\xi} - (1-r){\\varepsilon})\\t_{2})}\\;\\\\\nf((1-r)({\\xi} - r{\\varepsilon}) + r{\\eta}_{1}, \\t) \\; g((1-r){\\eta}_{1} + r({\\xi} + (1-r){\\varepsilon}), \n\\t_{2}).\n\\end{multline}\nNow we introduce new variables: ${\\varepsilon}_{1}={\\eta}_{1} - (1-r){\\varepsilon}  - {\\xi}$,\n ${\\varepsilon}_{2}={\\eta}_{1}  + r{\\varepsilon}  - {\\xi}$. This change of variables is \n invertible: ${\\eta}_{1}={\\xi} + r{\\varepsilon}_{1} + (1-r){\\varepsilon}_{2}$, ${\\varepsilon}={\\varepsilon}_{2}-{\\varepsilon}_{1}$, \n and its Berezinian is unity. In these variables the formula \n ~(\\ref{comp00}) becomes\n\\begin{multline}\n(f\\circ g)({\\xi},\\t)=(-i\\hbar)^{2n}\\int D{\\varepsilon}_{1}\\,D{\\varepsilon}_{2}\\,D\\t_{1}\\,D\\t_{2}\\; \ne^{\\frac{i}{\\hbar}({\\varepsilon}_{2}(\\t-\\t_{1}) + {\\varepsilon}_{1}(\\t_{2}-\\t))}\\;\\\\\nf({\\xi}+r{\\varepsilon}_{1}, \\t_{1})\\; g({\\xi}+(1-r){\\varepsilon}_{2}, \\t_{2}).\n\\end{multline}\nFinally, we substitute $\\t_{1}=\\t + \\tau_{1}$, $\\t_{2}=\\t + \\tau_{2}$ \nand obtain~(\\ref{comp}). Formulas ~(\\ref{comp0}, \\ref{comp1}) follow \nfrom ~(\\ref{comp}) by direct integration.\n\\end{proof}\n\n\\begin{co} For arbitrary $r\\in [0,1]$ \n\\begin{equation} \n(f\\circ \ng)({\\xi},\\t)=e^{-\\frac{i}{\\hbar}((1-r)\\der{}{\\t_{1}}\\der{}{{\\xi}_{2}} + \nr\\der{}{{\\xi}_{1}}\\der{}{\\t_{2}})}\\;f({\\xi}_{1},\\t_{1})\\;g({\\xi}_{2},\\t_{2})\n\\,_{\\left|\\parbox{2cm}{\\scriptsize $ \n\\begin{aligned} {\\xi}_{1}&:={\\xi},  & {\\quad } {\\xi}_{2}&:={\\xi} \\\\ \n\\t_{1}&:=\\t,  & { \\quad} \\t_{2}&:=\\t \\\\ \n\\end{aligned}$}\\right.}\n\\end{equation}\n\\end{co}\n\n\n\\begin{co} In the limit $\\hbar\\to 0$ the composition becomes \nordinary multiplication. For the commutator we have the following \nformula:\n\\begin{equation}\n\tf\\circ g-(-1)^{\\tilde f\\tilde g}g\\circ f= -i\\hbar\\,\\{f,g\\}\\,(1 + O(\\hbar)),\n\\end{equation}\nwhere the nonvanishing Poisson brackets for the  coordinates \n${\\xi}^a,\\t_a$ are:\n\\begin{equation}\n\t\\{\\t_a,{\\xi}^b\\}=\\{{\\xi}^b,\\t_a\\}=\\delta_a{}^b\n\\end{equation}\n(``canonical brackets'').\n\\end{co}\n\n\n\n\\subsection{Formulas for local traces}\nThe space $\\Lambda$ can be endowed with different ${\\mathbb Z_{2}}$-gradings. In \naddition to the natural grading by parity, \none can define the grading by the eigenvalues of an \narbitrary  operator $S$ such that $S^2=1$. \nA form $u$ is {\\it $S$-even\\\/} (resp., {\\it $S$-odd\\\/}) if $Su=u$ (resp., \n$Su=-u$\\\/).\nThe {\\it trivial grading\\\/} corresponds to $S=1$ (the identity \noperator). The operator $S$ defining the natural grading  is \n$S=(-1)^{P}$, where $P$ is the ``parity operator'', with eigenvalues \n$0, 1$. \n\n\\begin{lm}\\label{kerparity}\n\\begin{equation}\nk_{(-1)^{P}}({\\xi},{\\eta}) = \\sqrt{g}\\delta({\\xi}+{\\eta})\n\\end{equation}\n\\end{lm}\n\\begin{proof} $k_{(-1)^{P}}({\\xi},{\\eta})=(-1)^{n}((-1)^{P}\\delta_{{\\eta}})({\\xi})=\n(-1)^{n}\\sqrt{g}\\delta(-{\\xi}-{\\eta})= {\\sqrt{g}\\delta({\\xi}+{\\eta})}$ \n\\end{proof}\n \n\nAn operator $A$ that preserves the $S$-grading is called {\\it \n$S$-even\\\/}. This is equivalent to $AS=SA$. Similarly, $A$ is {\\it \n$S$-odd\\\/} if $AS=-SA$.\n\nNotable example of ${\\mathbb Z_{2}}$-grading is given by \nHodge  ``star'' operator.  It is convenient here to present a \n``super'' construction for the star.\nThe Riemannian metric induces a bilinear form on the \nspace $\\S{n}=\\Pi T^*_x M$. For ${\\xi},{\\eta}$ denote \n${\\xi}\\cdot{\\eta}:=g_{ab}{\\xi}^a{\\eta}^b$.\nWe  define the operator $*$ by the following formula:\n{\n\\renewcommand{\\C}{\\mathop{\\rm const}\\nolimits}\n\\begin{equation} \\label{star}\n\t*u({\\xi}):=\\C\\int_{\\S{n}}\\frac{D{\\eta}}{\\sqrt{g}}\\;e^{-it\\,{\\xi}\\cdot{\\eta}}\\; \nu({\\eta}).\n\\end{equation}\nHere $t$ is an arbitrary parameter, and $\\C$ stands for a normalizing factor, \nwhich can be \nchosen by convenience.\n\n\\begin{lm} \\label{starkvadrat} In components, if $u=u_{a_1,\\dots,a_k}\\,{\\xi}^{a_1}\\dots \n{\\xi}^{a_k}$, where we assume the coefficients skew-symmetric,   \n\\begin{equation}\n(*u)_{a_1,\\dots,a_{n-k}}=\n\\C\\,t^{n-k}\\,i^{(n-k)^2+n(n-1)}\\frac{{\\sqrt{g}}}{(n-k)!}\\,u^{b_1,\\dots,b_{k}}\\;\n{\\varepsilon}_{b_1,\\dots,b_k,a_1,\\dots,a_{n-k}}.\n\\end{equation}\nHere the indices are raised with the help of the metric, and \n${\\varepsilon}_{b_1,\\dots,b_k,a_1,\\dots,a_{n-k}}$ is the standard Levi-Civita \nsymbol. So the operation $*$ defined by (\\ref{star}) differs from  \nusual only by some factor, which depends on the degree of the \nform.\nFor the inverse operator the following formula:\n\\begin{equation}\n\t*^{-1}=(\\C)^{-2}\\,(it)^{-n}(-1)^{n(n-1)\/2}\\;*\n\\end{equation}\nholds for any $n$.\n\\end{lm}\n\nConsider the case $n=2m$. Then  the operator $*$ is even (in the sense of natural \ngrading).\nSet $\\C:=t^{-m}$. By Lemma~\\ref{starkvadrat}, $*^2=1$. So we can consider $*$-grading. \n$*$-Even forms are what is called ``self-dual'',  and $*$-odd are \n``anti self-dual''.\n}\n\nTo each choice of ${\\mathbb Z_{2}}$-grading corresponds the {\\it trace}, defined \nby the formula:\n\\begin{equation}\n\t\\mathop{\\rm tr}\\nolimits_S A:=\\mathop{\\rm tr}\\nolimits A_{00}-A_{11}\n\\end{equation}\nfor an $S$-even operator $A$, where the block decomposition \ncorresponds to the chosen grading. \nUp to a factor, $\\mathop{\\rm tr}\\nolimits_S$ is specified by the property that it \nannihilates ``$S$-commutators'' (commutators of operators w.r.t. the \nchosen ${\\mathbb Z_{2}}$-grading). \n\nLet us consider the traces corresponding to \n parity, trivial grading, and $*$-grading (the last one only for \neven-dimensional case). Denote them by $\\mathop{\\rm str}\\nolimits$, $\\mathop{\\rm tr}\\nolimits_1$ and $\\mathop{\\rm tr}\\nolimits_*$ \nrespectively. Obviously, $\\mathop{\\rm tr}\\nolimits_{S} A= \\mathop{\\rm tr}\\nolimits_{1} (AS)= \\mathop{\\rm tr}\\nolimits_{1} \n(SA)$ for any $S$. \n\n\\begin{thm} \\label{loctrace}\nFor any operator $A$, \n\\begin{equation}\\label{str}\n\t\\mathop{\\rm str}\\nolimits A= (-1)^{\\tilde A n}\\int\\limits_{\\S{n}}\\frac{D{\\xi}}{\\sqrt \ng}\\; k_A({\\xi},{\\xi})\n\t=(-i\\hbar)^n\\int\\limits_{\\S{2n}}\\!\\! D({\\xi},\\t)\\; \\sigma A({\\xi},\\t)\n\\end{equation}\n(independent of $r$),\n\\begin{equation}\\label{tr1}\n\t\\mathop{\\rm tr}\\nolimits_1 A =\n\t(-1)^{n\\tilde A}\\int\\limits_{\\S{n}}\\frac{D{\\xi}}{\\sqrt g}\\; k_A(-{\\xi},{\\xi})\n\t=(-i\\hbar)^n\\int\\limits_{\\S{2n}}D({\\xi},\\t) \\;\\;e^{-\\frac{2i}{\\hbar}{\\xi}\\t}\\;\\sigma \nA((2r-1){\\xi},\\t),\n\\end{equation}\nand\n\\begin{equation}\\label{trstar}\n\t\\mathop{\\rm tr}\\nolimits_* A=\n\tt^{-m}\\!\\!\\!\\int\\limits_{\\S{2m}\\times\\S{2m}}\\frac{D{\\xi} D{\\eta}}{g}\\;\n\te^{-it\\,{\\xi}\\cdot{\\eta}}\\; k_A({\\xi},{\\eta})\n\t=t^{-m}\\int\\limits_{\\S{2m}}\\frac{D{\\xi}}{\\sqrt g}\\; \\sigma \n\tA({\\xi},t\\hbar\\,{\\xi}_{\\#})\n\\end{equation}\n(independent of $r$), where ${\\xi}_{\\#a}=g_{ab} {\\xi}^{b}$.     \nThe last formula works for $n=2m$.\n\\end{thm}\n\\begin{proof}\nLet us prove the first equality in~(\\ref{str}). Consider \n$[A,B]=AB-(-1)^{\\tilde A\\tilde B} BA$. Applying Lemma~\\ref{kerprod} \nand restricting the kernel to to the diagonal, one can deduce\n$$\nk_{[A,B]}({\\xi},{\\xi})=\\int\\frac{D{\\xi}'}{\\sqrt{g}}\\,(-1)^{n\\tilde A}\\;(k_{A}({\\xi},{\\xi}')\\,\nk_{B}({\\xi}',{\\xi}) - (-1)^{n}\\,(k_{A}({\\xi}',{\\xi})\\, k_{B}({\\xi},{\\xi}')),\n$$\nwhich obviously gives zero after the integration w.r.t. ${\\xi}$. Thus the \nright hand side of our formula annihilates commutators, and hence it is \nproportional to the trace $\\mathop{\\rm str}\\nolimits$. To obtain the normalization factor, it \nsuffices to check some particular operator. Consider \n$A: u\\mapsto u(0)$. For it, $\\mathop{\\rm str}\\nolimits A =1$ and\n$k_{A}({\\xi},{\\eta})=\\sqrt{g}\\,{\\eta}^{n}\\ldots{\\eta}^{1}$, so \n$$\n\\int\\frac{D{\\xi}}{\\sqrt{g}}\\;k_{A}({\\xi},{\\xi})=\\int \nD{\\xi}\\;{\\xi}^{n}\\ldots{\\xi}^{1}=1=\\mathop{\\rm str}\\nolimits A.\n$$\nThus the normalization factor \nactually equals $1$. The expression for the trace via the symbol follows from the \nequality\n\\begin{equation}\nk_{\\hat f}({\\xi},{\\xi})=(-1)^{n\\tilde f}{(-i\\hbar)^n}\\int_{\\S{n}} \\sqrt{g}\\,\nD\\t\\; f({\\xi}, \\t)\n\\end{equation}\n(see ~(\\ref{kerhatf})). So the formula~(\\ref{str})  is completely \nproved.  To prove~(\\ref{tr1}), notice that $\\mathop{\\rm tr}\\nolimits_{1}A=\\mathop{\\rm str}\\nolimits ((-1)^{P}A)= \n\\mathop{\\rm str}\\nolimits (A(-1)^{P})$. By Lemmas~\\ref{kerprod} and \\ref{kerparity},\n$$\nk_{(-1)^{P}A}({\\xi},{\\eta})=\\int \\frac{D{\\xi}'}{\\sqrt{g}} \n\\,\\sqrt{g}\\delta({\\xi}+{\\xi}')\\, k_{A}({\\xi}',{\\eta}) = k_{A}(-{\\xi},{\\eta}). \n$$\nApplying the formula for $\\mathop{\\rm str}\\nolimits$, we obtain the first equality in \n~(\\ref{tr1}). Expressing $k_{A}(-{\\xi},{\\eta})$ via the symbol completes the \nproof of~(\\ref{tr1}). Finally, to obtain~(\\ref{trstar}), we \napply~(\\ref{tr1}) to the operator $*A$. From Lemma~\\ref{kerprod} and \nthe formula~(\\ref{star}), we obtain \n$$\nk_{*A}(-{\\xi},{\\xi}) = \n\\int\\frac{D{\\xi}'}{\\sqrt{g}}\\,t^{-m}\\;e^{it\\,{\\xi}\\cdot{\\xi}'}\\;k_{A}({\\xi}',{\\xi}). \n$$\nThus, \n\\begin{multline*}\n\\mathop{\\rm tr}\\nolimits_{1}(*A)=\\int\\frac{D{\\xi}\\,D{\\eta}}{g}\\,t^{-m}\\;e^{-it{\\xi}\\cdot{\\eta}}\\;k_{A}({\\xi},{\\eta}) \n=\\\\\n\\int\\frac{D{\\xi} \\,D{\\eta}}{g}\\,t^{-m}\\;e^{-it{\\xi}\\cdot{\\eta}}\\;\n\\int(-i\\hbar)^{2m}\\,\\sqrt{g}\\,D\\t\\;e^{\\frac{i}{\\hbar}({\\xi}-{\\eta})\\t}\\;\nf((1-r){\\xi} + r{\\eta}, \\t).\n\\end{multline*}\nSubstitute ${\\xi}={\\varepsilon}-r{\\zeta}$, ${\\eta}={\\varepsilon} + (1-r){\\zeta}$. The Berezinian of this \nchange of variables is unity. Now, $(1-r){\\xi} + r{\\eta}={\\varepsilon}$, ${\\xi}-{\\eta}=-{\\zeta}$, \nand the argument of the first exponential becomes $-it({\\varepsilon}-r{\\zeta})\\cdot \n({\\varepsilon} + (1-r){\\zeta})=-it({\\varepsilon}\\cdot(1-r){\\zeta} - r{\\zeta}\\cdot{\\varepsilon})= it{\\zeta}\\cdot{\\varepsilon}$. So,\n\\begin{multline*}\n\\mathop{\\rm tr}\\nolimits_{*}A=\\int\\frac{D{\\varepsilon}\\,D\\t\\,D{\\zeta}}{\\sqrt{g}}\\,t^{-m}(-i\\hbar)^{2m}\\;\ne^{it{\\zeta}\\cdot{\\varepsilon} -\\frac{i}{\\hbar}{\\zeta}\\t}\\;f({\\varepsilon},\\t)=\\\\\n\\int\\frac{D{\\varepsilon}\\,D\\t\\,D{\\zeta}}{\\sqrt{g}}\\,t^{-m}(-i\\hbar)^{2m}\\;\ne^{-\\frac{i}{\\hbar}{\\zeta}(\\t-t\\hbar\\,{\\varepsilon}_{\\#})}\\;f({\\varepsilon},\\t)=\\\\\n\\int\\frac{D{\\varepsilon}\\,D\\t}{\\sqrt{g}}\\,t^{-m}\\;\\delta(\\t-t\\hbar\\,{\\varepsilon}_{\\#})\\;f({\\varepsilon},\\t)=\n\\int\\frac{D{\\varepsilon}}{\\sqrt{g}}\\,t^{-m}\\;f({\\varepsilon}, t\\hbar\\,{\\varepsilon}_{\\#}),\n\\end{multline*}\nand the theorem is completely proved.\n\\end{proof}\n\n\n\\begin{rem} For $r=1\/2$ (the Weyl quantization), \nit follows from the formula~(\\ref{tr1}) that $\\mathop{\\rm tr}\\nolimits_{1} A=2^{n}\\,(\\sigma A) (0,0)$.\n\\end{rem}\n\n\n\\subsection{Global quantization}\nNow we can combine the quantization built in Section~\\ref{QS} with the \nabove ``fiberwise'' quantization.  \n\nWe describe forms on $M$ as \nfunctions on the supermanifold $\\hat M=\\Pi TM$. Locally they look as \nfunctions $u=u(x,{\\xi})$, where ${\\xi}^{a}$ are elements of a coframe at \n$x$. For the coordinate coframe, ${\\xi}^{a}=dx^{a}$. Operators on forms can \nbe described by  ``Schwartz  kernels'' of the appearance $K(x,{\\xi},y,{\\eta})$:\n\\begin{equation}\n(Au)(x,{\\xi}) = (-1)^{n\\tilde A}\\int_{\\hat M} D(y,{\\eta})\\;K(x,{\\xi},y,{\\eta})\\;u(y,{\\eta}),\n\\end{equation}\nfor ${\\xi}^{a}=dx^{a}$. In the\ncoordinate-free language, the kernel is a generalized form (de Rham's \ncurrent) on $M\\times M$, and $Au=\\pi_{2*} (K\\pi_{1}^{*}u)$, where \n$\\pi_{1},\\pi_{2}:M\\times M \\to M$ are projections. \n\n\n\n\n\nAfter certain simplifications, we \nobtain the following formulas.  Below the letter $T$ \nstands for the parallel transport along geodesics, w.r.t. \nthe Levi-Civita connection, in the bundles $T^{*}M$, $\\Pi TM$ and \n$T^{*}M\\Pi$.\n\n\n{\\sl For quantization.} Suppose we have a function $f=f(x,p,{\\xi},\\t)$.  \nGeometrically, this is a section of the pull-back \n$\\pi^*(\\Lambda(TM\\oplus T^*M))$, by $\\pi: T^{*}M\\to M$,   or a function \non the supermanifold $T^*M\\oplus\\Pi TM\\oplus T^*M\\Pi)$. Denote this \nsupermanifold by $N$ (this is a  bundle over $M$).\nTo this function we assign the following operator, acting on $\\Omega \n(M)$:\n\\begin{multline}\n\t(\\hat f u)(x,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int_N \nD(y,q,{\\eta},\\t)\\;\\a(x,y)\\mu(x,y)\\;e^{\\frac{i}{\\hbar}\\left(\\exp_y^{-1}x\\cdot \nq + (T(y,x){\\xi}-{\\eta})\\t\\right)}\\;\\\\\nf( {\\zs}, T({\\zs},y)q, \n(1-r)\\,T({\\zs},x){\\xi}  + r\\,T({\\zs},y){\\eta},\\\\  \nT({\\zs},y)\\t )\\;\\;u(y,{\\eta}).\n\\end{multline}\nEquivalent definition:\n\\begin{multline}\\label{totquant1}\n\t(\\hat f u)(x,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int D(v,p,{\\varepsilon},\\t)\\a(x,\\exp_x \nv)\\;e^{-\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)}\\;\\\\\nf(\\exp_x sv , T(\\exp_x sv,x)p,\n T(\\exp_x sv,x)({\\xi}  + r{\\varepsilon}), T(\\exp_x sv,x)\\t)\\\\ \\;u(\\exp_x \nv,T(\\exp_x v,x)({\\xi}+{\\varepsilon})).\n\\end{multline}\nHere integration is over the fiber $T_xM\\oplus T_x^*M\\oplus\\Pi \nT_xM\\oplus T_x^*M\\Pi)$.\n\n{\\sl For symbols.} Let $A$ be an operator on differential forms, with \nthe Schwartz kernel $K(x,{\\xi},y,{\\eta})$. \n(Here ${\\xi}^{a}$ are elements of a coframe at $x$, and \n${\\eta}^{a}$ are elements of a coframe at $y$.)\nThen its {\\it symbol} $\\sigma \nA=(\\sigma A)\\,(x,p,{\\xi},\\t)$ is defined by the formula\n\\begin{multline} \\label{fsymbol}\n\t(\\sigma A)(x,p,{\\xi},\\t):=\\int_{T_x M\\times \\Pi T_x \nM}D(v,{\\varepsilon})\\;\\a(\\exp_x(-sv),\\exp_x (1-s)v)\\; \\rho_s(x,v) \\\\\n\\;e^{\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)} \n\\;\\;\nK(\\exp_x (-sv),T(\\exp_x (-sv),x)({\\xi}-r{\\varepsilon}), \\exp_x (1-s)v, \n\\\\\nT(\\exp_x (1-s)v,x)({\\xi}+(1-r){\\varepsilon})).\n\\end{multline}\n\nThese formulas can be obtained from~(\\ref{quant}, \\ref{symbol}) \nand~(\\ref{squant}, \\ref{ssymb}) by more or less direct calculation. At \na certain step it is necessary to use Lemma~\\ref{spinrep} to  change \nthe points at which reference frames are taken, and to change \nvariables in the integrals.\n\n\nOur constructions depend on two real parameters $s,r\\in [0,1]$. \nNote also that without any difficulties the definitions of \nquantization and symbols can be extended from ``scalar-valued'' forms \nto forms taking values \nin an arbitrary vector bundle with a connection. Quite helpful for \npractical calculations is the appearance of the formula~(\\ref{totquant1}) \nin normal coordinates centered at $x$ (so $x^{a}=0$, \n$(\\exp_{x}v)^{a}=v^{a}$):\n\\begin{multline}\\label{totquantnorm}\n\t(\\hat f u)(0,{\\xi}):=\\frac{1}{(2\\pi i)^n}\\int D(v,p,{\\varepsilon},\\t)\\a(x,v)\\;e^{-\\frac{i}{\\hbar}(vp+{\\varepsilon}\\t)}\\;\\\\\nf(sv , T(sv,0) p,\n T(sv,0)({\\xi}  + r{\\varepsilon}), T(sv,0)\\t) \\;u(v,T(v,0)({\\xi}+{\\varepsilon})).\n\\end{multline}\n\n\n\nAs it was said, the symbols in our refined symbol calculus are \nfunctions on the supermanifold $N=T^{*}M\\oplus \\Pi TM\\oplus T^{*}M\\Pi$. \nRoughly speaking, they are tensor products of forms and multivectors on $M$, depending \nalso on a point $p$ in the cotangent space. Luckily, they can be given \na nicer description. Using the Levi-Civita connection on $M$, the supermanifold $N$ \ncan be identified with $\\widehat{T^{*}M}$. That means that our \nsymbols can be considered simply as forms on $T^{*}M$. For clarity, let us \nwrite down the change of coordinates on $N$ (assuming that we use \ncoordinate frames):\n\\begin{align}\n&\\left\\{ \n\\begin{aligned}\nx^{a}&=x^{a}(x')\\\\\np_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,p_{a'}\\\\\ndx^{a}&=dx^{a'}\\,\\der{x^{a}}{x^{a'}}(x')\\\\\n\\t_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,\\t_{a'}\n\\end{aligned}\n\\right.\\\\\n\\intertext{Compare it with  $\\widehat{T^{*}M}$:}\n&\\left\\{ \n\\begin{aligned}\nx^{a}&=x^{a}(x')\\\\\np_{a}&=\\der{x^{a'}}{x^{a}}(x(x'))\\,p_{a'}\\\\\ndx^{a}&=dx^{a'}\\,\\der{x^{a}}{x^{a'}}(x')\\\\\ndp_{a}&=d\\left(\\der{x^{a'}}{x^{a}}(x(x'))\\right)\\,p_{a'} + \\der{x^{a'}}{x^{a}}(x(x'))\\,dp_{a'}\n\\end{aligned}\n\\right.\n\\end{align}\nThe tempting idea to identify $\\t_{a}$ with $dp_{a}$ fails because of \nthe additional term in the transformation law for $dp_{a}$. However, \nusing the connection, we can identify $\\t_{a}$ with $\\nabla p_{a}$, \nwhere\n\\begin{equation}\n\\nabla p_{a}=dp_{a} - dx^{b}\\mathop{\\rm \\Gamma}\\nolimits_{ab}^{c}p_{c}.\n\\end{equation}\nNotice that the change of variables from $(x,p,dx,dp)$ to \n$(x,p,dx,\\nabla p)$ has the unit Berezinian, hence all our integrals \nover $(x,p,{\\xi},\\t)$ can be rewritten as integrals over $(x,p,dx,dp)$, \ni.e., as integrals of differential forms over the manifold $T^{*}M$.\n\n\\begin{rem} \\label{getz} In the paper~\\cite{Ge} Getzler proposed a \ncomplete symbol calculus for spinor fields, combining a symbol \ncalculus on manifolds with the ``Weyl'' isomorphism $\\L(V)\\to C(V)$ \n(where $\\L(V)$ and $C(V)$ are respectively exterior and Clifford \nalgebra of a Euclidean space $V$).  The resulting symbols were {\\sl \nhorizontal\\\/} forms on $T^{*}M$, i.e., functions  $f=f(x,p,{\\xi})$ in our notation. \nThe key element of that construction  was some nonobvious\nfiltration introduced into the algebra of symbols (by the total degree \nin $p_{a}$ and ${\\xi}^{b}$, in our notation). \nAs we showed \nin~\\cite{AS}, in the quantization language this was equivalent to a \nvery peculiar convention \n$\\hbar_{\\text{spinor}}=\\hbar_{\\text{usual}}^{2}$ (in~\\cite{Ge} \nno Planck constant was used, so this fact was not explicit there). \nThat strange convention was nevertheless essential for calculating the \nDirac index, see ~\\cite{AS}. Comparing with the complete symbol calculus \nconstructed it this section, we see that now {\\sl all\\\/} forms \non $T^{*}M$ appear as symbols, and  odd  and  even variables are on equal \nfooting (no discrimination w.r.t.  $\\hbar$).\n\\end{rem}\n\n\\subsection{Examples}\n\nThe exterior differential and the codifferential (or divergence), \nwhich is defined on Riemannian manifold, are given by the \nformulas:\n\\begin{align}\n\td&=dx^a\\der{}{x^a}={\\xi}^a\\,\\nabla_a  \\label{d}\\\\\n\t\\delta &=g^{ab}\\der{}{{\\xi}^a}\\,\\nabla_b.\\label{delta}\n\\end{align}\n(Notice that the second expression for $d$ in ~(\\ref{d}) is valid for arbitrary \nframe.) The covariant derivative of (inhomogeneous) form is\n\\begin{equation}\\label{nabla}\n\\nabla_{a}=\\der{}{x^{a}} - \\mathop{\\rm \\Gamma}\\nolimits_{ak}{}^{l}{\\xi}^{k}\\der{}{{\\xi}^{l}},\n\\end{equation}\nwhere $\\mathop{\\rm \\Gamma}\\nolimits_{ak}{}^{l}$ is the Christoffel symbol.\n\n\\begin{rem} In components: if $u=u_{a_1,\\dots,a_k}dx^{a_1}\\dots \ndx^{a_k}$, then\n\\begin{equation}\n\t(\\delta u)_{a_1,\\dots,a_{k-1}}=k\\,u_{aa_1,\\dots,a_{k-1}}{}^{;a}.\n\\end{equation}\n\\end{rem}\n\nDirect application of the formula~(\\ref{totquantnorm}) gives the \nfollowing results for the quantization of symbols ${\\xi}\\mathrm p={\\xi}^{a} \np_{a}$ and $\\t\\cdot\\mathrm p=g^{ab}\\t_{a} p_{b}$:\n\n\\begin{ex}\n\t\\begin{equation}\n\t\t\\widehat{{\\xi}\\mathrm p}=-i\\hbar\\,d\n\t\\end{equation}\n\\end{ex}\n\n\\begin{ex}\n\t\\begin{equation}\n\t\t\\widehat{\\t\\cdot\\mathrm p}=-\\hbar^2\\,\\delta\n\t\\end{equation}\n\\end{ex}\n\n\\subsection{Traces}\n\n\\begin{thm}\n For any operator $A$ acting on $\\Omega (M)$,\n\\begin{align}\n\\mathop{\\rm Str}\\nolimits A &= \\frac{1}{(2\\pi i)^n}\\int_{N}D(x,p,{\\xi},\\t)\\;\\sigma A(x,p,{\\xi},\\t) = \n\\frac{i^{n}}{(2\\pi)^{n}}\\int_{T^{*}M}\\;\\sigma A   \\label{supertrace}   \\\\\n\\intertext{(integral of differential form over $T^{*}M$), independent of \n$s$ and $r$,}\n\\mathop{\\rm Tr}\\nolimits_1 A &=\\frac{1}{(2\\pi i)^n}\\int_{N}D(x,p,{\\xi},\\t)\\;\\;e^{-\\frac{2i}{\\hbar}{\\xi}\\t}\\sigma \nA(x,p,(2r-1){\\xi},\\t),   \\label{trivtrace\n\\end{align}\nindependent of $s$.\nFor $n=2m$, and for any $A$,\n\\begin{equation}\\label{startrace}\n\\mathop{\\rm Tr}\\nolimits_*A=\\frac{1}{(2\\pi)^{2m}}\\int \\frac{D(x,p,{\\xi})}{\\sqrt{g}}\\;\\sigma \nA(x,p,{\\xi},\\hbar^{-1}\\,{\\xi}_{\\#}),\n\\end{equation}\nindependent of $s$ and $r$. Here $({\\xi}_{\\#})_a=g_{ab}{\\xi}^b$. (We set in \nthe last formula $t:=\\hbar^{-2}$.)\n\\end{thm}\n\\begin{proof} Immediately follows from Theorem~\\ref{trace} and  \nTheorem~\\ref{loctrace}.\\end{proof}\n\nNotice that these formulas contain no powers of $\\hbar$ in front of the \nintegrals, compared to~(\\ref{tra}) and ~(\\ref{str}, \\ref{tr1}, \n\\ref{trstar}). This comes naturally by cancellation of the powers of $\\hbar$  \n from the ``even''\nand the ``odd'' parts of our quantization. \n\n\\section{Natural Laplacians and their symbols}\n\n\\subsection{Weitzenb\\\"ock formula}\n\nFor any vector bundle with a connection over Riemannian $M$ there is \nan operator\n\\begin{equation}\n\t\\Delta=g^{ab}(\\nabla_a\\nabla_b-\\Gamma_{ab}{}^c\\nabla_c)\n\\end{equation}\n(here $\\nabla$ stands for the covariant derivative of the sections of \nthe bundle). This Laplacian is called  Laplace-Beltrami operator \nor Bochner Laplacian. In the case when the bundle is the exterior \nbundle $\\Lambda M =\\Lambda (T^{*}M)$, there is another natural \nconstruction of a Laplacian operator, namely the Hodge Laplacian \n$\\square=(d+\\delta)^2=d\\delta +\\delta d$. The relation between these \ntwo operators is  given by the ``Weitzenb\\\"ock formula''~\\cite[p.~397]{W} (see also \n~\\cite{Rh}). (This name \n is used, in general, for formulas relating any natural Laplacian \noperators in arbitrary bundle; by a ``Laplacian \noperator'' is meant an operator generalizing the usual Laplacian on \nfunctions in the Euclidean space, in the sense that in components  \nit's  the usual Laplacian applied  componentwise, plus \nsome lower-order terms.) We shall use it in the following form.\n\n\\begin{thm}\\label{Weitz}\n\\begin{align}\n\t\\square=(d+\\delta)^2&=\\Delta + \\mathop{\\rm Ric}\\nolimits_a{}^b\\,dx^a\\der{}{dx^b} + \nR_a{}^k{}_b{}^l\\,dx^adx^b\\der{}{dx^k}\\der{}{dx^l}  \\label{weitz1}\\\\\n&=\\Delta + \\mathop{\\rm Ric}\\nolimits_a{}^b\\,dx^a\\der{}{dx^b} + \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,dx^a dx^b\\der{}{dx^k}\\der{}{dx^l}, \\label{weitz2}\n\\end{align}\nwhere $R_{ab}{}^{kl}$ is the Riemann tensor, $\\mathop{\\rm Ric}\\nolimits_a{}^b=R_{ka}{}^{kb}$ \nis the Ricci tensor, indices are raised and lowered by the Riemannian \nmetric.\n\\end{thm}\n\\begin{proof} By the formulas~(\\ref{d}) and (\\ref{delta}),  $d+\\delta={\\hat \\gamma^{a}}\\, \\nabla_{a}$, \nwhere we define ${\\hat \\gamma^{a}}:={\\xi}^{a}+g^{ak}\\lder{}{{\\xi}^{k}}$. We need to find \n$(d+\\delta)^{2}=(1\/2)\\,[d+\\delta, d+\\delta]$. Directly:\n\\begin{multline}\n[d+\\delta, d+\\delta]=[{\\hat \\gamma^{a}}\\, \\nabla_{a},{\\hat \\gamma^{b}}\\, \\nabla_{b}]\n\\\\={\\hat \\gamma^{a}}\\,[\\nabla_{a},{\\hat \\gamma^{a}}]\\,\\nabla_{b} + [{\\hat \\gamma^{a}},{\\hat \\gamma^{b}}]\\,\\nabla_{a}\\nabla_{b} \n-{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}\\,[\\nabla_{a,}\\nabla_{b}]-{\\hat \\gamma^{b}}\\,[{\\hat \\gamma^{a}},\\nabla_{b}]\\,\\nabla_{a}.\n\\end{multline}\nObserve the following facts. First, $[{\\hat \\gamma^{a}},{\\hat \\gamma^{b}}]=2g^{ab}$. Second, \n$[\\nabla_{a,}\\nabla_{b}]=-R_{abk}{}^{l}{\\xi}^{k}\\lder{}{{\\xi}^{l}}$ \n(cp.(\\ref{nabla})). Third, in \nthe frame associated with normal coordinates centered at a given \npoint $x_{0}$, the operators ${\\hat \\gamma^{a}}$ and $\\nabla_{b}$ commute at \n$x_{0}$ (since the Christoffel symbols and partial derivatives of the \nmetric vanish at $x_{0}$). Let's calculate in these coordinates. \nThus, at  $x_{0}$,\n$$\n[d+\\delta, d+\\delta]=g^{ab}\\,\\nabla_{a}\\nabla_{b}+\n{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}\\,R_{abp}{}^{q}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n=\n2\\Delta + {\\hat \\gamma^{a}}{\\hat \\gamma^{b}}\\,R_{abp}{}^{q}{\\xi}^{p}\\der{}{{\\xi}^{q}}.\n$$\nSince (by direct computation)\n$$\n{\\hat \\gamma^{b}}{\\hat \\gamma^{a}}={\\xi}^{b}{\\xi}^{a}+ g^{ba} +({\\xi}^{b}g^{ak}-{\\xi}^{a}g^{bk})\\der{}{{\\xi}^{k}} + \ng^{bl}g^{ak}\\der{}{{\\xi}^{l}}\\der{}{{\\xi}^{k}},\n$$\nthen, using the properties of the Riemann tensor,\n\\begin{multline*}\n[d+\\delta, d+\\delta]=2\\Delta - 2 {\\xi}^{a}g^{bk}\\,R_{abp}{}^{q}\n\\der{}{{\\xi}^{k}}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n=2\\Delta -2 R_{a}{}^{k}{}_{p}{}^{q}\\,{\\xi}^{a}\\der{}{{\\xi}^{k}}{\\xi}^{p}\\der{}{{\\xi}^{q}}\n\\\\=2\\Delta + \n2 R_{a}{}^{k}{}_{p}{}^{q}\\,({\\xi}^{a}{\\xi}^{p}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{q}}\n-\\delta_{k}{}^{p}{\\xi}^{a}\\der{}{{\\xi}^{q}})\n\\\\=2\\left(\n\\Delta + R_{a}{}^{k}{}_{b}{}^{l}\\,{\\xi}^{a}{\\xi}^{b}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{l}}\n+ \\mathop{\\rm Ric}\\nolimits_{a}{}^{b}{\\xi}^{a}\\der{}{{\\xi}^{b}}\n\\right),\n\\end{multline*}\nand the formula~(\\ref{weitz1}) is proved. To deduce~(\\ref{weitz2}), one should \nraise all indices in $R_{a}{}^{k}{}_{b}{}^{l}$ (respectively lowering \nthem for ${\\xi}^{a}{\\xi}^{b}$), alternate in $a,b$ (the factor $1\/2$ \nappears) and apply the Ricci identity $R^{akbl}+R^{bakl}+ \n\\underbrace{R^{kbal}}_{-R^{bkal}}=0$.\n\\end{proof}\n\n\\begin{rem} Presentation of $d+\\delta$ as a ``Dirac'' operator which we \nused in the proof of Theorem~\\ref{Weitz} corresponds to the action on forms \nof the Clifford algebra $C(n)$,  induced by the Riemannian metric. \nOperators ${\\hat \\gamma^{a}}$ {\\sl do not\\\/} generate all endomorphisms of $\\L$. At the \nsame time, the quantization that we constructed in the previous subsection\nactually realizes forms as spinors for larger Clifford algebra \n$C(n,n)\\cong\\mathop{\\rm End}\\nolimits\\L$, which is independent of metric. So there are two \ndifferent Clifford module structures on the space $\\L$, for two \ndifferent Clifford algebras, which should not lead to a confusion.\n\\end{rem}\n\n\n\\begin{rem} Formulas~(\\ref{weitz1}, \\ref{weitz2}) can be  \nalso rewritten as $\\square = \\Delta + R$, with the operator \n$R=R_a{}^k{}_b{}^l{\\xi}^{a}\\der{}{{\\xi}^{k}}{\\xi}^{b}\\der{}{{\\xi}^{l}}$, \nsee~\\cite{Ros}. In such form the Weitzenb\\\"ock formula was used \nin~\\cite{A}. The ``normal'' form~(\\ref{weitz2}) is most \nsuitable for our purposes.\n\n\\end{rem}\n\\subsection{The symbol of the Hodge Laplacian}\n\n\\begin{thm} For any $s\\in [0,1]$,\n\\begin{multline} \\label{symlap}\n\t(\\sigma(\\square))\\,(x,p,{\\xi},\\t)=\\\\\n=-\\hbar^{-2}\\,\\left(\\mathrm p^2 + \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l\\right) + \ni\\hbar^{-1}\\,(1-2r)\\, \\mathop{\\rm Ric}\\nolimits_a{}^b\\,{\\xi}^a\\t_b + r(1-r) R\n\\end{multline}\n\\end{thm}\n\\begin{proof}\nWe apply  the\nWeitzenb\\\"ock formula~(\\ref{weitz2}).\nExample~\\ref{lapsymb} provides $\\sigma(\\Delta) = -\\hbar^{-2}\\, \\mathrm \np^2$, independent of $s$. All we need, is to calculate \nthe symbols of the remaining terms. \nDenote $A:=\\mathop{\\rm Ric}\\nolimits_a{}^b\\,{\\xi}^a\\lder{}{{\\xi}^b}$,\n$B:= R_{ab}{}^{kl}\\,{\\xi}^a {\\xi}^b\\lder{}{{\\xi}^k}\\lder{}{{\\xi}^l}$. For \nsimplicity we shall write below $R_a{}^b$ instead of $\\mathop{\\rm Ric}\\nolimits_a{}^b$. \nFirst we find the kernels:\n\\begin{align*}\nk_{A}({\\xi},{\\eta})&=R_a{}^b{\\xi}^{a}\\,\\der{}{{\\xi}^{b}}\\,\\delta({\\eta}-{\\xi})\\,\\sqrt{g},\\\\\nk_{B}({\\xi},{\\eta})&=R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\der{}{{\\xi}^{k}}\\der{}{{\\xi}^{l}}\\,\\delta({\\eta}-{\\xi})\\,\\sqrt{g}.\n\\end{align*}\nNow we change variables: ${\\xi}={\\zeta}-r{\\varepsilon}, {\\eta}={\\zeta}+(1-r){\\varepsilon}$, or ${\\varepsilon}={\\eta}-{\\xi}, \n{\\zeta}=(1-r){\\xi}+r{\\eta}$. As $\\lder{}{{\\xi}}=(1-r) \\lder{}{{\\zeta}} - \\lder{}{{\\varepsilon}}$, it follows \nthat\n\\begin{align}\nk_{A}({\\zeta}-r{\\varepsilon},{\\zeta}+(1-r){\\varepsilon})\n&=-R_a{}^b({\\zeta}^{a}-r{\\varepsilon}^{a})\\der{}{{\\varepsilon}^{b}}\\delta({\\varepsilon})\\,\\sqrt{g}, \n\\label{kA}\\\\\nk_{B}({\\zeta}-r{\\varepsilon},{\\zeta}+(1-r){\\varepsilon})\n&=R_{ab}{}^{kl}({\\zeta}^{a}-r{\\varepsilon}^{a})({\\zeta}^{b}-r{\\varepsilon}^{b})\\dder{}{{\\varepsilon}^{k}}{{\\varepsilon}^{l}}\n\\delta({\\varepsilon})\\,\\sqrt{g} \\label{kB}.\n\\end{align}\nSubstituting~(\\ref{kA}) and (\\ref{kB}) into~(\\ref{ssymb}) and integrating by \nparts, we obtain\n\\begin{multline*}\n(\\sigma A)({\\xi},\\t)=\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\der{}{{\\varepsilon}^{b}}\n\\left(\n-R_a{}^b\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})\n\\right)\n\\\\=\n\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\n\\left(\n-R_a{}^b\\,\\frac{i}{\\hbar}\\,\\t_{b}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})+\nR_a{}^b\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,r\\delta_{b}{}^{a}\n\\right) \n=\\frac{i}{\\hbar}\\,R_a{}^b\\,{\\xi}^{a}\\t_{b} + rR,\\\\\n\\shoveleft{\n(\\sigma B)({\\xi},\\t)=\n\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\dder{}{{\\varepsilon}^{k}}{{\\varepsilon}^{l}}\n\\left(\nR_{ab}{}^{kl}\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b})\n\\right)\n}\n\\\\= \\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\\der{}{{\\varepsilon}^{k}}\n\\left(\n\\frac{i}{\\hbar}\\,\\t_{l}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,R_{ab}{}^{kl}\\,\n({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b}) + \ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lb}{}^{kl}(-r)({\\xi}^{b}-r{\\varepsilon}^{b})\n\\right)\n\\\\=\\int D{\\varepsilon}\\,\\delta({\\varepsilon})\\,\n\\Bigl(\n-{\\Bigl(\\frac{i}{\\hbar}\\Bigr)}^{2}\\,\\t_{l}\\t_{k}\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,\nR_{ab}{}^{kl}\\,({\\xi}^{a}-r{\\varepsilon}^{a})({\\xi}^{b}-r{\\varepsilon}^{b}) - \n\\frac{i}{\\hbar}\\,\\t_{l}\n\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{kb}{}^{kl}\\cdot\n\\\\\n\\left.\n(-r)({\\xi}^{b}-r{\\varepsilon}^{b})+\n\\frac{i}{\\hbar}\\,\\t_{k}\n\\,e^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lb}{}^{kl}(-r)({\\xi}^{b}-r{\\varepsilon}^{b})+\ne^{\\frac{i}{\\hbar}{\\varepsilon}\\t}\\,2 R_{lk}{}^{kl}(-r)(-r)\n\\right)\n\\\\=\\Bigl(\\frac{i}{\\hbar}\\Bigr)^2 R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\n+\\frac{i}{\\hbar}\\left( -r2 R_{kb}{}^{kl}\\,{\\xi}^{b}\\t_{l} + r2 \n R_{lb}{}^{kl}\\,{\\xi}^{b}\\t_{k} \\right) + r^{2}\\,2R_{lk}{}^{kl}\n \\\\=\\Bigl(\\frac{i}{\\hbar}\\Bigr)^2 R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\n +\\frac{i}{\\hbar}(-4r)\\,R_{a}{}^{k}{\\xi}^{a}\\t_{k} - 2r^{2} R.\n\\end{multline*}\n(where we restored ${\\xi}$ in the argument).\nFinally we obtain\n\\begin{multline*}\n\\sigma\\square = \\sigma\\Delta + \\sigma A + \\frac{1}{2}\\sigma B\n\\\\=\n\\sigma\\Delta +i\\hbar^{-1}R_{a}{}^{b}{\\xi}^{a}\\t_{b} + rR +\n-\\hbar^{2}\\frac{1}{2} R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l} +\ni\\hbar^{-1}(-2r) R_{a}{}^{b}{\\xi}^{a}\\t_{b} -r^{2}R\n\\\\=\n-\\hbar^{2}\\left(\\mathrm p^{2} +  \n\\frac{1}{2} R_{ab}{}^{kl}\\,{\\xi}^{a}{\\xi}^{b}\\t_{k}\\t_{l}\\right)\n+i\\hbar^{-1}(1-2r) R_{a}{}^{b}{\\xi}^{a}\\t_{b}+r(1-r) R,\n\\end{multline*}\nwhich completes the proof.\n\\end{proof}\n\n\n\n\n\\section{Gauss-Bonnet-Chern Theorem}\n\nAs it is very well known, the index of any elliptic operator \n$A: \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0})\\to \\mathop{\\rm \\Gamma}\\nolimits(M,E_{1})$ on compact manifold $M$ can be \nexpressed as the supertrace of a suitable function of the corresponding ``Laplacian''\n$\\Delta_{A}=AA^{*}+A^{*}A : \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0}\\oplus E_{1}) \\to \\mathop{\\rm \\Gamma}\\nolimits(M, E_{0}\\oplus E_{1})$:\n\\begin{equation}\n\\mathop{\\rm index}\\nolimits A = \\mathop{\\rm str}\\nolimits \\varphi(\\Delta_{A}),\n\\end{equation}\nprovided the trace is absolutely convergent, and $\\varphi(0)=1$. In \nparticular, \n\\begin{equation}\n\\mathop{\\rm index}\\nolimits A = \\mathop{\\rm str}\\nolimits e^{t\\Delta_{A}},\n\\end{equation}\nfor suitable $t$. In this section we shall apply this formula \nto obtain the expression for the Euler characteristic of the \nRiemannian manifold $M$ via curvature, i.e., to deduce the \nGauss-Bonnet-Chern Theorem. Our main tool will be the formula for the \nsupertrace~(\\ref{supertrace}).\n\nConsider $D=d+\\delta: \\O^{ev}(M)\\to \\O^{od}(M)$. We have\n\\begin{equation}\\label{euler}\n\\chi(M)=\\mathop{\\rm index}\\nolimits D = \\mathop{\\rm str}\\nolimits e^{t\\square}\n\\end{equation}\nfor suitable $t$. Since\n\\begin{equation}\n\\sigma e^{t\\square}=e^{t\\sigma(\\square)}\\,(1+ O(\\hbar)) = \ne^{t(-\\hbar^{-2}(\\mathrm p^{2} +\\ldots)(1 + O(\\hbar))}\\,(1+ \nO(\\hbar))\n\\end{equation}\n(see ~(\\ref{symlap})), $t=\\hbar^{2}$ seems a good choice. So\n\\begin{equation}\n\\sigma (\\hbar^{2}\\square)= -\\mathrm p^2 - \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l + O(\\hbar),\n\\end{equation}\nand\n\\begin{equation}\n\\sigma e^{\\hbar^{2}\\square}=e^{\\sigma \n(\\hbar^{2}\\square)}\\,(1+O(\\hbar)) =e^{-\\mathrm p^2 - \n\\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l} + O(\\hbar).\n\\end{equation}\nPassing to the limit $\\hbar\\to 0$ (which is harmless since the \nsupertrace ~(\\ref{euler}) is independent of $\\hbar$), we obtain:\n\\begin{multline}\n\\chi(M)=\\mathop{\\rm index}\\nolimits D=\\mathop{\\rm str}\\nolimits e^{\\hbar^{2}\\square}=\n\\frac{1}{(2\\pi i)^n} \\int_{N} D(x,p,{\\xi},\\t)\\; \ne^{-\\mathrm p^2 - \\frac{1}{2}\\,R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b\\t_k\\t_l} \n\\\\=\\begin{cases}\n0, & n=2m+1\\\\\n{\\displaystyle \\frac{(\\sqrt{\\pi})^{2m}}{(2\\pi i)^{2m}}\\,\\int\\;D(x,{\\xi})\\;\n\\mathop{\\rm Pf}\\nolimits (R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b)}, & n=2m\n\\end{cases}\n\\\\=\n\\left\\{\n\\begin{array}{l}\n   0, \\qquad n=2m+1\\\\\n   {\\displaystyle (-1)^{m}\\int D(x,{\\xi})\\;\\mathop{\\rm Pf}\\nolimits \n   \\Bigl(%\n   \\frac{1}{2\\pi}\\underbrace{\\frac{1}{2}R_{ab}{}^{kl}\\,{\\xi}^a{\\xi}^b}%\n                  _{\\text{curvature $2$-form}\n   \\Bigr) { }}\n\\end{array}\n\\right.\n=\\begin{cases}\n0, \\qquad  n=2m+1\\\\\n{\\displaystyle\\int_{M} \\underbrace{\\mathop{\\rm Pf}\\nolimits \\left(\\frac{1}{2\\pi} \n\\frac{1}{2}R_{ab}{}^{kl}\\,dx^a\\,dx^b\\right)}_{\\text{Euler class of \n$TM$}} }.\n\\end{cases}\n\\end{multline}\nThis is the Gauss-Bonnet-Chern formula for the Euler \ncharacteristic.\n\nWe used the well-known formula for the \nPfaffian as a Gaussian integral:\n$$\n\\int_{\\S{n}}D\\t\\;e^{-\\frac{1}{2}Q^{ab}\\,\\t_{a}\\t_{b}}=\n\\begin{cases}\n0 & \\text{for $n=2m+1$}\\\\ \n\\mathop{\\rm Pf}\\nolimits Q & \\text{for $n=2m$}\n\\end{cases}\n$$\nwhere $Q^{ab}=-Q^{ba}$ (see, e.g.,~\\cite{GIT}).\n\n\\begin{rem} The fact that the formula for the trace~(\\ref{supertrace}) \ncontains no $\\hbar$  factors is crucial for our calculation. It \ncomes naturally in our symbol calculus. In the calculus for spinor \nfields~\\cite{AS} the similar cancellation of the powers of $\\hbar$ was \nachieved only by an artifial ``asymmetric'' convention for Planck constant.\n\\end{rem}\n\n\\section{Discussion}\nWith very little effort we were able to deduce the Gauss-Bonnet-Chern \nTheorem from the trace formula~(\\ref{supertrace}). Though we possess \nan analogous trace formula~(\\ref{startrace}) for the $*$-grading, more subtle \nanalysis of the symbol is needed in the case of the Hirzebruch \nsignature operator, because \nthis formula contains $\\hbar^{-1}$. \n\nWe can also discuss the results with the relation to  the  problem \nof quantization of differential algebras, in particular the algebras \nof forms. In a general setting (cp.~\\cite{wein1}), \nwe consider deformation of commutative \nmultiplication in an algebra $A$, of the form\n\\begin{equation}\nf\\mathbin{*_{\\hbar}} g =fg +(-i\\hbar)\\,\\{f,g\\} + \n(-i\\hbar)^{2}\\,\\mathop{B_{2}}\\nolimits (f,g) +\\ldots,\n\\end{equation}\nstarting from some Poisson bracket,\nand of the differential in $A$, of the form\n\\begin{equation} \\label{dh}\n{d_{\\hbar}} f= df + (-i\\hbar)\\,d_{1}f + \n(-i\\hbar)^{2}\\,d_{2}f+\\ldots,\n\\end{equation}\nall this being considered up to  ``gauge equivalences''\n$f\\mapsto f+(-i\\hbar)\\,a_{1}(f) +   \\ldots$. (Here $d_{k}$ and $a_{k}$ \nare differential operators on $A$, and $B_{k}$ are bidifferential \noperators.)\nIn the first order the derivation property for ${d_{\\hbar}}$ is equivalent to\n\\begin{equation}\\label{der}\nd\\{f,g\\}-\\{df,g\\} -(-1)^{\\tilde f}\\{f,dg\\}\n=-\\Bigl(d_{1}(fg)-d_{1}f\\,g -(-1)^{\\tilde f}f\\,d_{1}g\\Bigr)\n\\end{equation}\nThe obstruction to killing $d_{1}$ or, more generally, the first \nnonvanishing term $d_{n}$ in the expansion~(\\ref{dh}) , $n\\geq 1$,\n by a gauge transformation is the \ncohomology class $[d_{n}]$ in the complex  of operators on $A$ with \nthe differential $\\mathop{\\rm ad}\\nolimits d=[d,\\phantom{d}]$. \n\nConsider the case of forms. We hope \nto analyze the general picture elsewhere, and here we shall make just a few \nremarks.  First, there is a question whether the deformation should \nrespect the $\\mathbb Z$-grading or just ${\\mathbb Z_{2}}$-grading (parity). For the \nformer option, the induced Poisson bracket should be also $\\mathbb \nZ$-graded, for example the bracket of  $1$-forms should \nbe $2$-form (and not a function, for example). This is highly \nunlikely. Poisson structure obtained above has the canonical brackets\n\\begin{equation}\\label{brac}\n\\{p_{a},x^{b}\\}=\\delta_{a}{}^{b}, \\quad \\{\\t_{a},{\\xi}^{b}\\}=\\delta_{a}{}^{b}\n\\end{equation} \nin the coordinates $(x,p,{\\xi}=dx,\\t)$ and  is more complicated in \nthe coordinates  $(x,p,dx,dp)$. This structure  respects parity but by no means\n$\\mathbb Z$-grading. The second question concerns the relation between \nthe bracket and $d$. An easy computation produces the following formulas \nfor the action of $d$:\n\\begin{align*}\ndx^{a}&={\\xi}^{a}\\\\\ndp_{a}&=\\mathop{\\rm \\Gamma}\\nolimits_{ba}^{c}{\\xi}^{b}p_{c} + \\t_{a} \\\\\nd{\\xi}^{a}&=0\\\\\nd\\t_{a}&=\\mathop{\\rm \\Gamma}\\nolimits_{ba}^{c}{\\xi}^{b}\\t_{c} -\\frac{1}{2}R_{kla}{}^{c}{\\xi}^{k}{\\xi}^{l}p_{c}, \n\\end{align*}\nwhich resemble  Cartan's equations of structure.\nDirect check shows that $d$ does not respect the  brackets~(\\ref{brac}). (We \nomit the explicit formulas for $d\\{f,g\\}-\\{df,g\\} -(-1)^{\\tilde \nf}\\{f,dg\\}$, which contain both the Christoffel symbol and the Riemann \ntensor.) \nIt's not surprising, of course, because in our quantization we did not \ncare about $d$. The open questions are, if $d$ can be ``adjusted'' by \nadditional terms to satisfy more general condition~(\\ref{der}), or if some \nother quantization for $\\Omega(T^{*}M)$ can be found, which would better \n fit into the picture incorporating $d$.\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn 2006, it was suggested by Pendry et al. \\cite{pendrycloak} that an\nobject surrounded by a coating consisting of an exotic\nmaterial becomes invisible to electromagnetic\nwaves. This device was named ``invisibility cloak'' in reference to Harry Potter, the popular character of J.K. Rowling. Beside his famous cloak, the little wizard has other spells to go unnoticed. Among the most spectacular is the ``polyjuice potion'' that is able to turn somebody into anybody else's appearance \\cite{jkrowling}. In this paper, we do not present a potion but rather an optical device able to accomplish the same task, i.e. to give an arbitrary optical response chosen in advance to any other object placed inside the device. In fact, the principle is here very similar to the design of Pendry's invisibility cloak but, instead of geometrically transforming an empty domain, it is a region containing the object to be imitated that is transformed leading thus to a generalization of cloaking.\n\n\nIn this section, we present a generalization of cloaking able to arbitrarily transform the electromagnetic appearance of an object. The basic principle is to obtain the constitutive relations of the cloak by application of a space transformation to a non-empty region.\n\nA geometric transformation is given by a map $\\varphi$ from a space $N$ to a space $M$. For all our practical purposes, $M$ and $N$ will be here the whole or parts of $\\mathbb{R}^3$. Given a Cartesian coordinate system $\\mathbf{x}$ on $M$ and an arbitrary coordinate system $\\mathbf{x}'$ on $N$, $\\varphi: N \\rightarrow M$ is described by $\\mathbf{x}(\\mathbf{x'})$, i.e. $\\mathbf{x}$ given as function of $\\mathbf{x}'$. All the useful information, i.e. necessary to transform differential forms   (indeed the formalism of differential geometry is the most natural to write the Maxwell's equation \\cite{deschamps,burke,nicolet,bossavit} specially when quite general geometric transformations are involved)   and other covariant tensors, is contained in the Jacobian matrix field $\\mathbf{J}(\\mathbf{x'})=\\partial \\mathbf{x}(\\mathbf{x}')\/\\partial \\mathbf{x}' $.\n\nThe basic principle of transformation optics is that, when you have an electromagnetic system described by the tensor fields $\\underline{\\underline{\\varepsilon}}(\\mathbf{x})$ for the dielectric permittivity and $\\underline{\\underline{\\mu}}(\\mathbf{x})$ for the magnetic permeability in the space $M$,  if you replace your initial material properties by equivalent material properties given by the following rule \\cite{pcfbook,milton,twistedEPJ,twistedJWAves}:\n      \\begin{eqnarray}\n      \\underline{\\underline{\\varepsilon'}}(\\mathbf{x'}) =\n      \\mathbf{J}^{-1}(\\mathbf{x'})\\underline{\\underline{\\varepsilon}}(\\mathbf{x}(\\mathbf{x'}))\\mathbf{J}^{-T}(\\mathbf{x'})\\det(\\mathbf{J}(\\mathbf{x'})),  \\notag \\\\\n      \\underline{\\underline{\\mu'}}(\\mathbf{x'}) =\n      \\mathbf{J}^{-1}(\\mathbf{x'})\\underline{\\underline{\\mu}}(\\mathbf{x}(\\mathbf{x'}))\\mathbf{J}^{-T}(\\mathbf{x'})\\det(\\mathbf{J}(\\mathbf{x'})),\n      \\label{equivalence_rule}\n      \\end{eqnarray}\n ($\\mathbf{J}^{-T}$ is\nthe transposed of the inverse of $\\mathbf{J}$), you get an equivalent problem on $N$. Here, an equivalent problem means that the solution of the new problem on $N$, i.e. electromagnetic quantities described as differential forms, are the pulled back of the solution \\cite{nicolet} of the original problem on $M$ and that the same Maxwell's equations (i.e. as if we were in Cartesian coordinates or, more accurately, having the same form written with the exterior derivative) are still satisfied.\n\nIn the case of the cylindrical Pendry's map \\cite{pendrycloak,opl,compel_geo}, described by the transformation of the 2D cross section, the plane $\\mathbb{R}^2$ minus a disk $D_1$ of radius $R_1$ is mapped on the whole plane $\\mathbb{R}^2$ in such a way that a disk $D_2$ of radius $R_2 > R_1$, concentric with $D_1$, is the image of the annulus $D_2\\backslash D_1$ by a radial transformation. In cylindrical coordinates, this transformation is given by:\n\n\\begin{equation}\\begin{cases}  r = (r'-R_1) R_2\/(R_2-R_1) \\; \\mathrm{for} \\; R_1 \\leq r'\\leq R_2, \\; \\\\ \\theta = \\theta', \\; z = z'. \\;\n\\label{pendrysmap}\\end{cases}\\end{equation}\n\nAs for the outside of the disk $D_2$, the map between the two copies of $\\mathbb{R}^2\\backslash D_2$ is the identity map.\n\nThe material properties given by rule (\\ref{equivalence_rule}) corresponding to this transformation provide an ideal invisibility cloak: outside $D_2$, everything behaves as if we were in free space, including the propagation of electromagnetic waves across the cloak, and is completely independent of the content of $D_1$.\n\n\nNow, rule (\\ref{equivalence_rule}) may be applied to $D_2$ containing objects with arbitrary electromagnetic properties so that a region cloaked by this device is still completely hidden but has the appearance of the objects originally in $D_2$. We may call this optical effect masking \\cite{teixeira} or ``polyjuice'' effect.\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{juicemesh2.pdf}\n\\caption{\\label{mesh}  This figure shows a part of the triangular mesh used for the finite element modeling of the scattering problem of  Fig. \\ref{fig2}. The singular behavior of the permittivity and of the permeability requires a very fine mesh along the inner boundary of the cloak in order to achieve a satisfactory accuracy with the numerical model.  }}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering \\includegraphics[width=0.49\\textwidth]{etopim2.jpg}\n\\caption{\\label{fig1}  A conducting triangular cylinder is scattering cylindrical waves.}}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{etopim1.jpg}\n\\caption{\\label{fig2}  A triangular cylinder different from the one on Fig. \\ref{fig1} is surrounded by a cloak designed to reproduce the scattering pattern of the Fig. \\ref{fig1} triangular cylinder in spite of the change of diffracting object. Of course, the diffracting object inside the cloak may be arbitrary as far as it is small enough to fit inside the cloak}}\n\\end{figure}\n\n\n\\begin{figure}[h\n{\\centering  \\includegraphics[width=0.49\\textwidth]{compare.pdf}\n\\caption{\\label{fig3}   The value of the electric field (the real part of $E_z$) on a circle of radius $4\\lambda$ concentric with the cloak is represented as a function of the position angle $\\theta$ (increasing counterclockwise and with $\\theta=0$ corresponding to the point the most on the right). The three configurations considered here are the ones of Fig. \\ref{fig2} (coated), Fig. \\ref{fig1} (original), and the triangle of Fig. \\ref{fig2} without the coating (reversed).  } }\n\\end{figure}\n\n\nFigs. \\ref{fig1} and \\ref{fig2} show the effect of masking on a scattering structure. On Fig. \\ref{fig1}, a cylindrical TM wave emitted by a circular cylindrical antenna is scattered by a conducting triangular cylinder (the longest side of the cross section is $1.62\\lambda$ and $\\varepsilon_r=1 + 40 i $). The field map  represents the longitudinal electric field $E_z(x,y)$ and the outer boundary of the cloak is shown to ease the comparison with the masked case. On Fig. \\ref{fig2}, the same cylindrical TM wave is scattered by a masked triangular cylinder (but the diffracting object inside the cloak may be arbitrary as far as it is small enough to fit inside the cloak). This triangular cylinder is the symmetric of the previous one with respect to the horizontal plane containing the central fibre of the cylindrical antenna. This bare scatterer would therefore give the Fig. \\ref{fig1} image inverted upside-down but, here, this object is surrounded by a cloak in order to give the very same scattering as before. Indeed, on both sides, the electric fields outside the cloak limit are alike.\n\n  Fig. \\ref{fig3} highlights the different scattering patterns by displaying the value of $\\Re e(E_z)$ on a circle of radius $4 \\lambda$ located around the antenna-scatterer system in the three following cases: the case of Fig. \\ref{fig1} (original) with the triangle alone, the case of Fig. \\ref{fig2} (coated), and the triangle of Fig. \\ref{fig2} without the coating (reversed). It is obvious that the coating restores the field distribution independently of the object present in the central hole.\n\n\nThe numerical computation is performed using the finite element method (via the free GetDP \\cite{getdp} and Gmsh \\cite{gmsh} software tools). The mesh is made of 148,000 second order triangles   including the Perfectly Matched Layers used to truncate the computation domain . The singularity of $\\varepsilon$ and $\\mu$ requires a very fine mesh in the vicinity of the inner boundary of the cloak (see Fig. \\ref{mesh}) and is also responsible for the small discrepancies between the numerical model and a perfect cloak (see Fig. \\ref{fig2}) --- including the non zero field in the hole of the cloak.\n\nNote that a small technical problem arises in practice when rule (\\ref{equivalence_rule}) is applied: the material properties are defined piecewise on various domains and it is very useful to know explicitly the boundaries of these domains, e.g. to build the finite element mesh (see Fig. \\ref{mesh}). These boundaries are curves in the cross section and are thus contravariant objects. Therefore, their transformation requires the inverse map $\\varphi^{-1}$ from $M$ to $N$. Fortunately, map (\\ref{pendrysmap}) is very simple to invert.\n\nMore explicitly, for a given curve $\\mathbf{x}(t)$ of parameter $t$ in the initial Cartesian coordinates, its push forward by Pendry's map is:\n\\begin{equation}\n\\mathbf{x}'(t)=\\varphi^{-1}(\\mathbf{x}(t))=(\\frac{R_2-R_1}{R_2}+ \\frac{R_1}{\\|\\mathbf{x}(t) \\|})\\mathbf{x}(t),\n\\end{equation}\nwith the same variation of the parameter $t$. Note that the most common curves used in the design of devices, i.e. line segments and arc of circles, are transformed to less usual curves except   for radial segments (with respect to the center of the cloak) and arc of circles concentric with the cloak.\n\n On Fig. \\ref{fig2}, the image by $\\varphi^{-1}$ of the triangle of Fig. \\ref{fig1} is the curvilinear triangle inside the coating region of the cloak. In practice, this anamorphosis of the triangle is described by three splines interpolating each 40 points that are images of points of the segments by $\\varphi^{-1}$.\n\n\n\n\nTransformation optics do not offer only the possibility make optically disappear objects in invisibility cloaks but also to completely tune their optical signature i.e. to give them an arbitrary appearance.\n   The possibility to place an object inside the coating of the cloak and that it will therefore appear different was already considered in \\cite{opl} where a point source was shifted creating a mirage effect. A more general case is considered here since both the shape and the position of the object placed in the coating are modified .\n Note that if the object used to create the illusion is perfectly conducting and surrounds the central point of the cloak, its anamorphosis will surround the inner boundary of the cloak and it \\emph{de facto} suppresses the singular behavior of the material properties. In this particular case,   the present approach is similar to the the hiding under the carpet idea   of Li and Pendry \\cite{carpet} but here a bounded object replaces the infinite reflecting plane. This technique can be naturally extended to cloaks of arbitrary shapes \\cite{arbitrary}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $\\I\\subset\\R$ be an interval and $f\\colon\\I\\to\\R$. The \\emph{difference operator} is given by\n\\[\n \\Delta_hf(x)=f(x+h)-f(x)\n\\]\nfor any $x\\in\\I$ and $h\\in\\R$ \\st\\ $x+h\\in\\I$. In 1954 Wright~\\cite{Wri54} considered the class of functions~$f:\\I\\to\\R$, for which the function $\\Delta_h f(\\cdot)$ is non-decreasing for any $h>0$ small enough to guarantee that all the involved arguments belong to~$\\I$. This monotonicity property is equivalent to the condition\n\\begin{equation}\\label{eq:Wright}\n f\\bigl(tx+(1-t)y\\bigr)+f\\bigl((1-t)x+ty\\bigr)\\xle f(x)+f(y)\\quad \\big(x,y\\in\\I,\\;t\\in[0,1]\\big).\n\\end{equation}\nNowadays a~function fulfilling~\\eqref{eq:Wright} is called to be \\emph{Wright-convex}. For $t=\\frac{1}{2}$ we get immediately\n\\[\n f\\Bigl(\\frac{x+y}{2}\\Bigr)\\xle\\frac{f(x)+f(y)}{2}\\quad\\bigl(x,y\\in\\I\\bigr),\n\\]\nso any Wright-convex function is necessarily Jensen-convex. It was shown by Ng~\\cite{Ng87} that any Wright-convex function is a~sum of the convex function (in the usual sense) and the additive one. Such representation leads to the simple (and now classical) proof that the inclusion in question is proper. Namely, if $a\\colon\\R\\to\\R$ is a~discontinuous additive function and $f(x)=|a(x)|$, then $f$~is discontinuous and Jensen-convex. Hence the graph of~$f$ is not dense on the whole plane. But it is well-known (\\cf~\\eg~\\cite{Kuc09}) that the graphs of discontinuous additive functions (and also -- by Ng's representation -- the graphs of discontinuous Wright-convex functions) mapping~$\\R$ into~$\\R$ are dense on the whole plane. That is why~$f$ is not Wright-convex.\n\\par\\medskip\nRecently Nikodem, Rajba and W\u0105sowicz in the paper~\\cite{NikRajWas12JMAA} considered the related comparison problem for the classes of $n$-Wright-convex functions and $n$-Jensen-convex functions (the explanation is given below in this section). If $n\\in\\N$ is odd, they have shown that the first class is the proper subclass of the latter one. The example of $n$-Jensen-convex function which is not $n$-Wright-convex was $f\\colon\\R\\to\\R$ \\st\\ $f(x)=\\bigl(\\max\\{a(x),0\\}\\bigr)^n$, where $a\\colon\\R\\to\\R$ is the certain discontinuous additive function. Next P\\'ales~\\cite{Pal13} proved that this example works for any discontinuous additive function. The authors of~\\cite{NikRajWas12JMAA} did not solve the  problem, if~$n$ is even. The objective of the present paper is to give the complete solution by showing that the inclusion is proper for all $n\\in\\N$. Moreover, the construction of our example does not depend on the parity of~$n$. Then for odd~$n$ we have a~solution different from the ones given in the papers~\\cite{NikRajWas12JMAA,Pal13}.\n\\par\\medskip\nLet us recall the concepts of higher-order Jensen-convex (coming from Popoviciu~\\cite{Pop34}) and Wright-convex functions (introduced by Gil\\'anyi and P\\'ales~\\cite{GilPal01}).\n\\par\\medskip\nThe iterates of ~difference operator are we defined as usual: for $n\\in\\N$\n\\[\n \\Delta_{h_1\\dots h_nh_{n+1}}f(x)=\\Delta_{h_1\\dots h_n}\\bigl(\\Delta_{h_{n+1}}f(x)\\bigr)\n\\]\nprovided that all the involved arguments belong to~$\\I$. If $h_1=\\dots=h_n=h$, then we write \n\\[\n \\Delta_h^nf(x)=\\Delta_{h\\dots h}f(x).\n\\]\nIn particular, $\\Delta_h^1f(x)=\\Delta_hf(x)$. \nLet $n\\in\\N$ and $f\\colon\\I\\to\\R$ be a~function. We say that $f$ is \\emph{Wright-convex of order~$n$} (or $n$-\\emph{Wright-convex} for short), if\n\\[\n \\Delta_{h_1\\dots h_{n+1}}f(x)\\xge 0\n\\]\nfor any $x\\in\\I$ and $h_1,\\ldots,h_{n+1}>0$ \\st\\ $x+h_1+\\dots+h_{n+1}\\in\\I$. If the above inequality is required only for $h_1=\\dots=h_{n+1}=h>0$, precisely, if\n\\[\n \\Delta_h^{n+1}f(x)\\xge 0\n\\]\nfor all $x\\in I$ and $h>0$ such that $x+(n+1)h\\in\\I$, then~$f$ is called \\emph{Jensen-convex of order~$n$} ($n$-\\emph{Jensen-convex} for short). It is easy to see that $1$-Wright-convexity and $1$-Jensen-convexity are equivalent to Wright-convexity and Jensen-convexity, respectively.\n\\par\\medskip\nBy the above definitions any $n$-Wright-convex function is necessarily $n$-Jensen-convex. As we announced above, in this paper we show that this inclusion is proper.\n\n\\section{Main result}\nLet us start with two preparatory notes. The following equations hold for a~power function function $f(x)=x^n$ (\\cf~\\cite[Lemma 15.9.2]{Kuc09}):\n\\begin{equation}\\label{eq:delta-xn}\n \\Delta_{h_1\\ldots h_n}\\bigl(x^n\\bigr)=n!\\cdot\\prod_{i=1}^nh_i \\qquad\\text{and}\\qquad \\Delta_h^n\\bigl(x^n\\big)=n!\\cdot h^n.\n\\end{equation}\n\\par\\medskip\nThe formula below was given in~\\cite{NikRajWas12JMAA} (as a~part of the proof of Corollary~2.2).\n\\begin{lem}\\label{lemat2}\nLet $a\\colon\\R\\to\\R$ be an additive function, $g\\colon\\R\\to\\R$ and $n\\in\\N$.\nThen\n\\[\n \\Delta_{h_1\\dots h_n}(g\\circ a) (x)=\\Delta_{a(h_1)\\dots\\,a(h_n)}g\\bigl(a(x)\\bigr)\n\\]\nfor any $x,h_1,\\dots,h_n\\in\\R$.\n\\end{lem}\nNow we are in a position to state our main result.\n\\begin{thm}\\label{th_main}\nFor any $n\\in\\N$ the class of $n$-Wright-convex functions is a~proper subclass of the class of $n$-Jensen-convex functions.\n\\end{thm}\n\n\\begin{proof}\n Because every $n$-Wright-convex function is $n$-Jensen-convex, it is enough to construct an~$n$-Jensen-convex function $f\\colon\\R\\to\\R$ which is not $n$-Wright convex. To this end consider a~Hamel basis $H$ of the linear space $\\R$ over~$\\Q$ \\st~$h_0=1\\in H$. Any $x\\in\\R$ is uniquely represented as a linear combination\n\\[\n x=\\lambda_0h_0+\\sum_{t}\\lambda_th_t,\n\\]\nwhere $\\lambda_0,\\lambda_t\\in\\Q$, $h_t\\in H$. Obviously the functions $\\alpha(x)=\\lambda_0h_0$ and $\\beta(x)=x-\\alpha(x)$ are additive. Define the function $f\\colon\\R\\to\\R$ by\n\\[\n f(x)=\\bigl(\\alpha(x)\\bigr)^{n+1}+\\bigl(\\beta(x)\\bigr)^{n+1}.\n\\]\nSince the difference operator is additive, we infer by Lemma~\\ref{lemat2} and the equation~\\eqref{eq:delta-xn} that\n\\begin{multline}\\label{eq:dj}\n\\Delta_{h_1\\dots h_{n+1}}f(x)=\\Delta_{\\alpha(h_1)\\ldots\\alpha(h_{n+1})}\\bigl(\\alpha(x)\\bigr)^{n+1}+\\Delta_{\\beta(h_1)\\ldots\\beta(h_{n+1})}\\bigl(\\beta(x)\\bigr)^{n+1}\\\\\n=(n+1)!\\biggl(\\prod_{i=1}^{n+1}\\alpha(h_i)+\\prod_{i=1}^{n+1}\\beta(h_i)\\biggr).\n\\end{multline}\nIn particular,\n\\[\n \\Delta_h^{n+1}f(x)=(n+1)!\\Bigl(\\bigl(\\alpha(h)\\bigr)^{n+1}+\\bigl(\\beta(h)\\bigr)^{n+1}\\Bigr).\n\\]\nLet $h>0$. Because of the representation $h=\\alpha(h)+\\beta(h)$, it is easy to see that $\\bigl(\\alpha(h)\\bigr)^{n+1}+\\bigl(\\beta(h)\\bigr)^{n+1}>0$. Therefore $\\Delta_h^{n+1}f(x)>0$ which shows that~$f$ is $n$-Jensen-convex.\n\nTo prove that $f$ is not $n$-Wright-convex take now $h_1=-1+\\sqrt{2}$, $h_2=1$ and (if $n\\ge 2$) $h_3=\\dots=h_{n+1}=1+\\sqrt{2}$. Then $h_i>0$, $i=1,\\dots,n+1$. Moreover $\\alpha(h_1)=-1$, $\\alpha(h_i)=1$,  $i=2,\\dots,n+1$ and $\\beta(h_2)=0$. By~\\eqref{eq:dj} we obtain \n\\[\n \\Delta_{h_1\\ldots h_{n+1}}f(x)=-(n+1)!<0.\n\\]\nIt means that~$f$ is not $n$-Wright-convex.\n\\end{proof}\n\n\\section{The classes of higher order strongly Jensen-convex functions and Wright-convex functions}\n\nLet $n\\in\\N$ and $c>0$. A~function $f\\colon\\I\\to\\R$ is called \\emph{strongly Wright-convex of order $n$ with modulus $c$} (or strongly $n$-Wright-convex with modulus $c$) if\n\\[\n\\Delta_{h_1\\ldots h_{n+1}}f(x)\\xge c (n+1)!\\,h_1\\ldots h_{n+1}\n\\]\nfor all $x\\in I$ and $h_1,\\ldots,h_{n+1}>0$ such that $x+h_1+\\dots+h_{n+1}\\in\\I$. If the above inequality is required only for $h_1=\\dots=h_{n+1}=h>0$, precisely, if\n\\[\n \\Delta_{h}^{n+1}f(x)\\xge c(n+1)!\\,h^{n+1}\n\\]\nfor all $x\\in I$ and $h>0$ such that $x+(n+1)h\\in\\I$, then $f$ is called \\emph{strongly Jensen-convex of order $n$ with modulus~$c$} (or strongly $n$-Jensen-convex with modulus $c$) (\\cf~ \\cite{GerNik11}).\nFor $c=0$ we arrive at standard $n$-Wright-convex and $n$-Jensen-convex functions, respectively.\n\\par\\medskip\nThe following characterization of strongly $n$-Wright-convex functions was recently given in~\\cite{GilMerNikPal15}.\n\\begin{thm}\\label{tw4}\nLet $n\\in\\N$ and $c>0$. A function $f\\colon\\I\\to\\R$ is strongly $n$-Wright-convex with modulus~$c$ if and only if the function $g\\colon\\I\\to\\R$ \\st\\ $g(x)=f(x)-cx^{n+1}$, is $n$-Wright-convex.\n\\end{thm}\nIt is surprising that so far nobody wrote explicitly the similar characterization of strongly $n$-Jensen-convex functions. We fill this gap now.\n\\begin{thm}\\label{tw5}\nLet $n\\in\\N$ and $c>0$. A function $f\\colon\\I\\to\\R$ is strongly $n$-Jensen-convex with modulus~$c$ if and only if the function $g\\colon\\I\\to\\R$ \\st\\ $g(x)=f(x)-cx^{n+1}$, is $n$-Jensen-convex.\n\\end{thm}\n\\begin{proof}\nSuppose first, that the function $f$ is strongly $n$-Jensen-convex with modulus $c$. Applying~\\eqref{eq:delta-xn} to $g$ we infer that\n\\[\n\\Delta_{h}^{n+1}g(x)= \\Delta_{h}^{n+1}f(x) - \\Delta_{h}^{n+1}(c x^{n+1}) \\xge c (n+1)!\\,h^{n+1}- c (n+1)!\\,h^{n+1}=0,\n\\]\nwhence $g$ is $n$-Jensen-convex.\n\nConversely, if $g$ is an $n$-Jensen-convex with modulus~$c$, use~\\eqref{eq:delta-xn} for $f(x)=g(x)+cx^{n+1}$:\n\\[\n\\Delta_{h}^{n+1}f(x)= \\Delta_{h}^{n+1}g(x) + \\Delta_{h}^{n+1}(c x^{n+1}) \\xge 0+ c (n+1)!\\,h^{n+1}=c (n+1)!\\,h^{n+1}.\n\\]\nThis proves that~$f$ is strongly $n$-Jensen-convex.\n\\end{proof}\nThe following result we derive from Theorems \\ref{th_main}, \\ref{tw4}, \\ref{tw5}.\n\\begin{thm}\nLet $n\\in\\N$ and $c>0$. The class of strongly $n$-Wright-convex functions with modulus~$c$ is a~proper subclass of the class of strongly $n$-Jensen-convex functions with modulus~$c$.\n\\end{thm}\n\\begin{proof}\nIt is enough to show, that there exists a strongly $n$-Jensen--convex function with modulus $c$, which is not strongly $n$-Wright--convex with modulus $c$. By Theorem~\\ref{th_main} there exists a function~$g$, which is $n$-Jensen-convex and $g$ is not $n$-Wright-convex. Let $f(x)=g(x)+cx^{n+1}$. Since $g$ is $n$-Jensen-convex, it follows by Theorem~\\ref{tw5} that $f$~is strongly $n$-Jensen-convex with modulus~$c$. By Theorem~\\ref{tw4}~$f$ is not strongly $n$-Wright-convex with modulus~$c$ (because the function $g(x)=f(x)-cx^{n+1}$ is not $n$-Wright-convex). This concludes the proof.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\\label{se:intro}\n\nAfter the discovery of a Higgs boson at the Large Hadron Collider (LHC) \\cite{Aad:2012tfa,Chatrchyan:2012ufa} at CERN, the complete identification of this particle is ongoing. The  properties of the discovered particle, such as its couplings, are determined experimentally in order to fully identify its nature. For the endeavour of the identification of this particle, \ninput from the theory side is needed in form of precise predictions for the production and decay processes in the Standard Model (SM) as well as in its extensions that are to be tested. It is also crucial to provide reliable uncertainty estimates of the theoretical predictions. \nUnderestimating this uncertainty might lead to wrong conclusions.\nIn the SM, predictions and error estimates are well advanced, and in SM extensions they are\nconsolidating as well (see, e.g., the reviews in \nRefs.~\\cite{Dittmaier2011,Dittmaier2012,Dittmaier:2012nh,Heinemeyer:2013tqa,deFlorian:2016spz,%\nSpira:2016ztx}).\n\nOne of the simplest extensions of the SM \nis the Two-Higgs-Doublet Model (THDM) \\cite{Lee:1973iz,HHGuide:1990} where a second Higgs doublet is added \nto the SM field content. The underlying gauge group \n$\\mathrm{SU}(3)_{\\text{C}} \\times \\mathrm{SU}(2)_{\\text{W}} \\times \\mathrm{U}(1)_{\\text{Y}}$ as well as the fermion content of the SM are kept. After spontaneous symmetry breaking, there are five physical Higgs bosons where three of them are neutral and two are charged. In the CP-conserving case, which we consider, one of the neutral Higgs bosons is CP-odd and two are CP-even with one of them being SM-like. \n\nEven such a simple extension of the SM \ncan help solving some questions that are unanswered in the SM. For example, CP-violation in the Higgs sector could provide solutions to the problem of baryogenesis~\\cite{Turok:1990zg,Davies:1994id,Cline:1995dg,Fromme:2006cm}, and inert THDMs contain a dark matter candidate \\cite{PhysRevD.18.2574,Dolle:2009fn}. An even larger motivation comes from the embedding of the THDM into more complex models, such as axion \\cite{Kim:1986ax, Peccei:1977hh} or supersymmetric models \\cite{Haber:1984rc}. Some of the latter are promising candidates for a fundamental theory, and supersymmetric Higgs sectors contain a THDM (in which the doublets have opposite hypercharges). Even though the THDM is unlikely to be the fundamental theory of nature, it provides a rich phenomenology, which can be used in the search for a non-minimal Higgs sector without being limited by constraints from a \nmore fundamental theory.\n\nIn this sense, it is obvious that the THDM should be tested against data,\nand phenomenological studies have been performed recently, e.g., \n{in Refs.~\\cite{Celis:2013rcs,Celis:2013ixa,Chang:2013ona,Harlander:2013mla,%\nHespel:2014sla,Baglio:2014nea,Broggio:2014mna,%\nBernon:2015qea,Bernon:2015wef,%\nGoncalves:2016qhh,Dorsch:2016tab,%\nCacciapaglia:2016tlr,Cacchio:2016qyh,Aggleton:2016tdd,Han:2017pfo}.} \nIn order to provide precise predictions within this model, not only leading-order \n(LO), but also next-to-leading-order (NLO) contributions have to be taken into account. For the calculation of NLO contributions, a proper definition of a renormalization scheme is mandatory. There is no unique choice, and applying different renormalization schemes can help to estimate the theoretical uncertainty of the prediction that originates from the truncation of the perturbative series. \nThe renormalization of the THDM has already been tackled in several publications:\nFirst, in Ref.~\\cite{Santos:1996vt}, the fields and masses were renormalized in the on-shell scheme, \nhowever, the prescription given there does not cover all parameters.\nIn Refs.~\\cite{Kanemura:2004mg,LopezVal:2009qy}, a minimal field renormalization was applied, and the field mixing conditions were used to fix some of the mixing angles. \nIn view of an automation of NLO predictions within the THDM a tool was written by \nDegrande~\\cite{Degrande:2014vpa} \nwhere all finite rational terms and all\ndivergent terms are computed using on-shell conditions or conditions within the ``modified minimal subtraction scheme'' ($\\overline{\\mathrm{MS}}$). \nThough automation is very helpful, often specific problems occur depending on the model, the process, or the renormalization scheme considered, and, it might be necessary to solve these ``manually''. Specifically, spontaneously broken gauge theories with extended scalar sectors pose issues with the renormalization of vacuum expectation values and the related ``tadpoles'', \njeopardizing gauge independence and perturbative stability in predictions.\nRenormalization schemes employing a gauge-independent treatment of the tadpole terms were described recently in Refs.~\\cite{Krause:2016oke,Denner:2016etu,Krause:2016xku}. \n\nIn this paper, we perform the renormalization of various types of THDMs \n(Type I, Type II, ``lepton-specific'', and ``flipped''),\ndescribe four different renormalization schemes\n(for each type), \nand provide explicit results facilitating their application in NLO calculations. The comparison of results obtained in these renormalization schemes allows for checking their perturbative consistency, i.e.\\ whether the expansion point for the perturbation series is chosen well and no unphysically large corrections are introduced. Knowing in which parts of the parameter space a renormalization scheme leads to a stable perturbative behaviour  is important for the applicability of the scheme. \nIn addition, we investigate the dependence on the renormalization scale $\\mu_\\mathrm{r}$ which is introduced by defining some parameters via {$\\overline{\\mathrm{MS}}$}{} conditions. In order to investigate the $\\mu_\\mathrm{r}$ dependence consistently, we solve the renormalization group equations (RGEs) and include the running effects. We also make contact to different renormalization schemes suggested in the literature, including the recent formulations \\cite{Krause:2016oke,Denner:2016etu} \nwith gauge-independent treatments of tadpoles.%\n\\footnote{In our work we do not consider the ``tadpole-pinched'' scheme suggested\nin \\citere{Krause:2016oke}.\nFollowing the arguments of \\citeres{Denner:1994nn,Denner:1994xt}\nwe consider the ``pinch technique'' just as one of many physically\nequivalent choices to fix the gauge arbitrariness in off-shell\nquantities (related to the 't~Hooft--Feynman gauge of the\nquantum fields in the background-field gauge)\nrather than singling out ``its gauge-invariant part'' in any sense.}\nTo facilitate NLO calculations in practice, we have implemented our renormalization schemes\nfor the THDM into a \\textsc{FeynArts} \\cite{Hahn2001} model file, so that amplitudes and squared matrix elements can be generated straightforwardly. \nFinally, we apply the proposed renormalization schemes in the NLO calculation of the partial decay width of the \nlighter CP-even Higgs boson decaying into four fermions, $\\mathswitchr h \\to \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ\\to 4f$,\na process class that is a cornerstone in the experimental determination of Higgs-boson\ncouplings, \n{but for which electroweak corrections in the THDM are not yet known in the literature.}\nThe impact of NLO corrections on Higgs couplings in the THDM was, for instance, investigated more\nglobally in \\citeres{Kanemura:2014dja,Kanemura:2015mxa}.\n{However, a full set of electroweak corrections to all Higgs-boson decay processes in the THDM \ndoes not yet exist in the literature, so that current predictions (see, e.g., \\citere{Harlander:2013qxa})\nfor THDM Higgs analyses globally neglect electroweak higher-order effects.}\n{Our calculation, thus, contributes to overcome this shortcoming;\nelectroweak corrections to some $1\\to2$ particle decays of heavy\nHiggs bosons were presented in \\citere{Krause:2016oke}.}%\n\\footnote{Since we consider the decays of the light Higgs boson $\\mathswitchr h$ via\n$\\mathswitchr W$- or $\\mathswitchr Z$-boson pairs, where at least one of the gauge bosons\nis off its mass shell, we have to consider the full $1\\to4$ process\nwith all off-shell and decay effects, rendering a comparison to\nresults on $\\mathswitchr H\\to\\mathswitchr W\\PW\/\\mathswitchr Z\\PZ$ not meaningful.}\n\nThe structure of the paper is as follows. We introduce the four considered types of THDMs\nand our conventions in Sect.~\\ref{sec:THDM}, and the derivation of the counterterm Lagrangian is performed in Sect.~\\ref{sec:CTLagrangian}. Afterwards we fix the renormalization constants with renormalization conditions (Sect.~\\ref{sec:renconditions}). The on-shell conditions, where the renormalized parameters correspond to measurable quantities, are  described and applied in Sect.~\\ref{sec:onshellrenconditions}. The renormalization constants of parameters that do not directly correspond to physical quantities are fixed in the $\\overline{\\mathrm{MS}}$ scheme, so that they contain only the UV divergences and no finite terms. We describe different renormalization schemes based on different definitions of the $\\overline{\\mathrm{MS}}$-renormalized parameters  in Sect.~\\ref{sec:diffrenschemes}. \nThe RGEs of the {$\\overline{\\mathrm{MS}}$}{}-renormalized parameters \nare derived and numerically solved in Sect.~\\ref{sec:running}. \nThe implementation of the results into an automated matrix element generator is described in Sect.~\\ref{sec:FAmodelfile}, and numerical results for \nthe partial decay width $\\mathswitchr h\\rightarrow 4 f$ are \npresented in Sect.~\\ref{se:numerics}. \nFinally, we conclude in Sect.~\\ref{se:conclusion}, \nand further details on the renormalization prescription as well as \nsome counterterms are given in the appendix.\n\n\\section{The Two-Higgs-Doublet Model}\n\\label{sec:THDM}\n\nThe Lagrangian of the THDM,  $\\mathcal{L}_\\mathrm{THDM}$, is composed of the following parts,\n\\begin{align}\n\\mathcal{L}_\\mathrm{THDM}=\\mathcal{L}_\\mathrm{Gauge}+\\mathcal{L}_\\mathrm{Fermion}+\\mathcal{L}_\\mathrm{Higgs}+\\mathcal{L}_\\mathrm{Yukawa}+\\mathcal{L}_\\mathrm{Fix}+\\mathcal{L}_\\mathrm{Ghost}. \n\\label{eq:LTHDM}\n\\end{align}\nThe gauge, fermionic, gauge-fixing, and ghost parts \ncan be obtained in a straightforward way from the SM ones, e.g., \ngiven in Ref.~\\cite{Denner:1991kt}. The Higgs Lagrangian and the Yukawa couplings to the fermions are discussed in the following and are mostly affected by the additional degrees of freedom \nof the THDM. A very elaborate and complete discussion of the THDM Higgs and Yukawa Lagrangians, including general and specific cases, can, e.g., \nbe found in Refs.~\\cite{Gunion:1989we,Branco:2011iw}.\n\n\\subsection{The Higgs Lagrangian}\n\\label{sec:HiggsL}\n\nThe Higgs Lagrangian, $ \\mathcal{L}_\\mathrm{Higgs}$, contains the kinetic terms and a potential $V$,\n\\begin{align}\n  \\mathcal{L}_\\mathrm{Higgs}= (D_\\mu\\Phi_1)^\\dagger (D^\\mu \\Phi_1)+(D_\\mu\\Phi_2)^\\dagger (D^\\mu \\Phi_2)- V(\\Phi_1^\\dagger \\Phi_1,\\Phi_2^\\dagger \\Phi_2,\\Phi_2^\\dagger \\Phi_1,\\Phi_1^\\dagger \\Phi_2),\n\\label{eq:Lhiggs}\n\\end{align}\nwith the complex scalar doublets $\\Phi_{1,2}$ \nof hypercharge $Y_\\mathrm{W}=1$, \n\\begin{align}\n\\Phi_1=\\begin{pmatrix} \\phi_1^+ \\\\ \\phi_1^0\\end{pmatrix}, \\qquad \\Phi_2=\\begin{pmatrix}\\phi_2^+ \\\\ \\phi^0_2\\end{pmatrix},\n\\label{eq:doublets}\n\\end{align}\nand the covariant derivative \n\\begin{align}\n  D_\\mu=\\partial_\\mu \\mp \\mathrm{i} g_2 I^a_\\mathrm{W} W^a_\\mu +\\mathrm{i} g_1 \\frac{Y_\\mathrm{W}}{2}B_\\mu,\n\\label{eq:EWcovderivative}\n\\end{align}\nwhere $I^a_\\mathrm{W}$ ($a = 1,2,3$) are the generators of the weak isospin. \nThe SU(2) and \nU(1) gauge fields are denoted \n$W^a_\\mu$ and $B_\\mu$ with the corresponding gauge couplings $g_2$ and $g_1$, respectively.\nThe sign in the \n$g_2$~term is negative in the conventions of B\\\"ohm, Hollik and Spiesberger \n(BHS)~\\cite{Denner:1991kt,Bohm:1986rj} and positive in the convention of Haber and Kane \n(HK)~\\cite{Haber:1984rc}. We implemented both sign conventions, but used the former one as default. \nIn general, the potential involves all hermitian functions of the two doublets up to dimension four and can be parameterized in the most general case as follows \\cite{Wu1994,Branco:2011iw},\n\\begin{align}\nV=&\\,m^2_{11} \\Phi^{\\dagger}_1 \\Phi_1+ m^2_{22} \\Phi^\\dagger_2 \\Phi_2\n-\\left[m^2_{12} \\Phi^\\dagger_1 \\Phi_2 + h.c.\\right]\\nonumber\\\\\n&+ \\frac{1}{2} \\lambda_1 (\\Phi^\\dagger_1 \\Phi_1)^2+\\frac{1}{2} \\lambda_2 (\\Phi^\\dagger_2 \\Phi_2)^2+ \\lambda_3 (\\Phi^\\dagger_1 \\Phi_1)(\\Phi^\\dagger_2 \\Phi_2)+ \\lambda_4 (\\Phi^\\dagger_1 \\Phi_2)(\\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\\n&+\\left[\\frac{1}{2} \\lambda_5(\\Phi^\\dagger_1 \\Phi_2)^2\n+(\\lambda_6 \\Phi^\\dagger_1\\Phi_1+\\lambda_7 \\Phi^\\dagger_2 \\Phi_2)\\Phi^\\dagger_1 \\Phi_2+h.c.\\right]\\label{eq:THDMpot}.\n\\end{align}\nThe parameters $m_{11}^2,m_{22}^2,\\lambda_1,\\lambda_2,\\lambda_3,\\lambda_4$ are real, while the parameters $m_{12}^2,\\lambda_5,\\lambda_6,\\lambda_7$ are complex, yielding a total number of 14 real degrees of freedom for the potential. However, the component fields of the two Higgs doublets $\\Phi_1$ and $\\Phi_2$ do not correspond to mass eigenstates, and these doublets can be redefined using an SU(2) transformation without changing the physics, so that only 11 physical degrees of freedom remain~\\cite{Branco:2011iw}.\nFor each Higgs doublet we demand that the fields develop a\nvacuum expectation value (vev) in the neutral component,\n\\begin{align}\n  \\langle \\Phi_1 \\rangle&=\\langle 0| \\Phi_1|0\\rangle =\\begin{pmatrix}0 \\\\ \\frac{v_1}{\\sqrt{2} }  \\end{pmatrix},&\n  \\langle \\Phi_2 \\rangle=\\langle 0| \\Phi_2|0\\rangle =\\begin{pmatrix} 0 \\\\ \\frac{v_2}{\\sqrt{2}}  \\end{pmatrix}.\n\\end{align}\nIt is non-trivial that such a stable minimum of the potential exists, \nrestricting the allowed parameter space already strongly~\\cite{Ferreira:2004yd}. \nIn general, the vevs are complex (with a significant relative phase).\nThe Higgs doublets can be decomposed as follows,\n\\begin{align}\n\\Phi_1=\\begin{pmatrix} \\phi_1^+ \\\\ \\frac{1}{\\sqrt{2}}(\\eta_1+i \\chi_1+v_1)\\end{pmatrix}, && \\Phi_2=\\begin{pmatrix}\\phi_2^+ \\\\ \\frac{1}{\\sqrt{2}}(\\eta_2+i \\chi_2+v_2)\\end{pmatrix},\n\\label{eq:decom}\n\\end{align}\nwith the charged fields $\\phi_1^+,\\phi_2^+$, the neutral CP-even fields $\\eta_1,\\eta_2$, and the neutral CP-odd fields $\\chi_1,\\chi_2$.\n\n\\paragraph{Additional constraints:}\nSince the 11-dimensional parameter space of the potential is too large for early experimental analyses, we restrict the model in our analysis by imposing two additional conditions, motivated by experimental results:\n\\begin{itemize}\n\\item absence of flavour-changing neutral currents at tree level,\n\\item CP conservation in the Higgs sector (even though this holds only approximately).\n\\end{itemize}\nThe former requirement can be ensured by adding a discrete $\\mathbb{Z}_2$ symmetry $\\Phi_1 \\rightarrow - \\Phi_1$ (see Sect.~\\ref{sec:YukawaL}). This condition implies that the parameters $\\lambda_6$ and $\\lambda_7$ vanish.  Permitting operators of dimension two that\nviolate this  $\\mathbb{Z}_2$ symmetry softly,  non-zero values of  $m_{12}$ are still allowed~\\cite{HHGuide:1990, Gunion:2002zf}.\nConcerning the second condition, the potential is CP-conserving if and only if a basis of the Higgs doublets exists in which all parameters and the vevs are real~\\cite{Gunion:2005ja}. For our description we assume that a transformation to such a basis has been done already (if the parameters or vevs were initially complex), so that we only have to deal with real parameters. This renders $m_{12}$ and $\\lambda_5$ real.\nHowever, at higher orders in perturbation theory CP-breaking terms and complex phases in the Higgs sector are generated radiatively through loop contributions\ninvolving the quark mixing matrix. For our NLO analysis, this does not present a problem as they appear only beyond NLO in the specific processes we consider.\nIn addition we assume that a basis of the doublets is chosen in which $v_1,v_2>0$ (which is always possible as a redefinition $\\Phi_i \\to - \\Phi_i$ changes the sign of the vacuum expectation value).\nThe potential~\\eqref{eq:THDMpot} has then the following form,\n\\begin{align}\nV={}&m^2_{11} \\Phi^{\\dagger}_1 \\Phi_1+ m^2_{22} \\Phi^\\dagger_2 \\Phi_2\n-m^2_{12} (\\Phi^\\dagger_1 \\Phi_2 + \\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\ \n&+ \\frac{1}{2} \\lambda_1 (\\Phi^\\dagger_1 \\Phi_1)^2+\\frac{1}{2} \\lambda_2 (\\Phi^\\dagger_2 \\Phi_2)^2+ \\lambda_3 (\\Phi^\\dagger_1 \\Phi_1)(\\Phi^\\dagger_2 \\Phi_2)+ \\lambda_4 (\\Phi^\\dagger_1 \\Phi_2)(\\Phi^\\dagger_2 \\Phi_1)\\nonumber\\\\ \n&+\\frac{1}{2} \\lambda_5\\left[(\\Phi^\\dagger_1 \\Phi_2)^2+(\\Phi^\\dagger_2 \\Phi_1)^2\\right]. \n\\label{eq:lambdapara}\n\\end{align}\nExpanding the potential using the decomposition~\\eqref{eq:decom} and ordering terms with respect to powers of the fields, leads to the form\n\\begin{align}\nV={}& -t_{\\eta_1} \\eta_1 - t_{\\eta_2} \\eta_2\\nonumber\\\\\n&+\\frac{1}{2} \\,(\\eta_1,\\eta_2) \\mathbf{M}_\\eta \\begin{pmatrix}\\eta_1 \\\\ \\eta_2 \\end{pmatrix}\n+\\frac{1}{2} (\\chi_1,\\chi_2)\\,  \\mathbf{M}_\\chi  \\begin{pmatrix}\\chi_1 \\\\ \\chi_2 \\end{pmatrix}\n+(\\phi_1^+,\\phi^+_2) \\, \\mathbf{M}_\\phi \\begin{pmatrix}\\phi^-_1 \\\\ \\phi^-_2 \\end{pmatrix}+\\ldots,\\label{eq:potinmatrix}\n\\end{align}\nwith the tadpole terms proportional to the tadpole parameters $t_{\\eta_1}$, $t_{\\eta_2}$ and linear in the fields. The mass terms contain the mass matrices \n$\\mathbf{M}_\\eta$, $\\mathbf{M}_\\chi$, and $ \\mathbf{M}_\\phi$ and are quadratic in the CP-even, CP-odd, and charged Higgs-boson fields, respectively. Terms cubic or quartic in the fields are suppressed in the notation here. Only the neutral CP-even scalar fields can develop non-vanishing tadpole terms, since they carry the quantum numbers of the vacuum. Further, in the mass terms, only particles with the same quantum numbers can mix, so that the three different \ntypes of scalars (neutral CP-even, neutral CP-odd, and charged) do not mix with one another. Of course, through the cubic and quartic terms, which are not shown here, these particles interact with each other. The tadpole parameters are\n\\begin{subequations}\n\\begin{align}\nt_{\\eta_1}=- m^2_{11} v_1- \\lambda_1v_1^3\/2 + v_2 (m^2_{12}-\\lambda_{345}v_1 v_2\/2),\\\\\nt_{\\eta_2}=- m^2_{22} v_2- \\lambda_2v_2^3\/2 + v_1 (m^2_{12}-\\lambda_{345} v_1 v_2\/2),\n\\end{align}%\n\\label{eq:tadpoleterms}%\n\\end{subequations}%\nwhere we introduced the abbreviations $\\lambda_{ij\\ldots}=\\lambda_i+\\lambda_j+\\ldots$, and the mass matrices are given by\n\\begin{subequations}\n\\begin{align}\n\\mathbf{M}_\\eta&=\\begin{pmatrix}  m^2_{11}+ 3 \\lambda_1 v_1^2\/2+ \\lambda_{345} v_2^2\/2 & -m^2_{12}+  \\lambda_{345} v_1 v_2\\\\ -m^2_{12}+  \\lambda_{345} v_1 v_2&  m^2_{22}+ 3 \\lambda_2 v_2^2\/2+ \\lambda_{345} v_1^2\/2\\end{pmatrix},\n\\\\\n\\mathbf{M}_\\chi&=\\begin{pmatrix} m^2_{11}+ \\lambda_1 v_1^2\/2+ (\\lambda_{34}-\\lambda_{5}) v_2^2\/2 & -m^2_{12}+ \\lambda_{5} v_1 v_2\\\\ \n\\hspace{-0pt}-m^2_{12}+ \\lambda_{5} v_1 v_2& \n\\hspace{-0pt}m^2_{22}+ \\lambda_2 v_2^2\/2+ (\\lambda_{34}-\\lambda_{5}) v_1^2\/2\\end{pmatrix},\n\\\\\n\\mathbf{M}_\\phi&=\n\\begin{pmatrix} m^2_{11}+ \\lambda_1 v_1^2\/2+ \\lambda_{3} v_2^2\/2 \n& -m^2_{12}+ \\lambda_{45} v_1 v_2\/2 \\\\ \n-m^2_{12}+ \\lambda_{45} v_1 v_2\/2 & m^2_{22}+ \\lambda_2 v_2^2\/2+ \\lambda_{3} v_1^2\/2\n\\end{pmatrix}.\n\\label{eq:massmatricesfirst}\n\\end{align}\n\\end{subequations}\nThe fields can be transformed into their mass eigenstate basis via\n\\begin{subequations}\n\\label{eq:rotations}\n\\begin{align}\n\\label{eq:Hhrot}\n\\begin{pmatrix}\\eta_1 \\\\ \\eta_2\\end{pmatrix} &\n= \\begin{pmatrix}\\cos{\\alpha} & -\\sin{\\alpha}\\\\ \\sin{\\alpha}& \\cos{\\alpha}\\end{pmatrix} \\begin{pmatrix}H \\\\ h\\end{pmatrix},\n\\\\\n\\begin{pmatrix}\\chi_1 \\\\ \\chi_2\\end{pmatrix}&\n= \\begin{pmatrix}\\cos{\\beta_n} & -\\sin{\\beta_n}\\\\ \\sin{\\beta_n}& \\cos{\\beta_n}\\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0\\end{pmatrix},\n\\\\\n\\begin{pmatrix}\\phi^\\pm_1 \\\\ \\phi^\\pm_2\\end{pmatrix}&\n= \\begin{pmatrix}\\cos{\\beta_c} & -\\sin{\\beta_c}\\\\ \\sin{\\beta_c}& \\cos{\\beta_c}\\end{pmatrix} \\begin{pmatrix}G^\\pm \\\\ H^\\pm\\end{pmatrix}.\n\\end{align}\n\\end{subequations}\nwhere $h$, $H$ correspond to the CP-even, $A_0$ to the CP-odd, and $H^\\pm$ to the charged mass eigenstates\\footnote{In order to avoid a conflict in our notation, we define $\\alpha_\\mathrm{em}=e^2\/(4\\pi)$ as electromagnetic coupling constant and consistently keep the symbol $\\alpha$ for the rotation angle.}. The fields $G^\\pm$, \n$G_0$ correspond to the Goldstone bosons.\nAfter a rotation of the fields, the potential has the following form,\n\\begin{align}\n\\label{eq:NLObarepot}\nV=& -t_\\mathswitchr H H - t_\\mathswitchr h h\\nonumber\\\\*\n&+\\frac{1}{2} (H,h) \\begin{pmatrix}\\mathswitch {M_\\PH}^2 & M^2_{\\mathswitchr H\\mathswitchr h}\\\\ M^2_{\\mathswitchr H\\mathswitchr h}& \\mathswitch {M_\\Ph}^2\\end{pmatrix} \\begin{pmatrix}H \\\\ h \\end{pmatrix}\n+\\frac{1}{2} (G_0,A_0) \\begin{pmatrix}M^2_\\mathswitchr {G_0} & M^2_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\\\ M^2_{\\mathswitchr {G_0}\\mathswitchr {A_0}}& M^2_{\\mathswitchr {A_0}}\\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0 \\end{pmatrix}\\nonumber\\\\\n&+ (G^+,H^+) \\begin{pmatrix}M_\\mathswitchr {G^+}^2 & M^2_{\\mathswitchr {G}\\mathswitchr {H^+}}\\\\ M^2_{\\mathswitchr {G}\\mathswitchr {H^+}}& \\mathswitch {M_\\PHP}^2\\end{pmatrix} \\begin{pmatrix}G^- \\\\ H^- \\end{pmatrix}+ \\textrm{interaction terms} \n\\end{align}\nwith the \ntadpole parameters\n\\begin{align}\n\\label{eq:tadpoledefs}\nt_\\mathswitchr H=\\mathswitch {c_\\alpha} t_{\\eta_1}+\\mathswitch {s_\\alpha} t_{\\eta_2}, \\qquad\nt_\\mathswitchr h=-\\mathswitch {s_\\alpha} t_{\\eta_1}+\\mathswitch {c_\\alpha} t_{\\eta_2},\n\\end{align}\nwhere general abbreviations for the trigonometric functions $s_x\\equiv\\sin x $, $c_x\\equiv\\cos x$, $t_x\\equiv\\tan x$ are introduced.\nAfter the elimination of $m_{11}$, $m_{22}$ using the above equations, the entries of the mass matrices contain also the tadpole parameters $t_h$, $t_H$. \nUsing \n\\begin{align}\nv^2=v_1^2+v_2^2, \\qquad \n\\tan{\\beta} = \\frac{v_2}{v_1},\n\\label{eq:tanbeta}\n\\end{align}\nwe obtain for the mass parameters of the CP-even Higgs bosons,\n\\begin{subequations}\n\\label{eq:neutralmassdef}\n\\begin{align}\n\\mathswitch {M_\\PH}^2={}&\\frac{2s_{\\alpha-\\beta}^2}{s_{2\\beta}}  m^2_{12}\n+ \\frac{v^2}{2} \\left( 2\\lambda_1 c_\\beta^2 c_\\alpha^2 +2\\lambda_2 s_\\alpha^2 s_\\beta^2 +s_{2 \\alpha} s_{2\\beta} \\lambda_{345}\\right)\n-2t_\\mathswitchr H \\frac{s_\\alpha^3 c_\\beta+c_\\alpha^3 s_\\beta }{ v s_{2 \\beta }}\n-t_\\mathswitchr h\\frac{s_{2 \\alpha } s_{\\alpha -\\beta }}{v  s_{2 \\beta }},\n\\\\\n\\mathswitch {M_\\Ph}^2={}&\\frac{2c_{\\alpha - \\beta}^2}{s_{2 \\beta}} m^2_{12}\n+ \\frac{v^2}{2} \\left( 2\\lambda_1 c_\\beta^2 s_\\alpha^2 +2\\lambda_2 c_\\alpha^2 s_\\beta^2-s_{2 \\alpha} s_{2\\beta} \\lambda_{345}\\right)\n-t_\\mathswitchr H\\frac{s_{2 \\alpha } c_{\\alpha -\\beta }}{v s_{2 \\beta }}\n-2t_\\mathswitchr h\\frac{c_\\alpha^3 c_\\beta-s_\\alpha^3 s_\\beta }{ v s_{2\\beta }},\n\\\\\nM_{\\mathswitchr H\\mathswitchr h}^2={}& \\frac{s_{2(\\alpha-\\beta)}}{s_{2 \\beta}}m^2_{12}\n+ \\frac{v^2}{2} \\left[ s_{2\\alpha} (-c_\\beta^2 \\lambda_1+ s_\\beta^2 \\lambda_2)+ s_{2 \\beta} c_{2\\alpha} \\lambda_{345}\\right]\n-t_\\mathswitchr H\\frac{s_{2 \\alpha } s_{\\alpha -\\beta }}{v s_{2 \\beta }}\n-t_\\mathswitchr h\\frac{s_{2 \\alpha } c_{\\alpha -\\beta }}{v s_{2 \\beta }},\n\\label{eq:Hhmassmixing}\n\\end{align}\n\\end{subequations}\nthe ones of the mass matrix of the CP-odd Higgs fields result in\n\\begin{subequations}\n\\label{eq:cpmassdef}\n\\begin{align}\n\\mathswitch {M_\\PAO}^2&=2c_{\\beta-\\beta_n}^2\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big) \n- 2t_\\mathswitchr H \\frac{c_{\\beta_n}^2 c_\\beta s_\\alpha + s_{\\beta_n}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{c_{\\beta_n}^2 c_\\beta c_\\alpha - s_{\\beta_n}^2 s_\\beta s_\\alpha}{v s_{2\\beta}}, \n\\label{eq:A0massdef}\\\\\nM_{\\mathswitchr {G_0}}^2&=2s_{\\beta-\\beta_n}^2\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big)\n- 2t_\\mathswitchr H \\frac{s_{\\beta_n}^2 c_\\beta s_\\alpha + c_{\\beta_n}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{s_{\\beta_n}^2 c_\\beta c_\\alpha - c_{\\beta_n}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\\\\\nM_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2&=-s_{2(\\beta-\\beta_n)}\\Big(\\frac{m^2_{12}}{s_{2\\beta}}- \\frac{\\lambda_5 v^2}{2}\\Big)- t_\\mathswitchr H \\frac{s_{2\\beta_n} s_{\\alpha-\\beta}}{v s_{2\\beta}}- t_\\mathswitchr h \\frac{s_{2\\beta_n} c_{\\alpha-\\beta}}{v s_{2\\beta}}, \\label{eq:goldstonemassmixing}\n\\end{align}\n\\end{subequations}\nand the ones of the mass matrix of the charged Higgs-boson fields are\n\\begin{subequations}\n\\label{eq:chargedmassdef}\n\\begin{align}\n\\mathswitch {M_\\PHP}^2&=c_{\\beta-\\beta_c}^2\\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- 2t_\\mathswitchr H \\frac{c_{\\beta_c}^2 c_\\beta s_\\alpha + s_{\\beta_c}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{c_{\\beta_c}^2 c_\\beta c_\\alpha - s_{\\beta_c}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\n\\label{eq:Hpmmassdef}\\\\\nM_\\mathswitchr {G^+}^2&=s_{\\beta-\\beta_c}^2\\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- 2t_\\mathswitchr H \\frac{s_{\\beta_c}^2 c_\\beta s_\\alpha + c_{\\beta_c}^2 s_\\beta c_\\alpha}{v s_{2\\beta}}\n- 2t_\\mathswitchr h \\frac{s_{\\beta_c}^2 c_\\beta c_\\alpha - c_{\\beta_c}^2 s_\\beta s_\\alpha}{v s_{2\\beta}},\n\\\\\nM_{\\mathswitchr {G}\\mathswitchr {H^+}}^2&=-\\frac{s_{2(\\beta-\\beta_c)}}{2} \\Big[\\frac{2m^2_{12}}{s_{2\\beta}}-\\frac{v^2}{2}(\\lambda_4+\\lambda_5) \\Big]\n- t_\\mathswitchr H \\frac{s_{2\\beta_c} s_{\\alpha-\\beta}}{v s_{2 \\beta}}\n- t_\\mathswitchr h \\frac{s_{2\\beta_c} c_{\\alpha-\\beta}}{v s_{2 \\beta}}.\n\\label{eq:chargedgoldstonemassmixing}\n\\end{align}\n\\end{subequations}\nAt tree level we demand vanishing tadpole terms corresponding to $t_\\mathswitchr H=t_\\mathswitchr h=0$ and diagonal propagators, so that the mixing terms proportional to $M_{\\mathrm{Hh}}^2$, $M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$, $M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2$ vanish at LO. This yields $\\beta_c=\\beta_n=\\beta$ and fixes the angle $\\alpha$ as well.\nAt NLO such a diagonalization is not possible, as the propagators receive also mixing contributions from the field renormalization and from \n(momentum-dependent) one-loop diagrams, so that there is no distinct condition to define the mixing angles. Therefore we keep the bare mass mixing parameters $M_{\\mathrm{Hh}}^2$, $M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$, $M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2$ and the tadpole terms $t_\\mathswitchr H$, $t_\\mathswitchr h$ in this section, and \nspecify defining conditions for the bare parameters $\\alpha$, $\\beta_n$, $\\beta_c$ and the tadpole terms later.\nWith the above equations, $m_{12}$, $\\lambda_1$, $\\lambda_2$, $\\lambda_3$, $\\lambda_4$ can be traded for the masses of the physical bosons $M_\\mathswitchr H$, $M_\\mathswitchr h$, $M_\\mathswitchr {A_0}$, $\\mathswitch {M_\\PHP}$, and the mixing angle $\\alpha$. The parameter $\\lambda_5$ cannot be replaced by a mass or a mixing angle as it appears only in cubic and quartic Higgs couplings and acts like an additional coupling constant. Explicit relations can be obtained by inverting Eqs.~\\eqref{eq:neutralmassdef}, \n\\eqref{eq:A0massdef}, and~\\eqref{eq:Hpmmassdef}.\nThe other parameters are related to the masses by\n\\begin{subequations}\n\\label{eq:repara}\n\\begin{align}\n\\lambda_1={}&\\frac{1}{\\mathswitch {c_\\beta}^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\Ph}^2- M_{\\mathswitchr H\\mathswitchr h}^2 s_{2\\alpha}-\\sb^2 \\biggl(\\frac{\\mathswitch {M_\\PAO}^2}{ c_{\\beta-\\beta_n}^2}+\\lambda_5 v^2\\biggr) \\right]\n\\nonumber\\\\\n&+t_\\mathswitchr H\\frac{c_{{\\beta_{n}}} \\left(2 s_{\\beta } s_{{\\beta_{n}}} c_{\\alpha }+c_{\\alpha +\\beta }c_{\\beta_n}\\right)}{v^3 c_{\\beta }^2 c_{\\beta -{\\beta_{n}}}^2}\n -t_\\mathswitchr h \\frac{c_{\\beta_{n}} \\left(2 s_{\\beta } s_{\\beta_{n}} s_{\\alpha }+s_{\\alpha +\\beta }c_{\\beta_n}\\right)}{v^3 c_{\\beta }^2 c_{\\beta -\\beta_{n}}^2},\n\\\\\n\\lambda_2={}&\\frac{1}{\\sb^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\Ph}^2+\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\PH}^2+ M_{\\mathswitchr H\\mathswitchr h}^2 s_{2\\alpha}-\\mathswitch {c_\\beta}^2 \\biggl(\\frac{\\mathswitch {M_\\PAO}^2}{c_{\\beta-\\beta_n}^2}+\\lambda_5 v^2\\biggr) \\right]\\nonumber\\\\\n&+ t_\\mathswitchr H \\frac{s_{\\beta_n} (2 c_\\beta c_{\\beta_n} s_\\alpha- c_{\\alpha+\\beta} s_{\\beta_n})}{v^3 s_\\beta^2 c_{\\beta-\\beta_n}^2}+ t_\\mathswitchr h \\frac{s_{\\beta_n} (2 c_\\beta c_{\\beta_n} c_\\alpha+ s_{\\alpha+\\beta} s_{\\beta_n})}{v^3 s_\\beta^2 c_{\\beta-\\beta_n}^2},\n\\\\\n\\lambda_3={}&\\frac{1}{v^2 s_{2 \\beta}} \\left[s_{2\\alpha}(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2) + 2 c_{2\\alpha}M_{\\mathswitchr H\\mathswitchr h}^2\\right]-\\frac{\\mathswitch {M_\\PAO}^2}{v^2c_{\\beta-\\beta_n}^2}+\\frac{2 \\mathswitch {M_\\PHP}^2}{v^2c_{\\beta-\\beta_c}^2}-\\lambda_5\\nonumber \n\\\\\n&+\\frac{2t_\\mathswitchr H}{v^3 s_{2\\beta }} \n\\left[s_{\\beta } c_{\\alpha } \\bigg(\\frac{2 s^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{s^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)+c_{\\beta } s_{\\alpha }\n   \\bigg(\\frac{2 c^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{c^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)\\right]\n\\nonumber\\\\\n&+  \\frac{2t_\\mathswitchr h}{v^3 s_{2\\beta }} \n\\left[c_{\\beta } c_{\\alpha }\\bigg(\\frac{2 c^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}-\\frac{c^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}\\bigg)+s_{\\beta } s_{\\alpha }\n   \\bigg(\\frac{s^2_{\\beta_n}}{c_{\\beta -\\beta_n}^2}-\\frac{2 s^2_{\\beta_c}}{c_{\\beta -\\beta_c}^2}\\bigg)\\right]\n\\label{eq:Lambda3NLO}\\\\\n\\lambda_4={}&\\lambda_5+\\frac{2 \\mathswitch {M_\\PAO}^2}{v^2 c_{\\beta -\\beta _n}^2}-\\frac{2 \\mathswitch {M_\\PHP}^2}{v^2 c_{\\beta -\\beta _c}^2}+\\frac{2 t_\\mathrm{H} s_{\\beta _c-\\beta _n} \\left(s_{\\alpha +\\beta -\\beta _c-\\beta\n   _n}-s_{\\beta -\\alpha } c_{\\beta _c-\\beta _n}\\right)}{v^3 c_{\\beta -\\beta _c}^2 c_{\\beta -\\beta _n}^2}\\nonumber\\\\\n&+\\frac{2 t_\\mathrm{h} s_{\\beta _c-\\beta _n} \\left(c_{\\alpha +\\beta -\\beta _c-\\beta\n   _n}+c_{\\beta -\\alpha } c_{\\beta _c-\\beta _n}\\right)}{v^3 c_{\\beta -\\beta _c}^2 c_{\\beta -\\beta _n}^2},\\\\\nm^2_{12}={}&\\frac{1}{2}\\lambda_5 v^2 s_{2\\beta }\n+\\frac{\\mathswitch {M_\\PAO}^2 s_{2\\beta }}{2c_{\\beta -\\beta _n}^2}\n+\\frac{t_\\mathrm{H} \\left(s_{\\beta } c_{\\alpha } s_{\\beta _n}^2+c_{\\beta } s_{\\alpha } c_{\\beta_n}^2\\right)}{v c_{\\beta -\\beta _n}^2}\n+\\frac{t_\\mathrm{h} \\left(c_{\\beta } c_{\\alpha } c_{\\beta _n}^2-s_{\\beta }\n   s_{\\alpha } s_{\\beta _n}^2\\right)}{v c_{\\beta -\\beta _n}^2}.\\label{eq:m12NLO}\n\\end{align}\n\\end{subequations}\nThe tree-level relations are\neasily obtained by setting $\\beta_c=\\beta_n=\\beta$ and $t_\\mathswitchr H=t_\\mathswitchr h=M_{\\mathrm{Hh}}^2=M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2=M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2=0$,\n\\begin{subequations}\n\\begin{align}\n\\lambda_1&={}\\frac{1}{\\mathswitch {c_\\beta}^2 v^2} \\left[\\mathswitch {c_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {s_\\alpha}^2\n\\mathswitch {M_\\Ph}^2-\\sb^2 (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2) \\right],\\\\\n\\lambda_2&={}\\frac{1}{\\sb^2 v^2} \\left[\\mathswitch {s_\\alpha}^2 \\mathswitch {M_\\PH}^2+\\mathswitch {c_\\alpha}^2\n\\mathswitch {M_\\Ph}^2-\\mathswitch {c_\\beta}^2 (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2) \\right],\\\\\n\\lambda_3&={}\\frac{s_{2\\alpha}}{s_{2\\beta} v^2}\n(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)-\\frac{1}{v^2} (\\mathswitch {M_\\PAO}^2-2\\mathswitch {M_\\PHP}^2)-\\lambda_5,\n\\label{eq:lambda3tree}\\\\\n\\lambda_4&={}\\frac{2(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\PHP}^2)}{v^2}+\\lambda_5,\\\\\nm^2_{12}&={}\\frac{s_{2\\beta}}{2} (\\mathswitch {M_\\PAO}^2+\\lambda_5 v^2).\n\\end{align}\n\\end{subequations}\n\n\\paragraph{Parameters of the gauge sector:}\nMass terms of the gauge bosons arise through the interaction of the gauge bosons with the vevs, \nanalogous to the SM. After a rotation into fields corresponding to mass eigenstates, one obtains relations similar to the SM ones:\n\\begin{align}\n\\mathswitch {M_\\PW}=g_2 \\frac{v}{2},&& \\mathswitch {M_\\PZ}=\\frac{v}{2} \\sqrt{g_1^2+g_2^2},&& e= \\frac{g_1g_2}{\\sqrt{g_1^2+g_2^2}},\n\\label{eq:EWmassesTHDM}\n\\end{align}\nwhere the electric unit charge $e$ is identified with the coupling constant of the photon field $A_\\mu$ in the covariant derivative.\nInverting these relations and introducing the weak mixing angle $\\theta_W$ via $\\cos \\theta_W=g_2\/\\sqrt{g_1^2+g_2^2}$, one can replace $v$ and the gauge couplings $g_1$ and $g_2$ by\n\\begin{align}\nv&= \\frac{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}}{e},& g_1&=\\frac{e}{\\mathswitch {c_{\\scrs\\PW}}},& g_2&=\\frac{e}{\\mathswitch {s_{\\scrs\\PW}}}.\n\\label{eq:EWtrafo}\n\\end{align}\n\n\\paragraph{Mass parameterization:}\nThe relations~\\eqref{eq:tadpoledefs}, \n\\refeq{eq:tanbeta},\n(\\ref{eq:repara}a,b,d,e), and \\eqref{eq:EWtrafo} between the masses, angles, and the basic parameters can be used to reparameterize the Higgs Lagrangian and change the parameters\n\\begin{align}\n\\{p_{\\mathrm{basic}}\\}=\\{\\lambda_1,\\ldots,\\lambda_5,m^2_{11},m^2_{22},m^2_{12},v_1,v_2,g_1,g_2\\},\\label{eq:basicpara}\n\\end{align}\nin favour of the bare mass parameters including one parameter from scalar self-interactions\nwhich we take as $\\lambda_3$, \n\\begin{align}\n\\{p'_{\\mathrm{mass}}\\}=\\{\\mathswitch {M_\\PH},\\mathswitch {M_\\Ph},\\mathswitch {M_\\PAO},\\mathswitch {M_\\PHP},M_\\mathrm{W},\\mathswitch {M_\\PZ},e,\\lambda_5,\\lambda_3,\\beta, t_\\mathrm{H} ,t_\\mathrm{h}\\}\n\\label{eq:physparaNLOprime}\n\\end{align}\nAdditionally, one has to keep in mind that we keep the mixing parameters $\\alpha$, $\\beta_n$, and $\\beta_c$ generic, and they have to be fixed by additional conditions (which  will be given later).\nOne can use Eq.~(\\ref{eq:repara}c) to trade $\\lambda_3$ for $\\alpha$, in which case the mixing angle becomes a free parameter of the theory. Then, one obtains the parameter set\n\\begin{align}\n\\{p_{\\mathrm{mass}}\\}=\\{\\mathswitch {M_\\PH},\\mathswitch {M_\\Ph},\\mathswitch {M_\\PAO},\\mathswitch {M_\\PHP},M_\\mathrm{W},\\mathswitch {M_\\PZ},e,\\lambda_5,\\alpha,\\beta, t_\\mathrm{H} ,t_\\mathrm{h} \\}\n\\label{eq:physparaNLO}\n\\end{align}\n\n\\subsection{Yukawa couplings}\n\\label{sec:YukawaL}\n\nThe Higgs mechanism does not only give rise to the gauge-boson mass terms (which are determined by the vevs), \nbut via Yukawa couplings, it introduces\nmasses to chiral fermions. Since both Higgs doublets can couple to fermions, the general Yukawa couplings have the form\n\\begin{align}\n\\mathcal{L}_\\mathrm{Yukawa}=&-\\sum_{k=1,2}\\sum_{i,j} \\left(\\bar{L}'^{\\mathrm{L}}_i \\zeta^{l,k}_{ij} l'^{\\mathrm{R}}_j \\Phi_k+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{u,k}_{ij} u'^{\\mathrm{R}}_j \\tilde{\\Phi}_k+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{d,k}_{ij} d'^{\\mathrm{R}}_j \\Phi_k+h.c.\\right),\n\\label{LYuk}\n\\end{align}\nwith the mixing matrices $\\zeta^{f,k}$, $k=1,2$, in generation space for the \ngauge-invariant interactions with $\\Phi_1$ and $\\Phi_2$, respectively, and the generation indices $i, j = 1,2,3$.\nThe left-handed SU(2) doublets of quarks and leptons are denoted \n${Q}'^{\\mathrm{L}}=\\left(u'^{\\mathrm{L}},d'^{\\mathrm{L}}\\right)^{\\mathrm{T}}$ and \n${L}'^{\\mathrm{L}}=\\left(\\nu'^{\\mathrm{L}},l'^{\\mathrm{L}}\\right)^{\\mathrm{T}}$, \nwhile the right-handed up-type quark, down-type quark, and lepton singlets are  $u'^{\\mathrm{R}}$, $d'^{\\mathrm{R}}$, and $l'^{\\mathrm{R}}$, respectively. \nThe primes indicate that we deal with fields \nin the interaction basis here;\nfields without primes correspond to mass eigenstates.\nThe field $\\tilde{\\Phi}_k$, $k = 1, 2$, is the charge-conjugated field of $\\Phi_k$.\nSince, in the general THDM, there are two mass mixing matrices for each type $f$ of fermions, flavour-changing neutral currents (FCNC) can occur at tree level, which, however, are experimentally known to be strongly suppressed.\nAccording to the Paschos--Glashow--Weinberg theorem~\\cite{Glashow1977,Paschos1977a}, \nFCNC are absent at tree level\nif each type of fermion couples only to one of the Higgs doublets.\n\n This can be achieved by imposing an additional discrete $\\mathbb{Z}_2$ symmetry. It should be noted that the soft-$\\mathbb{Z}_2$-breaking term  proportional to $m_{12}$ in the Higgs potential does not introduce FCNC. \nThe Yukawa Lagrangian reduces then to\n\\begin{align}\n\\mathcal{L}_\\mathrm{Yukawa}=-\\sum_{i,j} \\left(\\bar{L}'^{\\mathrm{L}}_i \\zeta^{l}_{ij} l'^{\\mathrm{R}}_j \\Phi_{n_1}+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{u}_{ij} u'^{\\mathrm{R}}_j \\tilde{\\Phi}_{n_2}+\\bar{Q}'^{\\mathrm{L}}_i \\zeta^{d}_{ij} d'^{\\mathrm{R}}_j \\Phi_{n_3}+h.c.\\right),\n\\end{align}\nwith $n_i$ being either $1$ or $2$. \nDepending on the exact form of the symmetry,\none distinguishes four types of THDMs. In Type~I models, all fermions couple to one Higgs doublet (conventionally $\\Phi_2$, but this is equivalent to $\\Phi_1$ due to possible basis changes) which can be ensured by demanding a $\\Phi_1 \\rightarrow -\\Phi_1$ symmetry. In Type~II models, down-type fermions couple to the other doublet, which can be enforced by the symmetry $\\Phi_1 \\rightarrow -\\Phi_1$, $d'^{\\mathrm{R}}_j \\to- d'^{\\mathrm{R}}_j$, $l'^{\\mathrm{R}} \\to -l'^{\\mathrm{R}}$. The other two possibilities are called ``lepton-specific'' (Type~X) and ``flipped'' (Type~Y) models. An overview over the couplings and symmetries of the different models is given in Tab.~\\ref{tab:yukcoupl}.\n\\begin{table}\n  \\centering\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n&\\hspace{20pt} $u_i$\\hspace{20pt} &\\hspace{20pt} $d_i$\\hspace{20pt}  &\\hspace{20pt}  $e_i$\\hspace{20pt} & $\\mathbb{Z}_2$ symmetry \\\\\\hline\nType I & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_1 \\rightarrow - \\Phi_1$ \\\\\nType II& $\\Phi_2$ & $\\Phi_1$ & $\\Phi_1$ &$(\\Phi_1, d_i,e_i) \\rightarrow -(\\Phi_1, d_i,e_i)$\\\\\nLepton-specific & $\\Phi_2$ & $\\Phi_2$ & $\\Phi_1$ &$(\\Phi_1, e_i) \\rightarrow -(\\Phi_1,e_i)$ \\\\\nFlipped & $\\Phi_2$ & $\\Phi_1$ & $\\Phi_2$ & $(\\Phi_1, d_i) \\rightarrow -(\\Phi_1, d_i)$\\\\\\hline\n\\end{tabular}  \n  \\caption{Different types of the THDM having in common that only one of the Higgs doublet couples to each type of fermions. This can be achieved by imposing appropriate $\\mathbb{Z}_2$ symmetry charges to the fields. }\n\\label{tab:yukcoupl}\n\\end{table}\nFor each of the fermion types, a redefinition of the fields can be performed in order to get diagonal mass matrices, analogously to the SM case. Similar to the SM, \nthe coupling of fermions to the Z boson is flavour conserving, and a CKM matrix appears in the coupling to the charged gauge bosons. \nBy specifying the model type, the Higgs--fermion interaction is determined, and one can write them, \nwidely following the notation of \nRef.~\\cite{Aoki2009}, as\n\\begin{align}\n \\mathcal{L}_\\mathrm{Yukawa,int}=\n&- \\sum_i \\sum_{f=u,d,l} \\frac{m_{f,i}}{v}\\left( \\xi^f_\\mathswitchr h\\, \\bar{f_i}f_i h+ \\xi^f_\\mathswitchr H \\, \\bar{f}_if_iH\n-2\\mathrm{i} I^3_{\\mathswitchr W,f} \\xi^f_\\mathswitchr {A_0} \\, \\bar{f}_i\\gamma_5 f_i A_0\n- 2\\mathrm{i} I^3_{\\mathswitchr W,f} \\bar{f}_i\\gamma_5 f_i G_0\\right)\\nonumber\\\\\n&-\\sum_{i,j}\\biggl[\\frac{\\sqrt{2}V_{ij}}{v} \\bar{u}_i (-m_{u,i} \\xi^u_\\mathswitchr {A_0} \\omega_- +m_{d,j} \\xi^d_\\mathswitchr {A_0} \\omega_+)\\, d_j H^+ + h.c. \\biggr]\n\\nonumber\\\\\n&- \\sum_i \\biggl[ \\frac{\\sqrt{2}m_{l,i} \\xi^l_\\mathswitchr {A_0}}{v}\\, \\bar{\\nu}_i^{\\mathrm{L}} l_i^{\\mathrm{R}} H^+ + h.c. \\biggr]\n\\nonumber\\\\\n&-\\sum_{i,j}\\biggl[ \\frac{\\sqrt{2}V_{ij}}{v} \\bar{u}_i (-m_{u,i} \\omega_- +m_{d,j} \\omega_+)\\, d_j G^+ + h.c. \\biggr]\n- \\sum_i \\biggl[ \\frac{\\sqrt{2}m_{l,i} }{v}\\, \\bar{\\nu}_i^{\\mathrm{L}} l_i^{\\mathrm{R}} G^+ + h.c. \\biggr],\n\\label{eq:ffH}\n\\end{align}\nwhere $m_{l,i}$, $m_{u,i}$, and $m_{d,i}$ are the lepton, the up-type, and the down-type quark masses, respectively,\nand $V_{ij}$ are the coefficients of the CKM matrix.\nLeft- and right-handed fermion fields, $f^{\\mathrm{L}}$ and $f^{\\mathrm{R}}$, are obtained from the \ncorresponding Dirac spinor $f$ by applying the chirality projectors\n$\\omega_\\pm= ( 1\\pm \\gamma_5)\/2$, i.e.\\ $f=(\\omega_+ +\\omega_-)f=f^{\\mathrm{L}}+f^{\\mathrm{R}}$.\nThe coupling coefficients $\\xi^f_{\\mathswitchr H,\\mathswitchr h,\\mathswitchr {A_0}}$ are defined as the couplings relative to the canonical SM value of $m_f\/v$ and are shown in Tab.~\\ref{tab:yukint}. \n\\begin{table}\n  \\centering\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n&Type I & Type II &  Lepton-specific & Flipped\\\\\\hline\n$\\xi^l_\\mathswitchr H$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$  &$\\sin{\\alpha}\/\\sin{\\beta}$\\\\\n$\\xi^u_\\mathswitchr H$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ &$\\sin{\\alpha}\/\\sin{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr H$& $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ & $\\sin{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\cos{\\beta}$ \\\\\\hline\n$\\xi^l_\\mathswitchr h$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$& $\\cos{\\alpha}\/\\sin{\\beta}$  \\\\\n$\\xi^u_\\mathswitchr h$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ &$\\cos{\\alpha}\/\\sin{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr h$& $\\cos{\\alpha}\/\\sin{\\beta}$ & $-\\sin{\\alpha}\/\\cos{\\beta}$ & $\\cos{\\alpha}\/\\sin{\\beta}$ &$-\\sin{\\alpha}\/\\cos{\\beta}$  \\\\\\hline\n$\\xi^l_\\mathswitchr {A_0}$ & $\\cot{\\beta}$ & $-\\tan{\\beta}$ &$-\\tan{\\beta}$ & $\\cot{\\beta}$  \\\\\n$\\xi^u_\\mathswitchr {A_0}$ & $\\cot{\\beta}$ & $\\cot{\\beta}$ & $\\cot{\\beta}$&$\\cot{\\beta}$ \\\\\n$\\xi^d_\\mathswitchr {A_0}$& $\\cot{\\beta}$ & $-\\tan{\\beta}$ & $\\cot{\\beta}$ & $-\\tan{\\beta}$ \\\\\\hline\n\\end{tabular}  \n\\caption{The coupling strengths $\\xi^f$ of $\\mathswitchr H,\\mathswitchr h,\\mathswitchr {A_0}$ to the fermions relative to the SM value of $m_f\/v$, see Eq.~\\eqref{eq:ffH}. Note that the sign of $\\xi^f_\\mathswitchr {A_0}$ is defined relative to\nthe coupling of the Goldstone-boson field $G_0$ and the relation $\\beta_n=\\beta_c=\\beta$\nis used here.}\n\\label{tab:yukint}\n\\end{table}\nNote that we have used $\\beta_n=\\beta_c=\\beta$ in Eq.~\\refeq{eq:ffH} and \\refta{tab:yukint},\nwhich is most relevant in applications; the generalization to independent\n$\\beta_n$, $\\beta_c$, $\\beta$ is simple.\n\n\\section{The counterterm Lagrangian}\n\\label{sec:CTLagrangian}\n\nThe next step in calculating higher-order corrections is the renormalization of the theory. In this section we focus on electroweak corrections of $\\order{\\alpha_{\\mathrm{em}}}$. The QCD renormalization of the THDM is straightforward and completely analogous to the SM case, since all scalar degrees of freedom are colour singlets and do not interact strongly. \nIn the formulation of the basic Lagrangian in the previous section, we dealt with {\\it bare} \nparameters and fields. To distinguish those from {\\it renormalized} quantities, in the\nfollowing we indicate {\\it bare} quantities by subscripts ``0'' consistently.\nWe perform a multiplicative renormalization, i.e.\\ we split {\\it bare} quantities into\n{\\it renormalized} parts and corresponding counterterms,\nuse dimensional regularization,\nand allow for matrix-valued field renormalization constants in the case that there are several fields with the same quantum numbers.\nThe counterterm Lagrangian $\\delta \\mathcal{L}$ containing the full dependence on the renormalization constants can be split into several parts analogous to Eq.~\\eqref{eq:LTHDM},\n\\begin{align}\n\\label{eq:CTLagrangian}\n\\delta \\mathcal{L}_\\mathrm{THDM}\n=\\delta \\mathcal{L}_{\\mathrm{Gauge}}+\\delta \\mathcal{L}_{\\mathrm{Fermion}}+\\delta \\mathcal{L}_{\\mathrm{Higgs,kin}}-\\delta V_{\\mathrm{Higgs}}+\\delta \\mathcal{L}_{\\mathrm{Yukawa}},\n\\end{align}\nwhere the Higgs part of the Lagrangian $\\delta \\mathcal{L}_{\\mathrm{Higgs}}$ is split up into the kinetic part $\\delta \\mathcal{L}_{\\mathrm{Higgs,kin}}$ and the Higgs potential part $\\delta V_{\\mathrm{Higgs}}$.\nSince the gauge fixing is applied after renormalization, no gauge-fixing counterterms occur, and since ghost fields\noccur only in loop diagrams, a renormalization of the ghost sector is not necessary at NLO for the calculation of S-matrix elements. Though, for analyzing Slavnov--Taylor or Ward identities a complete renormalization procedure would be advisable. \n{Our renormalization procedure, thus, widely parallels the treatment described for the SM in \\cite{Denner:1991kt};\nan alternative variant that is based on the transformation of fields in the gauge eigenstate basis, as suggested\nfor the SM in \\citere{Bohm:1986rj} and for the MSSM in \\citere{Dabelstein:1994hb}, is described in\n\\citere{Altenkamp:Dis}.}\n\n\\subsection{Higgs potential}\n\\label{sec:CTpot}\nAccording to Eq.~\\eqref{eq:lambdapara}, the Higgs potential contains \n8 independent parameters which have to be renormalized. In addition, there are two vevs and two gauge couplings completing the set of input parameters of \\eqref{eq:basicpara}. \nWe have carried out different renormalization procedures and in this paper discuss the renormalization of the Lagrangian in the mass parameterization\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{(\\alph{enumi}}\n\\renewcommand{\\labelenumi}{\\theenumi)}\n\\item with renormalization of the mixing angles,\n\\item alternatively,\ntaking the mixing angles as dependent parameters and applying the field transformation after renormalization.\n\\end{enumerate}\nAdditionally we have performed a renormalization of the basic parameters and a \nsubsequent transformation to renormalization constants and parameters of the mass parameter set similar to \nDabelstein in the MSSM~\\cite{Dabelstein:1994hb}%\n\\footnote{Details about this method can be found in \\citere{Altenkamp:Dis}.}. The latter has been used, after changing to our \nconventions for field and parameter renormalization, to check the counterterm Lagrangian. \nIn the following, we give a detailed description of method (a), while method~(b) is \nbriefly described in App.~\\ref{App:renormassa}.\n\n\\subsubsection{Renormalization of the mixing angles}\n\\label{sec:renmixangles}\n\n{In this section we show that the counterterms of mixing angles \nthat are not used to replace another free parameter of the theory \nare redundant in the sense that they can be absorbed by \nfield renormalization constants.}\nWe sketch the argument for generic scalar fields $\\varphi_{1},\\varphi_{2}$, which are transformed into fields $h_1, h_2$ \ncorresponding to mass eigenstates by a rotation by the angle~$\\theta$,\n\\begin{align}\n\\begin{pmatrix}\\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}\n& = \\mathbf{R}_\\varphi(\\theta_{0}) \\begin{pmatrix}h_{1,0} \\\\ h_{2,0}\\end{pmatrix}\n=\\begin{pmatrix}c_{\\theta,0} & -s_{\\theta,0}\\\\ s_{\\theta,0}& c_{\\theta,0}\\end{pmatrix}\n\\begin{pmatrix}h_{1,0} \\\\ h_{2,0}\\end{pmatrix},\n\\label{eq:mixanglesrot}\n\\end{align}\nwhere we added subscripts ``0'' to indicate bare quantities.\nThe general argument can be applied to the neutral CP-even, the neutral \nCP-odd, and the charged Higgs fields of the THDM by replacing\n$h_1$, $h_2$ by $H$, $h$, or $G$, $A_0$ or $G^\\pm$, $H^\\pm$, respectively, and by substituting the angle $\\theta$ by \n$\\alpha$, $\\beta_n$, or $\\beta_c$.\nThe fields corresponding to mass eigenstates are renormalized using matrix-valued renormalization constants, so that the renormalization transformation reads\n\\begin{align}\n\\begin{pmatrix}h_{1,0} \\\\ h_{2,0} \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1 \\mathswitchr h_1} &\\frac{1}{2} \\delta Z_{\\mathswitchr h_1 \\mathswitchr h_2} \\\\\\frac{1}{2} \\delta Z_{\\mathswitchr h_2 \\mathswitchr h_1} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2 \\mathswitchr h_2} \\end{pmatrix} \\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix},&&\n  \\theta_{0} = \\theta + \\delta \\theta.\\label{eq:alpherendef}\n\\end{align}\nApplying this renormalization transformation to Eq.~\\eqref{eq:mixanglesrot} leads to\n\\begin{align}\n\\begin{pmatrix} \\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}={}\n& \\bigg[ \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  & \\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_2} \\\\ \\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n+\\begin{pmatrix}-s_{\\theta}  & -c_{\\theta} \\\\ c_{\\theta}& -s_{\\theta} \\end{pmatrix} \\delta \\theta \\bigg]\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}\n\\nonumber\\\\\n{}={}& \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  &\\frac{1}{2}(\\delta Z_{\\mathswitchr h_1\\mathswitchr h_2}- 2 \\delta \\theta) \n\\\\ \n\t\\frac{1}{2}(\\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}+ 2 \\delta \\theta)& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}.  \n\\label{eq:mixanglesrot2}\n\\end{align}\nOne can easily remove the dependence on the mixing angle \nwith a redefinition of the mixing \nfield renormalization constants by introducing\n\\begin{align}\n\\label{eq:redeffielren}\n \\delta \\tilde{Z}_{\\mathswitchr h_2\\mathswitchr h_1}= \\delta Z_{\\mathswitchr h_2\\mathswitchr h_1}+ 2 \\delta \\theta,\\qquad\n \\delta \\tilde{Z}_{\\mathswitchr h_1\\mathswitchr h_2}= \\delta Z_{\\mathswitchr h_1\\mathswitchr h_2}- 2 \\delta \\theta.\n \\end{align}\nThen, the Eq.~\\eqref{eq:mixanglesrot2} reads\n\\begin{align}\n\\begin{pmatrix} \\varphi_{1,0} \\\\ \\varphi_{2,0}\\end{pmatrix}=\n& \\begin{pmatrix}c_{\\theta}  & -s_{\\theta} \\\\ s_{\\theta}& c_{\\theta} \\end{pmatrix}\n\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_1\\mathswitchr h_1}  &\\frac{1}{2}\\delta \\tilde{Z}_{\\mathswitchr h_1\\mathswitchr h_2} \\\\ \n\t\\frac{1}{2}\\delta \\tilde{Z}_{\\mathswitchr h_2\\mathswitchr h_1}& 1+\\frac{1}{2}\\delta Z_{\\mathswitchr h_2\\mathswitchr h_2} \\end{pmatrix}\n\\begin{pmatrix}h_1 \\\\ h_2\\end{pmatrix}.\n\\end{align}\nObviously,\nthe dependence on $\\delta \\theta$ can always be removed from the Lagrangian \nby a redefinition of the field renormalization constants. \nAs a simple shift of the mixing field renormalization constant is performing the task, the renormalization of the mixing angle $\\theta$ can be seen as an additional field renormalization \n(as it is done, e.g., in Ref.~\\cite{Aoki2009}). \nThis argument is general and holds for any renormalization condition on $\\theta$.\nWithout loss of generality one can even assume that such a redefinition has already been performed and set $\\delta \\theta=0$ from the beginning, as done in method (b) in App.~\\ref{App:renormassa}.\nOf course, the bookkeeping of counterterms depends on the way $\\delta \\theta$ is treated.\nThis can be seen by considering the mass term of the potential. \nThe general bare mass term can be written using the rotation matrix $\\mathbf{R_\\varphi}(\\theta_{0})$ as\n\\begin{align}\nV_{\\mathswitchr h_1\\mathswitchr h_2} {}={} \n&\\frac{1}{2} \\,(h_{1,0}, h_{2,0})\\, \\mathbf{R}_\\varphi^{\\mathrm{T}}(\\theta_{0}) \\mathbf{M}_{\\varphi,0} \n\\mathbf{R}_\\varphi(\\theta_{0}) \\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix}\n\\nonumber\\\\\n{}={} &\\frac{1}{2} \\, (h_{1,0}, h_{2,0})\\, \n\\mathbf{R}_\\varphi^{\\mathrm{T}}(\\theta+\\delta \\theta) \n\\left(\\mathbf{M_\\varphi}+\\delta \\mathbf{M_\\varphi}\\right) \n\\mathbf{R_\\varphi}(\\theta+\\delta \\theta) \n\\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix}.\n\\end{align}\nThis expression can be expanded in terms of renormalized and counterterm contributions, yielding\n\\begin{align}\nV_{\\mathswitchr h_1\\mathswitchr h_2}=\n\\frac{1}{2} \\, (h_{1,0}, h_{2,0})\\, \\begin{pmatrix} \n\\hspace{-0pt}M_{\\mathswitchr h_1}^2+\\delta M_{\\mathswitchr h_1}^2 & \n\\hspace{-5pt}\\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1}^2)+f_{\\theta}(\\{ \\delta p\\}) \\\\ \\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1}^2)+f_{\\theta}(\\{ \\delta p\\}) & \n\\hspace{-0pt}M_{\\mathswitchr h_2}^2+\\delta M_{\\mathswitchr h_2}^2\\end{pmatrix} \\begin{pmatrix} h_{1,0}\\\\ h_{2,0} \\end{pmatrix},\n\\end{align}\nwhere we obtain off-diagonal terms from the counterterm $\\delta\\theta$\nof the mixing angle and from the renormalization of the mass matrix. The latter contribution depends on the independent counterterms $\\{ \\delta p\\}$ and is abbreviated by $f_{\\theta}(\\{ \\delta p\\})$.\nAt NLO, the mixing entry of the mass matrix reads\n\\begin{align}\n\\delta M_{\\mathswitchr h_1\\mathswitchr h_2}^2= \\delta \\theta (M_{\\mathswitchr h_2}^2-M_{\\mathswitchr h_1})+f_{\\theta}(\\{ \\delta p\\}), \\label{eq:NLOmassmixHh}\n\\end{align}\n{and since the renormalization of the {\\it redundant} mixing angle can be chosen freely,} \ncounterterm contributions in the Lagrangian can be shifted arbitrarily from mixing terms to mixing-angle counterterms.\n{Note that $f_{\\theta}(\\{ \\delta p\\})$ does not change\nby such redistributions, since it is fixed by the remaining renormalization \nconstants.}\n\n\\subsubsection{Renormalization with a diagonal mass matrix -- version (a)}\n\\label{sec:renormassb}\n\nIn this prescription, we use the angle $\\alpha$ as an independent parameter instead of $\\lambda_3$. \n{To define $\\alpha$ at NLO, we demand that \nthe mass matrix of the CP-even Higgs bosons (in the Lagrangian), written in terms of bare fields,\nis diagonal at all orders (i.e.\\ in terms of bare or renormalized parameters).}\nEquation~\\eqref{eq:Lambda3NLO} with\n\\begin{align}\\label{eq:MHhchoice2}\nM_{\\mathrm{Hh},0}^2 = 0 + \\delta M_{\\mathrm{Hh}}^2=0, \\qquad M_{\\mathrm{Hh}}^2 = 0\n\\end{align}\nthen defines the parameter $\\alpha$. \n{Note that (momentum-dependent)\nloop diagrams tend to destroy the diagonality \nof the matrix-valued two-point functions (inverse propagators)\nin the effective action as well.\nBelow,\nthe field renormalization will be chosen to compensate those loop effects  \nat the mass shells of the propagating particles.}\n{It is relation \\refeq{eq:MHhchoice2} that distinguishes \n$\\alpha$ from the case of a redundant mixing angle, such as $\\theta$ in the\nprevious section, which can be chosen freely or absorbed by field\nrenormalization constants.}\nThe relation between $\\alpha$ and $\\lambda_3$ of Eq.~\\eqref{eq:Lambda3NLO} is the same for bare and renormalized quantities \nand can be used to eliminate $\\lambda_3$ from the theory.\nFor each of the independent parameters of the mass parameter set \\eqref{eq:physparaNLO} we apply the renormalization transformation\n\\begin{align}\nM_{\\mathswitchr H,0}^2&=\\mathswitch {M_\\PH}^2+\\delta \\mathswitch {M_\\PH}^2,&M_{\\mathswitchr h,0}^2&= \\mathswitch {M_\\Ph}^2+\\delta \\mathswitch {M_\\Ph}^2,&M_{\\mathswitchr {A_0},0}^2 &= \\mathswitch {M_\\PAO}^2+\\delta \\mathswitch {M_\\PAO}^2,\\nonumber\\\\\nM_{\\mathswitchr {H^+},0}^2&= \\mathswitch {M_\\PHP}^2+\\delta \\mathswitch {M_\\PHP}^2,&M_{\\mathswitchr W,0}^2&= \\mathswitch {M_\\PW}^2+\\delta \\mathswitch {M_\\PW}^2,&M_{\\mathswitchr Z,0}^2&= \\mathswitch {M_\\PZ}^2+\\delta \\mathswitch {M_\\PZ}^2, \\nonumber\\\\\ne_0&= e+\\delta e,&\\lambda_{5,0} &= \\lambda_5+\\delta \\lambda_5&\\alpha_0 &= \\alpha+\\delta \\alpha,\\nonumber\\\\\n\\beta_0&= \\beta+\\delta \\beta,&t_{\\mathswitchr H,0}&= 0+\\delta t_\\mathswitchr H,&t_{\\mathswitchr h,0}&= 0+\\delta t_\\mathswitchr h, \\label{eq:physpararen2}\n\\end{align}\nso that the 12 parameter renormalization constants are\n\\begin{align}\n\\{\\delta p_\\mathrm{mass}\\}=\\{\\delta \\mathswitch {M_\\PH}^2, \\delta \\mathswitch {M_\\Ph}^2, \\delta \\mathswitch {M_\\PAO}^2, \\delta \\mathswitch {M_\\PHP}^2, \\delta \\mathswitch {M_\\PW}^2, \\delta \\mathswitch {M_\\PZ}^2, \\delta e, \\delta \\lambda_5, \\delta \\alpha,\\delta \\beta,\\delta t_\\mathswitchr H, \\delta t_\\mathswitchr h\\},\n\\label{eq:renconstsR}\n\\end{align}\ncorresponding to $\\{p_\\mathrm{mass}\\}$. \nIn this part we describe the commonly used tadpole renormalization (which is gauge dependent) and describe \na gauge-independent scheme in Sect.~\\ref{sec:FJscheme}.\n\nThe higher-order corrections of the mixing angles $\\beta_n$ and $\\beta_c$ are irrelevant according to Sect.~\\ref{sec:renmixangles} and we can choose\n\\begin{align}\n\\beta_{n,0}=\\beta_{c,0}= \\beta_0 = \\beta+\\delta \\beta,\n\\label{eq:mixanglect}\n\\end{align}\nwhich defines the mixing terms uniquely and ensures that the angles $\\beta_n,\\beta_c$, and $\\beta$ do not have to be distinguished at any order.\nFrom these conditions, we can compute the mass mixing terms from Eq.~\\eqref{eq:goldstonemassmixing} and Eq.~\\eqref{eq:chargedgoldstonemassmixing} to\n\\begin{align}\n M_{\\mathswitchr {G_0}\\mathswitchr {A_0},0}^2 &= 0+\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n=-e\\frac{ \n\\delta t_\\mathswitchr H s_{\\alpha -\\beta }+\n\\delta t_\\mathswitchr h c_{\\alpha -\\beta }\n}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}},&\\nonumber\\\\\n M_{\\mathswitchr {G}\\mathswitchr {H^+},0}^2 &= 0+\\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2\n=-e\\frac{ \n\\delta t_\\mathswitchr H s_{\\alpha -\\beta }+\n\\delta t_\\mathswitchr h c_{\\alpha -\\beta }\n}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}}. \\label{eq:Mabchoice2}\n\\end{align}\nThe field re\\-nor\\-ma\\-li\\-zation is performed \nfor each field corresponding to mass eigenstates,\n\\begin{align}\n\\begin{pmatrix}H_0 \\\\ h_0 \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_\\mathswitchr H &\\frac{1}{2} \\delta Z_{\\mathswitchr H\\mathswitchr h} \\\\\\frac{1}{2} \\delta Z_{\\mathswitchr h\\mathswitchr H} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr h} \\end{pmatrix} \\begin{pmatrix}H \\\\ h\\end{pmatrix},\\nonumber\\\\\n\\begin{pmatrix}G_{0,0} \\\\ A_{0,0} \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr {G_0}} &\\frac{1}{2}\\delta  Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}} \\\\\\frac{1}{2}\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}} &1+\\frac{1}{2}\\delta Z_\\mathswitchr {A_0} \\end{pmatrix} \\begin{pmatrix}G_0 \\\\ A_0\\end{pmatrix},\\nonumber\\\\\n\\begin{pmatrix}G^\\pm_0 \\\\ H^\\pm_0 \\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathrm{G}^+} &\\frac{1}{2}\\delta  Z_{\\mathrm{GH}^+} \\\\\\frac{1}{2}\\delta Z_{\\mathrm{HG}^+} &1+\\frac{1}{2}\\delta Z_{\\mathswitchr H^+} \\end{pmatrix} \\begin{pmatrix}G^\\pm \\\\ H^\\pm\\end{pmatrix},\\label{eq:physfieldren}\n\\end{align}\nwith the field renormalization constants $\\delta Z_\\mathswitchr H$, $\\delta Z_{\\mathswitchr H\\mathswitchr h}$, $\\delta Z_{\\mathswitchr h\\mathswitchr H}$, $\\delta Z_{\\mathswitchr h}$,  $\\delta Z_{\\mathswitchr {G_0}}$, \n$\\delta  Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}$, $\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}$, $\\delta Z_\\mathswitchr {A_0}$,  $\\delta Z_{\\mathrm{G}^+}$, $\\delta  Z_{\\mathrm{GH}^+}$, $\\delta Z_{\\mathrm{HG}^+}$, and $\\delta Z_{\\mathswitchr H^+}$ for the CP-even, the CP-odd, and the charged Higgs fields.\nWe denote the complete set of parameter and field renormalization constants with $\\{\\delta \\mathbf{R}_\\mathrm{mass}\\}$. All renormalization constants \nare of $\\order{\\alpha_{\\mathrm{em}}}$, i.e.\\ all contributions of $\\order{\\alpha_{\\mathrm{em}}^2}$ or higher are omitted. \n\n\\begin{sloppypar}\nApplying the renormalization transformations~\\refeq{eq:physpararen2}, \\refeq{eq:physfieldren} \nto the bare potential of Eq.~\\eqref{eq:NLObarepot} and linearizing in the renormalization constants results in \n$V(\\{p_\\mathrm{mass}\\})+\\delta V(\\{p_\\mathrm{mass}\\},\\{\\delta \\mathbf{R}_\\mathrm{mass}\\})$\nwith the LO potential as in Eq.~\\eqref{eq:NLObarepot}, but with renormalized quantities and the counterterm potential of $\\order{\\alpha_\\mathrm{em}}$,\n\\begin{align}\n\\delta V(\\{p_\\mathrm{mass}\\},\\{\\delta \\mathbf{R}_\\mathrm{mass}\\})=&- \\delta t_\\mathswitchr H H -\\delta t_\\mathswitchr h h \\nonumber\\\\\n&+\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\PH}^2 +\\delta Z_{\\mathswitchr H} \\mathswitch {M_\\PH}^2\\right) H^2 +\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\Ph}^2 +\\delta Z_{\\mathswitchr h} \\mathswitch {M_\\Ph}^2\\right) h^2 \\nonumber\\\\\n&+\\frac{1}{2} \\left(\\delta \\mathswitch {M_\\PAO}^2 +\\delta Z_{\\mathswitchr {A_0}} \\mathswitch {M_\\PAO}^2\\right) A_0^2 + \\left(\\delta \\mathswitch {M_\\PHP}^2+\\delta Z_{\\mathswitchr {H^+}} \\mathswitch {M_\\PHP}^2\\right)H^+ H^-\\nonumber\\\\\n&+ \\frac{e}{4 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}} \\left(\n-\\delta t_\\mathswitchr H c_{\\alpha-\\beta}\n+\\delta t_\\mathswitchr h s_{\\alpha-\\beta}\n\\right) (G_0^2 + 2 G^+ G^-)\\nonumber\\\\\n&+ \\frac{1}{2}\\left( \\mathswitch {M_\\PH}^2\\delta Z_{\\mathswitchr H\\mathswitchr h}+\\mathswitch {M_\\Ph}^2 \\delta Z_{\\mathswitchr h\\mathswitchr H}\\right) H h\\nonumber\\\\\n&+\\frac{1}{2} \\left( \\mathswitch {M_\\PAO}^2 \\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}} \n+ 2\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2 \\right) G_0 A_0\n\\nonumber\\\\\n&+ \\frac{1}{2} \\left( \\mathswitch {M_\\PHP}^2 \\delta Z_{\\mathswitchr H\\mathswitchr {G^+}} \n+2\\delta M_{\\mathswitchr {G}\\mathswitchr H+}^2 \\right)(H^+ G^-+G^+ H^-)\n\\nonumber \\\\\n&+ \\textrm{interaction terms}.\\label{eq:dVH2p}\n\\end{align}\nThe interaction terms are derived in\nthe same way, but they are very lengthy and not shown here. \n\\end{sloppypar}\n\nThe prescription for the field renormalization~\\eqref{eq:physfieldren} is non-minimal in the sense that a renormalization of the doublets with two renormalization constants,\n\\begin{align}\n\\Phi_{1,0} = Z_{\\mathswitchr H_1}^{1\/2} \\Phi_1& \\textstyle = \\Phi_1 \\left(1+\\frac{1}{2} \\delta Z_{\\mathswitchr H_1}\\right),\\nonumber\\\\\n\\Phi_{2,0} = Z_{\\mathswitchr H_2}^{1\/2} \\Phi_2& \\textstyle = \\Phi_2 \\left(1+\\frac{1}{2} \\delta Z_{\\mathswitchr H_2}\\right),\\label{eq:inputfieldren}\n\\end{align}\nactually would be\nsufficient to cancel the UV divergences. However, the prescription with matrix-valued renormalization constants allows us to renormalize each field on-shell. The UV-divergent parts of \nthe renormalization constants in Eq.~\\refeq{eq:physfieldren}\ncannot be independent, and relations between the UV-divergent parts of the two prescriptions exist. They can be obtained by applying the renormalization prescription \\eqref{eq:inputfieldren} to the left-hand side and \\eqref{eq:physfieldren} to the right-hand side of Eqs.~\\eqref{eq:rotations}, transforming thereafter the interaction states on the left-hand side to mass eigenstates and comparing both sides.\nThis results in\n\\begin{align}\n\\delta Z_\\mathswitchr h\\big|_\\mathrm{UV}&=s_{\\alpha}^2\\;\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+c_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr H\\big|_\\mathrm{UV}&=c_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+s_{\\alpha}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr H\\mathswitchr h}\\big|_\\mathrm{UV}&=s_{\\alpha} c_{\\alpha}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)+2 \\delta \\alpha\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr h\\mathswitchr H}\\big|_\\mathrm{UV}&=s_{\\alpha} c_{\\alpha}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)-2 \\delta \\alpha\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr {A_0}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr H^+}\\big|_\\mathrm{UV}=s_{\\beta}^2\\;\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+c_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_\\mathswitchr {G_0}\\big|_\\mathrm{UV}&=\\delta Z_{G^+}\\big|_\\mathrm{UV}=c_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+s_{\\beta}^2\\; \\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\big|_\\mathrm{UV}=s_{\\beta} c_{\\beta}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)+2 \\delta \\beta\\big|_\\mathrm{UV},\\nonumber\\\\\n\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}\\big|_\\mathrm{UV}&=\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}\\big|_\\mathrm{UV}=s_{\\beta} c_{\\beta}\\; \\left(-\\delta Z_{\\mathrm{H}_1}\\big|_\\mathrm{UV}+\\delta Z_{\\mathrm{H}_2}\\big|_\\mathrm{UV}\\right)-2 \\delta \\beta\\big|_\\mathrm{UV}.\\label{eq:fieldrenconstrel}\n\\end{align}\nWe will use these relations to derive UV-divergent parts for specific renormalization constants in Sect.~\\ref{sec:diffrenschemes}. In App.~\\ref{App:renormassa} we discuss a different choice of the mixing angles which is suited for \nthe renormalization with $\\lambda_3$ as an independent parameter. \n\n\\subsection{The Higgs kinetic part}\n\\label{sec:Hkinren}\n\nAfter expressing the Higgs kinetic term\n\\begin{align}\n \\mathcal{L}_\\mathrm{H,kin}=(D_\\mu\\Phi_1)^\\dagger (D^\\mu \\Phi_1)+(D_\\mu\\Phi_2)^\\dagger (D^\\mu \\Phi_2)\n\\end{align}\nin terms of\nbare physical fields, mixing angles, and parameters, one can apply the renorma\\-lization transformations \\eqref{eq:physpararen2} and~\\eqref{eq:physfieldren} to obtain the counterterm part of the kinetic Lagrangian which introduces \nscalar--vector mixing terms. The explicit terms are stated in App.~\\ref{App:SV-terms}.\n\n\\subsection{Fermionic and gauge parts}\n\\label{sec:fermgauge}\nSince the THDM extension of the SM does not affect the gauge and the fermion parts of the Lagrangian,  \nthe renormalization of these parts is identical to the SM case. \n{It is described in detail in Ref.~\\cite{Denner:1991kt} in \nBHS convention, which is included in the standard implementation of the \\textsc{FeynArts} package~\\cite{Hahn2001}.\nOther renormalization prescriptions\ncan, e.g., be found in \\citeres{Denner:2004bm,Kniehl:2009kk}.}\nTherefore, we here do not repeat the renormalization procedure of the CKM matrix, which does not change in the \ntransition from the SM to the THDM, and spell out the renormalization of the fermionic parts only for the\ncase where the CKM matrix is set to the unit matrix, i.e.\\ $V_{ij}=\\delta_{ij}$.\nThe transformation of the left- and right-handed fermions and of the gauge-boson fields are\n\\begin{align}\n f^\\sigma_{i,0} & = \\left(1 + \\textstyle\\frac{1}{2}\\delta Z^{f,\\sigma}_i\\right) f^\\sigma_i,\\qquad  \n\\rlap{$f=\\nu,l,u,d, \\quad \\sigma={\\mathrm{L}},{\\mathrm{R}}, \\quad i=1,2,3,$}\\\\\n \\begin{pmatrix}Z_0 \\\\ A_\\mathrm{bare}\\end{pmatrix}&=\\begin{pmatrix}1+\\frac{1}{2}\\delta Z_{\\mathswitchr Z\\PZ} &\\frac{1}{2} Z_{\\mathswitchr Z{\\mathrm{PA}}} \\\\\\frac{1}{2} \\delta Z_{{\\mathrm{PA}}\\mathswitchr Z} &1+\\frac{1}{2}\\delta Z_{{\\mathrm{PA}}\\PA} \\end{pmatrix} \\begin{pmatrix}Z \\\\ A\\end{pmatrix}, &\nW^\\pm_0 &= \\left(1 + \\textstyle\\frac{1}{2} \\delta Z_\\mathswitchr W\\right) W^\\pm,\n\\end{align}\nwhere the bare photon field is denoted $A_\\mathrm{bare}$ to distinguish it from the \nneutral CP-odd field~$A_0$.\nMixing between left-handed up- and down-type fermions does not occur, owing to charge conservation. Inserting this into the Lagrangian directly delivers the renormalized and the counterterm Lagrangians.\n\n\\subsection{Yukawa part}\n\\label{sec:YukRen}\n\nThe renormalization of the Yukawa sector is straightforward in Type I, II, lepton-specific, and flipped models and can be done by taking the Lagrangian of Eq.~\\eqref{eq:ffH}, \nreplacing the vev $v$, and applying the renormalization transformations of \nSect.~\\ref{sec:renormassb}, as well as a renormalization of the fermion masses,\n\\begin{align}\n  m_{f,i,0} &= m_{f,i} + \\delta m_{f,i}.\n  \\end{align}\nThe corresponding  counterterm couplings are stated in App.~\\ref{App:Yuk-CT}.\n\n\\section{Renormalization conditions}\n\\label{sec:renconditions}\n\nThe renormalization constants are fixed using on-shell conditions \nfor all parameters that are accessible by experiments. \nHowever, not all parameters of the THDM correspond to measurable quantities, so that we renormalize three parameters of the Higgs sector in the $\\overline{\\mathrm{MS}}$ scheme, where the renormalization constants only contain the standard UV divergence\n\\begin{align}\n  \\Delta_\\mathrm{UV}=\\frac{2}{4-D}-\\gamma_\\mathrm{E}+\\ln{4\\pi}=\\frac{1}{\\epsilon}-\\gamma_\\mathrm{E}+\\ln{4\\pi}\n\\end{align}\nin $D=4-2 \\epsilon$ dimensions and with the Euler--Mascheroni constant $\\gamma_\\mathrm{E}$.\nIn Sect.~\\ref{sec:diffrenschemes}, four different options, resulting in four different renormalization schemes, are presented.\nAn overview over the renormalization constants introduced in the previous section is shown in Tab.~\\ref{tab:renconsts}. In the following, we adapt the notation of Ref.~\\cite{Denner:1991kt}, i.e.\\ we use the same symbols for the renormalized and the corresponding unrenormalized \nGreen function, self-energies, etc., but denoting the renormalized quantities with a caret.\n\n\\begin{table}\n  \\centering\n\\begin{tabular}{|lrl|}\\hline\nParameters:&&\\\\\\hline\n&EW (3): & $\\delta \\mathswitch {M_\\PZ}^2,\\delta \\mathswitch {M_\\PW}^2,\\delta e, (\\delta \\mathswitch {c_{\\scrs\\PW}},\\delta \\mathswitch {s_{\\scrs\\PW}})$\\\\\n&fermion masses (9): & $\\delta m_{f,i},\\qquad f=l,u,d,\\quad i=1,2,3$\\\\%\\delta m_{u,i},\\delta m_{d,i}$\\\\\n&Higgs masses (4): & $\\delta \\mathswitch {M_\\PH}^2,\\delta \\mathswitch {M_\\Ph}^2,\\delta \\mathswitch {M_\\PAO}^2,\\delta \\mathswitch {M_\\PHP}^2$\\\\\n&Higgs potential (3): & $\\delta \\lambda_3 \\text{ or } \\delta \\alpha,\\delta \\lambda_5, \\delta \\beta$\\\\\n&tadpoles (2): & $\\delta t_\\mathswitchr H,\\delta t_\\mathswitchr h$\\\\\\hline\nFields:&&\\\\\\hline\n&EW (5): & $\\delta Z_\\mathswitchr W,\\delta Z_{\\mathswitchr Z\\PZ},\\delta Z_{\\mathswitchr Z{\\mathrm{PA}}},\\delta Z_{{\\mathrm{PA}}\\mathswitchr Z},\\delta Z_{{\\mathrm{PA}}\\PA}$\\\\\n&left-handed fermions (12): & $\\delta Z^{f,{\\mathrm{L}}}_i,\\qquad f=\\nu,l,u,d,\\quad i=1,2,3$ \\\\\n&right-handed fermions (9): & $\\delta Z^{f,{\\mathrm{R}}}_i,\\qquad f=l,u,d,\\quad i=1,2,3$ \\\\\n&Higgs (12): & $\\delta Z_{\\mathswitchr H},\\delta Z_{\\mathswitchr H\\mathswitchr h},\\delta Z_{\\mathswitchr h\\mathswitchr H},\\delta Z_\\mathswitchr h$\\\\\n&& $\\delta Z_{\\mathswitchr {A_0}},\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}},\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}},\\delta Z_\\mathswitchr {G_0}$\\\\\n&&$\\delta Z_{\\mathswitchr {H^+}},\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}},\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}},\\delta Z_{\\mathswitchr {G^+}}$\\\\\\hline\n\\end{tabular}  \n  \\caption{The renormalization constants used to describe the THDM, separated into sectors of parameter and field renormalization. The renormalization constants in parentheses are not independent, but useful for a better bookkeeping. The numbers in parentheses are the numbers of independent renormalization constants. In total there are 38 field and 20 parameter renormalization constants to fix. }\n\\label{tab:renconsts}\n\\end{table}%\n\n\\subsection{On-shell renormalization conditions}\n\\label{sec:onshellrenconditions}\n\n\\subsubsection{Higgs sector}\n\n\\paragraph{Tadpoles:}\nWe start with the \n(irreducible) renormalized one-point vertex function\n\\begin{align}\n\\hat\\Gamma^{\\mathswitchr H,\\mathswitchr h}=\n{\\mathrm{i}}\\hat{T}^{\\mathswitchr H,\\mathswitchr h}=\\,\\parbox{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpole1.eps}}\n\\hspace{20pt}.\n\\end{align}\nAt NLO the renormalized tadpole $\\hat{T}$ consists of a counterterm contribution $\\delta t$ and an unrenormalized one-loop irreducible one-point vertex function $T$\nresulting from the diagrams shown in Fig.~\\ref{fig:tadpoles}. \n\\begin{figure}\n\\vspace{-30pt}\n\\input{diagrams\/tadpoles}\n\\vspace{-40pt}\n\\caption{Generic tadpole diagrams. There is one diagram for each massive fermion $(f)$, \nscalar $(S)$, gauge-boson $(V)$, and ghost field $(u)$.}\n\\label{fig:tadpoles}\n\\end{figure}\nIn the conventional, but gauge-dependent tadpole treatment one demands that these two contributions cancel each other,\n\\begin{align}\n\\hat{T}_\\mathswitchr H&=\\delta t_\\mathswitchr H+T_\\mathswitchr H=0,&\\hat{T}_\\mathswitchr h&=\\delta t_\\mathswitchr h+T_\\mathswitchr h=0,\n\\label{eq:tadpoleren}\n\\end{align}\nwhich means that explicit tadpole diagrams can be omitted from the \nset of one-loop diagrams for any process. \nHowever, as a remnant of the tadpole diagrams the tadpole counter\\-terms \nappear also in various coupling \ncounterterms and need to be calculated. \nIt should be noted \nthat the condition on the tadpoles does not affect physical observables \nas long as physically equivalent renormalization conditions are imposed on\nthe input parameters. This is, in particular, the case for on-shell renormalization,\nwhere input parameters are tied to measurable quantities.\nThat means, changing the tadpole renormalization condition shifts contributions between \nGreen functions and counterterms and merely changes the bookkeeping,\nbut the dependence of predicted observables on renormalized input parameters remains the same.\nThe situation changes if an {$\\overline{\\mathrm{MS}}$}{} renormalization condition is used, where the counterterm\nis not fixed by a measurable quantity, \nbut by a divergence in a specific Green function, so that\nthe gauge-dependent tadpole terms can affect the relation between renormalized input parameters and\nobservables.\nThe gauge-independent treatment of tadpole contributions is based on a different renormalization condition and discussed in Sect.~\\ref{sec:FJscheme}. \n\\vspace{10pt}\n\n\\paragraph{Scalar self-energies:}\nFor scalars, the irreducible two-point functions  with momentum transfer $k$ are\n\\begin{align}\n\\hat{\\Gamma}^{{ab}}(k)&=\\quad \\parbox{3.0cm}{\n\\includegraphics[scale=1]{.\/diagrams\/SE1.eps}} \\quad\n=\\mathrm{i}\\delta_{{ab}} (k^2-M_{a}^2)+\\mathrm{i} \\hat{\\Sigma}^{{ab}}(k),\n\\label{eq:two-def}\n\\end{align}\nwhere both fields $a,b$ are incoming and $a,b=H,h,A_0,G_0,H^\\pm,G^\\pm$. The first term is the LO two-point vertex function, while the functions $\\hat{\\Sigma}^{{ab}}$ are the renormalized self-energies containing loop diagrams and counterterms. Generic diagrams contributing to the self-energies are shown in Fig.~\\ref{fig:selfenergies}. \n\\begin{figure}\n\\vspace{-10pt}\n\\input{diagrams\/selfenergies}\n\\vspace{-20pt}\n\\caption{Generic self-energy diagrams for the heavy, neutral CP-even Higgs self-energy, for other scalar self-energies the diagrams are analogous. Only massive particles contribute.}\n\\label{fig:selfenergies}\n\\end{figure}\nMixing occurs only between $H$ and $h$,\nbetween $A_0$ and $G_0$, and between $H^\\pm$ and $G^\\pm$.\nFor the neutral CP-even fields we obtain\n\\begin{subequations}\n\\label{eq:neutralSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr h\\Ph}(k^2)&= \\Sigma^{\\mathswitchr h\\Ph}(k^2) + \\delta Z_\\mathswitchr h(k^2-\\mathswitch {M_\\Ph}^2)  - \\delta \\mathswitch {M_\\Ph}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr H\\PH}(k^2)&= \\Sigma^{\\mathswitchr H\\PH}(k^2) + \\delta Z_\\mathswitchr H(k^2-\\mathswitch {M_\\PH}^2)  - \\delta \\mathswitch {M_\\PH}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(k^2)&= \\Sigma^{\\mathswitchr H\\mathswitchr h}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr H\\mathswitchr h}(k^2-\\mathswitch {M_\\PH}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr h\\mathswitchr H}(k^2-\\mathswitch {M_\\Ph}^2) - \\delta M_{\\mathswitchr H\\mathswitchr h}^2, \\label{eq:twopointHh}\n\\end{align}\n\\end{subequations}\nand for the CP-odd fields\n\\begin{subequations}\n\\label{eq:CPoddSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr {A_0}\\PAO}(k^2)&= \\Sigma^{\\mathswitchr {A_0}\\PAO}(k^2) + \\delta Z_{\\mathswitchr {A_0}}(k^2-\\mathswitch {M_\\PAO}^2)  - \\delta \\mathswitch {M_\\PAO}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr {G_0}\\PGO}(k^2)&= \\Sigma^{\\mathswitchr {G_0}\\PGO}(k^2) + \\delta Z_\\mathswitchr {G_0} k^2  - \\delta M_\\mathswitchr {G_0}^2,\\\\\n\\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2)&= \\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}(k^2-\\mathswitch {M_\\PAO}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}} \\;k^2 - \\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2.\n\\end{align}\n\\end{subequations}\nThe charged sector involves the following self-energies,\n\\begin{subequations}\n\\label{eq:chargedSE}\n\\begin{align}\n\\hat{\\Sigma}^{\\mathrm{H}^+ \\mathrm{H}^-}(k^2)&= \\Sigma^{\\mathrm{H}^+ \\mathrm{H}^-}(k^2) + \\delta Z_{\\mathswitchr {H^+}}(k^2-\\mathswitch {M_\\PHP}^2)  - \\delta \\mathswitch {M_\\PHP}^2,\\\\\n\\hat{\\Sigma}^{\\mathrm{G}^+\\mathrm{G}^-}(k^2)&= \\Sigma^{\\mathrm{G}^+\\mathrm{G}^-}(k^2) + \\delta Z_{\\mathrm{G}^+}\\; k^2  - \\delta M_\\mathswitchr {G^+}^2,\\\\\n\\hat{\\Sigma}^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2)&= \\Sigma^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2) + \\frac{1}{2}\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}(k^2-\\mathswitch {M_\\PHP}^2) +\\frac{1}{2}\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\; k^2 - \\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2.\n\\end{align}\n\\end{subequations}\nThe mass mixing constants are given in Eqs.~\\eqref{eq:MHhchoice2}  and \\eqref{eq:Mabchoice2}.\nOn these two-point functions we now impose our renormalization conditions. First, we  fix the renormalized mass parameters to the on-shell\nvalues, so that the zeros of the real parts of the one-particle-irreducible two-point functions are located at the squares of the physical masses:\n\\begin{align}\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2)&=0,& \\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2)&=0,\\nonumber\\\\\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2)&=0,&\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathrm{H}^+\\mathrm{H}^-}(\\mathswitch {M_\\PHP}^2)&=0.\n\\label{eq:massrenconddef}\n\\end{align}\nUsing Eqs.~\\eqref{eq:neutralSE}, \\eqref{eq:CPoddSE}, and \\eqref{eq:chargedSE}, fixes  the mass renormalization constants to\n\\begin{align}\n\\delta \\mathswitch {M_\\PH}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2),&\\delta \\mathswitch {M_\\Ph}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2),\\nonumber\\\\\n\\delta \\mathswitch {M_\\PAO}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2),&\\delta M_{\\mathswitchr {H^+}}^2&= \\mathrm{Re}\\, \\Sigma^{\\mathrm{H}^+\\mathrm{H}^-}(M_{\\mathswitchr H^+}^2).\n\\end{align}\nFor the propagators of the fields, we demand that the residues of the particle poles are not changed by higher-order corrections.\nThis  determines the \ndiagonal field renormalization constants by conditions on the one-particle-irreducible two-point functions,\n\\begin{align}\n\\lim_{k^2\\to \\mathswitch {M_\\PH}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr H\\PH}(k^2)}{k^2-\\mathswitch {M_\\PH}^2}&=-1,& \n\\lim_{k^2\\to \\mathswitch {M_\\Ph}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr h\\Ph}(k^2)}{k^2-\\mathswitch {M_\\Ph}^2}&=-1,\\nonumber\\\\\n\\lim_{k^2\\to \\mathswitch {M_\\PAO}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathswitchr {A_0}\\PAO}(k^2)}{k^2-\\mathswitch {M_\\PAO}^2}&=-1,&\n\\lim_{k^2\\to \\mathswitch {M_\\PHP}^2}\\mathrm{Re}\\,\\frac{\\mathrm{i}\\,\\hat{\\Gamma}^{\\mathrm{H}^+\\mathrm{H}^-}(k^2)}{k^2-\\mathswitch {M_\\PHP}^2}&=-1,\n\\end{align}\nwhich implies\n\\begin{align}\n\\delta Z_\\mathswitchr H&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr H\\PH}(\\mathswitch {M_\\PH}^2),\n&\\delta Z_\\mathswitchr h&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr h\\Ph}(\\mathswitch {M_\\Ph}^2), \\label{eq:dZHdef}\\nonumber\\\\\n\\delta Z_{\\mathswitchr {A_0}}&= -\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr {A_0}\\PAO}(\\mathswitch {M_\\PAO}^2),\n&\\delta Z_{\\mathrm{H}^+}&= -\\mathrm{Re}\\, \\Sigma'^{\\mathrm{H}^+\\mathrm{H}^-}(\\mathswitch {M_\\PHP}^2),\n\\end{align}\nwhere we introduced \n${\\Sigma}'(k^2)$ as the derivative w.r.t.\\  the argument $k^2$.\nTo fix the mixing renormalization constants, we enforce the condition that on-mass-shell fields do not mix, i.e.\n\\begin{align}\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)&=0,& \\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)&=0,\n\\nonumber\\\\\n\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)&=0,&\\mathrm{Re}\\,\\hat{\\Sigma}^{\\mathswitchr {G}^+\\mathswitchr H^-}(\\mathswitch {M_\\PHP}^2)&=0.\n\\label{eq:mixrencond}\n\\end{align}\nAfter inserting the renormalized self-energies we obtain\n\\begin{align}\n\\delta Z_{\\mathswitchr H\\mathswitchr h}&= 2 \\, \\frac{\\delta M_{\\mathswitchr H\\mathswitchr h}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2},\n&\\delta Z_{\\mathswitchr h\\mathswitchr H}&= 2 \\, \\frac{\\delta M_{\\mathswitchr H\\mathswitchr h}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2},\n\\nonumber\\\\\n\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}&= 2 \\, \\frac{\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)}{\\mathswitch {M_\\PAO}^2},\n&\\delta Z_{\\mathswitchr {G}\\mathrm{H}^+}&= 2 \\, \\frac{\\delta M_{\\mathswitchr {G}\\mathswitchr {H^+}}^2-\\mathrm{Re}\\,\\Sigma^{\\mathrm{H}^+\\mathrm{G}^-}(\\mathswitch {M_\\PHP}^2)}{\\mathswitch {M_\\PHP}^2}.\\label{eq:mixfieldrendef}\n\\end{align}\nSince Goldstone-boson fields \ndo not correspond to physical states, we do not render Green functions with external \nGoldstone bosons finite, so that we need not fix the constants \n$\\delta Z_{\\mathswitchr {G_0}}$, $\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}$, $\\delta Z_{\\mathswitchr {G^+}}$, $\\delta Z_{\\mathswitchr H\\mathswitchr {G^+}}$;\nwe could even set them to zero consistently.\nThe possible \n$\\mathswitchr Z\\mathswitchr {A_0}$ and $\\mathrm{W}^\\pm \\mathswitchr H^\\mp$ mixings\nvanish for physical on-shell gauge bosons due to the Lorentz structure of the two-point function \nand the fact that polarization vectors $\\varepsilon^\\mu$ are orthogonal to the corresponding momentum.\nUsing the convention\n\\begin{align}\n\\hat\\Gamma^{\\mathswitchr Z\\mathswitchr {A_0}}_\\mu(k) &= k_\\mu \\, \\hat\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)  = \nk_\\mu \\left[ \\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)-\\textstyle\\frac{1}{2}\\mathswitch {M_\\PZ}\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}\\right],\n\\nonumber\\\\\n\\hat\\Gamma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}_\\mu(k) &= k_\\mu \\, \\hat\\Sigma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)  = \nk_\\mu \\left[ \\Sigma^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)\\pm\\textstyle\\frac{\\mathrm{i}}{2}\\mathswitch {M_\\PW}\\delta Z_{\\mathswitchr {G}\\mathswitchr {H^+}}\\right],\n\\end{align}\nwhere all fields are incoming and $k$ is the incoming momentum of the gauge-boson fields,\nthe vector--scalar mixing self-energies obey\n\\begin{align}\n\\left.\\varepsilon^\\mu_\\mathswitchr Z k_\\mu \\,\\mathop{\\mathrm{Re}}\\nolimits\\,\\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)\\right|_{k^2= \\mathswitch {M_\\PZ}^2}=0,\n\\qquad &\n\\left.\\varepsilon^\\mu_\\mathswitchr W k_\\mu \\,\\mathop{\\mathrm{Re}}\\nolimits\\,\\hat{\\Sigma}^{\\mathswitchr W^\\pm\\mathswitchr H^\\mp}(k^2)\\right|_{k^2= \\mathswitch {M_\\PW}^2}=0.\n\\end{align}\nThe mixing self-energies on the other on-shell points \n$k^2=\\mathswitch {M_\\PAO}^2$ and $k^2=\\mathswitch {M_\\PHP}^2$, respectively, \nare connected to the mixing of \n$\\mathswitchr {A_0}$ or $\\mathswitchr H^\\pm$ with the Goldstone-boson fields of the\n$\\mathswitchr Z$ or the $\\mathswitchr W$ boson and can be calculated from a BRST symmetry \\cite{Becchi:1975nq}. The BRST variation of the Green functions of one anti-ghost and a Higgs field\n\\begin{align}\n \\delta_\\mathrm{BRST} \\langle 0 | T \\bar{u}^\\mathswitchr Z(x) A_0(y)|0\\rangle=0, &&\n \\delta_\\mathrm{BRST} \\langle 0 | T \\bar{u}^\\pm(x) H^\\pm(y)|0\\rangle=0,\n\\end{align}\nimplies Slavnov--Taylor identities. While the variation of the anti-ghost fields yields the gauge-fixing term, the variation of the Higgs fields introduces ghost contributions which \nvanish for on-shell momentum resulting in%\n\\footnote{A particularly simple, alternative \nway to derive these identities is to exploit the gauge invariance\nof the effective action in the background-field gauge, as \ndone in \\citere{Denner:1994xt} for the SM. The respective Ward identities for the background\nfields differ from the Slavnov--Taylor identities only by off-shell terms, which vanish\non the particle poles. Generalizing the derivation of \\citere{Denner:1994xt} to the THDM\nand adapting the results to our conventions for self-energies,\nthe desired Ward identities for the unrenormalized background fields read\n\\begin{align}\n0 &= k^2 {\\Sigma}^{\\hat\\mathswitchr Z\\hat{\\mathrm{PA}}_0}(k^2)+ \\mathswitch {M_\\PZ} {\\Sigma}^{\\hat{\\mathrm{PA}}_0\\hat\\mathswitchr {G}_0}(k^2)\n+\\frac{e}{2\\mathswitch {c_{\\scrs\\PW}}\\mathswitch {s_{\\scrs\\PW}}}\\left(T^{\\hat\\mathswitchr H}s_{\\beta-\\alpha}-T^{\\hat\\mathswitchr h}c_{\\beta-\\alpha}\\right),\n\\\\\n0 &= \nk^2 {\\Sigma}^{\\hat\\mathswitchr W^\\pm\\hat\\mathswitchr H^\\mp}(k^2)\\mp \\mathrm{i} \\mathswitch {M_\\PW} {\\Sigma}^{\\hat\\mathswitchr {G}^\\pm\\hat\\mathswitchr H^\\mp}(k^2)\n\\mp\\frac{\\mathrm{i} e}{2\\mathswitch {s_{\\scrs\\PW}}}\\left(T^{\\hat\\mathswitchr H}s_{\\beta-\\alpha}-T^{\\hat\\mathswitchr h}c_{\\beta-\\alpha}\\right),\n\\end{align}\nwhere the carets on the fields indicate background fields.\nSetting $k^2$ to $\\mathswitch {M_\\PAO}^2$ or $\\mathswitch {M_\\PHP}^2$, respectively, and adding the \nrelevant renormalization constants, directly leads to the identities \\refeq{eq:ZA-WI}\nand \\refeq{eq:WH-WI}.\n}\n\\begin{align}\n\\label{eq:ZA-WI}\n \\left[k^2 \\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)+ \\mathswitch {M_\\PZ} \\hat{\\Sigma}^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(k^2)\\right]_{k^2= \\mathswitch {M_\\PAO}^2}=0,\\\\\n \\left[k^2 \\hat{\\Sigma}^{\\mathrm{W}^\\pm\\mathswitchr H^\\mp}(k^2)\\mp \\mathrm{i} \\mathswitch {M_\\PW} \\hat{\\Sigma}^{\\mathrm{G}^\\pm\\mathrm{H}^\\mp}(k^2)\\right]_{k^2= \\mathswitch {M_\\PHP}^2}=0.\n\\label{eq:WH-WI}\n\\end{align}\nWe have verified these identities analytically and numerically.\nTogether with the renormalization condition of \nEq.~\\eqref{eq:mixrencond} we conclude that \n\\begin{align}\n\\hat{\\Sigma}^{\\mathswitchr Z\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)=0, \\quad \\hat{\\Sigma}^{\\mathrm{W}^\\pm\\mathswitchr H^\\mp}(\\mathswitch {M_\\PHP}^2)=0.\n\\end{align}\nThis set of renormalization conditions ensures that no on-shell two-point vertex function obtains any one-loop corrections,\nand the corresponding external\nself-energy diagrams do not have to be taken into account in any calculation. \n\n\\subsubsection{Electroweak sector}\n\nThe fixing of the renormalization constants of the electroweak sector is identical to the SM case. The mass renormalization constants are fixed in such a way that the squares of the masses correspond to the \n(real parts of the) locations of the poles of the gauge-boson propagators. The field renormalization constants are fixed by the conditions that residues of on-shell gauge-boson propagators  do not obtain higher-order corrections, and that on-shell gauge bosons do not mix. For a better bookkeeping we also keep the dependent renormalization constants $\\delta \\mathswitch {c_{\\scrs\\PW}}$ and $\\delta \\mathswitch {s_{\\scrs\\PW}}$ in our calculation. This results in~\\cite{Denner:1991kt}\n\\begin{align}\n\\delta \\mathswitch {M_\\PW}^2 &=\\mathrm{Re}\\, \\Sigma^W_\\mathrm{T}(\\mathswitch {M_\\PW}^2),&\\delta Z_{\\mathswitchr W}&=-\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr W}_\\mathrm{T}(\\mathswitch {M_\\PW}^2),\\nonumber\\\\\n\\delta \\mathswitch {M_\\PZ}^2&=\\mathrm{Re}\\, \\Sigma^{\\mathswitchr Z\\PZ}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2),\\nonumber\\\\\n\\delta Z_{\\mathswitchr Z\\PZ}&=-\\mathrm{Re}\\, \\Sigma'^{\\mathswitchr Z\\PZ}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2),&\\delta Z_{{\\mathrm{PA}}\\PA}&=-\\mathrm{Re}\\, \\Sigma'^{{\\mathrm{PA}}\\PA}_\\mathrm{T}(0),\\nonumber\\\\\n\\delta Z_{{\\mathrm{PA}}\\mathswitchr Z}&=-2\\mathrm{Re}\\, \\frac{\\Sigma^{{\\mathrm{PA}}\\mathswitchr Z}_\\mathrm{T}(\\mathswitch {M_\\PZ}^2)}{\\mathswitch {M_\\PZ}^2},&\n\\delta Z_{\\mathswitchr Z{\\mathrm{PA}}}&=2\\mathrm{Re}\\, \\frac{\\Sigma^{{\\mathrm{PA}}\\mathswitchr Z}_\\mathrm{T}(0)}{\\mathswitch {M_\\PZ}^2},\\nonumber\\\\\n\\delta \\mathswitch {c_{\\scrs\\PW}}&=\\frac{\\mathswitch {c_{\\scrs\\PW}}}{2}\\left(\\frac{\\delta \\mathswitch {M_\\PW}^2}{\\mathswitch {M_\\PW}^2}-\\frac{\\delta \\mathswitch {M_\\PZ}^2}{\\mathswitch {M_\\PZ}^2}\\right), &\\delta \\mathswitch {s_{\\scrs\\PW}}&= -\\frac{\\mathswitch {c_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}}\\delta \\mathswitch {c_{\\scrs\\PW}}.\n\\end{align}\nThe electric charge $e$ is defined via the $\\mathrm{ee} \\gamma $ coupling in the Thomson limit of on-shell external electrons and zero momentum transfer to the photon,\nwhich yields in BHS convention \\cite{Denner:1991kt}\n\\begin{align}\n\\delta Z_e = -\\frac{1}{2} \\left(\\delta Z_{{\\mathrm{PA}}\\PA}+ \\frac{\\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {c_{\\scrs\\PW}}} \\delta Z_{\\mathswitchr Z{\\mathrm{PA}}}\\right).\n\\label{eq:dZe}\n\\end{align}\n\n\\subsubsection{Fermions}\n\nThe renormalization conditions for the fermions are identical to the ones in the SM, described in detail in Ref.~\\cite{Denner:1991kt}. We demand that the (real parts of the locations of the)\npoles of the fermion propagators correspond to the squared fermion masses, \n and that on-shell fermion propagators do not obtain loop corrections. \nAssuming the CKM matrix \nequal to the unit matrix, the results for the renormalization constants simplify to\n\\begin{align}\n\\delta m_{f,i}&=\\frac{m_{f,i}}{2}\\mathrm{Re}\\,\\big[\\Sigma^{f,{\\mathrm{L}}}_i(m^2_{f,i})+\\Sigma^{f,{\\mathrm{R}}}_i(m^2_{f,i})+2 \\Sigma^{f,{\\mathrm{S}}}_i(m^2_{f,i})\\big],\n\\nonumber\\\\\n\\delta Z^{f,\\sigma}_i&=-\\mathrm{Re}\\, \\Sigma^{f,\\sigma}_i(m^2_{f,i})-m_{f,i}^2\\frac{\\partial}{\\partial k^2} \\mathrm{Re} \\big[\\Sigma_i^{f,{\\mathrm{L}}}(k^2)+\\Sigma^{f,{\\mathrm{R}}}_i(k^2)+2\\Sigma^{f,{\\mathrm{S}}}_i(k^2)\\big]\\Big|_{k^2=m_{f,i}^2}, \\quad \\sigma={\\mathrm{L}},{\\mathrm{R}},\n\\end{align}\nwhere we have \nused the usual \ndecomposition of the fermion self-energies into a left-handed, a right-handed, and a scalar part, $\\Sigma^{f,{\\mathrm{L}}}_i$, $\\Sigma^{f,{\\mathrm{R}}}_i$, and $ \\Sigma^{f,{\\mathrm{S}}}_i$, respectively.\nThe expressions for a non-trivial CKM matrix can be found in \\citere{Denner:1991kt}.\n\n\\subsection{\\boldmath{$\\overline{\\mathrm{MS}}$ renormalization conditions}}\n\\label{sec:diffrenschemes}\n\nIn the four renormalization schemes we \nare going to present, the imposed on-shell conditions are identical, and the differences only occur in the choice of different $\\overline{\\mathrm{MS}}$ conditions.\nThe parameters $\\alpha$ or $\\lambda_3$ governing the mixing of the CP-even Higgs bosons, and the parameters $\\beta$ and $\\lambda_5$ need to be fixed.\nA formulation of an on-shell condition for these parameters is not obvious.  \nOne could relate the parameters to \nsome physical processes, such as Higgs-boson decays, and demand that these processes do not \nreceive higher-order corrections. However, so far, no sign of further Higgs bosons has been observed, hence, there is no distinguished process, and such a prescription does not only require more calculational effort, but could introduce artificially large corrections to the corresponding parameters, which would spread to many other observables, as discussed in Refs.~\\cite{Freitas:2002um,Krause:2016oke}. Therefore, we choose to renormalize these parameters within the $\\overline{\\mathrm{MS}}$ scheme,\nthough different variables (such as $\\alpha$ or $\\lambda_3$) can be chosen to parameterize the model. Imposing an $\\overline{\\mathrm{MS}}$ condition on either of the parameters leads to differences in the calculation of observables. In addition, gauge-dependent definitions of $\\overline{\\mathrm{MS}}$-renormalized parameters spoil the gauge independence of the relations between input parameters and observables. However, gauge dependences might be even\nacceptable if the renormalization scheme yields stable results and a good convergence of the perturbation series. The price to pay is that subsequent calculations should be done in the same gauge\nor properly translated into another gauge. We will discuss different renormalization schemes based on different treatments of\n$\\alpha$ or $\\lambda_3$ parameterizing the CP-even Higgs-boson mixing, of the parameter $\\beta$, and of the Higgs coupling constant\n $\\lambda_5$ \nin the following. We begin with the so-called $\\overline{\\mathrm{MS}}(\\alpha)$ scheme.\n\n\\subsubsection{\\boldmath{$\\overline{\\mathrm{MS}}(\\alpha)$ scheme}}\n\\label{sec:alphaMSscheme}\nIn this scheme the independent parameter set is $\\{p_{\\mathrm{mass}}\\}$ of Eq.~\\eqref{eq:physparaNLO}, so that the parameters $\\beta$, $\\alpha$, and $\\lambda_5$ are renormalized in $\\overline{\\mathrm{MS}}$. The corresponding counterterm Lagrangian was derived in Sect.~\\ref{sec:renormassb}.\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta \\beta$}:}\n\nThe renormalization constant \n$\\delta \\tan{\\beta}=\\delta \\beta\/{c_\\beta^2}$\nof the mixing angle $\\beta$ is\nrelated to the renormalization constants of the vevs\nby demanding the defining relation $\\tan\\beta=v_2\/v_1$ for bare and renormalized \nquantities.\nIn {$\\overline{\\mathrm{MS}}$}{}, $\\delta \\beta$ can be most easily calculated using the minimal field\nrenormalization \\refeq{eq:inputfieldren} with the following renormalization transformation\nof the vevs,\n\\begin{align}\nv_{1,0} &=  Z_{\\mathswitchr H_1}^{1\/2}(v_1+\\delta \\overline{v}_1),&v_{2,0}&=Z_{\\mathswitchr H_2}^{1\/2}(v_2+\\delta \\overline{v}_2),\n \\label{eq:vevren}\n\\end{align}\nUsing the well-known relation~\\cite{Sperling:2013eva}\n\\begin{align}\n\\delta \\overline{v}_1\/v_1-\\delta \\overline{v}_2\/v_2=\\mathrm{finite},\\label{eq:vacrencond}\n\\end{align}\nthe general form of $\\delta\\beta$ in the {$\\overline{\\mathrm{MS}}$}{} scheme\n\\begin{align}\n\\delta \\beta\n&= \\frac{s_{2\\beta}}{2}\\left(\n\\frac{\\delta \\overline{v}_2}{v_2} -\\frac{\\delta \\overline{v}_1}{v_1}- \\frac{1}{2}\\delta Z_{\\mathswitchr H_1} + \\frac{1}{2} \\delta Z_{\\mathswitchr H_2} \n\\right)\\bigg|_\\mathrm{UV}\n\\end{align}\nsimplifies to \n\\begin{align}\n\\delta \\beta\n&=\\frac{s_{2\\beta}}{4}(-\\delta Z_{\\mathswitchr H_1}+\\delta Z_{\\mathswitchr H_2})\\big|_\\mathrm{UV}\n=\\frac{s_{2\\beta}}{4c_{2\\alpha}} \\left(\\delta Z_\\mathswitchr h -\\delta Z_\\mathswitchr H\\right)\\big|_\\mathrm{UV}\n=\\frac{s_{2\\beta}}{4s_{2\\alpha}} \\left(\\delta Z_{\\mathswitchr h\\mathswitchr H} +\\delta Z_{\\mathswitchr H\\mathswitchr h}\\right)\\big|_\\mathrm{UV},\n\\label{eq:tanbetarencond}\n\\end{align}\nwhere $\\big|_\\mathrm{UV}$ indicates that we take only the UV-divergent parts, which are proportional to \nthe standard divergence $\\Delta_\\mathrm{UV}$.\nThe explicit calculation of the UV-divergent terms of $\\delta Z_\\mathswitchr h,$ $\\delta Z_\\mathswitchr H$ according to Eqs.~\\eqref{eq:dZHdef} in \n't~Hooft-Feynman gauge reveals that only diagrams with closed fermion loops contribute to the counterterm, \n\\begin{align}\\label{eq:deltabeta}\n\\delta \\beta = -\\Delta_\\mathrm{UV}\\, \\frac{e^2}{64\\pi^2\\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2} \\sum_f c_f \\xi^f_\\mathswitchr {A_0} m_f^2,\n\\end{align}\nwith the colour factors $c_\\mathrm{quark}=3$, $c_\\mathrm{lepton}=1$ and the coupling coefficients $\\xi^f_{\\mathswitchr {A_0}}$ \nas defined in Tab.~\\ref{tab:yukint}. In the class of $R_\\xi$ gauges this result is gauge independent at one-loop order \\cite{Krause:2016oke,Denner:2016etu}.\n\n\\paragraph{Neutral Higgs mixing:}\nIn the neutral Higgs sector, relations between field renormalization constants can also be used to determine another parameter in $\\overline{\\mathrm{MS}}$.\nThe first four equations of Eqs.~\\eqref{eq:fieldrenconstrel} can be solved for $\\delta \\alpha$ in various\nways, e.g., yielding\n\\begin{align}\n\\label{eq:deltaalphaderiv}\n\\delta \\alpha\\big|_\\mathrm{UV}&= \\frac{1}{4} \\left(\\delta Z_{\\mathswitchr H\\mathswitchr h}-\\delta Z_{\\mathswitchr h\\mathswitchr H}\\right) \\big|_\\mathrm{UV}.\n\\end{align}\nThe field renormalization constants can be inserted according to Eqs.~\\eqref{eq:dZHdef}, \\eqref{eq:mixfieldrendef} using \n$\\delta M_{\\mathswitchr H\\mathswitchr h}^2 = 0$, thus \n\\begin{align}\n\\label{eq:deltaalphaderiv2}\n\\delta \\alpha&= \\left. \\mathrm{Re}\\frac{\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\\right|_\\mathrm{UV}.\n\\end{align}\nAn explicit calculation of the counterterm \nyields for the fermionic contribution, \n\\begin{align}\\label{eq:delalphaferm}\n\\delta \\alpha\\big|_\\mathrm{ferm} = \\Delta_\\mathrm{UV}\\, \\frac{e^2 s_{2\\alpha}}{64\\pi^2\\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)} \n\\sum_f c_f \\xi^f_\\mathswitchr {A_0} m_f^2 (\\mathswitch {M_\\PH}^2+\\mathswitch {M_\\Ph}^2-12m_f^2),\n\\end{align}\nand for the bosonic contribution\n\\begin{align}\\label{eq:delalphabos}\n\\delta \\alpha\\big|_\\mathrm{bos} = &\n-\\Delta_\\mathrm{UV} \\frac{\\lambda_5^2 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 }{8\\pi^2 e^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2\\beta}^2}\n   \\Big[s_{2 (\\alpha - 3 \\beta)} + 10 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}\\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{\\lambda_5 }{ 128\\pi^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2)  s_{2\\beta}^2} \\Big[\n   -4 \\mathswitch {M_\\PH}^2 (13 c_{2 \\alpha} + 2 c_{2 (\\alpha - 2 \\beta)} - 27 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 (13 c_{2 \\alpha} + 2 c_{2 (\\alpha - 2 \\beta)} + 27 c_{2 \\beta}) s_{2 \\alpha} \n  + 2 \\mathswitch {M_\\PHP}^2 (s_{2 (\\alpha - 3 \\beta)} - 6 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - \\mathswitch {M_\\PAO}^2 (7 s_{2 (\\alpha - 3 \\beta)} + 86 s_{2 (\\alpha - \\beta)} + 91 s_{2 (\\alpha + \\beta)})\n  + 4 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n\\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{e^2 }{ 1024\\pi^2 (\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) \\mathswitch {M_\\PW}^2  \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^2} \\Big[\n- 2 \\mathswitch {M_\\PH}^4 (-36 c_{2 \\alpha} + 5 c_{4 \\alpha - 2 \\beta} + 31 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PH}^2 (5 c_{4 \\alpha - 2 \\beta} - 29 c_{2 \\beta}) s_{2 \\alpha} \n- 2 \\mathswitch {M_\\Ph}^4 (36 c_{2 \\alpha} + 5 c_{4 \\alpha - 2 \\beta} + 31 c_{2 \\beta}) s_{2 \\alpha} \n\\nonumber\\\\\n& + 32 \\mathswitch {M_\\PHP}^4 s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n+ 2 \\mathswitch {M_\\PH}^2 \\mathswitch {M_\\PHP}^2 (3 s_{4 \\alpha} + 4 s_{2 (\\alpha - \\beta)} + s_{4 (\\alpha - \\beta)} \n\t+ 9 s_{4 \\beta} - 12 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - 2 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PHP}^2 (3 s_{4 \\alpha} -4 s_{2 (\\alpha - \\beta)} + s_{4 (\\alpha - \\beta)} \n\t+ 9 s_{4 \\beta} + 12 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& - 2 \\mathswitch {M_\\PAO}^4 (5 s_{2 (\\alpha - 3 \\beta)} + 42 s_{2 (\\alpha - \\beta)} + 41 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PH}^2 (-49 s_{4 \\alpha} + 112 s_{2 (\\alpha - \\beta)} - 7 s_{4 (\\alpha - \\beta)} \n\t+ s_{4 \\beta} + 96 s_{2 (\\alpha + \\beta)})\n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\Ph}^2 (49 s_{4 \\alpha} + 112 s_{2 (\\alpha - \\beta)} + 7 s_{4 (\\alpha - \\beta)} \n\t- s_{4 \\beta} + 96 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PHP}^2 (s_{2 (\\alpha - 3 \\beta)} - 6 s_{2 (\\alpha - \\beta)} + 13 s_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + 4 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 (\\alpha - \\beta)} s_{2 \\beta} \n\t((\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2 \\alpha} + 2 \\mathswitch {M_\\PAO}^2 s_{2 \\beta}) \n\\nonumber\\\\\n& + 48 (2 \\mathswitch {M_\\PW}^4 + \\mathswitch {M_\\PZ}^4) s_{2 (\\alpha - \\beta)} s_{2 \\beta}^2 \n\\Big].\n\\end{align}\nThis result, which is derived in 't~Hooft Feynman gauge, is gauge dependent \\cite{Krause:2016oke,Denner:2016etu}.\n\n\\paragraph{Higgs self-coupling:}\nThe Higgs self-coupling counterterm $\\delta \\lambda_5$ has to be fixed via a vertex correction. We define this renormalization constant  in $\\overline{\\mathrm{MS}}$ as well, as there is no distinguished process to fix it on-shell. Any 3- or 4-point vertex function with external Higgs bosons is suited to calculate the divergent terms.\nSince the $\\mathswitchr H\\mathswitchr {A_0}\\PAO$ vertex correction involves fewest diagrams, it is our preferred choice. The condition is\n\\begin{align}\n\\hat{\\Gamma}^{\\mathswitchr H\\mathswitchr {A_0}\\PAO}\\big|_\\mathrm{UV}=\n\\parbox{2.4cm}{\\includegraphics[scale=1]{.\/diagrams\/L5ren.eps}}\\left.\\rule{0cm}{1.1cm}\\right|_\\mathrm{UV}=0.\n\\end{align}\nSolving this equation for $\\delta \\lambda_5$ fixes this renormalization constant. The generic one-loop diagrams appearing in this vertex correction are shown in Fig.~\\ref{fig:dL5diags}, the contribution of the diagrams involving closed fermion loops is \n\\begin{align}\n\\label{eq:L5ferm}\n\\delta \\lambda_{5,\\mathrm{ferm}}=\\Delta_\\mathrm{UV} \n\\frac{e^2 \\lambda_5}{16\\pi^2 \\mathswitch {M_\\PW}^2\\mathswitch {s_{\\scrs\\PW}}^2}\\sum_f c_f \\left(1+\\frac{c_{2\\beta}}{s_{2\\beta}}\\xi_{\\mathswitchr {A_0}}^f\\right) m_f^2.\n\\end{align}\n\\begin{figure}\n\\vspace{-15pt}\n\\input{diagrams\/triangles}\n\\vspace{-20pt}\n\\caption{Generic diagrams contributing to the $HA_0A_0$ vertex correction used for the renormalization of $\\lambda_5$.}\n\\label{fig:dL5diags}\n\\end{figure}\nThe diagrams containing only bosons lead to \n\\begin{align}\n\\label{eq:L5bos}\n\\delta \\lambda_5\\big|_\\mathrm{bos}{}={}& {}\n\\Delta_\\mathrm{UV} \\frac{\\lambda_5}{32 \\pi^2}  \n\\left(2 \\lambda_1 + 2\\lambda_2+8\\lambda_3 +12 \\lambda_4- 9 g_2^2 - 3 g_1^2\\right)\n\\nonumber\\\\\n{}={}& - \\Delta_\\mathrm{UV}\\frac{\\lambda_5^2c^2_{2\\beta}}{4 \\pi^2 s^2_{2\\beta}}\n +\\Delta_\\mathrm{UV}\\frac{\\lambda_5 e^2}{64 \\pi^2\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^2} \\Big[\n  \\mathswitch {M_\\PH}^2 (2 + c_{2 (\\alpha - \\beta)} - 3 c_{2 (\\alpha + \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\Ph}^2 (2 - c_{2 (\\alpha - \\beta)} + 3 c_{2 (\\alpha + \\beta)})\n+ \\mathswitch {M_\\PAO}^2 (1 - 5 c_{4 \\beta})\n- 4 \\mathswitch {M_\\PHP}^2 s_{2 \\beta}^2\n- 6(2 \\mathswitch {M_\\PW}^2+\\mathswitch {M_\\PZ}^2) s_{2 \\beta}^2 \n\\Big].\n\\end{align}\nSince $\\lambda_5$ is a fundamental parameter of the Higgs potential, an $\\overline{\\mathrm{MS}}$  definition leads to a gauge-independent counterterm.\n\n\\subsubsection{\\boldmath{$\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme}}\n\\label{sec:lambda3MSscheme}\n\nIn this scheme, the independent parameter set is $\\{p'_{\\mathrm{mass}}\\}$ defined in Eq.~\\eqref{eq:physparaNLOprime}.  The renormalization of $\\beta$ and $\\lambda_5$ is identical to the previous renormalization scheme and not stated again, but now the parameter $\\lambda_3$ (instead of $\\alpha$) is an independent parameter being renormalized in $\\overline{\\mathrm{MS}}$. This has the advantage that this parameter is gauge independent, as it is a defining parameter of the basic \nparameterization of the Higgs potential\nand thus is safe against potentially gauge-dependent contributions appearing in relations between bare parameters. \nAs stated above, the {$\\overline{\\mathrm{MS}}$}{} renormalization of the \nparameter $\\beta$ generally breaks gauge \nindependence, but in $\\mathrm{R}_\\xi$ gauges the gauge dependence cancels at one loop \\cite{Krause:2016oke,Denner:2016etu}, \nso that this scheme yields gauge-independent results at NLO. \nWe take the counterterm potential of Sect.~\\ref{sec:renormassb}, but treat $\\delta \\alpha$ as a dependent counterterm. \nAs $\\alpha$ is a pure mixing angle, we choose to apply the renormalization prescription of Sect.~\\ref{sec:renormassb}, where the mixing angle diagonalizes the potential to all orders.\nThe relation between $\\delta \\alpha$ and the independent constants is given in Eq.~\\eqref{eq:NLOmassmixHh} with $ \\delta M_{\\mathswitchr H\\mathswitchr h}^2=0$,\n\\begin{align}\n\\label{eq:alphadef32}\n \\delta \\alpha=\\frac{f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})} {\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2},\n\\end{align}\nwhere $f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})$ can be obtained from Eq.~\\eqref{eq:Hhmassmixing} by applying the renormalization transformation of Eq.~\\eqref{eq:physpararen1} (which is identical to the renormalization transformation of Sect.~\\ref{sec:renormassb}, but renormalizing $\\lambda_3$ instead of $\\alpha$). This yields\n\\begin{align}\nf_\\alpha(\\{\\delta p'_\\mathrm{mass}\\})={}&\n\\frac{1}{2} t_{2 \\alpha } \\left(\\delta \\mathswitch {M_\\Ph}^2-\\delta \\mathswitch {M_\\PH}^2\\right)\n+\\frac{s_{2 \\beta } \\left(\\delta \\mathswitch {M_\\PAO}^2-2 \\delta \\mathswitch {M_\\PHP}^2\\right)}{2 c_{2 \\alpha }}\n\\nonumber\\\\\n&+\\frac{\\delta \\beta c_{2 \\beta } \\left(\\mathswitch {M_\\PH}^2- \\mathswitch {M_\\Ph}^2\\right) t_{2 \\alpha }}{s_{2 \\beta }}\n+\\frac{2 \\mathswitch {M_\\PW}^2 s_{2\\beta } (\\delta \\lambda_3+\\delta \\lambda_5) \\mathswitch {s_{\\scrs\\PW}}^2}{e^2 c_{2 \\alpha }}\n\\nonumber\\\\\n&+\\frac{s_{2 \\beta } \\left(\\mathswitch {M_\\PAO}^2-2 \\mathswitch {M_\\PHP}^2\\right)+(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2)s_{2\\alpha}}{c_{2\\alpha}}\n\\left( \\delta Z_e -\\frac{\\delta \\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}} -\\frac{\\delta \\mathswitch {M_\\PW}^2}{2\\mathswitch {M_\\PW}^2} \\right)\n\\nonumber\\\\\n&\n-\\frac{ e  \\left[\n\\delta t_\\mathswitchr H \\left(s_{\\alpha -3 \\beta }+3 s_{\\alpha +\\beta }\\right)\n+\\delta t_\\mathswitchr h \\left(c_{\\alpha -3 \\beta }+3 c_{\\alpha +\\beta } \\right) \n\\right]}{8 \\mathswitch {M_\\PW}\n   c_{2 \\alpha } \\mathswitch {s_{\\scrs\\PW}}}.\\label{eq:MHhdependence}\n\\end{align}\nThe UV-divergent term of $\\delta \\alpha$ has been calculated in Eq.~\\eqref{eq:deltaalphaderiv2}, and by renormalizing $\\delta \\lambda_3$ in $\\overline{\\mathrm{MS}}$ scheme, it is clear that the dependent $\\delta \\alpha$ must now have a finite part in addition. \nWe choose this finite term in such a way that the finite part in $\\delta \\lambda_3$ \n(which results from $\\delta \\lambda_3$ by setting $\\Delta_{\\mathrm{UV}}$ to zero) vanishes and obtain\n\\begin{align}\n\\label{eq:alphainL3MS}\n\\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} =\n\\left. \\mathrm{Re}\\frac{\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\\right|_\\mathrm{UV}\n+\\frac{f_\\alpha(\\{ \\delta p'_\\mathrm{mass}\\})} {\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\bigg|_\\mathrm{finite},\n\\end{align}\nwhere $\\delta \\lambda_3$ drops out as it has no finite part. The divergent part of $\\delta \\lambda_3$ can be calculated by solving Eq.~\\eqref{eq:alphadef32} and using the knowledge about the divergent parts of $\\delta \\alpha$ from Eqs.~\\eqref{eq:delalphaferm} and \\eqref{eq:delalphabos}. This results in \n\\begin{align}\n\\delta \\lambda_3= &\\biggl[\n\\frac{e^2 c_{2\\alpha}}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}} \n\\left(\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\mathrm{Re}\\,\\Sigma^{\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)\\right)\n-\\frac{\\delta \\beta e^2 s_{2 \\alpha} c_{2 \\beta} \\left(\\mathswitch {M_\\PH}^2- \\mathswitch {M_\\Ph}^2\\right) }{2 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2 \\beta }^2}\n\\nonumber\\\\\n& -\\delta \\lambda_5\n-\\frac{e^2 s_{2 \\alpha }}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}}  \\left(\\delta \\mathswitch {M_\\Ph}^2-\\delta \\mathswitch {M_\\PH}^2\\right)\n-\\frac{e^2 \\left(\\delta \\mathswitch {M_\\PAO}^2-2 \\delta \\mathswitch {M_\\PHP}^2\\right)}{4 \\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}\n\\nonumber\\\\\n&-\\frac{e^2\\left(s_{2\\beta}\\left(\\mathswitch {M_\\PAO}^2-2 \\mathswitch {M_\\PHP}^2\\right)+(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2) s_{2 \\alpha }\\right)}{2\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}}\n\\left(\\delta Z_e- \\frac{\\delta \\mathswitch {s_{\\scrs\\PW}}}{\\mathswitch {s_{\\scrs\\PW}}} -\\frac{\\delta \\mathswitch {M_\\PW}^2}{2 \\mathswitch {M_\\PW}^2} \\right)\n\\nonumber\\\\\n& +\\frac{ e^3  \\left[\n\\delta t_\\mathswitchr H \\left(s_{\\alpha -3 \\beta }+3 s_{\\alpha +\\beta }\\right)\n+\\delta t_\\mathswitchr h \\left(c_{\\alpha -3 \\beta }+3 c_{\\alpha +\\beta } \\right) \n\\right]}{16 \\mathswitch {M_\\PW}^3 \\mathswitch {s_{\\scrs\\PW}}^3 s_{2\\beta}}\\biggr]_{\\mathrm{UV}}.\n\\label{eq:deltaL3}\n\\end{align}\nThe fermionic contribution to $\\delta \\lambda_3$ is given by\n\\begin{align}\n\\nonumber\n\\delta \\lambda_3\\big|_\\mathrm{UV,ferm}\n& = -\\delta \\lambda_5\\big|_\\mathrm{UV,ferm}\n-\\Delta_\\mathrm{UV} \\frac{3e^4}{32\\pi^2\\mathswitch {M_\\PW}^4\\mathswitch {s_{\\scrs\\PW}}^4} \\sum_i\n\\left(\\xi_\\mathswitchr {A_0}^u-\\xi_\\mathswitchr {A_0}^d\\right)^2\nm_{u,i}^2 m_{d,i}^2\n\\\\ \n&\\quad\n- \\Delta_\\mathrm{UV} \\frac{e^4}{64\\pi^2\\mathswitch {M_\\PW}^4\\mathswitch {s_{\\scrs\\PW}}^4}\n\\sum_f c_f \\left(1+\\frac{c_{2\\beta}}{s_{2\\beta}}\\xi_\\mathswitchr {A_0}^f\\right) m_f^2\n\\left[\\mathswitch {M_\\PAO}^2-2\\mathswitch {M_\\PHP}^2+\\frac{s_{2\\alpha}}{s_{2\\beta}}(\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2)\\right]\n\\end{align}\nwith the massive fermions $f = \\mathswitchr e, \\dots,\\mathswitchr t$ and the generation index $i$. For the bosonic contribution we obtain\n\\begin{align}\n\\label{eq:L3bos}\n\\delta \\lambda_3\\big|_\\mathrm{UV,bos}\n= \\,& \\Delta_\\mathrm{UV}\\frac{1}{32 \\pi^2} \\big[(\\lambda_1 + \\lambda_2) (6 \\lambda_3 + 2 \\lambda_4) + 4 \\lambda_3^2 + 2 \\lambda_4^2 + 2 \\lambda_5^2\n-3 \\lambda_3 (3 g_2^2 + g_1^2 )\n\\nonumber\\\\\n& +\\textstyle\\frac{3}{4} (3 g_2^4 + g_1^4 - 2 g_2^2 g_1^2)\n\\big]\n\\nonumber\\\\\n= &\n\\,\\Delta_\\mathrm{UV} \\frac{\\lambda_5^2}{2 \\pi^2 s_{2\\beta}^2}\n+ \\Delta_\\mathrm{UV} \\frac{e^2 \\lambda_5}{128 \\mathswitch {M_\\PW}^2 \\pi^2 \\mathswitch {s_{\\scrs\\PW}}^2 s_{2\\beta}^3}\\Big[\n12 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 \\beta}^3 \n+ \\mathswitch {M_\\PAO}^2 (27 s_{2 \\beta} - s_{6 \\beta}) \n\\nonumber\\\\\n&+ 2 \\mathswitch {M_\\PHP}^2 (-19 s_{2 \\beta} + s_{6 \\beta}) \n + \\mathswitch {M_\\PH}^2 (-22 s_{2 \\alpha} - 3 s_{2 (\\alpha - 2 \\beta)} - 8 s_{2 \\beta} + s_{2 (\\alpha + 2 \\beta)}) \n\\nonumber\\\\\n& + \\mathswitch {M_\\Ph}^2 (22 s_{2 \\alpha} + 3 s_{2 (\\alpha - 2 \\beta)} - 8 s_{2 \\beta} - s_{2 (\\alpha + 2 \\beta)}) \\Big]\n\\nonumber\\\\\n& + \\Delta_\\mathrm{UV} \\frac{e^4}{256 \\mathswitch {M_\\PW}^4 \\pi^2 \\mathswitch {s_{\\scrs\\PW}}^4 s_{2\\beta}^3} \\Big[\n-2 \\mathswitch {M_\\PH}^4 (-3 + c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}  \n\\nonumber\\\\\n& + 4 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PH}^2 (c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}\n- 2 \\mathswitch {M_\\Ph}^4 (3 + c_{2 (\\alpha - \\beta)} + 2 c_{2 (\\alpha + \\beta)}) s_{2 \\alpha}  \n\\nonumber\\\\\n& - \\mathswitch {M_\\PAO}^4 (-7 s_{2 \\beta} + s_{6 \\beta}) \n- \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PH}^2 (11 s_{2 \\alpha} + s_{2 (\\alpha - 2 \\beta)} + 2 s_{2 \\beta})\n\\nonumber\\\\\n& + \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\Ph}^2 (11 s_{2 \\alpha} + s_{2 (\\alpha - 2 \\beta)} - 2 s_{2 \\beta})\n+ 12 \\mathswitch {M_\\PHP}^4 s_{2 \\beta}^3 \n+ 16 \\mathswitch {M_\\PH}^2 \\mathswitch {M_\\PHP}^2 s_{2 \\beta} s_{\\alpha + \\beta}^2\n\\nonumber\\\\\n&+ 16 \\mathswitch {M_\\Ph}^2 \\mathswitch {M_\\PHP}^2 c_{\\alpha + \\beta}^2 s_{2 \\beta}\n+ 2 \\mathswitch {M_\\PAO}^2 \\mathswitch {M_\\PHP}^2 (-11 s_{2 \\beta} + s_{6 \\beta})\n\\nonumber\\\\\n&+ 6 (2 \\mathswitch {M_\\PW}^2 + \\mathswitch {M_\\PZ}^2) s_{2 \\beta}^2 ((\\mathswitch {M_\\Ph}^2 - \\mathswitch {M_\\PH}^2) s_{2 \\alpha} + (\\mathswitch {M_\\PAO}^2 - 2 \\mathswitch {M_\\PHP}^2) s_{2 \\beta})\n\\nonumber\\\\\n&+ 6 (6 \\mathswitch {M_\\PW}^4 - 4 \\mathswitch {M_\\PW}^2 \\mathswitch {M_\\PZ}^2 + \\mathswitch {M_\\PZ}^4) s_{2 \\beta}^3\n\\Big].\n\\end{align}\n\n\\subsubsection{The FJ tadpole scheme}\n\\label{sec:FJscheme}\n\nSince tadpole loop contribution $T_S$ are gauge dependent \\cite{Degrassi:1992ff}, the connection among bare parameters potentially becomes gauge dependent \nif $\\delta t_S=-T_S$ enters the relations between bare parameters, as it is the case if renormalized tadpole parameters $t_S$ \nare forced to vanish. Note that these gauge dependences systematically cancel if on-shell renormalization conditions are employed, i.e.\\ if predictions \nfor observables are parameterized by directly measurable input parameters. If some input parameters are renormalized in the {$\\overline{\\mathrm{MS}}$}{} scheme this cancellation of gauge dependences does not take place anymore in general, and the gauge dependence is ma\\-ni\\-fest \nin relations between predicted observables and input parameters at NLO.\nIn the $\\overline{\\mathrm{MS}}(\\alpha)$ and the $\\overline{\\mathrm{MS}}(\\lambda_3)$ renormalization schemes, the bare de\\-finitions of $\\alpha$ and $\\beta$ contain tadpole terms \nleading to a gauge dependence (although the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme is gauge \nindependent at NLO in $R_\\xi$ gauges).\n \nFleischer and Jegerlehner \\cite{Fleischer:125234} proposed a renormalization scheme for the SM, referred to as the FJ scheme in the following, that preserves gauge independence for all bare parameters, including the \nmasses and mixing angles%\n\\footnote{A similar scheme, called $\\beta_h$~scheme, was suggested in \\citere{Actis:2006ra}.\nA comparison of that approach to the conventional {$\\overline{\\mathrm{MS}}$}{} and FJ schemes can be found in\n\\citere{Denner:2016etu}.}.\nIn this scheme, the parameters are defined in such a way that tadpole terms do not enter the definition of any bare parameter so that all relations among bare parameters remain gauge independent. \nThis can be achieved by demanding that {\\it bare} tadpole terms vanish,\n$t_{S,0}=0$,\nfor all fields $S$ with the quantum numbers of the vacuum. Since tadpole conditions have no effect on physical observables and change only the bookkeeping, such a procedure is possible. The disadvantage is that now tadpole diagrams have to be taken explicitly into account \nin all higher-order calculations. In particular, the one-particle reducible tadpole contributions destroy the simple relation between propagators and \ntwo-point functions. In the SM, the FJ scheme does not affect observables if all parameters are \nrenormalized using on-shell conditions---as usually done---except \nfor the strong coupling constant $\\alpha_\\mathrm{s}$, which is, however, directly related to the strong gauge coupling, a model defining parameter.\nA gauge-independent renormalization scheme for the THDM can be defined by applying the FJ prescription and imposing the {$\\overline{\\mathrm{MS}}$}{} condition on  mixing angles \\cite{Denner:2016etu, Krause:2016oke}.\nThe {\\it bare} physical parameters defined in the FJ scheme \ndiffer by NLO tadpole contributions (including divergent and finite terms)\nfrom the gauge-dependent definition of the bare parameters $\\{p_{\\mathrm{mass}}\\}$ given in Eq.~\\eqref{eq:physparaNLO}. \nExceptions are $e$ and the parameter $\\lambda_5$, which is a parameter of the basic potential and therefore gauge independent by construction. \nThe renormalization of $\\lambda_5$\nin {$\\overline{\\mathrm{MS}}$}{} is identical to the one in the previous schemes.\n\nIt should be noted that in Refs.~\\cite{Denner:2016etu, Krause:2016oke} $m_{12}^2$ is chosen as independent\nparameter in contrast to our choice of $\\lambda_5$. The latter, however, is closer to common practice used in\nthe MSSM~\\cite{Frank2007,Baro:2008bg}.\nMoreover, in  Refs.~\\cite{Krause:2016oke,Denner:2016etu}\ntadpole counterterms are reintroduced by shifting the Higgs fields according to $\\eta_i \\to \\eta_i + \\Delta v_{\\eta, i}$, $i = 1,2$, where \nthe constants $\\Delta v_{\\eta, i}$ can be chosen arbitrarily, since physical observables do not depend on this shift,\nwhich can be interpreted as an unobservable change of the integration variables in the path integral. In Refs.~\\cite{Krause:2016oke,Denner:2016etu}, this freedom of choice is exploited, \nand $\\Delta v_{\\eta, i}$ \nare chosen in such a way that the fields $\\eta_1$, $\\eta_2$ do not develop vevs at all order\n. This affects the form of the counterterm Lagrangian and the definition of the renormalization constants with the consequence that the formulae given in Eq.~\\eqref{eq:dVH2p} and Sect.\\ref{sec:onshellrenconditions} cannot be applied.\n\nWe have implemented the FJ scheme following the strategy of Ref.~\\cite{Denner:2016etu} by performing the shifts $\\eta_i \\to \\eta_i+ \\Delta v_{\\eta, i}$ and in an alternative, simpler (but physically equivalent) way. \nIn this simplified approach we keep the dependence of the Lagrangian in terms of gauge-dependent masses and couplings. In addition we keep the tadpole renormalization \ncondition~\\eqref{eq:tadpoleren}, so that the definitions of the renormalization constants of the on-shell parameters and the $Z$ factors according to Sect.~\\ref{sec:onshellrenconditions} remain valid (otherwise we   needed to take into account actual tadpole diagrams everywhere).\nIn this simplified approach the counterterms for $\\alpha$ and $\\beta$ which reproduce the\nresults in the FJ scheme result from the previously derived $\\delta\\alpha$ and $\\delta\\beta$\nby adding appropriate finite terms,\n\\begin{align}\n\\delta \\alpha\\big|_\\mathrm{FJ}&= \\delta \\alpha+\\mbox{finite terms},\\nonumber\\\\\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta +\\mbox{finite terms},\n\\end{align}\nwhich depend on the (finite parts of the) tadpole contributions $T_\\mathswitchr H$ and $T_\\mathswitchr h$.\n\nBefore performing the full calculations, we outline the strategy of the derivation of\nthose finite terms for $\\beta$; for $\\alpha$ everything works analogously.\nWe start by exploiting the fact that the form of the tadpole renormalization cannot\nchange physical results if all counterterms for independent parameters are determined \nby the same physical conditions. This means, as mentioned above, that we can simply define\nthe bare tadpoles to vanish, but this forces us to include all explicit tadpole\ncontributions to Green functions. \nWe indicate quantities in this variant by a superscript ``$t$'' in the following. \nWe get the same physical predictions in this ``$t$-variant'' if we use the\ncounterterm\n\\begin{align}\n\\delta \\beta^t &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\label{eq:dbt_def}\n\\end{align}\ninstead of $\\delta\\beta$,\nwhere $\\delta\\beta^t$ is calculated in the same way as $\\delta \\beta$, but with\ntadpole counterterms omitted and \nexplicit tadpole diagrams (including divergent and finite parts)\nin the occurring Green functions taken into account.\nNote that the {$\\overline{\\mathrm{MS}}$}{} prescription to include only divergent terms, which is employed to define\n$\\delta \\beta$, is not applied to the new tadpole contribution $\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)$.\nOtherwise the new $\\Delta\\beta^t$ terms could not be fully compensated by explicit \ntadpole contributions occuring elsewhere, so that\nthere would be differences in the renormalized amplitudes.\nIn fact, applying the {$\\overline{\\mathrm{MS}}$}{} prescription to $\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)$ as well\ndefines the FJ renormalization scheme,\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}.\n\\end{align}\nThe quantity $\\delta \\beta^t\\big|_\\mathrm{FJ}$ is the gauge-independent counterterm for $\\beta$\nintroduced in Ref.~\\cite{Denner:2016etu} which is to be used in the $t$-variant,\nwhere all explicit tadpole diagrams are included in Green functions \n(or equivalently are redistributed by the $\\Delta v$ shift as described Ref.~\\cite{Denner:2016etu}).\nWe can translate the FJ renormalization prescription back to our \nrenormalization scheme (with vanishing renormalized tadpoles) by the counterpart\nof Eq.~\\refeq{eq:dbt_def}, but now formulated in the FJ scheme,\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta\\big|_\\mathrm{FJ} +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h),\n\\label{eq:dbtFJ_def}\n\\end{align}\ni.e.\\ $\\delta \\beta\\big|_\\mathrm{FJ}$ is the counterterm for $\\beta$ to be used in our\ncounterterm Lagrangian in order to calculate renormalized amplitudes in the FJ scheme.\nCombining the above formulas, we obtain the finite difference between\n$\\delta \\beta$ in the (gauge-dependent) {$\\overline{\\mathrm{MS}}$}{} scheme and \n$\\delta \\beta\\big|_\\mathrm{FJ}$ in the (gauge-independent) FJ scheme,\n\\begin{align}\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta^t\\big|_\\mathrm{FJ} -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\nonumber\\\\\n&= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV} -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\n\\nonumber\\\\\n&= \\delta \\beta -\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}.\n\\end{align}\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta  \\beta\\big|_\\mathrm{FJ}$}:}\nWe begin our calculation of $\\delta\\beta|_\\mathrm{FJ}$ with an alternative \ncomputation of $\\delta \\beta$ in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme, because \nEq.~\\eqref{eq:vacrencond} cannot be applied in the FJ scheme.\nTo avoid the use of Eq.~\\eqref{eq:vacrencond},\nwe calculate the counterterm in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme from the field \nrenormalization constant by employing the last two equations of Eq.~\\eqref{eq:fieldrenconstrel}. \nThis results in\n\\begin{align}\n \\delta \\beta=\\frac{1}{4} \\left(\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}-\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}\\right)\\big|_\\mathrm{UV}\n=\\left.\\frac{\n2\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)-\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)\n}{2 \\mathswitch {M_\\PAO}^2}\\right|_\\mathrm{UV},\n \\label{eq:betaMSaltern}\n\\end{align}\nwith $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ as given in Eq.~\\eqref{eq:Mabchoice2}. In the second step,  $\\delta Z_{\\mathswitchr {G_0}\\mathswitchr {A_0}}$ from Eq.~\\eqref{eq:mixfieldrendef} and\n\\begin{align}\n\\delta Z_{\\mathswitchr {A_0}\\mathswitchr {G_0}}=2 \\, \\frac{-\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2+\\mathrm{Re}\\,\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)}{\\mathswitch {M_\\PAO}^2}\n\\label{eq:dZAG}\n\\end{align}\nhave been used. Equation~\\refeq{eq:dZAG} results from demanding finiteness of the\n$G_0A_0$ mixing self-energy at zero-momentum transfer, $k^2=0$, but actually any other \nvalue of $k^2$ would be possible as well, since we only have to remove all UV-divergent terms\nin the mixing. \nThe non-vanishing tadpole counter\\-terms in the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme are $\\delta t_S=-T_S$.\nIn the transition to the $t$-variant, $\\delta \\beta$ gets modified by two kind of terms:\nFirst, there are no tadpole counterterms, i.e.\\ the $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ term is absent,\nand second, there are explicit tadpole contributions to $\\Sigma^{\\mathswitchr {G_0}\\mathswitchr {A_0}}$.\nThis implies\n\\begin{align}\n\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h) &= \\delta\\beta^t-\\delta\\beta\n= -\\frac{\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2}{\\mathswitch {M_\\PAO}^2} \n-\\left.\\frac{\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {G_0}\\mathswitchr {A_0}}(\\mathswitch {M_\\PAO}^2)+\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {G_0}\\mathswitchr {A_0}}(0)}{2 \\mathswitch {M_\\PAO}^2}\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h},\n\\end{align}\nwhere the superscript ``$t$'' indicates that one-particle-reducible tadpole diagrams are \nincluded in the self-energies.\nThe subscript ``$T_\\mathswitchr H,T_\\mathswitchr h$'' means that only the those explicit tadpole contributions\nare taken into account here.\nInserting $\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2$ from Eq.~\\refeq{eq:Mabchoice2}\nand evaluating the (momentum-independent) tadpole diagrams for the $G_0A_0$ mixing,\n$\\Delta\\beta^t$ evaluates to\n\\begin{align}\n\\lefteqn{\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)} \n\\nonumber\\\\*\n&= -\\frac{1}{\\mathswitch {M_\\PAO}^2} \\left[\n\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n+\\left.\\mathrm{Re}\\,\\Sigma^{t,\\mathswitchr {A_0}\\mathswitchr {G_0}}(0)\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h} \\right]\n\\nonumber\\\\\n&=-\\frac{1}{ \\mathswitch {M_\\PAO}^2}\\left[\n\\delta M_{\\mathswitchr {G_0}\\mathswitchr {A_0}}^2\n+ \\parbox[c][1cm]{2.8cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEHAG1.eps}}+\\parbox[c][1cm]{2.8cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEhAG2.eps}}\\right]\n\\nonumber\\\\\n&=-\\frac{e}{2 \\mathswitch {M_\\PW} \\mathswitch {s_{\\scrs\\PW}}\\mathswitch {M_\\PAO}^2} \\left[\nT_\\mathswitchr H s_{\\alpha -\\beta }\n+T_\\mathswitchr h c_{\\alpha -\\beta }\n+ T_\\mathswitchr H \\frac{\\left(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\PH}^2\\right) s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2} \n+T_\\mathswitchr h \\frac{\\left(\\mathswitch {M_\\PAO}^2-\\mathswitch {M_\\Ph}^2\\right) c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2}\n\\right]\n\\nonumber\\\\\n&= -\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right).\n\\end{align}\nThe counterterm $\\delta\\beta^t\\big|_\\mathrm{FJ}$ of the FJ scheme in the $t$-variant,\nthus, reads\n\\begin{align}\n\\delta \\beta^t\\big|_\\mathrm{FJ} &= \\delta \\beta +\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}\n= \\delta \\beta \n-\\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{UV},\n\\label{eq:dbtFJ_expl}\n\\end{align}\nwhich is in agreement with Ref.~\\cite{Krause:2016oke,Denner:2016etu}. \nThis translates to our treatment of tadpoles as\n\\begin{align}\n\\label{eq:dbFJ_expl}\n\\delta \\beta\\big|_\\mathrm{FJ} &= \\delta \\beta \n-\\Delta\\beta^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n= \\delta \\beta\n+ \\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite},\n\\end{align}\nwhere again ``finite'' means that $\\Delta_\\mathrm{UV}$ is set to zero in the tadpole contribution.\nUsing this counterterm, it is possible to keep the form of the\ncounterterm Lagrangian derived \nin Sect.~\\ref{sec:CTLagrangian}\nto obtain results in the gauge-independent\nFJ scheme, although the above counterterm Lagrangian employs a gauge-dependent\n(but very convenient) tadpole renormalization.\n\nAlternatively, $\\delta\\beta$ could be fixed by an analogous consideration\nof the $\\mathswitchr Z\\mathswitchr {A_0}$ mixing, leading to \n\\begin{align}\n\\delta\\beta|_\\mathrm{UV}\n=\\left[ \\frac{s_{2\\beta}}{4c_{2\\alpha}}\\left(\\delta Z_{\\mathswitchr H}-\\delta Z_{\\mathswitchr h}\\right) \n+ \\frac{\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}(k^2)}{\\mathswitch {M_\\PZ}} \\right]_\\mathrm{UV},\n\\end{align}\nwhich is independent of $k^2$ and does not use relation~\\refeq{eq:vacrencond}.\nThe transition to the FJ scheme then simply amounts to replacing the\none-particle-irreducible self-energy $\\Sigma^{\\mathswitchr Z\\mathswitchr {A_0}}$ by\n$\\Sigma^{t,\\mathswitchr Z\\mathswitchr {A_0}}$, which includes tadpole diagrams. \nThe result $\\delta \\beta^t\\big|_\\mathrm{FJ}$ of this procedure\nis again given by Eq.~\\refeq{eq:dbtFJ_expl}, as it should be.\n\n\\paragraph{The renormalization constant \\boldmath{$\\delta \\alpha\\big|_\\mathrm{FJ}$}:}\nWe apply the same method to the renormalization constant \n$\\delta \\alpha\\big|_\\mathrm{FJ}$, starting from Eq.~\\refeq{eq:deltaalphaderiv2}.\nThe difference between $\\delta \\alpha$ and $\\delta \\alpha^t$ is entirely given\nby the explicit tadpole diagrams that appear in the change from \n$\\Sigma^{\\mathswitchr H\\mathswitchr h}$ to $\\Sigma^{t,\\mathswitchr H\\mathswitchr h}$ in Eq.~\\refeq{eq:deltaalphaderiv2},\n\\begin{align}\n\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h) &= \\delta\\alpha^t-\\delta\\alpha\n= \\left. \\mathrm{Re}\\frac{\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\PH}^2)+\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{2(\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2)}\n\\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h},\n\\end{align}\nwhich evaluates to\n\\begin{align}\n\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h) \n& = \\left. \\mathrm{Re}\\frac{\\Sigma^{t,\\mathswitchr H\\mathswitchr h}(\\mathswitch {M_\\Ph}^2)}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\right|_\\mathrm{T_\\mathswitchr H,T_\\mathswitchr h}\n=\\frac{1}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\left[\\parbox[c][1cm]{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEhHh2.eps}}+\n\\parbox[c][1cm]{2.5cm}{\\includegraphics[scale=1]{.\/diagrams\/tadpoleSEHHh1.eps}} \\right]\n\\nonumber\\\\\n&=\\frac{e}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right),\n\\end{align}\nwith the coupling factors of the $\\mathswitchr h\\mathswitchr H\\PH$ and $\\mathswitchr h\\Ph\\mathswitchr H$ vertices\n\\begin{subequations}\n\\begin{align}\nC_{\\mathswitchr h\\mathswitchr H\\PH} &=   \\frac{e s_{\\beta-\\alpha} }{2\\mathswitch {M_\\PW}\\mathswitch {s_{\\scrs\\PW}} s_{2\\beta}}\\biggl[ \n-(3s_{2\\alpha}+s_{2\\beta})\n\\left(\\mathswitch {M_\\PAO}^2 +4 \\lambda_5 \\frac{\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}{e^2}\\right)\n+ s_{2 \\alpha} \\left(\\mathswitch {M_\\Ph}^2 + 2 \\mathswitch {M_\\PH}^2\\right)  \\biggr], \n\\\\\nC_{\\mathswitchr h\\Ph\\mathswitchr H} &= \\frac{ec_{\\beta-\\alpha}}{2\\mathswitch {M_\\PW}\\mathswitch {s_{\\scrs\\PW}} s_{2\\beta}}  \\biggl[\n(3s_{2\\alpha}-s_{2\\beta})\n\\left( \\mathswitch {M_\\PAO}^2 + 4 \\lambda_5 \\frac{\\mathswitch {M_\\PW}^2 \\mathswitch {s_{\\scrs\\PW}}^2}{e^2}\\right)\n- s_{2\\alpha}\\left(2 \\mathswitch {M_\\Ph}^2  + \\mathswitch {M_\\PH}^2  \\right) \\biggr].\n\\end{align}\n\\end{subequations}\nThe counterterm $\\delta\\alpha^t\\big|_\\mathrm{FJ}$ of the FJ scheme in the $t$-variant,\nthus, reads\n\\begin{align}\n\\delta \\alpha^t\\big|_\\mathrm{FJ} &= \\delta \\alpha +\\Delta\\alpha^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{UV}\n= \\delta \\alpha \n+\\frac{e}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}+\nT_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\bigg|_\\mathrm{UV},\n\\label{eq:datFJ_expl}\n\\end{align}\nwhich is again in agreement with Ref.~\\cite{Krause:2016oke,Denner:2016etu}. \nThis translates to our treatment of tadpoles as\n\\begin{align}\n\\delta \\alpha\\big|_\\mathrm{FJ} &= \\delta \\alpha \n-\\Delta\\alpha^t(T_\\mathswitchr H ,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n= \\delta \\alpha\n+\\frac{e}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\bigg|_\\mathrm{finite}.\n\\end{align}\nConcerning the use of $\\delta \\alpha\\big|_\\mathrm{FJ}$ in our counterterm Lagrangian\nto obtain renormalized amplitudes in the gauge-independent FJ scheme,\nthe same comments made above for $\\delta\\beta\\big|_\\mathrm{FJ}$ apply.\n\n\\subsubsection{\\boldmath{The FJ($\\lambda_3$) scheme}}\n\\label{sec:FJL3scheme}\n\nIn the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme, the parameters $\\lambda_{3}$ \nand $\\lambda_{5}$ are defining parameters of the basic para\\-meterization and gauge independent by construction. Therefore, the condition on $\\delta \\beta$ is the only renormalization condition potentially being gauge dependent. To provide a fully\ngauge-independent renormalization scheme where $\\lambda_3$ is an independent quantity, we apply the FJ scheme to the parameter $\\beta$ and keep the renormalization of \n$\\lambda_{3}$ and $\\lambda_{5}$ \nas in the $\\overline{\\mathrm{MS}}(\\lambda_3)$. We call the resulting scheme the FJ($\\lambda_3$) scheme. The renormalization of the parameters reads:\n\\begin{align}\n \\delta \\beta\\big|_\\mathrm{FJ}\\text{ as in Eq.~\\eqref{eq:dbFJ_expl}},\\nonumber\\\\\n \\delta \\lambda_3 \\text{ as in Eqs.~\\eqref{eq:deltaL3}}{-}\\refeq{eq:L3bos}.\\nonumber\n\\end{align}\n\n\\subsection{Conversion between different renormalization schemes}\n\\label{sec:inputconversion}\n\nIn the previous section, we have\npresented four different renormalization schemes, which treat the mixing parameters differently. When observables calculated in different renormalization schemes are compared, particular care has to be taken that the input parameters are consistently translated from one scheme to the other. The bare values of identical independent\nparameters are equal and independent of the renormalization scheme. \nExemplarily, for a parameter $p$, the renormalized values $p^{(1)}$ and $p^{(2)}$ in two different renormalization schemes~1 and 2 are connected via the bare parameter $p_0$, \n\\begin{align}\np_0=p^{(1)}+\\delta p^{(1)} (p^{(1)})=p^{(2)} +\\delta p^{(2)} (p^{(2)}),\n\\end{align}\nwithin the considered order. \nIf $p$ is a dependent parameter in one or both schemes, it must be calculated from the independent renormalized para\\-meters and their counterterms from the relations between bare and renormalized quantities. For converting an input value from one scheme to another, one can solve for one renormalized quantity\n\\begin{align}\n\\label{eq:convertinputparam}\n p^{(1)}=p^{(2)} +\\delta p^{(2)} (p^{(2)})-\\delta p^{(1)} (p^{(1)}).\n\\end{align}\nAt NLO, this equation can be linearized by substituting the input value of $p^{(1)}$ by $p^{(2)}$ \nin the computation of the last counterterm. The differences to an exact solution are of higher order and beyond our desired NLO\naccuracy. However, large counterterms or small tree-level values can spoil the approximation so that in this case a proper solution using numerical techniques could improve the results. \nAnother benefit of a full solution of the implicit equation is the possibility that one can \nswitch to another scheme and back in a self-consistent way, while start and end scenarios\nin scheme~(1) do not exactly coincide when switching from scheme~(1) to (2) and back to (1)\nusing the linearized approximation.\nThe comparison of both methods allows for a consistency check of the computation and for an analysis of perturbative stability.\nWe have derived the Higgs mixing angles $\\alpha$ and $\\beta$ and their counterterm in all schemes. The finite parts of the gauge-dependent counterterms $\\delta \\alpha$, $\\delta \\beta$ are given here for the different renormalization schemes indicated by the respective index: \n\\begin{subequations}\n\\label{eq:fintermsalpha}\n\\begin{align}\n \\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha),\\mathrm{finite}} &= 0,\\\\\n \\delta \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3),\\mathrm{finite}} \n&= \\delta \\alpha\\big|_{\\mathrm{FJ}(\\lambda_3),\\mathrm{finite}} \n= \\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\PH}^2-\\mathswitch {M_\\Ph}^2}\\Big|_\\mathrm{finite},\\\\\n \\delta \\alpha\\big|_{\\mathrm{FJ}(\\alpha),\\mathrm{finite}} &= -\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}\n=\\left.\\frac{e}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2} \\left( \nT_\\mathswitchr H \\frac{C_{\\mathswitchr h\\mathswitchr H\\PH}}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{C_{\\mathswitchr h\\Ph\\mathswitchr H}}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite}.\n\\end{align}\n\\end{subequations}\nFor the angle $\\beta$ we obtain the following finite terms \nin the $\\overline{\\mathrm{MS}}$ and the FJ schemes,\n\\begin{subequations}\n\\label{eq:fintermsbeta}\n\\begin{align}\n \\delta \\beta\\big|_{\\overline{\\mathrm{MS}(\\alpha)},\\mathrm{finite}} \n&= \\delta \\beta\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3),\\mathrm{finite}} \n= 0,\\\\\n\\delta \\beta\\big|_{\\mathrm{FJ}(\\alpha),\\mathrm{finite}} \n&= \\delta \\beta\\big|_{\\mathrm{FJ(\\lambda_3)},\\mathrm{finite}} \n = -\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}=\\left.\\frac{e}{2 \\mathswitch {s_{\\scrs\\PW}} \\mathswitch {M_\\PW}} \\left(\nT_\\mathswitchr H \\frac{ s_{\\alpha -\\beta }}{\\mathswitch {M_\\PH}^2}\n+T_\\mathswitchr h \\frac{ c_{\\alpha -\\beta }}{\\mathswitch {M_\\Ph}^2} \n\\right)\\right|_\\mathrm{finite}.\n\\end{align}\n\\end{subequations}\nWith these formulae we can convert\nthe input variables for $\\alpha$ and $\\beta$\neasily into each other. \nFor instance, the conversion of the input values of $\\alpha$ and $\\beta$ \ndefined in the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme into \nthe other renormalization schemes reads\n\\begin{subequations}\n\\begin{align}\n\\alpha\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} \n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\left.\\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2}\\right|_\\mathrm{finite}, &\n\\beta\\big|_{\\overline{\\mathrm{MS}}(\\lambda_3)} &=\n\\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)},\n\\label{eq:convertinput1}%\n\\\\[.3em]\n\\alpha\\big|_{\\mathrm{FJ}(\\alpha)}\n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\alpha^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}, &\n\\beta\\big|_{\\mathrm{FJ}(\\alpha)}\n& = \\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite},\n\\label{eq:convertinput2}%\n\\\\[.3em]\n\\alpha\\big|_{\\mathrm{FJ}(\\lambda_3)}\n& = \\alpha\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\left.\\frac{f_\\alpha\\{\\delta p'_\\mathrm{mass}\\}}{\\mathswitch {M_\\Ph}^2-\\mathswitch {M_\\PH}^2}\\right|_\\mathrm{finite}, & \n\\beta\\big|_{\\mathrm{FJ}(\\lambda_3)}\n& = \\beta\\big|_{\\overline{\\mathrm{MS}}(\\alpha)}\n+\\Delta\\beta^t(T_\\mathswitchr H,T_\\mathswitchr h)\\big|_\\mathrm{finite}.\n\\label{eq:convertinput3}%\n\\end{align}\n\\end{subequations}%\nWithin a given scheme, $\\lambda_3$ and $\\alpha$ can be translated into each other\nusing the tree-level relation~\\refeq{eq:lambda3tree}.\nNote that, thus, the numerical values of $\\alpha$, $\\beta$, and $\\lambda_3$ corresponding\nto a given physical scenario of the THDM are different in different renormalization schemes.\nIn turn, fixing the input values in the four renormalization schemes to the same values\ncorresponds to different physical scenarios.\nIn particular, this means that the ``alignment limit'', in which $s_{\\beta-\\alpha}\\to1$\nso that $\\mathswitchr h$ is SM like \n{(see, e.g., \\citeres{Haber:1995be,Gunion:2002zf,Bernon:2015qea}),} is a notion that depends on the renormalization scheme\n(actually even on the scale choice in a given scheme).%\n\\footnote{In this brief account of results, we do not consider the (phenomenologically\ndisfavoured, though not excluded) possibility that the heavier CP-even Higgs boson $\\mathswitchr H$ is SM-like,\nwhich is discussed in detail in \\citere{Bernon:2015wef}.}\n\nExemplarily, the \nconversions of $c_{\\beta-\\alpha}$  from the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ scheme into the $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ$(\\alpha)$ (pink), and \nFJ$(\\lambda_3)$ (turquoise) schemes are shown in Fig.~\\ref{fig:plot_Umrechnung-von-alpha-LM}. The results of the transformations \nin the inverse directions are displayed in Fig.~\\ref{fig:plot_Umrechnung-nach-alpha-LM}, and all other conversions can be seen as a combination of the presented ones. \n\\begin{figure}\n\\centering\n\\subfigure[]{\n\\label{fig:plot_Umrechnung-von-alpha-LM}\n\\includegraphics{.\/diagrams\/plot_Umrechnung-von-alpha-LM.eps}\n}\n\\hspace{15pt}\n\\subfigure[]{\n\\label{fig:plot_Umrechnung-nach-alpha-LM}\n\\includegraphics{.\/diagrams\/plot_Umrechnung-nach-alpha-LM.eps}\n}\n\\vspace*{-.5em}\n\\caption{(a) Conversion of the value of $c_{\\beta-\\alpha}$ from $\\overline{\\mathrm{MS}}(\\alpha)$ \nto the $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ$(\\alpha)$ (pink), and FJ($\\lambda_3$) \nschemes (turquoise) for scenario~A. Panel~(b) shows the conversion to the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme using the same colour coding. \nThe solid lines are obtained \nby solving the implicit equations \\refeq{eq:convertinputparam} numerically, the dashed lines\ncorrespond to the linearized approximation.\nThe phenomenologically relevant region is highlighted in the centre.}\n\\label{fig:plotconversionA}%\n\\end{figure}%\nThe input values (defined before the conversion) correspond to the low-mass scenario called ``A''\nof a THDM of Type~I (based on a benchmark scenario of Ref.~\\cite{Haber2015})\nwith  \n\\begin{align}\n\\mathswitch {M_\\Ph}=125 \\text{ GeV}, \\quad\n \\mathswitch {M_\\PH}=300 \\text{ GeV}, \\quad \\mathswitch {M_\\PAO}=\\mathswitch {M_\\PHP}=460 \\text{ GeV}, \\quad \\lambda_5=-1.9,\\quad \\tan \\beta=2. \n\\label{eq:scenario}\n \\end{align}\nSpecifically, scenario~A is a scan in $c_{\\beta-\\alpha}$ in the mass parameterization, \nAa and Ab are points of the scan region used to analyze the scale dependence:\n\\begin{subequations}\n\\begin{align}\n\\mbox{A:}  \\quad \\cos{(\\beta-\\alpha)} &= -0.2\\ldots0.2, \\\\\n\\mbox{Aa:} \\quad \\cos{(\\beta-\\alpha)} &= +0.1, \\\\\n\\mbox{Ab:} \\quad \\cos{(\\beta-\\alpha)} &= -0.1. \n\\end{align}\n\\label{eq:cba_A}\n\\end{subequations}\nThe {$\\overline{\\mathrm{MS}}$}{} parameters are defined at the scale\n\\begin{align}\n\\label{eq:centralscale}\n \\mu_0=\\frac{1}{5}(\\mathswitch {M_\\Ph}+\\mathswitch {M_\\PH}+\\mathswitch {M_\\PAO}+2\\mathswitch {M_\\PHP}).\n\\end{align}\nThe motivation for this choice will become clear below.\nThe remaining input parameters for the SM part are given in \nApp.~\\ref{app:smparams}.\nIn both plots, we highlight the phenomenologically relevant region in the centre. \nThe solid lines are the result obtained \nby solving the implicit equations \\refeq{eq:convertinputparam} numerically,\nthe dashed lines correspond to a linearized conversion.\nAll curves show only minor conversion effects in the parameter values, i.e.\\ the \nsolution of the implicit equations\nagrees well with the approximate linearized conversion, affirming that the contributions of the\nhigher-order functions $\\Delta\\alpha^t$, $\\Delta\\beta^t$, and $f_\\alpha$ of \nEqs.~\\refeq{eq:convertinput1}--\\refeq{eq:convertinput3}\nare small, and perturbation theory is applicable. Since the values of the parameters change when going from one renormalization scheme to another, \nthe alignment limit does not persist in these transformations, i.e.\\\nin this scenario the alignment limit sensitively depends on the definition of the parameters at NLO.\n\n{For the schemes with $\\lambda_3$ as input parameter, some singular behaviour in the parameter\nconversion can be observed in the phenomenologically disfavoured region where $c_{\\beta-\\alpha}\\lsim-0.3$.\nThis artifact in the conversion appears when $c_{2\\alpha}\\to0$ (see, e.g., Eq.~\\refeq{eq:MHhdependence}),\nindicating the breakdown of the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and FJ($\\lambda_3$) schemes in such\nparameter regions. Already this case-specific study shows that stability issues of different\nrenormalization schemes have to be carefully carried out for all interesting parameter regions\nand that the applicability of a specific scheme in general does not cover the full THDM \nparameter space, a fact that was also pointed out in \\citere{Krause:2016oke}\nfor the THDM and that is known from NLO calculations in the MSSM (see, e.g., \\citeres{Heinemeyer:2010mm,Chatterjee:2011wc}).\nSpecifically, if the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and FJ($\\lambda_3$) schemes are not\napplicable in a region that might be favoured by future data analyses, it would\nbe desirable and straightforward to replace $\\lambda_3$ by $\\lambda_1$ or $\\lambda_2$\nas independent parameter, thereby defining analogous schemes like $\\overline{\\mathrm{MS}}(\\lambda_1)$, etc..\n\nTo address this issue properly, was our basic motivation to introduce and compare different renormaliztion\nschemes. We will continue this discussion in more detail in a forthcoming publication,\nwhere further THDM scenarios are considered.}\n\n\\section{\\boldmath{The running of the {$\\overline{\\mathrm{MS}}$}{} parameters}}\n\\label{sec:running}\n\nParameters renormalized in the $\\overline{\\mathrm{MS}}$ scheme depend on an unphysical renormalization scale $\\mu_\\mathrm{r}$. The one-loop $\\beta$-function of a parameter $p$  can be obtained from the UV-divergent parts of its \ncounterterm $\\delta p$, \n\\begin{align}\n\\beta_p(\\mu^2_\\mathrm{r})=\\frac{\\partial}{\\partial \\ln{\\mu_\\mathrm{r}^2}} p(\\mu_\\mathrm{r}^2)\n=\\frac{\\partial}{\\partial \\Delta_\\mathrm{UV}} \\delta p.\n\\label{eq:betafct}\n\\end{align}\nSince the renormalization constants are computed in a perturbative manner, the $\\beta$-functions have a perturbative expansion in the coupling parameters.\nNote that the last equality in Eq.~\\refeq{eq:betafct} holds in the FJ schemes\nfor $\\alpha$ and $\\beta$\nonly in the $t$-variant explained above, because the finite contributions\n$\\Delta\\alpha^t$ and $\\Delta\\beta^t$ depend on the scale $\\mu_\\mathrm{r}$.\n\nAs discussed in the previous sections, \nthe ratio of the vevs, $\\tan \\beta$, the Higgs mixing parameter $\\alpha$ or $\\lambda_3$, and the Higgs self-coupling $\\lambda_5$ are renormalized in the $\\overline{\\mathrm{MS}}$ scheme. For each renormalization scheme described in Sect.~\\ref{sec:diffrenschemes}, one obtains a set of coupled RGEs involving the $\\beta$-functions of the independent parameters. Therefore, the scale dependence varies when different schemes are applied. In the perturbative expansion of the $\\beta$-function we consider only the one-loop term, being second order in the coupling constants,\ne.g., in the {$\\overline{\\mathrm{MS}}$}$(\\alpha)$ scheme\n\\begin{align}\n\\beta_p(\\mu^2_\\mathrm{r})= A_p \\alpha_\\mathrm{em}+ B_p \\lambda_5+C_p \\lambda_5^2\/\\alpha_\\mathrm{em}.\n\\end{align}\nThe dependence on the strong coupling constant vanishes at one-loop order as the parameters renormalized in {$\\overline{\\mathrm{MS}}$}{} appear only in couplings of particles that do not interact strongly. The coefficients $A_p, B_p, C_p$ of the respective renormalized parameter can be easily read \nfrom the divergent terms which have been derived in the previous section. We have checked them against the $\\beta$-functions given for $\\lambda_{3}$ and $\\lambda_{5}$ in Ref.~\\cite{Branco:2011iw} and for $\\beta$ in Ref.~\\cite{Sperling:2013eva} (supersymmetric contributions need to be omitted).\n\nIn general, RGEs, which are a set of coupled differential equations, cannot be solved analytically. \nUsually numerical techniques, such as a Runge--Kutta method, need to be employed to solve the RGEs and to compute the values of the parameters at a desired scale.\nMoreover, we emphasize that the renormalization-group flow of a running parameter\ndepends on the renormalization scheme of the full set of independent parameters.\nThat means the fact that we use on-shell quantities, such as all the Higgs-boson masses,\nto fix most of the scalar self-couplings has a significant impact on the running\nof our {$\\overline{\\mathrm{MS}}$}{} parameters.\nThe renormalization-group flow in other schemes was, e.g., investigated in\n\\citeres{Cvetic:1997zd,Branco:2011iw,Bijnens:2011gd,Chakrabarty:2014aya,Ferreira:2015rha}.\n\nThe scale dependence of $c_{\\beta-\\alpha}$ \nfor $\\mu = 100{-}900$~GeV is plotted in Fig.~\\ref{fig:plotrunningA}, for the scenario defined in Eq.~\\eqref{eq:scenario} with $c_{\\beta-\\alpha}=0.1$  (l.h.s) and $c_{\\beta-\\alpha}=-0.1$ (r.h.s) and input values given at the central scale $\\mu_0$ stated in\nEq.~\\refeq{eq:centralscale}. \nWe observe that the choice of  the renormalization scheme has a large impact on the scale dependence. While the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme introduces only a mild running, the other schemes show a much stronger scale dependence, so that excluded and unphysical values of input parameters can be reached quickly. A similar observation has also been made in supersymmetric models for the parameter~$\\tan\\beta$~\\cite{Freitas:2002um}. \nGauge-dependent $\\overline{\\mathrm{MS}}$ schemes have a small scale dependence while replacing the parameters by gauge-independent ones like in the FJ schemes introduce additional terms in the $\\beta$-functions which induce a stronger scale dependence.\nIn Fig~\\ref{fig:running_cba-0.1-LM} one can also see that the curves for the $\\overline{\\mathrm{MS}}(\\lambda_3)$ and the FJ($\\lambda_3$) schemes terminate around 250 GeV. \n\\begin{figure}\n  \\centering\n  \\subfigure[]{\n\\label{fig:running_cba0.1-LM}\n\\includegraphics{.\/diagrams\/running_cba0.1-LM.eps}\n}\n\\hspace{15pt}\n\\subfigure[]{\n\\label{fig:running_cba-0.1-LM}\n\\includegraphics{.\/diagrams\/running_cba-0.1-LM.eps}\n}\n\\vspace*{-.5em}\n  \\caption{The running of $c_{\\beta-\\alpha}$ for the low-mass scenario~A with $c_{\\beta-\\alpha}=0.1$ (a) and $c_{\\beta-\\alpha}=-0.1$ (b) in the $\\overline{\\mathrm{MS}}(\\alpha)$ (blue), $\\overline{\\mathrm{MS}}(\\lambda_3)$ (green), FJ($\\alpha)$ (pink), and FJ($\\lambda_3$) (turquoise) schemes.}\n\\label{fig:plotrunningA}\n\\end{figure}\nAt this scale, the running of $\\lambda_3$ yields unphysical values for which \nEq.~\\refeq{eq:lambda3tree}\nwith the given Higgs masses becomes \noverconstrained, and no solution with $|s_{2\\alpha}|\\le1$ exists. This is unique to the $\\lambda_3$ running as only there an \nimplicit equation needs to be solved to obtain the input parameter $\\alpha$. For the other cases we prevent the angles from running out of their domain of definition by solving the running for the tangent function of the angles.\n\n\\section{Implementation into a \\textsc{FeynArts} Model File}\n\\label{sec:FAmodelfile}\nThe \\textsc{Mathematica} package \\textsc{FeynRules} (FR)~\\cite{Christensen2009} \nis a tool to generate Feynman rules from a given \nLagrangian, providing the possibility to produce model files in various output formats\nwhich can be employed by automated amplitude generators. \nWe have inserted the Lagrangian into FR in its internal notation to obtain\nthe corresponding counter\\-term Lagrangian after the renormalization transformations. \nBefore this insertion, we have computed and simplified  \nthe Higgs potential and the corresponding counterterm potential~\\eqref{eq:dVH2p} \nwith inhouse \\textsc{Mathematica} routines.\nUsing FR,\nthe tree-level and the counterterm Feynman rules as well as the renormalization conditions \nin the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ and {$\\overline{\\mathrm{MS}}$}{}($\\lambda_3$) schemes have been implemented into \na model file for the amplitude generator \\textsc{FeynArts} (FA)~\\cite{Hahn2001}. \nThe renormalization conditions of the FJ($\\alpha)$ and the FJ($\\lambda_3$) have not \nbeen included in the model file, because using \nEqs.~\\eqref{eq:fintermsalpha},~\\eqref{eq:fintermsbeta} it is straightforward to \nimplement the corresponding finite terms of $\\delta \\alpha$ and $\\delta \\beta$. \nWith such a model file, NLO amplitudes for any process can be generated in an automated way. \n\nThe FA NLO model file for the THDM, obtained with FR, has the following features:\n\\begin{itemize}\n\\item Type I, II, flipped, or lepton-specific THDM;\n\\item all tree-level and counterterm Feynman rules;\n\\item renormalization conditions according to the {$\\overline{\\mathrm{MS}}$}{}$(\\alpha)$ and {$\\overline{\\mathrm{MS}}$}{}$(\\lambda_3)$  schemes;\n\\item all renormalization constants are implemented additionally in $\\overline{\\mathrm{MS}}$ \nas well, which allows for fast checks of UV-finiteness;\n\\item BHS and HK conventions;\n\\item\nCKM matrix set to the unit matrix (the generalization is straightforward).\n\\end{itemize}\nThis model file has been tested intensively, including checks of UV-finiteness for \nseveral processes, both numerically and analytically. \nThis allows for the generation of amplitudes \n(and further processing with \\textsc{FormCalc} \\cite{Hahn1999}) for any process at the one-loop level, at any parameter point of the THDM.\nThe model file can be obtained from the authors upon request.\n\n\\section{\\boldmath{Numerical results for \\boldmath$ \\mathswitchr h \\to \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ \\to 4f$}}\n\\label{se:numerics}\n\nIn this section we present first results from the computation of the decay of the light, neutral \nCP-even Higgs boson of the THDM into four fermions at NLO. The computer program \n\\textsc{Prophecy4f}~\\cite{Bredenstein:2006rh,Bredenstein:2006nk,Bredenstein:2006ha}%\n\\footnote{\\tt http:\/\/prophecy4f.hepforge.org\/index.html}\nprovides a ``\\textbf{PROP}er description of the \\textbf{H}iggs d\\textbf{EC}a\\textbf{Y} into \\textbf{4 F}ermions'' and calculates \nobservables for the decay process $\\mathswitchr h {\\to} \\mathswitchr W\\PW\/\\mathswitchr Z\\PZ {\\to} 4 f$ at NLO EW+QCD in the SM. \nWe have extended this program to the calculation of the corresponding decay in the THDM in such a way that the usage of the program and its applicability as event\ngenerator basically remains the same. \n{Owing to the fact that LO and real-emission amplitudes in the THDM receive only the\nmultiplicative factor $s_{\\beta-\\alpha}$ with respect to the SM, the bremsstrahlung corrections as well\nas the treatment of infrared singularities could be taken over from the SM \ncalculation~\\cite{Bredenstein:2006rh,Bredenstein:2006ha}\nvia simple rescaling.}\nThe calculation in the THDM, the implementation in  \\textsc{Prophecy4f}, as well as results of the application will be described in detail in an upcoming publication.\nWe just mention that we employ the complex-mass scheme~\\cite{Denner:2005fg} to describe the \nW\/Z~resonances, as already done in \\citeres{Bredenstein:2006rh,Bredenstein:2006nk,Bredenstein:2006ha}\nfor the SM. Note that the W\/Z-boson masses as well as the weak mixing angle are consistently taken as complex quantities in the complex-mass scheme to guarantee gauge invariance of all amplitudes\nin resonant and non-resonant phase-space regions. \nConsequently all our renormalization constants of the THDM inherit imaginary parts from the complex\ninput values, but the impact of these spurious imaginary parts is beyond NLO and negligible\n(as in the SM). Moreover, we mention that the modified version of \\textsc{Prophecy4f}\nmakes use of the public \\textsc{Collier} library~\\cite{Denner:2016kdg} for the calculation \nof the one-loop integrals.\n{Apart from performing two independent loop calculations, we have verified\nour one-loop matrix elements by numerically comparing our results to\nthe ones obtained in \\citere{Denner:2016etu} for the related\n$\\mathswitchr W\\mathswitchr h\/\\mathswitchr Z\\mathswitchr h$ production channels (including $\\mathswitchr W\/\\mathswitchr Z$ decays)\nusing crossing symmetry.}\n\nIn this paper, we present first results in order to demonstrate the use and the \nself-consistency of our renormalization schemes, employing again the scenario\ninspired by the first benchmark scenario of Ref.~\\cite{Haber2015} where the additional \nHiggs bosons are not very heavy. \nThe input values of the THDM parameters \n{for a Type~I THDM}\nare given in Eqs.~\\refeq{eq:scenario} and \n\\refeq{eq:cba_A}.\nSince $c_{\\beta-\\alpha}$ is the only parameter of the THDM appearing at LO, \nour process is most sensitive to this parameter. We vary $c_{\\beta-\\alpha}$ \nin the range $[-0.2,+0.2]$ in scenario~A for the computation of \nthe partial decay width for $\\mathswitchr h\\to\\mathswitchr W\\PW\/\\mathswitchr Z\\PZ\\to4f$, $\\Gamma^{\\mathswitchr h\\to4f}_\\mathrm{THDM}$,\nwhich is obtained by summing the partial widths of the h~boson\nover all massless four-fermion final states $4f$.\nThe parameters of the SM part of the THDM are collected in App.~\\ref{app:smparams}.\n{Note that a non-trivial CKM matrix would not change our results,\nsince quark mass effects of the first two generations as well as mixing\nwith the third generation are completely negligible in the considered\ndecays.}\n\nTo perform scale variations we take two distinguished points named Aa and Ab with $c_{\\beta-\\alpha}=\\pm0.1$. \nFor the central renormalization scale we use the average mass \n$\\mu_0$ defined in Eq.~\\refeq{eq:centralscale}\nof all scalar degrees of freedom.\nThe scale $\\mu$ of $\\alpha_\\mathrm{s}$ is kept fixed at $\\mu=\\mathswitch {M_\\PZ}$ which is the appropriate scale for the QCD corrections (which are dominated by the hadronic W\/Z decays).\n\n\\subsection{Scale variation of the width}\n\\label{sec:LowMassscalevariation}\n\n{The running of the {$\\overline{\\mathrm{MS}}$}-renormalized parameters $\\alpha$ and $\\beta$ is\ninduced by the Higgs-boson self-energies (and some scalar vertex for $\\lambda_5$),\ni.e.\\ the relevant particles in the loops are all Higgs bosons, the W\/Z bosons,\nand the top quark. If all Higgs-boson masses are near the electroweak scale,\nsay $\\sim 100{-}200\\unskip\\,\\mathrm{GeV}$, \nwhere the W\/Z-boson and top-quark masses are located, \nthen the scale $\\mathswitch {M_\\Ph}$ turns out to be a reasonable scale, as expected.\n{However, if some heavy Higgs-boson masses increase to some generic mass\nscale $M_S$ and the mixing angle\n$\\beta-\\alpha$ stays away from the alignment limit, there is no \ndecoupling of heavy Higgs-boson effects, so that $M_S$ acts\nas generic UV cutoff scale appearing in logarithms $\\log(M_S\/\\mu_\\mathrm{r})$.\nThe renormalization scale $\\mu_\\mathrm{r}$ has to go up with $M_S$ to\navoid that the logarithm drives the correction unphysically large.\nThe optimal choice of $\\mu_\\mathrm{r}$, though, is somewhat empirical.}\nA good choice of the central scale $\\mu_0$ should come close to the\nstability point (plateau in the $\\mu_{r}$ variation) in the major part of THDM\nparameter space. Our choice \\eqref{eq:centralscale} of $\\mu_0$ \neffectively takes care of this and is eventually justified by the numerics.\n\nTo illustrate this and to estimate the theoretical uncertainties due to the\nresidual scale dependence, we compute the total width while the scale $\\mu_\\mathrm{r}$ is varied from $100{-}900\\unskip\\,\\mathrm{GeV}$.}\n{Results with central scale $\\mathswitch {M_\\Ph}$ are shown in App.~\\ref{app:mu0mh},\nproving that this would be not a good choice.}\nThe parameters $\\alpha$ and $\\beta$ are defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme, and to compute results in other renormalization schemes their values are converted using\nEqs.~\\refeq{eq:convertinput1}--\\refeq{eq:convertinput3},\nwhich are solved numerically without linearization.\nThereafter the scale is varied, the RGEs solved, and the width computed using the respective renormalization scheme. \nThe results are shown \nin Fig.~\\ref{fig:plotmuscan} at LO (dashed) and NLO EW (solid) for the benchmark points Aa and Ab.\n\\begin{figure}\n  \\centering\n  \\subfigure[]{\n\\label{fig:plot_MUSCAN-Aa-L3MS}\n\\includegraphics{.\/diagrams\/plot_MUSCAN-Aa-L3MS.eps}\n}\n\\hspace{15pt}\n  \\subfigure[]{\n\\label{fig:plot_MUSCAN-Ab-alphaMS}\n\\includegraphics{.\/diagrams\/plot_MUSCAN-Ab-L3MS.eps}\n}\n\\caption{The decay width for $\\mathswitchr h \\to 4f$ at LO \n(dashed) and NLO EW (solid) \nin dependence of the renormalization scale with $\\beta$ and $\\alpha$ defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. The result is computed in all four different renormalization schemes after converting the input at NLO (also for the LO curves)\nand displayed for the benchmark points Aa~(a) and Ab~(b) using the colour code of Fig.~\\ref{fig:plotrunningA}.  }\n\\label{fig:plotmuscan}\n\\end{figure}\nThe QCD corrections are not part of the EW scale variation and therefore omitted in these results. The benchmark point Aa shows almost textbook-like behaviour with the LO computation \nexhibiting a strong scale dependence for all renormalization schemes, resulting in sizable differences between the curves. However, each of the NLO curves shows a wide extremum with a large plateau, reducing the scale dependence drastically, as it is expected for NLO calculations. \nThe central scale $\\mu_\\mathrm{r}=(\\mathswitch {M_\\Ph}+\\mathswitch {M_\\PH}+\\mathswitch {M_\\PAO}+2\\mathswitch {M_\\PHP})\/5$ lies perfectly in the middle of the plateau regions motivating this scale choice. In contrast, the naive scale choice $\\mu_0=\\mathswitch {M_\\Ph}$ is not within the plateau region, leads to large, unphysical corrections, and should not be chosen. The breakdown of the FJ($\\alpha)$ curve for small scales can be explained by the running which becomes unstable for these values (see Fig.~\\ref{fig:running_cba0.1-LM}).\nFor all renormalization schemes, the plateaus coincide and the agreement between the renormalization schemes is improved at NLO w.r.t.~the LO results. This is expected, since results obtained with different  renormalization schemes should be equal up to higher-order terms, after the input parameters are properly converted. \nThe relative renormalization scheme dependence at the central scale,\n\\begin{align}\n\\Delta_\\mathrm{RS}=2 \\,\\frac{\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{max}(\\mu_0)-\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{min}(\\mu_0)}{\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{max}(\\mu_0)+\\Gamma^{\\mathswitchr h \\to 4f}_\\mathrm{min}(\\mu_0)},\n\\label{eq:renschemedep}\n\\end{align}\n expresses the dependence of the result on the renormalization scheme. It can be computed from the difference of the smallest and largest width \nin the four renormalization schemes normalized to their average. \nIn the calculation of $\\Delta_\\mathrm{RS}$, the full NLO EW+QCD corrections to the width\n$\\Gamma^{\\mathswitchr h \\to 4f}$ should be taken into account.\nIn Tab.~\\ref{tab:schemevar}, $\\Delta_\\mathrm{RS}$ is given at LO and NLO and confirms the reduction of the scheme dependence in the NLO calculation. In addition, as already perceived when the running was analyzed, the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme shows the smallest dependence on the renormalization scale, which attests a good absorption of further corrections into the NLO prediction.\\\\\n  \\begin{table}\n   \\centering\n   \\renewcommand{\\arraystretch}{1.2}\n \\begin{tabular}{|c|cc|}\\hline\n  \n  & Scenario Aa & Scenario Ab\\\\\n    $\\Delta^\\mathrm{LO}_\\mathrm{RS}$[\\%]       & 0.67& 0.84  \\\\\n    $\\Delta^\\mathrm{NLO}_\\mathrm{RS}$ [\\%]     & 0.08 & 0.34  \\\\\\hline\n  \\end{tabular}  \n   \\caption{The variation $\\Delta_\\mathrm{RS}$\nof the $\\mathswitchr h {\\to} 4f$ width using different renormalization schemes \nfor input parameters defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme.}\n \\label{tab:schemevar}\n \\end{table}%\nThe situation for the benchmark point Ab is more subtle. For negative values of $c_{\\beta-\\alpha}$ the truncation of the schemes involving $\\lambda_3$ at \n$\\mu_\\mathrm{r}=250{-}300$ GeV as well as the breakdown of the running of the FJ($\\alpha)$ scheme, which both were observed in the running in Fig.~\\ref{fig:running_cba-0.1-LM}, are also ma\\-ni\\-fest in the computation of the $\\mathswitchr h {\\to} 4f$ width.\nTherefore, the results vary much more, and the extrema with the plateau regions are not as distinct as for the benchmark point Aa. They are even missing for the truncated curves. Nevertheless, the situation improves at NLO. \nAs for scenario~Aa, the central scale choice of $\\mu_0$ \nis more appropriate in contrast than the choice of $\\mathswitch {M_\\Ph}$.\n \nFor both benchmark points, \nthe estimate of the theoretical uncertainties by varying the scale by a factor of two from the central value for an arbitrary renormalization scheme is generally not appropriate. \nA proper strategy would be to identify the renormalization schemes which yield reliable results, and to use only those to quantify the theoretical uncertainties from the scale variation. In addition, the renormalization scheme dependence of those schemes should be investigated. \nThis procedure should be performed for different parameter regions (and corresponding \nbenchmark points) separately, which is beyond the scope of this work. \n \n\\subsection{\\boldmath{$c_{\\beta-\\alpha}$} dependence}\n\\label{sec:cbascanA}\n\nThe decay width for $\\mathswitchr h \\to 4f$ in dependence of $c_{\\beta-\\alpha}$ in scenario~A is presented in Fig.~\\ref{fig:plotscbascanA} for all renormalization schemes with \nthe input values $\\alpha$ and $\\beta$ defined in the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. \n\\begin{figure}\n\\centering\n\\includegraphics{.\/diagrams\/plot_cbascan-A-diffschemes-L3MS.eps}\n\\caption{The decay width for $\\mathswitchr h \\to 4f$ at LO \n(dashed) and full NLO EW+QCD (solid) for scenario A in dependence of $c_{\\beta-\\alpha}$. \nThe input values are defined in the  $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme and \nare converted to the other schemes at NLO (also for the LO curves). \nThe results computed with different renormalization schemes are displayed with the colour code of Fig.~\\ref{fig:plotrunningA}, and the SM \n(with SM Higgs-boson mass $\\mathswitch {M_\\Ph}$) is shown for comparison in red.}\n\\label{fig:plotscbascanA}\n\\end{figure}%\nThe LO (dashed) and the full NLO EW+QCD total widths (solid) are computed in the different renormalization schemes after the NLO input conversion \n(without linearization) and using the constant default scale $\\mu_0$ of Eq.~\\eqref{eq:centralscale}. The SM values are illustrated in red. \nAt tree level the widths show the suppression w.r.t. to the SM with the factor $s_{\\beta-\\alpha}^2$ originating from the \n$\\mathswitchr H\\mathswitchr W\\PW$ and $\\mathswitchr H\\mathswitchr Z\\PZ$ couplings. The differences between the renormalization schemes are due to the conversion of the input. As the conversion induces NLO differences in the LO results, a pure LO computation is identical for all renormalization schemes as the conversion vanishes at this order \nand is represented by the LO curve of the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme. The suppression w.r.t.\\ the SM computation does not change at NLO, while the shape becomes slightly asymmetric,\nand the NLO results show a significantly better agreement between the renormalization schemes. \nDeviations of the THDM results from the SM expectations can be investigated when the SM Higgs-boson mass is identified with the mass \n$\\mathswitch {M_\\Ph}$ of the light CP-even Higgs boson~h \nof the THDM. The relative deviation of the full width from the SM is then\n\\begin{align}\n \\Delta_\\mathrm{SM}= \\frac{\\Gamma_\\mathrm{THDM}-\\Gamma_\\mathrm{SM}}{\\Gamma_\\mathrm{SM}},\n\\end{align}\nwhich is shown in Fig.~\\ref{fig:plot_cbascanrelSM-A-diffschemes-L3MS} at LO (dashed) and NLO (solid) in percent for parameters defined in the the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme.\n\\begin{figure}\n  \\centering\n\\includegraphics{.\/diagrams\/plot_cbascanrelSM-A-diffschemes-L3MS.eps}\n  \\caption{The relative difference of the decay width for $\\mathswitchr h \\to 4f$ in the THDM w.r.t.\\ the SM prediction at LO (dashed) and NLO EW+QCD (solid). \nThe input values are defined in the  $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme and \nare converted to the other schemes at NLO (also for the LO curves). \nThe results computed with different renormalization schemes are displayed with the colour code of Fig.~\\ref{fig:plotrunningA}.}\n\\label{fig:plot_cbascanrelSM-A-diffschemes-L3MS}\n\\end{figure}%\nThe SM exceeds the THDM widths at LO and NLO. \nThe LO  shape which is just given by \n$c_{\\beta-\\alpha}^2$ shows minor distortions due to the parameter conversions. At NLO, the shape is slightly distorted by an asymmetry of the EW corrections, and a small offset of $-0.5$\\% is visible even in the alignment limit where the diagrams including heavy Higgs bosons still contribute. The NLO computations show larger negative deviations, and this could be used to improve current exclusion bounds or increase their significance. Nevertheless, in the whole scan region the deviation from the SM is within 6\\% and for phenomenologically most interesting region with $|c_{\\beta-\\alpha}|<0.1$ even less than 2\\%, which is challenging for experiments to measure.\n\n\\section{Conclusions}\n\\label{se:conclusion}\n\nConfronting experimental results on Higgs precision observables with theory predictions\nwithin extensions of the SM, provides an important alternative to search for\nphysics beyond the SM, in addition to the search for new particles.\nThe THDM comprises an extended scalar sector with regard to the SM Higgs sector and allows for a comprehensive study of the impact of new scalar degrees of freedom without introducing new fundamental symmetries or other new theoretical structures. \n\nIn this article, we have considered the Type I, II, lepton-specific, and flipped versions of the \nTHDM. We have introduced four different renormalization schemes which employ directly\nmeasurable parameters such as masses as far as possible and make use of fields that \ndirectly correspond to mass eigenstates.\nIn all the schemes, the masses are defined via on-shell conditions, the electric charge is fixed via the Thomson limit, and the coupling $\\lambda_5$ is defined with the $\\overline{\\mathrm{MS}}$ prescription. The fields are also defined on-shell which is most convenient in applications. The renormalization schemes differ in the treatment of the coupling $\\lambda_3$ and the mixing angles $\\alpha$ and $\\beta$: In the $\\overline{\\mathrm{MS}}(\\alpha)$ scheme, $\\alpha$ and $\\beta$ are renormalized using $\\overline{\\mathrm{MS}}$ conditions. In the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme instead $\\lambda_3$ and $\\beta$ are $\\overline{\\mathrm{MS}}$-renormalized parameters. In addition to the conventional treatment of tadpole contributions, we have\nimplemented an alternative prescription suggested by Fleischer and Jegerlehner where the mixing angles $\\alpha$ and $\\beta$ obtain extra terms of tadpole contributions, rendering these schemes gauge independent to all orders. \nIt should, however, be noted that the $\\overline{\\mathrm{MS}}(\\lambda_3)$ scheme is also gauge independent at NLO in the \nclass of $R_\\xi$ gauges. \nWe have also discussed relations to renormalization procedures\nsuggested in the literature for the THDM. \n\nA comparison of these four different renormalization schemes allows for testing \nthe perturbative consistency, and, for the parameter regions and renormalization schemes fulfilling this test,  estimating the theoretical uncertainty due to the truncation of the perturbation series.\nTo further investigate the latter,\nwe have investigated the scale dependence, solving the corresponding RGEs. \nOne important observation is that it is crucial to be very careful and specific \nabout the definitions of the parameters applied, i.e.\\ the declaration of the renormalization\nscheme of the parameters is vital if one aims at precision.\nThis is already relevant in the formulation of\nbenchmark scenarios, because a conversion to a different scheme might alter the physical properties of the scenario significantly. For example, the alignment limit \nmay be reached with one specific set of parameters defined in a specific renormalization scheme, but converting these parameters consistently to parameters in a different renormalization scheme might shift the parameters away from the alignment limit.\n \nThe different renormalization schemes have been implemented into\na \\textsc{FeynArts} model file and are thus ready for applications.%\n\\footnote{The model file is restricted to a unit CKM matrix, but can be generalized\nto a non-trivial CKM matrix exactly as in the SM.}\nAs a first example, we have applied and tested the different schemes in the calculation of the decay width of a \nlight CP-even Higgs boson decaying into four massless fermions.\nWe discuss the dependence of the total $\\mathswitchr h{\\to}4f$ decay width on the renormalization scale\nand advocate a scale that is significantly higher than\nthe naive choice of $\\mu_\\mathrm{r} = \\mathswitch {M_\\Ph}$, taking care of the different mass scales in the\nTHDM Higgs sector.\nIn addition, results for various values of $\\cos (\\beta-\\alpha)$, a parameter entering the prediction already at LO, are presented. The deviations of the SM are relatively small, in the phenomenologically interesting region they are about \n$2{-}6\\%$---a challenge for future measurements. \n \nThe detailed\ndescription of the calculation of the decay width in the THDM \nand a survey of numerical results will be given in a forthcoming paper. \nThis includes a deeper investigation in the renormalization scale dependence and the comparison of different renormalization schemes \nfor more benchmark points\nas well as differential distributions.\n \n\n\\subsection*{Acknowledgements}\n\nWe would like to thank Ansgar Denner, Howard Haber and Jean-Nicolas Lang for helpful discussions\n{and especially Jean-Nicolas for an independent check of one-loop matrix elements\nagainst the crossing-related amplitudes used in \\citere{Denner:2016etu}.}\nHR's work is partially funded by the Danish National Research Foundation, grant number DNRF90. HR acknowledges also support by the National Science Foundation under Grant No. NSFPHY11-25915. \nWe thank the German Research Foundation~(DFG) and \nResearch Training Group GRK~2044 for the funding and the support and\nacknowledge support by the state of Baden--W\u00fcrttemberg through bwHPC and the \nDFG through grant no INST 39\/963-1 FUGG.\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nAs early as in 1927~\\cite{dir27}, Paul A, M. Dirac considered the problem\nof extending Heisenberg's uncertainty relations to the\nLorentz-covariant world.\nIn 1945~\\cite{dir45}, he attempted to construct the Lorentz\ngroup using the Gaussian wave function.\nIn 1949~\\cite{dir49},\nDirac pointed out the task of constructing relativistic dynamics\nis to construct a representation of the inhomogeneous Lorentz group.\nHe then wrote down the ten generators of this group and their closed set\nof commutation relations.   This set is known as the Lie algebra of\nthe Poincar\\'e group.\n\nIn 1963~\\cite{dir63}, Dirac considered two coupled harmonic oscillators\nand constructed an algebra leading to the Lie algebra for the $SO(3,2)$\nde Sitter group, which is the Lorentz group applicable to three space\ndimensions and two time-like variables.\n\nFrom the mathematical point of view, it is straight-forward to contract\none of those two time-like dimensions to construct $ISO(3,1)$ or the\nPoincar\\'e group.  This is what we present in this paper.  However,\nfrom the physical point of view, we are deriving the Poincar\\'e symmetry\nfor the Lorentz-covariant quantum world purely from the symmetries of\nHeisenberg's uncertainty relations.\n\nIn Sec.~\\ref{sp2}, it is noted that a one-dimensional uncertainty relation\ncontains the symmetry of the $Sp(2)$ group in the two-dimensional phase\nspace.  It is pointed out that this group, with three generators, is isomorphic\nto the Lorentz group applicable to two space dimensions and one time variable.\nWe can next consider another set with three additional generators.\n\nIn Sec.~\\ref{2osc}, we write those Heisenberg uncertainty relations in\nterm of step-up and step-down operators in the oscillator system.  It\nis then possible to consider the two coupled oscillator system\nwith the ten generators constructed by Dirac in 1963~\\cite{dir63}.\nIt is gratifying to note that this oscillator system can serve as the basic\nlanguage for the two-photon system of current interest~\\cite{yuen76,yurke86}.\n\nIn Sec.~\\ref{contraction}, we contract one of the time-like variables in\n$SO(3,2)$ to arrive at the inhomogeneous Lorentz group $ISO(3,1)$ or the\nPoincar\\'e group. In Sec.~\\ref{remarks}, we give some concluding\nremarks.\n\n\n\\section{Sp(2) Symmetry for the Single-variable Uncertainty Relation}\\label{sp2}\nIt is known that the symmetry of quantum mechanics and quantum field\ntheory is governed by the Poincar\\'e group~\\cite{dir49,dir62}.  The\nPoincar\\'e group means the inhomogeneous Lorentz group which includes\nthe Lorentz group applicable to the four-dimensional Minkowskian\nspace-time, plus space-time translations~\\cite{wig39}.\n\nThe question is whether this Poincar\\'e symmetry is derivable from\nHeisenberg's uncertainty relation, which takes the familiar form\n\\begin{equation}\\label{101}\n   \\left[x_{i}, p_{j}\\right] = i \\delta_{ij}.\n\\end{equation}\nThere are three commutation relations in this equation. Let us choose\none of them, and write it as\n\\begin{equation}\\label{102}\n   [x, p] = i .\n\\end{equation}\nThis commutation relation possesses the symmetry of the Poisson bracket in\nclassical mechanics~\\cite{arnold78,stern84}.  The best way to address this property\nis to use the Gaussian form for the Wigner function defined in the phase\nspace, which takes the\nform~\\cite{hkn88,kiwi90ajp,knp91}\n\\begin{equation}\\label{103}\nW(x,p) = \\frac{1}{\\pi} \\exp\\{-\\left(x^2 + p^2\\right)\\}.\n\\end{equation}\nThis distribution is concentrated in the circular region around the origin.\nLet us define the circle as\n\\begin{equation}\\label{105}\nx^{2} + p^{2} = 1.\n\\end {equation}\nWe can use the area of this circle in the phase space of $x$ and $p$ as the\nminimum uncertainty.  This uncertainty is preserved\nunder rotations in the phase space:\n\\begin{equation}\\label{107}\n\\pmatrix{\\cos\\theta & -\\sin\\theta \\cr \\sin\\theta & \\cos\\theta }\n\\pmatrix{ x \\cr p },\n\\end{equation}\nas well as the squeeze of the form\n\\begin{equation}\\label{108}\n\\pmatrix{e^{\\eta} &  0 \\cr 0 &  e^{-\\eta} }\n\\pmatrix{x \\cr p } .\n\\end{equation}\n\nThe rotation and the squeeze are generated by\n\\begin{equation}\\label{109}\nJ_{2} = - i\\left(x \\frac{\\partial}{\\partial p} - p\\frac{\\partial}{\\partial x } \\right),\n\\qquad\nK_{1} = -i \\left(x\\frac{\\partial}{\\partial x} - p \\frac{\\partial}{\\partial p} \\right),\n\\end{equation}\nrespectively.\nIf we take the commutation relation with these two operators, the result is\n\\begin{equation}\\label{111}\n\\left[J_{2}, K_{1}\\right] = -i K_{3},\n\\end{equation}\nwith\n\\begin{equation}\\label{113}\nK_{3} = -i \\left(x\\frac{\\partial}{\\partial p} +\np \\frac{\\partial}{\\partial x} \\right).\n\\end{equation}\nIndeed, these three generators form a closed set of commutation\nrelations:\n\\begin{equation}\\label{115}\n \\left[J_{2}, K_{1}\\right] = - iK_{3},  \\qquad\n \\left[J_{2}, K_{3}\\right] = iK_{1},   \\qquad\n \\left[K_{1}, K_{3}\\right] = iJ_{2}.\n\\end{equation}\nThis closed set is called the Lie algebra of the $Sp(2)$ group, isomorphic\nto the Lorentz group applicable to two space  and one time dimensions.\n\nLet us consider the Minkowskian space of $(x, y, z, t)$.  It is possible to\nwrite three four-by-four matrices satisfying the Lie algebra of Eq.(\\ref{115}).\nThe three four-by-four matrices satisfying this set\nof commutation relations are:\n\\begin{equation}\\label{119}\nJ_{2} = \\pmatrix{0 & 0 & i & 0 \\cr 0 & 0 & 0 & 0 \\cr i & 0 & 0 & 0 \\cr\n    0 & 0 & 0 & 0 },\\quad\nK_{1} = \\pmatrix{0 & 0 & 0 & i \\cr 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 \\cr\n    i & 0 & 0 & 0 },  \\quad\n K_{3} = \\pmatrix{0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & i \\cr\n    0 & 0 & i & 0 } .\n\\end{equation}\nHowever, these matrices have null second rows and null second columns.\nThus, they can generate Lorentz transformations applicable only to the\nthree-dimensional space of $(x,z,t)$, while the $y$ variable remains invariant.\n\n\n\n\\section{Two-oscillator System}\\label{2osc}\nIn order to generate Lorentz transformations applicable to the full\nMinkowskian space, along with $J_{2},K_{1}$, and $K_{3}$ we need two\nmore Heisenberg commutation relations.\nIndeed, Paul A. M. Dirac started this program in 1963~\\cite{dir63}.\nIt is possible to write the two uncertainty relations using two harmonic\noscillators as\n\\begin{equation}\\label{121}\n\\left[a_{i}, a^{\\dag}_{j}\\right] = \\delta_{ij} .\n\\end{equation}\nwith\n\\begin{equation}\\label{123}\n  a_{i} = \\frac{1}{\\sqrt{2}}\\left(x_{i} + ip_{i} \\right), \\qquad\n  a^{\\dag}_{i} = \\frac{1}{\\sqrt{2}}\\left(x_{i} - ip_{i} \\right),\n\\end{equation}\nand\n\\begin{equation}\\label{125}\n  x_{i} = \\frac{1}{\\sqrt{2}}\\left(a_{i} + a^{\\dag}_{i} \\right), \\qquad\n  p_{i} = \\frac{i}{\\sqrt{2}}\\left(a^{\\dag}_{i} - a_{i} \\right),\n\\end{equation}\n where $i$ and $j$ could be 1 or 2.\n\nMore recently in 1986, this two-oscillator system was considered\nby Yurke {\\it et al.}~\\cite{yurke86} in their study of two-mode\ninterferometers.  They considered first\n\\begin{equation}\\label{301}\n Q_{3} = \\frac{i}{2}\\left(a_{1}^{\\dag}a_{2}^{\\dag} - a_{1}a_{2}\\right),\n\\end{equation}\nwhich leads to the generation of the two-mode coherent state or the\nsqueezed state~\\cite{yuen76}.\n\nYurke {\\it et al.} then considered possible interferometers requiring\nthe following two additional operators.\n\\begin{equation}\\label{303}\nS_{3} = {1\\over 2}\\left(a^{\\dag}_{1}a_{1} + a_{2}a^{\\dag}_{2}\\right) ,\n\\qquad\nK_{3} = {1\\over 2}\\left(a^{\\dag}_{1}a^{\\dag}_{2} + a_{1}a_{2}\\right) .\n\\end{equation}\nThe three Hermitian operators from Eq.(\\ref{301}) and Eq.(\\ref{303})\nsatisfy the commutation relations\n\\begin{equation} \\label{305}\n\\left[K_{3}, Q_{3}\\right] = -iS_{3}, \\qquad\n\\left[Q_{3}, S_{3}\\right] = iK_{3}, \\qquad\n\\left[S_{3}, K_{3}\\right] = iQ_{3} .\n\\end{equation}\nThese relations are like those given in Eq.(\\ref{115}) for the Lorentz\ngroup applicable to two space-like and one time-like dimensions.\n\nIn addition, in the same paper~\\cite{yurke86}, Yurke {\\it et al.}\ndiscussed the possibility of constructing interferometers exhibiting\nthe symmetry generated by\n\\begin{equation}\\label{307}\n L_{1} = {1\\over 2}\\left(a^{\\dag}_{1}a_{2} + a^{\\dag}_{2}a_{1}\\right) , \\quad\n L_{2} = {1\\over 2i}\\left(a^{\\dag}_{1}a_{2} - a^{\\dag}_{2}a_{1}\\right), \\quad\n L_{3} = {1\\over 2}\\left(a^{\\dag}_{1}a_{1} - a^{\\dag}_{2}a_{2} \\right).\n\\end{equation}\nThese generators satisfy the closed set of commutation relations\n\\begin{equation}\\label{309}\n\\left[L_{i}, L_{j}\\right] = i\\epsilon_{ijk} L_{k} ,\n\\end{equation}\nand therefore define a Lie algebra which is the same as that for $SU(2)$\nor the three-dimensional rotation group.\n\nWe are then led to ask whether it is possible to construct a closed set\nof commutation relations with the six Hermitian operators from\nEqs.(\\ref{301},\\ref{303},\\ref{307}). It is not possible.  We have to\nadd four additional operators, namely\n\\begin{eqnarray}\\label{311}\n&{}& K_{1} = -{1\\over 4}\\left(a^{\\dag}_{1}a^{\\dag}_{1} + a_{1}a_{1} -\n  a^{\\dag}_{2}a^{\\dag}_{2} - a_{2}a_{2}\\right) ,   \\quad\nK_{2} = +{i\\over 4}\\left(a^{\\dag}_{1}a^{\\dag}_{1} - a_{1}a_{1} +\n  a^{\\dag}_{2}a^{\\dag}_{2} - a_{2}a_{2}\\right) ,   \\nonumber \\\\[1ex]\n&{}& Q_{1} = -{i\\over 4}\\left(a^{\\dag}_{1}a^{\\dag}_{1} - a_{1}a_{1} -\n  a^{\\dag}_{2}a^{\\dag}_{2} + a_{2}a_{2} \\right) ,   \\quad\nQ_{2} = -{1\\over 4}\\left(a^{\\dag}_{1}a^{\\dag}_{1} + a_{1}a_{1} +\n   a^{\\dag}_{2}a^{\\dag}_{2} + a_{2}a_{2} \\right) .\n\\end{eqnarray}\n\nThere are now ten operators from Eqs.(\\ref{301},\\ref{303},\\ref{307},\\ref{311}).\nIndeed, these ten operators satisfy the following closed set of commutation\nrelations.\n\\begin{eqnarray}\\label{313}\n&{}& [L_{i}, L_{j}] = i\\epsilon _{ijk} L_{k} ,\\quad\n[L_{i}, K_{j}] = i\\epsilon_{ijk} K_{k} ,  \\quad\n[L_{i}, Q_{j}] = i\\epsilon_{ijk} Q_{k} , \\nonumber\\\\[1ex]\n &{}& [K_{i}, K_{j}] = [Q_{i}, Q_{j}] = -i\\epsilon _{ijk} L_{k} , \\quad\n             [K_{i}, Q_{j}] = -i\\delta_{ij} S_{3},   \\nonumber\\\\[1ex]\n&{}&[L_{i}, S_{3}] = 0, \\quad [K_{i}, S_{3}] =  -iQ_{i},\\quad\n[Q_{i}, S_{3}] = iK_{i} .\n\\end{eqnarray}\nAs Dirac noted in 1963~\\cite{dir63}, this set is the same as the Lie\nalgebra for the $SO(3,2)$ de Sitter group, with ten generators. This\nis the Lorentz group applicable to the three-dimensional space with\ntwo time variables.  This group plays a very important role in\nspace-time symmetries.\n\n\nIn the same paper, Dirac pointed out that this set of commutation\nrelations serves as the Lie algebra for the four-dimensional\nsymplectic group commonly called $Sp(4)$, applicable to the systems\nof two one-dimensional particles, each with a two-dimensional phase\nspace.\n\n\nFor a dynamical system consisting of two pairs of canonical variables\n$x_{1}, p_{1}$ and $x_{2}, p_{2}$, we can use the four-dimensional\nspace with the coordinate variables defined as~\\cite{hkn95jmp}\n\\begin{equation}\\label{317}\n\\left(x_{1}, p_{1}, x_{2}, p_{2} \\right).\n\\end{equation}\nThen the four-by-four transformation matrix $M$ applicable to this\nfour-component vector is canonical if~\\cite{abra78,goldstein80}\n\\begin{equation}\\label{319}\nM J \\tilde{M} = J ,\n\\end{equation}\nwhere $\\tilde{M}$ is the transpose of the $M$ matrix,\nwith\n\\begin{equation}\\label{321}\nJ = \\pmatrix{0 & 1 & 0 & 0 \\cr -1 & 0 & 0 & 0 \\cr\n             0 & 0 & 0 & 1 \\cr 0 & 0 & -1 & 0 } .\n\\end{equation}\nAccording to this form of the $J$ matrix, the area of the phase space for\nthe $x_{1}$ and $p_{1}$ variables remains invariant, and the story is the\nsame for the phase space of $x_{2}$ and $p_{2}.$\n\nWe can then write the generators of the $Sp(4)$ group as\n\\begin{eqnarray}\\label{322}\n&{}& L_{1} = -\\frac{1}{2}\\pmatrix{0 & I\\cr I & 0 }\\sigma_{2}  , \\quad\nL_{2} = \\frac{i}{2} \\pmatrix{0 & -I \\cr I & 0} I , \\quad\nL_{3} = \\frac{1}{2}\\pmatrix{-I & 0 \\cr 0 & I}\\sigma_{2} , \\nonumber\\\\[2ex]\n&{}& S_{3} = \\frac{1}{2}\\pmatrix{I   & 0\\cr 0 & I} \\sigma_{2},\n\\end{eqnarray}\n\\noindent and\n\\begin{eqnarray}\\label{323}\n&{}&K_{1} = \\frac{i}{2}\\pmatrix{I  & 0 \\cr 0 & -I } \\sigma_{1}, \\quad\nK_{2} = \\frac{i}{2} \\pmatrix{I & 0 \\cr 0 & I } \\sigma_{3}, \\quad\nK_{3} = -\\frac{i}{2}\\pmatrix{0 & I \\cr I & 0 }  \\sigma_{1}, \\nonumber\\\\[2ex]\n&{}& Q_{1} = -\\frac{i}{2}\\pmatrix{I & 0 \\cr 0 & -I }\\sigma_{3}, \\quad\nQ_{2} = \\frac{i}{2}\\pmatrix{I & 0 \\cr 0 & I }\\sigma_{1} , \\quad\nQ_{3} = \\frac{i}{2}\\pmatrix{0 &  I \\cr I  & 0 } \\sigma_{3} ,\n\\end{eqnarray}\nwhere $I$ is the two-by-two identity matrix, while $\\sigma_{1},\n\\sigma_{2}$, and $\\sigma_{3}$ are the two-by-two Pauli matrices.  The four matrices\ngiven in Eq.(\\ref{322}) generate rotations, while those of Eq.(\\ref{323}) lead to\nsqueezes in the four-dimensional phase space.\n\nAs for the difference in methods used in Secs.~\\ref{sp2} and~\\ref{2osc},\nlet us look at the ten four-by-four matrices given in Eqs.(\\ref{322}) and (\\ref{323}).\nAmong these ten matrices, six of them are diagonal.  They are\n$S_{3}, L_{3}, K_{1}, K_{2} , Q_{1},$ and $Q_{2}$.\nIn the language of two harmonic oscillators, these generators do not\nmix up the first and second oscillators.   There are six of them because\neach operator has three generators for its own $Sp(2)$ symmetry.\nLet us consider the three generators, $S_{3}, K_{2}$, and $Q_{2}$.\nFor each oscillator, the generators consist of\n\\begin{equation}\n  \\sigma_{2}, \\quad  i\\sigma_{1} \\quad \\mbox{and} \\quad i\\sigma_{3} .\n\\end{equation}\nThese separable generators thus constitute the Lie algebra of Sp(2) group\nfor the one-oscillator system, which we discussed in Sec.~\\ref{sp2}.  Hence, the\none-oscillator system constitutes a subgroup of the two-oscillator system.\n\nThe off-diagonal matrix $L_{2}$ couples the first and second oscillators\nwithout changing the overall volume of the four-dimensional phase space.\nHowever, in order to construct the closed set of commutation relations,\nwe need the three additional generators: $L_{1} , K_{3},$ and $Q_{3}.$\nThe commutation relations given in Eq.(\\ref{313}) are clearly\nconsequences of Heisenberg's uncertainty relations.\n\n\n\n\n\n\n\\section{Contraction of SO(3,2) to ISO(3,1)}\\label{contraction}\n\n\nLet us next go back to the $SO(3,2)$ contents of this two-oscillator\nsystem~\\cite{dir63}.  There are three space-like coordinates $(x, y, z)$\nand two time-like coordinates $s$ and $t$.  It is thus possible to\nconstruct the five-dimensional space of $(x, y, z, t, s)$, and to consider\nfour-dimensional Minkowskian subspaces consisting of $(x, y, z, t)$\nand $(x, y, z, s)$.\n\nAs for the $s$ variable, we can make it longer or shorter, according\nto procedure of group contractions introduced first by In{\\\"o}n{\\\"u} and\nWigner~\\cite{inonu53}.  In this five-dimensional space, the boosts along the\n$x$ direction with respect to the $t$ and $s$ variables are generated by\n\\begin{equation}\\label{333}\nA_{x} = \\pmatrix{0 & 0 & 0 & i & 0 \\cr 0 & 0 & 0 & 0 & 0 \\cr\n0 & 0 & 0 & 0 & 0 \\cr i & 0 & 0 & 0 & 0 \\cr 0 & 0  & 0 & 0 & 0 }, \\qquad\nB_{x} = \\pmatrix{0 & 0 & 0 & 0 & i \\cr 0 & 0 & 0 & 0 & 0 \\cr\n0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 \\cr i & 0  & 0 & 0 & 0 },\n\\end{equation}\nrespectively.  The boost generators along the $y$ and $z$ directions\ntake similar forms.\n\nLet us then introduce the five-by-five contraction matrix~\\cite{kiwi87jmp,kiwi90jmp}\n\\begin{equation}\\label{335}\nC(\\epsilon) = \\pmatrix{1 & 0 & 0 & 0 & 0 \\cr 0 & 1 & 0 & 0 & 0 \\cr\n0 & 0 & 1 & 0 & 0 \\cr 0 & 0 & 0 & 1 & 0 \\cr 0 & 0  & 0 & 0 &  \\epsilon } .\n\\end{equation}\nThis matrix leaves the first four columns and rows invariant, and the\nfour-dimensional Minkowskian sub-space of $(x, y, z, t)$ stays invariant.\n\nAs for the boost with respect to the $s$ variable, according to the procedure\nspelled out in Ref.~\\cite{kiwi87jmp,kiwi90jmp}, the contracted boost generator becomes\n\\begin{equation} \\label{341}\n  B^{c}_{x} = \\lim_{\\epsilon\\rightarrow \\infty}\\frac{1}{\\epsilon}~\n  \\left[C^{-1}(\\epsilon)~B_{x}~C(\\epsilon)\\right] =\n   \\pmatrix{0 & 0 & 0 & 0 &  i \\cr 0 & 0 & 0 & 0 & 0 \\cr\n0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0  \\cr 0 & 0 & 0 & 0 & 0 } .\n\\end{equation}\nLikewise, $B^{c}_{y}$ and $B^{c}_{z}$ become\n\\begin{equation}\\label{343}\n     B^{c}_{y} = \\pmatrix{0 & 0 & 0 & 0 &  0 \\cr 0 & 0 & 0 & 0 & i \\cr\n     0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0  \\cr 0 & 0 & 0 & 0 & 0 },\n     \\qquad\n     B^{c}_{z} = \\pmatrix{0 & 0 & 0 & 0 &  0 \\cr 0 & 0 & 0 & 0 & 0 \\cr\n     0 & 0 & 0 & 0 & i \\cr 0 & 0 & 0 & 0 & 0  \\cr 0 & 0 & 0 & 0 & 0} ,\n\\end{equation}\nrespectively.\n\nAs for the $t$ direction, the transformation applicable to the $s$ and $t$\nvariables is a rotation, generated by\n\\begin{equation}\\label{345}\nB_{t} = \\pmatrix{0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 \\cr\n0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & i \\cr 0 & 0  & 0 & -i & 0 } .\n\\end{equation}\nThis matrix also becomes contracted to\n\\begin{equation}\\label{347}\nB^{c}_{t} = \\pmatrix{0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & 0 \\cr\n0 & 0 & 0 & 0 & 0 \\cr 0 & 0 & 0 & 0 & i \\cr 0 & 0  & 0 & 0 & 0 } .\n\\end{equation}\nThese contraction procedures are illustrated in Fig.~\\ref{contrac}.\n\n\n\n\\begin{figure\n\\centerline{\\includegraphics[scale=2.0]{contrac66.eps}}\n\\caption{Contraction of the $SO(3,2)$ group to the Poincar\\'e group.\nThe time-like $s$ coordinate is contracted with respect to the\nspace-like $x$ variable, and with respect to the time-like\nvariable $t$.}\\label{contrac}\n\\end{figure}\n\n\n\n\nThese four contracted generators lead to the five-by-five transformation\nmatrix\n\\begin{equation}\\label{349}\n\\exp\\left\\{-i\\left(aB^{c}_{x}+ bB^{c}_{y} + cB^{c}_{z} + dB^{c}_{t}\\right)\\right\\}\n= \\pmatrix{1 & 0 & 0 & 0 & a \\cr 0 & 1 & 0 & 0 & b \\cr\n0 & 0 & 1 & 0 & c \\cr 0 & 0 & 0 & 1 & d \\cr 0 & 0  & 0 & 0 & 1 } ,\n\\end{equation}\nperforming translations:\n\\begin{equation}\\label{351}\n     \\pmatrix{1 & 0 & 0 & 0 & a \\cr 0 & 1 & 0 & 0 & b \\cr\n0 & 0 & 1 & 0 & c \\cr 0 & 0 & 0 & 1 & d \\cr 0 & 0  & 0 & 0 & 1 }\n\\pmatrix{x \\cr y \\cr z \\cr t \\cr 1 } =\n\\pmatrix{x + a \\cr y + b \\cr z + c \\cr t + d \\cr 1 } .\n\\end{equation}\nThis matrix leaves the first four rows and columns invariant.  They are for\nthe Lorentz transformation applicable to the Minkowskian space of $(x, y, z, t)$.\n\nIn this way, the boosts along the $s$ direction become contracted to the\ntranslation.  This means the group $SO(3,2)$ derivable from the Heisenberg's\nuncertainty relations becomes the inhomogeneous Lorentz group governing the\nPoincar\\'e symmetry for quantum mechanics and quantum field\ntheory in the Lorentz-covariant world~\\cite{dir63,dir62}.\n\nThe group contraction has a long history in physics, starting from the 1953\npaper by In{\\\"o}n{\\\"u} and Wigner~\\cite{inonu53}.   It starts with a geometrical\nconcept.  Our earth is a sphere, but is is convenient to consider a flat surface\ntangent to a given point on the spherical surface of the earth.  This approximation\nis called the contraction of $SO(3)$ to $SE(2)$ or the two-dimensional Euclidean\ngroup with one rotational and two translational degrees of freedom.\n\nThis mathematical method was extended to the contraction of the $SO(3,1)$ Lorentz\ngroup to the three-dimensional Euclidean group.  More recently, Kim and Wigner\nconsidered a cylindrical surface tangent to the sphere~\\cite{kiwi87jmp,kiwi90jmp}\nat its equatorial belt.  This cylinder has one rotational degree of freedom and\none up-down translational degree of freedom.  It was shown that the rotation and\ntranslation correspond to the helicity and gauge degrees of freedom for massless\nparticles.\n\nSince the Lorentz $SO(3,1)$ is isomorphic to the $SL(2,c)$ group of two-by-two\nmatrices, we can ask whether it is possible to perform the same contraction\nprocedure in the regime of two-by-two matrices.  It does not appear possible\nto represent the $ISE(3)$ (inhomogeneous Euclidean group) with two-by-two\nmatrices.  Likewise, there seem to be difficulties in addressing the\nquestion of contracting $SO(3,2)$ to $ISO(3,1)$ within the frame work of\nthe four-by-four matrices of $Sp(4)$.\n\n\n\n\n\n\\section{Concluding Remarks}\\label{remarks}\n\nSpecial relativity and quantum mechanics served as the major\ntheoretical basis for modern physics for one hundred years.\nThey coexisted in harmony: quantum mechanics augmented by Lorentz\ncovariance when needed.\nIndeed, there have been attempts in the past to construct a\nLorentz-covariant quantum world by augmenting the Lorentz group\nto the uncertainty relations~\\cite{dir27,dir45,dir49,dir62,yuka53}.\nThere are recent papers on this subject~\\cite{fkr71,lowe83,bars09}.\nThere are also papers on group contractions including contractions\nof the $SO(3,2)$ group~\\cite{gilmore74,bohm84,bohm85}.\n\nIt is about time for us to examine whether  both of these two great\ntheories can be synthesized.  The first step toward this process is\nto find the common mathematical ground.  Before Newton, open orbits\n(comets) and closed orbits (planets) were treated differently, but\nNewton came up with one differential equation for both.  Before\nMaxwell, electricity and magnetism were different branches of physics.\nMaxwell's equations synthesized these two branches into one.  It is\nshown in this paper that the group $ISO(3,1)$ can be derived from\nthe algebra of quantum mechanics.\n\nIt is gratifying to note that the Poincar\\'e symmetry is derivable\nwithin the system of Heisenberg's uncertainty relations.  The\nprocedure included two coupled oscillators resulting in the $SO(3,2)$\nsymmetry~\\cite{dir63}, and the contraction of this $SO(3,2)$ to the\ninhomogeneous Lorentz group $ISO(3,1)$.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Experimental Procedure}\nWe study four experimental factors: the model, the batch size, the initialization point, and the learning rate. The model factor has three levels that correspond to the three models described above. The batch size will take the values corresponding to SGD-$1$, SGD-$200$, SGD-$500$, and Gradient Descent. The initialization point will be randomly selected from a uniform distribution on a ball whose radius will take values $\\lbrace 10^{-3},10^{-2},10^{-1}, 1 \\rbrace$ and is centered at either the sharpest minimizer or the flattest minimizer. The learning rates will be linear combinations of the upper and lower bounds. Specifically, if $u$ is the threshold for convergence and $l$ is the threshold for divergence, then the learning rates will be $1.5l$, $0.5(u+l)$, or $0.5u$. For each unique collection of the levels of the four factors, one hundred independent runs are executed with at most twenty iterations. For each run, the euclidean distances between the iterates and the minimizer near the initialization point are recorded. \n\n\\subsection{Results and Discussion}\nNote that the results Models 1, 2 and 3 are nearly identical for the purposes of our discussion, and so we will feature Model 1 only in our discussion below.\n\n\\begin{figure}[hbtp]\n\\centering \n\\includegraphics[width=\\textwidth]{st_fig1}\n\\caption[SGD-$k$ on Styblinski-Tang Near Flat Minimizer]{The behavior of SGD-$k$ on Model 1 for different batch sizes when initialized near the flat minimizer. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the flat minimizer for all runs of the specified batch size and specified learning rate.}\n\\label{figure-sgd:st-1}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering \n\\includegraphics[width=\\textwidth]{st_fig3}\n\\caption[SGD-$k$ on Styblinski-Tang Near Sharp Minimizer]{The behavior of SGD-$k$ on Model 1 for different batch sizes when initialized near the sharp minimizer. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the sharp minimizer for all runs of the specified batch size and specified learning rate.}\n\\label{figure-sgd:st-3}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{st_fig2}\n\\caption[Initialization and SGD-$k$ on Styblinski-Tang Near Flat Minimizer]{The behavior of SGD-$k$ on Model 1 for different starting radii when initialized near the flat minimizer. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the flat minimizer for all runs of the specified starting radius and specified learning rate.}\n\\label{figure-sgd:st-2}\n\\end{figure}\n\n\\begin{figure}[hbtp]\n\\centering \n\\includegraphics[width=\\textwidth]{st_fig4}\n\\caption[Initialization and SGD-$k$ on Styblinski-Tang Near Sharp Minimizer]{The behavior of SGD-$k$ on Model 1 for different starting radii when initialized near the sharp minimizer. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the sharp minimizer for all runs of the specified starting radius and specified learning rate.}\n\\label{figure-sgd:st-4}\n\\end{figure}\n\nFigure \\ref{figure-sgd:st-1} shows the distance between SGD-$k$ iterates and the flat minimizer on Model 1 for different batch sizes and different learning rates when SGD-$k$ is initialized near the flat minimizer. \nWe note that, regardless of batch size, the iterates are diverging from the flat minimizer and are converging to the corners of the feasible region for learning rates $1.5l$ and $0.5(u+l)$. \nOn the other hand, for the learning rate $0.5u$, we see stability for $k=1,200,500$ about the minimizer and we see convergence for GD (i.e., $k=\\infty$). \nSimilarly, Figure \\ref{figure-sgd:st-3} shows the distance between SGD-$k$ iterates and the sharp minimizer on Model 1 for different batch sizes and different learning rates when SGD-$k$ is initialized near the sharp minimizer. \nWe note that, regardless of batch size, the iterates are diverging from the sharp minimizer and converging to the corners of the feasible region for learning rates $1.5l$ and $0.5(u+l)$. On the other hand, for the learning rate $0.5u$, we see stability for $k=1,200,500$ about the sharp minimizer and we see convergence for GD (i.e., $k = \\infty$). \n\nTaking the results in Figures \\ref{figure-sgd:st-1} and \\ref{figure-sgd:st-3} together, we see that we are able to use our deterministic mechanism to find learning rates that either ensure divergence from or stability about a minimizer if we know its local geometric properties. Consequently, we again have evidence that our deterministic mechanism can correctly predict the behavior of SGD-$k$ for nonconvex problems. Moreover, when divergence does occur, Figures \\ref{figure-sgd:st-1} and \\ref{figure-sgd:st-3} also display exponential divergence as predicted by our deterministic mechanism.\n\nFor a different perspective, Figure \\ref{figure-sgd:st-2} shows the distance between SGD-$k$ iterates and the flat minimizer on Model 1 for different starting radii and different learning rates when SGD-$k$ is initialized near the flat minimizer. We note that, regardless of the starting radius, the iterate are diverging from the flat minimizer and are converging to the corners of the feasible region for learning rates $1.5l$ and $0.5(u+l)$. On the other hand, for the learning rate $0.5u$, we see stability and even convergence for all of the runs regardless of the starting radius. \nSimilarly, Figure \\ref{figure-sgd:st-4} shows the distance between SGD-$k$ iterates and the sharp minimizer on Model 1 for different starting radii and different learning rates when SGD-$k$ is initialized near the sharp minimizer. We note that, regardless of the starting radius, the iterates are diverging from the sharp minimizer and are converging to the corners of the feasible region for learning rates $1.5l$ and $0.5(u+l)$. On the other hand, for learning rate $0.5u$, we see stability about and even convergence to the sharp minimizer. \n\nTaking the results in Figures \\ref{figure-sgd:st-2} and \\ref{figure-sgd:st-4} together, we see that we are able to use our deterministic mechanism to find learning rates that either ensure divergence from, or stability about, a minimizer if we know its local geometric properties.  Consequently, we again have evidence that our deterministic mechanism can correctly predict the behavior of SGD-$k$ for nonconvex problems. Moreover, when divergence does occur, Figures \\ref{figure-sgd:st-1} and \\ref{figure-sgd:st-3} also display exponential divergence as predicted by our deterministic mechanism.\n\n\\subsection{Experimental Procedure}  \nWe study four experimental factors: the model, the learning method, the initialization point, and the learning rate. \nThe model factor has four levels as given by the four model objective functions shown in Figure \\ref{figure-sgd:qc-models}. The learning method has two levels which are SGD-$1$ or GD. \nThe initialization point also has two levels: either the initialization point is selected randomly with a uniform probability from a disc of radius $10^{-8}$ centered about the circular basin minimizer, or the initialization point is selected randomly with a uniform probability from a disc of radius $10^{-8}$ centered about the quadratic basin minimizer. The learning rate has levels that are conditional on the initialization point. \nIf the initialization point is near the minimizer of the circular basin, then the learning rate takes on values $10^{10}$, $5(10^{10})$, $10^{11}$, $5(10^{11})$, $10^{12}$, and $5(10^{12})$. \nIf the initialization point is near the minimizer of the circular basin, then the learning rate takes on values $1, 4, 16, 64, 256, 1024$. \n\nFor each unique collection of the levels of the four factors, one hundred independent runs are executed with at most twenty iterations. For each run, the euclidean distance between the iterates and the circular basin minimizer and the euclidean distance between the iterates and the quadratic basin minimizer are recorded. \n\n\\subsection{Results and Discussion}\nNote that the results of SGD-$k$ on Models 1 and 2 and Models 3 and 4 are similar when initialized around the circular basin minimizer, and the results of SGD-$k$ on Models 1 and 3 and Models 2 and 4 are similar when initialized around the quadratic basin minimizer. Hence, when we discuss the circular basin minimizer, we will compare Models 1 and 3, but we could have just as easily replaced Model 1 with Model 2 or Model 3 with Model 4 and the discussion would be identical. Similarly, when we discuss the quadratic basin minimizer, we will compare Models 1 and 2, but we could have replaced the results of Model 1 with Model 3 or Model 2 with Model 4 and the discussion would be identical.\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{cq_fig1}\n\\caption[SGD-$1$ Near Circle Basin Minimizer]{The behavior of SGD-$1$ on Models 1 and 3 when initialized near the circular minimum. The $y$-axis shows the distance (in logarithmic scale) between the estimates and the circular minimum for all runs of the specified model and the specified learning rate.}\n\\label{figure-sgd:cq-1}\n\\end{figure}\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{cq_fig2}\n\\caption[SGD-$1$ Near Quadratic Basin Minimizer]{The behavior of SGD-$1$ on Models 1 and 2 when initialized near the quadratic minimum. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the quadratic minimum for all runs of the specified model and the specified learning rate.}\n\\label{figure-sgd:cq-2}\n\\end{figure}\n\nFigure \\ref{figure-sgd:cq-1} shows the distance between the iterates and the circular basin minimizer for the one hundred independent runs of SGD-$1$ for the specified model and the specified learning rate when initialized near the circular basin minimizer. The learning rates that are displayed are the ones where a transition in the convergence-divergence behavior of the method occur for the specific model. Specifically, SGD-$1$ begins to diverge for learning rates between $5 \\times 10^{10}$ and $10^{11}$ for Model 1 and between $5\\times 10^{11}$ and  $10^{12}$ for Model 3. \nSimilarly, Figure \\ref{figure-sgd:cq-2} shows the distance between the iterates and the quadratic basin minimizer for the one hundred independent runs of SGD-$1$ for the specified model and the specified learning rate when initialized near the quadratic basin minimizer. SGD-$1$ begins to diverge for learning rates between $4$ and $16$ for Model 1 and between $256$ and $1024$ for Model 2. \n\nFrom these observations, we see that relatively flatter minimizers enjoy larger thresholds for divergence of SGD-$1$ in comparison to sharper minimizers. Moreover, while the bounds computed in Table \\ref{table:lb} are conservative (as we would expect), they are still rather informative, especially in the case of the quadratic basin. Thus, regarding questions (1) and (2) above, we see that while our thresholds are slightly conservative, we still correctly predict the expected behavior of the SGD-$k$ iterates. Moreover, regarding question (3), we observe that the iterates diverge from their respective minimizers in a deterministic way with an exponential rate. Again, this observations lends credence to our deterministic mechanism over the widely accepted stochastic mechanism.\n\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{cq_fig3}\n\\caption[GD Near Circle Basin Minimizer]{The behavior of GD on Models 1 and 3 when initialized near the circular basin minimizer. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the circular minimizer for all runs of the specified model and the specified learning rate.}\n\\label{figure-sgd:cq-3}\n\\end{figure}\n\nFigure \\ref{figure-sgd:cq-3} shows the distance between the iterates and the circular basin minimizer for the one hundred independent runs of gradient descent (GD) for the specified model and the specified learning rate when initialized near the circular basin minimizer. If we compare Figures \\ref{figure-sgd:cq-1} and \\ref{figure-sgd:cq-3}, we notice that, for Model 1 and learning rate $5 \\times 10^{10}$, the runs of GD converge whereas some of the runs for SGD-$1$ diverge. Although we do not report the results for all of the models or all of the learning rates, we note the boundary for divergence-convergence for GD are smaller than those of SGD-$1$. In light of our deterministic mechanism, this behavior is expected: as the batch-size, $k$, increases, the lower bound on the learning rates for divergence increases. Therefore, we should expect that at those boundary learning rates where some SGD-$1$ runs are stable and others diverge, for GD, we should only see stable runs, and, indeed, this is what the comparison of Figures \\ref{figure-sgd:cq-1} and \\ref{figure-sgd:cq-3} shows.\n\n\\begin{figure}[hbt]\n\\centering\n\\includegraphics[width=\\textwidth]{cq_fig4}\n\\caption[Comparing SGD-$1$ and GD Near Quadratic Basin Minimizer]{The behavior of SGD-$1$ and GD on Model 1 when initialized near the quadratic minimum for select learning rates. The $y$-axis shows the distance (in logarithmic scale) between the iterates and the circular minimizer for all runs of the specified method and the specified learning rate.}\n\\label{figure-sgd:cq-4}\n\\end{figure}\n\nFigure \\ref{figure-sgd:cq-4} shows the distance between the iterates and the circular basin minimizer for the one hundred independent runs of SGD-$1$ and the one hundred independent runs of GD for the specified method and the specified learning rate when initialized near the quadratic basin minimizer. In Figure \\ref{figure-sgd:cq-4}, we see that for the learning rates that lead to divergence from the quadratic basin minimizer (compare to the top three subplots of Figure \\ref{figure-sgd:cq-1}) are in some cases able to converge to the circular minimizer. Again, in light of our deterministic mechanism, this is expected: the learning rates shown in Figure \\ref{figure-sgd:cq-4} guarantee divergence from the quadratic minimizer and are sufficiently small that they can lead to convergence to the circular minimizer. However, we notice in Figure \\ref{figure-sgd:cq-4} that as the learning rate increases, even though the learning rate is below the divergence bound for the circular minimizer, the iterates for SGD-$1$ and GD are diverging from both the circular and quadratic minima and are converging to the corners of the feasible region.\n\nTo summarize, we see that convergence-divergence behavior of SGD-$k$ for the quadratic-circle sums nonconvex problem is captured by our deterministic mechanism. Although our estimated bounds are sometimes conservative, they generally divide the convergence and divergence regions of SGD-$k$. Moreover, as our deterministic mechanism predicts, we observed exponential divergence away from minimizers when the learning rate is above the divergence threshold. Again, this lends credence to our mechanism for divergence and provides evidence against the stochastic mechanism for SGD-$k$ iterates to ``escape'' sharp minimizers.\n\n\\subsection{A Generic Stochastic Quadratic Problem} \\label{subsection:stochastic-quadratic}\n\nWe will begin by defining the quadratic problem, derive some of its relevant properties, and discuss the two types of minimizers that can occur for the problem.\n\n\\begin{problem}[Quadratic Problem] \\label{problem-sgd:quadratic}\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space. Let $Q \\in \\mathbb{R}^{p \\times p}$ be a nonzero, symmetric, positive semidefinite random matrix and $r \\in \\mathbb{R}^p$ be a random vector in the image space of $Q$ (with probability one) such that (1) $\\E{Q} \\prec \\infty$, (2) $\\E{ Q \\E{Q} Q} \\prec \\infty$, (3) $\\E{Q \\E{Q} r} $ is finite, and (4) $\\E{r'Qr}$ is finite. Let $\\lbrace (Q_N,r_N): N \\in \\mathbb{N} \\rbrace$ be independent copies of $(Q,r)$. The quadratic problem is to use $\\lbrace (Q_N,r_N) : N \\in \\mathbb{N} \\rbrace$ to determine a $\\theta^*$ such that\n\\begin{equation} \\label{eqn-sgd-problem:quadratic-obj}\n\\theta^* \\in \\underset{\\theta \\in \\mathbb{R}^p}{\\mathrm{argmin}} \\frac{1}{2}\\theta'\\E{Q}\\theta + \\E{r}'\\theta.\n\\end{equation}\n\\end{problem}\n\nThere are several aspects of the formulation of Problem \\ref{problem-sgd:quadratic} worth highlighting. First, $Q$ is required to be symmetric and positive semidefinite, and $r$ is required to be in the image space of $Q$. Together, these requirements on $(Q,r)$ ensure that for each $\\omega$ in some probability one set in $\\mathcal{F}$, there exists a minimizer for $0.5 \\theta'Q\\theta + r'\\theta$. While this condition seems quite natural for the quadratic problem, it will cause some challenges for generalizing our results directly to the nonconvex case, which we will discuss further in \\S \\ref{subsection:deterministic-mechanism}. Second, we have required a series of assumptions on the moments of $(Q,r)$. As we will see in the analysis of SGD-$k$ on Problem \\ref{problem-sgd:quadratic}, these moment assumptions will be essential to the stability of the iterates. Finally, we have not required that the solution, $\\theta^*$, be unique. That is, we are not requiring a strongly convex problem. This generality is important as it will allow us to better approximate more exotic nonconvex problems. \n\nWe now establish some basic geometric properties about the quadratic problem. We first establish a standard, more convenient reformulation of the objective function in (\\ref{eqn-sgd-problem:quadratic-obj}). Then, we formulate some important geometric properties of the quadratic problem.\n\n\\begin{lemma} \\label{lemma-sgd:objective-reformulation}\nThe objective function in (\\ref{eqn-sgd-problem:quadratic-obj}) is equivalent to (up to an additive constant)\n\\begin{equation}\n\\frac{1}{2}(\\theta - \\theta^*)'\\E{Q}(\\theta - \\theta^*),\n\\end{equation}\nfor any $\\theta^*$ satisfying (\\ref{eqn-sgd-problem:quadratic-obj}). \n\\end{lemma}\n\\begin{proof}\nIf $\\theta^*$ is a minimizer of the objective in (\\ref{eqn-sgd-problem:quadratic-obj}), then $\\theta^*$ must be a stationary point of the objective function. This implies that $-\\E{Q}\\theta^* = \\E{r}$. Therefore, the objective in (\\ref{eqn-sgd-problem:quadratic-obj}) is, up to an additive constant,\n\\begin{equation}\n\\frac{1}{2} \\theta'\\E{Q}\\theta - \\theta' \\E{Q} \\theta^* + \\frac{1}{2}(\\theta^*)'\\E{Q} \\theta^* = \\frac{1}{2}(\\theta - \\theta^*)'\\E{Q}(\\theta - \\theta^*),\n\\end{equation} \n\\end{proof}\n\nBecause $Q$ is symmetric, positive semidefinite with probability one, $\\E{Q}$ will also be symmetric, positive semidefinite. That is, $m := \\texttt{rank}(\\E{Q}) \\leq p$. For future reference, we will denote the nonzero eigenvalues of $\\E{Q}$ by $\\lambda_1 \\geq \\cdots \\lambda_m > 0$. Beyond the eigenvalues, we will also need some higher order curvature information, namely, $s_Q$ and $t_Q$, which are given by\n\\begin{equation} \\label{eqn-sgd:tQ}\nt_Q = \\sup \\left\\lbrace \\frac{v'\\E{Q \\E{Q} Q}v - v' \\E{Q}^3 v}{v' \\E{Q} v} : v \\in \\mathbb{R}^p, ~ v'\\E{Q}v \\neq 0 \\right\\rbrace,\n\\end{equation}\nand\n\\begin{equation} \\label{eqn-sgd:sQ}\ns_Q = \\inf \\left\\lbrace \\frac{v'\\E{Q \\E{Q} Q}v - v' \\E{Q}^3 v}{v' \\E{Q} v} : v \\in \\mathbb{R}^p, ~ v'\\E{Q}v \\neq 0 \\right\\rbrace.\n\\end{equation}\nFrom the definitions in (\\ref{eqn-sgd:tQ}) and (\\ref{eqn-sgd:sQ}), it follows that $t_Q \\geq s_Q$. The next result establishes that for nontrivial problems, $s_Q > 0$. \n\n\\begin{lemma} \\label{lemma-sgd:higher-moment-bounds}\nLet $Q$ be as in (\\ref{problem-sgd:quadratic}). Then, the following properties hold.\n\\begin{enumerate}\n\\item If there is an $x \\in \\mathbb{R}^p$ such that $x' \\E{Q} x = 0$ then $x' \\E{ Q \\E{Q} Q} x = 0$. \n\\item $\\E{Q \\E{Q} Q} \\succeq \\E{Q}^3$. \n\\item If $\\Prb{Q = \\E{Q}} < 1$ then $s_Q > 0$. \n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\nFor (1), when $x=0$ the result follows trivially. Suppose then that there is an $x \\neq 0$ such that $x'\\E{Q}x = 0$. Then, since $Q \\succeq 0$, $x'Qx = 0$ almost surely. Therefore, $x$ is in the null space of all $Q$ on a probability one set. Therefore, $x'Q \\E{Q} Q x = 0$ with probability one. The conclusion follows.\n\nFor (2), note that for any $x \\in \\mathbb{R}^q$, the function $f(M) = x'M \\E{Q} Mx$ is convex over the space of all symmetric, positive semi-definite matrices. Therefore, by Jensen's inequality, $\\E{f(Q)} \\geq f(\\E{Q})$. Moreover, since $x$ is arbitrary, the conclusion follows.\n\nFor (3), note that Jensen's inequality holds with equality only if $f$ is affine or if $\\Prb{Q = \\E{Q}} = 1$. Therefore, if $\\Prb{Q = \\E{Q}} < 1$, then the second condition is ruled out. Moreover, if $\\Prb{Q = \\E{Q}} < 1$, then $\\Prb{Q = 0} < 1$. Thus, $\\exists x \\neq 0$ such that $f(Q) \\neq 0$, which rules out the first condition.  \n\\end{proof}\n\nNow, if $\\Prb{Q = \\E{Q}} = 1$ then there exists a solution to (\\ref{eqn-sgd-problem:quadratic-obj}) such that $Q\\theta^* + r = 0$ with probability one. This phenomenon of the existence of a single solution for all pairs $(Q,r)$ with probability one will play a special role, and motivates the following definition.\n\n\\begin{definition}[Homogeneous and Inhomogeneous Solutions]\nA solution to $\\theta^*$ satisfying (\\ref{eqn-sgd-problem:quadratic-obj}) is called homogeneous if\n\\begin{equation} \\label{eqn-sgd-definition:homogeneous}\n\\theta^* \\in \\mathrm{argmin}_{\\theta \\in \\mathbb{R}^p} \\frac{1}{2} \\theta'Q \\theta + r'\\theta \\text{ with probability one.}\n\\end{equation}\nOtherwise, the solution is called inhomogeneous.\n\\end{definition}\n\nIn the case of a homogeneous minimizer, the objective function in (\\ref{eqn-sgd-definition:homogeneous}) can be rewritten as (up to an additive constant)\n\\begin{equation}\n\\frac{1}{2} (\\theta - \\theta^*)' Q (\\theta - \\theta^*),\n\\end{equation}\nwhich follows from the same reasoning Lemma \\ref{lemma-sgd:objective-reformulation}. Importantly, SGD-$k$ will behave differently for problems with homogeneous minimizers and inhomogeneous minimizer. As we will show, for homogeneous minimizers, SGD-$k$ behaves rather similarly to classical gradient descent in terms of convergence and divergence, whereas for inhomogeneous minimizers, SGD-$k$ will behave markedly differently. To show these results, we will first define SGD-$k$ for the quadratic problem.\n\n\\begin{definition}[SGD-$k$] \\label{definition-sgd:sgd-k}\nLet $\\theta_0 \\in \\mathbb{R}^p$ be arbitrary. For Problem \\ref{problem-sgd:quadratic}, SGD-$k$ generates a sequence of iterates $\\lbrace \\theta_N : N \\in \\mathbb{N} \\rbrace$ defined by\n\\begin{equation} \\label{eqn-definition:sgd-k}\n\\theta_{N+1} = \\theta_N - \\frac{C_{N+1}}{k}\\sum_{j=Nk+1}^{N(k+1)} Q_j\\theta_N + r_j,\n\\end{equation} \nwhere $\\lbrace C_N: N \\in \\mathbb{N} \\rbrace$ is a sequence of scalars. \n\\end{definition}\n\nIn Definition \\ref{definition-sgd:sgd-k}, when $k = 1$, we have the usual notion of stochastic gradient descent and, as $k \\to \\infty$, we recover gradient descent. Therefore, this notion of SGD-$k$ will allow us to generate results that explore the complete range of stochastic to deterministic methods, and bridge the theory between stochastic gradient descent and gradient descent.\n\n\\subsection{SGD-$k$ on the Stochastic Quadratic Problem} \\label{subsection:analysis}\n\n\\begin{table}\n\\tbl{Summary of Notation}\n{\n\\begin{tabular}{@{}ll@{}} \\toprule\nNotation & Value \\\\ \\midrule\n$M$ & $\\E{Q \\E{Q} Q} - \\E{Q}^3$ \\\\\n$e_N$ & $(\\theta_N - \\theta^*)'\\E{Q}(\\theta_N - \\theta^*)$ \\\\\n$e_{N,l}$ & $(\\theta_N - \\theta^*)'\\E{Q}^l(\\theta_N - \\theta^*)$ for $l \\in \\mathbb{N}_{\\geq 2}$ \\\\\n$e_{N,M}$ & $(\\theta_N - \\theta^*)'M (\\theta_N - \\theta^*)$ \\\\ \n$m$ & $\\texttt{rank}( \\E{Q} )$ \\\\\n$\\lambda_1 \\geq \\cdots \\geq \\lambda_m > 0$ & Nonzero eigenvalues of $\\E{Q}$ \\\\ \n$t_Q, s_Q$ & Higher-order Curvature Parameters of $Q$ \\\\\\bottomrule\n\\end{tabular}\n}\n\\label{table:summary-of-notation}\n\\end{table}\n\nTable \\ref{table:summary-of-notation} summarizes the notation that will be used throughout the remainder of this work. With this notation, we now analyze the iterates generated by SGD-$k$ for Problem \\ref{problem-sgd:quadratic}. Our first step is to establish a relationship between $e_{N+1}$ and $e_N$, which is a standard computation using conditional expectations.\n\n\\begin{lemma} \\label{lemma-sgd:quadratic-recursion}\nLet $\\lbrace \\theta_N : N \\in \\mathbb{N} \\rbrace$ be the iterates of SGD-$k$ from Definition \\ref{definition-sgd:sgd-k}. Let $\\theta^*$ be a solution to the quadratic problem. Then, with probability one, \n\\begin{equation}\n\\begin{aligned}\n\\cond{e_{N+1}}{\\theta_N} &= e_{N} - 2C_{N+1}e_{N,2} + C_{N+1}^2 e_{N,3} + \\frac{C_{N+1}^2}{k} e_{N,M} \\\\\n\t\t\t\t\t\t&+ 2\\frac{C_{N+1}^2}{k}(\\theta_N - \\theta^*)' \\E{ Q \\E{Q} (Q\\theta^* + r)} \\\\\n\t\t\t\t\t\t&+ \\frac{C_{N+1}^2}{k} \\E{ (Q\\theta^* + r)' \\E{Q}(Q \\theta^* + r)}.\n\\end{aligned}\n\\end{equation}\nMoreover, if $\\theta^*$ is a homogeneous minimizer, then, with probability one,\n\\begin{equation}\n\\begin{aligned}\n\\cond{e_{N+1}}{\\theta_N} &= e_{N} - 2C_{N+1} e_{N,2} + C_{N+1}^2 e_{N,3} + \\frac{C_{N+1}^2}{k} e_{N,M}.\n\\end{aligned}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe result is a straightforward calculation from the properties of SGD-$k$ and the quadratic problem. In the case of the homogeneous minimizer, recall that $Q\\theta^* +r = 0$ with probability one.\n\\end{proof} \n\nOur second step is to find bounds on $\\cond{e_{N+1}}{\\theta_N}$ in terms of $e_N$. While this is a standard thing to do, our approach differs from the standard approach in two ways. First, not only will we find an upper bound, but we will also find a lower bound. Second, our lower bounds will be much more delicate than the usual strong convexity and Lipschitz gradient based arguments \\cite[see][\\S 4]{bottou2016}. This second step can be accomplished for the homogeneous case with the following two lemmas.\n\n\\begin{lemma} \\label{lemma-sgd:technical-systematic-component}\nIn the setting of Lemma \\ref{lemma-sgd:quadratic-recursion} and for $\\alpha_j \\geq 0$ for $j=0,1,2,3$, \n\\begin{equation} \\label{eqn-sgd-lemma:technical-systematic-component}\n\\begin{aligned}\n&\\alpha_0e_N - \\alpha_1 C_{N+1} e_{N,2} + \\alpha_2 C_{N+1}^2 e_{N,3} + \\alpha_3 C_{N+1}^2  e_{N,M}\n\\end{aligned}\n\\end{equation}\nis bounded above by\n\\begin{equation}\ne_N\\left[ \\alpha_0 - \\alpha_1 C_{N+1} \\lambda_j + \\alpha_2 C_{N+1}^2 \\lambda_j^2 + \\alpha_3 C_{N+1}^2 t_Q \\right], \n\\end{equation}\nwhere\n\\begin{equation}\nj = \\begin{cases}\n1 & C_{N+1} > \\frac{\\alpha_1}{\\alpha_2} \\frac{1}{\\lambda_1 + \\lambda_m} \\text{ or } C_{N+1} \\leq 0 \\\\\nm & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nMoreover, (\\ref{eqn-sgd-lemma:technical-systematic-component}) is bounded below by\n\\begin{equation}\ne_N\\left[ \\alpha_0 - \\alpha_1 C_{N+1} \\lambda_j + \\alpha_2 C_{N+1}^2 \\lambda_j^2 + \\alpha_3 C_{N+1}^2 s_Q \\right],\n\\end{equation}\nwhere \n\\begin{equation}\nj = \\begin{cases}\n1 & C_{N+1} \\in \\left(0 ,\\frac{\\alpha_1}{\\alpha_2} \\frac{1}{\\lambda_1 + \\lambda_2}\\right] \\\\\nl ~(l \\in \\lbrace 2,\\ldots,m-1 \\rbrace) & C_{N+1} \\in \\left( \\frac{\\alpha_1}{\\alpha_2} \\frac{1}{\\lambda_{l-1} + \\lambda_l}, \\frac{\\alpha_1}{\\alpha_2} \\frac{1}{\\lambda_l + \\lambda_{l+1}} \\right] \\\\\nm & C_{N+1} > \\frac{\\alpha_1}{\\alpha_2} \\frac{1}{\\lambda_{m-1} + \\lambda_m} \\text{ or } C_{N+1} \\leq 0.\n\\end{cases}\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWhen $e_N = 0 $, then $e_{N,j} =0$ for $j=2,3$ and $e_{N,M}=0$ by Lemma \\ref{lemma-sgd:higher-moment-bounds}. Therefore, if $e_N = 0$, the bounds hold trivially. So, we will assume that $e_N > 0$. \n\nLet $u_1,\\ldots,u_m$ be the orthonormal eigenvectors corresponding to eigenvalues $\\lambda_1,\\ldots,\\lambda_m$ of $\\E{Q}$. Then, \n\\begin{equation}\n\\frac{e_{N,j}}{e_N} = \\sum_{l=1}^m \\lambda_l^{j-1}\\frac{\\lambda_l (u_l'(\\theta_N - \\theta^*))^2}{e_N}.\n\\end{equation}\nDenote the ratio on the right hand side by $w_l$, and note that $\\lbrace w_l :l=1,\\ldots,m\\rbrace$ sum to one and are nonnegative. With this notation, we bound (\\ref{eqn-sgd-lemma:technical-systematic-component}) from above by\n\\begin{equation}\ne_N\\left[ \\alpha_0 - \\alpha_1 C_{N+1} \\sum_{l=1}^m \\lambda_l w_l + \\alpha_2 C_{N+1}^2 \\sum_{l=1}^m \\lambda_l^2 w_l + \\alpha_3 C_{N+1}^2 t_Q \\right],  \n\\end{equation}\nand from below by the same equation but with $t_Q$ replaced by $s_Q$. By the properties of $\\lbrace w_l : l=1,\\ldots, m \\rbrace$, we see that the bounds are composed of convex combinations of quadratics of the eigenvalues. Thus, for the upper bound, if we assign all of the weight to the eigenvalue that maximizes the polynomial $-\\alpha_1 C_{N+1} \\lambda + \\alpha_2 C_{N+1}^2 \\lambda^2$, then we will have the upper bound presented in the result. To compute the lower bounds, we do the analogous calculation.\n\\end{proof}\n\nUsing these two lemmas, we can conclude the following for the homogeneous case.\n\n\\begin{theorem}[Homogeneous Minimizer] \\label{theorem-sgd:homogeneous}\nLet $\\lbrace \\theta_N : N \\in \\mathbb{N} \\rbrace$ be the iterates of SGD-$k$ from Definition \\ref{definition-sgd:sgd-k}. Let $\\theta^*$ be a homogeneous solution to the quadratic problem.\n\nIf $0 < C_{N+1}$ and \n\\begin{equation} \\label{eqn-sgd-theorem:homogeneous-ub}\nC_{N+1} < \\begin{cases}\n\\frac{2\\lambda_1}{\\lambda_1^2 + t_Q\/k} & k > t_Q\/(\\lambda_1 \\lambda_m) \\\\\n\\frac{2\\lambda_m}{\\lambda_m^2 + t_Q\/k} & k \\leq t_Q\/(\\lambda_1 \\lambda_m),\n\\end{cases}\n\\end{equation}\nthen $\\E{e_{N+1}} < \\E{e_N}$. Moreover, if $\\lbrace C_{N+1} \\rbrace$ satisfy (\\ref{eqn-sgd-theorem:homogeneous-ub}) and are uniformly bounded away from zero, then $\\E{e_{N}}$ decays exponentially to zero.\n\nMoreover, there exists a deterministic constant $\\rho_N > 1$ such that $\\cond{e_{N+1}}{\\theta_N} > \\rho_N e_N$ with probability one, if either $C_{N+1} < 0$ or \n\\begin{equation} \\label{eqn-sgd-theorem:homogeneous-lb}\nC_{N+1} > \\frac{2\\lambda_j}{\\lambda_j^2 + s_Q\/k},\n\\end{equation}\nwhere\n\\begin{equation}\nj = \\begin{cases}\n1 & k \\leq \\frac{s_Q}{\\lambda_1\\lambda_2} \\\\\nl ~(l \\in \\lbrace 2,\\ldots,m-1 \\rbrace) & \\frac{s_Q}{\\lambda_l\\lambda_{l-1}} < k \\leq  \\frac{s_Q}{\\lambda_{l+1}\\lambda_l} \\\\\nm & k > \\frac{s_Q}{\\lambda_m \\lambda_{m-1}}.\n\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe upper and lower bounds follow from Lemma \\ref{lemma-sgd:quadratic-recursion} and Lemma \\ref{lemma-sgd:technical-systematic-component}, and observations that $k > t_Q\/(\\lambda_1 \\lambda_m)$ implies\n\\begin{equation} \\label{eqn-sgd-theorem:homogeneous-proof-1}\n\\frac{2\\lambda_j}{\\lambda_j^2 + t_Q\/k} > \\frac{2}{\\lambda_1 + \\lambda_m},\n\\end{equation}\nfor $j = 1,m$. For convergence, when $C_{N+1} > 0$ and strictly less than expression on the left hand side of (\\ref{eqn-sgd-theorem:homogeneous-proof-1}), the upper bound in Lemma \\ref{lemma-sgd:technical-systematic-component} is strictly less than one. Thus, if $\\lbrace C_{N+1} \\rbrace$ are uniformly bounded away from zero, the exponential convergence rate of $\\E{e_N}$ follows trivially.\n\nFor the lower bound, we make note of several facts. First, if $k \\geq \\frac{s_Q}{\\lambda_l \\lambda_{l+1}}$ then $k \\geq \\frac{s_Q}{\\lambda_j \\lambda_{j+1}}$ for $j=1,2,\\ldots,l$. Moreover, when $k \\geq \\frac{s_Q}{\\lambda_l \\lambda_{l+1}}$, then \n\\begin{equation}\n\\frac{2\\lambda_l}{\\lambda_l^2 + s_Q\/k} \\geq \\frac{2}{\\lambda_l + \\lambda_{l+1}}.\n\\end{equation}\nBy Lemma \\ref{lemma-sgd:technical-systematic-component}, the left hand side of this inequality is the lower bound on $C_{N+1}$ that guarantees that $\\E{e_{N+1}} > e_N$ with probability one. Therefore, whenever $k \\geq \\frac{s_Q}{\\lambda_l\\lambda_{l+1}}$, $\\cond{e_N+1}{\\theta_N}$ is not guaranteed to diverge. On the other hand, if $k < \\frac{s_Q}{\\lambda_l \\lambda_{l-1}}$ then $k < \\frac{s_Q}{\\lambda_j \\lambda_{j-1}}$ for $j=l,l+1,\\ldots,m$. Moreover, if $k < \\frac{s_Q}{\\lambda_l \\lambda_{l-1}}$ then\n\\begin{equation}\n\\frac{2\\lambda_l}{\\lambda_l^2 + s_Q\/k} < \\frac{2}{\\lambda_l + \\lambda_{l-1}}.\n\\end{equation}\nTherefore, since the left hand side of this inequality is the lower bound on $C_{N+1}$ that guarantees that there exists a deterministic scalar $ \\rho_N > 1$ such that $\\cond{e_{N+1}}{\\theta_N} > \\rho_N e_N$ with probability one, the lower bound is guaranteed to diverge for $C_{N+1}$ larger than the right hand side.\n\nThus, by the monotonicty of the eigenvalues, for the $l \\in \\lbrace 2,\\ldots,m-1 \\rbrace$ such that $\\frac{s_Q}{\\lambda_l \\lambda_{l-1}} < k \\leq \\frac{s_Q}{\\lambda_l\\lambda_{l+1}}$, $\\cond{e_{N+1}}{\\theta_N} > \\rho_N e_N$ if\n\\begin{equation}\nC_{N+1} > \\frac{2\\lambda_l}{\\lambda_l^2 + s_Q\/k}.\n\\end{equation}\nNote, we can handle the edge cases similarly.\n\\end{proof}\n\\begin{remark}\nBecause $\\rho_N$ is a deterministic scalar quantity, we can conclude that $\\E{e_{N+1}} > \\rho_N \\E{e_N}$. \n\\end{remark}\n\nUsing Theorem \\ref{theorem-sgd:homogeneous}, as $k \\to \\infty$, we recover Theorem \\ref{theorem:gd}. That is, Theorem \\ref{theorem-sgd:homogeneous} contains gradient descent as a special case. Moreover, Theorem \\ref{theorem-sgd:homogeneous} also captures the exponential divergence property that we observed for gradient descent (see Figure \\ref{figure:divergence-gd}). Theorem \\ref{theorem-sgd:homogeneous} also contains additional subtleties. First, it captures SGD-$k$'s distinct phases for convergence and divergence, which depend on the batch size and expected geometry of the quadratic problem as represented by the eigenvalues, $t_Q$ and $s_Q$. Thus, as opposed to classical gradient descent, SGD-$k$ requires a more complex convergence and divergence statement to capture these distinct phases. \n\nWe now state the analogous statement for the inhomogeneous problem. However, we will need the following additional lemmas to bound the behavior of the cross term in Lemma \\ref{lemma-sgd:quadratic-recursion}. Again, our analysis below is more delicate than the standard approach for analyzing SGD. \n\n\\begin{lemma} \\label{lemma-sgd:technical-pseudo-inverse}\nLet $Q$ and $r$ be as in Problem \\ref{problem-sgd:quadratic}. Then, letting $(\\cdot)^\\dagger$ denote the Moore-Penrose pseudo-inverse, \n\\begin{equation}\n\\E{ Q \\E{Q}(Q \\theta^* + r )} = \\E{Q}\\E{Q}^\\dagger\\E{ Q \\E{Q}(Q \\theta^* + r )} .\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nNote that for any $x \\in \\texttt{row}(\\E{Q})$, $x = \\E{Q}\\E{Q}^\\dagger x$. Thus, we must show that $\\E{Q \\E{Q} (Q \\theta^* + r)}$ is in $\\texttt{row}(\\E{Q})$. Recall, that if there exists a $v \\in \\mathbb{R}^p$ such that $\\E{Q}v = 0$, then $v' \\E{Q} v = 0$. In turn, $v'Qv = 0$ with probability one, which implies that $Q v = 0$ with probability one. Hence, $\\texttt{null}(\\E{Q}) \\subset \\texttt{null}( Q)$ with probability one. Let the set of probability one be denoted by $\\Omega'$. Then, \n\\begin{equation}\n\\texttt{row}( \\E{Q}) = \\texttt{null}(\\E{Q})^\\perp \\supset \\bigcup_{\\omega \\in \\Omega'} \\texttt{null}(Q)^\\perp = \\bigcup_{\\omega \\in \\Omega'} \\texttt{col}( Q) \\supset \\bigcup_{\\omega \\in \\Omega'} \\texttt{col}( Q \\E{Q} ),\n\\end{equation}\nwhere $\\texttt{row}(Q) = \\texttt{col}(Q)$ by symmetry. Therefore, $Q \\E{Q} (Q\\theta^* + r) \\in \\texttt{row}(\\E{Q})$ with probability one. Hence, its expectation is in $\\texttt{row}(\\E{Q})$. \n\\end{proof}\n\n\\begin{lemma} \\label{lemma-sgd:technical-cross-term-bound}\nUnder the setting of Lemma \\ref{lemma-sgd:quadratic-recursion}, for any $\\phi > 0$ and $j \\in \\mathbb{N}$,  \n\\begin{equation}\n\\begin{aligned}\n&2 \\left\\vert\\frac{C_{N+1}^2}{k} (\\theta_N - \\theta^*)' \\E{Q \\E{Q}(Q\\theta^* + r)}\\right\\vert \\\\\n&\\quad \\leq 2 \\frac{C_{N+1}^2}{k} \\norm{ \\E{Q}^{j\/2} (\\theta_N - \\theta^*)}_2 \\norm{(\\E{Q}^{j\/2})^\\dagger \\E{Q \\E{Q}(Q\\theta^* + r)}}_2 \\\\\n&\\quad \\leq \\frac{\\phi C_{N+1}^2}{k} (\\theta_N - \\theta^*)' \\E{Q}^j (\\theta_N - \\theta^*) \\\\\n&\\quad \\quad + \\frac{C_{N+1}^2}{\\phi k}\\E{(Q\\theta^* + r)' \\E{Q} Q} ( \\E{Q}^j)^\\dagger \\E{Q \\E{Q}(Q \\theta^* + r)},\n\\end{aligned}\n\\end{equation}\nfor any $N = 0,1,2,\\ldots$.\n\\end{lemma}\n\\begin{proof}\nWe will make use of three facts: Lemma \\ref{lemma-sgd:technical-pseudo-inverse}, H\\\"{o}lder's inequality, and the fact that for any $\\phi > 0$, $ \\varphi \\geq 0$ and $x \\in \\mathbb{R}$, $2 |\\varphi x| \\leq \\phi x^2 + \\phi^{-1} \\varphi^2.$ Using these facts, we have\n\\begin{equation}\n\\begin{aligned}\n&2 \\left\\vert\\frac{C_{N+1}^2}{k} (\\theta_N - \\theta^*)' \\E{Q \\E{Q}(Q\\theta^* + r)}\\right\\vert \\\\\n&\\quad = 2 \\left\\vert\\frac{C_{N+1}^2}{k} (\\theta_N - \\theta^*)'\\E{Q}^{j\/2}(\\E{Q}^{j\/2})^\\dagger \\E{Q \\E{Q}(Q\\theta^* + r)}\\right\\vert\\\\\n&\\quad \\leq 2 \\frac{C_{N+1}^2}{k} \\norm{ \\E{Q}^{j\/2} (\\theta_N - \\theta^*)}_2 \\norm{(\\E{Q}^{j\/2})^\\dagger \\E{Q \\E{Q}(Q\\theta^* + r)}}_2 \\\\\n&\\quad \\leq \\frac{\\phi C_{N+1}^2}{k} (\\theta_N - \\theta^*)' \\E{Q}^j (\\theta_N - \\theta^*) \\\\\n&\\quad +   \\frac{C_{N+1}^2}{\\phi k}\\E{(Q\\theta^* + r)' \\E{Q} Q} ( \\E{Q}^j)^\\dagger \\E{Q \\E{Q}(Q \\theta^* + r)}.\n\\end{aligned}\n\\end{equation}\n\\end{proof}\n\n\\begin{lemma} \\label{lemma-sgd:quadratic-recursion-ineq}\nUnder the setting of Lemma \\ref{lemma-sgd:quadratic-recursion},\n\\begin{equation}\n\\begin{aligned}\n\\cond{e_{N+1}}{\\theta_N} &\\leq e_{N} - 2C_{N+1} e_{N,2} + C_{N+1}^2 e_{N,3} + \\frac{C_{N+1}^2}{k} e_{N,M} + \\frac{C_{N+1}^2}{k} e_{N,3} \\\\\n\t\t\t\t\t\t&+ \\frac{C_{N+1}^2}{k} \\E{ (Q\\theta^* + r)' \\E{Q}(Q \\theta^* + r)} \\\\\n\t\t\t\t\t\t&+ \\frac{C_{N+1}^2}{k} \\E{(Q\\theta^*+r)'\\E{Q} Q} ( \\E{Q}^3)^\\dagger \\E{Q \\E{Q}(Q \\theta^* + r)}\n\\end{aligned}\n\\end{equation}\nNow, for any $\\varphi > 0$, $\\exists \\psi \\in \\mathbb{R}$ such that \n\\begin{equation}\n\\begin{aligned}\n\\cond{e_{N+1}}{\\theta_N} &\\geq \\left(1 - \\frac{\\varphi}{k} \\right) e_{N} - 2C_{N+1}e_{N,2}  + C_{N+1}^2 e_{N,3} + \\frac{C_{N+1}^2}{k} e_{N,M} \\\\\n\t\t\t\t\t\t&+ \\frac{C_{N+1}^2}{k} \\psi,\n\\end{aligned}\n\\end{equation}\nwhere $\\psi \\geq 0$ if\n\\begin{equation}\n\\varphi\\frac{\\E{ (Q\\theta^* + r)' \\E{Q}(Q \\theta^* + r)}}{\\E{(Q\\theta^*+r)'\\E{Q} Q} \\E{Q}^\\dagger \\E{Q \\E{Q}(Q \\theta^* + r)}} \\geq C_{N+1}^2.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWhen $C_{N+1} = 0,$ the statements hold by Lemma \\ref{lemma-sgd:quadratic-recursion}. Suppose $C_{N+1} \\neq 0$. Using Lemma \\ref{lemma-sgd:quadratic-recursion} and applying Lemma \\ref{lemma-sgd:technical-cross-term-bound} with $\\phi =1$ and $j=3$, the upper bound follows. Now, using Lemma \\ref{lemma-sgd:quadratic-recursion} and applying Lemma \\ref{lemma-sgd:technical-cross-term-bound} with $\\phi =  C_{N+1}^{-2} \\varphi$ for $\\varphi > 0$ and $j=1$, we have\n\\begin{equation}\n\\begin{aligned}\n\\cond{e_{N+1}}{\\theta_N} &\\geq \\left(1 - \\frac{\\varphi}{k} \\right) e_{N} - 2C_{N+1} e_{N,2} + C_{N+1}^2 e_{N,3} + \\frac{C_{N+1}^2}{k} e_{N,M}  \\\\\n\t\t\t\t\t\t&+ \\frac{C_{N+1}^2}{k} \\psi,\n\\end{aligned}\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{aligned}\n\\psi &=  - \\frac{C_{N+1}^2}{\\varphi}  \\E{(Q\\theta^*+r)'\\E{Q} Q}\\E{Q}^\\dagger \\E{Q \\E{Q}(Q \\theta^* + r)} \\\\\n&\\quad + \\E{ (Q\\theta^* + r)' \\E{Q}(Q \\theta^* + r)}.\n\\end{aligned}\n\\end{equation}\nTherefore, if $\\varphi$ is given as selected in the statement of the result, then $\\psi \\geq 0$.\n\\end{proof}\n\n\\begin{theorem}[Inhomogeneous Minimizer] \\label{theorem-sgd:inhomogeneous}\nLet $\\lbrace \\theta_N : N \\in \\mathbb{N} \\rbrace$ be the iterates of SGD-$k$ from Definition \\ref{definition-sgd:sgd-k}.  Suppose $\\theta^*$ is an inhomogeneous solution to the quadratic problem.\n\nIf $0 < C_{N+1}$ and\n\\begin{equation} \\label{eqn-theorem-sgd:inhomogeneous-ub}\nC_{N+1} < \\begin{cases}\n\\frac{2\\lambda_1^2}{(1 + 1\/k) \\lambda_1^2 + t_Q\/k } & k+1 > t_Q\/(\\lambda_1 \\lambda_m) \\\\\n\\frac{2\\lambda_m^2}{(1 + 1\/k) \\lambda_m^2 + t_Q\/k } & k+1 \\leq t_Q\/(\\lambda_1\\lambda_m), \n\\end{cases}\n\\end{equation}\nthen $\\E{e_{N+1}} < \\rho_N \\E{e_{N}} + C_{N+1}^2 \\frac{\\psi}{k}$, where $\\rho_N < 1$ and is uniformly bounded away from zero for all $N$, and $\\psi > 0$. Moreover, if $\\lbrace C_{N} : N \\in \\mathbb{N} \\rbrace$ are nonnegative, converge to zero and sum to infinity then $\\E{e_{N}} \\to 0$ as $N \\to \\infty$.\n\nFurthermore, suppose $C_{N+1} < 0$ or \n\\begin{equation} \\label{eqn-theorem-sgd:inhomogeneous-lb}\nC_{N+1} > \\frac{2(\\lambda_j + \\gamma)}{\\lambda_j^2 + s_Q\/k},\n\\end{equation} \nwhere $4\\gamma^2 \\in \\left(0, \\frac{s_Q}{k} \\right]$\nand \n\\begin{equation}\nj = \\begin{cases}\n1 & k \\leq \\frac{s_Q}{ \\lambda_1 \\lambda_2 + \\gamma(\\lambda_1 + \\lambda_2)} \\\\\nl ~(l \\in \\lbrace 2,\\ldots,m-1 \\rbrace) & \\frac{s_Q}{\\lambda_l\\lambda_{l-1} + \\gamma(\\lambda_l + \\lambda_{l-1})} < k \\leq \\frac{s_Q}{\\lambda_l \\lambda_{l+1} + \\gamma (\\lambda_l + \\lambda_{l+1})} \\\\\nm & k > \\frac{s_Q}{\\lambda_m \\lambda_{m-1} + \\gamma(\\lambda_m + \\lambda_{m-1})}.\n\\end{cases}\n\\end{equation}\nThen, $\\exists \\delta > 0$ such that if $\\norm{\\E{Q}(\\theta_{N}-\\theta^*)} < \\delta$ then $\\cond{e_{N+1}}{\\theta_N} \\geq \\rho_N e_{N} + C_{N+1}^2 \\psi$ with probability one, where $\\rho_N > 1$ and $\\psi > 0$ independently of $N$.\n\\end{theorem}\n\\begin{proof}\nFor the upper bound, we apply Lemmas \\ref{lemma-sgd:quadratic-recursion-ineq} and \\ref{lemma-sgd:technical-systematic-component} to conclude that\n\\begin{equation}\n\\cond{e_{N+1}}{\\theta_N} \\leq e_{N}\\left(1 - 2 C_{N+1}\\lambda_j + \\left(1 + \\frac{1}{k}\\right) C_{N+1}^2\\lambda_j^2 + C_{N+1}^2 \\frac{t_Q}{k} \\right) + C_{N+1}^2 \\frac{\\psi}{k},\n\\end{equation}\nwhere $\\psi > 0$ is independent of $N$ and\n\\begin{equation}\nj = \\begin{cases}\n1 & C_{N+1} \\leq 0 \\text{ or } \\frac{2}{(1 + 1\/k)(\\lambda_1 + \\lambda_m)} \\\\\nm & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nMoreover, the component multiplying $e_N$ is less than one if $C_{N+1} > 0$ and\n\\begin{equation} \\label{eqn-sgd-theorem:proof-inhomo-1}\nC_{N+1} < \\frac{2\\lambda_j}{\\left( 1 + \\frac{1}{k}\\right)\\lambda_j^2 + \\frac{t_Q}{k}}.\n\\end{equation}\nWhen $k + 1 > t_Q\/(\\lambda_1\\lambda_m)$, the right hand side of (\\ref{eqn-sgd-theorem:proof-inhomo-1}) is larger than $\\frac{2}{(1 + 1\/k)(\\lambda_1 + \\lambda_m)}$. The upper bound follows. Convergence follows by applying Lemma 1.5.2 of \\cite{bertsekas1999}.\n\nFor the lower bound, note that since $\\theta^*$ is inhomogeneous, $\\Prb{Q = \\E{Q}} < 1$. Therefore, $s_Q > 0$. Now, we apply Lemmas \\ref{lemma-sgd:quadratic-recursion-ineq} and \\ref{lemma-sgd:technical-systematic-component} to conclude that for any $\\gamma > 0$ there exists $\\psi_N$ such that\n\\begin{equation}\n\\cond{e_{N+1}}{\\theta_N} \\geq e_{N}\\left(\\left(1 - \\frac{4\\gamma^2}{\\lambda_j^2 + \\frac{1}{k}s_Q} \\right) - 2 C_{N+1}\\lambda_j + C_{N+1}^2\\lambda_j^2 + C_{N+1}^2 \\frac{s_Q}{k} \\right) + C_{N+1}^2 \\frac{\\psi_N}{k},\n\\end{equation}\nwhere $\\psi_N$ are nonnegative and uniformly bounded from zero for $C_{N+1}$ sufficiently small, and\n\\begin{equation}\nj = \\begin{cases}\n1 & C_{N+1} \\in \\left(0, \\frac{2}{\\lambda_1 + \\lambda_2} \\right] \\\\\nl ~(l \\in \\lbrace 2,\\ldots,m-1 \\rbrace) & C_{N+1} \\in \\left( \\frac{2}{\\lambda_{l-1} + \\lambda_l}, \\frac{2}{\\lambda_l + \\lambda_{l+1}} \\right] \\\\\nm & C_{N+1} \\leq 0 \\text{ or } \\frac{2}{\\lambda_m + \\lambda_{m-1}} < C_{N+1}.\n\\end{cases}\n\\end{equation}\nThe term multiplying $e_N$ can be rewritten as\n\\begin{equation} \\label{eqn-sgd-theorem:proof-inhomo-2}\n1 - \\frac{4\\gamma^2}{\\lambda_j^2 + \\frac{1}{k}s_Q} - \\frac{\\lambda_j^2}{\\lambda_j^2 + \\frac{1}{k}s_Q} + \\left(\\lambda_j^2 + \\frac{1}{k}s_Q\\right)\\left( C_{N+1} - \\frac{\\lambda_j}{\\lambda_j^2 + \\frac{1}{k}s_Q} \\right)^2,\n\\end{equation}\nwhich nonnegative when $4\\gamma^2$ is in the interval given by in the statement of the result. Moreover, if\n\\begin{equation} \\label{eqn-sgd-theorem:proof-inhomo-3}\nC_{N+1} > \\frac{2(\\lambda_j + \\gamma)}{\\lambda_j^2 + \\frac{s_Q}{k}},\n\\end{equation}\nthen\n\\begin{equation}\nC_{N+1}\t> \\frac{\\lambda_j}{\\lambda_j^2 + \\frac{s_Q}{k}} + \\frac{\\sqrt{\\lambda_j^2 + 4\\gamma^2}}{\\lambda_j^2 + \\frac{s_Q}{k}},\t\n\\end{equation}\nwhich implies\n\\begin{equation}\n\\left(C_{N+1} - \\frac{\\lambda_j}{\\lambda_j^2 + \\frac{s_Q}{k}} \\right)^2 > \\frac{\\lambda_j^2 + 4\\gamma^2}{\\left(\\lambda_j^2 + \\frac{s_Q}{k}\\right)^2}.\n\\end{equation}\nFrom this inequality, we conclude that if (\\ref{eqn-sgd-theorem:proof-inhomo-3}) holds then (\\ref{eqn-sgd-theorem:proof-inhomo-2}) is strictly larger than $1$. Lastly, for any $\\tilde{j} \\neq j$, \n\\begin{equation}\n\\frac{s_Q}{\\lambda_j \\lambda_{\\tilde{j}} + \\gamma(\\lambda_j + \\lambda_{\\tilde{j}})} > k\n\\end{equation}\nis equivalent to\n\\begin{equation}\n\\frac{2}{\\lambda_j + \\lambda_{\\tilde{j}}} > \\frac{2(\\lambda_j + \\gamma)}{\\lambda_j^2 + \\frac{1}{k}s_Q}.\n\\end{equation}\nWith these facts, the conclusion for the lower bound follows just as in the proof of Theorem \\ref{theorem-sgd:homogeneous}.\n\\end{proof}\n\nComparing Theorems \\ref{theorem-sgd:homogeneous} and \\ref{theorem-sgd:inhomogeneous}, we see that inhomogeneity exacts an additional price. To be specific, call the term multiplying $e_N$ in the bounds as the systematic component and the additive term as the variance component. First, the inhomogeneous case contains a variance component that was not present for the homogeneous case. Of course, this variance component is induced by the inhomogeneity: at each update, SGD-$k$ is minimizing an approximation to the quadratic problem that has a distinct solution from $\\theta^*$ with nonzero probability, which requires decaying the step sizes to remove the variability of solutions induced by these approximations. Second, the inhomogeneity shrinks the threshold for step sizes that ensure a reduction in the systematic component, and inflates the threshold for step sizes that ensure an increase in the systematic component.\n\nContinuing to compare the results for the homogeneous and inhomogeneous cases, we also observe that the systematic component diverges exponentially when the expected gradient is sufficiently small (less than $\\delta$). Indeed, this is the precise region in which such a result is important: it says that for step sizes that are too large, any estimate near the solution set is highly unstable. In other words, if we initialize SGD-$k$ near the solution set, our iterates will diverge exponentially fast from this region for $C_N$ above the divergence threshold.\n\nTo summarize, our analysis of the stochastic quadratic problem defined in Problem \\ref{problem-sgd:quadratic} is a generalization of the standard quadratic problem considered in classical numerical optimization. Moreover, our analysis of SGD-$k$ (see Definition \\ref{definition-sgd:sgd-k}) on the stochastic quadratic problem generalizes the results and properties observed for gradient descent. In particular, our analysis (see Theorems \\ref{theorem-sgd:homogeneous} and \\ref{theorem-sgd:inhomogeneous}) derive a threshold for step sizes above which the iterates will diverge, and this threshold includes the threshold for classical gradient descent. Moreover, our analysis predicts an exponential divergence of the iterates from the minimizer.  \n\n\\subsection{The Deterministic Mechanism} \\label{subsection:deterministic-mechanism}\n\nWe now use the insight developed from our analysis of the generic quadratic problem to formulate a deterministic mechanism for how SGD-$k$ ``escapes'' local minimizers. We will begin by stating the generic nonconvex problem, formalizing our deterministic mechanism, and discussing its implications for nonconvex optimization. We then briefly discuss the pitfalls of naive attempts to directly apply the theory developed in \\S \\ref{subsection:analysis} to arbitrary nonconvex problems.\n\n\\begin{problem}[Nonconvex Problem] \\label{problem:nonconvex}\nLet $(\\Omega,\\mathcal{F},\\mathbb{P})$ be a probability space, and \nlet $C^2(\\mathbb{R}^p,\\mathbb{R})$ be the set of twice continuously differentiable functions from $\\mathbb{R}^p$ to\nMoreover, let $f$ be a random quantity taking values in $C^2(\\mathbb{R}^p,\\mathbb{R})$. Moreover, suppose that\n\\begin{enumerate}\n\\item $\\E{f(\\theta)}$ exists for all $\\theta \\in \\mathbb{R}^p$, and is denoted by $F(\\theta)$.\n\\item Suppose $F$ is bounded from below. \n\\item Suppose that $F \\in C^2(\\mathbb{R}^p,\\mathbb{R})$, and $\\nabla F(\\theta) = \\E{ \\nabla f(\\theta) }$ and $\\nabla^2 F(\\theta) = \\E{ \\nabla^2 f(\\theta)}$. \n\\end{enumerate}\nLet $\\lbrace f_N: N \\in \\mathbb{N} \\rbrace$ be independent copies of $f$. The nonconvex problem is to use $\\lbrace f_N : N \\in \\mathbb{N} \\rbrace$ to determine a $\\theta^*$ that is a minimizer of $F(\\theta)$. \n\\end{problem}\n\nNote, while Problem \\ref{problem:nonconvex} focuses on twice differentiable functions, we often only have continuous functions in machine learning applications. However, for most machine learning applications, the continuous functions are only nondifferentiable at finitely many points, which can be approximated using twice continuously differentiable functions.\n\nBefore stating the deterministic mechanism, we will also need to extend our definition of SGD-$k$ to the generic nonconvex problem. Of course, this only requires replacing (\\ref{eqn-definition:sgd-k}) with \n\\begin{equation}\n\\theta_{N+1} = \\theta_N - \\frac{C_{N+1}}{k}\\sum_{j=Nk+1}^{N(k+1)} \\nabla f_j(\\theta_N).\n\\end{equation}\nWe now state our deterministic mechanism. In this statement, $B(\\theta,\\epsilon)$ denotes a ball around a point $\\theta \\in \\mathbb{R}^p$ of radius $\\epsilon > 0$.\n\n\\begin{definition}[Deterministic Mechanism] \\label{definition:deterministic mechanism}\nLet $\\lbrace \\theta_N : N \\in \\mathbb{N} \\rbrace$ be the iterates generated by SGD-$k$ with deterministic step sizes $\\lbrace C_N : N \\in \\mathbb{N} \\rbrace$ for the nonconvex problem defined in Problem \\ref{problem:nonconvex}. Suppose $\\theta^*$ is a local minimizer of Problem \\ref{problem:nonconvex}.\n\nFor an $\\epsilon > 0$ sufficiently small, let $\\lambda_1 \\geq \\cdots \\geq \\lambda_m > 0$ denote the nonzero eigenvalues of\n\\begin{equation} \n\\frac{1}{|B(\\theta^*,\\epsilon)|} \\int_{B(\\theta^*,\\epsilon)}\\nabla^2 F(\\theta) d\\theta,\n\\end{equation}\nand define $s_f$ to be the maximum between $0$ and \n\\begin{equation}\n\\frac{1}{|B(\\theta^*,\\epsilon)|}\\int_{B(\\theta^*,\\epsilon)}\\inf_{v'\\nabla^2 F(\\theta) v \\neq 0\n} \\left\\lbrace \\frac{v' \\left(\\E{ \\nabla^2 f(\\theta) \\nabla^2 F(\\theta) \\nabla^2 f(\\theta)} - (\\nabla^2 F(\\theta) )^3\\right) v}{v' \\nabla^2 F(\\theta) v}  \\right\\rbrace d\\theta.\n\\end{equation}\nThere is a $\\delta > 0$ such that if $\\theta_N \\in B(\\theta^*,\\delta) \\setminus \\lbrace \\theta^* \\rbrace$ and $C_{N+1} < 0$ or\n\\begin{equation}\nC_{N+1} > \\frac{2\\lambda_j}{\\lambda_j^2 + \\frac{1}{k}s_f},\n\\end{equation} \nwhere\n\\begin{equation}\nj = \\begin{cases}\n1 & k \\leq \\frac{s_f}{ \\lambda_1 \\lambda_2 } \\\\\nl ~(l \\in \\lbrace 2,\\ldots,m-1 \\rbrace) & \\frac{s_f}{\\lambda_l\\lambda_{l-1}} < k \\leq \\frac{s_f}{\\lambda_l \\lambda_{l+1}} \\\\\nm & k > \\frac{s_f}{\\lambda_m \\lambda_{m-1}},\n\\end{cases}\n\\end{equation}\nthen for some $\\rho_N > 1$, $\\cond{F(\\theta_{N+1}) - F(\\theta^*)}{\\theta_N} > \\rho_N ( F(\\theta_N) - F(\\theta^*) )$ with probability one.\n\\end{definition}\n\nFor nonconvex problems, our deterministic mechanism makes several predictions. First, suppose a minimizer's local geometry has several large eigenvalues. Then, for $k$ sufficiently small, SGD-$k$'s divergence from a local minimizer will be determined by these large eigenvalues. This prediction is indeed observed in practice: \\cite{keskar2016} noted that methods that were ``more stochastic'' (i.e., small $k$) diverged from regions that had only a fraction of relatively large eigenvalues. \n\nSecond, suppose for a given nonconvex function, we have finitely many local minimizers and, for each $k$, we are able to compute the threshold for each local minimizer (for some sufficiently small $\\epsilon > 0$, which may differ for each minimizer). Then, for identical choices in step sizes, $\\lbrace C_{N+1} : N \\in \\mathbb{N} \\rbrace$, the number of minimizers for which SGD-$k$ can converge to is a nondecreasing function of $k$. Moreover, SGD-$k$ and Gradient Descent will only converge to minimizers that can support the step sizes $\\lbrace C_{N+1} : N \\in \\mathbb{N} \\rbrace$, which implies that SGD-$k$ will prefer flatter minimizers in comparison to gradient descent. Again, this prediction is indeed observed in practice: the main conclusion of \\cite{keskar2016} is that SGD-$k$ with smaller $k$ converges to flatter minimizers in comparison to SGD-$k$ with large $k$. \n\nThird, our deterministic mechanism states that whenever the iterates are within some $\\delta$ ball of $\\theta^*$ with step sizes that are too large, we will observe exponential divergence. Demonstrating that exponential divergence does indeed occur in practice is the subject of \\S \\ref{section:homogeneous} and \\S \\ref{section:inhomogeneous}.\n\nHowever, before discussing these experiments, it is worth noting the challenges of generalizing the quadratic analysis to the nonconvex case. In terms of convergence, recall that a key ingredient in the local quadratic analysis for classical numerical optimization is stability: once iterates enter a basin of the minimizer, then the iterates will remain within this particular basin of the minimizer. However, for stochastic problems, this will not hold with probability one, especially when the gradient has a noise component with an unbounded support (e.g., linear regression with normally distributed errors). \n\nFortunately, attempts to generalize the divergence results will not require this stability property because divergence implies that the iterates will eventually escape from any such basin. The challenge for divergence, however, follows from the fact that the lower bounds are highly delicate, and any quadratic approximation will require somewhat unrealistic conditions on the remainder term in Taylor's theorem for the nonconvex problem. Thus, generalizing the quadratic case to the nonconvex case is something that we are actively pursuing. \n\\subsection{The Stochastic Mechanism}\n\nConsider using SGD on a one-dimensional stochastic nonconvex problem whose expected objective function is given by the graph in Figure \\ref{figure:one-dim-obj-A}. The stochasticity of the nonconvex problem is generated by adding independent symmetric, bounded random variables to the derivative evaluation at any point. That is, at any iteration, SGD's search direction is generated by the derivative of the expected objective function plus some symmetric, bounded random noise. \n\nIntuitively, the noise in the SGD search directions, by the size of the basins alone, will force the iterates to spend more time in the flatter minimizer on the left. That is, the noise of in SGD's search directions will increase an iterates probability of being in the flatter basin in comparison to the sharper basin. This is precisely stochastic mechanism.\n\n\\begin{figure}[bh]\n\\centering\n\\input{figure\/sgd_stochastic_mechanism_A}\n\\caption{The expected objective function for a simple one-dimensional nonconvex problem. The minimizer on the left is the flatter minimizer. The minimizer on the right is the sharper minimizer.}\n\\label{figure:one-dim-obj-A}\n\\end{figure}\n\nFor another example, consider a similar one-dimensional stochastic nonconvex problem whose expected objective function is given by the graph in Figure \\ref{figure:one-dim-obj-B}. In this example, the minimizer on the left is the flatter minimizer while the minimizer on the right is the sharper minimizer. Now the stochastic mechanism would predict that the probability of an iterate being in the basin of the sharper minimizer is actually greater than that of the flatter minimizer. \n\n\\begin{figure}\n\\centering\n\\input{figure\/sgd_stochastic_mechanism_B}\n\\caption{The expected objective function for a different one-dimensional nonconvex problem. The minimizer on the left is the flatter minimizer. The minimizer on the right is the sharper minimizer.}\n\\label{figure:one-dim-obj-B}\n\\end{figure} \n\nThus, the stochastic mechanism is not explaining why SGD prefers flatter minimizers, but rather it is explaining why SGD has a higher probability of being in basins with larger size. Moreover, while flatness and basin size might seem to be intuitively connected, as Figure \\ref{figure:one-dim-obj-B} shows, this is not necessarily the case. Therefore, according to this example, we might assume that flatness is an inappropriate term, and it is actually the volume of the minimizer that is important (based on the stochastic mechanism) in determining to which minimizers the SGD iterates converge.\n\nHowever, ignoring flatness or sharpness of a minimizer in favor of the size of its basin of attraction does not align with another experimental observation: in high-dimensional problems, SGD even avoids those sharper minimizers that had a small fraction of their eigenvalues termed relatively ``large'' \\cite{keskar2016}. Importantly, in such high-dimensions, the probability of the noise exploring those limited directions that correspond to the ``larger'' eigenvalues is low (e.g., consider isotropic noise in large dimensions). This suggests that, while the stochastic mechanism is intuitive and can explain why SGD may prefer large basins of attractions, it is insufficient to explain why SGD prefers flatter minimizers over sharper minimizers. Accordingly, what is an appropriate mechanism to describe how SGD iterates ``select'' minimizers?\n\n\\subsection{Main Contribution}\nIn this work, we derive and propose a deterministic mechanism based on the expected local geometry of a minimizer and the batch size of SGD to describe why SGD iterates will converge or diverge from the minimizer. As a consequence of our deterministic mechanism, we can explain why SGD prefers flatter minimizers to sharper minimizers in comparison to gradient descent, and we can explain the manner in which SGD will ``escape'' from certain minimizers. Importantly, we verify the predictions of our deterministic mechanism with numerical experiments on two nontrivial nonconvex problems. \n\n\\subsection{Outline}\n\nIn \\S \\ref{section:classical}, we begin by reviewing the classical mechanism by which gradient descent diverges from a quadratic basin. In \\S \\ref{section:stochastic-quadratic}, we will rigorously generalize the classical, deterministic mechanism to SGD-$k$ (i.e., SGD with a batch size of $k$) on a generic stochastic quadratic problem, and, using this rigorous analysis, we define our deterministic mechanism. In \\S \\ref{section:homogeneous} and \\S \\ref{section:inhomogeneous}, we numerically verify the predictions of our deterministic mechanism on two nontrivial, nonconvex problem. In \\S \\ref{section:conclusion}, we conclude this work. \n\n\n\\section{Introduction}\n\\input{section\/introduction}\n\n\\section{Classical Gradient Descent} \\label{section:classical}\n\\input{section\/classical_gd}\n\n\\section{A Deterministic Mechanism} \\label{section:stochastic-quadratic}\n\\input{section\/quadratic}\n\n\\section{Numerical Study of a Homogeneous Nonconvex Problem} \\label{section:homogeneous}\n\\input{section\/homogeneous}\n\n\\section{Numerical Study of an Inhomogeneous Nonconvex Problem} \\label{section:inhomogeneous}\n\\input{section\/inhomogeneous}\n\n\\section{Conclusion} \\label{section:conclusion}\n\\input{section\/conclusion}\n\n\\section*{Acknowledgements}\nWe would like to thank Rob Webber for pointing out the simple proof item 2 in Lemma \\ref{lemma-sgd:higher-moment-bounds}. We would also like to thank Mihai Anitescu for his general guidance throughout the preparation of this work.\n\n\\section*{Funding}\nThe author is supported by the NSF Research and Training Grant \\# 1547396.\n\n\\bibliographystyle{tfs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusion}\nIn this paper, we presented MasterGNN, a joint air quality and weather prediction model that explicitly models the correlations and interactions between two predictive tasks.\nSpecifically, we first proposed a heterogeneous recurrent graph neural network to capture spatiotemporal autocorrelation among air quality and weather monitoring stations. Then, we developed a multi-adversarial learning framework to resist observation noise propagation. In addition, an adaptive training strategy is devised to automatically balance the optimization of multiple discriminative losses in multi-adversarial learning. Extensive experimental results on two real-world datasets demonstrate that the performance of MasterGNN on both air quality and weather prediction tasks consistently outperforms seven state-of-the-art baselines.\n\\section{Experiments}\n\n\\begin{table}[t]\n\\centering\n\\begin{small}\n\\caption{Statistics of datasets.}\n\\label{table-data-stats}\n\\vspace{-6pt}\n\\begin{tabular}{c | c  c} \\hline\n{Data description} & Beijing & Shanghai \\\\ \\hline \\hline\n\\# of air quality stations & 35 & 76 \\\\ \n\\hline\n\\# of weather stations & 18 & 11\\\\ \n\\hline\n\\# of air quality observations & 409,853 & 1,331,520 \\\\ \n\\hline\n\\# of weather observations & 210,824 & 192,720\\\\ \n\\hline\n\\# of  POIs & 900,669 & 1,061,399\\\\ \n\\hline\n\\# of road segments & 812,195 & 768,336 \\\\ \n\\hline\n\\end{tabular}\n\\end{small}\n\\vspace{-7mm}\n\\end{table}\n\n\\begin{table*}[t]\n\\centering\n\\vspace{-10pt}\n\\begin{small}\n\\caption{Overall performance comparison of different approaches. A smaller value indicates a better performance.}\n\\vspace{-5pt}\n\\label{tab-eff}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c}%\n\\hline\n\\multirow{3}{*}{Model} & \\multicolumn{8}{|c}{MAE\/SMAPE}\\\\\n\\cline{2-9}\n\\multirow{3}{*}{} & \\multicolumn{4}{|c}{Beijing} & \\multicolumn{4}{|c}{Shanghai}\\\\\n\\cline{2-9}\n\\multirow{3}{*}{} & {AQI} & {Temperature} & {Humidity} & {Wind speed} & {AQI} & {Temperature} & {Humidity} & {Wind speed}\\\\\n\\hline\nARIMA & 54.97\/0.925 & 3.26\/0.475 & 16.42\/0.403 & 1.17\/0.637 & 34.36\/0.597 & 2.27\/0.394 & 11.96\/0.267 & 1.35\/0.653 \\\\\nLR  & 45.27\/0.732 & 2.39\/0.426 & 9.58\/0.296 & 1.04\/0.543 & 27.68\/0.462 & 1.93\/0.342 & 9.53\/0.254 & 1.21\/0.568 \\\\\nGBRT  & 38.36\/0.678 &  2.31\/0.397 & 8.43\/0.254 & 0.93\/0.489 & 24.13\/0.423 & 1.87\/0.329 & 8.91\/0.232 & 1.05\/0.497 \\\\\n\\hline\nDeepAir & 31.45\/0.613 & 2.21\/0.375 & 8.25\/0.246 & 0.82\/0.468 & 18.79\/0.297 & 1.79\/0.304 & 7.82\/0.209 & 0.93\/0.516 \\\\\nGeoMAN & 30.04\/0.595 & 2.05\/0.356 & 8.01\/0.223 & 0.71\/0.457 & 20.75\/0.348 & 1.56\/0.261 & 6.45\/0.153 & 0.82\/0.465 \\\\\nGC-DCRNN & 29.58\/0.586 & 2.14\/0.364 & 8.14\/0.235 & 0.73\/0.462 & 18.73\/0.305 & 1.61\/0.263 & 6.83\/0.186 & 0.78\/0.462 \\\\\nDUQ & 32.69\/0.624 & 2.02\/0.348 & 7.97\/0.217 & 0.69\/0.446 & 22.81\/0.364 & 1.42\/0.235 & 6.23\/0.136 & 0.67\/0.442 \\\\\n\\hline\\hline\nMasterGNN & \\textbf{27.45\/0.548} & \\textbf{1.87\/0.326} & \\textbf{7.25\/0.184}& \\textbf{0.64\/0.428} &\\textbf{16.51\/0.265} & \\textbf{1.25\/0.213} & \\textbf{5.64\/0.095}&\\textbf{0.61\/0.427}\\\\\n\\hline\n\\end{tabular}\n\\vspace{-8pt}\n\\end{small}\n\\end{table*}\n\n\\subsection{Experimental settings}\n\\subsubsection{Data description.}\nWe conduct experiments on two real-world datasets collected from Beijing and Shanghai, two metropolises in China. \nThe Beijing dataset is ranged from January 1, 2017, to April 1, 2018, and the Shanghai dataset is ranged from June 1, 2018, to June 1, 2020.\nBoth datasets include (1) air quality observations~(\\emph{i.e.},\\xspace PM2.5, PM10, O3, NO2, SO2, and CO), (2) weather observations and weather forecast~(\\emph{i.e.},\\xspace weather condition, temperature, pressure, humidity, wind speed and wind direction), and (3) urban contextual data~(\\emph{i.e.},\\xspace POI distribution and Road network distribution~\\cite{zhu2016days,liu2019hydra,liu2020polestar}). \nThe observation data of Beijing dataset is from KDD CUP 2018\\footnote{https:\/\/www.biendata.xyz\/competition\/kdd\\_2018\/data\/}, and the observation data of Shanghai dataset is crawled from China government websites\\footnote{http:\/\/www.cnemc.cn\/en\/}.\nAll air quality and weather observations are collected by every hour. \nWe associate POI and road network features to each station through an open API provided by an anonymized online map application.\nSame as existing studies~\\cite{yi2018deep,wang2019deep}, we focus on Air Quality Index~(AQI) for air quality prediction, which is derived by the Chinese AQI standard, and temperature, humidity and wind speed for weather prediction.\nWe split the data into training, validation, and testing data by the ratio of 8:1:1.\nThe statistics of two datasets are shown in Table~\\ref{tab-eff}.\n\n\\subsubsection{Implementation details.}\nOur model and all the deep learning baselines are implemented with PyTorch. All methods are evaluated on a Linux server with 8 NVIDIA Tesla P40 GPUs.\nWe set context-aware heterogeneous graph attention layers $l=2$.\nThe cell size of the GRU is set to $64$. \nWe set $\\epsilon=15$, and the hidden size of the multi-layer perceptron of each discriminator is fixed to $64$. All the learning parameters are initialized with a uniform distribution, and the model is trained by stochastic gradient descent with a learning rate $lr=0.00001$. The activation function used in the hidden layers is LeakyReLU ($\\alpha$=0.2).\nAll the baselines use the same input features as ours, except ARIMA excludes contextual features. \nEach numerical feature is normalized by Z-score. We set $T=72$ and the future time step $\\tau=48$.\nFor a fair comparison, we carefully fine-tuned the hyper-parameters of each baseline, and use one another historical observations as a part of feature input.\n\n\n\\subsubsection{Evaluation metrics.}\nWe use Mean Absolute Error (MAE) and Symmetric Mean Absolute Percentage Error (SMAPE), two widely used metrics~\\cite{luo2019accuair} for evaluation.\n\n\\subsubsection{Baselines.}\nWe compare the performance of MasterGNN with the following seven baselines:\n\n\\begin{itemize}\n    \\item \\textbf{ARIMA} ~\\cite{brockwell2002introduction} is a classic time series prediction method utilizes the moving average and auto-regressive component to estimate future trends.\n    \\item \\textbf{LR} uses linear regression ~\\cite{montgomery2012introduction} for joint prediction. We concatenate previous observations and contextual features as input.\n   \n    \\item \\textbf{GBRT} is a tree-based model widely used for regression tasks. We adopt an efficient version XGBoost~\\cite{chen2016xgboost}, and the input feature is same as LR.\n    \\item \\textbf{GeoMAN}~\\cite{liang2018geoman} is a multi-level attention-based network which integrates spatial and temporal attention for geo-sensory time series prediction.\n    \\item \\textbf{DeepAir}~\\cite{yi2018deep} is a deep learning method for air quality prediction. It combines spatial transformation and distributed fusion components to fuse various urban data.\n    \\item \\textbf{GC-DCRNN}~\\cite{lin2018exploiting} leverages a diffusion convolution recurrent neural network to model spatiotemporal dependencies for air quality prediction.\n    \\item \\textbf{DUQ}~\\cite{wang2019deep} is an advanced weather forecasting framework that improves numerical weather prediction by using uncertainty quantification.\n\\end{itemize}\n\n\\begin{figure*}[t]\n\t\\begin{minipage}{1.\\linewidth}\n\t\t\\centering\n\t\t\\vspace{0ex}\n\t\n\t\t\\subfigure[{\\small Effect of joint prediction.}]{\\label{copred}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/exp_copred.pdf}}\n\t\t\\subfigure[{\\small Effect of heterogeneous recurrent graph neural network.}]{\\label{heterst}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/exp_st.pdf}}\n\t\t\\subfigure[{\\small Effect of multi-adversarial learning.}]{\\label{adv}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/exp_adv.pdf}}\n\t\t\\subfigure[{\\small Effect of multi-task adaptive training.}]{\\label{opti}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/exp_ms.pdf}}\n\t\\end{minipage}\n\t\\vspace{-3ex}\n\t\\caption{Ablation study of MasterGNN on the Beijing dataset.} \n\t\\vspace{-7mm}\n\t\\label{parasensitivity}\n\\end{figure*}\n\n\\subsection{Overall performance}\nTable 2 reports the overall performance of MasterGNN and compared baselines on two datasets with respect to MAE and SMAPE. \nWe observe our approach consistently outperforms all the baselines using both metrics, which shows the superiority of MasterGNN.\nSpecifically, our model achieves (7.8\\%, 8.0\\%, 10.0\\%, 7.8\\%) and (6.9\\%, 6.7\\%, 17.9\\%, 4.2\\%) improvements beyond the best baseline on MAE and SMAPE on Beijing for (AQI, temperature, humidity, wind speed) prediction, respectively. Similarly, the improvement of MAE and SMAPE on Shanghai are (13.4\\%, 13.6\\%, 10.4\\%, 9.8\\%) and (12.1\\%, 10.3\\%, 43.2\\%, 3.5\\%), respectively. \nMoreover, we observe all deep learning based models outperform statistical learning based approaches by a large margin, which demonstrate the effectiveness of deep neural network for modeling spatiotemporal dependencies. \nRemarkably, GC-DCRNN performs better than all other baselines for air quality prediction, and DUQ consistently outperforms other baselines for weather prediction. \nHowever, they perform relatively poorly on the other task, which indicates their poor generalization capability and further demonstrate the advantage of joint prediction.\n\n\\begin{figure*}[t]\n\t\\begin{minipage}{1.\\linewidth}\n\t\t\\centering\n\t\t\\vspace{-10pt}\n\t\n\t\t\\subfigure[{\\small Effect of $T$.}]{\\label{length}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/para_t.pdf}}\n\t\t\\subfigure[{\\small Effect of $d$.}]{\\label{hidden_units}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/para_d.pdf}}\n\t\t\\subfigure[{\\small Effect of $\\epsilon$.}]{\\label{epsilon}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/para_epsilon.pdf}}\n\t\t\\subfigure[{\\small Effect of $\\tau$.}]{\\label{tau}\n\t\t\t\\includegraphics[width=0.24\\textwidth]{exp\/para_tau.pdf}}\n\t\\end{minipage}\n\t\\vspace{-3ex}\n\t\\caption{Parameter sensitivities on the Beijing dataset.} \n\t\\vspace{-3ex}\n\t\\label{parasensitivity}\n\\end{figure*}\n\n\\subsection{Ablation study}\nThen we study the effectiveness of each module in MasterGNN. Due to the page limit, we only report the results on Beijing by using SMAPE. The results on Beijing using MAE and on Shanghai using both metrics are similar.\n\n\\textbf{Effect of joint prediction}. To validate the effectiveness of joint prediction, we compare the following variants of MasterGNN:\n(1) SLE models two tasks independently without information sharing,\n(2) CXT models two tasks independently but uses the other observations as a part of contextual features. \nAs depicted in Figure \\ref{copred}, we observe a performance gain of CXT by adding the other observation features, but MasterGNN achieves the best performance.\nSuch results demonstrate the advantage of modeling the interactions between two tasks compared with simply adding the other observations as the feature input.\n\n\\textbf{Effect of heterogeneous recurrent graph neural network}. We evaluate three variants of HRGNN, (1) RS removes the spatial autocorrelation block from MasterGNN, (2) AP replaces GRU by average pooling, (3) RH handles air quality and weather observations homogeneously. As shown in Figure \\ref{heterst}, we observe MasterGNN achieves the best performance compared with other variants, demonstrating the benefits of incorporating spatial and temporal autocorrelations for joint air quality and weather prediction. \n\n\\textbf{Effect of multi-adversarial learning}.\nWe compare the following variants: (1) STD removes the macro discriminator, (2) SAD removes the micro temporal discriminator, (3) TAD removes the micro spatial discriminators. As shown in Figure \\ref{adv}, the macro discriminator is the most influential one, which indicates the importance of defending noises from the global view.\nOverall, MasterGNN consistently outperforms STD, SAD, and TAD on all tasks by integrating all discriminators, demonstrating the benefits of developing multiple discriminators for joint prediction. \n\n\\textbf{Effect of multi-task adaptive training}.\nTo verify the effect of multi-task adaptive training, we develop (1) Average Loss Minimization (ALM) uses the same weight for all discriminators, (2) Fixed Loss Minimization~(FLM) sets weighted but fixed importance based on the performance loss of MasterGNN when the corresponding discriminator is removed.\nAs shown in Figure \\ref{opti}, we observe performance degradation when we fix each discriminator's weight, either on average or weighted. \nOverall, adaptively re-weighting each discriminator's importance can guide the optimization direction of the generator and lead to better performance. \n\n\n\\subsection{Parameter sensitivity}\n\nWe further study the parameter sensitivity of MasterGNN.\nWe report SMAPE on the Beijing dataset. Each time we vary a parameter, we set others to their default values.\n\nFirst, we vary the input length $T$ from 6 to 96. The results are reported in Figure ~\\ref{length}. As the input length increases, the performance first increases and then gradually decreases. The main reason is that a short sequence cannot provide sufficient temporal periodicity and trend information. But too large input length may introduce noises for future prediction, leading to performance degradation.\n\nThen, we vary  $d$ from 8 to 128. The results are reported in Figure ~\\ref{hidden_units}. We can observe that the performance first increases and then remains stable. However, too large $d$ leads to extra computation overhead. Therefore, set the $d=64$ is enough to achieve a satisfactory result.\n\nAfter that, to test the impact of the distance threshold in HSG, we vary $\\epsilon$ from 6 to 20. The results are reported in Figure ~\\ref{epsilon}.\nAs $\\epsilon$ increases, we observe a performance first increase then slightly decrease, which is perhaps because a large $\\epsilon$ integrates too many stations, which introduces more noises during spatial autocorrelation modeling.\n\nFinally, we vary the prediction step $\\tau$ from 3 to 48. The results are reported in Figure ~\\ref{tau}. We observe the performance drops rapidly when $\\tau$ goes large. The reason perhaps is the uncertainty increases when $\\tau$ is large, and it is difficult for the machine learning model to quantify such uncertainty.\n\n\\subsection{Qualitative study}\n\\begin{figure}\n  \\centering\n \n  \\vspace{-4pt}\n  \\includegraphics[width=8cm]{exp\/casestudy.pdf}\n  \\vspace{-4pt}\n  \\caption{Visualization of the learned attention weight of neighboring air quality and weather stations.}\\label{fig:casestudy}\n  \\vspace{-7mm}\n\\end{figure}\n\nFinally, we qualitatively analyze why MasterGNN can yield better performance on both tasks. \nFigure~\\ref{fig:casestudy}~(a) shows the distribution of air quality and weather monitoring stations in the Haidian district, Beijing. \nWe perform a case study from 3:00 May 4, 2017, to 3:00 May 5, 2017.\nFirst, we take the air quality station $S_0$ as an illustrative example to show how weather stations help air quality prediction.\nIn MasterGNN, the attention score reflects the relative importance of each neighboring station for prediction.\nFigure~\\ref{fig:casestudy}~(b) visualizes the attention weights of eight neighboring weather stations of $S_0$.\nAs can be seen, the attention weights of weather stations $S_1$, $S_2$ and $S_4$ abruptly increase because a strong wind blows through this area, which results in notable air pollutant dispersion in the next five hours.\nClearly, the weather stations provide additional knowledge for air quality prediction.\nThen, we take the weather station $S_2$ to depict air quality stations' impact on weather prediction.\nFigure~\\ref{fig:casestudy}~(c) shows the attention weights of neighbouring station of $S_2$.\nWe observe the nearest air quality station $S_3$ is the most influential station during 5:00 May 4 2017 and 10:00 May 4 2017, while $S_7$ plays a more important role during 15:00 May 4 2017 and 21:00 May 4 2017, corresponding to notable air pollution concentration changes during these periods.\nThe above observations further validate the additional knowledge introduced by the joint modeling of heterogeneous stations for both predictive tasks.\n\n\\eat{\nFigure~\\ref{fig:casestudy} presents a local map of Beijing, several air quality and weather stations are scattered in geographical space. We perform a case study from 3:00 May 4, 2017 to 3:00 May 5, 2017. \nTo show how weather conditions impacts air quality, we take the air quality station $S_0$ as an example to visualize the attention weights with eight neighboring weather stations in Figure~\\ref{fig:casestudy}. The attention scores semantically represent the relative importance of each weather station. According to the attention weight matrix, the stations are not correlated before 22:00 May 4, 2017. Then, there came a strong wind and has lasted for 5 hours, the air quality dropped rapidly in this time period. We observe the attention weights increase when the wind blows strongly. Note that the nearest weather station $S_1$  have the largest attention scores, showing biggest impact to the selected air quality station. The attention weights gradually decreases with increasing distance. The above example have shown that the attention weights generated by our model are indeed meaningful in real world and can be easily interpreted.\nNext, we further illustrate how air quality impacts meteorology. As shown in Figure ~\\ref{fig:casestudy}, the attention weights of nearby air quality stations are higher than remote stations due to severe air pollution in this region. Another observation is that the attention weight of air quality to weather is lower than that of weather to air quality, this shows the influence of weather on air quality is much more significant.\nThe case study indicates that our model successfully captures the complex spatial correlation between the air quality stations and weather stations.\n}\n\\section{Introduction}\n\n\\begin{figure}\n  \\centering\n \n  \\vspace{-4pt}\n  \\includegraphics[width=8cm]{fig\/distribution.pdf}\n  \\vspace{-5pt}\n  \\caption{\n  Spatial distribution of air quality and weather monitoring stations in Beijing. Two types of stations are monitoring exclusively different but correlated air quality and weather conditions in different city locations.}\\label{dist}\n  \\vspace{-7mm}\n\\end{figure}\n\n\n\nWith the rapid development of the economy and urbanization, people are increasingly concerned about emerging public health risks and environmental sustainability.\nAs reported by the World Health Organization~(WHO), air pollution is the world's largest environmental health risk~\\cite{campbell2019climate}, and the changing weather profoundly affects the local economic development and people's daily life~\\cite{baklanov2018urban}.\nThus, accurate and timely air quality and weather predictions are of great importance to urban governance and human livelihood.\nIn the past years, massive sensor stations have been deployed for monitoring air quality~(\\emph{e.g.},\\xspace PM2.5 and PM10) or weather conditions~(\\emph{e.g.},\\xspace temperature and humidity) and many efforts have been made for air quality and weather predictions~\\cite{liang2018geoman,yi2018deep,wang2019deep}.\n\nHowever, existing air quality and weather prediction methods are designated for either air quality or weather prediction, perhaps with one another as a side input~\\cite{zheng2015forecasting,yi2018deep}, but overlook the intrinsic connections and interactions between two tasks.\nDifferent from existing studies, this work is motivated by the following two insights. First, air quality and weather predictions are two highly correlated tasks and can be mutually enhanced.\nFor example, accurately predicting the regional wind condition can help model the future dispersion and transport of air pollutants~\\cite{ding2017air}.\nAs another example, modeling the future concentration of aerosol pollutants~(\\emph{e.g.},\\xspace PM2.5 and PM10) also can help predict the local climate~(\\emph{e.g.},\\xspace temperature and humidity), since aerosol elements can scatter or absorb the incoming radiation~\\cite{hong2020weakening}.\nSecond, as illustrated in Figure~\\ref{dist}, the geospatially distributed air quality and weather monitoring stations provide additional hints to improve both predictive tasks.\nThe air quality and weather condition variations in different city locations reflect the urban dynamics and can be exploited to improve spatiotemporal autocorrelation modeling.\n\nIn this work, we investigate the joint prediction of air quality and weather conditions by explicitly modeling the correlations and interactions between two predictive tasks.\nHowever, two major challenges arise in achieving this goal.\n(1) \\emph{Observation heterogeneity}.\nAs illustrated in Figure \\ref{dist}, the geo-distributed air quality and weather stations are heterogeneous spatial objects that are monitoring exclusively different atmospheric conditions. \nExisting methods~\\cite{zheng2013u,wang2019deep} are initially designed to model homogeneous spatial objects~(\\emph{i.e.},\\xspace either air quality or weather stations), which are not suitable for joint air quality and weather predictions.\nTherefore, the first challenge is how to capture spatiotemporal autocorrelations among heterogeneous monitoring stations to mutually benefit air quality and weather prediction.\n(2) \\emph{Compounding observation error vulnerability}.\nIn practice, observations reported by monitoring stations are often noisy due to the sensor error and environmental interference~\\cite{yi2016st}.\nHowever, most existing prediction models~\\cite{zheng2015forecasting,liang2018geoman,yi2018deep} rely on spatiotemporal dependency modeling between stations, which is susceptible to local perturbations and noise propagation~\\cite{zugner2018adversarial,bengio2015scheduled}.\nMore severely, jointly modeling spatiotemporal autocorrelation among air quality and weather monitoring stations will further accumulate errors from both spatial and temporal domains.\nAs a result, it is challenging to learn robust representations to resist compounding observation error for joint air quality and weather predictions.\n\nTo tackle the above challenges, we propose the \\emph{\\textbf{M}ulti-\\textbf{a}dversarial \\textbf{s}patio\\textbf{te}mporal \\textbf{r}ecurrent \\textbf{G}raph \\textbf{N}eural \\textbf{N}etworks}~(\\textbf{MasterGNN}) for robust air quality and weather joint predictions.\nSpecifically, we first devise a heterogeneous recurrent graph neural network to simultaneously incorporate spatial and temporal autocorrelations among heterogeneous stations conditioned on dynamic urban contextual factors~(\\emph{e.g.},\\xspace POI distributions, traffics).\nThen, we propose a multi-adversarial learning framework to against noise propagation from both microscopic and macroscopic perspectives.\nBy proactively simulating perturbations and maximizing the divergence between target and fake observations, multiple discriminators dynamically regularize air quality and weather station representations to resist the propagation of observation noises~\\cite{durugkar2016generative}.\nMoreover, we introduce a multi-task adaptive training strategy to improve the joint prediction performance by automatically balancing the importance of multiple discriminators.\nFinally, we conduct extensive experiments on two real-world datasets collected from Beijing and Shanghai, and the proposed model consistently outperforms seven baselines on both air quality and weather prediction tasks.\n\\eat{\nTo sum up, our main contributions are as follows:\n\\begin{itemize}\n\t\\item We study the novel air quality and weather co-prediction problem by exploiting the inner-connection between air quality and weather conditions.\n\t\\item We propose a heterogeneous recurrent graph neural network to capture the spatiotemporal autocorrelation among heterogeneous monitoring stations.\n\t\\item We introduce a multi-adversarial learning framework coupled with a multi-objective optimization strategy to resist observation noises.\n\t\\item We conduct extensive experiments on two real-world datasets collected from Beijing and Shanghai. The proposed model outperforms seven baselines by at least 10.7\\% and 5.5\\% on air quality and weather prediction tasks, respectively.\n\\end{itemize}\n}\n\n\\section{Methodology}\nMasterGNN simultaneously makes air quality and weather predictions based on the following intuitions.\n\n\\textbf{Intuition 1: Heterogeneous spatiotemporal autocorrelation modeling}. The geo-distributed air quality and weather monitoring stations provide additional spatiotemporal hints for air quality and weather prediction. The model should be able to simultaneously incorporate spatial and temporal autocorrelations of such heterogeneous monitoring stations for joint air quality and weather prediction.\n\n\\textbf{Intuition 2: Robust representation learning}. \nThe spatiotemporal autocorrelation based joint prediction model is vulnerable to observation noises and suffers from compounding propagation errors. The model should be robust to error propagation from both spatial and temporal domains.\n\n\\textbf{Intuition 3: Adaptive model training}. Learning robust representations introduces extra optimization objectives, which guiding the predictive model to approximate the underlying data distribution from different semantic aspects. The model should be able to balance the optimization of multiple objectives for optimal prediction adaptively. \n\n\\subsection{Framework Overview}\n\n\\begin{figure}[t]\n  \\centering\n \n \n  \\includegraphics[width=7.5cm]{fig\/framework.pdf}\n  \\vspace{-0.1cm}\n  \\caption{An overview of MasterGNN.}\\label{framework}\n  \\vspace{-0.6cm}\n\\end{figure}\n\nFigure \\ref{framework} shows the architecture of our approach, including three major tasks: \n(i) modeling the spatiotemporal autocorrelation among air quality and weather stations for joint prediction; (ii) learning robust station representations via multi-adversarial learning; (iii) exploiting adaptive training strategy to ease the model learning.\nSpecifically, in the first task, we propose a \\emph{Heterogeneous Recurrent Graph Neural Network}~(HRGNN) to jointly incorporate spatial autocorrelation between heterogeneous monitoring stations and past-current temporal autocorrelation of each monitoring station.\nIn the second task, we develop a \\emph{Multi-Adversarial Learning} framework to resist the propagation of observation noises via (i) microscopic discriminators against adversarial attacks respectively from the spatial and temporal domain, and (ii) macroscopic discriminator against adversarial attacks from a global view of the city.\nIn the third task, we introduce a \\emph{Multi-Task Adaptive Training} strategy to automatically balance the optimization of multiple discriminative losses in multi-adversarial learning.\n\n\n\\subsection{Heterogeneous Recurrent Graph Neural Network}\n\n\n\\subsubsection{The base model.}\nWe adopt the encoder-decoder architecture~\\cite{zhang2020tkde,zhang2020semi} as the base model.\nSpecifically, the encoder step projects the historical observation sequence of all stations $\\mathbfcal{X}$ to a hidden state $\\mathbf{H}^t=f_{encoder}(W\\mathbfcal{X}+b)$, where $f_{encoder}(\\cdot)$ is parameterized by a neural network.\nIn the decoder step, we employ another neural network $f_{decoder}(\\cdot)$ to generate air quality and weather predictions $(\\hat{\\mathbf{Y}}^{t+1},\\hat{\\mathbf{Y}}^{t+2},\\dots,\\hat{\\mathbf{Y}}^{t+\\tau})=f_{decoder}(\\mathbf{H}^t, \\mathbf{C})$, where $\\tau$ is the future time steps we aim to predict, and $\\mathbf{C}$ are contextual features~\\cite{liu2020incorporating}.\nSimilar with conventional air quality or weather prediction models~\\cite{liang2018geoman,li2020weather}, the heterogeneous recurrent graph neural network aims to minimize the \\emph{Mean Square Error}~(MSE) between the target observations and predictions,\n\\begin{equation}\\label{equ:mse}\n\\mathcal{L}_g=\\frac{1}{\\tau}\\sum^{\\tau}_{i=1}||\\hat{\\mathbf{Y}}^{t+i}-\\mathbf{Y}^{t+i}||^2_2.\n\\end{equation}\n\n\\subsubsection{Incorporating spatial autocorrelation.}\nIn the spatial domain, the regional concentration of air pollutants and weather conditions are highly correlated and mutually influenced.\nInspired by the recent variants~\\cite{wang2019heterogeneous,zhang2019heterogeneous,liu2021vldb} of graph neural network on handling non-Euclidean semantic dependencies on heterogeneous graphs, we devise a context-aware heterogeneous graph attention block (CHAT) to model spatial interactions between heterogeneous stations.\nWe first construct the heterogeneous station graph to describe the spatial adjacency of each station.\n\n\\begin{myDef}\n\\textbf{Heterogeneous station graph}. A heterogeneous station graph (HSG) is defined as $\\mathcal{G}=\\{\\mathcal{V},\\mathcal{E}, \\psi \\}$, where $\\mathcal{V}=S$ is the set of monitoring stations, $\\psi$ is a mapping function indicates the type of each station, and $\\mathcal{E}$ is a set of directed edges indicating the connectivity among monitoring stations, defined as\n\\begin{equation}\ne_{ij}=\\left\\{\n\\begin{array}{lr}\n1,\\quad d_{ij}<\\epsilon\\\\\n0,\\quad otherwise\n\\end{array}\n\\right.,\n\\end{equation}\nwhere $d_{ij}$ is the spherical distance between station $s_i$ and station $s_j$, $\\epsilon$ is a distance threshold.\n\\end{myDef}\n\nDue to the heterogeneity of monitoring stations, there are two types of vertices~(air quality station $s^a$, weather station $s^w$) and four types of edges, \\emph{i.e.},\\xspace $\\Psi=\\{s^a$-$s^a$, $s^w$-$s^w$, $s^a$-$s^w$, $s^w$-$s^a\\}$. \nWe use $\\psi(i)$ to denote the station type of $s_i$, and $r \\in \\Psi$ to denote the semantic type of each edge.\n\nFormally, given an observation $\\mathbf{x}_i$ at a particular time step, we first devise a type-specific transformation layer to project heterogeneous observations into unified feature space,\n$\\widetilde{\\mathbf{x}}_i = \\mathbf{W}^{\\psi(i)} \\mathbf{x}_i$,\nwhere $\\widetilde{\\mathbf{x}}_i\\in\\mathcal{R}^d$ is a low-dimensional embedding vector, $\\mathbf{W}^{\\psi(i)}\\in\\mathcal{R}^{|\\mathbf{x}_i|\\times d}$ is a learnable weighted matrix shared by all monitoring stations of type $\\psi(i)$.\n\nThen, we introduce a type-dependent attention mechanism to quantify the non-linear correlation between homogeneous and heterogeneous stations under different contexts. Given a station pair $(s_i,s_j)$ which are connected by an edge of type $r$, the attention score is defined as\n\\begin{equation}\na^r_{ij} =\\frac{Attn(\\widetilde{\\mathbf{x}}_i,\\widetilde{\\mathbf{x}}_j,\\mathbf{c}_i,\\mathbf{c}_j,d_{ij})}{\\sum_{k\\in\\mathcal{N}^r_{i}}Attn(\\widetilde{\\mathbf{x}}_i,\\widetilde{\\mathbf{x}}_k,\\mathbf{c}_i,\\mathbf{c}_k,d_{ik})},\n\\end{equation}\nwhere $Attn(\\cdot)$ is a concatenation based attention function, $\\mathbf{c}_i,\\mathbf{c}_j\\in\\mathbf{C}$ are contextual features of station $s_i$ and $s_j$, $d_{ij}$ is the spherical distance between $s_i$ and $s_j$, and $\\mathcal{N}^r_i$ is the set of type-specific neighbor stations of $s_i$ in $\\mathcal{G}$.\nBased on $a^r_{ij}$, we define the context-aware heterogeneous graph convolution operation to update the type-wise station representation,\n\\begin{equation}\n\\widetilde{\\mathbf{x}}_i^{r\\prime} = GConv(\\widetilde{\\mathbf{x}}_{i}, r) = \\sigma(\\sum_{j\\in \\mathcal{N}^r_i}\\alpha^r_{ij}\\mathbf{W}^r\\widetilde{\\mathbf{x}}_j),\n\\end{equation}\nwhere $\\widetilde{\\mathbf{x}}^{r\\prime}_{i}$ is the aggregated node representation based on edge type $r$, $\\sigma$ is a non-linear activation function, $\\mathbf{W}^r\\in\\mathcal{R}^{d \\times d}$ is a learnable weighted matrix shared over all edges of type $r$.\nFinally, we obtain the updated station representation of $s_i$ by concatenating type-specific representations,\n\\begin{equation}\n\\widetilde{\\mathbf{x}}^{\\prime}_i = ||_{r\\in\\Psi}GConv(\\widetilde{\\mathbf{x}}_{i}, r),\n\\end{equation}\nwhere $||$ is the concatenation operation.\nNote that we can stack $l$ graph convolution layers to capture the spatial autocorrelation between $l$-hop heterogeneous stations.\n\n\\eat{\nConsider an air quality observation $\\mathbf{x}^a_i$ and a weather observation $\\mathbf{x}^w_j$ at a particular time step, we first employ a type-specific linear transformation layer to project heterogeneous observations into a unified feature space,\n\\begin{equation}\n\\mathbf{h}^a_i = \\mathbf{W}^a \\mathbf{x}^a_i,~~\\mathbf{h}^w_j = \\mathbf{W}^w \\mathbf{x}^w_j,\n\\end{equation}\nwhere $\\mathbf{h}^a_i\\in\\mathcal{R}^d$ and $\\mathbf{h}^w_j\\in\\mathcal{R}^d$ are low-dimensional embedding vectors, $\\mathbf{W}^a\\in\\mathcal{R}^{|\\mathbf{x}^a_i|\\times d}$ and $\\mathbf{W}^w\\in\\mathcal{R}^{|\\mathbf{x}^w_j|\\times d}$ are learnable weighted matrices shared by air quality and weather monitoring stations, respectively.\n\nWe use $\\psi_i$ to denote the node type of $v_i$, and $\\psi_{ij}$ to denote the edge type from $v_i$ to $v_j$.\n\nConsider an atmospheric observation $\\mathbf{x}_i$ at a particular time step, we first employ a type-specific linear transformation layer to project heterogeneous observations into a unified feature space,\n\\begin{equation}\n\\mathbf{h}_i = \\mathbf{W}_{\\psi_i} \\mathbf{x}_i,\n\\end{equation}\nwhere $\\mathbf{h}_i\\in\\mathcal{R}^d$ is a low-dimensional embedding vector, $\\mathbf{W}_{\\psi_i}\\in\\mathcal{R}^{|\\mathbf{X}_i|\\times d}$ is a learnable weighted matrix shared by all same-typed monitoring stations.\n}\n\n\\subsubsection{Incorporating temporal autocorrelation.}\nIn the temporal domain, the air quality and weather conditions also depend on previous observations.\nWe further extend the Gated Recurrent Units (GRU), a simple variant of recurrent neural network~(RNN), to integrate the temporal autocorrelation among heterogeneous observation sequences.\nConsider a station $s_i$, given the learned spatial representation $\\widetilde{\\mathbf{x}}^t_i$ at time step $t$, we denote the hidden state of $s_i$ at $t-1$ and $t$ as $\\mathbf{h}^{t-1}_i$ and $\\mathbf{h}^t_i$, respectively. The temporal autocorrelation between  $\\mathbf{h}^{t-1}_i$ and $\\mathbf{h}^t_i$ is modeled by\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n    &\\mathbf{h}^t_i=(1-\\mathbf{z}^t_i)\\odot \\mathbf{h}^{t-1}_i+\\mathbf{z}^t_i\\odot \\widetilde{\\mathbf{h}}^t_i\\\\\n    &\\mathbf{r}^{t}_i = \\sigma{(\\mathbf{W}^{\\psi(i)}_r[\\mathbf{h}^{t-1}_i\\parallel\\widetilde{\\mathbf{x}}^{t}_i]+\\mathbf{b}^{\\psi(i)}_r)}\\\\\n    &\\mathbf{z}^{t}_i = \\sigma(\\mathbf{W}^{\\psi(i)}_z[\\mathbf{h}^{t-1}_i\\parallel\\widetilde{\\mathbf{x}}^{t}_i]+\\mathbf{b}^{\\psi(i)}_z)\\\\\n    &\\widetilde{\\mathbf{h}}^{t}_i = \\tanh(\\mathbf{W}^{\\psi(i)}_{\\widetilde{h}}[\\mathbf{r}^{t}_i\\odot\\mathbf{h}^{t-1}_i\\parallel\\widetilde{\\mathbf{x}}^t_i]+\\mathbf{b}^{\\psi(i)}_{\\widetilde{h}})\n\\end{aligned},\n\\right.\n\\end{equation}\nwhere $\\mathbf{o}^t_i$, $\\mathbf{z}^t_i$ denote reset gate and update gate at time stamp t, $W_{o}^{\\psi(i)}$, $W_{z}^{\\psi(i)}$, $W_{\\widetilde{h}}^{\\psi(i)}$, $b_{o}^{\\psi(i)}$, $b_{z}^{\\psi(i)}$, $b_{\\widetilde{h}}^{\\psi(i)}$ are trainable parameters shared by all same-typed monitoring stations, and $\\odot$ represents the Hadardmard product. \n\n\\subsection{Multi-Adversarial Learning}\n\n\\begin{figure}[t]\n  \\centering\n \n \n  \\includegraphics[width=8.5cm]{fig\/adver_train.pdf}\n  \\vspace{-0.5cm}\n  \\caption{Discriminators in multi-adversarial learning.}\\label{adver_train}\n  \\vspace{-0.7cm}\n\\end{figure}\n\nWe further introduce the multi-adversarial learning framework to obtain robust station representations.\nBy proactively simulating and resisting noisy observations via a \\emph{minmax} game between the generator and discriminator~\\cite{goodfellow2014generative}, adversarial learning can make the generator more robust and generalize better to the underlying true distribution $p_{true}(\\mathbf{Y} | \\mathbfcal{X}, \\mathbf{C})$.\nCompared with standard adversarial learning, our multi-adversarial learning framework encourages the generator to make more consistent predictions from both spatial and temporal domains.\nSpecifically, we function the HRGNN $\\mathbf{\\phi}$ as the generator $\\hat{\\mathbf{Y}}=G(\\mathbfcal{X}, \\mathbf{C};\\mathbf{\\phi})$ for joint predictions.\nMoreover, we propose a set of distinct discriminators $\\{D_k(\\cdot;\\theta_k)\\}^K_{k=1}$ to distinguish the real observations and predictions from both microscopic and macroscopic perspectives, as detailed below.\n\n\\subsubsection{Microscopic discriminators.}\nMicroscopic discriminators aims at enforcing the generator to approximate the underlying spatial and temporal distributions, \\emph{i.e.},\\xspace pair-wise spatial autocorrelation and stepwise temporal autocorrelation.\n\n\\emph{Spatial discriminator}. From the spatial domain, the adversarial perturbation induces the compounding propagation error through the HSG, as illustrated in Figure~\\ref{adver_train}~(a). Consider a time step $t$, the spatial discriminator $D_s(\\mathbf{y}^t;\\theta_s)$ aims to maximize the accuracy of distinguishing city-wide real observations and predictions at the current time step,\n\\begin{equation}\\label{equ:dis1}\n\\mathcal{L}_{s}=\\log D_s(\\mathbf{y}^t;\\theta_s)+log(1-D_s(\\hat{\\mathbf{y}}^t;\\theta_s)),\n\\end{equation}\nwhere $D_s(\\mathbf{y}^t;\\theta_s)$ is parameterized by the context-aware heterogeneous graph attention block in HRGNN followed by a multi-layer perception, $\\mathbf{y}^t$ and $\\hat{\\mathbf{y}}^t$ are ground truth observations and predicted conditions of the city, respectively.\n\n\\emph{Temporal discriminator.} From the temporal domain, the adversarial perturbation induces accumulated error for each station from previous time steps. \nAs depicted in Figure~\\ref{adver_train}~(b), the temporal discriminator $D_t(\\mathbf{y}_i;\\theta_t)$ outputs a probability indicating how likely a observation sequence $\\mathbf{y}_i$ of a particular station $s_i$ is from the real data distribution $p_{true}(\\mathbf{y}_i | \\mathbf{x}_i, \\mathbf{c}_i)$ rather than the generator $p_{\\phi}(\\mathbf{y}_i | \\mathbf{x}_i, \\mathbf{c}_i)$.\n\\begin{equation}\\label{equ:dis2}\n\\mathcal{L}_{t}=\\log D_t(\\mathbf{y}_i;\\theta_t)+log(1-D_t(\\hat{\\mathbf{y}}_i;\\theta_t)).\n\\end{equation}\nDifferent from the spatial discriminator, $D_t(\\mathbf{y}_i;\\theta_t)$ is parameterized by the temporal block in HRGNN followed by a multi-layer perception. \n$\\mathbf{y}_i$ is a sequence of observations in past $T$ and future $\\tau$ time steps.\n\n\\subsubsection{Macroscopic discriminator.}\nAs illustrated in Figure~\\ref{adver_train}~(c), we further propose a macroscopic discriminator to capture the globally underlying distribution, denoted by $D_m(\\mathbf{Y};\\theta_m)$. In particular, the macroscopic discriminator aims to maximize the accuracy of distinguishing the ground truth observations and predictions generated from $G$ from a global view,\n\\begin{equation}\\label{equ:dis3}\n\\mathcal{L}_{m}=\\log D_m(\\mathbf{Y};\\theta_m)+log(1-D_m(G(\\mathbfcal{X},\\mathbf{C};\\mathbf{\\phi});\\theta_m)),\n\\end{equation}\nwhere $D_m(\\mathbf{Y};\\theta_m)$ is parameterized by a GRU followed by a multi-layer perception.\nNote that for efficiency concern, the input of $D_m(\\mathbf{Y};\\theta_m)$ is a simple concatenation of observations from all stations $S$ in past $T$ and future $\\tau$ time steps, but other  graph convolution operations are also applicable.\n\n\\eat{\nGiven the generator $G$ and discriminators $\\{D_i\\}^K_{i=1}$,\nthe minmax function of the multi-adversarial learning is\n\\begin{equation}\n\\begin{aligned}\n\\min_{G} \\max_{\\{D_i\\}^K_{i=1}} V(\\{D_i\\}^K_{i=1}&,G)=\\mathbb{E}_{\\mathbf{Y} \\sim p_{\\emph{true}}}[\\log D_i(\\mathbf{Y};\\theta_i)]+\\\\\n&\\mathbb{E}_{\\mathbfcal{X} \\sim p_{\\phi}}[\\log (1-D_i(G(\\mathbfcal{X},\\mathbf{C};\\phi)))].\n\\end{aligned}\n\\end{equation}\n}\n\n\\subsection{Multi-Task Adaptive Training}\nIt is widely recognized that adversarial training suffers from the unstable and mode collapse problem, where the generator is easy to over-optimized for a particular discriminator~\\cite{salimans2016improved,neyshabur2017stabilizing}.\nTo stabilize the multi-adversarial learning and achieve overall accuracy improvement, we introduce a multi-task adaptive training strategy to dynamically re-weight discriminative loss and enforce the generator to perform well in various spatiotemporal aspects.\n\nSpecifically, the objective of MasterGNN is to minimize \n\\begin{equation}\\label{equ:all-loss}\n\\mathcal{L}=\\mathcal{L}_g+\\sum^K_{i=1}\\lambda_i \\mathcal{L}_{d_i},\n\\end{equation}\nwhere $\\mathcal{L}_g$~(Equation~\\ref{equ:mse}) is the predictive loss, $\\mathcal{L}_D=\\{\\mathcal{L}_{d_i}\\}^K_{i=1}$ (Equation~\\ref{equ:dis1}-\\ref{equ:dis3}) are discriminative losses, and\n$\\lambda_i$ is the importance of the corresponding discriminative loss.\n\nSuppose $\\mathbf{H}_{i}$ and $\\hat{\\mathbf{H}}_{i}$ are intermediate hidden states in discriminator $D_i$ based on real observations $\\mathbf{Y}$ and predictions $\\hat{\\mathbf{Y}}$.\nWe measure the divergence between $\\mathbf{H}_{i}$ and $\\hat{\\mathbf{H}}_{i}$ by $\\gamma_i=sim(\\sigma(\\mathbf{H}_{i}),\\sigma(\\hat{\\mathbf{H}}_{i}))$,\nwhere $sim(\\cdot,\\cdot)$ is an Euclidean distance based similarity function, $\\sigma$ is the sigmoid function. \nIntuitively, $\\gamma_i$ reflects the hardness of $D_i$ to distinguish the real sample.\nIn each iteration, we re-weight discriminative losses by \n$\\lambda_i = \\frac{exp(\\gamma_i)}{\\sum^{K}_{k=1}exp(\\gamma_k)}$.\nIn this way, the generator pays more attention to discriminators with a larger room to improve, and will result in better prediction performance.\n\n\n\\eat{\nSpecifically, for the generative loss $\\mathcal{L}_g$ and $K$ discriminative losses $\\mathcal{L}_D=[\\mathcal{L}_1,\\mathcal{L}_2,...,\\mathcal{L}_K]$,\nwhere each $\\mathcal{L}_k=\\mathbb{E}_{\\mathbfcal{X} \\sim p_{\\phi}}[\\log(1-D_i(G(\\mathbfcal{X},\\mathbf{C};\\phi)))]$ is the loss provided by the i-th discriminator.\nWe use a dynamic weight $\\lambda_i$ to control the importance of discriminator $i$. A multi-task training strategy is designed to adaptively adjust $\\lambda_i$. Suppose $h_{i,f}$ and $h_{i,r}$ are the hidden state of discriminator $i$ derived from future predictions and ground truth observations, respectively. We first use a Sigmoid function to map the hidden state to the range [0,1]. Then we leverage a function to measure the similarity between $h_{i,f}$ and $h_{i,r}$, defined as follows\n\\begin{equation}\n\\gamma_i=sim(h_{i,f},h_{i,r}).\n\\end{equation}\nIn this paper, we use Euclidean Distance as similarity function. When $\\gamma_i$ becomes large, it is intuitive to conclude the discriminator $i$ can easily distinguish the predictions and ground truth observations, which means the generator performs poorly on discriminator $i$. We directly adopt $\\gamma_i$ as the weight to focus on gradients from discriminator $i$.\nWe employ a softmax function to convert the dynamic weights into a probability distribution.\n\\begin{equation}\n\\lambda_i = \\frac{exp(\\gamma_i)}{\\sum^{K}_{k=1}exp(\\gamma_k)}.\n\\end{equation}\nDuring training, our proposed strategy dynamically adjusts the influence of various discriminators. \nUpdates of generator are performed to minimize the following loss\n\\begin{equation}\n\\mathcal{L}=\\mathcal{L}_g+\\sum^K_{i=1}\\lambda_i \\mathcal{L}_{i},\n\\end{equation}\n\nWe update the multiple discriminators while keeping the parameters of the generator fixed. Each discriminator is minimized using the loss described in Eq.12.\n}\n\n\n\n\n\n\\section{Preliminaries}\n\n\n\nConsider a set of monitoring stations $S=S^a\\cup S^w$, where $S^a=\\{s^a_i\\}^m_{i=1}$ and $S^w=\\{s^w_j\\}^n_{j=1}$ are respectively air quality and weather station sets.\nEach station $s_i \\in S$ is associated with a geographical location $l_i$~(\\emph{i.e.},\\xspace latitude and longitude) and a set of time-invariant contextual features $\\mathbf{c}_i \\in \\mathbf{C}$.\n\n\\begin{myDef}\n\\textbf{Observations}. Given a monitoring station $s_i \\in S$, the observations of $s_i$ at time step $t$ are defined as $\\mathbf{x}^{t}_i$, which is a vector of air-quality or weather conditions, depending on the station type. \n\\end{myDef}\n\nNote that the observations of two types of monitoring stations are different~(\\emph{e.g.},\\xspace PM2.5 and CO in air quality stations, while temperature and humidity in weather stations) and the observation dimensionality of the air quality station and the weather station are also different. \nWe use $\\mathbfcal{X}=\\{\\mathbf{X}^{a,t}\\}^T_{t=1} \\cup \\{\\mathbf{X}^{w,t}\\}^T_{t=1}$ to denote time-dependent observations of all stations in a time period $T$, use $\\mathbf{X}^{a,t}=\\{\\mathbf{x}^{a,t}_1, \\mathbf{x}^{a,t}_2, \\dots, \\mathbf{x}^{a,t}_{m}\\}$ and $\\mathbf{X}^{w,t}=\\{\\mathbf{x}^{w,t}_1, \\mathbf{x}^{w,t}_2, \\dots, \\mathbf{x}^{w,t}_{n}\\}$ to respectively denote observations of air quality and weather stations at time step $t$. We use $\\mathbf{X}_i=\\{\\mathbf{x}^{1}_i, \\mathbf{x}^{2}_i, \\dots, \\mathbf{x}^{T}_i\\}$ to denote all observations of station $s_i$.\nIn the following, without ambiguity, we will omit the superscript and subscript.\n\n\\eat{\n\\begin{myDef}\n\\textbf{Heterogeneous Station Graph}. A heterogeneous station graph (HSG) is defined as $\\mathcal{G}=\\{\\mathcal{V},\\mathcal{E}, \\phi \\}$, where $\\mathcal{V}=S$ is the set of monitoring stations, $\\phi$ is a mapping set indicates the type of each station, and $\\mathcal{E}$ is a set of edges indicating the connectivity among monitoring stations, defined as\n\\begin{equation}\ne_{ij}=\\left\\{\n\\begin{array}{lr}\n1,\\quad dist(s_i,s_j)<\\epsilon\\\\\n0,\\quad otherwise\n\\end{array}\n\\right.,\n\\end{equation}\nwhere $dist(\\cdot,\\cdot)$ is the spherical distance between station $s_i$ and station $s_j$, $\\epsilon$ is a distance threshold.\n\\end{myDef}\n}\n\n\\begin{problem}\n\\textbf{Joint air quality and weather predictions}. \nGiven a set of monitoring stations $S$, contextual features $\\mathbf{C}$, and historical observations $\\mathbfcal{X}$, our goal at a time step $t$ is to simultaneously predict air quality and weather conditions for all $s_i \\in S$ over the next $\\tau$ time steps,\n\\begin{equation}\n(\\hat{\\mathbf{Y}}^{t+1},\\hat{\\mathbf{Y}}^{t+2},...,\\hat{\\mathbf{Y}}^{t+\\tau}) \\leftarrow \\mathcal{F}(\\mathbfcal{X}, \\mathbf{C}),\n\\end{equation}\nwhere $\\hat{\\mathbf{Y}}^{t+1}=\\hat{\\mathbf{Y}}^{a,t+1} \\cup \\hat{\\mathbf{Y}}^{w,t+1}$ is the estimated target observations of all stations at time step $t+1$, and $\\mathcal{F}(\\cdot)$ is the mapping function we aim to learn. Note that $\\hat{\\mathbf{y}}^{t+1}_i \\in \\hat{\\mathbf{Y}}^{t+1}$ is also station type dependent, \\emph{i.e.},\\xspace air quality or weather observations for corresponding stations.\n\\end{problem}\n\n\\section{Related work}\n\n\\textbf{Air quality and weather prediction}.\nExisting literature on air quality and weather prediction can be categorized into two classes.\n(1) \\emph{Numerical-based models} make predictions by simulating the dispersion of various air quality or weather elements based on physical laws~\\cite{lorenc1986analysis,vardoulakis2003modelling,liu2005prediction,richardson2007weather}.\n(2) \\emph{Learning-based models} utilize end-to-end machine learning methods to capture spatiotemporal correlations based on historical observations and various urban contextual data~(\\emph{e.g.},\\xspace POI distributions, traffics)~\\cite{chen2011comparison,zheng2013u,cheng2018neural}.\nRecently, many deep learning models~\\cite{yi2018deep,wang2019deep,lin2018exploiting} have been proposed to enhance the performance of air quality and weather prediction.\nBy leveraging the representation capacity of deep learning for spatiotemporal autocorrelation modeling, learning-based models usually achieve better prediction performance than numerical-based models.\nUnlike the above approaches, our method explicitly models the correlations and interactions between the air quality and weather prediction task and achieves a better performance.\n\n\n\\textbf{Adversarial Learning}.\nAdversarial learning~\\cite{goodfellow2014generative} is an emerging learning paradigm for better capturing the data distribution via a minmax game between a generator and a discriminator.\nIn the past years, adversarial learning has been widely applied to many real-world application domains, such as sequential recommendation~\\cite{yu2017seqgan} and graph learning~\\cite{Wang2018GraphGANGR}.\nRecently, we notice several multi-adversarial frameworks have been proposed for improving image generation tasks~\\cite{nguyen2017dual,hoang2018mgan,albuquerque2019multi}.\nInspired by the above studies, we extend the multi-adversarial learning paradigm to the environmental science domain and introduce an adaptive training strategy to improve the stability of adversarial learning.\n\n\\eat{\nleverages a deep distributed fusion network for air quality prediction. The study in ~\\cite{grover2015deep} model the statistics of a set of weather-related variables via a hybrid approach based on deep neural networks. More recently, DUQ~\\cite{wang2019deep} introduced a deep uncertainty quantification method for weather forecasting.\n\nare widely used in the earlier study, such as ARIMA \\cite{chen2011comparison}, SVM~\\cite{wang2014research}, and artificial neural networks~\\cite{brunelli2007two}, they fail to capture spatiotemporal dynamics from the original data. With the development of deep learning techniques, many deep learning models are proposed to solve air quality and weather forecasting tasks. For example, DeepAir~\\cite{yi2018deep} leverages a deep distributed fusion network for air quality prediction. The study in ~\\cite{grover2015deep} model the statistics of a set of weather-related variables via a hybrid approach based on deep neural networks. More recently, DUQ~\\cite{wang2019deep} introduced a deep uncertainty quantification method for weather forecasting.\n\n\n\\subsection{Graph neural network.}\nRecent years deep learning on graphs has gained much attention, which promotes the development of graph neural networks~\\cite{kipf2016semi}. For each node on the graph, graph neural networks learn a function to aggregate features of its neighbours and generate new node embedding, which encodes both feature information and local graph structure. For example, graph attention network (GAT)~\\cite{velivckovic2017graph} uses self-attention mechanism to select important neighbours adaptively and then aggregate them with different weights. Besides, researchers also extend graph neural networks on heterogeneous graphs~\\cite{zhang2019heterogeneous}. Graph neural networks are widely used in many applications, such as traffic flow prediction~\\cite{li2017diffusion}, ride-hailing demand forecasting~\\cite{geng2019spatiotemporal} and parking availability prediction~\\cite{zhang2020semi}.\n}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\\\}\n\\newcommand{\\varepsilon}{\\varepsilon}\n\n\\numberwithin{equation}{section}\n\\newcommand{{\\nabla}}{{\\nabla}}\n\\newcommand{{\\mathrm{c.c.}}}{{\\mathrm{c.c.}}}\n\n\\newcommand{{\\veps}}{{\\varepsilon}}\n\n\n\\newcommand{\\mathsf{Sp}}{\\mathsf{Sp}}\n\\newcommand{\\mathsf{SU}}{\\mathsf{SU}}\n\\newcommand{\\mathsf{SL}}{\\mathsf{SL}}\n\\newcommand{\\mathsf{GL}}{\\mathsf{GL}}\n\\newcommand{\\mathsf{SO}}{\\mathsf{SO}}\n\\newcommand{\\mathsf{U}}{\\mathsf{U}}\n\\newcommand{\\mathsf{S}}{\\mathsf{S}}\n\\newcommand{\\mathsf{PSU}}{\\mathsf{PSU}}\n\\newcommand{\\mathsf{PSL}}{\\mathsf{PSL}}\n\\newcommand{\\mathsf{OSp}}{\\mathsf{OSp}}\n\\newcommand{\\mathsf{Spin}}{\\mathsf{Spin}}\n\\newcommand{\\mathsf{Mat}}{\\mathsf{Mat}}\n\\newcommand{\\mathsf{ISO}}{\\mathsf{ISO}}\n\n\n\\begin{document}\n\n\\begin{titlepage}\n\\begin{flushright}\nOctober, 2020 \\\\\n\\end{flushright}\n\\vspace{5mm}\n\n\\begin{center}\n{\\Large \\bf \nMassless particles in five and higher dimensions}\n\\end{center}\n\n\\begin{center}\n\n{\\bf Sergei M. Kuzenko and Alec E. Pindur} \\\\\n\\vspace{5mm}\n\n\\footnotesize{\n{\\it Department of Physics M013, The University of Western Australia\\\\\n35 Stirling Highway, Perth W.A. 6009, Australia}}\n~\\\\\n\\vspace{2mm}\nEmail: \\texttt{ \nsergei.kuzenko@uwa.edu.au, 21504287@student.uwa.edu.au}\\\\\n\n\n\\end{center}\n\n\\begin{abstract}\n\\baselineskip=14pt\nWe describe a five-dimensional analogue of Wigner's  operator equation \n${\\mathbb W}_a = \\lambda  P_a$, where ${\\mathbb W}_a $ is the Pauli-Lubanski vector, \n$P_a$ the energy-momentum operator, and $\\lambda$ the helicity of a massless particle. \nHigher dimensional generalisations are also given.\n\\end{abstract}\n\\vspace{5mm}\n\n\n\n\\vfill\n\n\\vfill\n\\end{titlepage}\n\n\\newpage\n\\renewcommand{\\thefootnote}{\\arabic{footnote}}\n\\setcounter{footnote}{0}\n\n\n\\allowdisplaybreaks\n\n\n\\section{Introduction}\n\nThe unitary representations of the Poincar\\'e group in four dimensions were classified \nby Wigner in 1939 \\cite{Wigner}, see \\cite{Weinberg95} for a recent review. Our modern understanding of elementary particles is based on this classification. \n\nUnitary representations of the Poincar\\'e group $\\mathsf{ISO}_0(d-1, 1)$ in higher dimensions,\n$d>4$, have been studied in the literature, see, e.g., \\cite{BB}. However, there still remain some aspects that are not fully understood, see, e.g., \\cite{Weinberg2020}\nfor a recent discussion. In this note we analyse the \nirreducible massless representations of $\\mathsf{ISO}_0(4,1)$ with a finite (discrete) spin.\n\nWe recall that the Poincar\\'e algebra $\\mathfrak{iso} (d-1,1) $  in $d$ dimensions is characterised  by the commutation relations\\footnote{We make use of the mostly plus  Minkowski metric $\\eta_{ab}$ and normalise the Levi-Civita tensor $\\ve_{a_1 \\dots a_d}$ \nby $\\ve_{0 1\\dots d-1}=1$.} \n\\begin{subequations} \\label{PA}\n\\begin{eqnarray}\n \\big[P_{a},  P_{b}\\big] & = & 0 ~, \\\\\n \\big[J_{ab},  P_{c}\\big]  & = & {\\rm i}\\eta_{ac}P_{b} - {\\rm i}\\eta_{bc}P_a ~,\\\\\n \\big[J_{a b}, J_{c d}\\big]  & = &  {\\rm i} \\eta_{a c}J_{b d} - {\\rm i} \\eta_{a d} J_{b c} + {\\rm i} \\eta_{b d} J_{a c} - {\\rm i} \\eta_{bc} J_{a d} ~.\n\\end{eqnarray}\n\\end{subequations}\nIn any unitary representation of (the universal covering group of) the Poincar\\'e group, \nthe energy-momentum operator\n$P_a$ and the Lorentz generators $J_{ab}$ are Hermitian.\nFor every dimension $d$, the operator $P^a P_a$ is a Casimir operator. Other Casimir operators are dimension dependent. \n\nIn four dimensions, the second Casimir operator is \n${\\mathbb W}^a{\\mathbb W}_a $,\nwhere \n\\begin{equation}\n{\\mathbb W}^a=\\frac12 \\ve^{abcd} { J}_{bc}{ P}_d\n\\label{PLv}\n\\end{equation}\nis the Pauli-Lubanski vector.\nUsing the commutation relations \n \\eqref{PA}, it follows that \n the Pauli-Lubanski vector is translationally invariant, \n\\begin{subequations}\n\\begin{eqnarray}\n\\big[ P_a, {\\mathbb W}_b \\big]=0~,\n\\end{eqnarray}\nand possesses the following properties:\n\\begin{eqnarray}\n{\\mathbb W}^a{ P}_a&=&0 ~,  \\\\\n\\big[{ J}_{ab},{\\mathbb W}_c \\big]&=&{\\rm i} \\eta_{ac}{\\mathbb W}_b\n-{\\rm i}  \\eta_{bc}{\\mathbb W}_a~,\n\\\\ \n\\big[ {\\mathbb W}_a,{\\mathbb W}_b \\big]&=& {\\rm i}  \\ve_{abcd}{\\mathbb W}^c{P}^d~. \n\\end{eqnarray}\n\\end{subequations}\nThe irreducible massive representations are characterised by the conditions \n\\begin{subequations}\n\\begin{eqnarray}\n { P}^a{ P}_a &=&-m^2\\,{\\mathbbm 1}~, \\qquad m^2 > 0~, \n \\qquad \n {\\rm sign } \\,{ P}^0 >0~, \\label{1.4a}  \\\\\n {\\mathbb W}^a{\\mathbb W}_a &=& m^2s(s+1)\\,{\\mathbbm 1}~,  \n \\end{eqnarray} \n \\end{subequations}\nwhere the quantum number $s$ is called spin. Its possible values \nin different representations are \n$s= 0, 1\/2, 1, 3\/2 \\dots$. \nThe massless representations are characterised by the condition $P^aP_a =0$. \nFor the physically interesting massless representations, it holds that \n\\begin{eqnarray}\n{\\mathbb W}_a = \\lambda P_a~,\n\\label{1.5}\n\\end{eqnarray}\nwhere the parameter $\\lambda$ determines the representation and is called the helicity. Its possible values are $0, \\pm\\frac12, \\pm 1, $ and so on. The parameter $|\\lambda |$ is called the spin of a massless particle.\n\nIn this paper we present a generalisation of Wigner's equation \\eqref{1.5}\nto five and higher dimensions.\n\n\n\\section{Unitary representations of $\\mathsf{ISO}_0(4,1)$}\n\nThe five-dimensional analogue of \\eqref{PLv} is the  Pauli-Lubanski tensor\n\\begin{equation}\n\\mathbb{W}^{ab} = \\frac{1}{2}\\ve^{abcde}J_{cd}P_e~.\n\\label{2.1}\n\\end{equation}\nIt is translationally invariant, \n\\begin{eqnarray}\n\\big[  {\\mathbb W}_{ab} , P_c \\big] =0~,\n\\end{eqnarray}\nand possesses the following properties:\n\\begin{subequations}\n\\begin{eqnarray}\n\\mathbb{W}_{ab} P^b &=& 0~, \\\\\n\\big[\\mathbb{W}_{ab}, J_{cd}\\big] &=& {\\rm i} \\eta_{ac}\\mathbb{W}_{bd} \n- {\\rm i} \\eta_{ad}\\mathbb{W}_{bc}  - {\\rm i} \\eta_{bc}\\mathbb{W}_{ad} + {\\rm i} \\eta_{bd}\\mathbb{W}_{ac}~, \\\\\n \\big[\\mathbb{W}_{ab}, \\mathbb{W}_{cd}\\big] \n &=& {\\rm i}\\ve_{acdfg} {\\mathbb{W}_b}^f  P^{g} - {\\rm i} \\ve_{bcdfg} {\\mathbb{W}_a}^f P^{g}~.\n\\end{eqnarray}\n\\end{subequations}\nMaking use of ${\\mathbb W}_{ab}$ allows one to construct two Casimir operators, which are \n\\begin{equation}\n\\mathbb{W}_{ab} \\mathbb{W}^{ab}~,  \\qquad \n\\mathbb{H} := \\mathbb{W}^{ab}J_{ab}~.\n\\label{2.4}\n\\end{equation}\n\n\n\\subsection{Irreducible massive representations} \n\nThe irreducible massive representations of the Poincar\\'e group $\\mathsf{ISO}_0(4,1)$\nare characterised by two conditions\n\\begin{subequations}\\label{2.5}\n\\begin{eqnarray}\n \\frac{1}{8} \\Big( \\mathbb{W}^{ab}\\mathbb{W}_{ab} + m \\mathbb{H} \\Big) &=& \n m^2 s_1(s_1 + 1)\\mathbbm{1} ~,  \\\\\n\\frac{1}{8} \\Big( \\mathbb{W}^{ab}\\mathbb{W}_{ab} - m\\mathbb{H} \\Big) &=& \nm^2 s_2(s_2 + 1)\\mathbbm{1} ~,\n\\end{eqnarray}\n\\end{subequations}\nin addition to \\eqref{1.4a}.\nHere $s_1$ and $s_2$ are two spin values corresponding to the two $\\mathsf{SU}(2)$ subgroups of the universal covering group \n$\\mathsf{Spin}(4) \\cong \\mathsf{SU}(2) \\times \\mathsf{SU}(2)$\nof the little group.\\footnote{The equations \\eqref{2.5} were independently derived during the academic year 1992-93 by Arkady Segal and David Zinger, who were undergraduates at Tomsk State University at the time.}\n\n\n\\subsection{Irreducible massless representations} \\label{section2.2}\n\nIt turns out that all irreducible massless representations of \n$\\mathsf{ISO}_0(4,1)$ with a finite spin are characterised by the condition \n\\begin{eqnarray}\n\\varepsilon_{abcde}P^{c}\\mathbb{W}^{de} =0 \\quad \\Longleftrightarrow \\quad\nP^{[a} {\\mathbb W}^{bc]} =0~.\n\\label{main}\n\\end{eqnarray}\nBoth Casimir operators \\eqref{2.4} are equal to zero in these representations, \n$\\mathbb{W}_{ab} \\mathbb{W}^{ab}=0$  and   \n$ \\mathbb{W}^{ab}J_{ab} =0$.\n\n\nLet $ \\ket{p, \\sigma} $ be an orthonormal basis in the Hilbert space of one-particle states, where $p^a $ \ndenotes the momentum of a particle, $ P^a \\ket{p, \\sigma} = p^a  \\ket{p, \\sigma} $,\nand $\\sigma$ stands for the spin degrees of freedom.\nFor a massless particle, we choose as our standard 5-momentum $k^{a} = (E,0,0,0,E)$. On this eigenstate:\n\\begin{equation}\n\\mathbb{W}^{ab} \\ket{k, \\sigma} = \\frac{1}{2}\\varepsilon^{abcde}J_{cd}P_{e} \\ket{k, \n\\sigma} = \\frac{E}{2} \\left(\\varepsilon^{abcd4}J_{cd} - \\varepsilon^{abcd0}J_{cd} \\right) \\ket{k, \\sigma}~.\n\\end{equation}\nRunning through the elements of $\\mathbb{W}^{ab}$, one finds:\n\\begin{equation} \\label{eq:eq5}\n\\begin{split}\n\\mathbb{W}^{01} = \\mathbb{W}^{41} = -EJ_{23} ~,\\qquad & \\mathbb{W}^{12} = E(J_{30} +J_{34}) ~,\\\\\n\\mathbb{W}^{02} = \\mathbb{W}^{42} = -EJ_{31} ~,\\qquad & \\mathbb{W}^{23} = E(J_{10} +J_{14}) ~,\\\\\n\\mathbb{W}^{03} = \\mathbb{W}^{43} = -EJ_{12}~, \\qquad & \\mathbb{W}^{31} = E(J_{20} +J_{24}) ~,\\\\\n\\mathbb{W}^{04} = & \\ 0 ~.\n\\end{split} \n\\end{equation}\nIf we rescale these generators and define:\n\\begin{equation}\\label{eq:eq7}\n\\begin{split}\n\\mathbb{R}_1  \\equiv \\frac{1}{E}\\mathbb{W}^{23}~, \\qquad \\mathbb{R}_{2} \\equiv &  \\frac{1}{E}\\mathbb{W}^{31}~,  \\qquad \\mathbb{R}_{3} \\equiv \\frac{1}{E}\\mathbb{W}^{12} ~,\\\\\n\\qquad \\cJ_{i} \\equiv &-  \\frac{1}{E} \\mathbb{W}^{0i} ~,\n\\end{split}\n\\end{equation}\nthen these new operators satisfy:\n\\begin{equation}\n\\big[\\cJ_{i}, \\cJ_{j} \\big] = {\\rm i}\\varepsilon_{ijk}\\cJ_k~, \\qquad \n\\big[\\cJ_i , \\mathbb{R}_j \\big] = {\\rm i}\\varepsilon_{ijk} \\mathbb{R}_k~ , \n\\qquad \\big[\\mathbb{R}_i , \\mathbb{R}_j \\big] = 0~.\n\\end{equation}\nThese are the commutation relations for the three-dimensional Euclidean algebra, $\\mathfrak{iso}(3)$. The irreducible unitary representations of $\\mathfrak{iso}(3)$ are labelled by a continuous parameter $\\mu^2$, corresponding to the value the Casimir operator $\\mathbb{R}^{i} \\mathbb{R}_{i}$ takes. Since $\\mathbb{R}_{i}$ commute among themselves the operators can be simultaneously diagonalised, and the eigenvectors $\\ket{r_{i}}$ taken as a basis. However the only restriction on these is that $r_i r^i = \\mu^2$, which for non-zero $\\mu^2$ permits a continuous basis and is thus an infinite dimensional representation. Because we want only finite-dimensional representations, we must take:\n\\begin{equation} \\label{eq:eq6}\n\\mu^2 = 0 \\quad \\Longrightarrow \\quad \\mathbb{R}_{i} = 0 \\quad \\Longleftrightarrow \\quad J_{0i} = -J_{4i}~.\n\\end{equation}\nWe are therefore restricted to those representations in which the translation component is trivial, and so only the generators $\\cJ_i$ remain, which generate the algebra $\\mathfrak{so}(3)$. The algebra of the little group on massless representations is thus $\\mathfrak{so}(3) $ which is isomorphic to $\\mathfrak{su}(2)$. \nAs stated previously, the irreducible representations of $\\mathfrak{su}(2)$ are labelled  by a non-negative (half) integer $s$ and have a single Casimir operator $\\cJ^i \\cJ_i$ which takes the value $s(s+1)\\mathbbm{1}$. This analysis leads to \\eqref{main}.\n\nThe spin value of a massless representation can still be found using a `spin' operator.\nThe following relation holds on massless representations:\n\\begin{equation}\n \\mathbb{S}_a :=- \\frac{1}{4} \\varepsilon_{abcde} J^{bc} \\mathbb{W}^{de}  \n = \\cJ^2 P_{a} = s(s+1)P_a~,\n \\label{2.12}\n\\end{equation}\nwhere $\\cJ^2 = \\cJ^{i}\\cJ_{i}$ is the Casimir operator for the $\\mathfrak{so}(3)$ generators in \\eqref{eq:eq7}. The parameter $s$ is the spin of a massless particle.\nIts possible values in different representations are \n$s= 0, 1\/2, 1, $ and so on. Equation \\eqref{2.12} naturally holds for massless  spinor and vector fields \\cite{Pindur}.\n\n\n\nIn general, the operator ${\\mathbb S}_a$ is not translationally invariant,\n\\begin{equation}\n\\big[ \\mathbb{S}_b, P_{a} \\big] = \\frac{{\\rm i}}{2}\\varepsilon_{abcde}P^{c}\\mathbb{W}^{de} ~.\n\\end{equation}\nIt is only for the massless representations with finite spin that  the quantity on the right vanishes so that the spin operator commutes with the momentum operators. \nEquation \\eqref{2.12} is the five-dimensional analogue of the operator equation \\eqref{1.5}. Its consistency condition is \\eqref{main}.\\footnote{The consistency condition for \\eqref{1.5} is  $P^{[a} {\\mathbb W}^{b]} =0$, which is the four-dimensional counterpart of \\eqref{main}.}\n\n\n\\section{Generalisations} \n\nThe results of section \\ref{section2.2} can be generalised to $d>5$ dimensions. \nThe Pauli-Lubanski tensor \\eqref{2.1} turns into\n\\begin{eqnarray}\n\\mathbb{W}^{a_1 \\dots a_{d-3}} = \\frac{1}{2}\\ve^{a_1 \\dots a_{d-3} bc e}J_{bc}P_e~.\n\\end{eqnarray}\nThe condition \\eqref{main} is replaced with\n\\begin{eqnarray}\nP^{[a} {\\mathbb W}^{b_1 \\dots b_{d-3}]} =0~.\n\\label{maind}\n\\end{eqnarray}\nThis equation is very similar to another that has appeared in the literature\nusing the considerations of conformal invariance \n\\cite{Bracken,BrackenJ,Siegel,SiegelZ}. One readily checks that \\eqref{maind} is equivalent to \n\\begin{eqnarray}\nJ_{ab} P^2 + 2 J_{c[a} P_{b]} P^c =0 \\quad \\implies \\quad \nJ_{c[a} P_{b]} P^c =0 ~.\n\\label{maind2}\n\\end{eqnarray}\nThe latter is solved on the momentum eigenstates \nby $J_{ab} p^b \\propto p_a$, which is of the form considered\nin \\cite{Bracken,BrackenJ,Siegel,SiegelZ}.\\footnote{We are grateful to Warren Siegel for useful comments.}\n\n\nEquation \\eqref{maind} characterises all irreducible massless representations of \n$\\mathsf{ISO}_0(d-1,1)$ with a finite (discrete) spin. Finally, the spin equation \\eqref{2.12}\nturns into \n\\begin{eqnarray}\n \\mathbb{S}_a := \\frac{(-1)^d}{2 (d-3)!} \\varepsilon_{abc e_1 \\dots e_{d-3}} J^{bc} \\mathbb{W}^{e_1 \\dots e_{d-3}}  \n = \\cJ^2 P_{a} ~,\n\\end{eqnarray}\nwhere $\\cJ^2 = \\frac12 \\cJ^{ij} \\cJ_{ij} $ is the quadratic Casimir operator of  the algebra $\\mathfrak{so}(d-2)$, with $i,j=1, \\dots , d-2$.\nFor every irreducible massless representation of \n$\\mathsf{ISO}_0(d-1,1)$ with a finite spin, it holds that $\\cJ^2 \\propto {\\mathbbm 1}$. \n\nIt is possible to derive a five-dimensional analogue of the operator equation \ndefining the $\\cN=1$ superhelicity $\\kappa$  in four dimensions \\cite{BK}. \nThe latter has the form\\footnote{In the supersymmetric case, the conventions of \\cite{BK} are used, \nin particular the Levi-Civita tensor $\\ve_{abcd}$ is normalised by $\\ve_{0123}=-1$.}  \n\\begin{eqnarray}\n{\\mathbb L}_a = \\Big( \\kappa +\\frac 14 \\Big) P_a~, \n\\label{3.4}\n\\end{eqnarray}\nwhere the operator ${\\mathbb L}_a$ is defined by \n\\begin{eqnarray}\n{\\mathbb L}_a = {\\mathbb W}_a - \\frac{1}{16} (\\tilde \\sigma_a)^{\\ad \\alpha} \\big[ Q_\\alpha , \\bar Q_\\ad\\big] ~.\n\\label{3.5}\n\\end{eqnarray}\nThe fundamental properties of the operator ${\\mathbb L}_a $\n(the latter differs from the supersymmetric Pauli-Lubanksi vector \\cite{SS})\nare that it is translationally invariant and commutes with the supercharges $Q_\\alpha$ and $\\bar Q_\\ad$ in the massless representations of the $\\cN=1$ super-Poincar\\'e group.\\footnote{The irreducible massless\n representation of superhelicity $\\kappa$ is the direct sum of two irreducible massless Poincar\\'e representations corresponding to the helicity values $\\kappa$ and $\\kappa+\\frac12$.} \nThe superhelicity operator \\eqref{3.5} was generalised to higher dimensions\nin \\cite{PZ,AMT}. Generalisations of \\eqref{3.4} to five and higher dimension will be discussed elsewhere.\n\\\\\n\n\\noindent\n{\\bf Acknowledgements:}\\\\\nWe thank Warren Siegel for pointing out important references, and Michael Ponds for comments on the manuscript. \nSMK is grateful to Ioseph Buchbinder for email correspondence, and to Arkady Segal for discussions. The work of SMK work is supported in part by the Australian \nResearch Council, project No. DP200101944.\n\n\n\\begin{footnotesize}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecent advances on information technologies facilitate real-time message exchanges and decision-making among geographically dispersed strategic entities. This has boosted the emergence of new generation of networked systems; e.g., the smart grid and intelligent transportation systems. These networked systems share some common features: on one hand, the entities do not belong to a single authority and may pursue different or even competitive interests; on the other hand, each entity keeps private information which is unaccessible to others. It is of great interest to design practical mechanisms which allow for efficient coordination of self-interested entities and ensure network-wide performance. Game theory along with its distributed computation algorithms represents a promising tool to achieve the goal.\n\nIn many applications, distributed computation is executed in uncertain environments. For example, mobile robots are deployed in an operating environment where environmental distribution functions are unknown to robots in advance; e.g.,~\\cite{Stankovic.Johansson.Stipanovic:11,Zhu.Martinez:SICON13}. In traffic pricing, pricing policies of system operators may not be available to drivers. In optimal power flow control, the structural parameters of power systems are of national security interest and kept confidential from the public. The absence of such information makes game components; e.g., objective and constraint functions, inaccessible to players. Very recently, the informational constraint has been stimulating the investigation of \\emph{adaptive} algorithms, including~\\cite{Frihauf.Krstic.Basar:11,SL-MK:11,Stankovic.Johansson.Stipanovic:11} for continuous games and~\\cite{JRM-HPY-GA-JSS:06,Zhu.Martinez:SICON13} for discrete games.\n\n\\emph{Literature review.} Non-cooperative game theory has been widely used as a mathematical framework to reason about multiple selfish decision makers; see for instance~\\cite{Basar.Olsder:82}. These games\nhave found a variety of applications in economics, communication and robotics; see~\\cite{Altman.Basar:98,Chung.Hollinger.Isler:11,Dockner.Jorgensen.Long.Sorger:06,Frihauf.Krstic.Basar:11,Mitchell.Bayen.Tomlin:05}. In non-cooperative games, decision making of individuals is inherently distributed. Very recently, this attractive feature has been utilized to synthesize cooperative control schemes, and a partial reference list for this regard includes~\\cite{Arsie.Savla.ea:TAC09,Arslan.Marden.Shamma:07a,Li.Marden:11,Stankovic.Johansson.Stipanovic:11,Zhu.Martinez:SICON13}.\n\nThe set of papers more relevant to our work is concerned with generalized Nash games where strategy spaces are continuous and the actions of players are coupled through utility and constraint functions. Generalized Nash games are first formulated in~\\cite{Arrow.Debreu:54}. Since then, a great effort has been dedicated to studying the existence and structural properties of generalized Nash equilibria in; e.g.,~\\cite{Rosen:65} and the recent survey paper~\\cite{Facchinei.Kanzow:07}. A number of algorithms have been proposed to compute generalized Nash equilibria, including ODE-based methods~\\cite{SL-TB:87,Rosen:65}, nonlinear Gauss-Seidel-type approaches~\\cite{Pang.Scutari.Facchinei.Wang:08}, iterative primal-dual Tikhonov schemes~\\cite{Yin.Shanbhag.Mehta:11} and best-response dynamics~\\cite{Palomar.Eldar:10}.\n\nAs mentioned, the set of papers~\\cite{Frihauf.Krstic.Basar:11,SL-MK:11,JRM-HPY-GA-JSS:06,Stankovic.Johansson.Stipanovic:11,Zhu.Martinez:SICON13}\ninvestigates the \\emph{adaptiveness} of game theoretic learning algorithms. However, none of the papers mentioned in the last two paragraphs studies the \\emph{robustness} of the algorithms with respect to network unreliability; e.g., data transmission delays, quantization and dynamically changing topologies. In contrast, the robustness has been extensively studied for consensus and distributed optimization, including, to name a few,~\\cite{Jadbabaie.Lin.Morse:03,Nedic.Ozdaglar.Parrilo:08} for time-varying topologies,~\\cite{Rabbat.Nowak:05} for quantization and~\\cite{Munz.Papachristodoulou.Allgower:10} for time delays. Yet the adaptiveness issue has not been addressed in this group of papers.\n\n\n\\emph{Contributions.} In this paper, we aim to solve a class of generalized convex games over unreliable networks where the structures of component functions are unknown to the associated players. That is, we aim to simultaneously address the issues of adaptiveness and robustness for generalized convex games.\n\nIn the games, each player is associated with a convex objective function and subject to a private convex inequality constraint and a private convex constraint set. The component functions are assumed to be smooth and are unknown to the associated players. We investigate distributed first-order gradient-based computation algorithms for the following two scenarios:\n\n[Scenario One] The game map is pseudo-monotone and the maximum delay (equivalently, the maximum number of packet dropouts or link breaks) is bounded but unknown;\n\n[Scenario Two] The inequality constraints are absent, the (reduced) game map is strongly monotone and the maximum delay is known.\n\nInspired by simultaneous perturbation stochastic approximation for optimization in~\\cite{JS:03}, we utilize finite differences with diminishing approximation errors to estimate first-order gradients. We propose two distributed algorithms for the two scenarios and formally prove their asymptotic convergence. The comparison of the two proposed algorithms is given in Section~\\ref{sec:comparison}. The analysis integrates the tools from convex analysis, variational inequalities and simultaneous perturbation stochastic approximation. The algorithm performance is verified through demand response on the IEEE 30-bus Test System. A preliminary version of the current paper was published in~\\cite{Zhu.Frazzoli:CDC12} where the adaptiveness issue was not investigated.\n\n\n\\section{Problem formulation}\\label{sec:formulation}\n\nIn this section, we present the generalized convex game considered in the paper. It is followed by the notions and notations used throughout the paper.\n\n\\subsection{Generalized convex game}\n\nConsider the set of players $V \\triangleq \\{1,\\cdots,N\\}$ where the state of player~$i$ is denoted as $x^{[i]}\\in X_i\\subseteq \\mathds{R}^{n_i}$. The players are selfish and pursue different interests. In particular, given the joint state $x^{[-i]}\\in X_{-i} \\triangleq \\prod_{j\\neq i}X_j$ of its rivals\\footnote{We use the shorthand $-i\\triangleq V\\setminus\\{i\\}$ throughout the paper.}, each player~$i$ aims to solve the following program parameterized by $x^{[-i]}\\in X_{-i}$: \\begin{align}\\min_{x^{[i]}\\in X_i}f_i(x^{[i]},x^{[-i]}),\\quad {\\rm s.t.} \\quad G^{[i]}(x^{[i]},x^{[-i]})\\leq0,\\label{e10}\\end{align} where $f_i : \\mathds{R}^n\\rightarrow \\mathds{R}$ and $G^{[i]} : \\mathds{R}^n\\rightarrow\\mathds{R}^{m_i}$ with $n\\triangleq \\sum_{i\\in V}n_i$. In the remainder of the paper, we assume that the following properties about problem~\\eqref{e10} hold:\n\n\\begin{assumption} The maps  $f_i$ and $G^{[i]}$ are smooth, and the maps  $f_i(\\cdot,x^{[-i]})$ and $G^{[i]}(\\cdot,x^{[-i]})$ are convex in $x^{[i]}$. The set $X_i$ is convex and compact, and $X\\cap Y\\neq\\emptyset$ where $X \\triangleq \\prod_{i\\in V}X_i$ and $Y \\triangleq \\prod_{i\\in V}Y_i$ with $Y_i \\triangleq \\{x\\in X\\; |\\; G^{[i]}(x) \\leq 0\\}$.\\label{asm7}\n\\end{assumption}\n\n\n\nWe now proceed to provide an equivalent form of problem~\\eqref{e10}. To achieve this, we define the set-valued map $X_i^f : X_{-i} \\rightarrow 2^{X_i}$ as follows: \\begin{align*}X_i^f(x^{[-i]}) = \\{x^{[i]}\\in X_i \\; | \\; G^{[i]}(x^{[i]},x^{[-i]})\\leq 0\\}.\\end{align*} The set $X_i^f(x^{[-i]})$ represents the collection of feasible actions for player~$i$ when its opponents choose the joint state of $x^{[-i]}\\in X_{-i}$. With the map $X_i^f$, problem~\\eqref{e10} of player~$i$ is equivalent to the following one: \\begin{align}\\min_{x^{[i]}\\in X_i^f(x^{[-i]})}f_i(x^{[i]},x^{[-i]}).\\label{e11}\\end{align}\n\nGiven $x^{[-i]}\\in X_{-i}$, each player~$i$ aims to solve problem~\\eqref{e11}. The collection of such coupled optimization problems consists of the \\emph{generalized convex game} (for short, CVX). For the CVX game, we adopt the \\emph{generalized Nash equilibrium} (for short, GNE) as the solution notion which none of the players is willing to unilaterally deviate from:\n\n\\begin{definition} The joint state $\\tilde{x}\\in X\\cap Y$ is a generalized Nash equilibrium of the CVX game if the following holds: \\begin{align}&f_i(\\tilde{x})\\leq f_i(x^{[i]},\\tilde{x}^{[-i]}),\\quad \\forall x^{[i]}\\in X_i^f(\\tilde{x}^{[-i]}),\\quad \\forall i\\in V.\\nonumber\\end{align}\\label{def1}\n\\end{definition}\n\nDenote by $\\mathbb{X}_{\\rm CVX}$ the set of GNEs of the CVX game. The following lemma verifies the non-emptiness of $\\mathbb{X}_{\\rm CVX}$.\n\n\\begin{lemma} The set of generalized Nash equilibria of the CVX game is not empty, i.e., $\\mathbb{X}_{\\rm CVX} \\neq \\emptyset$.\\label{lem3}\n\\end{lemma}\n\n\\textbf{Proof:} Recall that $f_i$ is convex and $X\\cap Y$ is compact. Hence, $\\mathbb{X}_{\\rm CVX} \\neq \\emptyset$ is a direct result of~\\cite{Facchinei.Kanzow:07,Rosen:65}. \\oprocend\n\n\nIn the CVX game, the players desire to seek a GNE. It is noted that $f_i$, $G^{[i]}$, and $X_i$ are private information of player~$i$ and unaccessible to others. In order to compute a GNE, it becomes necessary that the players are inter-connected and able to communicate with each other to exchange their partial estimates of GNEs. The interconnection between players will be represented by a directed graph ${\\mathcal{G}} = (V,\\mathcal{E})$ where\n$\\mathcal{E}\\subset V\\times V\\setminus {\\rm diag}(V)$ is the set of\nedges. The neighbor relation is determined by the dependency of $f_i$ and\/or $G^{[i]}$ on $x^{[j]}$. In particular, $(i,j)\\in\\mathcal{E}$ if and only if $f_i$ and\/or $G^{[i]}$ depend upon $x^{[j]}$. Denote by $\\mathcal{N}_i^{\\rm IN} \\triangleq \\{j\\in V\\;|\\; (i,j)\\in\\mathcal{E}\\}$ the set of in-neighbors of\nplayer~$i$.\n\nIn this paper, we aim to develop distributed algorithms which allow for the computation of GNEs in the presence of the following two challenges.\n\\begin{enumerate}\n\\item Data transmissions between the players in $V$ are unreliable, and subject to transmission delays, packet dropouts and\/or link breaks. We let $x^{[j]}(k - \\tau^{[i]}_j(k))$ with $\\tau^{[i]}_j(k)\\geq0$ be either $(i)$ the outdated state of player~$j$ received by player~$i$ at time~$k$ due to transmission delays; or $(ii)$ the latest state of player~$j$ received by player~$i$ by time~$k$ due to packet dropouts and\/or link breaks. For example, if the link from player $j$ to player $i$ is broken at time $k$, then player~$i$ cannot receive the message $x^{[j]}(k)$ from player $j$. For this case, player~$i$ uses the more recent received message sent from player $j$ which is $x^{[j]}(k-\\tau_j^{[i]}(k))$. Denote by $\\Lambda_i(k)\\triangleq \\{x^{[j]}(k-\\tau^{[i]}_j(k))\\}_{j\\in\\mathcal{N}^{\\rm IN}_i}$ the set of latest states of $\\mathcal{N}_i^{\\rm IN}$ received by player~$i$ at time~$k$. Let $\\tau_{\\max} \\triangleq \\sup_{k\\geq0}\\max_{i\\in V}\\max_{j\\in\\mathcal{N}_i^{\\rm IN}}\\tau^{[i]}_j(k)$ be the maximum delay or the maximum number of successive packet dropouts.\n\\item Each player~$i$ is unaware of the structures of $f_i$ and $G^{[i]}$ but can observe their realizations. That is, if the players input the joint state $x$ into $f_i$, then player~$i$ can observe the realized value of $f_i(x)$.\n\\end{enumerate}\n\nThe disclosure of the value $f_i(x)$ is case dependent. In~\\cite{Stankovic.Johansson.Stipanovic:11,Zhu.Martinez:SICON13}, mobile sensors are unaware of environmental distribution functions but they can observe induced utilities via on-site measurements. This is an example of engineering systems. Section~\\ref{sec:simulations} will provide a concrete example of demand response of power networks where the system operator discloses realized values via communication given the decisions of end-users. This is an example of social systems.\n\n\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\nIn the CVX game, each player is subject to an inequality constraint. In optimization literature, Lagrangian relaxation is a widely used approach to handle inequality constraints. Following this vein, we will perform Lagrangian relaxation on the CVX game and obtain the unconstrained convex (for short, UC) game.\n\nIn the UC game, there are two sets of players: the set of primal players $V$ and the set of dual players $V_m \\triangleq \\{1,\\cdots,N\\}$. Define the following private Lagrangian $\\mathcal{H}_i : \\mathds{R}^n\\times\\mathds{R}^{m_i}_{\\geq0}\\rightarrow\\mathds{R}$: $\\mathcal{H}_i(x,\\mu^{[i]}) \\triangleq f_i(x) + \\langle \\mu^{[i]}, G^{[i]}(x)\\rangle$, for primal player~$i\\in V$ and dual player~$i\\in V_m$. Given any $x^{[-i]}\\in X_{-i}$ and $\\mu^{[i]}\\in\\mathds{R}^{m_i}_{\\geq0}$, each primal player~$i\\in V$ aims to minimize $\\mathcal{H}_i$ over $x^{[i]}\\in X_i$; i.e., $\\min_{x^{[i]}\\in X_i}\\mathcal{H}_i(x^{[i]},x^{[-i]},\\mu^{[i]})$, and, instead, the objective of the dual player $i\\in V_m$ is to maximize $\\mathcal{H}_i$ over $\\mu^{[i]}\\in M_i\\subseteq\\mathds{R}^{m_i}_{\\geq0}$; i.e., $\\min_{\\mu^{[i]}\\in M_i}-\\mathcal{H}_i(x,\\mu^{[i]})$ where the set $M_i\\subseteq\\mathds{R}^{m_i}_{\\geq0}$ is convex, non-empty and will be introduced in the sequel. We let $\\eta \\triangleq (x,\\mu)$ and the set $K \\triangleq X\\times M$ with $M \\triangleq \\prod_{i\\in V}M_i$. The above game among the players in $V\\cup V_m$ is referred to as the UC game parameterized by the set $M$. The set of $M$ will play an important role in determining the properties of the UC game and we will discuss the choice of $M$ later. The solution concept for the UC game parameterized by $M$ is the standard notion of Nash equilibrium given below:\n\n\\begin{definition} The joint state of $(\\tilde{x},\\tilde{\\mu}) \\in X\\times M$ is a Nash equilibrium of the UC game parameterized by $M$ if the following holds for all $i\\in V$: $\\mathcal{H}_i(\\tilde{x},\\tilde{\\mu}^{[i]})\\leq \\mathcal{H}_i(x^{[i]},\\tilde{x}^{[-i]},\\tilde{\\mu}^{[i]}),\\quad \\forall x^{[i]}\\in X_i$, and the following holds for all $i\\in V_m$: $\\mathcal{H}_i(\\tilde{x},\\mu^{[i]})\\leq \\mathcal{H}_i(\\tilde{x},\\tilde{\\mu}^{[i]}),\\quad \\forall \\mu^{[i]}\\in M_i$.\\label{def2}\n\\end{definition} Denote by $\\mathbb{X}_{\\rm UC}(M)$ the set of NEs of the UC game parameterized by $M$. We now proceed to illustrate the relation between the UC game and the CVX game. Before doing so, let us state the following boundedness assumption on the dual solutions:\n\n\\begin{assumption} There is a vector $\\vartheta \\triangleq (\\vartheta_i)_{i\\in V}\\in\\mathds{R}^N_{>0}$ such that for any $(\\tilde{x},\\tilde{\\mu})\\in \\mathbb{X}_{\\rm UC}(\\mathds{R}^m_{\\geq0})$,  $\\|\\tilde{\\mu}^{[i]}\\| \\leq \\vartheta_i$ for $i\\in V$.\\label{asm1}\n\\end{assumption}\n\n\\begin{remark} We would like to make a remark on Assumption~\\ref{asm1}. For convex optimization, it is shown in~\\cite{Hiriart.Lemarechal:96} that the Lagrangian multipliers are uniformly bounded under the standard Slater's condition. This boundedness property is used in~\\cite{Nedic.Ozdagalr:08} and further in~\\cite{Zhu.Martinez:09tac,Zhu.Martinez:10tac} to solve convex and non-convex programs in centralized and distributed manners. In this paper, Assumption~\\ref{asm1}, on one hand, ensures the existence of GNEs, and on the other hand, guarantees the boundedness of estimates and gradients for the case of unknown $\\tau_{\\max}$. We will discuss the verification of Assumption~\\ref{asm1} in Section~\\ref{sec:discussion}.\\oprocend\\label{rem6}\n\\end{remark}\n\nThe following proposition characterizes the relations between the UC game and the CVX game.\n\n\\begin{proposition} The following properties hold:\n\\begin{enumerate}\n\\item[(P1)] Consider any $(\\tilde{x}, \\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}(M)$. We have $\\tilde{x}\\in \\mathbb{X}_{\\rm CVX}$ if the following properties hold:\n\\begin{itemize}\n\\item  feasibility; i.e., $G^{[i]}(\\tilde{x})\\leq0$ for all $i\\in V$;\n\\item  slackness complementarity; i.e., $\\langle \\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle = 0$ for all $i\\in V$.\n\\end{itemize}\n\\item[(P2)] Suppose Assumption~\\ref{asm1} holds. Consider the UC game parameterized by $M$ with $M_i \\triangleq \\{\\mu^{[i]}\\in\\mathds{R}^{m_i}_{\\geq0}\\;|\\;\\|\\mu^{[i]}\\|\\leq \\vartheta_i+r_i\\}$ for some $r_i>0$. Then $\\mathbb{X}_{\\rm UC}(M)\\neq\\emptyset$. In addition, for any $(\\tilde{x}, \\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}(M)$, it holds that $\\tilde{x}\\in \\mathbb{X}_{\\rm CVX}$.\n\\end{enumerate} \\label{pro5}\n\\end{proposition}\n\n\\textbf{Proof:} The proof of (P1) is a slight extension of the results in~\\cite{Bertsekas:09} to our game setup. For the sake of completeness, we summarize the analysis here. Since $\\langle \\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle = 0$, we have the following relation: \\begin{align}f_i(\\tilde{x}) = \\mathcal{H}_i(\\tilde{x},\\tilde{\\mu}^{[i]}) - \\langle\\tilde{\\mu}^{[i]},G^{[i]}(\\tilde{x})\\rangle = \\mathcal{H}_i(\\tilde{x},\\tilde{\\mu}^{[i]}).\\label{e7}\\end{align} Since $(\\tilde{x},\\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}(M)$, then the following relation holds for all $x^{[i]}\\in X_i$: \\begin{align}\\mathcal{H}_i(\\tilde{x},\\tilde{\\mu}^{[i]})\n-\\mathcal{H}_i(x^{[i]},\\tilde{x}^{[-i]},\\tilde{\\mu}^{[i]}) \\leq 0.\\label{e17}\\end{align} Substitute~\\eqref{e7} into~\\eqref{e17}, and it renders the following for any $x^{[i]}\\in X_i$:\n\\begin{align}f_i(\\tilde{x}) \\leq f_i(x^{[i]},\\tilde{x}^{[-i]}) + \\langle\\tilde{\\mu}^{[i]},G^{[i]}(x^{[i]},\\tilde{x}^{[-i]})\\rangle,\\nonumber\\end{align} and thus $f_i(\\tilde{x}) \\leq f_i(x^{[i]},\\tilde{x}^{[-i]}),\\quad \\forall x^{[i]}\\in X_i^f(\\tilde{x}^{[-i]})$, by noting that $\\langle\\tilde{\\mu}^{[i]},G^{[i]}(x^{[i]},\\tilde{x}^{[-i]})\\rangle \\leq 0$ for any $ x^{[i]}\\in X_i^f(\\tilde{x}^{[-i]})$. Recall that $G^{[i]}(\\tilde{x})\\leq0$ for all $i\\in V$. The above arguments hold for all $i\\in V$, and thus it establishes that $\\tilde{x}\\in\\mathbb{X}_{\\rm CVX}$.\n\nWe now proceed to show (P2). Since $\\mu^{[i]}\\geq0$, $\\mathcal{H}_i$ is a positive combination of convex functions of $x^{[i]}$. Thus $\\mathcal{H}_i$ is convex in $x^{[i]}$. It is easy to see that $-\\mathcal{H}_i$ is convex (actually affine) in $x^{[i]}$. Since $X$ and $M$ are convex and compact, it follows that $\\mathbb{X}_{\\rm UC}(M) \\neq \\emptyset$ by the results on generalized convex games in; e.g.,~\\cite{Facchinei.Kanzow:07,Rosen:65}. Pick any $(\\tilde{x},\\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}(M)$ and then $\\|\\tilde{\\mu}^{[i]}\\| \\leq \\vartheta_i$ by Assumption~\\ref{asm1}. We then have the following relation: \\begin{align}{\\mathcal{H}}_i(\\tilde{x},\\tilde{\\mu}^{[i]})\n  -{\\mathcal{H}}_i(\\tilde{x},\\mu^{[i]}) \\geq 0,\\quad \\forall\\mu^{[i]}\\in M_i,\\nonumber\\end{align} and thus,\n\\begin{align}\\langle \\mu^{[i]}-\\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle \\leq 0,\\quad \\forall\\mu^{[i]}\\in M_i.\\label{e6}\\end{align}\n\nWe now proceed to verify by contradiction the feasibility of $G^{[i]}(\\tilde{x}) \\leq 0$ and the complementary slackness of $\\langle  \\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle = 0$. Assume that $G^{[i]}(\\tilde{x})_{\\ell} > 0$. We choose $\\mu^{[i]}$ such that $\\mu^{[i]}_{\\ell} = \\tilde{\\mu}^{[i]}_{\\ell} + \\pi r_i$ and $\\mu^{[i]}_{\\ell'} = 0$ for $\\ell' \\neq \\ell$ where $\\pi>0$ is sufficiently small such that $\\mu^{[i]}\\in M_i$. Then it follows from~\\eqref{e6} that $r_i G^{[i]}(\\tilde{x})_{\\ell} \\leq 0$ which is a contradiction. Hence, we have the feasibility of $G^{[i]}(\\tilde{x})\\leq 0$. Combine it with $\\tilde{\\mu}^{[i]}\\geq0$, and it renders that $\\langle \\tilde{\\mu}^{[i]}\\rangle, G^{[i]}(\\tilde{x})\\leq 0$. On the other hand, we let $\\mu^{[i]} = 0$ in the relation~\\eqref{e6} and have $\\langle \\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle \\geq 0$. Hence, it renders that $\\langle \\tilde{\\mu}^{[i]}, G^{[i]}(\\tilde{x})\\rangle = 0$. We reach the desired result by using the property (P1). \\oprocend\n\n\\subsection{Notations and notions}\\label{subsection:notations}\n\nThe vector $\\textbf{1}_n$ represents the column vector with $n$ ones. The vector $e^{[\\ell]}_{n}$ is the one in $\\mathds{R}^n$ whose $\\ell$-th coordinate is one and whose other coordinates are all zero. For any pair of vectors $a,b\\in\\mathds{R}^p$, the relation $a\\leq b$ means $a_{\\ell}\\leq b_{\\ell}$ for all $1\\leq\\ell\\leq p$. Since the function $f_i$ is continuous and $X_i$ is compact, then the following quantities are well-defined:\n$\\sigma_{i,\\min} = \\inf_{x\\in X_i}f_i(x),\\quad \\sigma_{i,\\max} = \\sup_{x\\in X_i}f_i(x),\\quad\n\\sigma_{\\min} = \\inf_{x\\in X}\\sum_{i\\in V}f_i(x),\\quad \\sigma_{\\max} = \\sup_{x\\in X}\\sum_{i\\in V}f_i(x)$. Given a non-negative scalar sequence $\\{\\alpha(k)\\}_{k\\geq0}$, it is \\emph{summable} if $\\sum_{k=0}^{\\infty}\\alpha(k)<+\\infty$ and \\emph{square summable} if $\\sum_{k=0}^{\\infty}\\alpha(k)^2<+\\infty$.\n\nIn the remainder of the paper, we will use some notions about \\emph{monotonicity}, and the readers are referred to~\\cite{Aubin.Frankowska:09,Facchinei.Pang:03} for detailed discussion. The mapping $F : Z \\rightarrow Z'$ is \\emph{strongly monotone} with constant $\\rho > 0$ over $Z$ if for each pair of $\\eta,\\eta'\\in Z$, the following holds: \\begin{align*}\\langle F(\\eta) - F(\\eta'), \\eta - \\eta' \\rangle \\geq \\rho \\|\\eta - \\eta'\\|^2.\\end{align*} The mapping $F : Z \\rightarrow Z'$ is \\emph{monotone} over $Z$ if for each pair of $\\eta,\\eta'\\in Z$, the following holds: \\begin{align*}\\langle F(\\eta) - F(\\eta'), \\eta - \\eta' \\rangle \\geq 0.\\end{align*} The mapping $F : Z \\rightarrow Z'$ is \\emph{pseudo-monotone} over $Z$ if for each pair of $\\eta,\\eta'\\in Z$, it holds that $\\langle F(\\eta'), \\eta - \\eta' \\rangle \\geq 0$ implies $\\langle F(\\eta), \\eta - \\eta' \\rangle \\geq 0$. It is known that strong monotonicity implies monotonicity, and monotonicity implies pseudo-monotonicity~\\cite{Facchinei.Pang:03}.\n\nGiven the non-empty, convex, and closed set $Z\\in {\\mathds{R}}^n$, the\nprojection operator onto $Z$, $P_Z: \\mathds{R}^n \\rightarrow Z$, is\ndefined as $P_Z[z] = {\\rm argmin}_{x\\in Z}\\|x-z\\|$. The following is the non-expansiveness of the projection operator; e.g.,~\\cite{Bertsekas:09}.\n\n\\begin{lemma} Let $Z$ be a non-empty, closed and convex set in ${\\mathds{R}}^n$. For any $z\\in{\\mathds{R}}^n$, the following holds for any $y\\in Z$: $\\|P_X[z] - y\n  \\|^2 \\leq \\| z - y \\|^2 - \\| P_X[z] - z\\|^2$. \\label{lem6}\n\\end{lemma}\n\nWe define $\\nabla_{x^{[i]}}\\mathcal{H}_i(x,\\mu^{[i]})$ as the gradient of the convex function $\\mathcal{H}_i(\\cdot,x^{[-i]},\\mu^{[i]})$ at $x^{[i]}$, and $\\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x,\\mu^{[i]}) = G^{[i]}(x)$ as the gradient of the concave (actually affine) function $\\mathcal{H}_i(x,\\cdot)$ at $\\mu^{[i]}$. Define $\\nabla \\Theta$ as the map of partial gradients of the players' objective functions: \\begin{align*}\\nabla \\Theta(\\eta) &\\triangleq [\\nabla_{x^{[1]}}\\mathcal{H}_1(x,\\mu^{[1]})^T\\cdots \\nabla_{x^{[N]}}\\mathcal{H}_N(x,\\mu^{[N]})^T\\\\ &\\nabla_{\\mu^{[1]}}-\\mathcal{H}_1(x,\\mu^{[1]})\\cdots \\nabla_{\\mu^{[N]}}-\\mathcal{H}_N(x,\\mu^{[N]})]^T.\\end{align*} The map $\\nabla \\Theta$ is referred to as the \\emph{game map}. It is well-known that $\\nabla\\Theta$ is monotone (resp. strongly monotone) over $K$ if and only if its Jacobian $\\nabla^2\\Theta(\\eta)$ is positive semi-definite (resp. positive definite) for all $\\eta\\in K$.\n\nFor $G^{[i]}(x)\\leq0$ being absent, we define $\\nabla_{x^{[i]}}f_i(x)$ as the gradient of the convex function $f_i(\\cdot,x^{[-i]})$ at $x^{[i]}$. Define $\\nabla \\Theta^r$ as the map of partial gradients of the players' objective functions: \\begin{align*}\\nabla \\Theta^r(x) \\triangleq [\\nabla_{x^{[1]}}f_1(x)^T\\cdots \\nabla_{x^{[N]}}f_N(x)^T]^T.\\end{align*} The map $\\nabla \\Theta^r$ is referred to as the \\emph{(reduced) game map} for $G^{[i]}(x)\\leq0$ being absent.\n\n\nGiven any convex and closed set $Z$, let $\\mathbb{P}_Z$ be the projection operator onto the set $Z$.\n\nThe following lemma is a direct result of the smoothness of the component functions and the compactness of the constraint sets.\n\n\\begin{lemma} The image set of the game map $\\nabla\\Theta$ is uniformly bounded over $K$. In addition, the game map $\\nabla\\Theta$ (resp. $\\nabla\\Theta^r$) is Lipschitz continuous over $K$ with constant $L_{\\Theta}$ (resp. $L_{\\Theta^r}$). \\label{lem4}\n\\end{lemma}\n\n\\section{Distributed computation algorithms}\\label{sec:algorithm}\n\nIn this section, we will study the scenarios mentioned in the introduction. Proposition~\\ref{pro5} reveals that it suffices to compute a Nash equilibrium in $\\mathbb{X}_{\\rm UC}$ if Assumption~\\ref{asm1} holds. In the remainder of this section, we will suppose Assumption~\\ref{asm1} holds, and then each dual player~$i$ can define the set of $M_i \\triangleq \\{\\mu^{[i]}\\in\\mathds{R}^{m_i}_{\\geq0}\\;|\\;\\|\\mu^{[i]}\\|\\leq \\vartheta_i+r_i\\}$ with $r_i>0$. Assumption~\\ref{asm1} will be justified in Section~\\ref{sec:assumption}.\n\n\\subsection{Scenario one: pseudo-monotone game map and unknown $\\tau_{\\max}$}\n\nIn this section, we synthesize a distributed first-order algorithm for the case where the quantity $\\tau_{\\max}$ is unknown and the game map $\\nabla \\Theta$ is merely pseudo-monotone.\n\nAlgorithm~\\ref{ta:algo1} is based on projected primal-dual gradient methods where each primal or dual player~$i$\\footnote{In practical implementation, the update rules of primal player~$i$ and dual player~$i$ are both executed by real player~$i$.} updates its own estimate by moving along its partial gradient with a certain step-size and projecting the estimate onto the local constraint set. Recall that $f_i$ and $G^{[i]}$ are unknown to player~$i$. Then player~$i$ cannot compute partial gradient $\\nabla_{x^{[i]}}\\mathcal{H}_i$. Inspired by simultaneous perturbation stochastic approximation in; e.g.,~\\cite{JS:03}, we use finite differences to approximate the partial gradient $\\nabla_{x^{[i]}}\\mathcal{H}_i$; i.e., for $\\ell = 1,\\cdots,n_i$, \\begin{align}&[\\nabla_{x^{[i]}}\\mathcal{H}_i(x(k),\\mu^{[i]}(k))]_{\\ell}\\nonumber\\\\\n&\\approx \\frac{1}{2c_i(k)}\\big(\\mathcal{H}_i(x^{[i]}(k)+c_i(k)e^{[\\ell]}_{n_i},x^{[-i]}(k),\\mu^{[i]}(k))\\nonumber\\\\\n&-\\mathcal{H}_i(x^{[i]}(k)-c_i(k)e^{[\\ell]}_{n_i},x^{[-i]}(k),\\mu^{[i]}(k))\\big),\\label{e51}\\end{align}\nwhere $-c_i(k)e^{[\\ell]}_{n_i}$ and $c_i(k)e^{[\\ell]}_{n_i}$ are two-way perturbations. In addition, $\\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x(k),\\mu^{[i]}(k)) = G^{[i]}(x(k))$. In Algorithm~\\ref{ta:algo1}, $\\mathcal{D}^{[i]}_x(k)$ (resp. $\\mathcal{D}^{[i]}_{\\mu}(k)$) is the right-hand side of~\\eqref{e51} (resp. $\\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x,\\mu^{[i]})$) with delayed estimates. Here we assume that player~$i$ can observe the \\emph{values} associated with $f_i$ and $G^{[i]}$ and thus $\\mathcal{D}^{[i]}_x(k)$ and $\\mathcal{D}^{[i]}_{\\mu}(k)$.\n\nThe scalar $c_i(k)>0$ is the perturbation magnitude. When it is small, the finite-difference approximation is close to the partial gradient. In order to asymptotically eliminate the error induced by the finite-difference approximation, $c_i(k)$ needs to be diminishing. The decreasing rate of $c_i(k)$ should match that of computation step-sizes $\\alpha(k)$. Otherwise, the convergence to Nash equilibrium may be prevented. This is captured by (A4) of Theorem~\\ref{the1}.\n\n\\begin{algorithm}[htbp]\\caption{Distributed gradient-based algorithm for Scenario one} \\label{ta:algo1}\n\\textbf{Require:} Each primal player $i\\in V$ chooses the initial state $x^{[i]}(0)\\in X_i$. And each dual player~$i\\in V_m$ defines the set $M_i$ and chooses the initial state $\\mu^{[i]}(0)\\in M_i$.\n\n\\textbf{Ensure:} At each $k \\ge 0$, each player in $V\\cup V_m$ executes the following steps:\n\n1. Each primal player $i\\in V$ updates its state according to the following rule:\\begin{align*} x^{[i]}(k+1) = \\mathbb{P}_{X_i}[x^{[i]}(k) - \\alpha(k)\\mathcal{D}^{[i]}_x(k)],\n\\end{align*} where the approximate gradient $\\mathcal{D}^{[i]}_x(k)$ is given by \\begin{align}&[\\mathcal{D}^{[i]}_x(k)]_{\\ell} = \\frac{1}{2c_i(k)}\n\\{f_i(x^{[i]}(k)+c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\nonumber\\\\\n&+ \\langle\\mu^{[i]}(k),G^{[i]}(x^{[i]}(k)+c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\rangle\\nonumber\\\\\n&-f_i(x^{[i]}(k)-c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\nonumber\\\\\n&- \\langle\\mu^{[i]}(k),G^{[i]}(x^{[i]}(k)\n-c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\rangle\\},\\label{e38}\\end{align} for $\\ell = 1,\\cdots,n_i$.\n\n2. Each dual player $i\\in V_m$ updates its state according to the following rule:\\begin{align*}\n\\mu^{[i]}(k+1) = \\mathbb{P}_{M_i}[\\mu^{[i]}(k) + \\alpha(k)\\mathcal{D}^{[i]}_{\\mu}(k)],\n\\end{align*} where the gradient $\\mathcal{D}^{[i]}_{\\mu}(k)$ is given by $\\mathcal{D}^{[i]}_{\\mu}(k) = \\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x^{[i]}(k),\\Lambda_i(k),\\mu^{[i]}(k)) = G^{[i]}(x^{[i]}(k),\\Lambda_i(k))$.\n\n3. Repeat for $k = k+1$.\n\\end{algorithm}\n\nThe following theorem demonstrates that Algorithm~\\ref{ta:algo1} is able to achieve a GNE from any initial state in $K$.\n\n\\begin{theorem} Suppose the following hold:\n\\begin{enumerate}\n\\item[(A1)] the quantity $\\tau_{\\max}$ is finite;\n\\item[(A2)] Assumptions~\\ref{asm7} and~\\ref{asm1} hold;\n\\item[(A3)] the sequence of $\\{\\alpha(k)\\}$ is positive, not summable but square summable;\n\\item[(A4)] the sequence of $\\displaystyle{\\{\\alpha(k)\\max_{i\\in V}c_i(k)\\}}$ is summable;\n\\item[(A5)] the game map $\\nabla\\Theta$ is pseudo-monotone over $K$.\n\\end{enumerate}\nFor any initial state $\\eta(0)\\in K$, the sequence of $\\{\\eta(k)\\}$ generated by Algorithm~\\ref{ta:algo1} converges to some $(\\tilde{x},\\tilde{\\mu})\\in{\\mathbb{X}}_{\\rm UC}$ where $\\tilde{x}\\in\\mathbb{X}_{\\rm CVX}$.\\label{the1}\n\\end{theorem}\n\n\\textbf{Proof:} First of all, it is noted that $K$ is compact since each $X_i$ and $X_i$ is compact in Assumption~\\ref{asm1} and (P2) in Proposition~\\ref{pro5} as well as $\\nabla\\Theta$ is uniformly bounded over $K$ in Lemma~\\ref{lem4}.\n\nThen we write Algorithm~\\ref{ta:algo1} in the following compact form:\n\\begin{align}\\eta(k+1) = \\mathbb{P}_K[\\eta(k)-\\alpha(k)\\mathcal{D}(k)],\\label{e19}\n\\end{align} where $\\mathcal{D}(k) \\triangleq ((\\mathcal{D}^{[i]}_x(k)^T)^T_{i\\in V},(-\\mathcal{D}^{[i]}_{\\mu}(k)^T)^T_{i\\in V_m})^T$ is subject to time delays and perturbations. Then pick any $\\eta\\in K$. It follows from the non-expansiveness of the projection operator $\\mathbb{P}_K$, Lemma~\\ref{lem6}, that for any $\\eta\\in K$, the following relation holds:\n\\begin{align}&\\|\\eta(k+1)-\\eta\\|^2\\nonumber\\\\\n&\\leq \\|\\eta(k)-\\alpha(k)\\mathcal{D}(k)-\\eta\\|^2-\\|\\mathcal{D}(k)\\|^2\\nonumber\\\\\n&\\leq \\|\\eta(k)-\\alpha(k)\\mathcal{D}(k)-\\eta\\|^2\\nonumber\\\\\n&= \\|\\eta(k)-\\eta\\|^2 - 2\\alpha(k)\\langle \\mathcal{D}(k), \\eta(k)-\\eta\\rangle\\nonumber\\\\\n&+ \\alpha(k)^2\\|\\mathcal{D}(k)\\|^2.\\label{e25}\n\\end{align}\n\nIt follows from~\\eqref{e25} that\n\\begin{align}&2\\alpha(k)\\langle \\mathcal{D}(k), \\eta(k)-\\eta\\rangle\\leq \\|\\eta(k) - \\eta\\|^2\\nonumber\\\\\n&- \\|\\eta(k+1) - \\eta\\|^2 +\\alpha(k)^2\\|\\mathcal{D}(k)\\|^2.\\label{e23}\\end{align}\n\nFor any $k\\geq0$ and $i$, we define the following: \\begin{align}&\\hat{\\mathcal{D}}^{[i]}_x(k) = \\nabla_{x^{[i]}}\\mathcal{H}_i(x^{[i]}(k),\\{x^{[j]}(k)\\}_{j\\in\\mathcal{N}^{\\rm IN}_i},\\mu^{[i]}(k)),\\nonumber\\\\\n&\\hat{\\mathcal{D}}^{[i]}_{\\mu}(k) \\triangleq \\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x^{[i]}(k),\\{x^{[j]}(k)\\}_{j\\in\\mathcal{N}^{\\rm IN}_i},\\mu^{[i]}(k)),\\nonumber\\\\\n&\\hat{\\mathcal{D}}(k) \\triangleq ((\\hat{\\mathcal{D}}^{[i]}_x(k)^T)^T_{i\\in V},(-\\hat{\\mathcal{D}}^{[i]}_{\\mu}(k)^T)^T_{i\\in V_m})^T,\\label{e16}\\end{align} which are the gradients evaluated at the delay-free states. So, the quantity $\\hat{\\mathcal{D}}(k)$ is free of time delays and perturbations.\n\nSimilarly, for any $k\\geq0$ and $i$, we define the following: \\begin{align}&\\tilde{\\mathcal{D}}^{[i]}_x(k) = \\nabla_{x^{[i]}}\\mathcal{H}_i(x^{[i]}(k),\\Lambda_i(k),\\mu^{[i]}(k)),\\nonumber\\\\\n&\\tilde{\\mathcal{D}}^{[i]}_{\\mu}(k) \\triangleq \\nabla_{\\mu^{[i]}}\\mathcal{H}_i(x^{[i]}(k),\\Lambda_i(k),\\mu^{[i]}(k)),\\nonumber\\\\\n&\\tilde{\\mathcal{D}}(k) \\triangleq ((\\tilde{\\mathcal{D}}^{[i]}_x(k)^T)^T_{i\\in V},(-\\tilde{\\mathcal{D}}^{[i]}_{\\mu}(k)^T)^T_{i\\in V_m})^T,\\nonumber\\end{align} which are the gradients evaluated at the delayed states. Then the quantity $\\tilde{\\mathcal{D}}(k)$ is free of perturbations but subject to time delays.\n\nWith the above notations at hand, the relation~\\eqref{e23} implies the following: \\begin{align}&2\\alpha(k)\\langle \\hat{\\mathcal{D}}(k), \\eta(k)-\\eta\\rangle \\leq \\alpha(k)^2\\|\\mathcal{D}(k)\\|^2 + \\|\\eta(k) - \\eta\\|^2\\nonumber\\\\&- \\|\\eta(k+1) - \\eta\\|^2 + 2\\alpha(k)\\langle\\tilde{\\mathcal{D}}(k)-\\mathcal{D}(k),\\eta(k)-\\eta\\rangle\\nonumber\\\\\n&+ 2\\alpha(k)\\langle\\hat{\\mathcal{D}}(k)-\\tilde{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\\nonumber\\\\\n&\\leq \\alpha(k)^2\\|\\mathcal{D}(k)\\|^2 + \\|\\eta(k) - \\eta\\|^2\\nonumber\\\\&- \\|\\eta(k+1) - \\eta\\|^2 + 2\\alpha(k)\\|\\tilde{\\mathcal{D}}(k)-\\mathcal{D}(k)\\| \\|\\eta(k)-\\eta\\|\\nonumber\\\\\n&+ 2\\alpha(k)\\|\\hat{\\mathcal{D}}(k)-\\tilde{\\mathcal{D}}(k)\\| \\|\\eta(k)-\\eta\\|.\\label{e30}\\end{align}\n\nNow let us examine the term with $\\|\\hat{\\mathcal{D}}(k)-\\tilde{\\mathcal{D}}(k)\\|$ on the right-hand side of~\\eqref{e30}. Since $x(k)\\in X$ and $X$ is convex and closed, it then follows from the non-expansiveness of the projection operator $\\mathbb{P}_X$, that \\begin{align}\\|\\eta(k+1)-\\eta(k)\\|\n&=\\|\\mathbb{P}_K[\\eta(k)-\\alpha(k)\\mathcal{D}(k)]-\\mathbb{P}_K[\\eta(k)]\\|\\nonumber\\\\\n&\\leq\\alpha(k)\\|\\mathcal{D}(k)\\|.\\label{e14}\\end{align} Consequently,\nit follows from the Lipschitiz continuity of the game map $\\nabla\\Theta$ and~\\eqref{e30},~\\eqref{e14} that \\begin{align} &\\|\\hat{\\mathcal{D}}(k)-\\tilde{\\mathcal{D}}(k)\\|\\nonumber\\\\\n&\\leq \\sum_{i\\in V}(\\|\\hat{\\mathcal{D}}^{[i]}_x(k)-\\tilde{\\mathcal{D}}^{[i]}_x(k)\\| + \\|\\hat{\\mathcal{D}}^{[i]}_{\\mu}(k)-\\tilde{\\mathcal{D}}^{[i]}_{\\mu}(k)\\|)\\nonumber\\\\\n&\\leq 2N L_{\\Theta}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\|\\eta(\\tau+1)-\\eta(\\tau)\\|\\nonumber\\\\\n&\\leq 2N L_{\\Theta}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\alpha(\\tau)\\|\\mathcal{D}(\\tau)\\|.\\label{e45}\\end{align}\n\nSince $K$ is compact and the image of $\\nabla\\Theta$ is uniformly compact, ~\\eqref{e45} implies that there is $\\Upsilon > 0$ such that \\begin{align}&2\\alpha(k)\\|\\hat{\\mathcal{D}}(k)-\\tilde{\\mathcal{D}}(k)\\|\\|\\eta(k)-{\\eta}\\|\\nonumber\\\\\n&\\leq 2NL_{\\Theta}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}2\\alpha(k)\\alpha(\\tau)\n\\|\\mathcal{D}(\\tau)\\|\\|\\eta(k)-{\\eta}\\|\\nonumber\\\\\n&\\leq 2N L_{\\Theta}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\big(\\alpha(k)^2+\\alpha(\\tau)^2\n\\|\\mathcal{D}(\\tau)\\|^2\\|\\eta(k)-{\\eta}\\|^2\\big)\\nonumber\\\\\n&\\leq 2N L_{\\Theta}\\tau_{\\max}\\alpha(k)^2 + \\Upsilon\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\alpha(\\tau)^2.\\label{e46}\\end{align}\n\nNow consider the term with $\\|\\tilde{\\mathcal{D}}(k)-\\mathcal{D}(k)\\|$ on the right-hand side in~\\eqref{e30}. Recall that $K$ is compact. By the Taylor expansion, we reach that \\begin{align}&\\mathcal{D}^{[i]}_x(k) =\\nabla_{x^{[i]}}f_i(x^{[i]}(k),\n\\Lambda_i(k))\\nonumber\\\\\n&+\\sum_{\\ell=1}^{m_i}\\mu^{[i]}_{\\ell}(k)\\nabla_{x^{[i]}}\nG^{[i]}_{\\ell}(x^{[i]}(k),\\Lambda_i(k))+O(c_i(k))\\textbf{1}_{n_i}.\\label{e21}\\end{align}\n\nWith the above relation, we have the following for $\\tilde{\\mathcal{D}}^{[i]}(k)-\\mathcal{D}^{[i]}(k)$:\n\\begin{align}\\tilde{\\mathcal{D}}^{[i]}_{\\mu}(k)-\\mathcal{D}^{[i]}_{\\mu}(k) = 0,\\quad\n\\tilde{\\mathcal{D}}^{[i]}_x(k)-\\mathcal{D}^{[i]}_x(k) = O(c_i(k))\\textbf{1}_{n_i}.\\label{e12}\\end{align}\n\nNotice that $\\|\\mathcal{D}(k)\\|$ is uniformly bounded. Substitute~\\eqref{e46} and~\\eqref{e12} into~\\eqref{e30}, sum over $[0,T]$, and it renders that there is some $\\Upsilon', \\Upsilon'' > 0$ such that the following estimate holds: \\begin{align}&2\\sum_{k=0}^T\\alpha(k)\\langle \\hat{\\mathcal{D}}(k), \\eta(k)-\\eta\\rangle\\nonumber\\\\\n&\\leq \\|\\eta(0) - \\eta\\|^2 - \\|\\eta(T+1) - \\eta\\|^2 + \\Upsilon'\\sum_{k=0}^T\\alpha(k)^2\\nonumber\\\\\n&+ \\Upsilon''\\sum_{k=0}^T\\alpha(k)\\max_{i\\in V}c_i(k).\\label{e26}\\end{align}\n\nSince $\\{\\alpha(k)^2\\}$ and $\\{\\alpha(k)\\max_{i\\in V}c_i(k)\\}$ are summable, then the right-hand side of~\\eqref{e26} is finite when we let $T\\rightarrow+\\infty$. We now show the following by contradiction: \\begin{align}\\liminf_{k\\rightarrow+\\infty}\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\\leq0.\\label{e34}\\end{align}\nAssume that there are $k_0\\geq0$ and $\\epsilon>0$ such that $\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle \\geq \\epsilon$ for all $k\\geq k_0$. Since $\\alpha(k) > 0$ and $\\{\\alpha(k)\\}$ is not summable, then we have the following: \\begin{align}&\\sum_{k=0}^{+\\infty}\\alpha(k)\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\\nonumber\\\\\n&\\geq \\sum_{k=0}^{k_0}\\alpha(k)\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\\nonumber\\\\\n&+ \\sum_{k=k_0+1}^{+\\infty}\\alpha(k)\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\\nonumber\\\\\n&\\geq \\sum_{k=0}^{k_0}\\alpha(k)\\langle \\hat{\\mathcal{D}}(k),\\eta(k)-\\eta\\rangle\n+ \\epsilon\\sum_{k=k_0+1}^{+\\infty}\\alpha(k)\\nonumber\\\\\n&\\geq +\\infty.\\nonumber\\end{align}\nWe then reach a contradiction, and thus~\\eqref{e34} holds. Equivalently, the following relation holds: \\begin{align*}\\limsup_{k\\rightarrow+\\infty}\\langle \\hat{\\mathcal{D}}(k),\\eta-\\eta(k)\\rangle\\geq0.\\end{align*} Since $\\nabla\\Theta$ is closed, the above relation implies that there is a limit point $\\tilde{\\eta}\\in K$ of the sequence $\\{\\eta(k)\\}$ such that the following holds: $\\langle \\nabla\\Theta(\\tilde{\\eta}),\\eta-\\tilde{\\eta}\\rangle\\geq0, \\quad \\forall\\eta\\in K$. Since $\\nabla\\Theta$ is pseudo-monotone, then we have the following relation: \\begin{align}\\langle \\nabla\\Theta(\\eta),\\eta-\\tilde{\\eta}\\rangle\\geq0, \\quad \\forall\\eta\\in K.\\label{e22}\\end{align}\n\nWe now set out to show the following by contradiction: \\begin{align}\\langle \\nabla\\Theta(\\tilde{\\eta}),\\eta-\\tilde{\\eta}\\rangle\\geq0,\\quad \\forall \\eta\\in K.\\label{e32}\\end{align} Assume that there is $\\hat{\\eta}\\in K$ such that the following holds: \\begin{align}\\langle \\nabla\\Theta(\\tilde{\\eta}),\\hat{\\eta}-\\tilde{\\eta}\\rangle < 0.\\label{e33}\\end{align} Now choose $\\varepsilon\\in(0,1)$, and define $\\eta_{\\varepsilon} \\triangleq \\hat{\\eta} + \\varepsilon(\\tilde{\\eta}-\\hat{\\eta})$. Since $K$ is convex, then we have $\\eta_{\\varepsilon}\\in K$. The following holds: \\begin{align}\\langle \\nabla\\Theta(\\eta_{\\varepsilon}),(1-\\varepsilon)(\\hat{\\eta}-\\tilde{\\eta})\\rangle = \\langle \\nabla\\Theta(\\eta_{\\varepsilon}),\\eta_{\\varepsilon}-\\tilde{\\eta}\\rangle\\geq0,\\label{e28}\n\\end{align} where in the inequality we use~\\eqref{e22}. It follows from~\\eqref{e28} that the following relation holds for any $\\varepsilon\\in(0,1)$: \\begin{align}\\langle \\nabla\\Theta(\\eta_{\\varepsilon}),\\hat{\\eta}-\\tilde{\\eta}\\rangle\\geq0,\\quad \\forall \\eta\\in K.\\label{e29}\n\\end{align} Since $\\nabla\\Theta$ is closed, letting $\\varepsilon\\rightarrow1$ in~\\eqref{e29} gives that: $\\langle \\nabla\\Theta(\\tilde{\\eta}),\\hat{\\eta}-\\tilde{\\eta}\\rangle\\geq0$, which contradicts~\\eqref{e33}. As a result, the relation~\\eqref{e32} holds. By~\\cite{Palomar.Eldar:10}, the relation~\\eqref{e32} implies that $\\tilde{\\eta}\\in\\mathbb{X}_{\\rm UC}$. We replay $\\eta$ by $\\eta(k)$ in~\\eqref{e32}, and establish the following:\n\\begin{align}\\langle \\nabla\\Theta(\\tilde{\\eta}),\\eta(k)-\\tilde{\\eta}\\rangle\\geq0.\\label{e31}\\end{align} Since $\\nabla\\Theta$ is pseudo-monotone, it follows from~\\eqref{e31} that\\begin{align}\\langle \\hat{\\mathcal{D}}(k), \\eta(k)-\\tilde{\\eta}\\rangle \\geq 0.\\label{e27}\\end{align} Replace $\\eta$ with $\\tilde{\\eta}$ in~\\eqref{e23}, apply~\\eqref{e27}, and it renders: \\begin{align}\\|\\eta(k+1) - \\tilde{\\eta}\\|^2 -\\|\\eta(k) - \\tilde{\\eta}\\|^2\n\\leq\\alpha(k)^2\\|\\mathcal{D}(k)\\|^2.\\label{e35}\\end{align} Sum up~\\eqref{e35} over $[s,k]$ and we have \\begin{align}\\|\\eta(k) - \\tilde{\\eta}\\|^2\n\\leq \\|\\eta(s) - \\tilde{\\eta}\\|^2 + \\sum_{\\tau=s}^{k-1}\\alpha(\\tau)^2\\|\\mathcal{D}(\\tau)\\|^2.\\label{e36}\\end{align}\nWe take the limits on $k$ first and then $s$ on both sides of~\\eqref{e36}. Since $\\{\\alpha(k)\\}$ is square summable and $\\{\\mathcal{D}(k)\\}$ is uniformly bounded, we have \\begin{align}\\limsup_{k\\rightarrow\\infty}\\|\\eta(k) - \\tilde{\\eta}\\|^2\n\\leq \\liminf_{s\\rightarrow\\infty}\\|\\eta(s) - \\tilde{\\eta}\\|^2.\\nonumber\\end{align} It implies that $\\{\\|\\eta(k) - \\tilde{\\eta}\\|\\}$ converges. Since $\\tilde{\\eta}$ is a limit point of $\\{\\eta(k)\\}$, $\\{\\eta(k)\\}$ converges to $\\tilde{\\eta}\\in\\mathbb{X}_{\\rm UC}$. Furthermore, we have $\\tilde{x}\\in\\mathbb{X}_{\\rm CVX}$ by (P2) in Proposition~\\ref{pro5}.\\oprocend\n\n\\subsection{Scenario two: strongly monotone reduced game map and known $\\tau_{\\max}$}\n\nIn the last section, the convergence of Algorithm~\\ref{ta:algo1} is ensured under the mild assumptions: $\\tau_{\\max}$ is unknown and the game map $\\nabla\\Theta$ is merely pseudo-monotone. This set of assumptions requires the usage of diminishing step-sizes. Note that diminishing step-sizes may cause a slow convergence rate. The shortcoming can be partially addressed by choosing a constant step-size when the inequality constraints are absent and (A5) in Theorem~\\ref{the1} is strengthened to the following one: \\begin{assumption} The map $\\nabla\\Theta^r$ is strongly monotone over $X$ with constant $\\rho$.\\label{asm5}\\end{assumption}\n\n\\begin{algorithm}[htbp]\\caption{The distributed gradient-based algorithm for Scenario two} \\label{ta:algo2}\n\n\\textbf{Require:} Each player in $V$ chooses the initial state $x^{[i]}(0)\\in X_i$.\n\n\\textbf{Ensure:} At each $k \\ge 0$, each player~$i\\in V$ executes the following steps:\n\n1. Each player $i\\in V$ updates its state according to the following rule:\\begin{align*} x^{[i]}(k+1) = \\mathbb{P}_{X_i}[x^{[i]}(k) - \\alpha D^{[i]}(k)],\\end{align*} where the approximate gradient $D^{[i]}(k)$ is given by: \\begin{align*}[D^{[i]}(k)]_{\\ell} &= \\frac{1}{2c_i(k)}\n\\{f_i(x^{[i]}(k)+c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\nonumber\\\\\n&-f_i(x^{[i]}(k)-c_i(k)e^{[\\ell]}_{n_i},\\Lambda_i(k))\\},\\end{align*} for $\\ell = 1,\\cdots,n_i$.\n\n2. Repeat for $k = k+1$.\n\\end{algorithm}\n\nAlgorithm~\\ref{ta:algo2} is proposed to address Scenario two. In particular, Algorithm~\\ref{ta:algo2} is similar to Algorithm~\\ref{ta:algo1}, but has two distinctions. Firstly, Algorithm~\\ref{ta:algo2} excludes the inequality constraints. Because the game map $\\nabla \\Theta$ cannot be strongly monotone due to $\\mathcal{H}_i$ being linear in $\\mu^{[i]}$. Secondly, thanks to the strong monotonicity of $\\nabla \\Theta$, a constant step-size replaces diminishing step-sizes. The following theorem summarizes the convergence properties of Algorithm~\\ref{ta:algo2}.\n\n\\begin{theorem} Suppose the following holds:\n\\begin{enumerate}\n\\item[(B1)] the quantity $0\\leq\\tau_{\\max}<\\frac{\\rho}{4NL_{\\Theta^r}}$ is known;\n\\item[(B2)] Assumptions~\\ref{asm7} and~\\ref{asm1} hold;\n\\item[(B3)] the step-size $\\alpha\\in(0,\\frac{\\rho-4NL_{\\Theta^r}\\tau_{\\max}}{L^2_{\\Theta^r}(2+16N^2\\tau_{\\max})})$;\n\\item[(B4)] The sequence $\\{\\max_{i\\in V}c_i(k)\\}$ is summable.\n\\item[(B5)] Assumption~\\ref{asm5} holds.\n\\end{enumerate}\nFor any initial state $x(0)\\in X$, the sequence of $\\{x(k)\\}$ generated by Algorithm~\\ref{ta:algo2} converges to $\\tilde{x}\\in\\mathbb{X}_{\\rm CVX}$.\\label{the2}\n\\end{theorem}\n\n\\textbf{Proof:} First of all, for any $k\\geq0$ and $i$, we define $\\hat{\\mathcal{D}}^{[i]}(k) \\triangleq \\nabla_{x^{[i]}}f_i(x^{[i]}(k),\\{x^{[j]}(k)\\}_{j\\in\\mathcal{N}^{\\rm IN}_i})$, which is the gradient evaluated at the delay-free states. So, the quantity $\\hat{\\mathcal{D}}(k)$ is free of time delays and perturbations. Similarly, for any $k\\geq0$ and $i$, we define $\\tilde{\\mathcal{D}}^{[i]}(k) \\triangleq \\nabla_{x^{[i]}}f_i(x^{[i]}(k),\\Lambda_i(k))$ which is the gradient evaluated at the delayed states. Then the quantity $\\tilde{\\mathcal{D}}(k)$ is free of perturbations but subject to time delays.\n\nWe then write Algorithm~\\ref{ta:algo2} in the following compact form: \\begin{align}x(k+1) = \\mathbb{P}_X[x(k)-\\alpha D(k)].\\label{e39}\n\\end{align} It is noticed that $\\tilde{x}\\in{\\mathbb{X}}_{\\rm CVX}$ is a fixed point of the operator $\\mathbb{P}_X[\\cdot-\\alpha \\nabla\\Theta^r(\\cdot)]$ for any $\\alpha > 0$; i.e., it holds that $\\tilde{x} = \\mathbb{P}_X[\\tilde{x}-\\alpha \\nabla\\Theta^r(\\tilde{x})]$. By this, we have the following relations: \\begin{align}&\\|x(k+1)-\\tilde{x}\\|^2\\nonumber\\\\\n&= \\|\\mathbb{P}_X[x(k)-\\alpha D(k)] - \\mathbb{P}_X[\\tilde{x}-\\alpha \\nabla\\Theta^r(\\tilde{x})]\\|^2\\nonumber\\\\\n&\\leq \\|(x(k) - \\tilde{x})-\\alpha(D(k)-\\nabla\\Theta^r(\\tilde{x}))\\|^2\\nonumber\\\\\n&= \\|x(k) - \\tilde{x}\\|^2-2\\alpha\\langle x(k) - \\tilde{x}, \\hat{D}(k)-\\nabla\\Theta^r(\\tilde{x})\\rangle\\nonumber\\\\\n&+ \\alpha^2\\|(D(k)-\\tilde{D}(k))+(\\tilde{D}(k)-\\hat{D}(k))+(\\hat{D}(k)-u)\\|^2\\nonumber\\\\\n&-2\\alpha\\langle x(k) - \\tilde{x}, D(k)-\\tilde{D}(k)\\rangle\\nonumber\\\\\n&-2\\alpha\\langle x(k) - \\tilde{x}, \\tilde{D}(k)-\\hat{D}(k)\\rangle\\nonumber\\\\\n&\\leq (1-2\\rho\\alpha+4L_{\\Theta^r}^2\\alpha^2)\\|x(k)-\\tilde{x}\\|^2\\nonumber\\\\\n&+ 4\\alpha^2(\\|D(k)-\\tilde{D}(k)\\|^2+\\|\\tilde{D}(k)-\\hat{D}(k)\\|^2)\\nonumber\\\\\n&+2\\alpha\\|x(k) - \\tilde{x}\\|(\\|D(k)-\\tilde{D}(k)\\|+\\|\\tilde{D}(k)-\\hat{D}(k)\\|),\\label{e42}\\end{align}\nwhere we use the non-expansiveness property of the projection operator $\\mathbb{P}_X$ in the first inequality, and the strong monotonicity and the Lipschitz continuity of $\\nabla\\Theta^r$ in the last inequality. For the term of $\\|\\hat{D}(k)-\\tilde{D}(k)\\|$ in~\\eqref{e42}, one can derive the following relations: \\begin{align}&\\|\\hat{D}(k)-\\tilde{D}(k)\\| \\leq  NL_{\\Theta^r}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\|x(\\tau+1)-x(\\tau)\\|\\nonumber\\\\\n&\\leq  NL_{\\Theta^r}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\big(\\|x(\\tau+1)-\\tilde{x}\\|\n+\\|x(\\tau)-\\tilde{x}\\|\\big)\\nonumber\\\\\n&\\leq  2NL_{\\Theta^r}\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\|x(\\tau)-\\tilde{x}\\|.\\label{e47}\\end{align}\n\nSimilar to~\\eqref{e12}, we have \\begin{align}\\tilde{D}^{[i]}(k)-D^{[i]}(k) = O(c_i(k))\\textbf{1}_{n_i}.\\label{e13}\\end{align}\n\nSubstituting~\\eqref{e47} and~\\eqref{e13} into~\\eqref{e42} yields: \\begin{align}&\\|x(k+1)-\\tilde{x}\\|^2\\nonumber\\\\\n&\\leq (1-2\\rho\\alpha+4L_{\\Theta^r}^2\\alpha^2+4NL_{\\Theta^r}\\alpha\\tau_{\\max})\n\\|x(k)-\\tilde{x}\\|^2\\nonumber\\\\\n&+ (32N^2L_{\\Theta^r}^2\\alpha^2+4NL_{\\Theta^r}\\alpha)\n\\sum_{\\tau=k-\\tau_{\\max}}^{k-1}\\|x(\\tau)-\\tilde{x}\\|^2\\nonumber\\\\\n&+4\\alpha^2(\\max_{i\\in V}c_i(k))^2 + 2\\alpha\\max_{i\\in V}c_i(k)\\|x(k)-\\tilde{x}\\|,\\label{e48}\\end{align}\nwhere we use the relation $2ab \\leq a^2 + b^2$.\n\nTo study the convergence of $x(k)$ in~\\eqref{e48}, we define the following notation: \\begin{align*}&\\psi(k) \\triangleq x(k) - \\tilde{x},\\quad\\|\\chi(k)\\| \\triangleq \\max_{k-\\tau_{\\max}\\leq \\tau \\leq k-1}\\|\\psi(\\tau)\\|,\\nonumber\\\\\n&e(k)\\triangleq 4\\alpha^2(\\max_{i\\in V}c_i(k))^2 + 2\\alpha\\max_{i\\in V}c_i(k)\\|x(k)-\\tilde{x}\\|.\\end{align*} Then~\\eqref{e48} is rewritten as follows: \\begin{align}\\|\\psi(k+1)\\|\\leq a\\|\\psi(k)\\|+b\\|\\chi(k)\\|+e(k).\\label{e43}\\end{align} where $a \\triangleq (1-2\\rho\\alpha+4L_{\\Theta^r}^2\\alpha^2+4NL_{\\Theta^r}\\alpha\\tau_{\\max})$, $b \\triangleq \\tau_{\\max}(32N^2L_{\\Theta^r}^2\\alpha^2+4NL_{\\Theta^r}\\alpha)$ and the sequence $\\{e(k)\\}$ is diminishing. From the recursion of~\\eqref{e43}, we can derive the following relation for any pair of $k>s\\geq0$: \\begin{align}&\\|\\psi(k)\\| \\leq a^{k-s}\\|\\psi(s)\\| + \\sum_{\\tau=s}^{k-1}a^{k-\\tau}b\\chi(\\tau) + \\sum_{\\tau=s}^{k-1}e(\\tau)\\nonumber\\\\\n&\\leq a^{k-s}\\|\\psi(s)\\| + b\\sup_{s\\leq\\tau\\leq k}\\|\\chi(\\tau)\\|\\sum_{\\tau=s}^{k-1}a^{k-\\tau} + \\sum_{\\tau=s}^{k-1}e(\\tau)\\nonumber\\\\\n&\\leq a^{k-s}\\|\\psi(s)\\| + \\frac{b}{1-a}\\sup_{s\\leq\\tau\\leq k}\\|\\chi(\\tau)\\| + \\sum_{\\tau=s}^{k-1}e(\\tau).\\label{e44}\\end{align} Recall that $\\{\\max_{i\\in V}c_i(k)\\}$ and thus $\\{(\\max_{i\\in V}c_i(k))^2\\}$ are summable. Take the limits on $k$ and $s$ at both sides of~\\eqref{e44}, and it gives the following relation: \\begin{align}\\limsup_{k\\rightarrow+\\infty}\\|\\psi(k)\\| \\leq \\frac{b}{1-a}\\limsup_{k\\rightarrow+\\infty}\\|\\chi(k)\\|.\\label{e49}\\end{align} On the other hand, one can see that \\begin{align}\\limsup_{k\\rightarrow+\\infty}\\|\\chi(k)\\| = \\limsup_{k\\rightarrow+\\infty}\\|\\psi(k)\\|.\\label{e50}\\end{align}\n\nSince $\\alpha\\in(0,\\frac{\\rho-4NL_{\\Theta^r}\\tau_{\\max}}{L^2_{\\Theta^r}(2+16N^2\\tau_{\\max})})$, then $\\frac{a}{1-b}<1$. The combination of~\\eqref{e49} and~\\eqref{e50} renders that \\begin{align}\\limsup_{k\\rightarrow+\\infty}\\|\\psi(k)\\| = 0.\\label{e40}\\end{align} Apparently, \\begin{align}\\liminf_{k\\rightarrow+\\infty}\\|\\psi(k)\\| \\geq 0.\\label{e41}\\end{align} The combination of~\\eqref{e40} and~\\eqref{e41} establishes the convergence of $\\{x(k)\\}$ to $\\tilde{x}$. It completes the proof.\\oprocend\n\n\\section{Discussions}\\label{sec:discussion}\n\n\\subsection{Comparison of two scenarios}\\label{sec:comparison}\n\nThe two proposed algorithms are complementary. Algorithm~\\ref{ta:algo1} can address inequality constraints, does not need to know $\\tau_{\\max}$ and merely requires the game map to be pseudo monotone. It comes with the price of potentially slow convergence due to the utilization of diminishing step-sizes. In contrast, Algorithm~\\ref{ta:algo2} cannot deal with inequality constraints, needs to know $\\tau_{\\max}$ and requires the game map to be strongly monotone. It comes with the benefit of potentially fast convergence due to the utilization of a constant step-size.\n\n\n\\subsection{Discussion on Assumption~\\ref{asm1}}\\label{sec:assumption}\n\nThe proposed algorithms rely upon Assumption~\\ref{asm1}. In what follows, we will provide two sufficient conditions for Assumption~\\ref{asm1}. Recall that $\\sigma_{i,\\min}$, $\\sigma_{i,\\max}$,\n$\\sigma_{\\min}$ and $\\sigma_{\\max}$ are defined in Section~\\ref{subsection:notations}.\n\n\\subsubsection{Global Slater vectors}\n\n\\begin{assumption} There exist $\\bar{x}\\in X$ and $\\sigma' > 0$ such that $\\|-G^{[i]}(\\bar{x})\\|_{\\infty} \\geq \\sigma'$ for all $i\\in V$.\\label{asm3}\n\\end{assumption}\n\nThe vector $\\bar{x}$ that satisfies Assumption~\\ref{asm3} is referred to as the \\emph{global} Slater vector. The existence of the global Slater vector ensures the boundedness of $\\mathbb{X}_{\\rm UC}(M)$ as follows.\n\n\\begin{lemma} If Assumption~\\ref{asm3} holds and $G^{[i]}_{\\ell}$ is convex in $x$, then for any $(\\tilde{x},\\tilde{\\mu})\\in \\mathbb{X}_{\\rm UC}(\\mathds{R}^m_{\\geq0})$, it holds that $\\|\\tilde{\\mu}^{[i]}\\|_{\\infty} \\leq \\frac{\\sigma_{\\max}-\\sigma_{\\min}}{\\sigma'}$ for $i\\in V$.\\label{lem2}\n\\end{lemma}\n\n\\textbf{Proof:} Pick any $\\tilde{\\eta}\\triangleq(\\tilde{x},\\tilde{\\mu})\\in \\mathbb{X}_{\\rm UC}(\\mathds{R}^m_{\\geq0})$. It holds that \\begin{align}\\langle \\nabla\\Theta(\\tilde{\\eta}),\\tilde{\\eta}-\\eta\\rangle\\leq0.\\label{e55}\\end{align}\nChoose $\\eta = (\\bar{x}^T,0^T)^T$ in~\\eqref{e55}, and we have\n\\begin{align}&0\\geq\\langle \\nabla\\Theta^r(\\tilde{x}),\\tilde{x}-\\bar{x}\\rangle - \\sum_{i\\in V}\\langle \\tilde{\\mu}^{[i]},G^{[i]}(\\tilde{x})\\rangle\\nonumber\\\\\n&+ \\sum_{i\\in V}\\sum_{\\ell=1}^{m_i}\\tilde{\\mu}_{\\ell}^{[i]}\\langle \\nabla_{x^{[i]}} G^{[i]}_{\\ell}(\\tilde{x}),\\tilde{x}^{[i]}-\\bar{x}^{[i]}\\rangle\\nonumber\\\\\n&\\geq \\langle \\nabla\\Theta^r(\\tilde{x}),\\tilde{x}-\\bar{x}\\rangle - \\sum_{i\\in V}\\langle \\tilde{\\mu}^{[i]},G^{[i]}(\\bar{x})\\rangle,\\label{e56}\n\\end{align} where the last inequality uses $\\tilde{\\mu}^{[i]}_{\\ell}\\geq0$ and Taylor theorem and the convexity of $G^{[i]}_{\\ell}$ in $x$. It follows from~\\eqref{e56} and Assumption~\\ref{asm3} that \\begin{align}\\|\\tilde{\\mu}^{[i]}\\|_{\\infty}\\leq\\frac{-\\langle \\nabla\\Theta^r(\\tilde{x}),\\tilde{x}-\\bar{x}\\rangle}{\\sigma'}\\leq \\frac{\\sigma_{\\max}-\\sigma_{\\min}}{\\sigma'}.\\nonumber\\end{align}\n\\oprocend\n\n\n\\subsubsection{Private Slater vectors}\n\nIn Lemma~\\ref{lem2}, each player has to access the global information of $\\bar{x}$, $\\sigma'$, $\\sigma_{\\min}$ and $\\sigma_{\\max}$. In what follows, we will derive a sufficient condition which only requires private information of each player~$i$ to determine an upper bound on $\\mu^{[i]}$.\n\n\\begin{assumption} For each $i\\in V$, there exists $\\sigma_i > 0$ and $\\bar{x}_i\\in X_i$ such that for any $(\\tilde{x},\\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}$, $\\|-G^{[i]}(\\bar{x}^{[i]},\\tilde{x}^{[-i]})\\|_{\\infty} \\geq \\sigma_i$ holds.\\label{asm2}\n\\end{assumption}\n\nThe vector $\\bar{x}^{[i]}$ that satisfies Assumption~\\ref{asm3} is referred to as the \\emph{private} Slater vector. One case where Assumption~\\ref{asm2} holds is that $G^{[i]}$ only depends upon $x^{[i]}$ and, for each $i\\in V$, there is $\\bar{x}^{[i]}\\in X_i$ such that $\\|-G^{[i]}(\\bar{x}^{[i]})\\|_{\\infty} \\geq \\sigma_i$ holds for some $\\sigma_i > 0$.\n\n\\begin{lemma} If Assumption~\\ref{asm2} holds, then for any $(\\tilde{x},\\tilde{\\mu})\\in \\mathbb{X}_{\\rm UC}(\\mathds{R}^m_{\\geq0})$, it holds that $\\|\\tilde{\\mu}^{[i]}\\|_{\\infty} \\leq \\frac{\\sigma_{i,\\max}-\\sigma_{i,\\min}}{\\sigma_i}$ for $i\\in V$.\\label{lem5}\n\\end{lemma}\n\n\\textbf{Proof:} Pick any $(\\tilde{x},\\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}(\\mathds{R}^m_{\\geq0})$. By Assumption~\\ref{asm2}, there is $\\bar{x}^{[i]}\\in X_i$ such that \\begin{align*}\\|-G^{[i]}(\\bar{x}^{[i]},\\tilde{x}^{[-i]})\\|_{\\infty} \\geq \\sigma.\\end{align*} Since $(\\tilde{x},\\tilde{\\mu})\\in\\mathbb{X}_{\\rm UC}$, we then have the following: \\begin{align}&\\mathcal{H}_i(\\tilde{x}^{[i]},\\tilde{x}^{[-i]},0)\\leq\n\\mathcal{H}_i(\\tilde{x}^{[i]},\\tilde{x}^{[-i]},\\tilde{\\mu}^{[i]})\\nonumber\\\\\n&\\leq\n\\mathcal{H}_i(\\bar{x}^{[i]},\\tilde{x}^{[-i]},\\tilde{\\mu}^{[i]})\\nonumber\\end{align} where in the first inequality we use $0\\in M_i$ and in the second inequality we use $\\bar{x}^{[i]}\\in X_i$. The above relations further render the following: \\begin{align}f_i(\\tilde{x})\\leq f_i(\\bar{x}^{[i]},\\tilde{x}^{[-i]}) + \\langle \\tilde{\\mu}^{[i]}, G^{[i]}(\\bar{x}^{[i]},\\tilde{x}^{[-i]})\\rangle.\\label{e24}\\end{align} Recall $\\|-G^{[i]}(\\bar{x}^{[i]},\\tilde{x}^{[-i]})\\|_{\\infty} \\geq \\sigma_i$. Then the relation~\\eqref{e24} implies the relation of $\\|\\tilde{\\mu}^{[i]}\\|_{\\infty} \\leq \\frac{\\sigma_{i,\\max}-\\sigma_{i,\\min}}{\\sigma_i}$.\\oprocend\n\n\n\n\n\n\n\\subsection{Future directions}\n\nThe current paper imposes several assumptions: (1) the compactness of $X_i$; (2) the smoothness of component functions; (3) identical $\\alpha(k)$ or $\\alpha$ for all the players; (4) the convexity of component functions and constraints. In particular, the compactness of $X_i$ ensures the uniform boundedness of partial gradients. The convexity guarantees that local optimum are also globally optimal in the sense of Nash equilibrium. It is of interest to relax these assumptions.\n\n\\section{Numerical simulations}\\label{sec:simulations}\n\nIn this section, we will provide a set of numerical simulations to verify the performance of our proposed algorithms. The procedure ${\\rm sample}(A)$ returns a uniform sample from the set $A$.\n\n\\subsection{Algorithm~\\ref{ta:algo1}}\\label{sec:simulations-alg1}\n\nConsider a power network which can be modeled as an interconnected graph $\\mathcal{G}_p\\triangleq \\{\\mathcal{V}_p,\\mathcal{E}_p\\}$ where each node $i\\in \\mathcal{V}_p$ represents a bus and each link in $\\mathcal{E}_p$ represents a branch. The buses in $\\mathcal{V}_p$ are indexed by $1,\\cdots,N$ and bus $1$ denotes the feeder which has a fixed voltage and flexible power injection. A lossless DC model is used to characterize the relation between power generations and loads at different buses and power flows across various branches. Each bus is connected to either a power load or supply and each load is associated with an end-user. The set of load buses is denoted by $\\mathcal{V}_l\\subseteq \\mathcal{V}_p$ and the set of supply buses is denoted by $\\mathcal{V}_s\\subseteq \\mathcal{V}_p$.\n\nThe maximum available power supply at bus $i\\in \\mathcal{V}_s$ is denoted by $S_i\\geq0$, and the intended power load at bus $i\\in \\mathcal{V}_l$ is denoted by $L_i\\geq0$. If the total power supply exceeds the total intended power load, then all the intended loads can be satisfied. Otherwise, some loads need to reduce. The reduced load of end-user~$i$ is denoted by $R_i\\in[0,L_i]$. The objective function of each end-user~$i$ is given by $f_i(R_i) = c_i R_i + p(\\textbf{1}^T_{|\\mathcal{V}_l|}(L-R))(L_i-R_i) - u_i(L_i-R_i)$. The quantity $c_iR_i$ represents the disutility induced by load reduction $R_i$ with $c_i>0$. The scalar $p(\\textbf{1}^T(L-R))$ is the charged price given the total actual load $\\textbf{1}^T(L-R)$. The value $u_i(L_i-R_i)$ stands for the benefit produced by load $L_i-R_i$.\n\nEach end-user~$i$ aims to minimize its own objective function $f_i(R_i)$ by adjusting $R_i$. Such decision making is subject to the physical constraints of the power grid. The first constraint is that the total actual load cannot exceed the maximum available supply; i.e., \\begin{align}\n\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R) \\leq \\textbf{1}_{|\\mathcal{V}_s|}^TS.\\label{e52}\\end{align} Another set of constraints are induced by the limitations of power flows on branches. Let $H_g\\in[-1,1]^{|\\mathcal{E}|\\times|\\mathcal{V}_g|}$ (resp. $H_l\\in[-1,1]^{|\\mathcal{E}|\\times|\\mathcal{V}_l|}$) be the generation (resp. load) shift factor matrix. For $H_g$, the $(i,\\ell)$ entry represents the power that is distributed on line $\\ell$ when $1MW$ is injected into bus $i$ and withdrawn at the reference bus. Denote by $f_e^{\\max}$ the maximum capacity of branch $e$. The power flow constraint for branch $e$ can be expressed as: \\begin{align}-f^{\\max}\\leq H_g S - H_l (L-R)\\leq f^{\\max},\\label{e53}\\end{align} where $f^{\\max}\\triangleq{\\rm col}[f_e^{\\max}]_{e\\in \\mathcal{E}_p}$, $S\\triangleq {\\rm col}[S_i]_{i\\in\\mathcal{V}_s}$, $L\\triangleq {\\rm col}[L_i]_{i\\in\\mathcal{V}_l}$ and $R\\triangleq {\\rm col}[R_i]_{i\\in\\mathcal{V}_l}$.\n\nThe above description defines a non-cooperative game among end-users in $\\mathcal{V}_l$ as follows:\n\\begin{align}&\\min_{R_i\\in [0,L_i]}c_i R_i + p(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))(L_i-R_i) - u_i(L_i-R_i)\\nonumber\\\\\n&{\\rm s.t.}\\quad \\textbf{1}_{|\\mathcal{V}_l|}^T(L-R) \\leq \\textbf{1}_{|\\mathcal{V}_s|}^TS\\nonumber\\\\\n&\\quad\\quad H_g S - H_l (L-R) - f^{\\max}\\leq 0\\nonumber\\\\\n&\\quad\\quad - H_g S + H_l (L-R) -f^{\\max}\\leq 0.\\label{e54}\n\\end{align}\n\nIn game~\\eqref{e54}, $p$ is the pricing policy of load serving entity (LSE). This is confidential information of LSE and should not be disclosed to end-users. So the objective function $f_i$ is partially unknown to end-user~$i$. In addition, the numerical values of generation and load shift factor matrices are of national security interest and are kept confidential from the public. Therefore, the power flow constraints are unknown to end-users. Assume that all the end-users can communicate with LSE. Given $R$, LSE broadcasts the price value $p(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))$, the power imbalance $\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R) - \\textbf{1}_{|\\mathcal{V}_s|}^TS$, and power limit violations $H_g S - H_l (L-R) - f^{\\max}$ and $- H_g S + H_l (L-R) -f^{\\max}$. In this way, end-users can access the values of $f_i$ and $G^{[i]}$ in Algorithm~\\ref{ta:algo1} without knowing their structures.\\footnote{Many networked engineering systems operate in a hierarchical structure; e.g., Internet, power grid and transportation systems, where a central system operator is placed at the top layer and end-users are placed at the bottom layer~\\cite{MC-SHL-ARC-JCD:07a,Wu.Moslehi.Bose:89}. Here we assume that LSE can communicate with all the end-users. This assumption is widely used in; e.g., network flow control~\\cite{LL-99}.}\n\nWe choose $u_i(L_i-R_i) = -\\frac{1}{2}a_i(L_i-R_i)^2+b_i(L_i-R_i)$ with $a_i>0$. For the pricing mechanism, we use (4) in~\\cite{Bulow.Peiderer:83}; i.e., $p(w) = \\beta w^{\\frac{1}{\\eta}}$ with $\\beta>0$, and $\\eta\\in(0,1)$. So $f_i(R) = c_iR_i + (L_i-R_i)\\beta (\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))^{\\frac{1}{\\eta}} +\\frac{1}{2}a_i(L_i-R_i)^2-b_i(L_i-R_i)$. One can compute the following: \\begin{align}\\frac{\\partial^2f_i}{\\partial^2R_i}&=a_i-2\\beta\\frac{1}{\\eta}\n(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))^{\\frac{1}{\\eta}-1}\\nonumber\\\\\n&-\\beta (L_i-R_i)\\frac{1}{\\eta}(\\frac{1}{\\eta}-1)(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))^{\\frac{1}{\\eta}-2},\\nonumber\\\\\n\\frac{\\partial^2f_i}{\\partial R_i \\partial R_j}&=-\\beta\\frac{1}{\\eta}\n(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))^{\\frac{1}{\\eta}-1}\\nonumber\\\\\n&-\\beta (L_i-R_i)\\frac{1}{\\eta}(\\frac{1}{\\eta}-1)(\\textbf{1}_{|\\mathcal{V}_l|}^T(L-R))^{\\frac{1}{\\eta}-2}.\\nonumber\\end{align}\n\nChoose $a_i$ such that $a_i > (N-2)\\beta\\frac{1}{\\eta}\n(\\textbf{1}_{|\\mathcal{V}_l|}^TL)^{\\frac{1}{\\eta}-1}+(N-1)\\beta L_i\\frac{1}{\\eta}(\\frac{1}{\\eta}-1)(\\textbf{1}_{|\\mathcal{V}_l|}^TL)^{\\frac{1}{\\eta}-2}>0$. Then the monotonicity property holds. Assume that there is a global Slater vector. Figure~\\ref{fig1} shows the simulation results for the IEEE $30$-bus Test System~\\cite{PSTCA} and the system parameters are adopted from MATPOWER~\\cite{MATPOWER}. The delays at each iteration are randomly chosen from $0$ to $10$.\n\n\\begin{figure*}[bh]\n  \\centering\n  \\includegraphics[width = \\linewidth]{DemandResponse}\n  \\caption{The estimates of Algorithm~\\ref{ta:algo1} at $30$ buses for $\\tau_{\\max} = 10$.}\\label{fig1}\n\\end{figure*}\n\n\n\\subsection{Algorithm~\\ref{ta:algo2}}\\label{sec:simulations-alg2}\n\nConsider $10$ players and their components functions are defined as follows: \\begin{align*}f_i(x) = \\frac{1}{2}((Q^{[i]})^Tx)^2 + (q^{[i]})^Tx,\\end{align*} where\n\\begin{itemize}\n\\item $X_i = [\\psi_i^m\\;\\;\\psi_i^M]$ with $\\psi_i^m = {\\rm sample}([-10\\;\\;-5])$ and $\\psi_i^M = {\\rm sample}([5\\;\\;10])$;\n\\item $Q^{[i]}\\in\\mathds{R}^{10}$ with $Q^{[i]}_i = 1$ and $Q^{[i]}_j = \\frac{1}{10}$ for $j\\neq i$;\n\\item $q^{[i]}\\in\\mathds{R}^{10}$ with $q^{[i]}_j = {\\rm sample}([-50\\;\\;50])$ for $j\\in V$.\n\\end{itemize}\n\nOne can verify that $\\rho = \\frac{1}{10}$ and $L_{\\Theta^r} = 2$ in Theorem~\\ref{the2}. By (B4), we have $\\varrho_b>1$. In the simulation, we choose $\\varrho_b = 3$. Figure~\\ref{fig4} presents the estimate evolution of the players.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width = \\linewidth]{Algorithm2-x}\n  \\caption{The estimates of Algorithm~\\ref{ta:algo2} for $\\tau_{\\max} = 10$.}\\label{fig4}\n\\end{figure}\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe have studied a set of distributed robust adaptive computation algorithms for a class of generalized convex games. We have shown their asymptotic convergence to Nash equilibrium in the presence of network unreliability and the uncertainties of component functions and illustrated the algorithm performance via numerical simulations.\n\n\n\n\\bibliographystyle{plain}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\\IEEEPARstart{L}{earning} convolutional operators from large datasets is a growing trend in signal\/image processing, computer vision, and machine learning.\nThe widely known \\emph{patch-domain} approaches for learning kernels (e.g., filter, dictionary, frame, and transform) extract patches from training signals for simple mathematical formulation and optimization, yielding (sparse) features of training signals \\cite{Aharon&Elad&Bruckstein:06TSP, Elad&Aharon:06TIP, Yaghoobi&etal:13TSP, Hawe&Kleinsteuber&Diepold:13TIP, Mairal&Bach&Ponce:14FTCGV, Cai&etal:14ACHA, Ravishankar&Bressler:15TSP, Pfister&Bresler:15SPIE, Coates&NG:bookCh}.\nDue to memory demands, using many overlapping patches across the training signals hinders using large datasets\nand building hierarchies on the features, \ne.g., deconvolutional neural networks \\cite{Zeiler&etal:10CVPR}, \nconvolutional neural network (CNN) \\cite{LeCun&etal:98ProcIEEE}, \nand multi-layer convolutional sparse coding \\cite{Papyan&Romano&Elad:17JMLR}.\nFor similar reasons, the memory requirement of patch-domain approaches discourages learned kernels from being applied to large-scale inverse problems. \n\n\n\n\nTo moderate these limitations of the patch-domain approach, the so-called \\emph{convolution} perspective has been recently introduced by learning filters and obtaining (sparse) representations directly from the original signals without storing many overlapping patches, e.g.,  convolutional dictionary learning (CDL) \\cite{Zeiler&etal:10CVPR, Bristow&etal:13CVPR, Heide&eta:15CVPR, Wohlberg:16TIP, Chun&Fessler:18TIP, Chun&Fessler:17SAMPTA}. \nFor large datasets, CDL using careful algorithmic designs \\cite{Chun&Fessler:18TIP} is more suitable for learning filters than patch-domain dictionary learning \\cite{Aharon&Elad&Bruckstein:06TSP}; in addition, CDL can learn translation-invariant filters without obtaining highly redundant sparse representations \\cite{Chun&Fessler:18TIP}.\nThe CDL method applies the convolution perspective for learning kernels within \\dquotes{synthesis} signal models.\nWithin \\dquotes{analysis} signal models, however, there exist no prior frameworks using the convolution perspective for learning convolutional operators, whereas patch-domain approaches for learning analysis kernels are introduced in \\cite{Yaghoobi&etal:13TSP, Hawe&Kleinsteuber&Diepold:13TIP, Cai&etal:14ACHA, Ravishankar&Bressler:15TSP, Pfister&Bresler:15SPIE}.\n(See brief descriptions about synthesis and analysis signal models in \\cite[Sec.~\\Romnum{1}]{Hawe&Kleinsteuber&Diepold:13TIP}.)\n\n\n\nResearchers interested in dictionary learning have actively studied the structures of kernels learned by the patch-domain approach \\cite{Yaghoobi&etal:13TSP, Hawe&Kleinsteuber&Diepold:13TIP, Cai&etal:14ACHA, Ravishankar&Bressler:15TSP, Pfister&Bresler:15SPIE, Barchiesi&Plumbley:13TSP, Bao&Cai&Ji:13ICCV, Ravishankar&Bressler:13ICASSP}. \nIn training CNNs (see Appendix~\\ref{sec:CNN}), however,\nthere has been less study of filter structures having non-convex constraints, e.g., orthogonality and unit-norm constraints in Section~\\ref{sec:CAOL}, although it is thought that diverse (i.e., incoherent) filters can improve performance for some applications, e.g., image recognition \\cite{Coates&NG:bookCh}.\nOn the application side, researchers have applied (deep) NNs to signal\/image recovery problems.\nRecent works combined model-based image reconstruction (MBIR) algorithm with image refining networks \\cite{Yang&etal:16NIPS, Zhang&etal:17CVPR, Chen&Pock:17PAMI, Chen&etal:17arXiv, Wu&etal:17Fully3D, Romano&Elad&Milanfar:17SJIS, Buzzard&etal:18SJIS, Chun&Fessler:18IVMSP, Chun&etal:18arXiv:momnet, Chun&etal:18Allerton}.\nIn these \\emph{iterative} NN methods, \nrefining NNs should satisfy the non-expansiveness for fixed-point convergence \n\\cite{Chun&etal:18arXiv:momnet}; \nhowever, their trainings lack consideration of filter diversity constraints, e.g., orthogonality constraint in Section~\\ref{sec:CAOL}, \nand thus it is unclear whether the trained NNs are nonexpansive mapping \\cite{Chun&etal:18Allerton}.\n\n\n\n\\begin{figure*}[!pt]\n\\centering\n\n\\begin{tabular}{c}\n\\includegraphics[trim={3cm 11.2cm 3cm 11.3cm},clip,scale=0.46]{.\/Fig_v1\/diagram_abstract.pdf}\n\\end{tabular}\n\n\\vspace{-0.25em}\n\\caption{A general flowchart from\nlearning sparsifying operators $\\cO$ to \nsolving inverse problems via MBIR using learned operators $\\cO^\\star$;\nsee Section~\\ref{sec:back}.\nFor the $l\\rth$ training sample $x_l$, \n$F(\\cO; x_l)$ measures its sparse representation or sparsification errors, and\nsparsity of its representation generated by $\\cO$.\n}\n\\label{diag:abstract}\n\\end{figure*}\n\n\n\n\nThis paper proposes \\textit{1)} a new \\textit{convolutional analysis operator learning} (CAOL) framework that learns an analysis sparsifying regularizer with the convolution perspective, and \\textit{2)} a new convergent \\textit{Block Proximal Extrapolated Gradient method using a Majorizer} (BPEG-M \\cite{Chun&Fessler:18TIP}) for solving block multi-nonconvex problems \\cite{Xu&Yin:17JSC}.\nTo learn diverse filters, we propose \\textit{a)} CAOL with an orthogonality constraint that enforces a tight-frame (TF) filter condition in convolutional perspectives, and \\textit{b)} CAOL with a regularizer that promotes filter diversity.\nBPEG-M with sharper majorizers converges significantly faster than the state-of-the-art technique, Block Proximal Gradient (BPG) method \\cite{Xu&Yin:17JSC} for CAOL.\nThis paper also introduces a new X-ray computational tomography (CT) MBIR model using a convolutional sparsifying regularizer learned via CAOL \\cite{Chun&Fessler:18Asilomar}.\n\n\n\n\nThe remainder of this paper is organized as follows. \nSection~\\ref{sec:back} reviews how learned regularizers can help solve inverse problems.\nSection~\\ref{sec:CAOL} proposes the two CAOL models.\nSection~\\ref{sec:reBPG-M} introduces BPEG-M with several generalizations, analyzes its convergence, and applies a momentum coefficient formula and restarting technique from \\cite{Chun&Fessler:18TIP}.\nSection~\\ref{sec:CAOL+BPGM} applies the proposed BPEG-M methods to the CAOL models, designs two majorization matrices, and describes memory flexibility and applicability of parallel computing to BPEG-M-based CAOL.\nSection~\\ref{sec:CTrecon} introduces the CT MBIR model using a convolutional regularizer learned via CAOL \\cite{Chun&Fessler:18Asilomar}, along with its properties, i.e., its mathematical relation to a convolutional autoencoder, the importance of TF filters, and its algorithmic role in signal recovery.\nSection~\\ref{sec:result} reports numerical experiments that show \\textit{1)} the importance of sharp majorization in accelerating BPEG-M, and \\textit{2)} the benefits of BPEG-M-based CAOL -- acceleration, convergence, and memory flexibility.\nAdditionally, Section~\\ref{sec:result} reports sparse-view CT experiments that show \\textit{3)} the CT MBIR using learned convolutional regularizers significantly improves the reconstruction quality compared to that using a conventional edge-preserving (EP) regularizer, and \\textit{4)} more and wider filters in a learned regularizer better preserves edges in reconstructed images.\nFinally, Appendix~\\ref{sec:CNN} mathematically formulates unsupervised training of CNNs via CAOL, and shows that its updates attained via BPEG-M correspond to the three important CNN operators.\nAppendix~\\ref{sec:egs} introduces some potential applications of CAOL to image processing, imaging, and computer vision.\n\n\n\n\n\n\n\n\\section{Backgrounds: MBIR Using \\emph{Learned} Regularizers} \\label{sec:back}\n\nTo recover a signal $x \\in \\bbC^{N'}$ from a data vector $y \\in \\bbC^m$, \none often considers the following MBIR optimization problem\n(Appendix~\\ref{sec:notation} provides mathematical notations): \n$\n\\argmin_{x \\in \\cX} f(x; y) + \\gamma \\, g(x),\n$\nwhere $\\cX$ is a feasible set,\n$f(x; y)$ is data fidelity function that models imaging physics (or image formation) and noise statistics, \n$\\gamma \\!>\\! 0$ is a regularization parameter,\nand $g(x)$ is a regularizer, such as total variation \\cite[\\S2--3]{Arridge&etal:19AN}. \nHowever, when inverse problems are extremely ill-conditioned, \nthe MBIR approach using hand-crafted regularizers $g(x)$ has limitations in recovering signals.\nAlternatively, there has been a growing trend in learning sparsifying regularizers\n(e.g., convolutional regularizers \\cite{Chun&Fessler:18TIP, Chun&Fessler:17SAMPTA, Chun&etal:19SPL, Chun&Fessler:18Asilomar, Crockett&etal:19CAMSAP}) \nfrom training datasets and applying the learned regularizers to the following MBIR problem \\cite{Arridge&etal:19AN}:\n\\be{\n\\label{eq:mbir:learn}\n\\argmin_{x \\in \\cX} f(x; y) + \\gamma g(x; \\cO^\\star),\n\\tag{B1}\n}\nwhere a learned regularizer $g(x; \\cO^\\star)$ quantifies consistency between \nany candidate $x$ and training data that is encapsulated in some trained sparsifying operators $\\cO^\\star$.\nThe diagram in Fig.~\\ref{diag:abstract} shows the general process from\ntraining sparsifying operators to solving inverse problems via \\R{eq:mbir:learn}.\nSuch models \\R{eq:mbir:learn} arise in a wide range of applications. \nSee some examples in Appendix~\\ref{sec:egs}.\n\nThis paper describes multiple aspects of learning convolutional regularizers. \nThe next section first starts with proposing a new convolutional regularizer.\n\n\n\n\n\n\\section{CAOL: Models \\textit{Learning} Convolutional Regularizers}  \\label{sec:CAOL}\n\nThe goal of CAOL is to find a set of filters that \\dquotes{best} sparsify a set of training images.\nCompared to hand-crafted regularizers,\nlearned convolutional regularizers \ncan better extract \\dquotes{true} features of estimated images and \nremove \\dquotes{noisy} features with thresholding operators.\nWe propose the following CAOL model:\n\\ea{\n\\label{sys:CAOL}\n\\argmin_{D = [d_1, \\ldots, d_K]} \\min_{\\{ z_{l,k} \\}} &~ F(D, \\{ z_{l,k} \\}) + \\beta g (D),  \\tag{P0}\n\\\\\nF(D, \\{ z_{l,k} \\}) & :=  \\sum_{l=1}^L \\sum_{k=1}^K \\frac{1}{2}  \\left\\| d_k \\circledast x_l - z_{l,k} \\right\\|_2^2 + \\alpha \\| z_{l,k} \\|_0, \\nn\n} \nwhere $\\circledast$ denotes a convolution operator (see details about boundary conditions in the supplementary material), $\\{ x_l \\in \\bbC^N : l =1,\\ldots,L \\}$ is a set of training images, $\\{ d_{k} \\in \\bbC^{R}: k = 1,\\ldots, K \\}$ is a set of convolutional kernels, $\\{ z_{l,k} \\in \\bbC^N : l = 1,\\ldots,L, k=1,\\ldots,K \\}$ is a set of sparse codes, and $g ( D )$ is a regularizer or constraint that encourages filter diversity or incoherence, $\\alpha \\!>\\! 0$ is a thresholding parameter controlling the sparsity of features $\\{  z_{l,k} \\}$, and $\\beta > 0$ is a regularization parameter for $g (D)$.\nWe group the $K$ filters into a matrix $D \\in \\bbC^{R \\times K}$:\n\\be{\n\\label{eq:D}\nD := \\left[ \\arraycolsep=3pt \\begin{array}{ccc} d_1 & \\ldots & d_K \\end{array} \\right].\n}\nFor simplicity, we fix the dimension for training signals, i.e., $\\{ x_l, z_{l,k} \\in \\bbC^N \\}$, but the proposed model \\R{sys:CAOL} can use training signals of different dimension, i.e., $\\{ x_l, z_{l,k} \\in \\bbC^{N_l} \\}$.\nFor sparse-view CT in particular, \nthe diagram in Fig.~\\ref{diag:caol-ctmbir} shows the process from\nCAOL \\R{sys:CAOL} to solving its inverse problem via\nMBIR using learned convolutional regularizers.\n\nThe following two subsections design the constraint or regularizer $g(D)$ \nto avoid redundant filters (without it, all filters could be identical).\n\n\n\n\n\n\n\\begin{figure*}[!pt]\n\\centering\n\n\\begin{tabular}{c}\n\\includegraphics[trim={2.2cm 9.5cm 2.2cm 9.9cm},clip,scale=0.46]{.\/Fig_v1\/diagram_caol-ctmbir.pdf}\n\\end{tabular}\n\n\\vspace{-0.25em}\n\\caption{A flowchart from CAOL \\R{sys:CAOL} to \nMBIR using a convolutional sparsifying regularizer learned via CAOL \\R{sys:CT&CAOL}\nin sparse-view CT.\nSee details of the CAOL process \\R{sys:CAOL} and its variants \\R{sys:CAOL:orth}--\\R{sys:CAOL:div}, \nand the CT MBIR process \\R{sys:CT&CAOL} in \nSection~\\ref{sec:CAOL} and Section~\\ref{sec:CTrecon}, respectively.\n}\n\\label{diag:caol-ctmbir}\n\\end{figure*}\n\n\n\n\n\n\n\\subsection{CAOL with Orthogonality Constraint} \\label{sec:CAOL:orth}\n\nWe first propose a CAOL model with a nonconvex orthogonality constraint on the filter matrix $D$ in \\R{eq:D}:\n\\be{\n\\label{sys:CAOL:orth}\n\\argmin_{D} \\min_{\\{ z_{l,k} \\}} ~ F(D, \\{ z_{l,k} \\}) \\quad \\mathrm{subj.~to} ~ D D^H = \\frac{1}{R} \\cdot I. \\tag{P1}\n} \nThe orthogonality condition $D D^H = \\frac{1}{R} I$ in \\R{sys:CAOL:orth} enforces a TF condition on the filters $\\{ d_k \\}$ in CAOL \\R{sys:CAOL}.\nProposition~\\ref{p:TFconst} below formally states this relation.\n\n\\prop{[Tight-frame filters]\\label{p:TFconst}\nFilters satisfying the orthogonality constraint \n$D D^H = \\frac{1}{R} I$ in \\R{sys:CAOL:orth} satisfy the following TF condition in a convolution perspective:\n\\be{\n\\label{eq:CAOL:TFcond}\n\\sum_{k=1}^K \\nm{ d_k \\circledast x }_2^2 = \\nm{ x }_2^2, \\quad \\forall x \\in \\bbC^N,\n}\nfor both circular and symmetric boundary conditions. \n}\n\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:p:TF} of the supplementary material.\n}\n\n\nProposition~\\ref{p:TFconst} corresponds to a TF result from patch-domain approaches; see~Section~\\ref{sec:prf:p:TF}. (Note that the patch-domain approach in \\cite[Prop.~3]{Cai&etal:14ACHA} requires $R = K$.)\nHowever, we constrain the filter dimension to be $R \\leq K$ to have an efficient solution for CAOL model \\R{sys:CAOL:orth}; see Proposition~\\ref{p:orth} later.\nThe following section proposes a more flexible CAOL model in terms of the filter dimensions $R$ and $K$.\n\n\n\n\n\\subsection{CAOL with Diversity Promoting Regularizer}\n\nAs an alternative to the CAOL model \\R{sys:CAOL:orth}, we propose a CAOL model with a diversity promoting regularizer and a nonconvex norm constraint on the filters $\\{ d_k \\}$:\n\\ea{\n\\label{sys:CAOL:div}\n\\argmin_{D} \\min_{\\{ z_{l,k} \\}} &~ F(D, \\{ z_{l,k} \\}) + \\frac{\\beta}{2} \\overbrace{\\left\\| D^H D - \\frac{1}{R} \\cdot I \\right\\|_{\\mathrm{F}}^2}^{\\mathrm{\\hbox{$=: g_{\\text{div}} (D)$}}}, \\nn\n\\\\\n\\mathrm{subject~to} ~&~ \\| d_{k} \\|_2^2 = \\frac{1}{R}, \\quad k = 1,\\ldots, K.  \\tag{P2}\n} \nIn the CAOL model \\R{sys:CAOL:div}, we consider the following:\n\\bulls{\n\\item The constraint in \\R{sys:CAOL:div} forces the learned filters $\\{ d_k \\}$ to have uniform energy. In addition, it avoids the \\dquotes{scale ambiguity} problem \\cite{Rem&Karin:10TIT}. \n\n\\item The regularizer in \\R{sys:CAOL:div}, $g_{\\text{div}} (D)$, promotes filter diversity, i.e., incoherence between $d_k$ and $\\{ d_{k'} : k' \\neq k \\}$, measured by $| \\ip{d_k}{d_{k'}} |^2$ for $k \\neq k'$.\n}\n\nWhen $R = K$ and $\\beta \\rightarrow \\infty$, the model \\R{sys:CAOL:div} becomes \\R{sys:CAOL:orth} since $D^H D = \\frac{1}{R} I$ implies $D D^H = \\frac{1}{R} I$ (for square matrices $A$ and $B$, if $AB=I$ then $BA = I$).\nThus \\R{sys:CAOL:div} generalizes \\R{sys:CAOL:orth} by relaxing the off-diagonal elements of the equality constraint in \\R{sys:CAOL:orth}. (In other words, when $R=K$, the orthogonality constraint in \\R{sys:CAOL:orth} enforces the TF condition and promotes the filter diversity.)\nOne price of this generalization is the extra tuning parameter $\\beta$.\n\n\n\n\\R{sys:CAOL:orth}--\\R{sys:CAOL:div} are challenging nonconvex optimization problems and block optimization approaches seem suitable. \nThe following section proposes a new block optimization method with momentum and majorizers, \nto rapidly solve the multiple block multi-nonconvex problems proposed in this paper, while guaranteeing convergence to critical points.\n\n\n\n\n\n\n\n\n\n\\section{BPEG-M: Solving Block Multi-Nonconvex Problems with Convergence Guarantees} \\label{sec:reBPG-M}\n\nThis section describes a new optimization approach, BPEG-M,\nfor solving block multi-nonconvex problems like \n\\textit{a)} CAOL \\R{sys:CAOL:orth}--\\R{sys:CAOL:div},\\footnote{\nA block coordinate descent algorithm can be applied to CAOL \\R{sys:CAOL:orth};\nhowever, its convergence guarantee in solving CAOL \\R{sys:CAOL:orth} is not yet known and might require stronger sufficient conditions than BPEG-M \\cite{Tseng:2001JOTA}.\n} \n\\textit{b)} CT MBIR \\R{sys:CT&CAOL} using learned convolutional regularizer via \\R{sys:CAOL:orth} (see Section~\\ref{sec:CTrecon}),\nand \\textit{c)} \\dquotes{hierarchical} CAOL \\R{sys:CNN:orth} (see Appendix~\\ref{sec:CNN}).\n\n\n\\subsection{BPEG-M -- Setup} \\label{sec:BPGM:setup}\n\nWe treat the variables of the underlying optimization problem either as a single block or multiple disjoint blocks. \nSpecifically, consider the following \\textit{block multi-nonconvex} optimization problem:\n\\ea{\n\\label{sys:multiConvx}\n\\min &~ F(x_1,\\ldots, x_B) := f(x_1,\\ldots,x_B) + \\sum_{b=1}^B g_b (x_b),\n}\nwhere variable $x$ is decomposed into $B$ blocks $x_1 ,\\ldots,x_B$ ($\\{ x_b \\in \\bbR^{n_b} : b=1,\\ldots,B \\}$), $f$ is assumed to be continuously differentiable, but functions $\\{ g_b : b = 1,\\ldots,B \\}$ are not necessarily differentiable. \nThe function $g_b$ can incorporate the constraint $x_b \\in \\cX_b$, by allowing any $g_b$ to be extended-valued, e.g., $g_b (x_b) = \\infty$ if $x_b \\notin \\cX_b$, for $b=1,\\ldots,B$.\nIt is standard to assume that both $f$ and $\\{ g_b \\}$ are closed and proper and the sets $\\{ \\cX_b \\}$ are closed and nonempty.\nWe do \\textit{not} assume that $f$, $\\{ g_b \\}$, or $\\{ \\cX_b \\}$ are convex. Importantly, $g_b$ can be a nonconvex $\\ell^p$ quasi-norm, $p \\in [0, 1)$.\nThe general block multi-convex problem in \\cite{Chun&Fessler:18TIP, Xu&Yin:13SIAM} is a special case of \\R{sys:multiConvx}.\n\n\nThe BPEG-M framework considers a more general concept than Lipschitz continuity of the gradient as follows:\n\n\\defn{[$M$-Lipschitz continuity] \\label{d:QM}\nA function $g: \\bbR^n \\rightarrow \\bbR^{n}$ is \\emph{$M$-Lipschitz continuous} on $\\bbR^n$ if there exist a (symmetric) positive definite matrix $M$ such that\n\\bes{\n\\nm{g(x) - g(y)}_{M^{-1}} \\leq \\nm{x - y}_{M}, \\quad \\forall x,y,\n}\nwhere $\\nm{x}_{M}^2 := x^T M x$.\n}\n\nLipschitz continuity is a special case of $M$-Lipschitz continuity with $M$ equal to a scaled identity matrix with \na Lipschitz constant of the gradient $\\nabla f$ (e.g., for $f(x) = \\frac{1}{2} \\| Ax - b\\|_2^2$, the (smallest) Lipschitz constant of $\\nabla f$ is the maximum eigenvalue of $A^T A$).\nIf the gradient of a function is $M$-Lipschitz continuous, then we obtain the following quadratic majorizer (i.e., surrogate function \\cite{Lange&Hunter&Yang:00JCGS, Jacobson&Fessler:07TIP}) at a given point $y$ without assuming convexity:\n\n\\lem{[Quadratic majorization (QM) via $M$-Lipschitz continuous gradients]\n\\label{l:QM}\nLet $f : \\bbR^n \\rightarrow \\bbR$. If $\\nabla f$ is $M$-Lipschitz continuous, then\n\\bes{\nf(x) \\leq f(y) + \\ip{\\nabla f(y)}{x-y} + \\frac{1}{2} \\nm{x - y}_M^2, \\quad \\forall x,y \\in \\bbR^n.\n}\n}\n\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:l:QM} of the supplementary material.\n}\n\nExploiting Definition~\\ref{d:QM} and Lemma~\\ref{l:QM}, the proposed method, BPEG-M, is given as follows.\nTo solve \\R{sys:multiConvx}, we minimize a majorizer of $F$ cyclically over each block $x_1,\\ldots,x_B$, while fixing the remaining blocks at their previously updated variables. Let $x_b^{(i+1)}$ be the value of $x_b$ after its $i\\rth$ update, and define\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\bes{\nf_b^{(i+1)}(x_b) := f \\Big( x_1^{(i+1)}, \\ldots, x_{b-1}^{(i+1)}, x_b, x_{b+1}^{(i)}, \\ldots, x_{B}^{(i)} \\Big), \\quad \\forall b,i.\n}\n\\endgroup\nAt the $b\\rth$ block of the $i\\rth$ iteration, we apply Lemma~\\ref{l:QM} to functional $f_b^{(i+1)}(x_b)$ with a $M_b^{(i+1)}$-Lipschitz continuous gradient, and minimize the majorized function.\\footnote{The quadratically majorized function allows a unique minimizer if $g_b^{(i+1)} (x_b)$ is convex and $\\cX_b^{(i+1)}$ is a convex set (note that $M_b^{(i+1)} \\!\\succ\\! 0$).} Specifically, BPEG-M uses the updates\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\fontsize{9.5pt}{11.4pt}\\selectfont\n\\allowdisplaybreaks\n\\ea{\nx_b^{(i+1)} \n& = \\argmin_{ x_b } \\hspace{0.1em} \\ip{ \\nabla_{x_b} f_b^{(i+1)} (\\acute{x}_b^{(i+1)}) }{ x_b - \\acute{x}_b^{(i+1)} } \\nn\n\\\\\n& \\hspace{4.5em} + \\frac{1}{2} \\nm{ x_b - \\acute{x}_b^{(i+1)} }_{\\widetilde{M}_b^{(i+1)}}^2 + g_b (x_b) \\label{update:x} \n\\\\\n&= \\argmin_{ x_b } \\hspace{0.1em} \\frac{1}{2} \\bigg\\| x_b - \\bigg( \\acute{x}_b^{(i+1)} - \\Big( \\widetilde{M}_b^{(i+1)} \\Big)^{\\!\\!\\!-1} \\nn\n\\\\\n& \\hspace{8.0em} \\cdot \\nabla_{x_b} f_b^{(i+1)} (\\acute{x}_b^{(i+1)}) \\bigg) \\bigg\\|_{\\widetilde{M}_b^{(i+1)}}^2 \\!\\! + g_b (x_b) \\nn\n\\\\\n&= \\mathrm{Prox}_{g_b}^{\\widetilde{M}_b^{(i+1)}} \\!\\! \\bigg( \\! \\underbrace{ \\acute{x}_b^{(i+1)} - \\Big( \\! \\widetilde{M}_b^{(i+1)} \\! \\Big)^{\\!\\!\\!-1} \\! \\nabla_{x_b} f_b^{(i+1)} (\\acute{x}_b^{(i+1)}) }_{\\text{\\emph{extrapolated gradient step using a majorizer} of $f_b^{(i+1)}$}} \\! \\bigg), \\nn\n}\n\\endgroup\nwhere\n\\be{\n\\label{update:xacute}\n\\acute{x}_b^{(i+1)} = x_{b}^{(i)} + E_b^{(i+1)} \\left( x_b^{(i)} - x_b^{(i-1)} \\right),\n}\nthe proximal operator is defined by\n\\bes{\n\\mathrm{Prox}_g^M (y) := \\argmin_{x} \\, \\frac{1}{2} \\nm{x - y}_M^2 + g(x),\n}\n$\\nabla f_b^{(i+1)} (\\acute{x}_b^{(i+1)})$ is the block-partial gradient of $f$ at $\\acute{x}_b^{(i+1)}$, an \\textit{upper-bounded majorization matrix} is updated by\n\\be{\n\\label{update:Mtilde}\n\\widetilde{M}_b^{(i+1)} = \\lambda_b \\cdot M_b^{(i+1)} \\succ 0, \\qquad \\lambda_b > 1,\n}\nand $M_b^{(i+1)} \\!\\in\\! \\bbR^{n_b \\times n_b}$ is a symmetric positive definite \\textit{majorization matrix} of $\\nabla f_b^{(i+1)}$.\nIn \\R{update:xacute}, the $\\bbR^{n_b \\times n_b}$ matrix $E_b^{(i+1)} \\succeq 0$ is an \\textit{extrapolation matrix} that accelerates convergence in solving block multi-convex problems \\cite{Chun&Fessler:18TIP}.\nWe design it in the following form:\n\\be{\n\\label{update:Eb}\nE_b^{(i+1)} = e_b^{(i)} \\cdot \\frac{\\delta (\\lambda_b - 1)}{2 (\\lambda_b + 1)} \\cdot \\left( M_b^{(i+1)} \\right)^{-1\/2} \\left( M_b^{(i)} \\right)^{1\/2}, \n}\nfor some $\\{ 0 \\leq e_b^{(i)} \\leq 1 : \\forall b, i \\}$ and $\\delta < 1$, to satisfy condition \\R{cond:Wb} below.\nIn general, choosing $\\lambda_b$ values in \\R{update:Mtilde}--\\R{update:Eb} to accelerate convergence is application-specific.\nAlgorithm~\\ref{alg:BPGM} summarizes these updates.\n\n\n\n\n\n\n\\begin{algorithm}[t!]\n\\caption{BPEG-M}\n\\label{alg:BPGM}\n\n\\begin{algorithmic}\n\\REQUIRE $\\{ x_b^{(0)} = x_b^{(-1)} : \\forall b \\}$, $\\{ E_b^{(i)} \\in [0, 1], \\forall b,i \\}$, $i=0$\n\n\\WHILE{a stopping criterion is not satisfied}\n\n\\FOR{$b = 1,\\ldots,B$}\n\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\fontsize{9.5pt}{11.4pt}\\selectfont\n\\STATE Calculate  $M_b^{(i+1)}$, $\\displaystyle \\widetilde{M}_b^{(i+1)}$ by \\R{update:Mtilde}, and $E_b^{(i+1)}$ by \\R{update:Eb}\n\\STATE $\\displaystyle \\acute{x}_b^{(i+1)} = \\, x_{b}^{(i)} + E_b^{(i+1)} \\! \\left( x_b^{(i)} - x_b^{(i-1)} \\right)$\n\\STATE $\\displaystyle x_b^{(i+1)} = \\, \\ldots$ \\\\ $\\displaystyle \\mathrm{Prox}_{g_b}^{\\widetilde{M}_b^{(i+1)}} \\!\\! \\bigg( \\!\\! \\acute{x}_b^{(i+1)} \\!-\\! \\Big( \\! \\widetilde{M}_b^{(i+1)} \\! \\Big)^{\\!\\!\\!\\!-1} \\!\\! \\nabla f_b^{(i+1)} (\\acute{x}_b^{(i+1)}) \\!\\! \\bigg)$\n\\endgroup\n\n\\ENDFOR\n\n\\STATE $i = i+1$\n\n\\ENDWHILE\n\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\n\n\nThe majorization matrices $M_b^{(i)}$ and $\\widetilde{M}_b^{(i+1)}$ in \\R{update:Mtilde} influence the convergence rate of BPEG-M.\nA tighter majorization matrix (i.e., a matrix giving tighter bounds in the sense of Lemma~\\ref{l:QM}) provided faster convergence rate \\cite[Lem.~1]{Fessler&etal:93TNS}, \\cite[Fig.~2--3]{Chun&Fessler:18TIP}.\nAn interesting observation in Algorithm~\\ref{alg:BPGM} is that there exists a tradeoff between majorization sharpness via \\R{update:Mtilde} and extrapolation effect via \\R{update:xacute} and \\R{update:Eb}. For example, increasing $\\lambda_b$ (e.g., $\\lambda_b = 2$) allows more extrapolation but results in looser majorization; setting $\\lambda_b \\rightarrow 1$ results in sharper majorization but provides less extrapolation.\n\n\n\\rem{\\label{r:BPGM}\nThe proposed BPEG-M framework -- with key updates \\R{update:x}--\\R{update:xacute} -- generalizes the BPG method \\cite{Xu&Yin:17JSC}, and has several benefits over BPG \\cite{Xu&Yin:17JSC} and BPEG-M introduced earlier in \\cite{Chun&Fessler:18TIP}:\n\\begin{itemize}\n\\item The BPG setup in \\cite{Xu&Yin:17JSC} is a particular case of BPEG-M using a scaled identity majorization matrix $M_b$ with a Lipschitz constant of $\\nabla f_b^{(i+1)} (\\acute{x}_b^{(i+1)})$. The BPEG-M framework can significantly accelerate convergence by allowing sharp majorization; see \\cite[Fig.~2--3]{Chun&Fessler:18TIP} and Fig.~\\ref{fig:Comp:diffBPGM}. \nThis generalization was first introduced for block multi-convex problems in \\cite{Chun&Fessler:18TIP}, \nbut the proposed BPEG-M in this paper addresses the more general problem, block multi-(non)convex optimization.\n\n\\item BPEG-M is useful for controlling the tradeoff between majorization sharpness and extrapolation effect in different blocks, by allowing each block to use different $\\lambda_b$ values. If tight majorization matrices can be designed for a certain block $b$, then it could be reasonable to maintain the majorization sharpness by setting $\\lambda_b$ very close to 1. \nWhen setting $\\lambda_b = 1 + \\epsilon$ (e.g., $\\epsilon$ is a machine epsilon) and using $E_b^{(i+1)} = 0$ (no extrapolation), solutions of the original and its upper-bounded problem become (almost) identical.  In such cases, it is unnecessary to solve the upper bounded problem \\R{update:x}, and the proposed BPEG-M framework allows using the solution of $f_b^{(i+1)}(x_b)$ without QM; see Section~\\ref{sec:CAOL:spCd}. This generalization was not considered in \\cite{Xu&Yin:17JSC}.\n\n\\item The condition for designing the extrapolation matrix \\R{update:Eb}, i.e., \\R{cond:Wb} in Assumption 3, is more general than that in \\cite[(9)]{Chun&Fessler:18TIP} (e.g., \\R{eg:Wb:eigen}).\nSpecifically, the matrices $E_b^{(i+1)}$ and $M_b^{(i+1)}$ in \\R{update:Eb} need not be diagonalized by the same basis.\n\\end{itemize}\n}\n\n\n\n\n\n\n\nThe first two generalizations lead to the question, \\dquotes{Under the sharp QM regime (i.e., having tight bounds in Lemma~\\ref{l:QM}), what is the best way in controlling $\\{ \\lambda_b \\}$ in \\R{update:Mtilde}--\\R{update:Eb} in Algorithm~\\ref{alg:BPGM}?}\nOur experiments show that, if sufficiently sharp majorizers are obtained for partial or all blocks, then giving more weight to sharp majorization provides faster convergence compared to emphasizing extrapolation; for example, $\\lambda_{b} = 1+\\epsilon$ gives faster convergence than $\\lambda_b = 2$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{BPEG-M -- Convergence Analysis} \\label{sec:convg-analysis}\n\nThis section analyzes the convergence of Algorithm~\\ref{alg:BPGM} under the following assumptions.\n\n\\begin{itemize}\n\\item[] {\\em Assumption 1)} $F$ is proper and lower bounded in $\\dom(F)$, $f$ is continuously differentiable, $g_b$ is proper lower semicontinuous, $\\forall b$.\\footnote{\n$F : \\bbR^n \\rightarrow (- \\infty, + \\infty]$ is proper if $\\dom F \\neq \\emptyset$. \n$F$ is lower bounded in $\\dom(F) := \\{ x : F(x) < \\infty \\}$ if $\\inf_{x \\in \\dom(F)} F(x) > -\\infty$.\n$F$ is lower semicontinuous at point $x_0$ if $\\liminf_{x \\rightarrow x_0} F(x) \\geq F(x_0)$.\n}\n\\R{sys:multiConvx} has a critical point $\\bar{x}$, i.e., $0 \\in \\partial F(\\bar{x})$,\nwhere $\\partial F(x)$ denotes the limiting subdifferential of $F$ at $x$ (see \\cite[\\S 1.9]{Kruger:03JMS}, \\cite[\\S 8]{Rockafellar&Wets:book}).\n\\item[] {\\em Assumption 2)} The block-partial gradients of $f$, $\\nabla f_b^{(i+1)}$, are $M_b^{(i+1)}$-Lipschitz continuous, i.e.,\n\\ea{\n\\label{eq:QMbound}\n&~ \\nm{ \\nabla_{x_b} f_b^{(i+1)} (u) - \\nabla_{x_b} f_b^{(i+1)} (v)  }_{\\big( \\! M_b^{(i+1)} \\! \\big)^{\\!-1}} \n\\nn\n\\\\\n& \\leq \\nm{u - v}_{M_b^{(i+1)}},\n}\nfor $u,v \\in \\bbR^{n_b}$, and (unscaled) majorization matrices satisfy $m_{b} I_{n_b} \\preceq M_b^{(i+1)}$ with $0 < m_{b} < \\infty$, $\\forall b, i$.\n\\item[] {\\em Assumption 3)} The extrapolation matrices $E_b^{(i+1)} \\succeq 0$ satisfy\n\\be{\n\\label{cond:Wb}\n\\Big( E_b^{(i+1)} \\Big)^T M_b^{(i+1)} E_b^{(i+1)} \\preceq \\frac{\\delta^2 (\\lambda_b - 1)^2}{4 (\\lambda_b + 1)^2} \\cdot M_b^{(i)},\n}\nfor any $\\delta < 1$, $\\forall b,i$.\n\\end{itemize}\n\n\nCondition \\R{cond:Wb} in Assumption 3 generalizes that in \\cite[Assumption~3]{Chun&Fessler:18TIP}.\nIf eigenspaces of $E_b^{(i+1)}$ and $M_b^{(i+1)}$ coincide (e.g., diagonal and circulant matrices), $\\forall i$ \\cite[Assumption~3]{Chun&Fessler:18TIP}, \\R{cond:Wb} becomes\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\be{\n\\label{eg:Wb:eigen}\nE_b^{(i+1)} \\preceq \\frac{\\delta (\\lambda_b - 1)}{2 (\\lambda_b + 1)} \\cdot  \\left( M_b^{(i)} \\right)^{1\/2} \\left( M_b^{(i+1)} \\right)^{-1\/2},\n} \n\\endgroup\nas similarly given in \\cite[(9)]{Chun&Fessler:18TIP}. \nThis generalization allows one to consider arbitrary structures of $M_b^{(i)}$ across iterations. \n\n\n\n\\lem{[Sequence bounds]\n\\label{l:seqBound}\nLet $\\{ \\widetilde{M}_b : b = 1,\\ldots,B \\}$ and $\\{ E_b : b = 1,\\ldots,B \\}$ be as in \\R{update:Mtilde}--\\R{update:Eb}, respectively. \nThe cost function decrease for the $i\\rth$ update satisfies:\n\\ea{\n\\label{eq:l:seqBound}\nF_b (x_b^{(i)}) - F_b (x_b^{(i+1)}) \n& \\geq \\frac{\\lambda_b - 1}{4} \\nm{ x_b^{(i)} - x_b^{(i+1)} }_{M_b^{(i+1)}}^2 \\nn\n\\\\\n& \\hspace{1.2em} - \\frac{ ( \\lambda_b - 1 ) \\delta^2}{4} \\nm{ x_b^{(i-1)} - x_b^{(i)} }_{ M_b^{(i)}}^2\n}\n}\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:l:seqBound} of the supplementary material.\n}\n\n \nLemma~\\ref{l:seqBound} generalizes \\cite[Lem.~1]{Xu&Yin:17JSC} using $\\{ \\lambda_b = 2 \\}$. \nTaking the majorization matrices in \\R{eq:l:seqBound} to be scaled identities with Lipschitz constants, i.e., $M_b^{(i+1)} \\!=\\! L_b^{(i+1)} \\cdot I$ and $M_b^{(i)} \\!=\\! L_b^{(i)} \\cdot I$, where $L_b^{(i+1)}$ and $L_b^{(i)}$ are Lipschitz constants, the bound \\R{eq:l:seqBound} becomes equivalent to that in \\cite[(13)]{Xu&Yin:17JSC}.\nNote that BPEG-M for block multi-convex problems in \\cite{Chun&Fessler:18TIP} can be viewed within BPEG-M in Algorithm~\\ref{alg:BPGM}, by similar reasons in \\cite[Rem.~2]{Xu&Yin:17JSC} -- bound \\R{eq:l:seqBound} holds for the block multi-convex problems by taking $E_b^{(i+1)}$ in \\R{eg:Wb:eigen} as $E_b^{(i+1)} \\preceq \\delta \\cdot ( M_b^{(i)} )^{1\/2} ( M_b^{(i+1)} )^{-1\/2}$ in \\cite[Prop.~3.2]{Chun&Fessler:18TIP}. \n\n\n\n\n\n\n\\prop{[Square summability]\n\\label{p:sqSum}\nLet $\\{ x^{(i+1)} : i \\geq 0 \\}$ be generated by Algorithm \\ref{alg:BPGM}.\nWe have\n\\be{\n\\label{eq:p:sqSum}\n\\sum_{i=0}^{\\infty} \\nm{ x^{(i)} - x^{(i+1)} }_2^2 < \\infty.\n}\n}\n\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:p:sqSum} of the supplementary material.\n}\n\n\n\n\n\nProposition~\\ref{p:sqSum} implies that\n\\be{\n\\label{eq:convg_to0}\n\\nm{ x^{(i)} - x^{(i+1)} }_2^2 \\rightarrow 0,\n}\nand \\R{eq:convg_to0} is used to prove the following theorem:\n\n\n\n\n\n\\thm{[A limit point is a critical point]\n\\label{t:subseqConv}\nUnder Assumptions 1--3, let $\\{ x^{(i+1)} : i \\geq 0 \\}$ be generated by Algorithm \\ref{alg:BPGM}.\nThen any limit point $\\bar{x}$ of $\\{ x^{(i+1)} : i \\geq 0 \\}$ is a critical point of \\R{sys:multiConvx}. \nIf the subsequence $\\{ x^{(i_j+1)} \\}$ converges to $\\bar{x}$, then\n\\bes{\n\\lim_{j \\rightarrow \\infty} F(x^{(i_j + 1 )}) = F(\\bar{x}).\n} \n}\n\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:t:subseqConv} of the supplementary material.\n}\n\nFinite limit points exist if the generated sequence $\\{ x^{(i+1)} : i \\geq 0 \\}$ is bounded; see, for example, \\cite[Lem.~3.2--3.3]{Bao&Ji&Shen:14ACHA}. For some applications, the boundedness of $\\{ x^{(i+1)} : i \\geq 0 \\}$ can be satisfied by choosing appropriate regularization parameters, e.g., \\cite{Chun&Fessler:18TIP}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Restarting BPEG-M} \\label{sec:reBPGM}\n\nBPG-type methods \\cite{Chun&Fessler:18TIP, Xu&Yin:17JSC, Xu&Yin:13SIAM} can be further accelerated by applying \\textit{1)} a momentum coefficient formula similar to those used in fast proximal gradient (FPG) methods \\cite{Beck&Teboulle:09SIAM, Nesterov:07CORE, Tseng:08techRep}, and\/or \\textit{2)} an adaptive momentum restarting scheme \\cite{ODonoghue&Candes:15FCM, Giselsson&Boyd:14CDC}; see \\cite{Chun&Fessler:18TIP}. \nThis section applies these two techniques to further accelerate BPEG-M in Algorithm~\\ref{alg:BPGM}.\n\nFirst, we apply the following increasing momentum-coefficient formula to \\R{update:Eb} \\cite{Beck&Teboulle:09SIAM}: \n\\be{\n\\label{eq:mom_coeff}\ne_b^{(i+1)} = \\frac{\\theta^{(i)}  - 1}{\\theta^{(i+1)}}, \\quad \\theta^{(i+1)} = \\frac{1 + \\sqrt{1 + 4 (\\theta^{(i)})^2}}{2}.\n}\nThis choice guarantees fast convergence of FPG method \\cite{Beck&Teboulle:09SIAM}.\nSecond, we apply a momentum restarting scheme \\cite{ODonoghue&Candes:15FCM, Giselsson&Boyd:14CDC}, when the following \\textit{gradient-mapping} criterion is met \\cite{Chun&Fessler:18TIP}:\n\\be{\n\\label{eq:restart:grad}\n\\cos \\! \\left( \\Theta \\! \\left( M_b^{(i+1)} \\!\\! \\left( \\acute{x}_b^{(i+1)} - x_b^{(i+1)} \\right),  x_b^{(i+1)} - x_b^{(i)} \\right) \\right) > \\omega,\n}\nwhere the angle between two nonzero real vectors $\\vartheta$ and $\\vartheta'$ is $\\Theta (\\vartheta, \\vartheta') := \\ip{\\vartheta}{\\vartheta'} \/ ( \\nm{\\vartheta}_2 \\nm{\\vartheta'}_2 )$ and $\\omega \\in [-1, 0]$.\nThis scheme restarts the algorithm whenever the momentum, i.e., $x_b^{(i+1)} - x_b^{(i)}$, is likely to lead the algorithm in an unhelpful direction, as measured by the gradient mapping at the $x_b^{(i+1)}$-update.\nWe refer to BPEG-M combined with the methods \\R{eq:mom_coeff}--\\R{eq:restart:grad} as restarting BPEG-M (reBPEG-M). Section~\\ref{sec:reBPGM-supp} in the supplementary material summarizes the updates of reBPEG-M.\n\nTo solve the block multi-nonconvex problems proposed in this paper (e.g., \\R{sys:CAOL:orth}--\\R{sys:CT&CAOL}), we apply reBPEG-M (a variant of Algorithm~\\ref{alg:BPGM}; see Algorithm~\\ref{alg:reBPGM}), promoting fast convergence to a critical point. \n\n\n\n\n\n\n\n\n\n\n\\section{Fast and Convergent CAOL via BPEG-M} \\label{sec:CAOL+BPGM}\n\nThis section applies the general BPEG-M approach to CAOL.\nThe CAOL models \\R{sys:CAOL:orth} and \\R{sys:CAOL:div} satisfy the assumptions of BPEG-M; see Assumption 1--3 in Section~\\ref{sec:convg-analysis}.\nCAOL models \\R{sys:CAOL:orth} and \\R{sys:CAOL:div} readily satisfy Assumption~1 of BPEG-M.\nTo show the continuously differentiability of $f$ and the lower boundedness of $F$, consider that \\textit{1)} $\\sum_{l} \\sum_{k} \\frac{1}{2}  \\left\\| d_k \\circledast x_l - z_{l,k} \\right\\|_2^2$ in \\R{sys:CAOL} is continuously differentiable with respect to $D$ and $\\{ z_{l,k} \\}$;\n\\textit{2)} the sequences $\\{ D^{(i+1)} \\}$ are bounded, because they are in the compact set $\\cD_{\\text{\\R{sys:CAOL:orth}}} = \\{ D: D D^H = \\frac{1}{R} I \\}$ and $\\cD_{\\text{\\R{sys:CAOL:div}}} = \\{ d_k : \\| d_k \\|_2^2 = \\frac{1}{R}, \\forall k \\}$ in \\R{sys:CAOL:orth} and \\R{sys:CAOL:div}, respectively; \nand \\textit{3)} the positive thresholding parameter $\\alpha$ ensures that the sequence $\\{ z_{l,k}^{(i+1)} \\}$ is bounded (otherwise the cost would diverge).\nIn addition, for both \\R{sys:CAOL:orth} and \\R{sys:CAOL:div}, the lower semicontinuity of regularizer $g_b$ holds, $\\forall b$. \nFor $D$-optimization, the indicator function of the sets $\\cD_{\\text{\\R{sys:CAOL:orth}}}$ and $\\cD_{\\text{\\R{sys:CAOL:div}}}$ is lower semicontinuous, because the sets are compact.\nFor $\\{z_{l,k}\\}$-optimization, the $\\ell^0$-quasi-norm is a lower semicontinuous function.\nAssumptions~2 and 3 are satisfied with the majorization matrix designs in this section -- see Sections~\\ref{sec:CAOL:filt}--\\ref{sec:CAOL:spCd} later -- and the extrapolation matrix design in \\R{update:Eb}, respectively.\n\nSince CAOL models \\R{sys:CAOL:orth} and \\R{sys:CAOL:div} satisfy the BPEG-M conditions, we solve \\R{sys:CAOL:orth} and \\R{sys:CAOL:div} by the reBPEG-M method with a two-block scheme, i.e., we alternatively update all filters $D$ and all sparse codes $\\{z_{l,k} : l = 1,\\ldots,L, k=1,\\ldots,K\\}$. \nSections~\\ref{sec:CAOL:filt} and \\ref{sec:CAOL:spCd} describe details of $D$-block and $\\{ z_{l,k} \\}$-block optimization within the BPEG-M framework, respectively.\nThe BPEG-M-based CAOL algorithm is particularly useful for learning convolutional regularizers from large datasets because of its memory flexibility and parallel computing applicability, as described in Section~\\ref{sec:CAOL:memory} and Sections~\\ref{sec:CAOL:filt}--\\ref{sec:CAOL:spCd}, respectively.\n\n\n\n\n\n\n\n\n\\subsection{Filter Update: $D$-Block Optimization} \\label{sec:CAOL:filt}\n\nWe first investigate the structure of the system matrix in the filter update for \\R{sys:CAOL}.\nThis is useful for \\textit{1)} accelerating majorization matrix computation in filter updates (e.g., Lemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI}) and \\textit{2)} applying $R \\!\\times\\! N$-sized adjoint operators (e.g., $\\Psi_l^H$ in \\R{eq:Psi_l} below) to an $N$-sized vector without needing the Fourier approach \\cite[Sec.~\\Romnum{5}-A]{Chun&Fessler:18TIP} that uses commutativity of convolution and Parseval's relation.\nGiven the current estimates of $\\{ z_{l,k} : l=1,\\ldots,L, k=1,\\ldots,K \\}$, the filter update problem of \\R{sys:CAOL} is equivalent to\n\\be{\n\\label{sys:filter}\n\\argmin_{\\{d_k\\}} \\frac{1}{2} \\sum_{k=1}^K \\sum_{l=1}^L \\left\\| \\Psi_l d_k - z_{l,k} \\right\\|_2^2 + \\beta g (D),\n}\nwhere $D$ is defined in \\R{eq:D}, $\\Psi_l \\in \\bbC^{N \\times R}$ is defined by\n\\be{\n\\label{eq:Psi_l}\n\\Psi_l := \\left[ \\begin{array}{ccc} P_{B_1} \\hat{x}_l & \\ldots & P_{B_R} \\hat{x}_l \\end{array} \\right],\n}\n$P_{B_r} \\in \\bbC^{N \\times \\hat{N}}$ is the $r\\rth$ (rectangular) selection matrix that selects $N$ rows corresponding to the indices $B_r = \\{ r, \\ldots, r+N-1 \\}$ from $I_{\\hat{N}}$, $\\{ \\hat{x}_l \\in \\bbC^{\\hat{N}} : l=1,\\ldots,L \\}$ is a set of padded training data, $\\hat{N} = N+R-1$.\nNote that applying $\\Psi_l^H$ in \\R{eq:Psi_l} to a vector of size $N$ is analogous to calculating cross-correlation between $\\hat{x}_l$ and the vector, i.e., $( \\Psi_l^H \\hat{z}_{l,k} )_{r} = \\sum_{n=1}^N \\hat{x}_{n+r-1}^{*} (\\hat{z}_{l,k})_n$, $r=1,\\ldots,R$.\nIn general, $\\hat{(\\cdot)}$ denotes a padded signal vector.\n\n\n\n\n\n\n\n\n\n\\subsubsection{Majorizer Design} \\label{sec:filt:maj}\n\n\n\nThis subsection designs multiple majorizers for the $D$-block optimization and compares their required computational complexity and tightness. \nThe next proposition considers the structure of $\\Psi_l$ in \\R{eq:Psi_l} to obtain the Hessian $\\sum_{l=1}^L \\Psi^H_l \\Psi_l \\in \\bbC^{R \\times R}$ in \\R{sys:filter} for an arbitrary boundary condition.\n\\prop{[Exact Hessian matrix $M_D$] \\label{p:MDexhess}\nThe following matrix $M_D \\in \\bbC^{R \\times R}$ is identical to $\\sum_{l=1}^L \\Psi^H_l \\Psi_l$:\n\\be{\n\\label{eq:soln:filterHess}\n\\left[ M_D \\right]_{r,r'} = \\sum_{l=1}^L \\ip{P_{B_r} \\hat{x}_l}{P_{B_{r'}} \\hat{x}_l}, \\quad r,r'  = 1,\\ldots,R.\n}\n}\n\n\nA sufficiently large number of training signals (with $N \\geq R$), $L$, can guarantee $M_D = \\sum_{l=1}^L \\Psi^H_l \\Psi_l \\succ 0$ in Proposition~\\ref{p:MDexhess}.\nThe drawback of using Proposition~\\ref{p:MDexhess} is its polynomial computational complexity, i.e., $O (L R^2 N)$ -- see Table \\ref{tab:MD:compt}. \nWhen $L$ (the number of training signals) or $N$ (the size of training signals) are large, the quadratic complexity with the size of filters -- $R^2$ -- can quickly increase the total computational costs when multiplied by $L$ and $N$. (The BPG setup in \\cite{Xu&Yin:17JSC} additionally requires $O (R^3)$ because it uses the eigendecomposition of \\R{eq:soln:filterHess} to calculate the Lipschitz constant.)\n\n\n\n\n\n\nConsidering CAOL problems \\R{sys:CAOL} themselves, different from CDL  \\cite{Chun&Fessler:18TIP, Chun&Fessler:17SAMPTA, Wohlberg:16TIP, Heide&eta:15CVPR, Bristow&etal:13CVPR}, the complexity $O (L R^2 N)$ in applying Proposition~\\ref{p:MDexhess} is reasonable. In BPEG-M-based CDL \\cite{Chun&Fessler:18TIP, Chun&Fessler:17SAMPTA}, a majorization matrix for kernel update is calculated every iteration because it depends on updated sparse codes; however, in CAOL, one can precompute $M_D$ via Proposition~\\ref{p:MDexhess} (or Lemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI} below) without needing to change it every kernel update.\nThe polynomial computational cost in applying Proposition~\\ref{p:MDexhess} becomes problematic only when the training signals change.\nExamples include  \\textit{1)} hierarchical CAOL, e.g., CNN in Appendix~\\ref{sec:CNN}, \\textit{2)} \\dquotes{adaptive-filter MBIR} particularly with high-dimensional signals \\cite{Cai&etal:14ACHA, Xu&etal:12TMI, Elad&Aharon:06TIP}, and \\textit{3)} online learning \\cite{Liu&etal:17arXiv, Mairal&etal:09ICML}. Therefore, we also describe a more efficiently computable majorization matrix at the cost of looser bounds (i.e., slower convergence; see Fig~\\ref{fig:Comp:diffBPGM}). \nApplying Lemma~\\ref{l:diag(|At|W|A|1)}, we first introduce a diagonal majorization matrix $M_D$ for the Hessian $\\sum_{l} \\Psi_l^H \\Psi_l$ in \\R{sys:filter}:\n\n\n\n\\begin{table}[!pt]\t\n\n\\centering\n\\renewcommand{\\arraystretch}{1.1}\n\t\n\\caption{Computational complexity of different majorization matrix designs for the filter update problem \\R{sys:filter}}\t\n\\label{tab:MD:compt}\n\t\n\\begin{tabular}{C{2.2cm}C{2cm}}\n\\hline \\hline\nLemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI} & Proposition~\\ref{p:MDexhess} \\\\ \n\\hline\n$O( L R N )$ & $O( L R^2 N )$ \\\\\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\lem{[Diagonal majorization matrix $M_D$] \\label{l:MDdiag}\nThe following matrix $M_D \\in \\bbC^{R \\times R}$ satisfies $M_D \\succeq \\sum_{l=1}^L \\Psi^H_l \\Psi_l$:\n\\be{\n\\label{eq:soln:filterMaj:loose}\nM_D = \\diag \\! \\left( \\sum_{l=1}^L | \\Psi_l^H | | \\Psi_l | 1_{R} \\right),\n}\nwhere $|\\cdot|$ takes the absolute values of the elements of a matrix. \n}\nThe majorization matrix design in Lemma~\\ref{l:MDdiag} is more efficient to compute than that in Proposition~\\ref{p:MDexhess}, because no $R^2$-factor is needed for calculating $M_D$ in Lemma~\\ref{l:MDdiag}, i.e., $O( L R N )$; see Table~\\ref{tab:MD:compt}. \nDesigning $M_D$ in Lemma~\\ref{l:MDdiag} takes fewer calculations than \\cite[Lem.~5.1]{Chun&Fessler:18TIP} using Fourier approaches, when $R \\!<\\! \\log ( \\hat{N} )$.\nUsing Lemma~\\ref{l:diag(|A|1)}, we next design a potentially sharper majorization matrix than \\R{eq:soln:filterMaj:loose}, while maintaining the cost $O( L R N )$:\n\\lem{[Scaled identity majorization matrix $M_D$] \\label{l:MDscaleI}\nThe following matrix $M_D \\in \\bbC^{R \\times R}$ satisfies $M_D \\succsim \\sum_{l=1}^L \\Psi^H_l \\Psi_l$:\n\\be{\n\\label{eq:soln:filterMaj}\nM_D = \\sum_{r=1}^R \\left| \\sum_{l=1}^L \\ip{P_{B_1} \\hat{x}_l}{P_{B_{r}} \\hat{x}_l} \\right| \\cdot I_R,\n}\nfor a circular boundary condition.\n}\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:l:MD} of the supplementary material.\n}\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\begin{tabular}{c}\n\\vspace{-0.25em}\\includegraphics[scale=0.5, trim=0 0.2em 2.2em 1.2em, clip]{.\/Fig_v1\/obj_detDvsrandD_fruit_v1.eps} \\\\\n{\\small (a) The fruit dataset ($L = 10$, $N = 100 \\!\\times\\! 100$)} \\\\\n\\vspace{-0.25em}\\includegraphics[scale=0.5, trim=0 0.2em 2.2em 1em, clip]{.\/Fig_v1\/obj_detDvsrandD_city_v1.eps}  \\\\\n{\\small (b) The city dataset ($L = 10$, $N = 100 \\!\\times\\! 100$)} \\\\\n\\end{tabular}\n\n\\vspace{-0.25em}\n\\caption{Cost minimization comparisons in CAOL \\R{sys:CAOL:orth} with different BPG-type algorithms and datasets ($R \\!=\\! K \\!=\\! 49$ and $\\alpha \\!=\\! 2.5 \\!\\times\\! 10^{-4}$; solution \\R{eq:soln:spCode:exact} was used for sparse code updates; BPG (Xu \\& Ying '17) \\cite{Xu&Yin:17JSC} used the maximum eigenvalue of Hessians for Lipschitz constants; the cross mark \\text{\\sffamily x} denotes a termination point).\nA sharper majorization leads to faster convergence of BPEG-M;\nfor all the training datasets considered in this paper, the majorization matrix in Proposition~\\ref{p:MDexhess} is sharper than those in Lemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI}. \n}\n\\label{fig:Comp:diffBPGM}\n\\end{figure}\n\n\n\n\\begin{figure}[!tp]\n\\centering\n\\begin{tabular}{c}\n\\vspace{-0.25em}\\includegraphics[scale=0.58, trim=0 0.2em 2em 1em, clip]{.\/Fig_v1\/obj_diffBPGM_fruit_paper_v2.eps} \\\\\n{\\small (a) The fruit dataset ($L = 10$, $N = 100 \\!\\times\\! 100$)} \\\\\n\\vspace{-0.25em}\\includegraphics[scale=0.58, trim=0 0.2em 2em 1em, clip]{.\/Fig_v1\/obj_diffBPGM_city_paper_v2.eps}  \\\\\n{\\small (b) The city dataset ($L = 10$, $N = 100 \\!\\times\\! 100$)} \\\\\n\\end{tabular}\n\n\\vspace{-0.25em}\n\\caption{Cost minimization comparisons in CAOL \\R{sys:CAOL:orth} with different BPEG-M algorithms and datasets (Lemma~\\ref{l:MDdiag} was used for $M_D$; $R \\!=\\! K \\!=\\! 49$; deterministic filter initialization and random sparse code initialization). \nUnder the sharp majorization regime, maintaining sharp majorization (i.e., $\\lambda_D \\!=\\! 1+\\epsilon$) provides faster convergence than giving more weight on extrapolation (i.e., $\\lambda_D \\!=\\! 2$).\n(The same behavior was found in sparse-view CT application \\cite[Fig.~3]{Chun&Fessler:18Asilomar}.)\nThere exist no differences in convergence between solution \\R{eq:soln:spCode:exact} and solution \\R{eq:soln:spCode} using $\\{ \\lambda_Z \\!=\\! 1+\\epsilon \\}$. \n}\n\\label{fig:Comp:exact_vs_approx}\n\\end{figure}\n\n\n\n\n\nFor all the training datasets used in this paper, we observed that the tightness of majorization matrices in Proposition~\\ref{p:MDexhess} and Lemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI} for the Hessian $\\sum_{l} \\Psi^H_l \\Psi_l$ is given by\n\\be{\n\\label{eq:MD:tight}\n\\sum_{l=1}^L \\Psi^H_l \\Psi_l = \\text{\\R{eq:soln:filterHess}} \\preceq \\text{\\R{eq:soln:filterMaj}} \\preceq \\text{\\R{eq:soln:filterMaj:loose}}.\n}\n(Note that \\R{eq:soln:filterHess}\\,$\\preceq$\\,\\R{eq:soln:filterMaj:loose} always holds regardless of training data.)\nFig.~\\ref{fig:Comp:diffBPGM} illustrates the effects of the majorizer sharpness in \\R{eq:MD:tight} on CAOL convergence rates.\nAs described in Section~\\ref{sec:BPGM:setup}, selecting $\\lambda_D$ (see \\R{sys:prox:filter:orth} and \\R{sys:prox:filter:div} below) controls the tradeoff between majorization sharpness and extrapolation effect. \nWe found that using fixed $\\lambda_D = 1+\\epsilon$ \ngives faster convergence than $\\lambda_D = 2 $; see Fig.~\\ref{fig:Comp:exact_vs_approx} (this behavior is more obvious in solving the CT MBIR model in \\R{sys:CT&CAOL} via BPEG-M -- see \\cite[Fig.~3]{Chun&Fessler:18Asilomar}).\nThe results in Fig.~\\ref{fig:Comp:exact_vs_approx} and \\cite[Fig.~3]{Chun&Fessler:18Asilomar} show that, under the sharp majorization regime, maintaining sharper majorization is more critical in accelerating the convergence of BPEG-M than giving more weight to extrapolation.\n\n\n\nSections \\ref{sec:prox:filter:orth} and \\ref{sec:prox:filter:div} below apply the majorization matrices designed in this section to proximal mappings of $D$-optimization in \\R{sys:CAOL:orth} and \\R{sys:CAOL:div}, respectively.\n\n\n\n\n\n\n\n\n\n\\subsubsection{Proximal Mapping with Orthogonality Constraint} \\label{sec:prox:filter:orth}\n\nThe corresponding proximal mapping problem of \\R{sys:filter} using the orthogonality constraint in \\R{sys:CAOL:orth} is given by\n\\ea{\n\\label{sys:prox:filter:orth}\n\\{ d_k^{(i+1)} \\} = \\argmin_{\\{ d_k \\}} &~ \n\\sum_{k=1}^K \\frac{1}{2}  \\left\\| d_k - \\nu_k^{(i+1)} \\right\\|_{\\widetilde{M}_D}^2, \\nn\n\\\\\n\\mathrm{subject~to} &~ D D^H = \\frac{1}{R} \\cdot I,\n}\nwhere\n\\ea{\n\\nu_{k}^{(i+1)} &= \\textstyle \\acute{d}_k^{(i+1)} - \\widetilde{M}_D^{-1} \\sum_{l=1}^L \\Psi_l^H \\Big( \\Psi_l \\acute{d}_k^{(i+1)} - z_{l,k} \\Big),\n\\label{eq:nuk} \n\\\\\n\\acute{d}_k^{(i+1)} &= d_k^{(i)} + E_D^{(i+1)} \\Big( d_k^{(i)} - d_k^{(i-1)} \\Big), \\label{eq:dk_ac}\n}\nfor $k=1,\\ldots,K$, and $\\widetilde{M}_D = \\lambda_D M_D$ by \\R{update:Mtilde}. \nOne can parallelize over $k = 1,\\ldots,K$ in computing $\\{ \\nu_k^{(i+1)} \\}$ in \\R{eq:nuk}.\nThe proposition below provides an optimal solution to \\R{sys:prox:filter:orth}:\n\n\\prop{\n\\label{p:orth}\nConsider the following constrained minimization problem:\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\ea{\n\\label{p:eq:TF}\n\\min_{D} ~ \\nm{ \\widetilde{M}_D^{1\/2} D  - \\widetilde{M}_D^{1\/2} \\mathcal{V} }_{\\mathrm{F}}^2, \\quad \n\\mathrm{subj.~to} ~ D D^H = \\frac{1}{R} \\cdot I,\n}\n\\endgroup\nwhere $D$ is given as \\R{eq:D}, $\\mathcal{V} = [\\nu_{1}^{(i+1)} \\cdots \\nu_{K}^{(i+1)}] \\in \\bbC^{R \\times K}$, $\\widetilde{M}_D = \\lambda_D M_D$, and $M_D \\in \\bbR^{R \\times R}$ is given by \\R{eq:soln:filterHess}, \\R{eq:soln:filterMaj:loose}, or \\R{eq:soln:filterMaj}.\nThe optimal solution to \\R{p:eq:TF} is given by\n\\bes{\n D^\\star = \\frac{1}{\\sqrt{R}} \\cdot U \\left[ \\arraycolsep=1pt \\begin{array}{cc} I_R, & 0_{R \\times (K-R)} \\end{array} \\right] V^H, \\quad \\mbox{for}~ R \\leq K,\n}\nwhere $\\widetilde{M}_D \\mathcal{V}$ has (full) singular value decomposition, $\\widetilde{M}_D \\mathcal{V} = U \\Lambda V^H$.\n}\n\\prf{\\renewcommand{\\qedsymbol}{}\nSee Section~\\ref{sec:prf:p:orth} of the supplementary material.\n}\n\n\nWhen using Proposition~\\ref{p:MDexhess}, $\\widetilde{M}_D \\nu_{k}^{(i+1)}$ of $\\widetilde{M}_D \\mathcal{V}$ in Proposition~\\ref{p:orth} simplifies to the following update:\n\\bes{\n\\widetilde{M}_D \\nu_{k}^{(i+1)} = (\\lambda_D - 1) M_D \\acute{d}_k^{(i+1)} + \\sum_{l=1}^L \\Psi_l^H z_{l,k}.\n}\nSimilar to obtaining $\\{ \\nu_k^{(i+1)} \\}$ in \\R{eq:nuk}, computing $\\{ \\widetilde{M}_D \\nu_k^{(i+1)} : k=1,\\ldots,K \\}$ is parallelizable over $k$.\n\n\n\n\n\n\n\n\n\\subsubsection{Proximal Mapping with Diversity Promoting Regularizer}  \\label{sec:prox:filter:div}\n\nThe corresponding proximal mapping problem of \\R{sys:filter} using the norm constraint and diversity promoting regularizer in \\R{sys:CAOL:div} is given by\n\\ea{\n\\label{sys:prox:filter:div}\n\\{ d_k^{(i+1)} \\} = \\argmin_{\\{ d_k \\}} &~ \n\\sum_{k=1}^K \\frac{1}{2}  \\left\\| d_k - \\nu_k^{(i+1)} \\right\\|_{\\widetilde{M}_D}^2 + \\frac{\\beta}{2} g_{\\text{div}}( D ), \\nn\n\\\\\n\\mathrm{subject~to} &~ \\left\\| d_k \\right\\|_2^2 = \\frac{1}{R}, \\quad k = 1,\\ldots, K,\n}\nwhere $g_{\\text{div}}( D )$, $\\nu_{k}^{(i+1)}$, and $\\acute{d}_k^{(i+1)}$ are given as in \\R{sys:CAOL:div}, \\R{eq:nuk}, and \\R{eq:dk_ac}, respectively.\nWe first decompose the regularization term $g_{\\text{div}}(D)$ as follows:\n\\begingroup\n\\allowdisplaybreaks\n\\ea{\n\\label{eq:DtD}\ng_{\\text{div}} (D)\n& = \\sum_{k=1}^K \\sum_{k'=1}^K \\big( d_k^H d_{k'} d_{k'}^H d_k - R^{-1} \\big) \\nn\n\\\\\n& = \\sum_{k=1}^K d_k^H \\bigg( \\sum_{k' \\neq k} d_{k'} d_{k'}^H \\bigg) d_k + \\big( d_k^H d_k - R^{-1} \\big)^2 \\nn\n\\\\\n& = \\sum_{k=1}^K d_k^H \\Gamma_k  d_k, \n}\n\\endgroup\nwhere the equality in \\R{eq:DtD} holds by using the constraint in \\R{sys:prox:filter:div}, and the Hermitian matrix $\\Gamma_k \\in \\bbC^{R \\times R}$ is defined by\n\\be{\n\\label{eq:Ek}\n\\Gamma_k := \\sum_{k' \\neq k} d_{k'} d_{k'}^H.\n}\nUsing \\R{eq:DtD} and \\R{eq:Ek}, we rewrite \\R{sys:prox:filter:div} as\n\\ea{\n\\label{sys:prox:filter2}\nd_k^{(i+1)} = \\argmin_{d_k} &~ \n\\frac{1}{2} \\left\\| d_k - \\nu_k^{(i)} \\right\\|_{\\widetilde{M}_D}^2 + \\frac{\\beta}{2} d_k^H \\Gamma_k d_k, \\nn\n\\\\\n\\mathrm{subject~to} &~ \\left\\| d_k \\right\\|_2^2 = \\frac{1}{R}, \\quad k = 1,\\ldots,K.\n}\nThis is a quadratically constrained quadratic program with $\\{ \\widetilde{M}_D + \\beta \\Gamma_k \\succ 0 : k = 1,\\ldots,K\\}$. \nWe apply an accelerated Newton's method to solve \\R{sys:prox:filter2}; see Section~\\ref{sec:Newton}.\nSimilar to solving \\R{sys:prox:filter:orth} in Section~\\ref{sec:prox:filter:orth}, solving \\R{sys:prox:filter:div} is a small-dimensional problem ($K$ separate problems of size $R$). \n\n\n\n\n\\subsection{Sparse Code Update: $\\{z_{l,k} \\}$-Block Optimization} \\label{sec:CAOL:spCd}\n\nGiven the current estimate of $D$, the sparse code update problem for \\R{sys:CAOL} is given by\n\\be{\n\\label{sys:spCode}\n\\argmin_{\\{ z_{l,k} \\}} \\sum_{l=1}^L \\sum_{k=1}^K \\frac{1}{2} \\left\\| d_k \\circledast x_{l} - z_{l,k} \\right\\|_2^2  +  \\alpha \\left\\| z_{l,k} \\right\\|_0.\n}\nThis problem separates readily, \nallowing parallel computation with $LK$ threads.\nAn optimal solution to \\R{sys:spCode} is efficiently obtained by the well-known hard thresholding:\n\\be{\n\\label{eq:soln:spCode:exact}\nz_{l,k}^{(i+1)}  = \\cH_{\\!\\sqrt{2\\alpha}} \\left( d_k \\circledast x_{l} \\right), \n}\nfor $k=1,\\ldots,K$ and $l=1,\\ldots,L$, where \n\\be{\n\\label{eq:def:hardthr}\n\\cH_a (x)_{n} := \\left\\{ \\begin{array}{cc} 0, & | x_{n} | < a_{n}, \\\\ x_{n}, & | x_{n} | \\geq a_{n}. \\end{array} \\right.\n}\nfor all $n$. \nConsidering $\\lambda_Z$ (in $\\widetilde{M}_Z = \\lambda_Z M_Z$) as $\\lambda_Z \\!\\rightarrow\\! 1$, the solution obtained by the BPEG-M approach becomes equivalent to \\R{eq:soln:spCode:exact}. \nTo show this, observe first that the BPEG-M-based solution (using $M_Z = I_{N}$) to \\R{sys:spCode} is obtained by\n\\ea{\n\\label{eq:soln:spCode}\nz_{l,k}^{(i+1)}  &= \\cH_{\\!\\sqrt{ \\frac{2\\alpha}{\\lambda_Z}}} \\Big( \\zeta_{l,k}^{(i+1)} \\Big),\n\\\\\n\\zeta_{l,k}^{(i+1)} &= \\left( 1 - \\lambda_Z^{-1} \\right) \\cdot \\acute{z}_{l,k}^{(i+1)} + \\lambda_Z^{-1} \\cdot d_k \\circledast x_{l},\n\\nn \\\\\n\\acute{z}_{l,k}^{(i+1)} &= z_{l,k}^{(i)} + E_Z^{(i+1)} \\Big( z_{l,k}^{(i)} - z_{l,k}^{(i-1)} \\Big). \\nn\n}\nThe downside of applying solution \\R{eq:soln:spCode} is that it would require additional memory to store the corresponding extrapolated points -- $\\{ \\acute{z}_{l,k}^{(i+1)} \\}$ -- and the memory grows with $N$, $L$, and $K$.\nConsidering the sharpness of the majorizer in \\R{sys:spCode}, i.e.,  $M_Z = I_{N}$, and the memory issue, it is reasonable to consider the solution \\R{eq:soln:spCode} with no extrapolation, i.e., $\\{ E_Z^{(i+1)} = 0 \\}$:\n\\bes{\nz_{l,k}^{(i+1)} \n= \\cH_{\\! \\sqrt{\\frac{2\\alpha}{\\lambda_Z}}} \\Big( (\\lambda_Z - 1)^{-1}\\lambda_Z \\cdot z_{l,k}^{(i)} + \\lambda_Z^{-1} \\cdot d_k \\circledast x_{l} \\Big)\n}\nbecoming equivalent to \\R{eq:soln:spCode:exact} as $\\lambda_Z \\!\\rightarrow\\! 1$.\n\n\n\n\n\n\nSolution \\R{eq:soln:spCode:exact} has two benefits over \\R{eq:soln:spCode}: compared to \\R{eq:soln:spCode}, \\R{eq:soln:spCode:exact} requires only half the memory to update all $z_{l,k}^{(i+1)}$ vectors and no additional computations related to $\\acute{z}_{l,k}^{(i+1)}$.\nWhile having these benefits, empirically \\R{eq:soln:spCode:exact} has equivalent convergence rates as \\R{eq:soln:spCode} using $\\{ \\lambda_Z \\!=\\! 1+\\epsilon \\}$; see Fig.~\\ref{fig:Comp:exact_vs_approx}. Throughout the paper, we solve the sparse coding problems (e.g., \\R{sys:spCode} and $\\{ z_k \\}$-block optimization in \\R{sys:CT&CAOL}) via optimal solutions in the form of \\R{eq:soln:spCode:exact}.\n\n\n\n\n\n\n\\subsection{Lower Memory Use than Patch-Domain Approaches} \\label{sec:CAOL:memory}\n\nThe convolution perspective in CAOL \\R{sys:CAOL} requires much less memory than conventional patch-domain approaches; \nthus, it is more suitable for learning filters from large datasets or applying the learned filters to high-dimensional MBIR problems.\nFirst, consider the training stage (e.g., \\R{sys:CAOL}).\nThe patch-domain approaches, e.g., \\cite{Cai&etal:14ACHA, Ravishankar&Bressler:15TSP, Aharon&Elad&Bruckstein:06TSP}, require about $R$ times more memory to store training signals. For example, 2D patches extracted by $\\sqrt{R} \\!\\times\\! \\sqrt{R}$-sized windows (with \\dquotes{stride} one and periodic boundaries \\cite{Cai&etal:14ACHA, Papyan&Romano&Elad:17JMLR}, as used in convolution) require about $R$ (e.g., $R = 64$ \\cite{Aharon&Elad&Bruckstein:06TSP, Ravishankar&Bressler:15TSP}) times more memory than storing the original image of size $\\sqrt{N} \\!\\times\\! \\sqrt{N}$. For $L$ training images, their memory usage dramatically increases with a factor $L R N$.\nThis becomes even more problematic in forming hierarchical representations, e.g., CNNs -- see Appendix~\\ref{sec:CNN}.\nUnlike the patch-domain approaches, the memory use of CAOL \\R{sys:CAOL} only depends on the $L N$-factor to store training signals.\nAs a result, the BPEG-M algorithm for CAOL \\R{sys:CAOL:orth} requires about two times less memory than the patch-domain approach \\cite{Cai&etal:14ACHA} (using BPEG-M).\nSee Table~\\ref{tab:AOL}-B.\n(Both the corresponding BPEG-M algorithms use identical computations per iteration that scale with $L R^2 N$; see Table~\\ref{tab:AOL}-A.)\n\n\n\n\n\nSecond, consider solving MBIR problems. Different from the training stage, the memory burden depends on how one applies the learned filters.\nIn \\cite{Pfister&Bresler:17ICASSP}, the learned filters are applied with the conventional convolutional operators -- e.g., $\\circledast$ in \\R{sys:CAOL} -- and, thus, there exists no additional memory burden. \nHowever, in \\cite{Elad&Aharon:06TIP, Chun&etal:17Fully3D, Zheng&etal:19TCI}, the $\\sqrt{R} \\!\\times\\! \\sqrt{R}$-sized learned kernels are applied with a matrix constructed by many overlapping patches extracted from the updated image at each iteration. \nIn adaptive-filter MBIR problems \\cite{Elad&Aharon:06TIP, Cai&etal:14ACHA, Pfister&Bresler:15SPIE}, the memory issue pervades the patch-domain approaches.\n\n\n\n\n\\begin{table}[!t]\t\n\n\\centering\n\\renewcommand{\\arraystretch}{1.1}\n\t\n\\caption{\nComparisons of computational complexity and memory usages between CAOL and patch-domain approach\n}\t\n\\label{tab:AOL}\n\t\n\\begin{tabular}{C{2.15cm}|C{2.85cm}|C{2.3cm}}\n\\hline \\hline\n\\multicolumn{3}{c}{A.~Computational complexity per BPEG-M iteration}\n\\\\\n\\hline\n& Filter update & Sparse code update\n\\\\\n\\hline\nCAOL \\R{sys:CAOL:orth} & $O(L K R N) + O(R^2 K)$ & $O(L K R N)$ \n\\\\ \n\\hline\nPatch-domain \\cite{Cai&etal:14ACHA}$^\\dagger$ & $O (L R^2 N) + O (R^3)$ & $O (L R^2 N)$\n\\\\\n\\hline \\hline\n\\end{tabular}\n\n\\vspace{0.75pc}\n\n\\begin{tabular}{C{2.15cm}|C{2.3cm}|C{2.3cm}}\n\\hline \\hline\n\\multicolumn{3}{c}{B.~Memory usage for BPEG-M algorithm}\n\\\\\n\\hline\n& Filter update & Sparse code update\n\\\\\n\\hline\nCAOL \\R{sys:CAOL:orth} & $O(LN) + O(RK)$ &  $O(L K N)$\n\\\\ \n\\hline\nPatch-domain \\cite{Cai&etal:14ACHA}$^\\dagger$ & $O(L R N) + O(R^2)$ & $O(L R N)$\n\\\\\n\\hline \\hline\n\\end{tabular}\n\n\\medskip\n\\begin{myquote}{0.1in}\n$^\\dagger$\nThe patch-domain approach \\cite{Cai&etal:14ACHA} considers the orthogonality constraint in \\R{sys:CAOL:orth} with $R \\!=\\! K$; see Section~\\ref{sec:CAOL:orth}.\nThe estimates consider all the extracted overlapping patches of size $R$ with the stride parameter $1$ and periodic boundaries, as used in convolution.\n\\end{myquote}\n\\end{table}\n\n\n\n\n\n\n\n\\section{Sparse-View CT MBIR using Convolutional Regularizer Learned via CAOL, and BPEG-M} \\label{sec:CTrecon}\n\nThis section introduces a specific example of applying the learned convolutional regularizer, i.e., $F(D^\\star, \\{ z_{l,k} \\})$ in \\R{sys:CAOL}, from a representative dataset to recover images in \\textit{extreme} imaging that collects highly undersampled or noisy measurements.\nWe choose a sparse-view CT application since it has interesting challenges in reconstructing images that include Poisson noise in measurements, nonuniform noise or resolution properties in reconstructed images, and complicated (or no) structures in the system matrices. \nFor CT, undersampling schemes can significantly reduce the radiation dose and cancer risk from CT scanning. \nThe proposed approach can be applied to other applications (by replacing the data fidelity and spatial strength regularization terms in \\R{sys:CT&CAOL} below).\n\n\n\nWe pre-learn TF filters $\\{ d_k^\\star \\in \\bbR^K : k = 1,\\ldots,K \\}$ via CAOL \\R{sys:CAOL:orth} with a set of high-quality (e.g., normal-dose) CT images $\\{ x_l : l = 1,\\ldots,L \\}$.\nTo reconstruct a linear attenuation coefficient image $x \\in \\bbR^{N'}$ from post-log measurement $y \\in \\bbR^m$ \\cite{Chun&Talavage:13Fully3D, Chun&etal:17Fully3D}, we apply the learned convolutional regularizer to CT MBIR and solve the following block multi-nonconvex problem \\cite{Chun&Fessler:18Asilomar, Crockett&etal:19CAMSAP}:\n\\begingroup\n\\allowdisplaybreaks\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\ea{\n\\label{sys:CT&CAOL}\n\\argmin_{x \\geq 0} & \\underbrace{ \\frac{1}{2} \\left\\| y - A x \\right\\|_W^2 }_{\\text{data fidelity $f(x;y)$}} + \n\\nn \\\\\n\\gamma \\cdot & \\underbrace{ \\min_{\\{ z_k \\}} \\sum_{k=1}^K \\frac{1}{2} \\left\\| d_{k}^\\star \\circledast x - z_k \\right\\|_2^2 \n+ \\alpha' \\sum_{n=1}^{N'} \\psi_{j} \\phi( (z_{k})_n ) }_{\\text{learned convolutional regularizer $g(x,\\{z_k\\}; \\{ d_k \\})$}}. \\tag{P3}\n}\n\\endgroup\n\\!\\!Here, $A \\in \\bbR^{m \\times N'}$ is a CT system matrix, $W \\in \\bbR^{m \\times m}$ is a (diagonal) weighting matrix with elements $\\{ W_{l,l} = \\rho_l^2 \/ ( \\rho_l + \\sigma^2 ) : l  = 1,\\ldots,m \\}$ based on a Poisson-Gaussian model for the pre-log measurements $\\rho \\in \\bbR^m$ with electronic readout noise variance $\\sigma^2$ \\cite{Chun&Talavage:13Fully3D, Chun&etal:17Fully3D, Zheng&etal:19TCI}, $\\psi \\in \\bbR^{N'}$ is a pre-tuned spatial strength regularization vector \\cite{Fessler&Rogers:96TIP} with non-negative elements\n$\\{ \\psi_{n} =  ( \\sum_{l=1}^m A_{l,n}^2 W_{l,l} )^{1\/2} \/ ( \\sum_{l=1}^m A_{l,n}^2 )^{1\/2} : n = 1,\\ldots,N' \\}$\\footnote{\nSee details of computing $\\{ A_{l,j}^2 : \\forall l,j \\}$ in \\cite{Chun&Fessler:18Asilomar}.\n} \nthat promotes uniform resolution or noise properties in the reconstructed image \\cite[Appx.]{Chun&etal:17Fully3D}, an indicator function $\\phi(a)$ is equal to $0$ if $a = 0$, and is $1$ otherwise, $z_k \\in \\bbR^{N'}$ is unknown sparse code for the $k\\text{th}$ filter, \nand $\\alpha' \\!>\\! 0$ is a thresholding parameter. \n\n\n\n\n\nWe solved \\R{sys:CT&CAOL} via reBPEG-M in Section~\\ref{sec:reBPG-M} with a two-block scheme \\cite{Chun&Fessler:18Asilomar}, and summarize the corresponding BPEG-M updates as\n\\begingroup\n\\allowdisplaybreaks\n\\ea{\n\\label{eq:soln:CT:bpgm}\nx^{(i+1)} &= \\bigg[ \\big( \\widetilde{M}_A + \\gamma I_R \\big)^{-1} \\cdot \\bigg( \\widetilde{M}_A \\eta^{(i+1)} + \n\\nn \\\\\n& \\hspace{1.95em} \\gamma \\sum_{k=1}^K ( P_f d_k^\\star ) \\circledast  \\cH_{\\!\\sqrt{2 \\alpha' \\psi}} \\big( d_k^\\star \\circledast x^{(i)} \\big) \\bigg) \\bigg]_{\\geq 0},\n}\nwhere \n\\ea{\n\\eta^{(i+1)} &= \\acute{x}^{(i+1)} - \\widetilde{M}_A^{-1} A^T W \\Big( A \\acute{x}^{(i+1)} -y \\Big),\n\\label{eq:soln:CT:bpgm:eta} \\\\\n\\acute{x}^{(i+1)} &= x^{(i)} + E_A^{(i+1)} \\Big( x^{(i)} - x^{(i-1)} \\Big),\n\\nn\n}\n\\endgroup\n$\\widetilde{M}_A = \\lambda_A M_A$ by \\R{update:Mtilde}, a diagonal majorization matrix $M_A \\succeq A^T W A$ is designed by Lemma~\\ref{l:diag(|At|W|A|1)}, \nand $P_f \\in \\bbC^{R \\times R}$ flips a column vector in the vertical direction (e.g., it rotates 2D filters by $180^\\circ$).\nInterpreting the update \\R{eq:soln:CT:bpgm} leads to the following two remarks:\n\n\n\n\n\n\n\n\\rem{\n\\label{r:autoenc}\nWhen the convolutional regularizer learned via CAOL \\R{sys:CAOL:orth} is applied to MBIR, it works as an autoencoding CNN: \n\\be{\n\\label{eq:autoenc}\n\\cM ( x ) = \\sum_{k=1}^K (P_f d_k^\\star) \\circledast \\cH_{\\!\\sqrt{2 \\alpha_k'}} \\left( d_k^\\star \\circledast x \\right)\n}\n(setting $\\psi = 1_{N'}$ and generalizing $\\alpha'$ to $\\{ \\alpha'_k : k=1,\\ldots,K \\}$ in \\R{sys:CT&CAOL}).\nThis is an explicit mathematical motivation\nfor constructing architectures of iterative regression CNNs for MBIR, \ne.g., BCD-Net \\cite{Chun&Fessler:18IVMSP, Chun&etal:19MICCAI, Lim&etal:19TMI, Lim&etal:18NSSMIC}\nand Momentum-Net \\cite{Chun&etal:18Allerton, Chun&etal:18arXiv:momnet}. \nParticularly when the learned filters $\\{ d_k^\\star \\}$ in \\R{eq:autoenc} satisfy the TF condition, they are useful for compacting energy of an input signal $x$ and removing unwanted features via the non-linear thresholding in \\R{eq:autoenc}.\n}\n\n\n\n\\rem{\n\\label{r:autoenc-bpgm}\nUpdate \\R{eq:soln:CT:bpgm} improves the solution $x^{(i+1)}$ by weighting between \\textit{a)} the extrapolated point considering the data fidelity, i.e., $\\eta^{(i+1)}$ in \\R{eq:soln:CT:bpgm:eta}, and \\textit{b)} the \\dquotes{refined} update via the ($\\psi$-weighting) convolutional autoencoder, i.e., $\\sum_{k} ( P_f d_k^\\star ) \\circledast  \\cH_{\\!\\sqrt{2 \\alpha' \\psi}} ( d_k^\\star \\circledast x^{(i)} )$.\n}\n\n\n\n\n\n\n\n\n\n\\section{Results and Discussion} \\label{sec:result}\n\n\\subsection{Experimental Setup} \\label{sec:exp}\n\nThis section examines the performance (e.g., scalability, convergence, and acceleration) and behaviors (e.g., effects of model parameters on filters structures and effects of dimensions of learned filter on MBIR performance) of the proposed CAOL algorithms and models, respectively.\n\n\\subsubsection{CAOL} \\label{sec:exp:CAOL}\n\nWe tested the introduced CAOL models\/algorithms for four datasets: \n\\textit{1)} the fruit dataset with $L = 10$ and $N = 100 \\!\\times\\! 100$ \\cite{Zeiler&etal:10CVPR}; \n\\textit{2)} the city dataset with $L = 10$ and $N = 100 \\!\\times\\! 100$ \\cite{Heide&eta:15CVPR}; \n\\textit{3)} the CT dataset of $L = 80$ and $N = 128 \\!\\times\\! 128$, created by dividing down-sampled $512 \\!\\times\\! 512$ XCAT phantom slices \\cite{Segars&etal:08MP} into $16$ sub-images \\cite{Olshausen&Field:96Nature, Bristow&etal:13CVPR} -- referred to the CT-(\\romnum{1}) dataset; \n\\textit{4)} the CT dataset of with $L = 10$ and $N = 512 \\!\\times\\! 512$ from down-sampled $512 \\!\\times\\! 512$ XCAT phantom slices \\cite{Segars&etal:08MP} -- referred to the CT-(\\romnum{2}) dataset.\nThe preprocessing includes intensity rescaling to $[0,1]$ \\cite{Zeiler&etal:10CVPR, Bristow&etal:13CVPR, Heide&eta:15CVPR} and\/or (global) mean substraction \\cite[\\S 2]{Jarrett&etal:09ICCV}, \\cite{Aharon&Elad&Bruckstein:06TSP}, as conventionally used in many sparse coding studies, e.g., \\cite{Aharon&Elad&Bruckstein:06TSP, Jarrett&etal:09ICCV, Zeiler&etal:10CVPR, Bristow&etal:13CVPR, Heide&eta:15CVPR}. \nFor the fruit and city datasets, we trained $K = 49$ filters of size $R = 7 \\!\\times\\! 7$.\nFor the CT dataset~(\\romnum{1}), we trained filters of size $R = 5 \\!\\times\\! 5$, with $K = 25$ or $K = 20$.\nFor CT reconstruction experiments, we learned the filters from the CT-(\\romnum{2}) dataset; however, we did not apply mean subtraction because it is not modeled in \\R{sys:CT&CAOL}. \n\nThe parameters for the BPEG-M algorithms were defined as follows.\\footnote{The remaining BPEG-M parameters not described here are identical to those in \\cite[\\Romnum{7}-A2]{Chun&Fessler:18TIP}.}\nWe set the regularization parameters $\\alpha, \\beta$ as follows:\n\\begin{itemize}\n\\item {CAOL \\R{sys:CAOL:orth}}: To investigate the effects of $\\alpha$, we tested \\R{sys:CAOL:orth} with different $\\alpha$'s in the case $R = K$. For the fruit and city datasets, we used $\\alpha = 2.5 \\!\\times\\! \\{ 10^{-5}, 10^{-4} \\}$; for the CT-(\\romnum{1}) dataset, we used $\\alpha = \\{ 10^{-4}, 2 \\!\\times\\! 10^{-3} \\}$. \nFor the CT-(\\romnum{2}) dataset (for CT reconstruction experiments), see details in \\cite[Sec.~\\Romnum{5}1]{Chun&Fessler:18Asilomar}.\n\n\\item {CAOL \\R{sys:CAOL:div}}: Once $\\alpha$ is fixed from the CAOL \\R{sys:CAOL:orth} experiments above, we tested \\R{sys:CAOL:div} with different $\\beta$'s to see its effects in the case $R > K$. For the CT-(\\romnum{1}) dataset, we fixed $\\alpha = 10^{-4}$, and used $\\beta = \\{ 5 \\!\\times\\! 10^6, 5 \\!\\times\\! 10^4 \\}$.\n\\end{itemize}\nWe set $\\lambda_D = 1+\\epsilon$ as the default.\nWe initialized filters in either deterministic or random ways.\nThe deterministic filter initialization follows that in \\cite[Sec.~3.4]{Cai&etal:14ACHA}.\nWhen filters were randomly initialized, we used a scaled one-vector for the first filter.\nWe initialize sparse codes mainly with a deterministic way that applies \\R{eq:soln:spCode:exact} based on $\\{ d_k^{(0)} \\}$.\nIf not specified, we used the random filter and deterministic sparse code initializations.\nFor BPG \\cite{Xu&Yin:17JSC}, we used the maximum eigenvalue of Hessians for Lipschitz constants in \\R{sys:filter}, and applied the gradient-based restarting scheme in Section~\\ref{sec:reBPGM}.\nWe terminated the iterations if the relative error stopping criterion (e.g., \\cite[(44)]{Chun&Fessler:18TIP}) is met before reaching the maximum number of iterations.\nWe set the tolerance value as $10^{-13}$ for the CAOL algorithms using Proposition~\\ref{p:MDexhess}, and $10^{-5}$ for those using Lemmas~\\ref{l:MDdiag}--\\ref{l:MDscaleI}, and the maximum number of iterations to $2 \\!\\times\\! 10^4$.\n\nThe CAOL experiments used the convolutional operator learning toolbox~\\cite{chun:19:convolt}.\n\n\n\n\n\\subsubsection{Sparse-View CT MBIR with Learned Convolutional Regularizer via CAOL} \\label{sec:exp:CT}\n\nWe simulated sparse-view sinograms of size $888 \\times 123$  (\\quotes{detectors or rays} $\\times$ \\quotes{regularly spaced projection views or angles}, where $984$ is the number of full views) with GE LightSpeed fan-beam geometry corresponding to a monoenergetic source with $10^5$ incident photons per ray and no background events, and electronic noise variance $\\sigma^2 \\!=\\! 5^2$. \nWe avoided an inverse crime in our imaging simulation and reconstructed images with a coarser grid with $\\Delta_x \\!=\\! \\Delta_y \\!=\\! 0.9766$ mm; see details in \\cite[Sec.~\\Romnum{5}-A2]{Chun&Fessler:18Asilomar}.\n\n\n\n\nFor EP MBIR, we finely tuned its regularization parameter to achieve both good root mean square error (RMSE) and structural similarity index measurement \\cite{Wang&etal:04TIP} values. \nFor the CT MBIR model \\R{sys:CT&CAOL}, we chose the model parameters $\\{ \\gamma, \\alpha' \\}$ that showed a good tradeoff between the data fidelity term and the learned convolutional regularizer, and set $\\lambda_A \\!=\\! 1+\\epsilon$.\nWe evaluated the reconstruction quality by the RMSE (in a modified Hounsfield unit, HU, where air is $0$ HU and water is $1000$ HU) in a region of interest.\nSee further details in \\cite[Sec.~\\Romnum{5}-A2]{Chun&Fessler:18Asilomar} and Fig.~\\ref{fig:CTrecon}.\n\n\nThe imaging simulation and reconstruction experiments used the Michigan image reconstruction toolbox~\\cite{fessler:16:irt}.\n\n\n\n\n\n\n\n\n\\subsection{CAOL with BPEG-M} \\label{sec:result:CAOL}\n\n\nUnder the sharp majorization regime (i.e., partial or all blocks have sufficiently tight bounds in Lemma~\\ref{l:QM}), the proposed convergence-guaranteed BPEG-M can achieve significantly faster CAOL convergence rates compared with the state-of-the-art BPG algorithm \\cite{Xu&Yin:17JSC} for solving block multi-nonconvex problems, by several generalizations of  BPG (see Remark~\\ref{r:BPGM}) and two majorization designs (see Proposition~\\ref{p:MDexhess} and Lemma~\\ref{l:MDscaleI}). See Fig.~\\ref{fig:Comp:diffBPGM}.\nIn controlling the tradeoff between majorization sharpness and extrapolation effect of BPEG-M (i.e., choosing $\\{ \\lambda_b \\}$ in \\R{update:Mtilde}--\\R{update:Eb}), maintaining majorization sharpness is more critical than gaining stronger extrapolation effects to accelerate convergence under the sharp majorization regime.\nSee Fig.~\\ref{fig:Comp:exact_vs_approx}. \n\n\nWhile using about two times less memory (see Table~\\ref{tab:AOL}), CAOL \\R{sys:CAOL} learns TF filters corresponding to those given by the patch-domain TF learning in \\cite[Fig.~2]{Cai&etal:14ACHA}.\nSee Section~\\ref{sec:CAOL:memory} and Fig.~\\ref{fig:filters_BPGM} with deterministic $\\{ d_k^{(0)} \\}$.\nNote that BPEG-M-based CAOL \\R{sys:CAOL} requires even less memory than BPEG-M-based CDL in \\cite{Chun&Fessler:18TIP}, by using exact sparse coding solutions (e.g., \\R{eq:soln:spCode:exact} and \\R{eq:soln:CT:bpgm}) without saving their extrapolated points.\nIn particular,  \nwhen tested with the large CT dataset of $\\{ L \\!=\\! 40, N \\!=\\! 512 \\!\\times\\! 512 \\}$,\nthe BPEG-M-based CAOL algorithm ran fine,\nwhile BPEG-M-based CDL \\cite{Chun&Fessler:18TIP} and patch-domain AOL \\cite{Cai&etal:14ACHA} were terminated due to exceeding available memory.\\footnote{\nTheir double-precision MATLAB implementations were tested on 3.3 GHz Intel Core i5 CPU with 32 GB RAM.\n}\nIn addition, the CAOL models \\R{sys:CAOL:orth} and \\R{sys:CAOL:div} are easily parallelizable with $K$ threads.\nCombining these results, the BPEG-M-based CAOL is a reasonable choice for learning filters from large training datasets. \nFinally, \\cite{Chun&etal:19SPL} shows theoretically how using many samples can improve CAOL, \naccentuating the benefits of the low memory usage of CAOL.\n\n\n\nThe effects of parameters for the CAOL models are shown as follows.\nIn CAOL~\\R{sys:CAOL:orth}, as the thresholding parameter $\\alpha$ increases, the learned filters have more elongated structures; see Figs.~\\ref{fig:CAOL:filters:CT}(a) and \\ref{fig:CAOL:filters:fruit&city&ct-ii}.\nIn CAOL~\\R{sys:CAOL:div}, when $\\alpha$ is fixed, increasing the filter diversity promoting regularizer $\\beta$ successfully lowers coherences between filters (e.g., $g_{\\text{div}}(D)$ in \\R{sys:CAOL:div}); see Fig.~\\ref{fig:CAOL:filters:CT}(b).\n\n\n\n\\begin{figure}[!t]\n\\small\\addtolength{\\tabcolsep}{-4pt}\n\\centering\n\n\\begin{tabular}{cc}\n\n\\vspace{-0.1em}\\hspace{-0.05em}\\includegraphics[scale=0.5, trim=2em 5.2em 0em 4em, clip]{.\/Fig\/filt_tfMexhess_CTeg1.eps} &\n\\vspace{-0.1em}\\hspace{-0.4em}\\includegraphics[scale=0.5, trim=2em 5.2em 0.1em 4em, clip]{.\/Fig\/filt_tfMexhess_CTeg2.eps} \\\\\n\n{\\small (a1) $\\alpha = 10^{-4}$} & {\\small (a2) $\\alpha = 2 \\!\\times\\! 10^{-3}$} \\\\\n\n\\multicolumn{2}{c}{\\small (a) Learned filters via CAOL \\R{sys:CAOL:orth} ($R = K = 25$)} \\\\\n\n\\includegraphics[scale=0.506, trim=2em 1.0em 0em 0.4em, clip]{.\/Fig\/filt_divMexhess_CTeg2-1.eps} &\n\\includegraphics[scale=0.506, trim=2em 1.0em 0em 0.4em, clip]{.\/Fig\/filt_divMexhess_CTeg1-1.eps} \\vspace{-0.3em} \\\\\n\n\\small{$g_{\\text{div}}(D) = \\mathbf{8.96 \\!\\times\\! 10^{-6}}$} & \\small{$g_{\\text{div}}(D) = \\mathbf{0.12}$} \\\\\n\n\\small{(b1) $\\alpha = 10^{-4}$, $\\beta = 5 \\!\\times\\! 10^{6}$} & {\\small (b2) $\\alpha = 10^{-4}$, $\\beta = 5 \\!\\times\\! 10^{4}$}  \\\\\n\n\\multicolumn{2}{c}{\\small (b) Learned filters via CAOL \\R{sys:CAOL:div} ($R = 25, K = 20$) } \n\n\\end{tabular}\n\n\\vspace{-0.25em}\n\\caption{Examples of learned filters with different CAOL models and parameters (Proposition~\\ref{p:MDexhess} was used for $M_D$; the CT-(\\romnum{1}) dataset with a symmetric boundary condition).}\n\\label{fig:CAOL:filters:CT}\n\\end{figure}\n\n\n\n\nIn adaptive MBIR (e.g., \\cite{Elad&Aharon:06TIP, Cai&etal:14ACHA, Pfister&Bresler:15SPIE}), one may apply adaptive image denoising \\cite{Donoho:95TIT, Donoho&Johnstone:95JASA, Chang&Yu&Vetterli:00TIP, Blu&Luisier:07TIP, Liu&etal:15CVPR, Pfister&Bresler:17ICASSP} to optimize thresholding parameters. \nHowever, if CAOL \\R{sys:CAOL} and testing the learned convolutional regularizer to MBIR (e.g., \\R{sys:CT&CAOL}) are separated, selecting \\dquotes{optimal} thresholding parameters in (unsupervised) CAOL is challenging -- similar to existing dictionary or analysis operator learning methods.\nOur strategy to select the thresholding parameter $\\alpha$ in CAOL~\\R{sys:CAOL:orth} (with $R=K$) is given as follows.\nWe first apply the first-order finite difference filters $\\{ d_k : \\| d_k \\|_2^2 = 1\/R, \\forall k \\}$ (e.g., $\\frac{1}{\\sqrt{2R}} [1, -1]^T$ in 1D) to all training signals and find their sparse representations, and then find $\\alpha_{\\mathrm{est}}$ that corresponds to the largest $95 (\\pm 1)\\%$ of non-zero elements of the sparsified training signals. This procedure defines the range $[ \\frac{1}{10} \\alpha_{\\mathrm{est}}, \\alpha_{\\mathrm{est}}]$ to select desirable $\\alpha^\\star$ and its corresponding filter $D^\\star$. We next ran CAOL~\\R{sys:CAOL:orth} with multiple $\\alpha$ values within this range. Selecting $\\{ \\alpha^\\star, D^\\star \\}$ depends on application.  \nFor CT MBIR, $D^\\star$ that both has (short) first-order finite difference filters and captures diverse (particularly diagonal) features of training signals, gave good RMSE values and well preserved edges; see Fig.~\\ref{fig:CAOL:filters:fruit&city&ct-ii}(c) and \\cite[Fig.~2]{Chun&Fessler:18Asilomar}. \n\n\n\n\n\n\\begin{figure*}[!t]\n\\centering\n\\small\\addtolength{\\tabcolsep}{-6.5pt}\n\\renewcommand{\\arraystretch}{0.1}\n\n    \\begin{tabular}{ccccc}\n\n\n      \\small{(a) Ground truth} \n      & \n      \\small{(b) Filtered back-projection} & \\small{(c) EP} \n      & \n      \\specialcell[c]{\\small (d) Proposed MBIR \\mbox{\\R{sys:CT&CAOL}}, \\\\ \\small with \\R{eq:autoenc} of $R \\!=\\! K \\!=\\! 25$} \n      & \n      \\specialcell[c]{\\small (e) Proposed MBIR \\mbox{\\R{sys:CT&CAOL}}, \\\\ \\small with \\R{eq:autoenc} of $R \\!=\\! K \\!=\\! 49$} \\\\\n       \n       \\begin{tikzpicture}\n       \t\t\\begin{scope}[spy using outlines={rectangle,yellow,magnification=1.25,size=15mm,connect spies}]\n       \t\t\t\\node {\\includegraphics[viewport=35 105 275 345, clip, width=3.4cm,height=3.4cm]{.\/Fig_CTrecon\/x_true.png}};\n\t\t\t\\spy on (0.8,-0.95) in node [left] at (-0.3,-2.2);\n                \t\t\\draw [->, thick, red] (0.1,0.85+0.3) -- (0.1,0.85);\n                \t\t\\draw [->, thick, red] (-0.65+0.3,0.1) -- (-0.65,0.1);\n                \t\t\\draw [->, thick, red] (-1.1,-1.3) -- (-1.1+0.3,-1.3);  \n\t\t\t\\draw [red] (-1.35,0.15) circle [radius=0.28];\n\t\t\t\\draw [red] (-1.35,-0.45) circle [radius=0.28];\n\t\t\\end{scope}\n        \\end{tikzpicture}   \n        &\n        \\begin{tikzpicture}\n        \t\t\\begin{scope}[spy using outlines={rectangle,yellow,magnification=1.25,size=15mm,connect spies}]\n       \t\t\t\\node {\\includegraphics[viewport=35 105 275 345, clip, width=3.4cm,height=3.4cm]{.\/Fig_CTrecon\/x123_fbp.png}};\n\t\t\t\\spy on (0.8,-0.95) in node [left] at (-0.3,-2.2);\n\t\t\t\\node [black] at (0.7,-1.95) {\\small $\\mathrm{RMSE} = 82.8$};\n\t\t\\end{scope}\n        \\end{tikzpicture}   \n        &\n        \\begin{tikzpicture}\n        \t\t\\begin{scope}[spy using outlines={rectangle,yellow,magnification=1.25,size=15mm,connect spies}]\n       \t\t\t\\node {\\includegraphics[viewport=35 105 275 345, clip, width=3.4cm,height=3.4cm]{.\/Fig_CTrecon\/x123_ep.png}};\n\t\t\t\\spy on (0.8,-0.95) in node [left] at (-0.3,-2.2);\n                \t\t\\draw [->, thick, red] (0.1,0.85+0.3) -- (0.1,0.85);\n                \t\t\\draw [->, thick, red] (-0.65+0.3,0.1) -- (-0.65,0.1);\n                \t\t\\draw [->, thick, red] (-1.1,-1.3) -- (-1.1+0.3,-1.3);  \n\t\t\t\\node [black] at (0.7,-1.95) {\\small $\\mathrm{RMSE} = 40.8$};\n\t\t\\end{scope}\n        \\end{tikzpicture}   \n        &\n        \\begin{tikzpicture}\n        \t\t\\begin{scope}[spy using outlines={rectangle,yellow,magnification=1.25,size=15mm,connect spies}]\n       \t\t\t\\node {\\includegraphics[viewport=35 105 275 345, clip, width=3.4cm,height=3.4cm]{.\/Fig_CTrecon_v1\/x123_caol5x5_old.png}};\n\t\t\t\\spy on (0.8,-0.95) in node [left] at (-0.3,-2.2);\n                \t\t\\draw [->, thick, red] (0.1,0.85+0.3) -- (0.1,0.85);\n                \t\t\\draw [->, thick, red] (-0.65+0.3,0.1) -- (-0.65,0.1);\n                \t\t\\draw [->, thick, red] (-1.1,-1.3) -- (-1.1+0.3,-1.3);  \n\t\t\t\\draw [red] (-1.35,0.15) circle [radius=0.28];\n\t\t\t\\draw [red] (-1.35,-0.45) circle [radius=0.28];\n\t\t\t\\node [blue] at (0.7,-1.95) {\\small $\\mathrm{RMSE} = 35.2$};\n\t\t\\end{scope}\n        \\end{tikzpicture}   \n        &\n        \\begin{tikzpicture}\n        \t\t\\begin{scope}[spy using outlines={rectangle,yellow,magnification=1.25,size=15mm,connect spies}]\n       \t\t\t\\node {\\includegraphics[viewport=35 105 275 345, clip, width=3.4cm,height=3.4cm]{Fig_CTrecon_v1\/x123_caol7x7_old.png}};\n\t\t\t\\spy on (0.8,-0.95) in node [left] at (-0.3,-2.2);\n                \t\t\\draw [->, thick, red] (0.1,0.85+0.3) -- (0.1,0.85);\n                \t\t\\draw [->, thick, red] (-0.65+0.3,0.1) -- (-0.65,0.1);\n                \t\t\\draw [->, thick, red] (-1.1,-1.3) -- (-1.1+0.3,-1.3);  \n\t\t\t\\draw [red] (-1.35,0.15) circle [radius=0.28];\n\t\t\t\\draw [red] (-1.35,-0.45) circle [radius=0.28];\n\t\t\t\\node [red] at (0.7,-1.95) {\\small $\\mathrm{RMSE} = 34.7$};\n\t\t\\end{scope}\n        \\end{tikzpicture}          \n\n    \\end{tabular}\n\n \\vspace{-0.25em}\n\\caption{\nComparisons of reconstructed images from different reconstruction methods for sparse-view CT ($123$ views ($12.5$\\% sampling); for the MBIR model \\R{sys:CT&CAOL}, convolutional regularizers were trained by CAOL \\R{sys:CAOL:orth} -- see \\cite[Fig.~2]{Chun&Fessler:18Asilomar}; display window is within $[800, 1200]$ HU) \\cite{Chun&Fessler:18Asilomar}.  \nThe MBIR model \\R{sys:CT&CAOL} using convolutional sparsifying regularizers trained via CAOL \\R{sys:CAOL:orth} shows higher image reconstruction accuracy compared to the EP reconstruction; see red arrows and magnified areas. \nFor the MBIR model \\R{sys:CT&CAOL}, the autoencoder (see Remark~\\ref{r:autoenc}) using the filter dimension $R \\!=\\! K \\!=\\! 49$ improves reconstruction accuracy of that using $R \\!=\\! K \\!=\\! 25$; compare the results in (d) and (e).\nIn particular, the larger dimensional filters improve the edge sharpness of reconstructed images; see circled areas. The corresponding error maps are shown in Fig.~\\ref{fig:CTrecon:err} of the supplementary material.\n}\n\\label{fig:CTrecon}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsection{Sparse-View CT MBIR with Learned Convolutional Sparsifying Regularizer (via CAOL) and BPEG-M} \\label{sec:result:CT}\n\n\n\nIn sparse-view CT using only $12.5$\\% of the full projections views, the CT MBIR \\R{sys:CT&CAOL} using the learned convolutional regularizer via CAOL \\R{sys:CAOL:orth} outperforms EP MBIR; it reduces RMSE by approximately $5.6$--$6.1$HU.\nSee the results in Figs.~\\ref{fig:CTrecon}(c)--(e).\nThe model \\R{sys:CT&CAOL} can better recover high-contrast regions (e.g., bones) -- see red arrows and magnified areas in Fig.~\\ref{fig:CTrecon}(c)--(e). \nNonetheless, the filters with $R\\!=\\!K\\!=\\!5^2$ in the ($\\psi$-weighting) autoencoding CNN, i.e., $\\sum_{k} ( P_f d_k^\\star ) \\circledast  \\cH_{\\!\\sqrt{2 \\alpha' \\psi}} ( d_k^\\star \\circledast (\\cdot) )$ in \\R{eq:autoenc}, can blur edges in low-contrast regions (e.g., soft tissues) while removing noise. See Fig.~\\ref{fig:CTrecon}(d) -- the blurry issues were similarly observed in \\cite{Chun&etal:17Fully3D, Zheng&etal:19TCI}. \nThe larger dimensional kernels (i.e., $R\\!=\\!K\\!=\\!7^2$) in the convolutional autoencoder can moderate this issue, while further reducing RMSE values; compare the results in Fig.~\\ref{fig:CTrecon}(d)--(e).\nIn particular, the larger dimensional convolutional kernels capture more diverse features -- see \\cite[Fig.~2]{Chun&Fessler:18Asilomar}) -- and the diverse features captured in kernels are useful to further improve the performance of the proposed MBIR model \\R{sys:CT&CAOL}. (The importance of diverse features in kernels was similarly observed in CT experiments with the learned autoencoders having a fixed kernel dimension; see Fig.~\\ref{fig:CAOL:filters:fruit&city&ct-ii}(c).)\nThe RMSE reduction over EP MBIR is comparable to that of CT MBIR \\R{sys:CT&CAOL} \nusing the $\\{ R, K \\!=\\! 8^2 \\}$-dimensional filters trained via the patch-domain AOL \\cite{Ravishankar&Bressler:15TSP}; \nhowever, at each BPEG-M iteration, this MBIR model using the trained (non-TF) filters via patch-domain AOL \\cite{Ravishankar&Bressler:15TSP} requires more computations than the proposed CT MBIR model \\R{sys:CT&CAOL} using the learned convolutional regularizer via CAOL \\R{sys:CAOL:orth}.\nSee related results and discussion in \nFig.~\\ref{fig:CTrecon:TL} and Section~\\ref{sec:result:supp}, respectively.  \n \n\n\n\nOn the algorithmic side, the BPEG-M framework can guarantee the convergence of CT MBIR \\R{sys:CT&CAOL}.\nUnder the sharp majorization regime in BPEG-M, maintaining the majorization sharpness is more critical than having stronger extrapolation effects -- see \\cite[Fig.~3]{Chun&Fessler:18Asilomar}, as similarly shown in CAOL experiments (see Section~\\ref{sec:result:CAOL}).\n\n\n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nDeveloping rapidly converging and memory-efficient CAOL engines is important, since it is a basic element in training CNNs in an unsupervised learning manner (see Appendix~\\ref{sec:CNN}). \nStudying structures of convolutional kernels is another fundamental issue, since it can avoid learning redundant filters or provide energy compaction properties to filters.\nThe proposed BPEG-M-based CAOL framework has several benefits.\nFirst, the orthogonality constraint and diversity promoting regularizer in CAOL are useful in learning filters with diverse structures.\nSecond, the proposed BPEG-M algorithm significantly accelerates CAOL over the state-of-the-art method, BPG \\cite{Xu&Yin:17JSC}, with our sufficiently sharp majorizer designs.\nThird, BPEG-M-based CAOL uses much less memory compared to patch-domain AOL methods \\cite{Yaghoobi&etal:13TSP, Hawe&Kleinsteuber&Diepold:13TIP, Ravishankar&Bressler:15TSP}, and easily allows parallel computing.\nFinally, the learned convolutional regularizer provides the autoencoding CNN architecture in MBIR, and outperforms EP reconstruction in sparse-view CT.\n\n\nSimilar to existing unsupervised synthesis or analysis operator learning methods, the biggest remaining challenge of CAOL is optimizing its model parameters. \nThis would become more challenging when one applies CAOL to train CNNs (see Appendix~\\ref{sec:CNN}).\nOur first future work is developing \\dquotes{task-driven} CAOL that is particularly useful to train thresholding values.\nOther future works include further acceleration of BPEG-M in Algorithm~\\ref{alg:BPGM}, designing sharper majorizers requiring only $O( L R N )$ for the filter update problem of CAOL~\\R{sys:CAOL}, and applying the CNN model learned via \\R{sys:CNN:orth} to MBIR. \n\n\n\n\n\n\n\n\n\n\n\\section*{Appendix}\n\\renewcommand{\\thesubsection}{\\Alph{subsection}}\n\n\\subsection{Training CNN in a unsupervised manner via CAOL} \\label{sec:CNN}\n\nThis section mathematically formulates an unsupervised training cost function for classical CNN (e.g., LeNet-5 \\cite{LeCun&etal:98ProcIEEE} and AlexNet \\cite{Krizhevsky&etal:12NIPS}) and solves the corresponding optimization problem, via the CAOL and BPEG-M frameworks studied in Sections~\\ref{sec:CAOL}--\\ref{sec:CAOL+BPGM}.\nWe model the three core modules of CNN: \\textit{1)} convolution, \\textit{2)} pooling, e.g., average \\cite{LeCun&etal:98ProcIEEE} or max \\cite{Jarrett&etal:09ICCV}, and \\textit{3)} thresholding, e.g., RELU \\cite{Nair&Hinton:10ICML}, while considering the TF filter condition in Proposition~\\ref{p:TFconst}. \nParticularly, the orthogonality constraint in CAOL \\R{sys:CAOL:orth} leads to a sharp majorizer, and BPEG-M is useful to train CNNs with convergence guarantees.\nNote that it is unclear how to train such diverse (or incoherent) filters described in Section~\\ref{sec:CAOL} by the most common CNN optimization method, the stochastic gradient method in which gradients are computed by back-propagation. The major challenges include \\textit{a)} the non-differentiable hard thresholding operator related to $\\ell^0$-norm in \\R{sys:CAOL}, \\textit{b)} the nonconvex filter constraints in \\R{sys:CAOL:orth} and \\R{sys:CAOL:div}, \\textit{c)} using the identical filters in both encoder and decoder (e.g., $W$ and $W^H$ in Section~\\ref{sec:prf:p:TF}), and \\textit{d)} vanishing gradients.\n\nFor simplicity, we consider a two-layer CNN with a single training image, but one can extend the CNN model \\R{sys:CNN:orth} (see below) to \\dquotes{deep} layers with multiple images. The first layer consists of \\textit{1c)} convolutional, \\textit{1t)} thresholding, and \\textit{1p)} pooling layers; the second layer consists of \\textit{2c)} convolutional and \\textit{2t)} thresholding layers.\nExtending CAOL \\R{sys:CAOL:orth}, we model two-layer CNN training as the following optimization problem:\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\allowdisplaybreaks\n\\ea{\n\\label{sys:CNN:orth}\n\\argmin_{\\{ d_{k}^{[1]}, d_{k,k'}^{[2]} \\}} \\min_{\\{ z_{k}^{[1]}, z_{k'}^{[2]} \\}}\n&~ \\sum_{k=1}^{K_1} \\frac{1}{2}  \\left\\| d_{k}^{[1]} \\circledast x - z_{k}^{[1]} \\right\\|_2^2 + \\alpha_1 \\left\\| z_{k}^{[1]} \\right\\|_0 \\nn\n\\\\\n+ &\\, \\frac{1}{2} \\left\\| \\left( \\sum_{k=1}^{K_1} \\left[ \\arraycolsep=2pt \\begin{array}{c} d_{k,1}^{[2]} \\circledast P z_{k}^{[1]} \\\\ \\vdots \\\\ d_{k,K_2}^{^{[2]}} \\circledast P z_{k}^{[1]} \\end{array} \\right] \\right) - \\left[ \\arraycolsep=2pt \\begin{array}{c} z_{1}^{[2]} \\\\ \\vdots \\\\ z_{K_2}^{[2]} \\end{array} \\right] \\right\\|_2^2 \\nn\n\\\\\n+ &\\, \\alpha_2 \\sum_{k'=1}^{K_2} \\left\\| z_{k'}^{[2]} \\right\\|_0 \\nn\n\\\\\n\\mathrm{subject~to} ~~~~&~ D^{[1]} \\big( D^{[1]} \\big)^H = \\frac{1}{R_1} \\cdot I, \\nn\n\\\\\n&~ D_k^{[2]} \\big( D_k^{[2]} \\big)^H = \\frac{1}{R_2} \\cdot I, \\quad k = 1,\\ldots,K_1,  \n\\tag{A1}\n} \n\\endgroup\nwhere $x \\in \\bbR^{N}$ is the training data, $\\{ d_{k}^{[1]} \\in \\bbR^{R_1} : k = 1,\\ldots,K_1 \\}$ is a set of filters in the first convolutional layer, $\\{ z_{k}^{[1]} \\in \\bbR^{N} : k = 1,\\ldots,K_1\\}$ is a set of features after the first thresholding layer, $\\{ d_{k,k'}^{[2]} \\in \\bbR^{R_2} : k' = 1,\\ldots,K_2 \\}$ is a set of filters for each of $\\{ z_{k}^{[1]} \\}$ in the second convolutional layer, $\\{ z_{k'}^{[2]} \\in \\bbR^{N\/\\omega} : k = 1,\\ldots,K_2\\}$ is a set of features after the second thresholding layer, $D^{[1]}$ and $\\{ D_k^{[2]} \\}$ are similarly given as in \\R{eq:D}, $P \\in \\bbR^{N\/\\omega \\times \\omega}$ denotes an average pooling \\cite{LeCun&etal:98ProcIEEE} operator (see its definition below), and $\\omega$ is the size of pooling window. \nThe superscripted number in the bracket of vectors and matrices denotes the $(\\cdot)\\rth$ layer.\nHere, we model a simple average pooling operator $P \\in \\bbR^{(N\/\\omega) \\times \\omega}$ by a block diagonal matrix with row vector $\\frac{1}{\\omega} 1_\\omega^T \\in \\bbR^{\\omega}$: $P := \\frac{1}{\\omega} \\bigoplus_{j=1}^{N\/\\omega} 1_\\omega^T$.\nWe obtain a majorization matrix of $P^T P$ by $P^T P \\preceq \\diag( P^T P 1_N ) = \\frac{1}{\\omega} I_N$ (using Lemma~\\ref{l:diag(|At|W|A|1)}).\nFor 2D case, the structure of $P$ changes, but $P^T P \\preceq \\frac{1}{\\omega} I_N$ holds.\n\n\n\n\n\n\n\nWe solve the CNN training model in \\R{sys:CNN:orth} via the BPEG-M techniques in Section~\\ref{sec:CAOL+BPGM}, and relate the solutions of \\R{sys:CNN:orth} and modules in the two-layer CNN training. The symbols in the following items denote the CNN modules.\n\\begin{itemize}\n[\\setlength{\\IEEElabelindent}{\\IEEEilabelindentA}]\n\\item[\\textit{1c})] Filters in the first layer, $\\{ d_{k}^{[1]} \\}$: Updating the filters is straightforward via the techniques in Section~\\ref{sec:prox:filter:orth}. \n\n\\item[\\textit{1t})] Features at the first layers, $\\{ z_{k}^{[1]} \\}$: Using BPEG-M with the $k\\rth$ set of TF filters $\\{ d_{k,k'}^{[2]} : k' \\}$ and $P^T P \\preceq \\frac{1}{\\omega} I_N$ (see above), the proximal mapping for $z_{k}^{[1]}$ is\n\\begingroup\n\\setlength{\\thinmuskip}{1.5mu}\n\\setlength{\\medmuskip}{2mu plus 1mu minus 2mu}\n\\setlength{\\thickmuskip}{2.5mu plus 2.5mu}\n\\be{\n\\label{eq:CNN:layer1:spCd}\n\\min_{z_{k}^{[1]}} \\frac{1}{2} \\left\\| d_{k}^{[1]} \\circledast x - z_{k}^{[1]} \\right\\|_2^2 + \\frac{1}{2\\omega'} \\left\\| z_{k}^{[1]} - \\zeta_{k}^{[k]} \\right\\|_2^2 + \\alpha_1  \\left\\| z_{k}^{[1]} \\right\\|_0,\n}\n\\endgroup\nwhere $\\omega' = \\omega \/ \\lambda_Z$ and $\\zeta_{k}^{[k]}$ is given by \\R{update:x}. Combining the first two quadratic terms in \\R{eq:CNN:layer1:spCd} into a single quadratic term leads to an optimal update for \\R{eq:CNN:layer1:spCd}:\n\\bes{\nz_{k}^{[1]} = \\cH_{\\! \\sqrt{ 2 \\frac{\\omega' \\alpha_1}{\\omega' + 1} }} \\left( d_{k}^{[1]} \\circledast x  + \\frac{1}{\\omega'} \\zeta_{k}^{[k]}  \\right), \\quad k \\in [K],\n} \nwhere the hard thresholding operator $\\cH_a (\\cdot)$ with a thresholding parameter $a$ is defined in \\R{eq:def:hardthr}.\n\n\n\\item[\\textit{1p})] Pooling, $P$: Applying the pooling operator $P$ to $\\{ z_{k}^{[1]} \\}$ gives input data -- $\\{ P z_{k}^{[1]} \\}$ -- to the second layer.\n\n\\item[\\textit{2c})] Filters in the second layer, $\\{ d_{k,k'}^{[2]} \\}$: We update the $k\\rth$ set filters $\\{ d_{k,k'}^{[2]} : \\forall k' \\}$ in a sequential way. Updating the $k\\rth$ set filters is straightforward via the techniques in Section~\\ref{sec:prox:filter:orth}. \n\n\\item[\\textit{2t})] Features at the second layers, $\\{  z_{k'}^{[2]} \\}$: The corresponding update is given by\n\\bes{\nz_{k'}^{[2]} = \\cH_{\\!\\sqrt{2 \\alpha_2}} \\left( \\sum_{k=1}^{K_1} d_{k,k'}^{[1]} \\circledast P z_{k}^{[1]} \\right), \\quad k' \\in [K_2].\n}\n\\end{itemize}\n\nConsidering the introduced mathematical formulation of training CNNs \\cite{LeCun&etal:98ProcIEEE} via CAOL, BPEG-M-based CAOL has potential to be a basic engine to rapidly train CNNs with big data (i.e., training data consisting of many (high-dimensional) signals).\n\n\n\n\\subsection{Examples of $\\{ f(x;y), \\cX \\}$ in MBIR model \\R{eq:mbir:learn} using learned regularizers} \\label{sec:egs}\n\n\nThis section introduces some potential applications \nof using MBIR model \\R{eq:mbir:learn} using learned regularizers\nin imaging processing, imaging, and computer vision.\nWe first consider quadratic data fidelity function in the form of $f(x;y) = \\frac{1}{2} \\| y - A x \\|_W^2$.\nExamples include\n\\begin{itemize}\n\\item Image debluring (with $W \\!=\\! I$ for simplicity), \nwhere $y$ is a blurred image, $A$ is a blurring operator, and $\\cX$ is a box constraint; \n\\item Image denoising (with $A \\!=\\! I$),\nwhere $y$ is a noisy image corrupted by additive white Gaussian noise (AWGN),\n$W$ is the inverse covariance matrix corresponding to AWGN statistics,\nand $\\cX$ is a box constraint;\n\\item Compressed sensing (with $\\{ W \\!=\\! I, \\cX \\!\\in\\! \\bbC^{N'} \\}$ for simplicity) \\cite{Chun&Adcock:17TIT, Chun&Adcock:18ACHA},\nwhere $y$ is a measurement vector, \nand $A$ is a compressed sensing operator,\ne.g., subgaussian random matrix, bounded orthonormal system, subsampled isometries, certain types of random convolutions;\n\\item Image inpainting (with $W \\!=\\! I$ for simplicity),\nwhere $y$ is an image with missing entries, $A$ is a masking operator, and $\\cX$ is a box constraint;\n\\item Light-field photography from focal stack data \nwith $f(x;y) = \\sum_{c} \\| y_c - \\sum_{s}  A_{c,s} x_{s} \\|_2^2$,\nwhere $y_c$ denotes measurements collected at the $c\\rth$ sensor, \n$A_{c,s}$ models camera imaging geometry at the $s\\rth$ angular position for the $c\\rth$ detector, \n$x_s$ denotes the $s\\rth$ sub-aperture image, $\\forall c,s$,\nand $\\cX$ is a box constraint  \\cite{Block&Chun&Fessler:18IVMSP, Chun&etal:18arXiv:momnet}.\n\\end{itemize}\nExamples that use nonlinear data fidelity function include \nimage classification using the logistic function \\cite{Mairal&etal:09NIPS}, \nmagnetic resonance imaging considering unknown magnetic field variation \\cite{Fessler:10SPM}, \nand positron emission tomography \\cite{Lim&etal:19TMI}.\n\n\n\n\n\\subsection{Notation} \\label{sec:notation}\n\nWe use $\\nm{\\cdot}_{p}$ to denote the $\\ell^p$-norm and write $\\ip{\\cdot}{\\cdot}$ for the standard inner product on $\\bbC^N$.  \nThe weighted $\\ell^2$-norm with a Hermitian positive definite matrix $A$ is denoted by $\\nm{\\cdot}_{A} = \\nm{ A^{1\/2} (\\cdot) }_2$.\n$\\nm{\\cdot}_{0}$ denotes the $\\ell^0$-quasi-norm, i.e., the number of nonzeros of a vector.  \nThe Frobenius norm of a matrix is denoted by $\\| \\cdot \\|_{\\mathrm{F}}$. \n$( \\cdot )^T$, $( \\cdot )^H$, and $( {\\cdot} )^*$ indicate the transpose, complex conjugate transpose (Hermitian transpose), and complex conjugate, respectively. \n$\\diag(\\cdot)$ denotes the conversion of a vector into a diagonal matrix or diagonal elements of a matrix into a vector.\n$\\bigoplus$ denotes the matrix direct sum of matrices.\n$[C]$ denotes the set $\\{1,2,\\ldots,C\\}$.  \nDistinct from the index $i$, we denote the imaginary unit $\\sqrt{-1}$ by $\\imath$. \nFor (self-adjoint) matrices $A,B \\in \\bbC^{N \\times N}$, the notation $B \\preceq A$ denotes that $A-B$ is a positive semi-definite matrix. \n\n\n\n\n\\section*{Acknowledgment}\n\nWe thank Xuehang Zheng for providing CT imaging simulation setup, and Dr. Jonghoon Jin for constructive feedback on CNNs.\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe Planck 2013 and 2015 data releases open new directions in precision cosmology with regard to a more advanced\ninvestigation of the statistical isotropy and non-Gaussianity of the cosmic microwave background (CMB)~\\cite{Planck1, Planck2, Planck_is, Planck_is2015}.\nWhile generally confirming the Gaussianity and the statistical isotropy of the CMB (for the relevant multipole domain $\\ell\\ge 50$), the Planck\nscience team confirmed the existence of a variety of anomalies in the temperature anisotropy on large angular scales ($\\ell\\le 50$)\npreviously seen in WMAP data~\\cite{Planck_is,Planck_is2015}.  Among these are the lack of power in the quadrupole component~\\cite{low_quadrupole} (see, however,~\\cite{wmap9b}\nand~\\cite{Planck_is}), the alignment of the quadrupole and octupole components~\\cite{Copi1, wmap9b, Planck_is}, the unusual symmetry of the\noctupole~\\cite{mhansen}, anisotropies in the temperature angular power spectrum~\\cite{WMAP7:powerspectra, Planck_is, Hansen2004, Eriksen2003},\n preferred directions~\\cite{Pref.Direction1, Pref.Direction2, Copi2, Planck_is, Planck_is2015, Akrami2014}, asymmetry in the power of even and odd modes~\\cite{parity1, parity2, wmap9b, Planck_is, Planck_is2015} and the Cold Spot~\\cite{vielva2003, Coldspot1, Planck_is, Planck_is2015}.\n Some of these anomalies are probably a consequence of the residuals of foreground effects that could be a major source of contamination in the primordial $E$- and $B$-modes of polarization (in this connection see~\\cite{Planck_bicep}).\n\nThe statistics of $B$-mode polarization that can be derived from ongoing and planned CMB experiments will be crucial for the determination of the\ncosmological gravitational waves associated with inflation~\\cite{inflation} at the range of multipoles $50\\le \\ell\\le 150$, closer to\nthe domain of interest for BICEP2 and Planck~\\cite{bicep,Planck_bicep}.   It seems likely that $B$-mode polarization in this range is affected\nby Galactic dust emission, the statistical properties of which are very poorly known. (For $\\ell >150$ we expect contamination of the $B$-modes\ndue to lensing effects the precise nature of which is also not fully understood.)  In the absence of such knowledge it is difficult to make \\emph{a priori}\nproposals for the best estimator of non-Gaussianity and statistical anisotropies in the derived $B$-modes due to possible contamination by foreground residuals.\nThus, we believe that it is of value to propose additional model-independent tests aimed at providing an improved quantitative understanding of the magnitude of non-Gaussianity in current CMB data.  Such tests would also be useful for the analysis of forthcoming CMB data sets.   In this paper\nwe propose use of the Kullback--Leibler (KL) divergence as such a test.  The goal of this paper is to illustrate the utility of the KL divergence in studying the properties of the CMB signal --- a Gaussian or almost Gaussian signal. The KL divergence is likely to be even more useful for very non-Gaussian cases such as the statistical behaviour of the Minkowski functionals for a single map or the pixel-pixel cross-correlation coefficient between two maps, when calculated in small areas.  We will consider such issues in a separate publication.\n\nThe one-point probability distribution function (PDF) would seem to be a reasonable starting point for the investigation of non-Gaussianity.\nSuch tests have been applied by the Planck team to a variety of derived CMB temperature maps including the SMICA, NILC, SEVEM and\nCommander maps~\\cite{Planck_is,Planck_is2015}.  In practice, these tests involve comparison of the various temperature fluctuation maps with an ensemble\nof simulated maps.  The CMB temperature is characterized by a power spectrum, $C_\\ell^\\text{Planck}$, which\ncorresponds to the Planck 2013 concordance \\LCDM\\ model with the cosmological parameters listed in~\\cite{Planck_cp}.\nThe simulated maps are obtained as Monte Carlo (MC) Gaussian draws on this power spectrum (in harmonic space).  When processed with the Planck component separation tool, the resulting simulation maps contain both CMB information as well as various residuals from foregrounds, the uncertainties of instrumental effects, etc.  For the Planck 2013 data release, the corresponding $10^3$ full focal plane simulations are referred to as the FFP7 simulations.\nThey reflect the intrinsic properties of the SMICA, NILC, SEVEM and Commander maps.  Differences between the FFP7 maps and the various empirical maps can provide useful information regarding non-Gaussianity.  In the following, we shall restrict our attention to the SMICA map.\n\nIn practice, the non-parametric Kolmogorov--Smirnov (KS) test is often used to assess the similarity of two distributions.  The KS test\ncharacterizes the difference between the two cumulative distribution functions (CDF) in terms of the maximum absolute deviation between them.  The KS estimator, $\\kappa$, is defined as\n\\begin{equation}\n\\kappa= \\sqrt{n} \\max[|F(x)-F_n(x)|] \\, ,\n\\label{eq:KS_definition}\n\\end{equation}\nwhere $F(x)$ is the theoretical expectation of the CDF and $F_n(x)$ is obtained from a data sample with $n$ elements.  Here, both $F(x)$ and $F_n(x)$ must  be normalized to the range $[0,1]$. Note that $F(x)$ is normally a continuous function or should at least be defined for all possible values of the data sample $F_n(x)$.   It is clear from Eq.~\\eqref{eq:KS_definition} that the KS estimator $\\kappa$ is local in the sense that its value will be determined at a point, $x$, where the PDFs corresponding to $F(x)$ and $F_n(x)$ cross.  We note that the use of PDFs in Eq.~\\eqref{eq:KS_definition} instead of CDFs would result in a\nmaximal sensitivity to the largest local anomaly.  \n\nUnlike the case of vectors (where the scalar product provides a standard measure), there is no generic ``best''\nmeasure for quantifying the similarity of two distributions.  Thus, we believe that it is also useful to consider the Kullback--Leibler divergence for two discrete probability distributions, $P$ and $Q$.  The KL divergence on the set of points $i$ is defined~\\cite{kl} as\n\\be \\label{eq:KL_definition}\nK(P\\|Q)=\\sum_i P_i\\log\\left(\\frac{P_i}{Q_i}\\right) \\, .\n\\ee\nIn other words, the KL divergence is the expectation value of the logarithmic difference between the two probability distributions as computed with\nweights of $P_i$.  Typically, $P$ represents the distribution of the data, while $Q$ represents a theoretical expectation of the data.  Unlike the\nKS test, the KL divergence is non-local.  Indeed, we shall indicate below that it is in a sense ``maximally'' non-local.  It is familiar in information\ntheory, where it represents the difference between the intrinsic entropy,\n$H_P$, of the distribution $P$ and the cross-entropy, $H_{PQ}$, between $P$ and $Q$,\n\\be\nK(P\\|Q)=H_{PQ}-H_P, \\qquad H_P=-\\sum_i P_i\\log P_i, \\qquad H_{PQ}=-\\sum_i P_i\\log Q_i \\, .\n\\label{eq:entropy}\n\\ee\nIn more practical terms, consider the most probable result of $N$ independent random draws on the distribution $P$.  When $N$ is large, the number of\ndraws at point $i$ is simply $n_i = N P_i$.  Now construct the probabilities, $\\Pi_P$ and $\\Pi_Q$, that this most probable result was drawn at random\non distribution $P$ or $Q$, respectively.  The KL divergence of Eq.~\\eqref{eq:KL_definition} is simply $N^{-1}\\log{(\\Pi_P \/ \\Pi_Q)}$.\n We note that simulations of the CMB\nmap, drawn independently in harmonic space, have correlations in pixel space.\n Nevertheless we regard this argument as motivation for applying the KL divergence to CMB pixels.\n\nThe main goal of this paper is to illustrate the implementation of the KL divergence for\nthe analysis of the statistical properties of the derived CMB maps in the low multipole domain as a complementary test to the methods listed in~\\cite{Planck_is,Planck_is2015}.  The structure of the paper is as follows. In Section~\\ref{sec:KL_divergence_and_Planck_data} we present some properties of the KL divergence. We also use it to analyze the Planck SMICA map and compare it to both the FFP7 set and to a purely Gaussian ensemble. In addition, we compare the two ensembles and compare the KL divergence to the KS test in the low multipole domain of the CMB map. In Section~\\ref{sec:discussion} we discuss the results. Note that the Planck papers~\\cite{Planck_is,Planck_is2015} test the Gaussianity of the one-point PDF by analyzing its variance, skewness and kurtosis. In this sense, the KL divergence is simply another test on the global shape of the PDF.  Here, we restrict our analysis by using the SMICA map and the corresponding simulations.  The extension of the method to $E$- and $B$-modes of polarization does not require any modification.\n\n\n\n\\section{KL divergence and Planck data}\n\\label{sec:KL_divergence_and_Planck_data}\n\n\\subsection{Preliminary remarks: Properties of the KL divergence}\n\\label{subsec:Preliminary_remarks_KL_divergence}\n\nAs noted above, the KL divergence provides a measure of the similarity of two known distributions.  In many cases, however, one of these\ndistributions is not known and must rather be approximated by the average PDF for a statistical ensemble of realizations of the random field.  The\nquestion then arises of how closely the resulting proxy reflects the properties of the true underlying distribution.  To offer some insight in this\nmatter, we consider a toy model based on a discrete Gaussian distribution, $P_k$, with\n\\be\nP_k \\sim \\exp{\\left[ -  k^2\/25 \\right]}\\, ,\n\\ee\nand $k$ an integer between $-10$ and $+10$ subject to the obvious normalization condition.  The mean value of\n$k$ is $0$ and the variance is approximately $25\/2$.  Suppose for simplicity that we define an individual data\nset as $N$ random draws on this distribution.  Each such data set can be regarded as a proxy, $Q_i$ for the underlying\ndistribution, and can be used to calculate the KL divergence $K(P\\|Q)$ defined in Eq.~\\eqref{eq:KL_definition}.  For a given value of $N$, we  repeat this\nprocess $M$ times and compute the average KL divergence, $\\overline{K}$, and the root-mean-square (RMS) deviation of the KL divergence, $\\Delta K = \\sqrt{\\overline{K^2} - \\overline{K}{}^2 }$.  The results of\nthis exercise are shown in Table~\\ref{tab:Change_N}, where we have used the common value of $M=1000$.\n\\begin{table}\n\t\\centering\n\t\\caption{Mean and RMS values of the KL divergence for a discrete Gaussian distribution.}\n\t\\vspace{10pt}\n\t\\begin{tabular}{rcccc}\n\t\t\\hline\n\t\t\\noalign{\\smallskip}\n\t\t\\multicolumn{1}{c}{$N$} & $\\overline{K}$ & $\\Delta K$ & $N\\overline{K}$ & $N\\Delta K$ \\\\\n\t\t\\hline\n\t\t 4000 & 0.002540 & 0.000796 & 10.160 & 3.18 \\\\\n\t\t 8000 & 0.001253 & 0.000413 & 10.024 & 3.30 \\\\\n\t\t16000 & 0.000636 & 0.000200 & 10.176 & 3.20 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{tab:Change_N}\n\\end{table}\nSeveral things seem clear.  Both the average value of the KL divergence and the RMS deviation from this average value vanish like $1\/N$ for large $N$. From general arguments,\nthe KL divergence cannot be negative.  Fig.~\\ref{fig:KL_hist_change_N} shows a histogram of the distribution of  KL divergences\nobtained with $M = 20000$ for the cases $N=100$ and $N=1000$.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\halfW]{KL_hist_change_N.ps}\n\t\\caption{Histogram of KL divergences with $M=20000$ for $N=100$ (\\emph{black})\n\tand $N=1000$ (\\emph{red}).  Note that the horizontal axis is measured in units of the\n\tcorresponding value of $\\overline{K}$.}\n\t\\label{fig:KL_hist_change_N}\n\\end{figure}\nThe KL divergences here are measured\nin units of the corresponding value of $\\overline{K}$.  The fact that these distributions scale like $1\/N$ is\nobvious.  These histograms suggest power law suppression near zero and Gaussian behaviour\nfor large values of the KL divergence.\n\n\nThe results shown in Fig.~\\ref{fig:KL_hist_change_N} are not specific to the KL divergence, and qualitatively\nsimilar results would be obtained for any measure chosen to describe the similarity of two distributions.\nAll that is required is that the measure chosen is always positive and vanishes when the distributions\nbeing compared are identical.  When drawn as here, each individual data set can be thought of\nas a combination of the ``exact'' distribution plus the amplitudes of $(N-1)$ ``fluctuations''.\\footnote{Due to\nthe fact that each data set contains exactly $N$ draws.}  Obviously, the amplitudes of every one of these\nfluctuations must be exactly zero if the measure is to be $K = 0$.  Of course, there are many combinations of the\nfluctuation amplitudes that will give any fixed non-zero value of the measure, and their number increases as\n$K$ grows from zero.  In contrast, for the case of only two options (e.g., ``heads'' and ``tails'') subject to a constraint\non the total number of draws, there is only a single degree of freedom.  In this case $K = 0$ is actually the most\nprobable value.\n\nA few additional remarks can help clarify the properties of the KL divergence.  Consider that each individual\ndata set is sufficiently large that $Q_i = P_i + \\delta P_i$ where $\\delta P_i$ is small.  Under these conditions,\nterms linear in $\\delta P_i$ vanish as a consequence of normalization, and the KL divergence is given simply as\n\\be\nK = \\frac{1}{2} \\, \\sum_i \\, \\frac{\\delta P_i^2}{P_i} \\, .\n\\label{eq:KL_small_fluctuation}\n\\ee\nWe see that the distributions $P$ and $Q$ are now treated symmetrically in spite of the asymmetry that is\napparent in Eq.~\\eqref{eq:KL_definition}.  Elementary arguments suggest that the average number of draws on bin $i$ will be\n$N P_i \\pm \\sqrt{N P_i}$.   The corresponding proxy for the underlying distribution will be $P_i \\pm \\sqrt{P_i \/ N}$.\nGiven this result, Eq.~\\eqref{eq:KL_small_fluctuation} suggests that, for fixed bin sizes and in the limit $N \\to \\infty$, each bin will make\na contribution to the KL divergence of roughly equal size.  It is hard to imagine a greater degree of non-locality. Moreover, in this limit the KL divergence is expected to be of order $N_b\/N$, where $N_b$ is the number of bins.  In other words, if we define\n\\be \\label{eq:alpha_definition}\n\\alpha=\\frac{N}{N_b}K\\, ,\n\\ee\nwe expect that $\\alpha\\sim\\mathcal{O}(1)$. This realization allows us to assign a rough scale for the expected KL divergence. If two given distributions yield a much larger value of $\\alpha$, we can conclude that they differ significantly from each other without resorting to comparisons with an ensemble. In the example shown in Table~\\ref{tab:Change_N} we are using $N_b=30$, meaning $\\alpha$ is small. This is to be expected since we are indeed using the true distribution to draw the data under examination.\n\n\n\\subsection{Preliminary remarks: Planck data}\n\\label{subsec:Preliminary_remarks_Planck_data}\n\nWe have performed the KL divergence test on the CMB map obtained  by the Planck collaboration~\\cite{Planck_cp} using the SMICA method. Since we are interested in the\nstatistical properties of the CMB map on large scales, we first degrade the map from its native \\textsc{HEALPix}~\\cite{HEALPix} resolution of $\\Nside=2048$\nto $\\Nside=32$. We then construct the convolution of this map with a Gaussian smoothing kernel of $5\\deg$ FWHM and retain only the harmonic coefficients with\n$\\ell \\le \\lmax=96$.  The SMICA map provides a useful estimation of the CMB temperature fluctuations for a very large fraction --- but not all --- of the sky. We use the SMICA inpainting mask to exclude heavily contaminated regions, mainly the Galactic plane. At the resolution considered here, the mask removes about 6\\% of the pixels, leaving the number of pixels under consideration to be $N=11565$. Since the analysis is performed in pixel space,\napplication of the mask is trivial.\n\nIn order to estimate the statistical significance of our results, we compare them to ensembles of realizations. In this work we use two different ensembles.\nThis is done in order to cross-check the significance estimations and also allows us to compare the two ensembles. The first of these ensembles is the\nFFP7 set described above.  We degrade and smooth the FFP7 maps in the same manner as we did the SMICA map.  We expect that the effects of detector noise will be minor on the large scales considered here.  To test this expectation we therefore also make use of the best-fit power spectrum,\n$C_\\ell^\\text{Planck}$~\\cite{Planck_cp}, to generate an ensemble of $10^3$ Gaussian random realizations free of residuals.  As in the\ncase of the FFP7 ensemble, we restrict the multipole domain to $\\lmax=96$ and smooth the harmonic coefficients with a Gaussian filter of $5\\deg$ FWHM.\nWe also multiply the coefficients with the pixel window function associated with an $\\Nside=32$ pixelization before converting them to\nan $\\Nside=32$ map.\n\nIn our analysis we calculate the KL divergence for the SMICA map in pixel space. We calculate a histogram of the temperature fluctuations in the\nunmasked pixels by taking bins of width $8\\uK$ in the range $[-200,200]\\uK$, meaning that the number of bins is $N_b=51$. Values outside this range are attributed to the edge bins. This histogram is taken as the $P$ probability distribution of Eq.~\\eqref{eq:KL_definition}. For $Q$, the expected distribution, we turn to the ensemble of simulations, either FFP7 or\nthe Gaussian realizations. We calculate the histogram for each simulation (using the same range and binning), and take $Q$ to be the mean of all histograms. These histograms are shown in Fig.~\\ref{subfig:maps_histograms} together with error bars showing the 5--95\\% range for the FFP7 set.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\halfW]{T_hist.ps}\n\t\t\\label{subfig:maps_histograms}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\halfW]{KL_hist.ps}\n\t\t\\label{subfig:KL_histograms}}\n\t \\caption{\\subref{subfig:maps_histograms}~Number of counts versus amplitude of the SMICA map (\\emph{red}), the FFP7 (\\emph{black}) and\n\tthe Gaussian (\\emph{blue}) ensembles. The error bars show the 5--95\\% range for the FFP7 set. \\subref{subfig:KL_histograms}~Histograms of normalized KL\n\tdivergence values for the FFP7 ensemble (\\emph{black}) and the Gaussian ensemble (\\emph{blue}). The values for the SMICA map are shown as red vertical\n\tlines, compared to the FFP7 mean distribution (\\emph{solid}) and to the Gaussian mean distribution (\\emph{dotted}).}\n\t\\label{fig:KLResults}\n\\end{figure}\nIt appears that the histogram of the SMICA map deviates from the reference histograms by~$\\approx2\\sigma$ primarily in the vicinity of the peak of the distribution. However, this estimation relies on a local feature. The KL divergence provides us with a recipe to sum all the deviations from the entire range of the distribution with appropriate weights.\n\nBefore using Eq.~\\eqref{eq:KL_definition} to calculate the KL divergence, it is necessary to pay particular attention to bins in which either $P$ or $Q$ has small values.\nThe case in which $P_i=0$ is not problematic since $P_i\\log P_i \\to 0$ in this limit.  Bins for which $Q_i=0$, however, should not be included since the\nKL divergence is logarithmically divergent as $Q_i \\to 0$.  Such a result is not unreasonable since it is impossible to draw to a bin if its probability is\nstrictly $0$.  In practice, however, small values of $Q_i$ are merely a consequence of the size of our ensemble.  We have chosen to ignore bins for\nwhich $Q_i<5$~pixels in order to minimize the sensitivity to small non-statistical fluctuations in the extreme tails of the $Q$ distribution.\n\n\n\\subsection{The Basic Results}\n\\label{subsec:The_Basic_Results}\n\nWe have calculated the KL divergences between the SMICA histogram and the histograms made from the FFP7 and the Gaussian ensembles. After normalization using the number of valid pixels, $N$, and the number of bins, $N_b$, in Eq.~\\eqref{eq:alpha_definition}, we have obtained $\\alpha=6.47$ and $6.28$, respectively. If these values were significantly larger than the expected order of magnitude, we would conclude that the distributions were in disagreement. Since this is not the case, we must compare the results to ensembles of values of $\\alpha$.\nIn order to calculate the $p$-values, we repeat the calculation of the KL divergence, replacing the distribution of the map, $P$, with that\nof each of the random simulations. This results in two histograms of normalized $K$ values, i.e.\\ $\\alpha$ values, for FFP7 (Gaussian) maps compared to the FFP7 (Gaussian) mean distribution,\nshown in Fig.~\\ref{subfig:KL_histograms}. It is evident that, as expected, $\\alpha\\lesssim10$ for most simulations. We find that $5.6\\%$ of the FFP7 simulations and $6.3\\%$ of the Gaussian simulations get a higher KL\ndivergence than the SMICA map. We see that the KL divergence of the SMICA map from the expected distribution is not significant. As expected, differences\nbetween the two reference ensembles, the FFP7 simulations and the pure Gaussian realizations, are quite small.\nIn order to demonstrate explicitly the similarity between the two ensembles with respect to the KL divergence, we have also tested each of the FFP7 simulations\nagainst the mean distribution of the Gaussian ensemble and vice versa. The results of this calculation are extremely similar to those shown in Fig.~\\ref{subfig:KL_histograms}, and we can conclude that the added complexity of the FFP7 simulations relative to that of simple Gaussian realizations plays a minor role at this resolution.\n\n\nIn addition to the KL divergence, and as a basis for comparison, we also use the KS test to compare between the histogram of the SMICA map and the mean histogram for each of the ensembles. The KS test, defined in Eq.~\\eqref{eq:KS_definition}, requires the use of the CDF. For the SMICA map, the CDF is calculated from the data without any binning. The reference CDF, however, is calculated by first fitting the mean histogram of the ensemble (either FFP7 or the Gaussian realizations) to a Gaussian, and then using the fitted parameters in the expression for a Gaussian CDF. As in the case of the KL divergence, for each ensemble, we compare the SMICA map to the mean histogram of the ensemble and also create a histogram of KS test values by taking each realization separately and comparing it to the mean. The results are shown in Fig.~\\ref{fig:KSResults}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=\\halfW]{KS_hist.ps}\n\t\\caption{Histograms of KS test values for the FFP7 ensemble (\\emph{black}) and the Gaussian ensemble (\\emph{blue}). The values for the SMICA map are shown as red vertical\nlines, compared to the FFP7 mean distribution (\\emph{solid}) and to the Gaussian mean distribution (\\emph{dotted}).}\n\t\\label{fig:KSResults}\n\\end{figure}\nThe KS test values we get when comparing the SMICA map to the FFP7 and Gaussian simulations are $\\kappa=8.32$ and $8.21$, respectively. The corresponding $p$-values are $3.0\\%$ and $2.6\\%$. Again we see that the results for the two ensembles are in good agreement. Moreover, while the $p$-values of the KS test are lower than those of the KL divergence, the SMICA map still appears to be consistent with the reference ensembles and not anomalous.\n\n\nAs we can see from Fig.~\\ref{subfig:maps_histograms}, there are well-defined temperature ranges in which the SMICA histogram is above or\nbelow the reference.  Thus, in Fig.~\\ref{fig:smica_regions} we plot the SMICA map, showing only the temperature range $|T|\\le50\\uK$ where the SMICA\nhistogram is above the reference and the temperature range $50\\uK\\le|T|\\le120\\uK$ where it is below.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\thirdW,angle=90]{Map50.ps}\n\t\t\\label{subfig:smica_-50_50}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\thirdW,angle=90]{Map120.ps}\n\t\t\\label{subfig:smica_abs_50_120}\n\t}\n\t\\caption{The SMICA map, showing only those regions where \\subref{subfig:smica_-50_50}~$|T|\\le50\\uK$ and \\subref{subfig:smica_abs_50_120}~$50\\uK\\le|T|\\le120\\uK$. The small Galactic mask used in the analysis appears as a thin horizontal gray line in the center of the maps, and the masked pixels are not included in any of the temperature ranges. In panel~\\subref{subfig:smica_abs_50_120}, the black curves mark the location of the ecliptic plane.}\n\t\\label{fig:smica_regions}\n\\end{figure}\nWe see that there is no apparent tendency for the contributions from either of these temperature ranges to be localized in specific regions of the sky. We do, however, pay special attention to the region of the ecliptic plane. As is apparent from fig.~3 of~\\cite{Planck15Maps}, the SMICA map is susceptible to contamination from foreground residuals in the region of the ecliptic. We therefore include in Fig.~\\ref{subfig:smica_abs_50_120} curves showing the location of the ecliptic plane and suggest that the number of cold spots in the ecliptic band might be unexpected. As it is not the focus of this work, we have not performed any quantitative analysis regarding the spatial distribution of hot or cold regions of the map. The maps in Fig.~\\ref{fig:smica_regions} provide\nan additional general indication that the SMICA temperature map is not anomalous. Nevertheless, we again emphasize that small foreground residuals, like those suspected to lie in the area of the ecliptic plane, while insignificant to the analysis of temperature fluctuations, can become extremely important when analyzing the CMB polarization pattern, specifically $B$-mode polarization. \n\n\n\n\\subsection{Interchangeability of $P$ and $Q$}\n\\label{sub:interchangeability_of_P_and_Q}\n\nAs has been noted above, the KL divergence is not symmetric with respect to the interchange of the distributions $P$ and $Q$ except in the limit $P \\to Q$.\nThis is a reminder of the fact that the KL divergence is not a true metric of the distance between $P$ and $Q$.   So far, we have followed the common practice of taking $P$ to be the distribution of the data and $Q$ the expected distribution~\\cite{kl}.  However, it is worth checking what happens when these roles are reversed.  We have performed two tests involving interchange of the two distributions. First, we simply calculate the KL divergence $K(Q\\|P)$, where $P$ is again the SMICA histogram and $Q$ is the mean histogram of the FFP7 ensemble.\\footnote{Note that with the roles reversed, bins with $P_i < 5$ are now ignored and bins with small values of $Q_i$ are all counted.} This value is then compared to the ensemble of values computed with $P$ replaced by each of the FFP7 maps. The\nresulting histogram, after normalization using Eq.~\\eqref{eq:alpha_definition}, is presented in Fig.~\\ref{subfig:PQ_and_QP} together with the histogram of $K(P\\|Q)$ (presented above) as reference.\n\\begin{figure}\n\t\\centering\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\halfW]{KL_QP_hist.ps}\n\t\t\\label{subfig:PQ_and_QP}\n\t}\n\t\\subfigure[]{\n\t\t\\includegraphics[width=\\halfW]{Delta_KL_hist.ps}\n\t\t\\label{subfig:PQ_-_QP}\n\t}\n\t\\caption{\\subref{subfig:PQ_and_QP}~Histograms for the usual KL divergence $K(P\\|Q)$ (\\emph{black}) and for the reversed divergence $K(Q\\|P)$\n\t(\\emph{blue}). The red and blue vertical lines are the values for the SMICA map for the normal and reversed tests, respectively. All $K$ values have been normalized using Eq.~\\eqref{eq:alpha_definition}.\n\\subref{subfig:PQ_-_QP}\n\t~Histogram of the normalized difference $\\alpha(P\\|Q)-\\alpha(Q\\|P)$. The vertical line is the value for the SMICA map.}\n\t\\label{fig:PQQP}\n\\end{figure}\nWe can see that the two histograms and the corresponding $p$-values are similar. The value $p = 6.5\\%$ was obtained for the reversed test; the\n$p$-value for the normal test is $5.6\\%$ as stated above.  It is apparent that, although similar, the reversed histogram is slightly but consistently shifted towards smaller $\\alpha$ values than the normal histogram. The value for the SMICA map is also lower for the reversed test.  However, the SMICA is a single map, and the shift between histograms only indicates a statistical shift for the whole ensemble. Therefore, the second test is to examine the difference $\\Delta\\alpha = \\alpha(P\\|Q)-\\alpha(Q\\|P)$ between the normal and reversed normalized KL divergences of the \\emph{same} map. Fig.~\\ref{subfig:PQ_-_QP} shows the resulting histogram for the FFP7 ensemble together with the value for SMICA. The SMICA map shows a highly standard $\\Delta\\alpha$, yielding a $p$-value of $49.4\\%$. A similar test performed versus the Gaussian ensemble gives very similar results.\n\nThe tendency of the KL divergences to become smaller when $P$ and $Q$ are interchanged can be understood easily. Since $P$ is calculated from a single map, it tends to fluctuate more than $Q$, which is the mean of all ensemble distributions. Therefore, when $P$ appears only inside the logarithm, as is the case in the reversed $K(Q\\|P)$, the fluctuations are suppressed relative to the normal $K(P\\|Q)$. We see here that the SMICA map not only shows the expected qualitative behavior upon interchanging $P$ and $Q$, it is also quantitatively shifted by the expected amount. While the KL divergence in general is not symmetric under the interchange of $P$ and $Q$, we conclude that when testing the one-dimensional temperature distribution of the CMB on large scales, reversing the two makes little difference.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nWe have discussed the applicability of the Kullback--Leibler divergence for the assessment of departures form Gaussianity of CMB temperature\nmaps on large scales.  We have illustrated this on the SMICA map, comparing it to both the set of FFP7 simulations and a set of $10^3$ Gaussian draws.\nWe have shown that it is consistent with each of these reference sets to a level of about 6\\%.  We have used the KL divergence to compare the FFP7 and Gaussian reference sets and have shown that they are in good agreement.  This suggests that the additional instrumental effects and foreground residuals included in the FFP7 simulations are unimportant on the scales considered here.  Since the KL divergence is not symmetric in $P$ and $Q$, we have performed tests to demonstrate that their interchange has little effect on these conclusions.  Finally, we have repeated these calculations using the Kolmogorov--Smirnov test.  The resulting\n$p$-value of about 3\\% suggests that the differences between the two tests are not large.  We note that there is no guarantee that these tests will always\ngive similar results.  For example, the KL divergence is likely to be far more sensitive than the KS test for situations where there are large relative differences in the small amplitude tails of the distributions.  We have also repeated all the tests on the CMB data of the 2015 release from Planck, which recently became publicly available.\\footnote{See the Planck Legacy Archive \\url{http:\/\/pla.esac.esa.int\/pla\/}.} The results on the 2015 data set are in very good agreement with those reported here.\n\nThe difficulty in devising tests for the assessment of non-Gaussianity of the temperature and polarization maps of the CMB lies in our ignorance of\nthe nature of the non-Gaussian residuals from foregrounds and systematic effects that could propagate to the maps.  In such circumstances, it seems advisable to adopt a procedure that uses as much information in the maps as possible.  With its connection to the intrinsic and cross-entropy of the distributions $P$ and $Q$,\nthe KL divergence would appear to be the natural choice.  Given the correlations between the pixels of the CMB, a consequence of a random draw in harmonic space, this is not necessarily the case.  However, the non-locality of the KL divergence and its sensitivity to the tails of the distributions still suggest that it is a valuable complement to the KS test and might be a useful alternative.  Indeed, one should utilize a variety of methods and tests to identify possible contamination of the cosmological product.  Obviously, \\emph{any\\\/} suggestion of an anomalous result would indicate the need for more sophisticated analyses to assess the quality of the CMB maps.\n\n\n\\acknowledgments\n\nWe would like to thank P. Naselsky for his enthusiastic support and encouragement of the work reported in this paper and for many valuable scientific discussions.\n\nThis work is based on observations obtained with Planck,\\footnote{\\url{http:\/\/www.esa.int\/Planck}} an ESA\nscience mission with instruments and contributions directly funded by ESA\nMember States, NASA, and Canada. The development of Planck has been\nsupported by: ESA; CNES and CNRS \/ INSU-IN2P3-INP (France); ASI, CNR, and\nINAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA\n(Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada);\nDTU Space (Denmark); SER\/SSO (Switzerland); RCN (Norway); SFI (Ireland);\nFCT\/MCTES (Portugal); and PRACE (EU). A description of the Planck Collaboration and a list of its members,\nincluding the technical or scientific activities in which they have been\ninvolved, can be found at the Planck web page.\\footnote{\\url{http:\/\/www.cosmos.esa.int\/web\/planck\/planck-collaboration}}\n\nWe acknowledge the use of the NASA Legacy Archive for Microwave Background Data Analysis (LAMBDA). Our data analysis made use of the GLESP package\\footnote{\\url{http:\/\/www.glesp.nbi.dk\/}}~\\cite{Glesp}, and of \\textsc{HEALPix}~\\cite{HEALPix}. This work is supported in part by Danmarks Grundforskningsfond which allowed the establishment of the Danish Discovery Center, FNU grants 272-06-0417, 272-07-0528 and 21-04-0355, the National Natural Science Foundation of China (Grant No.\\ 11033003), the National Natural Science Foundation for Young Scientists of China (Grant No.\\ 11203024) and the Youth Innovation Promotion Association, CAS.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAs part of the proof of his eponymous theorem~\\cite{Szemeredi75} on arithmetic progressions in dense sets of integers, Szemer\\'edi developed (a variant of what is now known as) the graph {\\em regularity lemma}~\\cite{Szemeredi78}.\nThe lemma roughly states that the vertex set of every graph can be partitioned into a bounded number of parts such that\nalmost all the bipartite graphs induced by pairs of parts in the partition are quasi-random.\nIn the past four decades this lemma has become one of the (if not the) most powerful tools in extremal combinatorics, with applications in many other areas of mathematics. We refer the reader to~\\cite{KomlosShSiSz02,RodlSc10}\nfor more background on the graph regularity lemma, its many variants and its numerous applications.\n\nPerhaps the most important and well-known application of the graph regularity lemma is the original\nproof of the {\\em triangle removal lemma}, which states that if an $n$-vertex graph $G$ contains only $o(n^3)$ triangles, then one can turn $G$ into a triangle-free graph by removing only $o(n^2)$ edges (see \\cite{ConlonFox13} for more details).\nIt was famously observed by Ruzsa and Szemer\\'edi~\\cite{RuzsaSz76} that the triangle removal lemma implies Roth's theorem~\\cite{Roth54}, the special case of Szemer\\'edi's theorem for $3$-term arithmetic progressions.\nThe problem of extending the triangle removal lemma to the hypergraph\nsetting was raised by Erd\\H{o}s, Frankl and R\\\"odl~\\cite{ErdosFrRo86}. One of the main motivations for obtaining such a result was the observation of Frankl and R\\\"odl~\\cite{FrankRo02} (see also~\\cite{Solymosi04}) that such a result would allow one to extend the Ruzsa--Szemer\\'edi~\\cite{RuzsaSz76} argument and thus obtain an alternative proof of Szemer\\'edi's theorem for progressions of arbitrary length.\n\nThe quest for a hypergraph regularity lemma, which would allow one to prove a hypergraph removal lemma, took about 20 years.\nThe first milestone was the result of Frankl and R\\\"odl~\\cite{FrankRo02}, who obtained a regularity lemma for $3$-uniform hypergraphs. About 10 years later, the approach of~\\cite{FrankRo02} was extended to hypergraphs of arbitrary uniformity by R\\\"odl, Skokan, Nagle and Schacht~\\cite{NagleRoSc06, RodlSk04}.\nAt the same time, Gowers~\\cite{Gowers07} obtained an alternative version of the regularity lemma for $k$-uniform hypergraphs (from now on we will use $k$-graphs instead of $k$-uniform hypergraphs).\nShortly after, Tao~\\cite{Tao06} and R\\\"odl and Schacht~\\cite{RodlSc07,RodlSc07-B} obtained two more versions of the lemma.\n\nAs it turned out, the main difficulty with obtaining a regularity lemma for $k$-graphs was defining the correct notion of hypergraph regularity that would:\n$(i)$ be strong enough to allow one to prove a counting lemma, and\n$(ii)$ be weak enough to be satisfied by every hypergraph (see the discussion in~\\cite{Gowers06} for more on this issue).\nAnd indeed, the above-mentioned variants of the hypergraph regularity lemma rely on four different notions of quasi-randomness, which\nto this date are still not known to be equivalent\\footnote{This should be contrasted with the setting of graphs in which\n(almost) all notions of quasi-randomness are not only known to be equivalent but even effectively equivalent. See e.g.~\\cite{ChungGrWi89}.} (see~\\cite{NaglePoRoSc09} for some partial results). What all of these proofs {\\em do} have in common however,\nis that they supply only Ackermann-type bounds for the size of a regular partition.\\footnote{Another variant of the hypergraph regularity lemma was obtained in~\\cite{ElekSz12}. This approach does not supply any quantitative bounds.} \nMore precisely, if we let $\\Ack_1(x)=2^x$ and then define $\\Ack_k(x)$ to be the $x$-times iterated\\footnote{$\\Ack_2(x)$ is thus a tower of exponents of height $x$,\n\t$\\Ack_3(x)$ is the so-called wowzer function, etc.} version of $\\Ack_{k-1}$, then all the above proofs guarantee to produce a regular partition of a $k$-graph whose order can be bounded from above by an $\\Ack_k$-type function.\n\nOne of the most important applications of the $k$-graph regularity lemma was that it gave the first explicit\nbounds for the multidimensional generalization of Szemer\\'edi's theorem, see~\\cite{Gowers07}. The original proof of this result, obtained by Furstenberg and Katznelson~\\cite{FurstenbergKa78}, relied on Ergodic Theory and thus supplied no quantitative bounds at all.\nExamining the reduction between these theorems~\\cite{Solymosi04} reveals that if one could improve the Ackermann-type bounds for the $k$-graph regularity\nlemma, by obtaining (say) $\\Ack_{k_0}$-type upper bounds (for all $k$), then one would obtain the first primitive recursive bounds for the\nmultidimensional generalization of Szemer\\'edi's theorem. Let us note that obtaining such bounds just for\nvan der Waerden's theorem~\\cite{Shelah89} and Szemer\\'edi's theorem~\\cite{Szemeredi75} (which are two special case) were open problems for many decades till\nthey were finally solved by Shelah~\\cite{Shelah89} and Gowers~\\cite{Gowers01}, respectively.\nFurther applications of the $k$-graph regularity lemma (and the hypergraph removal lemma in particular) are described in~\\cite{RodlNaSkScKo05} and~\\cite{RodlTeScTo06} as well as in R\\\"odl's recent ICM survey~\\cite{Rodl14}.\n\n\nA famous result of Gowers~\\cite{Gowers97} states that the $\\Ack_2$-type upper bounds for graph regularity\nare unavoidable. Several improvements~\\cite{FoxLo17},\nvariants~\\cite{ConlonFo12,KaliSh13,MoshkovitzSh18} and simplifications~\\cite{MoshkovitzSh16} of Gowers' lower bound were recently obtained, but no analogous lower bound was derived even for $3$-graph regularity.\nThe numerous applications of the hypergraph regularity lemma naturally lead to the question of whether one can improve upon the Ackermann-type\nbounds mentioned above and obtain primitive recursive bounds\nfor the $k$-graph regularity lemma. Tao~\\cite{Tao06-h} predicted that the answer to this question is negative, in the sense that one cannot obtain better than $\\Ack_k$-type upper bounds for the $k$-graph regularity lemma for every $k \\ge 2$.\nThe main result presented here and in the followup \\cite{MSk} confirms this prediction.\n\n\n\\begin{theo}{\\bf[Main result, informal statement]}\\label{thm:main-informal}\nThe following holds for every $k\\geq 2$: every regularity lemma for $k$-graphs satisfying some\nmild conditions can only guarantee to produce partitions of size bounded by an $\\Ack_k$-type function.\n\\end{theo}\n\nIn this paper we will focus on proving the key ingredient needed for obtaining Theorem~\\ref{thm:main-informal},\nstated as Lemma~\\ref{theo:core} in Subsection~\\ref{subsec:overview}, and on showing how it can be used in order to prove Theorem~\\ref{thm:main-informal}\nfor $k=3$. In a nutshell, the key idea is to use the graph construction given by Lemma~\\ref{theo:core} in order\nto construct a $3$-graph by taking a certain ``product'' of two graphs that are hard for graph regularity, in order to get\na $3$-graph that is hard for $3$-graph regularity. See the discussion following Lemma~\\ref{theo:core} in Subsection~\\ref{subsec:overview}.\nDealing with $k=3$ in this paper will allow us to present all the new ideas needed in order to actually prove Theorem~\\ref{thm:main-informal}\nfor arbitrary $k$, in the slightly friendlier setting of $3$-graphs.\nIn a followup paper~\\cite{MSk}, we will show how Lemma~\\ref{theo:core} can be used in order to prove\nTheorem~\\ref{thm:main-informal} for all $k \\ge 2$.\n\nIn this paper we will also show how to derive from Theorem~\\ref{thm:main-informal} tight lower bounds for\nthe $3$-graph regularity lemmas due to Frankl and R\\\"odl~\\cite{FrankRo02} and to Gowers~\\cite{Gowers06}.\n\n\\begin{coro}\\label{coro:FR-LB}\nThere is an $\\Ack_3$-type lower bound for the $3$-graph regularity lemmas of Frankl and R\\\"odl~\\cite{FrankRo02} and of Gowers~\\cite{Gowers06}.\n\\end{coro}\n\nIn \\cite{MSk} we will show how to derive from Theorem~\\ref{thm:main-informal} a tight lower bound for\nthe $k$-graph regularity lemma due to R\\\"odl and Schacht~\\cite{RodlSc07}.\n\n\\begin{coro}\\label{coro:RS-LB}\nThere is an $\\Ack_k$-type lower bound for the $k$-graph regularity lemma of R\\\"odl and Schacht~\\cite{RodlSc07}.\n\\end{coro}\n\n\nBefore getting into the gory details of the proof, let us informally discuss what we think are some\ninteresting aspects of the proof of Theorem \\ref{thm:main-informal}.\n\n\\paragraph{Why is it hard to ``step up''?}\nThe reason why the upper bound for graph regularity is of tower-type\nis that the process of constructing a regular partition of a graph proceeds by a sequence of steps, each increasing the size of the partition exponentially.\nThe main idea behind Gowers' lower bound for graph regularity~\\cite{Gowers97} is in ``reverse engineering'' the proof of the upper bound; in other words,\nin showing that (in some sense) the process of building the partition using a sequence of exponential refinements is unavoidable.\nNow, a common theme in all proofs of the hypergraph regularity lemma\nfor $k$-graphs is that they proceed by induction on $k$; that is, in the process of constructing a regular\npartition of the input $k$-graph $H$, the proof applies the $(k-1)$-graph regularity lemma on certain $(k-1)$-graphs\nderived from $H$. This is why one gets $\\Ack_k$-type upper bounds. So with~\\cite{Gowers97} in mind, one might guess\nthat in order to prove a matching lower bound one should ``reverse engineer'' the proof of the upper bound and show that such a process is unavoidable. However, this turns out to be false! As we argued in~\\cite{MoshkovitzSh18}, in order to prove an {\\em upper bound} for (say) $3$-graph regularity it is in fact enough to iterate a relaxed version of graph regularity which we call the ``sparse regular approximation lemma'' (SRAL for short).\nTherefore, in order to prove an $\\Ack_3$-type {\\em lower bound} for $3$-graph\nregularity one cannot simply ``step up'' an $\\Ack_2$-type lower bound for graph regularity. Indeed, a necessary condition\nwould be to prove an $\\Ack_2$-type lower bound for SRAL. See also the discussion following Lemma~\\ref{theo:core} in Subsection \\ref{subsec:overview}\non how do we actually use a graph construction in order to get a $3$-graph construction.\n\n\n\\paragraph{A new notion of graph\/hypergraph regularity:}\nIn a recent paper \\cite{MoshkovitzSh18} we proved an $\\Ack_2$-type\nlower bound for SRAL.\nAs it turned out, even this lower bound was not enough to allow us to step up the graph lower bound\ninto a $3$-graph lower bound. To remedy this, in the present paper we introduce an even weaker notion of graph\/hypergraph regularity\nwhich we call $\\langle \\d \\rangle$-regularity. This notion seems to be right at the correct level of ``strength'';\non the one hand, it is strong enough to allow one to prove $\\Ack_{k-1}$-type lower bounds for $(k-1)$-graph regularity, while at the\nsame time weak enough to allow one to induct, that is, to use it in order to then prove $\\Ack_{k}$-type lower bounds for $k$-graph regularity.\nAnother critical feature of our new notion of hypergraph regularity is that it has (almost) nothing to do with hypergraphs!\nA disconcerting aspect of all proofs of the hypergraph regularity lemma is that they involve a very complicated nested\/inductive structure.\nFurthermore, one has to introduce an elaborate hierarchy of constants that controls how regular one level of the partition is compared to\nthe previous one. What is thus nice about our new notion is that it involves only various kinds of instances of graph $\\langle \\d \\rangle$-regularity.\nAs a result, our proof is (relatively!) simple.\n\n\\paragraph{How do we find witnesses for $3$-graph irregularity?}\nThe key idea in Gowers' lower bound~\\cite{Gowers97} for graph regularity was in constructing a graph $G$, based on a sequence of\npartitions ${\\cal P}_1,{\\cal P}_2,\\ldots$ of $V(G)$, with the following\ninductive property: if a vertex partition $\\Z$ refines ${\\cal P}_i$\nbut does not refine ${\\cal P}_{i+1}$ then $\\Z$ is not $\\epsilon$-regular.\nThe key step of the proof of~\\cite{Gowers97} is in finding witnesses showing\nthat pairs of clusters of $\\Z$ are irregular. The main difficulty in extending this\nstrategy to $k$-graphs already reveals itself in the setting of $3$-graphs. In a nutshell, while\nin graphs, a witness to irregularity of a pair of clusters $A,B \\in \\Z$ is\n{\\em any} pair of large subsets $A' \\sub A$ and $B' \\sub B$, in the setting of $3$-graphs\nwe have to find three large edge-sets (called a {\\em triad}, see Section~\\ref{sec:FR}) that\nhave an additional property: they must together form a graph containing many triangles.\nIt thus seems quite hard to extend Gowers' approach already to the setting of $3$-graphs. By working\nwith the much weaker notion of $\\langle \\d \\rangle$-regularity, we circumvent this\nissue since two of the edges sets in our version of a triad are always complete bipartite graphs.\nSee Subsection~\\ref{subsec:definitions}.\n\n\\paragraph{What is then the meaning of Theorem \\ref{thm:main-informal}?}\nOur main result, stated formally as Theorem~\\ref{theo:main}, establishes an $\\Ack_3$-type lower bound\nfor $\\langle \\d \\rangle$-regularity of $3$-graphs, that is, for a specific new\nversion of the hypergraph regularity lemma. Therefore, we immediately get $\\Ack_3$-type lower bounds for\nany $3$-graph regularity lemma which is at least as strong as our new lemma, that is, for any lemma\nwhose requirements\/guarantees imply those that are needed in order to satisfy our new notion of regularity.\nIn particular, we will prove Corollary \\ref{coro:FR-LB} by showing that the regularity notions\nused in these lemmas are at least as strong as $\\langle \\d \\rangle$-regularity.\n\nIn \\cite{MSk} we will prove Theorem~\\ref{thm:main-informal} in its full generality by extending Theorem~\\ref{theo:main}\nto arbitrary $k$-graphs. This proof, though technically more involved, will be\nquite similar at its core to the way we derive Theorem~\\ref{theo:main} from Lemma~\\ref{theo:core} in the present paper.\nThe deduction of Corollary \\ref{coro:RS-LB}, which appears in \\cite{MSk}, will also turn out to be quite similar to the way\nCorollary \\ref{coro:FR-LB} is derived from Theorem~\\ref{theo:main} in the present paper\n\n\\paragraph{How strong is our lower bound?} Since Theorem \\ref{thm:main-informal} gives a lower bound for\n$\\langle \\d \\rangle$-regularity and Corollaries \\ref{coro:FR-LB} and \\ref{coro:RS-LB} show that\nthis notion is at least as weak as previously used notions of regularity, it is natural to ask:\n$(i)$ is this notion equivalent to one of the other notions? $(ii)$ is this notion strong enough for proving the\nhypergraph removal lemma, which was one of the main reasons for developing\nthe hypergraph regularity lemma? We will prove that the answer to both questions is {\\em negative} by showing that already for\ngraphs, $\\langle \\d \\rangle$-regularity (for $\\d$ a fixed constant) is not strong enough even for proving the triangle removal lemma.\nThis of course makes our lower bound even stronger as it already applies to a very weak notion of regularity.\nIn a nutshell, the proof proceeds by first taking a random tripartite graph, showing (using routine probabilistic\narguments) that with high probability the graph is $\\langle \\d \\rangle$-regular yet contains a small number of triangles.\nOne then shows that removing these triangles, and then taking a blowup of the resulting graph, gives a triangle-free graph of positive density that is $\\langle \\d \\rangle$-regular. The full details will appear in \\cite{MSk}.\n\n\n\\paragraph{How tight is our bound?} Roughly speaking, we will show that for a $k$-graph with $pn^k$ edges,\nevery $\\langle \\d \\rangle$-regular partition has order at least $\\Ack_k(\\log 1\/p)$. In a recent paper \\cite{MoshkovitzSh16}\nwe proved that in graphs, one can prove a matching $\\Ack_2(\\log 1\/p)$ upper bound, even for a slightly stronger notion than $\\langle \\d \\rangle$-regularity.\nThis allowed us to obtain a new proof of Fox's $\\Ack_2(\\log 1\/\\epsilon)$ upper bound for the graph removal lemma \\cite{Fox11} (since the stronger notion allows to count small subgraphs). We believe that it should\nbe possible to match our lower bounds with $\\Ack_k(\\log 1\/p)$ upper bounds (even for a slightly stronger notion analogous to the one used in \\cite{MoshkovitzSh16}). We think that it should be possible to deduce from such an upper bound an $\\Ack_k(\\log 1\/\\epsilon)$ upper bound\nfor the $k$-graph removal lemma. The best known bounds for this problem are (at least) $\\Ack_k(\\poly(1\/\\epsilon))$.\n\n\n\n\n\\subsection{Paper overview}\n\nIn Section~\\ref{sec:define} we will first define the new notion of hypergraph regularity, which we term $\\langle \\d \\rangle$-regularity,\nfor which we will prove our main lower bound. We will then give the formal version of Theorem~\\ref{thm:main-informal} (see Theorem~\\ref{theo:main}).\nThis will be followed by the statement of our core technical result, Lemma~\\ref{theo:core},\nand an overview of how this technical result is used in the proof of Theorem~\\ref{theo:main}.\nThe proof of Theorem~\\ref{theo:main} appears in Section~\\ref{sec:LB}. \nWe refer the reader to~\\cite{MSk} for the proof of Lemma~\\ref{theo:core}.\nIn Section \\ref{sec:FR} we prove Corollary~\\ref{coro:FR-LB}.\nIn Appendix~\\ref{sec:FR-appendix} we give the proof of certain technical claims missing from Section~\\ref{sec:FR}.\n\n\n\n\n\n\n\\section{$\\langle \\d \\rangle$-regularity and Proof Overview}\\label{sec:define}\n\n\nFormally, a \\emph{$3$-graph} is a pair $H=(V,E)$, where $V=V(H)$ is the vertex set and $E=E(H) \\sub \\binom{V}{3}$ is the edge set of $H$.\nThe number of edges of $H$ is denoted $e(H)$ (i.e., $e(H)=|E|$).\nThe $3$-graph $H$ is \\emph{$3$-partite} on (disjoint) vertex classes $(V_1,V_2,V_3)$ if every edge of $H$ has a vertex from each $V_i$.\nThe \\emph{density} of a $3$-partite $3$-graph $H$ is $e(H)\/\\prod_{i=1}^3 |V_i|$.\nFor a bipartite graph $G$, the set of edges of $G$ between disjoint vertex subsets $A$ and $B$ is denoted by $E_G(A,B)$; the density of $G$ between $A$ and $B$ is denoted by $d_G(A,B)=e_G(A,B)\/|A||B|$, where $e_G(A,B)=|E_G(A,B)|$. We use $d(A,B)$ if $G$ is clear from context.\nWhen it is clear from context, we sometimes identify a hypergraph with its edge set. In particular, we will write $V_1 \\times V_2$ for the complete bipartite graph on vertex classes $(V_1,V_2)$.\nFor partitions $\\P,\\Q$ of the same underlying set, we say that $\\Q$ \\emph{refines} $\\P$, denoted $\\Q \\prec \\P$, if every member of $\\Q$ is contained in a member of $\\P$.\nWe say that $\\P$ is \\emph{equitable} if all its members have the same size.\\footnote{In a regularity lemma one allows the parts to differ in size by at most $1$ so that it applies to all (hyper-)graphs. For our lower bound this is unnecessary.}\nWe use the notation $x \\pm \\e$ for a number lying in the interval $[x-\\e,\\,x+\\e]$.\n\nIn the following definition, and in the rest of the paper, we will sometimes identify a graph or a $3$-graph with its edge set when the vertex set is clear from context.\n\n\\begin{definition}[$2$-partition]\\label{def:2-partition}\nA \\emph{$2$-partition} $(\\Z,\\E)$ on a vertex set $V$ consists of a partition $\\Z$ of $V$ and a family of edge disjoint bipartite graphs\n$\\E$ so that:\n\\begin{itemize}\n\\item Every $E \\in \\E$ is a bipartite graph\nwhose two vertex sets are distinct $Z,Z' \\in \\Z$.\n\\item For every $Z \\neq Z' \\in \\Z$, the complete bipartite graph $Z \\times Z'$ is the union of graphs from $\\E$.\n\\end{itemize}\n\n\n\n\n\n\\end{definition}\n\nPut differently, a $2$-partition consists of vertex partition $\\Z$ and a collection of bipartite graphs $\\E$ such that\n$\\E$ is a refinement of the collection of complete bipartite graphs $\\{Z \\times Z' : Z \\neq Z' \\in \\Z \\}$.\n\n\n\\subsection{$\\langle \\d \\rangle$-regularity of graphs and hypergraphs}\\label{subsec:definitions}\n\nIn this subsection we define our new\\footnote{For $k=3$, related notions of regularity were studied in~\\cite{ReiherRoSc16,Towsner17}.} notion of $\\langle\\d\\rangle$-regularity, first for graphs and then for $3$-graphs in Definition~\\ref{def:k-reg} below.\nLet us first recall Szemer\\'edi's notion of $\\epsilon$-regularity.\nA bipartite graph on $(A,B)$ is \\emph{$\\e$-regular} if for all subsets $A' \\sub A$, $B' \\sub B$ with $|A'|\\ge\\e|A|$, $|B'|\\ge\\e|B|$ we have $|d(A',B') -d(A,B)| \\le \\e$.\nA vertex partition $\\P$ of a graph is $\\e$-regular\nif\nthe bipartite graph induced on each but at most $\\e|\\P|^2$ of the pairs $(A,B)$ with $A \\neq B \\in \\P$ is $\\e$-regular.\nSzemer\\'edi's graph regularity lemma says that every graph\nhas an $\\e$-regular equipartition of order at most some $\\Ack_2(\\poly(1\/\\e))$.\nWe now introduce a weaker notion of graph regularity which we will use throughout the paper.\n\\begin{definition}[graph $\\langle\\d\\rangle$-regularity]\\label{def:star-regular}\n\tA bipartite graph $G$ on $(A,B)$ is \\emph{$\\langle \\d \\rangle$-regular} if for all subsets $A' \\sub A$, $B' \\sub B$ with $|A'| \\ge \\d|A|$, $|B'|\\ge\\d|B|$ we have $d_G(A',B') \\ge \\frac12 d_G(A,B)$.\\\\\n\tA vertex partition $\\P$ of a graph $G$ is \\emph{$\\langle \\d \\rangle$-regular}\n\n\tif one can add\/remove at most $\\d \\cdot e(G)$ edges so that the bipartite graph induced on each $(A,B)$ with $A \\neq B \\in \\P$ is $\\langle \\d \\rangle$-regular.\n\n\\end{definition}\n\nFor the reader worried that in Definition~\\ref{def:star-regular} we merely replaced the $\\e$\nfrom the definition of $\\e$-regularity with $\\d$, we refer to the discussion following Theorem~\\ref{theo:main} below.\n\n\n\n\n\nThe definition of $\\langle\\d\\rangle$-regularity for hypergraphs involves the $\\langle\\d\\rangle$-regularity notion for graphs, applied to certain auxiliary graphs which are defined as follows.\n\n\n\n\n\\begin{definition}[The auxiliary graph $G_{H}^i$]\\label{def:aux}\nFor a $3$-partite $3$-graph $H$ on vertex classes $(V_1,V_2,V_3)$,\nwe define a bipartite graph $G_{H}^1$ on the vertex classes $(V_2 \\times V_3,\\,V_1)$ by\n$$E(G_{H}^1) = \\big\\{ ((v_2,v_3),v_1) \\,\\big\\vert\\, (v_1,v_2,v_3) \\in E(H) \\big\\} \\;.$$\nThe graphs $G_{H}^2$ and $G_{H}^3$ are defined in an analogous manner.\n\\end{definition}\n\nImportantly, for a $2$-partition (as defined in Definition~\\ref{def:2-partition}) to be $\\langle\\d\\rangle$-regular it must first satisfy a requirement on the regularity of its parts.\n\n\n\\begin{definition}[$\\langle\\d\\rangle$-good partition]\\label{def:k-good}\nA $2$-partition $(\\Z,\\E)$ on $V$ is \\emph{$\\langle\\d\\rangle$-good} if all bipartite graphs in $\\E$\n(between any two distinct vertex clusters of $\\Z$) are $\\langle \\d \\rangle$-regular.\n\\end{definition}\n\n\n\n\n\n\nFor a $2$-partition $(\\Z,\\E)$ of a $3$-partite $3$-graph on vertex classes $(V_1,V_2,V_3)$ with $\\Z \\prec \\{V_1,V_2,V_3\\}$, for every $1 \\le i \\le 3$ we denote $\\Z_i = \\{Z \\in \\Z \\,\\vert\\, Z \\sub V_i\\}$, and we denote $\\E_i = \\{E \\in \\E \\,\\vert\\, E \\sub V_j \\times V_k\\}$ where $\\{i,j,k\\}=\\{1,2,3\\}$.\nSo for example, $\\E_1$ is thus a partition of $V_2 \\times V_3$.\n\n\\begin{definition}[$\\langle\\d\\rangle$-regular partition]\\label{def:k-reg}\nLet $H$ be a $3$-partite $3$-graph on vertex classes $(V_1,V_2,V_3)$\nand $(\\Z,\\E)$ be a $\\langle \\d \\rangle$-good $2$-partition with  $\\Z \\prec \\{V_1,V_2,V_3\\}$.\nWe say that $(\\Z,\\E)$ is a \\emph{$\\langle \\d \\rangle$-regular} partition of $H$ if\nfor every $1 \\le i \\le 3$,\n$\\E_i \\cup \\Z_i$ is a $\\langle \\d \\rangle$-regular partition of $G_H^i$.\n\n\n\n\n\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\t\t\n\t\t\n\n\n\\subsection{Formal statement of the main result}\n\nWe are now ready to formally state our tight lower bound for $3$-graph $\\langle \\d \\rangle$-regularity (the formal version of\nTheorem~\\ref{thm:main-informal} above for $k=3$). Recall that we define the {\\em tower} functions $\\twr(x)$ to be a tower of exponents of height $x$,\nand then define the {\\em wowzer} function $\\wow(x)$ to be the $x$-times iterated tower function, that is\n$\\wow(x)= \\underbrace{\\twr(\\twr(\\cdots(\\twr(1))\\cdots))}_{x \\text{ times}}$.\n\n\\begin{theo}[Main result]\\label{theo:main}\nFor every $s \\in \\N$ there is a $3$-partite $3$-graph $H$ on vertex classes of equal size and of density at least $2^{-s}$,\nand a partition $\\V_0$ of $V(H)$ with $|\\V_0| \\le 2^{300}$, such that if\n$(\\Z,\\E)$ is a $\\langle 2^{-73} \\rangle$-regular partition of $H$ with\n$\\Z \\prec \\V_0$ then $|\\Z| \\ge \\wow(s)$.\n\n\n\\end{theo}\n\nLet us draw the reader's attention to an important and perhaps surprising aspect of Theorem~\\ref{theo:main}.\nAll the known tower-type lower bounds for graph regularity depend on the error parameter $\\epsilon$,\nthat is, they show the existence of graphs $G$ with the property that every $\\epsilon$-regular partition of $G$ is of order at least $\\Ack_2(\\poly(1\/\\e))$.\nThis should be contrasted with the fact that our lower bounds for $\\langle \\d \\rangle$-regularity holds for a {\\em fixed} error parameter $\\delta$.\nIndeed, instead of the dependence on the error parameter, our lower bound depends on the {\\em density} of the graph.\nThis delicate difference makes it possible for us to prove Theorem~\\ref{theo:main} by iterating the construction described in the next subsection.\n\n\n\\subsection{The core construction and proof overview}\\label{subsec:overview}\n\nThe graph construction in Lemma~\\ref{theo:core} below is the main technical result we will need in order to prove Theorem~\\ref{theo:main}.\nWe will first need to define ``approximate'' refinement (a notion that goes back to Gowers~\\cite{Gowers97}).\n\\begin{definition}[Approximate refinements]\nFor sets $S,T$ we write $S \\sub_\\b T$ if $|S \\sm T| < \\b|S|$.\nFor a partition $\\P$ we write $S \\in_\\b \\P$ if $S \\sub_\\b P$ for some $P \\in \\P$.\nFor partitions $\\P,\\Q$ of the same set of size $n$ we write $\\Q \\prec_\\b \\P$ if\n$$\\sum_{\\substack{Q \\in \\Q\\colon\\\\Q \\notin_\\b \\P}} |Q| \\le \\b n \\;.$$\n\\end{definition}\nNote that for $\\Q$ equitable, $\\Q \\prec_\\b \\P$ if and only if\nall but at most $\\b|\\Q|$ parts $Q \\in \\Q$ satisfy $Q \\in_\\b \\P$.\nWe note that throughout the paper we will only use approximate refinements with $\\b \\le 1\/2$, and so if $S \\in_\\b \\P$ then $S \\sub_\\b P$ for a unique $P \\in \\P$.\n\nWe stress that in Lemma~\\ref{theo:core} below we only use notions related to graphs. In particular, $\\langle \\d \\rangle$-regularity refers to Definition~\\ref{def:star-regular}.\n\n\\begin{lemma}[Core construction]\\label{theo:core}\nLet $\\Lside$ and $\\Rside$ be disjoint sets. Let\n$\\L_1 \\succ \\cdots \\succ \\L_s$ and $\\R_1 \\succ \\cdots \\succ \\R_s$ be two sequences of $s$ successively refined equipartitions of $\\Lside$ and $\\Rside$, respectively,\nthat satisfy for  every $i \\ge 1$ that:\n\\begin{enumerate}\n\\item\\label{item:core-minR}\n$|\\R_i|$ is a power of $2$ and $|\\R_1| \\ge 2^{200}$,\n\\item\\label{item:core-expR} $|\\R_{i+1}| \\ge 4|\\R_i|$  if $i < s$,\n\\item\\label{item:core-expL} $|\\L_i| = 2^{|\\R_i|\/2^{i+10}}$.\n\\end{enumerate}\nThen there exists a sequence of $s$ successively refined edge equipartitions $\\G_1 \\succ \\cdots \\succ \\G_s$ of $\\Lside \\times \\Rside$ such that for every $1 \\le j \\le s$, $|\\G_j|=2^j$, and the following holds for every $G \\in \\G_j$ and $\\d \\le 2^{-20}$.\nFor every $\\langle \\d \\rangle$-regular partition $\\P \\cup \\Q$ of $G$, where $\\P$ and $\\Q$ are partitions of $\\Lside$ and $\\Rside$, respectively, and every $1 \\le i \\le j$, if $\\Q \\prec_{2^{-9}} \\R_{i}$ then $\\P \\prec_{\\g} \\L_{i}$ with $\\g = \\max\\{2^{5}\\sqrt{\\d},\\, 32\/\\sqrt[6]{|\\R_1|} \\}$.\n\\end{lemma}\n\\begin{remark}\\label{remequi}\nEvery $G \\in \\G_j$ is a bipartite graph of density $2^{-j}$ since $\\G_j$ is equitable.\n\\end{remark}\n\nAs mentioned before, the proof of Lemma~\\ref{theo:core} appears in~\\cite{MSk}.\nLet us end this section by explaining the role Lemma~\\ref{theo:core} plays in the proof of Theorem \\ref{theo:main}.\n\n\\paragraph{Using graphs to construct $3$-graphs:}\nPerhaps the most surprising aspect of the proof of Theorem~\\ref{theo:main} is that in order to construct a $3$-graph we also use the graph construction of Lemma~\\ref{theo:core} in a somewhat unexpected way.\nIn this case, $\\Lside$ will be a complete bipartite graph and the $\\L_i$'s will be partitions of this complete bipartite graph themselves given by another application of Lemma~\\ref{theo:core}.\nThe partitions will be of wowzer-type growth, and the second application of Lemma~\\ref{theo:core} will ``multiply'' the graph partitions (given by the $\\L_i$'s) to\ngive a partition of the complete $3$-partite $3$-graph into $3$-graphs that are hard for $\\langle \\d \\rangle$-regularity.\nWe will take $H$ in Theorem~\\ref{theo:main} to be an arbitrary $3$-graph in this partition.\n\n\n\n\n\n\\paragraph{Why is Lemma~\\ref{theo:core} one-sided?}\nAs is evident from the statement of Lemma~\\ref{theo:core}, it is one-sided in nature; that is, under the premise that the partition $\\Q$\nrefines $\\R_i$ we may conclude that $\\P$ refines $\\L_i$.\nIt is natural to ask if one can do away with this assumption, that is, be able to show that\nunder the same assumptions $\\Q$ refines $\\R_i$ and $\\P$ refines $\\L_i$.\nAs we mentioned in the previous item, in order to prove a wowzer-type lower bound for $3$-graph\nregularity we have to apply Lemma~\\ref{theo:core} with a sequence of partitions that grows as a wowzer-type function.\nNow, in this setting, Lemma~\\ref{theo:core} does not hold without the one-sided assumption, because if it did, then\none would have been able to prove a wowzer-type lower bound for graph $\\langle \\d \\rangle$-regularity, and hence also for Szemer\\'edi's regularity lemma.\nPut differently, if one wishes to have a construction that holds with arbitrarily fast growing partition sizes, then\none has to introduce the one-sided assumption.\n\n\\paragraph{How do we remove the one-sided assumption?}\nThe proof of Theorem \\ref{theo:main} proceeds by first proving a one-sided version of Theorem \\ref{theo:main},\nstated as Lemma~\\ref{lemma:ind-k}. In order to get a construction that does not require such a one-sided assumption,\nwe will need one final trick; we will take $6$ clusters of\nvertices and arrange $6$ copies\nof this one-sided construction\nalong the $3$-edges of a cycle. This will give us a ``circle of implications'' that will eliminate the one-sided assumption.\nSee Subsection \\ref{subsec:pasting}.\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{theo:main}}\\label{sec:LB}\n\n\n\n\\renewcommand{\\k}{r}\n\n\\renewcommand{\\t}{t}\n\\newcommand{\\w}{w}\n\n\\newcommand{\\GG}{\\mathbf{G}}\n\\newcommand{\\FF}{\\mathbf{F}}\n\\newcommand{\\VV}{\\mathbf{V}}\n\n\n\\renewcommand{\\Hy}[1]{H_{{#1}}}\n\\renewcommand{\\A}{A}\n\n\\newcommand{\\subs}{\\subset_*}\n\\newcommand{\\pad}{P}\n\\renewcommand{\\K}{\\mathcal{K}}\n\\newcommand{\\U}{U}\n\n\n\\renewcommand{\\k}{k}\n\n\\renewcommand{\\K}{\\mathcal{K}}\n\n\\renewcommand{\\r}{k}\n\n\nThe purpose of this section is to prove the main result, Theorem~\\ref{theo:main}.\nThis section is self-contained save for the application of Lemma~\\ref{theo:core}.\nThe key step of the proof, stated as Lemma~\\ref{lemma:ind-k} and proved in Subsection \\ref{subsec:key}, relies on a subtle construction that uses Lemma~\\ref{theo:core} twice. This lemma only gives a ``one-sided'' lower bound for $3$-graph regularity, in the spirit of Lemma~\\ref{theo:core}.\nIn Subsection~\\ref{subsec:pasting} we show how to use Lemma~\\ref{lemma:ind-k} in order to complete the proof of Theorem~\\ref{theo:main}.\n\nWe first observe a simple yet crucial property of $2$-partitions, stated as Claim~\\ref{claim:uniform-refinement} below, which we will need later.\nThis property relates $\\d$-refinements of partitions and  $\\langle \\d \\rangle$-regularity of partitions,\nand relies\non Claim~\\ref{claim:refinement-union}. Here, as well as in the rest of this section, we will use\nthe definitions and notations introduced in Section \\ref{sec:define}.\nIn particular, recall that if a vertex partition $\\Z$ of vertex classes $(V_1,V_2,V_3)$ satisfies $\\Z \\prec \\{V_1,V_2,V_3\\}$,\nthen for every $1 \\le i \\le 3$ we denote $\\Z_i = \\{Z \\in \\Z \\,\\vert\\, Z \\sub V_i\\}$.\nMoreover, if a $2$-partition $(\\Z,\\E)$, satisfies $\\Z \\prec \\{V_1,V_2,V_3\\}$ we denote $\\E_i = \\{E \\in \\E \\,\\vert\\, E \\sub V_j \\times V_k\\}$ where $\\{i,j,k\\}=\\{1,2,3\\}$. We will first need the following easy claim regarding the union of $\\langle \\d\\rangle$-regular graphs.\n\n\n\\begin{claim}\\label{claim:star-union\n\tLet $G_1,\\ldots,G_\\ell$ be mutually edge-disjoint bipartite graphs on the same vertex classes $(Z,Z')$.\n\tIf every $G_i$ is $\\langle \\d \\rangle$-regular then $G=\\bigcup_{i=1}^\\ell G_i$ is also $\\langle \\d \\rangle$-regular.\n\\end{claim}\n\\begin{proof\n\n\tLet $S \\sub Z$, $S' \\sub Z'$ with $|S| \\ge \\d|Z|$, $|S'| \\ge \\d|Z'|$.\n\tThen\n\t$$d_G(S,S') = \\frac{e_G(S,S')}{|S||S'|}\n\t= \\sum_{i=1}^\\ell \\frac{e_{G_i}(S,S')}{|S||S'|}\n\t= \\sum_{i=1}^\\ell d_{G_i}(S,S')\n\t\\ge \\sum_{i=1}^\\ell \\frac12 d_{G_i}(Z,Z')\n\t= \\frac12 d_{G}(Z,Z') \\;,$$\n\twhere the second and last equalities follow from the mutual disjointness of the $G_i$, and the inequality follows from the $\\langle \\d \\rangle$-regularity of each $G_i$.\n\tThus, $G$ is $\\langle \\d \\rangle$-regular, as claimed.\n\\end{proof}\n\nWe use the following claim regarding approximate refinements.\n\n\\begin{claim}\\label{claim:refinement-union}\n\tIf $\\Q \\prec_\\d \\P$ then there exist $P \\in \\P$ and $Q$ that is a union of members of $\\Q$ such that $|P \\triangle Q| \\le 3\\d|P|$.\n\\end{claim}\n\\begin{proof}\n\tFor each $P\\in \\P$ let $\\Q(P) = \\{Q \\in \\Q \\colon Q \\sub_\\d P\\}$,\n\tand denote $P_\\Q = \\bigcup_{Q \\in \\Q(P)} Q$.\n\tWe have\n\t\\begin{align*}\n\t\\sum_{P \\in \\P} |P \\triangle P_\\Q|\n\t&= \\sum_{P \\in \\P} |P_\\Q \\sm P| + \\sum_{P \\in \\P} |P \\sm P_\\Q|\n\t= \\sum_{P \\in \\P} \\sum_{\\substack{Q \\in \\Q \\colon\\\\Q \\sub_\\d P}} |Q \\sm P|\n\t+ \\sum_{P \\in \\P} \\sum_{\\substack{Q \\in \\Q \\colon\\\\Q \\nsubseteq_\\d P}} |Q \\cap P| \\\\\n\t&\\le \\sum_{P \\in \\P} \\sum_{\\substack{Q \\in \\Q \\colon\\\\Q \\sub_\\d P}} \\d|Q|\n\t+ \\Big( \\sum_{\\substack{Q \\in \\Q\\colon\\\\Q \\notin_\\d \\P}} |Q|\n\t+ \\sum_{\\substack{Q \\in \\Q \\colon\\\\Q \\in_\\d \\P}} \\d|Q| \\Big)\n\t\\le 3\\d\\sum_{Q \\in \\Q} |Q|\n\t= 3\\d\\sum_{P \\in \\P} |P| \\;,\n\t\\end{align*}\n\twhere the last inequality uses the statement's assumption $\\Q \\prec_\\d \\P$ to bound the middle summand.\n\tBy averaging, there exists $P \\in \\P$ such that $|P \\triangle P_\\Q| \\le 3\\d|P|$, thus completing the proof.\n\\end{proof}\n\nThe property of $2$-partitions that we need is as follows.\n\\begin{claim}\\label{claim:uniform-refinement}\n\tLet $\\P=(\\Z,\\E)$ be a $2$-partition with  $\\Z \\prec \\{V_1,V_2,V_3\\}$,\n\tand let $\\G$ be a partition of $V_1\\times V_2$ with $\\E_3 \\prec_\\d \\G$.\n\tIf $(\\Z,\\E)$ is $\\langle \\d \\rangle$-good\n\tthen $\\Z_1 \\cup \\Z_2$ is a $\\langle 3\\d \\rangle$-regular partition of some $G \\in \\G$.\n\\end{claim}\n\n\\begin{proof}\n\tPut $\\E=\\E_3$.\n\tBy Claim~\\ref{claim:refinement-union}, since $\\E \\prec_\\d \\G$ there exist $G \\in \\G$ (a bipartite graph on $(V_1,V_2)$) and $G_\\E$ that is a union of members of $\\E$ (and thus also a bipartite graph on $(V_1,V_2)$) such that $|G \\triangle G_\\E| \\le 3\\d|G|$.\n\n\tLetting $Z_1 \\in \\Z_1$, $Z_2 \\in \\Z_2$, to complete the proof it suffices to show that the induced bipartite graph $G_\\E[Z_1,Z_2]$ is $\\langle \\d \\rangle$-regular (recall Definition~\\ref{def:star-regular}).\n\tBy Definition~\\ref{def:2-partition}, $G_\\E[Z_1,Z_2]$ is a union of bipartite graphs from $\\E$ on $(Z_1,Z_2)$.\n\tSince every graph in $\\E$ is $\\langle \\d \\rangle$-regular by the statement's assumption that $(\\Z,\\E)$ is $\\langle \\d \\rangle$-good (recall Definition~\\ref{def:k-good}), we have that $G_\\E[Z_1,Z_2]$ is a union of $\\langle \\d \\rangle$-regular bipartite graphs on $(Z_1,Z_2)$.\n\tBy Claim~\\ref{claim:star-union}, $G_\\E[Z_1,Z_2]$ is $\\langle \\d \\rangle$-regular as well, thus completing the proof.\n\\end{proof}\n\nWe will later need the following easy (but slightly tedious to state) claim.\n\\begin{claim}\\label{claim:restriction}\n\tLet $H$ be a $3$-partite $3$-graph on vertex classes $(V_1,V_2,V_3)$, and let $H'$ be the induced $3$-partite $3$-graph on vertex classes $(V_1',V_2',V_3')$ with $V_i' \\sub V_i$ and $\\a \\cdot e(H)$ edges.\n\tIf $(\\Z,\\E)$ is a $\\langle \\d \\rangle$-regular partition of $H$ with\n\t$\\Z \\prec \\bigcup_{i=1}^3 \\{V_i,\\,V_i \\sm V_i'\\}$\n\n\tthen its restriction $(\\Z',\\E')$ to $V(H')$ is a $\\langle \\d\/\\a \\rangle$-regular partition of $H'$.\n\\end{claim}\n\\begin{proof}\n\tRecall Definition~\\ref{def:k-reg}.\n\tClearly, $(\\Z',\\E')$ is $\\langle \\d \\rangle$-good. We will show that $\\E'_1 \\cup \\Z'_1$ is a $\\langle \\d\/\\a \\rangle$-regular partition of $G^1_{H'}$.\n\tThe argument for $G^2_{H'}$, $G^3_{H'}$ will be analogous, hence the proof would follow.\n\tObserve that $G^1_{H'}$ is an induced subgraph of $G^1_{H}$, namely, $G^1_{H'} = G^i_{H}[V_2' \\times V_3',\\, V_1']$.\n\tBy assumption, $e(H') = \\a e(H)$, and thus $e(G^1_{H'}) = \\a e(G^1_{H})$.\n\tBy the statement's assumption on $\\Z$ and since $\\E_1 \\cup \\Z_1$ is a $\\langle \\d \\rangle$-regular partition of $G^1_{H}$, we deduce---by adding\/removing at most $\\d e(G^1_{H}) =  (\\d\/\\a)e(G^1_{H'})$ edges of $G^1_{H'}$---that $\\E'_1 \\cup \\Z'_1$ is a $\\langle \\d\/\\a \\rangle$-regular partition of $G_{H'}^1$.\n\tAs explained above, this completes the proof.\n\\end{proof}\n\n\nFinally, we will need the following claim regarding approximate refinements.\n\\begin{claim}\\label{claim:refinement-size}\n\tIf $\\Q \\prec_{1\/2} \\P$ and $\\P$ is equitable then $|\\Q| \\ge \\frac14|\\P|$.\n\\end{claim}\n\\begin{proof}\n\tWe claim that the underlying set $U$ has a subset $U^*$ of size $|U^*|\\ge \\frac14|U|$ such that the partitions $\\Q^*=\\{Q \\cap U^* \\,\\vert\\, Q \\in \\Q \\} \\setminus \\{\\emptyset\\}$ and $\\P^*=\\{P \\cap U^* \\,\\vert\\, P \\in \\P \\} \\setminus \\{\\emptyset\\}$\n\n\tof $U^*$ satisfy $\\Q^* \\prec \\P^*$.\n\tIndeed, let $U^* = \\bigcup_{Q} Q \\cap P_Q$ where the union is over all $Q \\in \\Q$ satisfying $Q \\sub_{1\/2} P_Q$ for a (unique) $P_Q \\in \\P$.\n\n\tAs claimed, $|U^*| = \\sum_{Q \\in_{1\/2} \\P} |Q \\cap P_Q| \\ge \\sum_{Q \\in_{1\/2} \\P} \\frac12|Q| \\ge \\frac14|U|$, using $\\Q \\prec_{1\/2} \\P$ for the last inequality.\n\n\tNow, since $\\P$ is equitable, $|\\P^*| \\ge \\frac14|\\P|$.\n\n\tThus, $|\\Q| \\ge |\\Q^*| \\ge |\\P^*| \\ge \\frac14|\\P|$, as desired.\n\\end{proof}\n\n\n\\renewcommand{\\K}{k}\n\\renewcommand{\\w}{w}\n\n\\subsection{$3$-graph key argument}\\label{subsec:key}\n\n\nWe next introduce a few more definitions that are needed for the statement of Lemma \\ref{lemma:ind-k}.\nLet $e(i) = 2^{i+10}$. We define the following tower-type function $\\t\\colon\\N\\to\\N$;\n\\begin{equation}\\label{eq:t}\n\\t(i+1) = \\begin{cases}\n2^{\\t(i)\/e(i)}\t&\\text{if } i \\ge 1\\\\\n2^{250}\t&\\text{if } i = 0 \\;.\n\\end{cases}\n\\end{equation}\nIt is easy to prove, by induction on $i$, that $\\t(i) \\ge e(i)\\t(i-1)$ for $i \\ge 2$ (for the induction step,\n$t(i+1) \\ge 2^{\\t(i-1)} = t(i)^{e(i-1)}$, so $t(i+1)\/e(i+1) \\ge \\t(i)^{e(i-1)-i-11} \\ge \\t(i)$).\nThis means that $t$ is monotone increasing, and that $\\t$ is an integer power of $2$ (follows by induction as $t(i)\/e(i) \\ge 1$ is a positive power of $2$ and in particular an integer).\nWe record the following facts regarding $\\t$ for later use:\n\\begin{equation}\\label{eq:monotone}\n\\t(i) \\ge 4\\t(i-1)  \\quad\\text{ and }\\quad\n\\text{ $\\t(i)$ is a power of $2$} \\;.\n\\end{equation}\nFor a function $f:\\N\\to\\N$ with $f(i) \\ge i$ we denote\n\\begin{equation}\\label{eq:f*}\nf^*(i) = \\t\\big(f(i)\\big)\/e(i) \\;.\n\\end{equation}\nNote that $f^*(i)$ is indeed a positive integer (by the monotonicity of $\\t$, $f^*(i) \\ge \\t(i)\/e(i)$ is a positive power of $2$).\nIn fact, $f^*(i) \\ge f(i)$ (as $f^*(i) \\ge 4^{f(i)}\/e(i)$ using~(\\ref{eq:monotone})).\nWe recursively define the function $\\w\\colon\\N\\to\\N$ as follows;\n\\begin{equation}\\label{eq:Ak}\n\\w(i+1) = \\begin{cases}\n\\w^*(i)\t\t&\\text{if } i \\ge 1\\\\\n1\t\t&\\text{if } i = 0 \\;.\n\\end{cases}\n\\end{equation}\nIt is evident that $\\w$ is a wowzer-type function; in fact, one can check that:\n\\begin{equation}\\label{eq:A_k}\n\\w(i) \\ge \\wow(i) \\;.\n\\end{equation}\n\n\n\n\n\n\n\n\n\\begin{lemma}[Key argument]\\label{lemma:ind-k}\n\n\n\tLet $s \\in \\N$, let $\\Vside^1,\\Vside^2,\\Vside^3$ be mutually disjoint sets of equal size and let $\\V^1 \\succ\\cdots\\succ \\V^m$ be a sequence of\n\t$m=\\w^*(s)+1$ successive equitable refinements of $\\{\\Vside^1,\\Vside^2,\\Vside^3\\}$\n\twith $|\\V^i_1|=|\\V^i_2|=|\\V^i_3|=\\t(i)$ for every\\footnote{Since we assume that each $\\V^i$ refines $\\{\\Vside^1,\\Vside^2,\\Vside^3\\}$ then $\\V^i_1$ is (by the notation mentioned before Claim \\ref{claim:star-union}) the restriction of $\\V^i$ to $\\Vside^1$.} $1 \\leq i \\leq m$.\n\tThen there is a $3$-partite $3$-graph $H$ on $(\\Vside^1,\\Vside^2,\\Vside^3)$ of density $d(H)=2^{-s}$ satisfying the following property:\\\\\n\tIf $(\\Z,\\E)$ is a $\\langle 2^{-70} \\rangle$-regular partition of $H$ and for some\n\t$1 \\le i \\le \\w(s)$ $(< m)$ we have $\\Z_3 \\prec_{2^{-9}} \\V^i_3$ and $\\Z_2 \\prec_{2^{-9}} \\V^i_2$ then we also have $\\Z_1 \\prec_{2^{-9}} \\V^{i+1}_1$.\n\\end{lemma}\n\n\\begin{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\tPut $s':=\\w^*(s)$, so that $m = s'+1$.\n\tApply Lemma~\\ref{theo:core} with\n\t$$\\Lside=\\Vside^1,\\quad \\Rside=\\Vside^2 \\quad\\text{ and }\\quad \\V^2_1 \\succ \\cdots \\succ \\V^{s'+1}_1 ,\\quad \\V^1_2 \\succ \\cdots \\succ \\V^{s'}_2 \\;,$$\n\tand let\n\t\\begin{equation}\\label{eq:main-k-colors}\n\t\\G^1 \\succ \\cdots \\succ \\G^{s'} \\quad\\text{ with }\\quad |G^\\ell|=2^\\ell \\text{ for every } 1 \\le \\ell \\le s'\n\t\\end{equation}\n\tbe the resulting sequence of $s'$ successively refined equipartitions of $\\Vside^1 \\times \\Vside^2$.\n\n\n\n\n\t\n\t\\begin{prop}\\label{prop:main-k-hypo}\n\t\tLet $1 \\le \\ell \\le s'$ and $G \\in \\G^\\ell$.\n\t\tFor every $\\langle 2^{-28} \\rangle$-regular partition $\\Z_1 \\cup \\Z_2$ of $G$ (where $\\Z_1$ and $\\Z_2$ are partitions of $\\Vside^1$ and $\\Vside^2$, respectively) and every $1 \\le i \\le \\ell$,\n\t\tif $\\Z_2 \\prec_{2^{-9}} \\V^i_2$\n\t\tthen $\\Z_1 \\prec_{2^{-9}} \\V^{i+1}_1$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tFirst we need to verify that we may apply Lemma~\\ref{theo:core} as above.\n\t\tAssumptions~\\ref{item:core-minR},~\\ref{item:core-expR} in Lemma~\\ref{theo:core} hold by~(\\ref{eq:monotone}) and the fact that $|\\V^j_2|=\\t(j)$.\n\t\tAssumption~\\ref{item:core-expL} is satisfied since for every $1 \\le j \\le s$ we have\n\t\t$$|\\V^{j+1}_1| = \\t(j+1) = 2^{\\t(j)\/e(j)} = 2^{|\\V^{j}_2|\/e(j)} \\;,$$\n\t\twhere the second equality uses the definition of the function $\\t$ in~(\\ref{eq:t}).\n\t\tWe can thus use Lemma~\\ref{theo:core} to infer that the fact that $\\Z_2 \\prec_{2^{-9}} \\V^i_2$ implies that $\\Z_1 \\prec_x \\V^{i+1}_1$ with $x=\\max\\{2^{5}\\sqrt{2^{-28}},\\, 32\/\\sqrt[6]{\\t(1)} \\} = 2^{-9}$, using~(\\ref{eq:t}).\n\t\\end{proof}\n\t\n\n\n\n\n\n\n\n\n\n\n\t\n\tFor each $1 \\le j \\le s$ let\n\t\\begin{equation}\\label{eq:main-k-dfns}\n\t\\G^{(j)} = \\G^{\\w^*(j)}\n\t\\quad\\text{ and }\\quad\n\t\\V^{(j)} = \\V^{\\w(j)}_3 \\;.\n\t\\end{equation}\t\n\tAll these choices are well defined since $\\w^*(j)$ satisfies $1 \\le \\w^*(1) \\le \\w^*(j) \\le \\w^*(s) = s'$, and since $\\w(j)$ satisfies $1 \\le \\w(1) \\le \\w(j) \\le \\w(s) \\le m$. Observe that we have thus chosen two subsequences of $\\G^1,\\cdots,\\G^{s'}$ and $\\V^1_3,\\ldots,\\V^m_3$, each of length $s$.\n\tRecalling that each $\\G^{(j)}$ is a partition of $\\Vside^1 \\times \\Vside^2$, we now apply Lemma~\\ref{theo:core} again with\n\t$$\n\t\\Lside=\\Vside^1 \\times \\Vside^2,\\quad \\Rside=\\Vside^3 \\quad\\text{ and }\\quad \\G^{(1)} \\succ \\cdots \\succ \\G^{(s)}, \\quad \\V^{(1)} \\succ \\cdots \\succ \\V^{(s)} \\;.\n\t$$\n\tThe output of this application of Lemma~\\ref{theo:core} consists of a sequence of $s$ (successively refined)\n\tequipartitions of $(\\Vside^1 \\times \\Vside^2)\\times\\Vside^3$.\n\tWe can think of the $s$-th partition of this sequence as a collection of $2^s$ bipartite graphs on vertex sets\n\t$(\\Vside^1\\times\\Vside^{2},\\,\\Vside^3)$. For the rest of the proof let $G'$ be be any of these graphs.\n\tBy Remark \\ref{remequi} we have\n\t\\begin{equation}\\label{eq:ind-colors2}\n\td(G')=2^{-s} \\;.\n\t\\end{equation}\n\t\\begin{prop}\\label{prop:ind-prop2}\n\t\tFor every $\\langle 2^{-70} \\rangle$-regular partition $\\E \\cup \\V$ of $G'$ (where $\\E$ and $\\V$ are partitions of $\\Vside^1\\times\\Vside^{2}$ and $\\Vside^3$ respectively)\n\t\tand every $1 \\le j' \\le s$,\n\t\tif $\\V \\prec_{2^{-9}} \\V^{(j')}$ then $\\E \\prec_{2^{-30}} \\G^{(j')}$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tFirst we need to verify that we may apply Lemma~\\ref{theo:core} as above.\n\t\tNote that $|\\G^{(j)}|=2^{\\w^*(j)}$ by~(\\ref{eq:main-k-colors}) and (\\ref{eq:main-k-dfns}),\n\t\tand that $|\\V^{(j)}|=\\t(\\w(j))$ by (\\ref{eq:main-k-dfns}) and the statement's assumption that $|\\V^i_3|=\\t(i)$.\n\t\tTherefore,\n\t\t\\begin{equation}\\label{eq:main-k-orders}\n\t\t|\\G^{(j)}| = 2^{\\w^*(j)} = 2^{\\t(\\w(j))\/e(j)} = 2^{|\\V^{(j)}|\/e(j)} \\;,\n\t\n\t\t\\end{equation}\n\t\twhere the second equality relies on~(\\ref{eq:f*}).\n\t\tMoreover, note that $\\t(\\w(1)) = \\t(1) = 2^{300}$.\n\t\tNow, Assumptions~\\ref{item:core-minR} and~\\ref{item:core-expR} in Lemma~\\ref{theo:core} follow from the fact that $|\\V^{(j)}|=\\t(\\w(j))$, from~(\\ref{eq:monotone})\n\t\tand the fact that $|\\V^{(1)}| = \\t(\\w(1)) \\ge 2^{200}$ by~(\\ref{eq:Ak}). \t\n\t\n\t\tAssumption~\\ref{item:core-expL} follows from~(\\ref{eq:main-k-orders}).\n\t\tWe can thus use Lemma~\\ref{theo:core} to infer that the fact that $\\V \\prec_{2^{-9}} \\V^{(j')}$ implies that\n\t\t$\\E \\prec_x \\G^{(j')}$ with $x=\\max\\{2^{5}\\sqrt{2^{-70}},\\, 32\/\\sqrt[6]{\\t(\\w(1))} \\} = 2^{-30}$.\n\t\n\t\n\t\n\t\\end{proof}\n\t\n\tLet $H$ be the $3$-partite $3$-graph on vertex classes $(\\Vside^1,\\Vside^2,\\Vside^3)$ with edge set\n\t$$\n\tE(H) = \\big\\{ (v_1,v_2,v_3) \\,:\\, ((v_1,v_{2}),v_3) \\in E(G') \\big\\} \\;,\n\t$$\t\n\tand note that we have (recall Definition \\ref{def:aux})\n\t\\begin{equation}\\label{eqH}\n\tG'=G_{H}^3\\;.\n\t\\end{equation}\n\tWe now prove that $H$ satisfies the properties in the statement of the lemma.\n\n\n\n\n\tFirst, note that by~(\\ref{eq:ind-colors2}) and (\\ref{eqH}) we have $d(H)=2^{-s}$, as needed.\n\tAssume now that $i$ is such that\n\t\\begin{equation}\\label{eq:ind-i-assumption}\n\t1 \\le i \\le \\w(s)\n\t\\end{equation}\n\tand:\n\t\\begin{enumerate}\n\t\t\\item\\label{item:ind-reg}\n\t\t$(\\Z,\\E)$ is a $\\langle 2^{-70} \\rangle$-regular partition of $H$, and\n\t\t\\item\\label{item:ind-refine} $\\Z_3 \\prec_{2^{-9}} \\V^i_3$ and $\\Z_2 \\prec_{2^{-9}} \\V^i_2$.\n\t\\end{enumerate}\t\n\tWe need to show that\n\t\\begin{equation}\\label{eq:ind-goal}\n\t\\Z_1 \\prec_{2^{-9}} \\V^{i+1}_1 \\;.\n\t\\end{equation}\n\n\tSince Item~\\ref{item:ind-reg} guarantees that $(\\Z,\\E)$ is a $\\langle 2^{-70} \\rangle$-regular partition of $H$,\n\n\twe get from Definition~\\ref{def:k-reg} and (\\ref{eqH}) that\n\t\\begin{equation}\\label{eq:ind-reg}\n\t\\text{$\\E_3 \\cup \\Z_3$ is a $\\langle 2^{-70} \\rangle$-regular partition of } G'.\n\t\\end{equation}\n\tLet\n\t\\begin{equation}\\label{eq:ind-j'}\n\t1 \\le j' \\le s\n\t\\end{equation}\n\tbe the unique integer satisfying (the equality here is just (\\ref{eq:Ak}))\n\t\\begin{equation}\\label{eq:ind-sandwich}\n\t\\w(j') \\le i < \\w(j'+1) = \\w^*(j')\\;.\n\t\\end{equation}\n\tNote that (\\ref{eq:ind-j'}) holds due to~(\\ref{eq:ind-i-assumption}).\n\tRecalling~(\\ref{eq:main-k-dfns}),\n\tthe lower bound in~(\\ref{eq:ind-sandwich}) implies that $\\V^i_3 \\prec \\V^{\\w(j')} = \\V^{(j')}$.\n\tTherefore, the assumption $\\Z_3 \\prec_{2^{-9}} \\V^i_3$ in~\\ref{item:ind-refine} implies that\n\t\\begin{equation}\\label{eq:ind-Zk}\n\t\\Z_3 \\prec_{2^{-9}} \\V^{(j')} \\;.\n\t\\end{equation}\n\tApply Proposition~\\ref{prop:ind-prop2} on $G'$, using~(\\ref{eq:ind-reg}),~(\\ref{eq:ind-j'}) and~(\\ref{eq:ind-Zk}), to deduce that\n\t\\begin{equation}\\label{eq:ind-E}\n\t\\E_3 \\prec_{2^{-30}} \\G^{(j')} = \\G^{\\w^*(j')} \\;,\n\t\\end{equation}\n\twhere for the equality again recall~(\\ref{eq:main-k-dfns}).\n\tSince $(\\Z,\\E)$ is a $\\langle 2^{-70} \\rangle$-regular partition of $H$ (by Item~\\ref{item:ind-reg} above)\n\tit is in particular $\\langle 2^{-70} \\rangle$-good. By~(\\ref{eq:ind-E}) we may thus apply Claim~\\ref{claim:uniform-refinement}\n\tto conclude that\n\t\\begin{equation}\\label{eq:ind-reg2}\n\t\\Z_1 \\cup \\Z_2 \\text{ is a }\n\t\\langle 2^{-28} \\rangle\n\t\\text{-regular partition of some $G\\in\\G^{\\w^*(j')}$.}\n\t\\end{equation}\n\tBy~(\\ref{eq:ind-reg2}) we may apply Proposition~\\ref{prop:main-k-hypo} with $G$, $\\Z_1\\cup\\Z_2$, $\\ell=\\w^*(j')$ and $i$, observing (crucially)\n\tthat $i \\leq \\ell$ by (\\ref{eq:ind-sandwich}). We thus conclude that the fact $\\Z_2 \\prec_{2^{-9}} \\V^i_2$ (stated in~\\ref{item:ind-refine})\n\timplies that $\\Z_1 \\prec_{2^{-9}} \\V^{i+1}_1$, thus proving~(\\ref{eq:ind-goal}) and completing the proof.\n\\end{proof}\n\n\n\n\n\\subsection{Putting everything together}\\label{subsec:pasting}\n\n\nWe can now prove our main theorem, Theorem~\\ref{theo:main}, which we repeat here for convenience.\n\\addtocounter{theo}{-2}\n\\begin{theo}[Main theorem\n\tLet $s \\in \\N$.\n\tThere exists a $3$-partite $3$-graph $H$ on vertex classes of equal size and of density at least $2^{-s}$,\n\tand a partition $\\V_0$ of $V(H)$ with $|\\V_0| \\le 2^{300}$, such that if\n\t$(\\Z,\\E)$ is a $\\langle 2^{-73} \\rangle$-regular partition of $H$ with\n\t$\\Z \\prec \\V_0$ then $|\\Z| \\ge \\wow(s)$.\n\n\n\n\n\n\n\n\n\n\n\n\t\n\n\n\n\n\n\n\n\n\n\n\n\\end{theo}\n\\addtocounter{theo}{+1}\n\n\n\n\\begin{proof}\n\n\n\n\n\n\n\n\tLet the $3$-graph $B$ be the tight $6$-cycle; that is, $B$ is the $3$-graph on vertex classes $\\{0,1,\\ldots,5\\}$ with edge set $E(B)=\\{\\{0,1,2\\},\\{1,2,3\\},\\{2,3,4\\},\\{3,4,5\\},\\{4,5,0\\},\\{5,0,1\\}\\}$.\n\tNote that $B$ is $3$-partite with vertex classes $(\\{0,3\\},\\{1,4\\},\\{2,5\\}\\}$.\n\n\tPut $m=\\w^*(s-1)+1$ and let $n \\ge \\t(m)$.\n\tLet $\\Vside^0,\\ldots,\\Vside^{5}$ be $6$ mutually disjoint sets of size $n$ each.\n\tLet $\\V^1 \\succ\\cdots\\succ \\V^m$ be an arbitrary sequence of $m$ successive equitable refinements of $\\{\\Vside^0,\\ldots,\\Vside^{5}\\}$ with $|\\V^i_h|=\\t(i)$ for every $1 \\le i \\le m$ and $0 \\le h \\le 5$, which exists as $n$ is large enough.\n\n\tExtending the notation $\\Z_i$ (above Definition~\\ref{def:k-reg}), for every $0 \\le x \\le 5$\n\n\twe henceforth denote the restriction of the vertex partition $\\Z$ to $\\Vside^x$ by $\\Z_x = \\{Z \\in \\Z \\,\\vert\\, Z \\sub \\Vside^x\\}$.\t\n\tFor each edge $e=\\{x,x+1,x+2\\} \\in E(B)$ (here and henceforth when specifying an edge, the integers are implicitly taken modulo $6$)\n\tapply Lemma~\\ref{lemma:ind-k} with\n\t$$s-1,\\,\n\t\\Vside^{x},\\Vside^{x+1},\\Vside^{x+2}\n\t\\text{ and }\n\t(\\V^{1}_x \\cup \\V^1_{x+1} \\cup \\V^1_{x+2})\n\t\\succ\\cdots\\succ (\\V^{m}_{x}\\cup\\V^{m}_{x+1}\\cup\\V^{m}_{x+2}) \\;.$$\n\n\tLet $H_e$ denote the resulting $3$-partite $3$-graph on $(\\Vside^{x},\\Vside^{x+1},\\Vside^{x+2})$.\n\tNote that $d(H_e) = 2^{-(s-1)}$.\n\tMoreover, let\n\t$$c = 2^{-9} \\quad\\text{ and }\\quad K=\\w(s-1)+1 \\;.$$\n\tThen $H_e$ has the property that for every $\\langle 2^{-70} \\rangle$-regular partition $(\\Z',\\E')$ of $H_e$ and every $1 \\le i < K$,\n\n\n\n\n\n\n\n\t\\begin{equation}\\label{eq:paste-property}\n\t\\text{if $\\Z'_{x+2} \\prec_{c} \\V^i_{x+2}$ and $\\Z'_{x+1} \\prec_{c} \\V^i_{x+1}$ then $\\Z'_x \\prec_{c} \\V^{i+1}_x$.}\n\t\\end{equation}\t\n\tWe construct our $3$-graph on the vertex set $\\Vside:=\\Vside^0 \\cup\\cdots\\cup \\Vside^5$ as\n\t$E(H) = \\bigcup_{e} E(H_e)$; that is, $H$ is the edge-disjoint union of all six $3$-partite $3$-graphs $H_e$ constructed above.\n\tNote that $H$ is a $3$-partite $3$-graph (on vertex classes $(\\Vside^0 \\cup \\Vside^3,\\, \\Vside^1 \\cup \\Vside^4,\\, \\Vside^2 \\cup \\Vside^5))$ of density $\\frac68 2^{-(s-1)} \\ge 2^{-s}$, as needed.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tWe will later use the following fact.\n\t\\begin{prop}\\label{prop:restriction}\n\t\tLet $(\\Z,\\E)$ be an $\\langle 2^{-73}\\rangle$-regular partition of $H$ and let\n\t\t$e \\in E(B)$.\n\t\n\t\n\t\n\t\tIf $\\Z \\prec \\{\\Vside^0,\\ldots,\\Vside^{5}\\}$\n\t\n\t\tthen the restriction $(\\Z',\\E')$ of $(\\Z,\\E)$ to $V(H_e)$ is a $\\langle 2^{-70} \\rangle$-regular partition of $H_e$.\n\t\\end{prop}\n\t\\begin{proof}\n\t\tImmediate from Claim~\\ref{claim:restriction} using the fact that $e(H_e) = \\frac16 e(H)$.\n\t\n\t\\end{proof}\t\n\n\t\n\tNow, let $(\\Z,\\E)$ be a $\\langle 2^{-73} \\rangle$-regular partition of $H$\n\twith $\\Z \\prec \\V^1$.\n\tOur goal will be to show that\n\t\\begin{equation}\\label{eq:paste-goal}\n\t\\Z \\prec_{c} \\V^{K} \\;.\n\t\\end{equation}\n\tProving~(\\ref{eq:paste-goal}) would complete the proof, by setting $\\V_0$ in the statement to be $\\V^1$ here (notice $|\\V^1|=3\\t(1) \\le 2^{300}$ by~(\\ref{eq:t}));\n\tindeed, Claim~\\ref{claim:refinement-size} would imply that\n\t$$|\\Z| \\ge \\frac14|\\V^{K}| = \\frac14 \\cdot 6 \\cdot \\t(K)\n\t\\ge \\t(K)\n\t\\ge \\t(\\w(s-1))\n\t\\ge \\w(s)\n\t\\ge \\wow(s) \\;,$$\n\n\n\twhere the last inequality uses~$(\\ref{eq:A_k})$.\n\n\n\n\tAssume towards contradiction that $\\Z \\nprec_{c} \\V^{K}$. By averaging,\n\t\\begin{equation}\\label{eq:assumption}\n\t\\Z_h \\nprec_c \\V^{K}_h \\text{ for some } 0 \\le h \\le 5.\n\t\\end{equation}\n\tFor each $0 \\le h \\le 5$ let $1 \\le \\b(h) \\le K$ be the largest integer satisfying $\\Z_h \\prec_c \\V^{\\b(h)}_h$,\n\twhich is well defined since $\\Z_h \\prec_c \\V^1_h$,\n\tsince in fact $\\Z \\prec \\V^1$.\n\tPut $\\b^* = \\min_{0 \\le h \\le 5} \\b(h)$, and note that by~(\\ref{eq:assumption}),\n\t\\begin{equation}\\label{eq:paste-star}\n\t\\b^* < K \\;.\n\t\\end{equation}\n\tLet $0 \\le x \\le 5$ minimize $\\b$, that is, $\\b(x)=\\b^*$.\n\tTherefore:\n\t\\begin{equation}\\label{eqcontra}\n\t\\Z_{x+2} \\prec_c \\V^{\\b^*}_{x+2} \\mbox{~,~} \\Z_{x+1} \\prec_c \\V^{\\b^*}_{x+1} \\mbox{ and }\n\t\\Z_{x} \\nprec_c \\V^{\\b^*+1}_{x}.\t\t\n\t\\end{equation}\n\tLet $e=\\{x,x+1,x+2\\} \\in E(B)$.\n\tLet $(\\Z',\\E')$ be the restriction of $(\\Z,\\E)$ to $V(H_e)=\\Vside^{x} \\cup \\Vside^{x+1} \\cup \\Vside^{x+2}$, which is a $\\langle 2^{-70} \\rangle$-regular partition of $H_e$ by Proposition~\\ref{prop:restriction}. Since $\\Z'_x=\\Z_x$, $\\Z'_{x+1}=\\Z_{x+1}$, $\\Z'_{x+2}=\\Z_{x+2}$ we get\n\tfrom (\\ref{eqcontra}) a contradiction to~(\\ref{eq:paste-property}) with $i=\\beta^*$.\n\tWe have thus proved~(\\ref{eq:paste-goal}) and so the proof is complete.\n\\end{proof}\n\n\n\n\t\n\t\n\n\n\n\n\\section{Wowzer-type Lower Bounds for $3$-Graph Regularity Lemmas}\\label{sec:FR}\n\n\\renewcommand{\\K}{\\mathcal{K}}\n\nThe purpose of this section is to apply Theorem \\ref{theo:main} in order to prove Corollary~\\ref{coro:FR-LB},\nthus giving wowzer-type (i.e., $\\Ack_3$-type) lower bounds for the $3$-graph regularity lemmas of Frankl and R\\\"odl~\\cite{FrankRo02} and of Gowers~\\cite{Gowers06}.\nWe will start by giving the necessary definitions for Frankl and R\\\"odl's lemma\nand state our corresponding lower bound.\nNext we will state the necessary definitions for Gower's lemma and state our corresponding lower bound.\nThe formal proofs would then follow.\n\n\n\n\n\\subsection{Frankl and R\\\"odl's $3$-graph regularity}\n\n\n\n\n\n\n\\begin{definition}[$(\\ell,t,\\e_2)$-equipartition,~\\cite{FrankRo02}]\\label{def:ve-partition}\n\n\tAn \\emph{$(\\ell,t,\\e_2)$-equipartition} on a set $V$ is a $2$-partition $(\\Z,\\E)$ on $V$ where $\\Z$ is an equipartition of order $|\\Z|=t$ and every graph in $\\E$ is $\\e_2$-regular\\footnote{Here, and in several places in this section, we of course refer to the ``traditional'' notion of Szemer\\'edi's $\\e$-regularity, as defined at the beginning of Section \\ref{sec:define}. } of density $\\ell^{-1} \\pm \\e_2$.\n\\end{definition}\n\n\\begin{remark}\n\tIf $\\e_2 \\le \\frac12\\ell^{-1}$ then $(Z,\\E)$ has at most $2\\ell$ bipartite graphs between every pair of clusters of $\\Z$.\n\\end{remark}\n\n\n\nA \\emph{triad} of a $2$-partition $(\\Z,\\E)$ is any tripartite graph whose three vertex classes are in $\\Z$ and three edge sets are in $\\E$. We often identify a triad with a triple of its edge sets $(E_1,E_2,E_3)$.\nThe \\emph{density} of a triad $P$ in a $3$-graph $H$ is $d_H(P)=|E(H) \\cap T(P)|\/|T(P)|$ (and $0$ if $|T(P)|=0$).\nA \\emph{subtriad} of $P$ is any subgraph of $P$ on the same vertex classes.\n\n\n\n\n\n\n\n\\begin{definition}[$3$-graph $\\e$-regularity~\\cite{FrankRo02}]\\label{def:FR-reg}\n\n\tLet $H$ be a $3$-graph.\n\tA triad $P$ is \\emph{$\\e$-regular} in $H$ if every subtriad $P'$ with $|T(P')| \\ge \\e|T(P)|$ satisfies $|d_H(P')-d_H(P)| \\le \\e$.\\\\\n\tAn $(\\ell,t,\\e_2)$-equipartition $\\P$ on $V(H)$ is an \\emph{$\\e$-regular} partition of $H$ if\n\n\t$\\sum_P |T(P)| \\le \\e|V|^3$ where the sum is over all triads of $\\P$ that are not $\\e$-regular in $H$.\t\n\\end{definition}\n\nThe $3$-graph regularity of Frankl and R\\\"odl~\\cite{FrankRo02} states, very roughly, that for every $\\e>0$ and every function $\\e_2\\colon\\N\\to(0,1]$, every $3$-graph has an $\\e$-regular $(\\ell,t,\\e_2(\\ell))$-equipartition where $t,\\ell$ are bounded by a wowzer-type function.\nIn fact, the statement in~\\cite{FrankRo02} uses a considerably stronger notion of regularity of a partition than in Definition~\\ref{def:FR-reg} that involves an additional function $r(t,\\ell)$ which we shall not discuss here (as discussed in~\\cite{FrankRo02}, this stronger notion was crucial for allowing them to prove the $3$-graph removal lemma).\nOur lower bound below applies even to the weaker notion stated above, which corresponds to taking $r(t,\\ell)\\equiv 1$.\n\n\n\nUsing Theorem~\\ref{theo:main} we can deduce a wowzer-type \\emph{lower} bound for Frankl and R\\\"odl's $3$-graph regularity lemma.\nThe proof of this lower bound appears in Subsection~\\ref{subsec:FR-LB-proof}.\n\n\n\n\\begin{theo}[Lower bound for Frankl and R\\\"odl's regularity lemma]\\label{theo:FR-LB}\n\tPut $c = 2^{-400}$.\n\tFor every $s \\in \\N$ there exists a $3$-partite $3$-graph $H$ of density $p=2^{-s}$, and a partition $\\V_0$ of $V(H)$ with $|\\V_0| \\le 2^{300}$,\n\tsuch that\n\tif $(\\Z,\\E)$ is an $\\e$-regular $(\\ell,t,\\e_2(\\ell))$-equipartition of $H$,\n\twith $\\e \\le c p$, $\\e_2(\\ell) \\le c \\ell^{-3}$ and $\\Z \\prec \\V_0$, then $|\\Z| \\ge \\wow(s)$.\n\\end{theo}\n\n\\begin{remark}\n\tOne can easily remove the assumption $\\Z \\prec \\V_0$ by taking the common refinement of $\\Z$ with $\\V_0$ (and adjusting $\\E$ appropriately).\nSince $|\\V_0|=O(1)$ this has only a minor effect on the parameters $\\e,\\ell,t,\\e_2(\\ell)$ of the partition and thus\none gets essentially the same lower bound. We omit the details of this routine transformation.\n\\end{remark}\n\n\n\\subsection{Gowers' $3$-graph regularity}\n\nHere we consider the $3$-graph regularity Lemma due to Gowers~\\cite{Gowers06}.\n\n\n\\begin{definition}[$\\a$-quasirandomness, see Definition~6.3 in \\cite{Gowers06}]\\label{def:quasirandom}\n\tLet $H$ be a $3$-graph, and let $P=(E_0,E_1,E_2)$ be a triad with $d(E_0)=d(E_1)=d(E_2)=:d$ on vertex classes $(X,Y,Z)$ with $|X|=|Y|=|Z|=:n$. We say that $P$ is \\emph{$\\a$-quasirandom} in $H$ if\n\t$$\\sum_{x_0,x_1 \\in X}\\sum_{y_0,y_1 \\in Y}\\sum_{z_0,z_1 \\in Z} \\prod_{i,j,k\\in\\{0,1\\}} f(x_i,y_j,z_k) \\le \\a d^{12}n^6 \\;,$$\n\n\twhere\n\t$$f(x,y,z) =\n\t\\begin{cases}\n\t1-d_H(P)\t\t\t\t&\\text{if } (x,y,z) \\in T(P), (x,y,z) \\in E(H)\\\\\n\t-d_H(P) \t\t\t\t&\\text{if } (x,y,z) \\in T(P), (x,y,z) \\notin E(H)\\\\\n\t0 \t\t\t\t\t\t&\\text{if } (x,y,z) \\notin T(P) \\;.\n\t\\end{cases}$$\n\tAn $(\\ell,t,\\e_2)$-equipartition $\\P$ on $V(H)$ is an \\emph{$\\a$-quasirandom} partition of $H$ if\n\n\t$\\sum_P |T(P)| \\le \\a|V|^3$ where the sum is over all triads of $\\P$ that are not $\\a$-quasirandom in $H$.\t\n\\end{definition}\n\nThe $3$-graph regularity lemma of Gowers~\\cite{Gowers06} (see also~\\cite{NaglePoRoSc09}) can be equivalently phrased as stating that, very roughly, for every $\\a>0$ and every function $\\e_2\\colon\\N\\to(0,1]$, every $3$-graph has an $\\a$-quasirandom $(\\ell,t,\\e_2(\\ell))$-equipartition where $t,\\ell$ are bounded by a wowzer-type function.\n\nOne way to prove a wowzer-type lower bound for Gowers' $3$-graph regularity lemma is along similar lines to the proof of Theorem~\\ref{theo:FR-LB}.\nHowever, there is shorter proof using the fact that Gowers' notion of quasirandomness implies Frankl and R\\\"odl's notion of regularity.\nIn all that follows we make the rather trivial assumption that, in the notation above, $\\a,1\/\\ell \\le 1\/2$.\n\n\\begin{prop}[\\cite{NagleRoSc17}]\\label{prop:Schacht}\n\tThere is $C \\ge 1$ such that the following holds;\n\tif a triad $P=(E_0,E_1,E_2)$ is $\\e^C$-quasirandom and for every $0 \\le i \\le 2$ the bipartite graph $E_i$ is $d(E_i)^C$-regular then $P$ is $\\e$-regular.\t\n\n\n\n\n\n\\end{prop}\n\n\nOur lower bound for Gowers' $3$-graph regularity lemma is as follows.\n\n\n\\begin{theo}[Lower bound for Gowers' regularity lemma]\\label{theo:Gowers-LB}\n\n\n\tFor every $s \\in \\N$ there exists a $3$-partite $3$-graph $H$ of density $p=2^{-s}$, and a partition $\\V_0$ of $V(H)$ with $|\\V_0| \\le 2^{300}$,\n\tsuch that\n\tif $(\\Z,\\E)$ is an $\\a$-quasirandom $(\\ell,t,\\e_2(\\ell))$-equipartition of $H$,\n\n\twith $\\a \\le \\poly(p)$, $\\e_2(\\ell) \\le \\poly(1\/\\ell)$ and $\\Z \\prec \\V_0$, then $|\\Z| \\ge \\wow(s)$.\n\\end{theo}\n\n\n\n\n\n\\begin{proof\n\n\tGiven $s$, let $H$ and $\\V_0$ be as in Theorem~\\ref{theo:FR-LB}.\n\tLet $\\P=(\\Z,\\E)$ be an $\\a$-quasirandom $(\\ell,t,\\e_2(\\ell))$-equipartition of $H$ with $\\Z \\prec \\V_0$, $\\a \\le (cp)^C$ and $\\e_2(\\ell) \\le \\min\\{c\\ell^{-3},\\,(2\\ell)^{-C}\\}$, where $c$ and $C$ are as in Theorem~\\ref{theo:FR-LB} and Proposition~\\ref{prop:Schacht} respectively.\n\tWe will show that $\\P$ is a $cp$-regular partition of $H$,\n\n\twhich would complete the proof using Theorem~\\ref{theo:FR-LB} and the fact that $\\e_2 \\le c\\ell^{-3}$.\n\tLet $P=(E_0,E_1,E_2)$ be a triad of $\\P$ that is $\\a$-quasirandom in $H$.\n\tNote that, by our choice of $\\e_2(\\ell)$, for every $0 \\le i \\le 2$ we have $d(E_i) \\ge 1\/\\ell - \\e_2(\\ell) \\ge 1\/2\\ell$; thus, since $\\e_2(\\ell) \\le (1\/2\\ell)^{C} \\le d(E_i)^C$, we have that $E_i$ is $d(E_i)^C$-regular.\n\tApplying Proposition~\\ref{prop:Schacht} on $P$ we deduce that $P$ is $\\e$-regular with $\\e=\\a^{1\/C} \\le cp$.\n\tSince $\\P$ is an $\\a$-quasirandom partition of $H$ we have, by Definition~\\ref{def:quasirandom} and since $\\a \\le \\e$, that $\\P$ is an $\\e$-regular partition of $H$, as needed.\n\n\n\\end{proof}\n\n\n\\subsection{Proof of Theorem~\\ref{theo:FR-LB}}\\label{subsec:FR-LB-proof}\n\nThe proof of Theorem~\\ref{theo:FR-LB} will follow quite easily from Theorem~\\ref{theo:main} together with Claim~\\ref{claim:reduction} below.\nClaim~\\ref{claim:reduction} basically shows that a $\\langle \\d \\rangle$-regularity ``analogue'' of Frankl and R\\\"odl's notion of regularity implies graph $\\langle \\d \\rangle$-regularity.\nHere it will be convenient to say that a graph partition is \\emph{perfectly} $\\langle \\d \\rangle$-regular if all pairs of distinct clusters are $\\langle \\d \\rangle$-regular without modifying any of the graph's edges.\nFurthermore, we will henceforth abbreviate $t(P)=|T(P)|$ for a triad $P$.\nWe will only sketch the proof of Claim~\\ref{claim:reduction}, deferring the full details to the Appendix~\\ref{sec:FR-appendix}.\n\n\\begin{claim}\\label{claim:reduction}\n\tLet $H$ be a $3$-partite $3$-graph on vertex classes $(\\Aside,\\Bside,\\Cside)$,\n\tand let $(\\Z,\\E)$ be an $(\\ell,t,\\e_2)$-equipartition of $H$ with $\\Z \\prec \\{\\Aside,\\Bside,\\Cside\\}$ such that for every triad $P$ of $\\P$ and every subtriad $P'$ of $P$ with $t(P') \\ge \\d \\cdot t(P)$ we have $d_H(P') \\ge \\frac23 d_H(P)$.\n\n\tIf $\\e_2(\\ell) \\le (\\d^2\/88)\\ell^{-3}$\n\tthen $\\E_3 \\cup \\Z_3$ is a perfectly $\\langle 2\\sqrt{\\d} \\rangle$-regular partition of $G_H^3$.\n\\end{claim}\n\n\\begin{proof}[Proof (sketch):]\nWe remind the reader that the vertex classes of $G_H^3$ are $(\\Aside \\times \\Bside,\\, \\Cside)$ (recall Definition~\\ref{def:aux}), and that $\\E_3$ and $\\Z_3$ are the partition of $\\Aside\\times\\Bside$ induced by $\\E$ and the partition of $\\Cside$ induced by $\\Z$, respectively. Suppose $(\\Z,\\E)$ is as in the statement of the claim, and define $\\E'$ as follows:\nfor every $A \\in \\Z_1$ and $C \\in \\Z_3$, replace all the bipartite graphs between $A$ and $C$ with the complete bipartite graph $A \\times C$.\nDo the same for every $B \\in \\Z_2$ and $C \\in \\Z_3$ (we do {\\em not} change the partitions between $\\Aside$ and $\\Bside$). The simple (yet somewhat tedious to prove)\nobservation is that if all triads of $(\\Z,\\E)$ are regular then all triads of $(\\Z,\\E')$ are essentially as regular.\nOnce this observation is proved, the proof of the claim reduces to checking definitions. We thus defer the proof to Appendix~\\ref{sec:FR-appendix}.\n\\end{proof}\n\n\nUsing Claim~\\ref{claim:reduction}, we now prove our wowzer lower bound.\n\n\n\n\n\\begin{proof}[Proof of Theorem~\\ref{theo:FR-LB}]\n\tPut $\\a = 2^{-73}$.\n\n\tWe have\n\t\\begin{equation}\\label{eq:FR-LB-ineq}\n\tc = 2^{-400} \\le \\a^4\/1500 \\;.\n\t\\end{equation}\n\n\n\tGiven $s$, let $H$ and $\\V_0$ be as in Theorem~\\ref{theo:main}.\n\tLet $\\P=(\\Z,\\E)$ be an $\\e$-regular $(\\ell,t,\\e_2(\\ell))$-equipartition of $H$\n\twith $\\e \\le c p$, $\\e_2(\\ell) \\le c \\ell^{-3}$ and $\\Z \\prec \\V_0$.\n\tThus, the bound $|Z| \\ge \\wow(s)$ would follow from Theorem~\\ref{theo:main}\n\n\tif we show that $\\P$ is an  $\\langle \\a \\rangle$-regular partition of $H$.\n\n\n\tFirst we need to show that $\\P$ is $\\langle \\a \\rangle$-good (recall Definition~\\ref{def:k-good}). Let $E$ be a graph with $E \\in \\E$ on vertex classes $(Z,Z')$ (so $Z \\neq Z' \\in \\Z$).\n\tWe need to show that $E$ is $\\langle \\a \\rangle$-regular.\n\n\tSince $\\P$ is an $(\\ell,t,\\e_2(\\ell))$-equipartition we have\n\t(recall Definition~\\ref{def:ve-partition}) that $E$ is $\\e_2(\\ell)$-regular and $d(E) \\ge \\ell^{-1} - \\e_2(\\ell)$.\n\tThe statement's assumption on $\\e_2(\\ell)$ thus implies\n\t$d(E) \\ge 2\\e_2(\\ell)$.\n\tIt follows that for every $S \\sub Z$, $S' \\sub Z'$ with $|S| \\ge \\e_2(\\ell)|Z|$, $|S'| \\ge \\e_2(\\ell)|Z'|$ we have $d_E(S,S') \\ge d(E)-\\e_2(\\ell) \\ge \\frac12 d(E)$.\n\tThis proves that $E$ is $\\langle \\e_2(\\ell) \\rangle$-regular, and since $\\e_2(\\ell) \\le c \\le \\a$, that $E$ is $\\langle \\a \\rangle$-regular, as needed.\n\n\t\n\t\n\n\n\n\n\n\n\tIt remains to show that the $\\langle \\a \\rangle$-good $\\P$ is an $\\langle \\a \\rangle$-regular partition of $H$ (recall Definition~\\ref{def:k-reg}).\n\tBy symmetry, it suffices to show that $\\E_3 \\cup \\Z_3$ is an $\\langle \\a \\rangle$-regular partition of $G_{H}^3$.\n\tLet $H'$ be obtained from $H$ by removing all ($3$-)edges in triads of $\\P$ that are either not $\\e$-regular in $H$ or have density at most $3\\e$ in $H$.\n\tBy Definition~\\ref{def:FR-reg}, the number of edges removed from $H$ to obtain $H'$ is at most\n\t\\begin{equation}\\label{eq:FR-LB-modify}\n\t\\e|V(H)|^3 + 3\\e|V(H)|^3 \\le 4\\cdot c p |V(H)|^3\n\t\\le (\\a p\/27)|V(H)|^3 = \\a\\cdot e(H) \\;,\n\t\\end{equation}\n\n\twhere the second inequality uses ~(\\ref{eq:FR-LB-ineq}),\n\tand the equality uses the fact that all three vertex classes of $H$ are of the same size.\n\n\n\tThus, in $H'$, every non-empty triad of $\\P$ is $\\e$-regular and of density at least $3\\e$.\n\tPut $\\d = (\\a\/2)^2$.\n\tAgain by Definition~\\ref{def:FR-reg}, for every triad $P$ of $\\P$ and every subtraid $P'$ of $P$ with $t(P') \\ge \\d \\cdot t(P)$ ($\\ge \\e \\cdot t(P)$ by~(\\ref{eq:FR-LB-ineq})) we have $d_{H'}(P') \\ge d_{H'}(P)-\\e \\ge \\frac23 d_{H'}(P)$.\n\n\tIt follows from applying Claim~\\ref{claim:reduction} with $H'$ and $\\d$,\n\n\tusing~(\\ref{eq:FR-LB-ineq}),\n\n\n\tthat $\\E_3 \\cup \\Z_3$ is a perfectly $\\langle \\a \\rangle$-regular partition of $G_{H'}^3$.\n\tNote that~(\\ref{eq:FR-LB-modify}) implies that\n\tone can add\/remove at most $\\a \\cdot e(G_{H}^3)$ edges of $G_{H}^3$ to obtain $G_{H'}^3$.\n\tThus, $\\E_3 \\cup \\Z_3$ is an $\\langle \\a \\rangle$-regular partition of $G_{H}^3$, and as explained above, this completes the proof.\n\n\n\\end{proof}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDengue Fever is a mosquito-borne disease with a significant global burden. Half of the world's population and 129 countries are at risk of infection \\cite{brady2012refining}. Between 100 and 400 million infections are registered each year globally \\cite{bhatt2013global}.  Dengue presents as a flu-like illness with symptoms ranging from mild to severe. There are four serotypes, meaning up to four infections are possible for each person over their lifetime, with potentially life-threatening complications arising from severe Dengue. There is no specific treatment or universal vaccine, and the primary preventative measure is vector control. This underscores the importance of disease surveillance. Predictive models can help efficiently allocate public health resources to combat Dengue, with the goal of reducing the overall disease burden \\cite{who2020dengue}. Dengue incidence is associated with many different risk factors. These include climate conditions such as rainfall, extreme weather events, and temperature \\cite{chien2014impact, thu1998effect, fan2013identifying}, land use \\cite{kilpatrick2012drivers}, and poverty \\cite{mulligan2014dengue}. These factors may have nonlinear, context-specific, and time-variant effects on disease incidence, which poses a challenge to disease modeling. \n\nThe literature on Dengue forecasting is multi-disciplinary, uniting expertise from areas such as epidemiology, environmental science, computer science, and mathematics. Modeling frameworks include both theoretical and data-driven approaches. Compartmental models, for example, estimate the dynamics of Dengue infections in a population over time, and are based on extensive knowledge of the host, vector, and transmission process. Another simulation technique, agent based modeling is useful for estimating impacts of interventions, such as the release of sterile males to control the mosquito population \\cite{isidoro2009agent}. Statistical time series forecasting takes a more data-centric approach and is effective at modeling the highly auto-correlated nature of Dengue Fever \\cite{johansson2016evaluating, riley2020sarima_preprint}. Machine learning models leverage the increasing data availability on risk factors of disease spreading and offer non-parametric approaches that require less detailed knowledge of the disease and context \\cite{guo2017developing}. \n\nNeural networks are a subset of machine learning algorithms, which have made significant contributions to medicine and public health, including applications such as medical image analysis for disease diagnosis \\cite{qayyum2018medical}, identifying abnormalities in signals such as electrocardiographs (ECG) \\cite{papik1998application}, and optimizing decisions of health care providers, hospitals, and policy-makers \\cite{shahid2019applications}. \nNeural networks are also used to forecast diseases, including Malaria \\cite{zinszer2012scoping}, Influenza. \\cite{alessa2017review}, and Covid-19 \\cite{bullock2020mapping}. \n\nThis review examines the use of neural networks for Dengue Fever prediction. The objective is to summarize the existing literature and also provide an introduction to this still somewhat novel modeling technique. Our contributions are as follows:\n\\begin{itemize}\n\n    \\item We summarize the technical decisions made in the literature, including architecture selection and hyper-parameter tuning. \n    \\item We examine the data inputs (such as climate or population demographics) and best predictors of Dengue fever identified in specific contexts.\n    \\item We review the relative performance of different neural network architectures and comparator models, such as other machine learning techniques. \n\\end{itemize}\n\n\n\nTo the knowledge of the authors, no systematic review of the literature on neural networks applied to Dengue Fever prediction has yet been conducted. \nSiriyasatien et al (2018) \\cite{siriyasatien2018dengue} provide a broader review of data science models applied for Dengue Fever prediction. Racloz et al (2012) \\cite{racloz2012surveillance} review the literature on surveillance systems for Dengue Fever. The study finds that most papers use logistic or multiple regression to analyze Dengue risk factors. (Seasonal) Auto-Regressive Integrated Moving Average (S\/ARIMA) models have become a popular choice to incorporate auto-regressions but are not suitable for all data types. Though also apt at analyzing auto-regressive behavior, the study does not review any literature on neural networks or machine learning for Dengue surveillance. We aim to fill this knowledge gap in the present review.\n\n\n\n\n\n\\section{A Primer on Neural Networks}\n\nThis section gives an overview of some of the neural network models that are relevant to Dengue prediction. We explain the model first intuitively, then mathematically, and focus on feed-forward neural networks, which are used most often for Dengue prediction. Many high-quality textbooks and open source tutorials offer a deeper introduction to neural networks (for example \\cite{deng2014deep, haykin1999neural}). Several Python libraries include easy-to-use implementations of neural network models (for example TensorFlow \\cite{tensorflow}, scikit-learn \\cite{scikit-learn}, or Keras \\cite{chollet2015keras}). We also provide sample code for Dengue prediction in a Github repository\\footnote{https:\/\/github.com\/KRoster\/NN4Dengue}.\n\n\nLike other machine learning models, neural networks learn to execute specific supervised tasks, such as predicting the number of Dengue infections, based on a large set of labeled examples. Given a set of inputs, such as climate conditions, the model learns to estimate the output, such as next month's Dengue incidence. After making a first guess, the model looks at the correct answer and updates its parameters to reduce its error on the next iteration until its predictions are optimized.\n\nComputationally, neural networks are represented as a network of processing units (\"neurons\" or \"hidden units\") that transform and pass information from one layer to the next. During training, we distinguish between forward- and backward-propagation, depending on the direction of the information flow. During the forward-propagation step, input information passes between units, each time being transformed according to a set of weights and a non-linear activation function. The prediction error of the output relative to the true label is computed. Back-propagation is then used to reduce this error on the next iteration: The weights of each unit, the parameters that define how information is combined, are updated according to their influence on the prediction error. This combination of forward- and backward-propagation is repeated several times until the predictions are increasingly accurate.\n\nAfter this training phase, model performance is tested on a hold-out test set. Since the test set is not used to train the model, it can give a good indication of how well the model will generalize to new data, once applied in the real world.\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{\"nn_fully_annotated\".png}\n    \\caption{Sample feed-forward neural network architecture}\n    \\medskip\n    \\small\n    The figure shows a sample neural network architecture with 4 input features, 2 hidden layers with 5 and 2 nodes respectively and a single node in the output layer.\n    \\label{fig:nn annotated}\n\\end{figure}\n\nThere are many different neural network architectures that are designed for different kinds of inputs and tasks. The architecture determines how the units are connected, how information flows through the network, and how many weights need to be optimized. An example of a feed-forward network is illustrated in figure \\ref{fig:nn annotated}. It shows the forward propagation of information through the individual neurons of the network, which was described above. The information that is passed through a given neuron $j$ in layer $l$, called its activation $a_{j}^{[l]}$, is computed as the linear combination of weights $w_{ij}^{[l]}$, biases $b_{j}^{[l]}$, and inputs from the previous layer $a_{i}^{[l-1]}$, which is passed through a nonlinear activation function $g^{[l]}$: \n\n\\begin{flalign} \\label{eq forward prop}\n    a_{j}^{[l]} &= g^{[l]}(\\sum\\limits_{i=1}^{n_l} w_{ij}^{[l]} a_{i}^{[l-1]} + b_{j}^{[l]}) \\\\\n    \\text{with: } a_{j}^{[0]} &= X_{j} \\\\\n    \\hat{y} &= a^{[k]}\n\\end{flalign}\n\n\\noindent where ${n_l}$ is the number of neurons in layer $l$, $k$ is the number of layers (the input layer is counted as layer $0$), and $\\hat{y}$ is the prediction that is generated in the final layer.\n\n\nThe gradient of the loss with respect to a given weight $w_{ij}^{[l]}$ or bias  $b_{j}^{[l]}$  tells the model how the given parameter needs to be adjusted, with the learning rate $\\alpha$ determining the size of the adjustment:\n\n\\begin{flalign} \\label{eq gradient descent}\n    \\theta &= \\theta - \\alpha \\frac{d\\mathcal{L}}{d\\theta}\n\\end{flalign}\n\n\\noindent where $\\theta$ is a given set of weights or biases to be updated and $\\mathcal{L}$ is the loss.\n\n\n\nA benefit of neural networks is that parameters are updated by the model itself, not predefined by the researcher. They do not rely on strong assumptions and knowledge of the disease. Yet some hyper-parameters of neural network models must still be set by the researcher and are generally determined through a combination of domain knowledge and iterative experimentation. These hyper-parameters are tuned using a validation set or through cross-validation. Hyper-parameters may include the learning rate (by how much the weights are updated at each backward propagation step), the number of epochs (how often the forward and backward propagation steps are repeated), the mini-batch size (how many training examples are processed at each step), how weights are initialized (for example all zeros or random values), the number of hidden units in each layer, and the number of layers. Other important decisions include the choice of activation function in each layer, the choice of loss function, and the relative size of training, validation, and test sets. The optimal choices and ranges tested for these hyper-parameters depend, among other aspects, on the size and type of data available and the nature of the predictive task. One aim of this review is to summarize the choice of parameters deemed optimal by researchers in the existing literature to assist future researchers in the hyper-parameter tuning process. \n\nBesides feed-forward neural networks, this review includes two other categories of models used for Dengue forecasting. Recurrent neural networks (RNN) were developed for the analysis of sequence data, such as time series (sequences of observations in time) or text (sequences of words). They are cyclical (they contain loops) and process sequences of inputs together, giving the network a \"memory\" to remember historical feature states. RNNs are used in contexts where the evolution of input features matters for the target feature, where the prediction relies on information at multiple time steps. An example is machine translation, where the meaning of a word is influenced by its context, by the other words in the sentence. The number of Dengue cases may also depend on the sequence of risk factors or the sequence of previous Dengue incidence. Using time series data, RNNs may help capture the auto-correlation of the disease and lagged influence of risk factors, such as rainfall or travel patterns. Within the family of RNNs we distinguish different architectures, such as Long Short-Term Memory (LSTM) and Gated Recurrent Units (GRU) \\cite{hochreiter1997lstm, schmidhuber2015deep, lipton2015critical}. \n\nConvolutional neural networks (CNN) are another subclass of deep neural network models with applications in computer vision \\cite{lecun1990handwritten}, such as image segmentation and classification. Inputs into CNNs are three-dimensional - such as a two dimensional image with three color channels, which form a 3-D matrix of pixel values. The architectures of CNNs include different kinds of operations, such as convolution and pooling, which help identify key regions in the image that help make the final prediction. For example, the model may learn that the presence of a body of water (shiny pixels in a satellite image) influence the number of Dengue cases we should expect. Crucial in this example is that the type of relevant landscape feature is not defined by the researcher but learned by the model. However, CNNs may also be used for data processing prior to prediction, for example to classify land use (such as vegetation or urban areas) from satellite images, which are known to influence Dengue spread and which can then be fed into Dengue a separate predictive model.\n\n\nThe tasks executed by machine learning models in general, and neural networks in particular, fall into two different categories: classification and prediction. Classification is the process of assigning a categorical label to an example based on its inputs, such as determining whether a patient has Dengue fever or a different disease. Prediction is the process of forecasting a future state based on historical or current inputs, for example predicting next month's Dengue incidence from this month's rainfall. \n\nThis review is focused on predictive models of Dengue Fever at the population level. In addition to classic regression models, we include papers using classification for prediction approaches, which produce categorical instead of continuous outputs. For example, a model may predict whether a city will experience an outbreak (binary classification) or which risk category will prevail (multi-label classification). \n\n\n\n\n\n\n\n\n\\section{Methodology}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{\"PRISMA_2009Flow_filled_new\".pdf}\n    \\caption{PRISMA Flow Chart}\n    \\label{fig:prisma flow}\n\\end{figure}\n\nThis review follows the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) statement guidelines \\cite{prisma2009}. The PRISMA flow chart is represented in figure \\ref{fig:prisma flow}. A systematic search was conducted using Web of Science\/ Knowledge, Scopus (abstract, title, keywords), PubMed, and Science Direct (abstract, title) databases. References of papers appearing in the search results were also examined for relevant works. The searches were conducted between 21 July and 17 August 2020 and used the following search string:\n\\begin{center}\n    \"( deep learning OR neural network ) AND dengue\"\n\\end{center}\n\n\\begin{comment}\nThe databases yielded the following numbers of results:\n\n\\begin{itemize}\n    \\item Web of Science \/ Knowledge: 92 (abstract screened: 46) (full text review: 26)\n    \\item PubMed: 21 (abstract screened: 4) (full text review: 4)\n    \\item Scopus: 23 (abstract screened: 18) (full text review: 8)\n    \\item Science Direct: 13 (abstract screened: 3) (full text review: 3)\n    \\item references of other papers: 31 (abstract screened: 27) (full text review: 21)\n\\end{itemize}\n\n\\end{comment}\n\n\n\n\n\n\nThe papers were examined for inclusion in the review in three stages: by title, by abstract, and finally by full text. In each phase, the following inclusion and exclusion criteria were applied liberally to determine which papers would be considered in the next phase. If the information in the title and\/or abstract was inconclusive, the paper was included in the full-text review. Inclusion and exclusion criteria are as follows:\n\n\\begin{itemize}\n    \\item Studies must implement a neural network or deep learning technique, either as the main method or as a comparator model. Reviews of the literature on neural networks for Dengue forecasting would also be considered. \n    \\item Studies must predict Dengue fever incidence or risk. There are no restrictions on the type of target variable used. For example, the number of Dengue cases, Dengue incidence rate, or a binary Dengue risk variable are all accepted as target features.\n    \\item Studies must examine Dengue in a human population. Models for disease diagnosis of individuals are excluded. Studies modeling the location of vectors without relation to Dengue incidence are excluded. Studies examining animal hosts are excluded.\n    \\item As the search query is in English, only English language articles were identified and included in the review.\n\\end{itemize}\n\n\n\n\n\n\\section{Results}\nThis section summarizes the findings of the review. The full list of papers and their properties are provided in table 1 at the end of this section.\n\n\n\\subsection{Prediction Target}\nWithin the criteria of study selection, there was significant variation in the study specifications, including the formulation of the target feature. Most studies predict the future number of Dengue cases based on historical time series. However, there are some notable exceptions. Andersson et al (2019) \\cite{andersson2019combining} predict a static incidence rate at the neighborhood level in Rio de Janeiro, Brazil. Two studies predict the risk of a Dengue outbreak: Abeyrathna et al (2019) \\cite{abeyrathna2019scheme} formulate their task as a binary classification of 'outbreak' or 'no outbreak', while Anno et al (2019) \\cite{anno2019spatiotemporal} classify cities according to five risk categories. \n\nEarly forecasts provide more time for public health response but may have lower confidence. In most of the selected studies, researchers predict one period ahead corresponding to the measurement frequency of the data used. A single study predicted the present: Livelo and Cheng (2018) \\cite{livelo2018intelligent} use social media activity in the Philippines to predict the present situation of Dengue infection. They use a neural network model to classify tweets according to a set of different categories of Dengue-related topics. Their weekly Dengue index based on this Twitter data is correlated with actual Dengue case counts. Koh et al. (2018) \\cite{koh2018model} compare one and two-week forecast horizons. Soemsap et al. (2014) \\cite{soemsap2014forecasting} use a two-week horizon with weekly data inputs. Dharmawardhana et al (2017) \\cite{dharmawardana2017predictive} predict Dengue cases in Sri Lankan districts four weeks ahead using weekly data on cases, climate, human mobility, and vegetation indices. \nChakraborty et al. (2019) \\cite{Chakraborty2019forecasting} use the longest forecasting horizon. They use three different datasets of Dengue cases, two of which are weekly (Peru and Puerto Rico), and one which is measured monthly (Philippines). For the weekly data, they compare three and six month horizons. For the monthly data, they compare a horizon of six months with one year forecasts. For Peru and the Philippines, the best model changes with the prediction horizon.  \n\n\n\n\n\\subsection{Data Sources}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{\"articles_for_review_variables_used2\".pdf}\n    \\caption{Variable types used in the selected studies}\n    \\medskip\n    \\small\n    The figure illustrates how frequently studies included individual data types as well as combinations thereof. Each row represents a data type and the horizontal bars measure their frequency of inclusion in the selected papers. The columns represent the combinations of variables, and the vertical bars measure how often the given combinations occurred. \n    \\label{fig:variables}\n\\end{figure}\n\nBesides epidemiological data on the history of Dengue Fever, which is used in all studies, the most common predictors are meteorological variables (used in 14 studies) (see figure \\ref{fig:variables}). Additional data sources include: vegetation indices and other landscape features extracted from satellite images (in two studies), human mobility, specifically mobile phone data (used in one study), aerial and street view images (in one study), social media data, specifically from Twitter (one study), vector information (one study), and demography (one study). Three papers use only epidemiological data.\n\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{\"articles_for_review_distribution_of_geographic_scale\".png}\n    \\caption{Geographic granularity used in the selected studies}\n    \\label{fig:geographic granularity}\n\\end{figure}\n\nMost studies examine granular geographic regions, such as cities (nine studies) or districts (six studies).  Two studies even modeled Dengue at the sub-city (neighborhood) level (see figure \\ref{fig:geographic granularity}). Granular scales can be beneficial in two ways: they allow for local predictions and accordingly targeted public health response. They also generally result in larger training datasets with more observations than national or state-level aggregations, which plays to the strength of neural networks. The studies cover countries in both Asia and Latin America, which are considered the primary Dengue endemic regions \\cite{who2020dengue}. A total of 13 countries are included (see figure \\ref{fig:countries map}), with the Philippines and Brazil appearing most often. \n\n\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{\"articles_for_review_geographic_distribution\".png}\n    \\caption{Countries covered by the selected studies}\n    \\label{fig:countries map}\n\\end{figure}\n\n\n\n\n\\subsection{Model Selection}\n\n\\begin{figure}[!h]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{\"articles_for_review_models_used2\".pdf}\n    \\caption{Models used in the selected studies}\n    \\label{fig:models}\n    \\medskip\n    \\small\n    The figure illustrates how frequently studies used different forecasting models as well as combinations thereof. Each row represents an algorithm and the horizontal bars measure their frequency of inclusion in the selected papers. Green rows refer to neural network models, while blue bars show comparator models, such as machine learning algorithms. The columns illustrate which combinations of models were used, the frequency of which is measured by the vertical bars. \n\\end{figure}\n\n\nMost studies implement a shallow neural network with just one hidden layer. Four studies use deeper feed-forward architectures. Three studies employ recurrent neural networks, two of which used LSTM units and one used a GRU. Three studies implemented CNNs. \n\n\nThe neural network models were compared to a range of other models. Support Vector Machine (SVM) models were most common (six studies). Generalized Additive Models (GAMs), (S)ARIMA and linear autoregressive (AR) models occurred twice each. The remaining comparators were used in just one paper each: Gradient Boosting Machine (GBM), XGBoost, Linear regression, Poisson regression, Tsetlin Machine (TM), Cellular automata, a compartmental model, and a naive baseline. Figure \\ref{fig:models} shows how frequently different model types and combinations thereof were used in the selected literature. \n\nThe most common comparator model used is SVM. Evidence suggests that both SVM and neural networks perform well, and which is better depends on the context. In Xu et al (2020) \\cite{xu2020forecast}, an LSTM neural network outperforms an SVM model in predicting the number of Dengue fever cases in 20 Chinese cities. Transfer learning further increased the performance gains in cities with few overall Dengue cases. Abeyrathna et al (2019) \\cite{abeyrathna2019scheme} use a classification for prediction approach to identify whether districts in the Philippines are likely to experience a Dengue Fever outbreak. Whether SVM or the neural network model is better in the context of this study depends on the evaluation metric used: SVM has a higher precision, while ANN has a higher F1-score. Yusof and Mustaffa (2011) \\cite{yusof2011dengue} and Kesorn et al (2015) \\cite{kesorn2015morbidity} both find SVM to have a higher performance than the neural network. \n\nFour studies use a feed-forward network with at least two hidden layers. They provide varying evidence to the performance relative to shallow neural networks or other machine learning models. In \\cite{dharmawardana2017predictive}, a model with two hidden layers performs better than an XGBoost model in forecasting Dengue in Sri Lankan districts. In two studies (\\cite{abeyrathna2019scheme} and \\cite{baquero2018dengue}), the deep neural network provides better predictions than alternative neural network models, but is not the best model overall: In \\cite{baquero2018dengue}, the MLP with two hidden layers performs better than the LSTM-RNN to forecast Dengue in a Brazilian city, but a GAM is best overall. In \\cite{abeyrathna2019scheme}, a model with three hidden layers has lower errors than models with a single or five hidden layers. However, the best F1 score in classifying binary outbreak risk is achieved by a Tsetlin Machine. The final study \\cite{rahayu2019prediction} uses a model with three hidden layers but does not employ any comparator models.\n\n\nPerformance of the RNN models is also mixed. The two studies using LSTM-RNNs produce contradicting results despite having similar study designs. Baquero et al. (2018) \\cite{baquero2018dengue} and Xu et al. (2020) \\cite{xu2020forecast} both use LSTM-RNNs to forecast Dengue at the city level, compare their model to a feed-forward neural network and a GAM, use meteorological and epidemiological input features, and evaluate performance using RMSE. Xu et al. (2020) \\cite{xu2020forecast} find the LSTM to have the best performance relative to all comparator models, whereas in the study by Baquero et al. (2018) \\cite{baquero2018dengue}, the LSTM-ANN performs worse than the simple neural network and the best model overall is the GAM. \n\nThe three studies using CNNs do not provide comparisons to other types of models, though one study compares different CNN architectures. However, the key advantage of the examined CNN studies is the granular geographic resolution in predictions. The models take satellite and other images as inputs and can therefore generate forecasts at the sub-city level. Andersson et al. (2019) \\cite{andersson2019combining} compare different kinds of CNN architectures (based on DenseNet-161) for each of three input types, specifically satellite imagery, street view images, and the combination of the two. Their model predicts Dengue incidence rates at the neighborhood level in Rio de Janeiro, Brazil. Best performance is achieved when combining street and aerial images. Rehman et al. (2019) \\cite{rehman2019deep} implement a pre-trained CNN (based on U-Net architecture) to extract landscape features from satellite imagery, which they use in a compartmental SIR model to forecast Dengue in neighborhoods of two Pakistani cities. Anno et al. (2019) \\cite{anno2019spatiotemporal} use a CNN (based on AlexNet architecture) to classify five levels of outbreak risk in cities in Taiwan, based on images capturing sea surface temperatures. \n\n\n\n\n\n\n\n\\subsection{Architecture}\nArchitectures vary across the different models. The studies implementing shallow models with one hidden layer cumulatively tested 1-30 hidden units. Three studies (\\cite{wijekoon2014prediction, aburas2010dengue, soemsap2014forecasting}) tested a broad number of values across this range, and their optimal number of hidden units were related to the number of input features. Two studies \\cite{wijekoon2014prediction, aburas2010dengue} identified four hidden units as the optimum. Both used meteorological and epidemiological variables and both measured four different pieces of information. Wijekoon et al (2014) \\cite{wijekoon2014prediction} used temperature, rainfall, humidity and Dengue cases. Aburas et al (2010) \\cite{aburas2010dengue} included a total of seven features that measured the same information but with additional lags. The third study \\cite{soemsap2014forecasting} identified 25 nodes as the optimum, which matches the larger input size of 32 features. Instead of trying different values, Datoc et al (2016) \\cite{datoc2016forecasting} fixed the number of hidden units to the number of input features, and selected the best-performing model among eight models with three to five nodes. The best model used three nodes (corresponding to three input features). \n\nAbeyrathna et al (2019) \\cite{abeyrathna2019scheme} compared shallow and deep architectures. Their best neural network model had three layers with (20, 150, 100) units respectively. This model performed better than the two shallow models with five or 20 neurons in a single hidden layer. It also outperformed a five-hidden layer model of (20, 200, 150, 100, 50) topology. \n\nBaquero et al (2018) \\cite{baquero2018dengue} also used deep architectures with 2 layers with combinations of (10 or 20) neurons in the first layer, and (5 or 10) neurons in the second. They implemented both MLP and LSTM models. The best neural network model was an MLP with (20,10) neurons, though a Generalized Additive Model (GAM) and an ensemble both performed better than the neural networks. \n\nThe other LSTM model was implemented by Xu et al (2020) \\cite{xu2020forecast} with a different architecture. They used a single hidden layer with 64 neurons. Unlike Baquero et al (2018) \\cite{baquero2018dengue}, their model outperformed GAM as well as SVR and GBM. \n\n\n\n\\subsection{Model Evaluation}\nThe studies utilize a range of metrics to assess model performance, in part due to different prediction targets. The most common measures are Root Mean Squared Error (RMSE) and Pearson's correlation, which are each used in seven studies each. Mean Average Error (MAE), Mean Squared Error (MSE), and accuracy are also commonly used.\n\n\n\n\n\n\\section{Discussion}\n\nThe literature on neural networks applied to Dengue forecasting is still somewhat scarce, but the reviewed studies include some promising examples, where neural networks outperform other machine learning or statistical approaches. The literature therefore suggests that neural networks may be appropriate for inclusion in a set of candidate models when forecasting Dengue incidence. However there is variation as to whether neural network models or other approaches perform best. Future research may compare different model types and architectures on multiple datasets, to better understand how the ideal model varies by context, such as geography or data availability.\n\nThough many risk factors of Dengue have been identified, most neural network models limit inputs to meteorological and epidemiological data. Future studies may evaluate the value of alternative predictors, especially as NNs tend to deal well with high-dimensional problems. Along the same lines, there is space for more incorporation of non-causal predictors of Dengue fever from sources such as social media. Some papers in this review have leveraged these new predictors, specifically from Twitter \\cite{livelo2018intelligent}, Street view and satellite images \\cite{andersson2019combining}, mobile phone records \\cite{dharmawardana2017predictive}, and features derived from satellite imagery \\cite{dharmawardana2017predictive, rehman2019deep}. These data sources have also been effective with other machine learning models, for example Baidu search data to predict Dengue in a Chinese province \\cite{guo2017developing}. \n\nTransfer learning was implemented in \\cite{xu2020forecast}, where it provided promising results. The authors trained a model on a city with high Dengue incidence and used it to predict disease in lower-incidence geographies. This was the only study in this review that used transfer learning outside the context of pre-trained CNNs. This may present another avenue for further research, especially in locations where data is scarce. \n\n\\subsubsection*{Acknowledgments}\n\nKirstin Roster gratefully acknowledges support from the S\u00e3o Paulo Research Foundation (FAPESP) (grant number 2019\/26595-7). Francisco Rodrigues acknowledges partial support from CNPq (grant number 309266\/2019-0) and FAPESP (grant number 2019\/23293-0).\n\n\\begin{comment}\n\\begin{table}[]\n    \\centering\n   \n   \n    {\\small %\n    \\begin{tabular}{L{3cm} | L{1.5cm} | L{1.5cm} | L{2cm} | L{1.5cm} | L{3cm} }\n    \\toprule\n    Reference & Country & Geospatial scale &            Target  &   Prediction horizon & Input data \\\\\n    \\midrule\n    \n    \\bottomrule\n    \\end{tabular}\n    }%\n   \n\\end{table}\n\\end{comment}\n\n\n\\includepdf[pages={1-}, fitpaper]{\"Tableofincludedpapers_2021-05-31\".pdf}\n\n\n\\newpage\n\\nocite{*}\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSubmodular set functions are defined by the following condition for all pairs of sets $S,T$:\n$$ f(S \\cup T) + f(S \\cap T) \\leq f(S) + f(T),$$\nor equivalently by the property that the {\\em marginal value} of any element,\n$f_S(j) = f(S+j)-f(S)$, satisfies $f_T(j) \\leq f_S(j)$, whenever\n$j \\notin T \\supset S$. In addition, a set function is called monotone\nif $f(S) \\leq f(T)$ whenever $S \\subseteq T$.\nThroughout this paper, we assume that $f(S)$ is nonnegative.\n\nSubmodular functions have been studied in the context of combinatorial\noptimization since the 1970's, especially in connection with matroids\n\\cite{E70,E71,NWF78,NWF78II,NW78,W82a,W82b,L83,F97}.\nSubmodular functions appear mostly for the following two reasons:\n(i) submodularity arises naturally in various combinatorial settings,\nand many algorithmic applications use it either explicitly or implicitly;\n(ii) submodularity has a natural interpretation as the property\nof {\\em diminishing returns}, which defines an important class of\nutility\/valuation functions.\nSubmodularity as an abstract concept is both general enough to be useful for applications\nand it carries enough structure to allow strong positive results.\nA fundamental algorithmic result is that any submodular function\ncan be {\\em minimized} in strongly polynomial time \\cite{FFI01,Lex00}.\n\nIn contrast to submodular minimization, submodular maximization problems are\ntypically hard to solve exactly. Consider the classical problem\nof maximizing a monotone submodular function subject to a cardinality constraint,\n$\\max \\{f(S): |S|\\leq k\\}$. It is known that this problem admits a $(1-1\/e)$-approximation\n\\cite{NWF78} and this is optimal for two different reasons:\n(i) Given only black-box access to $f(S)$, we cannot achieve a better approximation,\nunless we ask exponentially many value queries \\cite{NW78}.\nThis holds even if we allow unlimited computational power.\n(ii) In certain special cases where $f(S)$ has a compact representation on the input,\nit is NP-hard to achieve an approximation better than $1-1\/e$ \\cite{Feige98}.\nThe reason why the hardness threshold is the same in both cases\nis apparently not well understood.\n\nAn optimal $(1-1\/e)$-approximation for the problem $\\max \\{f(S): |S| \\leq k\\}$ where $f$ is\nmonotone sumodular is achieved\nby a simple greedy algorithm \\cite{NWF78}. This seems to be rather coincidental; for\nother variants of submodular maximization, such as unconstrained (nonmonotone)\nsubmodular maximization \\cite{FMV07}, monotone submodular maximization subject\nto a matroid constraint \\cite{NWF78II,CCPV07,Vondrak08}, or submodular\nmaximization subject to linear constraints \\cite{KTS09,LMNS09},\ngreedy algorithms achieve suboptimal results. A tool which has proven useful\nin approaching these problems is the {\\em multilinear relaxation}.\n\n\\\n\n\\noindent{\\bf Multilinear relaxation.}\nLet us consider a discrete optimization problem $\\max \\{f(S): S \\in {\\cal F}\\}$, where\n$f:2^X \\rightarrow {\\boldmath R}$ is the objective function and ${\\cal F} \\subset 2^X$\nis the collection of feasible solutions. In case $f$ is a linear function,\n$f(S) = \\sum_{j \\in S} w_j$, it is natural to replace this problem by a linear\nprogramming problem. For a general set function $f(S)$, however, a linear\nrelaxation is not readily available. Instead, the following relaxation\nhas been proposed \\cite{CCPV07,Vondrak08,CCPV09}:\nFor ${\\bf x} \\in [0,1]^X$, let $\\hat{{\\bf x}}$ denote a random vector in $\\{0,1\\}^X$\nwhere each coordinate of $x_i$ is rounded independently to $1$ with probability $x_i$\nand $0$ otherwise.\\footnote{We denote vectors consistently in boldface $({\\bf x})$ and their coordinates in italics $(x_i)$.\nWe also identify vectors in $\\{0,1\\}^n$ with subsets of $[n]$ in a natural way.\n}\nWe define\n$$ F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})] = \\sum_{S \\subseteq X} f(S) \\prod_{i \\in S} x_i\n \\prod_{j \\notin S} (1-x_j).$$\nThis is the unique {\\em multilinear polynomial} which coincides with $f$ on $\\{0,1\\}$-vectors.\nWe remark that although we cannot compute the exact value of $F({\\bf x})$ for a given\n${\\bf x} \\in [0,1]^X$ (which would require querying $2^n$ possible values of $f(S)$),\nwe can compute $F({\\bf x})$ approximately, by random sampling. Sometimes this causes\ntechnical issues, which we also deal with in this paper.\n\nInstead of the discrete problem $\\max \\{f(S): S \\in {\\cal F}\\}$,\nwe consider a continuous optimization problem $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$,\nwhere $P({\\cal F})$ is the convex hull of characteristic vectors corresponding to ${\\cal F}$,\n$$ P({\\cal F})\n = \\left\\{ \\sum_{S \\in {\\cal F}} \\alpha_S {\\bf 1}_S:\n   \\sum_{S \\in {\\cal F}} \\alpha_S = 1, \\alpha_S \\geq 0 \\right\\}.$$\nThe reason why the extension $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$ is useful\nfor submodular maximization problems is that $F({\\bf x})$ has convexity properties\nthat allow one to solve the continuous problem $\\max \\{F({\\bf x}): {\\bf x} \\in P\\}$\n(within a constant factor) in a number of interesting cases.\nMoreover, fractional solutions can be often\nrounded to discrete ones without losing {\\em anything} in terms of the objective\nvalue. Then, our ability to solve the multilinear relaxation\ntranslates directly into an algorithm for the original problem.\nIn particular, this is true when the collection of feasible solutions\nforms a matroid.\n\n{\\em Pipage rounding} was originally developed by Ageev and Sviridenko\nfor rounding solutions in the bipartite matching polytope \\cite{AS04}.\nThe technique was adapted to matroid polytopes by Calinescu et al.\n \\cite{CCPV07}, who proved that for any submodular function $f(S)$\nand any ${\\bf x}$ in the matroid base polytope $B({\\cal M})$,\nthe fractional solution ${\\bf x}$ can be rounded to a base\n$B \\in {\\cal B}$ such that $f(B) \\geq F({\\bf x})$.\nThis approach leads to an optimal $(1-1\/e)$-approximation for\nthe Submodular Welfare Problem,\nand more generally for monotone submodular maximization subject\nto a matroid constraint \\cite{CCPV07,Vondrak08}.\nIt is also known that\nthe factor of $1-1\/e$ is optimal for the Submodular Welfare Problem\nboth in the NP framework \\cite{KLMM05} and in the value oracle model\n\\cite{MSV08}. Under the assumption that the submodular function $f(S)$\nhas {\\em curvature} $c$, there is a $\\frac{1}{c}(1-e^{-c})$-approximation\nand this is also optimal in the value oracle model \\cite{VKyoto08}.\nThe framework of pipage rounding can be also extended to nonmonotone submodular functions;\nthis presents some additional issues which we discuss in this paper.\n\nFor the problem of\nunconstrained (nonmonotone) submodular maximization, a $2\/5$-approximation\nwas developed in \\cite{FMV07}. This algorithm implicitly uses\nthe multilinear relaxation $\\max \\{F({\\bf x}): {\\bf x} \\in [0,1]^X\\}$.\nFor {\\em symmetric} submodular functions, it is shown in \\cite{FMV07}\nthat a uniformly random solution ${\\bf x} = (1\/2,\\ldots,1\/2)$ gives\n$F({\\bf x}) \\geq \\frac12 OPT$, and there is no better approximation algorithm\nin the value oracle model. Recently, a $1\/2$-approximation was found\nfor unconstrained maximization of a general nonnegative submodular function \\cite{BFNS12}.\nThis algorithm can be formulated in the multilinear relaxation framework, but also\nas a randomized combinatorial algorithm.\n\nUsing additional techniques, the multilinear relaxation can be also applied\nto submodular maximization with knapsack constraints\n($\\sum_{j \\in S} c_{ij} \\leq 1$). For the problem of maximizing a monotone\nsubmodular function subject to a constant number of knapsack constraints,\nthere is a $(1-1\/e-\\epsilon)$-approximation algorithm for any $\\epsilon > 0$ \\cite{KTS09}.\nFor maximizing a nonmonotone submodular function\nsubject to a constant number of knapsack constraints,\na $(1\/5-\\epsilon)$-approximation was designed in \\cite{LMNS09}.\n\nOne should mention that not all the best known results for submodular maximization\nhave been achieved using the multilinear relaxation. The greedy algorithm\nyields a $1\/(k+1)$-approximation for monotone submodular maximization\nsubject to $k$ matroid constraints \\cite{NWF78II}. Local search methods\nhave been used to improve this to a $1\/(k+\\epsilon)$-approximation,\nand to obtain a $1\/(k+1+1\/(k-1)+\\epsilon)$-approximation\nfor the same problem with a nonmonotone submodular function,\nfor any $\\epsilon>0$ \\cite{LMNS09,LSV09}.\nFor the problem of maximizing a nonmonotone submodular function\nover the {\\em bases} of a given matroid, local search yields\na $(1\/6-\\epsilon)$-approximation, assuming that the matroid contains two disjoint bases\n\\cite{LMNS09}.\n\n\n\\subsection{Our results}\n\nOur main contribution (Theorem~\\ref{thm:general-hardness}) \nis a general hardness construction that yields\ninapproximability results in the value oracle model in an automated\nway, based on what we call the {\\em symmetry gap} for some fixed instance.\nIn this generic fashion, we are able to replicate a number of previously\nknown hardness results (such as the optimality of the factors\n$1-1\/e$ and $1\/2$ mentioned above),\nand we also produce new hardness results using this construction\n(Theorem~\\ref{thm:matroid-bases}).\nOur construction helps explain the particular hardness thresholds obtained under\nvarious constraints, by exhibiting a small instance where the threshold can be\nseen as the gap between the optimal solution and the best symmetric solution.\nThe query complexity results in \\cite{FMV07,MSV08,VKyoto08}\ncan be seen in hindsight as special cases of Theorem~\\ref{thm:general-hardness},\nbut the construction in this paper is somewhat different and\ntechnically more involved than the previous proofs for particular cases.\n\n\\paragraph{Concrete results}\nBefore we proceed to describe our general hardness result, we present its implications\nfor two more concrete problems.\nWe also provide closely matching approximation algorithms for these two problems,\nbased on the multilinear relaxation. In the following, we assume that the objective function is given by a value oracle\nand the feasibility constraint is given by a membership oracle: a value oracle for $f$ returns the value $f(S)$ for a given set $S$,\nand a membership oracle for ${\\cal F}$ answers whether $S \\in {\\cal F}$ for a given set $S$.\n\nFirst, we consider the problem of maximizing a nonnegative (possibly nonmonotone) submodular function\nsubject to a \\emph{matroid base} constraint. (This generalizes for example the maximum bisection\nproblem in graphs.) We show that the approximability of this problem is related to base\npackings in the matroid. We use the following definition.\n\n\\begin{definition}\n\\label{def:fractional-base}\nFor a matroid ${\\cal M}$ with a collection of bases ${\\cal B}$, the fractional base\npacking number is the maximum possible value of $\\sum_{B \\in {\\cal B}} \\alpha_B$\nfor $\\alpha_B \\geq 0$ such that $\\sum_{B \\in {\\cal B}: j \\in B} \\alpha_B \\leq 1$ for\nevery element $j$.\n\\end{definition}\n\n\\noindent{\\bf Example.}\nConsider a uniform matroid with bases ${\\cal B} = \\{ B \\subset [n]: |B| = k \\}$. ($[n]$ denotes the set of integers\n$\\{1,2,\\cdots,n\\}$.)\nHere, we can take each base with a coefficient $\\alpha_B = 1 \/ {n-1 \\choose k-1}$, which satisfies\nthe condition $\\sum_{B \\in {\\cal B}: j \\in B} \\alpha_B \\leq 1$ for every element $j$ since every element\nis contained in ${n-1 \\choose k-1}$ bases. We obtain that the fractional packing number is at least\n${n \\choose k} \/ {n-1 \\choose k-1} = \\frac{n}{k}$. It is also easy to check that the fractional packing number\ncannot be larger than $\\frac{n}{k}$.\n\n\\begin{theorem}\n\\label{thm:matroid-bases}\nFor any $\\nu$ in the form $\\nu = \\frac{k}{k-1}, k \\geq 2$,\nand any fixed $\\epsilon>0$, a $(1-\\frac{1}{\\nu}+\\epsilon) = \\frac{1}{k}$-approximation for the problem\n$\\max \\{f(S): S \\in {\\cal B}\\}$, where $f(S)$ is a nonnegative submodular function,\nand ${\\cal B}$ is a collection of bases in a matroid with fractional packing number at least $\\nu$,\nwould require exponentially many value queries.\n\nOn the other hand, for any $\\nu \\in (1,2]$,\nthere is a randomized $\\frac{1}{2}(1-\\frac{1}{\\nu}-o(1))$-approximation\nfor the same problem.\n\\end{theorem}\n\nIn case the matroid contains two disjoint bases ($\\nu=2$), we obtain a\n$(\\frac14-o(1))$-approximation, improving the previously known factor of $\\frac16-o(1)$ \\cite{LMNS09}.\nIn the range of $\\nu \\in (1,2]$, our positive and negative results are within a factor of $2$.\nFor maximizing a submodular function over the bases of a general matroid, \nwe obtain the following.\n\n\\begin{corollary}\n\\label{coro:matroid-bases}\nFor the problem $\\max \\{f(S): S \\in {\\cal B}\\}$,  where $f(S)$ is a nonnegative submodular function,\nand ${\\cal B}$ is a collection of bases in a matroid, any constant-factor approximation requires\nan exponential number of value queries.\n\\end{corollary}\n\nWe also consider the problem of maximizing a nonnegative submodular function subject to a matroid independence\nconstraint. \n\n\\begin{theorem}\n\\label{thm:matroid-indep}\nFor any $\\epsilon>0$, a $(\\frac12 + \\epsilon)$-approximation for the problem $\\max \\{f(S): S \\in {\\cal I} \\}$,\nwhere $f(S)$ is a nonnegative submodular function, and ${\\cal I}$ is a collection of independent sets in a matroid,\nwould require exponentially many value queries.\n\nOn the other hand, there is a randomized $\\frac14 (-1+\\sqrt{5}-o(1)) \\simeq 0.309$-approximation for the same problem.\n\\end{theorem}\n\nOur algorithmic result improves a previously known $(\\frac14 - o(1))$-approximation \\cite{LMNS09}.\nThe hardness threshold follows from our general result, but also quite easily from \\cite{FMV07}.\n\n\n\\medskip\n\\noindent{\\bf Hardness from the symmetry gap.}\nNow we describe our general hardness result.\nConsider an instance $\\max \\{f(S): S \\in {\\cal F}\\}$ which exhibits a certain\ndegree of symmetry. This is formalized by the notion of a {\\em symmetry group}\n${\\cal G}$. We consider permutations $\\sigma \\in {\\bf S}(X)$ where ${\\bf S}(X)$\nis the symmetric group (of all permutations) on the ground set $X$.\nWe also use $\\sigma$ for the naturally induced mapping of subsets of $X$:\n$\\sigma(S) = \\{ \\sigma(i): i \\in S \\}$.\nWe say that the instance is invariant\nunder ${\\cal G} \\subset {\\bf S}(X) $, if for any $\\sigma \\in {\\cal G}$\nand any $S \\subseteq X$,\n$f(S) = f(\\sigma(S))$ and $S \\in {\\cal F} \\Leftrightarrow \\sigma(S) \\in {\\cal F}$.\nWe emphasize that even though we apply $\\sigma$ to sets, it must be derived\nfrom a permutation on $X$.\nFor ${\\bf x} \\in [0,1]^X$, we define the ``symmetrization of ${\\bf x}$'' as\n$$\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})],$$\nwhere $\\sigma \\in {\\cal G}$ is uniformly random and $\\sigma({\\bf x})$ denotes ${\\bf x}$ with coordinates permuted by $\\sigma$.\n\n\\medskip\n{\\bf Erratum:}\nThe main hardness result in the conference version of this paper \\cite{Vondrak09} was formulated\nfor an arbitrary feasibility constraint ${\\cal F}$, invariant under ${\\cal G}$.\nUnfortunately, this was an error and the theorem does not hold in that form\n --- an algorithm could gather some information from querying ${\\cal F}$, and combining this\nwith information obtained by querying the objective function $f$ it could possibly determine the hidden optimal solution.\nThe possibility of gathering information from the membership oracle for ${\\cal F}$ was neglected in the proof. (The reason for this\nwas probably that the feasibility constraints used in concrete applications of the theorem were very simple and indeed\ndid not provide any information about the optimum.)\nNevertheless, to correct this issue, one needs to impose a stronger symmetry constraint on ${\\cal F}$, namely\nthe condition that $S \\in {\\cal F}$ depends only on the symmetrized version of $S$, $\\overline{{\\bf 1}_S} = {\\bf E}_{\\sigma \\in {\\cal G}}[{\\bf 1}_{\\sigma(S)}]$.\nThis is the case in all the applications of the hardness theorem in \\cite{Vondrak09} and \\cite{OV11} and hence\nthese applications are not affected.\n\n\\begin{definition}\n\\label{def:total-sym}\nWe call an instance $\\max \\{f(S): S \\in {\\cal F}\\}$ on a ground set $X$ strongly symmetric with respect to a group of permutations ${\\cal G}$ on $X$, if $f(S) = f(\\sigma(S))$ for all $S \\subseteq X$ and $\\sigma \\in {\\cal G}$, and $S \\in {\\cal F} \\Leftrightarrow S' \\in {\\cal F}$ whenever ${\\bf E}_{\\sigma \\in {\\cal G}}[{\\bf 1}_{\\sigma(S)}] = {\\bf E}_{\\sigma \\in {\\cal G}}[{\\bf 1}_{\\sigma(S')}] $.\n\\end{definition}\n\n\\noindent{\\bf Example.} A cardinality constraint, ${\\cal F} = \\{S \\subseteq [n]: |S| \\leq k \\}$, is strongly symmetric with respect to all permutations, because the condition $S \\in {\\cal F}$ depends only on the symmetrized vector $\\overline{{\\bf 1}_S} = \\frac{|S|}{n} {\\bf 1}$.\nSimilarly, a partition matroid constraint, ${\\cal F} = \\{S \\subseteq [n]: |S \\cap X_i| \\leq k \\}$ for disjoint sets $X_i$, is also strongly symmetric.\nOn the other hand, consider a family of feasible solution ${\\cal F} = \\{ \\{1,2\\}, \\{2,3\\}, \\{3,4\\}, \\{4,1\\} \\}$. This family is invariant under a group generated by the cyclic rotation $1 \\rightarrow 2 \\rightarrow 3 \\rightarrow 4 \\rightarrow 1$. It is not strongly symmetric, because the condition $S \\in {\\cal F}$ does not depend only the symmetrized vector $\\overline{{\\bf 1}_S} = (\\frac14 |S|,\\frac14 |S|,\\frac14 |S|,\\frac14 |S|)$;\nsome pairs are feasible and others are not.\n\n\n\\\n\nNext, we define the symmetry gap as the ratio between the optimal solution\nof $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$ and the best {\\em symmetric} solution\nof this problem.\n\n\\begin{definition}[Symmetry gap]\nLet $\\max \\{ f(S): S \\in {\\cal F} \\}$ be an instance on a ground set $X$,\nwhich is strongly symmetric with respect to ${\\cal G} \\subset {\\bf S}(X)$.\nLet $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$ be the multilinear extension of $f(S)$\nand $P({\\cal F}) = \\mbox{conv}(\\{{\\bf 1}_I: I \\in {\\cal F} \\})$ the polytope\nassociated with $\\cal F$. Let $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$.\nThe symmetry gap of $\\max \\{ f(S): S \\in {\\cal F} \\}$ is defined as\n$\\gamma = \\overline{OPT} \/ OPT$ where\n$$OPT = \\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F})\\},$$\n$$\\overline{OPT} = \\max \\{F(\\bar{{\\bf x}}): {\\bf x} \\in P({\\cal F}) \\}.$$\n\\end{definition}\n\n\\noindent\nWe give examples of computing the symmetry gap in Section~\\ref{section:hardness-applications}.\nNext, we need to define the notion of a {\\em refinement} of an instance.\nThis is a natural way to extend a family of feasible sets to a larger ground set.\nIn particular, this operation preserves the types of constraints that we care about,\nsuch as cardinality constraints, matroid independence, and matroid base constraints.\n\n\\begin{definition}[Refinement]\nLet ${\\cal F} \\subseteq 2^X$, $|X|=k$ and $|N| = n$. We say that\n$\\tilde{{\\cal F}}\\subseteq 2^{N \\times X}$ is a refinement of $\\cal F$, if\n$$ \\tilde{{\\cal F}} = \\left\\{ \\tilde{S} \\subseteq N \\times X \\ \\big| \\ (x_1,\\ldots,x_k) \\in P({{\\cal F}})\n \\mbox{ where } x_j = \\frac{1}{n} |\\tilde{S} \\cap (N \\times \\{j\\})| \\right\\}. $$\n\\end{definition}\n\nIn other words, in the refined instance, each element $j \\in X$ is replaced by a set $N \\times \\{j\\}$.\nWe call this set the {\\em cluster} of elements corresponding to $j$.\nA set $\\tilde{S}$ is in $\\tilde{{\\cal F}}$ if and only if the fractions $x_j$ of the respective clusters that are intersected by $\\tilde{S}$ form a vector ${\\bf x} \\in P({\\cal F})$, i.e. a convex combination of sets in ${\\cal F}$.\n\nOur main result is that the symmetry gap for any strongly symmetric instance\ntranslates automatically into hardness of approximation for refined instances.\n(See Definition~\\ref{def:total-sym} for the notion of being ``strongly symmetric\".)\nWe emphasize that this is a query-complexity lower bound,\nand hence independent of assumptions such as $P \\neq NP$.\n\n\\begin{theorem}\n\\label{thm:general-hardness}\nLet $\\max \\{ f(S): S \\in {\\cal F} \\}$ be an instance of nonnegative\n(optionally monotone) submodular maximization, strongly symmetric with respect to ${\\cal G}$,\nwith symmetry gap $\\gamma = \\overline{OPT} \/ OPT$.\nLet $\\cal C$ be the class of instances $\\max \\{\\tilde{f}(S): S \\in \\tilde{{\\cal F}}\\}$\nwhere $\\tilde{f}$ is nonnegative (optionally monotone) submodular\nand $\\tilde{{\\cal F}}$ is a refinement of ${\\cal F}$.\nThen for every $\\epsilon > 0$,\nany (even randomized) $(1+\\epsilon) \\gamma$-approximation algorithm for the class $\\cal C$ would require\nexponentially many value queries to $\\tilde{f}(S)$.\n\\end{theorem}\n\nWe remark that the result holds even if the class $\\cal C$ is restricted\nto instances which are themselves symmetric under a group related to ${\\cal G}$\n(see the discussion in Section~\\ref{section:hardness-proof}, after the proofs of Theorem~\\ref{thm:general-hardness}\nand \\ref{thm:multilinear-hardness}).\nOn the algorithmic side, submodular maximization seems easier for symmetric instances\nand in this case we obtain optimal approximation factors, up to lower-order terms\n(see Section~\\ref{section:symmetric}).\n\nOur hardness construction yields impossibility results also for solving\nthe continuous problem $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F}) \\}$. In the case of matroid constraints,\nthis is easy to see, because an approximation to the continuous problem gives the same\napproximation factor for the discrete problem (by pipage rounding, see Appendix~\\ref{app:pipage}).\nHowever, this phenomenon\nis more general and we can show that the value of a symmetry gap translates into\nan inapproximability result for the multilinear optimization problem under any constraint\nsatisfying a symmetry condition. \n\n\\begin{theorem}\n\\label{thm:multilinear-hardness}\nLet $\\max \\{ f(S): S \\in {\\cal F} \\}$ be an instance of nonnegative\n(optionally monotone) submodular maximization, strongly symmetric with respect to ${\\cal G}$,\nwith symmetry gap $\\gamma = \\overline{OPT} \/ OPT$.\nLet $\\cal C$ be the class of instances $\\max \\{\\tilde{f}(S): S \\in \\tilde{{\\cal F}}\\}$\nwhere $\\tilde{f}$ is nonnegative (optionally monotone) submodular\nand $\\tilde{{\\cal F}}$ is a refinement of ${\\cal F}$.\nThen for every $\\epsilon > 0$,\nany (even randomized) $(1+\\epsilon) \\gamma$-approximation algorithm for the multilinear relaxation\n$\\max \\{\\tilde{F}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}})\\}$ of problems in $\\cal C$ would require\nexponentially many value queries to $\\tilde{f}(S)$.\n\\end{theorem}\n\n\n\\\n\n\\noindent{\\bf Additions to the conference version and follow-up work.}\nAn extended abstract of this work appeared in IEEE FOCS 2009 \\cite{Vondrak09}.\nAs mentioned above, the main theorem in \\cite{Vondrak09} suffers from a technical flaw.\nThis does not affect the applications, but the general theorem in \\cite{Vondrak09} is not correct.\nWe provide a corrected version of the main theorem with a complete proof\n(Theorem~\\ref{thm:general-hardness}) and we extend this hardness result to the problem of solving\n the multilinear relaxation (Theorem~\\ref{thm:multilinear-hardness}).\n\nSubsequently, further work has been done which exploits the symmetry gap concept. \nIn \\cite{OV11}, it has been proved using Theorem~\\ref{thm:general-hardness} that\nmaximizing a nonnegative submodular function subject to a matroid independence constraint\nwith a factor better than $0.478$ would require exponentially many queries. Even in the case of a cardinality constraint,\n$\\max \\{f(S): |S| \\leq k\\}$ cannot be approximated within a factor better than $0.491$ using subexponentially many\nqueries \\cite{OV11}.\nIn the case of a matroid base constraint, assuming that the fractional base packing number\nis $\\nu = \\frac{k}{k-1}$ for some $k \\geq 2$, there is no $(1-e^{-1\/k}+\\epsilon)$-approximation in the value oracle model \\cite{OV11}, improving the hardness of $(1-\\frac{1}{\\nu}+\\epsilon) = (\\frac{1}{k}+\\epsilon)$-approximation from this paper.\nThese applications are not affected by the flaw in \\cite{Vondrak09},\nand they are implied by the corrected version of Theorem~\\ref{thm:general-hardness} here.\n\nRecently \\cite{DV11}, it has been proved using the symmetry gap technique that combinatorial auctions with submodular bidders do not admit any truthful-in-expectation $1\/m^\\gamma$-approximation, where $m$ is the number of items and $\\gamma>0$ some absolute constant.\nThis is the first nontrivial hardness result for truthful-in-expectation mechanisms for combinatorial auctions;\nit separates the classes of monotone submodular functions and coverage functions,\nwhere a truthful-in-expectation $(1-1\/e)$-approximation is possible \\cite{DRY11}.\nThe proof is self-contained and does not formally refer to \\cite{Vondrak09}.\n\nMoreover, this hardness result for truthful-in-expectation mechanisms as well as the main hardness result in this paper\nhave been converted from the oracle setting to a computational complexity setting \\cite{DV12a,DV12b}.  This recent work\nshows that the hardness of approximation arising from symmetry gap is not limited to instances given by an oracle, but holds\nalso for instances encoded explicitly on the input, under a suitable complexity-theoretic assumption.\n\n\\\n\n\\noindent{\\bf Organization.}\nThe rest of the paper is organized as follows.\nIn Section~\\ref{section:hardness-applications}, we present applications\nof our main hardness result (Theorem~\\ref{thm:general-hardness}) to concrete cases,\nin particular we show how it implies the hardness statements in Theorem~\\ref{thm:matroid-indep}\nand \\ref{thm:matroid-bases}.\nIn Section~\\ref{section:hardness-proof}, we present the proofs of Theorem~\\ref{thm:general-hardness}\n and Theorem~\\ref{thm:multilinear-hardness}.\nIn Section~\\ref{section:algorithms}, we prove the algorithmic results\nin Theorem~\\ref{thm:matroid-indep} and \\ref{thm:matroid-bases}.\nIn Section~\\ref{section:symmetric}, we discuss the special case of symmetric instances.\nIn the Appendix,\nwe present a few basic facts concerning submodular functions,\nan extension of pipage rounding to matroid independence polytopes (rather than matroid base polytopes),\nand other technicalities that would hinder the main exposition.\n\n\n\n\\section{From symmetry to inapproximability: applications}\n\\label{section:hardness-applications}\n\nBefore we get into the proof of Theorem~\\ref{thm:general-hardness},\nlet us show how it can be applied to a number of specific problems.\nSome of these are hardness results that were proved previously by\nan ad-hoc method. The last application is a new one\n(Theorem~\\ref{thm:matroid-bases}).\n\n\\\n\n\\noindent{\\bf Nonmonotone submodular maximization.}\nLet $X = \\{1,2\\}$ and for any $S \\subseteq X$,\n$f(S) = 1$ if $|S|=1$, and $0$ otherwise.\nConsider the instance $\\max \\{ f(S): S \\subseteq X \\}$.\nIn other words, this is the Max Cut problem on the graph $K_2$.\nThis instance exhibits a simple symmetry, the group of all (two) permutations\non $\\{1,2\\}$. We get $OPT = F(1,0) = F(0,1) = 1$, while $\\overline{OPT} = F(1\/2,1\/2)\n = 1\/2$. Hence, the symmetry gap is $1\/2$.\n\n\\begin{figure}[here]\n\\begin{tikzpicture}[scale=.50]\n\n\\draw (-10,0) node {};\n\n\\filldraw [fill=gray,line width=1mm] (0,0) rectangle (4,2);\n\n\\filldraw [fill=white] (0.5,1) .. controls +(0,1) and +(0,1) .. (1.5,1)\n .. controls +(0,-1) and +(0,-1) .. (0.5,1);\n\n\n\\fill (1,1) circle (5pt);\n\\fill (3,1) circle (5pt);\n\\draw (1,1) -- (3,1);\n\n\\draw (-1,1) node {$X$};\n\n\\draw (7,1) node {$\\bar{x}_1 = \\bar{x}_2 = \\frac{1}{2}$};\n\n\\end{tikzpicture}\n\n\\caption{Symmetric instance for nonmonotone submodular maximization: Max Cut on the graph $K_2$.\nThe white set denotes the optimal solution, while $\\bar{{\\bf x}}$ is the (unique) symmetric solution.}\n\n\\end{figure}\n\n\nSince $f(S)$ is nonnegative submodular and there is no constraint on $S \\subseteq X$,\nthis will be the case for any refinement of the instance as well.\nTheorem~\\ref{thm:general-hardness} implies immediately the following: any algorithm achieving\na $(\\frac12 + \\epsilon)$-approximation for nonnegative (nonmonotone) submodular\nmaximization requires exponentially many value queries\n(which was previously known \\cite{FMV07}). Note that a ``trivial instance\" implies\na nontrivial hardness result. This is typically the case in applications of Theorem~\\ref{thm:general-hardness}.\n\nThe same symmetry gap holds if we impose some simple constraints:\nthe problems $\\max \\{ f(S): |S| \\leq 1 \\}$ and $\\max \\{ f(S): |S| = 1 \\}$\nhave the same symmetry gap as above. Hence, the hardness threshold of $1\/2$\nalso holds for nonmonotone submodular maximization under cardinality constraints\nof the type $|S| \\leq n\/2$, or $|S| = n\/2$. This proves the hardness part of\nTheorem~\\ref{thm:matroid-indep}. This can be derived quite easily\nfrom the construction of \\cite{FMV07} as well.\n\n\\\n\n\\noindent{\\bf Monotone submodular maximization.}\nLet $X = [k]$ and $f(S) = \\min \\{|S|, 1\\}$.\nConsider the instance $\\max \\{f(S): |S| \\leq 1 \\}$.\nThis instance is invariant under all permutations on $[k]$, the symmetric group ${\\bf S}_k$.\nNote that the instance is {\\em strongly symmetric} (Def.~\\ref{def:total-sym}) with respect to ${\\bf S}_k$,\nsince the feasibility constraint $|S| \\leq 1$ depends only on the symmetrized vector\n$\\overline{{\\bf 1}_S} = (\\frac{1}{k}|S|,\n\\ldots, \\frac{1}{k}|S|)$. We get $OPT = F(1,0,\\ldots,0) = 1$, while\n$\\overline{OPT} = F(1\/k,1\/k,\\ldots,1\/k) = 1 - (1-1\/k)^k$.\n\n\\iffalse\n\\begin{figure}[here]\n\\begin{tikzpicture}[scale=.50]\n\n\\draw (-8,0) node {};\n\n\\filldraw [fill=gray,line width=1mm] (0,0) rectangle (9,2);\n\n\\filldraw [fill=white] (0.5,1) .. controls +(0,1) and +(0,1) .. (1.5,1)\n .. controls +(0,-1) and +(0,-1) .. (0.5,1);\n\n\n\\fill (1,1) circle (5pt);\n\\fill (2,1) circle (5pt);\n\\fill (3,1) circle (5pt);\n\\fill (4,1) circle (5pt);\n\\fill (5,1) circle (5pt);\n\\fill (6,1) circle (5pt);\n\\fill (7,1) circle (5pt);\n\\fill (8,1) circle (5pt);\n\n\\draw [dotted] (1,1) -- (8,1);\n\n\\draw (-1,1) node {$X$};\n\n\\draw (11,1) node {$\\bar{x}_{i} = \\frac{1}{k}$};\n\n\\end{tikzpicture}\n\n\\caption{Symmetric instance for monotone submodular maximization.}\n\n\\end{figure}\n\\fi\n\n\n\nHere, $f(S)$ is monotone submodular and any refinement of $\\cal F$ is\na set system of the type $\\tilde{{\\cal F}} = \\{S: |S| \\leq \\ell \\}$.\nBased on our theorem, this implies that any approximation better than $1 - (1-1\/k)^k$\nfor monotone submodular maximization subject\nto a cardinality constraint would require exponentially many value queries.\nSince this holds for any fixed $k$, we get the same hardness\nresult for any $\\beta > \\lim_{k \\rightarrow \\infty} (1 - (1-1\/k)^k) = 1-1\/e$\n(which was previously known \\cite{NW78}).\n\n\\iffalse\n\n\\noindent{\\bf Submodular welfare maximization.}\nLet $X = [k] \\times [k]$,\n$ {\\cal F} = \\{ S: S$ contains at most $1$ pair $(i,j)$ for each $j \\}$,\nand $f(S) = |\\{ i: \\exists (i,j) \\in S \\}|$.\nConsider the instance $\\max \\{f(S): S \\in {\\cal F}\\}$.\nThis instance can be interpreted as an allocation problem of $k$ items to\n$k$ players. A set $S$ represents an assignment in the sense that\n$(i,j) \\in S$ if item $j$ is allocated to player $i$.\nA player is satisfied is she receives at least 1 item;\nthe objective function is the number of satisfied players.\nThis instance exhibits the symmetry of all permutations of the players\n(and also all permutations of the items, although we do not use it here).\nNote that the feasibility constraint $S \\in {\\cal F}$ depends only on the symmetrized\nvector $\\overline{{\\bf 1}_S}$ which averages out the allocation of each item\nacross all players. Therefore the instance is strongly symmetric with respect\nto permutations of players.\nAn optimum solution allocates each item to a different player, and $OPT = k$.\nThe symmetrized optimum allocates a $1\/k$-fraction of each item to each player,\nwhich gives $\\overline{OPT} = F(\\frac{1}{k},\\frac{1}{k},\\ldots,\\frac{1}{k}) =\n k(1-(1-\\frac{1}{k})^k)$.\n\n\\begin{figure}[here]\n\\begin{tikzpicture}[scale=.50]\n\n\\draw (-8,0) node {};\n\n\\filldraw [fill=gray,line width=1mm] (0,0) rectangle (8,2);\n\\filldraw [fill=white] (0.5,1) .. controls +(0,1) and +(0,1) .. (1.5,1)\n .. controls +(0,-1) and +(0,-1) .. (0.5,1);\n\\filldraw [fill=gray,line width=1mm] (0,2) rectangle (8,4);\n\\filldraw [fill=white] (2.5,3) .. controls +(0,1) and +(0,1) .. (3.5,3)\n .. controls +(0,-1) and +(0,-1) .. (2.5,3);\n\\filldraw [fill=gray,line width=1mm] (0,4) rectangle (8,6);\n\\filldraw [fill=white] (4.5,5) .. controls +(0,1) and +(0,1) .. (5.5,5)\n .. controls +(0,-1) and +(0,-1) .. (4.5,5);\n\\filldraw [fill=gray,line width=1mm] (0,6) rectangle (8,8);\n\\filldraw [fill=white] (6.5,7) .. controls +(0,1) and +(0,1) .. (7.5,7)\n .. controls +(0,-1) and +(0,-1) .. (6.5,7);\n\n\n\\fill (1,1) circle (5pt);\n\\fill (3,1) circle (5pt);\n\\fill (5,1) circle (5pt);\n\\fill (7,1) circle (5pt);\n\\fill (1,3) circle (5pt);\n\\fill (3,3) circle (5pt);\n\\fill (5,3) circle (5pt);\n\\fill (7,3) circle (5pt);\n\\fill (1,5) circle (5pt);\n\\fill (3,5) circle (5pt);\n\\fill (5,5) circle (5pt);\n\\fill (7,5) circle (5pt);\n\\fill (1,7) circle (5pt);\n\\fill (3,7) circle (5pt);\n\\fill (5,7) circle (5pt);\n\\fill (7,7) circle (5pt);\n\n\\draw [dotted] (1,1) -- (7,1);\n\\draw [dotted] (1,3) -- (7,3);\n\\draw [dotted] (1,5) -- (7,5);\n\\draw [dotted] (1,7) -- (7,7);\n\n\\draw (-1.5,4) node {players};\n\\draw (4,9) node {items};\n\n\\draw (10,4) node {$\\bar{x}_{ij} = \\frac{1}{k}$};\n\n\\end{tikzpicture}\n\n\\caption{Symmetric instance for submodular welfare maximization.}\n\n\\end{figure}\n\nA refinement of this instance can be interpreted as an allocation problem\nwhere we have $n$ copies of each item, we still have $k$ players,\nand the utility functions are monotone submodular. Our theorem implies\nthat for submodular welfare maximization with $k$ players, a better approximation factor\nthan $1 - (1-\\frac{1}{k})^k$ is impossible.\\footnote{We note that \nour theorem here assumes an oracle model where only the total value can be\nqueried for a given allocation. This is actually enough for the $(1-1\/e)$-approximation\nof \\cite{Vondrak08} to work. However, the hardness result holds even if each player's valuation\nfunction can be queried separately; this result was proved in \\cite{MSV08}.}\n\n\\fi\n\n\\\n\n\\noindent{\\bf Submodular maximization over matroid bases.}\nLet $X = A \\cup B$, $A = \\{a_1, \\ldots, a_k\\}$, $B = \\{b_1, \\ldots, b_k\\}$\nand ${\\cal F} = \\{ S: |S \\cap A| = 1 \\ \\& \\ |S \\cap B| = k-1 \\}$.\nWe define $f(S) = \\sum_{i=1}^{k} f_i(S)$ where\n$f_i(S) = 1$ if $a_i \\in S \\ \\& \\ b_i \\notin S$, and $0$ otherwise.\nThis instance can be viewed as a Maximum Directed Cut problem on a graph\nof $k$ disjoint arcs, under the constraint that exactly one arc tail\nand $k-1$ arc heads should be on the left-hand side ($S$).\nAn optimal solution is for example $S = \\{a_1, b_2, b_3, \\ldots, b_k \\}$,\nwhich gives $OPT = 1$.\nThe symmetry here is that we can apply the same permutation to $A$ and $B$\nsimultaneously. Again, the feasibility of a set $S$ depends only on the symmetrized vector $\\overline{{\\bf 1}_S}$:\nin fact $S \\in {\\cal F}$ if and only if $\\overline{{\\bf 1}_S} = (\\frac1k,\\ldots,\\frac1k,1-\\frac1k,\\ldots,1-\\frac1k)$.\nThere is a unique symmetric solution\n$\\bar{{\\bf x}} = (\\frac1k,\\ldots,\\frac1k,1-\\frac1k,\\ldots,1-\\frac1k)$, and\n$\\overline{OPT} = F(\\bar{{\\bf x}}) = {\\bf E}[f(\\hat{\\bar{{\\bf x}}})] = \\sum_{i=1}^{k}\n {\\bf E}[f_i(\\hat{\\bar{{\\bf x}}})] = \\frac1k$\n(since each arc appears in the directed cut induced by $\\hat{\\bar{{\\bf x}}}$ with probability $\\frac{1}{k^2}$).\n\n\\begin{figure}[here]\n\\begin{tikzpicture}[scale=.60]\n\n\\draw (-6,0) node {};\n\n\\filldraw [fill=gray,line width=1mm] (0,0) rectangle (9,2);\n\\filldraw [fill=gray,line width=1mm] (0,2) rectangle (9,4);\n\n\\filldraw [fill=white] (0.5,3) .. controls +(0,1) and +(0,1) .. (1.5,3)\n .. controls +(0,-1) and +(0,-1) .. (0.5,3);\n\\filldraw [fill=white] (1.5,1) .. controls +(0,1) and +(0,1) .. (8.5,1)\n .. controls +(0,-1) and +(0,-1) .. (1.5,1);\n\n\n\\fill (1,1) circle (5pt);\n\\fill (2,1) circle (5pt);\n\\fill (3,1) circle (5pt);\n\\fill (4,1) circle (5pt);\n\\fill (5,1) circle (5pt);\n\\fill (6,1) circle (5pt);\n\\fill (7,1) circle (5pt);\n\\fill (8,1) circle (5pt);\n\\fill (1,3) circle (5pt);\n\\fill (2,3) circle (5pt);\n\\fill (3,3) circle (5pt);\n\\fill (4,3) circle (5pt);\n\\fill (5,3) circle (5pt);\n\\fill (6,3) circle (5pt);\n\\fill (7,3) circle (5pt);\n\\fill (8,3) circle (5pt);\n\\draw[-latex] [line width=0.5mm] (1,3) -- (1,1);\n\\draw[-latex] [line width=0.5mm] (2,3) -- (2,1);\n\\draw[-latex] [line width=0.5mm] (3,3) -- (3,1);\n\\draw[-latex] [line width=0.5mm] (4,3) -- (4,1);\n\\draw[-latex] [line width=0.5mm] (5,3) -- (5,1);\n\\draw[-latex] [line width=0.5mm] (6,3) -- (6,1);\n\\draw[-latex] [line width=0.5mm] (7,3) -- (7,1);\n\\draw[-latex] [line width=0.5mm] (8,3) -- (8,1);\n\n\n\\draw (-1,1) node {$B$};\n\\draw (-1,3) node {$A$};\n\n\\draw (11,3) node {$\\bar{x}_{a_i} = \\frac{1}{k}$};\n\\draw (11,1) node {$\\bar{x}_{b_i} = 1 - \\frac{1}{k}$};\n\n\\end{tikzpicture}\n\n\\caption{Symmetric instance for submodular maximization over matroid bases.}\n\n\\end{figure}\nThe refined instances are instances of (nonmonotone) submodular maximization\nover the bases of a matroid, where the ground set is partitioned into $A \\cup B$\nand we should take a $\\frac1k$-fraction of $A$ and a $(1-\\frac1k)$-fraction of $B$.\n(This means that the fractional packing number of bases is $\\nu = \\frac{k}{k-1}$.)\nOur theorem implies that for this class of instances, an approximation better\nthan $1\/k$ is impossible - this proves the hardness part of Theorem~\\ref{thm:matroid-bases}.\n\n\\\n\nObserve that in all the cases mentioned above, the multilinear relaxation is equivalent\nto the original problem, in the sense that any fractional solution can be rounded\nwithout any loss in the objective value. This implies that the same hardness factors apply to\nsolving the multilinear relaxation of the respective problems. In particular, using the last result\n(for matroid bases), we obtain that the multilinear optimization problem $\\max \\{ F({\\bf x}): {\\bf x} \\in P \\}$\ndoes not admit a constant factor for nonnegative submodular functions and matroid base polytopes.\n(We remark that a $(1-1\/e)$-approximation can be achieved for any {\\em monotone} submodular function\nand any solvable polytope, i.e.~polytope over which we can optimize linear functions \\cite{Vondrak08}.)\n\nAs Theorem~\\ref{thm:multilinear-hardness} shows, this holds more generally - any symmetry gap\nconstruction gives an inapproximability result for solving the multilinear optimization problem\n$\\max \\{F({\\bf x}): {\\bf x} \\in P\\}$. This in fact implies limits on what hardness results we can\npossibly hope for using this technique. For instance, we cannot prove using the symmetry gap\nthat the monotone submodular maximization problem subject to the intersection of $k$ matroid constraints\ndoes not admit a constant factor  - because we would also prove that the respective multilinear relaxation\ndoes not admit such an approximation. But we know from \\cite{Vondrak08} that a $(1-1\/e)$-approximation\nis possible for the multilinear problem in this case.\n\nHence, the hardness arising from the symmetry gap is related to the difficulty of solving\nthe multilinear optimization problem rather than the difficulty of rounding a fractional solution.\nThus this technique is primarily suited to optimization problems where the multilinear optimization problem\ncaptures closely the original discrete problem.\n \n\n\n\n\\section{From symmetry to inapproximability: proof}\n\\label{section:hardness-proof}\n\n\\paragraph{The roadmap}\nAt a high level,\nour proof resembles the constructions of \\cite{FMV07,MSV08}.\nWe construct instances based on continuous functions $F({\\bf x})$, $G({\\bf x})$,\nwhose optima differ by a gap\nfor which we want to prove hardness. Then we show that after a certain perturbation,\nthe two instances are very hard to distinguish.\nThis paper generalizes the ideas of \\cite{FMV07,MSV08} and brings\ntwo new ingredients. First, we show that the functions\n$F({\\bf x}), G({\\bf x})$, which are ``pulled out of the hat'' in \\cite{FMV07,MSV08},\ncan be produced in a natural way from the multilinear relaxation\nof the respective problem, using the notion of a {\\em symmetry gap}.\nSecondly, the functions $F({\\bf x}), G({\\bf x})$ are perturbed in a way that makes\nthem indistinguishable and this forms the main technical part\nof the proof. In \\cite{FMV07}, this step is quite simple.\nIn \\cite{MSV08}, the perturbation is more complicated, but still relies\non properties of the functions $F({\\bf x}), G({\\bf x})$ specific to that application.\nThe construction that we present here (Lemma~\\ref{lemma:final-fix})\nuses the symmetry properties of a fixed instance in a generic fashion.\n\n\n\\\n\nFirst, let us present an outline of our construction. Given an instance\n$\\max \\{f(S): S \\in {\\cal F}\\}$ exhibiting a symmetry gap $\\gamma$,\nwe consider two smooth submodular\\footnote{\"Smooth submodularity\"\nmeans the condition $\\mixdiff{F}{x_i}{x_j} \\leq 0$ for all $i,j$.}\nfunctions, $F({\\bf x})$ and $G({\\bf x})$.\nThe first one is the multilinear extension $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$,\nwhile the second one is its symmetrized version $G({\\bf x}) = F(\\bar{{\\bf x}})$.\nWe modify these functions slightly so that we obtain\nfunctions $\\hat{F}({\\bf x})$ and $\\hat{G}({\\bf x})$ with the following property:\nFor any vector ${\\bf x}$ which is close to its symmetrized version\n$\\bar{{\\bf x}}$, $\\hat{F}({\\bf x}) = \\hat{G}({\\bf x})$.\nThe functions $\\hat{F}({\\bf x}), \\hat{G}({\\bf x})$ induce instances of\nsubmodular maximization on the refined ground sets. The way we define\ndiscrete instances based on $\\hat{F}({\\bf x}), \\hat{G}({\\bf x})$ is natural,\nusing the following lemma.\nEssentially, we interpret the fractional variables as fractions of clusters\nin the refined instance.\n\n\\begin{lemma}\n\\label{lemma:smooth-submodular}\nLet $F:[0,1]^X \\rightarrow {\\boldmath R}$,\n $N = [n]$, $n \\geq 1$, and define $f:2^{N \\times X} \\rightarrow {\\boldmath R}$\nso that $f(S) = F({\\bf x})$ where $x_i = \\frac{1}{n} |S \\cap (N \\times \\{i\\})|$. Then\n\\begin{enumerate}\n\\item If $\\partdiff{F}{x_i} \\geq 0$ everywhere for each $i$, then\n$f$ is monotone.\n\\item If  the first partial derivatives of $F$ are absolutely continuous\\footnote{\nA function $F:[0,1]^X \\rightarrow {\\boldmath R}$ is absolutely continuous,\nif $\\forall \\epsilon>0; \\exists \\delta>0; \\sum_{i=1}^{t} ||{\\bf x}_i-{\\bf y}_i|| < \\delta \\Rightarrow\n\\sum_{i=1}^{t} |F({\\bf x}_i) - F({\\bf y}_i)| < \\epsilon$.} and $\\mixdiff{F}{x_i}{x_j} \\leq 0$ almost everywhere for all $i,j$,\nthen $f$ is submodular.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst, assume $\\partdiff{F}{x_i} \\geq 0$ everywhere for all $i$. This implies that $F$ is nondecreasing in every coordinate,\ni.e. $F({\\bf x}) \\leq F({\\bf y})$ whenever ${\\bf x} \\leq {\\bf y}$. This means that $f(S) \\leq f(T)$ whenever $S \\subseteq T$.\n\nNext, assume $\\partdiff{F}{x_i}$ is absolutely continuous for each $i$ and $\\mixdiff{F}{x_j}{x_i} \\leq 0$ almost everywhere for all $i,j$.\nWe want to prove that $\\partdiff{F}{x_i} |_{\\bf x} \\geq \\partdiff{F}{x_i} |_{\\bf y}$ whenever ${\\bf x} \\leq {\\bf y}$, which implies that the marginal\nvalues of $f$ are nonincreasing.\n\nLet ${\\bf x} \\leq {\\bf y}$,\nfix $\\delta>0$ arbitrarily small, and pick ${\\bf x}',{\\bf y}'$ such that $||{\\bf x}'-{\\bf x}||<\\delta, ||{\\bf y}'-{\\bf y}||<\\delta, {\\bf x}' \\leq {\\bf y}'$ and\non the line segment $[{\\bf x}', {\\bf y}']$, we have $\\mixdiff{F}{x_j}{x_i} \\leq 0$ except for a set of (1-dimensional) measure zero. If such a pair of points\n${\\bf x}', {\\bf y}'$ does not exist, it means that there are sets $A,B$ of positive measure such that\n${\\bf x} \\in A, {\\bf y} \\in B$ and for any\n${\\bf x}' \\in A, {\\bf y}' \\in B$, the line segment $[{\\bf x}',{\\bf y}']$ contains a subset of positive (1-dimensional) measure where $\\mixdiff{F}{x_j}{x_i}$ \nfor some $j$ is positive or undefined. This would imply that $[0,1]^X$ contains a subset of positive measure where\n$\\mixdiff{F}{x_j}{x_i}$ for some $j$ is positive or undefined, which we assume is not the case.\n\nTherefore, there is a pair of points ${\\bf x}', {\\bf y}'$ as described above.\nWe compare $\\partdiff{F}{x_i} |_{{\\bf x}'}$ and $\\partdiff{F}{x_i} |_{{\\bf y}'}$\nby integrating along the line segment $[{\\bf x}', {\\bf y}']$. Since $\\partdiff{F}{x_i}$ is absolutely continuous and $\\partdiff{}{x_j} \\partdiff{F}{x_i} = \\mixdiff{F}{x_j}{x_i} \\leq 0$ along this line segment for all $j$ except for a set of measure zero, we obtain $\\partdiff{F}{x_i} |_{{\\bf x}'} \\geq \\partdiff{F}{x_i} |_{{\\bf y}'}$. This is true for ${\\bf x}', {\\bf y}'$ arbitrarily\nclose to ${\\bf x}, {\\bf y}$, and hence by continuity of $\\partdiff{F}{x_i}$, we get $\\partdiff{F}{x_i} |_{{\\bf x}} \\geq \\partdiff{F}{x_i} |_{{\\bf y}}$.\nThis implies that the marginal values of $f$ are nonincreasing.\n\\end{proof}\n\n\nThe way we construct $\\hat{F}({\\bf x}), \\hat{G}({\\bf x})$ is such that,\ngiven a large enough refinement of the ground set,\nit is impossible to distinguish the instances corresponding to\n$\\hat{F}({\\bf x})$ and $\\hat{G}({\\bf x})$. As we argue more precisely later,\nthis holds because if the ground set is large and labeled in a random way\n(considering the symmetry group of the instance), a query about a vector ${\\bf x}$\neffectively becomes a query about the symmetrized vector $\\bar{{\\bf x}}$.\nWe would like this property to imply that all queries with high probability fall\nin the region where $\\hat{F}({\\bf x}) = \\hat{G}({\\bf x})$ and the inability to distinguish between $\\hat{F}$\nand $\\hat{G}$ gives the hardness result that we want.\nThe following lemma gives the precise properties of $\\hat{F}({\\bf x})$\nand $\\hat{G}({\\bf x})$ that we need.\n\n\\begin{lemma}\n\\label{lemma:final-fix}\nConsider a function $f:2^X \\rightarrow {\\boldmath R}_+$ invariant under a group\nof permutations $\\cal G$ on the ground set $X$.\nLet $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$, $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$,\nand fix any $\\epsilon > 0$.\nThen there is $\\delta > 0$ and functions $\\hat{F}, \\hat{G}:[0,1]^X \\rightarrow {\\boldmath R}_+$\n(which are also symmetric with respect to ${\\cal G}$), satisfying:\n\\begin{enumerate}\n\\item For all ${\\bf x} \\in [0,1]^X$, $\\hat{G}({\\bf x}) = \\hat{F}(\\bar{{\\bf x}})$.\n\\item For all ${\\bf x} \\in [0,1]^X$, $|\\hat{F}({\\bf x}) - F({\\bf x})| \\leq \\epsilon$.\n\\item Whenever $||{\\bf x} - \\bar{{\\bf x}}||^2 \\leq \\delta$, $\\hat{F}({\\bf x}) = \\hat{G}({\\bf x})$\nand the value depends only on $\\bar{{\\bf x}}$.\n\\item The first partial derivatives of $\\hat{F}, \\hat{G}$ are absolutely continuous. \n\\item If $f$ is monotone, then $\\partdiff{\\hat{F}}{x_i} \\geq 0$ and\n   $\\partdiff{\\hat{G}}{x_i} \\geq 0$ everywhere.\n\\item If $f$ is submodular, then $\\mixdiff{\\hat{F}}{x_i}{x_j} \\leq 0$ and\n   $\\mixdiff{\\hat{G}}{x_i}{x_j} \\leq 0$ almost everywhere.\n\\end{enumerate}\n\\end{lemma}\n\nThe proof of this lemma is the main technical part of this paper\nand we defer it to the end of this section. Assuming this lemma,\nwe first finish the proof of the main theorem. We prove the following.\n\n\\begin{lemma}\n\\label{lemma:indistinguish}\nLet $\\hat{F}, \\hat{G}$ be the two functions provided by Lemma~\\ref{lemma:final-fix}.\nFor a parameter $n \\in {\\boldmath Z}_+$ and $N = [n]$, define two discrete functions\n$\\hat{f}, \\hat{g}: 2^{N \\times X} \\rightarrow {\\boldmath R}_+$ as follows:\nLet $\\sigma^{(i)}$ be an arbitrary permutation in ${\\cal G}$ for each $i \\in N$.\nFor every set $S \\subseteq N \\times X$, we define a vector $\\xi(S) \\in [0,1]^X$ by\n$$ \\xi_j(S) = \\frac{1}{n} \\left|\\{i \\in N: (i,\\sigma^{(i)}(j)) \\in S \\}\\right|.$$\nLet us define:\n$$ \\hat{f}(S) = \\hat{F}(\\xi(S)), \\ \\ \\ \\ \\ \\hat{g}(S) = \\hat{G}(\\xi(S)).$$\nIn addition, let $\\tilde{{\\cal F}} = \\{\\tilde{S}: \\xi(\\tilde{S}) \\in P({\\cal F})\\}$\nbe a feasibility constraint such that the condition $S \\in {\\cal F}$ depends\nonly on the symmetrized vector $\\overline{{\\bf 1}_S}$.\nThen deciding whether an instance given by value\/membership oracles is\n$\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}}\\}$ or $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$\n(even by a randomized algorithm, with any constant probability of success)\nrequires an exponential number of queries.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\sigma^{(i)} \\in {\\cal G}$ be chosen independently at random for each $i \\in N$ and consider the instances\n$\\max\\{\\hat{f}(S): S \\in \\tilde{{\\cal F}} \\}$, $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$ as described in the lemma. We show that\nevery deterministic algorithm will follow the same computation path and return the same answer on both instances,\nwith high probability. By Yao's principle, this means that every randomized algorithm returns\nthe same answer for the two instances with high probability, for some particular $\\sigma^{(i)} \\in {\\cal G}$.\n\nThe feasible sets in the refined instance, $S \\in \\tilde{{\\cal F}}$, are such that\nthe respective vector $\\xi(S)$ is in the polytope $P({{\\cal F}})$.\nSince the instance is strongly symmetric, the condition $S \\in {\\cal F}$ depends\nonly on the symmetrized vector $\\overline{{\\bf 1}_S}$. Hence, the condition $\\xi(S) \\in P({\\cal F})$\ndepends only on the symmetrized vector $\\overline{\\xi(S)}$. Therefore, $S \\in \\tilde{{\\cal F}} \\Leftrightarrow\n\\xi(S) \\in P({\\cal F}) \\Leftrightarrow \\overline{\\xi(S)} \\in P({\\cal F})$.\nWe have $\\overline{\\xi(S)}_j = {\\bf E}_{\\sigma \\in {\\cal G}}[\\frac{1}{n} \\left|\\{i \\in N: (i,\\sigma^{(i)}(\\sigma(j))) \\in S \\}\\right|].$ The distribution of $\\sigma^{(i)} \\circ \\sigma$ is the same as that of $\\sigma$, i.e.~uniform over ${\\cal G}$. Hence, $\\overline{\\xi(S)}_j$ and consequently the condition $S \\in \\tilde{{\\cal F}}$ does not depend on $\\sigma^{(i)}$ in any way.\nIntuitively, an algorithm cannot learn any information about the permutations $\\sigma^{(i)}$ by querying the feasibility oracle,\nsince the feasibility condition $S \\in \\tilde{{\\cal F}}$ does not depend on $\\sigma^{(i)}$ for any $i \\in N$.\n\nThe main part of the proof is to show that even queries to the objective function are unlikely to reveal\nany information about the permutations $\\sigma^{(i)}$.\nThe key observation is that for any fixed query $Q$ to the objective function,\nthe associated vector ${\\bf q} = \\xi(Q)$ is very likely to be close\nto its symmetrized version $\\bar{{\\bf q}}$.\nTo see this, consider a query $Q$. The associated vector ${\\bf q} = \\xi(Q)$\nis determined by\n$$ q_j = \\frac{1}{n} |\\{i \\in N: (i,\\sigma^{(i)}(j)) \\in Q\\}|\n = \\frac{1}{n} \\sum_{i=1}^{n} Q_{ij} $$\nwhere $Q_{ij}$ is an indicator variable of the event $(i,\\sigma^{(i)}(j)) \\in Q$.\nThis is a random event due to the randomness in $\\sigma^{(i)}$.\nWe have\n$$ {\\bf E}[Q_{ij}] = \\Pr[Q_{ij}=1] = \\Pr_{\\sigma^{(i)} \\in {\\cal G}}[(i,\\sigma^{(i)}(j)) \\in Q].$$\nAdding up these expectations over $i \\in N$, we get\n$$ \\sum_{i \\in N} {\\bf E}[Q_{ij}]\n = \\sum_{i \\in N} \\Pr_{\\sigma^{(i)} \\in {\\cal G}}[(i,\\sigma^{(i)}(j)) \\in Q] \n = {\\bf E}_{\\sigma \\in {\\cal G}} [|\\{i \\in N: (i,\\sigma(j)) \\in Q \\}|].$$\nFor the purposes of expectation, the independence of $\\sigma^{(1)},\n \\ldots, \\sigma^{(n)}$ is irrelevant and that is why we replace them by one random permutation $\\sigma$.\nOn the other hand, consider the symmetrized vector $\\bar{{\\bf q}}$, with coordinates\n$$ \\bar{q}_j = {\\bf E}_{\\sigma \\in {\\cal G}}[q_{\\sigma(j)}]\n = \\frac{1}{n} {\\bf E}_{\\sigma \\in {\\cal G}}[|\\{i \\in N: (i,\\sigma^{(i)}(\\sigma(j))) \\in Q\\}|]\n = \\frac{1}{n} {\\bf E}_{\\sigma \\in {\\cal G}}[|\\{i \\in N: (i,\\sigma(j)) \\in Q \\}|] $$\nusing the fact that the distribution of $\\sigma^{(i)} \\circ \\sigma$\nis the same as the distribution of $\\sigma$ - uniformly random over ${\\cal G}$.\nNote that the vector ${\\bf q}$ depends on the random permutations $\\sigma^{(i)}$\nbut the symmetrized vector $\\bar{{\\bf q}}$ does not; this will be also useful\nin the following. For now, we summarize that\n$$ \\bar{q}_j = \\frac{1}{n} \\sum_{i=1}^{n} {\\bf E}[Q_{ij}] = {\\bf E}[q_j].$$\nSince each permutation $\\sigma^{(i)}$ is chosen independently, the random variables\n$\\{Q_{ij}: 1 \\leq i \\leq n\\}$ are independent (for a fixed $j$).\nWe can apply Chernoff's bound (see e.g. \\cite{AlonSpencer}, Corollary A.1.7):\n$$ \\Pr\\left[\\left|\\sum_{i=1}^{n} Q_{ij}- \\sum_{i=1}^{n} {\\bf E}[Q_{ij}] \\right| > a \\right]\n < 2e^{-2a^2 \/ n}.$$\nUsing $q_j = \\frac{1}{n} \\sum_{i=1}^{n} Q_{ij}$,\n $\\bar{q}_j = \\frac{1}{n} \\sum_{i=1}^{n} {\\bf E}[Q_{ij}]$\nand setting $a = n \\sqrt{\\delta\/|X|}$, we obtain\n$$ \\Pr\\left[|q_j - \\bar{q}_j| > \\sqrt{{\\delta}\/{|X|}}\\right]\n < 2e^{-2 n \\delta \/ |X|}.$$\nBy the union bound,\n\\begin{equation}\n\\label{eq:D(q)}\n  \\Pr[||{\\bf q}-\\bar{{\\bf q}}||^2 > \\delta] \\leq \\sum_{j \\in X} \\Pr[|q_j-\\bar{q}_j|^2 > {\\delta}\/{|X|}] < 2|X| e^{-2n \\delta\/|X|}.\n\\end{equation}\nNote that while $\\delta$ and $|X|$ are constants, $n$ grows as the size\nof the refinement and hence the probability is exponentially small in the\nsize of the ground set $N \\times X$.\n\nDefine $D({\\bf q}) = ||{\\bf q}-\\bar{{\\bf q}}||^2$. As long as $D({\\bf q}) \\leq \\delta$ for every query issued by the algorithm,\nthe answers do not depend on the randomness of the input. This is because\nthen the values of $\\hat{F}({\\bf q})$ and $\\hat{G}({\\bf q})$ depend only on $\\bar{{\\bf q}}$,\nwhich is independent of the random permutations $\\sigma^{(i)}$,\nas we argued above.\nTherefore, assuming that $D({\\bf q}) \\leq \\delta$ for each query,\nthe algorithm will always follow the same path\nof computation and issue the same sequence of queries $\\cal S$.\n(Note that this is just a fixed sequence which can be written down\nbefore we started running the algorithm.)\nAssume that $|{\\cal S}| = e^{o(n)}$, i.e.,~the number of queries is subexponential in $n$.\nBy (\\ref{eq:D(q)}), using a union bound over all $Q \\in {\\cal S}$,\nit happens with probability $1 - e^{-\\Omega(n)}$ that ${D}({\\bf q}) = ||{\\bf q}-\\bar{{\\bf q}}||^2 \\leq \\delta$ for all $Q \\in {\\cal S}$.\n(Note that $\\delta$ and $|X|$ are constants here.)\nThen, the algorithm indeed follows this path of computation and gives the same answer.\nIn particular, the answer does not depend on whether the instance is $\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}}\\}$\nor $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}}\\}$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:general-hardness}]\nFix an $\\epsilon > 0$.\nGiven an instance $\\max \\{f(S): S \\in {\\cal F}\\}$\nstrongly symmetric under ${\\cal G}$, let $\\hat{F}, \\hat{G}: [0,1]^X \\rightarrow {\\boldmath R}$\nbe the two functions provided by Lemma~\\ref{lemma:final-fix}.\nWe choose a large number $n$ and consider a refinement $\\tilde{{\\cal F}}$\non the ground set $N \\times X$, where $N = [n]$.\nWe define discrete instances of submodular maximization\n$\\max \\{ \\hat{f}(S): S \\in \\tilde{{\\cal F}} \\}$ and $\\max \\{ \\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$.\nAs in Lemma~\\ref{lemma:indistinguish}, for each $i \\in N$\nwe choose a random permutation $\\sigma^{(i)} \\in {\\cal G}$.\nThis can be viewed as a random shuffle of the labeling of the ground set\nbefore we present it to an algorithm.\nFor every set $S \\subseteq N \\times X$, we define a vector $\\xi(S) \\in [0,1]^X$ by\n$$ \\xi_j(S) = \\frac{1}{n} \\left|\\{i \\in N: (i,\\sigma^{(i)}(j)) \\in S \\}\\right|.$$\nIn other words, $\\xi_j(S)$ measures the fraction of copies of element $j$\ncontained in $S$; however, for each $i$ the $i$-copies of all elements\nare shuffled by $\\sigma^{(i)}$. Next, we define\n$$ \\hat{f}(S) = \\hat{F}(\\xi(S)), \\ \\ \\ \\ \\ \\hat{g}(S) = \\hat{G}(\\xi(S)).$$\nWe claim that $\\hat{f}$ and $\\hat{g}$ are submodular (for any fixed $\\xi$).\nNote that the effect\nof $\\sigma^{(i)}$ is just a renaming (or shuffling) of the elements\nof $N \\times X$, and hence for the purpose of proving submodularity we can assume\nthat $\\sigma^{(i)}$ is the identity for all $i$. Then, $\\xi_j(S) = \\frac{1}{n}\n |S \\cap (N \\times \\{j\\})|$. Due to Lemma~\\ref{lemma:smooth-submodular},\nthe property $\\mixdiff{\\hat{F}}{x_i}{x_j} \\leq 0$ (almost everywhere) implies that $\\hat{f}$ is submodular.\nIn addition, if the original instance was monotone, then $\\partdiff{\\hat{F}}{x_j} \\geq 0$\nand $\\hat{f}$ is monotone. The same holds for $\\hat{g}$.\n\nThe value of $\\hat{g}(S)$ for any feasible solution $S \\in \\tilde{{\\cal F}}$\nis bounded by $\\hat{g}(S) = \\hat{G}(\\xi(S)) = \\hat{F}(\\overline{\\xi(S)})\n\\leq \\overline{OPT} + \\epsilon$.\nOn the other hand, let ${\\bf x}^*$ denote a point where the optimum of the continuous problem\n$\\max \\{\\hat{F}({\\bf x}): {\\bf x} \\in P({{\\cal F}}) \\}$ is attained, i.e. $\\hat{F}({\\bf x}^*) \\geq OPT - \\epsilon$.\nFor a large enough $n$, we can approximate the point ${\\bf x}^*$ arbitrarily closely\nby a rational vector with $n$ in the denominator,\nwhich corresponds to a discrete solution $S^* \\in \\tilde{{\\cal F}}$ whose\nvalue $\\hat{f}(S^*)$ is at least, say, $OPT - 2 \\epsilon$.\nHence, the ratio between the optima of the {\\em discrete} optimization\nproblems $\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}}\\}$ and $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$\ncan be made at most $\\frac{\\overline{OPT} + \\epsilon}{OPT - 2 \\epsilon}$, i.e. arbitrarily close\nto the symmetry gap $\\gamma = \\frac{\\overline{OPT}}{OPT}$.\n\nBy Lemma~\\ref{lemma:indistinguish}, distinguishing the two instances\n$\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}} \\}$  and $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$,\neven by a randomized algorithm, requires an exponential number of value queries.\nTherefore, we cannot estimate the optimum within a factor better than $\\gamma$.\n\\end{proof}\n\nNext, we prove Theorem~\\ref{thm:multilinear-hardness} (again assuming Lemma~\\ref{lemma:final-fix}),\ni.e.~an analogous hardness result for solving the multilinear optimization problem.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:multilinear-hardness}]\nGiven a symmetric instance $\\max \\{f(S): S \\in {\\cal F}\\}$ and $\\epsilon>0$, we construct\nrefined and modified instances $\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}} \\}$,\n $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\}$,\nderived from the continuous functions $\\hat{F}, \\hat{G}$ provided by Lemma~\\ref{lemma:final-fix},\nexactly as we did in the proof of Theorem~\\ref{thm:general-hardness}.\nLemma~\\ref{lemma:indistinguish} states that these two instances cannot be distinguished using a subexponential number of value queries. Furthermore, the gap between the two modified instances corresponds to the symmetry gap $\\gamma$ of the original instance: $\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}} \\} \\geq OPT - 2 \\epsilon$ and $\\max \\{\\hat{g}(S): S \\in \\tilde{{\\cal F}} \\} \\leq \\overline{OPT} + \\epsilon = \\gamma OPT + \\epsilon$.\n\nNow we consider the multilinear relaxations of the two refined instances, $\\max \\{\\check{F}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}})\\}$\nand $\\max \\{\\check{G}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}})\\}$. Note that $\\check{F}, \\check{G}$, (although related to $\\hat{F},\n\\hat{G}$) are not exactly the same as the functions $\\hat{F}, \\hat{G}$; in particular, they are defined\non a larger (refined) domain. However, we show that the gap between the optima of the two instances remains\nthe same.\n\nFirst, the value of $\\max \\{\\check{F}({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$ is at least the optimum of the discrete problem,\n$\\max \\{\\hat{f}(S): S \\in \\tilde{{\\cal F}} \\}$, which is at least $OPT - 2\\epsilon$ as in the proof of\nTheorem~\\ref{thm:general-hardness}. The value of $\\max \\{\\check{G}({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$ can be analyzed as follows. For any fractional solution ${\\bf x} \\in P({\\cal F})$, the value of $\\check{G}({\\bf x})$\nis the expectation ${\\bf E}[\\hat{g}(\\hat{{\\bf x}})]$, where $\\hat{{\\bf x}}$ is obtained by independently rounding the\ncoordinates of ${\\bf x}$ to $\\{0,1\\}$. Recall that $\\hat{g}$ is obtained by discretizing the continuous function\n$\\hat{G}$ (using Lemma~\\ref{lemma:smooth-submodular}). In particular, $\\hat{g}(S) = \\hat{G}(\\tilde{{\\bf x}})$ where\n$\\tilde{x}_i = \\frac{1}{n} |S \\cap (N \\times \\{i\\})|$ is the fraction of the respective cluster contained in $S$, and $|N| = n$ is the size of each cluster (the refinement parameter). If ${\\bf 1}_S = \\hat{{\\bf x}}$, i.e. $S$ is chosen by independent sampling with probabilities according to ${\\bf x}$, then for large $n$ the fractions $\\frac{1}{n}|S \\cap (N \\times \\{i\\})|$ will be strongly concentrated around their expectation. As $\\hat{G}$ is continuous, we get\n$\\lim_{n \\rightarrow \\infty} {\\bf E}[\\hat{g}(\\hat{{\\bf x}})] = \\lim_{n \\rightarrow \\infty} {\\bf E}[\\hat{G}(\\tilde{{\\bf x}})]\n = \\hat{G}({\\bf E}[\\tilde{{\\bf x}}]) = \\hat{G}(\\bar{{\\bf x}})$.\nHere, $\\bar{{\\bf x}}$ is the vector ${\\bf x}$ projected back to the original ground set $X$, where the coordinates\nof each cluster have been averaged. By construction of the refinement, if ${\\bf x} \\in P(\\tilde{{\\cal F}})$ then\n$\\bar{{\\bf x}}$ is in the polytope corresponding to the original instance, $P({\\cal F})$.\nTherefore, $\\hat{G}(\\bar{{\\bf x}}) \\leq\n\\max \\{\\hat{G}({\\bf x}): {\\bf x} \\in P({\\cal F}) \\} \\leq \\gamma OPT + \\epsilon$. For large enough $n$, this means that\n$\\max \\{\\check{G}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}}) \\} \\leq \\gamma OPT + 2 \\epsilon$. This holds for\nan arbitrarily small fixed $\\epsilon>0$, and hence the gap between the instances \n$\\max \\{\\check{F}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}})\\}$ and $\\max \\{\\check{G}({\\bf x}): {\\bf x} \\in P(\\tilde{{\\cal F}})\\}$\n(which cannot be distinguished) can be made arbitrarily close to $\\gamma$.\n\\end{proof}\n\n\n\\paragraph{Hardness for symmetric instances}\nWe remark that since Lemma~\\ref{lemma:final-fix} provides functions $\\hat{F}$ and $\\hat{G}$ symmetric\nunder ${\\cal G}$, the refined instances that we define are invariant with respect to\nthe following symmetries: permute the copies of each element in an arbitrary\nway, and permute the classes of copies according to any permutation\n$\\sigma \\in {\\cal G}$. This means that our hardness results also hold\nfor instances satisfying such symmetry properties.\n\n\\\n\nIt remains to prove Lemma~\\ref{lemma:final-fix}.\nBefore we move to the final construction of $\\hat{F}({\\bf x})$ and $\\hat{G}({\\bf x})$,\nwe construct as an intermediate step a function $\\tilde{F}({\\bf x})$ which is helpful\nin the analysis.\n\n\\\n\n\\paragraph{Construction}\nLet us construct a function $\\tilde{F}({\\bf x})$ which satisfies the following:\n\\begin{itemize}\n\\item For ${\\bf x}$ ``sufficiently close'' to $\\bar{{\\bf x}}$, $\\tilde{F}({\\bf x}) = G({\\bf x})$.\n\\item For ${\\bf x}$ ``sufficiently far away'' from $\\bar{{\\bf x}}$, $\\tilde{F}({\\bf x}) \\simeq F({\\bf x})$.\n\\item The function $\\tilde{F}({\\bf x})$ is ``approximately\" smooth submodular.\n\\end{itemize}\nOnce we have $\\tilde{F}({\\bf x})$, we can fix it to obtain a smooth submodular\nfunction $\\hat{F}({\\bf x})$, which is still close to the original function $F({\\bf x})$.\nWe also fix $G({\\bf x})$ in the same way, to obtain a function $\\hat{G}({\\bf x})$\nwhich is equal to $\\hat{F}({\\bf x})$ whenever ${\\bf x}$ is close to $\\bar{{\\bf x}}$. We defer\nthis step until the end.\n\nWe define $\\tilde{F}({\\bf x})$ as a convex linear combination\nof $F({\\bf x})$ and $G({\\bf x})$, guided by a ``smooth transition'' function, depending\non the distance of ${\\bf x}$ from $\\bar{{\\bf x}}$. The form that we use is the following:\\footnote{We remark\nthat a construction analogous to \\cite{MSV08}\nwould be $\\tilde{F}({\\bf x}) = F({\\bf x}) - \\phi(H({\\bf x}))$ where $H({\\bf x}) = F({\\bf x}) - G({\\bf x})$.\nWhile this makes the analysis easier in \\cite{MSV08}, it cannot be used in general.\nRoughly speaking, the problem is that in general the partial derivatives of $H({\\bf x})$\nare not bounded in any way by the value of $H({\\bf x})$.}\n$$ \\tilde{F}({\\bf x}) = (1-\\phi(D({\\bf x}))) F({\\bf x}) + \\phi(D({\\bf x})) G({\\bf x}) $$\nwhere $\\phi:{\\boldmath R}_+ \\rightarrow [0,1]$ is a suitable smooth function,\nand\n$$ D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 = \\sum_i (x_i - \\bar{x}_i)^2.$$\nThe idea is that when ${\\bf x}$ is close to $\\bar{{\\bf x}}$, $\\phi(D({\\bf x}))$ should be close to $1$,\ni.e. the convex linear combination should give most of the weight to $G({\\bf x})$.\nThe weight should shift gradually to $F({\\bf x})$ as ${\\bf x}$ gets further away from $\\bar{{\\bf x}}$.\nTherefore, we define $\\phi(t) = 1$ in a small interval $t \\in [0,\\delta]$,\nand $\\phi(t)$ tends to $0$ as $t$ increases.\nThis guarantees that $\\tilde{F}({\\bf x}) = G({\\bf x})$ whenever\n$D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 \\leq \\delta$.\nWe defer the precise construction of $\\phi(t)$ to Lemma~\\ref{lemma:phi-construction},\nafter we determine what properties we need from $\\phi(t)$.\nNote that regardless of the definition of $\\phi(t)$, $\\tilde{F}({\\bf x})$\nis symmetric with respect to ${\\cal G}$, since $F({\\bf x}), G({\\bf x})$ and $D({\\bf x})$ are.\n\n\\\n\n\\paragraph{Analysis of the construction}\nDue to the construction of $\\tilde{F}({\\bf x})$, it is clear that when\n$D({\\bf x}) = ||{\\bf x}-\\bar{{\\bf x}}||^2$ is small, $\\tilde{F}({\\bf x}) = G({\\bf x})$.\nWhen $D({\\bf x})$ is large, $\\tilde{F}({\\bf x}) \\simeq F({\\bf x})$.\nThe main issue, however, is whether we can say something about the first\nand second partial derivatives of $\\tilde{F}$. This is crucial for\nthe properties of monotonicity and submodularity, which we would like to preserve.\nLet us write $\\tilde{F}({\\bf x})$ as\n$$ \\tilde{F}({\\bf x}) = F({\\bf x}) - \\phi(D({\\bf x})) H({\\bf x}) $$\nwhere $H({\\bf x}) = F({\\bf x}) - G({\\bf x})$. By differentiating once, we get\n\\begin{equation}\n\\label{eq:partdiff}\n\\partdiff{\\tilde{F}}{x_i} = \\partdiff{F}{x_i} - \\phi(D({\\bf x})) \\partdiff{H}{x_i}\n - \\phi'(D({\\bf x})) \\partdiff{D}{x_i} H({\\bf x})\n\\end{equation}\nand by differentiating twice,\n\\begin{eqnarray}\n\\label{eq:mixdiff}\n \\mixdiff{\\tilde{F}}{x_i}{x_j} & = & \\mixdiff{F}{x_i}{x_j}\n- \\phi({D}({\\bf x})) \\mixdiff{H}{x_i}{x_j}\n - \\phi''({D}({\\bf x})) \\partdiff{{D}}{x_i} \\partdiff{{D}}{x_j} H({\\bf x}) \\\\\n & & - \\phi'({D}({\\bf x})) \\left( \\partdiff{{D}}{x_j} \\partdiff{H}{x_i}\n  + \\mixdiff{{D}}{x_i}{x_j} H({\\bf x}) + \\partdiff{{D}}{x_i} \\partdiff{H}{x_j} \\right) \\nonumber.\n\\end{eqnarray}\n\nThe first two terms on the right-hand sides of (\\ref{eq:partdiff}) and\n(\\ref{eq:mixdiff}) are not bothering us, because they form\nconvex linear combinations of the derivatives of $F({\\bf x})$ and $G({\\bf x})$,\nwhich have the properties that we need. The remaining terms might cause\nproblems, however, and we need to estimate them.\n\nOur strategy is to define $\\phi(t)$ in such a way that it eliminates\nthe influence of partial derivatives of $D$ and $H$ where they become too large.\nRoughly speaking, $D$ and $H$ have negligible partial derivatives when ${\\bf x}$\nis very close to $\\bar{{\\bf x}}$. As ${\\bf x}$ moves away from $\\bar{{\\bf x}}$, the partial\nderivatives grow but then the behavior of $\\phi(t)$ must be such that\ntheir influence is supressed.\n\nWe start with the following important claim.\\footnote{\nWe remind the reader that $\\nabla F$, the gradient of $F$, is a vector\nwhose coordinates are the first partial derivatives $\\partdiff{F}{x_i}$.\nWe denote by $\\nabla F |_{{\\bf x}}$ the gradient evaluated at ${\\bf x}$.}\n\n\\begin{lemma}\n\\label{lemma:grad-symmetry}\nAssume that $F:[0,1]^X \\rightarrow {\\boldmath R}$ is differentiable and invariant under a group of\npermutations of coordinates ${\\cal G}$. Let $G({\\bf x}) = F(\\bar{{\\bf x}})$,\nwhere $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$.\nThen for any ${\\bf x} \\in [0,1]^X$,\n $$ \\nabla{G}|_{\\bf x} = \\nabla{F}|_{\\bar{{\\bf x}}}.$$\n\\end{lemma}\n\n\\begin{proof}\nTo avoid confusion, we use ${\\bf x}$ for the arguments of the functions $F$ and $G$,\nand ${\\bf u}$, $\\bar{{\\bf u}}$, etc. for points where their partial derivatives are evaluated.\nTo rephrase, we want to prove that for any point ${\\bf u}$ and any coordinate $i$,\nthe partial derivatives of $F$ and $G$ evaluated at $\\bar{u}$ are equal:\n$\\partdiff{G}{x_i} \\Big|_{{\\bf x}={\\bf u}} = \\partdiff{F}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf u}}}$.\n\nFirst, consider $F({\\bf x})$. We assume that $F({\\bf x})$ is invariant under\na group of permutations of coordinates $\\cal G$,\ni.e. $F({\\bf x}) = F(\\sigma({\\bf x}))$ for any $\\sigma \\in {\\cal G}$.\nDifferentiating both sides at ${\\bf x}={\\bf u}$, we get by the chain rule:\n$$ \\partdiff{F}{x_i} \\Big|_{{\\bf x}={\\bf u}} =\n \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\sigma({\\bf u})} \\partdiff{}{x_i} (\\sigma({\\bf x}))_j\n = \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\sigma({\\bf u})} \\partdiff{x_{\\sigma(j)}}{x_i}. $$\nHere, $\\partdiff{x_{\\sigma(j)}}{x_i} = 1$ if $\\sigma(j) = i$, and $0$ otherwise.\nTherefore,\n$$ \\partdiff{F}{x_i} \\Big|_{{\\bf x}={\\bf u}} =\n \\partdiff{F}{x_{\\sigma^{-1}(i)}} \\Big|_{{\\bf x}=\\sigma({\\bf u})}. $$\nNow, if we evaluate the left-hand side at $\\bar{{\\bf u}}$, the right-hand side is evaluated\nat $\\sigma(\\bar{{\\bf u}}) = \\bar{{\\bf u}}$, and hence for any $i$ and any $\\sigma \\in {\\cal G}$,\n\\begin{equation}\n\\label{eq:F-inv}\n\\partdiff{F}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf u}}} = \\partdiff{F}{x_{\\sigma^{-1}(i)}} \\Big|_{{\\bf x}=\\bar{{\\bf u}}}.\n\\end{equation}\nTurning to $G({\\bf x}) = F(\\bar{{\\bf x}})$, let us write $\\partdiff{G}{x_i}$\nusing the chain rule:\n$$ \\partdiff{G}{x_i} \\Big|_{{\\bf x}={\\bf u}} = \\partdiff{}{x_i} F(\\bar{{\\bf x}}) \\Big|_{{\\bf x}={\\bf u}}\n = \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\bar{{\\bf u}}} \\cdot \\partdiff{\\bar{x}_j}{x_i}. $$\nWe have $\\bar{x}_j = {\\bf E}_{\\sigma \\in {\\cal G}}[{\\bf x}_{\\sigma(j)}]$, and\nso\n$$ \\partdiff{G}{x_i} \\Big|_{{\\bf x}={\\bf u}} = \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\bar{{\\bf u}}}\n \\cdot \\partdiff{}{x_i} {\\bf E}_{\\sigma \\in {\\cal G}}[x_{\\sigma(j)}] \n = {\\bf E}_{\\sigma \\in {\\cal G}} \\left[\\sum_j \\partdiff{F}{x_j} \\Big|_{x=\\bar{u}}\n \\cdot \\partdiff{x_{\\sigma(j)}}{x_i} \\right]. $$\nAgain, $\\partdiff{x_{\\sigma(j)}}{x_i} = 1$ if $\\sigma(j) = i$ and $0$ otherwise.\nConsequently, we obtain\n$$ \\partdiff{G}{x_i} \\Big|_{{\\bf x}={\\bf u}} = {\\bf E}_{\\sigma \\in {\\cal G}}\n \\left[ \\partdiff{F}{x_{\\sigma^{-1}(i)}} \\Big|_{{\\bf x}=\\bar{{\\bf u}}} \\right]\n = \\partdiff{F}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf u}}} $$\nwhere we used Eq. (\\ref{eq:F-inv}) to remove the dependence on $\\sigma \\in {\\cal G}$.\n\\end{proof}\n\n\nObserve that the symmetrization operation $\\bar{{\\bf x}}$ is idempotent, i.e.\n$\\bar{\\bar{{\\bf x}}} = \\bar{{\\bf x}}$.\nBecause of this, we also get $\\nabla{G}|_{\\bar{{\\bf x}}} = \\nabla{F}|_{\\bar{{\\bf x}}}$.\nNote that $G(\\bar{{\\bf x}}) = F(\\bar{{\\bf x}})$ follows from the definition,\nbut it is not obvious that the same holds for gradients, since their\ndefinition involves points where $G({\\bf x}) \\neq F({\\bf x})$. For second partial\nderivatives, the equality no longer holds, as can be seen from\na simple example such as $F(x_1,x_2) = 1 - (1-x_1)(1-x_2)$,\n$G(x_1,x_2) = 1 - (1-\\frac{x_1+x_2}{2})^2$.\n\nNext, we show that the functions $F({\\bf x})$ and $G({\\bf x})$ are very similar\nin the close vicinity of the region where $\\bar{{\\bf x}} = {\\bf x}$. Recall our definitions:\n$H({\\bf x}) = F({\\bf x}) - G({\\bf x})$, $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2$.\nBased on Lemma~\\ref{lemma:grad-symmetry}, we know that $H(\\bar{{\\bf x}}) = 0$\nand $\\nabla H|_{\\bar{{\\bf x}}} = 0$. In the following lemmas,\nwe present bounds on $H({\\bf x})$, $D({\\bf x})$ and their partial derivatives.\n\n\n\\begin{lemma}\n\\label{lemma:H-bounds}\nLet $f:2^X \\rightarrow [0,M]$ be invariant under a permutation group $\\cal G$.\nLet $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$,\n$D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2$ and $H({\\bf x}) = F({\\bf x}) - G({\\bf x})$ where $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$\nand $G({\\bf x}) = F(\\bar{{\\bf x}})$. Then\n\\begin{enumerate}\n\\item  $ |\\mixdiff{H}{x_i}{x_j}| \\leq 8M $ everywhere, for all $i,j$;\n\\item  $ ||\\nabla H({\\bf x})|| \\leq 8M|X| \\sqrt{D({\\bf x})}$;\n\\item  $ |H({\\bf x})| \\leq  8M|X| \\cdot D({\\bf x}). $\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst, let us get a bound on the second partial derivatives.\nAssuming without loss of generality $x_i=x_j=0$, we have\\footnote{${\\bf x} \\vee {\\bf y}$\ndenotes the coordinate-wise maximum, $({\\bf x} \\vee {\\bf y})_i = \\max \\{x_i,y_i\\}$ and\n${\\bf x} \\wedge {\\bf y}$ denotes the coordinate-wise minimum, $({\\bf x} \\wedge {\\bf y})_i = \\min \\{x_i,y_i\\}$.}\n$$ \\mixdiff{F}{x_i}{x_j} =  {\\bf E}[f(\\hat{{\\bf x}} \\vee ({\\bf e}_i+{\\bf e}_j)) -\n f(\\hat{{\\bf x}} \\vee {\\bf e}_i) - f(\\hat{{\\bf x}} \\vee {\\bf e}_j) + f(\\hat{{\\bf x}})] $$\n(see \\cite{Vondrak08}). Consequently,\n$$ \\Big| \\mixdiff{F}{x_i}{x_j} \\Big| \\leq 4 \\max |f(S)| = 4 M.$$\nIt is a little bit more involved to analyze $\\mixdiff{G}{x_i}{x_j}$.\nSince $G({\\bf x}) = F(\\bar{{\\bf x}})$ and $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$,\nwe get by the chain rule:\n$$ \\mixdiff{G}{x_i}{x_j} = \\sum_{k,\\ell} \\mixdiff{F}{x_k}{x_\\ell}\n \\partdiff{\\bar{x}_k}{x_i} \\partdiff{\\bar{x}_\\ell}{x_j} \n = {\\bf E}_{\\sigma,\\tau \\in {\\cal G}} \\left[ \\sum_{k,\\ell} \\mixdiff{F}{x_k}{x_\\ell}\n   \\partdiff{x_{\\sigma(k)}}{x_i} \\partdiff{x_{\\tau(\\ell)}}{x_j} \\right].$$\nIt is useful here to use the Kronecker symbol, $\\delta_{i,j}$,\nwhich is $1$ if $i=j$ and $0$ otherwise. Note that\n$\\partdiff{x_{\\sigma(k)}}{x_i} = \\delta_{i,\\sigma(k)} = \\delta_{\\sigma^{-1}(i),k}$,\netc. Using this notation, we get\n$$ \\mixdiff{G}{x_i}{x_j} = {\\bf E}_{\\sigma,\\tau \\in {\\cal G}} \\left[  \\sum_{k,\\ell}\n\\mixdiff{F}{x_k}{x_\\ell} \\delta_{\\sigma^{-1}(i),k} \\delta_{\\sigma^{-1}(j),\\ell} \\right] \n = {\\bf E}_{\\sigma,\\tau \\in {\\cal G}}\n \\left[ \\mixdiff{F}{x_{\\sigma^{-1}(i)}}{x_{\\tau^{-1}(j)}} \\right], $$\n$$ \\Big| \\mixdiff{G}{x_i}{x_j} \\Big| \\leq {\\bf E}_{\\sigma,\\tau \\in {\\cal G}}\n\\left[ \\Big| \\mixdiff{F}{x_{\\sigma^{-1}(i)}}{x_{\\tau^{-1}(j)}} \\Big| \\right]\n \\leq 4 M $$\nand therefore\n$$ \\Big| \\mixdiff{H}{x_i}{x_j} \\Big|\n = \\Big| \\mixdiff{F}{x_i}{x_j} - \\mixdiff{G}{x_i}{x_j} \\Big|\n \\leq 8 M.$$\n\nNext, we estimate $\\partdiff{H}{x_i}$ at a given point ${\\bf u}$, depending\non its distance from $\\bar{{\\bf u}}$. Consider\nthe line segment between $\\bar{{\\bf u}}$ and ${\\bf u}$. The function $H({\\bf x}) = F({\\bf x}) - G({\\bf x})$\nis $C_\\infty$-differentiable, and hence we can apply the mean value theorem\nto $\\partdiff{H}{x_i}$: There exists a point $\\tilde{{\\bf u}}$ on the line segment\n$[\\bar{{\\bf u}}, {\\bf u}]$ such that\n$$ \\partdiff{H}{x_i} \\Big|_{{\\bf x}={\\bf u}} - \\partdiff{H}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf u}}}\n = \\sum_j \\mixdiff{H}{x_j}{x_i} \\Big|_{{\\bf x}=\\tilde{{\\bf u}}} (u_j-\\bar{u}_j).$$\nRecall that $\\partdiff{H}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf u}}} = 0$.\nApplying the Cauchy-Schwartz inequality to the right-hand side, we get\n$$ \\left( \\partdiff{H}{x_i} \\Big|_{{\\bf x}={\\bf u}} \\right)^2\n \\leq \\sum_j \\left( \\mixdiff{H}{x_j}{x_i} \\Big|_{{\\bf x}=\\tilde{{\\bf u}}} \\right)^2\n  || {\\bf u} - \\bar{{\\bf u}} ||^2  \\leq (8M)^2 |X| ||{\\bf u} - \\bar{{\\bf u}}||^2.$$\nAdding up over all $i \\in X$, we obtain\n$$ || \\nabla H({\\bf u}) ||^2 = \\sum_{i}\\left( \\partdiff{H}{x_i} \\Big|_{{\\bf x}={\\bf u}} \\right)^2\n \\leq (8M|X|)^2 ||{\\bf u} - \\bar{{\\bf u}}||^2.$$\nFinally, we estimate the growth of $H({\\bf u})$. Again, by the mean value theorem,\nthere is a point $\\tilde{{\\bf u}}$ on the line segment $[\\bar{{\\bf u}},{\\bf u}]$, such that\n$$ H({\\bf u}) - H(\\bar{{\\bf u}}) = ({\\bf u} - \\bar{{\\bf u}}) \\cdot \\nabla H(\\tilde{{\\bf u}}).$$\nUsing $H(\\bar{{\\bf u}}) = 0$, the Cauchy-Schwartz inequality and the above bound\non $\\nabla H$,\n$$ (H({\\bf u}))^2 \\leq ||\\nabla H({\\tilde{{\\bf u}}})||^2  ||{\\bf u} - \\bar{{\\bf u}}||^2\n \\leq (8M|X|)^2 ||\\tilde{{\\bf u}}-\\bar{{\\bf u}}||^2 ||{\\bf u}-\\bar{{\\bf u}}||^2. $$\nClearly, $||\\tilde{{\\bf u}} - \\bar{{\\bf u}}|| \\leq ||{\\bf u} - \\bar{{\\bf u}}||$, and therefore\n$$ |H({\\bf u})| \\leq 8M|X| \\cdot ||{\\bf u}-\\bar{{\\bf u}}||^2.$$\n\\end{proof}\n\n\\begin{lemma}\n\\label{lemma:D-bounds}\nFor the function $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2$, we have\n\\begin{enumerate}\n\\item $\\nabla D = 2({\\bf x} - \\bar{{\\bf x}})$, and therefore\n $||\\nabla D|| = 2 \\sqrt{D({\\bf x})}$.\n\\item For all $i,j$, $|\\mixdiff{D}{x_i}{x_j}| \\leq 2$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet us write $D({\\bf x})$ as\n$$ D({\\bf x}) = \\sum_i (x_i - \\bar{x}_i)^2 = \\sum_i {\\bf E}_{\\sigma \\in {\\cal G}}[x_i - x_{\\sigma(i)}]\n {\\bf E}_{\\tau \\in {\\cal G}}[x_i - x_{\\tau(i)}]. $$\nTaking the first partial derivative,\n$$ \\partdiff{D}{x_j} = 2 \\sum_i {\\bf E}_{\\sigma \\in {\\cal G}}[x_i - x_{\\sigma(i)}]\n \\partdiff{}{x_j} {\\bf E}_{\\tau \\in {\\cal G}}[x_i - x_{\\tau(i)}].$$\nAs before, we have $\\partdiff{x_i}{x_j} = \\delta_{ij}$.\nUsing this notation, we get\n\\begin{eqnarray*}\n\\partdiff{D}{x_j} & = & 2 \\sum_i {\\bf E}_{\\sigma \\in {\\cal G}}[x_i - x_{\\sigma(i)}]\n {\\bf E}_{\\tau \\in {\\cal G}}[\\delta_{ij} - \\delta_{\\tau(i),j}] \\\\\n & = & 2 \\sum_i {\\bf E}_{\\sigma,\\tau \\in {\\cal G}}[(x_i - x_{\\sigma(i)})\n    (\\delta_{ij} - \\delta_{i,\\tau^{-1}(j)})] \\\\\n & = & 2 \\ {\\bf E}_{\\sigma,\\tau \\in {\\cal G}}[x_j - x_{\\sigma(j)} - x_{\\tau^{-1}(j)}\n    + x_{\\sigma(\\tau^{-1}(j))}].\n\\end{eqnarray*}\nSince the distributions of $\\sigma(j)$, $\\tau^{-1}(j)$ and $\\sigma(\\tau^{-1}(j))$\nare the same, we obtain\n$$ \\partdiff{D}{x_j} = 2 \\ {\\bf E}_{\\sigma \\in {\\cal G}}[x_j - x_{\\sigma(j)}]\n = 2(x_j - \\bar{x}_j) $$\nand\n$$ ||\\nabla D||^2 = \\sum_j \\Big| \\partdiff{D}{x_j} \\Big|^2\n = 4 \\sum_j (x_j - \\bar{x}_j)^2 = 4 D({\\bf x}).$$\n\nFinally, the second partial derivatives are\n$$ \\mixdiff{D}{x_i}{x_j} = 2 \\partdiff{}{x_i} (x_j - \\bar{x}_j)\n = 2 \\partdiff{}{x_i} {\\bf E}_{\\sigma \\in {\\cal G}}[x_j - x_{\\sigma(j)}]\n = 2 {\\bf E}_{\\sigma \\in {\\cal G}}[\\delta_{ij} - \\delta_{i,\\sigma(j)}] $$\nwhich is clearly bounded by $2$ in the absolute value.\n\\end{proof}\n\nNow we come back to $\\tilde{F}({\\bf x})$ and its partial derivatives.\nRecall equations (\\ref{eq:partdiff}) and (\\ref{eq:mixdiff}).\nThe problematic terms are those involving $\\phi'(D({\\bf x}))$ and $\\phi''(D({\\bf x}))$.\nUsing our bounds on $H({\\bf x})$, $D({\\bf x})$ and their derivatives, however,\nwe notice that $\\phi'(D({\\bf x}))$ always appears with factors on the order\nof $D({\\bf x})$ and $\\phi''(D({\\bf x}))$ appears with factors on the order of $(D({\\bf x}))^2$.\nThus, it is sufficient if $\\phi(t)$ is defined so that we have control\nover $t \\phi'(t)$ and $t^2 \\phi''(t)$. The following lemma describes\nthe function that we need.\n\n\\begin{lemma}\n\\label{lemma:phi-construction}\nFor any $\\alpha, \\beta > 0$, there is $\\delta \\in (0, \\beta)$\nand a function $\\phi:{\\boldmath R}_+ \\rightarrow [0,1]$ with an absolutely continuous first derivative\nsuch that\n\\begin{enumerate}\n\\item For $t \\leq \\delta$, $\\phi(t) = 1$.\n\\item For $t \\geq \\beta$, $\\phi(t) < e^{-1\/\\alpha}$.\n\\item For all $t \\geq 0$, $|t \\phi'(t)| \\leq 4 \\alpha$.\n\\item For almost all $t \\geq 0$, $|t^2 \\phi''(t)| \\leq 10 \\alpha$.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFirst, observe that if we prove the lemma for some particular value $\\beta>0$, we can also prove prove it\nfor any other value $\\tilde{\\beta}>0$, by modifying the function as follows: $\\tilde{\\phi}(t) = \\phi(\\beta t \/ \\beta')$.\nThis corresponds to a scaling of the parameter $t$ by $\\beta' \/ \\beta$. Observe that then $|t \\tilde{\\phi}'(t)| = |\\frac{\\beta}{\\beta'} t \\phi'(\\beta t \/ \\beta')| \\leq 4 \\alpha$ and $|t^2 \\tilde{\\phi}''(t)| = |(\\frac{\\beta}{\\beta'})^2 t^2 \\phi''(\\beta t \/ \\beta')| \\leq 10 \\alpha$, so the conditions are still satisfied.\n\nTherefore, we can assume without\nloss of generality that $\\beta>0$ is a value of our choice,\nfor example $\\beta = e^{1\/(2\\alpha^2)}+1$.\nIf we want to prove the result for a different value of $\\beta$,\nwe can just scale the argument $t$ and the constant $\\delta$\nby $\\beta \/ (e^{1\/(2\\alpha^2)}+1)$; the bounds on\n$t \\phi'(t)$ and $t^2 \\phi''(t)$ still hold.\n\nWe can assume that $\\alpha \\in (0,\\frac18)$ because for larger $\\alpha$,\nthe statement only gets weaker. As we argued, we can assume WLOG that\n$\\beta = e^{1\/(2\\alpha^2)}+1$. We set $\\delta = 1$ and\n$\\delta_2 = 1 + (1+\\alpha)^{-1\/2} \\leq 2$.\n(We remind the reader that in general these values will be scaled depending\non the actual value of $\\beta$.) We define the function as follows:\n\\begin{enumerate}\n\\item $\\phi(t) = 1$ for $t \\in [0,\\delta]$.\n\\item $\\phi(t) = 1 - \\alpha (t-1)^2$ for $t \\in [\\delta, \\delta_2]$.\n\\item $\\phi(t) = (1+\\alpha)^{-1-\\alpha} (t - 1)^{-2 \\alpha}$\n for $t \\in [\\delta_2, \\infty)$.\n\\end{enumerate}\n\n\\noindent\nLet's verify the properties of $\\phi(t)$.\nFor $t \\in [0,\\delta]$, we have\n$\\phi'(t) = \\phi''(t) = 0$. For $t \\in [\\delta,\\delta_2]$, we have\n$$ \\phi'(t) = -2\\alpha \\left(t-1 \\right), \\ \\ \\ \\ \\ \\ \n \\phi''(t) = -2\\alpha, $$\nand for $t \\in [\\delta_2, \\infty)$,\n$$ \\phi'(t) = -{2\\alpha}{(1+\\alpha)^{-1-\\alpha}} \\left( t-1 \\right)^{-2\\alpha-1}, $$\n$$ \\phi''(t) = {2\\alpha(1+2\\alpha)}{(1+\\alpha)^{-1-\\alpha}} \\left( t-1 \\right)^{-2\\alpha-2}. $$\nFirst, we check that the values and first derivatives agree at the breakpoints.\nFor $t = \\delta = 1$, we get $\\phi(1) = 1$ and $\\phi'(1) = 0$.\nFor $t = \\delta_2 = 1 + (1+\\alpha)^{-1\/2}$, we get\n$\\phi(\\delta_2) = (1+\\alpha)^{-1}$ and\n$\\phi'(\\delta_2) = -2 \\alpha (1+\\alpha)^{-1\/2}$.\nNext, we need to check is that $\\phi(t)$ is very small\nfor $t \\geq \\beta$. The function is decreasing for $t > \\beta$,\ntherefore it is enough to check $t = \\beta = e^{1\/(2\\alpha^2)}+1$:\n$$ \\phi\\left(\\beta \\right) = (1+\\alpha)^{-1-\\alpha} (\\beta - 1)^{-2\\alpha}\n  \\leq (\\beta-1)^{-2\\alpha} = e^{-1\/\\alpha}. $$\nThe derivative bounds are satisfied trivially for $t \\in [0,\\delta]$.\nFor $t \\in [\\delta, \\delta_2]$, using $t \\leq \\delta_2 = 1 + (1+\\alpha)^{-1\/2}$,\n$$ |t \\phi'(t)| = t \\cdot 2 \\alpha (t-1)\n \\leq 2 \\alpha (1 + (1+\\alpha)^{-1\/2}) (1+\\alpha)^{-1\/2} \\leq 4 \\alpha $$\nand using $\\alpha \\in (0,\\frac18)$,\n$$ |t^2 \\phi''(t)| = t^2 \\cdot 2 \\alpha \\leq 2 \\alpha (1 + (1+\\alpha)^{-1\/2})^2\n \\leq 8 \\alpha.$$\nFor $t \\in [\\delta_2,\\infty)$, using $t-1 \\geq (1+\\alpha)^{-1\/2}$,\n\\begin{eqnarray*}\n|t \\phi'(t)| & = & t \\cdot \\frac{2 \\alpha}{(1+\\alpha)^{1+\\alpha}} \\left(t-1 \\right)^{-2\\alpha-1} \n = \\frac{2 \\alpha}{1+\\alpha} \\left( (1 + \\alpha)\n\\left( t-1 \\right)^2 \\right)^{-\\alpha} \\frac{t}{t-1} \\\\\n& \\leq & \\frac{2 \\alpha}{1 + \\alpha} \\cdot \\frac{t}{t-1}\n \\leq \\frac{2 \\alpha}{1+\\alpha} \\cdot \\frac{1 + (1+\\alpha)^{-1\/2}}{(1+\\alpha)^{-1\/2}}\n = 2 \\alpha \\cdot \\frac{1 + (1+\\alpha)^{-1\/2}}{(1+\\alpha)^{1\/2}} \\leq 4 \\alpha\n\\end{eqnarray*}\nand finally, using $\\alpha \\in (0,\\frac18)$,\n\\begin{eqnarray*}\n |t^2 \\phi''(t)| & = & t^2 \\cdot \\frac{2\\alpha(1+2\\alpha)}{(1+\\alpha)^{1+\\alpha}}\n \\left( t-1 \\right)^{-2\\alpha-2} \n = \\frac{2\\alpha(1+2\\alpha)}{1+\\alpha}\n \\left( (1 + \\alpha) \\left( t-1 \\right)^2 \\right)^{-\\alpha}\n \\left( \\frac{t}{t-1} \\right)^2 \\\\\n& \\leq & \\frac{2\\alpha(1+2\\alpha)}{1+\\alpha} \\cdot\n \\left( \\frac{1 + (1+\\alpha)^{-1\/2}}{(1+\\alpha)^{-1\/2}} \\right)^2\n \\leq 8 \\alpha (1 + 2 \\alpha) \\leq 10 \\alpha.\n\\end{eqnarray*}\n\\end{proof}\n\nUsing Lemmas~\\ref{lemma:H-bounds}, \\ref{lemma:D-bounds} and \\ref{lemma:phi-construction},\nwe now prove bounds on the derivatives of $\\tilde{F}({\\bf x})$.\n\n\\begin{lemma}\n\\label{lemma:F-bounds}\nLet $ \\tilde{F}({\\bf x}) = (1-\\phi(D({\\bf x}))) F({\\bf x}) + \\phi(D({\\bf x})) G({\\bf x}) $\nwhere $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$, $f:2^X \\rightarrow [0,M]$,\n$G({\\bf x}) = F(\\bar{{\\bf x}})$, $D({\\bf x})=||{\\bf x}-\\bar{{\\bf x}}||^2$\nare as above and $\\phi(t)$ is provided by Lemma~\\ref{lemma:phi-construction}\nfor a given $\\alpha>0$.\nThen, whenever $\\mixdiff{F}{x_i}{x_j} \\leq 0$,\n$$\n \\mixdiff{\\tilde{F}}{x_i}{x_j}\n  \\leq  512 M |X| \\alpha. $$\nIf, in addition, $\\partdiff{F}{x_i} \\geq 0$, then\n$$\n\\partdiff{\\tilde{F}}{x_i}\n  \\geq  -64 M |X| \\alpha. $$\n\\end{lemma}\n\n\\begin{proof}\nWe have\n$ \\tilde{F}({\\bf x}) = F({\\bf x}) - \\phi(D({\\bf x})) H({\\bf x}) $\nwhere $H({\\bf x}) = F({\\bf x}) - G({\\bf x})$. By differentiating once, we get\n$$ \\partdiff{\\tilde{F}}{x_i} = \\partdiff{F}{x_i}\n - \\phi(D({\\bf x})) \\partdiff{H}{x_i} - \\phi'(D({\\bf x})) \\partdiff{D}{x_i} H({\\bf x}), $$\ni.e.\n$$ \\Big| \\partdiff{\\tilde{F}}{x_i} -\n \\left(\\partdiff{F}{x_i} - \\phi(D({\\bf x})) \\partdiff{H}{x_i} \\right) \\Big|\n = \\Big| \\phi'(D({\\bf x})) \\partdiff{D}{x_i} H({\\bf x}) \\Big|.$$\nBy Lemma~\\ref{lemma:H-bounds} and \\ref{lemma:D-bounds},\nwe have $|\\partdiff{D}{x_i}| = 2 |x_i - \\bar{x}_i| \\leq 2$\nand $|H({\\bf x})| \\leq 8M|X| \\cdot D({\\bf x})$. Therefore,\n$$ \\Big| \\partdiff{\\tilde{F}}{x_i} -\n \\left(\\partdiff{F}{x_i} - \\phi(D({\\bf x})) \\partdiff{H}{x_i} \\right) \\Big|\n \\leq 16 M |X| D({\\bf x}) \\cdot \\Big| \\phi'(D({\\bf x})) \\Big|.$$\nBy Lemma~\\ref{lemma:phi-construction}, $|D({\\bf x}) \\phi'(D({\\bf x}))| \\leq 4 \\alpha$, and hence\n$$ \\Big| \\partdiff{\\tilde{F}}{x_i} -\n \\left(\\partdiff{F}{x_i} - \\phi(D({\\bf x})) \\partdiff{H}{x_i} \\right) \\Big|\n \\leq 64 M |X| \\alpha.$$\nAssuming that $\\partdiff{F}{x_i} \\geq 0$, we also have $\\partdiff{G}{x_i} \\geq 0$\n(see Lemma~\\ref{lemma:grad-symmetry}) and therefore,\n$ \\partdiff{F}{x_i} - \\phi(D({\\bf x})) \\partdiff{H}{x_i}\n = (1-\\phi(D({\\bf x}))) \\partdiff{F}{x_i} + \\phi(D({\\bf x})) \\partdiff{G}{x_i} \\geq 0$.\nConsequently,\n$$ \\partdiff{\\tilde{F}}{x_i} \\geq - 64 M |X| \\alpha.$$\nBy differentiating $\\tilde{F}$ twice, we obtain\n\\begin{eqnarray*}\n\\mixdiff{\\tilde{F}}{x_i}{x_j} & = &\n \\mixdiff{F}{x_i}{x_j} - \\phi({D}({\\bf x})) \\mixdiff{H}{x_i}{x_j}\n - \\phi''({D}({\\bf x})) \\partdiff{{D}}{x_i} \\partdiff{{D}}{x_j} H({\\bf x}) \\\\\n& &  - \\phi'({D}({\\bf x})) \\left( \\partdiff{{D}}{x_j} \\partdiff{H}{x_i}\n + \\mixdiff{{D}}{x_i}{x_j} H({\\bf x})\n + \\partdiff{{D}}{x_i} \\partdiff{H}{x_j} \\right).\n\\end{eqnarray*}\nAgain, we use Lemma~\\ref{lemma:H-bounds} and \\ref{lemma:D-bounds}\nto bound $|H({\\bf x})| \\leq 8M|X| D({\\bf x})$, $|\\partdiff{H}{x_i}| \\leq 8M|X| \\sqrt{D({\\bf x})}$,\n $|\\mixdiff{H}{x_i}{x_j}| \\leq 8M$, $|\\partdiff{D}{x_i}| \\leq 2 \\sqrt{D({\\bf x})}$\nand $|\\mixdiff{D}{x_i}{x_j}| \\leq 2$. We get\n\\begin{eqnarray*}\n\\Big| \\mixdiff{\\tilde{F}}{x_i}{x_j} - \\mixdiff{F}{x_i}{x_j} +\n \\phi({D}({\\bf x})) \\mixdiff{H}{x_i}{x_j} \\Big|\n \\leq 32 M|X| \\Big| D^2({\\bf x}) \\phi''({D}({\\bf x})) \\Big|\n  + 48 M|X| \\Big| D({\\bf x}) \\phi'({D}({\\bf x})) \\Big|\n\\end{eqnarray*}\nObserve that $\\phi'(D({\\bf x}))$ appears with $D({\\bf x})$ and\n$\\phi''(D({\\bf x}))$ appears with $(D({\\bf x}))^2$.\nBy Lemma~\\ref{lemma:phi-construction}, $|D({\\bf x}) \\phi'(D({\\bf x}))| \\leq 4 \\alpha$\nand $|D^2({\\bf x}) \\phi''(D({\\bf x}))| \\leq 10 \\alpha$. Therefore,\n\\begin{eqnarray*}\n\\Big| \\mixdiff{\\tilde{F}}{x_i}{x_j} - \\left( \\mixdiff{F}{x_i}{x_j}\n - \\phi({D}({\\bf x})) \\mixdiff{H}{x_i}{x_j} \\right) \\Big|\n \\leq  320 M|X|\\alpha + 192 M|X| \\alpha  = 512 M |X| \\alpha.\n\\end{eqnarray*}\nIf $\\mixdiff{F}{x_i}{x_j} \\leq 0$ for all $i,j$, then also $\\mixdiff{G}{x_i}{x_j} =\n {\\bf E}_{\\sigma,\\tau \\in {\\cal G}}[\\mixdiff{F}{x_{\\sigma^{-1}(i)}}{x_{\\tau^{-1}(j)}}] \\leq 0$\n (see the proof of Lemma~\\ref{lemma:H-bounds}). Also, \n$\\mixdiff{F}{x_i}{x_j} - \\phi({D}({\\bf x})) \\mixdiff{H}{x_i}{x_j}\n = \\phi(D({\\bf x})) \\mixdiff{F}{x_i}{x_j} + (1-\\phi(D({\\bf x}))) \\mixdiff{G}{x_i}{x_j} \\leq 0$.\nWe obtain\n\\begin{eqnarray*}\n\\mixdiff{\\tilde{F}}{x_i}{x_j} \\leq 512 M |X| \\alpha.\n\\end{eqnarray*}\n\\end{proof}\n\nFinally, we can finish the proof of Lemma~\\ref{lemma:final-fix}.\n\n\\begin{proof}[Proof of Lemma~\\ref{lemma:final-fix}]\nLet $\\epsilon>0$ and $f:2^X \\rightarrow [0,M]$.\nWe choose $\\beta = \\frac{\\epsilon}{16 M|X|}$, so that $|H({\\bf x})| = |F({\\bf x}) - G({\\bf x})| \\leq \\epsilon\/2$\nwhenever $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 \\leq \\beta$\n(due to by Lemma~\\ref{lemma:H-bounds}, which states that $|H({\\bf x})| \\leq 8 M |X| D({\\bf x})$).\nAlso, let $\\alpha = \\frac{\\epsilon}{2000 M |X|^3}$.\nFor these values of $\\alpha,\\beta>0$,\nlet $\\delta > 0$ and $\\phi:{\\boldmath R}_+ \\rightarrow [0,1]$\nbe provided by Lemma~\\ref{lemma:phi-construction}.\nWe define\n$$ \\tilde{F}({\\bf x}) = (1-\\phi(D({\\bf x}))) F({\\bf x}) + \\phi(D({\\bf x})) G({\\bf x}). $$\nLemma~\\ref{lemma:F-bounds} provides bounds on the first and second\npartial derivatives of $\\tilde{F}({\\bf x})$.\nFinally, we modify $\\tilde{F}({\\bf x})$ so that it satisfies the required\nconditions (submodularity and optionally monotonicity).\nFor that purpose, we add a suitable multiple of the following function:\n$$ J({\\bf x}) = |X|^2 + 3|X| \\sum_{i \\in X} x_i - \\left(\\sum_{i \\in X} x_i \\right)^2.$$\nWe have $0 \\leq J({\\bf x}) \\leq 3|X|^2$,\n$\\partdiff{J}{x_i} = 3|X| - 2 \\sum_{i \\in X} x_i \\geq |X|$.\nFurther, $\\mixdiff{J}{x_i}{x_j} = -2$. Note also that $J(\\bar{{\\bf x}}) = J({\\bf x})$,\nsince $J({\\bf x})$ depends only on the sum of all coordinates $\\sum_{i \\in X} x_i$.\nTo make $\\tilde{F}({\\bf x})$ submodular and optionally monotone, we define:\n$$ \\hat{F}({\\bf x}) = \\tilde{F}({\\bf x}) + 256 M |X| \\alpha J({\\bf x}),$$\n$$ \\hat{G}({\\bf x}) = G({\\bf x}) + 256 M |X| \\alpha J({\\bf x}). $$\nWe verify the properties of $\\hat{F}({\\bf x})$ and $\\hat{G}({\\bf x})$:\n\\begin{enumerate}\n\\item\nFor any ${\\bf x} \\in P({\\cal F})$, we have\n\\begin{eqnarray*}\n\\hat{G}({\\bf x}) & = & G({\\bf x}) + 256 M |X| \\alpha J({\\bf x}) \\\\\n & = & F(\\bar{{\\bf x}}) + 256 M |X| \\alpha J(\\bar{{\\bf x}}) \\\\\n & = & \\hat{F}(\\bar{{\\bf x}}).\n\\end{eqnarray*}\n\n\\item When $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 \\geq \\beta$,\nLemma~\\ref{lemma:phi-construction} guarantees\nthat $0 \\leq \\phi(D({\\bf x})) < e^{-1\/\\alpha} \\leq \\alpha$ and\n\\begin{eqnarray*}\n|\\hat{F}({\\bf x}) - F({\\bf x})| & \\leq & \\phi(D({\\bf x})) |G({\\bf x}) - F({\\bf x})| + 256 M |X| \\alpha J({\\bf x}) \\\\\n & \\leq & \\alpha M + 768 M |X|^3 \\alpha \\\\\n & \\leq & \\epsilon\n\\end{eqnarray*}\nusing $0 \\leq F({\\bf x}), G({\\bf x}) \\leq M$, $|X| \\leq J({\\bf x}) \\leq 3|X|^2$ and $\\alpha = \\frac{\\epsilon}{2000 M |X|^3}$.\n\nWhen $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 < \\beta$, we chose the value of $\\beta$ so that\n$|G({\\bf x}) - F({\\bf x})| < \\epsilon\/2$ and so by the above,\n\\begin{eqnarray*}\n|\\hat{F}({\\bf x}) - F({\\bf x})| & \\leq & \\phi(D({\\bf x})) |G({\\bf x}) - F({\\bf x})| + 256 M |X| \\alpha J({\\bf x}) \\\\\n & \\leq & \\epsilon\/2 + 768 M |X|^3 \\alpha \\\\\n & \\leq & \\epsilon.\n\\end{eqnarray*}\n\n\\item Due to Lemma~\\ref{lemma:phi-construction}, $\\phi(t) = 1$ for $t \\in [0,\\delta]$.\nHence, whenever $D({\\bf x}) = ||{\\bf x} - \\bar{{\\bf x}}||^2 \\leq \\delta$, we have\n$ \\tilde{F}({\\bf x}) = G({\\bf x}) = F(\\bar{{\\bf x}})$, which depends only on $\\bar{{\\bf x}}$.\nAlso, we have $\\hat{F}({\\bf x}) = \\hat{G}({\\bf x}) = F(\\bar{{\\bf x}}) + 256 M |X| \\alpha J({\\bf x})$\nand again, $J({\\bf x})$ depends only on $\\bar{{\\bf x}}$ (in fact, only on the average\nof all coordinates of ${\\bf x}$). Therefore, $\\hat{F}({\\bf x})$ and $\\hat{G}({\\bf x})$\nin this case depend only on $\\bar{{\\bf x}}$.\n\n\\item The first partial derivatives of $\\hat{F}$ are given by the formula\n$$ \\partdiff{\\hat{F}}{x_i} = \\partdiff{F}{x_i}\n - \\phi(D({\\bf x})) \\partdiff{H}{x_i} - \\phi'(D({\\bf x})) \\partdiff{D}{x_i} H({\\bf x}) + 256 M|X|\\alpha \\partdiff{J}{x_i}. $$\nThe functions $F, H, D, J$ are infinitely differentiable, so the only possible issue is with $\\phi$.\nBy inspecting our construction of $\\phi$ (Lemma~\\ref{lemma:phi-construction}), we can see that it is piecewise \ninfinitely differentiable, and $\\phi'$ is continuous at the breakpoints. Therefore, it is also\nabsolutely continuous. This implies that $\\partdiff{\\hat{F}}{x_i}$ is absolutely continuous.\n\nThe function $\\hat{G}({\\bf x}) = F(\\bar{{\\bf x}}) + 256 M|X|\\alpha J({\\bf x})$ is infinitely differentiable,\nso its first partial derivatives are also absolutely continuous.\n\n\\item Assuming $\\partdiff{F}{x_i} \\geq 0$, we get $\\partdiff{\\tilde{F}}{x_i}\n \\geq - 64 M |X| \\alpha$ by Lemma~\\ref{lemma:F-bounds}. Using $\\partdiff{J}{x_i}\n \\geq |X|$, we get $\\partdiff{\\hat{F}}{x_i} = \\partdiff{\\tilde{F}}{x_i}\n + 256 M|X| \\alpha J({\\bf x}) \\geq 0$. The same holds for $\\partdiff{\\hat{G}}{x_i}$\nsince $\\partdiff{G}{x_i} \\geq 0$.\n\n\\item Assuming $\\mixdiff{F}{x_i}{x_j} \\leq 0$, we get $\\mixdiff{\\tilde{F}}{x_i}{x_j}\n \\leq 512 M |X| \\alpha$ by  Lemma~\\ref{lemma:F-bounds}. Using $\\mixdiff{J}{x_i}{x_j}\n = -2$, we get $\\mixdiff{\\hat{F}}{x_i}{x_j} = \\mixdiff{\\tilde{F}}{x_i}{x_j}\n + 256 M |X| \\alpha \\mixdiff{J}{x_i}{x_j} \\leq 0$. The same holds for\n$\\mixdiff{\\hat{G}}{x_i}{x_j}$ since $\\mixdiff{G}{x_i}{x_j} \\leq 0$.\n\\end{enumerate}\n\\end{proof}\n\nThis concludes the proofs of our main hardness results.\n\n\n\n\\section{Algorithms using the multilinear relaxation}\n\\label{section:algorithms}\n\n\nHere we turn to our algorithmic results. First, we discuss\nthe problem of maximizing a submodular (but not necessarily\nmonotone) function subject to a matroid independence constraint.\n\n\n\\subsection{Matroid independence constraint}\n\\label{section:submod-independent}\n\nConsider the problem $\\max \\{ f(S): S \\in {\\cal I} \\}$, where ${\\cal I}$ is the collection\nof independent sets in a matroid ${\\cal M}$.\nWe design an algorithm based on the multilinear relaxation of the problem,\n$\\max \\{ F({\\bf x}): {\\bf x} \\in P({\\cal M}) \\}$.\nOur algorithm can be seen as \"continuous local search\" in the matroid polytope $P({\\cal M})$,\nconstrained in addition by the box $[0,t]^X$\nfor some fixed $t \\in [0,1]$. The intuition is that this forces\nour local search to use fractional solutions that are more fuzzy\nthan integral solutions and therefore less likely to get stuck in a local\noptimum. On the other hand, restraining the search space too much would not\ngive us much freedom in searching for a good fractional point.\nThis leads to a tradeoff and an optimal choice of $t \\in [0,1]$\nwhich we leave for later.\n\nThe matroid polytope is defined as\n$ P({\\cal M}) = \\mbox{conv} \\{ {\\bf 1}_I: I \\in {\\cal I} \\} $.\nor equivalently \\cite{E70} as\n$ P({\\cal M}) = \\{ {\\bf x} \\geq 0: \\forall S; \\sum_{i \\in S} x_i \\leq r_{\\cal M}(S) \\}$,\nwhere $r_{\\cal M}(S)$ is the rank function of ${\\cal M}$.\nWe define\n$$ P_t({\\cal M}) = P({\\cal M}) \\cap [0,t]^X = \\{ {\\bf x} \\in P({\\cal M}): \\forall i; x_i \\leq t \\}.$$\nWe consider the problem $\\max \\{F({\\bf x}): {\\bf x} \\in P_t({\\cal M})\\}$.\nWe remind the reader that\n$F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$ denotes the multilinear extension.\nOur algorithm works as follows.\n\n\\\n\n\\noindent{\\bf Fractional local search in $P_t({\\cal M})$} \\\\\n (given $t = \\frac{r}{q}$, $r \\leq q$ integer)\n\\begin{enumerate}\n\\item Start with ${\\bf x} := (0,0,\\ldots,0)$. Fix $\\delta = 1\/q$.\n\\item If there is  $i,j \\in X$ and a direction ${\\bf v} \\in\n \\{{\\bf e}_j, -{\\bf e}_i, {\\bf e}_j - {\\bf e}_i \\}$ such that\n${\\bf x} + \\delta {\\bf v} \\in P_t({\\cal M})$ and $F({\\bf x} + \\delta {\\bf v}) > F({\\bf x})$,\nset ${\\bf x} := {\\bf x} + \\delta {\\bf v}$ and repeat.\n\\item If there is no such direction ${\\bf v}$, apply pipage rounding to ${\\bf x}$\nand return the resulting solution.\n\\end{enumerate}\n\n\\noindent{\\em Notes.}\nThe procedure as presented here would not run in polynomial\ntime. A modification which runs in polynomial time is that\nwe move to a new solution only if $F({\\bf x}+\\delta {\\bf v}) >\n F({\\bf x}) + \\frac{\\delta}{poly(n)} OPT$ (where we first get\na rough estimate of $OPT$ using previous methods).\nFor simplicity, we analyze the variant above\nand finally discuss why we can modify it without\nlosing too much in the approximation factor. We also defer\nthe question of how to estimate the value of $F({\\bf x})$\nto the end of this section.\n\nFor $t=1$, we have $\\delta=1$ and the procedure\nreduces to discrete local search. However, it is known that discrete\nlocal search alone does not give any approximation guarantee. With\nadditional modifications, an algorithm based on discrete local search\nachieves a $(\\frac14-o(1))$-approximation \\cite{LMNS09}.\n\nOur version of fractional local search avoids this issue and leads\ndirectly to a good fractional solution.  Throughout the algorithm,\nwe maintain ${\\bf x}$ as a linear combination of $q$ independent sets\nsuch that no element appears in more than $r$ of them. A local step\ncorresponds to an add\/remove\/switch operation preserving this condition.\n\nFinally, we use pipage rounding to convert a fractional solution into\nan integral one. As we show in Lemma~\\ref{lemma:pipage} (in the Appendix),\na modification of the technique from \\cite{CCPV07}\ncan be used to find an integral solution without any loss in the objective\nfunction.\n\n\\begin{theorem}\n\\label{thm:submod-matroid-approx}\nThe fractional local search algorithm for any fixed $t \\in [0,\\frac12 (3-\\sqrt{5})]$\nreturns a solution of value at least $(t - \\frac12 t^2) OPT$,\nwhere $OPT = \\max \\{f(S): S \\in {\\cal I}\\}$.\n\\end{theorem}\n\nWe remark that for $t=\\frac12(3-\\sqrt{5})$, we would obtain a $\\frac14(-1+\\sqrt{5})\n \\simeq 0.309$-approximation, improving the factor of $\\frac14$ \\cite{LMNS09}.\nThis is not a rational value, but we can pick a rational $t$\narbitrarily close to $\\frac12 (3-\\sqrt{5})$.\nFor values $t > \\frac12 (3-\\sqrt{5})$, our analysis does not yield\na better approximation factor.\n\nFirst, we discuss properties of the point found by the fractional local search algorithm.\n\n\\begin{lemma}\n\\label{lemma:local-opt}\nThe outcome of the fractional local search algorithm $x$ is\na ``fractional local optimum'' in the following sense.\n(All the partial derivatives are evaluated at $x$.)\n\\begin{itemize}\n\\item For any $i$ such that ${\\bf x} - \\delta {\\bf e}_i \\in P_t({\\cal M})$,\n$\\partdiff{F}{x_i} \\geq 0.$\n\\item For any $j$ such that ${\\bf x} + \\delta {\\bf e}_j \\in P_t({\\cal M})$,\n$\\partdiff{F}{x_j} \\leq 0.$\n\\item For any $i,j$ such that ${\\bf x} + \\delta ({\\bf e}_j - {\\bf e}_i) \\in P_t({\\cal M})$,\n$\\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0.$\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nWe use the property (see \\cite{CCPV07}) that along any direction ${\\bf v} = \\pm {\\bf e}_i$\nor ${\\bf v} = {\\bf e}_i - {\\bf e}_j$, the function $F({\\bf x} + \\lambda {\\bf v})$ is a convex\nfunction of $\\lambda$. Also, observe that if it is possible to move from ${\\bf x}$\nin the direction of ${\\bf v}$ by any nonzero amount, then it is possible to move\nby $\\delta {\\bf v}$, because all coordinates of ${\\bf x}$ are integer multiples of $\\delta$\nand all the constraints also have coefficients which are integer multiples of $\\delta$.\nTherefore, if $\\frac{dF}{d\\lambda} > 0$ and it is possible\nto move in the direction of ${\\bf v}$, we would get\n$F({\\bf x} + \\delta {\\bf v}) > F({\\bf x})$ and the fractional local search would continue.\n\nIf ${\\bf v} = -{\\bf e}_i$ and it is possible to move along $-{\\bf e}_i$,\nwe get $\\frac{dF}{d\\lambda} = -\\partdiff{F}{x_i} \\leq 0$. Similarly,\nif ${\\bf v} = {\\bf e}_j$ and it is possible to move along ${\\bf e}_j$,\nwe get $\\frac{dF}{d\\lambda} = \\partdiff{F}{x_j} \\leq 0$. Finally,\nif ${\\bf v} = {\\bf e}_j - {\\bf e}_i$ and it is possible to move along ${\\bf e}_j - {\\bf e}_i$,\nwe get $\\frac{dF}{d\\lambda} = \\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0$.\n\\end{proof}\n\nWe refer to the following exchange property for matroids\n(which follows easily from \\cite{Schrijver}, Corollary 39.12a; see also \\cite{LMNS09}).\n\n\\begin{lemma}\n\\label{lemma:exchange}\nIf $I, C \\in {\\cal I}$, then for any $j \\in C \\setminus I$,\nthere is $\\pi(j) \\subseteq I \\setminus C$, $|\\pi(j)| \\leq 1$,\nsuch that $I \\setminus \\pi(j) + j \\in {\\cal I}$.\nMoreover, the sets $\\pi(j)$ are disjoint\n(each $i \\in I \\setminus C$ appears at most once as $\\pi(j) = \\{i\\}$).\n\\end{lemma}\n\nUsing this, we prove a lemma about fractional local optima\nwhich generalizes Lemma 2.2 in \\cite{LMNS09}.\n\n\\begin{lemma}\n\\label{lemma:fractional-search}\nLet ${\\bf x}$ be the outcome of fractional local search over $P_t({\\cal M})$.\nLet $C \\in {\\cal I}$ be any independent set.\nLet $C' = \\{i \\in C: x_i < t\\}$. Then\n$$ 2 F({\\bf x}) \\geq F({\\bf x} \\vee {\\bf 1}_{C'}) + F({\\bf x} \\wedge {\\bf 1}_C).$$\n\\end{lemma}\n\nNote that for $t = 1$, the lemma reduces to\n$2 F({\\bf x}) \\geq F({\\bf x} \\vee {\\bf 1}_C) + F({\\bf x} \\wedge {\\bf 1}_C)$\n(similar to Lemma 2.2 in \\cite{LMNS09}).\nFor $t < 1$, however, it is necessary to replace $C$ by $C'$\nin the first expression, which becomes apparent in the proof.\nThe reason is that we do not have any information on $\\partdiff{F}{x_i}$\nfor coordinates where $x_i = t$.\n\n\\begin{proof}\nLet $C \\in {\\cal I}$ and assume ${\\bf x} \\in P_t({\\cal M})$ is a local optimum.\nSince ${\\bf x} \\in P({\\cal M})$, we can decompose it into a convex linear\ncombination of vertices of $P({\\cal M})$, ${\\bf x} = \\sum_{I \\in {\\cal I}} x_I {\\bf 1}_I$\nwhere $\\sum x_I = 1$..\nBy the smooth submodularity of $F({\\bf x})$ (see \\cite{Vondrak08}),\n\\begin{eqnarray*}\nF({\\bf x} \\vee {\\bf 1}_{C'}) - F({\\bf x}) \\leq \\sum_{j \\in C'} (1 - x_j) \\partdiff{F}{x_j} \n  =  \\sum_{j \\in C'} \\sum_{I: j \\notin I} x_I \\partdiff{F}{x_j}\n  = \\sum_I x_I \\sum_{j \\in C' \\setminus I} \\partdiff{F}{x_j}.\n\\end{eqnarray*}\nAll partial derivatives here are evaluated at ${\\bf x}$.\nOn the other hand, also by submodularity,\n\\begin{eqnarray*}\n F({\\bf x}) - F({\\bf x} \\wedge {\\bf 1}_C) \\geq \\sum_{i \\notin C} x_i \\partdiff{F}{x_i}\n = \\sum_{i \\notin C} \\sum_{I: i \\in I} x_I \\partdiff{F}{x_i}\n = \\sum_I x_I \\sum_{i \\in I \\setminus C} \\partdiff{F}{x_i}.\n\\end{eqnarray*}\nTo prove the lemma, it remains to prove the following.\n\n\\\n\n\\noindent {\\bf Claim.} Whenever $x_I > 0$,\n$\\sum_{j \\in C' \\setminus I} \\partdiff{F}{x_j}\n \\leq \\sum_{i \\in I \\setminus C} \\partdiff{F}{x_i}$.\n\n\\\n\n\\noindent {\\em Proof:}\nFor any $I \\in {\\cal I}$, we can apply\nLemma~\\ref{lemma:exchange} to get a mapping $\\pi$ such that\n$I \\setminus \\pi(j) + j \\in {\\cal I}$ for any $j \\in C \\setminus I$.\nNow, consider $j \\in C' \\setminus I$, i.e. $j \\in C \\setminus I$ and $x_j < t$.\n\nIf $\\pi(j) = \\emptyset$, is possible to move from ${\\bf x}$ in the direction\nof ${\\bf e}_j$, because $I + j \\in {\\cal I}$ and hence we can replace $I$ by $I+j$\n(or at least we can do this for some nonzero fraction of its coefficient)\nin the linear combination.\nBecause $x_j < t$, we can move by a nonzero amount inside $P_t({\\cal M})$.\nBy Lemma~\\ref{lemma:local-opt}, $\\partdiff{F}{x_j} \\leq 0$.\n\nSimilarly, if $\\pi(j) = \\{i\\}$, it is possible to move in the direction\nof ${\\bf e}_j - {\\bf e}_i$, because $I$ can be replaced by $I \\setminus \\pi(j) + i$\nfor some nonzero fraction of its coefficient. By Lemma~\\ref{lemma:local-opt},\nin this case $\\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0$.\n\nFinally, for any $i \\in I$ we have $x_i > 0$ and therefore we can decrease\n$x_i$ while staying inside $P_t({\\cal M})$. By Lemma~\\ref{lemma:local-opt},\nwe have $\\partdiff{F}{x_i} \\geq 0$ for all $i \\in I$.\nThis means\n$$ \\sum_{j \\in C' \\setminus I} \\partdiff{F}{x_j}\n \\leq  \\sum_{j \\in C' \\setminus I: \\pi(j) = \\emptyset} \\partdiff{F}{x_j} +\n  \\sum_{j \\in C' \\setminus I: \\pi(j) = \\{i\\}} \\partdiff{F}{x_i}\n \\leq \\sum_{i \\in I \\setminus C} \\partdiff{F}{x_i} $$\nusing the inequalities we derived above, and the fact that\neach $i \\in I \\setminus C$ appears at most once in $\\pi(j)$.\nThis proves the Claim, and hence the Lemma.\n\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{thm:submod-matroid-approx}.\n\n\\begin{proof}\nLet ${\\bf x}$ be the outcome of the fractional local search over $P_t({\\cal M})$.\nDefine $A = \\{ i: x_i = t \\}$.\nLet $C$ be the optimum solution and $C' = C \\setminus A = \\{ i \\in C: x_i < t\\}$.\nBy Lemma~\\ref{lemma:fractional-search},\n$$ 2 F({\\bf x}) \\geq F({\\bf x} \\vee {\\bf 1}_{C'}) + F({\\bf x} \\wedge {\\bf 1}_C).$$\nFirst, let's analyze $F({\\bf x} \\wedge {\\bf 1}_C)$. We apply Lemma~\\ref{lemma:rnd-threshold} (in the Appendix),\nwhich states that $F({\\bf x} \\wedge {\\bf 1}_C) \\geq {\\bf E}[f(T({\\bf x} \\wedge {\\bf 1}_C))]$. Here,\n$T({\\bf x} \\wedge {\\bf 1}_C)$ is a random threshold set corresponding to the\nvector ${\\bf x} \\wedge {\\bf 1}_C$, i.e.\n$$ T({\\bf x} \\wedge {\\bf 1}_C) = \\{ i: ({\\bf x} \\wedge {\\bf 1}_C)_i > \\lambda \\}\n = \\{ i \\in C: x_i > \\lambda \\} = T({\\bf x}) \\cap C $$\nwhere $\\lambda \\in [0,1]$ is uniformly random.\nEquivalently,\n$$ F({\\bf x} \\wedge {\\bf 1}_C) \\geq {\\bf E}[f(T({\\bf x}) \\cap C)].$$\nDue to the definition of a threshold set, with probability $t$ we have\n$\\lambda < t$ and $T({\\bf x})$ contains $A = \\{ i: x_i = t \\} = C \\setminus C'$.\nThen, $f(T({\\bf x}) \\cap C) + f(C') \\geq f(C)$ by submodularity.\nWe conclude that\n\\begin{equation}\n\\label{eq:1}\nF({\\bf x} \\wedge {\\bf 1}_C) \\geq t (f(C) - f(C')).\n\\end{equation}\nNext, let's analyze $F({\\bf x} \\vee {\\bf 1}_{C'})$. We consider the ground set\npartitioned into $X = C \\cup \\bar{C}$, and we apply Lemma~\\ref{lemma:submod-split} (in the Appendix).\n(Here, $\\bar{C}$ denotes $X \\setminus C$, the complement of $C$ inside $X$.)\nWe get\n$$ F({\\bf x} \\vee {\\bf 1}_{C'}) \\geq {\\bf E}[f((T_1({\\bf x} \\vee {\\bf 1}_{C'}) \\cap C)\n \\cup (T_2({\\bf x} \\vee {\\bf 1}_{C'}) \\cap \\bar{C}))].$$\nThe random threshold sets look as follows: $T_1({\\bf x} \\vee {\\bf 1}_{C'}) \\cap C\n = (T_1({\\bf x}) \\cup C') \\cap C$ is equal to $C$ with probability $t$,\nand equal to $C'$ otherwise. $T_2({\\bf x} \\vee {\\bf 1}_{C'}) \\cap \\bar{C}\n = T_2({\\bf x}) \\cap \\bar{C}$ is empty with probability $1-t$. \n(We ignore the contribution when $T_2({\\bf x}) \\cap \\bar{C} \\neq \\emptyset$.)\nBecause $T_1$ and $T_2$ are independently sampled, we get\n$$ F({\\bf x} \\vee {\\bf 1}_{C'}) \\geq t(1-t) f(C) + (1-t)^2 f(C').$$\nProvided that $t \\in [0,\\frac12 (3 - \\sqrt{5})]$, we have $t \\leq (1-t)^2$.\nThen, we can write\n\\begin{equation}\n\\label{eq:1'}\nF({\\bf x} \\vee {\\bf 1}_{C'}) \\geq t(1-t) f(C) + t f(C').\n\\end{equation}\nCombining equations (\\ref{eq:1}) and (\\ref{eq:1'}), we get\n\\begin{eqnarray*}\n& & F({\\bf x} \\vee {\\bf 1}_{C'})  +  F({\\bf x} \\wedge {\\bf 1}_C)\n \\geq t (f(C) - f(C')) + t(1-t) f(C) + t \\, f(C')\n = (2t - t^2) f(C).\n\\end{eqnarray*}\nTherefore,\n$$ F({\\bf x}) \\geq \\frac12 (F({\\bf x} \\vee {\\bf 1}_{C'}) + F({\\bf x} \\wedge {\\bf 1}_C))\n \\geq (t - \\frac12 t^2) f(C).$$\nFinally, we apply the pipage rounding technique which does not\nlose anything in terms of objective value (see Lemma~\\ref{lemma:pipage}).\n\\end{proof}\n\n\\paragraph{Technical remarks}\nIn each step of the algorithm, we need to estimate values of $F({\\bf x})$\nfor given ${\\bf x} \\in P_t({\\cal M})$. We accomplish this by using the expression\n$F({\\bf x}) = {\\bf E}[f(R({\\bf x}))]$ where $R({\\bf x})$ is a random set associated with ${\\bf x}$.\nBy standard bounds, if the values of $f(S)$ are in a range $[0,M]$,\nwe can achieve accuracy $M \/ poly(n)$ using a polynomial number of samples.\nWe use the fact that $OPT \\geq \\frac{1}{n} M$ (see\nLemma~\\ref{lemma:solution-value} in the Appendix) and therefore\nwe can achieve $OPT \/ poly(n)$ additive error in polynomial time.\n\nWe also relax the local step condition: we move to the next solution\nonly if $F({\\bf x} + \\delta {\\bf v}) > F({\\bf x}) + \\frac{\\delta}{poly(n)} OPT$\nfor a suitable polynomial in $n$. This way, we can only make a polynomial number of steps.\nWhen we terminate, the local optimality conditions (Lemma~\\ref{lemma:local-opt})\nare satisfied within an additive error of $OPT \/ poly(n)$,\nwhich yields a polynomially small error in the approximation bound.\n\n\n\\subsection{Matroid base constraint}\n\\label{section:submod-bases}\n\nLet us move on to the problem $\\max \\{f(S): S \\in {\\cal B}\\}$ where\n${\\cal B}$ are the bases of a matroid.\nFor a fixed $t \\in [0,1]$, let us consider an algorithm which can be seen\nas local search inside the base polytope $B({\\cal M})$, further constrained\nby the box $[0,t]^X$. The matroid base polytope is defined as\n$ B({\\cal M}) = \\mbox{conv} \\{ {\\bf 1}_B: B \\in {\\cal B} \\} $\nor equivalently \\cite{E70} as\n$ B({\\cal M}) = \\{ {\\bf x} \\geq 0: \\forall S \\subseteq X; \\sum_{i \\in S} x_i \\leq r_{\\cal M}(S),\n \\sum_{i \\in X} x_i = r_{\\cal M}(X) \\}, $\nwhere $r_{\\cal M}$ is the matroid rank function of ${\\cal M}$. Finally, we define\n$$ B_t({\\cal M}) = B({\\cal M}) \\cap [0,t]^X = \\{ {\\bf x} \\in B({\\cal M}): \\forall i \\in X; x_i \\leq t \\}.$$\nObserve that $B_t({\\cal M})$ is nonempty if and only if there is a convex linear\ncombination ${\\bf x} = \\sum_{B \\in {\\cal B}} \\xi_B {\\bf 1}_B$ such that $x_i \\in [0,t]$ for all $i$.\nThis is equivalent to saying that there is a linear combination (a fractional base packing)\n${\\bf x}' = \\sum_{B \\in {\\cal B}} \\xi'_B {\\bf 1}_B$ such that $x_i \\in [0,1]$\nand $\\sum \\xi'_B \\geq \\frac{1}{t}$, in other words the fractional base packing\nnumber is $\\nu \\geq \\frac{1}{t}$.  Since the optimal fractional packing of bases\nin a matroid can be found efficiently (see Corollary 42.7a in \\cite{Schrijver}, Volume B),\nwe can find efficiently the minimum $t \\in [\\frac12,1]$ such that $B_t({\\cal M}) \\neq \\emptyset$.\nThen, our algorithm is the following.\n\n\\\n\n\\noindent{\\bf Fractional local search in $B_t({\\cal M})$} \\\\\n (given $t = \\frac{r}{q}$, $r \\leq q$ integer)\n\\begin{enumerate}\n\\item Let $\\delta = \\frac{1}{q}$.\nAssume that ${\\bf x} \\in B_t({\\cal M})$; adjust ${\\bf x}$ (using pipage rounding)\nso that each $x_i$ is an integer multiple of $\\delta$.\nIn the following, this property will be maintained.\n\\item If there is a direction ${\\bf v} = {\\bf e}_j - {\\bf e}_i$ such that\n${\\bf x} + \\delta {\\bf v} \\in B_t({\\cal M})$ and $F({\\bf x} + \\delta {\\bf v}) > F({\\bf x})$,\nthen set ${\\bf x} := {\\bf x} + \\delta {\\bf v}$ and repeat.\n\\item If there is no such direction ${\\bf v}$, apply pipage rounding to ${\\bf x}$\nand return the resulting solution.\n\\end{enumerate}\n\n\\noindent{\\em Notes.}\nWe remark that the starting point can be found as a convex linear combination\nof $q$ bases, $x = \\frac{1}{q} \\sum_{i=1}^{q} {\\bf 1}_{B_i}$, such that\nno element appears in more than $r$ of them, using matroid union techniques (see Theorem 42.9 in \\cite{Schrijver}).\nIn the algorithm, we maintain this representation.\nThe local search step corresponds to switching a pair of elements in one base,\nunder the condition that no element is used in more than $r$ bases at the same time.\nFor now, we ignore the issues of estimating $F({\\bf x})$ and stopping the local search\nwithin polynomial time. We discuss this at the end of this section.\n\nFinally, we use pipage rounding to convert the fractional solution ${\\bf x}$\ninto an integral one of value at least $F({\\bf x})$ (Lemma~\\ref{lemma:base-pipage} in the Appendix).\nNote that it is not necessarily true that any of the bases in\na convex linear combination ${\\bf x} = \\sum \\xi_B {\\bf 1}_{B}$ achieves the value $F({\\bf x})$.\n\n\\begin{theorem}\n\\label{thm:submod-bases-approx}\nIf there is a fractional packing of $\\nu \\in [1,2]$ bases in ${\\cal M}$,\nthen the fractional local search algorithm with $t = \\frac{1}{\\nu}$\nreturns a solution of value at least $\\frac12 (1-t) \\ OPT.$\n\\end{theorem}\n\nFor example, assume that ${\\cal M}$ contains two disjoint bases $B_1, B_2$\n(which is the case considered in \\cite{LMNS09}). Then,\nthe algorithm can be used with $t = \\frac{1}{2}$ and\nand we obtain a $(\\frac{1}{4}-o(1))$-approximation, improving the\n$(\\frac{1}{6}-o(1))$-approximation from \\cite{LMNS09}.\nIf there is a fractional packing of more than 2 bases,\nour analysis still gives only a $(\\frac{1}{4}-o(1))$-approximation.\nIf the dual matroid ${\\cal M}^*$ admits a better fractional packing of bases,\nwe can consider the problem $\\max \\{f(\\bar{S}): S \\in {\\cal B}^* \\}$ which\nis equivalent. For a uniform matroid, ${\\cal B} = \\{B: |B|=k\\}$,\nthe fractional base packing number is either at least $2$ or the same\nholds for the dual matroid, ${\\cal B}^* = \\{B: |B|=n-k\\}$ (as noted in \\cite{LMNS09}).\nTherefore, we get a $(\\frac{1}{4}-o(1))$-approximation for any uniform matroid.\nThe value $t=1$ can be used for any matroid,\nbut it does not yield any approximation guarantee.\n\n\n\n\n\\paragraph{Analysis of the algorithm}\nWe turn to the properties of fractional local optima.\nWe will prove that the point ${\\bf x}$ found by the fractional local search algorithm\nsatisfies the following conditions that allow us to compare $F({\\bf x})$ to the actual\noptimum.\n\n\\begin{lemma}\n\\label{lemma:local-opt2}\nThe outcome of the fractional local search algorithm ${\\bf x}$\nis a ``fractional local optimum'' in the following sense.\n\\begin{itemize}\n\\item For any $i,j$ such that ${\\bf x} + \\delta ({\\bf e}_j - {\\bf e}_i) \\in B_t({\\cal M})$,\n$$\\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0.$$\n(The partial derivatives are evaluated at ${\\bf x}$.)\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nSimilarly to Lemma~\\ref{lemma:local-opt},\nobserve that the coordinates of ${\\bf x}$ are always integer multiples of $\\delta$,\ntherefore if it is possible to move from ${\\bf x}$ in the direction of\n${\\bf v} = {\\bf e}_j - {\\bf e}_i$ by any nonzero amount, then it is possible to\nmove by $\\delta {\\bf v}$.\nWe use the property that for any direction ${\\bf v} = {\\bf e}_j - {\\bf e}_i$,\nthe function $F({\\bf x} + \\lambda {\\bf v})$ is a convex function of $\\lambda$ \\cite{CCPV07}.\nTherefore, if $\\frac{dF}{d\\lambda} > 0$ and it is possible\nto move in the direction of ${\\bf v}$, we would get\n$F({\\bf x} + \\delta {\\bf v}) > F({\\bf x})$ and the fractional local search would continue.\nFor ${\\bf v} = {\\bf e}_j - {\\bf e}_i$, we get\n$$ \\frac{dF}{d\\lambda} = \\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0.$$\n\\end{proof}\n\nWe refer to the following exchange property for matroid bases\n(see \\cite{Schrijver}, Corollary 39.21a).\n\n\\begin{lemma}\n\\label{lemma:exchange2}\nFor any $B_1, B_2 \\in {\\cal B}$, there is a bijection $\\pi:B_1 \\setminus B_2\n \\rightarrow B_2 \\setminus B_1$ such that\n$\\forall i \\in B_1 \\setminus B_2$; $B_1-i+\\pi(i) \\in {\\cal M}$.\n\\end{lemma}\n\nUsing this, we prove a lemma about fractional local optima\nanalogous to Lemma 2.2 in \\cite{LMNS09}.\n\n\\begin{lemma}\n\\label{lemma:fractional-search2}\nLet ${\\bf x}$ be the outcome of fractional local search over $B_t({\\cal M})$.\nLet $C \\in {\\cal B}$ be any base. Then there is ${\\bf c} \\in [0,1]^X$ satisfying\n\\begin{itemize}\n\\item $c_i = t$, if $i \\in C$ and $x_i = t$\n\\item $c_i = 1$, if $i \\in C$ and $x_i < t$\n\\item $0 \\leq c_i \\leq x_i$, if $i \\notin C$\n\\end{itemize}\nsuch that\n$$ 2 F({\\bf x}) \\geq F({\\bf x} \\vee {\\bf c}) + F({\\bf x} \\wedge {\\bf c}).$$\n\\end{lemma}\n\nNote that for $t = 1$, we can set ${\\bf c} = {\\bf 1}_C$.\nHowever, in general we need this more complicated formulation.\nIntuitively, ${\\bf c}$ is obtained from ${\\bf x}$ by raising the variables $x_i, i \\in C$\nand decreasing $x_i$ for $i \\notin C$.\nHowever, we can only raise the variables $x_i, i \\in C$, where $x_i$ is below\nthe threshold $t$, otherwise we do not have any information about $\\partdiff{F}{x_i}$.\nAlso, we do not necessarily decrease all the variables outside of $C$ to zero.\n\n\\begin{proof}\nLet $C \\in {\\cal B}$ and assume ${\\bf x} \\in B_t({\\cal M})$ is a fractional local optimum.\nWe can decompose ${\\bf x}$ into a convex linear combination of vertices\nof $B({\\cal M})$, ${\\bf x} = \\sum \\xi_B {\\bf 1}_B$.\nBy Lemma~\\ref{lemma:exchange2}, for each base $B$ there is a bijection\n$\\pi_B:B \\setminus C \\rightarrow C \\setminus B$ such that\n$\\forall i \\in B \\setminus C$; $B - i + \\pi_B(i) \\in {\\cal B}$.\n\nWe define $C' = \\{i \\in C: x_i < t \\}$.\nThe reason we consider $C'$ is that\nif $x_i = t$, there is no room for an exchange step increasing $x_i$,\nand therefore Lemma~\\ref{lemma:local-opt2} does not give any information\nabout $\\partdiff{F}{x_i}$.\nWe construct the vector ${\\bf c}$ by starting from ${\\bf x}$, and for each $B$\nswapping the elements in $B \\setminus C$ for their image under $\\pi_B$,\nprovided it is in $C'$, until we raise the coordinates on $C'$ to $c_i=1$.\nFormally, we set $c_i = 1$ for $i \\in C'$, $c_i = t$ for $i \\in C \\setminus C'$,\nand for each $i \\notin C$, we define\n$$ c_i = x_i - \\sum_{B: i \\in B, \\pi_B(i) \\in C'} \\xi_B.$$\n\nIn the following, all partial derivatives are evaluated at ${\\bf x}$.\nBy the smooth submodularity of $F({\\bf x})$ (see \\cite{CCPV09}),\n\\begin{eqnarray}\n\\label{eq:2}\nF({\\bf x} \\vee {\\bf c}) - F({\\bf x}) & \\leq & \\sum_{j: c_j > x_j} (c_j - x_j) \\partdiff{F}{x_j}\n = \\sum_{j \\in C'} (1 - x_j) \\partdiff{F}{x_j}\n  = \\sum_B \\sum_{j \\in C' \\setminus B} \\xi_B \\partdiff{F}{x_j}\n\\end{eqnarray}\nbecause $\\sum_{B: j \\notin B} \\xi_B = 1 - x_j$ for any $j$.\nOn the other hand, also by smooth submodularity,\n\\begin{eqnarray*}\nF({\\bf x}) - F({\\bf x} \\wedge {\\bf c}) & \\geq & \\sum_{i: c_i < x_i} (x_i - c_i) \\partdiff{F}{x_i}\n = \\sum_{i \\notin C} (x_i - c_i) \\partdiff{F}{x_i}\n = \\sum_{i \\notin C} \\sum_{B: i \\in B, \\pi_B(i) \\in C'} \\xi_B \\partdiff{F}{x_i}\n\\end{eqnarray*}\nusing our definition of $c_i$. In the last sum,\nfor any nonzero contribution, we have $\\xi_B > 0$, $i \\in B$ and $j = \\pi_B(i) \\in C'$,\ni.e. $x_j < t$. Therefore it is possible to move in the direction ${\\bf e}_j - {\\bf e}_i$\n(we can switch from $B$ to $B-i+j$).\nBy Lemma~\\ref{lemma:local-opt2},\n$$ \\partdiff{F}{x_j} - \\partdiff{F}{x_i} \\leq 0.$$\nTherefore, we get\n\\begin{eqnarray}\nF({\\bf x}) - F({\\bf x} \\wedge {\\bf c}) & \\geq &\n\\sum_{i \\notin C} \\sum_{B: i \\in B, j=\\pi_B(i) \\in C'} \\xi_B \\partdiff{F}{x_j}\n\\label{eq:3}\n = \\sum_B \\sum_{i \\in B \\setminus C: j=\\pi_B(i) \\in C'} \\xi_B \\partdiff{F}{x_j}.\n\\end{eqnarray}\nBy the bijective property of $\\pi_B$, this is equal to\n$\\sum_B \\sum_{j \\in C' \\setminus B} \\xi_B \\partdiff{F}{x_j}$.\nPutting (\\ref{eq:2}) and (\\ref{eq:3}) together, we get\n$F({\\bf x} \\vee {\\bf c}) - F({\\bf x}) \\leq F({\\bf x}) - F({\\bf x} \\wedge {\\bf c})$.\n\\end{proof}\n\nNow we are ready to prove Theorem~\\ref{thm:submod-bases-approx}.\n\n\\begin{proof}\nAssuming that $B_t({\\cal M}) \\neq \\emptyset$, we can find a starting point ${\\bf x}_0 \\in B_t({\\cal M})$.\nFrom this point, we reach a fractional local optimum ${\\bf x} \\in B_t({\\cal M})$\n(see Lemma~\\ref{lemma:local-opt2}).\nWe want to compare $F({\\bf x})$ to the actual optimum; assume that $OPT = f(C)$.\n\nAs before, we define $C' = \\{i \\in C: x_i < t\\}$.\nBy Lemma~\\ref{lemma:fractional-search2}, we know that\nthe fractional local optimum satisfies:\n\\begin{equation}\n\\label{eq:4}\n2 F({\\bf x}) \\geq F({\\bf x} \\vee {\\bf c}) + F({\\bf x} \\wedge {\\bf c})\n\\end{equation}\nfor some vector ${\\bf c}$ such that $c_i = t$ for all $i \\in C \\setminus C'$,\n$c_i = 1$ for $i \\in C'$ and $0 \\leq c_i \\leq x_i$ for $i \\notin C$.\n\nFirst, let's analyze $F({\\bf x} \\vee {\\bf c})$. We have\n\\begin{itemize}\n\\item $({\\bf x} \\vee {\\bf c})_i = 1$ for all $i \\in C'$.\n\\item $({\\bf x} \\vee {\\bf c})_i = t$ for all $i \\in C \\setminus C'$.\n\\item $({\\bf x} \\vee {\\bf c})_i \\leq t$ for all $i \\notin C$.\n\\end{itemize}\nWe apply Lemma~\\ref{lemma:submod-split} to the partition $X = C \\cup \\bar{C}$.\nWe get\n$$ F({\\bf x} \\vee {\\bf c}) \\geq {\\bf E}[f((T_1({\\bf x} \\vee {\\bf c}) \\cap C) \\cup (T_2({\\bf x} \\vee {\\bf c}) \\cap \\bar{C}))] $$\nwhere $T_1({\\bf x})$ and $T_2({\\bf x})$ are independent threshold sets.\nBased on the information above, $T_1({\\bf x} \\vee {\\bf c}) \\cap C = C$ with probability $t$\nand $T_1({\\bf x} \\vee {\\bf c}) \\cap C = C'$ otherwise. On the other hand,\n$T_2({\\bf x} \\vee {\\bf c}) \\cap \\bar{C} = \\emptyset$ with probability at least $1-t$.\nThese two events are independent. We conclude that on the right-hand side,\nwe get $f(C)$ with probability at least $t(1-t)$, or $f(C')$ with probability\nat least $(1-t)^2$:\n\\begin{equation}\n\\label{eq:5}\nF({\\bf x} \\vee {\\bf c}) \\geq t(1-t) f(C) + (1-t)^2 f(C').\n\\end{equation}\nTurning to $F({\\bf x} \\wedge {\\bf c})$, we see that\n\\begin{itemize}\n\\item $({\\bf x} \\wedge {\\bf c})_i = x_i$ for all $i \\in C'$.\n\\item $({\\bf x} \\wedge {\\bf c})_i = t$ for all $i \\in C \\setminus C'$.\n\\item $({\\bf x} \\wedge {\\bf c})_i \\leq t$ for all $i \\notin C$.\n\\end{itemize}\nWe apply Lemma~\\ref{lemma:submod-split} to $X = C \\cup \\bar{C}$.\n$$ F({\\bf x} \\wedge {\\bf c}) \\geq {\\bf E}[f((T_1({\\bf x} \\wedge {\\bf c}) \\cap C) \\cup (T_2({\\bf x} \\wedge {\\bf c}) \\cap \\bar{C}))].$$\nWith probability $t$,  $T_1({\\bf x} \\wedge {\\bf c}) \\cap C$ contains $C \\setminus C'$\n(and maybe some elements of $C'$). In this case, $f(T_1({\\bf x} \\wedge {\\bf c}) \\cap C)\n \\geq f(C) - f(C')$ by submodularity.\nAlso, $T_2({\\bf x} \\wedge {\\bf c}) \\cap \\bar{C}$ is empty with probability at least $1-t$.\nAgain, these two events are independent.\nTherefore,\n$ F({\\bf x} \\wedge {\\bf c}) \\geq t(1-t) (f(C) - f(C')).$\nIf $f(C') > f(C)$, this bound is vacuous; otherwise, we can replace $t(1-t)$ by\n$(1-t)^2$, because $t \\geq 1\/2$. In any case,\n\\begin{equation}\n\\label{eq:6}\nF({\\bf x} \\wedge {\\bf c}) \\geq (1-t)^2 (f(C) - f(C')).\n\\end{equation}\nCombining (\\ref{eq:4}), (\\ref{eq:5}) and (\\ref{eq:6}),\n$$ F({\\bf x}) \\geq \\frac12 (F({\\bf x} \\vee {\\bf c}) + F({\\bf x} \\wedge {\\bf c}))\n \\geq \\frac12 (t(1-t) f(C) + (1-t)^2 f(C)) = \\frac12 (1-t) f(C).$$\n\\end{proof}\n\n\\paragraph{Technical remarks}\nAgain, we have to deal with the issues of estimating $F({\\bf x})$ and\nstopping the local search in polynomial time. We do this exactly\nas we did at the end of Section~\\ref{section:submod-independent}.\nOne issue to be careful about here is that if $f:2^X \\rightarrow [0,M]$,\nour estimates of $F({\\bf x})$ are within an additive error of $M \/ poly(n)$.\nIf the optimum value $OPT = \\max \\{f(S): S \\in {\\cal B}\\}$ is very small\ncompared to $M$, the error might be large compared to $OPT$ which would be\na problem. The optimum could in fact be very small in general.\nBut it holds that if ${\\cal M}$ contains no loops and co-loops\n(which can be eliminated easily), then $OPT \\geq \\frac{1}{n^2} M$\n(see Appendix~\\ref{app:base-value}). Then, our sampling errors are\non the order of $OPT \/ poly(n)$ which yields a $1\/poly(n)$ error\nin the approximation bound.\n\n\n\\section{Approximation for symmetric instances}\n\\label{section:symmetric}\n\nWe can achieve a better approximation assuming that the instance exhibits\na certain symmetry. This is the same kind of symmetry that we use in our hardness\nconstruction (Section~\\ref{section:hardness-proof}) and the hard instances exhibit\nthe same symmetry as well. It turns out that our approximation in this case\nmatches the hardness threshold up to lower order terms.\n\nSimilar to our hardness result, the symmetries that we consider\nhere are permutations of the ground set $X$, corresponding\nto permutations of coordinates in ${\\boldmath R}^X$. We start with some basic\nproperties which are helpful in analyzing symmetric instances.\n\n\\begin{lemma}\n\\label{lemma:grad-sym}\nAssume that $f:2^X \\rightarrow {\\boldmath R}$ is invariant with respect to\na group of permutations ${\\cal G}$ and $F({\\bf x}) = {\\bf E}[f(\\hat{{\\bf x}})]$.\nThen for any symmetrized vector $\\bar{{\\bf c}} = {\\bf E}_{\\sigma \\in {{\\cal G}}}[\\sigma({\\bf c})]$,\n$\\nabla F |_{\\bar{{\\bf c}}}$ is also symmetric w.r.t. ${\\cal G}$. I.e., for any $\\tau \\in {\\cal G}$,\n$$ \\tau(\\nabla F |_{{\\bf x}=\\bar{{\\bf c}}}) = \\nabla F |_{{\\bf x}=\\bar{{\\bf c}}}. $$\n\\end{lemma}\n\n\n\\begin{proof}\nSince $f(S)$ is invariant under ${\\cal G}$, so is $F({\\bf x})$,\ni.e. $F({\\bf x}) = F(\\tau({\\bf x}))$ for any $\\tau \\in {\\cal G}$.\nDifferentiating both sides at ${\\bf x}={\\bf c}$, we get by the chain rule:\n$$ \\partdiff{F}{x_i} \\Big|_{{\\bf x}={\\bf c}} =\n \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\tau({\\bf c})} \\partdiff{}{x_i} (\\tau(x))_j\n = \\sum_j \\partdiff{F}{x_j} \\Big|_{{\\bf x}=\\tau({\\bf c})} \\partdiff{x_{\\tau(j)}}{x_i}. $$\nHere, $\\partdiff{x_{\\tau(j)}}{x_i} = 1$ if $\\tau(j) = i$, and $0$ otherwise.\nTherefore,\n$$ \\partdiff{F}{x_i} \\Big|_{{\\bf x}={\\bf c}} =\n \\partdiff{F}{x_{\\tau^{-1}(i)}} \\Big|_{{\\bf x}=\\tau({\\bf c})}. $$\nNote that $\\tau(\\bar{{\\bf c}}) = {\\bf E}_{\\sigma \\in {\\cal G}}[\\tau(\\sigma({\\bf c}))]\n =  {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf c})] = \\bar{{\\bf c}}$\nsince the distribution of $\\tau \\circ \\sigma$ is equal to the distribution\nof $\\sigma$. Therefore,\n$$ \\partdiff{F}{x_i} \\Big|_{{\\bf x}=\\bar{{\\bf c}}} = \\partdiff{F}{x_{\\tau^{-1}(i)}}\n  \\Big|_{{\\bf x}=\\bar{{\\bf c}}} $$\nfor any $\\tau \\in {\\cal G}$.\n\\end{proof}\n\n\nNext, we prove that the ``symmetric optimum''\n$\\max \\{F(\\bar{{\\bf x}}): {\\bf x} \\in P({\\cal F})\\}$ gives a solution which is\na local optimum for the original instance $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$.\n(As we proved in Section~\\ref{section:hardness-proof}, in general\nwe cannot hope to find a better solution than the symmetric optimum.)\n\n\\begin{lemma}\n\\label{lemma:sym-local-opt}\nLet $f:2^X \\rightarrow {\\boldmath R}$ and ${\\cal F} \\subset 2^X$\nbe invariant with respect to a group of permutations $\\cal G$.\nLet $\\overline{OPT} = \\max \\{ F(\\bar{{\\bf x}}): {\\bf x} \\in P({\\cal F}) \\}$\nwhere $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$,\nand let ${\\bf x}_0$ be the symmetric point\nwhere $\\overline{OPT}$ is attained  ($\\bar{{\\bf x}}_0 = {\\bf x}_0$).\nThen ${\\bf x}_0$ is a local optimum for the problem\n$\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F}) \\}$, in the sense that\n$({\\bf x}-{\\bf x}_0) \\cdot \\nabla F |_{{\\bf x}_0} \\leq 0$ for any ${\\bf x} \\in P({\\cal F})$.\n\\end{lemma}\n\n\\begin{proof}\nAssume for the sake of contradiction that $({\\bf x}-{\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0} > 0$\nfor some ${\\bf x} \\in P({\\cal F})$. We use the symmetric properties of $f$ and ${\\cal F}$\nto show that $(\\bar{{\\bf x}} - {\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0} > 0$ as well.\nRecall that ${\\bf x}_0 = \\bar{{\\bf x}}_0$. We have\n$$ (\\bar{{\\bf x}} - {\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0}\n = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x}-{\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0}] \n = {\\bf E}_{\\sigma \\in {\\cal G}}[({\\bf x}-{\\bf x}_0) \\cdot \\sigma^{-1}(\\nabla F|_{{\\bf x}_0})]\n = ({\\bf x}-{\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0} > 0 $$\nusing Lemma~\\ref{lemma:grad-sym}.  Hence, there would be a direction\n$\\bar{{\\bf x}} - {\\bf x}_0$ along which an improvement can be obtained.\nBut then, consider a small $\\delta > 0$ such that\n${\\bf x}_1 = {\\bf x}_0 + \\delta (\\bar{{\\bf x}} - {\\bf x}_0) \\in P({\\cal F})$ and also\n$F({\\bf x}_1) > F({\\bf x}_0)$. The point ${\\bf x}_1$ is symmetric ($\\bar{{\\bf x}}_1 = {\\bf x}_1$)\nand hence it would contradict the assumption that $F({\\bf x}_0) = \\overline{OPT}$.\n\\end{proof}\n\n\n\\subsection{Submodular maximization over independent sets in a matroid}\n\nLet us derive an optimal approximation result for the problem\n$\\max \\{f(S): S \\in {\\cal I}\\}$ under the assumption that the instance is \"element-transitive\". \n\n\\begin{definition}\nFor a group ${\\cal G}$ of permutations on $X$, the orbit of an element $i \\in X$\nis the set $\\{ \\sigma(i): \\sigma \\in {\\cal G} \\}$.\n${\\cal G}$ is called element-transitive, if the orbit of any element is\nthe entire ground set $X$.\n\\end{definition}\n\nIn this case, we show that it is easy to achieve an optimal $(\\frac12-o(1))$-approximation\nfor a matroid independence constraint.\n\n\\begin{theorem}\n\\label{thm:sym-submod-independent}\nLet $\\max \\{f(S): S \\in {\\cal I}\\}$ be an instance symmetric with respect\nto an element-transitive group of permutations ${\\cal G}$. Let\n$\\overline{OPT} = \\max \\{F(\\bar{{\\bf x}}): {\\bf x} \\in P({\\cal M})\\}$\n where $\\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$.\nThen $\\overline{OPT} \\geq \\frac12 OPT$.\n\\end{theorem}\n\n\\begin{proof}\nLet $OPT = f(C)$.\nBy Lemma~\\ref{lemma:sym-local-opt}, $\\overline{OPT} = F({\\bf x}_0)$\nwhere ${\\bf x}_0$ is a local optimum for the problem $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal M})$.\nThis means it is also a local optimum in the sense of Lemma~\\ref{lemma:local-opt},\nwith $t=1$.\nBy Lemma~\\ref{lemma:fractional-search},\n$$ 2 F({\\bf x}_0) \\geq F({\\bf x}_0 \\vee {\\bf 1}_C) + F({\\bf x}_0 \\wedge {\\bf 1}_C).$$\nAlso, ${\\bf x}_0 = \\bar{{\\bf x}}_0$. As we are dealing with an element-transitive\ngroup of symmetries, this means all the coordinates of ${\\bf x}_0$ are equal,\n${\\bf x}_0 = (\\xi,\\xi,\\ldots,\\xi)$. Therefore, ${\\bf x}_0 \\vee {\\bf 1}_C$ is equal\nto $1$ on $C$ and $\\xi$ outside of $C$. By Lemma~\\ref{lemma:rnd-threshold} (in the Appendix),\n$$ F({\\bf x}_0 \\vee {\\bf 1}_C) \\geq (1-\\xi) f(C).$$\nSimilarly, ${\\bf x}_0 \\wedge {\\bf 1}_C$ is equal to $\\xi$ on $C$ and $0$ outside of $C$.\nBy Lemma~\\ref{lemma:rnd-threshold},\n$$ F({\\bf x}_0 \\wedge {\\bf 1}_C) \\geq \\xi f(C).$$\nCombining the two bounds,\n$$ 2 F({\\bf x}_0) \\geq F({\\bf x}_0 \\vee {\\bf 1}_C) + F({\\bf x}_0 \\wedge {\\bf 1}_C) \n \\geq (1-\\xi) f(C) + \\xi f(C) = f(C) = OPT.$$\n\\end{proof}\n\nSince all symmetric solutions ${\\bf x} = (\\xi,\\xi,\\ldots,\\xi)$ form\na 1-parameter family, and $F(\\xi,\\xi,\\ldots,\\xi)$ is a concave function,\nwe can search for the best symmetric solution (within any desired accuracy)\nby binary search. By standard techniques, we get the following.\n\n\\begin{corollary}\nThere is a $(\\frac12 - o(1))$-approximation (\"brute force\" search\nover symmetric solutions) for the problem $\\max \\{f(S): S \\in {\\cal I}\\}$\nfor instances symmetric under an element-transitive group of permutations.\n\\end{corollary}\n\nThe hard instances for submodular maximization subject to a matroid\nindependence constraint correspond to refinements of the\nMax Cut instance for the graph $K_2$ (Section~\\ref{section:hardness-applications}).\nIt is easy to see that such instances are element-transitive,\nand it follows from Section~\\ref{section:hardness-proof} that a $(\\frac12+\\epsilon)$-approximation\nfor such instances would require exponentially many value queries.\nTherefore, our approximation for element-transitive instances is optimal.\n\n\n\\subsection{Submodular maximization over bases}\n\nLet us come back to the problem of submodular maximization over the bases\nof matroid. The property that $\\overline{OPT}$ is a local optimum\nwith respect to the original problem $\\max \\{F({\\bf x}): {\\bf x} \\in P({\\cal F})\\}$\nis very useful in arguing about the value of $\\overline{OPT}$.\nWe already have tools to deal with local optima from\nSection~\\ref{section:submod-bases}. Here we prove the following.\n\n\\begin{lemma}\n\\label{lemma:base-local-opt}\nLet $B({\\cal M})$ be the matroid base polytope of ${\\cal M}$ and\n${\\bf x}_0 \\in B({\\cal M})$ a local maximum for the submodular maximization problem\n$\\max \\{F({\\bf x}): {\\bf x} \\in B({\\cal M})\\}$, in the sense that $({\\bf x}-{\\bf x}_0) \\cdot \\nabla F|_{{\\bf x}_0} \\leq 0$\nfor any ${\\bf x} \\in B({\\cal M})$. Assume in addition that ${\\bf x}_0 \\in [s,t]^X$. Then\n$$ F({\\bf x}_0) \\geq \\frac12 (1-t+s) \\cdot OPT.$$\n\\end{lemma}\n\n\\begin{proof}\nLet $OPT = \\max \\{ f(B): B \\in {\\cal B} \\} = f(C)$.\nWe assume that ${\\bf x}_0 \\in B({\\cal M})$ is a local optimum with respect\nto any direction ${\\bf x}-{\\bf x}_0$, ${\\bf x} \\in B({\\cal M})$, so it is also a local optimum with respect\nto the fractional local search in the sense of Lemma~\\ref{lemma:fractional-search2},\nwith $t=1$. The lemma implies that\n$$ 2 F({\\bf x}_0) \\geq F({\\bf x} \\vee {\\bf 1}_C) + F({\\bf x} \\wedge {\\bf 1}_C).$$\nBy assumption, the coordinates of ${\\bf x} \\vee {\\bf 1}_C$ are equal to $1$ on $C$\nand at most $t$ outside of $C$. With probability $1-t$, a random threshold in $[0,1]$\nfalls between $t$ and $1$, and Lemma~\\ref{lemma:rnd-threshold} (in the Appendix) implies that\n$$ F({\\bf x} \\vee {\\bf 1}_C) \\geq (1-t) \\cdot f(C).$$\nSimilarly, the coordinates of ${\\bf x} \\wedge {\\bf 1}_C$ are $0$ outside of $C$,\nand at least $s$ on $C$. A random threshold falls between $0$ and $s$\nwith probability $s$, and Lemma~\\ref{lemma:rnd-threshold} implies that\n$$ F({\\bf x} \\wedge {\\bf 1}_C) \\geq s \\cdot f(C).$$\nPutting these inequalities together, we get\n$$2 F({\\bf x}_0) \\geq F({\\bf x} \\vee {\\bf 1}_C) + F({\\bf x} \\wedge {\\bf 1}_C) \\geq (1-t+s) \\cdot f(C).$$\n\\end{proof}\n\n\\\n\n\\noindent\n{\\bf Totally symmetric instances.}\nThe application we have in mind here is a special case of submodular\nmaximization over the bases of a matroid, which we call {\\em totally symmetric}.\n\n\\begin{definition}\n\\label{def:totally-symmetric}\nWe call an instance $\\max \\{f(S): S \\in {\\cal F} \\}$ totally symmetric with respect\nto a group of permutations ${\\cal G}$, if both $f(S)$ and ${\\cal F}$ are invariant\nunder ${\\cal G}$ and moreover, there is a point ${\\bf c} \\in P({\\cal F})$ such that\n${\\bf c} = \\bar{{\\bf x}} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf x})]$ for every ${\\bf x} \\in P({\\cal F})$.\nWe call ${\\bf c}$ the center of the instance.\n\\end{definition}\n\nNote that this is indeed stronger than just being invariant under ${\\cal G}$.\nFor example, an instance on a ground set $X = X_1 \\cup X_2$ could be symmetric\nwith respect to any permutation of $X_1$ and any permutation of $X_2$.\nFor any ${\\bf x} \\in P({\\cal F})$, the symmetric vector $\\bar{{\\bf x}}$ is constant\non $X_1$ and constant on $X_2$. However, in a totally symmetric instance,\nthere should be a unique symmetric point.\n\n\\paragraph{Bases of partition matroids}\nA canonical example of a totally symmetric instance is as follows.\nLet $X = X_1 \\cup X_2 \\cup \\ldots \\cup X_m$ and let integers $k_1,\\ldots,k_m$\nbe given. This defines a partition matroid ${\\cal M} = (X,{\\cal B})$, whose bases are\n$$ {\\cal B} = \\{B: \\forall j; |B \\cap X_j| = k_j \\}.$$\nThe associated matroid base polytope is\n$$ B({\\cal M}) = \\{{\\bf x} \\geq 0: \\forall j; \\sum_{i \\in X_j} x_i = k_j \\}.$$\nLet ${\\cal G}$ be a group of permutations such that the orbit of each element\n$i \\in X_j$ is the entire set $X_j$.\nThis implies that for any ${\\bf x} \\in B({\\cal M})$,\n$\\bar{{\\bf x}}$ is the same vector ${\\bf c}$, with coordinates $k_j \/ |X_j|$ on $X_j$.\nIf $f(S)$ is also invariant under ${\\cal G}$, we have a totally symmetric\ninstance $\\max \\{f(S): S \\in {\\cal B} \\}$. \n\n\\paragraph{Example: welfare maximization}\nTo present a more concrete example, consider $X_j = \\{ a_{j1}, \\ldots, a_{jk} \\}$\nfor each $j \\in [m]$, a set of bases ${\\cal B} = \\{B: \\forall j; |B \\cap X_j| = 1\\}$,\nand an objective function in the form $f(S) = \\sum_{i=1}^{k} v(\\{j: a_{ji} \\in S\\})$,\nwhere $v:2^{[m]} \\rightarrow {\\boldmath R}_+$ is a submodular function.\nThis is a totally symmetric instance, which captures the welfare maximization\nproblem for combinatorial auctions where each player has the same valuation function.\n(Including element $a_{ji}$ in the solution corresponds to allocating item $j$ to player $i$;\nsee \\cite{CCPV09} for more details.) We remark that here we consider possibly\nnonmonotone submodular functions, which is not common for combinatorial auctions;\nnevertheless the problem still makes sense.\n\n\\\n\nWe show that for such instances, the center point achieves an improved approximation.\n\n\\begin{theorem}\n\\label{thm:sym-solution}\nLet $\\max \\{f(S): S \\in {\\cal B}\\}$ be a totally symmetric instance.\nLet the fractional packing number of bases be $\\nu$ and the fractional\npacking number of dual bases $\\nu^*$. Then the center point ${\\bf c}$\nsatisfies\n$$ F({\\bf c}) \\geq \\left(1 - \\frac{1}{2\\nu} - \\frac{1}{2\\nu^*} \\right) OPT.$$\n\\end{theorem}\n\nRecall that in the general case, we get a $\\frac12 (1-1\/\\nu-o(1))$-approximation\n(Theorem~\\ref{thm:submod-bases-approx}). By passing to the dual matroid,\nwe can also obtain a $\\frac12 (1-1\/\\nu^*-o(1))$-approximation,\nso in general, we know how to achieve a $\\frac12 (1-1\/\\max\\{\\nu,\\nu^*\\}-o(1))$-approximation.\nFor totally symmetric instances where $\\nu=\\nu^*$, we improve this\nto the optimal factor of $1-1\/\\nu$.\n\n\\begin{proof}\nSince there is a unique center ${\\bf c} = \\bar{{\\bf x}}$ for any ${\\bf x} \\in B({\\cal M})$,\nthis means this is also the symmetric optimum $F({\\bf c}) = \\max \\{ F(\\bar{{\\bf x}}): {\\bf x} \\in B({\\cal M})\\}$.\nDue to Lemma~\\ref{lemma:sym-local-opt}, ${\\bf c}$ is a local optimum\nfor the problem $\\max \\{F({\\bf x}): {\\bf x} \\in B({\\cal M})\\}$.\n\nBecause the fractional packing number of bases is $\\nu$, we have $c_i \\leq 1\/\\nu$\nfor all $i$. Similarly, because the fractional packing number of dual bases\n(complements of bases) is $\\nu^*$,\nwe have $1-c_i \\leq 1\/\\nu^*$. This means that ${\\bf c} \\in [1-1\/\\nu^*,1\/\\nu]$.\nLemma~\\ref{lemma:base-local-opt} implies that\n$$ 2 F({\\bf c}) \\geq \\left( 1-\\frac{1}{\\nu} + 1-\\frac{1}{\\nu^*} \\right) OPT.$$\n\\end{proof}\n\n\\begin{corollary}\nLet $\\max \\{f(S): S \\in {\\cal B}\\}$ be an instance\non a partition matroid where every base takes at least\nan $\\alpha$-fraction of each part, at most a $(1-\\alpha)$-fraction of each part,\nand the submodular function $f(S)$ is invariant under a group ${\\cal G}$\nwhere the orbit of each $i \\in X_j$ is $X_j$.\nThen, the center point ${\\bf c} = {\\bf E}_{\\sigma \\in {\\cal G}}[\\sigma({\\bf 1}_B)]$\n(equal for any $B \\in {\\cal B}$) satisfies $F({\\bf c}) \\geq \\alpha \\cdot OPT$.\n\\end{corollary}\n\n\\begin{proof}\nIf the orbit of any element $i \\in X_j$ is the entire set $X_j$,\nit also means that $\\sigma(i)$ for a random $\\sigma \\in {\\cal G}$\nis uniformly distributed over $X_j$ (by the transitive property of ${\\cal G}$).\nTherefore, symmetrizing any fractional vector ${\\bf x} \\in B({\\cal M})$ gives\nthe same vector $\\bar{{\\bf x}} = {\\bf c}$, where $c_i = k_j \/ |X_j|$ for $i \\in X_j$.\nAlso, our assumptions mean that the fractional packing number of bases\nis $1\/(1-\\alpha)$, and the fractional packing number of dual bases is\nalso $1\/(1-\\alpha)$.\nDue to Lemma~\\ref{thm:sym-solution}, the center ${\\bf c}$ satisfies\n$F({\\bf c}) \\geq \\alpha \\cdot OPT$.\n\\end{proof}\n\nThe hard instances for submodular maximization over matroid bases\nthat we describe in Section~\\ref{section:hardness-applications} are exactly of this form\n(see the last paragraph of Section~\\ref{section:hardness-applications}, with $\\alpha=1\/k$).\nThere is a unique symmetric solution,\n$x = (\\alpha,\\alpha,\\ldots,\\alpha, 1-\\alpha,1-\\alpha,\\ldots,1-\\alpha)$.\nThe fractional base packing number for these matroids is\n$\\nu = 1\/(1-\\alpha)$ and Theorem~\\ref{thm:general-hardness} implies that\nany $(\\alpha+\\epsilon) = (1-1\/\\nu+\\epsilon)$-approximation for such matroids would\nrequire exponentially many value queries. Therefore, our approximation\nin this special case is optimal.\n\n\n\n\\section*{Acknowledgment}\nThe author would like to thank Jon Lee and Maxim Sviridenko for helpful discussions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n In \\cite{Barrett-Suli}, the authors established the existence of global-in-time weak solutions to the Navier--Stokes--Fokker--Planck equations arising in the kinetic theory of dilute polymer solutions and describing a large class of bead-spring chain models with finitely extensible nonlinear elastic (FENE) type spring potentials. For $\\O\\subset \\R^3$ a bounded domain, the solvent density $\\vr$ and the solvent velocity field $\\vu$ satisfy the following equations in the space-time cylinder $(0,T]\\times \\O$, $T>0$:\n\\be\\label{i1}\n\\d_t \\vr + \\Div_x (\\vr \\vu) = 0,\n\\ee\n\\be\\label{i2}\n \\d_t (\\vr\\vu)+ \\Div_x (\\vr \\vu \\otimes \\vu) +\\nabla_x p(\\vr) -\n\\Div_x \\SSS =\\Div_x \\TT +\\vr\\, \\ff,\n\\ee\nwhich we assume here to be supplemented with the no-slip boundary condition\n\\be\\label{i4}\n\\vu=\\mathbf{0}\\quad \\mbox{on}\\ (0,T]\\times \\partial\\Omega.\n\\ee\n\nIgnoring the effect of temperature changes, we consider a barotropic pressure law\n\\be\\label{pressure}\np = p(\\vr), \\ p(\\vr) \\approx \\vr^\\gamma \\ \\mbox{for large values of}\\ \\vr.\n\\ee\nThe {\\em Newtonian stress tensor} $\\SSS$ is defined by\n\\be\\label{i3}\n\\SSS  = \\mu^S \\left( \\frac{\\nabla_x \\vu + \\nabla^{\\rm T}_x \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B (\\Div_x \\vu) \\II,\n\\ee\nwith the shear and bulk viscosity coefficients $\\mu^S$ and $\\mu^B$ defined below (see \\eqref{mu-eta1}). In contrast with \\cite{Barrett-Suli},  where the shear and bulk viscosity coefficients are taken to be constant, $\\mu^S >0$ and $\\mu^B \\geq 0$, we consider here the case when they are functions of the polymer number density. Such an extension requires nontrivial modifications of the method used in \\cite{Barrett-Suli}, and represents the main contribution of the present paper.\n\nIn \\emph{a bead-spring chain model} consisting of $K+1$ beads coupled with $K$ elastic springs representing a polymer chain,  the non-Newtonian elastic extra stress tensor $\\TT$ is defined by a version of the Kramers expression (cf. \\eqref{def-TT0} below), depending on the probability density function $\\psi$, which, in addition to $t$ and $x$, also depends on the conformation vector $(q_1^{\\rm T},\\dots q_K^{\\rm T})^{\\rm T} \\in \\R^{3K}$, with $q_i$ representing the $3$-component \\emph{conformation\/orientation vector} of the $i$th spring in the chain. Let $D:=D_1\\times \\cdots \\times D_K \\subset \\R^{3K}$ be the domain of admissible conformation vectors. Typically $D_i$ is the whole space $\\R^3$ or a bounded open ball centered at the origin $0$ in $\\R^3$, for each $i=1,\\dots,K$. When $K=1$, the model is referred to as the \\emph{dumbbell model}.  Here we focus on finitely extensible nonlinear (or, briefly, FENE-type) bead-spring-chain models where $D_i=B(0,b_i^{\\frac12})$, a ball centered at the origin $0$ in $\\mathbb{R}^3$ and of radius $b_i^{\\frac12}$, with $b_i>0$ for each $i \\in \\{1,\\dots, K\\}$. The {\\em extra-stress tensor} $\\TT$ is defined by the formula:\n\\be\\label{def-TT0}\n\\TT (\\psi)(t,x) := \\TT_1 (\\psi) (t,x) -\\left(\\int_{D\\times D} \\gamma(q,q')\\, \\psi(t,x,q)\\,\\psi(t,x,q') \\, \\dq\\,\\dq'\\right)\\II,\n\\ee\nwhere, similarly to \\cite{Barrett-Suli}, the interaction kernel $\\g$ is assumed to be a positive constant $\\gamma(q,q')\\equiv \\de>0.$\nConsequently,\n\\be\\label{def-TT}\n\\TT (\\psi) := \\TT_1 (\\psi)  -\\de \\left(\\int_{D}  \\psi \\,\\dq\\right)^2\\II.\n\\ee\nThe first part, $\\TT_1(\\psi)$, of $\\TT(\\psi)$ is given by the {\\em Kramers expression}\n\\be\\label{def-TT1}\n\\t_1(\\psi) := k \\left[\\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-(K+1) \\left(\\int_{D}  \\psi \\ \\dq\\right)\\II\\right],\n\\ee\nwhere $k>0$ is the product of the Boltzmann constant and the absolute temperature and\n\\be\\label{def-Ci}\n\\CC_{i}(\\psi)(t,x) := \\int_D \\psi(t,x,q)\\, U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg)\\, q_i q_i^{\\rm T}\\,\\dq, \\quad i=1,\\dots,K.\n\\ee\n\nIn the expression \\eqref{def-Ci}, the smooth functions $U_i : [0,\\frac{b_i}{2}) \\to [0,\\infty)$,  $i=1,\\dots,K,$ are the spring potentials satisfying\n$U_i(0)=0$, $\\lim_{s\\to \\frac{b_i}{2}-} U_i(s) =+\\infty$.\nWe introduce the {\\em partial Maxwellian} $M_i : D_i \\to [0,\\infty)$ by\n\\be\\label{def-Mi}\nM_i(q_i) := \\frac{1}{Z_i} {\\rm e}^{- U_i\\big(\\frac{|q_i|^2}{2}\\big)},\\quad \\mbox{where } Z_i:=\\int_{D_i} \\mathrm{e}^{- U_i\\big(\\frac{|p_i|^2}{2}\\big)}\\,{\\rm d}p_i.\n\\ee\nThe {\\em Maxwellian} $M : D \\to [0,\\infty)$ is then defined as the product of the $K$ partial Maxwellians: i.e., for any $q=(q_1^{\\rm T},\\dots, q_K^{\\rm T})^{\\rm T}$ in $D = D_1\\times\\cdots \\times D_K$, we have that\n\\[M(q) := \\prod_{i=1}^K M_i(q_i).\\]\nClearly,\n$\n\\int_D M(q)\\ \\dq=1.\n$\nBy direct calculations one verifies that, for any $i \\in \\{1,\\dots,K\\}$,\n\\be\\label{pt2-M}\nM(q)\\, \\nabla_{q_i} (M(q))^{-1} = - M(q)^{-1}\\, \\nabla_{q_i} M(q) = \\nabla_{q_i}\\left(U_i\\bigg(\\frac{|q_i|^2}{2}\\bigg)\\right) = U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg) q_i\n\\quad \\forall\\,q = (q_1^{\\rm T},\\dots,q_K^{\\rm T})^{\\rm T}\\in D.\n\\ee\n\nAs in \\cite{Barrett-Suli}, we shall suppose that, for any $i\\in\\{1,\\dots, K\\}$, there exist positive constants $c_{ij}, \\ j=1,\\dots,4$, and $\\th_i > 1$ such that\n\\ba\\label{pt1-Mi-Ui}\nc_{i1}\\left({\\rm dist}\\,(q_i,\\d D_i)\\right)^{\\th_i} \\leq M_i(q_i)\\leq c_{i2}\\left({\\rm dist}\\,(q_i,\\d D_i)\\right)^{\\th_i},\\quad\nc_{i3} \\leq \\left({\\rm dist}\\,(q_i,\\d D_i)\\right) U_i'\\left(\\frac{|q_i|^2}{2}\\right)\\leq c_{i4} \\quad \\forall\\,q_i\\in D_i.\n\\ea\nIt is then straightforward to deduce that\n\\be\\label{pt3-Mi-Ui}\n\\int_{D_i} \\left(1+ \\left( U_i\\bigg(\\frac{|q_i|^2}{2}\\bigg) \\right)^2+ \\left( U_i'\\bigg(\\frac{|q_i|^2}{2}\\bigg) \\right)^2  \\right) M_i(q_i)\\,\\dq_i <\\infty,\\quad i=1,\\dots, K.\n\\ee\n\n\n\nThe probability density function $\\psi$ satisfies the following {\\em Fokker--Planck equation} in $(0,T]\\times \\O\\times D$:\n\\ba\\label{eq-psi}\n\\d_t \\psi + \\Div_x (\\vu\\,\\psi) + \\sum_{i=1}^K \\Div_{q_i} \\left(  (\\nabla_x \\vu)\\, q_i\\, \\psi \\right)= \\e \\Delta_x \\psi+\\frac{1}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\, \\Div_{q_i}\\left( M \\nabla_{q_j} \\left( \\frac{\\psi}{M} \\right)\\right).\n\\ea\n\nThe \\emph{centre-of-mass diffusion} term $\\e \\Delta_x \\psi$ is generally of the form\n$\n\\e \\Delta_x \\left( \\frac{\\psi}{\\zeta(\\vr)} \\right),\n$\nwhich involves the {\\em drag coefficient} $\\zeta(\\cdot)$ depending on the fluid density $\\vr$. Here we assume that $\\zeta$ is a constant function, which is, for simplicity, taken to be identically $1$. The constant parameter $\\e$ is the {\\em centre-of-mass diffusion coefficient}, which is strictly positive.\nThe positive parameter $\\l$ is called the \\emph{Deborah number}; it characterizes the elastic relaxation property of the fluid. The constant matrix $A=(A_{ij})_{1\\leq i,j\\leq K}$, called the \\textit{Rouse matrix}, is symmetric and positive definite. We denote by $A_0$ the smallest eigenvalue of $A$; clearly, $A_0>0$. We refer to Section 1 of Barrett and S\\\"uli \\cite{Barrett-Suli1} for a derivation of the Fokker--Planck equation \\eqref{eq-psi}.\n\nThe Fokker--Planck equation needs to be supplemented by suitable boundary conditions. For any $i=1,\\dots, K$, let $\\d \\overline D_i:= D_1\\times \\cdots \\times D_{i-1} \\times \\d D_i \\times D_{i+1} \\cdots \\times D_K$ and suppose that\n\\ba\\label{boundary-psi}\n\\left(\\frac{1}{4\\lambda} \\sum_{j=1}^K A_{ij}\\, \\Div_{q_i}\\left(M \\nabla_{q_j} \\left( \\frac{\\psi}{M} \\right)\\right) -(\\nabla_x \\vu) \\, q_i\\, \\psi  \\right)\\cdot \\frac{q_i}{|q_i|}&=0 \\quad &&\\mbox{on}\\ (0,T]\\times \\O \\times \\d \\overline D_i,\\\\\n \\nabla_x\\psi \\cdot {\\bf n}&=0 \\quad &&\\mbox{on}\\ (0,T]\\times \\d\\O \\times D.\n\\ea\n\nFinally,  we introduce the {\\em polymer number density} $\\eta$ defined by\n\\be\\label{def-eta}\n\\eta(t,x) := \\int_D \\psi(t,x,q)\\,\\dq, \\qquad (t,x)\\in [0,T]\\times \\O.\n\\ee\nBy (formally) integrating the partial differential equation \\eqref{eq-psi} over $D$ and using the boundary condition in $\\eqref{boundary-psi}_1$, and by integrating the boundary condition $\\eqref{boundary-psi}_2$ over $D$, we deduce the following partial differential equation and boundary condition for the function $\\eta$:\n\\be\\label{eq-eta}\n\\d_t \\eta + \\Div_x (\\vu \\,\\eta) = \\e \\Delta_x \\eta\\quad \\mbox{in}\\ (0,T] \\times \\O;\\qquad \\nabla_x \\eta \\cdot {\\bf n}=0 \\quad \\mbox{on}\\  (0,T]\\times \\d\\O.\n\\ee\nBy noting \\eqref{def-eta} we see that the expression for the extra-stress tensor in \\eqref{def-TT} and \\eqref{def-TT1} can also be expressed as follows:\n\\be\\label{def-TT-f}\n\\TT (\\psi) := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta  + \\de\\,  \\eta^2\\right) \\II.\n\\ee\nAs has already been pointed out above, our main objective is to consider a class of models of this form where the viscosity coefficients $\\mu^S = \\mu^S(\\eta)$ and $\\mu^B = \\mu^B(\\eta)$\ndepend on $\\eta$.\n\n\n\n\\subsection{Dissipative (finite-energy) weak solutions}\n\nWe adopt the following hypotheses on the initial data:\n\\ba\\label{ini-data}\n&\\vr(0,\\cdot) = \\vr_0(\\cdot) \\ \\mbox{with}\\ \\vr_0 \\geq 0 \\ {\\rm a.e.} \\ \\mbox{in} \\ \\O, \\quad \\vr_0 \\in L^\\gamma(\\O) \\ \\mbox{with}\\ \\gamma>\\textstyle{\\frac32};\\\\\n&\\vu(0,\\cdot) = \\vu_0(\\cdot) \\in L^r(\\O;\\R^3) \\ \\mbox{for some $r>1$}\\ \\mbox{such that}\\ \\vr_0|\\vu_0|^2 \\in L^1(\\O);\\\\\n&\\psi(0,\\cdot) = \\psi_0(\\cdot) \\ \\mbox{with}\\ \\psi_0 \\geq 0 \\ \\mbox{a.e. in}  \\ \\O\\times D,\\quad  \\psi_0 \\left(\\log \\frac{\\psi_0}{M}\\right) \\in L^1(\\O\\times D);\\\\\n&\\eta(0,\\cdot)=\\int_{D}\\psi_0 \\,\\dq =:\\eta_0 \\in L^2 (\\O).\n\\ea\nWe deduce from $\\eqref{ini-data}_1$ and $\\eqref{ini-data}_2$ by using H\\\"older's inequality that $(\\vr\\vu)(0,\\cdot) = \\vr_0 \\vu_0 =\\sqrt{\\vr_0}\\sqrt{\\vr_0} \\vu_0 \\in L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3)$.\n\n\\begin{definition}\\label{def-weaksl} We say that $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy)\nweak solution in $(0,T]\\times \\O\\times D$ to the system of equations\n\\eqref{i1}--\\eqref{i3},  \\eqref{eq-psi}--\\eqref{def-TT-f}, supplemented by the initial data \\eqref{ini-data}, if:\n\\begin{itemize}\n\n\\item $\\vr \\geq 0 \\ {\\rm a.e.\\  in} \\ (0,T] \\times \\Omega,\\quad  \\vr \\in  C_w ([0,T];  L^\\gamma(\\Omega)),\\quad \\vu\\in L^{r}(0,T;W_0^{1,r}(\\Omega; \\R^3))\\quad \\mbox{for some $r>1$},$\n\\ba\\label{weak-est}\n&\\vr \\vu \\in C_w([0,T]; L^{\\frac{2 \\gamma}{ \\gamma + 1}}(\\Omega; \\R^3)),\\quad \\vr |\\vu|^2 \\in L^\\infty(0,T; L^{1}(\\Omega));\\\\\n&\\psi \\geq 0 \\ {\\rm a.e.\\  in} \\ (0,T] \\times \\Omega \\times D, \\quad \\psi \\in C_{w}([0,T];L^1(\\Omega\\times D)),\\\\\n& \\nabla_x \\psi\\in L^1((0,T)\\times \\O\\times D;\\R^3), \\quad  M \\nabla_{q} \\left(\\frac{\\psi}{M}\\right)\\in L^1((0,T)\\times \\O\\times D;\\R^{3K}),\\\\\n&\\eta =\\int_D \\psi\\ \\dq  \\ {\\rm a.e.\\  in} \\ (0,T] \\times \\Omega,\\quad \\eta \\in C_w ([0,T];  L^2(\\Omega)) \\cap L^2 (0,T;  W^{1,2}(\\Omega)),\\\\\n&\\TT (\\psi) := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta  + \\de\\,  \\eta^2\\right) \\II \\quad {\\rm a.e.\\  in} \\ (0,T] \\times \\Omega,\\quad  \\TT\\in L^1((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ea\n\n\n\\item For any $t \\in (0,T]$ and any test function $\\phi \\in C^\\infty([0,T] \\times \\Ov{\\Omega})$, one has\n\n\\be\\label{weak-form1}\n\\int_0^t\\intO{\\big[ \\vr \\partial_t \\phi + \\vr \\vu \\cdot \\Grad \\phi \\big]} \\,\\dt' =\n\\intO{\\vr(t, \\cdot) \\phi (t, \\cdot) } - \\intO{ \\vr_{0} \\phi (0, \\cdot) },\n\\ee\n\\be\\label{weak-form2}\n\\int_0^t \\intO{ \\big[ \\eta \\partial_t \\phi + \\eta \\vu \\cdot \\Grad \\phi - \\e \\nabla_x\\eta \\cdot \\nabla_x \\phi \\big]} \\, \\dt' =  \\intO{ \\eta(t, \\cdot) \\phi (t, \\cdot) } - \\intO{ \\eta_{0} \\phi (0, \\cdot) }.\n\\ee\n\n\\item For any $t \\in (0,T]$ and any test function $\\vvarphi \\in \\DC([0,T] \\times {\\Omega};\\R^3)$, one has\n\\ba\\label{weak-form3}\n&\\int_0^t \\intO{ \\big[ \\vr \\vu \\cdot \\partial_t \\vvarphi + (\\vr \\vu \\otimes \\vu) : \\Grad \\vvarphi  + p(\\vr)\\, \\Div_x \\vvarphi - \\SSS : \\Grad \\vvarphi \\big] } \\, \\dt'\\\\\n&= \\int_0^t \\intO{ \\TT : \\nabla_x\\vvarphi - \\vr\\, \\ff \\cdot \\vvarphi} \\, \\dt' + \\intO{ \\vr \\vu (t, \\cdot) \\cdot \\vvarphi (t, \\cdot) } - \\intO{ \\vr_{0} \\vu_{0} \\cdot \\vvarphi(0, \\cdot) },\n\\ea\nwhere $\\SSS$ is defined by \\eqref{i3}.\n\n\\item For any $t \\in (0,T]$ and any test function $\\phi \\in \\DC([0,T] \\times \\Ov{\\Omega}\\times D)$, one has\n\\ba\\label{weak-form4}\n&\\int_0^t \\intO{\\int_D \\bigg[ \\psi \\partial_t \\phi + \\psi \\vu \\cdot \\Grad \\phi  + \\sum_{i=1}^K (\\nabla_x \\vu)\\, q_i\\, \\psi \\cdot \\nabla_{q_i} \\phi - \\e\\nabla_x  \\psi \\cdot \\nabla_x \\phi \\bigg]\\,\\dq}\\,\\dt'\\\\\n&=\\frac{1}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\int_0^t \\intO{ \\int_D  M\\nabla_{q_j}\\left(\\frac{\\psi}{M}\\right)\\cdot \\nabla_{q_i}\\phi\\ \\dq} \\, \\dt' + \\intO{ \\int_D \\psi \\phi (t, \\cdot)- \\psi_{0}  \\phi(0, \\cdot)\\,\\dq }.\n\\ea\n\n\\item The continuity equation holds in the sense of renormalized solutions:\n\\be\\label{weak-renormal}\n\\d_t b(\\vr) +\\Div_x (b(\\vr)\\vu) + (b'(\\vr)\\vr - b(\\vr))\\,\\Div_x \\vu =0 \\quad \\mbox{in} \\ \\mathcal{D}' ((0,T]\\times \\O),\n\\ee\nfor any\n\\be\\label{cond-b-renormal}\nb\\in C^1([0,\\infty)),\\quad |b'(s)s|+| b(s)| \\leq c<\\infty \\qquad \\forall\\, s\\in [0,\\infty).\n\\ee\n\\item  Let $\\mathcal{F}(s) := s(\\log s -1)+1$ for $s>0$ and define $\\mathcal{F}(0):=\\lim_{s\\to 0+} \\mathcal{F}(s)=1$. For a.e. $t \\in (0,T]$, the\nfollowing \\emph{energy inequality} holds:\n\\ba\\label{energy}\n&\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr |\\vu|^2 +  P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\ \\dq\\bigg] (t, \\cdot) \\, \\dx\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S \\bigg|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu)\\, \\II\\, \\bigg|^2 +\\mu^B |\\Div_x \\vu|^2  } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{   |\\nabla_x \\eta|^2} \\, \\dt'\\\\\n&\\quad+ \\e\\, k \\int_0^t \\intO{ \\int_D M\\bigg|\\nabla_x \\sqrt{\\widetilde \\psi}\\, \\bigg|^2 \\, \\dq} \\,\\dt'\n+ \\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\bigg|\\nabla_{q}  \\sqrt{\\widetilde \\psi}\\,\\bigg|^2 \\, \\dq} \\, \\dt'\\\\\n&\\leq \\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 +  P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\, \\dq\\bigg]  \\dx + \\int_0^t\\int_\\O \\vr \\,\\ff \\cdot \\vu \\,\\,\\dx\\, \\dt',\n\\ea\n where we have set $P(\\vr) := \\vr \\int_1^\\vr p(z)\/ z^2 \\, {\\rm d}z$ and $\\widetilde \\psi:=\\frac{\\psi}{M}$.\n\n\\end{itemize}\n\n\\end{definition}\n\n\n\n\\begin{remark} Definition \\ref{def-weaksl} is fairly standard. The energy\ninequality (\\ref{energy}) identifies an important class of weak solutions, usually termed \\emph{dissipative (finite-energy)}.\nWe note that, given a \\emph{smooth} solution, by tedious but rather straightforward calculations one can obtain the following a priori bound (see (1.22) in \\cite{Barrett-Suli}):\n\\ba\\label{energy0}\n&\\int_{\\Omega} \\left[\n\\frac{1}{2} \\vr |\\vu|^2 +  P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\, \\dq\\right] (t, \\cdot) \\, \\dx+ \\int_0^t \\intO{ \\SSS : \\Grad \\vu\\, } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{   |\\nabla_x \\eta|^2}\\, \\dt'\\\\\n& \\quad + \\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi} \\,\\right|^2 \\dq} \\, \\dt' + \\frac{k}{4\\lambda} \\sum_{i=1}^K\\sum_{j=1}^K A_{ij}\\int_0^t \\intO{ \\int_D M \\nabla_{q_j}  \\sqrt{\\widetilde \\psi} \\cdot \\nabla_{q_i}  \\sqrt{\\widetilde \\psi} \\, \\dq} \\, \\dt' \\\\\n&\\leq \\int_{\\Omega} \\left[\n\\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 +  P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\, \\dq\\right]\\!\\dx + \\int_0^t\\int_\\O \\vr \\,\\ff \\cdot \\vu \\, \\dx\\,\\dt',\\nn\n\\ea\nwhich then implies (\\ref{energy}). Indeed, thanks to the form of the Newtonian stress tensor in \\eqref{i3}, direct calculations yield that\n$$\n\\SSS : \\Grad \\vu = \\mu^S \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right|^2 +\\mu^B |\\Div_x \\vu|^2.\n$$\nHence, by the positive definiteness of the Rouse matrix $A=(A_{ij})_{1\\leq i,j\\leq K}$, and recalling that the smallest eigenvalue of $A$ is $A_0>0$, we deduce \\eqref{energy}.\n\\end{remark}\n\n\\subsection{Assumptions and main results}\n\nWe shall suppose that both $\\mu^S$ and $\\mu^B$ are $C^1$ functions of the polymer number density $\\eta$, and we adopt the following assumptions: there exist positive constants $c_j, \\ j=1,\\dots,5$, and an $\\o\\in\\R$ such that\n\\be\\label{mu-eta1}\nc_{1} (1+\\eta)^\\o   \\leq \\mu^S(\\eta) \\leq c_2 (1+\\eta)^\\o, \\quad  |(\\mu^S)'(\\eta)| \\leq c_3+c_4 (1+\\eta)^{\\o-1}, \\quad 0   \\leq \\mu^B(\\eta) \\leq c_5 (1+\\eta)^\\o\n\\quad \\forall\\,\\eta\\geq 0.\n\\ee\nIn addition, for the sake of simplicity, we shall assume that\n\\be\\label{press}\np(\\vr) = a \\vr^\\gamma, \\quad a > 0, \\ \\gamma > \\frac32.\n\\ee\n\nAs the complete proof of the existence of dissipative weak solutions in the special case of constant viscosity coefficients is already very long and technical (cf. \\cite{Barrett-Suli}), in the more general setting of polymer-number-dependent viscosity coefficients considered here we shall confine ourselves to establishing \\emph{weak sequential stability} of the family of dissipative weak solutions, whose existence we shall assume; we shall however indicate in Section \\ref{End} the main steps of a possible complete existence proof in the case of polymer-number-density-dependent viscosity coefficients.\n\nAccordingly, the main result of the paper reads as follows.\n\n\\begin{theorem}[Weak Sequential Stability]\\label{theorem} Let $\\{(\\vr_n,\\vu_n,\\psi_n,\\eta_n)\\}_{n\\in \\N}$ be a sequence of dissipative (finite-energy) weak solutions in the sense of Definition \\ref{def-weaksl} associated with the initial data $\\{(\\vr_{0,n},\\vu_{0,n},\\psi_{0,n},\\eta_{0,n})\\}_{n\\in\\N}$ satisfying:\n\\ba\\label{cond-ini-n}\n&\\vr_{0,n} \\geq 0 \\ \\mbox{a.e. in} \\ \\O, \\quad \\vr_{0,n} \\to \\vr_0 \\ \\mbox{strongly in}\\ L^\\g(\\O);\\\\\n&\\vu_{0,n} \\to \\vu_0 \\ \\mbox{in}\\  L^r(\\O;\\R^3) \\ \\mbox{for some $r>1$}\\ \\mbox{such that}\\ \\vr_{0,n}|\\vu_{0,n}|^2 \\to \\vr_0|\\vu_0|^2 \\ \\mbox{strongly in}\\  L^1(\\O);\\\\\n& \\psi_{0,n} \\geq 0 \\ \\mbox{a.e. in} \\ \\O\\times D,\\quad \\psi_{0,n} \\to \\psi_{0},\\quad \\psi_{0,n} \\left(\\log \\frac{\\psi_{0,n}}{M}\\right) \\to \\psi_0 \\left(\\log \\frac{\\psi_0}{M}\\right) \\ \\mbox{strongly in}\\  L^1(\\O\\times D);\\\\\n&\\eta_{0,n} = \\int_{D}\\psi_{0,n} \\, \\dq \\to \\eta_0 \\ \\mbox{strongly in} \\ L^2 (\\O).\n\\ea\n\nLet $\\vc{f} \\in L^\\infty((0,T) \\times \\Omega;\\R^3)$.\nSuppose that the exponent $\\g$ in \\eqref{press} and the parameter $\\o$ in \\eqref{mu-eta1} satisfy\n\\be\\label{mu-eta2}\n0\\leq \\o < \\frac{5}{3}, \\quad \\g>\\frac{3}{2}\n\\ee\nor\n\\be\\label{mu-eta3}\n -\\frac{4}{3}<\\o\\leq 0, \\quad \\g>\\frac{6}{4+3\\o}.\n\\ee\nThen, there exists a subsequence (not indicated) such that\n$$\n(\\vr_n,\\vu_n,\\psi_n,\\eta_n) \\to (\\vr,\\vu,\\psi,\\eta) \\quad \\mbox{as $n\\to \\infty$, in the sense of distributions (at least weakly in}\\ L^1),\n$$\nwhere the limit $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl} associated with the initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$.\n\n\n\\end{theorem}\n\nBefore embarking on the proof of Theorem \\ref{theorem} two remarks are in order.\n\n\\begin{remark}\\label{rem-thm1} The strong convergence assumptions in \\eqref{cond-ini-n} imply that\n\\ba\\label{cond-ini-n1}\n\\vr_{0,n} \\vu_{0,n} \\to \\vr_{0} \\vu_{0}  \\quad \\mbox{strongly in}\\ L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^3),\\qquad \\eta_0 = \\int_D \\psi_0\\, \\dq  \\quad \\mbox{a.e. in}\\ \\O.\n\\ea\n\n\n\\end{remark}\n\n\\begin{remark}\\label{rem-thm2} It is important to note that we allow the viscosity coefficients $\\mu^B$ and $\\mu^S$ to decay to zero as the polymer number density tends to infinity; this is achieved at the expense of assuming a larger adiabatic exponent $\\g$; cf. \\eqref{mu-eta3}.\n\n\\end{remark}\n\n\n\nThe bulk of the rest of the paper is devoted to the proof of Theorem \\ref{theorem}. Comments on the possibility of carrying out a complete proof of the existence\nof dissipative (finite-energy) weak solutions are given in Section \\ref{End}.  Throughout the rest of the paper, if there is no specification, $c$ will denote a positive constant depending only on the length $T$ of the time interval and the following quantity associated with the initial data:\n$$\n\\sup_{n\\in \\N}\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_{0,n} |\\vu_{0,n} |^2 +  P(\\vr_{0,n}) + \\de\\, \\eta_{0,n}^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_{0,n}}{M}\\bigg)\\, \\dq\\bigg] \\dx.\n$$\nWe emphasize, however, that the value of $c$ may vary from line to line.\n\n\\section{Preliminaries}\n\nIn this section, we recall some concepts that will be used systematically throughout the rest of the paper, including Maxwellian-weighted Lebesgue and Sobolev spaces, embeddings of spaces of Banach-space-valued weakly-continuous functions, the Div-Curl lemma, and Riesz operators.\n\n\\subsection{Maxwellian-weighted spaces}\\label{sec:def-LMr-HM1}\nFor any $r \\in [1,\\infty)$, $L_M^r(D)$ denotes the Maxwellian-weighted Lebesgue space over $D$ with norm\n\\be\\label{def-LMD1}\n\\|u\\|_{L^r_M(D)}:=\\left(\\int_{ D} M |u(x)|^r\\, \\dq \\right)^{\\frac{1}{r}}.\n\\ee\nSimilarly, we define $L_M^r(\\O\\times D):=L^r(\\O;L^r_M(D))$ and the Maxwellian-weighted Sobolev spaces\n\\ba\\label{def-SMD2}\nH^1_M(D)&:=\\left\\{ u \\in L_{\\rm loc}^1( D) :  \\|u\\|_{H^1_M( D)}^2:=\\int_{ D} M\\big( |u|^2 + |\\nabla_q u|^2 \\big)\\, \\dq<\\infty \\right\\}.\\\\\nH^1_M(\\O\\times D)&:=\\left\\{ u \\in L_{\\rm loc}^1(\\O\\times D) :  \\|u\\|_{H^1_M(\\O\\times D)}^2:=\\int_{\\O\\times D} M\\big( |u|^2 + |\\nabla_x u|^2 +|\\nabla_q u|^2\\big)\\, \\dq \\ \\dx<\\infty \\right\\}.\\nonumber\n\\ea\nThe proof of the following lemma can be found in Appendix C and Appendix D in \\cite{Barrett-Suli2}.\n\\begin{lemma}\\label{lem:HM1-emb}  The normed spaces $L^r_M (D)$, $L_M^r(\\O\\times D)$, $H^1_M(D)$ and $H^1_M(\\O \\times D)$ are Banach spaces.\nThe embedding $H^1(\\O;L_M^2(D)) \\hookrightarrow L^6(\\O;L_M^2(D))$ is continuous, and the embeddings\n$H^1_M(D) \\hookrightarrow L_M^2(D)$, $H^1_M(\\O\\times D)  \\hookrightarrow L^2_M(\\O\\times D)$ are compact.\n\\end{lemma}\n\n\\subsection{On $C_w([0, T];X)$ type spaces}\n\nLet $X$ be a Banach space. We denote by $C_w([0, T];X)$ the set of all functions $u\\in L^\\infty(0, T;X)$ such that the mapping\n$t \\in  [0, T] \\mapsto \\langle \\phi,u(t)\\rangle_{X}\\in \\R$ is continuous on $[0, T]$ for all $\\phi \\in X'$. Here and throughout the paper, we use $X'$ to denote the\ndual space of $X$, and $\\langle \\cdot,\\cdot \\rangle_{X}$ to denote the duality pairing between $X'$ and $X$.\n\nWhenever $X$ has a predual $E$, in the sense that $E'=X$, we denote by $C_{w*}([0, T];X)$ the set of all functions $u\\in L^\\infty(0, T;X)$ such that the mapping $t \\in  [0, T] \\mapsto \\langle u(t),\\phi\\rangle_{E}\\in \\R$ is continuous on $[0, T]$ for all $\\phi \\in E$. We reproduce Lemma 3.1 from \\cite{Barrett-Suli}.\n\\begin{lemma}\\label{lem-Cw-Cw*}\nSuppose that $X$ and $Y$ are Banach spaces.\n\\begin{itemize}\n\\item[(i)] Assume that the space $X$ is reflexive and is continuously embedded in the space $Y$; then,\n$$ L^\\infty(0, T;X) \\cap C_w([0, T];Y) = C_w([0, T];X).$$\n\\item[(ii)] Assume that $X$ has a separable predual $E$ and $Y$ has a predual $F$ such that $F$ is continuously\nembedded in $E$; then,\n   $$ L^\\infty(0, T;X) \\cap C_{w*}([0, T];Y) = C_{w*}([0, T];X).$$\n\\end{itemize}\n\n\\end{lemma}\n\nWe recall an Arzel\\`{a}--Ascoli type result in $C_w([0,T];L^s(\\O))$. We refer to Lemma 6.2 in \\cite{N-book} for the proof.\n\\begin{lemma}\\label{lem-Cw} Let $r, s \\in (1,\\infty)$ and let $\\O$ be a bounded Lipschitz domain in $\\R^d, \\ d\\geq 2$. Suppose  $\\{g_n\\}_{n\\in \\N}$ is a sequence of functions in $C_w([0, T]; L^s(\\O))$ such that $\\{g_n\\}_{n\\in \\N}$ is bounded in $C([0, T]; W^{-1,r}(\\O)) \\cap \\ L^\\infty(0, T;L^s(\\O))$. Then, there exists a subsequence (not indicated) such that the following hold:\n\\begin{itemize}\n\\item[(i)] $g_n\\to g$ in $C_w([0, T]; L^s(\\O))$;\n\\item[(ii)] If, in addition, $r\\leq \\frac{d}{d-1}$, or $r>\\frac{d}{d-1}$ and $s>\\frac{d\\,r}{d+r}$, then $g_n\\to g$ strongly in $C([0, T]; W^{-1,r}(\\O))$.\n\\end{itemize}\n\\end{lemma}\n\n\\subsection{Div-Curl lemma}\n\nWe recall the celebrated Div-Curl lemma due to L. Tartar \\cite{L.Tartar}.\n\\begin{lemma}\\label{lem-div-curl2}\nLet $Q\\subset \\R^d$ be a domain and let $\\{({\\bf U}_n,{\\bf V}_n)\\}_{n\\in \\N}$ be a sequence of functions such that\n$$\n{\\bf U}_n \\to {\\bf U} \\ \\mbox{weakly in $L^p(Q;\\R^d)$}, \\quad {\\bf V}_n \\to {\\bf V} \\ \\mbox{weakly in $L^{q}(Q;\\R^d)$}, \\quad \\mbox{as $n\\to \\infty$},\n$$\nwhere\n$\n\\frac{1}{p}+\\frac 1q =\\frac{1}{r}<1.\n$\nSuppose in addition that, for some $s>0$,\n$$\n\\{\\Div \\,{\\bf U}_n\\}_{n\\in \\N} \\ \\mbox{is precompact in $W^{-1,s}(Q)$},\\quad \\{{\\rm curl} \\,{\\bf V}_n\\}_{n\\in \\N} \\ \\mbox{is precompact in $W^{-1,s}(Q;\\R^{d\\times d})$}.\n$$\nThen,\n$\n{\\bf U}_n \\cdot {\\bf V}_n \\to {\\bf U} \\cdot {\\bf V} \\ \\mbox{weakly in}\\ L^r(Q).\n$\n\n\\end{lemma}\n\n\\subsection{On Riesz type operators}\\label{sec:Riesz}\n\nThe Riesz operator $\\RR_{j}$, $1\\leq  j \\leq d$, in $\\R^d$ is defined as a Fourier integral operator with symbol $\\frac{\\xi_j}{|\\xi|}$. That is, for any $u\\in \\mathcal{S}'(\\R^d)$, where $\\mathcal{S}'(\\R^d)$ denotes the space of tempered distributions on $\\R^d$, $\\RR_j$ is defined by\n$$\n\\RR_{j}[u] := \\mathfrak{F}^{-1}\\left[\\frac{\\xi_j}{|\\xi|} \\mathfrak{F}[u]\\right],\n$$\nwhere $\\mathfrak{F}$ is the Fourier transform and $\\mathfrak{F}^{-1}$ is the inverse Fourier transform. We then define, for any $u\\in \\mathcal{S}'(\\R^d)$,\n$$\n\\RR_{ij} [u]:=\\RR_i \\circ \\RR_j u= \\mathfrak{F}^{-1}\\left[\\frac{\\xi_i\\xi_j}{|\\xi|^2} \\mathfrak{F}[u]\\right],\\quad 1\\leq i,j\\leq d.\n$$\n\n\nWe define $\\A_j$ by\n$\n\\A_{j} [u] :=- \\mathfrak{F}^{-1}\\left[\\frac{i\\xi_j}{|\\xi|^2} \\mathfrak{F}[u]\\right].\n$\nSince the derivative $\\d_j$ and the Laplacian $\\Delta$ can be seen as Fourier integral operators with symbols $i\\xi_j$ and $-|\\xi|^{-2}$, respectively, we can write\n$$\n\\RR_{ij} = \\d_i\\d_j{\\Delta}^{-1},\\quad  \\A_j = -\\d_{j} \\Delta^{-1}.\n$$\n\nLet the matrix-valued operator $\\RR$ and the vector-valued operator $\\A$ be defined by\n\\be\\label{def-Reize}\n\\RR=(\\RR_{ij})_{i,j=1}^d=(\\nabla\\otimes\\nabla) \\Delta^{-1},\\quad \\A:=(A_j)_{j=1}^d = - \\nabla \\Delta^{-1}.\n\\ee\nWe have\n$\n\\RR_{ij}= - \\d_i \\A_j,\\ \\sum_{j=1}^d \\RR_{jj}= -\\sum_{j}\\d_j\\A_j =\\II.\n$ By Theorem 1.55 and Theorem 1.57 in \\cite{N-book} the following result holds.\n\\begin{lemma}\\label{lem-Riesz1} For any $p \\in (1,\\infty)$ the operators $\\RR_j$ and $\\RR_{ij}$, $1\\leq i, j\\leq d$, are bounded from $L^p(\\R^d)$ to $L^p(\\R^d)$. That is,\nthere exists a positive constant $c=c(p,d)$ such that\n$$\n\\|\\RR_j [u] \\|_{L^p(\\R^d)}+  \\|\\RR_{ij} [u] \\|_{L^p(\\R^d)} \\leq c(p,d) \\|u \\|_{L^p(\\R^d)}\\qquad \\forall\\,u\\in L^p(\\R^d).\n$$\nMoreover, for any $u\\in L^p(\\R^d)$, $v\\in L^{p'}(\\R^d)$, $1<p<\\infty$, we have\n$$\n\\int_{\\R^d} \\RR_{ij}[u]\\,  v \\, \\dx=\\int_{\\R^d} u \\,\\RR_{ij}[v]\\, \\dx.\n$$\nFurther, for any $p \\in (1,d)$, $\\A_j$ is bounded from $L^p(\\R^d)$ to $L^{\\frac{dp}{d-p}}(\\R^d)$; that is, there exists a positive constant $c=c(p,d)$ such that\n$$\n\\|\\A_j [u] \\|_{L^{\\frac{dp}{d-p}}(\\R^d)} \\leq c(p,d) \\|u \\|_{L^p(\\R^d)}\\qquad \\forall\\,u\\in L^p(\\R^d).\n$$\n\\end{lemma}\n\n\n\n\nThe following commutator estimate is taken from Theorem 10.28 in \\cite{F-N-book}, which is in the spirit of Coifman and Meyer \\cite{CM}:\n\\begin{lemma}\\label{lem-Riesz3}\nLet $w\\in W^{1,r}(\\R^d)$ and $v\\in L^p (\\R^d)$ with\n$\n1<r<d,\\ 1<p<\\infty, \\ \\frac{1}{r}+\\frac 1p-\\frac 1d <1.\n$ Then, for any $s>1$ satisfying\n$\n\\frac{1}{r}+\\frac 1p-\\frac 1d < \\frac 1s < \\min\\{1,\\frac{1}{r}+\\frac 1p\\},\n$\nthere exists a constant $c=c(r,p,s,d)$ such that:\n$$\n\\| \\RR_{ij}[w\\,v] - w \\RR_{ij}[v] \\|_{W^{\\b,s}(\\R^d)}\\leq c\\, \\|w\\|_{W^{1,r}(\\R^d)} \\|v\\|_{L^{p}(\\R^d)},\n$$\nfor any $i, j \\in \\{1, \\dots, d\\}$, where $\\b\\in (0,1)$ satisfies\n$\n\\frac{\\b}{d} =\\frac 1d + \\frac 1s - \\frac{1}{r} - \\frac 1p.\n$\n\n\n\\end{lemma}\n\nIn Lemma \\ref{lem-Riesz3}, we used the fractional-order Sobolev space $W^{\\b,s}(\\R^d)$. Let $\\O$ be the whole space $\\R^d$ or a bounded Lipschitz domain in $\\R^d$. For any $\\b \\in (0,1)$ and $s\\in [1,\\infty) $, we define\n\\be\\label{def-frac-Sob}\n W^{\\b,s}(\\O):=\\left\\{u\\in L^s(\\O) : \\| u \\|_{W^{\\b,s}(\\O)}:= \\| u \\|_{L^{s}(\\O)} + \\left(\\int_\\O\\int_\\O  \\frac{|u(x)-u(y)|^s}{|x-y|^{d+\\b s}} \\ \\dx \\ {\\rm d}y\\right)^{\\frac{1}{s}} <\\infty\\right\\}.\n\\ee\nWe recall the following classical compact embedding theorem (see Theorem 7.1 in \\cite{NPV}).\n\\begin{lemma}\\label{lem-frac-sob} Let $\\O\\subset \\R^d$ be a bounded Lipschitz domain and suppose that $\\b \\in (0,1)$ and $s\\in [1,\\infty)$; then, the embedding\nof $W^{\\b,s}(\\O)$ into $L^s(\\O)$ is compact, i.e. $W^{\\b,s}(\\O)\\hookrightarrow \\hookrightarrow L^s(\\O)$.\n\n\\end{lemma}\n\n\n\n\\section{Uniform bounds}\\label{sec:estimate}\n\nLet $(\\vr,\\vu,\\psi,\\eta)$ be a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl} with initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$ satisfying \\eqref{ini-data}. This section is devoted to establishing bounds on $(\\vr,\\vu,\\psi,\\eta)$ under the hypotheses \\eqref{mu-eta1}--\\eqref{mu-eta3}.\n\n\n\\subsection{Gronwall's inequality and uniform bounds}\n\nWe begin by noting that\n\\ba\\label{ini-est2}\n\\int_{\\Omega}  \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\,\\dq \\,\\dx =\\int_{\\Omega}  \\int_D  \\bigg( \\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)-\\psi_0 +M\\bigg)\\,\\dq \\,\\dx.\n\\ea\nAs $ s\\log s \\geq s-1$ for all $s\\geq 0$, it follows that $\\psi_0\\log\\big(\\frac{\\psi_0}{M}\\big)\\geq \\psi_0 - M$, which then implies that\n$$\n\\left|\\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)-\\psi_0 +M\\right|\\leq 2 \\left|\\psi_0\\log\\bigg(\\frac{\\psi_0}{M}\\bigg)\\right|.\n$$\nThus, by \\eqref{ini-data}, we have that\n\\be\\label{ini-estf}\n\\int_\\O \\bigg[\\frac{1}{2} \\vr_{0} |\\vu_{0} |^2 +  P(\\vr_0) + \\de\\, \\eta_0^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_0}{M}\\bigg)\\,\\dq\\bigg]\\dx \\leq c.\n\\ee\n\nSince $\\ff\\in L^\\infty((0,T)\\times \\O;\\R^3)$ and\n$$\n|\\vr \\,\\ff \\cdot \\vu |\\leq |\\ff| \\, |\\sqrt \\vr|\\,| \\sqrt\\vr \\vu| \\leq |\\ff|\\left( \\vr + \\vr|\\vu|^2 \\right) \\leq |\\ff|\\left(1 + \\vr^\\g + \\vr|\\vu|^2 \\right),\n$$\nby using Gronwall's inequality we deduce from \\eqref{energy} and \\eqref{ini-estf} that, for a.e. $t\\in (0,T)$,\n\\ba\\label{energy-f}\n&\\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr |\\vu|^2 +  P(\\vr) +\\de\\, \\eta^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi)\\, \\dq\\bigg] (t, \\cdot) \\,\\dx\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta) \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right|^2 +\\mu^B(\\eta) |\\Div_x \\vu|^2  } \\, \\dt' + 2\\, \\e\\, \\de \\int_0^t \\intO{   |\\nabla_x \\eta|^2} \\, \\dt' \\\\\n&\\quad+\\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi}\\, \\right|^2\\dq} \\, \\dt+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q}  \\sqrt{\\widetilde \\psi}\\,\\right|^2\\dq} \\, \\dt' \\\\\n&\\leq c(1+T)\\, {\\rm e}^{ct}.\n\\ea\nIn the rest of this section, we shall establish additional bounds on the unknowns, one by one, by using \\eqref{energy-f}.\n\n\\subsection{Bounds on the fluid density and the polymer number density}\n\nFrom \\eqref{energy-f},  we have\n\\be\\label{est-r-eta1}\n\\vr\\in L^\\infty(0,T;L^\\gamma(\\O)), \\quad \\eta\\in L^\\infty(0,T;L^2(\\O))\\cap L^2(0,T;W^{1,2}(\\O)),  \\quad \\vr |\\vu|^2 \\in L^{\\infty}(0,T;L^{1}(\\O)).\n\\ee\nBy Sobolev embedding and interpolation it follows that\n\\be\\label{est-r-eta3}\n \\eta\\in L^\\infty(0,T;L^2(\\O)) \\cap L^2(0,T;L^6(\\O)) \\hookrightarrow L^{a}(0,T;L^{\\frac{6a}{3a-4}}(\\O)),\\quad 2\\leq a\\leq \\infty.\n \\ee\nThe bounds in \\eqref{est-r-eta1} then imply that\n\\be\\label{est-ru2}\n\\vr \\vu = \\sqrt{\\vr} \\sqrt{\\vr}\\vu \\in L^{\\infty}(0,T;L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3)).\n\\ee\n\n\\subsection{Bounds on the fluid velocity field}\\label{sec:est-u}\n\nBy \\eqref{energy-f} we deduce that\n\\be\\label{est-u1}\ng:= \\sqrt{\\mu^S(\\eta)} \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^2(\\O)).\n\\ee\nWe recall that $\\mu^S(\\eta)$ fulfills the hypotheses stated in \\eqref{mu-eta1} with $\\o$ satisfying \\eqref{mu-eta2} or \\eqref{mu-eta3}. We begin by considering the case when $\\o\\geq 0$; \\eqref{mu-eta1} then implies that $\\mu^{-1}$ is uniformly bounded. Thus,\n\\be\\label{est-u2}\n \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^2(\\O));\n\\ee\nwhence\nKorn's inequality (see \\cite{Dain}) with the no-slip boundary condition on $\\vu$ implies that\n\\be\\label{est-u4}\n|\\nabla_x \\vu| \\in L^2(0,T;L^2(\\O)), \\qquad \\mbox{and hence}\\qquad \\vu \\in L^2(0,T;W^{1,2}_0(\\O;\\R^3)).\n\\ee\nOn the other hand, when $\\o\\leq 0$ satisfies the constraint \\eqref{mu-eta3}, that is $-4\/3<\\o\\leq 0$, then, by \\eqref{mu-eta1} and \\eqref{est-r-eta3}, we have that\n\\be\\label{est-mu1}\n\\mu(\\eta)^{-1}\\in L^\\infty(0,T;L^{\\frac{2}{|\\o|}}(\\O)) \\cap L^{\\frac{2}{|\\o|}}(0,T;L^{\\frac{6}{|\\o|}}(\\O)) \\cap L^{\\frac{10}{3|\\o|}}((0,T)\\times \\O).\n\\ee\nThus, from \\eqref{est-u1}, we deduce that\n\\be\\label{est-u11}\n \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| \\in L^2(0,T;L^{\\frac{4}{2+|\\o|}}(\\O)) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{6+|\\o|}}(\\O))\\cap L^{\\frac{20}{10+3|\\o|}}((0,T)\\times \\O).\n\\ee\nHence, taking advantage of the no-slip boundary condition, we may use\nKorn's inequality to obtain\n\\be\\label{est-u13}\n\\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3)) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^3))\\cap L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^3)).\n\\ee\n\n\\subsection{Bounds on the Newtonian stress tensor}\n\nConsider first the case when \\eqref{mu-eta2} holds with $\\o\\geq 0$. Let us write\n$$\n\\mu^S(\\eta) \\left|\\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right| = g \\sqrt{\\mu^S(\\eta)},\n$$\nwhere $g\\in L^2((0,T)\\times \\O)$ is defined by \\eqref{est-u1}.  Thanks to \\eqref{mu-eta1}, \\eqref{est-r-eta3}, \\eqref{est-u4}, and by a similar argument as in the derivation of \\eqref{est-u13}, we obtain\n\\be\\label{est-SS1}\n\\SSS\\in L^2(0,T;L^{\\frac{4}{2+|\\o|}}(\\O;\\R^{3\\times 3})) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{6+|\\o|}}(\\O;\\R^{3\\times 3}))\\cap L^{\\frac{20}{10+3|\\o|}}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ee\nOn the other hand, when $\\o\\leq 0$, by observing that $\\sqrt{\\mu^S} + \\sqrt{\\mu^B}\\leq c<\\infty$ we deduce that\n\\be\\label{est-SS2}\n\\SSS\\in L^2((0,T)\\times\\O;\\R^{3\\times 3}).\n\\ee\n\n\n\\subsection{Bounds on the probability density function}\\label{sec:est-psi}\n\nIt follows from \\eqref{energy-f} that\n\\be\\label{est-psi1}\n\\mathcal{F}(\\tpsi) \\in L^\\infty(0,T;L^1_M(\\O\\times D)), \\quad  \\sqrt{\\tpsi} \\in L^2(0,T;H_M^1(\\O\\times D)),\n\\ee\nwhere we recall that $\\tpsi := \\psi\/M$. Consequently, by Sobolev embedding and thanks to Lemma \\ref{lem:HM1-emb} we then have that $\\sqrt{\\tpsi} \\in L^2(0,T;L^6(\\O;L^2_M(D)))$.\n\nIf $s>{\\rm e}^2$, then $s\\log s > 2s >2(s-1)$, which then implies that $\\mathcal{F}(s)=s\\log s -(s-1)>\\frac{1}{2}{s\\log s}$ for $s>{\\rm e}^2$. Thus,\n\\ba\\label{est-psi3}\n\\|\\tpsi\\log\\tpsi\\|_{L^1_M(\\O\\times D))}&=\\int_{\\O\\times D} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx = \\int_{0 \\leq \\tpsi \\leq {\\rm e}^2} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx +\\int_{\\tpsi > e^2} |\\tpsi\\log\\tpsi|\\,M\\,\\dq\\,\\dx \\\\\n&  \\leq  2 {\\rm e}^2\\int_{\\O\\times D}\\,M\\,\\dq\\,\\dx + 2 \\int_{\\tpsi > {\\rm e}^2} |\\mathcal{F}(\\tpsi)|\\, M\\,\\dq\\,\\dx \\leq c,\n\\ea\nwhere we have used \\eqref{pt1-Mi-Ui} and \\eqref{est-psi1}. This implies that\n\\be\\label{est-psi5}\n\\tpsi\\log\\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\psi\\log\\psi \\in L^\\infty(0,T;L^1(\\O\\times D)).\n\\ee\n\nClearly,\n\\be\\label{nabla-psi}\n\\nabla_{x}\\tpsi = 2 \\sqrt{\\tpsi} \\ \\nabla_{x}\\sqrt{\\tpsi},\\quad \\nabla_{q}\\tpsi = 2 \\sqrt{\\tpsi} \\ \\nabla_{q}\\sqrt{\\tpsi}.\n\\ee\nBy \\eqref{nabla-psi} and H\\\"older's inequality we therefore have that\n\\ba\\label{est-psi8}\n\\|\\nabla_{q,x} \\tpsi\\|_{L^1_M(D;\\R^{3(K+1)})} & = \\int_{D} | M\\,\\nabla_{q,x}\\tpsi | \\, \\dq = 2\\int_{D} M \\sqrt{\\tpsi} \\, \\left|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right|\\dq \\\\\n&\\leq 2 \\left(\\int_{D} M \\left|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right|^2 \\, \\dq \\right)^{\\frac 12}\\left(\\int_{D} M \\tpsi \\, \\dq \\right)^{\\frac 12}=2 \\left\\|\\nabla_{q,x}\\sqrt{\\tpsi}\\,\\right\\|_{L^2_M(D;\\R^{3(K+1)})} \\eta^{\\frac 12}.\n\\ea\nThus, we benefit from the estimates in \\eqref{est-r-eta3} for $\\eta$ and deduce that\n\\ba\\label{est-psi9}\n&\\nabla_{q,x} \\tpsi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1_M(D;\\R^{3(K+1)})))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1_M(D;\\R^{3(K+1)}))).\n\\ea\nBy observing that $\\|\\nabla_x \\tpsi\\|_{L^1_M(D;\\R^{3})}=\\|\\nabla_x\\psi\\|_{L^1(D;\\R^{3})}$ we obtain\n\\ba\\label{est-psi10}\n\\nabla_x \\psi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1(D;\\R^{3})))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1(D;\\R^{3}))).\n\\ea\n\n\n\n\\subsection{Bounds on the extra-stress tensor}\n\nWe recall that the extra-stress tensor can be expressed as (see \\eqref{weak-est})\n$$\\TT := k \\left(\\sum_{i=1}^K \\CC_i(\\psi)\\right)-\\left(k(K+1) \\eta  + \\de\\,  \\eta^2\\right) \\II. $$\nThanks to the bounds on $\\eta$ stated in \\eqref{est-r-eta1} and \\eqref{est-r-eta3}, we have that\n\\be\\label{est-TT1}\n\\eta^2\\in L^\\infty(0,T;L^1(\\O)) \\cap L^1(0,T;L^3(\\O)).\n\\ee\n\nAccording to \\eqref{pt1-Mi-Ui}, we have $M=0$ on $\\d D$. We then deduce by using \\eqref{def-Ci} and \\eqref{pt2-M} that\n\\ba\\label{est-TT2}\n\\CC_i(\\psi) &= \\CC_i(M\\tpsi) = \\int_D M \\tpsi\\, U_i'\\bigg(\\frac{q_i^2}{2}\\bigg)\\, q_i q_i^{\\rm T} \\, \\dq = - \\int_{D} \\tpsi\\, (\\nabla_{q_i} M)q_i^{\\rm T}\\, \\dq\\\\\n&=\\int_{D} M\\, (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq + \\left(\\int_{D} M  \\tpsi \\, \\dq\\right)\\II = \\int_{D} M (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq + \\eta \\II.\n\\ea\nThus, by \\eqref{nabla-psi}, H\\\"older's inequality implies that\n\\ba\\label{est-TT4}\n\\left|\\int_{D} M (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq \\right|  \\leq c \\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\, \\dq \\right)^{\\frac 12}\\left(\\int_{D} M \\tpsi \\, \\dq \\right)^{\\frac 12}= c \\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\, \\dq \\right)^{\\frac 12} \\eta ^{\\frac 12}.\\nn\n\\ea\nBy \\eqref{est-psi1} we have\n$$\n\\left(\\int_{D} M \\left(\\nabla_{q_i}\\sqrt{\\tpsi}\\right)^2 \\dq \\right)^{\\frac 12} \\in L^2(0,T;L^2(\\O)).\n$$\nThus, by \\eqref{est-r-eta3} and \\eqref{est-TT1}, we obtain\n\\be\\label{est-TTf}\n\\TT \\in L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3})).\n\\ee\n\n\\subsection{Higher integrability of the fluid density and the pressure}\\label{sec:est-higher-vr}\n\nFrom the energy inequality \\eqref{energy} we deduce that\n$\\vr\\in L^\\infty(0,T;L^\\g(\\O))$, which implies that $p(\\vr)\\in L^\\infty(0,T;L^1(\\O))$; unfortunately, $p(\\vr)$ is only in $L^1$ with respect to the spatial variable, $x$. In order to improve the\nintegrability of the pressure with respect to the spatial variable, one may use the so-called Bogovski{\\u{\\i}} operator (see \\cite{bog}), exactly as in \\cite{FNP, F-book, N-book}.\nWe recall the following lemma whose proof can be found in Chapter III of Galdi's book \\cite{Galdi-book}.\n\\begin{lemma}\\label{lem-bog}\n Let $p \\in (1,\\infty)$ and suppose that $\\O\\subset \\R^d$ is a bounded Lipschitz domain. Let $L^{p}_0(\\Omega)$ be the space of $L^p(\\Omega)$ functions with zero mean-value.\nThen, there exists a linear operator $\\mathcal{B}_\\O$ from $L_0^p(\\O)$ to $W_0^{1,p}(\\O;\\R^d)$ such that\n$$\n\\Div_x\\, \\mathcal{B}_\\O (f) =f \\quad \\mbox{in} \\ \\O; \\quad \\|\\mathcal{B}_\\O (f)\\|_{W_0^{1,p}(\\O;\\R^d)} \\leq c (d,p,\\O)\\, \\|f\\|_{L^p(\\O)}\\quad\\, \\forall\\,  f\\in L_0^p(\\O),\n$$\nwhere the constant $c$ depends only on $p$, $d$ and the Lipschitz character of $\\O$. If, in addition, $f=\\Div_x \\vg$ for some $\\vg \\in L^{q}, \\  1<q<\\infty, \\ \\vg \\cdot {\\bf n} = 0 $ on $\\d\\O$, then\n$$\n\\|\\mathcal{B}_\\O (f)\\|_{L^{q}(\\O;\\R^d)} \\leq c (d, q,\\O)\\, \\|\\vg \\|_{L^q(\\O;\\R^d)}.\n$$\n\n\\end{lemma}\n\nThe key idea in establishing higher integrability of $\\vr$ and $p(\\vr)$ is to choose the following test function in \\eqref{weak-form3}:\n\\be\\label{bog-test}\n\\varphi (t,x) : =  \\phi(t)\\, \\mathcal{B}_{\\O} \\bigg(S_\\e\\big[ b_{n}(\\vr) \\big] - \\frac{1}{|\\O|}\\int_\\O S_\\e\\big[ b_{n}(\\vr)\\big]\\, \\dx \\bigg),\n\\ee\nwhere $\\phi\\in C^\\infty_c((0,T))$ is a nonnegative test function, $S_\\e$ is the classical (Friedrichs) mollifier with respect to the spatial variable, and $\\{b_n(\\vr)\\}_{n\\in \\N}$ is an increasing sequence of $C^1$ functions satisfying \\eqref{cond-b-renormal}, which approximates the function $\\vr^\\th$. As in Lemma 2.1 in \\cite{Feireisl-2001} we have, for any $b$ satisfying \\eqref{cond-b-renormal},\n\\be\\label{weak-renormal1}\n\\d_t b(\\vr) +\\Div_x (b(\\vr)\\vu) + (b'(\\vr)\\vr-b(\\vr))\\,\\Div_x \\vu =0 \\ \\mbox{in} \\ \\mathcal{D}' ((0,T)\\times \\R^3),\n\\ee\nwhere the functions $\\vr$ and $\\vu$ are extended by zero outside $\\O$. For any $n$, $b_n(\\vr)$ satisfies \\eqref{weak-renormal1}. After tedious but rather straightforward calculations (see \\cite{Feireisl-2001,FNP} for details), one obtains, for $\\th>0$ sufficiently small, that\n\\be\\label{est-bog-p1}\n\\int_0^t\\int_\\O \\phi(t)\\, \\vr^\\g\\, b_{n}(\\vr)\\, \\dx \\, \\dt \\leq c\\qquad \\forall\\,n \\geq 0.\n\\ee\nLetting $n\\to \\infty$ in \\eqref{est-bog-p1} and approximating $1$ by $C_c^\\infty ((0,T))$ functions finally gives\n\\be\\label{est-bog-pf1}\n \\vr \\in L^{\\g+\\th}((0,T)\\times \\O),\\quad p(\\vr) \\in L^{1+\\frac{\\th}{\\g}}((0,T)\\times \\O) \\quad \\mbox{for some $\\th>0$}.\n\\ee\n\n\\medskip\n\n\n\\begin{remark}\\label{rem-bog-th}\n\n\n\nIn order to obtain \\eqref{est-bog-pf1} for some positive $\\th$, the conditions imposed in \\eqref{mu-eta2} and \\eqref{mu-eta3} can be relaxed. By careful analysis the following constraints are found to be sufficient:\n\\be\\label{mu-eta20}\n\\g>\\frac 32,\\ 0\\leq \\o < \\frac{10}{3}  \\quad \\mbox{or} \\quad -2<\\o\\leq 0,\\  \\g>\\frac{6}{4+3\\o}.\n\\ee\nThe more restrictive conditions featuring in \\eqref{mu-eta2} and \\eqref{mu-eta3} are needed later on, in Section \\ref{vis-flux}, in order to prove the so-called {\\rm effective viscous flux} equality (see Remark \\ref{rem-mu-eta}).\n\\end{remark}\n\n\\subsection{Bounds on the time derivative and continuity}\\label{sec:est-time}\n\nThis section is devoted to establishing bounds on the time derivatives $(\\d_t \\vr, \\ \\d_t\\eta, \\  \\d_t (\\vr \\vu), \\ \\d_t \\psi)$.\n\nAs $\\d_t \\vr = - \\Div_x(\\vr\\vu)$, the bound in \\eqref{est-ru2} implies that\n$\\d_t \\vr \\in L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O)).$ Then, by \\eqref{est-r-eta1} and Lemma \\ref{lem-Cw-Cw*}, we have that\n$$\n\\vr \\in C_w([0,T]; L^\\g(\\O)).\n$$\n\n\n\\medskip\n\nRecall that $\\d_t \\eta = - \\Div_x(\\eta\\vu) + \\e \\Delta_x \\eta$.  For $\\o\\geq 0$ satisfying \\eqref{mu-eta2}, we have $\\vu \\in L^2(0,T; W_0^{1,2}(\\O;\\R^3))$, which is embedded in $L^2(0,T; L^6(\\O;\\R^3))$. Thanks to \\eqref{est-r-eta1}, we then obtain $\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{3}{2}}(\\O)).$\nIf on the other hand $\\o\\leq 0$ satisfies \\eqref{mu-eta3}, then we have $\\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3)) \\hookrightarrow L^2(0,T; L^\\frac{12}{3|\\o|+2}(\\O;\\R^3))$. Consequently, $\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{12}{3|\\o|+8}}(\\O))$.\nThus, in both cases, by \\eqref{est-r-eta1} and Lemma \\ref{lem-Cw-Cw*}, we have\n$$\n\\eta \\in C_w([0,T]; L^2(\\O)).\n$$\n\n\\medskip\n\nNext, we recall that $\\d_t(\\vr\\vu)=- \\Div_x (\\vr \\vu \\otimes \\vu) - \\nabla_x p(\\vr) +\n\\Div_x \\SSS +\\Div_x \\TT +\\vr\\, \\ff$. We first consider the case with $\\o\\geq 0$ satisfying \\eqref{mu-eta2}. By the estimates in Section \\ref{sec:est-u} we have $\\vu \\in L^2(0,T; L^6(\\O;\\R^3))$. Since $\\vr\\in L^\\infty(0,T;L^\\g(\\O))$  with $\\g>\\frac{3}{2}$, together with \\eqref{est-r-eta1}, we obtain\n\\be\\label{est-ruu}\n\\vr \\vu \\otimes \\vu \\in L^1(0,T; L^{\\frac{3\\g}{\\g+3}}(\\O;\\R^{3\\times 3}))\\cap L^\\infty(0,T; L^{1}(\\O;\\R^{3\\times 3})) \\hookrightarrow L^r(0,T; L^{r}(\\O;\\R^{3\\times 3})) \\quad \\mbox{for some $r>1$}.\n\\ee\nBy \\eqref{est-SS1}, \\eqref{est-bog-pf1}, \\eqref{est-TTf}, together with \\eqref{est-ruu}, we deduce that\n\\be\\label{est-dt-ru}\n\\d_t (\\vr\\vu) \\in  L^r(0,T; W^{-1,r}(\\O;\\R^{3})) \\quad \\mbox{for some $r>1$}.\n\\ee\nFor the case with $\\o\\leq 0$ satisfying \\eqref{mu-eta3}, a similar argument gives the same result as in \\eqref{est-dt-ru}. Then, by \\eqref{est-ru2}, \\eqref{est-dt-ru} and Lemma \\ref{lem-Cw-Cw*}, we have in both cases that\n$$\\vr\\vu \\in C_w([0,T]; L^\\frac{2\\g}{\\g+1}(\\O;\\R^3)).$$\n\n\nBy the Fokker--Planck equation \\eqref{eq-psi} for $\\psi$, the estimates in \\eqref{est-psi9} and \\eqref{est-psi10}, direct calculations yield that\n\\be\\label{est-dt-psi}\n\\d_t \\psi \\in\nL^2(0,T; W^{s,2}(\\O\\times D)') \\quad \\mbox{with $s>1 + \\frac{3}{2}(K+1)$},\n\\ee\nwhere we have used that $W^{s,2}(\\Omega \\times D) \\hookrightarrow W^{1,\\infty}(\\Omega \\times D)$ for $s>1+\\frac{3}{2}(K+1)$.\nMoreover, by the estimates \\eqref{est-psi5} and \\eqref{est-dt-psi}, using Lemma \\ref{lem-Cw-Cw*} and the same argument as in Section 4.5 in \\cite{Barrett-Suli}, we deduce that  $$\\psi \\in C_w([0,T];L^1(\\O\\times D)).$$\n\n\n\n\n\\subsection{Summary}\n\nWe summarize the results obtained in this section. Let $(\\vr,\\vu,\\psi,\\eta)$ be a dissipative weak solution in the sense of Definition \\ref{def-weaksl} associated\nwith initial data $(\\vr_0,\\vu_0,\\psi_0,\\eta_0)$ satisfying \\eqref{ini-data}. Then,\n\\ba\\label{est-sum1}\n& \\vr\\in C_w([0,T);L^\\gamma(\\O))\\cap L^{\\g+\\th}((0,T)\\times \\O);\\quad  \\eta \\in C_w([0,T];L^2(\\O)) \\cap L^2(0,T;W^{1,2}(\\O));\\\\\n& \\vr |\\vu|^2 \\in L^{\\infty}(0,T;L^{1}(\\O)),\\quad \\vr \\vu \\in C_w([0,T];L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^3));\\quad  \\TT \\in L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3}));\\\\\n&\\vu \\in L^2(0,T;W^{1,2}_0(\\O;\\R^3)) \\ \\mbox{if $\\o\\geq 0$ satisfies \\eqref{mu-eta2}};\\\\\n& \\vu \\in L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^{3})) \\cap L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^{3}))\\cap L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^{3})) \\\\\n&\\qquad \\mbox{if $\\o\\leq 0$ satisfies \\eqref{mu-eta3}};\\\\\n&\\mathcal{F}(\\tpsi),  \\tpsi\\log\\tpsi, \\ \\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\sqrt{\\tpsi} \\in L^2(0,T;H_M^1(\\O\\times D)),\\\\\n&\\qquad \\nabla_{q,x} \\tpsi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1_M(D;\\R^{3(K+1)})))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1_M(D;\\R^{3(K+1)})));\\\\\n&\\mathcal{F}(\\psi),\\  \\psi\\log\\psi \\  \\in L^\\infty(0,T;L^1(\\O\\times D)),\\quad  \\psi \\in C_w([0,T];L^1(\\O\\times D)),\\\\\n &\\qquad \\nabla_x \\psi \\in L^2(0,T;L^{\\frac 43}(\\O;L^1(D;\\R^{3})))\\cap  L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;L^1(D;\\R^{3})));\n\\ea\nand\n\n\\begin{equation}\\label{est-sum2}\n\\begin{aligned}\n&\\d_t \\vr \\in L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O)), \\quad \\d_t (\\vr\\vu) \\in  L^r(0,T; W^{-1,r}(\\O;\\R^{3})) &&\\quad \\mbox{for some $r>1$};\\\\\n& \\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{3}{2}}(\\O)) &&\\quad \\mbox{if $\\o\\geq 0$ satisfies \\eqref{mu-eta2}};\\\\\n&\\d_t \\eta \\in L^2(0,T; W^{-1,\\frac{12}{3|\\o|+8}}(\\O)) &&\\quad \\mbox{if $\\o\\leq 0$ satisfies \\eqref{mu-eta3}};\\\\\n&\\d_t \\psi \\in  L^2(0,T; W^{s,2}(\\O\\times D)') &&\\quad \\mbox{with $s>1 + \\frac{3}{2}(K+1)$}.\n\\end{aligned}\n\\end{equation}\nIt is important to note that all of the above inclusions are consequences of bounds that depend only on the initial energy, the final time $T$, and the structural constants in the hypotheses imposed on the constitutive relations.\n\n\n\n\\section{Passing to the limit}\\label{sec:lim}\n\nLet $(\\vr_n,\\vu_n,\\psi_n,\\eta_n)_{n\\in \\N}$ be a sequence of the dissipative (finite-energy) weak solutions satisfying the assumptions in Theorem \\ref{theorem}. Then, the energy inequality \\eqref{energy} and \\eqref{energy-f} give, respectively,\n\\ba\\label{energy-n}\n&\\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_n |\\vu_n|^2 +  P(\\vr_n) +\\de\\, \\eta_n^2 + k \\int_D M \\mathcal{F} (\\psi_n)\\, \\dq\\bigg] (t, \\cdot) \\, \\dx + 2\\, \\e\\, \\de \\int_0^t \\intO{   |\\nabla_x \\eta_n|^2} \\, \\dt'\\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta_n) \\left|\\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right|^2 +\\mu^B(\\eta_n) |\\Div_x \\vu_n|^2  } \\, \\dt' \\\\\n&\\quad+ \\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\tpsi_n} \\right|^2 \\dq} \\, \\dt+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q} \\sqrt{\\tpsi_n}\\right|^2 \\dq} \\, \\dt'\\\\\n&\\leq \\int_{\\Omega} \\bigg[\n\\frac{1}{2} \\vr_{0,n} |\\vu_{0,n} |^2 +  P(\\vr_{0,n}) + \\de\\, \\eta_{0,n}^2 + k \\int_D M \\mathcal{F} \\bigg(\\frac{\\psi_{0,n}}{M}\\bigg)\\,\\dq\\bigg]\\dx + \\int_0^t\\int_\\O \\vr_n \\,\\ff \\cdot \\vu_n \\, \\dx\\, \\dt'\n\\ea\n and, therefore,\n \\ba\\label{energy-f-n}\n&\\int_{\\Omega} \\bigg[ \\frac{1}{2} \\vr_n |\\vu_n|^2 +  P(\\vr_n) +\\de\\, \\eta_n^2 + k \\int_D M \\mathcal{F} (\\widetilde \\psi_n)\\,\\dq\\bigg] (t, \\cdot) \\, \\dx + 2\\, \\e\\, \\de \\int_0^t \\intO{   |\\nabla_x \\eta_n|^2} \\, \\dt' \\\\\n&\\quad+ \\int_0^t \\intO{ \\mu^S(\\eta_n) \\left|\\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right|^2 +\\mu^B(\\eta_n) |\\Div_x \\vu_n|^2  } \\ \\dt' \\\\\n&\\quad+\\e\\, k \\int_0^t \\intO{ \\int_D M\\left|\\nabla_x \\sqrt{\\widetilde \\psi_n} \\right|^2\\dq} \\ \\dt'+\\frac{k\\,A_0}{4\\lambda} \\int_0^t \\intO{ \\int_D M \\left|\\nabla_{q}  \\sqrt{\\widetilde \\psi_n}\\right|^2\\dq} \\, \\dt' \\\\\n&\\leq c(1+T)\\, {\\rm e}^t.\n\\ea\nThus, from the results in Section \\ref{sec:estimate}, this solution sequence $(\\vr_n,\\vu_n,\\psi_n,\\eta_n)_{n\\in \\N}$ satisfies the uniform bounds  \\eqref{est-sum1} and \\eqref{est-sum2}.\n\nThe present section is devoted to studying the limit of this solution sequence. We remark that, throughout this section, the limits are taken up to subtractions of subsequences without identification.\n\n\n\\subsection{Convergence of the fluid density}\\label{sec:vr-lim}\n\nWe have that $\\{\\vr_n\\}_{n\\in \\N}$ is a sequence in $C_w([0,T];L^\\gamma(\\O))$ satisfying\n\\be\\label{vrn-est}\n\\sup_{n\\in \\N} \\left(\\|\\vr_n\\|_{L^\\infty(0,T;L^\\gamma(\\O))} +  \\|\\d_t \\vr_n \\|_{L^\\infty(0,T; W^{-1,\\frac{2\\g}{\\g+1}}(\\O))}  \\right) \\leq c.\n\\ee\nBy Sobolev embedding one has\n$$\nL^\\g(\\O) \\hookrightarrow W^{-1,\\frac{3\\g}{3-\\g}}(\\Omega) \\quad \\mbox{ if $\\g<3$};\\quad L^\\g(\\O) \\hookrightarrow W^{-1,s}(\\Omega) \\quad \\mbox{ for any $s \\in (1,\\infty)$, \\ if $\\g\\geq3$}.\n$$\nThus, by applying Lemma \\ref{lem-Cw}, we deduce that\n\\ba\\label{vrn-lim}\n& \\vr_n \\to \\vr \\quad \\mbox{in}\\  C_w([0,T];L^\\gamma(\\O));\\\\\n& \\vr_n \\to \\vr \\quad \\mbox{strongly in} \\  C([0,T];W^{-1,r}(\\O)) \\quad \\mbox{for any $r\\geq \\frac{3}{2}$ such that $\\frac{3r}{3+r}<\\g$}.\n\\ea\n\n\\subsection{Convergence of the fluid velocity field}\\label{sec:lim-u}\n\nBy Sobolev embedding, for the case $\\o\\geq 0$, we have\n\\be\\label{vun-limit1}\n\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\  L^2(0,T;W^{1,2}_0(\\O;\\R^3))\\  \\mbox{and in}\\  L^2(0,T;L^{6}(\\O;\\R^3)),\n\\ee\nwhile for the case $\\o\\leq 0$, we have\n\\ba\\label{vun-limit2}\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\  L^2(0,T;W_0^{1,\\frac{4}{2+|\\o|}}(\\O;\\R^3))\\  \\mbox{and weakly in}\\  L^2(0,T;L^{\\frac{12}{2+3|\\o|}}(\\O;\\R^3)),\\\\\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\  L^{\\frac{4}{2+|\\o|}}(0,T;W_0^{1,\\frac{12}{6+|\\o|}}(\\O;\\R^3))\\  \\mbox{and weakly in}\\  L^{\\frac{4}{2+|\\o|}}(0,T;L^{\\frac{12}{2+|\\o|}}(\\O;\\R^3)),\\\\\n&\\vu_n \\to \\vu \\quad \\mbox{weakly in}\\  L^{\\frac{20}{10+3|\\o|}}(0,T;W_0^{1,\\frac{20}{10+3|\\o|}}(\\O;\\R^3)) \\  \\mbox{and weakly in}\\  L^{\\frac{20}{10+3|\\o|}}(0,T;L^{\\frac{60}{10+9|\\o|}}(\\O;\\R^3)).\n\\ea\n\n\\subsection{Convergence of the nonlinear terms $\\vr_n\\vu_n$ and $\\vr_n \\vu_n\\otimes \\vu_n$}\\label{sec:lim-convective}\n\nWe first consider the case with $\\o\\geq 0$ satisfying \\eqref{mu-eta2}. Since $\\g>\\frac 32>\\frac 65$, we have, because of \\eqref{vrn-lim}, that\n\\ba\\label{vrn-lim1}\n \\vr_n \\to \\vr \\quad \\mbox{strongly in} \\  C([0,T];W^{-1,2}(\\O)).\n\\ea\nTogether with \\eqref{vun-limit1}, we have\n\\be\\label{vrvun-lim0}\n\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{in $\\mathcal{D}'((0,T)\\times \\O;\\R^{3})$.}\n\\ee\nBy observing the uniform estimate\n$$\n\\sup_{n \\in \\mathbb{N}}\\left(\\|\\vr_n\\vu_n\\|_{L^\\infty(0,T;L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^{3}))} + \\|\\vr_n\\vu_n\\|_{L^2(0,T;L^{\\frac{6\\g}{6+\\g}}(\\O;\\R^{3}))}\\right)\\leq c\n$$\nwe have\n\\be\\label{vrvun-lim1}\n\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{weakly* in} \\ L^\\infty(0,T;L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^{3})) \\ \\mbox{and weakly in} \\ L^2(0,T;L^{\\frac{6\\g}{6+\\g}}(\\O;\\R^{3})).\n\\ee\n\nMoreover, by \\eqref{est-sum1} and \\eqref{est-sum2}, we have $\\vr_n\\vu_n \\in C_w([0,T];L^{\\frac{2\\g}{\\g+1}}(\\O;\\R^d))$ and $\\{\\d_t (\\vr_n\\vu_n)\\}_{n\\in \\N}$ is uniformly bounded in $L^r(0,T;W^{-1,r}(\\O;\\R^3))$ for some $r>1$. By Lemma \\ref{lem-Cw} and the interpolation argument in Section \\ref{sec:vr-lim} we have\n\\ba\\label{vrvun-lim2}\n&\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{in}\\  C_w([0,T];L^{\\frac{2\\g}{1+\\g}}(\\O;\\R^3));\\\\\n&\\vr_n\\vu_n \\to \\vr\\vu \\quad \\mbox{strongly in} \\  C([0,T];W^{-1,r}(\\O;\\R^3)) \\quad \\mbox{for any $r\\geq \\frac{3}{2}$ such that $\\frac{3r}{3+r}<\\frac{2\\g}{1+\\g}$}.\n\\ea\n\nThe fact that $\\g>\\frac 32$ implies $\\frac{2\\g}{1+\\g} > \\frac{6}{5}$. This gives $\\vr_n\\vu_n \\to \\vr\\vu$ strongly in $C([0,T];W^{-1,2}(\\O;\\R^3))$. Together with \\eqref{vun-limit1}, we have for the sequence of convective terms:\n\\be\\label{vrvuvun-lim0}\n\\vr_n\\vu_n\\otimes \\vu_n \\to \\vr\\vu\\otimes \\vu \\quad \\mbox{ in $\\mathcal{D}'((0,T)\\times \\O;\\R^{3\\times 3})$.}\n\\ee\n\n\nThe case $\\o\\leq 0$ is dealt with similarly to the case $\\o\\geq 0$, and we obtain the results stated in \\eqref{vrvun-lim0} and \\eqref{vrvuvun-lim0}, so we omit the details. We merely remark that when  $\\o\\leq 0$, then we have weaker integrability for $\\vu$; this is, however, compensated by supposing stronger integrability for $\\vr$ through hypothesis \\eqref{mu-eta3}.\n\n We also remark that the hypotheses stated in \\eqref{mu-eta2} and \\eqref{mu-eta3} can be relaxed in this part of the analysis: the constraints stated in \\eqref{mu-eta20} in Remark \\ref{rem-bog-th} are sufficient.\n\n\\subsection{Convergence of the polymer number density}\\label{sec:lim-eta}\n\nWe begin by noting that the sequence $\\{\\eta_n\\}_{n\\in \\N}$ is contained in $C_w([0,T];L^2(\\O))$ and satisfies the bound\n\\be\\label{etan-est1}\n\\sup_{n \\in \\mathbb{N}} \\left(\\|\\eta_n\\|_{L^\\infty(0,T;L^2(\\O)) \\cap L^2(0,T;W^{1,2}(\\O))} + \\|\\d_t \\eta_n\\|_{L^2(0,T;W^{-1,\\frac{12}{3|\\o|+8}}(\\O))}\\right) \\leq c.\n\\ee\nThus, by Lemma \\ref{lem-Cw},\n\\be\\label{etan-lim1}\n\\eta_n \\to  \\eta \\quad \\mbox{strongly in}\\ C_w([0,T];L^2(\\O)) \\ \\mbox{and weakly in}\\ L^2(0,T;W^{1,2}(\\O)).\n\\ee\nThanks to the compact Sobolev embedding $W^{1,2}(\\O) \\hookrightarrow \\hookrightarrow L^q(\\O)$, $q<6$, the\nAubin--Lions--Simon compactness theorem (see \\cite{Lions-InC} or \\cite{Simon}) implies that\n\\be\\label{etan-lim2}\n\\eta_n \\to  \\eta \\quad \\mbox{strongly in}\\  L^2(0,T;L^q(\\O))\\quad \\mbox{for any $q<6$}.\n\\ee\nBy interpolation we also have\n\\be\\label{etan-lim3}\n\\eta_n \\to  \\eta \\quad \\mbox{strongly in}\\  L^q((0,T) \\times \\O)\\quad \\mbox{for any $q<\\frac{10}{3}$}.\n\\ee\n\nWe still need to show that $\\eta=\\int_D \\psi \\ \\dq$, where $\\psi$ is the limit of $\\psi_n$. This will be done later on, in Section  \\ref{sec:limit-psi}.\n\n\n\\subsection{Convergence of the Newtonian stress tensor}\\label{sec:lim-SS}\nWhen $0\\leq \\o  <\\frac{5}{3}$ as in \\eqref{mu-eta2}, by \\eqref{mu-eta1} and \\eqref{etan-lim3}, we have that\n\\be\\label{mu-lim1}\n(\\mu^S(\\eta_n),\\mu^B(\\eta_n)) \\to  (\\mu^S(\\eta),\\mu^B(\\eta))  \\quad \\mbox{strongly in}\\  L^{\\frac{10}{3\\o}}((0,T)\\times \\O).\n\\ee\nTogether with \\eqref{vun-limit1}, we deduce that\n\\be\\label{Newtonian-lim1}\n\\SSS_n  \\to \\SSS :=\\mu^S(\\eta) \\left( \\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B(\\eta) (\\Div_x \\vu) \\II\\quad {\\rm weakly\\  in} \\ L^{\\frac{10}{3\\o+5}}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ee\n\nOn the other hand, when $-\\frac{4}{3}<\\o\\leq 0$ as in \\eqref{mu-eta2}, then we have\n\\be\\label{mu-lim2}\n(\\mu^S(\\eta_n),\\mu^B(\\eta_n)) \\to  (\\mu^S(\\eta),\\mu^B(\\eta))  \\quad \\mbox{strongly in}\\  L^q(0,T;L^q(\\O)) \\quad \\mbox{for any $q<\\infty$}.\n\\ee\nThus, by \\eqref{vun-limit2}, we have, for any  $r<\\frac{20}{3\\o+10}$,\n\\ba\\label{Newtonian-lim10}\n\\SSS_n  \\to \\SSS :=\\mu^S(\\eta) \\left( \\frac{\\nabla \\vu + \\nabla^{\\rm T} \\vu}{2} - \\frac{1}{3} (\\Div_x \\vu) \\II \\right) + \\mu^B(\\eta) (\\Div_x \\vu) \\II\\quad\n{\\rm weakly~in} \\ L^{r}((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ea\n\n\nWe remark that, by rewriting\n\\ba\\label{Newtonian-est2}\n\\SSS_n  = \\sqrt{\\mu^S(\\eta_n)} \\sqrt{\\mu^S(\\eta_n)} \\left( \\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} - \\frac{1}{3} (\\Div_x \\vu_n) \\II \\right) + \\sqrt{\\mu^B(\\eta_n)}\\sqrt{\\mu^B(\\eta_n)} (\\Div_x \\vu_n) \\II,\n\\ea\nwe can relax the constraints in \\eqref{mu-eta2} and \\eqref{mu-eta3} as in \\eqref{mu-eta20} in Remark \\ref{rem-bog-th}.\n\n\\subsection{Convergence of the probability density function}\\label{sec:limit-psi}\n\nFirst of all, by the energy inequality \\eqref{energy-f-n} we have\n\\be\\label{est-psin1}\n\\sup_{n\\in\\N}\\left( \\left\\|\\mathcal{F}(\\tpsi_n)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} + \\left\\|\\sqrt{\\tpsi_n}\\,\\right\\|_{L^2(0,T;H_M^1(\\O\\times D))}  \\right)\\leq c.\n\\ee\n\n\nAs in Section 4 in \\cite{Barrett-Suli} and Section 5 in \\cite{Barrett-Suli4}, we use Dubinski{\\u\\i}'s compactness theorem (cf. \\cite{Dubin}; see also \\cite{Barrett-Suli3},  Theorem 3.1 or \\cite{Barrett-Suli}) by setting\n\\ba\\label{dubinskii-spaces}\n&X:= L^1_M(\\O\\times D),\\quad X_0:=\\{ \\vp\\in X : \\vp\\geq 0,\\ \\sqrt{\\vp} \\in H^1_M(\\O\\times D) \\},\\\\\n&X_1: = M^{-1} W^{s,2}(\\O\\times D)':=\\{ M^{-1} \\vp : \\vp \\in  W^{s,2}(\\O\\times D)'\\}  \\quad \\mbox{with $s>1+\\frac{3}{2}(K+1)$},\n\\ea\nwhere $X_0$ is a seminomed space (in the sense of Dubinski{\\u\\i}) with seminorm defined by\n$$\n[\\vp]_{X_0}: = \\|\\vp\\|_{X} + \\int_{\\O\\times D} M\\bigg( \\left|\\nabla_x \\sqrt{\\vp}\\,\\right|^2 + \\left|\\nabla_q \\sqrt{\\vp}\\,\\right|^2 \\bigg) \\, \\dq \\, \\dx.\n$$\n\nBy \\eqref{est-sum1} and \\eqref{est-sum2} we have that\n\\be\\label{psin-est1}\n\\mbox{$\\{\\widetilde \\psi_n\\}_{n\\in \\N}$ is uniformly bounded in $L^1(0,T;X_0)$\\quad and \\quad $\\{\\d_t \\widetilde \\psi_n\\}_{n\\in \\N}$ is uniformly bounded in $L^2(0,T;X_1)$}.\n\\ee\nThe continuity of the embedding $X\\hookrightarrow X_1$ and the compactness of the embedding $X_0 \\hookrightarrow X$ are shown in Section 5 in \\cite{Barrett-Suli4}. Then, by virtue of Dubinski{\\u\\i}'s compact embedding theorem, we have that\n\\be\\label{psin-lim1}\n\\tpsi_n \\to \\tpsi \\quad \\mbox{strongly in} \\ L^1(0,T;L_M^1(\\Omega\\times D)),\n\\ee\nwhich is equivalent to\n\\be\\label{psin-lim2}\n\\psi_n \\to \\psi \\quad \\mbox{strongly in} \\ L^1(0,T;L^1(\\Omega\\times D)).\n\\ee\nThis implies that\n\\be\\label{psin-lim3}\n\\eta_n = \\int_D \\psi_n \\,\\dq \\to \\int_D \\psi \\, \\dq \\quad \\mbox{strongly in} \\ L^1((0,T)\\times \\Omega);\n\\ee\nin addition, by the uniqueness of the limit,  the function $\\eta$ obtained in Section \\ref{sec:lim-eta} satisfies $\\eta = \\int_D \\psi \\, \\dq$.\n\n\n\nSince $\\mathcal{F}(\\tpsi)$ is nonnegative on $(0,T)\\times \\Omega \\times D$,  by applying Fatou's lemma we have that\n\\be\\label{psin-est3}\n \\left\\|\\mathcal{F}(\\tpsi)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} \\leq \\liminf_{n\\to \\infty} \\left\\|\\mathcal{F}(\\tpsi_n)\\right\\|_{L^\\infty(0,T;L^1_M(\\O\\times D))} \\leq c.\n\\ee\nBy the same technique as in Section \\ref{sec:est-psi} we deduce from \\eqref{psin-est3} that\n\\be\\label{psin-est4}\n\\tpsi\\log\\tpsi, \\ \\tpsi \\in L^\\infty(0,T;L^1_M(\\O\\times D)),\\quad \\psi\\log\\psi, \\ \\psi \\in L^\\infty(0,T;L^1(\\O\\times D)).\n\\ee\n\nBy \\eqref{psin-est1} we have\n\\be\\label{psin-est5}\n\\d_t \\psi \\in L^2(0,T;W^{s,2}(\\O\\times D)') \\quad \\mbox{for any $s>1 + \\frac{3}{2}(K+1)$}.\n\\ee\n\nFrom the estimates in \\eqref{psin-est4} and \\eqref{psin-est5}, by using the second part of Lemma \\ref{lem-Cw-Cw*} and the same argument as in Section 4.5 in \\cite{Barrett-Suli}, we deduce that\n$\n \\psi \\in C_w([0,T];L^1(\\O\\times D)).\n$\n\n\nAgain, we write $\\nabla_{q_i}\\tpsi_n = 2 \\sqrt{\\tpsi_n}\\nabla_{q_i}\\sqrt{\\tpsi_n}$.  By \\eqref{psin-lim1} we have\n$$\n\\sqrt{\\tpsi_n} \\to \\sqrt{\\tpsi} \\quad  \\mbox{strongly in} \\ L^2(0,T;L_M^2(\\Omega\\times D)).\n$$\nFurther, by \\eqref{est-psin1}, we have that\n$$\n\\nabla_q \\sqrt{\\tpsi_n} \\to \\nabla_q \\sqrt{\\tpsi} \\quad \\mbox{weakly in} \\ L^2(0,T;L_M^2(\\Omega\\times D;\\R^{3K})).\n$$\nThus,\n\\be\\label{psin-limf1}\n\\nabla_{q}\\tpsi_n \\to \\nabla_{q}\\tpsi \\quad \\mbox{weakly in} \\ L^1(0,T;L_M^1(\\Omega\\times D;\\R^{3K})).\n\\ee\n\n\nSimilarly, we also have that\n\\be\\label{psin-limf2}\n\\nabla_{x}\\tpsi_n \\to \\nabla_{x}\\tpsi \\quad \\mbox{weakly in} \\ L^1(0,T;L_M^1(\\Omega\\times D;\\R^{3})),\\quad\n\\nabla_{x}\\psi_n \\to \\nabla_{x}\\psi \\quad \\mbox{weakly in} \\ L^1(0,T;L^1(\\Omega\\times D;\\R^{3})).\n\\ee\n\n\n\\subsection{Convergence of the extra-stress tensor}\\label{sec:lim-TT}\n\nWe recall the formula for the extra-stress tensor. By \\eqref{weak-est} and \\eqref{est-TT2} we have that\n\\be\\label{TTn-est1}\n\\TT_n := k \\left(\\sum_{i=1}^K \\int_{D} M (\\nabla_{q_i}\\tpsi_n) q_i^{\\rm T}\\, \\dq \\right) - \\left(k \\,\\eta_n  + \\de\\,  \\eta_n^2\\right) \\II,\n \\ee\nwhich, because of \\eqref{est-TTf}, satisfies\n\\be\\label{TTn-est2}\n\\sup_{n\\in\\N}\\left(\\|\\TT_n\\|_{L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3}))} + \\|\\TT_n\\|_{L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3}))}\\right)<c.\n\\ee\nThus,\n\\be\\label{TTn-lim1}\n\\TT_n\\to \\bar \\TT \\quad \\mbox{weakly in} \\ L^2(0,T;L^{\\frac 43}(\\O;\\R^{3\\times 3})) \\cap L^{\\frac 43}(0,T;L^\\frac{12}{7}(\\O;\\R^{3\\times 3})).\n\\ee\n\nWe still need to show\n\\be\\label{TTn-lim2}\n\\bar\\TT = \\TT := k \\left(\\sum_{i=1}^K \\int_{D} M (\\nabla_{q_i}\\tpsi) q_i^{\\rm T}\\, \\dq \\right) - \\left(k \\,\\eta  + \\de\\,  \\eta^2\\right) \\II.\n\\ee\nIndeed, using the same argument as in Section 4.5 in \\cite{Barrett-Suli}, it follows that\n\\be\\label{TTn-lim-strf}\n\\TT_n \\to \\TT \\quad \\mbox{strongly in}\\  L^1((0,T)\\times \\O;\\R^{3\\times 3}).\n\\ee\nBy the uniform estimates \\eqref{TTn-est2} and interpolation we also have that\n\\be\\label{TTn-lim-strff}\n\\TT_n\\to \\TT \\quad \\mbox{strongly in } L^r((0,T)\\times \\O;\\R^{3\\times 3}) \\quad \\mbox{for any $r<\\frac{20}{13}$}.\n\\ee\n\n\n\\subsection{Convergence of the nonlinear terms in the Fokker--Panck equation}\\label{sec:lim-nonl-FP}\n\nIn this section, we study the limits associated with the nonlinear terms in the Fokker--Planck equation \\eqref{eq-psi}, which are $\\Div_x(\\vu_n \\psi_n)$ and  $\\Div_{q_i}\\!\\left((\\nabla_x \\vu_n )\\, q_i\\, \\psi_n\\right)$. We would like to show that\n\\be\\label{nonlin-FP-lim0}\n\\vu_n \\psi_n \\to  \\vu \\psi \\quad \\mbox{in $\\mathcal{D}'((0,T)\\times \\O \\times D;\\R^{3})$}\\quad\\mbox{and}\\quad \\left(\\nabla_x \\vu_n \\right) q_i\\psi_n \\to \\left(\\nabla_x \\vu \\right) q_i \\psi \\quad \\mbox{in}\\ \\mathcal{D}'((0,T)\\times \\O\\times D;\\R^{3\\times 3}).\n\\ee\n\nTo this end, let $\\vvarphi \\in C_c^\\infty((0,T)\\times \\O\\times D;\\R^3)$ be a test function. We then have that\n\\ba\\label{nonlin-FP-est1}\n&\\int_0^T\\int_{\\O\\times D}(\\vu_n \\psi_n - \\vu \\psi) \\cdot  \\vvarphi \\, \\dq\\, \\dx \\, \\dt \\\\\n&= \\int_0^T\\int_{\\O\\times D}(\\vu_n -\\vu)\\psi \\cdot \\vvarphi \\, \\dq\\, \\dx \\, \\dt  +  \\int_0^T\\int_{\\O\\times D}\\vu_n(\\psi_n-\\psi) \\cdot \\vvarphi \\, \\dq\\, \\dx \\, \\dt =:I_1^n+I_2^n.\n\\ea\n\nFor $I_1^n$, we have\n\\ba\\label{nonlin-FP-est2}\nI_1^n:= \\int_0^T\\int_{\\O\\times D}(\\vu_n -\\vu)\\psi \\cdot \\vvarphi \\, \\dq\\, \\dx \\, \\dt  = \\int_0^T\\int_{\\O}(\\vu_n -\\vu) \\cdot\\left(\\int_D \\psi \\, \\vvarphi \\, \\dq \\right) \\dx \\, \\dt.\n\\ea\nWe observe that\n\\be\\label{nonlin-FP-est3}\n\\left|\\int_D \\psi \\, \\vvarphi \\, \\dq \\right| \\leq c\\,\\int_D \\psi\\,  \\dq = c\\, \\eta \\in L^\\infty(0,T;L^2(\\O))\\cap L^2(0,T;L^6(\\O)).\n\\ee\nWe recall the weak convergence results for $\\vu_n$ stated in \\eqref{vun-limit1} and \\eqref{vun-limit2}; in particular we have that $\\vu_n \\to \\vu$ weakly in $L^2(0,T;L^{\\frac{12}{2+3|\\o|}}(\\O;\\R^3))$, with\n$$\n\\frac{2+3|\\o|}{12} + \\frac{1}{6} = \\frac{1}{3} + \\frac{|\\o|}{4} <1,\n$$\nunder hypothesis \\eqref{mu-eta2} or hypothesis \\eqref{mu-eta3}. Therefore,\n$\\lim_{n\\to \\infty} I_1^n = 0.$\n\n\n\\medskip\n\nFor $I_2^n$, we have\n\\ba\\label{nonlin-FP-est4}\nI_2^n:= \\int_0^T\\int_{\\O\\times D}\\vu_n (\\psi_n-\\psi) \\cdot \\vvarphi \\, \\dq\\, \\dx \\, \\dt = \\int_0^T\\int_{\\O} \\vu_n \\cdot \\left(\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\, \\dq \\right) \\dx \\,\\dt.\n\\ea\nBy the strong convergence of $\\psi_n$ stated in \\eqref{psin-lim2} we have that\n\\be\\label{nonlin-FP-lim2}\n\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\, \\dq \\to \\mathbf{0}\\quad \\mbox{a.e. in $(0,T)\\times \\O$}.\n\\ee\nSimilarly to \\eqref{nonlin-FP-est4}, we have that\n\\be\\label{nonlin-FP-est5}\n\\left|\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\, \\dq\\right| \\in L^\\infty(0,T;L^2(\\O))\\cap L^2(0,T;L^6(\\O)) \\hookrightarrow L^{\\frac{10}{3}}((0,T)\\times \\O).\n\\ee\nThanks to \\eqref{nonlin-FP-lim2} and \\eqref{nonlin-FP-est5}, Vitali's theorem implies that\n\\be\\label{nonlin-FP-lim3}\n\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\, \\dq  \\to 0 \\quad \\mbox{in}\\  L^{q}((0,T)\\times \\O;\\R^3) \\quad \\mbox{for any $q<\\frac{10}{3}$}.\n\\ee\nIt follows from \\eqref{vun-limit1} (for $\\omega \\geq 0$) and from \\eqref{vun-limit2} (for $\\omega \\leq 0$) that\n$\\lim_{n\\to \\infty} I_2^n = 0.$ Therefore,\n\\be\\label{nonlin-FP-limf1}\n\\vu_n \\psi_n \\to  \\vu \\psi \\quad \\mbox{in}\\ \\mathcal{D}'((0,T)\\times \\O\\times D;\\R^3).\n\\ee\n\n\\medskip\n\nTo prove the convergence of $(\\nabla_x \\vu_n)\\,q_i \\psi_n$ to $(\\nabla_x \\vu )\\,q_i\\psi$ in $\\mathcal{D}'((0,T)\\times \\O\\times D;\\R^3)$ one can proceed similarly. We point out that, when $\\o\\leq 0$, the requirement that $\\o > -\\frac{4}{3}$, appearing in \\eqref{mu-eta3}, is really needed. Indeed, for any test function $\\vvarphi$, just as in \\eqref{nonlin-FP-lim3}, we have that\n\\be\\label{nonlin-FP-lim5}\n\\int_D (\\psi_n-\\psi) \\, \\vvarphi \\,q_i \\ \\dq  \\to \\mathbf{0} \\quad \\mbox{in}\\  L^q(0,T;L^q(\\O;\\R^3)) \\quad \\mbox{for any $q<\\frac{10}{3}$}.\n\\ee\nAccording to the bounds established in Section \\ref{sec:lim-u} for the case $\\o\\leq 0$, the sequence $\\left\\{\\nabla_x \\vu_n\\right\\}_{n\\in \\N}$ is uniformly bounded in $L^{\\frac{20}{10+3|\\o|}}((0,T)\\times\\O;\\R^{3\\times 3})$. To deduce the desired convergence result, we therefore need that\n\\be\\label{mu-eta30}\n\\frac{10+3|\\o|}{20} + \\frac{3}{10}<1  \\Longleftrightarrow |\\o| < \\frac{4}{3}.\n\\ee\n\n\n\\subsection{Convergence of the fluid pressure}\nAs was shown in Section \\ref{sec:est-higher-vr} concerning higher integrability of the fluid density and pressure, we have that\n\\be\\label{pn-lim1}\np(\\vr_n) \\to \\overline{p(\\vr)} \\quad \\mbox{weakly in} \\ L^{1+\\frac{\\th}{\\g}}((0,T)\\times \\O).\n\\ee\n\nBy applying this, together with the convergence results from the previous sections, in particular the convergence results for the nonlinear terms stated in Section \\ref{sec:lim-convective}, Section \\ref{sec:lim-SS} and Section \\ref{sec:lim-TT}, we deduce that\n\\be\\label{i1-0}\n\\d_t \\vr + \\Div_x (\\vr \\vu) = 0 \\quad \\mbox{in}\\  \\mathcal{D}'((0,T)\\times \\O),\n\\ee\n\\be\\label{i2-0}\n \\d_t (\\vr\\vu)+ \\Div_x (\\vr \\vu \\otimes \\vu) +\\nabla_x \\overline{p(\\vr)} -\n\\Div_x \\SSS =\\Div_x \\TT +\\vr\\, \\ff\\quad \\mbox{in}\\  \\mathcal{D}'((0,T)\\times \\O;\\R^3),\n\\ee\nwhere the Newtonian stress tensor $\\SSS$  and the extra-stress tensor $\\TT$ are defined by \\eqref{Newtonian-lim1} and  \\eqref{TTn-lim2}, respectively.\n\n\nConcerning the convergence of the nonlinear terms in the Navier--Stokes--Fokker--Planck system with variable viscosity coefficients considered here, it remains to show that $p(\\vr) = \\overline{p(\\vr)},$\nwhich, because of the strict convexity of $p(\\cdot)$, is equivalent to the strong convergence of $\\vr_n$:\n\\be\\label{pn-limf2}\n\\vr_n \\to \\vr \\quad \\mbox{a.e. in} \\ (0,T)\\times \\O.\n\\ee\n\nThis is also one of the main difficulties in the study of global existence of weak solutions to the compressible Navier--Stokes equations (see \\cite{Lions-C, FNP,F-book,N-book}), where the so called \\emph{effective viscous flux} introduced by P. L. Lions \\cite{Lions-C} plays a crucial role. It turns out that the effective viscous flux as a whole is more regular than its components.\n  We will prove the following lemma, which is in the spirit of Proposition 5.1 in \\cite{Feireisl-2001}.\n\\begin{lemma}\\label{lem-viscous-flux}\nFor any $\\phi\\in \\mathcal{D}(0,T)$, $\\vp \\in \\mathcal{D}(\\O)$, we have that\n\\ba\\label{vis-flux}\\lim_{n\\to \\infty}\\int_0^T \\intO{\\phi(t)\\vp(x) \\left[p(\\vr_n) - \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt\\\\\n=\\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\overline{p(\\vr)} - \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt,\n\\ea\nwhere $T_k$ is a cut-off function defined by $T_k(\\cdot) :=  k \\,T(\\frac{\\cdot}{k})$ for some concave function $T\\in C^\\infty([0,\\infty))$ such that $T(s)=s$ for $0\\leq s\\leq 1$ and $T(s)=2$ for $s\\geq 3$. Here $\\overline{T_k(\\vr)}$ is the weak* limit of the sequence $\\{T_k(\\vr_n)\\}_{n\\in \\N}$ in $L^\\infty((0,T)\\times \\O)$ as $n$ goes to infinity.\n\\end{lemma}\n\n\nThe viscosity coefficients $\\mu^B$ and $\\mu^S$ in our case are not constant, which gives rise to additional complications in the proof of this lemma. We shall employ the commutator estimates stated in Lemma \\ref{lem-Riesz3} and the techniques used in Section 3.6.5 in \\cite{F-N-book} to overcome the resulting difficulties.\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem-viscous-flux}] As in the proof of the effective viscous flux lemma for the compressible Navier--Stokes equations with constant viscosity coefficients (see Proposition 5.1 in \\cite{Feireisl-2001} or Proposition 7.36 in \\cite{N-book}), we introduce the following test functions:\n\\be\\label{def-test-flux}\n\\vv_n(t,x) = \\phi(t) \\vp(x) \\A \\left[T_k(\\vr_n) \\right],\\quad \\vv(t,x) = \\phi(t) \\vp(x) \\A \\left[ \\overline{T_k(\\vr)}\\right],\n\\ee\nwhere $\\phi\\in C_c^\\infty(0,T), \\ \\vp \\in C_c^\\infty(\\O)$,  and $\\A$ is the Fourier integral operator introduced in \\eqref{def-Reize}. We remark that $\\vr_n$ and $\\vr$ are extended by zero outside of $\\O$ in \\eqref{def-test-flux}.\n\n\nTaking these $\\vv_n$ and $\\vv$ as test functions in the weak formulation \\eqref{weak-form3} and the weak formulation of \\eqref{i2-0}, respectively, results in\n\\begin{align}\n\\label{weak-vn}\n&\\int_0^T \\intO{ \\big[ \\vr_n \\vu_n \\cdot \\partial_t \\vv_n + (\\vr_n \\vu_n \\otimes \\vu_n) : \\Grad \\vv_n  + p(\\vr_n)\\, \\Div_x \\vv_n - \\SSS_n : \\Grad \\vv_n \\big]} \\,\\dt\n\\nonumber\\\\\n&\\hspace{3in}= \\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n  - \\vr_n\\, \\ff \\cdot \\vv_n} \\, \\dt,\\\\\n\\label{weak-v}\n&\\int_0^T \\intO{ \\left[ \\vr \\vu \\cdot \\partial_t \\vv + (\\vr \\vu \\otimes \\vu) : \\Grad \\vv  + \\overline{p(\\vr)}\\, \\Div_x \\vv - \\SSS : \\Grad \\vv \\right]}\\,\\dt\n\\nonumber\\\\\n&\\hspace{3in}= \\int_0^T \\intO{ \\TT : \\nabla_x \\vv  - \\vr\\, \\ff \\cdot \\vv} \\, \\dt.\n\\end{align}\n\nThe main idea is to pass to the limit $n\\to \\infty$ in \\eqref{weak-vn}, and compare the resulting limit with \\eqref{weak-v}, which will ultimately imply our desired result \\eqref{vis-flux}. Since, following the contributions of Lions \\cite{Lions-C} and Feireisl \\cite{F-book}, this type of argument is by now well understood, instead of including all of the details here we shall focus on the terms that do not appear in the case of the compressible Navier--Stokes equations with constant viscosity coefficients. These `new' terms are the following:\n\\ba\\label{flux-new-terms1}\n&\\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n} \\, \\dt,\\quad \\int_0^T \\intO{ \\TT : \\nabla_x \\vv } \\, \\dt,\\quad \\int_0^T \\intO{  \\SSS_n : \\Grad \\vv_n } \\, \\dt,\\quad \\int_0^T \\intO{  \\SSS : \\Grad \\vv } \\, \\dt,\\nn\n\\ea\nand the goal is to prove that\n\\ba\\label{flux-new-terms2}\n\\lim_{n\\infty}\\int_0^T \\intO{ \\TT_n : \\nabla_x \\vv_n} \\, \\dt = \\int_0^T \\intO{ \\TT : \\nabla_x \\vv } \\, \\dt,\n\\ea\nand\n\\ba\\label{flux-new-terms3}\n&\\lim_{n\\to\\infty}\\int_0^T \\intO{  \\SSS_n : \\Grad \\vv_n } \\, \\dt -  \\int_0^T \\intO{  \\SSS : \\Grad \\vv } \\, \\dt \\\\\n&= \\lim_{n\\to\\infty} \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt \\\\\n&\\quad - \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt.\n\\ea\n\nBy the strong convergence of $\\TT_n$ stated in \\eqref{TTn-lim-strff} in Section \\ref{sec:lim-TT} it is straightforward to show the convergence result \\eqref{flux-new-terms2}, so we only focus on proving \\eqref{flux-new-terms3}. We begin by noting that\n\\ba\\label{flux-new31}\n\\int_0^T \\intO{  \\SSS_n : \\Grad \\vv_n } \\, \\dt  = \\int_0^T \\phi(t)\\intO{  \\SSS_n :\\, \\left(\\nabla_x \\vp \\otimes \\A[ T_k(\\vr_n)]\\right)} \\, \\dt + \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\SSS_n : \\RR [ T_k(\\vr_n)]} \\, \\dt,\\nn\n\\ea\nwhere $\\RR$ is the Riesz operator defined in \\eqref{def-Reize} in Section \\ref{sec:Riesz}.\n\n\nFor any fixed $k\\in\\N$,  the sequence $\\{ T_k(\\vr_n)\\}_{n\\in \\N}$ is uniformly bounded in $ L^\\infty(0,T;L^r(\\R^3))$ for all $r \\in [1, \\infty]$. Thus, by Lemma \\ref{lem-Riesz1}, we have that\n\\be\\label{A-Tk-vrn-1}\n\\sup_{n\\in \\N} \\left\\| \\A[T_k(\\vr_n)]\\right\\|_{L^\\infty(0,T;W^{1,r}(\\O;\\R^3))} \\leq c\\quad \\mbox{for any $r \\in (1,\\infty)$}.\n\\ee\n\nObserving that $T_k(\\cdot)$ fulfills the properties in \\eqref{cond-b-renormal}, it follows from \\eqref{weak-renormal1} that\n\\be\\label{renomal-Tk}\n\\d_t T_k(\\vr_n) + \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) + \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu_n =0\\quad \\mbox{in $ \\mathcal{D}'((0,T)\\times \\R^3)$}.\n\\ee\nThis implies that\n\\ba\\label{A-Tk-vrn-2}\n\\d_t \\A[ T_k(\\vr_n)]= \\A[\\d_t T_k(\\vr_n)] = - \\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right] - \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu_n \\right].\n\\ea\nSince\n$\n\\A_j \\Div_x = -\\sum_{i=1}^3\\d_j \\d_i \\Delta^{-1}  = - \\sum_{i=1}^3\\RR_{ij}\n$\nare Riesz type operators, by Lemma \\ref{lem-Riesz1} we have that\n$$\n\\|\\A_j \\left[ \\Div_x \\vvarphi \\right]\\|_{L^r(\\R^3:\\R^3)}\\leq c \\, \\|\\vvarphi\\|_{L^r(\\R^3;\\R^3)} \\quad \\mbox{for any $\\vvarphi\\in L^r(\\R^3;\\R^3)$ and any $r \\in (1,\\infty)$}.\n$$\nBy using \\eqref{vun-limit1} and \\eqref{vun-limit2} we deduce that\n\\ba\\label{A-Tk-vrn-3}\n\\left\\|\\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right]\\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$}; \\\\\n\\left\\|\\A\\left[ \\Div_x\\left(T_k(\\vr_n)\\vu_n\\right) \\right]\\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad\\mbox{when $\\o\\leq 0$}.\n\\ea\nFurthermore, by Lemma \\ref{lem-Riesz1} in conjunction with \\eqref{vun-limit1} and \\eqref{vun-limit2} we deduce that\n\\ba\\label{A-Tk-vrn-4}\n\\left\\| \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu \\right] \\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$}; \\\\\n\\left\\| \\A\\left[ \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu \\right] \\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\leq 0$}.\n\\ea\nThus,\n\\ba\\label{A-Tk-vrn-5}\n\\left\\| \\d_t \\A[ T_k(\\vr_n)]  \\right\\|_{L^2(0,T;L^6(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\geq 0$};\\\\\n\\left\\|  \\d_t \\A[ T_k(\\vr_n)] \\right\\|_{L^2(0,T;L^{\\frac{12}{2+6|\\o|}}(\\O;\\R^3))} &\\leq c &&\\quad \\mbox{when $\\o\\leq 0$}.\n\\ea\nIt follows from the uniform estimates in \\eqref{A-Tk-vrn-1} and \\eqref{A-Tk-vrn-5} and the Aubin--Lions--Simon compactness theorem that\n\\ba\\label{A-Tk-vrn-f}\n\\A[T_k(\\vr_n)] \\to \\overline{\\A[T_k(\\vr)]} = \\A[\\overline{T_k(\\vr)}] \\quad \\mbox{strongly in $L^\\infty(0,T;L^r(\\O;\\R^3))$ \\quad for any $r\\in (1,\\infty)$}.\n\\ea\nTogether with the weak convergence of $\\SSS_n$ stated in \\eqref{Newtonian-lim1} and \\eqref{Newtonian-lim10}, we deduce that\n\\ba\\label{flux-new32}\n \\lim_{n\\to\\infty}\\int_0^T \\phi(t)\\intO{  \\SSS_n : (\\nabla_x \\vp \\otimes \\A[ T_k(\\vr_n)])} \\, \\dt = \\int_0^T \\phi(t)\\intO{  \\SSS : (\\nabla_x \\vp \\otimes \\A[ \\overline{T_k(\\vr)}])} \\, \\dt.\n\\ea\nTherefore, to show \\eqref{flux-new-terms3}, it suffices to prove that\n\\ba\\label{flux-new33}\n& \\lim_{n\\to \\infty} \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\SSS_n : \\RR [ T_k(\\vr_n)]} \\, \\dt -  \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\SSS : \\RR [ \\overline{T_k(\\vr)}]} \\, \\dt\\\\\n &= \\lim_{n\\to\\infty} \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta_n)}{3}-\\mu^B(\\eta_n)\\right) \\Div_x \\vu_n \\right] T_k(\\vr_n) }\\, \\dt \\\\\n&\\quad - \\int_0^T \\intO{\\phi(t)\\vp(x) \\left[ \\left(\\frac{2\\mu^S(\\eta)}{3}-\\mu^B(\\eta)\\right) \\Div_x \\vu\\right] \\overline{T_k(\\vr)} }\\, \\dt.\n\\ea\n\nBy \\eqref{i3} we have that\n\\ba\\label{flux-new34}\n&\\int_0^T \\phi(t)\\intO{ \\vp \\,  \\SSS_n : \\RR [ T_k(\\vr_n)]} \\ \\dt= \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\mu^S(\\eta_n) \\left( \\frac{\\nabla \\vu_n + \\nabla^{\\rm T} \\vu_n}{2} \\right) : \\RR [ T_k(\\vr_n)]} \\, \\dt\\\\\n&\\qquad + \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\left(\\mu^B(\\eta_n)-\\frac{\\mu^S(\\eta_n)}{3}\\right) (\\Div_x \\vu_n) \\II: \\RR [ T_k(\\vr_n)]} \\, \\dt\\\\\n& = \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\mu^S(\\eta_n) \\nabla \\vu_n  : \\RR [ T_k(\\vr_n)]} \\, \\dt + \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\left(\\mu^B(\\eta_n)-\\frac{\\mu^S(\\eta_n)}{3}\\right) (\\Div_x \\vu_n) \\, [ T_k(\\vr_n)]} \\, \\dt,\n\\ea\nwhere have we used the fact that $\\RR$ is symmetric (see Lemma \\ref{lem-Riesz1}) and that $\\II : \\RR = \\sum_{i=1}^3 \\RR_{ii}=\\II$.\nFurther, by Lemma \\ref{lem-Riesz1}, and noting that $\\sum_{i,j=1}^3 \\RR_{ij}\\partial_j \\vu_n^i = \\Div_x \\vu_n$, we have that\n\\ba\\label{flux-new35}\n& \\int_0^T \\phi(t)\\intO{ \\vp \\,  \\mu^S(\\eta_n) \\nabla \\vu_n  : \\RR [ T_k(\\vr_n)]} \\, \\dt = \\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3 \\RR_{ij} \\left[\\vp \\,  \\mu^S(\\eta_n)\\, \\d_j \\vu_n^i \\right] T_k(\\vr_n)} \\, \\dt\\\\\n&\\quad =\\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3 \\left( \\vp \\,  \\mu^S(\\eta_n)\\, \\RR_{ij} \\left[ \\d_j \\vu_n^i \\right] T_k(\\vr_n) + R_n^{ij} \\, T_k(\\vr_n) \\right)\\!} \\, \\dt \\\\\n&\\quad = \\int_0^T \\phi(t)\\intO{ \\left(\\vp \\,  \\mu^S(\\eta_n)\\, (\\Div_x \\vu_n)  \\, T_k(\\vr_n) + \\sum_{i,j=1}^3 R_n^{ij} \\, T_k(\\vr_n) \\right)\\!} \\, \\dt,\n\\ea\nwhere $R_n^{ij}$ is the commutator defined by $R_n^{ij}:= \\RR_{ij} \\left[\\vp \\,  \\mu^S(\\eta_n) \\d_j \\vu_n^i \\right]- \\vp \\,  \\mu^S(\\eta_n) \\RR_{ij} \\left[ \\d_j \\vu_n^i \\right].$ We will use Lemma \\ref{lem-Riesz3} to derive uniform bounds on $R_n^{ij}$. We first consider the case $0\\leq \\o < \\frac 53$. Then, by \\eqref{mu-eta1} and \\eqref{etan-est1}, the sequence\n$\\{\\nabla_x \\mu^S(\\eta_n)\\}_{n\\in\\N} = \\{(\\mu^S)'(\\eta_n) \\nabla_x \\eta_n\\}_{n\\in\\N}$ is uniformly bounded in $L^2(0,T;L^2(\\O;\\R^3))$ when $0\\leq \\o\\leq 1$, and in $L^{2}(0,T;L^\\frac{2}{\\o}(\\O;\\R^3))$ when $1\\leq \\o < \\frac 53$.\nThus,\n$$\n\\sup_{n\\in \\N}\\| \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,2}(\\O))} \\leq c \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\qquad \\sup_{n\\in \\N}\\| \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,\\frac{2}{\\o}}(\\O))} \\leq c \\quad \\mbox{when}\\ 1 \\leq \\o < \\frac 53.\n$$\n\nBy \\eqref{vun-limit1} the sequence $\\{\\nabla_x \\vu_n\\}_{n\\in\\N}$ is uniformly bounded in $ L^2(0,T; L^{2}(\\O;\\R^{3 \\times 3}))$. Furthermore, we note that\n\\ba\\label{mu-eta10}\n\\frac{1}{2} + \\frac{1}{2} -\\frac{1}{3} = \\frac{2}{3}< 1;\\quad \\frac{1}{2}+\\frac{\\o}{2} - \\frac{1}{3} =\\frac{1+3\\o}{6} < 1, \\quad \\mbox{as long as}\\ 0\\leq \\o < \\frac{5}{3}.\n\\ea\nHence, by Lemma \\ref{lem-Riesz3}, for any $s>1$ such that\n\\ba\\label{est-mun-2}\n \\frac{1}{s} >\\frac{1}{2}+\\frac{1}{2} - \\frac{1}{3}=\\frac{2}{3} \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\quad \\frac{1}{s} >\\frac{1}{2}+\\frac{\\o}{2} - \\frac{1}{3} =\\frac{1+3\\o}{6} \\quad \\mbox{when}\\ 1\\leq \\o < \\frac{5}{3},\n\\ea\nwe have, for $r_\\o=2$ when $0\\leq \\o\\leq 1$ and for $r_\\o=\\frac{2}{\\o}$ when $1\\leq \\o < \\frac{5}{3}$, that\n\\ba\\label{flux-commu-2}\n&\\left\\|R_n^{ij}\\right\\|_{W^{\\b,s}(\\R^3)} \\leq c\\, \\| \\vp \\mu^S(\\eta_n) \\|_{W^{1,r_\\o}(\\O)} \\|\\d_j \\vu_n^i\\|_{L^2(\\O)},\n\\ea\nwhere $\\b\\in (0,1)$ is such that\n\\ba\\label{est-mun-3}\n\\frac{\\b}{3} =\\frac 13 + \\frac 1s - 1 \\quad \\mbox{when}\\ 0\\leq \\o\\leq 1;\\quad \\frac{\\b}{3} =\\frac 13 + \\frac 1s - \\frac{1+\\o}{2}\\quad \\mbox{when}\\ 1\\leq \\o < \\frac 53.\n\\ea\nTherefore, for some $s>1$ and some $0<\\b<1$, we have the uniform estimate\n\\ba\\label{flux-commu-3}\n\\sup_{n\\in \\N} \\left\\|R_n^{ij} \\right\\|_{L^1(0,T;W^{\\b,s}(\\R^3))} \\leq c\\, \\sup_{n\\in \\N} \\| \\vp \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,r_\\o}(\\O))}\\, \\sup_{n\\in \\N} \\|\\d_j \\vu_n^i\\|_{L^2(0,T;L^2(\\O))}\\leq c.\n\\ea\nHence, by the strong convergence of $\\eta_n$ shown in Section \\ref{sec:lim-eta} we have that\n\\ba\\label{flux-commu-4}\nR_n^{ij} \\to R^{ij}:= \\RR_{ij} \\left[\\vp \\,  \\mu^S(\\eta) \\d_j \\vu^i \\right]- \\vp \\,  \\mu^S(\\eta) \\RR_{ij} \\left[ \\d_j \\vu^i \\right] \\quad \\mbox{weakly in} \\ L^{\\frac{10}{3\\o+5}}((0,T)\\times \\O).\n\\ea\n\nBy the boundedness of $T_k$ we have\n\\be\\label{Tk-weak}\nT_k(\\vr_n)\\to \\overline{T_k(\\vr)} \\quad \\mbox{weakly in $L^r((0,T)\\times \\O)$\\quad for any $1<r<\\infty$}.\n\\ee\nNext, we want to prove the convergence of the product\n\\ba\\label{flux-commu-5}\nR_n^{ij}\\, T_k(\\vr_n) \\to R^{ij} \\overline{T_k(\\vr)} \\quad \\mbox{weakly in} \\ L^{r}(0,T;L^{r}(\\O)) \\quad \\mbox{for any $r<\\frac{10}{3\\o+5}$}.\n\\ea\nWe shall use the Div-Curl lemma to this end in the time-space variables by setting\n\\be\\label{div-curl-set}\n{\\bf U}_n := (T_k(\\vr_n), T_k(\\vr_n) \\vu_n ),\\quad {\\bf V}_n := (R_n^{ij},0,0,0).\n\\ee\nBy \\eqref{renomal-Tk} we then have that\n\\be\\label{div-curl1}\n\\Div_{t,x} {\\bf U}_n = \\d_t T_k(\\vr_n) + \\Div_x\\!\\left(T_k(\\vr_n)\\vu_n\\right) = - \\left(T_k'(\\vr_n)\\vr_n-T_k(\\vr_n)\\right)\\Div_x \\vu_n,\n\\ee\nwhich is uniformly bounded in $L^2((0,T)\\times \\O)$ and, therefore,\n\\[\\{\\Div_{t,x} {\\bf U}_n\\}_{n \\in \\N}\\quad \\mbox{is precompact in $W^{-1,2}((0,T)\\times \\O)$}.\\]\nNext, on observing that ${\\rm curl}_{t,x}\\,  {\\bf V}_n$ does not include the time-derivative of $R_n^{ij}$, by \\eqref{flux-commu-3} and Lemma \\ref{lem-frac-sob} we have that\n\\[\n\\{{\\rm curl}_{t,x} {\\bf V}_n\\}_{n\\in \\N} \\quad \\mbox{is precompact in $W^{-1,s}((0,T)\\times \\O;\\R^{3\\times 3})$}.\n\\]\nThus, the Div-Curl lemma (Lemma \\ref{lem-div-curl2}) implies \\eqref{flux-commu-5}. This gives\n\\ba\\label{flux-new36}\n\\lim_{n\\to \\infty} \\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3  R_n^{ij} \\, T_k(\\vr_n) } \\ \\dt = \\int_0^T \\phi(t)\\intO{ \\sum_{i,j=1}^3  R^{ij} \\, T_k(\\vr) } \\ \\dt.\n\\ea\n\n\nWe shall now briefly summarize the proof of \\eqref{flux-new36} when $-\\frac{4}{3}<\\o\\leq 0$.   By \\eqref{mu-eta1} and \\eqref{etan-est1} the sequence $\\{ \\mu^S(\\eta_n)\\}_{n\\in\\N}$ is uniformly bounded in $L^2(0,T;W^{1,2}(\\O))$ when $\\o\\leq 0$. By \\eqref{vun-limit2} the sequence $\\{\\nabla_x \\vu_n\\}_{n\\in\\N}$ is uniformly bounded in $ L^2(0,T; L^{\\frac{4}{2+|\\o|}}(\\O);\\R^{3\\times 3})$. We have that\n\\ba\\label{mu-eta10-w<0}\n\\frac{1}{2}+\\frac{2+|\\o|}{4} - \\frac{1}{3} < 1, \\quad \\mbox{as long as}\\ -\\frac{4}{3}< \\o \\leq 0.\n\\ea\nThen, by Lemma \\ref{lem-Riesz3}, for any $s>1$ such that\n\\ba\\label{est-mun-2-w<0}\n& \\frac{1}{s} > \\frac{1}{2}+\\frac{2+|\\o|}{4} - \\frac{1}{3},\n\\ea\nwe have that\n\\ba\\label{flux-commu-2-w<0}\n&\\left\\|R_n^{ij}\\right\\|_{W^{\\b,s}(\\R^3)} \\leq c\\, \\| \\vp \\mu^S(\\eta_n) \\|_{W^{1,2}(\\O)} \\|\\d_j \\vu_n^i\\|_{L^\\frac{4}{2+|\\o|}(\\O)},\n\\ea\nwhere $\\b\\in (0,1)$ is such that\n\\ba\\label{est-mun-3-w<0}\n\\frac{\\b}{3} =\\frac 13 + \\frac 1s - \\frac{1}{2}-\\frac{2+|\\o|}{4}.\n\\ea\nTherefore, for some $s>1$ and some $\\b>0$, the following uniform estimate holds:\n\\ba\\label{flux-commu-3-w<0}\n\\sup_{n\\in \\N} \\left\\|R_n^{ij} \\right\\|_{L^1(0,T;W^{\\b,s}(\\R^3))} \\leq c\\, \\sup_{n\\in \\N} \\| \\vp \\mu^S(\\eta_n) \\|_{L^2(0,T;W^{1,2}(\\O))}\\,\\sup_{n\\in \\N}  \\|\\d_j \\vu_n^i\\|_{L^2(0,T;L^\\frac{4}{2+|\\o|}(\\O))}\\leq c.\n\\ea\nThanks to the strong convergence of $\\eta_n$ shown in Section \\ref{sec:lim-eta}, we have that\n\\ba\\label{flux-commu-4-w<0}\n R_n^{ij} \\to R^{ij}:= \\RR_{ij} \\left[\\vp \\,  \\mu^S(\\eta) \\d_j \\vu^i \\right]- \\vp \\,  \\mu^S(\\eta) \\RR_{ij} \\left[ \\d_j \\vu^i \\right], \\\\\n \\mbox{weakly in} \\ L^{r}((0,T)\\times \\O),\\ \\mbox{for any $r<\\frac{20}{10+3|\\o|}$}.\n\\ea\nThus, the Div-Curl lemma implies that\n\\ba\\label{flux-commu-5-w<0}\nR_n^{ij}\\, T_k(\\vr_n) \\to R^{ij} \\overline{T_k(\\vr_n)} \\quad \\mbox{weakly in} \\ L^{r}(0,T;L^{r}(\\O)) \\quad \\mbox{for any $r<\\frac{20}{10+3|\\o|}$},\n\\ea\nwhich, once again, implies \\eqref{flux-new36}.\n\n\\medskip\n\n\n\n\nFinally, from \\eqref{flux-new34}, \\eqref{flux-new35} and \\eqref{flux-new36}, we deduce the desired result \\eqref{flux-new33}. This implies \\eqref{flux-new-terms3} by using \\eqref{flux-new32}. At the same time, by tedious but, by now, standard calculations, as in the proof of the effective viscous flux lemma in \\cite{Lions-C, F-book, N-book}, we have that\n\\ba\\label{flux-new-terms3f}\n\\lim_{n\\to\\infty}\\int_0^T \\intO{  \\SSS_n : \\Grad \\vv_n } \\ \\dt -  \\int_0^T \\intO{  \\SSS : \\Grad \\vv } \\ \\dt =0.\n\\ea\nCombining \\eqref{flux-new-terms3f} with \\eqref{flux-new-terms3}, we obtain \\eqref{vis-flux} and complete the proof of Lemma \\ref{lem-viscous-flux}.\n\\end{proof}\n\n\n\\medskip\n\nWith Lemma \\ref{lem-viscous-flux} in hand, the proof of the strong convergence result \\eqref{pn-limf2} then proceeds along a well-understood route (see for example \\cite{Feireisl-2001}, from Section 6 to Section 8), so we shall not dwell on the details here. In particular, as in Proposition 7.1 in \\cite{Feireisl-2001}, the limit $(\\vr, \\vu)$ satisfies \\eqref{weak-renormal1} in the sense of renormalized weak solutions.\n\n\\medskip\n\nIt remains to show that the limit $(\\vr,\\vu,\\psi,\\eta)$ satisfies the energy inequality \\eqref{energy}. This is easily seen by noting the strong convergence assumption on the initial data in \\eqref{cond-ini-n} and passage to the limit $n\\to \\infty$ in \\eqref{energy-n}, which directly imply the energy\ninequality \\eqref{energy} \nby the application of Tonelli's sequential weak (weak*) lower semicontinuity theorem (cf. Theorem 10.16 in \\cite{RR-book}, for example,) to the terms appearing on the left-hand side of \\eqref{energy-n}; in particular, the sequential weak lower semicontinuity of the $L^p$ norm, $1< p<\\infty$, the sequential weak* lower semicontinuty of the $L^\\infty$ norm and inequality \\eqref{psin-est3} are used.\nThus we have shown that the limit $(\\vr,\\vu,\\psi,\\eta)$ is a dissipative (finite-energy) weak solution in the sense of Definition \\ref{def-weaksl}. That completes the proof of Theorem \\ref{theorem}.\n\n\\begin{remark}\\label{rem-mu-eta}\nThe constraint $\\o<\\frac{5}{3}$ for the case $\\o\\geq 0$ is crucially determined by \\eqref{mu-eta10} and the condition $\\frac{10}{3\\o+5}>1$, which appears in \\eqref{flux-commu-4}, while the constraint $\\o>-\\frac{4}{3}$ for the case $\\o\\leq 0$ is crucially determined by \\eqref{mu-eta10-w<0}. It is unclear whether,\nwith our present techniques at least, these restrictions on $\\o$ can be relaxed.\n\\end{remark}\n\n\n\n\\section{Conclusion: existence of dissipative weak solutions}\\label{sec:end}\n\n\\label{End}\n\nThe conclusions of Theorem \\ref{theorem} \\emph{do not}, of course, imply the existence of dissipative (finite-energy) weak solutions to the Navier--Stokes--Fokker--Planck system with polymer-number-density-dependent viscosity coefficients. A rigorous proof of the existence of dissipative weak solutions would require the following:\n\\begin{itemize}\n\n\\item a suitable approximation scheme, compatible with the energy inequality and the compactness arguments presented in this paper;\n\n\\item a rigorous proof of the existence of a solution to the approximation scheme;\n\n\\item proof of the convergence of the sequence of approximate solutions to a dissipative weak solution, mimicking the arguments presented in this paper.\n\n\\end{itemize}\n\nGiven the formal similarity of the present model to the one studied in \\cite{Barrett-Suli}, a natural approach would be to adjust the approximation scheme used\nin \\cite{Barrett-Suli}, based on time-discretization, in the case of the Navier--Stokes--Fokker--Planck system with constant viscosity coefficients (or, alternatively, to use a Galerkin approximation scheme in the spatial variables, similar to the one in \\cite{FNP}). The added technical difficulties, caused by the presence of the variable viscosity coefficients, can be handled exactly as in \\cite{AbFei}, where a similar scheme, based on time-discretization, was applied to a diffuse interface model with viscosity coefficients that depended on the concentration.\n\n\n\n\\section*{Acknowledgements}\n\\thispagestyle{empty}\n\nThe authors acknowledge the support of the project LL1202 in the programme ERC-CZ funded by the Ministry of Education, Youth and Sports of the Czech Republic.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe discovery of the standard model (SM) Higgs boson~\\cite{Chatrchyan:2012xdj,Aad:2012gk}, \nthe only known elementary scalar particle, has been the main result of the Large Hadron Collider\nphysics program. Remarkably, the measured Higgs boson mass was actually predicted with a good accuracy already in 1995~\\cite{Froggatt:1995rt}  by\napplying the principle of multiple point criticality (PMPC)~\\cite{Bennett:1993pj,Bennett:1996vy,Bennett:1996hx} to the Higgs boson effective potential. \nToday the SM Higgs criticality has been studied to three-loop accuracy, indicating that the present data prefers a meta-stable Higgs vacuum \nsufficiently close to criticality~\\cite{Buttazzo:2013uya}.\\footnote{This statement depends most strongly on the precise value of the top quark Yukawa coupling, as the critical point remains within $3\\sigma$ uncertainty of $y_t$. Since $y_t$ is not a directly measured quantity, the systematic uncertainty related to its derivation from collider top mass measurements may be presently underestimated~\\cite{Corcella:2019tgt}.}\nThis fact is the most interesting unexplained experimental outcome of collider physics so far.\n\n\nWithin the SM, there is no explanation as to why the SM parameters -- the gauge, Yukawa and scalar quartic couplings -- must take values leading to the Higgs criticality.\nTherefore, in the original proposal~\\cite{Froggatt:1995rt}, the existence of multiple degenerate vacua was argued to be analogous to the \nsimultaneous existence of multiple phases in thermodynamics. This statement is purely empirical  and cannot follow solely from the SM physics, nevertheless\nit may have some support from statistical systems in solid state physics~\\cite{Volovik:2003hn}.\nTo derive the PMPC from fundamental physical principles, Higgs criticality has been attributed to the proposed asymptotic safety of quantum gravity~\\cite{Shaposhnikov:2009pv}.\nMore recently Dvali~\\cite{Dvali:2011wk} revived the old argument by Zeldovich~\\cite{Zeldovich:1974py,Kobzarev:1974cp}  that physical systems with multiple vacua must all be degenerate because of the consistency of quantum field theory -- tunneling from a Minkowski vacuum to a lower one must be exponentially fast as in any theory possessing ghosts. \nThe Zeldovich-Dvali argument contradicts Coleman's semi-classical computation in the Euclidian space that the formation of vacuum bubbles is exponentially slow~\\cite{Coleman:1977py},  and has also been challenged by studies of different physical systems~\\cite{Garriga:2011we}. However, attempts to generalize Coleman's computation to the realistic case with zero-mass vacuum bubbles, which is the origin of the divergence, have failed and the Zeldovich-Dvali argument has not been disproved~\\cite{Gross:2020tph}. Altogether, the underlying fundamental physics behind the PMPC remains presently unclear: we accept it on empirical grounds.\n\nNevertheless, the phenomenological success of PMPC has motivated different studies of scalar potentials in extended models. Already the very first papers\nafter the Higgs boson discovery argued that if the SM is extended by a singlet scalar, the Higgs vacuum stability as well as criticality can be achieved due to the radiative corrections of the singlet~\\cite{Kadastik:2011aa}. Later the PMPC has been applied to the singlet~\\cite{Haba:2014sia,Kawana:2014zxa,Haba:2016gqx,McDowall:2019knq} and to the two Higgs doublet models (2HDM)~\\cite{Froggatt:2004st,Froggatt:2006zc,Froggatt:2008am,McDowall:2018ulq,Maniatis:2020wfz}.\nThe Higgs criticality may play an important role in understanding Higgs inflation~\\cite{Bezrukov:2007ep}. We do not make an attempt to form a comprehensive list of all possible applications of PMPC and limit ourselves to the above examples.\n\nThe aim of this work is to point out that so far all the applications of PMPC to scalar potentials, including the examples given above, deal with radiatively generated\nmultiple minima. However, if the degeneracy of multiple vacua is a fundamental property of Nature, all vacua of multi-scalar models -- including the ones \ndominated by {\\it tree-level} potential terms -- should be degenerate. We first argue that this formulation of PMPC should be correct, and after that exemplify its physics potential, analyzing the vacuum solutions of the scalar singlet dark matter (DM) model~\\cite{Silveira:1985rk,McDonald:1993ex,Burgess:2000yq,Cline:2013gha} (see~\\cite{Arcadi:2019lka} for a review).\nWe present three simple scalar DM models utilizing the PMPC: I) $\\mathbb{Z}_2$-symmetric real scalar DM model, II) $\\mathbb{Z}_2$-symmetric pseudo-Goldstone DM (pGDM) model and III) $\\mathbb{Z}_3$-symmetric pseudo-Goldstone DM model.\\footnote{In a more complicated pGDM model, all these global symmetries can be broken, save for the CP-like $S \\to S^{*}$ that stabilizes DM \\cite{Alanne:2020jwx}.} Our results will show that the allowed parameter spaces of those models are constrained and the PMPC \nwill, in principle, be testable if the DM of Universe is the singlet scalar. Generalizing this result, our work demonstrates that the PMPC might be a useful tool to discriminate\nbetween different models of new physics beyond the SM.\n\nThis work is organized as follows. In Section~\\ref{sec:PMPC} we generalize the concept of PMPC to the case of multiple scalar fields. In Sections~\\ref{sec:model1},~\\ref{sec:model2} and~\\ref{sec:model3} we analyze our example models. We conclude and discuss perspectives in Section~\\ref{sec:conc}.\n\n\n\n\\section{Tree-level PMPC}\n\\label{sec:PMPC}\n\nSo far the PMPC has been used to constrain model parameters by requiring that radiatively generated minimum of the\nscalar potential is degenerate with the electroweak vacuum. This was the case with the SM as well as with the scalar singlet and doublet\nextensions of the SM. Clearly, in multi-scalar models there can be several minima of the \nscalar potential already at tree level. If the PMPC follows from some underlying fundamental physics principle, all those vacua must be degenerate as well.\nThis is the concept we adopt and study in this work. Although we do not know which physics principle is behind the PMPC, we exemplify with the following discussion\nthat such principles may exist.\n\nAs Zeldovich pointed out in early 70ies~\\cite{Zeldovich:1974py,Kobzarev:1974cp}, if there exist two minima of the scalar potential with $V=0$ and $V<0$,\nthe vacuum decay bubble with mass $m = 0$ can appear with any initial velocity, giving rise to a divergent boost integral. It should be noticed that \nthere is no such a problem with the pair creation of ordinary particles. Instead, this problem is intrinsic to the theories with ghosts, {\\it i.e.}, to particles with negative energy. \nWhile the theories with ghosts are widely believed to be pathological precisely because of the exponentially fast vacuum decay, similar phenomenon\nshould appear in the Lorentz-invariant decay of the Minkowski vacuum, rendering this configuration unphysical. Dvali, who apparently was not aware of the \nZeldovich paper, independently make this claim in Ref.~\\cite{Dvali:2011wk}. More recently this setup was studied in theories of ghosts~\\cite{Gross:2020tph}, finding the same divergence.\n\n\nNeedless to say, this claim contradicts Coleman's result that the vacuum decay rate is finite and exponentially suppressed~\\cite{Coleman:1977py}. \nAs Coleman computed one bubble decay rate in Euclidean space, the authors of Ref.~\\cite{Gross:2020tph} wanted to generalize Coleman's computation to\nthe case of ghosts, but did not succeed. The Zeldovich-Dvali claim was also criticized in Ref.~\\cite{Garriga:2011we} which,\nhowever, studies a completely different physical situation -- particle pair production in uniform electric field. Whether this set-up is analogous to the case of \nvacuum decay is far from being obvious. It seems that the main source of different opinions is whether the divergent boost integral has to be taken or not,\nand there is no definitive answer at the moment. \n\nNevertheless, this discussion demonstrates that there exist attempts to derive PMPC from general physics principles.\nThe PMPC in this, broader formulation has been used to criticize certain supersymmetry breaking scenarios~\\cite{Bajc:2011iu}.\nIn this work we perform phenomenological analyses of implications of tree-level PMPC on multi-scalar DM models.\n\n\n\\section{Model I: $\\mathbb{Z}_2$ real singlet}\n\\label{sec:model1}\n\nThe real scalar singlet DM model extends the SM with one real scalar singlet which is stabilized by a $\\mathbb{Z}_2$ symmetry. \nThis is among the most popular DM models whose phenomenology has been thoroughly studied~\\cite{Davoudiasl:2004be,Ham:2004cf,OConnell:2006rsp,Patt:2006fw,Profumo:2007wc,Barger:2007im,He:2007tt,He:2008qm,Yaguna:2008hd,Lerner:2009xg,Farina:2009ez,Goudelis:2009zz,Profumo:2010kp,Guo:2010hq,Barger:2010mc,Arina:2010rb,Bandyopadhyay:2010cc,Drozd:2011aa,Djouadi:2011aa,Low:2011kp,Mambrini:2011ik,Espinosa:2011ax,Mambrini:2012ue,Djouadi:2012zc,Cheung:2012xb,Cline:2013gha,Urbano:2014hda,Endo:2014cca,Feng:2014vea,Duerr:2015bea,Duerr:2015mva,Beniwal:2015sdl,Cuoco:2016jqt,Escudero:2016gzx,Han:2016gyy,He:2016mls,Ko:2016xwd,Athron:2017kgt,Ghorbani:2018yfr}. Nevertheless, we shall show that applying the PMPC to this model scalar potential will open new perspectives in\nunderstanding the allowed parameter space of the model.\n\nThe scalar potential of the model is given by\n\\begin{equation}\nV=\\mu_h^2 \\lvert H \\rvert^{2} + \\frac{\\mu_s^2}{2}s^2+\\lambda_h \\lvert H \\rvert^{4}+\\frac{\\lambda_s}{4}s^4+\\frac{\\lambda_{hs}}{2} \\lvert H \\rvert^{2} s^2,\n\\end{equation}\nwhere $H$ is the SM Higgs doublet in unitary gauge:\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\n0\\\\\nh+v_h\n\\end{array}\n\\right).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV.\nThe scalar $s$, as the  dark matter candidate, must not acquire a non-zero VEV: then the scalar $s$ does not mix with the Higgs and remains stable due to the $\\mathbb{Z}_2$ symmetry. The freeze-out of the $s$ is independent of the self-coupling of the scalar singlet $\\lambda_s$ and its relic abundance is therefore set only by the Higgs portal coupling $\\lambda_{hs}$. \n\nThe potential may have three types of minima in the field space $(h,s)$: the Higgs vacuum $(v_h,0)$, singlet vacuum $(0,v_s)$ and a general or mixed vacuum $(v'_h,v'_s)$. Presently, the Universe is in the Higgs vacuum. According to the PMPC these minima are to be degenerate, if they exist simultaneously. Let us examine the three vacuum configurations more closely. \n\n\\subsection{The Higgs vacuum configuration $(v_h,0)$}\nThe Higgs vacuum $(v_h,0)$, in which we live, satisfies the following minimization condition:\n\\begin{equation}\n\\mu_h^2=-\\lambda_h v_h^2.\\label{Our vacuum minimization condition}\n\\end{equation} \nThe scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n2\\lambda_h v_h^2 & 0\\\\\n0 & \\frac{1}{2}\\lambda_{hs} v_h^2+\\mu_s^2\n\\end{array}\n\\right).\n\\end{equation}\n\n\n\\subsection{The singlet vacuum configuration $(0,v_s)$}\nThe minimization condition for the singlet vacuum $(0,v_s)$ is given by\n\\begin{equation}\n\\mu_s^2=-\\lambda_s v_s^2.\\label{singlet vacuum minimization condition}\n\\end{equation}\n\nIn this case, the scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n\\frac{1}{2}\\lambda_{hs} v_s^2+\\mu_h^2  & 0\\\\\n0 & 2\\lambda_s v_s^2\n\\end{array}\n\\right).\n\\end{equation}\n\n\n\n\\subsection{The mixed vacuum configuration $(v'_h,v'_s)$}\nThe minimization conditions for the mixed vacuum $(v'_h,v'_s)$  are given by\n\\begin{eqnarray}\n\\mu_h^2 & = & -\\lambda_h {v'_h}^2-\\frac{1}{2}\\lambda_{hs}{v'_s}^2\\\\\n\\mu_s^2 & = & -\\lambda_{s} {v'_s}^2-\\frac{1}{2}\\lambda_{hs} {v'_h}^2\n\\end{eqnarray}\nThe scalar mass matrix is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{cc}\n2\\lambda_{h} {v'_h}^2  & \\lambda_{hs} {v'_h}v'_s\\\\\n\\lambda_{hs} {v'_h}v'_s & 2\\lambda_{s} {v'_s}^2\n\\end{array}\n\\right),\n\\end{equation}\nleading to mixing of the Higgs doublet with the real singlet.\n\n\\subsection{Comparison of minima}\nThe current minimum configuration is the Higgs vacuum $(v_h,0)$, which allows dark matter and correctly breaks the electroweak symmetry. If the other vacuum configurations are to exist simultaneously with the $(v_h,0)$, they must be degenerate. Imposing that the $(0,v_s)$ vacuum configuration is degenerate to $(v_h,0)$,\n\\begin{equation}\nV(h=v_h,s=0)= V(h=0,s=v_s),\n\\end{equation}\nleads to the condition\n\\begin{equation}\\label{the constraint for our vacuum VS singlet vacuum}\nm_s^4-\\lambda_{hs} m_s^2 v_h^2 -\\lambda_s\\lambda_h v_h^4+\\frac{1}{4}\\lambda_{hs}^2 v_h^4=0.\n\\end{equation}\n\nDemanding that the $(v'_h,v'_s)$ vacuum configuration be degenerate to the present Higgs vacuum $(v_h,0)$ leads to a contradiction.  Therefore, according to the PMPC, there can at most only be two degenerate vacua: the Higgs vacuum $(v_h,0)$ and the singlet vacuum $(0,v_s)$. \n\nThe collider experiments impose constraints on the singlet $s$  when it is lighter than $m_h\/2$. Then the Higgs decay width to the $ss$ final state is \\cite{Cline:2013gha}\n\\begin{equation}\n\\Gamma (h\\to ss) =\\frac{\\sqrt{m_h^2-4m_s^2}}{32\\pi m_h^2}\\lambda_{hs}^2 v_h^2. \n\\end{equation}\nIn addition, the Higgs invisible branching ratio is then given by\n\\begin{equation}\n\\text{BR}_\\text{inv} = \\frac{\\Gamma_{h \\to ss}}{\\Gamma_{h \\to \\text{SM}} + \\Gamma_{h \\to ss}},\n\\end{equation}\nwhich is constrained to be below $0.24$ at $95\\%$ confidence level \\cite{Khachatryan:2016whc,ATLAS-CONF-2018-031} by direct measurements and below about $0.17$ by statistical fits of Higgs couplings \\cite{Giardino:2013bma,Belanger:2013xza}, which excludes singlet masses below $53$ GeV.\n\nThe portal coupling $\\lambda_{hs}$ is determined by demanding that the correct relic abundance $\\Omega_{c} h^{2} = 0.120$ \\cite{Aghanim:2018eyx} is produced in the freeze-out of the singlet. The PMPC in Eq.~(\\ref{the constraint for our vacuum VS singlet vacuum}) allows us to determine the singlet self-coupling $\\lambda_s$ as a function of the dark matter mass, as is presented in the Fig. \\ref{dark matter self-coupling as a function of dark matter mass}. The self-coupling $\\lambda_s$ becomes non-perturbative as the dark matter mass reaches electroweak scale. \n\nDirect detection also limits \nthe dark matter mass so that the only allowed mass region is near the Higgs resonance. The PMPC thus predicts that the dark matter mass must be in the resonance region. \n\n\\begin{figure*}[tb]\n\t\\begin{center}\n\t\\includegraphics[width=0.45\\linewidth]{RealSingletLowMassPlot.pdf}\n\t\\includegraphics[width=0.45\\linewidth]{RealSingletHighMassPlot.pdf}\n\t\\end{center}\n\t\\caption{\\label{dark matter self-coupling as a function of dark matter mass} The value of $\\mathbb{Z}_2$ real singlet DM self-coupling $\\lambda_s$ as a function of DM mass according to PMPC in Eq. \\eqref{the constraint for our vacuum VS singlet vacuum}. The portal coupling is fixed so that the correct relic abundance is obtained. Left panel: the self-coupling $\\lambda_s$ near the Higgs resonance, $m_s\\in [53,70]$ GeV. Right panel: the  self-coupling $\\lambda_s$ away from the resonance, $m_s\\in [65,1000]$ GeV.  }\n\\end{figure*}\n\n\n\n\n\n\\section{Model II: $\\mathbb{Z}_2$ pseudo-Goldstone DM}  \n\\label{sec:model2}\n\nThis  model is an extension of SM with a  complex scalar $S$ carrying a global $U(1)$ charge.   The most general scalar potential invariant under global $U(1)$ transformation is given by\n\\begin{equation}\\label{}\n\\begin{split}\n  V_0 &= \\mu_h^2 \\lvert H \\rvert^{2} + \\mu_s^2 \\lvert S \\rvert^{2} +\\lambda_h \\lvert H \\rvert^{4}\n  +\\lambda_s \\lvert S \\rvert^{4} \\\\\n  &+\\lambda_{hs} \\lvert H \\rvert^{2} \\lvert S \\rvert^{2},\n\\end{split}\n\\end{equation}\nwhere $H$ is the SM Higgs doublet. \nThe global $U(1)$ symmetry is spontaneously broken as the scalar $S$ acquires a non-zero vacuum expectation value. This would result in a massless Goldstone boson in the physical spectrum. In order to avoid this, the global $U(1)$ symmetry is explicitly broken into its discrete subgroup $\\mathbb{Z}_2$ by the following soft term:\n\\begin{equation}\nV_{\\mathbb{Z}_2}=-\\frac{\\mu^{\\prime 2}}{4}S^2+\\text{h.c},\n\\end{equation}\nso the full potential becomes\n\\begin{equation}\\label{Z2 scalar potential}\nV=V_0+V_{\\mathbb{Z}_2}.\n\\end{equation}\n\nThe model was first considered in \\cite{Barger:2008jx,Chiang:2017nmu}, but only in the Higgs resonance region. The suppression of the direct detection cross section of the pseudo-Goldstone DM was noted in \\cite{Gross:2017dan}; there has been large interest in the phenomenology of the model \\cite{Huitu:2018gbc,Azevedo:2018oxv,Azevedo:2018exj,Alanne:2018zjm,Cline:2019okt,Kannike:2019wsn}.\n\nThe parameter $\\mu^{\\prime 2}$ can be made real and positive by absorbing its phase into the field $S$. When $\\mu^{\\prime 2}$ is positive, then the VEV of $S$ is real. The reality of the VEV can be easily seen by writing the singlet in polar coordinates:\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}r e^{i\\theta}.\n\\end{equation}\nThe soft term now becomes\n\\begin{equation}\nV_{\\mathbb{Z}_2}=-\\frac{\\mu^{\\prime 2}}{4}r^2\\cos (2\\theta).\n\\end{equation}\nThe soft term is the only part of the potential that depends on the phase of the singlet $S$. Therefore the global minimum of the potential is the minimum of the soft term.  Because  $\\mu^{\\prime 2}>0$, the minimum is obtained when $\\cos(2\\theta)=1$. It follows that\n\\begin{equation}\n2\\theta =\\pm 2\\pi n\\Rightarrow \\theta=0,\\pi,\n\\end{equation}\nand therefore the VEV is real.\n\n\nWe parametrize the fields in the unitary gauge as:\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\left(\\begin{array}{c}\n0\\\\\nh+v_h\n\\end{array}\n\\right),\n\\end{equation}\nand\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}(v_s+s+i\\chi).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV and $v_s$ is real.\n\nThe scalar potential in Eq. (\\ref{Z2 scalar potential}) is invariant under two different   $\\mathbb{Z}_2$ symmetries,\n\\begin{equation}\nS\\to -S,\n\\end{equation}\nand the CP-like\n\\begin{equation} \nS\\to S^\\ast.\n\\end{equation}  \nAfter the symmetry breaking the latter symmetry becomes equivalent to $\\chi\\to -\\chi$, which stabilizes $\\chi$, making it a dark matter candidate. \n\nWe will now move on to study the vacuum structure of the potential in Eq. (\\ref{Z2 scalar potential}).\n\n \n\\subsection{Different vacua}\nThe potential in \\eqref{Z2 scalar potential} is bounded from below if\n\\begin{equation} \n\\lambda_h > 0,\\quad \\lambda_s > 0, \\quad 2\\sqrt{\\lambda_h \\lambda_s}+\\lambda_{hs} > 0, \n\\end{equation}\n There are three possible vacuum configurations in field space $(h,s,\\chi)$: the mixed vacuum $(v_h, v_s,0)$, the Higgs vacuum $(v'_h, 0, 0)$ and the singlet vacuum $(0, v'_s, 0)$. Let us study if there can be degenerate minima.\n\n\n\\subsubsection{Our vacuum $(v_h,v_s,0)$}\n\nIn order for the pseudo-Goldstone to be the DM candidate, our Universe has to reside in the mixed vacuum. This vacuum $(v_h,v_s,0)$ satisfies the following minimization conditions:\n\\begin{align}\n\\mu_h^2 \n& = -\\lambda_h v_h^2-\\frac{1}{2}\\lambda_{hs} v_s^2\\label{Our vacuum minimization condition h},\\\\\n\\mu_s^2 & = -\\lambda_s v_s^2-\\frac{1}{2}\\lambda_{hs} v_h^2+\\frac{1}{2}\\mu^{\\prime 2},\n\\label{Our vacuum minimization condition s}\n\\end{align}\nwhich can be inverted in favor of the VEVs:\n\\begin{align}\nv_h^2 & =  \\frac{-4\\lambda_s \\mu_h^2 + 2\\lambda_{hs}\\mu_s^2 -\\lambda_{hs}\\mu^{\\prime 2}}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2}>0\\label{solved VEVs 1},\\\\\nv_s^2 & = \\frac{-4\\lambda_h \\mu_s^2 + 2\\lambda_{hs}\\mu_h^2 -2\\lambda_{h}\\mu^{\\prime 2}}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2}>0.\\label{solved VEVs 2}\n\\end{align}\n\nThe mass matrix in the $(h,s,\\chi)$-basis is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s & 0\\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2 & 0\\\\\n0 & 0 & \\mu^{\\prime 2}\n\\end{pmatrix}.\n\\end{equation}\n\nThe determinant of the $2\\times 2$ CP-even block matrix has to be strictly positive in order for the mixed configuration to be a physical minimum, yielding the condition\n\\begin{equation}\\label{positive CP-even masses}\n4\\lambda_h\\lambda_s-\\lambda_{hs}^2 >0.\n\\end{equation}\nWe have chosen    the parameter $\\mu^{\\prime 2}$  to be positive and therefore all the masses will be positive. The mixed vacuum configuration $(v_h, v_s,0)$ is therefore the minimum to which the other configurations are compared.\n\nWhen one combines Eq. \\eqref{positive CP-even masses} with Eqs. \\eqref{solved VEVs 1} and \\eqref{solved VEVs 2}, the following inequalities are obtained:\n\\begin{align}\n-4\\lambda_s\\mu_h^2+2\\lambda_{hs}\\mu_s^2-\\lambda_{hs}\\mu^{\\prime 2} &>0,\\label{VEV positivity condition 1}\\\\\n-4\\lambda_h\\mu_s^2+2\\lambda_{hs}\\mu_h^2+2\\lambda_h\\mu^{\\prime 2} &>0.\\label{VEV positivity condition 2}\n\\end{align}\n\n\n\\subsubsection{The Higgs vacuum $(v'_h,0,0)$}\n\nWhile the Higgs vacuum $(v'_h,0,0)$ gives rise to correct electroweak symmetry breaking, in this case it is not the pseudo-Goldstone boson but the complex singlet $S$ that is the DM candidate, contrary to the present assumptions. The mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n-2\\mu_h^2 & 0 & 0\\\\\n0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2-\\frac{\\mu^{\\prime 2}}{2} & 0\\\\\n0 & 0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2+\\frac{\\mu^{\\prime 2}}{2}\n\\end{pmatrix}.\n\\end{equation}\nBy using Eq. (\\ref{VEV positivity condition 2}), one sees that the mass in the middle is negative. Therefore this vacuum configuration is not a minimum.\n\n\\subsubsection{The singlet vacuum $(0,v'_s,0)$}\n\nThe singlet vacuum $(0,v'_s,0)$ can in principle be a minimum. Let us compare, however the potential energy of this minimum to that of our mixed minimum $(v_h,v_s,0)$. Their difference is\n\\begin{equation}\n\\begin{split}\n  V(v_h,v_s,0)&-V(0,v'_s,0)  =-\\frac{1}{4}\\frac{1}{4\\lambda_h\\lambda_s-\\lambda_{hs}^2} \\frac{1}{\\lambda_s}\n  \\\\\n  &\\times \\left[\\frac{\\lambda_{hs}}{2}(2\\mu_s^2-\\mu^{\\prime 2})-2\\lambda_s\\mu_s^2\\right]^2<0,\n\\end{split}\n\\end{equation}\naccording to Eq.~(\\ref{positive CP-even masses}). Therefore, we have that\n\\begin{equation}\nV(v_h,v_s,0)<V(0,v'_s,0).\n\\end{equation}\nThis inequality is strict, and therefore the singlet minimum $(0,v'_s,0)$ cannot be degenerate with our  vacuum.\n\n\n\n\\subsection{Verdict on $\\mathbb{Z}_2$ pGDM from the PMPC}\nWe see that if we demand  that the mixed vacuum $(v_h,v_s,0)$ be a minimum of the scalar potential, the other vacuum configurations cannot be true minima or if they are, they cannot be degenerate with our minimum. This is in a agreement with \\cite{Azevedo:2018oxv}. Therefore, the singlet vacuum $(0,v'_s,0)$ cannot be a  minimum according to the PMPC. The PMPC allows for the $\\mathbb{Z}_2$ pseudo-Goldstone dark matter model only if $(0, v'_s, 0)$ is not a minimum.\n \n\n\n\\section{Model III:  $\\mathbb{Z}_3$ pseudo-Goldstone DM}\n\\label{sec:model3}\n\nThe $\\mathbb{Z}_3$ pseudo-Goldstone DM model is an extension of SM with a complex scalar $S$ carrying a global $U(1)$ charge. The most general  scalar potential invariant under a global $U(1)$ symmetry is\n\\begin{equation}\\label{lagrangian}\n\\begin{split}\nV_0 &= \\mu_h^2 H^\\dagger H+ \\mu_s^2 S^\\ast S+\\lambda_h (H^\\dagger H)^2+\\lambda_s(S^\\ast S)^2\n\\\\ \n&+\\lambda_{hs} (H^\\dagger H)(S^\\ast S),\n\\end{split}\n\\end{equation}\nwhere $H$ is the SM Higgs doublet. \nThe global $U(1)$ symmetry is spontaneously broken as the scalar $S$ acquires a non-zero vacuum expectation value. This would result in a massless Goldstone boson in the physical spectrum. In order to avoid this, the global $U(1)$ symmetry is explicitly broken into its discrete subgroup $\\mathbb{Z}_3$ by the following soft term:\n\\begin{equation}\nV_{\\mathbb{Z}_3}=\\frac{{\\mu'_s}}{2}S^3+\\text{h.c.},\n\\end{equation}\nso the full potential becomes\n\\begin{equation}\\label{Z3 scalar potential}\nV=V_0+V_{\\mathbb{Z}_3}.\n\\end{equation}\nThe parameter ${\\mu'_s}$ can be made real by absorbing its phase to the field $S$. When ${\\mu'_s}$ is negative the VEV of $S$ is real and positive.  This can  be easily seen by writing the singlet in polar coordinates:\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}r e^{i\\theta}.\n\\end{equation}\nThe soft term now becomes\n\\begin{equation}\nV_{\\mathbb{Z}_3}=\\frac{{\\mu'_s}}{2}\\frac{1}{2^{3\/2}}r^3\\cos (3\\theta).\n\\end{equation}\nThe soft term is the only part of the potential that depends on the phase. Therefore the global minimum of the potential is the minimum of the soft term. Let us solve the phase at the minimum. Since  ${\\mu'_s}<0$, the minimum is obtained when $\\cos(3\\theta)=1$, which implies that\n\\begin{equation}\n3\\theta =0+2\\pi n\\Rightarrow \\theta=0,\\frac{2\\pi}{3},-\\frac{2\\pi}{3}.\n\\end{equation}\nTherefore, the potential has a three-fold degenerate minimum. One of them is real and two are complex. The different vacua are related to each other by $\\mathbb{Z}_3$ transformations $S\\rightarrow e^{2i\\pi\/3} S$. The singlet VEV can always be chosen to be real. Note that with the choice, ${\\mu'_s}>0$, the VEV of $S$ is negative.\n\n\nWe now parametrize the scalar fields in unitary gauge as\n\\begin{equation}\nH=\\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n0\\\\\nh+v_h\n\\end{pmatrix},\n\\end{equation}\nand\n\\begin{equation}\nS=\\frac{1}{\\sqrt{2}}(v_s+s+i\\chi).\n\\end{equation}\nThe Higgs vacuum expectation value is $v_h=246$ GeV.\nThe pseudo-scalar $\\chi$ can be dark matter \\cite{Kannike:2019mzk}. The $\\mathbb{Z}_3$-symmetric complex singlet model has also been studied in the phase where  the $\\mathbb{Z}_3$ remains unbroken \\cite{Belanger:2012zr,Hektor:2019ote}.  \n\nThe scalar potential in Eq. (\\ref{Z3 scalar potential}) is invariant under the CP-like $\\mathbb{Z}_2$-symmetry\n\\begin{equation}\nS\\to S^\\ast.\n\\end{equation}\nThis symmetry is becomes \n\\begin{equation} \n\\chi\\to -\\chi,\n\\end{equation}\nwhen $S$ acquires VEV. This symmetry stabilizes $\\chi$, making it a dark matter candidate. \n\nWe next  study the vacuum structure of the model.\n \n\\subsection{Different vacua}\n\nThe potential in Eq. (\\ref{Z3 scalar potential}) is bounded from below, if\n\\begin{equation} \n\\lambda_h > 0,\\quad \\lambda_s > 0, \\quad 2\\sqrt{\\lambda_h \\lambda_s}+\\lambda_{hs} > 0. \n\\end{equation}\n  \nThere are three possible vacuum configurations in the field space $(h,s,\\chi)$: the mixed vacuum $(v_h,v_s,0)$, the Higgs vacuum $(v'_h,0,0)$ and the singlet vacuum $(0,v'_s,0)$.\n\n\n\n\n\\subsection{Our vacuum $(v_h,v_s,0)$}\n\nIn order for the pseudo-Goldstone to be the DM candidate, our Universe has to reside in the mixed vacuum. \nThe vacuum $(v_h,v_s,0)$ satisfies the following minimization conditions:\n\\begin{eqnarray}\n\\mu_h^2 \n& = & -\\lambda_h v_h^2-\\frac{1}{2}\\lambda_{hs} v_s^2,\n\\label{Our vacuum minimization condition h}\\\\\n\\mu_s^2 & = &-\\lambda_s v_s^2-\\frac{1}{2}\\lambda_{hs} v_h^2-\\frac{3}{2\\sqrt{2}}v_s{\\mu'_s}.\\label{Our vacuum minimization condition s}\n\\end{eqnarray} \nThe minimization conditions are now hard to invert in favor of VEVs as minimization condition of $\\mu_s^2$ contains a linear term in $v_s$, due to the soft cubic self-coupling of $S$.  The VEVs are solved to be:\n\\begin{align}\nv_s & =  \\frac{-\\frac{3\\mu'_s}{\\sqrt{2}}\\pm 2\\sqrt{\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)\\left(\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}-\\mu_s^2\\right)+\\frac{9\\mu_{s}^{\\prime 2}}{8}}}{\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)},\n\\label{vs our vacuum minimum}\\\\\nv_h & =  \\pm\\frac{\\sqrt{-\\lambda_{hs} v_s^2-2\\mu_h^2}}{\\sqrt{2}\\sqrt{\\lambda_h}}.\n\\end{align}\nWe choose the positive sign $v_s$-solution to be the VEV in our minimum. The negative sign $v_s$-solution is not a minimum as one of the scalar masses is necessarily negative in this case.\n\nThe mass matrix in the $(h,s,\\chi)$-basis is given by\n\\begin{equation}\\label{Z3 our minimum masses}\n\\mathcal{M}^2=\\left(\\begin{array}{ccc}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s & 0\\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2+\\frac{3}{2\\sqrt{2}}v_{s} \\mu'_s & 0\\\\\n0 & 0 & -\\frac{9v_s \\mu'_s}{2\\sqrt{2}}\n\\end{array}\n\\right).\n\\end{equation}\n\nThe $2 \\times 2$ block in the upper-left corner of Eq. \\eqref{Z3 our minimum masses} can be diagonalized by an orthogonal transformation\n\\begin{equation}\nU^T M^2 U = \\textrm{diag}(m_h^2,m_H^2),\n\\end{equation}\nwhere\n\\begin{equation}\nM^2=\\left(\\begin{array}{cc}\n2\\lambda_h v_h^2 & \\lambda_{hs} v_h v_s \\\\\n\\lambda_{hs} v_h v_s & 2\\lambda_s v_s^2+\\frac{3}{2\\sqrt{2}}vs \\mu'_s \n\\end{array}\n\\right),\n\\end{equation}\nand\n\\begin{equation}\nU=\\left(\\begin{array}{cc}\n\\cos\\theta & \\sin\\theta \\\\\n-\\sin\\theta & \\cos\\theta \n\\end{array}\n\\right).\n\\end{equation}\nThe mixing angle $\\theta$ is defined as\n\\begin{equation}\n\\tan 2\\theta =-\\frac{\\lambda_{hs}v_h v_s}{\\lambda_h v_h^2-\\lambda_s v_s^2-\\frac{3}{4\\sqrt{2}}v_s \\mu'_s}.\n\\end{equation}\nWith the knowledge that we must choose the positive sign solution of $v_s$, the parameters in the potential can be interchanged to more suitable ones. The potential parameters $\\mu_h^2$, $\\mu_s^2$, $\\lambda_h$, $\\lambda_s$, $\\lambda_{hs}$ and $\\mu'_s$ are replaced by $v_h$, $v_s$, $m_h$, $m_H$, $m_\\chi$ and the mixing angle $\\theta$:\n\\begin{align}\n\\mu_h^2 & = -\\frac{1}{4}(m_h^2+m_H^2)+\\frac{1}{4v_h}(m_H^2-m_h^2)\n \\notag\n \\\\\n &\\times (v_h\\cos 2\\theta -v_s\\sin 2\\theta),\\\\\n\\mu_s^2 & = -\\frac{1}{4}(m_h^2+m_H^2)+\\frac{1}{4v_s}(m_h^2-m_H^2)\n\\notag\n\\\\\n&\\times(v_s\\cos 2\\theta +v_h\\sin 2\\theta)+\\frac{1}{6}m_\\chi^2,\\\\\n\\lambda_h & = \\frac{m_h^2+m_H^2+(m_h^2-m_H^2)\\cos 2\\theta}{4 v_h^2},\\\\\n\\lambda_s & = \\frac{3(m_h^2+m_H^2)+3(m_H^2-m_h^2)\\cos 2\\theta+2m_\\chi^2}{12 v_s^2},\\\\\n\\lambda_{hs} & = -\\frac{(m_h^2-m_H^2)\\sin 2\\theta}{2 v_h v_s},\\\\\n\\mu'_s & = -\\frac{2\\sqrt{2}}{9}\\frac{m_\\chi^2}{v_s}.\n\\end{align}\n\n\\subsection{The Higgs vacuum $(v'_h,0,0)$}\n\nWhile the Higgs vacuum $(v'_h,0,0)$ gives rise to correct electroweak symmetry breaking, in this case it is not the pseudo-Goldstone boson but the complex singlet $S$ that is the DM candidate, contrary to the present assumptions.\nThe mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\begin{pmatrix}\n-2\\mu_h^2 & 0 & 0\\\\\n0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2 & 0\\\\\n0 & 0 & -\\frac{\\lambda_{hs}\\mu_h^2}{2\\lambda_h}+\\mu_s^2\n\\end{pmatrix}.\n\\end{equation}\nAll the masses can be positive simultaneously. This minimum can be degenerate with our mixed vacuum, leading into the condition\n\\begin{equation}\n\\mu^{\\prime 2}=\\frac{1}{\\lambda_h}\\left(4\\lambda_h\\lambda_s-\\lambda_{hs}^2\\right)\\left(-\\frac{\\lambda_{hs}}{2\\lambda_h}\\mu_h^2+\\mu^2_s\\right),\n\\end{equation}\nor equivalently,\n\\begin{equation}\\label{higgs only vacuum degeneracy}\nm_\\chi^2 = \\frac{9m_h^2 m_H^2}{m_h^2\\cos^2\\theta+m_H^2\\sin^2\\theta}.\n\\end{equation}\n\n\n\\subsection{The singlet vacuum $(0,v'_s,0)$}\nIn the singlet vacuum $(0,v'_s,0)$, only the complex singlet $S$ gets a VEV and electroweak symmetry is not broken.\nThe minimization condition is given by\n\\begin{equation}\n\\mu_s^2=-\\lambda_s {v'_s}^2-\\frac{3}{2\\sqrt{2}}v'_s \\mu'_s.\n\\end{equation}\nThe VEV $v'_s$ can be found to be\n\\begin{equation}\nv'_s=\\frac{-3\\mu'_s\\pm \\sqrt{-32\\lambda_s\\mu_s^2+9\\mu^{\\prime 2}}}{4\\sqrt{2}\\lambda_s}.\n\\end{equation}\n\nThe mass matrix in this vacuum is given by\n\\begin{equation}\n\\mathcal{M}^2=\\left(\\begin{array}{ccc}\n\\mu_h^2-\\frac{\\lambda_{hs}}{2 \\lambda_s}\\left[\\mu_s^2+\\frac{3\\mu'_s v'_s}{2\\sqrt{2}}\\right] & 0 & 0\\\\\n0 & -2\\mu_s^2-\\frac{3\\mu'_s v'_s}{2\\sqrt{2}} & 0\\\\\n0 & 0 & -\\frac{9\\mu'_s v'_s}{2\\sqrt{2}}\n\\end{array}\n\\right).\n\\end{equation}\n\nThis vacuum configuration does not seem to be able to be a minimum when the mixed vacuum $(v_h,v_s,0)$ and the Higgs vacuum $(v'_h,0,0)$ are degenerate. It is hard to show analytically, but all numerical evidence points to that.\n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{direct_detection}\n\\end{center}\n\\caption{Direct detection cross section for $\\mathbb{Z}_{3}$ pGDM. The XENON1T result \\cite{Aprile:2018dbl} is shown in solid line and prediction \\cite{Ni2017} for the XENONnT in dashed line. The red points are excluded by the Higgs invisible width or a too large Higgs total width.}\n\\label{fig:Z3:direct:detection}\n\\end{figure}\n\n\n\n\\subsection{Final verdict on $\\mathbb{Z}_3$ pGDM from the PMPC}\nThe two vacuum configurations $(v_h,v_s,0)$ and $(v'_h,0,0)$ can be minima and degenerate at the same time.  This degeneracy of these two minima will impose the condition in Eq. (\\ref{higgs only vacuum degeneracy}), which aproximately reads:\n\\begin{equation}\nm_\\chi\\approx 3 m_H,\n\\end{equation}\nso strictly speaking, $\\chi$ is on the borderline of not being a true pseudo-Goldstone boson.\nThe third vacuum configuration $(0,v'_s,0)$ does not seem to be a minimum when the $(v_h,v_s,0)$ and $(v'_h,0,0)$ are degenerate. This is also checked only numerically. \n\n\nWe also impose constraints from collider experiments. The fit of the Higgs signal strengths constrains the value of the mixing angle to $\\cos^2\\theta\\ge 0.9$~\\cite{Beacham:2019nyx}. If the mass of $x \\equiv \\chi \\text{ or }h_2$ is less than $m_h\/2$, then the Higgs decay width into this particle is given by \\cite{Kannike:2019mzk}\n\\begin{equation}\n  \\Gamma_{h \\to x x} = \\frac{g_{hxx}}{8 \\pi} \\sqrt{1 - 4 \\frac{m_x^2}{m_h^2}}\n\\end{equation}\nwith\n\\begin{align}\n  g_{h1 \\chi \\chi} &= \\frac{m_h^2 + m_\\chi^2}{v_S},\n  \\\\\n  g_{h1 h_2 h_2} &= \\frac{1}{v v_S} \\left[ \\left(\\frac{1}{2} m_h^2 + m_2^2\\right) (v \\cos \\theta + v_S \\sin \\theta) \\right.\n  \\notag\n  \\\\\n  & \\left. \n  + \\frac{1}{6} v m_\\chi^2 \\cos \\theta \\right] \\sin 2 \\theta.\n\\end{align}\nThe latest measurements of the width of an SM-like Higgs boson\ngives $\\Gamma^{h_1}_\\text{tot}\\,=3.2^{+2.8}_{-2.2}$ MeV, with 95\\% CL limit on\n$\\Gamma_\\text{tot} \\leq 9.16$ MeV \\cite{Sirunyan:2019twz}.\nIn addition, the Higgs invisible branching ratio is then given by\n\\begin{equation}\n\\text{BR}_\\text{inv} = \\frac{\\Gamma_{h \\to \\chi\\chi}}{\\Gamma_{h_1 \\to \\text{SM}} + \\Gamma_{h \\to \\chi\\chi} + \\Gamma_{h \\to h_2 h_2}},\n\\end{equation}\nwhich is constrained to be below $0.24$ at $95\\%$ confidence level \\cite{Khachatryan:2016whc,ATLAS-CONF-2018-031} by direct measurements and below about $0.17$ by statistical fits of Higgs couplings \\cite{Giardino:2013bma,Belanger:2013xza}. Decays of the resulting $h_{2}$ into the SM may be visible \\cite{Huitu:2018gbc}.\n\nA scan of the parameter space with the condition \\eqref{higgs only vacuum degeneracy} imposed is shown in Fig.~\\ref{fig:Z3:direct:detection}. We take the mass of the other CP-even scalar $m_{2} \\in [10, 1500]$ and the mixing angle $\\sin \\theta \\in \\pm [0.01, 0.3]$. The VEV $v_{s}$ is fitted from the DM relic density $\\Omega_{c} h^{2} = 0.120$ \\cite{Aghanim:2018eyx}. The points in red are excluded by the Higgs invisible width and\/or by an overly large Higgs total width.\n\n\\section{Discussion and Conclusion}\n\\label{sec:conc}\n\n\nIn this work we have argued that the principle of multiple point criticality must be generalized to all vacuum states, implying that all minima of a scalar potential \nmust be degenerate. Indeed, if the PMPC follows from some fundamental underlying principle, its validity must not depend on the particulars of the model.\nFor example, in the most studied case, one minimum is taken to be the SM EW Higgs minimum and another one the Planck-scale minimum generated by the running of Higgs quartic coupling. We argue that the principle must apply to all minima in multi-scalar models, whether they are generated radiatively or at tree level.\n\nPhenomenological implications of this formulation of the PMPC are straightforward but profound -- the PMPC becomes a tool to constrain the parameter space of \nmulti-scalar models. Since the scalar singlet models represent one of the most popular class of dark matter models, we exemplified the usefulness of the PMPC by studying how different realizations of scalar DM models are constrained  by imposing the PMPC requirements. \n\nIn the simplest possible model, studied in Section~\\ref{sec:model1}, the DM is a real scalar singlet stabilized by a $\\mathbb{Z}_2$ symmetry. The PMPC predicts that the singlet self-coupling hits a Landau pole at DM mass of about $300$~GeV, invalidating the model above that scale. Due to the stringent direct detection constraints, the DM mass in this model is predicted to be near the Higgs resonance.\n\nThe second model under investigation, the $\\mathbb{Z}_2$ symmetric pseudo-Goldstone DM model, studied in Section~\\ref{sec:model2}, cannot have degenerate minima at all. Therefore it cannot be  consistent with the PMPC requirements. \n\nThe third model under investigation, the $\\mathbb{Z}_3$ symmetric pseudo-Goldstone DM model, studied in Section~\\ref{sec:model3}, can have degenerate minima. Applying the PMPC in this case  leads to a simple relation between the DM and the second scalar masses and excludes DM masses below about $190$~GeV via the measured Higgs width. This model represents an example how the PMPC can constrain the phenomenology of multi-scalar models.\n\nIn conclusion, if the PMPC is indeed realized in Nature, which we have assumed in this work, it may lead to significant constraints and simplifications of the phenomenology of multi-scalar models. Further studies of degenerate tree-level minima in those models is worth of pursuing. \n\n\n\\vspace{5mm}\n\\noindent \\textbf{Acknowledgement.} We thank Gia Dvali, Luca Marzola, \nAlessandro Strumia, Daniele Teresi and Hardi Veerm{\\\"a}e for discussions on the origin of PMPC. \nThis work was supported by the Estonian Research Council grants PRG434, PRG803, PRG803, MOBTT5 and MOBTT86, and by the EU through the European Regional Development Fund CoE program TK133 ``The Dark Side of the Universe.\" \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf 1. Introduction.}\n\nIn two previous works, [\\putref{DandK2}], [\\putref{DandK1}], we have\ninvestigated the thermal quantities for scalar and spinor fields on the\nspace--time T$\\times$S$^d$, where $d$ is odd. The conformal scalar\ninternal energy, $E$, for example, is a linear combination of `partial'\nenergies, $\\epsilon_t,\\, t=1,2,\\ldots$, (see below). These quantities possess\na known behaviour under modular transformations and are central to\nelliptic function theory. In particular, a temperature inversion symmetry\ncan be exhibited.\n\nFrom basic elliptic properties, it was shown that $E$ could be expressed\nas a polynomial in $\\epsilon_2$ and $\\epsilon_3$, while, for the specific heat, and\nall higher derivatives, one has to include $\\epsilon_1$, preferably via a\nmodular covariant derivative. The free energy involves an integration and\nthings are not so simple as obstructions arise to simple modular\nbehaviour. It is this aspect that is explored in the present work. I use\nthis thermal angle just to motivate the introduction of various\nquantities, the analysis of which allows us to make contact with various\ntopics in the theory of modular forms. Some of the procedures can be\ngiven a physical terminology, which might be suggestive.\n\n\\section{\\bf 2. Mellin transforms and the period polynomial.}\n\nIt is advantageous to begin from the basic Mellin transform, used {\\it e.g. } by\nMalurkar and Hardy, for the Eisenstein series (or partial internal\nenergy, [\\putref{DandK1}] [\\putref{DandK2}]),\n  $$\\eqalign{\n  \\epsilon_t({\\mbox{{\\ssb\\char98}}})&=-{B_{2t}\\over 4t}+\n  \\sum_{n=1}^\\infty {n^{2t-1}q^{2n}\\over1-q^{2n}}\\cr\n   &={1\\over2}\\,\\zeta_R(1-2t)+{1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Gamma(s)\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,(2\\pi{\\mbox{{\\ssb\\char98}}})^{-s}\\,,}\n  \\eql{mell1}$$\nwhere $q=e^{i\\pi\\tau}$ and $c$ lies above $2t$. I am using\n${\\mbox{{\\ssb\\char98}}}=-i\\tau= \\beta\/2\\pi=1\/\\xi$ as yet another convenient variable.\nAlthough, physically, $\\mbox{{\\ssb\\char98}}$ is real, if we wish to consider general\nmodular transformations then we must allow it to become complex.\n\nThe Mellin transform has been used systematically in finite temperature\nfield theory and elsewhere in discussions of asymptotic limits. The\nsurvey by Elizalde {\\it et al}, [\\putref{Elizalde2}], is useful and I\nfurther mention, as being somewhat relevant here, Cardy [\\putref{Cardy}],\nKutasov and Larsen, [\\putref{KandL}]. Terras, [\\putref{Terras}] pp.55,229,\ncan be consulted for aspects of the mathematical side. \\marginnote{OTHERS} In\nnumber theory see Hardy and Littlewood, [\\putref{HandL}], and Landau,\n[\\putref{Landau,Landau2}].\n\nEquation (\\puteqn{mell1}) can be looked upon as a continuous expansion in\n${\\mbox{{\\ssb\\char98}}}$. Various limits in ${\\mbox{{\\ssb\\char98}}}$ are uncovered by displacing the\nintegration contour. The pole in the integrand at $s=2t$ corresponds to\nthe Planck term, while the remaining one at $s=0$ corresponds to minus\nthe Casimir value, which is the first term in (\\puteqn{mell1}). The high\ntemperature limit (${\\mbox{{\\ssb\\char98}}}\\to0$) follows by pushing the contour to the\nleft, beyond the Casimir pole, the contribution from which cancels the\nfirst term in (\\puteqn{mell1}) leaving just the effect of the Planck pole\nand a correction that tends to zero exponentially. Conversely, on moving\nthe contour all the way to the right only the zero temperature Casimir\nterm remains.\n\nThis cosmetic asymmetry can be avoided (if desired) by also extracting\nthe Planck term, defining,\n  $$\n  \\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})=\\epsilon_t(\\mbox{{\\ssb\\char98}})-{1\\over2}\\,\\zeta_R(1-2t)\n  \\big(1+({i\\mbox{{\\ssb\\char98}}})^{-2t}\\big)\\,,\n  \\eql{rege}$$\nand writing\n  $$\\eqalign{\n\\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})={1\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,\\Gamma(s)\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,(2\\pi{\\mbox{{\\ssb\\char98}}})^{-s}\\,,}\n  \\eql{mell2}$$\nwhere now $c$, at the moment, lies anywhere {\\it between} $0$ and $2t$.\n\nThe quantity in the integrand is the Hecke $L$--series of the normalised\nEisenstein modular form,\n  $$\n  L(G_t,s)=\\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,,\n  \\eql{eisl1}$$\nsatisfying the (typical) functional relation\n  $$\n  (2\\pi)^{-s}\\Gamma(s)\\,L(G_t,s)=(-1)^t(2\\pi)^{s-2t}\\Gamma(2t-s)\\,L(G_t,2t-s)\\,.\n  \\eql{gfr}$$\n\nI now consider a multiple integration with respect to ${\\mbox{{\\ssb\\char98}}}$ of the\ninternal energy. Actually for the free energy only one integration is\nneeded but the general case is mathematically important. So integrate\n$\\epsilon_t^{\\rm sub}$, $h$ times to give, using (\\puteqn{mell2}),\n  $$\\eqalign{\n{1\\over\\Gamma(h)}&\\int_\\mbox{{\\ssb\\char120}}^\\infty d{\\mbox{{\\ssb\\char98}}}\\,({\\mbox{{\\ssb\\char98}}}-\\mbox{{\\ssb\\char120}})^{h-1}\n\\,\\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})\\cr\n & ={1\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,{\\Gamma(s-h)\\over(2\\pi)^s}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{-s+h}\\,,}\n  \\eql{mint}$$\nwith $h<c<2t$.\n\nI have imposed the boundary condition that each successive integral\nvanishes at $\\mbox{{\\ssb\\char120}}=\\infty$, {\\it i.e. } at zero temperature.\n\nMore poles have been introduced into the integrand from the $\\Gamma$\nfunction and the contour has to be moved to the right to avoid them but\nwe should stop when they pass beyond the contour at $ {\\rm Re\\,}\ns=c=2t-\\delta,\\,0<\\delta<1$. Hence the maximum $h$ can be is $2t-1$.\n\nI now set $h$ equal to this maximum, $2t-1$, simply for the reason that\nit gives the ultimately recognisable, and important, quantity,\n  $$\\eqalign{\n{\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}})&\\equiv{2(2\\pi)^{2t}\\over\\Gamma(2t-1)}\n\\int_\\mbox{{\\ssb\\char120}}^\\infty d{\\mbox{{\\ssb\\char98}}}\\,({\\mbox{{\\ssb\\char98}}}-\\mbox{{\\ssb\\char120}})^{2t-2}\\, \\epsilon_t^{\\rm\nsub}({\\mbox{{\\ssb\\char98}}})\\cr\n & ={2\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,{\\Gamma(s-2t+1)\\over(2\\pi)^{s-2t}}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\, \\mbox{{\\ssb\\char120}}^{-s+2t-1}\\,,}\n  \\eql{mint2}$$\nwith $2t-1<c<2t$. The first line is recognised as a Weyl fractional\nintegral.\n\nThe significance of this quantity for statistical mechanics on spheres is\nadmittedly obscure because the internal energy is a sum of the $\\epsilon_t$\nfor different $t$, and the integral depends on $t$. Only for the\n3--sphere, where $t=2$ and $\\overline E=\\epsilon_2$, is there a direct\nrelation but even then $\\overline \\phi_4$ is a third integral of $E$\nhaving no particular thermal meaning. Nevertheless, its further analysis\nis not without interest as it connects with some salient mathematical\nconcepts.\n\nAccording to my general programme, interest lies in the behaviour under\nmodular transformations. My treatment is equivalent to that of Malurkar,\n[\\putref{Malurkar}], {\\it cf } also Guinand [\\putref{Guinand}]. In contrast to the\nderivatives of $E$, there are obstructions to the modular invariance of\nthe integrals and the {\\it period polynomials} or, rather, period {\\it\nfunctions}, provide a measure of these obstructions, as we will see.\n\nFirstly under translations, ${\\mbox{{\\ssb\\char98}}}\\to{\\mbox{{\\ssb\\char98}}}-i$, and, because of the\nPlanck subtraction,\n  $$ \\epsilon_{t}^{\\rm\nsub}({\\mbox{{\\ssb\\char98}}}-i)-\\epsilon_{t}^{\\rm\nsub}({\\mbox{{\\ssb\\char98}}})=(-)^{t+1}{1\\over2}\\,\\zeta_R(1-2t)\n\\bigg({1\\over({\\mbox{{\\ssb\\char98}}}-i)^{2t}}- {1\\over{\\mbox{{\\ssb\\char98}}}^{2t}}\\bigg)\\,,\n  \\eql{trans1}$$\nwhich implies, through (\\puteqn{mint2}), that\n  $$\n  {\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}}-i)-{\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}})\n  =2i{\\zeta_R(2t)\\over \\mbox{{\\ssb\\char120}}(\\mbox{{\\ssb\\char120}}-i)}\\,.\n  \\eql{trans}$$\n\nUnder inversion, $\\mbox{{\\ssb\\char120}}\\to1\/\\mbox{{\\ssb\\char120}}$, the integrand involves\n  $$\n  \\epsilon_t^{\\rm sub}(1\/{\\mbox{{\\ssb\\char98}}})=(-1)^t {\\mbox{{\\ssb\\char98}}}^{2t}\\,\\epsilon^{\\rm sub}_t({\\mbox{{\\ssb\\char98}}})\n  $$\nsince the regularising subtraction, (\\puteqn{rege}), maintains the inversion\nproperty,\n  $$\n  \\epsilon_t(1\/{\\mbox{{\\ssb\\char98}}})=(-1)^t {\\mbox{{\\ssb\\char98}}}^{2t}\\,\\epsilon_t({\\mbox{{\\ssb\\char98}}})\\,,\n  $$\nenjoyed by the full Eisenstein series in (\\puteqn{mell1}),\n\nTherefore,\n\n$$\\eqalign{\n  {\\overline\\phi}_{2t}(1\/\\mbox{{\\ssb\\char120}})=&{2(2\\pi)^{2t}\\over\\Gamma(2t-1)}\\int_{1\/\\mbox{{\\ssb\\char120}}}^{\\infty}\n  d\\mbox{{\\ssb\\char98}}\\,({\\mbox{{\\ssb\\char98}}}-1\/\\mbox{{\\ssb\\char120}})^{2t-2}\\,\\epsilon_t^{\\rm sub}({\\mbox{{\\ssb\\char98}}})\\cr\n  =&-{2(2\\pi)^{2t}\\over\\Gamma(2t-1)}\\int_\\mbox{{\\ssb\\char120}}^0\n  {d{\\mbox{{\\ssb\\char98}}}'\\over{\\mbox{{\\ssb\\char98}}}'^2}\\,\\big(1\/{\\mbox{{\\ssb\\char98}}}'-1\/\\mbox{{\\ssb\\char120}}\\big)^{2t-2}\\,\n\\epsilon^{\\rm sub}_t(1\/{\\mbox{{\\ssb\\char98}}}')\\cr\n  =&(-1)^t\\,\\mbox{{\\ssb\\char120}}^{2-2t}{2(2\\pi)^{2t}\\over\\Gamma(2t-1)}\\int_0^\\mbox{{\\ssb\\char120}}\n  d{\\mbox{{\\ssb\\char98}}}\\,(\\mbox{{\\ssb\\char120}}-{\\mbox{{\\ssb\\char98}}})^{2t-2}\\,\\epsilon^{\\rm sub}_t({\\mbox{{\\ssb\\char98}}})\\,.\n  }\n  \\eql{mint3}\n  $$\n\nThe amount by which ${\\overline\\phi}_{2t}$ violates the pure modular\ninversion property is the quantity,\n  $$\n  {\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}})-(i\\mbox{{\\ssb\\char120}})^{2t-2}\\,{\\overline\\phi}_{2t}(1\/\\mbox{{\\ssb\\char120}})\n\\equiv \\overline P_t(\\mbox{{\\ssb\\char120}})\\,,\n  \\eql{pee1}$$\nor, from (\\puteqn{mint2}) and (\\puteqn{mint3})\n  $$\\eqalign{\n  \\overline P_t(\\mbox{{\\ssb\\char120}})=&{2(2\\pi)^{2t}\\over\\Gamma(2t-1)}\n  \\int_0^{\\infty}d{\\mbox{{\\ssb\\char98}}}\\,\\,(\\mbox{{\\ssb\\char120}}-{\\mbox{{\\ssb\\char98}}})^{2t-2}\\,\n\\epsilon^{\\rm sub}_t({\\mbox{{\\ssb\\char98}}})}\\,.\n  \\eql{ppol}$$\n\nThe calculation of $\\overline P_t$ from the Mellin contour form goes as\nfollows. From (\\puteqn{mint2}),\n  $$\\eqalign{\n&{\\overline\\phi}_{2t}(1\/\\mbox{{\\ssb\\char120}})\\cr&={2\\over2\\pi i}\n\\int_{c-i\\infty}^{c+i\\infty}ds\\,{\\Gamma(s-2t+1)\\over(2\\pi)^{s-2t}}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{s-2t+1}\\cr\n&={2\\over \\mbox{{\\ssb\\char120}}^{2t-2}}{1\\over2\\pi i}\n\\int_{2t-c-i\\infty}^{2t-c+i\\infty}ds\\,{\\Gamma(1-s)\\over(2\\pi)^{-s}}\\,\n  \\zeta_R(2t-s)\\,\\zeta_R(1-s)\\,\\mbox{{\\ssb\\char120}}^{2t-s-1}\\,,}\n  \\eql{mint4}$$\nby setting $s\\to2t-s$.\n\nThe $\\zeta$--functional equation (one can also, more easily, use\n(\\puteqn{gfr})) leads to,\n  $$\\eqalign{\n  &\\zeta_R(s)\\,\\zeta_R(s-2t+1)\\cr&=\n  {2\\Gamma(1-s)\\over(2\\pi)^{1-s}}\\sin(\\pi s\/2)\\,\\zeta_R(1-s){2\\Gamma(2t-s)\n  \\over(2\\pi)^{2t-s}}\n  \\sin(\\pi(s-2t+1)\/2)\\zeta_R(2t-s)\\cr\n  &={4\\Gamma(1-s)\\Gamma(2t-s)\\over(2\\pi)^{2t+1-2s}}\\sin(\\pi s\/2)\\,\n  \\sin(\\pi(s-2t+1)\/2)\\zeta_R(1-s)\\zeta_R(2t-s)\\cr\n   &=(-1)^t{2\\Gamma(1-s)\\Gamma(2t-s)\\over(2\\pi)^{2t+1-2s}}\\sin(\\pi s)\n   \\,\\zeta_R(1-s)\\zeta_R(2t-s)\\cr\n    &={(-1)^{t-1}\\Gamma(1-s)\\over(2\\pi)^{2t-2s}\\Gamma(s-2t+1)}\\,\\zeta_R(1-s)\\zeta_R(2t-s)\\,,\n  }\n  $$\nand so (\\puteqn{mint4}) becomes,\n  $$\\eqalign{\n&{\\overline\\phi}_{2t}(1\/\\mbox{{\\ssb\\char120}})\\cr\n &={2(-1)^{t-1}\\over \\mbox{{\\ssb\\char120}}^{2t-2}}{1\\over2\\pi i}\n\\int_{2t-c-i\\infty}^{2t-c+i\\infty}ds\\,{\\Gamma(s-2t+1)\\over(2\\pi)^{s-2t}}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{2t-s-1}\\,.}\n  \\eql{mint5}$$\nI then find the alternative expression for the obstruction, $\\overline\nP_t(\\mbox{{\\ssb\\char120}})$, defined in (\\puteqn{pee1}) or (\\puteqn{ppol}),\n  $$\n  \\overline P_t(\\mbox{{\\ssb\\char120}})={1\\over\\pi i}\\bigg(\n\\int_{c-i\\infty}^{c+i\\infty}-\n\\int_{2t-c-i\\infty}^{2t-c+i\\infty}\\bigg)ds\\,{\\Gamma(s-2t+1)\\over(2\\pi)^{s-2t}}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{2t-s-1}\\,.\n  $$\nTranslating the contours so that they mutually cancel leaves the\ncontributions of the intervening poles {\\it i.e. } those between 0 and $2t$. Thus\n  $$\\eqalign{\n  \\overline P_t(\\mbox{{\\ssb\\char120}})&={1\\over\\pi i}\\oint_C ds\\,{\\Gamma(s-2t+1)\n  \\over(2\\pi)^{s-2t}}\\,\n  \\zeta_R(s)\\,\\zeta_R(s-2t+1)\\,\\mbox{{\\ssb\\char120}}^{2t-s-1}\\cr\n  &=(-1)^{t-1}{1\\over i}\\oint_C ds\\,\n  \\zeta_R(s)\\,\\zeta_R(2t-s)\\,{\\rm cosec\\,} (\\pi s\/2)\\mbox{{\\ssb\\char120}}^{2t-s-1}\\cr\n  &=2\\pi\\zeta_R(2t-1)\\big((i \\mbox{{\\ssb\\char120}})^{2t-2}-1\\big)-4i\n  \\sum_{j=1}^{t-1}\\zeta_R(2j)\\,\\zeta_R(2t-2j)(i \\mbox{{\\ssb\\char120}})^{2t-2j-1}\\cr\n  &=2\\pi\\zeta_R(2t-1)\\big((i \\mbox{{\\ssb\\char120}})^{2t-2}-1\\big)-i(-1)^t(2\\pi)^t\n  \\sum_{j=1}^{t-1}{B_{2j}\\,B_{2t-2j}\\over (2j)!(2t-2j)!}\\,(i \\mbox{{\\ssb\\char120}})^{2t-2j-1}\n  \\,,\n  }\n  \\eql{ppoly2}$$\nwhich is a polynomial of degree $(2t-2)$ and is the final, main result of\nthis section. The odd powers come from the poles of the cosec while the\ntwo even powers arise from the poles of the Riemann $\\zeta$--functions. $\\overline\nP_t(\\mbox{{\\ssb\\char120}})$ is real if $\\mbox{{\\ssb\\char120}}$ is.\n\nSetting $\\mbox{{\\ssb\\char120}}$ to zero in (\\puteqn{ppoly2}) and (\\puteqn{ppol}) produces the\nspecific formulae,\n  $$\\eqalign{\n  \\zeta_R(2t-1)&=-{2(2\\pi)^{2t-1}\\over\\Gamma(2t-1)}\\int_0^\\infty d\\mbox{{\\ssb\\char98}}\\,\n  \\mbox{{\\ssb\\char98}}^{2t-2}\\epsilon_t^{\\rm sub}(\\mbox{{\\ssb\\char98}})\\cr\n  &=-(-1)^t{2(2\\pi)^{2t-1}\\over\\Gamma(2t-1)}\\int_0^\\infty d\\mbox{{\\ssb\\char98}}\\,\n  \\epsilon_t^{\\rm sub}(\\mbox{{\\ssb\\char98}})\\,,\n  }\n  \\eql{even}\n  $$\nusing inversion. These can be checked numerically. More generally,\nexpanding (\\puteqn{ppol}) gives the other nonzero moments of $\\epsilon_t^{\\rm\nsub}(\\mbox{{\\ssb\\char98}})$,\n  $$\n  \\int_0^\\infty d\\mbox{{\\ssb\\char98}}\\,\\mbox{{\\ssb\\char98}}^{2j-1}\\epsilon_t^{\\rm\n  sub}(\\mbox{{\\ssb\\char98}})={(-1)^j\\over8j(t-j)}\\,B_{2j}\\,B_{2t-2j}\\,,\\quad 1<j<t-1\\,.\n  \\eql{moments}\n  $$\nThe periods given by Kohnen and Zagier, [\\putref{KandZ}], are identical to\n(\\puteqn{even}) and (\\puteqn{moments}) and I now have made contact with this\nmathematical notion.\n\nThe {\\it periods} of a {\\it cusp} modular form, ${\\cal F}$, of weight $2t$\nare the numbers ({\\it e.g. }\\ Lang [\\putref{Lang}]),\\footnote{ I prefer the\nnormalisation of Kohnen and Zagier, [\\putref{KandZ}].}\n  $$\n  r_n({\\cal F})=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,{\\cal F}(i\\mbox{{\\ssb\\char98}})\n  \\,\\mbox{{\\ssb\\char98}}^n\\,,\\quad 0\\le n\\le 2t-2\\,.\n  \\eql{periods}$$\nThe end points are cusps where ${\\cal F}$ vanishes (exponentially) to ensure\nconvergence. The restriction on $n$ is a polynomial one (see\n[\\putref{Lang}].)\n\nIt is important for the following to assemble the periods into a {\\it\nperiod polynomial}, which, as usual, is a handier quantity,\n  $$\\eqalign{\n  r_{\\cal F}(\\mbox{{\\ssb\\char120}})&=\\sum_{n=0}^{2t-2}(-1)^n\\comb{2t-2}{n}\\,r_n({\\cal F})\\,\n  \\mbox{{\\ssb\\char120}}^{2t-2-n} \\cr\n&=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,(\\mbox{{\\ssb\\char120}}-\\mbox{{\\ssb\\char98}})^{2t-2}{\\cal F}(i\\mbox{{\\ssb\\char98}})\n \\,.}\n  \\eql{ppol7}$$\n\nIn order to extend the notion to {\\it non}--cusp forms (such as\nEisenstein series) one course (see Zagier, [\\putref{Zagier}] and Kohnen and\nZagier, [\\putref{KandZ}] \\S4), is to continue the integer, $n$, into the\ncomplex plane by employing the Hecke $L$--series of ${\\cal F}$ via the Mellin\ntransform,\n  $$\n  L({\\cal F},s)={(2\\pi)^s\\over\\Gamma(s)}\\int_0^{\\infty}dt\\,\\big({\\cal F}(it)\n -{\\cal F}(i\\infty)\\big)\\,t^{s-1}\\,,\n  \\eql{ell}$$\nwhich converges for $s>2t-2$ and, continued in $s$, satisfies the\nfunctional relation,\n  $$\n  (2\\pi)^{-s}\\,\\Gamma(s)\\,L({\\cal F},s)=(-1)^t(2\\pi)^{s-2t}\\,\\Gamma(2t-s)\\,L({\\cal F},2t-s)\\,,\n  \\eql{lfr}$$\nby virtue of the (inversion) modularity of ${\\cal F}$, and conversely. This\nis usually attributed to Hecke, [\\putref{Hecke}] (but see the Appendix).\nTerras, [\\putref{Terras}] p.229, gives a useful pedagogical treatment of\nthis topic.\n\nThen the periods could be {\\it defined}, in general, by,\n[\\putref{KandZ}],[\\putref{Zagier}],\n  $$\\eqalign{\n  r_n({\\cal F})&=\\lim_{s\\to n+1}\\,{\\Gamma(s)\\over(2\\pi)^{s}}\\,L({\\cal F},s)\n  \\cr\n  &\\equiv\\lim_{s\\to n+1}L^*({\\cal F},s)\\,,\n  }\n  \\eql{periods2}$$\nbecause this coincides with the integral expression, (\\puteqn{periods}), for\ncusp forms.\n\nThe explicit values of $r_n$ ($n=0,\\ldots 2t-2$) for Eisenstein series\n(which is all that is required because of a decomposition theorem for\nmodular form space) are easily computed from (\\puteqn{eisl1}) and are given\nin [\\putref{KandZ}]. They agree with (\\puteqn{even}) and (\\puteqn{moments}), as\nalready noted.\n\nHowever, the Dirichlet series, $L^*({\\cal F},s)$ has poles at $s=0$ and\n$s=2t$. A means of incorporating these values is given by Zagier,\n[\\putref{Zagier}], using the period polynomial. One notes that the\nsummation in the cusp period polynomials can be extended,\n  $$\\eqalign{\n  r_{\\cal F}(\\mbox{{\\ssb\\char120}})&=\\sum_{n\\in\\,\\mbox{\\open\\char90}}(-1)^n\\comb{2t-2}{n}\\,r_n({\\cal F})\\,\n  \\mbox{{\\ssb\\char120}}^{2t-2-n} \\cr\n&=\\sum_{n\\in\\,\\mbox{\\open\\char90}}(-1)^n\\comb{2t-2}{n}\\,i^{n+1}\\,L^*({\\cal F},n+1)\\,\n\\mbox{{\\ssb\\char120}}^{2t-2-n}\\,,}\n  \\eql{ppol6}$$\nbecause the binomial coefficient vanishes outside the range\n$n=0,\\ldots,2t-2$. For {\\it non}--cusp forms, however, the sum extends\nfrom $-1$ to $n=2t-1$, the nonzero end values arising from the\nsingularities of $L^*({\\cal F},s)$ mentioned above. The resulting expression\ncan then be taken as a {\\it definition} of the period polynomial of a\nnon--cusp form.\n\nAgain, for the Eisenstein series, $G_{2t},\\,=\\epsilon_{t}$, the computation is\nstraightforward, [\\putref{Zagier}], and (\\puteqn{ppol6}) with (\\puteqn{eisl1})\nyields, ($n=2j-1$),\n  $$\\eqalign{\n  r_G(x)={(2t-2)!\\over2(2\\pi)^{2t}}\\bigg(2\\pi\n  \\zeta_R&(2t-1)\\,\\big((-1)^{t-1}\\mbox{{\\ssb\\char120}}^{2t-2}-1\\big)-\\cr\n  &(2\\pi)^t\\sum_{j=0}^t\\,(-1)^{t-j}{B_{2j}\\,B_{2t-2j}\\over(2j)!(2t-2j)!}\n  \\,\\mbox{{\\ssb\\char120}}^{2j-1}\\bigg)\\,,\n  }\n  \\eql{eperiodp}\n  $$\nwhich is $1\/\\mbox{{\\ssb\\char120}}$ times a polynomial. For comparison with\n[\\putref{Zagier}], Zagier's $X$ equals my $i\\mbox{{\\ssb\\char120}}$.\n\nI now comment on this construction. The periods, and period polynomials,\nfor non--cusp forms are defined by analogy to those of cusp forms, the\njustification being that the expressions, for ${\\cal F}(i\\infty)$ equal to\nzero, reduce to those for cusp forms. The means to attain this is not\nunique. Instead of the `zeta--function' regularisation, as in the\ndefinition of $L$, (\\puteqn{ell}), I employ the {\\it subtracted} form, as in\n(\\puteqn{rege}),\n  $$\n   {\\cal F}^{\\rm sub}\\equiv {\\cal F}-a_0-{a_0\\over\\tau^{2t}}\\,,\\quad\n   a_0={\\cal F}(i\\infty)\\,,\n  $$\nwhose moments are non--infinite and provide an alternative definition of\nthe periods of a non--cusp form which equally reduces to that for cusp\nforms when $a_0=0$. This subtraction maintains inversion modularity, but\nviolates translational, whereas the subtraction of just $a_0$ does the\nreverse. Making use of my earlier considerations of Eisenstein series,\nsee (\\puteqn{ppol}), I define the subtracted period polynomial as an\nintegral,\n  $$\n  r_{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char120}})=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,\n  (\\mbox{{\\ssb\\char98}}-\\mbox{{\\ssb\\char120}})^{2t-2}{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char98}})\\,.\n  $$\n\nFor Eisenstein series, (\\puteqn{ppol}) shows that $r_G^{\\rm sub}(\\mbox{{\\ssb\\char120}})$ is\nproportional to $\\overline P_t(\\mbox{{\\ssb\\char120}})$ whose explicit form is given in\n(\\puteqn{ppoly2}).\n\nIt is necessary to compare this definition with that of Zagier,\n(\\puteqn{eperiodp}). The difference is the absence of the `end point' terms\nproportional to $1\/\\mbox{{\\ssb\\char120}}$ and $\\mbox{{\\ssb\\char120}}^{2t-1}$. This can be remedied by\nincluding, in the terminology of this paper, the regularised\ncontributions from the subtracted terms, the Casimir and Planck terms,\nwhich can be accomplished simply by expanding the closed contour in\n(\\puteqn{ppoly2}) to include the corresponding poles at $s=0$ and $s=2t$.\nThis just extends the sum by the two end points giving the well known\nquantity,\n   $$\\eqalign{\n\\overline R_t(\\mbox{{\\ssb\\char120}})=2\\pi\\,\\zeta_R(2t-1)\\big((i \\mbox{{\\ssb\\char120}})^{2t-2}-1\\big)-4i\n  \\sum_{j=0}^t\\zeta_R(2j)\\,\\zeta_R(2t-2j)(i \\mbox{{\\ssb\\char120}})^{2t-2j-1}\n  \\,,}\n  \\eql{ppoly3}\n  $$\nwhich agrees, up to a factor, with Zagier's expression given in\n(\\puteqn{eperiodp}).\n\nFurthermore, the use of the subtracted form, ${\\cal F}^{\\rm sub}$, allows one\nto give a cleaner integral expression for the enlarged period polynomial\nthan that in [\\putref{Zagier}], {\\it i.e. },\n  $$\n  r_{\\cal F}^{\\rm enl}(\\mbox{{\\ssb\\char120}})=\\int_0^{\\infty}d\\mbox{{\\ssb\\char98}}\\,\n  (\\mbox{{\\ssb\\char98}}-\\mbox{{\\ssb\\char120}})^{2t-2}{\\cal F}^{\\rm sub}(\\mbox{{\\ssb\\char98}})+\n  {a_0\\over2t-1}\\big(\\mbox{{\\ssb\\char120}}^{2t-1}+\\mbox{{\\ssb\\char120}}^{-1}\\big)\\,.\n  $$\n\nWhile equivalent to the method in [\\putref{Zagier}], I believe the contour\napproach is neater and is capable of being extended to the general theory\nof periods. Zagier, [\\putref{Zagier}], shows that logistic simplifications\noccur in this theory when it is extended to include non--cusp forms such\nas the Eisenstein series.\n\nAs an historical point, it will be noticed that the above expressions\nhave been available since the time of Ramanujan, with contour derivations\nby Malurkar, [\\putref{Malurkar}], and Guinand, [\\putref{Guinand}]. In order\nto investigate this point further, I return to the mode sum expressions\nfor the thermodynamic related quantities.\n\n\\section{ \\bf 3. Back to summations.}\nI start with the Eisenstein series forms and work the development to\nparallel the quantities of the previous section. This allows me to\nintroduce the useful calculations of Smart, [\\putref{Smart}], on the\nEpstein function (which will arise later) using its relation with\nEisenstein series.\n\nThe subtracted partial energy $\\epsilon_t^{\\rm sub}$ in (\\puteqn{rege})\ncorresponds to the the `regularised' Eisenstein series,\n  $$\\eqalign{\n  G_t^{\\,\\rm sub}(\\tau)&=\\sumdasht{m=-\\infty}{\\infty}\\,\\sumdasht{n=-\\infty}{\\infty}\n  {1\\over (m\\tau+n)^{2t}}\\cr\n  &=\\,G_t(\\tau)-2\\zeta_R(2t)(1+\\tau^{-2t})\\,,}\n  \\eql{geesub}$$\nThe relation is, [\\putref{DandK1}],\n  $$\n \\epsilon^{\\rm sub}_t(\\tau)={(2t-1)!\\over2(2\\pi i)^{2t}}\\, \\overline\n G^{\\,\\rm sub}_t(\\tau)\\,.\n  \\eql{inten5}$$\n\nNow, corresponding to (\\puteqn{mint2}), integrate (\\puteqn{geesub}) with\nrespect to $\\tau$ from $\\tau$ to $i\\infty$, $2t-1$ times to give,\nformally,\n  $$\n  {1\\over(2t-1)!}\\,\\phi_{2t}(\\tau)\\,,\n  \\eql{geeint}$$\nwhere $\\phi$ is defined by, [\\putref{Smart}], eqn.(2.11),\n  $$\n  \\phi_{2t}(\\tau)=\\sumdasht{m=-\\infty}{\\infty}\\,\\sumdasht{n=-\\infty}{\\infty}\n  {1\\over m^{2t-1}}{1\\over m\\tau+n}\\,.\n  \\eql{phis}$$\n\nThe behaviour of the series $\\phi_{2k}(\\tau)$ under modular\ntransformations, in particular $\\tau\\to-1\/\\tau$, has been especially\nconsidered by Smart, [\\putref{Smart}], who refers to the $\\phi$'s as `{\\it\nmodular forms of weight $2-2t$ with rational period functions}'.\n\nUsing the partial fraction identity (see also Hurwitz, [\\putref{Hurwitz}],\nEisenstein, [\\putref{Eisenstein}]),\n  $$\n     {1\\over m^{2t-1}}{1\\over m\\tau+n}=\n     -{\\tau^{2t-1}\\over n^{2t-1}}{1\\over m\\tau+n}+\\sum_{j=1}^{2t-1}(-1)^{j+1}\n     {\\tau^{2t-j-1}\\over m^j\\,n^{2t-j}}\\,,\n  \\eql{pf}$$\none finds easily the inversion relation, [\\putref{Smart}], eqn.(1.10b),\n  $$\n  \\phi_{2t}(\\tau)-\\tau^{2t-2}\\phi_{2t}(-1\/\\tau)=-4\\sum_{j=1}^{t-1}\n     \\tau^{2t-2j-1}\\zeta_R(2j)\\,\\zeta_R(2t-2j)\\,.\n  \\eql{sm1}$$\nAlso, under translations,\n  $$\n  \\phi_{2t}(\\tau+1)-\\phi_{2t}(\\tau)=2{\\zeta_R(2t)\\over\\tau(\\tau+1)}\\,.\n  \\eql{sm2}$$\n\nUp to a simple factor, one might expect $\\phi_{2t}$ to be the same as\n${\\overline\\phi}_{2t}$ in (\\puteqn{mint2}). However the cocycle functions in\n(\\puteqn{sm1}) and (\\puteqn{ppoly2}) differ by the two {\\it even} powers. I\nnote that these are related to the $j=1$ and $j=2t-1$ terms in the sum in\n(\\puteqn{pf}) which have gone out in the passage to the difference,\n(\\puteqn{sm1}), by antisymmetry. However the summation over $m$,\nrespectively $n$, for these values of $j$ is conditionally convergent and\nis therefore subject to ambiguity, that is to say, a choice has to be\nmade, as in the discussion of the Eisenstein series, $G_2$. The\nregularised Mellin transform approach results in a different definition\nof the series (\\puteqn{phis}) to that used by Smart, [\\putref{Smart}],p.3. The\nrelation is easily found by taking the even powers in (\\puteqn{ppoly2}) over\nto the left and then we see that one has the relation, (further remarks\nare given later),\n  $$\n  {\\overline\\phi}_{2t}(\\mbox{{\\ssb\\char120}})=\ni\\phi_{2t}(\\tau)-2\\pi \\zeta_R(2t-1)\\,,\\quad \\mbox{{\\ssb\\char120}}=-i\\tau\\,,\n  \\eql{connec}\n  $$\nalso agreeing with (\\puteqn{trans}) and (\\puteqn{sm2}).\n\nSmart, [\\putref{Smart}], also gives the relation with a known Lambert\nseries as\n  $$\\eqalign{\n  S_t(\\tau)\\equiv\\sum_{m=1}^\\infty{1\\over m^{2t-1}}{q^{2m}\\over1-q^{2m}}\n  &=-{1\\over4\\pi i}\\,\\phi_{2t}(\\tau)-{1\\over2\\pi i\\tau}\\,\n\\zeta_R(2t)-{1\\over2}  \\zeta_R(2t-1)\\cr\n&={1\\over4\\pi}\\,\\big(\\overline\\phi_{2t}(\\mbox{{\\ssb\\char120}})+{2\\over\\mbox{{\\ssb\\char120}}}\\zeta_R(2t)\\big)\\cr\n&\\equiv{1\\over4\\pi}\\,\\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}})\\,,\n}\n  \\eql{ls}$$\nwhere $\\overline\\psi_{2t}$ is the function having $\\overline R_{2t}(\\mbox{{\\ssb\\char120}})$,\n(\\puteqn{ppoly3}), as its inversion cocycle function,\n  $$\n  {\\overline\\psi}_{2t}(\\mbox{{\\ssb\\char120}})-(i\\mbox{{\\ssb\\char120}})^{2t-2}\\,{\\overline\\psi}_{2t}(1\/\\mbox{{\\ssb\\char120}})\n= \\overline R_{2t}(\\mbox{{\\ssb\\char120}})\\,,\n  \\eql{are1}$$\nas is easily confirmed.\n\nThe introduction of $\\overline\\psi$ has restored translational\ninvariance, (the Planck term has been put back in (\\puteqn{ls})),\n  $$\n  \\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}}-i)- \\overline\\psi_{2t}(\\mbox{{\\ssb\\char120}})=0\\,.\n  \\eql{per}$$\nIt is almost convention to consider modular integrals, for example, to be\nperiodic ({\\it e.g. } Knopp, [\\putref{Knopp}]). See some later comments.\n\nEquations (\\puteqn{are1}) and (\\puteqn{ls})  imply an inversion relation for\nthe series $S_t$ which has a certain history, some of which is detailed\nby Berndt, [\\putref{Berndt}], p.153, who says that it was first written\ndown by Ramanujan. As is clear, it is quite the same as the expression\nfor the Eisenstein period polynomials.\n\nThe Lambert series, $S_t(\\tau)$, has been considered by Apostol,\n[\\putref{Apostol}], Berndt, [\\putref{Berndt}], Grosswald, [\\putref{Grosswald}],\nand Smart [\\putref{Smart}], for example. It can be related to statistical\nmechanics in the following way.\n\nOne can resum the statistical expression for the partial free energy in a\nstandard manner, beginning, ($q=e^{\\pi i\\tau}=e^{-\\pi\/\\xi}$),\n  $$\n  f_t=\\epsilon_{t,0}-{\\xi\\over2\\pi}\\sum_{n=1}^\\infty n^{2t-2}\\log(1-q^{2n})\n  =\\epsilon_{t,0}-{\\xi\\over2\\pi}\\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}\\,.\n  \\eql{ssum}$$\n\nThe scaled entropy is defined by,\n  $$\n  s_t(\\xi)={\\partial f_t(\\xi)\\over\\partial \\xi}=-{1\\over2\\pi}\\bigg(1-{2\\pi\\over\\xi}D\n\\bigg) \\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}\\,.\n  \\eql{scent}$$\n\nConcentrate now on the summation and rewrite it,\n  $$\n  \\sum_{m,n=1}^\\infty{n^{2t-2}\\over m}q^{2mn}=D^{2t-2}\n\\sum_{m=1}^\\infty{1\\over m^{2t-1}}{q^{2m}\\over1-q^{2m}}\\,,\n  \\eql{sum9}$$\nin terms of $S_t(\\tau)$, (\\puteqn{ls}).\n\nThis Lambert series  has been discussed in connection with generalised\nDedekind $\\eta$--functions by Berndt, [\\putref{Berndt}], and it is worth\nnoting that, in a certain sense, $\\log \\eta$ is a modular form of zero\nweight, in agreement with the results $\\epsilon_1=-D\\log\\eta$,\n[\\putref{DandK2}], which could therefore be written,\n  $\n  \\epsilon_1=-{\\cal D}\\log\\eta\\,,\n  $\nin terms of the covariant derivative, ${\\cal D}$.\n\nThe question, not answered here, is whether there is an `expansion' or\nrepresentation theorem for modular {\\it integrals} combining that for\nmodular forms (of {\\it any} real weight, see [\\putref{Rankin}]). It is not\nlikely that one can find a pure elliptic formula. As evidence I cite the\nparticular evaluations of Smart who gives at the lemniscate point\n($\\tau=i,\\mbox{{\\ssb\\char120}}=1$),\n  $$\n  \\overline\\psi_{4}(1)={7\\pi^4\\over90}-2\\pi\\zeta_R(3)\\,,\n  $$\nand, more generally, if $2t=0({\\rm mod} 4)$,\n  $$\n  S_t(i)={\\zeta_R(2t)\\over2\\pi}-{\\zeta_R(2t-1)\\over2}+{1\\over2\\pi}\n  \\sum_{j=1}^{t-1}(-1)^{j+1}\\zeta_R(2t-2j)\\zeta_R(2j)\\,,\n  $$\na result actually due to Lerch which follows from (\\puteqn{are1}) with\n(\\puteqn{ppoly3}).\n\nAs a final general comment in this section, it is always possible to\nobtain the required quantities as (infinite) $q$-series by direct\nintegration, but these do not count as closed forms. For example, one\nintegration yields the $q$-series for the partial free energy in terms of\nthe arithmetic functions, $\\sigma_k(m)$, (the sum of the $k$-th powers of\nthe divisors of $m$),\n  $$\n  f_t=\\epsilon_{t,0}+{1\\over\\beta}\\sum_{m=1}^\\infty\\sigma_{2t-1}(m)\\,{q^{2m}\\over m}\\,,\n  $$\nwhich amounts only to doing the $n$-summation in ({\\puteqn{ssum}), or an\nintegration of the standard relation, (\\puteqn{mell1}),\n  $$\n  \\epsilon_t=-{B_{2t}\\over4t}+\\sum_{m=1}^\\infty \\sigma_{2t-1}(m)\\,q^{2m}\\,.\n  $$\n\nOne also has the formula, {\\it e.g. } [\\putref{Rad,Glaisher}],\n  $$\n  S_t(\\tau)=\\sum_{m=1}^\\infty \\sigma_{-2t+1}(m)\\,q^{2m}=\n  \\sum_{m=1}^\\infty {\\sigma_{2t-1}(m)\\over m^{2t-1}}\\,q^{2m}\\,,\\quad\\forall\\, t\\,,\n  \\eql{Ssi}$$\nin agreement with the result of a $(2t-1)$-fold integration.\n\\section{\\bf 4. Another approach via modular integrals.}\nThe period functions are cocycle functions associated with Eichler\ncohomology and modular integrals. These notions put a different slant on\nthe preceding formulae. In effect one begins again.\n\nThe best place to start is the important (but particular) analytical result\nof Bol, [\\putref{Bol}], which states that, for {\\it any} reasonable function,\n$\\varphi(\\tau)$,\n  $$\n  \\big(D^{r+1}\\varphi\\big)(\\gamma\\tau)=\n  (c\\tau+d)^{r+2}D^{r+1}\\big((c\\tau+d)^r\\,\\varphi(\\gamma\\tau)\\big)\\,,\n  \\eql{bol}$$\nwhere $r$ is a nonnegative number, and I use,\n  $$(D^{r+1}\\varphi)(\\tau)\\equiv \\bigg(q{d\\over dq}\\bigg)^{r+1}\\varphi(\\tau)\\,,\n  $$\nwith,\n  $$\n   \\gamma=\\left(\\matrix{*&*\\cr\n                   c&d}\\right)\\in {\\rm SL}(2,\\mbox{\\open\\char82})\\,.\n  $$\nThis shows that, {\\it if} $\\varphi$ is an automorphic form of weight\n$-r$, then $D^{r+1}\\,\\varphi$ is an automorphic form of weight $2+r$, see\nalso Petersson, [\\putref{Petersson}]. Note that it is the ordinary\nderivative that enters here. The proof of (\\puteqn{bol}) follows by\ninduction.\n\nThe classic discussion of {\\it modular integrals} can proceed by asking\nfor the inverse of Bol's result. That is, given a form of weight $2+r$,\n$\\rho(\\tau)$, as a source, can one integrate the differential equation,\n  $$\n  D^{r+1}\\varphi=\\rho\n  \\eql{de}$$\nto obtain $\\varphi$, a form of weight $-r$? Clearly this will not be so\nin general because of the freedom introduced by constants of integration.\nThat is, one would expect $\\varphi$ to behave as a form of weight $-r$ up\nto an additional piece, the general form of which can be determined from\nthe differential equation (\\puteqn{de}) and Bol's theorem, (\\puteqn{bol}).\n\nGenerally we would write (ignoring any multiplier systems for\nsimplicity),\n  $$\n  \\varphi(\\gamma\\tau)=(c\\tau+d)^{-r}\\big(\\varphi(\\tau)+P(\\gamma,\\tau)\\big)\\,,\n  \\eql{co2}$$\nwhere $P(\\gamma,\\tau)$ has to be a polynomial in $\\tau$ of order $\\le r$,\nThe inverse to Bol's result is that, if any $\\varphi$ satisfies\n(\\puteqn{co2}), then $\\rho$, of (\\puteqn{de}), is a form of weight $(2+r)$.\nThis follows most simply by substituting (\\puteqn{co2}) into (\\puteqn{bol}).\n\n$P$ must satisfy the 1--cocycle consistency condition, [\\putref{Eichler}],\n  $$\\eqalign{\n   P(\\gamma_1\\gamma_2,\\tau)&=(c_2\\tau+d_2)^r\\,P(\\gamma_1,\\gamma_2\\tau)+P(\\gamma_\n   2,\\tau)\\cr\n   &\\equiv P(\\gamma_1,\\tau)\\big|\\gamma_2+P(\\gamma_2,\\tau)\\,.\n   }\n  \\eql{cocycle}$$\n\n$\\varphi$ is known as an {\\it automorphic} or {\\it Eichler integral}, of\nweight $-r$, and $P$ is the {\\it associated period polynomial},\n[\\putref{Eichler,HandK}]. I have also introduced the useful, standard\n`stroke' operator, $\\big|\\gamma$, sometimes denoted $\\big|[\\gamma]$.\n\nThe {\\it general} solution (equivalently the indefinite integral) of the\ndifferential equation (\\puteqn{de}) is a particular integral plus the\ncomplementary function (a zero mode) {\\it i.e. }\n  $$\n  \\varphi(\\tau)={(2\\pi i)^{r+1}\\over\\Gamma(r+1)}\n  \\int_{\\tau_0}^\\tau d\\sigma\\, (\\tau-\\sigma)^{r}\\,\\rho(\\sigma)+\\Theta(\\tau)\\,,\n  \\eql{gensol}\n  $$\nwhere $\\Theta(\\tau)$ is a polynomial of degree $r$. The integral, in the\nupper half plane, is path independent if $\\rho$ has weakish analytic\nproperties and, if desired, the solution can be made independent of the\nlower limit by suitable adjustment of $\\Theta(\\tau)$. In elementary fashion,\n$\\Theta(\\tau)$ is determined by the first $r$ derivatives of $\\varphi(\\tau)$\nat $\\tau_0$.\n\nAll this is general analysis. In modular applications to SL(2,$\\mbox{\\open\\char90}$),\n$\\tau_0$ is chosen as the cusp $i\\infty$ and the limits in (\\puteqn{gensol})\nreversed. In terms of Fourier expansions, a holomorphic cusp form source\nis\n  $$\n  \\rho(\\tau)=\\sum_{m=1}^\\infty \\rho_m\\,q^{2m}\\,,\n  \\eql{fe1}$$\nand the particular Eichler integral of $\\rho$, is, from (\\puteqn{gensol}),\n  $$\n  \\varphi(\\tau)=\\sum_{m=1}^\\infty {\\rho_m\\over m^{r+1}}\\,q^{2m}\\,,\n  \\eql{fe2}$$\nwhich has the same periodicity as $\\rho$, a result of choosing\n$\\tau_0=i\\infty$.\n\nThe problem in the present Eisenstein case has already been noted. The\nEisenstein series, $G_t(\\tau)$ does not vanish at $i\\infty$ (or $0$). It\nis a `non--cusp' form and the direct integration of (\\puteqn{de}) meets\nobstacles at an elementary level. This can be appreciated simply from the\nappearance of $1\/m$ factors during integration, and $m$ can be zero in\n$G_t$. This fits in because the $m=0$ terms in $G_t$ are those making\n$G_t$ nonzero at $i\\infty$, the zero temperature value. Saying it yet\nagain, the Fourier transform of $G_t$ begins with an $m=0$ term, the\nCasimir contribution, so (\\puteqn{fe2}) makes no immediate sense.\n\nOne way of avoiding these obstructions, is to regularise $G_t$ by\ndropping the offending term(s) as in (\\puteqn{geesub}) and consider the\ndifferential equation,\n  $$\\eqalign{\n  D^{2t-1}\\,\\varphi_{2t}=-4\\pi\\epsilon_t^{\\rm sub}\\,,}\n  \\eql{gee}$$\nwhich has the particular integral, $\\varphi_{2t}=\\overline\\phi_{2t}$\nwhere,\n  $$\n  \\overline\\phi_{2t}(\\tau)=-4\\pi{(2\\pi i)^{2t-1}\\over\\Gamma(2t-1)}\n  \\int_\\tau^{i\\infty}d\\tau'\\,(\\tau'-\\tau)^{2t-2}\n  \\,\\epsilon_t^{\\rm sub}(\\tau')\\,,\n  $$\nthe constants and boundary conditions being chosen so as to reproduce\n(\\puteqn{mint2}). The upper limit is a cusp. Smart's function, $\\phi_{2t}$,\nresults if one adds a complementary constant (and multiplies by $i$).\n(See (\\puteqn{connec}).)\n\nBecause of the subtraction, the quantity $\\epsilon_t^{\\rm sub}(\\tau)$ is not\nexactly a modular form, rather one has, expressed slightly generally,\n($r=2t-2$),\n  $$\n  \\epsilon(\\gamma\\tau)=(c\\tau+d)^{r+2}\\big(\\epsilon(\\tau)+E(\\gamma,\\tau)\\big)\n  \\eql{submod}\n  $$\nand also\n  $$\n  \\phi(\\gamma\\tau)=(c\\tau+d)^{-r}\\big(\\phi(\\tau)+P(\\gamma,\\tau)\\big)\\,,\n  \\eql{co3}\n  $$\nwhich are related by the same differential equation, (\\puteqn{de}),\n  $$\n  D^{r+1}\\phi=-4\\pi\\epsilon\\,.\n  $$\nBol's formula, (\\puteqn{bol}),  relates $P$ and $E$,\n  $$\n  D^{r+1}P=-4\\pi E\\,,\n  \\eql{bol2}\n  $$\nand these relations generalise the usual ones applicable to cusp forms.\nThe cocycle formula (\\puteqn{cocycle}) still holds of course, as (\\puteqn{co3})\nis the same as (\\puteqn{co2}).\n\nI remark that $\\overline\\phi$ and $\\epsilon^{\\rm sub}$, not being periodic, do\nnot quite possess Fourier expansions, as in (\\puteqn{fe1}) and (\\puteqn{fe2}).\nHowever $\\overline\\psi$, in view of (\\puteqn{per}), does,\n  $$\n  \\overline\\psi(\\mbox{{\\ssb\\char120}})=\\sum_{m=1}^\\infty \\psi_m\\,q^{2m}\\,,\n  $$\n\nBecause $P$, this time, satisfies (\\puteqn{bol2}), it is not required to be\na polynomial. Generally, since the modular group is generated \\footnote{\nBeware that some authors, {\\it e.g. } Knopp, swap the usage of the symbols $S$\nand $T$.} by $S$ and $T$, $P(\\gamma,\\tau)$ is a {\\it rational period\nfunction}. In the present case, however, $P(S,\\tau)$ {\\it is} a\npolynomial, given by (\\puteqn{ppoly2}),\n  $$\n  P(S,\\tau)=\\overline P_t(\\tau)\\,,\n  \\eql{pt}$$\nwhile the (non--zero) $P(T,\\tau)$, which follows from (\\puteqn{trans}),\n  $$\n  P(T,\\tau)=2\\,\\zeta_R(2t)\\bigg({1\\over \\tau}-{1\\over\\tau+1}\\bigg)\\,,\n  \\eql{ps}$$\nis a ratio of polynomials. Of course, $P(\\pm{\\bf 1},\\tau)=0$ and I remark\nthat the polynomial $P(S,\\tau)$ runs from $\\tau^0$ to $\\tau^{2t-2}$.\n\nAs a simple check of the algebra, set $\\gamma_1=\\gamma_2=S$ in the cocycle\nrelation, (\\puteqn{cocycle}). This gives,\n  $$\n P(S,\\tau)\\big|(1+S)\\equiv P(S,\\tau)+\\tau^{r}P(S,S\\tau)=\n P(S,\\tau)+\\tau^{r}P(S,-1\/\\tau)=0\n  $$\nwhich, for $r=2t-2$, agrees with the explicit form, (\\puteqn{ppoly2}).\n\nAlso, as examples,\n  $$\\eqalign{\n  P(TS,\\tau)=&2\\,\\zeta_R(2t){\\tau^{2t}\\over1-\\tau}+\\overline P_t(\\tau)\\cr\n  P(ST,\\tau)=&{2\\,\\zeta_R(2t)\\over\\tau(1+\\tau)}+\\overline P_t(1+\\tau)\\,.}\n  $$\nI note that the period function appears to have poles at $-1$, $0$, $+1$\nand $i\\infty$.\n\nFrom (\\puteqn{submod}) and the form of $\\epsilon^{\\rm sub}$, (\\puteqn{rege}), one\nfinds\n  $$\n  E(S,\\tau)=0\\,,\\quad E(T,\\tau)=\\bigg({1\\over(\\tau+1)^{2t}}\n  -{1\\over\\tau^{2t}}\\bigg)\n  $$\nwhich are consistent with (\\puteqn{ps}) and (\\puteqn{pt}).\n\nFor completeness, and to aid comparison, this is a convenient place to\noutline some standard elementary things about period polynomials, mostly\nfor SL$(2,\\mbox{\\open\\char90})$. Two basic `polynomials' are the inversion one,\n$P(S,\\tau)$, and the translation one, $P(T,\\tau)$. For cusp forms, the\nlatter vanishes because of periodicity, $P(T,\\tau)=0$, [\\putref{Knopp}],\ncorresponding to the periodicity of the modular integrals, {\\it cf }\n(\\puteqn{fe1}), (\\puteqn{fe2}). In this case, $P(S,\\tau)$ satisfies the\nEichler--Shimura relations\n  $$\\eqalign{\n  P(S,\\tau)\\big|(1+S)&=0 \\cr\n  P(S,\\tau)\\big|(1+TS+(TS)^2)&=0\n  }\n  \\eql{ES}\n  $$\nwith $S^2=1$ and $(TS)^3=1$. A standard way of proving these uses the\ncomplex integral form for $P$, {\\it e.g. } [\\putref{Zagier}], but they  can also\nfollow, algebraically from the cocycle relation as I now show. (See also\nKnopp, [\\putref{Knopp3}], for slightly different details.)\n\nFor the first identity, just set, as above, $\\gamma_1=1$ and $\\gamma_2=S$ in\n(\\puteqn{cocycle}) and it falls straight out. The second identity is\nslightly more work. Using (\\puteqn{cocycle}) twice, one has, dropping the\n$\\tau$ argument temporarily and noting $(TS)^2=(TS)^{-1}$,\n  $$\\eqalign{\n  P(S)\\big|\\big(1+&TS+(TS)^2\\big)=\\cr\n  &P(S)+P(STS)-P(TS)+P(SS^{-1}T^{-1})-P(TSTS)\\,.\n  }\n  \\eql{ES2}\n  $$\n\nFurther, setting $\\gamma_1=T$ in the cocycle condition, and using the\nassumption that $P(T)=0$, yields\n  $$\n  P(T\\gamma)=P(\\gamma)\\,.\n  \\eql{pt0}\n  $$\nOne then sees that the first and third terms on the right--hand side of\n(\\puteqn{ES2}) cancel, as do the second and fifth. The fourth term vanishes\non its own because (\\puteqn{pt0}) shows that $P(T^{-1})=0$, and (\\puteqn{ES})\nfollows.\n\nIn one development, the notion of modular integral has been somewhat\ndivorced from that of integration. One says that a {\\it modular integral}\nis any function (with appropriate analyticity properties) that obeys the\nquasi--modular relation (\\puteqn{co2}), where the {\\it period function} $P$\nmust still, of course, satisfy the cocycle condition (\\puteqn{cocycle}).\nMotivation is then provided to look for modular integrals with {\\it\nrational} period functions, {\\it i.e. } ratios of polynomials. In this case it\nseems to be conventional to {\\it assume} that the translational period\nfunction, $P(T,\\tau)$, vanishes, [\\putref{Knopp}], so that both\nEichler--Shimura relations, ({\\puteqn{ES}), still hold. There is a certain\ninterest in computing such rational period functions and investigating\ntheir zeros and poles. The easiest introduction to this topic is by\nKnopp, [\\putref{Knopp3}]. It is, however, somewhat peripheral to our\nconsiderations which are more concerned with non--cusp forms {\\it i.e. } the\nEisenstein series, {\\it e.g. }\\ [\\putref{Zagier}].\n\n\\section{\\bf 5. Green function distributional description.}\n\nDeveloping the formalism a little further, it is possible to give the\ndiscussion of Eichler integrals a distributional or Green function look.\nFor this purpose, it is better to use the `real' variable form and consider\nthe differential equation,\n  $$\n  {d^{2t-1}\\over d{\\mbox{{\\ssb\\char120}}}^{2t-1}}\\,\\psi(\\mbox{{\\ssb\\char120}})=\\rho(\\mbox{{\\ssb\\char120}})\\,,\n  \\eql{rde}$$\nfor $\\mbox{{\\ssb\\char120}}>0$ with solution vanishing at $\\infty$, {\\it cf } (\\puteqn{mint2}),\n  $$\n  \\psi(\\mbox{{\\ssb\\char120}})=\n\\int_{0^-}^\\infty d{\\mbox{{\\ssb\\char120}}'}\\,\\Phi_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,\\rho({\\mbox{{\\ssb\\char120}}'})\n  \\eql{soln}$$\nwhere the generalised function, $\\Phi_\\alpha(\\mbox{{\\ssb\\char120}})$, is\n $$\n\\Phi_\\alpha(\\mbox{{\\ssb\\char120}})={\\mbox{{\\ssb\\char120}}^{\\alpha-1}_+\\over\\Gamma(\\alpha)}\\,.\n \\eql{genp}$$\nThe {\\it generalised} function $\\mbox{{\\ssb\\char120}}^\\alpha_+$ is concentrated on the positive\n$\\mbox{{\\ssb\\char120}}$-axis, {\\it i.e. } $\\mbox{{\\ssb\\char120}}^\\alpha_+$ is equal to $\\mbox{{\\ssb\\char120}}^\\alpha$ for $\\mbox{{\\ssb\\char120}}\\ge0$ and is\nzero for $\\mbox{{\\ssb\\char120}}<0$, {\\it e.g. } Gelfand and Shilov, [\\putref{GandS}] I,\\S5.5.\n$\\Phi_{2t-1}$ acts as a Green function satisfying,\n  $$\n  {d^{2t-1}\\over d{\\mbox{{\\ssb\\char120}}}^{2t-1}}\\,\\Phi_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})=\\delta(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,.\n  $$\nThe formal, distributional equivalent is obtained from the convolution, valid\nfor all $\\alpha$ and $\\beta$,\n  $$\n  \\Phi_\\alpha*\\Phi_\\beta=\\Phi_{\\alpha+\\beta}\\,,\n \\eql{conv}$$\nby setting $\\alpha=-\\beta=2t-1$ and noting\n  $$\n\\Phi_{-k}(\\mbox{{\\ssb\\char120}})=\\delta^{(k)}(\\mbox{{\\ssb\\char120}})\\,,\\quad k=0,1,2,\\ldots.\n \\eql{deriv}$$\n\nAn important formula is the Fourier transform of $\\Phi$, [\\putref{GandS}]\nI.p.360,\n  $$\n  \\int_0^\\infty d\\mbox{{\\ssb\\char120}}\\, \\Phi_\\alpha(\\mbox{{\\ssb\\char120}})\\,e^{i\\sigma \\mbox{{\\ssb\\char120}}}\n  ={1\\over (\\sigma+i0)^\\alpha}\\,e^{i\\pi\\alpha\/2}\\,,\n  \\eql{ftphi}$$\nand the inverse, which is a (continuous) eigenfunction expansion,\n  $$\n  \\Phi_\\alpha(\\mbox{{\\ssb\\char120}})=e^{i\\pi\\alpha\/2}{1\\over2\\pi}\\int_{-\\infty}^\\infty d\\sigma\\,\n  {e^{-i\\sigma \\mbox{{\\ssb\\char120}}}\\over(\\sigma+i0)^\\alpha}\\,.\n  \\eql{invft}$$\n\nIn the present case, the source $\\rho$ is assumed to be periodic  under\n$\\mbox{{\\ssb\\char120}}\\to\\mbox{{\\ssb\\char120}}-i$ (the Planck term is not removed) and to converge as\n$\\mbox{{\\ssb\\char120}}\\to\\infty$. So one has the `Fourier' series,\n  $$\n  \\rho(\\mbox{{\\ssb\\char120}})=\\sum_{m=1}^\\infty \\rho_m\\, e^{-2m\\pi\\mbox{{\\ssb\\char120}}}\\,,\\quad \\mbox{{\\ssb\\char120}}>0\\,.\n  \\eql{ftrho}$$\nSubstitution of this and (\\puteqn{invft}) into (\\puteqn{soln}) gives,\n  $$\n  \\psi(\\mbox{{\\ssb\\char120}})=(-1)^{t+1}{1\\over2\\pi}\\sum_{m=1}^\\infty\\rho_m\n \\int_{-\\infty}^\\infty d\\sigma\\,\n  {e^{i\\sigma\\mbox{{\\ssb\\char120}}}\\over(\\sigma+i0)^{2t-1}(\\sigma-2m\\pi i)}\\,.\n  \\eql{psi1}\n  $$\nClosing the contour in the upper half $\\sigma$--plane I obtain, $\\mbox{{\\ssb\\char120}}>0$,\n  $$\n  \\psi(\\mbox{{\\ssb\\char120}})={1\\over(2\\pi)^{2t-1}}\\sum_{m=1}^\\infty\n  {\\rho_m\\over m^{2t-1}}\\,e^{-2m\\pi\\mbox{{\\ssb\\char120}}}\\,,\n  \\eql{soln2}$$\nwhich, it is no surprise to see, is just (\\puteqn{fe2}) derived by a much\nlonger route but one on which we encounter some handy information. It is\nseen also that the source periodicity has been transmitted to the solution.\n\nIncluding an $m=0$ term in the Fourier series, (\\puteqn{ftrho}), still gives\nconvergence at infinity, but of course the solution, (\\puteqn{soln2}),\nbreaks down, as has already been mentioned. As with all Green function\napproaches, when this happens the corresponding eigenfunction has to be\nremoved from the source. This can be done formally by redefining the\nGreen function to exclude this function which is, in the present case, a\nconstant zero mode. Altering the contour in the Fourier integral\n(\\puteqn{invft}) gives a modified Green function,\n  $$\n  \\Phi^{\\rm sub}_{2t-1}(\\mbox{{\\ssb\\char120}})\\equiv(-1)^{t}{1\\over2\\pi i}\n  \\int_{-\\infty+i\\delta}^{\\infty+i\\delta} d\\sigma\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{2t-1}}\\,,\\quad 0<\\delta<2\\,,\n  \\eql{phisub}$$\nwhich automatically takes out any constant part of the source. This can\nbe seen by going back to (\\puteqn{psi1}) and replacing the integral by one\nfrom $-\\infty+i\\delta$ to $\\infty+i\\delta$,\n  $$\n  \\psi(\\mbox{{\\ssb\\char120}})=(-1)^{t+1}{1\\over2\\pi}\\sum_{m=1}^\\infty\\rho_m\n \\int_{-\\infty+i\\delta}^{\\infty+i\\delta} d\\sigma\\,\n  {e^{i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{2t-1}(\\sigma-2m\\pi i)}\\,.\n  \\eql{psi2}\n  $$\nFor $ 0<\\delta<2$, if $\\mbox{{\\ssb\\char120}}>0$, closing the contour in the upper half plane\nstill yields (\\puteqn{soln2}). However now, even if the source Fourier\nseries, (\\puteqn{ftrho}), is extended to the constant term, $m=0$, the\nresult is again (\\puteqn{soln2}). This should be denoted by $\\psi^{\\rm\nsub}(\\mbox{{\\ssb\\char120}})$ and one can write\n  $$\n  \\psi^{\\rm sub}(\\mbox{{\\ssb\\char120}})=\n\\int_{0^-}^\\infty d{\\mbox{{\\ssb\\char120}}'}\\,\\Phi^{\\rm\nsub}_{2t-1}(\\mbox{{\\ssb\\char120}}'-\\mbox{{\\ssb\\char120}})\\,\\rho({\\mbox{{\\ssb\\char120}}'})\\,.\n  \\eql{soln3}$$\n\nRetaining the general structure, (\\puteqn{invft}), for any $\\alpha$, one can\ndefine for any $\\alpha$,\n  $$\n  \\Phi^{\\rm sub}_\\alpha(\\mbox{{\\ssb\\char120}})\\equiv e^{i\\pi\\alpha\/2}{1\\over2\\pi}\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\n  {e^{-i\\sigma \\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha}\\,.\n  \\eql{invft2}$$\nThen $\\Phi^{\\rm sub}_\\alpha(\\mbox{{\\ssb\\char120}})$ is still concentrated on the positive\n$\\mbox{{\\ssb\\char120}}$--axis and the convolution (\\puteqn{conv}) also remains valid,\n  $$\n  \\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta =\\Phi^{\\rm sub}_{\\alpha+\\beta}\n  \\eql{conv2}\n  $$\ntogether with (\\puteqn{deriv}),\n  $$\n  \\Phi^{\\rm sub}_{-k}=\\delta^{(k)}\\,.\n  \\eql{deriv2}\n  $$\n\nTo check these statements, firstly, if $\\mbox{{\\ssb\\char120}}<0$, closing the contour in\n(\\puteqn{invft2}) in the upper half plane yields zero as the only\nsingularity is at $\\sigma=0$. Then\n  $$\\eqalign{\n  \\big(\\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta\\big) (\\mbox{{\\ssb\\char120}})=\n {e^{i\\pi(\\alpha+\\beta)\/2}\\over(2\\pi)^2}\\int_{-\\infty}^\\infty d\\mbox{{\\ssb\\char120}}'\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma'\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}-i\\mbox{{\\ssb\\char120}}'(\\sigma'-\\sigma)}\\over\\sigma^\\alpha\\sigma'^\\beta}\n  }\n  $$\nwhere the lower limit on the $\\mbox{{\\ssb\\char120}}'$ integral is allowed because\n$\\Phi_\\beta^{\\rm sub}(\\mbox{{\\ssb\\char120}}')$ is concentrated on the positive $\\mbox{{\\ssb\\char120}}'$ axis.\n\nSetting $\\sigma=s+i\\delta$, $\\sigma'=s'+i\\delta$, I get, in detail,\n  $$\\eqalign{\n \\big(\\Phi^{\\rm sub}_\\alpha* \\Phi^{\\rm sub}_\\beta\\big) (\\mbox{{\\ssb\\char120}})&= e^{i\\pi(\\alpha+\\beta)\/2}{1\\over(2\\pi)^2}\\int_{-\\infty}^\\infty d\\mbox{{\\ssb\\char120}}'\n\\int_{-\\infty}^{\\infty}ds\\,\\int_{-\\infty}^{\\infty}ds'\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}-i\\mbox{{\\ssb\\char120}}'(s'-s)}\\over\\sigma^\\alpha\\sigma'^\\beta}\\cr\n  &= e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty}^{\\infty}ds\\,\\int_{-\\infty}^{\\infty}ds'\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha\\sigma'^\\beta}\\,\\delta(s-s')\\cr\n   &=e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty}^{\\infty}ds\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^\\alpha\\sigma^\\beta}=e^{i\\pi(\\alpha+\\beta)\/2}{1\\over2\\pi}\n\\int_{-\\infty+i\\delta}^{\\infty+i\\delta}d\\sigma\\,\n  {e^{-i\\sigma\\mbox{{\\ssb\\char120}}}\\over\\sigma^{\\alpha+\\beta}}\\cr\n  &=\\Phi^{\\rm sub}_{\\alpha+\\beta}(\\mbox{{\\ssb\\char120}})\\,.\n  }\n  $$\n\n  Equation (\\puteqn{deriv2}) follows either from (\\puteqn{conv2}) or\n  (\\puteqn{invft2}) since the integrand has no singularity, and so one can\n  set $\\delta=0$.\n\n\\subsection{\\it Eichler cohomology.}\nAs a side remark, the general solution to (\\puteqn{rde}) consists of the\nparticular integral, (\\puteqn{soln}), plus a solution of the homogeneous\nequation, {\\it i.e. } a zero mode, which in this case is a polynomial,\n$\\Theta(\\mbox{{\\ssb\\char120}})$, of degree at most $2t-2$. The contribution of this to the\ncocycle polynomial, $P$ of (\\puteqn{co2}), constitutes an element of the set\nof {\\it coboundaries} and a cohomology of polynomials can be set up\n(Eichler cohomology). Working with the cohomology classes removes the\nconstants of integration ambiguity in the solution of (\\puteqn{rde}). That\nis, there is a one-to-one correspondence between $\\rho$ and an element of\nthe cohomology group denoted $H^1(\\Gamma,\\Pi_r)$, where $\\Pi_r$ is the\nvector space of polynomials of degree $\\le r=2t-2$ (for cusp forms) and\n$\\Gamma$ is (here) the modular group. The development of these ideas, while\nsignificant generally, is not required here.\n\n\n\\section{\\bf 6. Epstein approach to thermal quantities.}\n\nAn `earlier' form of the free energy, or effective action, follows from\nthe thermal $\\zeta$--function, which, in this case is related to the Epstein $\\zeta$--function.\nStarting from this I will be able to make contact with the Eisenstein\nformulation and derive useful relations. Using the Epstein function is\nequivalent to the non-holomorphic Eisenstein series. As a rule the\nanalysis is harder. In reality one is doing twice the necessary work.\n\nIn some ways we are going backwards. The use of holomorphic forms, {\\it e.g. }\n(\\puteqn{inten5}), should be considered a simplification available through\nworking with an explicit square root of the Laplacian.\n\nAgain splitting up the degeneracy, one is led to define a `partial'\n$\\zeta$--function,\n  $$\n  \\zeta_t(s,\\beta)={i\\over2\\beta}\\sumdasht{{m,n=\\atop-\\infty}}\n{\\infty}{n^{2t-2}\\over\n \\big(4\\pi^2m^2\/\\beta^2+n^2\/a^2\\big)^s}\\,,\n  \\eql{epthzf}$$\nand the related free energy,\n  $$\n  \\overline F_t=-{\\xi\\over8\\pi}\\lim_{s\\to0}{1\\over s}\\,\n  \\sumdasht{{m,n=\\atop-\\infty}}{\\infty}{n^{2t-2}\\over\n \\big(m^2+n^2\\xi^{-2}\\big)^s}\\,.\n  \\eql{eff}$$\n\nI have rescaled the free energy by defining, as above and as in\n[\\putref{DandK1}], $\\overline F_t=a F_t$ and have also taken a preliminary\nlimit of $s\\to0$ in an extracted factor of $(2\\pi\/\\beta)^{2s}$. This will\nbe justified in a moment.\n  $$\n  \\overline F_t=(-1)^{t-1}{\\xi\\over8\\pi}\\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\n  \\lim_{s\\to0}{\\Gamma(s-t+1)\\over\\Gamma( s+1)}\\,\n  \\sumdasht{{m,n=\\atop-\\infty}}{\\infty}{1\\over\n \\big(m^2+n^2\\xi^{-2}\\big)^{s-t+1}}\\,.\n  \\eql{eff2}$$\n\nNot surprisingly, the degeneracy factor is replaced by a derivative and the\nresulting sum is precisely an Epstein $\\zeta$--function. This has been used before {\\it e.g. }\n[\\putref{Kennedy}]. The point is that there is no pole at $s=0$, which\njustifies the limit. In terms of the Epstein function, $Z_2$,\n  $$\n  \\overline F_t=(-1)^{t-1}{\\xi\\over8\\pi}\\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\n  \\lim_{s\\to0}\\,{\\Gamma(s-t+1)\\over\\Gamma( s+1)}\\,Z_2(2s-2t+2,A)\\,,\n  \\eql{eff3}$$\nwhere $A$ is the matrix (the `modulus'),\n  $$\n  A=\\left(\\matrix{1&0\\cr\n             0&\\xi^{-2}}\\right)\\,.\n  $$\nThe idea now is to use the standard functional relation satisfied by\n$Z_2(s,A)$, repeated here for convenience,\n  $$\n  Z_2(2s,A)={\\pi^{2s-1}\\over\\sqrt{{\\rm det\\,} A}}{\\Gamma(1-s)\\over\\Gamma(s)}\n  \\,Z_2(2-2s,A^{-1})\\,,\n  \\eql{freln}$$\nto give\n  $$\n  \\overline F_t={\\xi\\over8\\pi}\\bigg(-{d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\\xi\n  \\lim_{s\\to0}\\pi^{2s-2t+1}\\,{\\Gamma(t-s)\\over\\Gamma( s+1)}\\,Z_2(2t-2s,A^{-1})\\,.\n  $$\n$Z_2(2s,A)$ has a pole at $s=1$ only, so working at $t>1$ to begin with,\none has\n  $$\n  \\overline F_t=\\xi{(-1)^{t-1}\\Gamma(t)\\over8\\pi^{2t}}\n  \\bigg({d\\over d(1\/\\xi^2)}\\bigg)^{t-1}\\xi\n  \\,\\,Z_2(2t,A^{-1})\\,.\n  \\eql{epder}$$\nAs the simplest case, set $t=2$ giving the three-sphere. Then\n   $$\n  \\overline F_3={\\xi\\over8\\pi^4}{d\\over d(1\/\\xi^2)}\\,\\xi\\,Z_2(4,A^{-1})=\n  -{\\xi^4\\over16\\pi^4}{d\\over d\\xi}\\,\\xi\\,Z_2(4,A^{-1})\\,,\n  \\eql{3free1}$$\nagreeing with the result in [\\putref{Kennedy}], allowing for the changes in\nnotation.\n\nThis formula can be taken further as shown by Epstein [\\putref{Epstein}]\np.633 who gives,\n  $$\n  Z_2(4, A^{-1})={\\pi^4\\over45}+\\pi\\xi^{-3}\\zeta_R(3)+2\\pi\\xi^{-3}\\chi(q')\n  +4\\pi^2\\xi^{-2}\\,D'\\,\\chi(q')\\,,\n  \\eql{epz2}$$\nwhere $q'=e^{-\\pi\\xi}$ and\n  $$\n  \\chi(q')=S_2(-1\/\\tau)=\\sum_{m=1}^\\infty {q'^{2m}\\over m^3(1-q'^{2m})}\\,.\n  $$\n\nEquation (\\puteqn{3free1}) with (\\puteqn{epz2}) is appropriate for the high\ntemperature limit since $q'$ is the `inverse' temperature Boltzmann factor.\nThe low temperature form follows by simple homogeneous scaling of the\nEpstein $\\zeta$--function, which gives,\n  $$\n  \\,Z_2(2s,A^{-1})=\\xi^{-2s}Z_2(2s,A),\n  \\eql{scale}$$\nand therefore the equivalent form,\n  $$\n  Z_2(4, A^{-1})={\\pi^4\\over45}\\xi^{-4}+\\pi\\xi^{-1}\\zeta_R(3)+2\\pi\\xi^{-1}\\chi(q)\n  +4\\pi^2\\xi^{-2}\\,D\\,\\chi(q)\\,,\n  \\eql{epz21}$$\nwhere $q=e^{-\\pi\/\\xi}$.\n\nThe low temperature form (\\puteqn{epz21}) yields a somewhat simpler expression\nfor the free energy, (\\puteqn{3free1}), as the $\\zeta_R(3)$ term goes out and\nthere is some cancellation that produces,\n  $$\n  \\overline F_3={1\\over240}+{\\xi\\over2\\pi}D^2\\,\\chi(q)\\,.\n  $$\nI have thus regained, at some length, the summation form (\\puteqn{ssum})\nwith (\\puteqn{sum9}).\n\nAll that has been done is to rederive, in a special case and in a\ndetailed way, the standard statistical mode sum, the general form of\nwhich was obtained, in this fashion, in [\\putref{DandK,Dow2}] and we are no\nfurther forward in finding a closed form for the free energy, which is\none of my aims.\n\nThe equivalence of (\\puteqn{epz2}) and (\\puteqn{epz21}), derived here rather\ntrivially (granted Epstein's calculation), is a known inversion identity,\nand can also be derived from (\\puteqn{sm1}). The exact identity is written\nout in Katayama [\\putref{Kata}]. His derivation seems to be similar to that\nvia the Epstein function. Clearly the higher odd sphere expressions will\ninvolve the Riemann $\\zeta$--function\\ at positive odd integers. Katayama writes out\nthe one appropriate for the (partial) five-sphere.\n\nSmart, [\\putref{Smart}], also derives these identities, which is not\nsurprising since his work is concerned with the evaluation of the Epstein\n$\\zeta$--function\\ at integer arguments, through which he is led to the forms\n$\\phi_{2k}$. We see that Epstein had already arrived at the same\ndevelopment. Consult also Bodendiek and Halbritter, [\\putref{BandH}], for a\nrelated treatment.\n\n\\section{\\bf 7. The Epstein--Kober--Selberg--Chowla formula.}\nSmart makes use of the, oft quoted, paper of Selberg and Chowla,\n[\\putref{SandC}], which is concerned, partly, with the evaluation of the\nEpstein function via the Kronecker limit formula \\footnote{Landau,\n[\\putref{Landau2}], gives some interesting history of this famous formula.}\nand elegantly yields some of the finite forms for the complete elliptic\nintegral used in [\\putref{DandK1}]. The important expression for the\nEpstein function quoted by Selberg and Chowla, [\\putref{SandC2}], and\nderived by them in [\\putref{SandC}], (equ.(6)), in terms of $K$--Bessel\nfunctions, is essentially the same as that already given by Epstein,\n[\\putref{Epstein}], p.631, equn.(12), p.622, equn.(16). Rankin\n[\\putref{Rankin3}] obtained the same result and it also occurs in Bateman\nand Grosswald, [\\putref{BandG}]. These are the standard references. See\nalso Terras, [\\putref{Terras}] p.209. Smart, [\\putref{Smart}], employs this\nformula to split the Epstein function into (simpler) holomorphic and\nanti--holomorphic parts using the explicit form of the Bessel function,\n$K_{t-1\/2}$, which henceforth disappears from his analysis. Other\nexpressions were given by Deuring [\\putref{Deuring}] and Mordell,\n[\\putref{Mordell}].\n\nAn early, and little known, derivation is that by Kober, [\\putref{Kober}].\nFollowing, and generalising, Watson, [\\putref{Watson}], he obtains, {\\it\nstarting} from the Bessel function end, the formula (I retain his\nnotation),\n  $$\\eqalign{\n  {1\\over\\sqrt u}\\,&\\Delta^{(2w+1)\/4}{\\Gamma(w+1\/2)\\over8\\pi^{w+1\/2}}Z_2(a,b,c;w+1\/2)\\cr\n&={u^{-w}\\over4}{\\Gamma(w)\\over\\pi^{w}}\\,\\zeta_R(2w)\n+{u^{w}\\over4}{\\Gamma(w+1\/2)\\over\\pi^{w+1\/2}}\\,\\zeta_R(2w+1)+\\,\\cr\n  &\\hspace{************}\n  \\sum_{n=1}^\\infty \\sigma_{2w}(n)n^{-w}\\cos(2\\pi vn)K_{w}(2\\pi un)\\,,}\n  \\eql{kob1}$$\nwhere the quadratic form in the Epstein function, $Z_2$, is\n$am_1^2+2bm_1m_2+cm_2^2$ and\n  $$\n  \\Delta=ac-b^2\\equiv a^2\\,u^2\\,,\\quad v\\equiv{b\\over a}\\,.\n  $$\nEquation (\\puteqn{kob1}) is identical to the corresponding one in\n[\\putref{Epstein}] and [\\putref{SandC}] and therefore, with justice, should\nbe called the Epstein--Kober formula. It is the {\\it Fourier expansion}\nof the Epstein function in $v$. For the sum of squares case, $v=0$, the\nresult is rederived by Guinand [\\putref{Guinand2}] in the same way.\n\nKober also discusses behaviour under `reciprocal' transformations,\n$u\\to1\/u$. In particular the diagonal case, $v=0$, when one can write\n$u=\\mbox{{\\ssb\\char120}}$.\n\nIn this way of doing things, the Bessel function can be eliminated using\nthe standard formula\n  $$\n  \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^{s+w}}=\\zeta_R(s+w)\\,\\zeta_R(s-w)\\,,\n  \\eql{sig}$$\nand the Mellin transform (Heaviside's integral),\n  $$\n  {1\\over4}\\pi^{-s}\\Gamma\\big({s+w\\over2}\\big)\\,\\Gamma\\big({s-w\\over2}\\big)=\n  \\int_0^\\infty dy \\,y^{s-1}\\,K_w(2\\pi y)\\,,\\quad {\\rm Re\\,} s>1+|{\\rm Re\\,} w|\\,,\n  \\eql{hi}$$\nso that,\n  $$\n  \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w} K_{w}(2\\pi un)\n=\n{1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds\\,u^{-s}\\xi(s+w)\\,\\xi(s-w)\\,,\n  \\eql{epmell}$$\nwith $c>1+w$ and where,\n  $$\\xi(2w)\\equiv{1\\over2}\\pi^{-w}\\,\\Gamma(w)\\,\\zeta_R(2w)\\,.\n  $$\nThis is the same as Deuring's expression, [\\putref{Deuring}]\np.589.\n\nThe integrand possesses a functional equation which follows most easily\nfrom that for the Riemann $\\zeta$--function, $\\xi(2w)=\\xi(1-2w)$, and is,\n  $$\n  \\xi(s+w)\\,\\xi(s-w)\\equiv f_w(s)=f_w(1-s)\\,.\n  \\eql{ffrel}$$\n\nUsing (\\puteqn{ffrel}) one could {\\it derive} the functional equation for the\nEpstein $\\zeta$--function\\ in (\\puteqn{kob1}). The usual derivation employs the inversion\nrelation for generalised $\\theta$--functions, [\\putref{Epstein,Krazer}].\n\nFurthermore, it is possible to reverse the argument and use (\\puteqn{epmell})\nto obtain (\\puteqn{kob1}) (for $v=0$). The diagonal Epstein function is\nparticularly easy to deal with being the $\\zeta$--function\\ on the torus,\nS$^1\\times$S$^1$. It can be expressed very simply in terms of the $\\zeta$--functions\\ on\nthe circle factors (these are Riemann $\\zeta$--functions\\ ) and the result is exactly\n(\\puteqn{kob1}) with (\\puteqn{epmell}) {\\it without} the intervention of the\nBessel function. The individual Riemann $\\zeta$--functions\\ in (\\puteqn{kob1}) arise from\nadjusting the zero modes. All this is standard and can easily be\ngeneralised to higher dimensions.\n\n\nA natural step is to proceed as with the holomorphic modular forms and ask\nfor the corresponding period polynomials which will follow directly from\n(\\puteqn{epmell}) and (\\puteqn{ffrel}) in the standard manner.\n\nThere are four poles in the relevant strip, $-c\\le s\\le c$, and\na straightforward contour calculation produces the result,\n  $$\\eqalign{\n  \\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w}K_{w}(2\\pi n u)-&\n  {1\\over u}\\sum_{n=1}^\\infty {\\sigma_{2w}(n)\\over n^w}K_{w}(2\\pi n\/u)\\cr\n  &={1\\over2}\\xi(2w)(u^{w-1}-u^{-w})+{1\\over2}\\xi(-2w)(u^{-w-1}-u^{w})\\,,}\n  \\eql{ppk}$$\nThe right hand side of (\\puteqn{ppk}) might be termed a period function.\n\nThis formula was obtained by Guinand, [\\putref{Guinand2}], but using the\ncomplete equation (\\puteqn{kob1}), with $v=0$, and properties of the Epstein\nfunction. This is unnecessary as I have just shown.\n\nIn order to make a connection with the period polynomials encountered\nearlier,{\\it e.g. } (\\puteqn{are1}), one introduces the formula for the $K$-Bessel\nfunction, written as,\n  $$\n  K_{t-1\/2}(x)=\\sqrt{2\\over \\pi}\\,x^{t-1\/2}\\,(-1)^{t-1}\\,\\bigg({1\\over x}\n{d\\over dx}\\bigg)^{t-1}{e^{-x}\\over x}\\,,\\quad t\\in\\mbox{\\open\\char90}\\,.\n  \\eql{kbess}$$\nThen, somewhat similarly to [\\putref{Smart}],\n  $$\\eqalign{\n  \\sum_{n=1}^\\infty {\\sigma_{2t-1}(n)\\over n^{t-1\/2}}K_{t-1\/2}(2\\pi n u)&=\n  \\sqrt{2\\over\\pi}{(-1)^{t-1}\\over(2\\pi)^{t-1\/2}}\\bigg({1\\over u}\n  {d\\over du}\\bigg)^{t-1}\n  \\sum_{n=1}^\\infty{\\sigma_{2t-1}(n)\\over n^{2t-1}}\\,q^{2n}\\cr\n  &=\\sqrt{2\\over\\pi}{(-1)^{t-1}\\over(2\\pi)^{t-1\/2}}\n  \\bigg({1\\over u}{d\\over du}\\bigg)^{t-1}\n  S_t(iu)\\,,}\n  $$\nso that, for the left--hand side of (\\puteqn{ppk}),  one finds,\n  $$\n  (-1)^{t-1}{1\\over{\\pi}^t}\\bigg(\\bigg({d\\over du^2}\\bigg)^{t-1}\n  S_t(iu)-{1\\over u}\\bigg({d\\over d(1\/u^2)}\\bigg)^{t-1}S_t(i\/u)\\bigg)\\,.\n \\eql{krelp} $$\nIt is not obvious from (\\puteqn{are1}) that this reduces so nicely to\n(\\puteqn{ppk}), and I leave this as a question mark.\n\nFurther expressions involving Bessel functions appear in the Appendix.\n\\begin{ignore}\n\nThis can be turned around in the following way. Write (\\puteqn{kbess})\nexplicitly in the form,\\marginnote{THis is wrong}\n  $$\n e^{-x}\\sum_{j=0}^{t-1}\n  {(t-1+j)!\\over j!(t-1-j)!}\\,(2x)^{-j-1\/2}=\\pi^{-1\/2}\\,K_{t-1\/2}(x)\\,,\n  $$\nand solve this for $(2x)^{-j-1\/2}$, $0\\le j\\le t-1$,\n  $$\n  e^{-x}\\,(2x)^{-j-1\/2}={1\\over\\sqrt\\pi}\\sum_{t=0}^{j-1}{()!\\over()!}\n  \\,K_{t-1\/2}(x)\\,.\n  $$\nNow set $x=2\\pi n u$ and get\n  $$\n  {e^{-2\\pi nu}\\over n^{j+1\/2}}=(2\\pi)^{j+1\/2}\n  {1\\over\\sqrt\\pi}\\sum_{t=0}^{j-1}{()!\\over()!}\\,K_{t-1\/2}(2\\pi nu)\n  $$\n\\end{ignore}\n\\begin{ignore}\n\\section{\\bf More hyperbolic sums.}\nIn terms of the hyperbolic expression, (\\puteqn{speen}), the quantity\ncorresponding to the $c_t$ is,\n  $$\n  {\\partial\\over\\partial\\beta}\\,E(\\beta)=-{1\\over2^{d+1}}\\sum_{m=1}^\\infty m\\bigg({(d-1)^2\n\\cosh(m\\beta\/2)\\over\\sinh^{d-1}(m\\beta\/2)}+{d(d+1)\\cosh(m\\beta\/2)\n\\over\\sinh^{d+1}(m\\beta\/2)}\\bigg)\\,,\n  \\eql{speen2}$$\nwhich means that it is also possible to evaluate the sums,\n  $$\n  \\sum_{m=1}^\\infty {m^{2p+1}\\cosh(mx)\\over\\sinh^d(mx)}\n  $$\nand\n  $$\n  \\sum_{m=1}^\\infty {m^{2p}\\over\\sinh^d(mx)}\\,,\n  $$\nat special values of $x$. Cases with $d=1$ have been considered by Zucker,\n[\\putref{zucker2}] and Ling, [\\putref{Ling2}] and can follow from expansion of\nthe dn function.\n\\section{\\bf The fermion case. More to be done.}\n\nIn [\\putref{DandK1}] it was shown that just as for bosons, the fermion\ninternal energies on all spheres can be expressed in terms of just two\nquantities which could be taken to be the first two partial energies,\n$\\eta_1$ and $\\eta_2$. This was proved without recourse to the boson result\nand in a different manner. The recursion is not similar to (\\puteqn{recurs})\nbut has a cubic form. One might therefore reasonably expect an equation\ncorresponding to (\\puteqn{shr}) to exist, but in a different form and derived\nby a different route, possibly from the Jacobi function ds $u$.\n\nA preliminary fact, that it is useful to confirm, starts from the\nexpression for the partial spinor energy on the circle given in\n[\\putref{DandK1}],\n  $$\n  \\eta_1(\\beta)=-{1\\over24} \\mbox{{\\goth\\char75}}^2\\,(1-2k^2)\\,,\n  $$\nwhich, using (\\puteqn{diffrel}), can be integrated to give,\n  $$\n   \\int d\\beta\\, \\eta_1(\\beta)={1\\over12}\\int d\\kappa {1-2\\kappa\\over\\kappa(1-\\kappa)}=\n  {1\\over12}\\log(\\kappa\\ka'\/16)\\,,\n  $$\nwhere $\\kappa=k^2$ and a constant of integration has been chosen to make the\nfree energy and internal energies agree at absolute zero.\n\nI have, in this roundabout way, derived another of Jacobi's formulae,\n  $$\n  q^{1\/24}\\prod_{j=0}^\\infty(1+q^{2j+1})=2^{1\/6}(kk')^{-1\/12}\\,,\n  $$\nthe logarithm of which gives the mode sum form of the circle fermion\nfree energy, as is well known in various contexts.\n\nWe can also see the difficulty if we attempt the same approach for $\\eta_2$\n  $$\\eqalign{\n   -\\int d\\beta\\,\\eta_2(\\beta)&={1\\over120}\\int d\\kappa\\,\\mbox{{\\goth\\char75}}^2(\\kappa)\\,\n{(7+8\\kappa-8\\kappa^2)\\over\\kappa(1-\\kappa)}\\cr\n&={1\\over120}\\int d\\kappa\\,\\mbox{{\\goth\\char75}}^2(\\kappa)\\,\\bigg({7\\over\\kappa}\n+{7\\over\\kappa'}-8\\bigg)\\,,}\n  $$\nand the integral is not obvious.  I merely note the hypergeometric\nrepresentation,\n  $$\n  \\mbox{{\\goth\\char75}}(\\kappa)=F\\big({1\\over2},{1\\over2},1; \\kappa\\big)={1\\over\\pi\\kappa^{1\/4}}\\,\n  Q_{-1\/2}\\big((\\kappa+1)\/2\\sqrt\\kappa\\big)\\,.\n  $$\nA similar thing holds in the boson case of course.\n\\end{ignore}\n\\section{\\bf 8. Conclusion}\n\nIt does not seem possible to ellipticise integrals of the internal\nenergy, in particular the free energy (except for the circle), because\nthe modular properties are more complicated possessing nonzero cocycle\nfunctions. These are related to a known Lambert series connected with the\nEisenstein series and Zagier's expression for the period functions of\nthese is more neatly re--computed by a contour method. The use of the\nfully subtracted series $\\epsilon_t^{\\rm sub}$ allows a more compact\ntreatment.\n\nSufficient historical material has been exhibited to indicate that the\nSelberg--Chowla formula should be renamed the Epstein--Kober formula.\n\\newpage\n\\section{\\bf Appendix, Dirichlet Series.}\n\nIt is possible to give a fairly broad context to the inversion behaviour\n({\\it e.g. } modular properties), without too much restriction, in terms of\nDirichlet series. Although these ideas, by now classic, are associated\nwith Hecke, and have been expounded by Ogg [\\putref{Ogg,Ogg2}], for\nexample, the simplest case occurs earlier in other places. I mention\nKoshliakov, [\\putref{Kosh}], whose discussion I now paraphrase both for\nhistorical justice and interest.\n\nDefine two ordinary Dirichlet series (`zeta functions'),\n  $$\n  \\phi(s)=\\sum_{n=1}^\\infty {a_n\\over n^s}\\,,\\,\\,({\\rm Re\\,} s>\\nu_a)\\,,\\quad\n  \\psi(s)=\\sum_{n=1}^\\infty {b_n\\over n^s}\\,,\\,\\,({\\rm Re\\,} s>\\nu_b>\\nu_a)\\,,\n  $$\nfor some $\\nu$'s depending on the asymptotics of $a_n$ and $b_n$, and\n{\\it assume} that the $a_n$ and $b_n$ are such that the corresponding\n`cylinder kernels' satisfy the `formula of transformation',\n  $$\n  \\sum_{n=0}^\\infty a_n\\,e^{-nb\\rho}={a\\over\\rho^\\nu}\n  \\sum_{n=0}^\\infty b_n\\,e^{-nb\/\\rho}\\,,\\quad \\nu=\\nu_a\\,,\\quad \\rho>0\\,,\n  \\eql{trans2}$$\nfor fixed $a$ and $b$, where the numbers (not necessarily integers)\nof `zero modes' are taken to be,\n  $$\n  a_0=-\\phi(0)\\,\\quad b_0=-\\psi(0)\\,.\n  $$\nThen Koshliakov, [\\putref{Kosh}], shows the equivalence of (\\puteqn{trans2})\nwith\n  $$\n  a\\,{\\Gamma(\\nu-s)\\,\\psi(\\nu-s)\\over b^{\\nu-s}}={\\Gamma(s)\\,\\phi(s)\\over b^s}\\,,\n  \\eql{frel}$$\nby brute force using the summation function, $\\sigma(z)$, given by the\nBessel function expression,\n  $$\n  \\sigma(z)=-2ab\\,z^{(\\nu-1)\/2}\\,\\sum_{n=1}^\\infty {b_n\\over n^{(\\nu-1)\/2}}\\,\n  K_{\\nu-1}\\big(2b\\sqrt{nz}\\big)\\,,\n  $$\nand Heaviside's formula, (\\puteqn{hi}). $\\sigma(z)$ has the important property\nof possessing poles at $z=1,2,\\ldots$ with residues $a_1,a_2,\\ldots$ and\nalso has a cut along the negative $x$--axis with discontinuity\n$ab^\\nu2\\pi i\\psi(0)(-x)^{\\nu-1}\/\\Gamma(\\nu)$, $z=x+iy$.\n\nKoshliakov shows that $\\phi(s)$ is a single--valued holomorphic function\nhaving a first-order pole at $s=\\nu$,\n  $$\n  \\phi(s)\\sim-\\psi(0){ab^\\nu\\over\\Gamma(\\nu)}{1\\over s-\\nu}\\,,\n  \\eql{phipole}$$\nand it is then a theorem that (\\puteqn{frel}) together with (\\puteqn{phipole}) is\nequivalent to (\\puteqn{trans2}), as can also be established easily, and\ninstructively, by Mellin transform. It follows, symmetrically, that\n$\\psi(s)$ is a single--valued holomorphic function having a first-order\npole at $s=\\nu$,\n  $$\n  \\psi(s)\\sim-\\phi(0){b^\\nu\\over a\\Gamma(\\nu)}{1\\over s-\\nu}\\,.\n  $$\n\nKoshliakov also gives an integral form of $\\phi(s)$,\n  $$\n  \\phi(s)=-{\\sin\\pi s\\over\\pi}\\int_0^\\infty dx\\, {\\sigma(x)\\over x^s}\n  \\,,\\quad {\\rm Re\\,} s<0\\,,\n  \\eql{intphi}$$\nshowing that $\\phi(s)$ vanishes at negative integers in agreement with\n(\\puteqn{frel}).\n\nHecke, [\\putref{Hecke}], derives the same equivalence, but only for\n$a_n=b_n$, and eight years later. The more general case can also be found\nin the standard references Ogg [\\putref{Ogg,Ogg2}] and Weil,\n[\\putref{Weil2}], and no doubt elsewhere. A related formula involving\nRamanujan's `reciprocal functions' occurs in Hardy and Littlewood,\n[\\putref{HandL}].\n\nDefining cylinder--kernels, or theta--series, including the zero modes,\nby,\n  $$\n  \\Phi(\\beta)=\\sum_{n=0}^\\infty a_n\\,e^{-n\\beta}\\,,\\quad\n  \\Psi(\\beta)=\\sum_{n=0}^\\infty b_n\\,e^{-n\\beta}\\,,\n  $$\nthe reciprocal relation, (\\puteqn{trans2}), reads,\n  $$\n  \\Phi(b\\rho)={a\\over\\rho^\\nu}\\,\\Psi(b\/\\rho)\\,,\n  \\eql{recip}\n  $$\n(continued from the regions of convergence in $s$) and one has, in\nstandard fashion ({\\it cf } (\\puteqn{ell})),\n  $$\n  \\phi(s)={1\\over\\Gamma(s)}\\int_0^\\infty d\\beta\\,\\beta^{s-1}\\,\\big(\\Phi(\\beta)-a_0\\big)\\,,\n  \\quad\n  \\psi(s)={1\\over\\Gamma(s)}\\int_0^\\infty d\\beta\\,\\beta^{s-1}\n  \\,\\big(\\Psi(\\beta)-b_0\\big)\\,.\n  $$\n\nAs an example, the Dirichlet series corresponding to the classical\nEisenstein series is, from (\\puteqn{mell1}) or (\\puteqn{sig}),\n  $$\n  \\sum_{m=1}^\\infty \\sigma_{2t-1}(m)\\,m^{-s}=\\zeta_R(s)\\,\\zeta_R(s+1-2t)\\,,\n  $$\nand this is a case discussed by Koshliakov, [\\putref{Kosh}] p.19, with\n$a_n=b_n=\\sigma_{2t-1}(n), \\nu=2t,a=(-1)^t,b=2\\pi,\\phi(0)=\\psi(0)=-B_{2t}\/4t$.\n\nKoshliakov also considers Jacobi elliptic cases that yield reciprocal\nidentities covered earlier by Glaisher [\\putref{Glaisher}].\n\n\\subsection{\\it Generalised Dirichlet series.}\nI have described above the classic Dirichlet set up. A certain, and\nsometimes only apparent, generalisation is obtained by replacing $n$ by\narbitrary sequences, $\\lambda_m$ and $\\mu_n$, and defining new $\\phi$ and\n$\\psi$ by\n  $$\n  \\phi(s)=\\sum_{m=1}^\\infty {a_m\\over \\lambda_m^s}\\,,\\quad\\quad\n  \\psi(s)=\\sum_{n=1}^\\infty {b_n\\over \\mu_n^s}\\,,\n  \\eql{newdir}$$\nwhich satisfy, {\\it by definition}, the functional equation\n(one of many possible),\n  $$\n  \\Gamma(\\delta-s)\\,\\psi(\\delta-s)=\\Gamma(s)\\,\\phi(s)\\,,\\quad \\delta\\in\\mbox{\\open\\char82}\\,,\n  \\eql{frel2}$$\nwith corresponding heat--kernels,\n  $$\n  \\Phi(\\beta)=\\sum_{m=1}^\\infty a_m\\,e^{-\\lambda_m\\beta}\\,,\\quad\n  \\Psi(\\beta)=\\sum_{n=1}^\\infty b_n\\,e^{-\\mu_n\\beta}\\,.\n  \\eql{hkz}$$\n\nBochner, [\\putref{Bochner}], (see also Chandrasekharan and Narasimhan,\n[\\putref{CandN}], Knopp, [\\putref{Knopp2}] ), derives, by Mellin transforms,\nthe `modular relation',\n  $$\n  \\Phi(\\beta)-\\beta^{-\\delta}\\,\\Psi\\big({1\\over\\beta}\\big)=B(\\beta)\\,,\n  \\eql{modrel}$$\nwhere the `residual function' $B(\\beta)$ is\n  $$\n  B(\\beta)={1\\over2\\pi i}\\int_{C({\\cal S})} ds\\,\\chi(s)\\,\\beta^{-s}=\\sum_{s\\in{\\cal S}}\\beta^{-s}\n  {\\rm Res}\\,\\chi(s)\\,.\n  $$\n$\\chi(s)$ is the joint continuation of the two sides of (\\puteqn{frel2}) and\nis assumed to have only simple poles for singularities confined to a\ncompact region, ${\\cal S}$, of the $s$--plane, usually on the real axis.\n$C({\\cal S})$ is a loop surrounding ${\\cal S}$. I especially note that if there\nexist higher--order poles then integer powers of $\\log \\beta$ occur.\n\nIn certain situations, {\\it e.g. } Koshliakov's, $B(\\beta)$ can be naturally\namalgamated with\nthe left--hand side to give an exact modular relation like (\\puteqn{trans2}),\n{\\it e.g. } Bochner,\n[\\putref{Bochner}] p.342. Typically one would include zero modes $\\lambda_0=0$\nand $\\mu_0=0$ in (\\puteqn{hkz}), but not in (\\puteqn{newdir}). (There seems to be\na misprint on p.342 of [\\putref{Bochner}] which has $\\lambda_0=\\mu_0=1$.)\n\nThere is another formula connected with this analysis that is useful. It\nis concerned with the representation of the {\\it modified} series,\n  $$\n  \\phi(s,w)=\\sum_{n=1}^\\infty {a_n\\over (\\lambda_n+w^2)^s}\\,\\quad{\\rm Re\\,} w>0.\n  \\eql{asser}$$\n\nThere are many reasons why we should wish to add the number $w^2$. For\nexample it might correspond to a mass or simply be added for extra\nflexibility as when calculating heat--kernel expansion coefficients via\nresolvents or it might be another part of the denominator as in Epstein's\nderivation of (\\puteqn{kob1}); see [\\putref{DandKi}].\n\nThe representation alluded to is, (Berndt [\\putref{Berndt4}]),\n  $$\n  \\Gamma(s)\\,\\phi(s,w)=R(s,w)+2\\sum_{n=1}^\\infty b_n\\,\\bigg({\\mu_n\\over w^2}\n  \\bigg)^{(s-\\delta)\/2}\\,K_{s-\\delta}\\big(2w\\sqrt\\mu_n\\big)\\,,\n  \\eql{berndt}$$\nwhere $s$ is such that the summation converges absolutely and $R(s,w)$ is\ngiven by\n  $$\n  R(s,w)=\\sum_{s'\\in\\,{\\cal S}} \\Gamma(s-s')\\,w^{2s'-2s}{\\rm Res}\\,\\, \\chi(s')\\,.\n  \\eql{resi}$$\n\nBerndt's second proof of (\\puteqn{berndt}) proceeds via the\ncylinder--kernels, $\\Phi$ and $\\Psi$. I prefer his first, Mellin, proof,\n[\\putref{Berndt6}]. Of course, the information employed is the same. I give\nthis proof out of interest.\n\nThe Mellin transform relation between $\\phi(s)$ and the $\\phi(s,w)$ of\n(\\puteqn{asser}), is\n  $$\n  \\Gamma(s)\\,\\phi(s,w)={1\\over2\\pi i}\\int_{c-i\\infty}^{c+i\\infty}ds'\\,w^{2s'-2s}\n  \\Gamma(s-s')\\Gamma(s')\\,\\phi(s')\n  $$\nwhere $c$ is chosen sufficiently positive to make both $\\phi(s)$ and\n$\\psi(s)$ converge. The vertical contour is now moved to the left so as\nto run from $\\delta-c-i\\infty$ to $\\delta-c+i\\infty$ picking up the residues,\n$R(s,w)$, (\\puteqn{resi}), on the way. The region, ${\\cal S}$, lies between\nthese vertical lines and also the horizontal pieces contribute zero.\n\nOn the shifted vertical contour make the coordinate change $s'\\to\\delta-s'$\nand use (\\puteqn{frel2}) {\\it i.e. } $\\chi(\\delta-s')=\\Gamma(s')\\psi(s')$, valid in this\nrange, and so for this part I find,\n  $$\\eqalign{\n  \\sum_n b_n {1\\over2\\pi i}&\\int_{c-i\\infty}^{c+i\\infty}ds'\\,\\Gamma(s'-\\delta+s)\n  \\Gamma(s')\\mu_n^{-s'} w^{2\\delta-2s'-2s}\\cr\n&=2\\sum_{n=1}^\\infty b_n\\,\\bigg({\\mu_n\\over w^2}\n  \\bigg)^{(s-\\delta)\/2}\\,K_{s-\\delta}\\big(2w\\sqrt\\mu_n\\big)\\,,}\n  $$\non using (\\puteqn{hi}). Hence (\\puteqn{berndt}) has been established.\n\nIn specific situations (\\puteqn{berndt}) is quite familiar in the $\\zeta$--function\\ area\nand in  background and finite--temperature quantum field theory.\n\nAn illustrative example is the simplest, diagonal Epstein function,\n   $$\n   \\phi(s)=Z_p(s)\\equiv\\sumdasht{{\\bf m}=-\\infty}{\\infty}\n  {1\\over({\\bf m.m})^s}\\,,\\quad {\\bf m}=(m_1,m_2,\\ldots,m_p)\\,,\n   \\eql{diagep}$$\nwhich obeys the functional equation (Epstein [\\putref{Epstein}]),\n   $$\n   \\Gamma(s)\\,Z_p(s)=\\pi^{2s-p\/2}\\,\\Gamma(p\/2-s)\\,Z_p(p\/2-s)\\,,\n   \\eql{epfunc}$$\nso that, see (\\puteqn{frel2}), $\\delta=p\/2$, $a_n=1$, $\\lambda_m = {\\bf m.m}$,\n$b_n=\\pi^{p\/2}$, $\\mu_n=\\pi^2\\,{\\bf m.m}$.\n\n$\\Gamma(s)\\,Z_p(s)$ has poles at $s=0$ and $s=p\/2$ and (\\puteqn{berndt}) gives\nfor the modified, or inhomogeneous, or `massive' Epstein function,\n$Z_p(s,w)$,\n  $$\\eqalign{\n  \\sumdasht{{\\bf m}=-\\infty}{\\infty}\n  {1\\over({\\bf m.m}+w^2)^s}=-w^{-2s}&+{\\Gamma(p\/2)\\Gamma(s-p\/2)\\over\\Gamma(s)}\n  \\,w^{p-2s}\\cr\n & +{2\\pi^s\\over\\Gamma(s)}\\sumdasht{{\\bf m}=-\\infty}{\\infty}\n  \\bigg({|{\\bf m}|\\over w}\\bigg)^{s-p\/2}\\,\n  K_{s-p\/2}\\big(2w\\pi|{\\bf m}|\\big)\\,,}\n  \\eql{modep}$$\nwhere the dash means to omit the ${\\bf m=0}$ term. Note that the $w^{-2s}$ could be\nincluded in the left--hand sum if this were extended to include ${\\bf m=0}$.\n\nThe right--hand side can be taken as the continuation of the left--hand\nside showing the single pole at $s=p\/2$, as is correct and may be proved in\nother ways.\n\nActually (\\puteqn{modep}) is shown much more directly using the Jacobi\ninversion relation, which is, of course, how Epstein obtained\n(\\puteqn{epfunc}). Equivalent to Jacobi inversion is Poisson summation.\n\nEquations like (\\puteqn{modep}) also occur in lattice summations. Some references,\nbut none earlier than those already quoted, are given in the review by Glasser\nand Zucker [\\putref{GandZ}] p.109.\n\nA similar formula holds for the general Epstein function which may, or may\nnot, have a pole at $s=p\/2$ and may, or may not, vanish at $s=0$. The works\n[\\putref{Kirsten1,Elizalde2,Elizalde3}], for example, can be consulted for details,\nsome further references and physical applications.\n\nWhen $p=1$, $\\phi(s)=2\\zeta_R(2s)$ and I recover an earlier, well--known\nformula, [\\putref{Watson,Kober}], which yields, after putting $w=u\\,m_2$\nand summing over $m_2$, the formula for the Epstein $\\zeta$--function\\ mentioned\nearlier; see (\\puteqn{kob1}).\n\nIt is left as an exercise to show that the non-diagonal formula,\n(\\puteqn{kob1}), can be obtained in the same way using  reciprocal relations\nfor the Hurwitz $\\zeta$--function, or, equivalently, the Lipshitz formula ({\\it cf } Epstein,\n[\\putref{Epstein}], Kober [\\putref{Kober}] p.622).\n\nAs an application of (\\puteqn{modep}) it might be helpful if I present a\nstandard derivation of the usual statistical free energy mode sum from\nthe thermal $\\zeta$--function\\ expression, for a reasonably general system. The\nstarting point is the determinant form,\n  $$\n  F={i\\over2}\\lim_{s\\to0}{\\zeta(s,\\beta)\\over s}\n  =-{1\\over2\\beta}\\lim_{s\\to0}{1\\over s}\n  \\sumdasht{{m=-\\infty\\atop n}}\n{\\infty}{d_n\\over\n \\big(\\omega_n^2+4\\pi^2m^2\/\\beta^2\\big)^s}\\,,\n  \\eql{eff2}$$\nwhere $\\omega_n^2$ and $d_n$ are the eigenvalues and degeneracies of the\nappropriate operator (related to the Laplacian) on the spatial section,\n${\\cal M}$. The $\\omega_n$ are the single--particle energies. The dash here means\nthat the denominator should never be zero ({\\it cf } (\\puteqn{eff})) but the sum\nincludes $m=0$. For simplicity it is assumed that there is no spatial zero\nmode and so the dash can be removed.\n\nNow consider (\\puteqn{modep}) for $p=1$. Include the $w^{-2s}$ with the\nleft--hand sum, set $w=\\beta\\omega_n\/2\\pi$, multiply by the degeneracy, $d_n$\nand sum over $n$. The limit (\\puteqn{eff2}) is then easily taken since\n$s\\Gamma(s)=\\Gamma(s+1)$ and only the $K_{1\/2}$ Bessel appears. There is\nnothing deep about this calculation and the result is a standard form,\n  $$\\eqalign{\n  F=&{1\\over2}\\,\\zeta_{\\cal M}(-1\/2)+{1\\over\\beta}\\sum_n d_n\n  \\sum_{m=1}^\\infty{e^{-m\\omega_n\\beta}\\over m}\\cr\n  =&{1\\over2}\\,\\zeta_{\\cal M}(-1\/2)+{1\\over\\beta}\\sum_n d_n\\log(1-e^{-\\omega_n\\beta})\\,.}\n  \\eql{modesum}$$\nTo avoid specifically field theoretic problems, it has been assumed that\n$\\zeta_{\\cal M}(-1\/2)$ exists. This is true for the cases discussed in this\npaper.\n\nThe result, (\\puteqn{modesum}), is an analogue of the (first) Kronecker\nlimit--formula applied to the generalised torus S$^1\\times{\\cal M}$; see\n[\\putref{DandA}]. The original formula relates to ${\\cal M}=$S$^1$ and, in this\ncase, there exists a functional relation, (\\puteqn{frel2}), so that the\nlimit--formula is often expressed in terms of the remainder about the\npole in $\\zeta(s,\\beta)$ at $s=1$. In the general case this is not possible.\n\nFinally, if one introduces the standard arithmetic quantity $r_p(n)$,\nequal to the number of representations of the integer $n$ as the sum of\n$p$ squares, then the diagonal Epstein function, (\\puteqn{diagep}), is\nobtained by writing,\n  $$\n  \\phi(s)=Z_p(s)=\\sum_n{r_{p}(n)\\over n^s}\\,,\n  $$\nand is treated as such by Koshliakov who also gives the `exact', product\nforms of $Z_p(s)$ for $p=2$ and $p=4$.\n\n\\begin{ignore}\nThe other useful formula is a generalisation of one occurring in the\ntheory of Riesz means which has already been encountered, essentially in\n(\\puteqn{mint}) and (\\puteqn{soln}).\n\nWith the same notation as above, one has, for $h\\ge0$,\n  $$\n  {1\\over\\Gamma(h+1)}\\sum_{\\lambda_n}a_n\\,\\theta(x-\\lambda_n)\\,(x-\\lambda_n)^h\n  ={1\\over2\\pi i}\\int_{c-i\\infty}\n  ^{c+i\\infty}ds\\,{\\Gamma(s)\\,\\phi(s)\\,x^{s+h}\\over\\Gamma(h+1+s)}\\,,\n  $$\nwhere $\\theta$ is Heaviside's step function.\n\\end{ignore}\n\\newpage\n\\section{\\bf References.}\n\n\\begin{putreferences}\n   \\reference{RandA}{Rao,M.B. and Ayyar,M.V. \\jims{15}{1923\/24}{150}.}\n  \\reference{DandK2}{Dowker,J.S. and Kirsten,K. {\\it Elliptic aspects of\n  statistical mechanics on spheres} ArXiv:0807.1995.}\n  \\reference{DandK1}{Dowker,J.S. and Kirsten,K. \\np {638}{2002}{405}.}\n  \\reference{GandS}{Gel'fand, I.M. and Shilov, G.E. {\\it Generalised Functions}, vol.1\n  (Academic Press, New York, 1964). }\n  \\reference{DandA}{Dowker,J.S. and Apps, J.S. \\cqg{12}{1995}{1363}.}\n  \\reference{Weil}{Weil,A., {\\it Elliptic functions according to Eisenstein and\n  Kronecker}, Springer, Berlin, 1976.}\n  \\reference{Ling}{Ling,C-H. {\\it SIAM J.Math.Anal.} {\\bf5} (1974) 551.}\n  \\reference{Ling2}{Ling,C-H. {\\it J.Math.Anal.Appl.}(1988).}\n \\reference{BMO}{Brevik,I., Milton,K.A. and Odintsov, S.D. {\\it Entropy bounds in\n $R\\times S^3$ geometries}. hep-th\/0202048.}\n \\reference{KandL}{Kutasov,D. and Larsen,F. {\\it JHEP} 0101 (2001) 1.}\n \\reference{KPS}{Klemm,D., Petkou,A.C. and Siopsis {\\it Entropy\n bounds, monoticity properties and scaling in CFT's}. hep-th\/0101076.}\n \\reference{DandC}{Dowker,J.S. and Critchley,R. \\prD{15}{1976}{1484}.}\n \\reference{AandD}{Al'taie, M.B. and Dowker, J.S. \\prD{18}{1978}{3557}.}\n \\reference{Dow1}{Dowker,J.S. \\prD{37}{1988}{558}.}\n \\reference{Dow3}{Dowker,J.S.\\prD{28}{1983}{3013}.}\n \\reference{DandK}{Dowker,J.S. and Kennedy,G. \\jpa{11}{1978}{895}.}\n \\reference{Dow2}{Dowker,J.S. \\cqg{1}{1984}{359}.}\n \\reference{DandKi}{Dowker,J.S. and Kirsten, K. {\\it Comm. in Anal. and Geom.\n  }{\\bf7}(1999) 641.}\n \\reference{DandKe}{Dowker,J.S. and Kennedy,G.\n \\jpa{11}{1978}{895}.} \\reference{Gibbons}{Gibbons,G.W. \\pl{60A}{1977}{385}.}\n \\reference{Cardy}{Cardy,J.L. \\np{366}{1991}{403}.}\n \\reference{ChandD}{Chang,P. and\n  Dowker,J.S. \\np{395}{1993}{407}.}\n  \\reference{DandC2}{Dowker,J.S. and\n  Critchley,R. \\prD{13}{1976}{224}.}\n \\reference{Camporesi}{Camporesi,R.\n \\prp{196}{1990}{1}.}\n \\reference{BandM}{Brown,L.S. and Maclay,G.J.\n \\pr{184}{1969}{1272}.}\n \\reference{CandD}{Candelas,P. and Dowker,J.S.\n \\prD{19}{1979}{2902}.}\n  \\reference{Unwin1}{Unwin,S.D. Thesis. University of\n Manchester. 1979.} \\reference{Unwin2}{Unwin,S.D. \\jpa{13}{1980}{313}.}\n \\reference{DandB}{Dowker,J.S. and Banach,R. \\jpa{11}{1979}{}.}\n \\reference{Obhukov}{Obhukov,Yu.N. \\pl{109B}{1982}{195}.}\n \\reference{Kennedy}{Kennedy,G. \\prD{23}{1981}{2884}.}\n \\reference{CandT}{Copeland,E.\n and Toms,D.J. \\np {255}{1985}{201}.} \\reference{ELV}{Elizalde,E., Lygren, M.\n and Vassilevich, D.V. \\jmp {37}{1996}{3105}.}\n \\reference{Malurkar}{Malurkar,S.L. {\\it J.Ind.Math.Soc} {\\bf16} (1925\/26) 130.}\n \\reference{Glaisher}{Glaisher,J.W.L. {\\it Messenger of Math.} {\\bf18} (1889)\n  1.}\n \\reference{Anderson}{Anderson,A. \\prD{37}{1988}{536}.}\n \\reference{CandA}{Cappelli,A. and D'Appollonio,\\pl{487B}{2000}{87}.}\n \\reference{Wot}{Wotzasek,C. \\jpa{23}{1990}{1627}.}\n \\reference{RandT}{Ravndal,F. and Tollesen,D. \\prD{40}{1989}{4191}.}\n \\reference{SandT}{Santos,F.C. and Tort,A.C. \\pl{482B}{2000}{323}.}\n \\reference{FandO}{Fukushima,K. and Ohta,K. {\\it Physica} {\\bf A299} (2001) 455.}\n \\reference{GandP}{Gibbons,G.W. and Perry,M. \\prs{358}{1978}{467}.}\n \\reference{Dow4}{Dowker,J.S. {\\it Zero modes, entropy bounds and partition\n functions.} hep-th\\break \/0203026.}\n  \\reference{Rad}{Rademacher,H. {\\it Topics in analytic number theory,}\n (Springer- Verlag,  Berlin, 1973).}\n  \\reference{Halphen}{Halphen,G.-H. {\\it Trait\\'e des Fonctions Elliptiques}, Vol 1,\n  Gauthier-Villars, Paris, 1886.}\n  \\reference{CandW}{Cahn,R.S. and Wolf,J.A. {\\it Comm.Mat.Helv.} {\\bf 51} (1976) 1.}\n  \\reference{Berndt}{Berndt,B.C. \\rmjm{7}{1977}{147}.}\n  \\reference{Hurwitz}{Hurwitz,A. \\ma{18}{1881}{528}.}\n  \\reference{Hurwitz2}{Hurwitz,A. {\\it Mathematische Werke} Vol.I. Basel,\n  Birkhauser, 1932.}\n  \\reference{Berndt2}{Berndt,B.C. \\jram{303\/304}{1978}{332}.}\n  \\reference{RandA}{Rao,M.B. and Ayyar,M.V. \\jims{15}{1923\/24}{150}.}\n  \\reference{Hardy}{Hardy,G.H. \\jlms{3}{1928}{238}.}\n  \\reference{TandM}{Tannery,J. and Molk,J. {\\it Fonctions Elliptiques},\n   Gauthier-Villars, Paris, 1893--1902.}\n  \\reference{schwarz}{Schwarz,H.-A. {\\it Formeln und Lehrs\\\"atzen zum Gebrauche..},\n  Springer 1893.(The first edition was 1885.) The French translation by\n  Henri Pad\\'e is, Gauthier-Villars, Paris, 1894}\n  \\reference{Hancock}{Hancock,H. {\\it Theory of elliptic functions}, Vol I.\n   Wiley, New York 1910.}\n  \\reference{watson}{Watson,G.N. \\jlms{3}{1928}{216}.}\n  \\reference{MandO}{Magnus,W. and Oberhettinger,F. {\\it Formeln und S\\\"atze},\n  Springer-Verlag, Berlin 1948.}\n  \\reference{Klein}{Klein,F. {\\it }.}\n  \\reference{AandL}{Appell,P. and Lacour,E. {\\it Fonctions Elliptiques},\n  Gauthier-Villars,\n  Paris, 1897.}\n  \\reference{HandC}{Hurwitz,A. and Courant,C. {\\it Allgemeine Funktionentheorie},\n  Springer,\n  Berlin, 1922.}\n  \\reference{WandW}{Whittaker,E.T. and Watson,G.N. {\\it Modern analysis},\n  Cambridge 1927.}\n  \\reference{SandC}{Selberg,A. and Chowla,S. \\jram{227}{1967}{86}. }\n  \\reference{zucker}{Zucker,I.J. {\\it Math.Proc.Camb.Phil.Soc} {\\bf 82 }(1977) 111.}\n  \\reference{glasser}{Glasser,M.L. {\\it Maths.of Comp.} {\\bf 25} (1971) 533.}\n  \\reference{GandW}{Glasser, M.L. and Wood,V.E. {\\it Maths of Comp.} {\\bf 25} (1971)\n  535.}\n  \\reference{greenhill}{Greenhill,A,G. {\\it The Applications of Elliptic\n  Functions}, MacMillan, London, 1892.}\n  \\reference{Weierstrass}{Weierstrass,K. {\\it J.f.Mathematik (Crelle)}\n  {\\bf 52} (1856) 346.}\n  \\reference{Weierstrass2}{Weierstrass,K. {\\it Mathematische Werke} Vol.I,p.1,\n  Mayer u. M\\\"uller, Berlin, 1894.}\n  \\reference{Fricke}{Fricke,R. {\\it Die Elliptische Funktionen und Ihre Anwendungen},\n    Teubner, Leipzig. 1915, 1922.}\n  \\reference{Konig}{K\\\"onigsberger,L. {\\it Vorlesungen \\\"uber die Theorie der\n Elliptischen Funktionen},  \\break Teubner, Leipzig, 1874.}\n  \\reference{Milne}{Milne,S.C. {\\it The Ramanujan Journal} {\\bf 6} (2002) 7-149.}\n  \\reference{Schlomilch}{Schl\\\"omilch,O. {\\it Ber. Verh. K. Sachs. Gesell. Wiss.\n  Leipzig}  {\\bf 29} (1877) 101-105; {\\it Compendium der h\\\"oheren Analysis},\n  Bd.II, 3rd Edn, Vieweg, Brunswick, 1878.}\n  \\reference{BandB}{Briot,C. and Bouquet,C. {\\it Th\\`eorie des Fonctions Elliptiques},\n  Gauthier-Villars, Paris, 1875.}\n  \\reference{Dumont}{Dumont,D. \\aim {41}{1981}{1}.}\n  \\reference{Andre}{Andr\\'e,D. {\\it Ann.\\'Ecole Normale Superior} {\\bf 6} (1877) 265;\n  {\\it J.Math.Pures et Appl.} {\\bf 5} (1878) 31.}\n  \\reference{Raman}{Ramanujan,S. {\\it Trans.Camb.Phil.Soc.} {\\bf 22} (1916) 159;\n {\\it Collected Papers}, Cambridge, 1927}\n  \\reference{Weber}{Weber,H.M. {\\it Lehrbuch der Algebra} Bd.III, Vieweg,\n  Brunswick 190  3.}\n  \\reference{Weber2}{Weber,H.M. {\\it Elliptische Funktionen und algebraische Zahlen},\n  Vieweg, Brunswick 1891.}\n  \\reference{ZandR}{Zucker,I.J. and Robertson,M.M.\n  {\\it Math.Proc.Camb.Phil.Soc} {\\bf 95 }(1984) 5.}\n  \\reference{JandZ1}{Joyce,G.S. and Zucker,I.J.\n  {\\it Math.Proc.Camb.Phil.Soc} {\\bf 109 }(1991) 257.}\n  \\reference{JandZ2}{Zucker,I.J. and Joyce.G.S.\n  {\\it Math.Proc.Camb.Phil.Soc} {\\bf 131 }(2001) 309.}\n  \\reference{zucker2}{Zucker,I.J. {\\it SIAM J.Math.Anal.} {\\bf 10} (1979) 192,}\n  \\reference{BandZ}{Borwein,J.M. and Zucker,I.J. {\\it IMA J.Math.Anal.} {\\bf 12}\n  (1992) 519.}\n  \\reference{Cox}{Cox,D.A. {\\it Primes of the form $x^2+n\\,y^2$}, Wiley, New York,\n  1989.}\n  \\reference{BandCh}{Berndt,B.C. and Chan,H.H. {\\it Mathematika} {\\bf42} (1995) 278.}\n  \\reference{EandT}{Elizalde,R. and Tort.hep-th\/}\n  \\reference{KandS}{Kiyek,K. and Schmidt,H. {\\it Arch.Math.} {\\bf 18} (1967) 438.}\n  \\reference{Oshima}{Oshima,K. \\prD{46}{1992}{4765}.}\n  \\reference{greenhill2}{Greenhill,A.G. \\plms{19} {1888} {301}.}\n  \\reference{Russell}{Russell,R. \\plms{19} {1888} {91}.}\n  \\reference{BandB}{Borwein,J.M. and Borwein,P.B. {\\it Pi and the AGM}, Wiley,\n  New York, 1998.}\n  \\reference{Resnikoff}{Resnikoff,H.L. \\tams{124}{1966}{334}.}\n  \\reference{vandp}{Van der Pol, B. {\\it Indag.Math.} {\\bf18} (1951) 261,272.}\n  \\reference{Rankin}{Rankin,R.A. {\\it Modular forms} (CUP, Cambridge, 1977).}\n  \\reference{Rankin2}{Rankin,R.A. {\\it Proc. Roy.Soc. Edin.} {\\bf76 A} (1976) 107.}\n  \\reference{Skoruppa}{Skoruppa,N-P. {\\it J.of Number Th.} {\\bf43} (1993) 68 .}\n  \\reference{Down}{Dowker.J.S. \\np {104}{2002}{153}.}\n  \\reference{Eichler}{Eichler,M. \\mz {67}{1957}{267}.}\n  \\reference{Zagier}{Zagier,D. \\invm{104}{1991}{449}.}\n   \\reference{KandZ}{Kohnen, W. and Zagier,D. {\\it Modular forms with rational periods.}\n   in {\\it Modular Forms} ed. by Rankin, R.A. (Ellis Horwood, Chichester, 1984).}\n  \\reference{Lang}{Lang,S. {\\it Modular Forms}, (Springer, Berlin, 1976).}\n  \\reference{Kosh}{Koshliakov,N.S. {\\it Mess.of Math.} {\\bf 58} (1928) 1.}\n  \\reference{BandH}{Bodendiek, R. and Halbritter,U. \\amsh{38}{1972}{147}.}\n  \\reference{Smart}{Smart,L.R., \\pgma{14}{1973}{1}.}\n  \\reference{Grosswald}{Grosswald,E. {\\it Acta. Arith.} {\\bf 21} (1972) 25.}\n  \\reference{Kata}{Katayama,K. {\\it Acta Arith.} {\\bf 22} (1973) 149.}\n  \\reference{Ogg}{Ogg,A. {\\it Modular forms and Dirichlet series} (Benjamin, New York,\n   1969).}\n  \\reference{Bol}{Bol,G. \\amsh{16}{1949}{1}.}\n  \\reference{Epstein}{Epstein,P. \\ma{56}{1902}{615}.}\n  \\reference{Petersson}{Petersson, H. \\ma {116}{1939}{401}.}\n  \\reference{Serre}{Serre,J-P. {\\it A Course in Arithmetic}, Springer, New York,\n  1973.}\n  \\reference{Schoenberg}{Schoenberg,B., {\\it Elliptic Modular Functions},\n  Springer, Berlin, 1974.}\n  \\reference{Apostol}{Apostol,T.M. \\dmj {17}{1950}{147}.}\n  \\reference{Ogg2}{Ogg,A. {\\it Lecture Notes in Math.} {\\bf 320} (1973) 1.}\n  \\reference{Knopp}{Knopp,M.I. \\dmj {45} {1978}{47}.}\n  \\reference{Knopp2}{Knopp,M.I. \\invm {}{1994}{361}.}\n  \\reference{Knopp3}{Knopp,M.I. {\\it Lecture Notes in Mathematics} {\\bf 1383}(1989) 111 .}\n  \\reference{LandZ}{Lewis,J. and Zagier,D. \\aom{153}{2001}{191}.}\n  \\reference{HandK}{Husseini, S.Y. and Knopp, M.I. {\\it Illinois.J.Math.} {\\bf 15} (1971)\n  565.}\n  \\reference{Kober}{Kober,H. \\mz{39}{1934-5}{609}.}\n  \\reference{HandL}{Hardy,G.H. and Littlewood, \\am{41}{1917}{119}.}\n  \\reference{Watson}{Watson,G.N. \\qjm{2}{1931}{300}.}\n  \\reference{SandC2}{Chowla,S. and Selberg,A. {\\it Proc.Nat.Acad.} {\\bf 35}\n  (1949) 371.}\n  \\reference{Landau}{Landau, E. {\\it Lehre von der Verteilung der Primzahlen},\n  (Teubner, Leipzig, 1909).}\n  \\reference{Berndt4}{Berndt,B.C. \\tams {146}{1969}{323}.}\n  \\reference{Berndt3}{Berndt,B.C. \\tams {}{}{}.}\n  \\reference{Bochner}{Bochner,S. \\aom{53}{1951}{332}.}\n  \\reference{Weil2}{Weil,A.\\ma{168}{1967}{}.}\n  \\reference{CandN}{Chandrasekharan,K. and Narasimhan,R. \\aom{74}{1961}{1}.}\n  \\reference{Rankin3}{Rankin,R.A. {\\it Proc.Glas.Math.Assoc.} {\\bf 1} (1953) 149.}\n  \\reference{Berndt6}{Berndt,B.C. {\\it Trans.Edin.Math.Soc} {\\bf 10} (1967) 309.}\n  \\reference{Elizalde}{Elizalde,E. {\\it Ten Physical Applications of Spectral\n  Zeta Function Theory}, \\break (Springer, Berlin, 1995).}\n  \\reference{Allen}{Allen,B., Folacci,A. and Gibbons,G.W. \\pl{189}{1987}{304}.}\n  \\reference{Krazer}{Krazer,A. {\\it Lehrbuch der Thetafunktionen} (Teubner,\n  Leipzig, 1903)}\n  \\reference{Elizalde3}{Elizalde,E. {\\it J.Comp.and Appl. Math.} {\\bf 118} (2000) 125.}\n  \\reference{Elizalde2}{Elizalde,E., Odintsov.S.D, Romeo, A. and Bytsenko, A.A and\n  Zerbini,S.\n  {\\it Zeta function regularisation}, (World Scientific, Singapore, 1994).}\n  \\reference{Eisenstein}{Eisenstein,B. \\jram {35}{1847}{153}.}\n  \\reference{Hecke}{Hecke,E. \\ma{112}{1936}{664}.}\n  \\reference{Terras}{Terras,A. {\\it Harmonic analysis on Symmetric Spaces} (Springer,\n  New York, 1985).}\n  \\reference{BandG}{Bateman,P.T. and Grosswald,E. {\\it Acta Arith.} {\\bf 9} (1964) 365.}\n  \\reference{Deuring}{Deuring,M. \\aom{38}{1937}{585}.}\n  \\reference{Guinand2}{Guinand, A.P. \\qjm{6}{1955}{156}.}\n  \\reference{Guinand}{Guinand, A.P. \\qjm {15}{1944}{11}.}\n  \\reference{Minak}{Minakshisundaram.}\n  \\reference{Mordell}{Mordell,J. \\qjm {1}{1930}{77}.}\n  \\reference{GandZ}{Glasser,M.L. and Zucker,I.J. ``Lattice Sums\" in Theoretical\n  Chemistry, Advances and Perspectives, vol.5, ed. D.Henderson, New York,\n  1986, 67  .}\n  \\reference{Landau2}{Landau,E. \\jram{}{1903}{64}.}\n  \\reference{Kirsten1}{Kirsten,K. \\jmp{35}{1994}{459}.}\n  \\reference{Sommer}{Sommer,J. {\\it Vorlesungen \\\"uber Zahlentheorie} (1907,Teubner,Leipzig).\n  French edition 1913 .}\n  \\reference{Reid}{Reid,L.W. {\\it Theory of Algebraic Numbers}, (1910,MacMillan,New York).}\n\\end{putreferences}\n\\bye\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nLet $n \\geq 1$ be an integer and $\\xi \\in \\lR$.\nWe denote by $w_n(\\xi)$ (resp.\\ $w_n ^ {*}(\\xi)$) the supremum of the real numbers $w$ (resp.\\ $w ^ {*}$) which satisfy\n\\begin{equation*}\n\t0 < \\abs{P(\\xi)} \\leq H(P)^{- w} \\quad (\\text{resp.\\ } 0 < \\abs{\\xi - \\alpha} \\leq H(\\alpha)^{- w ^ {*} - 1})\n\\end{equation*}\nfor infinitely many integer polynomials $P(X)$ of degree at most $n$ (resp.\\ algebraic numbers $\\alpha \\in \\lC$ of degree at most $n$).\nHere, $H(P)$ is defined to be the maximum of the absolute values of the coefficients of $P(X)$ and $H(\\alpha)$ is equal to $H(Q)$, where $Q(X)$ is the minimal polynomial of $\\alpha$ over $\\lZ$.\nThe functions $w_n$ and $w_n ^ {*}$ are called \\textit{Diophatine exponents}.\nLiouville  proved that $w_1 (\\alpha) \\leq \\deg \\alpha - 1$ for any algebraic real number $\\alpha$.\nRoth \\cite{Roth55} improved Liouville Theorem, that is, he showed that $w_1 (\\alpha) = 1$ for all algebraic irrational real numbers $\\alpha$.\nFor higher Diophantine exponents, from the Schmidt Subspace Theorem, it is known that\n\\begin{equation*}\n\tw_n (\\alpha ) = w_n ^ {*} (\\alpha ) = \\min \\{ n, d-1 \\},\n\\end{equation*}\nwhere $n \\geq 1$ is an integer and $\\alpha$ is an algebraic real number of degree $d$ (see \\cite[Section 3]{Bugeaud04}).\n\nLet $p$ be a prime, $q=p^s$, where $s \\geq 1$ is an integer and $\\lF_q$ be the finite field containing $q$ elements.\nWe denote by $\\lF_q [T], \\lF_q (T), $ and $\\lF_q ((T ^ {- 1}))$ respectively the ring of polynomials, the field of rational functions, the field of Laurent series in $T ^ {- 1}$ over $\\lF_q$.\nWe consider analogues of Diophantine exponents $w_n$ and $w_n ^ {*}$ for Laurent series (see Section \\ref{sec:notation} for the precisely definition).\nAs an analogue of Liouville Theorem, Mahler \\cite{Mahler49} showed $w_1 (\\alpha) \\leq \\deg \\alpha - 1$ for any algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$.\nHowever, there exist counter examples of analogue of Roth Theorem for Laurent series over a finite field.\nLet $r_1 = p ^ {t_1}$, where $t_1 \\geq 1$ is an integer.\nHe proved that the Laurent series $\\alpha_1 := \\sum_{n = 0}^{\\infty}T ^ {- r_1 ^ n}$ is algebraic of degree $r_1$ with $w_1 (\\alpha_1) = r_1 - 1$.\nAfter that many authors investigated rational approximations (e.g.\\ \\cite{Firicel13, Mathan92, Schmidt00, Thakur99}) and algebraic approximations (e.g.\\ \\cite{Ooto18, Thakur11, Thakur13}) for algebraic Laurent series.\nIn particular, the second author (\\cite{Ooto18}) proved that, for any rational number $w > 2n-1$, there exists an algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$ such that\n\\begin{equation*}\n\tw_1(\\alpha) = w_1^{*}(\\alpha) = \\ldots = w_n(\\alpha) = w_n^{*}(\\alpha) = w.\n\\end{equation*}\nThe first purpose of this paper is to consider Problem 2.2 in \\cite{Ooto18}.\n\n\n\n\\begin{prob}\\label{prob:main1}\n\tIs it true that\n\t\\begin{equation*}\n\t\tw_n(\\alpha) = w_n^{*}(\\alpha)\n\t\\end{equation*}\n\tfor an integer $n \\geq 1$ and an algebraic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$?\n\\end{prob}\n\n\n\nNote that it is well-known and easy to see that $w_1(\\xi) = w_1^{*}(\\xi)$ for all $\\xi \\in \\lF_q((T^{-1}))$.\nWe give a negative answer of Problem \\ref{prob:main1} for any $n \\geq 2$ and $q \\geq 4$.\n\n\n\n\\begin{thm}\\label{thm:neq_alg}\n\tLet $n \\geq 2$ be an integer, $q = p^s$, and $r = q^t$ with $s, t \\geq 1$ be integers and \n\t\\begin{equation*}\n\t\tr \\geq \\frac{3n + 2 + \\sqrt{9n^2 + 4n + 4}}{2}.\n\t\\end{equation*}\n\tIf $q \\geq 4$, then there exist algebraic Laurent series $\\alpha \\in \\lF_q ((T ^ {- 1}))$ with $\\deg \\alpha = r+1$ such that\n\t\\begin{equation*}\n\t\tw_n (\\alpha) \\neq w_n ^ {*} (\\alpha)\n\t\\end{equation*}\n\\end{thm}\n\n\n\nLet $\\alpha$ be in $\\lF_q ((T ^ {- 1}))$ and $r = p ^ t$, where $t \\geq 0$ is an integer.\nWe call $\\alpha$ \\textit{hyperquadratic} if $\\alpha$ is irrational and there exists $(A, B, C, D) \\in \\lF_q [T] ^ 4 \\setminus \\{ {\\bf 0} \\}$ such that\n\\begin{equation}\\label{eq:hyperquad}\n\tA \\alpha^{r+1} + B \\alpha ^ r + C \\alpha + D = 0.\n\\end{equation}\nFor example, quadratic Laurent series are hyperquadratic, $\\alpha_1$ is hyperquadratic and satisfies $T \\alpha_1^r - T \\alpha_1 + 1 = 0$.\nBaum and Sweet \\cite{Baum76} started a study of hyperquadratic continued fractions in characteristic two.\nIn 1986, Mills and Robbins \\cite{Mills86} showed the existence of hyperquadratic continued fractions with all partial quotient of degree one in odd characteristic with a prime field.\nWe  refer the reader to \\cite{Lasjaunias172} for a survey of recent works of hyperquadratic continued fractions.\nWe observe that the degree of hyperquadratic Laurent series which satisfies \\eqref{eq:hyperquad} is less than or equal to $r+1$.\nHowever, the exact degree of these is open.\nIn this paper, we discuss the following problem.\n\n\n\n\\begin{prob}\n\tGiven a hyperquadratic Laurent series $\\alpha \\in \\lF_q((T^{-1}))$, determine the exact degree of $\\alpha$.\n\\end{prob}\n\n\n\nIn order to determine the exact degree, we approach to quadratic approximations.\nSince $w_2(\\alpha) \\leq \\deg \\alpha - 1$ for $\\alpha$ as in \\eqref{eq:hyperquad} (see Lemma \\ref{lem:alg}), if $w_2(\\alpha) > r-1$, then $\\deg \\alpha = r+1$.\nIn this paper, we deal with two families for hyperquadratic continued fractions introduced in \\cite{Lasjaunias02, Lasjaunias17}.\nWe determine the exact degree of these families in particular cases.\n\nThis paper is organized as follows.\nIn Section \\ref{sec:notation}, we recall notations which are used in this paper.\nIn Section \\ref{sec:main}, we state main results of quadratic approximation for hyperquadratic continued fractions and applications of the results.\nIn Section \\ref{sec:lemma}, we give lemmas for the proof of main results.\nIn Section \\ref{sec:proof}, we prove the main results.\n\n\n\n\n\n\n\\section{Notations}\\label{sec:notation}\n\n\n\nA nonzero Laurent series $\\xi \\in \\lF_q((T ^ {- 1}))$ is represented by $\\xi = \\sum_{n = N}^{\\infty}a_n T ^ {- n}$, where $N \\in \\lZ, a_n \\in \\lF_q,$ and $a_N \\neq 0$.\nThe field $\\lF_q ((T ^ {- 1}))$ has a non-Archimedean absolute value $\\abs{\\xi} = q ^ {- N}$ and $\\abs{0} = 0$.\nThe absolute value can be uniquely extended to the algebraic closure of $\\lF_q ((T ^ {- 1}))$ and we also write $\\abs{\\cdot}$ for the extended absolute value.\n\nThe {\\itshape height} of $P(X) \\in (\\lF_q [T])[X]$, denoted by $H(P)$, is defined to be the maximum of the absolute values of the coefficients of $P(X)$.\nFor $\\alpha \\in \\overline{\\lF_q (T)}$, there exists a unique non-constant, irreducible, primitive polynomial $P(X) \\in (\\lF _q [T]) [X]$ whose leading coefficients are monic polynomials in $T$ such that $P(\\alpha) = 0$. \nThe polynomial $P(X)$ is called the {\\itshape minimal polynomial} of $\\alpha$.\nThe {\\itshape height} (resp.\\ the {\\itshape degree}, the {\\itshape inseparable degree}) of $\\alpha$, denoted by $H(\\alpha)$ (resp.\\ $\\deg \\alpha$, $\\insep \\alpha$), is defined to be the height of $P(X)$ (resp.\\ the degree of $P(X)$, the inseparable degree of $P(X)$).\nLet $n \\geq 1$ be an integer and $\\xi$ be in $\\lF_q (( T ^ {- 1}))$.\nWe denote by $w_n (\\xi)$ (resp.\\ $w_n ^ {*} (\\xi)$) the supremum of the real numbers $w$ (resp.\\ $w ^ {*}$) which satisfy \n\\begin{equation*}\n\t0 < \\abs{P(\\xi )} \\leq H(P) ^ {- w} \\quad (\\text{resp.\\ } 0 < \\abs{\\xi - \\alpha } \\leq H(\\alpha) ^ {- w ^ {*} - 1})\n\\end{equation*}\nfor infinitely many $P(X) \\in (\\lF_q [T]) [X]$ of degree at most $n$ (resp.\\ $\\alpha \\in \\overline{\\lF_q (T)}$ of degree at most $n$).\n\nLet $\\alpha \\in \\overline{\\lF_q (T)}$ be a quadratic number.\nIf $\\insep \\alpha = 1$, let $\\alpha ' \\neq \\alpha$ be the Galois conjugate of $\\alpha$.\nIf $\\insep \\alpha = 2$, let $\\alpha ' = \\alpha $.\n\nA Laurent series $\\xi \\in \\lF_q ((T ^ {- 1}))$ can be expressed as a continued fraction:\n\\begin{equation*}\n\t\\xi = a_0 + \\cfrac{1}{a_1 + \\cfrac{1}{a_2 + \\cfrac{1}{\\cdots }}},\n\\end{equation*}\nwhere $a_n \\in \\lF_q [T]$ for all $n \\geq 0$ and $\\deg _T a_m \\geq 1$ for all $m \\geq 1$.\nWe write the expansion $\\xi = [a_0, a_1, a_2, \\ldots ]$.\nThe continued fraction expansion of $\\xi$ is finite if and only if $\\xi$ is in $\\lF_q (T)$.\nIt is known that the continued fraction expansion of $\\xi$ is ultimately periodic if and only if $\\xi$ is quadratic (see Th\\'{e}or\\`{e}me 4 in \\cite[CHAPITRE IV]{Mathan70}).\nWe define sequences $(p_n)_{n \\geq -1}$ and $(q_n)_{n \\geq -1}$ by\n\\begin{equation*}\n\t\\begin{cases}\n\t\tp_{- 1} = 1, \\ p_0 = a_0, \\ p_n = a_n p_{n - 1} + p_{n - 2},\\ n \\geq 1,\\\\\n\t\tq_{- 1} = 0,\\ q_0 = 1,\\ q_n = a_n q_{n - 1} + q_{n - 2},\\ n \\geq 1.\n\t\\end{cases}\n\\end{equation*}\nWe call $(p_n \/ q_n)_{n \\geq 0}$ the {\\itshape convergent sequence} of $\\xi$ and $p_n \/ q_n$ the \\textit{$n$-th convergent} of $\\xi$.\n\nLet  $n \\geq 1$ be an integer and $W, V$ be finite words.\nWe denote by $W \\bigoplus V$ the word of concatenation of $W$ and $V$.\nWe put $W ^ {[n]} := W \\bigoplus \\cdots \\bigoplus W$ ($n$ times) and $\\overline{W} := W\\bigoplus W\\bigoplus \\cdots \\bigoplus W\\bigoplus \\cdots $ (infinitely many times).\nWe write $W ^ {[0]}$ the empty word.\n\nLet $A, B$ be real numbers with $B \\neq 0$.\nWe write $A \\ll B$ (resp.\\ $A \\ll_a B$) if $\\abs{A} \\leq C \\abs{B}$ for some constant (resp.\\ some constant depending at most on $a$) $C > 0$.\nWe write $A \\asymp B$ (resp.\\ $A \\asymp_a B$) if $A \\ll B$ and $B \\ll A$ (resp.\\ $A \\ll_a B$ and $B \\ll_a A$) hold.\n\n\n\n\n\n\n\\section{Main results}\\label{sec:main}\n\n\n\nWe recall a family of hyperquadratic continued fractions.\nLet $s, t \\geq 1$ be integers and we put $q := p ^ s$ and $r := q ^ t$.\nLet  $k \\geq 0$ be an integer and $\\lambda$ be in $\\lF_q ^ {*}$.\nIf $p \\neq 2$, then we assume that $\\lambda \\neq 2$ and put $\\mu := 2 - \\lambda$.\nWe define $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ by\n\\begin{equation}\\label{eq:firstfamily}\n\t\\Theta_k ^ {t} (\\lambda)\n\t=\n\t\\begin{cases}\n\t\t[0, T ^ {[k]}, \\bigoplus_{i \\geq 1} (T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k+1]}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t[0, T ^ {[k]} \\bigoplus_{i \\geq 1} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}] & \\text{if}\\ p = 2.\n\t\\end{cases}\n\\end{equation}\nLasjaunias and Ruch \\cite{Lasjaunias02} proved that $\\Theta_k ^ {t} (\\lambda)$ is hyperquadratic and  satisfies the algebraic equation\n\\begin{equation}\\label{eq:firstfamilyequation}\n\t\\begin{cases}\n\t\tq_k X ^ {r + 1} - p_k X ^ r + (\\lambda \\mu ) ^ {(r - 1) \/ 2} q_{k + r} X - (\\lambda \\mu) ^ {(r - 1) \/ 2} p_{k + r} = 0 \\quad \\text{if}\\ p \\neq 2,\\\\\n\t\tq_k X ^ {r + 1} - p_k X ^ r + q_{k + r} X - p_{k + r} = 0 \\quad \\text{if}\\ p = 2,\n\t\\end{cases}\n\\end{equation}\nwhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\Theta_k ^ {t} (\\lambda)$.\nNote that if $\\lambda = 1$, then $\\Theta_k ^ {t} (\\lambda)$ is ultimately periodic continued fraction, that is, $\\Theta_k ^ {t} (\\lambda)$ is quadratic.\n\nOur first main result is the following.\n\n\n\n\\begin{thm}\\label{thm:main1}\n\tLet $d \\geq 2$ be an integer and $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ be as in \\eqref{eq:firstfamily}.\n\tAssume that $\\lambda \\neq 1$.\n\tThen we have\n\t\\begin{gather}\\label{eq:w2*}\n\t\tw_2 ^ {*} (\\Theta_k ^{t} (\\lambda)) \\geq \\max \\left\\lbrace \\frac{r - 1}{k + 1}, r - 1 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right\\rbrace ,\\\\\n\t\t\\label{eq:w2}\n\t\tw_2 (\\Theta_k ^ {t} (\\lambda)) \\geq \\max \\left\\lbrace \\frac{r - 1}{k + 1} + 1, r - \\frac{r(r - 1)(r - 3)}{2r(k + 1)+(r - 1)(r - 2)}\\right\\rbrace .\n\t\\end{gather}\n\tIf\n\t\\begin{equation}\\label{eq:maincond1}\n\t\tk \\leq \\frac{2(r - 1)}{d + \\sqrt{d ^ 2 + 4(2r - 1)d + 4}} - 1,\n\t\\end{equation}\n\tthen we have\n\t\\begin{equation}\\label{eq:mainequal1}\n\t\tw_n ^ {*} ( \\Theta_k ^ {t} (\\lambda)) = \\frac{r - 1}{k + 1}, \\quad w_n (\\Theta_k ^ {t} (\\lambda)) = \\frac{r - 1}{k + 1} + 1\n\t\\end{equation}\n\tfor all integers $2 \\leq n \\leq d$.\n\tIf\n\t\\begin{equation}\\label{eq:maincond2}\n\t\t2rd \\leq \\left( r - 2 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right) \\left( r - d - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)}\\right) ,\n\t\\end{equation}\n\tthen we have\n\t\\begin{gather}\\label{eq:mainequal2}\n\t\tw_n ^ {*} (\\Theta_k ^ {t} (\\lambda)) = r - 1 - \\frac{r(r - 1)(r - 4)}{2r(k + 1) + (r - 1)(r - 2)},\\\\\n\t\t\\label{eq:mainequal22}\n\t\tw_n (\\Theta_k ^ {t} (\\lambda )) = r - \\frac{r(r - 1)(r - 3)}{2r(k + 1) + (r - 1)(r - 2)}\n\t\\end{gather}\n\tfor all integers $2 \\leq n \\leq d$.\n\\end{thm}\n\n\n\n\\begin{rem}\n\tIt is well-known and easy to see that for any $\\xi \\in \\lF_q ((T ^ {- 1}))$,\n\t\\begin{equation}\\label{eq:w1_determine}\n\t\tw_1 (\\xi) = w_1 ^ {*} (\\xi) = \\limsup_{n \\longrightarrow \\infty } \\frac{\\deg q_{n + 1}}{\\deg q_n},\n\t\\end{equation}\n\twhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\xi$.\n\tSince $\\Theta_k ^ {t}(\\lambda)$ has a bounded partial quotient, we have $w_1 (\\Theta_k ^ {t}(\\lambda)) = w_1 ^ {*} (\\Theta_k ^ {t} (\\lambda)) = 1$.\n\tIf\n\t\\begin{equation*}\n\t\tr > \\frac{3d + 2 + \\sqrt{9d ^ 2 + 4d + 4}}{2},\n\t\\end{equation*}\n\tthere exists an effectively computable positive constant $C_1 (r, d)$, depending only on $r$ and $d$, such that we have \\eqref{eq:mainequal2} and \\eqref{eq:mainequal22} for all $k \\geq C_1 (r, d)$.\n\\end{rem}\n\n\n\nThe key point of the proof of Theorem \\ref{thm:main1} is to find two sequences of very good quadratic approximation, that is, ultimately periodic continued fractions.\n\nAs an application of Theorem \\ref{thm:main1}, we give a sufficient condition to determine the exact degree of $\\Theta_k ^ {t} (\\lambda)$.\n\n\n\n\\begin{cor}\\label{cor:main1}\n\tLet $\\Theta_k ^ {t} (\\lambda) \\in \\lF_q ((T ^ {- 1}))$ be as in \\eqref{eq:firstfamily}.\n\tAssume that $\\lambda \\neq 1$.\n\tIf\n\t\\begin{equation}\\label{eq:main1_cor_cond}\n\t\tk = 0 \\ \\text{or}\\ k > \\frac{(r ^ 2 - 4r + 2)(r - 1)}{2r} - 1,\n\t\\end{equation}\n\tthen we have $\\deg \\Theta_k ^ {t} (\\lambda) = r + 1$.\n\\end{cor}\n\n\n\nWe recall another family of hyperquadratic continued fractions.\nWe define a sequence $(F_n)_{n \\geq 0}$ of polynomials in $\\lF_q [T]$ by\n\\begin{equation}\\label{eq:fib}\n\tF_0 = 1, \\quad F_1 = T, \\quad F_{n + 1} = TF_n + F_{n - 1}\\quad \\text{for}\\ n \\geq 1.\n\\end{equation}\nNote that the sequence can be regarded as the analogue of the Fibonacci sequence.\nLet $s, t \\geq 1$ be integers and we put $q := 2 ^ s, r := 2 ^ t$.\nLet $\\ell \\geq 1$ be an integer, ${\\bm \\lambda} = (\\lambda_1, \\ldots , \\lambda_{\\ell}) \\in (\\lF_q ^ {*}) ^ {\\ell}, {\\bm \\varepsilon} = (\\varepsilon_1, \\varepsilon_2) \\in (\\lF_q ^ {*}) ^ 2$, and $(p_n \/ q_n)_{0 \\leq n \\leq \\ell - 1}$ be the convergent sequence of the continued fraction $[\\lambda_1 T, \\ldots , \\lambda_{\\ell}T]$.\nWe consider the algebraic equation\n\\begin{equation}\\label{eq:char2eq}\n\tq_{\\ell - 1} X ^ {r + 1} + p_{\\ell - 1} X ^ r + (\\varepsilon_1 q_{\\ell - 2} F_{r-1} + \\varepsilon_2 q_{\\ell - 1} F_{r - 2}) X + \\varepsilon_1 p_{\\ell - 2} F_{r - 1} + \\varepsilon_2 p_{\\ell  - 1} F_{r - 2} = 0.\n\\end{equation}\nLasjaunias \\cite{Lasjaunias17} proved that \\eqref{eq:char2eq} has a unique root $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ in $\\lF_q ((T ^ {- 1}))$ with $\\abs{\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})} \\geq q$, $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is hyperquadratic and its continued fraction expansion is the following:\n\\begin{equation}\\label{eq:secondfamily}\n\t\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}) = [\\lambda_1 T, \\lambda_2 T, \\ldots ],\n\\end{equation}\nwhere a sequence $(\\lambda_n)_{n \\geq 1}$ satisfy\n\\begin{gather*}\n\t\\lambda_{\\ell + r m + 1} = (\\varepsilon_2 \/ \\varepsilon_1) \\varepsilon_2 ^ {(- 1) ^ {m + 1}} \\lambda_{m + 1} ^ r,\\\\\n\t\\lambda_{\\ell + r m + i} = (\\varepsilon_1 \/ \\varepsilon_2 ) ^ {(- 1) ^ i} \\quad \\text{for}\\ m \\geq 0 \\ \\text{and}\\ 2 \\leq i \\leq r.\n\\end{gather*}\nNote that if $\\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_j = 1$ for all $1\\leq j \\leq \\ell$, then $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is ultimately periodic continued fraction, that is, $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ is quadratic.\n\nIn this paper, we consider the continued fractions $\\Phi_{\\ell } ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ in particular cases.\n\n\n\\begin{thm}\\label{thm:main2}\n\tLet $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ be as in \\eqref{eq:secondfamily}, $m \\geq 1, d \\geq 2$ be integers with $m \\leq \\ell$, and $\\lambda$ be in $\\lF_q ^ {*}$ with $\\lambda \\neq 1$.\n\t\\begin{enumerate}\n\t\t\\item[(1)] Assume that $\\lambda _1 = \\lambda , \\lambda _i = 1$ for all $2 \\leq i \\leq m, \\varepsilon_1 = \\varepsilon_2 = 1$ and if $m < \\ell $, then $\\lambda_{m + 1} \\neq 1$.\n\t\tThen we have\n\t\t\\begin{equation}\\label{eq:main22}\n\t\t\tw_2 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}))\\geq \\frac{m}{\\ell}(r - 1),\\quad \n\t\t\tw_2 (\\Phi_{\\ell} ^ t ({\\bm \\lambda }; {\\bm \\varepsilon})) \\geq 1 + \\frac{m}{\\ell} (r - 1).\n\t\t\\end{equation}\n\t\tIf\n\t\t\\begin{equation}\\label{eq:maincond3}\n\t\t\t\\frac{\\ell}{m} \\leq \\frac{2(r - 1)}{d + \\sqrt{d ^ 2 + 4(2r - 1)d + 4}},\n\t\t\\end{equation}\n\t\tthen we have\n\t\t\\begin{equation}\\label{eq:mainequal3}\n\t\t\tw_n ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = \\frac{m}{\\ell}(r - 1), \\quad \n\t\t\tw_n (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = 1 + \\frac{m}{\\ell}(r - 1)\n\t\t\\end{equation}\n\t\tfor all integers $2 \\leq n \\leq d$.\n\t\t\n\t\t\\item[(2)] Assume that $r$ is a power of $q, \\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_i = \\lambda$ for all $1 \\leq i \\leq \\ell$.\n\t\tThen we have\n\t\t\\begin{gather}\\label{eq:mainequal6}\n\t\t\tw_2 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) \\geq r - 1 - \\frac{r(r - 1)(r - 4)}{2\\ell r + (r - 1)(r - 2)},\\\\\n\t\t\t\\label{eq:mainequal66}\n\t\t\tw_2 (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) \\geq r - \\frac{r(r - 1)(r - 3)}{2\\ell r + (r - 1)(r - 2)}.\n\t\t\\end{gather}\n\t\tIf\n\t\t\\begin{equation}\\label{eq:maincond7}\n\t\t\t2 r d \\leq \\left( r - 2 - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)}\\right) \\left( r - d - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)}\\right) ,\n\t\t\\end{equation}\n\t\tthen we have\n\t\t\\begin{gather}\\label{eq:mainequal7}\n\t\t\tw_n ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = r - 1 - \\frac{r(r - 1)(r - 4)}{2 \\ell r + (r - 1)(r - 2)},\\\\\n\t\t\t\\label{eq:mainequal77}\n\t\t\tw_n (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = r - \\frac{r (r - 1)(r - 3)}{2 \\ell r + (r - 1)(r - 2)}\n\t\t\\end{gather}\n\t\tfor all integers $2 \\leq n \\leq d$.\n\t\\end{enumerate}\n\\end{thm}\n\n\n\\begin{rem}\n\tBy \\eqref{eq:w1_determine}, we have $w_1 (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = w_1 ^ {*} (\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})) = 1$.\n\tIf\n\t\\begin{equation*}\n\t\tr > \\frac{3d + 2 + \\sqrt{9d ^ 2 + 4d + 4}}{2},\n\t\\end{equation*}\n\tthere exists an effectively computable positive constant $C_2 (r, d)$, depending only on $r$ and $d$, such that we have \\eqref{eq:mainequal7} and \\eqref{eq:mainequal77} for all $\\ell \\geq C_2 (r, d)$.\n\\end{rem}\n\n\n\n\\begin{cor}\\label{cor:main2}\n\tLet $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$ be as in \\eqref{eq:secondfamily}, $m \\geq 1, d \\geq 2$ be integers with $m \\leq \\ell$, and $\\lambda$ be in $\\lF_q ^ {*}$ with $\\lambda \\neq 1$.\n\t\\begin{enumerate}\n\t\t\\item[(1)] Assume that $\\lambda _1 = \\lambda , \\lambda _i = 1$ for all $2 \\leq i \\leq m, \\varepsilon_1 = \\varepsilon_2 = 1$ and if $m < \\ell $, then $\\lambda_{m + 1} \\neq 1$.\n\t\tIf\n\t\t\\begin{equation*}\n\t\t\t\\frac{m}{\\ell} > \\frac{r - 2}{r - 1},\n\t\t\\end{equation*}\n\t\tthen we have $\\deg \\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon}) = r + 1$.\n\t\t\n\t\t\\item[(2)] Assume that $r$ is a power of $q, \\varepsilon_1 = \\varepsilon_2 = 1$ and $\\lambda_i = \\lambda$ for all $1 \\leq i \\leq \\ell$.\n\t\tIf\n\t\t\\begin{equation*}\n\t\t\t\\ell > \\frac{(r ^ 2 - 4r + 2)(r - 1)}{2r},\n\t\t\\end{equation*}\n\t\tthen we have $\\deg \\Phi_{\\ell} ^ t({\\bm \\lambda}; {\\bm \\varepsilon}) = r + 1$.\n\t\\end{enumerate}\n\\end{cor}\n\n\n\nIn the last part of this section, we mention a problem associated to Problem \\ref{prob:main1}, Corollary \\ref{cor:main1} and \\ref{cor:main2}.\n\n\n\n\\begin{prob}\n\tLet $n\\geq 2$ be an integer.\n\tDetermine the set of all values taken by $w_n - w_n ^ {*}$ over the set of algebraic Laurent series.\n\\end{prob}\n\n\n\n\n\n\n\\section{Preliminaries}\\label{sec:lemma}\n\n\n\nIn this section, we gather lemmas for the proof of main results.\n\n\n\n\\begin{lem}\\label{lem:alg}{\\rm (\\cite[Theorem 5.2]{Ooto17}).}\n\tLet $n \\geq 1$ be an integer and $\\alpha \\in \\lF_q ((T ^ {- 1}))$ be an algebraic Laurent series.\n\tThen we have \n\t\\begin{equation*}\n\t\tw_n ^ {*} (\\alpha), w_n (\\alpha) \\leq \\deg \\alpha - 1.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nLemma \\ref{lem:contidif} and \\ref{lem:conti.lem} are immediately seen.\n\n\n\n\\begin{lem}\\label{lem:contidif}\n\tLet $\\xi = [0, a_1, a_2, \\ldots ], \\zeta =[0, b_1, b_2, \\ldots ]$ be in $\\lF_q ((T ^ {- 1}))$.\n\tAssume that there exists an integer $k \\geq 1$ such that $a_n = b_n$ for all $1 \\leq n \\leq k$ and $a_{k + 1} \\neq  b_{k + 1}$.\n\tThen we have\n\t\\begin{equation*}\n\t|\\xi - \\zeta| = \\frac{\\abs{a_{k + 1} - b_{k + 1}}}{\\abs{a_{k + 1} b_{k + 1}}\\abs{q_k} ^ 2},\n\t\\end{equation*}\n\twhere $(p_n \/ q_n)_{n \\geq 0}$ is the convergent sequence of $\\xi$.\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:conti.lem}\n\tLet $\\xi = [a_0, a_1, \\ldots ]$ be in $\\lF_q ((T ^ {- 1}))$ and $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\xi$.\n\tThen, for any  $n \\geq 1$, we have\n\t\\begin{equation}\\label{enum:q}\n\t\t|q_n| = |a_1| |a_2| \\cdots |a_n|.\n\t\\end{equation}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:conj2}{\\rm (\\cite[Lemma 4.6]{Ooto17}).}\n\tLet $r, s \\geq 1$ be integers and $\\alpha = [0, a_1,\\ldots ,a_r, \\overline{a_{r + 1}, \\ldots , a_{r + s}}] \\in \\lF_q ((T ^ {- 1}))$ be an ultimately periodic continued fraction with $a_r \\neq a_{r + s}$.\n\tLet $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\alpha$.\n\tThen we have\n\t\\begin{equation*}\n\t\t\\frac{\\min (|a_r|, |a_{r + s}| )}{|q_r| ^ 2} \\leq |\\alpha - \\alpha '| \\leq \\frac{|a_r a_{r + s}|}{|q_r| ^ 2}.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following lemma is well-known and easy to see.\n\n\n\n\\begin{lem}\\label{lem:quadheightupper}\n\tLet $r \\geq 0, s \\geq 1$ be integers and $\\alpha = [0, a_1, \\ldots , a_r, \\overline{a_{r + 1}, \\ldots , a_{r + s}}] \\in \\lF_q ((T ^ {- 1}))$ be an ultimately periodic continued fraction.\n\tLet $(p_n \/ q_n)_{n \\geq 0}$ be the convergent sequence of $\\xi$.\n\tThen we have $H(\\alpha) \\leq |q_r q_{r + s}|$.\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:quadheight}\n\tLet $b, c, d \\in \\lF_q [T]$ be distinct polynomials with $\\deg_T b, \\deg_T c, \\deg_T d \\geq 1$.\n\tLet $n, m, \\ell \\geq 1$ be integers and $a_1, \\ldots , a_{n - 1} \\in \\lF_q [T]$ be polynomials with $\\deg_T a_i \\geq 1$ for all $1 \\leq i \\leq n - 1$.\n\tWe put\n\t\\begin{gather*}\n\t\t\\alpha_1 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b}], \\quad \n\t\t\\alpha_2 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b ^ {[m]}, d}],\\\\\n\t\t\\alpha_3 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{b ^ {[m]}, c, b ^ {[\\ell]}}], \\quad\n\t\t\\alpha_4 := [0, a_1, \\ldots , a_{n - 1}, c, \\overline{(b, d) ^ {[m]}, c, (b, d) ^ {[\\ell]}}].\n\t\\end{gather*}\n\tLet $(p_{i, k} \/ q_{i, k})_{k \\geq 0}$ be the convergent sequence of $\\xi_i$ for $i = 1, 2, 3, 4$.\n\tThen we have\n\t\\begin{gather}\\label{eq:quadheight12}\n\t\tH(\\alpha_1) \\asymp_{b, c} \\abs{q_{1, n}} ^ 2, \\quad\n\t\tH(\\alpha_2) \\asymp _{b, c, d} \\abs{q_{2, n} q_{2, n + m + 1}},\\\\\n\t\t\\label{eq:quadheight3}\n\t\tH(\\alpha_3) \\asymp_{b, c} \\abs{q_{3, n} q_{3, n + m + \\ell + 1}}, \\quad\n\t\tH(\\alpha_4) \\asymp_{b, c, d} \\abs{q_{4, n} q_{4, n + 2m + 2\\ell + 1}}.\n\t\\end{gather}\n\\end{lem}\n\n\n\n\\begin{proof}\n\tSee Lemma 4.7 in \\cite{Ooto17} for the proof of \\eqref{eq:quadheight12}.\n\t\n\tIt follows from Lemma \\ref{lem:quadheightupper} that $H(\\alpha_3) \\leq |q_{3, n} q_{3, n + m + \\ell + 1}|$.\n\tWe put \n\t\\begin{equation*}\n\t\t\\beta_3 :=[0, a_1, \\ldots , a_{n - 1}, c, b ^ {[m]}, c, \\overline{b}].\n\t\\end{equation*}\n\t\n\tLet $P_3(X)$ and $Q_3(X)$ be the minimal polynomial of $\\alpha_3$ and $\\beta_3$, respectively.\n\tSince $P_3$ and $Q_3$ do not have a common root, we have\n\t\\begin{equation*}\n\t\t1 \\leq |\\Res(P_3, Q_3)| \\leq H(\\beta_3) ^ 2 H(\\alpha_3) ^ 2 |\\alpha_3 - \\beta_3| |\\alpha_3' - \\beta_3| |\\alpha_3 - \\beta_3'| |\\alpha_3' - \\beta_3'|.\n\t\\end{equation*}\n\tBy Lemma \\ref{lem:contidif}, \\ref{lem:conti.lem} and \\ref{lem:conj2}, we obtain\n\t\\begin{gather*}\n\t\t|\\alpha_3 - \\beta_3| \\ll_{b, c} |q_{3, n + 2m + \\ell + 1}| ^ {- 2}, \\quad \n\t\t|\\alpha_3 - \\beta_3'| \\ll_{b, c} |q_{3, n + m + 1}| ^ {- 2},\\\\\n\t\t|\\alpha_3' - \\beta_3|, |\\alpha_3' - \\beta_3'| \\ll_{b, c} |q_{3, n}| ^ {- 2}.\n\t\\end{gather*}\n\tTherefore, by Lemma \\ref{lem:conti.lem} and \\ref{lem:quadheightupper}, we have $H(\\alpha_3) \\gg_{b, c} |q_{3, n} q_{3, n + m + \\ell + 1}|$.\n\t\n\tIn the same way to the above proof, we obtain $H(\\alpha_4) \\asymp_{b, c, d} \\abs{q_{4, n} q_{4, n + 2m + 2\\ell + 1}}$.\n\\end{proof}\n\n\n\n\\begin{lem}\\label{lem:wnwn*}{\\rm (\\cite[Proposition 5.6]{Ooto17}).}\n\tLet $n \\geq 1$ be an integer and $\\xi \\in \\lF_q ((T ^ {- 1}))$.\n\tThen we have \n\t\\begin{equation*}\n\t\tw_n ^ {*} (\\xi) \\leq w_n (\\xi).\n\t\\end{equation*}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:wnlower}\n\tLet $n, m \\geq 1$ be integers with $n \\geq m$ and $\\xi \\in \\lF_q ((T ^ {- 1}))$ be not algebraic of degree at most $m$.\n\tThen we have $w_n (\\xi) \\geq m$.\n\\end{lem}\n\n\n\n\\begin{proof}\n\tSince the sequence $(w_k (\\xi))_{k \\geq 1}$ is increasing, we have $w_n (\\xi) \\geq m$ (see e.g.\\ \\cite[page~200]{Bugeaud04}).\n\\end{proof}\n\n\n\nThe following lemma is well-known and immediately seen.\n\n\n\n\\begin{lem}\\label{lem:height}\n\tLet $P(X)$ be in $(\\lF_q [T]) [X]$.\n\tAssume that $P(X)$ can be factorized as\n\t\\begin{equation*}\n\t\tP(X) = A \\prod_{i = 1}^{n} (X - \\alpha_i),\n\t\\end{equation*}\n\twhere $A \\in \\lF_q [T]$ and $\\alpha_i \\in \\overline{\\lF_q (T)}$ for $1 \\leq i \\leq n$.\n\tThen we have\n\t\\begin{equation*}\n\t\tH(P) = |A| \\prod_{i = 1}^{n} \\max (1, |\\alpha_i|).\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following lemma is immediately seen by the definition of discriminant.\n\n\n\n\\begin{lem}\\label{lem:Galois}\n\tLet $\\alpha \\in \\overline{\\lF_q (T)}$ be a quadratic number.\n\tIf $\\alpha \\neq \\alpha'$, then we have\n\t\\begin{equation*}\n\t\t|\\alpha - \\alpha'| \\geq H(\\alpha) ^ {- 1}.\n\t\\end{equation*}\n\\end{lem}\n\n\n\nThe following two lemmas are key lemmas for proving Theorem \\ref{thm:main1} and \\ref{thm:main2}.\n\n\n\n\\begin{lem}\\label{lem:bestquad}{\\rm (\\cite[Lemma 3.19]{Ooto18}).}\n\tLet $d \\geq 2$ be an integer.\n\tLet $\\xi$ be in $\\lF_q ((T ^ {- 1}))$, $\\theta , \\delta$ be positive numbers, and $\\varepsilon$ be a non-negative number.\n\tAssume that there exist a sequence $(\\alpha_j)_{j \\geq 1}$ and positive number $c$ such that for any $j \\geq 1$, $\\alpha_j \\in \\overline{\\lF_q (T)}$ is quadratic with $0 < |\\alpha_j - \\alpha_j'| \\leq c$ and $\\xi \\neq \\alpha_j$,  $(H(\\alpha_j))_{j \\geq 1}$ is a divergent increasing sequence, and\n\t\\begin{gather*}\n\t\t\\limsup_{k \\rightarrow \\infty } \\frac{\\log H(\\alpha_{k + 1})}{\\log H(\\alpha_k)} \\leq \\theta ,\\\\\n\t\t\\lim_{k \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_k|}{\\log H(\\alpha_k)} = d + \\delta , \\quad \\lim_{k \\rightarrow \\infty } \\frac{- \\log |\\alpha_k - \\alpha_k'|}{\\log H(\\alpha_k)} = \\varepsilon .\n\t\\end{gather*}\n\tIf $2 d \\theta \\leq (d - 2 + \\delta ) \\delta $, then we have for all $2 \\leq n \\leq d$,\n\t\\begin{equation}\\label{eq:bestquad_eq}\n\t\tw_n ^ {*} (\\xi) = d - 1 + \\delta , \\quad w_n(\\xi) = d - 1 + \\delta + \\varepsilon .\n\t\\end{equation}\n\\end{lem}\n\n\n\n\\begin{lem}\\label{lem:bestquad2}\n\tLet $\\xi$ be in $\\lF_q ((T ^ {- 1})) \\setminus \\lF_q (T)$.\n\tAssume that there exists a sequence $(\\alpha_j)_{j \\geq 1}$ such that for any $j \\geq 1$, $\\alpha_j \\in \\overline{\\lF_q (T)}$ is quadratic with $\\alpha_j \\neq \\alpha_j'$ and $\\xi \\neq \\alpha_j$, and $(H(\\alpha_j))_{j \\geq 1}$ is a divergent increasing sequence.\n\tIf there exist limits of the sequences \n\t\\begin{equation*}\n\t\t\\left( \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)}\\right) _{j \\geq 1}\\ \\text{and}\\ \\left( \\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)}\\right) _{j \\geq 1},\n\t\\end{equation*}\n\tthen we have\n\t\\begin{gather}\\label{eq:quad1}\n\t\tw_2 ^ {*} (\\xi) \\geq \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)} - 1,\\\\\n\t\t\\label{eq:quad2}\n\t\tw_2 (\\xi) \\geq \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)} + \\lim_{j \\rightarrow \\infty }\\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)} - 1.\n\t\\end{gather}\n\\end{lem}\n\n\n\n\\begin{proof}\n\tThe inequality \\eqref{eq:quad1} is immediately seen.\n\tIn what follows, we show \\eqref{eq:quad2}.\n\tWe put\n\t\\begin{equation*}\n\t\t\\gamma_1 := \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\xi - \\alpha_j|}{\\log H(\\alpha_j)}, \\quad \\gamma_2 := \\lim_{j \\rightarrow \\infty } \\frac{- \\log |\\alpha_j - \\alpha_j'|}{\\log H(\\alpha_j)}.\n\t\\end{equation*}\n\t\n\tWe may assume that $\\gamma_1 > 1 $ and $\\gamma_2 > 0$.\n\tIn fact, if $\\gamma_2 \\leq 0$, then we have \\eqref{eq:quad2} by Lemma \\ref{lem:wnwn*}.\n\tBy Lemma \\ref{lem:Galois}, we have $\\gamma_2 \\leq 1$, which implies $\\gamma_2 < \\gamma_1$.\n\tIf $\\gamma_1 \\leq 1$, then we obtain \\eqref{eq:quad2} by Lemma \\ref{lem:wnlower}.\n\t\n\tThen we obtain\n\t\\begin{equation*}\n\t\t|\\xi - \\alpha_j| < |\\alpha_j - \\alpha_j'| \\leq 1\n\t\\end{equation*}\n\tfor all sufficiently large $j$.\n\tTherefore, we have\n\t\\begin{gather*}\n\t\t\\max (1, |\\xi|) = \\max (1, |\\alpha_j|) = \\max (1, |\\alpha_j'|),\\\\\n\t\t|\\xi - \\alpha_j'| = |\\alpha_j - \\alpha_j'|\n\t\\end{gather*}\n\tfor all sufficiently large $j$.\n\tIt follows from Lemma \\ref{lem:height} that\n\t\\begin{equation*}\n\t\tH(P_j) = |A_j| \\max (1, |\\xi|) ^ 2\n\t\\end{equation*}\n\tfor all sufficiently large $j$, where $P_j (X) = A_j (X - \\alpha_j) (X - \\alpha_j ')$ is the minimal polynomial of $\\alpha_j$.\n\tHence, we have\n\t\\begin{equation*}\n\t\t|P_j (\\xi)| = H(P_j) |\\xi - \\alpha_j| |\\alpha_j - \\alpha_j'| \\max (1, |\\xi|) ^ {- 2}\n\t\\end{equation*}\n\tfor all sufficiently large $j$.\n\tThus, we obtain\n\t\\begin{equation*}\n\t\t\\lim_{j\\rightarrow \\infty }\\frac{- \\log |P_j (\\xi)|}{\\log H(P_j)} = \\gamma_1 + \\gamma_2 - 1,\n\t\\end{equation*}\n\twhich implies \\eqref{eq:quad2}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Proof of Main results}\\label{sec:proof}\n\n\n\nIn this section, we prove the main results, that is, Theorem \\ref{thm:neq_alg}, \\ref{thm:main1}, \\ref{thm:main2}, Corollary \\ref{cor:main1} and \\ref{cor:main2}.\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main1}]\n\tFor an integer $n \\geq 3$, we put\n\t\\begin{gather*}\n\t\t\\alpha_n :=\n\t\t\\begin{cases}\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1}(T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, T, \\overline{\\lambda T, \\mu T}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1}(T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, T, \\overline{\\lambda T}] & \\text{if}\\ p = 2,\n\t\t\\end{cases}\\\\\n\t\t\\beta_n :=\n\t\t\\begin{cases}\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1} (T, (\\lambda T, \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, \\overline{T, (\\lambda T, \\mu T) ^ {[(r ^ n - 1) \/ 2]}}] & \\text{if}\\ p \\neq 2,\\\\\n\t\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 1} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, \\overline{T, \\lambda T ^ {[r ^ n - 1]}}] & \\text{if}\\ p = 2.\n\t\t\\end{cases}\n\t\\end{gather*}\n\tThen we have\n\t\\begin{align*}\n\t\t\\beta_n =\n\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 2}(T, (\\lambda T, & \\mu T) ^ {[(r ^ i - 1) \/ 2]}) ^ {[k + 1]}, (T, (\\lambda T, \\mu T) ^ {[(r ^ {n - 1} - 1) \/ 2]})^{[k]},\\\\\n\t\t& T, \\overline{(\\lambda T, \\mu T) ^ {[(r ^ {n - 1} - 1) \/ 2]}, T, (\\lambda T, \\mu T) ^ {[(r ^ n - r ^ {n - 1}) \/ 2]}}]\t\n\t\\end{align*}\n\tif $p \\neq 2$, and\n\t\\begin{align*}\n\t\t\\beta_n\t=\n\t\t[0, T ^ {[k]}, \\bigoplus_{1 \\leq i \\leq n - 2} (T, \\lambda T ^ {[r ^ i - 1]}) ^ {[k + 1]}, (T, \\lambda T ^ {[r ^ {n - 1} - 1]}) ^ {[k]}, T, \\overline{\\lambda T ^ {[r ^ {n - 1} - 1]}, T, \\lambda T ^ {[r ^ n - r ^ {n - 1}]}}]\n\t\\end{align*}\n\tif $p = 2$.\n\tIt follows from Lemma \\ref{lem:conti.lem} and \\ref{lem:quadheight} that\n\t\\begin{equation*}\n\t\tH(\\alpha_n) \\asymp q ^ {2 (k + 1) (r ^ n - 1) \/ (r - 1)}, \\quad\n\t\tH(\\beta_n) \\asymp q ^ {2 (k + 1) (r ^ n - 1) \/ (r - 1) + r ^ n - 2 r ^ {n - 1}}.\n\t\\end{equation*}\n\tSince $\\Theta_k ^ {t} (\\lambda)$ and $\\alpha_n$ have the same first $((k + 1) \\sum_{i = 0} ^ {n - 1} r ^ i + r ^ n - 1)$-th partial quotients, while the next partial quotient are different, we have\n\t\\begin{equation*}\n\t\t\\abs{\\Theta_k ^ {t} (\\lambda) - \\alpha_n} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1) - 2 r ^ n}\n\t\\end{equation*}\n\tby Lemma \\ref{lem:contidif} and \\ref{lem:conti.lem}.\n\tSimilary, since $\\Theta_k ^ {t} (\\lambda)$ and $\\beta_n$ have the same first $((k + 1) \\sum_{i = 0} ^ {n} r ^ i + r ^ n - 1)$-th partial quotients, while the next partial quotient are different, we get\n\t\\begin{equation*}\n\t\t\\abs{\\Theta_k ^ {t} (\\lambda) - \\beta_n} \\asymp q ^ {- 2 (k + 1) (r ^ {n + 1} - 1) \/ (r - 1) - 2 r ^ n}.\n\t\\end{equation*}\n\tBy Lemma \\ref{lem:conti.lem} and \\ref{lem:conj2}, we obtain\n\t\\begin{equation*}\n\t\t\\abs{\\alpha_n - \\alpha_n'} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1)},\\quad\n\t\t\\abs{\\beta_n - \\beta_n'} \\asymp q ^ {- 2 (k + 1) (r ^ n - 1) \/ (r - 1) + 2 r ^ {n - 1}}.\n\t\\end{equation*}\n\tTherefore, we deduce that\n\t\\begin{gather*}\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{\\log H(\\alpha_{n + 1})}{\\log H(\\alpha_n)} = \\lim_{n \\rightarrow \\infty } \\frac{\\log H(\\beta_{n + 1})}{\\log H(\\beta_n)} = r,\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\Theta_k ^ {t} (\\lambda) - \\alpha_n}}{\\log H(\\alpha_n)} = 1 + \\frac{r - 1}{k + 1},\\quad \n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\alpha_n - \\alpha_n'}}{\\log H(\\alpha_n)} = 1,\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\Theta_k ^ {t} (\\lambda) - \\beta_n}}{\\log H(\\beta_n)} = r - \\frac{r (r - 1)(r - 4)}{2 r (k + 1) + (r - 1) (r - 2)},\\\\\n\t\t\\lim_{n \\rightarrow \\infty } \\frac{- \\log \\abs{\\beta_n - \\beta_n'}}{\\log H(\\beta_n)} = 1 - \\frac{r (r - 1)}{2 r (k + 1) + (r - 1)(r - 2)}.\n\t\\end{gather*}\n\tHence, we obtain \\eqref{eq:w2*} and \\eqref{eq:w2} by Lemma \\ref{lem:bestquad2}.\n\t\n\tIf \\eqref{eq:maincond1} holds, then we heve\n\t\\begin{equation*}\n\t\t2 r d \\leq \\left( \\frac{r - 1}{k + 1} - 1\\right) \\left( \\frac{r - 1}{k + 1} - d + 1 \\right) .\n\t\\end{equation*}\n\tTherefore, by Lemma \\ref{lem:bestquad}, we obtain \\eqref{eq:mainequal1}.\n\t\n\tSimilaly, if \\eqref{eq:maincond2} holds, then we heve \\eqref{eq:mainequal2} and \\eqref{eq:mainequal22} by Lemma \\ref{lem:bestquad}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main1}]\n\tIt follows from \\eqref{eq:hyperquad} that $\\deg \\Theta_k ^ {t} (\\lambda) \\leq r+1$.\n\tIf \\eqref{eq:main1_cor_cond} holds, then we have $w_2(\\Theta_k ^ {t} (\\lambda)) > r-1$ by Theorem \\ref{thm:main1}.\n\tTherefore, by Lemma \\ref{lem:alg}, we obtain $\\deg \\Theta_k ^ {t} (\\lambda) = r+1$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:neq_alg}]\n\tIt follows from $q \\geq 4$ that we can take $\\lambda \\in \\lF_q ^ {*}$ with $\\lambda \\neq 1$ and $\\lambda \\neq 2$.\n\tLet $n \\geq 2$ be integers.\n\tSince $r \\geq (3n + 2 + \\sqrt{9n^2 + 4n + 4})\/2$, we take $r = p^t$, where $t \\geq 0$ is an integer with\n\t\\begin{equation*}\n\t\t0 \\leq \\frac{2(r - 1)}{n + \\sqrt{n ^ 2 + 4(2r - 1)n + 4}} - 1.\n\t\\end{equation*}\n\tThen, by Theorem \\ref{thm:main1} and Corollary \\ref{cor:main1}, we have $w_n(\\Theta_0 ^ {t} (\\lambda)) \\neq w_n^{*}(\\Theta_0 ^ {t} (\\lambda))$ and $\\deg \\Theta_0 ^ {t} (\\lambda) = r + 1$.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main2}]\n\tSince $w_n (\\xi) = w_n (\\xi^{- 1})$ and $w_n ^ {*} (\\xi) = w_n ^ {*} (\\xi ^{- 1})$ for all $0 \\neq \\xi \\in \\lF_q ((T ^ {- 1}))$ and $n \\geq 1$, we consider $\\Phi := \\Phi_{\\ell} ^ t({\\bm \\lambda}; {\\bm \\varepsilon}) ^ {- 1} = [0, \\lambda_1T, \\lambda_2T, \\ldots ]$ instead of $\\Phi_{\\ell} ^ t ({\\bm \\lambda}; {\\bm \\varepsilon})$.\n\tBy the assumption, we obtain\n\t\\begin{equation*}\n\t\t\\lambda_n\n\t\t=\n\t\t\\begin{cases}\n\t\t\t\\lambda_{i + 1} ^ {r ^ j} & \\text{if}\\ n = 1 + \\ell \\sum_{h = 0} ^ {j - 1} r ^ h + i r ^ j \\ \\text{for some}\\ j \\geq 0, 0 \\leq i < \\ell ,\\\\\n\t\t\t1 & \\text{otherwise}.\n\t\t\\end{cases}\n\t\\end{equation*}\n\t\n\tFirst, we prove (1).\n\tFor an integer $n \\geq 1$, we put\n\t\\begin{equation*}\n\t\t\\alpha_n := [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n)}T, \\overline{T}],\n\t\\end{equation*}\n\twhere $i(n) = 1 + \\ell \\sum_{j = 0} ^ {n - 1} r ^ j$.\n\tThen we have $\\lambda_{i(n)} = \\lambda ^ {r ^ n}$.\n\tIn a similar way to the proof of Theorem \\ref{thm:main1}, we obtain\n\t\\begin{gather*}\n\t\\abs{\\Phi - \\alpha_n} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) - 2 m r ^ n},\\quad \n\t\\abs{\\alpha_n - \\alpha_n'} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1)},\\\\\n\tH(\\alpha_n) \\asymp q ^ {2 \\ell (r ^ n - 1) \/ (r - 1)}.\n\t\\end{gather*}\n\tTherefore, we have \\eqref{eq:main22} by Lemma \\ref{lem:bestquad2}.\n\tIf \\eqref{eq:maincond3} holds, then we obtain \\eqref{eq:mainequal3} by Lemma \\ref{lem:bestquad}.\n\t\n\tNext, we prove (2).\n\tFor an integer $n \\geq 1$, we put\n\t\\begin{equation*}\n\t\\beta_n := [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n) - 1}T, \\overline{\\lambda ^ {r ^ n}T, T ^ {[r ^ n - 1]}}],\n\t\\end{equation*}\n\tThen we have\n\t\\begin{equation*}\n\t\\beta_n = [0, \\lambda_1T, \\lambda_2T, \\ldots ,\\lambda_{i(n) - r ^ {n - 1}}T, \\overline{T ^ {[r ^ {n - 1} - 1]},\\lambda T, T ^ {[r ^ n - r ^ {n - 1}]}}],\n\t\\end{equation*}\n\tand $\\lambda_{i(n) - r ^ {n - 1}}=\\lambda$.\n\tIn a similar way to the proof of Theorem \\ref{thm:main1}, we obtain\n\t\\begin{gather*}\n\t\\abs{\\Phi - \\beta_n} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) - 2 m r ^ n},\\quad \n\t\\abs{\\beta_n - \\beta_n'} \\asymp q ^ {- 2 \\ell (r ^ n - 1) \/ (r - 1) + 2 r ^ {n - 1}},\\\\\n\tH(\\beta_n) \\asymp q ^{2 \\ell (r ^ n - 1) \/ (r - 1) + r ^ n - 2 r ^ {n - 1}}.\n\t\\end{gather*}\n\tTherefore, we have \\eqref{eq:mainequal6} and \\eqref{eq:mainequal66} by Lemma \\ref{lem:bestquad2}.\n\tIf \\eqref{eq:maincond7} holds, then we obtain \\eqref{eq:mainequal7} and \\eqref{eq:mainequal77} by Lemma \\ref{lem:bestquad}.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Corollary \\ref{cor:main2}]\n\tIn the same way to the proof Corollary \\ref{cor:main1}, we prove Corollary \\ref{cor:main2}.\n\\end{proof}\n\n\n\n\n\n\\subsection*{Acknowledgements}\nThe author would like to thank the referee for helpful comments.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn the eighteenth century, the famous astronomer William Herschel showed us the\npowerful method of star count to understand our own Milky Way \\citep{Hers1785}.\nThe technique has been used since then by generations of astronomers. With\ngreat improvement on data collection over the years, more and more details of\nour Milky Way were unfolded. However, the characteristic scales of smooth\nGalactic structures (i.e., disks and halo) obtained by previous studies does\nnot converge to a common value as an outcome of improving data collection\n(see Table~\\ref{table_ref}). The spread of values is attributed to the\ndegeneracy of Galactic model parameters (i.e., same star count data could be\nfitted equally well by different Galactic models) \\citep{Chen2001,\nSiegel2002, Juric2008, Bilir2008}.\nThis is due to the different sky regions\nand limiting magnitudes (i.e., limiting volumes) used in these studies\n\\citep{Siegel2002, Karaali2004, Bilir2006a, Bilir2006b, Juric2008}. On the\ncontrary, \\citet{Bilir2008, Yaz2010} did not show such degeneracy in determining\nthe Galactic model parameters. This controversy is still\nactively debated.\nTherefore, systematic all sky surveys with deeper limiting magnitude and wider sky\nregion, such as the Two Micron All Sky Survey \\citep[2MASS;][]{Skrutskie2006},\nthe Sloan Digital Sky Survey \\citep[SDSS;][]{York2000}, the Panoramic Survey\nTelescope \\& Rapid Response System \\citep[Pan-Starrs;][]{Kaiser2002} and the\nGAIA mission \\citep{Perryman2001} could provide a good opportunity for us to\nstudy our Galaxy from a global\nperspective. Free from limited sky\nfields, astronomers can acquire many more information from the stellar distribution\nof these surveys.\n\nOn Galactic structure study, besides the simple and smooth\ntwo-component model \\citep{Bahcall1980} or the three-component model\n\\citep{Gilmore1985}, many more structures have been discovered, such as\ninner bars in the Galactic center\n\\citep{Alves2000, Hammersley2000, vanLoon2003, Nishiyama2005, Cabrera2008},\nand flares and warps \\citep{Lopez2002, Robin2003, Momany2006, Reyle2009},\nwhich has been contributed the variation of disk model parameters with\nGalactic longitude \\citep{Cabrera2007, Bilir2008, Yaz2010}.\nMoreover, the overdensities in the halo, such as Sagittarius\n\\citep{Majewski2003}, Triangulum-Andromeda \\citep{Rocha2004, Majewski2004},\nVirgo \\citep{Juric2008}, and in the outer disk, such as Canis Major\n\\citep{Martin2004}, Monoceros \\citep{Newberg2002,Rocha2003}, show the\ncomplexity of the Milky Way. The formation history of our Galaxy is more\ncomplicated than what we thought previously.\n\nOn stellar luminosity function\nstudy, a recent study by \\citet{Chang2010} using the $K_s$ band star count of\n2MASS point source catalog \\citep[2MASS PSC;][]{Cutri2003} verified for\nthe first time the universality hypothesis of the luminosity function (i.e.,\nthe common practice of assuming one luminosity function for the entire Milky\nWay).\n\nWe are interested in the global smooth structure of our Galaxy. In view of the\nexisting fine structures (e.g., flares, warps, overdensities\\dots etc.), the\nstructure of the smooth components can be determined either by including these\nfine structures in a grand Galaxy model or by avoiding the sky area\n``contaminated'' with these features. The first method demands a complex model,\nwhich involves many more structure parameters, and needs high computing power\nto accomplish the fitting task. \\citet{Lopez2002} is a good example of this\nmethod using 2MASS data. The second method is clearly simpler but needs\njustifications. We observe that (1) the fine structures (e.g., inner bars,\nflares and warps), which have observable contribution on 2MASS star count data,\nare all confined in the Galactic plane region.\n(2) The overdensities or substructures in the outer disk region and the halo are\ndifficult to identify in general.\nTheir contribution is negligible in the 2MASS star count data.\nHere we quote \\citet{Majewski2004} on halo substructure:\n``This substructure is typically subtle and obscured by a substantial\nforeground veil of disk stars, eliciting its presence requires strategies that\noptimize the substructure signal compared to the foreground noise.''\n\nWe prefer the second method in this paper and only use Galactic latitude\n$|b|>30^{\\circ}$ 2MASS $K_s$ band star\ncount data to obtain the structure parameters of a three-component model.\nIn addition, the influence of the near infrared extinction in these\nregions is small. This also allows us to use a simpler extinction model for\ncorrection. We describe our model in section 2 and the analysis method in\nsection 3. Section 4 provides the results and a discussion.\n\n\n\n\\section{The Milky Way Model}\nWe adopt a three-component model for the smooth stellar distribution of the\nMilky Way. It comprises a thin disk, a thick disk and an oblate halo\n\\citep{Bahcall1980, Gilmore1983}. The total stellar density $n(R,Z)$\nat a location $(R,Z)$\nis the sum of the thin disk $D_1$, the think disk $D_2$ and the\nhalo $H$,\n\\begin{equation}\\label{density}\n  n(R,Z)=n_0\\left[D_1(R,Z)+D_2(R,Z)+H(R,Z)\\right]\\,,\n\\end{equation}\nwhere $R$ is the galactocentric distance on the Galactic plane, $Z$ is the\ndistance from the the Galactic mid-plane and $n_0$ is the local stellar\ndensity of the thin disk at the solar neighborhood.\n\nThe stellar distribution of the thin disk $D_1$ and the thick disk $D_2$\ndecreases exponentially along $R$ and $Z$ (the so called double exponential\ndisk),\n\\begin{equation}\\label{Di}\n  D_i(R,Z)=f_i\\exp\\left[-\\,{(R-R_\\odot)\\over H_{ri}}-\\,{(|Z|-|Z_\\odot|)\\over H_{zi}}\\right]\\,,\n\\end{equation}\nwhere $(R_\\odot,Z_\\odot)$ is the location of the Sun, $H_{ri}$ is the\nscale-length, $H_{zi}$ is the scale-height, and $f_i$ is the density ratio to\nthe thin disk at the solar neighborhood. The subscript $i=1$ stands for the\nthin disk (thus $f_1=1$) and $i=2$ for thick disk. We adopted $R_\\odot=8$ kpc\nin our model \\citep{Reid1993}.\n\nThe halo is a power law decay oblate spheroid flattening in the $Z$ direction,\n\\begin{equation}\\label{SH}\n  H(R,Z)=f_h\\left[R^2+(Z\/\\kappa)^2\\over R_\\odot^2+(Z_\\odot\/\\kappa)^2\\right]^{-p\/2}\\,,\n\\end{equation}\nwhere $\\kappa$ is the axis ratio, $p$ is the power index and $f_h$ is the local\nhalo-to-thin disk density ratio.\n\n\\citet{Chang2010} showed that the whole sky $K_s$ band luminosity function can\nbe well approximated by a single power law with a power law index\n$\\gamma=1.85\\pm0.035$, a bright cutoff at $M_b=-7.86\\pm0.60$ and a faint cutoff\nat $M_f=6.88\\pm0.66$. We adopt this luminosity function in the following analysis.\nIn normalized form, it is\n\\begin{equation}\\label{LF}\n  \\psi(M_{K_s})={2\\log_{\\rm e}10\\ (\\gamma-1)\\over\n  5\\left[10^{2(\\gamma-1)M_{f}\/5}-10^{2(\\gamma-1)M_{b}\/5}\\right]}\\,10^{2(\\gamma-1)M\/5}\\,.\n\\end{equation}\nNote that $\\psi(M)$ includes all luminosity classes.\nIn our analysis the observing magnitude range is $5 \\le K_s \\le 14$ mag.\nThe corresponding distances\nof bright cutoff, faint cutoff and $M_{K_s}=0$ are 3.4 kpc to 216 kpc, 4 pc to\n265 pc and 0.1 kpc to 6.3 kpc, respectively.\n\nAlthough we only use NIR data in the medium and high Galactic latitude regions,\nwe still need to correct possible interstellar extinction. We adopt the new\nCOBE\/IRAS result \\citep{Chen1999} and convert it to the $K_s$ band extinction\nby $A_{K_s}\/E(B-V)=0.367$ \\citep{Schlegel1998}. This extinction model is then\napplied to our simulation data. The extinction values of most of our analyzed\nregions are $A_{K_s}< 0.03$.\n\n\n\n\\section{The Data and Analysis Method}\n\\subsection{The 2MASS Data}\nThe 2MASS Point Source Catalog \\citep[2MASS PSC,][]{Cutri2003} is employed to\ncarry out the $K_s$ band differential star count for the entire Milky Way. We\ndivide the whole sky into 8192 nodes according to level 5 Hierarchical\nTriangular Mesh \\citep[HTM,][]{Kunszt2001}. The level 5 HTM samples the whole\nsky in roughly equal area with an average angular distance about 2 degrees\nbetween any two neighboring nodes. The amount of stars within 1 degree radius\nof each node (i.e., each node covers $\\pi$ square degree) is then retrieved with\na bin size $K_s$=0.5 mag via 2MASS online data service\n\\citep[][catalog]{Cutri2003}. Our selection criterion is: the object must be\ndetected in all $J, H, K_s$ bands and has signal-to-noise ratio $\\ge 5$. Since\nthe limiting magnitude of 2MASS $K_s$ band is 14.3 mag, which has 10\nsignal-to-noise ratio and 99\\% completeness \\citep[see Table~1\nin][]{Skrutskie2006}, and $K_s \\le 5$ mag objects have relatively large\nphotometric error, we only compare 2MASS data with our simulation data from\n$K_s=5$ to $14$ mag.\n\nIn order to minimize the effects coming from the close-to-Galactic-plane fine\nstructures (e.g., flares, warps, arms, budge and bars\\dots etc., which have\nconsiderable contribution to the star count data), and the relatively complex\nextinction correction at the low Galactic latitude region,\nwe avoid low galactic latitudes, and only consider data in Galactic\nlatitude $|b|>30^{\\circ}$. Although several\noverdensities in the halo (such as Sagittarius, Triangulum-Andromeda,\nVirgo\\dots etc.) were identified, they cannot be\npicked up\nfrom the\noverwhelming foreground field stars on star count data without additional\ninformation \\citep[e.g., color, distance, metallicity\\dots\netc.,][]{Majewski2004}. Therefore, their contribution to the 2MASS $K_s$ band\ndifferential star count is negligible and will not affect our result. We also\nexclude the areas around Large and Small Magellanic Clouds for their\nsignificant stellar population.\n\n\n\\subsection{The Analysis Method}\\label{method}\nThe Maximum Likelihood Method (Bienayme et al. 1987) is applied to compare the\n$K_s$ band 2MASS differential star counts and the simulation data to search for\nthe best-fit structure parameters of the three-component Milky Way model. Our\nfitting strategy is as follows:\n\\begin{enumerate}\\label{fitting_steps}\n\\item Take $R_\\odot=8$ kpc;\n\\item choose one $Z_\\odot$ and work out the maximum likelihood value by fitting the 9 parameters\n$(n_0,H_{z1},H_{r1},f_2,H_{z2},H_{r2},f_h,\\kappa,p)$;\n\\item repeat step 2 for other $Z_\\odot$;\n\\item pick the $Z_\\odot$ corresponds to the maximum of the maximum likelihood values in step 3;\n\\item repeat steps 2 to 4 for finer grid size of the 9-parameter fit and a narrower range of\n$Z_\\odot$ around the one found in step 4;\n\\item the uncertainty is estimated by adding Poisson noise on the simulation data to see\nhow the likelihood varies. The difference of the likelihoods of 500\nrealizations of the same model, differed by the Poisson statistics only,\ngives a range of likelihood around the maximum likelihood that defines the\nconfidence level.\n\\end{enumerate}\nTable~\\ref{para_space} lists our searching parameter space and the finest grid\nsize we used. The key parameters in our study are $n_0$, $H_{z1}$, $f_2$ and\n$H_{z2}$ (see Eqs. (1)-(2)). The first two play a primary role on the variation\nof the likelihood value and the latter two play a secondary role. The other\nfive parameters $H_{r1}$, $H_{r2}$, $f_h$, $\\kappa$ and $p$ (see Eqs. (1)-(3))\nare non-key parameters, which play a minor role and do not affect the\nlikelihood value as much as the key parameters.\n\n\\section{The Results and Discussion}\nTable~\\ref{result} lists our best-fit results with the corresponding\nuncertainties. Fig.~\\ref{2Dcon} shows contour plots of likelihood against\ndifferent pairs of parameters. The contour changes dramatically along the key\nparameters $n_0$, $H_{z1}$, $f_2$ and $H_{z2}$, but relatively mild along other\nparameters $H_{r1}$, $H_{r2}$, $f_h$, $p$ and $\\kappa$.\nThis indicates the importance of the key parameters in determining the best-fit result.\n\nDegeneracies exist between some pairs of key parameters, such as $(n_0,\nH_{z1})$, $(n_0, f_2)$, $(H_{z1}, f_2)$\\dots etc. Here degeneracy means that\nthe likelihood value stays almost the same when the pairs of parameters change\ntogether in a particular way. Similar degeneracy between the local\nthick-to-thin disk ratio $f_2$ and the scale-height of the thick disk $H_{z2}$\nhas been reported in \\citet{Chen2001,Siegel2002,Juric2008,Bilir2008}.\nConsequently, it is possible that different combinations of parameters can be\nregarded as `acceptable' fitting. For example, if we choose a higher local\nstellar density $n_0$, then we can pick a smaller $H_{z1}$ such that the\nlikelihood value is very close to the maximum likelihood value and assign it as\nthe `best-match' scale-height of the thin disk. Therefore, a thin light color\ndiagonal strip shows on the $n_0$ against $H_{z1}$ contour plot in\nFig.~\\ref{2Dcon}. Besides, similar trends happen in other pairs of key\nparameters. When one parameter of the pair is higher, we can get a similar\nlikelihood value by lowering the other parameter (see the corresponding\ntwo-parameter contour plots of key parameters in Fig.~\\ref{2Dcon}). This\nanti-correlation is not unexpected. The number of stars along the line of sight\nin the model increases when any one of the key parameters increases. Thus for a\ngiven observed number of stars, an increase in one key parameter can be\ncompensated by a decrease in the other. Perhaps this is the reason that our\nbest-fit scale-height of the thin disk, $H_{z1}=360$ pc, is somewhat larger\nthan the reported values, $H_{z1}=285$ pc, in \\citet{Lopez2002} study (they\nalso use star count of 2MASS to obtain Galactic model parameters). Our best-fit\nlocal stellar density from $K_s=$-8 to 6.5 mag is $n_0\\sim 0.030$ star\/pc$^3$,\nwhich is about half of $\\sim 0.056$ star\/pc$^3$ of the corresponding value\ncited in \\citet{Eaton1984}, and a bit lower than $\\sim 0.032$ star\/pc$^3$ of\nthe corresponding value cited in \\citet{Lopez2002}. As a result, our fitting\ntends to choose a larger scale-height of the thin disk. If we force our local\nstellar density to be comparable with that of \\citet{Lopez2002}, then the\ncorresponding `best-fit' scale-height of the thin disk would be $\\sim$320 pc,\nwhich is closer to their result. If we choose even higher local stellar\ndensity, then the `best-fit' scale-height of the thin disk would be made\nbetween 200 to 300 pc, which is similar to the most of recent studies (see\nTable~\\ref{table_ref}). Moreover, we do not apply binarism correction in our\nanalysis, and it has been shown that scale-length and scale-height might be\nunderestimated without binarism correction \\citep{Siegel2002, Juric2008,\nIvezic2008, Yaz2010}.\n\n\nFor our purpose, we deem that in order to lift the degeneracy it is crucial to\nhave a reliable near infrared luminosity function by observation or a near\ninfrared local stellar density. Unfortunately, a systematic study in this\ndirection is yet to come. Some related studies, such as synthetic luminosity\nfunction \\citep[see e.g.,][]{Girardi2005} or luminosity function transformed\nfrom optical observation \\citep[see e.g.,][]{Wainscoat1992} do exist, but some\nuncertainties still need to be settled (e.g., the initial mass function,\nmass-luminosity relation for NIR and color transformation between different\nwavelengths\\dots etc.). Once the `true' local stellar density is known, the\n`true' structure of the Milky Way would be revealed.\n\n\n\nIn order to see how good the agreement between 2MASS data and our best-fit\nmodel, we show an all sky map of the ratios of observed to predicted integrated\nstar count from $K_s=$5 to 14 mag for each node as a function of position on\nthe sky in Fig.~\\ref{comparison}, and the color indicates the values of the\nratios. We do not see obvious deviation in the Galactic latitude\n$|b|>30^{\\circ}$ areas, but only in the Large Magellanic Cloud and the Small\nMagellanic Cloud areas. For comfirmation, we plot integrated star count from\n$K_s=$5 to 14 mag along Galactic longitude and latitude for $|b|>30^{\\circ}$\nareas in Figs.~\\ref{surf_b} \\& \\ref{surf_l}. We see 2MASS data and our best-fit\nmodel agree well and only some small deviations in the nodes at the\nanti-Galactic center $b\\sim30^{\\circ}$ areas. The significant spikes in\nFigs.~\\ref{surf_b} \\& \\ref{surf_l} are due to the populations of the Large\nMagellanic Cloud and the Small Magellanic Cloud. For testing how these small\ndeviations affect the key parameters selection, we exclude these small\ndeviations and re-analyze the data with fixed non-key parameters. It shows only\nthe scale-height of thick disk shifts slightly but still within our error\nestimation. Besides, we do not see any significant differences of reported\noverdensities in sky regions with $|b|>30^{\\circ}$. Thus, we conclude that the\nthree-component model can describe the Milky Way structure sufficiently well\nfor high Galactic latitude regions and the single power law luminosity function\nof \\citet{Chang2010} is a good approximation as well. Since our main purpose is\nto search for the global Galactic model parameters, we do not try to explore\nthe best-fit result for each node individually as what \\citet{Cabrera2007,\nBilir2008, Yaz2010} have done in their studies to seek the variations of disk\nmodel parameters with Galactic longitude. Instead, we treat the differences of\n2MASS data to our best-fit model as deviations from a global smooth\ndistribution. We believe that similar variations in disk model parameters will\nbe obtained if we take similar analysis procedure (i.e., searching best-fit\nparameters for each node), but this is beyond the scope of this work.\nBecause we do not consider flares, warps and other overdensities in our\nmodel, there are some discrepancies between 2MASS data and the model in the low\nGalactic latitude regions (see Fig.~\\ref{comparison}). These fine structures\nmake the star distribution more fluffy in the vertical direction toward the\nedge of the Milky Way. Hence the discrepancy between 2MASS data and the model\nincreases vertically towards the anti-Galactic center region. In addition, the\nabsence of Galactic bulge in our model contributes to the large discrepancy in\nGalactic center areas. The difference in the low Galactic latitude region needs\nmore delicate analysis to rectify \\citep[see, e.g.,][]{Lopez2002, Momany2006}.\n\n\\subsection{Summary}\nIn summary, we set forth to study the global smooth structure of the Milky Way\nby a three-component stellar distribution model which comprises two double\nexponential disks (one thin and one thick) and an oblate halo. The $K_s$ band\n2MASS star count is used to determine the structure parameters. To avoid the\ncomplication introduced by the fine structures and complex extinction\ncorrection close to Galactic plane, we use only Galactic latitude\n$|b|>30^{\\circ}$ data. There are 10 parameters in the model, but only four of\nthem play the dominant  role in the fitting process. They are the local stellar\ndensity of the thin disk $n_0$, the local density ratio of thick-to-thin disk\n$f_2$, and the scale-height of the thin and thick disks $H_{z1}$ and $H_{z2}$.\nThe best-fit result is listed in Table~\\ref{result}. In short the scale-height\nof the thin and thick disks are $360\\pm 10$ pc and $1020\\pm 30$ pc,\nrespectively; the scale-length of the thin disk is $3.7\\pm 1.0$ kpc and that of\nthick disk is $5.0\\pm 1.0$ kpc (the uncertainty in scale-length is large\nbecause it is not very sensitive to high latitude data.) The local stellar\ndensity ratio of thick-to-thin disk and halo-to-thin disk are $7\\pm 1$\\% and\n$0.20\\pm 0.10$\\%, respectively. The local stellar density of the thin disk is\n$0.030\\pm 0.002$ stars\/pc$^3$.\n\nAn all sky comparison of the 2MASS data to our best-fit model is shown in\nFig.~\\ref{comparison}. A good agreement in the Galactic latitude\n$|b|>30^{\\circ}$ areas is expected from our fitting procedure. In low Galactic\nlatitude regions, fine structures (such as flares, warps\\dots etc.) increase\nthe effective scale-height towards the edge of the Milky way. This is reflected\nin the fan-like increase in discrepancy towards the anti-Galactic center\nregions.\n\nDegeneracy (i.e., different combinations of parameters give similar likelihood\nvalues) is found in pairs of key parameters\n(see Fig.~\\ref{2Dcon}). Thus different combinations of parameters may fit the\ndata almost as good as the best-fit one, and these are all legitimate\n`acceptable' fitting in view of the uncertainty. Therefore, accompanying our\nlower local stellar density $0.030$ stars\/pc$^3$ (from $K_s=-$8 to 6.5 mag) is\na higher thin disk scale-height $360$ pc. In the context of NIR star count, the\nNIR luminosity function or the NIR local stellar density is imperative to\ndetermine the scale-height and other Milky Way structure parameters. We hope\nthat systematic study on the luminosity function and the local stellar density\nin near infrared will be available in the near future.\n\n\n\\acknowledgements\nWe acknowledge the use of the Two Micron All Sky Survey Point\nSource Catalog (2MASS PSC).\nWe would like to thank the anonymous referee, whose advice greatly improves the\npaper.\nCMK is grateful to S. Kwok and K.S. Cheng for their\nhospitality during his stay at the Department of Physics, Faculty of Science,\nUniversity of Hong Kong. This work is supported in part by the National Science\nCouncil of Taiwan under the grants NSC-98-2923-M-008-001-MY3 and\nNSC-99-2112-M-008-015-MY3.\n\n\n\\begin{deluxetable}{lllllllll}\n\\tabletypesize{\\tiny}  \\setlength{\\tabcolsep}{0.03in} \\tablecaption{Previous\nGalactic models. The parentheses are the corrected values for binarism. The\nasterisk denotes the power-law index replacing Re. References indicating with\nthe original result table mean Galactic longitude or limiting magnitude\ndependent Galactic model parameters. \\label{table_ref}} \\tablewidth{0pt}\n\\tablehead{ \\colhead{$H_{z1}$ (pc)} & \\colhead{$H_{r1}$ (kpc)} & \\colhead{$f_2$\n(\\%)} & \\colhead{$H_{z2}$ (kpc)} & \\colhead{$H_{r2}$ (kpc)} &\n\\colhead{$f_h$(\\%)} & \\colhead{Re(S) (kpc)} & \\colhead{$\\kappa$} &\n\\colhead{Reference}} \\startdata\n310 - 325   &    -  &   0.0125-0.025    &   1.92 - 2.39 &    -  &    -  &    -  &    -  &   \\citet{Yoshii1982}  \\\\\n300 &    -  &   0.02    &   1.45    &    -  &   -   &   -   &   -    &   \\citet{Gilmore1983} \\\\\n325 &    -  &   0.02    &   1.3     &    -  &   0.002   &   3   &   0.85    &   \\citet{Gilmore1984} \\\\\n280 &    -  &   0.0028  &   1.9 &    -  &   0.0012  &    -  &    -  &   \\citet{Tritton1984} \\\\\n125-475 &    -  &   0.016   &   1.18 - 2.21 &    -  &   0.0013  &    3.1*  &   0.8 &   \\citet{Robin1986}   \\\\\n300 &    -  &   0.02    &   1   &    -  &   0.001   &    -  &   0.85    &   \\citet{delRio1987}  \\\\\n285 &    -  &   0.015   &   1.3 - 1.5   &    -  &   0.002   &   2.36    &   Flat    &   \\citet{Fenkart1987} \\\\\n325 &    -  &   0.0224  &   0.95    &    -  &   0.001   &   2.9 &   0.9 &   \\citet{Yoshii1987}  \\\\\n249 &    -  &   0.041   &   1   &    -  &   0.002   &   3   &   0.85    &   \\citet{Kuijken1989} \\\\\n350 &   3.8 &   0.019   &   0.9 &   3.8 &   0.0011  &   2.7 &   0.84    &   \\citet{Yamagata1992}    \\\\\n290 &    -  &    -  &   0.86    &    -  &    -  &   4   &    -  &   \\citet{vonHippel1993}   \\\\\n325 &    -  &   0.020-0.025  &   1.6-1.4 &    -  &   0.0015  &   2.67 &   0.8 &   \\citet{ReidMajewski1993}    \\\\\n325 &   3.2 &   0.019   &   0.98    &   4.3 &   0.0024  &   3.3 &   0.48    &   \\citet{Larsen1996}  \\\\\n250-270 &   2.5 &   0.056   &   0.76    &   2.8 &   0.0015  &   2.44 - 2.75*    &   0.60 - 0.85 &   \\citet{Robin1996, Robin2000}    \\\\\n260 &   2.3 &   0.074   &   0.76    &   3 &    -  &    -  &    -  &   \\citet{Ojha1996}    \\\\\n290 &   4   &   0.059   &   0.91    &   3   &   0.0005  &   2.69    &   0.84    &   \\citet{Buser1998, Buser1999}    \\\\\n240 &   - &   0.061   &   0.79    &   - &    -  &    -  &    -  &   \\citet{Ojha1999}    \\\\\n280\/267  &   -    &   0.02    &   1.26\/1.29   &    -  &    -  &   2.99*   &   0.63    &   \\citet{Phleps2000}  \\\\\n330 &   2.25    &   0.065 - 0.13    &   0.58 - 0.75 &   3.5 &   0.0013  &    -  &   0.55    &   \\citet{Chen2001}    \\\\\n-   &   2.8 &   3.5     &   0.86        &   3.7     & -     & -         & -     & \\citet{Ojha2001} \\\\\n280(350)    &   2 - 2.5 &   0.06 - 0.10 &   0.7 - 1.0 (0.9 - 1.2)   &   3 - 4  &   0.0015  &    -  &   0.50 - 0.70 &   \\citet{Siegel2002}  \\\\\n285 &   1.97    &    -  &    -  &    -  &    -  &    -  &    -  &   \\citet[][]{Lopez2002}   \\\\\n -  &   3.5 &   0.02-0.03   &   0.9 &   4.7 &   0.002-0.003 &   4.3 &   0.5-0.6 &   \\citet{Larsen2003}  \\\\\n320 &    -  &   0.07    &   0.64    &    -  &   0.00125 &    -  &   0.6 &   \\citet{Du2003}  \\\\\n265-495 &   -  &   0.052-0.098 &   0.805-0.970  &    -  &   0.0002-0.0015   &    -  &   0.6-0.8 &   \\citet[][Table 16]{Karaali2004} \\\\\n268 &   2.1 &   0.11    &   1.06    &   3.04    &    -  &    -  &    -  &   \\citet{Cabrera2005} \\\\\n300  &   -    &   0.04-0.10    &   0.9   &    -  &    -  &   3\/2.5*   &   1\/0.6    &   \\citet{Phleps2005}  \\\\\n220 &   1.9 &    -  &    -  &    -  &    -  &    -  &    -  &   \\citet{Bilir2006a}  \\\\\n160-360 &    -  &   0.033-0.076 &   0.84-0.87   &    -  &   0.0004-0.0006   &    -  &   0.06-0.08   &   \\citet[][Table 15]{Bilir2006b}  \\\\\n301\/259  &    -  &   0.087\/0.055    &   0.58\/0.93    &    -  &   0.001   &    -  &   0.74    &   \\citet[][Table 5]{Bilir2006c}   \\\\\n220-320 &    -  &   0.01-0.07   &   0.6-1.1 &    -  &   0.00125 &    -  &   $>$0.4    &   \\citet{Du2006}  \\\\\n206\/198  &    -  &   0.16\/0.10    &   0.49\/0.58    &    -  &    -  &    -  &   0.45    &   \\citet{Ak2007}  \\\\\n140-269 &    -  &   0.062-0.145   &   0.80-1.16   &    -  &    -  &    -  &    -  &   \\citet[][Table 1]{Cabrera2007}   \\\\\n220-360 &   1.65-2.52   &   0.027-0.099   &   0.62-1.03   &   2.3-4.0 &   0.0001-0.0022   &    -  &   0.25-0.85   &   \\citet[][Table 3]{Karaali2007} \\\\\n167-200 &    -  &   0.055-0.151 &   0.55-0.72   &    -  &   0.0007-0.0019    &    -  &   0.53-0.76   &   \\citet[][Table 1]{Bilir2008}    \\\\\n245(300)    &   2.15(2.6)   &   0.13(0.12)  &   0.743(0.900)    &   3.261(3.600)    &   0.0051  &   2.77*   &   0.64    &   \\citet{Juric2008}   \\\\\n325-369 &   1.00-1.68   &   0.0640-0.0659   &   0.860-0.952 &   2.65-5.49   &   0.0033-0.0039   &    -  &   0.0489-0.0654   &   \\citet[][Table 1]{Yaz2010}  \\\\\n360 &   3.7 &   0.07    &   1.02    &   5   &   0.002   &   2.6*    &   0.55    &   This Work   \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\\begin{deluxetable}{lcc}\n\\tabletypesize{\\scriptsize} \\tablecaption{The searching parameter space.\n\\label{para_space}} \\tablewidth{0pt} \\tablehead{ \\colhead{} & \\colhead{Range} &\n\\colhead{Grid size}} \\startdata\nThin Disk &  \\\\\n\\quad $H_{r1}$ & 1.0-6.0 kpc  & 100 pc \\\\\n\\quad $H_{z1}$ & 200-450 pc   & 10 pc \\\\\n\\quad $n_0$    & 0.02-0.04 stars\/pc$^3$  & 0.002 stars\/pc$^3$ \\\\\n\\quad $Z_\\odot$ & 11-35 pc    & 2.5 pc \\\\\nThick Disk & \\\\\n\\quad $H_{r2}$ & 3.0-7.0 kpc & 200 pc \\\\\n\\quad $H_{z2}$ & 400-1200 pc & 20 pc \\\\\n\\quad $f_2$ & 0-20\\% & 1\\% \\\\\nSpheroid & \\\\\n\\quad $\\kappa$ & 0.1-1.0  & 0.05  \\\\\n\\quad $p$ & 2.3-3.3 & 0.1  \\\\\n\\quad $f_h$ & 0-0.45\\% & 0.05\\% \\\\\n\\enddata\n\\end{deluxetable}\n\n\\begin{deluxetable}{lcc}\n\\tabletypesize{\\scriptsize} \\tablecaption{The best-fit Milky Way model.\n\\label{result}} \\tablewidth{0pt} \\tablehead{ \\colhead{} & \\colhead{Value} &\n\\colhead{Uncertainty}} \\startdata\nThin Disk &  \\\\\n\\quad $H_{r1}$ & 3.7 kpc &  1.0 kpc \\\\\n\\quad $H_{z1}$ & 360 pc  &  10 pc   \\\\\n\\quad $n_0$    & 0.030 stars\/pc$^3$    & 0.002 stars\/pc$^3$    \\\\\n\\quad $Z_\\odot$ & 25 pc   & 5 pc \\\\\nThick Disk & \\\\\n\\quad $H_{r2}$ & 5.0 kpc & 1.0 kpc \\\\\n\\quad $H_{z2}$ & 1020 pc & 30 pc \\\\\n\\quad $f_2$    & 7\\%     & 1 \\% \\\\\nSpheroid & \\\\\n\\quad $\\kappa$ & 0.55        & 0.15  \\\\\n\\quad $p$   & 2.6        & 0.6  \\\\\n\\quad $f_h$ & 0.20\\%     & 0.10\\% \\\\\n\\enddata\n\\end{deluxetable}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{section-introduction}\nGiven positive integers $m$ and $n$ with $m  - n\\equiv 0 \\bmod 8$, let\n$\\mathrm{II}_{m,n}$ denote the unique even self-dual lattice of signature\n$(m,n)$; see \\cite{CS:SPLAG} chapter 26, 27.\nFor basic definitions about lattices, see\n\\cite{REB:L} or \\cite{CS:SPLAG}. All the lattices considered here are\nnonsingular unless otherwise stated. A bilinear form is usually denoted by $\\ip{\\;}{\\;}$.\nThe lattice $\\mathrm{II}_{1,1}$ is sometimes called a {\\it hyperbolic cell}.\nThe even self-dual Lorentzian lattices are $\\mathrm{II}_{8n + 1, 1}$. Among these,\nthe lattice $L = \\mathrm{II}_{25,1}$ stands out\nas exceptional  (see \\cite{CS:SPLAG} or \\cite{REB:Thesis}, \\cite{REB:L}, \nfor a wealth of information on $L$).\nOf course this is related to the many exceptional properties of the\nLeech lattice $\\Lambda$. \nIndeed $ L \\simeq \\Lambda \\oplus \\mathrm{II}_{1,1}$.\nLet $L(2)$ denote the set of norm $2$ vectors (also called {\\it roots}) of $L$.\nThe roots form a single orbit under the automorphism group $\\op{Aut}(L)$.\nIn this article, we study the Poincare-Weierstrass series\n\\begin{equation*}\nE_L(z,s) = E(z,s) = \\sum_{r \\in L(2) } \\ip{r}{z}^{-s}.\n\\end{equation*}\nHere $z$ is a negative norm vector in $L \\otimes \\mathbb{R}$ and\n $s$ is a complex\nnumber.\nThe function $E(z,s)$ is real analytic in $z$. \nComplex analytic versions of this function\nhave appeared in Looijenga's study of compactifications of\ncomplex ball quotients (see the functions $F^{(l)}_{\\mathcal{O}}$ in lemma 5.4 of \\cite{L:CDBA1}). \nThe functions $F^{(l)}_{\\mathcal{O}}$ are analogous to the \nEisenstein series $\\sum_{(m,n) \\neq (0,0)} (m \\tau + n)^{-2l}$ whereas\n $E(z,s)$ is analogous to real analytic Eisenstein series.\n\\par\nThe infinite series for $E(z,s)$ converges for $\\op{Re}(s) > 25$\n(For a conceptual proof, see lemma 5.4 of \\cite{L:CDBA1}.\nFor the sake of completeness, we have included an elementary\nproof in appendix \\ref{l-convergence}).\nSo, for fixed $s$ with $\\op{Re}(s) > 25$, we obtain a real analytic function\n$z \\mapsto E(z,s)$ on hyperbolic space\n$\\mathcal{H}^{25}$.\nBy definition, $E(z,s)$ is an automorphic function invariant under $\\op{Aut}(L)$ with singularities\nalong the mirrors of the reflection group of $L$. \nOur objective is to compute the Fourier series for $E(z,s)$ and prove that\n$E(z,s)$ can be analytically continued to a meromorphic function on $\\op{Re}(s) > 25\/2$.\n\\par\nMuch of what we say about $E_{\\mathrm{II}_{25,1}}(z,s)$\ncan probably be generalized for all the even self-dual \nLorentzian lattices $\\mathrm{II}_{ 8 n + 1, 1}$\n(at least if we assume $n \\geq 3$, which would guarantee the existence of\na positive definite even self-dual lattice of rank $8n$ with no roots). \nHowever, in section \\ref{section-II251} and \\ref{section-Fourier-expansion}, while\nstudying $E_L(z,s)$, \nwe have decided to restrict to the example $L = \\mathrm{II}_{25,1}$ for several reasons.\nFirst, it allows us to keep the exposition relatively simple.\nSecond, in view of moonshine, this is the most important example. \nThird, because of special properties of the Leech lattice, in particular, \nbecause of Conway's beautiful description of the automorphism group of $L$, \nthis is also the nicest example. \n\\par\nTo describe the form of the Fourier expansion of $E(z,s)$, we need to first\nbriefly recall Conway's description of the {\\it reflection group} $R(L)$. By definition,\n$R(L)$ is the subgroup of $\\op{Aut}(L)$ generated by reflections in the roots of $L$.\nBoth $\\op{Aut}(L)$ and $R(L)$ act on\nthe hyperbolic space\n $ B(L) = \\lbrace x \\in L \\otimes \\mathbb{R} \\colon x^2 < 0 \\rbrace\/ \\mathbb{R}^*\n \\simeq \\mathcal{H}^{25}$.\nTo describe a fundamental domain of $R(L)$ acting on $\\mathcal{H}^{25}$,\nchoose a {\\it Leech cusp} $\\rho$:  this means that $\\rho$ is a primitive \nnorm zero vector of $L$ and $\\rho^{\\bot}\/\\rho \\simeq \\Lambda$.\nThis lets us split a hyperbolic cell from $L$ and \nidentify $L = \\Lambda \\oplus \\mathrm{II}_{1,1}$. So we write $v \\in L$ in the form\n$v = (\\lambda; m,n)$ with $\\lambda \\in \\Lambda$ and $m,n \\in \\mathbb{Z}$\nwith $v^2 = \\lambda^2 - 2 m n$. In this co-ordinate system $\\rho = (0; 0, 1)$.\nThe Leech cusp $\\rho$ determines a point in $\\partial \\mathcal{H}^{25}$,\nalso denoted by $\\rho$. The action of  $R(L)$ on $\\mathcal{H}^{25}$\nhas a unique fundamental domain $C$ whose closure contains $\\rho$.\nWe say that $C$ is the {\\it Weyl chamber ``around\" $\\rho$}.\nThe walls of $C$ are in bijection with the vectors of Leech lattice. The angles\nbetween these walls are determined by the inner products of the corresponding Leech lattice\nvectors. This lets one describe $R(L)$ explicitly as a Coxeter group so\nthat\n{\\it ``the Leech lattice is the Dynkin diagram for $R(L)$\"}.\n(see \\ref{t-C} or \\cite{C:Aut26}, \\cite{REB:Thesis} for details).\n\\par\nBecause the Leech lattice has no roots, there are no mirrors of $R(L)$\nthat pass through $\\rho$. The subgroup $\\mathbb{T}$\nof $\\op{Aut}(L)$ that fix $\\rho$ and act trivially on\n$\\rho^{\\bot}\/\\rho$ is a free abelian group isomorphic to $\\Lambda$.\nWe call $\\mathbb{T}$ the group of {\\it translations}. \nWe compute the Fourier series of $E(z,s)$ at $\\rho$\nwith respect to the group of translations; see \\ref{topic-continuous-group-of-translations},\n\\ref{topic-discrete-group-of-translations}, \\ref{topic-setup-for-Fourier-series-computation}\n for details.\nFix positive real numbers $h,k$ such that $2h^2\/k < 1$. Let $z \\in L \\otimes \\mathbb{R}$ \nbe a vector of norm $-k$ having the form $z = (v h; h, *)$, where\n$v \\in \\Lambda \\otimes \\mathbb{R}$  and the last coordinate of $z$ is determined by the condition\n$z^2 = -k$. Then $z$ (or rather its image in $\\mathcal{H}^{25}$)\nis contained in the Weyl chamber $C$. So $z$ does not lie on any mirror.\nThe Fourier series of $E(z,s)$ has the form\n\\begin{equation*}\nE((vh; h, *),s) = \\sum_{\\lambda \\in \\Lambda} a_{\\lambda}(k,h,s)\\exp( 2 \\pi i \\ip{\\lambda}{v}).\n\\end{equation*}\nEach coefficient in turn is an infinite series of the form\n\\begin{equation*}\na_{\\lambda}(k,h,s) = \\sum_{n = 1}^{\\infty} j_{\\lambda, n} n^{-s} g_{\\lambda}(s,1\/n)\n\\end{equation*}\nwhere \n\\begin{equation*}\nj_{\\lambda, n} = \\sum_{l \\in \\Lambda\/n \\Lambda : l^2\/2 \\equiv 1 \\bmod n} \n\\exp( 2 \\pi i \\ip{l}{\\lambda}\/n) \n\\end{equation*}\nand $g_{\\lambda}(s,y)$ is some function that is \nentire in $s$ and analytic for $y$ in a disc of radius bigger than $1$\nand exponentially decaying as $\\abs{\\lambda}  \\to \\infty$; see theorem\n\\ref{th-fourier-coeffients}.\n\\par\nSection \\ref{section-j} is devoted to studying exponential sums of the form $j_{\\lambda, n}$\nfor general even self-dual lattices of arbitrary signature. \nThe results of section \\ref{section-j} and related results in appendix \\ref{appendix-gauss-sums}\nare probably known to experts. But since we could not find a reference,\nand we think these results may have applications in other contexts,\nwe have decided to include the details. In theorem \\ref{th-j} we compute\nthe sums $j_{\\lambda,n}$ in terms of Kloosterman sums.\nWeil's bound on Kloosterman sums implies that the infinite series for $a_{\\lambda}(k,h,s)$\nconverges for $\\op{Re}(s) > 25\/2$\nand it follows that $E(z,s)$ can be meromorphically continued in this half plane; \nsee theorem \\ref{th-analytic-continuation-of-E}.\nFurther, using analytic continuation of Kloosterman sum zeta functions,\nwe prove that the Dirichlet series $\\sum_{n} j_{\\lambda,n} n^{-s}$ and\nthe individual Fourier coefficients $a_{\\lambda}(k,h,s)$ have\nmeromorphic continuation to the whole $s$-plane \n(see \\ref{l-analytic-continuation-of-Dirichlet-series-of-j} and\n \\ref{remark-analytic-continuation-of-Fourier-coefficients}).\nHowever, since we are unable to find bounds on these functions obtained after\nanalytic continuation, we cannot prove continuation of $E(z,s)$ beyond\nthe line $\\op{Re}(s) = 25\/2$.\n\\par\nWe have two motivations for studying $E(z,s)$. \nThe first motivation is to find out if the residue of $E(z,s)$ at the ``first pole\"\nis related to some of the automorphic forms of type $O(25,1)$ constructed by taking Borcherds lift\n(see \\cite{REB:Aut}, section 10, in particular, the set up of example 10.7).\nThe second motivation is that the ``complex analytic versions of $E(z,s)$\" are\nrelevant in projective uniformization of a $13$ dimensional complex  ball quotient\nwhose fundamental group is conjectured to be related to the monster simple group \\cite{Allcock-monstrous}.\nThese complex analytic versions of $E(z,s)$ are instances of the functions considered in \\cite{L:CDBA1}\nand they are defined as\n\\begin{equation*}\nE^{(l)}(z) = \\sum \\nolimits_{r \\in L(3) } \\ip{z}{r}^{-l}\n\\end{equation*}\nwhere $L$ is the unique\nhermitian $\\mathbb{Z}[e^{2 \\pi i\/3}]$-lattice of signature $(13,1)$ satisfying $\\sqrt{-3} L^{\\vee} = L$,\nthe sum is over $L(3)$ which is the set of roots of $L$, the exponent\n$l \\in 6 \\mathbb{N}$ and $z$ is a negative norm\nvector of $L \\otimes_{\\mathbb{Z}[e^{2 \\pi i\/3}]} \\mathbb{C}$. \nSo $E^{(l)}(z)$ is a section of the tautological line bundle on the complex $13$-ball\n$B(L) = \\lbrace v \\in L \\otimes_{\\mathbb{Z}[e^{2 \\pi i\/3}]} \\mathbb{C} \\colon v^2 < 0 \\rbrace\/\\mathbb{C}^*$.\nThus $E^{(l)}(z)$ are meromorphic automorphic forms of type $U(13,1)$\ninvariant under $\\op{Aut}(L)$ and having poles along\nthe mirrors of the complex hyperbolic reflection group $R(L)$. So \n\\begin{equation*}\nz \\mapsto [E^{(30)}(z)\/E^{(36)}(z) : E^{(36)}(z)\/  E^{(42)}(z) : E^{(42)}(z)\/E^{(48)}(z) :  \\dotsb]\n\\end{equation*}\nis a meromorphic map from the ball quotient \n$ (B(L) - \\lbrace \\text{mirrors of $R(L)$} \\rbrace) \/ \\op{Aut}(L)$ to the projective space.\nAllcock's monstrous proposal states that the orbifold fundamental group of this ball quotient\nsurjects onto the monster; see  \\cite{Allcock-monstrous}, \\cite{AB-braidlike} and\nthe references in there for more details. The calculation of Fourier series of $E^{(l)}(z)$ is\ncomplicated by the fact that the group of translations is a discrete Heisenberg group\nrather than an abelian group. We think that the analysis of $E(z,s)$ is a warm-up exercise\nfor studying the functions $E^{(l)}(z)$.\n\\par\n{\\bf Acknowledgement:} I would like to thank Prof. Richard Borcherds\nfor many stimulating conversations at the beginning of this work.\nI would like to thank Prof. Eric Weber and Prof. Manjunath Krishnapur for their help in a \ncouple of proofs.\n\\section{Some exponential sums related to even lattices}\n\\label{section-j}\n\\begin{topic}{\\bf Notation: }\n\\label{topic-notation}\nWe fix some notation that will be used throughout this section.\nLet $p$ be a prime. Let $q = p^r$ for some integer $r \\geq 1$.\nLet $M$ be a nonsingular even lattice of rank $m$.\nExcept for lemma \\ref{l-bound-on-magnitude-of-j},\nwe do not assume in this section that $M$ is necessarily positive definite.\n Let $n, d \\in \\mathbb{Z}$ and $n \\geq 1$.\nWe abbreviate $M\/n = M\/n M$.\nWe define\n\\begin{equation*}\nM_n(d) = \\lbrace l \\in M\/n : l^2\/2 \\equiv d \\bmod n \\rbrace.\n\\end{equation*}\nLet $\\e(x) = \\exp(2 \\pi i x)$. Let\n$\\lambda \\in M$. The first goal of this section is to prove theorem \\ref{th-j} which lets us\ncalculate some exponential sums of the form \n\\begin{equation*}\nj_{\\lambda, n}^M(d) = j_{\\lambda, n}(d)  = \\sum_{l \\in M_n(d)} \\e (\\ip{l}{\\lambda}\/n)\n\\end{equation*}\nin terms of Kloosterman sums \n\\begin{equation*}\nS(a,b, n) = \\sum_{r \\in \\mathbb{Z}\/n : \\op{gcd}(r,n)=1 } \\e((ar + b \\bar{r})\/n)\n\\end{equation*}\nand Jordan totient function $J_k(n) = n^k \\prod_{p \\mid n} (1 - p^{-k} )$. \nAs usual,  $\\bar{r}$ is the inverse of $r$ modulo $n$, \nthat is, $\\bar{r}$ denotes any integer such that $r \\bar{r} \\equiv 1 \\bmod n$.\nIn the definition of $J_k(n)$, the product is over all prime divisors of $n$.\nLet $j_{\\lambda, n} = j_{\\lambda, n}(1)$.\nAs a consequence of \\ref{th-j}, we obtain bounds on $\\abs{j_{\\lambda, n}}$\nin \\ref{l-bound-on-magnitude-of-j} and\nshow in \\ref{l-analytic-continuation-of-Dirichlet-series-of-j}\nthat the Dirichlet series $\\sum_{n = 1}^{\\infty} j_{\\lambda, n} n^{-s}$\ncan be analytically continued to a meromorphic function on the whole complex\nplane.\n\\par\nGiven $\\lambda \\in M$, let $c(\\lambda)$ be the largest positive integer such that\n$\\lambda \\in c(\\lambda) M$. So $\\lambda$ is a primitive vector of $M$\nif and only if $c(\\lambda) = 1$.\nWe say $n$ divides $\\lambda$ (resp. $n$ and $\\lambda$ are relatively prime)\nif $n$ divides $c(\\lambda)$ (resp. if $n$ and $c(\\lambda)$ are relatively prime).\nIf $p$ is a prime, we let $v_p(\\lambda)$ be the $p$-valuation of $c(\\lambda)$.\n\\end{topic}\n\\begin{theorem} Assume the setup of \\ref{topic-notation}.\n\\par\n(a) If $r, n$ are relatively prime positive integers,  then\n$M_{n r}(d) \\simeq M_r(d) \\times M_n(d)$ and \n$j_{\\lambda, nr}(d) = j_{\\bar{n} \\lambda, r}(d)  j_{\\bar{r} \\lambda, n}(d)$.\n\\par\n(b) If $n \\mid \\lambda$, then $j_{\\lambda, n} (d) = j_{0, n} (d) = \\abs{M_n(d)}$.\nIf $p \\nmid d$, \nthen  $j_{ p \\lambda, p q}(d)  = p^{m-1} j_{\\lambda, q}  (d)$.\n\\par\n(c) Assume that $M$ is self-dual and that $n$ and $\\lambda$ are relatively prime.\nThen\n$j_{\\lambda, n}(d)  = n^{(m\/2) - 1} S(d, \\lambda^2\/2, n)$. \n\\par\n(d) Assume $M$ is self-dual. Then\n$ j_{0, n}(1) = n^{(m\/2) - 1} J_{m\/2}(n)$.\n\\label{th-j}\n\\end{theorem}\nFor later application,\nwe need to calculate $j_{\\lambda,n} = j_{\\lambda, n}(1)$ for an even self-dual\nlattice. Theorem \\ref{th-j} is  sufficient for this purpose. \nFirst, because of part (a), it suffices to calculate\n$j_{\\lambda, q}$ when $q$ is a prime power.\nNext, part (b) lets us reduce to the case when $p \\nmid \\lambda$ or $\\lambda = 0$.\nFinally, parts (c) and (d) handle these two cases respectively.\nProofs of parts (c) and (d) require calculation of some quadratic Gauss sums.\nThese  calculations have been moved back to appendix \\ref{appendix-gauss-sums}.\n\\begin{proof}[proof of \\ref{th-j}(a)]\nChoose $\\bar{n}, \\bar{r} \\in \\mathbb{Z}$ such that $\\bar{n} n + \\bar{r} r = 1$.\nFrom Chinese remainder theorem we have  mutually inverse isomorphisms \n$\\pi: M\/nr \\to M\/n \\times M\/r$ and $\\phi: M\/n \\times M\/r \\to M\/nr$ given by\n\\begin{equation*}\n\\pi( l \\bmod nr) = (l \\bmod n, l \\bmod r), \\;\\;\n\\phi(l_1 \\bmod n , l_2 \\bmod r) = ( \\bar{n} n l_2 + \\bar{r} r l_1) \\bmod n r\n\\end{equation*}\nfor $l , l_1, l_2 \\in M$.\nPick $v = (l_1 \\bmod n, l_2 \\bmod r) \\in M_n(d) \\times M_r(d)$.\nThen $\\phi(v) =  (l \\bmod n r)$ where $l =  \\bar{n} n l_2 + \\bar{r} r l_1$. We have\n\\begin{equation*}\nl^2\/2\n \\equiv  \\bar{n}^2 n^2 (l_2^2\/2) +  \\bar{r}^2  r^2(l_1^2\/2) \n \\equiv  \\bar{n}^2 n^2 d + \\bar{r}^2 r^2 d \n\\equiv d \\bmod  n r.\n\\end{equation*}\nSo $\\phi(v) \\in M_{ n r }(d)$. This proves $\\phi(M_n (d) \\times M_r(d)) \\subseteq M_{n r}(d)$.\nClearly $\\pi (M_{n r}(d)) \\subseteq M_n (d) \\times M_r(d)$. So\nrestrictions of $\\pi$ and $\\phi$ yield mutually inverse\nisomorphisms between $M_{n r}(d)$ and $M_n (d) \\times M_r(d)$. \nNow let \n$(l_1,l_2)$\nrun over $M_n(d) \\times M_r(d)$.\nThen $\\phi(l_1,l_2)$\nruns over $M_{n r}(d)$.\nIt follows that\n\\begin{equation*}\nj_{\\lambda, n r}(d) \n= \\sum_{l_1 , l_2} \\e\\bigl(\\tfrac{\\ip{ \\bar{n} n l_2 + \\bar{r} r l_1 }{\\lambda}}{n r} \\bigr)  \n= \\sum_{l_1 , l_2} \\e \\bigl(\\tfrac{\\ip{  l_2}{\\bar{n} \\lambda}}{ r} \\bigr) \n\\e\\bigl( \\tfrac{\\ip{l_1}{ \\bar{r} \\lambda}}{n} \\bigr)\n= j_{\\bar{n} \\lambda, r}(d) j_{\\bar{r} \\lambda, n} (d).\n\\qedhere\n\\end{equation*}\n\\end{proof}\nPart (b) of theorem \\ref{th-j} can be proved using a Hensel's lemma type argument. We isolate\nthis argument in the proof of the next lemma.\n\\begin{lemma} Assume the setup of \\ref{topic-notation}.\nFurther, assume that $M$ is self-dual and $p \\nmid d$. Then\nthe natural projection $\\pi: M\/p q \\to M\/q$ maps\n$M_{p q } (d)$ onto $M_{q} (d)$.\nFor each $\\bar{l} \\in M_q(d)$, there exists $p^{m-1}$ elements \n $l \\in M_{p q}(d)$ such that $\\pi(l) = \\bar{l}$. \n\\label{l-Hensel}\n\\end{lemma}\n\\begin{proof} Clearly $\\pi(M_{p q}(d) ) \\subseteq M_q(d)$.\nGiven $\\bar{u} \\in M_{q} (d)$, \nfix $u \\in M$ such that  $\\bar{u} = u \\bmod qM$.\nThe lifts of $\\bar{u}$ to $M\/p q$ are of the form\n$(u + q x) \\bmod p q M$\n where $x$ runs over a full set of coset representatives for $M\/p$.\nSince $\\bar{u} \\in M_{q}(d)$, we have $\\tfrac{1}{2} u^2 \\equiv d \\bmod q$. \nNote that $(u + q x) \\bmod p q M \\in M_{p q}(d)$ if and only if\n\\begin{equation*}\n \\tfrac{1}{2} u^2 + q \\ip{u}{x} \\equiv d \\bmod p q, \n\\end{equation*}\nor equivalently,\n\\begin{equation}\n\\ip{u}{x} \\equiv q^{-1} (d - \\tfrac{1}{2} u^2) \\bmod p.\n\\label{eq-ipux}\n\\end{equation}\nSince $M$ is self-dual, the bilinear form $\\ip{x}{y} \\bmod p$ on the $\\mathbb{F}_p$ -vector space\n$M\/p$ is non-degenerate. \nNote that $(u \\bmod p M)$ is a nonzero vector in $M\/p$ Since\n$\\tfrac{1}{2} u^2 \\equiv d \\bmod q$ and $p \\nmid d$.\nSo the set of all possible $x \\in M\/p$ satisfying\nequation \\eqref{eq-ipux} forms an affine hyperplane in the $m$ dimensional \n$\\mathbb{F}_p$ -vector space $M\/p$. \n\\end{proof}\n\\begin{proof}[proof of theorem \\ref{th-j}(b)]\nThe first claim is obvious from the definition of $j_{\\lambda, n}(d)$. \nThe second claim follows from  lemma \\ref{l-Hensel}, since\n\\begin{equation*}\nj_{p \\lambda, p q}(d) \n= \\sum_{ l \\in M_{p q}(d)} \\e( \\ip{l}{\\lambda}\/ q ) \n= p^{m - 1} \\sum_{ \\bar{l} \\in M_{q} (d)} \\e( \\ip{\\bar{l}}{\\lambda}\/ q ) \n= p^{m-1} j_{\\lambda, q}  (d) .\n\\end{equation*}\n\\end{proof}\nGiven a function $f : \\mathbb{Z}\/n \\to \\mathbb{C}$, let \n$(\\mathcal{F} f)(x) = \\sum_{y \\in \\mathbb{Z}\/n} f(y) \\e(x y\/n)$\ndenote its Fourier transform.\nLemma \\ref{l-Fj} calculates the Fourier transform of \n$d \\mapsto j_{\\lambda, q}(d)$.\nThen we recover $j_{\\lambda, q}(d)$ by using the Fourier\ninversion formula $(\\mathcal{F}^2 f)(x) = n f(-x)$.\n\\begin{lemma}\nAssume the setup of \\ref{topic-notation} and that $M$ is self-dual.\n Let $\\lambda \\in M$.\n\\par\n(a) We have  \n\\begin{equation}\n\\mathcal{F} j_{\\lambda, q}(c) =\n\\sum_{l \\in M \/q }  \\e ( (2 \\ip{l}{\\lambda} + c l^2)\/2q ).\n\\label{eq-Fj}\n\\end{equation}\n\\par\n(b) Assume $p \\nmid \\lambda$. If $p \\mid c$, then $\\mathcal{F} j_{\\lambda, q} (c) = 0$. \nIf $p \\nmid c$, then\n\\begin{equation}\n\\mathcal{F} j_{\\lambda, q} (c) =  \\e( - \\bar{c} \\lambda^2\/2 q ) q^{m\/2}.\n\\label{eq-Fj-rel-prime-case}\n\\end{equation}\n\\label{l-Fj}\n\\end{lemma}\n\\begin{proof} \n(a) Note that $M\/q$ is the disjoint union of $M_q(1), M_q(2), \\dotsb, M_q(q)$.\nPart (a) now follows since\n\\begin{equation*}\n \\mathcal{F} j_{\\lambda, q}(c) \n= \\sum_{d = 1}^q \\sum_{\\substack{l \\in M \/q \\\\ l^2\/2 \\equiv d \\bmod q}}\n\\e\\bigl( \\tfrac{\\ip{l}{\\lambda}}{q} \\bigr) \\e \\bigl(\\tfrac{c d}{q} \\bigr)\n= \\sum_{l \\in M \/q } \n\\e\\bigl( \\tfrac{\\ip{l}{\\lambda}}{q} + \\tfrac{c l^2}{2q} \\bigr).\n\\end{equation*}\n\\par\n(b)  If $\\op{gcd}(c,q) = 1$, we can complete squares in equation \\eqref{eq-Fj} to get\n\\begin{equation*}\n \\mathcal{F} j_{\\lambda, q}(c) \n=  \\sum_{l \\in M\/q }  \\e\\bigl( \\tfrac{-\\bar{c} \\lambda^2}{2q } + \\tfrac{c(l + \\bar{c}\\lambda)^2}{2q} \\bigr)\n= \\e \\bigl( \\tfrac{-\\bar{c} \\lambda^2}{2q} \\bigr) \\sum_{l \\in M \/q }  \\e \\bigl( \\tfrac{c l^2}{2q} \\bigr).\n\\end{equation*}\nEquation \\eqref{eq-Fj-rel-prime-case} now follows from the lemma \\ref{l-theta-q} in the appendix.\n\\par\nNow assume that $c = p c_1$ for some $c_1 \\in \\mathbb{Z}$.\nLet $u$ run over a set of coset representatives for $M\/p^{r-1} $\nand $x$ run over a set of coset representatives of $M\/ p $. Then $(u + p^{r-1} x)$ runs over\nover a set of coset representatives of $M\/p^r$. \nSo, substituting $ l = (u + p^{r-1} x)$  in equation \n\\eqref{eq-Fj}, we  \nobtain\n\\begin{align*}\n  \\mathcal{F} j_{ \\lambda, p^r}(c) \n = \\sum_{u \\in M\/p^{r-1} } \n\\e \\bigl( \\tfrac{\\ip{u}{\\lambda}}{p^r} + \\tfrac{c_1 u^2}{2p^{r-1} } \\bigr) \\sum_{x \\in M\/p} \n\\e \\bigl( \\tfrac{\\ip{x }{\\lambda}}{ p} \\bigr). \n\\end{align*}\nIf $p \\nmid \\lambda$, then\n$x \\mapsto \\ip{x }{\\lambda}\/ p$ is a nontrivial character on $M\/p$, since $M$ is self-dual. \nSo the inner sum in the last expression vanishes.\n\\end{proof}\n\\begin{proof}[proof of theorem \\ref{th-j}(c)]\nWhen $n = q$ is  a prime power, the formula for $j_{\\lambda, n}(d)$ follows from \nlemma \\ref{eq-Fj-rel-prime-case} using\nFourier inversion. The\ngeneral case follows from this using multiplicativity property of $j_{\\lambda, n}$ \nand Kloosterman sums (see \\ref{th-j}(a) and \\cite{IK:ANT} respectively).\n\\end{proof}\n\n\\begin{proof}[proof of theorem \\ref{th-j}(d)]\nBoth sides of the formula we want to prove are multiplicative. So it is enough to \nprove the formula when $n = q = p^r$ is a prime power.\nFrom equation \\eqref{eq-Fj} and lemma \\ref{l-theta-q},\nwe have\n\\begin{equation*}\n\\mathcal{F} j_{0,p}(c) = \\sum_{l \\in M\/p } \\e( cl^2\/ 2p) \n= \\begin{cases} p^{ m\/2} & \\text{\\; if \\;} 1 \\leq c < p \\\\\np^m & \\text{\\; if \\;} c = p.\n\\end{cases}\n\\end{equation*}\nUsing Fourier inversion, we find  \n\\begin{equation*}\np j_{0,p} (1) \n=  \\sum_{c = 1}^p \\mathcal{F} j_{0,p}(c) \\e(-c\/p) \n= (p^m - p^{m\/2}). \n\\end{equation*}\nLemma \\ref{l-Hensel} implies\n$j_{0,pq}(1) =  \\abs{J_{p q}(1)} = p^{m-1}  \\abs{J_{q }(1)} = p^{m-1} j_{0,q}(1)$.\nPart (d) of \\ref{th-j} follows from this.\n\\end{proof}\nRecall the notation $j_{\\lambda,n} = j_{\\lambda,n}(1)$.\n\\begin{lemma} \nAssume that $M$ is self-dual.\nLet $u = \\min \\lbrace v_p(n), v_p(\\lambda) \\rbrace > 0$.\nThen \n\\begin{equation*}\n j_{\\lambda, n} =  \\begin{cases}\np^{(m-1)u} j_{p^{-u} \\lambda, p^{-u} n}  &\\text{\\; if \\;}  v_p(\\lambda) < v_p(n), \\\\\n p^{(m-1)u} (1 - p^{-m\/2}) j_{p^{-u} \\lambda, p^{-u} n}  & \\text{\\; if \\;} v_p(\\lambda) \\geq v_p(n).\n\\end{cases}\n\\end{equation*}\n\\label{l-cancel-common-factor-in-j}\n\\end{lemma}\n\\begin{proof} This is a straight forward computation using \\ref{th-j} (a), (b) in the first case and \\ref{th-j} (a), (d) in the second case.\n\\end{proof}\n\\begin{lemma} Assume $M$ is positive definite and self-dual.\nFix $\\epsilon > 0$. Then there exists a constant $C_{\\epsilon}$ such \nthat\n\\begin{equation*}\n\\abs{j_{n, \\lambda}} <  C_{\\epsilon} \\abs{\\lambda}^{(m-1)\/2 } n^{\\epsilon  + (m-1)\/2 }\n\\text{\\; for all \\;} n \\in \\mathbb{N} \\text{\\; and all \\;} \\lambda \\in M - \\lbrace  0 \\rbrace.\n\\end{equation*}\n\\label{l-bound-on-magnitude-of-j}\n\\end{lemma}\n\\begin{proof}\nSince $M$ is positive definite and integral, $\\lambda^2  \\geq \\op{gcd}(n, \\lambda)^2$. So\nit suffices to prove an inequality of the form\n\\begin{equation}\n\\abs{j_{n, \\lambda}} <  C_{\\epsilon} (n \\op{gcd}(n, \\lambda))^{(m-1)\/2} n^{\\epsilon}.\n\\label{eq-bound-on-j}\n\\end{equation}\nWe prove \\eqref{eq-bound-on-j} by induction on $\\op{gcd}(n,\\lambda)$.\nIf $\\op{gcd}(n, \\lambda) = 1$, then \\eqref{eq-bound-on-j} follows \nfrom \\ref{th-j}(c) and Weil's bound for Kloosterman sum (see \\cite{IK:ANT}, p. 280)\n\\begin{equation*}\n\\abs{S(a,b,n) } \\leq \\bigl( \\sum\\nolimits_{d \\mid n} 1 \\bigr) \\op{gcd}(a,b,n)^{1\/2} n^{1\/2} \n\\end{equation*}\nand the estimate $\\sum_{d \\mid n} 1 = o( n^{\\epsilon})$ for any $\\epsilon > 0$.\n\\par \nIf $\\op{gcd}(n, \\lambda) > 1$, then choose a prime $p$ dividing $\\op{gcd}(n, \\lambda)$. \nLemma \\ref{l-cancel-common-factor-in-j}\nimplies that \n$\\abs{j_{\\lambda, n} } \\leq p^{(m-1)u} \\abs{j_{p^{-u} \\lambda, p^{-u} n}(d) }$\nwhere $u  = \\min \\lbrace v_p(n), v_p(\\lambda)\\rbrace$.\nNow one can induct.\n\\end{proof}\nWe end this section by showing that the Dirichlet series\n\\begin{equation*}\nL(j_{ \\lambda}, s) = \\sum_{n \\in \\mathbb{N}} j_{\\lambda, n} n^{-s}\n\\end{equation*}\ncan be meromorphically continued to $\\mathbb{C}$ when the lattice $M$ is self-dual. \nThis follows from Selberg's theorem on analytic continuation\nof Kloosterman sum zeta function. The specific result we need is\nrecorded in the next lemma. \n\\begin{lemma}\nThe series $\\sum_{n \\colon \\op{gcd}(n, k) = 1} S(a,b, n) n^{-s}$ can be analytically continued to a meromorphic function on the\nwhole $s$-plane. It is analytic for $\\op{Re}(s) > 1$, except possibly for a finite set of poles on the real segment $1 < s < 2$.\n\\label{l-Selberg}\n\\end{lemma}\n\\begin{proof} \n In the notation used in\n equation (3.9) of \\cite{AS:FCM}, the Dirichlet series\n $F(d) =  \\sum_{n \\equiv 0 \\bmod d} S(a,b, n) n^{-s}$ is \n equal to $Z(s\/2,a,b,  \\chi, \\Gamma_0(N))$ for the trivial multiplier \n system $\\chi = 1$. By results of \\cite{AS:FCM}\n the series $F(d)$ can be analytically continued to a meromorphic function on $\\mathbb{C}$\n (also see \\cite{DI:KS} for details).\nThe lemma follows since\n$\\sum_{n : \\op{gcd}(n,k) = 1 } S(a,b, n) n^{-s} = \\sum_{d \\mid k} \\mu(d) F(d)$.\n\\end{proof}\n\\begin{lemma} \nAssume the setup of \\ref{topic-notation}. Further, assume that $M$ is self-dual.\nLet $T$ be a finite set of primes. Let $\\mathbb{N}_T$ be the set of all positive integers $n$\nsuch that $ p \\nmid n$ for all $p \\in T$.\nLet $\\lambda \\in M - \\lbrace  0 \\rbrace$. Then the Dirichlet series\n\\begin{equation*}\nL_T( j_{\\lambda}, s) = \\sum_{n \\in \\mathbb{N}_T } j_{\\lambda, n} n^{-s}\n\\end{equation*}\ncan be analytically\ncontinued to a meromorphic function on the whole $s$-plane.\nFor $\\lambda  = 0$, we have \n$L(j_{ 0} , s) = \\zeta( s + 1 - m)\/ \\zeta( s + 1 - (m\/2))$.\n\\par\nIn particular, $L(j_{\\lambda}, s)$ has analytic continuation as a meromorphic function on \n the whole $s$-plane for all $\\lambda \\in M$.\n\\label{l-analytic-continuation-of-Dirichlet-series-of-j}\n\\end{lemma}\n\\begin{proof}\nLet $P(\\lambda)$ be the set of prime divisors of $\\lambda$. \nWe shall prove the lemma by induction on $\\abs{P(\\lambda) - T }$.\nFirst, assume that $T \\supseteq P(\\lambda)$. \nIf $n \\in \\mathbb{N}_T$, then $n$ and $\\lambda$ are relatively prime.\nSo by theorem \\ref{th-j}(c), we have\n\\begin{equation*}\nL_T( j_{\\lambda}, s) =  \\sum_{n \\in \\mathbb{N}_T} S(1, \\lambda^2\/2, n) n^{(m\/2) - 1 - s}.\n\\end{equation*}\nNote that $\\mathbb{N}_T = \\lbrace n \\in \\mathbb{N} \\colon \\op{gcd}(n, k_T) =1 \\rbrace$\nwhere $k_T$ is the product of all the primes in $T$.\nSo \\ref{l-Selberg} proves the lemma in the case $\\abs{ P(\\lambda) - T } = 0$\n\\par\nNow fix a $\\lambda \\in M$ and  a finite set of primes $T$ such that\n$P(\\lambda) - T \\neq \\emptyset$. We want to prove the lemma for $(\\lambda , T)$.\nAssume by induction that the lemma holds for all $(\\lambda', T')$\nsuch that $\\abs{ P(\\lambda') - T' } <  \\abs{ P(\\lambda) - T} $. \nChoose a prime $p \\in P(\\lambda) - T$. Let $T' = T \\cup \\lbrace p \\rbrace$.\nWrite $\\lambda = p^r \\mu$  with $p \\nmid \\mu$. \nWrite\n$L_T(j_{ \\lambda}, s)  = A_{\\infty} + \\sum_{t = 0}^{r} A_t$\nwhere\n\\begin{equation*}\nA_t =  \\sum_{n \\in \\mathbb{N}_T : v_p(n) = t} j_{\\lambda, n} n^{-s} \n\\text{\\; and \\;} \nA_{\\infty} =  \\sum_{n \\in \\mathbb{N}_T : v_p(n) > r} j_{\\lambda, n} n^{-s}.\n\\end{equation*}\nWe shall argue that each $A_j$ can be analytically continued.\nFist consider the sum $A_t$ with $t \\leq r$. \nSo let $n \\in \\mathbb{N}_T$ such that $v_p(n) = t \\leq r$. Let $ n_1 = p^{-t} n$.\nUsing lemma \\ref{l-cancel-common-factor-in-j}, we obtain\n\\begin{equation*}\nA_t \n= \\sum_{n_1 \\in \\mathbb{N}_{ T' } }  j_{0,p^t} \\cdot  j_{p^{-t} \\lambda, n_1} (p^t n_1)^{-s}\n= p^{-t s}  j_{0,p^t}   \\cdot L_{T'} (j_{ p^{-t} \\lambda} , s).\n\\end{equation*}\nNote that $\\abs{ P(p^{-t} \\lambda) - T' }= \\abs{ (P(\\lambda) - T) - \\lbrace p \\rbrace } \n= \\abs{P(\\lambda) - T} - 1$. \nSo $A_t$ has an analytic continuation by induction hypothesis.\n\\par\nNow consider the sum $A_{\\infty}$. Let $n \\in \\mathbb{N}_T$ such that $v_p(n)  > r$. \nLemma \\ref{l-cancel-common-factor-in-j} implies\n\\begin{equation*}\nj_{\\lambda, n} \n= p^{(m-1) r}  j_{\\mu, p^{-r} n}.\n\\end{equation*}\nWrite $n_2 = p^{-r} n$. Note that as $n$ varies over\n$\\mathbb{N}_T \\cap \\lbrace k \\in \\mathbb{N} \\colon v_{p}(k) > r \\rbrace$,\nthe number $n_2$ varies over $\\mathbb{N}_T - \\mathbb{N}_{T'}$.\nIt follows that\n\\begin{equation*}\nA_{\\infty} \n= p^{(m-1) r} \\sum_{n_2 \\in \\mathbb{N}_{T} - \\mathbb{N}_{T'} } j_{\\mu, n_2} (p^{r} n_2)^{-s}\n= p^{(m - 1 - s) r} (  L_{T} ( j_{ \\mu}, s) - L_{T' } ( j_{ \\mu}, s) ).\n\\end{equation*}\nNote that $\\abs{P(\\mu) - T} = \\abs{P(\\mu) - T' } = \\abs{P(\\lambda) - T} - 1$. So\nby induction hypothesis, it follows that\n$A_{\\infty}$ has analytic continuation. This proves the lemma for $\\lambda \\neq 0$.\n\\par\nFinally, for $\\lambda = 0$, theorem \\ref{th-j}(d) implies\n$L(j_0,s) = L( J_{m\/2}, s + 1 - (m\/2))$.\nSince $\\sum_{d \\mid n} J_{k} (n) = n^{k}$, we have\n$L(J_{k}, w) = \\zeta( w - k)\/\\zeta(w)$. \n\\end{proof}\n\\section{The even self-dual lattice of signature (25,1)}\n\\label{section-II251}\nIn this section, we collect a few facts about the Leech lattice and $\\mathrm{II}_{25,1}$ that\nwe would need for our calculations in section \\ref{section-Fourier-expansion}. \nFor more details see \\cite{C:Aut26} or \\cite{REB:Thesis}.\n Our sign conventions about Lorentzian \nlattices are consistent with \\cite{REB:Thesis}.\n\\begin{topic}{\\bf Leech cusp and Weyl chamber ``containing\" a Leech cusp:} \n\\label{t-C}\nAs in the introduction, Let $L  = \\Lambda \\oplus \\mathrm{II}_{1,1} \\simeq \\mathrm{II}_{25,1}$\nwhere $\\Lambda$ is the Leech lattice and $\\mathrm{II}_{1,1}$ is a hyperbolic cell.\nWrite $V = L \\otimes \\mathbb{R}$. \nIf $l \\in L$, we shall write $l = (\\lambda; m,n)$ with $\\lambda \\in \\Lambda$ and $m,n \\in \\mathbb{Z}$.\nThen $l^2 = \\lambda^2 - 2 m n$. \nLet $\\rho = (0; 0, 1) \\in L$. Then $\\rho$ is a primitive norm zero vector in $L$ such that \n$\\rho^{\\bot}\/\\rho \\simeq \\Lambda$, that is, $\\rho$ is a Leech cusp. \nIf $v \\in V$, we define its {\\it height} by\n\\begin{equation*}\n\\op{ht}(v) = - \\ip{v}{\\rho}.\n\\end{equation*}\nSo the height of $( *; h, *)$ is equal to $h$.\nThe automorphism group of $L$ acts on the hyperbolic space $\\mathcal{H}^{25}$.\nAs a concrete model of $\\mathcal{H}^{25}$, we take\n\\begin{equation*}\n\\mathcal{H}^{25} = \\lbrace v \\in V \\colon v^2 = -1, \\op{ht}(v)  > 0 \\rbrace.\n\\end{equation*}\nSo if $v \\in \\mathcal{H}^{25}$, then it has the form $ v = (x; m, (x^2 + 1)\/2m) $ for some $x \\in \\Lambda \\otimes \\mathbb{R}$ and some $m > 0 $.\nIf $c \\in \\mathbb{R}$, define\n \\begin{equation*}\n B_{\\rho}(c) =  \\lbrace v \\in \\mathcal{H}^{25} \\colon  \\op{ht}(v) < c \\rbrace.\n \\end{equation*}\nThe sets of the form $B_{\\rho}(c)$ are called {\\it (open) horoballs around $\\rho$}.\nGiven $x, y \\in \\mathcal{H}^{25}$, we say that $x$ is closer to $\\rho$ than $y$ if\n$\\op{ht}(x) < \\op{ht}(y)$. This is equivalent to saying that $d(x, B) < d(y,B)$ where\n $B$ is a small horoball around $\\rho$ that does not meet $\\lbrace x, y \\rbrace$.\n\\par\nA {\\it root} of a lattice is a norm $2$ lattice vector. Let $r$ be a root of $L$.\nWe say that $r$ is a {\\it positive root} if $\\op{ht}(r) > 0$. \nThe hyperplane $r^{\\bot}$ determines a \nhyperplane in $\\mathcal{H}^{25}$, called the {\\it mirror} of the root $r$.\nSince the Leech lattice has no roots, \n$\\rho$ does not lie on the mirror of any root of $L$.\nThe positive roots of $L$ corresponding to the mirrors closest to $\\rho$ are the roots of height $1$. These\nare called the {\\it Leech roots} (or {\\it simple roots}) and are parametrized by the vectors of the Leech lattice.\nFor each $\\lambda \\in \\Lambda$, we have a Leech root \n\\begin{equation*}\ns_{\\lambda} =  ( \\lambda; 1, \\tfrac{\\lambda^2}{2} - 1).\n\\end{equation*}\nThere is a unique Weyl chamber $C$ of the reflection group $R(L)$ in $\\mathcal{H}^{25}$ \ncontaining $\\rho$ in its closure.\nConway proved that\n\\begin{equation*}\nC = \\lbrace x \\in \\mathcal{H}^{25} \\colon \\ip{x}{s_{\\lambda}} < 0 \\text{\\; for all \\;} \\lambda \\in \\Lambda \\rbrace.\n\\end{equation*}\nAll these inequalities are necessary to define $C$, that is, \nthe walls of $C$ are in bijection with the mirrors\northogonal to the Leech roots. \n\\end{topic}\n\\begin{lemma}\nThe horoball $B_{\\rho}(1\/\\sqrt{2})$ is contained in the Weyl chamber $C$.\n\\label{l-small-horoballs-fit-in-Weyl-chamber}\n\\end{lemma}\n\\begin{proof}\nTake $v = (\\alpha; h, \\tfrac{\\alpha^2 +1}{2 h} ) \\in B_{\\rho}(1\/\\sqrt{2})$. Then $h^2 < \\tfrac{1}{2}$.\nWriting out $\\ip{s_{\\lambda}}{v}$ and completing square one obtains\n\\begin{equation*}\n\\ip{s_{\\lambda}}{v} = -\\tfrac{h}{2} ( \\lambda - \\tfrac{\\alpha}{h} )^2 + h - \\tfrac{1}{2 h}.\n\\end{equation*}\nNow $h^2< \\tfrac{1}{2}$ implies $h - \\tfrac{1}{2h} < 0$. So\n$\\ip{s_{\\lambda}}{v} < 0$ for all $\\lambda \\in \\Lambda$.\n\\end{proof}\n\\begin{topic}{\\bf The translations:}\nLet $\\mathbb{T}_{\\mathbb{R}}$ be the automorphisms of $V$ that fix $\\rho$ and act trivially on\n$\\rho^{\\bot}\/\\rho$.\nOne verifies that $\\mathbb{T}_{\\mathbb{R}} = \\lbrace T_v : v \\in \\Lambda \\otimes \\mathbb{R} \\rbrace$ where \n\\begin{equation*}\nT_v(l; a,b) = (l + av; a, b + \\ip{l}{v} + av^2\/2).\n\\end{equation*}\nThe group $\\mathbb{T}_{\\mathbb{R}}$ is naturally isomorphic to the additive group\n$\\Lambda \\otimes \\mathbb{R}$ via $ T_v \\mapsto v$.\nLet \n\\begin{equation*}\n\\mathbb{T} = \\mathbb{T}_{\\mathbb{R}} \\cap \\Aut(L) = \\lbrace T_{\\mu} : \\mu \\in \\Lambda \\rbrace,\n\\end{equation*}\n that is,\n$\\mathbb{T}$ consists of the automorphisms of $L$ that fix $\\rho$ and act trivially on $\\rho^{\\bot}\/\\rho$. \n\\end{topic}\n\\begin{topic}{\\bf The orbits of the continuous group of translations acting on $V$: }\n\\label{topic-continuous-group-of-translations}\nThe orbits of action of $\\mathbb{T}_{\\mathbb{R}}$ on $V$ are as follows:\nThe subspace $\\rho^{\\bot}$ splits into the zero dimensional orbit $\\lbrace \\rho \\rbrace$\nand one dimensional orbits $(l,0,0) + \\mathbb{R} \\rho$, for\n$l \\in \\Lambda - \\lbrace 0 \\rbrace$.\nOn the complement of $\\rho^{\\bot}$, the action of $\\mathbb{T}_{\\mathbb{R}}$ is free, so we have\n$24$ dimensional orbits. \nFor each $h, k \\in \\mathbb{R}$, we have a free orbit $S_{k,h}$ defined by\n\\begin{equation*}\nS_{k,h}  = \\mathbb{T}_{\\mathbb{R}}( 0;h,k\/2h)= \\lbrace v \\in V: v^2 = -k , \\op{ht}(v) = h \\rbrace.\n\\end{equation*}\nThe map \n\\begin{equation}\n\\phi: \\Lambda \\otimes \\mathbb{R} \\to S_{k,h} \\text{\\; defined by \\;} \\phi(v) = T_v(0;h,k\/2h)\n\\label{eq-phi}\n\\end{equation}\nis an isomorphism with inverse given by $ (l, h, *) \\mapsto l\/h$. \nUnder this identification, the action of $\\mathbb{T}$ on $S_{ k , h}$ corresponds to \nthe action of $\\Lambda$ on $\\Lambda \\otimes \\mathbb{R}$ by translation.\nSo $S_{k,h }\/\\mathbb{T}$ can be identified with the torus $(\\Lambda \\otimes \\mathbb{R})\/  \\Lambda$.\n\\end{topic}\n\\begin{topic}{\\bf The orbits of the discrete group of translations acting on the roots: }\n\\label{topic-discrete-group-of-translations}\nSince $\\Lambda$ does not have any roots, all roots of $L$ have nonzero height.\nFix $n \\geq 1$.\nA root $r \\in L(2)$ of height $n$ has the form $r = (l; n, \\tfrac{l^2\/2 - 1}{n} )$.\nSo, given $l \\in \\Lambda$, there is a root of the form $(l; n , *)$\nif and only if $l^2\/2 \\equiv 1 \\bmod n$. \nAs in \\ref{topic-notation}, we write\n\\begin{equation*}\n\\Lambda_n(1) = \\lbrace \\bar{l} \\in \\Lambda\/n : \\bar{l}^2\/2 \\equiv 1 \\bmod n \\rbrace.\n\\end{equation*}\nFor each coset $\\bar{l} \\in \\Lambda_n(1)$,\nlift $\\bar{l}$ to a lattice vector $l$ and fix once and for all a root\n$r_{n,\\bar{l}} =(l;n,*)$ (recall: the last coordinate of $r_{n , \\bar{l}}$ is determined by\nthe condition $r_{n,\\bar{l}}^2 = 2$).\nNow one verifies easily that the set of roots of height $n$\nis the disjoint union of the free $\\mathbb{T}$-orbits $\\mathbb{T} r_{n, \\bar{l}}$,\nparametrized by $\\bar{l} \\in \\Lambda_n(1)$. \n\\end{topic}\nThe lemma below is immediate corollary of \\ref{l-small-horoballs-fit-in-Weyl-chamber}.\n\\begin{corollary}\nLet $h, k \\in \\mathbb{R}_{>0}$ such that $ 2 h^2\/k < 1$. If $z \\in S_{k,h}$, then  \n$z\/\\sqrt{k}$  is contained in the Weyl chamber $C$.\n\\label{l-choice-of-h-k}\n\\end{corollary}\n\\section{Fourier series expansion of the Poincare series}\n\\label{section-Fourier-expansion}\n\\begin{topic}\n\\label{topic-setup-for-Fourier-series-computation}\nWe continue with the setup of section \\ref{section-II251}. \nFix a branch of logarithm on positive\nhalf plane so that $(-1)^s = \\e(s\/2)$.\nAs stated in the introduction,\nthe infinite series $E(z,s) = \\sum_{ r \\in L(2)}  \\ip{r}{z}^{-s}$ \nconverges for $\\op{Re}(s) > 25$ and  defines an analytic function in $s$\ninvariant under $\\Aut(L)$. \nFix, $k, h > 0$ such that \n\\begin{equation*}\n\\nu = 2h^2\/k < 1.\n\\end{equation*}\nThen \\ref{l-choice-of-h-k} implies that $S_{k,h}$ does not meet any hyperplane\northogonal to the roots of $L$.\nSo the restriction of $E(z,s)$ to\n$S_{k,h} \\simeq \\Lambda \\otimes \\mathbb{R} \\simeq \\mathbb{R}^{24}$ (see \\ref{topic-continuous-group-of-translations})\nis well defined and invariant under the group of translation $\\mathbb{T} \\simeq \\Lambda \\simeq \\mathbb{Z}^{24}$.\nThus $E(z,s)$ defines a function on the torus $S_{k ,h}\/\\mathbb{T}$.\nWe shall now calculate the Fourier series of this function.\nWe call this the Fourier series of $E(z,s)$ at the Leech cusp $\\rho$.\nThe Fourier series has the form:\n\\begin{equation}\nE( z ,s) = \\sum_{\\lambda \\in \\Lambda} a_{\\lambda} (k,h,s)\n\\e ( \\ip{ \\lambda}{ \\phi^{-1}(z)} )     \n\\label{eq-Fourier-series-of-E}\n\\end{equation}\nwhere $\\phi : \\Lambda \\otimes \\mathbb{R} \\xrightarrow[]{\\sim} S_{k,h}$ is as in equation \\eqref{eq-phi}.\nThe Fourier coefficients are given by\n\\begin{equation*}\na_{\\lambda}(k, h ,s) \n= \\int_{v \\in (\\Lambda \\otimes \\mathbb{R})\/\\Lambda} E(\\phi(v),s)   \n\\e ( -\\ip{\\lambda}{v}  ) d v.\n\\end{equation*}\n\\end{topic}\nTheorem \\ref{th-fourier-coeffients} gives a formula for the Fourier coefficients. The formula\ninvolves the character sums $j_{n, \\lambda}  = j_{n, \\lambda}^{\\Lambda}(1)$\nfrom section \\ref{section-j} and the numbers\n\\begin{equation*}\nc_n = \\sqrt{\\tfrac{k}{h^2} - \\tfrac{2}{n^2} }.\n\\end{equation*}\nNote that our choice $2 h^2\/k < 1$ ensures that $c_n^2  > 0$ for all $n \\geq 1$.\n\\begin{theorem}\nWrite $a^*_{\\lambda}(k, h, s) = (h\/2 \\pi)^s \\Gamma(s)  a_{\\lambda}(k,h,s)$.\nThen one has\n\\begin{equation*}\n a_{\\lambda}^*(k,h,s) \n=\n\\begin{cases} 2 (1 + \\e(-\\tfrac{s}{2} ))\\abs{\\lambda}^{s - 12}\n\\sum_{n \\geq 1}  j_{\\lambda, n} n^{-s} c_{n}^{12-s} K_{12 - s}(2 \\pi \\abs{\\lambda} c_{n} ) \n& \\text{ if \\;} \\lambda \\neq 0, \\\\\n& \\\\\n(1 + \\e(-\\tfrac{s}{2} )) \\Gamma(s - 12) \\pi^{12 - s} \\sum_{n \\geq 1}  j_{0, n} n^{-s}\nc_n^{24 - 2 s}\n& \\text{ if \\;} \\lambda = 0.\n\\end{cases}\n\\end{equation*}\n\\label{th-fourier-coeffients}\n\\end{theorem}\n\\begin{proof}\nTake $z \\in S_{k,h}$. Then \\ref{l-choice-of-h-k} implies that $z\/ \\sqrt{k}$ belongs to the Weyl chamber $C$.\nSo by Conway's theorem (see \\cite{REB:L}) $z$ and $\\rho$ are on the same side of $r^{\\bot}$ for every root $r \\in L(2)$.\nSo if $r$ is a positive root, then $-\\ip{r}{z} >  0$.\nUsing this and the decomposition of $L(2)$ into $\\mathbb{T}$-orbits from  \\ref{topic-discrete-group-of-translations}, we obtain\n\\begin{align*}\nE(z,s)\n= (1 + (-1)^{-s}) \\sum_{r : \\op{ht}(r) > 0} (-\\ip{r}{z})^{-s}  \n= (1 + \\e(-\\tfrac{s}{2} )) \n\\sum_{\\substack{ n \\geq 1 \\\\ \\bar{l} \\in \\Lambda_n(1) } }\n\\sum_{T \\in \\mathbb{T}} (-\\ip{T r_{n,\\bar{l}}}{ z})^{-s}.\n\\end{align*}\nSo\n\\begin{equation*}\na_{\\lambda}(k, h ,s) \n=(1 + \\e(-\\tfrac{s}{2} )) \\int_{v \\in (\\Lambda \\otimes \\mathbb{R})\/ \\Lambda} \n \\sum_{n , \\bar{l} }  \\sum_{ \\mu \\in \\Lambda}\n(-\\ip{r_{n,\\bar{l}}}{ T_{\\mu} \\phi( v ) })^{-s} \\e( - \\ip{\\lambda}{v + \\mu} ) d v.  \n\\end{equation*}\nSince $T_{\\mu} \\phi(v) = \\phi(v + \\mu)$,\nwe can change the above expression into an integral over the whole vector space\n$\\Lambda \\otimes \\mathbb{R}$ to get:\n\\begin{equation*}\na_{\\lambda}(k,h,s)= (1 + \\e(-\\tfrac{s}{2} )) \\sum_{n \\geq 1}  \\sum_{\\bar{l} \\in \\Lambda_n(1)}\n\\int_{v \\in \\Lambda \\otimes \\mathbb{R} } (- \\ip{r_{n,\\bar{l}} }{ \\phi(v) })^{-s}\n\\e (-  \\ip{\\lambda}{v} ) d v. \n\\end{equation*}\nRecall that $r_{n,\\bar{l}}=(l;n,\\tfrac{l^2 - 2}{2n})$\nand $\\phi(v) = (vh ;h, \\tfrac{ (vh)^2 + k}{2h}  )$.\nExpanding the expression for $-\\ip{r_{n,\\bar{l}} }{ \\phi(v) }$\nand completing square we get\n\\begin{equation*}\n-\\ip{r_{n,\\bar{l}} }{ \\phi(v) } = \\tfrac{n h}{2}( c_n^2 +(v -\\tfrac{ l}{n}  )^2 )\n\\end{equation*}\nwhere\n$c_n^2 = (\\tfrac{k}{h^2} - \\tfrac{2}{n^2} )$.\nWe substitute $  u = v - \\tfrac{l}{n} $ in the integral to get\n\\begin{equation*}\n a_{\\lambda}(k,h,s)\n= (1 + \\e(-\\tfrac{s}{2} ))\\sum_{n \\geq 1}  \\sum_{\\bar{l} \\in \\Lambda_n(1)}\n\\int_{\\Lambda \\otimes \\mathbb{R}} (\\tfrac{n h}{2})^{-s} (c_n^2 + u^2)^{-s}\n\\e (- \\ip{\\lambda}{ u + \\tfrac{l}{n}} ) du \n\\end{equation*}\nor\n\\begin{equation}\na_{\\lambda} (k, h, s)\n=(1 + \\e(-\\tfrac{s}{2} ))(\\tfrac{2}{h})^s  \\sum_{n \\geq 1}  j_{\\lambda, n} n^{-s}\n\\int_{\\Lambda \\otimes \\mathbb{R}}  (c_n^2 + u^2)^{-s}\n\\e(- \\ip{\\lambda}{u}) du\n\\label{eq-integral-expression-for-fourier-coeff}\n\\end{equation}\nwhere\n\\begin{equation*}\nj_{\\lambda, n} = j_{\\lambda, n}^{\\Lambda}\n= \\sum_{l \\in \\Lambda_n(1)} \\e( -\\ip{l}{\\lambda}\/n)\n= \\sum_{l \\in \\Lambda_n(1)} \\e(\\ip{l}{\\lambda}\/n).\n\\end{equation*}\nNote that the two sums are equal since $l \\mapsto -l$ is an involution of $\\Lambda_n(1)$.\n\\par\nAssume $\\lambda \\neq 0$.\nLet $\\op{Gram}(\\Lambda)$ be a Gram matrix of the  $\\Lambda$. So\n$u^2 = u' \\op{Gram}(\\Lambda) u$.\nLet $\\mu = \\op{Gram}(\\Lambda)^{1\/2} \\lambda$.\nSo $ \\mu^2 =  \\lambda^2$.\nUse the substitution $v = \\op{Gram}(\\Lambda)^{1\/2} u $\nin the integral in \\eqref{eq-integral-expression-for-fourier-coeff} to find\n\\begin{equation*}\nI =  \\int_{\\Lambda \\otimes \\mathbb{R}}  ( c_n^2 + u^2)^{-s}\n\\e (- \\ip{ u }{\\lambda}) du\n=   \\int_{\\mathbb{R}^{24}}  (c_n^2 + v^2)^{-s}\n\\e ( -(v \\cdot \\mu) ) dv.\n\\end{equation*}\nThus $I$ is the Fourier transform of a radial function.\nLet $r = \\abs{v}$, $\\xi = v\/\\abs{v}$ and $d v = r^{23} d r d \\omega(\\xi)$, where\n$\\omega$ is the standard measure on $S^{23}$. Changing to polar co-ordinates we have\n\\begin{equation*}\nI = \\int_{r \\in \\mathbb{R}_{+}}  r^{23} (c_n^2 + r^2)^{-s} \\int_{\\xi \\in S^{23}} \n\\e( (-r\\abs{\\mu}) (\\xi \\cdot \\mu\/\\abs{\\mu} )  ) d \\omega(\\xi) d r   \n\\end{equation*}\nThe integral over the sphere gives a Bessel function of the first kind.\nUsing lemma 9.10.2 of \\cite{AAR:SF}, we evaluate the integral over the sphere to get\n\\begin{equation*}\nI = \\int_0^{\\infty}  (c_n^2 + r^2)^{-s} r^{23} 2\\pi J_{11}(2 \\pi \\abs{\\lambda} r )(r\\abs{\\lambda})^{-11} d r.   \n\\end{equation*}\nSubstitute $x = 2 \\pi \\abs{\\lambda} r$  to get\n\\begin{equation*}\nI = (2 \\pi \\abs{\\lambda})^{2 s - 12} \\abs{\\lambda}^{-12} \n\\int_0^{\\infty}  x^{12} ((2 \\pi \\abs{\\lambda} c_n)^2 + x^2)^{-s}   J_{11}(x ) d x.\n\\end{equation*}\nThe last integral can be expressed in terms of the modified Bessel function $K_{\\nu}$. For\n $\\re(s) > 23\/4$, one has\n\\begin{equation*}\nI = \n(2 \\pi \\abs{\\lambda})^{2 s - 12} \\abs{\\lambda}^{-12} \n(2 \\pi \\abs{\\lambda} c_n)^{12 - s} K_{12 - s} (2 \\pi \\abs{\\lambda} c_n) \/ (2^{s-1} \\Gamma(s)).\n\\end{equation*}\n(see the table of Hankel transforms in \n\\cite{EMOT:TOIT}, vol II. p. 24, \nand substitute $y=1$, $\\nu = 11$ and $s = \\mu + 1 $ in formula (20)).  \nSubstituting in equation \\eqref{eq-integral-expression-for-fourier-coeff} and\nsimplifying, we get the formula for the Fourier coefficients \nfor $\\lambda \\neq 0$.\n\\par\nFor $\\lambda = 0$, from equation \\eqref{eq-integral-expression-for-fourier-coeff}\nwe get\n\\begin{equation*}\na_0(k, h, s)\n=(1 + \\e(-\\tfrac{s}{2} ))(\\tfrac{2}{h})^s  \\sum_{n \\geq 1}  j_{0, n} n^{-s}\n\\int_{\\Lambda \\otimes \\mathbb{R}}  (c_n^2 + u^2)^{-s} du \\\\\n\\end{equation*}\nBy changing to polar coordinates and changing variable $r = c_n x$, we obtain\n\\begin{equation*}\na_{0}(k,h,s) = (1 + \\e(-\\tfrac{s}{2} ))(\\tfrac{2}{h})^s \\op{vol}(S^{23}) \\sum_{n \\geq 1}  j_{0, n} n^{-s} c_n^{24 - 2s} \n\\int_0^{\\infty}   x^{23} (1 + x^2)^{-s} dx\n\\end{equation*}\nThe last integral is equal to $ \\Gamma(12) \\Gamma(s -12)\/ 2 \\Gamma(s)$\nfor $\\op{Re}(s) > 12$.\n(see the table of Mellin transforms in \n\\cite{EMOT:TOIT} Vol I, p. 311 and substitute $h = 1, \\alpha = 1$ $s  = 24$ in formula (30)).\nSubstituting the value of the integral and $\\op{vol}(S^{23}) = 2 \\pi^{12}\/\\Gamma(12)$\nand simplifying, we get the formula for $a_0( k,h,s)$.\n\\end{proof}\nUsing theorem \\ref{th-fourier-coeffients}, we now prove that\n$E(z,s)$ can be analytically continued. \nThe argument is similar to an argument used to prove analytic continuation\nof real analytic Eisenstein series (see \\cite{B:A}, p. 68-69).\nRecall that we write $\\nu = 2 h^2\/k < 1$.\nNote that $c_n^2 = (2\/\\nu) ( 1 - \\nu\/n^2)$.\n\\begin{theorem}\nThe series $E(z,s)$ can be analytically continued to a meromorphic function\non the half plane $\\op{Re}(s) > 25\/2$. The only poles of $E(z,s)$\nin this half plane comes from the Fourier coefficient $a_0(k,h,s)$.\n\\label{th-analytic-continuation-of-E}\n\\end{theorem}\n\\begin{proof}\nFrom  \\ref{l-analytic-continuation-of-Dirichlet-series-of-j}, \nwe know $\\sum_n j_{0, n} n^{-s} = \\zeta(s - 23)\/\\zeta(s - 11)$; so this Dirichlet series\nhas meromorphic continuation to $\\mathbb{C}$. \nLemma \n\\ref{l-analytic-continuation-of-perturbation} \n(applied with $f(n) = j_{0,n}$ and $g(s,y) = (1 - \\nu y^2)^{12 - s}$)\nimplies that\nthe infinite series for $a_0(k,h,s)$ given in theorem \\ref{th-fourier-coeffients} has an \nmeromorphic continuation to $\\mathbb{C}$. Now let $\\lambda \\neq 0$.\nThe bound on $j_{\\lambda,n}$ from lemma \\ref{l-bound-on-magnitude-of-j}\nshows that the infinite series for $a_{\\lambda}(k,h,s)$ in \\ref{th-fourier-coeffients}\nconverges absolutely and uniformly on compacta, for $\\op{Re}(s) >25\/2$. \nSo each Fourier coefficient defines an analytic function on $\\op{Re}(s) > 25\/2$.\nLet  $s $ vary on a compact subset $A$ of $\\lbrace z : \\op{Re}(z) > 25\/2 \\rbrace$.\nLet $\\kappa = \\sqrt{2(1  - \\nu)\/\\nu} > 0$. \nNote that $c_n \\geq  \\kappa$ for all $n$.\nSo \n\\begin{equation*}\n\\abs{ c_n^{12 - s} } = (1\/c_n)^{\\op{Re}(s - 12)} \\leq (1\/\\kappa)^{\\op{Re}(s - 12)}\n\\text{\\; and \\;} \ne^{- \\pi \\abs{ \\lambda} c_n}  \\leq e^{- \\pi \\kappa \\abs{\\lambda}}.\n\\end{equation*}\nThe Macdonald Bessel function $K_s(y)$ decays exponentially as $y \\to \\infty$.\nIt is convenient to use the bound $\\abs{K_s(y)} \\leq e^{-y\/2} K_{\\op{Re}(s)} (2)$\nfor $y > 4$ (see \\cite{B:A}, p. 66). \nFix  $\\epsilon >0$ such that $\\op{Re}(s) > 2 \\epsilon + 25\/2$ for all $s \\in A$.\nFrom \\ref{l-bound-on-magnitude-of-j},\nwe know that there exists a constant $C_{\\epsilon}$ such that\n$\\abs{j_{\\lambda,n}} \\leq C_{\\epsilon} \\abs{\\lambda}^{23\/2} n^{23\/2 + \\epsilon}$. \nUsing the expression for $a_{\\lambda}(k,h,s)$ from \\ref{th-fourier-coeffients} and \nthe bound for $\\abs{j_{n,\\lambda}}$, we find\n \\begin{align*}\n \\abs{a_{\\lambda}(k,h,s)}\n\n\n\n\n\n \\leq\n C(s) \\abs{\\lambda}^{\\op{Re}(s) - 1\/2} e^{- \\pi \\kappa \\abs{ \\lambda}}\n \\end{align*}\nwhere \n$C(s)$   is a fixed positive real valued\n continuous function on $ A$ \n that does not depend on $\\lambda$.\n Since $C(s)$ and $\\op{Re}(s)$ \n stay bounded on a compact set,\n$\\abs{a_{\\lambda}(k,h,s)}$ decays exponentially as a function of $\\abs{\\lambda}$ as\n $\\abs{\\lambda} \\to \\infty$. It follows that the Fourier series for $E(z,s)$ given in \n equation \\eqref{eq-Fourier-series-of-E} converges\n absolutely and\n uniformly on compact subsets of $\\op{Re}(s) > 25\/2$.\n \\end{proof}\n\\begin{remark}\n\\label{remark-analytic-continuation-of-Fourier-coefficients}\nLemma \\ref{l-analytic-continuation-of-perturbation} applied with\n$f(n) = j_{n, \\lambda}$ and\n\\begin{equation*}\ng(s, y) = \\sqrt{1 - \\nu y^2}^{12-s} K_{12 - s}( 2 \\pi \\abs{\\lambda} \\sqrt{2\/\\nu} \\sqrt{1 - \\nu y^2} )\n\\end{equation*}\nimplies that each Fourier coefficient $a_{\\lambda}(k,h,s)$ has\nanalytic continuation to a meromorphic function on the whole $s$-plane. But we are not able \nto establish how these extended functions decay as $\\abs{\\lambda} \\to \\infty$.\nSo we are unable to prove analytic continuation of $E(z,s)$ beyond\nthe line $\\op{Re}(s) = 25\/2$.\n\\end{remark}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDoping in organic semiconductors is an ubiquitous phenomenon appearing when donor and acceptor molecules are combined together, and crucially determines the electronic, optical, and transport properties of the resulting materials~\\cite{aziz+07am,yim+08am,ping+10jpcl,gao+13jmcc,ping-nehe13prb,duon+13oe,salz+13prl,mend+13acie, yang+14ami, mend+15ncom,salz+16acr,fuze+16jpcl,jaco-moul17am,duva+18jpcc,beye+19cm}.\nTwo main mechanisms have been identified to be responsible for doping in organic semiconductors: Integer charge transfer (ICT) manifests itself when an electron is transferred from the donor to the acceptor, leading to the formation of ion pairs. \nPartial charge transfer occurs upon electronic hybridization of the frontier orbitals of the donor and the acceptor, giving rise to a charge-transfer complex (CTC). \nWhile a general consensus has been established around these two processes, recent studies showing that ICT and CTC can coexist in the same molecular material~\\cite{jaco+18mh,neel+18jpcl} have reignited the debate about doping in organic semiconductors.\nOne of the open questions in this respect concerns the role of the donor conjugation length in determining the electronic structure and the optical properties of these systems.\nA recent study by Mendez \\textit{et al.}~\\cite{mend+15ncom} shows that ICT appears when the strong electron acceptor 2,3,5,6-tetrafluoro-7,7,8,8-tetracyano-quinodimethane (F4TCNQ) is blended with the poly-3-hexylthiophene (P3HT) polymer, (see also Refs.~\\citenum{yim+08am,ping-nehe13prb,duon+13oe,karp+17mm,scho+17afm}), while CTCs are formed when the same acceptor is coupled to the quarterthiophene oligomer.\nThese findings suggest a fundamental relation between the doping mechanism and the length of the donor chain.\nOn the way to tackle this problem in its general complexity, it is relevant to first understand how the conjugation length of the donor oligomers impacts the electronic structure and the optical excitations of complexes exhibiting only partial charge transfer.\nRationalizing this behavior in the presence of one specific doping mechanism is essential to set the stage for further theoretical and experimental studies.\n\nWe perform a first-principles study based on density-functional theory (DFT) and many-body perturbation theory (MBPT), which offer state-of-the-art accuracy and insight into the electronic and optical properties of the investigated systems~\\cite{blas-atta11apl,baum+12jctc,fabe+14ptrsa,nied+15afm}.\nWe focus on CTCs formed by thiophene oligomers (nT) $p$-doped by F4TCNQ.\nWe consider thiophene chains with an even number of rings ranging from 4T, which is the shortest oligomer typically investigated in experiments~\\cite{aziz+07am,ping+10jpcl,mend+15ncom}, up to 10T.\nAll systems are modeled as molecular dimers \\textit{in vacuo}~\\cite{zhu+11cm,gao+13jmcc,mend+13acie,mend+15ncom}, with the donor and the acceptor molecules in the $\\pi$-$\\pi$ stacking arrangement~\\cite{zhu+11cm,gao+13jmcc}.\nIn this way, we are able to systematically rationalize the role of the donor length on the electronic and optical properties of CTCs which differ from each other only by the size of the nT oligomers.\nStarting from the level alignment of the individual components, we relate the band-gap reduction upon increasing donor length to the variation of the ionization energy of the system.\nIn the analysis of the optical properties we examine how the absorption onset and the spectral density are affected by the size of the nT oligomer.\nWe finally discuss the character of the lowest-energy excitations in terms of electron and hole density and demonstrate that their behavior is critically sensitive to the donor conjugation length.\n\n\\section{Theoretical Background and Computational Details}\nEquilibrium geometries of the individual donor and acceptor molecules as well as of the CTCs are computed from DFT with the all-electron package FHI-aims~\\cite{blum+09cpc} adopting tight integration grids, TIER2 basis sets~\\cite{havu+09jcp}, and the Perdew-Burke-Ernzerhof~\\cite{perd+96prl} generalized-gradient approximation for the exchange-correlation functional.\nVan der Waals interactions, that play an essential role in $\\pi$-$\\pi$ stacked systems, are accounted for by means of the Tkatchenko-Scheffler scheme~\\cite{tkat-sche09prl}. \nAtomic positions are relaxed until the Hellmann-Feynman forces are smaller than 10$^{-3}$ eV\/\\AA{}.\nThe acceptor molecule adsorbed in the middle of the donor chain gives rise to the most stable configuration, in agreement with previous first-principles results~\\cite{zhu+11cm}.\nFurther details and discussion about the geometry of the CTCs are reported in the Supporting Information. \n\nThe electronic structure and the optical properties of the considered systems are computed with the MOLGW code~\\cite{brun+16cpc}.\nGaussian-type cc-pVDZ basis sets~\\cite{brun12jcp} are used including the frozen-core approximation. \nThe resolution-of-identity approximation is also employed~\\cite{weig+02}. \nThe $GW$ approximation in the perturbative approach $G_0W_0$ is adopted to calculate the quasi-particle (QP) correction to the single-particle levels starting from the solution of the Kohn-Sham equations.\nThe range-separated hybrid functional CAM-B3LYP~\\cite{yana+04cpl} is chosen to obtain a reliable starting point for the $G_0W_0$ calculations~\\cite{fabe+13jcp,brun+13jctc}.\nOptical absorption spectra and (bound) electron-hole pairs are computed from the solution of the Bethe-Salpeter equation (BSE), the equation of motion for the two-particle polarizability, $L = L_0 + L_0 \\, K \\, L$~\\cite{brun+16cpc}.\nThe so-called BSE kernel $K$ includes the repulsive exchange term given by the bare Coulomb potential $\\bar{v}$ and the statically screened electron-hole Coulomb attraction $W$. \nFor further details about this formalism, see, \\textit{e.g.}, Refs.~\\citenum{fabe+13jcp,cocc-drax15prb,brun+16cpc}.\nWe solve the BSE in the Tamm-Dancoff approximation (TDA), which does not impact the nature of the excitations analyzed below.\nMinor effects on the excitation energies and oscillator strengths in line with previous works~\\cite{palu+09jcp,baum+12jctc1,rang+17jcp} are discussed in the Supporting Information (see Figure S6 and Table S1).\nThe character and spatial distribution of the electron-hole pairs are obtained from the analysis of the hole and electron densities, which for the $\\lambda^{th}$ excitation are defined as~\\cite{cocc+11jpcl,deco+14jpcc}\n\\begin{equation}\n\\rho_{h}^{\\lambda} (\\textbf{r})=\\sum_{\\alpha \\beta} A_{\\alpha \\beta}^{\\lambda} \\rvert \\phi_{\\alpha}(\\textbf{r})\\lvert^{2}\n\\label{eq:h}\n\\end{equation}\nand\n\\begin{equation}\n\\rho_{e}^{\\lambda} (\\textbf{r})=\\sum_{\\alpha \\beta} A_{\\alpha \\beta}^{\\lambda} \\rvert \\phi_{\\beta}(\\textbf{r})\\lvert^{2},\n\\label{eq:e}\n\\end{equation}\nrespectively.\nThe coefficients $A_{\\alpha \\beta}^{\\lambda}$ are the square of the normalized BSE eigenvectors that act as weighting coefficients of each transition between occupied $(\\phi_{\\alpha})$ and unoccupied $(\\phi_{\\beta})$ QP states contributing to the $\\lambda^{th}$ excitation.\n\n\\section{Results and Discussion}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{level_alignment_DFT.png}\n\\caption{Energy level alignment between the isolated 4T (left) and F4TCNQ (right) computed from DFT (CAM-B3LYP functional). The HOMO and the LUMO of the resulting complex  are shown in the the middle panel. For comparison, the HOMO of the donor and the LUMO of the acceptor are also depicted.}\n\\label{level_alignment}\n\\end{figure}\n\nWe start our analysis by considering the electronic structure of the CTC formed by the 4T oligomer doped by F4TCNQ, depicted in Figure~\\ref{level_alignment}.\nThe energy values defined therein are computed with respect to the vacuum level set to zero.\nBy first inspecting the electronic energy levels of the individual molecules, we notice that they form a \\textit{staggered} or \\textit{type-II} level alignment, with the lowest-unoccupied molecular orbital (LUMO) of the acceptor lying within the gap of the donor.\nIn the resulting 4T-F4TCNQ complex, the highest-occupied molecular orbital (HOMO) is energetically lower than the HOMO of the donor, while the LUMO is higher than the LUMO of the acceptor (see Figure~\\ref{level_alignment}, middle panel).\nIn agreement with the known behavior of CTCs~\\cite{zhu+11cm,mend+13acie,gao+13jmcc,mend+15ncom}, the band gap of the complex is almost 1 eV larger than the energy difference between the HOMO of 4T and the LUMO of F4TCNQ.\nThe HOMO and LUMO of the CTC exhibit the typical features of bonding and anti-bonding states, respectively, due to the $\\pi$-$\\pi$ coupling between the components: The HOMO is seamlessly extended across the donor and the acceptor molecules, while the LUMO distribution is characterized by an evident nodal plane between the 4T and the F4TCNQ (see Figure~\\ref{level_alignment}, middle panel).\nThe character of the frontier orbitals is an intrinsic property of the CTCs, independent of modeling them as dimers \\textit{in vacuo}.\nBonding and anti-bonding states are formed also when the donor\/acceptor complexes are represented by periodic stacks -- see Figure S5 in the Supporting Information, and Ref.~\\citenum{beye+19cm}.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.48\\textwidth]{level_alignment_g0w0.png}\n\\caption{Energy level alignment computed from $G_{0}W_{0}$ on top of DFT (CAM-B3LYP functional) for the 4T-, 6T-, 8T-, and 10T-F4TCNQ complexes and their individual components.}\n\\label{Elec_all}\n\\end{figure}\n\nLevel alignment and band gaps analyzed above are computed from DFT, employing the hybrid functional CAM-B3LYP.\nAn improved picture can be obtained from MBPT including the QP correction computed from $G_0W_0$.\nIn Figure~\\ref{Elec_all} we report the results of these calculations for all the investigated systems, showing that CTCs are formed also upon increasing donor length, with the qualitative trend for the gaps and the frontier levels being unchanged compared to the case of 4T-F4TCNQ discussed above. \nAs expected upon the inclusion of the electronic self-energy, the size of the band gap of the individual molecules and of the resulting CTCs is increased by about 1 eV. \nWe first consider again the case of the 4T-F4TCNQ complex, shown on the top panel of Figure~\\ref{Elec_all}.\nIn the $G_0W_0$ energy levels of both donor and acceptor separately, the LUMO is shifted up by almost 1 eV while the QP correction to the HOMO is one order of magnitude smaller and moves towards more negative energies.\nAs a result, the energy difference between the HOMO of the donor and the LUMO of the acceptor increases from 1.68 eV in DFT (Figure~\\ref{level_alignment}) to 2.87 eV in $G_0W_0$ (Figure~\\ref{Elec_all}).\nThe CTC exhibits a similar although less pronounced behavior, such that the QP gap of 4T-F4TCNQ is equal to 3.56 eV.\nInspecting now Figure~\\ref{Elec_all} in full we notice that the QP gap decreases by about 0.8 eV going from 4T-F4TCNQ to 10T-F4TCNQ.\nThe band gap reduction in the CTCs is less significant than in the isolated thiophene oligomers with increasing length~\\cite{fabi+05jpca,sun-auts14jctc,cocc-drax15prb}, revealing that the hybridization between donor and acceptor cannot be trivially predicted from the level alignment of the individual constituents. \nYet, it is clear from Figure~\\ref{level_alignment} that the decrease of the electronic gaps in the CTCs with varying nT size is mainly due to the up-shift of the HOMO level energy with the LUMO remaining almost constant.\nAs the orbital energies are computed with the vacuum level set to zero, this finding implies that the electron affinity of the CTCs is almost unaffected by the increasing donor length, while the ionization energy is critically influenced by this parameter.\nThis behavior is not surprising, since the examined systems differ only by the length of the donor doped by the same electron acceptor.\nOn the other hand, this result is not obvious \\textit{a priori} considering the hybridized character of the HOMO and the LUMO in all CTCs.\nFrom this analysis it is possible to deduce the first relevant result of this work: The increasing donor conjugation length leads to a sizable decrease in the band gaps of the CTCs (0.8 eV going from 4T-F4TCNQ to 10T-F4TCNQ) which stems almost entirely from the reduction of the ionization energy.\nThe physical rationale of this behavior can be deduced from the spatial distribution of the frontier orbitals (see Figure S4 in the Supporting Information). \nWhile the bonding and anti-bonding character of the HOMO and the LUMO is shared by all CTCs, the delocalization of the highest-occupied orbital is enhanced upon the increasing donor length.\nOn the contrary, even in the longest nT chains, the LUMO remains confined on the F4TCNQ and on the few thiophene rings interacting with the acceptor through $\\pi$-$\\pi$ coupling.\nThe similar trend given by $G_0W_0$ (Figure~\\ref{Elec_all}) and DFT results (Figure~\\ref{level_alignment} and Figure S3 in the Supporting Information) suggests that quantum confinement and electronic hybridization are mainly responsible for the variation of the gap due to the up-shift of the HOMO with almost constant LUMO energies.\nDifferent from dimers formed by the same moiety (see \\textit{e.g.}, Ref.~\\citenum{hoga+13jcp}), here many-body effects, such as the electronic screening and the self-energy, improve the accuracy of orbital energies and electronic gaps, but do not qualitatively alter the picture given by (hybrid) DFT.\n\n\\begin{figure} \n\\centering\n\\includegraphics[width=0.5\\textwidth]{camb3lyp_spectra.png}\n\\caption{Optical absorption spectra of the 4T-, 6T-, 8T-, and 10T-F4TCNQ complexes. The first three bright peaks are labeled as P$_{1}$, P$_{2}$, and P$_{3}$, and the energy of the QP gaps ($E_{gap}$) is marked by vertical dashed lines. }\n\\label{spectra}\n\\end{figure}\n\nWe now turn to the optical properties of the CTCs.\nIn all complexes, the first excitation is dipole-allowed and gives rise to the weak peak labeled P$_1$ (see Figure~\\ref{spectra}).\nRegardless of the donor length, it has very similar energy ranging between 1.1 eV and 1.2 eV and comparable oscillator strength.\nThis behavior can be better understood by analyzing the character of this excitation in terms of orbital contributions.\nIn Table~\\ref{CTC-composition} we report the composition of P$_1$ in the 4T-F4TCNQ and 8T-F4TCNQ complexes (details about the other CTCs are given in the Supporting Information, Table S2) and in Figure~\\ref{h-e} we display the spatial distribution of the corresponding electron and hole densities (Eqs.~\\ref{eq:h} and~\\ref{eq:e}).\nIn all complexes the first excitation stems from the HOMO-LUMO transition, such that the hole and the electron components of P$_1$ exhibit the same character and spatial extension of the frontier orbitals.\nThis has two main consequences: In all systems the lowest-energy excitation has a Frenkel-like character localized within the CTC. \nMoreover, the comparable overlap between the HOMO and the LUMO (and equivalently of the hole and the electron density) in all complexes explains why P$_1$ has almost constant oscillator strength going from 4T-F4TCNQ to 10T-F4TCNQ.\nWe also recall that P$_1$ retains approximately the same energy in all CTCs irrespective of the donor length, in contrast with the decrease of the electronic gaps in the presence of longer nT chains discussed above. \nTo clarify this point, we take advantage of the adopted formalism where optical excitations are computed including the electron-hole correlation.\nIn this framework, the difference between the quasi-particle HOMO-LUMO gap and the excitation energy of P$_1$ can be interpreted as the exciton binding energy ($E_b$).\nBy comparing the excitation energies in Table~\\ref{CTC-composition} (and Table S2 in the Supporting Information) with the QP gaps in Figure~\\ref{Elec_all}, it is evident that $E_b$ decreases as the length of the donor oligomer increases.\nThe binding energy reduction of P$_1$ upon increasing donor length is therefore comparable with the decrease of the band gap (see Figure~\\ref{Elec_all}), such that these two quantities end up compensating each other.\nIn the presence of longer nT chains the Coulomb screening is enhanced and the quantum confinement of the electron-hole pairs is concomitantly reduced.\nThe Coulomb screening is solely embedded in the direct term of the BSE while quantum confinement affects also the bare exchange interaction between the electron and the hole~\\cite{hoga+13jcp,cocc-drax15prb}. \nDue to these intertwined effects and to the reduction of the QP gaps discussed above, P$_1$ appears at the same energy in all the examined CTCs, regardless of the donor conjugation length.\n\n\\begin{table*} \n\\centering\n\\caption{Energies, oscillator strength (OS), and composition in terms of transitions between molecular orbitals of the first three bright excitations in the 4T-F4TCNQ and 8T-F4TCNQ complexes. HOMO and LUMO are abbreviated by H and L, respectively.}\n\\label{CTC-composition}%\n\\begin{tabular}{lcccl}\n\\hline\n& Excitation & Energy [eV]& OS &  Composition \\\\ \\hline\n4T-F4TCNQ   & P$_1$  & 1.23 & 0.19& H $\\rightarrow$ L \\\\ \n& P$_2$ & 2.02 & 0.19  & H-1 $\\rightarrow$ L \\\\ \n& P$_3$  & 2.72 & 0.10 & H $\\rightarrow$ L+1  \\\\         \n& & & & H-2 $\\rightarrow$ L \\\\ \\hline \n8T-F4TCNQ & P$_1$ & 1.08 & 0.17 & H $\\rightarrow$ L  \\\\ \n& P$_2$ & 1.39 & 0.55 & H-1 $\\rightarrow$ L \\\\ \n& P$_3$ & 2.26 & 2.00 & H $\\rightarrow$ L+1 \\\\ \\hline\t\\end{tabular}\n\\end{table*}\n\nAbove the absorption onset a number of peaks appears in all the spectra shown in Figure~\\ref{spectra}.\nIn the following, we focus on the second and third bright excitations, P$_2$ and P$_3$, and analyze their behavior. \nIn the spectrum of 4T-F4TCNQ both P$_2$ and P$_3$ have comparable oscillator strength with respect to each other and to P$_1$. \nThe electron and hole densities of these three excitations are indeed very similar, corresponding in all cases to Frenkel excitons localized within the CTC (see Figure~\\ref{h-e}). \nTheir different composition in terms of single QP transitions (see Table~\\ref{CTC-composition}) reveals that in this complex, where the sizes of the donor and the acceptor are comparable, also the HOMO-1 and the LUMO+1 are hybridized (further details in the Supporting Information, see Figure S4).\nThis picture drawn by our first-principles results is consistent with the experimental data available for the 4T-F4TCNQ complex.\nIn Ref.~\\citenum{mend+15ncom} the optical density of this CTC is reported: The sharp maximum at the absorption onset is followed by a dip and by a broader and more intense peak around 2.4 eV. \nTwo findings are particularly significant: First, the energy separation between the first and second maxima in the measured spectra is consistent with the computed energy difference between P$_1$ and P$_3$ shown in Figure~\\ref{spectra} and Table~\\ref{CTC-composition}. \nSecond, the relative intensity between the two maxima recorded in the experiment follows the same trend resulting in the calculated spectrum. \nAbsolute values of excitation energies and binding energies are not quantitatively comparable between our theoretical results and the experiment, as the CTCs are modeled here by dimers \\textit{in vacuo}. \nThus, only relative trends can be correlated.\nWe also note that in organic films the exciton binding energy may be affected by the crystalline structure of the doped material, which in turns may influence the hole delocalization -- see, \\textit{e.g.}, Ref.~\\citenum{li+17prm}.\n\\begin{figure*} \n\\centering\n\\includegraphics[width=1.0\\textwidth]{hole-elec.png}\n\\caption{Electron (e) and hole (h) densities of the first (P$_{1}$), second (P$_{2}$) and third (P$_{3}$) optically active excitations of each complex. Isosurfaces are plotted with a cutoff of 0.002 \\AA{}$^{-3}$.}\n\\label{h-e}\n\\end{figure*}\n\nThe scenario changes substantially when the CTCs are formed by oligothiophenes with six rings or more: P$_2$ and P$_3$ have significantly higher oscillator strength than P$_1$ (see Figure~\\ref{spectra}) and their character varies accordingly (see Figure~\\ref{h-e}). \nThe orbital composition of these two excitations is analogous to the case of 4T-F4TCNQ (see Table~\\ref{CTC-composition}), demonstrating that the influence of the donor length manifests itself primarily in the character of the orbitals.\nIn these CTCs with enhanced size mismatch between the donor and the acceptor the orbitals below the HOMO and above the LUMO remain localized only on the nT oligomer (see Supporting Information, Figure S4): Being considerably longer than F4TCNQ, 6T, 8T, and 10T have higher density of states around the gap. \nAs a consequence, P$_2$, which stems from the transition between the HOMO-1 and the LUMO in all CTCs, becomes a charge-transfer exciton with the hole localized on the donor and the electron almost completely spread on the acceptor. \nThe enhanced space separation between electron and hole densities is responsible for the relative low oscillator strength of P$_2$ compared to higher-energy peaks.\nOn the other hand, both the electron and the hole in P$_3$ tend to be increasingly localized on the nT chain as the size of the latter grows, giving rise to a Frenkel exciton within the donor. \nThe significant increase of the oscillator strength of P$_3$ upon increasing length of the thiophene oligomers is therefore explained by the larger overlap between the electron and the hole components of this excitation. \nFinally, by inspecting all in all the spectra shown in Figure~\\ref{spectra}, we notice that different from P$_1$, but in line with the behavior of the electronic gaps, both P$_2$ and P$_3$ red-shift upon increasing donor length and so do also the peaks at higher energies.\nThe concomitant rise of their oscillator strength results in increased spectral density in the visible region.\nThe relevance of this result is apparent: Even in case only partial charge transfer occurs in doped organic semiconductors, photo-excited charge carriers can be effectively generated upon efficient absorption of visible light and enhanced electron-hole separation.\n\nBefore concluding, it is instructive to discuss the trends obtained by our results in light of the available experimental findings on nT-F4TCNQ charge-transfer complexes.\nThe optical spectrum measured for 4T-F4TCNQ with 50$\\%$ doping concentration and reported in Ref.~\\citenum{mend+15ncom} is dominated by a bright peak at the onset which is interpreted as a signature of CTC formation.\nThe character of $P_1$ discussed above is fully compatible with this analysis.\nMoreover, the experimental spectrum is characterized by a sharp increase of the oscillator strength about 2 eV above the optical gap, which is also in line with the results shown in Figure~\\ref{spectra}.\nDue to the lack of measurements on CTCs formed by longer oligomers than 4T, it is not possible to extend this comparison to the other systems investigated in this work.\nHowever, considering the tendency of P3HT to bend, forming linear units consisting of at least half a dozen of thiophene rings, it is reasonable to discuss our results in light of measurements performed on P3HT-F4TCNQ blends where partial charge transfer is locally observed~\\cite{jaco+18mh,neel+18jpcl}.\nThe measured spectra reveal a broad absorption band with a maximum between 2.0 eV and 2.5 eV, which is assigned to transitions between hybridized states and thus interpreted as a signature of CTC formation~\\cite{jaco+18mh,neel+18jpcl}. \nThis feature can be related to the peak P$_3$ in the spectra of 8T-F4TCNQ and 10T-F4TCNQ shown in Figure~\\ref{spectra}, taking into account the expected blue-shift of a few hundred meV compared to experiments induced by modelling the CTCs as dimers \\textit{in vacuo}.\nNew experiments on nT-F4TCNQ CTCs with longer donor chains than 4T are certainly needed to corroborate this analysis.\nA quantitative comparison with future measurements will be enabled by the inclusion in the calculated spectra of an effective screening term accounting for the polarizable environment surrounding the doped species in the crystalline phase.\n\n\\section{Summary and Conclusions}\nIn summary, we have investigated from first-principles many-body theory how the donor conjugation length impacts the electronic structure and the optical excitations of CTCs formed by thiophene oligomers doped by the strong electron acceptor F4TCNQ.\nWe have found that all systems are characterized by hybridized frontier orbitals regardless of the size of the thiophene oligomers. \nThe electronic gaps decrease upon increasing donor length solely due to the reduction of the ionization energy, with the electron affinity remaining almost constant in all CTCs. \nThe optical gaps follow a different trend, being at approximately the same energy irrespective of the donor length.\nTogether with the comparable oscillator strength of the first absorption peak, this finding implies that whenever a nT-F4TCNQ charge-transfer complex is formed, it absorbs the same about of light at approximately the same energy independently of the donor length. \nThe size of the nT oligomer influences the strength and character of the higher-energy excitations. \nWhile in the 4T-F4TCNQ complex the first three bright excitations retain the same oscillator strength, in the presence of thiophene oligomers with six rings or more, higher-energy peaks gain significant spectral weight relative to the absorption onset and experience a sizable red-shift.\nAs a result, the optical density in the visible region is significantly enhanced upon increasing donor length.\nThe nature of the excitations changes accordingly. \nIn 4T-F4TCNQ the first three excitations have the same Frenkel-like character within the CTC.\nConversely, in the complexes formed by longer nT chains the second bright peak corresponds to a charge-transfer-like excitation and the third one to a Frenkel exciton with increasing localization on the donor as the size of the latter grows.\n\nOur work and the rationale elaborated therein offer important insight into the electronic structure and optical excitations of charge-transfer complexes.\nWhile doping in organic semiconductors remains a debated topic, our findings unravel fundamental structure-property relations in the presence of partial charge transfer, which is one of the leading doping mechanisms in molecular materials.\nAs such, our results represent an important step forward towards a deeper understanding of the opto-electronic properties of doped organic semiconductors.\n\n\\begin{acknowledgement}\nInspiring discussions with Arrigo Calzolari, Andreas Optiz, and Dieter Neher are gratefully acknowledged. \nThis work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 182087777 - SFB 951 and HE 5866\/2-1.\n\\end{acknowledgement}\n\n\n\\begin{suppinfo}\nAdditional details about the structural, electronic, and optical properties of the CTCs are reported in the Supporting Information.\n\\end{suppinfo}\n\\newpage\n\n\\section*{Graphical TOC Entry}\n\\begin{figure}%\n\\centering\n\\includegraphics[height=4.45cm]{TOC_NEW_C.png}\n\\end{figure}\n\\newpage\n\\providecommand{\\latin}[1]{#1}\n\\providecommand*\\mcitethebibliography{\\thebibliography}\n\\csname @ifundefined\\endcsname{endmcitethebibliography}\n  {\\let\\endmcitethebibliography\\endthebibliography}{}\n\\begin{mcitethebibliography}{45}\n\\providecommand*\\natexlab[1]{#1}\n\\providecommand*\\mciteSetBstSublistMode[1]{}\n\\providecommand*\\mciteSetBstMaxWidthForm[2]{}\n\\providecommand*\\mciteBstWouldAddEndPuncttrue\n  {\\def\\unskip.}{\\unskip.}}\n\\providecommand*\\mciteBstWouldAddEndPunctfalse\n  {\\let\\unskip.}\\relax}\n\\providecommand*\\mciteSetBstMidEndSepPunct[3]{}\n\\providecommand*\\mciteSetBstSublistLabelBeginEnd[3]{}\n\\providecommand*\\unskip.}{}\n\\mciteSetBstSublistMode{f}\n\\mciteSetBstMaxWidthForm{subitem}{(\\alph{mcitesubitemcount})}\n\\mciteSetBstSublistLabelBeginEnd\n  {\\mcitemaxwidthsubitemform\\space}\n  {\\relax}\n  {\\relax}\n\n\\bibitem[Aziz \\latin{et~al.}(2007)Aziz, Vollmer, Eisebitt, Eberhardt, Pingel,\n  Neher, and Koch]{aziz+07am}\nAziz,~E.; Vollmer,~A.; Eisebitt,~S.; Eberhardt,~W.; Pingel,~P.; Neher,~D.;\n  Koch,~N. Localized Charge Transfer in a Molecularly Doped Conducting Polymer.\n  \\emph{Adv.~Mater.~} \\textbf{2007}, \\emph{19}, 3257--3260\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Yim \\latin{et~al.}(2008)Yim, Whiting, Murphy, Halls, Burroughes,\n  Friend, and Kim]{yim+08am}\nYim,~K.-H.; Whiting,~G.~L.; Murphy,~C.~E.; Halls,~J. J.~M.; Burroughes,~J.~H.;\n  Friend,~R.~H.; Kim,~J.-S. Controlling Electrical Properties of Conjugated\n  Polymers via a Solution-Based p-Type Doping. \\emph{Adv.~Mater.~}\n  \\textbf{2008}, \\emph{20}, 3319--3324\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Pingel \\latin{et~al.}(2010)Pingel, Zhu, Park, Vogel, Janietz, Kim,\n  Rabe, Br\\'{e}das, and Koch]{ping+10jpcl}\nPingel,~P.; Zhu,~L.; Park,~K.~S.; Vogel,~J.-O.; Janietz,~S.; Kim,~E.-G.;\n  Rabe,~J.~P.; Br\\'{e}das,~J.-L.; Koch,~N. Charge-Transfer Localization in\n  Molecularly Doped Thiophene-Based Donor Polymers. \\emph{J.~Phys.~Chem.~Lett.}\n  \\textbf{2010}, \\emph{1}, 2037--2041\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Gao \\latin{et~al.}(2013)Gao, Roehling, Li, Guo, Moul\\'{e}, and\n  Grey]{gao+13jmcc}\nGao,~J.; Roehling,~J.~D.; Li,~Y.; Guo,~H.; Moul\\'{e},~A.~J.; Grey,~J.~K. The\n  Effect of 2,3,5,6-tetrafluoro-7,7,8,8-tetracyanoquinodimethane Charge\n  Transfer Dopants on the Conformation and Aggregation of\n  Poly(3-hexylthiophene). \\emph{J.~Mater.~Chem.~C} \\textbf{2013}, \\emph{1},\n  5638--5646\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Pingel and Neher(2013)Pingel, and Neher]{ping-nehe13prb}\nPingel,~P.; Neher,~D. Comprehensive Picture of $p$-Type Doping of P3HT with the\n  Molecular Acceptor F${}_{4}$TCNQ. \\emph{Phys.~Rev.~B} \\textbf{2013},\n  \\emph{87}, 115209\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Duong \\latin{et~al.}(2013)Duong, Wang, Antono, Toney, and\n  Salleo]{duon+13oe}\nDuong,~D.~T.; Wang,~C.; Antono,~E.; Toney,~M.~F.; Salleo,~A. The Chemical and\n  Structural Origin of Efficient p-Type Doping in P3HT. \\emph{Org.~Electron.~}\n  \\textbf{2013}, \\emph{14}, 1330 -- 1336\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Salzmann \\latin{et~al.}(2012)Salzmann, Heimel, Duhm, Oehzelt, Pingel,\n  George, Schnegg, Lips, Blum, Vollmer, and Koch]{salz+13prl}\nSalzmann,~I.; Heimel,~G.; Duhm,~S.; Oehzelt,~M.; Pingel,~P.; George,~B.~M.;\n  Schnegg,~A.; Lips,~K.; Blum,~R.-P.; Vollmer,~A. \\latin{et~al.}\n  Intermolecular Hybridization Governs Molecular Electrical Doping.\n  \\emph{Phys.~Rev.~Lett.~} \\textbf{2012}, \\emph{108}, 035502\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[M\\'{e}ndez \\latin{et~al.}(2013)M\\'{e}ndez, Heimel, Opitz, Sauer,\n  Barkowski, Oehzelt, Soeda, Okamoto, Takeya, Arlin, Balandier, Geerts, Koch,\n  and Salzmann]{mend+13acie}\nM\\'{e}ndez,~H.; Heimel,~G.; Opitz,~A.; Sauer,~K.; Barkowski,~P.; Oehzelt,~M.;\n  Soeda,~J.; Okamoto,~T.; Takeya,~J.; Arlin,~J.-B. \\latin{et~al.}  Doping of\n  Organic Semiconductors: Impact of Dopant Strength and Electronic Coupling.\n  \\emph{Angew.~Chem.~Int.~Ed.} \\textbf{2013}, \\emph{52}, 7751--7755\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Yang \\latin{et~al.}(2014)Yang, Li, Duhm, Tang, Kera, and\n  Ueno]{yang+14ami}\nYang,~J.; Li,~Y.; Duhm,~S.; Tang,~J.; Kera,~S.; Ueno,~N. Molecular\n  Structure-Dependent Charge Injection and Doping Efficiencies of Organic\n  Semiconductors: Impact of Side Chain Substitution. \\emph{Adv.~Mater.~}\n  \\textbf{2014}, \\emph{1}, 1300128\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[M\\'{e}ndez \\latin{et~al.}(2015)M\\'{e}ndez, Heimel, Winkler, Frisch,\n  Opitz, Sauer, Wegner, Oehzelt, R\\\"{o}thel, Duhm, T\\\"{o}bbens, Koch, and\n  Salzmann]{mend+15ncom}\nM\\'{e}ndez,~H.; Heimel,~G.; Winkler,~S.; Frisch,~J.; Opitz,~A.; Sauer,~K.;\n  Wegner,~B.; Oehzelt,~M.; R\\\"{o}thel,~C.; Duhm,~S. \\latin{et~al.}\n  Charge-Transfer Crystallites as Molecular Electrical Dopants. \\emph{Nature\n  Commun.} \\textbf{2015}, \\emph{6}, 8560\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Salzmann \\latin{et~al.}(2016)Salzmann, Heimel, Oehzelt, Winkler, and\n  Koch]{salz+16acr}\nSalzmann,~I.; Heimel,~G.; Oehzelt,~M.; Winkler,~S.; Koch,~N. Molecular\n  Electrical Doping of Organic Semiconductors: Fundamental Mechanisms and\n  Emerging Dopant Design Rules. \\emph{Acc.~Chem.~Res.~} \\textbf{2016},\n  \\emph{49}, 370--378\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Fuzell \\latin{et~al.}(2016)Fuzell, Jacobs, Ackling, Harrelson, Huang,\n  Larsen, and Moul\\'e]{fuze+16jpcl}\nFuzell,~J.; Jacobs,~I.~E.; Ackling,~S.; Harrelson,~T.~F.; Huang,~D.~M.;\n  Larsen,~D.; Moul\\'e,~A.~J. Optical Dedoping Mechanism for P3HT: F4TCNQ\n  Mixtures. \\emph{J.~Phys.~Chem.~Lett.} \\textbf{2016}, \\emph{7},\n  4297--4303\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Jacobs and Moul\\'{e}(2017)Jacobs, and Moul\\'{e}]{jaco-moul17am}\nJacobs,~I.~E.; Moul\\'{e},~A.~J. Controlling Molecular Doping in Organic\n  Semiconductors. \\emph{Adv.~Mater.~} \\textbf{2017}, \\emph{29}, 1703063\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Duva \\latin{et~al.}(2018)Duva, Pithan, Zeiser, Reisz, Dieterle,\n  Hofferberth, Beyer, Bogula, Opitz, Kowarik, Hinderhofer, Gerlach, and\n  Schreiber]{duva+18jpcc}\nDuva,~G.; Pithan,~L.; Zeiser,~C.; Reisz,~B.; Dieterle,~J.; Hofferberth,~B.;\n  Beyer,~P.; Bogula,~L.; Opitz,~A.; Kowarik,~S. \\latin{et~al.}  Thin-Film\n  Texture and Optical Properties of Donor\/Acceptor Complexes.\n  Diindenoperylene\/F6TCNNQ vs Alpha-Sexithiophene\/F6TCNNQ.\n  \\emph{J.~Phys.~Chem.~C} \\textbf{2018}, \\emph{122}, 18705--18714\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Beyer \\latin{et~al.}(2019)Beyer, Pham, Peter, Koch, Meister,\n  Br\\\"utting, Grubert, Hecht, Nabok, Cocchi, Draxl, and Opitz]{beye+19cm}\nBeyer,~P.; Pham,~D.; Peter,~C.; Koch,~N.; Meister,~E.; Br\\\"utting,~W.;\n  Grubert,~L.; Hecht,~S.; Nabok,~D.; Cocchi,~C. \\latin{et~al.}  State of Matter\n  Dependent Charge Transfer Interactions between Planar Molecules for Doping\n  Applications. \\emph{Chem.~Mater.~} \\textbf{2019},\n  \\emph{10.1021\/acs.chemmater.8b01447}\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Jacobs \\latin{et~al.}(2018)Jacobs, Cendra, Harrelson, Bedolla~Valdez,\n  Faller, Salleo, and Moul\\'{e}]{jaco+18mh}\nJacobs,~I.~E.; Cendra,~C.; Harrelson,~T.~F.; Bedolla~Valdez,~Z.~I.; Faller,~R.;\n  Salleo,~A.; Moul\\'{e},~A.~J. Polymorphism Controls the Degree of Charge\n  Transfer in a Molecularly Doped Semiconducting Polymer. \\emph{Mater.~Horiz.~}\n  \\textbf{2018}, \\emph{5}, 655--660\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Neelamraju \\latin{et~al.}(2018)Neelamraju, Watts, Pemberton, and\n  Ratcliff]{neel+18jpcl}\nNeelamraju,~B.; Watts,~K.~E.; Pemberton,~J.~E.; Ratcliff,~E.~L. Correlation of\n  Coexistent Charge Transfer States in F4TCNQ-Doped P3HT with Microstructure.\n  \\emph{J.~Phys.~Chem.~Lett.} \\textbf{2018}, \\emph{9}, 6871--6877\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Karpov \\latin{et~al.}(2017)Karpov, Erdmann, Stamm, Lappan, Guskova,\n  Malanin, Raguzin, Beryozkina, Bakulev, G\\\"{u}nther, Gemming, Seifert,\n  Hambsch, Mannsfeld, Voit, and Kiriy]{karp+17mm}\nKarpov,~Y.; Erdmann,~T.; Stamm,~M.; Lappan,~U.; Guskova,~O.; Malanin,~M.;\n  Raguzin,~I.; Beryozkina,~T.; Bakulev,~V.; G\\\"{u}nther,~F. \\latin{et~al.}\n  Molecular Doping of a High Mobility Diketopyrrolopyrrole--Dithienylthieno [3,\n  2-b] thiophene Donor--Acceptor Copolymer with F6TCNNQ. \\emph{Macromolecules}\n  \\textbf{2017}, \\emph{50}, 914--926\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Scholes \\latin{et~al.}(2017)Scholes, Yee, Lindemuth, Kang, Onorato,\n  Ghosh, Luscombe, Spano, Tolbert, and Schwartz]{scho+17afm}\nScholes,~D.~T.; Yee,~P.~Y.; Lindemuth,~J.~R.; Kang,~H.; Onorato,~J.; Ghosh,~R.;\n  Luscombe,~C.~K.; Spano,~F.~C.; Tolbert,~S.~H.; Schwartz,~B.~J. The Effects of\n  Crystallinity on Charge Transport and the Structure of Sequentially Processed\n  F4TCNQ-Doped Conjugated Polymer Films. \\emph{Adv.~Funct.~Mater.~}\n  \\textbf{2017}, \\emph{27}, 1702654\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Blase and Attaccalite(2011)Blase, and Attaccalite]{blas-atta11apl}\nBlase,~X.; Attaccalite,~C. Charge-Transfer Excitations in Molecular\n  Donor--Acceptor Complexes within the Many-Body Bethe-Salpeter Approach.\n  \\emph{Appl.~Phys.~Lett.~} \\textbf{2011}, \\emph{99}, 171909\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Baumeier \\latin{et~al.}(2012)Baumeier, Andrienko, and\n  Rohlfing]{baum+12jctc}\nBaumeier,~B.; Andrienko,~D.; Rohlfing,~M. Frenkel and Charge-Transfer\n  Excitations in Donor--Acceptor Complexes from Many-Body Green's Functions\n  Theory. \\emph{J.~Chem.~Theory.~Comput.~} \\textbf{2012}, \\emph{8},\n  2790--2795\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Faber \\latin{et~al.}(2014)Faber, Boulanger, Attaccalite, Duchemin, and\n  Blase]{fabe+14ptrsa}\nFaber,~C.; Boulanger,~P.; Attaccalite,~C.; Duchemin,~I.; Blase,~X. Excited\n  States Properties of Organic Molecules: From Density Functional Theory to the\n  GW and Bethe--Salpeter Green's Function Formalisms. \\emph{Phil. Trans. R.\n  Soc. A} \\textbf{2014}, \\emph{372}, 20130271\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Niedzialek \\latin{et~al.}(2015)Niedzialek, Duchemin, de~Queiroz,\n  Osella, Rao, Friend, Blase, K{\\\"u}mmel, and Beljonne]{nied+15afm}\nNiedzialek,~D.; Duchemin,~I.; de~Queiroz,~T.~B.; Osella,~S.; Rao,~A.;\n  Friend,~R.; Blase,~X.; K{\\\"u}mmel,~S.; Beljonne,~D. First Principles\n  Calculations of Charge Transfer Excitations in Polymer--Fullerene Complexes:\n  Influence of Excess Energy. \\emph{Adv.~Funct.~Mater.~} \\textbf{2015},\n  \\emph{25}, 1972--1984\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Zhu \\latin{et~al.}(2011)Zhu, Kim, Yi, and Br\\'{e}das]{zhu+11cm}\nZhu,~L.; Kim,~E.-G.; Yi,~Y.; Br\\'{e}das,~J.-L. Charge Transfer in Molecular\n  Complexes with 2,3,5,6-Tetrafluoro-7,7,8,8-tetracyanoquinodimethane\n  (F4-TCNQ): A Density Functional Theory Study. \\emph{Chem.~Mater.~}\n  \\textbf{2011}, \\emph{23}, 5149--5159\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Blum \\latin{et~al.}(2009)Blum, Gehrke, Hanke, Havu, Havu, Ren, Reuter,\n  and Scheffler]{blum+09cpc}\nBlum,~V.; Gehrke,~R.; Hanke,~F.; Havu,~P.; Havu,~V.; Ren,~X.; Reuter,~K.;\n  Scheffler,~M. Ab initio Molecular Simulations with Numeric Atom-Centered\n  Orbitals. \\emph{Comput.~Phys.~Commun.~} \\textbf{2009}, \\emph{180}, 2175 --\n  2196\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Havu \\latin{et~al.}(2009)Havu, Blum, Havu, and Scheffler]{havu+09jcp}\nHavu,~V.; Blum,~V.; Havu,~P.; Scheffler,~M. Efficient O(N) Integration for\n  All-Electron Electronic Etructure Calculation Using Numeric Basis Functions.\n  \\emph{J.~Comp.~Phys.~} \\textbf{2009}, \\emph{228}, 8367 -- 8379\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Perdew \\latin{et~al.}(1996)Perdew, Burke, and Ernzerhof]{perd+96prl}\nPerdew,~J.~P.; Burke,~K.; Ernzerhof,~M. Generalized Gradient Approximation Made\n  Simple. \\emph{Phys.~Rev.~Lett.~} \\textbf{1996}, \\emph{77}, 3865--3868\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Tkatchenko and Scheffler(2009)Tkatchenko, and\n  Scheffler]{tkat-sche09prl}\nTkatchenko,~A.; Scheffler,~M. Accurate Molecular Van Der Waals Interactions\n  from Ground-State Electron Density and Free-Atom Reference Data.\n  \\emph{Phys.~Rev.~Lett.~} \\textbf{2009}, \\emph{102}, 073005\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Bruneval \\latin{et~al.}(2016)Bruneval, Rangel, Hamed, Shao, Yang, and\n  Neaton]{brun+16cpc}\nBruneval,~F.; Rangel,~T.; Hamed,~S.~M.; Shao,~M.; Yang,~C.; Neaton,~J.~B. molgw\n  1: Many-Body Perturbation Theory Software for Atoms, Molecules, and Clusters.\n  \\emph{Comput.~Phys.~Commun.~} \\textbf{2016}, \\emph{208}, 149 -- 161\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Bruneval(2012)]{brun12jcp}\nBruneval,~F. Ionization Energy of Atoms Obtained from GW Self-Energy or from\n  Random Phase Approximation Total Energies. \\emph{J.~Chem.~Phys.~}\n  \\textbf{2012}, \\emph{136}, 194107\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Weigend \\latin{et~al.}(2002)Weigend, K\\\"{o}hn, and\n  H\\\"{a}ttig]{weig+02}\nWeigend,~F.; K\\\"{o}hn,~A.; H\\\"{a}ttig,~C. Efficient Use of the Correlation\n  Consistent Basis Sets in Resolution of the Identity MP2 Calculations.\n  \\emph{J.~Chem.~Phys.~} \\textbf{2002}, \\emph{116}, 3175--3183\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Yanai \\latin{et~al.}(2004)Yanai, Tew, and Handy]{yana+04cpl}\nYanai,~T.; Tew,~D.~P.; Handy,~N.~C. A New Hybrid Exchange-Correlation\n  Functional Using the Coulomb-Attenuating Method (CAM-B3LYP).\n  \\emph{Chem.~Phys.~Lett.~} \\textbf{2004}, \\emph{393}, 51 -- 57\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Faber \\latin{et~al.}(2013)Faber, Boulanger, Duchemin, Attaccalite, and\n  Blase]{fabe+13jcp}\nFaber,~C.; Boulanger,~P.; Duchemin,~I.; Attaccalite,~C.; Blase,~X. Many-body\n  Green's Function GW and Bethe-Salpeter Study of the Optical Excitations in a\n  Paradigmatic Model Dipeptide. \\emph{J.~Chem.~Phys.~} \\textbf{2013},\n  \\emph{139}, 194308\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Bruneval and Marques(2013)Bruneval, and Marques]{brun+13jctc}\nBruneval,~F.; Marques,~M. A.~L. Benchmarking the Starting Points of the GW\n  Approximation for Molecules. \\emph{J.~Chem.~Theory.~Comput.~} \\textbf{2013},\n  \\emph{9}, 324--329\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Cocchi and Draxl(2015)Cocchi, and Draxl]{cocc-drax15prb}\nCocchi,~C.; Draxl,~C. Optical Spectra from Molecules to Crystals: Insight from\n  Many-Body Perturbation Theory. \\emph{Phys.~Rev.~B} \\textbf{2015}, \\emph{92},\n  205126\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Palummo \\latin{et~al.}(2009)Palummo, Hogan, Sottile, Bagal{\\'a}, and\n  Rubio]{palu+09jcp}\nPalummo,~M.; Hogan,~C.; Sottile,~F.; Bagal{\\'a},~P.; Rubio,~A. Ab initio\n  Electronic and Optical Spectra of Free-Base Porphyrins: The Role of\n  Electronic Correlation. \\emph{J.~Chem.~Phys.~} \\textbf{2009}, \\emph{131},\n  08B607\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Baumeier \\latin{et~al.}(2012)Baumeier, Andrienko, Ma, and\n  Rohlfing]{baum+12jctc1}\nBaumeier,~B.; Andrienko,~D.; Ma,~Y.; Rohlfing,~M. Excited states of\n  Dicyanovinyl-Substituted Oligothiophenes from Many-Body Green's Functions\n  Theory. \\emph{J.~Chem.~Theory.~Comput.~} \\textbf{2012}, \\emph{8},\n  997--1002\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Rangel \\latin{et~al.}(2017)Rangel, Hamed, Bruneval, and\n  Neaton]{rang+17jcp}\nRangel,~T.; Hamed,~S.~M.; Bruneval,~F.; Neaton,~J.~B. An Assessment of\n  Low-Lying Excitation Energies and Triplet Instabilities of Organic Molecules\n  with an Ab initio Bethe-Salpeter Equation Approach and the Tamm-Dancoff\n  Approximation. \\emph{J.~Chem.~Phys.~} \\textbf{2017}, \\emph{146}, 194108\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Cocchi \\latin{et~al.}(2011)Cocchi, Prezzi, Ruini, Caldas, and\n  Molinari]{cocc+11jpcl}\nCocchi,~C.; Prezzi,~D.; Ruini,~A.; Caldas,~M.~J.; Molinari,~E. Optical\n  Properties and Charge-Transfer Excitations in Edge-Functionalized\n  All-Graphene Nanojunctions. \\emph{J.~Phys.~Chem.~Lett.} \\textbf{2011},\n  \\emph{2}, 1315--1319\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[De~Corato \\latin{et~al.}(2014)De~Corato, Cocchi, Prezzi, Caldas,\n  Molinari, and Ruini]{deco+14jpcc}\nDe~Corato,~M.; Cocchi,~C.; Prezzi,~D.; Caldas,~M.~J.; Molinari,~E.; Ruini,~A.\n  Optical Properties of Bilayer Graphene Nanoflakes. \\emph{J.~Phys.~Chem.~C}\n  \\textbf{2014}, \\emph{118}, 23219--23225\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Fabiano \\latin{et~al.}(2005)Fabiano, Della~Sala, Cingolani, Weimer,\n  and G\\\"{o}rling]{fabi+05jpca}\nFabiano,~E.; Della~Sala,~F.; Cingolani,~R.; Weimer,~M.; G\\\"{o}rling,~A.\n  Theoretical Study of Singlet and Triplet Excitation Energies in\n  Oligothiophenes. \\emph{J.~Phys.~Chem.~A} \\textbf{2005}, \\emph{109},\n  3078--3085\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Sun and Autschbach(2014)Sun, and Autschbach]{sun-auts14jctc}\nSun,~H.; Autschbach,~J. Electronic Energy Gaps for $\\pi$-conjugated Oligomers\n  and Polymers Calculated with Density Functional Theory.\n  \\emph{J.~Chem.~Theory.~Comput.~} \\textbf{2014}, \\emph{10}, 1035--1047\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Hogan \\latin{et~al.}(2013)Hogan, Palummo, Gierschner, and\n  Rubio]{hoga+13jcp}\nHogan,~C.; Palummo,~M.; Gierschner,~J.; Rubio,~A. Correlation Effects in the\n  Optical Spectra of Porphyrin Oligomer Chains: Exciton Confinement and Length\n  Dependence. \\emph{J.~Chem.~Phys.~} \\textbf{2013}, \\emph{138}, 01B607\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\bibitem[Li \\latin{et~al.}(2017)Li, D'Avino, Pershin, Jacquemin, Duchemin,\n  Beljonne, and Blase]{li+17prm}\nLi,~J.; D'Avino,~G.; Pershin,~A.; Jacquemin,~D.; Duchemin,~I.; Beljonne,~D.;\n  Blase,~X. Correlated Electron-Hole Mechanism for Molecular Doping in Organic\n  Semiconductors. \\textbf{2017}, \\emph{1}, 025602\\relax\n\\mciteBstWouldAddEndPuncttrue\n\\mciteSetBstMidEndSepPunct{\\mcitedefaultmidpunct}\n{\\mcitedefaultendpunct}{\\mcitedefaultseppunct}\\relax\n\\unskip.}\n\\end{mcitethebibliography}\n\n\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDespite the fact that ultra-high energy (UHE) cosmic rays are known\nfor decades, their sources and  the acceleration mechanisms are\nstill under debate. Sources are only detectable by $\\gamma$-rays produced\nin interactions near the sources.\nIn the very-high energy (VHE) range near 1\\,TeV \nmore and more $\\gamma$-ray sources are revealed by the imaging Cherenkov technique.\nOn the contrary, no clear source detections have been made in the UHE domain\nabove about 100\\,TeV, except perhaps a few episodic cases. \n\nMainly for reasons of the required power, the dominant sources of\ncosmic rays up to about 100\\,TeV and probably up to the {\\em knee}\nof the cosmic-ray energy spectrum at a few $10^{15}$\\,eV are believed\nto be supernova remnants in the Sedov phase. The change of the spectrum\nnear the knee presumably reflects a change in the origin and the\ntakeover of another, yet unclear type of sources at energies above\nthe knee. A change in the cosmic-ray propagation with a decreasing\nGalactic containment has also been considered. In either case,\nthe change in the slope of the spectrum should be accompanied by a change \nin the mass composition of cosmic rays. In the case of a change of\nsources across the knee, the composition could change dramatically.\nA much less spectacular change of the composition is expected\nin the case of decreasing containment.\n\nDirect measurements well below the knee\n\\cite{Asakimori-1993ab,Ichimura-1993b}\nshow indeed a substantial change in the mass composition\nalready at energies around 100\\,TeV. In particular, the fraction\nof protons seems to diminish with increasing energy.\nDue to their small collection\nareas the balloon-borne direct experiments run out of statistics\nabove several hundred TeV. Indirect methods using ground-based experiments\nare, so far, not able to classify individual cosmic rays unambiguously\nby their mass. Such methods are more appropriate for evaluating some\naverage mass number. Most notably, the results of the Fly's Eye\ngroup \\cite{Gaisser-1993} indicate a rather heavy composition well above \nthe knee. If taken at face value, their results represent a composition of\nmainly very heavy nuclei, like iron, at $10^{17}$\\,eV.\n\nExperiments measuring the composition right at the knee of the\nspectrum obtained either ambiguous or even conflicting results.\nResults with no significant change\n\\cite{Zhu-1990,Ahlen-1992b} or a slight increase of heavy elements\n\\cite{Khristiansen-1994} have been reported. Other \ngroups found more significant increases\nof heavy elements \\cite{Ren-1988a,Freudenreich-1990,Mitsui-1995} \nor, on the other hand, predominantly protons \\cite{Cebula-1990}.\nFurther measurements are needed, if the cosmic-ray composition\nshould help to resolve the question where cosmic rays are accelerated.\n\nThe method proposed in this paper should be relatively easy to implement\nat sites where an air-shower array with an angular resolution below\none degree already exists. The method is mainly a measurement of the\nlongitudinal shower development by the angles of muons with respect to\nthe shower axis. It does not require to measure many muons in a single \nshower because it uses only the {\\em inclusive}\nangular distributions, nor is accurate timing required.\nIndeed, if several muons are measured in one event, they are treated\nseparately.\n\nNot only the average muon angles but also the average electron angles \nwith respect to the shower axis are sensitive to the primary\ncomposition. Because the detector response \nis, in general, better understood for muon tracks than for electron tracks,\nthis paper focuses mainly on the muons.\nUsing the tracking detectors and the air-shower array, the\nmethod can be supplemented by traditional methods like the\naverage $\\mu\/e$ ratio or the muon lateral distribution.\nAlthough one can think of complex experiments dedicated\nto measuring the cosmic-ray composition, like KASKADE \\cite{Rebel-1993},\nwhere more pieces of information are collected, the\npurpose of this paper is to demonstrate that even \nthe muon angles alone provide significant information about\nthe composition.\n\nThe proposed method relies on shower simulations to obtain\nan absolute value of an average mass number, just like\nother indirect methods. If suitable tracking detectors are used,\nit has the advantage that essentially\nall required detector parameters for the simulation can be\nobtained from the measured data as a function of shower size.\nTherefore, it can be particularly sensitive to changes in the composition\nacross the measured shower size spectrum, even if a comparison\nwith simulations using different interaction models may yield\ndifferent absolute values.\n\nAn analysis of data taken with ten CRT detectors \n-- tracking detectors of 2.5\\,m$^2$ sensitive area each \\cite{CRT-NIM-1} --\nand the HEGRA air-shower array \\cite{Fonseca-1995} on La Palma is\nin progress \\cite{future-CRT-results}. In the data of CRT and HEGRA\nboth muon and electron tracks are analysed. Although the intention of\nthis paper is to promote the method to other air-shower experiments,\nthe simulations shown in this paper are specific to the combination of the\nCRT and HEGRA experiments. These simulations take into account the\nresponse of both components -- the tracking detectors and the\nair-shower array -- in much detail. After an outline of the method,\nthe specific simulations are described to demonstrate that the\nmethod can be very well applied with existing detector technology.\nExperimental requirements for application of the method and\nlimitations by shower simulations are shown in a more general context,\nnot specific to CRT and HEGRA.\n\n\n\\section{Outline of the method}\n\nFirst measurements of the angles of muons with respect to the\ncorresponding air showers or, in alternative terms, of the apparent\nheight of origin of the muons were done in the 1960s \\cite{Earnshaw-1973}.\nAt that time neither the angular resolutions of air-shower arrays\nnor the accuracy of shower simulations were appropriate to use\nsuch measurements for investigating the mass composition of\nprimary cosmic rays. For an early comparison of measured\nand simulated height of muon origin see \\cite{Gaisser-1978}.\n\nThe proposed method is based on the fact that showers initiated\nby heavy primaries, like iron nuclei, develop on average\nat smaller atmospheric depths, i.e.\\ higher altitudes, than proton showers.\nMuons from showers of heavy primaries arrive, therefore, at smaller\nangles with respect to the shower axis for any given core distance.\nThe proposed method is not intended to measure individual primaries\nbut to measure accurately an average number of the composition, like \n$\\langle\\ln A\\rangle$, the average logarithm of the primary mass numbers.\nMeasuring many tracks in a single shower is, therefore, not necessary\nand only helps to improve statistics.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.6\\textwidth\n\\hc{\\epsffile{geo_rad_tang.eps}}\n\\caption[Geometry of radial and tangential angles]{The\ngeometry of radial and tangential angles between the\nshower axis, e.g.\\ as reconstructed from air-shower data,\nand the measured track of a particle.}\n\\label{fig:geometry}\n\\end{figure}\n\nThe angles of muons or other particles with respect to the shower axis are \nbest expressed in terms of their {\\em radial angle\\\/} and {\\em tangential angle\\\/}\ncomponents. The radial angle is the projection onto the plane defined\nby the geometry of the shower axis and the location of the muon detector.\nThroughout this paper, a negative radial angle is assigned to\na particle flying away from the shower axis. In fact, at large\ncore distances most particles have such a negative radial angle.\nThe tangential angle (also called the transverse angle) is the projection\nonto the plane perpendicular to the `radial' plane and parallel to the\nshower axis (see figure~\\ref{fig:geometry}). \nFor almost vertical\nshowers a more convenient definition with projections into vertical\nplanes defined only by the positions of the shower core and the\ndetector is essentially equivalent. \n\nThe radial projection can be\nused to derive an {\\em apparent height of origin\\\/} of a particle\n(figure~\\ref{fig:geo-height}). Although radial angle and apparent\nheight of origin are almost equivalent, the radial angle is the\nmore robust variable.\nThis is due to the fact that\na small scattering or experimental smearing around zero radial\nangle corresponds to $\\pm\\infty$ apparent height.\nIn addition, the derived height for each muon depends directly on the\nresolution of the core position. In particular, the last point turns out \nto be a major drawback of an analysis in terms of the apparent height.\nThe core position resolution is usually not very precisely known\nas a function of shower size, zenith angle, and position within the\narray. An analysis in terms of the apparent height is, for this\nreason, limited to core distances much larger than the core position\nresolution.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.6\\textwidth\n\\hc{\\epsffile{geo_height.eps}}\n\\caption[Apparent height of origin]{An apparent\nheight of origin can be derived from the track projected onto\nthe radial plane.}\n\\label{fig:geo-height}\n\\end{figure}\n\nThe distribution of tangential angles (see figure \\ref{fig:tang+rad})\nis symmetric and its width is\ndominated by multiple scattering and the lateral distribution of the\nparent particle generation. For the muons in UHE air showers the\nintrinsic distribution is very narrow, apart from tails due\nto some low-energy muons. This distribution shows\nlittle difference between proton- and iron-initiated showers,\neven for an ideal angular resolution.\nUnder certain experimental constraints the tangential angle\ncan be used to measure the combined angular resolution of\nair-shower array and tracking detectors.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.8\\textwidth\n\\hc{\\epsffile{tan-rad-ideal.eps}}\n\\caption[Distributions of tangential and radial muon angles]{Distributions\nof tangential and radial angles of shower muons in simulated \nproton and iron showers, respectively,\nwith $N_e>10000$, at core distances of 200--250~m. \nPerfect angular resolution but a realistic detection\nefficiency as appropriate for CRT detectors is assumed.}\n\\label{fig:tang+rad}\n\\end{figure}\n\nThe radial angle, representing the longitudinal shower development,\nis the quantity sensitive to the composition. The {\\em average}\n(which may be mean, median, or mode) radial angle depends in a\ncharacteristic way on the core distance. \nTo compare the simulated average radial-angle curves\nwith measurements requires to know the combined angular resolution of the\nair-shower array and tracking detectors and to some extent also the\ncore position resolution of the array. However, the average radial angle\nat any given core distance changes very little with shower size or zenith angle.\nThe figures in this paper are intended as examples. They are based on\nsimulations using the CORSIKA code (see section \\ref{sect:shower+det-sim}).\nThe simulations assume an experimental resolution as observed\nfor CRT detectors at the HEGRA array \\cite{CRT-NIM-2}. \nSee figure~\\ref{fig:rad-ang-primaries}\nfor the median radial angles of different primary types.\nAt any core distance the average radial angle is almost a linear function\nof $\\langle\\ln A\\rangle$, where $A$ is the mass number of the primary.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.8\\textwidth\n\\hc{\\epsffile{rad-ang-primaries.eps}}\n\\caption[Radial angle of muons for different primaries]{Median radial angle of \nmuons in showers initiated by different primaries, for a minimum shower size of \n10$^4$ and angular resolutions and detector responses as \nappropriate for CRT detectors and the HEGRA scintillator array \n(see section \\ref{sect:shower+det-sim}).}\n\\label{fig:rad-ang-primaries}\n\\end{figure}\n\nAlthough a similar difference between showers initiated by protons or\niron nuclei is also present in the radial-angle curves of electrons,\nthe electron component is less usefull for composition studies than muons.\nFirst of all, a very detailed simulation of the behaviour of\nthe tracking detectors to electrons, gammas, muons, and hadrons \nis required to compare with experimental data. \nFor muons the detector is generally much better understood,\nalthough punch-throughs of other particles have to be considered.\n\nThe second reason in favour of muons is based on the shower selection\nintroduced by the air-shower array. To achieve the same shower size\n($N_e$) at ground level, showers initiated by heavy primaries require\na larger primary energy than proton showers. Due to the steep spectrum,\nconventional air-shower arrays will, therefore, mainly see proton showers\nat any shower size, even if protons account for only half of the\nmass composition at a given energy per particle. Muons can almost\ncompensate for this bias because, at the same shower size, a proton\nshower contains less muons than a shower of a heavy primary.\n\nBecause of the compensation of the selection bias by the larger number\nof muons for heavy primaries the quantity\n\\begin{equation}\n\\Lambda_\\mu = \\frac{1}{\\sum w_i}\\sum_i \\,w_i\\,\\frac{\\langle\\alpha_i\\rangle -\n  \\langle\\alpha_{i,{\\rm p}}\\rangle}{\\langle\\alpha_{i,{\\rm Fe}}\\rangle -\n  \\langle\\alpha_{i,{\\rm p}}\\rangle}\n\\end{equation}\nis even for a mixed composition essentially proportional to $\\langle\\ln A\\rangle$:\n\\begin{equation}\n   \\langle\\ln A \\rangle \\approx \\Lambda_\\mu\\,\\ln 56.\n\\label{eq:lnA-Lambda}\n\\end{equation}\nIn this context $\\langle\\alpha_i\\rangle$ is the measured (or simulated) \nmedian radial angle in the core distance interval $i$, \n$\\langle\\alpha_{i,{\\rm p}}\\rangle$ and $\\langle\\alpha_{i,{\\rm Fe}}\\rangle$\nare the expected median values for pure protons and pure iron \nnuclei from the simulations, and the weights $w_i$ take\nthe statistical accuracy of measured as well as simulated average values\ninto account. In the simplest possible formulation, ignoring statistical\nerrors of the simulation and correlations of errors in different\nradial bins and taking into account only $\\sigma(\\langle\\alpha_i\\rangle)$\nas the measured accuracy of $\\langle\\alpha_i\\rangle$,\n\\begin{equation}\n   w_i = \\Bigl( \\sigma(\\langle\\alpha_i\\rangle) \\;\/\\; \n  ({\\langle\\alpha_{i,{\\rm Fe}}\\rangle -\n  \\langle\\alpha_{i,{\\rm p}}\\rangle}) \\Bigr)^{-2}.\n\\end{equation}\n\nFor compositions as reported by direct measurements, equation\n\\ref{eq:lnA-Lambda}\nholds to\nbetter than 0.05 at all investigated shower size intervals --\nat least for CRT and HEGRA to which the simulations\ncorrespond. Although this relation is quite coincidental,\nit is expected to hold as well for most other existing\nair-shower arrays in combination with tracking detectors\nof a low energy threshold for muons.\nIn the case of extreme compositions like\npure helium, however, the relation may be wrong by up to 0.26\nin terms of $\\langle\\ln A\\rangle$.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.8\\textwidth\n\\hc{\\epsffile{rad-ang-composite.eps}}\n\\caption[Radial angle of muons for different compositions]{Median radial \nangle of muons in showers\nfor different energy-independent model compositions (pure protons, pure iron,\nand two compositions as measured by the JACEE collaboration \n\\cite{Asakimori-1993ab}).\nSimulations for $N_e>10^4$, as in figure \\ref{fig:rad-ang-primaries}.}\n\\label{fig:mu-rad-ang-composite}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.8\\textwidth\n\\hc{\\epsffile{elec-rad-ang-composite.eps}}\n\\caption[Radial angle of electrons for different compositions]{Simulated \nmedian radial angle of particles identified as electrons,\nfor the same model compositions as in figure~\\ref{fig:mu-rad-ang-composite}.\n$N_e>10^4$, as in figure \\ref{fig:rad-ang-primaries}.}\n\\label{fig:elec-rad-ang-composite}\n\\end{figure}\n\nFigure \\ref{fig:mu-rad-ang-composite} shows that the median (or mean) radial\nangle of muons is indeed sensitive to changes in the composition\nas indicated by direct measurements \\cite{Asakimori-1993ab} below\nthe knee. Even the electron radial angles can be used for that\npurpose (see figure \\ref{fig:elec-rad-ang-composite}) but\na more accurate knowledge of the detector response is required in that case.\nIn addition, $\\Lambda_e$ (defined in analogy to $\\Lambda_\\mu$), is not\nproportional to $\\langle\\ln A\\rangle$ but is more sensitive to the\nfraction of light nuclei than to heavy nuclei.\n\n\n\\section{Shower and detector simulations}\n\\label{sect:shower+det-sim}\n\nThe scope of this paper is beyond the measurements\ncarried out with CRT detectors and the HEGRA array.\nAlthough the simulations presented here serve only as examples,\nexperimental effects of CRT detectors and the\nHEG\\-RA scintillator array are carefully taken into account. It may\nbe instructive to see which of these effects turn out to be\nthe most relevant ones.\n\nThe CORSIKA simulation code \\cite{CORSIKA-1992,Knapp-1993}\nwas used in version 4.068 for all shower simulations presented in this paper.\nExcept for tests of the accuracy of the shower simulations (see section\n\\ref{sect:shower-accuracy}) the VENUS hadron-nucleus and nucleus-nucleus\ninteraction code \\cite{Werner-1993} was used for high-energy\nhadronic interactions, GHEISHA \\cite{Fesefeldt-1985} for \nhadronic interactions below 80~GeV, and EGS4 \\cite{Nelson-1985} for\nthe electromagnetic component -- all in the framework of CORSIKA.\nThe simulations take also the geomagnetic field at La Palma into account.\nA total of 5580 showers (1860 proton showers and 930 showers\nfor each of the other four types of primaries, He, N, Mg, and Fe) were generated \nisotropically in the zenith angle range 0$^\\circ$ to 32$^\\circ$.\nThe energy ranges were selected to cover approximately the same\nshower sizes, from 20~TeV to 2~PeV for protons increasing to\n40~TeV to 4~PeV for iron nuclei. Showers were generated with a\n$E^{-1.7}$ differential energy spectrum to have sufficient\nnumbers of showers at all energies and weighted by $E^{-1}$ for the\nanalysis. The analysis was restricted to shower sizes where\ncontributions from showers outside the simulated energy ranges\nare negligible. \n\nConcerning the CRT detectors \\cite{CRT-NIM-1}, it should be \nmentioned that muons are identified as pairs of tracks in two\ndrift chambers, one above and one below a 10~cm thick iron plate.\nThe measured angles between the two tracks of an identified muon are\nrequired to be less than 2.5$^\\circ$ in two perpendicular projections. \nElectrons are identified as non-muon tracks in the upper drift chamber.\nResults of very detailed simulations of the response of CRT detectors to various\ntypes of particles (electrons, gammas, muons, pions, and protons) -- as\noutlined in \\cite{CRT-NIM-2} and described in detail in \\cite{Zink-1995} --\nand the measured detector performance \\cite{CRT-NIM-2}\nwere carefully parametrized. This includes detection efficiencies\nand angular resolutions as functions of particle energies, zenith angles,\nand track densities -- independent\nfor each particle type and for the identification as an electron track \nand as a muon track. As a consequence, the simulations include\nnot only punch-through of electrons but also of hadrons.\nIt should be noted that the energy dependence for\ndetecting genuine muons is fully described by multiple\nscattering and energy loss in the iron plate \\cite{CRT-NIM-2}.\nCRT detectors are mainly sensitive to muons above about\n1~GeV. Electrons, on the other hand, are detected above some 10~MeV.\nThe effect of multiple scattering in the detector container\nand the iron plate are included in the angular resolutions.\n\nFor a detailed simulation of the HEGRA array see \\cite{Martinez-1995}.\nFor the simulations presented here, a much simpler approach is\nsufficient because the simulations are restricted to shower\nsizes where the HEGRA array is fully efficient, independent of\nshower age, core position, and zenith angle.\nThe resolutions for shower sizes and core positions as obtained\nby the detailed simulations were parametrized. \nThe HEGRA angular resolution is implemented in the simulations\nin such a way that the combined\nangular resolution of CRT and HEGRA -- as measured by the muon\ntangential angles as a function of shower size and zenith angle \n\\cite{CRT-NIM-2} -- are fully reproduced.\n\nIt may be instructive to note that the median radial-angle curves\npresented in this paper are not significantly altered by\nmodifying the assumed angular resolution\nwithin the experimental limits. Modifying the assumed\ncore position resolution has an impact on the median radial\nangles only within a few ten meters from the core. \nPunch-throughs, although included in some detail, turned out\nto be of minor importance with CRT detectors, due to their\nexcellent punch-through rejection. The most important\nexperimental parameters in the simulations are the energy \ndependence of the detection efficiencies and the combined angular resolution\nof tracking detectors and air-shower array. In the case of CRT detectors\nand the HEGRA array both are very well known for muons and even\nsufficiently well known for electrons. \n\nIt appears very reasonable that the response of other types of \ntracking detectors and other air-shower arrays can also be \nunderstood well enough to take advantage of the radial-angle method. \nHowever, a few experimental requirements as outlined in section\n\\ref{sect:requirements} should be observed for that purpose.\n\n\n\\section{Accuracy of shower simulations}\n\\label{sect:shower-accuracy}\n\nApart from the detector response, a major uncertainty in the\ninterpretation of any indirect measurement of the\ncosmic-ray mass composition is due to the limited accuracy of\nthe shower simulations. Despite much progress in recent\nyears, the nucleus-nucleus and hadron-nucleus interaction models in \nair-shower simulations lead to non-negligible theoretical\nuncertainties. The CORSIKA simulation code\nattempts a particularly detailed and accurate shower simulation. \nIn the version 4.068, as used for this paper, it offers several\nindependent interaction models.\n\n\\begin{figure}[htbp]\n\\epsfxsize=0.8\\textwidth\n\\hc{\\epsffile{rad-ang-syserr.eps}}\n\\caption[Radial angle of muons for interaction models]{Median radial \nangle of muons in proton and iron showers with different interaction\nmodels (10000\\,$<$\\,$N_e$\\,$<$\\,20000, with fixed height of first interaction\nto keep shower-to-shower fluctuations small). 1: protons with\nVENUS\/GHEISHA, 2: protons with DPM, 3: iron with VENUS\/GHEISHA,\n4(5): iron with DPM and no (full) fragmentation of the spectators\nin projectile nuclei.}\n\\label{fig:rad-ang-syserr}\n\\end{figure}\n\nResults for different\ninteraction models, using the VENUS and GHEI\\-SHA \noptions on one side and DPM on the other side, and for\ndifferent fragmentation (evaporation) of the projectile nuclei have\nbeen compared in the energy range $10^{14}$--$10^{15}$\\,eV.\nSystematic differences in the average shower sizes of up\nto 20\\% are found and also differences in the\nshower fluctuations. If the median radial-angle curves are compared\nin the same ranges of shower sizes, the differences between models are\nvery small within core distances up to about 200\\,m, increasing to\nabout 10\\% of the proton-iron difference at a core distance of 400\\,m \n(figure~\\ref{fig:rad-ang-syserr}).\nThat increase of systematic uncertainties is due to the fact that \nhard interactions resulting in high-$p_t$ secondaries are not yet\nincluded very accurately.\nWith all interaction models the shower-size dependence of the\nradial-angle curve is small. A 30\\% systematic error in the\nrelation of simulated to measured shower sizes would result\nonly in a systematic error of the median radial angles of 3\\% of\nthe proton-iron difference.\nMoreover, this dependence is essentially \nthe same for different models.\n\nAnother uncertainty in the longitudinal shower development is\ndue to the sparse particle physics data in the very forward rapidity \nregion at high energies. Until more such data is available and \nincorporated into the interaction models it seems desirable to have also\nindirect composition measurements in an energy range overlapping with \ndirect measurements. When searching only for a change of the\ncomposition near the knee of the cosmic-ray spectrum, the poor\nknowledge of the very forward region should not be a limitation.\nDifferent parametrizations of parton densities which pose problems\nfor shower simulations at extremely high energies \\cite{Capdevielle-1995}\nare not of concern near $10^{15}$\\,eV primary energy.\n\n\n\n\\section{Experimental requirements}\n\\label{sect:requirements}\n\nTo exploit the described method for composition studies several\nexperimental requirements should be met. These are requirements for\nthe {\\em particle identification} with a tracking detector and \nthe {\\em resolution} of angles and positions.\n\nBecause the particles of the electromagnetic shower component, \n$e^\\pm$ and $\\gamma$, have angular distributions much different from\nthose of muons, a good electron rejection is important. Hadrons\nonly matter very close to the shower core. Due to the broad\nmuon lateral distribution it is generally not such a problem\nto identify muons at large core distances but more of a problem\nto reject random, non-shower muons. This can be easily solved\nby a timing information of a few hundred nanoseconds accuracy\nand by the angular correlation of shower muons with the shower axis.\nAnother requirement for the\nmuon identification is due to the fact that the radial angles\nare well correlated with the muon momentum, with higher-energy\nmuons usually coming from further up in the atmosphere and being\nbetter aligned with the shower axis. The muon identification should,\ntherefore, be well understood also as a function of momentum and\nzenith angle.\nFor a high muon-energy threshold, a very good angular resolution\nis required, while for a low threshold the multiple scattering\nand magnetic deflection of muons cannot be neglected.\n\nThe combined angular resolution of air-shower array\nand tracking detector should be better than about 1$^\\circ$\nin each projection but has to be known as a function of shower size\nto compare measurements with simulations. Definitely sufficient would be\na combined resolution of some 0.5$^\\circ$ if the muon-energy \nthreshold is as low as about 1~GeV.\nThis would be difficult to achieve with a tracking detector\nentirely below thick shielding material which also scatters the muons.\nA good uniformity of the resolution over azimuth angles would\nbe of advantage for the important detector calibration. Only with a\nuniform resolution can the angular resolution be derived from\nthe tangential angle distribution of the same data set which is\nused to compare the radial angles with simulations. Otherwise,\nshower-size dependent systematic errors can arise. \n\nThe alignment of the tracking detectors with respect to the\nreference frame as defined by showers reconstructed from array data\nshould be known to better than about 0.1$^\\circ$. This may require some\ncareful calibrations but is certainly feasible. \n\nThe core position resolution which also enters into the simulations \nhas to be determined from the air-shower array alone. Keeping \nselection effects of the air-shower array as small as possible\ndemands to select only showers well contained in the array\nand with shower sizes well above the trigger and reconstruction\nthresholds. For many of the existing air-shower arrays that\nwould correspond to threshold energies of some 80--200\\,TeV,\njust below where direct measurements are running out of statistics.\nReasonable errors in the shower size and energy calibrations of the array \ndo not compromise the method as long as the same experimental resolution\nis used in the simulations as seen in the data to be compared with\nthese simulations.\n\nWith the 25~m$^2$ sensitive area of the CRT detectors the\nstatistics with data of about one year would be the limit for \nprimary energies above several $10^{16}$~eV. A few hundred\nsquare meters area would be sufficient\nin order to achieve\nsome overlap with the Fly's Eye composition measurements.\n\nAll these requirements can be fulfilled with present\ntechnology for large-area tracking detectors and existing air-shower\narrays. Although simultaneous precise (nanosecond) timing\ninformation of the muons could supplement the described method\n(see for example \\cite{Danilova-1994}),\nprecise timing is not required and might be difficult to accomplish\nwith large-area detectors.\n\n\n\\section{Concluding remarks}\n\nThe average radial angle of particles are sensitive to the\ncosmic-ray mass composition. Muons have, among other particles in air\nshowers, many advantages. Their intrinsic angular distribution is\nvery narrow and, thus, the tangential-angle distribution can be\nused to calibrate the radial-angle resolution as a function of\nshower size. Muons can be distinguished from other particle types\nvery well. The larger number of muons in showers initiated by \nheavy primaries compensates for the $N_e$ shower selection bias\nwhich favours light primaries.\n\nLike any other\nindirect measurement of some average of the cosmic-ray composition,\nthat method requires comparison of measured data with simulations.\nCompared to many other indirect methods, the muon radial-angle method\nhas two major advantages: First, the most important detector effects\nto be included in the simulation can be measured very well and, second,\nsystematic errors, for example due to the interaction model,\ncan be easily checked by comparing measurements at low energies\n(a few hundred TeV) with simulations for directly measured\ncompositions as in \\cite{Asakimori-1993ab}. \n\nDespite some systematic uncertainties in the shower simulations\nan average mass number ($\\langle\\ln A\\rangle$) can be derived\nand compared with direct measurements well below the knee. \nRegardless of that, the muon radial-angle method can be very\nsensitive to changes in the composition. The radial-angle\nmethod would not require a very large experimental effort if\nexisting air-shower arrays are supplemented with suitable\ntracking detectors.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section*{Ideas that we want to include}\n\n\\input{abstract}\\\\\n\\noindent\\textbf{Key Words}: BUGS\/JAGS; Capture-recapture; Data augmentation; Heterogeneity; Multinomial; Reversible jump MCMC.\n\n\\section{Introduction}\nInference about $N$, the unknown size of a population, based on capture-recapture models is of enormous interest, with ecology and epidemiology just two areas in which there are many applications.  This is a specific case of a more general class of problems in which the index of a multinomial model is unknown.  A sub-class of models of particular interest are the heterogeneity models in which capture probabilities vary among individuals.  \\cite{Link2003} has shown that inference about $N$ within this class of models is sensitive to the choice of model for describing individual variation.\n\nA drawback of much available software for fitting $M_h$ is that analyses are restricted to default choices of model for the individual capture probabilities.  Restricting analyses based on available computer code seems undesirable and an advantage of Bayesian approaches is that hierarchical modelling and data augmentation implemented via Markov chain Monte Carlo (MCMC) provide a natural framework for general model fitting.\n\nA commonly used dataset for illustrating model $M_h$ is the snowshoe hare data provided by \\cite{Otis1978} in which 68 individuals were captured in six days of trapping with encounter frequencies $f\\bm = (25, 22, 13, 5, 1, 2)'$ where $f_j \\; (i=1, \\ldots,k)$ is the number of hares caught $i$ times during the $k=6$ study.  Define $N$ as the population size, $y_i$ as the number of times that individual $i$ was caught and $p_i$ as the corresponding capture probability.  To model these using a program such as BUGS \\cite[]{Lunn2000} or JAGS  \\cite[]{Plummer2003} we would like to be able to specify the model simply as\n\\[\n y_i \\sim Bin(k, p_i)\n\\]\nfor $i$ in $1:N$ (herein, unless we specify the two programs separately, we will use BUGS to refer to BUGS and JAGS jointly).  To complete the model specification we would then specify a suitable distribution for $p_i$, hyperpriors for any parameters of this distribution and a prior for $N$.  However two related things prevent us from doing this: (1) BUGS does not allow the index of the loop to be stochastic and (2) the dimension of $p$ changes each time we update $N$ to a new value.\n\n\\cite{Royle2007} provide a neat solution that is easily implemented in BUGS based on a superpopulation.  However, a drawback of their model and code is that $N$ does not appear in the likelihood; rather it is a derived parameter.  This makes specification of priors on $N$ more of a challenge than if $N$ appears explicitly in the likelihood.  \n\nHere we contrast approaches to fitting $M_h$ using the models of \\cite{King2008} and \\cite{Royle2007} when using Bayesian methods.  The approach taken by \\cite{King2008}, as well as \\cite{Durban2005,Schofield2008} and \\cite{Wright2009} for similar models, is to consider a trans-dimensional (TD) algorithm \\cite[]{Carlin1995,Green1995} such as the reversible jump MCMC (RJMCMC) algorithm \\cite[]{Green1995}.  In contrast, \\cite{Royle2007} as well as \\cite{Royle2008,Link2010,Schofield2010} use a data augmentation (DA) approach \\cite[]{Tanner1987} to fit the model.  \n\nOur purpose is to:\n\\begin{itemize}\n \\item[1.] Clarify the distinction between the different likelihoods used to fit $M_h$ models by MCMC.\n \\item[2.] Provide simple BUGS code for $M_h$ that explicitly includes $N$ as a parameter.\n \\item[3.] Consider the comment made by \\cite{Royle2007} that their approach is not a case of trans-dimensional MCMC and fundamentally differs from approaches that use RJMCMC.\n\\end{itemize}\n\n\n\\section{$M_h$ Likelihoods}\\label{sect:comparelike}\nLikelihoods for model $M_{h}$ include an integrated likelihood \\cite[]{Burnham1978,Otis1978}, a ``full'' likelihood \\cite[]{Pledger2000, King2008}, or a super-population likelihood \\cite[]{Royle2007}.  The three most obvious differences between these approaches are (i) whether or not $N$ is explicitly included in the likelihood, (ii) the presence or absence of combinatorial terms involving $N$, and (iii) the presence or absence of a superpopulation (with corresponding value $M$) in the model.  The likelihood of \\cite{Royle2007} does not include $N$, or combinatorial terms involving $N$, but does include $M$.  The full likelihood approach of \\cite{Pledger2000} explicitly includes $N$ and associated combinatorial terms, does not include $M$ and treats the vector of capture probabilities $\\bm{p}$ as parameters.  \\cite{Burnham1978} explicitly includes $N$, does not include $M$ and models $\\bm p$ as random effects; combinatorial terms appear once the capture probabilities are integrated out of the likelihood.\n\n\n\\begin{flushright}{\\bf Table 1 about here} \\end{flushright}\n\\subsection{Burnham's Integrated Likelihood}\nA complete data model can be written as $\\Pr(\\bm X = \\bm x | \\bm P = \\bm p)$; as shorthand we use $[\\bm x | \\bm p]$.   Both $\\bm x$ and $\\bm p$ are unobserved, although we know values of some rows of $\\bm x$ through the observed data matrix $\\bm x^{obs}$.  Following \\cite{Burnham1978} we can write the model as \n\\begin{eqnarray}\n [\\bm x | \\bm p] &=& \\prod_{i=1}^N \\prod_{j=1}^k p_i^{x_{ij}}(1-p_i)^{1-x_{ij}} \\label{eq:XgivenP}\n\n\\end{eqnarray}\nunder the assumption that, conditional on $\\bm P = \\bm p$, capture events are independent among individuals and occasions.  Conditional on $N$, which is treated as a parameter, the number of rows of $\\bm x$ are now known.  \n\nAs noted by  \\cite{Burnham1978} this model is over-parameterized and not useful for estimation.  The usual remedy for over-parameterization is to model $\\bm P$ as random effects sampled from a distribution with $pdf$ $f_{\\bm P}(\\bm p | \\theta_p)$ for $\\bm P \\in \\mathbb{R}_{[0,1]}^N$.  We then integrate over $\\bm p$ to obtain the integrated likelihood given by \\cite{Burnham1978}; this is the observed data likelihood \\cite[]{Gelman2004}.\n\n\\subsection{Complete Data Likelihood (CDL)}\nThe model (\\ref{eq:XgivenP}) is underspecified.\nAdding the random effects distribution for $\\bm P$ we can write a CDL as\n\\begin{eqnarray}\n [\\bm x | \\bm p][\\bm p | \\theta_p] = [\\bm p |\\theta_p] \\times \\prod_{i=1}^N \\prod_{j=1}^k p_i^{x_{ij}}(1-p_i)^{1-x_{ij}} \\label{eq:CDL}.\n\\end{eqnarray}\nAs before, both $\\bm x$ and $\\bm p$ are unobserved.  Associated with $\\bm x^{obs}$ is a subspace of ${\\mathcal S}$, which we denote by $\\Omega(\\bm x)$, of capture history matrices that are identical in appearance to $\\bm x^{obs}$.  The only differences among elements of $\\Omega(\\bm x)$ are that the row indices are permuted.  \n\nUnder the assumption that $x_i | p_i$ and $x_j | p_j$ are independent for all $i \\neq j$, it follows that the $x$'s are exchangeable given the $p$'s and that the elements in $\\Omega(\\bm x)$ have identical joint probabilities across their rows.  A heuristic explanation is that once we have drawn $(x_i, p_i)$ pairs from their joint distribution they are linked -- when we permute the $x$'s the $p$'s get permuted with them.\n\nIt now follows that\n\\begin{eqnarray}\n [N, \\bm p | \\bm x^{obs}, \\theta_p] &\\propto&  [\\bm x^{obs} | \\bm p][\\bm p | \\theta_p] \\nonumber \\\\\n &=&\\frac{N!}{\\prod_{h \\in {\\mathcal H}}z_h!} \\times \\prod_{i=1}^N [p_i | \\theta_p] \\prod_{j=1}^k p_i^{x_{ij}}(1-p_i)^{1-x_{ij}} \\label{eq:CDL2}\n\\end{eqnarray}\nis also a likelihood where $\\frac{N!}{\\prod_{h \\in {\\mathcal H}} z_h!}$ is the dimension of $\\Omega(\\bm x)$.  The likelihood (\\ref{eq:CDL2}) is not the observed data likelihood (ODL) as it still contains unobserved elements.  If we integrate across the random effects we obtain the ODL given by \\cite{Burnham1978}.  However, the likelihood (\\ref{eq:CDL2}) is useful in Bayesian inference carried out by MCMC in that explicit modelling in terms of latent components can be a useful device for improving mixing.\n\n\n\\subsection{Super Population CDL}\n\\cite{Royle2007} describe a CDL in terms of a super-population $M$ (essentially an upper bound on the population size $N$), with $\\bm X$ now an $M$ by $k$ matrix.  The idea is that we replace $N$ in the likelihood with the outcomes $\\bm w$ of the random vector $\\bm W$ of dimension $M$ where $W_{i}$ takes the value $1$ if individual $i$ is included in the population and $0$ otherwise such that $N=\\sum_{i=1}^{M}W_{i}$.   By assumption, $\\bm W$ and $\\bm P$ are independent given their parameters.  The complete model on which the \\cite{Royle2007} approach is based can be written as\n\\begin{eqnarray}\n [\\bm x, \\bm w, \\bm p | \\alpha_p,\\psi] &=& [\\bm w | \\psi][\\bm p | \\bm w, \\alpha_p][\\bm x |\\bm w, \\bm p] \\nonumber \\\\\n &=& [\\bm w | \\psi][\\bm p | \\alpha_p][\\bm x |\\bm w, \\bm p] \\nonumber\\\\\n &=&\\prod_{i=1}^M \\prod_{j=1}^t (p_iw_i)^{x_{ij}} (1-p_iw_i)^{1-x_{ij}} f_P(p_i) \\psi^{w_i}(1-\\psi)^{1-w_i} \\nonumber \\\\\n &=&\\prod_{i=1}^M (p_iw_i)^{y_i} (1-p_iw_i)^{t-y_i} f_P(p_i) \\psi^{w_i}(1-\\psi)^{1-w_i} \\label{eq:spCDL}.\n\\end{eqnarray}\nThe sample space ${\\mathcal S}$ for $\\bm x$ is now of dimension $2^{Mt}$ and the sample space ${\\mathcal W}$ for $\\bm w$ of dimension $2^M$.  The sample space for $\\bm p$ is given by ${\\mathcal P} =\\mathbb{R}_{[0,1]}^M$. \n\nDenoting the $cdf$ for $p$ by $F(p) = F_{\\theta_p}(p)$ and integrating over $\\bm p$ and $\\bm w$, we obtain\n\\[\n [\\bm x^{obs}, N | \\alpha_p, \\psi, M] = \\frac{N!}{\\prod_{h \\neq \\bm 0}z_h!(N-n)!} \\prod_{j=0}^t \\theta_{jF}^{f_j} \\times \\binom{M}{N}\\psi^N(1-\\psi)^{M-N},\n\\]\nwhere $\\theta_{jF} = \\int_0^1 p^j(1-p)^{t-j}dF(p).$ That is, we have Burnham's marginal likelihood multiplied by the prior for $N$ induced by the model $W_i \\sim Bern(\\psi)$. \n\nInference in the two approaches will be identical for the appropriate choice of prior on $\\psi$ in the \\cite{Royle2007} likelihood.  For example, \n\\[\n f(\\psi) \\propto 1\n\\]\ncorresponds to a discrete uniform prior on $N$ as shown by \\cite{Royle2007}.  Another choice could be:\n\\[\n f(\\psi) \\propto \\frac{1}{\\psi}\n\\]\nwhich leads to $N\\propto \\frac{1}{N}$ but with the upper bound $M$.  This is Jeffrey's prior for $N$ but truncated to the range $[0, M]$.  In general, the prior on $N$ induced in the super-population approach is the marginal distribution for $N$ arising from a binomial mixture, with mixing distribution given by the prior on $\\psi$.  While it is relatively easy to construct reference priors of the form given above, constructing general priors on $N$ that are not binomial mixtures will be more difficult.  We do not wish to overstate this problem as the binomial mixtures provide a flexible class of priors including the beta-binomial and the negative-binomial as a limiting case.    \n\n\\section{Gibbs Sampling and Model $M_h$}\n\\cite{Durban2005},\\cite{King2008} and \\cite{Schofield2008} all used a trans-dimensional (RJMCMC) approach to fit $M_h$ using the CDL described above. \\cite{Wright2009} used a similar approach in an application where individual identities were uncertain.  In this approach, sampling from the full conditional for $N$ includes an explicit reversible-jump step in the sense of \\cite{Green1995}.  In contrast, the reversible-jump step is unecessary in the super-population approach of \\cite{Royle2007}. \n\nTo describe what we call the ``standard approach'' to Gibbs sampling we start with the complete data likelihood (\\ref{eq:CDL}).  Although we describe model fitting from a Bayesian stand-point, we could implement a Frequentist approach using an EM-algorithm instead of Gibbs sampling.\n\nFor simplicity we assume $\\theta_{p}$ is known. A Gibbs sampler can be constructed by alternating sampling from the following distributions,\n\\begin{enumerate}\n \\item The full conditional distribution for $p_{i}$, $\\forall i$:\n      \\[\n       [p_{i}|x,N,\\theta_{p}] \\propto p_{i}^{x_{ij}}(1-p_{i})^{1-x_{ij}}f(p|\\theta_{p}).\n      \\]\n      Depending on $[p|\\theta_{p}]$, we can either sample from this distribution directly, or use another a simulation based sampler, such as the Metropolis-Hastings algorithm or rejection sampling.\n  \\item The full conditional distribution for $N$:\n       \\[\n       [N|x,p] \\propto\\frac{N!}{(N-n)!}\\prod_{i=1}^{N}\\prod_{j=1}^{k}p_{i}^{x_{ij}}(1-p_{i})^{1-x_{ij}}f(N).\n      \\]\n      This update is more difficult since the value of $N$ changes the dimension of $p$, so we use a TD algorithm.  Following \\cite{Schofield2008} we use a special case of the RJMCMC algorithm.  \n      We first propose a new value $N^{\\star}$ from $J(N^{\\star}|N)$ and specify $p^{\\star}$ to be a vector of length $N^{\\star}$ corresponding to the capture probabilities for the $N^{\\star}$ individuals.  We set $p^{\\star}_{i}=p_{i}$ for all $i = 1,\\ldots,N$ and generate $p^{\\star}_{i}$ for $i = N+1,\\ldots,N^{\\star}$ from the prior distribution $[p|\\theta_{p}]$.  Generating $p^{\\star}$ in such a way simplifies the RJMCMC algorithm.  The acceptance probability is\n      \\[\n       q = \\frac{N^{\\star}!(N-n)!}{(N^{\\star}-n)!N!}\\prod_{i=N+1}^{N^{\\star}}(1-p^{\\star}_{i})^{k}\\frac{J(N|N^{\\star})}{J(N^{\\star}|N)}\\frac{f(N^{\\star})}{f(N)}.\n      \\]\n      If our proposed value $N^{\\star}<N$ we obtain $p^{\\star}$ by removing the last $N-N^{\\star}$ rows of $p$.  The acceptance probability becomes\n      \\[\n       q = \\frac{N^{\\star}!(N-n)!}{(N^{\\star}-n)!N!}\\frac{1}{\\prod_{i=N^{\\star}+1}^{N}(1-p_{i})^{k}}\\frac{J(N|N^{\\star})}{J(N^{\\star}|N)}\\frac{f(N^{\\star})}{f(N)}I(N^{\\star}\\geq n),\n      \\]\n      where $I(\\cdot)$ is an indicator function.  Importantly, we note that the full RJMCMC expression has been reduced to a standard Metropolis-Hastings like expression that consists of a likelihood ratio $\\times$ jumping distribution ratio $\\times$ prior ratio.\n\\end{enumerate}\nIf we also wish to estimate $\\theta_{p}$, we simply need to (i) specify a prior distribution for $\\theta_{p}$, and (ii) sample from the full conditional distribution for $\\theta_{p}$ in turn.  Although we have never found it necessary, more complicated RJMCMC setups could be specified in order to speed up the RJMCMC algorithm.  For more extensive details of RJMCMC we direct the reader to \\cite{Gelman2004}.  \n\nA strength of this approach is the ease with which we can incorporate extensions to the model.  For example, if the tags are themselves uncertain we need to make only one adjustment to the algorithm above: we add a step where we sample from the full conditional distribution for all of the uncertain tags (see \\citealt{Wright2009} for details).  We do not need to alter the steps we already have because conditional on $x$ the terms $y_{h}!$ are fixed and cancel out of the above distributions.  With $N$ explicit in the model as a parameter it is also easy to include different priors for $N$, including hierarchical priors, in the usual way.  For example, if  mark-recapture is carried out across distinct subpopulations it would be straightforward to model $N$ for each population in terms of an appropriate spatial model.\n\nOne perceived weakness of this TD approach is that it is difficult to incorporate into BUGS, in contrast to the super population approach \\cite[]{Royle2007}.   However, this is not the case, so long as we specify a prior for $N$ that has an upper bound $M$ \\cite[]{Durban2005}.  We provide BUGS code for model $M_{h}$ (Figure \\ref{fig:rjbug}) that is just as simple and requires similar programming effort to the code given by \\cite{Royle2007}.   A full description of the model including the data and initial values files are available for both BUGS and JAGS at \\url{www.maramatanga.com}.  \nWe also provide alternative model\/data files for model $M_{h}$ that allow for easier specification of the data, albeit with a longer model statement.  \n\nNote that specifying an upper limit is required \\emph{only} for easy implementation in BUGS; there is no superpopulation in the \\cite{Royle2007} sense.  We can achieve an approximation to any distributions for $N$ with support on the non-negative integers by including a large $M$ and truncating (JAGS) or censoring (BUGS) the distribution at $M$ (although very large values for $M$ will slow model fitting).  We find that both (i) the time taken to run the BUGS code and (ii) the mixing of the MCMC output is approximately equivalent to the corresponding superpopulation code.   However, we note that many things can substantially influence both the time taken for the code to run and the mixing performance.  These include (i) how we parameterize the model on $N$, (ii) the sampling algorithms the programs choose and (iii) which software package and version we use.   We find that in general for models of this type, JAGS takes longer to run than BUGS but mixes better.  \n\nWe illustrate the code in Figure \\ref{fig:rjbug} with model $M_{h}$ using the snowshoe hare data.\nThe model was fitted in JAGS 2.1.0 and density plots of $N$ (Figure \\ref{fig:JAGSout}) can be compared to those in \\cite{Royle2007} and \\cite{Link2010}.\n \n\\begin{flushright}{\\bf Figure 1 about here} \\end{flushright}\n\\begin{flushright}{\\bf Figure 2 about here} \\end{flushright}\n\n\\subsection{Super Population approach and RJMCMC}\nThe starting point for the super population model is the complete data likelihood specified in (\\ref{eq:spCDL}).  To complete the specification of the model we specify a prior for $\\psi$, $f(\\psi)$, as well as a random effects distribution for $p$. \\cite{Royle2007} provide BUGS code for fitting this model.  The CDL that their BUGS code provides, referred to as a data augmentation (DA) algorithm,  is identical to that in (\\ref{eq:spCDL}) with the exception that \\cite{Royle2007} fix the first $n$ values of $W_i$ at 1.  Data augmentation now occurs over a reduced subspace for $\\bm W$; that in which $W_1 = W_2=\\ldots=W_n$ = 1.  However, the effect on the reduced CDL is simply the omission of the $\\prod_h z_h!$ terms.  Therefore, the likelihood is equivalent, although as we discuss below a consequence of conditioning on this one order $\\bm W$ is that the observed $z_h$'s cannot be modelled in the presence of tag-reading uncertainty.\n\n\\cite{Royle2007} comment that the DA algorithm is an estimation approach which fundamentally differs from other TD algorithms which are ``model-selection approaches'' since the use of a TD algorithm treats each value $N$ as a separate model.  Here we argue that this difference is semantic, and that in an algorithmic sense the two approaches are equivalent since they can be set up to give the same Gibbs sampler\n\nTo see the equivalence, we start with a superpopulation model similar to the that of \\cite{Royle2007}.  The only difference is that we treat the capture probability as undefined for any individual that is not in the population, that is, $p_{i}$ is undefined when $w_{i}=0$.  It is clear that this model is TD, in the sense that changing the value of $w_{i}$ changes the dimension of the capture probability vector $\\bm{p}$ whose length is defined by $N=\\sum_{i} w_{i}$.  \nThis setup has $2^{M-n}$ possible ``models'' (by possible models we refer to models with non-zero probability).  That is, $N$ is censored by $n$ and has an upper limit of $M$ so that there are $M-n$ pseudo-individuals who were unobserved.  Each of these $2^{M-n}$ models describes the different ways that the $M-n$ unobserved individuals could potentially be included or excluded from the population.  To complete the TD specification we require a number of additional components.  Using the language of \\cite{Gelman2004} pg 338-339, we require\n\\begin{enumerate}\n  \\item Auxiliary random variables that make the parameter space equivalent for each of these $2^{M-n}$ models.  In particular, we need to ensure that we have $M$ parameters (equal to the dimension of $p$ if all $M$ individuals are included in the population).  For each of the $2^{M-n}$ models we need a distribution that describes how to generate the auxiliary random variables when moving to another model. \n  \\item A set of bijections that describe how the set of parameters and auxiliary variables in one model relates to the set of parameters and auxiliary variables in another model.\n  \\item Prior model probabilities.  \n\\end{enumerate}\nWe specify these components as follows:\n\\begin{enumerate}\n  \\item Set the distribution for the auxiliary variables for each pair of models to be the prior distribution $f(p|\\theta_{p})$.  This simplifies the algorithm since this distribution is the same for each of the auxiliary random variables in every model.\n  \\item Specify identity bijections between each pair of models.\n  \\item Set the prior model probability as proportional to $\\psi^{N}(1-\\psi)^{M-N}$ for a model with $N$ individuals in the population and include a hierarchical $Be(1,1)$ prior distribution for $\\psi$.\n\\end{enumerate}\n\nIf we implement this scheme using a TD Gibbs sampler \\cite[e.g.][pg. 146]{Link2010}, where in each iteration we choose between pairs of these models that correspond to $w_{i}=0$ or $w_{i}=1$, then the resulting TD algorithm has the same full conditional distributions as the DA algorithm described in \\cite{Royle2007}.  In this sense, the TD algorithm described above is exactly the same as the equivalent DA algorithm and any distinctions can be thought of as semantic. \n\nAny difference that does exist between the two approaches lies in the interpretation that can be placed on supplemental variables.  In the fixed-dimension DA approach, the $M-N$ supplemental capture probabilities correspond to phantom individuals that exist in a sense, but only in the `otherworld' that is described  by the superpopulation. Of necessity, these capture probabilities have the same prior as any other capture probability in the model.\nIn the TD approach described above, as well as the standard TD approach to $M_h$, the supplemental variables have only a dimension-matching role.  Their distribution is chosen solely to ensure an efficient Gibbs sampler and they have no interpretation as parameters within the context of the model\n\nIt is not surprising that the two methods correspond to the same Gibbs sampler, since a crucial aspect of the RJMCMC algorithm as laid out by \\cite{Green1995} is the use of auxiliary variables to match the dimension of the various models.  In this case, we can think of RJMCMC as using DA to match dimensions.  So long as we choose these auxiliary variables appropriates, as we did above, we are able to make the DA scheme of \\cite{Royle2007} correspond to a RJMCMC sheme by only considering RJMCMC algorithms that use identity bijections between models.\n\n\n\n\\section{Discussion}\\label{sect:discuss}\nWhile the super-population model as implemented by \\cite{Royle2007} is simple and appropriate for most analyses we emphasize that extensions may be difficult or problematic to incorporate in this modeling framework.  The BUGS code we have provided for the standard approach offers an easy solution if other hierarchical priors are of interest for $N$.  \n\nExtensions to include changes such as relaxing the assumption of error-free tag reading \\cite[e.g.,][]{Wright2009} are easy to include in the standard approach provided an explicit reversible-jump step is included in the Gibbs sampling algorithm.  It is also important in this context that we do not condition on one ordering of the $\\bm x$ (or equivalently $\\bm y_i$ = the number of times individual $i$ is caught) as is done in the BUGS code of \\cite{Royle2007} as well as the BUGS code in Figure \\ref{fig:rjbug}.  Irrespective of the model we choose to use, care needs to be taken to ensure the combinatorial term is correct for the given model.  For the full likelihood approach, this means that we include the full combinatorial term, as shown in (\\ref{eq:CDL2}), in the model when we sample from the full conditional distributions for all unknowns, including the uncertain tags (see \\cite{Wright2009} for an example).  It is less clear how we would proceed in the superpopulation approach in the presence of tag error.  \n\nBeing able to specify the model without having to explicitly include the combinatorial terms is one of the advantages of the superpopulation approach.  However, as we have shown, while correct for the standard model, conditioning on $W_{1}=1,\\ldots,W_{n}=1$ results in an marginal likelihood that has the incorrect combinatorial term if there is error in tag reading.  In order to obtain valid inference in this setting, additional adjustments to the superpopulation model will be required to ensure that the marginal likelihood is correct.  \n\nWe emphasise that although we have included an upper limit $M$ on $N$ in our BUGS code, this is not required for the general TD algorithm.  Also, during general implementation we only need sample values for auxiliary variables that are involved in the change of dimension from $N$ to $N^*$ rather than the $M-n$ sampled by the BUGS code. This can result in substantial gains in efficiency when $M$ is much larger than a typical value for $N$, for example, when there is skew in the posterior for $N$.\n\nWe believe that the main advantage of the super-population approach as advocated by \\cite{Royle2007} is the fast and easy implementation of the model in BUGS. However, an unappealing aspect of this approach is that $N$ is not explicit in the likelihood and its use appears to be restricted to implied priors for $N$ that are based on binomial mixture models.  BUGS code for fast and easy implementation of the standard model can also be written as we have shown.\n\nAlthough we have focused here on the mark-recapture model $M_{h}$, the insights we offer are not restricted to $M_h$ but extend to other closed and open population capture-recapture models, occupancy modeling, distance sampling, and any other problem that involves inference about an unobserved number of draws from a series of categorical distributions each with a distinct probability vector.  \n\\bibliographystyle{asa}\n\\input{base.bbl}\n\n\\clearpage\n\\begin{table}[p]\n \\begin{tabular}{lp{14cm}}\n \\hline\n &\\multicolumn{1}{c}{Definition}\\\\\n \\cline{1-2}\\\\\n    $k$ & the number of occasions on which the population is sampled.\\\\\n   $N$ & population size.\\\\\n   $M$ & superpopulation size ($M \\geq N$).\\\\\n   $\\bm P$ & an $N \\times 1$ random vector of capture probabilities with a realisation denoted by $\\bm p$.\\\\\n   ${\\mathcal P}$ & the sample space for $\\bm P$.\\\\\n   $\\bm X$ & a random capture history matrix of dimension $N \\times k$ with with $X_{ij}$ the indicator for capture of individual $i$ $(i=1,\\ldots,N)$ in sample $j$ $(j=1,\\ldots,k$.  We use $\\bm x$ to denote a realisation of $\\bm X$ and $\\bm x^{obs}$ a value observed after sampling; $\\bm x^{obs}$ includes $N-n$ rows of zeros which are treated as observed conditional on $N$.\\\\  \n   ${\\mathcal S}$ & the sample space for $\\bm X$.\\\\\n   $\\bm W$ & a random vector with value $1$ if the $i$th member of the superpopulation is one of the $N$ individuals that are in the study population.  We denote a realisation of $W$ by $\\bm w$.\\\\\n   ${\\mathcal W}$ & the sample space for $\\bm W$.\\\\\n   $z_h$ & the number of individuals with capture history $h \\in {\\mathcal H}$, where ${\\mathcal H}  = \\{ 11\\cdots11,11\\cdots10,\\ldots,00\\cdots00\\}$.  We also refer to the null history $00\\ldots00$ as $\\bm 0$.\\\\\n   \\hline\n \\end{tabular}\n \\caption{Notation}\n\\end{table}\n\n\n\\clearpage\n\n\n\\begin{figure}[p]\n\\begin{verbatim}\n1.  model {\n2.    mu ~ dlogis(0,1)\n3.    t ~ dt(0,1,2)\n4.    tau<-abs(t*5)\n5.    for(i in 1:M){\n6.      logit(p[i]) <- lp[i]\n7.      lp[i] ~ dnorm(mu,tau)\n8.      y[i] ~ dbin(pi[i],k)\n9.      pi[i] <- p[i]*w[i]\n10.     w[i] <- step(N-i)\n11.   }\n12. N ~ dcat(pee[1:M])\n13. n ~ dbin(0.00001,N)\n14. }\n\\end{verbatim}\nExplanation of the BUGS code:\n\\begin{description}\n  \\item[Line 2--4 \\& 6--7] Distribution $f(p|\\theta_{p})$, including prior distributions on $\\theta_{p} = (\\mu,\\tau)$.  \n  \\item[Line 8--10] Modeling the observed capture histories -- {\\tt y[i]} is the number of times individual $i$ was caught during $k$ samples.  We could model the capture indicators structure {\\tt x[i,j]} if we prefer.\n  \\item[Line 12] Specifying $f(N)$.  Note that the starting value for \\texttt{N}$>n$.  Here we chose to use a discrete uniform prior on $N$ so the values \\texttt{pee[i]} are specified as data and are all set to $\\frac{1}{M}$.  Note that (i) there are many different ways we can parameterize this distribution and that this may affect the mixing speed as well as the time taken to run the code.\n  \\item[Line 13] Including the term $\\propto \\frac{N!}{(N-n)!}$ using the approximation of \\cite{Durban2005} who treat $n$ as binomially distributed with index $N$ and success probability $\\epsilon \\approx 0$.  We assume that tags are correctly read and so do not include $\\prod y_{h}!$.  Another approach to including the normalizing constant is to include a variable {\\tt zero=0} and then substitute line 13 with two lines: (i) {\\tt zeros\\~{}dpois(lam)} and (ii) {\\tt lam <- -(logfact(N)-logfact(N-n)-logfact(M))}.\n\\end{description}\nNote that this model statement is appropriate for both BUGS and JAGS.  We provide alternative model statements and the data and initial values required for both BUGS and JAGS for each of these modeling statements at \\url{www.maramatanga.com}.\n\\caption{BUGS Code For Model Mh Using RJMCMC}\n\\label{fig:rjbug}\n\\end{figure}\n\n\n\\begin{figure}[htbp]\n\\begin{minipage}{\\textwidth}\n  \\begin{center}\n    \\includegraphics[angle=270,width = \\textwidth]{Nplot.eps} \n    \\caption{Traceplot and Posterior Density for $N$ for the snowshoe hare data fitted in JAGS using the code in figure \\ref{fig:rjbug}.}\n    \\label{fig:JAGSout}\n  \\end{center}\n\\end{minipage}\n\\end{figure}\n\n\\end{document}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe polaron is a model of an electron interacting with its self-induced polarization field of the underlying crystal. The description of the polarization as a quantum field corresponds to the Fr\\\"ohlich model \\cite{frohlich}.  In the  classical approximation, on the other hand, the dynamics of a polaron is described by the Landau--Pekar (LP) equations.  For $(\\psi_t, \\varphi_t) \\in H^1(\\mathbb{R}^{3})\\times L^2(\\mathbb{R}^{3})$, where $\\psi_t$ is the electron wave function and $\\varphi_t$  denotes the phonon field, these equations read in suitable units\n\\begin{equation}\\label{eq:LP}\n\\begin{aligned}\ni \\partial_t \\psi_t  & = h_{ \\varphi_t  }  \\psi_t,  \\\\\ni \\alpha^2 \\partial_t \\varphi_t  & = \\varphi_t + \\sigma_{\\psi_t},\n\\end{aligned}\n\\end{equation}\nwhere $h_{\\varphi}$ is the Schr\\\"odinger operator \n\\begin{equation}\nh_{\\varphi} = - \\Delta  + V_\\varphi\n\\end{equation}\nwith potential \n\\begin{equation}\nV_\\varphi(x) = 2 (2\\pi)^{3\/2} \\Re\\, [ ( - \\Delta)^{-1\/2} \\varphi ] (x)= 4 (2\\pi)^{5\/2} |x|^{-2} * \\Re\\, \\varphi ,\n\\end{equation}\nand\n\\begin{equation}\n\\sigma_{\\psi} (x) = (2\\pi)^{3\/2}\\left[ (- \\Delta)^{-1\/2} \\vert \\psi \\vert^2\\right]  (x) = 2(2\\pi)^{5\/2}  |x|^{-2} * |\\psi|^2 .\n\\end{equation}\nThe parameter $\\alpha>0$ quantifies  the strength of the coupling of the electron's charge to the polarization field. \n\nThe LP equations can be derived from the dynamics generated by the (quantum) Fr\\\"ohlich Hamiltonian for suitable initial states in the strong coupling limit $\\alpha\\to \\infty$ \\cite{LMRSS} (see also \\cite{FS,FG,Griesemer,LRSS,M} for earlier results on this problem). One of the outstanding open problems concerns the polaron's effective mass \\cite{LS,seiringer2020polaron,spohn1987effective}: due to the interaction with the polarization field, the electron effectively becomes heavier and behaves like a particle with a larger mass. This mass increases with the coupling $\\alpha$, and is expected to diverge as $\\alpha^4$ as $\\alpha\\to\\infty$. A precise asymptotic formula was obtained by Landau and Pekar \\cite{ landau1948effective} based on the classical approximation, and hence it is natural to ask to what extent the derivation of the LP equations in \\cite{LMRSS} allows to draw conclusions on the effective mass problem.\n\nIt is, however, far from obvious how to rigorously obtain the effective mass even on the classical level, i.e., from the LP equations \\eqref{eq:LP}. A heuristic derivation, reviewed in Section \\ref{sec:TW} below, considers traveling wave solutions of \\eqref{eq:LP} for non-zero velocity $v\\in \\mathbb{R}^3$, and expands the corresponding energy for small $v$. The existence of such solutions remains unclear, however, and we in fact conjecture that no such solutions exist for non-zero $v$. This is related to the fact the energy functional corresponding to \\eqref{eq:LP} (given in Eq. \\eqref{def:G} below) does not dominate the total momentum, and a computation of the ground state energy as a function of the (conserved) total momentum would simply yield a constant function (corresponding to an infinite effective mass). Due to the vanishing of the sound velocity in the medium, a moving electron can be expected to be slowed down to zero speed by emitting radiation. (See \\cite{Fr1,Fr2} for the study of a similar effect in a model of a classical particle coupled to a field.) \n\nIn this paper, we provide a novel definition of the effective mass for the LP equations. We shall argue that all low energy states have a well-defined notion of (initial) velocity, and hence we can minimize the energy functional among states with given velocity. Expanding the resulting energy-velocity relation for small velocity gives a definition of the effective mass, which coincides with the prediction by Landau and Pekar \\cite{landau1948effective}. \n\n\\subsection{Structure of the paper}\n In Section \\ref{sec:result}, we explain our rigorous approach to derive the energy-velocity relation of the system, allowing for a precise definition and computation of the effective mass. After introducing some notation and recalling fundamental properties of the Pekar energy functional in Section \\ref{sec:prel},  we identify in Section \\ref{sec:vel} a set of initial data for the LP equations for which it is possible  to define the position, and consequently the velocity, at any time. We then arrive at an energy-velocity relation by defining $E(v)$ in Section \\ref{sec:initial} as the minimal energy among all admissible initial states of fixed initial velocity $v$. Finally, in Section \\ref{sec:thm} we state our main result, an expansion of $E(v)$ for small velocities $v$, allowing for the computation the effective mass of the system.  \n\nSection \\ref{sec:proof-thm} contains the proof of our main result, Theorem \\ref{thm:freephase}.  \n\nIn Section \\ref{sec:approaches} we discuss the formal approach to the effective mass via traveling waves. Moreover, we investigate an alternative definition of the effective mass, through an alternative notion of velocity of low-energy states.\n\n\n\n\n\\iffalse\n\nOur main object of study is the effective mass of the polaron in this classical approximation. Several computations are present in the physics literature concerning this quantity \\cite{alexandrov2010advances,landau1948effective,spohn1987effective},  which, however, are all based on heuristic arguments. In particular, it is conjectured that the effective mass diverges as $\\alpha^4$ as $\\alpha \\to \\infty$, with an explicit prefactor related to the Pekar minimizer $\\varphi_{\\rm P}$, i.e.\n\\begin{align}\n\\label{eq:mass-pred}\n\\lim_{\\alpha \\rightarrow \\infty} \\alpha^{-4} \\; m_{\\text{eff}}  =  \\frac{2}{3}\\| \\nabla \\varphi_{\\rm P} \\|^2 \\; .\n\\end{align}\nThis work provides a rigorous definition and computation of the effective mass, verifying \\eqref{eq:mass-pred}. Our approach relies on the introduction of a well-defined notion of position and velocity of the system and a perturbative expansion of the energy functional associated to the Landau-Pekar equations. These allow to derive an explicit energy-velocity relation for the system, yielding the exact value of the effective mass of the system. \n\n\\fi\n\n\n\\section{Main Results}\n\\label{sec:result}\n\n\\subsection{Preliminaries}\n\\label{sec:prel} We start by introducing further notation and recalling some known results. \nThe classical energy functional corresponding to the Landau--Pekar equations \\eqref{eq:LP} is defined on $H^1(\\mathbb{R}^{3})\\times L^2(\\mathbb{R}^{3})$ as \n\\begin{equation}\\label{def:G}\n\\mathcal{G} ( \\psi, \\varphi ) = \\langle \\psi, h_\\varphi \\psi \\rangle + \\| \\varphi \\|_2^2 \\quad \\text{for} \\quad \\|\\psi\\|_2=1 .\n\\end{equation}\nEquipped with the symplectic form $\\frac 1{2i} \\int d\\psi \\wedge d\\bar\\psi + \\frac{\\alpha^2}{2i} \\int d\\varphi\\wedge d\\bar\\varphi$, it defines a dynamical system leading to the LP equations \\eqref{eq:LP}.  Moreover, $\\mathcal{G}$ is conserved along solutions of \\eqref{eq:LP}.\n\nIt was proved in \\cite{lieb1977existence} that the Pekar ground state energy \n\\begin{align}\n\\min \\mathcal{G}(\\psi,\\varphi) =: e_{\\rm P} \n\\end{align}\nis attained for the Pekar minimizers $( \\psi_{\\rm P}, \\varphi_{\\rm P})$, which are radial functions in $C^{\\infty}(\\mathbb{R}^{3})$ satisfying $\\psi_{\\rm P}>0$, $\\varphi_{\\rm P}=-\\sigma_{\\psi_{\\rm P}}$ and $\\psi_{\\rm P}=\\psi_{\\varphi_{\\rm P}}$, where  $\\psi_{\\varphi}$ denotes the ground state of $h_{\\varphi}$ whenever it exists. Moreover, this minimizer is unique up to the symmetries of the problem, i.e., translation-invariance and multiplication of $\\psi$ by a phase. \nWe shall denote \n\\begin{align}\n\\label{eq:def-HP}\nH_{\\rm P} = h_{\\varphi_{\\rm P}} - \\mu_{\\rm P}  \\quad \\text{with} \\quad \\mu_{\\rm P}=\\infspec h_{\\varphi_{\\rm P}} .\n\\end{align}  \n\nAssociated to $\\mathcal{G}$, there are the two functionals \n\\begin{align}\n\\label{eq:EandF}\n\\mathcal{E}(\\psi):=\\inf_{\\varphi\\in L^2(\\mathbb{R}^{3})} \\mathcal{G}(\\psi,\\varphi), \\quad \\mathcal{F}(\\varphi):=\\inf_{\\psi\\in H^1(\\mathbb{R}^{3}) \\atop \\|\\psi\\|_2=1} \\mathcal{G}(\\psi,\\varphi)\n\\end{align}\nand clearly $e_{\\rm P}=\\min \\mathcal{G}(\\psi,\\varphi)=\\min\\mathcal{E}(\\psi)=\\min \\mathcal{F}(\\varphi)$. \nWe also define the manifolds of minimizers \n\\begin{align}\n\\label{eq:PekEn}\n\\mathcal{M}_{\\mathcal{G}}:=\\{(\\psi,\\varphi) \\,|\\, \\mathcal{G}(\\psi,\\varphi)&=e_{\\rm P}\\}, \\ \\mathcal{M}_{\\mathcal{E}}:=\\left\\{\\psi \\,|\\, \\mathcal{E}(\\psi)=e_{\\rm P}\\right\\}, \\ \\mathcal{M}_{\\mathcal{F}}:=\\left\\{\\varphi \\,|\\, \\mathcal{F}(\\varphi)=e_{\\rm P}\\right\\}.\n\\end{align}\nThe results  in \\cite{lieb1977existence} imply that we can write these in terms of the Pekar minimizers $(\\psi_{\\rm P}, \\varphi_{\\rm P})$ as \n\\begin{align}\n\\mathcal{M}_{\\mathcal{G}}=\\{(e^{i\\theta}& \\psi_{\\rm P}^y,\\varphi_{\\rm P}^y) \\,|\\, \\theta \\in [0,2\\pi), \\, y \\in \\mathbb{R}^{3} \\},\\nonumber\\\\\n\\mathcal{M}_{\\mathcal{E}}=\\{&e^{i\\theta} \\psi_{\\rm P}^y \\,|\\, \\theta \\in [0,2\\pi), \\, y \\in \\mathbb{R}^{3}\\},\\nonumber\\\\\n&\\mathcal{M}_{\\mathcal{F}}=\\{\\varphi_{\\rm P}^y \\,|\\, y \\in \\mathbb{R}^{3}\\}\n\\end{align}\nwhere  $f^y:=f(\\cdot-y)$ for any function $f$.  Furthermore, it can be deduced from the results in  \\cite{lenzmann2009uniqueness}  that the energy functionals $\\mathcal{F}$ and $\\mathcal{E}$ are both coercive (see \\cite[Lemmas~2.6 and 2.7]{feliciangeli2021persistence}), i.e., there exists $C>0$ such that\n\\begin{align}\n\\label{eq:coercivity}\n\\mathcal{F}(\\varphi)\\geq e_{\\rm P}+C\\dist^2_{L^2}(\\varphi,\\mathcal{M}_{\\mathcal{F}}), \\quad \\mathcal{E}(\\psi)\\geq e_{\\rm P}+ C \\dist^2_{H^1}(\\psi,\\mathcal{M}_{\\mathcal{E}}).\n\\end{align} \n\n\nThe following Lemma on properties of the projection onto  the manifold $\\mathcal{M}_{\\mathcal{F}}$ will be important for our analysis below. Its proof will be given in Appendix \\ref{app:projection}.\n\n\\begin{lem}\n\t\\label{lem:minEFproj}\n\tThere exists $\\delta>0$ such that the $L^2$-projection onto $\\mathcal{M}_{\\mathcal{F}}$, is well-defined (i.e., unique) on \n\t\\begin{align}\n\t(\\mathcal{M}_{\\mathcal{F}})_{\\delta}&:=\\{\\varphi\\in L^2(\\mathbb{R}^{3})\\;|\\; \\dist_{L^2}(\\varphi, \\mathcal{M}_{\\mathcal{F}})\\leq \\delta\\} \\; .\n\t\\end{align}\n\tFor any $\\varphi \\in (\\mathcal{M}_{\\mathcal{F}})_{\\delta}$,  we define $z_\\varphi \\in \\mathbb{R}^3$ via \n\t\\begin{align}\n\tP_{L^2}^{\\mathcal{M}_{\\mathcal{F}}}(\\varphi)&=\\varphi_{\\rm P}^{z_{\\varphi}} \\; .\n\t\\end{align}\n\tThen $z_{\\varphi}$ is a differentiable function from $(\\mathcal{M}_{\\mathcal{F}})_{\\delta}$ to $\\mathbb{R}^{3}$ and its partial derivative in the direction $\\eta \\in L^2(\\mathbb{R}^3)$  is given by \n\t\\begin{align}\n\t \\partial_t z_{\\varphi + t \\eta} \\restriction_{t=0}\n\t = A_\\varphi^{-1} \\langle \\Re \\eta \\vert  \\nabla \\varphi_P^{z_\\varphi} \\rangle,\n\t\\end{align} \n\twhere $A$ is the invertible matrix defined  for any $\\varphi \\in(\\mathcal{M}_{\\mathcal{F}})_{\\delta}$  by $A_{i,j} :=  - \\Re \\langle \\varphi \\vert \\partial_i \\partial_j \\varphi_P^{z_\\varphi} \\rangle$. \n\\end{lem}  \n\n\n\\begin{rem} \\label{rem:E} Likewise, it can be shown that the $H^1$- (resp. $L^2$-) projection onto $\\mathcal{M}_{\\mathcal{E}}$ have similar properties: There exists $\\delta >0 $ such that the $H^1$- (resp. the $L^2$-) projection onto $\\mathcal{M}_{\\mathcal{E}}$ \n\t\\begin{align}\n\t\\label{eq:proj-E}\n\tP_{H^1}^{\\mathcal{M}_{\\mathcal{E}}}(\\psi)=e^{i\\theta_{\\psi}}\\psi_{\\rm P}^{y_{\\psi}}, \\quad\\Big(\\text{resp.}  \\ P_{L^2}^{\\mathcal{M}_{\\mathcal{E}}}(\\psi)&=e^{i\\theta'_{\\psi}}\\psi_{\\rm P}^{y'_{\\psi}} \\Big) \n\t\\end{align}\n\tis well-defined on the set $(\\mathcal{M}_{\\mathcal{E}})^{H^1}_{\\delta}:=\\{\\psi\\in L^2(\\mathbb{R}^{3})\\;|\\; \\dist_{H^1}(\\psi, \\mathcal{M}_{\\mathcal{E}})\\leq \\delta\\}$ (resp.  $(\\mathcal{M}_{\\mathcal{E}})^{L^2}_{\\delta}:=\\{\\psi\\in L^2(\\mathbb{R}^{3})\\;|\\; \\dist_{L^2}(\\psi, \\mathcal{M}_{\\mathcal{E}})\\leq \\delta\\}$) and the functions $y_\\psi, \\theta_\\psi$ (resp.  $y'_\\psi, \\theta'_\\psi$) defined through  \\eqref{eq:proj-E} \nare differentiable functions from  $(\\mathcal{M}_{\\mathcal{E}})^{H^1}_{\\delta}$ (resp.  $(\\mathcal{M}_{\\mathcal{E}})^{L^2}_{\\delta}$)  to $\\mathbb{R} \/ (2\\pi \\mathbb{Z})$ and $\\mathbb{R}^{3}$.  \n\\end{rem}\n\n\\subsection{Position and velocity of solutions} \\label{sec:vel}\n\nIn this section, we give a meaning to the notion of position, and therefore velocity, for solutions of the Landau--Pekar equations (at least for a class of initial data). There is a natural way of defining, given $\\psi_t$, the position of the electron at time $t$, which is simply given by \n\\begin{align}\n\\label{eq:defXel}\n\\text{X}_{\\text{el}}(t):=\\expval{x}{\\psi_t}.\n\\end{align}\nThis yields, by straightforward computations using \\eqref{eq:LP}, that\n\\begin{align}\\label{vel}\n\\text{V}_{\\text{el}}(t):=\\frac{d}{dt}\\text{X}_{\\text{el}}(t)=2\\expval{-i\\nabla}{\\psi_t}.\n\\end{align} \nNote that \\eqref{vel} is always well-defined for $\\psi\\in H^1(\\mathbb{R}^3)$, even although \\eqref{eq:defXel} not necessarily is. \n\nFor the phonon field, the situation is more complicated as $\\varphi$ cannot be interpreted as a probability distribution over positions. This calls for a different approach. By \\eqref{eq:coercivity}, Lemma \\ref{lem:minEFproj} and the conservation of $\\mathcal{G}$ along solutions of \\eqref{eq:LP}, we know that there exists $\\delta^*$ such that for any initial condition $(\\psi_0,\\varphi_0)$ such that \n\\begin{align}\n\\label{eq:IDuniqueproj}\n\\mathcal{G}(\\psi_0,\\varphi_0)\\leq e_{\\rm P}+\\delta^*,\n\\end{align}\n$\\varphi_t$\nadmits a unique $L^2$-projection $\\varphi_{\\rm P}^{z(t)}$ onto $\\mathcal{M}_{\\mathcal{F}}$ for all times.\nWe use this to define \n\\begin{align}\n\\label{eq:defXph}\n\\text{X}_{\\text{ph}}(t):=z(t), \\quad \\text{V}_{\\text{ph}}:=\\frac{d}{dt} \\text{X}_{\\text{ph}}(t)=\\dot{z}(t).\n\\end{align}\nNote that $\\text{X}_{\\text{ph}}(t)$ is indeed differentiable by Lemma \\ref{lem:minEFproj} and the differentiability of the LP dynamics. At this point, for any initial data satisfying \\eqref{eq:IDuniqueproj}, we have a well-defined notion of position and velocity for all times, admittedly in a much less explicit form for the phonon field.\n\n\\subsection{Initial conditions of velocity $v$}\n\\label{sec:initial}\n \nFor any $v\\in \\mathbb{R}^{3}$ (or at least for $|v|$ sufficiently small), we are now interested in considering all initial conditions $(\\psi_0,\\varphi_0)$ whose solutions have instantaneous velocity $v$ at $t=0$ (both in the electron and in the phonon coordinate) and to then minimize the functional $\\mathcal{G}$ over such states. This will give us an explicit relation between the energy and the velocity of the system, allowing us to define the effective mass of the polaron in the classical setting defined by the Landau--Pekar equations.\n\nNote that by radial symmetry of the problem only the absolute value of the velocity, and not its direction, affects our analysis. Hence, for $v\\in \\mathbb{R}$, we consider  initial conditions $(\\psi_0,\\varphi_0)$ such that\n\\begin{itemize}\n\t\\item[(i)] $(\\psi_0, \\varphi_0) \\in H^1 (\\mathbb{R}^3) \\times L^2 ( \\mathbb{R}^3) $ with $\\| \\psi_0\\|_2=1$ and such that \\eqref{eq:IDuniqueproj} is satisfied,\n\t\\item[(ii)] $\\text{V}_{\\text{el}}(0)=\\text{V}_{\\text{ph}}(0)=v (1,0,0)$. \n\\end{itemize}\nThe set of admissible initial conditions of velocity $v\\in \\mathbb{R}$ can hence be compactly written as\n\\begin{align}\n\\label{eq:defI}\nI_v:= \\lbrace (\\psi_0, \\varphi_0) \\, \\vert \\,  \\text{(i), (ii) are satisfied} \\rbrace.\n\\end{align}\nWe will show below that it is non-empty for small enough $v$. \n\n\\subsection{Expansion of the energy} \\label{sec:thm} In order to compute the effective mass of the polaron, we now minimize the energy $\\mathcal{G}$ over the set $I_v$.\nTo this end, we define the energy\n\\begin{align}\\label{def:Ev}\nE(v) :=\\inf_{(\\psi_0,\\varphi_0)\\in I_v} \\mathcal{G}(\\psi_0,\\varphi_0).\n\\end{align}\nThe following theorem gives an expansion of $E(v)$ for sufficiently small velocities $v$.  Its proof will be given in Section \\ref{sec:proof-thm}.\n\n\\begin{theo}\n\t\\label{thm:freephase}\n\tAs $v \\rightarrow 0$  we have \n\t\\begin{equation}\n\t\\label{eq:thm}\n\tE(v)=e_{\\rm P}+v^2 \\left(  \\frac{1}{4}+ \\frac{\\alpha^4}{3} \\| \\nabla  \\varphi_{\\rm P} \\|_2^2 \\right) + O(v^3).\t\n\t\\end{equation}\n\\end{theo}\n\nSince the kinetic energy of a particle of mass $m$ and velocity $v$ equals $m v^2\/2$, \\eqref{eq:thm} identifies the effective mass of the system as\n\\begin{align}\n\\label{eq:OurEffMass}\nm_{\\text{eff}}= \\lim_{v \\rightarrow 0} \\frac{E(v) - e_{\\rm P}}{v^2\/2} = \\frac 1 2 +\\frac {2\\alpha^4} 3 \\| \\nabla \\varphi_{\\rm P} \\|_2^2  .\n\\end{align}\nThe first term $1\/2$ is simply the bare mass of the election, while the second term $\\frac {2\\alpha^4} 3 \\| \\nabla \\varphi_{\\rm P} \\|_2^2$ corresponds to the additional mass acquired through the interaction with the  phonon field, and agrees with the prediction in \\cite{landau1948effective}.\n\n\n\\begin{rem}[Traveling waves]\n\\label{rem:TW}\nThe heuristic computations contained in the physics literature concerning $m_{\\text{eff}}$ \\cite{alexandrov2010advances,landau1948effective}\nall rely, in one way or another, on the existence of traveling wave solutions of the LP equations of velocity $v$ (at least for sufficiently small velocity), i.e. solutions with initial data $(\\psi_v,\\varphi_v)$ such that \n\\begin{align}\n(\\psi_t(x),\\varphi_t(x))=(e^{-ie_{v}t}\\psi_v(x-vt),\\varphi_v(x-vt))\n\\end{align}\nfor suitable $e_v \\in \\mathbb{R}$. \nSuch solutions would allow to define the energy of the system at velocity $v$ as $E^{\\text{TW}}(v) = \\mathcal{G}( \\psi_v, \\varphi_v)$, and a perturbative calculation  (discussed in Section \\ref{sec:TW} below) yields indeed\n\\begin{align}\n\\label{eq:mass-predTW}\n\\lim_{v \\rightarrow 0} \\frac{E^{\\text{TW}}(v) - e_{\\rm P}}{v^2\/2} = \\frac 1 2 +\\frac {2\\alpha^4} 3 \\| \\nabla \\varphi_{\\rm P} \\|^2, \n\\end{align} \nin agreement with \\eqref{eq:OurEffMass}. \n Unfortunately, this approach turns out to be only formal, and we conjecture traveling wave solutions to not exist for any $\\alpha>0$, $v>0$.\n\\end{rem} \n\n\n\\iffalse\n\\begin{rem}[The Fr\\\"ohlich polaron model]\n\t\nWe remark that the true dynamics of the polaron is given by its quantum mechanical description based on the Fr\\\"ohlich Hamiltonian, denoted in the following by $H^F_{\\alpha}$, which models the polarization field as quantized phonons (we refer to \\cite{seiringer2020polaron} and references therein for the definition of $H^F_{\\alpha}$ and a comprehensive discussion of its properties and the model). It features a dimensionless coupling constant $\\alpha$, modeling the strength of the interaction between the electron and the crystal.  In the strong coupling regime $\\alpha\\gg 1$,  the quantum field displays classical behavior.  In particular, it was shown (see \\cite{FS,FG,Griesemer,LMRSS,LRSS,M}) that the LP equations provide an effective description of its dynamics for large $\\alpha$. Therefore, we keep this model in mind as a motivation for our work, even if the analysis that we carry out on the classical level says little about the full quantum problem. \n\nNote that, in this context, very little is known about the effective mass of the system $m^F_{\\text{eff}}$. One approach to define it, relies on deriving a momentum-energy relation of the system (rather than the velocity-energy relation we opt for at the classical level). This approach exploits the translational invariance of the Hamiltonian, which allows to write it in its momentum fiber-integral decomposition as $H^F_\\alpha = \\int_{\\mathbb{R}^3}^\\oplus H^P_\\alpha \\; dP$, arriving at the relation $E(P) = \\inf \\text{spec}H^P_\\alpha $, which implicitly defines $m_{\\text{eff}}^{F}$. The only rigorous result concerning this definition is contained in \\cite{LS}, where it is shown that $m_{\\text{eff}}^{F}\\to \\infty$ as $\\alpha\\to \\infty$. \n\\end{rem}\n\\fi\n\n\n\n\\begin{rem} \\label{rem:AlternatePos}\nIn Section \\ref{sec:vel}, we used the standard approach from quantum mechanics to define the electron's position \\eqref{eq:defXel} and velocity  \\eqref{vel}. We could, instead, use also for the electron a similar approach to the one we use for the phonon field (i.e. \\eqref{eq:defXph}) through the projection onto the manifold of minimizers $\\mathcal{M}_{\\mathcal{E}}$.  A natural question is whether one  obtains the same effective mass using this different notion of position. In Section \\ref{sec:diff-vel}, we show that, in fact, this alternate definition yields a different effective mass equal to \n\\begin{align}\n\\widetilde{m}_{\\text{eff}} = \\frac{2 \\| \\nabla \\psi_{\\rm P} \\|_2^4}{3 \\|\\nabla \\varphi_{\\rm P} \\|_2^2 } +\\frac{2\\alpha^4}{3} \\| \\nabla \\varphi_{\\rm P} \\|_2^2  .\n\\end{align}\nThis coincides with \\eqref{eq:OurEffMass} and \\eqref{eq:mass-predTW} for large $\\alpha$ (hence still confirming the prediction in \\cite{landau1948effective}), but differs  in the $O(1)$ term. In fact, as we discuss in Section \\ref{sec:diff-vel}, one has $\\widetilde{m}_{\\text{eff}} < m_{\\text{eff}}$. \n\\end{rem}\n\n\\section{Proof of Theorem~\\ref{thm:freephase}}\\label{sec:proof-thm}\n\n\n\n Let us denote $\\delta_1=\\psi_0-\\psi_{\\rm P}$ and $\\delta_2=\\varphi_0-\\varphi_{\\rm P}$. Expanding $\\mathcal{G}$ in \\eqref{def:G} and using that $\\varphi_{\\rm P}=-\\sigma_{\\psi_{\\rm P}}$ we find\n\\begin{align}\n\\mathcal{G}(\\psi_0,\\varphi_0) & =\\mathcal{G}(\\psi_{\\rm P}+\\delta_1,\\varphi_{\\rm P}+\\delta_2) \\nonumber\\\\\n& =e_{\\rm P}+2\\bra{\\psi_{\\rm P}}h_{\\varphi_{\\rm P}}\\ket{\\Re \\delta_1}\\nonumber\\\\\n&\\quad +\\expval{h_{\\varphi_{\\rm P}}}{\\delta_1}+2\\bra{\\Re \\delta_1}V_{\\delta_2}\\ket{\\psi_{\\rm P}}+\\|\\delta_2\\|_2^2 +\\expval{V_{\\delta_2}}{\\delta_1}. \\label{eq:expansion-G}\n\\end{align}\nSince $\\psi_0$ is normalized, we have\n\\begin{align}\\label{pdd}\n1=\\|\\psi_0\\|_2^2=\\|\\psi_{\\rm P}+\\delta_1\\|_2^2=1+\\|\\delta_1\\|_2^2+2\\bra{\\psi_{\\rm P}}\\ket{\\Re \\delta_1}\\iff 2\\bra{\\psi_{\\rm P}}\\ket{\\Re \\delta_1}=-\\|\\delta_1\\|_2^2.\n\\end{align}\nHence\n\\begin{align}\n2\\bra{\\psi_{\\rm P}}h_{\\varphi_{\\rm P}}\\ket{\\Re \\delta_1}=2\\mu_{\\rm P}\\bra{\\psi_{\\rm P}}\\ket{\\Re \\delta_1}=-\\mu_{\\rm P}\\|\\delta_1\\|_2^2,\n\\end{align}\nand  using $\\| V_{\\delta_2} \\delta_1 \\|_2 \\leq C \\| \\delta_2\\|_2 \\; \\|\\delta_1 \\|_{H^1}$ (see, e.g., \\cite[Lemma III.2]{LRSS}) we arrive at\n\\begin{align}\n\\mathcal{G}(\\psi_0,\\varphi_0)=e_{\\rm P}+\\expval{H_{\\rm P}}{\\delta_1}+2\\bra{\\Re \\delta_1}V_{\\delta_2}\\ket{\\psi_{\\rm P}}+\\|\\delta_2\\|_2^2+O(\\|\\delta_2\\|_2 \\|\\delta_1\\|_{H^1}^2).\n\\end{align}\nBy completing the square, we have\n\\begin{align}\n&\\|\\Re\\delta_2\\|_2^2+2\\bra{\\Re \\delta_1}V_{\\delta_2}\\ket{\\psi_{\\rm P}}\\nonumber\\\\\n&=\\|\\Re \\delta_2 +2 (2\\pi)^{3\/2}(-\\Delta)^{-1\/2} (\\psi_{\\rm P}\\Re \\delta_1)\\|_2^2-4 (2\\pi)^3\\expval{\\psi_{\\rm P}(-\\Delta)^{-1} \\psi_{\\rm P}}{\\Re \\delta_1}\n\\end{align}\nand therefore\n\\begin{align}\\label{36}\n\\mathcal{G}(\\psi_0,\\varphi_0) &= e_{\\rm P}+\\expval{H_{\\rm P}}{\\Im\\psi_0}+\\|\\Im \\varphi_0\\|_2^2 \\nonumber \\\\ & \\quad +\\|\\Re \\delta_2 +2(2\\pi)^{3\/2}(-\\Delta)^{-1\/2} (\\psi_{\\rm P}\\Re \\delta_1)\\|_2^2\\nonumber\\\\\n&\\quad +\\expval{H_{\\rm P}-4X_{\\rm P}}{\\Re \\delta_1}+O(\\|\\delta_2\\|_2 \\|\\delta_1\\|^2_{H^1}) ,\n\\end{align} \nwhere $X_{\\rm P}$ is the operator with integral kernel $X_{\\rm P}(x,y):= (2\\pi)^3\\psi_{\\rm P}(x)(-\\Delta)^{-1}(x,y) \\psi_{\\rm P}(y)$. Since $X_{\\rm P}$ is bounded, and $\\| P_{\\psi_{\\rm P}} \\Re\\delta_1\\| = \\|\\delta_1\\|_2^2\/2$ by \\eqref{pdd} (with $P_{\\psi_{\\rm P}} = |\\psi_{\\rm P}\\rangle\\langle \\psi_{\\rm P}|$ the rank one projection onto $\\psi_{\\rm P}$), \nwe also have \n\\begin{align}\n\\mathcal{G}(\\psi_0,\\varphi_0)& =e_{\\rm P}+\\expval{H_{\\rm P}}{\\Im\\psi_0}+\\|\\Im\\varphi_0\\|_2^2 \\nonumber \\\\ & \\quad +\\|\\Re \\delta_2 +2(2\\pi)^{3\/2} (-\\Delta)^{-1\/2} (\\psi_{\\rm P}\\Re \\delta_1)\\|_2^2\\nonumber\\\\\n &\\quad+\\expval{Q(H_{\\rm P}-4X)Q}{\\Re \\delta_1}+O(\\|\\delta_2\\|_2\\|\\delta_1\\|^2_{H^1})+O(\\|\\delta_1\\|_{L^2}^3) , \\label{eq:expressionG}\n\\end{align}\nwhere $Q=1\\!\\!1-P_{\\psi_{\\rm P}}$. \n\n\n\n\\textbf{Upper Bound:} For sufficiently small $v$,  we use as a trial state\n\\begin{align}\n\\label{eq:trialstate}\n(\\bar\\psi_0,\\bar\\varphi_0)=\\left( f_v \\psi_{\\rm P}+i g_v H_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P} , \\; \\varphi_{\\rm P}+iv\\alpha^2 \\partial_1 \\varphi_{\\rm P}\\right)\n\\end{align}\nwith $f_v, g_v >0$ given by \n\\begin{align}\n\\label{eq:defFG}\nf_v^2 := \\frac{2 v^2 \\|H_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}\\|_2^2 }{1 - \\sqrt{1-4v^2 \\|H_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}\\|_2^2 } }, \\;  g_v^2 := \\frac{ 1 - \\sqrt{1-4v^2 \\|H_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}\\|_2^2 } }{2 \\|H_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}\\|_2^2 }  .\n\\end{align}\nNote that $\\partial_1 \\psi_{\\rm P}$ is orthogonal to $\\psi_{\\rm P}$, hence $H_{\\rm P}^{-1} \\partial_1 \\psi_{\\rm P}$ is well-defined. \nWe begin by showing that \\eqref{eq:trialstate} is an element of the set of admissible initial data $I_v$ in \\eqref{eq:defI}. To prove that $(\\bar\\psi_0,\\bar\\varphi_0)$ satisfies (i), we only need to check that $\\bar\\psi_0$ is normalized (which follows easily from \\eqref{eq:defFG}) as the  condition \\eqref{eq:IDuniqueproj}  will follow a posteriori from the energy bound we shall derive. We now proceed to show that $(\\bar\\psi_0,\\bar\\varphi_0)$ satisfies (ii). For the electron, using that  $H_{\\rm P}^{-1} \\partial_j \\psi_{\\rm P}= -x_j \\psi_{\\rm P} \/2  $ (which can be checked by applying $H_P$ and using that $[H_P,x_1]=-2 \\partial_1$) and  consequently that \n\\begin{align}\n\\label{eq:sandwich}\n\\bra{\\partial_i \\psi_{\\rm P}}H_{\\rm P}^{-1}\\ket{\\partial_j \\psi_{\\rm P}}=\\delta_{ij}\/ 4 \n\\end{align} \nsince $\\psi_{\\rm P}$ is radial, \nwe can conclude that\n\\begin{align}\n\\label{eq:trial-v}\n-2\\expval{i \\partial_j }{\\bar \\psi_0}=4 f_v g_v  \\langle H_{\\rm P}^{-1} \\partial_1 \\psi_{\\rm P} \\vert \\partial_j \\psi_{\\rm P} \\rangle  =v \\delta_{j1}  ,\n\\end{align}\ni.e., that $\\text{V}_{\\text{el}}(0)=v(1,0,0) $, as required.\n\nFor the phonons, we first note that $\\text{X}_{\\text{ph}}(0)=0$, since $\\Re \\bar{\\varphi}_0 = \\varphi_{\\rm P}$. Next,  we derive a relation for the velocity of the phonons $\\text{V}_{\\text{ph}}(t) = \\dot{z}(t)$ in terms of their position $\\text{X}_{\\text{ph}} (t) =z(t)$ for general time $t$.  Since \n\\begin{align}\n\\label{eq:defZ}\n\\min_z\\|\\varphi_t -\\varphi_{\\rm P}^z\\|_2^2=\\|\\varphi_t-\\varphi_{\\rm P}^{z(t)}\\|_2^2,\n\\end{align}\nthe position $z(t)$ necessarily  has to satisfy \n\\begin{align}\n\\label{eq:orthPhon}\n\\Re \\bra{\\varphi_t}\\ket{(u\\cdot \\nabla) \\varphi_{\\rm P}^{z(t)}}=0 \\quad \\text{for all} \\quad u \\in \\mathbb{S}^2 \\iff \\Re \\varphi_t \\perp \\spn\\{\\nabla \\varphi_{\\rm P}^{z(t)}\\}.\n\\end{align}\nDifferentiating \\eqref{eq:orthPhon} w.r.t. time, using \\eqref{eq:LP} and evaluating the resulting expression at $t=0$,  we arrive at \n\\begin{align}\n\\label{eq:defVph2}\n0=&\\Re\\bra{-i\\alpha^{-2} (u\\cdot \\nabla)(\\bar\\varphi_0+\\sigma_{\\bar\\psi_0})}\\ket{\\varphi_{\\rm P}}-\\Re\\bra{(\\dot{z}(0)\\cdot\\nabla)\\bar\\varphi_0}\\ket{(u\\cdot \\nabla)\\varphi_{\\rm P}}\\nonumber\\\\\n=&\\bra{-\\alpha^{-2} \\Im\\bar\\varphi_0}\\ket{(u\\cdot \\nabla)\\varphi_{\\rm P}}-\\bra{(\\dot{z}(0)\\cdot\\nabla)\\Re \\bar\\varphi_0}\\ket{(u\\cdot \\nabla)\\varphi_{\\rm P}}\\nonumber\\\\\n=&-\\bra{v\\partial_1 \\varphi_{\\rm P}}\\ket{(u\\cdot \\nabla)\\varphi_{\\rm P}}-\\bra{(\\dot{z}(0)\\cdot\\nabla)\\varphi_{\\rm P}}\\ket{(u\\cdot \\nabla)\\varphi_{\\rm P}},\n\\end{align}\nwhich the velocity $\\dot{z} (0)$ has to satisfy for all $u\\in \\mathbb{S}^2$, given its position $\\text{X}_{\\text{ph}}(0)=z (0)=0$.  By invertibility of the coefficient matrix,  \\eqref{eq:defVph2} has the unique solution $\\dot{z}(0)=v(1,0,0)$, and we indeed conclude that $\\text{V}_{\\text{ph}} (0) = v (1,0,0)$.\n\nWe now evaluate $\\mathcal{G}(\\bar\\psi_0,\\bar\\varphi_0)$. Since $f_v = 1 + O(v^2), g_v = v+ O(v^3)$, using \\eqref{eq:expressionG} and \\eqref{eq:sandwich} we find\n\\begin{align}\nE(v)\\leq \\mathcal{G}(\\bar\\psi_0,\\bar\\varphi_0)=e_{\\rm P}+v^2 \\left(  \\frac 1 4+\\alpha^4 \\| \\partial_1 \\varphi_{\\rm P} \\|_2^2 \\right) + O(v^3)\n\\end{align}\nverifying on the one hand \\eqref{eq:IDuniqueproj} for sufficiently small $v$, and on the other hand the r.h.s. of \\eqref{eq:thm} as an upper bound on $E(v)$ (using that $\\varphi_{\\rm P}$ is radial).\n\n\n\n\\textbf{Lower Bound:} We first observe that to derive a lower bound on $E(v)$ we can w.l.o.g. restrict to initial conditions $(\\psi_0,\\varphi_0)$  satisfying  additionally\n\\begin{align}\n\\label{eq:centeringI}\nP_{L^2}^{\\mathcal{M}_{\\mathcal{E}}}(\\psi_0)>0, \\\\\n\\label{eq:centeringII} \\text{X}_{\\text{ph}}(0)=0.\n\\end{align} \nThis simply follows from the invariance of $\\mathcal{G}$ under translations (of both $\\psi$ and $\\varphi$) and under changes of phase of $\\psi$.  \nMoreover, by the upper bound derived in the first step of this proof and the coercivity of $\\mathcal{E}$ and $\\mathcal{F}$ in \\eqref{eq:coercivity}, we conclude that it is sufficient to minimize over elements of $I_v$ such that $\\dist_{H^1}(\\psi_0, \\mathcal{M}_{\\mathcal{E}})=O(v)=\\dist_{L^2}(\\varphi_0,\\mathcal{M}_{\\mathcal{F}})$  for small $v$.  Since the $L^2$-projection of $\\varphi_0$ is $\\varphi_{\\rm P}$ by \\eqref{eq:centeringII}, it immediately follows that $\\|\\delta_2\\|_2=O(v)$. We now proceed to show that necessarily also $\\|\\delta_1\\|_{H^1}=O(v)$.\nLet $y',y\\in \\mathbb{R}^{3}$ and $\\theta \\in[0,2\\pi)$ be such that \n\\begin{align}\n\\label{eq:defTheta}\nP_{L^2}^{\\mathcal{M}_{\\mathcal{E}}}(\\psi_0)=\\psi_{\\rm P}^{y'}, \\quad P_{H^1}^{\\mathcal{M}_{\\mathcal{E}}}(\\psi_0)=e^{i\\theta}\\psi_{\\rm P}^y,\n\\end{align}\nwhere we recall that the $L^2$-projection is assumed to be positive by \\eqref{eq:centeringI}. Combining the upper bound derived in the first step with  \\cite[Eq.~(53)]{feliciangeli2021persistence}, we get\n\\begin{align}\n\\| \\varphi_0 - \\varphi_{\\rm P}^y \\|_2^2 \\leq C \\left( \\mathcal{G} (\\psi_0, \\varphi_0)  - e_{\\rm P} \\right)\\leq Cv^2.\n\\end{align}\nThere exist $\\delta, C_1, C_2>0$ such that  \n\\begin{align}\n\\| \\varphi_{\\rm P} - \\varphi_{\\rm P}^y \\|_2\\geq\\begin{cases}\nC_1\\vert y \\vert \\| \\nabla \\varphi_{\\rm P} \\|_2, & |y|\\leq \\delta\\\\\nC_2 & |y|>\\delta\n\\end{cases},\n\\end{align}\nand this allows to conclude that $|y| = O(v)$. In other words,  assuming  centering w.r.t. to translations in the phonon coordinate (i.e. \\eqref{eq:centeringII}) forces, at low energies, also the centering w.r.t. translations in the electron coordinate, at least approximately. At this point, it is also easy to verify that $\\theta=O(v)$ (and, as an aside, that $|y'|=O(v)$), since, by the upper bound derived in the first step and the coercivity of $\\mathcal{E}$, we have\n\\begin{align}\n\\|\\psi_{\\rm P}^{y'}-e^{i\\theta} \\psi_{\\rm P}^y\\|_2\\leq \\|\\psi_{\\rm P}^{y'}-\\psi_0\\|_{2}+\\|e^{i\\theta} \\psi_{\\rm P}^y-\\psi_0\\|_{2}=O(v).\n\\end{align}\nIn particular, we conclude that\n\\begin{align}\\label{aa}\n\\| \\delta_1 \\|_{H^1} \\leq \\| e^{i\\theta}\\psi_{\\rm P}^{y} - \\psi_0 \\|_{H^1}+ \\| \\psi_{\\rm P} - e^{i\\theta}\\psi_{\\rm P}^{y} \\|_{H^1} = O(v).\n\\end{align} \n\nUsing again \\eqref{eq:expressionG} and that $Q(H_{\\rm P}-4X_{\\rm P})Q\\geq 0$, we conclude that for any $(\\psi_0,\\varphi_0)\\in I_v$ satisfying \n\\eqref{eq:centeringI} and \\eqref{eq:centeringII}, as well as $\\mathcal{G}(\\psi_0,\\varphi_0) \\leq e_{\\rm P} + O(v^2)$, we have\n\\begin{align}\n\\label{eq:firstLB}\n\\mathcal{G}(\\psi_0,\\varphi_0)\\geq e_{\\rm P}+\\expval{H_{\\rm P}}{\\Im\\psi_0}+\\|\\Im\\varphi_0\\|_2^2+O(v^3).\n\\end{align}\nBy arguing as in \\eqref{eq:defVph2}, we see that the conditions $\\text{X}_{\\text{ph}}(0)=0$ and $\\text{V}_{\\text{ph}}(0)=v$ imply that \n\\begin{align}\nP_{\\nabla \\varphi_{\\rm P}} (\\Im \\varphi_0 + v\\alpha^2 \\partial_1 \\Re\\varphi_0)=0 ,\n\\end{align}\nwhere $P_{\\nabla \\varphi_{\\rm P}}$ denotes the projection onto the span of $\\partial_j \\varphi_{\\rm P}$, $1\\leq j\\leq 3$. \nSince $P_{\\nabla \\varphi_{\\rm P}} \\partial_1$ is a bounded operator, and $\\|\\delta_2\\|_2=O(v)$, we find \n\\begin{equation}\\label{325}\n\\|\\Im\\varphi_0\\|_2^2 \\geq \\| P_{\\nabla \\varphi_{\\rm P}} \\Im\\varphi_0\\|_2^2 = v^2 \\alpha^4 \\| \\partial_1 \\varphi_{\\rm P} + P_{\\nabla \\varphi_{\\rm P}} \\partial_1 \\Re \\delta_2 \\|_2^2 \\geq v^2 \\alpha^4 \\| \\partial_1 \\varphi_{\\rm P}\\|_2^2 - O(v^3) .\n\\end{equation}\nWe are left with giving a lower bound on $\\expval{H_{\\rm P}}{\\Im\\psi_0}$, \nunder the condition that   \n\\begin{align}\n2\\expval{-i\\nabla}{\\psi_0}=4\\bra{\\Im \\psi_0}\\ket{\\nabla\\Re\\psi_0}=v(1,0,0).\n\\end{align}\nWe already argued in \\eqref{aa} that  $\\|\\psi_0-\\psi_{\\rm P}\\|_{H^1}=O(v)$, and therefore\n\\begin{align}\n\\label{eq:constranit-Im}\n4\\bra{\\Im \\psi_0}\\ket{\\nabla\\psi_{\\rm P}}=v(1,0,0)+O(v^2).\n\\end{align}\nCompleting the square, we find \n\\begin{align}\n\\label{eq:completing-square}\n\\expval{H_{\\rm P}}{\\Im\\psi_0} & = \\expval{H_{\\rm P}^{-1}}{H_{\\rm P}\\Im\\psi_0 - v \\partial_1 \\psi_{\\rm P}} \\notag \\\\\n& \\quad + 2v \\bra{\\Im\\psi_0}\\ket{\\partial_1  \\psi_{\\rm P}} - v^2 \\expval{H_{\\rm P}^{-1}}{ \\partial_1  \\psi_{\\rm P}} \\notag \\\\\n&  \\geq  2v \\bra{\\Im\\psi_0}\\ket{\\partial_1  \\psi_{\\rm P}} - v^2 \\expval{H_{\\rm P}^{-1}}{ \\partial_1 \\psi_{\\rm P}} .\n\\end{align}\nFrom the constraint \\eqref{eq:constranit-Im} and \\eqref{eq:sandwich}, it thus follows that\n\\begin{align}\\label{329}\n\\expval{H_{\\rm P}}{\\Im\\psi_0} &\\geq  \\frac{v^2}{4}  + O(v^3).\n\\end{align}\nBy combining \\eqref{eq:firstLB}, \\eqref{325} and \\eqref{329}, we arrive at the final lower bound \n\\begin{align}\nE(v) \\geq e_{\\rm P}+ v^2\\left(\\frac 1 4 + \\alpha^4 \\|\\partial_1\\varphi_{\\rm P}\\|_2^2\\right)+O(v^3) .\n\\end{align}\nAgain,  since $\\varphi_{\\rm P}$ is radial, this is of the desired form, and hence the proof is complete.\n\\hfill\\qed\n\n\n\\section{Further Considerations}\n\\label{sec:approaches}\n\nIn this section, we carry out the details related to Remarks \\ref{rem:TW} and \\ref{rem:AlternatePos}. \n\n\\subsection{Effective mass through traveling wave solutions}\n\\label{sec:TW}\n\nA possible way of formalizing the derivation of the effective mass in \\cite{alexandrov2010advances,landau1948effective}  relies on traveling wave solutions of the Landau--Pekar equations.  \nA traveling wave of velocity $v\\in \\mathbb{R}^{3}$\nis a solution $(\\psi_t, \\varphi_t)$ of \\eqref{eq:LP}\nof the form \n\t\\begin{align}\n\t\\label{eq:TWdef}\n\t(\\psi_t, \\varphi_t) = (e^{-i e_v t} \\psi_v^{\\text{TW}} ( \\cdot - vt ), \\varphi_v^{\\text{TW}} ( \\cdot - vt ) ) \n \\end{align} \n for all $t\\in \\mathbb{R}$, with $e_v\\in \\mathbb{R}$ defining a suitable phase factor.  As before,  by rotation invariance we can restrict our attention to velocities of the form $v (1,0,0)$ with $v\\in \\mathbb{R}$ in the following.\n \nNote that in the case $\\alpha =0$, where $\\varphi_t = - \\sigma_{\\psi_t}$ for all $t\\in\\mathbb{R}$, the LP equations simplify to a non-linear Schr\\\"odinger equation (also known as Choquard equation). In this case,  a traveling wave is given by $\\psi_v^{\\text{TW}} (x)= e^{i x_1 v \/2 } \\psi_{\\rm P} ( x)$ with $e_v= \\mu_{\\rm P} + \\frac{v^2}{4} $, and its energy can be computed to be \n\\begin{align}\n\\mathcal{G}\\left( \\psi_v^{\\text{TW}}, -\\sigma_{\\psi_v^{\\text{TW}}} \\right) = e_{\\rm P} + \\frac{v^2}{4}, \\label{eq:Ev-expansion-alpha-0}\n\\end{align}\nyielding an effective mass $m=1\/2$ at $\\alpha=0$. For the case $\\alpha > 0$, on the other hand, we conjecture that there are no traveling wave solutions of the form \\eqref{eq:TWdef}.  \n \n \\begin{Conj}\n For $\\alpha>0$, there are no solutions to the LP equations \\eqref{eq:LP} of the form \\eqref{eq:TWdef} with $v\\neq 0$.\n \\end{Conj}\n \n If one {\\em assumes} the existence of traveling wave solutions, at least for small $v$, one can predict an effective mass that agrees with our formula \\eqref{eq:OurEffMass}, as we shall now demonstrate.\n\nFrom the LP equations \\eqref{eq:LP} one easily sees that  a traveling wave solution needs to satisfy \n\\begin{equation}\\label{eq:LPTW}\n\\begin{aligned}\n-i v \\partial_1  \\psi_v^{\\text{TW}}  & =\\left(  h_{ \\varphi_v^{\\text{TW}}} + e_v \\right)   \\psi_v^{\\text{TW}}  \\\\\n-i \\alpha^2 v \\partial_1 \\varphi_v^{\\text{TW}}  & = \\varphi_v^{\\text{TW}} + \\sigma_{\\psi_v^{\\text{TW}}} .\n\\end{aligned}\n\\end{equation}\nWe shall denote by $E^{\\text{TW}}(v)$ the  energy of the traveling wave as a function of the velocity $v\\in \\mathbb{R}$, i.e. \n\\begin{align}\n\\label{eq:Evdef}\nE^{\\text{TW}}(v):=\\mathcal{G}(\\psi_v^{\\text{TW}},\\varphi_v^{\\text{TW}}) .\n\\end{align}\nIn the following, we assume that $e_v=\\mu_{\\rm P}+O(v^2)$ and that the traveling wave is of the form \n\\begin{align}\n\\label{eq:ansatz_TW}\n(\\psi_v^{\\text{TW}}, \\varphi_v^{\\text{TW}}) =   \\left( \\frac{\\psi_{\\rm P} + v \\xi_v}{\\| \\psi_{\\rm P} + v \\xi_v \\|_2}, \\varphi_{\\rm P} + v \\eta_v \\right) ,\n\\end{align}  \nwith both $\\xi_v$ and $\\eta_v$ bounded in $v$ and converging to some $(\\xi,\\eta)$ as $v\\to 0$.  In other words, we assume that the traveling waves have a suitable differentiability in $v$, at least for small $v$, and converge to the standing wave solution $(e^{-i \\mu_{\\rm P} t}\\psi_{\\rm P}, \\varphi_{\\rm P})$ for $v=0$. W.l.o.g. we may also assume that $\\xi_v$ is orthogonal to $\\psi_{\\rm P}$. \n\nWe can then use that \n\\begin{align}\n\\label{eq:exp_norm}\n\\frac{1}{\\| \\psi_{\\rm P} + v \\xi_v \\|_2^2} &= 1- v^2  \\frac{ \\| \\xi_v \\|_2^2}{\\| \\psi_{\\rm P} + v \\xi_v\\|_2^2}\n= 1 - v^2 \\| \\xi \\|_2^2 + o (v^2) \n\\end{align}\nin order to linearize the traveling wave equations \\eqref{eq:TWdef}, obtaining that $(\\xi, \\eta)$ solves\n\\begin{align}\n\\label{eq:xieta}\n\\begin{pmatrix}\ni \\partial_1 \\psi_{\\rm P} \\\\ i \\alpha^2 \\partial_1 \\varphi_{\\rm P} \n\\end{pmatrix}= \n\\begin{pmatrix}\nH_{\\rm P} & 2(2\\pi)^{3\/2}\\psi_{\\rm P}(-\\Delta)^{-1\/2} \\Re\\\\ 2(2\\pi)^{3\/2}(-\\Delta)^{-1\/2} \\psi_{\\rm P}   \\Re & 1\n\\end{pmatrix}\n\\begin{pmatrix}\n\\xi \\\\ \\eta\n\\end{pmatrix},\n\\end{align}\nwhere $H_{\\rm P}=h_{\\varphi_{\\rm P}}-\\mu_{\\rm P}$, as defined in \\eqref{eq:def-HP}. Splitting into real and imaginary parts, we equivalently find \n\\begin{align}\n&H_{\\rm P} \\Im \\xi =  \\partial_1 \\psi_{\\rm P} \\label{eq:ne_Imxi}\\\\\n&\\Im \\eta =\\alpha^2 \\partial_1 \\varphi_{\\rm P} \\label{eq:ne_Imeta} \\\\\n&H_{\\rm P} \\Re \\xi + 2 (2 \\pi)^{3\/2}  \\psi_{\\rm P} \\left( - \\Delta \\right)^{-1\/2} \\Re \\eta  = 0 \\label{eq:ne_Rexi}\\\\\n&2 (2 \\pi)^{3\/2} \\left( -\\Delta \\right)^{-1\/2} \\psi_{\\rm P} \\Re \\xi + \\Re \\eta  =0. \\label{eq:ne_Reeta}\n\\end{align}\nCombining \\eqref{eq:ne_Rexi} and \\eqref{eq:ne_Reeta} gives $(H_{\\rm P} - 4 X_{\\rm P}) \\Re \\xi =0$, with $X_{\\rm P}$ defined after \\eqref{36}. It was shown in  \\cite{lenzmann2009uniqueness} that the kernel of $H_{\\rm P} - 4 X_{\\rm P}$ is spanned by $\\nabla \\psi_{\\rm P}$, hence $\\Re \\xi\\in \\spn \\{\\nabla \\psi_{\\rm P}\\}$. Eq, \n\\eqref{eq:ne_Reeta} then implies that $\\Re \\eta\\in \\spn \\{ \\nabla \\varphi_{\\rm P}\\}$. \n\nUsing these equations and \\eqref{eq:exp_norm} in  the expansion \\eqref{eq:expressionG}, it is  straightforward to obtain\n\\begin{align}\n\\label{eq:EvexpansionTW}\nE^{\\text{TW}}(v) = e_{\\rm P} + v^2 \\left( \\frac 14 + \\alpha^4 \\| \\partial_1 \\varphi_{\\rm P} \\|_2^2  \\right) + o( v^2), \n\\end{align}\nwhich agrees with \\eqref{eq:Ev-expansion-alpha-0} for the case $\\alpha =0$, and also with \\eqref{eq:thm} to leading order in $v$. In particular, \\eqref{eq:mass-predTW} holds.\n\n\\subsection{Effective mass with alternative definition for the electron's velocity} \n\\label{sec:diff-vel}\nIn this Section, we discuss a different approach to the definition of the effective mass. This approach is based on an alternative way of defining the electron's position and velocity. While in Section \\ref{sec:vel} we use the standard definition from quantum mechanics,  here we use a definition similar to the one of the phonons' position and velocity \\eqref{eq:defXph}.  For this purpose, we recall Remark \\ref{rem:E} and that $\\delta^*$ has been chosen such that the condition $\\mathcal{E} (\\psi_0) \\leq \\mathcal{G}( \\psi_0, \\varphi_0) \\leq e_{\\rm P} + \\delta^*$ ensures that for all times there exists a unique $L^2$-projection $e^{i\\theta(t)}\\psi_{\\rm P}^{y(t)}$ of  $\\psi_t$ onto the manifold $\\mathcal{M}_\\mathcal{E}$.  Then, we define the electron's position and velocity by\n\\begin{align}\n\\label{eq:def-tildeX}\n\\widetilde{\\text{X}}_{\\text{el}}(t) = y(t), \\quad \\widetilde{\\text{V}}_{\\text{el}}(t) = \\dot{y}(t).\n\\end{align}\nSimilarly to the conditions (i) and (ii) in Section \\ref{sec:vel}, we define the set of admissible initial data as  \n\\begin{align}\n\\widetilde{I}_v = \\lbrace (\\psi_0, \\varphi_0) \\, \\vert \\,  \\text{(i),(ii') are satisfied} \\rbrace\n\\end{align}\nwhere \n\\begin{itemize}\n\t\\item[(ii')] $\\widetilde{\\text{V}}_{\\text{el}}(t) =\\text{V}_{\\text{ph}}(0)=v (1,0,0)$. \n\\end{itemize}\n\nNote that we are leaving the parameter $\\dot{\\theta}(0)$ free, which in this case is also relevant. In other words, we have \n\\begin{align}\n\\widetilde{I}_v = \\cup_{\\kappa\\in \\mathbb{R}} \\widetilde{I}_{v,\\kappa},\n\\end{align} \nwhere\n\\begin{align}\n\\widetilde{I}_{v,\\kappa}=\\lbrace (\\psi_0, \\varphi_0) \\, \\vert \\,  \\text{(i),(ii') are satisfied and}\\; \\dot{\\theta}(0)=\\kappa \\rbrace.\n\\end{align}\nMinimizing now the energy over all states of the set $\\widetilde{I}_v$\n\\begin{align}\n\\widetilde{E}(v):=\\inf_{(\\psi_0,\\varphi_0)\\in \\widetilde{I}_v} \\mathcal{G}(\\psi_0,\\varphi_0),\n\\end{align}\nleads to an energy expansion in $v$ that differs from the one of Theorem \\ref{thm:freephase} in its second order. \n\n\\begin{prop}\n\t\\label{rmk:EV}\n\tAs $v \\rightarrow 0$,  we have \n\t\\begin{align}\n\t\\label{eq:Evtilde}\n\t\\widetilde{E}(v) =e_{\\rm P}+v^2 \\left(  \\frac{\\| \\nabla \\psi_{\\rm P} \\|_2^4}{3 \\|\\nabla \\varphi_{\\rm P} \\|_2^2}  + \\frac{\\alpha^4}{3} \\| \\nabla \\varphi_{\\rm P} \\|_2^2  \\right) + O(v^3).\t\n\t\\end{align}\n\\end{prop}\n\nThe energy expansion in \\eqref{eq:Evtilde} leads to the effective mass \n\\begin{align}\n \\widetilde{m}_{\\text{eff}} = \\lim_{v \\rightarrow 0} \\frac{\\widetilde{E}(v) -e_{\\rm P}}{v^2\/2} = \\frac{2 \\| \\nabla \\psi_{\\rm P} \\|_2^4}{3 \\|\\nabla \\varphi_{\\rm P} \\|_2^2 } + \\frac{2\\alpha^4}{3} \\| \\nabla \\varphi_{\\rm P} \\|_2^2 \n\\end{align} \nwhich agrees with \\eqref{eq:OurEffMass} and \\eqref{eq:mass-predTW} in leading order for large $\\alpha$ only (and thus still confirms the Landau--Pekar prediction \\cite{landau1948effective}), but differs in the $O(1)$ term.  In fact, it turns out that $\\widetilde{m}_{\\text{eff}} < m_{\\text{eff}}$ with $m_{\\text{eff}}$ defined in \\eqref{eq:mass-predTW}.   \n\nThis follows from the observation that the trial state\n\\begin{align}\n\\label{eq:trialstate-mass}\n(\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0)=\\left(\\frac{f_v \\psi_{\\rm P}+ivH_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}}{\\|f_v \\psi_{\\rm P}+ivH_{\\rm P}^{-1}\\partial_1 \\psi_{\\rm P}\\|}, \\varphi_{\\rm P}+iv\\alpha^2 \\partial_1 \\varphi_{\\rm P}\\right)\n\\end{align}\nwith $f_v = \\frac{1}{2}\\left( 1 + \\sqrt{1 - v^2\/(4 \\| \\partial_1 \\psi_{\\rm P} \\|_2^2)} \\right)$ (which coincides up to terms of order $v^2$ with the trial state \\eqref{eq:trialstate}) is an element of $\\widetilde{I}_{v,\\bar \\kappa}$ for $\\bar \\kappa= - \\mu_{\\rm P}+4\\|\\partial_1 \\psi_{\\rm P}\\|_2^2(f_v-1)$ and is such that $\\mathcal{G} (\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0) = e_{\\rm P} + m_{\\text{eff}}\\;  v^2\/2 + O(v^3)$.  Thus,  $\\widetilde{m}_{\\text{eff}} \\leq m_{\\text{eff}}$ and equality holds if and only if equality (up to terms $o(v^2)$) holds in \\eqref{eq:completing-square2}. This is the case if and only if \n\\begin{align} \nQ_{\\psi_{\\rm P}} \\left( \\Im\\widetilde{\\psi}_0 - cv  \\partial_1 \\psi_{\\rm P}\\right) = o(v).\n\\end{align}\nUsing \\eqref{eq:trialstate-mass}, equality holds if and only if\n\\begin{align} \n 0 = H_{\\rm P}^{-1}\\partial_1 \\psi_P -  c  \\partial_1 \\psi_{\\rm P} = -\\left( x_1\/2 +  c  \\partial_1 \\right)  \\psi_{\\rm P} \n\\end{align}\ni.e. , recalling the radiality of $\\psi_{\\rm P}$,  if and only if $\\psi_{\\rm P}$ is a Gaussian with variance $\\sigma^2 = 1\/(2c)$. Since $\\psi_{\\rm P}$ satisfies the Euler--Lagrange equation\n\\begin{align}\nH_{\\rm P} \\psi_{\\rm P}=0 \\iff V_{\\varphi_{\\rm P}} \\psi_{\\rm P}=(-\\Delta+\\mu_P) \\psi_{\\rm P},\n\\end{align}\nit cannot be a Gaussian and therefore $\\tilde{m}_{\\text{eff}}<m_{\\text{eff}}$.\n\n\nWe present only a sketch of the proof of Proposition \\ref{rmk:EV}, since it uses very similar arguments as the proof of Theorem \\ref{thm:freephase}.  \n\n\\begin{proof}[Sketch of Proof of Proposition \\ref{rmk:EV}]  \\textbf{Upper bound}: We use the alternative trial state \n\\begin{align}\n(\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0)=\\left(\\frac{f_v \\psi_{\\rm P}+ivc\\partial_1 \\psi_{\\rm P}}{\\|f_v \\psi_{\\rm P}+ivc\\partial_1 \\psi_{\\rm P}\\|}, \\varphi_{\\rm P}+iv\\alpha^2 \\partial_1 \\varphi_{\\rm P}\\right),\n\\end{align}\nwith \n\\begin{align}\nf_v:=\\frac{1+\\sqrt{1+4c^2v^2\\|\\partial_1 \\psi_{\\rm P}\\|^2}}2, \\quad c:=\\frac{\\|\\partial_1 \\psi_{\\rm P}\\|^2}{\\|\\partial_1\\varphi_{\\rm P}\\|^2} . \n\\end{align}\nWith similar arguments as in the previous section, one can verify that $(\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0)\\in \\widetilde{ I}_v$, in particular  $(\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0)\\in \\widetilde{ I}_{v, \\kappa}$ with $\\kappa = -\\mu_{\\rm P}+\\frac{-1+\\sqrt{1+4 c^2v^2\\|\\partial_1 \\psi_{\\rm P}\\|^2}}{2c}$.\n\nNote that, similarly to \\eqref{eq:defVph2}, one can derive necessary conditions for the velocities  $\\dot{y}(0), \\dot{\\theta}(0)$ (using $\\widetilde{\\text{X}}_{\\text{el}} (0) =0, \\theta (0) =0$),  namely \n\\begin{align}\n\\label{eq:necessary1}\n&\\bra{[h_{\\Re\\widetilde{\\varphi}_0}+\\dot{\\theta}(0)] \\Im \\widetilde{\\psi}_0- \\dot{y}(0) \\cdot \\nabla  \\Re \\widetilde{\\psi}_0}\\ket{(u\\cdot \\nabla) \\psi_{\\rm P}}=0 \\quad \\text{for all} \\quad u \\in \\mathbb{S}^2 \n\\end{align}\nand \n\\begin{align}\n\\label{eq:necessary2}\n&\\bra{\\psi_{\\rm P}}\\ket{(h_{\\Re\\widetilde{ \\varphi}_0}+\\dot{\\theta}(0)) \\Re \\widetilde{\\psi}_0+\\dot{y}(0) \\cdot \\nabla \\Im \\widetilde{\\psi}_0}=0  .\n\\end{align}\nStraightforward computations then show that\n\\begin{align}\n\\widetilde{E}(v)\\leq \\mathcal{G}(\\widetilde{\\psi}_0,\\widetilde{\\varphi}_0)=e_{\\rm P}+v^2 \\left(\\frac{\\| \\partial_1 \\psi_{\\rm P} \\|_2^4}{\\|\\partial_1 \\varphi_{\\rm P} \\|_2^2} +  \\alpha^4 \\| \\partial_1 \\varphi_{\\rm P} \\|_2^2  \\right) + O(v^3).\n\\end{align}\n\n\\textbf{Lower bound}: We proceed similarly to the lower bound in the previous section.  First, we assume w.l.o.g.  that\n\\begin{align}\nP_{L^2}^{\\mathcal{M}_{\\mathcal{E}}} ( \\psi_0 ) = \\psi_{\\rm P}^{y(0)}, \\quad P_{L^2}^{\\mathcal{M}_{\\mathcal{F}}} ( \\varphi_0 ) = \\varphi_{\\rm P} ,\n\\end{align}\ni.e.,  centering with respect to translations and changes of phase.\nWe can then substitute the two conditions of (ii') and the conditions for $\\psi_{\\rm P}^{y(0)} $ (resp.  $\\varphi_{\\rm P}$)  to be the $L^2$-projection of $\\psi_0$ (resp. $\\varphi_0$) onto $\\mathcal{M}_\\mathcal{E}$ (resp.  $\\mathcal{M}_\\mathcal{F}$) with their analogue necessary conditions (whose computations proceed along the lines of \\eqref{eq:necessary1} and \\eqref{eq:necessary2}).  With this discussion, we are left with the task of minimizing $\\mathcal{G}$ over the set\n\\begin{align}\n\\widetilde{I}_v':=\\bigcup_{\\kappa \\in \\mathbb{R}} \\widetilde{I}'_{v,\\kappa}\n\\end{align}\nwith \n\\begin{align}\n\\label{eq:def-tildeI}\n\\widetilde{I}'_{v,\\kappa}:=\\Big\\{(\\psi_0,\\varphi_0) \\in\\widetilde{ I}^*\\,|\n&P_{\\nabla \\psi_{\\rm P}^{y(0)}}\\left[(h_{\\Re \\varphi_0}+\\kappa) \\Im \\psi_0 - v\\partial_1 \\Re \\psi_0\\right]=0 ,\\nonumber\\\\\n&P_{\\psi_{\\rm P}^{y(0)}}\\left[(h_{\\Re \\varphi_0}+\\kappa) \\Re \\psi_0+v\\partial_1 \\Im \\psi_0 \\right]=0,\\nonumber\\\\\n&P_{\\nabla \\varphi_{\\rm P}} (\\Im \\varphi_0-v\\alpha^2 \\partial_1 \\Re\\varphi_0)=0\\Big\\}.\n\\end{align}\nand \n\\begin{align}\n\\widetilde{I}^*:=\\left\\{(\\psi_0,\\varphi_0) \\,|\\, \\mathcal{G}(\\psi_0,\\varphi_0)\\leq e_{\\rm P}+\\delta^*,\\,\\,\\|\\psi_0\\|_2^2=1, \\, \\Re \\psi_0 \\perp \\nabla \\psi_{\\rm P}^{y(0)},\\, \\Re \\varphi_0 \\perp \\nabla \\varphi_{\\rm P}\\right\\}.\n\\end{align}\n\n\nAs in the previous section, one can argue by coercivity of $\\mathcal{E}$ and $\\mathcal{F}$ and the upper bound that it is possible to restrict to initial conditions such that $ \\| \\delta_2 \\|_2,  \\| \\delta_1 \\|_{H^1}, y(0) $ are all $O(v)$.  Moreover, the second constraint of the r.h.s.  of \\eqref{eq:def-tildeI} shows that $\\kappa = -\\mu_{\\rm P} + O(v)$. Thus, we are left with minimizing $\\mathcal{G}$ over the set \n\\begin{align}\n\\label{eq:tilde-I''}\n\\widetilde{I}''_v:=\\widetilde{I}_v'\\cap \\left\\{\\kappa+\\mu_{\\rm P} =O(v),\\, \\|\\delta_1\\|_{H^1}=O(v),\\,\\|\\delta_2\\|_2=O(v)\\right\\}.\n\\end{align}\nThe lower bound is proven in the same way as before.  But instead of the constraint \\eqref{eq:constranit-Im}, this time we need to minimize w.r.t. \n\\begin{align}\n\\label{eq:constraint7}\nP_{\\nabla \\psi_{\\rm P}^{y(0)}}\\left[(h_{\\Re \\varphi_0}+\\kappa) \\Im \\psi_0 - v\\partial_1 \\Re \\psi_0\\right]=0. \n\\end{align}\nSince  $\\kappa+\\mu_{\\rm P}, y_0, \\|\\delta_1\\|_{H^1}$ and $\\|\\delta_2\\|_2$ are all order $v$ and  $\\psi_{\\rm P} \\in C_0^\\infty (\\mathbb{R}^3)$ (and these facts also allow to infer that $\\psi_{\\rm P}^{y(0)} = \\psi_{\\rm P} + O(v)$), the constraint \\eqref{eq:constraint7} can be written as \n\\begin{align}\n\\langle \\nabla \\psi_{\\rm P} \\vert H_{\\rm P} \\vert \\Im \\psi_0 \\rangle = v \\| \\partial_1 \\psi_P \\|_2^2 (1,0,0) + O(v^2).\n\\end{align}\nDenoting $c= \\| \\partial_1 \\psi_P \\|_2^2\/\\| \\partial_1 \\varphi_P \\|_2^2$, we complete the square \n\\begin{align}\n\\expval{H_{\\rm P}}{\\Im\\psi_0} & = \\expval{H_{\\rm P}}{\\Im\\psi_0 - v c  \\partial_1 \\psi_{\\rm P}} \\notag \\\\\n& \\quad + 2v c \\bra{\\Im\\psi_0}\\ket{H_{\\rm P} \\vert \\partial_1  \\psi_{\\rm P}} - c^2  v^2  \\expval{H_{\\rm P}}{  \\partial_1 \\psi_{\\rm P}}\\notag \\\\\n&  \\geq  2 c v \\bra{\\Im\\psi_0}\\ket{ H_{\\rm P} \\partial_1  \\psi_{\\rm P}} -c^2  v^2 \\expval{H_{\\rm P}}{  \\partial_1 \\psi_{\\rm P}}  .  \\label{eq:completing-square2}\n\\end{align}\n With the constraint \\eqref{eq:constraint7} and  $\\langle \\partial_i \\psi_{\\rm P}\\vert H_{\\rm P} \\vert \\partial_j \\psi_{\\rm P} \\rangle = \\delta_{i,j}\\| \\partial_j \\varphi_{\\rm P} \\|_2^2$, we arrive at \\eqref{eq:Evtilde}. \n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIn this paper, we are concerned with numerically solving convection-diffusion equations of the form\n\\begin{equation}\\label{CDeqn}\n\\begin{cases}\n    u_t + \\nabla\\cdot\\mathbf{F}(u) = \\epsilon\\Delta u + g(\\mathbf{x},t),&\\mathbf{x}\\in\\mathcal{D},\\quad t>0,\\\\\n    u(\\mathbf{x},t=0)=u_0(\\mathbf{x}),&\\mathbf{x}\\in\\mathcal{D},\n\\end{cases}\n\\end{equation}\nwhere $\\epsilon\\geq 0$. We propose an Eulerian-Lagrangian Runge-Kutta finite volume (EL-RK-FV) scheme utilizing weighted essentially non-oscillatory (WENO) schemes with adaptive order (WENO-AO) for spatial reconstruction. The proposed method is designed for one-dimensional problems of the form \\eqref{CDeqn} and extended to higher-dimensional problems via dimensional splitting.\\\\\n\\indent Eulerian-Lagrangian (EL) and semi-Lagrangian (SL) schemes \\cite{Celia1990,Russell2002,Xiu2001} have proven to be computationally effective when solving hyperbolic problems because of their ability to employ high spatial resolution schemes while admitting very large CFL with numerical stability. Generally speaking, an EL or SL method involves working on a background grid so that high-order spatial resolutions can be used (the Eulerian part), and tracing characteristics of the cell boundaries backward\/forward in time to relax the CFL constraint (the Lagrangian part). Such methods have been developed in a wide variety of frameworks: discontinuous Galerkin \\cite{Cai2017,Ding2020,Qiu2011b,Rossmanith2011}, finite difference \\cite{Carrillo2007,Chen2021,Huot2003,Li2022,Qiu2010,Qiu2011,Xiong2019}, finite volume \\cite{Abreu2017,Benkhaldoun2015,Crouseilles2010,Filbet2001,Huang2012,Huang2017}. Although the SL and EL frameworks are similar in spirit, the SL framework assumes exact characteristic tracing and hence poses difficulties when considering nonlinear problems. Two other methods similar to EL and SL methods are the arbitrary Lagrangian-Eulerian (ALE) methods where an arbitrary mesh velocity not necessarily aligned with the fluid velocity is defined \\cite{Boscheri2014,Boscheri2013,Boscheri2015,Boscheri2017,Donea2004,Hirt1974,Peery2000}, and moving mesh methods where the PDE is first evolved in time and then followed by some mesh-redistribution procedure \\cite{Li2001,Li1997,Luo2019,Russell2002,Stockie2001,Tang2003}. The main difference between these two methods and the previously mentioned methods is that they move the mesh adaptively to focus resolving the solution around sharp transitions; whereas EL and SL methods evolve the equation by following characteristics.\\\\\n\\indent Our goal in this paper is to develop a new high-order EL method in the finite volume framework using method-of-lines (MOL) RK time discretizations. Finite volume methods are attractive since they are naturally mass conservative, easy to physically interpret, and modifiable for nonuniform grids. Similar to the recent developments made by Huang, Arbogast, and Qiu \\cite{Huang2012,Huang2017}, we use approximate characteristics to define a traceback space-time region, and then use WENO reconstructions to evaluate the modified flux. Huang and Arbogast developed a re-averaging technique that allows high-order reconstruction of the solution at arbitrary points by applying a standard WENO scheme \\cite{Cockburn1997,Shu2009} over a uniform reconstruction grid that is defined separately. They then used a natural continuous extension \\cite{Zennaro1986} of Runge-Kutta schemes, which requires the solution at several Gaussian nodes of the interval $[t^n,t^{n+1}]$, to evolve the solution along the approximate characteristics. \\\\\n\\indent The novelty of our proposed method is twofold: (1) the partition of space-time regions formed by linear approximations of the characteristic curves, and (2) integrating the differential equation over the partitioned space-time regions, followed by rewriting the space-time integral form of the equation into a spatial-integral time-differential form. In this way, a MOL RK type method can be directly applied for time discretization, thus avoiding the need to use a natural continuous extension of RK schemes. To be more precise, we construct linear approximate characteristics by using the Rankine-Hugoniot jump condition to define the traceback space-time regions. If the linear approximate characteristics are defined with zero velocity, then the proposed EL-RK-FV scheme reduces to the standard RK-FV method. Whereas, when linear space-time curves adequately approximate the exact characteristics, a large time stepping size is still permitted. We use WENO-AO to perform a solution remapping of the uniform cell averages onto the possibly nonuniform traceback cells. The recently developed WENO-AO schemes \\cite{Arbogast2020,Balsara2016,Balsara2020} are robust and guarantee the existence of the linear weights at arbitrary points. We note that Chen, et al. used WENO-AO schemes in the SL framework \\cite{Chen2021}, and Huang and Arbogast have recently used WENO-AO schemes in the Eulerian framework \\cite{Arbogast2019,Arbogast2020}. RK methods are used to evolve the MOL system along the approximate characteristics. Explicit RK methods, such as the strong stability-preserving (SSP) RK methods \\cite{Gottlieb2001}, are used for convection equations; and implicit-explicit (IMEX) RK methods \\cite{Ascher1997,Conde2017,Higueras2014} are used for convection-diffusion equations. In the latter case, the non-stiff convective term is treated explicitly and the stiff diffusive term is treated implicitly. Dimensional splitting is used to extend the one-dimensional algorithm to solve multi-dimensional problems. The proposed method is high-order accurate, capable of resolving discontinuities without oscillations, mass conservative, and stable with large time stepping sizes.\\\\\n\\indent The paper is organized as follows. We discuss the EL-RK-FV algorithm for pure convection problems in Section 2. In Section 3, we discuss the EL-RK-FV algorithm, coupled with IMEX RK schemes, for convection-diffusion equations. Numerical performance of the EL-RK-FV algorithm is shown in Section 4 by applying the algorithm to several linear and nonlinear test problems. Conclusions are made in Section 5. Appendices are listed after the references.\n\n\n\n\n\n\\section{The EL-RK-FV method for pure convection problems}\n\nThe spirit of the EL-RK-FV method is best demonstrated by starting with a pure convection problem in one dimension, i.e., equation \\eqref{CDeqn} with $\\epsilon=0$ and $g(x)=0$. Let the flux be denoted $f(x)$. We first discuss the formulation of the scheme in Section~\\ref{sec2.1}, followed by discussion of high-order spatial reconstruction in Section~\\ref{sec2.2} and time discretization in Section~\\ref{sec2.3}.\n\n\\subsection{\nScheme formulation\n}\n\\label{sec2.1}\n\nWe discretize the spatial domain $[a,b]$ into $N_x$ intervals with $N_x+1$ uniformly distributed nodes\n\\[a=x_{\\frac{1}{2}}<x_{\\frac{3}{2}}<...<x_{N_x-\\frac{1}{2}}<x_{N_x+\\frac{1}{2}}=b.\\]\nDefine the cells $I_j\\coloneqq[x_{j-\\frac{1}{2}},x_{j+\\frac{1}{2}}]$ with centers $x_j=(x_{j-\\frac{1}{2}}+x_{j+\\frac{1}{2}})\/2$ and widths $\\Delta x_j=\\Delta x$ for $j=1,...,N_x$. We let\n\\begin{equation}\n    \\label{CFL}\n    \\Delta t = \\frac{CFL\\Delta x}{\\text{max}|f'(u)|},\n\\end{equation}\nwhere $CFL$ defines the time stepping size. In contrast to Eulerian methods, which evolve the solution on a stationary mesh, our EL algorithm proposes tracing the characteristics \\textit{backwards} in time from $t^{n+1}$ to $t^n$ to partition a set of space-time regions based on the computational grid. Since tracing characteristics in the nonlinear case is often nontrivial, we consider computing \\textit{approximations of characteristics} that are linear space-time curves. In particular, the \\textit{approximate characteristic speeds} at nodes $x_{j+\\frac{1}{2}}$ and time $t^{n+1}$ are defined using the Rankine-Hugoniot jump condition,\n\\begin{equation}\\label{RHjump}\n\t\\nu_{j+\\frac{1}{2}} = \n\t\\begin{cases}\n\t\t\\mathlarger{\\frac{f(\\overline{u}_{j+1}) - f(\\overline{u}_j)}{\\overline{u}_{j+1} - \\overline{u}_j}},&\\overline{u}_{j+1}\\neq \\overline{u}_j,\\\\\n\t\tf'(\\overline{u}_j),&\\overline{u}_{j+1}=\\overline{u}_j,\n\t\\end{cases}\n\\end{equation}\nwhere $\\overline{u}_j$ and $\\overline{u}_{j+1}$ are the cell averages at time $t^n$. In practice we tested if $|\\overline{u}_{j+1}-\\overline{u}_j|<\\epsilon$, with $\\epsilon=1.0e-8$, in which case we took $\\overline{u}$ to be the average of $\\overline{u}^-$ and $\\overline{u}^+$. As seen in Figure \\ref{spacetime}, the (upstream) traceback nodes can be defined by\n\\begin{equation}\n    x_{j+\\frac{1}{2}}^*\\coloneqq x_{j+\\frac{1}{2}} - \\nu_{j+\\frac{1}{2}}\\Delta t,\\quad j=1,...,N.\n\\end{equation}\nDefine $\\tilde{x}_{j+\\frac{1}{2}}(t)=x^*_{j+\\frac{1}{2}}+\\nu_{j+\\frac{1}{2}}(t-t^n)$ with $t^n\\leq t\\leq t^{n+1}$, for $j=0, 1,2,...,N$. The approximate characteristics are given by the space-time tracelines\n\\[\\mathcal{S}_{left}\\coloneqq\\{(\\tilde{x}_{j-\\frac{1}{2}}(t),t)\\text{ : }t^n\\leq t\\leq t^{n+1}\\}\\quad\\text{and}\\quad\\mathcal{S}_{right}\\coloneqq\\{(\\tilde{x}_{j+\\frac{1}{2}}(t),t)\\text{ : }t^n\\leq t\\leq t^{n+1}\\}.\\]\nNote that $x_{j+\\frac{1}{2}}^*=\\tilde{x}_{j+\\frac{1}{2}}(t^n)$, $x_{j+\\frac{1}{2}}=\\tilde{x}_{j+\\frac{1}{2}}(t^{n+1})$, and the (upstream) traceback cells $I_j^*=\\tilde{I}_j(t^n)$ are in general nonuniform. We define the space-time domain $\\Omega_j$ as the region bounded by $I_j$, $I_j^*$, $\\mathcal{S}_{left}$, and $\\mathcal{S}_{right}$.\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{.\/images_spacetime\/spacetime0.png}\n\t\\caption{The space-time region $\\Omega_j$.}\n\t\\label{spacetime}\n\\end{figure}\n\\\\\nWith the constructed space-time region $\\Omega_j$, we rewrite\nthe one-dimensional pure convection problem in divergence form $\\nabla_{t,x}\\cdot(u,f(u))^T=0$, integrate it over $\\Omega_j$, and apply the divergence theorem to get\n\\begin{align}\n\\begin{split}\\label{semidiscrete0}\n\t&\\int_{I_j}{u(x,t^{n+1})dx} - \\int_{I_j^*}{u(x,t^n)dx}\\\\\n\t=&- \\left[\\int_{t^n}^{t^{n+1}}{\\big(f(u(\\tilde{x}_{j+\\frac{1}{2}}(t),t))-\\nu_{j+\\frac{1}{2}}u(\\tilde{x}_{j+\\frac{1}{2}}(t),t)\\big)dt} - \\int_{t^n}^{t^{n+1}}{\\big(f(u(\\tilde{x}_{j-\\frac{1}{2}}(t),t))-\\nu_{j-\\frac{1}{2}}u(\\tilde{x}_{j-\\frac{1}{2}}(t),t)\\big)dt}\\right].\n\\end{split}\n\\end{align}\nWe rewrite the time-integral form \\eqref{semidiscrete0} into the time-differential form\nto get\n\\begin{equation}\\label{semidiscrete1}\n\t\\frac{d}{dt}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = -\\Big[F_{j+\\frac{1}{2}}(t) - F_{j-\\frac{1}{2}}(t)\\Big],\n\\end{equation}\n\n\\noindent where $F_{j+\\frac{1}{2}}(t)\\coloneqq f(u(\\tilde{x}_{j+\\frac{1}{2}}(t),t))-\\nu_{j+\\frac{1}{2}}u(\\tilde{x}_{j+\\frac{1}{2}}(t),t)$ is called the \\textit{modified flux function}. Choosing any appropriate monotone numerical flux function\n\\begin{equation}\n    \\label{numflux}\n    \\hat{F}_{j+\\frac{1}{2}}(t) = \\hat{F}_{j+\\frac{1}{2}}(u_{j+\\frac{1}{2}}^{-},u_{j+\\frac{1}{2}}^{+};t)\n\\end{equation}\n(e.g., Lax-Friedrichs flux) for the modified flux function, equation \\eqref{semidiscrete1} can be rewritten as the following semi-discrete finite volume scheme:\n\\begin{equation}\\label{semidiscrete2}\n\t\\frac{d}{dt}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = -\\Big[\\hat{F}_{j+\\frac{1}{2}}(t) - \\hat{F}_{j-\\frac{1}{2}}(t)\\Big].\n\\end{equation}\nHere, the starting condition is obtained by a solution remapping onto a trackback grid, discussed in Section~\\ref{sec2.2}; $\\hat{F}_{j+\\frac{1}{2}}(t)$ are computed from \\eqref{numflux} with reconstructed values from neighboring cell averages at time $t$, discussed in Section~\\ref{sec2.3}. We evolve equation \\eqref{semidiscrete2} by the MOL approach with explicit RK schemes, discussed in Section~\\ref{sec2.4}.\n\n\\subsection{Solution remapping onto a traceback grid}\n\\label{sec2.2}\n\nReferring back to Figure \\ref{spacetime} and equation \\eqref{semidiscrete0}, we see that the solution will need to be projected onto the traceback grid in order to compute the \\textit{traceback cell averages} \n\\begin{equation}\n    \\tilde{u}_j^n \\coloneqq \\frac{1}{\\Delta x_j^*}\\int_{I_j^*}{u(x,t^n)dx},\\qquad j=1,2,...,N,\n\\end{equation}\nthat is, the starting condition for \\eqref{semidiscrete2}. Hence, we desire a procedure for an \\textit{integral reconstruction} that --in general-- uses known uniform cell averages $\\{ \\overline{u}_j(t)\\text{ : }j=1,2,...,N\\}$ to approximate the desired cell averages $\\{ \\tilde{u}_j(t)\\text{ : }j=1,2,...,N \\}$ defined by\n\\begin{equation}\n    \\tilde{u}_j(t) \\coloneqq \\frac{1}{\\Delta\\tilde{x}_j(t)}\\int_{\\tilde{I}_j(t)}{u(x,t)dx},\\qquad j=1,2,...,N.\n\\end{equation}\nUnless otherwise stated, overlines (e.g., $\\overline{u}_j(t)$) denote \\textit{uniform} cell averages, and tildes (e.g., $\\tilde{u}_j(t)$) denote \\textit{nonuniform} cell averages.\\\\\n\\ \\\\\nSince discontinuities and sharp gradients can occur when solving pure convection problems, we use high-resolution schemes to control spurious oscillations, e.g., weighted essential non-oscillatory (WENO) methods \\cite{Balsara2016,Cockburn1997,Levy1999,Shu2009}. Again, we assume to only be given the uniform cell averages at some time $t$. One might apply the well known WENO procedure presented in \\cite{Cockburn1997,Shu2009} to obtain the \\textit{reconstruction polynomials} $\\mathcal{R}_j(x\\in I_j)$ in terms of the neighboring uniform cell averages (at time $t$) $\\overline{u}_i$, $i=j-p,...,j+q$. Referring to Figure \\ref{spacetime} for the sake of demonstration, we might then approximate the cell averages $\\tilde{u}_j^*$ by\n\\begin{equation}\n    \\label{WENO}\n    \\tilde{u}_j^* \\approx \\frac{1}{\\Delta x_j^*}\\left(\\int_{I_j^*\\cap I_{j-1}}{\\mathcal{R}_{j-1}(x)dx} + \\int_{I_j^*\\cap I_{j}}{\\mathcal{R}_{j}(x)dx}\\right).\n\\end{equation}\nHowever, the linear weights in the WENO reconstruction are not guaranteed to exist or be positive at arbitrary points. To alleviate this issue, we instead use the WENO schemes with adaptive order (WENO-AO) presented in \\cite{Balsara2016} since the linear weights exist at arbitrary points.\\\\\n\\ \\\\\nThe overarching idea of WENO-AO methods is to provide high order accuracy for smooth solutions over a large center stencil and adaptively reduce to lower order accuracy when the solution does not permit the high order accuracy. This is done by creating a nonlinear hybridization between a large center stencil with high order accuracy, and very stable lower order WENO schemes (e.g., CWENO schemes \\cite{Levy1999}). Aside from the high order accuracy and existence of linear weights at arbitrary points, the robustness of these WENO-AO schemes is particularly attractive for our purposes. The authors in \\cite{Balsara2016} write WENO-AO($p,r$) to denote an adaptive order that is at best $p$th order (from the large center stencil) and at worst $r$th order (from the stable lower order stencils). For our purposes we use WENO-AO(5,3). The end product of WENO and WENO-AO methods is ultimately a reconstruction polynomial $\\mathcal{R}_j(x\\in I_j)$ that we shall use for reconstruction.\\\\\n\\ \\\\\nEquation \\eqref{WENO} is valid when using WENO-AO reconstruction polynomials $\\mathcal{R}_j(x\\in I_j)$. In general, if the traceback points $\\tilde{x}_{j-\\frac{1}{2}}(t)$ and $\\tilde{x}_{j+\\frac{1}{2}}(t)$ reside in cells $I_{\\ell}$ and $I_r$ respectively, then\n\\begin{equation}\n    \\label{WENOAO}\n    \\int_{\\tilde{I}_j(t)}{u(x,t)dx} \\approx \\int_{\\tilde{I}_j(t)\\cap I_{\\ell}}{\\mathcal{R}_{\\ell}(x)dx} + \\Delta x\\sum\\limits_{i=\\ell+1}^{r-1}{\\overline{u}_i(t)} + \\int_{\\tilde{I}_j(t)\\cap I_{r}}{\\mathcal{R}_{r}(x)dx}.\n\\end{equation}\nBelow, we summarize the integral reconstruction procedure using WENO-AO as the Algorithm 1.\\\\\n\n\\noindent\\xdashthick\\\\\n\\textbf{Algorithm 1.} Integral reconstruction using WENO-AO\\\\\n\\xdashthin\\\\\n\\textbf{Input:} uniform cell averages $\\overline{u}_j(t)$ on the background grid of nodes $x_{j+\\frac{1}{2}}$.\\\\\n\\textbf{Output:} possibly nonuniform cell averages $\\tilde{u}_j(t)$ on the traceback grid of nodes $\\tilde{x}_{j+\\frac{1}{2}}(t)$.\\\\\n\\par\\qquad \\textbf{do $j=1,2,...,N$}\n\\par\\qquad\\qquad Locate the uniform background cells that $\\tilde{x}_{j-\\frac{1}{2}}(t)$ and $\\tilde{x}_{j+\\frac{1}{2}}(t)$ reside in.\n\\par\\qquad\\qquad Call these cells $I_{\\ell}$ and $I_r$, respectively.\n\\par\\qquad\\qquad\\textbf{if $\\ell=r$}\n\\par\\qquad\\qquad\\qquad Compute $\\mathlarger{\\tilde{u}_j(t) \\approx \\frac{1}{\\Delta\\tilde{x}_j(t)}\\int_{\\tilde{I}_j(t)}{\\mathcal{R}_{\\ell}(x)dx}}$.\n\\par\\qquad\\qquad\\textbf{else}\n\\par\\qquad\\qquad\\qquad\\textbf{do $k=\\ell,r$}\n\\par\\qquad\\qquad\\qquad\\qquad Compute $\\mathlarger{\\int_{\\tilde{I}_j(t)\\cap I_k}{\\mathcal{R}_k(x)dx}}$.\n\\par\\qquad\\qquad\\qquad\\textbf{end do}\n\\par\\qquad\\qquad\\qquad Compute $\\mathlarger{\\tilde{u}_j(t) \\approx \\frac{1}{\\Delta\\tilde{x}_j(t)}\\left(\\int_{\\tilde{I}_j(t)\\cap I_{\\ell}}{\\mathcal{R}_{\\ell}(x)dx} + \\Delta x\\sum\\limits_{i=\\ell+1}^{r-1}{\\overline{u}_i(t)} + \\int_{\\tilde{I}_j(t)\\cap I_{r}}{\\mathcal{R}_{r}(x)dx}\\right)}$.\n\\par\\qquad\\qquad\\textbf{end if}\n\\par\\qquad\\textbf{end do}\\\\\n\\xdashthick\n\n\n\\subsection{Reconstruction of point values}\n\\label{sec2.3}\n\nReferring back to Figure \\ref{spacetime} and equations \\eqref{semidiscrete0}-\\eqref{semidiscrete2}, we also need to reconstruct the point values $u(\\tilde{x}^-_{j+\\frac{1}{2}}(t),t)$ and $u(\\tilde{x}^+_{j+\\frac{1}{2}}(t),t)$ for the modified flux function. However, the WENO-AO schemes in \\cite{Balsara2016} assume a uniform grid for convenience and efficiency. We wish to avoid using nonuniform WENO methods since the linear weights need to be recomputed every step and will quickly become expensive. However, nonuniform WENO methods for two-dimensional problems \\cite{Balsara2020,Zhu2017,Zhu2017b,Zhu2019,Zhu2018} might become a more reasonable and realistic choice when dealing with non-splitting algorithms. Yet, we still need to reconstruct the left and right limits on the generally nonuniform traceback grid. We propose using the continuous piecewise-linear transformation from the nonuniform grid consisting of nodes $\\tilde{x}_{j+\\frac{1}{2}}(t)$ to the uniform background grid; performing a WENO-AO reconstruction on the uniform grid; and mapping back to the nonuniform grid to obtain the desired limits. We call this the \\textit{nonuniform-to-uniform transformation}. Given a fixed $t\\in[t^n,t^{n+1}]$, consider the linear bijection $\\phi_j:\\tilde{I}_j(t)\\rightarrow I_j$ for $j=1,2,...,N$. Letting $x\\in\\tilde{I}_j(t)$ and $\\xi\\in I_j$,\n\\begin{equation}\n\\label{eq: transform}\n    \\int_{\\tilde{I}_j(t)}{u(x,t)dx} = |J|\\int_{I_j}{u(x(\\xi),t)d\\xi},\n\\end{equation}\nwhere $|J|(\\xi) = |dx\/d\\xi|(\\xi)$ is the Jacobian of the bijection $\\phi_j$. In particular, the Jacobian is constant for the linear interpolant,\n\\begin{equation}\n    \\label{}\n    |J|=\\left|\\frac{d}{d\\xi}\\left(\\frac{\\Delta\\tilde{x}_j(t)}{\\Delta x}\\xi - \\frac{x_{j+\\frac{1}{2}}\\tilde{x}_{j-\\frac{1}{2}}(t) + x_{j-\\frac{1}{2}}\\tilde{x}_{j+\\frac{1}{2}}(t)}{\\Delta x}\\right)\\right| = \\frac{\\Delta\\tilde{x}_j(t)}{\\Delta x}.\n\\end{equation}\nSince we want to apply WENO-AO over the uniform grid, we define the \\textit{auxiliary uniform cell averages} as\n\\begin{equation}\n    \\check{u}_j(t) \\coloneqq \\frac{1}{\\Delta x}\\int_{I_j}{u(x(\\xi),t)d\\xi} = \\frac{1}{\\Delta x|J|}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = \\frac{1}{\\Delta\\tilde{x}_j(t)}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = \\tilde{u}_j(t).\n\\end{equation}\nNote that we have shown the auxiliary uniform cell averages (on the uniform background grid) are identical to the nonuniform cell averages. Hence, under this continuous piecewise-linear mapping, we can directly use the nonuniform cell averages at time $t$ in the (uniform) WENO-AO procedure to obtain the left and right limits $u(\\tilde{x}^-_{j+\\frac{1}{2}}(t),t)$ and $u(\\tilde{x}^+_{j+\\frac{1}{2}}(t),t)$. We further note that the high-order accuracy is preserved with such a strategy only when there is a smooth mapping with equation \\eqref{eq: transform}. Theoretical justification for the existence of such a mapping is highly nontrivial. Yet, our numerical tests verify that high-order spatial accuracy is achieved, when the characteristic field is smooth.\\\\\n\n\\noindent \\xdashthick\\\\\n\\textbf{Algorithm 2.} Reconstruction using WENO-AO with the nonuniform-to-uniform transformation\\\\\n\\xdashthin\\\\\n\\textbf{Input:} nonuniform cell averages $\\tilde{u}_j(t)$ on the traceback grid.\\\\\n\\textbf{Output:} left and right limits $u(\\tilde{x}^-_{j+\\frac{1}{2}}(t),t)$ and $u(\\tilde{x}^+_{j+\\frac{1}{2}}(t),t)$.\\\\\n\\par\\qquad \\textbf{do $j=1,2,...,N$}\n\\par\\qquad\\qquad Calculate the (uniform) WENO-AO reconstruction polynomial $\\mathcal{R}_j(x\\in I_j)$ in terms of\n\\par\\qquad\\qquad the neighboring auxiliary uniform cell averages $\\check{u}_i(t)=\\tilde{u}_i(t)$, $i=j-p,...,j+q$.\n\\par\\qquad\\qquad Compute the left limit $u(\\tilde{x}^-_{j+\\frac{1}{2}}(t),t) \\approx \\mathcal{R}_j(x_{j+\\frac{1}{2}})$.\n\\par\\qquad\\qquad Compute the right limit $u(\\tilde{x}^+_{j-\\frac{1}{2}}(t),t) \\approx \\mathcal{R}_j(x_{j-\\frac{1}{2}})$.\n\\par\\qquad\\textbf{end do}\\\\\n\\xdashthick\n\n\\subsection{Time evolution with explicit Runge-Kutta methods}\n\\label{sec2.4}\n\nAlgorithms 1 and 2 now allow us to perform (integral) reconstruction on a traceback grid consisting of nodes $\\tilde{x}_{j+\\frac{1}{2}}(t)$. With these two tools we can evolve equation \\eqref{semidiscrete2} using any explicit RK method. Recall that our primary goal is to achieve high order accuracy while also taking large time steps. In our numerical tests we use WENO-AO(5,3) for the spatial approximation; although higher order WENO-AO methods can certainly be used. As such, we would like to use high-order time stepping methods. In the cases where the solution is smooth, the standard fourth-order RK method suffices. However, if the solution is discontinuous (e.g., a travelling Heaviside step function), then we require an explicit SSP RK method \\cite{Gottlieb2001}. Explicit SSP RK methods are especially attractive when numerically solving hyperbolic conservation laws because they maintain strong stability while also achieving high order accuracy in time. When applicable, we use the optimal three-stage, third-order explicit SSP RK method outlined below for demonstrative purposes.\n\n\\begin{align}\\label{SSPRK3}\n\\begin{split}\n\t\\Delta\\tilde{x}_j^{(0)}\\tilde{u}_j^{(0)} &= \\Delta x_j^*\\overline{u}_j^*,\\\\\n\t\\Delta\\tilde{x}_j^{(1)}\\tilde{u}_j^{(1)} &= \\Delta\\tilde{x}_j^{(0)}\\tilde{u}_j^{(0)} - \\Delta t\\big(\\hat{F}_{j+\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)-\\hat{F}_{j-\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)\\big),\\\\\n\t\\Delta\\tilde{x}_j^{(2)}\\tilde{u}_j^{(2)} &= \\Delta\\tilde{x}_j^{(0)}\\tilde{u}_j^{(0)} - \\frac{\\Delta t}{4}\\Big[\\big(\\hat{F}_{j+\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)-\\hat{F}_{j-\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)\\big)\\\\\n\t&\\qquad\\qquad\\qquad\\qquad+ \\big(\\hat{F}_{j+\\frac{1}{2}}^{(1)}(u^-,u^+;t^{n+1})-\\hat{F}_{j-\\frac{1}{2}}^{(1)}(u^-,u^+;t^{n+1})\\big)\\Big],\\\\\n\t\\Delta x_j\\overline{u}_j^{n+1} &= \\Delta\\tilde{x}_j^{(0)}\\tilde{u}_j^{(0)} - \\frac{2\\Delta t}{3}\\big(\\hat{F}_{j+\\frac{1}{2}}^{(2)}(u^-,u^+;t^{n+\\frac{1}{2}}) - \\hat{F}_{j-\\frac{1}{2}}^{(2)}(u^-,u^+;t^{n+\\frac{1}{2}})\\big)\\\\\n\t&\\qquad\\qquad\\qquad\\qquad- \\frac{\\Delta t}{6}\\Big[\\big(\\hat{F}_{j+\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)-\\hat{F}_{j-\\frac{1}{2}}^{(0)}(u^-,u^+;t^n)\\big)\\\\\n\t&\\qquad\\qquad\\qquad\\qquad+ \\big(\\hat{F}_{j+\\frac{1}{2}}^{(1)}(u^-,u^+;t^{n+1})-\\hat{F}_{j-\\frac{1}{2}}^{(1)}(u^-,u^+;t^{n+1})\\big)\\Big],\n\\end{split}\n\\end{align}\n\n\\noindent where $\\hat{F}_{j\\pm\\frac{1}{2}}^{(k)}(u^-,u^+;t)$ denotes using the cell averages $\\tilde{u}_j^{(k)}$ from stage $k$ to approximate the limits $u_{j\\pm\\frac{1}{2}}^{-}$ and $u_{j\\pm\\frac{1}{2}}^{+}$ in the numerical flux function at time $t$ using Algorithms 1 and 2. The Butcher tables for other explicit RK methods are provided in Appendix A.\n\n\\begin{remark}\nIf the approximate characteristics are defined such that $\\nu_{j+\\frac{1}{2}}=0$ for all $j$, then the EL-RK-FV scheme reduces to the standard RK-FV scheme \\cite{Shu2009}. Referring to Figure \\ref{spacetime}, the approximate characteristics would be vertical lines.\n\\end{remark}\n\n\\subsection{Two-dimensional problems}\n\\label{sec2.5}\n\nWe now consider the two-dimensional equation\n\\begin{equation}\n\\label{2Dscalar}\n    u_t + f(u)_x + g(u)_y = 0.\n\\end{equation}\nThe spatial domain is discretized into $N_xN_y$ cells, $I_{i,j}\\coloneqq I_i\\times I_j$ with centers $x_{i,j}=((x_{i-\\frac{1}{2}}+x_{i+\\frac{1}{2}})\/2,(y_{j-\\frac{1}{2}}+y_{j+\\frac{1}{2}})\/2)$, where the spatial discretizations in $x$ and $y$ are respectively given by\n\\[0=x_{\\frac{1}{2}}<x_{\\frac{3}{2}}<...<x_{N_x-\\frac{1}{2}}<x_{N_x+\\frac{1}{2}}=2\\pi\\qquad\\text{and}\\qquad 0=y_{\\frac{1}{2}}<y_{\\frac{3}{2}}<...<y_{N_y-\\frac{1}{2}}<y_{N_y+\\frac{1}{2}}=2\\pi.\\]\nThe CFL number is defined as\n\\begin{equation}\n    \\label{CFL2D}\n    \\Delta t = \\frac{CFL}{\\mathlarger{\\frac{\\max{|f'(u)|}}{\\Delta x} + \\frac{\\max{|g'(u)|}}{\\Delta y}}},\n\\end{equation}\nand the uniform cell averages at time $t^n$ are defined by\n\\begin{equation}\n    \\tilde{\\bar{u}}_{i,j}^n\\coloneqq \\frac{1}{\\Delta x\\Delta y}\\int_{I_{i,j}}{u(x,y,t^n)dxdy}.\n\\end{equation}\nIn the two-dimensional case, we also want to compute the \\textit{interval averages} over the interval $I_i$ at a fixed $y$, or over the interval $I_j$ at a fixed $x$. Define the uniform \\textit{interval averages} at time $t^n$ with one variable fixed by\n\\begin{subequations}\n\\begin{equation}\n    \\bar{u}_{i|y}^n\\coloneqq \\frac{1}{\\Delta x}\\int_{I_i}{u(x,y,t^n)dx},\n\\end{equation}\n\\begin{equation}\n    \\tilde{u}_{j|x}^n\\coloneqq \\frac{1}{\\Delta y}\\int_{I_j}{u(x,y,t^n)dy}.\n\\end{equation}\n\\end{subequations}\nNote that \\textit{only in this subsection} will the superscript tilde denote uniform interval averages in $y$, not nonuniform interval averages.\\\\\n\\ \\\\\nDimensional splitting methods are commonly used to solve two (or higher) dimensional problems, with second-order Strang splitting being a popular choice. Dimensional splitting methods solve equation \\eqref{2Dscalar} by alternating between solving the easier problems\n\\begin{subequations}\n\\begin{equation}\\label{Strang1}\n\tu_t + f(u)_x = 0,\n\\end{equation}\n\\begin{equation}\\label{Strang2}\n\tu_t + g(u)_y = 0.\n\\end{equation}\n\\end{subequations}\nStrang splitting uses $\\tilde{\\bar{u}}_{i,j}^n$ to solve \\eqref{Strang1} over a half time step $\\Delta t\/2$ for intermediate solution $\\tilde{\\bar{u}}_{i,j}^*$; uses $\\tilde{\\bar{u}}_{i,j}^*$ to solve \\eqref{Strang2}  over a full time step $\\Delta t$ for intermediate solution $\\tilde{\\bar{u}}_{i,j}^{**}$; and uses $\\tilde{\\bar{u}}_{i,j}^{**}$ to solve \\eqref{Strang1} over another half time step $\\Delta t\/2$ for solution $\\tilde{\\bar{u}}_{i,j}^{n+1}$. As such, we can reduce the two-dimensional problem to solving several (easier) one-dimensional problems. Since the one-dimensional EL-RK-FV algorithm evolves interval averages, we need a way to go between two-dimensional cell averages and one-dimensional interval averages. We note that the nonlinear weights in the WENO-AO method \\cite{Balsara2016} are the same for all nodes within the same cell.\n\n\\subsubsection{Going from\/to cell averages to\/from interval averages}\n\nConsider the interval averages as functions of $y$ and $x$, respectively defined by\n\\begin{subequations}\n\\begin{equation}\n    \\psi_i(y) \\coloneqq \\bar{u}_{i|y}^n = \\frac{1}{\\Delta x}\\int_{I_i}{u(x,y,t^n)dx},\n\\end{equation}\n\\begin{equation}\n    \\phi_j(x) \\coloneqq \\tilde{u}_{j|x}^n = \\frac{1}{\\Delta y}\\int_{I_j}{u(x,y,t^n)dy}.\n\\end{equation}\n\\end{subequations}\n For any given $i=1,2,...,N_x$ or $j=1,2,...,N_y$, consider the respective Gauss-Legendre quadrature nodes $\\{x_k^{(i)}\\in I_i\\text{ : }k=1,...,K\\}$ and $\\{y_l^{(j)}\\in I_j\\text{ : }l=1,...,L\\}$. Observing that the cell averages at time $t^n$ can be expressed as the interval averages of $\\psi(y)$ and $\\phi(x)$,\n \\begin{equation}\n     \\tilde{\\bar{u}}_{i,j}^n = \\frac{1}{\\Delta y}\\int_{I_j}{\\psi_i(y)dy} = \\frac{1}{\\Delta x}\\int_{I_i}{\\phi_j(x)dx},\n \\end{equation}\nwe can use WENO-AO to go from cell averages to interval averages evaluated at the Gauss-Legendre nodes,\n\\begin{subequations}\n\\begin{equation}\n\t\\tilde{\\bar{u}}_{i,j}^n\\longrightarrow\\psi_i(y_l^{(j)})=\\bar{u}_{i|l}^n,\n\\end{equation}\n\\begin{equation}\n\t\\tilde{\\bar{u}}_{i,j}^n\\longrightarrow\\phi_j(x_k^{(i)})=\\tilde{u}_{j|k}^n.\n\\end{equation}\n\\end{subequations}\n\n\\noindent Algorithm 3 presents how to implement WENO-AO to go from cell averages to $x-$interval averages at the fixed $y-$Gauss-Legendre nodes. A similar algorithm can be done to go from cell averages to $y-$interval averages at the fixed $x-$Gauss-Legendre nodes.\\\\\n\n\\noindent\\xdashthick\\\\\n\\textbf{Algorithm 3.} Going from cell averages to $x-$interval averages\\\\\n\\xdashthin\\\\\n\\textbf{Input:} uniform cell averages $\\tilde{\\bar{u}}_{i,j}$.\\\\\n\\textbf{Output:} uniform $x-$interval averages $\\bar{u}_{i|l}$ at fixed Gauss-Legendre nodes $\\{y_l^{(j)}\\in I_j\\text{ : }l=1,...,L\\}$.\\\\\n\\par\\qquad\\textbf{do $i=1,2,...,N_x$}\n\\par\\qquad\\qquad \\textbf{do $j=1,2,...,N_y$}\n\\par\\qquad\\qquad\\qquad Calculate the WENO-AO reconstruction polynomial $\\mathcal{R}_j^{(i)}(y\\in I_j)$ in terms of the\n\\par\\qquad\\qquad\\qquad neighboring averages $\\tilde{\\bar{u}}_{i,k}$, $k=j-p,...,j+q$.\n\\par\\qquad\\qquad\\qquad\\textbf{do $l=1,...,L$}\n\\par\\qquad\\qquad\\qquad\\qquad Store $\\bar{u}_{i|l} = \\psi_i(y_l^{(j)}) \\approx \\mathcal{R}_j^{(i)}(y_l^{(j)})$.\n\\par\\qquad\\qquad\\qquad\\textbf{end do}\n\\par\\qquad\\qquad\\textbf{end do}\n\\par\\qquad\\textbf{end do}\n\\par\\qquad The uniform $x-$interval averages at a fixed Gauss-Legendre node $y_l^{(j)}$ are $\\{\\bar{u}_{i|l} = \\psi_i(y_l^{(j)})\\text{ : }l=1,2,...,N_x\\}$.\\\\\n\\xdashthick\n\\\\\n\\ \\\\\nComputing the cell averages from the interval averages at the Gauss-Legendre nodes is straightforward using a Gaussian quadrature. Let $\\xi$ and $w$ denote the standard Gauss-Legendre nodes and weights over the interval $[-1,1]$, respectively. Without loss of generality, we can go from $x-$interval averages to cell averages.\n\\begin{align}\n\\begin{split}\n    \\label{intervaltocell1}\n\t\\tilde{\\bar{u}}_{i,j} &= \\frac{1}{\\Delta y}\\int_{I_j}{\\psi(y)dy}\\\\\n\t&= \\frac{1}{2}\\int_{-1}^{1}{\\psi\\Big(y_{j-\\frac{1}{2}}+\\frac{\\Delta y}{2}(y'+1)\\Big)dy'}\\\\\n\t&\\approx \\frac{1}{2}\\sum\\limits_{l=1}^{L}{w_l\\psi\\Big(y_{j-\\frac{1}{2}}+\\frac{\\Delta y}{2}(\\xi_l+1)\\Big)}\\\\\n\t&= \\frac{1}{2}\\sum\\limits_{l=1}^{L}{w_l\\psi(y_l^{(j)})},\\qquad\\text{where $y_l^{(j)}=y_{j-\\frac{1}{2}}+\\frac{\\Delta y}{2}(\\xi_l+1)$.}\n\\end{split}\n\\end{align}\n\n\\subsubsection{Strang splitting}\n\nFor demonstrative purposes, we outline the EL-RK-FV algorithm for two-dimensional problems via Strang splitting. Since Strang splitting is only second-order in time, higher-order splitting methods might be preferred. We present the fourth-order splitting method developed by Forest and Ruth \\cite{Forest1990} and Yoshida \\cite{Yoshida1990} in Appendix B; Rossmanith and Seal used this fourth-order splitting method in the semi-Lagrangian framework in \\cite{Rossmanith2011}.\\\\\n\n\\noindent\\xdashthick\\\\\n\\textbf{Algorithm 4.} Strang splitting for the EL-RK-FV method\\\\\n\\xdashthin\\\\\n\\textbf{Input:} uniform cell averages $\\tilde{\\bar{u}}_{i,j}^n$.\\\\\n\\textbf{Output:} uniform cell averages $\\tilde{\\bar{u}}_{i,j}^{n+1}$.\\\\\n\\par\\qquad\\textbf{Step 1 ($x$-direction).} Solve equation \\eqref{Strang1} for a half time step $\\Delta t\/2$; that is, over $[t^n,t^{n+1\/2}]$.\n\\par\\qquad Use Algorithm 3 to get $x-$interval averages; that is, $\\tilde{\\bar{u}}_{i,j}^n\\longrightarrow\\bar{u}_{i|l}^n$.\n\\par\\qquad\\textbf{do $l=1,2,...,N_y\\cdot L$}\n\\par\\qquad\\qquad For each Gauss-Legendre node, update the $x-$interval averages by applying the 1D algorithm;\n\\par\\qquad\\qquad that is, $\\bar{u}_{i|l}^n\\longrightarrow\\bar{u}_{i|l}^*$.\n\\par\\qquad\\textbf{end do}\n\\par\\qquad Use equation \\eqref{intervaltocell1} to get updated cell averages; that is, $\\bar{u}_{i|l}^*\\longrightarrow\\tilde{\\bar{u}}_{i,j}^*$.\\\\\n\n\\par\\qquad\\textbf{Step 2 ($y$-direction).} Solve equation \\eqref{Strang2} for a full time step $\\Delta t$; that is, over $[t^n,t^{n+1}]$.\n\\par\\qquad Use Algorithm 3 analogue to get $y-$interval averages; that is, $\\tilde{\\bar{u}}_{i,j}^*\\longrightarrow\\tilde{u}_{j|k}^*$.\n\\par\\qquad\\textbf{do $k=1,2,...,N_x\\cdot K$}\n\\par\\qquad\\qquad For each Gauss-Legendre node, update the $y-$interval averages by applying the 1D algorithm;\n\\par\\qquad\\qquad that is, $\\tilde{u}_{j|k}^{*}\\longrightarrow\\tilde{u}_{j|k}^{**}$.\n\\par\\qquad\\textbf{end do}\n\\par\\qquad Use equation \\eqref{intervaltocell1} analogue to get updated cell averages; that is, $\\tilde{u}_{j|k}^{**}\\longrightarrow\\tilde{\\bar{u}}_{i,j}^{**}$.\\\\\n\n\\par\\qquad\\textbf{Step 3 ($x$-direction).} Solve equation \\eqref{Strang1} for a half time step $\\Delta t\/2$; that is, over $[t^{n+1\/2},t^{n+1}]$.\n\\par\\qquad Use Algorithm 3 to get $x-$interval averages; that is, $\\tilde{\\bar{u}}_{i,j}^{**}\\longrightarrow\\bar{u}_{i|l}^{**}$.\n\\par\\qquad\\textbf{do $l=1,2,...,N_y\\cdot L$}\n\\par\\qquad\\qquad For each Gauss-Legendre node, update the $x-$interval averages by applying the 1D algorithm;\n\\par\\qquad\\qquad that is, $\\bar{u}_{i|l}^{**}\\longrightarrow\\bar{u}_{i|l}^{n+1}$.\n\\par\\qquad\\textbf{end do}\n\\par\\qquad Use equation \\eqref{intervaltocell1} to get updated cell averages; that is, $\\bar{u}_{i|l}^{n+1}\\longrightarrow\\tilde{\\bar{u}}_{i,j}^{n+1}$.\n\\\\\n\\xdashthick\n\n\n\n\n\\section{The EL-RK-FV method for convection-diffusion equations}\n\nThroughout this section, overlines (e.g., $\\overline{u}_j(t)$) denote \\textit{uniform} cell averages, and tildes (e.g., $\\tilde{u}_j(t)$) denote \\textit{nonuniform} cell averages. We discuss the EL-RK-FV algorithm for the convection-diffusion equation \\eqref{CDeqn}. Since we use dimensional splitting methods for two-dimensional problems, it suffices to discuss the 1D EL-RK-FV algorithm for problems of the form\n\\begin{equation}\n    \\label{1Dconvectiondiffusion}\n    u_t + f(u)_x = \\epsilon u_{xx} + g(x,t),\n\\end{equation}\nwhere we impose periodic boundary conditions. Rewriting equation \\eqref{1Dconvectiondiffusion} in divergence form, integrating over the same space-time region $\\Omega_j$ as in the $\\epsilon=0$ case, applying the divergence theorem and fundamental theorem of calculus, integrating over $[t^n,t^{n+1}]$, and converting to the time-differential form, we get the semi-discrete formulation\n\\begin{equation}\n    \\label{semidiscrete3}\n    \\frac{d}{dt}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = -\\left[\\hat{F}_{j+\\frac{1}{2}}(t)-\\hat{F}_{j-\\frac{1}{2}}(t)\\right] + \\epsilon\\int_{\\tilde{I}_j(t)}{u_{xx}(x,t)dx} + \\int_{\\tilde{I}_j(t)}{g(x,t)dx},\n\\end{equation}\nwhere $\\hat{F}_{j+\\frac{1}{2}}(t)$ is any monotone numerical flux (e.g., Lax-Friedrichs flux) and $F_{j+\\frac{1}{2}}(t)$ is the modified flux function defined in \\eqref{semidiscrete1}.\n\n\\subsection{Computing the uniform cell averages of $u_{xx}$}\n\nWhen evolving equation \\eqref{semidiscrete3} we will need to compute the uniform cell averages of $u_{xx}(x,t)$, defined by\n\\begin{equation}\n    \\overline{u}_{xx,j}(t)\\coloneqq \\frac{1}{\\Delta x}\\int_{I_j}{u_{xx}(x,t)dx}.\n\\end{equation}\nWe use the centered five-point stencil $\\{j-2,j-1,j,j+1,j+2\\}$ for linear reconstruction. Let $P(x\\in I_j)$ be the linear reconstruction polynomial over the centered five-point stencil. After some tedious algebra, we find that the uniform cell averages $\\overline{u}_{xx,j}(t)$ can be expressed in terms of the uniform cell averages $\\overline{u}_j(t)$ with fourth-order accuracy using the equation\n\\begin{equation}\n    \\label{5stencil}\n    \\overline{u}_{xx,j}(t) = \\frac{1}{\\Delta x^2}\n    \\begin{bmatrix*}\n        -\\frac{1}{12}&\\frac{4}{3}&-\\frac{5}{2}&\\frac{4}{3}&-\\frac{1}{12}\n    \\end{bmatrix*}\n    \\begin{bmatrix*}\n        \\overline{u}_{j-2}(t)\\\\\n        \\overline{u}_{j-1}(t)\\\\\n        \\overline{u}_{j}(t)\\\\\n        \\overline{u}_{j+1}(t)\\\\\n        \\overline{u}_{j+2}(t)\n    \\end{bmatrix*}.\n\\end{equation}\n\n\\noindent For implementation it is easier to express equation \\eqref{5stencil} in matrix form (assuming periodic or zero boundary conditions). We denote this matrix $\\mathbf{D}_4$ in Algorithm 5. Computing the nonuniform cell averages $\\tilde{u}_{xx,j}(t)$ can now be done using Algorithm 1.\n\n\\subsection{Time evolution with implicit-explicit Runge-Kutta methods}\n\nWe can split equation \\eqref{1Dconvectiondiffusion} into its non-stiff and stiff terms,\n\\begin{equation}\n    \\label{FG}\n    u_t = \\mathcal{F}(u;x,t) + \\mathcal{G}(u;x,t),\n\\end{equation}\nwhere $\\mathcal{F}(u;x,t)=-f(u)_x$ is the non-stiff convective term, and $\\mathcal{G}(u;x,t)=\\epsilon u_{xx}+g(x,t)$ is the stiff diffusive (and source) term. Discretization methods used for such problems are implicit-explicit (IMEX) RK schemes \\cite{Ascher1997,Conde2017,Higueras2014}. The intuition behind these schemes is straightforward -- evolve the non-stiff term explicitly, and evolve the stiff term implicitly. As such, each stage in the RK method will involve explicitly evaluating the non-stiff term, and solving a linear system due to the stiff term.\\\\\n\\ \\\\\nAll that's needed are the possibly nonuniform cell averages at each RK stage $\\mu=1,...,s$ over the space-time regions $\\Omega_j$. There are two approaches one can take to approximate the possibly nonuniform cell averages at each RK stage: (1) have a single space-time partition for multiple RK stages and directly update the solution to the possibly nonuniform traceback grid at each intermediate RK stage, or (2) have multiple space-time partitions for intermediate RK stages, and compute the uniform cell averages at time $t^{(\\mu)}$ and project them onto the possibly nonuniform traceback grid via Algorithm 1. Although the first approach is more intuitive, it is computationally expensive since the system to be solved (from the implicit part) will depend on the measure of each cell. We choose the second approach, as computing the uniform cell averages at time $t^{(\\mu)}$ requires solving a system dependent only on the background uniform mesh size $\\Delta x$. Since we choose the second approach, we will need to: (1) solve for the \\textit{uniform} cell averages at each consecutive stage, and (2) project the uniform cell averages onto the possibly nonuniform traceback grid via Algorithm 1. Recall the uniform cell averages $\\overline{u}_{xx,j}^{(\\mu)}$ can be computed from the uniform cell averages $\\overline{u}_{j}^{(\\mu)}$ using equation \\eqref{5stencil}.\\\\\n\\ \\\\\nFor simplicity, we only use the IMEX RK schemes outlined in \\cite{Ascher1997}. As per the notation used by Ascher, et al., \\textit{IMEX($s$,$\\sigma$,$p$)} denotes using an $s-$stage diagonally-implicit Runge-Kutta (DIRK) scheme for $\\mathcal{G}(u;x,t)$, using a $\\sigma-$stage explicit RK scheme for $\\mathcal{F}(u;x,t)$, and being of combined order $p$. Consider the semi-discrete formulation \\eqref{semidiscrete3} rewritten as\n\\begin{equation}\n    \\frac{d}{dt}\\int_{\\tilde{I}_j(t)}{u(x,t)dx} = \\mathcal{F}(u;t) + \\mathcal{G}(u;t),\n\\end{equation}\nwhere we redefine $\\mathcal{F}(u;t)=-\\left[\\hat{F}_{j+\\frac{1}{2}}(t)-\\hat{F}_{j-\\frac{1}{2}}(t)\\right]$ and $\\mathcal{G}(u;t) =  \\epsilon\\int_{\\tilde{I}_j(t)}{u_{xx}(x,t)dx} + \\int_{\\tilde{I}_j(t)}{g(x,t)dx}$. The IMEX($s$,$\\sigma$,$p$) RK schemes are expressed with two Butcher tables: one for the implicit RK method, and another for the explicit RK method.\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{Implicit Scheme}\n    \\begin{tabular}{c|lllll}\n        0&0&0&0&$\\hdots$&0\\\\\n        $c_1$&0&$a_{11}$&0&$\\hdots$&0\\\\\n        $c_2$&0&$a_{21}$&$a_{22}$&$\\hdots$&0\\\\\n        $\\vdots$&$\\vdots$&$\\vdots$&$\\vdots$&$\\ddots$&$\\vdots$\\\\\n        $c_s$&0&$a_{s1}$&$a_{s2}$&$\\hdots$&$a_{ss}$\\\\\n        \\hline\n        &0&$b_1$&$b_2$&$\\hdots$&$b_s$\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{Explicit Scheme}\n    \\begin{tabular}{c|lllll}\n        0&0&0&0&$\\hdots$&0\\\\\n        $c_1$&$\\hat{a}_{21}$&0&0&$\\hdots$&0\\\\\n        $c_2$&$\\hat{a}_{31}$&$\\hat{a}_{32}$&0&$\\hdots$&0\\\\\n        $\\vdots$&$\\vdots$&$\\vdots$&$\\vdots$&$\\ddots$&$\\vdots$\\\\\n        $c_s$&$\\hat{a}_{\\sigma 1}$&$\\hat{a}_{\\sigma 2}$&$\\hat{a}_{\\sigma 3}$&$\\hdots$&0\\\\\n        \\hline\n        &$\\hat{b}_1$&$\\hat{b}_2$&$\\hat{b}_3$&$\\hdots$&$\\hat{b}_{\\sigma}$\n        \\end{tabular}\n    \\end{minipage}\n\\end{table}\n\n\\noindent Defining $U^{(\\mu)}\\coloneqq\\int_{\\tilde{I}_j(t^{(\\mu)})}{u(x,t^{(\\mu)})dx}$, the IMEX($s$,$\\sigma$,$p$) scheme over the space-time region $\\Omega_j$ is as follows:\n\\begin{subequations}\n\\begin{align}\n\\begin{split}\n\\label{IMEX}\n    U^{n+1} &= U^n + \\Delta t\\sum\\limits_{\\mu=1}^{s}{b_{\\mu}K_{\\mu}} + \\Delta t\\sum\\limits_{\\mu=1}^{\\sigma}{\\hat{b}_{\\mu}\\hat{K}_{\\mu}},\n\\end{split}\\\\\n\\begin{split}\n\\label{K}\n    K_{\\mu} &= \\mathcal{G}(U^{(\\mu)};t^{(\\mu)}),\\qquad \\mu=1,2,...,s,\n\\end{split}\\\\\n\\begin{split}\n\\label{Khat1}\n    \\hat{K}_1 &= \\mathcal{F}(U^n;t^n),\n\\end{split}\\\\\n\\begin{split}\n\\label{Khat}\n    \\hat{K}_{\\mu+1} &= \\mathcal{F}(U^{(\\mu)};t^{(\\mu)}),\\qquad \\mu=1,2,...,s.\n\\end{split}\n\\end{align}\n\\end{subequations}\n\n\\noindent More precisely,\n\\begin{subequations}\n\\begin{align}\n\\begin{split}\n    K_{\\mu} &= \\epsilon\\int_{\\tilde{I}_j(t^{(\\mu)})}{u_{xx}(x,t^{(\\mu)})dx} + \\int_{\\tilde{I}_j(t^{(\\mu)})}{g(x,t^{(\\mu)})dx},\n\\end{split}\\\\\n\\begin{split}\n    \\hat{K}_{\\mu+1} &= -\\left[\\hat{F}_{j+\\frac{1}{2}}(t^{(\\mu)}) - \\hat{F}_{j-\\frac{1}{2}}(t^{(\\mu)})\\right].\n\\end{split}\n\\end{align}\n\\end{subequations}\nBased on the IMEX RK method, the solution $U^{(\\mu)}$ over the traceback grid can be approximated by\n\\begin{equation}\n\\label{Um}\n    U^{(\\mu)} = U^{n} + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu}{a_{\\mu,\\nu}K_{\\nu}} + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu}{\\hat{a}_{\\mu+1,\\nu}\\hat{K}_{\\nu}},\\qquad \\mu=1,2,...,s.\n\\end{equation}\nRecall that we choose \\textit{not} to directly update the solution over $\\Omega_j$. Instead, we opt to solve for the uniform cell averages at each RK stage and project them onto the corresponding possibly nonuniform traceback grid. We define sub-space-time regions ${}_{\\mu}\\Omega_j$ for the $\\mu$-th stage of the IMEX RK scheme, as shown in Figures \\ref{spacetime1} and \\ref{spacetime2}. The space-time region ${}_{\\mu}\\Omega_j$ traces back to time $t^n$ (using the same approximate characteristic speeds as in $\\Omega_j$) from time $t^{(\\mu)}$. Hence, at time $t^{(\\mu)}$ on the sub-space-time region ${}_{\\mu}\\Omega_j$ the grid is uniform. Lower left subscript $\\mu$ denotes values confined to the sub-space-time region ${}_{\\mu}\\Omega_j$. For example, ${}_{2}U^n$ are the possibly nonuniform definite integrals (and hence the possibly nonuniform cell averages) at time $t^n$ over the traceback cell in ${}_{2}\\Omega_j$, as seen in Figure \\ref{spacetime2}.\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n\t\\includegraphics[width=\\textwidth]{.\/images_spacetime\/spacetime1.png}\n\t\\caption{The space-time region ${}_{1}\\Omega_j$.}\n\t\\label{spacetime1}\n\\end{minipage}\n\\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n\t\\includegraphics[width=\\textwidth]{.\/images_spacetime\/spacetime2.png}\n\t\\caption{The space-time region ${}_{2}\\Omega_j$.}\n\t\\label{spacetime2}\n\\end{minipage}\n\\end{figure}\n\n\\noindent The desired uniform cell averages $\\overline{u}_j^{(\\mu)} = {}_{\\mu}U^{(\\mu)}\/\\Delta x$ at stage $\\mu$ can be obtained by computing the first $\\mu-$stages of the IMEX RK scheme over the sub-space-time region ${}_{\\mu}\\Omega_j$, given below.\n\\begin{subequations}\n\\label{IMEXsubstage}\n\\begin{align}\n\\begin{split}\n    {}_{\\mu}U^{(\\mu)}& = {}_{\\mu}U^n + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu}{(a_{\\mu,\\nu})({}_{\\mu}K_{\\nu})} + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu}{(\\hat{a}_{\\mu+1,\\nu})({}_{\\mu}\\hat{K}_{\\nu})}\n\\end{split}\\\\\n\\begin{split}\n    {}_{\\mu}K_{\\nu} &= \\mathcal{G}({}_{\\mu}U^{(\\nu)};t^{(\\nu)}),\\qquad \\nu=1,2,...,\\mu,\n\\end{split}\\\\\n\\begin{split}\n    {}_{\\mu}\\hat{K}_1 &= \\mathcal{F}({}_{\\mu}U^n;t^n),\n\\end{split}\\\\\n\\begin{split}\n {}_{\\mu}\\hat{K}_{\\nu+1} &= \\mathcal{F}({}_{\\mu}U^{(\\nu)};t^{(\\nu)}),\\qquad \\nu=1,2,...,\\mu.\n\\end{split}\n\\end{align}\n\\end{subequations}\n\\noindent More precisely,\n\\begin{subequations}\n\\begin{align}\n\\begin{split}\n    {}_{\\mu}K_1 &= \\epsilon\\int_{{}_{\\mu}\\tilde{I}_j(t^{(\\nu)})}{u_{xx}(x,t^{(\\nu)})dx} + \\int_{{}_{\\mu}\\tilde{I}_j(t^{(\\nu)})}{g(x,t^{(\\nu)})dx},\n\\end{split}\\\\\n\\begin{split}\n    {}_{\\mu}\\hat{K}_{\\nu+1} &= -\\left[\\hat{F}_{j+\\frac{1}{2}}(t^{(\\nu)}) - \\hat{F}_{j-\\frac{1}{2}}(t^{(\\nu)})\\right].\n\\end{split}\n\\end{align}\n\\end{subequations}\nNote that ${}_{\\mu}\\hat{K}_{\\nu+1}$ uses the cell averages restricted to the sub-space-time regions ${}_{\\mu}\\Omega_j$. Further note that equation \\eqref{IMEXsubstage} can recycle and reuse the uniform cell averages already computed during stages $1,2,...,\\mu-1$. Simply project the uniform cell averages ($\\overline{u}_j^{(\\nu)}$ or $\\overline{u}_{xx,j}^{(\\nu)}$, $\\nu=1,2,...,\\mu-1$) onto the traceback grids formed by the space-time regions ${}_{\\mu}\\Omega_{j}$. Although IMEX RK schemes are straightforward, it is easy to get lost in the notation. To help demonstrate the EL-RK-FV algorithm coupled with an IMEX RK scheme, we present the IMEX(2,2,2) case in Appendix D. The Butcher tables for other IMEX RK schemes are included in Appendix C. There are IMEX SSP RK schemes that can be found in \\cite{Conde2017,Higueras2014}.\\\\\n\\clearpage\n\\noindent\\xdashthick\\\\\n\\textbf{Algorithm 5.} EL-RK-FV algorithm coupled with an IMEX RK scheme \\cite{Ascher1997}, $IMEX(s,\\sigma,p)$\\\\\n\\xdashthin\\\\\n\\textbf{Input:} uniform cell averages $\\overline{u}_j^n$.\\\\\n\\textbf{Output:} uniform cell averages $\\overline{u}_j^{n+1}$.\\\\\n\\par\\qquad Compute the possibly nonuniform traceback cell averages $\\tilde{u}_{j}(t^n)$ using Algorithm 1.\n\\par\\qquad Store $\\hat{K}_1 = \\mathcal{F}(U^n;t^n)$. This is the traceback grid in $\\Omega_j$.\n\\par\\qquad\\textbf{do $\\mu=1,2,...,s$}\n\\par\\qquad\\qquad 1. Compute the uniform cell averages $\\overline{u}_j^{(\\mu)}$ at time $t^{(\\mu)}$ with equation \\eqref{IMEXsubstage} by:\n\\par\\qquad\\qquad\\qquad i. Restrict yourself to the space-time region ${}_{\\mu}\\Omega_j$.\n\\par\\qquad\\qquad\\qquad ii. Compute the values ${}_{\\mu}K_{\\nu}$ and ${}_{\\mu}\\hat{K}_{\\nu+1}$ at each stage $\\nu=1,2,...,\\mu-1$ using Algorithm 1\n\\par\\qquad\\qquad\\qquad to compute the necessary nonuniform cell averages, and Algorithm 2 for the modified flux\n\\par\\qquad\\qquad\\qquad function. Definite integrals of $g(x,t)$ can be computed using a Gaussian quadrature.\n\\par\\qquad\\qquad\\qquad iii. Recalling equation \\eqref{5stencil}, solve equation \\eqref{IMEXsubstage} by setting it up as the linear system\n\\[\\left(\\mathbf{I} - \\frac{a_{\\mu,\\mu}\\epsilon\\Delta t}{\\Delta x^2}\\mathbf{D}_4\\right){}_{\\mu}\\vv{U}^{(\\mu)} = {}_{\\mu}\\vv{U}^n + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu-1}{(a_{\\mu,\\nu})({}_{\\mu}\\vv{K}_{\\nu})} + \\Delta t\\sum\\limits_{\\nu=1}^{\\mu}{(\\hat{a}_{\\mu+1,\\nu})({}_{\\mu}\\vv{\\hat{K}}_{\\nu})} + a_{\\mu,\\mu}\\Delta t\\vv{g}(x,t^{(\\mu)}),\\]\n\\par\\qquad\\qquad\\qquad where $\\vv{g}_j(x,t^{(\\mu)}) = \\int_{I_j}{g(x,t^{(\\mu)})dx}$.\n\\par\\qquad\\qquad\\qquad iv. Store the uniform cell averages $\\overline{u}_j^{(\\mu)} = {}_{\\mu}U_j^{(\\mu)}\/\\Delta x$.\n\\par\\qquad\\qquad 2. Compute and store the uniform cell averages $\\overline{u}_{xx,j}^{(\\mu)}$ using equation \\eqref{5stencil}.\n\\par\\qquad\\qquad 3. Compute the possibly nonuniform traceback cell averages $\\tilde{u}_j^{(\\mu)}$ using Algorithm 1.\n\\par\\qquad\\qquad 4. Compute the possibly nonuniform traceback cell averages $\\tilde{u}_{xx,j}^{(\\mu)}$ using Algorithm 1.\n\\par\\qquad\\qquad 5. Compute and store $K_{\\mu}$ and $\\hat{K}_{\\mu+1}$. Note that we are back on the possibly nonuniform\n\\par\\qquad\\qquad traceback grid consisting of cells $\\tilde{I}_j(t^{(\\mu)})$.\n\\par\\qquad\\textbf{end do}\\\\\n\\par\\qquad Compute $U^{n+1} = U^n + \\Delta t\\sum\\limits_{\\mu=1}^{s}{b_{\\mu}K_{\\mu}} + \\Delta t\\sum\\limits_{\\mu=1}^{\\sigma}{\\hat{b}_{\\mu}\\hat{K}_{\\mu}}$.\n\\par\\qquad Compute the uniform cell averages $\\overline{u}_j^{n+1} = U_j^{n+1}\/\\Delta x$.\\\\\n\\xdashthick\n\n\\subsection{Mass conservation}\n\nWe now show that the 1D EL-RK-FV algorithm is mass conservative when coupled with any IMEX RK scheme in \\cite{Ascher1997}. Since we extend the EL-RK-FV algorithm to higher dimensions via dimensional splitting in this paper, showing mass conservation for the 1D problem suffices. Note that mass conservation of the $\\epsilon=0$ case can be proved just as easily.\n\n\\begin{prop}\n(Mass conservative). The 1D EL-RK-FV algorithm coupled with any IMEX RK scheme in \\cite{Ascher1997} for (non)linear convection-diffusion equations is mass conservative, assuming the source term $g(x,t)=0$ and periodic boundary conditions.\n\\end{prop}\n\n\\begin{proof}\nMaking use of the semi-discrete form of the convection-diffusion equation,\n\\begin{align}\n\\begin{split}\n    \\int_{I_j}{u(x,t^{n+1})dx} &= \\int_{\\tilde{I}_j(t^n)}{u(x,t^n)dx} - \\int_{t^n}^{t^{n+1}}{\\left\\{\\hat{F}_{j+\\frac{1}{2}}(t)-\\hat{F}_{j-\\frac{1}{2}}(t) - \\epsilon\\int_{\\tilde{I}_j(t)}{u_{xx}(x,t)dx}\\right\\}dt}\\\\\n    &= \\int_{\\tilde{I}_j(t^n)}{u(x,t^n)dx} - \\int_{t^n}^{t^{n+1}}{\\left\\{\\hat{F}_{j+\\frac{1}{2}}(t)-\\hat{F}_{j-\\frac{1}{2}}(t) - \\epsilon\\left(u_x(\\tilde{x}_{j+\\frac{1}{2}}(t),t) - u_x(\\tilde{x}_{j-\\frac{1}{2}}(t),t)\\right)\\right\\}dt}.\n\\end{split}\n\\end{align}\nSumming over all $j=1,2,...,N_x$ and making use of the periodic boundary conditions,\n\\begin{equation}\n    \\int_a^b{u(x,t^{n+1})dx} = \\int_a^b{u(x,t^n)dx}.\n\\end{equation}\n\\end{proof}\n\n\n\n\n\\section{Numerical tests}\n\nIn this section, we present results applying the proposed EL-RK-FV algorithm to various benchmark problems. In particular, we include error tables and error plots to investigate the spatial and temporal convergence of the algorithm. Mass conservation is also numerically verified by applying the proposed algorithm to the 0D2V (zero dimensions in physical space and two dimensions in velocity space) Leonard-Bernstein Fokker-Planck equation. We assume a uniform mesh, apply WENO-AO(5,3) in Algorithms 1 and 2, use three Gauss-Legendre nodes in Algorithm 3, and use the fourth-order approximation given by equation \\eqref{5stencil} for the diffusive term. Unless otherwise stated, for the time-stepping we use the fourth-order RK method for pure convection problems, and IMEX(2,3,3) for convection-diffusion equations. We also use second-order Strang splitting for two-dimensional convection-diffusion equations. Although higher-order splitting methods can be used for pure convection problems, it is well-known that negative time integration can lead to significant instabilities when dealing with a diffusive term.\\\\\n\n\\noindent There are three sources of error: spatial approximation, time-stepping, and splitting. Depending on the CFL number and test problem, these three sources of error will influence the observed order of convergence. We compute the $L^1$, $L^2$, and $L^{\\infty}$ errors (in one-dimension),\n\\begin{subequations}\n\\begin{equation}\n    \\norm{\\overline{u}-\\overline{u}_{exact}}_1 = \\Delta x\\sum\\limits_{j=1}^{N_x}{|\\overline{u}_j-\\overline{u}_{exact,j}|}\n\\end{equation}\n\\begin{equation}\n    \\norm{\\overline{u}-\\overline{u}_{exact}}_2 = \\Delta x\\sqrt{\\sum\\limits_{j=1}^{N_x}{|\\overline{u}_j-\\overline{u}_{exact,j}|^2}}\n\\end{equation}\n\\begin{equation}\n    \\norm{\\overline{u}-\\overline{u}_{exact}}_{\\infty} = \\max\\limits_{1\\leq j\\leq N_x}{|\\overline{u}_j-\\overline{u}_{exact,j}|}\n\\end{equation}\n\\end{subequations}\n\n\\noindent Note that for the norms defined above, $\\norm{\\overline{u}-\\overline{u}_{exact}}_1\\leq  |\\mathcal{D}|\\norm{\\overline{u}-\\overline{u}_{exact}}_{\\infty}$.\n\n\n\\subsection{Pure convection problems: one-dimensional tests}\n\n\\begin{example}(1D transport with constant coefficient)\n\\begin{equation}\n    \\label{1Dtransport_constant}\n    u_t + u_x = 0,\\qquad x\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution $u(x,t)=\\sin{(x-t)}$. The errors provided in Table \\ref{1Dtransport_constant_table} verify the convergence of the EL-RK-FV method when using WENO-AO(5,3) and forward Euler. As expected, we see fifth-order convergence despite the large CFL number. There is no temporal error for the convective part since the characteristics are traced exactly and hence $F_{j+\\frac{1}{2}}(t) = 0$ for all $t\\in[t^n,t^{n+1}]$ and $j=1,2,...,N_x$.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1Dtransport_constant} with forward Euler at $T_f=1$.}\n\\label{1Dtransport_constant_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&1.09E-08&-&4.83E-09&-&2.86E-09&-\\\\\n100&3.34E-10&5.03&1.48E-10&5.03&8.34E-11&5.10\\\\\n200&9.86E-12&5.08&4.37E-12&5.08&2.51E-12&5.06\\\\\n400&2.80E-13&5.14&1.43E-13&4.94&1.91E-13&3.72\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\n\n\n\\begin{example}(1D transport with variable coefficient in space)\n\\begin{equation}\n    \\label{1Dtransport_variablespace}\n    u_t + (\\sin{(x)}u)_x = 0,\\qquad x\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution\n\\[u(x,t) = \\frac{\\sin{(2\\arctan{(e^{-t}\\tan{(x\/2)})})}}{\\sin{(x)}}.\\]\nAs seen in Table \\ref{1Dtransport_variablespace_table}, we observe the high-order convergence. As the CFL number (and hence the time step) increases, the temporal error starts to play more of a role, as evidenced by the fourth-order convergence. We verify the high-order temporal convergence in Figure \\ref{1Dtransport_variablespace_plot} by fixing the mesh $N_x=400$ and varying the CFL from 0.2 to 20.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1Dtransport_variablespace} with RK4 at $T_f=1$.}\n\\label{1Dtransport_variablespace_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline\n\\multicolumn{7}{|c|}{$CFL=0.3$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&2.76E-04&-&2.53E-04&-&3.42E-04&-\\\\\n100&3.05E-06&6.50&2.53E-06&6.64&2.94E-06&6.87\\\\\n200&9.78E-08&4.96&7.90E-08&5.00&9.86E-08&4.90\\\\\n400&3.24E-09&4.91&2.59E-09&4.93&3.23E-09&4.93\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&6.81E-02&-&4.11E-02&-&4.51E-02&-\\\\\n100&1.18E-03&5.85&6.23E-04&6.04&6.38E-04&6.15\\\\\n200&5.89E-05&4.32&3.63E-05&4.10&5.46E-05&3.54\\\\\n400&3.71E-06&3.99&2.57E-06&3.82&4.31E-06&3.66\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\\begin{example}(1D transport with variable coefficient in time)\n\\begin{equation}\n    \\label{1Dtransport_variabletime}\n    u_t + \\left(\\frac{u}{t+1}\\right)_x = 0,\\qquad x\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution $u(x,t)=\\text{exp}(-5(x-\\log{(t+1)}-\\pi)^2)$. Periodic boundary conditions are a valid assumption for sufficiently thin Gaussian curves and small enough times. The expected high-order convergence for both small and large CFL numbers is seen in Table \\ref{1Dtransport_variabletime_table}. Fixing the mesh $N_x=400$ and varying the CFL number from 0.2 to 20, fifth-order convergence in time is seen in Figure \\ref{1Dtransport_variabletime_plot}. Observe that there are two optimal CFL numbers for this mesh.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1Dtransport_variabletime} with RK4 at $T_f=1$.}\n\\label{1Dtransport_variabletime_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&6.36E-03&-&5.80E-03&-&8.23E-03&-\\\\\n100&2.37E-04&4.75&2.28E-04&4.67&5.33E-04&3.95\\\\\n200&1.71E-06&7.12&1.30E-06&7.46&2.36E-06&7.82\\\\\n400&4.40E-08&5.28&3.84E-08&5.08&6.02E-08&5.29\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&6.71E-02&-&5.72E-02&-&8.15E-02&-\\\\\n100&7.74E-04&6.44&6.48E-04&6.46&1.06E-03&6.27\\\\\n200&1.52E-05&5.67&1.26E-05&5.69&1.76E-05&5.91\\\\\n400&9.65E-07&3.98&8.11E-07&3.96&9.61E-07&4.20\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_hyperbolic_1D\/errorplot_variablespace_5s4t_Nx400_T0,5.png}\n    \\caption{RK4, Final time $T_f=0.5$.}\n    \\label{1Dtransport_variablespace_plot}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_hyperbolic_1D\/errorplot_variabletime_5s4t_Nx400_T0,5.png}\n    \\caption{RK4, Final time $T_f=0.5$.}\n    \\label{1Dtransport_variabletime_plot}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Hyperbolic conservation laws: two-dimensional tests}\n\n\n\\begin{example}(2D transport with constant coefficient)\n\\begin{equation}\n    \\label{2Dtransport_constant}\n    u_t + u_x + u_y = 0,\\qquad x,y\\in[-\\pi,\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution $u(x,y,t)=\\sin{(x+y-2t)}$. The expected high-order convergence is shown in Table \\ref{2Dtransport_constant_table} when using Strang splitting. \n\n Just like equation \\eqref{1Dtransport_constant}, there is no temporal error for the convective part since the characteristics are traced exactly. \n\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2Dtransport_constant} with forward Euler and $CFL=8$ at $T_f=1$.}\n\\label{2Dtransport_constant_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{Strang splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&4.24E-09&-&7.49E-10&-&1.69E-10&-\\\\\n200&1.27E-10&5.05&2.25E-11&5.05&5.15E-12&5.03\\\\\n300&1.60E-11&5.11&2.84E-12&5.11&8.23E-13&4.52\\\\\n400&3.57E-12&5.21&6.88E-13&4.92&3.68E-13&2.80\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\n\n\\begin{example}(Rigid body rotation)\n\\begin{equation}\n    \\label{2Dtransport_rigidbody}\n    u_t - yu_x + xu_y = 0,\\qquad x,y\\in[-\\pi,\\pi]\n\\end{equation}\nwith periodic boundary conditions. We choose the exact solution $u(x,y,t)=u(x,y,t=0)=\\text{exp}(-3(x^2+y^2))$ for convergence tests. The convergence results are presented in Tables \\ref{2Dtransport_rigidbody_table1} and \\ref{2Dtransport_rigidbody_table2}. Strang splitting dominates the error and we observe the expected second-order convergence. Whereas, the spatial error dominates when using fourth-order splitting as evidenced by the fifth-order convergence. The error plot using a fixed mesh $N_x=N_y=200$ and varying the CFL number from 0.1 to 50 is shown in Figure \\ref{rigidplot1}. Second-order convergence in time is observed when using Strang splitting, and fourth-order convergence is observed when using fourth-order splitting. We note that comparable convergence results were observed for the non-symmetric initial condition $u(x,y,t=0)=\\text{exp}(-3x^2-2y^2)$. To demonstrate the effectiveness of WENO-AO controlling spurious oscillations we choose the initial condition $u(x,y,t=0)=1$ if $x,y\\in[-\\pi\/2,\\pi\/2]$; $u(x,y,t=0)=0$ otherwise. With a fixed mesh $N_x=N_y=100$ and $CFL=2.2$, we compute the solution up to time $T_f=2\\pi$ using SSP RK3. The discontinuities are smoothed out and no spurious oscillations occur.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2Dtransport_rigidbody} with RK4 and $CFL=0.95$ at $T_f=0.5$.}\n\\label{2Dtransport_rigidbody_table1}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{Strang splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&3.57E-05&-&1.54E-05&-&2.43E-05&-\\\\\n200&1.83E-06&4.29&9.94E-07&3.95&1.23E-06&4.31\\\\\n300&8.01E-07&2.04&4.35E-07&2.04&4.72E-07&2.36\\\\\n400&4.50E-07&2.01&2.44E-07&2.01&2.55E-07&2.13\\\\\n\\hline\n\\multicolumn{7}{|c|}{Fourth-order splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&1.05E-04&-&4.66E-05&-&8.95E-05&-\\\\\n200&1.08E-06&6.60&6.13E-07&6.25&7.74E-07&6.85\\\\\n300&1.39E-07&5.06&8.12E-08&4.99&1.02E-07&5.00\\\\\n400&3.32E-08&4.99&1.93E-08&4.99&2.43E-08&4.99\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2Dtransport_rigidbody} with RK4 and $CFL=8$ at $T_f=0.5$.}\n\\label{2Dtransport_rigidbody_table2}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{Strang splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&4.97E-04&-&2.68E-04&-&2.78E-04&-\\\\\n200&1.24E-04&2.00&6.74E-05&1.99&6.86E-05&2.02\\\\\n300&5.59E-05&1.97&3.03E-05&1.97&3.08E-05&1.97\\\\\n400&3.20E-05&1.94&1.73E-05&1.94&1.76E-05&1.94\\\\\n\\hline\n\\multicolumn{7}{|c|}{Fourth-order splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&4.95E-05&-&2.46E-05&-&6.82E-05&-\\\\\n200&4.84E-07&6.68&2.92E-07&6.39&4.77E-07&7.16\\\\\n300&6.17E-08&5.08&3.86E-08&4.99&6.19E-08&5.03\\\\\n400&1.47E-08&4.99&9.22E-09&4.88&1.47E-08&5.00\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/rotation_Nx=Ny=200_T0,5_RK4.png}\n    \\caption{Error plot for \\eqref{2Dtransport_rigidbody} with RK4 at $T_f=0.5$. $N_x=N_y=200$.}\n    \\label{rigidplot1}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/rigid_2split_NxNy100_CFL2,2.png}\n    \\caption{Plot of the numerical solution to \\eqref{2Dtransport_rigidbody} with SSP RK3 and $CFL=2.2$ at $T_f=2\\pi$. $N_x=N_y=100$.}\n    \\label{rigidplot2}\n\\end{minipage}\n\\end{figure}\n\n\n\n\n\\begin{example}(Swirling deformation)\n\\begin{equation}\n    \\label{2Dtransport_swirling}\n    u_t - \\left(\\cos^2{(x\/2)}\\sin{(y)}g(t)u\\right)_x + \\left(\\sin{(x)}\\cos^2{(y\/2)}g(t)u\\right)_y = 0,\\qquad x,y\\in[-\\pi,\\pi]\n\\end{equation}\nWhen testing convergence we set $g(t)=\\cos{(\\pi t\/T_f)}\\pi$ and choose the initial condition to be the smooth (with $C^5$ smoothness) cosine bell\n\\begin{equation}\n    \\label{cosbell}\n    u(x,y,t=0)=\n\\begin{cases}\n    r_0^b\\cos^6{\\left(\\frac{r^b(x,y)\\pi}{2r_0^b}\\right)},&\\text{if\\ }r^b(x,y)<r_0^b,\\\\\n    0,&\\text{otherwise,}\n\\end{cases}\n\\end{equation}\nwhere $r_0^b=0.3\\pi$ and $r^b(x,y)=\\sqrt{(x-x_0^b)^2+(y-y_0^b)^2}$ with $(x_0^b,y_0^b)=(0.3\\pi,0)$. The convergence results under spatial mesh refinement are presented in Tables \\ref{2Dtransport_swirling_table1} and \\ref{2Dtransport_swirling_table2}. Surprisingly, high-order convergence is observed in all test cases, even for the large CFL number of 8. In particular, we observed super-convergence for $CFL=8$ when using Strang splitting. We got comparable convergence results when letting the initial condition be: (1) a cosine bell of $C^3$ smoothness, and (2) the cosine bell \\eqref{cosbell} but with $x_0^b=0.6\\pi$ and the width in the $x-$direction scaled by a factor of 1\/2.\\\\\n\\ \\\\\nThe temporal orders of convergence are shown in Figure \\ref{swirlplot1} using a fixed mesh $N_x=N_y=200$ and varying the CFL number from 0.1 to 25. When using Strang splitting the temporal convergence switches from second-order to third-order, indicating that for very large CFL numbers the splitting error does not dominate the time-stepping error as much. Fourth-order convergence is observed when using fourth-order splitting. To demonstrate the effectiveness of WENO-AO in controlling spurious oscillations we set $g(t)=1$ and choose the initial condition \\cite{Leveque1996}\n\\begin{equation}\nu(x,y,t=0) = \n\\begin{cases}\n    1,&\\text{if\\ }r^b(x,y)<r_0^b,\\\\\n    0,&\\text{otherwise,}\n\\end{cases}\n\\end{equation}\nwhere $r_0^b=8\\pi\/5$ and $r^b(x,y)=\\sqrt{(x-x_0^b)^2+(y-y_0^b)^2}$ with $(x_0^b,y_0^b)=(\\pi,\\pi)$. With a fixed mesh $N_x=N_y=100$ and $CFL=8$, we compute the solution up to time $T_f=5\\pi$ using SSP RK3 and Strang splitting. The discontinuities are smoothed out and no spurious oscillations occur.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2Dtransport_swirling} with RK4 and $CFL=0.95$ at $T_f=1.5$.}\n\\label{2Dtransport_swirling_table1}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{Strang splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&5.17E-03&-&6.11E-03&-&2.04E-02&-\\\\\n200&1.69E-04&4.94&1.69E-04&5.18&4.80E-04&5.41\\\\\n300&3.12E-05&4.16&3.85E-05&3.64&1.41E-04&3.01\\\\\n400&8.29E-06&4.61&1.12E-05&4.66&4.66E-05&3.86\\\\\n\\hline\n\\multicolumn{7}{|c|}{Fourth-order splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&1.42E-02&-&1.73E-02&-&5.25E-02&-\\\\\n200&3.98E-04&5.16&3.70E-04&5.54&9.15E-04&5.84\\\\\n300&7.69E-05&4.06&9.02E-05&3.48&3.24E-04&2.56\\\\\n400&2.20E-05&4.35&2.89E-05&3.96&1.17E-04&3.52\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2Dtransport_swirling} with RK4 and $CFL=8$ at $T_f=1.5$.}\n\\label{2Dtransport_swirling_table2}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{Strang splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&1.90E-03&-&2.11E-03&-&6.83E-03&-\\\\\n200&1.02E-04&4.23&8.88E-05&4.57&2.47E-04&4.79\\\\\n300&1.90E-05&4.13&1.79E-05&3.94&6.36E-05&3.35\\\\\n400&2.82E-06&6.63&4.11E-06&5.12&1.98E-05&4.05\\\\\n\\hline\n\\multicolumn{7}{|c|}{Fourth-order splitting}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n100&3.89E-03&-&4.31E-03&-&1.43E-02&-\\\\\n200&1.30E-04&4.90&1.34E-04&5.01&4.42E-04&5.01\\\\\n300&2.41E-05&4.16&3.32E-05&3.44&1.40E-04&2.84\\\\\n400&6.54E-06&4.53&1.02E-05&4.10&4.81E-05&3.70\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/swirl_NxNy200_T1,5.png}\n    \\caption{Error plot for \\eqref{2Dtransport_swirling} with RK4 at $T_f=1.5$. $N_x=N_y=200$.}\n    \\label{swirlplot1}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/swirl_hurricane_2split_NxNy100_CFL8_T5pi.png}\n    \\caption{Plot of the numerical solution to \\eqref{2Dtransport_swirling} with $g(t)=1$, SSP RK3 and $CFL=8$ at $T_f=5\\pi$. $N_x=N_y=100$.}\n    \\label{swirlplot2}\n\\end{minipage}\n\\end{figure}\n\n\n\n\\subsection{Convection-diffusion equations: one-dimensional tests}\n\n\n\\begin{example}(1D equation with constant coefficient)\n\\begin{equation}\n    \\label{1D_CD_constant}\n    u_t + u_x = \\epsilon u_{xx},\\qquad x\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution $u(x,t)=\\sin{(x-t)}\\text{exp}(-\\epsilon t)$. We set $\\epsilon=1$. The convergence results under spatial mesh refinement are shown in Table \\ref{1D_CD_constant_table} for $CFL=0.95$ and $CFL=8$. In both cases we observe the expected third-order convergence since we are using IMEX(2,3,3) for the time-stepping. Note that there is no temporal error for the convective part since the characteristics are traced exactly and hence $F_{j+\\frac{1}{2}}(t)=0$ for all $t\\in[t^n,t^{n+1}]$ and $j=1,2,...,N_x$. Figure \\ref{1D_CD_constant_plot} shows the expected third-order convergence in time using fixed mesh $N_x=400$ and varying the CFL number from 0.1 to 15.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1D_CD_constant} with IMEX(2,3,3) at $T_f=1$.}\n\\label{1D_CD_constant_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&1.87E-04&-&8.26E-05&-&4.66E-05&-\\\\\n100&2.55E-05&2.87&1.13E-05&2.87&6.36E-06&2.87\\\\\n200&3.34E-06&2.93&1.48E-06&2.93&8.35E-07&2.93\\\\\n400&4.31E-07&2.95&1.91E-07&2.95&1.08E-07&2.95\\\\\n800&5.44E-08&2.99&2.41E-08&2.99&1.36E-08&2.99\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&6.87E-02&-&3.04E-02&-&1.72E-02&-\\\\\n100&1.09E-02&2.66&4.82E-03&2.66&2.72E-03&2.66\\\\\n200&1.63E-03&2.74&7.22E-04&2.74&4.07E-04&2.74\\\\\n400&2.27E-04&2.84&1.01E-04&2.84&5.69E-05&2.84\\\\\n800&3.03E-05&2.91&1.34E-05&2.91&7.57E-06&2.91\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{example}(1D equation with variable coefficient)\n\\begin{equation}\n    \\label{1D_CD_variable}\n    u_t + (\\sin{(x)}u)_x = \\epsilon u_{xx} + g,\\qquad x\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and $g(x,t)=\\sin{(2x)}\\text{exp}(-\\epsilon t)$ and exact solution $u(x,t)=\\sin{(x)}\\text{exp}(-\\epsilon t)$. We set $\\epsilon=1$. Table \\ref{1D_CD_variable_table} shows the convergence results under spatial mesh refinement. Third-order convergence in space is observed for $CFL=0.95$. Whereas, the convergence for $CFL=8$ is roughly order 2.6 since the time-stepping start to dominate. We note that the order of convergence for IMEX(2,3,3) under increasing the CFL number dips slightly below three for larger CFL numbers. We use fixed mesh $N_x=400$ and vary the CFL number from 0.1 to 15 for the error plot showing the temporal order of convergence in Figure \\ref{1D_CD_variable_plot}.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1D_CD_variable} with IMEX(2,3,3) at $T_f=1$.}\n\\label{1D_CD_variable_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&2.10E-03&-&9.99E-04&-&7.70E-04&-\\\\\n100&3.99E-04&2.39&1.88E-04&2.41&1.44E-04&2.42\\\\\n200&6.41E-05&2.64&2.99E-05&2.65&2.27E-05&2.66\\\\\n400&9.84E-06&2.70&4.57E-06&2.71&3.44E-06&2.73\\\\\n800&1.30E-06&2.92&6.04E-07&2.92&4.53E-07&2.92\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&7.50E-01&-&4.34E-01&-&3.92E-01&-\\\\\n100&1.26E-01&2.58&6.60E-02&2.72&6.63E-02&2.56\\\\\n200&1.74E-02&2.85&8.31E-03&2.99&6.50E-03&3.35\\\\\n400&3.09E-03&2.50&1.46E-03&2.51&1.12E-03&2.53\\\\\n800&5.03E-04&2.62&2.36E-04&2.63&1.81E-04&2.63\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_convdiff_1D\/errorplot_linear_4s3t_IMEX233_Nx400_T0,5.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 1$, Final time $T_f=0.5$.}\n    \\label{1D_CD_constant_plot}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_convdiff_1D\/errorplot_variable_4s3t_IMEX233_Nx400_T0,5.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 1$, Final time $T_f=0.5$.}\n    \\label{1D_CD_variable_plot}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{example}(1D viscous Burgers' equation)\n\\begin{equation}\n    \\label{1D_viscous_burgers}\n    u_t + \\left(\\frac{u^2}{2}\\right) = \\epsilon u_{xx},\\qquad x\\in[0,2]\n\\end{equation}\nwith periodic boundary conditions. As in \\cite{Seydaoglu2016}, we set $\\epsilon=0.1$ and choose the initial condition $u(x,t=0)=0.2\\sin{(\\pi x)}$. The exact solution is\n\\[u(x,t) = 2\\epsilon\\pi\\frac{\\sum\\limits_{n=1}^{\\infty}{c_n\\text{exp}(-n^2\\pi^2\\epsilon t)n\\sin{(n\\pi x)}}}{c_0 + \\sum\\limits_{n=1}^{\\infty}{c_n\\text{exp}(-n^2\\pi^2\\epsilon t)\\cos{(n\\pi x)}}},\\]\nwhere $c_0 = \\int_0^1{\\text{exp}(-(1-\\cos{(\\pi x)})\/(10\\pi\\epsilon))dx}$ and $c_n=2\\int_0^1{\\text{exp}(-(1-\\cos{(\\pi x)})\/(10\\pi\\epsilon))\\cos{(n\\pi x)}dx}$ for $n=1,2,3,...$. We computed the first ten Fourier coefficients in Mathematica$^{\\text{\\textregistered}}$ for the exact solution; the eleventh Fourier coefficient was less than machine precision.\\\\\n\\ \\\\\nTable \\ref{1D_viscous_burgers_table} shows the expected orders of convergence under spatial mesh refinement. Third-order convergence is observed for $CFL=0.95$. Whereas, the convergence for $CFL=8$ is slightly below order three since the time-stepping error starts to dominate. W note that the order of convergence for IMEX(2,3,3) under increasing the CFL number dips slightly below three for larger CFL numbers. The error plot in Figure \\ref{1D_viscous_burgers_plot} showing third-order convergence in time uses mesh $N_x=400$ and CFL numbers varying from 0.1 to 15.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1D_viscous_burgers} with IMEX(2,3,3) at $T_f=1$.}\n\\label{1D_viscous_burgers_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&9.50E-05&-&8.43E-05&-&1.17E-04&-\\\\\n100&1.48E-05&2.68&1.32E-05&3.67&1.87E-05&2.65\\\\\n200&2.42E-06&2.62&2.20E-06&2.59&3.21E-06&2.55\\\\\n400&3.27E-07&2.88&3.00E-07&2.87&4.42E-07&2.86\\\\\n800&4.28E-08&2.94&3.95E-08&2.93&5.86E-08&2.92\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&1.26E-02&-&1.31E-02&-&2.24E-02&-\\\\\n100&2.79E-03&2.17&2.50E-03&2.38&3.70E-03&2.60\\\\\n200&4.73E-04&2.56&4.16E-04&2.59&5.64E-04&2.71\\\\\n400&1.28E-04&1.89&1.15E-04&1.85&1.64E-04&1.78\\\\\n800&1.97E-05&2.70&1.78E-05&2.70&2.56E-05&2.68\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\\begin{example}(The 0D1V Leonard-Bernstein (linearized) Fokker-Planck equation)\n\\begin{equation}\n    \\label{1DFP}\n    f_t -\\frac{1}{\\epsilon}((v_x - \\overline{v}_x)f)_{v_x}= \\frac{1}{\\epsilon}Df_{v_xv_x},\\qquad v_x\\in[-2\\pi,2\\pi]\n\\end{equation}\nwith zero boundary conditions and equilibrium solution the Maxwellian\n\\begin{equation}\n    f_M(v_x) = \\frac{n}{\\sqrt{2\\pi RT}}\\text{exp}\\left(-\\frac{(v_x - \\overline{v}_x)^2}{2RT}\\right),\n\\end{equation}\nwhere $\\epsilon=1$, gas constant $R=1\/6$, temperature $T=3$, thermal velocity $v_{th}=\\sqrt{2RT}=\\sqrt{2D}=1$, number density $n=\\pi$, and bulk velocity $\\overline{v}_x=0$. These quantities were chosen for scaling convenience. When testing convergence we set the initial distribution $f(v_x,t=0)=f_M(v_x)$. Table \\ref{1DFP_table} shows the convergence results, for which we use IMEX(4,4,3) for the time-stepping; we show the results using IMEX(4,4,3) because it gave slightly better convergence than IMEX(2,3,3). We observe fourth-order convergence under spatial mesh refinement for $CFL=0.95$. Whereas, for $CFL=8$ the time-stepping error starts to dominate and we observe third-order convergence. The error plot in Figure \\ref{1DFP_plot} showing third-order convergence in time uses a fixed mesh $N_{v_x}=400$ and CFL numbers varying from 0.1 to 15. We note that although high-order convergence is observed, the proposed EL-RK-FV algorithm is not equilibrium-preserving.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{1DFP} with IMEX(4,4,3) at $T_f=1$.}\n\\label{1DFP_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_{v_x}$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&8.02E-04&-&5.19E-04&-&5.65E-04&-\\\\\n100&6.12E-05&3.71&3.83E-05&3.76&4.27E-05&3.73\\\\\n200&4.41E-06&3.79&2.63E-06&3.87&2.84E-06&3.91\\\\\n400&2.99E-07&3.88&1.73E-07&3.93&1.80E-07&3.98\\\\\n800&2.03E-08&3.88&1.13E-08&3.94&1.10E-08&4.04\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_{v_x}$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&9.33E-03&-&4.52E-03&-&3.63E-03&-\\\\\n100&1.34E-03&2.80&6.60E-04&2.78&5.63E-04&2.69\\\\\n200&1.91E-04&2.81&9.57E-05&2.79&8.15E-05&2.79\\\\\n400&3.21E-05&2.57&1.61E-05&2.57&1.34E-05&2.61\\\\\n800&4.13E-06&2.96&2.09E-06&2.95&1.74E-06&2.94\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_convdiff_1D\/errorplot_burgers_4s3t_IMEX233_Nx400_T0,5.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 0.1$, Final time $T_f=0.5$.}\n    \\label{1D_viscous_burgers_plot}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots_convdiff_1D\/errorplot_FP_4s3t_IMEX443_Nx400_T0,5.png}\n    \\caption{IMEX(4,4,3), Final time $T_f=0.5$.}\n    \\label{1DFP_plot}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Convection-diffusion equations: two-dimensional tests}\n\n\n\\begin{example}(2D equation with constant coefficient)\n\\begin{equation}\n    \\label{2D_CD_constant}\n    u_t + u_x + u_y = \\epsilon(u_{xx} + u_{yy}),\\qquad x,y\\in[0,2\\pi]\n\\end{equation}\nwith periodic boundary conditions and exact solution $u(x,y,t)=\\text{exp}(-2\\epsilon t)\\sin{(x+y-2t)}$. We set $\\epsilon=1$. Third-order convergence under spatial mesh refinement is seen in Table \\ref{2D_CD_constant_table} for $CFL=0.95$ and $CFL=8$. As with equation \\eqref{1D_CD_constant}, there is no temporal error for the convective part since the characteristics are traced exactly. Note that the error is larger for $CFL=8$ than $CFL=0.95$ since this problem also has diffusion. Figure \\ref{2D_CD_constant_plot} shows the third-order convergence in time using fixed mesh $N_x=N_y=400$ and varying the CFL number from 6 to 20.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2D_CD_constant} with IMEX(2,3,3) and Strang splitting at $T_f=0.5$.}\n\\label{2D_CD_constant_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&6.74E-05&-&1.19E-05&-&2.68E-06&-\\\\\n100&1.02E-05&2.72&1.81E-06&2.72&4.07E-07&2.72\\\\\n200&1.41E-06&2.86&2.49E-07&2.86&5.62E-08&2.86\\\\\n400&1.86E-07&2.92&3.29E-08&2.92&7.41E-09&2.92\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&3.92E-02&-&6.94E-03&-&1.56E-03&-\\\\\n100&5.82E-03&2.75&1.03E-03&2.75&2.32E-04&2.75\\\\\n200&8.09E-04&2.85&1.43E-04&2.85&3.22E-05&2.85\\\\\n400&1.07E-04&2.92&1.90E-05&2.92&4.27E-06&2.92\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\\begin{example}(Rigid body rotation with diffusion)\n\\begin{equation}\n    \\label{2D_CD_rigidbody}\n    u_t - yu_x + xu_y = \\epsilon(u_{xx} + u_{yy}) + g,\\qquad x,y\\in[-2\\pi,2\\pi]\n\\end{equation}\nwith periodic boundary conditions, $g(x,y,t)=(6\\epsilon-4xy-4\\epsilon(x^2+9y^2))\\text{exp}(-(x^2+3y^2+2\\epsilon t))$, and exact solution $u(x,y,t)=\\text{exp}(-(x^2+3y^2+2\\epsilon t))$. We set $\\epsilon=1$. Table \\ref{2D_CD_rigidbody_table2} shows the order of convergence when using IMEX(4,4,3). We use IMEX(4,4,3) instead of IMEX(2,3,3) because the latter choice, along with the Strang splitting, showed an order of convergence less than two for large CFL numbers. The expected second-order convergence in time (due to Strang splitting) is seen in Figure \\ref{2D_CD_rigidbody_plot} assumes fixed mesh $N_x=N_y=400$ and CFL numbers varying from 6 to 20.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2D_CD_rigidbody} with IMEX(4,4,3) and Strang splitting at $T_f=0.5$.}\n\\label{2D_CD_rigidbody_table2}\n\\begin{tabular}{|*{7}{c|}} \n\\hline\n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&3.22E-03&-&1.42E-03&-&1.44E-03&-\\\\\n100&3.92E-04&3.04&1.75E-04&3.02&2.00E-04&2.85\\\\\n200&7.27E-05&2.43&3.14E-05&2.47&3.54E-05&2.50\\\\\n400&1.65E-05&2.14&7.11E-06&2.15&7.80E-06&2.18\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=5$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&2.21E-02&-&8.81E-03&-&7.44E-03&-\\\\\n100&5.98E-03&1.89&2.48E-03&1.83&2.46E-03&1.60\\\\\n200&1.61E-03&1.89&6.81E-04&1.86&7.14E-04&1.79\\\\\n400&4.24E-04&1.93&1.81E-04&1.91&1.94E-04&1.88\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/errorplot_linear_4s3t2t_IMEX233_NxNy400_T0,5.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 1$, Final time $T_f=0.5$.}\n    \\label{2D_CD_constant_plot}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/errorplot_rigidbody_4s3t2t_IMEX443_NxNy400_T0,1.png}\n    \\caption{IMEX(4,4,3), $\\epsilon = 1$, Final time $T_f=0.1$.}\n    \\label{2D_CD_rigidbody_plot}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{example}(Swirling deformation with diffusion)\n\\begin{equation}\n    \\label{2D_CD_swirling}\n    u_t - \\left(\\cos^2{(x\/2)}\\sin{(y)}f(t)u\\right)_x + \\left(\\sin{(x)}\\cos^2{(y\/2)}f(t)u\\right)_y = \\epsilon(u_{xx} + u_{yy}),\\qquad x,y\\in[-\\pi,\\pi]\n\\end{equation}\nWhen testing the convergence we set $f(t)=\\cos{(\\pi t\/T_f)}\\pi$, $\\epsilon=1$, and choose the initial condition to be the cosine bell in equation \\eqref{cosbell}. Since there is no analytic solution, we use a reference solution computed with a mesh size of $400\\times 400$ and $CFL=0.1$. The convergence results under spatial mesh refinement are presented in Table \\ref{2D_CD_swirling_table2}. The splitting error seems to dominate the time-stepping error for $CFL=0.95$ as evidenced by the apparent second-order convergence. Whereas, the time-stepping error seems to contribute more for $CFL=8$. Due to the interplay between the time-stepping and splitting errors, the temporal order 2.4 is also observed in Figure \\ref{2D_CD_swirling_plot}, for which we use fixed mesh $N_x=N_y=400$ and CFL numbers varying from 6 to 20.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2D_CD_swirling} with IMEX(2,3,3) and Strang splitting at $T_f=0.1$.}\n\\label{2D_CD_swirling_table2}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&4.98E-04&-&2.57E-04&-&3.38E-04&-\\\\\n100&9.13E-05&2.45&4.58E-05&2.49&6.05E-05&2.48\\\\\n200&1.91E-05&2.26&9.56E-06&2.26&1.22E-05&2.31\\\\\n400&4.48E-06&2.09&2.19E-06&2.13&2.61E-06&2.23\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&5.93E-02&-&3.60E-02&-&5.82E-02&-\\\\\n100&2.05E-02&1.53&1.16E-02&1.63&1.65E-02&1.82\\\\\n200&3.14E-03&2.71&1.67E-03&2.80&2.04E-03&3.01\\\\\n400&6.08E-04&2.37&3.13E-04&2.41&4.16E-04&2.30\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\n\\begin{example}(2D viscous Burgers' equation)\n\\begin{equation}\n    \\label{2D_viscousburgers}\n    u_t + \\left(\\frac{u^2}{2}\\right)_x + \\left(\\frac{u^2}{2}\\right)_y = \\epsilon(u_{xx} + u_{yy}) + g,\\qquad x,y\\in[-\\pi,\\pi]\n\\end{equation}\nwith periodic boundary conditions. As in \\cite{Wang2016}, we set $\\epsilon=0.1$, $g(x,y,t)=\\text{exp}(-4\\epsilon t)\\sin{(2(x+y))}$, and suppose the solution $u(x,y,t)=\\text{exp}(-2\\epsilon t)\\sin{(x+y)}$. The convergence results are presented in Table \\ref{2D_viscousburgers_table2}. The splitting error seems to dominate the time-stepping error for $CFL=0.95$ as evidenced by the second-order convergence. Whereas, the time-stepping error seems to contribute more for $CFL=8$ since the order is between two and three. The temporal order of convergence in the $L^1$ norm is roughly 2.3, as seen in Figure \\ref{2D_viscousburgers_plot}, for which we use fixed mesh $N_x=N_y=400$ and CFL numbers varying from 6 to 20.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2D_viscousburgers} with IMEX(2,3,3) and Strang splitting at $T_f=0.5$.}\n\\label{2D_viscousburgers_table2}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&3.01E-04&-&5.21E-05&-&1.19E-05&-\\\\\n100&7.12E-05&2.08&1.30E-05&2.00&3.42E-06&1.80\\\\\n200&1.76E-05&2.02&3.34E-06&1.96&9.42E-07&1.86\\\\\n400&4.53E-06&1.95&8.60E-07&1.96&2.49E-07&1.92\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_x=N_y$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&8.11E-02&-&1.60E-02&-&5.80E-03&-\\\\\n100&1.02E-02&2.99&2.00E-03&3.00&7.78E-04&2.90\\\\\n200&1.61E-03&2.66&3.01E-04&2.74&9.81E-05&2.99\\\\\n400&3.47E-04&2.21&5.97E-05&2.33&1.35E-05&2.86\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/errorplot_swirling_4s3t2t_IMEX233_NxNy400_T0,1_refsolnCFL0,1.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 1$, Final time $T_f=0.1$.}\n    \\label{2D_CD_swirling_plot}\n\\end{minipage}\n\\begin{minipage}{0.03\\linewidth}\n    \\centering\n    \\ \n\\end{minipage}\n\\begin{minipage}{0.48\\linewidth}\n    \\centering\n    \\includegraphics[width=\\textwidth]{images_errorplots\/errorplot_burgers_4s3t2t_IMEX233_NxNy400_T0,2.png}\n    \\caption{IMEX(2,3,3), $\\epsilon = 0.1$, Final time $T_f=0.2$.}\n    \\label{2D_viscousburgers_plot}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{example}(The 0D2V Leonard-Bernstein (linearized) Fokker-Planck equation)\n\\begin{equation}\n    \\label{2DFP}\n    f_t -\\frac{1}{\\epsilon}((v_x - \\overline{v}_x)f)_{v_x} - \\frac{1}{\\epsilon}((v_y - \\overline{v}_y)f)_{v_y}= \\frac{1}{\\epsilon}D(f_{v_xv_x} + f_{v_yv_y}),\\qquad v_x,v_y\\in[-2\\pi,2\\pi]\n\\end{equation}\nwith zero boundary conditions and equilibrium solution the Maxwellian\n\\begin{equation}\n    f_M(v_x,v_y) = \\frac{n}{2\\pi RT}\\text{exp}\\left(-\\frac{(v_x - \\overline{v}_x)^2 + (v_y-\\overline{v}_y)^2}{2RT}\\right),\n\\end{equation}\nwhere $\\epsilon=1$, gas constant $R=1\/6$, temperature $T=3$, thermal velocity $v_{th}=\\sqrt{2RT}=\\sqrt{2D}=1$, number density $n=\\pi$, and bulk velocities $\\overline{v}_x=\\overline{v}_y=0$. These quantities were chosen for scaling convenience. When testing the spatial and temporal orders of accuracy we set the initial distribution $f(v_x,v_y,t=0)=f_M(v_x,v_y)$. Table \\ref{2DFP_table} shows the convergence results under spatial mesh refinement. We observe higher fourth-order convergence in space for $CFL=0.95$. The time-stepping and splitting errors start to dominate the spatial error for larger CFL numbers, as observed for $CFL=8$. Figure \\ref{2DFP_plot} shows the temporal order of convergence is roughly 2.6 for fixed mesh $N_{v_x}=N_{v_y}=400$ and CFL numbers varying from 6 to 20. We again note the interplay between the third-order time-stepping and second-order splitting.\\\\\n\\ \\\\\nWhen testing for relaxation of the system, we choose the initial distribution $f(v_x,v_y,t=0)=f_{M1}(v_x,v_y)+f_{M2}(v_x,v_y)$, that is, the sum of two randomly generated Maxwellians such that the total macro-parameters are preserved. The number density, bulk velocities, and temperature of each Maxwellian are listed in Table \\ref{2maxwells}. We set $\\overline{v}_y=0$ so that the two generated Maxwellians are shifted only along the $v_x$ axis.\\\\\n\n\\begin{table}[h!] \n\\begin{center}\n\\label{}\n\\begin{tabular}{|c|c|c|} \n\\hline \n&$f_{M1}$&$f_{M2}$\\\\\n\\hline\n$n$&1.990964530353041&1.150628123236752\\\\\n\\hline\n$\\overline{v}_x$&0.4979792385268875&-0.8616676237412346\\\\\n\\hline\n$\\overline{v}_y$&0&0\\\\\n\\hline\n$T$&2.46518981703837&0.4107062104302872\\\\\n\\hline\n\\end{tabular} \n\\caption{$n=\\pi$, $\\overline{\\mathbf{v}}=\\mathbf{0}$, and $T=3$.}\n\\label{2maxwells}\n\\end{center} \n\\end{table}\n\\noindent The macro-parameters we want to conserve are number density, bulk velocity, and temperature, which in two dimensions are respectively given by\n\\begin{subequations}\n\\begin{equation}\n    n = \\int_{-\\infty}^{\\infty}{\\int_{-\\infty}^{\\infty}{f(\\mathbf{v})dv_ydv_x}},\n\\end{equation}\n\\begin{equation}\n    \\overline{\\mathbf{v}} = \\frac{1}{n}\\int_{-\\infty}^{\\infty}{\\int_{-\\infty}^{\\infty}{\\mathbf{v}f(\\mathbf{v})dv_ydv_x}},\n\\end{equation}\n\\begin{equation}\n    T = \\frac{1}{2Rn}\\int_{-\\infty}^{\\infty}{\\int_{-\\infty}^{\\infty}{(\\mathbf{v}-\\overline{\\mathbf{v}})^2f(\\mathbf{v})dv_ydv_x}}.\n\\end{equation}\n\\end{subequations}\nFigures \\ref{FPtwoMaxwell_images}(f) and \\ref{FPtwoMaxwell_snapshots} show the solution using fixed mesh $N_{v_x}=N_{v_y}=200$ and $CFL=6$. Although we computed the solution up to time $T_f=20$, there was no difference (to the naked eye) after time $t=3$. Although the solution appears to reach equilibrium, we again note that the proposed EL-RK-FV algorithm is not equilibrium-preserving. Figure \\ref{FPtwoMaxwell_images}(a) verifies mass conservation, but Figure \\ref{FPtwoMaxwell_images}(b) implies that the numerical solution has some negative values and is not positivity-preserving. Referring to Figure \\ref{FPtwoMaxwell_images}, momentum and energy are not conserved. As seen in Figure \\ref{FPtwoMaxwell_images}(d), the bulk velocity in the $v_y$-direction is on the order of machine epsilon because we constructed the two Maxwellians in Table \\ref{2maxwells} such that $\\overline{v}_{M1,y}=\\overline{v}_{M2,y}=0$. Hence, there is no drift velocity in the $v_y$-direction.\n\\end{example}\n\n\\begin{table}[h!] \n\\begin{center}\n\\caption{Convergence study with spatial mesh refinement for equation \\eqref{2DFP} with IMEX(2,3,3) and Strang splitting at $T_f=0.5$.}\n\\label{2DFP_table}\n\\begin{tabular}{|*{7}{c|}} \n\\hline \n\\multicolumn{7}{|c|}{$CFL=0.95$}\\\\\n\\hline\n$N_{v_x}=N_{v_y}$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&9.07E-04&-&4.22E-04&-&5.49E-04&-\\\\\n100&7.19E-05&3.66&3.15E-05&3.74&4.36E-05&3.66\\\\\n200&5.35E-06&3.75&2.15E-06&3.87&2.93E-06&3.89\\\\\n400&3.54E-07&3.92&1.37E-07&3.98&1.82E-07&4.01\\\\\n\\hline\n\\multicolumn{7}{|c|}{$CFL=8$}\\\\\n\\hline\n$N_{v_x}=N_{v_y}$&$L^1$ Error&Order&$L^2$ Error&Order&$L^{\\infty}$ Error&Order\\\\\n\\hline\n50&5.70E-03&-&1.84E-03&-&1.26E-03&-\\\\\n100&1.08E-03&2.40&3.53E-04&2.39&2.82E-04&2.16\\\\\n200&1.69E-04&2.67&5.68E-05&2.64&5.02E-05&2.49\\\\\n400&2.73E-05&2.63&9.30E-06&2.61&8.63E-06&2.54\\\\\n\\hline\n\\end{tabular} \n\\end{center} \n\\end{table} \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.49\\textwidth]{images_errorplots\/errorplot_fokkerplancktrial1_4s3t2t_IMEX233_NxNy400_T0,1.png}\n\\caption{IMEX(2,3,3), Final time $T_f=0.1$.}\n\\label{2DFP_plot}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/n_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(a)}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/vybar_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(d)}\n\\end{minipage}\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/L1_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(b)}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/temp_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(e)}\n\\end{minipage}\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/vxbar_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(c)}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\caption*{(f)}\n\\end{minipage}\n\\caption{Figures (a)-(e): Relative macro-parameters for equation \\eqref{2DFP} with initial distribution of two Maxwellians defined by Table \\ref{2maxwells}. Mesh $N_{v_x}=N_{v_y}=200$, $CFL=6$. Figure (f): The initial distribution.}\n\\label{FPtwoMaxwell_images}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0,15_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0,60_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n\\end{minipage}\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0,30_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0,75_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n\\end{minipage}\n\\begin{minipage}[b]{0.33\\linewidth}\n    \\centering\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T0,45_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n    \\includegraphics[width=1.05\\textwidth]{images_FPtwoMaxwell\/T3_FP_twoMaxwell_T20_NxNy200_CFL6_4s3t2t_IMEX233.png}\n\\end{minipage}\n\\caption{Various snapshots of the numerical solution to equation \\eqref{2DFP} with initial distribution of two Maxwellians defined by Table \\ref{2maxwells}. Mesh $N_{v_x}=N_{v_y}=200$, $CFL=6$. Times: 0.15, 0.30, 0.45, 0.60, 0.75, 3.}\n\\label{FPtwoMaxwell_snapshots}\n\\end{figure}\n\n\n\n\n\\section{Conclusion}\n\nIn this paper, we proposed a new EL-RK-FV method for solving convection and  convection-diffusion equations. Whereas SL methods require solving for the exact characteristics, which is often highly nontrivial for nonlinear problems, our EL method computes linear space-time curves as the approximate characteristics. WENO-AO schemes allowed us to perform spatial reconstruction at arbitrary points which was essential since the traceback grid was not necessarily the (uniform) background grid. By working with the time-differential form, we could use a method-of-lines approach. Explicit RK methods were used for pure convection problems, and IMEX RK methods were used for convection-diffusion equations. Dimensional splitting was used for higher-dimensional problems. Several one- and two-dimensional test problems demonstrated the algorithm's robustness, high-order accuracy, and ability to allow extra large time steps. Ongoing and future work includes modifying the algorithm to handle shocks and rarefaction waves (the authors already have promising results that will be written in another paper), and developing a non-splitting version of the EL-RK-FV algorithm.\n\n\n\n\\section*{Acknowledgements}\n\nResearch is supported by NSF grant NSF-DMS-1818924 and NSF-DMS-2111253, Air Force Office of Scientific Research FA9550-18-1-0257 and University of Delaware.\nThe authors would like to thank William Taitano (Air Force Research Laboratory) and Alexander Alekseenko (California State University at Northridge) for their help in constructing the Leonard-Bernstein Fokker-Planck equation test problems. Further thanks goes to Robert Martin (Army Research Laboratory) and Alexander Alekseenko for their mentorship during Joseph Nakao's summer internship at the Air Force Research Laboratory.\n\n\\clearpage\n\n\\section*{Appendices}\n\n\\subsection*{Appendix A. Butcher Tables for Explicit RK Methods}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{SSP RK3}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        1&1&0&0\\\\\n        1\/2&1\/4&1\/4&0\\\\\n        \\hline\n        &1\/6&1\/6&2\/3\\\\\n    \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{RK4}\n    \\begin{tabular}{c|llll}\n        0&0&0&0&0\\\\\n        1\/2&1\/2&0&0&0\\\\\n        1\/2&0&1\/2&0&0\\\\\n        1&0&0&1&0\\\\\n        \\hline\n        &1\/6&1\/3&1\/3&1\/6\\\\\n    \\end{tabular}\n    \\end{minipage}\n\\end{table}\n\n\\subsection*{Appendix B. Fourth-order operator splitting}\nDefine two constants\n\\[\\gamma_1=\\frac{1}{2-2^{1\/3}}\\approx 1.351207191959658\\quad\\text{and}\\quad\\gamma_2=\\frac{-2^{1\/3}}{2-2^{1\/3}}\\approx -1.702414383919315.\\]\nGiven constants $\\gamma_1$ and $\\gamma_2$, the fourth-order splitting method in \\cite{Forest1990,Yoshida1990} has seven stages, compared to the three stages required for Strang splitting.\\\\\n\\ \\\\\n\\textbf{Step 1 ($x-$direction).} Solve equation \\eqref{Strang1} over a time step $\\gamma_1\\Delta t\/2$.\\\\\n\\textbf{Step 2 ($y-$direction).} Solve equation \\eqref{Strang2} over a time step $\\gamma_1\\Delta t$.\\\\\n\\textbf{Step 3 ($x-$direction).} Solve equation \\eqref{Strang1} over a time step $(\\gamma_1+\\gamma_2)\\Delta t\/2$.\\\\\n\\textbf{Step 4 ($y-$direction).} Solve equation \\eqref{Strang2} over a time step $\\gamma_2\\Delta t$.\\\\\n\\textbf{Step 5 ($x-$direction).} Solve equation \\eqref{Strang1} over a time step $(\\gamma_1+\\gamma_2)\\Delta t\/2$.\\\\\n\\textbf{Step 6 ($y-$direction).} Solve equation \\eqref{Strang2} over a time step $\\gamma_1\\Delta t$.\\\\\n\\textbf{Step 7 ($x-$direction).} Solve equation \\eqref{Strang1} over a time step $\\gamma_1\\Delta t\/2$.\\\\\n\\ \\\\\nNote that steps 2 and 6 require steps larger than $\\Delta t$, and steps 3, 4, and 5 require steps backwards in time.\n\n\n\\subsection*{Appendix C. Butcher Tables for IMEX RK Methods}\n\nAll IMEX RK schemes included in this appendix are taken from \\cite{Ascher1997}. By construction, each IMEX RK scheme has slightly different properties that are better suited from different problems. Some schemes might have better damping properties and stability regions, be stiffly accurate, etc. In this appendix we opt to pad the implicit Butcher tables with zeros.\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(1,1,1) -- Implicit Table}\n    \\begin{tabular}{c|ll}\n        0&0&0\\\\\n        1&0&1\\\\\n        \\hline\n        &0&1\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(1,1,1) -- Explicit Table}\n    \\begin{tabular}{c|ll}\n        0&0&0\\\\\n        1&1&0\\\\\n        \\hline\n        &1&0\n        \\end{tabular}\n    \\end{minipage}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(1,2,2) -- Implicit Table}\n    \\begin{tabular}{c|ll}\n        0&0&0\\\\\n        1\/2&0&1\/2\\\\\n        \\hline\n        &0&1\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(1,2,2) -- Explicit Table}\n    \\begin{tabular}{c|ll}\n        0&0&0\\\\\n        1\/2&1\/2&0\\\\\n        \\hline\n        &0&1\n        \\end{tabular}\n    \\end{minipage}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,2,2) -- Implicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&0&$\\gamma$&0\\\\\n        1&0&$1-\\gamma$&$\\gamma$\\\\\n        \\hline\n        &0&$1-\\gamma$&$\\gamma$\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,2,2) -- Explicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&$\\gamma$&0&0\\\\\n        1&$\\delta$&$1-\\delta$&0\\\\\n        \\hline\n        &$\\delta$&$1-\\delta$&0\n        \\end{tabular}\n    \\end{minipage}\n    \\caption*{Let $\\gamma=1-\\sqrt{2}\/2$ and $\\delta = 1 - 1\/(2\\gamma)$.}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,3,3) -- Implicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&0&$\\gamma$&0\\\\\n        $1-\\gamma$&0&$1-2\\gamma$&$\\gamma$\\\\\n        \\hline\n        &0&1\/2&1\/2\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,3,3) -- Explicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&$\\gamma$&0&0\\\\\n        $1-\\gamma$&$\\gamma-1$&$2(1-\\gamma)$&0\\\\\n        \\hline\n        &0&1\/2&1\/2\n        \\end{tabular}\n    \\end{minipage}\n    \\caption*{Let $\\gamma=(3+\\sqrt{3})\/6$.}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,3,2) -- Implicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&0&$\\gamma$&0\\\\\n        1&0&$1-\\gamma$&$\\gamma$\\\\\n        \\hline\n        &0&$1-\\gamma$&$\\gamma$\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(2,3,2) -- Explicit Table}\n    \\begin{tabular}{c|lll}\n        0&0&0&0\\\\\n        $\\gamma$&$\\gamma$&0&0\\\\\n        1&$\\delta$&$1-\\delta$&0\\\\\n        \\hline\n        &0&$1-\\gamma$&$\\gamma$\n        \\end{tabular}\n    \\end{minipage}\n    \\caption*{Let $\\gamma=(2-\\sqrt{2})\/2$ and $\\delta = -2\\sqrt{2}\/3$.}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n   \n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(3,4,3) -- Implicit Table}\n    \\begin{tabular}{c|llll}\n        0&0&0&0&0\\\\\n        $\\gamma$&0&$\\gamma$&0&0\\\\\n        0.717933&0&0.282067&$\\gamma$&0\\\\\n        1&0&1.208497&-0.644363&$\\gamma$\\\\\n        \\hline\n        &0&1.208497&-0.644363&$\\gamma$\n    \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(3,4,3) -- Explicit Table}\n    \\begin{tabular}{c|llll}\n        0&0&0&0&0\\\\\n        $\\gamma$&$\\gamma$&0&0&0\\\\\n        0.717933&0.321279&0.396654&0&0\\\\\n        1&-0.105858&0.552929&0.552929&0\\\\\n        \\hline\n        &0&1.208497&-0.644363&$\\gamma$\n    \\end{tabular}\n    \\end{minipage}\n    \\caption*{Let $\\gamma=0.435867$.}\n\\end{table}\n\n\\begin{table}[h!]\n    \\centering\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(4,4,3) -- Implicit Table}\n    \\begin{tabular}{c|lllll}\n        0&0&0&0&0&0\\\\\n        1\/2&0&1\/2&0&0&0\\\\\n        2\/3&0&1\/6&1\/2&0&0\\\\\n        1\/2&0&-1\/2&1\/2&1\/2&0\\\\\n        1&0&3\/2&-3\/2&1\/2&1\/2\\\\\n        \\hline\n        &0&3\/2&-3\/2&1\/2&1\/2\n        \\end{tabular}\n    \\end{minipage}\n    \\begin{minipage}[b]{0.49\\linewidth}\n    \\centering\n    \\caption*{IMEX(4,4,3) -- Explicit Table}\n    \\begin{tabular}{c|lllll}\n        0&0&0&0&0&0\\\\\n        1\/2&1\/2&0&0&0&0\\\\\n        2\/3&11\/18&1\/18&0&0&0\\\\\n        1\/2&5\/6&-5\/6&1\/2&0&0\\\\\n        1&1\/4&7\/4&3\/4&-7\/4&0\\\\\n        \\hline\n        &1\/4&7\/4&3\/4&-7\/4&0\n        \\end{tabular}\n    \\end{minipage}\n\\end{table}\n\n\\clearpage\n\\subsection*{Appendix D. An illustrative example with IMEX(2,2,2)}\n\nIn this section, we couple the EL-RK-FV algorithm with IMEX(2,2,2), that is, two-stage implicit, two-stage explicit, and of combined order two. This scheme is L-stable and uses a second order DIRK method. The Butcher tables are given in Appendix C. Figure \\ref{IMEX222} shows the lone sub-space-time region ${}_{1}\\Omega_j$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.5\\textwidth]{.\/images_spacetime\/spacetimeIMEX222.png}\n\t\\caption{The space-time region ${}_{1}\\Omega_j$ for IMEX(2,2,2).}\n\t\\label{IMEX222}\n\\end{figure}\n\n\\noindent\\textbf{Step 0a.} Compute the approximate characteristic speeds using equation \\eqref{RHjump}. After defining the space-time region $\\Omega_j$, compute the possibly nonuniform traceback cell averages $\\tilde{u}_j(t^n)$ using Algorithm 1.\\\\\n\\textbf{Step 0b.} Use the possibly nonuniform traceback cell averages $\\tilde{u}_j(t^n)$ in Algorithm 2 to compute $\\hat{K}_1 = \\mathcal{F}(U^n;t^n))$.\\\\\n\\textbf{Step 1a.} Using the same approximate characteristic speeds from step 1a, define the sub-space-time region ${}_{1}\\Omega_j$ as seen in Figure \\ref{IMEX222}. Compute the possibly nonuniform traceback cell averages ${}_{1}\\tilde{u}_j^n$ using Algorithm 1.\\\\\n\\textbf{Step 1b.} Use the possibly nonuniform traceback cell averages ${}_{1}\\tilde{u}_j^n$ in Algorithm 2 to compute ${}_{1}\\hat{K}_1 = \\mathcal{F}({}_{1}U^n;t^n))$.\\\\\n\\textbf{Step 1c.} Recalling equation \\eqref{5stencil}, solve equation \\eqref{IMEXsubstage} by solving the linear system\n\\begin{equation}\n    \\left(\\mathbf{I}-\\frac{\\gamma\\epsilon\\Delta t}{\\Delta x^2}\\mathbf{D}_4\\right){}_{1}\\vv{U}^{(1)} = {}_{1}\\vv{U}^n + \\gamma\\Delta t{}_{1}\\vv{\\hat{K}}_1 + \\gamma\\Delta t\\vv{g}(x,t^{(1)}),\n\\end{equation}\nwhere $\\vv{g}_j(x,t^{(1)}) = \\int_{I_j}{g(x,t^{(1)})dx}$ can be computed with a Gaussian quadrature.\\\\\n\\textbf{Step 1d.} Compute the uniform cell averages $\\overline{u}_j^{(1)} = {}_{1}U_j^{(1)}\/\\Delta x$.\\\\\n\\textbf{Step 1e.} Compute the uniform cell averages $\\overline{u}_{xx,j}^{(1)}$ using equation \\eqref{5stencil},\n\\begin{equation}\n    \\vv{\\overline{u}}_{xx}^{(1)} = \\frac{1}{\\Delta x^2}\\mathbf{D}_4\\vv{\\overline{u}}^{(1)}.\n\\end{equation}\n\\textbf{Step 1f.} Compute the possibly nonuniform traceback cell averages $\\tilde{u}_j^{(1)}$ and $\\tilde{u}_{xx,j}^{(1)}$ (we are now in the space-time region $\\Omega_j$) using Algorithm 1.\\\\\n\\textbf{Step 1g.} Compute $K_1 = \\mathcal{G}(U^{(1)};t^{(1)})$,\n\\begin{equation}\n    K_1 = \\epsilon\\Delta\\tilde{x}_j^{(1)}\\tilde{u}_{xx,j}^{(1)} + \\int_{\\tilde{I}_j(t^{(1)})}{g(x,t^{(1)})dx},\n\\end{equation}\nwhere the definite integral involving $g(x,t)$ can be evaluated using a Gaussian quadrature.\\\\\n\\textbf{Step 1h.} Use the possibly nonuniform traceback cell averages $\\tilde{u}_j^{(1)}$ in Algorithm 2 to compute $\\hat{K}_2 = \\mathcal{F}(U^{(1)};t^{(1)})$.\\\\\n\\textbf{Step 2.} Recalling equation \\eqref{5stencil}, solve equation \\eqref{IMEX} by solving the linear system\n\\begin{equation}\n    \\left(\\mathbf{I}-\\frac{\\gamma\\epsilon\\Delta t}{\\Delta x^2}\\mathbf{D}_4\\right)\\vv{U}^{n+1} = \\vv{U}^n + (1-\\gamma)\\Delta t\\vv{\\hat{K}}_1 + \\Delta t(\\delta\\vv{\\hat{K}}_1 + (1-\\delta)\\vv{\\hat{K}}_2) + \\gamma\\Delta t\\vv{g}(x,t^{n+1}),\n\\end{equation}\nwhere $\\vv{g}_j(x,t^{n+1}) = \\int_{I_j}{g(x,t^{n+1})dx}$ can be computed with a Gaussian quadrature.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{Sec:Introduction}\n\nThe $N$-vector model with $O(N)$ group symmetry plays an important role in the understanding of crucial aspects of renormalization group (RG) flows: In four dimensions, it exhibits a Landau pole and corresponds to a trivial theory \\cite{Aizenman:1981du,Frohlich:1982tw,Luscher:1987ay,*Luscher:1987ek,*Luscher:1988uq,Rosten:2008ts}. In other words, as an interacting model it is only valid over a finite range of scales, thus constituting an effective low-energy theory. This could affect the possible range of validity of the standard model of particle physics \\cite{Dashen:1983ts,Callaway:1983zd,Kuti:1987nr,Luscher:1988gc,Kogut:1988sf,Gockeler:1997dn} and could also play a role in cosmology as, e.g., many inflationary models probably share this feature. On the other hand, in three dimensions the theory exhibits an important example of an interacting RG fixed point \\cite{LeGuillou:1977ju,Guida:1998bx}. Such fixed points are crucial in the understanding of scaling and universality in critical phenomena \\cite{Wilson:1973jj,Wegner:1976bk,Pelissetto:2000ek} and, more recently, they have been of considerable interest, e.g., in the problem of the ultraviolet (UV) completion of gravity \\cite{Weinberg:1980gg, Reuter:2012id}. On a more technical level, well-known examples such as, e.g., the infrared (IR) attractive Wilson-Fisher fixed point (FP) in the $O(N)$ model may provide an important benchmark test for nonperturbative methods, which one may then apply to other problems of interest (see, e.g., Ref.\\ \\cite{Delamotte:2004zg}).\n\nExtending the $O(N_1)$ vector model by a coupling to another $O(N_2)$ symmetric vector field leads to complex dynamics that has been discussed extensively in the context of multicritical phenomena and systems with competing order parameters \\cite{Fisher:1974zz,Kosterlitz:1976zza,Aharony:2002,Aharony:2002zz,Calabrese:2002bm,Folk:2008mi}. Such a theory is characterized by an $O(N_{1})\\oplus O(N_{2})$ symmetry which admits a number of interacting (IR attractive) FPs. These travel through the coupling space of the model as the numbers of field components $N_{1}$ and $N_{2}$ are varied. At particular values of $N_1$ and $N_2$, two of these FPs can collide and exchange their stability properties, cf.\\ Fig.\\ \\ref{Fig:collision}. In this context, an IR stable FP is defined as featuring only two positive critical exponents, as this corresponds to the number of relevant couplings that need to be tuned in order to approach the FP. When two FPs collide, a FP with three positive critical exponents trades one of them for a negative exponent, while the second FP picks up the additional relevant direction and becomes unstable. As a consequence, it turns out that for every combination of $N_1$ and $N_2$ there is exactly one stable FP. Of course, this statement assumes that one considers renormalization group trajectories within a single domain of attraction. In general, the parameter\/coupling space of the model will allow for separate domains, where different FPs might exist, and may or may not be stable. \n\n\\begin{figure}[!h]\n\\includegraphics[width=0.7\\linewidth]{collisionsketch.pdf}\n\\caption{\\label{Fig:collision}We show a sketch of the value of the third largest critical exponent, $\\theta_3$, of three different FPs (solid, dotted, and dashed lines) in the $O(N_1)\\oplus O(N_2)$ model at fixed $N_1$ as a function of $N_2$. The regime where a FP is stable is indicated by the labels FP1, FP2 and FP3. As the coordinates of two different FPs coincide at $N_2^{\\ast}$ and at $N_2^{\\ast\\ast}$, these FPs exchange their stability properties, and $\\theta_{3}$ changes its sign if evaluated beyond the stable regime. An explicit calculation showing this situation can be found in Ref.\\ \\cite{Eichhorn:2013zza}.}\n\\end{figure}\n\nIn this work, we will for the first time provide a comprehensive analysis of a model with a coupling to an additional field, with a resulting $O(N_1) \\oplus O(N_2) \\oplus O(N_3)$ symmetry. Note that the $O(N_1) \\oplus O(N_2)$ symmetry acts trivially on the third vector, and similarly the first two fields transform as singlets under the $O(N_3)$ symmetry. At first, one might expect that this model will exhibit very similar behavior to that already encountered in the case of the two-field models, and will feature a single stable FP with three relevant directions at every value of $(N_1, N_2, N_3)$. In this study, we present evidence for a rather different behavior, where FPs exhibit a large number of relevant eigendirections in a given range of values for the number of field components. This leads to the absence of stable FP solutions in a part of the parameter space. In the following, we will argue that this is a generic feature of multifield models and is due to a significantly increased number of possible mixed interactions compared to the single-field or two-field models. This behavior is akin to the absence of FPs in the low-energy effective models for phases of strongly-interacting matter \\cite{Basile:2005hw,Vicari:2007ma}, or frustrated spin systems \\cite{Pelissetto:2000ne,Tissier:2002zz}. In both cases, one observes the absence of stable FP solutions beyond some critical number of field components, indicative of a first order phase transition (see, e.g., Ref.\\ \\cite{Bak:1976}).\n\nThe presence of competing orders can change the nature of a continuous transition or even drive it to be first order. Although there are numerous examples in the literature, this is probably best illustrated using the example of the two-field model \\cite{Fisher:1974zz,Kosterlitz:1976zza,Aharony:2002,Aharony:2002zz,Calabrese:2002bm,Folk:2008mi}. Already at the mean-field level one observes a suppression of the coexistence region in the presence of a strong repulsive interaction between the two competing fields \\cite{She:2010}. If the coupling assumes a critical value, the coexistence region vanishes and the second-order lines merge to a first order transition. This dramatic change in the phase diagram marks a change in the universality class of the adjacent multicritical point. Fluctuations will favor either one of these scenarios (corresponding to a tetracritical or bicritical point) as long as the associated fixed point is within its domain of attraction.\\footnote{This is not the case if the interactions are sufficiently strong and the total number of field components $N = N_{1} + N_{2}$ is larger than some critical value. Then, instead of a multicritical point one observes a genuine first order transition in the phase diagram of the model.} Multifield models feature a different possibility: fluctuation-mediated interactions might not only affect the universality class of the multicritical point, but they might even render it unstable, thus allowing no IR convergent RG trajectory. The difference between two- and multifield models lies in the distinct RG flow topologies and mechanisms of stability trading between different fixed points, which leads to the absence of a stable fixed point. We will investigate these properties in detail in the following sections, as well as the general behavior of systems with a large number of interacting sectors.\n\nThe main motivation for this study is to gain an improved understanding of generalized Wilson-Fisher universality classes (multicritical points) and to understand how these might show up in the phase diagram of systems with multiple order parameters. Previous work, in the context of systems with two competing order parameters, has led to the understanding that the effect of fluctuations plays an important role in addressing the stability of a particular continuous transition \\cite{Fisher:1974zz,Kosterlitz:1976zza,Aharony:2002,Aharony:2002zz,Calabrese:2002bm,Folk:2008mi}. The $SO(5)$ theory of high-$T_c$ superconductivity \\cite{Calabrese:2002bm,Demler:2004} provides one example, where fluctuations of the order parameters are seen to alter the stability properties of the associated fixed point and rule out such a theory as an effective IR description in the region where both order parameters become critical. On the other hand, it has been pointed out that the interplay of two competing order parameters might explain the presence of first order transitions or spatially inhomogeneous phases that exhibit finite wavevector ordering near quantum criticality \\cite{She:2010}. Here, we argue that first order transition might be a generic scenario for systems that feature a large number of competing phases.\n\n\nThe outline of this paper is as follows: In Sec.~\\ref{Sec:Model} we present the model under consideration in detail. In Sec.~\\ref{Sec:Fixed-point analysis} we then explain the results of our study, discussing numerical results and scaling relations for several different FPs. In the Appendix we present the renormalization group flow equations for these models in $d$ dimensions, both in a local potential approximation (LPA) and including anomalous dimensions. Sections \\ref{Sec:Model} and \\ref{Sec:Fixed-point analysis} are self-contained, and can be read without referring to the technical details of our study.\n\n\n\\section{Model}\n\\label{Sec:Model}\n\nWe consider a model with three different bosonic fields, $\\phi_1,\\phi_2$, and $\\phi_3$, with $N_1,N_2$, and $N_3$ field components, respectively. We derive the $\\beta$ functions from the nonperturbative functional flow equation for the (Euclidean) scale dependent effective action $\\Gamma_{k}$ \\cite{Wetterich:1992yh}, (see Appendix for details, and reviews, e.g., Ref.\\ \\cite{Berges:2000ew,Polonyi:2001se,Pawlowski:2005xe,Gies:2006wv,Delamotte:2007pf,Rosten:2010vm,Metzner:2011cw}). This method has been shown to yield results in very good agreement with those obtained from the $\\epsilon$-expansion and lattice simulations in the case of the $O(N)$ Wilson-Fisher FP, see, e.g., Ref.\\ \\cite{Canet:2003qd, Bervillier:2007rc,Litim:2010tt,Codello:2012ec} and the $O(N_1)\\oplus O(N_2)$ FPs \\cite{Eichhorn:2013zza}. To leading order in the derivative expansion \\cite{Morris:1994au, *Morris:1994ie, *Morris:1996kn, *Morris:1996xq} our \\textit{ansatz} for $\\Gamma_{k}$ reads\n\\begin{eqnarray}\n\\Gamma_{k} &=& \\int\\! d^dx \\left( \\sum_{I=1}^3 Z_{I} \\left( \\partial_{\\mu} \\phi_{I} \\right)^{2} + U_k (\\phi_{1}, \\phi_{2}, \\phi_{3})\\right) ~, \n\\label{Eq:EffectiveAction}\n\\end{eqnarray}\nwhere $\\phi_{I}^{a}$, $a = 1, \\ldots, N_{I}$, and $\\phi_{I}^{2} \\equiv \\phi_{I}^{a} \\phi_{I}^{a}$. Here, we have introduced the scale dependent effective potential\n\\begin{equation}\n\\hspace{-5pt} U_k = \\sum_{l,m,n} \\frac{\\bar{\\lambda}_{l,m,n}}{l!\\, m!\\, n!} \\left( \\bar{\\rho}_{I} - \\bar{\\kappa}_{1} \\right)^{l} \\left( \\bar{\\rho}_{2} - \\bar{\\kappa}_{2} \\right)^{m} \\left( \\bar{\\rho}_{3} - \\bar{\\kappa}_{3} \\right)^n , \n\\label{Eq:EffectivePotential}\n\\end{equation}\nwhich we have written in terms of the invariants $\\bar{\\rho}_{I} = \\frac{1}{2} \\phi_{I}^{2}$, thereby making the $O(N_1) \\oplus O(N_2) \\oplus O(N_3)$ symmetry manifest. The parameter $k$ defines an infrared momentum cutoff scale, on which the parameters and couplings depend. For brevity we do not indicate the scale dependence explicitly, i.e., $\\bar{\\lambda}_{l,m,n} = \\bar{\\lambda}_{l,m,n} (k)$. Similarly, scale dependent wavefunction renormalization factors are simply denoted by $Z_{I}$. We expand the scale dependent effective potential $U_{k}$ around (possibly) nonvanishing scale dependent minima for the fields, $\\bar\\kappa_{I}$.\n\nFor the identification of scaling solutions, we introduce dimensionless renormalized couplings, given by\n\\begin{eqnarray}\nu_k &=& U_k k^{-d} ~, \\nonumber\\\\ \n\\kappa_{I} &=& Z_I k^{2-d}\\bar{\\kappa}_{I} ~, \\nonumber\\\\ \n\\rho_{I} &=& {Z_I k^{2-d}} \\bar{\\rho}_{I} ~,\\nonumber\\\\\n\\lambda_{l,m,n} &=& \\bar{\\lambda}_{l,m,n} Z_{1}^{-l}\\, Z_{2}^{-m}\\, Z_{3}^{-n} k^{-d + (l+m+n) (d-2)} ~.\n\\end{eqnarray}\nWe truncate the coupling space to a finite-dimensional subspace of the form Eqs.\\ \\eqref{Eq:EffectiveAction} and \\eqref{Eq:EffectivePotential}, which includes all relevant operators, i.e., those with a positive critical exponent at the FP of interest. Including field monomials up to order $4$, $6$, and $8$, defines the local potential approximation, LPA 4\/$4+\\eta$, LPA 6\/$6+\\eta$, and the LPA 8\/$8+\\eta$, respectively (depending on the inclusion of a scale dependent wavefunction renormalization, $\\partial_{t} Z_{I} \\neq 0$).\n\nIn order to distinguish physically meaningful from spurious FPs arising within a given truncation, we demand that a FP can be continued to higher orders in the truncation, and universal quantities, e.g., critical exponents, show signs of convergence. Further, corrections to canonical scaling should not be too large, as otherwise we would not expect our truncation to be reliable. Moreover, we demand that all eigenvalues of $\\left. \\left(\\frac{\\partial^2 u_k}{\\partial \\rho_I \\partial \\rho_J} \\right)\\right|_{\\rho_I = \\kappa_I}$ are non-negative. If this condition is violated, the expansion point for the effective potential does not correspond to its true minimum, and critical exponents evaluated around this point will show poor convergence properties. The parameter $\\Delta \\equiv \\left. \\det \\left(\\frac{\\partial^2 u_k}{\\partial \\rho_I \\partial \\rho_J} \\right) \\right|_{\\rho_I = \\kappa_I}$ serves to separate the space of couplings into different (not necessarily bounded) domains of attraction. Within such a domain there exists at most \\textit{one} IR stable FP, characterized by the strength of correlations \\cite{Vicari:2006xr}. In the following, we will be interested specifically in IR scaling solutions in the $\\Delta \\geq 0$ domain, corresponding to a minimum of the effective potential.\n\nWith these preliminaries and definitions we now turn to analyze the fixed-point structure of this model.\n\n\n\\section{Fixed-point analysis}\n\\label{Sec:Fixed-point analysis}\n\nFor generic multifield models with $\\bigoplus_{I} O(N_{I})$ symmetry, a number of FPs and their stability properties can be deduced from the existence of the $O(N)$ Wilson-Fisher FP. These FPs are typically characterized by an enhancement of symmetry.\n\n\\begin{itemize}[leftmargin=*]\n\n\\item[] The \\textit{isotropic fixed point (IFP)} shows maximal symmetry enhancement: All couplings at a given order in the fields take the same value, i.e., in the three-field model we have $\\left. \\lambda_{l,m,n} \\right|_{l+m+n = 2} \\equiv \\lambda_{2}$, and similarly for higher order couplings. It is characterized by $O(N)$ symmetry, where $N \\equiv \\sum N_{I}$. Accordingly, it features additional massless Goldstone modes, even in the case of an underlying discrete symmetry, e.g., with $Z_{2} \\oplus Z_{2} \\oplus Z_{2}$ symmetry.\n\n\\item[] The \\textit{decoupled fixed point (DFP)} is characterized by vanishing couplings between different sectors of the theory. In a model with three fields this implies $\\lambda_{l,m,n} = 0$ if $l,m \\neq 0$, $l,n \\neq 0$, or $m,n \\neq 0$. The values of the couplings in each sector approach those of the corresponding $O(N_{I})$ Wilson-Fisher FP. However, while the action at that FP is fully decoupled, critical exponents that relate to mixed couplings are nontrivial.\n\n\\item[] The \\textit{decoupled isotropic fixed point (DIFP)} occurs for the first time in a model with three fields: It is characterized by a partial enhancement of symmetry, as two fields remain fully coupled and the couplings in those sectors become degenerate. Simultaneously, the third field decouples completely and its couplings approach the corresponding values of the Wilson-Fisher FP. There exist three realizations of this FP, as any of the three sectors can be the one to decouple. For generic multifield models, a set of different DIFPs exists, where any number of the fields decouple, and the couplings in the remaining sectors show a symmetry enhancement.\n\n\\item[] We may additionally infer the existence of another class of FPs from the knowledge of the anisotropic scaling solution in the two-field model with $O(N_{1})\\oplus O(N_{2})$ symmetry. In general, any FP of the two-field model can be extended to the three-field model as a partially decoupled FP, where the third field decouples from the other two and the fixed-point values of its couplings are given by those of the Wilson-Fisher fixed point. In particular, this applies to the \\textit{biconical} fixed-point solution in the two-field model, which we identify as the \\textit{decoupled biconical fixed point (DBFP)}.\n\n\\end{itemize}\n\nIn analogy to the two-field case, we will refer to a FP in the three-field model as stable, if it features three relevant directions. This terminology relates to the requirement that these three parameters need to be tuned to reach the multicritical point.\n\nOur model, cf.\\ Eqs.\\ \\eqref{Eq:EffectiveAction} and \\eqref{Eq:EffectivePotential}, contains nine running couplings in the potential when we take into account all operators up to fourth order in the fields, that is, the associated parameters $\\kappa_I$ and couplings $\\lambda_{l,m,n}$ with $l + m + n = 2$. They give rise to the nine largest critical exponents of the model\\footnote{These can be calculated from the stability matrix\n\\begin{equation}\n\\mathit{\\Theta}_{i,j} = \\left. \\frac{\\partial \\beta_{g_i}}{\\partial g_j} \\right|_{\\textrm{FP}} ~.\n\\end{equation}\nHere, the $g_i$ label all the (dimensionless renormalized) running couplings\/parameters and $\\beta_{g_i}$ define the corresponding beta functions. The critical exponents are then given by:\n\\begin{equation}\n\\theta_{i} ~ \\in - {\\rm spec} (\\mathit{\\Theta}) ~.\n\\end{equation}\n}.\nGoing to higher orders in the expansion of the effective potential, the number of running couplings $\\lambda_{l,m,n}$ increases. Accordingly, the number of critical exponents will increase, but those subleading critical exponents will be irrelevant. The leading order critical exponents will typically receive corrections from the additional higher-order couplings and will therefore vary with the order of the truncation. \n\n\n\\subsection{Isotropic fixed point}\n\\label{SubSec:Isotropic fixed point}\n\nTo determine the critical exponents at the IFP in multifield models, it is crucial to realize that a subset of those is determined by the $O(N)$ Wilson-Fisher exponents. These correspond to the directions in theory space that respect that full symmetry, i.e., those directions that span the Wilson-Fisher theory space. Additional directions in the full theory space break (a subgroup) of the enhanced $O(N)$ symmetry, and their associated critical exponents are therefore not associated with the Wilson-Fisher critical exponents, see Fig.~\\ref{IFP_illustration}. \n\n\\begin{figure}[!t]\n\\includegraphics[width=0.7\\linewidth, height=120pt, clip=true, trim= 10cm 8cm 0cm 0cm]{illustration_symmetry.pdf}\n\\caption{\\label{IFP_illustration}This illustration of the two-dimensional subspace of the two-field coupling space shows that the Wilson-Fisher theory space is a one-dimensional subspace, corresponding to the $\\lambda_{2,0} = \\lambda_{0,2}$ line. The critical exponent along this direction corresponds to the largest Wilson-Fisher critical exponent. Another eigendirection of the stability matrix does not respect the enhanced symmetry of the FP.}\n\\end{figure}\n\nIt accordingly follows, that of the nine largest critical exponents at the IFP, two are determined by the scaling exponents of the $O(N)$ Wilson-Fisher FP. Among the additional critical exponents, one can observe a degeneracy, which can be understood from the following considerations:\n\n\\begin{table*}[!t]\n\\renewcommand{\\arraystretch}{1.2}\n\\renewcommand{\\tabcolsep}{6pt}\n\\begin{tabular}{c | c c c c c}\n$N = \\sum N_{I}$ & $ \\theta_{1} = \\theta_{2} = y_{2,2}$ & $\\theta_{3} = \\frac{1}{\\nu}$ & $\\theta_{4} = \\theta_{5} = \\theta_{6} = y_{4,4}$ & $\\theta_{7} = \\theta_{8} = y_{4,2}$ & $\\theta_{9} = -\\omega$ \\\\\n\\hline\\hline\n3 & 1.790 & 1.362 & 0.086 & -0.380 & -0.756 \\\\\n4 & 1.818 & 1.292 & 0.196 & -0.324 & -0.775 \\\\\n5 & 1.842 & 1.240 & 0.289 & -0.283 &-0.797 \\\\ \n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:IFP}Critical indices for the IFP in LPA 12 including anomalous dimensions. Our notation corresponds to the one introduced in Ref.\\ \\cite{Eichhorn:2013zza}.}\n\\end{table*}\n\nSufficiently close to $d = 4$, i.e., in the vicinity of the noninteracting fixed point, the relevant perturbations at the Wilson-Fisher FP are determined by the spin-$l$ representations of the $O(N)$ symmetry group \\cite{Wegner:1972zz} (see also Ref.\\ \\cite{Calabrese:2002bm}). Here, we will assume that such a classification of perturbations also holds for arbitrary dimensions, and only operators up to quartic order need to be taken into account. Defining the $N$-component field $\\Phi=(\\phi_1,\\phi_2,\\phi_3)$, we find the following eigendirections of the stability matrix at the IFP in the three-field model:\n\n\\begin{itemize}[leftmargin=*]\n\n\\item[] A scalar quadratic perturbation at the IFP $\\sim m^{2} \\Phi^{2}$, where $P_{2,0} = \\Phi^2$, defines the critical exponent $\\nu$ related to the divergence of the correlation length, i.e., $[m^{2}] = \\frac{1}{\\nu}$. This critical exponent is thus always positive, corresponding to one relevant direction. \n\n\\item[] From the quadratic perturbation $P^{a b}_{2,2} = \\Phi^{a} \\Phi^{b} - \\frac{1}{N} \\delta^{a b} \\Phi^{2}$ in the spin-$1$ representation of the $O(N)$ symmetry group, we can construct an $O(N)$ invariant operator by a suitable contraction of indices where, e.g., $\\mathcal{P}_{2,2} = \\phi_{I}^{2} - \\frac{N_{I}}{N} \\Phi^{2}$, $I = 1,2,3$. Then, the perturbation $\\sim v \\mathcal{P}_{2,2} $ defines the critical exponent $y_{2,2} = [v] = d - [\\mathcal{P}_{2,2}]$. For the three-field model, two independent operators of that form can be constructed. Thus the corresponding critical exponent shows a two-fold degeneracy in the scaling spectrum. We emphasize that these critical exponents are identical to those evaluated for the $O(N)$ symmetric IFP in the two-field or anisotropic $N$-vector models, see, e.g., Ref.\\ \\cite{Calabrese:2002bm, Folk:2008mi, 2011PhRvB..84l5136H, Eichhorn:2013zza}. These exponents are always positive, adding two further relevant directions at the FP.\n\n\\item[] A scalar quartic perturbation $\\sim u \\Phi^{4}$, where $P_{4,0} = \\Phi^{4}$, which is irrelevant at the IFP and defines the Wegner critical exponent $\\omega$, yields a negative critical exponent, i.e., $[u] = y_{4,0} = - \\omega$. \n\n\\item[] A quartic operator in the spin-$1$ representation of the $O(N)$ symmetry group: $P_{4,2}^{a b} = \\Phi^{2} P_{2,2}^{a b}$ can be contracted to define the exponent $y_{4,2} = d - [\\mathcal{P}_{4,2}]$, which is also given by the value calculated in the two-field model and shows a two-fold degeneracy in the three-field case.\n\n\\item[] A quartic perturbation in the spin-$2$ representation of the $O(N)$ symmetry group is given by\n\\begin{eqnarray}\n\\hspace{8pt} P_{4,4}^{a b c d} &=& \\Phi^{a} \\Phi^{b} \\Phi^{c} \\Phi^{d} \\nonumber - \\frac{1}{N + 4} \\Phi^{2} \\left( \\Phi^{a} \\Phi^{b} \\delta^{c d} + p(a,b,c,d) \\right) \\nonumber \\\\ && +\\: \\frac{1}{(N + 2) (N + 4)} ( \\Phi^{2} )^{2} \\left( \\delta^{a b} \\delta^{c d} + p(a,b,c,d) \\right) ~. \\nonumber\\\\ &&\n\\end{eqnarray}\nNote, that $p(a,b,c,d)$ denotes inequivalent permutation of the indices on the preceding operator, e.g., $\\Phi^{a} \\Phi^{b} \\delta^{c d} \\rightarrow \\Phi^{c} \\Phi^{d} \\delta^{a b} + \\ldots$ or $\\delta^{a b} \\delta^{c d} \\rightarrow \\delta^{a c} \\delta^{b d} + \\ldots$ The corresponding perturbation defines the critical exponent $y_{4,4} = d - [\\mathcal{P}_{4,4}]$, which becomes negative for $N = 3$, cf.\\ Fig.\\ \\ref{Fig:IFP}. This critical exponent with three-fold degeneracy is again determined by the two-field model, see, e.g., Ref.\\ \\cite{Calabrese:2002bm, Eichhorn:2013zza}.\n\\end{itemize}\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.8\\linewidth]{IFP_N1.pdf}\\\\\n\\includegraphics[width=0.8\\linewidth]{IFP_N.pdf}\n\\caption{\\label{Fig:IFP}(Color online) We show the fourth-largest critical exponent at the IFP as a function of $N_1$, with $N_2=N_3=1$ (upper panel) and as a function of $N_{1} = N_{2} = N_{3}$ (lower panel), with the LPA 4 result (green circles), LPA 6 (blue squares) and LPA 8 (red diamonds). The LPA clearly converges rapidly. As this exponent is positive beyond the LPA 4 and shows a three-fold degeneracy, the IFP is characterized by a total of six relevant directions.}\n\\end{figure}\n\nOur explicit numerical results within the LPA up to 8th order of the three-field model confirm this picture. We may therefore directly exploit the LPA to order 12 including anomalous dimensions within the two-field model to obtain the corresponding exponents. Note that a comparable computation in the full three-field model is quite demanding due to the large number of couplings between different sectors of the model. Using our results for the two-field case, see Ref.\\ \\cite{Eichhorn:2013zza}, we can accordingly determine the nine largest critical exponents of the model, cf.\\ Tab.\\ \\ref{Tab:IFP}. In general the IFP shows a large number of relevant parameters that require tuning to approach the FP. In fact, the IFP is unstable for any integer combination of field components $(N_1,N_2,N_3)$. It is clear that this pattern will persist to generic multifield models where additional fields are coupled to the system -- for each additional sector the number of relevant directions at the IFP increases (at least) by three.\n\n\n\\subsection{Decoupled fixed point}\n\\label{SubSec:Decoupled fixed point}\n\nAt the DFP, the nonvanishing couplings, i.e., the mass parameters and self-couplings lead to one relevant and one irrelevant direction in each sector. The associated critical exponents are those of the corresponding $O(N_{I})$ Wilson-Fisher FP. While the mixed couplings such as $\\lambda_{1,1,0}$ vanish at the FP, the corresponding critical exponents are nontrivial. This follows, as the FP is an interacting FP, and these residual interactions affect scaling dimensions of operators even if the corresponding coupling vanishes. In other words, contributions \\mbox{$\\sim \\lambda_{1,1,0} \\lambda_{2,0,0}$} in the $\\beta$ functions yield nonvanishing entries in the stability matrix even if $\\lambda_{1,1,0} = \\lambda_{1,0,1} = \\ldots = 0$.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{4pt}\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$& $N_2$&$N_3$ & $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1& 1& 1& 1.571 & 1.571 &1.571 & 0.142 & 0.142& 0.142\\\\\n2&1&1&1.459 & 1.571 & 1.571 & 0.030 & 0.030 & 0.142\\\\\n3&1&1&1.367 & 1.571 & 1.571 & -0.062 & -0.062 & 0.142\\\\\n4&1&1&1.296 & 1.571 &1.571 & -0.133& -0.133& 0.142\\\\\n\\hline\n2&2&1 & 1.459 & 1.459 & 1.571 & -0.082 & 0.030& 0.030\\\\\n3&2&1& 1.367 & 1.459 & 1.571 & -0.174 & -0.062& 0.030\\\\\n4&2&1&1.296 & 1.459&1.571& -0.245& -0.133 & 0.030\\\\\n3&3&1&1.367 &1.367&1.571& -0.266& -0.062 & -0.062\\\\\n\\hline\n2&2&2& 1.459 & 1.459 & 1.459 &-0.082 & -0.082 & -0.082\\\\\n3&2&2& 1.367 & 1.459 & 1.459& -0.174 & -0.174 & -0.082\\\\ \n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:DFPa}We list the six largest critical exponents as a function of $N_1, N_2, N_3$ at the DFP, employing results from the LPA 12 including anomalous dimensions from \\cite{Eichhorn:2013zza} and using the scaling relations Eqs.\\ \\eqref{Eq:ScalingRelationsDFPa} -- \\eqref{Eq:ScalingRelationsDFPc}.}\n\\end{table}\n\nAt the DFP, the eigendirections corresponding to the six largest critical exponents can be determined using a scaling relation: The quartic couplings $\\lambda_{1,1,0}, \\lambda_{1,0,1}$, and $\\lambda_{0,1,1}$ correspond to eigendirections of the FP with critical exponents\n\\begin{eqnarray}\n\\theta_{4} &=& \\frac{1}{\\nu_1} + \\frac{1}{\\nu_2}-d ~, \\label{Eq:ScalingRelationsDFPa} \\\\\n\\theta_{5} &=& \\frac{1}{\\nu_1} + \\frac{1}{\\nu_3}-d ~, \\label{Eq:ScalingRelationsDFPb} \\\\\n\\theta_{6} &=& \\frac{1}{\\nu_2} + \\frac{1}{\\nu_3}-d ~, \\label{Eq:ScalingRelationsDFPc}\n\\end{eqnarray}\nwhere $\\nu_{I} = \\frac{1}{\\theta_{I}}$, $I = 1,2,3$. These scaling relations can be  motivated as follows \\cite{Aharony:1976,Aharony:2002}: At the DFP, the decoupling of the three sectors implies $[\\phi_I^{2} \\phi_{J}^{2}] = [\\phi_{I}^{2}] + [\\phi_{J}^{2}]$. Furthermore, the scaling dimensions of $\\phi_{I}^{2}$ are -- due to the decoupling -- determined by the Wilson-Fisher critical exponents, such that $[\\phi_{I}^{2}] = - \\frac{1}{\\nu_{I}} + d$. The relations Eqs.\\ \\eqref{Eq:ScalingRelationsDFPa} -- \\eqref{Eq:ScalingRelationsDFPc} follow directly.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{4pt}\n\\textbf{LPA 8}\n\\vskip 2pt\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ & $N_3$ & $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1 & 1 & 1 & 1.537 & 1.537 & 1.537 & 0.067 & 0.067 & 0.067\\\\\n2 & 1 & 1 & 1.399 & 1.537 & 1.537 & -0.057 & -0.057 & 0.067\\\\\n3 & 2 & 1 & 1.306 & 1.399 & 1.537 & -0.275 & -0.150& -0.057\\\\\n\\hline \\hline\n & & & \\multicolumn{2}{c}{$\\Delta \\theta_{4}$} & \\multicolumn{2}{c}{$\\Delta \\theta_{5}$} & \\multicolumn{2}{c}{$\\Delta \\theta_{6}$} \\\\ \\hline \\hline\n1 & 1 & 1 & \\multicolumn{2}{c}{0.007} & \\multicolumn{2}{c}{0.007} & \\multicolumn{2}{c}{0.007}\\\\\n2 & 1 & 1 & \\multicolumn{2}{c}{-0.007} & \\multicolumn{2}{c}{-0.007} & \\multicolumn{2}{c}{0.007}\\\\\n3 & 2 & 1 & \\multicolumn{2}{c}{-0.020} & \\multicolumn{2}{c}{-0.007} & \\multicolumn{2}{c}{-0.007}\\\\ \\hline \\hline\n\\end{tabular}\n\\vskip 10pt\n\\textbf{LPA 8}$\\mathbf{+\\eta}$\n\\vskip 2pt\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ & $N_3$ & $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\ \\hline\\hline\n1 & 1 & 1 & 1.564 & 1.564 & 1.564 & 0.080 & 0.080 & 0.080\\\\\n2 & 1 & 1 & 1.447 & 1.564 & 1.564 & -0.028 & -0.028 & 0.080\\\\\n3 & 2 & 1 & 1.359 & 1.447 & 1.564 & -0.220 & -0.112 & -0.028\\\\\n\\hline \\hline\n & & & \\multicolumn{2}{c}{$\\Delta \\theta_{4}$} & \\multicolumn{2}{c}{$\\Delta \\theta_{5}$} & \\multicolumn{2}{c}{$\\Delta \\theta_{6}$} \\\\ \\hline \\hline\n1 & 1 & 1 & \\multicolumn{2}{c}{0.047} & \\multicolumn{2}{c}{0.047} & \\multicolumn{2}{c}{0.047}\\\\\n2 & 1 & 1 & \\multicolumn{2}{c}{0.038} & \\multicolumn{2}{c}{0.038} & \\multicolumn{2}{c}{0.047}\\\\\n3 & 2 & 1 & \\multicolumn{2}{c}{0.027} & \\multicolumn{2}{c}{0.035} & \\multicolumn{2}{c}{0.038}\\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:DFPb}We list the six largest critical exponents as a function of $N_{1}, N_{2}, N_{3}$ at the DFP, within the LPA 8 and LPA 8 with anomalous dimensions (LPA 8$+ \\eta$). We do not exploit the scaling relations, but instead evaluate all critical exponents explicitly from the $\\beta$ functions at the DFP. We check the scaling relation explicitly and give the deviation to this order of the truncation. The inclusion of the anomalous dimensions leads to a slightly larger violation of the scaling relation.}\n\\end{table}\n\n{ We observe that the violation of the scaling relation is slightly larger when anomalous dimensions are taken into account. This is not necessarily surprising, as the main effect of a running wavefunction renormalization is \\emph{not} to give a sizable improvement in the value of the critical exponents, but instead to provide a first reasonable estimate of the value of $\\eta$ itself. We expect that an enlargement of our truncation, including a field-dependent wavefunction renormalization (or, in other words, momentum-dependent interaction terms), will improve the situation.} \n\n\\begin{figure}[!t]\n\\includegraphics[width=0.8\\linewidth]{DFP_N.pdf}\n\\caption{\\label{Fig:DFP}(Color online) We show the fourth-largest critical exponent at the DFP as a function of $N_{1} = N_{2} = N_{3}$, with the LPA 4 result (green circles), LPA 6 (blue squares) and LPA 8 (red diamonds). The LPA clearly converges rapidly.}\n\\end{figure}\n\nAccordingly the stability of this FP can be determined completely from a knowledge of the Wilson-Fisher FP. Employing a LPA to order 12, including anomalous dimensions, we arrive at the results given in Tab.~\\ref{Tab:DFPa}, cf.\\ Ref.\\ \\cite{Eichhorn:2013zza}. Our results obtained within the LPA 8 for the three-field model show reasonable agreement with results deduced from the scaling relations, cf.\\ Fig.\\ \\ref{Fig:DFP}. In fact, we may check the quantitative accuracy of the scaling relations explicitly, without referencing the results from the $O(N)$ model. We simply calculate the deviations $\\Delta \\theta_{4} = \\theta_1 + \\theta_2 - d - \\theta_4$, $\\Delta \\theta_{5} = \\theta_1 + \\theta_3 - d - \\theta_5$, and $\\Delta \\theta_{6} = \\theta_2 + \\theta_3 - d - \\theta_6$, shown in Tab.\\ \\ref{Tab:DFPb} for the given data sets.\n\n{Our explicit numerical results in LPA 8 deviate from the results inferred from the LPA 12 in the two-field model in some cases. There, we expect that an enlarged truncation in the three-field model will give results in full agreement with those deduced from the two-field case. As the number of couplings grows very substantially with the truncation, such an explicit check is beyond the scope of this work. Where LPA 8 and LPA 12 $+ \\eta$ results deviate, the latter are more trustworthy.}\n\n\\subsection{Decoupled isotropic fixed point}\n\\label{SubSec:Decoupled isotropic fixed point}\n\nFor three interacting fields, we observe a new FP, where only one of the fields decouples, while the other two sectors show an enhancement of symmetry. For the following discussion, we will assume that it is the $\\phi_1$-sector that decouples, while the remaining sectors have a $O(N_2+N_3)$ symmetry. Two other possible DIFPs exist, for which one of the other subsectors decouples, respectively.\n\nClearly, one positive critical exponent is inherited from the $O(N_1)$ symmetric and another one from the $O(N_2+N_3)$ symmetric Wilson-Fisher scaling spectrum. Two critical exponents that are relevant for the stability properties of this FP follow from the critical exponents of the spin-$1$ and spin-$2$ perturbations of the two-field isotropic $O(N_{2} + N_{3})$ FP, i.e., $y_{2,2}$ and $y_{4,4}$. While $y_{2,2}$ is always positive, $y_{4,4}$ becomes positive for $N_2 + N_3 > 2$. We therefore conclude, that the DIFP can only be stable for $N_2 = N_3 = 1$.\n\nThere are two further exponents that we need to consider to establish the stability of the DIFP solution. Both follow from a scaling relation exploiting the decoupling of the $1$-sector: At the FP, the operators $\\mathcal{O}_1=\\phi_1^2 (\\phi_2^2+\\phi_3^2)$ and $\\mathcal{O}_2 = \\phi_1^2 \\left(\\phi_2^2 - \\frac{N_2}{N_2+N_3}(\\phi_2^2+\\phi_3^2) \\right)$ correspond to eigendirections of the stability matrix. As we know from the Wilson-Fisher FP that the scaling dimension $[\\phi_I^2] = -\\frac{1}{\\nu_I}+d$, we deduce that\n\\begin{equation}\n[\\mathcal{O}_1] = -\\frac{1}{\\nu_1} - \\frac{1}{\\nu_{2+3}} + 2d, \n\\end{equation}\nand accordingly the corresponding critical exponent is given by\n\\begin{equation}\n\\theta_{6} = \\frac{1}{\\nu_{1}} + \\frac{1}{\\nu_{2+3}} - d ~.\n\\end{equation}\nSimilarly, we deduce for the second operator that the corresponding critical exponent is given by\n\\begin{equation}\n\\theta_{4} = \\frac{1}{\\nu_{1}} + y_{2,2} - d ~,\n\\end{equation}\nwhere $y_{2,2}$ is the scaling dimension of the coupling belonging to $\\phi_2^2 - \\frac{N_2}{N_2+N_3}(\\phi_2^2+\\phi_3^2)$ in the two-field case, cf.\\ Ref.\\ \\cite{Eichhorn:2013zza}. The first relation is the one that arises for a two-field DFP, and gives a negative critical exponent (for values of $N_{1} > 1$).\n\nAt fixed $N_{2} = N_{3} = 1$, it is the critical exponent $\\theta_{4}$ that decides about the stability of the FP. Using results from the two-field case (LPA 12 including an anomalous dimension, cf.\\ Ref.\\ \\cite{Eichhorn:2013zza}) to obtain $\\theta_{1} = y_{2,2}$, $\\theta_{2} = \\frac{1}{\\nu_{1}}$, and $\\theta_{3} = \\frac{1}{\\nu_{2+3}}$, we arrive at the results shown in Tab.~\\ref{Tab:DIFP}. The DIFP is the stable FP for $N_{1} \\geq 6$ and $N_{2} = N_{3} = 1$.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{4pt}\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ & $N_3$& $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1 & 1 & 1 & 1.765 & 1.571 & 1.459 & 0.336 & -0.042 & 0.030 \\\\ \n2 & 1 & 1 & 1.765 & 1.459 & 1.459 & 0.224 & -0.042 & -0.092 \\\\\n3 & 1 & 1 & 1.765 & 1.367 & 1.459 & 0.132 & -0.042 & -0.174 \\\\\n4 & 1 & 1 & 1.765 & 1.296 & 1.459 & 0.061 & -0.042 & -0.245 \\\\\n5 & 1 & 1 & 1.765 & 1.242& 1.459 & 0.007 & -0.042 & -0.299 \\\\\n6 & 1 & 1 & 1.765 & 1.203 & 1.459 & -0.032 & -0.042 & -0.338 \\\\ \n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:DIFP}Critical exponents at the DIFP, using the LPA 12 including anomalous dimensions, and employing the above scaling relations.}\n\\end{table}\n\nNote that the derivation of the scaling relations is based on the assumption that the operators corresponding to these couplings are eigenoperators of the stability matrix. This property is, to the best of our knowledge, an assumption in $d = 3$ \\cite{Calabrese:2002bm}, and is usually not true within a truncation of the RG flow. Nevertheless, the stability properties are not incompatible with explicit numerical results within the LPA 8, where the transition to stability occurs already at $N_1=5$. { Explicitly, the critical exponents in LPA 8 at $N_1=5$ and $N_2=N_3=1$ read $\\theta_1=1.783, \\theta_2=1.193, \\theta_3=1.399, \\theta_4=-0.024,\\theta_5=-0.027$ and $\\theta_6=-0.395$.}\n\n\\vskip 5pt\n\n\n\\subsection{Decoupled biconical fixed point}\n\\label{SubSec:Decoupled biconical fixed point}\n\n\\begin{table}[!h]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{2.7pt}\n\\begin{tabular}{c c c | c c c c c c}\n$N_1$ & $N_2$ &$N_3$& $\\theta_{1}$ & $\\theta_{2}$ & $\\theta_{3}$ & $\\theta_{4}$ & $\\theta_{5}$ & $\\theta_{6}$ \\\\\n\\hline\\hline\n1 & 1.2 & 1 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.537} & 0.285 & -0.075 & {\\bf -0.005} \\\\\n1 & 1.4 & 1 & {\\bf 1.535} & {\\bf 1.448} & \\textit{1.537} & 0.105 & -0.015 & {\\bf -0.010} \\\\\n1 & 1.5 & 1 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.537} & 0.068 & 0.001 & {\\bf -0.001} \\\\ \n1 & 1.2 & 2 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.399} & 0.161 & -0.200 & {\\bf -0.005} \\\\\n1 & 1.4 & 2 & {\\bf 1.535} & {\\bf 1.448} & \\textit{1.399} & -0.020 & -0.140 & {\\bf -0.010} \\\\\n1 & 1.5 & 2 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.399} & -0.057 & -0.124 & {\\bf -0.001} \\\\\n1.2 & 1 & 5 & {\\bf 1.753} & {\\bf 1.381} & \\textit{1.193} & -0.051 & -0.413 & {\\bf -0.005} \\\\\n1.5 & 1 & 5 & {\\bf 1.537} & {\\bf 1.462} & \\textit{1.193} & -0.270 & -0.338 & {\\bf -0.001} \\\\\n\\hline\\hline\n\\end{tabular}\n\\caption{\\label{Tab:BFP}Critical exponents at the DBFP in the LPA 8. Critical exponents emphasized in \\textbf{bold} font correspond to those of the two-field BFP, while those in \\textit{italic} arise in the sector with $O(N_3)$ group symmetry. Here, we list the DBFP for values of the field components $N_{I}$, where it is characterized by $\\Delta > 0$ (see text).}\n\\end{table}\n\nBeyond the isotropic and decoupled FP solutions, the two-field models feature another scaling solution, which is the biconical FP \\cite{PhysRevLett.33.813, Calabrese:2002bm, Folk:2008mi}. It is stable only in a restricted parameter region of these models, where $\\left. \\Delta^{\\textrm{2-field}} \\right|_{\\textrm{BFP}} = \\lambda_{2,0} \\lambda_{0,2} - \\lambda_{1,1}^2 > 0$, as it transfers the stability from the IFP to the DFP. Certainly, this FP should similarly manifest itself in the three-field case. While one of the three sectors decouples, the two remaining sectors should feature nondegenerate couplings, and we expect that in a given range of the parameter space such a decoupled BFP will be stable. \n\n{ To obtain as precise results as possible, we should make use of all methods available to us. In fact, results obtained using an $\\epsilon$-expansion around $d=4$ in the two-field case, allow us to infer the stability of the decoupled biconical fixed point in the three field case in one important instance: }\nFrom Refs.\\ \\cite{Folk:2008mi, Calabrese:2002bm}, we know that the biconical FP is stable for $N_1 = 1, N_2 = 2$ in two-field models (and similarly when the sectors are interchanged). Combined with the pattern in Tab.\\ \\ref{Tab:BFP} for the additional critical exponents in the three-field model, we conjecture that the DBFP is stable for $N_1 = 1, N_2 = 2, N_3\\geq 2$ (up to a permutation of the three sectors). To calculate the corresponding critical exponents directly in the three-field model, we expect that an extended truncation will be necessary, taking into account a field-dependent wavefunction renormalization.\n\n\n\\subsection{Search for further stable fixed points}\n\\label{SubSec:Search for further stable fixed points}\n\nWe summarize our results obtained so far in Fig.~\\ref{Fig:StabilityPlot}. The figure shows the stable FP solution for the corresponding values of field components, $(N_1, N_2, N_3)$. Apparently, no stable FP exists in the range $N_{1} < 6$, $N_{2} = N_{3}=1$ (up to a permutation of the sectors) that can be derived from the known scaling solutions in the one- and two-field models. This motivates an independent analysis of fully coupled FPs in the three-field model, which we describe in the following sections.\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.8\\linewidth]{stability_cube_conjectured_modified.pdf}\n\\caption{\\label{Fig:StabilityPlot}(Color online) We show the stable DIFP (large blue dots), the stable DFP (small orange dots) and points without a stable FP (small gray dots) using the LPA 12$+\\eta$ results. We also include points where the DBFP is conjectured to be stable (middle-sized green dots).}\n\\end{figure}\n\n\n\\subsubsection{Stability trading between stable FPs}\n\\label{SubSec:Stability trading between stable FPs}\n\nGenerally, the $\\beta$ functions are non-polynomial functions of a large number of couplings in the multifield models. In the local potential type approximations the number of parameters and couplings increases from 9 to 34 if the order of the truncation is changed from 4 to 8. Thus, finding FPs in the multifield models becomes a highly nontrivial search for zeros of the $\\beta$ functions in a high-dimensional parameter space. In the following, we consider strategies to identify new FP solutions using a simple example. Consider the following $\\beta$ function which is expanded in terms of the coupling $g$ (assuming that higher than quadratic terms are zero):\n\\begin{equation}\n\\beta_g = g ( c + g ),\n\\end{equation}\nwhere $c$ is a function of the parameters of the model (e.g., dimensionality, number of field components, etc.) and possibly other couplings in a given truncation of the theory. Note that such a form captures the essential properties of typical fixed points, as it allows both for a trivial Gaussian FP and a nontrivial interacting FP, as a function of the parameter $c$. The critical exponent at a fixed point is given by\n\\begin{equation}\n\\theta = - \\left. \\frac{\\partial\\beta_g}{\\partial g} \\right|_{\\textrm{FP}} = \\left. - c - 2 g \\right|_{\\rm FP}. \n\\end{equation}\nAssuming that it is the exponent $\\theta$ that decides about the stability of the FP, we may distinguish the following scenarios: For $c < 0$, the interacting FP at $g = -c$ is infrared stable, whereas the Gaussian FP is unstable. As the parameter $c$ increases towards positive values (as a function of, e.g., $N_I$) the two FPs will approach each other. At $c = 0$, both FPs collide and exchange their stability properties. Moving apart again for $c > 0$, the interacting FP has become the unstable one, whereas the noninteracting FP is stable, cf.\\ Fig.\\ \\ref{Fig:StabilityTrading}. This simple example demonstrates that FPs will typically change their stability when they collide, as was also observed in Ref.\\ \\cite{Eichhorn:2013zza} in the two-field model with $O(N_{1})\\oplus O(N_{2})$ symmetry.\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.8\\linewidth]{stability_trading3.pdf}\n\\caption{\\label{Fig:StabilityTrading}To illustrate the mechanism how stability properties are exchanged between FPs, we examine the $\\beta$-function $\\beta = g(c + g)$ for varying parameter $c$ describing the position of the nontrivial FP. As the FPs pass by one another, the local derivatives $\\theta = -\\beta'(g)$ exchange their sign indicating an interchange of their IR stability properties. }\n\\end{figure}\n\n\nInspired by the above stability-trading mechanism\nwe may devise a strategy to identify new FP solutions. Our search for FPs will concentrate on the vicinity of points in coupling space, where a known FP loses its stability.\n\n\n\\subsubsection{Fully coupled fixed points in the three-field model}\n\\label{SubSec:Fully coupled fixed points in the three-field model}\n\nIt turns out that the three-field model works in a different way from the mechanism described above, which applies in the two-field case: Within the LPA 4, we observe that the IFP, DFP, and DIFP have partially overlapping stability regions. No similar behavior occurs in two-field models, where stability regions of different FPs always touch, but never overlap, due to the above stability-trading mechanism. However, this changes dramatically as we include anomalous dimensions: While the IFP inhabits the same points, the DIFP is now only stable for $N_1 \\geq 4$, $N_2 = N_3=1$ etc.\\ Extensive numerical searches did not reveal a stable FP for $N_1 =3$, $N_2 = N_3=1$, and similarly for the cases where the sectors are interchanged. A similar result holds for higher orders of the LPA: As there are a larger number of independent operators that serve as a basis for the LPA, and thus potentially relevant directions in the three-field model, the stability exchange mechanism may not be captured by the simple model considered above. To elucidate the differences in the three-field model, we will focus on results obtained within the LPA to 8th order. \n\n\\begin{figure}[!t]\n\\includegraphics[width=\\linewidth]{BFPDFPcollision.pdf}\n\\caption{\\label{BFPDFPcollision}We plot $\\lambda_{1,0,1}, \\lambda_{0,1,1}$, and $\\lambda_{1,1,0}$ as a function of $N_I$ for the three DBFPs. At $N_I=1.244$ they each pass through the origin of that coordinate system, where the DFP sits. For $N_I>1.244$, the three BFPs move away from each other towards more negative values of the mixed couplings.}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.8\\linewidth]{BFPDFPthetas.pdf}\n\\includegraphics[width=0.8\\linewidth]{BFPDFPcoords.pdf}\n\\caption{\\label{BFPDFPthetas}We plot the fifth-largest critical exponent at the DBFPs (blue points of increasing value) and the corresponding critical exponent at the DFP (red points of decreasing value) as a function of $N_I$ (upper panel). Below, we show the coordinates of the couplings $\\lambda_{2,0,0}$ and $\\lambda_{0,2,0}$ at the DBFP (blue diamonds and black squares) and the couplings $\\lambda_{2,0,0}= \\lambda_{0,2,0} = \\lambda_{0,0,2}$ at the DFP (red dots) in the vicinity of the collision point.}\n\\end{figure}\n\nAs a first example, let us consider the point $N_1 = N_2 = N_3 \\approx 1.25$. Here, the DFP is stable, but it gains three additional relevant directions around $N_1 = N_2 = N_3 \\approx 1.244$. As within the two-field model, this point is marked by a collision with a decoupled biconical FP (DBFP). The main difference is that within a three-field model, three generalizations of the BFP exist, cf.\\ Sec.\\ \\ref{SubSec:Decoupled biconical fixed point}. For $N_I =1.25$, $I = 1,2,3$, all of these FPs feature five relevant directions. Toward smaller $N_I$, all three DBFPs approach the DFP, and simultaneously collide with it at $N_I \\approx 1.244$. At this point, the DFP gains one relevant direction from each of the three DBFPs, which subsequently feature only four relevant exponents, cf.\\ Fig.\\ \\ref{BFPDFPcollision} and Fig.~\\ref{BFPDFPthetas}. Thus this FP loses stability in a fixed-point collision. The central difference to the two-field case lies in the fact that the symmetry of the model which forces the DFP to collide with three other FPs simultaneously (for $N_1 = N_2 = N_3$ there is an exchange symmetry $N_1 \\leftrightarrow N_2 \\leftrightarrow N_3$). As each of them starts off with five relevant directions, the collision does not produce a stable FP for $N_I \\leq 1.24$, but instead leaves behind three FPs with four relevant directions each.\n\nAs a second example of new behavior in three-field models, we consider the point where the DBFP collides with the DIFP in the LPA 8. We fix $N_2 =1$ and $N_3=4$: Then the DBFP has five positive critical exponents at $N_1=1$. Going to larger values of $N_1$, it collides with the DIFP at $N_1 \\approx 1.6$. During this collision, the DIFP becomes unstable, and the DBFP gains one negative critical exponent, cf.\\ Fig.\\ \\ref{BFPDIFP}. Similar to the previous scenario, this FP collision is not sufficient to make the DBFP stable. Following the DBFP to even larger values of $N_1$, it undergoes another collision, this time being hit by two other FPs simultaneously. This is a novel feature that is not observed in simple two-field models. Starting in the region $N_1 < 1.2$, these FPs do not seem to exist for real values of the couplings -- at least no sign of them showed up in extensive numerical searches. They can be thought of as being created at the collision. Following this collision, they quickly move away from the collision point for increasing values of $N_{1}$. Each of the newly created FPs features four relevant critical exponents, while the DBFP is stable, cf.\\ Fig.\\ \\ref{BFPcollision}. Both new FPs are anisotropic, and define a new universality class that occurs for the first time if three fields are coupled, cf.\\ Tab.\\ \\ref{Tab:AFP}.\n\n\\begin{figure}[!h]\n\\includegraphics[width=0.9\\linewidth]{BFPcoords.pdf}\\\\\n\\includegraphics[width=0.9\\linewidth]{BFPthetas.pdf}\\\\\n\\includegraphics[width=0.9\\linewidth]{DIFPtheta.pdf}\n\\caption{\\label{BFPDIFP}(Color online) Here, we show the couplings $\\lambda_{2,0,0}$, $\\lambda_{0,2,0}$, and $\\lambda_{1,1,0}$ at the DBFP (blue circles, purple squares, cyan diamonds) and $\\lambda_{2,0,0} = \\lambda_{0,2,0} = \\lambda_{1,1,0}$ (red triangles) at the DIFP (uppermost panel). We show the fourth- and fifth-largest critical exponent (middle panel) at the DBFP as a function of $N_1$ for $N_2=1, N_3=4$. Around $N_1 \\approx 1.16$, the three couplings are clearly degenerate, as expected for a collision with the DIFP. At the same point, the fourth critical exponent crosses zero and becomes negative. Simultaneously, the fifth critical exponent at the DIFP (lower panel) crosses zero and becomes positive.}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\includegraphics[width=\\linewidth]{AFPcollision.pdf}\\\\\n\\includegraphics[width=\\linewidth]{AFPtheta.pdf}\\\\\n\\caption{\\label{BFPcollision}Here, we show the couplings $\\lambda_{1,0,1}$ and $\\lambda_{0,1,1}$ at the two anisotropic FPs. As a function of $N_1$ they converge towards zero, which is where the DBFP is sitting. The collision occurs at $N_1 \\approx 1.2$, which is where the fifth critical exponent of these two FPs (which numerically is nearly the same for both) approaches zero.}\n\\end{figure}\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{2.7pt}\n\\begin{tabular}{ccc|ccc|ccc|c}\n$N_1$ & $N_2$ & $N_3$ & $\\lambda_{2,0,0}$ & $\\lambda_{0,2,0}$ & $\\lambda_{0,0,2}$ & $\\lambda_{1,1,0}$ & $\\lambda_{1,0,1}$ & $\\lambda_{0,1,1}$ & $\\lambda_{1,1,1}$\\\\ \\hline \\hline\n2 & 2 & 2 & 6.2 & 6.4& 6.4 & 1.8 & 1.8 & -2.2 & -2.6\\\\\n2 & 1 & 2 & 6.5 & 7.5 & 6.7 & 1.2 & 0.5 & -2.1 & -1.4\\\\\n2 & 1 & 3 & 6.2 & 7.0 & 5.8 & 3.2 & 0.7 & -1.8 & -2.2\\\\\n2 & 1 & 4 & 6.0 & 6.5 & 5.1 & 4.6 & 0.6 & -1.3 & -1.6\\\\ \\hline \\hline\n\\end{tabular}\n\\vskip 4pt\n\\renewcommand{\\arraystretch}{1.4}\n\\renewcommand{\\tabcolsep}{6.8pt}\n\\begin{tabular}{ccc|ccc|ccc}\n{} & {} & {} & $\\theta_1$ & $\\theta_2$ & $\\theta_3$ & $\\theta_4$ & $\\theta_5$ & $\\theta_6$ \\\\ \\hline \\hline\n2 & 2 & 2 & 1.70 & 1.33 & 1.29 & 0.20 & -0.17 & -0.24\\\\\n2 & 1 & 2 & 1.62 & 1.40 & 1.35 & 0.08 & -0.05 & -0.23\\\\\n2 & 1 & 3 & 1.71 & 1.34 & 1.28 & 0.15 & -0.10 & -0.30\\\\\n2 & 1 & 4 & 1.77 & 1.32 & 1.23 & 0.16 & -0.07 & -0.35\\\\ \\hline \\hline\n\\end{tabular}\n\\caption{\\label{Tab:AFP}We list selected FP values and critical exponents of the first anisotropic FP in the LPA 8, restricting ourselves to couplings at fourth order in the fields, and additionally giving the value of the coupling associated with the three-field operator $\\sim \\phi_1^2 \\phi_2^2 \\phi_3^2$.}\n\\end{table}\n\nDue to the complicated dynamics of FPs in the parameter space of three-field models, it seems that these models do not necessarily feature a stable FP solution for all values of the model parameters. One might now wonder, why FPs of three-field models show such a disproportionate increase in the number of relevant directions for small values of $N_I$. While the Wilson-Fisher FP has one relevant direction, and two-field models always feature FPs with only two relevant directions, three-field models show FPs with more than three relevant directions. We conjecture that this is related to the fact that the transition from a two-field to a three-field model implies the existence of more than one additional new class of operators: While the transition from one to two fields only adds the mixed interactions $\\sim \\left( \\phi_1^{2} \\right)^{m} \\left( \\phi_2^{2} \\right)^{n}$, the transition to three-field models features three different classes of mixed interactions $\\sim \\left( \\phi_I^{2} \\right)^{m} \\left( \\phi_J^{2} \\right)^{n}$, $I,J = 1,2,3$, in addition to the new three-field couplings $\\sim \\left( \\phi_1^{2}\\right)^{l} \\left( \\phi_2^{2} \\right)^{m} \\left( \\phi_3^{2} \\right)^{n}$. These seem to play a particularly important role for small $N_I$ and imply that the IR scaling properties cannot be accounted for by only three free relevant parameters. Note that this does not imply that the corresponding additional couplings need to become relevant. Since these operators do not necessarily correspond to eigendirections of the RG, they may mix with other operators, and yield corrections to the scaling spectrum.\n\nApplying our results to determine the properties of possible multicritical points in phase diagrams for systems with three competing order parameters, we may conclude that models with small $N_I$ will typically feature a first order, rather than a second order (multicritical) transition. In particular, this applies to the phenomenologically relevant model of three interacting $Z_{2}$-Ising fields, i.e., $Z_{2} \\oplus Z_{2} \\oplus Z_{2}$ symmetry.\n\n\n\\section{Summary and conclusions}\n\\label{Sec:Summary and conclusions}\n\nHere, we present a renormalization group study of IR stable FP solutions in three-field models with $O(N_1) \\oplus O(N_2) \\oplus O(N_3)$ symmetry. Our main results regarding the existence of stable FPs are summarized in Fig.~\\ref{Fig:StabilityPlot}. Models in this class exhibit FPs that generalize the Wilson-Fisher FP, falling into three distinct categories, each characterized by the degree of symmetry-enhancement. We find a decoupled FP, a partially isotropic FP solution, and a fully isotropic FP solution. Their scaling spectrum can be deduced partially by considering perturbations around the single- and two-field models with $O(N)$ and $O(N_{1})\\oplus O(N_{2})$ symmetry, respectively. We proceed by deriving scaling relations between different critical exponents to discover the stability properties of nontrivial FPs in the three-field case. Apart from the generalized Wilson-Fisher scaling solutions, we identify a decoupled biconical FP whose scaling properties are partly inherited from the BFP in the $O(N_{1})\\oplus O(N_{2})$ symmetric model. As a main result of this work, we find that these FPs all show a significantly larger number of relevant critical exponents than in the two-field case, in the region of small $N_1$, $N_2$, and $N_3$. We tentatively connect this result to the existence of a large number of mixed interactions, and further conjecture, that similar results will hold for models with $n>3$ interacting fields.\n\nSummarizing our results, we find no IR stable FP for a small number of fields ($N_{1} < 6$, $N_{2} = N_{3} = 1$, up to permutations of the fields) in $d = 3$ dimensions. This result is certainly unexpected, as there is no evidence for similar behavior in coupled two-field models. While, in principle, we cannot exclude the possibility that stable FPs exist in that region of parameter space, we find no evidence for their existence in extensive numerical searches for FPs of the nonperturbative $\\beta$ functions. The identification of FPs in the three-field field models with $O(N_{1})\\oplus O(N_{2})\\oplus O(N_{3})$ is in general a difficult problem, since the search has to proceed through a high-dimensional coupling space. Nevertheless, the understanding of basic stability transitions between different FPs serves as a guiding principle to single out possible candidates for nontrivial FPs. Quite generally, in coupled-field models, stability seems to be inherited from single mergers or collisions of different FPs. Searches around such stability transition points have not yielded any FP that carries over the stability properties from the IFP (stable at small, noninteger values of $N_{I}<1$) to the decoupled FP. This indicates that the dynamics of FPs in $O(N_{1})\\oplus O(N_{2})\\oplus O(N_{3})$ symmetric three-field, or general $\\bigoplus_{I} O(N_{I})$ symmetric multifield models is very different from that encountered in the simpler $O(N_1)\\oplus O(N_2)$-type models.\n\nFrom these results, we may conclude that models with phenomenological relevance such as, e.g., the $Z_{2} \\oplus Z_{2} \\oplus Z_{2}$ symmetric model will not feature a multicritical point in its phase diagram. Certainly, it is challenging to find three parameters that are accessible experimentally, and may be tuned to the multicritical point. This would be necessary to quantify the scaling behavior close to the corresponding FP, or to show the absence of such a transition. Nevertheless, it is conceivable that in the context of ultracold atomic systems such a control of the system might be achievable \\cite{doi:10.1146\/annurev-conmatphys-070909-104059}. \n\nFinally, let us comment on the general applicability of our results to other systems of interest. The renormalization group flow equations are derived for general $d$ Euclidean dimensions and can be applied to $d = 2$, relevant for critical behavior of low-dimensional condensed-matter systems, and $d = 4$, for multifield models of inflation, as well as possible extensions of the Higgs sector of the standard model. We leave this for future work.\n\n\\begin{acknowledgements}\nWe thank P.~Calabrese for discussions.\nResearch at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. This work is supported by DOE grant No.\\ DE-FG0201ER41195. M.M.S. is supported by the grant ERC-AdG-290623 and DFG grant FOR 723. A.E. would like to thank the University of Heidelberg and the University of Illinois at Chicago for hospitality during the early stages of this work.\n\\end{acknowledgements}\n\n\\begin{appendix}\n\\label{Appendix}\n\n\\section{Functional renormalization group}\n\\label{Sec:Functional renormalization group}\n\nThe nonperturbative functional renormalization group \\cite{Wetterich:1992yh,Berges:2000ew,Polonyi:2001se,Pawlowski:2005xe,Gies:2006wv,Delamotte:2007pf} defines a functional flow for the scale dependent effective action $\\Gamma_{k}$, which interpolates between the microscopic action $S$ defined at some ultraviolet cutoff scale $k = \\Lambda$ and the full effective action $\\Gamma_{k\\rightarrow 0} = \\Gamma$, when the renormalization group parameter $k$ is removed. It is given by\n\\begin{equation}\n\\partial_t \\Gamma_{k} = \\frac{1}{2} \\Tr \\int\\! \\frac{d^{d}q}{(2\\pi)^{d}} \\left(\\Gamma_k^{(2)} (q) + R_k(q) \\right)^{-1} \\partial_t R_k(q) ~,\n\\label{Eq:WetterichEquation}\n\\end{equation}\nwhere the logarithmic scale derivative is written in terms of the parameter $t = \\ln (k\/\\Lambda)$, and the second functional derivative $\\Gamma_k^{(2)}(p,q) = \\frac{\\delta^2 \\Gamma_k}{\\delta \\chi(-p) \\delta \\chi(q)}$, and $\\Gamma^{(2)}(q) (2\\pi)^{d} \\delta^{(d)}(p-q) \\equiv \\Gamma^{(2)} (p,q)$. Here, $\\chi$ denotes the complete field content of our model and the trace $\\Tr$ denotes a summation over internal degrees of freedom, i.e., both fields and field components. The regulator function $R_{k}$ implements a mass-like cutoff and regulates the infrared divergences. We take $R_{I J} (q) = R_{I}(q) \\delta_{I J}$, $I, J = 1,2,3$, while the momentum dependence is given by $R_{I}(q) = Z_{I} (k^{2} - q^{2}) \\theta(k^{2} - q^{2})$, where the wavefunction renormalization is scale dependent. This choice is referred to as the optimized regulator \\cite{Litim:2000ci, Litim:2001up} which allows us to derive fully analytic expressions for the nonperturbative $\\beta$ functions, and it is thus a convenient choice to identify possible scaling solutions.\n\n\\onecolumngrid\n\\section{Scale dependent effective potential}\n\\label{Sec:Scale dependent effective potential}\n\nWe derive the renormalization group flow equation for the effective potential by plugging our \\textit{ansatz} Eq.\\ \\eqref{Eq:EffectivePotential} into the flow equation \\eqref{Eq:WetterichEquation} and projecting the right-hand-side onto a constant field configuration:\n\\begin{eqnarray}\n\\partial_t u_k &=& -d u_k +\\sum_I (d-2+ \\eta_I) \\rho_I \\partial_{\\rho_{I}} u_{k} + { 2 v_{d} \\sum_{I} \\Big\\{ (N_I-1) l_{0}^{(I)} \\left( \\partial_{\\rho_{I}} u_{k} \\right)} \\nonumber\\\\ && { +\\: l_{R}^{(I)} \\left( \\left\\{ \\partial_{\\rho_{J}} u_{k} + 2 \\rho_{J} \\partial_{\\rho_{J}}^{2} u_{k} \\right\\}, \\left\\{ 4 \\rho_{J} \\rho_{K} \\partial_{\\rho_{J}} \\partial_{\\rho_{K}} u_{k} \\right\\} \\right)} \\Big\\} ~. \n\\label{Eq:FlowEquationPotential}\n\\end{eqnarray}\nHere, $v_{d}^{-1} = 2^{d+1} \\pi^{\\frac{d}{2}} \\Gamma\\left(\\frac{d}{2}\\right)$ arises from the volume integration, and the anomalous dimensions are defined as $\\eta_I = - \\partial_t \\ln Z_I$. In the following the notation follows Sec.\\ \\ref{Sec:Model}. The threshold functions $l_{0}^{(I)}$ and $l_{R}^{(I)}$ (see, e.g., \\cite{Braun:2011pp}) define the diagrammatic contributions to the renormalization group flow of the scale dependent effective potential, where the upper index indicates the corresponding sector. Using an optimized regulator function \\cite{Litim:2000ci,*Litim:2001up} the threshold functions take the following form:\n\\begin{eqnarray}\nl_{0}^{(I)}(w_{I}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{I}}{d+2}\\right)\\frac{1}{1+w_{I}} ~, \\\\\n\\hspace{-20pt} l_{R}^{(1)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{1}}{d+2}\\right) \\frac{\\left( 1 + w_{2} \\right) \\left( 1 + w_{3} \\right) - \\delta_{2,3}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~, \\\\\n\\hspace{-20pt} l_{R}^{(2)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{2}}{d+2}\\right) \n\\frac{\\left( 1 + w_{1} \\right) \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~, \\\\\n\\hspace{-20pt} l_{R}^{(3)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{3}}{d+2}\\right) \n\\frac{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right) - \\delta_{1,2}^{2}}{2 \\delta_{1,2} \\delta_{1,3} \\delta_{2,3} \n- \\delta_{1,2}^{2} \\left( 1 + w_{3} \\right) - \\delta_{1,3}^{2} \\left( 1 + w_{2} \\right) - \\delta_{2,3}^{2} \\left( 1 + w_{1} \\right)\n+ \\prod_{J} \\left( 1 + w_{J} \\right)} ~.\n\\end{eqnarray}\nNote, that the nature of the interactions is such, that only the radial modes $l_{R}^{(I)}$ are affected by the couplings, e.g., $\\delta_{1,2}^{2} = 4 \\kappa_{1} \\kappa_{2} \\lambda_{1,1,0}^{2}$, and equivalent couplings between the remaining sectors. One may easily check that in the limit of vanishing couplings, $\\delta_{1,3} \\rightarrow 0$ and $\\delta_{2,3} \\rightarrow 0$, the radial contributions $l_{R}^{(I)}$ in the $(1,2)$ sectors reduce to the threshold functions that were already derived in the two-field model \\cite{Eichhorn:2013zza}:\n\\begin{eqnarray}\n\\hspace{-20pt} l_{0}^{(I)}( w_{I} ) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{I}}{d+2}\\right)\\frac{1}{1+w_{I}} ~, \n\\label{Eq:ThresholdFunctionL0} \\\\\n\\hspace{-20pt} l_{R}^{(1)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{1}}{d+2}\\right) \\frac{1 + w_{2}}{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right)- \\delta_{1,2}^{2}} ~, \\\\\n\\hspace{-20pt} l_{R}^{(2)}(\\{ w_{I} \\}, \\{ \\delta_{I,J}^{2} \\}) &=& \\frac{2}{d} \\left( 1 - \\frac{\\eta_{2}}{d+2}\\right) \\frac{1 + w_{1}}{\\left( 1 + w_{1} \\right) \\left( 1 + w_{2} \\right)- \\delta_{1,2}^{2}} ~.\n\\label{Eq:ThresholdFunctionLTildeR}\n\\end{eqnarray}\nFrom \\eqref{Eq:FlowEquationPotential} the flow equations for the couplings are derived by the differentiation with respect to the fields and successive projection onto a nonvanishing constant background field configuration $\\rho_{I} = \\kappa_{I}$, defined by the minimum of the effective potential. For some FPs it might be necessary to employ an expansion point where one or several of the $\\kappa_{I}$ are vanishing. In this case $\\beta_{\\kappa_{3}} \\equiv 0$. For a detailed discussion of this issue, we refer to Ref.\\ \\cite{Eichhorn:2013zza}. \n\nWe obtain the $\\beta$ functions for the couplings $\\lambda_{l,m,n}$, $l + m + n \\geq 2$:\n\\begin{equation}\n\\beta_{\\lambda_{l,m,n}} = \\left. \\left(\\frac{\\partial^{l+m+n}}{\\partial \\rho_1^{l} \\partial \\rho_2^{m} \\partial \\rho_3^{n}} \\partial_t u_k + \\beta_{\\kappa_1} \\frac{\\partial^{(l+1)+m+n} u_k}{\\partial \\rho_{1}^{l+1} \\partial \\rho_2^m \\partial \\rho_3^n} + \\beta_{\\kappa_2} \\frac{\\partial^{l+(m+1)+n} u_k}{\\partial \\rho_{1}^{l} \\partial \\rho_2^{m+1} \\partial \\rho_3^n} + \\beta_{\\kappa_3} \\frac{\\partial^{l+m+(n+1)} u_k}{\\partial \\rho_{1}^{l} \\partial \\rho_2^m \\partial \\rho_3^{n+1}} \\right) \\right|_{\\rho_I = \\kappa_I} ~,\n\\end{equation}\nwhere $\\beta_{\\lambda_{l,m,n}} \\equiv \\partial_{t} \\lambda_{l,m,n}$. The $\\beta$ functions for the scale dependent dimensionless field expectation values $\\kappa_{I}$, $I = 1,2,3$, are given by\n\\begin{eqnarray}\n\\beta_{\\kappa_1} &=& \\frac{1}{\\Delta} \\left\\{\n- \\Delta_{2,3} \\partial_{\\rho_1} \\partial_t u_k + \\left(\\lambda_{1,1,0} \\lambda_{0,0,2} - \\lambda_{1,0,1} \\lambda_{0,1,1}\\right) \\partial_{\\rho_2} \\partial_t u_k \\right. \\nonumber \\\\ && \\left.\\left. +\\: \\left(\\lambda_{1,0,1} \\lambda_{0,2,0} - \\lambda_{1,1,0} \\lambda_{0,1,1}\\right) \n\\partial_{\\rho_3} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~, \n\\end{eqnarray}\n\\begin{eqnarray}\n\\beta_{\\kappa_2} &=& \\frac{1}{\\Delta} \\left\\{ - \\Delta_{1,3} \\partial_{\\rho_2} \\partial_t u_k + \\left( \\lambda_{2,0,0} \\lambda_{0,1,1} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\right) \\partial_{\\rho_3} \\partial_t u_k \\right. \\nonumber\\\\ \n&& \\left.\\left. +\\: \\left( \\lambda_{0,0,2} \\lambda_{1,1,0} - \\lambda_{1,0,1} \\lambda_{0,1,1} \\right) \\partial_{\\rho_1} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~, \\\\\n\\beta_{\\kappa_3} &=& \\frac{1}{\\Delta} \\left\\{\n- \\Delta_{1,2} \\partial_{\\rho_3} \\partial_t u_k  + \\left( \\lambda_{2,0,0} \\lambda_{0,1,1} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\right) \\partial_{\\rho_2} \\partial_t u_k \\right. \\nonumber\\\\ && \\left.\\left. +\\: \\left( \\lambda_{0,2,0} \\lambda_{1,0,1} - \\lambda_{1,1,0} \\lambda_{0,1,1}\\right) \\partial_{\\rho_1} \\partial_t u_k \\right\\} \\right|_{\\rho_I = \\kappa_I} ~. \n\\end{eqnarray}\nHere, we have defined the coupling parameter\n\\begin{equation}\n\\Delta_{1,2} = \\lambda_{2,0,0} \\lambda_{0,2,0} - \\lambda_{1,1,0}^{2} ~,\n\\end{equation}\nand equivalently $\\Delta_{1,3}$ and $\\Delta_{2,3}$ (defined from the remaining quartic couplings in the three-field model), as well as the parameter\n\\begin{eqnarray}\n\\Delta &=& - 2 \\left( \\lambda_{2,0,0} \\lambda_{0,2,0} \\lambda_{0,0,2} - \\lambda_{1,1,0} \\lambda_{1,0,1} \\lambda_{0,1,1} \\right) + \\lambda_{0,0,2} \\Delta_{1,2} + \\lambda_{0,2,0} \\Delta_{1,3} + \\lambda_{2,0,0} \\Delta_{2,3} ~.\n\\end{eqnarray}\nThese parameters quantify the symmetry enhancement properties of the system. In particular, for certain symmetry enhanced FPs, these quantities vanish exactly.\n\n\\section{Wavefunction renormalization and anomalous dimensions}\n\\label{Sec:Wavefunction renormalization and anomalous dimensions}\n\nTo determine the scale dependence of the field independent renormalization factor $Z_{I}$ from the functional flow equation \\eqref{Eq:WetterichEquation} we perform a projection of the flow onto operators of the type $\\sim \\left( \\partial_{\\mu} \\phi_{I} \\right)^{2} $. This yields the scale dependence of the coefficient for the corresponding operator in the effective action. We have\n\\begin{equation}\n\\partial_{t} Z_{I} = \\frac{(2\\pi)^{d}}{\\delta^{(d)}(0)} \\lim_{Q\\rightarrow 0} \\frac{\\partial}{\\partial Q^{2}} \\frac{\\delta^{2}}{\\delta \\varphi_{I}(-Q) \\delta \\varphi_{I}(Q)} \\partial_{t} \\Gamma_{k} ~,\n\\label{Eq:RenormalizationFactorProjection}\n\\end{equation}\nwhere the functional derivatives are taken with respect to the Nambu-Goldstone (NG) degrees of freedom $\\varphi_{I}$ in the $I$-sector. Note, there is no summation implied over the $I$-index. For details of the derivation in the context of the $O(N)$ vector model, we refer to \\cite{Tetradis:1993ts,Berges:2000ew}. \n\nThe anomalous dimensions are defined via the scaling contribution to the wavefunction renormalization, i.e., $\\eta_{I} = - \\partial_{t} \\ln Z_{I}$, and take the following form in the three-field model:\n\\begin{eqnarray}\n\\eta_{1} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{2} \\lambda_{1,1,0}^2 + \\kappa_{3} \\lambda_{1,0,1}^2 + \\kappa_{1} \\left( \\lambda_{2,0,0} + 2 \\kappa_{2} \\Delta_{1,2} + 2 \\kappa_{3} \\Delta_{1,3} + 4 \\kappa_{2} \\kappa_{3} \\Delta \\right)^2 \\right. \\nonumber\\\\ && \\left. +\\: 4 \\kappa_{2} \\kappa_{3} \\left( \\Delta_{2,3} \\left( \\lambda_{2,0,0} + \\kappa_{2} \\Delta_{1,2} + \\kappa_{3} \\Delta_{1,3} \\right) - \\Delta \\left(1 + \\kappa_{2} \\lambda_{0,2,0} + \\kappa_{3} \\lambda_{0,0,2} \\right) \\right) \\right\\} ~, \n\\label{Eq:Eta1} \\\\\n\\eta_{2} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{1} \\lambda_{1,1,0}^{2} + \\kappa_{3} \\lambda_{0,1,1}^2 + \\kappa_{2} \\left( \\lambda_{0,2,0} + 2 \\kappa_{1} \\Delta_{1,2} + 2 \\kappa_{3} \\Delta_{2,3} + 4 \\kappa_{1} \\kappa_{3} \\Delta \\right)^2 \\right. \\nonumber\\\\ \n&& \\left. +\\: 4 \\kappa_{1} \\kappa_{3} \\left( \\Delta_{1,3} \\left( \\lambda_{0,2,0} + \\kappa_{1} \\Delta_{1,2} + \\kappa_{3} \\Delta_{2,3} \\right) - \\Delta \\left(1 + \\lambda_{2,0,0} \\kappa_{1} + \\lambda_{0,0,2} \\kappa_{3} \\right) \\right) \\right\\} ~, \n\\label{Eq:Eta2} \\\\\n\\eta_{3} &=& \\frac{16 v_{d}}{d} \\Xi^{-1} \\left\\{ \\kappa_{1} \\lambda_{1,0,1}^2 + \\kappa_{2} \\lambda_{0,1,1}^2 + \\kappa_{3} \\left( \\lambda_{0,0,2} + 2 \\kappa_{1} \\Delta_{1,3} + 2 \\kappa_{2} \\Delta_{2,3} + 4 \\kappa_{1} \\kappa_{2} \\Delta \\right)^2 \\right. \\nonumber\\\\ \n&& \\left. +\\: 4 \\kappa_{1} \\kappa_{2} \\left( \\Delta_{1,2} \\left( \\lambda_{0,0,2} + \\kappa_{1} \\Delta_{1,3} + \\kappa_{2} \\Delta_{2,3} \\right) - \\Delta \\left(1 + \\lambda_{2,0,0} \\kappa_{1} + \\lambda_{0,2,0} \\kappa_{2} \\right) \\right) \\right\\} ~,\n\\label{Eq:Eta3}\n\\end{eqnarray} \nwhere the prefactor in the above expressions is defined as:\n\\begin{eqnarray}\n\\Xi &=& \\Big(1 + 2 \\sum_{I} \\lambda_{I} \\kappa_{I} + 4 \\sum_{I < J} \\kappa_{I} \\kappa_{J} \\Delta_{I,J} + 8 \\kappa_{1} \\kappa_{2} \\kappa_{3} \\Delta \\Big)^2 ~.\n\\end{eqnarray}\nNote, that Eqs.\\ \\eqref{Eq:Eta1} -- \\eqref{Eq:Eta3} reduce to the two-field results in the limit $\\delta_{1,3}^{2} = \\delta_{2,3}^{2} = 0$ (cf.\\ Appendix \\ref{Sec:Scale dependent effective potential})\n\\begin{eqnarray}\n\\hspace{-20pt} \\eta_{1} &=& \\frac{16 v_{d}}{d} \\frac{ \\kappa_{2} \\lambda_{1,1}^2 + \\kappa_{1} \\left(\\lambda_{2,0} + 2 \\kappa_{2} \\Delta_{1,2} \\right)^2}{\\left(1 + 2 \\kappa_{1} \\lambda_{2,0} + 2 \\kappa_{2} \\lambda_{0,2} + 4 \\kappa_{1} \\kappa_{2} \\Delta_{1,2} \\right)^2} ~, \\\\\n\\hspace{-20pt} \\eta_{2} &=& \\frac{16 v_{d}}{d} \\frac{ \\kappa_{1} \\lambda_{1,1}^2 + \\kappa_{2} \\left(\\lambda_{0,2} + 2 \\kappa_{1} \\Delta_{1,2} \\right)^2}{\\left(1 + 2 \\kappa_{1} \\lambda_{2,0} + 2 \\kappa_{2} \\lambda_{0,2} + 4 \\kappa_{1} \\kappa_{2} \\Delta_{1,2} \\right)^2} ~,\n\\end{eqnarray}\nwhere we have written $\\lambda_{1,1} \\equiv \\lambda_{1,1,0}$, etc., as the $3$-sector effectively decouples in this case. From here, it is easy to identify the decoupling and symmetry enhancement scenarios for the spectrum of anomalous dimensions, by considering the proper limits of the couplings between different sectors. These results for the two-field case are equivalent to the anomalous dimensions given in Refs.\\ \\cite{Friederich:2010hr,Wetzel:2013}.\n\\twocolumngrid\n\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Approach}\n\\label{sec:approach}\nDetermining whether the bottleneck router in a network path is employing a drop-tail or AQM queuing scheme is fundamentally a classification problem, which has been rigorously studied in Machine Learning literature. To this end, we employ a data-driven approach\nthat entails generating random network configurations of varying complexity, simulating the generated scenarios using Mininet \\cite{mininet}, and training a classification algorithm on the obtained data. Our approach contains three main components: (i) randomized network topology generation and simulation, (ii) feature engineering to encode relevant information, and (iii) training of the classification algorithm. \n\n\\begin{figure}[!htb]\n  \\centering\n  \\includegraphics[width=\\linewidth]{figures\/topology}\n  \\centering\n   \\includegraphics[width=\\linewidth]{figures\/legend}\n\\caption{A random network topology generated by our framework. Edge weights denote link capacities.}\n\\label{fig:topology}\n\\end{figure}\n\n\\subsection{Random Topology Generation}\nProperly training a machine learning algorithm to simultaneously achieve high prediction accuracy and generalization to different kinds of network configurations requires a large and diverse set of training data. Hence, we opted to leverage randomization in order to generate a wide array of sufficiently diverse network topologies, which we subsequently simulated using Mininet. In particular, we developed a framework for constructing varied network configurations. \n\nAn example of a random network topology $G = (V, E)$ generated by our framework is depicted in Fig.~\\ref{fig:topology} and is constructed as follows. We initialize a \\textit{path graph}, i.e., a graph defined by a path from the source node $s$ to the sink node $g$, $\\pi = (s, v_1, \\ldots, v_n, g)$, consisting of $n \\sim \\mathcal{U}(3,5)$ \\footnote{$\\mathrm{U}(a,b)$ denotes a continuous uniform distribution, whereas $\\mathcal{U}(a,b)$ denotes a discrete uniform distribution.} switches, i.e., routers. For each switch along the path $v \\in \\{v_1,\\ldots,v_n\\}$, a random Erd\\\"{o}s-R\\'{e}nyi graph $G_\\text{rand}(n_p, p)$ \\cite{erdos} characterized by the uniformly drawn number of nodes $n_p \\sim \\mathcal{U}(1,5)$ and connectivity probability $p \\sim \\mathrm{U}(0,1)$ is generated such that each node is a switch or a host with equal probability; the largest connected component $G_\\text{conn} \\subseteq G_\\text{rand}$ is identified and connected to switch $v$ by an edge (i.e., network link). Additional $h \\sim \\mathcal{U}(1,5)$ hosts are generated and connected to vertex $v$.\n\nRandomly chosen values for link delays and capacities are generated to diversify the network configuration. For each link $e \\in E$, we set a random link delay $d \\in \\mathcal{U}(10,100)$ and link capacity $c \\in \\mathcal{U}(10,1000)$. To increase network complexity and simulate scenarios with bottlenecks caused by competing flows, we add auxiliary flows, i.e., paths, from randomly chosen host nodes to the sink node. The randomly generated link capacities are adjusted accordingly so that there exists a single bottleneck router along the path $\\pi$ from source to sink, which is subsequently labeled as either Tail Drop or AQM (denoting the queuing scheme to be simulated).\\footnote{The ranges used for the uniform distributions can be easily altered to generate random networks of desired complexity.}\n\n\\subsection{Feature Engineering}\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[Tail Drop]{%\n  \\includegraphics[width=\\linewidth]{figures\/Basic-2.pdf}%\n\\label{fig:tail-drop}}\n\\subfigure[PIE]{%\n   \\includegraphics[width=\\linewidth]{figures\/PIE-2.pdf}%\n\\label{fig:pie}}\n\\caption{RTT data obtained from two distinct Mininet simulations of the same network topology with the exception of the queuing scheme used by the bottleneck router: (a) Tail Drop and (b) PIE (AQM).}\n\\label{fig:data}\n\\end{figure}\nThe randomly generated network topologies are simulated using Mininet to obtain training data in the form of RTT and CWND information over time, as further detailed in Sect.~\\ref{subsect:training}. Figs.~\\ref{fig:tail-drop} and \\ref{fig:pie} depict the RTT data gathered from a 20 second Mininet simulation of the topology shown in Fig.~\\ref{fig:topology}. In order to construct a vector of distinguishing features, we examined a multitude of diversely-ranged RTT and CWND figures, similar to the figures shown above. Our observations indicated that high amount of noise was present in the data, therefore, we used the Exponentially Weighted Moving Average (EWMA) filter with parameter $\\alpha=0.3$ to smoothen the resulting RTT and CWND data.\n\nOur observations reaffirmed the inherent differences in the methodology of Tail Drop and AQM schemes: Tail Drop does not drop packets until the queue is full, which leads to a pronounced sawtooth pattern in the RTT graph (Fig.~\\ref{fig:tail-drop}), i.e., lengthy periods of linear increase in the RTT plot followed by a sharp drop. PIE, on the other hand, preemptively (and probabilistically) drops packets, which results in relatively short bursts of increasing and decreasing RTT values. \n\nWe combined our insights from experimental observations and the inherent differences in Tail Drop and AQM schemes to construct a vector of distinguishing features. Our feature vector was 72 dimensional (36 features each for RTT and CWND) that were computed for each simulated time series. Examples of particularly distinguishing features that constituted our feature vector include: statistics regarding the first and second order gradient (e.g., mean, variance, maximum), the number of local minima \\& maxima, the $L_2$-norm squared total variation, the maximum sum subsequence, and the mean and variances of the local maxima. Fig.~\\ref{fig:features} shows a complete list of the features that were used, ranked by descending importance.\n\n\\subsection{Training}\n\\label{subsect:training}\nTo train our classifier, we generated 1,100 different network topologies of varying complexity and heterogeneity so that balanced, labeled data relevant to both Tail Drop and AQM schemes were obtained. For each random network topology, we instantiated two consecutive, distinct Mininet simulations by varying the queue management technique on the bottleneck between Tail Drop and PIE. We simulated a long running TCP flow on each network from our known source to the sink and auxiliary flows from other randomly generated hosts to their respective sinks for 20 seconds. From the simulation, we gathered the resulting congestion window and RTT over time using \\verb|tcpprobe| and \\verb|ping| respectively. This data, as exemplified by the RTT plot in Fig.~\\ref{fig:data}, were then used as labeled training data in our classifier.\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe presented a data-driven approach that leveraged a Neural Network for reliably determining whether a bottleneck router along a given path employs a Tail Drop or AQM scheme. We developed a framework for generating random network topologies and used Mininet to simulate a wide range of scenarios and obtain diverse training data. We identified distinguishing features by analyzing the intrinsic differences between AQM and Tail Drop and quantified the respective importance of each feature. Our feature importance analysis indicated that the most salient features were statistics pertaining to the gradient and RTT time-series data.\n\nOur algorithm attained an accuracy of 97\\% when evaluated against test data and generalized and performed well against data from highly complex topologies that it had not been exposed to before. We conjecture that our algorithm has potential to generalize to and provide reliable and accurate predictions in complex, real-world networks. Future work includes further training and evaluation on even more complex topologies and real-world data. We also plan to extend our method for classification of different AQM interfaces such as RED and PIE, in addition to the Tail Drop scheme.\n\\section{Contributions}\n\\emph{Cenk Baykal} developed the algorithm used to generate the random network configurations and helped develop the features that were used to classify the queueing scheme. \n\\\\\n\n\\emph{Wilko Schwarting} worked on the classification algorithm and evaluated state-of-the-art classification methods and conducted hyper-parameter optimization.\n\\\\\n\n\\emph{Alex Wallar} developed the Mininet codebase that was used to simulate the randomly generated network configurations and obtain data.\n\n\\section{Evaluation}\n\\label{sec:evaluation}\n\nUsing the the data obtained from our simulations, we trained and benchmarked state-of-the-art classification algorithms such as Support Vector Machines (SVMs) \\cite{svm}, Neural Networks (NNs) \\cite{hornik1989multilayer}, Logistic Regression \\cite{logistic}, and Random Forests, among others. We conducted k-fold cross validation (with $k = 10$) for each classifier on multiple data sets and found Neural Networks to be consistently outperforming all other tested classification techniques.\n\nHaving converged on using NNs as our classification algorithm, we employed a randomized search to optimize hyper-parameters by drawing from a previously specified parameter space distribution.\nThe parameter space included 1) NN architecture, i.e., the number of layers (1-4) and neurons (2-20) per layer,\n2) $L_2$-Regularization parameter $\\alpha$ penalizing large connection weights, and\n3) solvers, including Stochastic Gradient Descent (SGD), ADAM, and Limited-memory Broyden Fletcher Goldfarb Shanno (LBFGS) algorithms.\nThe randomized search evaluates each parameter tuple with a score consisting of the average accuracy over 100 stratified cross-validations with $90\\%$ training data and $10\\%$ test data, to avoid biased parameter selection from overfitting. After 10,000 evaluations, the optimal hyper-parameters were found to be a NN topology with 1 hidden layer consisting of 14 neurons, a regularization of $\\alpha = 0.5e-10$, and the LBFGS solver.\n\n\\subsection{Training Results}\nFrom the 2,200 labeled data points that were generated, 200 were left out of the training to be used later for testing. After training the classifier and optimizing its hyper-parameters, we tested how well the classification generalized by evaluating its performance on the left-out data set. Fig.~\\ref{fig:cm-left-out} shows a confusion matrix summarizing the results of this classification test. On a per-category basis, we achieve 94\\% and 99\\% prediction accuracy for correctly classifying the bottleneck router as Tail Drop or PIE respectively. Overall, we were able to accurately classify whether the bottleneck router was utilizing a Tail Drop or PIE as its queuing scheme 97\\% of the time. This can be seen as an improvement over current state-of-the-art AQM detection algorithms, such as the recently proposed method by Bideh et al. \\cite{tada}, which achieved an overall classification accuracy of 73\\% on a serial bottleneck within a relatively simple network topology. In addition to achieving a higher classification accuracy than that of previous work, we also note that our algorithm was evaluated on significantly more complex and diverse network configurations, whereas the algorithm presented in \\cite{tada} was evaluated against a single topology consisting of solely 3 hosts and 3 routers in total.\n\nTo thoroughly assess the dexterity of our algorithm in generating reliable predictions, we evaluated the classification accuracy of our algorithm on a data-set obtained from simulations of highly complex topologies with significantly more hosts, routers, and flows than the topologies that were used to train our algorithm. We emphasize the fact that our classifier was \\emph{not} trained on any data from these more complicated scenarios.\n\nDespite the challenges of this classification task, our classifier produced accurate labels for Tail Drop and PIE 80\\% and 65\\% of the time respectively, for an average classification accuracy of 73\\% overall. The confusion matrix for these tests is shown in Fig.~\\ref{fig:cm-complex}. This evaluation is evidence for our classifier's ability in generalizing well even to data obtained from highly complex network configurations that it was not explicitly trained to handle beforehand. Our favorable results demonstrate our algorithm's potential to be applied to real-world settings and achieve high prediction accuracy even against highly complex network configurations.\n\n\\begin{figure}[t!]\n\\centering\n\\subfigure[Left Out Data]{%\n\\includegraphics[width=0.49\\columnwidth]{figures\/cm_left_out}%\n\\label{fig:cm-left-out}}\n\\subfigure[Complex Topologies]{%\n\\includegraphics[width=0.49\\columnwidth]{figures\/cm_complex}%\n\\label{fig:cm-complex}}\n\\caption{Confusion matrices for the two testing scenarios, (a) data left out of the training with the same topology complexity and (b) data generated using more complex topologies than used for training}\n\\label{fig:cm}\n\\end{figure}\n\n\\subsection{Feature Importance}\nAs introduced in Sec.~\\ref{sec:approach}, our feature vector for a single data point was 72 dimensional, with 36 features each for RTT and CWND. The 72 distinguishing features for each data point mentioned in  Sec.~\\ref{sec:approach} were motivated by our knowledge of the profound differences between Tail Drop and AQM schemes, as we covered in class \\cite{class}. Fig.~\\ref{fig:features} depicts a quantification of feature importance for each of the 72 features. By inspecting the highest ranking features, we obtain the insight that features pertaining to gradient-like statistics were the most distinguishing features and that data about the RTT over time were more helpful than the CWND data.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=\\columnwidth]{figures\/feature_importances}\n\\caption{Plot showing each feature's importance for classification. Note the high importance scores of features pertaining to RTT information and gradients' statistics.}\n\\label{fig:features}\n\\end{figure}\n\n\n\\section{Problem Definition}\n\\label{sec:problem}\nActive Queue Management (AQM) schemes, such as PIE \\cite{pie} and RED \\cite{red}, have been particularly effective in combating bufferbloat and attenuating network congestion. Despite their widespread success and improved performance over traditional Tail Drop queuing schemes, \nthe extent of AQM deployment in contemporary networks is not known \\cite{tada}. Detecting the presence of AQMs in a network has potential to not only provide insight into a network's internal characteristics (i.e., network tomography), but also facilitate network optimization \\footnote{e.g., in the case that a transport protocol is known to perform poorly over an AQM or certain subset of AQMs \\cite{tada}.}.\n\nIn this paper, we address the problem of determining whether a bottleneck router on a given network path is using an AQM or a drop-tail scheme. We assume that we are given a source-to-sink path of interest -along which a bottleneck router exists- and data regarding the Round-Trip Times (RTT) and Congestion Window (CWND) sizes with respect to this flow. We develop a reliable classification algorithm that solely uses RTT and CWND information pertaining to a single flow to classify the queuing scheme, Tail Drop or AQM, used by the bottleneck router. We evaluate our method and present results that demonstrate our algorithm's highly accurate classification ability across a wide array of complex network topologies and configurations.","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\t\\label{S:11}\n\t\n\tChemometric classification methods are often used to discriminate high-dimensional chemical signatures of unknown samples to determine their most likely class label \\cite{patternrec}. These methods have proven to be valuable for the fields of archeaometry and forensics where the origin, or provenance, of a manufactured item is often of interest\\cite{patternrec, obsidian, crypto, shipwreck, butter, heroin} because the chemical data, especially those obtained from neutron activation analysis (NAA) \\cite{sayre,NAArev}, tend to be highly multivariate. The high dimensionality of data poses a challenge for understanding sample relationships because it cannot be easily visualized or interpreted directly. While, these methods can succeed at classifying samples, they do so by including information from the class labels into the model, and they provide little insight about trends or patterns found in the data with the absence of labels. Information pertaining to which samples are most similar to one another between classes, or which are different from the rest of the samples within a given source class, without the influence of label discrimination is not available from classification methods. \n\t\n\tExploratory methods are often used to reveal trends and other patterns hidden in data without the use of class labels \\cite{patternrec}. Exploratory data analysis (EDA) is a form of data analysis which encompasses a number of visual exploration methods. Success in an EDA study depends on the creativity of the analyst as much as on the technique. Although there is no strict definition of EDA, it has been stated that, ``Exploratory data analysis isolates patterns and features of the data and reveals these forcefully to the analyst\" \\cite{tukeybook}, and also that ``Exploratory data analysis' is an attitude, a state of flexibility, a willingness to look for those things that we believe are not there, as well as those we believe to be there\"\\cite{tukeybooktwo}. The coupling of EDA methods with preprocessing and\/or feature selection can aid an analyst in the discovery of patterns, or the lack thereof, in data. The information gained from EDA may be used with feature selection, preprocessing and modeling methods to iteratively improve a data analysis pipeline.\n\t\n\tThe two predominant types of EDA focus on dimension reduction and clustering. These methods have been extensively used for the EDA of archeometric data \\cite{crypto, shipwreck,  butter, cluster} because high dimensional data are not readily visualized. Dimension reduction techniques such as principal component analysis (PCA), discriminant analysis, or exploratory projection pursuit are used to change the basis of the data to one based on lower-dimensional projections \\cite{patternrec}. Unfortunately, these methods do not offer a guarantee that the resulting 2-D or 3-D projections will offer meaningful information about class relationships or sample-class relationships, due to the presence of class overlap or class heterogeneity\\cite{patternrec}. Unsupervised classification as implemented in clustering algorithms suffers from the opposite problem clustering methods readily create groupings of high-dimensional data but tend to offer little information about the cluster assignments themselves. Clustering methods have been known to incorrectly associate archeological samples depending on the method used and on the chemistry of the samples \\cite{cluster}. Information about sample or class relationships may be inferred by employing many techniques, but the resulting information is often hidden, and typically key relationships in the data cannot be seen by an analyst \\cite{cluster}.\n\t\n\tThis work reports a new approach, which we call the uncharted forest, to the visualization and measuring of relationships within and between classes of data. This approach has elements of a clustering algorithm in that it groups similar samples with one another, but is also similar to dimension reduction methods in that it outputs a single heat map which can be interpreted to reveal information about the samples. Uncharted forest analysis uses a partitioning method that is related to the sample partitioning approach used in decision trees but, it does not use class labels like most tree methods do \\cite{CARTBook}. Instead, the uncharted forest analysis explores how samples relate to one another under the context of univariate variance partitions. Although the method is unsupervised, we show that when the results are overlaid with external class labels, the method can be used to investigate sample relationships as they pertain to class labels. We demonstrate that this technique can be used as a tool for exploratory data analysis to visualize class or cluster associations, sample-sample associations, class heterogeneity, and uninformative classes. The utility of uncharted forest analysis is demonstrated on two classification datasets and two provenance datasets. Additionally, two empirical clustering metrics are compared with two of the metrics obtained by uncharted forest analysis on another provenance dataset.\n\t\n\t\n\t\\section{Theory}\n\tTo motivate the use of an unsupervised tree ensemble, a brief review of supervised trees and bagged tree ensembles as used in classification is provided here. A complete review of these methods can be found elsewhere \\cite{CARTBook,BreimanRF}. \n\t\n\t\\subsection{Supervised Classification Trees}\n\tBefore describing the mechanisms by which supervised decision trees are used to partition data, a survey of the vocabulary common to these methods is presented. Classification trees are a type of supervised classifier, which means that the assignment of new data to a class label requires that each sample the tree was trained on also has a label. The aim of developing a classification tree is the establishment of a set of sequential rules that can be applied to label a sample based on its features. Supervised decision trees are collections of many binary decisions, where each decision is made at one of three locations: roots, branches, or terminal nodes\/leaves. These three locations are displayed in their hierarchical ordering in Figure 1. A root is the first decision made in the tree. Branch nodes indicate later, non-terminal decisions. Terminal nodes indicate where a branch has been terminated and where final decisions that assign class labels are made. \n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=0.40\\linewidth]{Figure1}\n\t\t\\caption{A block diagram showing the relationships between roots, branches and terminal nodes via arrows.}\n\t\\end{figure}\n\t\n\tEvery sample from the training set is partitioned at a branch or node based on whether a selected variable for the sample has a value that is greater than or less than a specified value. The combination of a variable and value that partitions a set of samples is often referred to as a decision boundary. Decision boundaries are obtained by exhaustively searching each variable and finding the threshold value which affords the highest gain \\cite{CARTBook}. Here, gain is a metric-dependent measure of how well a decision separates the available samples according to their class labels. Metrics that can be used to minimize class label impurities at terminal nodes, such as the Gini impurity, informational entropy, and classification accuracy are commonly used to train supervised classification trees by finding a set of decision boundaries \\cite{CARTBook}. \n\t\n\tFor example, if the samples available at a branch node are better separated into their respective classes by application of some decision, two new branches are made at that node. If not, the partitioning terminates, the node becomes a terminal node, and the decision tree ceases growth using those samples. In this way, each observation is sorted from the root of a tree through branches and eventually to terminal nodes, based on whether creating a new decision boundary better separates samples by known class labels\\cite{CARTBook}. \t\n\t\n\tOne advantage to the use of decision trees is that the classifier that results can be easily understood by examining the decision boundaries that partition the data and the relation of these boundaries to class labels. However, the predictive performance of classification trees on new samples is generally poor relative to other models because the partitions are sensitive to noise in the variables \\cite{CARTez}. Tree-based classifiers also tend to overfit the data, and some sort of branch removal, or pruning step, is usually employed to reduce the complexity of the tree \\cite{CARTBook}. \n\t\n\t\\subsection{Random Forest Classification}\n\tThe random forest classifer is a multiclass classification technique that utilizes the results from an ensemble, or collection of supervised decision trees \\cite{BreimanRF} to assign class labels. In this technique, a user-specified number of supervised decision trees is created, each overfit to a selection of bootstrapped samples and random selections of variables. Random forests make use of the complexity of many, biased decision trees to make robust classification assignments by using the average class designation assigned by every tree to predict overall class membership for new data. This approach to ensemble model building is referred to as bootstrap aggregating, or bagging \\cite{bagging}. The average of many, complex trees tends to perform well in practice \\cite{BreimanRF}.\n\t\n\tRandom forest classification has some distinct advantages relative to many other classifiers. It is robust to outliers because sample selection at branches is bootstrapped, and the effect of a few outlying samples does not affect its cost function significantly \\cite{BreimanRF}. Random forest models are also robust to noise in variables because many classifiers, each with randomly selected variables, are used to decide on class memberships. Another notable advantage to random forest classification is that there is no requirement for linearity in class boundaries. Random forest models, unlike methods such as linear discriminant analysis or support vector machines, are not based on the hypothesis that classes are linearly separable \\cite{linearmethods,kernel}. Random forest classification also requires relatively few hyperparameters and works without the need for significant tuning of the classifier on many kinds of data \\cite{CARTez}. \n\t\n\t\\subsection{Uncharted Forest Tree}\n\tThe concept of unsupervised decision tree modeling is relatively new to chemometric applications. Only a few approaches using unsupervised trees have been reported. An approach reported by Khudanpur, et al. is based on an algorithm similar to the one reported here, but their approach uses the Kullback-Leibler distance to find sets of unsupervised partitions that are optimal with respect to an information theoretic measure\n\t\\cite{InformationApp}. Other methods for developing unsupervised decision trees are based on the use of an assumed class label for each observation; the trees are trained in a manner that is similar to that used for training a random forest classifier \\cite{UnsupRF, CLTrees}. \n\t\n\tOur approach to unsupervised decision trees, the uncharted forest tree algorithm, focuses on intuitive concepts from statistics or machine learning rather than on information theory, to allow for easy interpretation. The trees can be created without the need for class labels because the gain function that is optimized in the construction of each tree relies only on information in the data matrix, not on the labels. This algorithm does not utilize the usual supervised metrics for optimization such as the Gini importance or entropy \\cite{BreimanRF}. Instead, the tree hierarchies used in uncharted forest are constructed from decision boundaries that reduce measures of spread in a given variable. The reduction of spread is common in pattern recognition and chemometrics; it is the mechanism underlying techniques such as K-means clustering, and principal component analysis for dimension reduction \\cite{patternrec}. Metrics based on changes in the variance, median absolute deviation (MAD), or mean absolute deviation (AD) can also be used for creating the decision boundaries used in an uncharted forest tree. If variance is used, however, univariate Gaussian probabilities may be calculated from the samples on either side of a decision boundary. We prefer the variance metric because it is related to several established figures of merit for clustering, as we describe later. The gain metric based on changes in variance that we use is\n\t\n\n\t\\begin{equation}\n\tgain = 1 - \\frac{Var(Branch_1) - Var(Branch_2)} {Var(Parent Node)}\n\t\\end{equation}\n\t\n\t\\noindent where the sample data in either partition are denoted as $Branch_1$ and $Branch_2$, and the sample data prior to introducing the partition is contained in the previous - or parent - node, which has been denoted as, $Parent Node$. The gain in the unsupervised case can then be seen as the loss of variance ($Var$) in the data resulting from the construction of a given decision boundary. The gain in the unsupervised case can then be seen as the loss of variance in the data resulting from the construction of a given decision boundary, relative to that in the previous - or parent - node. Similar gain equations can be written for measures of spread based on AD and MAD. These are not considered in this work because they do not have direct comparisons to clustering metrics. \n\t\n\tThe spread in data can almost always be reduced by partitioning data around an available decision boundary. Thus, uncharted forest trees require a user-defined parameter, the tree depth, to limit the number of possible partitioning decisions made. Tree depth can be defined as the maximum number of branches that are counted from the root node to a given terminal node. For exploratory analysis of classification or cluster data, a reasonable depth may be obtained from the rounded base 2 logarithm of either the number of known classes or the number of clusters selected. The number of classes\/clusters available is often known information; for example, in most provenance studies the number of classes is decided upon by the number of sampling sites used. In our implementation the minimum node size, or when branch growth is ceased after the number of samples becomes fewer than a given number, has been set to five samples because five samples allows for a calculation of variance that is not undersized but still accommodates a small cluster\/class size. Pseudocode that depicts the recursive algorithm for constructing an uncharted decision tree is shown in Algorithm 1.\n\t\n\t\n\n\t\n\t\\begin{algorithm}\n\t\t\\caption{Uncharted Forest Decision Tree}\\label{unchartedtree}\n\t\t\\begin{algorithmic}[1]\n\t\t\t\\Procedure{  }{}\n\t\t\t\\State for each variable in variables\n\t\t\t\\State \\ \\ \\ \\ Calculate gain. (Eqn. 1)\n\t\t\t\\State \\ \\ \\ \\ Store highest gain.\n\t\t\t\\State end for\n\t\t\t\\State Partition samples at largest gain.\n\t\t\t\\If {branch depth $\\geq$ number of branches} \n\t\t\t\\State Stop growth.\n\t\t\t\\Else\n\t\t\t\\State For each partition \\textbf{goto} 2\n\t\t\t\\EndIf\n\t\t\t\\State \\textbf{end if}\n\t\t\t\\EndProcedure\n\t\t\\end{algorithmic}\n\t\\end{algorithm}\n\t\n\tThe uncharted forest decision tree partitions a set of data by finding decision boundaries that reduce the spread in the data. It functions as a kind of clustering, where the clusters result from divisive partitioning of spread in the data. Unfortunately, unsupervised trees of the form described above often have marginal utility in practice. These trees can be used for clustering but, like a single supervised tree, the individual uncharted forest decision trees are sensitive to noise effects in the data. The variables in a set of data have many contributions to their spreads, and as a consequence, the partitions that result from a single unsupervised tree tend to be uninformative.\n\t\n\t\\subsection{Uncharted Forest}\n\tOne solution to the uninformative partitions created by a single uncharted forest decision tree is to create an ensemble of unsupervised trees, an ``uncharted forest\". The approach is similar to that taken in the random forest classifier. For each tree in this uncharted forest, the variables used are randomly selected, so that potentially relevant information within an individual variable is not obscured by other contributions to its spread or by another variable with much larger gain in the reduction of spread. The algorithm for the uncharted forest method follows that of the random forest algorithm \\cite{BreimanRF}, except that the supervised classification and regression tree models are replaced with the uncharted forest decision tree and the samples are not bootstrapped.\n\t\n\tBecause construction of the trees is unsupervised, the output of the uncharted ensemble algorithm differs from that of the random forest. An uncharted forest returns a square matrix whose dimensionality is that of the number of observations, so that each row or column vector stores information about a single observation and how it relates to the other samples. The matrix is populated in the following manner: when samples are partitioned into a terminal node together, the row and column representative of each of these samples each gain a +1 count to account for their associations with one another. Similarly the row and column of any samples that were not present at a terminal node with those samples each receive a -1 count. The associations formed by random chance become less pronounced by including the -1 penalty on observations that were not grouped with one another at their respective terminal nodes. After row-wise normalization, the matrix possesses diagonal symmetry. In the case of variance-only partitions, a Gaussian probability for node membership may be used instead of $\\pm$ 1. \n\t\n\n\t\n\t\\subsection{Figures of Merit}\n\tWe refer to the matrix generated by the tree ensemble as a probabilistic affinity matrix, because of its resemblance to the distance-based affinity matrices utilized by clustering methods \\cite{spectral}. Each element in the matrix represents the row-wise, likelihood that a given row (observation) is associated with a given column (another or the same observation) at a terminal node in an uncharted forest tree. In order to maintain the above interpretation, a ramp function is applied to every matrix element (p), max(0, p), to remove any negative entries. In this way, the diagonal elements have a sampled probability of association of 1.0, because there is a 100\\% likelihood that a given sample will be associated with itself at each terminal node where it was assigned. \n\t\n\tWe introduce two schemes that allow for the probability values in the matrix to be quantified for exploratory analysis. Two figures of merit can be calculated to measure how often samples between- or within- clusters\/classes are partitioned together. The submatrix of the probability affinity matrix (P) that contains the associations between two clusters\/classes i and j, can be quantified by what we define as the interaction quotient (IQ),\n\t\n\n\t\\begin{equation}\n\tIQ_{i,j} =  \\frac{\\sum_{c=j_{min}}^{j_{max}}  \\sum_{r=i_{min}}^{i_{max}}   P_{r,c}} {(j_{max} - j_{min}))(i_{max} - i_{min}) }\\end{equation}\n\t\n\t\\noindent where the min and max subscripts denote the contigious row\/column beginning and end for cluster\/class indices within the matrix. The IQ is simply the sum of all probabilities inside of the submatrix that is defined by the rows shared from cluster\/class $i$, and the columns shared by cluster\/classes $j$ normalized by the maximum possible probabilities for association within that submatrix. Similarly, we define the self association quotient (SAQ) as a measure of within- class or cluster sample associations. The SAQ metric is calculated from the probabilistic affinity matrix for the special case where $i = j$. For purposes of visualization, the matrix blocks from which these metrics are calculated are displayed below,\n\t\n\t\\[\n\t\\begin{bmatrix}\n\tSAQ_{1} & IQ_{1,2} & IQ_{1,3} & \\dots  & IQ_{1,j} \\\\\n\tIQ_{2,1} & SAQ_{2} & IQ_{2,3}  & \\dots  & IQ_{2,j} \\\\\n\t\\vdots & \\vdots & \\vdots & \\ddots & \\vdots \\\\\n\tIQ_{i,1} & IQ_{i,2} & IQ_{i,3} & \\dots  & SAQ_{i}\n\t\\end{bmatrix}\n\t\\]\n\t\n\tThe total interaction quotient (TIQ) which describes the extent of the total interactions between the samples of all class\/cluster assignments can be expressed by the sum of probabilities within each IQ block divided by the number of matrix entries in those blocks(n),\n\t\n\n\t{\\centering\n\t\t\\begin{equation} TIQ = \\sum_{i}  \\sum_{j}   IQ_{i,j} \/ n   \\ \\ \\ \\ \\ provided \\ \\ i \\neq j \\end{equation} \\par\n\t}\n\t\n\tIn an analogous manner, the total self association quotient (TSAQ) can be used to describe how often the samples in all class\/clusters are partitioned with samples in their same respective class\/cluster by uncharted forest,\n\t\n\n\t\\begin{equation}TSAQ = \\sum_{i}   SAQ_{i} \/ n \\end{equation}\n\t\n\tThe metrics SAQ and IQ provide a means to assess the likelihoods that samples are within the same terminal nodes after variance partitioning with reference to class\/cluster ownership. SAQ can provide a measure of class\/cluster heterogeneity by indicating how likely it is that the samples inside of a given label are partitioned together. The IQ provides a measure of how often data were associated to the samples that belong to a different class\/cluster label. The IQ measure can therefore be used as an aid to identify classes in the data that do not associate with samples from other classes of data, or classes which are similar to one another. These measures can also be made on a sample-wise basis to discover outliers which may either be between class\/cluster labels or are not associated with any class\/cluster label at all.\n\t\n\t\n\t\\section{Scope of the Uncharted Forest Method}\n\tThere is one restriction to the use of uncharted forest analysis that should be noted. The associations that a uncharted forest analysis can describe are often not meaningful without data ordered by class or cluster labels, despite that it is an unsupervised technique. The hypothesized membership of samples allow interpretations of the data as it pertains to class\/cluster labels. In some cases, data have serial correlations, such as in time series, or variables from spectroscopic instruments, and the probabilistic affinity matrix appears well-ordered. However, with uncorrelated, labelless data, the probabilistic affinity matrix most often appears similar to random noise. One interesting feature of the probabilistic affinity matrix is that it is invariant to the ordering of data. This is a useful attribute when the matrix could display seemingly random disjoint probabilities, because the order of the rows and columns of the matrix can be changed to permit a more meaningful interpretation, like a cluster for example, without running the algorithm again. The studies explored in this work address the case where class or cluster labels were available for provenance and classification data.\n\t\n\t\\section{Selected Examples}\n\tIn this work, five datasets were investigated by uncharted forest analysis. The first, the well-known iris dataset, is presented as an instructional example to demonstrate how the technique can be used to examine data with class labels, and how it compares to the widely used principal components analysis method. The well-known ARCH obsidian data is presented to compare the sample-sample associations obtained by uncharted forest analysis with two supervised classification methods. The Nan'ao One shipwreck and cryptocrystalline sillicate datasets are presented to compare the results from uncharted forest analysis to those of published works that used other data analysis methods on the same data. The fifth dataset, the butter data, is used to compare the new method to established variance-based partitioning methods and to show that the TIQ and TSAQ figures of merit are similar to well-established, variance-based clustering metrics.\n\t\n\t\\subsection{Iris Data EDA}\n\tThe iris dataset contains measurements of 150 iris flower samples. Both the widths and lengths of the sepals and petals of each flower were measured. Each sample belongs to either the virginica, versicolor, or setosa species \\cite{iris1, iris2}. The iris data is widely regarded as a standard dataset for introducing classification problems and algorithms because it features two elements common to classification problems: it possesses a linearly separable class (setosa) and two classes with overlapping boundaries (virginica, versicolor).\n\t\n\tWhen the uncharted forest algorithm was applied to the iris data using 100 trees, a depth of 4, and using the variance metric, the probabilistic affinity matrix indicated associations between several samples of virginica and versicolor. The row-wise total interaction quotients for the samples in the setosa class were $\\ll$ 5\\%; this suggested that the setosa class was well separated from the others. Four samples that were greater than 3$\\sigma$ from the mean row-wise IQ were found in both the virginica and versicolor classes. This result implies that, by making consecutive decision boundaries to reduce variance, the virginica and versicolor classes had several samples that appeared to share the same class label space. \n\t\n\tPrincipal components analysis was employed to reduce the dimensions of the iris data and to visually assess which samples uncharted forest analysis indicated as associated with other classes. When the iris data were projected onto the first two principal component axes, five samples which had significant row-wise IQ's were located between the virginica and versicolor classes, as can be seen in Figure 2. A conventional random forest model with 500 trees was then trained on the entire dataset. The random forest model misclassified 5 samples. Of those 5 samples, 3 were the samples with row-wise IQ values that exceeded 3$\\sigma$. The heat map from uncharted forest algorithm highlighted many of the samples that the random forest classifier had difficulty classifying due to class overlap.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\linewidth]{Figure2}\n\t\t\\label{fig:Ng1} \n\t\t\n\t\t\\caption{A heat map of the probabilistic affinity matrix of the Iris data with class labels (top left). The row-wise IQ values for the corresponding heat-map with the 3$\\sigma$ vertical line (top right). Plot of the PCA scores for the iris data. Samples denoted with an X, are the samples with row-wise IQ values above the 3$\\sigma$ line (bottom).}\n\t\\end{figure}\n\t\n\t\\subsection{Obsidian Data Pattern Recognition}\n\tThis dataset contains X-ray fluorescence measurements made on 12 obsidian artifacts and 63 obsidian samples that were collected from four locations in Northern California. The four source locations were from quarries located near: Mount Konocti, Borax Lake, Glass Mountain (Napa County), and Anadel (Sonoma County). We used this dataset demonstrate that the sample-sample associations afforded by uncharted forest analysis could be used to establish the likely origin of the artifacts from measured quantities of the following trace elements: iron, titanium, barium, calcium, potassium, manganese, rubidium, strontium, yttrium, and zirconium \\cite{Obsidian}. This dataset has previously been demonstrated as a test-bed for unsupervised and supervised learning methods in chemometrics \\cite{kowalski}.\n\t\n\tThe unknown samples were assigned source labels based on an uncharted forest model made with 200 trees and a tree depth of 2, as well as a nearest neighbor classification (k = 1) after PCA projection onto the first two components of the unknown samples The assignments of the artifact samples to their predicted source classes can be seen in Figure 3. The PCA was created from standard normal variate-scaled obsidian source data that had the X-ray fluorescence measurements of iron, strontium, and manganese removed due to correlation with other variables. The uncharted forest analysis assignments were made from the row-wise IQ values of the rows in the probabalistic affinity matrix, which corresponded to the twelve unknown samples via a maximum voting scheme. The voting scheme was performed to remove analyst bias and to automate the assignment of samples to suspected classes, but the results were equivalent to our visual assessment. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.95\\linewidth]{Figure3}\n\t\t\n\t\t\\caption{The probability affinity matrix (left) and the PCA scores plot obtained from the first two principal components of the standard normal variate-scaled ARCH data (right). The unknown artifact samples are denoted with a `?', and the four source classes of the obsidian data are labeled as follows: Mount Konocti (1), Borax Lake (2), Glass Mountain (3), and Anadel (4). }\n\t\\end{figure}\n\t\n\tThe source assignments of the artifacts from the uncharted forest analysis with the voting scheme differed from those made from the nearest neighbor (1-NN) PCA classification model by only one sample. For a sense of comparison, an ordinary random forest classification model with 200 trees was also trained on the source data. Again, the assignments made by the maximum vote uncharted forest analysis differed by only one sample from those of the random forest model. However, the classification produced by the nearest neighbor classifier differed from the assignments made by the random forest model by two samples. None of the models assigned any of the artifact samples to the Anadel source class. This result is consistent with a classification analysis performed by a linear learning machine \\cite{kowalski}.\n\t\n\tSimilar assignments between unknown samples and assigned source classes were obtained by the uncharted forest method and the two supervised classification models. This finding suggested that the unsupervised partitioning method used in the uncharted forest method with a maximum voting scheme can offer sample-sample associations that are comparable to that found using supervised methods.\n\t\n\t\n\t\\subsection{Nan'ao One Shipwreck Data EDA}\n\tThis dataset was selected because it was relatively high in dimension, and because it possessed complex class labels and sample membership. Twelve porcelain samples were collected from a shipwreck and compared, by using NAA measurements of 27 elements, with samples from several nearby kilns. The data were scaled by applying a base ten logarithm prior to analysis as suggested by the authors \\cite{shipwreck}. The goal of the study was to determine which kilns were most likely used to create the blue and white porcelain artifacts that accounted for 95\\% of the samples found on the wreck \\cite{shipwreck}.\n\t\n\tAn uncharted forest analysis using 30 trees and a tree depth of 2 was used to investigate the source and artifact data, as shown in Figure 4. It was found that two of the artifact samples had high associations with source samples obtained from the Jingdezhen kiln, and no associations with other source samples. This result implied that those samples could be assigned to the Jingdezhen kiln. Zhu et al.'s PCA analysis agreed with the assignment of those two samples to the Jingdezhen kiln \\cite{shipwreck}. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.65\\linewidth]{Figure4}\n\t\t\n\t\t\\caption{Probability affinity matrix of the three source classes and the shipwreck samples. }\n\t\\end{figure}\n\t\n\tThe probabilistic affinity matrix also indicated that samples from the Jingdezhen kiln may be composed of two subclasses. Our PCA analysis did not readily show that the Jingdezhen kiln could be represented as two subclasses. It was found that the scores of the subclassess that were identified by uncharted forest analysis were linearly separable after projection from the fourth and fifth principal components. Manual assessment of the 729 pairs plots showed that the two subclasses identified by uncharted forest were linearly separable in the Lutetium measurements. The base-10 logarithmically scaled Lutetium variable was was used in 16\\% of all decision boundaries made from the random selections of variables and was the third most used variable for constructing decision boundaries in all of the uncharted forest trees. In every instance the decision boundaries were created at -0.569, and this boundary was never imposed at a root node. A plot of normalized kernel density estimates of Lutetium, displayed in Figure 5, showed that the samples from these two possible subclasses were well-separated by the Lutetium decision boundary. This information was readily accessible by uncharted forest analysis, whereas, by the other EDA methods we did not know that this division of samples in the Jingdezhen class may be of interest, or that it existed. The reason why the Jingdezhen class has two well-separated distributions of Lu measurements is not known, but this finding could be of value for future investigations or to provide information about appropriate sample selection that accounts for the sources of variation, as is necessary in the training of classification models. \n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Figure5}\n\t\t\n\t\t\\caption{ Kernel density estimates for the base-10 logarithmically scaled Lutetium concentrations in the speculated Jingdezhen 1 and 2 subclasses as well as in the other samples. The kernel density estimates were created using a bandwidth of 0.04 and were normalized relative to the kernel estimates of the entirety of the data. The decision boundary created by the uncharted forest algorithm is shown as a vertical line. }\n\t\\end{figure}\n\t\n\tUncharted forest analysis revealed that the remaining artifact samples were found to be most associated with those from the Zhangzou province. However, there was notable interaction between samples from the Zhangzou province and three source samples from the Dapu kiln. Similarly, Zhu et al. found that the remaining samples appeared to be from the Zhangzou province, and that several samples were outside of the 90\\% confidence bounds of their principal component analysis \\cite{shipwreck}. \n\t\n\tTo make the partitions of uncharted forest more specific to the samples believed to be in the Zhangzou province, both the source and artifact samples attributed to the Jingdezhen kiln were removed. The source samples collected from the Dapu kiln were also removed to focus the partitioning only on the kilns in the Zhangzou province. The goal of this analysis was to assess the samples obtained from specific kilns within the Zhangzou province as they related to the shipwreck, rather than assuming the Zhangzou province was a homogeneous collection of source kilns. The probabilistic affinity matrix, shown in Figure 6, produced by uncharted forest analysis with 75 trees and a tree depth of 4 displayed no associations (IQ = 0) between the samples from the shipwreck and the Hauzilou and Tiankeng kilns. This finding suggests that both kilns were unlikely sources for the artifacts, a finding that was not observed in the original study\\cite{shipwreck}.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.65\\linewidth]{Figure6}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix after the fragmentation of the Zhangzou province into its respective source kilns and the removal of the two artifact samples associated with the Jingdezhen kiln, and the sources from the Jingdezhen kiln and the Dapu kiln. }\n\t\\end{figure}\n\t\n\t\n\tAlthough the additional analysis did not clearly display which individual kilns in the Zhangzou province were most likely associated with the artifact samples, it did discount two unlikely locations. In a provenance study it is also of interest to know where samples likely did not originate from, because that information can be used for the design of future studies.\n\t\n\t\\subsection{Cryptocrystalline Sillicate Data EDA}\n\tHoard et al. investigated the geological source of several cryptocrystalline sillicate artifacts that appeared identical by eye and by microscope \\cite{crypto}. Artifacts were collected in Eckles, Kansas and at three sites in Nebraska (Signal Butte I\/II\/III). Eleven trace elements in the artifacts were quantified by NAA and compared to source materials collected from Flattop Butte, Gering formation, Nelson Butte, Table Mountain, and from West Horse Creek. \n\t\n\tThe Gering formation TIQ was calculated to be $\\ll$ 5\\% by an uncharted forest model that had 100 trees and a tree depth of 4. This result suggested that the Gering formation was not commonly partitioned with samples outside of its assigned class label, in agreement with the authors' key finding that the Gering formation was not a likely source for the artifacts\\cite{crypto}. This finding is a clear example of the identification of an uninformative class by uncharted forest analysis. This finding was corroborated from the scores obtained by a PCA constructed using only the source classes. However, the PCA scores plots were uninformative with regards to obtaining other information Hoard et al. had described using canonical discriminant analysis (CDA), or that observed via the probabilistic affinity matrix obtained from uncharted forest analysis. The score plots of the first four principal components showed nondistinct overlap of almost all of the samples belonging to the classes, even after the CDA-separable Gering formation class was removed. \n\t\n\tThe probabilistic affinity matrix obtained by uncharted forest analysis using 100 trees and a tree depth of 4 with only the Eckles artifact class is displayed in Figure 7. This Figure clearly shows that the most likely source for the samples collected at the Eckles site was the Flattop Butte. This finding by uncharted forest concurred with the CDA analysis performed by Hoard et al \\cite{crypto}. \n\t\n\n\n\n\n\t\n\n\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Figure7}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix from uncharted forest clustering of the cryptocrystalline sillicate data with only the Eckles Kansas artifact and the source classes included. Class boundaries are shown by the overlays of vertical and horizontal lines. }\n\t\\end{figure}\n\t\n\tHowever, unlike the CDA analysis performed by Hoard et al. or the PCA analysis explored in this study, an uncharted forest analysis displayed class heterogeneity. A very clear example of a class with heterogeneous structure, shown in Figure 8, was observed via uncharted forest analysis using 100 trees and a tree depth of 4, by comparing associations between artifacts from the Signal Butte III site and the source locations. Seven of the twelve samples collected from the Signal Butte III site were most frequently partitioned with two samples collected from Table Mountain, not the Signal Butte III samples. Those two samples that were collected from Table Mountain did not appear to be associated with the rest of the data that was labeled as Table Mountain. This instance of class heterogeneity may be indicative of a missing source class, the introduction of a sampling error, or of another anomaly in the Table Mountain data. Hoard et al. was unable to determine the likely origin for several of the samples that were collected from the Signal Butte III site; CDA indicated that the samples from Signal Butte III were tightly grouped \\cite{crypto}. Uncharted forest analysis, on the other hand, indicated that the samples may actually be from two groups, one not well-represented in the NAA data, and the other most like the samples obtained from Table Mountain.\n\t\n\t\\begin{figure}[H]\n\t\t\\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Figure8}\n\t\t\n\t\t\\caption{The probabilistic affinity matrix from uncharted forest clustering of the cryptocrystalline sillicate data, with overlaid class boundaries. Heterogeneity within the first 7 samples collected from the Signal Butte III site can be observed in that they only associate with two samples collected from Table Mountain. These samples do not associate with the rest of the data from Table Mountain. }\n\t\\end{figure}\n\t\n\t\n\t\\subsection{Geographical Butter Data Clustering Analysis}\n\tThe butter dataset consists of 78 butter samples collected from 17 countries. The goal of this study was to assess the viability of determining the origin of butter samples from their chemical signatures \\cite{butter}. Butter samples were analyzed by inductively coupled plasma mass spectrometry (ICP-MS) and isotope-ratio mass spectrometry using isotopic ratios of $^{13}$C, $^{15}$N, $^{18}$O, and $^{34}$S \\cite{butter}. Of the 17 countries included in the data, 8 were represented by two samples or less; these data were removed. A total of 65 samples were retained for analysis, of which 44.6\\% were of German origin. A significant proportion of the measurements of $^{34}$S in butter proteins were lower than the detection limit \\cite{butter}. To make use of the information in this variable, the samples with $^{34}$S values lower than the detection limit were imputed with zero concentrations. \n\t\n\tIn this study, the butter dataset was used to compare the variance partitions made by the uncharted forest algorithm with two variance-based clustering algorithms. K-means clustering and hierarchical clustering with a Ward's criterion were applied to the butter data, using a range of clusters (2-7), with 15 replicates. An uncharted forest analysis with 200 trees and a tree depth of 5 was used to assess each replicate of the clustering methods. K-means clustering optimizes the location of cluster partitions by minimizing within-cluster variance in an iterative fashion \\cite{kmeans}. Ward's clustering was employed with a loss function that minimizes within-cluster variance by forming clusters using an agglomerative approach\\cite{ward}. The goal of this study was to assess how the TIQ and TSAQ metrics compared to within- and between-cluster variance metrics, respectively. \n\t\n\tConfidence intervals of the correlation coefficients between uncharted forest metrics and the empirical cluster variances for both clustering methods were created by Fisher transforming the correlation coefficients, calculating the confidence intervals as usual, and transforming back \\cite{Fisher}. All intervals reported here were computed at 99\\% confidence. TSAQ and within-cluster variance were found to be strongly correlated: K-means clustering had a correlation of 0.90 $\\pm$ 0.12 with TSAQ, and Ward's clustering had a correlation of 0.94 $\\pm$ 0.05. This result strongly suggested that the relationship between TSAQ and the within-cluster variance metric were well approximated by a linear relationship. The intervals for the correlation coefficients between TIQ and between-cluster variances were -0.50 $\\pm$ 0.07 for K-means clustering and -0.60 $\\pm$ 0.05 for Ward's clustering. Overall, the correlation between TIQ and between-cluster variance of either clustering method was moderate; this finding indicated that TIQ provides information that is only somewhat similar in direction to between-cluster variance.\n\t\n\tThe negative correlation between TIQ and between cluster variances was expected, because as the number of clusters increases, the cluster centroid distances also increase, and so do the between cluster variances. However, as cluster distances increase, the associations found by uncharted forest analysis between the samples inside of those clusters would be expected to become less prevalent, because the clusters are becoming more specific to the groups in the data; thus, TIQ values would be expected to decrease. The magnitude of the correlations were not near-perfect (-1.0) because TIQ and TSAQ metrics result from randomly created partitions and the uncharted forest algorithm does not partition data the same way that either K-means or Ward clustering does. However, the fact that the correlation was correct in sign suggests that the partitioning mechanism is overall variance-based, as intended.\n\t\n\t\n\t\\section{Conclusion}\n\tWe have presented a novel adaptation to the random forest paradigm which extends it to unsupervised exploratory data analysis. The method allows for high- and low-dimensional data to be represented as a matrix where every entry is a sampled probability-like value that represents the likelihood that a given sample (row) resides in the same terminal node as the other samples (column). This exploratory data analysis tool and its associated metrics are shown to be a informative aid for the analysis of classification and provenance data. Uncharted forest analysis was able to provide information related to uninformative class labels, class heterogeneity, and sample-wise relationships. \n\t\n\tThe uncharted forest analysis method using variance partitions was explored by comparing its measurable results with both between- and within- cluster variances of two known variance based clustering algorithms. It was shown that the uncharted forest algorithm can provide metrics (TIQ, TSAQ) that are correlated to global clustering statistics. However, the true utility of the method is found in the visualization of sample-wise information, and how it relates to cluster\/class-wise associations. As a proof-of-concept though, when sample-wise variance partitions were used in uncharted forest analyses, the probabilistic affinity matrix generated metrics that were correlated to rigorously defined statistical measures of spread for clustered data. \n\t\n\tVery similar findings to the classification based provenance studies performed by Zhu et al. and Hoard et al. were obtained from exploratory uncharted forest analysis. This is of interest because the uncharted forest algorithm is unsupervised, but with an overlay of class labels, it was possible to visually infer class labels for unknown samples in high-dimensional, multiclass datasets. Most importantly, uncharted forest analysis was able to display information about the data that was not readily apparent from the classification methods used in the aforementioned studies and other exploratory data analysis methods. The uncharted forest method was shown to be an EDA technique that can provide information that is relevant to the domains of exploratory data analysis, clustering, and classification.\n\t\n\t\\section{Acknowledgments}\n\tThis work was supported by the United States National Science Foundation grant 1506853.\n\t\n\t\\section{References}\n\n\n\n\t\n\n\n\t\n\n\n\n\n\n\n\n\t\n\n\t\n\t\\bibliographystyle{model1-num-names}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\nIn this work, we study local problems on regular trees from three different perspectives. \n\n\\textit{First,} we consider the perspective of distributed algorithms. In distributed computing, the studied setup is a network of computers where each computer can only communicate with its neighbors. Roughly speaking, the question of interest in this area is which problems can be solved with only a few rounds of communication in the underlying network. \n\n\\textit{Second}, we consider the perspective of (finitary) factors of iid processes.\nIn probability, random processes model systems that appear to vary in a random manner.\nThese include Bernoulli processes, Random walks etc.\nA particular, well-studied, example is the Ising model.\n\n\\textit{Third}, we investigate the perspective of descriptive combinatorics. The goal of this area is to understand which constructions on infinite graphs can be performed without using the so-called axiom of choice.\n\nAlthough many of the questions of interest asked in these three areas are quite similar to each other, no systematic connections were known until an insightful paper of Bernshteyn~\\cite{Bernshteyn2021LLL} who showed that results from distributed computing can automatically imply results in descriptive combinatorics.\nIn this work, we show that the connections between the three areas run much deeper than previously known, both in terms of techniques and in terms of complexity classes.\nIn fact, our work suggests that it is quite useful to consider all three perspectives as part of a common theory, and we will attempt to present our results accordingly.\nWe refer the reader to \\cref{fig:big_picture_trees} for a partial overview of the rich connections between the three perspectives, some of which are proven in this paper. \n\nIn this work, we focus on the case where the graph under consideration is a regular tree.\nDespite its simplistic appearance, regular trees play an important role in each of the three areas, as we will substantiate at the end of this section.\nTo already provide an example, in the area of distributed algorithms, almost all locality lower bounds are achieved on regular trees.\nMoreover, when regarding lower bounds, the property that they already apply on regular trees actually \\emph{strengthens} the result---a fact that is quite relevant for our work as our main contribution regarding the transfer of techniques between the areas is a new lower bound technique in the area of distributed computation that is an adaptation and generalization of a technique from descriptive combinatorics.\nRegarding our results about the relations between complexity classes from the three areas, we note that such connections are also studied in the context of paths and grids in two parallel papers \\cite{grebik_rozhon2021LCL_on_paths,grebik_rozhon2021toasts_and_tails}.\n\nIn the remainder of this section, we give a high-level overview of the three areas that we study.\nThe purpose of these overviews is to provide the reader with a comprehensive picture of the studied settings that can also serve as a starting point for delving deeper into selected topics in those areas---in order to follow our paper, it is not necessary to obtain a detailed understanding of the results and connections presented in the overviews. Necessary technical details will be provided in \\cref{sec:preliminaries}.\nMoreover, in \\cref{sec:contribution}, we present our contributions in detail.\n\n\n\n\n\n\\paragraph{Distributed Computing}\nThe definition of the $\\mathsf{LOCAL}$ model of distributed computing by Linial~\\cite{linial92LOCAL} was motivated by the desire to understand distributed algorithms in huge networks. \nAs an example, consider a huge network of wifi routers. Let us think of two routers as connected by an edge if they are close enough to exchange messages. \nIt is desirable that such close-by routers communicate with user devices on different channels to avoid interference.\nIn graph-theoretic language, we want to properly color the underlying network. \nEven if we are allowed a color palette with $\\Delta+1$ colors where $\\Delta$ denotes the maximum degree of the graph (which would admit a simple greedy algorithm in a sequential setting), the problem remains highly interesting in the distributed setting, as, ideally, each vertex decides on its output color after only a few rounds of communication with its neighbors, which does not allow for a simple greedy solution. \n\nThe $\\mathsf{LOCAL}$ model of distributed computing formalizes this setup: we have a large network, where each vertex knows the network's size, $n$, and perhaps some other parameters like the maximum degree $\\Delta$. In the case of randomized algorithms, each vertex has access to a private random bit string, while in the case of deterministic algorithms, each vertex is equipped with a unique identifier from a range polynomial in the size $n$ of the network. \nIn one round, each vertex can exchange any message with its neighbors and can perform an arbitrary computation. The goal is to find a solution to a given problem in as few communication rounds as possible. \nAs the allowed message size is unbounded, a $t$-round $\\mathsf{LOCAL}$ algorithm can be equivalently described as a function that maps $t$-hop neighborhoods to outputs---the output of a vertex is then simply the output its $t$-hop neighborhood\nmapped to by this function.\nAn algorithm is correct if and only if the collection of outputs at all vertices constitutes a correct solution to the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. \nThe case of trees is a highlight of the theory: it is known that any local problem (a class of natural problems we will define later) belongs to one of only very few complexity classes.\nMore precisely, for any local problem, its randomized local complexity is either $O(1), \\Theta(\\log^* n), \\Theta(\\log\\log n), \\Theta(\\log n)$, or $\\Theta(n^{1\/k})$ for some $k \\in \\mathbb{N}$. Moreover, the deterministic complexity is always the same as the randomized one, except for the case $\\Theta(\\log\\log n)$, for which the corresponding deterministic complexity is $\\Theta(\\log n)$ (see \\cref{fig:big_picture_trees}). \n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/big_picture_trees.pdf}\n    \\caption{Complexity classes on regular trees considered in the three areas of distributed computing, factors of iid processes\/uniform algorithms, and descriptive combinatorics. The left part shows complexity classes of distributed computing. We use the shorthand $\\mathsf{LOCAL}$ if it does not matter whether we talk about the deterministic or randomized complexity. These two notions differ only for the class of problems of randomized local complexity $O(\\log\\log n)$, which have deterministic complexity $O(\\log n)$. \\\\\n    The uniform complexity classes of sublogarithmic complexity are in correspondence to appropriate classes in the randomized local complexity model, as proven in \\cref{sec:separating_examples}. On the other hand, the class fiid is very similar to the class $\\mathsf{MEASURE}$ from descriptive combinatorics. The equivalence of the class $\\mathsf{CONTINUOUS}$ and $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$ is marked with a dashed arrow as it was proven in case the tree is generated by a group action (think of the tree being equipped with an additional $\\Delta$-edge coloring). The inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ however clearly holds also in our setting. The class $\\mathsf{BOREL}$ is incomparable with $\\mathsf{RLOCAL}(O(\\log\\log n))$, as proven in \\cref{sec:separating_examples}.   }\n    \\label{fig:big_picture_trees}\n\\end{figure}\n\n\n\\todo{thm 5.1 - 5.3}\n\n\n\n\n\\paragraph{(Finitary) Factors of iid Processes and Uniform Algorithms}\n\nIn recent years, \\emph{factors of iid (fiid) processes} on trees attracted a lot of attention in combinatorics, probability, ergodic theory and statistical physics \\cite{ABGGMP,ArankaFeriLaci,BackSzeg,BackVir,backhausz,backhausz2,BGHV,Ball,Bowen1,Bowen2,CPRR,csoka2015invariant,GabLyo,GamarnikSudan,GGP,HarangiVirag,HolroydPoisson,HLS,HPPS,HolroydPeres,HoppenWormald,Kun,RahmanPercolation,RahVir,LyonsTrees,Mester,schmidt2016circular,Timar1,Timar2}.\nIntuitively, factors of iid processes are randomized algorithms on, e.g., infinite $\\Delta$-regular trees, where each vertex outputs a solution to a problem after it explores random strings on vertices of the whole tree.\nAs an example, consider the \\emph{perfect matching} problem.\nAn easy parity argument shows that perfect matching cannot be solved by any local randomized algorithm on finite trees.\nHowever, if we allow a small fraction of vertices not to be matched, then, by a result of Nguyen and Onak \\cite{nguyen_onak2008matching} (see also \\cite{ElekLippner}), there is a constant-round randomized algorithm that produces such a matching on high-girth graphs (where the constant depends on the fraction of unmatched vertices that we allow).\nThis result can also be deduced from a result of Lyons and Nazarov \\cite{lyons2011perfect}, who showed that perfect matching can be described as a factor of iid process on an infinite $\\Delta$-regular tree.\nThe high-level idea behind this connection is that high-girth graphs approximate the infinite $\\Delta$-regular tree and constant-round local algorithms approximate factors of iid processes.\nThis correspondence is formalized in the notion of \\emph{Benjamini-Schramm} or \\emph{local-global} convergence \\cite{benjamini2011recurrence, hatamilovaszszegedy}.\nWe note that getting only ``approximate'' solutions, that is, solutions where a small fraction of vertices does not have to satisfy the constraints of a given problem, is intrinsic in this correspondence.\nRegardless, there are many techniques, such as entropy inequality \\cite{backhausz} or correlation decay \\cite{backhausz2}, and particular results such as the aforementioned perfect matching problem \\cite{lyons2011perfect} that provide lower and upper bounds, respectively, in our setting as well.\nWe refer the reader to \\cite{LyonsTrees,BGHV} for a comprehensive summary of the field. \n\n\nIn this paper, we mostly consider a stronger condition than fiid, namely so-called \\emph{finitary} factors of iid (ffiid) processes that are studied in the same context as fiid \\cite{holroyd2017one_dependent_coloring,HolroydSchrammWilson2017FinitaryColoring,Spinka}. \nPerhaps surprisingly, the notion of ffiid is identical to the notion of so-called uniform distributed randomized algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring,grebik_rozhon2021toasts_and_tails} that we now describe. \nWe define an uniform local algorithm as a randomized local algorithm that does not know the size of the graph $n$ -- this enables us to run such an algorithm on infinite graphs, where there is no $n$. \nMore precisely, we require that each vertex eventually outputs a solution that is compatible with the output in its neighborhood, but the time until the vertex finishes is a potentially unbounded random variable. \nAs in the case of classical randomized algorithms, we can now measure the \\emph{uniform complexity} of an uniform local algorithm (known as the tail decay of ffiid \\cite{HolroydSchrammWilson2017FinitaryColoring}). \nThe uniform complexity of an algorithm is defined as the function $t(\\varepsilon)$ such that the probability that the algorithm run on a specific vertex needs to see outside its $t(\\varepsilon)$-hop neighborhood is at most $\\varepsilon$. \nAs in the case of classical local complexity, there is a whole hierarchy of possible uniform complexities (see \\cref{fig:big_picture_trees}). \n\nWe remark that uniform distributed local algorithms can be regarded as Las Vegas algorithms. The output will always be correct; there is however no fixed guarantee at what point all vertices have computed their final output. On the other hand, a randomized distributed local algorithm can be viewed as a Monte Carlo algorithm as it needs to produce an output after a fixed number of rounds, though the produced output might be incorrect. \n\n\n\n\n\n\n\n\n\\paragraph{Descriptive Combinatorics}\n   \n   \n\t\n\tThe Banach-Tarski paradox states that a three-dimensional ball of unit volume can be decomposed into finitely many pieces that can be moved by isometries (distance preserving transformations such as rotations and translations) to form two three-dimensional balls each of them with unit volume(!).\n    The graph theoretic problem lurking behind this paradox is the following: fix finitely many isometries of $\\mathbb{R}^3$ and then consider a graph where $x$ and $y$ are connected if there is an isometry that sends $x$ to $y$.\n\tThen our task becomes to find a perfect matching in the appropriate subgraph of this graph -- namely, the bipartite subgraph where one partition contains points of the first ball and the other contains points of the other two balls. \n\tBanach and Tarski have shown that, with a suitably chosen set of isometries, the axiom of choice implies the existence of such a matching. \n\tIn contrast, since isometries preserve the Lebesgue measure, the pieces in the decomposition cannot be Lebesgue measurable. \n\tSurprisingly, Dougherty and Foreman \\cite{doughertyforeman} proved that the pieces in the Banach-Tarski paradox can have the \\emph{Baire property}. \n\tThe Baire property is a topological analogue of being Lebesgue measurable; a subset of $\\mathbb{R}^3$ is said to have the Baire property if its difference from some open set is topologically negligible.\n\t\n       \n\t    \n\t\n\t\n    Recently, results similar to the Banach-Tarski paradox that lie on the border of combinatorics, logic, group theory, and ergodic theory led to an emergence of a new field often called \\textit{descriptive} or \\textit{measurable combinatorics}. \n    The field focuses on the connection between the discrete and continuous and is largely concerned with the investigation of graph-theoretic concepts. \n    The usual setup in descriptive combinatorics is that we have a graph with uncountably many connected components, each being a countable graph of bounded degree. \n    For example, in case of the Banach-Tarski paradox, the vertices of the underlying graph are the points of the three balls, edges correspond to isometries, and the degree of each vertex is bounded by the number of chosen isometries. \n    Some of the most beautiful results related to the field include  \\cite{laczk, marksunger,measurablesquare,doughertyforeman,marks2016baire,gaboriau,KST,DetMarks,millerreducibility,conley2020borel,csokagrabowski,Bernshteyn2021LLL}, see  \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} for recent surveys. \n    \n    Importantly, in many of these results, including the Banach-Tarski paradox, graphs where each component is an infinite $\\Delta$-regular tree appear naturally. \n    Oftentimes, questions considered in descriptive combinatorics lead to constructing a solution to a local problem in the underlying uncountable graph (in the case of Banach-Tarski, the local problem is perfect matching).  \n    The construction needs to be such that the solution of the problem has some additional regularity properties. For example in the case of Banach-Tarski, a solution is possible when the regularity condition is the Baire property, but not if it is Lebesgue measurability.\n    In fact, together with Borel measurability these are the most prominent regularity conditions studied in descriptive combinatorics.\n   \n    The corresponding complexity classes of local problems that always admit a solution with the respective regularity property are $\\mathsf{BOREL},\\mathsf{MEASURE},\\mathsf{BAIRE}$ (See \\cref{fig:big_picture_trees}). \n    In this paper, we moreover consider the setting where each connected component of the underlying graph is a $\\Delta$-regular tree. \n   \n   \n    \n    \n    \n   \n\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\n\t    The connection between distributed computing and descriptive combinatorics arises from the fact that in descriptive combinatorics we care about constructions that do not use the axiom of choice. \n\t    In the distributed language, the axiom of choice corresponds to leader election, that is, the constructions in descriptive combinatorics do not allow picking exactly one point in every component.  \n\t   \n\t    To get some intuition about the power of the complexity class $\\mathsf{BOREL}$, we note that Borel constructions allow us to alternate countably many times the following two operations. \n\t    First, any local algorithm with  constant local complexity can be run. \n\t    Second, we have an oracle that provides a maximal independent set (MIS) on any graph that can be constructed locally from the information computed so far \\cite{KST}. \n        Note that from the speedup result of \\cite{chang_kopelowitz_pettie2019exp_separation} we get that every local problem with local complexity $O(\\log^* n)$ can be solved by constant local constructions and \\emph{one} call to such an MIS oracle. This implies the inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ in \\cref{fig:big_picture_trees} proven in the insightful paper of Bernshteyn \\cite{Bernshteyn2021LLL}.  \n        The relationship of the class $\\mathsf{MEASURE}$ (and of the class $\\mathsf{fiid}$ from the discussion of factors) to the class $\\mathsf{BOREL}$ is analogous to the relationship of randomized distributed algorithms to deterministic distributed algorithms. \n\n       \n        \n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\n\n\n\n\n\n\\paragraph{Local Problems on Regular Trees}\n\nAfter introducing the three areas of interest in this work, we conclude the section by briefly discussing the kinds of problems we focus on, which are local problems on regular trees.\nMore precisely, we study \\emph{locally checkable labeling (LCL)} problems, which are a class of problems, where the correctness of the solution can be checked locally.\nExamples include classical problems from combinatorics such as proper vertex coloring, proper edge coloring, perfect matching, and maximal independent set.\nOne main goal of this paper is to understand possible complexity classes of LCLs without inputs on infinite $\\Delta$-regular trees and their finite analogues.\nWe refer the reader to \\cref{sec:preliminaries} for a precise definition of a finite $\\Delta$-regular tree.\n\nThe motivation for studying regular trees in this work stems from different sources: (a) infinite $\\Delta$-regular trees are studied in the area of ergodic theory \\cite{Bowen1,Bowen2}, random processes \\cite{backhausz,backhausz2,lyons2011perfect} and descriptive combinatorics \\cite{DetMarks,BrooksMeas}, (b) many lower bounds in distributed computing are proven in regular trees \\cite{balliu_et_al:binary_lcl,balliu2019LB,brandt_etal2016LLL,brandt19automatic,Brandt20tightLBmatching,ChangHLPU20,goeoes14_PODC}, and (c) connecting and comparing the techniques of the three areas in this simple setting reveals already deep connections, see \\cref{sec:contribution}.\n\n\n\n\n\n\n\n\n\n\\section{Our Contributions}\n\\label{sec:contribution}\n\nWe believe that our main contribution is presenting all three perspectives as part of a common theory. Our technical contribution is split into three main parts. \n\n\n\n\n\\subsection{Generalization of Marks' Technique}\\label{sec:intromarks}\nIn Section \\ref{sec:marks} we extend the Borel determinacy technique of Marks \\cite{DetMarks}, which was used to prove the nonexistence of Borel $\\Delta$-colorings and perfect matchings, to a  broader class of problems, and \nadapt the extended technique to the distributed setting, thereby obtaining a simple method for proving distributed lower bounds.\nThis method is the first lower bound technique for distributed computing using ideas coming from descriptive combinatorics (see \\cite{BernshteynVizing} for a distributed computing upper bound motivated by descriptive combinatorics).\nMoreover, we show how to use the developed techniques to obtain both $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}$ lower bounds for local problems from a natural class, called homomorphism problems.\nOur key technical ingredient for obtaining the mentioned techniques and results is a novel technique based on the notion of an \\emph{ID graph}.\nWe note that a very similar concept to the ID graph was independently discovered by \\cite{id_graph}. \n\n\\paragraph{Marks' technique}\nIn the following we give an introduction to Marks' technique by going through a variant of his proof~\\cite{DetMarks,Marks_Coloring} that shows that $\\Delta$-coloring has deterministic local complexity $\\Omega(\\log n)$. The proof already works in the case where the considered regular tree comes with an input $\\Delta$-edge coloring. In this case, the output color of a given vertex $u$ can be interpreted as that $u$ ``grabs'' the incident edge of that color. The problem is hence equivalent to the \\emph{edge grabbing} problem where every vertex is required to grab an incident edge such that no two vertices grab the same edge. \n\nWe first show the lower bound in the case that vertices do not have unique identifiers but instead are properly colored with $L > \\Delta$ colors. \nSuppose there is an algorithm $\\mathcal{A}$ solving the edge grabbing problem with local complexity $t(n) = o(\\log n)$, and consider a tree rooted at vertex $u$ of depth $t(n)$; such a tree has less than $n$ vertices, for large enough $n$.\nAssume that $u$ has input color $\\sigma \\in [L]$, and consider, for some fixed edge color $\\alpha$, the edge $e$ that is incident to $u$ and has color $\\alpha$.\nTwo players, Alice and Bob, are playing the following game. In the $i$-th round, Alice colors the vertices at distance $i$ from $u$ in the subtree reachable via edge $e$ with colors from $[L]$. Then, Bob colors all other vertices at distance $i$ from $u$ with colors from $[L]$ (see \\cref{fig:game}).\nConsider the output of $u$ when executing $\\mathcal{A}$ on the obtained colored tree.\nBob wins the game if $u$ grabs the edge $e$, and Alice wins otherwise. \n\nNote that either Alice or Bob has a winning strategy. \nGiven the color $\\sigma$ of $u$, if, for each edge color $\\alpha$, Alice has a winning strategy in the game corresponding to the pair $(\\sigma, \\alpha)$, then we can create $\\Delta$ copies of Alice and let them play their strategy on each subtree of $u$, telling them that the colors chosen by the other Alices are what Bob played. The result is a coloring of the input tree such that $u$, by definition, does not pick any edge, contradicting the fact that $\\mathcal{A}$ provides a valid solution!\nSo for every $\\sigma$ there is at least one $\\alpha$ such that Bob has a winning strategy for the game corresponding to $(\\sigma, \\alpha)$. By the pigeonhole principle, there are two colors $\\sigma_1, \\sigma_2$, such that Bob has a winning strategy for both pairs $(\\sigma_1, \\alpha)$ and $(\\sigma_2, \\alpha)$. \nBut now we can imagine a tree rooted in an edge between vertices $u_1, u_2$ that are colored with colors $\\sigma_1, \\sigma_2$. We can now take two copies of Bob, one playing at $u_1$ and the other playing at $u_2$ and let them battle it out, telling each copy that the other color from $\\{\\sigma_1, \\sigma_2\\}$ and whatever the other copy plays are the moves of Alice. The resulting coloring has the property that both $u_1$ and $u_2$, when executing $\\mathcal{A}$ on the obtained colored tree, grab the edge between them, a contradiction that finishes the proof!\n\n\\paragraph{The ID graph}\nThe downside of the proof is that it does not work in the model with unique identifiers (where the players' moves consist in assigning identifiers instead of colors), since gluing copies of the same player could result in an identifier assignment where the identifiers are not unique. \nOne possible remedy is to conduct the whole proof in the context of Borel graphs as was done by Marks. This proves an even stronger statement, namely that $\\Delta$-coloring is not in the class $\\mathsf{BOREL}$, but requires additional ad-hoc tricks and a pretty heavy set theoretic tool---Martin's celebrated Borel determinacy theorem~\\cite{martin} stating that even for infinite two-player games one of the players has to have a winning strategy if the payoff set is Borel. \nThe ID graph enables us to adapt the proof (and its generalization that we develop in \\cref{sec:marks}) to the distributed setting, where the fact that one of the players has a winning strategy is obvious. \nMoreover, we use an infinite version of the ID graph to generalize Marks' technique also in the Borel setting. \n\nHere is how it works: \nThe ID graph is a specific graph whose vertices are the possible unique input identifiers (for input graphs of size $n$), that is, numbers from $[n^{O(1)}]$. Its edges are colored with colors from $[\\Delta]$ and its girth is $\\Omega(\\log n)$. When we define the game between Alice and Bob, we require them to label vertices with identifiers in such a way that whenever a new vertex is labeled with identifier $i$, and its already labeled neighbor has identifier $j$, then $ij$ is an edge in the ID graph.\nMoreover, the color of edge $ij$ in the ID graph is required to be the same as the color of the edge between the vertices labeled $i$ and $j$ in the tree where the game is played. \nIt is straightforward to check that these conditions enforce that even if we let several copies of the same player play, the resulting tree is labeled with unique identifiers. Hence, the same argument as above now finally proves that the deterministic local complexity of $\\Delta$-coloring is  $\\Omega (\\log n)$. \n\nWe note that our ID graph technique is of independent interest and may have applications in many different contexts. \nTo give an example from distributed computing, consider the proof of the deterministic $\\Omega(\\log n)$-round lower bound for $\\Delta$-coloring developed by the distributed community~\\cite{brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation}, which is based on the celebrated round elimination technique. \nEven though the result is deterministic, the proof is quite technical due to the fact that it relies on examining randomized algorithms, for reasons similar to the reasons why Marks' proof does not apply directly to the setting with unique identifiers. \nFortunately, it can be again streamlined with the use of the ID graph technique. \nMoreover, in a follow-up work, the ID graph technique has already led to a new lower bound for the Lov\u00e1sz local lemma~\\cite{brandt_grunau_rozhon2021LLL_in_LCA} in the area of Local Computation Algorithms (which is part of the realm of sequential computation), thereby giving further evidence for the versatility of the technique. \n\n\n\\paragraph{Marks vs. Round Elimination}\nIt is quite insightful to compare Marks' technique (and our generalization of it) with the powerful round elimination technique~\\cite{brandt19automatic}, which has been responsible for all locality lower bounds of the last years~\\cite{brandt_etal2016LLL, balliu2019LB, brandt19automatic, balliu2019hardness_homogeneous, balliu_et_al:binary_lcl, Brandt20tightLBmatching, balliu20rulingset, ChangHLPU20, balliu21dominatingset}. \nWhile, on the surface, Marks' approach developed for the Borel world may seem quite different from the round elimination technique, there are actually striking similarities between the two methods.\nOn a high level, in the round elimination technique, the following argument is used to prove lower bounds in the LOCAL model:\nIf a $T$-round algorithm exists for a problem $\\Pi_0$ of interest, then there exists a $(T-1)$-round algorithm for some problem $\\Pi_1$ that can be obtained from $\\Pi_0$ in a mechanical manner.\nBy applying this step iteratively, we obtain a problem $\\Pi_t$ that can be solved in $0$ rounds; by showing that there is no $0$-algorithm for $\\Pi_t$ (which is easy to do if $\\Pi_t$ is known), a $(T+1)$-round lower bound for $\\Pi_0$ is obtained.\n\nThe interesting part regarding the relation to Marks' technique is how the ($T-i-1$)-round algorithms $\\mathcal{A}'$ are obtained from the ($T-i$)-round algorithms $\\mathcal{A}$ in the round elimination framework: in order to execute $\\mathcal{A}'$, each vertex $v$, being aware of its ($T-i-1$)-hop neighborhood, essentially asks whether, for all possible extensions of its view by one hop along a chosen incident edge, there exists some extension of its view by one hop along all other incident edges such that $\\mathcal{A}$, executed on the obtained ($T-i$)-hop neighborhood, returns a certain output at $v$, and then bases its output on the obtained answer.\nIt turns out that the vertex sets corresponding to these two extensions correspond precisely to two moves of the two players in the game(s) played in Marks' approach: more precisely, in round $T-i$ of a game corresponding to the considered vertex $v$ and the chosen incident edge, the move of Alice consists in labeling the vertices corresponding to the first extension, and the move of Bob consists in labeling the vertices corresponding to the second extension.\n\nHowever, despite the similarities, the two techniques (at least in their current forms) have their own strengths and weaknesses and are interestingly different in that there are local problems that we know how to obtain good lower bounds for with one technique but not the other, and vice versa.\nFinding provable connections between the two techniques is an exciting research direction that we expect to lead to a better understanding of the possibilities and limitations of both techniques.\n\nIn \\cref{sec:marks} we use our generalized and adapted version of Marks' technique to prove new lower bounds for so-called homomorphism problems. \nHomomorphism problems are a class of local problems that generalizes coloring problems---each vertex is to be colored with some color and there are constraints on which colors are allowed to be adjacent. \nThe constraints can be viewed as a graph---in the case of coloring this graph is a clique. \nIn general, whenever the underlying graph of the homomorphism problem is $\\Delta$-colorable, its deterministic local complexity is $\\Omega(\\log n)$, because solving the problem would imply that we can solve $\\Delta$-coloring too (in the same runtime). \nIt seems plausible that homomorphism problems of this kind are the only hard, i.e., $\\Omega(\\log n)$, homomorphism problems. However, our generalization of Marks' technique asserts that this is not true. \n\n\\begin{theorem}\nThere are homomorphism problems whose deterministic local complexity on trees is $\\Omega(\\log n)$ such that the chromatic number of the underlying graph is $2\\Delta-2$.\n\\end{theorem}\n\nIt is not known how to prove the same lower bounds using round elimination\\footnote{Indeed, the descriptions of the problems have comparably large numbers of labels and do not behave like so-called ``fixed points'' (i.e., nicely) under round elimination, which suggests that it is hard to find a round elimination proof with the currently known approaches.}; in fact, as far as we know, these problems are the only known examples of problems on $\\Delta$-regular trees for which a lower bound is known to hold but currently not achievable by round elimination. \nProving the same lower bounds via round elimination is an exciting open problem. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Separation of Various Complexity Classes}\n\\label{sec:introseparation}\n\\paragraph{Uniform Complexity Landscape}\nWe investigate the connection between randomized and uniform distributed local algorithms, where uniform algorithms are equivalent to the studied notion of finitary factors of iid. \nFirst, it is simple to observe that local problems with uniform complexity $t(\\varepsilon)$ have randomized complexity $t(1 \/ n^{O(1)})$ -- by definition, every vertex knows its local output after that many rounds with probability $1 - 1\/n^{O(1)}$. The result thus follows by a union bound over the $n$ vertices of the input graph.\n\nOn the other hand, we observe that on $\\Delta$-regular trees the implication also goes in the opposite direction in the following sense. Every problem that has a randomized complexity of $t(n) = o(\\log n)$ has an uniform complexity of $O(t(1\/\\varepsilon))$. \n\nOne could naively assume that this equivalence also holds for higher complexities, but this is not the case. Consider for example the $3$-coloring problem.\nIt is well-known in the distributed community that $3$-coloring a tree can be solved deterministically in $O(\\log n)$ rounds  using the rake-and-compress decomposition~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,MillerR89}.\nOn the other hand, there is no uniform algorithm for $3$-coloring a tree. \nIf there were such an uniform algorithm, we could run it on any graph with large enough girth and color $99\\%$ of its vertices with three colors. This in turn would imply that the high-girth graph has an independent set of size at least $0.99 \\cdot n\/3$. This is a contradiction with the fact that there exists high-girth graphs with a much smaller independence number~\\cite{bollobas}. \n\nInterestingly, the characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that \\emph{any} uniform distributed algorithm can be ``sped up'' to a deterministic local $O(\\log n)$ complexity, as we prove in~\\cref{thm:MainBaireLog}. \n\nWe show that there are local problems that can be solved by an uniform algorithm but only with a complexity of $\\Omega(\\log 1\/\\varepsilon)$. Namely, the problem of constructing a $2$-hop perfect matching on infinite $\\Delta$-regular trees for $\\Delta \\ge 3$ has an uniform local complexity between $\\Omega(\\log 1\/\\varepsilon)$ and $O(\\operatorname{poly}\\log 1\/\\varepsilon)$. \nFormally, this proves the following theorem. \n\n\\begin{theorem}\n$\\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon)) \\subsetneq \\mathsf{OLOCAL}(O(\\operatorname{poly}\\log 1\/\\varepsilon))$. \n\\end{theorem}\n\nThe uniform  algorithm for this problem is based on a so-called one-ended forest decomposition introduced in \\cite{BrooksMeas} in the descriptive combinatorics context. In a one-ended forest decomposition, each vertex selects exactly one of its neighbors as its parent by orienting the corresponding edge outwards. This defines a decomposition of the vertices into infinite trees. We refer to such a decomposition as a one-ended forest decomposition if the subtree rooted at each vertex only contains finitely many vertices. \nHaving computed such a decomposition, $2$-hop perfect matching can be solved inductively starting from the leaf vertices of each tree.\n\nWe leave the understanding of the uniform complexity landscape in the regime $\\Omega(\\log 1\/\\varepsilon)$ as an exciting open problem. In particular, does there exist a function $g(\\varepsilon)$ such that each local problem that can be solved by an uniform algorithm has an uniform complexity of $O(g(\\varepsilon))$?\n\n\n\n\\paragraph{Relationship of Distributed Classes with Descriptive Combinatorics}\n\nBernshteyn recently proved that $\\mathsf{LOCAL}(\\log^* n) \\subseteq \\mathsf{BOREL}$ \\cite{Bernshteyn2021LLL}. That is, each local problem with a deterministic $\\mathsf{LOCAL}$ complexity of $O(\\log^* n)$ also admits a Borel-measurable solution. A natural question to ask is whether the converse also holds. \nIndeed, it is known that $\\mathsf{LOCAL}(\\log^* n) = \\mathsf{BOREL}$ on paths with no additional input \\cite{grebik_rozhon2021LCL_on_paths}. \nWe show that on regular trees the situation is different. \nOn one hand, a characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that $\\mathsf{BOREL} \\subseteq \\mathsf{BAIRE} \\subseteq \\mathsf{LOCAL}(O(\\log n))$. \nOn the other hand, we show that this result cannot be strengthened by proving the following result. \n\n\\begin{theorem}\n$\\mathsf{BOREL} \\subsetneq \\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\nThat is, there exists a local problem that admits a Borel-measurable solution but cannot be solved with a (randomized) $\\mathsf{LOCAL}$ algorithm running in a sublogarithmic number of rounds. \n\nLet us sketch a weaker separation, namely that $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n)) \\neq \\emptyset$. Consider a version of $\\Delta$-coloring where a subset of vertices can be left uncolored. However, the subgraph induced by the uncolored vertices needs to be a collection of doubly-infinite paths (in finite trees, this means each path needs to end in a leaf vertex). \nThe nonexistence of a fast distributed algorithm for this problem essentially follows from the celebrated $\\Omega(\\log n)$ deterministic lower bound for $\\Delta$-coloring of~\\cite{brandt_etal2016LLL}. \nOn the other hand, the problem allows a Borel solution. First, sequentially compute $\\Delta-2$ maximal independent sets, each time coloring all vertices in the MIS with the same color, followed by removing all the\ncolored vertices from the graph. In that way, a total of $\\Delta-2$ colors are used. Moreover, each\nuncolored vertex has at most $2$ uncolored neighbors. \nThis implies that the set of uncolored vertices forms a disjoint union of finite paths, one ended\ninfinite paths and doubly infinite paths. The first two classes can be colored inductively with two\nadditional colors, starting at one endpoint of each path in a Borel way (namely it can be done by making use of the countably many MISes in larger and larger powers of the input graph). \nHence, in the end only doubly infinite paths are left uncolored, as desired.  \n\nTo show the stronger separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{RLOCAL}(o(\\log n))$ we use a variation of the\n$2$-hop perfect matching problem. In this variation, some of the vertices can be left unmatched, but similar as in the variant of the $\\Delta$-coloring problem described above, the graph induced by all the unmatched vertices needs to satisfy some additional constraints.\n\nWe conclude the paragraph by noting that the separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}(O(\\log^* n))$  is not as simple as it may look in the following sense. \nThis is because problems typically studied in the $\\mathsf{LOCAL}$ model with a $\\mathsf{LOCAL}$ complexity of $\\omega(\\log^* n)$ like $\\Delta$-coloring and perfect matching also do not admit a Borel-measurable solution due to the technique of Marks \\cite{DetMarks} that we discussed in \\cref{sec:intromarks}.\n\n\n\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$} \n\nWe already discussed that one of complexity classes studied in descriptive combinatorics is the class $\\mathsf{BAIRE}$.\nRecently, Bernshteyn proved \\cite{Bernshteyn_work_in_progress} that all local problems that are in the complexity class $\\mathsf{BAIRE}, \\mathsf{MEASURE}$ or $\\mathsf{fiid}$ have to satisfy a simple combinatorial condition which we call being $\\ell$-full. \nOn the other hand, all $\\ell$-full problems allow a $\\mathsf{BAIRE}$ solution \\cite{Bernshteyn_work_in_progress}.\nThis implies a complete combinatorial characterization of the class $\\mathsf{BAIRE}$. \nWe defer the formal definition of $\\ell$-fullness to Section \\ref{sec:baire} as it requires a formal definition of a local problem. \nInformally speaking, in the context of vertex labeling problems, a problem is $\\ell$-full if we can choose a subset $S$ of the labels with the following property.\nWhenever we label two endpoints of a path of at least $\\ell$ vertices with two labels from $S$, we can extend the labeling with labels from $S$ to the whole path such that the overall labeling is valid.\nFor example, proper $3$-coloring  is $3$-full with $S=\\{1,2,3\\}$ because for any path of three vertices such that its both endpoints are colored arbitrarily, we can color the middle vertex so that the overall coloring is proper. \nOn the other hand, proper $2$-coloring is not $\\ell$-full for any $\\ell$. \n\nWe complement this result as follows. \nFirst, we prove that any $\\ell$-full problem has local complexity $O(\\log n)$, thus proving that all complexity classes considered in the areas of factors of iids and descriptive combinatorics from \\cref{fig:big_picture_trees} are contained in $\\mathsf{LOCAL}(O(\\log n))$. \nIn particular, this implies that the existence of \\emph{any} uniform algorithm implies a local distributed algorithm for the same problem of local complexity $O(\\log n)$. \nWe obtain this result via the well-known rake-and-compress decomposition~\\cite{MillerR89}.  \n\n\nOn the other hand, we prove that any problem in the class $\\mathsf{LOCAL}(O(\\log n))$ satisfies the $\\ell$-full condition. \nThe proof combines a machinery  developed by Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}\nwith additional nontrivial ideas.\nIn this proof we construct recursively a sequence of sets of rooted, layered, and partially labeled trees, where the partial labeling is computed by simulating any given $O(\\log n)$-round distributed algorithm, \nand then the set  $S$ meeting the $\\ell$-full condition is constructed by considering all possible extensions of the partial labeling to complete correct labeling of these trees.\n\nThis result implies the following equality:\n\\begin{theorem}\n$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$. \n\\end{theorem}\n\nThis equality is surprising in that the definitions of the two classes do not seem to have much in common on the first glance!\nMoreover, the proof of the equality relies on nontrivial results in both distributed algorithms (the technique of Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}) and descriptive combinatorics (the fact that a hierarchical decomposition, so-called toast, can be constructed in $\\mathsf{BAIRE}$, \\cite{conleymillerbound}, see Proposition \\ref{pr:BaireToast}). \n\n\n\n\n\n\n\n\n\n\n\n\nThe combinatorial characterization of the local complexity class $\\mathsf{LOCAL}(O(\\log n))$ on $\\Delta$-regular trees is interesting from the perspective of distributed computing alone. \nThis result can be seen as a part of a large research program aiming at classification of possible local complexities on various graph classes~\\cite{balliu2020almost_global_problems,brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,Chang_paths_cycles,balliu2021_rooted_trees,balliu_et_al:binary_lcl,balliu2018new_classes-loglog*-log*}. \nThat is, we wish not only to understand possible complexity classes (see the left part of \\cref{fig:big_picture_trees} for possible local complexity classes on regular trees), but also to find combinatorial characterizations of problems in those classes that allow us to  efficiently decide for a given problem which class it belongs to.  \nUnfortunately, even for grids with input labels, it is \\emph{undecidable} whether a given local problem can be solved in $O(1)$ rounds~\\cite{naorstockmeyer,brandt_grids}, since\nlocal problems on grids can be used to simulate a Turing machine.\nThis undecidability result does not apply to paths and trees, hence for these graph classes it is still hopeful that we can find simple and useful characterizations for different classes of distributed problems. \n\nIn particular, on paths it is decidable what classes a given local problem belongs to, for all classes coming from the three areas considered here, and this holds even if we allow inputs~\\cite{Chang_paths_cycles,grebik_rozhon2021LCL_on_paths}. \nThe situation becomes much more complicated when we consider trees. \nRecently, classification results on trees were obtained for so-called binary-labeling problems \\cite{balliu_et_al:binary_lcl}. \nMore recently, a complete classification was obtained in the case of \\emph{rooted regular trees}~\\cite{balliu2021_rooted_trees}. \nAlthough their algorithm takes exponential time in the worst case, the authors provided a practical implementation fast enough to classify many typical problems of interest. \n\nMuch less is known for general, \\emph{unoriented} trees, with an arbitrary number of labels. \nIn general, deciding the optimal distributed complexity for a local problem on  bounded-degree trees is $\\mathsf{EXPTIME}$-hard~\\cite{chang2020n1k_speedups},  such a hardness result does not rule out the possibility for having a simple and polynomial-time characterization for the case of \\emph{regular trees}, where there is no input and the constraints are placed only on degree-$\\Delta$ vertices. \nIndeed, it was stated in~\\cite{balliu2021_rooted_trees} as an open question to find such a characterization. Our characterization of $\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$ by $\\ell$-full problems makes progress in better understanding the distributed complexity classes on trees and towards answering this open question. \n\n\n\n\n\n\n\n\\paragraph{Roadmap}\n\\hypertarget{par:roadmap}{}\nIn \\cref{sec:preliminaries}, we define formally all the three setups we consider in the paper. \nIn \\cref{sec:marks} we discuss the lower bound technique of Marks and the new concept of an ID graph.\nNext, in \\cref{sec:separating_examples} we prove some basic results about the uniform complexity classes and give examples of problems separating some classes from \\cref{fig:big_picture_trees}. \nFinally, in \\cref{sec:baire} we prove that a problem admits a Baire measurable solution if and only if it admits a distributed algorithm of local complexity $O(\\log n)$.\n\nThe individual sections can be read largely independently of each other.\nMoreover, most of our results that are specific to only one of the three areas can be understood without reading the parts of the paper that concern the other areas.\nWe encourage the reader interested mainly in one of the areas to skip the respective parts.\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nIn this section, we explain the setup we work with, the main definitions and results. \nThe class of graphs that we consider in this work are either infinite $\\Delta$-regular trees, or their finite analogue that we define formally in \\cref{subsec:local_problems}.\n\nWe sometimes explicitly assume $\\Delta > 2$. \nThe case $\\Delta = 2$, that is, studying paths, behaves differently and seems much easier to understand \\cite{grebik_rozhon2021LCL_on_paths}. \nUnless stated otherwise, we do not consider any additional structure on the graphs, but sometimes it is natural to work with trees with an input $\\Delta$-edge-coloring. \n\n\n\\subsection{Local Problems on $\\Delta$-regular trees}\n\\label{subsec:local_problems}\n\nThe problems we study in this work are locally checkable labeling (LCL) problems, which, roughly speaking, are problems that can be described via local constraints that have to be satisfied in a suitable neighborhood of each vertex.\nIn the context of distributed algorithms, these problems were introduced in the seminal work by Naor and Stockmeyer~\\cite{naorstockmeyer}, and have been studied extensively since.\nIn the modern formulation introduced in~\\cite{brandt19automatic}, instead of labeling vertices or edges, LCL problems are described by labeling half-edges, i.e., pairs of a vertex and an incident edge.\nThis formulation is very general in that it not only captures vertex and edge labeling problems, but also others such as orientation problems, or combinations of all of these types.\nBefore we can provide this general definition of an LCL, we need to introduce some definitions.\nWe start by formalizing the notion of a half-edge.\n\n\\begin{definition}[Half-edge]\n\\label{def:half-edge}\n    A \\emph{half-edge} is a pair $(v, e)$ where $v$ is a vertex, and $e$ an edge incident to $v$.\n    We say that a half-edge $(v, e)$ is \\emph{incident} to a vertex $w$ if $w = v$, we say that a vertex $w$ is \\emph{contained} in a half edge $(v,e)$ if $w=v$, and we say that $(v, e)$ \\emph{belongs} to an edge $e'$ if $e' = e$.\n    We denote the set of all half-edges of a graph $G$ by $H(G)$.\n    A \\emph{half-edge labeling} is a function $c \\colon H(G) \\to \\Sigma$ that assigns to each half-edge an element from some label set $\\Sigma$.\n\\end{definition}\n\nIn order to speak about finite $\\Delta$-regular trees, we need to consider slightly modified definition of a graph.\nWe think of each vertex to be contained in $\\Delta$-many half-edges, however, not every half edge belongs to an actual edge of the graph.\nThat is half-edges are pairs $(v,e)$, but $e$ is formally not a pair of vertices.\nSometimes we refer to these half-edges as \\emph{virtual} half-edges.\nWe include a formal definition to avoid confusions. See also \\cref{fig:Delta_reg_tree}. \n\n\n\\begin{definition}[$\\Delta$-regular trees]\n    A tree $T$, finite or infinite is a \\emph{$\\Delta$-regular tree} if either it is infinite and $T=T_\\Delta$, where $T_\\Delta$ is the unique infinite $\\Delta$-regular tree, that is each vertex has exactly $\\Delta$-many neighbors, or it is finite of maximum degree $\\Delta$ and each vertex $v\\in T$ of degree $d\\le \\Delta$ is contained in $(\\Delta-d)$-many virtual half-edges.\n   \n    \n    Formally, we can view $T$ as a triplet $(V(T), E(T), H(T))$, where $(V(T),E(T))$ is a tree of maximum degree $\\Delta$ and $H(T)$ consists of real half-edges, that is pairs $(v,e)$, where $v\\in V(T)$, $e\\in E(T)$ and $e$ is incident to $v$, together with some virtual edges, in the case when $T$ is finite, such that each vertex $v\\in V(T)$ is contained in exactly $\\Delta$-many half-edges (real or virtual).\n   \n   \n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig\/Delta_reg_tree.pdf}\n    \\caption{A $3$-regular tree on 15 vertices. Every vertex has $3$ half-edges but not all half-edges lead to a different vertex. }\n    \\label{fig:Delta_reg_tree}\n\\end{figure}\n\nAs we are considering trees in this work, each LCL problem can be described in a specific form that provides two lists, one describing all label combinations that are allowed on the half-edges incident to a vertex, and the other describing all label combinations that are allowed on the two half-edges belonging to an edge.\\footnote{Every problem that can be described in the form given by Naor and Stockmeyer~\\cite{naorstockmeyer} can be equivalently described as an LCL problem in this list form, by simply requiring each output label on some half-edge $h$ to encode all output labels in a suitably large (constant) neighborhood of $h$ in the form given in~\\cite{naorstockmeyer}.}\nWe arrive at the following definition for LCLs on $\\Delta$-regular trees.\\footnote{Note that the defined LCL problems do not allow the correctness of the output to depend on input labels.}\n\n\\begin{definition}[LCLs on $\\Delta$-regular trees]\\label{def:LCL_trees}\nA \\emph{locally checkable labeling problem}, or \\emph{LCL} for short, is a triple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, where $\\Sigma$ is a finite set of labels, $\\mathcal{V}$ is a subset of unordered cardinality-$\\Delta$ multisets\\footnote{Recall that a multiset is a modification of the concept of sets, where repetition is allowed.} of labels from $\\Sigma$ , and $\\mathcal{E}$ is a subset of unordered cardinality-$2$ multisets of labels from $\\Sigma$.\n\nWe call $\\mathcal{V}$ and $\\mathcal{E}$ the \\emph{vertex constraint} and \\emph{edge constraint} of $\\Pi$, respectively.\nMoreover, we call each multiset contained in $\\mathcal{V}$ a \\emph{vertex configuration} of $\\Pi$, and each multiset contained in $\\mathcal{E}$ an \\emph{edge configuration} of $\\Pi$.\n\nLet $T$ be a $\\Delta$-regular tree and $c:H(T)\\to \\Sigma$ a half-edge labeling of $T$ with labels from $\\Sigma$.\nWe say that $c$ is a \\emph{$\\Pi$-coloring}, or, equivalently, a correct solution for $\\Pi$, if, for each vertex $v$ of $T$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$ of $T$, the cardinality-$2$ multiset of labels assigned to the half-edges belonging to $e$ is an element of $\\mathcal{E}$.\n\\end{definition}\n\nAn equivalent way to define our setting would be to consider $\\Delta$-regular trees as commonly defined, that is, there are vertices of degree $\\Delta$ and vertices of degree $1$, i.e., leaves.\nIn the corresponding definition of LCL one would consider leaves as unconstrained w.r.t. the vertex constraint, i.e., in the above definition of a correct solution the condition ``for each vertex  $v$'' is replaced by ``for each non-leaf vertex $v$''.\nEquivalently, we could allow arbitrary trees of maximum degree $\\Delta$ as input graphs, but, for vertices of degree $< \\Delta$, we require the multiset of labels assigned to the half-edges to be extendable to some cardinality-$\\Delta$ multiset in $\\mathcal{V}$.\nWhen it helps the exposition of our ideas and is clear from the context, we may make use of these different but equivalent perspectives.\n\nWe illustrate the difference between our setting and ``standard setting'' without virtual half-edges on the perfect matching problem.\nA standard definition of the perfect matching problem is that some edges are picked in such a way that each vertex is covered by exactly one edge.\nIt is easy to see that there is no local algorithm to solve this problem on the class of finite trees (without virtual half-edges), this is a simple parity argument.\nHowever, in our setting, every vertex needs to pick exactly one half-edge (real or virtual) in such a way that both endpoints of each edge are either picked or not picked.\nWe remark that in our setting it is not difficult to see that (if $\\Delta>2$), then this problem can be solved by a local deterministic algorithm of local complexity $O(\\log(n))$.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The $\\mathsf{LOCAL}$ model}\n\nIn this section, we define local algorithms and local complexity.\nWe discuss the general relation between the classical local complexity and the uniform local complexity. Recall that when we talk about distributed algorithm on $\\Delta$-regular trees, the algorithm has access to $n$, the size of the tree. The measure of complexity is the classical local complexity. On the other hand, when we talk about uniform distributed algorithms on $\\Delta$-regular trees, we talk about an infinite $\\Delta$-regular tree and the measure of complexity is the uniform local complexity. \nWe start with the classical notions.\nRecall that $B(v,t)$ is the $t$-hop neighborhood of a vertex $v$ in an underlying graph $G$.\n\n\\begin{definition}[Local algorithm]\n\\label{def:local_algorithm}\nA \\emph{distributed local algorithm} $\\mathcal{A}$ of local complexity $t(n)$ is a function defined on all possible $t(n)$-hop neighborhoods of a vertex. Applying an algorithm $\\mathcal{A}$ on an input graph $G$ means that the function is applied to a $t(n)$-hop neighborhood $B(u, t(n))$ of each vertex $u$ of $G$. The output of the function is a labeling of the half-edges around the given vertex. The algorithm also takes as input the size of the input graph $n$. \n\\end{definition}\n\n\\begin{definition}[Local complexity]\n\\label{def:local_complexity}\nWe say that an LCL problem $\\Pi$ has a deterministic local complexity $t(n)$ if there is a local algorithm $\\mathcal{A}$ of local complexity $t(n)$ such that when run on the input graph $G$, with each of its vertices having a unique identifier from $[n^{O(1)}]$, $\\mathcal{A}$ always returns a valid solution to $\\Pi$.  \nWe also say $\\Pi \\in \\mathsf{DLOCAL}( O(t(n))$. \n\nThe randomized complexity is define analogously, but instead of unique identifiers, each vertex of $G$ has an infinite random string. The solution of $\\mathcal{A}$ needs to be a valid solution with probability $1 - 1\/n$. \nWe also say $\\Pi \\in \\mathsf{RLOCAL}( O(t(n))$. \n\\end{definition}\n\nWhenever we talk about \\emph{local} complexity on $\\Delta$-regular trees, we always tacitly think about the class of finite $\\Delta$-regular trees. We use the notation $\\mathsf{LOCAL}(O(t(n)))$ whenever the deterministic and the randomized complexity of the problems are the same up to constant factor (cf. \\cref{thm:basicLOCAL}). \n\n\\paragraph{Classification of Local Problems on $\\Delta$-regular Trees}\n\nThere is a lot of work aiming to classify all possible local complexities of LCL problems on bounded degree trees. This classification was recently finished.\nEven though the results in the literature are stated for the classical notion of finite trees, we note that it is easy to check that the same picture emerges if we restrict ourselves to $\\Delta$-regular trees according to our definition.\n\n\\begin{theorem}[Classification of local complexities of LCLs on $\\Delta$-regular trees \\cite{naorstockmeyer,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,balliu2020almost_global_problems,balliu2019hardness_homogeneous,brandt_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:basicLOCAL}\nLet $\\Pi$ be an LCL problem. Then the deterministic\/randomized local complexity of $\\Pi$ on $\\Delta$-regular trees is one of the following:\n\\begin{enumerate}\n    \\item $O(1)$,\n    \\item $\\Theta(\\log^*n)$,\n    \\item $\\Theta(\\log n)$ deterministic and $\\Theta(\\log\\log n)$ randomized,\n    \\item $\\Theta(\\log n)$,\n    \\item $\\Theta(n^{1\/k})$ for $k \\in \\mathbb{N}$.\n\\end{enumerate}\n\\end{theorem}\n\nWe do not know whether the complexity classes $\\Theta(n^{1\/k})$ are non-empty on \\emph{$\\Delta$-regular} trees. The $2 \\frac{1}{2}$ coloring problems $\\mathcal{P}_k$  are known to have complexity $\\Theta(n^{1\/k})$ on \\emph{bounded-degree} trees~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. However, as the correctness criterion of $\\mathcal{P}_k$ depends on the degree of vertices, it does not fit into the class of LCL problems that we consider in \\cref{def:LCL_trees}. \nNevertheless, as can be seen in \\cref{fig:big_picture_trees}, all classes of factors of iid and from descriptive combinatorics are already contained in the class $\\mathsf{LOCAL}(O(\\log n))$. \n\n\n\n\\paragraph{Uniform Algorithms}\nAs we mentioned before, when talking about local complexities, we always have in mind that the underlying graph is finite.\nIn particular, the corresponding algorithm knows the size of the graph.\nOn infinite $\\Delta$-regular trees, or infinite graphs in general, we use the following notion \\cite{HolroydSchrammWilson2017FinitaryColoring,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}. \n\n\\begin{definition}\nAn \\emph{uniform local algorithm} $\\mathcal{A}$ is a function that is defined on all possible (finite) neighborhoods of a vertex.\nFor some neighborhoods it outputs a special symbol $\\emptyset$ instead of a labeling of the half-edges around the central vertex.\nApplying $\\mathcal{A}$ on a graph $G$ means that for each vertex $u$ of $G$ the function is applied to $B(u,t)$, where $t$ is the minimal number such that $\\mathcal{A}(u,t)\\not=\\emptyset$.\nWe call $t$ the coding radius of $\\mathcal{A}$, and denote it, as a function on vertices, as $R_\\mathcal{A}$.\n\\end{definition}\n\nWe define the corresponding notion of \\emph{uniform local complexity} for infinite $\\Delta$-regular trees where each vertex is assigned an infinite random string.\n\n\\begin{definition}[Uniform local complexity \\cite{HolroydSchrammWilson2017FinitaryColoring}]\n\\label{def:uniform_complexity}\nWe say that the \\emph{uniform local (randomized) complexity} of an LCL problem $\\Pi$ is $t(\\varepsilon)$ if there is an uniform local algorithm $\\mathcal{A}$ such that the following hold on the infinite $\\Delta$-regular tree.\nRecall that $R_\\mathcal{A}$ is the random variable measuring the coding radius of a vertex $u$, that is, the distance $\\mathcal{A}$ needs to look at to decide the answer for $u$. Then, for any $0 < \\varepsilon < 1$:\n\\[\n\\P(R_\\mathcal{A} \\ge t(\\varepsilon)) \\leq \\varepsilon.\n\\]\nWe also say $\\Pi \\in \\mathsf{OLOCAL}(O(t(\\varepsilon))$.\n\\end{definition}\n\nWe finish by stating the following lemma that bounds the uniform complexity of concatenation of two uniform algorithm (we need to be little bit more careful and cannot just add the complexites up). \n\n\\begin{lemma}[Sequential Composition]\n\\label{lem:sequential_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_1(\\varepsilon)$ and $t_2(\\varepsilon)$, respectively. \nLet $A$ be the sequential composition of $A_1$ and $A_2$. That is, $A$ needs to know the output that $A_2$ computes when the local input at every vertex is equal to the output of the algorithm $A_1$ at that vertex. Then, $t_A(\\varepsilon) \\leq t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$.\n\\end{lemma}\n\\begin{proof}\nConsider some arbitrary vertex $u$. Let $E_1$ denote the event that the coding radius of $\\mathcal{A}_1$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right)$ for all vertices in the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$. As the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$ contains at most $\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}$ vertices, a union bound implies that $P(E_1) \\geq 1 - \\varepsilon\/2$. Moreover, let $E_2$ denote the event that the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$ is at most $t_{A_2}(\\varepsilon\/2)$. By definition, $P(E_2) \\geq 1 - \\varepsilon\/2$. Moreover, if both events $E_1$ and $E_2$ occur, which happens with  probability at least $1 - \\varepsilon$, then the coding radius of algorithm $\\mathcal{A}$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$, thus finishing the proof.\n \\end{proof}\n\n\n\\subsection{Descriptive combinatorics}\n\\label{subsec:descriptivecombinatorics}\n\nBefore we define formally the descriptive combinatorics complexity classes, we give a high-level overview on their connection to distributing computing for the readers more familiar with the latter.\n\nThe complexity class that partially captures deterministic local complexity classes is called $\\mathsf{BOREL}$ (see also Remark \\ref{rem:cont}).\nFirst note that by a result of Kechris, Solecki and Todor\\v{c}evi\\'c \\cite{KST} the \\emph{maximal independent set problem} (with any parameter $r\\in \\mathbb{N}$) is in this class for any bounded degree graph.\\footnote{That is, it is possible to find a Borel maximal independent set, i.e., a maximal independent set which is, moreover, a Borel subset of the vertex set.}\nIn particular, this yields that $\\mathsf{BOREL}$ contains the class $\\mathsf{LOCAL}(O(\\log^* n))$ by the characterization of \\cite{chang_kopelowitz_pettie2019exp_separation}, see \\cite{Bernshteyn2021LLL}. Moreover, as mentioned before, $\\mathsf{BOREL}$ is closed under countably many iterations of the operations of finding maximal independent set (for some parameter that might grow) and of applying a constant local rule that takes into account what has been constructed previously.\\footnote{It is in fact an open problem, whether this captures fully the class $\\mathsf{BOREL}$. However, note that an affirmative answer to this question would yield that problems can be solved in an ``effective\" manner in the Borel context, which is known not to be the case in unbounded degree graphs \\cite{todorvcevic2021complexity}.}\n\nTo get a better grasp of what this means, consider for example the proper vertex $2$-coloring problem on half-lines. It is clear that no local algorithm can solve this problem.\nHowever, as it is possible to determine the starting vertex after countably many iterations of the maximal independent set operation, we conclude that this problem is in the class $\\mathsf{BOREL}$.\nThe idea that $\\mathsf{BOREL}$ can compute some unbounded, global, information will be implicitly used in all the construction in \\cref{sec:separating_examples} that separate $\\mathsf{BOREL}$ from local classes.\n\nThe intuition behind the class $\\mathsf{MEASURE}$ is that it relates in the same way to the class $\\mathsf{BOREL}$, as randomized local algorithms relate to deterministic ones.\nIn particular, the operations that are allowed in the class $\\mathsf{MEASURE}$ are the same as in the class $\\mathsf{BOREL}$ but the solution of a given LCL can be incorrect on a measure zero set.\n\nThe class $\\mathsf{BAIRE}$ can be considered as a topological equivalent of the measure theoretic class $\\mathsf{MEASURE}$, that is, a solution can be incorrect on a topologically negligible set.\nThe main difference between the classes $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$ is that in the later there is a hierarchical decomposition that is called \\emph{toast}.\n(Note that this phenomenon is present in the case of $\\mathsf{MEASURE}$ exactly on so-called amenable graphs. It is also tightly connected with the notion of hyperfiniteness \\cite{conleymillerbound,gao2015forcing}.)\nThe independence of colorings on a tree together with this structure allows for a combinatorial characterization of the class $\\mathsf{BAIRE}$, which was proven by Bernshteyn \\cite{Bernshteyn_work_in_progress}, see also \\cref{sec:baire}.\n\n\n\n\nNext we formulate the precise definitions.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey,Bernshteyn2021LLL,kechrisclassical}, or to \\cite[Introduction,~Section~4.1]{grebik_rozhon2021LCL_on_paths} and \\cite[Section~7.1,~7.2]{grebik_rozhon2021toasts_and_tails} for intuitive examples and standard facts of descriptive set theory.\nIn particular, we do not define here the notion standard Borel\/probability space, a Polish topology, a Borel probability measure, Baire property etc.\n\nLet $\\mathcal{G}$ be a Borel graph of bounded maximum degree on a standard Borel space $X$.\nIn this paper we consider exclusively acyclic $\\Delta$-regular Borel graphs and we refer to them as \\emph{$\\Delta$-regular Borel forests}.\nIt is easy to see that the set of half-edges (see Definition \\ref{def:half-edge}) is naturally a standard Borel space, we denote this set by $H(\\mathcal{G})$.\nThus, it makes sense to consider Borel labelings of $H(\\mathcal{G})$.\nMoreover, if $\\mathcal{G}$ is a $\\Delta$-regular Borel forest and $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is an LCL, we can also decide whether a coloring $f:\\mathcal{H}(\\mathcal{G})\\to \\Sigma$ is a solution to $\\Pi$ as in Definition \\ref{def:LCL_trees}. \nSimilarly, we say that the coloring $f$ solves $\\Pi$, e.g., on a $\\mu$-conull set for some Borel probability measure $\\mu$ on $X$ if there is a Borel set $C\\subseteq X$ such that $\\mu(C)=1$, the vertex constraints are satisfied around every $x\\in C$ and the edge constraints are satisfied for every $x,y\\in C$ that form an edge in $\\mathcal{G}$.\n\n\\begin{definition}[Descriptive classes]\n\\label{def:descriptive}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL.\nWe say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every compatible Polish topology $\\tau$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\n\nWe say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every Borel probability measure $\\mu$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\n\\end{definition}\n\nThe following claim follows directly from the definition.\n\n\\begin{claim}\\label{cl:basicDC}\nWe have $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{claim}\n\nRecently Bernshteyn \\cite{Bernshteyn2021LLL,Bernshteyn2021local=cont} proved several results that connect distributed computing with descriptive combinatorics.\nUsing the language of complexity classes we can formulate some of the result as inclusions in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}[\\cite{Bernshteyn2021LLL}]\nWe have\n\\begin{itemize}\n    \\item $\\mathsf{LOCAL}(O(\\log^*(n)))\\subseteq \\mathsf{BOREL}$,\n    \\item $\\mathsf{RLOCAL}(o(\\log(n))) \\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{itemize}\nIn fact, these inclusions hold for any reasonable class of bounded degree graphs.\n\\end{theorem}\n\n\\begin{remark}\n\\label{rem:cont}\nThe ``truly'' local class in descriptive combinatorics is the class $\\mathsf{CONTINUOUS}$.\nEven though we do not define this class here\\footnote{To define the class $\\mathsf{CONTINUOUS}$, rather than asking for a continuous solution on all possible $\\Delta$-regular Borel graphs, one has to restrict to a smaller family of graphs, otherwise the class trivializes. To define precisely this family is somewhat inconvenient, and not necessary for our arguments.}, and we refer the reader to \\cite{Bernshteyn2021local=cont,GJKS,grebik_rozhon2021toasts_and_tails} for the definition and discussions in various cases, we mention that the inclusion \\ \\[\\mathsf{LOCAL}(O(\\log^* n))\\subseteq \\mathsf{CONTINUOUS} \\tag{*}\\] holds in most reasonable classes of bounded degree graphs, see \\cite{Bernshteyn2021LLL}.\nThis also applies to our situation\nRecently, it was shown by Bernshteyn \\cite{Bernshteyn2021local=cont}, and independently by Seward \\cite{Seward_personal}, that $(*)$ can be reversed for Cayley graphs of finitely generated groups.\nThis includes, e.g., natural $\\Delta$-regular trees with proper edge $\\Delta$-coloring, as this is a Cayely graph of free product of $\\Delta$-many $\\mathbb{Z}_2$ induced by the standard generating set.\nIt is, however, not clear whether it $(*)$ can be reversed in our situation, i.e., $\\Delta$-regular trees without any additional labels.\n\n\n\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Random processes}\n\nWe start with an intutitve description of fiid processes. Let $T_\\Delta$ be the infinite $\\Delta$-regular tree.\nInformally, \\emph{factors of iid processes (fiid)} on $T_\\Delta$ are infinite analogues of local randomized algorithms in the following way.\nLet $\\Pi$ be an LCL and $u\\in T_\\Delta$.\nIn order to solve $\\Pi$, we are allowed to explore the whole graph, and the random strings that are assigned to vertices, and then output a labeling of half-edges around $u$.\nIf such an assignment is a measurable function and produces a $\\Pi$-coloring almost surely, then we say that $\\Pi$ is in the class $\\mathsf{fiid}$.\nMoreover, if every vertex needs to explore only finite neighborhood to output a solution, then we say that $\\Pi$ is in the class $\\mathsf{ffiid}$.\nSuch processes are called \\emph{finitary} fiid (\\emph{ffiid}). There is also an intimate connection between ffiid and uniform algorithms.\nThis is explained in \\cite[Section 2.2]{grebik_rozhon2021toasts_and_tails}.\nInformally, an ffiid process that almost surely solves $\\Pi$ is, in the language of distributed computing, an uniform local algorithm that solves $\\Pi$.\nThis allows us to talk about uniform local complexity of an ffiid.\nIn the rest of the paper we interchange both notions freely with slight preference for the distributed computing language.\n\nNow we define formally these classes, using the language of probability.\nWe denote by $\\operatorname{Aut}(T_\\Delta)$ the automorphism group of $T_\\Delta$.\nAn \\emph{iid process} on $T_\\Delta$ is a collection of iid random variables $Y=\\{Y_v\\}_{v\\in V(T_\\Delta)}$ indexed by vertices, or edges, half-edges etc, of $T_\\Delta$ such that their joint distribution is invariant under $\\operatorname{Aut}(T_\\Delta)$.\nWe say that $X$ is a \\emph{factor of iid process (fiid)} if $X=F(Y)$, where $F$ is a measurable $\\operatorname{Aut}(T_\\Delta)$-invariant map and $Y$ is an iid process on $T_\\Delta$.\\footnote{We always assume $Y=(2^\\mathbb{N})^{T_\\Delta}$ endowed with the product Lebesgue measure.}\nMoreover, we say that $X$ is a \\emph{finitary factor if iid process (ffiid)} if $F$ depends with probability $1$ on a finite (but random) radius around each vertex.\nWe denote as $R_F$ the random variable that assigns minimal such radius to a given vertex, and call it the \\emph{coding radius} of $F$.\nWe denote the space of all $\\Pi$-colorings of $T_\\Delta$ as $X_\\Pi$.\nThis is a subspace of $\\Sigma^{H(T_\\Delta)}$ that is invariant under $\\operatorname{Aut}(T_\\Delta)$.\n\n\\begin{definition}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{fiid}$ ($\\mathsf{ffiid}$) if there is an fiid (ffiid) process $X$ that almost surely produces elements of $X_\\Pi$.\n\nEquivalently, we can define $\\mathsf{ffiid} = \\bigcup_f \\mathsf{OLOCAL}(f(\\varepsilon))$ where $f$ ranges over all functions. That is, $\\mathsf{ffiid}$ is the class of problems solvable by any uniform distributed algorithm. \n\n\\end{definition}\n\n\nIt is obvious that $\\mathsf{ffiid}\\subseteq \\mathsf{fiid}$.\nThe natural connection between descriptive combinatorics and random processes is formulated by the inclusion $\\mathsf{MEASURE}\\subseteq \\mathsf{fiid}$.\nWhile this inclusion is trivially satisfied, e.g., in the case of $\\Delta$-regular trees with proper edge $\\Delta$-coloring, in our situation we need a routine argument that we include for the sake of completeness in \\cref{app:ktulu}.\n\n\\begin{restatable}{lemma}{FIIDinMEASURE}\\label{thm:FIIDinMEASURE}\nLet $\\Pi$ be an LCL such that $\\Pi\\in \\mathsf{MEASURE}$.\nThen $\\Pi\\in \\mathsf{fiid}$.\n\\end{restatable}\n\n\n\n\n\\subsection{Specific Local Problems}\n\nHere we list some well-known local problems that were studied in all three considered areas. We describe the results known about these problems with respect to classes from \\cref{fig:big_picture_trees}. \n\n\\paragraph{Edge Grabbing}\nWe start by recalling the well-known problem of edge grabbing (a close variant of the problem is known as sinkless orientation\\cite{brandt_etal2016LLL}). \nIn this problem, every vertex should mark one of its half-edges (that is, grab an edge) in such a way that no edge can be marked by two vertices. \nIt is known that $\\Pi_{\\text{edgegrab}}  \\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_etal2016LLL} but $\\Pi_{\\text{edgegrab}} \\in \\mathsf{RLOCAL}(O(\\log\\log n))$.\n\nSimilarly, $\\Pi_{\\text{edgegrab}} \\not \\in \\mathsf{BOREL}$ by \\cite{DetMarks}, but $\\Pi_{\\text{edgegrab}}\\in \\mathsf{MEASURE}$ by \\cite{BrooksMeas}: to see the former, just note that if a $\\Delta$-regular tree admits proper $\\Delta$-colorings of both edges and vertices, every vertex can grab an edge of the same color as the color of the vertex. Thus, $\\Pi_{\\text{edgegrab}}\\in \\mathsf{BOREL}$ would yield that $\\Delta$-regular Borel forests with Borel proper edge-colorings admit a Borel proper $\\Delta$-coloring, contradicting \\cite{DetMarks}.\n\nThis completes the complexity characterization of $\\Pi_{\\text{edgegrab}}$ as well as the proper vertex $\\Delta$-coloring with respect to classes in \\cref{fig:big_picture_trees}. \n\n\n\n\n\n\\paragraph{Perfect Matching}\nAnother notorious LCL problem, whose position in \\cref{fig:big_picture_trees} is, however, not completely understood, is the perfect matching problem $\\Pi_{\\text{pm}}$.\nRecall that the perfect matching problem $\\Pi_{\\text{pm}}$ asks for a matching that covers all vertices of the input tree.\\footnote{In our formalism this means that around each vertex exactly one half-edge is picked.}\nIt is known that $\\Pi_{\\text{pm}} \\in \\mathsf{fiid}$ \\cite{lyons2011perfect}, and it is easy to see that $\\Pi_{\\text{pm}} \\not\\in \\mathsf{RLOCAL}(O(\\log\\log(n)))$ (we will prove a stronger result in \\cref{pr:AddingLineLLL}).\nMarks proved \\cite{DetMarks} that it is not in $\\mathsf{BOREL}$, even when the underlying tree admits a Borel bipartition.\nIt is not clear if $\\Pi_{\\text{pm}}$ is in $\\mathsf{ffiid}$, nor whether it is in $\\mathsf{MEASURE}$.\n\n\n\n\n\n\\paragraph{Graph Homomorphism}\nWe end our discussion with LCLs that correspond to graph homomorphisms (see also the discussion in Section \\ref{sec:marks}).\nThese are also known as edge constraint LCLs.\nLet $G$ be a finite graph.\nThen we define $\\Pi_G$ to be the LCL that asks for a homomorphism from the input tree to $G$, that is, vertices are labeled with vertices of $G$ in such a way that edge relations are preserved.\nThere are not many positive results except for the class $\\mathsf{BAIRE}$.\nIt follows from the result of Bernshteyn \\cite{Bernshteyn_work_in_progress} (see \\cref{sec:baire}) that $\\Pi_G\\in \\mathsf{BAIRE}$ if and only if $G$ is not bipartite.\nThe main negative results can be summarized as follows.\nAn easy consequence of the result of Marks \\cite{DetMarks} is that if $\\chi(G)\\le \\Delta$, then $\\Pi_G\\not \\in \\mathsf{BOREL}$.\nIn this paper, we describe examples of graphs of chromatic number up to $2\\Delta-2$ such that the corresponding homomorphism problem is not in $\\mathsf{BOREL}$, see \\cref{sec:marks}.\nThe theory of \\emph{entropy inequalities} see \\cite{backhausz} implies that if $G$ is a cycle on more than $9$ vertices, then $\\Pi_G\\not\\in \\mathsf{fiid}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalization of Marks' technique}\n\\label{sec:marks}\n\nIn this section, we first develop a new way of proving lower bounds in the $\\mathsf{LOCAL}$ model based on a generalization of a technique of Marks \\cite{DetMarks}. Then, we use ideas arising in the finitary setting---connected to the adaptation of Marks' technique---to obtain new results back in the Borel context.\nFor an introduction to Marks' technique and a high-level discussion about the challenges in adapting the technique to the standard distributed setting as well as our solution via the new notion of an ID graph, we refer the reader to \\cref{sec:intromarks}.\n\n\\begin{comment}\nIn \\cite{DetMarks}, Marks showed that there exist $\\Delta$-regular Borel forests that do not admit a Borel proper $\\Delta$-coloring. As it was indicated above, this is a striking difference between $\\mathsf{MEASURE}$ or $\\mathsf{BAIRE}$ and $\\mathsf{BOREL}$, as such colorings exist in the former classes, see \\cite{BrooksMeas}. Since prior to Marks' breakthrough (essentially) any lower bound on Borel chromatic numbers utilized measure theoretic or Baire category arguments, his approach was (and had to be) based on a new technique: \\emph{determinacy of infinite games}. \n\nFix a payoff set $A \\subseteq \\mathbb{R}$. Consider the infinite game in which two players alternatingly determine digits of a real number. The first player wins iff the resulting real is in $A$. A rather counter-intuitive fact of set theory is that there exists a set $A$ for which none of the players has a winning strategy, i.e., the game is not determined. However, a deep theorem of Martin \\cite{martin} says that if $A$ is Borel, then one of the players does have such a strategy. \nMarks has constructed $\\Delta$-regular Borel forests and games with Borel payoff sets, for which the existence of a winning strategy for either of the players contradicts the existence of a Borel proper $\\Delta$-coloring. \n\nIn what follows, we generalize and adapt his games to the finitary setting, thus obtaining new lower bounds for LOCAL algorithms; then we use ideas inspired by the LOCAL technique of ID graphs to prove a new theorem in the Borel context. \n\nNote that Marks considered Borel versions of certain locally checkable problems (e.g., coloring, edge grabbing, perfect matching). As discussed above, for these problems, lower bounds were independently obtained via the round elimination technique in the LOCAL context, see e.g., \\cite{brandt19automatic}\\todo{cite more stuff??-Z}. Remarkably, the core ideas behind Brandt's and Marks' argument exhibit a surprising degree of similarity. \n\n\\todo{add the story about going from borel to local, and then back from local to borel}\n\nHowever, our extension of Marks' technique yields lower bounds for fairly natural problems that were resistant to round elimination.\nMoreover, as with our approach the lower bound is obtained by checking whether the considered LCL satisfies a certain combinatorial property called \\emph{playability condition} (see Definition~\\ref{def:Playability}), we can use it to obtain lower bounds for problems that have complicated descriptions---something that the round elimination technique in its current form is ill-suited for. \\todo{Maybe this sentence should rather be somewhere in the intro?}\n\n\n\n\\end{comment}\n\n\n\n\n\n\n\nThe setting we consider in this section is $\\Delta$-regular trees that come with a proper $\\Delta$-edge coloring with colors from $[\\Delta]$.\nAll lower bounds discussed already hold under these restrictions (and therefore also in any more general setting).\n\nRecall that an LCL $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is given by specifying a list of allowed vertex configurations and a list of allowed edge configurations (see Definition \\ref{def:LCL_trees}).\nTo make our lower bounds as widely applicable as possible, we replace the latter list by a separate list for each of the $\\Delta$ edge colors; in other words, we consider LCLs where the correctness of the output is allowed to depend on the input that is provided by the edge colors.\nHence, formally, in this section, an LCL is more generally defined: it is a tuple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ where $\\Sigma$ and $\\mathcal{V}$ are as before, while $\\mathcal{E} = (\\mathcal{E}_{\\alpha})_{\\alpha \\in [\\Delta]}$ is now a $\\Delta$-tuple of sets $\\mathcal{E}_{\\alpha}$ consisting of cardinality-$2$ multisets.\nSimilarly as before, a half-edge labeling (see Definition \\ref{def:descriptive} for the Borel context) with labels from $\\Sigma$ is a correct solution for $\\Pi$ if, for each vertex $v$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$, the multiset of labels assigned to the half-edges belonging to $e$ is contained in $\\mathcal{E}_{\\alpha}$, where $\\alpha$ is the color of the edge $e$.\n\nThe idea of our approach is to identify a condition that an LCL $\\Pi$ necessarily has to satisfy if it is solvable in $O(\\log^* n)$ rounds in the $\\mathsf{LOCAL}$ model.\nShowing that $\\Pi$ does not satisfy this condition then immediately implies an $\\Omega(\\log n)$ deterministic and $\\Omega(\\log \\log n)$ randomized lower bound, by Theorem~\\ref{thm:basicLOCAL}.\n\nIn order to define our condition we need to introduce the notion of a configuration graph: a \\emph{configuration graph} is a $\\Delta$-tuple $\\mathbb{P}=(\\mathbb{P}_\\alpha)_{\\alpha\\in \\Delta}$ of graphs, where the vertex set of each of the graphs $\\mathbb{P}_\\alpha$ is the set of subsets of $\\Sigma$, and there is an edge in $\\mathbb{P}_\\alpha$ connecting two vertices $S,T$ if and only if there are $\\mathtt{a}\\in S$ and $\\mathtt{b}\\in T$ such that $\\{\\mathtt{a},\\mathtt{b}\\}\\in {\\mathcal{E}}_\\alpha$, note that loops are allowed.\n(Naturally, we will consider any two vertices of different $\\mathbb{P}_\\alpha$ to be distinct, even if they correspond to the same subset of $\\Sigma$.)\n\nNow we are set to define the aforementioned condition. The intuition behind the playability condition is the following: assume that there exists a local algorithm $\\mathcal{A}$ that solves $\\Pi$ using the $t$ neighbourhood of a given vertex. We are going to define a family of two player games. The game will depend on some $S \\subseteq \\Sigma$. $\\operatorname{Alice}$ and $\\operatorname{Bob}$  will assign labels (or IDs) to the vertices in the $t$-neighbourhood (in some way specified later on, depending on $\\alpha \\in [\\Delta]$). When the assignment is complete, we evaluate $\\mathcal{A}$ on the root of the obtained labelled graph, this way obtaining an element $s$ of $\\Sigma$, and decide who is the winner based on $s \n\\in S$ or not. Naturally, it depends on $S$ and $\\alpha$, which player has a winning strategy. This gives rise to a two coloring of vertices of $\\mathbb{P}_\\alpha$  by colors $\\operatorname{Alice}$ and $\\operatorname{Bob}$. The failure of the playability condition will guarantee that using a strategy stealing argument one can derive a contradiction.\n\n\n\n\n\\begin{definition}[Playability condition]\\label{def:Playability}\nWe say that an LCL $\\Pi$ is \\emph{playable} if for every $\\alpha\\in [\\Delta]$ there is a coloring $\\Lambda_{\\alpha}$ of the vertices of $\\mathbb{P}_{\\alpha}$ with two colors $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$ such that the following conditions are satisfied:\n\\begin{enumerate}\n    \\item [(A)] For any tuple $(S_{\\alpha})_{\\alpha \\in [\\Delta]} \\in V(\\mathbb{P}_1) \\times \\dots \\times V(\\mathbb{P}_{\\Delta})$ satisfying $\\Lambda_\\alpha(S_\\alpha)=\\operatorname{Alice}$ for each $\\alpha \\in [\\Delta]$, there exist $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ such that $\\{\\mathtt{a}_\\alpha\\}_{\\alpha\\in \\Delta}\\in {\\mathcal{V}}$, and\n    \\item [(B)] for any $\\alpha \\in [\\Delta]$, and any tuple $(S,T) \\in V(\\mathbb{P}_{\\alpha}) \\times V(\\mathbb{P}_{\\alpha})$ satisfying $\\Lambda_\\alpha(S)=\\Lambda_\\alpha(T)=\\operatorname{Bob}$, we have that $(S,T)$ is an edge of $\\mathbb{P}_{\\alpha}$.\n\\end{enumerate}\n\n\\end{definition}\n\nOur aim in this section is to show the following general results.\n\n\\begin{theorem}\\label{th:MainLOCAL}\nLet $\\Pi$ be an LCL that is not playable.\nThen $\\Pi$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nUsing ideas from the proof of this result, we can formulate an analogous theorem in the Borel context.\nLet us mention that while \\cref{th:MainLOCAL} is a consequence of \\cref{thm:MainBorel} by \\cite{Bernshteyn2021LLL}, we prefer to state the theorems separately.\nThis is because the proof of the Borel version of the theorem uses heavily complex results such as the Borel determinacy theorem, theory of local-global limits and some set-theoretical considerations about measurable sets.\nThis is in a stark contrast with the proof of \\cref{th:MainLOCAL} that uses `merely' the existence of large girth graphs and determinacy of finite games.\nHowever, the ideas surrounding these concepts are the same in both cases.\n\n\\begin{theorem}\\label{thm:MainBorel}\nLet $\\Pi$ be an LCL that is not playable\nThen $\\Pi$ is not in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\nThe main application of these abstract results is to find lower bounds for \\emph{(graph) homomorphism LCLs} aka \\emph{edge constraint LCLs}.\nWe already defined this class in \\cref{subsec:local_problems} but we recall the definition in the formalism of this section to avoid any confusion.\nLet $G$ be a finite graph.\nThen $\\Pi_G=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is defined by letting\n\\begin{enumerate}\n\\item \\label{p:vertex} ${\\mathcal{V}}=\\{({\\mathtt{a}},\\dots,{\\mathtt{a}}):\\mathtt{a}\\in \\Sigma\\}$,\n\\item $\\Sigma=V(G)$,\n\\item $\\forall \\alpha \\in [\\Delta] \\ (\\mathcal{E}_\\alpha=E(G)).$\n\\end{enumerate}\nAn LCL for which \\eqref{p:vertex} holds is called a \\emph{vertex-LCL}. There is a one-to-one correspondence between vertex-LCLs for which $\\forall \n\\alpha,\\beta \\ (\\mathcal{E}_\\alpha=\\mathcal{E}_\\beta)$ and LCLs of the form $\\Pi_G$. (Indeed, to vertex-LCLs with this property one can associate a graph $G$ whose vertices are the labels in $\\Sigma$ and where two vertices $\\ell, \\ell'$ are connected by an edge if and only if $\\{ \\ell, \\ell' \\} \\in \\mathcal{E}$.)\nNote that if $\\Pi$ is a vertex-LCL, then condition (A) in Definition~\\ref{def:Playability} is equivalent to the statement that no tuple $\\{S_\\alpha\\}_{\\alpha\\in \\Delta}$ that satisfies the assumption of (A) can cover $\\Sigma$, i.e., that $\\Sigma \\nsubseteq S_1 \\cup \\dots \\cup S_{\\Delta}$ for such a tuple.\nMoreover, if $\\Pi_G$ is a homomorphism LCL, then $\\mathbb{P}_\\alpha=\\mathbb{P}_{\\beta}$ for every $\\alpha,\\beta\\in \\Delta$.\n\nThe simplest example of a graph homomorphism LCL is given by setting $G$ to be the clique on $k$ vertices; the associated LCL $\\Pi_G$ is simply the problem of finding a proper $k$-coloring of the vertices of the input graph.\nNow we can easily derive Marks' original result from our general theorem.\n\n\\begin{corollary}[Marks \\cite{DetMarks}]\\label{cor:detmarks}\nLet $G$ be a finite graph that has chromatic number at most $\\Delta$. Then $\\Pi_G$ is not playable. In particular, there is a $\\Delta$-regular Borel forest that have Borel chromatic number $\\Delta+1$. \n\\end{corollary}\n\\begin{proof}\nLet $A_1,\\dots, A_\\Delta$ be some independent sets that cover $G$, and $\\Lambda_1, \\dots, \\Lambda_{\\Delta}$ arbitrary colorings of the vertices of $\\mathbb{P}_1, \\dots, \\mathbb{P}_{\\Delta}$, respectively, with colors from $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$.\nIt follows that $\\Lambda_\\alpha(A_\\alpha)=\\operatorname{Alice}$ for every $\\alpha\\in \\Delta$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated with $S=T=A_\\alpha$.\nBut then condition (A) does not hold.\n\\end{proof}\n\nAs our main application we describe graphs with chromatic number larger than $\\Delta$ such that $\\Pi_G$ is not playable.\nThis rules out the hope that the complexity of $\\Pi_G$ is connected with chromatic number being larger than $\\Delta$.\nIn Section \\ref{subsec:homlcls} we show the following.\nNote that the case $k=\\Delta$ is \\cref{cor:detmarks}.\n\n\\begin{theorem}\n\\label{th:homomorphism}\nLet $\\Delta > 2$ and $\\Delta < k \\leq 2\\Delta-2$. There exists a graph $G_k$ with $\\chi(G_k)= k$, such that $\\Pi_{G_k}$ is not playable and $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log n ))$. In particular, $\\Pi_{G_k}\\not\\in \\mathsf{BOREL}$ and $\\Pi_{G_k}\\not\\in\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\nInterestingly, recent results connected to counterexamples to Hedetniemi's conjecture yield the same statement asymptotically, as $\\Delta \\to \\infty$ (see Remark \\ref{r:hedet}).\n\n\\begin{remark}\nIt can be shown that for $\\Delta=3$ both the Chv\\' atal and Gr\\\" otsch graphs are suitable examples for $k=4$.\n\\end{remark}\n\n\\begin{remark}\nAs another application of his technique Marks showed in \\cite{DetMarks} that there is a Borel $\\Delta$-regular forest that does not admit Borel perfect matching.\nThis holds even when we assume that the forest is Borel bipartite, i.e., it has Borel chromatic number $2$.\nIn order to show this result Marks works with free joins of colored hyperedges, that is, Cayley graphs of free products of cyclic groups.\nOne should think of two types of triangles ($3$-hyperedges) that are joined in such a way that every vertex is contained in both types and there are no cycles.\nWe remark that the playability condition can be formulated in this setting.\nSimilarly, one can derive a version of \\cref{thm:MainBorel}.\nHowever, we do not have any application of this generalization.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\subsection{Applications of playability to homomorphism LCLs}\\label{subsec:homlcls}\n\nIn this section we find the graph examples from \\cref{th:homomorphism}.\nFirst we introduce a condition $\\Delta$-(*) that is a weaker condition than having chromatic number at most $\\Delta$, but still implies that the homomorphism LCL is not playable. Then, we will show that--similarly to the way the complete graph on $\\Delta$-many vertices, $K_\\Delta$, is maximal among graphs of chromatic number $\\leq \\Delta$--there exists a maximal graph (under homorphisms) with property $\\Delta$-(*). Recall that we assume $\\Delta >2$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.6\\textwidth]{fig\/h3.pdf}\n    \\caption{The maximal graph with the property $\\Delta$-(*) for $\\Delta=3$.\n    \\label{fig:h3}\n\\end{figure}\n\n\n\n\n\\begin{definition}[Property $\\Delta$-(*)]\nLet $\\Delta>2$ and $G=(V,E)$ be a finite graph.\nWe say that $G$ satisfies property $\\Delta$-(*) if there are sets $S_0,S_1\\subseteq V$ such that $G$ restricted to $V \\setminus S_i$ has chromatic number at most $(\\Delta-1)$ for $i\\in \\{0,1\\}$, and there is no edge between $S_0$ and $S_1$.\n\\end{definition}\n\nNote that $\\chi(G) \\leq \\Delta$ implies $\\Delta$-(*): indeed, if $A_1,\\dots,A_\\Delta$ are independent sets that cover $V(G)$, we can set $S_0=S_1=A_1$.\n\nOn the other hand, we claim that if $G$ satisfies $\\Delta$-(*) then $\\chi(G)\\le 2\\Delta-2$. In order to see this, take $S_0,S_1 \\subseteq V(G)$ witnessing $\\Delta$-(*). Then, as there is no edge between $S_0$ and $S_1$, so in particular, between $S_0 \\setminus S_1$ and $S_0 \\cap S_1$, it follows that the chromatic number of $G$'s restriction to $S_0$ is $\\leq \\Delta-1$. But then we can construct proper $\\Delta-1$-colorings of $S_0$ and $V(G) \\setminus S_0$, which shows our claim.\n\n\n\\begin{proposition}\\label{pr:CONdition*}\nLet $G$ be a graph satisfying $\\Delta$-(*).\nThen $\\Pi_G$ is not playable.\n\\end{proposition}\n\\begin{proof}\nFix $S_0,S_1$ as in the definition of $\\Delta$-(*) and assume for a contradiction that colorings $\\Lambda_1, \\dots, \\Lambda_\\Delta$ as described in Definition~\\ref{def:Playability} exist.\nBy Property $\\Delta$-(*), there exist independent sets $A_1, \\dots, A_{\\Delta-1}$ such that $S_0$ together with the $A_i$ covers $G$, i.e., such that $S_0 \\cup A_1 \\cup \\dots \\cup A_{\\Delta-1} = V(G)$.\nFor each $\\alpha \\in [\\Delta-1]$, we must have $\\Lambda_\\alpha(A_{\\alpha})=\\operatorname{Alice}$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated.\nConsequently $\\Lambda_{\\Delta}(S_0)=\\operatorname{Bob}$, otherwise condition (A) is violated.\nSimilarly $\\Lambda_\\Delta(S_1)=\\operatorname{Bob}$.\nThis shows that $\\Lambda_\\Delta$ does not satisfy condition (B) with $S = S_0$ and $T = S_1$.\n\\end{proof}\n\nNext we describe maximal examples of graphs that satisfy the condition $\\Delta$-(*). That is, we define a graph $H_\\Delta$ that satisfies $\\Delta$-(*), its chromatic number is $2\\Delta-2$ and every other graph that satisfies $\\Delta$-(*) admits a homomorphism in $H_\\Delta$.\nThe graph $H_{3}$ is depicted in \\cref{fig:h3}.\n\nRecall that the (categorical) product $G \\times H$ of graphs $G,H$ is the graph on $V(G) \\times V(H)$, such that $((g,h),(g',h')) \\in E(G \\times H)$ iff $(g,g') \\in E(G)$ and $(h,h') \\in E(H)$.\n  \n  Write $P$ for the product $K_{\\Delta-1}\\times K_{\\Delta-1}$. Let $V_0$ and $V_1$ be vertex disjoint copies of $K_{\\Delta-1}$.\nWe think of vertices in $V_i$ and $P$ as having labels from $[\\Delta-1]$ and $[\\Delta-1] \\times [\\Delta-1]$, respectively.\nThe graph $H_\\Delta$ is the disjoint union of $V_0$, $V_1$, $P$ and an extra vertex $\\dagger$ that is connected by an edge to every vertex in $P$, and additionally, if $v$ is a vertex in $V_0$ with label $i\\in [\\Delta-1]$, then we connect it by an edge with $(i',j)\\in P$ for every $i'\\not=i$ and $j\\in [\\Delta-1]$, and if $v$ is a vertex in $V_1$ with label $j\\in \\Delta-1$, then we connect it by an edge with $(i,j')\\in P$ for every $j'\\not=j$ and $i \\in [\\Delta-1]$.\n\n\n\\begin{proposition}\\label{pr:MinimalExample}\n\\begin{enumerate}\n    \\item \\label{prop:hdelta}$H_\\Delta$ satisfies $\\Delta$-(*).\n    \\item \\label{prop:hdeltachrom} $\\chi(H_\\Delta)=2\\Delta-2$.\n    \\item \\label{prop:hdeltamax} A graph $G$ satisfies $\\Delta$-(*) if and only if it admits a homomorphism to $H_\\Delta$.\n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\n\\eqref{prop:hdelta} Set $S_0=V(V_0)\\cup \\{\\dagger\\}$ and $S_1=V(V_1)\\cup \\{\\dagger\\}$.\nBy the definition there are no edges between $S_0$ and $S_1$.\nConsider now, e.g., $V(H_\\Delta)\\setminus S_0$.\nLet $A_j$ consist of all elements in $P$ that have second coordinate equal to $j$ together with the vertex in $V_1$ that has the label $j$.\nBy the definition, the set $A_i$ is independent and $\\bigcup_{i\\in [\\Delta-1]}A_i$ covers $H_\\Delta\\setminus S_0$, and similarly for $S_1$.\n\n\\eqref{prop:hdeltachrom} By \\eqref{prop:hdelta} and the claim after the definition of $\\Delta$-(*), it is enough to show that $\\chi(H_\\Delta)\\geq 2\\Delta-2$. Towards a contradiction, assume that $c$ is a proper coloring of $H_\\Delta$ with $<2\\Delta-2$-many colors. Note the vertex $\\dagger$ guarantees that  $|c(V(P))|\\leq 2\\Delta-4$, and also $\\Delta-1 \\leq |c(V(P))|$.\n\nFirst we claim that there are no indices $i,j\\in [\\Delta-1]$ (even with $i=j$) such that $c(i,r)\\not=c(i,s)$ and $c(r,j)\\not=c(s,j)$ for every $s\\not=r$: indeed, otherwise, by the definition of $P$ we had $c(i,r)\\not=c(s,j)$ for every $r,s$ unless $(i,r)=(s,j)$, which would the upper bound on the size of $c(V(P))$.\n\nTherefore, without loss of generality, we may assume that for every $i\\in [\\Delta-1]$ there is a color $\\alpha_i$ and two indices $j_i \\neq j'_i$ such that $c(i,j_i)=c(i,j'_i)=\\alpha_i$. It follows form the definition of $P$ and $j_i \\neq j'_i$ that\n $\\alpha_i\\not=\\alpha_{i'}$ whenever $i\\not= i'$. \n \n Moreover, note that any vertex in $V_1$ is connected to at least one of the vertices $(i,j_i)$ and $(i,j'_i)$, hence none of the colors $\\{\\alpha_i\\}_{i \\in [\\Delta-1]}$ can appear on $V_1$. Consequently, since $V_1$ is isomorphic to $K_{\\Delta-1}$ we need to use at least $\\Delta-1$ additional colors, a contradiction.\n\n\\eqref{prop:hdeltamax} First note that if $G$ admits a homomorphism into $H_\\Delta$, then the pullbacks of the sets witnessing  $\\Delta$-(*) will witness that $G$ has $\\Delta$-(*).\n\nConversely, let $G$ be a graph that satisfies $\\Delta$-(*).\nFix the corresponding sets $S_0,S_1$ together with $(\\Delta-1)$-colorings $c_0,c_1$ of their complements.\nWe construct a homomorphism $\\Theta$ from $G$ to $H_\\Delta$. Let\n\\[\\Theta(v)=\n\\begin{cases} \\dagger &\\mbox{if } v \\in S_0 \\cap S_1, \\\\\nc_0(v) & \\mbox{if } v \\in S_1 \\setminus S_0, \\\\\nc_1(v) & \\mbox{if } v \\in S_0 \\setminus S_1,\n\\\\\n(c_0(v),c_1(v)) & \\mbox{if } v \\not \\in S_0 \\cup S_1.\n\\end{cases} \\]\n\n\nObserve that $S=S_0\\cap S_1$ is an independent set such that there is no edge between $S$ and $S_0 \\cup S_1$.\nUsing this observation, one easily checks case-by-case that $\\Theta$ is indeed a homomorphism.\n\n \n \n   \n   \n   \n   \n\\end{proof}\n\nNow, combining what we have so far, we can easily prove Theorem \\ref{th:homomorphism}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:homomorphism}]\nIt follows that $\\Pi_{H_\\Delta}$ is not playable from \\cref{pr:MinimalExample}, \\cref{pr:CONdition*}. It is easy to see that if for a graph $G$ the LCL $\\Pi_G$ is not playable then $\\Pi_{G'}$ is not playable for every subgraph $G'$ of $G$. Since erasing a vertex decreases the chromatic number with at most one, for each $k\\leq 2\\Delta-2$ there is a subgraph $G_k$ of $H_\\Delta$ with $\\chi(G_k)=k$, such that $\\Pi_{G_k}$ is not playable.\n\nIt follows from Theorems \\ref{th:MainLOCAL} and \\ref{thm:MainBorel} that there is a $\\Delta$-regular Borel forest that admits no Borel homomorphism to any graph of $G_k$ and that $\\Pi_{G_k} \\not \\in \\mathsf{LOCAL}(O(\\log^*(n)))$.\n\nFinally, note that if $k \\geq \\Delta$ then $G_k$ can be chosen so that it contains $K_\\Delta$, yielding $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log(n))$.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:hedet}Recall that Hedetniemi's conjecture is the statement that if $G,H$ are finite graphs then $\\chi(G \\times H)=\\min\\{\\chi(G),\\chi(H)\\}$. This conjecture has been recently disproven by Shitov \\cite{shitov}, and strong counterexamples have been constructed later (see, \\cite{tardif2019note,zhu2021note}). We claim that these imply for $\\varepsilon>0$ the existence of finite graphs $H$ with $\\chi(H) \\geq (2-\\varepsilon)\\Delta$ to which $\\Delta$-regular Borel forests cannot have a homomorphism in BOREL, for every large enough $\\Delta$. Indeed, if a $\\Delta$-regular Borel forest admitted a Borel homomorphism to each finite graph of chromatic number at least $(2-\\varepsilon)\\Delta$, it would have such a homomorphism to their product as well. Thus, we would obtain that the chromatic number of the product of any graphs of chromatic number $(2-\\varepsilon)\\Delta$ is at least $\\Delta+1$. This contradicts Zhu's result \\cite{zhu2021note}, which states that the chromatic number of the product of graphs with chromatic number $n$ can drop to $\\approx \\frac{n}{2}$.\n\n\\end{remark}\n\n\\begin{remark}\nA natural attempt to construct graphs with large girth and not playable homomorphisms problem would be to consider random $d$-regular graphs of size $n$ for a large enough $n$. However, it is not hard to see that setting $\\Lambda(A)=\\operatorname{Alice}$ if and only if $|A|<\\frac{n}{d}$ shows that this approach cannot work.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{th:MainLOCAL}}\\label{sec:marksproof}\n\nIn this section we prove Theorem \\ref{th:MainLOCAL}, by applying Marks' game technique in the LOCAL setting. In order to define our games, we will need certain auxiliary graphs, the so-called \\emph{ID graphs}. \nThe purpose of these graphs is to define a ``playground'' for the games that we consider.\nNamely, vertices in the game are labeled by vertices from the ID graph in such a way that the at the end we obtain a homomorphism from our underlying graph to the ID graph. \n\n\n\n\n\\begin{definition}\nA pair $\\idgraphcolored{n}{t}{r}=(\\idgraph{n}{t}{r},c)$ is called an \\emph{ID graph}, if\n\\begin{enumerate}\n    \\item \\label{p:idgirth} $\\idgraph{n}{t}{r}$ is graph with girth at least $2t+2$,\n    \\item \\label{p:idsize} $|V(\\idgraph{n}{t}{r})|\\le n$,\n    \\item \\label{p:idlabels} $c$ is a $\\Delta$-edge-coloring of $\\idgraph{n}{t}{r}$, such that every vertex is adjacent to at least one edge of each color,\n    \\item \\label{p:idratio} for each $\\alpha \\in [\\Delta]$ the ratio of a maximal independent set of $\\idgraph{n}{t}{r}^{\\alpha}=(V(\\idgraph{n}{t}{r}), E(\\idgraph{n}{t}{r}) \\cap c^{-1}(\\alpha))$ is at most $r$ (i.e., $\\idgraph{n}{t}{r}^\\alpha$ is the graph formed by $\\alpha$-colored edges).\n\\end{enumerate}\n\\end{definition}\n\nBefore we define the game we show that ID graphs exist.\n\n\\begin{proposition}\\label{pr:IDgraph}\nLet $t_n\\in o(\\log(n))$, $r>0$, $\\Delta\\geq 2$.\nThen there is an ID graph $\\idgraphcolored{n}{t_n}{r}$ for every $n\\in \\mathbb{N}$ sufficiently large.\n\\end{proposition}\n\\begin{proof}\nWe use the \\emph{configuration model} for regular random graphs, see \\cite{wormald1999models}.\nThis model is defined as follows.\nLet $n$ be even.\nThen a $d$-regular random sample on $n$-vertices is just a union of $d$-many independent uniform random perfect matchings.\nNote that in this model we allow parallel edges.\n\nIt was proved by Bollob\\'as \\cite{bollobas} that the independence ratio of a random $d$-regular graph is at most $\\frac{2\\log(d)}{d}$ a.a.s.\nMoreover, this quantity is concentrated by an easy application of McDiarmid's result \\cite{mcdiarmid2002concentration}, i.e., \n\\begin{equation}\n    \\P\\left(\\left|X-\\textrm{\\textbf{E}}(X)\\right|\\ge \\sqrt{n}\\right)<2\\exp\\left(-\\frac{n}{d}\\right),\n\\end{equation}\nwhere $X$ is the random variable that counts the size of a maximal independent set.\nTherefore for fixed $n$ large enough we have that the independence ratio of a random sample is at most $\\frac{3\\log(d)}{d}$ with probability at least $\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)$.\n\n\nPick a $d$ large enough such that $\\frac{3\\log(d)}{d}<r$.\nNow for an $n$ large enough take $\\Delta$-many independent samples of random $d$-regular graphs according to the configuration model.\nNote that this is a random sample from the configuration model for $\\Delta d$-regular graphs.\nWe define $c$ to be equal to $\\alpha\\in [\\Delta]$ on edges of the $\\alpha$-th sample.\nThen condition (4) is satisfied with probability at least\n$$\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta.$$\n\nIt remains to show that the girth condition is satisfied.\nRecall that we assume $t_n\\in o(\\log n)$.\nThen we have $(\\Delta d-1)^{2t_n-1}\\in o(n)$.\nUsing \\cite[Corollary~1]{mckayshortcycles} we have that the probability of having girth at least $t_n$ is\n\\begin{equation}\n\\begin{split}\n    \\exp\\left(-\\sum_{a=3}^{t_n}\\frac{(\\Delta d-1)^a}{2a}+o(1)\\right)\\ge & \\exp\\left(-(\\Delta d)^{t_n}\\right) \\\\\n    \\ge & \\exp(-o(n)) \\\\\n    > & 1-\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta\n\\end{split}\n\\end{equation}\nas $n\\to \\infty$.\nThis shows that there exists such a graph $\\idgraphcolored{t_n}{n}{r}$ with non-zero probability.\n\\end{proof}\n\n\n\n\nNext, we define the games. As mentioned before, the games are going to depend on the following parameters: an algorithm $\\mathcal{A}_n$ of local complexity $t\\in o(\\log(n))$, an ID graph $\\idgraphcolored{n}{t}{r}$, $\\alpha\\in \\Delta$, $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $S \\subseteq \\Sigma$.\n(We will view $\\mathcal{A}_n$, $\\idgraphcolored{n}{t}{r}$ as fixed, and the rest of the parameters as variables).\n\nThe game\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$$\nis defined as follows: two players, $\\operatorname{Alice}$ and $\\operatorname{Bob}$ assign labels to the vertices of a rooted $\\Delta$-regular tree of diameter $t$. The labels are vertices of $\\idgraph{n}{t}{r}$ and the root is labeled $\\sigma$. In the $k$-th round, where $0<k\\le t$, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphcolored{n}{t}{r}$. (For example, if it is $\\operatorname{Alice}$'s turn to label a neighbor of some vertex $v$, that has been assigned a label $\\rho\\in V(\\idgraph{n}{t}{r})$ in the previous round, along an edge that has color $\\beta$, then the allowed labels are only those that span a $\\beta$ edge with $\\rho$ in $\\idgraph{n}{t}{r}$).\n\nBy property \\eqref{p:idsize} of the ID graph, we can fix an injective map from $V(\\idgraph{n}{t}{r})$ to $[n]$.\nNow, we say that $\\operatorname{Alice}$ wins an instance of the game iff $\\mathcal{A}_n$ applied to the produced labeling of the rooted tree does {\\bf not} produce an element of $S$ on the half edge that starts in the root and has edge color $\\alpha$.\nNote that this is well defined thanks to our assumption on the girth of $\\idgraph{n}{t}{r}$.\nLet us define $\\Lambda^{\\sigma}_{\\alpha}(S)$ to be $\\operatorname{Alice}$ or $\\operatorname{Bob}$ depending on who has a winning strategy in the game $$\\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma,S].$$\nSince the game is finite, we have that\n$$\\Lambda^\\sigma_\\alpha:\\mathbb{P}_{\\alpha}\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$$\nis well-defined for all $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $\\alpha\\in [\\Delta]$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[]{fig\/gamepick-figure0.pdf}\n    \\caption{The game $\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$}\n    \\label{fig:game}\n\\end{figure}\n\n\\begin{proposition}\\label{pr:(A)satisfied}\nLet $\\mathcal{A}_n$ be an algorithm of local complexity $t\\in o(\\log(n))$ that solves $\\Pi$ and $\\sigma\\in V(\\idgraph{n}{t}{r})$.\nThen $(\\Lambda^{\\sigma}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^\\sigma_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$.\nLetting these strategies play against each other in the obvious way, produces a labeling of the tree.\nSince $\\mathcal{A}_n$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at the root and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\nWe are ready to prove Theorem~\\ref{th:MainLOCAL}. \n\\begin{proof}[Proof of Theorem \\ref{th:MainLOCAL}]\nLet $(\\mathcal{A}_n)_{n\\in \\mathbb{N}}$ be a sequence of algorithms of local complexity $t_n\\in o(\\log(n))$ that solve $\\Pi$ and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\nBy \\cref{pr:IDgraph} there exists an ID graph $\\mathbf{H}_{n,t,r}$. Thus, the games above are well-defined and we can construct the functions $(\\Lambda^\\sigma_\\alpha)_{\\alpha\\in \\Delta}$. Since there are only $N$ possibilities for such a sequence, there exists a set $X \\subseteq V(\\idgraph{n}{t}{r})$ of relative size greater than $r$, such that for all $\\sigma \\in X$ the sequence of functions $(\\Lambda^\\sigma_\\alpha)_{\\alpha\\in \\Delta}$ is the same.\n\nSince $\\Pi$ is not playable, we can find an edge color $\\alpha\\in \\Delta$ and sets $S,T\\in \\mathbb{P}_\\alpha$ such that $\\Lambda^\\sigma_{\\alpha}$ does not satisfy (B) from \\cref{def:Playability} with $S,T$ for every $\\sigma\\in X$.\nNote that this is because (A) is always satisfied by \\cref{pr:(A)satisfied}.\n\nBy property \\eqref{p:idratio} of the ID graph, there exist $\\sigma_0,\\sigma_1\\in X$ that span an $\\alpha$ edge in $\\idgraph{n}{t}{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma_0,S], \\ \\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma_1,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $\\sigma_0,\\sigma_1$.\nThis produces a labeling of the vertices that have distance at most $t$ from either of the endpoints of the edge (intuitively, the tree ``rooted'' at this edge of ``diameter'' $t+\\frac{1}{2}$).\nNow applying $\\mathcal{A}_n$ on the vertex with label $\\sigma_0$ produces $\\mathtt{a}_0\\in S$ on the the half edge that starts at this vertex and has color $\\alpha$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}_n$ does not solve $\\Pi$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:MainBorel}}\n\\label{subsec:local_to_borel}\n\nWe show how to use an infinite analogue of the ID graph to prove our main result in the Borel context, \\cref{thm:MainBorel}.\nFinding such a graph\/graphing is in fact much easier in this context.\n\n\\begin{definition}\nLet $r>0$. $\\idgraphingcolored{r}=(\\idgraphing{r},c)$ is an \\emph{ID graphing}, if\n\n\\begin{enumerate}\n    \\item $\\idgraphing{r}$ is an acyclic locally finite Borel graphing on a standard probability measure space $(X,\\mu)$,\n    \\item \\label{p:idgraphinglabels} $c$ is a Borel $\\Delta$-edge-coloring of $\\idgraphing{r}$, such that every vertex is adjacent to at least one edge of each color,\n    \\item \\label{p:idgraphingratio} for each $\\alpha \\in [\\Delta]$ the $\\mu$-measure of a maximal independent set of $\\idgraphing{r}^{\\alpha}=(V(\\idgraphing{r}), E(\\idgraphing{r}) \\cap c^{-1}(\\alpha))$ is at most $r$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\n\\label{p:idgraphing}\nFor each $r>0$ there exists an ID graphing $\\idgraphingcolored{r}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\idgraphing{r}$ be a \\emph{local-global limit} of the random graphs constructed in \\cref{pr:IDgraph} (see, e.g., \\cite{hatamilovaszszegedy} for the basic results about local-global convergence).\nIt is not hard to check that this limit satisfies the required properties.\n\\end{proof}\n\n\nNow we are ready to prove the theorem. The proof will closely follow the argument given in the proof of the LOCAL version, i.e., Theorem \\ref{th:MainLOCAL}, but can be understood without reading the latter.\n\n\\begin{proof}[Proof of \\cref{thm:MainBorel}]\nLet $\\Pi$ be an LCL that is not playable, and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\n\nWe define a Borel acyclic $\\Delta$-regular graph $\\mathcal{G}$, with edges properly colored by $\\Delta$, that does not admit a Borel solution of $\\Pi$.\nVertices of $\\mathcal{G}$ are pairs $(x,A)$, where $x\\in X$ is a vertex of $\\idgraphing{r}$ and $A$ is a countable subgraph of $\\idgraphing{r}$ that is a $\\Delta$-regular tree that contains $x$ and the edge coloring of $\\idgraphingcolored{r}$ induces a proper edge coloring of $A$.\nWe say that $(x,A)$ and $(y,B)$ are connected by an $\\alpha$-edge in $\\mathcal{G}$ if $A=B$, $x,y$ are adjacent in $A$ and the edge that connects them has color $\\alpha\\in \\Delta$.\n\nSuppose for a contradiction that $\\mathcal{A}$ is a Borel function that solves $\\Pi$ on $\\mathcal{G}$.\n\nNext, we define a family of games parametrized by $\\alpha \\in [\\Delta]$, $x \\in V(\\idgraphing{r})$ and $S \\subseteq \\Sigma$. For the reader familiar with Marks' construction, let us point out that for a fixed $x$, the games are analogues to the ones he defines, with the following differences: allowed moves are vertices of the ID graphing $\\idgraphing{r}$ and restricted by its edge relation, and the winning condition is defined by a set of labels, not just merely one label.\n\nSo, the game\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$$ \nis defined as follows: $\\operatorname{Alice}$ and $\\operatorname{Bob}$ alternatingly label vertices of a $\\Delta$-regular rooted tree. The root is labelled by $x$, and the labels come from $V(\\idgraphing{r})$. In the $k$-th round, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see  \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphingcolored{r}$. \n\nIt follows from the acyclicity of $\\idgraphing{r}$ that a play of a game determines a $\\Delta$-regular rooted subtree of $\\idgraphing{r}$ to which the restriction of the edge-coloring is proper. That is, it determines a vertex $(x,A)$ of $\\mathcal{G}$. Let $\\operatorname{Alice}$ win iff the output of $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$ is \\textbf{not} in $S$.\n\nDefine the function $\\Lambda^x_{\\alpha}:\\mathbb{P}_\\alpha\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$ assigning the player to some $S \\in V(\\mathbb{P}_\\alpha)$ who has a winning strategy in $\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$.\nNote that, since $\\idgraphing{r}$ is locally finite, each player has only finitely many choices at each position. Thus, it follows from Borel Determinacy Theorem that $\\Lambda^x_{\\alpha}$ is well defined.\n\nNow we show the analogue of Proposition \\ref{pr:(A)satisfied}.\n\n\\begin{proposition}\\label{pr:Borel(A)satisfied}\n$(\\Lambda^{x}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^x_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$. Letting these strategies play against each other in the obvious way, produces a vertex $(x,A)$ of $\\mathcal{G}$.\nSince $\\mathcal{A}$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at $(x,A)$ and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\n Let $f$ be \n defined by $$x\\mapsto (\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}.$$ Note that $f$ has a finite range. Using the fact that the allowed moves for each player can be determined in a Borel way, uniformly in $x$, it is not hard to see that for each element $s$ in the range, $f^{-1}(s)$ is in the algebra generated by sets that can be obtained by applying game quantifiers to Borel sets (for the definition see \\cite[Section 20.D]{kechrisclassical}). It has been shown by Solovay \\cite{Solovay} and independently by Fenstad-Norman \\cite{fenstadnorman} that such sets are provably $\\Delta^1_2$, and consequently, measurable. Therefore, $f$ is a measurable map.\n \n As the number of sequences of functions $(\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}$ is $\\leq N$, by the choice of $r$, there exists a Borel set $Y$ with $\\mu(Y) > r$, such that $f$ is constant on $Y$.\n \nSince $\\Pi$ is not playable Proposition \\ref{pr:Borel(A)satisfied} gives that there is an $\\alpha\\in \\Delta$ and $S,T\\in \\mathbb{P}_\\alpha$ that violate condition (B).\nRecall that the measure of an independent Borel set of $\\idgraphing{r}^\\alpha$ is at most $r$.\nThat means that there are $x,y\\in Y$ such that $(x,y)$ is an $\\alpha$-edge in $\\idgraphing{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S], \\ \\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,y,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $x,y$.\nThis produces a labeling of a $\\Delta$-regular tree $A$ with labels from $\\idgraphing{r}$ such that $(x,A)$ and $(y,A)$ span an $\\alpha$-edge in $\\idgraphing{r}$.\nApplying $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$, we obtain $\\mathtt{a}_0\\in S$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$ for $(y,A)$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}$ does not produce a Borel solution to $\\Pi$.\n\\end{proof}\n\n\\begin{remark}\nOne can give an alternative proof of the existence of a Borel $\\Delta$-regular forest that does not admit a Borel homomorphism to $G_k$ that avoids using the graphing $\\idgraphing{r}$ and uses the original example described by Marks \\cite{DetMarks} instead.\nThe reason is that the example graphs $G_k$ contain $K_\\Delta$, as discussed in the proof of \\cref{th:homomorphism}.\nThis takes care of the ``non-free'' as in the argument of Marks.\nThen it is enough to use the pigeonhole principle on $\\mathbb{N}$ instead of the independence ratio reasoning.\n\\end{remark}\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\\section{Separation Of Various Complexity Classes}\n\\label{sec:separating_examples}\n\nIn this section we provide examples of several problems that separate some classes from \\cref{fig:big_picture_trees}.\nThe examples show two things. First, we have $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon) \\not= \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$. \nThis shows that there are problems such that their worst case complexity is at least $\\Theta(\\log n)$ on finite $\\Delta$-regular trees, but their average local complexity is constant. \nSecond, we show that there are problems in the class $\\mathsf{BOREL}$ that are not in the class $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{RLOCAL}(o(\\log n))$, on $\\Delta$-regular trees. This shows that one cannot in general hope that results from (Borel) measurable combinatorics can be turned into very efficient (sublogarithmic) distributed algorithms. \n\n\\subsection{Preliminaries}\n\\label{subsec:preliminaries_separating}\n\nWe first show that there is a strong connection between randomized local complexities and uniform local complexities. Afterwards, we introduce a generic construction that turns any LCL into a new LCL. We later use this generic transformation to construct an LCL that is contained in the set $\\mathsf{BOREL} \\setminus \\mathsf{RLOCAL}(O(\\log \\log n))$.\n\n\\paragraph{Uniform vs Local Randomized Complexity} \n\nWe will now discuss the connections between uniform and randomized complexities. \nNote that the easy part of the connection between the two concepts is turning uniform local algorithms into randomized ones, as formalized in the following proposition. \n\n\\begin{proposition}\n\\label{prop:bee_gees}\nWe have $\\mathsf{OLOCAL}(t(\\varepsilon)) \\subseteq \\mathsf{RLOCAL}(t(1\/n^{O(1)}))$. \n\\end{proposition}\n\\begin{proof}\nWe claim that an uniform local algorithm $\\mathcal{A}$ with an uniform local complexity of $t(\\varepsilon)$ can be turned into a local randomized algorithm $\\mathcal{A}'$ with a local complexity  of $t(1\/n^{O(1)})$. \n\nThe algorithm $\\mathcal{A}'$ simulates $\\mathcal{A}$ on an infinite $\\Delta$-regular tree -- each vertex $u$ of degree less than $\\Delta$ in the original tree pretends that the tree continues past its virtual half-edges and the random bits of $u$ are used to simulate the random bits in this virtual subtree. \nChoosing $\\varepsilon = 1\/n^{O(1)}$, one gets that the probability of $\\mathcal{A}'$ needing to look further than $t(\\varepsilon)$ for any vertex is bounded by $1\/n^{O(1)}$, as needed in the definition of the randomized local complexity. \n\\end{proof}\n\n\nOn the other hand, we will use the following proposition from \\cite{grebik_rozhon2021toasts_and_tails}. It informally states that the existence of \\emph{any} uniform local algorithm together with the existence of a sufficiently fast randomized local algorithm for a given LCL $\\Pi$ directly implies an upper bound on the uniform complexity of $\\Pi$.\n\\begin{proposition}[\\cite{grebik_rozhon2021toasts_and_tails}]\n\\label{prop:local->uniform}\nLet $\\mathcal{A}$ be an uniform local algorithm solving an LCL problem $\\Pi$ such that its \\emph{randomized} local complexity on finite $\\Delta$-regular trees is $t(n)$ for $t(n) = o(\\log n)$. \nThen, the uniform local complexity of $\\mathcal{A}$ on infinite $\\Delta$-regular trees is $O(t(1\/\\varepsilon))$.\n\\end{proposition}\n\n\\cref{prop:local->uniform} makes our life simpler, since we can take a known randomized distributed local algorithm with local complexity $g(n) = o(\\log n)$ and check if it works without the knowledge of $n$. If yes, this automatically implies that the  uniform local complexity of the algorithm is $h(\\varepsilon) = O(g(1\/\\varepsilon))$. \nIn particular, combining \\cref{prop:local->uniform} with the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup} and verifying that the algorithms of \nFischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and  Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} work without the knowledge of $n$, we obtain the following result. \n\n\n\\begin{theorem}\n\\label{thm:classification_local_uniform_trees}\nWe have:\n\\begin{itemize}\n    \\item $\\mathsf{LOCAL}(O(1)) = \\mathsf{OLOCAL}(O(1))$\n    \\item $\\mathsf{RLOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$\n    \\item $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$\n\\end{itemize}\nMoreover, there are no other possible uniform local complexities for $t(\\varepsilon) = o(\\log 1\/\\varepsilon)$. \n\\end{theorem}\nWe note that the first two items are proven in \\cite{grebik_rozhon2021toasts_and_tails}.\nFor the third item it suffices by known reductions to find an uniform algorithm\nsolving a version of the distributed Lov\\'asz Local Lemma (LLL) on so-called tree-structured dependency graphs considered in~\\cite{ChangHLPU20}.\n\n\\begin{proof}[Proof sketch]\nThe proof follows from \\cref{prop:bee_gees} and the following ideas. \nThe first item follows from the fact that any local algorithm with local complexity $O(1)$ can simply be made uniform. More specifically, there exists a constant $n_0$ --- depending only on the problem $\\Pi$ --- such that the algorithm, being told the size of the graph is $n_0$, is correct on \\emph{any} graph of size $n \\ge n_0$. \n\nFor the second item, by the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup}, it suffices to check that there is an uniform distributed $(\\Delta + 1)$-coloring algorithm with uniform local complexity $O(\\log^* 1\/\\varepsilon)$. Such an algorithm was given in \\cite{HolroydSchrammWilson2017FinitaryColoring}, or follows from the work in \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} and \\cref{prop:local->uniform}. \n\nSimilarly, for the third item, by the work of \\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} it suffices to check that there is an uniform distributed algorithm for a specific LLL problem on trees\nwith uniform local complexity $O(\\log\\log 1\/\\varepsilon)$.\nSuch an algorithm can be obtained by combining the randomized \\emph{pre-shattering} algorithm of Fischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and the deterministic \\emph{post-shattering} algorithm of Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} in a \\emph{graph shattering} framework~\\cite{barenboim2016locality}, which solves the LLL problem \nwith local complexity $O(\\log\\log n)$.\nBy \\cref{prop:local->uniform}, it suffices to check that this algorithm can be made to work even if it does not know the size of the graph $n$. We defer the details to \\cref{sec:LLL}.\nThis finishes the proof of \\cref{thm:classification_local_uniform_trees}. \n\\end{proof}\n\n\n\n\n\n\n\n\\paragraph{Adding Paths}\nBefore we proceed to show some separation results, we define a certain construction that turns any LCL problem $\\Pi$ into a new LCL problem $\\overline{\\Pi}$ with the following property. If the original problem $\\Pi$ cannot be solved by a fast local algorithm, then the same holds for $\\overline{\\Pi}$. However, $\\overline{\\Pi}$ might be strictly easier to solve than $\\Pi$ for $\\mathsf{BOREL}$ constructions. \n\n\\begin{definition}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL. \nWe define an LCL $\\overline{\\Pi}=(\\Sigma',\\mathcal{V}',\\mathcal{E}')$ as follows.\nLet $\\Sigma'$ be $\\Sigma$ together with one new label.\nLet $\\mathcal{V}'$ be the union of $\\mathcal{V}$ together with any cardinality-$\\Delta$ multiset that contains the new label exactly two times.\nLet $\\mathcal{E}'$ be the union of $\\mathcal{E}$ together with the cardinality-$2$ multiset that contains the new label twice.\n\\end{definition}\n\nIn other words, the new label determines doubly infinite lines in infinite $\\Delta$-regular trees, or lines that start and end in virtual half-edges in finite $\\Delta$-regular trees.\nMoreover, a vertex that is on such a line does not have to satisfy any other vertex constraint.\nWe call these vertices \\emph{line-vertices} and each edge on a line a \\emph{line-edge}.\n\n\n\\begin{proposition}\n\\label{pr:AddingLineLLL}\nLet $\\Pi$ be an LCL problem such that $\\overline{\\Pi}\\in \\mathsf{RLOCAL}(t(n))$ for $t(n)  = o(\\log(n))$. Then also $\\Pi \\in \\mathsf{RLOCAL}(O(t(n)))$.\n\\end{proposition}\n\\begin{proof}\nLet $\\overline{\\mathcal{A}}$ be a randomized $\\mathsf{LOCAL}$ algorithm that solves $\\overline{\\Pi}$ in $t(n) = o(\\log n)$ rounds with probability at least $1 - 1\/n^C$ for some sufficiently large constant $C$. We will construct an algorithm $\\mathcal{A}$ for $\\Pi$ with complexity $t(n)$ that is correct with probability $1 - \\frac{4}{n^{C\/3 - 2}}$. The success probability of $\\mathcal{A}$ can then be boosted by ``lying to it'' that the number of vertices is $n^{O(1)}$ instead of $n$; this increases the running time by at most a constant factor and boosts the success probability back to $1 - 1\/n^C$. \n\nConsider a $\\Delta$-regular rooted finite tree $T$ of depth $10t(n)$ and let $u$ be its root vertex. \nNote that $|T| \\le \\Delta^{10t(n)+1} < n$, for $n$ large enough. \n\n\nWe start by proving that when running $\\overline{\\mathcal{A}}$ on the tree $T$, then $u$ is most likely not a line-vertex. This observation then allows us to turn $\\overline{\\mathcal{A}}$ into an algorithm $\\mathcal{A}$ that solves $\\Pi$ on any $\\Delta$-regular input tree.\nLet $X$ be the indicator of $\\mathcal{A}$ marking $u$ as a line-vertex. Moreover, for $i \\in [\\Delta]$, let $Y_i$ be the indicator variable for the following event. The number of line edges in the $i$-th subtree of $u$ with one endpoint at depth $5t(n)$ and the other endpoint at depth $5t(n) + 1$ is odd.\n\nBy a simple parity argument we can relate the value of $X$ with the values of $Y_1,Y_2,\\ldots,Y_\\Delta$ as follows. If $X = 0$, that is, $u$ is not a line-vertex, then all of the $Y_i$'s have to be $0$, as each path in the tree $T$ is completely contained in one of the $\\Delta$ subtrees of $u$. On the other hand, if $u$ is a line-vertex, then there exists exactly one path, the one containing $u$, that is not completely contained in one of the $\\Delta$ subtrees of $u$. This in turn implies that exactly two of the $\\Delta$ variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are equal to $1$.\n\nThe random variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are identically distributed and mutually independent. Hence, if $\\P(Y_i = 1) > \\frac{1}{n^{C\/3}}$, then the probability that there are at least $3$ $Y_i$'s equal to $1$ is strictly greater than $\\left(\\frac{1}{n^{C\/3}}\\right)^3 = \\frac{1}{n^C}$. This is a contradiction, as in that case $\\overline{\\mathcal{A}}$ does not produce a valid output, which according to our assumption happens with probability at most $\\frac{1}{n^C}$. \n\nThus, we can conclude that $\\P(Y_i = 1) \\leq \\frac{1}{n^{C\/3}}$. By a union bound, this implies that all of the $Y_i$'s are zero with probability at least $\\frac{1}{n^{C\/3 - 1}}$, and in that case $u$ is not a line-vertex. \n\nFinally, the algorithm $\\mathcal{A}$ simulates $\\overline{\\mathcal{A}}$ as if the neighborhood of each vertex was a $\\Delta$-regular branching tree up to depth at least $t(n)$. \n\nIt remains to analyze the probability that $\\mathcal{A}$ produces a valid solution for the LCL problem $\\Pi$. To that end, let $v$ denote an arbitrary vertex of the input tree. The probability that the output of $v$'s half edges satisfy the vertex constraint of the LCL problem $\\overline{\\Pi}$ is at least $1 - 1\/n^C$. Moreover, in the case that $v$ is not a line-vertex, which happens with probability at least $1 - \\frac{1}{n^{C\/3 - 1}}$, the output of $v$'s half edges even satisfy the vertex constraint of the LCL problem $\\Pi$. Hence, by a union bound the vertex constraint of the LCL problem $\\Pi$ around $v$ is satisfied with probability at least $1 - \\frac{2}{n^{C\/3 - 1}}$. With exactly the same reasoning, one can argue that the probability that the output of $\\mathcal{A}$ satisfies the edge constraint of the LCL problem $\\Pi$ at a given edge is also at least $1 - \\frac{2}{n^{C\/3 - 1}}$. Finally, doing a union bound over the $n$ vertex constraints and $n-1$ edge constraints it follows that $\\mathcal{A}$ produces a valid solution for $\\Pi$ with probability at least $1 - \\frac{4}{n^{C\/3 - 2}}$.\n\\end{proof}\n\n\\begin{remark}\nThe ultimate way how to solve LCLs of the form $\\overline{\\Pi}$ is to find a spanning forest of lines.\nThis is because once we have a spanning forest of lines, then we might declare every vertex to be a line-vertex and label every half-edge that is contained on a line with the new symbol in $\\overline{\\Pi}$.\nThe remaining half-edges can be labeled arbitrarily in such a way that the edge constraints are satisfied.\n\nPut otherwise, once we are able to construct a spanning forest of lines and $\\mathcal{E}\\not=\\emptyset$, where $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, then $\\overline{\\Pi}$ can be solved.\nMoreover, in that case the complexity of $\\overline{\\Pi}$ is upper bounded by the complexity of finding the spanning forest of lines.\n\nLyons and Nazarov \\cite{lyons2011perfect} showed that $\\Pi_{\\text{pm}}\\in \\mathsf{fiid}$ and it is discussed in  \\cite{LyonsTrees} that the construction can be iterated $(\\Delta-2)$-many times. After removing each edge that is contained in one of the $\\Delta-2$ perfect matchings each vertex has a degree of two.\nHence, it is possible to construct a spanning forest of lines as fiid.\nConsequently, $\\overline{\\Pi}\\in \\mathsf{fiid}$ for every $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ with $\\mathcal{E}\\not=\\emptyset$.\n\\end{remark}\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$}\n\nWe now formalize the proof that $\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$ from \\cref{sec:introseparation}. \nRecall that $\\Pi_{\\chi,\\Delta}$ is the proper vertex $\\Delta$-coloring problem.\nThe following claim directly follows by  \\cref{pr:AddingLineLLL} together with the fact that $\\Pi_{\\chi,\\Delta}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{claim}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{claim}\n\nOn the other hand we show that adding lines helps to find a Borel solution.\nThis already provides a simple example of a problem in the set $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{proposition}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}} \\in \\mathsf{BOREL}$. \n\\end{proposition}\n\\begin{proof}\nBy \\cite{KST}, every Borel graph of finite maximum degree admits a Borel maximal independent set.\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular acyclic graph on a standard Borel space $X$.\nBy an iterative application of the fact above we find Borel sets $A_1,\\dots,A_{\\Delta-2}$ such that $A_1$ is a maximal independent set in $\\mathcal{G}$ and $A_i$ is a maximal independent set in $\\mathcal{G}\\setminus \\bigcup_{j<i}A_j$ for every $i>1$.\nWe think of $A_1,\\dots,A_{\\Delta-2}$ as color classes for the first $\\Delta-2$ colors.\nLet $B=X\\setminus \\bigcup_{i\\in [\\Delta-2]}A_i$ and let $\\mathcal{H}$ be the subgraph of $\\mathcal{G}$ determined by $B$.\nIt is easy to see that the maximum degree of $\\mathcal{H}$ is $2$.\nIn particular, $\\mathcal{H}$ consists of finite paths, one-ended infinite paths or doubly infinite paths.\nWe use the extra label in $\\overline{\\Pi_{\\chi,\\Delta}}$ to mark the doubly infinite paths.\nIt remains to use the remaining $2$ colors in $[\\Delta]$ to define a proper vertex $2$-coloring of the finite and one-ended paths.\nIt is a standard argument that this can be done in a Borel way and it is easy to verify that, altogether, we found a Borel $\\overline{\\Pi_{\\chi,\\Delta}}$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\n\n\n\\subsection{Examples and Lower Bound}\n\nIn this subsection we define two LCLs and show that they are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\nIn the next subsection we show that one of them is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and the other in the class $\\mathsf{BOREL}$.\nBoth examples that we present are based on the following relaxation of the perfect matching problem.\n\n\\begin{definition}[Perfect matching in power-$2$ graph]\nLet $\\Pi^2_{\\text{pm}}$ be the perfect matching problem in the power-$2$ graph, i.e., in the graph that we obtain from the input graph by adding an edge between any two vertices that have distance at most $2$ in the input graph\n\\end{definition}\n\nWe show that $\\Pi^2_{\\text{pm}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$. \\cref{pr:AddingLineLLL} then directly implies that $\\overline{\\Pi^2_{\\text{pm}}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$ as well.\nOn the other hand, we later use a one-ended spanning forest decomposition to show that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and a one or two-ended spanning forest decomposition to show that $\\overline{\\Pi^2_{\\text{pm}}}$ is in $\\mathsf{BOREL}$.\n\n\nWe first show the lower bound result.\nThe proof is based on a simple parity argument as in the proof of \\cref{pr:AddingLineLLL}.\n\n\\begin{theorem}\nThe problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that the theorem statement does not hold for $\\Pi^2_{\\text{pm}}$.\nThen, by \\cref{thm:basicLOCAL}, there is a distributed randomized algorithm solving $\\Pi^2_{\\text{pm}}$ with probability at least $1 - 1\/n$. By telling the algorithm that the size of the input graph is $n^{2\\Delta}$ instead of $n$, we can further boost the success probability to $1 - 1\/n^{2\\Delta}$. The resulting round complexity is at most a constant factor larger, as $\\Delta = O(1)$.\nWe can furthermore assume that the resulting algorithm $\\mathcal{A}$ will always output for each vertex $v$ a vertex $M(v) \\neq v$ in $v$'s $2$-hop neighborhood, even if $\\mathcal{A}$ fails to produce a valid solution. The vertex $M(v)$ is the vertex $v$ decides to be matched to, and if $\\mathcal{A}$ produces a valid solution it holds that $M(M(v)) = v$.\n\nConsider a fully branching $\\Delta$-regular tree $T$ of depth $10t(n)$. Note that $|T| < n$ for $n$ large enough. Let $u$ be the root of $T$ and $T_i$ denote the $i$-th subtree of $u$ (so that $u \\not\\in T_i$).\nLet $S_i$ denote the set of vertices $v$ in $T_i$ such that both $v$ and $M(v)$ have distance at most $2t(n)$ from $u$.\nNote that $M(v)$ is not necessarily a vertex of $T_i$.\nLet $Y_i$ be the indicator of whether $S_i$ is even.\n\nObserve that, if $\\mathcal{A}$ does not fail on the given input graph, then the definition of the $S_i$ implies that every vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is matched to a vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$.\nHence, the number of vertices in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is even, which implies that we cannot have $Y_1 = Y_2 = \\ldots = Y_\\Delta = 1$ unless $\\mathcal{A}$ fails.\nNote that $Y_i$ depends only on the output of $\\mathcal{A}$ at the vertices in $T_i$ that have a distance of precisely $2t(n) - 1$ or $2t(n)$ to $u$ (as all other vertices in $T_i$ are guaranteed to be in $S_i$). \nHence, the events $Y_i = 1$, $i \\in [\\Delta]$, are independent, and, since $\\P(Y_1 = 1) = \\ldots = \\P(Y_{\\Delta} = 1)$ (as all vertices in all $T_i$ see the same topology in their $t(n)$-hop view), we obtain $(\\P(Y_1 = 1))^{\\Delta} \\le 1\/n^{2\\Delta}$, which implies $\\P(Y_i = 1) \\le 1\/n^2$, for any $i \\in [\\Delta]$.\n\n\nHence, with probability at least $1 - \\Delta \/n^2$ we have $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, by a union bound.\nLet $v_1, v_2, \\dots, v_\\Delta$ denote the neighbors of $u$ with $v_i$ being the root of subtree $T_i$.\nNote that if $\\mathcal{A}$ does not fail and $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, then it has to be the case that $u$ is matched to a vertex in some subtree $T_i$ (not necessarily $v_i$), while any vertex from $N_{-i}(u) := \\{v_1, v_2, \\dots, v_{i-1}, v_{i+1}, \\dots, v_\\Delta\\}$ is matched with another vertex from $N_{-i}(u)$ (hence, we already get that $\\Delta$ needs to be odd).\nThis is because each subtree needs to have at least one vertex matched to a different subtree (since $|S_i|$ is odd for each $i \\in [\\Delta]$).\nIf $M(u)$ is a vertex in $T_i$ and, for any $v \\in N_{-i}(u)$, we have $M(v) \\in N_{-i}(u)$, we say that $u$ orients the edge going to $v_i$ outwards.\n\nConsider a path $(u_0 = u, u_1, \\dots, u_k)$, where $k = 2t(n) + 3$.\nBy the argument provided above, the probability that $u$ orients an edge outwards is at least $1 - \\Delta\/n^2 - 1\/n^{2\\Delta}$, and, if $u$ orients an edge outwards, the probability that this edge is not $uu_1$ is $(\\Delta - 1)\/\\Delta$, by symmetry.\nHence, we obtain (for sufficiently large $n$) that $\\P(\\mathcal{E}_1) \\geq 1\/2$, where $\\mathcal{E}_1$ denotes the event that vertex $u$ orients an edge different from $uu_1$ outwards.\nWith an analogous argument, we obtain that $\\P(\\mathcal{E}_2) \\geq 1\/2$, where $\\mathcal{E}_2$ denotes the event that vertex $u_k$ orients an edge different from $u_ku_{k-1}$ outwards.\nSince $\\mathcal{E}_1$ and $\\mathcal{E}_2$ are independent (due to the fact that the distance between any neighbor of $u$ and any neighbor of $u_k$ is at least $2t(n) + 1$), we obtain $\\P(\\mathcal{E}_1 \\cap \\mathcal{E}_2) \\geq 1\/4$.\nMoreover, the probability that all vertices in $\\{ u_1, \\dots, u_{k-1} \\}$ orient an edge outwards is at least $1 - (2t(n)+2)(\\Delta\/n^2 + 1\/n^{2\\Delta})$, which is at least $4\/5$ for sufficiently large $n$.\nThus, by a union bound, there is a probability of at least $1 - 3\/4 - 1\/5 - 1\/n^{2\\Delta}$ that $\\mathcal{A}$ does not fail, all vertices in $(u, u_1, \\dots, u_k)$ orient an edge outwards, and the outward oriented edges chosen by $u$ and $u_k$ are not $uu_1$ and $u_ku_{k-1}$, respectively.\nFor sufficiently large $n$, the indicated probability is strictly larger than $0$.\nWe will obtain a contradiction and conclude the proof for $\\Pi^2_{\\text{pm}}$ by showing that there is no correct solution with the described properties.\n\nAssume in the following that the solution provided by $\\mathcal{A}$ is correct and $u, u_1, \\dots, u_k$ satisfy the mentioned properties.\nSince $u$ orients an edge different from $uu_1$ outwards, $u_1$ must be matched to a neighbor of $u$, which implies that $u_1$ orients edge $u_1u$ outwards.\nUsing an analogous argumentation, we obtain inductively that $u_j$ orients edge $u_ju_{j-1}$ outwards, for all $2 \\leq j \\leq k$.\nHowever, this yields a contradiction to the fact that the edge that $u_k$ orients outwards is different from $u_ku_{k-1}$, concluding the proof (for $\\Pi^2_{\\text{pm}}$).\n\nThe theorem statement for $\\overline{\\Pi^2_{\\text{pm}}}$ follows by applying \\cref{pr:AddingLineLLL}.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Upper Bounds Using Forest Decompositions}\n\nWe prove an upper bound for the problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ defined in the previous subsection.\nWe use a technique of decomposing the input graph in a spanning forest with some additional properties, i.e., one or two ended. This technique was used in \\cite{BrooksMeas} to prove Brooks' theorem in a measurable context.\nNamely, they proved that if $\\mathcal{G}$ is a Borel $\\Delta$-regular acyclic graph and $\\mu$ is a Borel probability measure, then it is possible to erase some edges of $\\mathcal{G}$ in such a way that the remaining graph is a one-ended spanning forest on a $\\mu$-conull set.\nIt is not hard to see that one can solve $\\Pi^2_{\\text{pm}}$ on one-ended trees in an inductive manner, starting with the leaf vertices. Consequently we have $\\Pi^2_{\\text{pm}}\\in \\mathsf{MEASURE}$.\nWe provide a quantitative version of this result and thereby show that the problem of constructing a one-ended forest is contained in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$.\nA variation of the construction shows that it is possible to find a one or two-ended spanning forest in a Borel way.\nThis allows us to show that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\nAs an application of the decomposition technique we show that Vizing's theorem, i.e., proper $\\Delta+1$ edge coloring $\\Pi_{\\chi',\\Delta+1}$, for $\\Delta=3$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\n\n\\subsubsection{Uniform Complexity of the One-Forest Decomposition }\nWe start with the definition of a one-ended spanning forest.\nIt can be viewed as a variation of the edge grabbing problem $\\Pi_{\\text{edgegrab}}$.\nNamely, if a vertex $v$ grabs an incident edge $e$, then we think of $e$ as being oriented away from $v$ and $v$ selecting the other endpoint of $e$ as its parent.\nSuppose that $\\mathcal{T}$ is a solution of $\\Pi_{\\text{edgegrab}}$ on $T_\\Delta$.\nWe denote with $\\mathcal{T}^{\\leftarrow}(v)$ the subtree with root $v$.\n\n\n\\begin{definition}[One-ended spanning forest]\nWe say that a solution $\\mathcal{T}$ of $\\Pi_{\\text{edgegrab}}$ is a \\emph{one-ended spanning forest} if $\\mathcal{T}^{\\leftarrow}(v)$ is finite for every $v\\in T_\\Delta$ (see \\cref{fig:one_ended}). \n\n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{fig\/one_ended.pdf}\n    \\caption{An example of a one-ended forest decomposition of an infinite 3-regular tree. }\n    \\label{fig:one_ended}\n\\end{figure}\n\n\nNote that on an infinite $\\Delta$-regular tree every connected component of a one-ended spanning forest must be infinite.\nFurthermore, recall that we discussed above that it is possible to construct a one-ended spanning forest in $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nNext we formulate our quantitative version.\n\n\n\n\n\n\\begin{restatable}{theorem}{oneendedtreetail}\n\\label{thm:one_ended_forest}\nThe one-ended spanning forest can be constructed by an uniform local algorithm  with an uniform local complexity of $O(\\operatorname{poly}\\log1\/\\varepsilon)$. \nMore precisely, there is an uniform distributed algorithm $\\mathcal{A}$ that computes a one-ended spanning forest and if we define $R(v)$ to be the smallest coding radius that allows $v$ to compute $\\mathcal{T}^{\\leftarrow}(v)$, then\n\\[\nP(R(v) > O(\\operatorname{poly} \\log 1\/\\varepsilon)) \\le \\varepsilon. \n\\]\n\\end{restatable}\n\nThe formal proof of \\cref{thm:one_ended_forest} can be found in \\cref{sec:formalproof}.\nWe first show the main applications of the result.\nNamely, we show that $\\Pi^2_{\\text{pm}}$ can be solved inductively starting from the leaves of the one-ended trees. This proves that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$ and hence $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon) \\setminus \\mathsf{RLOCAL}(o(\\log n))$ is non-empty.\n\n\\begin{theorem}\\label{thm:twomatchFFIID}\nThe problem $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$ and $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\\end{theorem}\n\\begin{proof}\nFirst we show that every infinite one-ended directed tree $T$ admits an inductive solution from the leaves to the direction of infinity. More precisely, we define a power-$2$ perfect matching on the infinite one-ended directed tree $T$ such that the following holds. Let $v$ be an arbitrary vertex and $T^{\\leftarrow}(v)$ the finite subtree rooted at $v$. Then, all vertices in $T^{\\leftarrow}(v)$, with the possible exception of $v$, are matched to a vertex in $T^{\\leftarrow}(v)$. Moreover, for each vertex in $T^{\\leftarrow}(v) \\setminus \\{v\\}$, it is possible to determine to which vertex it is matched by only considering the subtree $T^{\\leftarrow}(v)$. \nWe show that it is possible to define a perfect matching in that way by induction on the height of the subtree $T^{\\leftarrow}(v)$. \n\nWe first start with the leaves and declare them as unmatched in the first step of the induction.\n\nNow, consider an arbitrary vertex $v$ with $R \\geq 1$ children that we denote by $v_1, v_2, \\ldots, v_R$.\nBy the induction hypothesis, all vertices in $T^{\\leftarrow}(v_i) \\setminus \\{v_i\\}$ are matched with a vertex in $T^{\\leftarrow}(v_i)$ for $i \\in [R]$ and it is possible to compute that matching given $T^{\\leftarrow}(v)$. However, some of the children of $v$ are possibly still unmatched. If the number of unmatched children of $v$ is even, then we simply pair them up in an arbitrary but fixed way and we leave $v$ unmatched. Otherwise, if the number of unmatched children of $v$ is odd, we pair up $v$ with one of its unmatched children and we pair up all the remaining unmatched children of $v$ in an arbitrary but fixed way. This construction clearly satisfies the criteria stated above. In particular, the resulting matching is a perfect matching, as every vertex gets matched eventually.\n\n\nBy \\cref{thm:one_ended_forest} there is an uniform distributed algorithm of complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$ that computes a one-ended spanning forest $\\mathcal{T}$ on $T_\\Delta$.\nMoreover, every vertex $v\\in T_\\Delta$ can compute where it is matched with an uniform local complexity of $O(\\operatorname{poly} \\log 1\/\\varepsilon)$.\nThis follows from the discussion above, as a vertex $v$ can determine where it is matched once it has computed $\\mathcal{T}^{\\leftarrow}(w)$ for each $w$ in its neighborhood.\nHence, $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\nSimilarly, the one-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$ can be constructed in the class $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nBy the discussion above, this immediately implies that $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$. \n\\end{proof}\n\n\n\n\n\\subsubsection{Decomposition in $\\mathsf{BOREL}$}\nNext, we show that a similar, but weaker, decomposition can be done in a Borel way.\nNamely, we say that a subset of edges of an acyclic infinite graph $G$, denoted by $\\mathcal{T}$, is a \\emph{one or two-ended spanning forest} if every vertex $v\\in G$ is contained in an infinite connected component of $\\mathcal{T}$ and each infinite connected component of $\\mathcal{T}$ has exactly one or two directions to infinity.\nLet $S$ be such a connected component.\nNote that if $S$ has one end, then we can solve $\\Pi^2_{\\text{pm}}$ on $S$ as above.\nIf $S$ has two ends, then $S$ contains a doubly infinite path $P$ with the property that erasing $P$ splits $S\\setminus P$ into finite connected components.\nIn order to solve $\\overline{\\Pi^2_{\\text{pm}}}$ we simply solve $\\Pi^2_{\\text{pm}}$ on the finite connected components of $S\\setminus P$ (with some of the vertices in $S\\setminus P$ being potentially matched with vertices on the path $P$) and declare vertices on $P$ to be line vertices.\n\nThe high-level idea to find a one or two-ended spanning forest is to do the same construction as in the measure case and understand what happens with edges that do not disappear after countably many steps.\nA slight modification in the construction guarantees that what remains are doubly infinite paths.\n\n\\begin{restatable}{theorem}{oneendedforestborel}\n\\label{thm:OneTwoEndBorel}\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular forest. \nThen there is a Borel one or two-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$.\n\\end{restatable}\n\nThe proof of \\cref{thm:OneTwoEndBorel} can be found in \\cref{sec:oneortwo}.\nWe remark that the notation used in the proof is close to the notation in the proof that gives a measurable construction of a one-ended spanning forest in \\cite{BrooksMeas}.\nNext, we show more formally how \\cref{thm:OneTwoEndBorel} implies that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\n\n\\begin{theorem}\nThe problem $\\overline{\\Pi^2_{\\text{pm}}}$ is in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathcal{T}$ be the one or two-ended spanning forest given by \\cref{thm:OneTwoEndBorel} and $S$ be a connected component of $\\mathcal{T}$.\nIf $S$ is one-ended, then we use the first part of the proof of \\cref{thm:twomatchFFIID} to find a solution of $\\Pi^2_{\\text{pm}}$ on $S$.\nSince deciding that $S$ is one-ended as well as computing an orientation towards infinity can be done in a Borel way, this yields a Borel labeling.\n\nSuppose that $S$ is two-ended.\nThen, there exists a doubly infinite path $P$ in $S$ with the property that the connected components of $S\\setminus P$ are finite.\nMoreover, it is possible to detect $P$ in a Borel way.\nThat is, declaring the vertices on $P$ to be line vertices and using the special symbol in $\\overline{\\Pi^2_{\\text{pm}}}$ for the half-edges on $P$ yields a Borel measurable labeling.\nLet $C\\not=\\emptyset$ be one of the finite components in $S\\setminus P$ and $v\\in C$ be the vertex of distance $1$ from $P$.\nOrient the edges in $C$ towards $v$, this can be done in a Borel way since $v$ is uniquely determined for $C$.\nThen the first part of the proof of \\cref{thm:twomatchFFIID} shows that one can inductively find a solution to $\\Pi^2_{\\text{pm}}$ on $C$ in such a way that all vertices, possibly up to $v$, are matched.\nIf $v$ is matched, then we are done.\nIf $v$ is not matched, then we add the edge that connects $v$ with $P$ to the matching. Note that it is possible that multiple vertices are matched with the same path vertex, but this is not a problem according to the definition of $\\overline{\\Pi^2_{\\text{pm}}}$. \nIt follows that this defines a Borel function on $H(\\mathcal{G})$ that solves $\\overline{\\Pi^2_{\\text{pm}}}$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Vizing's Theorem for $\\Delta=3$}\nFinding a measurable or local version of Vizing's Theorem, i.e., proper edge $(\\Delta+1)$-coloring $\\Pi_{\\chi',\\Delta+1}$, was studied recently in \\cite{grebik2020measurable,weilacher2021borel,BernshteynVizing}.\nIt is however not known, even on trees, whether $\\Pi_{\\chi',\\Delta+1}$ is in $\\mathsf{RLOCAL}(O(\\log\\log n))$.\nHere we use the one-ended forest construction to show that $\\Pi_{\\chi',\\Delta+1}$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ for $\\Delta=3$.\n\n\n\n\\iffalse\n\\paragraph{Composition of Uniform Local Algorithms}\nWhen designing algorithms, one often composes several simple algorithms to obtain a more sophisticated algorithm. For example, an algorithm $\\mathcal{A}$ might first invoke algorithm $\\mathcal{A}_1$ and after $\\mathcal{A}_1$ finished, $\\mathcal{A}$ invokes algorithm $\\mathcal{A}_2$ with the input of $\\mathcal{A}_2$ being equal to the output of $\\mathcal{A}_1$. Then, the final output of $\\mathcal{A}$ is simply the output of $\\mathcal{A}_2$. We call such a composition a sequential composition. In the context of the classic deterministic local model, the round complexity of $\\mathcal{A}$ is simply the sum of the round complexities of the two algorithms $\\mathcal{A}_1$ and $\\mathcal{A}_2$ that are used as a subroutine. Here, we prove two such composition results in the context of uniform local algorithms. These general results will help us in \\cref{app:one_ended} to analyze the uniform complexity of an uniform algorithm that computes a so-called one-ended forest in a principled manner.\n\n\n\n\\begin{lemma}[Parallel Composition]\n\\label{lem:parallel_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_{A_1}(\\varepsilon)$ and $t_{A_2}(\\varepsilon)$, respectively. \nLet $A$ be the parallel composition of $A_1$ and $A_2$. That is, the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is the smallest radius such that $A$ knows the local answer of both $A_1$ and $A_2$ at $u$. \nThen, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$.\n\\end{lemma}\n\\begin{proof}\nThe probability that the coding radius of algorithm $\\mathcal{A}_1$ at vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2$ and the same holds for the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$. Hence, by a simple union bound, the probability that the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon$. \\\\ Hence, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$, as desired.\n\\end{proof}\n\n\\fi\n\n\n\n\\begin{proposition}\nLet $\\Delta=3$. We have $\\Pi_{\\chi',\\Delta+1} \\in\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\\end{proposition}\n\\begin{proof}[Proof sketch.]\nBy \\cref{thm:one_ended_forest}, we can compute a one-ended forest decomposition $\\mathcal{T}$ with uniform local complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \nNote that every vertex has at least one edge in $\\mathcal{T}$, hence the edges in $G \\setminus \\mathcal{T}$ form paths. \nThese paths can be $3$-edge colored by using the uniform version of Linial's coloring algorithm. This algorithm has an uniform local complexity of $O(\\log^* 1\/\\varepsilon)$. By \\cref{lem:sequential_composition}, the overall complexity is $O(\\operatorname{poly} \\log (\\Delta^{\\log^* 1\/\\varepsilon} \/ \\varepsilon) + \\log^* 1\/\\varepsilon)) = O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \n\nFinally, we color the edges of $T$. We start from the leaves and color the edges inductively. In particular, whenever we consider a vertex $v$ we color the at most two edges in $T^{\\leftarrow}(v)$ below $v$. \nNote that there always is at least one uncolored edge going from $v$ in the direction of $T$. Hence, we can color the at most two edges below $v$ in $T^{\\leftarrow}(v)$ greedily -- each one neighbors with at most $3$ colored edges at any time.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:one_ended_forest}} \n\\label{sec:formalproof}\n\n\nIn this section we formally prove \\cref{thm:one_ended_forest}. \n\n\n\\oneendedtreetail*\n\nWe follow the construction of the one-ended spanning forest in \\cite{BrooksMeas}.\nThe construction proceeds in rounds. Before each round, some subset of the vertices have already decided which incident edge to grab, and we refer to these vertices as settled vertices. All the remaining vertices are called unsettled. The goal in each round is to make a large fraction of the unsettled vertices settled. To achieve this goal, our construction relies on certain properties that the graph induced by all the unsettled vertices satisfies. \n\nOne important such property is that the graph induced by all the unsettled vertices is expanding. That is, the number of vertices contained in the neighborhood around a given vertex grows exponentially with the radius. The intuitive reason why this is a desirable property is the following. Our algorithm will select a subset of the unsettled vertices and clusters each unsettled vertex to the closest selected vertex. As the graph is expanding and any two vertices in the selected subset are sufficiently far away, there will be a lot of edges leaving a given cluster. For most of these inter-cluster edges, both of the endpoints will become settled. This in turn allows one to give a lower bound on the fraction of vertices that become settled in each cluster.\n\nTo ensure that the graph induced by all the unsettled vertices is expanding, a first condition we impose on the graph induced by all the unsettled vertices is that it has a minimum degree of at least $2$. While this condition is a first step in the right direction, it does not completely suffice to ensure the desired expansion, as an infinite path has a minimum degree of $2$ but does not expand sufficiently. Hence, our algorithm will keep track of a special subset of the unsettled vertices, the so-called hub vertices. Each hub vertex has a degree of $\\Delta$ in the graph induced by all the unsettled vertices. That is, all of the neighbors of a hub vertex are unsettled as well. Moreover, each unsettled vertex has a bounded distance to the closest hub vertex, where the specific upper bound on the distance to the closest hub vertex increases with each round. As we assume $\\Delta > 2$, the conditions stated above suffice to show that the graph induced by all the unsettled vertices expands.\n\nNext, we explain in more detail how a vertex decides which edge to grab. Concretely, if a vertex becomes settled, it grabs an incident edge such that the other endpoint of that edge is strictly closer to the closest unsettled vertex as the vertex itself. For example, if a vertex becomes settled and there is exactly one neighbor that is still unsettled, then the vertex will grab the edge that the vertex shares with its unsettled neighbor. Grabbing edges in that way, one can show two things. First, for a settled vertex $v$, the set $T^{\\leftarrow}(v)$ of vertices behind $v$ does not change once $v$ becomes settled. The intuitive reason for this is that the directed edges point towards the unsettled vertices and therefore the unsettled vertices lie before $v$ and not after $v$. Moreover, one can also show that $T^{\\leftarrow}(v)$ only contains finitely many vertices. The reason for this is that at the moment a vertex $v$ becomes settled, there exists an unsettled vertex that is sufficiently close to $v$, where the exact upper bound on that distance again depends on the specific round in which $v$ becomes settled. \n\nWhat remains to be discussed is at which moment a vertex decides to become settled. As written above, in each round the algorithm considers a subset of the unsettled vertices and each unsettled vertex is clustered to the closest vertex in that subset. This subset only contains hub vertices and it corresponds to an MIS on the graph with the vertex set being equal to the set of hub vertices and where two hub vertices are connected by an edge if they are sufficiently close in the graph induced by all the unsettled vertices. Now, each cluster center decides to connect to exactly $\\Delta$ different neighboring clusters, one in each of its $\\Delta$ subtrees. Remember that as each cluster center is a hub vertex, all of its neighbors are unsettled as well. Now, each unsettled vertex becomes settled except for those that lie on the unique path between two cluster centers such that one of the cluster centers decided to connect to the other one.\n\n\n\n\n\n\n\\paragraph{Construction}\nThe algorithm proceeds in rounds. After each round, a vertex is either settled or unsettled and a settled vertex remains settled in subsequent rounds. Moreover, some unsettled vertices are so-called hub vertices. We denote the set of unsettled, settled and hub vertices after the $i$-th round with $U_i$, $S_i$ and $H_i$, respectively. We set $U_0 = H_0 = V$ prior to the first round and we always set $S_i = V \\setminus U_i$.  \\cref{fig:constr1} illustrates the situation after round $i$.  Moreover, we denote with $O_i$ a partial orientation of the edges of the infinite $\\Delta$-regular input tree $T$. In the beginning, $O_0$ corresponds to the partial orientation with no oriented edges. Each vertex is incident to at most $1$ outwards oriented edge in $O_i$. If a vertex is incident to an outwards oriented edge in $O_i$, then the other endpoint of the edge will be its parent in the one-ended forest decomposition. For each vertex $v$ in $S_i$, we denote with $T_{v,i}$ the smallest set that satisfies the following conditions. First, $v \\in T_{v,i}$. Moreover, if $u \\in T_{v,i}$ and $\\{w,u\\}$ is an edge that according to $O_i$ is oriented from $w$ to $u$, then $w \\in T_{v,i}$. We later show that $T_{v,i}$ contains exactly those vertices that are contained in the subtree $\\mathcal{T}^\\leftarrow(v)$.\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics{fig\/one_ended_construction.pdf}\n    \\caption{The picture illustrates the situation after some round $i$ of the construction. The vertices on the paths are all contained in $U_i$, while the vertices corresponding to the big black dots are additionally contained in $H_i$. Note that the length of each path is at least $d_i$. However, the distance between $u$ and $v$ in the original graph can be much smaller.}\n    \\label{fig:constr1}\n\\end{figure}\n\n\nAfter the $i$-th round, the construction satisfies the following invariants.\n\n\n\n\\begin{enumerate}\n    \n    \\item For each $v \\in S_{i-1}$, we have $T_{v,i} = T_{v,i-1}$.\n    \n    \\item Let $v$ be an arbitrary vertex in $S_i$. Then, $T_{v,i}$ contains finitely many vertices and furthermore $T_{v,i} \\subseteq S_i$.\n    \n    \\item The minimum degree of the graph $T[U_i]$ is at least $2$ and each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n    \n    \\item Each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$.\n\\end{enumerate}\n\nWe now describe how to compute $U_i, S_i$, $H_i$ and $O_i$. The construction during the $i$-th round is illustrated in $\\cref{fig:constr2}$. Note that we can assume that the invariants stated above are satisfied after the $(i-1)$-th round. In the $i$-th round we have a parameter $d_i$ that we later set to $2^{2^i}$.\n\n\\begin{enumerate}\n    \\item $H_i$ is a subset of $H_{i-1}$ that satisfies the following property. No two vertices in $H_i$ have a distance of at most $d_i$ in $T[U_{i-1}]$. Moreover, each vertex in $H_{i-1} \\setminus H_i$ has a distance of at most $d_i$ to the closest vertex in $H_i$ in the graph $T[U_{i-1}]$. Note that we can compute $H_i$ by computing an MIS in the graph with vertex set $H_{i-1}$ and where two vertices are connected iff they have a distance of at most $d_i$ in the graph $T[U_{i-1}]$. Hence, we can use Ghaffari's MIS algorithm \\cite{ghaffari2016MIS} to compute the set $H_i$.\n    \\item Next, we describe how to compute $U_i$.  \n    We assign each vertex $u \\in U_{i-1}$ to the closest vertex in the graph $T[U_{i-1}]$ that is contained in $H_i$, with ties being broken arbitrarily. We note that there exists a node in $H_i$ with a distance of at most $(\\sum_{j=0}^{i-1} d_j) + d_i$ to $u$. To see why, note that Invariant $(4)$ from round $i - 1$ implies that there exists a vertex $w$ in $H_{i-1}$ with a distance of at most $\\sum_{j=0}^{i-1} d_j$ to $u$. We are done if $w$ is also contained in $H_i$. If not, then it follows from the way we compute $H_i$ that there exists a vertex in $H_i$ with a distance of at most $d_i$ to $w$, but then the triangle inequality implies that the distance from $u$ to that vertex is at most $(\\sum_{j=0}^{i-1} d_j) + d_i$, as desired. For each vertex $v \\in H_i$, we denote with $C(v)$ the set of all vertices in $U_{i-1}$ that got assigned to $v$. Now, let $E_v$ denote the set of edges that have exactly one endpoint in $C(v)$. We can partition $E_v$ into $E_{v,1} \\sqcup E_{v,2} \\sqcup \\ldots \\sqcup E_{v,\\Delta}$, where for $\\ell \\in [\\Delta]$, $E_{v,\\ell}$ contains all the edges in $E_v$ that are contained in the $\\ell$-th subtree of $v$. \n    Invariants $(3)$ and $(4)$ after the $(i-1)$-th round imply that $E_{v,\\ell} \\neq \\emptyset$. Moreover, $E_{v,\\ell}$ contains only finitely many edges, as we have shown above that the cluster radius is upper bounded by $\\sum_{j=0}^{i} d_j$.\n    \n    Now, for $\\ell \\in [\\Delta]$, we choose an edge $e_\\ell$ uniformly at random from the set $E_{v,\\ell}$. Let $u_\\ell \\neq v$ be the unique vertex in $H_i$ such that one endpoint of $e_\\ell$ is contained in $C(u_\\ell)$. We denote with $P_{v,\\ell}$ the set of vertices that are contained in the unique path between $v$ and $u_\\ell$. Finally, we set $U_i = \\bigcup_{v \\in H_i, \\ell \\in [\\Delta]} P_{v,\\ell}$.\n    \\item It remains to describe how to compute the partial orientation $O_i$. All edges that are oriented in $O_{i-1}$ will be oriented in the same direction in $O_i$. Additionally, we orient for each vertex $u$ that got settled in the $i$-th round, i.e., $u \\in S_i \\cap U_{i-1}$, exactly one incident edge away from $u$. Let $w$ be the closest vertex to $u$ in $T[U_{i-1}]$ that is contained in $U_i$, with ties being broken arbitrarily. We note that it follows from the discussions above that the distance between $u$ and $w$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. We now orient the edge incident to $u$ that is on the unique path between $u$ and $w$ outwards. We note that this orientation is well-defined in the sense that we only orient edges that were not oriented before and that we don't have an edge such that both endpoints of that edge want to orient the edge outwards.\n\\end{enumerate}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig\/one_ended_construction2.pdf}\n    \\caption{The picture illustrates the situation during some round $i$ of the construction. The vertices on all paths are contained in $U_{i-1}$ and the vertices corresponding to the big black dots are contained in $H_{i-1}$. The vertices $v$, $u_1$, $u_2$ and $u_3$ are the only vertices in the picture that are contained in $H_i$. All the vertices on the red paths remain unsettled during round $i$ while all the vertices on the black paths become settled. Each vertex that becomes settled during the $i$-th round orients one incident edge outwards, illustrated by the green edges. Each green edge is oriented towards the closest vertex that is still unsettled after the $i$-th round. We note that some of the edges in the picture are oriented away from $v$.}\n    \\label{fig:constr2}\n\\end{figure}\n\n\nWe now prove by induction that our procedure satisfies all the invariants.\nPrior to the first round, all invariants are trivially satisfied. Hence, it remains to show that the invariants hold after the $i$-th round, given that the invariants hold after the $(i-1)$-th round. \n\n\\begin{enumerate}\n\n    \\item Let $v \\in S_{i-1}$ be arbitrary. As $O_i$ is an extension of $O_{i-1}$, it follows from the definition of $T_{v,i-1}$ and $T_{v,i}$ that $T_{v,i-1} \\subseteq T_{v,i}$. Now assume that $T_{v,i} \\not \\subseteq T_{v,i-1}$. This would imply the existence of an edge $\\{u,w\\}$ such that $u \\in T_{v,i-1}$ and the edge was oriented during the $i$-th round from $w$ to $u$. As $u \\in T_{v,i-1}$, Invariant $(2)$ from the $(i-1)$-th round implies $u \\in S_{i-1}$. This is a contradiction as during the $i$-th round only edges with both endpoints in $U_{i-1}$ can be oriented. Therefore it holds that $T_{v,i} = T_{v,i-1}$, as desired.\n\n    \\item Let $v \\in S_i$ be arbitrary. If it also holds that $v \\in S_{i-1}$, then the invariant from round $i-1$ implies that $T_{v,i-1} = T_{v,i}$ contains finitely many vertices and $T_{v,i-1} \\subseteq S_{i-1} \\subseteq S_i$. Thus, it suffices to consider the case that $v \\in S_i \\cap U_{i-1}$. Note that $T_{v,i} \\cap U_i = \\emptyset$. Otherwise there would exist an edge that is oriented away from a vertex in $U_i$ according to $O_i$, but this cannot happen according to the algorithm description. Hence, $T_{v,i} \\subseteq S_i$. Now, for the sake of contradiction, assume that $T_{v,i}$ contains infinitely many vertices. From the definition of $T_{v,i}$, this implies that there exists a sequence of vertices $(v_k)_{k \\geq 1}$ with $v = v_1$ such that for each $k \\geq 1$, $\\{v_{k+1},v_k\\}$ is an edge in $T$ that is oriented from $v_{k+1}$ to $v_k$ according to $O_i$. For each $k \\geq 1$ we have $v_k \\in S_i$. We furthermore know that $v_k \\in U_{i-1}$, as otherwise $T_{v_k,i} = T_{v_k,i-1}$ would contain infinitely many vertices, a contradiction. From the way we orient the edges, a simple induction proof implies that the closest vertex of $v_k$ in the graph $T[U_{i-1}]$ that is contained in $U_i$ has a distance of at least $k$. However, we previously discussed that the distance between $v_k$ and the closest vertex in $U_i$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. This is a contradiction. Hence, $T_{v,i}$ contains finitely many vertices.\n    \n    \\item It follows directly from the description of how to compute $U_i$ and $H_i$ that the minimum degree of the graph $T[U_i]$ is at least $2$ and that each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n    \n    \\item Let $u \\in U_i$ be arbitrary. We need to show that $u$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$. Let $v$ be the vertex with $u \\in C(v)$. We know that there is no vertex in $H_i$ that is closer to $u$ in $T[U_{i-1}]$ than $v$. Hence, the distance between $u$ and $v$ is at most $\\sum_{j=0}^i d_j$ in the graph $T[U_{i-1}]$. Moreover, from the description of how we compute $U_i$, it follows that all the vertices on the unique path between $u$ and $v$ are contained in $U_i$. Hence, each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$, as desired.\n    \\end{enumerate}\n\nWe derive an upper bound on the coding radius of the algorithm in three steps. First, for each $i \\in \\mathbb{N}$ and $\\varepsilon > 0$, we derive an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. Given this upper bound on the coding radius, we can use the sequential composition lemma (\\cref{lem:sequential_composition}) and a simple induction proof to give an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, this time without providing any additional input. \n\nSecond, we analyze after how many rounds a given vertex is settled with probability at least $1- \\varepsilon$ for a given $\\varepsilon > 0$.  \n\nFinally, we combine these two upper bounds to prove  \\cref{thm:one_ended_forest}.\n \n \n\\begin{lemma}\n\\label{lem:tail1}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $g_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. It holds that $g_i(\\varepsilon) = O(d^2_i \\log (\\Delta\/\\varepsilon))$.\n\\end{lemma}\n\n\\begin{proof}\n When running Ghaffari's uniform MIS algorithm on some graph $G'$ with maximum degree $\\Delta'$, each vertex knows with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not after $O(\\log (\\Delta') + \\log (1\/\\varepsilon))$ communication rounds in $G'$ (\\cite{ghaffari2016MIS}, Theorem 1.1). For the MIS computed during the $i$-th round, $\\Delta' =\\Delta^{O(d_i)}$ and each communication round in $G'$ can be simulated with $O(d_i)$ communication rounds in the tree $T$. Hence, to know for a given vertex with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not, it suffices consider the $O(d_i^2\\log(\\Delta) + d_i\\log(1\/\\varepsilon))$-hop neighborhood around that vertex. Now, let $u$ be an arbitrary vertex. In order to compute in which of the sets $S_i, U_i$ and $H_i$ $u$ is contained in and for each incident edge of $u$ whether and how it is oriented in $O_i$, we not only need to know whether $u$ is in the MIS or not. However, it suffices if we know for all the vertices in the $O(d_i)$-hop neighborhood of $u$ whether they are contained in the MIS or not (on top of knowing for each vertex in the neighborhood in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$). Hence, by a simple union bound over the at most $\\Delta^{O(d_i)}$ vertices in the $O(d_i)$-hop neighborhood around $u$, we obtain $g_i(\\varepsilon) = O(d_i) + O(d_i^2\\log(\\Delta) + d_i\\log(\\Delta^{O(d_i)}\/\\varepsilon)) = O(d_i^2 \\log(\\Delta\/\\varepsilon))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:tail2}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $h_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$. Then, there exists a constant $c$ independent of $\\Delta$ such that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$.\n\\end{lemma}\n\n\\begin{proof}\nBy prove the statement by induction on $i$. For a large enough constant $c$, it holds that $h_1(\\varepsilon) \\leq 2^{2^{3}} \\cdot(c\\log(\\Delta)) \\log(\\Delta\/\\varepsilon)$. Now, consider some arbitrary $i$ and assume that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$ for some large enough constant $c$. We show that this implies $h_{i+1}(\\varepsilon) \\leq  2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon)$.\nBy the sequential composition lemma (\\cref{lem:sequential_composition}) and assuming that $c$ is large enough, we have\n\n\\begin{align*}\n  h_{i+1}(\\varepsilon) &\\leq h_{i}((\\varepsilon\/2)\/\\Delta^{g_{i+1}(\\varepsilon\/2) + 1}) + g_{i+1}(\\varepsilon\/2)  \\\\\n                &\\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) + g_{i+1}(\\varepsilon\/2)  \\\\\n                &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) \\\\\n                &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}\/(\\varepsilon\/2)) \\\\\n                &\\leq 4 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}) \\\\\n                &\\leq (4\/10) \\cdot 2^{2^{i+2}} \\cdot 2^{2^{i+2}}  (c \\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon) \\\\\n                &\\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon),\n\\end{align*}\n\nas desired.\n\n\\end{proof}\n\n\n \n \n\n\\begin{lemma}\\label{lm:tail0}\nLet $u$ be an arbitrary vertex. For each $\\varepsilon \\in (0,0.01]$, let $f(\\varepsilon)$ denote the smallest $i \\in \\mathbb{N}$ such that $u$ is settled after the $i$-th round with probability at least $1 - \\varepsilon$. There exists a fixed $c \\in \\mathbb{R}$ independent of $\\Delta$ such that $f(\\varepsilon) \\leq \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $i \\in \\mathbb{N}$ be arbitrary. We show that a given vertex is settled after the $i$-th round with probability at least $1 - O(1\/\\Delta^{\\Omega(d_i\/d_{i-1})})$. For the sake of analysis, we run the algorithm on a finite $\\Delta$-regular high-girth graph instead of an infinite $\\Delta$-regular tree. For now, we additionally assume that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. That is, we assume that for each vertex the coding radius to compute all its local information after the $i$-th round is much smaller than the girth of the graph.  \n\nWith this assumption, we give a deterministic upper bound on the fraction of vertices that are not settled after the $i$-th round. On the one hand, the number of vertices that are not settled after the $i$-th round can be upper bounded by $|H_i| \\cdot \\Delta \\cdot O(d_i)$. On the other hand, we will show that the fraction of vertices that are contained in $H_i$ is upper bounded by $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. We do this by showing that $C(v)$ contains $\\Delta^{\\Omega(d_i\/d_{i-1})}$ many vertices for a given $v \\in H_i$. Combining these two bounds directly implies that the fraction of unsettled vertices is smaller than \n\n\\[\\frac{\\Delta \\cdot O(d_i)}{\\Delta^{\\Omega(d_i\/d_{i-1})}} = 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}.\\]\n\n\nLet $v \\in H_i$ be arbitrary. We show that $|C(v)| = \\Delta^{\\Omega(d_i\/d_{i-1})}$. Using Invariants (3) and (4) together with a simple induction argument, one can show that there are at least $(\\Delta-1)^{\\lfloor  D \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor}$ vertices contained in $H_{i-1}$ and whose distance to $v$ is at most $D$ in the graph induced by the vertices in $U_{i-1}$. Furthermore, from the way we defined the clustering $C(v)$ and the fact that two vertices in $H_i$ have a distance of at least $d_i$ in the graph $T[U_{i-1}]$, it follows that all vertices in $U_{i-1}$ having a distance of at most $d_i\/2 -1$ to $v$  are contained in the cluster $C(v)$. Hence, the total number of vertices in $C(v)$ is at least $(\\Delta-1)^{\\lfloor  (d_i\/2 - 1) \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor} = \\Delta^{\\Omega(d_i\/d_{i-1})}$, as promised.\n\nNow we remove the assumption that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. If a vertex does realize that we don't run the algorithm on an infinite $\\Delta$-regular tree, then we consider the vertex as being unsettled after the $i$-th round. By considering graphs with increasing girth, we can make the expected fraction of vertices that realize that we are not on an infinite $\\Delta$-regular tree arbitrarily small.  Combining this observation with the previous discussion, this implies that the expected fraction of vertices that are not settled after the $i$-th round is at most $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. By symmetry, each vertex has the same probability of being settled after the $i$-th round. Hence, the probability that a given vertex is settled after the $i$-th round when run on a graph with sufficiently large girth is $1 - 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$, and the same holds for each vertex when we run the algorithm on an infinite $\\Delta$-regular tree. Thus, there exists a constant $c$ such that the probability that a given vertex is unsettled after the $i$-th round is at most $1\/\\Delta^{c \\cdot d_i\/d_{i-1}} = 1\/\\Delta^{c \\cdot 2^{2^{i-1}}}$. Setting $i = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$ finishes the proof.\n\\end{proof}\n\n\n\n\n\nWe are now finally ready to finish the proof of \\cref{thm:one_ended_forest}. Let $u$ be an arbitrary vertex and $\\varepsilon \\in (0,0.01]$. We need to compute an upper bound on the coding radius that is necessary for $u$ to know all the vertices in $\\mathcal{T}^{\\leftarrow}(u)$ with probability at least $1-\\varepsilon$. To compute such an upper bound for the  required coding radius it suffices to find an $i^*$ and an $R^*$ such that the following holds. First, $u$ is settled after $i^*$ rounds with probability at least $1 - \\varepsilon\/2$. Second, $u$ knows for each edge in its $O(d_{i^*})$-hop neighborhood whether it is oriented in the partial orientation $O_{i^*}$, and if yes, in which direction, by only considering its $R^*$-hop neighborhood with probability at least $1 - \\varepsilon\/2$.  By a union bound, both of these events occur with probability at least $1-\\varepsilon$. Moreover, if both events occur $u$ knows all the vertices in the set $\\mathcal{T}^{\\leftarrow}(u)$. The reason is as follows. If $u$ is settled after round $i^*$, then it follows from the previous analysis that only vertices in the $O(d_{i^*})$-hop neighborhood of $u$ can be contained in $\\mathcal{T}^{\\leftarrow}(u)$. Moreover, for each vertex in its $O(d_{i^*})$-hop neighborhood, $u$ can determine if the vertex is contained in $\\mathcal{T}^{\\leftarrow}(u)$ if it knows all the edge orientations on the unique path between itself and that vertex after the $i^*$-th round. Hence, it remains to find concrete values for $i^*$ and $R^*$. According to \\cref{lm:tail0}, we can choose $i^* = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta (2\/\\varepsilon)\\rceil$ for some large enough constant $c$. Moreover, it follows from a union bound that all vertices in the $O(d_{i^*})$-hop neighborhood around $u$ know with probability at least $1-\\varepsilon\/2$ the orientation of all its incident edges according to $O_{i^*}$ by only considering their $h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})})$-hop neighborhood. Hence, we can set \n\n\\begin{align*}\nR^* &= O(d_{i^*}) + h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})}) \\\\\n    &\\leq O(d_{i^*}) + 2^{2^{i^* + 2}} \\cdot (c \\log(\\Delta))^{i^*}\\log(\\Delta^{O(d_{i^*})}\/\\varepsilon) \\\\\n    &= O(2^{2^{i^*}}\\cdot 2^{2^{i^* + 2}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= O(2^{2^{i^* + 3}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= O(2^{2^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 5}} (c \\log(\\Delta))^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 2}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= \\log(1\/\\varepsilon)^{32} \\cdot \\log(\\Delta\/\\varepsilon) \\cdot \\log\\log(1\/\\varepsilon)^{O(\\log\\log(\\Delta))}.\n\\end{align*}\n\nAs $\\Delta = O(1)$, it therefore holds that\n\n\\[P(R(v)) = \\operatorname{poly}(\\log(1\/\\varepsilon)) \\geq 1 - \\varepsilon, \\]\n\nas desired.\n\n\n\n\n\n\n\\section{$\\mathsf{BAIRE} = \\mathsf{LOCAL}(O(\\log n))$}\n\\label{sec:baire}\n\nIn this section, we show that on $\\Delta$-regular trees the classes $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$ are the same.\nAt first glance, this result looks rather counter-intuitive.\nThis is because in finite $\\Delta$-regular trees every vertex can see a leaf of distance $O(\\log(n))$, while there are no leaves at all in an infinite $\\Delta$-regular tree.\nHowever, there is an intuitive reasons why these classes are the same: in both setups there is a technique to decompose an input graph into a hierarchy of subsets. Furthermore, the existence of a solution that is defined inductively with respect to these decompositions can be characterized by the same combinatorial condition of Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nWe start with a high-level overview of the decomposition techniques used in both contexts.\n\n\n\\paragraph{{\\sf Rake}\\ and {\\sf Compress}}\nThe hierarchical decomposition in the context of distributed computing is based on a variant of a decomposition algorithm of\nMiller and Reif~\\cite{MillerR89}.\nTheir original decomposition algorithm works as follows.\nStart with a tree $T$, and repeatedly apply the following two operations alternately: {\\sf Rake}\\ (remove all degree-1 vertices) and {\\sf Compress}\\ (remove all degree-2 vertices).\nThen $O(\\log n)$ iterations suffice to remove all vertices in $T$~\\cite{MillerR89}. To view it another way, this produces a decomposition of the vertex set $V$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$, where $\\VR{i}$ is the set of vertices removed during the $i$-th {\\sf Rake}\\ operation and $\\VC{i}$ is the set of vertices removed during the $i$-th {\\sf Compress}\\ operation. We will use a variant~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} of this decomposition in the proof of \\cref{thm:ellfull_to_logn}.\n\n\n\n\nVariants of this decomposition turned out to be useful in designing  $\\mathsf{LOCAL}$ algorithms~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,ChangHLPU20,chang2020n1k_speedups}. \nIn our context, we assume that the given LCL satisfies a certain combinatorial condition and then find a solution inductively, in the reversed order of the construction of the decomposition.\nNamely, in the {\\sf Rake}\\ step we want to be able to existentially extend the inductive partial solution to all relative degree $1$-vertices (each $v \\in \\VR{i}$ has degree at most 1 in the subgraph induced by $\\VR{i}  \\cup \\cdots \\cup \\VR{L}$) and in the {\\sf Compress}\\ step we want to extend the inductive partial solution to paths with endpoints labeled from the induction (the vertices in $\\VC{i}$ form degree-2 paths in the subgraph induced by $\\VC{i} \\cup \\cdots \\cup \\VR{L}$).\n\n\n\\paragraph{$\\TOAST$}\nFinding a hierarchical decomposition in the context of descriptive combinatorics is tightly connected with the notion of \\emph{Borel hyperfiniteness}.\nUnderstanding what Borel graphs are Borel hyperfinite is a major theme in descriptive set theory \\cite{doughertyjacksonkechris,gaojackson,conley2020borel}.\nIt is known that grids, and generally polynomial growth graphs are hyperfinite, while, e.g., acyclic graphs are not in general hyperfinite \\cite{jackson2002countable}.\nA strengthening of hyperfiniteness that is of interest to us is called a \\emph{toast} \\cite{gao2015forcing,conleymillerbound}.\nA $q$-toast, where $q\\in \\mathbb{N}$, of a graph $G$ is a collection $\\mathcal{D}$ of fintie subsets of $G$ with the property that (i) every pair of vertices is covered by an element of $\\mathcal{D}$ and (ii) the boundaries of every $D\\not=E\\in \\mathcal{D}$ are at least $q$ apart.\nThe idea to use a toast structure to solve LCLs appears in \\cite{conleymillerbound} and has many applications since then \\cite{gao2015forcing,marksunger}.\nThis approach has been formalized in \\cite{grebik_rozhon2021toasts_and_tails}, where the authors introduce $\\TOAST$ algorithms.\nRoughly speaking, an LCL $\\Pi$ admits a $\\TOAST$ algorithm if there is $q\\in \\mathbb{N}$ and a partial extending function (the function is given a finite subset of a tree that is partially colored and outputs an extension of this coloring on the whole finite subset) that has the property that whenever it is applied inductively to a $q$-toast, then it produces a $\\Pi$-coloring.\nAn advantage of this approach is that once we know that a given Borel graph admits, e.g., a Borel toast structure and a given LCL $\\Pi$ admits a $\\TOAST$ algorithm, then we may conclude that $\\Pi$ is in the class $\\mathsf{BOREL}$.\nSimilarly for $\\mathsf{MEASURE}$, $\\mathsf{BAIRE}$ or $\\mathsf{OLOCAL}$, we refer the reader to \\cite{grebik_rozhon2021toasts_and_tails} for more details and results concerning grids.\n\nIn the case of trees there is no way of constructing a Borel toast in general, however, it is a result of Hjorth and Kechris \\cite{hjorth1996borel} that every Borel graph is hyperfinite on a comeager set for every compatible Polish topology.\nA direct consequence of \\cite[Lemma~3.1]{marks2016baire} together with a standard construction of toast via Voronoi cells gives the following strengthening to toast.\nWe include a sketch of the proof for completeness.\n\n\\begin{restatable}{proposition}{BaireToast}\n\\label{pr:BaireToast}\nLet $\\mathcal{G}$ be a Borel graph on a Polish space $(X,\\tau)$ with degree bounded by $\\Delta\\in \\mathbb{N}$.\nThen for every $q>0$ there is a Borel $\\mathcal{G}$-invariant $\\tau$-comeager set $C$ on which $\\mathcal{G}$ admits a Borel $q$-toast.\n\\end{restatable}\n\\begin{proof}[Proof sketch]\nLet $\\{A_n\\}_{n\\in \\mathbb{N}}$ be a sequence of MIS with parameter $f(n)$ as in \\cite[Lemma~3.1]{marks2016baire} for a sufficiently fast growing function $f(n)$, e.g., $f(n)=(2q)^{n^2}$.\nThen $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ is a Borel $\\tau$-comeager set that is $\\mathcal{G}$-invariant.\nWe produce a toast structure in a standard way, e.g., see~\\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\nLet $B_n(x)$ denote the ball of radius $f(n)\/3$ and $R_n(x)$ the ball of radius $f(n)\/4$ around $x\\in A_n$.\nIteratively, define cells $\\mathcal{D}_n=\\{C_{n}(x)\\}_{x\\in A_n}$ as follows.\nSet $C_1(x)=R_1(x)$.\nSuppose that $\\mathcal{D}_n$ has been defined and set\n\\begin{itemize}\n\t\\item $H^{n+1}(x,n+1):=R_{n+1}(x)$ for every $x\\in A_{n+1}$,\n\t\\item if $1\\le i\\le n$ and $\\left\\{H^{n+1}(x,i+1)\\right\\}_{x\\in A_{k+1}}$ has been defined, then we put\n\t$$H^{n+1}(x,i)=\\bigcup \\left\\{B_i(y):H^{n+1}(x,i+1)\\cap B_i(y)\\not=\\emptyset\\right\\}$$\n\tfor every $x\\in A_{n+1}$,\n\t\\item set $C_{n+1}(x):=H^{n+1}(x,1)$ for every $x\\in A_{n+1}$, this defines $\\mathcal{D}_{n+1}$.\n\\end{itemize}\nThe fact that $\\mathcal{D}=\\bigcup_{n\\in\\mathbb{N}} \\mathcal{D}_n$ is a $q$-toast on $C$ follows from the fact that $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ together with the fact that the boundaries are $q$ separated which can be shown as in \\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\\end{proof}\n\n\n\nTherefore to understand LCLs in the class $\\mathsf{BAIRE}$ we need understand what LCLs on trees admit $\\TOAST$ algorithm.\nIt turns out that these notions are equivalent, again by using the combinatorial characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} that we now discuss.\n\n\n\\paragraph{Combinatorial Condition -- $\\ell$-full set}\nIn both decompositions, described above, we need to extend a partial coloring along paths that have their endpoints colored from the inductive step.\nThe precise formulation of the combinatorial condition that captures this demand was extracted by Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nHe proved that it characterizes the class $\\mathsf{BAIRE}$ for Cayley graphs of virtually free groups.\nNote that this class contains, e.g., $\\Delta$-regular trees with a proper edge $\\Delta$-coloring.\n\n\\begin{definition}[Combinatorial condition -- an $\\ell$-full set]\n\\label{def:ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL and $\\ell\\geq 2$.\nA set $\\mathcal{V}' \\subseteq \\mathcal{V}$ is \\emph{$\\ell$-full} whenever the following is satisfied.\nTake a path with at least $\\ell$ vertices, and add half-edges to it so that each vertex has degree $\\Delta$.\nTake any $c_1, c_2 \\in \\mathcal{V}'$ and label arbitrarily the half-edges around the endpoints with $c_1$ and $c_2$, respectively.\nThen there is a way to label the half-edges around the remaining $\\ell-2$ vertices with configurations from $\\mathcal{V}'$ such that all the $\\ell-1$ edges on the path have valid edge configuration on them. \n\\end{definition}\n\nNow we are ready to formulate the result that combines Bernshteyn's result \\cite{Bernshteyn_work_in_progress} (equivalence between (1.) and (2.), and the moreover part) with the main results of this section.\nThis also shows the remaining implications in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}\\label{thm:MainBaireLog}\nLet $\\Pi$ be an LCL on regular trees. \nThen the following are equivalent:\n\\begin{enumerate}\n    \\item $\\Pi\\in \\mathsf{BAIRE}$,\n    \\item $\\Pi$ admits a $\\TOAST$ algorithm,\n    \\item $\\Pi$ admits an $\\ell$-full set,\n    \\item $\\Pi\\in \\mathsf{LOCAL}(O(\\log(n)))$.\n\\end{enumerate}\nMoreover, any of the equivalent conditions is necessary for $\\Pi\\in \\mathsf{fiid}$.\n\\end{theorem}\n\nNext we discuss the proof of \\cref{thm:MainBaireLog}.\nWe refer the reader to Bernshteyn's paper \\cite{Bernshteyn_work_in_progress} for full proofs in the case of $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$, here we only sketch the argument for completeness.\nWe also note that instead of using the toast construction, he used a path decomposition of acyclic graphs of Conley, Marks and Unger \\cite{conley2020measurable}.\n\n\n\\subsection{Sufficiency}\n\nWe start by showing that the combinatorial condition is sufficient for $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$.\nNamely, it follows from the next results together with \\cref{pr:BaireToast} that (2.) implies all the other conditions in \\cref{thm:MainBaireLog}.\nAs discussed above the main idea is to color inductively along the decompositions.\n\n\\begin{proposition}\\label{pr:toastable}\nLet $\\Pi=(\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL that admits $\\ell$-full set $\\mathcal{V}'\\subseteq \\mathcal{V}$ for some $\\ell>0$.\nThen $\\Pi$ admits a $\\TOAST$ algorithm that produces a $\\Pi$-coloring for every $(2\\ell+2)$-toast $\\mathcal{D}$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\nOur aim is to build a partial extending function.\nSet $q:=2\\ell+2$.\nLet $E$ be a piece in a $q$-toast $\\mathcal{D}$ and suppose that $D_1,\\dots,D_k\\in \\mathcal{D}$ are subsets of $E$ such that the boundaries are separated.\nSuppose, moreover, that we have defined inductively a coloring of half-edges of vertices in $D=\\bigcup D_i$ using only vertex configurations from $\\mathcal{V}'$ such that every edge configuration $\\mathcal{E}$ is satisfied for every edge in $D$.\n\nWe handle each connected component of $E\\setminus D$ separately.\nLet $A$ be one of them.\nLet $u\\in A$ be a boundary vertex of $E$.\nSuch an vertex exists since every vertex in $E$ has degree $\\Delta$.\nThe distance of $u$ and any $D_i$ is at least $2\\ell+2$ for every $i\\in [k]$.\nWe orient all the edges from $A$ towards $u$.\nMoreover if $v_i\\in A$ is a boundary vertex of some $D_i$ we assign to $v_i$ a path $V_i$ of length $\\ell$ towards $u$.\nNote that $V_i$ and $V_j$ have distance at least $1$, in particular, are disjoint for $i\\not=j\\in [k]$ .\nNow,  until you encounter some path $V_i$, color any in manner half-edges of vertices in $A$ inductively starting at $u$ in such a way that edge configurations $\\mathcal{E}$ are satisfied on every edge and only vertex configurations from $\\mathcal{V}'$ are used.\nUse the definition of $\\ell$-full set to find a coloring of any such $V_i$ and continue in a similar manner until the whole $A$ is colored.\n\\end{proof}\n\n\n\n\n\\begin{proposition}[$\\ell$-full $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log n))$]\\label{thm:ellfull_to_logn}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL with an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$. Then $\\Pi$ can be solved in $O(\\log n)$ rounds in  $\\mathsf{LOCAL}$.\n\\end{proposition}\n\\begin{proof}\nThe proof  uses a variant of the rake-and-compress decomposition considered  in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\paragraph{The Decomposition}\nThe decomposition is parameterized an integer $\\ell' \\geq 1$, and it decomposes the vertices of $T$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$. We write $\\GC{i}$ to denote the subtree induced by the vertices  $\\left(\\bigcup_{j=i+1}^{L} \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$. Similarly, $\\GR{i}$ is the subtree induced by the vertices  \n$\\left(\\bigcup_{j=i}^L \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$.\nThe sets $\\VR{i}$ and $\\VC{i}$ are required to satisfy the following requirements. \n\\begin{itemize}\n    \\item Each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\n    \\item Each $v \\in \\VC{i}$ has degree exactly two in the graph $\\GC{i}$. Moreover, the $\\VC{i}$-vertices in $\\GC{i}$ form paths with $s$ vertices, with $\\ell' \\leq s \\leq 2 \\ell'$.\n\\end{itemize}\nFor any given constant $\\ell' \\geq 1$, it was shown in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} that such a decomposition of a tree $T$ can be computed in $O(\\log n)$ rounds. See \\cref{fig:rake_and_compress} for  an example of such a decomposition with $\\ell' = 4$.\n\n\\paragraph{The Algorithm}\nGiven such a decomposition with $\\ell' = \\max\\{1, \\ell - 2\\}$, $\\Pi$ can be solved in $O(\\log n)$ rounds by labeling the vertices in this order: $\\VR{L}$, $\\VC{L-1}$, $\\VR{L-1}$, $\\ldots$, $\\VR{1}$, as follows. The algorithm only uses the vertex configurations in the $\\ell$-full set $\\mathcal{V}'$.\n\n\\paragraph{Labeling $\\VR{i}$}\nSuppose  all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VC{i}$ have been labeled using $\\mathcal{V}'$.  Recall that each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\nIf $v \\in \\VR{i}$ has no neighbor in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$, then we can label the half edges surrounding $v$ by any $c \\in \\mathcal{V}'$.\nOtherwise, $v \\in \\VR{i}$ has exactly one neighbor $u$ in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$. Suppose the vertex configuration of $u$ is $c$, where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a} \\in c$.  A simple observation from the definition of $\\ell$-full sets is that for any $c \\in \\mathcal{V}'$ and any $\\mathtt{a} \\in c$, there exist $c' \\in \\mathcal{V}'$ and $\\mathtt{a}' \\in c'$ in such a way that $\\{\\mathtt{a}, \\mathtt{a}'\\} \\in \\mathcal{E}$.\nHence we can label the half edges surrounding $v$ by  $c' \\in \\mathcal{V}'$  where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a}' \\in c'$. \n\n\n\\paragraph{Labeling $\\VC{i}$} Suppose  all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VR{i+1}$ have been labeled using $\\mathcal{V}'$. Recall that the   $\\VC{i}$-vertices in $\\GC{i}$ form degree-2 paths $P=(v_1, v_2, \\ldots, v_s)$, with $\\ell' \\leq s \\leq 2 \\ell'$.\nLet $P'= (x, v_1, v_2, \\ldots, v_s, y)$ be the path resulting from appending to $P$ the neighbors of the two end-points of $P$ in $\\GC{i}$. The two vertices  $x$ and $y$ are in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VR{i+1}$, so they have been assigned half-edge labels using $\\mathcal{V}'$. Since  $P'$ contains at least $\\ell'+2 \\geq \\ell$ vertices, the definition of $\\ell$-full sets  ensures that we can label $v_1, v_2, \\ldots, v_s$ using vertex configurations in $\\mathcal{V}'$ in such a way that the half-edge labels on $\\{x, v_1\\}, \\{v_1, v_2\\}, \\ldots, \\{v_s, y\\}$ are all in $\\mathcal{E}$.\n\\end{proof}\n\n\n\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .8\\textwidth]{fig\/y1.pdf}\n    \\caption{The variant of the rake-and-compress  decomposition used in the proof of \\cref{thm:ellfull_to_logn}. }\n    \\label{fig:rake_and_compress}\n\\end{figure}\n\n\n\n\\subsection{Necessity}\n\nWe start by sketching that (2.) in \\cref{thm:MainBaireLog} is necessary for $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$.\n\n\n\\begin{theorem}[Bernshteyn \\cite{Bernshteyn_work_in_progress}]\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL and suppose that $\\Pi\\in \\mathsf{BAIRE}$ or $\\Pi\\in \\mathsf{fiid}$.\nThen $\\Pi$ admits an $\\ell$-full set $\\mathcal{V}'\\subseteq\\mathcal{V}$ for some $\\ell>0$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\n\n\nWe start with $\\mathsf{BAIRE}$.\nSuppose that every Borel acyclic $\\Delta$-regular graph admits a Borel solution on a $\\tau$-comeager set for every compatible Polish topology $\\tau$.\nIn particular, this holds for the Borel graph induced by the standard generators of the free product of $\\Delta$-copies of $\\mathbb{Z}_2$ on the free part of the shift action on the alphabet $\\{0,1\\}$ endowed with the product topology.\nLet $F$ be such a solution.\nWrite $\\mathcal{V}'\\subseteq \\mathcal{V}$ for the configurations of half-edge labels around vertices that $F$ outputs on a non-meager set.\nLet $C$ be a comeager set on which $F$ is continuous. Then, every element of $\\mathcal{V}'$ is encoded by some finite window in the shift on $C$, that is, for each element there are a $k \\in \\mathbb{N}$ and function $s:B(1,k) \\to \\{0,1\\}$ such that $F$ is constant on the set $N_s \\cap C$ (where $N_s$ is the basic open neighbourhood determined by $s$, and $B(1,k)$ is the $k$-neighbourhood of the identity in the Cayley graph of the group).\nSince $\\mathcal{V}'$ is finite, we can take $t>0$ to be the maximum of such $k$'s.\nIt follows by standard arguments that $\\mathcal{V}'$ is $\\ell$-full for $\\ell>2t+1$.\n\nA similar argument works for the $\\mathsf{fiid}$, however, for the sake of brevity, we sketch a shorter argument that uses the fact that there must be a correlation decay for factors of iid's. Let $\\Pi\\in \\mathsf{fiid}$.\nThat is, there is an $\\operatorname{Aut}(T)$-equivariant measurable function from iid's on $T$ (without colored edges this time) into the space of $\\Pi$-colorings.\nLet $\\mathcal{V}'$ be the set of half-edges configurations around vertices that have non-zero probability to appear.\nLet $u,v\\in T$ be vertices of distance $k_0\\in \\mathbb{N}$.\nBy \\cite{BGHV} the correlation between the configurations around $u$ and $v$ tends to $0$ as $k_0\\to \\infty$.\nThis means that if the distance is big enough, then all possible pairs of $\\mathcal{V}'$ configurations need to appear.\n\\end{proof}\n\nTo finish the proof of \\cref{thm:MainBaireLog} we need to demonstrate the following theorem.  Note that  $\\mathsf{LOCAL}(n^{o(1)}) =  \\mathsf{LOCAL}(O(\\log n))$ according to the $\\omega(\\log n)$ -- $n^{o(1)}$ complexity gap~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\begin{restatable}{theorem}{logcomb}\\label{thm:logn_to_ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL solvable in  $\\mathsf{LOCAL}(n^{o(1)})$ rounds.\nThen there exists an $\\ell$-full set  $\\mathcal{V}' \\subseteq \\mathcal{V}$ for some $\\ell \\geq 2$.\n\\end{restatable}\n\n\nThe rest of the section is devoted to the proof of \\cref{thm:logn_to_ellfull}. We start with the high-level idea of the proof.\nA natural attempt for showing $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full is to simply take any $\\mathsf{LOCAL}(n^{o(1)})$ algorithm $\\mathcal{A}$ solving $\\Pi$, and then take $\\mathcal{V}'$ to be all vertex configurations that can possibly occur in an output of $\\mathcal{A}$. It is not hard to see that this approach does not work in general, because the algorithm might use a special strategy to label vertices with degree smaller than $\\Delta$. Specifically, there might be some vertex configuration $c$ used by $\\mathcal{A}$ so that some $\\mathtt{a} \\in c$ will only be used to label \\emph{virtual} half edges. It will be problematic to include $c$ in $\\mathcal{V}'$.\n\nTo cope with this issue, we do not deal with general bounded-degree trees. Instead, we construct recursively a sequence $(W^\\ast_1, W^\\ast_2, \\ldots, W^\\ast_L)$ of sets of rooted, layered, and \\emph{partially labeled} tree in a special manner. A tree $T$ is included in $W^\\ast_i$ if it can be constructed by gluing a multiset of rooted trees in  $W^\\ast_{i-1}$ and a new root vertex $r$ in a certain fixed manner. A vertex is said to be in layer $i$ if it is introduced during the $i$-th step of the construction, i.e., it is introduced as the root $r$ during the construction of  $W^\\ast_i$ from $W^\\ast_{i-1}$. All remaining vertices are said to be in layer 0.\n\n\nWe  show that each $T \\in W^\\ast_L$ admits a correct labeling that extends the given partial labeling, as these partial labelings are computed by a simulation of $\\mathcal{A}$.\nMoreover, in these correct labelings, the variety of possible configurations of half-edge labels around vertices in different non-zero layers is the same for each layer.\nThis includes vertices of non-zero layer whose half-edges are labeled by the given partial labeling.\nWe simply pick $\\mathcal{V}'$ to be the set of all configurations of half-edge labels around vertices that can appear in a non-zero layer in a correct labeling of a tree $T \\in W^\\ast_L$.\nOur construction ensures that each $c \\in \\mathcal{V}'$ appears as the labeling of some degree-$\\Delta$ vertex in some tree that we consider.\n\n\nThe proof that $\\mathcal{V}'$ is an $\\ell$-full set is based on finding paths  using vertices of non-zero layers connecting  two vertices with any two  vertex configurations in  $\\mathcal{V}'$ in different lengths. These paths exist because the way rooted trees in $W^\\ast_{i-1}$ are glued together in the construction of $W^\\ast_{i}$ is sufficiently flexible.\nThe reason that we need $\\mathcal{A}$ to have complexity $\\mathsf{LOCAL}(n^{o(1)})$ is that the construction of the trees can be parameterized by a number $w$ so that all the trees have size polynomial in $w$ and the vertices needed to be assigned labeling are at least distance $w$ apart from each other. Since the number of rounds of $\\mathcal{A}$ executed on trees of size $w^{O(1)}$ is much less than $w$, each labeling assignment can be calculated locally and independently. \nThe construction of the trees as well as the analysis are based on a machinery developed in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n Specifically, we will consider the equivalence relation $\\overset{\\star}{\\sim}$ defined in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} and prove some of its properties, including a pumping lemma for bipolar trees. The exact definition of  $\\overset{\\star}{\\sim}$  in this paper is  different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} because the mathematical formalism describing LCL problems in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\nAfter that, we will consider a procedure for gluing trees parameterized by a labeling function $f$ similar to the one used in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. \nWe will apply this procedure iteratively to generate a set of trees. We will show that the desired  $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ can be constructed by considering the set of all possible correct labeling of these trees. \n \n\n\\paragraph{The Equivalence Relation $\\overset{\\star}{\\sim}$} \nWe consider trees with a list of designated vertices $v_1, v_2, \\ldots, v_k$ called \\emph{poles}. A \\emph{rooted tree} is a tree with one pole $r$, and a \\emph{bipolar tree} is a tree with two poles $s$ and $t$. \nFor a tree $T$ with its poles $S=(v_1, v_2, \\ldots, v_k)$ with $\\deg(v_i) = d_i < \\Delta$, we denote by $h_{(T,S)}$ the function that maps each choice of the \\emph{virtual} half-edge labeling surrounding the poles of $T$ to $\\textsf{YES}$ or $\\textsf{NO}$, indicating whether such a partial labeling can be completed into a correct complete labeling of $T$. \nMore specifically, consider \n\\[X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots,  \\mathcal{I}_k),\\]\nwhere $\\mathcal{I}_i$ is a size-$(\\Delta - d_i)$ multiset of labels in $\\Sigma$, for each $1 \\leq i \\leq k$. Then $h_{(T,S)}(X) = \\textsf{YES}$ if there is a correct labeling of $T$ such that the $\\Delta - d_i$ virtual half-edge labels surrounding $v_i$ are labeled by $\\mathcal{I}_i$,  for each $1 \\leq i \\leq k$. This definition can be generalized to the case $T$ is already partially labeled in the sense that some of the half-edge labels have been fixed. In this case, $h_{(T,S)}(X) = \\textsf{NO}$ whenever $X$ is incompatible with the given partial labeling.\n\nLet $T$ be a tree with poles $S = (v_1, v_2, \\ldots, v_k)$ and let $T'$ be another tree with poles $S' = (v_1', v_2', \\ldots, v_k')$ such that $\\deg(v_i) = \\deg(v_i') = d_i$ for each $1 \\leq i \\leq k$.\nThen we write $T_1 \\overset{\\star}{\\sim} T_2$ if $h_{(T,S)} = h_{(T',S')}$.\n\nGiven an LCL problem $\\Pi$, it is clear that the number of equivalence classes of rooted trees and bipolar trees w.r.t.~$\\overset{\\star}{\\sim}$ is finite.\nFor a rooted tree $T$,  denote by $\\textsf{Class}_1(T)$ the equivalence class of $T$. For a bipolar tree $H$, denote by $\\texttt{type}(H)$ the equivalence class of $H$.\n\n\n\n\n\\paragraph{Subtree Replacement}\nThe following lemma provides a sufficient condition that the equivalence class of a tree $T$ is invariant of the equivalence class of its subtree $T'$.\nWe note that a real half edge in $T$ might become virtual in its subtree $T'$.\nConsider a vertex $v$ in $T'$ and its neighbor $u$ that is in $T$ but not in $T'$. Then the half edge $(v, \\{u,v\\})$  is real in $T$ and virtual in $T'$.\n\n\n\\begin{lemma}[Replacing subtrees]\\label{lem:replace}\nLet $T$ be a tree with poles $S$.\nLet $T'$ be a connected subtree of $T$ induced by $U \\subseteq V$, where $V$ is the set of vertices in $T$.\nWe identify a list of designated vertices $S' = (v_1', v_2', \\ldots, v_k')$ in $U$ satisfying the following two conditions to be the poles of $T'$.\n\\begin{itemize}\n    \\item $S \\cap U \\subseteq S'$.\n    \\item Each edge $e=\\{u,v\\}$ connecting $u \\in U$ and $ v \\in V \\setminus U$ must satisfy $u \\in S'$.\n\\end{itemize}\nLet $T''$ be another tree with poles $S'' = (v_1'', v_2'', \\ldots, v_k'')$ that is in the same equivalence class as $T'$.\nLet $T^\\ast$ be the result of replacing $T'$ by $T''$ in $T$ by identifying $v_i' = v_i''$ for each $1 \\leq i \\leq k$.\nThen $T^\\ast$ is in the same equivalence class as $T$. \n\\end{lemma}\n\\begin{proof}\nTo prove the lemma, by symmetry, it suffices to show that starting from any correct labeling $\\mathcal{L}$ of $T$, it is possible to find a correct labeling  $\\mathcal{L}^\\ast$ of $T^\\ast$ in such a way that the multiset of the\nvirtual half-edge labels surrounding each pole in $S$ remain the same.\n\nSuch a correct labeling $\\mathcal{L}^\\ast$  of $T^\\ast$ is constructed as follows. If $v \\in V \\setminus U$, then we simply adopt the given labeling $\\mathcal{L}$ of $v$ in $T$.\nNext, consider the vertices in $U$.\nSet $X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k)$ to be the one compatible with the labeling $\\mathcal{L}$ of $T$ restricted to the subtree $T'$ in the sense that $\\mathcal{I}_i$ is the multiset of the virtual half-edge labels surrounding the pole $v_i'$ of $T'$, for each $1 \\leq i \\leq k$. We must have $h_{T',S'}(X) = \\textsf{YES}$.\nSince $T'$ and $T''$ are in the same equivalence class, we have $h_{T'',S''}(X) = \\textsf{YES}$, and so we can find a correct labeling $\\mathcal{L}''$ of $T''$ that is also compatible with $X$. Combining this labeling $\\mathcal{L}''$ of the vertices $U$ in $T''$ with the  labeling  $\\mathcal{L}$ of the vertices  $V \\setminus U$ in $T$, we obtain a desired  labeling  $\\mathcal{L}^\\ast$ of $T^\\ast$.\n\nWe verify that $\\mathcal{L}^\\ast$  gives a correct labeling of $T^\\ast$. Clearly, the size-$\\Delta$ multiset that labels each vertex $v \\in V$ is in $\\mathcal{V}$, by the correctness of $\\mathcal{L}$  and $\\mathcal{L}''$. Consider any edge $e=\\{u,v\\}$ in  $T^\\ast$. Similarly, if $\\{u,v\\} \\subseteq V \\setminus U$ or $\\{u,v\\} \\cap (V \\setminus U) = \\emptyset$, then  the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$  and $\\mathcal{L}''$. For the case that $e=\\{u,v\\}$ connects a vertex $u \\notin V$ and a vertex $v \\in V \\setminus U$, we must have $u \\in S'$ by the lemma statement. Therefore, the label of the half edge $(u,e)$ is the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$  by our choice of $\\mathcal{L}''$. Thus, the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$.\n\nWe verify that  the \nvirtual half-edge labels surrounding each pole in $S$ are the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$. Consider a pole $v \\in S$. If $v \\in V \\setminus U$, then the labeling of $v$ is clearly the same in both $\\mathcal{L}$  and $\\mathcal{L}^\\ast$.\nIf $v \\notin V\\setminus U$, then the condition $S \\cap U \\subseteq S'$ in the statement implies that $v \\in S'$. In this case, the way we pick $\\mathcal{L}''$ ensures that the \nvirtual half-edge labels surrounding $v \\in S'$ are the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$.\n\\end{proof}\n\nIn view of the proof of \\cref{lem:replace}, as long as the conditions in \\cref{lem:replace} are met, we are able to abstract out a subtree by its equivalence class when reasoning about correct labelings of a tree. This observation will be applied repeatedly in the subsequent discussion.\n\n\n\\paragraph{A Pumping Lemma} We will prove a pumping lemma of bipolar trees using \\cref{lem:replace}. \nSuppose $T_i$ is a tree with a root $r_i$ for each $1 \\leq i \\leq k$, then $H=(T_1, T_2, \\ldots, T_k)$ denotes the bipolar tree resulting from concatenating the roots $r_1, r_2, \\ldots, r_k$ into a path $(r_1, r_2, \\ldots, r_k)$ and setting the two poles of $H$ by $s = r_1$ and $t = r_k$.\nA simple consequence of \\cref{lem:replace} is that $\\texttt{type}(H)$ is determined by  $\\textsf{Class}_1(T_1), \\textsf{Class}_1(T_2), \\ldots, \\textsf{Class}_1(T_k)$.\nWe have the following pumping lemma. \n\n\\begin{lemma}[Pumping lemma]\\label{lem:pump}\nThere exists a finite number $ {\\ell_{\\operatorname{pump}}}  > 0$ such that as long as $k \\geq  {\\ell_{\\operatorname{pump}}} $, any  bipolar tree $H=(T_1, T_2, \\ldots, T_k)$ can be decomposed into $H = X \\circ Y \\circ Z$ with $0 < |Y| < k$ so that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nSet $ {\\ell_{\\operatorname{pump}}} $ to be the number of equivalence classes for bipolar trees plus one. By the pigeon hole principle, there exist $1 \\leq a < b \\leq k$ such that $(T_1, T_2, \\ldots, T_a)$ and $(T_1, T_2, \\ldots, T_b)$ are in the same equivalence class. Set $X = (T_1, T_2, \\ldots, T_a)$, $Y = (T_{a+1}, T_{a+2}, \\ldots, T_b)$, and $Z = (T_{b+1}, T_{b+2}, \\ldots, T_k)$. As we already know that $\\texttt{type}(X) = \\texttt{type}(X\\circ Y)$, \\cref{lem:replace} implies that \n$\\texttt{type}(X\\circ Y) = \\texttt{type}(X^2\\circ Y)$ by replacing $X$ by $X\\circ Y$ in the bipolar tree $X\\circ Y$.\nSimilarly, $\\texttt{type}(X \\circ Y^i)$ is the same for for each $i \\geq 0$. Applying \\cref{lem:replace} again to replace $X \\circ Y$ by $X \\circ Y^i$ in $H = X \\circ Y \\circ Z$, we conclude that  $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{proof}\n\n\n\n\\paragraph{A Procedure for Gluing Trees} Suppose that we have a set of rooted trees $W$. We  devise a procedure that generates a new set of rooted trees by gluing the rooted trees in $W$ together. \nThis procedure is parameterized by a labeling function $f$. Consider a bipolar tree \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] \nwhere $T^m$ is formed by attaching the roots $r_1, r_2, \\ldots, r_{\\Delta - 2}$ of the rooted trees $T_1^m, T_2^m, \\ldots, T_{\\Delta -2}^m$ to the root $r^m$ of $T^m$.\n\n\nThe labeling function $f$ assigns the half-edge labels surrounding $r^m$ based on \n\\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r)\\]\nwhere $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$.\n\nWe write  $T^{m}_\\ast$ to denote the result of applying $f$ to label the root $r^m$ of $T^{m}$ in the bipolar tree $H$, and we write \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] to denote the result of applying $f$ to $H$.  We make the following observation.\n\n\\begin{lemma}[Property of $f$]\\label{lem:labeling_function}\nThe two equivalence classes\n$\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H_\\ast)$ are determined by \\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:replace}, once the half-edge labelings of the root $r^m$ of $T^{m}$ is fixed, $\\textsf{Class}_1(T_\\ast^m)$ is determined by $\\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m)$. \nTherefore, indeed $\\textsf{Class}_1(T_\\ast^m)$ is determined by\n\\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\nSimilarly, applying  \\cref{lem:replace} to the decomposition of $H_\\ast$ into $H^l$, $T_\\ast^m$, and $H^r$, we infer that  $\\texttt{type}(H_\\ast)$ depends only on $\\texttt{type}(H^l)$, $\\textsf{Class}_1(T_\\ast^m)$, and $\\texttt{type}(H^r)$.\n\\end{proof}\n\nThe three sets of trees $X_f(W)$, $Y_f(W)$, and $Z_f(W)$ are constructed as follows.\n\n\\begin{itemize}\n    \\item $X_f(W)$ is the set of all rooted trees  resulting from appending $\\Delta-2$ arbitrary rooted trees $T_1, T_2, \\ldots, T_{\\Delta-2}$ in $W$ to a new root vertex $r$.\n    \\item  $Y_f(W)$ is the set of bipolar trees constructed as follows. For each choice of $2 {\\ell_{\\operatorname{pump}}}  +1$ rooted trees $T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r$ from $X_f(W)$, concatenate them into a bipolar tree \n    \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\]  let $H_\\ast$ be the result of applying the labeling function $f$ to $H$, and then add $H_\\ast$ to $Y_f(W)$.\n    \\item $Z_f(W)$ is the set of rooted trees  constructed as follows. For each \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_f(W),\\] add $(T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m)$ to $Z_f(W)$,  where we set the root of $T_1^l$ as the root, and  add $(T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to $Z_f(W)$, where we set the root  of ${T_{ {\\ell_{\\operatorname{pump}}} }^r}$ as the root. \n\\end{itemize}\n\nWe write $\\textsf{Class}_1(S) = \\bigcup_{T \\in S} \\{\\textsf{Class}_1(T)\\}$ and  $\\texttt{type}(S) = \\bigcup_{H \\in S} \\{\\texttt{type}(H)\\}$, and \nwe make the following observation.\n\n\\begin{lemma}[Property of  $X_f(W)$, $Y_f(W)$, and $Z_f(W)$]\\label{lem:sets_of_trees}\nThe three sets of equivalence classes $\\textsf{Class}_1(X_f(W))$, $\\texttt{type}(Y_f(W))$, and $\\textsf{Class}_1(Z_f(W))$ depend only on $\\textsf{Class}_1(W)$ and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nThis is a simple consequence of \\cref{lem:replace,lem:labeling_function}.\n\\end{proof}\n\n\\paragraph{A Fixed Point $W^\\ast$}\nGiven a fixed labeling function $f$,\nwe want to find a set of rooted trees $W^\\ast$ that is a \\emph{fixed point} for the procedure $Z_f$ in the sense that \\[\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast).\\] \nTo find such a set $W^\\ast$, we construct a two-dimensional array of rooted trees $\\{W_{i,j}\\}$, as follows.\n\n\\begin{itemize}\n    \\item For the base case, $W_{1,1}$ consists of only the one-vertex rooted tree.\n    \\item Given that $W_{i,j}$ as been constructed, we define $W_{i,j+1} = Z_f(W_{i,j})$.\n    \\item Given that $W_{i,j}$ for all positive integers $j$ have been constructed,  $W_{i+1,1}$ is defined as follows. Pick $b_i$ as the smallest index such that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$ for some $1 \\leq a_i < b_i$. By the pigeon hole principle, the index $b_i$ exists, and it is upper bounded by $2^C$, where $C$ is the number of equivalence classes for rooted trees. We set \n    \\[W_{i+1,1} = W_{i, a_i} \\cup W_{i, a_i + 1} \\cup \\cdots \\cup W_{i, b_i - 1}.\\]\n\\end{itemize}\n\nWe show that the  sequence $\\textsf{Class}_1(W_{i,a_i}), \\textsf{Class}_1(W_{i,a_i+1}), \\textsf{Class}_1(W_{i,a_i+2}), \\ldots$ is periodic with a period $c_i = b_i - a_i$.\n\n\\begin{lemma}\\label{lem:W-aux-1}\nFor any $i \\geq 1$ and for any $j \\geq a_i$, we have $\\textsf{Class}_1(W_{i,j}) = \\textsf{Class}_1(W_{i,j+c_i})$, where $c_i = b_i - a_i$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:sets_of_trees}, $\\textsf{Class}_1(W_{i,j})$  depends only on $\\textsf{Class}_1(W_{i,j-1})$. Hence the lemma follows from the fact that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$.\n\\end{proof}\n\nNext, we show that $\\textsf{Class}_1(W_{2,1}) \\subseteq \\textsf{Class}_1(W_{3, 1}) \\subseteq \\textsf{Class}_1(W_{4, 1}) \\subseteq  \\cdots$.\n\n\\begin{lemma}\\label{lem:W-aux-2}\nFor any $i \\geq 2$, we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$, and so $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$.\n\\end{lemma}\n\\begin{proof}\nSince $W_{i,1} = \\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}$, we have \\[\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l} \\subseteq W_{i,j}\\] according to the procedure of constructing $W_{i,j}$. By \\cref{lem:W-aux-1}, \\[  \\textsf{Class}_1\\left(\\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}\\right) = \\textsf{Class}_1\\left(\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l}\\right)\\]  for all $j \\geq 1$, and so we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$. The claim $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$ follows from the fact that $W_{i+1,1} = \\bigcup_{a_{i} \\leq l \\leq b_{i} - 1} W_{i, l}$.\n\\end{proof}\n\nSet $i^\\ast$ to be the smallest index $i \\geq 2$ such that $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$. By the pigeon hole principle and \\cref{lem:W-aux-2}, the index $i^\\ast$ exists, and it is upper bounded by $C$, the number  of equivalence classes for rooted trees. We set \\[W^\\ast = W_{i^\\ast, 1}.\\] \n\nThe following lemma shows that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$, as needed.\n\n\\begin{lemma}\\label{lem:W-aux-3}\nFor any $i \\geq 2$, if $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$, then $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:W-aux-1} and the way we construct $W_{i+1, 1}$, we have \\[\\textsf{Class}_1(W_{i, j}) \\subseteq \\textsf{Class}_1(W_{i+1, 1}) \\ \\ \\text{for each} \\ \\ j \\geq a_i.\\]\nBy \\cref{lem:W-aux-2}, we have  \\[\\textsf{Class}_1(W_{i, 1}) \\subseteq \\textsf{Class}_1(W_{i, j}) \\ \\ \\text{for each} \\ \\ j > 1.\\]\nTherefore, $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$ implies that $\\textsf{Class}_1(W_{i, j}) = \\textsf{Class}_1(W_{i,1})$ for each $j \\geq a_i$. Hence we must have $\\textsf{Class}_1(Z_f(W_{i,1})) = \\textsf{Class}_1(W_{i,1})$, and so $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{proof}\n\nWe remark that simply selecting $W^\\ast$ to be any set such that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$ is not enough for our purpose. As we will later see, it is crucial that the set $W^\\ast$ is constructed by iteratively applying the function $Z_f$ and taking the union of previously constructed sets.\n\n\n\n\n\n\n\n\\paragraph{A Sequence of Sets of Trees}\nWe define $W_1^\\ast = W^\\ast$ and $W_i^\\ast = Z_f(W_{i-1}^\\ast)$ for each $1 <  i \\leq L$, where $L$ is some sufficiently large number to be determined.\n\nThe way we choose $W^\\ast$ guarantees that $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W^\\ast)$  for all $1 \\leq i \\leq L$. \nFor convenience, we write $X_i^\\ast = X_f(W_i^\\ast)$ and $Y_i^\\ast = Y_f(W_i^\\ast)$.\nSimilarly, $\\textsf{Class}_1(X_i^\\ast)$ is the same for all $1 \\leq i \\leq L$ and $\\texttt{type}(Y_i^\\ast)$ is the same for all $1 \\leq i \\leq L$.\n\n\nOur analysis will rely on the assumption that all rooted trees in $W^\\ast$ admit correct labelings. Whether this is true depends only on $\\textsf{Class}_1(W^\\ast)$, which depends only on the labeling function $f$. We say that $f$ is \\emph{feasible} if it leads to a set $W^\\ast$ where all the rooted trees therein admit correct labelings. The proof that a feasible labeling function $f$ exists is deferred.\n\nThe assumption that $f$ is feasible implies that all trees in $W_i^\\ast$, $X_i^\\ast$, and $Y_i^\\ast$, for all $1 \\leq i \\leq L$, admit correct labelings. All rooted trees in $X_i^\\ast$ admit correct labelings because they are subtrees of the rooted trees in $W_{i+1}^\\ast$. A correct labeling of any bipolar tree $H \\in Y_i^\\ast$ can be obtained by combining any correct labelings of the two rooted trees in $W_{i+1}^\\ast$ resulting from $H$.\n\n\n\n\n\\paragraph{Layers of Vertices}\nWe assign a layer number $\\lambda(v)$ to each vertex $v$ in a tree  based on the step that $v$ is introduced in the construction \\[W_1^\\ast \\rightarrow W_2^\\ast \\rightarrow \\cdots \\rightarrow W_L^\\ast.\\]\n\nIf a vertex $v$ is introduced as the root vertex of a tree in $X_i^\\ast = X_f(W_i^\\ast)$, then we say that the layer number of $v$ is $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. A vertex $v$ has  $\\lambda(v) = 0$ if it belongs to a tree in $W_1^\\ast$.\n\n\nFor any vertex $v$ with $\\lambda(v) = i$ in a tree $T \\in X_j^\\ast$ with $i \\leq j$, we write $T_v$ to denote the subtree of $T$ such that $T_v \\in X_i^\\ast$ where $v$ is the root of $T_v$. \n\n\n \nWe construct a sequence of sets $R_1, R_2, \\ldots, R_L$ as follows.\nWe go over all rooted trees $T \\in X_L^\\ast$, all possible correct labeling $\\mathcal{L}$ of $T$, and all vertices $v$ in $T$ with $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. Suppose that the $\\Delta-2$ rooted trees in the construction of $T_v \\in X_i^\\ast = X_f(W_i^\\ast)$ are $T_1$, $T_2$, $\\ldots$, $T_{\\Delta-2}$, and let $r_i$ be the root of $T_i$. Consider the following parameters.\n\\begin{itemize}\n    \\item $c_i = \\textsf{Class}_1(T_i)$.\n    \\item $\\mathtt{a}_i$ is the real half-edge label of $v$ in $T_v$ for the edge $\\{v, r_i\\}$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n    \\item $\\mathcal{I}$ is the size-$2$ multiset of virtual half-edge labels of  $v$ in $T_v$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$.\n\n\n\n\n\\begin{lemma}\\label{lem:Rsets1}\nFor each $1 \\leq i < L$, $R_i$ is determined by $R_{i+1}$.\n\\end{lemma}\n\\begin{proof}\nWe consider the following alternative way of constructing $R_i$ from $R_{i+1}$.\nEach rooted tree $T'$ in $W_{i+1}^\\ast = Z_f(W_i^\\ast)$ can be described as follows.\n\\begin{itemize}\n    \\item Start with a path $(r_1, r_2, \\ldots, r_{ {\\ell_{\\operatorname{pump}}}  + 1})$, where $r_1$ is the root of $T'$.\n    \\item For each $1 \\leq j \\leq  {\\ell_{\\operatorname{pump}}} +1$, append $\\Delta-2$ rooted trees $T_{j,1}, T_{j,2}, \\ldots, T_{j, \\Delta-2} \\in W_i^\\ast$ to $r_j$.\n    \\item Assign the labels to the half edges surrounding $r_{ {\\ell_{\\operatorname{pump}}}  + 1}$ according to the labeling function $f$.\n\\end{itemize}\n\n Now, consider the function $\\phi$ that maps each equivalence class $c$ for rooted trees to a subset of $\\Sigma$ defined as follows:\n$\\mathtt{a} \\in \\phi(c)$ if there exist \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{i+1}\\] and $1 \\leq j \\leq \\Delta -2$ such that $c = c_j$ and $\\mathtt{a} = \\mathtt{a}_j$.\n\nWe go over all possible $T' \\in W_{i+1}^\\ast = Z_f(W_i^\\ast)$.\nNote that the root $r$ of $T'$ has exactly one virtual half edge. For each $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$, we go over all  possible correct labelings $\\mathcal{L}$ of $T'$ where the virtual half edge of $r$ is labeled $\\mathtt{b}$. For each $1 \\leq j \\leq  {\\ell_{\\operatorname{pump}}} +1$, consider the following parameters.\n\\begin{itemize}\n    \\item $c_l = \\textsf{Class}_1(T_{j,l})$.\n    \\item $\\mathtt{a}_l$ is the half-edge label of $r_j$ for the edge $\\{r_j, r_{j,l}\\}$.\n    \\item $\\mathcal{I}$ is the size-$2$ multiset of the remaining two half-edge labels of  $v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$. \n\nThis construction of $R_i$ is equivalent to the original construction of $R_i$ because a correct labeling of $T' \\in W_i^\\ast$ can be extended to a correct labeling of a tree $T \\in X_L^\\ast$ that contains $T'$ as a subtree if and only if the virtual half-edge label of the root $r$ of $T'$ is $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$.\n\n\nIt is clear that this construction of $R_i$ only depends on $\\textsf{Class}_1(W_i^\\ast)$, the labeling function $f$, and the function $\\phi$, which depends only on $R_{i+1}$. Since the labeling function $f$ is fixed and $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$, we conclude that $R_{i}$ depends only on $R_{i+1}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Rsets2}\nWe have $R_1 \\subseteq R_2 \\subseteq \\cdots \\subseteq R_L$.\n\\end{lemma}\n\\begin{proof}\nFor the base case, we show that $R_{L-1} \\subseteq R_L$. In fact, our proof will show that $R_{i} \\subseteq R_L$ for each $1 \\leq i < L$. Consider any \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_i.\\] \nThen there is a rooted tree $T \\in X_i^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex so that if we label the half edges surrounding the root vertex according to  $\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}$, and  $\\mathcal{I}$, then this partial labeling can be completed into a correct labeling of $T$.\n\nBecause $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W_L^\\ast)$, there is also a rooted tree $T' \\in X_L^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex. Therefore, if we label the root vertex of $T'$ in the same way as we do for $T$, then this partial labeling can also be completed into a correct labeling of $T'$. Hence we must have \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_L.\\] \n\n\nNow, suppose that we already have $R_i \\subseteq R_{i+1}$ for some $1 < i < L$. We will show that  $R_{i-1} \\subseteq R_{i}$. \nDenote by $\\phi_{i}$ and $\\phi_{i+1}$ the function $\\phi$ in \\cref{lem:Rsets1} constructed from  $R_{i}$ and $R_{i+1}$. \nWe have $\\phi_i(c) \\subseteq \\phi_{i+1}(c)$ for each equivalence class $c$, because $R_i \\subseteq R_{i+1}$. Therefore, in view of the alternative construction described in the proof of \\cref{lem:Rsets1}, we have $R_{i-1} \\subseteq R_{i}$.\n\\end{proof}\n\n\n\n\\paragraph{The Set of Vertex Configurations $\\mathcal{V}'$}\nBy \\cref{lem:Rsets1,lem:Rsets2}, if we pick $L$ to be sufficient large, we can have  $R_1 = R_2 = R_3$.\nMore specifically, if we pick \n\\[L \\geq C^{\\Delta-2} \\cdot \\binom{|\\Sigma|+1}{|\\Sigma|-1} + 3,\\]\nthen there exists an index $3 \\leq i \\leq L$ such that $R_i = R_{i-1}$, implying that $R_1 = R_2 = R_3$. Here $C$ is the number of equivalence classes for rooted trees and $\\binom{|\\Sigma|+1}{|\\Sigma|-1}$ is the number of size-$2$ multisets of elements from $\\Sigma$.\n\n\n\nThe set $\\mathcal{V}'$ is defined by including all size-$\\Delta$ multisets $\\{\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}\\} \\cup \\mathcal{I}$ such that  \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{1}\\] for some $c_1, c_2, \\ldots, c_{\\Delta-2}$.\n\n\n\n\\paragraph{The Set $\\mathcal{V}'$  is $\\ell$-full}\nTo show that the set $\\mathcal{V}'$  is $\\ell$-full, we  consider the subset $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$ defined by the set of vertex configurations used by the labeling function $f$ in the construction of $Y_f(W_i^\\ast)$. The definition of $\\mathcal{V}^\\ast$ is invariant of $i$ as $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$. Clearly, for each $x \\in \\mathcal{V}^\\ast$, there is a rooted tree $T \\in W_2^\\ast$ where the root of a subtree $T' \\in X_1^\\ast$ of $T$ has its size-$\\Delta$ multiset of half-edge labels fixed to be $x$ by $f$, and so  $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$.\n\nFor notational simplicity, we write $x \\overset{k}{\\leftrightarrow} x'$ if there is a correct labeling of a $k$-vertex path $(v_1, v_2, \\ldots, v_k)$ using only vertex configurations in $\\mathcal{V}'$ so that the vertex configuration of $v_1$ is $x$ and the vertex configuration of $v_k$ is $x'$. If it is further required that the half-edge label of $v_1$ for the edge $\\{v_1, v_2\\}$ is $\\mathtt{a} \\in x$, then we write $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$. The notation $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{a}')$ is defined similarly.\n\n\n\\begin{lemma}\\label{lem:find_path_1}\nFor any $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$ and $\\mathtt{a} \\in x$, there exist a vertex configuration $x' \\in \\mathcal{V}^\\ast$ and a number $2 \\leq k \\leq 2  {\\ell_{\\operatorname{pump}}}  + 1$ such that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$.\n\\end{lemma}\n\\begin{proof}\nLet $T \\in W_L^\\ast$ be chosen so that there is a correct labeling $\\mathcal{L}$ where $x$ is a vertex configuration of some vertex $v$ with $\\lambda(v) = 2$.\n\nTo prove the lemma, it suffices to show that for each of the $\\Delta$ neighbors $u$ of $v$, it is possible to find a path $P = (v,u, \\ldots, w)$ meeting the following conditions.\n\\begin{itemize}\n    \\item $w$ is a vertex whose half-edge labels have been fixed by $f$. This ensures that the vertex configuration of $w$ is in $\\mathcal{V}^\\ast$.\n    \\item All vertices in $P$ are within layers 1,2, and 3. This ensures that the vertex configuration of all vertices in $P$ are in $\\mathcal{V}'$.\n    \\item The number of vertices $k$ in $P$ satisfies $2 \\leq k \\leq 2  {\\ell_{\\operatorname{pump}}}  + 1$. \n\\end{itemize}\n\nWe divide the proof into three cases. Refer to \\cref{fig:ell_full_construction1} for an illustration, where squares are vertices whose half-edge labels have been fixed by $f$.\n\n\\paragraph{Case 1} Consider the subtree $T_v \\in X_2^\\ast$ of $T$ whose root is $v$. In view of the construction of the set $X_f(W_2^\\ast)$, $v$ has $\\Delta-2$ children $u_1, u_2, \\ldots, u_{\\Delta-2}$ in $T_v$, where the subtree $T_i$ rooted at $u_i$ is a rooted tree in $W_2^\\ast$. \n\nFor each $1 \\leq i \\leq \\Delta-2$, according to the structure of the trees in the set $W_2^\\ast = Z_f(W_1^\\ast)$, there is a path $(u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ in $T_i$ containing only layer-1 vertices, where the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed by $f$. Hence $P = (v, u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ is a desired path with $k =  {\\ell_{\\operatorname{pump}}}  + 2$ vertices.\n    \n   \n    \n   \n    \n\\paragraph{Case 2}  Consider the subtree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ that contains $v$ in $T$.\nSimilarly, according to the structure of the trees in the set $W_3^\\ast = Z_f(W_2^\\ast)$, there is a path $(r = w_1', w_2', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T'$ containing only layer-2 vertices so that $r$ is the root of $T'$, $v = w_i'$ for some $1 \\leq i' \\leq  {\\ell_{\\operatorname{pump}}} +1$, and the  half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}'$ have been fixed by $f$.\nSince  $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$, we have $v \\neq w_{ {\\ell_{\\operatorname{pump}}} +1}'$. Hence $P = (v = w_i', w_{i+1}', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ is a desired path with $2 \\leq k \\leq  {\\ell_{\\operatorname{pump}}}  + 1$ vertices.\n\n    \n    \n \n\n\\paragraph{Case 3} There is only one remaining neighbor of $v$ to consider. In view of the construction of $X_f(W_3^\\ast)$, there is a layer-3 vertex $v'$ adjacent to the vertex $r$, the root of the tree $T' \\in W_3^\\ast$ considered in the previous case. If the half-edge labels of $v'$ have been fixed by $f$, then $P = (v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$ is a desired path.\nOtherwise, similar to the analysis in the previous case, we can find a path $P' = (v', \\ldots, w)$ connecting $v'$ to a vertex $w$ whose half-edge labels  have been fixed by $f$. All vertices in $P'$ are of layer-3, and the number of vertices in $P'$ is within $[2,  {\\ell_{\\operatorname{pump}}}  + 1]$. \nCombining $P'$ with the path $(v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$, we obtain the desired path $P$ whose number of vertices satisfies $2 \\leq k  \\leq  2 {\\ell_{\\operatorname{pump}}}  + 1$.\n   \n   \n\\end{proof}\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .75\\textwidth]{fig\/y2.pdf}\n    \\caption{An illustration for the proof of \\cref{lem:find_path_1}.}\n    \\label{fig:ell_full_construction1}\n\\end{figure}\n\nFor \\cref{lem:find_path_2}, note that if $\\mathtt{a}$ appears more than once in the multiset $x$, then we still have $\\mathtt{a} \\in x \\setminus \\{\\mathtt{a}\\}$.\n\n\n\\begin{lemma}\\label{lem:find_path_2}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, and $0 \\leq t \\leq  {\\ell_{\\operatorname{pump}}}  - 1$, we have  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n with $k = 2  {\\ell_{\\operatorname{pump}}}  + 4 + t$.\n\\end{lemma}\n\\begin{proof}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, we can find a rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ such that the path $(v_1$, $v_2$, $\\ldots$, $v_{ {\\ell_{\\operatorname{pump}}}  +1})$ of the layer-1 vertices in  $T_{x, \\mathtt{a}}$ where $v_1 = r$ is the root of   $T_{x, \\mathtt{a}}$ satisfies the property that the half-edge labels of $v_{ {\\ell_{\\operatorname{pump}}}  +1}$ have been fixed to $x$ by $f$ where the half-edge label for $\\{v_{ {\\ell_{\\operatorname{pump}}} }, v_{ {\\ell_{\\operatorname{pump}}} +1}\\}$ is in $x \\setminus \\{\\mathtt{a}\\}$.\n\nNow, observe that for any choice of $(\\Delta-2)( {\\ell_{\\operatorname{pump}}} +1)$ rooted trees \\[\\{T_{i,j}\\}_{1 \\leq i \\leq  {\\ell_{\\operatorname{pump}}} +1, 1 \\leq j \\leq \\Delta-2}\\] in $W_2^\\ast$, there is a rooted tree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$  formed by appending $T_{i, 1}$, $T_{i,2}$, $\\ldots$, $T_{i, \\Delta-2}$ to $u_i$ in the path $(u_1$, $u_2$, $\\ldots$, $u_{ {\\ell_{\\operatorname{pump}}} +1})$ and fixing the half-edge labeling of $u_{ {\\ell_{\\operatorname{pump}}} +1}$ by $f$. All vertices in $(u_1, u_2, \\ldots, u_{ {\\ell_{\\operatorname{pump}}} +1})$ are of layer-2.\n\nWe choose any $T' \\in W_2^\\ast$ with $T_{i,j} = T_{x, \\mathtt{a}}$ and $T_{i',j'} = T_{x', \\mathtt{a}'}$ such that $i' - i = t + 1$.\nThe possible range of $t$ is $[0,  {\\ell_{\\operatorname{pump}}}  - 1]$.\nConsider any $T \\in W_L^\\ast$ that contains $T'$ as its subtree, and consider any correct labeling of $T$.\nThen the path $P$ resulting from concatenating the path $(v_{ {\\ell_{\\operatorname{pump}}}  +1}, v_{ {\\ell_{\\operatorname{pump}}} }, \\ldots, v_1)$ in $T_{x, \\mathtt{a}}$, the path $(u_i, u_{i+1}, \\ldots, u_{i'})$, and the path $(v_1', v_2', \\ldots, v_{ {\\ell_{\\operatorname{pump}}}  +1}')$ in $T_{x', \\mathtt{a}'}$\nshows that $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ \nfor $k = ( {\\ell_{\\operatorname{pump}}} +1) + (t+2) + ( {\\ell_{\\operatorname{pump}}} +1) = 2 {\\ell_{\\operatorname{pump}}}  + 4 + t$. See \\cref{fig:ell_full_construction2} for an illustration.\n\nFor the rest of the proof, we show that the desired rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ exists for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$.\nFor any $x \\in \\mathcal{V}^\\ast$, we can find a bipolar tree\n\\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_1^\\ast = Y_f(W_1^\\ast)\\]\nsuch that the vertex configuration of the root $r^m$ of  $T_\\ast^m$ is fixed to be $x$ by the labeling function $f$.\nThen, for any $\\mathtt{a} \\in x$, at least one of the two rooted trees in $W_2^\\ast = Z_f(W_1^\\ast)$ resulting from cutting $H_\\ast$ satisfies the desired requirement. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .65\\textwidth]{fig\/y3.pdf}\n    \\caption{An illustration for the proof of \\cref{lem:find_path_2}.}\n    \\label{fig:ell_full_construction2}\n\\end{figure} \n\nIn the subsequent discussion, the length of a path refers to the number of edges in a path. In particular, the length of a $k$-vertex path is $k-1$.\n\nThe following lemma is proved by iteratively applying \\cref{lem:find_path_2} via intermediate vertex configurations $\\tilde{x} \\in \\mathcal{V}^\\ast$. \n\n\\begin{lemma}\\label{lem:find_path_3}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$,  we have  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n for all $k \\geq 6  {\\ell_{\\operatorname{pump}}}  + 10$.\n\\end{lemma}\n\\begin{proof}\nBy applying \\cref{lem:find_path_2} for three times,\nwe infer that \\cref{lem:find_path_2}  also works for any path length $k - 1 = (k_1 - 1) + (k_2 - 1) + (k_3 - 1)$, where $k_i - 1 = 2  {\\ell_{\\operatorname{pump}}}  + 3 + t_i$ for $0 \\leq t_i \\leq  {\\ell_{\\operatorname{pump}}} -1$. \n\nSpecifically, we first apply \\cref{lem:find_path_2} to find a path  realizing $(x,\\mathtt{b}) \\overset{k_1}{\\leftrightarrow} \\tilde{x}$ for some  $\\tilde{x} \\in \\mathcal{V}^\\ast$, and for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$.\nLet $\\tilde{\\mathtt{a}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the path. \nWe extend the length of this path by $k_2 - 1$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{b}}) \\overset{k_2}{\\leftrightarrow} \\tilde{x}$, for some $\\tilde{\\mathtt{b}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$.\nFinally, let $\\tilde{\\mathtt{c}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the current path. \nWe extend the length of the current path by $k_3$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{d}}) \\overset{k_3}{\\leftrightarrow} (x', \\mathtt{b}')$, for some $\\tilde{\\mathtt{d}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{x}}\\}$, and for some $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\nThe resulting path realizes  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\n\n\nTherefore,  \\cref{lem:find_path_2} also works with path length of the form $k-1 = 6 {\\ell_{\\operatorname{pump}}}  + 9  + t$, for any $t \\in [0, 3 {\\ell_{\\operatorname{pump}}}  - 3]$. \n\nAny $k-1 \\geq 6  {\\ell_{\\operatorname{pump}}}  + 9$ can be written as $k-1 = b(2 {\\ell_{\\operatorname{pump}}}  +3) + (6 {\\ell_{\\operatorname{pump}}}  + 9 + t)$, for some $t \\in [0, 3 {\\ell_{\\operatorname{pump}}}  - 3]$ and some integer $b \\geq 0$.\nTherefore,  similar to the above, we can find a path showing $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ by first   applying \\cref{lem:find_path_2} with  path length\n$2 {\\ell_{\\operatorname{pump}}}  +3$ for $b$ times, and then applying the above variant of \\cref{lem:find_path_2} to extend the path length by $6 {\\ell_{\\operatorname{pump}}}  + 9  + t$.\n\\end{proof}\n\nWe show that \\cref{lem:find_path_1,lem:find_path_3} imply that $\\mathcal{V}'$  is $\\ell$-full for some $\\ell$.  \n\n\\begin{lemma}[$\\mathcal{V}'$  is $\\ell$-full] \\label{lem:find_path_4}\nThe set $\\mathcal{V}'$  is $\\ell$-full for $\\ell = 10  {\\ell_{\\operatorname{pump}}}  + 10$. \n\\end{lemma}\n\\begin{proof}\nWe show that for any target path length $k-1 \\geq 10  {\\ell_{\\operatorname{pump}}}  + 9$, and for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\n\nBy \\cref{lem:find_path_1}, there exists a vertex configuration $\\tilde{x} \\in \\mathcal{V}^\\ast$ so that we can find a path $P$ realizing $(x,\\mathtt{a}) \\overset{\\ell_1}{\\leftrightarrow} \\tilde{x}$ for some path length  $1 \\leq (\\ell_1-1) \\leq 2  {\\ell_{\\operatorname{pump}}} $.\nSimilarly, there exists a vertex configuration $\\tilde{x}' \\in \\mathcal{V}^\\ast$ so that we can find a path $P'$ realizing $(x',\\mathtt{a}') \\overset{\\ell_2}{\\leftrightarrow} \\tilde{x}'$ for some path length   $1 \\leq  (\\ell_2-1) \\leq 2  {\\ell_{\\operatorname{pump}}} $. \n\n\nLet $\\tilde{\\mathtt{a}}$ be the real half-edge label for $\\tilde{x}$ in $P$, and let $\\tilde{\\mathtt{a}}'$ be the real half-edge label for $\\tilde{x}'$ in $P'$.\nWe apply \\cref{lem:find_path_3} to find a path $\\tilde{P}$ realizing  $(x,\\mathtt{b}) \\overset{\\tilde{\\ell}}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$ and $\\mathtt{b}' \\in \\tilde{x}' \\setminus \\{\\tilde{\\mathtt{a}}'\\}$ with path length  \n\\[\\tilde{\\ell}-1 = (k-1) - (\\ell_1-1) - (\\ell_2-1) \\geq  6  {\\ell_{\\operatorname{pump}}}  + 9.\\] \nThe path formed by concatenating $P_1$, $\\tilde{P}$, and $P_2$ shows that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\\end{proof}\n\n\n\n\n\n\n   \n\n\n\n\\paragraph{A Feasible Labeling Function $f$ exists} We show that a feasible labeling function $f$ exists given that $\\Pi$ can be solved in $\\mathsf{LOCAL}(n^{o(1)})$ rounds. We will construct a labeling function $f$ in such a way  each equivalence class in $\\textsf{Class}_1(W^\\ast)$ contains only rooted trees that admit legal labeling. That is, for each  $c \\in \\textsf{Class}_1(W^\\ast)$, the mapping $h$ associated with $c$ satisfies $h(X) = \\textsf{YES}$ for some $X$.\n\n\n\nWe will consider a series of modifications in the construction \\[W  \\rightarrow  X_f(W) \\rightarrow Y_f(W) \\rightarrow Z_f(W)\\] that do not alter the sets of equivalence classes in these sets, conditioning on the assumption that all trees that we have processed admit correct labelings. That is, whether $f$ is feasible is invariant of the modifications.\n\n\n\\paragraph{Applying the Pumping Lemma} Let $w > 0$ be some target length for the pumping lemma. In the construction of $Y_f(W)$, when we process a bipolar tree \n\\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] we apply  \\cref{lem:pump} to the two subtrees $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to obtain two new bipolar trees $H_+^l$ and $H_+^r$.\nThe $s$-$t$ path in the new trees $H_+^l$ and $H_+^r$ contains $w + x$ vertices, for some $0 \\leq x <  {\\ell_{\\operatorname{pump}}} $. The equivalence classes do not change, that is, $\\texttt{type}(H^l) = \\texttt{type}(H_+^l)$ and $\\texttt{type}(H^r) = \\texttt{type}(H_+^r)$.\n\n\nWe replace $H^l$ by $H_+^l$ and replace $H^r$ by $H_+^r$ in the bipolar tree $H$.\nRecall that the outcome of applying the labeling function $f$ to the root  $r$ of the rooted tree $T_{ {\\ell_{\\operatorname{pump}}} +1}$ depends only on \\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] so applying the pumping lemma to $H^l$ and $H^r$ during the construction of $Y_f(W)$ does not alter $\\textsf{Class}_1(T_\\ast^m)$ and  $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$  and its middle rooted tree $T_\\ast^m$, by \\cref{lem:labeling_function}.\n\n\n\n\\paragraph{Reusing Previous Trees} During the construction of the fixed point $W^\\ast$, we remember all bipolar trees to which we have applied the feasible function $f$.\n\nDuring the construction of $Y_f(W)$. Suppose that we are about to process a bipolar tree  $H$, and there is already some  other bipolar tree $\\tilde{H}$  to which we have applied $f$ before so that \n$\\texttt{type}(H^l)=\\texttt{type}(\\tilde{H}^l)$,  $\\textsf{Class}_1(T_{i}^m) = \\textsf{Class}_1(\\tilde{T}_{i}^m)$ for each $1 \\leq i \\leq \\Delta-2$, and $\\texttt{type}(H^r)=\\texttt{type}(\\tilde{H}^r)$.\nThen we replace $H$ by $\\tilde{H}$, and then we process $\\tilde{H}$ instead.\n\nBy \\cref{lem:labeling_function}, this modification does not alter $\\textsf{Class}_1(T_\\ast^m)$ and  $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$.\n\n\\paragraph{Not Cutting Bipolar Trees} We consider the following different construction of $Z_f(W)$ from $Y_f(W)$. For each $H^\\ast \\in Y_f(W)$, we simply add two copies of $H^\\ast$ to $Z_f(W)$, one of them has $r = s$ and the other one has $r = t$.\nThat is, we do not cut the bipolar tree $H^\\ast$, as in the original construction of $Z_f(W)$.\n\nIn general, this modification might alter $\\textsf{Class}_1(Z_f(W))$.\nHowever, it is not hard to see that if all  trees in $Y_f(W)$ already admit correct labelings, then  $\\textsf{Class}_1(Z_f(W))$ does not alter after the modification.\n\nSpecifically, let $T$ be a rooted tree in  $Z_f(W)$ with the above modification. \nThen $T$ is identical to a bipolar tree  $H^\\ast=(T_1, T_2, \\ldots, T_k) \\in Y_f(W)$, where there exists some $1 < i < k$ such that the root $r_i$ of $T_i$ has its half-edge labels fixed by $f$.\nSuppose that the root of $T$ is the root $r_1$ of $T_1$.\nIf we do not have the above modification, then the rooted tree $T'$ added to $Z_f(W)$ corresponding to $T$ is $(T_1, T_2, \\ldots, T_i)$, where the root of $T'$ is also $r_1$. \n\nGiven the assumption that all bipolar trees in $Y_f(W)$ admit correct labelings, it is not hard to see that $\\textsf{Class}_1(T') = \\textsf{Class}_1(T)$. For any correct labeling $\\mathcal{L}'$ of $T'$, we can extend the labeling to a correct labeling $\\mathcal{L}$ of $T$ by labeling the remaining vertices according to any arbitrary correct labeling of $H^\\ast$. This is possible because the labeling of $r_i$ has been fixed. For any correct labeling $\\mathcal{L}$ of $T$, we can obtain a correct labeling $\\mathcal{L}'$ of $T'$ by restricting $\\mathcal{L}$ to  $T'$.\n\n\n\n\n\n\n\\paragraph{Simulating a $\\mathsf{LOCAL}$ Algorithm} We will show that there is a labeling function $f$ that makes all the rooted trees in $W^\\ast$ to admit correct solutions, where we apply the above three modifications in the construction of $W^\\ast$. In view of the above discussion, such a function $f$ is feasible.\n\nThe construction of $W^\\ast$ involves only a finite number $k$ of iterations of $Z_f$ applied to some previously constructed rooted trees. If we view the target length $w$ of the pumping lemma as a variable, then the size of a tree in $W^\\ast$ can be upper bounded by $O(w^{k})$. \n\n\nSuppose that we are given an arbitrary $n^{o(1)}$-round $\\mathsf{LOCAL}$ algorithm $\\mathcal{A}$ that solves $\\Pi$. It is known~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups} that randomness does not help for LCL problems on bounded-degree trees with round complexity $\\Omega(\\log n)$, so we assume that \n$\\mathcal{A}$ is deterministic.\n\nThe runtime of $\\mathcal{A}$ on a tree of size \n$O(w^{k})$ can be made to be at most $t = w\/10$ if $w$ is chosen to be sufficiently large. \n\nWe pick the labeling function $f$ as follows. \nWhenever we encounter a new bipolar tree $H$ to which we need to apply $f$, we simulate  $\\mathcal{A}$ locally at the root $r^m$ of the middle rooted tree $T^m$ in $H$. Here we assume that the pumping lemma was applied before simulating $\\mathcal{A}$. Moreover, when we do the simulation, we assume that the number of vertices is $n = O(w^{k})$. \n\nTo make the simulation possible, we just locally generate arbitrary distinct identifiers in the radius-$t$ neighborhood $S$ of $r^m$. This is possible because $t = w\/10$  is so small that for any vertex $v$ in any tree $T$ constructed by recursively applying $Z_f$ for at most $k$ iterations, the radius-$(t+1)$ neighborhood of $v$ intersects at most one such $S$-set. Therefore, it is possible to complete the identifier assignment to the rest of the vertices in $T$ so that for any edge $\\{u,v\\}$, the set of identifiers seen by $u$ and $v$ are distinct in an execution of a $t$-round algorithm. \n\nThus, by the correctness of $\\mathcal{A}$, the labeling of all edges $\\{u,v\\}$ are configurations in $\\mathcal{E}$ and the labeling of all vertices $v$ are configurations in $\\mathcal{V}$.\nOur choice of $f$ implies that all trees in $W^\\ast$ admit correct labeling, and so $f$ is feasible. Combined with \\cref{lem:find_path_4}, we have proved that  $\\mathsf{LOCAL}(n^{o(1)})$  $\\Rightarrow$ $\\ell$-full.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\\label{sec:concludingRemarksTrees}\n\nHere we collect some open problems we find interesting. \n\n\n\\begin{enumerate}\n    \\item What is the relation of $\\mathsf{ffiid}, \\mathsf{MEASURE}, \\mathsf{fiid}$? In particular is perfect matching in the class $\\mathsf{ffiid}$ or $\\mathsf{MEASURE}$? \n    \\item Are there LCL problems whose uniform complexity is nontrivial and slower than $\\operatorname{poly}\\log 1\/\\varepsilon$?\n   \n   \n   \n   \n   \n   \n    \\item Suppose that $\\Pi_G$ is a homomorphism LCL, is it true that $\\Pi_G\\in\\mathsf{BOREL}$ if and only if $\\Pi_G\\in O(\\log^*(n))$?\n    \\item Are there finite graphs of chromatic number bigger than $2\\Delta-2$ such that the corresponding homomorphism LCL is not in $\\mathsf{BOREL}$? See also Remark \\ref{r:hedet}.\n\\end{enumerate}\n\n\\subsection*{Acknowledgements} \nWe would like to thank Anton Bernshteyn, Endre Cs\u00f3ka, Mohsen Ghaffari, Jan Hladk\u00fd, Steve Jackson, Alexander Kechris, Edward Krohne, Oleg Pikhurko, Brandon Seward, Jukka Suomela, and Yufan Zheng for insightful discussions. \nWe especially thank Yufan Zheng for   suggesting potentially hard instances of homomorphism LCL problems.\n\n\n\\bibliographystyle{alpha}\n\n\\section{Introduction}\n\nIn this paper, we study local problems on grid graphs from three different perspectives: (a) the perspective of the theory of distributed algorithms, (b) the perspective of the theory of random processes, (c) the perspective of the field of descriptive combinatorics. \nAlthough different communities in these three fields ask very similar questions, no systematic connections between them were known, until the insightful work of Bernshteyn \\cite{Bernshteyn2021LLL} who applied results from the field (a) to the field (c). In this work, our most important results follow by applying techniques from the field (a) to (b), but perhaps more importantly, we present all three different perspectives as a part of one common theory (see \\cref{fig:big_picture_grids}). \n\nIn this introductory section, we first quickly present the three fields and define the object of interest, local problems, a class of problems that contains basic problems like vertex or edge coloring, matchings, and others. \nThen, we briefly present our results, most of which can be seen as inclusions between different classes of local problems shown in \\cref{fig:big_picture_grids}. \n\n\\paragraph{Distributed Computing}\n\nThe $\\mathsf{LOCAL}$ model of computing \\cite{linial1987LOCAL} was motivated by understanding distributed algorithms in huge networks. \nAs an example, consider the network of all wifi routers, where two routers are connected if they are close enough to exchange messages. In this case, they should better communicate with user devices on different channels, so as to avoid interference. In the graph-theoretic language, we want to color the network. Even if the maximum degree is $\\Delta$ and we want to color it with $\\Delta+1$ colors, the problem remains nontrivial, because the decision of each node should be done after few communication rounds with its neighbors. \n\nThe $\\mathsf{LOCAL}$ model formalizes this setup: we have a large network, with each node knowing its size, $n$, and perhaps some other parameter like the maximum degree $\\Delta$. In case of randomized algorithms, each node has an access to a random string, while in case of deterministic algorithms, each node starts with a unique identifier from a range of size polynomial in $n$. \nIn one round, each node can exchange any message with its neighbors and can perform an arbitrary computation. We want to find a solution to a problem in as few communication rounds as possible. \nImportantly, there is an equivalent view of $t$-rounds $\\mathsf{LOCAL}$ algorithms: such an algorithm is simply a function that maps $t$-hop neighbourhoods to the final output. An algorithm is correct if and only if applying this function to each node solves the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. One example: if the input graph is the $d$-dimensional torus (think of the grid $\\mathbb{Z}^d$ and identify nodes with the same remainder after dividing the coordinate with $n^{1\/d}$), a simple picture emerges. Any local problem (we define those formally later) is solvable with local complexity $O(1)$, $\\Theta(\\log^* n)$, or we need to see the whole graph to solve it, i.e., the local complexity is $\\Theta(n^{1\/d})$ (see \\cref{thm:classification_of_local_problems_on_grids}). For example, a proper coloring of nodes with $4$ colors has local complexity $\\Theta(\\log^* n)$, while coloring with $3$ colors has already complexity $\\Theta(n^{1\/d})$. \n\n\\paragraph{Finitary factors of iid processes}\n\\todo{zeptat se Spinky}\nConsider the following example of the so-called anti-ferromagnetic Ising model. \nImagine atoms arranged in a grid $\\mathbb{Z}^3$. Each atom can be in one of two states and there is a preference between neighboring atoms to be in a different state. We can formalize this by defining a potential function $V(x,y)$ indicating whether $x$ and $y$ are equal.  \nThe Ising model models the behavior of atoms in the grid as follows: for the potential $V$ and inverse temperature $\\beta$, if we look at the atoms, we find them in a state $\\Lambda$ with probability proportional to $\\exp(-\\beta \\sum_{u \\sim v \\in E(\\mathbb{Z}^d)} V(\\Lambda(u), \\Lambda(v)))$. If $\\beta$ is close to zero, we sample essentially from the uniform distribution. If $\\beta$ is infinite, we sample one of the two $2$-colorings of the grid. \n\nTo understand better the behavior of phase transitions, one can ask the following question. For what values of $\\beta$ can we think of the distribution $\\mu$ over states from the Ising model as follows: there is some underlying random variable $X(u)$ at every node $u$, with these variables being independent and identically distributed. We can think of $\\mu$ as being generated by a procedure, where each node explores the random variables in its neighborhood until it decides on its state, which happens with probability $1$. If this view of the underlying distribution is possible, we say it is a finitary factor of an iid process (ffiid). \nIntuitively, being an ffiid justifies the intuition that there is a correlation decay between different sites of the grid. To understand ffiids, we try to analyze solutions to which local problems can be sampled from an ffiid distribution (2-coloring of the grid is not among them). \n\nAlthough this definition was raised in a different context then distributed computing, it is actually almost identical to the definition of a distributed algorithm! The differences are two. First, the ffiid procedure does not know the value of $n$ since, in fact, it makes most sense to consider it in infinite graphs (like the infinite grid $\\mathbb{Z}^d$). Second, the natural measure of the complexity is different: it is the speed of the decay of the random variable corresponding to the time until the local algorithm finds the solution. \nNext, note that the algorithm has to return a correct solution with probability $1$, while the usual randomized local algorithm can err with a small probability. We call ffiids \\emph{oblivious local algorithms} to keep the language of distributed computing (see \\cref{sec:preliminaries}).\nWe define the class $\\mathsf{TAIL}(f(r))$ as the class of problems solvable by an oblivious local algorithm if the probability of the needed radius being bigger than $r$ is bounded by $1\/f(r)$ (see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Descriptive Combinatorics}\n\nA famous Banach-Tarski theorem states that one can split a three-dimensional unit volume ball into five pieces, translate and rotate each piece and end up with two unit volume balls. \nThis theorem serves as an introductory warning in the beginner measure theory courses: not every subset of $\\mathbb{R}^3$ can be measurable.\nSimilarly surprising was when in 1990 Laczkovich answered the famous Circle Squaring Problem of Tarski\\todo{cite laczkovich}: A square of unit area can be decomposed into finitely many pieces such that one can translate each of them to get a circle of unit area.\nUnlike in Banach-Tarski paradox, this result was improved in recent years to make the pieces measurable in various sense \\cite{OlegCircle, Circle, JordanCircle}.\n\n\nThis theorem and its subsequent strengthenings are the highlights of a field nowadays called descriptive combinatorics \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} that has close connections to distributed computing as was shown in an insightful paper by Bernshteyn \\cite{Bernshteyn2021LLL}. \nGoing back to the Circle Squaring Problem, the proofs of its versions first choose a clever set of constantly many translations $\\tau_1, \\dots, \\tau_k$. Then, one constructs a bipartite graph between the vertices of the square and the cycle and connects two vertices $x, y$ if there is a translation $\\tau_i$ mapping the point $x$ in the square to the point $y$ in the cycle. The problem can now be formulated as a perfect matching problem and this is how combinatorics enters the picture. In fact, versions of a matching or flow problems are then solved on the grid graph $\\mathbb{Z}^d$ using a trick called \\emph{toast construction} also used in the area of the study of ffiids \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}. \n\nWe discuss the descriptive combinatorics classes in a more detail in the expository \\cref{sec:OtherClasses} but we will not need them in the main body. Let us now give an intuitive definition of the class $\\mathsf{BOREL}$. \nBy a result of Chang and Pettie \\cite{chang2016exp_separation}, we know that the maximal independent set problem is complete for the class $\\mathsf{LOCAL}(O(\\log^* n))$ in the sense that this problem can be solved in $O(\\log^* n)$ rounds and any problem in that class can be solved in constant number of rounds if we are given access to an oracle computing MIS (we make this precise in \\cref{rem:chang_pettie_decomposition}). Imagine now that the input graph is infinite and our local algorithm has access to MIS oracle and local computations but now both can be used countably many times instead of just once or, equivalently, fixed number of times as is the case with the class $\\mathsf{LOCAL}(O(\\log^* n))$. That is, we can think of an algorithm that constructs MIS of larger and larger powers of the input graph and uses this infinite sequence to solve an LCL. Those constructions are in the class $\\mathsf{BOREL}$ from \\cref{fig:big_picture_grids}. In fact, the exact relationship between this definition and full power of $\\mathsf{BOREL}$ is not clear.\nHowever, it should give an idea about the difference between descriptive combinatorics and distributed computing -- from the descriptive combinatorics perspective, reasonable classes are closed under repeating basic operations countably many times.\n\n\n\n\n\\paragraph{Locally Checkable Problems on Grids as a ``Rosetta Stone''}\n\nLocally checkable problems (LCLs) are graph problems where a correctness of a solution can be checked locally. That is, there is a set of local constraints such that if they are satisfied at every node, the solution to a problem is correct. \nAn examples of such constraints include vertex coloring, edge coloring, or perfect matching. We also allow inputs on each node, so the list coloring is also an example of LCL (given that the total number of colors in all lists is finite). One should think of this class as an analogue of the class NP from complexity theory. \n\nIn this paper, we study locally checkable problems on grid graphs in the three settings introduced above. This approach reveals number of connections. For example, in \\cref{fig:big_picture_grids}, we can see that the two classes of local complexities $\\mathsf{LOCAL}(O(1))$ and $\\mathsf{LOCAL}(O(\\log^* n))$ are equivalent to different natural classes from the perspective of factors of iid and descriptive combinatorics. On the other hand, those two worlds are richer. Many of our discussions in fact apply to graphs of bounded growth, but the grids serve as the simplest, but still quite nontrivial model in which one can try to understand all three settings together (the case of lines is understood quite well \\cite{grebik_rozhon2021LCL_on_paths} and we study aspects of the connection for trees in a parallel paper \\cite{brandt_chang_grebik_grunau_rozhon_vidnyaszky2021LCLs_on_trees_descriptive}). \n\n\\begin{landscape}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 1.5\\textwidth]{fig\/big_picture_grids.png}\n    \\caption{The big picture of local complexity classes on grids from the perspective of distributed algorithms, factors of iid, and descriptive combinatorics. Canonical example of each class is on the left side. }\n    \\label{fig:big_picture_grids}\n\\end{figure}\n\\end{landscape}\n\n\n\\subsection{Our Contribution }\n\nOur contribution can be mostly summed up as several new arrows in \\cref{fig:big_picture_grids}. In general,  this clarifies several basic questions regarding the connections of the three fields and helps to view all three fields as part of a general theory of locality. \n \n\n\nFor example, consider the following question from the paper of Brandt et al. \\cite{brandt_grids} that considered LCL problems on grids: \n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colorings and distributed complexity, it remains unclear how to directly translate results from one setting to the other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al., and does our complexity classification imply answers to the open questions they pose?\n}\n\\end{quote}\nWe show how to 'directly translate' results between the setting of distributed algorithms and finitary factors of iid in \\cref{subsec:preliminaries_tail,subsec:speedup_warmup}. This indeed answers two open questions of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\nTo understand the connection between the world of factors of iids and descriptive combinatorics, we formalize the notion of toast construction used in both setups. We prove general results connecting the toast constructions with oblivious local algorithms and this enables us to understand the $\\mathsf{TAIL}$ complexity  of a number of concrete problems like vertex $3$-coloring, edge $4$-coloring, perfect matchings and others. \n\nIn general, we hope that the concept of $\\mathsf{TAIL}$ complexities will find other usecases. We discuss in \\cref{subsec:congestLCA} how it relates to studied notions of oblivious algorithms, energy complexity, or its relevance to the $\\mathsf{LCA}$ model.\\todo{Oleg se ptal jestli uz ma tady vedet co je LCA a jestli to nekde neprehlid}\nIn fact, the $\\mathsf{TAIL}$ complexity approach was already considered e.g. in the celebrated MIS paper of \\cite{ghaffari2016MIS} (see the introduction of his paper) and the $\\mathsf{TAIL}$ bound on the MIS algorithm there may be the reason of its wide applicability. \n\nWe now list concrete results that we prove (or they follow from the work of others, see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Classification Theorems for $\\mathsf{TAIL}$}\n\nWe prove several results that one can view as arrows in \\cref{fig:big_picture_grids}. Those can be viewed as an analogy to the line of work aiming for the classification of $\\mathsf{LOCAL}$ complexities discussed in \\cref{subsec:preliminaries_local}. \n\n\\begin{itemize}\n    \\item $\\mathsf{TAIL}(\\mathrm{tower}(\\omega(r))) = \\mathsf{LOCAL}(O(1))$\\hspace*{1cm} We show that any oblivious local algorithm with tail decay faster than tower function can be in fact solved in $O(1)$ local complexity. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) = \\mathsf{CONTINUOUS}$\\hspace*{1cm} There are three natural definitions of classes of basic symmetry breaking problems in all three considered fields and they are the same \\cite{GJKS,HolroydSchrammWilson2017FinitaryColoring}. \n    As a direct application one gets that from the three proofs that $3$-coloring of grids is not in $\\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, not in $ \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ \\cite{HolroydSchrammWilson2017FinitaryColoring} and not in $ \\mathsf{CONTINUOUS}$ \\cite{GJKS} any one implies the others. These three classes are the same in a much more general setup (see \\cref{subsec:CONT}). \n    \\item $\\mathsf{TAIL}(\\omega(r^{d-1})) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$\\hspace*{1cm} We prove that any problem that can be solved with an oblivious local algorithm with tail decay faster than $r^{d-1}$, can in fact be solved with a tower tail decay. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item Similarly to the time hierarchy theorems of the $\\mathsf{LOCAL}$ model \\cite{chang2017time_hierarchy}, we study the range of possible $\\mathsf{TAIL}$ complexities and give examples of problems LCL problems with $\\mathsf{TAIL}$-complexities $\\Theta(r^{j - o(1)})$ for any $1 \\le j \\le d-1$. \n\\end{itemize}\n\n\\paragraph{$\\TOAST$ constructions}\nWe formally define $\\TOAST$ algorithms (we mentioned earlier that these constructions are used in descriptive combinatorics and the theory of random processes) and prove results about them.  \nLet us first explain the toast constructions in a more detail. A $q$-toast in the infinite graph $\\mathbb{Z}^d$ is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that (a) for any two nodes of the graph we can find a piece that contains both,\\todo{it is enough that every vertex is contained in a piece} (b) boundaries of any two different pieces are at least distance  $q$ apart. \nThat is, imagine a collection of bigger and bigger pieces that form a laminar family (only inclusions, not intersections) that, in the limit, covers the whole infinite graph.\n\nThis structure can be built in the descriptive combinatorics setups \\cite{FirstToast,GJKS,Circle} but also with oblivious local algorithms \\cite{HolroydSchrammWilson2017FinitaryColoring,Spinka}. In fact, we show that one can construct the structure with  $1\/r^{1-o(1)}$ $\\mathsf{TAIL}$ decay in \\cref{thm:toast_in_tail}. This is best possible up to the $o(1)$ loss in the exponent. \nIf one has an inductive procedure building the toast, we can try to solve an LCL problem during the construction. That is, whenever a new piece of the toast is constructed, we define how the solution of the problem looks like at this piece. This is what we call a $\\TOAST$ algorithm. \nThere is a randomized variant of the $\\TOAST$ algorithm and a lazy variant. In the lazy variant one can postpone the decision to color given piece, but it needs to be colored at some point in the future. \nNext, we prove several relations between the $\\TOAST$ classes (see \\cref{fig:big_picture_grids}).  \n\n\\begin{itemize}\n    \\item $\\TOAST \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$\\hspace*{1cm} This result allows us to translate toast-algorithm constructions used in descriptive combinatorics to the $\\mathsf{TAIL}$-setting. This includes problems like vertex $3$-coloring (this answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}), edge $2d$-coloring, or perfect matching. \n    \\item $\\RTOAST = \\TOAST$\\hspace*{1cm} This derandomization result for example implies on the first glance surprising inclusion $\\mathsf{FINDEP} \\subseteq \\mathsf{BOREL}$ -- these two classes are discussed in \\cref{sec:OtherClasses}. \n    \\item $\\mathsf{TAIL} = \\RLTOAST$\\hspace*{1cm} Here, $\\mathsf{TAIL}$ is defined as the union of $\\mathsf{TAIL}(f)$ over all functions $f \\in \\omega(1)$. This result shows how the language of toast constructions can in fact characterize the whole hierarchy of constructions by oblivious local algorithms. \n    \\item We show that the problem of $2d$ edge coloring of $\\mathbb{Z}^d$ is in $\\TOAST$, hence in $\\mathsf{BOREL}$ which answers the question of Gao et al \\cite{GJKS}. Independently, this was proven by \\cite{ArankaFeriLaci}. \n\\end{itemize}\n\n\\subsection{Roadmap}\n\nIn \\cref{sec:preliminaries} we recall the basic definitions regarding the $\\mathsf{LOCAL}$ model and explain the formal connection between finitary factors of iids. \nNext, in \\cref{sec:speedup_theorems} we prove the basic speedup theorems for $\\mathsf{TAIL}$. \nIn \\cref{sec:Toast} we define $\\TOAST$ algorithms and relate it to the $\\mathsf{TAIL}$ model. \nIn \\cref{sec:SpecificProblems} we discuss what our results mean for specific local problems and in \\cref{sec:Inter} we construct problems with $r^{j-o(1)}$ $\\mathsf{TAIL}$ complexities for $1 \\le j \\le d-1$. Finally, in \\cref{sec:OtherClasses} we discuss connection of our work to other classes of problems and finish with open questions in \\cref{sec:open_problems}. \n\n\n\n\n\n\\begin{comment}\nHonza - Introduction\n\nA \\emph{locally checkable problems (LCL) $\\Pi$ on a family of graphs $\\mathbf{G}$} is a procedure that checks whether a given ``candidate'' vertex coloring of $G\\in \\mathbf{G}$ satisfies a local constraints given by $\\Pi$\\footnote{\nFormally, $\\Pi$ is a triplet $\\Pi=(b,t,\\mathcal{P})$, where $b$ is a set of colors, $t\\in \\mathbb{N}$ is the locality of $\\Pi$ and $\\mathcal{P}$ is a set of allowed configurations around each vertex of locality at most $t$.}.\nMore concretely, given a graph $G=(V(G),E(G))\\in \\mathbf{G}$ and a ``candidate'' coloring $c:V(G)\\to b$, we say that $c$ is a \\emph{$\\Pi$-coloring} if for every $v\\in V(G)$ the restriction of $c$ to $t$-neighborhood of $v$ is in $\\mathcal{P}$, where $t$ is the locality of $\\Pi$ and $\\mathcal{P}$ is the set of constraints of $\\Pi$.\n\nIn this paper the family $\\mathbf{G}$ always refers to graphs that look locally like the standard Cayley graph of $\\mathbb{Z}^d$, $\\mathbb{Z}^d$, for some $d\\ge 1$.\nThis includes $\\mathbb{Z}^d$ as well as various finite $d$-dimensional tori with oriented edges that are labeled by the standard generators of $\\mathbb{Z}^d$.\nThe dimension $d$ is always assumed to be bigger than $1$, since for $d=1$ there is a full classification of LCL problems.\nWe omit mentioning ${\\bf G}$ and refer to all LCL problems on the family ${\\bf G}$ simply as \\emph{LCL problems (on $\n\\mathbb{Z}^d$)}.\nWhen we want to stress that a given LCL problem is on $\\mathbb{Z}^d$ we write $d$ in the superscript, i.e., $\\Pi^d$.\n\nWe study a complexity of LCL problems in various settings.\nIn the most abstract terms, a complexity of a given LCL problem $\\Pi$ is measured in terms of how difficult it is to \\emph{transform} a \\emph{given labeling} of a graph into a $\\Pi$-coloring.\nThe different contexts varies in what is considered to be an allowed starting labeling and what is allowed transformation.\nMost relevant for our investigation is the following probability notion.\nStart with a graph $G$ and assign to each vertex $v\\in V(G)$ a real number from $[0,1]$ uniformly and independently, i.e., \\emph{iid labeling} of $G$.\nA function $F$ that transforms these iid labels in a measurable and equivaraint way into a different coloring is called \\emph{factor of iid (fiid)}.\nHere equivariant refers to the group of all symmetries of $G$.\nIt is not hard to see that the group of all symmetries of $\\mathbb{Z}^d$ is the group $\\mathbb{Z}^d$ itself that acts by shift-translations.\nA fiid $F$ on $\\mathbb{Z}^d$ is called \\emph{finitary fiid (ffiid)} if for almost every iid labeling of $\\mathbb{Z}^d$ the  color $F(x)({\\bf 0}_d)$ only depends on the labels of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, if $x$ and $x'$ agrees on this neighborhood, then $F(x)({\\bf 0}_d)=F(x')({\\bf 0}_d)$.\nGiven ffiid $F$, we write $R_\\mathcal{A}$ for the random variable that codes minimum diameter of such neighborhood and we refer to it as the coding radius of $F$.\n\nWe say that a ffiid $F$ \\emph{solves an LCL problem $\\Pi$} if it produces a $\\Pi$-coloring for almost every iid labeling of $\\mathbb{Z}^d$.\nan LCL problem $\\Pi$ is in the class $\\mathsf{TAIL}$ if $\\Pi$ admits a ffiid solution.\nFollowing Holroyd, Schramm and Wilson \\cite{HolroydSchrammWilson2017FinitaryColoring}, the complexity of LCL problems in $\\mathsf{TAIL}$ can be measured on a finer scale by the properties of the coding radius $R_\\mathcal{A}$, where $F$ runs over all ffiid's that solve the given problem.\nSimilarly, we define when fiid $F$ solves $\\Pi$ and the class $\\mathsf{fiid}$.\n\nBefore we move to distributed algorithm we recall what is known about the notorious class of LCL problems that are the \\emph{proper vertex $k$-colorings of $\\mathbb{Z}^d$}, $\\Pi^d_{\\chi,k}$, where $k>1$, in the context of ffiid.\nIt is known that if $d>1$ and $k>3$, then $\\Pi^d_{\\chi,k}$ admits ffiid solution with a tower tail decay, see Holroyd et.al~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nIf $d>0$ and $k=2$, then $\\Pi^d_{\\chi,2}$ is not even in $\\mathsf{fiid}$.\nThe most interesting case is when $d>1$ and $k=3$.\nIt was shown by Holroyd et al~\\cite{HolroydSchrammWilson2017FinitaryColoring} that in that case $\\Pi^d_{\\chi,3}$ is in the class $\\mathsf{TAIL}$ and every ffiid that solves $\\Pi^d_{\\chi,3}$ must have infinite the $2$-moment of its coding radius. \nMoreover, they found a ffiid that solve $\\Pi^d_{\\chi,3}$ and has finite some moments of its coding radius.\nHere we strengthen their result by showing that ${\\bf M}(\\Pi^d_{\\chi,3})=1$ and every ffiid solution of $\\Pi^d_{\\chi,3}$ has infinite the $1$-moment of its coding radius.\n\nIn the context of deterministic distributed computing the setting is the following.\nStart with finite graph $G$ on $n$-vertices (in our case $n$ is big enough and $G$ is a torus) and assign to every vertex of $G$ its unique identifier from $[n]$.\\footnote{In generality, the identifier could be assigned from some domain that has size polynomial in $n$.}\nA \\emph{local algorithm $\\mathcal{A}_n$ of locality $t_n$} is a rule that computes color of a vertex $v$ based only on its $t$-neighborhood and the unique identifiers in this neighborhood.\nWe say that $\\mathcal{A}_n$ \\emph{solves} $\\Pi$ if for every (admissible)\\footnote{As a torus $G$ needs to be wider than the locality of $\\mathcal{A}_n$} graph $G$ on $n$-vertices and any assignment of unique identifiers to $G$ the rule $\\mathcal{A}_n$ produces a $\\Pi$-coloring.\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if there is a sequence of local rules \\todo{co je to local rule?}$\\{\\mathcal{A}_n,t_n\\}_{n\\in \\mathbb{N}}$ such that $\\mathcal{A}_n$ has locality $t_n$, solves $\\Pi$ for every $n\\in\\mathbb{N}$ and $n\\mapsto t_n$ is in the class $O(\\log^*(n))$.\nThe reason why we picked $\\log^*(n)$ is that there is a complete classification of LCL problems on oriented labeled grids.\nRoughly speaking, an LCL problem $\\Pi$ is either \\emph{trivial}, i.e., there is a single color that produces $\\Pi$-coloring, \\emph{local}, i.e., there is an algorithm that needs locality $\\Theta(\\log^*(n))$, or \\emph{global}, i.e., every local algorithm that solves $\\Pi$ needs to see essentially the whole graph.\\footnote{Same classification holds in the context of randomized distributed algorithms.}\n\nWe use this classification to provide several speed-up results in the context of ffiid solutions.\nWe note that the connection between distributed computing and probability setting was previously observed by several authors.\nFor example, it follows easily from the result of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring} that local coloring problems that are not global in the distributed setting admit ffiid solution with tower tail decay.\nIn fact, Brandt et al. \\cite{brandt_grids} asked:\n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colourings\nand distributed complexity, it remains unclear how to directly translate results from one setting to\nthe other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al. [24], and does our complexity classification imply answers to the open\nquestions they pose?\n}\n\\end{quote}\nAs one of our main result we address their question.\n\nThe most abstract setting that we consider is Borel combinatorics.\nIn this context one studies graphs that additionally posses a completely metrizable and separable topological structure (or a standard Borel structure) on its vertex set.\nIt follows from the work of Seward and Tucker-Drob \\cite{ST} and Gao, Jackson, Krohne and Seward \\cite{GJKS} that the canonical object to investigate is the graph structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, the free part of the shift action $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$, induced by the standard generating set of $\\mathbb{Z}^d$. \nThe topological and Borel structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is the one inherited from the product topology on $2^{\\mathbb{Z}^d}$.\nWe say that an LCL problem $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous and equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ into the space of $\\Pi$-colorings.\nThe class $\\mathsf{BOREL}$ is defined similarly, only the requirement on the map to be continuous is relaxed to be merely Borel measurable.\nBernshteyn \\cite{Bernshteyn2021LLL} proved that LCL problems that are not global in the distributed setting admit continuous and therefore Borel solution.\nIn another words, he showed that the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ is contained in $\\mathsf{CONTINUOUS}$.\nIt is not hard to show that the other inclusion follows from the Twelve Tile Theorem of Gao et al \\cite{GJKS}, i.e., the class $\\mathsf{CONTINUOUS}$ in fact coincide with the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nHere we investigate in greater detail what is the connection between existence of a Borel solution and a tail behavior of ffiid.\nIn fact all known Borel constructions uses the following structure that can be rephrased without mentioning topology or Borel measurability.\n\nA \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ such that\n\\begin{itemize}\n    \\item for every $g,h\\in \\mathbb{Z}^d$ there is $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$.\n\\end{itemize}\nWe say that an LCL problem $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and an equivariant rule $\\Omega$ that extends partial colorings such that if we apply $\\Omega$ on every $q$-toast $\\mathcal{D}$ inductively on the depth, then we produce a $\\Pi$-coloring.\nIt is not known whether the classes $\\mathsf{BOREL}$ and $\\TOAST$ coincide, however, it follows from the work of Gao et al.~\\cite{GJKS2} that $\\TOAST\\subseteq \\mathsf{BOREL}$.\nThis is proved by constructing a toast structure in a Borel way on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, see Marks and Unger~\\cite{Circle} for application and proof of this fact.\nThe specific problems that belong to the class $\\TOAST$ include $3$-vertex coloring, perfect matching, $2d$-edge coloring, Schreier decoration (for non-oriented grids), some rectangular tilings, space filling curve etc.\nThese results were proved by many people in slightly different language, \\cite{GJKS2, ArankaFeriLaci, spencer_personal}.\nWe utilize here all the constructions and show that these all belong to $\\TOAST$.\n\nWe connect the classes $\\TOAST$ and $\\mathsf{TAIL}$ as follows.\nWe show that for every $q\\in \\mathbb{N}$ and $\\epsilon>0$ there is a ffiid $F$ that produces $q$-toast that has the $(1-\\epsilon)$-moment finite.\nA slightly stronger result implies that, in fact, ${\\bf M}(\\Pi)\\ge 1$, whenever an LCL problem $\\Pi$ is in the class $\\TOAST$.\nThis shows that ${\\bf M}(\\Pi)\\ge 1$ for all the problems mentioned above.\nWe note that in the special case when $d=2$ and the corresponding LCL problem $\\Pi$ does not admit ffiid solution with a tower tail decay, we deduce that ${\\bf M}(\\Pi)=1$.\nThis follows from our speed-up result in the context of ffiid.\n\nTraditionally there are two more classes that are investigated in Borel combinatorics, $\\mathsf{BAIRE}$ and $\\mathsf{MEASURE}$.\nTo define the class $\\mathsf{BAIRE}$ we replace the topological requirement on continuity by \\emph{Baire measurability}, i.e., we require a solution to be continuous only on a \\emph{comeager} set, see the work of Bernshteyn \\cite{Bernshteyn2021LLL}.\nThe class $\\mathsf{MEASURE}$ generalizes the setting of fiid.\nThe goal here is to find a Borel solution on a conull set for a given Borel graph that is endowed with a Borel probability measure.\nThis direction has been recently investigated by Bernshteyn \\cite{Bernshteyn2021LLL} for general graphs, where he proved a measurable version of Lovasz's Local Lemma.\nSee \\cref{fig:big_picture_grids} for all the relations that are known to us.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/big_picture_grids.png}\n    \\caption{Big picture of lcl for $\\mathbb{Z}^d$}\n   \n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{The Results}\n\nIn this subsection we formulate our main results.\nThe formulation uses mostly the language of ffiid because the tail behavior of the coding radius allows to compare problems on a quantitative and, as it turns out, rich scale.\nOne should always re-interpret the result with the help of Figure~\\ref{fig:big_picture_grids} to different contexts.\nIronically enough, the techniques that are most valuable when investigating ffiid solutions come either from local algorithms or Borel combinatorics, where such a rich and quantitative distinction is not possible.\nRecall that in the distributed algorithms there is a trichotomy of LCL problems, i.e., trivial, local or global, and in Borel combinatorics we distinguish mainly between trivial, continuous, Borel, and global problems.\n\n\nWe start with results that use ideas from distributed algorithms.\nFirst, we prove that LCL problems that are not global in distributed algorithms are precisely problems that admit ffiid solution with a tower tail decay.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nThen $\\Pi$ is not global in the local world, i.e., there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds, if and only if there is an ffiid solution $F$ of $\\Pi$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{tower(Cr)}$$\nfor some $C>0$ and large enough $r\\in \\mathbb{N}$.\n\\end{theorem*}\n\nThe translation between ffiid and distributed algorithms allows to deduce the following result that answers problem (ii) of Holroyd at el.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf $\\Pi$ admits an ffiid solution $F$ that has faster tail decay than any tower function, then $\\Pi$ is trivial, i.e., there is a color such that a constant coloring with this color is a correct solution.\n\\end{theorem*}\n\nAs a last application of ideas from the distributed algorithms, we show the following speed-up theorem for ffiid.\n\n\\begin{theorem*}\nLet $d,t\\in \\mathbb{N}$.\nThere is $C_{d,t}>0$ such that if an LCL problem $\\Pi$ admits an ffiid solution $F$ such that\n$$\\P(R_\\mathcal{A}>r)\\le C_{d,t}r^{d-1},$$\nthen there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds and solves $\\Pi$.\n\\end{theorem*}\n\nTo understand better LCL problems that are global in the distributed algorithms or admit no continuous solution in the context of Borel combinatorics we introduce the following numerical invariant.\nThe inspiration comes from the paper of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{definition*}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an ffiid solution $F$ of $\\Pi$ such that $R_F$ has finite the $\\alpha$-moment.\n\\end{definition*}\n\nCombination of previous results immediately implies the following strengthening of the upper bound on the necessary infinite moment of ffiid solution of the proper vertex $3$-coloring problem, $\\Pi^d_{\\chi,3}$, from Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nMoreover, this answers their problem (iii).\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf ${\\bf M}(\\Pi)>d-1$, then $\\Pi$ admits an ffiid solution with tower tail decay.\n\\end{theorem*}\n\nAs a first application of ideas from Borel combinatorics, i.e., the toast structure, we compute the optimal tail decay for $\\Pi^d_{\\chi,3}$, thus answering a problem (i) of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d>1$ and $\\Pi^d_{\\chi,3}$ be the proper vertex $3$-coloring problem.\nThen ${\\bf M}(\\Pi^d_{\\chi,3})=1$.\n\\end{theorem*}\n\nA high level explanation of a toast construction in the context of ffiid says that if $r\\in \\mathbb{N}$ is given, then the fraction of vertices not covered by first $r$-layers of the toast is at most $\\frac{1}{r}$.\nSimilar construction that uses only $1$-dimensional skeleton of some rectangular tiling of $\\mathbb{Z}^d$ improve this upper bound to $\\frac{1}{r^{d-1}}$.\nWe show that this type of structure is enough to find a perfect matching in a bigger graph then $\\mathbb{Z}^d$, a graph where we moreover add diagonal edges in every $2$-dimensional hyperplane.\nThis yields an existence of intermediate problems from the point of view of ${\\bf M}$.\n\n\\begin{theorem*}\nLet $2\\le \\ell\\le d\\in \\mathbb{N}$.\nThen there is an LCL problem $\\Pi$ on $\\mathbb{Z}^d$ that satisfies ${\\bf M}(\\Pi)=\\ell-1$.\n\\end{theorem*}\n\nAs a last result we summarize what is known about specific problems.\nSince we work mainly with oriented grids we state everything in this context, however, some of the results can be deduced for non-oriented grids as well, see Section~\\ref{sec:SpecificProblems}.\nIn particular, we answer Question~6.5 of Gao et al.~\\cite{GJKS} about Borel edge chroamtic number of $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\n\n\\begin{theorem*}\nLet $d\\ge 2$ and $\\Pi$ be any of the following LCL problems:\n\\begin{enumerate}\n    \\item proper vertex $3$-coloring, \\cite{GJKS}, \\cite{spencer_personal},\n    \\item proper edge $2d$-coloring,\n    \\item rectangular tiling with rectangles from some set $\\mathcal{R}$ that satisfies \\cref{eq:GCD}, \\cite{Unger},\n    \\item perfect matching, \\cite{GJKS}.\n\\end{enumerate}\nThen $\\Pi$ is in the class $\\TOAST$.\nConsequently, $\\Pi$ is in the class $\\mathsf{BOREL}$ and satisfies $1\\le {\\bf M}(\\Pi)$.\n\\end{theorem*}\n\n\n\nThe paper is organized as follows...\n\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nLet $d\\in \\mathbb{N}$ be fixed.\nThe class of graphs that we consider in this work contains mostly the infinite grid $\\mathbb{Z}^d$, that is $\\mathbb{Z}^d$ with the Cayley graph structure given by the standard set of generators \n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}. $$\nWe also used the finite variant of the graph: the torus of side length $\\Theta(n^{1\/d})$ with edges oriented and labeled by $S$.\nUsually, when we talk about a graph $G$ we mean one of the graphs above and it follows from the context whether we mean finite or infinite graph. \n\nGiven a graph $G$, we denote as $V(G)$ the set of vertices, as $E(G)$ the set of edges and as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. When $G$ is the infinite grid $\\mathbb{Z}^d$, then we leave the subscripts blank.\nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the \\emph{isomorphism type} of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften we do not mention the labels and tacitly assume that they are understood from the context.\nAlso, the label set might depend on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\n\\begin{comment}\n\nWe always consider the group $\\mathbb{Z}^d$ with the standard set of generators $S$, i.e.,\n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}.$$\nThe \\emph{Cayley graph} $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right):=\\operatorname{Cay}\\left(\\mathbb{Z}^d,S\\right)$ of $\\mathbb{Z}^d$ is then defined on the vertex set $\\mathbb{Z}^d$ with oriented and $S$-labeled edges \n$$\\left\\{(g,s+g):s\\in S, g\\in \\mathbb{Z}^d\\right\\}=E\\left(\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)\\right).$$\nWe denote the identity in $\\mathbb{Z}^d$ as ${\\bf 0}_d$. \nWhen we talk about $\\mathbb{Z}^d$ next, we mean formally the appropriate Cayley graph with labelled edges, that is, we think of the graph together with its consistent orientation. \n\n\nA \\emph{graph} $G$ is a pair $(V(G),E(G))$, where $V(G)$ is a vertex set and $E(G)$ consists of oriented and $S$-labeled edges.\nGiven a graph $G$, we denote as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. Usually, $G$ is just the grid $\\mathbb{Z}^d$, then we leave the subscript blank. \nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the isomorphism type of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\n\\todo{vsechny strany toru maji delku n na 1\/d; zjevne vsechny problemy maji lokalitu nejvyse n na 1\/d}\n\n\\end{comment}\n\n\n\\begin{comment}\nWe write $\\operatorname{\\bf FG}^{S}$ for the set of all graphs that look in their $2$-neighborhoods as $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$, i.e., a graph $G=(V(G),E(G))$ is in $\\operatorname{\\bf FG}^{S}$ if $[\\mathcal{B}_G(v,2)]=\\left[\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,2)\\right]$ for every $v\\in V(G)$.\nNote that $\\mathbb{Z}^d\\in \\operatorname{\\bf FG}^{S}$ and there are finite graphs in $\\operatorname{\\bf FG}^{S}$, as an example take any torus that is big enough.\nWrite $\\operatorname{\\bf FG}_{\\bullet}^{S}$ for the rooted graphs of the form\n$$(\\mathcal{B}_G(v,r),v)$$\nwhere $G\\in \\operatorname{\\bf FG}^{S}$, $v\\in V(G)$ and $r\\in \\mathbb{N}$.\nWe say that a rooted graph $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ has \\emph{diameter at most $m\\in \\mathbb{N}$} if $(\\mathcal{B}_G(v,m),v)=(G,v)$.\nNote that, in particular, we have $\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,r)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ for every $r\\in \\mathbb{N}$.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften the label set $D$ depends on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\\end{comment}\n\n\n\n\\subsection{LCLs on $\\mathbb{Z}^d$}\n\nOur goal is to understand locally checkable problems (LCL) on $\\mathbb{Z}^d$. \n\n\\begin{definition}[Locally checkable problems \\cite{naorstockmeyer}]\nA \\emph{locally checkable problems (LCL)} on $\\mathbb{Z}^d$ is a quadruple $\\Pi=(\\Sigma_{in}, \\Sigma_{out},t,\\mathcal{P})$ where $\\Sigma_{in}$ and $\\Sigma_{out}$ are finite sets, $t\\in \\mathbb{N}$ and $\\mathcal{P}$ is an assignment\n$$[H_c,w]\\mapsto \\mathcal{P}([H_c,w])\\in \\{0,1\\},$$\nwhere $(H,w)$ is a rooted grid of radius at most $t$ together with an input and output coloring $c: V(H)\\to \\Sigma_{in}\\times\\Sigma_{out}$. \n \nGiven a graph $G$ together with a coloring $c:V(G)\\to \\Sigma_{in}\\times\\Sigma_{out}$, we say that $c$ is a \\emph{$\\Pi$-coloring} if\n$$\\mathcal{P}([\\mathcal{B}_{G_c}(v,t),v])=1$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\n\n\n\\begin{comment}\n\nClasses of functions\n\nBefore we proceed to definition of local algorithms and their complexity we recall here the relevant notions concerning functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nImportantly, we often use asymptotic notation as usual in Computer Science, e.g., we define the class of functions $O(f(n))$ as the union over all functions $g:\\mathbb{N} \\rightarrow \\mathbb{N}$ such that there exists $C > 0$ such that for all $n$ we have $g(n) \\le C f(n)$. Other symbols $\\Omega, \\Theta, o, \\omega$ are defined analogously and their usage is analogous (see e.g. \\cite{?}). \n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{LOCAL model}\n\\label{subsec:preliminaries_local}\n\nHere we formally introduce the $\\mathsf{LOCAL}$ model for both deterministic and randomized algorithms and state the classification of LCLs on $\\mathbb{Z}^d$ from the $\\mathsf{LOCAL}$ point of view.\nInformally, a local algorithm $\\mathcal{A}$ of radius $r\\in \\mathbb{N}$ is a rule that takes a labeled graph $G_c$ and then re-labels each vertex $v\\in V(G)$ depending only on the isomorphism type of its neighborhood $[\\mathcal{B}_{G_c}(v,r),v]$ with a new label from some finite set $b$.\n\n\\paragraph{Local algorithms}\n\nHere, we formally define the complexity classes $\\mathsf{LOCAL}(\\mathcal{F})$ of deterministic local algorithms, where $\\mathcal{F}$ is a class of functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nThe somewhat unintuitive part of the definition is that the algorithm takes as input \\emph{unique identifiers}. Usually, they are taken from some range of values that is polynomial in $n$, the number of vertices of the input graph. However, in most scenarios this is equivalent to the assumption that they are from range $\\{1,\\dots n\\}$, i.e., every identifier is used exactly once. We also use this definition here, for simplicity.\nIn all forthcoming definition we have some fixed LCL $\\Pi$ in mind.\n\n\\begin{definition}[A deterministic local algorithm for $\\mathbb{Z}^d$]\n\\label{def:local_algorithm\nLet $\\Sigma_{out}$ be a finite set.\nA \\emph{local (deterministic) algorithm $\\mathcal{A}$} is a sequence of pairs $\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$, where $\\ell_n\\in \\mathbb{N}$ and $\\mathcal{A}_n$ is an assignment, such that\n$$[\\mathcal{B}_{G_\\ell}(v,\\ell_n),v]\\mapsto \\mathcal{A}_n([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])\\in \\Sigma_{out}$$\nfor every $\\Sigma_{in}$-labeled graph $G$ (torus) on $n$ vertices, $v\\in V(G)$ and a bijection $\\ell:V(G)\\to [n]$.\nIn another words, $\\mathcal{A}_n$ is a function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with injectively labeled vertices from the set $\\{1,\\dots, n\\}$ and with range $b$. \n\nGiven such a $G$ and a bijection $\\ell:V(G)\\to |V(G)|$, we define\n$$\\mathcal{A}(G_\\ell)(v):=\\mathcal{A}_{|V(G)|}([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define the deterministic complexity of a given local coloring problem $\\Pi$.\n\n\\begin{definition}[Complexity of deterministic local algorithm for $\\mathbb{Z}^d$]\nWe say that a local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ \\emph{solves} an LCL $\\Pi=(\\Sigma_{in},\\Sigma_{out},t,\\mathcal{P})$ if $\\mathcal{A}(G_l)$ is a $\\Pi$-coloring of $G$ for every finite $\\Sigma_{in}$-labeled graph $G$ and any bijection $\\ell:V(G)\\to |V(G)|$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\nA randomized local algorithm and its complexity is defined analogously but instead of labeling with unique identifiers each node has an iid random string and the algorithm succeeds with probability $1 - 1\/n$.\n\nCrucial for our investigation are so-called \\emph{oblivious local algorithms}.\nRoughly speaking an algorithm is \\emph{oblivious} if it does not have knowledge of $n$ (see \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} for details). \nSince we do not consider deterministic oblivious algorithms, by an oblivious local algorithm we mean an oblivious randomized local algorithm.\n\n\\begin{definition}[Oblivious local algorithm]\n\\label{def:uniform_local}\nA \\emph{oblivious local algorithm} $\\mathcal{A}$ is a pair $(\\mathcal{C},f)$, where $\\mathcal{C}$ is a (measurable) collection of rooted graphs labeled with $\\Sigma_{in}$ and real numbers from $[0,1]$, and $f: \\mathcal{C} \\rightarrow \\Sigma_{out}$ is a measurable assignment.\n\nWe define $\\mathcal{A}(G)$ to be the labeling computed by $\\mathcal{A}$ on $G$, where for a given node $u$ we have $\\mathcal{A}(G)(u)=g([B_G(u, t), u])$ where $t$ is the smallest integer such that $[B_G(u, t),u] \\in \\mathcal{C}$.\n\\end{definition}\n\nNote that since there is no $n$ in the definition of oblivious local algorithm, the algorithm is correct if it always finishes with probability $1$, while randomized algorithms can have small probability of error dependent on $n$. Hence, one should think about such algorithms as an analogue of Las Vegas algorithms, while the classical randomized local algorithms correspond to Monte Carlo algorithms. \nAn advantage of not knowing $n$ is that it is possible to run $\\mathcal{A}$ on, e.g., the infinite grid $\\mathbb{Z}^d$ instead of just a torus. Usually, these two models give the same results but for toast algorithms that we consider later, there is a difference: While we define the algorithms for an infinite grid, for example the $3$-coloring algorithm makes sense also for a finite torus, while the perfect matching algorithm does not make sense in the finite world for the simple reason that the input graph can have an odd number of vertices. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Randomized local algorithms}\\todo{merge with previous paragraph}\n\nThe main difference between deterministic and randomized model is that the input of the local rule in the randomized case are graphs with vertices labeled uniformly independently at random by elements of the interval $[0,1]$ and we require that the output coloring is correct everywhere with high probability.\nThe complexity of local algorithm is then defined as the locality it needs.\n\n\\begin{definition}[Randomized local algorithm for $\\mathbb{Z}^d$]\nLet $b$ be a set.\nWe say that $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$ is a randomized local algorithm for the class $\\operatorname{\\bf FG}^{S}$ if $\\ell_n\\in \\mathbb{N}$ and \n$$[\\mathcal{B}_{G_l}(v,\\ell_n),v]\\mapsto \\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])\\in b$$\nis a measurable assignment, where $G\\in \\operatorname{\\bf FG}^{S}$, $|V(G)|=n$, $v\\in V(G)$ and $\\ell:V(G)\\to [0,1]$.\nIn another words, $\\mathbb{A}_n$ is a measurable function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with vertices labeled by $[0,1]$ and with range $b$. \n\nGiven $G\\in \\operatorname{\\bf FG}^{S}$ such that $n=|V(G)|$ and a map $\\ell:V(G)\\to [0,1]$, we define\n$$\\mathbb{A}(G_l)(v):=\\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define when a randomized algorithm solves an LCL $\\Pi$ and the randomized complexity classes.\n\n\\begin{definition}[Complexity of randomized local algorithm for $\\mathbb{Z}^d$]\nLet $\\Pi=(b,t,\\mathcal{P})$ be an LCL.\nWe say that a randomized algorithm $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ solves $\\Pi$ if \n$$\\lambda^{\\oplus n}\\left(\\left\\{\\ell\\in [0,1]^{V(G)}:\\mathbb{A}(G_l) \\ \\operatorname{is \\ a} \\ \\Pi\\operatorname{-coloring} \\right\\}\\right)>1-\\frac{1}{n}$$\nholds for every $n\\in \\mathbb{N}$ and $G\\in \\operatorname{\\bf FG}^{S}$ such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\\end{comment}\n\n\\begin{comment}\n\\label{sec:clocal_definition}\n\nThe last class of local algorithm that we introduce serves on one hand as a main technical tool for deterministic speed up tricks and on the other hand allows a smooth translation of local algorithm into the measurable setting.\nIntuitively, an algorithm is in the class $\\mathsf{C-LOCAL}$ if it only needs a fixed finite locality (not depending on the size of a graph) given some coloring of the vertices with fixed number of colors (not depending on the size of a graph).\n\nLet $s\\in \\mathbb{N}$ and $G$ be a graph.\nDenote as $G^s$ the $s$-power graph of $G$, i.e., $V(G^s)=V(G)$ and $(x,y)\\in E(G^s)$ if and only if $0<d_G(x,y)\\le s$.\nNote that if $G\\in \\operatorname{\\bf FG}^{S}$, then the maximum degree of a vertex in $G^s$ is bounded by $\\Delta_s:=\\left|\\mathcal{B}({\\bf 0}_d,s)\\right|$.\nWe say that $c:V(G)\\to m$ is a proper $s$-power vertex coloring if it is a proper vertex coloring of the graph $G^s$.\nNote that it follows from a standard greedy construction that it is possible to find an $s$-power coloring of every $G\\in \\operatorname{\\bf FG}^{S}$ with $(\\Delta_s+1)$-many colors.\n\n\\begin{definition}[$\\mathsf{C-LOCAL}(O(1))$]\nWe say that an LCL $\\Pi=(b,t,\\mathcal{P})$ is in the class $\\mathsf{C-LOCAL}(O(1))$ if there are $r,\\ell\\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ with domain that consists of rooted graphs $(H,w)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ of diameter at most $\\ell$ together with an injective map $d:V(H)\\to (\\Delta_{r+t}+1)$, i.e.,\n$$[H_d,w]\\mapsto \\mathcal{A}([H_d,w])\\in b,$$\nsuch that $\\mathcal{A}(G_c)$ is a $\\Pi$-coloring for every $G\\in \\operatorname{\\bf FG}^{S}$ that satisfies \n$$[\\mathcal{B}_{G}(v,r+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,r+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$ and $c:V(G)\\to (\\Delta_{r+t}+1)$, a proper $(r+t)$-power vertex coloring of the graph $G^{r+t}$, where\n$$\\mathcal{A}(G_c)(v):=\\mathcal{A}([\\mathcal{B}_{G_c}(v,\\ell),v])\\in b$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\\end{comment}\n\n\n\n\n\n\\paragraph{Classification of Local Problems on $\\mathbb{Z}^d$}\n\nIn this section, we state a special case of \\emph{Classification of LCLs}. This is an ongoing effort in the area of distributed algorithms and our current understanding emerges as a corollary of many results that follow from several recent lines of work on local algorithms including \\cite{naorstockmeyer, brandt,chang2016exp_separation,chang2017time_hierarchy, ghaffari2017complexity, ghaffari2018derandomizing,balliu2018new_classes-loglog*-log*, RozhonG19, ghaffari2019distributed, ghaffari_grunau_rozhon2020improved_network_decomposition,chang2020n1k_speedups,balliu2020almost_global_problems,brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and many others.  \nFor our purposes we will need only a special case of the Classification for grids. Fortunately, here (and on trees) all possible local complexities of an LCL are completely understood. This result is stated in \\cref{thm:classification_of_local_problems_on_grids}. \n\n\\begin{comment}\nRecall that an LCL $\\Pi=(b,t,\\mathcal{P})$ is satisfiable \\todo{do we need it?} if there is $x\\in b^\\Gamma$ that is a $\\Pi$-coloring of $\\operatorname{Cay}(\\Gamma,S)$ and it is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant $\\alpha$ coloring is a $\\Pi$-coloring.\n\\end{comment}\n\n\\begin{comment}\n\\begin{definition}[Global problems]\n\\label{def:global_problem}\nWe say that a satisfiable LCL problem $\\Pi$ is \\emph{global for deterministic (randomized) algorithms} if for every deterministic (randomized) local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ that solves $\\Pi$ we have $\\{\\ell_n\\}_{n\\in\\mathbb{N}}\\in \\Omega(n^{\\frac{1}{d}})$.\n\\end{definition}\n\\end{comment}\n\n\n\n\n\\begin{theorem}[Classification of LCLs on $\\mathbb{Z}^d$ \\cite{naorstockmeyer, chang2016exp_separation, chang2017time_hierarchy, brandt_grids, brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:classification_of_local_problems_on_grids}\nLet $\\Pi$ be an LCL.\nThen one of the following is true. \n\\begin{enumerate}\n\\item $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial,\n\\item $\\Pi \\in \\mathsf{LOCAL}(\\Theta(\\log^* n)), \\Pi \\in \\mathsf{RLOCAL}(\\Theta(\\log^* n))$,\n\\item $\\Pi$ is not in $\\mathsf{LOCAL}(o(n^{1\/d}))$ or $\\mathsf{RLOCAL}(o(n^{1\/d}))$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}[\\cite{chang2017time_hierarchy,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}]\n\\label{rem:chang_pettie_decomposition}\nIn fact, any LCL in $\\mathsf{LOCAL}(O(\\log^* n))$ can be solved as follows. First, we compute a maximal independent set (MIS) of a large enough power of the input graph, and then we run a local algorithm with just $O(1)$ local complexity. \\footnote{So in a sense, MIS is a complete problem for the class $\\mathsf{LOCAL}(O(\\log^* n))$. }\nMoreover, the algorithm in points (1) and (2) can be oblivious and the algorithm in point (1) does not need to access the input unique  identifiers. \n\\end{remark}\n\n\\begin{remark}\nThe problem of checking whether $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ is undecidable.  \n\\end{remark}\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities}\n\\label{subsec:preliminaries_tail}\n\nLet $\\mathcal{A}=(\\mathcal{C},f)$ be an oblivious local algorithm.\nWe denote as $R_\\mathcal{A}$ the \\emph{coding radius} of $\\mathcal{A}$, that is a random variable on $\\mathbb{N}$ that corresponds to the minimal diameter that a vertex $v$ needs to look at to find its neighborhood that is an element of $\\mathcal{C}$.\nNote that the randomness refers to the real number labels.\nTherefore when we write, e.g.,\n$$\\P(R_\\mathcal{A} > r),$$\nwhat we mean is the following. \nFirst, for a given rooted neighborhood $(G,v)$ with a fixed $\\Sigma_{in}$-labeling we ask what is the probability that, after labeling $G$ with real numbers uniformly at random, that $(G,v)$ is in $\\mathcal{C}$. \nSecond, we take the supremum over all such values over all $\\Sigma_{in}$-labels.\n\n\\begin{definition}[$\\mathsf{TAIL}(f)$]\n\\label{def:tail}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{TAIL}(f)$ for some function $f:\\mathbb{N}\\to \\mathbb{N}$ if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that\n$$\\P(R_\\mathcal{A} > r)\\le \\frac{1}{f(r)}$$\nfor every large enough $r\\in \\mathbb{N}$. \n\nWe say that $\\Pi \\in \\mathsf{TAIL}$ if there exists $f \\in \\omega(1)$ such that $\\Pi \\in \\mathsf{TAIL}(f)$. \nWe say that $\\Pi$ is solved by a \\emph{block factor} if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that for some $r_0$ we have $\\P(R_\\mathcal{A} > r_0) = 0$.\n\\end{definition}\n\nHere things are made simple because we think of the algorithm as being run on an infinite graph so nothing depends on $n$ in this complexity measure (while it is a crucial parameter in the classical local complexity definition). \n\nWe also use the asymptotic notation when talking about $\\mathsf{TAIL}$ complexities. For example, an important class $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ is defined as the class of LCLs $\\Pi$ such that there is a constant $C\\in \\mathbb{N}$ and oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ that satisfies\n$$\\P(R_\\mathcal{A}> r)\\le \\frac{1}{\\mathrm{tower}(Cr)}$$\nwhere $\\mathrm{tower}(x)=\\exp\\circ\\dots\\circ \\exp(1)$ is $\\lfloor x\\rfloor$-many times iterated exponential.\n\nA coarser way to measure the complexity of a problem $\\Pi$ is to ask what moments of $R_\\mathcal{A}$ could be finite, when $\\mathcal{A}$ is an oblivious local algorithm.\nFor example, the first moment finite corresponds to the fact that average time until an algorithm finishes is defined (for finite graphs this is independent on the size of the graph). \nIn particular, we investigate the question what is the least infinite moment for any oblivious local algorithm to solve a given problem $\\Pi$, as this language was used in \\cite{HolroydSchrammWilson2017FinitaryColoring}. \nRecall that $R_\\mathcal{A}$ has a \\emph{finite $\\alpha$-moment}, where $\\alpha\\in [0,+\\infty)$, if\n$$\\mathbb{E}\\left(\\left(R_\\mathcal{A}\\right)^\\alpha\\right)=\\sum_{r \\in \\mathbb{N}} r^\\alpha \\cdot \\P(R_\\mathcal{A} = r) <+\\infty.$$\n\n\\begin{definition}\nLet $\\Pi$ be an LCL.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an oblivious $\\mathcal{A}$ that solves $\\Pi$ and has the $\\alpha$-moment of $R_\\mathcal{A}$ finite.\n\\end{definition}\n\nThe $\\alpha$ moment being finite roughly corresponds to $\\Omega(r^{\\alpha})$ $\\mathsf{TAIL}$ complexity. In fact we now recall the following observation.\n\n\\begin{fact}\n\\label{fact:moment_implies_tail}\nSuppose that the coding radius of an oblivious local algorithm $\\mathcal{A}$ has the $\\alpha$-moment finite.\nThen for every $C>0$ we have\n$$\\P(R_\\mathcal{A} > r) \\le C\\frac{1}{r^\\alpha}$$\nfor large enough $r\\in \\mathbb{N}$.\n\\end{fact}\n\\begin{proof}\nWe prove the contrapositive. Suppose the tail of the coding radius $R$ satisfies $\\P(R > r_i) > C \\cdot \\frac{1}{r^\\alpha_i}$ for some $C > 0$ and an infinite sequence $(r_i)_{i\\in \\mathbb{N}}$. \nAs the sequence $\\P(R> r_i)$ goes to zero, we may pass to a subsequence such that for each $i\\in \\mathbb{N}$ we have $\\P(R > r_{i+1}) \\le \\frac{1}{2}\\P(R > r_i)$. We call the new subsequence still $(r_i)_{i\\in \\mathbb{N}}$.  \nWe have \n\\begin{align*}\n\\mathbb{E}(R^\\alpha) &\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot (\\P(R> r_i) - \\P(R > r_{i+1}))\\\\\n&\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{1}{2}\\P(R > r_i)\n\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{C}{2} \\cdot \\frac{1}{r_i^\\alpha}\n= +\\infty\n\\end{align*}\nwhich finishes the proof. \n\\end{proof}\n\nFinally, an important fact is that a small $\\mathsf{TAIL}$ complexity of an oblivious randomized local algorithm implies that it has small randomized complexity (on finite graphs). This is because if we run an oblivious algorithm with tail decay $1\/f(r)$ on an $n$-node graph and mark its run as a failure if the coding radius exceeds $f^{-1}(n^2)$, the failure at some node happens only with  probability $1\/n$. This was formally proven in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and here we only state the result. \n\n\\begin{lemma}\n\\label{lem:tail->local}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with $\\mathsf{TAIL}$ complexity $f(r)$. Then the randomized complexity of $\\mathcal{A}$ is $O(f^{-1}(n^2))$. \n\\end{lemma}\n\nIn fact, one can prove that for $o(\\log n)$ local complexities the two notions are the same. We refer to the reader to the discussion in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} for more details. \n\n\\begin{theorem}[\\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:tail=uniform}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with randomized complexity $f(n) = o(\\log n)$. Then this algorithm in fact never returns an incorrect solution and its tail complexity is $\\Omega(f^{-1}(r))$. \n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\subsection{Subshifts of finite type}\n\\label{subsec:subshifts}\n\nHere we connect our definition of LCLs and oblivious local algorithms with the well-studied notions of subshifts of finite type and finitary factors of iid labelings.\n\nTo define subshifts of finite type we restrict ourselves to LCLs without input.\nThat is $\\Sigma_{in}$ is empty or contains only one element.\nMoreover, for convenience we set $b:=\\Sigma_{out}$.\nThen an LCL $\\Pi$ is of the form $\\Pi=(b,t,\\mathcal{P})$.\nWe define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nThat is we consider all possible labelings of $\\mathbb{Z}^d$ by elements from $b$ and then include in $X_{\\Pi}$ only those that satisfies the finite constraints $\\mathcal{P}$.\nViewing $\\mathbb{Z}^d$ as a group, there is a natural shift action of $\\mathbb{Z}^d$ on (the Cayley graph) $\\mathbb{Z}^d$.\nNamely, $g\\in \\mathbb{Z}^d$ shifts $h$ to $g+h$.\nThis action naturally ascend to an action $\\cdot $ on $b^{\\mathbb{Z}^d}$ as\n$$(g\\cdot x)(h)=x(g+h)$$\nwhere $x\\in b^{\\mathbb{Z}^d}$.\nIt is not hard to see (since $\\mathcal{P}$ is defined on isomorphism types and not rooted in any particular vertex) that $X_\\Pi$ is invariant under the shift map.\nThen we say that $X_\\Pi$ together with the shift structure is a \\emph{subshift of finite type}.\n\nA \\emph{finitary factor of iid process (ffiid) $F$} is a function that takes as an input $\\mathbb{Z}^d$ labeled with real numbers from $[0,1]$, independently according to Lebesgue measure, and outputs a different labeling of $\\mathbb{Z}^d$, say with elements of some set $b$. \nThis labeling satisfies the following: (a) $F$ is equivariant, that is for every $g\\in \\mathbb{Z}^d$ shifting the real labels with $g$ and applying $F$ is the same as applying $F$ and then shifting with $g$, (b) the value of $F(x)({\\bf 0})$ depends almost surely on  the real labels on some random but finite neighborhood of ${\\bf 0}$.\nSimilarly as for oblivious local algorithms one defines the random variable $R_F$ that encodes minimal such neighborhood.\nWe say that ffiid $F$ solves $\\Pi$ if $F(x)\\in X_{\\Pi}$ with probability $1$.\n\nThe difference between ffiid and oblivious local algorithms is just formal.\nNamely, given an oblivious local algorithm $\\mathcal{A}$, we can turn it into an ffiid $F_\\mathcal{A}$ simply by running it simultaneously in every vertex of $\\mathbb{Z}^d$.\nWith probability $1$ this produces a labeling of $\\mathbb{Z}^d$ and it is easy to see that both conditions from the definition of ffiid are satisfied.\nOn the other hand, if $F$ is ffiid, then we collect the finite neighborhoods that determine the value at ${\\bf 0}$ together with the decision about the label.\nThis defines a pair $\\mathcal{A}_F=(\\mathcal{C},f)$ that satisfies the definition of an oblivious local algorithm.\nIt follows that the coding radius $R_\\mathcal{A}$, resp $R_F$, is the same after the translations.\nWe chose to work with the language of distributed algorithms as we borrow the techniques mainly from that area. \n\n\\begin{remark}\nIf in (b), in the definition of ffiid, we only demand that the value at ${\\bf 0}$ depends measurably on the labeling, i.e., $F$ is allowed to see the labels at the whole grid, then we say that $F$ is a \\emph{factor of iid labels (fiid)}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{fiid}$, if there is a fiid $F$ that produces element of $X_\\Pi$ almost surely.\nIt is easy to see that $\\mathsf{TAIL}\\subseteq \\mathsf{fiid}$.\nWe remark that in the setting without inputs we have $\\mathsf{TAIL}=\\mathsf{fiid}$ on $\\mathbb{Z}$ and it is unknown, however, still plausible, whether this holds for $d>1$.\nFor general LCLs with inputs (where the definition of $\\mathsf{fiid}$ is extended appropriately) we have $\\mathsf{TAIL}\\not=\\mathsf{fiid}$ even for $d=1$, see \\cite{grebik_rozhon2021LCL_on_paths}.\n\\end{remark}\n\n\\begin{comment}\nGiven an LCL $\\Pi=(b,t,\\mathcal{P})$, we define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nWe say that an LCL $\\Pi$ is \\emph{satisfiable} if $\\emptyset\\not=X_\\Pi$, i.e., if there is a $\\Pi$-coloring of $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$.\nWe say that an LCL $\\Pi$ is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant function $\\alpha$ is a $\\Pi$-coloring.\n\nThere is a canonical (right-)shift action $\\cdot$ of $\\mathbb{Z}^d$ on $b^{\\mathbb{Z}^d}$ (and in general on any $X^{\\mathbb{Z}^d}$, where $X$ is a set) that is defined as\n$$(g\\cdot x)(h)=x(h + g)$$\nfor every $g,h\\in \\mathbb{Z}^d$ and $x\\in b^{\\mathbb{Z}^d}$.\nIt is easy to see that $X_\\Pi$ is invariant under this action, i.e., $x\\in X_\\Pi$ implies $g\\cdot x\\in X_\\Pi$ for every $g\\in \\mathbb{Z}^d$.\nMoreover, since $b$ is a finite set, the product topology turns $b^\\Gamma$ into a compact metric space and it is easy to see $X_\\Pi$ is closed subspace.\nThe set $X_\\Pi$ together with the shift action is called a \\emph{sub-shift of finite type}.\n\nSuppose that $\\mathbb{Z}^d$ has an action $\\star_Y$ and $\\star_Z$ on spaces $Y$ and $Z$, respectively. \nWe say that a function $f:Y\\to Z$ is \\emph{equivariant} if $f(g\\star_Y x)=g\\star_Z f(x)$ for every $g\\in \\mathbb{Z}^d$ and $x\\in Y$.\n\nGiven an LCL $\\Pi$, our aim is to study complexity of finitary factors of iid that produce elements in $X_\\Pi$ almost surely.\nWe say that $\\Pi$ admits a \\emph{fiid solution} if there is a measurable equivariant map $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ that is defined on a $\\mu$-conull set, where $\\mu$ is the product measure on $[0,1]^{\\mathbb{Z}^d}$.\n\nWrite $\\varphi_F:[0,1]^{\\mathbb{Z}^d}\\to b$ for the composition of $F$ with the projection to ${\\bf 0}_d$-coordinate, i.e., \n$$x\\mapsto \\varphi_F(x)=F(x)({\\bf 0}_d).$$\nWe say that $F$ is a \\emph{finitary fiid (ffiid)} if for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the value of $\\varphi_F$ depends only on values of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, there is a random variable $R_F$ that takes values in $\\mathbb{N}$, is defined $\\mu$-almost everywhere and satisfies for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the following: if $y\\in [0,1]^{\\mathbb{Z}^d}$ is such that\n$$x\\upharpoonright \\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}\\left({\\bf 0}_d,R_\\varphi(x)\\right)=y\\upharpoonright \\mathcal{B}\\left({\\bf 0}_d,R_\\varphi(x)\\right),$$\nthen $\\varphi_F(x)=\\varphi_F(y)$.\n\\todo{nejak to zakoncit tim ze to nebudeme potrebovat}\n\\end{comment}\n\n\n\n\n\\begin{comment}\n\nWarm-up: Equivalence of $\\mathsf{LOCAL}(\\log^* n)$, $\\mathsf{TAIL}(\\mathrm{tower})$, $\\mathsf{CONTINUOUS}$\n\nRecall that $\\mathsf{LOCAL}(O(\\log^* n))$ denotes the class of satisfiable LCLs that satisfy (1) or (2) in Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nIn this section we show that every problem from $\\mathsf{LOCAL}(O(\\log^* n))$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nIn the proof we use the fact that $\\mathsf{LOCAL}=\\mathsf{C-LOCAL}(O(1))$, see~Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\n\nWe let $\\Delta_{s}:=|\\mathcal{B}({\\bf 0}_d,s)|$, where $s\\in \\mathbb{N}$.\nDenote as $\\mathcal{M}_s$ the subspace of $(\\Delta_s+1)^{\\mathbb{Z}^d}$ that consists of all proper vertex $(\\Delta_s+1)$-colorings of the $s$-power graph of $\\mathbb{Z}^d$.\nIt is easy to see that $\\mathcal{M}_s$ is nonempty, $\\mathbb{Z}^d$-invariant and compact.\nIn fact it is a subshift of finite type.\nThe following result holds for any graph of bounded degree, as was proved in \\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem}\\cite[Theorem~6]{HolroydSchrammWilson2017FinitaryColoring}\\label{th:ColorTower}\nLet $s\\in\\mathbb{N}$.\nThen there is an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to \\mathcal{M}_s$ and a constant $C_s>0$ such that\n$$\\mu(R_F>r)<\\frac{1}{tower(C_sr)}.$$\nIn particular, the LCL of coloring the $s$-power graph of $\\mathbb{Z}^d$ with $(\\Delta_s+1)$-many colors is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\n\n\\begin{proposition}\\label{pr:LOCALandCOLORING}\nLet $\\Pi=(b,t,\\mathcal{P})$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to \\mathcal{C}_{\\Pi}$ for some $s\\in \\mathbb{N}$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nBy Theorem~\\ref{thm:classification_of_local_problems_on_grids}, we have that $\\Pi$ is in the class $\\mathsf{C-LOCAL}(O(1))$.\nFix $r,\\ell\\in \\mathbb{N}$ and $\\mathcal{A}$ that witnesses this fact.\nLet $x\\in \\mathcal{M}_{r+t}$.\nThen define\n$$\\varphi(x)(h):=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_x}(h,\\ell),h]).$$\nIt is clear that $\\varphi$ is well defined, because $\\ell<r-t$ and consequently $x\\upharpoonright \\mathcal{B}(h,\\ell)$ is injective.\nSince $\\ell$ is fixed we see that $\\varphi$ is continuous map, moreover, it is equivariant becuase\n\\begin{equation*}\n\\begin{split}\ng\\cdot \\varphi(x)(h) & = \\varphi(x)(h + g)=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_{ x}}(h+g,\\ell),h+g]) \\\\\n& = \\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_{(g\\cdot x)}}(h,\\ell),h])=\\varphi(g\\cdot x)(h)\n\\end{split}  \n\\end{equation*}\nholds for every $g,h\\in \\mathbb{Z}^d$ and $x\\in \\mathcal{M}_{r+t}$.\n\nIt remains to show that $\\varphi(x)\\in X_\\Pi$, or equivalently that $\\varphi(x)$ is a $\\Pi$-coloring, for every $x\\in \\mathcal{M}_{r+t}$.\nBy equivariance, it is enough to show that\n$$\\mathcal{P}\\left(\\left[\\mathcal{B}_{\\mathbb{Z}^d_{\\varphi(x)}}({\\bf 0}_d,t),{\\bf 0}_d\\right]\\right)=1.$$\nFix $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ that has diameter bigger than $r+t$ and an isomorphism $\\kappa:(\\mathcal{B}_G(v,r+t),v)\\to \\mathcal{B}({\\bf 0}_d,r+t),{\\bf 0}_d)$, e.g., $G$ is a $d$-dimensional torus that is large enough.\nExtend the pull-back of the $(\\Delta_{r+t}+1)$-coloring $x$ that is given via $\\kappa$ to a coloring $c$ that is defined on the whole graph $G$ in such a way that that it is still a coloring of the $(r+t)$-power graph, that is possible simply by a greedy algorithm.\nBy $\\ell<r-t$, we have $\\mathcal{A}(G_c)(\\kappa^{-1}(h))=\\varphi(x)(h)$, whenever $h\\in \\mathcal{B}({\\bf 0}_d,t)$.\nSince $\\mathcal{A}(G_c)$ is a $\\Pi$-coloring we conclude that $\\varphi(x)\\in X_\\Pi$.\n\\end{proof}\n\n\\begin{corollary}[$\\mathsf{LOCAL}\\subseteq \\mathsf{TAIL}(\\tow(\\Omega(r)))$]\n\\label{cor:local<tower}\nLet $\\Pi$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{corollary}\n\\begin{proof}\nBy Proposition~\\ref{pr:LOCALandCOLORING}, there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to X_\\Pi$ for some $s\\in \\mathbb{N}$.\nComposing $\\varphi$ with $f$, that is obtained by Theorem~\\ref{th:ColorTower}, yields a measurable equivariant function $F=\\varphi\\circ f:[0,1]^\\Gamma\\to X_\\Pi$.\nLet $\\ell$ be the locality of $\\varphi$ and $C_s\\in \\mathbb{N}$ from Thereom~\\ref{th:ColorTower}.\nUsing the union bound, we have\n$$\\mu(R_F>r)<\\frac{\\Delta_{\\ell}}{tower(C_s(r-\\ell))}\\le \\frac{1}{tower(Dr)}$$\nfor $r\\in \\mathbb{N}$ large enough and some constant $D\\in (0,+\\infty)$.\n\\end{proof}\n\n\n\n\n\ngives that\n\n\n\n\n\n\n\n\n\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))$ $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log^* n))$\n\n\nFirst we show how to get randomized local algorithm from ffiid in a qualitative way.\nWe note that similar results holds for any bounded degree graph.\n\n\\begin{theorem}\\label{th:TailToRandom}\nLet $\\Pi$ be an LCL problem that is in the class $\\mathsf{TAIL}(f)$ for some $f:\\mathbb{N}\\to \\mathbb{N}$.\nSet $g(n)=\\min\\{m\\in \\mathbb{N}:f(m)\\ge n^2\\}$ and suppose that $g\\in o(n^{\\frac{1}{d}})$.\nThen $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$.\n\\end{theorem}\n\\begin{proof}\nFix an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ such that\n$$\\mu(R_F>r)\\le \\frac{1}{f(r)}$$\nfor large enough $r\\in \\mathbb{N}$.\nDenote as $\\varphi_F$ the finitary rule and as $\\operatorname{dom}(\\varphi_F)$ its domain.\nThat is $\\alpha\\in \\operatorname{dom}(\\varphi_F)$ if $\\alpha\\in [0,1]^{V(\\mathcal{B}({\\bf 0}_d,s))}$ for some $s\\in \\mathbb{N}$ and satisfies\n$$F(x)=F(y)$$\nfor every extensions $\\alpha\\sqsubseteq x,y\\in [0,1]^{\\mathbb{Z}^d}$.\nIn that case we set $\\varphi_F(\\alpha)=F(x)$ for any such extension $\\alpha\\sqsubseteq x$.\n\nDefine $\\ell_n:=g(n)$.\nLet $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ be of a radius at most $\\ell_n$.\nConsider any map $\\ell:V(G)\\to [0,1]$.\nDefine\n$$\\mathbb{A}_n([G_l,v])=\\varphi_F(\\ell)$$\nin the case when $\\ell\\in \\operatorname{dom}(\\varphi_F)$.\nOtherwise let $\\mathbb{A}_n([G_l,v])$ to be any element in $b$.\nIt is easy to verify that this defines a randomized local algorithm.\n\nLet $G\\in \\operatorname{\\bf FG}^{S}$ be such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nholds for every $v\\in V(G)$.\nNote that by the assumption on $g$ such graphs do exist, e.g., large enough torus.\nFor every $v\\in V(G)$ define the event\n$$A_v=\\left\\{\\ell\\in [0,1]^{V(G)}:\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n)\\not \\in \\operatorname{dom}(\\varphi_F)\\right\\}.$$\nNote that if $\\ell\\not \\in \\bigcup_{v\\in V(G)} A_v$, then $\\mathbb{A}(G_l)$ is a $\\Pi$-coloring.\nTo see this pick $v\\in V(G)$ and any extension $x\\in [0,1]^{\\mathbb{Z}^d}$ of $\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n+t)$ that is in the domain of $F$, where we interpret $v$ as the origin ${\\bf 0}_d$.\nThen $F(x)\\in X_\\Pi$ and\n$$F(x)\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\simeq\\mathbb{A}(G_l)\\upharpoonright \\mathcal{B}_G(v,t).$$\nConsequently, we have $\\mathcal{P}([\\mathcal{B}_{G_{\\mathbb{A}(G_l)}}(v,t),v])=1$.\n\nLet $v\\in V(G)$.\nWe have $\\ell\\in A_v$ if and only if $R_F(x)>\\ell_n$ for every $x\\in [0,1]^\\Gamma$ that extends $\\ell$, i.e., every $x\\in [0,1]^{\\mathbb{Z}^d}$ that satisfies\n$$[B_{G_l}(v,\\ell_n),v]=[B_{\\mathbb{Z}^d_x}({\\bf 0}_d,\\ell_n),{\\bf 0}_d].$$ \nThis implies that\n$$\\lambda^{\\oplus n}(A_v)=\\mu\\left(\\left\\{x\\in [0,1]^{\\mathbb{Z}^d}:R_F(x)>\\ell_n\\right\\}\\right)\\le \\frac{1}{f(\\ell_n)}\\le \\frac{1}{n^2}.$$\nA union bound then shows that $\\lambda^{\\oplus n}\\left(\\bigcup_{v\\in V(G)}A_v\\right)\\le \\frac{1}{n}$ and the proof is finished.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:tower=local}\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Corollary~\\ref{cor:local<tower} that $\\mathsf{LOCAL}\\subseteq \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nBy Theorem~\\ref{th:TailToRandom} we have that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$ where\n$$g(n)=\\min\\{m\\in \\mathbb{N}:tower(Cm)\\ge n^2\\}$$\nfor some $C\\in\\mathbb{N}$.\nSince $g$ is in $O(\\log^*(n))$, we have that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ by Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nThis finishes the proof.\n\\end{proof}\n\n\\end{comment}\n\n\n\\section{Speed-up Theorems for $\\mathsf{TAIL}$}\n\\label{sec:speedup_theorems}\n\nRecall the classification \\cref{thm:classification_of_local_problems_on_grids}. We are interested in building the analogous classification for $\\mathsf{TAIL}$ complexities. \nIn this section we first show that already our basic translations from \\cref{subsec:preliminaries_tail} give several interesting results and in \\cref{subsec:tail_d-1_speedup} we prove a nontrivial speedup result.\n\n\\subsection{Warm-up Results}\n\\label{subsec:speedup_warmup}\n\nFirst, we observe that already \\cref{thm:classification_of_local_problems_on_grids}, together with \\cref{lem:tail->local} itself has the following implications: \n\n\\begin{corollary}\n\\label{cor:block_factor}\n$\\Pi$ can be solved by a block factor (recall \\cref{def:tail}) if and only if $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \n\\end{corollary}\n\\begin{proof}\nThis follows from \\cref{thm:classification_of_local_problems_on_grids,rem:chang_pettie_decomposition}. \n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:BelowTOWERtrivial}\nIf $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\omega(r)))$, then $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in\\mathsf{LOCAL}(o(\\log^* n))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies the rest. \n\\end{proof}\n\nIn particular, Corollary~\\ref{cor:BelowTOWERtrivial} answers in negative Problem~(ii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a nontrivial local problem (without input labels) with super-tower function tail. \n\nThe second corollary is the following:\n\n\\begin{corollary}\n\\label{cor:local=tower}\n$\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ if and only if $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$. \n\\end{corollary}\n\\begin{proof}\nIf $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$, \\cref{lem:tail->local} yields $\\Pi \\in \\mathsf{RLOCAL}(O(\\log^* n))$. The other direction (special case of \\cref{thm:tail=uniform}) follows from \\cref{rem:chang_pettie_decomposition} that essentially states that it suffices to solve the independent set problem in the power graph. But this follows from a version of Linial's algorithm from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\\end{proof}\n\nFinally, we get the following. \n\\begin{corollary}\n\\label{cor:Below_d_local}\nIf $\\Pi \\in \\mathsf{TAIL}(\\omega(r^{2d}))$, then $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$.  \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in \\mathsf{LOCAL}(o(n^{1\/d}))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies that $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ which is equivalent to $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ by \\cref{cor:local=tower}. \n\\end{proof}\n\nCorollary~\\ref{cor:Below_d_local} answers in negative Problem~(iii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a problem with all its moments finite but with sub-tower function decay.   Recall that \\cref{fact:moment_implies_tail} implies that if the $2d$-th moment is finite, then the tail is $\\omega(r^{2d})$ and, hence, the problem can in fact be solved with a tower tail decay.  \nHowever, the $2d$ is not the right dependency and in the next section we prove a tight version of this result.\n\\subsection{Tight Grid Speedup for $\\mathsf{TAIL}$}\n\\label{subsec:tail_d-1_speedup}\n\nWhile \\cref{cor:Below_d_local} already answers the Problem of \\cite{HolroydSchrammWilson2017FinitaryColoring}, we now improve the speedup from the tail of $\\omega(r^{d-1})$. This result is tight (up to $o(1)$ in the exponent, see the discussion in \\cref{subsec:toast_construction} and \\cref{cor:toast_lower_bound}) and it is the first nontrivial difference between the $\\mathsf{TAIL}$ and $\\mathsf{LOCAL}$ model that we encountered so far. \n\n\n\n\n\\begin{theorem}\n\\label{thm:grid_speedup}\nLet $d,t\\in \\mathbb{N}$.\nFor $d \\ge 2$ there is $C_{d,t}>0$ such that if an LCL $\\Pi$ is in $\\mathsf{TAIL}(C_{d,t}r^{d-1})$, then $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\nSimilarly, for $d=1$ we have that a local coloring problem $\\Pi$ admits a solution by an oblivious random algorithm (it is in $\\mathsf{TAIL}(f(r))$ for any function $f = \\omega(1)$) if and only if $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\nNote that \\cref{fact:moment_implies_tail} hence shows that ${\\bf M}(\\Pi) > d-1$ already implies that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$. \n\nThe proof is a simple adaptation of Lov\u00e1sz local lemma speedup results such as  \\cite{chang2017time_hierarchy,fischer2017sublogarithmic}. \nThe proof goes as follows. Split the grid into boxes of side length roughly $r_0$ for some constant $r_0$. This can be done with $O(\\log^* (n))$ local complexity (or, equivalently, with tower tail decay). \n\nNext, consider a box $B$ of side length $r_0$ and volume $r_0^d$. In classical speedup argument we would like to argue that we can deterministically set the randomness in nodes of $B$ and its immediate neighborhood so that we can simulate the original oblivious local algorithm $\\mathcal{A}$ with that randomness and it finishes in at most $r_0$ steps. \nIn fact, ``all nodes of $B$ finish after at most $r_0$ steps'' is an event that happens with constant probability if the tail decay of $\\mathcal{A}$ is $r^d$, since then each node fails with probability $1\/r_0^d$ and there are $r_0^d$ nodes in $B$. \n\nHowever, we observe that it is enough to consider only the boundary of $B$ with $O(r_0^{d-1})$ nodes and argue as above only for the boundary nodes, hence the speedup from $r^{d-1}$ tail decay. \nThe reason for this is that although the coding radius of $\\mathcal{A}$ inside $B$ may be very large, just the fact that there \\emph{exists} a way of filling in the solution to the inside of $B$, given the solution on its boundary, is enough for our purposes. The new algorithm simply simulates $\\mathcal{A}$ only on the boundaries and then fills in the rest. \n\n\\begin{proof}\n\n\nWe will prove the result for $d \\ge 2$, the case $d=1$ is completely analogous and, in fact, simpler. \nSet $C=C_{d,t}:=d\\cdot t\\cdot(2\\Delta)^{2\\Delta}$, where $\\Delta=20^d$.\nSuppose that there is an oblivious local algorithm $\\mathcal{A}$ that explores the neighborhood of a given vertex $u$ and outputs a color of $u$ after seeing a radius $R$ such that \\begin{equation}\\label{eq:tail}\n\\P(R > r)\\le \\frac{1}{C r^{d-1}}\n\\end{equation}\nfor $r\\in \\mathbb{N}$ large enough.\nLet $r_0\\in \\mathbb{N}$ be large enough such that \\eqref{eq:tail} is satisfied.\nWrite $\\Box_{r_0}$ for the local coloring problem of tiling the input graph (which is either a large enough torus or an infinite grid $\\mathbb{Z}^d$) into boxes with length sizes in the set $\\{r_0,r_0+1\\}$ such that, additionally, each box $B$ is colored with a color from $[\\Delta+1]$ which is unique among all boxes $B'$ with distance at most $3(r_0+t)$. \n\nThe problem of tiling with boxes of side lengths in $\\{r_0, r_0+1\\}$ can be solved in $O(\\log^* n)$ local complexity \\cite[Theorem~3.1]{GaoJackson} but we only sketch the reason here: Find a maximal independent set of a large enough power of the input graph $\\mathbb{Z}^d$. Then, each node $u$ in the MIS collects a Voronoi cell of all non-MIS nodes such that $u$ is their closest MIS node. As the shape of the Voronoi cell of $u$ depends only on other MIS vertices in a constant distance from $u$, and their number can be bounded as a function of $d$ (see the ``volume'' argument given next), one can discretize each Voronoi cell into a number of large enough rectangular boxes. Finally, every rectangular box with all side lengths of length at least $r^2$ can be decomposed into rectangular boxes with all side lengths either $r$ or $r+1$. \n\nNote that in additional $O(\\log^* n)$ local steps, the boxes can then be colored with $(\\Delta+1)$-many colors in such a way that boxes with distance at most $3(r_0+t)$ must have different colors.\nTo see this, we need to argue that there are at most $\\Delta$ many boxes at distance at most $3(r_0+t)$ from a given box $B$.\nThis holds because if $B,B'$ are two boxes and $d(B,B')\\le 3(r_0+t)$, then $B'$ is contained in the cube centered around the center of $B$ of side length $10(r_0+t)$.\nThen a volume argument shows that there are at most $\\Delta\\ge \\frac{(10(r_0+t))^d}{r_0^d}$ many such boxes $B'$ (when $B$ is fixed).\n\nWe show now that there is a local algorithm $\\mathcal{A'}$ that solves $\\Pi$ when given the input graph $G$ together with a solution of $\\Box_{r_0}$ as an input.\nThis clearly implies that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\n\n\nLet $\\{B_i\\}_{i\\in I}$ be the input solution of $\\Box_{r_0}$, together with the coloring $B_i\\mapsto c(B_i)\\in [\\Delta+1]$.\nWrite $V(B)$ for the vertices of $B$ and $\\partial B$ for the vertices of $B$ that have distance at most $t$ from the boundary of $B$.\nWe have $|\\partial B|< d\\cdot t \\cdot (r_0+1)^{d-1}$.\nDefine $N(B)$ to be the set of rectangles of distance at most $(r_0+t)$ from $B$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width= .9\\textwidth]{fig\/tail_LLL.eps}\n    \\caption{The picture shows one block $B$ with a red boundary $\\partial B$ of length $t$. Boxes in $N(B)$, that is with distance at most $(r_0+t)$ from $B$ are dark grey. }\n    \\label{fig:tail_LLL}\n\\end{figure}\n\nWe are now going to describe how $\\mathcal{A}'$ iterates over the boxes in the order of their colors and for each node $u$ in the box $B$ it carefully fixes a random string $\\mathfrak{r}(u)$. \nThis is done in such a way that after fixing the randomness in all nodes, we can apply the original algorithm $\\mathcal{A}$ to each vertex in $\\partial B$ for each $B$ with the randomness given by $\\mathfrak{r}$ and the local algorithm at $u \\in \\partial B$ will not need to see more than its $r$-hop neighborhood. \n\nGiven a partial function $\\mathfrak{r_\\ell}:V(G)\\to [0,1]$ corresponding to nodes such that their randomness was already fixed, every box $B$ can compute a probability $\\mathbb{P}(\\fE_{\\ell}(B))$, where $\\fE_{\\ell}(B)$ is the (bad) event that there is a vertex in $\\partial B$ that needs to look further than $r_0$ to apply the local algorithm $\\mathcal{A}$.\nNote that for computing $\\mathbb{P}(\\fE_{\\ell}(B))$ it is enough to know the $r_0$-neighborhood of $B$.\n\nThe algorithm $\\mathcal{A}$ has $\\Delta+1$ rounds.\nIn the $0$-th round, we set $\\mathfrak{r}_0$ to be the empty function.\nNote that the probability that a vertex needs to look further than $r_0$ in the algorithm $\\mathcal{A}$ is smaller than $\\frac{1}{C r_0^{d-1}}$ by \\eqref{eq:tail}.\nThe union bound gives $\\mathbb{P}(\\fE_0(B))<\\frac{1}{(2\\Delta)^{2\\Delta}}$ for every $B$.\n\nIn each step $\\ell\\in [\\Delta+1]$ we define $\\mathfrak{r}_B:V(B)\\to [0,1]$ for every $B$ such that $c(B)=\\ell$ and put $\\mathfrak{r}_l$ to be the union of all $\\mathfrak{r}_D$ that have been defined so far, i.e., for $D$ such that $c(D)\\le \\ell$.\nWe make sure that the following is satisfied for every $B$:\n\\begin{enumerate}\n\t\\item [(a)] the assignment $\\mathfrak{r}_B$ depends only on $2(r_0+t)$-neighborhood of $B$ (together with previously defined $\\mathfrak{r}_{\\ell-1}$ on that neighborhood) whenever $c(B)=\\ell$,\n\t\\item [(b)] $\\mathbb{P}(\\fE_l(B))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$,\n\t\\item [(c)] restriction of $\\mathfrak{r}_l$ to $N(B)$ can be extend to the whole $\\mathbb{Z}^d$ such that $\\mathcal{A}$ is defined at every vertex, where we view $N(B)$ embedded into $\\mathbb{Z}^d$.\n\\end{enumerate}\nWe show that this gives the required algorithm.\nIt is easy to see that by (a) this defines a local construction that runs in constantly many rounds and produces a function $\\mathfrak{r}:=\\mathfrak{r}_{\\Delta+1}:V(G)\\to [0,1]$.\nBy (b) every vertex in $\\partial B$ for some $B$ looks into its $r_0$ neighborhood and use $\\mathcal{A}$ to choose its color according to $\\mathfrak{r}$.\nFinally, by (c) it is possible to assign colors in $B\\setminus \\partial B$ for every $B$ to satisfy the condition of $\\Pi$ within each $B$.\nPicking minimal such solution for each $B\\setminus \\partial B$ finishes the description of the algorithm.\n\nWe show how to proceed from $0\\le \\ell<\\Delta+1$ to $\\ell+1$.\nSuppose that $\\mathfrak{r}_l$ satisfies the conditions above.\nPick $B$ such that $c(B)=\\ell+1$.\nNote that $\\mathfrak{r}_B$ that we want to find can influence rectangles $D\\in N(B)$ and these are influenced by the function $\\mathfrak{r}_l$ at distance at most $r_0$ from them.\nAlso given any $D$ there is at most one $B\\in N(D)$ such that $c(B)=\\ell+1$. \nWe make the decision $\\mathfrak{r}_B$ based on $2r_0$ neighborhood around $B$ and the restriction of $\\mathfrak{r}_l$ there, this guarantees that condition (a) is satisfied.\n\nFix $D\\in N(B)$ and pick $\\mathfrak{r}_B$ uniformly at random.\nWrite $\\mathfrak{S}_{B,D}$ for the random variable that computes the probability of $\\fE_{\\mathfrak{r}_B\\cup \\mathfrak{r}_l}(D)$.\nWe have $\\mathbb{E}(\\mathfrak{S}_{B,D})=\\mathbb{P}(\\fE_l(D))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$.\nBy Markov inequality we have\n$$\\mathbb{P}\\left(\\mathfrak{S}_{B,D}\\ge 2\\Delta \\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}\\right)<\\frac{1}{2\\Delta}.$$\nSince $|N(B)|\\le \\Delta$ we have that with non-zero probability $\\mathfrak{r}_B$ satisfies $\\mathfrak{S}_{B,D}<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell-1}}$ for every $D\\in N(B)$.\nSince $\\mathcal{A}$ is defined almost everywhere we find $\\mathfrak{r}_B$ that satisfies (b) and (c) above.\nThis finishes the proof. \n\\end{proof}\n\nAs we already mentioned, the proof is in fact an instance of the famous Lov\u00e1sz Local Lemma problem. This problem is extensively studied in the distributed computing literature \\cite{chang2016exp_separation, fischer2017sublogarithmic, brandt_maus_uitto2019tightLLL, brandt_grunau_rozhon2020tightLLL} as well as descriptive combinatorics \\cite{KunLLL,OLegLLL,MarksLLL,Bernshteyn2021LLL}.\nOur usage corresponds to a simple LLL algorithm for the so-called exponential criteria setup (see \\cite{fischer2017sublogarithmic} and \\cite{Bernshteyn2021local=cont}). \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\\section{$\\TOAST$: General Results}\n\\label{sec:Toast}\n\nIn this section we introduce the toast construction. This construction is very standard in descriptive combinatorics \\cite{FirstToast,GJKS,Circle} and the theory of random processes (there it is often called \\emph{the hierarchy}) \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}.\n\nWhile the construction itself is well-known, in this paper we study it as a class of problems by itself. An analogy to our work is perhaps the paper of \\cite{ghaffari2018derandomizing} that defines the class $\\mathsf{S-LOCAL}$ and shows how to derandomize it.\n\n\n\nInformally, the basic toast construction works as follows. There is an adversary that has access to the (infinite) input grid $\\mathbb{Z}^d$. The adversary is sequentially choosing some connecting pieces of the grid and gives them to our algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ needs to label all vertices in the piece $D$ once it is given $D$. \nNote that $\\mathcal{A}$ does not know where exactly $D$ lies in $\\mathbb{Z}^d$ and its decision cannot depend on the precise position of $D$ in the input graph. In general, the piece $D$ can be a superset of some smaller pieces that were already colored by $\\mathcal{A}$. In this case, $\\mathcal{A}$ needs to extend their coloring to the whole $D$. This game is played for countable number of steps until the whole grid is colored. The algorithm solves $\\Pi$ if it never happens during the procedure that it cannot extend the labeling to $D$ in a valid manner. \n\n\n\n\n\nThe construction sketched above is what we call a spontaneous deterministic toast or just $\\TOAST$ algorithm. \nIn the next subsection we give a formal definition of $\\TOAST$ algorithms and we follow by proving relations between $\\TOAST$ algorithms and $\\mathsf{TAIL}$ complexities. \n\n\n\\subsection{Formal Definition of $\\TOAST$ classes}\n\\label{subsec:toast_definition}\nWe now give a formal definition of our four $\\TOAST$ classes. First we define the hereditary structure on which the $\\TOAST$ algorithm is interpreted. This is also called a toast (we use lowercase letters) and is parameterized by the minimum distance of boundaries of different pieces of the toast.  \nThis structure also corresponds to the adversary that asks our algorithm to extend a partial solution to $\\Pi$ to bigger and bigger pieces used in the informal introduction. \n\n\\begin{definition}[$q$-toast on $\\mathbb{Z}^d$]\nFor a $q >0$, a \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that\n\\begin{itemize}\n    \\item for every $g,h\\in \\mathbb{Z}^d$ there is a piece $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$, where $\\partial D$ is a boundary of a set $D$.\n\\end{itemize}\n\\end{definition}\n\nGiven a $q$-toast $\\mathcal{D}$, the \\emph{rank} of its piece $D \\in \\mathcal{D}$ is the level in the toast hierarchy that $D$ belongs to in $\\mathcal{D}$. Formally, we define a \\emph{rank function $\\mathfrak{r}_\\mathcal{D}$ for the toast $\\mathcal{D}$} as $\\mathfrak{r}_\\mathcal{D}:\\mathcal{D}\\to \\mathbb{N}$, inductively as\n$$\\mathfrak{r}_\\mathcal{D}(D)=\\max\\{\\mathfrak{r}_{\\mathcal{D}}(E)+1:E\\subseteq D, \\ E\\in \\mathcal{D}\\}$$ \nfor every $D\\in \\mathcal{D}$.\n\nNext, $\\TOAST$ algorithms are described in the following language of extending functions. \n\n\\begin{definition}[$\\TOAST$ algorithm]\n\\label{def:extending_function}\nWe say that a function $\\mathcal{A}$ is a \\emph{$\\TOAST$ algorithm} for a local problem $\\Pi = (\\Sigma_{in},\\Sigma_{out}, t, \\mathcal{P})$ if \n\\begin{enumerate}\n    \\item [(a)] the domain of $\\mathcal{A}$ consists of pairs $(A, c)$, where $A$ is a finite connected $\\Sigma_{in}$-labeled subset of $\\mathbb{Z}^d$ and $c$ is a partial map from $A$ to $\\Sigma_{out}$,\n    \\item [(b)] $\\mathcal{A}(A,c)$ is a partial coloring from $A$ to $\\Sigma_{out}$ that extends $c$, i.e., $\\mathcal{A}(A,c)\\upharpoonright \\operatorname{dom}(c)=c$, and that depends only on the isomorphism type of $(A,c)$ (not on the position in $\\mathbb{Z}^d$).  \n   \n\\end{enumerate}\nA \\emph{randomized $\\TOAST$ algorithm} is moreover given as an additional input a labeling of $A$ with real numbers from $[0,1]$ (corresponding to a stream of random bits).  \nFinally, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c)) =A$ for every pair $(A,c)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous (randomized) $\\TOAST$ algorithm}.\n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/3-col.eps}\n    \\caption{A $q$-toast and an example of an extending function for $3$-coloring. Observe that our extending function does not give a correct algorithm for $3$-coloring since the piece on the right is contained in an even bigger piece and the $3$-coloring constructed on the right cannot be extended to the node marked with an arrow}\n    \\label{fig:toast_example}\n\\end{figure}\n\n\nIt is easy to see that a $\\TOAST$ algorithm $\\mathcal{A}$ can be used to color ($\\Sigma_{in}$-labeled) $\\mathbb{Z}^d$ inductively on the rank $\\mathfrak{r}_\\mathcal{D}$ of any $q$-toast $\\mathcal{D}$ for every $q\\in \\mathbb{N}$. \nNamely, define $\\mathcal{A}(\\mathcal{D})$ to be the partial coloring of $\\mathbb{Z}^d$ that is the union of $\\mathcal{A}(D)$, where $D\\in \\mathcal{D}$ and $\\mathcal{A}(D)$ is defined inductively on $\\mathfrak{r}_\\mathcal{D}$.\nFirst put $\\mathcal{A}(D):=\\mathcal{A}(D,\\emptyset)$ whenever $\\mathfrak{r}_\\mathcal{D}(D)=0$.\nHaving defined $\\mathcal{A}(D)$ for every $D\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D)=k$, we define\n$$\\mathcal{A}(D'):=\\mathcal{A}\\left(\\bigcup\\{E\\in \\mathcal{D}:E\\subseteq D'\\},\\bigcup\\{\\mathcal{A}(E):E\\subseteq D', \\ E\\in \\mathcal{D}\\}\\right).$$\nfor every $D'\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D')=k+1$. That is, we let $\\mathcal{A}$ to extend the partial solution to $\\Pi$ to bigger and bigger pieces of the toast $\\mathcal{D}$. \nObserve that if $\\mathcal{A}$ is a spontaneous $\\TOAST$ function, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D}))=\\mathbb{Z}^d$ for every $q$-toast $\\mathcal{D}$ and every $q\\in \\mathbb{N}$.\n\n\\paragraph{$\\TOAST$ Algorithms}\nWe are now ready to define the four types of toast constructions that we are going to use (cf. \\cref{fig:big_picture_grids}). \n\n\\begin{definition}[$\\LTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\LTOAST$ if there is $q>0$ and an $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\nMoreover, we say that $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\\end{definition}\n\nBefore we define a randomized version of these notions, we make several comments.\nFirst note that $\\TOAST\\subseteq \\LTOAST$.\nBy (c), if we construct a ($q$-)toast using an oblivious local algorithm, then we get instantly that every problem from the class $\\TOAST$ is in the class $\\mathsf{TAIL}$, see Section~\\ref{sec:Toast}.\nSimilar reasoning applies to the class $\\mathsf{BOREL}$, see Section~\\ref{sec:OtherClasses}.\nWe remark that we are not aware of any LCL that is not in the class $\\TOAST$.\n\n\\begin{comment}\n\\begin{definition}[Randomized toast algorithm]\n\\label{def:extending_function}\nLet $b$ be a set.\nWe say that a function $\\mathcal{A}$ is a randomized $\\TOAST$ algorithm if \n\\begin{enumerate}\n    \\item [(a)] the domain of $\\mathcal{A}$ consists of triplets $(A, c, \\ell)$, where $A$ is a finite connected subset of $\\mathbb{Z}^d$, $c$ is a partial coloring of $A$ and $\\ell:A\\to [0,1]$. \n    \\item [(b)] $\\mathcal{A}(A,c,\\ell)$ is a partial coloring of $A$ that extends $c$, i.e., $\\mathcal{A}(A,c,\\ell)\\upharpoonright \\operatorname{dom}(c)=c$.  \n    \\item [(c)] the function $\\mathcal{A}$ is invariant with respect to the shift action of $\\mathbb{Z}^d$ on $[0,1]^{\\mathbb{Z}^d)}$. \n\\end{enumerate}\nMoreover, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c,\\ell)) =A$ for every triplet $(A,c,\\ell)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous randomized toast algorithm}.\n\\end{definition}\n\nSimilarly as above, we define $\\mathcal{A}(\\mathcal{D},\\ell)$ for every $q$-toast $\\mathcal{D}$, $q\\in \\mathbb{N}$ and $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nObserve that if $\\mathcal{A}$ is a spontaneous randomized toast algorithm, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D},\\ell))=\\mathbb{Z}^d$.\n\\end{comment}\n\n\\begin{definition}[$\\RLTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\RLTOAST$ if there is $q>0$ and a randomized $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$ with probability $1$.\nThat is, after $\\Sigma_{in}$-labeling is fixed, with probability $1$ an assignment of real numbers from $[0,1]$ to $\\mathbb{Z}^d$ successfully produces a $\\Pi$-coloring against all possible toasts $\\mathcal{D}$ when $\\mathcal{A}$ is applied.\n\nMoreover, we say that $\\Pi$ is in the class $\\RTOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$ as above.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Construction of $\\TOAST$}\n\\label{subsec:toast_construction_in_tail_sketch}\n\n\nThe class $\\TOAST$ (or $\\RTOAST$) is especially useful since in \\cref{sec:SpecificProblems} we discuss how many local problems that are not in class $O(\\log^* n)$ but on the other hand are not ``clearly hard'' like vertex $2$-coloring are in the class $\\TOAST$. Importantly, in \\cref{subsec:toast_construction}  we show how one can construct the toast decomposition with an oblivious local algorithm in $\\mathsf{TAIL}(r^{1 - O(1\/\\sqrt{\\log r})})$, which yields the following theorem.\n\n\\begin{restatable}{theorem}{toastconstruction}\n\\label{thm:toast_in_tail}\nLet $\\Pi$ be an LCL that is in the class $\\TOAST$.\nThen \\[\n\\Pi \\in \\mathsf{TAIL}\\left(\\frac{r}{2^{O(\\sqrt{\\log r})}}\\right) = \\mathsf{TAIL}\\left(r^{1-o(1)}\\right)\n\\]\nand, hence, ${\\bf M}(\\Pi)\\ge 1$.\n\\end{restatable}\n\nWe observe in \\cref{cor:toast_lower_bound} that faster then linear tail decay is not possible. \n\n\\paragraph{Proof Sketch}\n\nHere we provide an intuition for the proof that is otherwise technical. We first check that maximal independent sets of power graphs of the grid $G = \\mathbb{Z}^d$ can be constructed effectively, e.g. because of the MIS algorithm of Ghaffari \\cite{ghaffari2016MIS}. The slowdown by this construction can be neglected in this high-level overview.  \n\nWe then take a sequence $r_k = 2^{k^2}$ and construct sparser and sparser MIS in graphs $G^{r_k}, k = 1, 2, \\dots$. After constructing MIS of $G^{r_k}$, we construct Voronoi cells induced by this MIS. These Voronoi cells $V_1, V_2, \\dots$ will be, roughly, the pieces of rank $k$, but we need to ensure that all the boundaries are at least $q$ hops apart. \n\nTo do so, we first shrink each Voronoi cells by $t_k = O(\\sum_{j < k} r_j) \\ll r_k$ hops (grey boundary area in \\cref{fig:toast_construction}) and get new, smaller, cells $C_1, C_2, \\dots$. The final sequence of pieces $H_1, H_2, \\dots$ is then defined inductively, namely to define $H_i$ we start with $C_1$, then add all Voronoi cells from $k-1$-th iteration that $C_1$ intersects, then add all Voronoi cells from $k_2$-th iteration that the new, bigger, cluster intersects, \\dots\nThe definition of $t_k$ ensures that the final pieces constructed after all $k$ iterations are sufficiently far from each other, and also compatible with the smaller rank pieces. In fact, we always have $H_i \\subseteq V_i$. \n\nIt remains to argue that after $k$ steps of this process, only a small fraction of nodes remains uncovered. This is done roughly as follows. The vertices from $V_i \\setminus C_i$ that are not covered by $C_i$s are roughly $t_k \/ r_k \\approx A r_{k-1} \/ r_k$ fraction of vertices where $A$ is some (large) constant. By induction, $1 - A r_{k-2} \/ r_{k-1}$ fraction of those vertices were covered in step $k-1$ and so on. The fraction of remaining vertices is then $\\frac{A r_{k-1}}{r_{k}} \\cdot \\frac{A r_{k-2}}{r_{k-1}} \\dots \\frac{A }{r_1} = \\frac{A^k}{r_k}$. This fraction is asymptotically almost equal to $1\/r_k$, which is the desired tail behavior. However, the additional term $A^k$ is the reason why the final tail decay for $r$ such that $r_k \\le r \\le r_{k+1} = 2^{(k+1)^2}$ is bounded by $2^{O(k)}\/2^{k^2} \\le \\left( \\frac{1}{r} \\right)^{\\frac{k^2 - O(k)}{(k+1)^2}} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/k)} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/\\sqrt{\\log r})}  $, as needed.  \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/toast_construction.eps}\n    \\caption{A step during the toast construction. Voronoi cell $V$ is first shrinked to $C$ and we get the final piece $H$ by taking the union of smaller-level Voronoi cells intersecting $C$ (in this picture, there is only one lower level but the construction is recursive in general). }\n    \\label{fig:toast_construction}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\RTOAST = \\TOAST$}\n\\label{subsec:rtoast=toast}\n\n\n\nIn this section we show that randomness does not help spontaneous toast algorithms.\n\n\\begin{restatable}{theorem}{toastrtoast}\\label{thm:derandomOfToast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RTOAST$ if and only if $\\Pi$ is in the class $\\TOAST$.\n\\end{restatable}\n\nThe proof goes as follows. We start with a random toast algorithm $\\mathcal{A}$ and make it deterministic as follows. When the deterministic toast algorithm gets a piece $D$ of rank $1$, it considers all possible outputs that the algorithm $\\mathcal{A}$ could produce given the input random string and it uses any one of them that $\\mathcal{A}$ would produce with nonzero probability. \nBy induction, whenever $\\mathcal{A}'$ later gets a piece $D$ containing some pieces of smaller rank with a  partial solution that $\\mathcal{A}'$ already produced, the original algorithm $\\mathcal{A}$ could have produced the same partial solution with nonzero probability, hence $\\mathcal{A}$ can extend the coloring in a valid manner and $\\mathcal{A}'$ deterministically choosing any such valid extension that $\\mathcal{A}$ would use with nonzero probability. \nThis shows that $\\mathcal{A}'$ also solves $\\Pi$. \nA formal proof follows. \n\n\\begin{proof}\n\nIt is easy to see that $\\TOAST\\subseteq \\RTOAST$.\nSuppose that $\\Pi$ is in the class $\\RTOAST$ and fix a witnessing $q\\in \\mathbb{N}$ together with a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$.\nWe may assume that $t$, the locality of $\\Pi$, satisfies $t\\le q$.\n\nLet $(A,c)$ be such that $A\\subseteq \\mathbb{Z}^d$ is finite and connected and $\\operatorname{dom}(c)\\subseteq A$.\nWe say that $(A,c)$ comes from a \\emph{partial random $q$-toast} if there is a collection $\\mathcal{D}$ of subsets of $A$ together with an event $\\mathcal{F}$ on $\\operatorname{dom}(c)$ such that\n\\begin{itemize}\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$,\n    \\item $A\\in \\mathcal{D}$,\n    \\item $\\bigcup_{E\\in \\mathcal{D}\\setminus \\{A\\}}E=\\operatorname{dom}(c)$,\n    \\item $\\mathcal{F}$ has non-zero probability and if we apply inductively $\\mathcal{A}$ on $\\operatorname{dom}(c)$ we obtain $c$.\n\\end{itemize}\nIf $(A,c)$ comes from a partial random $q$-toast, then we let $\\mathcal{E}(A,c)$ to be any event $\\mathcal{F}$ as above.\nOtherwise we put $\\mathcal{E}(A,c)$ to be any event on $\\operatorname{dom}(c)$ of non-zero probability.\nIt is easy to see that $\\mathcal{E}$ does not depend on the position of $A$ in $\\mathbb{Z}^d$.\nThis is because $\\mathcal{A}$ is such.\n\nWe now define a spontaneous toast algorithm $\\mathcal{A}'$.\nGiven $(A,c)$, we let $\\mathcal{F}(A,c)$ to be any event on $A$ that has non-zero probability, satisfies\n$$\\mathcal{F}(A,c)\\upharpoonright \\operatorname{dom}(c)\\subseteq \\mathcal{E}(A,c)$$\nand $\\mathcal{A}(A,c)$ gives the same outcome on $\\mathcal{F}(A,c)$.\nLet $\\mathcal{A}'(A,c)$ be equal to this outcome on $\\mathcal{F}(A,c)$.\nIt is clear that $\\mathcal{A}'$ is a spontaneous $\\TOAST$ algorithm.\nObserve that if $(A,c)$ comes from a partial random $q$-toast, then $(A,\\mathcal{A}'(A,c))$ comes from a partial random $q$-toast.\nNamely, use the same $q$-toast witness $\\mathcal{D}$ as for $(A,c)$ together with the event $\\mathcal{F}(A,c)$.\n\nIt remains to show that $\\mathcal{A}'(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\nInduction on $\\mathfrak{r}_{\\mathcal{D}}$ shows that $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast for every $D\\in \\mathcal{D}$.\nThis is easy when $\\mathfrak{r}_{\\mathcal{D}}(D)=0$ and follows from the observation above in general.\nLet $v\\in \\mathbb{Z}^d$ and pick a minimal $D\\in \\mathcal{D}$ such that there is $D'\\in \\mathcal{D}$ that satisfies $v\\in D'$ and $D'\\subseteq D$.\nSince $t\\le q$ we have that $\\mathcal{B}(v,t)\\subseteq D$.\nSince $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast it follows that there is a non-zero event $\\mathcal{F}$ such that \n$$\\mathcal{A}'(D)\\upharpoonright \\mathcal{B}(v,t)=\\mathcal{A}(\\mathcal{D})\\upharpoonright \\mathcal{B}(v,t)$$\non $\\mathcal{F}$.\nThis shows that the constraint of $\\Pi$ at $v$ is satisfied and the proof is finished.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL} = \\RLTOAST$}\n\\label{subsec:tail=randomlazytoast}\n\nIn this section we prove that the class $\\RLTOAST$ in fact covers the whole class $\\mathsf{TAIL} = \\bigcup_{f = \\omega(1)} \\mathsf{TAIL}(f(r))$. \n\n\\begin{restatable}{theorem}{tailrltoast}\n\\label{thm:tail=rltoast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RLTOAST$ if and only if $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{restatable}\n\nWe first give an informal proof. Suppose that there is a random lazy $\\TOAST$ algorithm $\\mathcal{A}$ for $\\Pi$. We construct an oblivious algorithm with nontrivial tail behavior $\\mathcal{A}'$ for $\\Pi$ by constructing a toast via \\cref{thm:toast_in_tail} and during the construction, each node moreover simulates the algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ finishes after some number of steps with probability $1$, hence $\\mathcal{A}'$ finishes, too, with probability $1$ for each node. \n\nOn the other hand, suppose that there is oblivious local algorithm $\\mathcal{A}'$ for $\\Pi$. \nThen we can construct a lazy randomized toast algorithm $\\mathcal{A}$ that simply simulates $\\mathcal{A}'$ for every node. This is possible since as the pieces around a given vertex $u$ that has gets are growing bigger and bigger, the radius around $u$ that $\\mathcal{A}$ sees also grows, until it is large enough and $\\mathcal{A}$ sees the whole coding radius around $u$ and applies $\\mathcal{A}'$ then. \n\nA formal proof follows. \n\n\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\RLTOAST$, i.e, there is $q\\in\\mathbb{N}$ together with a randomized toast algorithm $\\mathcal{A}$.\nIt follows from Theorem~\\ref{thm:toast moment} that there is an oblivious local algorithm $F$ that produces $q$-toast with probability $1$.\n\nDefine an oblivious local algorithm $\\mathcal{A}'$ as follows.\nLet $\\mathcal{B}(v,r)$ be a $\\Sigma_{in}$-labeled neighborhood of $v$.\nAssign real numbers from $[0,1]$ to $\\mathcal{B}(v,r)$ and run $F$ and $\\mathcal{A}$ on $\\mathcal{B}(v,r)$.\nLet $a_{v,r}$ be the supremum over all $\\Sigma_{in}$-labelings of $\\mathcal{B}(v,r)$ of the probability that $\\mathcal{A}'$ fails.\nIf $a_{v,r}\\to 0$ as $r \\to \\infty$, then $\\mathcal{A}'$ is an oblivious local algorithm.\n\nSuppose not, i.e., $a_{v,r}>a>0$.\nSince $\\Sigma_{in}$ is finite, we find by a compactness argument a labeling of $\\mathbb{Z}^d$ with $\\Sigma_{in}$ such that the event $\\mathcal{F}_r$ that $\\mathcal{A}'$ fails on $\\mathcal{B}(v,r)$ with the restricted labeling has probability bigger than $a$.\nNote that since the labeling is fixed, we have $\\mathcal{F}_r\\supseteq \\mathcal{F}_{r+1}$.\nSet $\\mathcal{F}=\\bigcap_{r\\in \\mathbb{N}} \\mathcal{F}_r$.\nThen by $\\sigma$-additivity we have that the probability of the event $\\mathcal{F}$ is bigger than $a>0$.\nHowever, $F$ produces toast almost surely on $\\mathcal{F}$ and $\\mathcal{A}$ produces $\\Pi$-coloring with probability $1$ for any given toast.\nThis contradicts the fact that probability of $\\mathcal{F}$ is not $0$.\n\n\n\n\\begin{comment}\nIn another words, with probability $1$ any node $v$ can reconstruct bigger and bigger piece of the toast $\\mathcal{D}$ when exploring its neighborhood.\nIn particular, $v$ will eventually became element of some piece of the toast and $\\mathcal{A}$ has to label $v$ with $\\Sigma_{out}$ after finitely many steps.\nThis defines an oblivious local algorithm $\\mathcal{B}$.\n\n\n\nPick a labeling $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ such that $F(\\ell)$ is a $q$-toast and $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for $q$-toast $\\mathcal{D}$.\nNote that the picked labeling satisfies these conditions with probability $1$.\n\nDefine $G(\\ell)=\\mathcal{A}(F(\\ell),\\ell)$.\nIt remains to observe that $G$ is an oblivious local algorithm.\nNote that $G(\\ell)({\\bf 0}_d)=\\mathcal{A}(F(\\ell),\\ell)({\\bf 0}_d)$ depends on some $D\\in \\mathcal{D}$ such that ${\\bf 0}_d\\in D$, $\\{E\\subseteq D:E\\in \\mathcal{D}\\}$ and $\\ell\\upharpoonright D$.\nSince $F$ is an oblivious local algorithm, we see that $G$ is too.\nConsequently, $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{comment}\n\nThe idea to prove the converse is to ignore the $q$-toast part and wait for the coding radius.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}$.\nBy the definition there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ and an unbounded function $f$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{f(r)}.$$\nGiven an input labeling of $\\mathbb{Z}^d$ by $\\Sigma_{in}$, we see by the definition of $R_\\mathcal{A}$ that every node $v$ must finish after finitely many steps with probability $1$.\nTherefore it is enough to wait until this finite neighborhood of $v$ becomes a subset of some piece of a given toast $\\mathcal{D}$.\nThis happens with probability $1$ independently of the given toast $\\mathcal{D}$.\n\\end{proof}\n\n\\begin{comment}\nLet $(A,c,\\ell)$ be a triplet as in definition of randomized toast algorithm.\nTake $v\\in A\\setminus \\operatorname{dom}(c)$, and suppose that $v={\\bf 0}_d$.\nOutput $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)$ if and only if there is $t\\in \\mathbb{N}$ such that $\\mathcal{B}({\\bf 0}_d,t)\\subseteq A$ and $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)$ corresponds to some coding neighborhood of $F$.\nThat is to say, any extension $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\sqsubseteq x\\in [0,1]^{\\mathbb{Z}^d}$ outputs the same color at $F(x)({\\bf 0}_d)$.\nIn that case output $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)=F(x)({\\bf 0}_d)$.\n\nWe show that $\\mathcal{A}$ produces a $\\Pi$-coloring for $\\mu$-almost every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $1$-toast $\\mathcal{D}$.\nLet $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ be such that $F$ computes a $\\Pi$-coloring an need a finite coding radius for every $v\\in \\mathbb{Z}^d$.\nNote that almost every labeling $\\ell$ satisfies this condition.\nTake a $1$-toast $\\mathcal{D}$.\nIt is easy to see that $\\mathcal{A}(\\mathcal{D},\\ell)$ does not depend on $\\mathcal{D}$ and that $\\mathcal{A}(\\mathcal{D},\\ell)=F(\\ell)$, i.e., $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring.\nThis finishes the proof.\n\\end{comment}\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{$\\TOAST$: Specific Problems}\\label{sec:SpecificProblems}\n\\todo{explain that some of this is work of others and we show it to substantiate definition of toast}\n\nIn this section we focus on specific problems that are not in the class $\\mathsf{LOCAL}(O(\\log^* n))$ but they admit a solution by a (spontaneous) toast algorithm. Recall, that proving that $\\Pi \\in \\TOAST$ implies that there is an oblivious local algorithm with $1\/r^{1-o(1)}$ tail decay via \\cref{thm:toast_in_tail} and we discuss in \\cref{subsec:descriptive_combinatorics_discussion} that by a result of \\cite{GJKS2} it also implies $\\Pi\\in \\mathsf{BOREL}$. \n\nThe problems we discuss are the following: (a) vertex coloring, (b) edge coloring, (c) tiling the grid with boxes from a prescribed set (this includes, e.g., a perfect matching problem). \n\n\n\n\n\n\nAll these problems were recently studied in the literature from different points of view, see \\cite{ArankaFeriLaci, HolroydSchrammWilson2017FinitaryColoring, GJKS, spencer_personal}.\nIn all cases the main technical tool, sometimes in a slight disguise, is the toast construction.\nOur aim here is to put these problems into a unified context.\nWe note that sometimes the results in the aforementioned literature are stronger or work in a bigger generality (for example, all results discussed here can also be proven to hold on unoriented grids as discussed in \\cite{ArankaFeriLaci}, or the constructed coloring could be aperiodic \\cite{spencer_personal}, etc)\nIn what follows we only give a ``proof by picture'', sketch the description of the corresponding spontaneous toast algorithm and refer the reader to other papers for more details\/stronger results.\n\nThroughout this section we assume that $d>1$.\nWe refer the reader to \\cref{sec:OtherClasses} for definition of classes $\\mathsf{CONTINUOUS}$, $\\mathsf{BOREL}$, etc.\n\n\n\nThe main contribution if this section is the following:\n\\begin{enumerate}\n    \\item The fact that the problem of vertex $3$-coloring is in $\\TOAST$ together with the speed-up \\cref{thm:grid_speedup} answers a question (i) from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item We observe that the problem of edge $2d$-coloring the grid $\\mathbb{Z}^d$ is in $\\TOAST$, thereby answering a question of \\cite{GJKS} coming from the area of descriptive combinatorics. Independently, this was proven in \\cite{ArankaFeriLaci}. \n\\end{enumerate}\n\nIn general, our approach gives that if an LCL $\\Pi$ is in the class $\\TOAST\\setminus \\mathsf{LOCAL}(O(\\log^*(n)))$, then $1\\le {\\bf M}(\\Pi)\\le d-1$ by \\cref{thm:toast_in_tail} and \\cref{thm:grid_speedup}.\nThe examples in Section~\\ref{sec:Inter} show that there are LCLs with ${\\bf M}(\\Pi)=\\ell$ for every natural number $1\\le \\ell\\le d-1$, but for most natural LCLs we do not know the exact value of ${\\bf M}(\\Pi)$, unless $d=2$.\n\n\n\\subsection{Vertex Coloring}\n\\label{subsec:3coloring}\n\nLet $d>1$.\nRecall that $\\Pi^d_{\\chi,3}$ is the \\emph{proper vertex $3$-coloring problem}.\nThis problem was studied extensively in all contexts.\nIt is known that $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{CONTINUOUS}$ \\cite{GJKS}, $\\Pi^d_{\\chi,3}\\in \\mathsf{BOREL}$ \\cite{GJKS2} and $0<{\\bf M}(\\Pi^d_{\\chi,3})\\le 2$ \\cite{HolroydSchrammWilson2017FinitaryColoring}.\nA version of the following theorem is proven in \\cite{GJKS2}, here we give a simple ``proof by picture''. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:3in toast}\nLet $d>1$.\nThe LCL $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{proof}[Proof Sketch]\nThe toast algorithm tries to color all nodes in a given piece with two colors, blue and red. However, it can happen that two partial colorings of small pieces that have different parity in a bigger piece have to be connected (see \\cref{fig:3-coloring}). \nIn this case, we take the small pieces colored with incorrect parity and ``cover'' their border with the third color: green. From the inside of a small piece, the green color corresponds to one of the main two colors, while from the outside it corresponds to the other color. This enables us to switch the color parity. \n\\end{proof}\n\nAs a consequence of results from \\cref{sec:Toast} we can now fully determine the moment behavior of the problem which answers a question from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\n\\begin{corollary}\nLet $d>1$.\nEvery oblivious local algorithm that solves $\\Pi^d_{\\chi,3}$ has the $1$-moment infinite, but there is an oblivious local algorithm solving $\\Pi^d_{\\chi,3}$ with all $(1-\\varepsilon)$-moments finite. \nIn particular, ${\\bf M}(\\Pi^d_{\\chi,3})=1$. \n\\end{corollary}\n\\begin{proof}\nBy \\cref{thm:3in toast}, $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\nIt follows from \\cref{thm:toast_in_tail} that ${\\bf M}(\\Pi^d_{\\chi,3})\\ge 1$.\nBy the speedup \\cref{thm:grid_speedup} and the fact that $3$-coloring was proven not to be in class $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\tow(\\Omega(r))) = \\mathsf{CONTINUOUS}$ \\cite{brandt_grids,HolroydSchrammWilson2017FinitaryColoring,GJKS}, we have that every oblivious local algorithm $\\mathcal{A}$ solving $\\Pi^2_{\\chi,3}$ must have the $1$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{\\chi,3})=1$.\nIt remains to observe that every vertex $3$-coloring of $\\mathbb{Z}^d$ yields a vertex $3$-coloring of $\\mathbb{Z}^2$ by forgetting the last $(d-2)$ dimensions.\nConsequently, every oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi^d_{\\chi,3}$ defines an oblivious local algorithm $\\mathcal{A}'$ that solves $\\Pi^2_{\\chi,3}$ with the same tail behavior.\nThis finishes the proof.\n\\end{proof}\n\nThe $3$-coloring discussion above also implies that the toast itself cannot be constructed faster then with linear tail, otherwise $3$-coloring could be constructed with such a decay, too. \n\\begin{corollary}\n\\label{cor:toast_lower_bound}\nThere is a constant $C > 0$ such that a $q$-toast cannot be constructed in $\\mathsf{TAIL}(C r)$ on any $\\mathbb{Z}^d$.\nHere, by constructing a $q$-toast we mean that the coding radius of each node $u$ is such that $u$ can describe the minimal toast piece $D \\in \\mathcal{D}$ such that $u \\in D$. \n\\end{corollary}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/3-col-real.eps}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already vertex-colored. First step is to match the parity of colors with respect to the bigger piece. Then color the bigger piece.}\n    \\label{fig:3-coloring}\n\\end{figure}\n\n\n\\subsection{Edge Coloring}\n\\label{subsec:edgecoloring}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/4-edge-col.png}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already edge-colored. First step is to match the parity of colors with respect to the bigger piece. This is done by adding a three layered gadget in the problematic directions. After that the coloring is extended to the bigger piece.}\n    \\label{fig:4-edge-coloring}\n\\end{figure}\\todo{dopsat}\nLet $d>1$.\nWe denote as $\\Pi^d_{\\chi',2d}$ the \\emph{proper edge $2d$-coloring problem}.\nObserve that a solution to $\\Pi^d_{\\chi',2d}$ yields a solution to the perfect matching problem $\\Pi^d_{pm}$ (because $\\mathbb{Z}^d$ is $2d$-regular).\nThis shows that $\\Pi^d_{\\chi',2d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, because the perfect matching problem clearly cannot be solved with a local algorithm: if it was, we could find a perfect matching in every large enough finite torus, which is clearly impossible since it can have an odd number of vertices \\cite{brandt_grids}. \n\n\n\n\\begin{theorem}[See also \\cite{ArankaFeriLaci}]\\label{thm:edge toast}\nLet $d>1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\TOAST$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nThe spontaneous toast algorithm is illustrated for $d=2$ in \\cref{fig:4-edge-coloring}.\nThe ``interface'' used by the toast algorithm between smaller and bigger pieces is: smaller pieces pretend they are colored such that in the vertical direction violet and green edges alternate, while in the horizontal direction blue and red edges alternate. If all small pieces in a bigger one have the same parity, this interface is naturally extended to the bigger piece. Hence, we need to work out how to fix pieces with bad horizontal\/vertical parity, similarly to \\cref{thm:3in toast}. \n\n\\cref{fig:4-edge-coloring} shows how to fix the vertical parity, adding the highlighted gadget in the middle picture above and below given pieces keeps its interface the same but switches the parity. The same can be done in the horizontal direction. \n\nTo show the result for $d>2$ it is enough to realize that switching parity in one dimension can be done with a help of one extra additional dimension, while fixing the parity in the remaining dimensions.\nConsequently, this allows to iterate the previous argument at most $d$-many times to fix all the wrong parities.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $d>1$.\nThen $1 \\le {\\bf M}(\\Pi^d_{\\chi',2d}) \\le d-1$. \nIn particular, ${\\bf M}(\\Pi^2_{\\chi',4})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{\\chi',2d})$, where $d>2$.\n\nThe following corollary answers a question of \\cite{GJKS}. See \\cref{sec:OtherClasses} for the definition of class $\\mathsf{BOREL}$. \n\\begin{corollary}\nLet $d\\ge 1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\mathsf{BOREL}$.\nIn particular, if $\\mathbb{Z}^d$ is a free Borel action and $\\mathcal{G}$ is the Borel graph generated by the standard generators of $\\mathbb{Z}^d$, then $\\chi'_\\mathcal{B}(\\mathcal{G})=2d$, i.e., the Borel edge chromatic number is equal to $2d$.\n\\end{corollary}\n\n\n\\begin{remark}[Decomposing grids into finite cycles]\nBencs, Hru\u0161kov\u00e1, T\u00f3th \\cite{ArankaFeriLaci} showed recently that there is a $\\TOAST$ algorithm that decomposes an unoriented grid into $d$-many families each consisting of finite pairwise disjoint cycles.\nHaving this structure immediately gives the existence of edge coloring with $2d$-colors as well as Schreier decoration.\nSince every $\\TOAST$ algorithms imply Borel solution on graphs that look locally like unoriented grids \\cite{AsymptoticDim}, this result implies that all these problems are in the class $\\mathsf{BOREL}$.\nWe refer the reader to \\cite{ArankaFeriLaci} for more details.\n\\end{remark}\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Decomposing grids into finite cycles} \\todo{Jsem pro zkratit to na jeden odstavec}\n\nWe sketch how to use toast structure to decompose $\\mathbb{Z}^d$ into $d$-many families each consisting of finite pairwise disjoint cycles, formally, a collection $\\{\\mathcal{C}_i\\}_{i\\in [d]}$ such that $\\mathcal{C}_i$ is family of pairwise vertex disjoint cycles for every $i\\in [d]$ and $\\bigcup_{i\\in [d]} \\mathcal{C}_i=\\mathbb{Z}^d$.\nNote that finding such a decomposition is not an LCL because we do not have bound on the length of the cycles.\nOn the other hand, if we allow infinite paths, then such problem is LCL problem and we denote it as $\\Pi^d_{\\operatorname{cyc},d}$.\nMoreover, it is easy to see that since we work with oriented grids $\\Pi^d_{\\operatorname{cyc},d}$ is in the class $\\mathsf{LOCAL}(O(1))$.\n\nWe show that there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite for every $37$-toast $\\mathcal{D}$.\nMoreover, it is easy to see that our toastable algorithm also works for non-oriented grids, where the LCL problem $\\Pi^d_{\\operatorname{cyc},d}$ is not trivial.\nThe combinatorial construction is inspired by result of Bencs, Hruskova and Toth \\cite{ArankaFeriLaci} who originally proved similar result for $d=2$.\nAfter reading their paper we realized that the argument goes through for any dimension $d>1$ and we refer the reader to their paper for a detailed proof of this fact.\nHere we only sketch the main idea.\n\n\\begin{theorem}\nLet $d>1$.\nThen there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring for every $37$-toast $\\mathcal{D}$ with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite.\n\\end{theorem}\n\n\\begin{remark}\nA straightforward use of the toastable algorithm together with the fact that Borel graphs with finite Borel asymptotic dimension admit Borel toast structure (see~\\cite{AsymptoticDim}) gives a decomposition of any Borel graph that looks locally like a $d$-dimensional grid (oriented or non-oriented) into $d$-many Borel families of edges each consisting of pairwise disjoint cycles.\nSince each such cycle is necessarily of even length, we get immediately that such graphs admit Borel Schreier decoration, Borel balanced orientation and their Borel chromatic index is $2d$.\n\\end{remark}\n\n\\begin{proof}[Proof Sketch]\nLet $d>1$.\nIt is easy to see that decomposing $\\mathbb{Z}^d$ into $d$ families of cycles (possibly infinite) is the same as coloring the edges with $d$-colors, such that every vertex $v\\in \\mathbb{Z}^d$ has exactly two adjacent edges of each color.\nLet $\\{\\alpha_1,\\dots,\\alpha_d\\}$ be the set of the colors.\n\nAn \\emph{(edge) coloring} of a connected finite set $A\\subseteq \\mathbb{Z}^d$ is a coloring of all edges that are adjacent to $A$.\nWe say that an edge $e$ is a \\emph{boundary edge}, if only one vertex of $e$ is contained in $A$.\nA coloring $c$ is \\emph{suitable}, if after choosing a reference point and orientation of $A$ we have that parallel boundary edges in the $i$-th direction are assigned the color $\\alpha_i$.\n\nThere is a constant $q\\in \\mathbb{N}$ such that for every $i\\not=j\\in [d]$, finite connected set $A\\subseteq \\mathbb{Z}^d$ and a suitable edge coloring $c$, there is a suitable edge coloring $c'$ of $A':=\\mathcal{B}(A,q)$ that extends $c$ such that \n\\begin{itemize}\n    \\item every $c'$-cycle of color $\\alpha_i$ that contains a vertex from $A$ is finite for every $i\\in [d]$,\n    \\item the $i$-th and $j$-th directions of $A$ and $A'$ are swapped, e.g., parallel $A'$-boundary edges in the $i$-th direction from the point of view of $A$ are $c'$-colored with $\\alpha_j$,\n    \\item other directions remain the same.\n\\end{itemize}\nIt is easy to see that iterating this extension possibly $2d$-many times produces a spontaneous toast algorithm $\\mathcal{A}$ with all the desired properties.\n\nIt remains to sketch how to find such extension $c'$.\nFirst, since $c$ is suitable and we want to swap $i$-th and $j$-th direction, we may assume that $d=2$.\nSecond, we may assume that $A$ is a union of rectangles, such that each side has even number of vertices adjacent to a perpendicular boundary edge.\nThis could be done as follows, pick a reference point and consider decomposition of $\\operatorname{Cay}(\\mathbb{Z}^2)$ into a square of a fixed even number of vertices on a side.\nTake the union of all such squares that intersect $A$.\nThird, see Figure~\\ref{fig:SwapingColors} how to swap colors and close all cycles coming out of $A$.\nThis finishes the sketch.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsection{Tiling with Rectangles}\n\\label{subsec:rectangle_tiling}\n\n\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles.\nWe denote as $\\Pi^d_{\\mathcal{R}}$ the problem of \\emph{tiling $\\mathbb{Z}^d$ with elements of $\\mathcal{R}$}.\nIt is not fully understood when the LCL $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ (or equivalently $\\mathsf{CONTINUOUS}$).\nCurrently the only known condition implying that $\\Pi^d_{\\mathcal{R}}$ is \\emph{not} in $\\mathsf{LOCAL}(O(\\log^* n))$ is when GCD of the volumes of elements of $\\mathcal{R}$ is not $1$.\nMinimal unknown example is when $\\mathcal{R}=\\{2\\times 2, 2\\times 3,3\\times 3\\}$. \nFor the other complexity classes the situation is different.\nGiven a collection $\\mathcal{R}$ we define a condition \\eqref{eq:GCD} as\n\\begin{equation}\n\\label{eq:GCD}\\tag{$\\dagger$}\n    \\text{GCD of side lengths in the direction of} \\ i \\text{-th generator is} \\ 1 \\ \\text{for every} \\ i\\in [d].\n\\end{equation}\nIt follows from the same argument as for vertex $2$-coloring, that if $\\mathcal{R}$ does not satisfy \\eqref{eq:GCD}, then $\\Pi^d_{\\mathcal{R}}$ is a global problem, i.e., it is not in any  class from \\cref{fig:big_picture_grids}.\nOn the other hand, the condition \\eqref{eq:GCD} is sufficient for a spontaneous toast algorithm. For example, this implies $\\Pi^2_{\\{2\\times 3,3\\times 2\\}}\\in \\mathsf{BOREL} \\setminus \\mathsf{CONTINUOUS}$.\nA version of the following theorem was announced in \\cite{spencer_personal}, here we give a simple ``proof by picture''.\n\n\\begin{theorem}[\\cite{spencer_personal, stephen}]\n\\label{thm:tilings}\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles that satisfy \\cref{eq:GCD}.\nThen $\\Pi^d_{\\mathcal{R}}$ is in the class $\\TOAST$.\nConsequently, ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\ge 1$.\n\\end{theorem}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{fig\/rectangle_filling.eps}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already tiled. First step is to match the parity of the tiling with respect to the bigger piece. Then tile the bigger piece.}\n    \\label{fig:Tiling}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nThe two-dimensional version of the construction is shown in \\cref{fig:Tiling}. The interface of pieces this time is that they cover themselves with boxes of side length $L$ which is the least common multiple of all the lengths of basic boxes. As in the previous constructions, we just need to work out how to ``shift'' each piece so that all small pieces agree on the same tiling with boxes of side length $L$ which can then be extended to the whole piece. This can be done as in \\cref{fig:Tiling}. \n\\end{proof}\n\n\n\\begin{corollary}\nLet $d=2$ and $\\mathcal{R}$ be a finite collection of $2$-dimensional rectangles.\nThen $\\Pi^2_{\\mathcal{R}}$ is either in the class $\\mathsf{LOCAL}(O(\\log^* n))$, satisfies ${\\bf M}(\\Pi^2_{\\mathcal{R}})=1$, or is global (that is, in no class in \\cref{fig:big_picture_grids}).\n\\end{corollary}\n\nFor any $d$, \\cref{thm:tilings} implies that if a tiling problem is ``hard'' (in no class in \\cref{fig:big_picture_grids}), it is because of a simple reason: a projection to one dimension gives a periodic problem. It would be interesting to understand whether there are problems that are hard but their solutions do not have any periodicity in them (see \\cref{sec:open_problems}). \n\nWe do not know if there is $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$.\nLet $d\\ge \\ell>0$ and denote as $\\mathcal{R}_\\ell$ the restriction of rectangles to first $\\ell$-dimensions.\nAs in the case of proper $3$-vertex coloring problem, we have that ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le {\\bf M}(\\Pi^\\ell_{\\mathcal{R}_{\\ell}})$.\nIn particular, if, e.g., $\\Pi^d_{\\mathcal{R}_2}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, then ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le 1$.\n\n\\begin{remark}\nIn the context of non-oriented grids we need to assume that $\\mathcal{R}$ is invariant under swapping coordinates. \nSimilar argument, then shows that $\\Pi^d_\\mathcal{R}\\in \\TOAST$ if and only if $\\mathcal{R}$ satisfies \\eqref{eq:GCD}.\nCombined with the fact that there is a Borel toast on such Borel graphs (see~\\cite{AsymptoticDim}), we deduce that \\eqref{eq:GCD} for $\\mathcal{R}$ is equivalent with existence of Borel tiling with rectangles from $\\mathcal{R}$.\n\\end{remark}\n\n\n\\todo{2x3 a 3x2 se daji poskladat z elek, coz odpovida nejake partition kruznice}\n\n\n\n\n\n\\paragraph{Perfect matching}\n\nLet $d>1$.\nA special case of a tiling problem is the \\emph{perfect matching problem} $\\Pi^d_{pm}$.\nSince there are arbitrarily large tori of odd size we conclude that $\\Pi^d_{pm}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\nConsequently, $\\Pi^d_{pm}\\not \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ by \\cref{thm:tail=uniform}, and it follows from the Twelve Tile Theorem of Gao et al \\cite[Theorem 5.5]{GJKS} that $\\Pi^d_{pm}\\not \\in \\mathsf{CONTINUOUS}$.\nSimilarly as with the vertex $3$-coloring problem, the proof of $\\Pi^d_{pm}\\in \\mathsf{BOREL}$, where $d>1$, from \\cite{GJKS2} uses the toast construction and, in our notation, yields the following result. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:perfect_matching_intoast}\nLet $d>1$.\nThe LCL $\\Pi^d_{pm}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{corollary}\nLet $d>1$.\nThen ${\\bf M}(\\Pi^d_{pm})\\ge 1$ and every ffiid $F$ that solves $\\Pi^d_{\\chi,3}$ has the $(d-1)$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{pm})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{pm})$, where $d>2$.\nIn fact, this problem is similar to intermediate problems that we study in \\cref{sec:Inter}. \n\n\n\n\n\n\n\n\n\n\n\n\\renewcommand{\\pm}[2]{\\operatorname{PM}^{#1, #2}_\\boxtimes}\n\\renewcommand{\\pm}[1]{\\operatorname{PM}^{#1}_\\boxtimes}\n\\newcommand{\\mathbb{Z}^d_\\boxtimes}{\\mathbb{Z}^d_\\boxtimes}\n\n\\section{Intermediate Problems}\\label{sec:Inter}\n\nIn this section we construct local problems with $\\mathsf{TAIL}$ complexities $\\Theta(r^{j-o(1)})$ for $1 \\le j \\le d-1$. \nWe first define the problem $\\pm{d}$: this is just a perfect matching problem, but in the graph $\\mathbb{Z}^d_\\boxtimes$ where two different nodes $u = (u_1, \\dots, u_d)$ and $v = (v_1, \\dots, v_d)$ are connected with an edge if $\\sum_{1 \\le i \\le d} |u_i - v_i| \\le 2$. That is, we also add some ``diagonal edges'' to the grid. We do not know whether perfect matching on the original graph (Cayley graph of $\\mathbb{Z}^d$) is also an intermediate problem (that is, not $\\mathsf{LOCAL}$ but solvable faster than $\\TOAST$). \n\nWe will show that the tail complexity of $\\pm{d}$ is $\\Theta(r^{d - 1  - o(1)})$. \nAfter this is shown, the hierarchy of intermediate problems is constructed as follows. \nConsider the problem $\\pm{d,\\ell}$. This is a problem on $\\mathbb{Z}^d$ where one is asked to solve $\\pm{d}$ in every hyperplane $\\mathbb{Z}^\\ell$ given by the first $\\ell$ coordinates in the input graph. \nThe $\\mathsf{TAIL}$ complexity of $\\pm{d,\\ell}$ is clearly the same as the complexity of $\\pm{\\ell}$, that is, $\\Theta(r^{\\ell-1-o(1)})$. \nHence, we can focus just on the problem $\\pm{d}$ from now on. \n\n\\begin{claim}\\label{cl:PM is global}\nLet $2\\le d\\in \\mathbb{N}$.\nThen the local coloring problem $\\pm{d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nVia \\cref{thm:grid_speedup} this is equivalent to $\\pm{d} \\not\\in \\mathsf{TAIL}(Cr^{d-1})$ for large enough $C$. \n\\end{claim}\n\nThis follows from the fact that if there is a local algorithm, then in particular there exists a solution on every large enough torus. But this is not the case if the torus has an odd number of vertices. \n\nSecond, we need to show how to solve the problem $\\pm{d}$ faster than with a $\\TOAST$ construction algorithm. \n\n\\begin{restatable}{theorem}{intermediate}\n\\label{thm:intermediate}\nThe problem $\\pm{d}$ can be solved with an oblivious local algorithm of tail complexity $r^{d-1-O(\\frac{1}{\\sqrt{\\log r}})}$. \n\\end{restatable}\n\nHere we give only a proof sketch, the formal proof is deferred to \\cref{app:intermediate_problems}. \n\n\\begin{proof}[Proof Sketch]\nWe only discuss the case $d=3$ (and the actual proof differs a bit from this exposition due to technical reasons). As usual, start by tiling the grid with boxes of side length roughly $r_0$. \nThe \\emph{skeleton} $\\mathcal{C}_0$ is the set of nodes we get as the union of the $12$ edges of each box. \nIts \\emph{corner points} $X_0 \\subseteq \\mathcal{C}_0$ is the union of the $8$ corners of each box. \n\n\nWe start by matching vertices inside each box that are not in the skeleton $\\mathcal{C}_0$. It is easy to see that we can match all inside vertices, maybe except of one node per box that we call a \\emph{pivot}. We can ensure that any pivot is a neighbor of a vertex from $\\mathcal{C}_0$. \n\nWe then proceed similarly to the toast construction of \\cref{thm:toast_in_tail} and build a sequence of sparser and sparser independent sets that induce sparser and sparser subskeletons of the original skeleton.\nMore concretely, we choose a sequence $r_k = 2^{O(k^2)}$ and in the $k$-th step we construct a new set of corner points $X_k \\subseteq X_{k-1}$ such that any two corner points are at least $r_k$ hops apart but any point of $\\mathbb{Z}^d$ has a corner point at distance at most $2r_k$ (in the metric induced by $\\mathbb{Z}^d$). \nThis set of corner points $X_k$ induces a subskeleton $\\mathcal{C}_{k} \\subseteq \\mathcal{C}_{k-1}$ by connecting nearby points of $X_k$ with shortest paths between them in $\\mathcal{C}_{k-1}$. \n\nAfter constructing $\\mathcal{C}_k$, the algorithm solves the matching problem on all nodes of $\\mathcal{C}_{k-1} \\setminus \\mathcal{C}_k$. This is done by orienting those nodes towards the closest $\\mathcal{C}_k$ node in the metric induced by the subgraph $\\mathcal{C}_k$. \nThen, the matching is found on connected components that have radius bounded by $r_k$. There is perhaps only one point in each component, a pivot, that neighbors with $\\mathcal{C}_{k}$ and remains unmatched. \n\nAfter the $k$-th step of this algorithm, only nodes in $\\mathcal{C}_k$ (and few pivots) remain unmatched. Hence, it remains to argue that only roughly a $1\/r_k^{d-1}$ fraction of nodes survives to $\\mathcal{C}_k$. This should be very intuitive -- think about the skeleton $\\mathcal{C}_k$ as roughly looking as the skeleton generated by boxes of radius $r_k$. This is in fact imprecise, because in each step the distance between corner points of $X_k$ in $\\mathcal{C}_k$ may be stretched by a constant factor with respect to their distance in the original skeleton $\\mathcal{C}_{k-1}$. This is the reason why the final volume of $\\mathcal{C}_k$ is bounded only by $2^{O(k)} \/ r_k^{d-1}$ and by a similar computation as in \\cref{thm:toast_in_tail} we compute that the tail decay of the construction is $2^{O(\\sqrt{\\log r})}\/{r^{d-1}}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Other Classes of Problems}\\label{sec:OtherClasses}\n\nIn this mostly expository section we discuss other classes of problems from \\cref{fig:big_picture_grids} and how they relate to our results from previous sections. \n\n\n\n\\subsection{$\\mathsf{CONTINUOUS}$}\\label{subsec:CONT}\n\nHere we explain the fact that an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if and only if it is in the class $\\mathsf{CONTINUOUS}$.\nFor $\\mathbb{Z}^d$, the model $\\mathsf{CONTINUOUS}$ was extensively studied by Gao, Jackson, Krohne, Tucker-Drob, Seward and others, see \\cite{GaoJackson,GJKS,ST,ColorinSeward}.\nIn fact, the connection with local algorithms is implicit in \\cite{GJKS} under the name Twelve Tiles Theorem \\cite[Theorem~5.5]{GJKS}.\nRecently, independently Bernshteyn \\cite{Bernshteyn2021local=cont} and Seward \\cite{Seward_personal} showed that the same holds in bigger generality for all Cayley graphs of finitely generated groups.\nOur aim here is to formally introduce the model $\\mathsf{CONTINUOUS}$ and give an informal overview of the argument for $\\mathbb{Z}^d$ that follows the lines of Gao et al \\cite{GJKS}.\nThe reasons not to simply put reference to the aforementioned papers are two.\nFirst, we work with slightly more general LCLs that have inputs.\nSecond, our goal is to introduce all classes in a way accessible for  a broader audience. \n\nThe setup of the $\\mathsf{CONTINUOUS}$ model is somewhat similar to the deterministic $\\mathsf{LOCAL}$ model: there is no randomness in the definition and there is a certain symmetry breaking input on the nodes. Specifically, each vertex is labelled with one label from the set $\\{0, 1\\}$; the space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$. \nHowever, for the same reason as why we insist on the uniqueness of the identifiers in the $\\mathsf{LOCAL}$ model, we insist that the input $\\{0,1\\}$ labeling breaks all symmetries. Namely, we require that there is no $g$ such that shifting all elements by $g$ preserves the labeling. We write $\\operatorname{Free}(2^{\\mathbb{Z}^d}) \\subseteq 2^{\\mathbb{Z}^d}$ for this subspace. \nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},f)$, where $\\mathcal{C}$ is a collection of $\\{0,1\\} \\times \\Sigma_{in}$-labeled neighborhoods and $f$ is a function that assigns to each element of $\\mathcal{C}$ an output label. It needs to satisfy the following. Suppose we fix a node $v$ and the input $\\{0,1\\}$-labeling $\\Lambda$ of $\\mathbb{Z}^d$ satisfies $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$. If we let $v$ explore its neighborhood with an oblivious local algorithm, then we encounter an element from $\\mathcal{C}$ after finitely many steps. \nWe write $\\mathcal{A}(\\Lambda)(v)$, or simply $\\mathcal{A}(v)$, when $\\Lambda$ is understood, for the corresponding output.\n\n\\begin{comment}\n\\vspace*{5cm}\nTo break symmetries on $\\mathbb{Z}^d$ in the setting of random processes one considers iid labels on vertices.\nThe event that some symmetries remain has measure $0$ and we simply ignore such labelings.\nIn the continuous settings we label vertices with $2:=\\{0,1\\}$.\nThe space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$ and we write $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ for the space of labelings that break all symmetries.\nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},g)$, where $\\mathcal{C}$ is a collection of ${0,1},\\Sigma_{in}$-labeled neighborhoods and $g$ is a function that assigns to each element of $\\mathcal{C}$ an output label, such that if we let a node $v$ explore its labeled neighborhood $x$, then it encounters element from $\\mathcal{C}$ after finitely many steps whenever $x\\in \\Sigma_{in}^{\\mathbb{Z}^d}\\times \\operatorname{Free}(2^{\\mathbb{Z}^d})$, i.e., $x$ breaks all symmetries.\nWe write $\\mathcal{A}(x)(v)$ for the corresponding output.\n\\end{comment}\n\n\\begin{remark}\nThis is equivalent with the definition given in e.g. \\cite{GJKS} in the same way as the correspondence between oblivious local algorithms and ffiid, see \\cref{subsec:subshifts}.\nThe fact that the collection $\\mathcal{C}$ is dense, i.e., every node finishes exploring its neighborhoods in finite time, means that $\\mathcal{A}({\\bf 0})$ is \\emph{continuous} with respect to the product topology and defined in the whole space $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, the value computed at $v$ does not depend on the position of $v\\in \\mathbb{Z}^d$ but only on the isomorphism type of the labels, this is the same as saying that $\\mathcal{A}(\\_)$ is \\emph{equivariant}.\n\\end{remark}\n\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous algorithm $\\mathcal{A}$ that outputs a solution for every $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nNote that, in general, we cannot hope for the diameters of the neighborhoods in $\\mathcal{C}$ to be uniformly bounded.\nIn fact, if that happens then $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \nThis is because there are elements in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ with arbitrarily large regions with constant $0$ label.\nApplying $\\mathcal{A}$ on that regions gives an output label that depends only on $\\Sigma_{in}$ after bounded number of steps, hence we get a local algorithm with constant local complexity.\n\nThe main idea to connect this context with distributed computing is to search for a labeling in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ where any $\\mathcal{A}$ necessarily has a uniformly bounded diameter.\nSuch a labeling does exist!\nIn \\cite{GJKS} these labelings are called \\emph{hyperaperiodic}.\nTo show that hyperaperiodic labelings exist on $\\mathbb{Z}^d$ is not that difficult, unlike in the general case for all countable groups \\cite{ColorinSeward}. \nIn fact there is enough flexibility to encode in a given hyperaperiodic labeling some structure in such a way that the new labeling remains hyperaperiodic.\n\nWe now give a definition of a hyperaperiodic labeling. \nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}^d$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $t\\in \\mathbb{Z}^d$ we have \n$$\\Lambda(t+S) \\not= \\Lambda(g+t+S)$$\nThis notation means that the $\\Lambda$-labeling of $S$ differs from the $\\Lambda$-labeling of $S$ shifted by $g$ everywhere. \nIn another words, given some translation $g\\in \\mathbb{Z}^d$, there is a uniform witness $S$ for the fact that $\\Lambda$ is nowhere $g$-periodic.\nIt follows from a standard compactness argument that any continuous algorithm $\\mathcal{A}$ used on a hyperaperiodic labeling finishes after number of steps that is uniformly bounded by some constant that depends only on $\\mathcal{A}$. \n\n\n\n\\begin{comment}\n\\vspace*{5cm}\n\nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $s\\in \\mathbb{Z}^d$ we have \n$$x(s+t)\\not=x(g+s+t)$$\nfor some $t\\in S$.\nIn another words, given a shift $g\\in \\mathbb{Z}^d$ there is a uniform witness $S$ for the fact that $x$ is nowhere $g$-periodic.\nIt follows from some standard compactness argument that every hyperaperiodic element needs uniformly bounded neighborhoods for each continuous algorithm $\\mathcal{A}$, the bound of course depends on the algorithm. \n\\end{comment}\n\n\\begin{theorem}[\\cite{GJKS,Bernshteyn2021local=cont}]\nLet $d>0$.\nThen an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{CONTINUOUS}$ if and only if it is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{fig\/continuous.eps}\n    \\caption{On the left side we have examples of a labeled local tiling configurations compared with a possible continuity window $w$. These are then sparsely embedded into $\\mathbb{Z}^2$.}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nOne direction was observed by Bernshteyn \\cite{Bernshteyn2021local=cont} and follows from the fact that there is continuous algorithm that produces a maximal $r$-independent set for every $r\\in \\mathbb{N}$ (recall \\cref{thm:classification_of_local_problems_on_grids}).\nThis is implicit, even though in a different language, already in Gao and Jackson \\cite{GaoJackson}.\n\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm for $\\Pi$. \nWe will carefully construct a certain hyperaperiodic element $\\Lambda$ and let $\\mathcal{A}$ solve $\\Pi$ on it. \nWe know that there is a certain number $w$ such that $\\mathcal{A}$ solves $\\Pi$ on $\\Lambda$ with local complexity $w$. \nWe note that even though we do not know the value $w$ in advance, it depends on $\\Lambda$ that is due to be constructed, the flexibility of the construction guarantees to ``encode'' in $\\Lambda$ all possible scenarios.\nSee the discussion at the end of the argument. \n\nThe overall approach is as follows. \nOur local algorithm $\\mathcal{A}'$ first tiles the plane with  tiles of length much bigger than $w$ with $O(\\log^* n)$ local complexity. We cannot tile the plane in a periodic manner in $O(\\log^* n)$ steps, but it can be done with a constant number of different tiles such that the side length of each tile is either $r$ or $r+1$ for some $r \\gg w$ (this construction is from \\cite[Theorem~3.1]{GaoJackson}, see also the proof of \\cref{thm:grid_speedup}). \nFor each tile $T$, $\\mathcal{A}'$ labels it with certain $\\{0,1\\}$ labeling. \nThis labeling depends only on the shape of the tile.\nWe make sure that not only $T$ with its labeling, but also $T$ together with all possible local configurations of tiles around it and corresponding labelings appears in $\\Lambda$.\nIn particular, as $r\\gg t,w$ ($t$ is the radius necessary to check the solution $\\Pi$), we have that the $(w+t)$-hop neighborhood of any node is labelled in a way that also appears somewhere in the labeling $\\Lambda$.\nAfter labeling each tile, $\\mathcal{A}'$ simulates $\\mathcal{A}$. We know that $\\mathcal{A}$ finishes after $w$ steps on $\\Lambda$ and is correct there, hence $\\mathcal{A}'$ is correct and has local complexity $O(\\log^* n)$. What remains is to describe is how to get the labeling of each tile.  \n\nWe use the fact that there is a lot of flexibility when constructing a hyperaperiodic labeling. \nWe define the hyperaperiodic labeling $\\Lambda$ of the input graph $\\mathbb{Z}^d$ in such a way that, after fixing $r$, each one of constantly many local configurations of $\\{r,r+1\\}$-tiles are embedded in $\\Lambda$, each one in fact in many copies that exhaust possible $\\Sigma_{in}$-labelings.\n\nThis is done as follows.\nFix any hyperaperiodic element $\\Lambda'$, these exist by \\cite[Lemma~2.8]{GJKS}.\nGiven a tile $T$, that depends on $r$, $\\{0,1\\}$-label it by any $T$-pattern that appears in $\\Lambda'$.\nConsider the collection of all possible local configurations of such labeled tiles that are moreover labeled by $\\Sigma_{in}$.\nThis collection is finite. Since we do not know $w$ in advance, do this for any value of $r$. This yields a countable collection of possible configurations. \nEmbed this collection sparsely into $\\Lambda'$, \\cref{fig:my_label}. Extend the $\\Sigma_{in}$-labeling arbitrarily. \nThis defines $\\Lambda$.\nIt is routine to check that $\\Lambda$ is hyperaperiodic, see \\cite[Lemma~5.9]{GJKS}.\n\n\n\nIt remains to recall that since $\\Lambda$ is hyperaperiodic, there is $w\\in \\mathbb{N}$ such that $\\mathcal{A}$ has uniformly bounded locality by $w$ on $\\Lambda$.\nPick $r\\gg w,t$.\nThis works as required since any $w+t$-hop neighborhood $\\mathcal{A}'$ can encounter is present in $\\Lambda$.\n\\end{proof}\n\n\\begin{comment}\n\\vspace*{4cm}\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm.\nA high-level strategy for $d=2$ is as follows.\nThere are 12 type of tiles that are parametrized by natural numbers $n,p,q\\in \\mathbb{N}$, i.e., for fixed $n,p,q$ there are $12$ tiles.\nIf $n,p,q$ are fixed, then there is a local deterministic algorithm that tiles given, large enough, torus with these tiles in such a way that the boundary parts matches.\\todo{picture here ad reference to the nice pictures in \\cite{GJKS}}\nIt is not clear at this point how large parameters we should choose.\nThe trick is to encode all such possible tiles for all $n,p,q\\in \\mathbb{N}$ in one hyperaperiodic element $x$.\nThe encoding of the tiles in $x$ guarantees that whenever the regions \\todo{picture} are the same for the fixed parameters $n,p,q$, then $\\mathcal{A}$ outputs the same label.\nSince $x$ is hyperaperiodic, we have that $\\mathcal{A}$ stops after $T$-many steps to output $\\mathcal{A}(x)(v)$ for some $T\\in \\mathbb{N}$.\nWe choose $n,p,q$ according to $T$ in such a way that for any vertex at any region there is a tile such that the $T$-neighborhood of this vertex is fully contained in the tile.\nConsequently, this defines local rule once a given, large enough, torus is tiled into these tiles of 12 types.\n\nThis description exactly follows the proof of (1) $\\rightarrow$ (2) in \\cite[Theorem~5.5]{GJKS}, where (1) is the same as saying that there is a continuous algorithm and (2) fixes the coloring of the tiles for some parameters $n,p,q$.\nHowever, we note that we are skipping some technical details about additional parameter $w$ etc.\nThe additional step to produce a local algorithm by tiling a large enough torus into these 12 tiles is standard.\nIn fact, this is proved in \\cite[Theorem~5.10]{GJKS}, again in different language, to show (2) $\\rightarrow$ (1) in \\cite[Theorem~5.5]{GJKS}.\n\nThe main difference that we encounter are the input labels.\nWe explain how to handle them and hint, simultaneously, what we mean by encoding tiles in hyperaperiodic element, again ignoring a technical parameter $w$, see \\cite[Figures~13,14,etc]{GJKS}.\nFix a hyperaperiodic element $y$.\nFor $n,p,q\\in \\mathbb{N}$ and a tile $G$, see \\eqref{fig:???}, assign to each region labels from $\\{0,1\\}$ in such a way that (a) same regions get the same pattern, (b) this pattern is part of $y$.\nConsider moreover any possible input labeling of such $\\{0,1\\}$-labeled tiles.\nNow place these labeled tiles, for every $n,p,q\\in \\mathbb{N}$, sparsely in $y$ (Figure ???) and extend the input labeling on them to the whole $\\mathbb{Z}^2$, this defines $x$.\nIt is not hard to see that $x$ remains hyperaperiodic.\n\nIn higher dimension, i.e., when $d>2$, the same argument works as well with the only difference that there are more than 12 types of tiles.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\\subsection{Descriptive Combinatorics Classes}\n\\label{subsec:descriptive_combinatorics_discussion}\n\nIn this section we define the classes of problems that are studied in descriptive combinatorics.\nThere are two equivalent ways how the notion of a \\emph{Borel graph} can be defined.\nThe less abstract one is as follows.\nConsider the unit interval $[0,1]$ with the $\\sigma$-algebra of Borel sets, that is the smallest algebra of subsets that contains sub-intervals and is closed under countable unions and complements.\nLet $A,B\\subseteq [0,1]$ be Borel sets and consider a bijection $\\varphi:A \\to B$ between them.\nWe say that $\\varphi$ is \\emph{Borel measurable} if $\\varphi^{-1}(D\\cap B)$ is a Borel set for every Borel set $D$.\nNow, let $\\mathcal{G}$ be a graph with vertex set $[0,1]$ and assume that the degree of every vertex is bounded by some $\\Delta<\\infty$.\nWe say that $\\mathcal{G}$ is a \\emph{Borel graph} if the edge relation is induced by Borel bijections, that is there are sets $A_1,\\dots, A_\\ell$, $B_1,\\dots, B_\\ell$ and Borel bijections $\\varphi_1,\\dots, \\varphi_\\ell$ such that $(x,y)$ is an edge in $\\mathcal{G}$ if and only if there is $1\\le i\\le \\ell$ such that either $\\varphi_i(x)=y$ or $\\varphi_i(y)=x$.\n\nThe more abstract way how to define Borel graphs uses the notion of a \\emph{standard Borel space}.\nFor example $[0,1]$ with the $\\sigma$-algebra of Borel sets, when we forget that it comes from the standard topology, is a standard Borel space.\nFormally, a standard Borel space is a pair $(X,\\mathcal{B})$, where $X$ is a set and $\\mathcal{B}$ is a $\\sigma$-algebra that comes from some separable and complete metric structure on $X$.\nA uniformly bounded degree graph $\\mathcal{G}$ on $(X,\\mathcal{B})$ is Borel if it satisfies the same condition as in the case of the particular space $[0,1]$.\n\nIt is easy to see that with this definition all the basic constructions from graph theory are \\emph{relatively} Borel, e.g. if $A$ is a Borel set of vertices, then $N_\\mathcal{G}(A)$ the neighbors of elements in $A$ form a Borel set.\nAlso note that since we assume that the degree is uniformly bounded, every connected component in $\\mathcal{G}$ is at most countable.\nTypical questions that are studied in this context are Borel versions of standard notions, e.g., \\emph{Borel} chromatic number, \\emph{Borel} chromatic index, \\emph{Borel} perfect matching etc.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey} for more details.\n\nThe most relevant Borel graphs for us are the ones that are induced by a free Borel action of $\\mathbb{Z}^d$.\nThat is, there are $d$-many aperiodic Borel bijections $\\varphi_i:X\\to X$ that pairwise commute, where \\emph{aperiodic} means that $\\varphi_i^k(x)\\not=x$ for every $k>0$, $i\\in [d]$ and $x\\in X$.\nThen the Borel (labeled) graph $\\mathcal{G}$ that is induced by this action consists of edges of the form $(x,\\varphi_i(x))$, where $x\\in X$ and $i\\in [d]$.\nThe most important result in this context is that $\\mathcal{G}$ admits a \\emph{Borel toast structure}.\nThis follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof.\nWe note that it is not understood in general what Borel graphs admit a Borel toast structure.\nThis is connected with the notoriously difficult question about Borel hyperfiniteness, see \\cite{AsymptoticDim}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if there is a Borel measurable $\\Pi$-coloring of every Borel graph $\\mathcal{G}$ as above, i.e., induced by a free Borel action of $\\mathbb{Z}^d$.\nEquivalently, it is enough to decide whether $\\Pi$ can be solved on the canonical Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, if $\\Pi$ has some inputs $\\Sigma_{in}$, then we consider any Borel labeling $X\\to \\Sigma_{in}$ and then ask if there is a Borel solution that satisfies all the constraints on $\\Sigma_{in}\\times \\Sigma_{out}$.\n\n\\begin{theorem}[Basically \\cite{GJKS}]\nLet $\\Pi$ be an LCL that is in the class $\\LTOAST$.\nThen $\\Pi\\in\\mathsf{BOREL}$.\n\\end{theorem}\n\nIn fact, all the LCL problems without input that we are aware of and that are in the class $\\mathsf{BOREL}$ are in fact in the class $\\TOAST$.\nIt is an open problem to decide whether it is always the case.\n\nAnother interesting perspective how to see Borel constructions is as a countably infinite analogue of $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nNamely, suppose that you are allowed to inductively (a) construct Borel MIS with some parameter (b) do local $O(1)$-construction that depends on the previously build structure.\nIt is clear that objects that are build in this way are Borel, however, it is not clear if this uses the whole potential of Borel sets to find Borel solutions of LCLs.\n\nAs a last remark we note that traditionally in descriptive combinatorics, people consider relaxations of Borel constructions.\nThis is usually done via ignoring how a solution behave on some negligible Borel set.\nProminent examples are \\emph{measure $0$ sets} for some Borel probability measure on $(X,\\mathcal{B})$ or \\emph{meager sets} for some compatible complete metric on $(X,\\mathcal{B})$.\nWe denote the classes of problems that admit such a solution for \\emph{every} Borel probability measure or compatible complete metric as $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$, respectively.\nWithout going into any details we mention that on general graphs these classes are much richer than $\\mathsf{BOREL}$, see e.g. \\cite{DetMarks,BrooksMeas}.\nHowever, it is still plausible that on $\\mathbb{Z}^d$ we have $\\mathsf{BOREL}=\\mathsf{BAIRE}=\\mathsf{MEASURE}$.\nIt is well-know that this is true in $\\mathbb{Z}$ \\cite{grebik_rozhon2021LCL_on_paths}.\n\n\n\\begin{comment}\nIn both cases we are allowed to neglect some small sets, where we either do not care about the solution or are allowed to use the axiom of choice.\nSuppose that $\\mathcal{G}$ is a Borel graph on a standard Borel space $(X,\\mathcal{B})$ and $\\Pi$ is an LCL.\nSuppose, in addition, that there is a Borel probability measure $\\mu$ on $(X,\\mathcal{B})$, that is a $\\sigma$-additive measure on $\\mathcal{B}$, see \\cite[Section~?]{Kec}.\nWe say that $\\Pi$ admits \n\n\n\n\n\n\n\n\n\n\nand show the arrows that are left to complete Figure~\\ref{fig:big_picture_grids}.\nRecall that a \\emph{Borel graph} $\\mathcal{G}$ is a triplet $\\mathcal{G}=(X,\\mathcal{B},E)$, where $(X,\\mathcal{B})$ is a standard Borel space, $(X,E)$ is a graph and $E\\subseteq X\\times X$ is an element of the product $\\sigma$-algebra.\nFor the purposes of this paper we always assume that $\\mathcal{G}$ is induced by a free Borel action of $\\mathbb{Z}^d$ for some $d>0$, i.e., there is a Borel action $\\mathbb{Z}^d\\curvearrowright X$ with trivial stabilizers such that $\\mathcal{G}$ is induced by the standard generators of $\\mathbb{Z}^d$.\n\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ belongs to the class $\\mathsf{BOREL}$ if there is a Borel $\\Pi$-coloring of every such Borel graph $\\mathcal{G}$.\nMoreover, we say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{Borel probability measure} $\\mu$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\nSimilarly, we say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{compatible Polish topology} $\\tau$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\nIt follows directly from the definition that $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE},\\mathsf{BAIRE}$.\n\\end{comment}\n\n\n\n\\begin{comment}\nNext we follow Gao et al. \\cite{GJKS}.\nDenote as $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ the free part of the Bernoulli shift $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$ endowed with the inherited product topology.\nThen it is easy to see that $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is a Polish space and that the action is continuous.\nConsider the Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ induced by the standard set of generators.\nThis graph is universal in the sense that an LCL $\\Pi$ is in the class $\\mathsf{BOREL}$ if and only if there is a Borel $\\Pi$-coloring of $\\mathcal{G}$, this follows from result of Seward and Tucker-Drob~\\cite{?}.\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a $\\Pi$-coloring of $\\mathcal{G}$ that is continuous.\nThis is the same as saying that that there is a continuous equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ to $X_\\Pi$.\nThe following is a corollary of the Twelve Tiles Theorem of Gao et al. \\cite{GJKS}.\n\n\\begin{theorem}\\cite[Theorem 5.5]{GJKS}\n$\\mathsf{CONTINUOUS}=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nWe note that in the general case of all groups this equality was shown by Bernshteyn\\cite{?} and, independently, by Seward. \n\nIt remains to describe how toastable algorithms give Borel solutions.\nAs an immediate corollary we get that all the problems from Section~\\ref{sec:SpecificProblems} are in the class $\\mathsf{BOREL}$.\n\n\\begin{theorem}\n$\\LTOAST \\subseteq \\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\LTOAST$ and let $f$ be an toast algorithm with parameter $k$.\nIt follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof, that every Borel graph $\\mathcal{G}$ (that is induced by a free action of $\\mathbb{Z}^d$) admits a Borel $k$-toast for every $k\\in \\mathbb{N}$.\nIt is a standard fact that using $f$ inductively on the depth of the toast produces a Borel $\\Pi$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\\subsection{Finitely Dependent Processes}\n\nIn this section we use the notation that was developed in \\cref{subsec:subshifts}, also we restrict our attention to LCLs without input.\nLet $\\Pi=(b,t,\\mathcal{P})$ be such an LCL.\nRecall that $X_\\Pi$ is the space of all $\\Pi$-colorings.\n\nWe say that a distribution $\\mu$ on $b^{\\mathbb{Z}^d}$ is \\emph{finitely dependent} if it is shift-invariant and there is $k\\in \\mathbb{N}$ such that the events $\\mu\\upharpoonright A$ and $\\mu \\upharpoonright B$ are independent for every finite set $A,B\\subseteq \\mathbb{Z}^d$ such that $d(A,B)\\ge k$.\nNote that shift-invariance is the same as saying that the distribution $\\mu\\upharpoonright A$ does not depend on the position of $A$ in $\\mathbb{Z}^d$ but rather on its shape.\n\nHere we study the following situation:\nSuppose that there is a finitely dependent process $\\mu$ that is concentrated on $X_\\Pi$.\nIn that case we say that $\\Pi$ is in the class $\\mathsf{FINDEP}$. \nIn other words, $\\Pi \\in \\mathsf{FINDEP}$ if there is a distribution from which we can sample solutions of $\\Pi$ such that if we restrict the distribution to any two sufficient far sets, the corresponding restrictions of the solution of $\\Pi$ that we sample are independent. \n\nIt is easy to see that all problems in $\\mathsf{LOCAL}(O(1))$ (that is, problems solvable by a block factor) are finitely dependent, but examples of finitely dependent processes outside of this class were rare and unnatural, see the discussion in \\cite{FinDepCol}, until Holroyd and Liggett \\cite{FinDepCol} provided an elegant construction for 3- and 4-coloring of a path and, more generally, for distance coloring of a grid with constantly many colors. This implies the following result.\n\n\\begin{theorem}[\\cite{FinDepCol}]\n$\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{FINDEP}$ (see \\cref{fig:big_picture_grids}).\n\\end{theorem} \n\nOn the other hand, recently Spinka \\cite{Spinka} proved that any finitely dependent process can be described as a finitary factor of iid process.\\footnote{His result works in a bigger generality for finitely generated amenable groups.}\nUsing \\cref{thm:toast_in_tail}, we sketch how this can be done, we keep the language of LCLs.\n\n\\begin{theorem}[\\cite{Spinka}]\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$, i.e., $\\Pi\\in \\mathsf{FINDEP}$.\nThen there is a finitary factor of iid $F$ (equivalently an oblivious local algorithm $\\mathcal{A}$) with tail decay $1\/r^{1- O(\\sqrt{\\log r})}$ such that the distribution induced by $F$ is equal to $\\mu$.\nIn fact, $\\Pi$ is in the class $\\TOAST$. \n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nSuppose that every vertex is assigned uniformly two reals from $[0,1]$.\nUse the first string to produce a $k$-toast as in \\cref{thm:toast moment}, where $k$ is from the definition of $\\mu$.\n\nHaving this use the second random string to sample from $\\mu$ inductively along the constructed $k$-toast.\nIt is easy to see that this produces ffiid $F$ that induces back the distribution $\\mu$.\nIn fact, it could be stated separately that the second part describes a randomized $\\TOAST$ algorithm: always samples from $\\mu$ given the coloring on smaller pieces that was already fixed. The finitely dependent condition is exactly saying that the joint distribution over the solution on the small pieces of $\\mu$ is the same as the distribution we sampled from.\nNow use \\cref{thm:derandomOfToast} to show that $\\Pi\\in \\TOAST$. \n\\end{proof}\n\nIn our formalism, $\\mathsf{FINDEP} \\subseteq \\TOAST$, or equivalently we get $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) \\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$.\nIt might be tempting to thing that perhaps $\\mathsf{FINDEP} =\\TOAST$.\nIt is however a result of Holroyd et al \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring} that proper vertex $3$-coloring, $\\Pi^{d}_{3,\\chi}$, is not in $\\mathsf{FINDEP}$ for every $d>1$.\nThe exact relationship between these classes remain mysterious.\n\n\n\n\\begin{comment}\n\\begin{theorem}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nThen there is finitely dependent process $\\mu$ on $X_\\Pi$.\n\\end{theorem}\n\\begin{proof}\nIt follows from Theorem~\\ref{thm:classification_of_local_problems_on_grids} that there is $r,\\ell \\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ of locality at most $\\ell$ that produces a $\\Pi$-coloring for every $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$.\n\nSuppose that $\\mu_{(r+t)}$ is a finitely dependent $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$ process with locality $k\\in \\mathbb{N}$.\nWe claim that composing $\\mu_{(r+t)}$ with $\\mathcal{A}$ yields a finitely dependent process with locality $(2\\ell+k)$.\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite set.\nThen the output of $\\mathcal{A}$ depends on $\\mathcal{B}(A,\\ell)$.\nTherefore if $A,B\\subseteq \\mathbb{Z}^d$ satisfy $d(A,B)>2\\ell+k$, then\n$$d(\\mathcal{B}(A,\\ell),\\mathcal{B}(B,\\ell))>k$$\nand we conclude that the outputs of $\\mathcal{A}$ on $A$ and $B$ are independent.\n\nIt remains to find $\\mu_{r}$ finitely dependent $r$-power proper vertex coloring of $\\mathbb{Z}^d$ process for every $r>0$.\nThis is done by Holroyd and Liggett \\cite[Theorem~20]{FinDepCol}.\n{\\color{red} or something like that, they use a lot of colors fr $r$-power coloring...}\n\\end{proof}\n\nGiven a fiid $F$ we define $F^*\\lambda$ to be the push-forward of the product measure via $F$.\nSince $F$ is equivariant it follows that $F^*\\lambda$ is shift-invariant.\n\nThe following theorem was essentially proven by Spinka\\cite{?}. His result nicely fits to our definitions in \\cref{sec:Toast}. \n\\begin{theorem}\n\n\\end{theorem}\n\n\n\\begin{theorem}\nLet $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$.\nThen for every $\\epsilon>0$ there is a ffiid $F$ such that $F^*\\lambda=\\mu$ and the tail of $F$ has the $(1-\\epsilon)$-moment finite.\nMoreover, $\\Pi$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{remark}\nMaybe the random toast construction works for any finitely dependent processes not just LCL.\n\nThere is the notion of strongly irreducible shifts of finite type, some comment about it...\n\nDefine $\\mathsf{FINDEP}$ to be the class of all LCL problems that admit a finitely dependent distribution $\\mu$.\nIt follows that $\\mathsf{TAIL}(\\tow(\\Omega(r)))\\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TOAST}$ and we have $\\mathsf{FINDEP}\\not = \\TOAST$ because proper vertex $3$-coloring $\\Pi^d_{\\chi,3}$ is not in $\\mathsf{FINDEP}$ for every $d>1$, by \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring}.\n\\end{remark}\n\n\n\\begin{proof}\nWe define a spontaneous randomized toast algorithm $\\Omega$ such that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $k$-toast $\\mathcal{D}$ and every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nThis already implies the moreover part by Theorem~\\ref{thm:derandomOfToast}.\n\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite subset of $\\mathbb{Z}^d$.\nWrite $\\mathbb{P}_A$ for the distribution of $\\Pi$-patterns of $\\mu$ on $A$, i.e., $\\mathbb{P}_A=\\mu\\upharpoonright A$.\nSuppose now that $(A,c,\\ell)$ are given, where $A\\subseteq \\mathbb{Z}^d$ is finite subset, $c$ is a partial coloring of $A$ and $\\ell\\in [0,1]^{A}$.\nConsider the conditional distribution $\\mathbb{P}_{A,c}$ of $\\mathbb{P}_A$ given the event $c$.\nIf $\\mathbb{P}_{A,c}$ is non-trivial, then define $\\Omega$ such that $\\Omega(A,c,{-})$ induces the same distribution as $\\mathbb{P}_{A,c}$ when $\\ell$ is fixed on $\\operatorname{dom}(c)$.\nOtherwise define $\\Omega$ arbitrarily.\n\nLet $\\mathcal{D}$ be a $k$-toast.\nWe show that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $\\Omega(\\mathcal{D},{-})$ produces $\\mu$.\nNote that the first claim shows that $\\Pi$ is in the class $\\RTOAST$, while the second produces $\\mu$ as ffiid as follows.\nSuppose that every $v\\in \\mathbb{Z}^d$ is assigned two independent reals $l_{v,0}$ and $l_{v,1}$.\nUse the first real to produce a $k$-toast as in Theorem~\\ref{thm:toast moment} and use the second real together with the produced toast as an input to $\\Omega$.\nIt follows from the independence of $l_{v,0}$ and $l_{v,1}$ that this produces $\\mu$.\n\nWe show that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nfor every $D\\in \\mathcal{D}$ inductively on $\\mathfrak{r}_{\\mathcal{D}}$.\nIf $\\mathfrak{r}_{\\mathcal{D}}(D)=0$, then it follows from the definition.\nSuppose that $\\mathfrak{r}_{\\mathcal{D}}(D)>0$ and let $E_1,\\dots, E_t\\in \\mathcal{D}$ be an enumeration of $E\\subset D$.\nIt follows from the inductive hypothesis together with the assumption that $\\mu$ is $k$-independent and $\\mathcal{D}$ is a $k$-toast that \n$$\\Omega(\\mathcal{D},{-})\\upharpoonright \\bigcup_{i\\le t}E_i=\\mu\\upharpoonright \\bigcup_{i\\le t} E_i.$$\nNow when $\\ell$, and therefore a partial coloring $c$, is fixed on $\\bigcup_{i\\le t}E_i=\\operatorname{dom}(c)$, the definition $\\Omega(D,c,{-})$ extends $c$ in such a way that $\\mathbb{P}_{D,c}=\\mu\\upharpoonright \\operatorname{dom}(c)$.\nSince this holds for any such assignment $\\ell$ of real numbers on $\\bigcup_{i\\le t}E_i$ we deduce that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nand the proof is finished.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities in Distributed and Centralised Models}\n\\label{subsec:congestLCA}\n\nAlthough we are not aware of a systematic work on $\\mathsf{TAIL}$ complexities in the distributed algorithms community, similar notions like oblivious algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}, node-average complexity \\cite{feuilloley2017node_avg_complexity_of_col_is_const,barenboim2018avg_complexity,chatterjee2020sleeping_MIS} or energy complexity \\cite{chang2018energy_complexity_exp_separation,chang2020energy_complexity_bfs} are studied. \nThe breakthrough MIS algorithm of Ghaffari \\cite{ghaffari2016MIS} may have found many applications exactly because of the fact that its basis is an algorithm with exponential tail behavior. \n\nWe discuss two applications of algorithms with nontrivial $\\mathsf{TAIL}$ complexities. First, the node-average complexity literature studies the average coding radius that a node needs to answer. Whenever an oblivious algorithm has its first moment finite, we have an algorithm with constant average coding radius. \nWith a little more work, one can see that the result from \\cref{thm:3in toast}, that is $3$-coloring of grids has $\\mathsf{TAIL}$ complexity $r^{1-O(1\/\\sqrt{\\log r})}$  can be adapted to the setting of finite grids where it gives a $3$-coloring algorithm with average complexity $2^{O(\\sqrt{\\log n})}$. \nFor nonconstant $\\Delta$, the situation is more complicated. In \\cite{barenboim2018avg_complexity} it was shown that the Luby's $\\Delta+1$ coloring algorithm has tail $2^{\\Omega(r)}$ even for nonconstant $\\Delta$, but the question of whether MIS can have constant average coding radius in nonconstant degree graphs is open \\cite{chatterjee2020sleeping_MIS}. \n\nAnother application of the $\\mathsf{TAIL}$ bound approach comes from the popular centralised $\\mathsf{LCA}$ model \\cite{lca11,lca12}: there one has a huge graph and instead of solving some problem, e.g., coloring, on it directly, we are only required to answer queries about the color of some given nodes. It is known that, as in the $\\mathsf{LOCAL}$ model, one can solve $\\Delta+1$ coloring with $O(\\log^* n)$ queries on constant degree graphs by a variant of Linial's algorithm \\cite{even03_conflictfree}. But using the tail version of the Linial's algorithm instead, we automatically get that $O(\\log^* n)$ is only a worst-case complexity guarantee and on average we expect only $O(1)$ queries until we know the color of a given node. That is, the amortized complexity of a query is in fact constant. The same holds e.g. for the $O(\\log n)$ query algorithm for LLL from \\cite{brandt_grunau_rozhon2021LLL_in_LCA}. \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\\section{Open Questions}\n\\label{sec:open_problems}\n\nHere we collect some open problems that we find interesting. \n\n\\begin{enumerate}\n    \\item Is it true that $\\RLTOAST = \\LTOAST$? If yes, this implies $\\mathsf{TAIL} \\subseteq \\mathsf{BOREL}$. \n    \\item Is it true that $\\RLTOAST = \\RTOAST$? If yes, this implies $\\mathsf{TAIL} = \\mathsf{TAIL}(r^{1-o(1)})$ and answers the fourth question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\t\\item Can the tail complexity $r \/ 2^{O(\\sqrt{\\log r})}$ in the construction of toast be improved to $r \/ \\operatorname{poly} \\log (r)$? The same can be asked for the complexity of intermediate problems from \\cref{sec:Inter}. \n\t\t\\item Is there a local coloring problem $\\Pi$ such that ${\\bf M}(\\Pi)$ is non-integer value or $0$, i.e., ${\\bf M}(\\Pi)\\in [0,d-1]\\setminus \\{1,2,\\dots, d-1\\}$?\n\t\t\\item Can you find an example of a local problem without input labels which does not belong to any class in \\cref{fig:big_picture_grids} but the solutions to this problem are not ``periodic'' (this cannot happen for $d=1$). Maybe (a version of) Penrose tilings can be helpful there?\n\t\\item What is the value ${\\bf M}(\\Pi^d_{pm})$, where $\\Pi^d_{pm}$ is the problem of finding a perfect matching of $\\mathbb{Z}^d$, where $d>2$?\n    \\item Let $d>1$.\n\tFor what finite collections of $d$-dimensional rectangles $\\mathcal{R}$ is $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$? Is there a  finite collection of $d$-dimensional rectangles $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$?\n\t\\item Is it true (on $\\mathbb{Z}^d$) that $\\mathsf{BOREL}=\\mathsf{MEASURE}$ or $\\mathsf{BOREL} = \\mathsf{BAIRE}$?\n\\end{enumerate}\n\n\\section*{Acknowledgement}\nWe would like to thank Ferenc Bencs, Anton Bernshteyn, Sebastian Brandt, Yi-Jun Chang, Mohsen Ghaffari, Aranka Hru\u0161kov\u00e1, Stephen Jackson, Oleg Pikhurko, Brandon Sewart, Yinon Spinka, L\u00e1szlo T\u00f3th, and Zolt\u00e1n Vidny\u00e1szky for many engaging discussions. \n\nThe first author was supported by Leverhulme Research Project Grant RPG-2018-424. \nThis project has received funding from the European Research Council (ERC) under the European\nUnions Horizon 2020 research and innovation programme (grant agreement No. 853109). \n\n\\bibliographystyle{alpha}\n\n\\section{Introduction}\n\nIn this paper, we study local problems on grid graphs from three different perspectives: (a) the perspective of the theory of distributed algorithms, (b) the perspective of the theory of random processes, (c) the perspective of the field of descriptive combinatorics. \nAlthough different communities in these three fields ask very similar questions, no systematic connections between them were known, until the insightful work of Bernshteyn \\cite{Bernshteyn2021LLL} who applied results from the field (a) to the field (c). In this work, our most important results follow by applying techniques from the field (a) to (b), but perhaps more importantly, we present all three different perspectives as a part of one common theory (see \\cref{fig:big_picture_grids}). \n\nIn this introductory section, we first quickly present the three fields and define the object of interest, local problems, a class of problems that contains basic problems like vertex or edge coloring, matchings, and others. \nThen, we briefly present our results, most of which can be seen as inclusions between different classes of local problems shown in \\cref{fig:big_picture_grids}. \n\n\\paragraph{Distributed Computing}\n\nThe $\\mathsf{LOCAL}$ model of computing \\cite{linial1987LOCAL} was motivated by understanding distributed algorithms in huge networks. \nAs an example, consider the network of all wifi routers, where two routers are connected if they are close enough to exchange messages. In this case, they should better communicate with user devices on different channels, so as to avoid interference. In the graph-theoretic language, we want to color the network. Even if the maximum degree is $\\Delta$ and we want to color it with $\\Delta+1$ colors, the problem remains nontrivial, because the decision of each node should be done after few communication rounds with its neighbors. \n\nThe $\\mathsf{LOCAL}$ model formalizes this setup: we have a large network, with each node knowing its size, $n$, and perhaps some other parameter like the maximum degree $\\Delta$. In case of randomized algorithms, each node has an access to a random string, while in case of deterministic algorithms, each node starts with a unique identifier from a range of size polynomial in $n$. \nIn one round, each node can exchange any message with its neighbors and can perform an arbitrary computation. We want to find a solution to a problem in as few communication rounds as possible. \nImportantly, there is an equivalent view of $t$-rounds $\\mathsf{LOCAL}$ algorithms: such an algorithm is simply a function that maps $t$-hop neighbourhoods to the final output. An algorithm is correct if and only if applying this function to each node solves the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. One example: if the input graph is the $d$-dimensional torus (think of the grid $\\mathbb{Z}^d$ and identify nodes with the same remainder after dividing the coordinate with $n^{1\/d}$), a simple picture emerges. Any local problem (we define those formally later) is solvable with local complexity $O(1)$, $\\Theta(\\log^* n)$, or we need to see the whole graph to solve it, i.e., the local complexity is $\\Theta(n^{1\/d})$ (see \\cref{thm:classification_of_local_problems_on_grids}). For example, a proper coloring of nodes with $4$ colors has local complexity $\\Theta(\\log^* n)$, while coloring with $3$ colors has already complexity $\\Theta(n^{1\/d})$. \n\n\\paragraph{Finitary factors of iid processes}\n\\todo{zeptat se Spinky}\nConsider the following example of the so-called anti-ferromagnetic Ising model. \nImagine atoms arranged in a grid $\\mathbb{Z}^3$. Each atom can be in one of two states and there is a preference between neighboring atoms to be in a different state. We can formalize this by defining a potential function $V(x,y)$ indicating whether $x$ and $y$ are equal.  \nThe Ising model models the behavior of atoms in the grid as follows: for the potential $V$ and inverse temperature $\\beta$, if we look at the atoms, we find them in a state $\\Lambda$ with probability proportional to $\\exp(-\\beta \\sum_{u \\sim v \\in E(\\mathbb{Z}^d)} V(\\Lambda(u), \\Lambda(v)))$. If $\\beta$ is close to zero, we sample essentially from the uniform distribution. If $\\beta$ is infinite, we sample one of the two $2$-colorings of the grid. \n\nTo understand better the behavior of phase transitions, one can ask the following question. For what values of $\\beta$ can we think of the distribution $\\mu$ over states from the Ising model as follows: there is some underlying random variable $X(u)$ at every node $u$, with these variables being independent and identically distributed. We can think of $\\mu$ as being generated by a procedure, where each node explores the random variables in its neighborhood until it decides on its state, which happens with probability $1$. If this view of the underlying distribution is possible, we say it is a finitary factor of an iid process (ffiid). \nIntuitively, being an ffiid justifies the intuition that there is a correlation decay between different sites of the grid. To understand ffiids, we try to analyze solutions to which local problems can be sampled from an ffiid distribution (2-coloring of the grid is not among them). \n\nAlthough this definition was raised in a different context then distributed computing, it is actually almost identical to the definition of a distributed algorithm! The differences are two. First, the ffiid procedure does not know the value of $n$ since, in fact, it makes most sense to consider it in infinite graphs (like the infinite grid $\\mathbb{Z}^d$). Second, the natural measure of the complexity is different: it is the speed of the decay of the random variable corresponding to the time until the local algorithm finds the solution. \nNext, note that the algorithm has to return a correct solution with probability $1$, while the usual randomized local algorithm can err with a small probability. We call ffiids \\emph{oblivious local algorithms} to keep the language of distributed computing (see \\cref{sec:preliminaries}).\nWe define the class $\\mathsf{TAIL}(f(r))$ as the class of problems solvable by an oblivious local algorithm if the probability of the needed radius being bigger than $r$ is bounded by $1\/f(r)$ (see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Descriptive Combinatorics}\n\nA famous Banach-Tarski theorem states that one can split a three-dimensional unit volume ball into five pieces, translate and rotate each piece and end up with two unit volume balls. \nThis theorem serves as an introductory warning in the beginner measure theory courses: not every subset of $\\mathbb{R}^3$ can be measurable.\nSimilarly surprising was when in 1990 Laczkovich answered the famous Circle Squaring Problem of Tarski\\todo{cite laczkovich}: A square of unit area can be decomposed into finitely many pieces such that one can translate each of them to get a circle of unit area.\nUnlike in Banach-Tarski paradox, this result was improved in recent years to make the pieces measurable in various sense \\cite{OlegCircle, Circle, JordanCircle}.\n\n\nThis theorem and its subsequent strengthenings are the highlights of a field nowadays called descriptive combinatorics \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} that has close connections to distributed computing as was shown in an insightful paper by Bernshteyn \\cite{Bernshteyn2021LLL}. \nGoing back to the Circle Squaring Problem, the proofs of its versions first choose a clever set of constantly many translations $\\tau_1, \\dots, \\tau_k$. Then, one constructs a bipartite graph between the vertices of the square and the cycle and connects two vertices $x, y$ if there is a translation $\\tau_i$ mapping the point $x$ in the square to the point $y$ in the cycle. The problem can now be formulated as a perfect matching problem and this is how combinatorics enters the picture. In fact, versions of a matching or flow problems are then solved on the grid graph $\\mathbb{Z}^d$ using a trick called \\emph{toast construction} also used in the area of the study of ffiids \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}. \n\nWe discuss the descriptive combinatorics classes in a more detail in the expository \\cref{sec:OtherClasses} but we will not need them in the main body. Let us now give an intuitive definition of the class $\\mathsf{BOREL}$. \nBy a result of Chang and Pettie \\cite{chang2016exp_separation}, we know that the maximal independent set problem is complete for the class $\\mathsf{LOCAL}(O(\\log^* n))$ in the sense that this problem can be solved in $O(\\log^* n)$ rounds and any problem in that class can be solved in constant number of rounds if we are given access to an oracle computing MIS (we make this precise in \\cref{rem:chang_pettie_decomposition}). Imagine now that the input graph is infinite and our local algorithm has access to MIS oracle and local computations but now both can be used countably many times instead of just once or, equivalently, fixed number of times as is the case with the class $\\mathsf{LOCAL}(O(\\log^* n))$. That is, we can think of an algorithm that constructs MIS of larger and larger powers of the input graph and uses this infinite sequence to solve an LCL. Those constructions are in the class $\\mathsf{BOREL}$ from \\cref{fig:big_picture_grids}. In fact, the exact relationship between this definition and full power of $\\mathsf{BOREL}$ is not clear.\nHowever, it should give an idea about the difference between descriptive combinatorics and distributed computing -- from the descriptive combinatorics perspective, reasonable classes are closed under repeating basic operations countably many times.\n\n\n\n\n\\paragraph{Locally Checkable Problems on Grids as a ``Rosetta Stone''}\n\nLocally checkable problems (LCLs) are graph problems where a correctness of a solution can be checked locally. That is, there is a set of local constraints such that if they are satisfied at every node, the solution to a problem is correct. \nAn examples of such constraints include vertex coloring, edge coloring, or perfect matching. We also allow inputs on each node, so the list coloring is also an example of LCL (given that the total number of colors in all lists is finite). One should think of this class as an analogue of the class NP from complexity theory. \n\nIn this paper, we study locally checkable problems on grid graphs in the three settings introduced above. This approach reveals number of connections. For example, in \\cref{fig:big_picture_grids}, we can see that the two classes of local complexities $\\mathsf{LOCAL}(O(1))$ and $\\mathsf{LOCAL}(O(\\log^* n))$ are equivalent to different natural classes from the perspective of factors of iid and descriptive combinatorics. On the other hand, those two worlds are richer. Many of our discussions in fact apply to graphs of bounded growth, but the grids serve as the simplest, but still quite nontrivial model in which one can try to understand all three settings together (the case of lines is understood quite well \\cite{grebik_rozhon2021LCL_on_paths} and we study aspects of the connection for trees in a parallel paper \\cite{brandt_chang_grebik_grunau_rozhon_vidnyaszky2021LCLs_on_trees_descriptive}). \n\n\\begin{landscape}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = 1.5\\textwidth]{fig\/big_picture_grids.png}\n    \\caption{The big picture of local complexity classes on grids from the perspective of distributed algorithms, factors of iid, and descriptive combinatorics. Canonical example of each class is on the left side. }\n    \\label{fig:big_picture_grids}\n\\end{figure}\n\\end{landscape}\n\n\n\\subsection{Our Contribution }\n\nOur contribution can be mostly summed up as several new arrows in \\cref{fig:big_picture_grids}. In general,  this clarifies several basic questions regarding the connections of the three fields and helps to view all three fields as part of a general theory of locality. \n \n\n\nFor example, consider the following question from the paper of Brandt et al. \\cite{brandt_grids} that considered LCL problems on grids: \n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colorings and distributed complexity, it remains unclear how to directly translate results from one setting to the other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al., and does our complexity classification imply answers to the open questions they pose?\n}\n\\end{quote}\nWe show how to 'directly translate' results between the setting of distributed algorithms and finitary factors of iid in \\cref{subsec:preliminaries_tail,subsec:speedup_warmup}. This indeed answers two open questions of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\nTo understand the connection between the world of factors of iids and descriptive combinatorics, we formalize the notion of toast construction used in both setups. We prove general results connecting the toast constructions with oblivious local algorithms and this enables us to understand the $\\mathsf{TAIL}$ complexity  of a number of concrete problems like vertex $3$-coloring, edge $4$-coloring, perfect matchings and others. \n\nIn general, we hope that the concept of $\\mathsf{TAIL}$ complexities will find other usecases. We discuss in \\cref{subsec:congestLCA} how it relates to studied notions of oblivious algorithms, energy complexity, or its relevance to the $\\mathsf{LCA}$ model.\\todo{Oleg se ptal jestli uz ma tady vedet co je LCA a jestli to nekde neprehlid}\nIn fact, the $\\mathsf{TAIL}$ complexity approach was already considered e.g. in the celebrated MIS paper of \\cite{ghaffari2016MIS} (see the introduction of his paper) and the $\\mathsf{TAIL}$ bound on the MIS algorithm there may be the reason of its wide applicability. \n\nWe now list concrete results that we prove (or they follow from the work of others, see \\cref{fig:big_picture_grids}). \n\n\\paragraph{Classification Theorems for $\\mathsf{TAIL}$}\n\nWe prove several results that one can view as arrows in \\cref{fig:big_picture_grids}. Those can be viewed as an analogy to the line of work aiming for the classification of $\\mathsf{LOCAL}$ complexities discussed in \\cref{subsec:preliminaries_local}. \n\n\\begin{itemize}\n    \\item $\\mathsf{TAIL}(\\mathrm{tower}(\\omega(r))) = \\mathsf{LOCAL}(O(1))$\\hspace*{1cm} We show that any oblivious local algorithm with tail decay faster than tower function can be in fact solved in $O(1)$ local complexity. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) = \\mathsf{CONTINUOUS}$\\hspace*{1cm} There are three natural definitions of classes of basic symmetry breaking problems in all three considered fields and they are the same \\cite{GJKS,HolroydSchrammWilson2017FinitaryColoring}. \n    As a direct application one gets that from the three proofs that $3$-coloring of grids is not in $\\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, not in $ \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ \\cite{HolroydSchrammWilson2017FinitaryColoring} and not in $ \\mathsf{CONTINUOUS}$ \\cite{GJKS} any one implies the others. These three classes are the same in a much more general setup (see \\cref{subsec:CONT}). \n    \\item $\\mathsf{TAIL}(\\omega(r^{d-1})) = \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$\\hspace*{1cm} We prove that any problem that can be solved with an oblivious local algorithm with tail decay faster than $r^{d-1}$, can in fact be solved with a tower tail decay. This answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item Similarly to the time hierarchy theorems of the $\\mathsf{LOCAL}$ model \\cite{chang2017time_hierarchy}, we study the range of possible $\\mathsf{TAIL}$ complexities and give examples of problems LCL problems with $\\mathsf{TAIL}$-complexities $\\Theta(r^{j - o(1)})$ for any $1 \\le j \\le d-1$. \n\\end{itemize}\n\n\\paragraph{$\\TOAST$ constructions}\nWe formally define $\\TOAST$ algorithms (we mentioned earlier that these constructions are used in descriptive combinatorics and the theory of random processes) and prove results about them.  \nLet us first explain the toast constructions in a more detail. A $q$-toast in the infinite graph $\\mathbb{Z}^d$ is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that (a) for any two nodes of the graph we can find a piece that contains both,\\todo{it is enough that every vertex is contained in a piece} (b) boundaries of any two different pieces are at least distance  $q$ apart. \nThat is, imagine a collection of bigger and bigger pieces that form a laminar family (only inclusions, not intersections) that, in the limit, covers the whole infinite graph.\n\nThis structure can be built in the descriptive combinatorics setups \\cite{FirstToast,GJKS,Circle} but also with oblivious local algorithms \\cite{HolroydSchrammWilson2017FinitaryColoring,Spinka}. In fact, we show that one can construct the structure with  $1\/r^{1-o(1)}$ $\\mathsf{TAIL}$ decay in \\cref{thm:toast_in_tail}. This is best possible up to the $o(1)$ loss in the exponent. \nIf one has an inductive procedure building the toast, we can try to solve an LCL problem during the construction. That is, whenever a new piece of the toast is constructed, we define how the solution of the problem looks like at this piece. This is what we call a $\\TOAST$ algorithm. \nThere is a randomized variant of the $\\TOAST$ algorithm and a lazy variant. In the lazy variant one can postpone the decision to color given piece, but it needs to be colored at some point in the future. \nNext, we prove several relations between the $\\TOAST$ classes (see \\cref{fig:big_picture_grids}).  \n\n\\begin{itemize}\n    \\item $\\TOAST \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$\\hspace*{1cm} This result allows us to translate toast-algorithm constructions used in descriptive combinatorics to the $\\mathsf{TAIL}$-setting. This includes problems like vertex $3$-coloring (this answers a question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}), edge $2d$-coloring, or perfect matching. \n    \\item $\\RTOAST = \\TOAST$\\hspace*{1cm} This derandomization result for example implies on the first glance surprising inclusion $\\mathsf{FINDEP} \\subseteq \\mathsf{BOREL}$ -- these two classes are discussed in \\cref{sec:OtherClasses}. \n    \\item $\\mathsf{TAIL} = \\RLTOAST$\\hspace*{1cm} Here, $\\mathsf{TAIL}$ is defined as the union of $\\mathsf{TAIL}(f)$ over all functions $f \\in \\omega(1)$. This result shows how the language of toast constructions can in fact characterize the whole hierarchy of constructions by oblivious local algorithms. \n    \\item We show that the problem of $2d$ edge coloring of $\\mathbb{Z}^d$ is in $\\TOAST$, hence in $\\mathsf{BOREL}$ which answers the question of Gao et al \\cite{GJKS}. Independently, this was proven by \\cite{ArankaFeriLaci}. \n\\end{itemize}\n\n\\subsection{Roadmap}\n\nIn \\cref{sec:preliminaries} we recall the basic definitions regarding the $\\mathsf{LOCAL}$ model and explain the formal connection between finitary factors of iids. \nNext, in \\cref{sec:speedup_theorems} we prove the basic speedup theorems for $\\mathsf{TAIL}$. \nIn \\cref{sec:Toast} we define $\\TOAST$ algorithms and relate it to the $\\mathsf{TAIL}$ model. \nIn \\cref{sec:SpecificProblems} we discuss what our results mean for specific local problems and in \\cref{sec:Inter} we construct problems with $r^{j-o(1)}$ $\\mathsf{TAIL}$ complexities for $1 \\le j \\le d-1$. Finally, in \\cref{sec:OtherClasses} we discuss connection of our work to other classes of problems and finish with open questions in \\cref{sec:open_problems}. \n\n\n\n\n\n\\begin{comment}\nHonza - Introduction\n\nA \\emph{locally checkable problems (LCL) $\\Pi$ on a family of graphs $\\mathbf{G}$} is a procedure that checks whether a given ``candidate'' vertex coloring of $G\\in \\mathbf{G}$ satisfies a local constraints given by $\\Pi$\\footnote{\nFormally, $\\Pi$ is a triplet $\\Pi=(b,t,\\mathcal{P})$, where $b$ is a set of colors, $t\\in \\mathbb{N}$ is the locality of $\\Pi$ and $\\mathcal{P}$ is a set of allowed configurations around each vertex of locality at most $t$.}.\nMore concretely, given a graph $G=(V(G),E(G))\\in \\mathbf{G}$ and a ``candidate'' coloring $c:V(G)\\to b$, we say that $c$ is a \\emph{$\\Pi$-coloring} if for every $v\\in V(G)$ the restriction of $c$ to $t$-neighborhood of $v$ is in $\\mathcal{P}$, where $t$ is the locality of $\\Pi$ and $\\mathcal{P}$ is the set of constraints of $\\Pi$.\n\nIn this paper the family $\\mathbf{G}$ always refers to graphs that look locally like the standard Cayley graph of $\\mathbb{Z}^d$, $\\mathbb{Z}^d$, for some $d\\ge 1$.\nThis includes $\\mathbb{Z}^d$ as well as various finite $d$-dimensional tori with oriented edges that are labeled by the standard generators of $\\mathbb{Z}^d$.\nThe dimension $d$ is always assumed to be bigger than $1$, since for $d=1$ there is a full classification of LCL problems.\nWe omit mentioning ${\\bf G}$ and refer to all LCL problems on the family ${\\bf G}$ simply as \\emph{LCL problems (on $\n\\mathbb{Z}^d$)}.\nWhen we want to stress that a given LCL problem is on $\\mathbb{Z}^d$ we write $d$ in the superscript, i.e., $\\Pi^d$.\n\nWe study a complexity of LCL problems in various settings.\nIn the most abstract terms, a complexity of a given LCL problem $\\Pi$ is measured in terms of how difficult it is to \\emph{transform} a \\emph{given labeling} of a graph into a $\\Pi$-coloring.\nThe different contexts varies in what is considered to be an allowed starting labeling and what is allowed transformation.\nMost relevant for our investigation is the following probability notion.\nStart with a graph $G$ and assign to each vertex $v\\in V(G)$ a real number from $[0,1]$ uniformly and independently, i.e., \\emph{iid labeling} of $G$.\nA function $F$ that transforms these iid labels in a measurable and equivaraint way into a different coloring is called \\emph{factor of iid (fiid)}.\nHere equivariant refers to the group of all symmetries of $G$.\nIt is not hard to see that the group of all symmetries of $\\mathbb{Z}^d$ is the group $\\mathbb{Z}^d$ itself that acts by shift-translations.\nA fiid $F$ on $\\mathbb{Z}^d$ is called \\emph{finitary fiid (ffiid)} if for almost every iid labeling of $\\mathbb{Z}^d$ the  color $F(x)({\\bf 0}_d)$ only depends on the labels of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, if $x$ and $x'$ agrees on this neighborhood, then $F(x)({\\bf 0}_d)=F(x')({\\bf 0}_d)$.\nGiven ffiid $F$, we write $R_\\mathcal{A}$ for the random variable that codes minimum diameter of such neighborhood and we refer to it as the coding radius of $F$.\n\nWe say that a ffiid $F$ \\emph{solves an LCL problem $\\Pi$} if it produces a $\\Pi$-coloring for almost every iid labeling of $\\mathbb{Z}^d$.\nan LCL problem $\\Pi$ is in the class $\\mathsf{TAIL}$ if $\\Pi$ admits a ffiid solution.\nFollowing Holroyd, Schramm and Wilson \\cite{HolroydSchrammWilson2017FinitaryColoring}, the complexity of LCL problems in $\\mathsf{TAIL}$ can be measured on a finer scale by the properties of the coding radius $R_\\mathcal{A}$, where $F$ runs over all ffiid's that solve the given problem.\nSimilarly, we define when fiid $F$ solves $\\Pi$ and the class $\\mathsf{fiid}$.\n\nBefore we move to distributed algorithm we recall what is known about the notorious class of LCL problems that are the \\emph{proper vertex $k$-colorings of $\\mathbb{Z}^d$}, $\\Pi^d_{\\chi,k}$, where $k>1$, in the context of ffiid.\nIt is known that if $d>1$ and $k>3$, then $\\Pi^d_{\\chi,k}$ admits ffiid solution with a tower tail decay, see Holroyd et.al~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nIf $d>0$ and $k=2$, then $\\Pi^d_{\\chi,2}$ is not even in $\\mathsf{fiid}$.\nThe most interesting case is when $d>1$ and $k=3$.\nIt was shown by Holroyd et al~\\cite{HolroydSchrammWilson2017FinitaryColoring} that in that case $\\Pi^d_{\\chi,3}$ is in the class $\\mathsf{TAIL}$ and every ffiid that solves $\\Pi^d_{\\chi,3}$ must have infinite the $2$-moment of its coding radius. \nMoreover, they found a ffiid that solve $\\Pi^d_{\\chi,3}$ and has finite some moments of its coding radius.\nHere we strengthen their result by showing that ${\\bf M}(\\Pi^d_{\\chi,3})=1$ and every ffiid solution of $\\Pi^d_{\\chi,3}$ has infinite the $1$-moment of its coding radius.\n\nIn the context of deterministic distributed computing the setting is the following.\nStart with finite graph $G$ on $n$-vertices (in our case $n$ is big enough and $G$ is a torus) and assign to every vertex of $G$ its unique identifier from $[n]$.\\footnote{In generality, the identifier could be assigned from some domain that has size polynomial in $n$.}\nA \\emph{local algorithm $\\mathcal{A}_n$ of locality $t_n$} is a rule that computes color of a vertex $v$ based only on its $t$-neighborhood and the unique identifiers in this neighborhood.\nWe say that $\\mathcal{A}_n$ \\emph{solves} $\\Pi$ if for every (admissible)\\footnote{As a torus $G$ needs to be wider than the locality of $\\mathcal{A}_n$} graph $G$ on $n$-vertices and any assignment of unique identifiers to $G$ the rule $\\mathcal{A}_n$ produces a $\\Pi$-coloring.\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if there is a sequence of local rules \\todo{co je to local rule?}$\\{\\mathcal{A}_n,t_n\\}_{n\\in \\mathbb{N}}$ such that $\\mathcal{A}_n$ has locality $t_n$, solves $\\Pi$ for every $n\\in\\mathbb{N}$ and $n\\mapsto t_n$ is in the class $O(\\log^*(n))$.\nThe reason why we picked $\\log^*(n)$ is that there is a complete classification of LCL problems on oriented labeled grids.\nRoughly speaking, an LCL problem $\\Pi$ is either \\emph{trivial}, i.e., there is a single color that produces $\\Pi$-coloring, \\emph{local}, i.e., there is an algorithm that needs locality $\\Theta(\\log^*(n))$, or \\emph{global}, i.e., every local algorithm that solves $\\Pi$ needs to see essentially the whole graph.\\footnote{Same classification holds in the context of randomized distributed algorithms.}\n\nWe use this classification to provide several speed-up results in the context of ffiid solutions.\nWe note that the connection between distributed computing and probability setting was previously observed by several authors.\nFor example, it follows easily from the result of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring} that local coloring problems that are not global in the distributed setting admit ffiid solution with tower tail decay.\nIn fact, Brandt et al. \\cite{brandt_grids} asked:\n\\begin{quote}\n\\emph{\nHowever, despite the fact that techniques seem to translate between finitary colourings\nand distributed complexity, it remains unclear how to directly translate results from one setting to\nthe other in a black-box manner; for instance, can we derive the lower bound for 3-colouring from\nthe results of Holroyd et al. [24], and does our complexity classification imply answers to the open\nquestions they pose?\n}\n\\end{quote}\nAs one of our main result we address their question.\n\nThe most abstract setting that we consider is Borel combinatorics.\nIn this context one studies graphs that additionally posses a completely metrizable and separable topological structure (or a standard Borel structure) on its vertex set.\nIt follows from the work of Seward and Tucker-Drob \\cite{ST} and Gao, Jackson, Krohne and Seward \\cite{GJKS} that the canonical object to investigate is the graph structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, the free part of the shift action $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$, induced by the standard generating set of $\\mathbb{Z}^d$. \nThe topological and Borel structure on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is the one inherited from the product topology on $2^{\\mathbb{Z}^d}$.\nWe say that an LCL problem $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous and equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ into the space of $\\Pi$-colorings.\nThe class $\\mathsf{BOREL}$ is defined similarly, only the requirement on the map to be continuous is relaxed to be merely Borel measurable.\nBernshteyn \\cite{Bernshteyn2021LLL} proved that LCL problems that are not global in the distributed setting admit continuous and therefore Borel solution.\nIn another words, he showed that the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ is contained in $\\mathsf{CONTINUOUS}$.\nIt is not hard to show that the other inclusion follows from the Twelve Tile Theorem of Gao et al \\cite{GJKS}, i.e., the class $\\mathsf{CONTINUOUS}$ in fact coincide with the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nHere we investigate in greater detail what is the connection between existence of a Borel solution and a tail behavior of ffiid.\nIn fact all known Borel constructions uses the following structure that can be rephrased without mentioning topology or Borel measurability.\n\nA \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ such that\n\\begin{itemize}\n    \\item for every $g,h\\in \\mathbb{Z}^d$ there is $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$.\n\\end{itemize}\nWe say that an LCL problem $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and an equivariant rule $\\Omega$ that extends partial colorings such that if we apply $\\Omega$ on every $q$-toast $\\mathcal{D}$ inductively on the depth, then we produce a $\\Pi$-coloring.\nIt is not known whether the classes $\\mathsf{BOREL}$ and $\\TOAST$ coincide, however, it follows from the work of Gao et al.~\\cite{GJKS2} that $\\TOAST\\subseteq \\mathsf{BOREL}$.\nThis is proved by constructing a toast structure in a Borel way on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$, see Marks and Unger~\\cite{Circle} for application and proof of this fact.\nThe specific problems that belong to the class $\\TOAST$ include $3$-vertex coloring, perfect matching, $2d$-edge coloring, Schreier decoration (for non-oriented grids), some rectangular tilings, space filling curve etc.\nThese results were proved by many people in slightly different language, \\cite{GJKS2, ArankaFeriLaci, spencer_personal}.\nWe utilize here all the constructions and show that these all belong to $\\TOAST$.\n\nWe connect the classes $\\TOAST$ and $\\mathsf{TAIL}$ as follows.\nWe show that for every $q\\in \\mathbb{N}$ and $\\epsilon>0$ there is a ffiid $F$ that produces $q$-toast that has the $(1-\\epsilon)$-moment finite.\nA slightly stronger result implies that, in fact, ${\\bf M}(\\Pi)\\ge 1$, whenever an LCL problem $\\Pi$ is in the class $\\TOAST$.\nThis shows that ${\\bf M}(\\Pi)\\ge 1$ for all the problems mentioned above.\nWe note that in the special case when $d=2$ and the corresponding LCL problem $\\Pi$ does not admit ffiid solution with a tower tail decay, we deduce that ${\\bf M}(\\Pi)=1$.\nThis follows from our speed-up result in the context of ffiid.\n\nTraditionally there are two more classes that are investigated in Borel combinatorics, $\\mathsf{BAIRE}$ and $\\mathsf{MEASURE}$.\nTo define the class $\\mathsf{BAIRE}$ we replace the topological requirement on continuity by \\emph{Baire measurability}, i.e., we require a solution to be continuous only on a \\emph{comeager} set, see the work of Bernshteyn \\cite{Bernshteyn2021LLL}.\nThe class $\\mathsf{MEASURE}$ generalizes the setting of fiid.\nThe goal here is to find a Borel solution on a conull set for a given Borel graph that is endowed with a Borel probability measure.\nThis direction has been recently investigated by Bernshteyn \\cite{Bernshteyn2021LLL} for general graphs, where he proved a measurable version of Lovasz's Local Lemma.\nSee \\cref{fig:big_picture_grids} for all the relations that are known to us.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/big_picture_grids.png}\n    \\caption{Big picture of lcl for $\\mathbb{Z}^d$}\n   \n\\end{figure}\n\n\n\n\n\n\n\n\\subsection{The Results}\n\nIn this subsection we formulate our main results.\nThe formulation uses mostly the language of ffiid because the tail behavior of the coding radius allows to compare problems on a quantitative and, as it turns out, rich scale.\nOne should always re-interpret the result with the help of Figure~\\ref{fig:big_picture_grids} to different contexts.\nIronically enough, the techniques that are most valuable when investigating ffiid solutions come either from local algorithms or Borel combinatorics, where such a rich and quantitative distinction is not possible.\nRecall that in the distributed algorithms there is a trichotomy of LCL problems, i.e., trivial, local or global, and in Borel combinatorics we distinguish mainly between trivial, continuous, Borel, and global problems.\n\n\nWe start with results that use ideas from distributed algorithms.\nFirst, we prove that LCL problems that are not global in distributed algorithms are precisely problems that admit ffiid solution with a tower tail decay.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nThen $\\Pi$ is not global in the local world, i.e., there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds, if and only if there is an ffiid solution $F$ of $\\Pi$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{tower(Cr)}$$\nfor some $C>0$ and large enough $r\\in \\mathbb{N}$.\n\\end{theorem*}\n\nThe translation between ffiid and distributed algorithms allows to deduce the following result that answers problem (ii) of Holroyd at el.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf $\\Pi$ admits an ffiid solution $F$ that has faster tail decay than any tower function, then $\\Pi$ is trivial, i.e., there is a color such that a constant coloring with this color is a correct solution.\n\\end{theorem*}\n\nAs a last application of ideas from the distributed algorithms, we show the following speed-up theorem for ffiid.\n\n\\begin{theorem*}\nLet $d,t\\in \\mathbb{N}$.\nThere is $C_{d,t}>0$ such that if an LCL problem $\\Pi$ admits an ffiid solution $F$ such that\n$$\\P(R_\\mathcal{A}>r)\\le C_{d,t}r^{d-1},$$\nthen there is a deterministic algorithm in the local model that runs in $O(\\log^*(n))$ rounds and solves $\\Pi$.\n\\end{theorem*}\n\nTo understand better LCL problems that are global in the distributed algorithms or admit no continuous solution in the context of Borel combinatorics we introduce the following numerical invariant.\nThe inspiration comes from the paper of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{definition*}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an ffiid solution $F$ of $\\Pi$ such that $R_F$ has finite the $\\alpha$-moment.\n\\end{definition*}\n\nCombination of previous results immediately implies the following strengthening of the upper bound on the necessary infinite moment of ffiid solution of the proper vertex $3$-coloring problem, $\\Pi^d_{\\chi,3}$, from Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\nMoreover, this answers their problem (iii).\n\n\\begin{theorem*}\nLet $d\\in \\mathbb{N}$ and $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$.\nIf ${\\bf M}(\\Pi)>d-1$, then $\\Pi$ admits an ffiid solution with tower tail decay.\n\\end{theorem*}\n\nAs a first application of ideas from Borel combinatorics, i.e., the toast structure, we compute the optimal tail decay for $\\Pi^d_{\\chi,3}$, thus answering a problem (i) of Holroyd et al.~\\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem*}\nLet $d>1$ and $\\Pi^d_{\\chi,3}$ be the proper vertex $3$-coloring problem.\nThen ${\\bf M}(\\Pi^d_{\\chi,3})=1$.\n\\end{theorem*}\n\nA high level explanation of a toast construction in the context of ffiid says that if $r\\in \\mathbb{N}$ is given, then the fraction of vertices not covered by first $r$-layers of the toast is at most $\\frac{1}{r}$.\nSimilar construction that uses only $1$-dimensional skeleton of some rectangular tiling of $\\mathbb{Z}^d$ improve this upper bound to $\\frac{1}{r^{d-1}}$.\nWe show that this type of structure is enough to find a perfect matching in a bigger graph then $\\mathbb{Z}^d$, a graph where we moreover add diagonal edges in every $2$-dimensional hyperplane.\nThis yields an existence of intermediate problems from the point of view of ${\\bf M}$.\n\n\\begin{theorem*}\nLet $2\\le \\ell\\le d\\in \\mathbb{N}$.\nThen there is an LCL problem $\\Pi$ on $\\mathbb{Z}^d$ that satisfies ${\\bf M}(\\Pi)=\\ell-1$.\n\\end{theorem*}\n\nAs a last result we summarize what is known about specific problems.\nSince we work mainly with oriented grids we state everything in this context, however, some of the results can be deduced for non-oriented grids as well, see Section~\\ref{sec:SpecificProblems}.\nIn particular, we answer Question~6.5 of Gao et al.~\\cite{GJKS} about Borel edge chroamtic number of $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\n\n\\begin{theorem*}\nLet $d\\ge 2$ and $\\Pi$ be any of the following LCL problems:\n\\begin{enumerate}\n    \\item proper vertex $3$-coloring, \\cite{GJKS}, \\cite{spencer_personal},\n    \\item proper edge $2d$-coloring,\n    \\item rectangular tiling with rectangles from some set $\\mathcal{R}$ that satisfies \\cref{eq:GCD}, \\cite{Unger},\n    \\item perfect matching, \\cite{GJKS}.\n\\end{enumerate}\nThen $\\Pi$ is in the class $\\TOAST$.\nConsequently, $\\Pi$ is in the class $\\mathsf{BOREL}$ and satisfies $1\\le {\\bf M}(\\Pi)$.\n\\end{theorem*}\n\n\n\nThe paper is organized as follows...\n\n\n\\end{comment}\n\n\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nLet $d\\in \\mathbb{N}$ be fixed.\nThe class of graphs that we consider in this work contains mostly the infinite grid $\\mathbb{Z}^d$, that is $\\mathbb{Z}^d$ with the Cayley graph structure given by the standard set of generators \n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}. $$\nWe also used the finite variant of the graph: the torus of side length $\\Theta(n^{1\/d})$ with edges oriented and labeled by $S$.\nUsually, when we talk about a graph $G$ we mean one of the graphs above and it follows from the context whether we mean finite or infinite graph. \n\nGiven a graph $G$, we denote as $V(G)$ the set of vertices, as $E(G)$ the set of edges and as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. When $G$ is the infinite grid $\\mathbb{Z}^d$, then we leave the subscripts blank.\nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the \\emph{isomorphism type} of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften we do not mention the labels and tacitly assume that they are understood from the context.\nAlso, the label set might depend on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\n\\begin{comment}\n\nWe always consider the group $\\mathbb{Z}^d$ with the standard set of generators $S$, i.e.,\n$$S=\\{(1,0,\\dots, 0),(0,1,\\dots, 0), \\dots (0,\\dots,0,1)\\}.$$\nThe \\emph{Cayley graph} $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right):=\\operatorname{Cay}\\left(\\mathbb{Z}^d,S\\right)$ of $\\mathbb{Z}^d$ is then defined on the vertex set $\\mathbb{Z}^d$ with oriented and $S$-labeled edges \n$$\\left\\{(g,s+g):s\\in S, g\\in \\mathbb{Z}^d\\right\\}=E\\left(\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)\\right).$$\nWe denote the identity in $\\mathbb{Z}^d$ as ${\\bf 0}_d$. \nWhen we talk about $\\mathbb{Z}^d$ next, we mean formally the appropriate Cayley graph with labelled edges, that is, we think of the graph together with its consistent orientation. \n\n\nA \\emph{graph} $G$ is a pair $(V(G),E(G))$, where $V(G)$ is a vertex set and $E(G)$ consists of oriented and $S$-labeled edges.\nGiven a graph $G$, we denote as $d_G$ the \\emph{graph distance} on $G$, i.e., $d_G(v,w)$ is the minimal number of edges that connects $v$ and $w$.\nWe write $\\mathcal{B}_G(v,r)$ for the \\emph{ball of radius $r\\in \\mathbb{N}$} around $v\\in V(G)$. Usually, $G$ is just the grid $\\mathbb{Z}^d$, then we leave the subscript blank. \nA \\emph{rooted graph} is a pair $(G,v)$ where $G$ is a graph and $v\\in V(G)$.\nWe denote as $[G,v]$ the isomorphism type of the rooted graph $(G,v)$, where $(G,v)$ and $(G',v')$ are isomorphic if there is a bijection $\\varphi:V(G)\\to V(G')$ that sends $v$ to $v'$ and preserves edge orientation and labeling.\n\n\\todo{vsechny strany toru maji delku n na 1\/d; zjevne vsechny problemy maji lokalitu nejvyse n na 1\/d}\n\n\\end{comment}\n\n\n\\begin{comment}\nWe write $\\operatorname{\\bf FG}^{S}$ for the set of all graphs that look in their $2$-neighborhoods as $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$, i.e., a graph $G=(V(G),E(G))$ is in $\\operatorname{\\bf FG}^{S}$ if $[\\mathcal{B}_G(v,2)]=\\left[\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,2)\\right]$ for every $v\\in V(G)$.\nNote that $\\mathbb{Z}^d\\in \\operatorname{\\bf FG}^{S}$ and there are finite graphs in $\\operatorname{\\bf FG}^{S}$, as an example take any torus that is big enough.\nWrite $\\operatorname{\\bf FG}_{\\bullet}^{S}$ for the rooted graphs of the form\n$$(\\mathcal{B}_G(v,r),v)$$\nwhere $G\\in \\operatorname{\\bf FG}^{S}$, $v\\in V(G)$ and $r\\in \\mathbb{N}$.\nWe say that a rooted graph $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ has \\emph{diameter at most $m\\in \\mathbb{N}$} if $(\\mathcal{B}_G(v,m),v)=(G,v)$.\nNote that, in particular, we have $\\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}({\\bf 0}_d,r)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ for every $r\\in \\mathbb{N}$.\n\nLet $D$ be a set, $G$ a graph and $c:V(G)\\to D$ be a labeling of vertices.\nThen we write $G_c=(G,c)$ for the pair graph + labeling.\nSimilarly we denote as $(G_c,v)$ the rooted graph $(G,v)$ with the labeling $c$, $[G_c,v]$ its isomorphism class and $\\mathcal{B}_{G_c}(v,r)$ the labeled ball of radius $r\\in \\mathbb{N}$ around $v$, where $v\\in V(G)$.\nOften the label set $D$ depends on the size of of the vertex set, e.g., unique labels from $[|V(G)|]=\\{1,\\dots,|V(G)|\\}$ etc.\n\\end{comment}\n\n\n\n\\subsection{LCLs on $\\mathbb{Z}^d$}\n\nOur goal is to understand locally checkable problems (LCL) on $\\mathbb{Z}^d$. \n\n\\begin{definition}[Locally checkable problems \\cite{naorstockmeyer}]\nA \\emph{locally checkable problems (LCL)} on $\\mathbb{Z}^d$ is a quadruple $\\Pi=(\\Sigma_{in}, \\Sigma_{out},t,\\mathcal{P})$ where $\\Sigma_{in}$ and $\\Sigma_{out}$ are finite sets, $t\\in \\mathbb{N}$ and $\\mathcal{P}$ is an assignment\n$$[H_c,w]\\mapsto \\mathcal{P}([H_c,w])\\in \\{0,1\\},$$\nwhere $(H,w)$ is a rooted grid of radius at most $t$ together with an input and output coloring $c: V(H)\\to \\Sigma_{in}\\times\\Sigma_{out}$. \n \nGiven a graph $G$ together with a coloring $c:V(G)\\to \\Sigma_{in}\\times\\Sigma_{out}$, we say that $c$ is a \\emph{$\\Pi$-coloring} if\n$$\\mathcal{P}([\\mathcal{B}_{G_c}(v,t),v])=1$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\n\n\n\\begin{comment}\n\nClasses of functions\n\nBefore we proceed to definition of local algorithms and their complexity we recall here the relevant notions concerning functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nImportantly, we often use asymptotic notation as usual in Computer Science, e.g., we define the class of functions $O(f(n))$ as the union over all functions $g:\\mathbb{N} \\rightarrow \\mathbb{N}$ such that there exists $C > 0$ such that for all $n$ we have $g(n) \\le C f(n)$. Other symbols $\\Omega, \\Theta, o, \\omega$ are defined analogously and their usage is analogous (see e.g. \\cite{?}). \n\\end{comment}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{LOCAL model}\n\\label{subsec:preliminaries_local}\n\nHere we formally introduce the $\\mathsf{LOCAL}$ model for both deterministic and randomized algorithms and state the classification of LCLs on $\\mathbb{Z}^d$ from the $\\mathsf{LOCAL}$ point of view.\nInformally, a local algorithm $\\mathcal{A}$ of radius $r\\in \\mathbb{N}$ is a rule that takes a labeled graph $G_c$ and then re-labels each vertex $v\\in V(G)$ depending only on the isomorphism type of its neighborhood $[\\mathcal{B}_{G_c}(v,r),v]$ with a new label from some finite set $b$.\n\n\\paragraph{Local algorithms}\n\nHere, we formally define the complexity classes $\\mathsf{LOCAL}(\\mathcal{F})$ of deterministic local algorithms, where $\\mathcal{F}$ is a class of functions from $\\mathbb{N}$ to $\\mathbb{N}$. \nThe somewhat unintuitive part of the definition is that the algorithm takes as input \\emph{unique identifiers}. Usually, they are taken from some range of values that is polynomial in $n$, the number of vertices of the input graph. However, in most scenarios this is equivalent to the assumption that they are from range $\\{1,\\dots n\\}$, i.e., every identifier is used exactly once. We also use this definition here, for simplicity.\nIn all forthcoming definition we have some fixed LCL $\\Pi$ in mind.\n\n\\begin{definition}[A deterministic local algorithm for $\\mathbb{Z}^d$]\n\\label{def:local_algorithm\nLet $\\Sigma_{out}$ be a finite set.\nA \\emph{local (deterministic) algorithm $\\mathcal{A}$} is a sequence of pairs $\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$, where $\\ell_n\\in \\mathbb{N}$ and $\\mathcal{A}_n$ is an assignment, such that\n$$[\\mathcal{B}_{G_\\ell}(v,\\ell_n),v]\\mapsto \\mathcal{A}_n([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])\\in \\Sigma_{out}$$\nfor every $\\Sigma_{in}$-labeled graph $G$ (torus) on $n$ vertices, $v\\in V(G)$ and a bijection $\\ell:V(G)\\to [n]$.\nIn another words, $\\mathcal{A}_n$ is a function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with injectively labeled vertices from the set $\\{1,\\dots, n\\}$ and with range $b$. \n\nGiven such a $G$ and a bijection $\\ell:V(G)\\to |V(G)|$, we define\n$$\\mathcal{A}(G_\\ell)(v):=\\mathcal{A}_{|V(G)|}([\\mathcal{B}_{G_\\ell}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define the deterministic complexity of a given local coloring problem $\\Pi$.\n\n\\begin{definition}[Complexity of deterministic local algorithm for $\\mathbb{Z}^d$]\nWe say that a local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ \\emph{solves} an LCL $\\Pi=(\\Sigma_{in},\\Sigma_{out},t,\\mathcal{P})$ if $\\mathcal{A}(G_l)$ is a $\\Pi$-coloring of $G$ for every finite $\\Sigma_{in}$-labeled graph $G$ and any bijection $\\ell:V(G)\\to |V(G)|$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\nA randomized local algorithm and its complexity is defined analogously but instead of labeling with unique identifiers each node has an iid random string and the algorithm succeeds with probability $1 - 1\/n$.\n\nCrucial for our investigation are so-called \\emph{oblivious local algorithms}.\nRoughly speaking an algorithm is \\emph{oblivious} if it does not have knowledge of $n$ (see \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} for details). \nSince we do not consider deterministic oblivious algorithms, by an oblivious local algorithm we mean an oblivious randomized local algorithm.\n\n\\begin{definition}[Oblivious local algorithm]\n\\label{def:uniform_local}\nA \\emph{oblivious local algorithm} $\\mathcal{A}$ is a pair $(\\mathcal{C},f)$, where $\\mathcal{C}$ is a (measurable) collection of rooted graphs labeled with $\\Sigma_{in}$ and real numbers from $[0,1]$, and $f: \\mathcal{C} \\rightarrow \\Sigma_{out}$ is a measurable assignment.\n\nWe define $\\mathcal{A}(G)$ to be the labeling computed by $\\mathcal{A}$ on $G$, where for a given node $u$ we have $\\mathcal{A}(G)(u)=g([B_G(u, t), u])$ where $t$ is the smallest integer such that $[B_G(u, t),u] \\in \\mathcal{C}$.\n\\end{definition}\n\nNote that since there is no $n$ in the definition of oblivious local algorithm, the algorithm is correct if it always finishes with probability $1$, while randomized algorithms can have small probability of error dependent on $n$. Hence, one should think about such algorithms as an analogue of Las Vegas algorithms, while the classical randomized local algorithms correspond to Monte Carlo algorithms. \nAn advantage of not knowing $n$ is that it is possible to run $\\mathcal{A}$ on, e.g., the infinite grid $\\mathbb{Z}^d$ instead of just a torus. Usually, these two models give the same results but for toast algorithms that we consider later, there is a difference: While we define the algorithms for an infinite grid, for example the $3$-coloring algorithm makes sense also for a finite torus, while the perfect matching algorithm does not make sense in the finite world for the simple reason that the input graph can have an odd number of vertices. \n\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Randomized local algorithms}\\todo{merge with previous paragraph}\n\nThe main difference between deterministic and randomized model is that the input of the local rule in the randomized case are graphs with vertices labeled uniformly independently at random by elements of the interval $[0,1]$ and we require that the output coloring is correct everywhere with high probability.\nThe complexity of local algorithm is then defined as the locality it needs.\n\n\\begin{definition}[Randomized local algorithm for $\\mathbb{Z}^d$]\nLet $b$ be a set.\nWe say that $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in\\mathbb{N}}$ is a randomized local algorithm for the class $\\operatorname{\\bf FG}^{S}$ if $\\ell_n\\in \\mathbb{N}$ and \n$$[\\mathcal{B}_{G_l}(v,\\ell_n),v]\\mapsto \\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])\\in b$$\nis a measurable assignment, where $G\\in \\operatorname{\\bf FG}^{S}$, $|V(G)|=n$, $v\\in V(G)$ and $\\ell:V(G)\\to [0,1]$.\nIn another words, $\\mathbb{A}_n$ is a measurable function that is defined on isomorphism classes of rooted graphs of diameter at most $\\ell_n$ with vertices labeled by $[0,1]$ and with range $b$. \n\nGiven $G\\in \\operatorname{\\bf FG}^{S}$ such that $n=|V(G)|$ and a map $\\ell:V(G)\\to [0,1]$, we define\n$$\\mathbb{A}(G_l)(v):=\\mathbb{A}_n([\\mathcal{B}_{G_l}(v,\\ell_n),v])$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\nNext we define when a randomized algorithm solves an LCL $\\Pi$ and the randomized complexity classes.\n\n\\begin{definition}[Complexity of randomized local algorithm for $\\mathbb{Z}^d$]\nLet $\\Pi=(b,t,\\mathcal{P})$ be an LCL.\nWe say that a randomized algorithm $\\mathbb{A}=\\{\\mathbb{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ solves $\\Pi$ if \n$$\\lambda^{\\oplus n}\\left(\\left\\{\\ell\\in [0,1]^{V(G)}:\\mathbb{A}(G_l) \\ \\operatorname{is \\ a} \\ \\Pi\\operatorname{-coloring} \\right\\}\\right)>1-\\frac{1}{n}$$\nholds for every $n\\in \\mathbb{N}$ and $G\\in \\operatorname{\\bf FG}^{S}$ such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(\\mathcal{F})$, where $\\mathcal{F}$ is a class of functions, if $n\\mapsto \\ell_n$ is in $\\mathcal{F}$.\n\\end{definition}\n\\end{comment}\n\n\\begin{comment}\n\\label{sec:clocal_definition}\n\nThe last class of local algorithm that we introduce serves on one hand as a main technical tool for deterministic speed up tricks and on the other hand allows a smooth translation of local algorithm into the measurable setting.\nIntuitively, an algorithm is in the class $\\mathsf{C-LOCAL}$ if it only needs a fixed finite locality (not depending on the size of a graph) given some coloring of the vertices with fixed number of colors (not depending on the size of a graph).\n\nLet $s\\in \\mathbb{N}$ and $G$ be a graph.\nDenote as $G^s$ the $s$-power graph of $G$, i.e., $V(G^s)=V(G)$ and $(x,y)\\in E(G^s)$ if and only if $0<d_G(x,y)\\le s$.\nNote that if $G\\in \\operatorname{\\bf FG}^{S}$, then the maximum degree of a vertex in $G^s$ is bounded by $\\Delta_s:=\\left|\\mathcal{B}({\\bf 0}_d,s)\\right|$.\nWe say that $c:V(G)\\to m$ is a proper $s$-power vertex coloring if it is a proper vertex coloring of the graph $G^s$.\nNote that it follows from a standard greedy construction that it is possible to find an $s$-power coloring of every $G\\in \\operatorname{\\bf FG}^{S}$ with $(\\Delta_s+1)$-many colors.\n\n\\begin{definition}[$\\mathsf{C-LOCAL}(O(1))$]\nWe say that an LCL $\\Pi=(b,t,\\mathcal{P})$ is in the class $\\mathsf{C-LOCAL}(O(1))$ if there are $r,\\ell\\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ with domain that consists of rooted graphs $(H,w)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ of diameter at most $\\ell$ together with an injective map $d:V(H)\\to (\\Delta_{r+t}+1)$, i.e.,\n$$[H_d,w]\\mapsto \\mathcal{A}([H_d,w])\\in b,$$\nsuch that $\\mathcal{A}(G_c)$ is a $\\Pi$-coloring for every $G\\in \\operatorname{\\bf FG}^{S}$ that satisfies \n$$[\\mathcal{B}_{G}(v,r+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,r+t),{\\bf 0}_d\\right]$$\nfor every $v\\in V(G)$ and $c:V(G)\\to (\\Delta_{r+t}+1)$, a proper $(r+t)$-power vertex coloring of the graph $G^{r+t}$, where\n$$\\mathcal{A}(G_c)(v):=\\mathcal{A}([\\mathcal{B}_{G_c}(v,\\ell),v])\\in b$$\nfor every $v\\in V(G)$.\n\\end{definition}\n\\end{comment}\n\n\n\n\n\n\\paragraph{Classification of Local Problems on $\\mathbb{Z}^d$}\n\nIn this section, we state a special case of \\emph{Classification of LCLs}. This is an ongoing effort in the area of distributed algorithms and our current understanding emerges as a corollary of many results that follow from several recent lines of work on local algorithms including \\cite{naorstockmeyer, brandt,chang2016exp_separation,chang2017time_hierarchy, ghaffari2017complexity, ghaffari2018derandomizing,balliu2018new_classes-loglog*-log*, RozhonG19, ghaffari2019distributed, ghaffari_grunau_rozhon2020improved_network_decomposition,chang2020n1k_speedups,balliu2020almost_global_problems,brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and many others.  \nFor our purposes we will need only a special case of the Classification for grids. Fortunately, here (and on trees) all possible local complexities of an LCL are completely understood. This result is stated in \\cref{thm:classification_of_local_problems_on_grids}. \n\n\\begin{comment}\nRecall that an LCL $\\Pi=(b,t,\\mathcal{P})$ is satisfiable \\todo{do we need it?} if there is $x\\in b^\\Gamma$ that is a $\\Pi$-coloring of $\\operatorname{Cay}(\\Gamma,S)$ and it is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant $\\alpha$ coloring is a $\\Pi$-coloring.\n\\end{comment}\n\n\\begin{comment}\n\\begin{definition}[Global problems]\n\\label{def:global_problem}\nWe say that a satisfiable LCL problem $\\Pi$ is \\emph{global for deterministic (randomized) algorithms} if for every deterministic (randomized) local algorithm $\\mathcal{A}=\\{\\mathcal{A}_n,\\ell_n\\}_{n\\in \\mathbb{N}}$ that solves $\\Pi$ we have $\\{\\ell_n\\}_{n\\in\\mathbb{N}}\\in \\Omega(n^{\\frac{1}{d}})$.\n\\end{definition}\n\\end{comment}\n\n\n\n\n\\begin{theorem}[Classification of LCLs on $\\mathbb{Z}^d$ \\cite{naorstockmeyer, chang2016exp_separation, chang2017time_hierarchy, brandt_grids, brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:classification_of_local_problems_on_grids}\nLet $\\Pi$ be an LCL.\nThen one of the following is true. \n\\begin{enumerate}\n\\item $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial,\n\\item $\\Pi \\in \\mathsf{LOCAL}(\\Theta(\\log^* n)), \\Pi \\in \\mathsf{RLOCAL}(\\Theta(\\log^* n))$,\n\\item $\\Pi$ is not in $\\mathsf{LOCAL}(o(n^{1\/d}))$ or $\\mathsf{RLOCAL}(o(n^{1\/d}))$.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}[\\cite{chang2017time_hierarchy,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}]\n\\label{rem:chang_pettie_decomposition}\nIn fact, any LCL in $\\mathsf{LOCAL}(O(\\log^* n))$ can be solved as follows. First, we compute a maximal independent set (MIS) of a large enough power of the input graph, and then we run a local algorithm with just $O(1)$ local complexity. \\footnote{So in a sense, MIS is a complete problem for the class $\\mathsf{LOCAL}(O(\\log^* n))$. }\nMoreover, the algorithm in points (1) and (2) can be oblivious and the algorithm in point (1) does not need to access the input unique  identifiers. \n\\end{remark}\n\n\\begin{remark}\nThe problem of checking whether $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ is undecidable.  \n\\end{remark}\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities}\n\\label{subsec:preliminaries_tail}\n\nLet $\\mathcal{A}=(\\mathcal{C},f)$ be an oblivious local algorithm.\nWe denote as $R_\\mathcal{A}$ the \\emph{coding radius} of $\\mathcal{A}$, that is a random variable on $\\mathbb{N}$ that corresponds to the minimal diameter that a vertex $v$ needs to look at to find its neighborhood that is an element of $\\mathcal{C}$.\nNote that the randomness refers to the real number labels.\nTherefore when we write, e.g.,\n$$\\P(R_\\mathcal{A} > r),$$\nwhat we mean is the following. \nFirst, for a given rooted neighborhood $(G,v)$ with a fixed $\\Sigma_{in}$-labeling we ask what is the probability that, after labeling $G$ with real numbers uniformly at random, that $(G,v)$ is in $\\mathcal{C}$. \nSecond, we take the supremum over all such values over all $\\Sigma_{in}$-labels.\n\n\\begin{definition}[$\\mathsf{TAIL}(f)$]\n\\label{def:tail}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{TAIL}(f)$ for some function $f:\\mathbb{N}\\to \\mathbb{N}$ if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that\n$$\\P(R_\\mathcal{A} > r)\\le \\frac{1}{f(r)}$$\nfor every large enough $r\\in \\mathbb{N}$. \n\nWe say that $\\Pi \\in \\mathsf{TAIL}$ if there exists $f \\in \\omega(1)$ such that $\\Pi \\in \\mathsf{TAIL}(f)$. \nWe say that $\\Pi$ is solved by a \\emph{block factor} if there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ such that for some $r_0$ we have $\\P(R_\\mathcal{A} > r_0) = 0$.\n\\end{definition}\n\nHere things are made simple because we think of the algorithm as being run on an infinite graph so nothing depends on $n$ in this complexity measure (while it is a crucial parameter in the classical local complexity definition). \n\nWe also use the asymptotic notation when talking about $\\mathsf{TAIL}$ complexities. For example, an important class $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ is defined as the class of LCLs $\\Pi$ such that there is a constant $C\\in \\mathbb{N}$ and oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ that satisfies\n$$\\P(R_\\mathcal{A}> r)\\le \\frac{1}{\\mathrm{tower}(Cr)}$$\nwhere $\\mathrm{tower}(x)=\\exp\\circ\\dots\\circ \\exp(1)$ is $\\lfloor x\\rfloor$-many times iterated exponential.\n\nA coarser way to measure the complexity of a problem $\\Pi$ is to ask what moments of $R_\\mathcal{A}$ could be finite, when $\\mathcal{A}$ is an oblivious local algorithm.\nFor example, the first moment finite corresponds to the fact that average time until an algorithm finishes is defined (for finite graphs this is independent on the size of the graph). \nIn particular, we investigate the question what is the least infinite moment for any oblivious local algorithm to solve a given problem $\\Pi$, as this language was used in \\cite{HolroydSchrammWilson2017FinitaryColoring}. \nRecall that $R_\\mathcal{A}$ has a \\emph{finite $\\alpha$-moment}, where $\\alpha\\in [0,+\\infty)$, if\n$$\\mathbb{E}\\left(\\left(R_\\mathcal{A}\\right)^\\alpha\\right)=\\sum_{r \\in \\mathbb{N}} r^\\alpha \\cdot \\P(R_\\mathcal{A} = r) <+\\infty.$$\n\n\\begin{definition}\nLet $\\Pi$ be an LCL.\nWe define ${\\bf M}(\\Pi)\\in [0,+\\infty]$ to be the supremum over all $\\alpha\\in [0,+\\infty)$ such that there is an oblivious $\\mathcal{A}$ that solves $\\Pi$ and has the $\\alpha$-moment of $R_\\mathcal{A}$ finite.\n\\end{definition}\n\nThe $\\alpha$ moment being finite roughly corresponds to $\\Omega(r^{\\alpha})$ $\\mathsf{TAIL}$ complexity. In fact we now recall the following observation.\n\n\\begin{fact}\n\\label{fact:moment_implies_tail}\nSuppose that the coding radius of an oblivious local algorithm $\\mathcal{A}$ has the $\\alpha$-moment finite.\nThen for every $C>0$ we have\n$$\\P(R_\\mathcal{A} > r) \\le C\\frac{1}{r^\\alpha}$$\nfor large enough $r\\in \\mathbb{N}$.\n\\end{fact}\n\\begin{proof}\nWe prove the contrapositive. Suppose the tail of the coding radius $R$ satisfies $\\P(R > r_i) > C \\cdot \\frac{1}{r^\\alpha_i}$ for some $C > 0$ and an infinite sequence $(r_i)_{i\\in \\mathbb{N}}$. \nAs the sequence $\\P(R> r_i)$ goes to zero, we may pass to a subsequence such that for each $i\\in \\mathbb{N}$ we have $\\P(R > r_{i+1}) \\le \\frac{1}{2}\\P(R > r_i)$. We call the new subsequence still $(r_i)_{i\\in \\mathbb{N}}$.  \nWe have \n\\begin{align*}\n\\mathbb{E}(R^\\alpha) &\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot (\\P(R> r_i) - \\P(R > r_{i+1}))\\\\\n&\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{1}{2}\\P(R > r_i)\n\\ge \\sum_{i \\in \\mathbb{N}} r_i^\\alpha \\cdot \n\\frac{C}{2} \\cdot \\frac{1}{r_i^\\alpha}\n= +\\infty\n\\end{align*}\nwhich finishes the proof. \n\\end{proof}\n\nFinally, an important fact is that a small $\\mathsf{TAIL}$ complexity of an oblivious randomized local algorithm implies that it has small randomized complexity (on finite graphs). This is because if we run an oblivious algorithm with tail decay $1\/f(r)$ on an $n$-node graph and mark its run as a failure if the coding radius exceeds $f^{-1}(n^2)$, the failure at some node happens only with  probability $1\/n$. This was formally proven in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} and here we only state the result. \n\n\\begin{lemma}\n\\label{lem:tail->local}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with $\\mathsf{TAIL}$ complexity $f(r)$. Then the randomized complexity of $\\mathcal{A}$ is $O(f^{-1}(n^2))$. \n\\end{lemma}\n\nIn fact, one can prove that for $o(\\log n)$ local complexities the two notions are the same. We refer to the reader to the discussion in \\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids} for more details. \n\n\\begin{theorem}[\\cite{brandt_grebik_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:tail=uniform}\nLet $\\mathcal{A}$ be an oblivious local randomized algorithm with randomized complexity $f(n) = o(\\log n)$. Then this algorithm in fact never returns an incorrect solution and its tail complexity is $\\Omega(f^{-1}(r))$. \n\\end{theorem}\n\n\n\n\n\n\n\n\n\n\\subsection{Subshifts of finite type}\n\\label{subsec:subshifts}\n\nHere we connect our definition of LCLs and oblivious local algorithms with the well-studied notions of subshifts of finite type and finitary factors of iid labelings.\n\nTo define subshifts of finite type we restrict ourselves to LCLs without input.\nThat is $\\Sigma_{in}$ is empty or contains only one element.\nMoreover, for convenience we set $b:=\\Sigma_{out}$.\nThen an LCL $\\Pi$ is of the form $\\Pi=(b,t,\\mathcal{P})$.\nWe define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nThat is we consider all possible labelings of $\\mathbb{Z}^d$ by elements from $b$ and then include in $X_{\\Pi}$ only those that satisfies the finite constraints $\\mathcal{P}$.\nViewing $\\mathbb{Z}^d$ as a group, there is a natural shift action of $\\mathbb{Z}^d$ on (the Cayley graph) $\\mathbb{Z}^d$.\nNamely, $g\\in \\mathbb{Z}^d$ shifts $h$ to $g+h$.\nThis action naturally ascend to an action $\\cdot $ on $b^{\\mathbb{Z}^d}$ as\n$$(g\\cdot x)(h)=x(g+h)$$\nwhere $x\\in b^{\\mathbb{Z}^d}$.\nIt is not hard to see (since $\\mathcal{P}$ is defined on isomorphism types and not rooted in any particular vertex) that $X_\\Pi$ is invariant under the shift map.\nThen we say that $X_\\Pi$ together with the shift structure is a \\emph{subshift of finite type}.\n\nA \\emph{finitary factor of iid process (ffiid) $F$} is a function that takes as an input $\\mathbb{Z}^d$ labeled with real numbers from $[0,1]$, independently according to Lebesgue measure, and outputs a different labeling of $\\mathbb{Z}^d$, say with elements of some set $b$. \nThis labeling satisfies the following: (a) $F$ is equivariant, that is for every $g\\in \\mathbb{Z}^d$ shifting the real labels with $g$ and applying $F$ is the same as applying $F$ and then shifting with $g$, (b) the value of $F(x)({\\bf 0})$ depends almost surely on  the real labels on some random but finite neighborhood of ${\\bf 0}$.\nSimilarly as for oblivious local algorithms one defines the random variable $R_F$ that encodes minimal such neighborhood.\nWe say that ffiid $F$ solves $\\Pi$ if $F(x)\\in X_{\\Pi}$ with probability $1$.\n\nThe difference between ffiid and oblivious local algorithms is just formal.\nNamely, given an oblivious local algorithm $\\mathcal{A}$, we can turn it into an ffiid $F_\\mathcal{A}$ simply by running it simultaneously in every vertex of $\\mathbb{Z}^d$.\nWith probability $1$ this produces a labeling of $\\mathbb{Z}^d$ and it is easy to see that both conditions from the definition of ffiid are satisfied.\nOn the other hand, if $F$ is ffiid, then we collect the finite neighborhoods that determine the value at ${\\bf 0}$ together with the decision about the label.\nThis defines a pair $\\mathcal{A}_F=(\\mathcal{C},f)$ that satisfies the definition of an oblivious local algorithm.\nIt follows that the coding radius $R_\\mathcal{A}$, resp $R_F$, is the same after the translations.\nWe chose to work with the language of distributed algorithms as we borrow the techniques mainly from that area. \n\n\\begin{remark}\nIf in (b), in the definition of ffiid, we only demand that the value at ${\\bf 0}$ depends measurably on the labeling, i.e., $F$ is allowed to see the labels at the whole grid, then we say that $F$ is a \\emph{factor of iid labels (fiid)}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{fiid}$, if there is a fiid $F$ that produces element of $X_\\Pi$ almost surely.\nIt is easy to see that $\\mathsf{TAIL}\\subseteq \\mathsf{fiid}$.\nWe remark that in the setting without inputs we have $\\mathsf{TAIL}=\\mathsf{fiid}$ on $\\mathbb{Z}$ and it is unknown, however, still plausible, whether this holds for $d>1$.\nFor general LCLs with inputs (where the definition of $\\mathsf{fiid}$ is extended appropriately) we have $\\mathsf{TAIL}\\not=\\mathsf{fiid}$ even for $d=1$, see \\cite{grebik_rozhon2021LCL_on_paths}.\n\\end{remark}\n\n\\begin{comment}\nGiven an LCL $\\Pi=(b,t,\\mathcal{P})$, we define $X_\\Pi\\subseteq b^{\\mathbb{Z}^d}$ to be the space of all $\\Pi$-colorings.\nWe say that an LCL $\\Pi$ is \\emph{satisfiable} if $\\emptyset\\not=X_\\Pi$, i.e., if there is a $\\Pi$-coloring of $\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)$.\nWe say that an LCL $\\Pi$ is \\emph{trivial} if there is $\\alpha\\in b$ such that the constant function $\\alpha$ is a $\\Pi$-coloring.\n\nThere is a canonical (right-)shift action $\\cdot$ of $\\mathbb{Z}^d$ on $b^{\\mathbb{Z}^d}$ (and in general on any $X^{\\mathbb{Z}^d}$, where $X$ is a set) that is defined as\n$$(g\\cdot x)(h)=x(h + g)$$\nfor every $g,h\\in \\mathbb{Z}^d$ and $x\\in b^{\\mathbb{Z}^d}$.\nIt is easy to see that $X_\\Pi$ is invariant under this action, i.e., $x\\in X_\\Pi$ implies $g\\cdot x\\in X_\\Pi$ for every $g\\in \\mathbb{Z}^d$.\nMoreover, since $b$ is a finite set, the product topology turns $b^\\Gamma$ into a compact metric space and it is easy to see $X_\\Pi$ is closed subspace.\nThe set $X_\\Pi$ together with the shift action is called a \\emph{sub-shift of finite type}.\n\nSuppose that $\\mathbb{Z}^d$ has an action $\\star_Y$ and $\\star_Z$ on spaces $Y$ and $Z$, respectively. \nWe say that a function $f:Y\\to Z$ is \\emph{equivariant} if $f(g\\star_Y x)=g\\star_Z f(x)$ for every $g\\in \\mathbb{Z}^d$ and $x\\in Y$.\n\nGiven an LCL $\\Pi$, our aim is to study complexity of finitary factors of iid that produce elements in $X_\\Pi$ almost surely.\nWe say that $\\Pi$ admits a \\emph{fiid solution} if there is a measurable equivariant map $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ that is defined on a $\\mu$-conull set, where $\\mu$ is the product measure on $[0,1]^{\\mathbb{Z}^d}$.\n\nWrite $\\varphi_F:[0,1]^{\\mathbb{Z}^d}\\to b$ for the composition of $F$ with the projection to ${\\bf 0}_d$-coordinate, i.e., \n$$x\\mapsto \\varphi_F(x)=F(x)({\\bf 0}_d).$$\nWe say that $F$ is a \\emph{finitary fiid (ffiid)} if for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the value of $\\varphi_F$ depends only on values of $x$ on some finite neighborhood of ${\\bf 0}_d$.\nThat is to say, there is a random variable $R_F$ that takes values in $\\mathbb{N}$, is defined $\\mu$-almost everywhere and satisfies for $\\mu$-almost every $x\\in [0,1]^{\\mathbb{Z}^d}$ the following: if $y\\in [0,1]^{\\mathbb{Z}^d}$ is such that\n$$x\\upharpoonright \\mathcal{B}_{\\operatorname{Cay}\\left(\\mathbb{Z}^d\\right)}\\left({\\bf 0}_d,R_\\varphi(x)\\right)=y\\upharpoonright \\mathcal{B}\\left({\\bf 0}_d,R_\\varphi(x)\\right),$$\nthen $\\varphi_F(x)=\\varphi_F(y)$.\n\\todo{nejak to zakoncit tim ze to nebudeme potrebovat}\n\\end{comment}\n\n\n\n\n\\begin{comment}\n\nWarm-up: Equivalence of $\\mathsf{LOCAL}(\\log^* n)$, $\\mathsf{TAIL}(\\mathrm{tower})$, $\\mathsf{CONTINUOUS}$\n\nRecall that $\\mathsf{LOCAL}(O(\\log^* n))$ denotes the class of satisfiable LCLs that satisfy (1) or (2) in Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nIn this section we show that every problem from $\\mathsf{LOCAL}(O(\\log^* n))$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nIn the proof we use the fact that $\\mathsf{LOCAL}=\\mathsf{C-LOCAL}(O(1))$, see~Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\n\nWe let $\\Delta_{s}:=|\\mathcal{B}({\\bf 0}_d,s)|$, where $s\\in \\mathbb{N}$.\nDenote as $\\mathcal{M}_s$ the subspace of $(\\Delta_s+1)^{\\mathbb{Z}^d}$ that consists of all proper vertex $(\\Delta_s+1)$-colorings of the $s$-power graph of $\\mathbb{Z}^d$.\nIt is easy to see that $\\mathcal{M}_s$ is nonempty, $\\mathbb{Z}^d$-invariant and compact.\nIn fact it is a subshift of finite type.\nThe following result holds for any graph of bounded degree, as was proved in \\cite{HolroydSchrammWilson2017FinitaryColoring}.\n\n\\begin{theorem}\\cite[Theorem~6]{HolroydSchrammWilson2017FinitaryColoring}\\label{th:ColorTower}\nLet $s\\in\\mathbb{N}$.\nThen there is an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to \\mathcal{M}_s$ and a constant $C_s>0$ such that\n$$\\mu(R_F>r)<\\frac{1}{tower(C_sr)}.$$\nIn particular, the LCL of coloring the $s$-power graph of $\\mathbb{Z}^d$ with $(\\Delta_s+1)$-many colors is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\n\n\\begin{proposition}\\label{pr:LOCALandCOLORING}\nLet $\\Pi=(b,t,\\mathcal{P})$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to \\mathcal{C}_{\\Pi}$ for some $s\\in \\mathbb{N}$.\n\\end{proposition}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nBy Theorem~\\ref{thm:classification_of_local_problems_on_grids}, we have that $\\Pi$ is in the class $\\mathsf{C-LOCAL}(O(1))$.\nFix $r,\\ell\\in \\mathbb{N}$ and $\\mathcal{A}$ that witnesses this fact.\nLet $x\\in \\mathcal{M}_{r+t}$.\nThen define\n$$\\varphi(x)(h):=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_x}(h,\\ell),h]).$$\nIt is clear that $\\varphi$ is well defined, because $\\ell<r-t$ and consequently $x\\upharpoonright \\mathcal{B}(h,\\ell)$ is injective.\nSince $\\ell$ is fixed we see that $\\varphi$ is continuous map, moreover, it is equivariant becuase\n\\begin{equation*}\n\\begin{split}\ng\\cdot \\varphi(x)(h) & = \\varphi(x)(h + g)=\\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_{ x}}(h+g,\\ell),h+g]) \\\\\n& = \\mathcal{A}([\\mathcal{B}_{\\mathbb{Z}^d_{(g\\cdot x)}}(h,\\ell),h])=\\varphi(g\\cdot x)(h)\n\\end{split}  \n\\end{equation*}\nholds for every $g,h\\in \\mathbb{Z}^d$ and $x\\in \\mathcal{M}_{r+t}$.\n\nIt remains to show that $\\varphi(x)\\in X_\\Pi$, or equivalently that $\\varphi(x)$ is a $\\Pi$-coloring, for every $x\\in \\mathcal{M}_{r+t}$.\nBy equivariance, it is enough to show that\n$$\\mathcal{P}\\left(\\left[\\mathcal{B}_{\\mathbb{Z}^d_{\\varphi(x)}}({\\bf 0}_d,t),{\\bf 0}_d\\right]\\right)=1.$$\nFix $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ that has diameter bigger than $r+t$ and an isomorphism $\\kappa:(\\mathcal{B}_G(v,r+t),v)\\to \\mathcal{B}({\\bf 0}_d,r+t),{\\bf 0}_d)$, e.g., $G$ is a $d$-dimensional torus that is large enough.\nExtend the pull-back of the $(\\Delta_{r+t}+1)$-coloring $x$ that is given via $\\kappa$ to a coloring $c$ that is defined on the whole graph $G$ in such a way that that it is still a coloring of the $(r+t)$-power graph, that is possible simply by a greedy algorithm.\nBy $\\ell<r-t$, we have $\\mathcal{A}(G_c)(\\kappa^{-1}(h))=\\varphi(x)(h)$, whenever $h\\in \\mathcal{B}({\\bf 0}_d,t)$.\nSince $\\mathcal{A}(G_c)$ is a $\\Pi$-coloring we conclude that $\\varphi(x)\\in X_\\Pi$.\n\\end{proof}\n\n\\begin{corollary}[$\\mathsf{LOCAL}\\subseteq \\mathsf{TAIL}(\\tow(\\Omega(r)))$]\n\\label{cor:local<tower}\nLet $\\Pi$ be a satisfiable LCL problem that is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nThen $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{corollary}\n\\begin{proof}\nBy Proposition~\\ref{pr:LOCALandCOLORING}, there is a continuous equivariant function $\\varphi:\\mathcal{M}_s\\to X_\\Pi$ for some $s\\in \\mathbb{N}$.\nComposing $\\varphi$ with $f$, that is obtained by Theorem~\\ref{th:ColorTower}, yields a measurable equivariant function $F=\\varphi\\circ f:[0,1]^\\Gamma\\to X_\\Pi$.\nLet $\\ell$ be the locality of $\\varphi$ and $C_s\\in \\mathbb{N}$ from Thereom~\\ref{th:ColorTower}.\nUsing the union bound, we have\n$$\\mu(R_F>r)<\\frac{\\Delta_{\\ell}}{tower(C_s(r-\\ell))}\\le \\frac{1}{tower(Dr)}$$\nfor $r\\in \\mathbb{N}$ large enough and some constant $D\\in (0,+\\infty)$.\n\\end{proof}\n\n\n\n\n\ngives that\n\n\n\n\n\n\n\n\n\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))$ $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log^* n))$\n\n\nFirst we show how to get randomized local algorithm from ffiid in a qualitative way.\nWe note that similar results holds for any bounded degree graph.\n\n\\begin{theorem}\\label{th:TailToRandom}\nLet $\\Pi$ be an LCL problem that is in the class $\\mathsf{TAIL}(f)$ for some $f:\\mathbb{N}\\to \\mathbb{N}$.\nSet $g(n)=\\min\\{m\\in \\mathbb{N}:f(m)\\ge n^2\\}$ and suppose that $g\\in o(n^{\\frac{1}{d}})$.\nThen $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$.\n\\end{theorem}\n\\begin{proof}\nFix an ffiid $F:[0,1]^{\\mathbb{Z}^d}\\to X_\\Pi$ such that\n$$\\mu(R_F>r)\\le \\frac{1}{f(r)}$$\nfor large enough $r\\in \\mathbb{N}$.\nDenote as $\\varphi_F$ the finitary rule and as $\\operatorname{dom}(\\varphi_F)$ its domain.\nThat is $\\alpha\\in \\operatorname{dom}(\\varphi_F)$ if $\\alpha\\in [0,1]^{V(\\mathcal{B}({\\bf 0}_d,s))}$ for some $s\\in \\mathbb{N}$ and satisfies\n$$F(x)=F(y)$$\nfor every extensions $\\alpha\\sqsubseteq x,y\\in [0,1]^{\\mathbb{Z}^d}$.\nIn that case we set $\\varphi_F(\\alpha)=F(x)$ for any such extension $\\alpha\\sqsubseteq x$.\n\nDefine $\\ell_n:=g(n)$.\nLet $(G,v)\\in \\operatorname{\\bf FG}_{\\bullet}^{S}$ be of a radius at most $\\ell_n$.\nConsider any map $\\ell:V(G)\\to [0,1]$.\nDefine\n$$\\mathbb{A}_n([G_l,v])=\\varphi_F(\\ell)$$\nin the case when $\\ell\\in \\operatorname{dom}(\\varphi_F)$.\nOtherwise let $\\mathbb{A}_n([G_l,v])$ to be any element in $b$.\nIt is easy to verify that this defines a randomized local algorithm.\n\nLet $G\\in \\operatorname{\\bf FG}^{S}$ be such that $|V(G)|=n$ and\n$$[\\mathcal{B}_{G}(v,\\ell_n+t),v]=\\left[\\mathcal{B}({\\bf 0}_d,\\ell_n+t),{\\bf 0}_d\\right]$$\nholds for every $v\\in V(G)$.\nNote that by the assumption on $g$ such graphs do exist, e.g., large enough torus.\nFor every $v\\in V(G)$ define the event\n$$A_v=\\left\\{\\ell\\in [0,1]^{V(G)}:\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n)\\not \\in \\operatorname{dom}(\\varphi_F)\\right\\}.$$\nNote that if $\\ell\\not \\in \\bigcup_{v\\in V(G)} A_v$, then $\\mathbb{A}(G_l)$ is a $\\Pi$-coloring.\nTo see this pick $v\\in V(G)$ and any extension $x\\in [0,1]^{\\mathbb{Z}^d}$ of $\\ell\\upharpoonright \\mathcal{B}_G(v,\\ell_n+t)$ that is in the domain of $F$, where we interpret $v$ as the origin ${\\bf 0}_d$.\nThen $F(x)\\in X_\\Pi$ and\n$$F(x)\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\simeq\\mathbb{A}(G_l)\\upharpoonright \\mathcal{B}_G(v,t).$$\nConsequently, we have $\\mathcal{P}([\\mathcal{B}_{G_{\\mathbb{A}(G_l)}}(v,t),v])=1$.\n\nLet $v\\in V(G)$.\nWe have $\\ell\\in A_v$ if and only if $R_F(x)>\\ell_n$ for every $x\\in [0,1]^\\Gamma$ that extends $\\ell$, i.e., every $x\\in [0,1]^{\\mathbb{Z}^d}$ that satisfies\n$$[B_{G_l}(v,\\ell_n),v]=[B_{\\mathbb{Z}^d_x}({\\bf 0}_d,\\ell_n),{\\bf 0}_d].$$ \nThis implies that\n$$\\lambda^{\\oplus n}(A_v)=\\mu\\left(\\left\\{x\\in [0,1]^{\\mathbb{Z}^d}:R_F(x)>\\ell_n\\right\\}\\right)\\le \\frac{1}{f(\\ell_n)}\\le \\frac{1}{n^2}.$$\nA union bound then shows that $\\lambda^{\\oplus n}\\left(\\bigcup_{v\\in V(G)}A_v\\right)\\le \\frac{1}{n}$ and the proof is finished.\n\\end{proof}\n\n\n\n\\begin{corollary}\\label{cor:tower=local}\n$\\mathsf{TAIL}(\\tow(\\Omega(r)))=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{corollary}\n\\begin{proof}\nIt follows from Corollary~\\ref{cor:local<tower} that $\\mathsf{LOCAL}\\subseteq \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nBy Theorem~\\ref{th:TailToRandom} we have that $\\Pi$ is in the class $\\mathsf{R-LOCAL}(O(g(n)))$ where\n$$g(n)=\\min\\{m\\in \\mathbb{N}:tower(Cm)\\ge n^2\\}$$\nfor some $C\\in\\mathbb{N}$.\nSince $g$ is in $O(\\log^*(n))$, we have that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ by Theorem~\\ref{thm:classification_of_local_problems_on_grids}.\nThis finishes the proof.\n\\end{proof}\n\n\\end{comment}\n\n\n\\section{Speed-up Theorems for $\\mathsf{TAIL}$}\n\\label{sec:speedup_theorems}\n\nRecall the classification \\cref{thm:classification_of_local_problems_on_grids}. We are interested in building the analogous classification for $\\mathsf{TAIL}$ complexities. \nIn this section we first show that already our basic translations from \\cref{subsec:preliminaries_tail} give several interesting results and in \\cref{subsec:tail_d-1_speedup} we prove a nontrivial speedup result.\n\n\\subsection{Warm-up Results}\n\\label{subsec:speedup_warmup}\n\nFirst, we observe that already \\cref{thm:classification_of_local_problems_on_grids}, together with \\cref{lem:tail->local} itself has the following implications: \n\n\\begin{corollary}\n\\label{cor:block_factor}\n$\\Pi$ can be solved by a block factor (recall \\cref{def:tail}) if and only if $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \n\\end{corollary}\n\\begin{proof}\nThis follows from \\cref{thm:classification_of_local_problems_on_grids,rem:chang_pettie_decomposition}. \n\\end{proof}\n\n\\begin{corollary}\n\\label{cor:BelowTOWERtrivial}\nIf $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\omega(r)))$, then $\\Pi \\in \\mathsf{LOCAL}(O(1))$ and if $\\Sigma_{in} = \\emptyset$ then $\\Pi$ is trivial. \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in\\mathsf{LOCAL}(o(\\log^* n))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies the rest. \n\\end{proof}\n\nIn particular, Corollary~\\ref{cor:BelowTOWERtrivial} answers in negative Problem~(ii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a nontrivial local problem (without input labels) with super-tower function tail. \n\nThe second corollary is the following:\n\n\\begin{corollary}\n\\label{cor:local=tower}\n$\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ if and only if $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$. \n\\end{corollary}\n\\begin{proof}\nIf $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$, \\cref{lem:tail->local} yields $\\Pi \\in \\mathsf{RLOCAL}(O(\\log^* n))$. The other direction (special case of \\cref{thm:tail=uniform}) follows from \\cref{rem:chang_pettie_decomposition} that essentially states that it suffices to solve the independent set problem in the power graph. But this follows from a version of Linial's algorithm from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\\end{proof}\n\nFinally, we get the following. \n\\begin{corollary}\n\\label{cor:Below_d_local}\nIf $\\Pi \\in \\mathsf{TAIL}(\\omega(r^{2d}))$, then $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$.  \n\\end{corollary}\n\\begin{proof}\n\\cref{lem:tail->local} gives that $\\Pi \\in \\mathsf{LOCAL}(o(n^{1\/d}))$ and \\cref{thm:classification_of_local_problems_on_grids} then implies that $\\Pi \\in \\mathsf{LOCAL}(O(\\log^* n))$ which is equivalent to $\\Pi \\in \\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r)))$ by \\cref{cor:local=tower}. \n\\end{proof}\n\nCorollary~\\ref{cor:Below_d_local} answers in negative Problem~(iii) in \\cite{HolroydSchrammWilson2017FinitaryColoring} that asked about the existence of a problem with all its moments finite but with sub-tower function decay.   Recall that \\cref{fact:moment_implies_tail} implies that if the $2d$-th moment is finite, then the tail is $\\omega(r^{2d})$ and, hence, the problem can in fact be solved with a tower tail decay.  \nHowever, the $2d$ is not the right dependency and in the next section we prove a tight version of this result.\n\\subsection{Tight Grid Speedup for $\\mathsf{TAIL}$}\n\\label{subsec:tail_d-1_speedup}\n\nWhile \\cref{cor:Below_d_local} already answers the Problem of \\cite{HolroydSchrammWilson2017FinitaryColoring}, we now improve the speedup from the tail of $\\omega(r^{d-1})$. This result is tight (up to $o(1)$ in the exponent, see the discussion in \\cref{subsec:toast_construction} and \\cref{cor:toast_lower_bound}) and it is the first nontrivial difference between the $\\mathsf{TAIL}$ and $\\mathsf{LOCAL}$ model that we encountered so far. \n\n\n\n\n\\begin{theorem}\n\\label{thm:grid_speedup}\nLet $d,t\\in \\mathbb{N}$.\nFor $d \\ge 2$ there is $C_{d,t}>0$ such that if an LCL $\\Pi$ is in $\\mathsf{TAIL}(C_{d,t}r^{d-1})$, then $\\Pi \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\nSimilarly, for $d=1$ we have that a local coloring problem $\\Pi$ admits a solution by an oblivious random algorithm (it is in $\\mathsf{TAIL}(f(r))$ for any function $f = \\omega(1)$) if and only if $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\n\\end{theorem}\nNote that \\cref{fact:moment_implies_tail} hence shows that ${\\bf M}(\\Pi) > d-1$ already implies that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$. \n\nThe proof is a simple adaptation of Lov\u00e1sz local lemma speedup results such as  \\cite{chang2017time_hierarchy,fischer2017sublogarithmic}. \nThe proof goes as follows. Split the grid into boxes of side length roughly $r_0$ for some constant $r_0$. This can be done with $O(\\log^* (n))$ local complexity (or, equivalently, with tower tail decay). \n\nNext, consider a box $B$ of side length $r_0$ and volume $r_0^d$. In classical speedup argument we would like to argue that we can deterministically set the randomness in nodes of $B$ and its immediate neighborhood so that we can simulate the original oblivious local algorithm $\\mathcal{A}$ with that randomness and it finishes in at most $r_0$ steps. \nIn fact, ``all nodes of $B$ finish after at most $r_0$ steps'' is an event that happens with constant probability if the tail decay of $\\mathcal{A}$ is $r^d$, since then each node fails with probability $1\/r_0^d$ and there are $r_0^d$ nodes in $B$. \n\nHowever, we observe that it is enough to consider only the boundary of $B$ with $O(r_0^{d-1})$ nodes and argue as above only for the boundary nodes, hence the speedup from $r^{d-1}$ tail decay. \nThe reason for this is that although the coding radius of $\\mathcal{A}$ inside $B$ may be very large, just the fact that there \\emph{exists} a way of filling in the solution to the inside of $B$, given the solution on its boundary, is enough for our purposes. The new algorithm simply simulates $\\mathcal{A}$ only on the boundaries and then fills in the rest. \n\n\\begin{proof}\n\n\nWe will prove the result for $d \\ge 2$, the case $d=1$ is completely analogous and, in fact, simpler. \nSet $C=C_{d,t}:=d\\cdot t\\cdot(2\\Delta)^{2\\Delta}$, where $\\Delta=20^d$.\nSuppose that there is an oblivious local algorithm $\\mathcal{A}$ that explores the neighborhood of a given vertex $u$ and outputs a color of $u$ after seeing a radius $R$ such that \\begin{equation}\\label{eq:tail}\n\\P(R > r)\\le \\frac{1}{C r^{d-1}}\n\\end{equation}\nfor $r\\in \\mathbb{N}$ large enough.\nLet $r_0\\in \\mathbb{N}$ be large enough such that \\eqref{eq:tail} is satisfied.\nWrite $\\Box_{r_0}$ for the local coloring problem of tiling the input graph (which is either a large enough torus or an infinite grid $\\mathbb{Z}^d$) into boxes with length sizes in the set $\\{r_0,r_0+1\\}$ such that, additionally, each box $B$ is colored with a color from $[\\Delta+1]$ which is unique among all boxes $B'$ with distance at most $3(r_0+t)$. \n\nThe problem of tiling with boxes of side lengths in $\\{r_0, r_0+1\\}$ can be solved in $O(\\log^* n)$ local complexity \\cite[Theorem~3.1]{GaoJackson} but we only sketch the reason here: Find a maximal independent set of a large enough power of the input graph $\\mathbb{Z}^d$. Then, each node $u$ in the MIS collects a Voronoi cell of all non-MIS nodes such that $u$ is their closest MIS node. As the shape of the Voronoi cell of $u$ depends only on other MIS vertices in a constant distance from $u$, and their number can be bounded as a function of $d$ (see the ``volume'' argument given next), one can discretize each Voronoi cell into a number of large enough rectangular boxes. Finally, every rectangular box with all side lengths of length at least $r^2$ can be decomposed into rectangular boxes with all side lengths either $r$ or $r+1$. \n\nNote that in additional $O(\\log^* n)$ local steps, the boxes can then be colored with $(\\Delta+1)$-many colors in such a way that boxes with distance at most $3(r_0+t)$ must have different colors.\nTo see this, we need to argue that there are at most $\\Delta$ many boxes at distance at most $3(r_0+t)$ from a given box $B$.\nThis holds because if $B,B'$ are two boxes and $d(B,B')\\le 3(r_0+t)$, then $B'$ is contained in the cube centered around the center of $B$ of side length $10(r_0+t)$.\nThen a volume argument shows that there are at most $\\Delta\\ge \\frac{(10(r_0+t))^d}{r_0^d}$ many such boxes $B'$ (when $B$ is fixed).\n\nWe show now that there is a local algorithm $\\mathcal{A'}$ that solves $\\Pi$ when given the input graph $G$ together with a solution of $\\Box_{r_0}$ as an input.\nThis clearly implies that $\\Pi$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\n\n\nLet $\\{B_i\\}_{i\\in I}$ be the input solution of $\\Box_{r_0}$, together with the coloring $B_i\\mapsto c(B_i)\\in [\\Delta+1]$.\nWrite $V(B)$ for the vertices of $B$ and $\\partial B$ for the vertices of $B$ that have distance at most $t$ from the boundary of $B$.\nWe have $|\\partial B|< d\\cdot t \\cdot (r_0+1)^{d-1}$.\nDefine $N(B)$ to be the set of rectangles of distance at most $(r_0+t)$ from $B$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width= .9\\textwidth]{fig\/tail_LLL.eps}\n    \\caption{The picture shows one block $B$ with a red boundary $\\partial B$ of length $t$. Boxes in $N(B)$, that is with distance at most $(r_0+t)$ from $B$ are dark grey. }\n    \\label{fig:tail_LLL}\n\\end{figure}\n\nWe are now going to describe how $\\mathcal{A}'$ iterates over the boxes in the order of their colors and for each node $u$ in the box $B$ it carefully fixes a random string $\\mathfrak{r}(u)$. \nThis is done in such a way that after fixing the randomness in all nodes, we can apply the original algorithm $\\mathcal{A}$ to each vertex in $\\partial B$ for each $B$ with the randomness given by $\\mathfrak{r}$ and the local algorithm at $u \\in \\partial B$ will not need to see more than its $r$-hop neighborhood. \n\nGiven a partial function $\\mathfrak{r_\\ell}:V(G)\\to [0,1]$ corresponding to nodes such that their randomness was already fixed, every box $B$ can compute a probability $\\mathbb{P}(\\fE_{\\ell}(B))$, where $\\fE_{\\ell}(B)$ is the (bad) event that there is a vertex in $\\partial B$ that needs to look further than $r_0$ to apply the local algorithm $\\mathcal{A}$.\nNote that for computing $\\mathbb{P}(\\fE_{\\ell}(B))$ it is enough to know the $r_0$-neighborhood of $B$.\n\nThe algorithm $\\mathcal{A}$ has $\\Delta+1$ rounds.\nIn the $0$-th round, we set $\\mathfrak{r}_0$ to be the empty function.\nNote that the probability that a vertex needs to look further than $r_0$ in the algorithm $\\mathcal{A}$ is smaller than $\\frac{1}{C r_0^{d-1}}$ by \\eqref{eq:tail}.\nThe union bound gives $\\mathbb{P}(\\fE_0(B))<\\frac{1}{(2\\Delta)^{2\\Delta}}$ for every $B$.\n\nIn each step $\\ell\\in [\\Delta+1]$ we define $\\mathfrak{r}_B:V(B)\\to [0,1]$ for every $B$ such that $c(B)=\\ell$ and put $\\mathfrak{r}_l$ to be the union of all $\\mathfrak{r}_D$ that have been defined so far, i.e., for $D$ such that $c(D)\\le \\ell$.\nWe make sure that the following is satisfied for every $B$:\n\\begin{enumerate}\n\t\\item [(a)] the assignment $\\mathfrak{r}_B$ depends only on $2(r_0+t)$-neighborhood of $B$ (together with previously defined $\\mathfrak{r}_{\\ell-1}$ on that neighborhood) whenever $c(B)=\\ell$,\n\t\\item [(b)] $\\mathbb{P}(\\fE_l(B))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$,\n\t\\item [(c)] restriction of $\\mathfrak{r}_l$ to $N(B)$ can be extend to the whole $\\mathbb{Z}^d$ such that $\\mathcal{A}$ is defined at every vertex, where we view $N(B)$ embedded into $\\mathbb{Z}^d$.\n\\end{enumerate}\nWe show that this gives the required algorithm.\nIt is easy to see that by (a) this defines a local construction that runs in constantly many rounds and produces a function $\\mathfrak{r}:=\\mathfrak{r}_{\\Delta+1}:V(G)\\to [0,1]$.\nBy (b) every vertex in $\\partial B$ for some $B$ looks into its $r_0$ neighborhood and use $\\mathcal{A}$ to choose its color according to $\\mathfrak{r}$.\nFinally, by (c) it is possible to assign colors in $B\\setminus \\partial B$ for every $B$ to satisfy the condition of $\\Pi$ within each $B$.\nPicking minimal such solution for each $B\\setminus \\partial B$ finishes the description of the algorithm.\n\nWe show how to proceed from $0\\le \\ell<\\Delta+1$ to $\\ell+1$.\nSuppose that $\\mathfrak{r}_l$ satisfies the conditions above.\nPick $B$ such that $c(B)=\\ell+1$.\nNote that $\\mathfrak{r}_B$ that we want to find can influence rectangles $D\\in N(B)$ and these are influenced by the function $\\mathfrak{r}_l$ at distance at most $r_0$ from them.\nAlso given any $D$ there is at most one $B\\in N(D)$ such that $c(B)=\\ell+1$. \nWe make the decision $\\mathfrak{r}_B$ based on $2r_0$ neighborhood around $B$ and the restriction of $\\mathfrak{r}_l$ there, this guarantees that condition (a) is satisfied.\n\nFix $D\\in N(B)$ and pick $\\mathfrak{r}_B$ uniformly at random.\nWrite $\\mathfrak{S}_{B,D}$ for the random variable that computes the probability of $\\fE_{\\mathfrak{r}_B\\cup \\mathfrak{r}_l}(D)$.\nWe have $\\mathbb{E}(\\mathfrak{S}_{B,D})=\\mathbb{P}(\\fE_l(D))<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}$.\nBy Markov inequality we have\n$$\\mathbb{P}\\left(\\mathfrak{S}_{B,D}\\ge 2\\Delta \\frac{1}{(2\\Delta)^{2\\Delta-\\ell}}\\right)<\\frac{1}{2\\Delta}.$$\nSince $|N(B)|\\le \\Delta$ we have that with non-zero probability $\\mathfrak{r}_B$ satisfies $\\mathfrak{S}_{B,D}<\\frac{1}{(2\\Delta)^{2\\Delta-\\ell-1}}$ for every $D\\in N(B)$.\nSince $\\mathcal{A}$ is defined almost everywhere we find $\\mathfrak{r}_B$ that satisfies (b) and (c) above.\nThis finishes the proof. \n\\end{proof}\n\nAs we already mentioned, the proof is in fact an instance of the famous Lov\u00e1sz Local Lemma problem. This problem is extensively studied in the distributed computing literature \\cite{chang2016exp_separation, fischer2017sublogarithmic, brandt_maus_uitto2019tightLLL, brandt_grunau_rozhon2020tightLLL} as well as descriptive combinatorics \\cite{KunLLL,OLegLLL,MarksLLL,Bernshteyn2021LLL}.\nOur usage corresponds to a simple LLL algorithm for the so-called exponential criteria setup (see \\cite{fischer2017sublogarithmic} and \\cite{Bernshteyn2021local=cont}). \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\\section{$\\TOAST$: General Results}\n\\label{sec:Toast}\n\nIn this section we introduce the toast construction. This construction is very standard in descriptive combinatorics \\cite{FirstToast,GJKS,Circle} and the theory of random processes (there it is often called \\emph{the hierarchy}) \\cite{HolroydSchrammWilson2017FinitaryColoring,ArankaFeriLaci,Spinka}.\n\nWhile the construction itself is well-known, in this paper we study it as a class of problems by itself. An analogy to our work is perhaps the paper of \\cite{ghaffari2018derandomizing} that defines the class $\\mathsf{S-LOCAL}$ and shows how to derandomize it.\n\n\n\nInformally, the basic toast construction works as follows. There is an adversary that has access to the (infinite) input grid $\\mathbb{Z}^d$. The adversary is sequentially choosing some connecting pieces of the grid and gives them to our algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ needs to label all vertices in the piece $D$ once it is given $D$. \nNote that $\\mathcal{A}$ does not know where exactly $D$ lies in $\\mathbb{Z}^d$ and its decision cannot depend on the precise position of $D$ in the input graph. In general, the piece $D$ can be a superset of some smaller pieces that were already colored by $\\mathcal{A}$. In this case, $\\mathcal{A}$ needs to extend their coloring to the whole $D$. This game is played for countable number of steps until the whole grid is colored. The algorithm solves $\\Pi$ if it never happens during the procedure that it cannot extend the labeling to $D$ in a valid manner. \n\n\n\n\n\nThe construction sketched above is what we call a spontaneous deterministic toast or just $\\TOAST$ algorithm. \nIn the next subsection we give a formal definition of $\\TOAST$ algorithms and we follow by proving relations between $\\TOAST$ algorithms and $\\mathsf{TAIL}$ complexities. \n\n\n\\subsection{Formal Definition of $\\TOAST$ classes}\n\\label{subsec:toast_definition}\nWe now give a formal definition of our four $\\TOAST$ classes. First we define the hereditary structure on which the $\\TOAST$ algorithm is interpreted. This is also called a toast (we use lowercase letters) and is parameterized by the minimum distance of boundaries of different pieces of the toast.  \nThis structure also corresponds to the adversary that asks our algorithm to extend a partial solution to $\\Pi$ to bigger and bigger pieces used in the informal introduction. \n\n\\begin{definition}[$q$-toast on $\\mathbb{Z}^d$]\nFor a $q >0$, a \\emph{$q$-toast on $\\mathbb{Z}^d$} is a collection $\\mathcal{D}$ of finite connected subsets of $\\mathbb{Z}^d$ that we call \\emph{pieces} such that\n\\begin{itemize}\n    \\item for every $g,h\\in \\mathbb{Z}^d$ there is a piece $D\\in\\mathcal{D}$ such that $g,h\\in D$,\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$, where $\\partial D$ is a boundary of a set $D$.\n\\end{itemize}\n\\end{definition}\n\nGiven a $q$-toast $\\mathcal{D}$, the \\emph{rank} of its piece $D \\in \\mathcal{D}$ is the level in the toast hierarchy that $D$ belongs to in $\\mathcal{D}$. Formally, we define a \\emph{rank function $\\mathfrak{r}_\\mathcal{D}$ for the toast $\\mathcal{D}$} as $\\mathfrak{r}_\\mathcal{D}:\\mathcal{D}\\to \\mathbb{N}$, inductively as\n$$\\mathfrak{r}_\\mathcal{D}(D)=\\max\\{\\mathfrak{r}_{\\mathcal{D}}(E)+1:E\\subseteq D, \\ E\\in \\mathcal{D}\\}$$ \nfor every $D\\in \\mathcal{D}$.\n\nNext, $\\TOAST$ algorithms are described in the following language of extending functions. \n\n\\begin{definition}[$\\TOAST$ algorithm]\n\\label{def:extending_function}\nWe say that a function $\\mathcal{A}$ is a \\emph{$\\TOAST$ algorithm} for a local problem $\\Pi = (\\Sigma_{in},\\Sigma_{out}, t, \\mathcal{P})$ if \n\\begin{enumerate}\n    \\item [(a)] the domain of $\\mathcal{A}$ consists of pairs $(A, c)$, where $A$ is a finite connected $\\Sigma_{in}$-labeled subset of $\\mathbb{Z}^d$ and $c$ is a partial map from $A$ to $\\Sigma_{out}$,\n    \\item [(b)] $\\mathcal{A}(A,c)$ is a partial coloring from $A$ to $\\Sigma_{out}$ that extends $c$, i.e., $\\mathcal{A}(A,c)\\upharpoonright \\operatorname{dom}(c)=c$, and that depends only on the isomorphism type of $(A,c)$ (not on the position in $\\mathbb{Z}^d$).  \n   \n\\end{enumerate}\nA \\emph{randomized $\\TOAST$ algorithm} is moreover given as an additional input a labeling of $A$ with real numbers from $[0,1]$ (corresponding to a stream of random bits).  \nFinally, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c)) =A$ for every pair $(A,c)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous (randomized) $\\TOAST$ algorithm}.\n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/3-col.eps}\n    \\caption{A $q$-toast and an example of an extending function for $3$-coloring. Observe that our extending function does not give a correct algorithm for $3$-coloring since the piece on the right is contained in an even bigger piece and the $3$-coloring constructed on the right cannot be extended to the node marked with an arrow}\n    \\label{fig:toast_example}\n\\end{figure}\n\n\nIt is easy to see that a $\\TOAST$ algorithm $\\mathcal{A}$ can be used to color ($\\Sigma_{in}$-labeled) $\\mathbb{Z}^d$ inductively on the rank $\\mathfrak{r}_\\mathcal{D}$ of any $q$-toast $\\mathcal{D}$ for every $q\\in \\mathbb{N}$. \nNamely, define $\\mathcal{A}(\\mathcal{D})$ to be the partial coloring of $\\mathbb{Z}^d$ that is the union of $\\mathcal{A}(D)$, where $D\\in \\mathcal{D}$ and $\\mathcal{A}(D)$ is defined inductively on $\\mathfrak{r}_\\mathcal{D}$.\nFirst put $\\mathcal{A}(D):=\\mathcal{A}(D,\\emptyset)$ whenever $\\mathfrak{r}_\\mathcal{D}(D)=0$.\nHaving defined $\\mathcal{A}(D)$ for every $D\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D)=k$, we define\n$$\\mathcal{A}(D'):=\\mathcal{A}\\left(\\bigcup\\{E\\in \\mathcal{D}:E\\subseteq D'\\},\\bigcup\\{\\mathcal{A}(E):E\\subseteq D', \\ E\\in \\mathcal{D}\\}\\right).$$\nfor every $D'\\in \\mathcal{D}$ such that $\\mathfrak{r}_\\mathcal{D}(D')=k+1$. That is, we let $\\mathcal{A}$ to extend the partial solution to $\\Pi$ to bigger and bigger pieces of the toast $\\mathcal{D}$. \nObserve that if $\\mathcal{A}$ is a spontaneous $\\TOAST$ function, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D}))=\\mathbb{Z}^d$ for every $q$-toast $\\mathcal{D}$ and every $q\\in \\mathbb{N}$.\n\n\\paragraph{$\\TOAST$ Algorithms}\nWe are now ready to define the four types of toast constructions that we are going to use (cf. \\cref{fig:big_picture_grids}). \n\n\\begin{definition}[$\\LTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\LTOAST$ if there is $q>0$ and an $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\nMoreover, we say that $\\Pi$ is in the class $\\TOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\n\\end{definition}\n\nBefore we define a randomized version of these notions, we make several comments.\nFirst note that $\\TOAST\\subseteq \\LTOAST$.\nBy (c), if we construct a ($q$-)toast using an oblivious local algorithm, then we get instantly that every problem from the class $\\TOAST$ is in the class $\\mathsf{TAIL}$, see Section~\\ref{sec:Toast}.\nSimilar reasoning applies to the class $\\mathsf{BOREL}$, see Section~\\ref{sec:OtherClasses}.\nWe remark that we are not aware of any LCL that is not in the class $\\TOAST$.\n\n\\begin{comment}\n\\begin{definition}[Randomized toast algorithm]\n\\label{def:extending_function}\nLet $b$ be a set.\nWe say that a function $\\mathcal{A}$ is a randomized $\\TOAST$ algorithm if \n\\begin{enumerate}\n    \\item [(a)] the domain of $\\mathcal{A}$ consists of triplets $(A, c, \\ell)$, where $A$ is a finite connected subset of $\\mathbb{Z}^d$, $c$ is a partial coloring of $A$ and $\\ell:A\\to [0,1]$. \n    \\item [(b)] $\\mathcal{A}(A,c,\\ell)$ is a partial coloring of $A$ that extends $c$, i.e., $\\mathcal{A}(A,c,\\ell)\\upharpoonright \\operatorname{dom}(c)=c$.  \n    \\item [(c)] the function $\\mathcal{A}$ is invariant with respect to the shift action of $\\mathbb{Z}^d$ on $[0,1]^{\\mathbb{Z}^d)}$. \n\\end{enumerate}\nMoreover, if $\\mathcal{A}$ has the property that $\\operatorname{dom}(\\mathcal{A}(A,c,\\ell)) =A$ for every triplet $(A,c,\\ell)$, then we say that $\\mathcal{A}$ is a \\emph{spontaneous randomized toast algorithm}.\n\\end{definition}\n\nSimilarly as above, we define $\\mathcal{A}(\\mathcal{D},\\ell)$ for every $q$-toast $\\mathcal{D}$, $q\\in \\mathbb{N}$ and $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nObserve that if $\\mathcal{A}$ is a spontaneous randomized toast algorithm, then $\\operatorname{dom}(\\mathcal{A}(\\mathcal{D},\\ell))=\\mathbb{Z}^d$.\n\\end{comment}\n\n\\begin{definition}[$\\RLTOAST$]\nLet $\\Pi$ be an LCL.\nWe say that $\\Pi$ is in the class $\\RLTOAST$ if there is $q>0$ and a randomized $\\TOAST$ algorithm $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$ with probability $1$.\nThat is, after $\\Sigma_{in}$-labeling is fixed, with probability $1$ an assignment of real numbers from $[0,1]$ to $\\mathbb{Z}^d$ successfully produces a $\\Pi$-coloring against all possible toasts $\\mathcal{D}$ when $\\mathcal{A}$ is applied.\n\nMoreover, we say that $\\Pi$ is in the class $\\RTOAST$ if there is $q\\in \\mathbb{N}$ and a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$ as above.\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Construction of $\\TOAST$}\n\\label{subsec:toast_construction_in_tail_sketch}\n\n\nThe class $\\TOAST$ (or $\\RTOAST$) is especially useful since in \\cref{sec:SpecificProblems} we discuss how many local problems that are not in class $O(\\log^* n)$ but on the other hand are not ``clearly hard'' like vertex $2$-coloring are in the class $\\TOAST$. Importantly, in \\cref{subsec:toast_construction}  we show how one can construct the toast decomposition with an oblivious local algorithm in $\\mathsf{TAIL}(r^{1 - O(1\/\\sqrt{\\log r})})$, which yields the following theorem.\n\n\\begin{restatable}{theorem}{toastconstruction}\n\\label{thm:toast_in_tail}\nLet $\\Pi$ be an LCL that is in the class $\\TOAST$.\nThen \\[\n\\Pi \\in \\mathsf{TAIL}\\left(\\frac{r}{2^{O(\\sqrt{\\log r})}}\\right) = \\mathsf{TAIL}\\left(r^{1-o(1)}\\right)\n\\]\nand, hence, ${\\bf M}(\\Pi)\\ge 1$.\n\\end{restatable}\n\nWe observe in \\cref{cor:toast_lower_bound} that faster then linear tail decay is not possible. \n\n\\paragraph{Proof Sketch}\n\nHere we provide an intuition for the proof that is otherwise technical. We first check that maximal independent sets of power graphs of the grid $G = \\mathbb{Z}^d$ can be constructed effectively, e.g. because of the MIS algorithm of Ghaffari \\cite{ghaffari2016MIS}. The slowdown by this construction can be neglected in this high-level overview.  \n\nWe then take a sequence $r_k = 2^{k^2}$ and construct sparser and sparser MIS in graphs $G^{r_k}, k = 1, 2, \\dots$. After constructing MIS of $G^{r_k}$, we construct Voronoi cells induced by this MIS. These Voronoi cells $V_1, V_2, \\dots$ will be, roughly, the pieces of rank $k$, but we need to ensure that all the boundaries are at least $q$ hops apart. \n\nTo do so, we first shrink each Voronoi cells by $t_k = O(\\sum_{j < k} r_j) \\ll r_k$ hops (grey boundary area in \\cref{fig:toast_construction}) and get new, smaller, cells $C_1, C_2, \\dots$. The final sequence of pieces $H_1, H_2, \\dots$ is then defined inductively, namely to define $H_i$ we start with $C_1$, then add all Voronoi cells from $k-1$-th iteration that $C_1$ intersects, then add all Voronoi cells from $k_2$-th iteration that the new, bigger, cluster intersects, \\dots\nThe definition of $t_k$ ensures that the final pieces constructed after all $k$ iterations are sufficiently far from each other, and also compatible with the smaller rank pieces. In fact, we always have $H_i \\subseteq V_i$. \n\nIt remains to argue that after $k$ steps of this process, only a small fraction of nodes remains uncovered. This is done roughly as follows. The vertices from $V_i \\setminus C_i$ that are not covered by $C_i$s are roughly $t_k \/ r_k \\approx A r_{k-1} \/ r_k$ fraction of vertices where $A$ is some (large) constant. By induction, $1 - A r_{k-2} \/ r_{k-1}$ fraction of those vertices were covered in step $k-1$ and so on. The fraction of remaining vertices is then $\\frac{A r_{k-1}}{r_{k}} \\cdot \\frac{A r_{k-2}}{r_{k-1}} \\dots \\frac{A }{r_1} = \\frac{A^k}{r_k}$. This fraction is asymptotically almost equal to $1\/r_k$, which is the desired tail behavior. However, the additional term $A^k$ is the reason why the final tail decay for $r$ such that $r_k \\le r \\le r_{k+1} = 2^{(k+1)^2}$ is bounded by $2^{O(k)}\/2^{k^2} \\le \\left( \\frac{1}{r} \\right)^{\\frac{k^2 - O(k)}{(k+1)^2}} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/k)} = \\left( \\frac{1}{r} \\right)^{1 - O(1\/\\sqrt{\\log r})}  $, as needed.  \n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/toast_construction.eps}\n    \\caption{A step during the toast construction. Voronoi cell $V$ is first shrinked to $C$ and we get the final piece $H$ by taking the union of smaller-level Voronoi cells intersecting $C$ (in this picture, there is only one lower level but the construction is recursive in general). }\n    \\label{fig:toast_construction}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\subsection{$\\RTOAST = \\TOAST$}\n\\label{subsec:rtoast=toast}\n\n\n\nIn this section we show that randomness does not help spontaneous toast algorithms.\n\n\\begin{restatable}{theorem}{toastrtoast}\\label{thm:derandomOfToast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RTOAST$ if and only if $\\Pi$ is in the class $\\TOAST$.\n\\end{restatable}\n\nThe proof goes as follows. We start with a random toast algorithm $\\mathcal{A}$ and make it deterministic as follows. When the deterministic toast algorithm gets a piece $D$ of rank $1$, it considers all possible outputs that the algorithm $\\mathcal{A}$ could produce given the input random string and it uses any one of them that $\\mathcal{A}$ would produce with nonzero probability. \nBy induction, whenever $\\mathcal{A}'$ later gets a piece $D$ containing some pieces of smaller rank with a  partial solution that $\\mathcal{A}'$ already produced, the original algorithm $\\mathcal{A}$ could have produced the same partial solution with nonzero probability, hence $\\mathcal{A}$ can extend the coloring in a valid manner and $\\mathcal{A}'$ deterministically choosing any such valid extension that $\\mathcal{A}$ would use with nonzero probability. \nThis shows that $\\mathcal{A}'$ also solves $\\Pi$. \nA formal proof follows. \n\n\\begin{proof}\n\nIt is easy to see that $\\TOAST\\subseteq \\RTOAST$.\nSuppose that $\\Pi$ is in the class $\\RTOAST$ and fix a witnessing $q\\in \\mathbb{N}$ together with a spontaneous randomized $\\TOAST$ algorithm $\\mathcal{A}$.\nWe may assume that $t$, the locality of $\\Pi$, satisfies $t\\le q$.\n\nLet $(A,c)$ be such that $A\\subseteq \\mathbb{Z}^d$ is finite and connected and $\\operatorname{dom}(c)\\subseteq A$.\nWe say that $(A,c)$ comes from a \\emph{partial random $q$-toast} if there is a collection $\\mathcal{D}$ of subsets of $A$ together with an event $\\mathcal{F}$ on $\\operatorname{dom}(c)$ such that\n\\begin{itemize}\n    \\item if $D,E\\in \\mathcal{D}$ and $d(\\partial D,\\partial E)\\le q$, then $D=E$,\n    \\item $A\\in \\mathcal{D}$,\n    \\item $\\bigcup_{E\\in \\mathcal{D}\\setminus \\{A\\}}E=\\operatorname{dom}(c)$,\n    \\item $\\mathcal{F}$ has non-zero probability and if we apply inductively $\\mathcal{A}$ on $\\operatorname{dom}(c)$ we obtain $c$.\n\\end{itemize}\nIf $(A,c)$ comes from a partial random $q$-toast, then we let $\\mathcal{E}(A,c)$ to be any event $\\mathcal{F}$ as above.\nOtherwise we put $\\mathcal{E}(A,c)$ to be any event on $\\operatorname{dom}(c)$ of non-zero probability.\nIt is easy to see that $\\mathcal{E}$ does not depend on the position of $A$ in $\\mathbb{Z}^d$.\nThis is because $\\mathcal{A}$ is such.\n\nWe now define a spontaneous toast algorithm $\\mathcal{A}'$.\nGiven $(A,c)$, we let $\\mathcal{F}(A,c)$ to be any event on $A$ that has non-zero probability, satisfies\n$$\\mathcal{F}(A,c)\\upharpoonright \\operatorname{dom}(c)\\subseteq \\mathcal{E}(A,c)$$\nand $\\mathcal{A}(A,c)$ gives the same outcome on $\\mathcal{F}(A,c)$.\nLet $\\mathcal{A}'(A,c)$ be equal to this outcome on $\\mathcal{F}(A,c)$.\nIt is clear that $\\mathcal{A}'$ is a spontaneous $\\TOAST$ algorithm.\nObserve that if $(A,c)$ comes from a partial random $q$-toast, then $(A,\\mathcal{A}'(A,c))$ comes from a partial random $q$-toast.\nNamely, use the same $q$-toast witness $\\mathcal{D}$ as for $(A,c)$ together with the event $\\mathcal{F}(A,c)$.\n\nIt remains to show that $\\mathcal{A}'(\\mathcal{D})$ is a $\\Pi$-coloring for every $q$-toast $\\mathcal{D}$.\nInduction on $\\mathfrak{r}_{\\mathcal{D}}$ shows that $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast for every $D\\in \\mathcal{D}$.\nThis is easy when $\\mathfrak{r}_{\\mathcal{D}}(D)=0$ and follows from the observation above in general.\nLet $v\\in \\mathbb{Z}^d$ and pick a minimal $D\\in \\mathcal{D}$ such that there is $D'\\in \\mathcal{D}$ that satisfies $v\\in D'$ and $D'\\subseteq D$.\nSince $t\\le q$ we have that $\\mathcal{B}(v,t)\\subseteq D$.\nSince $(D,\\mathcal{A}'(D))$ comes from a partial random $q$-toast it follows that there is a non-zero event $\\mathcal{F}$ such that \n$$\\mathcal{A}'(D)\\upharpoonright \\mathcal{B}(v,t)=\\mathcal{A}(\\mathcal{D})\\upharpoonright \\mathcal{B}(v,t)$$\non $\\mathcal{F}$.\nThis shows that the constraint of $\\Pi$ at $v$ is satisfied and the proof is finished.\n\\end{proof}\n\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL} = \\RLTOAST$}\n\\label{subsec:tail=randomlazytoast}\n\nIn this section we prove that the class $\\RLTOAST$ in fact covers the whole class $\\mathsf{TAIL} = \\bigcup_{f = \\omega(1)} \\mathsf{TAIL}(f(r))$. \n\n\\begin{restatable}{theorem}{tailrltoast}\n\\label{thm:tail=rltoast}\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ is in the class $\\RLTOAST$ if and only if $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{restatable}\n\nWe first give an informal proof. Suppose that there is a random lazy $\\TOAST$ algorithm $\\mathcal{A}$ for $\\Pi$. We construct an oblivious algorithm with nontrivial tail behavior $\\mathcal{A}'$ for $\\Pi$ by constructing a toast via \\cref{thm:toast_in_tail} and during the construction, each node moreover simulates the algorithm $\\mathcal{A}$. The algorithm $\\mathcal{A}$ finishes after some number of steps with probability $1$, hence $\\mathcal{A}'$ finishes, too, with probability $1$ for each node. \n\nOn the other hand, suppose that there is oblivious local algorithm $\\mathcal{A}'$ for $\\Pi$. \nThen we can construct a lazy randomized toast algorithm $\\mathcal{A}$ that simply simulates $\\mathcal{A}'$ for every node. This is possible since as the pieces around a given vertex $u$ that has gets are growing bigger and bigger, the radius around $u$ that $\\mathcal{A}$ sees also grows, until it is large enough and $\\mathcal{A}$ sees the whole coding radius around $u$ and applies $\\mathcal{A}'$ then. \n\nA formal proof follows. \n\n\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\RLTOAST$, i.e, there is $q\\in\\mathbb{N}$ together with a randomized toast algorithm $\\mathcal{A}$.\nIt follows from Theorem~\\ref{thm:toast moment} that there is an oblivious local algorithm $F$ that produces $q$-toast with probability $1$.\n\nDefine an oblivious local algorithm $\\mathcal{A}'$ as follows.\nLet $\\mathcal{B}(v,r)$ be a $\\Sigma_{in}$-labeled neighborhood of $v$.\nAssign real numbers from $[0,1]$ to $\\mathcal{B}(v,r)$ and run $F$ and $\\mathcal{A}$ on $\\mathcal{B}(v,r)$.\nLet $a_{v,r}$ be the supremum over all $\\Sigma_{in}$-labelings of $\\mathcal{B}(v,r)$ of the probability that $\\mathcal{A}'$ fails.\nIf $a_{v,r}\\to 0$ as $r \\to \\infty$, then $\\mathcal{A}'$ is an oblivious local algorithm.\n\nSuppose not, i.e., $a_{v,r}>a>0$.\nSince $\\Sigma_{in}$ is finite, we find by a compactness argument a labeling of $\\mathbb{Z}^d$ with $\\Sigma_{in}$ such that the event $\\mathcal{F}_r$ that $\\mathcal{A}'$ fails on $\\mathcal{B}(v,r)$ with the restricted labeling has probability bigger than $a$.\nNote that since the labeling is fixed, we have $\\mathcal{F}_r\\supseteq \\mathcal{F}_{r+1}$.\nSet $\\mathcal{F}=\\bigcap_{r\\in \\mathbb{N}} \\mathcal{F}_r$.\nThen by $\\sigma$-additivity we have that the probability of the event $\\mathcal{F}$ is bigger than $a>0$.\nHowever, $F$ produces toast almost surely on $\\mathcal{F}$ and $\\mathcal{A}$ produces $\\Pi$-coloring with probability $1$ for any given toast.\nThis contradicts the fact that probability of $\\mathcal{F}$ is not $0$.\n\n\n\n\\begin{comment}\nIn another words, with probability $1$ any node $v$ can reconstruct bigger and bigger piece of the toast $\\mathcal{D}$ when exploring its neighborhood.\nIn particular, $v$ will eventually became element of some piece of the toast and $\\mathcal{A}$ has to label $v$ with $\\Sigma_{out}$ after finitely many steps.\nThis defines an oblivious local algorithm $\\mathcal{B}$.\n\n\n\nPick a labeling $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ such that $F(\\ell)$ is a $q$-toast and $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for $q$-toast $\\mathcal{D}$.\nNote that the picked labeling satisfies these conditions with probability $1$.\n\nDefine $G(\\ell)=\\mathcal{A}(F(\\ell),\\ell)$.\nIt remains to observe that $G$ is an oblivious local algorithm.\nNote that $G(\\ell)({\\bf 0}_d)=\\mathcal{A}(F(\\ell),\\ell)({\\bf 0}_d)$ depends on some $D\\in \\mathcal{D}$ such that ${\\bf 0}_d\\in D$, $\\{E\\subseteq D:E\\in \\mathcal{D}\\}$ and $\\ell\\upharpoonright D$.\nSince $F$ is an oblivious local algorithm, we see that $G$ is too.\nConsequently, $\\Pi$ is in the class $\\mathsf{TAIL}$.\n\\end{comment}\n\nThe idea to prove the converse is to ignore the $q$-toast part and wait for the coding radius.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}$.\nBy the definition there is an oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi$ and an unbounded function $f$ such that\n$$\\P(R_\\mathcal{A}>r)\\le \\frac{1}{f(r)}.$$\nGiven an input labeling of $\\mathbb{Z}^d$ by $\\Sigma_{in}$, we see by the definition of $R_\\mathcal{A}$ that every node $v$ must finish after finitely many steps with probability $1$.\nTherefore it is enough to wait until this finite neighborhood of $v$ becomes a subset of some piece of a given toast $\\mathcal{D}$.\nThis happens with probability $1$ independently of the given toast $\\mathcal{D}$.\n\\end{proof}\n\n\\begin{comment}\nLet $(A,c,\\ell)$ be a triplet as in definition of randomized toast algorithm.\nTake $v\\in A\\setminus \\operatorname{dom}(c)$, and suppose that $v={\\bf 0}_d$.\nOutput $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)$ if and only if there is $t\\in \\mathbb{N}$ such that $\\mathcal{B}({\\bf 0}_d,t)\\subseteq A$ and $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)$ corresponds to some coding neighborhood of $F$.\nThat is to say, any extension $\\ell\\upharpoonright \\mathcal{B}({\\bf 0}_d,t)\\sqsubseteq x\\in [0,1]^{\\mathbb{Z}^d}$ outputs the same color at $F(x)({\\bf 0}_d)$.\nIn that case output $\\mathcal{A}(A,c,\\ell)({\\bf 0}_d)=F(x)({\\bf 0}_d)$.\n\nWe show that $\\mathcal{A}$ produces a $\\Pi$-coloring for $\\mu$-almost every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $1$-toast $\\mathcal{D}$.\nLet $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ be such that $F$ computes a $\\Pi$-coloring an need a finite coding radius for every $v\\in \\mathbb{Z}^d$.\nNote that almost every labeling $\\ell$ satisfies this condition.\nTake a $1$-toast $\\mathcal{D}$.\nIt is easy to see that $\\mathcal{A}(\\mathcal{D},\\ell)$ does not depend on $\\mathcal{D}$ and that $\\mathcal{A}(\\mathcal{D},\\ell)=F(\\ell)$, i.e., $\\mathcal{A}(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring.\nThis finishes the proof.\n\\end{comment}\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{$\\TOAST$: Specific Problems}\\label{sec:SpecificProblems}\n\\todo{explain that some of this is work of others and we show it to substantiate definition of toast}\n\nIn this section we focus on specific problems that are not in the class $\\mathsf{LOCAL}(O(\\log^* n))$ but they admit a solution by a (spontaneous) toast algorithm. Recall, that proving that $\\Pi \\in \\TOAST$ implies that there is an oblivious local algorithm with $1\/r^{1-o(1)}$ tail decay via \\cref{thm:toast_in_tail} and we discuss in \\cref{subsec:descriptive_combinatorics_discussion} that by a result of \\cite{GJKS2} it also implies $\\Pi\\in \\mathsf{BOREL}$. \n\nThe problems we discuss are the following: (a) vertex coloring, (b) edge coloring, (c) tiling the grid with boxes from a prescribed set (this includes, e.g., a perfect matching problem). \n\n\n\n\n\n\nAll these problems were recently studied in the literature from different points of view, see \\cite{ArankaFeriLaci, HolroydSchrammWilson2017FinitaryColoring, GJKS, spencer_personal}.\nIn all cases the main technical tool, sometimes in a slight disguise, is the toast construction.\nOur aim here is to put these problems into a unified context.\nWe note that sometimes the results in the aforementioned literature are stronger or work in a bigger generality (for example, all results discussed here can also be proven to hold on unoriented grids as discussed in \\cite{ArankaFeriLaci}, or the constructed coloring could be aperiodic \\cite{spencer_personal}, etc)\nIn what follows we only give a ``proof by picture'', sketch the description of the corresponding spontaneous toast algorithm and refer the reader to other papers for more details\/stronger results.\n\nThroughout this section we assume that $d>1$.\nWe refer the reader to \\cref{sec:OtherClasses} for definition of classes $\\mathsf{CONTINUOUS}$, $\\mathsf{BOREL}$, etc.\n\n\n\nThe main contribution if this section is the following:\n\\begin{enumerate}\n    \\item The fact that the problem of vertex $3$-coloring is in $\\TOAST$ together with the speed-up \\cref{thm:grid_speedup} answers a question (i) from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n    \\item We observe that the problem of edge $2d$-coloring the grid $\\mathbb{Z}^d$ is in $\\TOAST$, thereby answering a question of \\cite{GJKS} coming from the area of descriptive combinatorics. Independently, this was proven in \\cite{ArankaFeriLaci}. \n\\end{enumerate}\n\nIn general, our approach gives that if an LCL $\\Pi$ is in the class $\\TOAST\\setminus \\mathsf{LOCAL}(O(\\log^*(n)))$, then $1\\le {\\bf M}(\\Pi)\\le d-1$ by \\cref{thm:toast_in_tail} and \\cref{thm:grid_speedup}.\nThe examples in Section~\\ref{sec:Inter} show that there are LCLs with ${\\bf M}(\\Pi)=\\ell$ for every natural number $1\\le \\ell\\le d-1$, but for most natural LCLs we do not know the exact value of ${\\bf M}(\\Pi)$, unless $d=2$.\n\n\n\\subsection{Vertex Coloring}\n\\label{subsec:3coloring}\n\nLet $d>1$.\nRecall that $\\Pi^d_{\\chi,3}$ is the \\emph{proper vertex $3$-coloring problem}.\nThis problem was studied extensively in all contexts.\nIt is known that $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_grids}, $\\Pi^d_{\\chi,3}\\not\\in \\mathsf{CONTINUOUS}$ \\cite{GJKS}, $\\Pi^d_{\\chi,3}\\in \\mathsf{BOREL}$ \\cite{GJKS2} and $0<{\\bf M}(\\Pi^d_{\\chi,3})\\le 2$ \\cite{HolroydSchrammWilson2017FinitaryColoring}.\nA version of the following theorem is proven in \\cite{GJKS2}, here we give a simple ``proof by picture''. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:3in toast}\nLet $d>1$.\nThe LCL $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{proof}[Proof Sketch]\nThe toast algorithm tries to color all nodes in a given piece with two colors, blue and red. However, it can happen that two partial colorings of small pieces that have different parity in a bigger piece have to be connected (see \\cref{fig:3-coloring}). \nIn this case, we take the small pieces colored with incorrect parity and ``cover'' their border with the third color: green. From the inside of a small piece, the green color corresponds to one of the main two colors, while from the outside it corresponds to the other color. This enables us to switch the color parity. \n\\end{proof}\n\nAs a consequence of results from \\cref{sec:Toast} we can now fully determine the moment behavior of the problem which answers a question from \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\n\\begin{corollary}\nLet $d>1$.\nEvery oblivious local algorithm that solves $\\Pi^d_{\\chi,3}$ has the $1$-moment infinite, but there is an oblivious local algorithm solving $\\Pi^d_{\\chi,3}$ with all $(1-\\varepsilon)$-moments finite. \nIn particular, ${\\bf M}(\\Pi^d_{\\chi,3})=1$. \n\\end{corollary}\n\\begin{proof}\nBy \\cref{thm:3in toast}, $\\Pi^d_{\\chi,3}$ is in the class $\\TOAST$.\nIt follows from \\cref{thm:toast_in_tail} that ${\\bf M}(\\Pi^d_{\\chi,3})\\ge 1$.\nBy the speedup \\cref{thm:grid_speedup} and the fact that $3$-coloring was proven not to be in class $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{TAIL}(\\tow(\\Omega(r))) = \\mathsf{CONTINUOUS}$ \\cite{brandt_grids,HolroydSchrammWilson2017FinitaryColoring,GJKS}, we have that every oblivious local algorithm $\\mathcal{A}$ solving $\\Pi^2_{\\chi,3}$ must have the $1$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{\\chi,3})=1$.\nIt remains to observe that every vertex $3$-coloring of $\\mathbb{Z}^d$ yields a vertex $3$-coloring of $\\mathbb{Z}^2$ by forgetting the last $(d-2)$ dimensions.\nConsequently, every oblivious local algorithm $\\mathcal{A}$ that solves $\\Pi^d_{\\chi,3}$ defines an oblivious local algorithm $\\mathcal{A}'$ that solves $\\Pi^2_{\\chi,3}$ with the same tail behavior.\nThis finishes the proof.\n\\end{proof}\n\nThe $3$-coloring discussion above also implies that the toast itself cannot be constructed faster then with linear tail, otherwise $3$-coloring could be constructed with such a decay, too. \n\\begin{corollary}\n\\label{cor:toast_lower_bound}\nThere is a constant $C > 0$ such that a $q$-toast cannot be constructed in $\\mathsf{TAIL}(C r)$ on any $\\mathbb{Z}^d$.\nHere, by constructing a $q$-toast we mean that the coding radius of each node $u$ is such that $u$ can describe the minimal toast piece $D \\in \\mathcal{D}$ such that $u \\in D$. \n\\end{corollary}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/3-col-real.eps}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already vertex-colored. First step is to match the parity of colors with respect to the bigger piece. Then color the bigger piece.}\n    \\label{fig:3-coloring}\n\\end{figure}\n\n\n\\subsection{Edge Coloring}\n\\label{subsec:edgecoloring}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width = .95\\textwidth]{fig\/4-edge-col.png}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already edge-colored. First step is to match the parity of colors with respect to the bigger piece. This is done by adding a three layered gadget in the problematic directions. After that the coloring is extended to the bigger piece.}\n    \\label{fig:4-edge-coloring}\n\\end{figure}\\todo{dopsat}\nLet $d>1$.\nWe denote as $\\Pi^d_{\\chi',2d}$ the \\emph{proper edge $2d$-coloring problem}.\nObserve that a solution to $\\Pi^d_{\\chi',2d}$ yields a solution to the perfect matching problem $\\Pi^d_{pm}$ (because $\\mathbb{Z}^d$ is $2d$-regular).\nThis shows that $\\Pi^d_{\\chi',2d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, because the perfect matching problem clearly cannot be solved with a local algorithm: if it was, we could find a perfect matching in every large enough finite torus, which is clearly impossible since it can have an odd number of vertices \\cite{brandt_grids}. \n\n\n\n\\begin{theorem}[See also \\cite{ArankaFeriLaci}]\\label{thm:edge toast}\nLet $d>1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\TOAST$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nThe spontaneous toast algorithm is illustrated for $d=2$ in \\cref{fig:4-edge-coloring}.\nThe ``interface'' used by the toast algorithm between smaller and bigger pieces is: smaller pieces pretend they are colored such that in the vertical direction violet and green edges alternate, while in the horizontal direction blue and red edges alternate. If all small pieces in a bigger one have the same parity, this interface is naturally extended to the bigger piece. Hence, we need to work out how to fix pieces with bad horizontal\/vertical parity, similarly to \\cref{thm:3in toast}. \n\n\\cref{fig:4-edge-coloring} shows how to fix the vertical parity, adding the highlighted gadget in the middle picture above and below given pieces keeps its interface the same but switches the parity. The same can be done in the horizontal direction. \n\nTo show the result for $d>2$ it is enough to realize that switching parity in one dimension can be done with a help of one extra additional dimension, while fixing the parity in the remaining dimensions.\nConsequently, this allows to iterate the previous argument at most $d$-many times to fix all the wrong parities.\n\\end{proof}\n\n\n\\begin{corollary}\nLet $d>1$.\nThen $1 \\le {\\bf M}(\\Pi^d_{\\chi',2d}) \\le d-1$. \nIn particular, ${\\bf M}(\\Pi^2_{\\chi',4})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{\\chi',2d})$, where $d>2$.\n\nThe following corollary answers a question of \\cite{GJKS}. See \\cref{sec:OtherClasses} for the definition of class $\\mathsf{BOREL}$. \n\\begin{corollary}\nLet $d\\ge 1$.\nThen $\\Pi^d_{\\chi',2d}$ is in the class $\\mathsf{BOREL}$.\nIn particular, if $\\mathbb{Z}^d$ is a free Borel action and $\\mathcal{G}$ is the Borel graph generated by the standard generators of $\\mathbb{Z}^d$, then $\\chi'_\\mathcal{B}(\\mathcal{G})=2d$, i.e., the Borel edge chromatic number is equal to $2d$.\n\\end{corollary}\n\n\n\\begin{remark}[Decomposing grids into finite cycles]\nBencs, Hru\u0161kov\u00e1, T\u00f3th \\cite{ArankaFeriLaci} showed recently that there is a $\\TOAST$ algorithm that decomposes an unoriented grid into $d$-many families each consisting of finite pairwise disjoint cycles.\nHaving this structure immediately gives the existence of edge coloring with $2d$-colors as well as Schreier decoration.\nSince every $\\TOAST$ algorithms imply Borel solution on graphs that look locally like unoriented grids \\cite{AsymptoticDim}, this result implies that all these problems are in the class $\\mathsf{BOREL}$.\nWe refer the reader to \\cite{ArankaFeriLaci} for more details.\n\\end{remark}\n\n\n\n\n\n\n\n\\begin{comment}\n\\paragraph{Decomposing grids into finite cycles} \\todo{Jsem pro zkratit to na jeden odstavec}\n\nWe sketch how to use toast structure to decompose $\\mathbb{Z}^d$ into $d$-many families each consisting of finite pairwise disjoint cycles, formally, a collection $\\{\\mathcal{C}_i\\}_{i\\in [d]}$ such that $\\mathcal{C}_i$ is family of pairwise vertex disjoint cycles for every $i\\in [d]$ and $\\bigcup_{i\\in [d]} \\mathcal{C}_i=\\mathbb{Z}^d$.\nNote that finding such a decomposition is not an LCL because we do not have bound on the length of the cycles.\nOn the other hand, if we allow infinite paths, then such problem is LCL problem and we denote it as $\\Pi^d_{\\operatorname{cyc},d}$.\nMoreover, it is easy to see that since we work with oriented grids $\\Pi^d_{\\operatorname{cyc},d}$ is in the class $\\mathsf{LOCAL}(O(1))$.\n\nWe show that there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite for every $37$-toast $\\mathcal{D}$.\nMoreover, it is easy to see that our toastable algorithm also works for non-oriented grids, where the LCL problem $\\Pi^d_{\\operatorname{cyc},d}$ is not trivial.\nThe combinatorial construction is inspired by result of Bencs, Hruskova and Toth \\cite{ArankaFeriLaci} who originally proved similar result for $d=2$.\nAfter reading their paper we realized that the argument goes through for any dimension $d>1$ and we refer the reader to their paper for a detailed proof of this fact.\nHere we only sketch the main idea.\n\n\\begin{theorem}\nLet $d>1$.\nThen there is a spontaneous extension function $\\mathcal{A}$ such that $\\mathcal{A}(\\mathcal{D})$ is always a $\\Pi^d_{\\operatorname{cyc},d}$-coloring for every $37$-toast $\\mathcal{D}$ with the additional property that all the cycles in $\\mathcal{A}(\\mathcal{D})$ are finite.\n\\end{theorem}\n\n\\begin{remark}\nA straightforward use of the toastable algorithm together with the fact that Borel graphs with finite Borel asymptotic dimension admit Borel toast structure (see~\\cite{AsymptoticDim}) gives a decomposition of any Borel graph that looks locally like a $d$-dimensional grid (oriented or non-oriented) into $d$-many Borel families of edges each consisting of pairwise disjoint cycles.\nSince each such cycle is necessarily of even length, we get immediately that such graphs admit Borel Schreier decoration, Borel balanced orientation and their Borel chromatic index is $2d$.\n\\end{remark}\n\n\\begin{proof}[Proof Sketch]\nLet $d>1$.\nIt is easy to see that decomposing $\\mathbb{Z}^d$ into $d$ families of cycles (possibly infinite) is the same as coloring the edges with $d$-colors, such that every vertex $v\\in \\mathbb{Z}^d$ has exactly two adjacent edges of each color.\nLet $\\{\\alpha_1,\\dots,\\alpha_d\\}$ be the set of the colors.\n\nAn \\emph{(edge) coloring} of a connected finite set $A\\subseteq \\mathbb{Z}^d$ is a coloring of all edges that are adjacent to $A$.\nWe say that an edge $e$ is a \\emph{boundary edge}, if only one vertex of $e$ is contained in $A$.\nA coloring $c$ is \\emph{suitable}, if after choosing a reference point and orientation of $A$ we have that parallel boundary edges in the $i$-th direction are assigned the color $\\alpha_i$.\n\nThere is a constant $q\\in \\mathbb{N}$ such that for every $i\\not=j\\in [d]$, finite connected set $A\\subseteq \\mathbb{Z}^d$ and a suitable edge coloring $c$, there is a suitable edge coloring $c'$ of $A':=\\mathcal{B}(A,q)$ that extends $c$ such that \n\\begin{itemize}\n    \\item every $c'$-cycle of color $\\alpha_i$ that contains a vertex from $A$ is finite for every $i\\in [d]$,\n    \\item the $i$-th and $j$-th directions of $A$ and $A'$ are swapped, e.g., parallel $A'$-boundary edges in the $i$-th direction from the point of view of $A$ are $c'$-colored with $\\alpha_j$,\n    \\item other directions remain the same.\n\\end{itemize}\nIt is easy to see that iterating this extension possibly $2d$-many times produces a spontaneous toast algorithm $\\mathcal{A}$ with all the desired properties.\n\nIt remains to sketch how to find such extension $c'$.\nFirst, since $c$ is suitable and we want to swap $i$-th and $j$-th direction, we may assume that $d=2$.\nSecond, we may assume that $A$ is a union of rectangles, such that each side has even number of vertices adjacent to a perpendicular boundary edge.\nThis could be done as follows, pick a reference point and consider decomposition of $\\operatorname{Cay}(\\mathbb{Z}^2)$ into a square of a fixed even number of vertices on a side.\nTake the union of all such squares that intersect $A$.\nThird, see Figure~\\ref{fig:SwapingColors} how to swap colors and close all cycles coming out of $A$.\nThis finishes the sketch.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\\subsection{Tiling with Rectangles}\n\\label{subsec:rectangle_tiling}\n\n\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles.\nWe denote as $\\Pi^d_{\\mathcal{R}}$ the problem of \\emph{tiling $\\mathbb{Z}^d$ with elements of $\\mathcal{R}$}.\nIt is not fully understood when the LCL $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$ (or equivalently $\\mathsf{CONTINUOUS}$).\nCurrently the only known condition implying that $\\Pi^d_{\\mathcal{R}}$ is \\emph{not} in $\\mathsf{LOCAL}(O(\\log^* n))$ is when GCD of the volumes of elements of $\\mathcal{R}$ is not $1$.\nMinimal unknown example is when $\\mathcal{R}=\\{2\\times 2, 2\\times 3,3\\times 3\\}$. \nFor the other complexity classes the situation is different.\nGiven a collection $\\mathcal{R}$ we define a condition \\eqref{eq:GCD} as\n\\begin{equation}\n\\label{eq:GCD}\\tag{$\\dagger$}\n    \\text{GCD of side lengths in the direction of} \\ i \\text{-th generator is} \\ 1 \\ \\text{for every} \\ i\\in [d].\n\\end{equation}\nIt follows from the same argument as for vertex $2$-coloring, that if $\\mathcal{R}$ does not satisfy \\eqref{eq:GCD}, then $\\Pi^d_{\\mathcal{R}}$ is a global problem, i.e., it is not in any  class from \\cref{fig:big_picture_grids}.\nOn the other hand, the condition \\eqref{eq:GCD} is sufficient for a spontaneous toast algorithm. For example, this implies $\\Pi^2_{\\{2\\times 3,3\\times 2\\}}\\in \\mathsf{BOREL} \\setminus \\mathsf{CONTINUOUS}$.\nA version of the following theorem was announced in \\cite{spencer_personal}, here we give a simple ``proof by picture''.\n\n\\begin{theorem}[\\cite{spencer_personal, stephen}]\n\\label{thm:tilings}\nLet $d>1$ and $\\mathcal{R}$ be a finite collection of $d$-dimensional rectangles that satisfy \\cref{eq:GCD}.\nThen $\\Pi^d_{\\mathcal{R}}$ is in the class $\\TOAST$.\nConsequently, ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\ge 1$.\n\\end{theorem}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{fig\/rectangle_filling.eps}\n    \\caption{The $\\TOAST$ algorithm is given small pieces of a toast that are already tiled. First step is to match the parity of the tiling with respect to the bigger piece. Then tile the bigger piece.}\n    \\label{fig:Tiling}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nThe two-dimensional version of the construction is shown in \\cref{fig:Tiling}. The interface of pieces this time is that they cover themselves with boxes of side length $L$ which is the least common multiple of all the lengths of basic boxes. As in the previous constructions, we just need to work out how to ``shift'' each piece so that all small pieces agree on the same tiling with boxes of side length $L$ which can then be extended to the whole piece. This can be done as in \\cref{fig:Tiling}. \n\\end{proof}\n\n\n\\begin{corollary}\nLet $d=2$ and $\\mathcal{R}$ be a finite collection of $2$-dimensional rectangles.\nThen $\\Pi^2_{\\mathcal{R}}$ is either in the class $\\mathsf{LOCAL}(O(\\log^* n))$, satisfies ${\\bf M}(\\Pi^2_{\\mathcal{R}})=1$, or is global (that is, in no class in \\cref{fig:big_picture_grids}).\n\\end{corollary}\n\nFor any $d$, \\cref{thm:tilings} implies that if a tiling problem is ``hard'' (in no class in \\cref{fig:big_picture_grids}), it is because of a simple reason: a projection to one dimension gives a periodic problem. It would be interesting to understand whether there are problems that are hard but their solutions do not have any periodicity in them (see \\cref{sec:open_problems}). \n\nWe do not know if there is $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$.\nLet $d\\ge \\ell>0$ and denote as $\\mathcal{R}_\\ell$ the restriction of rectangles to first $\\ell$-dimensions.\nAs in the case of proper $3$-vertex coloring problem, we have that ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le {\\bf M}(\\Pi^\\ell_{\\mathcal{R}_{\\ell}})$.\nIn particular, if, e.g., $\\Pi^d_{\\mathcal{R}_2}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$, then ${\\bf M}(\\Pi^d_{\\mathcal{R}})\\le 1$.\n\n\\begin{remark}\nIn the context of non-oriented grids we need to assume that $\\mathcal{R}$ is invariant under swapping coordinates. \nSimilar argument, then shows that $\\Pi^d_\\mathcal{R}\\in \\TOAST$ if and only if $\\mathcal{R}$ satisfies \\eqref{eq:GCD}.\nCombined with the fact that there is a Borel toast on such Borel graphs (see~\\cite{AsymptoticDim}), we deduce that \\eqref{eq:GCD} for $\\mathcal{R}$ is equivalent with existence of Borel tiling with rectangles from $\\mathcal{R}$.\n\\end{remark}\n\n\n\\todo{2x3 a 3x2 se daji poskladat z elek, coz odpovida nejake partition kruznice}\n\n\n\n\n\n\\paragraph{Perfect matching}\n\nLet $d>1$.\nA special case of a tiling problem is the \\emph{perfect matching problem} $\\Pi^d_{pm}$.\nSince there are arbitrarily large tori of odd size we conclude that $\\Pi^d_{pm}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\nConsequently, $\\Pi^d_{pm}\\not \\in \\mathsf{TAIL}(\\tow(\\Omega(r)))$ by \\cref{thm:tail=uniform}, and it follows from the Twelve Tile Theorem of Gao et al \\cite[Theorem 5.5]{GJKS} that $\\Pi^d_{pm}\\not \\in \\mathsf{CONTINUOUS}$.\nSimilarly as with the vertex $3$-coloring problem, the proof of $\\Pi^d_{pm}\\in \\mathsf{BOREL}$, where $d>1$, from \\cite{GJKS2} uses the toast construction and, in our notation, yields the following result. \n\n\\begin{theorem}[\\cite{GJKS2}]\\label{thm:perfect_matching_intoast}\nLet $d>1$.\nThe LCL $\\Pi^d_{pm}$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{corollary}\nLet $d>1$.\nThen ${\\bf M}(\\Pi^d_{pm})\\ge 1$ and every ffiid $F$ that solves $\\Pi^d_{\\chi,3}$ has the $(d-1)$-moment infinite.\nIn particular, ${\\bf M}(\\Pi^2_{pm})=1$.\n\\end{corollary}\n\nWe leave as an open problem to determine the exact value of ${\\bf M}(\\Pi^d_{pm})$, where $d>2$.\nIn fact, this problem is similar to intermediate problems that we study in \\cref{sec:Inter}. \n\n\n\n\n\n\n\n\n\n\n\n\\renewcommand{\\pm}[2]{\\operatorname{PM}^{#1, #2}_\\boxtimes}\n\\renewcommand{\\pm}[1]{\\operatorname{PM}^{#1}_\\boxtimes}\n\\newcommand{\\mathbb{Z}^d_\\boxtimes}{\\mathbb{Z}^d_\\boxtimes}\n\n\\section{Intermediate Problems}\\label{sec:Inter}\n\nIn this section we construct local problems with $\\mathsf{TAIL}$ complexities $\\Theta(r^{j-o(1)})$ for $1 \\le j \\le d-1$. \nWe first define the problem $\\pm{d}$: this is just a perfect matching problem, but in the graph $\\mathbb{Z}^d_\\boxtimes$ where two different nodes $u = (u_1, \\dots, u_d)$ and $v = (v_1, \\dots, v_d)$ are connected with an edge if $\\sum_{1 \\le i \\le d} |u_i - v_i| \\le 2$. That is, we also add some ``diagonal edges'' to the grid. We do not know whether perfect matching on the original graph (Cayley graph of $\\mathbb{Z}^d$) is also an intermediate problem (that is, not $\\mathsf{LOCAL}$ but solvable faster than $\\TOAST$). \n\nWe will show that the tail complexity of $\\pm{d}$ is $\\Theta(r^{d - 1  - o(1)})$. \nAfter this is shown, the hierarchy of intermediate problems is constructed as follows. \nConsider the problem $\\pm{d,\\ell}$. This is a problem on $\\mathbb{Z}^d$ where one is asked to solve $\\pm{d}$ in every hyperplane $\\mathbb{Z}^\\ell$ given by the first $\\ell$ coordinates in the input graph. \nThe $\\mathsf{TAIL}$ complexity of $\\pm{d,\\ell}$ is clearly the same as the complexity of $\\pm{\\ell}$, that is, $\\Theta(r^{\\ell-1-o(1)})$. \nHence, we can focus just on the problem $\\pm{d}$ from now on. \n\n\\begin{claim}\\label{cl:PM is global}\nLet $2\\le d\\in \\mathbb{N}$.\nThen the local coloring problem $\\pm{d}$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\nVia \\cref{thm:grid_speedup} this is equivalent to $\\pm{d} \\not\\in \\mathsf{TAIL}(Cr^{d-1})$ for large enough $C$. \n\\end{claim}\n\nThis follows from the fact that if there is a local algorithm, then in particular there exists a solution on every large enough torus. But this is not the case if the torus has an odd number of vertices. \n\nSecond, we need to show how to solve the problem $\\pm{d}$ faster than with a $\\TOAST$ construction algorithm. \n\n\\begin{restatable}{theorem}{intermediate}\n\\label{thm:intermediate}\nThe problem $\\pm{d}$ can be solved with an oblivious local algorithm of tail complexity $r^{d-1-O(\\frac{1}{\\sqrt{\\log r}})}$. \n\\end{restatable}\n\nHere we give only a proof sketch, the formal proof is deferred to \\cref{app:intermediate_problems}. \n\n\\begin{proof}[Proof Sketch]\nWe only discuss the case $d=3$ (and the actual proof differs a bit from this exposition due to technical reasons). As usual, start by tiling the grid with boxes of side length roughly $r_0$. \nThe \\emph{skeleton} $\\mathcal{C}_0$ is the set of nodes we get as the union of the $12$ edges of each box. \nIts \\emph{corner points} $X_0 \\subseteq \\mathcal{C}_0$ is the union of the $8$ corners of each box. \n\n\nWe start by matching vertices inside each box that are not in the skeleton $\\mathcal{C}_0$. It is easy to see that we can match all inside vertices, maybe except of one node per box that we call a \\emph{pivot}. We can ensure that any pivot is a neighbor of a vertex from $\\mathcal{C}_0$. \n\nWe then proceed similarly to the toast construction of \\cref{thm:toast_in_tail} and build a sequence of sparser and sparser independent sets that induce sparser and sparser subskeletons of the original skeleton.\nMore concretely, we choose a sequence $r_k = 2^{O(k^2)}$ and in the $k$-th step we construct a new set of corner points $X_k \\subseteq X_{k-1}$ such that any two corner points are at least $r_k$ hops apart but any point of $\\mathbb{Z}^d$ has a corner point at distance at most $2r_k$ (in the metric induced by $\\mathbb{Z}^d$). \nThis set of corner points $X_k$ induces a subskeleton $\\mathcal{C}_{k} \\subseteq \\mathcal{C}_{k-1}$ by connecting nearby points of $X_k$ with shortest paths between them in $\\mathcal{C}_{k-1}$. \n\nAfter constructing $\\mathcal{C}_k$, the algorithm solves the matching problem on all nodes of $\\mathcal{C}_{k-1} \\setminus \\mathcal{C}_k$. This is done by orienting those nodes towards the closest $\\mathcal{C}_k$ node in the metric induced by the subgraph $\\mathcal{C}_k$. \nThen, the matching is found on connected components that have radius bounded by $r_k$. There is perhaps only one point in each component, a pivot, that neighbors with $\\mathcal{C}_{k}$ and remains unmatched. \n\nAfter the $k$-th step of this algorithm, only nodes in $\\mathcal{C}_k$ (and few pivots) remain unmatched. Hence, it remains to argue that only roughly a $1\/r_k^{d-1}$ fraction of nodes survives to $\\mathcal{C}_k$. This should be very intuitive -- think about the skeleton $\\mathcal{C}_k$ as roughly looking as the skeleton generated by boxes of radius $r_k$. This is in fact imprecise, because in each step the distance between corner points of $X_k$ in $\\mathcal{C}_k$ may be stretched by a constant factor with respect to their distance in the original skeleton $\\mathcal{C}_{k-1}$. This is the reason why the final volume of $\\mathcal{C}_k$ is bounded only by $2^{O(k)} \/ r_k^{d-1}$ and by a similar computation as in \\cref{thm:toast_in_tail} we compute that the tail decay of the construction is $2^{O(\\sqrt{\\log r})}\/{r^{d-1}}$. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Other Classes of Problems}\\label{sec:OtherClasses}\n\nIn this mostly expository section we discuss other classes of problems from \\cref{fig:big_picture_grids} and how they relate to our results from previous sections. \n\n\n\n\\subsection{$\\mathsf{CONTINUOUS}$}\\label{subsec:CONT}\n\nHere we explain the fact that an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$ if and only if it is in the class $\\mathsf{CONTINUOUS}$.\nFor $\\mathbb{Z}^d$, the model $\\mathsf{CONTINUOUS}$ was extensively studied by Gao, Jackson, Krohne, Tucker-Drob, Seward and others, see \\cite{GaoJackson,GJKS,ST,ColorinSeward}.\nIn fact, the connection with local algorithms is implicit in \\cite{GJKS} under the name Twelve Tiles Theorem \\cite[Theorem~5.5]{GJKS}.\nRecently, independently Bernshteyn \\cite{Bernshteyn2021local=cont} and Seward \\cite{Seward_personal} showed that the same holds in bigger generality for all Cayley graphs of finitely generated groups.\nOur aim here is to formally introduce the model $\\mathsf{CONTINUOUS}$ and give an informal overview of the argument for $\\mathbb{Z}^d$ that follows the lines of Gao et al \\cite{GJKS}.\nThe reasons not to simply put reference to the aforementioned papers are two.\nFirst, we work with slightly more general LCLs that have inputs.\nSecond, our goal is to introduce all classes in a way accessible for  a broader audience. \n\nThe setup of the $\\mathsf{CONTINUOUS}$ model is somewhat similar to the deterministic $\\mathsf{LOCAL}$ model: there is no randomness in the definition and there is a certain symmetry breaking input on the nodes. Specifically, each vertex is labelled with one label from the set $\\{0, 1\\}$; the space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$. \nHowever, for the same reason as why we insist on the uniqueness of the identifiers in the $\\mathsf{LOCAL}$ model, we insist that the input $\\{0,1\\}$ labeling breaks all symmetries. Namely, we require that there is no $g$ such that shifting all elements by $g$ preserves the labeling. We write $\\operatorname{Free}(2^{\\mathbb{Z}^d}) \\subseteq 2^{\\mathbb{Z}^d}$ for this subspace. \nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},f)$, where $\\mathcal{C}$ is a collection of $\\{0,1\\} \\times \\Sigma_{in}$-labeled neighborhoods and $f$ is a function that assigns to each element of $\\mathcal{C}$ an output label. It needs to satisfy the following. Suppose we fix a node $v$ and the input $\\{0,1\\}$-labeling $\\Lambda$ of $\\mathbb{Z}^d$ satisfies $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$. If we let $v$ explore its neighborhood with an oblivious local algorithm, then we encounter an element from $\\mathcal{C}$ after finitely many steps. \nWe write $\\mathcal{A}(\\Lambda)(v)$, or simply $\\mathcal{A}(v)$, when $\\Lambda$ is understood, for the corresponding output.\n\n\\begin{comment}\n\\vspace*{5cm}\nTo break symmetries on $\\mathbb{Z}^d$ in the setting of random processes one considers iid labels on vertices.\nThe event that some symmetries remain has measure $0$ and we simply ignore such labelings.\nIn the continuous settings we label vertices with $2:=\\{0,1\\}$.\nThe space of all such labelings is denoted as $2^{\\mathbb{Z}^d}$ and we write $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ for the space of labelings that break all symmetries.\nA \\emph{continuous algorithm} is a pair $\\mathcal{A}=(\\mathcal{C},g)$, where $\\mathcal{C}$ is a collection of ${0,1},\\Sigma_{in}$-labeled neighborhoods and $g$ is a function that assigns to each element of $\\mathcal{C}$ an output label, such that if we let a node $v$ explore its labeled neighborhood $x$, then it encounters element from $\\mathcal{C}$ after finitely many steps whenever $x\\in \\Sigma_{in}^{\\mathbb{Z}^d}\\times \\operatorname{Free}(2^{\\mathbb{Z}^d})$, i.e., $x$ breaks all symmetries.\nWe write $\\mathcal{A}(x)(v)$ for the corresponding output.\n\\end{comment}\n\n\\begin{remark}\nThis is equivalent with the definition given in e.g. \\cite{GJKS} in the same way as the correspondence between oblivious local algorithms and ffiid, see \\cref{subsec:subshifts}.\nThe fact that the collection $\\mathcal{C}$ is dense, i.e., every node finishes exploring its neighborhoods in finite time, means that $\\mathcal{A}({\\bf 0})$ is \\emph{continuous} with respect to the product topology and defined in the whole space $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, the value computed at $v$ does not depend on the position of $v\\in \\mathbb{Z}^d$ but only on the isomorphism type of the labels, this is the same as saying that $\\mathcal{A}(\\_)$ is \\emph{equivariant}.\n\\end{remark}\n\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a continuous algorithm $\\mathcal{A}$ that outputs a solution for every $\\Lambda \\in \\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nNote that, in general, we cannot hope for the diameters of the neighborhoods in $\\mathcal{C}$ to be uniformly bounded.\nIn fact, if that happens then $\\Pi \\in \\mathsf{LOCAL}(O(1))$. \nThis is because there are elements in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ with arbitrarily large regions with constant $0$ label.\nApplying $\\mathcal{A}$ on that regions gives an output label that depends only on $\\Sigma_{in}$ after bounded number of steps, hence we get a local algorithm with constant local complexity.\n\nThe main idea to connect this context with distributed computing is to search for a labeling in $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ where any $\\mathcal{A}$ necessarily has a uniformly bounded diameter.\nSuch a labeling does exist!\nIn \\cite{GJKS} these labelings are called \\emph{hyperaperiodic}.\nTo show that hyperaperiodic labelings exist on $\\mathbb{Z}^d$ is not that difficult, unlike in the general case for all countable groups \\cite{ColorinSeward}. \nIn fact there is enough flexibility to encode in a given hyperaperiodic labeling some structure in such a way that the new labeling remains hyperaperiodic.\n\nWe now give a definition of a hyperaperiodic labeling. \nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}^d$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $t\\in \\mathbb{Z}^d$ we have \n$$\\Lambda(t+S) \\not= \\Lambda(g+t+S)$$\nThis notation means that the $\\Lambda$-labeling of $S$ differs from the $\\Lambda$-labeling of $S$ shifted by $g$ everywhere. \nIn another words, given some translation $g\\in \\mathbb{Z}^d$, there is a uniform witness $S$ for the fact that $\\Lambda$ is nowhere $g$-periodic.\nIt follows from a standard compactness argument that any continuous algorithm $\\mathcal{A}$ used on a hyperaperiodic labeling finishes after number of steps that is uniformly bounded by some constant that depends only on $\\mathcal{A}$. \n\n\n\n\\begin{comment}\n\\vspace*{5cm}\n\nLet $\\Lambda \\in 2^{\\mathbb{Z}^d}$ be a labeling of $\\mathbb{Z}^d$.\nWe say that $\\Lambda$ is \\emph{hyperaperiodic} if for every $g\\in \\mathbb{Z}$ there is a finite set $S\\subseteq \\mathbb{Z}^d$ such that \nfor every $s\\in \\mathbb{Z}^d$ we have \n$$x(s+t)\\not=x(g+s+t)$$\nfor some $t\\in S$.\nIn another words, given a shift $g\\in \\mathbb{Z}^d$ there is a uniform witness $S$ for the fact that $x$ is nowhere $g$-periodic.\nIt follows from some standard compactness argument that every hyperaperiodic element needs uniformly bounded neighborhoods for each continuous algorithm $\\mathcal{A}$, the bound of course depends on the algorithm. \n\\end{comment}\n\n\\begin{theorem}[\\cite{GJKS,Bernshteyn2021local=cont}]\nLet $d>0$.\nThen an LCL $\\Pi$ on $\\mathbb{Z}^d$ is in the class $\\mathsf{CONTINUOUS}$ if and only if it is in the class $\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\textwidth]{fig\/continuous.eps}\n    \\caption{On the left side we have examples of a labeled local tiling configurations compared with a possible continuity window $w$. These are then sparsely embedded into $\\mathbb{Z}^2$.}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\begin{proof}[Proof Sketch]\nOne direction was observed by Bernshteyn \\cite{Bernshteyn2021local=cont} and follows from the fact that there is continuous algorithm that produces a maximal $r$-independent set for every $r\\in \\mathbb{N}$ (recall \\cref{thm:classification_of_local_problems_on_grids}).\nThis is implicit, even though in a different language, already in Gao and Jackson \\cite{GaoJackson}.\n\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm for $\\Pi$. \nWe will carefully construct a certain hyperaperiodic element $\\Lambda$ and let $\\mathcal{A}$ solve $\\Pi$ on it. \nWe know that there is a certain number $w$ such that $\\mathcal{A}$ solves $\\Pi$ on $\\Lambda$ with local complexity $w$. \nWe note that even though we do not know the value $w$ in advance, it depends on $\\Lambda$ that is due to be constructed, the flexibility of the construction guarantees to ``encode'' in $\\Lambda$ all possible scenarios.\nSee the discussion at the end of the argument. \n\nThe overall approach is as follows. \nOur local algorithm $\\mathcal{A}'$ first tiles the plane with  tiles of length much bigger than $w$ with $O(\\log^* n)$ local complexity. We cannot tile the plane in a periodic manner in $O(\\log^* n)$ steps, but it can be done with a constant number of different tiles such that the side length of each tile is either $r$ or $r+1$ for some $r \\gg w$ (this construction is from \\cite[Theorem~3.1]{GaoJackson}, see also the proof of \\cref{thm:grid_speedup}). \nFor each tile $T$, $\\mathcal{A}'$ labels it with certain $\\{0,1\\}$ labeling. \nThis labeling depends only on the shape of the tile.\nWe make sure that not only $T$ with its labeling, but also $T$ together with all possible local configurations of tiles around it and corresponding labelings appears in $\\Lambda$.\nIn particular, as $r\\gg t,w$ ($t$ is the radius necessary to check the solution $\\Pi$), we have that the $(w+t)$-hop neighborhood of any node is labelled in a way that also appears somewhere in the labeling $\\Lambda$.\nAfter labeling each tile, $\\mathcal{A}'$ simulates $\\mathcal{A}$. We know that $\\mathcal{A}$ finishes after $w$ steps on $\\Lambda$ and is correct there, hence $\\mathcal{A}'$ is correct and has local complexity $O(\\log^* n)$. What remains is to describe is how to get the labeling of each tile.  \n\nWe use the fact that there is a lot of flexibility when constructing a hyperaperiodic labeling. \nWe define the hyperaperiodic labeling $\\Lambda$ of the input graph $\\mathbb{Z}^d$ in such a way that, after fixing $r$, each one of constantly many local configurations of $\\{r,r+1\\}$-tiles are embedded in $\\Lambda$, each one in fact in many copies that exhaust possible $\\Sigma_{in}$-labelings.\n\nThis is done as follows.\nFix any hyperaperiodic element $\\Lambda'$, these exist by \\cite[Lemma~2.8]{GJKS}.\nGiven a tile $T$, that depends on $r$, $\\{0,1\\}$-label it by any $T$-pattern that appears in $\\Lambda'$.\nConsider the collection of all possible local configurations of such labeled tiles that are moreover labeled by $\\Sigma_{in}$.\nThis collection is finite. Since we do not know $w$ in advance, do this for any value of $r$. This yields a countable collection of possible configurations. \nEmbed this collection sparsely into $\\Lambda'$, \\cref{fig:my_label}. Extend the $\\Sigma_{in}$-labeling arbitrarily. \nThis defines $\\Lambda$.\nIt is routine to check that $\\Lambda$ is hyperaperiodic, see \\cite[Lemma~5.9]{GJKS}.\n\n\n\nIt remains to recall that since $\\Lambda$ is hyperaperiodic, there is $w\\in \\mathbb{N}$ such that $\\mathcal{A}$ has uniformly bounded locality by $w$ on $\\Lambda$.\nPick $r\\gg w,t$.\nThis works as required since any $w+t$-hop neighborhood $\\mathcal{A}'$ can encounter is present in $\\Lambda$.\n\\end{proof}\n\n\\begin{comment}\n\\vspace*{4cm}\nNext, suppose that $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ and write $\\mathcal{A}$ for the continuous algorithm.\nA high-level strategy for $d=2$ is as follows.\nThere are 12 type of tiles that are parametrized by natural numbers $n,p,q\\in \\mathbb{N}$, i.e., for fixed $n,p,q$ there are $12$ tiles.\nIf $n,p,q$ are fixed, then there is a local deterministic algorithm that tiles given, large enough, torus with these tiles in such a way that the boundary parts matches.\\todo{picture here ad reference to the nice pictures in \\cite{GJKS}}\nIt is not clear at this point how large parameters we should choose.\nThe trick is to encode all such possible tiles for all $n,p,q\\in \\mathbb{N}$ in one hyperaperiodic element $x$.\nThe encoding of the tiles in $x$ guarantees that whenever the regions \\todo{picture} are the same for the fixed parameters $n,p,q$, then $\\mathcal{A}$ outputs the same label.\nSince $x$ is hyperaperiodic, we have that $\\mathcal{A}$ stops after $T$-many steps to output $\\mathcal{A}(x)(v)$ for some $T\\in \\mathbb{N}$.\nWe choose $n,p,q$ according to $T$ in such a way that for any vertex at any region there is a tile such that the $T$-neighborhood of this vertex is fully contained in the tile.\nConsequently, this defines local rule once a given, large enough, torus is tiled into these tiles of 12 types.\n\nThis description exactly follows the proof of (1) $\\rightarrow$ (2) in \\cite[Theorem~5.5]{GJKS}, where (1) is the same as saying that there is a continuous algorithm and (2) fixes the coloring of the tiles for some parameters $n,p,q$.\nHowever, we note that we are skipping some technical details about additional parameter $w$ etc.\nThe additional step to produce a local algorithm by tiling a large enough torus into these 12 tiles is standard.\nIn fact, this is proved in \\cite[Theorem~5.10]{GJKS}, again in different language, to show (2) $\\rightarrow$ (1) in \\cite[Theorem~5.5]{GJKS}.\n\nThe main difference that we encounter are the input labels.\nWe explain how to handle them and hint, simultaneously, what we mean by encoding tiles in hyperaperiodic element, again ignoring a technical parameter $w$, see \\cite[Figures~13,14,etc]{GJKS}.\nFix a hyperaperiodic element $y$.\nFor $n,p,q\\in \\mathbb{N}$ and a tile $G$, see \\eqref{fig:???}, assign to each region labels from $\\{0,1\\}$ in such a way that (a) same regions get the same pattern, (b) this pattern is part of $y$.\nConsider moreover any possible input labeling of such $\\{0,1\\}$-labeled tiles.\nNow place these labeled tiles, for every $n,p,q\\in \\mathbb{N}$, sparsely in $y$ (Figure ???) and extend the input labeling on them to the whole $\\mathbb{Z}^2$, this defines $x$.\nIt is not hard to see that $x$ remains hyperaperiodic.\n\nIn higher dimension, i.e., when $d>2$, the same argument works as well with the only difference that there are more than 12 types of tiles.\n\\end{comment}\n\n\n\n\n\n\n\n\n\n\\subsection{Descriptive Combinatorics Classes}\n\\label{subsec:descriptive_combinatorics_discussion}\n\nIn this section we define the classes of problems that are studied in descriptive combinatorics.\nThere are two equivalent ways how the notion of a \\emph{Borel graph} can be defined.\nThe less abstract one is as follows.\nConsider the unit interval $[0,1]$ with the $\\sigma$-algebra of Borel sets, that is the smallest algebra of subsets that contains sub-intervals and is closed under countable unions and complements.\nLet $A,B\\subseteq [0,1]$ be Borel sets and consider a bijection $\\varphi:A \\to B$ between them.\nWe say that $\\varphi$ is \\emph{Borel measurable} if $\\varphi^{-1}(D\\cap B)$ is a Borel set for every Borel set $D$.\nNow, let $\\mathcal{G}$ be a graph with vertex set $[0,1]$ and assume that the degree of every vertex is bounded by some $\\Delta<\\infty$.\nWe say that $\\mathcal{G}$ is a \\emph{Borel graph} if the edge relation is induced by Borel bijections, that is there are sets $A_1,\\dots, A_\\ell$, $B_1,\\dots, B_\\ell$ and Borel bijections $\\varphi_1,\\dots, \\varphi_\\ell$ such that $(x,y)$ is an edge in $\\mathcal{G}$ if and only if there is $1\\le i\\le \\ell$ such that either $\\varphi_i(x)=y$ or $\\varphi_i(y)=x$.\n\nThe more abstract way how to define Borel graphs uses the notion of a \\emph{standard Borel space}.\nFor example $[0,1]$ with the $\\sigma$-algebra of Borel sets, when we forget that it comes from the standard topology, is a standard Borel space.\nFormally, a standard Borel space is a pair $(X,\\mathcal{B})$, where $X$ is a set and $\\mathcal{B}$ is a $\\sigma$-algebra that comes from some separable and complete metric structure on $X$.\nA uniformly bounded degree graph $\\mathcal{G}$ on $(X,\\mathcal{B})$ is Borel if it satisfies the same condition as in the case of the particular space $[0,1]$.\n\nIt is easy to see that with this definition all the basic constructions from graph theory are \\emph{relatively} Borel, e.g. if $A$ is a Borel set of vertices, then $N_\\mathcal{G}(A)$ the neighbors of elements in $A$ form a Borel set.\nAlso note that since we assume that the degree is uniformly bounded, every connected component in $\\mathcal{G}$ is at most countable.\nTypical questions that are studied in this context are Borel versions of standard notions, e.g., \\emph{Borel} chromatic number, \\emph{Borel} chromatic index, \\emph{Borel} perfect matching etc.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey} for more details.\n\nThe most relevant Borel graphs for us are the ones that are induced by a free Borel action of $\\mathbb{Z}^d$.\nThat is, there are $d$-many aperiodic Borel bijections $\\varphi_i:X\\to X$ that pairwise commute, where \\emph{aperiodic} means that $\\varphi_i^k(x)\\not=x$ for every $k>0$, $i\\in [d]$ and $x\\in X$.\nThen the Borel (labeled) graph $\\mathcal{G}$ that is induced by this action consists of edges of the form $(x,\\varphi_i(x))$, where $x\\in X$ and $i\\in [d]$.\nThe most important result in this context is that $\\mathcal{G}$ admits a \\emph{Borel toast structure}.\nThis follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof.\nWe note that it is not understood in general what Borel graphs admit a Borel toast structure.\nThis is connected with the notoriously difficult question about Borel hyperfiniteness, see \\cite{AsymptoticDim}.\n\nGiven an LCL $\\Pi$, we say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if there is a Borel measurable $\\Pi$-coloring of every Borel graph $\\mathcal{G}$ as above, i.e., induced by a free Borel action of $\\mathbb{Z}^d$.\nEquivalently, it is enough to decide whether $\\Pi$ can be solved on the canonical Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$.\nMoreover, if $\\Pi$ has some inputs $\\Sigma_{in}$, then we consider any Borel labeling $X\\to \\Sigma_{in}$ and then ask if there is a Borel solution that satisfies all the constraints on $\\Sigma_{in}\\times \\Sigma_{out}$.\n\n\\begin{theorem}[Basically \\cite{GJKS}]\nLet $\\Pi$ be an LCL that is in the class $\\LTOAST$.\nThen $\\Pi\\in\\mathsf{BOREL}$.\n\\end{theorem}\n\nIn fact, all the LCL problems without input that we are aware of and that are in the class $\\mathsf{BOREL}$ are in fact in the class $\\TOAST$.\nIt is an open problem to decide whether it is always the case.\n\nAnother interesting perspective how to see Borel constructions is as a countably infinite analogue of $\\mathsf{LOCAL}(O(\\log^*(n)))$.\nNamely, suppose that you are allowed to inductively (a) construct Borel MIS with some parameter (b) do local $O(1)$-construction that depends on the previously build structure.\nIt is clear that objects that are build in this way are Borel, however, it is not clear if this uses the whole potential of Borel sets to find Borel solutions of LCLs.\n\nAs a last remark we note that traditionally in descriptive combinatorics, people consider relaxations of Borel constructions.\nThis is usually done via ignoring how a solution behave on some negligible Borel set.\nProminent examples are \\emph{measure $0$ sets} for some Borel probability measure on $(X,\\mathcal{B})$ or \\emph{meager sets} for some compatible complete metric on $(X,\\mathcal{B})$.\nWe denote the classes of problems that admit such a solution for \\emph{every} Borel probability measure or compatible complete metric as $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$, respectively.\nWithout going into any details we mention that on general graphs these classes are much richer than $\\mathsf{BOREL}$, see e.g. \\cite{DetMarks,BrooksMeas}.\nHowever, it is still plausible that on $\\mathbb{Z}^d$ we have $\\mathsf{BOREL}=\\mathsf{BAIRE}=\\mathsf{MEASURE}$.\nIt is well-know that this is true in $\\mathbb{Z}$ \\cite{grebik_rozhon2021LCL_on_paths}.\n\n\n\\begin{comment}\nIn both cases we are allowed to neglect some small sets, where we either do not care about the solution or are allowed to use the axiom of choice.\nSuppose that $\\mathcal{G}$ is a Borel graph on a standard Borel space $(X,\\mathcal{B})$ and $\\Pi$ is an LCL.\nSuppose, in addition, that there is a Borel probability measure $\\mu$ on $(X,\\mathcal{B})$, that is a $\\sigma$-additive measure on $\\mathcal{B}$, see \\cite[Section~?]{Kec}.\nWe say that $\\Pi$ admits \n\n\n\n\n\n\n\n\n\n\nand show the arrows that are left to complete Figure~\\ref{fig:big_picture_grids}.\nRecall that a \\emph{Borel graph} $\\mathcal{G}$ is a triplet $\\mathcal{G}=(X,\\mathcal{B},E)$, where $(X,\\mathcal{B})$ is a standard Borel space, $(X,E)$ is a graph and $E\\subseteq X\\times X$ is an element of the product $\\sigma$-algebra.\nFor the purposes of this paper we always assume that $\\mathcal{G}$ is induced by a free Borel action of $\\mathbb{Z}^d$ for some $d>0$, i.e., there is a Borel action $\\mathbb{Z}^d\\curvearrowright X$ with trivial stabilizers such that $\\mathcal{G}$ is induced by the standard generators of $\\mathbb{Z}^d$.\n\nLet $\\Pi$ be an LCL.\nThen $\\Pi$ belongs to the class $\\mathsf{BOREL}$ if there is a Borel $\\Pi$-coloring of every such Borel graph $\\mathcal{G}$.\nMoreover, we say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{Borel probability measure} $\\mu$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\nSimilarly, we say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every such Borel graph $\\mathcal{G}$ and for every \\emph{compatible Polish topology} $\\tau$ on $(X,\\mathcal{B})$ there is a Borel $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\nIt follows directly from the definition that $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE},\\mathsf{BAIRE}$.\n\\end{comment}\n\n\n\n\\begin{comment}\nNext we follow Gao et al. \\cite{GJKS}.\nDenote as $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ the free part of the Bernoulli shift $\\mathbb{Z}^d\\curvearrowright 2^{\\mathbb{Z}^d}$ endowed with the inherited product topology.\nThen it is easy to see that $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ is a Polish space and that the action is continuous.\nConsider the Borel graph $\\mathcal{G}$ on $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ induced by the standard set of generators.\nThis graph is universal in the sense that an LCL $\\Pi$ is in the class $\\mathsf{BOREL}$ if and only if there is a Borel $\\Pi$-coloring of $\\mathcal{G}$, this follows from result of Seward and Tucker-Drob~\\cite{?}.\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{CONTINUOUS}$ if there is a $\\Pi$-coloring of $\\mathcal{G}$ that is continuous.\nThis is the same as saying that that there is a continuous equivariant map from $\\operatorname{Free}(2^{\\mathbb{Z}^d})$ to $X_\\Pi$.\nThe following is a corollary of the Twelve Tiles Theorem of Gao et al. \\cite{GJKS}.\n\n\\begin{theorem}\\cite[Theorem 5.5]{GJKS}\n$\\mathsf{CONTINUOUS}=\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nWe note that in the general case of all groups this equality was shown by Bernshteyn\\cite{?} and, independently, by Seward. \n\nIt remains to describe how toastable algorithms give Borel solutions.\nAs an immediate corollary we get that all the problems from Section~\\ref{sec:SpecificProblems} are in the class $\\mathsf{BOREL}$.\n\n\\begin{theorem}\n$\\LTOAST \\subseteq \\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\nSuppose that $\\Pi$ is in the class $\\LTOAST$ and let $f$ be an toast algorithm with parameter $k$.\nIt follows from work of Gao et al \\cite{GJKS2}, see Marks and Unger \\cite{Circle} for proof, that every Borel graph $\\mathcal{G}$ (that is induced by a free action of $\\mathbb{Z}^d$) admits a Borel $k$-toast for every $k\\in \\mathbb{N}$.\nIt is a standard fact that using $f$ inductively on the depth of the toast produces a Borel $\\Pi$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\\subsection{Finitely Dependent Processes}\n\nIn this section we use the notation that was developed in \\cref{subsec:subshifts}, also we restrict our attention to LCLs without input.\nLet $\\Pi=(b,t,\\mathcal{P})$ be such an LCL.\nRecall that $X_\\Pi$ is the space of all $\\Pi$-colorings.\n\nWe say that a distribution $\\mu$ on $b^{\\mathbb{Z}^d}$ is \\emph{finitely dependent} if it is shift-invariant and there is $k\\in \\mathbb{N}$ such that the events $\\mu\\upharpoonright A$ and $\\mu \\upharpoonright B$ are independent for every finite set $A,B\\subseteq \\mathbb{Z}^d$ such that $d(A,B)\\ge k$.\nNote that shift-invariance is the same as saying that the distribution $\\mu\\upharpoonright A$ does not depend on the position of $A$ in $\\mathbb{Z}^d$ but rather on its shape.\n\nHere we study the following situation:\nSuppose that there is a finitely dependent process $\\mu$ that is concentrated on $X_\\Pi$.\nIn that case we say that $\\Pi$ is in the class $\\mathsf{FINDEP}$. \nIn other words, $\\Pi \\in \\mathsf{FINDEP}$ if there is a distribution from which we can sample solutions of $\\Pi$ such that if we restrict the distribution to any two sufficient far sets, the corresponding restrictions of the solution of $\\Pi$ that we sample are independent. \n\nIt is easy to see that all problems in $\\mathsf{LOCAL}(O(1))$ (that is, problems solvable by a block factor) are finitely dependent, but examples of finitely dependent processes outside of this class were rare and unnatural, see the discussion in \\cite{FinDepCol}, until Holroyd and Liggett \\cite{FinDepCol} provided an elegant construction for 3- and 4-coloring of a path and, more generally, for distance coloring of a grid with constantly many colors. This implies the following result.\n\n\\begin{theorem}[\\cite{FinDepCol}]\n$\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{FINDEP}$ (see \\cref{fig:big_picture_grids}).\n\\end{theorem} \n\nOn the other hand, recently Spinka \\cite{Spinka} proved that any finitely dependent process can be described as a finitary factor of iid process.\\footnote{His result works in a bigger generality for finitely generated amenable groups.}\nUsing \\cref{thm:toast_in_tail}, we sketch how this can be done, we keep the language of LCLs.\n\n\\begin{theorem}[\\cite{Spinka}]\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$, i.e., $\\Pi\\in \\mathsf{FINDEP}$.\nThen there is a finitary factor of iid $F$ (equivalently an oblivious local algorithm $\\mathcal{A}$) with tail decay $1\/r^{1- O(\\sqrt{\\log r})}$ such that the distribution induced by $F$ is equal to $\\mu$.\nIn fact, $\\Pi$ is in the class $\\TOAST$. \n\\end{theorem}\n\\begin{proof}[Proof Sketch]\nSuppose that every vertex is assigned uniformly two reals from $[0,1]$.\nUse the first string to produce a $k$-toast as in \\cref{thm:toast moment}, where $k$ is from the definition of $\\mu$.\n\nHaving this use the second random string to sample from $\\mu$ inductively along the constructed $k$-toast.\nIt is easy to see that this produces ffiid $F$ that induces back the distribution $\\mu$.\nIn fact, it could be stated separately that the second part describes a randomized $\\TOAST$ algorithm: always samples from $\\mu$ given the coloring on smaller pieces that was already fixed. The finitely dependent condition is exactly saying that the joint distribution over the solution on the small pieces of $\\mu$ is the same as the distribution we sampled from.\nNow use \\cref{thm:derandomOfToast} to show that $\\Pi\\in \\TOAST$. \n\\end{proof}\n\nIn our formalism, $\\mathsf{FINDEP} \\subseteq \\TOAST$, or equivalently we get $\\mathsf{TAIL}(\\mathrm{tower}(\\Omega(r))) \\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TAIL}(r^{1-o(1)})$.\nIt might be tempting to thing that perhaps $\\mathsf{FINDEP} =\\TOAST$.\nIt is however a result of Holroyd et al \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring} that proper vertex $3$-coloring, $\\Pi^{d}_{3,\\chi}$, is not in $\\mathsf{FINDEP}$ for every $d>1$.\nThe exact relationship between these classes remain mysterious.\n\n\n\n\\begin{comment}\n\\begin{theorem}\nLet $\\Pi$ be an LCL on $\\mathbb{Z}^d$.\nSuppose that $\\Pi$ is in the class $\\mathsf{TAIL}(\\tow(\\Omega(r)))$.\nThen there is finitely dependent process $\\mu$ on $X_\\Pi$.\n\\end{theorem}\n\\begin{proof}\nIt follows from Theorem~\\ref{thm:classification_of_local_problems_on_grids} that there is $r,\\ell \\in \\mathbb{N}$ such that $\\ell\\le r-t$ and a rule $\\mathcal{A}$ of locality at most $\\ell$ that produces a $\\Pi$-coloring for every $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$.\n\nSuppose that $\\mu_{(r+t)}$ is a finitely dependent $(r+t)$-power proper vertex coloring of $\\mathbb{Z}^d$ process with locality $k\\in \\mathbb{N}$.\nWe claim that composing $\\mu_{(r+t)}$ with $\\mathcal{A}$ yields a finitely dependent process with locality $(2\\ell+k)$.\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite set.\nThen the output of $\\mathcal{A}$ depends on $\\mathcal{B}(A,\\ell)$.\nTherefore if $A,B\\subseteq \\mathbb{Z}^d$ satisfy $d(A,B)>2\\ell+k$, then\n$$d(\\mathcal{B}(A,\\ell),\\mathcal{B}(B,\\ell))>k$$\nand we conclude that the outputs of $\\mathcal{A}$ on $A$ and $B$ are independent.\n\nIt remains to find $\\mu_{r}$ finitely dependent $r$-power proper vertex coloring of $\\mathbb{Z}^d$ process for every $r>0$.\nThis is done by Holroyd and Liggett \\cite[Theorem~20]{FinDepCol}.\n{\\color{red} or something like that, they use a lot of colors fr $r$-power coloring...}\n\\end{proof}\n\nGiven a fiid $F$ we define $F^*\\lambda$ to be the push-forward of the product measure via $F$.\nSince $F$ is equivariant it follows that $F^*\\lambda$ is shift-invariant.\n\nThe following theorem was essentially proven by Spinka\\cite{?}. His result nicely fits to our definitions in \\cref{sec:Toast}. \n\\begin{theorem}\n\n\\end{theorem}\n\n\n\\begin{theorem}\nLet $\\Pi$ be an LCL problem on $\\mathbb{Z}^d$ and $\\mu$ be finitely dependent process on $X_\\Pi$.\nThen for every $\\epsilon>0$ there is a ffiid $F$ such that $F^*\\lambda=\\mu$ and the tail of $F$ has the $(1-\\epsilon)$-moment finite.\nMoreover, $\\Pi$ is in the class $\\TOAST$.\n\\end{theorem}\n\n\\begin{remark}\nMaybe the random toast construction works for any finitely dependent processes not just LCL.\n\nThere is the notion of strongly irreducible shifts of finite type, some comment about it...\n\nDefine $\\mathsf{FINDEP}$ to be the class of all LCL problems that admit a finitely dependent distribution $\\mu$.\nIt follows that $\\mathsf{TAIL}(\\tow(\\Omega(r)))\\subseteq \\mathsf{FINDEP} \\subseteq \\mathsf{TOAST}$ and we have $\\mathsf{FINDEP}\\not = \\TOAST$ because proper vertex $3$-coloring $\\Pi^d_{\\chi,3}$ is not in $\\mathsf{FINDEP}$ for every $d>1$, by \\cite[Corollary~25]{HolroydSchrammWilson2017FinitaryColoring}.\n\\end{remark}\n\n\n\\begin{proof}\nWe define a spontaneous randomized toast algorithm $\\Omega$ such that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $k$-toast $\\mathcal{D}$ and every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$.\nThis already implies the moreover part by Theorem~\\ref{thm:derandomOfToast}.\n\nLet $A\\subseteq \\mathbb{Z}^d$ be a finite subset of $\\mathbb{Z}^d$.\nWrite $\\mathbb{P}_A$ for the distribution of $\\Pi$-patterns of $\\mu$ on $A$, i.e., $\\mathbb{P}_A=\\mu\\upharpoonright A$.\nSuppose now that $(A,c,\\ell)$ are given, where $A\\subseteq \\mathbb{Z}^d$ is finite subset, $c$ is a partial coloring of $A$ and $\\ell\\in [0,1]^{A}$.\nConsider the conditional distribution $\\mathbb{P}_{A,c}$ of $\\mathbb{P}_A$ given the event $c$.\nIf $\\mathbb{P}_{A,c}$ is non-trivial, then define $\\Omega$ such that $\\Omega(A,c,{-})$ induces the same distribution as $\\mathbb{P}_{A,c}$ when $\\ell$ is fixed on $\\operatorname{dom}(c)$.\nOtherwise define $\\Omega$ arbitrarily.\n\nLet $\\mathcal{D}$ be a $k$-toast.\nWe show that $\\Omega(\\mathcal{D},\\ell)$ is a $\\Pi$-coloring for every $\\ell\\in [0,1]^{\\mathbb{Z}^d}$ and $\\Omega(\\mathcal{D},{-})$ produces $\\mu$.\nNote that the first claim shows that $\\Pi$ is in the class $\\RTOAST$, while the second produces $\\mu$ as ffiid as follows.\nSuppose that every $v\\in \\mathbb{Z}^d$ is assigned two independent reals $l_{v,0}$ and $l_{v,1}$.\nUse the first real to produce a $k$-toast as in Theorem~\\ref{thm:toast moment} and use the second real together with the produced toast as an input to $\\Omega$.\nIt follows from the independence of $l_{v,0}$ and $l_{v,1}$ that this produces $\\mu$.\n\nWe show that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nfor every $D\\in \\mathcal{D}$ inductively on $\\mathfrak{r}_{\\mathcal{D}}$.\nIf $\\mathfrak{r}_{\\mathcal{D}}(D)=0$, then it follows from the definition.\nSuppose that $\\mathfrak{r}_{\\mathcal{D}}(D)>0$ and let $E_1,\\dots, E_t\\in \\mathcal{D}$ be an enumeration of $E\\subset D$.\nIt follows from the inductive hypothesis together with the assumption that $\\mu$ is $k$-independent and $\\mathcal{D}$ is a $k$-toast that \n$$\\Omega(\\mathcal{D},{-})\\upharpoonright \\bigcup_{i\\le t}E_i=\\mu\\upharpoonright \\bigcup_{i\\le t} E_i.$$\nNow when $\\ell$, and therefore a partial coloring $c$, is fixed on $\\bigcup_{i\\le t}E_i=\\operatorname{dom}(c)$, the definition $\\Omega(D,c,{-})$ extends $c$ in such a way that $\\mathbb{P}_{D,c}=\\mu\\upharpoonright \\operatorname{dom}(c)$.\nSince this holds for any such assignment $\\ell$ of real numbers on $\\bigcup_{i\\le t}E_i$ we deduce that\n$$\\Omega(\\mathcal{D},{-})\\upharpoonright D=\\mu\\upharpoonright D$$\nand the proof is finished.\n\\end{proof}\n\\end{comment}\n\n\n\n\n\n\n\\subsection{$\\mathsf{TAIL}$ Complexities in Distributed and Centralised Models}\n\\label{subsec:congestLCA}\n\nAlthough we are not aware of a systematic work on $\\mathsf{TAIL}$ complexities in the distributed algorithms community, similar notions like oblivious algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}, node-average complexity \\cite{feuilloley2017node_avg_complexity_of_col_is_const,barenboim2018avg_complexity,chatterjee2020sleeping_MIS} or energy complexity \\cite{chang2018energy_complexity_exp_separation,chang2020energy_complexity_bfs} are studied. \nThe breakthrough MIS algorithm of Ghaffari \\cite{ghaffari2016MIS} may have found many applications exactly because of the fact that its basis is an algorithm with exponential tail behavior. \n\nWe discuss two applications of algorithms with nontrivial $\\mathsf{TAIL}$ complexities. First, the node-average complexity literature studies the average coding radius that a node needs to answer. Whenever an oblivious algorithm has its first moment finite, we have an algorithm with constant average coding radius. \nWith a little more work, one can see that the result from \\cref{thm:3in toast}, that is $3$-coloring of grids has $\\mathsf{TAIL}$ complexity $r^{1-O(1\/\\sqrt{\\log r})}$  can be adapted to the setting of finite grids where it gives a $3$-coloring algorithm with average complexity $2^{O(\\sqrt{\\log n})}$. \nFor nonconstant $\\Delta$, the situation is more complicated. In \\cite{barenboim2018avg_complexity} it was shown that the Luby's $\\Delta+1$ coloring algorithm has tail $2^{\\Omega(r)}$ even for nonconstant $\\Delta$, but the question of whether MIS can have constant average coding radius in nonconstant degree graphs is open \\cite{chatterjee2020sleeping_MIS}. \n\nAnother application of the $\\mathsf{TAIL}$ bound approach comes from the popular centralised $\\mathsf{LCA}$ model \\cite{lca11,lca12}: there one has a huge graph and instead of solving some problem, e.g., coloring, on it directly, we are only required to answer queries about the color of some given nodes. It is known that, as in the $\\mathsf{LOCAL}$ model, one can solve $\\Delta+1$ coloring with $O(\\log^* n)$ queries on constant degree graphs by a variant of Linial's algorithm \\cite{even03_conflictfree}. But using the tail version of the Linial's algorithm instead, we automatically get that $O(\\log^* n)$ is only a worst-case complexity guarantee and on average we expect only $O(1)$ queries until we know the color of a given node. That is, the amortized complexity of a query is in fact constant. The same holds e.g. for the $O(\\log n)$ query algorithm for LLL from \\cite{brandt_grunau_rozhon2021LLL_in_LCA}. \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\\section{Open Questions}\n\\label{sec:open_problems}\n\nHere we collect some open problems that we find interesting. \n\n\\begin{enumerate}\n    \\item Is it true that $\\RLTOAST = \\LTOAST$? If yes, this implies $\\mathsf{TAIL} \\subseteq \\mathsf{BOREL}$. \n    \\item Is it true that $\\RLTOAST = \\RTOAST$? If yes, this implies $\\mathsf{TAIL} = \\mathsf{TAIL}(r^{1-o(1)})$ and answers the fourth question of Holroyd et al. \\cite{HolroydSchrammWilson2017FinitaryColoring}. \n\t\\item Can the tail complexity $r \/ 2^{O(\\sqrt{\\log r})}$ in the construction of toast be improved to $r \/ \\operatorname{poly} \\log (r)$? The same can be asked for the complexity of intermediate problems from \\cref{sec:Inter}. \n\t\t\\item Is there a local coloring problem $\\Pi$ such that ${\\bf M}(\\Pi)$ is non-integer value or $0$, i.e., ${\\bf M}(\\Pi)\\in [0,d-1]\\setminus \\{1,2,\\dots, d-1\\}$?\n\t\t\\item Can you find an example of a local problem without input labels which does not belong to any class in \\cref{fig:big_picture_grids} but the solutions to this problem are not ``periodic'' (this cannot happen for $d=1$). Maybe (a version of) Penrose tilings can be helpful there?\n\t\\item What is the value ${\\bf M}(\\Pi^d_{pm})$, where $\\Pi^d_{pm}$ is the problem of finding a perfect matching of $\\mathbb{Z}^d$, where $d>2$?\n    \\item Let $d>1$.\n\tFor what finite collections of $d$-dimensional rectangles $\\mathcal{R}$ is $\\Pi^d_{\\mathcal{R}}$ is in the class $\\mathsf{LOCAL}(O(\\log^* n))$? Is there a  finite collection of $d$-dimensional rectangles $\\mathcal{R}$ that would satisfy $1< {\\bf M}(\\Pi^d_{\\mathcal{R}})\\le d-1$?\n\t\\item Is it true (on $\\mathbb{Z}^d$) that $\\mathsf{BOREL}=\\mathsf{MEASURE}$ or $\\mathsf{BOREL} = \\mathsf{BAIRE}$?\n\\end{enumerate}\n\n\\section*{Acknowledgement}\nWe would like to thank Ferenc Bencs, Anton Bernshteyn, Sebastian Brandt, Yi-Jun Chang, Mohsen Ghaffari, Aranka Hru\u0161kov\u00e1, Stephen Jackson, Oleg Pikhurko, Brandon Sewart, Yinon Spinka, L\u00e1szlo T\u00f3th, and Zolt\u00e1n Vidny\u00e1szky for many engaging discussions. \n\nThe first author was supported by Leverhulme Research Project Grant RPG-2018-424. \nThis project has received funding from the European Research Council (ERC) under the European\nUnions Horizon 2020 research and innovation programme (grant agreement No. 853109). \n\n\\bibliographystyle{alpha}\n\n\n\\section{Introduction}\n\\label{sec:intro}\nIn this work, we study local problems on regular trees from three different perspectives. \n\n\\textit{First,} we consider the perspective of distributed algorithms. In distributed computing, the studied setup is a network of computers where each computer can only communicate with its neighbors. Roughly speaking, the question of interest in this area is which problems can be solved with only a few rounds of communication in the underlying network. \n\n\\textit{Second}, we consider the perspective of (finitary) factors of iid processes.\nIn probability, random processes model systems that appear to vary in a random manner.\nThese include Bernoulli processes, Random walks etc.\nA particular, well-studied, example is the Ising model.\n\n\\textit{Third}, we investigate the perspective of descriptive combinatorics. The goal of this area is to understand which constructions on infinite graphs can be performed without using the so-called axiom of choice.\n\nAlthough many of the questions of interest asked in these three areas are quite similar to each other, no systematic connections were known until an insightful paper of Bernshteyn~\\cite{Bernshteyn2021LLL} who showed that results from distributed computing can automatically imply results in descriptive combinatorics.\nIn this work, we show that the connections between the three areas run much deeper than previously known, both in terms of techniques and in terms of complexity classes.\nIn fact, our work suggests that it is quite useful to consider all three perspectives as part of a common theory, and we will attempt to present our results accordingly.\nWe refer the reader to \\cref{fig:big_picture_trees} for a partial overview of the rich connections between the three perspectives, some of which are proven in this paper. \n\nIn this work, we focus on the case where the graph under consideration is a regular tree.\nDespite its simplistic appearance, regular trees play an important role in each of the three areas, as we will substantiate at the end of this section.\nTo already provide an example, in the area of distributed algorithms, almost all locality lower bounds are achieved on regular trees.\nMoreover, when regarding lower bounds, the property that they already apply on regular trees actually \\emph{strengthens} the result---a fact that is quite relevant for our work as our main contribution regarding the transfer of techniques between the areas is a new lower bound technique in the area of distributed computation that is an adaptation and generalization of a technique from descriptive combinatorics.\nRegarding our results about the relations between complexity classes from the three areas, we note that such connections are also studied in the context of paths and grids in two parallel papers \\cite{grebik_rozhon2021LCL_on_paths,grebik_rozhon2021toasts_and_tails}.\n\nIn the remainder of this section, we give a high-level overview of the three areas that we study.\nThe purpose of these overviews is to provide the reader with a comprehensive picture of the studied settings that can also serve as a starting point for delving deeper into selected topics in those areas---in order to follow our paper, it is not necessary to obtain a detailed understanding of the results and connections presented in the overviews. Necessary technical details will be provided in \\cref{sec:preliminaries}.\nMoreover, in \\cref{sec:contribution}, we present our contributions in detail.\n\n\n\n\n\n\\paragraph{Distributed Computing}\nThe definition of the $\\mathsf{LOCAL}$ model of distributed computing by Linial~\\cite{linial92LOCAL} was motivated by the desire to understand distributed algorithms in huge networks. \nAs an example, consider a huge network of wifi routers. Let us think of two routers as connected by an edge if they are close enough to exchange messages. \nIt is desirable that such close-by routers communicate with user devices on different channels to avoid interference.\nIn graph-theoretic language, we want to properly color the underlying network. \nEven if we are allowed a color palette with $\\Delta+1$ colors where $\\Delta$ denotes the maximum degree of the graph (which would admit a simple greedy algorithm in a sequential setting), the problem remains highly interesting in the distributed setting, as, ideally, each vertex decides on its output color after only a few rounds of communication with its neighbors, which does not allow for a simple greedy solution. \n\nThe $\\mathsf{LOCAL}$ model of distributed computing formalizes this setup: we have a large network, where each vertex knows the network's size, $n$, and perhaps some other parameters like the maximum degree $\\Delta$. In the case of randomized algorithms, each vertex has access to a private random bit string, while in the case of deterministic algorithms, each vertex is equipped with a unique identifier from a range polynomial in the size $n$ of the network. \nIn one round, each vertex can exchange any message with its neighbors and can perform an arbitrary computation. The goal is to find a solution to a given problem in as few communication rounds as possible. \nAs the allowed message size is unbounded, a $t$-round $\\mathsf{LOCAL}$ algorithm can be equivalently described as a function that maps $t$-hop neighborhoods to outputs---the output of a vertex is then simply the output its $t$-hop neighborhood\nmapped to by this function.\nAn algorithm is correct if and only if the collection of outputs at all vertices constitutes a correct solution to the problem. \n\nThere is a rich theory of distributed algorithms and the local complexity of many problems is understood. \nThe case of trees is a highlight of the theory: it is known that any local problem (a class of natural problems we will define later) belongs to one of only very few complexity classes.\nMore precisely, for any local problem, its randomized local complexity is either $O(1), \\Theta(\\log^* n), \\Theta(\\log\\log n), \\Theta(\\log n)$, or $\\Theta(n^{1\/k})$ for some $k \\in \\mathbb{N}$. Moreover, the deterministic complexity is always the same as the randomized one, except for the case $\\Theta(\\log\\log n)$, for which the corresponding deterministic complexity is $\\Theta(\\log n)$ (see \\cref{fig:big_picture_trees}). \n\n\n\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig\/big_picture_trees.pdf}\n    \\caption{Complexity classes on regular trees considered in the three areas of distributed computing, factors of iid processes\/uniform algorithms, and descriptive combinatorics. The left part shows complexity classes of distributed computing. We use the shorthand $\\mathsf{LOCAL}$ if it does not matter whether we talk about the deterministic or randomized complexity. These two notions differ only for the class of problems of randomized local complexity $O(\\log\\log n)$, which have deterministic complexity $O(\\log n)$. \\\\\n    The uniform complexity classes of sublogarithmic complexity are in correspondence to appropriate classes in the randomized local complexity model, as proven in \\cref{sec:separating_examples}. On the other hand, the class fiid is very similar to the class $\\mathsf{MEASURE}$ from descriptive combinatorics. The equivalence of the class $\\mathsf{CONTINUOUS}$ and $\\mathsf{LOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$ is marked with a dashed arrow as it was proven in case the tree is generated by a group action (think of the tree being equipped with an additional $\\Delta$-edge coloring). The inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ however clearly holds also in our setting. The class $\\mathsf{BOREL}$ is incomparable with $\\mathsf{RLOCAL}(O(\\log\\log n))$, as proven in \\cref{sec:separating_examples}.   }\n    \\label{fig:big_picture_trees}\n\\end{figure}\n\n\n\\todo{thm 5.1 - 5.3}\n\n\n\n\n\\paragraph{(Finitary) Factors of iid Processes and Uniform Algorithms}\n\nIn recent years, \\emph{factors of iid (fiid) processes} on trees attracted a lot of attention in combinatorics, probability, ergodic theory and statistical physics \\cite{ABGGMP,ArankaFeriLaci,BackSzeg,BackVir,backhausz,backhausz2,BGHV,Ball,Bowen1,Bowen2,CPRR,csoka2015invariant,GabLyo,GamarnikSudan,GGP,HarangiVirag,HolroydPoisson,HLS,HPPS,HolroydPeres,HoppenWormald,Kun,RahmanPercolation,RahVir,LyonsTrees,Mester,schmidt2016circular,Timar1,Timar2}.\nIntuitively, factors of iid processes are randomized algorithms on, e.g., infinite $\\Delta$-regular trees, where each vertex outputs a solution to a problem after it explores random strings on vertices of the whole tree.\nAs an example, consider the \\emph{perfect matching} problem.\nAn easy parity argument shows that perfect matching cannot be solved by any local randomized algorithm on finite trees.\nHowever, if we allow a small fraction of vertices not to be matched, then, by a result of Nguyen and Onak \\cite{nguyen_onak2008matching} (see also \\cite{ElekLippner}), there is a constant-round randomized algorithm that produces such a matching on high-girth graphs (where the constant depends on the fraction of unmatched vertices that we allow).\nThis result can also be deduced from a result of Lyons and Nazarov \\cite{lyons2011perfect}, who showed that perfect matching can be described as a factor of iid process on an infinite $\\Delta$-regular tree.\nThe high-level idea behind this connection is that high-girth graphs approximate the infinite $\\Delta$-regular tree and constant-round local algorithms approximate factors of iid processes.\nThis correspondence is formalized in the notion of \\emph{Benjamini-Schramm} or \\emph{local-global} convergence \\cite{benjamini2011recurrence, hatamilovaszszegedy}.\nWe note that getting only ``approximate'' solutions, that is, solutions where a small fraction of vertices does not have to satisfy the constraints of a given problem, is intrinsic in this correspondence.\nRegardless, there are many techniques, such as entropy inequality \\cite{backhausz} or correlation decay \\cite{backhausz2}, and particular results such as the aforementioned perfect matching problem \\cite{lyons2011perfect} that provide lower and upper bounds, respectively, in our setting as well.\nWe refer the reader to \\cite{LyonsTrees,BGHV} for a comprehensive summary of the field. \n\n\nIn this paper, we mostly consider a stronger condition than fiid, namely so-called \\emph{finitary} factors of iid (ffiid) processes that are studied in the same context as fiid \\cite{holroyd2017one_dependent_coloring,HolroydSchrammWilson2017FinitaryColoring,Spinka}. \nPerhaps surprisingly, the notion of ffiid is identical to the notion of so-called uniform distributed randomized algorithms \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring,grebik_rozhon2021toasts_and_tails} that we now describe. \nWe define an uniform local algorithm as a randomized local algorithm that does not know the size of the graph $n$ -- this enables us to run such an algorithm on infinite graphs, where there is no $n$. \nMore precisely, we require that each vertex eventually outputs a solution that is compatible with the output in its neighborhood, but the time until the vertex finishes is a potentially unbounded random variable. \nAs in the case of classical randomized algorithms, we can now measure the \\emph{uniform complexity} of an uniform local algorithm (known as the tail decay of ffiid \\cite{HolroydSchrammWilson2017FinitaryColoring}). \nThe uniform complexity of an algorithm is defined as the function $t(\\varepsilon)$ such that the probability that the algorithm run on a specific vertex needs to see outside its $t(\\varepsilon)$-hop neighborhood is at most $\\varepsilon$. \nAs in the case of classical local complexity, there is a whole hierarchy of possible uniform complexities (see \\cref{fig:big_picture_trees}). \n\nWe remark that uniform distributed local algorithms can be regarded as Las Vegas algorithms. The output will always be correct; there is however no fixed guarantee at what point all vertices have computed their final output. On the other hand, a randomized distributed local algorithm can be viewed as a Monte Carlo algorithm as it needs to produce an output after a fixed number of rounds, though the produced output might be incorrect. \n\n\n\n\n\n\n\n\n\\paragraph{Descriptive Combinatorics}\n   \n   \n\t\n\tThe Banach-Tarski paradox states that a three-dimensional ball of unit volume can be decomposed into finitely many pieces that can be moved by isometries (distance preserving transformations such as rotations and translations) to form two three-dimensional balls each of them with unit volume(!).\n    The graph theoretic problem lurking behind this paradox is the following: fix finitely many isometries of $\\mathbb{R}^3$ and then consider a graph where $x$ and $y$ are connected if there is an isometry that sends $x$ to $y$.\n\tThen our task becomes to find a perfect matching in the appropriate subgraph of this graph -- namely, the bipartite subgraph where one partition contains points of the first ball and the other contains points of the other two balls. \n\tBanach and Tarski have shown that, with a suitably chosen set of isometries, the axiom of choice implies the existence of such a matching. \n\tIn contrast, since isometries preserve the Lebesgue measure, the pieces in the decomposition cannot be Lebesgue measurable. \n\tSurprisingly, Dougherty and Foreman \\cite{doughertyforeman} proved that the pieces in the Banach-Tarski paradox can have the \\emph{Baire property}. \n\tThe Baire property is a topological analogue of being Lebesgue measurable; a subset of $\\mathbb{R}^3$ is said to have the Baire property if its difference from some open set is topologically negligible.\n\t\n       \n\t    \n\t\n\t\n    Recently, results similar to the Banach-Tarski paradox that lie on the border of combinatorics, logic, group theory, and ergodic theory led to an emergence of a new field often called \\textit{descriptive} or \\textit{measurable combinatorics}. \n    The field focuses on the connection between the discrete and continuous and is largely concerned with the investigation of graph-theoretic concepts. \n    The usual setup in descriptive combinatorics is that we have a graph with uncountably many connected components, each being a countable graph of bounded degree. \n    For example, in case of the Banach-Tarski paradox, the vertices of the underlying graph are the points of the three balls, edges correspond to isometries, and the degree of each vertex is bounded by the number of chosen isometries. \n    Some of the most beautiful results related to the field include  \\cite{laczk, marksunger,measurablesquare,doughertyforeman,marks2016baire,gaboriau,KST,DetMarks,millerreducibility,conley2020borel,csokagrabowski,Bernshteyn2021LLL}, see  \\cite{kechris_marks2016descriptive_comb_survey,pikhurko2021descriptive_comb_survey} for recent surveys. \n    \n    Importantly, in many of these results, including the Banach-Tarski paradox, graphs where each component is an infinite $\\Delta$-regular tree appear naturally. \n    Oftentimes, questions considered in descriptive combinatorics lead to constructing a solution to a local problem in the underlying uncountable graph (in the case of Banach-Tarski, the local problem is perfect matching).  \n    The construction needs to be such that the solution of the problem has some additional regularity properties. For example in the case of Banach-Tarski, a solution is possible when the regularity condition is the Baire property, but not if it is Lebesgue measurability.\n    In fact, together with Borel measurability these are the most prominent regularity conditions studied in descriptive combinatorics.\n   \n    The corresponding complexity classes of local problems that always admit a solution with the respective regularity property are $\\mathsf{BOREL},\\mathsf{MEASURE},\\mathsf{BAIRE}$ (See \\cref{fig:big_picture_trees}). \n    In this paper, we moreover consider the setting where each connected component of the underlying graph is a $\\Delta$-regular tree. \n   \n   \n    \n    \n    \n   \n\t\n\t\n\t\t\n\t\n\t\t\n\t\t\n\n\t    The connection between distributed computing and descriptive combinatorics arises from the fact that in descriptive combinatorics we care about constructions that do not use the axiom of choice. \n\t    In the distributed language, the axiom of choice corresponds to leader election, that is, the constructions in descriptive combinatorics do not allow picking exactly one point in every component.  \n\t   \n\t    To get some intuition about the power of the complexity class $\\mathsf{BOREL}$, we note that Borel constructions allow us to alternate countably many times the following two operations. \n\t    First, any local algorithm with  constant local complexity can be run. \n\t    Second, we have an oracle that provides a maximal independent set (MIS) on any graph that can be constructed locally from the information computed so far \\cite{KST}. \n        Note that from the speedup result of \\cite{chang_kopelowitz_pettie2019exp_separation} we get that every local problem with local complexity $O(\\log^* n)$ can be solved by constant local constructions and \\emph{one} call to such an MIS oracle. This implies the inclusion $\\mathsf{LOCAL}(O(\\log^* n)) \\subseteq \\mathsf{BOREL}$ in \\cref{fig:big_picture_trees} proven in the insightful paper of Bernshteyn \\cite{Bernshteyn2021LLL}.  \n        The relationship of the class $\\mathsf{MEASURE}$ (and of the class $\\mathsf{fiid}$ from the discussion of factors) to the class $\\mathsf{BOREL}$ is analogous to the relationship of randomized distributed algorithms to deterministic distributed algorithms. \n\n       \n        \n\n\n\n\n\n\n\t\n\n\t\n\t\n\t\n\n\n\n\n\n\n\\paragraph{Local Problems on Regular Trees}\n\nAfter introducing the three areas of interest in this work, we conclude the section by briefly discussing the kinds of problems we focus on, which are local problems on regular trees.\nMore precisely, we study \\emph{locally checkable labeling (LCL)} problems, which are a class of problems, where the correctness of the solution can be checked locally.\nExamples include classical problems from combinatorics such as proper vertex coloring, proper edge coloring, perfect matching, and maximal independent set.\nOne main goal of this paper is to understand possible complexity classes of LCLs without inputs on infinite $\\Delta$-regular trees and their finite analogues.\nWe refer the reader to \\cref{sec:preliminaries} for a precise definition of a finite $\\Delta$-regular tree.\n\nThe motivation for studying regular trees in this work stems from different sources: (a) infinite $\\Delta$-regular trees are studied in the area of ergodic theory \\cite{Bowen1,Bowen2}, random processes \\cite{backhausz,backhausz2,lyons2011perfect} and descriptive combinatorics \\cite{DetMarks,BrooksMeas}, (b) many lower bounds in distributed computing are proven in regular trees \\cite{balliu_et_al:binary_lcl,balliu2019LB,brandt_etal2016LLL,brandt19automatic,Brandt20tightLBmatching,ChangHLPU20,goeoes14_PODC}, and (c) connecting and comparing the techniques of the three areas in this simple setting reveals already deep connections, see \\cref{sec:contribution}.\n\n\n\n\n\n\n\n\n\n\\section{Our Contributions}\n\\label{sec:contribution}\n\nWe believe that our main contribution is presenting all three perspectives as part of a common theory. Our technical contribution is split into three main parts. \n\n\n\n\n\\subsection{Generalization of Marks' Technique}\\label{sec:intromarks}\nIn Section \\ref{sec:marks} we extend the Borel determinacy technique of Marks \\cite{DetMarks}, which was used to prove the nonexistence of Borel $\\Delta$-colorings and perfect matchings, to a  broader class of problems, and \nadapt the extended technique to the distributed setting, thereby obtaining a simple method for proving distributed lower bounds.\nThis method is the first lower bound technique for distributed computing using ideas coming from descriptive combinatorics (see \\cite{BernshteynVizing} for a distributed computing upper bound motivated by descriptive combinatorics).\nMoreover, we show how to use the developed techniques to obtain both $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}$ lower bounds for local problems from a natural class, called homomorphism problems.\nOur key technical ingredient for obtaining the mentioned techniques and results is a novel technique based on the notion of an \\emph{ID graph}.\nWe note that a very similar concept to the ID graph was independently discovered by \\cite{id_graph}. \n\n\\paragraph{Marks' technique}\nIn the following we give an introduction to Marks' technique by going through a variant of his proof~\\cite{DetMarks,Marks_Coloring} that shows that $\\Delta$-coloring has deterministic local complexity $\\Omega(\\log n)$. The proof already works in the case where the considered regular tree comes with an input $\\Delta$-edge coloring. In this case, the output color of a given vertex $u$ can be interpreted as that $u$ ``grabs'' the incident edge of that color. The problem is hence equivalent to the \\emph{edge grabbing} problem where every vertex is required to grab an incident edge such that no two vertices grab the same edge. \n\nWe first show the lower bound in the case that vertices do not have unique identifiers but instead are properly colored with $L > \\Delta$ colors. \nSuppose there is an algorithm $\\mathcal{A}$ solving the edge grabbing problem with local complexity $t(n) = o(\\log n)$, and consider a tree rooted at vertex $u$ of depth $t(n)$; such a tree has less than $n$ vertices, for large enough $n$.\nAssume that $u$ has input color $\\sigma \\in [L]$, and consider, for some fixed edge color $\\alpha$, the edge $e$ that is incident to $u$ and has color $\\alpha$.\nTwo players, Alice and Bob, are playing the following game. In the $i$-th round, Alice colors the vertices at distance $i$ from $u$ in the subtree reachable via edge $e$ with colors from $[L]$. Then, Bob colors all other vertices at distance $i$ from $u$ with colors from $[L]$ (see \\cref{fig:game}).\nConsider the output of $u$ when executing $\\mathcal{A}$ on the obtained colored tree.\nBob wins the game if $u$ grabs the edge $e$, and Alice wins otherwise. \n\nNote that either Alice or Bob has a winning strategy. \nGiven the color $\\sigma$ of $u$, if, for each edge color $\\alpha$, Alice has a winning strategy in the game corresponding to the pair $(\\sigma, \\alpha)$, then we can create $\\Delta$ copies of Alice and let them play their strategy on each subtree of $u$, telling them that the colors chosen by the other Alices are what Bob played. The result is a coloring of the input tree such that $u$, by definition, does not pick any edge, contradicting the fact that $\\mathcal{A}$ provides a valid solution!\nSo for every $\\sigma$ there is at least one $\\alpha$ such that Bob has a winning strategy for the game corresponding to $(\\sigma, \\alpha)$. By the pigeonhole principle, there are two colors $\\sigma_1, \\sigma_2$, such that Bob has a winning strategy for both pairs $(\\sigma_1, \\alpha)$ and $(\\sigma_2, \\alpha)$. \nBut now we can imagine a tree rooted in an edge between vertices $u_1, u_2$ that are colored with colors $\\sigma_1, \\sigma_2$. We can now take two copies of Bob, one playing at $u_1$ and the other playing at $u_2$ and let them battle it out, telling each copy that the other color from $\\{\\sigma_1, \\sigma_2\\}$ and whatever the other copy plays are the moves of Alice. The resulting coloring has the property that both $u_1$ and $u_2$, when executing $\\mathcal{A}$ on the obtained colored tree, grab the edge between them, a contradiction that finishes the proof!\n\n\\paragraph{The ID graph}\nThe downside of the proof is that it does not work in the model with unique identifiers (where the players' moves consist in assigning identifiers instead of colors), since gluing copies of the same player could result in an identifier assignment where the identifiers are not unique. \nOne possible remedy is to conduct the whole proof in the context of Borel graphs as was done by Marks. This proves an even stronger statement, namely that $\\Delta$-coloring is not in the class $\\mathsf{BOREL}$, but requires additional ad-hoc tricks and a pretty heavy set theoretic tool---Martin's celebrated Borel determinacy theorem~\\cite{martin} stating that even for infinite two-player games one of the players has to have a winning strategy if the payoff set is Borel. \nThe ID graph enables us to adapt the proof (and its generalization that we develop in \\cref{sec:marks}) to the distributed setting, where the fact that one of the players has a winning strategy is obvious. \nMoreover, we use an infinite version of the ID graph to generalize Marks' technique also in the Borel setting. \n\nHere is how it works: \nThe ID graph is a specific graph whose vertices are the possible unique input identifiers (for input graphs of size $n$), that is, numbers from $[n^{O(1)}]$. Its edges are colored with colors from $[\\Delta]$ and its girth is $\\Omega(\\log n)$. When we define the game between Alice and Bob, we require them to label vertices with identifiers in such a way that whenever a new vertex is labeled with identifier $i$, and its already labeled neighbor has identifier $j$, then $ij$ is an edge in the ID graph.\nMoreover, the color of edge $ij$ in the ID graph is required to be the same as the color of the edge between the vertices labeled $i$ and $j$ in the tree where the game is played. \nIt is straightforward to check that these conditions enforce that even if we let several copies of the same player play, the resulting tree is labeled with unique identifiers. Hence, the same argument as above now finally proves that the deterministic local complexity of $\\Delta$-coloring is  $\\Omega (\\log n)$. \n\nWe note that our ID graph technique is of independent interest and may have applications in many different contexts. \nTo give an example from distributed computing, consider the proof of the deterministic $\\Omega(\\log n)$-round lower bound for $\\Delta$-coloring developed by the distributed community~\\cite{brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation}, which is based on the celebrated round elimination technique. \nEven though the result is deterministic, the proof is quite technical due to the fact that it relies on examining randomized algorithms, for reasons similar to the reasons why Marks' proof does not apply directly to the setting with unique identifiers. \nFortunately, it can be again streamlined with the use of the ID graph technique. \nMoreover, in a follow-up work, the ID graph technique has already led to a new lower bound for the Lov\u00e1sz local lemma~\\cite{brandt_grunau_rozhon2021LLL_in_LCA} in the area of Local Computation Algorithms (which is part of the realm of sequential computation), thereby giving further evidence for the versatility of the technique. \n\n\n\\paragraph{Marks vs. Round Elimination}\nIt is quite insightful to compare Marks' technique (and our generalization of it) with the powerful round elimination technique~\\cite{brandt19automatic}, which has been responsible for all locality lower bounds of the last years~\\cite{brandt_etal2016LLL, balliu2019LB, brandt19automatic, balliu2019hardness_homogeneous, balliu_et_al:binary_lcl, Brandt20tightLBmatching, balliu20rulingset, ChangHLPU20, balliu21dominatingset}. \nWhile, on the surface, Marks' approach developed for the Borel world may seem quite different from the round elimination technique, there are actually striking similarities between the two methods.\nOn a high level, in the round elimination technique, the following argument is used to prove lower bounds in the LOCAL model:\nIf a $T$-round algorithm exists for a problem $\\Pi_0$ of interest, then there exists a $(T-1)$-round algorithm for some problem $\\Pi_1$ that can be obtained from $\\Pi_0$ in a mechanical manner.\nBy applying this step iteratively, we obtain a problem $\\Pi_t$ that can be solved in $0$ rounds; by showing that there is no $0$-algorithm for $\\Pi_t$ (which is easy to do if $\\Pi_t$ is known), a $(T+1)$-round lower bound for $\\Pi_0$ is obtained.\n\nThe interesting part regarding the relation to Marks' technique is how the ($T-i-1$)-round algorithms $\\mathcal{A}'$ are obtained from the ($T-i$)-round algorithms $\\mathcal{A}$ in the round elimination framework: in order to execute $\\mathcal{A}'$, each vertex $v$, being aware of its ($T-i-1$)-hop neighborhood, essentially asks whether, for all possible extensions of its view by one hop along a chosen incident edge, there exists some extension of its view by one hop along all other incident edges such that $\\mathcal{A}$, executed on the obtained ($T-i$)-hop neighborhood, returns a certain output at $v$, and then bases its output on the obtained answer.\nIt turns out that the vertex sets corresponding to these two extensions correspond precisely to two moves of the two players in the game(s) played in Marks' approach: more precisely, in round $T-i$ of a game corresponding to the considered vertex $v$ and the chosen incident edge, the move of Alice consists in labeling the vertices corresponding to the first extension, and the move of Bob consists in labeling the vertices corresponding to the second extension.\n\nHowever, despite the similarities, the two techniques (at least in their current forms) have their own strengths and weaknesses and are interestingly different in that there are local problems that we know how to obtain good lower bounds for with one technique but not the other, and vice versa.\nFinding provable connections between the two techniques is an exciting research direction that we expect to lead to a better understanding of the possibilities and limitations of both techniques.\n\nIn \\cref{sec:marks} we use our generalized and adapted version of Marks' technique to prove new lower bounds for so-called homomorphism problems. \nHomomorphism problems are a class of local problems that generalizes coloring problems---each vertex is to be colored with some color and there are constraints on which colors are allowed to be adjacent. \nThe constraints can be viewed as a graph---in the case of coloring this graph is a clique. \nIn general, whenever the underlying graph of the homomorphism problem is $\\Delta$-colorable, its deterministic local complexity is $\\Omega(\\log n)$, because solving the problem would imply that we can solve $\\Delta$-coloring too (in the same runtime). \nIt seems plausible that homomorphism problems of this kind are the only hard, i.e., $\\Omega(\\log n)$, homomorphism problems. However, our generalization of Marks' technique asserts that this is not true. \n\n\\begin{theorem}\nThere are homomorphism problems whose deterministic local complexity on trees is $\\Omega(\\log n)$ such that the chromatic number of the underlying graph is $2\\Delta-2$.\n\\end{theorem}\n\nIt is not known how to prove the same lower bounds using round elimination\\footnote{Indeed, the descriptions of the problems have comparably large numbers of labels and do not behave like so-called ``fixed points'' (i.e., nicely) under round elimination, which suggests that it is hard to find a round elimination proof with the currently known approaches.}; in fact, as far as we know, these problems are the only known examples of problems on $\\Delta$-regular trees for which a lower bound is known to hold but currently not achievable by round elimination. \nProving the same lower bounds via round elimination is an exciting open problem. \n\n\n\n\n\n\n\n\n\n\n\\subsection{Separation of Various Complexity Classes}\n\\label{sec:introseparation}\n\\paragraph{Uniform Complexity Landscape}\nWe investigate the connection between randomized and uniform distributed local algorithms, where uniform algorithms are equivalent to the studied notion of finitary factors of iid. \nFirst, it is simple to observe that local problems with uniform complexity $t(\\varepsilon)$ have randomized complexity $t(1 \/ n^{O(1)})$ -- by definition, every vertex knows its local output after that many rounds with probability $1 - 1\/n^{O(1)}$. The result thus follows by a union bound over the $n$ vertices of the input graph.\n\nOn the other hand, we observe that on $\\Delta$-regular trees the implication also goes in the opposite direction in the following sense. Every problem that has a randomized complexity of $t(n) = o(\\log n)$ has an uniform complexity of $O(t(1\/\\varepsilon))$. \n\nOne could naively assume that this equivalence also holds for higher complexities, but this is not the case. Consider for example the $3$-coloring problem.\nIt is well-known in the distributed community that $3$-coloring a tree can be solved deterministically in $O(\\log n)$ rounds  using the rake-and-compress decomposition~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,MillerR89}.\nOn the other hand, there is no uniform algorithm for $3$-coloring a tree. \nIf there were such an uniform algorithm, we could run it on any graph with large enough girth and color $99\\%$ of its vertices with three colors. This in turn would imply that the high-girth graph has an independent set of size at least $0.99 \\cdot n\/3$. This is a contradiction with the fact that there exists high-girth graphs with a much smaller independence number~\\cite{bollobas}. \n\nInterestingly, the characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that \\emph{any} uniform distributed algorithm can be ``sped up'' to a deterministic local $O(\\log n)$ complexity, as we prove in~\\cref{thm:MainBaireLog}. \n\nWe show that there are local problems that can be solved by an uniform algorithm but only with a complexity of $\\Omega(\\log 1\/\\varepsilon)$. Namely, the problem of constructing a $2$-hop perfect matching on infinite $\\Delta$-regular trees for $\\Delta \\ge 3$ has an uniform local complexity between $\\Omega(\\log 1\/\\varepsilon)$ and $O(\\operatorname{poly}\\log 1\/\\varepsilon)$. \nFormally, this proves the following theorem. \n\n\\begin{theorem}\n$\\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon)) \\subsetneq \\mathsf{OLOCAL}(O(\\operatorname{poly}\\log 1\/\\varepsilon))$. \n\\end{theorem}\n\nThe uniform  algorithm for this problem is based on a so-called one-ended forest decomposition introduced in \\cite{BrooksMeas} in the descriptive combinatorics context. In a one-ended forest decomposition, each vertex selects exactly one of its neighbors as its parent by orienting the corresponding edge outwards. This defines a decomposition of the vertices into infinite trees. We refer to such a decomposition as a one-ended forest decomposition if the subtree rooted at each vertex only contains finitely many vertices. \nHaving computed such a decomposition, $2$-hop perfect matching can be solved inductively starting from the leaf vertices of each tree.\n\nWe leave the understanding of the uniform complexity landscape in the regime $\\Omega(\\log 1\/\\varepsilon)$ as an exciting open problem. In particular, does there exist a function $g(\\varepsilon)$ such that each local problem that can be solved by an uniform algorithm has an uniform complexity of $O(g(\\varepsilon))$?\n\n\n\n\\paragraph{Relationship of Distributed Classes with Descriptive Combinatorics}\n\nBernshteyn recently proved that $\\mathsf{LOCAL}(\\log^* n) \\subseteq \\mathsf{BOREL}$ \\cite{Bernshteyn2021LLL}. That is, each local problem with a deterministic $\\mathsf{LOCAL}$ complexity of $O(\\log^* n)$ also admits a Borel-measurable solution. A natural question to ask is whether the converse also holds. \nIndeed, it is known that $\\mathsf{LOCAL}(\\log^* n) = \\mathsf{BOREL}$ on paths with no additional input \\cite{grebik_rozhon2021LCL_on_paths}. \nWe show that on regular trees the situation is different. \nOn one hand, a characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} implies that $\\mathsf{BOREL} \\subseteq \\mathsf{BAIRE} \\subseteq \\mathsf{LOCAL}(O(\\log n))$. \nOn the other hand, we show that this result cannot be strengthened by proving the following result. \n\n\\begin{theorem}\n$\\mathsf{BOREL} \\subsetneq \\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\nThat is, there exists a local problem that admits a Borel-measurable solution but cannot be solved with a (randomized) $\\mathsf{LOCAL}$ algorithm running in a sublogarithmic number of rounds. \n\nLet us sketch a weaker separation, namely that $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n)) \\neq \\emptyset$. Consider a version of $\\Delta$-coloring where a subset of vertices can be left uncolored. However, the subgraph induced by the uncolored vertices needs to be a collection of doubly-infinite paths (in finite trees, this means each path needs to end in a leaf vertex). \nThe nonexistence of a fast distributed algorithm for this problem essentially follows from the celebrated $\\Omega(\\log n)$ deterministic lower bound for $\\Delta$-coloring of~\\cite{brandt_etal2016LLL}. \nOn the other hand, the problem allows a Borel solution. First, sequentially compute $\\Delta-2$ maximal independent sets, each time coloring all vertices in the MIS with the same color, followed by removing all the\ncolored vertices from the graph. In that way, a total of $\\Delta-2$ colors are used. Moreover, each\nuncolored vertex has at most $2$ uncolored neighbors. \nThis implies that the set of uncolored vertices forms a disjoint union of finite paths, one ended\ninfinite paths and doubly infinite paths. The first two classes can be colored inductively with two\nadditional colors, starting at one endpoint of each path in a Borel way (namely it can be done by making use of the countably many MISes in larger and larger powers of the input graph). \nHence, in the end only doubly infinite paths are left uncolored, as desired.  \n\nTo show the stronger separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{RLOCAL}(o(\\log n))$ we use a variation of the\n$2$-hop perfect matching problem. In this variation, some of the vertices can be left unmatched, but similar as in the variant of the $\\Delta$-coloring problem described above, the graph induced by all the unmatched vertices needs to satisfy some additional constraints.\n\nWe conclude the paragraph by noting that the separation between the classes $\\mathsf{BOREL}$ and $\\mathsf{LOCAL}(O(\\log^* n))$  is not as simple as it may look in the following sense. \nThis is because problems typically studied in the $\\mathsf{LOCAL}$ model with a $\\mathsf{LOCAL}$ complexity of $\\omega(\\log^* n)$ like $\\Delta$-coloring and perfect matching also do not admit a Borel-measurable solution due to the technique of Marks \\cite{DetMarks} that we discussed in \\cref{sec:intromarks}.\n\n\n\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$} \n\nWe already discussed that one of complexity classes studied in descriptive combinatorics is the class $\\mathsf{BAIRE}$.\nRecently, Bernshteyn proved \\cite{Bernshteyn_work_in_progress} that all local problems that are in the complexity class $\\mathsf{BAIRE}, \\mathsf{MEASURE}$ or $\\mathsf{fiid}$ have to satisfy a simple combinatorial condition which we call being $\\ell$-full. \nOn the other hand, all $\\ell$-full problems allow a $\\mathsf{BAIRE}$ solution \\cite{Bernshteyn_work_in_progress}.\nThis implies a complete combinatorial characterization of the class $\\mathsf{BAIRE}$. \nWe defer the formal definition of $\\ell$-fullness to Section \\ref{sec:baire} as it requires a formal definition of a local problem. \nInformally speaking, in the context of vertex labeling problems, a problem is $\\ell$-full if we can choose a subset $S$ of the labels with the following property.\nWhenever we label two endpoints of a path of at least $\\ell$ vertices with two labels from $S$, we can extend the labeling with labels from $S$ to the whole path such that the overall labeling is valid.\nFor example, proper $3$-coloring  is $3$-full with $S=\\{1,2,3\\}$ because for any path of three vertices such that its both endpoints are colored arbitrarily, we can color the middle vertex so that the overall coloring is proper. \nOn the other hand, proper $2$-coloring is not $\\ell$-full for any $\\ell$. \n\nWe complement this result as follows. \nFirst, we prove that any $\\ell$-full problem has local complexity $O(\\log n)$, thus proving that all complexity classes considered in the areas of factors of iids and descriptive combinatorics from \\cref{fig:big_picture_trees} are contained in $\\mathsf{LOCAL}(O(\\log n))$. \nIn particular, this implies that the existence of \\emph{any} uniform algorithm implies a local distributed algorithm for the same problem of local complexity $O(\\log n)$. \nWe obtain this result via the well-known rake-and-compress decomposition~\\cite{MillerR89}.  \n\n\nOn the other hand, we prove that any problem in the class $\\mathsf{LOCAL}(O(\\log n))$ satisfies the $\\ell$-full condition. \nThe proof combines a machinery  developed by Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}\nwith additional nontrivial ideas.\nIn this proof we construct recursively a sequence of sets of rooted, layered, and partially labeled trees, where the partial labeling is computed by simulating any given $O(\\log n)$-round distributed algorithm, \nand then the set  $S$ meeting the $\\ell$-full condition is constructed by considering all possible extensions of the partial labeling to complete correct labeling of these trees.\n\nThis result implies the following equality:\n\\begin{theorem}\n$\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$. \n\\end{theorem}\n\nThis equality is surprising in that the definitions of the two classes do not seem to have much in common on the first glance!\nMoreover, the proof of the equality relies on nontrivial results in both distributed algorithms (the technique of Chang and Pettie~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}) and descriptive combinatorics (the fact that a hierarchical decomposition, so-called toast, can be constructed in $\\mathsf{BAIRE}$, \\cite{conleymillerbound}, see Proposition \\ref{pr:BaireToast}). \n\n\n\n\n\n\n\n\n\n\n\n\nThe combinatorial characterization of the local complexity class $\\mathsf{LOCAL}(O(\\log n))$ on $\\Delta$-regular trees is interesting from the perspective of distributed computing alone. \nThis result can be seen as a part of a large research program aiming at classification of possible local complexities on various graph classes~\\cite{balliu2020almost_global_problems,brandt_etal2016LLL,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,Chang_paths_cycles,balliu2021_rooted_trees,balliu_et_al:binary_lcl,balliu2018new_classes-loglog*-log*}. \nThat is, we wish not only to understand possible complexity classes (see the left part of \\cref{fig:big_picture_trees} for possible local complexity classes on regular trees), but also to find combinatorial characterizations of problems in those classes that allow us to  efficiently decide for a given problem which class it belongs to.  \nUnfortunately, even for grids with input labels, it is \\emph{undecidable} whether a given local problem can be solved in $O(1)$ rounds~\\cite{naorstockmeyer,brandt_grids}, since\nlocal problems on grids can be used to simulate a Turing machine.\nThis undecidability result does not apply to paths and trees, hence for these graph classes it is still hopeful that we can find simple and useful characterizations for different classes of distributed problems. \n\nIn particular, on paths it is decidable what classes a given local problem belongs to, for all classes coming from the three areas considered here, and this holds even if we allow inputs~\\cite{Chang_paths_cycles,grebik_rozhon2021LCL_on_paths}. \nThe situation becomes much more complicated when we consider trees. \nRecently, classification results on trees were obtained for so-called binary-labeling problems \\cite{balliu_et_al:binary_lcl}. \nMore recently, a complete classification was obtained in the case of \\emph{rooted regular trees}~\\cite{balliu2021_rooted_trees}. \nAlthough their algorithm takes exponential time in the worst case, the authors provided a practical implementation fast enough to classify many typical problems of interest. \n\nMuch less is known for general, \\emph{unoriented} trees, with an arbitrary number of labels. \nIn general, deciding the optimal distributed complexity for a local problem on  bounded-degree trees is $\\mathsf{EXPTIME}$-hard~\\cite{chang2020n1k_speedups},  such a hardness result does not rule out the possibility for having a simple and polynomial-time characterization for the case of \\emph{regular trees}, where there is no input and the constraints are placed only on degree-$\\Delta$ vertices. \nIndeed, it was stated in~\\cite{balliu2021_rooted_trees} as an open question to find such a characterization. Our characterization of $\\mathsf{LOCAL}(O(\\log n)) = \\mathsf{BAIRE}$ by $\\ell$-full problems makes progress in better understanding the distributed complexity classes on trees and towards answering this open question. \n\n\n\n\n\n\n\n\\paragraph{Roadmap}\n\\hypertarget{par:roadmap}{}\nIn \\cref{sec:preliminaries}, we define formally all the three setups we consider in the paper. \nIn \\cref{sec:marks} we discuss the lower bound technique of Marks and the new concept of an ID graph.\nNext, in \\cref{sec:separating_examples} we prove some basic results about the uniform complexity classes and give examples of problems separating some classes from \\cref{fig:big_picture_trees}. \nFinally, in \\cref{sec:baire} we prove that a problem admits a Baire measurable solution if and only if it admits a distributed algorithm of local complexity $O(\\log n)$.\n\nThe individual sections can be read largely independently of each other.\nMoreover, most of our results that are specific to only one of the three areas can be understood without reading the parts of the paper that concern the other areas.\nWe encourage the reader interested mainly in one of the areas to skip the respective parts.\n\n\n\n\n\\section{Preliminaries}\n\\label{sec:preliminaries}\n\nIn this section, we explain the setup we work with, the main definitions and results. \nThe class of graphs that we consider in this work are either infinite $\\Delta$-regular trees, or their finite analogue that we define formally in \\cref{subsec:local_problems}.\n\nWe sometimes explicitly assume $\\Delta > 2$. \nThe case $\\Delta = 2$, that is, studying paths, behaves differently and seems much easier to understand \\cite{grebik_rozhon2021LCL_on_paths}. \nUnless stated otherwise, we do not consider any additional structure on the graphs, but sometimes it is natural to work with trees with an input $\\Delta$-edge-coloring. \n\n\n\\subsection{Local Problems on $\\Delta$-regular trees}\n\\label{subsec:local_problems}\n\nThe problems we study in this work are locally checkable labeling (LCL) problems, which, roughly speaking, are problems that can be described via local constraints that have to be satisfied in a suitable neighborhood of each vertex.\nIn the context of distributed algorithms, these problems were introduced in the seminal work by Naor and Stockmeyer~\\cite{naorstockmeyer}, and have been studied extensively since.\nIn the modern formulation introduced in~\\cite{brandt19automatic}, instead of labeling vertices or edges, LCL problems are described by labeling half-edges, i.e., pairs of a vertex and an incident edge.\nThis formulation is very general in that it not only captures vertex and edge labeling problems, but also others such as orientation problems, or combinations of all of these types.\nBefore we can provide this general definition of an LCL, we need to introduce some definitions.\nWe start by formalizing the notion of a half-edge.\n\n\\begin{definition}[Half-edge]\n\\label{def:half-edge}\n    A \\emph{half-edge} is a pair $(v, e)$ where $v$ is a vertex, and $e$ an edge incident to $v$.\n    We say that a half-edge $(v, e)$ is \\emph{incident} to a vertex $w$ if $w = v$, we say that a vertex $w$ is \\emph{contained} in a half edge $(v,e)$ if $w=v$, and we say that $(v, e)$ \\emph{belongs} to an edge $e'$ if $e' = e$.\n    We denote the set of all half-edges of a graph $G$ by $H(G)$.\n    A \\emph{half-edge labeling} is a function $c \\colon H(G) \\to \\Sigma$ that assigns to each half-edge an element from some label set $\\Sigma$.\n\\end{definition}\n\nIn order to speak about finite $\\Delta$-regular trees, we need to consider slightly modified definition of a graph.\nWe think of each vertex to be contained in $\\Delta$-many half-edges, however, not every half edge belongs to an actual edge of the graph.\nThat is half-edges are pairs $(v,e)$, but $e$ is formally not a pair of vertices.\nSometimes we refer to these half-edges as \\emph{virtual} half-edges.\nWe include a formal definition to avoid confusions. See also \\cref{fig:Delta_reg_tree}. \n\n\n\\begin{definition}[$\\Delta$-regular trees]\n    A tree $T$, finite or infinite is a \\emph{$\\Delta$-regular tree} if either it is infinite and $T=T_\\Delta$, where $T_\\Delta$ is the unique infinite $\\Delta$-regular tree, that is each vertex has exactly $\\Delta$-many neighbors, or it is finite of maximum degree $\\Delta$ and each vertex $v\\in T$ of degree $d\\le \\Delta$ is contained in $(\\Delta-d)$-many virtual half-edges.\n   \n    \n    Formally, we can view $T$ as a triplet $(V(T), E(T), H(T))$, where $(V(T),E(T))$ is a tree of maximum degree $\\Delta$ and $H(T)$ consists of real half-edges, that is pairs $(v,e)$, where $v\\in V(T)$, $e\\in E(T)$ and $e$ is incident to $v$, together with some virtual edges, in the case when $T$ is finite, such that each vertex $v\\in V(T)$ is contained in exactly $\\Delta$-many half-edges (real or virtual).\n   \n   \n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig\/Delta_reg_tree.pdf}\n    \\caption{A $3$-regular tree on 15 vertices. Every vertex has $3$ half-edges but not all half-edges lead to a different vertex. }\n    \\label{fig:Delta_reg_tree}\n\\end{figure}\n\nAs we are considering trees in this work, each LCL problem can be described in a specific form that provides two lists, one describing all label combinations that are allowed on the half-edges incident to a vertex, and the other describing all label combinations that are allowed on the two half-edges belonging to an edge.\\footnote{Every problem that can be described in the form given by Naor and Stockmeyer~\\cite{naorstockmeyer} can be equivalently described as an LCL problem in this list form, by simply requiring each output label on some half-edge $h$ to encode all output labels in a suitably large (constant) neighborhood of $h$ in the form given in~\\cite{naorstockmeyer}.}\nWe arrive at the following definition for LCLs on $\\Delta$-regular trees.\\footnote{Note that the defined LCL problems do not allow the correctness of the output to depend on input labels.}\n\n\\begin{definition}[LCLs on $\\Delta$-regular trees]\\label{def:LCL_trees}\nA \\emph{locally checkable labeling problem}, or \\emph{LCL} for short, is a triple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, where $\\Sigma$ is a finite set of labels, $\\mathcal{V}$ is a subset of unordered cardinality-$\\Delta$ multisets\\footnote{Recall that a multiset is a modification of the concept of sets, where repetition is allowed.} of labels from $\\Sigma$ , and $\\mathcal{E}$ is a subset of unordered cardinality-$2$ multisets of labels from $\\Sigma$.\n\nWe call $\\mathcal{V}$ and $\\mathcal{E}$ the \\emph{vertex constraint} and \\emph{edge constraint} of $\\Pi$, respectively.\nMoreover, we call each multiset contained in $\\mathcal{V}$ a \\emph{vertex configuration} of $\\Pi$, and each multiset contained in $\\mathcal{E}$ an \\emph{edge configuration} of $\\Pi$.\n\nLet $T$ be a $\\Delta$-regular tree and $c:H(T)\\to \\Sigma$ a half-edge labeling of $T$ with labels from $\\Sigma$.\nWe say that $c$ is a \\emph{$\\Pi$-coloring}, or, equivalently, a correct solution for $\\Pi$, if, for each vertex $v$ of $T$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$ of $T$, the cardinality-$2$ multiset of labels assigned to the half-edges belonging to $e$ is an element of $\\mathcal{E}$.\n\\end{definition}\n\nAn equivalent way to define our setting would be to consider $\\Delta$-regular trees as commonly defined, that is, there are vertices of degree $\\Delta$ and vertices of degree $1$, i.e., leaves.\nIn the corresponding definition of LCL one would consider leaves as unconstrained w.r.t. the vertex constraint, i.e., in the above definition of a correct solution the condition ``for each vertex  $v$'' is replaced by ``for each non-leaf vertex $v$''.\nEquivalently, we could allow arbitrary trees of maximum degree $\\Delta$ as input graphs, but, for vertices of degree $< \\Delta$, we require the multiset of labels assigned to the half-edges to be extendable to some cardinality-$\\Delta$ multiset in $\\mathcal{V}$.\nWhen it helps the exposition of our ideas and is clear from the context, we may make use of these different but equivalent perspectives.\n\nWe illustrate the difference between our setting and ``standard setting'' without virtual half-edges on the perfect matching problem.\nA standard definition of the perfect matching problem is that some edges are picked in such a way that each vertex is covered by exactly one edge.\nIt is easy to see that there is no local algorithm to solve this problem on the class of finite trees (without virtual half-edges), this is a simple parity argument.\nHowever, in our setting, every vertex needs to pick exactly one half-edge (real or virtual) in such a way that both endpoints of each edge are either picked or not picked.\nWe remark that in our setting it is not difficult to see that (if $\\Delta>2$), then this problem can be solved by a local deterministic algorithm of local complexity $O(\\log(n))$.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The $\\mathsf{LOCAL}$ model}\n\nIn this section, we define local algorithms and local complexity.\nWe discuss the general relation between the classical local complexity and the uniform local complexity. Recall that when we talk about distributed algorithm on $\\Delta$-regular trees, the algorithm has access to $n$, the size of the tree. The measure of complexity is the classical local complexity. On the other hand, when we talk about uniform distributed algorithms on $\\Delta$-regular trees, we talk about an infinite $\\Delta$-regular tree and the measure of complexity is the uniform local complexity. \nWe start with the classical notions.\nRecall that $B(v,t)$ is the $t$-hop neighborhood of a vertex $v$ in an underlying graph $G$.\n\n\\begin{definition}[Local algorithm]\n\\label{def:local_algorithm}\nA \\emph{distributed local algorithm} $\\mathcal{A}$ of local complexity $t(n)$ is a function defined on all possible $t(n)$-hop neighborhoods of a vertex. Applying an algorithm $\\mathcal{A}$ on an input graph $G$ means that the function is applied to a $t(n)$-hop neighborhood $B(u, t(n))$ of each vertex $u$ of $G$. The output of the function is a labeling of the half-edges around the given vertex. The algorithm also takes as input the size of the input graph $n$. \n\\end{definition}\n\n\\begin{definition}[Local complexity]\n\\label{def:local_complexity}\nWe say that an LCL problem $\\Pi$ has a deterministic local complexity $t(n)$ if there is a local algorithm $\\mathcal{A}$ of local complexity $t(n)$ such that when run on the input graph $G$, with each of its vertices having a unique identifier from $[n^{O(1)}]$, $\\mathcal{A}$ always returns a valid solution to $\\Pi$.  \nWe also say $\\Pi \\in \\mathsf{DLOCAL}( O(t(n))$. \n\nThe randomized complexity is define analogously, but instead of unique identifiers, each vertex of $G$ has an infinite random string. The solution of $\\mathcal{A}$ needs to be a valid solution with probability $1 - 1\/n$. \nWe also say $\\Pi \\in \\mathsf{RLOCAL}( O(t(n))$. \n\\end{definition}\n\nWhenever we talk about \\emph{local} complexity on $\\Delta$-regular trees, we always tacitly think about the class of finite $\\Delta$-regular trees. We use the notation $\\mathsf{LOCAL}(O(t(n)))$ whenever the deterministic and the randomized complexity of the problems are the same up to constant factor (cf. \\cref{thm:basicLOCAL}). \n\n\\paragraph{Classification of Local Problems on $\\Delta$-regular Trees}\n\nThere is a lot of work aiming to classify all possible local complexities of LCL problems on bounded degree trees. This classification was recently finished.\nEven though the results in the literature are stated for the classical notion of finite trees, we note that it is easy to check that the same picture emerges if we restrict ourselves to $\\Delta$-regular trees according to our definition.\n\n\\begin{theorem}[Classification of local complexities of LCLs on $\\Delta$-regular trees \\cite{naorstockmeyer,chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups,balliu2020almost_global_problems,balliu2019hardness_homogeneous,brandt_grunau_rozhon2021classification_of_lcls_trees_and_grids}]\n\\label{thm:basicLOCAL}\nLet $\\Pi$ be an LCL problem. Then the deterministic\/randomized local complexity of $\\Pi$ on $\\Delta$-regular trees is one of the following:\n\\begin{enumerate}\n    \\item $O(1)$,\n    \\item $\\Theta(\\log^*n)$,\n    \\item $\\Theta(\\log n)$ deterministic and $\\Theta(\\log\\log n)$ randomized,\n    \\item $\\Theta(\\log n)$,\n    \\item $\\Theta(n^{1\/k})$ for $k \\in \\mathbb{N}$.\n\\end{enumerate}\n\\end{theorem}\n\nWe do not know whether the complexity classes $\\Theta(n^{1\/k})$ are non-empty on \\emph{$\\Delta$-regular} trees. The $2 \\frac{1}{2}$ coloring problems $\\mathcal{P}_k$  are known to have complexity $\\Theta(n^{1\/k})$ on \\emph{bounded-degree} trees~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. However, as the correctness criterion of $\\mathcal{P}_k$ depends on the degree of vertices, it does not fit into the class of LCL problems that we consider in \\cref{def:LCL_trees}. \nNevertheless, as can be seen in \\cref{fig:big_picture_trees}, all classes of factors of iid and from descriptive combinatorics are already contained in the class $\\mathsf{LOCAL}(O(\\log n))$. \n\n\n\n\\paragraph{Uniform Algorithms}\nAs we mentioned before, when talking about local complexities, we always have in mind that the underlying graph is finite.\nIn particular, the corresponding algorithm knows the size of the graph.\nOn infinite $\\Delta$-regular trees, or infinite graphs in general, we use the following notion \\cite{HolroydSchrammWilson2017FinitaryColoring,Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring}. \n\n\\begin{definition}\nAn \\emph{uniform local algorithm} $\\mathcal{A}$ is a function that is defined on all possible (finite) neighborhoods of a vertex.\nFor some neighborhoods it outputs a special symbol $\\emptyset$ instead of a labeling of the half-edges around the central vertex.\nApplying $\\mathcal{A}$ on a graph $G$ means that for each vertex $u$ of $G$ the function is applied to $B(u,t)$, where $t$ is the minimal number such that $\\mathcal{A}(u,t)\\not=\\emptyset$.\nWe call $t$ the coding radius of $\\mathcal{A}$, and denote it, as a function on vertices, as $R_\\mathcal{A}$.\n\\end{definition}\n\nWe define the corresponding notion of \\emph{uniform local complexity} for infinite $\\Delta$-regular trees where each vertex is assigned an infinite random string.\n\n\\begin{definition}[Uniform local complexity \\cite{HolroydSchrammWilson2017FinitaryColoring}]\n\\label{def:uniform_complexity}\nWe say that the \\emph{uniform local (randomized) complexity} of an LCL problem $\\Pi$ is $t(\\varepsilon)$ if there is an uniform local algorithm $\\mathcal{A}$ such that the following hold on the infinite $\\Delta$-regular tree.\nRecall that $R_\\mathcal{A}$ is the random variable measuring the coding radius of a vertex $u$, that is, the distance $\\mathcal{A}$ needs to look at to decide the answer for $u$. Then, for any $0 < \\varepsilon < 1$:\n\\[\n\\P(R_\\mathcal{A} \\ge t(\\varepsilon)) \\leq \\varepsilon.\n\\]\nWe also say $\\Pi \\in \\mathsf{OLOCAL}(O(t(\\varepsilon))$.\n\\end{definition}\n\nWe finish by stating the following lemma that bounds the uniform complexity of concatenation of two uniform algorithm (we need to be little bit more careful and cannot just add the complexites up). \n\n\\begin{lemma}[Sequential Composition]\n\\label{lem:sequential_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_1(\\varepsilon)$ and $t_2(\\varepsilon)$, respectively. \nLet $A$ be the sequential composition of $A_1$ and $A_2$. That is, $A$ needs to know the output that $A_2$ computes when the local input at every vertex is equal to the output of the algorithm $A_1$ at that vertex. Then, $t_A(\\varepsilon) \\leq t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$.\n\\end{lemma}\n\\begin{proof}\nConsider some arbitrary vertex $u$. Let $E_1$ denote the event that the coding radius of $\\mathcal{A}_1$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right)$ for all vertices in the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$. As the $t_{A_2}(\\varepsilon\/2)$-hop neighborhood around $u$ contains at most $\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}$ vertices, a union bound implies that $P(E_1) \\geq 1 - \\varepsilon\/2$. Moreover, let $E_2$ denote the event that the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$ is at most $t_{A_2}(\\varepsilon\/2)$. By definition, $P(E_2) \\geq 1 - \\varepsilon\/2$. Moreover, if both events $E_1$ and $E_2$ occur, which happens with  probability at least $1 - \\varepsilon$, then the coding radius of algorithm $\\mathcal{A}$ is at most $t_{A_1} \\left( \\frac{\\varepsilon\/2}{\\Delta^{t_{A_2}(\\varepsilon\/2) + 1}}\\right) + t_{A_2}(\\varepsilon\/2)$, thus finishing the proof.\n \\end{proof}\n\n\n\\subsection{Descriptive combinatorics}\n\\label{subsec:descriptivecombinatorics}\n\nBefore we define formally the descriptive combinatorics complexity classes, we give a high-level overview on their connection to distributing computing for the readers more familiar with the latter.\n\nThe complexity class that partially captures deterministic local complexity classes is called $\\mathsf{BOREL}$ (see also Remark \\ref{rem:cont}).\nFirst note that by a result of Kechris, Solecki and Todor\\v{c}evi\\'c \\cite{KST} the \\emph{maximal independent set problem} (with any parameter $r\\in \\mathbb{N}$) is in this class for any bounded degree graph.\\footnote{That is, it is possible to find a Borel maximal independent set, i.e., a maximal independent set which is, moreover, a Borel subset of the vertex set.}\nIn particular, this yields that $\\mathsf{BOREL}$ contains the class $\\mathsf{LOCAL}(O(\\log^* n))$ by the characterization of \\cite{chang_kopelowitz_pettie2019exp_separation}, see \\cite{Bernshteyn2021LLL}. Moreover, as mentioned before, $\\mathsf{BOREL}$ is closed under countably many iterations of the operations of finding maximal independent set (for some parameter that might grow) and of applying a constant local rule that takes into account what has been constructed previously.\\footnote{It is in fact an open problem, whether this captures fully the class $\\mathsf{BOREL}$. However, note that an affirmative answer to this question would yield that problems can be solved in an ``effective\" manner in the Borel context, which is known not to be the case in unbounded degree graphs \\cite{todorvcevic2021complexity}.}\n\nTo get a better grasp of what this means, consider for example the proper vertex $2$-coloring problem on half-lines. It is clear that no local algorithm can solve this problem.\nHowever, as it is possible to determine the starting vertex after countably many iterations of the maximal independent set operation, we conclude that this problem is in the class $\\mathsf{BOREL}$.\nThe idea that $\\mathsf{BOREL}$ can compute some unbounded, global, information will be implicitly used in all the construction in \\cref{sec:separating_examples} that separate $\\mathsf{BOREL}$ from local classes.\n\nThe intuition behind the class $\\mathsf{MEASURE}$ is that it relates in the same way to the class $\\mathsf{BOREL}$, as randomized local algorithms relate to deterministic ones.\nIn particular, the operations that are allowed in the class $\\mathsf{MEASURE}$ are the same as in the class $\\mathsf{BOREL}$ but the solution of a given LCL can be incorrect on a measure zero set.\n\nThe class $\\mathsf{BAIRE}$ can be considered as a topological equivalent of the measure theoretic class $\\mathsf{MEASURE}$, that is, a solution can be incorrect on a topologically negligible set.\nThe main difference between the classes $\\mathsf{MEASURE}$ and $\\mathsf{BAIRE}$ is that in the later there is a hierarchical decomposition that is called \\emph{toast}.\n(Note that this phenomenon is present in the case of $\\mathsf{MEASURE}$ exactly on so-called amenable graphs. It is also tightly connected with the notion of hyperfiniteness \\cite{conleymillerbound,gao2015forcing}.)\nThe independence of colorings on a tree together with this structure allows for a combinatorial characterization of the class $\\mathsf{BAIRE}$, which was proven by Bernshteyn \\cite{Bernshteyn_work_in_progress}, see also \\cref{sec:baire}.\n\n\n\n\nNext we formulate the precise definitions.\nWe refer the reader to \\cite{pikhurko2021descriptive_comb_survey,kechris_marks2016descriptive_comb_survey,Bernshteyn2021LLL,kechrisclassical}, or to \\cite[Introduction,~Section~4.1]{grebik_rozhon2021LCL_on_paths} and \\cite[Section~7.1,~7.2]{grebik_rozhon2021toasts_and_tails} for intuitive examples and standard facts of descriptive set theory.\nIn particular, we do not define here the notion standard Borel\/probability space, a Polish topology, a Borel probability measure, Baire property etc.\n\nLet $\\mathcal{G}$ be a Borel graph of bounded maximum degree on a standard Borel space $X$.\nIn this paper we consider exclusively acyclic $\\Delta$-regular Borel graphs and we refer to them as \\emph{$\\Delta$-regular Borel forests}.\nIt is easy to see that the set of half-edges (see Definition \\ref{def:half-edge}) is naturally a standard Borel space, we denote this set by $H(\\mathcal{G})$.\nThus, it makes sense to consider Borel labelings of $H(\\mathcal{G})$.\nMoreover, if $\\mathcal{G}$ is a $\\Delta$-regular Borel forest and $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is an LCL, we can also decide whether a coloring $f:\\mathcal{H}(\\mathcal{G})\\to \\Sigma$ is a solution to $\\Pi$ as in Definition \\ref{def:LCL_trees}. \nSimilarly, we say that the coloring $f$ solves $\\Pi$, e.g., on a $\\mu$-conull set for some Borel probability measure $\\mu$ on $X$ if there is a Borel set $C\\subseteq X$ such that $\\mu(C)=1$, the vertex constraints are satisfied around every $x\\in C$ and the edge constraints are satisfied for every $x,y\\in C$ that form an edge in $\\mathcal{G}$.\n\n\\begin{definition}[Descriptive classes]\n\\label{def:descriptive}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL.\nWe say that $\\Pi$ is in the class $\\mathsf{BOREL}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$.\n\nWe say that $\\Pi$ is in the class $\\mathsf{BAIRE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every compatible Polish topology $\\tau$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\tau$-comeager set.\n\nWe say that $\\Pi$ is in the class $\\mathsf{MEASURE}$ if for every acyclic $\\Delta$-regular Borel graph $\\mathcal{G}$ on a standard Borel space $X$ and every Borel probability measure $\\mu$ on $X$, there is a Borel function $f:H(\\mathcal{G})\\to \\Sigma$ that is a $\\Pi$-coloring of $\\mathcal{G}$ on a $\\mu$-conull set.\n\\end{definition}\n\nThe following claim follows directly from the definition.\n\n\\begin{claim}\\label{cl:basicDC}\nWe have $\\mathsf{BOREL}\\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{claim}\n\nRecently Bernshteyn \\cite{Bernshteyn2021LLL,Bernshteyn2021local=cont} proved several results that connect distributed computing with descriptive combinatorics.\nUsing the language of complexity classes we can formulate some of the result as inclusions in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}[\\cite{Bernshteyn2021LLL}]\nWe have\n\\begin{itemize}\n    \\item $\\mathsf{LOCAL}(O(\\log^*(n)))\\subseteq \\mathsf{BOREL}$,\n    \\item $\\mathsf{RLOCAL}(o(\\log(n))) \\subseteq \\mathsf{MEASURE}, \\mathsf{BAIRE}$.\n\\end{itemize}\nIn fact, these inclusions hold for any reasonable class of bounded degree graphs.\n\\end{theorem}\n\n\\begin{remark}\n\\label{rem:cont}\nThe ``truly'' local class in descriptive combinatorics is the class $\\mathsf{CONTINUOUS}$.\nEven though we do not define this class here\\footnote{To define the class $\\mathsf{CONTINUOUS}$, rather than asking for a continuous solution on all possible $\\Delta$-regular Borel graphs, one has to restrict to a smaller family of graphs, otherwise the class trivializes. To define precisely this family is somewhat inconvenient, and not necessary for our arguments.}, and we refer the reader to \\cite{Bernshteyn2021local=cont,GJKS,grebik_rozhon2021toasts_and_tails} for the definition and discussions in various cases, we mention that the inclusion \\ \\[\\mathsf{LOCAL}(O(\\log^* n))\\subseteq \\mathsf{CONTINUOUS} \\tag{*}\\] holds in most reasonable classes of bounded degree graphs, see \\cite{Bernshteyn2021LLL}.\nThis also applies to our situation\nRecently, it was shown by Bernshteyn \\cite{Bernshteyn2021local=cont}, and independently by Seward \\cite{Seward_personal}, that $(*)$ can be reversed for Cayley graphs of finitely generated groups.\nThis includes, e.g., natural $\\Delta$-regular trees with proper edge $\\Delta$-coloring, as this is a Cayely graph of free product of $\\Delta$-many $\\mathbb{Z}_2$ induced by the standard generating set.\nIt is, however, not clear whether it $(*)$ can be reversed in our situation, i.e., $\\Delta$-regular trees without any additional labels.\n\n\n\n\\end{remark}\n\n\n\n\n\n\n\\subsection{Random processes}\n\nWe start with an intutitve description of fiid processes. Let $T_\\Delta$ be the infinite $\\Delta$-regular tree.\nInformally, \\emph{factors of iid processes (fiid)} on $T_\\Delta$ are infinite analogues of local randomized algorithms in the following way.\nLet $\\Pi$ be an LCL and $u\\in T_\\Delta$.\nIn order to solve $\\Pi$, we are allowed to explore the whole graph, and the random strings that are assigned to vertices, and then output a labeling of half-edges around $u$.\nIf such an assignment is a measurable function and produces a $\\Pi$-coloring almost surely, then we say that $\\Pi$ is in the class $\\mathsf{fiid}$.\nMoreover, if every vertex needs to explore only finite neighborhood to output a solution, then we say that $\\Pi$ is in the class $\\mathsf{ffiid}$.\nSuch processes are called \\emph{finitary} fiid (\\emph{ffiid}). There is also an intimate connection between ffiid and uniform algorithms.\nThis is explained in \\cite[Section 2.2]{grebik_rozhon2021toasts_and_tails}.\nInformally, an ffiid process that almost surely solves $\\Pi$ is, in the language of distributed computing, an uniform local algorithm that solves $\\Pi$.\nThis allows us to talk about uniform local complexity of an ffiid.\nIn the rest of the paper we interchange both notions freely with slight preference for the distributed computing language.\n\nNow we define formally these classes, using the language of probability.\nWe denote by $\\operatorname{Aut}(T_\\Delta)$ the automorphism group of $T_\\Delta$.\nAn \\emph{iid process} on $T_\\Delta$ is a collection of iid random variables $Y=\\{Y_v\\}_{v\\in V(T_\\Delta)}$ indexed by vertices, or edges, half-edges etc, of $T_\\Delta$ such that their joint distribution is invariant under $\\operatorname{Aut}(T_\\Delta)$.\nWe say that $X$ is a \\emph{factor of iid process (fiid)} if $X=F(Y)$, where $F$ is a measurable $\\operatorname{Aut}(T_\\Delta)$-invariant map and $Y$ is an iid process on $T_\\Delta$.\\footnote{We always assume $Y=(2^\\mathbb{N})^{T_\\Delta}$ endowed with the product Lebesgue measure.}\nMoreover, we say that $X$ is a \\emph{finitary factor if iid process (ffiid)} if $F$ depends with probability $1$ on a finite (but random) radius around each vertex.\nWe denote as $R_F$ the random variable that assigns minimal such radius to a given vertex, and call it the \\emph{coding radius} of $F$.\nWe denote the space of all $\\Pi$-colorings of $T_\\Delta$ as $X_\\Pi$.\nThis is a subspace of $\\Sigma^{H(T_\\Delta)}$ that is invariant under $\\operatorname{Aut}(T_\\Delta)$.\n\n\\begin{definition}\nWe say that an LCL $\\Pi$ is in the class $\\mathsf{fiid}$ ($\\mathsf{ffiid}$) if there is an fiid (ffiid) process $X$ that almost surely produces elements of $X_\\Pi$.\n\nEquivalently, we can define $\\mathsf{ffiid} = \\bigcup_f \\mathsf{OLOCAL}(f(\\varepsilon))$ where $f$ ranges over all functions. That is, $\\mathsf{ffiid}$ is the class of problems solvable by any uniform distributed algorithm. \n\n\\end{definition}\n\n\nIt is obvious that $\\mathsf{ffiid}\\subseteq \\mathsf{fiid}$.\nThe natural connection between descriptive combinatorics and random processes is formulated by the inclusion $\\mathsf{MEASURE}\\subseteq \\mathsf{fiid}$.\nWhile this inclusion is trivially satisfied, e.g., in the case of $\\Delta$-regular trees with proper edge $\\Delta$-coloring, in our situation we need a routine argument that we include for the sake of completeness in \\cref{app:ktulu}.\n\n\\begin{restatable}{lemma}{FIIDinMEASURE}\\label{thm:FIIDinMEASURE}\nLet $\\Pi$ be an LCL such that $\\Pi\\in \\mathsf{MEASURE}$.\nThen $\\Pi\\in \\mathsf{fiid}$.\n\\end{restatable}\n\n\n\n\n\\subsection{Specific Local Problems}\n\nHere we list some well-known local problems that were studied in all three considered areas. We describe the results known about these problems with respect to classes from \\cref{fig:big_picture_trees}. \n\n\\paragraph{Edge Grabbing}\nWe start by recalling the well-known problem of edge grabbing (a close variant of the problem is known as sinkless orientation\\cite{brandt_etal2016LLL}). \nIn this problem, every vertex should mark one of its half-edges (that is, grab an edge) in such a way that no edge can be marked by two vertices. \nIt is known that $\\Pi_{\\text{edgegrab}}  \\not\\in \\mathsf{LOCAL}(O(\\log^* n))$ \\cite{brandt_etal2016LLL} but $\\Pi_{\\text{edgegrab}} \\in \\mathsf{RLOCAL}(O(\\log\\log n))$.\n\nSimilarly, $\\Pi_{\\text{edgegrab}} \\not \\in \\mathsf{BOREL}$ by \\cite{DetMarks}, but $\\Pi_{\\text{edgegrab}}\\in \\mathsf{MEASURE}$ by \\cite{BrooksMeas}: to see the former, just note that if a $\\Delta$-regular tree admits proper $\\Delta$-colorings of both edges and vertices, every vertex can grab an edge of the same color as the color of the vertex. Thus, $\\Pi_{\\text{edgegrab}}\\in \\mathsf{BOREL}$ would yield that $\\Delta$-regular Borel forests with Borel proper edge-colorings admit a Borel proper $\\Delta$-coloring, contradicting \\cite{DetMarks}.\n\nThis completes the complexity characterization of $\\Pi_{\\text{edgegrab}}$ as well as the proper vertex $\\Delta$-coloring with respect to classes in \\cref{fig:big_picture_trees}. \n\n\n\n\n\n\\paragraph{Perfect Matching}\nAnother notorious LCL problem, whose position in \\cref{fig:big_picture_trees} is, however, not completely understood, is the perfect matching problem $\\Pi_{\\text{pm}}$.\nRecall that the perfect matching problem $\\Pi_{\\text{pm}}$ asks for a matching that covers all vertices of the input tree.\\footnote{In our formalism this means that around each vertex exactly one half-edge is picked.}\nIt is known that $\\Pi_{\\text{pm}} \\in \\mathsf{fiid}$ \\cite{lyons2011perfect}, and it is easy to see that $\\Pi_{\\text{pm}} \\not\\in \\mathsf{RLOCAL}(O(\\log\\log(n)))$ (we will prove a stronger result in \\cref{pr:AddingLineLLL}).\nMarks proved \\cite{DetMarks} that it is not in $\\mathsf{BOREL}$, even when the underlying tree admits a Borel bipartition.\nIt is not clear if $\\Pi_{\\text{pm}}$ is in $\\mathsf{ffiid}$, nor whether it is in $\\mathsf{MEASURE}$.\n\n\n\n\n\n\\paragraph{Graph Homomorphism}\nWe end our discussion with LCLs that correspond to graph homomorphisms (see also the discussion in Section \\ref{sec:marks}).\nThese are also known as edge constraint LCLs.\nLet $G$ be a finite graph.\nThen we define $\\Pi_G$ to be the LCL that asks for a homomorphism from the input tree to $G$, that is, vertices are labeled with vertices of $G$ in such a way that edge relations are preserved.\nThere are not many positive results except for the class $\\mathsf{BAIRE}$.\nIt follows from the result of Bernshteyn \\cite{Bernshteyn_work_in_progress} (see \\cref{sec:baire}) that $\\Pi_G\\in \\mathsf{BAIRE}$ if and only if $G$ is not bipartite.\nThe main negative results can be summarized as follows.\nAn easy consequence of the result of Marks \\cite{DetMarks} is that if $\\chi(G)\\le \\Delta$, then $\\Pi_G\\not \\in \\mathsf{BOREL}$.\nIn this paper, we describe examples of graphs of chromatic number up to $2\\Delta-2$ such that the corresponding homomorphism problem is not in $\\mathsf{BOREL}$, see \\cref{sec:marks}.\nThe theory of \\emph{entropy inequalities} see \\cite{backhausz} implies that if $G$ is a cycle on more than $9$ vertices, then $\\Pi_G\\not\\in \\mathsf{fiid}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Generalization of Marks' technique}\n\\label{sec:marks}\n\nIn this section, we first develop a new way of proving lower bounds in the $\\mathsf{LOCAL}$ model based on a generalization of a technique of Marks \\cite{DetMarks}. Then, we use ideas arising in the finitary setting---connected to the adaptation of Marks' technique---to obtain new results back in the Borel context.\nFor an introduction to Marks' technique and a high-level discussion about the challenges in adapting the technique to the standard distributed setting as well as our solution via the new notion of an ID graph, we refer the reader to \\cref{sec:intromarks}.\n\n\\begin{comment}\nIn \\cite{DetMarks}, Marks showed that there exist $\\Delta$-regular Borel forests that do not admit a Borel proper $\\Delta$-coloring. As it was indicated above, this is a striking difference between $\\mathsf{MEASURE}$ or $\\mathsf{BAIRE}$ and $\\mathsf{BOREL}$, as such colorings exist in the former classes, see \\cite{BrooksMeas}. Since prior to Marks' breakthrough (essentially) any lower bound on Borel chromatic numbers utilized measure theoretic or Baire category arguments, his approach was (and had to be) based on a new technique: \\emph{determinacy of infinite games}. \n\nFix a payoff set $A \\subseteq \\mathbb{R}$. Consider the infinite game in which two players alternatingly determine digits of a real number. The first player wins iff the resulting real is in $A$. A rather counter-intuitive fact of set theory is that there exists a set $A$ for which none of the players has a winning strategy, i.e., the game is not determined. However, a deep theorem of Martin \\cite{martin} says that if $A$ is Borel, then one of the players does have such a strategy. \nMarks has constructed $\\Delta$-regular Borel forests and games with Borel payoff sets, for which the existence of a winning strategy for either of the players contradicts the existence of a Borel proper $\\Delta$-coloring. \n\nIn what follows, we generalize and adapt his games to the finitary setting, thus obtaining new lower bounds for LOCAL algorithms; then we use ideas inspired by the LOCAL technique of ID graphs to prove a new theorem in the Borel context. \n\nNote that Marks considered Borel versions of certain locally checkable problems (e.g., coloring, edge grabbing, perfect matching). As discussed above, for these problems, lower bounds were independently obtained via the round elimination technique in the LOCAL context, see e.g., \\cite{brandt19automatic}\\todo{cite more stuff??-Z}. Remarkably, the core ideas behind Brandt's and Marks' argument exhibit a surprising degree of similarity. \n\n\\todo{add the story about going from borel to local, and then back from local to borel}\n\nHowever, our extension of Marks' technique yields lower bounds for fairly natural problems that were resistant to round elimination.\nMoreover, as with our approach the lower bound is obtained by checking whether the considered LCL satisfies a certain combinatorial property called \\emph{playability condition} (see Definition~\\ref{def:Playability}), we can use it to obtain lower bounds for problems that have complicated descriptions---something that the round elimination technique in its current form is ill-suited for. \\todo{Maybe this sentence should rather be somewhere in the intro?}\n\n\n\n\\end{comment}\n\n\n\n\n\n\n\nThe setting we consider in this section is $\\Delta$-regular trees that come with a proper $\\Delta$-edge coloring with colors from $[\\Delta]$.\nAll lower bounds discussed already hold under these restrictions (and therefore also in any more general setting).\n\nRecall that an LCL $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is given by specifying a list of allowed vertex configurations and a list of allowed edge configurations (see Definition \\ref{def:LCL_trees}).\nTo make our lower bounds as widely applicable as possible, we replace the latter list by a separate list for each of the $\\Delta$ edge colors; in other words, we consider LCLs where the correctness of the output is allowed to depend on the input that is provided by the edge colors.\nHence, formally, in this section, an LCL is more generally defined: it is a tuple $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ where $\\Sigma$ and $\\mathcal{V}$ are as before, while $\\mathcal{E} = (\\mathcal{E}_{\\alpha})_{\\alpha \\in [\\Delta]}$ is now a $\\Delta$-tuple of sets $\\mathcal{E}_{\\alpha}$ consisting of cardinality-$2$ multisets.\nSimilarly as before, a half-edge labeling (see Definition \\ref{def:descriptive} for the Borel context) with labels from $\\Sigma$ is a correct solution for $\\Pi$ if, for each vertex $v$, the multiset of labels assigned to the half-edges incident to $v$ is contained in $\\mathcal{V}$, and, for each edge $e$, the multiset of labels assigned to the half-edges belonging to $e$ is contained in $\\mathcal{E}_{\\alpha}$, where $\\alpha$ is the color of the edge $e$.\n\nThe idea of our approach is to identify a condition that an LCL $\\Pi$ necessarily has to satisfy if it is solvable in $O(\\log^* n)$ rounds in the $\\mathsf{LOCAL}$ model.\nShowing that $\\Pi$ does not satisfy this condition then immediately implies an $\\Omega(\\log n)$ deterministic and $\\Omega(\\log \\log n)$ randomized lower bound, by Theorem~\\ref{thm:basicLOCAL}.\n\nIn order to define our condition we need to introduce the notion of a configuration graph: a \\emph{configuration graph} is a $\\Delta$-tuple $\\mathbb{P}=(\\mathbb{P}_\\alpha)_{\\alpha\\in \\Delta}$ of graphs, where the vertex set of each of the graphs $\\mathbb{P}_\\alpha$ is the set of subsets of $\\Sigma$, and there is an edge in $\\mathbb{P}_\\alpha$ connecting two vertices $S,T$ if and only if there are $\\mathtt{a}\\in S$ and $\\mathtt{b}\\in T$ such that $\\{\\mathtt{a},\\mathtt{b}\\}\\in {\\mathcal{E}}_\\alpha$, note that loops are allowed.\n(Naturally, we will consider any two vertices of different $\\mathbb{P}_\\alpha$ to be distinct, even if they correspond to the same subset of $\\Sigma$.)\n\nNow we are set to define the aforementioned condition. The intuition behind the playability condition is the following: assume that there exists a local algorithm $\\mathcal{A}$ that solves $\\Pi$ using the $t$ neighbourhood of a given vertex. We are going to define a family of two player games. The game will depend on some $S \\subseteq \\Sigma$. $\\operatorname{Alice}$ and $\\operatorname{Bob}$  will assign labels (or IDs) to the vertices in the $t$-neighbourhood (in some way specified later on, depending on $\\alpha \\in [\\Delta]$). When the assignment is complete, we evaluate $\\mathcal{A}$ on the root of the obtained labelled graph, this way obtaining an element $s$ of $\\Sigma$, and decide who is the winner based on $s \n\\in S$ or not. Naturally, it depends on $S$ and $\\alpha$, which player has a winning strategy. This gives rise to a two coloring of vertices of $\\mathbb{P}_\\alpha$  by colors $\\operatorname{Alice}$ and $\\operatorname{Bob}$. The failure of the playability condition will guarantee that using a strategy stealing argument one can derive a contradiction.\n\n\n\n\n\\begin{definition}[Playability condition]\\label{def:Playability}\nWe say that an LCL $\\Pi$ is \\emph{playable} if for every $\\alpha\\in [\\Delta]$ there is a coloring $\\Lambda_{\\alpha}$ of the vertices of $\\mathbb{P}_{\\alpha}$ with two colors $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$ such that the following conditions are satisfied:\n\\begin{enumerate}\n    \\item [(A)] For any tuple $(S_{\\alpha})_{\\alpha \\in [\\Delta]} \\in V(\\mathbb{P}_1) \\times \\dots \\times V(\\mathbb{P}_{\\Delta})$ satisfying $\\Lambda_\\alpha(S_\\alpha)=\\operatorname{Alice}$ for each $\\alpha \\in [\\Delta]$, there exist $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ such that $\\{\\mathtt{a}_\\alpha\\}_{\\alpha\\in \\Delta}\\in {\\mathcal{V}}$, and\n    \\item [(B)] for any $\\alpha \\in [\\Delta]$, and any tuple $(S,T) \\in V(\\mathbb{P}_{\\alpha}) \\times V(\\mathbb{P}_{\\alpha})$ satisfying $\\Lambda_\\alpha(S)=\\Lambda_\\alpha(T)=\\operatorname{Bob}$, we have that $(S,T)$ is an edge of $\\mathbb{P}_{\\alpha}$.\n\\end{enumerate}\n\n\\end{definition}\n\nOur aim in this section is to show the following general results.\n\n\\begin{theorem}\\label{th:MainLOCAL}\nLet $\\Pi$ be an LCL that is not playable.\nThen $\\Pi$ is not in the class $\\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{theorem}\n\nUsing ideas from the proof of this result, we can formulate an analogous theorem in the Borel context.\nLet us mention that while \\cref{th:MainLOCAL} is a consequence of \\cref{thm:MainBorel} by \\cite{Bernshteyn2021LLL}, we prefer to state the theorems separately.\nThis is because the proof of the Borel version of the theorem uses heavily complex results such as the Borel determinacy theorem, theory of local-global limits and some set-theoretical considerations about measurable sets.\nThis is in a stark contrast with the proof of \\cref{th:MainLOCAL} that uses `merely' the existence of large girth graphs and determinacy of finite games.\nHowever, the ideas surrounding these concepts are the same in both cases.\n\n\\begin{theorem}\\label{thm:MainBorel}\nLet $\\Pi$ be an LCL that is not playable\nThen $\\Pi$ is not in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\nThe main application of these abstract results is to find lower bounds for \\emph{(graph) homomorphism LCLs} aka \\emph{edge constraint LCLs}.\nWe already defined this class in \\cref{subsec:local_problems} but we recall the definition in the formalism of this section to avoid any confusion.\nLet $G$ be a finite graph.\nThen $\\Pi_G=(\\Sigma,\\mathcal{V},\\mathcal{E})$ is defined by letting\n\\begin{enumerate}\n\\item \\label{p:vertex} ${\\mathcal{V}}=\\{({\\mathtt{a}},\\dots,{\\mathtt{a}}):\\mathtt{a}\\in \\Sigma\\}$,\n\\item $\\Sigma=V(G)$,\n\\item $\\forall \\alpha \\in [\\Delta] \\ (\\mathcal{E}_\\alpha=E(G)).$\n\\end{enumerate}\nAn LCL for which \\eqref{p:vertex} holds is called a \\emph{vertex-LCL}. There is a one-to-one correspondence between vertex-LCLs for which $\\forall \n\\alpha,\\beta \\ (\\mathcal{E}_\\alpha=\\mathcal{E}_\\beta)$ and LCLs of the form $\\Pi_G$. (Indeed, to vertex-LCLs with this property one can associate a graph $G$ whose vertices are the labels in $\\Sigma$ and where two vertices $\\ell, \\ell'$ are connected by an edge if and only if $\\{ \\ell, \\ell' \\} \\in \\mathcal{E}$.)\nNote that if $\\Pi$ is a vertex-LCL, then condition (A) in Definition~\\ref{def:Playability} is equivalent to the statement that no tuple $\\{S_\\alpha\\}_{\\alpha\\in \\Delta}$ that satisfies the assumption of (A) can cover $\\Sigma$, i.e., that $\\Sigma \\nsubseteq S_1 \\cup \\dots \\cup S_{\\Delta}$ for such a tuple.\nMoreover, if $\\Pi_G$ is a homomorphism LCL, then $\\mathbb{P}_\\alpha=\\mathbb{P}_{\\beta}$ for every $\\alpha,\\beta\\in \\Delta$.\n\nThe simplest example of a graph homomorphism LCL is given by setting $G$ to be the clique on $k$ vertices; the associated LCL $\\Pi_G$ is simply the problem of finding a proper $k$-coloring of the vertices of the input graph.\nNow we can easily derive Marks' original result from our general theorem.\n\n\\begin{corollary}[Marks \\cite{DetMarks}]\\label{cor:detmarks}\nLet $G$ be a finite graph that has chromatic number at most $\\Delta$. Then $\\Pi_G$ is not playable. In particular, there is a $\\Delta$-regular Borel forest that have Borel chromatic number $\\Delta+1$. \n\\end{corollary}\n\\begin{proof}\nLet $A_1,\\dots, A_\\Delta$ be some independent sets that cover $G$, and $\\Lambda_1, \\dots, \\Lambda_{\\Delta}$ arbitrary colorings of the vertices of $\\mathbb{P}_1, \\dots, \\mathbb{P}_{\\Delta}$, respectively, with colors from $\\{\\operatorname{Alice},\\operatorname{Bob}\\}$.\nIt follows that $\\Lambda_\\alpha(A_\\alpha)=\\operatorname{Alice}$ for every $\\alpha\\in \\Delta$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated with $S=T=A_\\alpha$.\nBut then condition (A) does not hold.\n\\end{proof}\n\nAs our main application we describe graphs with chromatic number larger than $\\Delta$ such that $\\Pi_G$ is not playable.\nThis rules out the hope that the complexity of $\\Pi_G$ is connected with chromatic number being larger than $\\Delta$.\nIn Section \\ref{subsec:homlcls} we show the following.\nNote that the case $k=\\Delta$ is \\cref{cor:detmarks}.\n\n\\begin{theorem}\n\\label{th:homomorphism}\nLet $\\Delta > 2$ and $\\Delta < k \\leq 2\\Delta-2$. There exists a graph $G_k$ with $\\chi(G_k)= k$, such that $\\Pi_{G_k}$ is not playable and $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log n ))$. In particular, $\\Pi_{G_k}\\not\\in \\mathsf{BOREL}$ and $\\Pi_{G_k}\\not\\in\\mathsf{LOCAL}(O(\\log^*(n)))$.\n\\end{theorem}\n\nInterestingly, recent results connected to counterexamples to Hedetniemi's conjecture yield the same statement asymptotically, as $\\Delta \\to \\infty$ (see Remark \\ref{r:hedet}).\n\n\\begin{remark}\nIt can be shown that for $\\Delta=3$ both the Chv\\' atal and Gr\\\" otsch graphs are suitable examples for $k=4$.\n\\end{remark}\n\n\\begin{remark}\nAs another application of his technique Marks showed in \\cite{DetMarks} that there is a Borel $\\Delta$-regular forest that does not admit Borel perfect matching.\nThis holds even when we assume that the forest is Borel bipartite, i.e., it has Borel chromatic number $2$.\nIn order to show this result Marks works with free joins of colored hyperedges, that is, Cayley graphs of free products of cyclic groups.\nOne should think of two types of triangles ($3$-hyperedges) that are joined in such a way that every vertex is contained in both types and there are no cycles.\nWe remark that the playability condition can be formulated in this setting.\nSimilarly, one can derive a version of \\cref{thm:MainBorel}.\nHowever, we do not have any application of this generalization.\n\\end{remark}\n\n\n\n\n\n\n\n\n\\subsection{Applications of playability to homomorphism LCLs}\\label{subsec:homlcls}\n\nIn this section we find the graph examples from \\cref{th:homomorphism}.\nFirst we introduce a condition $\\Delta$-(*) that is a weaker condition than having chromatic number at most $\\Delta$, but still implies that the homomorphism LCL is not playable. Then, we will show that--similarly to the way the complete graph on $\\Delta$-many vertices, $K_\\Delta$, is maximal among graphs of chromatic number $\\leq \\Delta$--there exists a maximal graph (under homorphisms) with property $\\Delta$-(*). Recall that we assume $\\Delta >2$.\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.6\\textwidth]{fig\/h3.pdf}\n    \\caption{The maximal graph with the property $\\Delta$-(*) for $\\Delta=3$.\n    \\label{fig:h3}\n\\end{figure}\n\n\n\n\n\\begin{definition}[Property $\\Delta$-(*)]\nLet $\\Delta>2$ and $G=(V,E)$ be a finite graph.\nWe say that $G$ satisfies property $\\Delta$-(*) if there are sets $S_0,S_1\\subseteq V$ such that $G$ restricted to $V \\setminus S_i$ has chromatic number at most $(\\Delta-1)$ for $i\\in \\{0,1\\}$, and there is no edge between $S_0$ and $S_1$.\n\\end{definition}\n\nNote that $\\chi(G) \\leq \\Delta$ implies $\\Delta$-(*): indeed, if $A_1,\\dots,A_\\Delta$ are independent sets that cover $V(G)$, we can set $S_0=S_1=A_1$.\n\nOn the other hand, we claim that if $G$ satisfies $\\Delta$-(*) then $\\chi(G)\\le 2\\Delta-2$. In order to see this, take $S_0,S_1 \\subseteq V(G)$ witnessing $\\Delta$-(*). Then, as there is no edge between $S_0$ and $S_1$, so in particular, between $S_0 \\setminus S_1$ and $S_0 \\cap S_1$, it follows that the chromatic number of $G$'s restriction to $S_0$ is $\\leq \\Delta-1$. But then we can construct proper $\\Delta-1$-colorings of $S_0$ and $V(G) \\setminus S_0$, which shows our claim.\n\n\n\\begin{proposition}\\label{pr:CONdition*}\nLet $G$ be a graph satisfying $\\Delta$-(*).\nThen $\\Pi_G$ is not playable.\n\\end{proposition}\n\\begin{proof}\nFix $S_0,S_1$ as in the definition of $\\Delta$-(*) and assume for a contradiction that colorings $\\Lambda_1, \\dots, \\Lambda_\\Delta$ as described in Definition~\\ref{def:Playability} exist.\nBy Property $\\Delta$-(*), there exist independent sets $A_1, \\dots, A_{\\Delta-1}$ such that $S_0$ together with the $A_i$ covers $G$, i.e., such that $S_0 \\cup A_1 \\cup \\dots \\cup A_{\\Delta-1} = V(G)$.\nFor each $\\alpha \\in [\\Delta-1]$, we must have $\\Lambda_\\alpha(A_{\\alpha})=\\operatorname{Alice}$, since otherwise condition (B) in Definition~\\ref{def:Playability} is violated.\nConsequently $\\Lambda_{\\Delta}(S_0)=\\operatorname{Bob}$, otherwise condition (A) is violated.\nSimilarly $\\Lambda_\\Delta(S_1)=\\operatorname{Bob}$.\nThis shows that $\\Lambda_\\Delta$ does not satisfy condition (B) with $S = S_0$ and $T = S_1$.\n\\end{proof}\n\nNext we describe maximal examples of graphs that satisfy the condition $\\Delta$-(*). That is, we define a graph $H_\\Delta$ that satisfies $\\Delta$-(*), its chromatic number is $2\\Delta-2$ and every other graph that satisfies $\\Delta$-(*) admits a homomorphism in $H_\\Delta$.\nThe graph $H_{3}$ is depicted in \\cref{fig:h3}.\n\nRecall that the (categorical) product $G \\times H$ of graphs $G,H$ is the graph on $V(G) \\times V(H)$, such that $((g,h),(g',h')) \\in E(G \\times H)$ iff $(g,g') \\in E(G)$ and $(h,h') \\in E(H)$.\n  \n  Write $P$ for the product $K_{\\Delta-1}\\times K_{\\Delta-1}$. Let $V_0$ and $V_1$ be vertex disjoint copies of $K_{\\Delta-1}$.\nWe think of vertices in $V_i$ and $P$ as having labels from $[\\Delta-1]$ and $[\\Delta-1] \\times [\\Delta-1]$, respectively.\nThe graph $H_\\Delta$ is the disjoint union of $V_0$, $V_1$, $P$ and an extra vertex $\\dagger$ that is connected by an edge to every vertex in $P$, and additionally, if $v$ is a vertex in $V_0$ with label $i\\in [\\Delta-1]$, then we connect it by an edge with $(i',j)\\in P$ for every $i'\\not=i$ and $j\\in [\\Delta-1]$, and if $v$ is a vertex in $V_1$ with label $j\\in \\Delta-1$, then we connect it by an edge with $(i,j')\\in P$ for every $j'\\not=j$ and $i \\in [\\Delta-1]$.\n\n\n\\begin{proposition}\\label{pr:MinimalExample}\n\\begin{enumerate}\n    \\item \\label{prop:hdelta}$H_\\Delta$ satisfies $\\Delta$-(*).\n    \\item \\label{prop:hdeltachrom} $\\chi(H_\\Delta)=2\\Delta-2$.\n    \\item \\label{prop:hdeltamax} A graph $G$ satisfies $\\Delta$-(*) if and only if it admits a homomorphism to $H_\\Delta$.\n\\end{enumerate}\n\n\\end{proposition}\n\\begin{proof}\n\\eqref{prop:hdelta} Set $S_0=V(V_0)\\cup \\{\\dagger\\}$ and $S_1=V(V_1)\\cup \\{\\dagger\\}$.\nBy the definition there are no edges between $S_0$ and $S_1$.\nConsider now, e.g., $V(H_\\Delta)\\setminus S_0$.\nLet $A_j$ consist of all elements in $P$ that have second coordinate equal to $j$ together with the vertex in $V_1$ that has the label $j$.\nBy the definition, the set $A_i$ is independent and $\\bigcup_{i\\in [\\Delta-1]}A_i$ covers $H_\\Delta\\setminus S_0$, and similarly for $S_1$.\n\n\\eqref{prop:hdeltachrom} By \\eqref{prop:hdelta} and the claim after the definition of $\\Delta$-(*), it is enough to show that $\\chi(H_\\Delta)\\geq 2\\Delta-2$. Towards a contradiction, assume that $c$ is a proper coloring of $H_\\Delta$ with $<2\\Delta-2$-many colors. Note the vertex $\\dagger$ guarantees that  $|c(V(P))|\\leq 2\\Delta-4$, and also $\\Delta-1 \\leq |c(V(P))|$.\n\nFirst we claim that there are no indices $i,j\\in [\\Delta-1]$ (even with $i=j$) such that $c(i,r)\\not=c(i,s)$ and $c(r,j)\\not=c(s,j)$ for every $s\\not=r$: indeed, otherwise, by the definition of $P$ we had $c(i,r)\\not=c(s,j)$ for every $r,s$ unless $(i,r)=(s,j)$, which would the upper bound on the size of $c(V(P))$.\n\nTherefore, without loss of generality, we may assume that for every $i\\in [\\Delta-1]$ there is a color $\\alpha_i$ and two indices $j_i \\neq j'_i$ such that $c(i,j_i)=c(i,j'_i)=\\alpha_i$. It follows form the definition of $P$ and $j_i \\neq j'_i$ that\n $\\alpha_i\\not=\\alpha_{i'}$ whenever $i\\not= i'$. \n \n Moreover, note that any vertex in $V_1$ is connected to at least one of the vertices $(i,j_i)$ and $(i,j'_i)$, hence none of the colors $\\{\\alpha_i\\}_{i \\in [\\Delta-1]}$ can appear on $V_1$. Consequently, since $V_1$ is isomorphic to $K_{\\Delta-1}$ we need to use at least $\\Delta-1$ additional colors, a contradiction.\n\n\\eqref{prop:hdeltamax} First note that if $G$ admits a homomorphism into $H_\\Delta$, then the pullbacks of the sets witnessing  $\\Delta$-(*) will witness that $G$ has $\\Delta$-(*).\n\nConversely, let $G$ be a graph that satisfies $\\Delta$-(*).\nFix the corresponding sets $S_0,S_1$ together with $(\\Delta-1)$-colorings $c_0,c_1$ of their complements.\nWe construct a homomorphism $\\Theta$ from $G$ to $H_\\Delta$. Let\n\\[\\Theta(v)=\n\\begin{cases} \\dagger &\\mbox{if } v \\in S_0 \\cap S_1, \\\\\nc_0(v) & \\mbox{if } v \\in S_1 \\setminus S_0, \\\\\nc_1(v) & \\mbox{if } v \\in S_0 \\setminus S_1,\n\\\\\n(c_0(v),c_1(v)) & \\mbox{if } v \\not \\in S_0 \\cup S_1.\n\\end{cases} \\]\n\n\nObserve that $S=S_0\\cap S_1$ is an independent set such that there is no edge between $S$ and $S_0 \\cup S_1$.\nUsing this observation, one easily checks case-by-case that $\\Theta$ is indeed a homomorphism.\n\n \n \n   \n   \n   \n   \n\\end{proof}\n\nNow, combining what we have so far, we can easily prove Theorem \\ref{th:homomorphism}.\n\n\n\\begin{proof}[Proof of Theorem \\ref{th:homomorphism}]\nIt follows that $\\Pi_{H_\\Delta}$ is not playable from \\cref{pr:MinimalExample}, \\cref{pr:CONdition*}. It is easy to see that if for a graph $G$ the LCL $\\Pi_G$ is not playable then $\\Pi_{G'}$ is not playable for every subgraph $G'$ of $G$. Since erasing a vertex decreases the chromatic number with at most one, for each $k\\leq 2\\Delta-2$ there is a subgraph $G_k$ of $H_\\Delta$ with $\\chi(G_k)=k$, such that $\\Pi_{G_k}$ is not playable.\n\nIt follows from Theorems \\ref{th:MainLOCAL} and \\ref{thm:MainBorel} that there is a $\\Delta$-regular Borel forest that admits no Borel homomorphism to any graph of $G_k$ and that $\\Pi_{G_k} \\not \\in \\mathsf{LOCAL}(O(\\log^*(n)))$.\n\nFinally, note that if $k \\geq \\Delta$ then $G_k$ can be chosen so that it contains $K_\\Delta$, yielding $\\Pi_{G_k} \\in \\mathsf{RLOCAL}(O(\\log\\log(n))$.\n\\end{proof}\n\n\\begin{remark}\n\\label{r:hedet}Recall that Hedetniemi's conjecture is the statement that if $G,H$ are finite graphs then $\\chi(G \\times H)=\\min\\{\\chi(G),\\chi(H)\\}$. This conjecture has been recently disproven by Shitov \\cite{shitov}, and strong counterexamples have been constructed later (see, \\cite{tardif2019note,zhu2021note}). We claim that these imply for $\\varepsilon>0$ the existence of finite graphs $H$ with $\\chi(H) \\geq (2-\\varepsilon)\\Delta$ to which $\\Delta$-regular Borel forests cannot have a homomorphism in BOREL, for every large enough $\\Delta$. Indeed, if a $\\Delta$-regular Borel forest admitted a Borel homomorphism to each finite graph of chromatic number at least $(2-\\varepsilon)\\Delta$, it would have such a homomorphism to their product as well. Thus, we would obtain that the chromatic number of the product of any graphs of chromatic number $(2-\\varepsilon)\\Delta$ is at least $\\Delta+1$. This contradicts Zhu's result \\cite{zhu2021note}, which states that the chromatic number of the product of graphs with chromatic number $n$ can drop to $\\approx \\frac{n}{2}$.\n\n\\end{remark}\n\n\\begin{remark}\nA natural attempt to construct graphs with large girth and not playable homomorphisms problem would be to consider random $d$-regular graphs of size $n$ for a large enough $n$. However, it is not hard to see that setting $\\Lambda(A)=\\operatorname{Alice}$ if and only if $|A|<\\frac{n}{d}$ shows that this approach cannot work.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{th:MainLOCAL}}\\label{sec:marksproof}\n\nIn this section we prove Theorem \\ref{th:MainLOCAL}, by applying Marks' game technique in the LOCAL setting. In order to define our games, we will need certain auxiliary graphs, the so-called \\emph{ID graphs}. \nThe purpose of these graphs is to define a ``playground'' for the games that we consider.\nNamely, vertices in the game are labeled by vertices from the ID graph in such a way that the at the end we obtain a homomorphism from our underlying graph to the ID graph. \n\n\n\n\n\\begin{definition}\nA pair $\\idgraphcolored{n}{t}{r}=(\\idgraph{n}{t}{r},c)$ is called an \\emph{ID graph}, if\n\\begin{enumerate}\n    \\item \\label{p:idgirth} $\\idgraph{n}{t}{r}$ is graph with girth at least $2t+2$,\n    \\item \\label{p:idsize} $|V(\\idgraph{n}{t}{r})|\\le n$,\n    \\item \\label{p:idlabels} $c$ is a $\\Delta$-edge-coloring of $\\idgraph{n}{t}{r}$, such that every vertex is adjacent to at least one edge of each color,\n    \\item \\label{p:idratio} for each $\\alpha \\in [\\Delta]$ the ratio of a maximal independent set of $\\idgraph{n}{t}{r}^{\\alpha}=(V(\\idgraph{n}{t}{r}), E(\\idgraph{n}{t}{r}) \\cap c^{-1}(\\alpha))$ is at most $r$ (i.e., $\\idgraph{n}{t}{r}^\\alpha$ is the graph formed by $\\alpha$-colored edges).\n\\end{enumerate}\n\\end{definition}\n\nBefore we define the game we show that ID graphs exist.\n\n\\begin{proposition}\\label{pr:IDgraph}\nLet $t_n\\in o(\\log(n))$, $r>0$, $\\Delta\\geq 2$.\nThen there is an ID graph $\\idgraphcolored{n}{t_n}{r}$ for every $n\\in \\mathbb{N}$ sufficiently large.\n\\end{proposition}\n\\begin{proof}\nWe use the \\emph{configuration model} for regular random graphs, see \\cite{wormald1999models}.\nThis model is defined as follows.\nLet $n$ be even.\nThen a $d$-regular random sample on $n$-vertices is just a union of $d$-many independent uniform random perfect matchings.\nNote that in this model we allow parallel edges.\n\nIt was proved by Bollob\\'as \\cite{bollobas} that the independence ratio of a random $d$-regular graph is at most $\\frac{2\\log(d)}{d}$ a.a.s.\nMoreover, this quantity is concentrated by an easy application of McDiarmid's result \\cite{mcdiarmid2002concentration}, i.e., \n\\begin{equation}\n    \\P\\left(\\left|X-\\textrm{\\textbf{E}}(X)\\right|\\ge \\sqrt{n}\\right)<2\\exp\\left(-\\frac{n}{d}\\right),\n\\end{equation}\nwhere $X$ is the random variable that counts the size of a maximal independent set.\nTherefore for fixed $n$ large enough we have that the independence ratio of a random sample is at most $\\frac{3\\log(d)}{d}$ with probability at least $\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)$.\n\n\nPick a $d$ large enough such that $\\frac{3\\log(d)}{d}<r$.\nNow for an $n$ large enough take $\\Delta$-many independent samples of random $d$-regular graphs according to the configuration model.\nNote that this is a random sample from the configuration model for $\\Delta d$-regular graphs.\nWe define $c$ to be equal to $\\alpha\\in [\\Delta]$ on edges of the $\\alpha$-th sample.\nThen condition (4) is satisfied with probability at least\n$$\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta.$$\n\nIt remains to show that the girth condition is satisfied.\nRecall that we assume $t_n\\in o(\\log n)$.\nThen we have $(\\Delta d-1)^{2t_n-1}\\in o(n)$.\nUsing \\cite[Corollary~1]{mckayshortcycles} we have that the probability of having girth at least $t_n$ is\n\\begin{equation}\n\\begin{split}\n    \\exp\\left(-\\sum_{a=3}^{t_n}\\frac{(\\Delta d-1)^a}{2a}+o(1)\\right)\\ge & \\exp\\left(-(\\Delta d)^{t_n}\\right) \\\\\n    \\ge & \\exp(-o(n)) \\\\\n    > & 1-\\left(1-2\\exp\\left(-\\frac{n}{d}\\right)\\right)^\\Delta\n\\end{split}\n\\end{equation}\nas $n\\to \\infty$.\nThis shows that there exists such a graph $\\idgraphcolored{t_n}{n}{r}$ with non-zero probability.\n\\end{proof}\n\n\n\n\nNext, we define the games. As mentioned before, the games are going to depend on the following parameters: an algorithm $\\mathcal{A}_n$ of local complexity $t\\in o(\\log(n))$, an ID graph $\\idgraphcolored{n}{t}{r}$, $\\alpha\\in \\Delta$, $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $S \\subseteq \\Sigma$.\n(We will view $\\mathcal{A}_n$, $\\idgraphcolored{n}{t}{r}$ as fixed, and the rest of the parameters as variables).\n\nThe game\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$$\nis defined as follows: two players, $\\operatorname{Alice}$ and $\\operatorname{Bob}$ assign labels to the vertices of a rooted $\\Delta$-regular tree of diameter $t$. The labels are vertices of $\\idgraph{n}{t}{r}$ and the root is labeled $\\sigma$. In the $k$-th round, where $0<k\\le t$, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphcolored{n}{t}{r}$. (For example, if it is $\\operatorname{Alice}$'s turn to label a neighbor of some vertex $v$, that has been assigned a label $\\rho\\in V(\\idgraph{n}{t}{r})$ in the previous round, along an edge that has color $\\beta$, then the allowed labels are only those that span a $\\beta$ edge with $\\rho$ in $\\idgraph{n}{t}{r}$).\n\nBy property \\eqref{p:idsize} of the ID graph, we can fix an injective map from $V(\\idgraph{n}{t}{r})$ to $[n]$.\nNow, we say that $\\operatorname{Alice}$ wins an instance of the game iff $\\mathcal{A}_n$ applied to the produced labeling of the rooted tree does {\\bf not} produce an element of $S$ on the half edge that starts in the root and has edge color $\\alpha$.\nNote that this is well defined thanks to our assumption on the girth of $\\idgraph{n}{t}{r}$.\nLet us define $\\Lambda^{\\sigma}_{\\alpha}(S)$ to be $\\operatorname{Alice}$ or $\\operatorname{Bob}$ depending on who has a winning strategy in the game $$\\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma,S].$$\nSince the game is finite, we have that\n$$\\Lambda^\\sigma_\\alpha:\\mathbb{P}_{\\alpha}\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$$\nis well-defined for all $\\sigma\\in V(\\idgraph{n}{t}{r})$ and $\\alpha\\in [\\Delta]$.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[]{fig\/gamepick-figure0.pdf}\n    \\caption{The game $\\mathbb{G}(\\mathcal{A}_n,n,t,\\idgraphcolored{n}{t}{r})[\\alpha,\\sigma,S]$}\n    \\label{fig:game}\n\\end{figure}\n\n\\begin{proposition}\\label{pr:(A)satisfied}\nLet $\\mathcal{A}_n$ be an algorithm of local complexity $t\\in o(\\log(n))$ that solves $\\Pi$ and $\\sigma\\in V(\\idgraph{n}{t}{r})$.\nThen $(\\Lambda^{\\sigma}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^\\sigma_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$.\nLetting these strategies play against each other in the obvious way, produces a labeling of the tree.\nSince $\\mathcal{A}_n$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at the root and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\nWe are ready to prove Theorem~\\ref{th:MainLOCAL}. \n\\begin{proof}[Proof of Theorem \\ref{th:MainLOCAL}]\nLet $(\\mathcal{A}_n)_{n\\in \\mathbb{N}}$ be a sequence of algorithms of local complexity $t_n\\in o(\\log(n))$ that solve $\\Pi$ and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\nBy \\cref{pr:IDgraph} there exists an ID graph $\\mathbf{H}_{n,t,r}$. Thus, the games above are well-defined and we can construct the functions $(\\Lambda^\\sigma_\\alpha)_{\\alpha\\in \\Delta}$. Since there are only $N$ possibilities for such a sequence, there exists a set $X \\subseteq V(\\idgraph{n}{t}{r})$ of relative size greater than $r$, such that for all $\\sigma \\in X$ the sequence of functions $(\\Lambda^\\sigma_\\alpha)_{\\alpha\\in \\Delta}$ is the same.\n\nSince $\\Pi$ is not playable, we can find an edge color $\\alpha\\in \\Delta$ and sets $S,T\\in \\mathbb{P}_\\alpha$ such that $\\Lambda^\\sigma_{\\alpha}$ does not satisfy (B) from \\cref{def:Playability} with $S,T$ for every $\\sigma\\in X$.\nNote that this is because (A) is always satisfied by \\cref{pr:(A)satisfied}.\n\nBy property \\eqref{p:idratio} of the ID graph, there exist $\\sigma_0,\\sigma_1\\in X$ that span an $\\alpha$ edge in $\\idgraph{n}{t}{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma_0,S], \\ \\mathbb{G}(\\mathcal{A}_n,n,t,\\mathbf{H}_{n,t,r})[\\alpha,\\sigma_1,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $\\sigma_0,\\sigma_1$.\nThis produces a labeling of the vertices that have distance at most $t$ from either of the endpoints of the edge (intuitively, the tree ``rooted'' at this edge of ``diameter'' $t+\\frac{1}{2}$).\nNow applying $\\mathcal{A}_n$ on the vertex with label $\\sigma_0$ produces $\\mathtt{a}_0\\in S$ on the the half edge that starts at this vertex and has color $\\alpha$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}_n$ does not solve $\\Pi$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:MainBorel}}\n\\label{subsec:local_to_borel}\n\nWe show how to use an infinite analogue of the ID graph to prove our main result in the Borel context, \\cref{thm:MainBorel}.\nFinding such a graph\/graphing is in fact much easier in this context.\n\n\\begin{definition}\nLet $r>0$. $\\idgraphingcolored{r}=(\\idgraphing{r},c)$ is an \\emph{ID graphing}, if\n\n\\begin{enumerate}\n    \\item $\\idgraphing{r}$ is an acyclic locally finite Borel graphing on a standard probability measure space $(X,\\mu)$,\n    \\item \\label{p:idgraphinglabels} $c$ is a Borel $\\Delta$-edge-coloring of $\\idgraphing{r}$, such that every vertex is adjacent to at least one edge of each color,\n    \\item \\label{p:idgraphingratio} for each $\\alpha \\in [\\Delta]$ the $\\mu$-measure of a maximal independent set of $\\idgraphing{r}^{\\alpha}=(V(\\idgraphing{r}), E(\\idgraphing{r}) \\cap c^{-1}(\\alpha))$ is at most $r$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{proposition}\n\\label{p:idgraphing}\nFor each $r>0$ there exists an ID graphing $\\idgraphingcolored{r}$.\n\\end{proposition}\n\\begin{proof}\nLet $\\idgraphing{r}$ be a \\emph{local-global limit} of the random graphs constructed in \\cref{pr:IDgraph} (see, e.g., \\cite{hatamilovaszszegedy} for the basic results about local-global convergence).\nIt is not hard to check that this limit satisfies the required properties.\n\\end{proof}\n\n\nNow we are ready to prove the theorem. The proof will closely follow the argument given in the proof of the LOCAL version, i.e., Theorem \\ref{th:MainLOCAL}, but can be understood without reading the latter.\n\n\\begin{proof}[Proof of \\cref{thm:MainBorel}]\nLet $\\Pi$ be an LCL that is not playable, and $N\\in \\mathbb{N}$ be the number of all possible colorings of vertices of $(\\mathbb{P}_{\\alpha})_{\\alpha\\in \\Delta}$ with two colors, that is, $N=2^{\\sum_\\alpha |V(\\mathbb{P}_\\alpha)|}$.\nSet $r:=\\frac{1}{N+1}$.\n\nWe define a Borel acyclic $\\Delta$-regular graph $\\mathcal{G}$, with edges properly colored by $\\Delta$, that does not admit a Borel solution of $\\Pi$.\nVertices of $\\mathcal{G}$ are pairs $(x,A)$, where $x\\in X$ is a vertex of $\\idgraphing{r}$ and $A$ is a countable subgraph of $\\idgraphing{r}$ that is a $\\Delta$-regular tree that contains $x$ and the edge coloring of $\\idgraphingcolored{r}$ induces a proper edge coloring of $A$.\nWe say that $(x,A)$ and $(y,B)$ are connected by an $\\alpha$-edge in $\\mathcal{G}$ if $A=B$, $x,y$ are adjacent in $A$ and the edge that connects them has color $\\alpha\\in \\Delta$.\n\nSuppose for a contradiction that $\\mathcal{A}$ is a Borel function that solves $\\Pi$ on $\\mathcal{G}$.\n\nNext, we define a family of games parametrized by $\\alpha \\in [\\Delta]$, $x \\in V(\\idgraphing{r})$ and $S \\subseteq \\Sigma$. For the reader familiar with Marks' construction, let us point out that for a fixed $x$, the games are analogues to the ones he defines, with the following differences: allowed moves are vertices of the ID graphing $\\idgraphing{r}$ and restricted by its edge relation, and the winning condition is defined by a set of labels, not just merely one label.\n\nSo, the game\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$$ \nis defined as follows: $\\operatorname{Alice}$ and $\\operatorname{Bob}$ alternatingly label vertices of a $\\Delta$-regular rooted tree. The root is labelled by $x$, and the labels come from $V(\\idgraphing{r})$. In the $k$-th round, first $\\operatorname{Alice}$ labels vertices of distance $k$ from the root on the side of the $\\alpha$ edge.\nAfter that, $\\operatorname{Bob}$ labels all remaining vertices of distance $k$, etc (see  \\cref{fig:game}). We also require the assignment of labels to give rise to an edge-color preserving homomorphism to $\\idgraphingcolored{r}$. \n\nIt follows from the acyclicity of $\\idgraphing{r}$ that a play of a game determines a $\\Delta$-regular rooted subtree of $\\idgraphing{r}$ to which the restriction of the edge-coloring is proper. That is, it determines a vertex $(x,A)$ of $\\mathcal{G}$. Let $\\operatorname{Alice}$ win iff the output of $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$ is \\textbf{not} in $S$.\n\nDefine the function $\\Lambda^x_{\\alpha}:\\mathbb{P}_\\alpha\\to \\{\\operatorname{Alice},\\operatorname{Bob}\\}$ assigning the player to some $S \\in V(\\mathbb{P}_\\alpha)$ who has a winning strategy in $\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S]$.\nNote that, since $\\idgraphing{r}$ is locally finite, each player has only finitely many choices at each position. Thus, it follows from Borel Determinacy Theorem that $\\Lambda^x_{\\alpha}$ is well defined.\n\nNow we show the analogue of Proposition \\ref{pr:(A)satisfied}.\n\n\\begin{proposition}\\label{pr:Borel(A)satisfied}\n$(\\Lambda^{x}_{\\alpha})_{\\alpha\\in \\Delta}$ satisfies (A) in \\cref{def:Playability}.\n\\end{proposition}\n\\begin{proof}\nAssume that $(S_\\alpha)_{\\alpha\\in \\Delta}$ is such that $\\Lambda^x_\\alpha(S_\\alpha)=\\operatorname{Alice}$.\nThis means that $\\operatorname{Alice}$ has winning strategy in all the games corresponding to $S_\\alpha$. Letting these strategies play against each other in the obvious way, produces a vertex $(x,A)$ of $\\mathcal{G}$.\nSince $\\mathcal{A}$ solves $\\Pi$ it has to output labeling of half edges $(\\mathtt{a}_{\\alpha})_{\\alpha\\in \\Delta}\\in \\mathcal{N}$, where $\\mathtt{a}_\\alpha$ is a label on the half edge that start at $(x,A)$ and has color $\\alpha$.\nNote that we must have $\\mathtt{a}_\\alpha\\not\\in S_\\alpha$ by the definition of winning strategy for $\\operatorname{Alice}$.\nThis shows that (A) of Definition \\ref{def:Playability} holds.\n\\end{proof}\n\n Let $f$ be \n defined by $$x\\mapsto (\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}.$$ Note that $f$ has a finite range. Using the fact that the allowed moves for each player can be determined in a Borel way, uniformly in $x$, it is not hard to see that for each element $s$ in the range, $f^{-1}(s)$ is in the algebra generated by sets that can be obtained by applying game quantifiers to Borel sets (for the definition see \\cite[Section 20.D]{kechrisclassical}). It has been shown by Solovay \\cite{Solovay} and independently by Fenstad-Norman \\cite{fenstadnorman} that such sets are provably $\\Delta^1_2$, and consequently, measurable. Therefore, $f$ is a measurable map.\n \n As the number of sequences of functions $(\\Lambda^x_{\\alpha})_{\\alpha\\in\\Delta}$ is $\\leq N$, by the choice of $r$, there exists a Borel set $Y$ with $\\mu(Y) > r$, such that $f$ is constant on $Y$.\n \nSince $\\Pi$ is not playable Proposition \\ref{pr:Borel(A)satisfied} gives that there is an $\\alpha\\in \\Delta$ and $S,T\\in \\mathbb{P}_\\alpha$ that violate condition (B).\nRecall that the measure of an independent Borel set of $\\idgraphing{r}^\\alpha$ is at most $r$.\nThat means that there are $x,y\\in Y$ such that $(x,y)$ is an $\\alpha$-edge in $\\idgraphing{r}$.\nWe let the winning strategies of $\\operatorname{Bob}$ in the games\n$$\\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,x,S], \\ \\mathbb{G}(\\mathcal{A},\\idgraphingcolored{r})[\\alpha,y,T],$$\nplay against each other, where we start with an $\\alpha$ edge with endpoints labeled by $x,y$.\nThis produces a labeling of a $\\Delta$-regular tree $A$ with labels from $\\idgraphing{r}$ such that $(x,A)$ and $(y,A)$ span an $\\alpha$-edge in $\\idgraphing{r}$.\nApplying $\\mathcal{A}$ on the half-edge determined by $(x,A)$ and the color $\\alpha$, we obtain $\\mathtt{a}_0\\in S$.\nSimilarly, we produce $\\mathtt{a}_1\\in T$ for $(y,A)$.\nHowever, $(\\mathtt{a}_0,\\mathtt{a}_1)\\not\\in {\\bf E}_\\alpha$ by the definition of an edge in $\\mathbb{P}_{\\alpha}$.\nThis shows that $\\mathcal{A}$ does not produce a Borel solution to $\\Pi$.\n\\end{proof}\n\n\\begin{remark}\nOne can give an alternative proof of the existence of a Borel $\\Delta$-regular forest that does not admit a Borel homomorphism to $G_k$ that avoids using the graphing $\\idgraphing{r}$ and uses the original example described by Marks \\cite{DetMarks} instead.\nThe reason is that the example graphs $G_k$ contain $K_\\Delta$, as discussed in the proof of \\cref{th:homomorphism}.\nThis takes care of the ``non-free'' as in the argument of Marks.\nThen it is enough to use the pigeonhole principle on $\\mathbb{N}$ instead of the independence ratio reasoning.\n\\end{remark}\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\\section{Separation Of Various Complexity Classes}\n\\label{sec:separating_examples}\n\nIn this section we provide examples of several problems that separate some classes from \\cref{fig:big_picture_trees}.\nThe examples show two things. First, we have $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon) \\not= \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$. \nThis shows that there are problems such that their worst case complexity is at least $\\Theta(\\log n)$ on finite $\\Delta$-regular trees, but their average local complexity is constant. \nSecond, we show that there are problems in the class $\\mathsf{BOREL}$ that are not in the class $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{RLOCAL}(o(\\log n))$, on $\\Delta$-regular trees. This shows that one cannot in general hope that results from (Borel) measurable combinatorics can be turned into very efficient (sublogarithmic) distributed algorithms. \n\n\\subsection{Preliminaries}\n\\label{subsec:preliminaries_separating}\n\nWe first show that there is a strong connection between randomized local complexities and uniform local complexities. Afterwards, we introduce a generic construction that turns any LCL into a new LCL. We later use this generic transformation to construct an LCL that is contained in the set $\\mathsf{BOREL} \\setminus \\mathsf{RLOCAL}(O(\\log \\log n))$.\n\n\\paragraph{Uniform vs Local Randomized Complexity} \n\nWe will now discuss the connections between uniform and randomized complexities. \nNote that the easy part of the connection between the two concepts is turning uniform local algorithms into randomized ones, as formalized in the following proposition. \n\n\\begin{proposition}\n\\label{prop:bee_gees}\nWe have $\\mathsf{OLOCAL}(t(\\varepsilon)) \\subseteq \\mathsf{RLOCAL}(t(1\/n^{O(1)}))$. \n\\end{proposition}\n\\begin{proof}\nWe claim that an uniform local algorithm $\\mathcal{A}$ with an uniform local complexity of $t(\\varepsilon)$ can be turned into a local randomized algorithm $\\mathcal{A}'$ with a local complexity  of $t(1\/n^{O(1)})$. \n\nThe algorithm $\\mathcal{A}'$ simulates $\\mathcal{A}$ on an infinite $\\Delta$-regular tree -- each vertex $u$ of degree less than $\\Delta$ in the original tree pretends that the tree continues past its virtual half-edges and the random bits of $u$ are used to simulate the random bits in this virtual subtree. \nChoosing $\\varepsilon = 1\/n^{O(1)}$, one gets that the probability of $\\mathcal{A}'$ needing to look further than $t(\\varepsilon)$ for any vertex is bounded by $1\/n^{O(1)}$, as needed in the definition of the randomized local complexity. \n\\end{proof}\n\n\nOn the other hand, we will use the following proposition from \\cite{grebik_rozhon2021toasts_and_tails}. It informally states that the existence of \\emph{any} uniform local algorithm together with the existence of a sufficiently fast randomized local algorithm for a given LCL $\\Pi$ directly implies an upper bound on the uniform complexity of $\\Pi$.\n\\begin{proposition}[\\cite{grebik_rozhon2021toasts_and_tails}]\n\\label{prop:local->uniform}\nLet $\\mathcal{A}$ be an uniform local algorithm solving an LCL problem $\\Pi$ such that its \\emph{randomized} local complexity on finite $\\Delta$-regular trees is $t(n)$ for $t(n) = o(\\log n)$. \nThen, the uniform local complexity of $\\mathcal{A}$ on infinite $\\Delta$-regular trees is $O(t(1\/\\varepsilon))$.\n\\end{proposition}\n\n\\cref{prop:local->uniform} makes our life simpler, since we can take a known randomized distributed local algorithm with local complexity $g(n) = o(\\log n)$ and check if it works without the knowledge of $n$. If yes, this automatically implies that the  uniform local complexity of the algorithm is $h(\\varepsilon) = O(g(1\/\\varepsilon))$. \nIn particular, combining \\cref{prop:local->uniform} with the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup} and verifying that the algorithms of \nFischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and  Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} work without the knowledge of $n$, we obtain the following result. \n\n\n\\begin{theorem}\n\\label{thm:classification_local_uniform_trees}\nWe have:\n\\begin{itemize}\n    \\item $\\mathsf{LOCAL}(O(1)) = \\mathsf{OLOCAL}(O(1))$\n    \\item $\\mathsf{RLOCAL}(O(\\log^* n)) = \\mathsf{OLOCAL}(O(\\log^* 1\/\\varepsilon))$\n    \\item $\\mathsf{RLOCAL}(O(\\log\\log n)) = \\mathsf{OLOCAL}(O(\\log\\log 1\/\\varepsilon))$\n\\end{itemize}\nMoreover, there are no other possible uniform local complexities for $t(\\varepsilon) = o(\\log 1\/\\varepsilon)$. \n\\end{theorem}\nWe note that the first two items are proven in \\cite{grebik_rozhon2021toasts_and_tails}.\nFor the third item it suffices by known reductions to find an uniform algorithm\nsolving a version of the distributed Lov\\'asz Local Lemma (LLL) on so-called tree-structured dependency graphs considered in~\\cite{ChangHLPU20}.\n\n\\begin{proof}[Proof sketch]\nThe proof follows from \\cref{prop:bee_gees} and the following ideas. \nThe first item follows from the fact that any local algorithm with local complexity $O(1)$ can simply be made uniform. More specifically, there exists a constant $n_0$ --- depending only on the problem $\\Pi$ --- such that the algorithm, being told the size of the graph is $n_0$, is correct on \\emph{any} graph of size $n \\ge n_0$. \n\nFor the second item, by the work of \\cite{chang_kopelowitz_pettie2019exp_separation,chang_pettie2019time_hierarchy_trees_rand_speedup}, it suffices to check that there is an uniform distributed $(\\Delta + 1)$-coloring algorithm with uniform local complexity $O(\\log^* 1\/\\varepsilon)$. Such an algorithm was given in \\cite{HolroydSchrammWilson2017FinitaryColoring}, or follows from the work in \\cite{Korman_Sereni_Viennot2012Pruning_algorithms_+_oblivious_coloring} and \\cref{prop:local->uniform}. \n\nSimilarly, for the third item, by the work of \\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} it suffices to check that there is an uniform distributed algorithm for a specific LLL problem on trees\nwith uniform local complexity $O(\\log\\log 1\/\\varepsilon)$.\nSuch an algorithm can be obtained by combining the randomized \\emph{pre-shattering} algorithm of Fischer and Ghaffari~\\cite[Section 3.1.1]{fischer-ghaffari-lll} and the deterministic \\emph{post-shattering} algorithm of Chang~et~al.~\\cite[Section 5.1]{ChangHLPU20} in a \\emph{graph shattering} framework~\\cite{barenboim2016locality}, which solves the LLL problem \nwith local complexity $O(\\log\\log n)$.\nBy \\cref{prop:local->uniform}, it suffices to check that this algorithm can be made to work even if it does not know the size of the graph $n$. We defer the details to \\cref{sec:LLL}.\nThis finishes the proof of \\cref{thm:classification_local_uniform_trees}. \n\\end{proof}\n\n\n\n\n\n\n\n\\paragraph{Adding Paths}\nBefore we proceed to show some separation results, we define a certain construction that turns any LCL problem $\\Pi$ into a new LCL problem $\\overline{\\Pi}$ with the following property. If the original problem $\\Pi$ cannot be solved by a fast local algorithm, then the same holds for $\\overline{\\Pi}$. However, $\\overline{\\Pi}$ might be strictly easier to solve than $\\Pi$ for $\\mathsf{BOREL}$ constructions. \n\n\\begin{definition}\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL. \nWe define an LCL $\\overline{\\Pi}=(\\Sigma',\\mathcal{V}',\\mathcal{E}')$ as follows.\nLet $\\Sigma'$ be $\\Sigma$ together with one new label.\nLet $\\mathcal{V}'$ be the union of $\\mathcal{V}$ together with any cardinality-$\\Delta$ multiset that contains the new label exactly two times.\nLet $\\mathcal{E}'$ be the union of $\\mathcal{E}$ together with the cardinality-$2$ multiset that contains the new label twice.\n\\end{definition}\n\nIn other words, the new label determines doubly infinite lines in infinite $\\Delta$-regular trees, or lines that start and end in virtual half-edges in finite $\\Delta$-regular trees.\nMoreover, a vertex that is on such a line does not have to satisfy any other vertex constraint.\nWe call these vertices \\emph{line-vertices} and each edge on a line a \\emph{line-edge}.\n\n\n\\begin{proposition}\n\\label{pr:AddingLineLLL}\nLet $\\Pi$ be an LCL problem such that $\\overline{\\Pi}\\in \\mathsf{RLOCAL}(t(n))$ for $t(n)  = o(\\log(n))$. Then also $\\Pi \\in \\mathsf{RLOCAL}(O(t(n)))$.\n\\end{proposition}\n\\begin{proof}\nLet $\\overline{\\mathcal{A}}$ be a randomized $\\mathsf{LOCAL}$ algorithm that solves $\\overline{\\Pi}$ in $t(n) = o(\\log n)$ rounds with probability at least $1 - 1\/n^C$ for some sufficiently large constant $C$. We will construct an algorithm $\\mathcal{A}$ for $\\Pi$ with complexity $t(n)$ that is correct with probability $1 - \\frac{4}{n^{C\/3 - 2}}$. The success probability of $\\mathcal{A}$ can then be boosted by ``lying to it'' that the number of vertices is $n^{O(1)}$ instead of $n$; this increases the running time by at most a constant factor and boosts the success probability back to $1 - 1\/n^C$. \n\nConsider a $\\Delta$-regular rooted finite tree $T$ of depth $10t(n)$ and let $u$ be its root vertex. \nNote that $|T| \\le \\Delta^{10t(n)+1} < n$, for $n$ large enough. \n\n\nWe start by proving that when running $\\overline{\\mathcal{A}}$ on the tree $T$, then $u$ is most likely not a line-vertex. This observation then allows us to turn $\\overline{\\mathcal{A}}$ into an algorithm $\\mathcal{A}$ that solves $\\Pi$ on any $\\Delta$-regular input tree.\nLet $X$ be the indicator of $\\mathcal{A}$ marking $u$ as a line-vertex. Moreover, for $i \\in [\\Delta]$, let $Y_i$ be the indicator variable for the following event. The number of line edges in the $i$-th subtree of $u$ with one endpoint at depth $5t(n)$ and the other endpoint at depth $5t(n) + 1$ is odd.\n\nBy a simple parity argument we can relate the value of $X$ with the values of $Y_1,Y_2,\\ldots,Y_\\Delta$ as follows. If $X = 0$, that is, $u$ is not a line-vertex, then all of the $Y_i$'s have to be $0$, as each path in the tree $T$ is completely contained in one of the $\\Delta$ subtrees of $u$. On the other hand, if $u$ is a line-vertex, then there exists exactly one path, the one containing $u$, that is not completely contained in one of the $\\Delta$ subtrees of $u$. This in turn implies that exactly two of the $\\Delta$ variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are equal to $1$.\n\nThe random variables $Y_1,Y_2,\\ldots,Y_\\Delta$ are identically distributed and mutually independent. Hence, if $\\P(Y_i = 1) > \\frac{1}{n^{C\/3}}$, then the probability that there are at least $3$ $Y_i$'s equal to $1$ is strictly greater than $\\left(\\frac{1}{n^{C\/3}}\\right)^3 = \\frac{1}{n^C}$. This is a contradiction, as in that case $\\overline{\\mathcal{A}}$ does not produce a valid output, which according to our assumption happens with probability at most $\\frac{1}{n^C}$. \n\nThus, we can conclude that $\\P(Y_i = 1) \\leq \\frac{1}{n^{C\/3}}$. By a union bound, this implies that all of the $Y_i$'s are zero with probability at least $\\frac{1}{n^{C\/3 - 1}}$, and in that case $u$ is not a line-vertex. \n\nFinally, the algorithm $\\mathcal{A}$ simulates $\\overline{\\mathcal{A}}$ as if the neighborhood of each vertex was a $\\Delta$-regular branching tree up to depth at least $t(n)$. \n\nIt remains to analyze the probability that $\\mathcal{A}$ produces a valid solution for the LCL problem $\\Pi$. To that end, let $v$ denote an arbitrary vertex of the input tree. The probability that the output of $v$'s half edges satisfy the vertex constraint of the LCL problem $\\overline{\\Pi}$ is at least $1 - 1\/n^C$. Moreover, in the case that $v$ is not a line-vertex, which happens with probability at least $1 - \\frac{1}{n^{C\/3 - 1}}$, the output of $v$'s half edges even satisfy the vertex constraint of the LCL problem $\\Pi$. Hence, by a union bound the vertex constraint of the LCL problem $\\Pi$ around $v$ is satisfied with probability at least $1 - \\frac{2}{n^{C\/3 - 1}}$. With exactly the same reasoning, one can argue that the probability that the output of $\\mathcal{A}$ satisfies the edge constraint of the LCL problem $\\Pi$ at a given edge is also at least $1 - \\frac{2}{n^{C\/3 - 1}}$. Finally, doing a union bound over the $n$ vertex constraints and $n-1$ edge constraints it follows that $\\mathcal{A}$ produces a valid solution for $\\Pi$ with probability at least $1 - \\frac{4}{n^{C\/3 - 2}}$.\n\\end{proof}\n\n\\begin{remark}\nThe ultimate way how to solve LCLs of the form $\\overline{\\Pi}$ is to find a spanning forest of lines.\nThis is because once we have a spanning forest of lines, then we might declare every vertex to be a line-vertex and label every half-edge that is contained on a line with the new symbol in $\\overline{\\Pi}$.\nThe remaining half-edges can be labeled arbitrarily in such a way that the edge constraints are satisfied.\n\nPut otherwise, once we are able to construct a spanning forest of lines and $\\mathcal{E}\\not=\\emptyset$, where $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$, then $\\overline{\\Pi}$ can be solved.\nMoreover, in that case the complexity of $\\overline{\\Pi}$ is upper bounded by the complexity of finding the spanning forest of lines.\n\nLyons and Nazarov \\cite{lyons2011perfect} showed that $\\Pi_{\\text{pm}}\\in \\mathsf{fiid}$ and it is discussed in  \\cite{LyonsTrees} that the construction can be iterated $(\\Delta-2)$-many times. After removing each edge that is contained in one of the $\\Delta-2$ perfect matchings each vertex has a degree of two.\nHence, it is possible to construct a spanning forest of lines as fiid.\nConsequently, $\\overline{\\Pi}\\in \\mathsf{fiid}$ for every $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ with $\\mathcal{E}\\not=\\emptyset$.\n\\end{remark}\n\n\n\\subsection{$\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$}\n\nWe now formalize the proof that $\\mathsf{LOCAL}(O(\\log^* n)) \\not= \\mathsf{BOREL}$ from \\cref{sec:introseparation}. \nRecall that $\\Pi_{\\chi,\\Delta}$ is the proper vertex $\\Delta$-coloring problem.\nThe following claim directly follows by  \\cref{pr:AddingLineLLL} together with the fact that $\\Pi_{\\chi,\\Delta}\\not\\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{claim}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}}\\not \\in \\mathsf{LOCAL}(O(\\log^* n))$.\n\\end{claim}\n\nOn the other hand we show that adding lines helps to find a Borel solution.\nThis already provides a simple example of a problem in the set $\\mathsf{BOREL} \\setminus \\mathsf{LOCAL}(O(\\log^* n))$.\n\n\\begin{proposition}\nWe have $\\overline{\\Pi_{\\chi,\\Delta}} \\in \\mathsf{BOREL}$. \n\\end{proposition}\n\\begin{proof}\nBy \\cite{KST}, every Borel graph of finite maximum degree admits a Borel maximal independent set.\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular acyclic graph on a standard Borel space $X$.\nBy an iterative application of the fact above we find Borel sets $A_1,\\dots,A_{\\Delta-2}$ such that $A_1$ is a maximal independent set in $\\mathcal{G}$ and $A_i$ is a maximal independent set in $\\mathcal{G}\\setminus \\bigcup_{j<i}A_j$ for every $i>1$.\nWe think of $A_1,\\dots,A_{\\Delta-2}$ as color classes for the first $\\Delta-2$ colors.\nLet $B=X\\setminus \\bigcup_{i\\in [\\Delta-2]}A_i$ and let $\\mathcal{H}$ be the subgraph of $\\mathcal{G}$ determined by $B$.\nIt is easy to see that the maximum degree of $\\mathcal{H}$ is $2$.\nIn particular, $\\mathcal{H}$ consists of finite paths, one-ended infinite paths or doubly infinite paths.\nWe use the extra label in $\\overline{\\Pi_{\\chi,\\Delta}}$ to mark the doubly infinite paths.\nIt remains to use the remaining $2$ colors in $[\\Delta]$ to define a proper vertex $2$-coloring of the finite and one-ended paths.\nIt is a standard argument that this can be done in a Borel way and it is easy to verify that, altogether, we found a Borel $\\overline{\\Pi_{\\chi,\\Delta}}$-coloring of $\\mathcal{G}$.\n\\end{proof}\n\n\n\n\\subsection{Examples and Lower Bound}\n\nIn this subsection we define two LCLs and show that they are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\nIn the next subsection we show that one of them is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and the other in the class $\\mathsf{BOREL}$.\nBoth examples that we present are based on the following relaxation of the perfect matching problem.\n\n\\begin{definition}[Perfect matching in power-$2$ graph]\nLet $\\Pi^2_{\\text{pm}}$ be the perfect matching problem in the power-$2$ graph, i.e., in the graph that we obtain from the input graph by adding an edge between any two vertices that have distance at most $2$ in the input graph\n\\end{definition}\n\nWe show that $\\Pi^2_{\\text{pm}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$. \\cref{pr:AddingLineLLL} then directly implies that $\\overline{\\Pi^2_{\\text{pm}}}$ is not contained in $\\mathsf{RLOCAL}(o(\\log n))$ as well.\nOn the other hand, we later use a one-ended spanning forest decomposition to show that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ and a one or two-ended spanning forest decomposition to show that $\\overline{\\Pi^2_{\\text{pm}}}$ is in $\\mathsf{BOREL}$.\n\n\nWe first show the lower bound result.\nThe proof is based on a simple parity argument as in the proof of \\cref{pr:AddingLineLLL}.\n\n\\begin{theorem}\nThe problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ are not in the class $\\mathsf{RLOCAL}(o(\\log n))$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose that the theorem statement does not hold for $\\Pi^2_{\\text{pm}}$.\nThen, by \\cref{thm:basicLOCAL}, there is a distributed randomized algorithm solving $\\Pi^2_{\\text{pm}}$ with probability at least $1 - 1\/n$. By telling the algorithm that the size of the input graph is $n^{2\\Delta}$ instead of $n$, we can further boost the success probability to $1 - 1\/n^{2\\Delta}$. The resulting round complexity is at most a constant factor larger, as $\\Delta = O(1)$.\nWe can furthermore assume that the resulting algorithm $\\mathcal{A}$ will always output for each vertex $v$ a vertex $M(v) \\neq v$ in $v$'s $2$-hop neighborhood, even if $\\mathcal{A}$ fails to produce a valid solution. The vertex $M(v)$ is the vertex $v$ decides to be matched to, and if $\\mathcal{A}$ produces a valid solution it holds that $M(M(v)) = v$.\n\nConsider a fully branching $\\Delta$-regular tree $T$ of depth $10t(n)$. Note that $|T| < n$ for $n$ large enough. Let $u$ be the root of $T$ and $T_i$ denote the $i$-th subtree of $u$ (so that $u \\not\\in T_i$).\nLet $S_i$ denote the set of vertices $v$ in $T_i$ such that both $v$ and $M(v)$ have distance at most $2t(n)$ from $u$.\nNote that $M(v)$ is not necessarily a vertex of $T_i$.\nLet $Y_i$ be the indicator of whether $S_i$ is even.\n\nObserve that, if $\\mathcal{A}$ does not fail on the given input graph, then the definition of the $S_i$ implies that every vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is matched to a vertex in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$.\nHence, the number of vertices in $\\{ u \\} \\cup \\bigcup_{i \\in [\\Delta]} S_i$ is even, which implies that we cannot have $Y_1 = Y_2 = \\ldots = Y_\\Delta = 1$ unless $\\mathcal{A}$ fails.\nNote that $Y_i$ depends only on the output of $\\mathcal{A}$ at the vertices in $T_i$ that have a distance of precisely $2t(n) - 1$ or $2t(n)$ to $u$ (as all other vertices in $T_i$ are guaranteed to be in $S_i$). \nHence, the events $Y_i = 1$, $i \\in [\\Delta]$, are independent, and, since $\\P(Y_1 = 1) = \\ldots = \\P(Y_{\\Delta} = 1)$ (as all vertices in all $T_i$ see the same topology in their $t(n)$-hop view), we obtain $(\\P(Y_1 = 1))^{\\Delta} \\le 1\/n^{2\\Delta}$, which implies $\\P(Y_i = 1) \\le 1\/n^2$, for any $i \\in [\\Delta]$.\n\n\nHence, with probability at least $1 - \\Delta \/n^2$ we have $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, by a union bound.\nLet $v_1, v_2, \\dots, v_\\Delta$ denote the neighbors of $u$ with $v_i$ being the root of subtree $T_i$.\nNote that if $\\mathcal{A}$ does not fail and $Y_1=Y_2=\\ldots = Y_\\Delta = 0$, then it has to be the case that $u$ is matched to a vertex in some subtree $T_i$ (not necessarily $v_i$), while any vertex from $N_{-i}(u) := \\{v_1, v_2, \\dots, v_{i-1}, v_{i+1}, \\dots, v_\\Delta\\}$ is matched with another vertex from $N_{-i}(u)$ (hence, we already get that $\\Delta$ needs to be odd).\nThis is because each subtree needs to have at least one vertex matched to a different subtree (since $|S_i|$ is odd for each $i \\in [\\Delta]$).\nIf $M(u)$ is a vertex in $T_i$ and, for any $v \\in N_{-i}(u)$, we have $M(v) \\in N_{-i}(u)$, we say that $u$ orients the edge going to $v_i$ outwards.\n\nConsider a path $(u_0 = u, u_1, \\dots, u_k)$, where $k = 2t(n) + 3$.\nBy the argument provided above, the probability that $u$ orients an edge outwards is at least $1 - \\Delta\/n^2 - 1\/n^{2\\Delta}$, and, if $u$ orients an edge outwards, the probability that this edge is not $uu_1$ is $(\\Delta - 1)\/\\Delta$, by symmetry.\nHence, we obtain (for sufficiently large $n$) that $\\P(\\mathcal{E}_1) \\geq 1\/2$, where $\\mathcal{E}_1$ denotes the event that vertex $u$ orients an edge different from $uu_1$ outwards.\nWith an analogous argument, we obtain that $\\P(\\mathcal{E}_2) \\geq 1\/2$, where $\\mathcal{E}_2$ denotes the event that vertex $u_k$ orients an edge different from $u_ku_{k-1}$ outwards.\nSince $\\mathcal{E}_1$ and $\\mathcal{E}_2$ are independent (due to the fact that the distance between any neighbor of $u$ and any neighbor of $u_k$ is at least $2t(n) + 1$), we obtain $\\P(\\mathcal{E}_1 \\cap \\mathcal{E}_2) \\geq 1\/4$.\nMoreover, the probability that all vertices in $\\{ u_1, \\dots, u_{k-1} \\}$ orient an edge outwards is at least $1 - (2t(n)+2)(\\Delta\/n^2 + 1\/n^{2\\Delta})$, which is at least $4\/5$ for sufficiently large $n$.\nThus, by a union bound, there is a probability of at least $1 - 3\/4 - 1\/5 - 1\/n^{2\\Delta}$ that $\\mathcal{A}$ does not fail, all vertices in $(u, u_1, \\dots, u_k)$ orient an edge outwards, and the outward oriented edges chosen by $u$ and $u_k$ are not $uu_1$ and $u_ku_{k-1}$, respectively.\nFor sufficiently large $n$, the indicated probability is strictly larger than $0$.\nWe will obtain a contradiction and conclude the proof for $\\Pi^2_{\\text{pm}}$ by showing that there is no correct solution with the described properties.\n\nAssume in the following that the solution provided by $\\mathcal{A}$ is correct and $u, u_1, \\dots, u_k$ satisfy the mentioned properties.\nSince $u$ orients an edge different from $uu_1$ outwards, $u_1$ must be matched to a neighbor of $u$, which implies that $u_1$ orients edge $u_1u$ outwards.\nUsing an analogous argumentation, we obtain inductively that $u_j$ orients edge $u_ju_{j-1}$ outwards, for all $2 \\leq j \\leq k$.\nHowever, this yields a contradiction to the fact that the edge that $u_k$ orients outwards is different from $u_ku_{k-1}$, concluding the proof (for $\\Pi^2_{\\text{pm}}$).\n\nThe theorem statement for $\\overline{\\Pi^2_{\\text{pm}}}$ follows by applying \\cref{pr:AddingLineLLL}.\n\n\n\\end{proof}\n\n\n\n\n\n\\subsection{Upper Bounds Using Forest Decompositions}\n\nWe prove an upper bound for the problems $\\Pi^2_{\\text{pm}}$ and $\\overline{\\Pi^2_{\\text{pm}}}$ defined in the previous subsection.\nWe use a technique of decomposing the input graph in a spanning forest with some additional properties, i.e., one or two ended. This technique was used in \\cite{BrooksMeas} to prove Brooks' theorem in a measurable context.\nNamely, they proved that if $\\mathcal{G}$ is a Borel $\\Delta$-regular acyclic graph and $\\mu$ is a Borel probability measure, then it is possible to erase some edges of $\\mathcal{G}$ in such a way that the remaining graph is a one-ended spanning forest on a $\\mu$-conull set.\nIt is not hard to see that one can solve $\\Pi^2_{\\text{pm}}$ on one-ended trees in an inductive manner, starting with the leaf vertices. Consequently we have $\\Pi^2_{\\text{pm}}\\in \\mathsf{MEASURE}$.\nWe provide a quantitative version of this result and thereby show that the problem of constructing a one-ended forest is contained in $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$.\nA variation of the construction shows that it is possible to find a one or two-ended spanning forest in a Borel way.\nThis allows us to show that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\nAs an application of the decomposition technique we show that Vizing's theorem, i.e., proper $\\Delta+1$ edge coloring $\\Pi_{\\chi',\\Delta+1}$, for $\\Delta=3$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\n\n\\subsubsection{Uniform Complexity of the One-Forest Decomposition }\nWe start with the definition of a one-ended spanning forest.\nIt can be viewed as a variation of the edge grabbing problem $\\Pi_{\\text{edgegrab}}$.\nNamely, if a vertex $v$ grabs an incident edge $e$, then we think of $e$ as being oriented away from $v$ and $v$ selecting the other endpoint of $e$ as its parent.\nSuppose that $\\mathcal{T}$ is a solution of $\\Pi_{\\text{edgegrab}}$ on $T_\\Delta$.\nWe denote with $\\mathcal{T}^{\\leftarrow}(v)$ the subtree with root $v$.\n\n\n\\begin{definition}[One-ended spanning forest]\nWe say that a solution $\\mathcal{T}$ of $\\Pi_{\\text{edgegrab}}$ is a \\emph{one-ended spanning forest} if $\\mathcal{T}^{\\leftarrow}(v)$ is finite for every $v\\in T_\\Delta$ (see \\cref{fig:one_ended}). \n\n\\end{definition}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.4\\textwidth]{fig\/one_ended.pdf}\n    \\caption{An example of a one-ended forest decomposition of an infinite 3-regular tree. }\n    \\label{fig:one_ended}\n\\end{figure}\n\n\nNote that on an infinite $\\Delta$-regular tree every connected component of a one-ended spanning forest must be infinite.\nFurthermore, recall that we discussed above that it is possible to construct a one-ended spanning forest in $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nNext we formulate our quantitative version.\n\n\n\n\n\n\\begin{restatable}{theorem}{oneendedtreetail}\n\\label{thm:one_ended_forest}\nThe one-ended spanning forest can be constructed by an uniform local algorithm  with an uniform local complexity of $O(\\operatorname{poly}\\log1\/\\varepsilon)$. \nMore precisely, there is an uniform distributed algorithm $\\mathcal{A}$ that computes a one-ended spanning forest and if we define $R(v)$ to be the smallest coding radius that allows $v$ to compute $\\mathcal{T}^{\\leftarrow}(v)$, then\n\\[\nP(R(v) > O(\\operatorname{poly} \\log 1\/\\varepsilon)) \\le \\varepsilon. \n\\]\n\\end{restatable}\n\nThe formal proof of \\cref{thm:one_ended_forest} can be found in \\cref{sec:formalproof}.\nWe first show the main applications of the result.\nNamely, we show that $\\Pi^2_{\\text{pm}}$ can be solved inductively starting from the leaves of the one-ended trees. This proves that $\\Pi^2_{\\text{pm}}$ is in $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$ and hence $\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon) \\setminus \\mathsf{RLOCAL}(o(\\log n))$ is non-empty.\n\n\\begin{theorem}\\label{thm:twomatchFFIID}\nThe problem $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$ and $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\\end{theorem}\n\\begin{proof}\nFirst we show that every infinite one-ended directed tree $T$ admits an inductive solution from the leaves to the direction of infinity. More precisely, we define a power-$2$ perfect matching on the infinite one-ended directed tree $T$ such that the following holds. Let $v$ be an arbitrary vertex and $T^{\\leftarrow}(v)$ the finite subtree rooted at $v$. Then, all vertices in $T^{\\leftarrow}(v)$, with the possible exception of $v$, are matched to a vertex in $T^{\\leftarrow}(v)$. Moreover, for each vertex in $T^{\\leftarrow}(v) \\setminus \\{v\\}$, it is possible to determine to which vertex it is matched by only considering the subtree $T^{\\leftarrow}(v)$. \nWe show that it is possible to define a perfect matching in that way by induction on the height of the subtree $T^{\\leftarrow}(v)$. \n\nWe first start with the leaves and declare them as unmatched in the first step of the induction.\n\nNow, consider an arbitrary vertex $v$ with $R \\geq 1$ children that we denote by $v_1, v_2, \\ldots, v_R$.\nBy the induction hypothesis, all vertices in $T^{\\leftarrow}(v_i) \\setminus \\{v_i\\}$ are matched with a vertex in $T^{\\leftarrow}(v_i)$ for $i \\in [R]$ and it is possible to compute that matching given $T^{\\leftarrow}(v)$. However, some of the children of $v$ are possibly still unmatched. If the number of unmatched children of $v$ is even, then we simply pair them up in an arbitrary but fixed way and we leave $v$ unmatched. Otherwise, if the number of unmatched children of $v$ is odd, we pair up $v$ with one of its unmatched children and we pair up all the remaining unmatched children of $v$ in an arbitrary but fixed way. This construction clearly satisfies the criteria stated above. In particular, the resulting matching is a perfect matching, as every vertex gets matched eventually.\n\n\nBy \\cref{thm:one_ended_forest} there is an uniform distributed algorithm of complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$ that computes a one-ended spanning forest $\\mathcal{T}$ on $T_\\Delta$.\nMoreover, every vertex $v\\in T_\\Delta$ can compute where it is matched with an uniform local complexity of $O(\\operatorname{poly} \\log 1\/\\varepsilon)$.\nThis follows from the discussion above, as a vertex $v$ can determine where it is matched once it has computed $\\mathcal{T}^{\\leftarrow}(w)$ for each $w$ in its neighborhood.\nHence, $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{OLOCAL}(O(\\operatorname{poly} \\log 1\/\\varepsilon))$.\n\nSimilarly, the one-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$ can be constructed in the class $\\mathsf{MEASURE}$ by \\cite{BrooksMeas}.\nBy the discussion above, this immediately implies that $\\Pi^2_{\\text{pm}}$ is in the class $\\mathsf{MEASURE}$. \n\\end{proof}\n\n\n\n\n\\subsubsection{Decomposition in $\\mathsf{BOREL}$}\nNext, we show that a similar, but weaker, decomposition can be done in a Borel way.\nNamely, we say that a subset of edges of an acyclic infinite graph $G$, denoted by $\\mathcal{T}$, is a \\emph{one or two-ended spanning forest} if every vertex $v\\in G$ is contained in an infinite connected component of $\\mathcal{T}$ and each infinite connected component of $\\mathcal{T}$ has exactly one or two directions to infinity.\nLet $S$ be such a connected component.\nNote that if $S$ has one end, then we can solve $\\Pi^2_{\\text{pm}}$ on $S$ as above.\nIf $S$ has two ends, then $S$ contains a doubly infinite path $P$ with the property that erasing $P$ splits $S\\setminus P$ into finite connected components.\nIn order to solve $\\overline{\\Pi^2_{\\text{pm}}}$ we simply solve $\\Pi^2_{\\text{pm}}$ on the finite connected components of $S\\setminus P$ (with some of the vertices in $S\\setminus P$ being potentially matched with vertices on the path $P$) and declare vertices on $P$ to be line vertices.\n\nThe high-level idea to find a one or two-ended spanning forest is to do the same construction as in the measure case and understand what happens with edges that do not disappear after countably many steps.\nA slight modification in the construction guarantees that what remains are doubly infinite paths.\n\n\\begin{restatable}{theorem}{oneendedforestborel}\n\\label{thm:OneTwoEndBorel}\nLet $\\mathcal{G}$ be a Borel $\\Delta$-regular forest. \nThen there is a Borel one or two-ended spanning forest $\\mathcal{T}\\subseteq \\mathcal{G}$.\n\\end{restatable}\n\nThe proof of \\cref{thm:OneTwoEndBorel} can be found in \\cref{sec:oneortwo}.\nWe remark that the notation used in the proof is close to the notation in the proof that gives a measurable construction of a one-ended spanning forest in \\cite{BrooksMeas}.\nNext, we show more formally how \\cref{thm:OneTwoEndBorel} implies that $\\overline{\\Pi^2_{\\text{pm}}}\\in \\mathsf{BOREL}$.\n\n\\begin{theorem}\nThe problem $\\overline{\\Pi^2_{\\text{pm}}}$ is in the class $\\mathsf{BOREL}$.\n\\end{theorem}\n\\begin{proof}\n\nLet $\\mathcal{T}$ be the one or two-ended spanning forest given by \\cref{thm:OneTwoEndBorel} and $S$ be a connected component of $\\mathcal{T}$.\nIf $S$ is one-ended, then we use the first part of the proof of \\cref{thm:twomatchFFIID} to find a solution of $\\Pi^2_{\\text{pm}}$ on $S$.\nSince deciding that $S$ is one-ended as well as computing an orientation towards infinity can be done in a Borel way, this yields a Borel labeling.\n\nSuppose that $S$ is two-ended.\nThen, there exists a doubly infinite path $P$ in $S$ with the property that the connected components of $S\\setminus P$ are finite.\nMoreover, it is possible to detect $P$ in a Borel way.\nThat is, declaring the vertices on $P$ to be line vertices and using the special symbol in $\\overline{\\Pi^2_{\\text{pm}}}$ for the half-edges on $P$ yields a Borel measurable labeling.\nLet $C\\not=\\emptyset$ be one of the finite components in $S\\setminus P$ and $v\\in C$ be the vertex of distance $1$ from $P$.\nOrient the edges in $C$ towards $v$, this can be done in a Borel way since $v$ is uniquely determined for $C$.\nThen the first part of the proof of \\cref{thm:twomatchFFIID} shows that one can inductively find a solution to $\\Pi^2_{\\text{pm}}$ on $C$ in such a way that all vertices, possibly up to $v$, are matched.\nIf $v$ is matched, then we are done.\nIf $v$ is not matched, then we add the edge that connects $v$ with $P$ to the matching. Note that it is possible that multiple vertices are matched with the same path vertex, but this is not a problem according to the definition of $\\overline{\\Pi^2_{\\text{pm}}}$. \nIt follows that this defines a Borel function on $H(\\mathcal{G})$ that solves $\\overline{\\Pi^2_{\\text{pm}}}$.\n\\end{proof}\n\n\n\n\n\n\\subsubsection{Vizing's Theorem for $\\Delta=3$}\nFinding a measurable or local version of Vizing's Theorem, i.e., proper edge $(\\Delta+1)$-coloring $\\Pi_{\\chi',\\Delta+1}$, was studied recently in \\cite{grebik2020measurable,weilacher2021borel,BernshteynVizing}.\nIt is however not known, even on trees, whether $\\Pi_{\\chi',\\Delta+1}$ is in $\\mathsf{RLOCAL}(O(\\log\\log n))$.\nHere we use the one-ended forest construction to show that $\\Pi_{\\chi',\\Delta+1}$ is in the class $\\mathsf{OLOCAL}(\\operatorname{poly}\\log 1\/\\varepsilon)$ for $\\Delta=3$.\n\n\n\n\\iffalse\n\\paragraph{Composition of Uniform Local Algorithms}\nWhen designing algorithms, one often composes several simple algorithms to obtain a more sophisticated algorithm. For example, an algorithm $\\mathcal{A}$ might first invoke algorithm $\\mathcal{A}_1$ and after $\\mathcal{A}_1$ finished, $\\mathcal{A}$ invokes algorithm $\\mathcal{A}_2$ with the input of $\\mathcal{A}_2$ being equal to the output of $\\mathcal{A}_1$. Then, the final output of $\\mathcal{A}$ is simply the output of $\\mathcal{A}_2$. We call such a composition a sequential composition. In the context of the classic deterministic local model, the round complexity of $\\mathcal{A}$ is simply the sum of the round complexities of the two algorithms $\\mathcal{A}_1$ and $\\mathcal{A}_2$ that are used as a subroutine. Here, we prove two such composition results in the context of uniform local algorithms. These general results will help us in \\cref{app:one_ended} to analyze the uniform complexity of an uniform algorithm that computes a so-called one-ended forest in a principled manner.\n\n\n\n\\begin{lemma}[Parallel Composition]\n\\label{lem:parallel_composition}\nLet $A_1$ and $A_2$ be two distributed uniform algorithms with an uniform local complexity of $t_{A_1}(\\varepsilon)$ and $t_{A_2}(\\varepsilon)$, respectively. \nLet $A$ be the parallel composition of $A_1$ and $A_2$. That is, the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is the smallest radius such that $A$ knows the local answer of both $A_1$ and $A_2$ at $u$. \nThen, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$.\n\\end{lemma}\n\\begin{proof}\nThe probability that the coding radius of algorithm $\\mathcal{A}_1$ at vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2$ and the same holds for the coding radius of algorithm $\\mathcal{A}_2$ at vertex $u$. Hence, by a simple union bound, the probability that the coding radius of algorithm $\\mathcal{A}$ at a vertex $u$ is larger than $\\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$ is at most $\\varepsilon\/2 + \\varepsilon\/2 = \\varepsilon$. \\\\ Hence, $t_A(\\varepsilon) \\leq \\max(t_{A_1}(\\varepsilon\/2), t_{A_2}(\\varepsilon\/2))$, as desired.\n\\end{proof}\n\n\\fi\n\n\n\n\\begin{proposition}\nLet $\\Delta=3$. We have $\\Pi_{\\chi',\\Delta+1} \\in\\mathsf{OLOCAL}(\\operatorname{poly} \\log 1\/\\varepsilon)$.\n\\end{proposition}\n\\begin{proof}[Proof sketch.]\nBy \\cref{thm:one_ended_forest}, we can compute a one-ended forest decomposition $\\mathcal{T}$ with uniform local complexity $O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \nNote that every vertex has at least one edge in $\\mathcal{T}$, hence the edges in $G \\setminus \\mathcal{T}$ form paths. \nThese paths can be $3$-edge colored by using the uniform version of Linial's coloring algorithm. This algorithm has an uniform local complexity of $O(\\log^* 1\/\\varepsilon)$. By \\cref{lem:sequential_composition}, the overall complexity is $O(\\operatorname{poly} \\log (\\Delta^{\\log^* 1\/\\varepsilon} \/ \\varepsilon) + \\log^* 1\/\\varepsilon)) = O(\\operatorname{poly} \\log 1\/\\varepsilon)$. \n\nFinally, we color the edges of $T$. We start from the leaves and color the edges inductively. In particular, whenever we consider a vertex $v$ we color the at most two edges in $T^{\\leftarrow}(v)$ below $v$. \nNote that there always is at least one uncolored edge going from $v$ in the direction of $T$. Hence, we can color the at most two edges below $v$ in $T^{\\leftarrow}(v)$ greedily -- each one neighbors with at most $3$ colored edges at any time.  \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Proof of \\cref{thm:one_ended_forest}} \n\\label{sec:formalproof}\n\n\nIn this section we formally prove \\cref{thm:one_ended_forest}. \n\n\n\\oneendedtreetail*\n\nWe follow the construction of the one-ended spanning forest in \\cite{BrooksMeas}.\nThe construction proceeds in rounds. Before each round, some subset of the vertices have already decided which incident edge to grab, and we refer to these vertices as settled vertices. All the remaining vertices are called unsettled. The goal in each round is to make a large fraction of the unsettled vertices settled. To achieve this goal, our construction relies on certain properties that the graph induced by all the unsettled vertices satisfies. \n\nOne important such property is that the graph induced by all the unsettled vertices is expanding. That is, the number of vertices contained in the neighborhood around a given vertex grows exponentially with the radius. The intuitive reason why this is a desirable property is the following. Our algorithm will select a subset of the unsettled vertices and clusters each unsettled vertex to the closest selected vertex. As the graph is expanding and any two vertices in the selected subset are sufficiently far away, there will be a lot of edges leaving a given cluster. For most of these inter-cluster edges, both of the endpoints will become settled. This in turn allows one to give a lower bound on the fraction of vertices that become settled in each cluster.\n\nTo ensure that the graph induced by all the unsettled vertices is expanding, a first condition we impose on the graph induced by all the unsettled vertices is that it has a minimum degree of at least $2$. While this condition is a first step in the right direction, it does not completely suffice to ensure the desired expansion, as an infinite path has a minimum degree of $2$ but does not expand sufficiently. Hence, our algorithm will keep track of a special subset of the unsettled vertices, the so-called hub vertices. Each hub vertex has a degree of $\\Delta$ in the graph induced by all the unsettled vertices. That is, all of the neighbors of a hub vertex are unsettled as well. Moreover, each unsettled vertex has a bounded distance to the closest hub vertex, where the specific upper bound on the distance to the closest hub vertex increases with each round. As we assume $\\Delta > 2$, the conditions stated above suffice to show that the graph induced by all the unsettled vertices expands.\n\nNext, we explain in more detail how a vertex decides which edge to grab. Concretely, if a vertex becomes settled, it grabs an incident edge such that the other endpoint of that edge is strictly closer to the closest unsettled vertex as the vertex itself. For example, if a vertex becomes settled and there is exactly one neighbor that is still unsettled, then the vertex will grab the edge that the vertex shares with its unsettled neighbor. Grabbing edges in that way, one can show two things. First, for a settled vertex $v$, the set $T^{\\leftarrow}(v)$ of vertices behind $v$ does not change once $v$ becomes settled. The intuitive reason for this is that the directed edges point towards the unsettled vertices and therefore the unsettled vertices lie before $v$ and not after $v$. Moreover, one can also show that $T^{\\leftarrow}(v)$ only contains finitely many vertices. The reason for this is that at the moment a vertex $v$ becomes settled, there exists an unsettled vertex that is sufficiently close to $v$, where the exact upper bound on that distance again depends on the specific round in which $v$ becomes settled. \n\nWhat remains to be discussed is at which moment a vertex decides to become settled. As written above, in each round the algorithm considers a subset of the unsettled vertices and each unsettled vertex is clustered to the closest vertex in that subset. This subset only contains hub vertices and it corresponds to an MIS on the graph with the vertex set being equal to the set of hub vertices and where two hub vertices are connected by an edge if they are sufficiently close in the graph induced by all the unsettled vertices. Now, each cluster center decides to connect to exactly $\\Delta$ different neighboring clusters, one in each of its $\\Delta$ subtrees. Remember that as each cluster center is a hub vertex, all of its neighbors are unsettled as well. Now, each unsettled vertex becomes settled except for those that lie on the unique path between two cluster centers such that one of the cluster centers decided to connect to the other one.\n\n\n\n\n\n\n\\paragraph{Construction}\nThe algorithm proceeds in rounds. After each round, a vertex is either settled or unsettled and a settled vertex remains settled in subsequent rounds. Moreover, some unsettled vertices are so-called hub vertices. We denote the set of unsettled, settled and hub vertices after the $i$-th round with $U_i$, $S_i$ and $H_i$, respectively. We set $U_0 = H_0 = V$ prior to the first round and we always set $S_i = V \\setminus U_i$.  \\cref{fig:constr1} illustrates the situation after round $i$.  Moreover, we denote with $O_i$ a partial orientation of the edges of the infinite $\\Delta$-regular input tree $T$. In the beginning, $O_0$ corresponds to the partial orientation with no oriented edges. Each vertex is incident to at most $1$ outwards oriented edge in $O_i$. If a vertex is incident to an outwards oriented edge in $O_i$, then the other endpoint of the edge will be its parent in the one-ended forest decomposition. For each vertex $v$ in $S_i$, we denote with $T_{v,i}$ the smallest set that satisfies the following conditions. First, $v \\in T_{v,i}$. Moreover, if $u \\in T_{v,i}$ and $\\{w,u\\}$ is an edge that according to $O_i$ is oriented from $w$ to $u$, then $w \\in T_{v,i}$. We later show that $T_{v,i}$ contains exactly those vertices that are contained in the subtree $\\mathcal{T}^\\leftarrow(v)$.\n\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics{fig\/one_ended_construction.pdf}\n    \\caption{The picture illustrates the situation after some round $i$ of the construction. The vertices on the paths are all contained in $U_i$, while the vertices corresponding to the big black dots are additionally contained in $H_i$. Note that the length of each path is at least $d_i$. However, the distance between $u$ and $v$ in the original graph can be much smaller.}\n    \\label{fig:constr1}\n\\end{figure}\n\n\nAfter the $i$-th round, the construction satisfies the following invariants.\n\n\n\n\\begin{enumerate}\n    \n    \\item For each $v \\in S_{i-1}$, we have $T_{v,i} = T_{v,i-1}$.\n    \n    \\item Let $v$ be an arbitrary vertex in $S_i$. Then, $T_{v,i}$ contains finitely many vertices and furthermore $T_{v,i} \\subseteq S_i$.\n    \n    \\item The minimum degree of the graph $T[U_i]$ is at least $2$ and each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n    \n    \\item Each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$.\n\\end{enumerate}\n\nWe now describe how to compute $U_i, S_i$, $H_i$ and $O_i$. The construction during the $i$-th round is illustrated in $\\cref{fig:constr2}$. Note that we can assume that the invariants stated above are satisfied after the $(i-1)$-th round. In the $i$-th round we have a parameter $d_i$ that we later set to $2^{2^i}$.\n\n\\begin{enumerate}\n    \\item $H_i$ is a subset of $H_{i-1}$ that satisfies the following property. No two vertices in $H_i$ have a distance of at most $d_i$ in $T[U_{i-1}]$. Moreover, each vertex in $H_{i-1} \\setminus H_i$ has a distance of at most $d_i$ to the closest vertex in $H_i$ in the graph $T[U_{i-1}]$. Note that we can compute $H_i$ by computing an MIS in the graph with vertex set $H_{i-1}$ and where two vertices are connected iff they have a distance of at most $d_i$ in the graph $T[U_{i-1}]$. Hence, we can use Ghaffari's MIS algorithm \\cite{ghaffari2016MIS} to compute the set $H_i$.\n    \\item Next, we describe how to compute $U_i$.  \n    We assign each vertex $u \\in U_{i-1}$ to the closest vertex in the graph $T[U_{i-1}]$ that is contained in $H_i$, with ties being broken arbitrarily. We note that there exists a node in $H_i$ with a distance of at most $(\\sum_{j=0}^{i-1} d_j) + d_i$ to $u$. To see why, note that Invariant $(4)$ from round $i - 1$ implies that there exists a vertex $w$ in $H_{i-1}$ with a distance of at most $\\sum_{j=0}^{i-1} d_j$ to $u$. We are done if $w$ is also contained in $H_i$. If not, then it follows from the way we compute $H_i$ that there exists a vertex in $H_i$ with a distance of at most $d_i$ to $w$, but then the triangle inequality implies that the distance from $u$ to that vertex is at most $(\\sum_{j=0}^{i-1} d_j) + d_i$, as desired. For each vertex $v \\in H_i$, we denote with $C(v)$ the set of all vertices in $U_{i-1}$ that got assigned to $v$. Now, let $E_v$ denote the set of edges that have exactly one endpoint in $C(v)$. We can partition $E_v$ into $E_{v,1} \\sqcup E_{v,2} \\sqcup \\ldots \\sqcup E_{v,\\Delta}$, where for $\\ell \\in [\\Delta]$, $E_{v,\\ell}$ contains all the edges in $E_v$ that are contained in the $\\ell$-th subtree of $v$. \n    Invariants $(3)$ and $(4)$ after the $(i-1)$-th round imply that $E_{v,\\ell} \\neq \\emptyset$. Moreover, $E_{v,\\ell}$ contains only finitely many edges, as we have shown above that the cluster radius is upper bounded by $\\sum_{j=0}^{i} d_j$.\n    \n    Now, for $\\ell \\in [\\Delta]$, we choose an edge $e_\\ell$ uniformly at random from the set $E_{v,\\ell}$. Let $u_\\ell \\neq v$ be the unique vertex in $H_i$ such that one endpoint of $e_\\ell$ is contained in $C(u_\\ell)$. We denote with $P_{v,\\ell}$ the set of vertices that are contained in the unique path between $v$ and $u_\\ell$. Finally, we set $U_i = \\bigcup_{v \\in H_i, \\ell \\in [\\Delta]} P_{v,\\ell}$.\n    \\item It remains to describe how to compute the partial orientation $O_i$. All edges that are oriented in $O_{i-1}$ will be oriented in the same direction in $O_i$. Additionally, we orient for each vertex $u$ that got settled in the $i$-th round, i.e., $u \\in S_i \\cap U_{i-1}$, exactly one incident edge away from $u$. Let $w$ be the closest vertex to $u$ in $T[U_{i-1}]$ that is contained in $U_i$, with ties being broken arbitrarily. We note that it follows from the discussions above that the distance between $u$ and $w$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. We now orient the edge incident to $u$ that is on the unique path between $u$ and $w$ outwards. We note that this orientation is well-defined in the sense that we only orient edges that were not oriented before and that we don't have an edge such that both endpoints of that edge want to orient the edge outwards.\n\\end{enumerate}\n\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{fig\/one_ended_construction2.pdf}\n    \\caption{The picture illustrates the situation during some round $i$ of the construction. The vertices on all paths are contained in $U_{i-1}$ and the vertices corresponding to the big black dots are contained in $H_{i-1}$. The vertices $v$, $u_1$, $u_2$ and $u_3$ are the only vertices in the picture that are contained in $H_i$. All the vertices on the red paths remain unsettled during round $i$ while all the vertices on the black paths become settled. Each vertex that becomes settled during the $i$-th round orients one incident edge outwards, illustrated by the green edges. Each green edge is oriented towards the closest vertex that is still unsettled after the $i$-th round. We note that some of the edges in the picture are oriented away from $v$.}\n    \\label{fig:constr2}\n\\end{figure}\n\n\nWe now prove by induction that our procedure satisfies all the invariants.\nPrior to the first round, all invariants are trivially satisfied. Hence, it remains to show that the invariants hold after the $i$-th round, given that the invariants hold after the $(i-1)$-th round. \n\n\\begin{enumerate}\n\n    \\item Let $v \\in S_{i-1}$ be arbitrary. As $O_i$ is an extension of $O_{i-1}$, it follows from the definition of $T_{v,i-1}$ and $T_{v,i}$ that $T_{v,i-1} \\subseteq T_{v,i}$. Now assume that $T_{v,i} \\not \\subseteq T_{v,i-1}$. This would imply the existence of an edge $\\{u,w\\}$ such that $u \\in T_{v,i-1}$ and the edge was oriented during the $i$-th round from $w$ to $u$. As $u \\in T_{v,i-1}$, Invariant $(2)$ from the $(i-1)$-th round implies $u \\in S_{i-1}$. This is a contradiction as during the $i$-th round only edges with both endpoints in $U_{i-1}$ can be oriented. Therefore it holds that $T_{v,i} = T_{v,i-1}$, as desired.\n\n    \\item Let $v \\in S_i$ be arbitrary. If it also holds that $v \\in S_{i-1}$, then the invariant from round $i-1$ implies that $T_{v,i-1} = T_{v,i}$ contains finitely many vertices and $T_{v,i-1} \\subseteq S_{i-1} \\subseteq S_i$. Thus, it suffices to consider the case that $v \\in S_i \\cap U_{i-1}$. Note that $T_{v,i} \\cap U_i = \\emptyset$. Otherwise there would exist an edge that is oriented away from a vertex in $U_i$ according to $O_i$, but this cannot happen according to the algorithm description. Hence, $T_{v,i} \\subseteq S_i$. Now, for the sake of contradiction, assume that $T_{v,i}$ contains infinitely many vertices. From the definition of $T_{v,i}$, this implies that there exists a sequence of vertices $(v_k)_{k \\geq 1}$ with $v = v_1$ such that for each $k \\geq 1$, $\\{v_{k+1},v_k\\}$ is an edge in $T$ that is oriented from $v_{k+1}$ to $v_k$ according to $O_i$. For each $k \\geq 1$ we have $v_k \\in S_i$. We furthermore know that $v_k \\in U_{i-1}$, as otherwise $T_{v_k,i} = T_{v_k,i-1}$ would contain infinitely many vertices, a contradiction. From the way we orient the edges, a simple induction proof implies that the closest vertex of $v_k$ in the graph $T[U_{i-1}]$ that is contained in $U_i$ has a distance of at least $k$. However, we previously discussed that the distance between $v_k$ and the closest vertex in $U_i$ in the graph $T[U_{i-1}]$ is at most $\\sum_{j=0}^{i} d_j$. This is a contradiction. Hence, $T_{v,i}$ contains finitely many vertices.\n    \n    \\item It follows directly from the description of how to compute $U_i$ and $H_i$ that the minimum degree of the graph $T[U_i]$ is at least $2$ and that each vertex in $H_i$ has a degree of $\\Delta$ in $T[U_i]$.\n    \n    \\item Let $u \\in U_i$ be arbitrary. We need to show that $u$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$. Let $v$ be the vertex with $u \\in C(v)$. We know that there is no vertex in $H_i$ that is closer to $u$ in $T[U_{i-1}]$ than $v$. Hence, the distance between $u$ and $v$ is at most $\\sum_{j=0}^i d_j$ in the graph $T[U_{i-1}]$. Moreover, from the description of how we compute $U_i$, it follows that all the vertices on the unique path between $u$ and $v$ are contained in $U_i$. Hence, each vertex in $U_i$ has a distance of at most $\\sum_{j=0}^i d_j$ to the closest vertex in $H_i$ in the graph $T[U_i]$, as desired.\n    \\end{enumerate}\n\nWe derive an upper bound on the coding radius of the algorithm in three steps. First, for each $i \\in \\mathbb{N}$ and $\\varepsilon > 0$, we derive an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. Given this upper bound on the coding radius, we can use the sequential composition lemma (\\cref{lem:sequential_composition}) and a simple induction proof to give an upper bound on the coding radius for computing with probability at least $1 - \\varepsilon$ for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, this time without providing any additional input. \n\nSecond, we analyze after how many rounds a given vertex is settled with probability at least $1- \\varepsilon$ for a given $\\varepsilon > 0$.  \n\nFinally, we combine these two upper bounds to prove  \\cref{thm:one_ended_forest}.\n \n \n\\begin{lemma}\n\\label{lem:tail1}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $g_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$, given that each vertex receives as additional input in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$. It holds that $g_i(\\varepsilon) = O(d^2_i \\log (\\Delta\/\\varepsilon))$.\n\\end{lemma}\n\n\\begin{proof}\n When running Ghaffari's uniform MIS algorithm on some graph $G'$ with maximum degree $\\Delta'$, each vertex knows with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not after $O(\\log (\\Delta') + \\log (1\/\\varepsilon))$ communication rounds in $G'$ (\\cite{ghaffari2016MIS}, Theorem 1.1). For the MIS computed during the $i$-th round, $\\Delta' =\\Delta^{O(d_i)}$ and each communication round in $G'$ can be simulated with $O(d_i)$ communication rounds in the tree $T$. Hence, to know for a given vertex with probability at least $1-\\varepsilon$ whether it is contained in the MIS or not, it suffices consider the $O(d_i^2\\log(\\Delta) + d_i\\log(1\/\\varepsilon))$-hop neighborhood around that vertex. Now, let $u$ be an arbitrary vertex. In order to compute in which of the sets $S_i, U_i$ and $H_i$ $u$ is contained in and for each incident edge of $u$ whether and how it is oriented in $O_i$, we not only need to know whether $u$ is in the MIS or not. However, it suffices if we know for all the vertices in the $O(d_i)$-hop neighborhood of $u$ whether they are contained in the MIS or not (on top of knowing for each vertex in the neighborhood in which of the sets $S_{i-1}, U_{i-1}$ and $H_{i-1}$ it is contained in and for each incident edge whether and how it is oriented in $O_{i-1}$). Hence, by a simple union bound over the at most $\\Delta^{O(d_i)}$ vertices in the $O(d_i)$-hop neighborhood around $u$, we obtain $g_i(\\varepsilon) = O(d_i) + O(d_i^2\\log(\\Delta) + d_i\\log(\\Delta^{O(d_i)}\/\\varepsilon)) = O(d_i^2 \\log(\\Delta\/\\varepsilon))$.\n\\end{proof}\n\n\\begin{lemma}\n\\label{lem:tail2}\nFor each $i \\in \\mathbb{N}$ and $\\varepsilon \\in (0,0.01]$, let $h_i(\\varepsilon)$ denote the smallest coding radius such that with probability at least $1 - \\varepsilon$ one knows for a given vertex in which of the sets $S_i, U_i$ and $H_i$ it is contained in and for each incident edge whether and how it is oriented in $O_i$. Then, there exists a constant $c$ independent of $\\Delta$ such that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$.\n\\end{lemma}\n\n\\begin{proof}\nBy prove the statement by induction on $i$. For a large enough constant $c$, it holds that $h_1(\\varepsilon) \\leq 2^{2^{3}} \\cdot(c\\log(\\Delta)) \\log(\\Delta\/\\varepsilon)$. Now, consider some arbitrary $i$ and assume that $h_i(\\varepsilon) \\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta\/\\varepsilon)$ for some large enough constant $c$. We show that this implies $h_{i+1}(\\varepsilon) \\leq  2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon)$.\nBy the sequential composition lemma (\\cref{lem:sequential_composition}) and assuming that $c$ is large enough, we have\n\n\\begin{align*}\n  h_{i+1}(\\varepsilon) &\\leq h_{i}((\\varepsilon\/2)\/\\Delta^{g_{i+1}(\\varepsilon\/2) + 1}) + g_{i+1}(\\varepsilon\/2)  \\\\\n                &\\leq 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) + g_{i+1}(\\varepsilon\/2)  \\\\\n                &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta \\cdot \\Delta^{g_{i+1}(\\varepsilon\/2) + 1}\/(\\varepsilon\/2)) \\\\\n                &\\leq 2 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}\/(\\varepsilon\/2)) \\\\\n                &\\leq 4 \\cdot 2^{2^{i+2}} \\cdot(c\\log(\\Delta))^i \\log(\\Delta^{(c\/10) \\cdot 2^{2^{i+2}}\\log(\\Delta\/\\varepsilon)}) \\\\\n                &\\leq (4\/10) \\cdot 2^{2^{i+2}} \\cdot 2^{2^{i+2}}  (c \\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon) \\\\\n                &\\leq 2^{2^{(i+1)+2}} \\cdot(c\\log(\\Delta))^{i+1} \\log(\\Delta\/\\varepsilon),\n\\end{align*}\n\nas desired.\n\n\\end{proof}\n\n\n \n \n\n\\begin{lemma}\\label{lm:tail0}\nLet $u$ be an arbitrary vertex. For each $\\varepsilon \\in (0,0.01]$, let $f(\\varepsilon)$ denote the smallest $i \\in \\mathbb{N}$ such that $u$ is settled after the $i$-th round with probability at least $1 - \\varepsilon$. There exists a fixed $c \\in \\mathbb{R}$ independent of $\\Delta$ such that $f(\\varepsilon) \\leq \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$.\n\\end{lemma}\n\n\\begin{proof}\n\nLet $i \\in \\mathbb{N}$ be arbitrary. We show that a given vertex is settled after the $i$-th round with probability at least $1 - O(1\/\\Delta^{\\Omega(d_i\/d_{i-1})})$. For the sake of analysis, we run the algorithm on a finite $\\Delta$-regular high-girth graph instead of an infinite $\\Delta$-regular tree. For now, we additionally assume that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. That is, we assume that for each vertex the coding radius to compute all its local information after the $i$-th round is much smaller than the girth of the graph.  \n\nWith this assumption, we give a deterministic upper bound on the fraction of vertices that are not settled after the $i$-th round. On the one hand, the number of vertices that are not settled after the $i$-th round can be upper bounded by $|H_i| \\cdot \\Delta \\cdot O(d_i)$. On the other hand, we will show that the fraction of vertices that are contained in $H_i$ is upper bounded by $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. We do this by showing that $C(v)$ contains $\\Delta^{\\Omega(d_i\/d_{i-1})}$ many vertices for a given $v \\in H_i$. Combining these two bounds directly implies that the fraction of unsettled vertices is smaller than \n\n\\[\\frac{\\Delta \\cdot O(d_i)}{\\Delta^{\\Omega(d_i\/d_{i-1})}} = 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}.\\]\n\n\nLet $v \\in H_i$ be arbitrary. We show that $|C(v)| = \\Delta^{\\Omega(d_i\/d_{i-1})}$. Using Invariants (3) and (4) together with a simple induction argument, one can show that there are at least $(\\Delta-1)^{\\lfloor  D \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor}$ vertices contained in $H_{i-1}$ and whose distance to $v$ is at most $D$ in the graph induced by the vertices in $U_{i-1}$. Furthermore, from the way we defined the clustering $C(v)$ and the fact that two vertices in $H_i$ have a distance of at least $d_i$ in the graph $T[U_{i-1}]$, it follows that all vertices in $U_{i-1}$ having a distance of at most $d_i\/2 -1$ to $v$  are contained in the cluster $C(v)$. Hence, the total number of vertices in $C(v)$ is at least $(\\Delta-1)^{\\lfloor  (d_i\/2 - 1) \/ ((2\\sum_{j=0}^{i-1}d_{j}) + 1)\\rfloor} = \\Delta^{\\Omega(d_i\/d_{i-1})}$, as promised.\n\nNow we remove the assumption that no vertex realizes that we don't run the algorithm on an infinite $\\Delta$-regular tree. If a vertex does realize that we don't run the algorithm on an infinite $\\Delta$-regular tree, then we consider the vertex as being unsettled after the $i$-th round. By considering graphs with increasing girth, we can make the expected fraction of vertices that realize that we are not on an infinite $\\Delta$-regular tree arbitrarily small.  Combining this observation with the previous discussion, this implies that the expected fraction of vertices that are not settled after the $i$-th round is at most $1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$. By symmetry, each vertex has the same probability of being settled after the $i$-th round. Hence, the probability that a given vertex is settled after the $i$-th round when run on a graph with sufficiently large girth is $1 - 1\/\\Delta^{\\Omega(d_i\/d_{i-1})}$, and the same holds for each vertex when we run the algorithm on an infinite $\\Delta$-regular tree. Thus, there exists a constant $c$ such that the probability that a given vertex is unsettled after the $i$-th round is at most $1\/\\Delta^{c \\cdot d_i\/d_{i-1}} = 1\/\\Delta^{c \\cdot 2^{2^{i-1}}}$. Setting $i = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta(1\/\\varepsilon)\\rceil$ finishes the proof.\n\\end{proof}\n\n\n\n\n\nWe are now finally ready to finish the proof of \\cref{thm:one_ended_forest}. Let $u$ be an arbitrary vertex and $\\varepsilon \\in (0,0.01]$. We need to compute an upper bound on the coding radius that is necessary for $u$ to know all the vertices in $\\mathcal{T}^{\\leftarrow}(u)$ with probability at least $1-\\varepsilon$. To compute such an upper bound for the  required coding radius it suffices to find an $i^*$ and an $R^*$ such that the following holds. First, $u$ is settled after $i^*$ rounds with probability at least $1 - \\varepsilon\/2$. Second, $u$ knows for each edge in its $O(d_{i^*})$-hop neighborhood whether it is oriented in the partial orientation $O_{i^*}$, and if yes, in which direction, by only considering its $R^*$-hop neighborhood with probability at least $1 - \\varepsilon\/2$.  By a union bound, both of these events occur with probability at least $1-\\varepsilon$. Moreover, if both events occur $u$ knows all the vertices in the set $\\mathcal{T}^{\\leftarrow}(u)$. The reason is as follows. If $u$ is settled after round $i^*$, then it follows from the previous analysis that only vertices in the $O(d_{i^*})$-hop neighborhood of $u$ can be contained in $\\mathcal{T}^{\\leftarrow}(u)$. Moreover, for each vertex in its $O(d_{i^*})$-hop neighborhood, $u$ can determine if the vertex is contained in $\\mathcal{T}^{\\leftarrow}(u)$ if it knows all the edge orientations on the unique path between itself and that vertex after the $i^*$-th round. Hence, it remains to find concrete values for $i^*$ and $R^*$. According to \\cref{lm:tail0}, we can choose $i^* = \\lceil 1 + \\log \\log \\frac{1}{c} \\log_\\Delta (2\/\\varepsilon)\\rceil$ for some large enough constant $c$. Moreover, it follows from a union bound that all vertices in the $O(d_{i^*})$-hop neighborhood around $u$ know with probability at least $1-\\varepsilon\/2$ the orientation of all its incident edges according to $O_{i^*}$ by only considering their $h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})})$-hop neighborhood. Hence, we can set \n\n\\begin{align*}\nR^* &= O(d_{i^*}) + h_{i^*}((\\varepsilon\/2)\/\\Delta^{O(d_{i^*})}) \\\\\n    &\\leq O(d_{i^*}) + 2^{2^{i^* + 2}} \\cdot (c \\log(\\Delta))^{i^*}\\log(\\Delta^{O(d_{i^*})}\/\\varepsilon) \\\\\n    &= O(2^{2^{i^*}}\\cdot 2^{2^{i^* + 2}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= O(2^{2^{i^* + 3}} (c \\log(\\Delta))^{i^*}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= O(2^{2^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 5}} (c \\log(\\Delta))^{\\log \\log \\frac{1}{c} \\log (2\/\\varepsilon) + 2}\\log(\\Delta\/\\varepsilon)) \\\\\n    &= \\log(1\/\\varepsilon)^{32} \\cdot \\log(\\Delta\/\\varepsilon) \\cdot \\log\\log(1\/\\varepsilon)^{O(\\log\\log(\\Delta))}.\n\\end{align*}\n\nAs $\\Delta = O(1)$, it therefore holds that\n\n\\[P(R(v)) = \\operatorname{poly}(\\log(1\/\\varepsilon)) \\geq 1 - \\varepsilon, \\]\n\nas desired.\n\n\n\n\n\n\n\\section{$\\mathsf{BAIRE} = \\mathsf{LOCAL}(O(\\log n))$}\n\\label{sec:baire}\n\nIn this section, we show that on $\\Delta$-regular trees the classes $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$ are the same.\nAt first glance, this result looks rather counter-intuitive.\nThis is because in finite $\\Delta$-regular trees every vertex can see a leaf of distance $O(\\log(n))$, while there are no leaves at all in an infinite $\\Delta$-regular tree.\nHowever, there is an intuitive reasons why these classes are the same: in both setups there is a technique to decompose an input graph into a hierarchy of subsets. Furthermore, the existence of a solution that is defined inductively with respect to these decompositions can be characterized by the same combinatorial condition of Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nWe start with a high-level overview of the decomposition techniques used in both contexts.\n\n\n\\paragraph{{\\sf Rake}\\ and {\\sf Compress}}\nThe hierarchical decomposition in the context of distributed computing is based on a variant of a decomposition algorithm of\nMiller and Reif~\\cite{MillerR89}.\nTheir original decomposition algorithm works as follows.\nStart with a tree $T$, and repeatedly apply the following two operations alternately: {\\sf Rake}\\ (remove all degree-1 vertices) and {\\sf Compress}\\ (remove all degree-2 vertices).\nThen $O(\\log n)$ iterations suffice to remove all vertices in $T$~\\cite{MillerR89}. To view it another way, this produces a decomposition of the vertex set $V$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$, where $\\VR{i}$ is the set of vertices removed during the $i$-th {\\sf Rake}\\ operation and $\\VC{i}$ is the set of vertices removed during the $i$-th {\\sf Compress}\\ operation. We will use a variant~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} of this decomposition in the proof of \\cref{thm:ellfull_to_logn}.\n\n\n\n\nVariants of this decomposition turned out to be useful in designing  $\\mathsf{LOCAL}$ algorithms~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,ChangHLPU20,chang2020n1k_speedups}. \nIn our context, we assume that the given LCL satisfies a certain combinatorial condition and then find a solution inductively, in the reversed order of the construction of the decomposition.\nNamely, in the {\\sf Rake}\\ step we want to be able to existentially extend the inductive partial solution to all relative degree $1$-vertices (each $v \\in \\VR{i}$ has degree at most 1 in the subgraph induced by $\\VR{i}  \\cup \\cdots \\cup \\VR{L}$) and in the {\\sf Compress}\\ step we want to extend the inductive partial solution to paths with endpoints labeled from the induction (the vertices in $\\VC{i}$ form degree-2 paths in the subgraph induced by $\\VC{i} \\cup \\cdots \\cup \\VR{L}$).\n\n\n\\paragraph{$\\TOAST$}\nFinding a hierarchical decomposition in the context of descriptive combinatorics is tightly connected with the notion of \\emph{Borel hyperfiniteness}.\nUnderstanding what Borel graphs are Borel hyperfinite is a major theme in descriptive set theory \\cite{doughertyjacksonkechris,gaojackson,conley2020borel}.\nIt is known that grids, and generally polynomial growth graphs are hyperfinite, while, e.g., acyclic graphs are not in general hyperfinite \\cite{jackson2002countable}.\nA strengthening of hyperfiniteness that is of interest to us is called a \\emph{toast} \\cite{gao2015forcing,conleymillerbound}.\nA $q$-toast, where $q\\in \\mathbb{N}$, of a graph $G$ is a collection $\\mathcal{D}$ of fintie subsets of $G$ with the property that (i) every pair of vertices is covered by an element of $\\mathcal{D}$ and (ii) the boundaries of every $D\\not=E\\in \\mathcal{D}$ are at least $q$ apart.\nThe idea to use a toast structure to solve LCLs appears in \\cite{conleymillerbound} and has many applications since then \\cite{gao2015forcing,marksunger}.\nThis approach has been formalized in \\cite{grebik_rozhon2021toasts_and_tails}, where the authors introduce $\\TOAST$ algorithms.\nRoughly speaking, an LCL $\\Pi$ admits a $\\TOAST$ algorithm if there is $q\\in \\mathbb{N}$ and a partial extending function (the function is given a finite subset of a tree that is partially colored and outputs an extension of this coloring on the whole finite subset) that has the property that whenever it is applied inductively to a $q$-toast, then it produces a $\\Pi$-coloring.\nAn advantage of this approach is that once we know that a given Borel graph admits, e.g., a Borel toast structure and a given LCL $\\Pi$ admits a $\\TOAST$ algorithm, then we may conclude that $\\Pi$ is in the class $\\mathsf{BOREL}$.\nSimilarly for $\\mathsf{MEASURE}$, $\\mathsf{BAIRE}$ or $\\mathsf{OLOCAL}$, we refer the reader to \\cite{grebik_rozhon2021toasts_and_tails} for more details and results concerning grids.\n\nIn the case of trees there is no way of constructing a Borel toast in general, however, it is a result of Hjorth and Kechris \\cite{hjorth1996borel} that every Borel graph is hyperfinite on a comeager set for every compatible Polish topology.\nA direct consequence of \\cite[Lemma~3.1]{marks2016baire} together with a standard construction of toast via Voronoi cells gives the following strengthening to toast.\nWe include a sketch of the proof for completeness.\n\n\\begin{restatable}{proposition}{BaireToast}\n\\label{pr:BaireToast}\nLet $\\mathcal{G}$ be a Borel graph on a Polish space $(X,\\tau)$ with degree bounded by $\\Delta\\in \\mathbb{N}$.\nThen for every $q>0$ there is a Borel $\\mathcal{G}$-invariant $\\tau$-comeager set $C$ on which $\\mathcal{G}$ admits a Borel $q$-toast.\n\\end{restatable}\n\\begin{proof}[Proof sketch]\nLet $\\{A_n\\}_{n\\in \\mathbb{N}}$ be a sequence of MIS with parameter $f(n)$ as in \\cite[Lemma~3.1]{marks2016baire} for a sufficiently fast growing function $f(n)$, e.g., $f(n)=(2q)^{n^2}$.\nThen $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ is a Borel $\\tau$-comeager set that is $\\mathcal{G}$-invariant.\nWe produce a toast structure in a standard way, e.g., see~\\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\nLet $B_n(x)$ denote the ball of radius $f(n)\/3$ and $R_n(x)$ the ball of radius $f(n)\/4$ around $x\\in A_n$.\nIteratively, define cells $\\mathcal{D}_n=\\{C_{n}(x)\\}_{x\\in A_n}$ as follows.\nSet $C_1(x)=R_1(x)$.\nSuppose that $\\mathcal{D}_n$ has been defined and set\n\\begin{itemize}\n\t\\item $H^{n+1}(x,n+1):=R_{n+1}(x)$ for every $x\\in A_{n+1}$,\n\t\\item if $1\\le i\\le n$ and $\\left\\{H^{n+1}(x,i+1)\\right\\}_{x\\in A_{k+1}}$ has been defined, then we put\n\t$$H^{n+1}(x,i)=\\bigcup \\left\\{B_i(y):H^{n+1}(x,i+1)\\cap B_i(y)\\not=\\emptyset\\right\\}$$\n\tfor every $x\\in A_{n+1}$,\n\t\\item set $C_{n+1}(x):=H^{n+1}(x,1)$ for every $x\\in A_{n+1}$, this defines $\\mathcal{D}_{n+1}$.\n\\end{itemize}\nThe fact that $\\mathcal{D}=\\bigcup_{n\\in\\mathbb{N}} \\mathcal{D}_n$ is a $q$-toast on $C$ follows from the fact that $C=\\bigcup_{n\\in \\mathbb{N}}A_n$ together with the fact that the boundaries are $q$ separated which can be shown as in \\cite[Appendix~A]{grebik_rozhon2021toasts_and_tails}.\n\\end{proof}\n\n\n\nTherefore to understand LCLs in the class $\\mathsf{BAIRE}$ we need understand what LCLs on trees admit $\\TOAST$ algorithm.\nIt turns out that these notions are equivalent, again by using the combinatorial characterization of Bernshteyn \\cite{Bernshteyn_work_in_progress} that we now discuss.\n\n\n\\paragraph{Combinatorial Condition -- $\\ell$-full set}\nIn both decompositions, described above, we need to extend a partial coloring along paths that have their endpoints colored from the inductive step.\nThe precise formulation of the combinatorial condition that captures this demand was extracted by Bernshteyn \\cite{Bernshteyn_work_in_progress}.\nHe proved that it characterizes the class $\\mathsf{BAIRE}$ for Cayley graphs of virtually free groups.\nNote that this class contains, e.g., $\\Delta$-regular trees with a proper edge $\\Delta$-coloring.\n\n\\begin{definition}[Combinatorial condition -- an $\\ell$-full set]\n\\label{def:ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL and $\\ell\\geq 2$.\nA set $\\mathcal{V}' \\subseteq \\mathcal{V}$ is \\emph{$\\ell$-full} whenever the following is satisfied.\nTake a path with at least $\\ell$ vertices, and add half-edges to it so that each vertex has degree $\\Delta$.\nTake any $c_1, c_2 \\in \\mathcal{V}'$ and label arbitrarily the half-edges around the endpoints with $c_1$ and $c_2$, respectively.\nThen there is a way to label the half-edges around the remaining $\\ell-2$ vertices with configurations from $\\mathcal{V}'$ such that all the $\\ell-1$ edges on the path have valid edge configuration on them. \n\\end{definition}\n\nNow we are ready to formulate the result that combines Bernshteyn's result \\cite{Bernshteyn_work_in_progress} (equivalence between (1.) and (2.), and the moreover part) with the main results of this section.\nThis also shows the remaining implications in \\cref{fig:big_picture_trees}.\n\n\\begin{theorem}\\label{thm:MainBaireLog}\nLet $\\Pi$ be an LCL on regular trees. \nThen the following are equivalent:\n\\begin{enumerate}\n    \\item $\\Pi\\in \\mathsf{BAIRE}$,\n    \\item $\\Pi$ admits a $\\TOAST$ algorithm,\n    \\item $\\Pi$ admits an $\\ell$-full set,\n    \\item $\\Pi\\in \\mathsf{LOCAL}(O(\\log(n)))$.\n\\end{enumerate}\nMoreover, any of the equivalent conditions is necessary for $\\Pi\\in \\mathsf{fiid}$.\n\\end{theorem}\n\nNext we discuss the proof of \\cref{thm:MainBaireLog}.\nWe refer the reader to Bernshteyn's paper \\cite{Bernshteyn_work_in_progress} for full proofs in the case of $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$, here we only sketch the argument for completeness.\nWe also note that instead of using the toast construction, he used a path decomposition of acyclic graphs of Conley, Marks and Unger \\cite{conley2020measurable}.\n\n\n\\subsection{Sufficiency}\n\nWe start by showing that the combinatorial condition is sufficient for $\\mathsf{BAIRE}$ and $\\mathsf{LOCAL}(O(\\log(n)))$.\nNamely, it follows from the next results together with \\cref{pr:BaireToast} that (2.) implies all the other conditions in \\cref{thm:MainBaireLog}.\nAs discussed above the main idea is to color inductively along the decompositions.\n\n\\begin{proposition}\\label{pr:toastable}\nLet $\\Pi=(\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL that admits $\\ell$-full set $\\mathcal{V}'\\subseteq \\mathcal{V}$ for some $\\ell>0$.\nThen $\\Pi$ admits a $\\TOAST$ algorithm that produces a $\\Pi$-coloring for every $(2\\ell+2)$-toast $\\mathcal{D}$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\nOur aim is to build a partial extending function.\nSet $q:=2\\ell+2$.\nLet $E$ be a piece in a $q$-toast $\\mathcal{D}$ and suppose that $D_1,\\dots,D_k\\in \\mathcal{D}$ are subsets of $E$ such that the boundaries are separated.\nSuppose, moreover, that we have defined inductively a coloring of half-edges of vertices in $D=\\bigcup D_i$ using only vertex configurations from $\\mathcal{V}'$ such that every edge configuration $\\mathcal{E}$ is satisfied for every edge in $D$.\n\nWe handle each connected component of $E\\setminus D$ separately.\nLet $A$ be one of them.\nLet $u\\in A$ be a boundary vertex of $E$.\nSuch an vertex exists since every vertex in $E$ has degree $\\Delta$.\nThe distance of $u$ and any $D_i$ is at least $2\\ell+2$ for every $i\\in [k]$.\nWe orient all the edges from $A$ towards $u$.\nMoreover if $v_i\\in A$ is a boundary vertex of some $D_i$ we assign to $v_i$ a path $V_i$ of length $\\ell$ towards $u$.\nNote that $V_i$ and $V_j$ have distance at least $1$, in particular, are disjoint for $i\\not=j\\in [k]$ .\nNow,  until you encounter some path $V_i$, color any in manner half-edges of vertices in $A$ inductively starting at $u$ in such a way that edge configurations $\\mathcal{E}$ are satisfied on every edge and only vertex configurations from $\\mathcal{V}'$ are used.\nUse the definition of $\\ell$-full set to find a coloring of any such $V_i$ and continue in a similar manner until the whole $A$ is colored.\n\\end{proof}\n\n\n\n\n\\begin{proposition}[$\\ell$-full $\\Rightarrow$ $\\mathsf{LOCAL}(O(\\log n))$]\\label{thm:ellfull_to_logn}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL with an $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$. Then $\\Pi$ can be solved in $O(\\log n)$ rounds in  $\\mathsf{LOCAL}$.\n\\end{proposition}\n\\begin{proof}\nThe proof  uses a variant of the rake-and-compress decomposition considered  in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\paragraph{The Decomposition}\nThe decomposition is parameterized an integer $\\ell' \\geq 1$, and it decomposes the vertices of $T$ into $2L - 1$ layers \\[V = \\VR{1} \\cup \\VC{1} \\cup \\VR{2} \\cup \\VC{2} \\cup \\VR{3} \\cup \\VC{3} \\cup \\cdots \\cup \\VR{L},\\] \nwith $L= O(\\log n)$. We write $\\GC{i}$ to denote the subtree induced by the vertices  $\\left(\\bigcup_{j=i+1}^{L} \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$. Similarly, $\\GR{i}$ is the subtree induced by the vertices  \n$\\left(\\bigcup_{j=i}^L \\VR{j}\\right) \\cup \\left( \\bigcup_{j=i}^{L-1} \\VC{j}\\right)$.\nThe sets $\\VR{i}$ and $\\VC{i}$ are required to satisfy the following requirements. \n\\begin{itemize}\n    \\item Each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\n    \\item Each $v \\in \\VC{i}$ has degree exactly two in the graph $\\GC{i}$. Moreover, the $\\VC{i}$-vertices in $\\GC{i}$ form paths with $s$ vertices, with $\\ell' \\leq s \\leq 2 \\ell'$.\n\\end{itemize}\nFor any given constant $\\ell' \\geq 1$, it was shown in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} that such a decomposition of a tree $T$ can be computed in $O(\\log n)$ rounds. See \\cref{fig:rake_and_compress} for  an example of such a decomposition with $\\ell' = 4$.\n\n\\paragraph{The Algorithm}\nGiven such a decomposition with $\\ell' = \\max\\{1, \\ell - 2\\}$, $\\Pi$ can be solved in $O(\\log n)$ rounds by labeling the vertices in this order: $\\VR{L}$, $\\VC{L-1}$, $\\VR{L-1}$, $\\ldots$, $\\VR{1}$, as follows. The algorithm only uses the vertex configurations in the $\\ell$-full set $\\mathcal{V}'$.\n\n\\paragraph{Labeling $\\VR{i}$}\nSuppose  all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VC{i}$ have been labeled using $\\mathcal{V}'$.  Recall that each $v \\in \\VR{i}$ has degree at most one in the graph $\\GR{i}$.\nIf $v \\in \\VR{i}$ has no neighbor in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$, then we can label the half edges surrounding $v$ by any $c \\in \\mathcal{V}'$.\nOtherwise, $v \\in \\VR{i}$ has exactly one neighbor $u$ in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VC{i}$. Suppose the vertex configuration of $u$ is $c$, where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a} \\in c$.  A simple observation from the definition of $\\ell$-full sets is that for any $c \\in \\mathcal{V}'$ and any $\\mathtt{a} \\in c$, there exist $c' \\in \\mathcal{V}'$ and $\\mathtt{a}' \\in c'$ in such a way that $\\{\\mathtt{a}, \\mathtt{a}'\\} \\in \\mathcal{E}$.\nHence we can label the half edges surrounding $v$ by  $c' \\in \\mathcal{V}'$  where the half-edge label on $\\{u,v\\}$ is $\\mathtt{a}' \\in c'$. \n\n\n\\paragraph{Labeling $\\VC{i}$} Suppose  all vertices in $\\VR{L}, \\VC{L-1}, \\VR{L-1}, \\ldots, \\VR{i+1}$ have been labeled using $\\mathcal{V}'$. Recall that the   $\\VC{i}$-vertices in $\\GC{i}$ form degree-2 paths $P=(v_1, v_2, \\ldots, v_s)$, with $\\ell' \\leq s \\leq 2 \\ell'$.\nLet $P'= (x, v_1, v_2, \\ldots, v_s, y)$ be the path resulting from appending to $P$ the neighbors of the two end-points of $P$ in $\\GC{i}$. The two vertices  $x$ and $y$ are in $\\VR{L} \\cup \\VC{L-1} \\cup \\VR{L-1} \\cup \\cdots \\cup \\VR{i+1}$, so they have been assigned half-edge labels using $\\mathcal{V}'$. Since  $P'$ contains at least $\\ell'+2 \\geq \\ell$ vertices, the definition of $\\ell$-full sets  ensures that we can label $v_1, v_2, \\ldots, v_s$ using vertex configurations in $\\mathcal{V}'$ in such a way that the half-edge labels on $\\{x, v_1\\}, \\{v_1, v_2\\}, \\ldots, \\{v_s, y\\}$ are all in $\\mathcal{E}$.\n\\end{proof}\n\n\n\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .8\\textwidth]{fig\/y1.pdf}\n    \\caption{The variant of the rake-and-compress  decomposition used in the proof of \\cref{thm:ellfull_to_logn}. }\n    \\label{fig:rake_and_compress}\n\\end{figure}\n\n\n\n\\subsection{Necessity}\n\nWe start by sketching that (2.) in \\cref{thm:MainBaireLog} is necessary for $\\mathsf{BAIRE}$ and $\\mathsf{fiid}$.\n\n\n\\begin{theorem}[Bernshteyn \\cite{Bernshteyn_work_in_progress}]\nLet $\\Pi=(\\Sigma,\\mathcal{V},\\mathcal{E})$ be an LCL and suppose that $\\Pi\\in \\mathsf{BAIRE}$ or $\\Pi\\in \\mathsf{fiid}$.\nThen $\\Pi$ admits an $\\ell$-full set $\\mathcal{V}'\\subseteq\\mathcal{V}$ for some $\\ell>0$.\n\\end{theorem}\n\\begin{proof}[Proof Sketch]\n\n\nWe start with $\\mathsf{BAIRE}$.\nSuppose that every Borel acyclic $\\Delta$-regular graph admits a Borel solution on a $\\tau$-comeager set for every compatible Polish topology $\\tau$.\nIn particular, this holds for the Borel graph induced by the standard generators of the free product of $\\Delta$-copies of $\\mathbb{Z}_2$ on the free part of the shift action on the alphabet $\\{0,1\\}$ endowed with the product topology.\nLet $F$ be such a solution.\nWrite $\\mathcal{V}'\\subseteq \\mathcal{V}$ for the configurations of half-edge labels around vertices that $F$ outputs on a non-meager set.\nLet $C$ be a comeager set on which $F$ is continuous. Then, every element of $\\mathcal{V}'$ is encoded by some finite window in the shift on $C$, that is, for each element there are a $k \\in \\mathbb{N}$ and function $s:B(1,k) \\to \\{0,1\\}$ such that $F$ is constant on the set $N_s \\cap C$ (where $N_s$ is the basic open neighbourhood determined by $s$, and $B(1,k)$ is the $k$-neighbourhood of the identity in the Cayley graph of the group).\nSince $\\mathcal{V}'$ is finite, we can take $t>0$ to be the maximum of such $k$'s.\nIt follows by standard arguments that $\\mathcal{V}'$ is $\\ell$-full for $\\ell>2t+1$.\n\nA similar argument works for the $\\mathsf{fiid}$, however, for the sake of brevity, we sketch a shorter argument that uses the fact that there must be a correlation decay for factors of iid's. Let $\\Pi\\in \\mathsf{fiid}$.\nThat is, there is an $\\operatorname{Aut}(T)$-equivariant measurable function from iid's on $T$ (without colored edges this time) into the space of $\\Pi$-colorings.\nLet $\\mathcal{V}'$ be the set of half-edges configurations around vertices that have non-zero probability to appear.\nLet $u,v\\in T$ be vertices of distance $k_0\\in \\mathbb{N}$.\nBy \\cite{BGHV} the correlation between the configurations around $u$ and $v$ tends to $0$ as $k_0\\to \\infty$.\nThis means that if the distance is big enough, then all possible pairs of $\\mathcal{V}'$ configurations need to appear.\n\\end{proof}\n\nTo finish the proof of \\cref{thm:MainBaireLog} we need to demonstrate the following theorem.  Note that  $\\mathsf{LOCAL}(n^{o(1)}) =  \\mathsf{LOCAL}(O(\\log n))$ according to the $\\omega(\\log n)$ -- $n^{o(1)}$ complexity gap~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n\n\\begin{restatable}{theorem}{logcomb}\\label{thm:logn_to_ellfull}\nLet $\\Pi = (\\Sigma, \\mathcal{V}, \\mathcal{E})$ be an LCL solvable in  $\\mathsf{LOCAL}(n^{o(1)})$ rounds.\nThen there exists an $\\ell$-full set  $\\mathcal{V}' \\subseteq \\mathcal{V}$ for some $\\ell \\geq 2$.\n\\end{restatable}\n\n\nThe rest of the section is devoted to the proof of \\cref{thm:logn_to_ellfull}. We start with the high-level idea of the proof.\nA natural attempt for showing $\\mathsf{LOCAL}(n^{o(1)})$ $\\Rightarrow$ $\\ell$-full is to simply take any $\\mathsf{LOCAL}(n^{o(1)})$ algorithm $\\mathcal{A}$ solving $\\Pi$, and then take $\\mathcal{V}'$ to be all vertex configurations that can possibly occur in an output of $\\mathcal{A}$. It is not hard to see that this approach does not work in general, because the algorithm might use a special strategy to label vertices with degree smaller than $\\Delta$. Specifically, there might be some vertex configuration $c$ used by $\\mathcal{A}$ so that some $\\mathtt{a} \\in c$ will only be used to label \\emph{virtual} half edges. It will be problematic to include $c$ in $\\mathcal{V}'$.\n\nTo cope with this issue, we do not deal with general bounded-degree trees. Instead, we construct recursively a sequence $(W^\\ast_1, W^\\ast_2, \\ldots, W^\\ast_L)$ of sets of rooted, layered, and \\emph{partially labeled} tree in a special manner. A tree $T$ is included in $W^\\ast_i$ if it can be constructed by gluing a multiset of rooted trees in  $W^\\ast_{i-1}$ and a new root vertex $r$ in a certain fixed manner. A vertex is said to be in layer $i$ if it is introduced during the $i$-th step of the construction, i.e., it is introduced as the root $r$ during the construction of  $W^\\ast_i$ from $W^\\ast_{i-1}$. All remaining vertices are said to be in layer 0.\n\n\nWe  show that each $T \\in W^\\ast_L$ admits a correct labeling that extends the given partial labeling, as these partial labelings are computed by a simulation of $\\mathcal{A}$.\nMoreover, in these correct labelings, the variety of possible configurations of half-edge labels around vertices in different non-zero layers is the same for each layer.\nThis includes vertices of non-zero layer whose half-edges are labeled by the given partial labeling.\nWe simply pick $\\mathcal{V}'$ to be the set of all configurations of half-edge labels around vertices that can appear in a non-zero layer in a correct labeling of a tree $T \\in W^\\ast_L$.\nOur construction ensures that each $c \\in \\mathcal{V}'$ appears as the labeling of some degree-$\\Delta$ vertex in some tree that we consider.\n\n\nThe proof that $\\mathcal{V}'$ is an $\\ell$-full set is based on finding paths  using vertices of non-zero layers connecting  two vertices with any two  vertex configurations in  $\\mathcal{V}'$ in different lengths. These paths exist because the way rooted trees in $W^\\ast_{i-1}$ are glued together in the construction of $W^\\ast_{i}$ is sufficiently flexible.\nThe reason that we need $\\mathcal{A}$ to have complexity $\\mathsf{LOCAL}(n^{o(1)})$ is that the construction of the trees can be parameterized by a number $w$ so that all the trees have size polynomial in $w$ and the vertices needed to be assigned labeling are at least distance $w$ apart from each other. Since the number of rounds of $\\mathcal{A}$ executed on trees of size $w^{O(1)}$ is much less than $w$, each labeling assignment can be calculated locally and independently. \nThe construction of the trees as well as the analysis are based on a machinery developed in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\n Specifically, we will consider the equivalence relation $\\overset{\\star}{\\sim}$ defined in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} and prove some of its properties, including a pumping lemma for bipolar trees. The exact definition of  $\\overset{\\star}{\\sim}$  in this paper is  different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup} because the mathematical formalism describing LCL problems in this paper is different from the one in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}.\nAfter that, we will consider a procedure for gluing trees parameterized by a labeling function $f$ similar to the one used in~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup}. \nWe will apply this procedure iteratively to generate a set of trees. We will show that the desired  $\\ell$-full set $\\mathcal{V}' \\subseteq \\mathcal{V}$ can be constructed by considering the set of all possible correct labeling of these trees. \n \n\n\\paragraph{The Equivalence Relation $\\overset{\\star}{\\sim}$} \nWe consider trees with a list of designated vertices $v_1, v_2, \\ldots, v_k$ called \\emph{poles}. A \\emph{rooted tree} is a tree with one pole $r$, and a \\emph{bipolar tree} is a tree with two poles $s$ and $t$. \nFor a tree $T$ with its poles $S=(v_1, v_2, \\ldots, v_k)$ with $\\deg(v_i) = d_i < \\Delta$, we denote by $h_{(T,S)}$ the function that maps each choice of the \\emph{virtual} half-edge labeling surrounding the poles of $T$ to $\\textsf{YES}$ or $\\textsf{NO}$, indicating whether such a partial labeling can be completed into a correct complete labeling of $T$. \nMore specifically, consider \n\\[X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots,  \\mathcal{I}_k),\\]\nwhere $\\mathcal{I}_i$ is a size-$(\\Delta - d_i)$ multiset of labels in $\\Sigma$, for each $1 \\leq i \\leq k$. Then $h_{(T,S)}(X) = \\textsf{YES}$ if there is a correct labeling of $T$ such that the $\\Delta - d_i$ virtual half-edge labels surrounding $v_i$ are labeled by $\\mathcal{I}_i$,  for each $1 \\leq i \\leq k$. This definition can be generalized to the case $T$ is already partially labeled in the sense that some of the half-edge labels have been fixed. In this case, $h_{(T,S)}(X) = \\textsf{NO}$ whenever $X$ is incompatible with the given partial labeling.\n\nLet $T$ be a tree with poles $S = (v_1, v_2, \\ldots, v_k)$ and let $T'$ be another tree with poles $S' = (v_1', v_2', \\ldots, v_k')$ such that $\\deg(v_i) = \\deg(v_i') = d_i$ for each $1 \\leq i \\leq k$.\nThen we write $T_1 \\overset{\\star}{\\sim} T_2$ if $h_{(T,S)} = h_{(T',S')}$.\n\nGiven an LCL problem $\\Pi$, it is clear that the number of equivalence classes of rooted trees and bipolar trees w.r.t.~$\\overset{\\star}{\\sim}$ is finite.\nFor a rooted tree $T$,  denote by $\\textsf{Class}_1(T)$ the equivalence class of $T$. For a bipolar tree $H$, denote by $\\texttt{type}(H)$ the equivalence class of $H$.\n\n\n\n\n\\paragraph{Subtree Replacement}\nThe following lemma provides a sufficient condition that the equivalence class of a tree $T$ is invariant of the equivalence class of its subtree $T'$.\nWe note that a real half edge in $T$ might become virtual in its subtree $T'$.\nConsider a vertex $v$ in $T'$ and its neighbor $u$ that is in $T$ but not in $T'$. Then the half edge $(v, \\{u,v\\})$  is real in $T$ and virtual in $T'$.\n\n\n\\begin{lemma}[Replacing subtrees]\\label{lem:replace}\nLet $T$ be a tree with poles $S$.\nLet $T'$ be a connected subtree of $T$ induced by $U \\subseteq V$, where $V$ is the set of vertices in $T$.\nWe identify a list of designated vertices $S' = (v_1', v_2', \\ldots, v_k')$ in $U$ satisfying the following two conditions to be the poles of $T'$.\n\\begin{itemize}\n    \\item $S \\cap U \\subseteq S'$.\n    \\item Each edge $e=\\{u,v\\}$ connecting $u \\in U$ and $ v \\in V \\setminus U$ must satisfy $u \\in S'$.\n\\end{itemize}\nLet $T''$ be another tree with poles $S'' = (v_1'', v_2'', \\ldots, v_k'')$ that is in the same equivalence class as $T'$.\nLet $T^\\ast$ be the result of replacing $T'$ by $T''$ in $T$ by identifying $v_i' = v_i''$ for each $1 \\leq i \\leq k$.\nThen $T^\\ast$ is in the same equivalence class as $T$. \n\\end{lemma}\n\\begin{proof}\nTo prove the lemma, by symmetry, it suffices to show that starting from any correct labeling $\\mathcal{L}$ of $T$, it is possible to find a correct labeling  $\\mathcal{L}^\\ast$ of $T^\\ast$ in such a way that the multiset of the\nvirtual half-edge labels surrounding each pole in $S$ remain the same.\n\nSuch a correct labeling $\\mathcal{L}^\\ast$  of $T^\\ast$ is constructed as follows. If $v \\in V \\setminus U$, then we simply adopt the given labeling $\\mathcal{L}$ of $v$ in $T$.\nNext, consider the vertices in $U$.\nSet $X=(\\mathcal{I}_1, \\mathcal{I}_2, \\ldots, \\mathcal{I}_k)$ to be the one compatible with the labeling $\\mathcal{L}$ of $T$ restricted to the subtree $T'$ in the sense that $\\mathcal{I}_i$ is the multiset of the virtual half-edge labels surrounding the pole $v_i'$ of $T'$, for each $1 \\leq i \\leq k$. We must have $h_{T',S'}(X) = \\textsf{YES}$.\nSince $T'$ and $T''$ are in the same equivalence class, we have $h_{T'',S''}(X) = \\textsf{YES}$, and so we can find a correct labeling $\\mathcal{L}''$ of $T''$ that is also compatible with $X$. Combining this labeling $\\mathcal{L}''$ of the vertices $U$ in $T''$ with the  labeling  $\\mathcal{L}$ of the vertices  $V \\setminus U$ in $T$, we obtain a desired  labeling  $\\mathcal{L}^\\ast$ of $T^\\ast$.\n\nWe verify that $\\mathcal{L}^\\ast$  gives a correct labeling of $T^\\ast$. Clearly, the size-$\\Delta$ multiset that labels each vertex $v \\in V$ is in $\\mathcal{V}$, by the correctness of $\\mathcal{L}$  and $\\mathcal{L}''$. Consider any edge $e=\\{u,v\\}$ in  $T^\\ast$. Similarly, if $\\{u,v\\} \\subseteq V \\setminus U$ or $\\{u,v\\} \\cap (V \\setminus U) = \\emptyset$, then  the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$  and $\\mathcal{L}''$. For the case that $e=\\{u,v\\}$ connects a vertex $u \\notin V$ and a vertex $v \\in V \\setminus U$, we must have $u \\in S'$ by the lemma statement. Therefore, the label of the half edge $(u,e)$ is the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$  by our choice of $\\mathcal{L}''$. Thus, the size-$2$ multiset that labels $e$ is in $\\mathcal{E}$, by the correctness of $\\mathcal{L}$.\n\nWe verify that  the \nvirtual half-edge labels surrounding each pole in $S$ are the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$. Consider a pole $v \\in S$. If $v \\in V \\setminus U$, then the labeling of $v$ is clearly the same in both $\\mathcal{L}$  and $\\mathcal{L}^\\ast$.\nIf $v \\notin V\\setminus U$, then the condition $S \\cap U \\subseteq S'$ in the statement implies that $v \\in S'$. In this case, the way we pick $\\mathcal{L}''$ ensures that the \nvirtual half-edge labels surrounding $v \\in S'$ are the same in both  $\\mathcal{L}$  and $\\mathcal{L}^\\ast$.\n\\end{proof}\n\nIn view of the proof of \\cref{lem:replace}, as long as the conditions in \\cref{lem:replace} are met, we are able to abstract out a subtree by its equivalence class when reasoning about correct labelings of a tree. This observation will be applied repeatedly in the subsequent discussion.\n\n\n\\paragraph{A Pumping Lemma} We will prove a pumping lemma of bipolar trees using \\cref{lem:replace}. \nSuppose $T_i$ is a tree with a root $r_i$ for each $1 \\leq i \\leq k$, then $H=(T_1, T_2, \\ldots, T_k)$ denotes the bipolar tree resulting from concatenating the roots $r_1, r_2, \\ldots, r_k$ into a path $(r_1, r_2, \\ldots, r_k)$ and setting the two poles of $H$ by $s = r_1$ and $t = r_k$.\nA simple consequence of \\cref{lem:replace} is that $\\texttt{type}(H)$ is determined by  $\\textsf{Class}_1(T_1), \\textsf{Class}_1(T_2), \\ldots, \\textsf{Class}_1(T_k)$.\nWe have the following pumping lemma. \n\n\\begin{lemma}[Pumping lemma]\\label{lem:pump}\nThere exists a finite number $ {\\ell_{\\operatorname{pump}}}  > 0$ such that as long as $k \\geq  {\\ell_{\\operatorname{pump}}} $, any  bipolar tree $H=(T_1, T_2, \\ldots, T_k)$ can be decomposed into $H = X \\circ Y \\circ Z$ with $0 < |Y| < k$ so that $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{lemma}\n\\begin{proof}\nSet $ {\\ell_{\\operatorname{pump}}} $ to be the number of equivalence classes for bipolar trees plus one. By the pigeon hole principle, there exist $1 \\leq a < b \\leq k$ such that $(T_1, T_2, \\ldots, T_a)$ and $(T_1, T_2, \\ldots, T_b)$ are in the same equivalence class. Set $X = (T_1, T_2, \\ldots, T_a)$, $Y = (T_{a+1}, T_{a+2}, \\ldots, T_b)$, and $Z = (T_{b+1}, T_{b+2}, \\ldots, T_k)$. As we already know that $\\texttt{type}(X) = \\texttt{type}(X\\circ Y)$, \\cref{lem:replace} implies that \n$\\texttt{type}(X\\circ Y) = \\texttt{type}(X^2\\circ Y)$ by replacing $X$ by $X\\circ Y$ in the bipolar tree $X\\circ Y$.\nSimilarly, $\\texttt{type}(X \\circ Y^i)$ is the same for for each $i \\geq 0$. Applying \\cref{lem:replace} again to replace $X \\circ Y$ by $X \\circ Y^i$ in $H = X \\circ Y \\circ Z$, we conclude that  $X \\circ Y^i \\circ Z$ is in the same equivalence class as $H$ for each $i \\geq 0$. \n\\end{proof}\n\n\n\n\\paragraph{A Procedure for Gluing Trees} Suppose that we have a set of rooted trees $W$. We  devise a procedure that generates a new set of rooted trees by gluing the rooted trees in $W$ together. \nThis procedure is parameterized by a labeling function $f$. Consider a bipolar tree \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] \nwhere $T^m$ is formed by attaching the roots $r_1, r_2, \\ldots, r_{\\Delta - 2}$ of the rooted trees $T_1^m, T_2^m, \\ldots, T_{\\Delta -2}^m$ to the root $r^m$ of $T^m$.\n\n\nThe labeling function $f$ assigns the half-edge labels surrounding $r^m$ based on \n\\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r)\\]\nwhere $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$.\n\nWe write  $T^{m}_\\ast$ to denote the result of applying $f$ to label the root $r^m$ of $T^{m}$ in the bipolar tree $H$, and we write \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)\\] to denote the result of applying $f$ to $H$.  We make the following observation.\n\n\\begin{lemma}[Property of $f$]\\label{lem:labeling_function}\nThe two equivalence classes\n$\\textsf{Class}_1(T_\\ast^m)$ and $\\texttt{type}(H_\\ast)$ are determined by \\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:replace}, once the half-edge labelings of the root $r^m$ of $T^{m}$ is fixed, $\\textsf{Class}_1(T_\\ast^m)$ is determined by $\\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m)$. \nTherefore, indeed $\\textsf{Class}_1(T_\\ast^m)$ is determined by\n\\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] and the labeling function $f$.\nSimilarly, applying  \\cref{lem:replace} to the decomposition of $H_\\ast$ into $H^l$, $T_\\ast^m$, and $H^r$, we infer that  $\\texttt{type}(H_\\ast)$ depends only on $\\texttt{type}(H^l)$, $\\textsf{Class}_1(T_\\ast^m)$, and $\\texttt{type}(H^r)$.\n\\end{proof}\n\nThe three sets of trees $X_f(W)$, $Y_f(W)$, and $Z_f(W)$ are constructed as follows.\n\n\\begin{itemize}\n    \\item $X_f(W)$ is the set of all rooted trees  resulting from appending $\\Delta-2$ arbitrary rooted trees $T_1, T_2, \\ldots, T_{\\Delta-2}$ in $W$ to a new root vertex $r$.\n    \\item  $Y_f(W)$ is the set of bipolar trees constructed as follows. For each choice of $2 {\\ell_{\\operatorname{pump}}}  +1$ rooted trees $T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r$ from $X_f(W)$, concatenate them into a bipolar tree \n    \\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\]  let $H_\\ast$ be the result of applying the labeling function $f$ to $H$, and then add $H_\\ast$ to $Y_f(W)$.\n    \\item $Z_f(W)$ is the set of rooted trees  constructed as follows. For each \\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_f(W),\\] add $(T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m)$ to $Z_f(W)$,  where we set the root of $T_1^l$ as the root, and  add $(T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to $Z_f(W)$, where we set the root  of ${T_{ {\\ell_{\\operatorname{pump}}} }^r}$ as the root. \n\\end{itemize}\n\nWe write $\\textsf{Class}_1(S) = \\bigcup_{T \\in S} \\{\\textsf{Class}_1(T)\\}$ and  $\\texttt{type}(S) = \\bigcup_{H \\in S} \\{\\texttt{type}(H)\\}$, and \nwe make the following observation.\n\n\\begin{lemma}[Property of  $X_f(W)$, $Y_f(W)$, and $Z_f(W)$]\\label{lem:sets_of_trees}\nThe three sets of equivalence classes $\\textsf{Class}_1(X_f(W))$, $\\texttt{type}(Y_f(W))$, and $\\textsf{Class}_1(Z_f(W))$ depend only on $\\textsf{Class}_1(W)$ and the labeling function $f$.\n\\end{lemma}\n\\begin{proof}\nThis is a simple consequence of \\cref{lem:replace,lem:labeling_function}.\n\\end{proof}\n\n\\paragraph{A Fixed Point $W^\\ast$}\nGiven a fixed labeling function $f$,\nwe want to find a set of rooted trees $W^\\ast$ that is a \\emph{fixed point} for the procedure $Z_f$ in the sense that \\[\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast).\\] \nTo find such a set $W^\\ast$, we construct a two-dimensional array of rooted trees $\\{W_{i,j}\\}$, as follows.\n\n\\begin{itemize}\n    \\item For the base case, $W_{1,1}$ consists of only the one-vertex rooted tree.\n    \\item Given that $W_{i,j}$ as been constructed, we define $W_{i,j+1} = Z_f(W_{i,j})$.\n    \\item Given that $W_{i,j}$ for all positive integers $j$ have been constructed,  $W_{i+1,1}$ is defined as follows. Pick $b_i$ as the smallest index such that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$ for some $1 \\leq a_i < b_i$. By the pigeon hole principle, the index $b_i$ exists, and it is upper bounded by $2^C$, where $C$ is the number of equivalence classes for rooted trees. We set \n    \\[W_{i+1,1} = W_{i, a_i} \\cup W_{i, a_i + 1} \\cup \\cdots \\cup W_{i, b_i - 1}.\\]\n\\end{itemize}\n\nWe show that the  sequence $\\textsf{Class}_1(W_{i,a_i}), \\textsf{Class}_1(W_{i,a_i+1}), \\textsf{Class}_1(W_{i,a_i+2}), \\ldots$ is periodic with a period $c_i = b_i - a_i$.\n\n\\begin{lemma}\\label{lem:W-aux-1}\nFor any $i \\geq 1$ and for any $j \\geq a_i$, we have $\\textsf{Class}_1(W_{i,j}) = \\textsf{Class}_1(W_{i,j+c_i})$, where $c_i = b_i - a_i$.\n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:sets_of_trees}, $\\textsf{Class}_1(W_{i,j})$  depends only on $\\textsf{Class}_1(W_{i,j-1})$. Hence the lemma follows from the fact that $\\textsf{Class}_1(W_{i, b_i}) = \\textsf{Class}_1(W_{i, a_i})$.\n\\end{proof}\n\nNext, we show that $\\textsf{Class}_1(W_{2,1}) \\subseteq \\textsf{Class}_1(W_{3, 1}) \\subseteq \\textsf{Class}_1(W_{4, 1}) \\subseteq  \\cdots$.\n\n\\begin{lemma}\\label{lem:W-aux-2}\nFor any $i \\geq 2$, we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$, and so $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$.\n\\end{lemma}\n\\begin{proof}\nSince $W_{i,1} = \\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}$, we have \\[\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l} \\subseteq W_{i,j}\\] according to the procedure of constructing $W_{i,j}$. By \\cref{lem:W-aux-1}, \\[  \\textsf{Class}_1\\left(\\bigcup_{a_{i-1} \\leq l \\leq b_{i-1} - 1} W_{i-1, l}\\right) = \\textsf{Class}_1\\left(\\bigcup_{a_{i-1}+j-1 \\leq l \\leq b_{i-1} +j} W_{i-1, l}\\right)\\]  for all $j \\geq 1$, and so we have $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i,j})$ for each $j > 1$. The claim $\\textsf{Class}_1(W_{i,1}) \\subseteq \\textsf{Class}_1(W_{i+1, 1})$ follows from the fact that $W_{i+1,1} = \\bigcup_{a_{i} \\leq l \\leq b_{i} - 1} W_{i, l}$.\n\\end{proof}\n\nSet $i^\\ast$ to be the smallest index $i \\geq 2$ such that $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$. By the pigeon hole principle and \\cref{lem:W-aux-2}, the index $i^\\ast$ exists, and it is upper bounded by $C$, the number  of equivalence classes for rooted trees. We set \\[W^\\ast = W_{i^\\ast, 1}.\\] \n\nThe following lemma shows that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$, as needed.\n\n\\begin{lemma}\\label{lem:W-aux-3}\nFor any $i \\geq 2$, if $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$, then $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{lemma}\n\\begin{proof}\nBy \\cref{lem:W-aux-1} and the way we construct $W_{i+1, 1}$, we have \\[\\textsf{Class}_1(W_{i, j}) \\subseteq \\textsf{Class}_1(W_{i+1, 1}) \\ \\ \\text{for each} \\ \\ j \\geq a_i.\\]\nBy \\cref{lem:W-aux-2}, we have  \\[\\textsf{Class}_1(W_{i, 1}) \\subseteq \\textsf{Class}_1(W_{i, j}) \\ \\ \\text{for each} \\ \\ j > 1.\\]\nTherefore, $\\textsf{Class}_1(W_{i,1}) = \\textsf{Class}_1(W_{i+1, 1})$ implies that $\\textsf{Class}_1(W_{i, j}) = \\textsf{Class}_1(W_{i,1})$ for each $j \\geq a_i$. Hence we must have $\\textsf{Class}_1(Z_f(W_{i,1})) = \\textsf{Class}_1(W_{i,1})$, and so $\\textsf{Class}_1(W_{i,j})$ is the same for all $j \\geq 1$. \n\\end{proof}\n\nWe remark that simply selecting $W^\\ast$ to be any set such that $\\textsf{Class}_1(Z_f(W^\\ast)) = \\textsf{Class}_1(W^\\ast)$ is not enough for our purpose. As we will later see, it is crucial that the set $W^\\ast$ is constructed by iteratively applying the function $Z_f$ and taking the union of previously constructed sets.\n\n\n\n\n\n\n\n\\paragraph{A Sequence of Sets of Trees}\nWe define $W_1^\\ast = W^\\ast$ and $W_i^\\ast = Z_f(W_{i-1}^\\ast)$ for each $1 <  i \\leq L$, where $L$ is some sufficiently large number to be determined.\n\nThe way we choose $W^\\ast$ guarantees that $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W^\\ast)$  for all $1 \\leq i \\leq L$. \nFor convenience, we write $X_i^\\ast = X_f(W_i^\\ast)$ and $Y_i^\\ast = Y_f(W_i^\\ast)$.\nSimilarly, $\\textsf{Class}_1(X_i^\\ast)$ is the same for all $1 \\leq i \\leq L$ and $\\texttt{type}(Y_i^\\ast)$ is the same for all $1 \\leq i \\leq L$.\n\n\nOur analysis will rely on the assumption that all rooted trees in $W^\\ast$ admit correct labelings. Whether this is true depends only on $\\textsf{Class}_1(W^\\ast)$, which depends only on the labeling function $f$. We say that $f$ is \\emph{feasible} if it leads to a set $W^\\ast$ where all the rooted trees therein admit correct labelings. The proof that a feasible labeling function $f$ exists is deferred.\n\nThe assumption that $f$ is feasible implies that all trees in $W_i^\\ast$, $X_i^\\ast$, and $Y_i^\\ast$, for all $1 \\leq i \\leq L$, admit correct labelings. All rooted trees in $X_i^\\ast$ admit correct labelings because they are subtrees of the rooted trees in $W_{i+1}^\\ast$. A correct labeling of any bipolar tree $H \\in Y_i^\\ast$ can be obtained by combining any correct labelings of the two rooted trees in $W_{i+1}^\\ast$ resulting from $H$.\n\n\n\n\n\\paragraph{Layers of Vertices}\nWe assign a layer number $\\lambda(v)$ to each vertex $v$ in a tree  based on the step that $v$ is introduced in the construction \\[W_1^\\ast \\rightarrow W_2^\\ast \\rightarrow \\cdots \\rightarrow W_L^\\ast.\\]\n\nIf a vertex $v$ is introduced as the root vertex of a tree in $X_i^\\ast = X_f(W_i^\\ast)$, then we say that the layer number of $v$ is $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. A vertex $v$ has  $\\lambda(v) = 0$ if it belongs to a tree in $W_1^\\ast$.\n\n\nFor any vertex $v$ with $\\lambda(v) = i$ in a tree $T \\in X_j^\\ast$ with $i \\leq j$, we write $T_v$ to denote the subtree of $T$ such that $T_v \\in X_i^\\ast$ where $v$ is the root of $T_v$. \n\n\n \nWe construct a sequence of sets $R_1, R_2, \\ldots, R_L$ as follows.\nWe go over all rooted trees $T \\in X_L^\\ast$, all possible correct labeling $\\mathcal{L}$ of $T$, and all vertices $v$ in $T$ with $\\lambda(v) = i \\in \\{1,2, \\ldots, L\\}$. Suppose that the $\\Delta-2$ rooted trees in the construction of $T_v \\in X_i^\\ast = X_f(W_i^\\ast)$ are $T_1$, $T_2$, $\\ldots$, $T_{\\Delta-2}$, and let $r_i$ be the root of $T_i$. Consider the following parameters.\n\\begin{itemize}\n    \\item $c_i = \\textsf{Class}_1(T_i)$.\n    \\item $\\mathtt{a}_i$ is the real half-edge label of $v$ in $T_v$ for the edge $\\{v, r_i\\}$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n    \\item $\\mathcal{I}$ is the size-$2$ multiset of virtual half-edge labels of  $v$ in $T_v$, under the correct labeling $\\mathcal{L}$ of $T$ restricted to $T_v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$.\n\n\n\n\n\\begin{lemma}\\label{lem:Rsets1}\nFor each $1 \\leq i < L$, $R_i$ is determined by $R_{i+1}$.\n\\end{lemma}\n\\begin{proof}\nWe consider the following alternative way of constructing $R_i$ from $R_{i+1}$.\nEach rooted tree $T'$ in $W_{i+1}^\\ast = Z_f(W_i^\\ast)$ can be described as follows.\n\\begin{itemize}\n    \\item Start with a path $(r_1, r_2, \\ldots, r_{ {\\ell_{\\operatorname{pump}}}  + 1})$, where $r_1$ is the root of $T'$.\n    \\item For each $1 \\leq j \\leq  {\\ell_{\\operatorname{pump}}} +1$, append $\\Delta-2$ rooted trees $T_{j,1}, T_{j,2}, \\ldots, T_{j, \\Delta-2} \\in W_i^\\ast$ to $r_j$.\n    \\item Assign the labels to the half edges surrounding $r_{ {\\ell_{\\operatorname{pump}}}  + 1}$ according to the labeling function $f$.\n\\end{itemize}\n\n Now, consider the function $\\phi$ that maps each equivalence class $c$ for rooted trees to a subset of $\\Sigma$ defined as follows:\n$\\mathtt{a} \\in \\phi(c)$ if there exist \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{i+1}\\] and $1 \\leq j \\leq \\Delta -2$ such that $c = c_j$ and $\\mathtt{a} = \\mathtt{a}_j$.\n\nWe go over all possible $T' \\in W_{i+1}^\\ast = Z_f(W_i^\\ast)$.\nNote that the root $r$ of $T'$ has exactly one virtual half edge. For each $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$, we go over all  possible correct labelings $\\mathcal{L}$ of $T'$ where the virtual half edge of $r$ is labeled $\\mathtt{b}$. For each $1 \\leq j \\leq  {\\ell_{\\operatorname{pump}}} +1$, consider the following parameters.\n\\begin{itemize}\n    \\item $c_l = \\textsf{Class}_1(T_{j,l})$.\n    \\item $\\mathtt{a}_l$ is the half-edge label of $r_j$ for the edge $\\{r_j, r_{j,l}\\}$.\n    \\item $\\mathcal{I}$ is the size-$2$ multiset of the remaining two half-edge labels of  $v$.\n\\end{itemize}\nThen we add \n$(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I})$ to $R_i$. \n\nThis construction of $R_i$ is equivalent to the original construction of $R_i$ because a correct labeling of $T' \\in W_i^\\ast$ can be extended to a correct labeling of a tree $T \\in X_L^\\ast$ that contains $T'$ as a subtree if and only if the virtual half-edge label of the root $r$ of $T'$ is $\\mathtt{b} \\in \\Sigma$ such that $\\{\\mathtt{a}, \\mathtt{b}\\} \\in \\mathcal{E}$ for some $\\mathtt{a} \\in \\phi(\\textsf{Class}_1(T'))$.\n\n\nIt is clear that this construction of $R_i$ only depends on $\\textsf{Class}_1(W_i^\\ast)$, the labeling function $f$, and the function $\\phi$, which depends only on $R_{i+1}$. Since the labeling function $f$ is fixed and $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$, we conclude that $R_{i}$ depends only on $R_{i+1}$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Rsets2}\nWe have $R_1 \\subseteq R_2 \\subseteq \\cdots \\subseteq R_L$.\n\\end{lemma}\n\\begin{proof}\nFor the base case, we show that $R_{L-1} \\subseteq R_L$. In fact, our proof will show that $R_{i} \\subseteq R_L$ for each $1 \\leq i < L$. Consider any \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_i.\\] \nThen there is a rooted tree $T \\in X_i^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex so that if we label the half edges surrounding the root vertex according to  $\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}$, and  $\\mathcal{I}$, then this partial labeling can be completed into a correct labeling of $T$.\n\nBecause $\\textsf{Class}_1(W_i^\\ast) = \\textsf{Class}_1(W_L^\\ast)$, there is also a rooted tree $T' \\in X_L^\\ast$ that is formed by attaching $\\Delta-2$ rooted trees of equivalence classes $c_1, c_2, \\ldots, c_{\\Delta-2}$ to the root vertex. Therefore, if we label the root vertex of $T'$ in the same way as we do for $T$, then this partial labeling can also be completed into a correct labeling of $T'$. Hence we must have \n\\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_L.\\] \n\n\nNow, suppose that we already have $R_i \\subseteq R_{i+1}$ for some $1 < i < L$. We will show that  $R_{i-1} \\subseteq R_{i}$. \nDenote by $\\phi_{i}$ and $\\phi_{i+1}$ the function $\\phi$ in \\cref{lem:Rsets1} constructed from  $R_{i}$ and $R_{i+1}$. \nWe have $\\phi_i(c) \\subseteq \\phi_{i+1}(c)$ for each equivalence class $c$, because $R_i \\subseteq R_{i+1}$. Therefore, in view of the alternative construction described in the proof of \\cref{lem:Rsets1}, we have $R_{i-1} \\subseteq R_{i}$.\n\\end{proof}\n\n\n\n\\paragraph{The Set of Vertex Configurations $\\mathcal{V}'$}\nBy \\cref{lem:Rsets1,lem:Rsets2}, if we pick $L$ to be sufficient large, we can have  $R_1 = R_2 = R_3$.\nMore specifically, if we pick \n\\[L \\geq C^{\\Delta-2} \\cdot \\binom{|\\Sigma|+1}{|\\Sigma|-1} + 3,\\]\nthen there exists an index $3 \\leq i \\leq L$ such that $R_i = R_{i-1}$, implying that $R_1 = R_2 = R_3$. Here $C$ is the number of equivalence classes for rooted trees and $\\binom{|\\Sigma|+1}{|\\Sigma|-1}$ is the number of size-$2$ multisets of elements from $\\Sigma$.\n\n\n\nThe set $\\mathcal{V}'$ is defined by including all size-$\\Delta$ multisets $\\{\\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}\\} \\cup \\mathcal{I}$ such that  \\[(c_1, c_2, \\ldots, c_{\\Delta-2}, \\mathtt{a}_1, \\mathtt{a}_2, \\ldots, \\mathtt{a}_{\\Delta-2}, \\mathcal{I}) \\in R_{1}\\] for some $c_1, c_2, \\ldots, c_{\\Delta-2}$.\n\n\n\n\\paragraph{The Set $\\mathcal{V}'$  is $\\ell$-full}\nTo show that the set $\\mathcal{V}'$  is $\\ell$-full, we  consider the subset $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$ defined by the set of vertex configurations used by the labeling function $f$ in the construction of $Y_f(W_i^\\ast)$. The definition of $\\mathcal{V}^\\ast$ is invariant of $i$ as $\\textsf{Class}_1(W_i^\\ast)$ is the same for all $i$. Clearly, for each $x \\in \\mathcal{V}^\\ast$, there is a rooted tree $T \\in W_2^\\ast$ where the root of a subtree $T' \\in X_1^\\ast$ of $T$ has its size-$\\Delta$ multiset of half-edge labels fixed to be $x$ by $f$, and so  $\\mathcal{V}^\\ast \\subseteq \\mathcal{V}'$.\n\nFor notational simplicity, we write $x \\overset{k}{\\leftrightarrow} x'$ if there is a correct labeling of a $k$-vertex path $(v_1, v_2, \\ldots, v_k)$ using only vertex configurations in $\\mathcal{V}'$ so that the vertex configuration of $v_1$ is $x$ and the vertex configuration of $v_k$ is $x'$. If it is further required that the half-edge label of $v_1$ for the edge $\\{v_1, v_2\\}$ is $\\mathtt{a} \\in x$, then we write $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$. The notation $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{a}')$ is defined similarly.\n\n\n\\begin{lemma}\\label{lem:find_path_1}\nFor any $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$ and $\\mathtt{a} \\in x$, there exist a vertex configuration $x' \\in \\mathcal{V}^\\ast$ and a number $2 \\leq k \\leq 2  {\\ell_{\\operatorname{pump}}}  + 1$ such that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} x'$.\n\\end{lemma}\n\\begin{proof}\nLet $T \\in W_L^\\ast$ be chosen so that there is a correct labeling $\\mathcal{L}$ where $x$ is a vertex configuration of some vertex $v$ with $\\lambda(v) = 2$.\n\nTo prove the lemma, it suffices to show that for each of the $\\Delta$ neighbors $u$ of $v$, it is possible to find a path $P = (v,u, \\ldots, w)$ meeting the following conditions.\n\\begin{itemize}\n    \\item $w$ is a vertex whose half-edge labels have been fixed by $f$. This ensures that the vertex configuration of $w$ is in $\\mathcal{V}^\\ast$.\n    \\item All vertices in $P$ are within layers 1,2, and 3. This ensures that the vertex configuration of all vertices in $P$ are in $\\mathcal{V}'$.\n    \\item The number of vertices $k$ in $P$ satisfies $2 \\leq k \\leq 2  {\\ell_{\\operatorname{pump}}}  + 1$. \n\\end{itemize}\n\nWe divide the proof into three cases. Refer to \\cref{fig:ell_full_construction1} for an illustration, where squares are vertices whose half-edge labels have been fixed by $f$.\n\n\\paragraph{Case 1} Consider the subtree $T_v \\in X_2^\\ast$ of $T$ whose root is $v$. In view of the construction of the set $X_f(W_2^\\ast)$, $v$ has $\\Delta-2$ children $u_1, u_2, \\ldots, u_{\\Delta-2}$ in $T_v$, where the subtree $T_i$ rooted at $u_i$ is a rooted tree in $W_2^\\ast$. \n\nFor each $1 \\leq i \\leq \\Delta-2$, according to the structure of the trees in the set $W_2^\\ast = Z_f(W_1^\\ast)$, there is a path $(u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ in $T_i$ containing only layer-1 vertices, where the half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}$ have been fixed by $f$. Hence $P = (v, u_i = w_1, w_2, \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1})$ is a desired path with $k =  {\\ell_{\\operatorname{pump}}}  + 2$ vertices.\n    \n   \n    \n   \n    \n\\paragraph{Case 2}  Consider the subtree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$ that contains $v$ in $T$.\nSimilarly, according to the structure of the trees in the set $W_3^\\ast = Z_f(W_2^\\ast)$, there is a path $(r = w_1', w_2', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ in $T'$ containing only layer-2 vertices so that $r$ is the root of $T'$, $v = w_i'$ for some $1 \\leq i' \\leq  {\\ell_{\\operatorname{pump}}} +1$, and the  half-edge labels of $w_{ {\\ell_{\\operatorname{pump}}} +1}'$ have been fixed by $f$.\nSince  $x \\in \\mathcal{V}' \\setminus \\mathcal{V}^\\ast$, we have $v \\neq w_{ {\\ell_{\\operatorname{pump}}} +1}'$. Hence $P = (v = w_i', w_{i+1}', \\ldots, w_{ {\\ell_{\\operatorname{pump}}} +1}')$ is a desired path with $2 \\leq k \\leq  {\\ell_{\\operatorname{pump}}}  + 1$ vertices.\n\n    \n    \n \n\n\\paragraph{Case 3} There is only one remaining neighbor of $v$ to consider. In view of the construction of $X_f(W_3^\\ast)$, there is a layer-3 vertex $v'$ adjacent to the vertex $r$, the root of the tree $T' \\in W_3^\\ast$ considered in the previous case. If the half-edge labels of $v'$ have been fixed by $f$, then $P = (v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$ is a desired path.\nOtherwise, similar to the analysis in the previous case, we can find a path $P' = (v', \\ldots, w)$ connecting $v'$ to a vertex $w$ whose half-edge labels  have been fixed by $f$. All vertices in $P'$ are of layer-3, and the number of vertices in $P'$ is within $[2,  {\\ell_{\\operatorname{pump}}}  + 1]$. \nCombining $P'$ with the path $(v = w_i', w_{i-1}', \\ldots, w_{1}'=r, v')$, we obtain the desired path $P$ whose number of vertices satisfies $2 \\leq k  \\leq  2 {\\ell_{\\operatorname{pump}}}  + 1$.\n   \n   \n\\end{proof}\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .75\\textwidth]{fig\/y2.pdf}\n    \\caption{An illustration for the proof of \\cref{lem:find_path_1}.}\n    \\label{fig:ell_full_construction1}\n\\end{figure}\n\nFor \\cref{lem:find_path_2}, note that if $\\mathtt{a}$ appears more than once in the multiset $x$, then we still have $\\mathtt{a} \\in x \\setminus \\{\\mathtt{a}\\}$.\n\n\n\\begin{lemma}\\label{lem:find_path_2}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, and $0 \\leq t \\leq  {\\ell_{\\operatorname{pump}}}  - 1$, we have  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n with $k = 2  {\\ell_{\\operatorname{pump}}}  + 4 + t$.\n\\end{lemma}\n\\begin{proof}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$, we can find a rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ such that the path $(v_1$, $v_2$, $\\ldots$, $v_{ {\\ell_{\\operatorname{pump}}}  +1})$ of the layer-1 vertices in  $T_{x, \\mathtt{a}}$ where $v_1 = r$ is the root of   $T_{x, \\mathtt{a}}$ satisfies the property that the half-edge labels of $v_{ {\\ell_{\\operatorname{pump}}}  +1}$ have been fixed to $x$ by $f$ where the half-edge label for $\\{v_{ {\\ell_{\\operatorname{pump}}} }, v_{ {\\ell_{\\operatorname{pump}}} +1}\\}$ is in $x \\setminus \\{\\mathtt{a}\\}$.\n\nNow, observe that for any choice of $(\\Delta-2)( {\\ell_{\\operatorname{pump}}} +1)$ rooted trees \\[\\{T_{i,j}\\}_{1 \\leq i \\leq  {\\ell_{\\operatorname{pump}}} +1, 1 \\leq j \\leq \\Delta-2}\\] in $W_2^\\ast$, there is a rooted tree $T' \\in W_3^\\ast = Z_f(W_2^\\ast)$  formed by appending $T_{i, 1}$, $T_{i,2}$, $\\ldots$, $T_{i, \\Delta-2}$ to $u_i$ in the path $(u_1$, $u_2$, $\\ldots$, $u_{ {\\ell_{\\operatorname{pump}}} +1})$ and fixing the half-edge labeling of $u_{ {\\ell_{\\operatorname{pump}}} +1}$ by $f$. All vertices in $(u_1, u_2, \\ldots, u_{ {\\ell_{\\operatorname{pump}}} +1})$ are of layer-2.\n\nWe choose any $T' \\in W_2^\\ast$ with $T_{i,j} = T_{x, \\mathtt{a}}$ and $T_{i',j'} = T_{x', \\mathtt{a}'}$ such that $i' - i = t + 1$.\nThe possible range of $t$ is $[0,  {\\ell_{\\operatorname{pump}}}  - 1]$.\nConsider any $T \\in W_L^\\ast$ that contains $T'$ as its subtree, and consider any correct labeling of $T$.\nThen the path $P$ resulting from concatenating the path $(v_{ {\\ell_{\\operatorname{pump}}}  +1}, v_{ {\\ell_{\\operatorname{pump}}} }, \\ldots, v_1)$ in $T_{x, \\mathtt{a}}$, the path $(u_i, u_{i+1}, \\ldots, u_{i'})$, and the path $(v_1', v_2', \\ldots, v_{ {\\ell_{\\operatorname{pump}}}  +1}')$ in $T_{x', \\mathtt{a}'}$\nshows that $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ \nfor $k = ( {\\ell_{\\operatorname{pump}}} +1) + (t+2) + ( {\\ell_{\\operatorname{pump}}} +1) = 2 {\\ell_{\\operatorname{pump}}}  + 4 + t$. See \\cref{fig:ell_full_construction2} for an illustration.\n\nFor the rest of the proof, we show that the desired rooted tree $T_{x, \\mathtt{a}} \\in W_2^\\ast$ exists for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$.\nFor any $x \\in \\mathcal{V}^\\ast$, we can find a bipolar tree\n\\[H_\\ast = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T_\\ast^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r) \\in Y_1^\\ast = Y_f(W_1^\\ast)\\]\nsuch that the vertex configuration of the root $r^m$ of  $T_\\ast^m$ is fixed to be $x$ by the labeling function $f$.\nThen, for any $\\mathtt{a} \\in x$, at least one of the two rooted trees in $W_2^\\ast = Z_f(W_1^\\ast)$ resulting from cutting $H_\\ast$ satisfies the desired requirement. \n\\end{proof}\n\n\n\\begin{figure}[h!]\n    \\centering\n    \\includegraphics[width= .65\\textwidth]{fig\/y3.pdf}\n    \\caption{An illustration for the proof of \\cref{lem:find_path_2}.}\n    \\label{fig:ell_full_construction2}\n\\end{figure} \n\nIn the subsequent discussion, the length of a path refers to the number of edges in a path. In particular, the length of a $k$-vertex path is $k-1$.\n\nThe following lemma is proved by iteratively applying \\cref{lem:find_path_2} via intermediate vertex configurations $\\tilde{x} \\in \\mathcal{V}^\\ast$. \n\n\\begin{lemma}\\label{lem:find_path_3}\nFor any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$,  we have  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$\n for all $k \\geq 6  {\\ell_{\\operatorname{pump}}}  + 10$.\n\\end{lemma}\n\\begin{proof}\nBy applying \\cref{lem:find_path_2} for three times,\nwe infer that \\cref{lem:find_path_2}  also works for any path length $k - 1 = (k_1 - 1) + (k_2 - 1) + (k_3 - 1)$, where $k_i - 1 = 2  {\\ell_{\\operatorname{pump}}}  + 3 + t_i$ for $0 \\leq t_i \\leq  {\\ell_{\\operatorname{pump}}} -1$. \n\nSpecifically, we first apply \\cref{lem:find_path_2} to find a path  realizing $(x,\\mathtt{b}) \\overset{k_1}{\\leftrightarrow} \\tilde{x}$ for some  $\\tilde{x} \\in \\mathcal{V}^\\ast$, and for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$.\nLet $\\tilde{\\mathtt{a}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the path. \nWe extend the length of this path by $k_2 - 1$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{b}}) \\overset{k_2}{\\leftrightarrow} \\tilde{x}$, for some $\\tilde{\\mathtt{b}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$.\nFinally, let $\\tilde{\\mathtt{c}} \\in \\tilde{x}$ be the real half-edge label used in the last vertex of the current path. \nWe extend the length of the current path by $k_3$ by attaching to it the path realizing $(\\tilde{x},\\tilde{\\mathtt{d}}) \\overset{k_3}{\\leftrightarrow} (x', \\mathtt{b}')$, for some $\\tilde{\\mathtt{d}} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{x}}\\}$, and for some $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\nThe resulting path realizes  $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in x \\setminus \\{\\mathtt{a}\\}$ and $\\mathtt{b}' \\in x' \\setminus \\{\\mathtt{a}'\\}$.\n\n\nTherefore,  \\cref{lem:find_path_2} also works with path length of the form $k-1 = 6 {\\ell_{\\operatorname{pump}}}  + 9  + t$, for any $t \\in [0, 3 {\\ell_{\\operatorname{pump}}}  - 3]$. \n\nAny $k-1 \\geq 6  {\\ell_{\\operatorname{pump}}}  + 9$ can be written as $k-1 = b(2 {\\ell_{\\operatorname{pump}}}  +3) + (6 {\\ell_{\\operatorname{pump}}}  + 9 + t)$, for some $t \\in [0, 3 {\\ell_{\\operatorname{pump}}}  - 3]$ and some integer $b \\geq 0$.\nTherefore,  similar to the above, we can find a path showing $(x,\\mathtt{b}) \\overset{k}{\\leftrightarrow} (x', \\mathtt{b}')$ by first   applying \\cref{lem:find_path_2} with  path length\n$2 {\\ell_{\\operatorname{pump}}}  +3$ for $b$ times, and then applying the above variant of \\cref{lem:find_path_2} to extend the path length by $6 {\\ell_{\\operatorname{pump}}}  + 9  + t$.\n\\end{proof}\n\nWe show that \\cref{lem:find_path_1,lem:find_path_3} imply that $\\mathcal{V}'$  is $\\ell$-full for some $\\ell$.  \n\n\\begin{lemma}[$\\mathcal{V}'$  is $\\ell$-full] \\label{lem:find_path_4}\nThe set $\\mathcal{V}'$  is $\\ell$-full for $\\ell = 10  {\\ell_{\\operatorname{pump}}}  + 10$. \n\\end{lemma}\n\\begin{proof}\nWe show that for any target path length $k-1 \\geq 10  {\\ell_{\\operatorname{pump}}}  + 9$, and for any $\\mathtt{a} \\in x \\in \\mathcal{V}^\\ast$ and $\\mathtt{a}' \\in x' \\in \\mathcal{V}^\\ast$, we have $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\n\nBy \\cref{lem:find_path_1}, there exists a vertex configuration $\\tilde{x} \\in \\mathcal{V}^\\ast$ so that we can find a path $P$ realizing $(x,\\mathtt{a}) \\overset{\\ell_1}{\\leftrightarrow} \\tilde{x}$ for some path length  $1 \\leq (\\ell_1-1) \\leq 2  {\\ell_{\\operatorname{pump}}} $.\nSimilarly, there exists a vertex configuration $\\tilde{x}' \\in \\mathcal{V}^\\ast$ so that we can find a path $P'$ realizing $(x',\\mathtt{a}') \\overset{\\ell_2}{\\leftrightarrow} \\tilde{x}'$ for some path length   $1 \\leq  (\\ell_2-1) \\leq 2  {\\ell_{\\operatorname{pump}}} $. \n\n\nLet $\\tilde{\\mathtt{a}}$ be the real half-edge label for $\\tilde{x}$ in $P$, and let $\\tilde{\\mathtt{a}}'$ be the real half-edge label for $\\tilde{x}'$ in $P'$.\nWe apply \\cref{lem:find_path_3} to find a path $\\tilde{P}$ realizing  $(x,\\mathtt{b}) \\overset{\\tilde{\\ell}}{\\leftrightarrow} (x', \\mathtt{b}')$ for some $\\mathtt{b} \\in \\tilde{x} \\setminus \\{\\tilde{\\mathtt{a}}\\}$ and $\\mathtt{b}' \\in \\tilde{x}' \\setminus \\{\\tilde{\\mathtt{a}}'\\}$ with path length  \n\\[\\tilde{\\ell}-1 = (k-1) - (\\ell_1-1) - (\\ell_2-1) \\geq  6  {\\ell_{\\operatorname{pump}}}  + 9.\\] \nThe path formed by concatenating $P_1$, $\\tilde{P}$, and $P_2$ shows that $(x,\\mathtt{a}) \\overset{k}{\\leftrightarrow} (x',\\mathtt{a}')$.\n\\end{proof}\n\n\n\n\n\n\n   \n\n\n\n\\paragraph{A Feasible Labeling Function $f$ exists} We show that a feasible labeling function $f$ exists given that $\\Pi$ can be solved in $\\mathsf{LOCAL}(n^{o(1)})$ rounds. We will construct a labeling function $f$ in such a way  each equivalence class in $\\textsf{Class}_1(W^\\ast)$ contains only rooted trees that admit legal labeling. That is, for each  $c \\in \\textsf{Class}_1(W^\\ast)$, the mapping $h$ associated with $c$ satisfies $h(X) = \\textsf{YES}$ for some $X$.\n\n\n\nWe will consider a series of modifications in the construction \\[W  \\rightarrow  X_f(W) \\rightarrow Y_f(W) \\rightarrow Z_f(W)\\] that do not alter the sets of equivalence classes in these sets, conditioning on the assumption that all trees that we have processed admit correct labelings. That is, whether $f$ is feasible is invariant of the modifications.\n\n\n\\paragraph{Applying the Pumping Lemma} Let $w > 0$ be some target length for the pumping lemma. In the construction of $Y_f(W)$, when we process a bipolar tree \n\\[H = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l, T^m, T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r),\\] we apply  \\cref{lem:pump} to the two subtrees $H^l = (T_1^l, T_2^l, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^l)$ and $H^r = (T_1^r, T_2^r, \\ldots, T_{ {\\ell_{\\operatorname{pump}}} }^r)$ to obtain two new bipolar trees $H_+^l$ and $H_+^r$.\nThe $s$-$t$ path in the new trees $H_+^l$ and $H_+^r$ contains $w + x$ vertices, for some $0 \\leq x <  {\\ell_{\\operatorname{pump}}} $. The equivalence classes do not change, that is, $\\texttt{type}(H^l) = \\texttt{type}(H_+^l)$ and $\\texttt{type}(H^r) = \\texttt{type}(H_+^r)$.\n\n\nWe replace $H^l$ by $H_+^l$ and replace $H^r$ by $H_+^r$ in the bipolar tree $H$.\nRecall that the outcome of applying the labeling function $f$ to the root  $r$ of the rooted tree $T_{ {\\ell_{\\operatorname{pump}}} +1}$ depends only on \\[\\texttt{type}(H^l),  \\textsf{Class}_1(T_{1}^m), \\textsf{Class}_1(T_{2}^m), \\ldots, \\textsf{Class}_1(T_{\\Delta - 2}^m), \\texttt{type}(H^r),\\] so applying the pumping lemma to $H^l$ and $H^r$ during the construction of $Y_f(W)$ does not alter $\\textsf{Class}_1(T_\\ast^m)$ and  $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$  and its middle rooted tree $T_\\ast^m$, by \\cref{lem:labeling_function}.\n\n\n\n\\paragraph{Reusing Previous Trees} During the construction of the fixed point $W^\\ast$, we remember all bipolar trees to which we have applied the feasible function $f$.\n\nDuring the construction of $Y_f(W)$. Suppose that we are about to process a bipolar tree  $H$, and there is already some  other bipolar tree $\\tilde{H}$  to which we have applied $f$ before so that \n$\\texttt{type}(H^l)=\\texttt{type}(\\tilde{H}^l)$,  $\\textsf{Class}_1(T_{i}^m) = \\textsf{Class}_1(\\tilde{T}_{i}^m)$ for each $1 \\leq i \\leq \\Delta-2$, and $\\texttt{type}(H^r)=\\texttt{type}(\\tilde{H}^r)$.\nThen we replace $H$ by $\\tilde{H}$, and then we process $\\tilde{H}$ instead.\n\nBy \\cref{lem:labeling_function}, this modification does not alter $\\textsf{Class}_1(T_\\ast^m)$ and  $\\texttt{type}(H^\\ast)$ for the resulting bipolar tree $H^\\ast$.\n\n\\paragraph{Not Cutting Bipolar Trees} We consider the following different construction of $Z_f(W)$ from $Y_f(W)$. For each $H^\\ast \\in Y_f(W)$, we simply add two copies of $H^\\ast$ to $Z_f(W)$, one of them has $r = s$ and the other one has $r = t$.\nThat is, we do not cut the bipolar tree $H^\\ast$, as in the original construction of $Z_f(W)$.\n\nIn general, this modification might alter $\\textsf{Class}_1(Z_f(W))$.\nHowever, it is not hard to see that if all  trees in $Y_f(W)$ already admit correct labelings, then  $\\textsf{Class}_1(Z_f(W))$ does not alter after the modification.\n\nSpecifically, let $T$ be a rooted tree in  $Z_f(W)$ with the above modification. \nThen $T$ is identical to a bipolar tree  $H^\\ast=(T_1, T_2, \\ldots, T_k) \\in Y_f(W)$, where there exists some $1 < i < k$ such that the root $r_i$ of $T_i$ has its half-edge labels fixed by $f$.\nSuppose that the root of $T$ is the root $r_1$ of $T_1$.\nIf we do not have the above modification, then the rooted tree $T'$ added to $Z_f(W)$ corresponding to $T$ is $(T_1, T_2, \\ldots, T_i)$, where the root of $T'$ is also $r_1$. \n\nGiven the assumption that all bipolar trees in $Y_f(W)$ admit correct labelings, it is not hard to see that $\\textsf{Class}_1(T') = \\textsf{Class}_1(T)$. For any correct labeling $\\mathcal{L}'$ of $T'$, we can extend the labeling to a correct labeling $\\mathcal{L}$ of $T$ by labeling the remaining vertices according to any arbitrary correct labeling of $H^\\ast$. This is possible because the labeling of $r_i$ has been fixed. For any correct labeling $\\mathcal{L}$ of $T$, we can obtain a correct labeling $\\mathcal{L}'$ of $T'$ by restricting $\\mathcal{L}$ to  $T'$.\n\n\n\n\n\n\n\\paragraph{Simulating a $\\mathsf{LOCAL}$ Algorithm} We will show that there is a labeling function $f$ that makes all the rooted trees in $W^\\ast$ to admit correct solutions, where we apply the above three modifications in the construction of $W^\\ast$. In view of the above discussion, such a function $f$ is feasible.\n\nThe construction of $W^\\ast$ involves only a finite number $k$ of iterations of $Z_f$ applied to some previously constructed rooted trees. If we view the target length $w$ of the pumping lemma as a variable, then the size of a tree in $W^\\ast$ can be upper bounded by $O(w^{k})$. \n\n\nSuppose that we are given an arbitrary $n^{o(1)}$-round $\\mathsf{LOCAL}$ algorithm $\\mathcal{A}$ that solves $\\Pi$. It is known~\\cite{chang_pettie2019time_hierarchy_trees_rand_speedup,chang2020n1k_speedups} that randomness does not help for LCL problems on bounded-degree trees with round complexity $\\Omega(\\log n)$, so we assume that \n$\\mathcal{A}$ is deterministic.\n\nThe runtime of $\\mathcal{A}$ on a tree of size \n$O(w^{k})$ can be made to be at most $t = w\/10$ if $w$ is chosen to be sufficiently large. \n\nWe pick the labeling function $f$ as follows. \nWhenever we encounter a new bipolar tree $H$ to which we need to apply $f$, we simulate  $\\mathcal{A}$ locally at the root $r^m$ of the middle rooted tree $T^m$ in $H$. Here we assume that the pumping lemma was applied before simulating $\\mathcal{A}$. Moreover, when we do the simulation, we assume that the number of vertices is $n = O(w^{k})$. \n\nTo make the simulation possible, we just locally generate arbitrary distinct identifiers in the radius-$t$ neighborhood $S$ of $r^m$. This is possible because $t = w\/10$  is so small that for any vertex $v$ in any tree $T$ constructed by recursively applying $Z_f$ for at most $k$ iterations, the radius-$(t+1)$ neighborhood of $v$ intersects at most one such $S$-set. Therefore, it is possible to complete the identifier assignment to the rest of the vertices in $T$ so that for any edge $\\{u,v\\}$, the set of identifiers seen by $u$ and $v$ are distinct in an execution of a $t$-round algorithm. \n\nThus, by the correctness of $\\mathcal{A}$, the labeling of all edges $\\{u,v\\}$ are configurations in $\\mathcal{E}$ and the labeling of all vertices $v$ are configurations in $\\mathcal{V}$.\nOur choice of $f$ implies that all trees in $W^\\ast$ admit correct labeling, and so $f$ is feasible. Combined with \\cref{lem:find_path_4}, we have proved that  $\\mathsf{LOCAL}(n^{o(1)})$  $\\Rightarrow$ $\\ell$-full.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Concluding Remarks}\\label{sec:concludingRemarksTrees}\n\nHere we collect some open problems we find interesting. \n\n\n\\begin{enumerate}\n    \\item What is the relation of $\\mathsf{ffiid}, \\mathsf{MEASURE}, \\mathsf{fiid}$? In particular is perfect matching in the class $\\mathsf{ffiid}$ or $\\mathsf{MEASURE}$? \n    \\item Are there LCL problems whose uniform complexity is nontrivial and slower than $\\operatorname{poly}\\log 1\/\\varepsilon$?\n   \n   \n   \n   \n   \n   \n    \\item Suppose that $\\Pi_G$ is a homomorphism LCL, is it true that $\\Pi_G\\in\\mathsf{BOREL}$ if and only if $\\Pi_G\\in O(\\log^*(n))$?\n    \\item Are there finite graphs of chromatic number bigger than $2\\Delta-2$ such that the corresponding homomorphism LCL is not in $\\mathsf{BOREL}$? See also Remark \\ref{r:hedet}.\n\\end{enumerate}\n\n\\subsection*{Acknowledgements} \nWe would like to thank Anton Bernshteyn, Endre Cs\u00f3ka, Mohsen Ghaffari, Jan Hladk\u00fd, Steve Jackson, Alexander Kechris, Edward Krohne, Oleg Pikhurko, Brandon Seward, Jukka Suomela, and Yufan Zheng for insightful discussions. \nWe especially thank Yufan Zheng for   suggesting potentially hard instances of homomorphism LCL problems.\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\\section{Introduction}\n\\label{sec:intro}\n\\noindent The problem of wealth allocation is a key driver for past and present economic studies. Since the seminal contribution of \\cite{markowitz.1952}, both the industry and the researchers, have been highly involved in finding the optimal way to improve the performance of allocation strategies. During last decades, the Markowitz's approach has been widely criticised for the assumptions behind its simple mean--variance framework. In particular, the most critical points regard the homoskedasticity assumption and, in general, the Gaussian assumption for the joint returns distribution.\nRecently, researchers and practitioners found that the departure from the simple Markowitz's mean--variance approach, considering, for example, higher moments, can provide substantial improvements of the asset allocation strategies in terms of cumulative returns, see, e.g., \\cite{jondeau_rockinger.2003}, \\cite{lai_etal.2006}, \\cite{harvey_etal.2010} and \\cite{holly_etal.2011}. Furthermore, as for the conditional second moment, there have also been several attempts to model the conditional skewness and kurtosis of financial time series.\nIndeed, starting from the seminal contribution of \\cite{hansen.1994}, in the last two decades, the financial econometric literature has been focusing on modelling time--varying higher--order moments. One of the main issues  is to figure out the correct way to introduce time--variation in skewness and kurtosis. Most important contributions in this sense are those of \\cite{hansen.1994}, \\cite{theodossiou_1998}, \\cite{harvey_siddique.1999}, \\cite{jondeau_rockinger.2003}, \\cite{deluca_etal.2006} and \\cite{jondeau_rockinger.2012}, who extend the model of \\cite{hansen.1994} and the class of ARCH models of \\cite{engle.1982} and \\cite{bollerslev.1986}. However, when dealing with higher--order conditional moments' dynamics in a multivariate context, the complexity increases principally for the high number of parameters needed and the major concern involves the overall tractability of the resulting model. As regards the co--skewness, main results are referred to \\cite{deluca_etal.2006}, \\cite{deluca_loperfido.2012} and \\cite{franceschini_loperfido.2010}, while for the co--kurtosis, up to our knowledge, there are not any contributions that impose a conditional dynamics on the fourth moment.\\newline\n\\indent The problem of dealing with higher order co--moments is strictly related to the general issue of describing the dependence structure affecting the multivariate financial time series. Since the seminal contribution of \\cite{sklar.1959}, copula models have been effectively used as an alternative flexible tool to the classical multivariate Gaussian models. Indeed, in the last years, copulas theory has played a central role for the financial econometric literature, as widely documented by \\cite{mcneil_etal.2005} and \\cite{joe.2014}. A comprehensive treatments of recent developments in this field can be found in the recent book of \\cite{durante_sempi.2015}. Furthermore, starting from the seminal contribution of \\cite{patton.2006a} and \\cite{jondeau_rockinger.2006b}, copula theory has been further extended to cope with dynamic evolving dependence structures. Time--varying copulas have been successfully employed in empirical works such as in \\cite{rodriguez.2007}, \\cite{delirasalvatierra_patton.2015} and \\cite{bernardi_catania.2015a}.\\newline\n\\indent An effective way to improve the models in sample fit as well as their forecast ability, is to include exogenous information. Nevertheless, most econometric models that aim to explain the joint dependence between economic factors, only consider endogenous information. By the way, it is natural to believe that, the dynamics affecting the relations between industries, and more generally between economies, are highly affected by exogenous, as well as, endogenous factors. It is worth noting that, the economic and financial literature focusing on univariate models, makes extensive use of exogenous information, see, e.g., \\cite{pagan.1971} and \\cite{baillie_1980} in the ARMAX context and \\cite{hwang_satchell.2005} for the GARCHX volatility model. Their empirical results confirm the superior ability of the models exploiting exogenous information, as compare with more simple alternatives. However, despite the documented usefulness of accounting for exogenous information in univariate time series models, in the financial econometric literature that focuses on the dynamic dependence modelling, there are only few examples of multivariate models dealing with exogenous information, see, e.g., \\cite{delirasalvatierra_patton.2015}. This lack is principally related to the difficulty of effectively including such information in such a way to retain model tractability, especially when the dimension of the observed variables is greater than 2. However, the contribution of including exogenous information in order to describe the dependence structure of economies or assets cannot be neglected. Let think about how the recent Global Financial Crises (GFC) of 2007--2008 influenced the dependence between economies. It is undoubtedly true that such a dramatic event drastically changed the choices of wealth allocation of economic agents. Indeed, capitals have moved away from the financial sector, and more generally from speculative instruments, such as options and futures, opening positions on less riskier assets such as short term government bonds or liquidity funds. In this reallocation of resources, it is obvious that, the behaviour of the sovereign interest rates dynamics, for examples, plays a central role in determining how much of the total wealth is reallocated, and hence how the dependence structure of the remaining assets is affected.\\newline\n\\indent In this paper, we propose a new Markov switching dynamic copula model being also able to deal with exogenous information in a fully multivariate setting using elliptical copulas. More precisely, the tractability induced by the copula methodology in dealing with marginal and joint separability is gathered with a new dynamic conditional dependence structure that relies on the presence of exogenous covariates and the inclusion of a latent Markovian structure, in order to enhance the model ability to account for time--varying higher moments. In this way, we are able to incorporate relevant information such as third and fourth conditional moments, as well as economic factors, in a portfolio optimisation framework. Throughout the paper, we will distinguish the marginal modelisation from the dependence structure linking individual asset into a multivariate framework. This distinction is available thanks to the use of copulas, and more precisely, to the \\cite{sklar.1959} Theorem and to the Inference Function for Margins (IFM) technique of \\cite{godambe_1960} and \\cite{mcleish_small.1988}.\nAs regards the univariate specification, we develop a new flexible parametric model accounting for time--varying conditional moments up to the fourth of the univariate distribution. In particular, we rely on the Generalised Autoregressive Score (GAS), also known as Dynamic Conditional Score (DCS), framework recently proposed by \\cite{creal_etal.2013} and \\cite{harvey.2013} in order to update the dynamics of time--varying parameters using the scaled score of the conditional distribution. This specification allows us to take account of a variety of well know stylised facts affecting the financial time series, like time varying moments, skewness and excess of kurtosis, as widely discussed by \\cite{mcneil_etal.2005}. Particularly interesting is the ability of the employed univariate model to approximate the unknown dynamics of the conditional density of returns by means of a filter based on the scaled score of the conditional distribution. This approach has been recently proved to be consistent with the goal of describing time variation for densities' parameters, see \\citep{koopman_etal.2015}.\n\\indent As stated before, the multivariate modelisation is carried on by using a copula function that links the dependence structure of each univariate series. This solution has been proved to provide highly flexible and reliable models for multivariate financial time series analysis, see e.g. \\cite{patton.2006b}, \\cite{joe.2014}, and the references quoted therein. Moreover, the seminal work of \\cite{patton.2006a}, also extend the original IFM two step estimator to the time--varying copula parameters framework. This fundamental result allows to effectively model the time--varying behaviour of linear as well as non--linear dependence patterns of financial returns. As discussed by \\cite{mcneil_etal.2005}, the flexibility in modelling linear and non--linear dependence patterns is one of the main stylised fact affecting multivariate financial time series. Indeed, as a direct consequence, econometric models dealing with multivariate financial time series, are required to account for this aspect, see, e.g., \\cite{delirasalvatierra_patton.2015}. For this reason, we propose a new specification for the evolution of the conditional correlation matrix of elliptical copulas relying on the famous Dynamic Conditional Correlation (DCC) specification of \\cite{engle.2002} and \\cite{tse_tsui.2002}, and the extension of \\cite{cappiello_etal.2006}. Moreover, as in \\cite{billio_caporin.2005} and, more recently, \\cite{bernardi_catania.2015a}, we also allow for the dynamic evolution of the conditional correlation matrix to depend upon the realisation of a first order Markovian process. More precisely, our model allows for different dynamic behaviours of the underlying dependence process in each specific state of the nature, as in \\cite{billio_caporin.2005} and \\cite{bernardi_catania.2015a}, but it additionally includes exogenous regressors in the dependence structure conditional to the latent state. The proposed model can effectively describe tranquil as well as turbulent periods affecting the financial markets, as for example, different phases of the economic or financial cycles. The opportunity to include exogenous covariates into the correlation dynamics is motivated by the observation that, in nowadays financial markets, economic agents usually base their decisions about wealth allocation conditionally on a huge amount of available information coming from multiple sources, which cannot be neglected. Summarising, our new specification is specifically tailored to model time variation in higher order conditional moments and possibly nonlinear time variation in the dependence structure of financial time series including the ability to account for exogenous information.\\newline\n\\indent The empirical part of the paper considers a panel of financial returns in a portfolio optimisation framework using higher order moments of the predictive conditional joint distribution. More precisely, in a similar way as \\cite{jondeau_rockinger.2006a}, we consider conditional portfolio moments into the expected utility of a rational investor by means of a Taylor expansion around the future wealth. Our empirical findings support our model since it outperform famous alternatives, like the DCC model. Specifically, the new specification improves the performance of allocation strategies based on the maximisation of the expected utility.\\newline\n\\indent The remaining of the paper is organised as follows. Section \\ref{sec:model} introduces the marginal and the joint dynamic model specifications and discusses their main features. Section \\ref{sec:asset_allocation} describes the asset allocation problem. Section \\ref{sec:empirical_study} deals with the empirical study and Section \\ref{sec:conclusion} concludes.\n\\section{The Model}\n\\label{sec:model}\n\\noindent In this paper, we propose to estimate the parameters governing the copula function and the associated dependence structure separately from those controlling for the dynamic evolution of the marginal distributions. This estimation technique, known as Inference Function for Marginals (IFM), has been proved to be feasible in large parametric spaces and asymptotically consistent for time varying copulas, see, e.g., \\cite{patton.2006a}. One of the main advantages of using the IFM estimating procedure concerns its ability to separate the univariate financial returns stylised facts from those regarding the multivariate distribution. This latter aspect, allows the researcher to focus on multivariate dependence structure only in a second moment, when the most appropriate model to describe all the empirical regularities of the observed univariate series has been selected and estimated. Whenever the marginal models are able to capture all the univariate empirical regularities of the marginal series, the only interestingly phenomenon to study is the remaining dependence structure. It follows that, the choice of an appropriate model that acts as a filter for the univariate series plays an important role for the subsequent multivariate analysis, see, e.g., \\cite{mcneil_etal.2005}.\\newline\n\\subsection{Marignal models}\n\\label{sec:marginal_model}\n\\noindent The financial econometrics literature offers several valid alternatives to model the conditional distribution of financial returns. In a fully parametric setting, following the same arguments presented by \\cite{cox.1981}, the first issue is to choose between the classes of observation driven and parameter driven models. The former can be well represented by the GARCH family of \\cite{engle.1982} and \\cite{bollerslev.1986}, while the latter is typically associated with the class of state space models; see for example \\cite{harvey_proietti.2005} and \\cite{durbin_koopman.2012}. However, the Generalised Autoregressive Score (GAS) framework, recently introduced by \\cite{creal_etal.2013} and \\cite{harvey.2013} is gaining lots of consideration by econometricians in many fields of time series analysis. Under the \\cite{cox.1981} classification, the GAS models can be considered as a class of observation driven models, with the usual consequences of having a closed form expression for the likelihood and ease of evaluation. However, as noted by \\cite{koopman_etal.2015}, the class of GAS models is similar to the class of state space models in the sense that both approaches make use of the information coming from the entire conditional distribution instead of using only that coming from the conditional expected value, as for example in the GARCH framework.\nIndeed, the key feature of GAS models is that the score of the conditional density is used as the forcing variable into the updating equation of a time--varying parameter. In the recent financial and econometrics literature, many reasons have been argued to support the adoption of score--based rules as a general updating mechanism of time--varying parameters, see, e.g., \\cite{harvey.2013} and \\cite{creal_etal.2013}, just to quote a few of them. Moreover, the flexibility of the GAS framework makes this class of models nested with a huge amount of famous econometrics models such as, for example, some of the ARCH--type models of \\cite{engle.1982} and \\cite{bollerslev.1986} for volatility modelling, and also the MEM, ACD and ACI models of \\cite{engle.2002b}, \\cite{engle_gallo.2006}, \\cite{engle_russell.1998} and \\cite{russell.1999}, respectively. Finally, one of the practical implications of using this framework in order to update the time--varying parameters, is that it avoids the problem of using a non--adequate forcing variable when the choice of it is not so obvious as for dynamic copula models; see, e.g., \\cite{bernardi_catania.2015a}.\\newline\n\\indent Formally, we assume that the $i$--th return at time $t$, for $t=1,2,\\dots,T$, is conditionally distributed according to the filtration $\\mathcal{F}_{i,t-1}=\\sigma\\left(y_{i,1},y_{i,2},\\dots,y_{i,t-1}\\right)$ as\n\\begin{equation}\ny_{i,t}\\sim\\mathcal{AST}\\left(y_{i,t};\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\mid\\mathcal{F}_{t-1}\\right),\n\\end{equation}\nwhere $\\mathcal{AST}\\left(y_{i,t};\\cdot\\right)$ denotes the Asymmetric Student--t distribution with equal left and right tail decay of \\cite{zhu_galbraith.2010} with time varying location $\\mu_{i,t}\\in\\mathbb{R}$, scale $\\sigma_{i,t}\\in\\mathbb{R}^+$, shape $\\nu_{i,t}\\in\\left(4,\\infty\\right)$ and skewness $\\gamma_{i,t}\\in\\left(0,1\\right)$ parameters and density given by\n\\begin{equation}\nf_\\mathcal{AST}\\left(y_{i,t};\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)=\\begin{cases}\n\\frac{1}{\\sigma_{i,t}}\\left[1+\\frac{1}{\\nu_{i,t}}\\left(\\frac{y_{i,t}-\\mu_{i,t}}{2\\gamma_{i,t}\\sigma_{i,t} K\\left(\\nu_{i,t}\\right)}\\right)\\right]^{-\\frac{\\left(\\nu_{i,t}+1\\right)}{2}},\\quad& y_{i,t}\\le\\mu_{i,t}\\nonumber\\\\\n\\frac{1}{\\sigma_{i,t}}\\left[1+\\frac{1}{\\nu_{i,t}}\\left(\\frac{y_{i,t}-\\mu_{i,t}}{2(1-\\gamma_{i,t})\\sigma_{i,t} K\\left(\\nu_{i,t}\\right)}\\right)\\right]^{-\\frac{\\left(\\nu_{i,t}+1\\right)}{2}},\\quad& y_{i,t}>\\mu_{i,t},\\nonumber\\\\\n\\end{cases}\n\\end{equation}\nwhere $K\\left(x\\right) = \\Gamma\\left(\\left(x+1\\right)\/2\\right)\/\\left[\\sqrt{\\pi x}\\Gamma\\left(x\/2\\right)\\right]$ and $\\Gamma\\left(\\cdot\\right)$ is the gamma function. We also define $\\mathbf{\\theta}_{i,t} = \\left(\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)^\\prime\\in\\mathbb{R}\\times\\mathbb{R}^+\\times\\left(0,1\\right)\\times\\left(4,\\infty\\right)$ to be a vector containing all the time--varying parameters. Furthermore, let $h:\\mathbb{R}^4\\to\\mathbb{R}\\times\\mathbb{R}^+\\times\\left(0,1\\right)\\times\\left(4,\\infty\\right)$ be a continuous, $\\mathcal{F}_{t-1}$ measurable, twice differentiable, vector--valued mapping function, such that $\\mathbf{\\theta}_t = h\\left(\\tilde{\\mathbf{\\theta}}_{i,t}\\right)$, where $\\tilde\\mathbf{\\theta}_{i,t}\\in\\mathbb{R}^4$ is a proper reparametrisation of $\\mathbf{\\theta}_{i,t}$, for all $t=1,2,\\dots,T$. In our context, a convenient choice for $h\\left(\\cdot\\right)$ is\n\\begin{equation}\nh\\left(\\tilde{\\mathbf{\\theta}}_{i,t}\\right) = \\begin{cases}\n\\mu_{i,t}=\\mu_{i,t}\\\\\n\\sigma_{i,t}=\\exp\\left(\\tilde\\sigma_{i,t}\\right)\\\\\n\\gamma_{i,t}=\\left[1 + \\exp\\left(-\\tilde\\gamma_{i,t}\\right)\\right]^{-1}\\\\\n\\nu_{i,t}=4+\\exp\\left(\\tilde\\nu_{i,t}\\right).\n\\end{cases}\n\\end{equation}\nWe let the time--varying reparameterised vector $\\tilde\\mathbf{\\theta}_{i,t}=\\left(\\mu_{i,t},\\tilde\\sigma_{i,t},\\tilde\\gamma_{i,t},\\tilde\\nu_{i,t}\\right)^\\prime$ to be updated using the score of the conditional distribution of $y_{i,t}$, exploiting the GAS dynamic where the Fisher information matrix of the conditional distribution is used as scaling matrix, as suggested by \\cite{creal_etal.2013} and \\cite{harvey.2013}. Specifically, we consider the following updating mechanism for the parameter dynnamics\n\\begin{align}\n \\begin{pmatrix}\n \\mu_{i,t+1}\\\\\n  \\tilde\\sigma_{i,t+1}\\\\\n  \\tilde\\gamma_{i,t+1}\\\\\n  \\tilde\\nu_{i,t+1}\n \\end{pmatrix} =\n  \\begin{pmatrix}\n \\omega_{\\mu_i}\\\\\n \\omega_{\\sigma_i}\\\\\n \\omega_{\\gamma_i}\\\\\n \\omega_{\\nu_i}\n \\end{pmatrix} +\n   \\begin{pmatrix}\n \\alpha_{\\mu_i} & 0 & 0 & 0\\\\\n  0 & \\alpha_{\\sigma_i} & 0 & 0\\\\\n  0 & 0 & \\alpha_{\\gamma_i} & 0 \\\\\n  0 & 0 & 0 & \\alpha_{\\nu_i}\n \\end{pmatrix} \\tilde{s}_{i,t}  +\n    \\begin{pmatrix}\n \\beta_{\\mu_i} & 0 & 0 & 0\\\\\n  0 & \\beta_{\\sigma_i} & 0 & 0\\\\\n  0 & 0 & \\beta_{\\gamma_i} & 0 \\\\\n  0 & 0 & 0 & \\beta_{\\nu_i}\n \\end{pmatrix}\n \\begin{pmatrix}\n \\mu_{i,t}\\\\\n  \\tilde\\sigma_{i,t}\\\\\n  \\tilde\\gamma_{i,t}\\\\\n  \\tilde\\nu_{i,t}\n \\end{pmatrix},\n \\label{eq:gas_empDyn}\n\\end{align}\nwhere $\\tilde s_t = \\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right)^{-1}\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right)^{-1}\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ is the scaled score of the reparameterised conditional distribution of $y_{i,t}$. The quantity $\\tilde s_t$ depends on the product of three matrices defined as\n\\begin{align}\n\\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right) &= \\frac{\\partial\\mathrm{h}\\left(\\tilde{\\boldsymbol{\\theta}}_{i,t}\\right)}{\\partial\\tilde\\mathbf{\\theta}_{i,t}}\\\\\n\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right) & = \\mathbb{E}\\left[\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)\\times\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)^\\prime\\mid\\mathcal{F}_{t-1}\\right]\\\\\n\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right) &= \\frac{\\partial\\ln f_\\mathcal{AST}\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)}{\\partial{\\mathbf{\\theta}_{i,t}}},\n\\end{align}\nwhere $\\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right)$ is the Jacobian of the mapping function $h\\left(\\cdot\\right)$, i.e.\n\\begin{equation}\n  \\mathcal{J}\\left(\\tilde\\mathbf{\\theta}_{i,t}\\right) = \\begin{pmatrix}\n                                                  1 & 0 & 0 & 0 \\\\\n                                                  0 & \\exp\\{\\tilde\\sigma_{i,t}\\} & 0 & 0 \\\\\n                                                  0 & 0 & -\\exp\\{-\\tilde\\gamma_{i,t}\\}\\left(1+\\exp\\{-\\tilde\\gamma_{i,t}\\}\\right)^{-2} & 0 \\\\\n                                                  0 & 0 & 0 & \\exp\\{\\tilde\\nu_{i,t}\\},\n                                                \\end{pmatrix},\n\\end{equation}\nwhere $\\mathcal{I}\\left(\\mathbf{\\theta}_{i,t}\\right)$ and $\\nabla\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ are the Fisher information matrix and the score of the conditional AST distribution, respectively. For the parameterisation here considered, the Fisher information matrix of the AST distribution is reported in \\cite{zhu_galbraith.2010}, while the score is reported in \\ref{sec:ScoreAndFisher}.\\newline\n\\indent It is worth noting that, in equation \\eqref{eq:gas_empDyn}, a diagonal structure for the matrix of coefficient is considered. Obviously, this is a convenient choice in order to reduce the number of model parameters, but other solutions are possible. In order to guarantee the stationarity of the process defined in equation \\eqref{eq:gas_empDyn} we only need to impose the constraint that all the autoregressive coefficients $\\beta_k$, $k\\in\\left(\\mu,\\sigma,\\gamma,\\nu\\right)$ would be in modulus less then one, since, under a correct model specification, the sequence of scores $\\{\\tilde s_t, t>0\\}$ is a martingale difference.\n\\subsection{Multivariate model}\n\\label{sec:multivariate_model}\n\\noindent Dynamic time--varying dependence structures of financial assets have been deeply investigated in the recent developments of financial econometrics. One of the most successful and powerful model has been the Dynamic Conditional Correlation (DCC) model of \\cite{engle.2002} and \\cite{tse_tsui.2002} which allows for time variation in the correlation of assets returns. In the last few years many models were also been proposed in order to improve the flexibility of the original DCC, see, e.g., \\cite{bauwens_etal.2006} for a complete surveys of these recent developments. The success of the DCC model is due to its two stage estimation procedure which mainly relies on the multivariate Gaussian assumption. However, when the Gaussian assumption is violated, the separability of the log likelihood does not hold anymore and the only viable alternative in order to remain with a feasible model is to adopt a Quasi--MLE estimation approach, see, e.g., \\cite{engle_sheppard.2001}. Despite its enormous popularity, only few works have been focused on the inclusion of exogenous information in a DCC framework in order to let the covariance co--movements to depend on covariates. Among those, \\cite{vargas.2008} firstly introduced exogenous covariates into the Asymmetric Generalised DCC (AGDCC) model of \\cite{cappiello_etal.2006}, while \\cite{chou_liao.2008} follow a similar approach on a simpler DCC model. However, both the empirical applications of the aforementioned papers, only consider a bivariate setting under the Gaussian assumption for the innovation terms. Besides the tractability of the resulting model, one of the main difficulties of including exogenous covariates into a DCC model, is to guarantee the positiveness of the variance--covariance matrix at each point in time. A common solution relies on the Constrained Maximum Likelihood (CML) or Penalised Maximum Likelihood (PML) estimation approaches. In the context of bivariate time--varying copulas, only \\cite{delirasalvatierra_patton.2015} consider the inclusion of exogenous information to model the correlation matrix dynamic. However, \\cite{delirasalvatierra_patton.2015} do not consider a DCC--type dynamic evolution of the copula parameters, but, they instead rely on the GAS framework of \\cite{creal_etal.2013} and \\cite{harvey.2013}. Moreover, their model cannot be used in a fully multivariate setting when considering elliptical copulas such as the Gaussian and the Student--t. In what follows, we introduce our Student--t dynamic Markov--Switching (MS) copula model where the copula dependence matrix evolves according to a MS--DCC model which also depends on exogenous covariates.\\newline\n\\indent Let $\\mathbf{u}_t=\\left(u_{1,t},u_{2,t},\\dots,u_{N,t}\\right)^\\prime$ with $u_{j,t}=F_{\\mathcal{AST}}\\left(y_{i,t};\\mathbf{\\theta}_{i,t}\\right)$ be the Probability Integral Transformation (PIT) of $y_{i,t}$ according to the conditional AST distribution $F_{\\mathcal{AST}}\\left(\\cdot\\right)$ with parameters $\\mathbf{\\theta}_{i,t}=\\left(\\mu_{i,t},\\sigma_{i,t},\\gamma_{i,t},\\nu_{i,t}\\right)^\\prime$, and assume that\n\\begin{equation}\n\\mathbf{u}_t\\mid S_t=s\\sim\\mathcal{C}_T\\left(\\mathbf{u}_t;\\mathbf{R}_t^s,\\nu_\\mathsf{c}^s\\right),\n\\end{equation}\nfor $t=1,2,\\dots,T$, where $\\mathbf{R}_t^s$ is the time--varying, regime dependent correlation matrix at time $t$ of the pseudo--observation $\\mathbf{u}_t$, and $\\nu_\\mathsf{c}^s$ is the degree of freedom parameter that is also subject to the realisation of the first order Markov chain. According to the DCC dynamics the conditional correlation matrix $\\mathbf{R}_t^s$ can be decomposed in the following way\n\\begin{align}\n\\mathbf{R}_t^s={\\mathbf{D}_t^s}^{-1}\\mathbf{C}_t^s{\\mathbf{D}_t^s}^{-1},\n\\end{align}\nwhere $\\mathbf{D}_t^{s}$ is a diagonal matrix containing the square root of the diagonal elements of $\\mathbf{C}_t^s$. We further assume that the latent states $s=1,2,\\dots,L$ are driven by a Markov process $S_t$, for $t=1,2,\\dots,T$ defined on the discrete space $\\Omega=\\left\\{1,2,\\dots,L\\right\\}$ with transition probability matrix $\\mathbf{Q}=\\left\\{q_{l,k}\\right\\}$, where $q_{l,k}=\\mathbb{P}\\left(S_t=k\\mid S_{t-1}=l\\right)$, $\\forall l,k\\in\\Omega$ is the probability that state $k$ is visited at time $t$ given that at time $t-1$ the chain was in state $l$, and initial probabilities vector $\\mbox{\\boldmath $\\delta$}=\\left(\\delta_1,\\delta_2,\\dots,\\delta_L\\right)^\\prime$, $\\delta_l=\\mathbb{P}\\left(S_1=l\\right)$, i.e., the probability of being in state $l=\\left\\{1,2,\\dots,L\\right\\}$ at time 1. For an up to date review of HMMs, see, e.g., \\cite{cappe_etal.2005}, \\cite{zucchini_macdonald.2009} and Dymarski \\citeyearpar{dymarski.2011}. The following dynamics is imposed on the state dependent variance--covariance matrix\n\\begin{align}\n\\mathbf{C}_{t+1}^s&= \\left(\\bar{\\mathbf{C}}-\\mathbf{A}^s\\bar{\\mathbf{C}}{\\mathbf{A}^{s}}^\\prime-\\mathbf{B}^s\\bar{\\mathbf{C}}{\\mathbf{B}^{s}}^\\prime-\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\bar{\\mathbf{X}}\\right)\n+\\mathbf{A}^s\\bXi_t{\\mathbf{A}^{s}}^\\prime+\\mathbf{B}^s\\mathbf{C}_t^s{\\mathbf{B}^{s}}^\\prime+{\\mathbf{K}\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\mathbf{X}_t,\n\\label{eq:corr_dynamic}\n\\end{align}\nfor $s=1,2,\\dots,L$, where $\\mathbf{A}^s$ and $\\mathbf{B}^s$ are diagonal matrices with elements $\\mathbf{A}^s=\\left\\{\\sqrt{\\alpha_i^s}\\right\\}$, $\\mathbf{B}^s=\\left\\{\\sqrt{\\beta_i^s}\\right\\}$, for $i=1,2,\\dots,N$, $s=1,2,\\dots,L$, $\\mathbf{K}$ is a $N\\times N$ matrix of ones, $\\mbox{\\boldmath $\\xi$}^s$ is a $p\\times 1$ vector containing the coefficients associated to the exogenous variables $\\mathbf{X}_t$. Here, $\\bar\\mathbf{C}$ denotes the empirical variance--covariance matrix of the pseudo--observations $\\mathbf{u}_t$ and $\\bar\\mathbf{X}$ is a $p\\times 1$ vector containing the empirical mean of the regressors, which are introduced in equation \\eqref{eq:corr_dynamic} in order to enforce the mean--reversion of the process. Moreover, in order to guarantee the stationarity of the process defined in equation \\eqref{eq:corr_dynamic}, we impose $\\alpha_i^s + \\beta_i^s<1$ for all $i=1,2,\\dots,N$ and $s=1,2,\\dots,L$. Moreover, we need to ensure that $\\mathbf{C}_{t}^s$ is positive defined matrix for each $t=1,2,\\dots,T$ and across all the possible states of nature $s=1,2,\\dots,L$.\nThe only unspecified quantity in equation \\eqref{eq:corr_dynamic}, is the forcing variable $\\bXi_t$ which is set to the moving average variance--covariance matrix of length $m$\n\\begin{equation}\n\\bXi_t = m^{-1}\\sum_{j=0}^m \\mathbf{u}_{t-m+j}\\mathbf{u}_{t-m+j}^\\prime - \\bar\\mathbf{u}_{m,j}\\bar\\mathbf{u}_{m,j}^\\prime,\n\\end{equation}\nwhere $\\bar\\mathbf{u}_{m,j} = m^{-1}\\sum_{j=0}^m \\mathbf{u}_{t-m+j}$ represents the vector containing the simple mean of the pseudo observations across the period $\\left\\{t-m;t\\right\\}$. The proposed model is similar those of \\cite{vargas.2008} and \\cite{chou_liao.2008} but differs for the chosen forcing variable $\\bXi_t$ which updates the conditional variance--covariance matrix. The proposed updating scheme makes use of a\nrolling window variance--covariance matrix as forcing variable by which we can get a smooth time evolution of the correlation.\nAs regards this aspect, the proposed correlation dynamics generalises the approach of \\cite{patton.2006b} and \\cite{jondeau_rockinger.2006b} in a fully multivariate setting.\\newline\n\\indent More interesting, the model here proposed, allows for the dynamic of the variance--covariance matrix to be also subject to the realisation of a first order Markov process. Using a different approach, \\cite{bernardi_catania.2015a} show that including this regime dependent behaviour into the variance--covariance dynamics, really helps in describing the time--varying dependence of equity indexes. Indeed, ad documented, for example, by \\cite{pelletier.2006}, \\cite{guidolin_timmermann.2008}, \\cite{ang_bekaert.2002}, \\cite{bernardi_catania.2015a}, there is strong evidence of the presence of different regimes affecting the dependence structure of equity assets. Obviously, the model proposed in equation \\eqref{eq:corr_dynamic} can be easily generalised to account for the leverage effect in a similar way as for the AGDCC of \\cite{cappiello_etal.2006}\n\\begin{align}\n\\mathbf{C}_{t+1}^s =&\\left(\\bar\\mathbf{C}-\\mathbf{A}^s\\bar\\mathbf{C}{\\mathbf{A}^{s}}^\\prime-\\mathbf{B}^s\\bar\\mathbf{C}{\\mathbf{B}^{s}}^\\prime-\\mbox{\\boldmath $\\Gamma$}^s\\mathrm{\\bar\\mathbf{N}}^s{\\mbox{\\boldmath $\\Gamma$}^{s}}^\\prime-\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\bar\\mathbf{X} \\right)\\nonumber\\\\\n&\\qquad\\qquad\\qquad+\\mathbf{A}^s\\bXi_t{\\mathbf{A}^{s}}^\\prime+\\mathbf{B}^s\\mathbf{C}_t^s{\\mathbf{B}^{s}}^\\prime+\\mbox{\\boldmath $\\Gamma$}^s\\boldsymbol\\eta_t^s{\\boldsymbol\\eta_t^s}^\\prime{\\mbox{\\boldmath $\\Gamma$}^{s}}^\\prime+\\mathbf{K}{\\mbox{\\boldmath $\\xi$}^{s}}^\\prime\\mathbf{X}_t,\n\\label{eq:Gdynamic}\n\\end{align}\nwhere $\\mbox{\\boldmath $\\Gamma$}^s=\\left\\{\\sqrt{\\gamma_i^s}\\right\\}$, $i=1,2,\\dots,N$ is a diagonal matrix, and $\\boldsymbol\\eta_t^s$ is a vector whose $i$--th component is given by $\\bbone\\left(x_{i,t}^s<0\\right)$ where $x_{i,t}^s=\\mathcal{T}_{\\nu_\\mathsf{c}^s}^{-1}\\left(u_{it}\\right)$ for $i=1,2,\\dots,N$, and $\\mathcal{T}_{\\nu_\\mathsf{c}^s}^{-1}\\left(\\cdot\\right)$ denotes the inverse cumulative density function (cdf) of a standardised Student--t distribution with $\\nu_\\mathsf{c}^s$ degree of freedom, and $\\bbone\\left(\\cdot\\right)$ represents the indicator function.\nIn equation \\eqref{eq:Gdynamic}, $\\mathrm{\\bar\\mathbf{N}}^s$ is the unconditional empirical average of the matrices $\\boldsymbol\\eta_t^s{\\boldsymbol\\eta_t^s}^\\prime$, $t=1,2,\\dots,T$. Several constraints must be imposed in order to avoid explosive patterns in the conditional variance--covariance matrix dynamic in equation \\eqref{eq:Gdynamic}, see, e.g., \\cite{cappiello_etal.2006}. Since in our empirical investigation we do not find any substantial improvement in favour of model \\eqref{eq:Gdynamic} with respect to model \\eqref{eq:corr_dynamic}, throughout the paper we will continue to refer to the specification without leverage effect. Moreover, since the number of parameters in the model in described in equation \\eqref{eq:corr_dynamic} grows linearly with respect to the number of asset $N$, sometimes it would be convenient to set $\\alpha_1^s = \\alpha_2^s = \\dots = \\alpha_N^s=\\alpha^s$ and $\\beta_1^s = \\beta_2^s = \\dots = \\beta_N^s=\\beta^s$. We name this constrained specification `` simple'', while we refer to the aforementioned general case with the name `` generalised''.\\newline\n\\indent To conclude the model specification, we report the multivariate density function of the random vector $\\mathbf{y}_t=\\left(y_{1,t},y_{2,t},\\dots,y_{N,t}\\right)^\\prime$ which is given by\n\\begin{align}\n\\mathbf{y}_{t+1}\\mid\\mathbf{y}_{t}\\sim h\\left(\\mathbf{y}_{t+1};\\mathbf{\\theta}_{t+1},\\mathbf{X}_{t},\\mbox{\\boldmath $\\Delta$}_{t+1}^s, s=1,\\dots,L\\right),\n\\label{eq:multivariate_density}\n\\end{align}\nwhere $\\mathbf{\\theta}_t=\\left(\\mathbf{\\theta}_{1,t}^\\prime,\\mathbf{\\theta}_{2,t}^\\prime,\\dots,\\mathbf{\\theta}_{N,t}^\\prime\\right)^\\prime$, and\n\\begin{align}\nh\\left(\\mathbf{y}_{y+1};\\cdot\\right)=&\\prod_{i=1}^N f_{\\mathcal{AST}}\\left(y_{i,t+1};\\mathbf{\\theta}_{i,t+1}\\right)\n\\sum_{s=1}^L\\pi_{t+1\\vert t}^{(s)} c_T\\left(F_{\\mathcal{AST}}\\left(\\mathbf{y}_{t+1};\\mathbf{\\theta}_{i,t+1}\\right),\\mbox{\\boldmath $\\Delta$}_{t+1}^s,\\mathbf{X}_t\\right)\\nonumber,\n\\end{align}\nwith $\\mbox{\\boldmath $\\Delta$}_{t+1}^s=\\left(\\mathbf{R}_{t+1}^s,\\nu_{\\mathsf{c}}^s\\right)$ and the mixing weight $\\pi_{t+1\\vert t}^{(s)}$ are\n\\begin{equation}\n\\pi_{t+1\\vert T}^{(s)}=\\sum_{m=1}^L {q}_{m,s}\\mathbb{P}\\left(S_{t+1}=m\\mid\\mathbf{y}_{1:t},\\mathbf{X}_{1:t}\\right),\\nonumber\n\\end{equation}\nfor $s=1,2,\\dots,L$, and $\\mathbb{P}\\left(S_{t+1}=m\\mid\\mathbf{r}_{1:t},\\mathbf{X}_{1:t}\\right)$ can be evaluated using the well known FFBS algorithm detailed in \\cite{fruhwirth_schnatter.2006}. Here, the $i$--th marginal density and probability functions are represented by $f_\\mathcal{AST}\\left(\\cdot\\right)$ and $F_\\mathcal{AST}\\left(\\cdot\\right)$ while the copula probability density is denoted by $c_T\\left(\\cdot\\right)$.\\newline\n\\indent Concerning the estimation of the proposed model, unfortunately, given the presence of the exogenous covariates, we cannot rely on the usual results for DCC models such as, for examples, those of \\cite{engle.2002} and \\cite{cappiello_etal.2006}, while, the Constrained ML (CML) approach used by \\cite{vargas.2008} and \\cite{chou_liao.2008} in a similar context, requires the evaluation of highly nonlinear constraint during the maximisation procedure. We follow a different approach which consists on penalising the log--likelihood function (PML) in the second step of the IFM procedure, whenever the variance--covariance matrix is not positive defined. However, this solution is not optimal since it introduces strongly nonlinearities in the likelihood shape, and it may result in possible local optima solutions of the maximisation problem. Hence, good starting values are required.\nIn the second step of the IFM procedure, we made use of the Expectation--Maximisation algorithm of \\cite{dempster_etal.1977} in order to deal with the markovian structure characterising the latent states. Further details about the employed estimation technique can be found in \\cite{bernardi_catania.2015a}.\n\\section{Asset Allocation Strategy}\n\\label{sec:asset_allocation}\n\\noindent In this section, we tailor the `` distribution timing\\qmcsp approach to the portfolio optimisation problem of \\cite{jondeau_rockinger.2012} to the dynamic MS copula approach introduced in the previous Section. Specifically, we consider a rational investor having full information about the whole distribution of his\/her future wealth and builds a dynamic asset allocation strategy by maximising his\/her expected utility. It is worth noting that, higher moments play a fundamental role during the utility maximisation process.\nFormally, let $y_{p,t+1} = \\sum_{i=1}^N\\lambda_{i,t\\vert t+1} y_{i,t+1}$ be the future portfolio return, where $\\lambda_{i,t\\vert t+1}$ denotes the portion of today wealth that the investor is willing to allocate in the $i$--th risky asset, with $\\sum_{i=1}^N \\lambda_{i,t\\vert t+1}=1$, then $W_{t+1} = 1 + y_{p,t+1}$ represents the future portfolio wealth at time $t+1$\\footnote{We will always consider an initial wealth of 1\\$. This choice will not affect the portfolio decision given our assumption on the investor's utility function.}. The vectors, $\\mathbf{y}_{t+1} = \\left(y_{1,t+1},y_{2,t+1},\\dots,y_{N,t+1}\\right)^\\prime$ and $\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}=\\left(\\lambda_{1,t\\vert t+1},\\lambda_{2,t\\vert t+1},\\dots,\\lambda_{N,t\\vert t+1}\\right)^\\prime$ contain all the assets in which the investor can open a speculative position at time $t$, which is subsequently held for the period $\\left[t,t+1\\right)$ and the portfolio weights, respectively. The portfolio optimisation problem can then be written as follows\n\\begin{equation}\n\\arg\\max_{\\left\\{\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right\\}}\\mathbb{E}_t\\left[\\mathcal{U}\\left(1 + \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime \\mathbf{y}_{t+1}\\right)\\right],\\nonumber\n\\label{eq:maximisation}\n\\end{equation}\nwhere $\\mathcal{U}\\left(W_{t+1}\\right)$ denotes the investor utility function and the expectation must be taken with respect to the future portfolio return distribution. Clearly, this problem does not have a closed--form solution given that $\\mathbf{y}_{t+1}$ is conditionally distributed according to equation \\eqref{eq:multivariate_density}. Moreover, when $N$ is large, numerical techniques such as, for example, Monte Carlo integration and quadrature rules become infeasible. Nevertheless, the maximisation problem can be easily solved by taking a Taylor expansion of the utility function around the wealth at time $t$, $W_t$, i.e., by considering\n\\begin{equation}\n\\mathcal{U}\\left(W_{t+1}\\right) = \\sum_{k=0}^\\infty\\frac{\\mathcal{U}^{\\left(k\\right)}\\left(W_t\\right)}{k!}\\left(W_{t+1} - W_t\\right)^k,\n\\label{eq:taylor_expansion}\n\\end{equation}\nwhere $\\mathcal{U}^{\\left(k\\right)}$ denotes the $k$--th derivative of the utility function with respect to its argument. Taking the expectation at time $t$ of both sides of equation \\eqref{eq:taylor_expansion} we get\n\\begin{equation}\n\\mathbb{E}_t\\left[\\mathcal{U}\\left(W_{t+1}\\right)\\right] =\\sum_{k=0}^\\infty\\frac{\\mathcal{U}^{\\left(k\\right)}\\left(W_t\\right)}{k!}m_{p,t+1}^{\\left(k\\right)},\n\\label{eq:taylor_expansion_portfolio}\n\\end{equation}\nwhere $m_{p,t+1}^{\\left(k\\right)}=\\mathbb{E}\\left[r_{p,t+1}^k\\right]=\\mathbb{E}\\left[\\left(W_{t+1}-W_t\\right)^k\\right]$ denotes the $k$--th non central moment of the portfolio return distribution at time $t+1$. If we consider a fourth--order Taylor expansion, i.e., we truncate equation \\eqref{eq:taylor_expansion_portfolio} at $\\bar{k}=4$, the first four central moments of the portfolio distribution are immediately available as a function of the corresponding non--central moments and the portfolio weights. More precisely, the centred moments of the portfolio returns, at time $t+1$, are\n\\begin{align}\n\\mu_{p,t+1} &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{1,t+1}\\nonumber\\\\\n\\sigma_{p,t+1}^2 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{2,t+1}\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\nonumber\\\\\ns_{p,t+1}^3 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{3,t+1}\\left(\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right)\\nonumber\\\\\nk_{p,t+1}^4 &= \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}^\\prime M_{4,t+1}\\left(\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\otimes \\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}\\right),\\nonumber\n\\end{align}\nwhere $\\otimes$ stands for the Kronecker product, and the corresponding non--centred moments required by equation \\eqref{eq:taylor_expansion_portfolio} can be obtained as\n\\begin{align}\nm_{p,t+1}^{\\left(1\\right)} &= \\mu_{p,t+1} \\nonumber\\\\\nm_{p,t+1}^{\\left(2\\right)} &= \\sigma_{p,t+1}^2 + \\mu_{p,t+1}^2 \\nonumber\\\\\nm_{p,t+1}^{\\left(3\\right)} &= s_{p,t+1}^3  + 3\\sigma_{p,t+1}^2\\mu_{p,t+1} + \\mu_{p,t+1}^3\\nonumber\\\\\nm_{p,t+1}^{\\left(4\\right)} &= k_{p,t+1}^4 + 4s_{p,t+1}^3\\mu_{p,t+1} + 6\\sigma_{p,t+1}^2\\mu_{p,t+1}^2 + \\mu_{p,t+1}^4.\\nonumber\n\\end{align}\nNote that $M_{1,t+1}$, $M_{2,t+1}$, $M_{3,t+1}$, $M_{4,t+1}$ are the mean, the variance--covariance matrix, the co--skewness and co--kurtosis of the conditional joint density function in equation \\eqref{eq:multivariate_density}, respectively. Here, the original $\\left(N,N,N\\right)$ and $\\left(N,N,N,N\\right)$ matrices of co--skewness and co--kurtosis were transformed into the $\\left(N,N^2\\right)$ and  $\\left(N,N^3\\right)$ matrices, respectively, as suggested by \\cite{deathayde_flores.2004} by slicing each $\\left(N,N\\right)$ layer and pasting them, in the same order, sideways.\\newline\n\\indent Having defined the general portfolio optimisation framework, the only quantity that remains unspecified is the specific utility function the rational investor maximises. We assume that our agent makes his\/her investment choices according to a Constant Relative Risk Aversion (CRRA) utility function defined as\n\\begin{equation}\n\\mathcal{U}_{t+1}\\left(W_{t+1}\\right) = \\begin{cases} \\frac{W_{t+1}^{1-\\upsilon}}{1-\\upsilon}, & \\mbox{if } \\upsilon>1 \\\\ \\log\\left(W_{t+1}\\right), & \\mbox{if } \\upsilon=1, \\end{cases}\n\\label{eq:utility}\n\\end{equation}\nwhere $\\upsilon$ is the coefficient of relative risk aversion, which is assumed to be constant with respect to the wealth $W_{t+1}$. For CRRA utility function, equation \\eqref{eq:taylor_expansion_portfolio} truncated at the fourth order becomes\n\\begin{align}\n\\mathbb{E}_t\\left[\\mathcal{U}\\left(W_{t+1}\\right)\\right]&\\approx\\frac{1}{1-\\upsilon} + m_{p,t+1}^{\\left(1\\right)} - \\frac{\\upsilon}{2} m_{p,t+1}^{\\left(2\\right)}\n+\\frac{\\upsilon\\left(\\upsilon + 1\\right)}{6}  m_{p,t+1}^{\\left(3\\right)} -\n\\frac{\\upsilon\\left(\\upsilon + 1\\right)\\left(\\upsilon + 2\\right)}{24} m_{p,t+1}^{\\left(4\\right)},\n\\label{eq:utility_final}\n\\end{align}\nwhere the approximation order is $O\\left(5\\right)$. The closed--form expression for the expected utility makes the maximisation of equation \\eqref{eq:utility_final} with respect to the portfolio weights, straightforward. However, unfortunately, there is no closed--form solution for the moments of order two to four of the multivariate joint density function and we need to resort to numerical procedures. \\ref{sec:appendix_Moments} describes the method used to approximate the moments of the predictive joint density.\n\\section{Empirical study}\n\\label{sec:empirical_study}\n\\noindent In this section we apply the model and the portfolio optimisation methodology introduced in previous sections to deliver a set of optimal portfolio weights. To this end, we consider the same data set of \\cite{jondeau_rockinger.2006a} which consists of five of the major international equity indexes, namely the Standard \\& Poors 500 (SPX), the Nikkei 225 (N225), the FTSE 100 (FTSE), the DAX30 (GDAXI), and the CAC40 (FCHI). At each time $t$ over the out of sample period, the rational investor holding the portfolio is expected to choose the optimal allocation of the wealth by maximising his\/her expected utility using the argument presented in Section \\ref{sec:asset_allocation}. The illustration of the empirical results firstly focus on the marginal distribution, then deals with the multivariate density estimation and forecast and concludes with the presentation of the optimal portfolio allocations.\n\\subsection{Data}\n\\label{sec:data}\n\\noindent The data set consists of 1449 weekly returns spanning the period from the 8--th January, 1988, to the 9--th October, 2015. Starting from this series, the last 449 observations spanning the period from the 9--th March, 2007 to the end of the sample, are used to perform the out of sample analysis and to backtest the portfolio strategy, while the fist 1000 observation are used to fit the model and assess its in--sample performance. Table \\ref{tab:Index_data_summary_stat} reports descriptive statistics of the indexes returns over the in--sample and out--of--sample periods. As expected, we find strong evidence of departure from normality, since all the considered series appear to be leptokurtic and skewed. Moreover, the \\cite{jarque_bera.1980} statistic test strongly rejects the null hypothesis of normality for all the considered series at the 1\\% confidence level. It is worth noting that, the departure from normality, appears to be stronger during the out--of--sample period. As discussed for example by \\cite{shiller_2012}, this empirical evidence can be considered as an effect of the recent GFC of 2007--2008 that affected the overall economy. Table \\ref{tab:Dependence} reports the linear correlation and the Kendall $\\tau$ unconditional estimates. All the coefficients seem to be substantially different between the two subperiods, suggesting a time varying behaviour of the dependence structure that characterise our series. To further investigate this aspect, we perform the LMC test for constant versus time--varying conditional correlation of \\cite{tse.2000} on the returns belonging to the whole period. In our case the LMC test is distributed according to a $\\chi_{10}^2$ with a critical value of about 23.21 at the 1\\% confidence level, while the test statistic is about 997 which strongly adverses the null hypothesis of time invariant correlation. Moreover, the lower triangular parts of Table \\ref{tab:Dependence} also gives an insights about the changes in the tail of the joint probability distribution. These findings strongly support for the Student--t copula DCC model with exogenous regressors introduced in the previous sections \\ref{sec:marginal_model}--\\ref{sec:multivariate_model}, being able to describe the evolution of the whole joint distribution of assets returns.\\newline\n\\indent As previously discussed, one of the main feature of the proposed model, concerns its ability to easily incorporate exogenous information in the state dependent correlation dynamics. Consequently, to control for the general economic conditions, we use observations of the following macroeconomic regressors as suggested by \\cite{adrian_brunnermeier.2011}, \\cite{chao_etal.2012} and \\cite{bernardi_etal.2015}:\n\\begin{enumerate}\n\\item[(I)] the weekly change in the three--month Treasury Bill rate (3MTB);\n\\item[(II)] the weekly change in the slope of the yield curve (TERMSPR), measured by the difference of the 10-year Treasury rate and the 3--month Treasury Bill rate;\n\\item[(III)] the weekly change in the credit spread (CREDSPR) between 10--year BAA rated bonds and the 10-year Treasury rate.\n\\end{enumerate}\nHistorical data for all the considered exogenous regressors are freely available on the Federal Reserve Bank of St. Louis (FRED) web site.\n\\subsection{Parameters estimation and goodness--of--fit results}\n\\label{sec:parameters_est}\n\\noindent The empirical experiment is conducted as follows. First, we estimate the marginals and the MS copula DCC models over the in--sample period, and then we make a one--step ahead rolling forecast of the multivariate joint density of the returns for the entire out of sample period. During the rolling forecast exercise, the rational investor takes an investment decision in terms of wealth allocation, at each point in time $t+s$ conditional to the information set available at time $t+s-1$, for $s=1,2,\\dots,448$. For both the marginal and copula models we re--estimate the model parameters each 24 observations, corresponding to six months using a fixed moving window.\\newline\n\\indent Table \\ref{tab:Marginal_estimates} reports the estimated coefficients for each GAS--AST marginal model. We find the scores coefficients associated with the scale $\\left(\\sigma\\right)$ and shape $\\left(\\nu\\right)$ parameters strongly significant indicating that our marginal model is able to capture the changes in the volatility patterns as well as in the tails of the conditional marginal distributions. On the contrary, the scores coefficients associated with the location $\\left(\\mu\\right)$ and the skewness $\\left(\\gamma\\right)$ parameters of the conditional AST distribution are not statistically different from zero. Moreover, we find that the shape and scale parameters of the AST distribution display high persistence, while the location and skewness parameters do not display such behaviour. The coefficients related to the scaled scores have quite similar magnitude across all the assets, indicating that the GAS updating mechanism adapts quite well to the different characteristics of the returns.\\newline\n\\indent Before moving to the copula specification, we asses the goodness of fit of the marginal models. In particular, we want to test if the PITs of the estimated marginal densities are independently and identically distributed uniformly on the unit interval. To this end, we perform the same test employed by \\cite{vlaar_palm.1993}, \\cite{jondeau_rockinger.2006b}, and \\cite{tay_etal.1998}. The test of the iid Uniform$\\left(0,1\\right)$ assumption consists of two parts.\nThe first part concerns the independent assumption, and it tests if all the conditional moments of the data, up to the fourth one, have been accounted for by the model, while the second part checks if the AST assumption is reliable by testing if the PITs are Uniform over the inverval $\\left(0,1\\right)$. In order to test if the PITs are independently and identically distributed, we define $\\hat u_{i,t}=F_\\mathcal{AST}\\left(y_{i,t},\\hat\\mathbf{\\theta}_{i,t}\\right)$ as the PIT of return $i$ at time $t$ and $\\bar{u}_i = T^{-1}\\sum_{t=1}^T \\hat u_{i,t}$ as the empirical average of the $i$--th PIT series, then we examine the serial correlation of $\\left(\\hat u_{i,t} - \\bar{u}_i\\right)^k$ for $k=1,2,\\dots,4$ by regressing each $\\left(\\hat u_{i,t} - \\bar{u}_i\\right)$ for $i=1,2,\\dots,N$ on its own lags up to the order 20.\nThe hypothesis of no serial correlation can be tested using a Lagrange multiplier test defined by the statistics $\\left(T-20\\right)R^2$ where $R^2$ is the coefficient of determination of the regression. This test is distributed according to a $\\chi^2\\left(20\\right)$ with a critic value of about 31.4 at the 5\\% confidence level. The first four columns of Table \\ref{tab:Uniform_test}, named ${\\rm DGT}-{\\rm AR}^{\\left(k\\right)}$, report the test for $k=1,2,3,4$. Our results confirm that, for almost all the considered time series and all the moments of the univariate conditional distributions, the marginal models are able to effectively account for the dynamic behaviour of the series. For what concerns the Uniform assumption of the PITs, we again employ the test suggested by \\cite{tay_etal.1998} which consists in splitting the empirical distribution of $\\hat u_{i,t}$ into $G$ bins and test whether the empirical and the theoretical distributions significantly differ on each bin. More precisely, let us define $n_{q,i}$ as the number of $\\hat u_{i,t}$ belonging to the bin $q$, it can be shown that\n\\begin{equation}\n\\xi_i=\\frac{\\sum_{q=1}^G\\left(n_i-\\mathbb{E} n_i\\right)^2}{\\mathbb{E} n_i} \\sim \\chi ^2\\left(G-1\\right),\n\\end{equation}\nwhere $G$ is the number of bins. In our case, since we have estimated the model parameters, the asymptotic distribution of $\\xi_i$ is bracketed between a $\\chi ^2\\left(G-1\\right)$ and $\\chi ^2\\left(G-m-1\\right)$ where $m$ is the number of estimated parameters.\nIn order to be consistent with the results reported by \\cite{tay_etal.1998} and \\cite{jondeau_rockinger.2006b}, we choose $G=20$ bins using a $\\chi^2\\left(19\\right)$ distribution for $\\xi_i,\\quad i=1,2,\\dots, N$, see also \\cite{vlaar_palm.1993}. The test statistic of the Uniform test are reported in the last column of Table \\ref{tab:Uniform_test} and are named $\\mathrm{DGT-H}\\left(20\\right)$.\nWe can observe that, for each series, the PITs are uniformly distributed over the interval $(0,1)$ at a confidence level lower then $1\\%$.\nThese findings definitely confirm the adequacy of our assumption on the innovation term and the conditional returns dynamic.\\newline\n\\indent We now move to study the dependence structure of the series using the proposed copula model. As previously discussed, various parameterisations of the proposed model for the conditional dependence structure of assets returns can be employed. Given the available data set, several different levels of flexibility are possible. Model specification concerns the selection of either the number of states, or the `` simple'', or `` generalised'', specifications of the joint model discussed in Section \\ref{sec:multivariate_model}, or the inclusion of the information coming from the exogenous data.\nIn particular, the choice of the number of regimes, in the HMM literature is an open question. As argued by \\cite{lindsay.1983} and \\cite{bohning.2000}, the number of states can be either estimated or tested. However given the large number of model combinations we are considering here, we decide to follow the most common practice of choosing the number of states according to the BIC, see e.g. \\cite{cappe_etal.2005}, \\cite{fruhwirth_schnatter.2006} and \\cite{zucchini_macdonald.2009}. We chose the best model according to the BIC since we want to penalise more the models with a high number of parameters. Moreover, since the marginal models are invariant to the choice of the possible dependence specifications, we report the log--likelihood and the information criteria only for the copula specifications.\nTable \\ref{tab:Model_Choice} reports the log--likelihood evaluated at its optimum, the BIC and the AIC for different combinations of number of states, type of dynamic and inclusion of exogenous covariates. According to the BIC, the best model is the '' simple\\qmcsp DCC dynamic specification with two regimes and the inclusion of the exogenous covariates. The BIC clearly select the model with the highest ability to represent different levels of the dependence structure affecting the data, and to react on the exogenous information depending on the state of the world. Table \\ref{tab:CopulaCoef} reports the estimated coefficients for the best fitting model.\nWe can observe that both regimes are characterised by strong persistence in the dynamic of the conditional dependence, however the dynamics react differently to the new endogenous and exogenous information. More precisely, we can observe that in the second regime, the coefficient $\\mathbf{A}^s$, $s=2$, which measures the reaction to new endogenous information, is much higher than the same coefficient in the other regime. This means that the second regime reacts more quickly, compared to the first one, when a change occurs in the market dependence structure. On the contrary, we observe a similar persistent behaviour of the correlation matrix within the two regimes. For what concerns the estimated state specific degree of freedom, we note that the second regime is characterised by a smaller coefficient indicating higher positive as well as negative tail dependence compared to the second one. These finding is also consistent with the empirical evidence reported by \\cite{chollete_etal.2009}, \\cite{bernardi_etal.2015} and \\cite{bernardi_catania.2015a}, in related studies.\nLooking at the estimated coefficients associated to the exogenous covariates, we can observe different behaviours of the conditional correlation matrix, depending on the regimes. In particular, we note that the returns dependence, conditional of being in the first regime, appears to react less to an absolute increase of the considered exogenous covariates. On the contrary, the dependence structure conditional of being in the second regime, displays a stronger reaction to the macroeconomic factors here considered.\nThe estimated transition probabilities of the unobserved Markov chain are $p_{11}=0.984$ and $p_{22}=0.987$, indicating quite strong persistence and an expected duration of the first and second states of about 61.5 and 75.9 weeks, respectively. Finally, we found that the dependence behaviour of the considered US market returns switched according to the 2000--2003 dot--com bubble and the recent 2008--2010 Global Financial Crisis.\n\\subsection{Portfolio allocation}\n\\noindent Now we consider the portfolio allocation problem. As stated in the Introduction, we focus on a rational investor who maximise his\/her expected utility. Specifically, once the models parameters have been estimated over the in sample period, the investor forecasts the one--step ahead joint distribution of his\/her assets return for each time $t$ of the out--of--sample period, and then he\/she allocates the available wealth among those assets accordingly to the portfolio allocation procedure discussed in Section \\ref{sec:asset_allocation}. We assume an initial wealth of 1\\textdollar, an allocation strategy that allows for either long and short positions and a risk free rate equal to zero. The choice of the level of the initial wealth is otherwise irrelevant because it does not influence the risk aversion, and hence, the allocation strategy under the defined CRRA utility function reported in equation \\eqref{eq:utility}. Moreover, in order to evaluate the influence of the level of risk aversion in the portfolio allocation strategy, we consider different levels of relative risk aversion, i.e., $\\upsilon=3,7,10,20$. In order to asses the performance of the proposed model from a portfolio optimisation viewpoint, we run an horse race with five common alternatives:\n\\begin{itemize}\n\\item[-] the `` Equally Weighted portfolio'', (EW) strategy, where the weights associated to each asset are kept fixed and equal to $1\/N$ during the validation period;\n\\item[-] the `` Minimum Variance portfolio'', (MV) strategy, where the weights associated to each asset are kept fixed and equal to those minimising the portfolio variance during the in--sample period;\n\\item[-] the `` DCC\\qmcsp strategy, where the one--step ahead forecast of the conditional mean and the conditional variance--covariance matrix of the joint distribution of assets return are performed using the DCC$\\left(1,1\\right)$ model of \\cite{engle.2002}, whose parameters are estimated by Quasi--ML, see, e.g., \\cite{francq_zakoian.2011}. The weights of each asset are then chosen by maximising equation \\eqref{eq:maximisation} truncated at the second order;\n\\item[-] the `` Naive Mean--Variance'', (NMV) and `` Naive Higher--Moments'', (NHM) strategies, where the optimal weights are chosen again by maximising equation \\eqref{eq:maximisation}, as in the case of the `` DCC\\qmcsp strategy, and the first four moments of the joint distribution of assets return are estimated using a rolling window approach.\n\\end{itemize}\nHereafter, we label the optimal investment strategy according to the model detailed in Section \\ref{sec:model}, Flexible Dynamic Dependence Model (FDDM).\\newline\n\\indent One easy way to compare different portfolios performance in terms of realised utility is to consider the so called `` management fee\\qmcsp approach. The management fee approach relies on the definition of the amount of money, $\\vartheta$, a rational investor is willing to pay, or gain, in order to switch from the portfolio that he\/she is currently holding on a given period, to a different one, and it coincides with the solution of the the following equation\n\\begin{equation}\nS^{-1}\\sum_{t=F+1}^{F+S}\\mathcal{U}\\left(1+\\mbox{\\boldmath $\\lambda$}^{\\mathcal{A}\\prime}_{t\\vert t+1}\\mathbf{y}_{t+1}\\right) = S^{-1}\\sum_{t=F+1}^{F+S}\\mathcal{U}\\left(1+\\mbox{\\boldmath $\\lambda$}^{\\mathcal{B}\\prime}_{t\\vert t+1}\\mathbf{y}_{t+1}-\\vartheta\\right),\n\\label{eq:managementFee}\n\\end{equation}\nwhere $F$ and $S$ are the length of the in--sample and out--of--sample periods, respectively, and, as before, $\\mbox{\\boldmath $\\lambda$}_{t\\vert t+1}$ denotes the vector of amounts of wealth $\\lambda_{j,t\\vert t+1}$, the investor is allocating to each asset $j=1,2,\\dots,N$ during the period $\\left(t,t+1\\right]$. From equation \\eqref{eq:managementFee}, it is easy to see that if $\\vartheta>0$, the investor is willing to pay a positive amount of money in order to switch from portfolio $\\mathcal{A}$ to portfolio $\\mathcal{B}$. On the contrary, if $\\vartheta<0$, the investor is going to ask for a higher return from portfolio $\\mathcal{B}$ in order to compensate the loss of utility he\/she would experience for switching from $\\mathcal{A}$ to $\\mathcal{B}$. Finally, if $\\vartheta=0$ the two portfolios give exactly the same utility to the investor, leaving he\/she indifferent between the two options. Table \\ref{tab:MF_mSR} reports the evaluated management fees for all the possible couples made by he FDDM strategy and a competitor, under different values of relative risk aversion coefficient $\\upsilon=\\left(3,5,10,20\\right)$. More precisely, looking at equation \\eqref{eq:managementFee}, we fix $\\mathcal{B}$ equal to the FDDM strategy, and we let $\\mathcal{A}$ to vary among the competing alternatives, i.e., the strategy a $\\mathcal{A}$ is selected among the alternatives $\\{\\mathrm{EW},\\mathrm{MV},\\mathrm{DCC},\\mathrm{NMV},\\mathrm{NHM}\\}$.\nAs a consequence, if the reported management fee is positive, then the optimal portfolio based on the proposed model is better compared to the specific competitor, otherwise the alternative strategy is preferred. Another useful indicator to compare portfolio strategies is the Sharpe ratio. However, since this indicator does not result in a measure of outperformance over alternative strategies with different levels of risk, we employ the so called modified Sharpe Ratio (mSR) introduced by \\cite{graham_harvey.1997} which is defined as\n\\begin{equation}\n\\mathrm{mSR}\\left(\\mathcal{A},\\mathcal{B}\\right)=\\frac{\\sigma_{\\mathcal{A}}}{\\sigma_\\mathcal{B}}\\mu_\\mathcal{B} - \\mu_\\mathcal{A},\n\\label{eq:modSharpeRatio}\n\\end{equation}\nwhere it immediately follows that if $\\mathrm{mSR}\\left(\\mathcal{A},\\mathcal{B}\\right)>0$, then model $\\mathcal{B}$ is preferred to model $\\mathcal{A}$ under a simple mean--variance preference ordering.\\newline\n\\indent Table \\ref{tab:MF_mSR} reports the management fee and the mSR for all the considered alternatives against the FDMM strategy. We note that, almost all the management fees and the Sharpe ratios are positive and significantly different from zero, indicating economic value in favour of the FDDM strategy. Indeed, concerning the evaluated management fees, we found that the FDDM strategy is almost always preferred by a rational investors with arbitrary level of risk aversion. The only exceptions are when the FDDM strategy is compared with the DCC strategy assuming an high degree of risk aversion $\\left(\\upsilon=3\\right)$ and with the EW strategy assuming a low degree of risk aversion $\\left(\\upsilon=20\\right)$. The results associated with the modified Sharpe ratio are in line with those reported by the management fees analysis. The only difference is that, when the degree of risk aversion is low, $\\left(\\upsilon=20\\right)$, the statistical significance of the modified Sharpe ratio vanishes, indicating that there is no evidence of outperformance between the FDDM and the alternative strategies.\nTable \\ref{tab:avg_weights} reports the average portfolio weights over the whole out of sample period associated with each strategy. We note that the positions taken by the FDDM and DCC strategies are quite homogeneous in terms of which asset to buy and which asset to sell. Indeed, the N225 and the FCHI indexes are always shorted by the FDDM and DCC strategies. These finding suggest that the signal coming from the two strategies is similar, demonstrating that a truly dynamic model is important when investment positions are concerned. We also note that, the weights associated with the NMV, NHM, and MVP are very similar, independently from the assumed degree of risk aversion of the investor. Looking at the descriptive statistics of the indexes given in Table \\ref{tab:Index_data_summary_stat} we can observe that the assets with the higher (in absolute value) wealth allocation, according to the FDDM strategy, are those who, ex--post, result in lower levels of kurtosis. On the contrary, the indexes with the lower exposition, report high kurtosis during the out--of--sample period. More interesting, looking again at Table \\ref{tab:Index_data_summary_stat}, we note that, during the in--sample period, the levels of kurtosis among assets were quite homogeneous, which means that our model is able to effectively forecast the changes in the behaviour of the joint density.\nMore precisely, on the one hand, the GAS updating mechanism is suited to effectively move the parameters through the direction of higher probability mass, and, on the other hand, the Markov switching dynamic copula model helps in capturing the evolution of the dependence structure during time and across states.\n\\section{Conclusion}\n\\label{sec:conclusion}\n\\noindent In this paper we propose a new flexible copula model being able to account for a wide variety of stylised facts affecting multivariate financial time series. Specifically, we allow the dependence structure affecting the considered financial time series to depend either on the realisation of a first order Markov chain and on exogenous covariates that may represent economic factors such as, interest rates, GDP growth rates, as well as financial indexes such as realised volatility indexes. The proposed solution is highly flexible, since the dependence dynamics account for either observable and unobservable latent factors.\nAnother interesting contribution concerns the marginal model specification. We develop a novel GAS filter \\citep{creal_etal.2013, harvey.2013} based on the Asymmetric Student--t distribution of \\cite{zhu_galbraith.2010} where all the location, scale and shape parameters evolves according to a first order transition dynamic based on the scaled score of the conditional distribution. In this way we allow for time variation in the first four moments of the conditional marginal distribution.\nParameters estimation is performed by exploiting the two step Inference Function for Margins procedure detailed in \\cite{patton.2006a} for conditional copulas, where in the second step we made use of the Expectation--Maximisation algorithm of \\cite{dempster_etal.1977}. Further details about the employed estimation technique can be found in \\cite{bernardi_catania.2015a}.\\newline\n\\indent The proposed model is applied to predict the evolution of the weekly returns of five major international stock indexes, previously considered by \\cite{jondeau_rockinger.2006a}, over the period 1988--2015. In the empirical application, in sample and out of sample performances of the proposed model are deeply investigated. The in sample analysis, reveals statistical evidence for nonlinearities and highlights the relevance of the inclusion of exogenous covariates in explaining the dependence structure affecting the indexes returns. Furthermore, our analysis confirms that our model can be effectively used to describe and predict various aspects of the conditional joint distribution of equity returns, such as time varying means, variances, co--skewness and co--kurtosis. Out of sample model performances are evaluated by solving an optimisation problem where a rational investor with power utility function takes future investment positions according to the predictive density of assets returns.\nSpecifically, we compare the optimal portfolio strategy implied by the proposed model with that of several static and dynamic alternatives. The portfolio optimisation strategy is based on the Taylor expansion of a CRRA utility function of the rational investor and involves the evaluation of the portfolio moments up to the fourth order. Since higher order moments of the joint distribution are not analytically known we resort to numerical integration. Our empirical results suggest that the proposed model outperforms competitors in terms of economic value.\n\\section*{Acknowledgments}\n\\noindent This research is supported by the Italian Ministry of Research PRIN 2013--2015, ``Multivariate Statistical Methods for Risk Assessment'' (MISURA), and by the ``Carlo Giannini Research Fellowship'', the ``Centro Interuniversitario di Econometria'' (CIdE) and ``UniCredit Foundation''. We would like to express our sincere thanks to all the participants to the Bozen -- Risk School, Bozen, September 2015,  for their constructive comments that greatly contributed to improving the final version of the paper. A special thank goes also to Juri Marcucci and to all the participants of the 2015 Workshop in Econometrics and Empirical Economics (WEEE) held at the SADiBA, Perugia 27--28 August 2015, organised by the CIdE and the Bank of Italy.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLarge number of inflationary cosmology models is based on scalar fields, which \nplay an important role in the particle physics theories. The inflation is \nproduced by an homogeneous scalar field, dubbed  inflaton, which under suitable \nconditions may lead to an early-time accelerated expansion. Following the first \nproposal of Guth~\\cite{Guth} and Sato~\\cite{Sato}, in the last years many \ninflationary models based on scalar fields ( and inspired by modified gravity \ntheories, string theories, quantum effects in the hot universe, etc) have been \nproposed.\n\nTypically the magnitude of scalar field is very large at the beginning of the \ninflation and\nthen it rolls down towards a potential minimum where the inflation ends (see \nRef.~\\cite{chaotic} as an example of chaotic inflation). In other models the \nfield can fall in a potential hole where it starts to oscillate and the \nrehating processes take place~\\cite{buca1, buca2, buca3, buca4}.\nSome more complicated models are based on a phase transition between two scalar \nfields: they are the so-called hybrid or double inflation models~\\cite{ibrida1, \nibrida2}.\nFor the introduction to the dynamics of inflation see Ref.~\\cite{Linde} and \nRefs.~\\cite{revinflazione}--\\cite{rr2}.\n\nRecently, cosmological and astrophysical data~\\cite{cosmdata} seem to confirm \nthe predictions of Starobinsky inflationary model~\\cite{Staro}. Such a model is \nbased on the account of $R^2$-term as the correction in the Einstein equations. \nThis quadratic correction emerges in the Planck epoch and plays a fundamental \nrole in the high curvature limit, when the early-time acceleration takes place. \nSuch theory is conformally equivalent to a scalar-tensor theory in the Einstein \nframe, where the inflaton drives the expansion in a quasi-de Sitter space-time \nand slowly moves to the end of inflation, when the reheating \nprocesses~\\cite{r1,r2,r3} start.\nSuch  inflationary model has been recently revisited in many works. Among them, \nin Ref.~\\cite{genStar} a superconformal generalization of such a model in \nsuperconformal theory has been investigated, and in \nRefs.~\\cite{SS1}--\\cite{SS2} other applications based on the spontaneous \nbreaking of conformal invariance\nand on the scale-invariant extensions of Starobinsky model have been presented.\nIn Ref.~\\cite{cinesi}, a generalization of the Starobinsky model represented by \na polynomial correction of the Einstein gravity of the type $c_1 R^2+c_2 R^n$ \nhas been studied.\n\nIn this paper, we will concentrate on inflation caused by scalar curvature \ncorrections to Einstein gravity, namely,\nwe will consider the so-called $F(R)$-gravity, whose action is in the form of \n$F(R)=R+f(R)$, being $f(R)$ a function of the Ricci scalar (for recent reviews \non modified gravity,\nsee Refs.~\\cite{Review-Nojiri-Odintsov} --\\cite{SebRev}  and \nRef.\\cite{others}). This kind of corrections may occur due to quantum effects \nin the\nhot universe or maybe motivated by the ultraviolet completation of quantum \ntheory of gravity. Our aim is to investigate which kinds of viable inflation \ncan be realized in the contest of $F(R)$-gravity beyond the Starobinsky model, \nwhose dynamics is governed in the Einstein frame by a potential of the type \n$V(\\sigma)\\sim (1-\\exp[-\\sigma])^2$, being $\\sigma$ the inflaton.\n\nThe paper is organized in the following way. In Sections {\\bf 2}--{\\bf 3}, we \nwill revisit the conformal transformations which permit to pass from the Jordan \nframe to the Einstein one and we will recall the dynamics of the viable \ninflation. In Section {\\bf 4}, we will study inflation for the general class of \nscalar potentials of the type $V(\\sigma)\\sim\\exp[n\\sigma]$, $n$ being a general \nparameter,\nperforming the analysis in the Einstein frame and therefore reconstructing the \n$F(R)$-gravity theories which correspond to the given potentials. Viable \ninflation must be consistent with the last Planck data (spectral index, \ntensor-to-scalar ratio...) and must correspond to  Einstein theory corrections \nwhich emerge only at mass scales larger than the Planck one, namely at high \ncurvatures. We will see the conditions on $n$ for which slow-roll conditions \nare satisfied, and we will reconstruct the form of the $F(R)$-models during \nearly-time acceleration and their form at small curvaures. Some specific \nexamples are presented.\nIn Section {\\bf 5}, following the recent success of\nhigher derivative gravity, we will revisit and study in detail the specific \nclass of models $F(R)=R+(R+R_0)^n$. The analysis in the\nEinstein frame reveals that $n$ must be very close to two in order to realize a \nviable inflation for large and negative values of the scalar field,\nbut other possibilities are allowed\nby bounding the field in a different way.\nIn particular, when $n>2$, the de Sitter solution emerges, but in order to \nstudy the exit from inflation is necessary to analyze the theory in the Jordan \nframe, where perturbations make possible an early time acceleration with a \nsufficient amount of inflation.\nSome summary and outlook are given in Section {\\bf 6}.\nTechnical details and further considerations are presented in Appendixes.\n\n\n\\paragraph*{} We shall use units in which $c=\\hbar=k_{\\mathrm{B}}=1$,\n$c\\,,\\hbar\\,,k_{\\mathrm{B}}$  being respectively the speed of light, the\nPlanck and Boltzmann constants.  Moreover we shall denote by $G_N$\nthe gravitational constant and  by $M_{\\mathrm{Pl}} =G_{N}^{-1\/2} =1.2\n\\times 10^{19}$GeV the Planck mass.\nFinally, we shall set   $\\kappa^2\\equiv 8 \\pi G_{N}$.\n\n\n\n\\section{Conformal transformations}\n\nIn scalar-tensor theories of gravity, a scalar field coupled to the metric \nappears in the action.\nThe first scalar-tensor theory was proposed by Brans \\& Dicke in \n1961~\\cite{BransDicke} in the attempt to incorporate the Mach's principle into \nthe theory of gravity, but today the interest to such theories is related with \nthe possibility to reproduce the primordial acceleration of the inflationary \nuniverse.\n\nIn principle, a modified gravity theory can be rewritten in scalar-tensor or \nEinstein frame\nform.\nLet us start by considering the general action of $F(R)$-modified gravity\n\\begin{equation}\nI = \\int_\\mathcal{M} d^4 x \\sqrt{-g} \\left[ \\frac{F(R)}{2\\kappa^2}\\right]\\,,\n\\label{action}\n\\end{equation}\nwhere $F(R)$ is a function of the Ricci scalar $R$, $g$ is the determinant of \nthe metric tensor $g_{\\mu\\nu}$ and\n$\\mathcal{M}$ is the space-time manifold.\nNow we introduce the field $A$ into\n~(\\ref{action}),\\\\\n\\phantom{line}\n\\begin{equation}\nI_{JF}=\\frac{1}{2\\kappa^{2}}\\int_{\\mathcal{M}}\\sqrt{-g}\\left[\nF_A(A) \\, (R-A)+F(A)\\right] d^{4}x\\label{JordanFrame}\\,.\n\\end{equation}\n\\phantom{line}\\\\\nHere, `$JF$' means `Jordan frame', and $F_A(A)$ denotes the derivative of \n$F(A)$ with respect to $A$.  By making the variation\nwith respect to $A$, we immediatly obtain $A=R$, such that (\\ref{JordanFrame}) \nis equivalent to (\\ref{action}) .\nWe define the scalar field $\\sigma$ (which in fact encodes the new degree of \nfreedom in the theory, namely the scalaron or inflaton) as\n\\begin{equation}\n\\sigma := -\\sqrt{\\frac{3}{2\\kappa^2}}\\ln [F_A(A)]\\,.\\label{sigma}\n\\end{equation}\nBy considering the conformal transformation of the metric,\n\\begin{equation}\n\\tilde g_{\\mu\\nu}=\\mathrm{e}^{-\\sigma}g_{\\mu\\nu}\\label{conforme}\\,,\n\\end{equation}\nwe finally get the `Einstein frame' action\n\\phantom{line}\n\\begin{eqnarray}\nI_{EF} &=& \\int_{\\mathcal{M}} d^4 x \\sqrt{-\\tilde{g}} \\left\\{ \n\\frac{\\tilde{R}}{2\\kappa^2} -\n\\frac{1}{2}\\left(\\frac{F_{AA}(A)}{F_A(A)}\\right)^2\n\\tilde{g}^{\\mu\\nu}\\partial_\\mu A \\partial_\\nu A - \n\\frac{1}{2\\kappa^2}\\left[\\frac{A}{F_A(A)}\n- \\frac{F(A)}{F_A(A)^2}\\right]\\right\\} \\nonumber\\\\ \\nonumber\\\\\n&=&\\int_{\\mathcal{M}} d^4 x \\sqrt{-\\tilde{g}} \\left( \n\\frac{\\tilde{R}}{2\\kappa^2} -\n\\frac{1}{2}\\tilde{g}^{\\mu\\nu}\n\\partial_\\mu \\sigma \\partial_\\nu \\sigma - \nV(\\sigma)\\right)\\,,\\label{EinsteinFrame}\n\\end{eqnarray}\n\\phantom{line}\\\\\nwhere $\\tilde{R}$ denotes the Ricci scalar evaluated in the conformal metric \n$\\tilde{g}_{\\mu\\nu}$ and $\\tilde{g}$ is the determinant of the conformal \nmetric, namely $\\tilde{g}=\\mathrm{e}^{-4\\sigma}g$.\nFurthermore, one has\\\\\n\\phantom{line}\n\\begin{equation}\nV(\\sigma)\\equiv\\frac{A}{F'(A)} - \n\\frac{F(A)}{F'(A)^2}=\\frac{1}{2\\kappa^2}\\left\\{\\mathrm{e}^{\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\nR\\left(\\mathrm{e}^{-\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\\right)\n-\\mathrm{e}^{2\\left(\\sqrt{2\\kappa^2\/3}\\right) \n\\sigma}F\\left[R\\left(\\mathrm{e}^{-\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}\\right)\\right]\n\\right\\}\\,,\\label{V}\n\\end{equation}\n\\phantom{line}\\\\\nbeing $R(\\mathrm{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma})$ the solution of\nEq.~(\\ref{sigma}) with $A=R$, namely $R$ a function of \n$\\mathrm{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}$.\nIn what follows, we will omit the tilde to denote all the quantities evaluated \nin the Einstein frame.\n\nNote that string-inspired inflationary models also contain canonical \nand\/or tachyon scalar (for recent discussion see Refs.~\\cite{Samiinfl}--\\cite{Jamil:2013nca}). \n\n\n\\section{Dynamics of inflation}\n\nIn this Section, for the sake of completeness, we will recall the well-known \nfacts on inflation. The energy density and pressure of the inflaton $\\sigma$ \nare given by\n\\begin{equation}\n\\rho_\\sigma=\\frac{\\dot\\sigma^2}{2}+V(\\sigma)\\,,\\quad \np_\\sigma=\\frac{\\dot\\sigma^2}{2}-V(\\sigma)\\,,\n\\end{equation}\nwhere the dot is the derivative with respect to the cosmological time. The \nFriedmann equations in the presence of  $\\sigma$ read\n\\begin{equation}\n\\frac{3 \nH^2}{\\kappa^2}=\\frac{\\dot\\sigma^2}{2}+V(\\sigma)\\,,\\quad-\\frac{1}{\\kappa^2}\\left(2\\dot \nH+3 H^2\\right)=\\frac{\\dot\\sigma^2}{2}-V(\\sigma)\\,,\n\\end{equation}\nand the energy conservation law coincides with the equation of motion for \n$\\sigma$ and reads\n\\begin{equation}\n\\ddot\\sigma+3H\\dot\\sigma=-V'(\\sigma)\\,,\n\\end{equation}\nwhere the prime denotes the derivative of the potential with respect to \n$\\sigma$.\nFrom the Friedmann equations we obtain\n\\begin{equation}\n\\dot{H}=-\\frac{\\kappa^2}{2}\\left( \\rho_\\sigma+ p_\\sigma \n\\right)=-\\frac{\\kappa^2}{2} \\dot\\sigma^2\\,.\n\\end{equation}\nOn the other hand, the acceleration can be expressed as\n\\begin{equation}\n\\frac{\\ddot{a}}{a}= H^2+\\dot{H}=H^2\\left(1-\\epsilon \\right)\\,,\n\\end{equation}\nwhere we have introduced the ``slow roll'' parameter\n\\begin{equation}\n\\epsilon=-\\frac{\\dot{H}}{H^2}\\,.\n\\end{equation}\nThis parameter may be expressed as a function of the inflaton as\n\\begin{equation}\n\\epsilon=\\frac{\\kappa^2 \\dot\\sigma^2}{2 H^2} \\,.\n\\end{equation}\nWe also have\n\\begin{equation}\n\\frac{\\ddot{a}}{a}=\\frac{\\kappa^2}{3}\\left(V-\\dot\\sigma^2 \\right)\\,.\n\\end{equation}\nThus, the  condition to have an acceleration is $\\epsilon <1$ or $\\dot\\sigma^2 \n< V(\\sigma)$. There is another slow roll parameter defined by\n\\begin{equation}\n\\eta=-\\frac{\\ddot{H}}{2H\\dot{H}}=\\epsilon-\\frac{1}{2 \\epsilon H}\\dot\\epsilon \n\\,.\n\\end{equation}\nAs a function of the inflaton one has\n\\begin{equation}\n\\eta=-\\frac{\\ddot\\sigma}{H\\dot\\sigma} \\,.\n\\end{equation}\nFor the inflation to occur and persist for a convenient amount of time, a quasi \nde Sitter space is required, namely $\\dot H$ has to be very small, and, as a \nresult,  also the two slow roll parameters have to be very  small,  and one has\n\\begin{equation}\n\\dot\\sigma^2\\ll V(\\sigma)\\,,\n\\end{equation}\nnamely the kinetic energy of the field has to be small during the inflation. As \na result, the Friedmann equations  reduce to\n\\begin{equation}\n\\frac{3H^2}{\\kappa^2}\\simeq V(\\sigma)\\,,\n\\quad\n3H\\dot\\sigma\\simeq -V'(\\sigma)\\,.\\label{EOMs}\n\\end{equation}\nIt is easy to show that within this slow roll regime,\n  the slow roll parameters may be expressed as function of the inflaton \npotential as\n\\begin{equation}\n\\epsilon=\\frac{1}{2\\kappa^2}\\left(\\frac{V'(\\sigma)}{V(\\sigma)}\\right)^2\\,,\\quad \n\\eta=\\frac{1}{\\kappa^2}\\left(\\frac{V''(\\sigma)}{V(\\sigma)}\\right)\\,.\\label{slow}\n\\end{equation}\nInflation ends when $\\epsilon\\,,|\\eta|\\sim 1$. A useful quantity which \ndescribes the amount of inflation is e-foldings number $N$ defined by\n\\begin{equation}\nN\\equiv\\ln \\frac{a_f}{a_i}=\\int_{t_i}^{t_e} H dt\\simeq \n\\kappa^2\\int^{\\sigma_i}_{\\sigma_e}\\frac{V(\\sigma)}{V'(\\sigma)}d\\sigma\\,,\\label{N}\n\\end{equation}\nwhere the indices $i,f$ are referred to the quantities at the beginning and the \nend of inflation, respectively. The required e-foldings number for inflation is \nat least $N\\simeq 60$.\nThe amplitude of the primordial scalar power spectrum is\n\\begin{equation}\n\\Delta_{\\mathcal R}^2=\\frac{\\kappa^4 V}{24\\pi^2\\epsilon}\\,,\\label{spectrum}\n\\end{equation}\nand for slow roll inflation the spectral index $n_s$ and the tensor-to-scalar \nratio are given by\n\\begin{equation}\nn_s=1-6\\epsilon+2\\eta\\,,\\quad r=16\\epsilon\\,.\\label{indexes}\n\\end{equation}\nThe last Planck data constrain these quantities as\n\\begin{equation}\nn_s=0.9603\\pm0.0073\\,,\\quad r<0.11\\,.\\label{data}\n\\end{equation}\n\n\n\\section{Reconstruction of $F(R)$ theory from the scalar potential and analysis \nof the inflation in the Einstein frame}\n\nIn this Section, we\nwill study some classes of scalar potential which produce inflation in the \nEinstein frame. The aim is to generalize the Starobinsky model\nby considering different behaviour of the scalar potential as \n$V(\\sigma)\\sim\\exp[n\\sqrt{2\\kappa^2\/3}\\sigma]$, where $n$ is the parameter on \nwhich it depends the dynamics of the inflation (slow roll parameters, spectral \nindices...). This analysis is motivated by the possibility to reconstruct \nsuitable $F(R)$ corrections to General Relativity in the corresponding Jordan \nframe. By `suitable' we mean corrections that vanish at mass scales smaller \nthan the Planck mass $M_{\\text{Pl}}$ and give rise to corrections only in the high \ncurvature limits, namely during the inflationary period.\nEvery scalar potential will be confronted with cosmological data.\n\nIn order to reconstruct the $F(R)$-gravity which corresponds to a given \npotential, we may start from Eq.~(\\ref{V}). By dividing such equation to \n$\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\right]$, and then by taking the derivative with respect to \n$R$, we get\n\\begin{equation}\nR \nF_R(R)=-2\\kappa^2\\sqrt{\\frac{3}{2\\kappa^2}}\\frac{d}{d\\sigma}\\left(\\frac{V(\\sigma)}{e^{2\\left(\\sqrt{2\\kappa^2\/3}\\right)\\sigma}}\\right)\\,.\\label{start}\n\\end{equation}\nAs a result, giving the explicit form of the potential $V(\\sigma),$ thanks to \nthe relation (\\ref{sigma}), we obtain an equation for $F_R(R)$, and therefore \nthe $F(R)$-gravity model in the Jordan frame.  In this process, one introduces \nthe integration constant, which has to be fixed by\nrequiring that  Eq.~(\\ref{V}) holds true.\n\n\\subsection{$V(\\sigma)\\sim c_0+c_1\\exp[\\sigma]+c_2\\exp[2\\sigma]$: $R+ \nR^2+\\Lambda $-models\\label{4.1}}\n\nLet us start with the following inflationary potential\n\\begin{equation}\nV(\\sigma)=\\left[c_0+c_1\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}+c_2\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}\n\\right]\\,.\\label{cucu}\n\\end{equation}\nThis is the simplest example and is the minimal generalization of the \nStarobinsky model.\nEquation (\\ref{start}) gives\n\\begin{equation}\n2c_0F_R^2+c_1F_R-RF_R=0\\,.\\label{req}\n\\end{equation}\nAssuming $F_R \\neq 0$, one gets\n\\begin{equation}\nF_R(R)=-\\frac{c_1}{2c_0}+\\frac{R}{2c_0}\\,.\\label{FF}\n\\end{equation}\nThus, the corresponding Lagrangian of $F(R)$ gravity is given by\n\\begin{equation}\nF(R)=-\\frac{c_1}{2c_0}R+\\frac{R^2}{4c_0}+\\Lambda\\,.\\label{Staro2}\n\\end{equation}\nHere, $\\Lambda$ is a constant of integration, which can be  determined  by \nequation (\\ref{V}). The result is\n\\begin{equation}\n\\Lambda=\\frac{c_1^2}{4c_0}-c_2\\,.\\label{L}\n\\end{equation}\nAn important remark is in order. In order to have the correct Einstein-Hilbert \nterm, we must put $-c_1\/(2c_0)=1$. Thus, we have the class \nof modified quadratic models depending on two constants\n\\begin{equation}\nF(R)=R+\\frac{R^2}{4c_0}+c_0-c_2\\,.\\label{Staro3}\n\\end{equation}\nFurthermore, in the specific case  $c_2=0$, one has\n  \\begin{equation}\nF(R)=R+\\frac{R^2}{4c_0}+ c_0\\,.\\label{cogno}\n\\end{equation}\nThis is an interesting model, and in the Appendix B, we will study its static \nsperically symmetric solutions.\n\nThe other interesting case is the vanishing  of cosmological constant, namely\n\\begin{equation}\nc_2=c_0.\\label{L1}\n\\end{equation}\nSince we are assuming $c_1\/(2c_0)=1$, it also follows \n$c_2=-c_1\/2$. As a consequence,   we recover the\nStarobinsky model $F(R)=R+\\frac{R^2}{4c_0}$, and related Einstein frame \npotential\n\\begin{equation}\nV(\\sigma)=c_0 \\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\n\\right)^2\\,.\\label{VStaro}\n\\end{equation}\nThe model  (\\ref{Staro3}) is the extension of the Starobinsky model to the case \nwith cosmological constant and gives a viable inflation.\nIn what follows, we denote\n\\begin{equation}\nc_0=\\frac{\\gamma}{4\\kappa^2}\\,.\n\\end{equation}\nThe initial value of the inflaton is large (and negative) and it rolls down \ntoward the potential minimum at $\\sigma\\rightarrow 0^-$, \n$V(0)=-\\gamma\/(4\\kappa^2)<0$. During inflation ($\\sigma\\rightarrow-\\infty$) \nequations (\\ref{EOMs}) lead to\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{12}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{\\gamma}{\\sqrt{6\\kappa^2}}\\right)\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\\label{copy}\n\\end{equation}\nIt means, that a quasi de Sitter solution can be realized.\nWe must require\n\\begin{equation}\n\\gamma\\propto M^2\\,,\\quad M\\ll M_P\\,,\\label{gammacond}\n\\end{equation}\nwhere $M_P$ is the Planck mass. As a consequence, the corrections of Eintein\ngravity emerge at high curvature and one gets the accelerated expansion with \n$H\\propto\\sqrt{\\gamma}$.\nThe scalar field behaves as\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{3}{2\\kappa^2}}\\ln\\left[\\frac{1}{3}\\sqrt{\\frac{2\\gamma}{3}}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. If at this time \n$|\\sigma|$ is very large, the slow roll paramters (\\ref{slow}),\n\\begin{equation}\n\\epsilon=\\frac{4}{3}\\frac{1}{(2-\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma})^2}\\simeq \n0\\,,\\quad|\\eta|=\\frac{4}{3}\\frac{1}{|2-\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}|}\\simeq \n0\\,,\\label{star}\n\\end{equation}\nare very small and the field moves slowly.\nThe inflation ends when such parameters are of the order of unit, namely at\n$\\sigma_e\\simeq-0.17\\sqrt{3\/(2\\kappa^2)}$.\nThe e-foldings number can be evaluated from (\\ref{N}) and reads \n($\\sigma_i\\gg\\sigma_e$),\n\\begin{equation}\nN\\simeq\\frac{3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}}{4}\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\\frac{1}{4}\\sqrt{\\frac{2\\gamma}{3}}t_0\\,.\n\\end{equation}\nThe inflation ends at $t_e=t_0-3\\sqrt{3\/(2\\alpha)}\\exp[0.17]$. For example, in \norder to obtain $N=60$, we must require \n$\\sigma_i\\simeq-\\sqrt{3\/(2\\kappa^2)}4.38\\simeq 1.07 M_{pl}$. Moreover, we may \nexpress the e-foldings numbers as\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{4N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nThe amplitude of primordial power spectrum (\\ref{spectrum}) is\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma \nN^2}{72\\pi^2}\\propto\\frac{\\kappa^2 M^2 N^2}{72\\pi^2}\\ll \\frac{\\kappa^2M_P^2 \nN^2}{72\\pi^2}\\,,\n\\end{equation}\nand the indexes (\\ref{indexes}) result to be\n\\begin{equation}\nn_s\\simeq1-\\frac{2}{N}\\,,\\quad r\\simeq\\frac{12}{N^2}\\,.\n\\end{equation}\nSince  we have $n_s>1-\\sqrt{0.11\/3}\\simeq 0.809$ when $r<0.11\\,,n_s<1$, we see \nthat these indices are compatible with (\\ref{data}). For example, for $N=60$, \none has $n_s=0.967$ and $r=0.003$.\nWe stress that this behaviour is the one of Starobinsky model, with a different \nminimum of the potential and a cosmological constant in the Jordan frame. The \nappeareance of a cosmological constant at large curvature needs some \nexplanation. It can be originated by some quantum effects or may be supported \nby a modified gravity term which makes it to vanish at small curvatures (see \nfor examples the so called `two step-models' in \nRefs.~\\cite{twostep1}--\\cite{twostep2}). However, if the cosmological constant \nis set equal to zero (or, if necessary, is set equal to an other value), the feature of the model in Einstein frame does not \nchange during the inflation: we will see that this double possibility, namely, \ntaking a cosmological constant in the Jordan frame, or taking an additional \nterm $\\propto\\exp[2\\sqrt{2\\kappa^2\/3}\\sigma]$ in the Einstein frame, does not \nmodify the proprieties of the scalar potentials during the early time \nacceleration.\n\n\n\\subsection{$V(\\sigma)\\sim\\gamma\\exp[-n\\sigma]\\,,n>0$: $c_0 \nR^{\\frac{n+2}{n+1}}$-models\\label{case}}\n\nAs a second example, we consider the following  potential\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}\\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\right)+\\frac{\\gamma}{\\kappa^2}\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\n\\end{equation}\nbeing $\\gamma\\,,n>0$ constants. This potential possesses a minimum in which the \nscalar field may fall at the end of inflation. Since for large and negative \nvalues of the scalar field the potential is not flat, we do not expect a de \nSitter universe, but if the slow roll conditions are\nsatisfied, we can obtain an acceleration with a sufficient amount of inflation.\n\nBy using our reconstruction,\nfrom (\\ref{start}) one derives\n\\begin{equation}\nF_R(R)-\\frac{1}{2}+\\frac{\\gamma}{2\\alpha}(2+n) \nF_R(R)^{1+n}=\\frac{R}{4\\alpha}\\,.\\label{pippo}\n\\end{equation}\nIn principle, this equation admitts many solutions. At the perturbative level, \nit is easy to find that, by putting\n\\begin{equation}\n\\alpha=-\\gamma(n+2)\\,,\\label{alphacond}\n\\end{equation}\nwe obtain\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq 1+c_1 R+c_2 R^2+c_3 R^3+...\\,,\\label{funzioncina}\n\\end{equation}\nwhen $|c_1 R|\\,,|c_2 R^2|\\,, |c_3 R^3|...\\ll 1$  (see Appendix A). It means \nthat, since \n$c_1\\propto\\gamma^{-1}\\,,c_2\\propto\\gamma^{-2}\\,,c_3\\propto\\gamma^{-3}...$,\nif $\\gamma$ satisfies (\\ref{gammacond}), we recover the Einstein's gravity when \n$R\\ll\\gamma$ and the theory is an high curvature correction to General \nRelativity. Moreover, when $R\\gg\\gamma$, the asymptotic solution of \nEq.~(\\ref{pippo}) with (\\ref{alphacond}) is given by\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq\\left(\\frac{1}{4(n+2)}\\right)^\\frac{1}{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{1}{n+1}}\\,,\\quad\nF(R\\gg\\gamma)\\simeq\\gamma\\left(\\frac{n+1}{n+2}\\right)\\left(\\frac{1}{4(n+2)}\\right)^\\frac{1}{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{n+2}{n+1}}\\,.\\label{expression}\n\\end{equation}\nHere, one important comment is required. In the Einstein frame, \n$R_{EF}\\sim\\gamma$ during inflation, but the corresponding curvature in the \nJordan frame is $R_{JF}\\simeq \\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}R_{EF}$ (we \nmay neglect the kinetic energy of scalar field in the slow roll approximation), \nsuch that $R_{JF}\\gg\\gamma$ when $\\sigma\\rightarrow-\\infty$ and expression \n(\\ref{expression}), which is evaluated in the Jordan frame, effectively is \nvalid for inflation\n(for $n\\rightarrow 0$ we recover $F(R)\\sim R^2$ in the Jordan frame).\n\nThe potential finally reads\n\\begin{equation}\nV(\\sigma)=-\\frac{\\gamma(n+2)}{\\kappa^2}\\left(1-\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\right)+\\frac{\\gamma}{\\kappa^2}\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\n\\end{equation}\nThis potential has a minimum ($V'(\\sigma_{min})=0$) at \n$\\sigma_{min}=-\\sqrt{3\/(2\\kappa^2)}\\log[(n+2)\/n]\/(n+1)$, and one gets\n\\begin{equation}\nV(\\sigma_{min})=\\frac{\\gamma}{\\kappa^2}\\left(n^{\\frac{1}{n+1}}(n+2)^{\\frac{n}{n+1}}+\\left(\\frac{n+2}{n}\\right)^{\\frac{1}{n+1}}-(n+2)\\right)>0\\,,\\quad\\gamma\\,,n>0\\,.\n\\end{equation}\nWhen $\\sigma\\rightarrow-\\infty$ (large curvature), the potential goes to \ninfinity, and when $\\sigma\\rightarrow 0^-$, $V(\\sigma)=\\gamma\/\\kappa^2$.\nSince the slow roll parameters are given by\n\\begin{eqnarray}\n\\epsilon&=&\\frac{\\left(n-(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}\\right)^2}{3\\left(\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}-(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}-1\n\\right)^2}\\,,\\nonumber\\\\ \\nonumber\\\\\n|\\eta|&=&\\frac{2}{3}\\frac{n^2+(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}}{|1+(n+2)\\text{e}^{(n+1)\\sqrt{2\\kappa^2\/3}\\sigma}-\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\n|}\\,,\n\\end{eqnarray}\none has $\\epsilon(\\sigma\\rightarrow-\\infty)\\simeq n^2\/3$ and\n$|\\eta(\\sigma\\rightarrow-\\infty)|\\simeq2n^2\/3$, which implies $0<n\\ll 1$.\nThe EOMs (\\ref{EOMs})  in the slow roll limit read\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{3}\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\quad\n3H\\dot\\sigma\\simeq\\sqrt{\\frac{2}{3\\kappa^2}}\\gamma\\,n\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\n\\end{equation}\nThe solution for the scalar field is\n\\begin{equation}\n\\sigma=\\frac{2}{n}\\sqrt{\\frac{3}{2\\kappa^2}}\\ln\\left[\\frac{n^2}{3}\\sqrt{\\frac{\\gamma}{3}}(t_0+t)\\right]\\,,\n\\label{SC}\n\\end{equation}\nwhere $t_0$ is bounded to be very small at the beginning of the inflation, such \nthat the field is negative and its magnitude very large. The solution for the \nHubble parameter finally reads\n\\begin{equation}\nH=\\frac{3}{n^2(t_0+t)}\\,,\\quad\\frac{\\ddot a}{a}=H^2+\\dot \nH=\\frac{3}{n^2(t_0+t)^2}\\left(\\frac{3}{n^2}-1\\right)>0\\,,\\quad (n<\\sqrt{3})\n\\end{equation}\nand we have an acceleration as soon as $\\epsilon<1$. Despite to the fact that \nin this kind of models the acceleration is smaller than in the de Sitter \nuniverse, the slow roll paramters can be small enough\nto justify our slow roll approximations. A direct evaluation of the ratio of \nkinetic energy  of the field and potential leads to \n$\\left(\\dot\\sigma^2\/2\\right)\/V(\\sigma)=n^2\/9$, which is much smaller than one \nwhen $n\\ll 1$. The inflation ends when the slow roll paramters are on the order \nof unit, before the minimum of the potential. Note that, by definition, since \n$V'(\\sigma_{min})=0$, one has $\\epsilon(\\sigma_{min})=0$, which corresponds to \nthe minimum of the slow roll paramter. However, before to this point, since the \nslow roll parameters behave as\n\\begin{equation}\n\\epsilon\\simeq\\frac{n^2}{3\\left(\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}-1\n\\right)^2}\\,,\\quad\n|\\eta|\\simeq\\frac{2}{3}\\frac{n^2}{|1-\n(n+2)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\n|}\\,,\n\\end{equation}\nwe find that $\\epsilon\\,,|\\eta|\\simeq 1$ when \n$\\sigma\\simeq-\\sqrt{3\/(2\\kappa^2)}\\log[(6+3n)\/(3-n\\sqrt{3})]\/n$.\nFinally, from Eq.~(\\ref{N}), we get the $N$-foldings number of inflation,\n\\begin{equation}\nN\\simeq -\\frac{1}{n}\\sqrt{\\frac{3\\kappa^2}{2}}\\sigma\n\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq-\\frac{3}{n^2}\\ln\\left[\\frac{n^2}{3}\\sqrt{\\frac{\\gamma}{3}}t_0\\right]\\,.\n\\end{equation}\nWhen the inflation ends, the field falls in the minimum of the potential and \nstarts to oscillate.\nThe reheating process takes place. The amplitude of primordial power spectrum \n(\\ref{spectrum}) and the spectral indexes (\\ref{indexes}) can be written as\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma \n\\text{e}^{\\frac{2}{3}N\\,n^2}}{8\\pi^2n^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{2n^2}{3}\\,,\\quad r\\simeq\\frac{16n^2}{3}\\,.\n\\end{equation}\nThe corrections to $n_s$ and $r$ are on the order of $\\exp\\left[-2n^2 \nN\/3\\right]\\ll 1$.\nThese indexes are compatible with (\\ref{data}) when\n\\begin{equation}\nn\\sim\\frac{1}{10}\\,,\\frac{2}{10}\\,.\n\\end{equation}\nThese are the typical values of $n$ which make the scalar potential (\\ref{V3}) \nable to reproduce a viable inflationary scenario. The result suggests that only \nthe models\nclose to $R^2$-gravity are able to produce this kind of inflation.\n\nIn general, $F_R(R)$ in Eq.~(\\ref{funzioncina}) may lead to a cosmological \nconstant proportional to $\\gamma$ in the $F(R)$-model. However, we can set it \nequal to zero, adding a suitable term in\nthe potential ($\\ref{V3}$) proportional to\n$\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, which\nchanges much slowler than\n$\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ when $0<n$, and the dynamics of \nthe inflation is the same of above.\n\n\n\\subsection{$V(\\sigma)\\sim3\\gamma\/4-\\gamma\\exp[\\sigma\/2]$: $R\/2+c_1 R^2+c_2 \n(R+R_0)^{3\/2}$-models}\n\nWe continue our analysis constructing potentials for de Sitter universe during \ninflation,\nbut with a different behaviour with respect to the Starobinsky one (which \ndecreases as $\\exp[-\\sigma]$). We propose the potential as\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}\\,,\\label{V2}\n\\end{equation}\nbeing $\\alpha,\\gamma>0$, as usually, constants. It follows\n\\footnote{Here, we exclude the solution with the minus sign in front of the \nsquare root which leads to an imaginary value of $\\sqrt{F_R}$ when $R$ is \nreal.}\n\\begin{equation}\nF_R(R)=\\frac{9\\gamma^2+8R\\alpha+3\\sqrt{16 \nR\\alpha\\gamma^2+9\\gamma^4}}{32\\alpha^2}\\,.\n\\end{equation}\nSince we must require $\\alpha\\,,\\gamma\\gg 1$ and at small curvature we want to \nrecover the Einstein gravity ($F_R=1$), we set\n$\\alpha=3\\gamma\/4$, $\\gamma>0$, and we get\n\\begin{equation}\nF_R(R)=\\frac{1}{2}+\\frac{1}{3\\gamma}R+\\frac{\\sqrt{3}}{6}\\sqrt{4R\/\\gamma+3}\\,.\\label{FFFF}\n\\end{equation}\nThus, from Eq.~(\\ref{V}) we obtain\n\\begin{equation}\nF(R)=\\frac{R}{2}+\\frac{R^2}{6\\gamma}+\\frac{\\sqrt{3}}{36}\\left(4R\/\\gamma+3\\right)^{3\/2}+\\frac{\\gamma}{4}\\,.\\label{zippozap}\n\\end{equation}\nHere, we stress that the conformal transformation gives for the Ricci scalar\n\\begin{equation}\nR=3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}\\left(1+\n\\text{e}^{\\sqrt{2\\kappa^2\/3\/2}\\sigma\/2}\\right)\\,,\\,\n3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma}\\left(1-\n\\text{e}^{\\sqrt{2\\kappa^2\/3\/2}\\sigma\/2}\\right)\\,,\n\\end{equation}\nbut only the second one leads to our potential (namely, is the one wich emerges \nfrom our reconstruction). For $R\\ll\\gamma$, the model reads $F(R\\ll\\gamma)\\simeq R+\\gamma\/2$. If we want to recover the General Relativity action $F(R\\ll\\gamma)\\simeq R$, we must set the cosmological constant, namely the last term in (\\ref{zippozap}), equal to $-\\gamma\/4$: in this case\nthe scalar potential is\n\\begin{equation}\nV(\\sigma)=\\frac{3\\gamma}{4\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}\n-\\gamma\\frac{\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}}{4\\kappa^2}\\,.\\label{V2bis}\n\\end{equation}\nLet us analyze the possibility to reproduce inflation from the potential \n(\\ref{V2}), which finally reads\n\\begin{equation}\nV(\\sigma)=\\left(\\frac{3}{4\\kappa^2}-\\frac{\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}}\n{\\kappa^2}\\right)\\gamma\\,.\n\\end{equation}\nThe initial value of the inflaton is large (and negative) and it rolls down \ntoward the potential minimum at $\\sigma\\rightarrow 0^-$, \n$V(0)=-\\gamma\/(4\\kappa^2)<0$. When $\\sigma\\rightarrow-\\infty$ the EOMs in the \nslow roll limit read\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{4}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{\\gamma}{\\sqrt{6\\kappa^2}}\\right)\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma\/2}\\,.\n\\end{equation}\nIt means, that $\\gamma$ must satisfy condition (\\ref{gammacond}), such that the \ninflation takes place at the Plank epoch.\nThe de Sitter expansion can be realized and the field behaves as\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{6}{\\kappa^2}}\\ln\\left[\\frac{\\sqrt{\\gamma}}{9}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. If at this time the \nmagnitude of $\\sigma$ is very large, the slow roll parameters (\\ref{slow})\n\\begin{equation}\n\\epsilon=\\frac{4}{3}\\frac{1}{\\left(4-3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}\\right)^2}\\simeq\n0\\,,\\quad|\\eta|=\\frac{2}{3}\\frac{1}{|4-3\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}|}\\simeq \n0\\,,\n\\end{equation}\nare small and the field moves slowly.\nThe inflation ends at\n$\\sigma_e\\simeq-0.12\\sqrt{3\/(2\\kappa^2)}$, when the slow roll parameters are of \nthe order of unit.\nThe e-foldings number can be evaluated from (\\ref{N}) and read,\n\\begin{equation}\nN\\simeq\\frac{9\\text{e}^{-\\sqrt{2\\kappa^2\/3}\\sigma\/2}}{2}\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\\frac{\\sqrt{\\gamma}}{2}t_0\\,.\n\\end{equation}\nAs a consequence, the slow roll parameters can be written as\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nThe amplitude of primordial power spectrum (\\ref{spectrum}) and the spectral \nindexes (\\ref{indexes}) are given by\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma N^2}{96\\pi^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{1}{N}\\,,\\quad r\\simeq\\frac{48}{N^2}\\,.\n\\end{equation}\nSince from these formulas we have $n_s>1-\\sqrt{0.11\/48}\\simeq 0.9521$ when \n$r<0.11\\,,n_s<1$, we see that these expressions are compatible with \n(\\ref{data}). For example, for $N=60$, one has $n_s=0.967$ and $r=0.013$.\n\nWe finish this Subsection with some considerations on the potential \n($\\ref{V2bis}$), which corresponds to the model with cosmological constant equal to $-\\gamma\/4$. \nSince the term $\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ changes slower than\n$\\exp\\left[\\sqrt{2\\kappa^2\/3}\\sigma\/2\\right]$, the dynamics of inflation is the \nsame of above. In this case, when the inflaton exits from the slow roll region, \nit falls in the minimum of the potential located at $V(0)=-\\gamma\/(8\\kappa^2)$.\n\n\n\\subsection{$V(\\sigma)\\sim\\gamma(2-n)\/2-\\gamma\\exp[n\\sigma]\\,,0<n<1$: $c_1 \nR^2+c_2 R^{2-n}$-models}\n\nNow, we would like to investigate some general features of the\ninflationary potential\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\label{V3}\n\\end{equation}\nwhere $0<\\alpha\\,,\\gamma$ and $0<n<2$. The above potential is explicitly \nconstructed to give the de Sitter solution in the slow roll limit when \n$\\sigma\\rightarrow-\\infty$. Equation (\\ref{start}) leads to\n\\begin{equation}\nF_R(R)+\\frac{\\gamma}{2\\alpha}F_R(R)^{1-n}(n-2)=\\frac{R}{4\\alpha}\\,.\\label{pippo2}\n\\end{equation}\nAt the perturbative level, it is easy to see that, by choosing\n\\begin{equation}\n\\alpha=\\frac{\\gamma(2-n)}{2}>0\\,,\\label{alphaalpha}\n\\end{equation}\nif $\\gamma$ satisfies (\\ref{gammacond}),\nat small curvature one gets\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq 1+c_1 R+c_2 R^2+c_3 R^3+...\\,,\\label{espansione}\n\\end{equation}\nwith \n$c_1\\propto\\gamma^{-1}\\,,c_2\\propto\\gamma^{-2}\\,,c_3\\propto\\gamma^{-3}...$, \nsuch that  our theory is the high curvature correction to General Relativity \n(see Appendix A).\nFor example, in the previous Subsection we have seen an exact solution for the \ncase $n=1\/2$. Since $1\\ll \\gamma$, when $R\/\\gamma\\ll 1$ we can expand $F_R(R)$ \nin (\\ref{FFFF}) as\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq \n1+\\frac{2}{3\\gamma}R-\\frac{R^2}{9\\gamma^2}+...\\,,\\label{primeF}\n\\end{equation}\nwhich returns to be (\\ref{espansione}) by using the coefficients in Appendix A.\nOn the other side, when $R\\gg\\gamma$, the asymptotic solution of \nEq.~(\\ref{pippo2}) with (\\ref{alphaalpha}) is given by\n\\begin{eqnarray}\nF_R(R\\gg\\gamma)&\\simeq&\\left(\\frac{R}{2\\gamma(2-n)}\\right)+\\left(\\frac{R}{2\\gamma(2-n)}\\right)^{1-n}\\,,\\nonumber\\\\\\nonumber\\\\\nF(R\\gg\\gamma)&\\simeq&\\frac{1}{2}\\left(\\frac{R^2}{2\\gamma(2-n)}\\right)+\\frac{1}{2-n}\\left(\\frac{1}{2\\gamma(2-n)}\\right)^{1-n}R^{2-n}\\,.\n\\label{megastana}\n\\end{eqnarray}\nAlso in this case, by taking the exact solution of the prevoius Section in the \nhigh curvature limit,\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq\\frac{R}{3\\gamma}+\\sqrt{\\frac{R}{3\\gamma}}\\,,\n\\end{equation}\nwe can verify the consistence of expression (\\ref{megastana}) for $n=1\/2$.\n\nAs an other example, let us consider the case $n=1\/3$. The reconstruction leads \nto\\\\\n\\phantom{line}\\\\\n\\begin{equation}\nF_R(R)=\\frac{1}{3}+\\frac{3}{10}\\left(\\frac{R}{\\gamma}\\right)\n+\\frac{2}{3\\times \n5^{1\/3}}\\left[9\\left(\\frac{R}{\\gamma}\\right)+5\\right]\\frac{1}{\\Delta^{1\/3}}+\\frac{1}{6\\times\n5^{2\/3}}\\Delta^{1\/3}\\,,\n\\end{equation}\n\\phantom{line}\\\\\nwhere\n\\phantom{line}\n\\begin{equation}\n\\Delta=200+243\\left(\\frac{R}{\\gamma}\\right)^2+540\\left(\\frac{R}{\\gamma}\\right)+27\\sqrt{81\\left(\\frac{R}{\\gamma}\\right)^4+40\\left(\\frac{R}{\\gamma}\\right)^3}\\,.\n\\end{equation}\n\\phantom{line}\\\\\nAt small curvature, it is easy to find\n\\begin{equation}\nF_R(R\\ll\\gamma)\\simeq1+\\frac{9}{10}\\left(\\frac{R}{\\gamma}\\right)+...\\,,\n\\end{equation}\nand in the high curvature limit this model has the following structure,\n\\begin{equation}\nF_R(R\\gg\\gamma)\\simeq \n\\frac{3}{10}\\left(\\frac{R}{\\gamma}\\right)+\\left(\\frac{3}{10}\\right)^{\\frac{2}{3}}\\left(\\frac{R}{\\gamma}\\right)^{\\frac{2}{3}}\\,,\n\\end{equation}\nwhich corresponds to (\\ref{megastana}) with $n=1\/3$.\nFinally, the model (\\ref{FF}) is the limiting case of $n\\rightarrow 1$.\n\nLet us analyze this class of potentials. When the magnitude of the inflaton is \nlarge, the scalar potential (\\ref{V3}) with\n$\\alpha=\\gamma(2-n)\/2$,\n\\begin{equation}\nV(\\sigma)=\\frac{\\gamma(2-n)}{2\\kappa^2}-\\frac{\\gamma}{\\kappa^2}\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\label{n<2}\n\\end{equation}\nbehaves as $V(\\sigma)\\simeq\\gamma(2-n)\/(2\\kappa^2)$, and the EOMs in the slow \nroll limit read\n\\begin{equation}\nH^2\\simeq\\left(\\frac{\\gamma(2-n)}{6}\\right)\\,,\n\\quad\n3H\\dot\\sigma\\simeq \n\\left(\\frac{n\\gamma\\sqrt{2}}{\\sqrt{3\\kappa^2}}\\right)\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\,.\n\\end{equation}\nAs a consequence, the field results to be\n\\begin{equation}\n\\sigma\\simeq \n-\\sqrt{\\frac{3}{2\\kappa^2}}\\frac{1}{n}\\ln\\left[\\frac{2\\sqrt{2}}{3\\sqrt{3}}\\frac{n^2\\sqrt{\\gamma}}{\\sqrt{2-n}}(t_0-t)\\right]\\,,\n\\end{equation}\nwhere $t_0$ is bounded at the beginning of the inflation. When \n$\\sigma\\rightarrow-\\infty$, the slow roll parameters became\n\\begin{equation}\n\\epsilon=\\frac{4n^2}{3}\\frac{1}{\\left(2+(n-2)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}\\right)^2}\\simeq\n0\\,,\n\\quad\n|\\eta|=\\frac{4n^2}{3}\\frac{1}{|2+(n-2)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}|}\\simeq \n0\\,,\n\\end{equation}\nand are very small.\nThe inflation ends at\n$\\sigma_e\\simeq-(1\/n)\\sqrt{3\/(2\\kappa^2)}\\log\\left[(\\gamma\/\\alpha)(1-n\/\\sqrt{3})\\right]$,\n\nsuch that the slow roll parameters are of the order of unit and the field \nreaches the minimum of the potential at $V(0)= -\\gamma\/(4\\kappa^2)$.\nThe e-foldings number (\\ref{N}) is given by\n\\begin{equation}\nN\\simeq\\frac{3(2-n)\\text{e}^{-n\\sqrt{2\\kappa^2\/3}\\sigma}}{4n^2}\\Big\n\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\n\\sqrt{\\frac{2(2-n)\\gamma}{6}}t_0\\,,\n\\end{equation}\nand\n\\begin{equation}\n\\epsilon\\simeq\\frac{3}{4n^2N^2}\\,,\\quad|\\eta|\\simeq\\frac{1}{N}\\,.\n\\end{equation}\nAs a consequence, the amplitude of primordial power spectrum and the spectral \nindexes read\n\\begin{equation}\n\\Delta_{\\mathcal R}^2\\simeq\\frac{\\kappa^2\\gamma(2-n)n^2 N^2}{36\\pi^2}\\,,\n\\quad\nn_s\\simeq1-\\frac{1}{N}\\,,\\quad r\\simeq\\frac{12}{n^2 N^2}\\,.\n\\end{equation}\nSince $n_s>1-(\\sqrt{0.11\/12})n\\simeq 1-(0.0957)n$ when $r<0.11\\,,n_s<1$, one \nhas that\n\\begin{equation}\n0.3386<n<1\\,,\n\\end{equation}\nin order to make the spectral indexes compatible with (\\ref{indexes}). It \nmeans, that the potential (\\ref{n<2}) can reproduce a viable inflation with at \nleast $n=1\/3$, namely we are considering models in the form $F(R\\gg \nM_{Pl})\\simeq c_1 R^2+c_2 R^\\zeta$, with $1<\\zeta<5\/3$.\n\nAlso in this case, $F_R(R)$ in Eq.~(\\ref{primeF}) may lead to a cosmological \nconstant proportional to $\\gamma$ in the $F(R)$-model. However, we can set it \nequal to zero, aquiring an additional term in\nthe potential ($\\ref{V3}$) proportional to\n$\\exp\\left[2\\sqrt{2\\kappa^2\/3}\\sigma\\right]$.\nThis term\nchanges slower than\n$\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ when $0<n<2$, and the dynamics of \nthe inflation is the same of above.\\\\\n\\\\\nTo conclude this Section, we add some comments about the scalar potentials \ncontaining exponential terms like $\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$ \nwith $n>1$. Since in this case Eq.~(\\ref{alphaalpha}) could lead to $\\alpha<0$ \n(when $n>2$) making the potential unable to reproduce inflation, we may \ngeneralize the potential to the following form\n\\begin{equation}\nV(\\sigma)=\\frac{\\alpha}{\\kappa^2}\\left(1-\n\\text{e}^{n\\sqrt{2\\kappa^2\/3}\\sigma}\\right)\n-\\frac{\\gamma}{\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\\label{V4}\n\\end{equation}\nwhere $\\alpha\\,,\\gamma>0$ and $n>1$.\nNow, in order to recover the Einstein gravity at small curvature, we have to \nput\n\\begin{equation}\n\\alpha=\\frac{\\gamma}{2(n-1)}>0\\,.\n\\end{equation}\nIn the slow roll limit ($\\sigma\\rightarrow-\\infty$) the EOMs read\n\\begin{equation}\nH^2\\simeq\\frac{\\gamma}{6(n-1)}\\,,\n\\quad\n3H\\dot\\sigma\\simeq \\left(\\frac{\\sqrt{2}\\gamma}{\\sqrt{3\\kappa^2}}\\right)\n\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\n\\end{equation}\nand the analysis of inflation results to be the same of $R^2$-models (with or \nwithout the cosmological constant term).\\\\\n\\\\\nHere, we can give some comments about the results of our investigation. Scalar \ninflationary potentials wich satisfy viable conditions for realistic primordial \nacceleration in the contest of large scalar curvature  corrections to General \nRelativity can be classified in two classes. In the first one, the scalar \npotential behaves as \n$V(\\sigma)\\sim\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]\\,,n>0$, and produces \nacceleration with a sufficient amount of inflation only for $n$ very close to \nzero, namely the $F(R)$-model must be close to the Starobinsky one. In the \nsecond class, $V(\\sigma)\\sim\\alpha \n-\\gamma\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, $n>0$, and a quasi de \nSitter solution emerges during inflation. Cosmological data are satisfied if \n$n>1\/3$. This kind of scalar potentials is originated from large scalar \ncurvature corrections of the type $F(R)\\sim c_1 R^2+c_2 R^\\zeta$, with \n$1<\\zeta<5\/3$ when $1\/3<n<1$. Finally, the cases with $1<n$ show in fact the \nsame behaviour of $R^2$-model during inflation.\n\n\n\\section{$R+\\alpha (R+R_0)^n$-models}\n\nIn this Section, we will consider power law corrections to Einstein gravity in \nthe form\n\\begin{equation}\nF(R)=R+\\alpha (R+R_0)^n+\\Lambda\\,,\\quad n>1\\,,\n\\label{modellino}\n\\end{equation}\nwhere $\\alpha>0$ is a (dimensional) constant parameter and $R_0\\,,\\Lambda$ are \ntwo (cosmological) constants introduced to generalize the model. This class of \nmodels, following the success of quadratic correction, have been often analyzed \nin literature in order to reproduce the dynamics of inflation and\nrecently, in Ref.~\\cite{cinesi}, the cases of polynomial corrections added to \nthe Starobinsky model have been investigated.\nIn what follows, at first we would like to study the\ninflation in the Einstein frame given by (\\ref{modellino}) with $n\\neq2$. We \nwill see that inflation for large magnitude values of the field is realized for \n$n\\lesssim2$. However, other possibilities are allowed by considering \nintermediate values of the field. In this case, the de Sitter solution may \nemerge in the models with $n>2$, but in order to study the exit from inflation \nis necessary to analyze the theory in the Jordan frame, where perturbations \nmake possible an early time acceleration with a sufficient amount of \n$N$-foldings number: that is the aim of the second part of this Section.\n\nLet us start from the potential\n  in the scalar field representation of (\\ref{modellino}),\n\\begin{eqnarray}\nV(\\sigma)&=&\\frac{1}{2\\kappa^2}\\left\\{\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\left(1-\\text{e}^{\\sqrt{(2\\kappa^2\/3)}\\sigma}\\right)^{\\frac{n}{n-1}}\\text{e}^{\\frac{(n-2)}{(n-1)}\\sqrt{(2\\kappa^2\/3)}\\sigma}\\left[\\frac{n-1}{n}\\right]\\right.\\nonumber\\\\\n&&\\left.+R_0\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\left(\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}-1\\right)\n-\\Lambda\\text{e}^{2\\sqrt{2\\kappa^2\/3}\\sigma}\\right\\}\\,.\n\\end{eqnarray}\nWe immediatly see that only if $n=2$, when $\\sigma\\rightarrow-\\infty$, one gets \n$V(\\sigma)\\sim\\text{const}$ and obtains the de Sitter solution (for $n=2$, \n$\\Lambda=R_0=0$ we recover (\\ref{VStaro}),\n$V(\\sigma)=(\\exp[\\sqrt{2\\kappa^2\/3}\\sigma]-1)^2\/(8\\alpha\\kappa^2)$).\nOn the other hand, even if $1<n<2$, the scalar field starts from a maximum of \nthe potential and falls toward the minimum at $\\sigma\\rightarrow 0^-$: it \ncorresponds to the case of \n$V(\\sigma\\rightarrow-\\infty)\\sim\\exp[-(2-n)\/(n-1)\\sigma]$ which may be used to \nreproduce an accelerating expansion if the slow roll limits are satisfied.\n\nThe derivatives of the potential read\n\\begin{eqnarray}\nV'(\\sigma)&=&\\left[-\\frac{1}{2\\kappa^2}\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\n\\left(1-\\text{e}^{\\sqrt{(2\\kappa^2\/3)}\\sigma}\\right)^{\\frac{1}{n-1}}\\text{e}^{\\frac{(n-2)}{(n-1)}\n\\sqrt{(2\\kappa^2\/3)}\\sigma}\n+2V(\\sigma)\\right.\n\\nonumber\\\\\n&&\\left.+\\frac{R_0}{2\\kappa^2}\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\right]\\left(\\sqrt{\\frac{2\\kappa^2}{3}}\\right)\\,,\n\\end{eqnarray}\n\\begin{eqnarray}\nV''(\\sigma)&=&\\frac{1}{3}\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\left[\\left(\\text{e}^{-\\sqrt{(2\\kappa^2\/3)}\\sigma}-1\\right)^{\\frac{2-n}{n-1}}-3\\left(\\text{e}^{-\\sqrt{(2\\kappa^2\/3)}\\sigma}-1\\right)^\\frac{1}{n-1}\\text{e}^{\\sqrt{(2\\kappa^2\/3)}\\sigma}\\right]\n\\nonumber\\\\\n&&+4V(\\sigma)\\left(\\frac{2\\kappa^2}{3}\\right)+R_0\\text{e}^{\\sqrt{2\\kappa^2\/3}\\sigma}\\,,\n\\end{eqnarray}\nand the slow roll parameters (\\ref{slow}) are derived as\n\\begin{equation}\n\\epsilon=\\frac{1}{3}\\left[2-\\frac{n}{(n-1)}\\frac{1}{y}+f(y)\\right]^2\\,,\\quad\\eta=\\frac{2n}{3(n-1)}\\left[\\frac{1}{y^2}-\\frac{3}{y}\\right]+\\frac{8}{3}+g(y)\\,,\n\\end{equation}\nwhere we have put\n\\begin{equation}\ny=\\left(1-\\text{e}^{\\sqrt{(\\kappa^2\/3)}\\sigma}\\right)\\,,\\quad\nf(y)=\\frac{\\left[R_0\\left(\\frac{2n-1}{n-1}\\right)-\\frac{\\Lambda(1-y)}{y}\\left(\\frac{n}{n-1}\\right)\\right]}\n{\\left[\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\frac{y^{\\frac{n}{n-1}}}{(1-y)^{\\frac{1}{n-1}}}\\left(\\frac{n-1}{n}\\right)+R_0\ny-\\Lambda(1-y)\\right]}\\,,\\nonumber\n\\end{equation}\n\\begin{equation}\ng(y)=2f(y)+\\frac{\\frac{2}{3y}\\left(\\frac{n}{n-1}\\right)\\left[\\frac{\\Lambda(1-y)}{y}-R_0\\right]}\n{\\left[\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\frac{y^{\\frac{n}{n-1}}}{(1-y)^{\\frac{1}{n-1}}}\\left(\\frac{n-1}{n}\\right)+R_0\ny-\\Lambda(1-y)\\right]}\\,.\n\\end{equation}\nWe have that $y<1$ for negative values of the field. If $\\Lambda=R_0=0$, \n$f(y)=g(y)=0$ and\nfor $n=2$ we recover (\\ref{star}). For large and negative values of the field, \nnamely when $y\\rightarrow 1^-$, one finds\n\\begin{equation}\n\\epsilon\\simeq\\frac{(2-n)^2}{3(n-1)^2}\\,,\\quad|\\eta|\\simeq\\frac{4|n-2|}{3(n-1)}\\,,\n\\end{equation}\nand $n$ has to be\n\\begin{equation}\n\\frac{2+\\sqrt{3}}{1+\\sqrt{3}}\\simeq1.36<n<2\\,.\\label{nn}\n\\end{equation}\nHere, we remember that we are considering only the cases $n<2$ whose\nEOMs in the slow roll limit read,\n\\begin{equation}\nH^2\\simeq\n\\frac{1}{6}\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\text{e}^{\\frac{(n-2)}{(n-1)}\\sqrt{(2\\kappa^2\/3)}\\sigma}\\left[\\frac{n-1}{n}\\right]\n\\,,\\quad\n3H\\dot\\sigma\\simeq\n\\sqrt{\\frac{1}{6\\kappa^2}}\\left(\\frac{1}{\\alpha \nn}\\right)^{\\frac{1}{n-1}}\\text{e}^{\\frac{(n-2)}{(n-1)}\\sqrt{(2\\kappa^2\/3)}\\sigma}\\left[\\frac{2-n}{n}\\right]\\,.\n\\end{equation}\nThe solution for the field is given by\n\\begin{equation}\n\\sigma=2\\sqrt{\\frac{3}{2\\kappa^2}}\\left[\\frac{n-1}{2-n}\\right]\\ln\\left[\n\\sqrt{\\frac{2}{3n}}\\frac{(2-n)^2}{6(n-1)^{3\/2}}\\left(\\frac{1}{n\\alpha}\\right)^{\\frac{1}{2(n-1)}}\n(t+t_0)\\right]\\,.\\label{bimbo}\n\\end{equation}\nSince $1<n<2$, $\\sigma$ is negative for $t_0>0$ very small bounded at the \nbeginning of inflation.\nThe Hubble parameter reads\n\\begin{equation}\nH=\\frac{3(n-1)^2}{(2-n)^2}\\frac{1}{(t_0+t)}\\,,\\quad\\frac{\\ddot \na}{a}=\\frac{3(n-1)^2((n+1)^2-2)}{(n-2)^4}\\frac{1}{(t_0+t)^2}>0\\,,\n\\end{equation}\nand we have an acceleration with decreasing Hubble parameter and curvature\n\\begin{equation}\nR=\\frac{3(n-1)^2\\left(3(n-1)^2-(2-n)\\right)}{(2-n)^4(t_0+t)}\\,.\n\\end{equation}\nThe $N$-foldings number of inflation is given by\n\\begin{equation}\nN\\simeq -\\frac{(n-1)}{(2-n)}\\sqrt{\\frac{3\\kappa^2}{2}}\\sigma\n\\Big\\vert^{\\sigma_i}_{\\sigma_e}\\simeq\n-3\\left(\\frac{n-1}{2-n}\\right)^2\\ln\\left[\n\\sqrt{\\frac{2}{3n}}\\frac{(2-n)^2}{6(n-1)^{3\/2}}\\left(\\frac{1}{n\\alpha}\\right)^{\\frac{1}{2(n-1)}}\nt_0\\right]\\,.\n\\end{equation}\nWhen the inflation ends, the field reaches the minimum of the potential. \nWe note that the cosmological constants introduced in the model do not play \nany role at inflationary stage, and change the behaviour of the potential only \nat the end of the inflation.\nIf \n$\\Lambda=R_0=0$, it is easy to see that $V(0)=0$, otherwise the potential \npossesses a minimum before $\\sigma=0$ (in the same way of \\S~\\ref{case}), where \nthe field falls and starts to oscillate.\n\nThe amplitude of primordial power spectrum (\\ref{spectrum}) and the spectral \nindexes (\\ref{indexes}) are\n\\begin{equation}\n\\Delta_{\\mathcal \nR}^2\\simeq\\frac{(n-1)^3\\kappa^2\\text{e}^{\\frac{2}{3}N\\frac{(n-2)^2}{(n-1)^2}}}{16\\pi^2(2-n)^2n}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}\\,,\n\\quad\nn_s\\simeq1-\\frac{8(2-n)}{3(n-1)}\\,,\\quad r\\simeq\\frac{16(2-n)^2}{3(n-1)^2}\\,,\n\\end{equation}\nwhere we have taken into account that $n$ is close to two. In order to make \nthese indexes compatible with (\\ref{data}), we must require\n$ n\\sim 1.8\\,, 1.9$,\nsuch that, as a matter of fact, the quadratic corrections with $n=2$ appear to be \nthe only possibilities of the type (\\ref{modellino}) able to reproduce a viable \ninflation in the Einstein frame. However, it is interesting to extend our \ninvestigation to the Jordan frame.\n\n\\subsection{Inflation in the Jordan frame}\n\nIn the first part of this Section, we have seen how model (\\ref{modellino}) can \nproduce inflation in the Einstein frame representation. We have an early time \nacceleration followed by the end of inflation and the slow roll conditions are \nsatisfied if $n$ is very close to two.\nFor the sake of simplicity, in this Subsection, we \nwill put $\\Lambda=R_0=0$ in (\\ref{modellino}).\nLet us return to the Jordan frame.\n\nThe Friedmann equation for a generic $F(R)$-model read (in vacuum)\n\\begin{equation}\n\\left(R F_R(R)- F(R)\\right)-6H^2 F_R(R)-6H\\dot F_R(R)=0\\,,\\label{eq}\n\\end{equation}\nsuch that for (\\ref{modellino}) one derives\n\\begin{equation}\n\\alpha(R)^{n-1}\\left[R(1-n)\\right]-6H^2\\left[1+\\alpha \nn(R)^{n-1}\\right]=6H\\alpha n\\,(n-1)(R)^{n-2}\\dot R\\,.\\label{First}\n\\end{equation}\nIn the high curvature limit (it means, for large and negative values of the \nscalar field $\\sigma$ in the Einstein frame) we may consider $\\alpha \nn(R)^{n-1}\\gg 1$,\n\\begin{equation}\nH^2\\simeq\\frac{(n-1)}{6n}\\left[R-\\frac{6n\\,H\\,\\dot R}{R}\\right]\\,.\n\\end{equation}\nDuring inflation, the Hubble parameter $H$ evolves slowly and we can consider \nthe anologous of the slow roll parameterers which have the same\nformal dependence of the Einstein frame, namely we require  $|\\dot H|\/H^2\\ll \n1$, $|\\ddot H\/(H\\dot H)|\\ll 1$. Thus, we get\n\\begin{equation}\n\\frac{\\dot \nH}{H^2}\\simeq-\\left[\\frac{4-2n}{n+4(n-1)^2}\\right]\\,,\\quad\\frac{\\ddot a}{a}=H^2\n+\\dot H=1-\\frac{4-2n}{n+4(n-1)^2}\\,.\n\\end{equation}\nIt follows that the expansion is accelerated only if\n\\begin{equation}\nn>\\frac{5}{4}=1.25\\,.\n\\end{equation}\nThis value is close to the one given by the left side of (\\ref{nn}). Moreover, \nfor $n>2$, we see that $\\dot H>0$ and the Hubble parameter, and therefore the \ncurvature, grows up with the time and the physics of Standard Model is not \nreached. This result is in agreement with the behaviour of the model discussed \nin the Einstein frame: when $n>2$ and the acceleration emerges in high \ncurvature limit, the field cannot reach a minimum of the potential for small \nvalues and we do not exit from inflation. We could arrive to this conclusion \nalso by looking for Eq.~(\\ref{bimbo}): when $n>2$, $t_0$ has to be very large \nat the beginning of inflation and the scalar field grows up with the time. \nHowever, it does not mean that models with $n>2$ do not produce inflation. In \nscalar field representation we start at very large and negative values of the \nfield (it means, that in Jordan framework we can ignore the $R$-term in the \naction). It may be interesting to see what happen for intermediate values of \nthe field. We analyze the problem in the Jordan frame. From Eq.~(\\ref{eq}) we \nhave the de Sitter condition at $R_0=12H_0^2$, $H_0$ constant,  namely\n\\begin{equation}\n2F(R_0)-R_0F_R(R_0)=0\\,,\\label{trace}\n\\end{equation}\nwhich leads in our case,\n\\begin{equation}\nR_0=\\left(\\frac{1}{\\alpha(n-2)}\\right)^{\\frac{1}{n-1}}\\,.\\label{RdS}\n\\end{equation}\nThe model with $n=2$ does not possess an exact de Sitter solution, but if \n$n>2$, we can realize it. Here, we remember that $\\alpha>0$.\nIn the Einstein frame this solution corresponds to\n\\begin{equation}\n\\sigma_{max}=-\\sqrt{\\frac{3}{2\\kappa^2}}\\log\\left[\\frac{2n-2}{n-2}\\right]\\,,\n\\end{equation}\nwhere we have used (\\ref{sigma}). We note that $\\sigma<0$ only if $n>2$ (we are \nassuming $n>1$); otherwise, if $n<2$, this expression becomes imaginary and \nmeaningless.\nThis solution corresponds to a maximum of the potential, because of\n\\begin{equation}\nV'(\\sigma_{max})=0\\,,\\quad \nV''(\\sigma_{max})=\\frac{1}{6}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}\n\\frac{n-2}{n-1}\\left(\\frac{n}{n-2}\\right)^{\\frac{(2-n)}{(n-1)}}>0\\,.\n\\end{equation}\nWhen $n>2$, the potential in Einstein frame has a maximum. The scalar field \nproduces the de Sitter solution, but the field does not evolve with the time \n($\\dot\\sigma=0$) and we do not have a natural exit form inflation. However, by \nmaking use of the perturbative theory, we may study the stability of the model \nin the Jordan frame.\n\nBy perturbating Eq.~(\\ref{eq}) as $R\\rightarrow R+\\delta R$, $|\\delta R|\\ll 1$ \naround the de Sitter solution, we get\n\\begin{equation}\n\\left(\\Box-m^2\\right)\\delta R\\simeq 0\\,,\n\\quad\nm^2=\\frac{1}{3}\\left(\\frac{F'(R)}{F''(R)}-R\n\\right)\\,.\n\\end{equation}\nHere, $m^2$ is the effective mass of the scalaron, namely the new degree of \nfreedom\nintroduced by the modified gravity through $F_R(R)$, which is proportional to \nthe opposite of the inflaton, namely $-\\sigma$. As a consequence, $m^2$ is \nproportional to the opposite of the scalar potential of the inflaton and when \n$m^2<0$ the solution is unstable. In our case, we get on the de Sitter \nsolution,\n\\begin{equation}\nm^2=-(n-2)^{\\frac{n-2}{n-1}}\\left(\\frac{1}{\\alpha\\,n}\\right)^{\\frac{1}{n-1}}<0\\,,\\quad \n2<n\\,.\\label{inst1}\n\\end{equation}\nIt means, that a small perturbation can cause the exit from inflation. Now, the \nquestion is: in which direction the inflaton moves due to a perturbation and \nwhich kind of perturbation we need to have a correct $N$-foldings number? To \nreproduce a viable comsology, we expect that the inflaton moves toward the \nminimum of the potential at $V(0)=0$, such that the small curvature regime can \nbe reached and the cosmology of Standard Model takes place, and $N$ must be at \nleast $N\\sim 60$. In principle, the inflaton can also moves to \n$\\sigma\\rightarrow-\\infty$ (large curvature regime), for which the potential \ntends also to zero.\n\nIn hot universe scenario,\nwe must take into account also the presence of\nultrarelativistic matter\/radiation, whose energy density is given by\n\\begin{equation}\n\\rho_r=\\rho_{r(0)}\\,a(t)^{-4}\\,.\n\\end{equation}\nHere, $\\rho_{r(0)}$ is a constant bounded at the beginning of inflation. Thus, \nEq.~(\\ref{First}) reads\n\\begin{equation}\n\\frac{3H^2}{\\kappa^22}=\\rho_{MG}+\\rho_r\\,,\\quad\\rho_{MG}=\n\\frac{1}{2\\kappa^2}\\left[\\alpha(R)^{n-1}\\left[R(1-n)\\right]-6H^2\\alpha \nn(R)^{n-1}-6H\\alpha n\\,(n-1)(R)^{n-2}\\dot R\\right]\\,,\\label{F1}\n\\end{equation}\nwhere $\\rho_{MG}$ encodes the amount of energy density given by the correction \nterm $R^n$ to Eintein gravity. For simplicity, we introduce the red shift \nparameter $z=-1+1\/a(t)$. We also remember that $d\/dt=-(z+1)H(z)d\/dz$.\n\nLet us define~\\cite{HuSaw, Bamba}\n\\begin{equation}\ny_H (z)\\equiv\\frac{\\rho_{\\mathrm{MG}}}{M^2\/\\kappa^2}=\\frac{3H^2}{M^2}\n-\\tilde\\chi (z+1)^{4}\\,,\\label{yy}\n\\end{equation}\nwhere $M^2$ is a suitable dimensional constant, such that $[M^2]=[H^2]$ and \n$\\tilde\\chi=\\rho_{r(0)}\\kappa^2\/M^2$.\nBy taking into account that the Ricci scalar $R(z)=\\left[12H(z)^2-6(z+1)H(z) d \nH(z)\/dz\\right]$ results to be\n\\begin{equation}\nR=M^2\\left(4y_H-(z+1)\\frac{d y_H(z)}{dz}\\right)\\,,\n\\end{equation}\nwe derive from (\\ref{First}),\n\\begin{eqnarray}\n&&\n\\frac{d^2y_H(z)}{dz^2}-\\frac{dy_H}{dz}\\frac{1}{z+1}\\left\\{3-\\frac{f_R(R)}{2M^2f_{RR}(R)\\left[y_H(z)+\\tilde\\chi(z+1)^4\\right]}\\right\\}\n\\nonumber\\\\\n&&\n{}+\\frac{y_H(z)}{(z+1)^2}\\left\\{\\frac{1-f_R(R)}{M^2f_{RR}(R)\\left[y_H(z)+\\tilde\\chi(z+1)^4\\right]}\\right\\}\n\\nonumber\\\\\n&&\n{}+\\frac{2f_R(R)\\tilde\\chi(z+1)^4+f(R)\/M^2}{(z+1)^2\\,2M^2f_{RR}(R)\\left[y_H(z)+\\tilde\\chi(z+1)^4\\right]}=0\\,,\n\\label{superEq}\n\\end{eqnarray}\nwhere in our case\n\\begin{equation}\nf(R)=\\alpha R^n\\,,\\quad f_R(R)=\\alpha\\,n R^{n-1}\\,,\\quad \nf_{RR}=\\alpha\\,n(n-1)R^{n-2}\\,.\n\\label{zazam}\n\\end{equation}\nLet us study the perturbations around the de Sitter solution (\\ref{RdS}), \nnamely\n\\begin{equation}\ny_{H}(z)\\simeq y_{0}+y_1(z)\\,,\\label{yexpansion}\\quad \ny_0=\\frac{1}{4M^2}\\left(\\frac{1}{\\alpha(n-2)}\\right)^{\\frac{1}{n-1}}\\,,\n\\end{equation}\nwhere\n$|y_1(z)\/y_0|\\ll 1$. Here, we have put $R_0=4M^2y_0$ in the de Sitter universe.\nBy assuming the contribution of ultrarelativistic matter much smaller than \n$y_0$ at the beginning of inflation, Eq.~(\\ref{superEq}) becomes, at first \norder in $y_1(z)$,\n\\begin{equation}\n\\frac{d^2 y_1(z)}{d z^2}-\\frac{2}{(z+1)}\\frac{d y_1(z)}{d z}+\\frac{1}{(z+1)^2}\n\\left(-4+\\frac{4(1+f_R(R))}{Rf_{RR}(R)}\\right)\ny_1(z)\\simeq0\\,,\\label{superEqbis}\n\\end{equation}\nwhere we have also used condition (\\ref{trace}). Thus, the solution for $y_1$ \nis given by\n\\begin{equation}\ny_1(z)=C_0(z+1)^x\\,,\\quad\nx=\\frac{1}{2}\\left(3-\\sqrt{25-\\frac{16(1+f_R(R))}{R f_{RR}(R)}}\\right)\\,,\n\\end{equation}\n$C_0$ being constant.\nHere, we do not consider the solution with the plus sign in front of the square \nroot of $x$, since it does not cause any instability. On the other hand, by \nmaking use of (\\ref{zazam}), one has on the de Sitter solution (\\ref{RdS})\n\\begin{equation}\nx=\\frac{1}{2}\\left(3-\\sqrt{\\frac{25n^2-57n+32}{n(n-1)}}\\right)<0\\,,\\quad n>2\\,,\n\\end{equation}\nsince it is easy to demonstrate that $x<0$ if $0<16(n-2)\/n$, that is always \ntrue when $2<n$ and we recover condition (\\ref{inst1}).\nAs a consequence, the perturbation $y_1(z)$ grows up in expanding universe as\n\\begin{equation}\ny_1(z)=y_1(z_\\mathrm{i})\\left[\\frac{(z+1)}{(z_\\mathrm{i}+1)}\\right]^x\\,.\\label{yyy}\n\\end{equation}\nHere, we have considered $C_0=y_1(z_\\mathrm{i})\/(z_\\mathrm{i}+1)^x$, $z_i$ \nbeing the redshift at the beginning of inflation where perturbation is bounded.\nWhen $y_1(z)$ is on the same order of $y_0$, the\ninflation ends. A classical perturbation on the (vacuum) de Sitter solution may \nbe given by\nthe ultrarelativistic matter in (\\ref{yy}), such that Eq.~(\\ref{yyy}) reads\n\\begin{equation}\ny_1(z)=-\\tilde\\chi(z_i+1)^4\\left[\\frac{(z+1)}{(z_\\mathrm{i}+1)}\\right]^x\\,,\\quad \ny_1(z_i)=-\\tilde\\chi(z_\\text{i}+1)^4\\,.\n\\end{equation}\nThus, the $N$-foldings number during inflation is\n\\begin{equation}\nN=\\log \\left[\\frac{z_\\mathrm{i}+1}{z_\\mathrm{e}+1}\\right]\n\\simeq \\frac{1}{x}\\log\\left[\\frac{\\tilde\\chi(z_{\\text{i}}+1)^4}{y_0}\\right]\\,.\n\\label{Nfolding}\n\\end{equation}\nHere, $z_e$ denotes the red shift at the end of inflation and we have \nconsidered $y_1(z_\\text{e})\\simeq-y_H$. As small $x$ is, much unstable is \ninflation. It means, that also a small initial perturbation may give arise to a \nlarge $N$-foldings. We can write $\\tilde\\chi$ as\n\\begin{equation}\n\\tilde\\chi=\\frac{y_0}{(z_i+1)^4}\\times\\delta\\,,\\quad\\delta\\ll 1\\,,\n\\end{equation}\nsuch that\n\\begin{equation}\n\\delta=\\text{e}^{x\\,N}\\,.\\label{delta}\n\\end{equation}\nIn order to obtain an $N$-foldings of $70$, for $n=3$ ($x\\simeq-0.393$) we need \n$\\delta\\sim10^{-12}$, for $n=4$ ($x\\simeq-0.562$) we need $\\delta\\sim10^{-18}$,\nfor $n=5$ ($x\\simeq-0.656$) we need $\\delta\\sim10^{-20}$,\nfor $n=10$ ($x\\simeq-0.835$) it is enough $\\delta\\sim10^{-26}$.\n\nMoreover, the behaviour of Ricci scalar is given by\n\\begin{equation}\nR=4y_0\\,M^2+M^2\\left(4y_1-(z+1)\\frac{dy_1}{dz}\\right)\\simeq 4y_0M^2-y_0\\delta \nM^2\n\\left[4-x\\right]\\left(\\frac{z+1}{z_\\text{i}+1}\\right)^x\\,.\n\\end{equation}\nSince $x>0$, the Ricci scalar decreases with the red shift and the physics of \nStandard Model can emerge.\nIn Ref.~\\cite{nostrum} numerical calculations have been executed in different \ninflationary models\nprovided by the specific $R^n$-term with $2<n<3$, which makes inflation \nunstable and\nin the presence of ultrarelativistic matter, which leads to the exit from the \ninflationary stage toward the physics of Standard Model.\n\nIn conclusion, we can say that large scalar curvature corrections to Einstein \ngravity of the type $R^n$, $n>2$, may represent a valid inflationary scenario. \nIn this case, the curvature is bounded at a specific value at the beginning of \ninflation, and the de Sitter space-time is an exact solution of the model, \nwhich results to be highly unstable. From (\\ref{delta}) we see how extremely \nsmall perturbations in hot universe give the correct $N$-foldings number and \nthe exit from inflation.\nFinally, it is important to note that positive energy density of perturbations \nbrings the curvature to decrease during the early-time acceleration making the \nmodel consistent with the observable evolution of our universe.\n\n\n\\section{Conclusions}\n\n\nThe attention that recently has been paid to modified theories of gravity is \ncaused by the idea  of unified description of early-time and late-time cosmic \nacceleration~\\cite{Review-Nojiri-Odintsov}. Moreover,\nthe gravitational action of such a kind of theories may describe quantum \neffects in hot universe scenario, and the last cosmological data seems to be in \nfavour of quadratic corrections to General Relativity during this phase.\n\nIn this paper, we have investigated some feautures of $F(R)$-modified gravity \nmodels for inflationary cosmology, by performing our analysis in the Jordan and \nin the Einstein framework.\n\nAt first, we have studied\ninflation for the class of scalar potentials of the type \n$V(\\sigma)\\sim\\exp[n\\sigma]$, $n$ being a general parameter, in Einstein frame. \nAs a matter of fact, for such a kind of models is possible to reconstruct  the \n$F(R)$-gravity theories which correspond to the given potentials.\nSince\nviable inflation must be consistent with the last Planck data, the potentials \nhave been carefully alalyzed, by finding the conditions on the parameters which \nmake possible the early-time acceleration according with $N$-foldings, spectral\nindex and tensor-to-scalar ratio coming from observations.\nWe have derived the form of the $F(R)$-models at the large and small curvature \nlimits, demonstrating that these\nmodels in the Jordan frame correspond to corrections to Einstein gravity which \nemerge only at mass scales larger than the Planck mass.\nThe investigated potentials can be classified in two classes. In the first one, \nthe scalar potential behaves as \n$V(\\sigma)\\sim\\exp\\left[n\\sqrt{2\\kappa^2\/3}\\sigma\\right]\\,,n>0$, and produces \nacceleration with a sufficient amount of inflation only for $n$ very close to \nzero, namely the $F(R)$-model must be closed to the Starobinsky one. In the \nsecond class, $V(\\sigma)\\sim\\alpha \n-\\gamma\\exp\\left[-n\\sqrt{2\\kappa^2\/3}\\sigma\\right]$, $n>0$, and the quasi de \nSitter solution emerges during viable inflation when $n>1\/3$. This kind of \nscalar potentials is originated from large curvature corrections of the type \n$F(R)\\sim c_1 R^2+c_2 R^\\zeta$, with $1<\\zeta<5\/3$ when $1\/3<n<1$, or from \nquadratic curvature corrections when $n>1$.\nIt is interesting to note that $R^2$-term which induces the early-time \ninflation is also responsible for removal of finite-time future \nsingularity  in $F(R)$-gravity unifying the inflation with dark energy~\\cite{OdSing}.\n\nIn the second part of the paper, we have studied in detail the specific class \nof models $F(R)=R+(R+R_0)^n$. The analysis in the\nEinstein frame reveals that $n$ must be very close to two in order to realize a \nviable inflation for large and negative values of the scalar field,\nbut other possibilities are allowed\nby starting from intermediate values of the field.\nTo be specific, when $n>2$, the de Sitter solution emerges, but in order to \nstudy the exit from inflation, since in this case the de Sitter space-time is \nan exact solution and the field does not move and exit from inflation in a \nnatural way, is necessary to analyze the theory in the Jordan frame, where \nperturbations make possible an early time acceleration with a sufficient amount \nof inflation. Moreover, we can explicitly demonstrate\nthat the curvature decreases making the model consistent with the historical \nevolution of our universe.\n\n\n\n\\section*{Acknowledgments}\nWe would like to thank M. Sami for useful discussions and valuable suggestions.\nThe work by SDO has been supported in part by MINECO (Spain), project \nFIS2010-15640, by AGAUR (Generalitat de Catalunya), contract 2009SGR-994 and by \nproject 2.1839.2011 of MES (Russia).\n\n\n\n\n\\section*{Appendix A}\nLet us consider the equation\n\\begin{equation}\nF_R(R)-\\frac{1}{2}-\\frac{F_R(R)^{1+n}}{2} =-\\frac{R}{4\\gamma(n+2)}\\,,\n\\label{EqA0}\\end{equation}\nwith $n\\,,\\gamma>0$ (it corresponds to (\\ref{pippo}) with (\\ref{alphacond})).\nWe want to find solutions of the above equation as a function of $R$.\nA simple possibility is looking for regular solutions as a power series of $R$, \nnamely\nsetting\n\\begin{equation*}\nF_R(R)=1+c_1R+c_2R^2+c_3R^3+...\\,,\n\\end{equation*}\nwhere the constants $c_{1,2,3...}$ have to be determined. At first, we are \ninterested in solutions where $|c_{1,2,3...}R^{1,2,3...}|\\ll 1$, namely we \nanalyze the limit at small curvature. Thus,\nplugging this Ansatz in the above equation one has\n\\begin{equation*}\n(1+c_1R+c_2R^2+c_3R^3+...)-\\frac{1}{2}-\\frac{1}{2}\\left(1+c_1R+c_2R^2+ \nc_3R^3+... \\right)^{1+n}=\n-\\frac{R}{4\\gamma(n+2)}\\,.\n\\end{equation*}\nPutting\n\\begin{equation*}\nX=+c_1R+c_2R^2+ c_3R^3+... \\,,\n\\end{equation*}\nand assuming  $|X|\\ll 1$, the  following expansion holds\n\\begin{equation*}\n(1+X)^{1+n}=1+(1+n)X+\\frac{(1+n)n}{2}X^2+\\frac{n(n+1)(n-1)}{3!}X^3+...\\,.\n\\end{equation*}\nAs a consequence, one arrives at the recursive  relations\n\\begin{equation*}\nc_1=\\frac{1}{2\\gamma(n-1)(n+2)}\\,,\\quad\n   c_2=-\\frac{(1+n)n\\,c_1^2}{2(n-1)}\\,,\\quad\n     c_3=-\\frac{(1+n)c_1^3}{6}\\,, ...\\,,\n\\end{equation*}\nfrom which it follows\n\\begin{equation*}\nc_1 R\\propto\\frac{R}{\\gamma}\\,,\\quad\n    c_2 R^2\\propto\\frac{R}{\\gamma^2}\\,,\\quad\n       c_3 R^3\\propto\\frac{R^3}{\\gamma^3}\\,,...\n\\end{equation*}\n  This approximation is valid when $R\\ll\\gamma$.\nOn the other side,  when $R\\gg\\gamma$  we can  check for the solutions in the \nform\n\\begin{equation*}\nF_R(R)=c_0\\left(\\frac{R}{\\gamma}\\right)^\\zeta\\,,\\quad\\zeta<1\\,.\n\\end{equation*}\nIn such a case from (\\ref{EqA0}) we get\n\\begin{equation*}\nc_0^{1+n}\\left(\\frac{R}{\\gamma}\\right)^{\\zeta(1+n)}\\simeq\\frac{1}{4(n+2)}\\left(\\frac{R}{\\gamma}\\right)\\,,\n\\end{equation*}\nand as a consequence\n\\begin{equation*}\nc_0=\\left(\\frac{1}{4(n+2)}\\right)^{\\frac{1}{1+n}}\\,,\\quad\\zeta=\\frac{1}{1+n}<1\\,.\n\\end{equation*}\nNow let us consider  the following equation (it corresponds to \\ref{pippo2} \nwith \\ref{alphaalpha}),\n\\begin{equation}\nF_R(R)-F_R(R)^{1-n} =\\frac{R}{2\\gamma(2-n)}\\,,\n\\label{EqA1}\\end{equation}\nwhere $\\gamma>0$ and $0<n<1$.\nThe solution for $R\\ll\\gamma$ is given by\n\\begin{equation*}\nF_R(R)=1+c_1R+c_2R^2+c_3R^3+...\\,,\n\\end{equation*}\nwith\n\\begin{equation*}\nc_1=\\frac{1}{2\\gamma\\,n(2-n)}\\,,\\quad\nc_2=-\\frac{n c_1^2}{2}\\,,\\quad\nc_3=\\frac{(1-n)(n+1)c_1^3}{6}\\,, ...\n\\end{equation*}\nOn the other hand, when $R\\gg\\gamma$ one can expand $F_R(R)$ as\n\\begin{equation*}\nF_R(R)=\\beta_1\\left(\\frac{R}{\\gamma}\\right)^{\\alpha_1}+\n\\beta_2 \\left(\\frac{R}{\\gamma}\\right)^{\\alpha_2}\\,,\\quad\\alpha_1>\\alpha_2\\,.\n\\end{equation*}\nSince $0<n<1$ at the first order in $R\/\\gamma$ one finds\n\\begin{equation*}\n\\beta_1\\left(\\frac{R}{\\gamma}\\right)^{\\alpha_1}\\simeq\\frac{R}{2\\gamma(2-n)}\\,,\n\\end{equation*}\nand so\n\\begin{equation*}\n\\alpha_1=1\\,,\\quad\\beta_1=\\left(\\frac{1}{2\\gamma(2-n)}\\right)\\,.\n\\end{equation*}\nMoreover, by using (\\ref{EqA1}) again  we obtain\n\\begin{equation*}\n\\alpha_2=1-n\\,,\\quad\\beta_2=\\left(\\frac{1}{2\\gamma(2-n)}\\right)^{1-n}\\,.\n\\end{equation*}\nAs a final result, we can conclude that  at high curvatures the model\nunder considertion  reads\n\\begin{equation*}\nF_R(R)=\\left(\\frac{R}{2\\gamma(2-n)}\\right)+\\left(\\frac{R}{2\\gamma(2-n)}\\right)^{1-n}\\,.\n\\end{equation*}\n\n\\section*{Appendix B}\nAs is well known, the Einstein-Hilbert Lagrangian density modified by a\nquadratic term $R^2$ and by  a cosmological constant term has the\nSchwarzshild-de Sitter solution,\nand this is the most general spherically symmetric, static solution if the\nparameters which enter in the modified Lagrangian are arbitrary (unrelated).\n\nHere we shall show that there exists a particular choice of such\nparameters for which the model has a more general spherically\nsymmetric, static solution, which is formally identical to the one\ncorresponding to the Reissner-Nordstr\\\"om-de Sitter black hole.\n\nTo this aim we consider the Lagrangian density\n\\begin{equation*}\nF(R)=R+\\frac{R^2}{4c_0}+c_0\\,,\n\\end{equation*}\nwhich has been obtained from the reconstruction process  described in \\S \n\\ref{4.1}.\nThis Lagrangian depends on one free parameter only.\n\nNow it is easy to verify that the most general spherically symmetric,\nstatic solution of field equations in vacuum is given by\n\\begin{equation*}\nds^2=-A(r)dt^2+ \\frac{dr^2}{B(r)}+r^2d\\Omega\\,,\\quad\\quad \nA(r)=B(r)=1-\\frac{a}{r}-\\frac{c_0}{6}r^2+\\frac{b}{r^2}\\,,\n\\end{equation*}\nwhere $d\\Omega$  is the metric on the two sphere, while\n  $a$ and $b$ are arbitrary constants.\nThe physical interpretation of such two  constants of integration is not an \neasy task.\nThe presence of the $1\/r^2$ term is quite interesting and it will be discussed \nelsewhere.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSignificant progress has been made in recent years on the\nclassification of crossed product C*-algebra{s} arising from\nfinite dimensional minimal dynamical systems.\nThe long unpublished\npreprint \\cite{QLinPhDiff} of Q.~Lin and N.~C.\\  Phillips\n(see also the\nsurvey articles \\cite{QLinPh1} and \\cite{QLinPh2}) provides a\nthorough description of the transformation group C*-algebra{s}\narising from minimal diffeomorphisms of finite dimensional smooth\ncompact manifolds in terms of a direct limit decomposition.\nIn\n\\cite{HLinPh} and \\cite{TomsWinter}, it is shown that crossed\nproducts arising from minimal homeomorphisms of infinite compact\nmetrizable spaces with finite covering dimension are classified by\ntheir ordered K-theory in the presence of sufficiently many projections\n(for instance, when projections separate traces).\nIn\n\\cite{TomsWinter} it is further proved that crossed products by such\nminimal homeomorphisms have finite nuclear dimension, and hence\nabsorb the Jiang-Su algebra ${\\mathcal{Z}}$ tensorially\n(that is, are ${\\mathcal{Z}}$-stable).\nFinally, G.~A.\\  Elliott and Z.~Niu (\\cite{EllNiu})\nhave shown that crossed products by minimal\nhomeomorphisms of compact metric spaces with mean dimension zero\n(including all minimal\nhomeomorphisms of finite dimensional compact metric spaces)\nare ${\\mathcal{Z}}$-stable,\nfrom which it follows that they are classifiable\nin the sense of the Elliott program by Corollary D of\nTheorem A of \\cite{CETWW}.\n\nNot as much is known for crossed products of\nC*-algebra{s} of the form $C (X, D)$ for a\nnoncommutative C*-algebra~$D$.\nHua (\\cite{Hua}) has shown\nthat such crossed products have tracial rank zero\nwhen $X$ is the Cantor set, $D$ has tracial rank zero,\nthe action on $X$ is minimal,\nand some additional K-theoretic assumptions are made.\nIn this paper we consider the structure of crossed products of\nthe form $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$,\nin which $X$ is a compact metric space,\n$D$ is simple unital a C*-algebra, and $\\alpha$ is an automorphism\nof $C (X, D)$ which ``lies over'' a minimal homeomorphism\n(as described in Definition~\\ref{D_4Y12_OverX}).\nIn Section~\\ref{Sec_Over}, we describe the types of actions on\n$C (X, D)$ which will be of interest here.\nIn Section~\\ref{Sec_OrbBreak} we introduce the generalization\nof the orbit breaking subalgebras of \\cite{PhLg}\nfor actions on $C (X, D)$,\nand show that these are large in various senses\ndefined in \\cite{PhLg}.\nSection~\\ref{RecStruct} introduces a\n``$D$-fibered'' generalization of the recursive subhomogeneous\nalgebras in \\cite{PhRsha1}, then demonstrates (following a\ndevelopment analogous to that in \\cite{QLinPhDiff}) that our\norbit breaking subalgebras have such a recursive structure.\nIn Section~\\ref{StructAY} we use the results of\nSections~\\ref{Sec_OrbBreak} and~\\ref{RecStruct} to obtain\nstronger structural properties for the orbit breaking\nsubalgebras, and deduce structural\nproperties of the crossed product from those of the\norbit breaking subalgebra under appropriate additional\nassumptions.\nIn Section~\\ref{Sec_9929_MinProd} we\nestablish minimality for products of certain Denjoy\nhomeomorphisms that will be used to produce examples.\nSection~\\ref{Sec_InfRed} gives a large collection of\nexamples of crossed products for which we can use the\ntheory developed here to deduce structural properties\nthat do not seem accessible using previously known methods.\nFinally, in Section~\\ref{Sec_Qs} we pose some\nopen questions for further research.\n\nWe recall Cuntz comparison and the Cuntz semigroup.\nFor a much fuller discussion,\nin a form useful for work with large subalgebras,\nwe refer to Section~1 of~\\cite{PhLg}.\n\n\\begin{ntn}\\label{N_9Y30_CzComp}\nIf $A$ is a C*-algebra{} and $a, b \\in M_{\\infty} (A)_{+}$,\nwe write $a \\precsim_A b$\nto mean that $a$ is Cuntz subequivalent to~$b$ over~$A$,\nthat is, there is a sequence $(v_n)_{n = 1}^{\\infty}$\nin $M_{\\infty} (A)_{+}$\nsuch that\n$\\limi{n} v_n b v_n^* = a$.\nWe write $a \\sim_A b$\nto mean that $a$ is Cuntz equivalent to~$b$ over~$A$,\nthat is, $a \\precsim_A b$ and $b \\precsim_A a$.\n\\end{ntn}\n\nWe specify~$A$ in the notation\nbecause Cuntz subequivalence with respect to proper\nsubalgebras will play a key role.\n\nThe Cuntz semigroup $W (A)$ is then defined to be the set of\nCuntz equivalence classes $M_{\\infty} (A)_{+} \/ \\sim_A$,\nwith addition given by direct sum and order coming from\nCuntz subequivalence.\n\n\\begin{ntn}\\label{ntn1-181221D}\nFor any C*-algebra{} $A$, we denote the set of normalized $2$-quasitraces\non~$A$ by ${\\operatorname{QT}} (A)$.\nWe use the word quasitrace to mean normalized $2$-quasitrace.\n\\end{ntn}\n\nFor a C*-algebra~$A$,\nthe topology on ${\\mathrm{Aut}} (A)$\nis always pointwise convergence in the norm of~$A$.\nThat is, $x \\mapsto \\alpha_x$ is continuous{}\nif and only if{} $x \\mapsto \\alpha_x (a)$ is continuous{} for all $a \\in A$.\n(This is the usual topology.)\nTo be explicit,\nwe point out that $\\alpha \\mapsto \\alpha^{-1}$ is continuous{} in this topology,\nas can be seen from the equation\n\\[\n\\| \\alpha^{-1} (a) - \\beta^{-1} (a) \\|\n  = \\bigl\\| \\beta^{-1}\n      \\bigl( \\alpha (\\alpha^{-1} (a)) - \\beta (af^{-1} (a)) \\bigr) \\bigr\\|\n\\]\nfor $a \\in A$ and $\\alpha, \\beta \\in {\\mathrm{Aut}} (A)$,\nand the fact that $\\beta^{-1}$ is isometric.\n\n\\begin{ntn}\\label{N_0108_DimX}\nFor a compact Hausdorff space~$X$,\nwe denote the covering dimension of~$X$\n(Definition 3.1.1 of~\\cite{Prs}) by $\\dim (X)$.\nIf $h \\colon X \\to X$ is a homeomorphism,\nits mean dimension\n(Definition 2.6 of~\\cite{LndWs})\nis denoted by $\\mathrm{mdim} (h)$.\n\\end{ntn}\n\n\\section{Preliminaries on actions on $C (X, D)$ lying over actions\n  on~$X$}\\label{Sec_Over}\n\nIn the section,\nwe give a few basic facts about actions of\ngroups on C*-algebra{s} of the form $C (X, D)$\nwhich ``lie over'' actions on~$X$.\nWe also introduce several technical conditions which\nwill be needed as hypotheses later,\nand give some cases in which they are automatically satisfied.\n\n\\begin{ntn}\\label{N_5418_ActCX}\nLet $G$ be a locally compact group and let $X$ be a\nlocally compact Hausdorff space~$X$ on which $G$ acts.\nWe take the corresponding action $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X))$\nto be be given by\n$\\alpha_g (f) (x) = f (g^{-1} x)$ for $f \\in C (X)$,\n$g \\in G$, and $x \\in X$.\nFor a homeomorphism{} $h \\colon X \\to X$,\nthis means that the corresponding action of~${\\mathbb{Z}}$ on $C_0 (X)$\nis generated by\nthe automorphism $\\alpha (f) = f \\circ h^{-1}$ for $f \\in C_0 (X)$.\n\\end{ntn}\n\n\\begin{dfn}\\label{D_4Y12_OverX}\nLet $X$ be a locally compact Hausdorff space,\nlet $G$ be a topological group,\nand let $D$ be a C*-algebra.\nLet $(g, x) \\mapsto g x$ be an action of $G$ on~$X$,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X, D))$\nbe an action of $G$ on $C_0 (X, D)$.\nWe say that\n{\\emph{$\\alpha$ lies over the action $(g, x) \\mapsto g x$}}\nif there exists a function\n$(g, x) \\mapsto \\alpha_{g, x}$ from $G \\times X$ to ${\\mathrm{Aut}} (D)$\nsuch that $\\alpha_g (a) (x) = \\alpha_{g, x} (a (g^{-1} x) )$\nfor all $g \\in G$, $x \\in X$,\nand $a \\in C_0 (X, D)$.\n\nWe say that an automorphism $\\alpha$ of $C_0 (X, D)$\n{\\emph{lies over}} a homeomorphism{} $h \\colon X \\to X$\nif the action generated by $\\alpha$\nlies over the action generated by~$h$.\n\\end{dfn}\n\nIn Definition~\\ref{D_4Y12_OverX},\nfor $g \\in G$ and $d \\in D$,\nthe function $x \\mapsto \\alpha_{g, x} (d)$\nmust be continuous.\nThe following elementary lemma,\nwhich will be used without comment,\nshows that if $G$ is discrete\nthen this is the only continuity condition that is needed.\nIf $G$ is not discrete,\nthere are additional continuity conditions.\n\n\\begin{lem}\\label{L_9Y30_MixCont}\nLet $D$ be a C*-algebra,\nlet $X$ be a locally compact Hausdorff space,\nand let $x \\mapsto \\alpha_{x}$\nbe a continuous function{} from $X$ to ${\\mathrm{Aut}} (D)$.\nThe for every $a \\in C_0 (X, D)$,\nthe function $b (x) = \\alpha_x (a (x))$ is also in $C_0 (X, D)$.\n\\end{lem}\n\n\\begin{proof}\nIt is immediate that $b$ vanishes at infinity.\nFor continuity,\nlet $x_0 \\in X$ and let $\\varepsilon > 0$.\nChoose an open set $U \\S X$ such that $x_0 \\in U$ and\nfor all $x \\in U$ we have\n\\[\n\\| a (x) - a (x_0) \\| < \\frac{\\varepsilon}{2}\n\\qquad {\\mbox{and}} \\qquad\n\\| \\alpha_x (a (x_0)) - \\alpha_{x_0} (a (x_0)) \\| < \\frac{\\varepsilon}{2}.\n\\]\nThen, using $\\| \\alpha_x \\| = 1$ for all $x \\in X$,\none sees that $x \\in U$\nimplies $\\| \\alpha_x (a (x)) - \\alpha_{x_0} (a (x_0)) \\| < \\varepsilon$.\n\\end{proof}\n\nFor any group~$G$,\nthere are also algebraic conditions\nrelating the automorphisms $\\alpha_{g, x}$,\ncoming from the requirement that $g \\mapsto \\alpha_g$\nbe a group homomorphism.\nIf $G = {\\mathbb{Z}}$,\nthen we really need only the function $x \\mapsto \\alpha_{1, x}$.\nFor reference,\nwe give the relevant statement as a lemma.\n\n\\begin{lem}\\label{L_4Y12_GetAuto}\nLet $X$ be a locally compact Hausdorff space,\nlet $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a C*-algebra.\nThen there is a one to one correspondence between actions of ${\\mathbb{Z}}$ on\n$C_0 (X, D)$\nthat lie over $h$ and\ncontinuous functions from $X$ to ${\\mathrm{Aut}} (D)$,\ngiven as follows.\n\nFor any function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$,\nthere is an automorphism $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$\ngiven by $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$,\nand this automorphism lies over~$h$.\n\nConversely,\nif $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$ lies over~$h$,\nthen there is a function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$\nand such that $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\n\\end{lem}\n\n\\begin{proof}\nUsing Lemma~\\ref{L_9Y30_MixCont}, this is immediate.\n\\end{proof}\n\nThere is a conflict in the notation in Lemma~\\ref{L_4Y12_GetAuto}:\nif $n \\in {\\mathbb{Z}}$\nthen $\\alpha_n$\nis one of the automorphisms in the action on $C_0 (X, D)$\n(namely $\\alpha_n$),\nwhile if $x \\in X$\nthen $\\alpha_x \\in {\\mathrm{Aut}} (D)$.\nWe use this notation anyway to avoid having more letters.\nTo distinguish the two uses,\ntake $\\alpha$ to be the action\nand write $x \\mapsto \\alpha_x$\nwhen the function from $X$ to ${\\mathrm{Aut}} (D)$ is intended.\n\nIf $D$ is prime,\nthen every action on $C_0 (X, D)$\nlies over an action of $G$ on~$X$.\n\n\\begin{lem}\\label{N_4Y12_IfDSimple}\nLet $X$ be a locally compact Hausdorff space,\nlet $D$ be a prime C*-algebra,\nlet $G$ be a topological group,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X, D))$\nbe an action of $G$ on $C_0 (X, D)$.\nThen there exists an action of $G$ on~$X$\nsuch that $\\alpha$ lies over this action.\n\\end{lem}\n\n\\begin{proof}\nThe action of $G$ on~$X$\nis obtained from the identification\n$X \\cong {\\mathrm{Prim}} ( C_0 (X, D) )$.\n\\end{proof}\n\n\\begin{prp}\\label{P_4Y15_CPSimple}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nthe action is minimal and\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple.\n\\end{prp}\n\n\\begin{proof}\nFor any C*-algebra~$A$,\nlet ${\\widehat{A}}$ be the space of unitary equivalence classes\nof irreducible representations of~$A$,\nwith the hull-kernel topology.\nSince the primitive ideals\nof $C (X, D)$ are exactly the kernels of the point evaluations,\nthere is an obvious map $q \\colon \\CXDHat{X}{D} \\to X$,\nand the open sets in $\\CXDHat{X}{D}$ are exactly\nthe sets $q^{-1} (U)$ for open sets $U \\subset X$.\nIt is now immediate that for every finite set\n$S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in \\CXDHat{X}{D} \\colon\n  {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in $\\CXDHat{X}{D}$.\nThat is, the action of $G$ on $\\CXDHat{X}{D}$\nis topologically free in the sense of Definition~1 of~\\cite{AS}.\n\nLet $J \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nbe a nonzero ideal.\nLet\n\\[\n\\pi \\colon C^* \\big( G, \\, C (X, D), \\, \\alpha \\big)\n      \\to C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)\n\\]\nbe the quotient map.\nTheorem~1 of~\\cite{AS}\nimplies that $\\pi^{-1} (J)$ has nonzero intersection\nwith the canonical copy of $C (X, D)$\nin $C^* \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nTherefore $J$ has nonzero intersection\nwith the canonical copy of $C (X, D)$\nin $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nSince $D$ is simple,\nthis intersection has the form $C_0 (U, D)$\nfor some nonempty open set $U \\subset X$.\n\nSince the action of $G$ on $X$ is minimal and $X$ is compact,\nthere exist $n \\in {\\mathbb{Z}}_{> 0}$ and $g_1, g_2, \\ldots, g_n \\in {\\mathbb{Z}}_{> 0}$\nsuch that the sets\n$g_1^{-1} U, \\, g_2^{-1} U, \\, \\ldots, g_n^{-1} U$ cover~$X$.\nChoose\n\\[\nf_1, f_2, \\ldots, f_n\n \\in C (X)\n \\subset C (X, D)\n \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)\n\\]\nsuch that ${\\mathrm{supp}} (f_k) \\subset g_k^{-1} U$ for $k = 1, 2, \\ldots, n$\nand $\\sum_{k = 1}^n f_k = 1$.\nFor $k = 1, 2, \\ldots, n$, the functions\n$\\alpha_{g_k} (f_k)$\nare in $C_0 (U) \\subset C_0 (U, D) \\subset J$,\nso\n\\[\n1 = \\sum_{k = 1}^n f_k\n  = \\sum_{k = 1}^n u_{g_k}^* \\alpha_{g_k} (f_k) u_{g_k}\n  \\in J.\n\\]\nSo $J = C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\n\\end{proof}\n\n\\begin{prp}\\label{P_5418_CPStFin}\nAssume the hypotheses of Proposition~\\ref{P_4Y15_CPSimple},\nand in addition assume that\n$G$ is amenable\nand $D$ has a tracial state.\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nhas a tracial state\nand is stably finite.\n\\end{prp}\n\n\\begin{proof}\nSince $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple},\nit suffices to show that\n$C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nhas a tracial state.\nWe know that the tracial state space\n${\\operatorname{T}} (C (X, D))$ is nonempty,\nsince one can compose a tracial state on~$D$\nwith a point evaluation\n$C (X, D) \\to D$.\nSince $G$ is amenable,\ncombining Theorem 2.2.1 and 3.3.1 of \\cite{Gr}\nshows that $C (X, D)$ has a $G$-invariant tracial state~$\\tau$.\nStandard methods show that the composition of $\\tau$\nwith the conditional expectation\nfrom $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nto $C (X, D)$\nis a tracial state on\n$C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\n\\end{proof}\n\nThe following condition is a technical hypothesis\nwhich we need for the proof of the main large subalgebra result\n(Theorem~\\ref{T_5418_AYLg}).\n\n\\begin{dfn}\\label{D_4Y12_PsPer}\nLet $D$ be a C*-algebra,\nand let $S \\subset {\\mathrm{Aut}} (D)$.\nWe say that $S$ is {\\emph{pseudoperiodic}}\nif for every $a \\in D_{+} \\setminus \\{ 0 \\}$\nthere is $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that for every $\\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\}$\nwe have $b \\precsim \\alpha (a)$.\n\\end{dfn}\n\nThe interpretation of pseudoperiodicity is roughly as follows.\nSuppose $S \\subset {\\mathrm{Aut}} (D)$ is pseudoperidic.\nThen there is no sequence $(\\alpha_n)_{n \\in {\\mathbb{Z}}_{> 0}}$\nin $S$ for which there is a nonzero element\n$\\eta \\in W (D)$\nsuch that the sequence $(\\alpha_{n} (\\eta))_{n \\in {\\mathbb{Z}}_{> 0}}$\nbecomes arbitrarily small in $W (D)$\nin a heuristic sense.\n\nWe give some conditions which imply pseudoperiodicity.\n\n\\begin{lem}\\label{L_4Y12_ApproxInn}\nLet $D$ be a unital C*-algebra.\nThen the set of approximately inner automorphisms of $D$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nIt suffices to prove that\n$a \\precsim \\alpha (a)$ for every approximately inner automorphism\n$\\alpha \\in {\\mathrm{Aut}} (D)$.\nTo see this, let $\\varepsilon > 0$ be arbitrary,\nand use approximate innerness\nto chose a unitary $u \\in D$ such that\n$\\| u \\alpha (a) u^* - a \\| < \\varepsilon$.\n\\end{proof}\n\n\\begin{lem}\\label{Lem1_190201D}\nLet $D$ be a simple C*-algebra.\nLet $S \\S {\\mathrm{Aut}} (D)$\nbe a subset which is compact in the topology\nof pointwise convergence in the norm on~$D$.\nThen $S$ is pseudoperiodic\nin the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nWithout loss of generality{} $\\| a \\| = 1$.\nFor $\\beta \\in S \\cup \\{ {\\mathrm{id}}_D \\}$ set\n\\[\nU_{\\beta}\n = \\left\\{ \\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\} \\colon\n    \\| \\alpha (a) - \\beta (a) \\| < \\frac{1}{2} \\right\\}.\n\\]\nBy compactness,\nthere are $\\beta_1, \\beta_2, \\ldots, \\beta_n \\in S \\cup \\{ {\\mathrm{id}}_D \\}$\nsuch that $U_{\\beta_1}, U_{\\beta_2}, \\ldots, U_{\\beta_n}$\ncover $S \\cup \\{ {\\mathrm{id}}_D \\}$.\nSince $\\bigl( \\beta (a) - \\frac{1}{2} \\bigr)_{+} \\neq 0$\nfor all $\\beta \\in {\\mathrm{Aut}} (D)$,\nby Lemma~2.6 of~\\cite{PhLg} there is $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim \\bigl( \\beta_k (a) - \\frac{1}{2} \\bigr)_{+}$\nfor $j = 1, 2, \\ldots, n$.\nLet $\\alpha \\in S \\cup \\{ {\\mathrm{id}}_D \\}$.\nChoose $k \\in \\{ 1, 2, \\ldots, n \\}$ such that $\\alpha \\in U_{\\beta_k}$.\nThen $\\| \\beta_k (a) - \\alpha (a) \\| < \\frac{1}{2}$,\nso\n$b \\precsim \\bigl( \\beta_k (a) - \\frac{1}{2} \\bigr)_{+}\n   \\precsim \\alpha (a)$.\n\\end{proof}\n\nThe following result will not be used,\nsince large subalgebras are not used in our proofs\nwhen $D$ is purely infinite.\nIt is included as a further example of pseudoperiodicity.\n\n\\begin{lem}\\label{L_9212_PISCA}\nLet $D$ be a purely infinite simple C*-algebra.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nLet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nThen $a \\precsim b$ for all $b \\in D_{+} \\setminus \\{ 0 \\}$.\nIn particular,\n$a \\precsim \\alpha (a)$ for all $\\alpha \\in {\\mathrm{Aut}} (D)$.\n\\end{proof}\n\n\\begin{lem}\\label{L_4Y12_TrPrsv}\nLet $D$ be a simple unital C*-algebra{}\nwhich has strict comparison of positive elements.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nIf $D$ is finite dimensional,\nthe conclusion is immediate.\nOtherwise,\nlet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nWithout loss of generality{} $\\| a \\| \\leq 1$.\nThen $\\tau (a) \\leq d_{\\tau} (a)$\nfor all $\\tau \\in {\\operatorname{QT}} (D)$.\nMoreover,\nsince ${\\operatorname{QT}} (D)$ is compact and $D$ is simple,\nthe number\n$\\delta = \\inf_{\\tau \\in {\\operatorname{QT}} (D)} \\tau (a)$\nsatisfies $\\delta > 0$.\nUse Corollary~2.5 of~\\cite{PhLg}\nto find\n$b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n$d_{\\tau} (\\langle b \\rangle ) < \\delta$\nfor all $\\tau \\in {\\operatorname{QT}} (D)$.\nThen\nfor every $\\tau \\in {\\operatorname{QT}} (D)$,\nusing $\\tau \\circ \\alpha \\in {\\operatorname{QT}} (D)$\nat the second step,\nwe have\n\\[\nd_{\\tau} (b)\n < \\delta\n \\leq (\\tau \\circ \\alpha) (a)\n \\leq d_{\\tau} ( \\alpha (a) ).\n\\]\nThe strict comparison hypothesis therefore implies\nthat $b \\precsim_A \\alpha (a)$.\n\\end{proof}\n\n\\begin{dfn}\\label{Def2-181221D}\nLet $A$ be a C*-algebra.\nWe say that the\n{\\emph{order on projections over $A$\nis determined by quasitraces}}\nif whenever $p,q \\in M_{\\infty} (A)$ are projections such that\n$\\tau (p) < \\tau(q) $\nfor all $\\tau \\in \\operatorname{QT} (A)$, then $p \\precsim q$.\n\\end{dfn}\n\n\\begin{lem}\\label{L_5417_SP}\nLet $D$ be a simple unital C*-algebra{}\nwith Property~(SP)\nand such that\nthe order on projections over~$A$ is determined by quasitraces.\nThen ${\\mathrm{Aut}} (D)$\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{lem}\n\n\\begin{proof}\nIf $D$ is finite dimensional,\nthe conclusion is immediate.\nOtherwise,\nlet $a \\in D_{+} \\setminus \\{ 0 \\}$.\nChoose a nonzero projection{} $p \\in {\\overline{a D a}}$.\nSince ${\\operatorname{QT}} (D)$ is compact and $D$ is simple,\nthe number\n$\\delta = \\inf_{\\tau \\in {\\operatorname{QT}} (D)} \\tau (p)$\nsatisfies $\\delta > 0$.\nChoose $n \\in {\\mathbb{Z}}_{> 0}$\nsuch that $\\frac{1}{n} < \\delta$.\nUse Lemma~2.3 of~\\cite{PhLg}\nto choose a unitary $u \\in A$\nand a nonzero positive element $b \\in A$\nsuch that the elements\n\\[\nb, \\, u b u^{-1}, \\, u^2 b u^{-2}, \\, \\ldots, \\, u^n b u^{-n}\n\\]\nare pairwise orthogonal.\nChoose a nonzero projection{} $q \\in {\\overline{b D b}}$.\nThen\nfor every $\\tau \\in {\\operatorname{QT}} (D)$,\nusing $\\tau \\circ \\alpha \\in {\\operatorname{QT}} (D)$\nat the third step,\nwe have\n\\[\n\\tau (q)\n \\leq \\frac{1}{n + 1}\n < \\delta\n \\leq (\\tau \\circ \\alpha) (p).\n\\]\nThe strict comparison hypothesis therefore implies\nthat $q$ is Murray-von Neumann equivalent{} to a subprojection of~$\\alpha (p)$.\nIt follows that $q \\precsim_A \\alpha (a)$.\n\\end{proof}\n\n\nThe following definition is intended only for convenience\nin this paper.\n(The condition occurs several times as a hypothesis,\nand is awkward to state.)\n\n\\begin{dfn}\\label{D_4Y16_PsPrGen}\nLet $X$ be a locally compact Hausdorff space,\nlet $G$ be a topological group,\nand let $D$ be a C*-algebra.\nLet $\\alpha \\colon G \\to {\\mathrm{Aut}} (C_0 (X, D))$\nbe an action of $G$ on $C_0 (X, D)$\nwhich lies over an action of $G$ on~$X$,\nand let $(g, x) \\mapsto \\alpha_{g, x} \\in {\\mathrm{Aut}} (D)$\nbe as in Definition~\\ref{D_4Y12_OverX}.\nWe say that $\\alpha$ is\n{\\emph{pseudoperiodically generated}}\nif\n\\[\n\\big\\{ \\alpha_{g, x} \\colon {\\mbox{$g \\in G$ and $x \\in X$}} \\big\\}\n\\]\nis pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}.\n\\end{dfn}\n\n\\begin{lem}\\label{L_4Y16_WhenPPG}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAs in Lemma~\\ref{L_4Y12_GetAuto},\nlet $( \\alpha_{x} )_{x \\in X}$\nbe the family in ${\\mathrm{Aut}} (D)$\nsuch that $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\nSuppose that the subgroup $H$ of ${\\mathrm{Aut}} (D)$ generated by\n$\\{ \\alpha_x \\colon x \\in X \\}$ is pseudoperiodic.\nThen the action of ${\\mathbb{Z}}$ generated by $\\alpha$\nis pseudoperiodically generated.\n\\end{lem}\n\n\\begin{proof}\nOne checks that if $(n, x) \\mapsto \\alpha_{n, x} \\in {\\mathrm{Aut}} (D)$\nis determined\n(following the notation of Definition~\\ref{D_4Y12_OverX})\nby $\\alpha^n (a) (x) = \\alpha_{n, x} (a (h^{-n} (x))$,\nthen $\\alpha_{n, x} \\in H$ for all $n \\in {\\mathbb{Z}}$ and $x \\in X$.\n\\end{proof}\n\n\\section{The orbit breaking subalgebra for a nonempty set meeting\neach orbit at most once}\\label{Sec_OrbBreak}\n\n\\indent\nLet $h \\colon X \\to X$ be a homeomorphism{}\nof a compact Hausdorff space~$X$,\nand let $D$ be a simple unital C*-algebra.\nFor $Y \\subset X$ closed,\nfollowing Putnam~\\cite{Pt1},\nin Definition~7.3 of~\\cite{PhLg}\nwe defined the $Y$-orbit breaking subalgebra\n$C^* ({\\mathbb{Z}}, X, h)_Y \\subset C^* ({\\mathbb{Z}}, X, h)$.\nHere,\nfor an automorphism $\\alpha \\in {\\mathrm{Aut}} ( C (X, D) )$\nwhich lies over~$h$\nwe define $C^* ({\\mathbb{Z}},  C (X, D), \\alpha)_Y$.\nWe prove that if $X$ is infinite,\n$h$~is minimal,\n$Y$ intersects each orbit at most once,\nand an additional technical condition is satisfied\n(namely, that the action of ${\\mathbb{Z}}$ generated by~$\\alpha$\nis pseudoperiodically generated),\nthen $C^* ({\\mathbb{Z}},  C (X, D), \\alpha)_Y$\nis a large subalgebra of $C^* ({\\mathbb{Z}},  C (X, D), \\alpha)$\nof crossed product type,\nin the sense of Definition~4.9 of~\\cite{PhLg}.\nThis is a generalization\nof Theorem~7.10 of~\\cite{PhLg}.\n\n\n\\begin{ntn}\\label{N_4Y12_CPN}\nLet $G$ be a discrete group,\nlet $A$ be a C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (A)$ be an action of $G$ on~$A$.\nWe identify $A$ with a subalgebra of $C^*_{\\mathrm{r}} (G, A, \\alpha)$\nin the standard way.\nWe let $u_g \\in M (C^*_{\\mathrm{r}} (G, A, \\alpha))$\nbe the standard unitary\ncorresponding to $g \\in G$.\nWhen $G = {\\mathbb{Z}}$, we write just $u$ for the unitary~$u_1$\ncorresponding to the generator $1 \\in {\\mathbb{Z}}$.\nWe let $A [G]$ denote the dense *-subalgebra\nof $C^*_{\\mathrm{r}} (G, A, \\alpha)$\nconsisting of sums $\\sum_{g \\in S} a_g u_g$\nwith $S \\subset G$ finite and $a_g \\in A$ for $g \\in S$.\nWe may always assume $1 \\in S$.\nWe let $E_{\\alpha} \\colon C^*_{\\mathrm{r}} (G, A, \\alpha) \\to A$\ndenote the standard conditional expectation,\ndefined on $A [G]$ by\n$E_{\\alpha} \\left( \\sum_{g \\in S} a_g u_g \\right) = a_1$.\nWhen $\\alpha$ is understood, we just write~$E$.\n\nWhen $G$ acts on a compact Hausdorff space~$X$,\nwe use obvious analogs of this notation for $C^*_{\\mathrm{r}} (G, X)$,\nwith the action as in Notation~\\ref{N_5418_ActCX}.\nIn particular,\nif $G = {\\mathbb{Z}}$ and the action of generated by a homeomorphism{}\n$h \\colon X \\to X$,\nwe have $u f u^* = f \\circ h^{-1}$.\n\\end{ntn}\n\n\\begin{ntn}\\label{N_4Y12_C0UD}\nFor a locally compact Hausdorff space~$X$\nand a C*-algebra~$D$,\nwe identify $C_0 (X, D) = C_0 (X) \\otimes D$\nin the standard way.\nFor an open subset $U \\subset X$,\nwe use the abbreviation\n\\[\nC_0 (U, D)\n = \\big\\{ a \\in C_0 (X, D) \\colon\n     {\\mbox{$a (x) = 0$ for all $x \\in X \\setminus U$}} \\big\\}\n \\subset C_0 (X, D).\n\\]\nThis subalgebra is of course canonically isomorphic to\nthe usual algebra $C_0 (U, D)$ when $U$ is considered\nas a locally compact Hausdorff space{} in its own right.\n\\end{ntn}\n\nIn particular,\nif $Y \\subset X$ is closed, then\n\\begin{equation}\\label{Eq_4Y12_C0XYD}\nC_0 (X \\setminus Y, \\, D)\n = \\big\\{ a \\in C_0 (X, D) \\colon\n  {\\mbox{$a (x) = 0$ for all $x \\in Y$}} \\big\\}.\n\\end{equation}\n\nThe following definition is the analog of\nDefinition~7.3 of~\\cite{PhLg}.\n\n\\begin{dfn}\\label{D_4Y12_OrbSubalg}\nLet $X$ be a locally compact Hausdorff space,\nlet $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} ( C_0 (X, D))$\nbe an automorphism which lies over~$h$.\nLet $Y \\subset X$ be a nonempty closed subset,\nand, following~(\\ref{Eq_4Y12_C0XYD}), define\n\\[\nC^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)_Y\n = C^* \\big(  C_0 (X, D), \\, C_0 (X \\setminus Y, \\, D) u \\big)\n  \\subset C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big).\n\\]\nWe call it the {\\emph{$Y$-orbit breaking subalgebra}}\nof $C^* \\big( {\\mathbb{Z}},  C_0 (X, D), \\alpha \\big)$.\n\\end{dfn}\n\nWe show that if $Y$ intersects each orbit of~$h$ at most once,\nand the action of ${\\mathbb{Z}}$ generated by~$\\alpha$\nis pseudoperiodically generated,\nthen $C^* ({\\mathbb{Z}},  C_0 (X, D), \\alpha)_Y$\nis a large subalgebra of $C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)$\nof crossed product type.\n\nThe following lemma is the analog of Proposition~7.5 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_AYStruct}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet\n\\[\nu \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)\n\\qquad {\\mbox{and}} \\qquad\nE_{\\alpha} \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\to C (X, D)\n\\]\nbe as in Notation~\\ref{N_4Y12_CPN}.\nLet $Y \\subset X$ be a nonempty closed subset.\nFor $n \\in {\\mathbb{Z}}$, set\n\\[\nY_n = \\begin{cases}\n   \\bigcup_{j = 0}^{n - 1} h^j (Y)    & \\hspace{3em} n > 0\n        \\\\\n   \\varnothing                        & \\hspace{3em} n = 0\n       \\\\\n   \\bigcup_{j = 1}^{- n} h^{-j} (Y)   & \\hspace{3em} n < 0.\n\\end{cases}\n\\]\nThen\n\\begin{align}\\label{Eq:2816CharOB}\n& C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n\\\\\n& = \\big\\{ a \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\colon\n    {\\mbox{$E_{\\alpha} (a u^{-n}) \\in C_0 (X \\setminus Y_n, \\, D)$\n          for all $n \\in {\\mathbb{Z}}$}} \\big\\}\n       \\notag\n\\end{align}\nand\n\\begin{equation}\\label{Eq:2816Dense}\n{\\overline{C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\cap\nC (X, D) [{\\mathbb{Z}}]}}\n = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y.\n\\end{equation}\n\\end{lem}\n\n\\begin{proof}\nBy Lemma~\\ref{L_4Y12_GetAuto},\nthere exists a function $x \\mapsto \\alpha_{x}$\nfrom $X$ to ${\\mathrm{Aut}} (D)$\nsuch that $x \\mapsto \\alpha_{x} (d)$ is continuous{} for all $d \\in D$\nand which satisfies $\\alpha (a) (x) = \\alpha_x (a (h^{-1} (x))$\nfor all $a \\in C_0 (X, D)$ and $x \\in X$.\n\nMost of the proof of Proposition~7.5 of~\\cite{PhLg}\ngoes through with only the obvious changes.\nIn analogy with that proof,\ndefine\n\\[\nB = \\big\\{ a \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\colon\n    {\\mbox{$E_{\\alpha} (a u^{-n}) \\in C_0 (X \\setminus Y_n, \\, D)$\n          for all $n \\in {\\mathbb{Z}}$}} \\big\\}\n\\]\nand\n\\[\nB_0 = B \\cap C (X) [{\\mathbb{Z}}].\n\\]\nThen $B_0$ is dense in~$B$ by the same reasoning as in~\\cite{PhLg}\n(using Ces\\`{a}ro means and Theorem VIII.2.2 of~\\cite{Dv}).\n\nThe proof of Proposition~7.5 of~\\cite{PhLg}\nshows that\nwhen $0 \\leq m \\leq n$ and also when $0 \\geq m \\geq n$,\nwe have $Y_m \\subset Y_n$,\nthat for $n \\in {\\mathbb{Z}}$,\nwe have\n\\begin{equation}\\label{Eq_4Y14_hnyn}\nh^{-n} (Y_n) = Y_{-n},\n\\end{equation}\nand that for $m, n \\in {\\mathbb{Z}}$,\nwe have $Y_{m + n} \\subset Y_m \\cup h^m (Y_n)$.\nIt then follows,\nas in~\\cite{PhLg},\nthat $B_0$ is a *-algebra,\nand that\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\subset {\\overline{B_0}} = B$.\n\nWe next claim that for all $n \\in {\\mathbb{Z}}$\nand $f \\in C_0 (X \\setminus Y_n, \\, D)$,\nwe have $f u^n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nThe changes to the proof of Proposition~7.5 of~\\cite{PhLg}\nat this point are more substantial.\n\nFor $n = 0$ the claim is trivial.\nLet $n > 0$\nand let $f \\in C_0 (X \\setminus Y_n, \\, D)$.\nDefine $b = (f^* f)^{1 \/ (2 n)}$.\nLet $s \\in C_0 (X \\setminus Y_n, \\, D)''$\nbe the partial isometry in the polar decomposition of~$f$,\nso that $f = s (f^* f)^{1 \/ 2} = s b^n$.\nIt follows from Proposition~1.3 of~\\cite{Cu0}\nthat the element $a_0 = s b$ is in $C_0 (X \\setminus Y_n, \\, D)$.\nMoreover,\n$a_0 (f^* f)^{\\frac{1}{2} - \\frac{1}{2 n}} = f$.\nDefine $a_1 \\in C (X, D)$\nby $a_1 (x) = \\alpha_{h (x)}^{-1} \\big( b (h (x)) \\big)$\nfor $x \\in X$,\nand for $k = 1, 2, \\ldots, n - 2$ inductively define\n$a_{k + 1} \\in C (X, D)$\nby $a_{k + 1} (x) = \\alpha_{h (x)}^{-1} \\big( a_k (h (x)) \\big)$\nfor $x \\in X$.\nThe definition of $Y_n$\nimplies that\n$a_1, a_2, \\ldots, a_{n - 1} \\in C_0 (X \\setminus Y, \\, D)$,\nand we already have\n$a_0 \\in C_0 (X \\setminus Y_n, \\, D) \\S C_0 (X \\setminus Y, \\, D)$.\nTherefore the element\n\\[\na = (a_0 u) (a_1 u) \\cdots (a_{n - 1} u)\n\\]\nis in $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nFor $k = 1, 2, \\ldots, n - 2$ and $x \\in X$, we have\n\\[\n\\alpha (a_{k + 1} ) (x)\n = \\alpha_x (a_{k + 1} (h^{-1} (x))\n = \\alpha_x \\big( \\alpha_{x}^{-1} ( a_k (x) ) \\big)\n = a_k (x),\n\\]\nso $\\alpha (a_{k + 1} ) = a_k$.\nSimilarly, $\\alpha (a_1) = b$.\nNow\n\\begin{align*}\na\n& = a_0 (u a_1 u^{-1}) (u^2 a_2 u^{-2})\n    \\cdots \\big( u^{n - 1} a_{n - 1} u^{- (n - 1)} \\big) u^n\n  \\\\\n& = a_0 \\alpha (a_1) \\alpha^2 (a_2)\n    \\cdots \\alpha^{n - 1} (a_{n - 1}) u^n\n  = (s b) b^{n - 1} u^n\n  = f u^n.\n\\end{align*}\nSo $f u^n \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nFinally,\nsuppose $n < 0$,\nand let $f \\in C_0 (X \\setminus Y_n, \\, D)$.\nIt follows from~(\\ref{Eq_4Y14_hnyn})\nthat $f \\circ h^n \\in C_0 (X \\setminus Y_{- n}, \\, D)$,\nwhence also $(f \\circ h^n)^* \\in C_0 (X \\setminus Y_{- n}, \\, D)$.\nSince $- n > 0$,\nwe therefore get\n\\[\nf u^n = \\big( u^{-n} f^* \\big)^*\n      = \\big( (f \\circ h^n)^* u^{- n} \\big)^*\n      \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y.\n\\]\nThe claim is proved.\n\nIt now follows that\n$B_0 \\subset C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nCombining this result with ${\\overline{B_0}} = B$\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\subset B$,\nwe get $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y = B$.\n\\end{proof}\n\nThe following lemma is the analog of Lemma~7.8 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_78}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nFollowing Notation~\\ref{N_4Y12_CPN},\nlet\n$a \\in C (X, D) [G]\n \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nand let $\\varepsilon > 0$.\nThen there exists $f \\in C (X) \\subset C (X, D)$\nsuch that\n\\[\n0 \\leq f \\leq 1,\n\\qquad\nf a^* a f \\in C (X, D),\n\\qquad {\\mbox{and}} \\qquad\n\\| f a^* a f \\| \\geq \\| E_{\\alpha} (a^* a) \\| - \\varepsilon.\n\\]\n\\end{lem}\n\n\\begin{proof}\nWe follow the proof of Lemma~7.8 of~\\cite{PhLg}.\nSet $b = a^* a$.\nThere are a finite set $T \\subset G$\nwith $1 \\in T$,\nand elements $b_g \\in C (X, D)$ for $g \\in T$,\nsuch that $b = \\sum_{g \\in T} b_g u_g$.\nIf $\\| b_1 \\| \\leq \\varepsilon$ take $f = 0$.\nOtherwise, define\n\\[\nU = \\big\\{ x \\in X \\colon\n   \\| b_1 (x) \\| > \\| E (a^* a) \\| - \\varepsilon \\big\\},\n\\]\nwhich is a nonempty open subset of~$X$.\nChoose $V$, $W$, $f$, and $x_0$\nas in the proof of Lemma~7.8 of~\\cite{PhLg}.\nThen,\nas there, $f b f = f b_1 f$.\nMoreover,\n\\[\n\\| f b_1 f \\|\n \\geq f (x_0) \\| b_1 (x_0) \\| f (x_0)\n = \\| b_1 (x_0) \\|\n > \\| E_{\\alpha} (a^* a) \\| - \\varepsilon.\n\\]\nThis completes the proof.\n\\end{proof}\n\nThe following lemma is the analog of Lemma~7.9 of~\\cite{PhLg}.\n\n\\begin{lem}\\label{L_4X21_79}\nLet $G$, $X$, the action of $G$ on $X$,\n$D$, and $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe as in Lemma~\\ref{L_4X21_78}.\nLet $B \\subset C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nbe a unital subalgebra\nsuch that,\nfollowing Notation~\\ref{N_4Y12_CPN},\n\\begin{enumerate}\n\\item\\label{L_4X21_79:CXI}\n$C (X, D) \\subset B$.\n\\item\\label{L_4X21_79:FSD}\n$B \\cap C (X, D)[G]$ is dense in~$B$.\n\\end{enumerate}\nLet $a \\in B_{+} \\setminus \\{ 0 \\}$.\nThen there exists $b \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim_{B} a$.\n\\end{lem}\n\n\\begin{proof}\nWe follow the proof of Lemma~7.9 of~\\cite{PhLg},\nusing our Lemma~\\ref{L_4X21_78}\nin place of Lemma~7.8 of~\\cite{PhLg},\nexcept that $c \\in B \\cap C (X, D)[G]$.\nThe element $(f c^* c f - 2 \\varepsilon)_{+}$\nwe obtain now satisfies\n$(f c^* c f - 2 \\varepsilon)_{+} \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nand $(f c^* c f - 2 \\varepsilon)_{+} \\precsim_{B} a$.\n\\end{proof}\n\nIn Lemma~\\ref{L_4X21_79},\nwe really want to have $b \\in C (X)_{+} \\setminus \\{ 0 \\}$.\nWhen $G = {\\mathbb{Z}}$ and under the pseudoperiodicity hypothesis\nof \\Def{D_4Y16_PsPrGen}, this is possible.\n\n\\begin{lem}\\label{L_4Y11_Comp}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact set\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nSet $B = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nThen for every $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nthere exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset B$\nsuch that $f \\precsim_B a$.\n\\end{lem}\n\n\\begin{proof}\nUse Kirchberg's Slice Lemma (Lemma 4.1.9 of~\\cite{Rrd})\nto find $g_0 \\in C (X)_{+} \\setminus \\{ 0 \\}$\nand $d_0 \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_5417_Zero}\ng_0 \\otimes d_0 \\precsim_{C (X, D)} a.\n\\end{equation}\nAs in Definition~\\ref{D_4Y12_OverX},\nlet $(m, x) \\mapsto \\alpha_{m, x}$\nbe the function ${\\mathbb{Z}} \\times X \\to {\\mathrm{Aut}} (D)$\nsuch that $\\alpha^m (a) (x) = \\alpha_{m, x} (a (h^{-m} (x))$\nfor $m \\in {\\mathbb{Z}}$, $a \\in C_0 (X, D)$, and $x \\in X$.\nSince the action generated by $\\alpha$\nis pseudoperiodically generated,\nthere exists $d_1\\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_5418_PsPer}\n\\alpha_{m, x} (d_1) \\precsim d_0\n\\end{equation}\nfor all $x \\in X$ and $m \\in {\\mathbb{Z}}$.\nWithout loss of generality{} $\\| d_1 \\| = 1$.\nSet $d = \\big( d_1 - \\tfrac{1}{2} \\big)_{+}$.\nUse Corollary 1.14 of~\\cite{PhLg}\nto find $k \\in {\\mathbb{Z}}_{> 0}$ and $w_1, w_2, \\ldots, w_k \\in D$\nsuch that\n\\begin{equation}\\label{Eq_5418_Sum1}\n\\sum_{j = 1}^k w_j d w_j^* = 1.\n\\end{equation}\n\nThe set $X \\setminus Y$ is dense in~$X$,\nso there is $x_0 \\in X \\setminus Y$ such that $g_0 (x_0) \\neq 0$.\nChoose $g_1 \\in C (X)_{+}$ such that $g_1 (x_0) = 1$\nand $g_1 |_Y = 0$.\nThen\n\\begin{equation}\\label{Eq_5417_One}\ng_1 g_0 \\otimes d_0\n \\precsim_{C (X, D)} g_0 \\otimes d_0.\n\\end{equation}\nSet\n$U_0 = \\big\\{ x \\in X \\colon (g_1 g_0) (x) \\neq 0 \\big\\}$.\nChoose a nonempty open set $U \\subset X$\nsuch that ${\\overline{U}} \\subset U_0$.\nSet\n\\begin{equation}\\label{Eq_9Z01_NewLabel}\n\\rho = \\inf_{x \\in {\\overline{U}}} (g_0 g_1) (x).\n\\end{equation}\nThen $\\rho > 0$.\nThe set\n\\[\nX \\setminus \\bigcup_{n \\in {\\mathbb{Z}}} h^n (Y)\n = \\bigcap_{n \\in {\\mathbb{Z}}} h^n (X \\setminus Y)\n\\]\nis dense by the Baire Category Theorem.\nSo we can choose\n$y_0 \\in U \\cap \\left( X \\setminus\n\\bigcup_{n \\in {\\mathbb{Z}}} h^n (Y) \\right)$.\nThe forward orbits $\\{ h^n (y_0) \\colon n \\geq N \\}$\nof $y_0$ are all dense,\nso there exist $n_1, n_2, \\ldots, n_k \\in {\\mathbb{Z}}_{\\geq 0}$\nwith $0 = n_1 < n_2 < \\cdots< n_k$\nand $h^{n_j} (y_0) \\in U$ for $j = 1, 2, \\ldots, k$.\n\nChoose an open set $V$ containing $y_0$ which is so small that\nthe following hold:\n\\begin{enumerate}\n\\item\\label{L_4Y11_Comp_Cont}\n${\\overline{V}} \\subset X \\setminus\n\\bigcup_{m = 0}^{n_k} h^{- m} (Y)$.\n\\item\\label{L_4Y11_Comp_InU}\n$h^{n_j} ( {\\overline{V}} ) \\subset U$\nfor $j = 1, 2, \\ldots, k$.\n\\item\\label{L_4Y11_Comp_Disj}\nThe sets\n${\\overline{V}}, \\, h ( {\\overline{V}} ), \\, \\ldots, \\,\n   h^{n_k} ( {\\overline{V}} )$\nare disjoint.\n\\item\\label{L_4Y11_Comp_Small}\nFor $y \\in V$ and $m = 0, 1, \\ldots, n_k$,\nwe have\n\\[\n\\bigl\\| \\alpha_{m, h^m (y)} (d_1) - \\alpha_{m, h^m (y_0)} (d_1) \\bigr\\|\n< \\frac{1}{4}.\n\\]\n\\setcounter{TmpEnumi}{\\value{enumi}}\n\\end{enumerate}\nChoose $f_0, f \\in C (X)_{+}$\nsuch that:\n\\begin{enumerate}\n\\setcounter{enumi}{\\value{TmpEnumi}}\n\\item\\label{L_4Y11_Comp_1}\n$\\| f_0 \\|, \\, \\| f \\| \\leq 1$.\n\\item\\label{L_4Y11_Comp_Supp}\n${\\mathrm{supp}} (f_0) \\subset V$.\n\\item\\label{L_4Y11_Comp_b0b}\n$f_0 f = f$.\n\\item\\label{L_4Y11_Comp_Aty0}\n$f (y_0) = 1$.\n\\end{enumerate}\nFor $m = 0, 1, \\ldots, n_k - 1$,\ndefine $c_m \\in C (X, D)$\nby $c_m (x) = f ( h^{- m} (x)) \\alpha_{m, x} (d)$\nfor $x \\in X$.\nThus\n\\begin{equation}\\label{Eq_5417_cm}\nc_m = \\alpha^m (f \\otimes d).\n\\end{equation}\nAlso set\n\\begin{equation}\\label{Eq_5418_DefBt}\n\\beta_m = \\alpha_{m, h^m (y_0)}.\n\\end{equation}\n\nWe claim that for $m = 0, 1, \\ldots, n_k$\nwe have\n\\[\nc_m \\precsim_{C (X, D)} (f \\circ h^{-m}) \\otimes \\beta_m (d_1).\n\\]\n\nTo prove the claim, let $\\varepsilon > 0$.\nSet $L = h^{m} ( {\\mathrm{supp}} (f_0) )$.\nDefine $b \\in C (L, D)$\nby $b (x) = \\alpha_{m, x} (d_1)$ for $x \\in L$.\nSince $L \\subset h^m (V)$,\ncondition~(\\ref{L_4Y11_Comp_Small})\nimplies that\nfor $x \\in L$\nwe have $\\| \\alpha_{m, x} (d_1) - \\beta_m (d_1) \\| < \\frac{1}{2}$.\nIn $C (L, D)$ we therefore get\n$\\| b - 1 \\otimes \\beta_m (d_1) \\|\n  \\leq \\frac{1}{4} < \\frac{1}{2}$.\nSo\n\\[\n\\bigl( b - \\tfrac{1}{2} \\bigr)_{+}\n  \\precsim_{C (L, D)} 1 \\otimes \\beta_m (d_1).\n\\]\nTherefore there exists $v_0 \\in C (L, D)$\nsuch that\n\\[\n\\big\\| v_0^* ( 1 \\otimes \\beta_m (d_1) ) v_0\n   - \\bigl( b - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n < \\frac{\\varepsilon}{2}.\n\\]\nDefine $v, a \\in C (X, D)$ by\n\\[\nv (x) = \\begin{cases}\n   f_0 (h^{- m} (x)) v_0 (x) & x \\in L\n        \\\\\n   0                     & x \\not\\in L\n\\end{cases}\n\\qquad {\\mbox{and}} \\qquad\na = v^* \\bigl[ (f \\circ h^{-m}) \\otimes \\beta_m (d_1) \\bigr] v.\n\\]\nWe show that $\\| a - c_m \\| < \\varepsilon$.\n\nFor $x \\in X \\setminus L$,\nwe have $a (x) = c_m (x) = 0$.\nFor $x \\in L$,\nusing $f_0 f = f$ at the second step\nand using $d = \\bigl( d_1 - \\tfrac{1}{2} \\bigr)_{+}$\nand the definition of~$b$ at the third step,\nwe get\n\\begin{align*}\n& \\| a (x) - c_m (x) \\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = \\big\\| f_0 (h^{- m} (x))^2 f (h^{- m} (x)) v_0 (x)^*\n          \\beta_m (d_1) v_0 (x)\n       - f (h^{- m} (x)) \\alpha_{m, x} (d) \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = \\big\\| f (h^{- m} (x))\n         [ v_0 (x)^* \\beta_m (d_1) v_0 (x) - \\alpha_{m, x} (d) ] \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = f (h^{- m} (x))\n       \\big\\| v_0 (x)^* \\beta_m (d_1) v_0 (x)\n              - \\bigl( b (x) - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  \\leq f (h^{- m} (x))\n        \\big\\| v_0^* ( 1 \\otimes \\beta_m (d_1) ) v_0\n            - \\bigl( b - \\tfrac{1}{2} \\bigr)_{+} \\big\\|\n  < \\frac{\\varepsilon}{2}.\n\\end{align*}\nTaking the supremum over $x \\in X$ gives\n$\\| a - c_m \\| \\leq \\frac{\\varepsilon}{2} < \\varepsilon$.\nThus\n\\[\n\\big\\| v^* [ (f \\circ h^{-m}) \\otimes \\beta_m (d_1) ] v - c_m \\big\\|\n < \\varepsilon.\n\\]\nSince $\\varepsilon > 0$ is arbitrary,\nthe claim follows.\n\nFor $m = 0, 1, \\ldots, n_k$,\nthe functions $f \\circ h^{-m}$ are orthogonal\nsince the sets $h^m ( {\\overline{V}} )$ are disjoint.\nThe claim therefore implies that\n\\begin{equation}\\label{Eq_5417_Two}\n\\sum_{j = 1}^k c_{n_j}\n \\precsim_{C (X, D)}\n     \\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1).\n\\end{equation}\n\nWe now claim that\n\\begin{equation}\\label{Eq_5417_Three}\nf \\otimes 1 \\precsim_{B} \\sum_{j = 1}^k c_{n_j}.\n\\end{equation}\nTo prove this claim,\nfor $j = 1, 2, \\ldots, k$\ndefine\n\\[\nv_j = [(f \\circ h^{-n_j}) \\otimes 1] u^{n_j} (1 \\otimes w_j)^*\n    \\in C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr).\n\\]\nCombining (\\ref{L_4Y11_Comp_Supp}) and~(\\ref{L_4Y11_Comp_Cont}),\nwe see that $f_0$ vanishes in particular on the sets\n\\[\nh^{-1} ( Y ), \\, h^{-2} ( Y ), \\, \\ldots, \\, h^{- n_j} ( Y ),\n\\]\nwhence $f_0 \\circ h^{- n_j}$ vanishes on the sets\n\\[\nY, \\, h ( Y ), \\, \\ldots, \\, h^{n_j - 1} ( Y ).\n\\]\nSo $v_j \\in B$ by\n\\Lem{L_4X21_AYStruct}.\nUsing (\\ref{Eq_5417_cm}) at the first step\nand $f_0 f = f$ at the last step,\nwe calculate:\n\\begin{align*}\nv_j^* c_{n_j} v_j\n& = (1 \\otimes w_j) u^{- n_j} [(f \\circ h^{-n_j}) \\otimes 1]\n     \\alpha^{n_j} (f \\otimes d) [(f \\circ h^{-n_j}) \\otimes 1]\n     u^{n_j} (1 \\otimes w_j)^*\n\\\\\n& = (1 \\otimes w_j) (f_0 \\otimes 1) (f \\otimes d)\n     (f_0 \\otimes 1) (1 \\otimes w_j^*)\n  = f \\otimes w_j d w_j^*.\n\\end{align*}\nWe apply (\\ref{Eq_5418_Sum1}) at the first step\nand use orthogonality\nof $c_0, c_1, \\ldots, c_{n_k}$\nand Lemma 1.4(12) of~\\cite{PhLg} at the second step,\nto get\n\\[\nf \\otimes 1\n = \\sum_{j = 1}^k f \\otimes w_j d w_j^*\n \\precsim_B \\sum_{j = 1}^k c_{n_j}.\n\\]\nThis proves the claim~(\\ref{Eq_5417_Three}).\n\nWe next claim that\n\\begin{equation}\\label{Eq_5417_Four}\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)} g_0 g_1 \\otimes d.\n\\end{equation}\nTo prove this,\ncombine (\\ref{Eq_5418_PsPer}), (\\ref{Eq_5418_DefBt}),\nand Lemma 1.11 of~\\cite{PhLg}\nto get\n\\[\n(f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)} (f \\circ h^{-n_j}) \\otimes d_0\n\\]\nfor $j = 1, 2, \\ldots, k$.\nSince the functions\n$f \\circ h^{-n_j}$ are orthogonal,\nit follows that\n\\[\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes \\beta_{n_j} (d_1)\n \\precsim_{C (X, D)}\n \\left( \\sum_{j = 1}^k (f \\circ h^{-n_j}) \\right) \\otimes d_0.\n\\]\nUsing orthogonality\nof the functions\n$f \\circ h^{-n_j}$ again,\ntogether with\n\\[\n{\\mathrm{supp}} (f \\circ h^{-n_j}) \\subset h^{n_j} ( {\\overline{V}} )\n \\subset U\n\\qquad {\\mbox{and}} \\qquad\n0 \\leq f \\circ h^{-n_j} \\leq 1\n\\]\nfor $j = 1, 2, \\ldots, k$,\nwe see that\n$0 \\leq \\sum_{j = 1}^k f \\circ h^{-n_j} \\leq 1$\nand,\nby~(\\ref{L_4Y11_Comp_InU}), that this sum is supported in~$U$.\nSince $g_0 g_1 \\geq \\rho$ on ${\\overline{U}}$\nby~(\\ref{Eq_9Z01_NewLabel}),\nwe get the first step in the following computation;\nthe second step is clear:\n\\[\n\\sum_{j = 1}^k (f \\circ h^{-n_j}) \\otimes d_0\n \\leq \\frac{1}{\\rho} ( g_0 g_1 \\otimes d_0 )\n \\sim_{C (X, D)} g_0 g_1 \\otimes d_0.\n\\]\nThe claim is proved.\n\nNow combine\n(\\ref{Eq_5417_Zero}),\n(\\ref{Eq_5417_One}),\n(\\ref{Eq_5417_Four}),\n(\\ref{Eq_5417_Two}),\nand\n(\\ref{Eq_5417_Three})\nto get $f \\otimes 1 \\precsim_{B} a$.\n\\end{proof}\n\n\\begin{cor}\\label{C_5418_CompCX}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact set\n(possibly empty)\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nSet $B = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$.\nLet $a \\in B_{+} \\setminus \\{ 0 \\}$.\nThen there exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset B$\nsuch that $f \\precsim_B a$.\n\\end{cor}\n\n\\begin{proof}\n\\Lem{L_4X21_AYStruct}\nimplies that $B$ satisfies the hypotheses\nof \\Lem{L_4X21_79}\nwith $G = {\\mathbb{Z}}$.\nTherefore,\nby \\Lem{L_4X21_79},\nthere exists $b \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nsuch that $b \\precsim_{B} a$.\nNow \\Lem{L_4Y11_Comp}\nprovides $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\precsim_B a$.\n\\end{proof}\n\nWe can now prove the analog of Theorem~7.10 of~\\cite{PhLg}.\n\n\\begin{thm}\\label{T_5418_AYLg}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a simple unital C*-algebra{} which has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type\nin the sense of Definition~4.9 of~\\cite{PhLg}.\n\\end{thm}\n\n\\begin{proof}\nWe verify the hypotheses of\nProposition~4.11 of~\\cite{PhLg}.\nWe follow Notation~\\ref{N_4Y12_CPN}.\nSet\n\\[\nA = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr),\n\\qquad\nB = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y,\n\\]\n\\[\nC = C (X, D),\n\\qquad {\\mbox{and}} \\qquad\nG = \\{ u \\}.\n\\]\n\nThe algebra $A$ is simple\nby Proposition~\\ref{P_4Y15_CPSimple}\nand finite by Proposition~\\ref{P_5418_CPStFin}.\nIn particular,\ncondition~(1)\nin Proposition 4.11 of~\\cite{PhLg} holds.\n\nWe next verify condition~(2)\nin Proposition 4.11 of~\\cite{PhLg}.\nAll parts are obvious except~(2d).\nSo let $a \\in A_{+} \\setminus \\{ 0 \\}$\nand $b \\in B_{+} \\setminus \\{ 0 \\}$.\nApply Corollary~\\ref{C_5418_CompCX} twice,\nthe first time with $Y = \\varnothing$ and $a$ as given\nand the second time with $Y$ as given\nand with $b$ in place of~$a$.\nWe get $a_0, b_0 \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $a_0 \\precsim_{A} a$ and $b_0 \\precsim_{B} b$.\n\nWe can now argue as in the corresponding part of the\nproof of Theorem~7.10 of~\\cite{PhLg}\nto find $f \\in C (X)$\nsuch that\n\\[\nf \\precsim_{C (X)} a_0 \\precsim_{A} a\n\\qquad {\\mbox{and}} \\qquad\nf \\precsim_{B} b_0 \\precsim_{B} b,\n\\]\ncompleting the proof of condition~(2d).\nAlternatively,\n$a_0, b_0\n \\in C^* ({\\mathbb{Z}}, X, h)_Y\n \\subset C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$,\nand $C^* ({\\mathbb{Z}}, X, h)_Y$ is a large subalgebra of $C^* ( {\\mathbb{Z}}, X, h)$\nof crossed product type\nby Theorem~7.10 of~\\cite{PhLg},\nso the existence of $f$\nfollows from condition~(2d)\nin Proposition 4.11 of~\\cite{PhLg}.\n\nWe next verify condition~(3)\nin Proposition 4.11 of~\\cite{PhLg}.\nLet $m \\in {\\mathbb{Z}}_{> 0}$,\nlet $a_1, a_2, \\ldots, a_m \\in A$,\nlet $\\varepsilon > 0$,\nand let $b \\in B_{+} \\setminus \\{ 0 \\}$.\nWe follow the corresponding part of the\nproof of Theorem~7.10 of~\\cite{PhLg}.\nChoose $c_1, c_2, \\ldots, c_m \\in C (X, D) [{\\mathbb{Z}}]$\nsuch that $\\| c_j - a_j \\| < \\varepsilon$\nfor $j = 1, 2, \\ldots, m$.\n(This estimate is condition~(3b).)\nChoose $N \\in {\\mathbb{Z}}_{> 0}$\nsuch that for $j = 1, 2, \\ldots, m$\nthere are $c_{j, l} \\in C (X, D)$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$\nwith\n\\[\nc_j = \\sum_{l = -N}^N c_{j, l} u^l.\n\\]\n\nApply Corollary~\\ref{C_5418_CompCX} to~$B$,\nto find $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\precsim_{B} b$.\nSet $U = \\{ x \\in X \\colon f (x) \\neq 0 \\}$,\nand choose nonempty disjoint open sets\n$U_l \\subset U$ for $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$.\nFor each such~$l$,\nuse Lemma~7.7 of~\\cite{PhLg}\nto choose $f_l, r_l \\in C (X)_{+}$\nsuch that $r_l (x) = 1$ for all $x \\in h^l (Y)$,\nsuch that $0 \\leq r_l \\leq 1$,\nsuch that ${\\mathrm{supp}} (f_l) \\subset U_l$,\nand such that $r_l \\precsim_{C^* ({\\mathbb{Z}}, X, h)_Y} f_l$.\nThen also $r_l \\precsim_{B} f_l$.\n\nChoose an open set $W$ containing~$Y$\nsuch that\n\\[\nh^{-N} (W), \\, h^{- N + 1} (W),\n \\, \\ldots, \\, h^{N - 1} (W), \\, h^{N} (W)\n\\]\nare disjoint,\nand choose $r \\in C (X)$ such that $0 \\leq r \\leq 1$,\n$r (x) = 1$ for all $x \\in Y$,\nand ${\\mathrm{supp}} (r) \\subset W$.\nSet\n\\[\ng_0 = r \\cdot \\prod_{l = - N}^{N} r_l \\circ h^{l}.\n\\]\nSet $g_l = g_0 \\circ h^{- l}$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$.\nThen $0 \\leq g_l \\leq r_l \\leq 1$.\nSet $g = \\sum_{l = - N}^{N} g_l$.\nThe supports of the functions $g_l$ are disjoint,\nso $0 \\leq g \\leq 1$.\nThis is condition~(3a)\nin Proposition 4.11 of~\\cite{PhLg}.\nThe proof of Theorem~7.10 of~\\cite{PhLg}\nshows that\n$g \\precsim_{C^* ({\\mathbb{Z}}, X, h)_Y} f \\precsim_B b$.\nSince $C^* ({\\mathbb{Z}}, X, h)_Y \\subset B$,\nit follows that $g \\precsim_B b$.\nThis is condition~(3d)\nin Proposition 4.11 of~\\cite{PhLg}.\n\nIt remains to verify condition~(3c)\nin Proposition 4.11 of~\\cite{PhLg}.\nThis is done the same way as\nthe corresponding part of the proof of Theorem~7.10 of~\\cite{PhLg},\nexcept using \\Lem{L_4X21_AYStruct}\nin place of Proposition~7.5 of~\\cite{PhLg}.\n\\end{proof}\n\nThe following corollary is the analog of Corollary~7.11\nof~\\cite{PhLg}.\n\n\\begin{cor}\\label{C_5418_AYStabLg}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal\nhomeomorphism, let $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a stably large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin the sense of Definition~5.1 of~\\cite{PhLg}.\n\\end{cor}\n\n\\begin{proof}\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is stably finite\n(by Proposition~\\ref{P_5418_CPStFin}),\nwe can combine Theorem~\\ref{T_5418_AYLg},\nProposition~4.10 of~\\cite{PhLg},\nand Corollary~5.8 of~\\cite{PhLg}.\n\\end{proof}\n\n\\begin{cor}\\label{AyCentLarge}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal\nhomeomorphism, let $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nAssume that the action generated by $\\alpha$\nis pseudoperiodically generated.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin the sense of Definition~3.2 of~\\cite{ArPh}.\n\\end{cor}\n\n\\begin{proof}\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is stably finite\n(by Proposition~\\ref{P_5418_CPStFin}),\nwe can combine Theorem~\\ref{T_5418_AYLg}\nand Theorem~4.6 of~\\cite{ArPh}.\n\\end{proof}\n\nWe conclude by giving some conditions on $D$ and~$\\alpha$ which\nguarantee the hypotheses of Corollary~\\ref{AyCentLarge}.\nThese are more natural to consider than the awkward\npseudoperiodicity hypothesis.\n\n\\begin{cor}\\label{AyCentLargeConds}\nLet $X$ be a compact metric space,\nlet $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $Y \\subset X$ be a compact subset\nsuch that $h^{n} (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\set{0}$,\nlet $D$ be a simple unital C*-algebra{}\nwhich has a tracial state,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $x \\mapsto \\alpha_{x}$\nbe the corresponding map from $X$ to ${\\mathrm{Aut}} (D)$,\nas in Lemma~\\ref{L_4Y12_GetAuto}.\nAssume one of the following conditions holds.\n\\begin{enumerate}\n\\item\\label{9212_AyCentLargeConds_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9212_AyCentLargeConds_StrCmp}\n$D$ has strict comparison of positive elements.\n\\item\\label{9212_AyCentLargeConds_SP}\n$D$ has property (SP) and the order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9212_AyCentLargeConds_CptGen}\nThe set\n$\\set{ \\alpha_{x} \\colon x \\in X}$\nis contained in a subgroup of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nThen\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\n\\end{cor}\n\n\\begin{proof}\nWe claim that, under any of the conditions\n(\\ref{9212_AyCentLargeConds_AppInn}),\n(\\ref{9212_AyCentLargeConds_StrCmp}),\n(\\ref{9212_AyCentLargeConds_SP}),\nor (\\ref{9212_AyCentLargeConds_CptGen}),\nthe set $\\set{ \\alpha_{x} \\colon x \\in X}$\nis contained in a pseudoperiodic subgroup of ${\\mathrm{Aut}} (D)$.\nFor~(\\ref{9212_AyCentLargeConds_AppInn}),\nthis follows from Lemma~\\ref{L_4Y12_ApproxInn};\nfor~(\\ref{9212_AyCentLargeConds_StrCmp}),\nfrom Lemma~\\ref{L_4Y12_TrPrsv};\nfor~(\\ref{9212_AyCentLargeConds_SP}),\nfrom Lemma~\\ref{L_5417_SP};\nand for~(\\ref{9212_AyCentLargeConds_CptGen}),\nfrom Lemma~\\ref{Lem1_190201D}.\nNow apply Lemma~\\ref{L_4Y16_WhenPPG}\nto see that\nthe hypotheses of Corollary~\\ref{AyCentLarge} are satisfied.\n\\end{proof}\n\nThe case in which $D$ is purely infinite and simple is\nmuch easier.\nWe give a direct proof, not depending on large subalgebras,\nthat $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis purely infinite simple for any discrete group~$G$.\nIt is still true,\nand will be proved below (Proposition \\ref{T_9921_PICase}),\nthat, under the other hypotheses of Theorem~\\ref{T_5418_AYLg}\nthe subalgebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type.\nThis fact seems potentially useful,\nbut does not help with the analysis of any of the examples\nin Section~\\ref{Sec_InfRed}.\n\n\\begin{thm}\\label{T_0101_PICase}\nLet $G$ be a discrete group,\nlet $X$ be a compact space,\nand suppose $G$ acts on $X$ in such a way that\nthe action is minimal and\nfor every finite set $S \\subset G \\setminus \\{ 1 \\}$,\nthe set\n\\[\n\\big\\{ x \\in X \\colon {\\mbox{$g x \\neq x$ for all $g \\in S$}} \\big\\}\n\\]\nis dense in~$X$.\nLet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over the given action of $G$ on $X$\n(in the sense of Definition~\\ref{D_4Y12_OverX}).\nThen $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis purely infinite simple.\n\\end{thm}\n\nWe saw in Proposition~\\ref{P_4Y15_CPSimple}\nthat $C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$\nis simple,\nbut we won't use that in this proof.\n\n\\begin{proof}[Proof of Theorem~\\ref{T_0101_PICase}]\nFor convenience of notation,\nset $A = C^*_{\\mathrm{r}} \\big( G, \\, C (X, D), \\, \\alpha \\big)$.\nWe also freely identify $C (X, D)$ with $C (X) \\otimes D$.\nWe prove the result by showing that $1_A \\precsim a$\nfor all $a \\in A_{+} \\setminus \\{ 0 \\}$.\nBy Lemma~\\ref{L_4X21_79}\n(taking $B$ there to be~$A$),\nwe can assume that $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$.\n\nUse Kirchberg's Slice Lemma (Lemma 4.1.9 of~\\cite{Rrd})\nto find $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nand $b \\in D_{+} \\setminus \\{ 0 \\}$\nsuch that\n\\begin{equation}\\label{Eq_0201_Slice}\nf \\otimes b \\precsim_{C (X, D)} a.\n\\end{equation}\nWithout loss of generality{} $\\| f \\| = 1$.\nSince $D$ is purely infinite and simple,\nwe have $1_D \\precsim_{D} b$, and so it follows that\n\\begin{equation}\\label{Eq_0201_InD}\nf \\otimes 1_D \\precsim_{D} f \\otimes b.\n\\end{equation}\nSet $U = \\bigl\\{ x \\in X \\colon f (x) > \\frac{1}{2} \\bigr\\}$.\nBy minimality of the action,\nthe sets $g U$ for $g \\in G$ cover~$X$.\nSo there are $n \\in {\\mathbb{Z}}_{> 0}$ and $g_1, g_2, \\ldots, g_n \\in G$\nsuch that the sets $g_1 U, g_2 U, \\ldots, g_n U$ cover~$X$.\nThe function $x \\mapsto \\sum_{k = 1}^n f (g_k^{-1} x)$\nis strictly positive on~$X$.\nUsing this fact at the first step, pure infiniteness of~$D$\nat the second last step,\nand (\\ref{Eq_0201_InD}) and~(\\ref{Eq_0201_Slice})\nat the last step,\nwe get\n\\begin{align*}\n1_A\n& \\precsim_{C (X)} \\sum_{k = 1}^n \\alpha_{g_k} (f \\otimes 1_D)\n\\\\\n& \\precsim_{C (X)} {\\mathrm{diag}} \\bigl( \\alpha_{g_1} (f \\otimes 1_D), \\,\n       \\alpha_{g_2} (f \\otimes 1_D), \\, \\ldots, \\,\n        \\alpha_{g_n} (f \\otimes 1_D) \\bigr)\n\\\\\n& = {\\mathrm{diag}} \\bigl( u_{g_1} (f \\otimes 1_D) u_{g_1}^*, \\,\n       u_{g_2} (f \\otimes 1_D) u_{g_2}^*, \\, \\ldots, \\,\n        u_{g_n} (f \\otimes 1_D) u_{g_n}^* \\bigr)\n\\\\\n& \\sim_{A} {\\mathrm{diag}} \\bigl( f \\otimes 1_D, \\,\n       f \\otimes 1_D, \\, \\ldots, \\,\n        f \\otimes 1_D \\bigr)\n  = f \\otimes 1_{M_n (D)}\n  \\precsim_{C (X, D)} f \\otimes 1_D\n  \\precsim_{A} a.\n\\end{align*}\nThis completes the proof.\n\\end{proof}\n\nAs promised,\nwe now give a result on large subalgebras when $G = {\\mathbb{Z}}$.\nWe use two lemmas,\nthe first of which has a similar proof to that of\nTheorem~\\ref{T_0101_PICase}.\n\n\\begin{lem}\\label{L_0101_SubalgPI}\nLet $X$ be a compact metric space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be a compact subset\nsuch that $h^n (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\{ 0 \\}$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis purely infinite and simple.\n\\end{lem}\n\n\\begin{proof}\nFor convenience of notation,\nset $B = C^* \\big( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\big)_Y$.\nWe also freely identify $C (X, D)$ with $C (X) \\otimes D$.\nWe prove the result by showing that $1_A \\precsim a$\nfor all $a \\in A_{+} \\setminus \\{ 0 \\}$.\nBy Lemma~\\ref{L_4X21_79}\nand Lemma~\\ref{L_4X21_AYStruct},\nwe can assume that $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$.\nUsing Kirchberg's Slice Lemma\nand pure infiniteness of~$D$\nas in the proof of Theorem~\\ref{T_0101_PICase},\nthere is $f \\in C (X)_{+} \\setminus \\{ 0 \\}$\nsuch that $f \\otimes 1_D \\precsim_{C (X, D)} a$.\n\nWe now claim that for every $x \\in X$ there is $l_x \\in C (X)_{+}$\nsuch that $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$\nand $l_x (x) > \\frac{1}{2}$.\nGiven this,\nthe proof is finished as in the last computation\nin the proof of\nTheorem~\\ref{T_0101_PICase}.\n\nTo prove the claim,\nset $U = \\bigl\\{ x \\in X \\colon f (x) > \\frac{1}{2} \\bigr\\}$.\nIf $x \\in U$,\ntake $l_x = f$.\n\nSuppose next that $x \\not\\in U$\nand $x \\not\\in \\bigcup_{m = 0}^{\\infty} h^m (Y)$.\nBy minimality of~$h$,\nthere is $n \\in {\\mathbb{Z}}_{> 0}$ such that $x \\in h^{n} (U)$.\nChoose $r \\in C (X)_{+}$ such that $r (y) = 0$\nfor all $y \\in \\bigcup_{m = 0}^{n - 1} h^m (Y)$\nand $r (x) = 1$.\nThen, letting $u \\in C^* \\big( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\big)$\nbe the standard unitary (as in Notation~\\ref{ntn1-181221D}),\nwe have $(r \\otimes 1_D) u^n \\in B$\nby Lemma~\\ref{L_4X21_AYStruct}.\nTaking $l_x = r^2 \\cdot (f \\circ h^{- n})$,\nwe have $l_x (x) = f ( h^{- n} (x)) > \\frac{1}{2}$.\nAlso, $(r u^n) (f \\otimes 1_D) (r u^n)^* = l_x \\otimes 1_D$,\nso $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$.\n\nFinally, suppose $x \\not\\in U$\nand $x \\in \\bigcup_{m = 0}^{\\infty} h^m (Y)$.\nThen $x \\not\\in \\bigcup_{m = 1}^{\\infty} h^{- m} (Y)$.\nBy minimality of~$h$,\nthere is $n \\in {\\mathbb{Z}}_{> 0}$ such that $x \\in h^{- n} (U)$.\nChoose $r \\in C (X)_{+}$ such that $r (y) = 0$\nfor all $y \\in \\bigcup_{m = 1}^{n} h^{- m} (Y)$\nand $r (x) = 1$.\nThen $(r \\otimes 1_D) u^{- n} \\in B$\nby Lemma~\\ref{L_4X21_AYStruct}.\nTaking $l_x = r^2 \\cdot (f \\circ h^{n})$,\nwe have $l_x (x) = f ( h^{n} (x)) > \\frac{1}{2}$.\nAlso, $(r u^{- n}) (f \\otimes 1_D) (r u^{- n})^* = l_x \\otimes 1_D$,\nso $l_x \\otimes 1_D \\precsim_{B} f \\otimes 1_D$.\nThis completes the proof.\n\\end{proof}\n\n\\begin{lem}\\label{L_9Z01_KeepLarge}\nLet $X$ be a compact Hausdorff space,\nlet $G$ be a discrete group,\nand let $D$ be a C*-algebra.\nLet $(g, x) \\mapsto g x$ be an action of $G$ on~$X$,\nand let $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$\nbe an action of $G$ on $C (X, D)$\nwhich lies over $(g, x) \\mapsto g x$\nin the sense of Definition~\\ref{D_4Y12_OverX}.\nThen for every\n$b \\in C^*_{\\mathrm{r}} \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$\nand every $\\varepsilon > 0$,\nthere is a finite set $T \\S G$ and a nonempty open set $V \\S X$\nsuch that,\nwhenever $x \\in V$\nand $f \\in C (X)$ satisfy $0 \\leq f \\leq 1$\nand $f (g x) = 1$ for all $g \\in T$,\nthen $\\| f b \\| > \\| b \\| - \\varepsilon$.\n\\end{lem}\n\nIf $D$ is not unital, then the product $f b$ is realized via\nthe inclusion of $C (X)$ in\n$M \\bigl( C^* \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr) \\bigr)$.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_9Z01_KeepLarge}]\nLet $(g, x) \\mapsto \\alpha_{g, x}$ be as in Definition~\\ref{D_4Y12_OverX}.\n\nFix a faithful representation $\\rho$ of $D$ on a Hilbert space~$H$.\nThen for every $x \\in X$ there is a representation\n$\\rho \\circ {\\mathrm{ev}}_x \\colon C (X, D) \\to L (H)$.\nLet $(v_x, \\pi_x)$ be the corresponding regular covariant\nrepresentation of $\\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$\non $l^2 (G, H)$,\nand let\n$\\sigma_x \\colon C^* \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr) \\to l^2 (G, H)$\nbe its integrated form.\nWe identify $l^2 (G, H)$ with $l^2 (G) \\otimes H$,\nand we write $\\delta_g \\in l^2 (G)$ for the standard basis vector\ncorresponding to $g \\in G$.\nFor later use, we recall that if $S_1, S_2 \\S G$ are finite sets,\n$d \\in C (X, D) [G]$\nis given as $d = \\sum_{g \\in S_1} d_g u_g$\nwith $d_g \\in C (X, D)$ for $g \\in S_1$,\nand $\\xi \\in l^2 (G) \\otimes H$ has the form\n$\\xi = \\sum_{h \\in S_2} \\delta_h \\otimes \\xi_h$\nwith $\\xi_h \\in H$ for $h \\in S_2$,\nthen\n\\begin{equation}\\label{Eq_9X01_IndRepFormula}\n\\sigma_x (d) \\xi\n = \\sum_{g \\in S_1} \\sum_{h \\in S_2}\n   \\delta_{g h} \\otimes\n       \\rho \\bigl( \\alpha_{h^{-1} g^{-1}, x} (d_g (g h x)) \\bigr) \\xi_h.\n\\end{equation}\n\nThe representation $\\bigoplus_{x \\in X} \\rho \\circ {\\mathrm{ev}}_x$\nis a faithful representation of $C (X, D)$,\nso $\\bigoplus_{x \\in X} \\sigma_x$\nis a faithful representation of\n$C^*_{\\mathrm{r}} \\bigl( G, \\, C (X, D), \\, \\alpha \\bigr)$.\nTherefore there exists $x_0 \\in X$ such that\n$\\| \\sigma_{x_0} (b) \\| > \\| b \\| - \\frac{\\varepsilon}{5}$.\nChoose $c \\in C (X, D) [G]$ such that\n$\\| b - c \\| < \\frac{\\varepsilon}{5}$.\nIn particular,\n$\\| \\sigma_{x_0} (c) \\| > \\| b \\| - \\frac{2 \\varepsilon}{5}$.\nChoose $\\xi \\in l^2 (G, H)$ with finite support such that\n$\\| \\xi \\| = 1$ and\n$\\| \\sigma_{x_0} (c) \\xi \\| > \\| b \\| - \\frac{3 \\varepsilon}{5}$.\nWrite $c = \\sum_{g \\in S_1} c_g u_g$\nand $\\xi = \\sum_{h \\in S_2} \\delta_h \\otimes \\xi_h$\nwith $S_1, S_2 \\S G$ finite,\n$c_g \\in C (X, D)$ for $g \\in S_1$,\nand $\\xi_h \\in H$ for $h \\in S_2$.\nUse Lemma~\\ref{L_9Y30_MixCont} to choose\nan open set $V \\S X$ such that $x_0 \\in V$\nand such that for all $x \\in V$, $g \\in S_1$, and $h \\in S_2$, we have\n\\begin{equation}\\label{Eq_9Z01_Choice}\n\\bigl\\| \\alpha_{h^{-1} g^{-1}, x} (c_g (g h x))\n    - \\alpha_{h^{-1} g^{-1}, x_0} (c_g (g h x_0) ) \\bigr\\|\n  < \\frac{\\varepsilon}{5 {\\mathrm{card}} (S_1) {\\mathrm{card}} (S_2) + 1}.\n\\end{equation}\nSet\n\\[\nT = \\bigl\\{ g h \\colon {\\mbox{$g \\in S_1$ and $h \\in S_2$}} \\bigr\\}.\n\\]\nNow let $x \\in V$,\nand suppose $f \\in C (X)$ satisfies $0 \\leq f \\leq 1$\nand $f (g x) = 1$ for all $g \\in T$.\nThe condition on~$f$\nand the formula~(\\ref{Eq_9X01_IndRepFormula})\nimply that $\\sigma_x (f c) \\xi = \\sigma_x (c) \\xi$.\nApplying~(\\ref{Eq_9X01_IndRepFormula}) again,\nand using~(\\ref{Eq_9Z01_Choice}) at the second step,\nwe get\n\\begin{equation}\\label{Eq_9Z01_xx0}\n\\| \\sigma_x (c) \\xi - \\sigma_{x_0} (c) \\xi \\|\n \\leq \\sum_{g \\in S_1} \\sum_{h \\in S_2}\n   \\bigl\\| \\alpha_{h^{-1} g^{-1}, x} (c_g (g h x))\n    - \\alpha_{h^{-1} g^{-1}, x_0} (c_g (g h x_0) ) \\bigr\\|\n < \\frac{\\varepsilon}{5}.\n\\end{equation}\nTherefore, using $\\| f \\| \\leq 1$ at the second step\nand $\\| b - c \\| < \\frac{\\varepsilon}{5}$ and the second and fifth steps,\nas well as (\\ref{Eq_9Z01_xx0}) at the fourth step,\n\\begin{align*}\n\\| f b \\|\n& \\geq \\| \\sigma_x (f b) \\|\n  > \\| \\sigma_x (f c) \\xi \\| - \\frac{\\varepsilon}{5}\n  = \\| \\sigma_x (c) \\xi \\| - \\frac{\\varepsilon}{5}\n  > \\| \\sigma_{x_0} (c) \\xi \\| - \\frac{2 \\varepsilon}{5}\n\\\\\n& > \\| \\sigma_{x_0} (b) \\xi \\| - \\frac{3 \\varepsilon}{5}\n  > \\| \\sigma_{x_0} (b) \\| - \\frac{4 \\varepsilon}{5}\n  > \\| \\sigma_{x_0} (b) \\| - \\frac{4 \\varepsilon}{5}\n  > \\| b \\| - \\varepsilon,\n\\end{align*}\nas desired.\n\\end{proof}\n\n\\begin{prp}\\label{T_9921_PICase}\nLet $X$ be a compact metric space,\nlet $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $Y \\subset X$ be a compact subset\nsuch that $h^{n} (Y) \\cap Y = \\varnothing$\nfor all $n \\in {\\mathbb{Z}} \\setminus \\set{0}$,\nlet $D$ be a purely infinite simple unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nThen\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nof crossed product type.\n\\end{prp}\n\n\\begin{proof}\nWe verify directly the conditions of Definition~4.9 of~\\cite{PhLg}.\nWe follow Notation~\\ref{N_4Y12_CPN}.\nSet\n\\[\nA = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr),\n\\qquad\nB = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y,\n\\]\n\\[\nC = C (X, D),\n\\qquad {\\mbox{and}} \\qquad\nG = \\{ u \\}.\n\\]\nAll parts of condition~(1)\nin Definition~4.9 of~\\cite{PhLg}\nare obvious.\n\nWe verify condition~(2) there.\nLet $m \\in {\\mathbb{Z}}_{> 0}$,\nlet $a_1, a_2, \\ldots, a_m \\in A$,\nlet $\\varepsilon > 0$,\nlet $b_1 \\in A_{+}$ satisfy $\\| b_1 \\| = 1$,\nand let $b_2 \\in B_{+} \\setminus \\{ 0 ]$.\nWe will part of the verification of\ncondition~(3)\nin Proposition 4.11 of~\\cite{PhLg}\nin the proof of Theorem~\\ref{T_5418_AYLg}.\nChoose $c_1, c_2, \\ldots, c_m \\in C (X, D) [{\\mathbb{Z}}]$\nsuch that $\\| c_j - a_j \\| < \\varepsilon$\nfor $j = 1, 2, \\ldots, m$.\n(This estimate is condition~(2b).)\nChoose $N \\in {\\mathbb{Z}}_{> 0}$\nsuch that for $j = 1, 2, \\ldots, m$\nthere are $c_{j, l} \\in C (X, D)$\nfor $l = -N, \\, - N + 1, \\, \\ldots, \\, N - 1, \\, N$\nwith\n\\[\nc_j = \\sum_{l = -N}^N c_{j, l} u^l.\n\\]\nApply Lemma~\\ref{L_9Z01_KeepLarge},\ngetting a finite set $T \\S {\\mathbb{Z}}$ and a nonempty open set $U \\S X$\nsuch that,\nwhenever $x \\in U$\nand $f \\in C (X)$ satisfy $0 \\leq f \\leq 1$\nand $f (h^k (x)) = 1$ for all $k \\in T$,\nthen $\\| f b_1^{1 \/ 2} \\| > 1 - \\frac{\\varepsilon}{2}$.\nDefine\n\\[\nS = \\bigl\\{ n - k \\colon\n      {\\mbox{$k \\in T$ and $n \\in [- N, N] \\cap {\\mathbb{Z}}$}} \\bigr\\},\n\\]\nwhich is a finite subset of~${\\mathbb{Z}}$.\nThen $\\bigcup_{n \\in S} h^n (Y)$ is nowhere dense in~$X$,\nso there exists $x_0 \\in W$\nsuch that $x_0 \\not\\in \\bigcup_{n \\in S} h^n (Y)$.\nThis choice implies that\n\\[\n\\{ h^k (x_0) \\colon k \\in T \\} \\cap \\bigcup_{n = - N}^{N} h^n (Y)\n  = \\varnothing,\n\\]\nso there is $g \\in C (X)$\nsuch that $0 \\leq g \\leq 1$,\n$g (y) = 1$ for all $y \\in \\bigcup_{n = - N}^{N} h^n (Y)$,\nand $g (h^k (x)) = 0$ for all $k \\in T$.\n\nCondition~(1a) in Definition~4.9 of~\\cite{PhLg}\nholds by construction.\nThe proof that $(1 - g) c_j \\in B$ for $j = 1, 2, \\ldots, m$\n(condition~(2c) in Definition~4.9 of~\\cite{PhLg})\nis the same as at the end of the proof of Theorem~\\ref{T_5418_AYLg}:\nfollow\nthe corresponding part of the proof of Theorem~7.10 of~\\cite{PhLg},\nexcept using \\Lem{L_4X21_AYStruct}\nin place of Proposition~7.5 of~\\cite{PhLg}.\nLemma~\\ref{L_0101_SubalgPI}\nimplies $1_A \\precsim_A b_1$ and $1_A \\precsim_B b_2$,\nso the requirements $g \\precsim_A b_1$ and $g \\precsim_B b_2$\n(condition~(2d) in Definition~4.9 of~\\cite{PhLg})\nfollow immediately.\nFor condition~(2e) in Definition~4.9 of~\\cite{PhLg},\nwe estimate, using $1 - g (h^k (x_0)) = 1$\nfor all $k \\in T$ at the second step:\n\\[\n\\bigl\\| (1 - g) b_1 (1 - g) \\bigr\\|\n = \\| (1 - g) b_1^{1 \/ 2} \\|^2\n > \\bigl( 1 - \\frac{\\varepsilon}{2} \\bigr)^2\n > \\varepsilon.\n\\]\nThis completes the proof.\n\\end{proof}\n\n\\section{Recursive structure for orbit breaking\n subalgebras}\\label{RecStruct}\n\nIn this section, under appropriate conditions\nwe will show that the orbit breaking subalgebras\nof Definition~\\ref{D_4Y12_OrbSubalg}\nhave a tractable recursive structure,\nas subalgebras of certain homogeneous algebras\ntensored with $D$. We start by introducing the\nformalism for a generalization of the recursive\nsubhomogeneous algebras introduced in\n\\cite{PhRsha1} that were crucial for the analysis in\n\\cite{QLinPhDiff} and \\cite{HLinPh}.\n\n\\begin{dfn}\\label{pullback}\nLet $A$, $B$, and $C$ be C*-algebra{s}, and let $\\varphi \\colon\nA \\to C$ and $\\psi \\colon B \\to C$ be homomorphisms.\nThen the associated {\\emph{pullback C*-algebra{}}} $A\n\\oplus_{C, \\varphi, \\psi} B$ is defined by\n\\[\nA \\oplus_{C, \\varphi, \\psi} B\n = \\bset{ (a, b) \\in A \\oplus B \\colon \\varphi (a) = \\psi (b) }.\n\\]\n\\end{dfn}\n\nWe frequently write $A \\oplus_{C} B$ when the maps $\\varphi$\nand $\\psi$ are understood.\nIf $C = 0$ we just get $A \\oplus B$.\n\n\\begin{dfn}\\label{RSHA}\nLet $D$ be a simple unital C*-algebra.\nThe class of\n{\\emph{recursive subhomogeneous algebras over $D$}} is\nthe smallest class $\\mathcal{R}$ of C*-algebra{s} that is\nclosed under isomorphism and such that:\n\\begin{enumerate}\n\\item\\label{9223_RSHA_1}\nIf $X$ is a compact Hausdorff space and $n \\geq 1$,\nthen $C (X, M_{n} (D)) \\in \\mathcal{R}$.\n\\item\\label{9223_RSHA_2}\nIf $B \\in \\mathcal{R}$, $X$ is compact Hausdorff,\n$n \\geq 1$, $X^{(0)} \\subset X$ is closed (possibly empty),\n$\\varphi \\colon B \\to C (X^{(0)}, \\, M_{n} (D))$\nis any unital homomorphism\n(the zero homomorphism{} if $X^{(0)}$ is empty), and\n$\\rho \\colon C (X, M_{n} (D)) \\to C (X^{(0)}, \\, M_{n} (D))$ is the\nrestriction homomorphism, then the pullback\n\\begin{align*}\n& B \\oplus_{C (X^{(0)}, \\, M_{n} (D)), \\, \\varphi, \\, \\rho)} C (X, M_{n} (D))\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n  = \\bset{ (b, f) \\in B \\oplus C (X, M_{n} (D)) \\colon\n        \\varphi (b) = f |_{(X^{(0)}} }\n\\end{align*}\nis in $\\mathcal{R}$.\n\\end{enumerate}\n\\end{dfn}\n\nTaking $D = {\\mathbb{C}}$ in this definition gives the usual definition\nfor the class of recursive subhomogeneous algebras.\n(See \\cite{PhRsha1}.)\nThis definition makes sense for any unital C*-algebra~$D$,\nbut it is not clear whether it is appropriate in this generality.\n\n\\begin{dfn}\\label{RSHAMachinery}\nWe adopt the following standard notation for recursive\nsubhomogeneous algebras over $D$.\nThe definition implies\nthat if $R$ is any recursive subhomogeneous algebra over\n$D$, then $R$ may be written in the form\n\\[\nR \\cong\n  \\left[ \\cdots \\left[ \\left[ C_{0} \\oplus_{C_{1}^{(0)}, \\varphi_1, \\rho_1}\n              C_{1} \\right]\n    \\oplus_{C_{2}^{(0)}, \\varphi_2, \\rho_2} \\right] \\cdots \\right]\n   \\oplus_{C_{l}^{(0)}, \\varphi_l, \\rho_l} C_{l},\n\\]\nwith $C_{k} = C (X_{k}, \\, M_{n (k)} (D))$ for compact Hausdorff\nspaces $X_{k}$ and positive integers $n (k)$, and with\n$C_{k}^{(0)} = C \\bigl( X_{k}^{(0)}, \\, M_{n (k)} (D) \\bigr)$\nfor compact\nsubsets $X_{k}^{(0)} \\subset X_{k}$ (possibly empty),\nwhere the maps $\\rho_{k} \\colon C_{k} \\to C_{k}^{(0)}$ are\nalways the restriction maps.\nAn expression of this type for\n$R$ will be referred to as a {\\emph{decomposition}} of~$R$,\nand the notation that appears here will be referred to as the\n{\\emph{standard notation for a decomposition}}.\nWe associate the following objects to this decomposition.\n\\begin{enumerate}\n\\item\\label{9223_RSHAMachinery_1}\nIts {\\emph{length}} $l$.\n\\item\\label{9223_RSHAMachinery_2}\nThe {\\emph{$k$-th stage algebra}}\n\\[\nR^{(k)}\n = \\left[ \\cdots \\left[ \\left[ C_{0} \\oplus_{C_{1}^{(0)}}\n      C_{1} \\right] \\oplus_{C_{2}^{(0)}} C_{2} \\right] \\cdots \\right]\n      \\oplus_{C_{k}^{(0)}} C_{k}.\n\\]\n\\item\\label{9223_RSHAMachinery_3}\nIts {\\emph{base spaces}} $X_{0},\\ldots,X_{l}$ and {\\emph{total space}}\n$X = \\coprod_{k = 0}^{l} X_{k}$.\n\\item\\label{9223_RSHAMachinery_4}\nIts {\\emph{matrix sizes}} $n (0),\\ldots,n (l)$ and {\\emph{matrix\nsize function}} $m \\colon X \\to {\\mathbb{Z}}_{\\geq 0}$ defined by $m(x) = n (k)$\nwhen $x \\in X_{k}$.\n(This is called the {\\emph{matrix size of $R$ at~$x$}}.)\n\\item\\label{9223_RSHAMachinery_5}\nIts {\\emph{minimum matrix size}} $\\min_{k} n (k)$ and\n{\\emph{maximum matrix size}} $\\max_{k} n (k)$.\n\\item\\label{9223_RSHAMachinery_6}\nIts {\\emph{topological dimension}}\n$\\dim (X) = \\max_{k} \\dim (X_{k})$\nand {\\emph{topological dimension function}}\n$d \\colon X \\to {\\mathbb{Z}}_{\\geq 0}$, defined by $d(x) = \\dim (X_{k})$\nfor $x \\in X_{k}$.\n(This is called the {\\emph{topological dimension of $R$ at $x$}}.)\n\\item\\label{9223_RSHAMachinery_7}\nIts {\\emph{standard representation}}\n$\\sigma\n = \\sigma_{R} \\colon R\n   \\to \\bigoplus_{k = 0}^{l} C (X_{k}, \\, M_{n (k)} (D))$,\ndefined by\nforgetting the restriction to a subalgebra in each of the fibered\nproducts in the decomposition.\n\\item\\label{9223_RSHAMachinery_8}\nThe associated {\\emph{evaluation maps}}\n${\\mathrm{ev}}_{x} \\colon R \\to M_{n (k)} (D)$,\ndefined to be the restriction of the usual evaluation\nmap on $\\bigoplus_{k = 0}^{l} C (X_{k}, \\, M_{n (k)} (D))$ to $R$,\nwhen $R$\nis identified with a subalgebra of this algebra through the standard\nrepresentation $\\sigma_{R}$.\n\\end{enumerate}\n\\end{dfn}\n\nA decomposition of an algebra $R$ as a\nrecursive subhomogeneous algebra over $D$ may be highly\nnonunique, which in the case $D = {\\mathbb{C}}$ is clear from examples\nin~\\cite{PhRsha1}.\nMoreover, the matrix sizes are not uniquely\ndetermined even if all the other data has already been chosen,\nwhich is easily seen through\nthe example of the $2^{\\infty}$~UHF algebra.\n\n\\begin{ntn}\\label{StronglySelfAbs}\nThroughout, we let $\\mathcal{E}$ denote a separable, unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}:\nthere\nexists an isomorphism $\\mathcal{E} \\cong \\mathcal{E} \\otimes \\mathcal{E}$ that is unitarily\nequivalent to $a \\mapsto a \\otimes 1_{\\mathcal{E}}$.\n\\end{ntn}\n\nExamples include the Cuntz algebras\n$\\mathcal{O}_{2}$ and $\\mathcal{O}_{\\infty}$,\nand the Jiang-Su algebra ${\\mathcal{Z}}$,\nwhich is a simple, separable, unital, infinite dimensional,\nstrongly selfabsorbing, nuclear C*-algebra{} having the same\nElliott invariant\nas the complex numbers ${\\mathbb{C}}$.\n(See \\cite{JiangSu} for its construction).\n\n\\begin{dfn}\\label{DStableDefn}\nAdopt Notation~\\ref{StronglySelfAbs}.\nA separable C*-algebra{} $D$ is called {\\emph{$\\mathcal{E}$-stable}}\nif there is\nan isomorphism $D \\otimes \\mathcal{E} \\cong D$.\n\\end{dfn}\n\nIt is clear that if $D$ is $\\mathcal{E}$-stable, then so are $M_{n} (D)$\nand $C (X, D)$.\nWith appropriate assumptions on the algebra $D$, we can give some\nresults about the $\\mathcal{E}$-stability of recursive subhomogeneous algebras\nover $C$.\n\n\\begin{lem}\\label{DStablePullback}\nAdopt Notation~\\ref{StronglySelfAbs}.\nLet $A$, $B$, and $C$ be separable unital C*-algebra{s}\n(allowing $C = 0$),\nand let $\\varphi \\colon A \\to C$ and $\\psi \\colon B \\to C$\nbe unital homomorphisms.\nLet $P = A \\oplus_{C, \\varphi, \\psi} B$ be the associated pullback\n(Definition~\\ref{pullback}).\nIf $\\psi$\nis surjective and both $A$ and $B$ are $\\mathcal{E}$-stable, then $P$ is\n$\\mathcal{E}$-stable.\n\\end{lem}\n\n\\begin{proof}\nDefine\n$\\pi \\colon P \\to A$ by $\\pi (a, b) = a$.\nThere is an isomorphism\n$\\iota \\colon \\mathrm{Ker} (\\psi) \\to \\mathrm{Ker} (\\pi)$\ngiven by $\\iota (b) = (0, b)$ for $b \\in \\mathrm{Ker} (\\psi)$.\nMoreover, surjectivity of $\\psi$ is easily seen to imply\nsurjectivity of $\\pi$.\nWe obtain an exact\nsequence\n\\[\n0 \\longrightarrow \\mathrm{Ker} (\\psi)\n \\stackrel{\\iota}{\\longrightarrow} P\n \\stackrel{\\pi}{\\longrightarrow} A\n \\longrightarrow 0.\n\\]\nNow $\\mathrm{Ker} (\\varphi)$ is $\\mathcal{E}$-stable by Corollary 3.1 of\n\\cite{TomsWinter2}.\nSince $B$ is also $\\mathcal{E}$-stable, Theorem 4.3 of\n\\cite{TomsWinter2} implies that $P$ is $\\mathcal{E}$-stable.\n\\end{proof}\n\n\n\\begin{prp}\\label{DStableRSHA}\nAdopt Notation~\\ref{StronglySelfAbs}.\nLet $D$ be a simple separable $\\mathcal{E}$-stable C*-algebra,\nand let $R$ be a recursive subhomogeneous algebra over $D$.\nThen $R$ is $\\mathcal{E}$-stable.\n\\end{prp}\n\n\\begin{proof}\nWe proceed by induction on the length of a decomposition for $R$ as\na recursive subhomogeneous algebra over $D$.\nThe base case is when\n$R = C (X, M_{n} (D))$, and this is $\\mathcal{E}$-stable whenever $D$ is\n$\\mathcal{E}$-stable.\nFor the inductive step, we may assume that\nthere are an $\\mathcal{E}$-stable unital C*-algebra~$R'$,\n$n \\in {\\mathbb{Z}}_{> 0}$, a compact Hausdorff space~$X$,\na closed subset $X^{(0)} \\subset X$,\nand a unital homomorphism\n$\\varphi \\colon R' \\to C \\bigl( X^{(0)}, \\, M_{n} (D) \\bigr)$\nsuch that\n\\[\nR = \\bset{ (a, f) \\in R' \\oplus C (X, M_{n} (D))\n   \\colon \\varphi (a) = f |_{X^{(0)}}  }\n\\]\nSince $f \\mapsto f |_{X^{(0)}}$ is surjective and both\n$R'$ and $C (X, M_{n} (D))$\nare $\\mathcal{E}$-stable, it follows from Lemma~\\ref{DStablePullback}\nthat $R$ is $\\mathcal{E}$-stable,\nwhich completes the induction.\n\\end{proof}\n\n\\begin{prp}\\label{AyDirLim}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed, and let\n$Y_{1} \\supset Y_{2} \\supset \\cdots$\nbe closed subsets of $X$ such that $\\bigcap_{n = 1}^{\\infty}\nY_{n} = Y$.\nThen\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n = {\\overline{ {\\ts{ {\\ds{\\bigcup}}_{n = 1}^{\\infty} }} A_{Y_{n}} }}\n \\cong \\varinjlim A_{Y_{n}}.\n\\]\n\\end{prp}\n\n\\begin{proof}\nThe proof is easy, and is omitted.\n\\end{proof}\n\nThe results in the remainder of this section are\nmostly generalizations of ones in Section 1 of\n\\cite{QLinPhDiff}.\nWe follow a slightly more modern development,\nadapted from Section 11.3 of \\cite{PhNotes}, since \\cite{QLinPhDiff}\nhas not been published.\n(The proofs in \\cite{PhNotes} are mostly\ntaken from \\cite{QLinPhDiff}, with some changes in notational\nconventions.)\nMost proofs go through with changes only to the notation,\nsuch as replacing the action of a minimal homeomorphism\n$h$ with the action $\\alpha$ on $C (X, D)$.\nThe biggest technical\ndifferences are in the proof of Lemma~\\ref{L_4X21_AYStruct}.\nWe begin with the well-known Rokhlin tower construction.\n\n\\begin{ntn}\\label{ModifiedRokhlinTower}\nLet $X$ be a compact Hausdorff space{} and let $h \\colon X \\to X$ be a minimal homeomorphism.\nLet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nFor $y \\in Y$, define\n$r (y) = \\inf \\bigl( \\set{m \\geq 1 \\colon h^{m} (y) \\in Y} \\bigr)$.\nThen $\\sup_{y \\in Y} r (y) < \\infty$\n(see Lemma \\ref{RokhlinTowerProps}(\\ref{9Z26_new})),\nso there are finitely many\ndistinct values $n (0) < n (1) < \\cdots < n (l)$ in the range of~$r$.\nFor $k = 0, 1, \\ldots l$, set\n\\[\nY_{k} = \\overline{ \\set{ y \\in Y \\colon r (y) = n (k) } }\n\\qquad {\\mbox{and}} \\qquad\nY_{k}^{\\circ}\n = {\\mathrm{int}} \\bigl( \\bset{ y \\in Y \\colon r (y) = n (k) } \\bigr).\n\\]\n\\end{ntn}\n\nWe warn that, while\n$Y_{k}^{\\circ} \\subset {\\mathrm{int}} (Y_k)$,\nthe inclusion may be proper.\n\n\\begin{lem}\\label{RokhlinTowerProps}\nLet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nThen:\n\\begin{enumerate}\n\\item\\label{9Z26_new}\n$\\sup_{y \\in Y} r (y) < \\infty$.\n\\item\\label{Item_RokhlinTowerProps_1}\nThe sets $h^{j} (Y_{k}^{\\circ})$ are pairwise disjoint for $0\n\\leq k \\leq l$ and $1 \\leq j \\leq n (k)$.\n\\item\\label{Item_RokhlinTowerProps_2}\n$\\bigcup_{k = 0}^{l} Y_{k} = Y$.\n\\item\\label{Item_RokhlinTowerProps_3}\n$\\bigcup_{k = 0}^{l} \\bigcup_{j= 0}^{n (k)-1} h^{j} (Y_{k}) = X$.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nThis is Lemma 11.3.5 of~\\cite{PhNotes}.\n\\end{proof}\n\nWe first consider the specific situation of the structure of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nwhen $X$ is the Cantor set.\nThis is needed later.\n\n\\begin{lem}\\label{CantorSetCase}\nAdopt Notation \\ref{ModifiedRokhlinTower}, let $X$ be the Cantor\nset, and let $Y \\subset X$ be a nonempty compact open subset.\nThen\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\cong \\bigoplus_{k = 0}^{l} C (Y_{k}, \\, M_{n (k)} (D)).\n\\]\n\\end{lem}\n\n\\begin{proof}\nThis is a straightforward adaption of the proof of Lemma 11.2.20\nof~\\cite{PhNotes}.\n\\end{proof}\n\nIf we set $B_{k} = C (Y_{k}, M_{n (k)})$, then $B_{k}$ is an\nAF algebra for each~$k$\n(since $Y_{k}$ is totally disconnected), and hence\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a direct sum of what might be called ``AF $D$-algebras''.\n\n\\begin{prp}\\label{BanachSpaceDirSum}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nlet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nFollowing the notation of Lemma~\\ref{L_4X21_AYStruct}, there is\n$N \\in {\\mathbb{Z}}_{> 0}$ such that $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nhas the Banach space direct sum decomposition\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n = \\bigoplus_{n = - N}^{N} C_{0} (X \\setminus Y_{n}, \\, D) u^{n}.\n\\]\n\\end{prp}\n\n\\begin{proof}\nThe proof is the same as that of Corollary 11.3.7 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{prp}\\label{AyHom}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nlet $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nand adopt Notation \\ref{ModifiedRokhlinTower}.\nDefine the unitary $s_{k} \\in C (Y_{k}, M_{n (k)} (D))$ by\n\\[\ns_{k} = \\left[ \\begin{array}{ccccccc}\n0 & 0 & \\cdots & \\cdots & 0 & 0 & 1 \\\\\n1 & 0 & \\cdots & \\cdots & 0 & 0 & 0 \\\\\n0 & 1 & \\cdots & \\cdots & 0 & 0 & 0 \\\\\n\\vdots & \\vdots & \\ddots &  & \\vdots & \\vdots & \\vdots \\\\\n\\vdots & \\vdots & & \\ddots & \\vdots & \\vdots & \\vdots \\\\\n0 & 0 & \\cdots & \\cdots & 1 & 0 & 0 \\\\\n0 & 0 & \\cdots & \\cdots & 0 & 1 & 0 \\end{array} \\right].\n\\]\nFor $k = 0, 1, \\ldots, l$ there is a unique linear map\n\\[\n\\gamma_{k} \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n \\to C (Y_{k}, M_{n (k)} (D))\n\\]\nsuch that for $m = 0, 1, \\ldots, n (k) - 1$\nand $f \\in C_{0} (X \\setminus Z_{m}, D)$ we have:\n\\begin{enumerate}\n\\item\\label{9222_AyHom_1}\n$\\gamma_{k} (f u^{m}) = {\\mathrm{diag}} \\bigl( f |_{Y_{k}}, \\,\n  \\alpha^{-1} (f) |_{Y_{k}}, \\,\n  \\ldots, \\, \\alpha^{-[n (k)-1]} (f) |_{Y_{k}} \\bigr) \\cdot s_{k}^{m}$.\n\\item\\label{9222_AyHom_2}\n$\\gamma_{k} (u^{-m} f) = s_{k}^{-m} \\cdot\n  {\\mathrm{diag}} \\bigl( f |_{Y_{k}}, \\, \\alpha^{-1} (f)\n  |_{Y_{k}}, \\, \\ldots, \\,\n   \\alpha^{-[n (k)-1]} (f) |_{Y_{k}} \\bigr)$.\n\\end{enumerate}\nMoreover, the map\n\\[\n\\gamma \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n      \\to \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))\n\\]\ngiven by $\\gamma (a) = (\\gamma_{0} (a), \\gamma_{1} (a), \\ldots, \\gamma_{l} (a))$\nis a\n$*$-homomorphism.\n\\end{prp}\n\n\\begin{proof}\nThe computations in this proof are analogous to those in\nthe proof of Proposition 11.3.9 of~\\cite{PhNotes}, using\nLemma~\\ref{L_4X21_AYStruct}\nin place of Proposition 11.3.6 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{lem}\\label{SubdiagProjnDecomp}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nlet $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$,\nand let $Y \\subset X$ be a closed set with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nAdopt Notation \\ref{ModifiedRokhlinTower}.\nIdentify $C (Y_{k}, \\, M_{n (k)} (D))$\nwith $M_{n (k)} (C (Y_{k}, D))$ in the\nobvious way.\nDefine maps\n\\[\nE_{k}^{(m)} \\colon C (Y_{k}, M_{n (k)} (D))\n\\to C (Y_{k}, M_{n (k)} (D))\n\\]\nby $E_{k}^{(m)} (b)_{m+j,j} = b_{m+j,j}$\nfor $1 \\leq j \\leq n (k) - m$ (if $m \\geq 0$)\nand for $-m + 1 \\leq j \\leq n (k)$ (if $m \\leq 0$),\nand $E_{k}^{(m)} (b)_{i,j} = 0$ for\nall other pairs $(i,j)$.\n(Thus, $E_{k}^{(m)}$ is the projection map\non the $m$-th subdiagonal.)\nWrite\n\\[\nG_{m} = \\bigoplus_{k = 0}^{l} E_{k}^{(m)} (C (Y_{k}, M_{n (k)} (D)).\n\\]\nThen:\n\\begin{enumerate}\n\\item\\label{SubdiagProjnDecomp_DirSum}\nThere is a Banach space direct sum decomposition\n\\[\n\\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D)) =\n\\bigoplus_{m=-n (l)}^{n (l)} G_{m}.\n\\]\n\\item\\label{SubdiagProjnDecomp_gmIs}\nFor $k = 0, 1,\\ldots,l$, $m \\geq 0$,\n$f \\in C_{0} (X \\setminus Z_{m}, D)$,\nand $x \\in Y_{k}$,\nthe expression $\\gamma_{k} (f u^{m}) (x)$\nis given by the following matrix, in which the first nonzero\nentry is in row $m+1$:\n\\[\n\\gamma_{k} (f u^{m}) (x)\n = \\left[ \\begin{array}{ccccccc} 0 & 0 & \\cdots &\n\\cdots  & \\cdots & \\cdots & 0\n\\\\\n\\vdots & \\vdots & & & & & \\vdots \\\\\n0 & 0 & & & & & \\vdots\n\\\\\n\\alpha^{-m} (f) (x) & 0 & & & & & \\vdots \\\\\n0 & \\alpha^{-[m+1]} (f) (x) & & & & & \\vdots\n\\\\\n\\vdots & & \\ddots & & &\n& \\vdots\n\\\\\n0 & \\cdots & \\cdots & \\alpha^{-[n (k)-1]} (f) (x) & 0\n& \\cdots & 0\n\\end{array} \\right].\n\\]\n\\item\\label{SubdiagProjnDecomp_3}\nFor $m \\geq 0$ and $f \\in C_{0} (X \\setminus Z_{m}, D)$, we\nhave\n\\[\n\\gamma_{k} (f u^{m}) \\in E_{k}^{(m)} (C (Y_{k}, M_{n (k)} (D))),\n\\qquad\n\\gamma (f u^{m}) \\in G_{m},\n\\]\n\\[\n\\gamma_{k} (u^{-m} f) \\in E_{k}^{(-m)} (C (Y_{k}, M_{n (k)} (D))),\n\\qquad {\\mbox{and}} \\qquad\n\\gamma (u^{-m} f) \\in G_{-m}.\n\\]\n\\item\\label{SubdiagProjnDecomp_Compatible}\nThe homomorphism $\\gamma$ is compatible with the\ndirect sum decomposition of Proposition~\\ref{BanachSpaceDirSum}\non its domain and the direct sum decomposition of part (1) on its\ncodomain.\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nThe direct sum decomposition is essentially immediate from the\ndefinition of the maps $E_{k}^{(m)}$, while the other statements\nfollow from Proposition~\\ref{BanachSpaceDirSum} and some\nstraightforward matrix calculations as in the proof of Lemma 11.3.15\nof~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{cor}\\label{GammaInj}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen\nthe homomorphism $\\gamma$ of Proposition~\\ref{BanachSpaceDirSum}\nis injective.\n\\end{cor}\n\n\\begin{proof}\nThe proof is the same as the proof of Lemma 11.3.17 of~\\cite{PhNotes}.\n\\end{proof}\n\n\\begin{lem}\\label{ByMemberCond}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nLet\n\\[\nb = (b_{0}, b_{1}, \\ldots, b_{l})\n  \\in \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D)).\n\\]\nThen\n$b \\in \\gamma \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr)$\nif and only if whenever\n\\begin{itemize}\n\\item\n$r > 0$,\n\\item\n$k,t_{1},\\ldots,t_{r} \\in \\set{0,\\ldots,l}$,\n\\item\n$n (t_{1}) + n (t_{2}) + \\cdots + n (t_{r}) = n (k)$,\n\\item\n$x \\in (Y_{k} \\setminus Y_{k}^{\\circ}) \\cap\nY_{t_{1}} \\cap h^{-n (t_{1})} (Y_{t_{2}}) \\cap \\cdots \\cap\nh^{-[n (t_{1}) + \\cdots + n (t_{r - 1}) ]} (Y_{t_{r}})$,\n\\end{itemize}\nthen $b_{k} (x)$ is given by the block\ndiagonal matrix\n\\[\nb_{k} (x) = \\left[ \\begin{array}{cccc}\nb_{t_{1}} (x) & & &\n\\\\\n& \\alpha^{-n (t_{1})} (b_{t_{2}}) (x) & &\n\\\\\n& &\n\\ddots &\n\\\\\n& & & \\alpha^{-[n (t_{1}) + \\cdots + n (t_{r - 1})]}\n(b_{t_{r}}) (x) \\end{array} \\right].\n\\]\n\\end{lem}\n\n\\begin{proof}\nThe proof is analogous (with appropriate changes to notation\nand exponents) to the proof of Lemma 11.3.18 of~\\cite{PhNotes}.\n\\end{proof}\n\nWe are now in position to give a decomposition of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nas a recursive subhomogeneous algebra over $D$.\n\n\\begin{thm}\\label{AyRSHA}\nLet $X$ be a compact Hausdorff space, let $h \\colon X \\to X$ be a minimal homeomorphism,\nlet $D$ be a unital C*-algebra,\nand let $\\alpha \\in {\\mathrm{Aut}} (C (X, D))$ lie over~$h$.\nLet $Y \\subset X$ be closed with ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nAdopt Notation \\ref{ModifiedRokhlinTower} and the notation of\nProposition \\ref{AyHom}.\nThen the homomorphism\n\\[\n\\gamma \\colon C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n    \\to \\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))\n\\]\nof Proposition \\ref{AyHom}\ninduces an isomorphism of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nwith the recursive subhomogeneous algebra over~$D$ defined,\nin the notation of Definition \\ref{RSHAMachinery}, as follows:\n\\begin{enumerate}\n\\item\\label{9223_AyRSHA_1}\n$l$ and $n (0),n (1),\\ldots,n (l)$ are as in Notation\n\\ref{ModifiedRokhlinTower}.\n\\item\\label{9223_AyRSHA_2}\n$X_{k} = Y_{k}$ for $0 \\leq k \\leq l$.\n\\item\\label{9223_AyRSHA_3}\n$X_{k}^{(0)} = Y_{k} \\cap \\bigcup_{j= 0}^{k - 1} Y_{j}$.\n\\item\\label{9223_AyRSHA_4}\nFor $x \\in X_{k}^{(0)}$ and $b = (b_{0}, b_{1},\\ldots, b_{k - 1})$\nin the image in $\\bigoplus_{j= 0}^{k - 1} C (Y_{j}, M_{n (j)} (D))$\nof the $k - 1$ stage algebra\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y^{(k - 1)}$, whenever\n\\[\nx \\in (Y_{k} \\setminus Y_{k}^{\\circ}) \\cap Y_{t_{1}} \\cap\nh^{-n (t_{1})} (Y_{t_{2}}) \\cap \\cdots \\cap h^{-[n (t_{1}) + \\cdots +\nn (t_{r - 1}]} (Y_{t_{r}})\n\\]\nwith\n$n (t_{1}) + n (t_{2}) + \\cdots + n (t_{r}) = n (k)$,\nthen $\\varphi_{k} (b (x))$ is given by the block\ndiagonal matrix\n\\[\n\\varphi_{k} (b (x))\n = \\left[ \\begin{array}{cccc}\nb_{t_{1}} (x) & & &\n\\\\\n& \\alpha^{-n (t_{1})} (b_{t_{2}}) (x) & &\n\\\\\n& & \\ddots &\n\\\\\n& & & \\alpha^{-[n (t_{1}) + \\cdots + n (t_{r - 1})]}\n(b_{t_{r}}) (x) \\end{array} \\right].\n\\]\n\\item\\label{9223_AyRSHA_5}\n$\\rho_{k}$ is the restriction map.\n\\end{enumerate}\nThe topological dimension of this decomposition is $\\dim (X)$, and\nthe standard representation of\n$\\sigma \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr)$\nis the inclusion map in\n$\\bigoplus_{k = 0}^{l} C (Y_{k}, M_{n (k)} (D))$.\n\\end{thm}\n\n\\begin{proof}\nThe proof is analogous to the proof of Theorem 11.3.19\nof~\\cite{PhNotes}, while the proof of the statement regarding the\ntopological dimension is the same as for the proof given\nin~\\cite{PhNotes} for the part of Theorem 11.3.14 that is not\nincluded in Theorem 11.3.19.\n\\end{proof}\n\n\\section{Structural properties of the orbit breaking subalgebra\n and the crossed product}\\label{StructAY}\n\nIn this section,\nwe give some general theorems on the structure of C*-algebra{s}\nof the form $C^* \\big( {\\mathbb{Z}}, \\, C_0 (X, D), \\, \\alpha \\big)$,\nwhich we derive from results on the structure of\norbit breaking subalgebras.\nWe give many explicit examples in Section~\\ref{Sec_InfRed}.\n\n\\begin{cnv}\\label{N_9215_AAY_ntn}\nIn this section,\nas in Definition~\\ref{D_4Y12_OrbSubalg}\nbut with additional restrictions,\n$X$ will be an infinite compact metric space,\n$h \\colon X \\to X$ will be a minimal homeomorphism,\n$D$ will be a simple unital C*-algebra,\nand $\\alpha \\in {\\mathrm{Aut}} ( C (X, D))$\nwill be an automorphism which lies over~$h$.\n\\end{cnv}\n\nFor a few results, simplicity is not needed.\n\n\\begin{prp}\\label{AyRSHADirLim}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nFor any nonempty closed set $Y \\subset X$,\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is a direct limit of\nrecursive subhomogeneous algebras over~$D$\nof the form $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_{Y_n}$\nfor closed subsets $Y_n \\S X$ with ${\\mathrm{int}} (Y_n) \\neq \\varnothing$.\n\\end{prp}\n\n\\begin{proof}\nGiven $Y \\subset X$ closed, choose a sequence\n$(Y_{n})_{n = 1}^{\\infty}$ of closed subsets of $X$\nwith ${\\mathrm{int}} (Y_{n}) \\neq \\varnothing$ and\n$Y_{n + 1} \\subset Y_{n}$ for $n \\in {\\mathbb{Z}}_{> 0}$,\nand with $\\bigcap_{n = 1}^{\\infty} Y_{n} = Y$.\nThat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is\na direct limit of separable recursive subhomogeneous algebras\nover $D$ follows by applying Theorem~\\ref{AyRSHA} and\nProposition~\\ref{AyDirLim}.\n\\end{proof}\n\n\\begin{prp}\\label{AYDStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $\\mathcal{E}$ be a separable unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}.\nAssume that $D$ is separable and $\\mathcal{E}$-stable.\nLet $Y \\subset X$ be closed and nonempty.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is $\\mathcal{E}$-stable.\n\\end{prp}\n\n\\begin{proof}\nIf ${\\mathrm{int}} (Y) \\neq \\varnothing$,\nthis follows immediately from Theorem \\ref{AyRSHA} and\nProposition \\ref{DStableRSHA}.\nThe case of a general nonempty closed set\nnow follows from Proposition~\\ref{AyRSHADirLim} above\nand Corollary 3.4 of~\\cite{TomsWinter2}.\n\\end{proof}\n\nIn particular, we get ${\\mathcal{Z}}$-stability.\n\n\\begin{cor}\\label{EStableGivesZStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $\\mathcal{E}$ be a separable unital C*-algebra{}\nthat is strongly selfabsorbing in the sense of \\cite{TomsWinter2}.\nAssume that $D$ is separable and $\\mathcal{E}$-stable.\nLet $Y \\subset X$ be closed and nonempty.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is ${\\mathcal{Z}}$-stable.\n\\end{cor}\n\n\\begin{proof}\nStrongly selfabsorbing C*-algebra{s} are ${\\mathcal{Z}}$-stable\nby Theorem~3.1 of~\\cite{WinterZStab},\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is $\\mathcal{E}$-stable by\nProposition~\\ref{AYDStable}, so the conclusion is immediate.\n\\end{proof}\n\nWe now turn to ${\\mathcal{Z}}$-stability\nof the crossed product $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\nWhen $D$ is nuclear, it can be obtained from the results of\n\\cite{ABP-Zstab}.\nHowever, if nuclearity is not assumed, then the\nmain conclusion of that paper only implies\nthat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is tracially\n${\\mathcal{Z}}$-stable (in the sense of~\\cite{HirOr}).\n\n\\begin{thm}\\label{ATraciallyZStable}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nAssume that $D$ is a simple separable unital ${\\mathcal{Z}}$-stable\nC*-algebra{} which has a quasitrace.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis tracially ${\\mathcal{Z}}$-stable.\nIf, in addition, $D$ is nuclear,\nthen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is ${\\mathcal{Z}}$-stable.\n\\end{thm}\n\n\\begin{proof}\nTheorem 3.3 of~\\cite{HirOr} implies that $D$ has strict comparison\nof positive elements, and so\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra\nof $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ by\nCorollary \\ref{AyCentLargeConds}(\\ref{9212_AyCentLargeConds_StrCmp}).\nAlso, $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis ${\\mathcal{Z}}$-stable by Corollary~\\ref{EStableGivesZStable},\nso Theorem~2.2 of~\\cite{ABP-Zstab} implies\nthat $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\ntracially ${\\mathcal{Z}}$-stable.\nIf $D$ is nuclear, then it is ${\\mathcal{Z}}$-stable\nby Theorem 4.1 of~\\cite{HirOr},\nand $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is nuclear.\nSo ${\\mathcal{Z}}$-stability of $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nfollows from Theorem 2.3\nof~\\cite{ABP-Zstab}.\n\\end{proof}\n\n\\begin{cor}\nAdopt Convention~\\ref{N_9215_AAY_ntn}.\nLet $D$ be a simple separable unital nuclear ${\\mathcal{Z}}$-stable\nC*-algebra{} which has a quasitrace.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ has nuclear\ndimension at most 1.\n\\end{cor}\n\n\\begin{proof}\nThis follows immediately from Theorem~\\ref{ATraciallyZStable} above\nand Theorem~B of~\\cite{CETWW}.\n\\end{proof}\n\nCorollary~\\ref{EStableGivesZStable} and its consequences\nrequire no\nassumption about the underlying dynamical system other than minimality.\nIn particular, if $D$ is ${\\mathcal{Z}}$-stable then\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ is\n${\\mathcal{Z}}$-stable when $X$ is infinite dimensional, and even when\n$\\mathrm{mdim} (X, h)$ (as in Notation~\\ref{N_0108_DimX}) is strictly positive.\nIf moreover $D$ is nuclear\nthen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\n${\\mathcal{Z}}$-stable.\nThus, crossed products of the form\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ are\n${\\mathcal{Z}}$-stable\nwhenever $h$ is minimal and $D$ is\nsimple, separable, unital, nuclear, and ${\\mathcal{Z}}$-stable,\nregardless of anything else about the underlying dynamics.\n\nIf $D$ is not assumed to be ${\\mathcal{Z}}$-stable,\nthen we must make assumptions about the dynamical system\n$(X, h)$ besides just minimality to expect\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$ to have\ntractable structure.\nThe hypotheses in Proposition~\\ref{TsrRR0A_Y},\nTheorem~\\ref{P_9Z26_XisCantor},\nProposition~\\ref{P_9Z21_dim1},\nand Theorem~\\ref{T_9222_Tsr1AndRR0_CrPrd}\n(which also include conditions on~$D$)\nare surely much stronger than needed.\nThey have the advantage that the proofs can easily by obtained from\nresults already in the literature.\n\n\\begin{prp}\\label{TsrRR0A_Y}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $X$ is the Cantor set.\nLet $Y \\subset X$ be closed and nonempty.\n\\begin{enumerate}\n\\item\\label{TsrRR0A_Y_tsr}\nIf ${\\mathrm{tsr}} (D) = 1$, then\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 1$.\n\\item\\label{TsrRR0A_Y_RR}\nIf ${\\mathrm{RR}} (D) = 0$, then\n${\\mathrm{RR}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 0$.\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nFirst suppose that ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen, from Lemma~\\ref{CantorSetCase} and the remark immediately\nafter it,\nwe see there are are AF algebras $B_{0}, B_{1}, \\ldots, B_{k}$\nsuch that\n\\[\nC^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y\n  \\cong \\bigoplus_{k = 0}^{l} B_{k} \\otimes D.\n\\]\nThe result follows immediately.\n\nSince stable rank one and real rank zero are preserved by direct limits,\nthe general case now follows from\nProposition~\\ref{AyRSHADirLim}.\n\\end{proof}\n\n\\begin{thm}\\label{P_9Z26_XisCantor}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $X$ is the Cantor set.\nFurther assume that one of the following conditions holds:\n\\begin{enumerate}\n\\item\\label{9217_GetTsr1_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9217_GetTsr1_StrCmp}\n$D$ has strict comparison of positive elements.\n\\item\\label{9217_GetTsr1_SP}\n$D$ has property (SP) and the order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9217_GetTsr1_CptGen}\nThe set\n$\\bigl\\{ \\alpha_x^n \\colon {\\mbox{$x \\in X$ and $n \\in {\\mathbb{Z}}$}} \\bigr\\}$\nis contained in a subset of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nIf ${\\mathrm{tsr}} (D) = 1$, then\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\bigr) = 1$,\nand if also ${\\mathrm{RR}} (D) = 0$, then\n${\\mathrm{RR}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\bigr) = 0$.\n\\end{thm}\n\nThe action in this theorem has the Rokhlin property.\nHowever, it seems to be unknown whether the crossed product of\neven a\nsimple unital C*-algebra{} with stable rank one by a Rokhlin automorphism\nstill has stable rank one.\nSee Problem~\\ref{Pb_9922_RokhlinTsr} and the discussion afterwards.\n\nWe also emphasize that the conclusion for real rank zero\nis only valid if stable rank one is also assumed.\nThe reason is the stable rank one hypothesis\nin Theorem~6.4 of~\\cite{ArPh}.\nThat theorem likely holds without stable rank one,\nbut this has not yet been proved.\nFor now, this is no great loss,\nsince, other than purely infinite simple C*-algebra{s}\n(which are covered by our Theorem~\\ref{T_0101_PICase}),\nthere are no known examples of simple unital C*-algebra{s} which have\nreal rank zero but not stable rank one.\n\n\\begin{proof}[Proof of Theorem~\\ref{P_9Z26_XisCantor}]\nChoose any one point subset $Y \\S X$.\nSince ${\\mathrm{tsr}} (D) = 1$,\nProposition \\ref{TsrRR0A_Y}(\\ref{TsrRR0A_Y_tsr}) implies\nthat the subalgebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\n(see Definition~\\ref{D_4Y12_OrbSubalg})\nhas stable rank one.\nIf also ${\\mathrm{RR}} (D) = 0$,\nthen Proposition \\ref{TsrRR0A_Y}(\\ref{TsrRR0A_Y_RR})  implies\n${\\mathrm{tsr}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\big) = 1$.\nNow $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y$\nis a centrally large subalgebra of\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nby the appropriate part of Corollary \\ref{AyCentLargeConds}.\nSo ${\\mathrm{tsr}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\big) = 1$\nby Theorem 6.3 of~\\cite{ArPh},\nand if ${\\mathrm{RR}} (D) = 0$, then Theorem~6.4 of~\\cite{ArPh}\ngives ${\\mathrm{RR}} \\big( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr) \\big) = 0$.\n\\end{proof}\n\nWith more restrictive assumptions on~$D$,\nenough is known to get results when $\\dim (X) = 1$.\n\n\\begin{prp}\\label{P_9027_FibPrdT_sr1}\nLet $A$, $B$, and $C$ be C*-algebra{s},\nand let $\\varphi \\colon A \\to C$ and $\\psi \\colon B \\to C$ be homomorphism{s}.\nLet $D = A \\oplus_{C, \\varphi, \\psi} B$ (Definition~\\ref{pullback}).\nAssume that $\\psi$ is surjective,\nand that $A$ and $B$ have stable rank one.\nThen $D$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nSet $J = {\\operatorname{Ker}} (\\psi)$.\nWe have a commutative diagram with exact rows (maps defined below):\n\\[\n\\begin{CD}\n0 @>>> J @>{\\lambda}>> D @>{\\pi}>> A @>>> 0  \\\\\n& &  @VV{{\\mathrm{id}}_B}V  @VV{\\sigma}V  @VV{\\gamma}V          \\\\\n0 @>>> J @>{\\iota}>> B @>{\\kappa}>> B\/J @>>> 0.\n\\end{CD}\n\\]\nHere $\\iota$ is the inclusion of $J$ in~$B$\nand $\\kappa$ is the quotient map,\n$\\pi \\colon D \\to A$ and $\\sigma \\colon D \\to B$\nare the restrictions of the coordinate projections\n$(a, b) \\to a$ and $(a, b) \\to b$,\nand $\\lambda (b) = (b, 0)$ for $b \\in J$.\nFor $a \\in A$,\ndefine $\\gamma (a)$ by\nfirst choosing $b \\in B$ such that $\\psi (b) = \\varphi (a)$,\nand then setting $\\gamma (a) = \\kappa (b)$.\nIt is easy to check that $\\gamma$ is well defined,\nand that the diagram commutes.\nAll parts of exactness are immediate except that\nsurjectivity of $\\pi$ follows from\nsurjectivity of~$\\psi$.\n\nThe algebra $A$ has stable rank one by hypothesis,\nand $J$ has stable rank one by Theorem~4.4 of~\\cite{Rffl6}.\nNext,\n${\\mathrm{ind}} \\colon K_1 (B \/ J) \\to K_0 (J)$ is the zero map\nby Corollary~2 of~\\cite{Nag},\nso $K_0 (\\iota)$ is injective by the long exact sequence in\nK-theory for the bottom row.\nSince $K_0 (\\iota) = K_0 (\\sigma) \\circ K_0 (\\lambda)$,\nit follows that $K_0 (\\lambda)$ is injective.\nTherefore ${\\mathrm{ind}} \\colon K_1 (A) \\to K_0 (J)$ is the zero map\nby the long exact sequence in\nK-theory for the top row.\nNow $D$ has stable rank one by Corollary~2 of~\\cite{Nag}.\n\\end{proof}\n\nWe state for convenient reference the result on the stable rank\nof direct limits.\n\n\\begin{lem}\\label{L_0111_tsr_lim}\nLet $(A_{\\lambda})_{\\lambda \\in \\Lambda}$ be a direct system of C*-algebra{s}.\nThen\n${\\mathrm{tsr}} \\bigl( \\varinjlim_{\\lambda} A_{\\lambda} \\bigr)\n  \\leq \\liminf_{\\lambda} {\\mathrm{tsr}} (A_{\\lambda})$.\n\\end{lem}\n\nWe do not assume that the maps of the direct system are injective.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_0111_tsr_lim}]\nWhen $\\Lambda = {\\mathbb{Z}}_{> 0}$,\nthis is Theorem~5.1 of~\\cite{Rffl6}.\nThe proof for general direct limits is the same.\n\\end{proof}\n\n\\begin{prp}\\label{P_9027_CXD_Tsr1}\nLet $D$ be a simple unital C*-algebra{}\nwith stable rank one and real rank zero,\nand suppose that $K_1 (D) = 0$.\nLet $X$ be a compact Hausdorff space{} with $\\dim (X) \\leq 1$.\nThen $C (X, D)$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nIf $X$ is a point, this is trivial.\nIf $X = [0, 1]$,\nit follows from Theorem~4.3 of~\\cite{NagOsPh2001}.\nIf $X$ is a one dimensional finite complex,\nthe result follows from these facts\nand Proposition~\\ref{P_9027_FibPrdT_sr1}\nby induction on the number of cells.\n\nNow suppose $X$ is a compact metric space.\nBy Corollary 5.2.6 of~\\cite{SkiK},\nthere is an inverse system $(X_n)_{n \\in {\\mathbb{Z}}_{\\geq 0}}$\nof one dimensional finite complexes\nsuch that $X \\cong \\varprojlim_n X_n$.\nThen $C (X, D) \\cong \\varinjlim_n C (X_n, D)$.\nSince $C (X_n, D)$ has stable rank one for all $n \\in {\\mathbb{Z}}_{\\geq 0}$,\nit follows from Lemma~\\ref{L_0111_tsr_lim}\nthat $C (X, D)$ has stable rank one.\n\nFinally,\nlet $X$ be a general compact Hausdorff space{} with $\\dim (X) \\leq 1$.\nBy Theorem~1 of~\\cite{Mdsc}\n(in Section~3 of that paper),\nthere is an inverse system $(X_{\\lambda})_{\\lambda \\in \\Lambda}$\nof one dimensional compact metric space{s}\nsuch that $X \\cong \\varprojlim_{\\lambda} X_{\\lambda}$.\nIt now follows that $C (X, D)$ has stable rank one\nby the same reasoning as in the previous paragraph.\n\\end{proof}\n\n\\begin{prp}\\label{P_9027_Rsha_Tsr1}\nLet $D$ be a simple unital C*-algebra.\nLet $R$ be a recursive subhomogeneous C*-algebra{} over~$D$\n(Definition~\\ref{RSHA})\nwith topological dimension at most~$1$\n(Definition \\ref{RSHAMachinery}(\\ref{9223_RSHAMachinery_6})).\nIf $D$ has stable rank one and real rank zero,\nand $K_1 (D) = 0$,\nthen $R$ has stable rank one.\n\\end{prp}\n\n\\begin{proof}\nThe proof is the same as that of Proposition~\\ref{DStableRSHA},\nexcept using Proposition~\\ref{P_9027_FibPrdT_sr1}\nin place of Lemma~\\ref{DStablePullback},\nand using Proposition~\\ref{P_9027_CXD_Tsr1}.\n\\end{proof}\n\n\\begin{prp}\\label{P_9Z21_dim1}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $\\dim (X) \\leq 1$,\nthat $D$ has stable rank one and real rank zero,\nand that $K_1 (D) = 0$.\nLet $Y \\subset X$ be closed and nonempty.\nThen\n${\\mathrm{tsr}} \\bigl( C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)_Y \\bigr) = 1$.\n\\end{prp}\n\n\\begin{proof}\nFirst suppose that ${\\mathrm{int}} (Y) \\neq \\varnothing$.\nThen the recursive subhomogeneous algebra over~$D$\nin Theorem~\\ref{AyRSHA}\nhas base spaces $Y_k$ which,\nfollowing Notation~\\ref{ModifiedRokhlinTower},\nare closed subsets of~$X$.\nBy Proposition 3.1.3 of~\\cite{Prs},\nthey therefore have covering dimension at most~$1$.\nThe result now follows from Proposition~\\ref{P_9027_Rsha_Tsr1}.\n\nStable rank one is preserved by direct limits\n(Lemma~\\ref{L_0111_tsr_lim}),\nso the general case now follows from\nProposition~\\ref{AyRSHADirLim}.\n\\end{proof}\n\nThe obvious examples of minimal homeomorphism{s}\nof one dimensional spaces are irrational rotations on~$S^1$,\nbut there are others.\nSee, for example, \\cite{GjdJhn},\nand the work on minimal homeomorphism{s} of the product of the Cantor set\nand the circle in \\cite{LnMti1} and~\\cite{LnMti3}.\n\n\\begin{thm}\\label{T_9222_Tsr1AndRR0_CrPrd}\nAdopt Convention~\\ref{N_9215_AAY_ntn},\nand assume that $\\dim (X) \\leq 1$,\nthat $D$ has stable rank one and real rank zero,\nand that $K_1 (D) = 0$.\nFurther assume that one of the following conditions holds:\n\\begin{enumerate}\n\\item\\label{9927_AyCentLargeConds_AppInn}\nAll elements of $\\set{ \\alpha_{x} \\colon x \\in X} \\subset {\\mathrm{Aut}} (D)$\nare approximately inner.\n\\item\\label{9927_AyCentLargeConds_SP}\nThe order on projections over~$D$\nis determined by quasitraces.\n\\item\\label{9927_AyCentLargeConds_CptGen}\nThe set\n$\\bigl\\{ \\alpha_x^n \\colon {\\mbox{$x \\in X$ and $n \\in {\\mathbb{Z}}$}} \\bigr\\}$\nis contained in a subset of ${\\mathrm{Aut}} (D)$\nwhich is compact in the topology\nof pointwise convergence in the norm on~$D$.\n\\end{enumerate}\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nhas stable rank one.\n\\end{thm}\n\n\\begin{proof}\nCombine Theorem 6.3 of~\\cite{ArPh},\nCorollary~\\ref{AyCentLargeConds},\nand Proposition~\\ref{P_9027_Rsha_Tsr1}.\n\\end{proof}\n\n\\section{Minimality of products of Denjoy homeomorphisms\n of the Cantor set}\\label{Sec_9929_MinProd}\n\n\\indent\nFor use in examples in the next section,\nwe prove minimality of products of Denjoy homeomorphisms\nwith rationally independent rotation numbers.\nWe have not found this result in the literature.\nThe method we use here is not the traditional approach\nto this kind of problem,\nbut we hope it will be useful elsewhere.\n\n\\begin{ntn}\\label{N_9X04_Rotation}\nFor $\\theta \\in {\\mathbb{R}}$ we let $r_{\\theta} \\colon S^1 \\to S^1$\ndenote the rotation by $2 \\pi \\theta$,\nthat is, $r_{\\theta} (\\zeta) = \\exp (2 \\pi i \\theta)  \\zeta$\nfor $\\zeta \\in S^1$.\n\\end{ntn}\n\n\n\\begin{prp}\\label{P_4X04_ProdRotation}\nLet $I$ be a set,\nand let $(\\theta_i)_{i \\in I}$ be a family of elements of~${\\mathbb{R}}$.\nSuppose that $1$ and the numbers $\\theta_i$ for $i \\in I$ are\nlinearly independent over~${\\mathbb{Q}}$.\nThen the product $r \\colon (S^1)^{I} \\to (S^1)^{I}$\nof the rotations $r_{\\theta_i}$ is minimal.\n\\end{prp}\n\n\\begin{proof}\nIf $I$ is finite,\nthis is Proposition 1.4.1 of~\\cite{KtHsb}.\nThe general case follows from the finite case by taking the inverse\nlimit of the products over finite subsets of~$I$,\nby the discussion after Proposition II.4 of~\\cite{Frtg}.\nThe proof is is written for actions of the semigroup~${\\mathbb{Z}}_{\\geq 0}$\nand for countable inverse limits,\nbut the proof for actions of ${\\mathbb{Z}}$ and arbitrary inverse limits\nis the same.\n\\end{proof}\n\n\\begin{dfn}\\label{D_9929_StrOpen}\nLet $X$ and $Y$ be topological spaces,\nand let $g \\colon Y \\to X$.\nLet $y \\in Y$.\nWe say that $g$ is {\\emph{strictly open}} at~$y$\nif\n\\[\n\\bigl\\{ g^{-1} (U) \\colon\n {\\mbox{$U \\S X$ is open and $g (y) \\in U$}} \\bigr\\}\n\\]\nis a neighborhood base for $y$ in~$Y$.\n\\end{dfn}\n\nSome of the theory below works without assuming $g$ is continuous,\nso we do not require that neighborhood{s} be open.\n\nThe next lemma gives an alternate interpretation of strict openness.\n\n\\begin{lem}\\label{L_9929_WhenStrOpen}\nIn the situation in Definition~\\ref{D_9929_StrOpen},\nassume that $g$ is continuous{} and surjective, $Y$ is compact,\nand $X$ and $Y$ are Hausdorff.\nThen $g$ is strictly open at~$y$ if and only if{}\nthe following conditions hold:\n\\begin{enumerate}\n\\item\\label{Item_D_9929_Inj}\n$g^{-1} (\\{ g (y) \\}) = \\{ y \\}$.\n\\item\\label{Item_D_9929_Open}\nFor every open set $V \\S Y$ with $y \\in V$,\nthe set $g (V)$ is a neighborhood{} of $g (y)$.\n\\end{enumerate}\n\\end{lem}\n\nThe lemma is false without compactness of~$Y$.\nTake\n\\[\nX = [0, 1]\n\\qquad {\\mbox{and}} \\qquad\nY = \\{ (0, 0) \\} \\cup \\bigl( (0, 1]\\times [0, 1] \\bigr) \\S [0, 1]^2,\n\\]\nlet $g$ be projection to the first coordinate,\nand take $y = (0, 0)$.\nThen (\\ref{Item_D_9929_Inj}) and~(\\ref{Item_D_9929_Open}) hold,\nbut $g$ is not strictly open at~$y$.\n\n\\begin{proof}[Proof of Lemma~\\ref{L_9929_WhenStrOpen}]\nAssume that $g$ is strictly open at~$y$.\nTo prove~(\\ref{Item_D_9929_Inj}),\nlet $z \\in Y \\setminus \\{ y \\}$,\nchoose $V \\S Y$ open with $y \\in V$ and $z \\not\\in V$,\nand choose $U \\S X$ open such that $g (y) \\in U$\nand $g^{-1} (U) \\S V$.\nThen $z \\not\\in g^{-1} (U)$, so $g (z) \\neq g (y)$.\n\nTo prove~(\\ref{Item_D_9929_Open}),\nlet $V \\S Y$ be open with $y \\in V$.\nChoose $U \\S X$ open such that $g (y) \\in U$\nand $g^{-1} (U) \\S V$.\nSurjectivity of~$g$ implies $U \\S g ( g^{-1} (U) ) \\S g (V)$,\nso $g (V)$ is a neighborhood{} of $g (y)$.\n\nFor the converse,\nassume~(\\ref{Item_D_9929_Open}),\nand suppose that $g$ is not strictly open at~$y$.\nLet ${\\mathcal{N}}$ be the set of open subsets of~$X$ which contain $g (y)$,\nordered by reverse inclusion.\nSince $g$ is continuous,\nfailure of strict openness means we can\nchoose $V \\S Y$ open with $y \\in V$\nand such that for every $U \\in {\\mathcal{N}}$,\nwe have $g^{-1} (U) \\not\\S V$.\nSo there is $z_U \\in Y$ such that $z_U \\not\\in V$ but $g (z_U) \\in U$.\nThe net $(z_U)_{U \\in {\\mathcal{N}}}$\nsatisfies $g (z_U) \\to g (y)$.\nChoose a subnet $(t_{\\lambda})_{\\lambda \\in \\Lambda}$\nsuch that $t = \\lim_{\\lambda} t_{\\lambda}$ exists.\nThen $t \\not\\in V$.\nAlso $g (t_{\\lambda}) \\to g (t)$ by continuity and $g (t_{\\lambda}) \\to g (y)$\nby construction, so $g (t) = g (y)$.\nWe have contradicted~(\\ref{Item_D_9929_Inj}).\n\\end{proof}\n\n\\begin{lem}\\label{L_9929_ExtMin}\nLet $X$ and $Y$ be compact Hausdorff space{s},\nand let $h \\colon X \\to X$ and $k \\colon Y \\to Y$ be homeomorphism{s}.\nLet $g \\colon Y \\to X$ be continuous and surjective,\nand satisfy $g \\circ k = h \\circ g$.\nSuppose that $h$ is minimal and there is a dense subset $S \\S Y$\nsuch that $g$ is strictly open at every point of~$S$.\nThen $k$ is minimal.\n\\end{lem}\n\n\\begin{proof}\nIt is enough to show that for every $y \\in Y$ and every\nnonempty open set $V \\S Y$,\nthere is $n \\in {\\mathbb{Z}}$ such that $k^n (y) \\in V$.\nChoose $z \\in S \\cap V$.\nSince $g$ is strictly open at~$z$,\nthere is an open set $U \\S X$ such that $g (z) \\in U$\nand $g^{-1} (U) \\S V$.\nSince $h$ is minimal,\nthere is $n \\in {\\mathbb{Z}}$ such that $h^n (g (y)) \\in U$.\nThen $g (k^n (y)) \\in U$,\nso $k^n (y) \\in g^{-1} (U) \\S V$.\n\\end{proof}\n\nIt is well known that Lemma~\\ref{L_9929_ExtMin} fails\nwithout strict openness,\neven if one assumes that $g^{-1} (x)$ is finite for all $x \\in X$\nand has only one point for a dense subset of points $x \\in X$.\nThe following example was suggested by B.~Weiss.\n\n\\begin{exa}\\label{E_9X04_NeedStrict}\nFix $\\theta \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$.\nTake $X = S^1$ and take $h = r_{\\theta}$\n(rotation by $2 \\pi \\theta$; Notation~\\ref{N_9X04_Rotation}).\nDefine $Y \\S S^1 \\times [0, 1]$ to be\n\\[\nY = \\bigl( S^1 \\times \\{ 0 \\} \\bigr)\n \\cup \\left\\{ \\left( e^{2 \\pi i n \\theta}, \\, \\frac{1}{| n | + 1} \\right)\n     \\colon n \\in {\\mathbb{Z}} \\right\\},\n\\]\nand define $k \\colon Y \\to Y$ to be\n$k (\\zeta, 0) = (r_{\\theta} (\\zeta), \\, 0)$ for $\\zeta \\in S^1$ and\n\\[\nk \\left( e^{2 \\pi i n \\theta}, \\, \\frac{1}{| n | + 1} \\right)\n   = \\left( e^{2 \\pi i (n + 1) \\theta}, \\, \\frac{1}{| n + 1 | + 1} \\right)\n\\]\nfor $n \\in {\\mathbb{Z}}$.\nLet $g \\colon Y \\to X$ be projection to the first coordinate.\nThen $g \\circ k = h \\circ g$,\n$g$ is surjective,\n$h$ is minimal,\n$k$ is not minimal,\n$g^{-1} (x)$ has at most two points for every $x \\in X$,\nand $g^{-1} (x)$ has one point  for all but countably many $x \\in X$.\n\\end{exa}\n\nThe following two easy lemmas will be used in the construction of\nexamples.\n\n\\begin{lem}\\label{L_9X04_Prod}\nLet $I$ be a nonempty set,\nand for $i \\in I$ let $g_i \\colon X_i \\to Y_i$\nand $y_i \\in Y_i$ be as in Definition~\\ref{D_9929_StrOpen},\nwith $g_i$ strictly open at~$y_i$.\nSet\n\\[\nX = \\prod_{i \\in I} X_i\n\\qquad {\\mbox{and}} \\qquad\nY = \\prod_{i \\in I} Y_i,\n\\]\nlet $g \\colon Y \\to X$ be given by $g (z) = ( g_i (z_i))_{i \\in I}$\nfor $z = (z_i)_{i \\in I} \\in Y$,\nand set $y = (y_i)_{i \\in I}$.\nThen $g$ is strictly open at~$y$.\n\\end{lem}\n\n\\begin{proof}\nIt is obvious that $g^{-1} ( \\{ g (y) \\} ) = \\{ y \\}$.\nNow let $V \\S Y$ be open with $y \\in V$.\nWe need an open set $U \\S X$\nsuch that $g (y) \\in U$, $g^{-1} (U)$ is a neighborhood{} of~$y$,\nand $g^{-1} (U) \\S V$.\nWithout loss of generality{} there are open subsets $V_i \\S Y_i$\nand a finite set $F \\S I$\nsuch that $V = \\prod_{i \\in I} V_i$,\n$V_i = Y_i$ for all $i \\in I \\setminus F$,\nand $y_i \\in V_i$ for all $i \\in I$.\nFor $i \\in F$ choose an open set $U_i \\S X_i$ with $g_i (y_i) \\in U_i$\nand such that $g_i^{-1} (U_i) \\S V_i$.\nSet $U_i = X_i$ for $i \\in I \\setminus F$.\nSince $g_i^{-1} (U_i)$ is a neighborhood{} of~$y_i$\nfor all $i \\in I$\nand $U_i = X_i$ for $i \\in I \\setminus F$,\nthe set $g^{- 1} (U) = \\prod_{i \\in I} g_i^{-1} (U_i)$\nis a neighborhood{} of~$y$.\nSo the set $U = \\prod_{i \\in I} U_i$ is the required set.\n\\end{proof}\n\n\\begin{lem}\\label{L_9X04_Subsp}\nIn the situation in Definition~\\ref{D_9929_StrOpen},\nassume that $g$ is strictly open at~$y$.\nLet $Y_0 \\S Y$ be a subspace such that $y \\in Y_0$.\nThen $g |_{Y_0}$ is strictly open at~$y$.\n\\end{lem}\n\n\\begin{proof}\nThe result is immediate from the definition of the subspace\ntopology.\n\\end{proof}\n\n\\begin{lem}\\label{L_9929_S1toS1}\nLet $g \\colon S^1 \\to S^1$ be continuous{} and surjective.\nSuppose that there are disjoint closed arcs\n$I_1, I_2, \\ldots \\S S^1$\nsuch that:\n\\begin{enumerate}\n\\item\\label{Item_L_9929_S1toS1_Const}\nFor $n \\in {\\mathbb{Z}}_{> 0}$, the function $g$ is constant on~$I_n$.\n\\item\\label{Item_L_9929_S1toS1_NI}\n$\\bigcup_{n = 1}^{\\infty} I_n$ is dense in~$S^1$.\n\\item\\label{Item_L_9929_S1toS1_Inj}\nWith $R = S^1 \\setminus \\bigcup_{n = 1}^{\\infty} I_n$,\nthe restriction $g |_R$ is injective.\n\\end{enumerate}\nThen $g$ is strictly open at every point of~$R$.\n\\end{lem}\n\n\\begin{proof}\nFor distinct $\\lambda, \\zeta \\in S^1$,\nwe denote the open, closed, and half open arcs from $\\lambda$ to~$\\zeta$\nusing interval notation:\n$(\\lambda, \\zeta)$, $[\\lambda, \\zeta]$, $[\\lambda, \\zeta)$, and $(\\lambda, \\zeta]$.\nFurther write $I_n = [\\beta_n, \\gamma_n]$ with $\\beta_n, \\gamma_n \\in S^1$,\nand set $B = \\{ \\beta_n \\colon n \\in {\\mathbb{Z}}_{> 0} \\}$\nand $C = \\{ \\gamma_n \\colon n \\in {\\mathbb{Z}}_{> 0} \\}$.\n\nThe set $g (S^1 \\setminus R)$ is countable\nand $g$ is surjective, so $S^1 \\setminus g (R)$ is countable.\n\nFirst,\nwe claim that if $E, F \\S S^1$ are disjoint,\nthen ${\\mathrm{int}} (g (E)) \\cap {\\mathrm{int}} (g (F)) = \\varnothing$.\nIf the claim is false,\nthen $g (E) \\cap g (F)$ is uncountable.\nNow $g (E) \\setminus g (E \\cap R) \\S g (S^1 \\setminus R)$\nwhich is countable, so $g (E) \\setminus g (E \\cap R)$ is countable.\nSimilarly $g (F) \\setminus g (F \\cap R)$ is countable.\nSo $g (E \\cap R) \\cap g (F \\cap R)$ is uncountable.\nThis contradicts $E \\cap F = \\varnothing$ and injectivity of $g |_R$.\n\nSecond, we claim that if $J \\S S^1$ is any nonempty open arc such that\nthere is no $n \\in {\\mathbb{Z}}_{> 0}$ with ${\\overline{J}} \\S I_n$,\nthen $R \\cap J \\neq \\varnothing$.\nSuppose the claim is false.\nWrite $J = (\\lambda, \\zeta)$ with $\\lambda, \\zeta \\in S^1$.\nDefine\n\\[\nS = \\begin{cases}\n   \\varnothing & \\hspace*{1em}        \\lambda \\not\\in R\n        \\\\\n   \\{ \\lambda \\} & \\hspace*{1em} \\lambda \\in R\n\\end{cases}\n\\qquad {\\mbox{and}} \\qquad\nT = \\begin{cases}\n   \\varnothing & \\hspace*{1em}        \\zeta \\not\\in R\n        \\\\\n   \\{ \\zeta \\} & \\hspace*{1em} \\zeta \\in R.\n\\end{cases}\n\\]\nThen ${\\overline{J}}$ is the disjoint union of the closed sets\n\\[\nS, \\, T, \\, {\\overline{J}} \\cap I_1, \\, {\\overline{J}} \\cap I_2, \\ldots.\n\\]\nSuppose there is $n \\in {\\mathbb{Z}}_{> 0}$ such that ${\\overline{J}} \\cap I_m = \\varnothing$\nfor all $m \\neq n$.\nSince ${\\overline{J}} \\not\\subset I_n$,\nthere is a nonempty open arc $L \\S J$\nsuch that $L \\cap I_n = \\varnothing$,\nwhence $L \\cap I_m = \\varnothing$ for all $m \\in {\\mathbb{Z}}_{> 0}$.\nThis contradicts~(\\ref{Item_L_9929_S1toS1_NI}).\nThus, at least two of\nthe sets ${\\overline{J}} \\cap I_n$ are nonempty.\nBut, according to\nTheorem~6 in Part III of Section~47 (in Chapter~5) of~\\cite{Krtw},\na connected compact Hausdorff space{} cannot be the disjoint union of closed subsets\n$E_1, E_2, \\ldots$ with at least two of them nonempty.\nThis contradiction proves the claim.\n\nThird, we claim that, under the same hypotheses, $R \\cap J$\nis infinite.\nAgain write $J = (\\lambda, \\zeta)$ with $\\lambda, \\zeta \\in S^1$.\nApply the previous claim\nto choose $\\rho_1 \\in R \\cap J$ and set $J_1 = (\\rho_1, \\zeta)$.\nThen $\\rho_1 \\not\\in \\bigcup_{n = 1}^{\\infty} I_n$\nso there is no $n \\in {\\mathbb{Z}}_{> 0}$ with ${\\overline{J_1}} \\S I_n$.\nApply the previous claim again, choosing $\\rho_2 \\in R \\cap J_1$,\nand showing that $J_2 = (\\rho_2, \\zeta)$\nsatisfies ${\\overline{J_1}} \\not\\S I_n$ for all $n \\in {\\mathbb{Z}}_{> 0}$.\nProceed inductively.\n\nFourth, we claim that if $\\lambda \\in R \\cup C$ and $\\zeta \\in R \\cup B$\nare distinct,\nthen $g \\bigl( (\\lambda, \\zeta) \\bigr)$ is an open arc.\nFor the proof, set $J = (\\lambda, \\zeta)$.\nWe can assume that $g (J) \\neq S^1$.\nSince $g (J)$ is connected, there are $\\mu, \\nu \\in S^1$\nsuch that $g (J)$ is one of\n$(\\mu, \\nu)$, $[\\mu, \\nu]$\n(in this case we allow $\\mu = \\nu$), $[\\mu, \\nu)$, or $(\\mu, \\nu]$.\nSuppose $\\mu \\in g (J)$; we will obtain a contradiction.\nChoose $\\rho \\in J$ such that $g (\\rho) = \\mu$.\nThere are two cases: $\\rho \\in R$ and $\\rho \\in \\bigcup_{n = 1}^{\\infty} I_n$.\n\nIn the first case, let $L_0$ and $L_1$ be the nonempty arcs\n$L_0 = (\\lambda, \\rho)$ and $L_1 = (\\rho, \\zeta)$.\nFor $j = 0, 1$,\nsince $\\rho \\in {\\overline{L_j}}$,\nthe arc $L_j$ satisfies the hypotheses of the second claim.\nSo $R \\cap L_j$ is infinite by the third claim.\nSince $g |_R$ is injective, $g (L_j)$ is infinite.\nClearly $\\mu \\in {\\overline{g (L_j)}} \\S [\\mu, \\nu]$.\nAlso, $g (L_j)$ is an arc by connectedness.\nSo there is $\\omega_j \\in (\\mu, \\nu)$ such that $(\\mu, \\omega_j) \\S g (L_j)$.\nThis implies ${\\mathrm{int}} ( g (L_0) ) \\cap {\\mathrm{int}} ( g (L_1) ) \\neq \\varnothing$,\ncontradicting the first claim.\n\nIn the second case,\nlet $n \\in {\\mathbb{Z}}_{> 0}$ be the integer such that $\\rho \\in I_n$.\nBoth $\\lambda \\in [\\beta_n, \\gamma_n)$ and $\\zeta \\in (\\beta_n, \\gamma_n]$\nare impossible, but $[\\beta_n, \\gamma_n] \\cap (\\lambda, \\zeta) \\neq \\varnothing$,\nso $[\\beta_n, \\gamma_n] \\subset (\\lambda, \\zeta)$.\nTherefore the arcs $L_0 = (\\lambda, \\beta_n)$ and $L_1 = (\\gamma_n, \\zeta)$\nare nonempty and do not intersect~$I_n$.\nIf there is $m \\in {\\mathbb{Z}}_{> 0}$ such that ${\\overline{L_0}} \\S I_m$,\nthen clearly $m \\neq n$,\nbut then $\\beta_n \\in I_m \\cap I_n$,\ncontradicting $I_m \\cap I_n = \\varnothing$.\nSo no such $m$ exists,\nwhence $L_0$ satisfies the hypotheses of the second claim.\nSimilarly $L_1$ satisfies the hypotheses of the second claim.\nSince $g$ is constant on~$I_n$,\nwe have $g (\\beta_n) = g (\\gamma_n) = g (\\rho) = \\mu$.\nNow we get a contradiction in the same way as in the first case.\n\nSo $\\mu \\not\\in g (J)$.\nSimilarly $\\nu \\not\\in g (J)$.\nThe claim is proved.\n\nFifth,\nwe claim that if $L \\S S^1$ is an open arc,\nthen $g^{-1} (L)$ is an open arc.\nTo prove the claim,\nsince $g^{-1} (L)$ is open,\nthere are an index set $S$\nand disjoint nonempty open arcs $J_s$ for $s \\in S$\nsuch that $g^{-1} (L) = \\bigcup_{s \\in S} J_s$,\nand there are $\\lambda_s, \\zeta_s \\in S^1$\nsuch that $J_s = (\\lambda_s, \\zeta_s)$.\n\nLet $s \\in S$.\nSuppose $\\lambda_s \\in I_n$ for some $n \\in {\\mathbb{Z}}_{> 0}$.\nSince $g$ is constant on~$I_n$,\neither $I_n \\cap g^{-1} (L) = \\varnothing$\nor $I_n \\S g^{-1} (L)$.\nIn the second case, $I_n \\cup (\\lambda_s, \\zeta_s)$\nis a connected subset of $g^{-1} (L)$.\nSince $(\\lambda_s, \\zeta_s)$ is a maximal connected subset of $g^{-1} (L)$,\nwe have $I_n \\S (\\lambda_s, \\zeta_s)$.\nThis contradicts $\\lambda_s \\in I_n$.\nSo the first case must apply, and therefore $\\lambda_s = \\gamma_n$.\nCombining this with the possibility\n$\\lambda_s \\not\\in \\bigcup_{n = 1}^{\\infty} I_n$,\nwe conclude that $\\lambda_s \\in R \\cup C$.\nSimilarly, $\\zeta_s \\in R \\cup B$.\nBy the previous claim,\n$g \\bigl( (\\lambda_s, \\zeta_s) \\bigr)$ is open.\n\nThe first claim now implies that the sets\n$g \\bigl( (\\lambda_s, \\zeta_s) \\bigr)$ are all disjoint.\nSince $L$ is connected,\nit follows that ${\\mathrm{card}} (S) = 1$,\nwhich implies the claim.\n\nTo prove the lemma,\nlet $\\eta \\in R$ and let $V \\S S^1$ be open with $\\eta \\in V$.\nWe need an open set $U \\S S^1$\nsuch that $g (\\eta) \\in U$ and $g^{-1} (U) \\S V$.\nChoose $\\lambda_0, \\zeta_0 \\in S^1$\nsuch that $\\eta \\in (\\lambda_0, \\zeta_0) \\S V$.\nUse the second claim to choose $\\lambda_1 \\in (\\lambda_0, \\eta) \\cap R$\nand $\\zeta_1 \\in (\\eta, \\zeta_0) \\cap R$.\nThen $U = g \\bigl( (\\lambda_1, \\zeta_1) \\bigr)$ is open by\nthe fourth claim,\nand by the fifth claim there are $\\lambda_2, \\zeta_2 \\in S^1$\nsuch that $g^{-1} (U) = (\\lambda_2, \\zeta_2)$.\nIf $\\lambda_1 \\in g^{-1} (U)$,\nthen $(\\lambda_2, \\lambda_1)$ is a nonempty open arc contained in $g^{-1} (U)$\nwith $\\lambda_1 \\in R$.\nThe third claim and injectivity of $g |_R$\nimply that $g \\bigl( (\\lambda_2, \\lambda_1) \\bigr)$\nis not a point.\nSince $g \\bigl( (\\lambda_2, \\lambda_1) \\bigr)$ is connected,\nit has nonempty interior.\nIt is contained in $U = g \\bigl( (\\lambda_1, \\zeta_1) \\bigr)$,\ncontradicting the first claim.\nSo $\\lambda_1 \\not\\in (\\lambda_2, \\zeta_2)$.\nSimilarly $\\zeta_1 \\not\\in (\\lambda_2, \\zeta_2)$.\nSince $\\eta \\in (\\lambda_2, \\zeta_2)$,\n$\\eta \\in (\\lambda_1, \\zeta_1)$, and $(\\lambda_2, \\zeta_2)$ is connected,\nit now follows that $(\\lambda_2, \\zeta_2) \\S (\\lambda_1, \\zeta_1)$.\n(In fact, we have equality, since $g \\bigl( (\\lambda_1, \\zeta_1) \\bigr) = U$.)\nThus\n\\[\ng^{-1} (U)\n = (\\lambda_2, \\zeta_2)\n \\S (\\lambda_1, \\zeta_1)\n \\S (\\lambda_0, \\zeta_0)\n \\S V.\n\\]\nThis completes the proof.\n\\end{proof}\n\nWe have not seen a term in the literature for the following\nclass of homeomorphism{s}.\n\n\\begin{dfn}\\label{D_9X04_RestrDj}\nA {\\emph{restricted Denjoy homeomorphism}}\nis a homeomorphism of the Cantor set which is conjugate to\nthe restriction and corestriction of a Denjoy homeomorphism of~$S^1$,\nin the sense of Definition~3.3 of~\\cite{PtSmdSk},\nto the unique minimal set of Proposition~3.4 of~\\cite{PtSmdSk}.\nThe {\\emph{rotation number}}\nof a restricted Denjoy homeomorphism is the rotation number,\nas at the beginning of Section~3 of~\\cite{PtSmdSk},\nof the Denjoy homeomorphism of~$S^1$\nof which it is the restriction;\nthis number is well defined by Remark~3 at the end of Section~3\nof~\\cite{PtSmdSk}.\n\\end{dfn}\n\n\\begin{lem}\\label{L_9X04_DjSemiCj}\nLet $X$ be the Cantor set,\nand let $k \\colon X \\to X$ be a restricted Denjoy homeomorphism\nwith rotation number~$\\theta$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nThen there are a continuous{} surjective map $g \\colon X \\to S^1$\nand a dense subset $R \\S X$ such that\n$g$ is strictly open at~$x$ for every $x \\in R$\nand, following Notation~\\ref{N_9X04_Rotation},\n$g \\circ h = r_{\\theta} \\circ g$.\n\\end{lem}\n\n\\begin{proof}\nWe may assume that $k_0 \\colon S^1 \\to S^1$ is a Denjoy homeomorphism\nas in Definition~3.3 of~\\cite{PtSmdSk},\nthat $X \\S S^1$ is the unique minimal set for~$k_0$,\nand that $k$ is the restriction and corestriction of $k_0$ to~$X$.\nCorollary~3.2 of~\\cite{PtSmdSk}\ngives $h \\colon S^1 \\to S^1$\nsuch that $h \\circ k_0 = r_{\\theta} \\circ h$.\nSet $g = h |_X$.\nSince $X \\S S^1$ is compact and invariant and $r_{\\theta}$ is minimal,\n$g$ must be surjective.\nThe discussion after Proposition~3.4 of~\\cite{PtSmdSk}\nimplies that $h$ satisfies the hypotheses of Lemma~\\ref{L_9929_S1toS1},\nwith $X = S^1 \\setminus \\bigcup_{n = 1}^{\\infty} {\\mathrm{int}} (I_n)$.\nMoreover, the set $R$ of Lemma~\\ref{L_9929_S1toS1}\nis nonempty by\nTheorem~6 in Part III of Section~47 (in Chapter~5) of~\\cite{Krtw},\nand $k$-invariant,\nhence dense in~$X$.\nBy Lemma~\\ref{L_9929_S1toS1} and Lemma~\\ref{L_9X04_Subsp},\n$g$ is strictly open at every point of~$R$.\n\\end{proof}\n\n\\begin{prp}\\label{P_4X04_ProdRstDj}\nLet $I$ be a set,\nlet $(\\theta_i)_{i \\in I}$ be a family of elements of~${\\mathbb{R}}$,\nand for $i \\in I$ let $k_i \\colon X_i \\to X_i$\nbe a restricted Denjoy homeomorphism with rotation number~$\\theta_i$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nSuppose that $1$ and the numbers $\\theta_i$ for $i \\in I$ are\nlinearly independent over~${\\mathbb{Q}}$.\nThen the product $k \\colon \\prod_{i \\in I} X_i \\to \\prod_{i \\in I} X_i$\nof the homeomorphism{s} $k_i$ is minimal.\n\\end{prp}\n\n\\begin{proof}\nFor $i \\in I$,\nlet $g_i \\colon X_i \\to S^1$ be as in Lemma~\\ref{L_9X04_DjSemiCj}.\nBy Lemma~\\ref{L_9X04_DjSemiCj},\nthere is a dense subset $R_i \\S X_i$\nsuch that $g_i$ is strictly open at every point of~$R_i$.\nLet $g \\colon \\prod_{i \\in I} X_i \\to (S^1)^{I}$\nbe the product of the maps~$g_i$.\nLemma~\\ref{L_9X04_Prod} implies that\n$g$ is strictly open at every point of $\\prod_{i \\in I} R_i$,\nthis set is dense,\nand the product $r \\colon (S^1)^{I} \\to (S^1)^{I}$\nof the rotations $r_{\\theta_i}$ is minimal\nby Proposition~\\ref{P_4X04_ProdRotation}.\nSo $k$ is minimal\nby Lemma~\\ref{L_9929_ExtMin}.\n\\end{proof}\n\n\n\\section{Examples}\\label{Sec_InfRed}\n\n\\indent\nTo take full advantage of Proposition~\\ref{AyRSHADirLim},\nwe need information on the structure of simple\ndirect limits of recursive subhomogeneous algebras\nover the algebra $D$ which appears there.\nFor example,\nwe need conditions for such direct limits to have stable rank~$1$\n(as originally done for $D = {\\mathbb{C}}$ in~\\cite{DNNP})\nand to have real rank zero\n(as originally done for $D = {\\mathbb{C}}$ in~\\cite{BDR}).\nWe intend to study this problem in a future paper.\nEven without generalizations of those theorems,\nthree cases are accessible now.\nThey are $h \\colon X \\to X$ wild\n(for example, with strictly positive mean dimension,\nas in Notation~\\ref{N_0108_DimX})\nbut with $D$ being ${\\mathcal{Z}}$-stable;\n$X$ is the Cantor set\nand $D$ has stable rank one but is otherwise wild;\nand $X = S^1$,\n$h$ is an irrational rotation,\nand $D$ has stable rank one, real rank zero,\nand trivial $K_1$-group,\nbut is otherwise wild.\n\nWe also give examples for Theorem~\\ref{T_0101_PICase},\nalthough only for $G = {\\mathbb{Z}}$.\n\nIn the first type of example,\nif $X$ is finite dimensional{} then the action has a\nhigher dimensional Rokhlin property,\nand results on crossed products by such actions can be applied.\nAs far as we know, however,\nthere are no previously known general theorems\nwhich apply when $X$ is infinite dimensional.\nIn the second type of example,\nthe action has the Rokhlin property.\nSince the algebra is not simple,\nresults of~\\cite{OP1} can't be applied,\nand, when $D$ is not ${\\mathcal{Z}}$-stable\nand does not have finite nuclear dimension,\nTheorems 4.1 and~5.8 of~\\cite{HWZ} can't be applied.\nWe know of no previous general theorems which apply in this case.\nActions in the third type of example\nare further away from known results:\nthe situation is like for the second type of example,\nbut the Rokhlin property must be weakened to\nfinite Rokhlin dimension with commuting towers.\n\nSince we will use quasifree automorphisms of reduced C*-algebras\nof free groups\nand the free shift on $C^*_{\\mathrm{r}} (F_{\\infty})$\nin several examples,\nwe establish notation for them separately.\n\n\\begin{ntn}\\label{N_9214_QFreeFn}\nFor $n \\in {\\mathbb{Z}}_{> 0} \\cup \\{ \\infty \\}$,\nwe let $F_n$ denote the free group on $n$~generators.\nWe let $u_1, u_2, \\ldots, u_n \\in C^*_{\\mathrm{r}} (F_n)$\n(when $n = \\infty$,\nfor $u_1, u_2, \\ldots \\in C^*_{\\mathrm{r}} (F_{\\infty})$)\nbe the ``standard'' unitaries in $C^*_{\\mathrm{r}} (F_n)$,\nobtained as the images of the standard generators of~$F_n$.\nFor $\\zeta = (\\zeta_1, \\zeta_2, \\ldots, \\zeta_n) \\in (S^1)^n$\n(when $n = \\infty$,\nfor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}_{> 0}} \\in (S^1)^{{\\mathbb{Z}}_{> 0}}$),\nwe let $\\varphi_{\\zeta} \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_n) )$\nbe the (quasifree) automorphism\ndetermined by $\\varphi_{\\zeta} (u_k) = \\zeta_k u_k$\nfor $k = 1, 2, \\ldots, n$\n(when $n = \\infty$,\nfor $k = 1, 2, \\ldots)$.\nIt is well known that $\\zeta \\mapsto \\varphi_{\\zeta}$ is continuous.\n\\end{ntn}\n\n\\begin{ntn}\\label{N_9214_FreeShift}\nTake the standard generators of the free group $F_{\\infty}$\nto be indexed by~${\\mathbb{Z}}$,\nand for $n \\in {\\mathbb{Z}}$ let $u_n \\in C^*_{\\mathrm{r}} (F_{\\infty})$\nbe the unitary obtained as the image of the corresponding\ngenerator of~$F_{\\infty}$.\nWe denote by $\\sigma$ the free shift on $C^*_{\\mathrm{r}} (F_{\\infty})$,\nthat is, the automorphism\n$\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\ndetermined by $\\sigma (u_n) = u_{n + 1}$\nfor $n \\in {\\mathbb{Z}}$.\n\\end{ntn}\n\n\\begin{exa}\\label{E_9214_ShiftAndUHF}\nSet $X_0 = (S^1)^{{\\mathbb{Z}}}$.\nLet $h_0 \\colon X_0 \\to X_0$\nbe the (forwards) shift,\ndefined for $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X_0$ by\n$h_0 (\\zeta) = (\\zeta_{n - 1})_{n \\in {\\mathbb{Z}}}$.\nLet $D = \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}} M_2$\nbe the $2^{\\infty}$~UHF algebra.\nChoose a bijection $\\sigma \\colon {\\mathbb{Z}}_{> 0} \\to {\\mathbb{Z}}$,\nand for $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X_0$\ndefine\n\\[\n\\alpha_{\\zeta}^{(0)}\n = \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}}\n  {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       1 & 0 \\\\\n       0 & \\zeta_{\\sigma (n)}\n     \\end{matrix} \\right) \\right)\n \\in {\\mathrm{Aut}} (D).\n\\]\nThen $\\zeta \\mapsto \\alpha_{\\zeta}^{(0)}$ is continuous.\n\nWe have $\\mathrm{mdim} (h_0) = 1$ (see Notation~\\ref{N_0108_DimX})\nby Proposition~3.3 of~\\cite{LndWs},\nand we can use $X_0$, $h_0$, $D$, and $\\zeta \\mapsto \\alpha^{(0)}_{\\zeta}$\nin Lemma~\\ref{L_4Y12_GetAuto}.\nHowever, $h_0$ is not minimal.\nWe therefore proceed as follows.\nIdentify $[0, 1]$ with a closed arc in~$S^1$,\nsay via $\\lambda \\mapsto \\exp (\\pi i \\lambda)$.\nUse this identification to identify\n$[0, 1]^{{\\mathbb{Z}}}$ with a closed subset of $X_0 = (S^1)^{{\\mathbb{Z}}}$.\nThis identification is equivariant when both spaces\nare equipped with the shift homeomorphism{s}.\nLet $X \\S X_0$ and $h = h_0 |_{X} \\colon X \\to X$\nbe the minimal subshift\nin~\\cite{GlKr},\nwhich can be taken to have mean dimension\narbitrarily close to~$1$.\nFor $\\zeta \\in X$ let $\\alpha_{\\zeta} = \\alpha_{\\zeta}^{(0)}$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThen $\\alpha$ lies over the free minimal action of ${\\mathbb{Z}}$ on~$X$\ngenerated by~$h$.\n\nThe crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is simple\nby Proposition~\\ref{P_4Y15_CPSimple},\nand is nuclear,\nso it is ${\\mathcal{Z}}$-stable by Theorem~\\ref{ATraciallyZStable}.\n\\end{exa}\n\nThe action in Example~\\ref{E_9214_ShiftAndUHF}\ncan also be described as follows.\nRealize $D$ as the algebra of the canonical anticommutation\nrelations on generators $a_k$ for $k \\in {\\mathbb{Z}}_{> 0}$,\nfollowing Section~5.1 of~\\cite{Brtl}.\nThen $\\alpha_{\\zeta}^{(0)}$ is the gauge automorphism\n$\\alpha_{\\zeta}^{(0)} (a_k) = \\zeta_{\\sigma (k)} a_k$.\nIn~\\cite{Brtl}, see Section~5.1 and the proof of Lemma~5.2.\n\nThe next example is a slightly different version,\nwith larger mean dimension.\n\n\\begin{exa}\\label{E_9Z26_ShiftAndUHF_v2}\nLet $X_0$, $h_0$, $D$, and $\\zeta \\mapsto \\alpha_{\\zeta}^{(0)}$\nbe as in Example~\\ref{E_9214_ShiftAndUHF}.\nWe will, however, use the homeomorphism{} $h_0^2$ of~$X_0$,\nwhich is the shift on $(S^1 \\times S^1)^{{\\mathbb{Z}}}$.\nLet $(X, h)$ be the minimal subspace of the shift on $([0, 1]^2)^{{\\mathbb{Z}}}$\nconstructed in Proposition 3.5 of \\cite{LndWs},\nwhich satisfies $\\mathrm{mdim} (h) > 1$ (Notation~\\ref{N_0108_DimX}).\nUse an embedding of $[0, 1]^2$ in~$(S^1)^2$\nto choose an equivariant homeomorphism{} $g$\nfrom $(X, h)$ to an invariant closed subset of $(X_0, h_0^2)$.\nFor $x \\in X$ let $\\alpha_{x} = \\alpha_{g (x)}^{(0)}$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThe crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is again\nsimple by Proposition~\\ref{P_4Y15_CPSimple},\nso ${\\mathcal{Z}}$-stable by Theorem~\\ref{ATraciallyZStable}.\n\\end{exa}\n\n\\begin{exa}\\label{E_9214_ShiftAndCStFI_2}\nLet $X$, $h$, $D$, and $\\zeta \\mapsto \\alpha_{\\zeta}$ be as in\nExample~\\ref{E_9214_ShiftAndUHF}.\nDefine $E = D \\otimes C^*_{\\mathrm{r}} (F_{\\infty})$,\nand, following Notation~\\ref{N_9214_QFreeFn},\nfor $\\zeta \\in X$ define\n$\\beta_{\\zeta} = \\alpha_{\\zeta} \\otimes \\varphi_{\\zeta}$.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action $\\beta \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, E))$\nwhich lies over the free minimal action of ${\\mathbb{Z}}$ on~$X$\ngenerated by~$h$.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ is tracially ${\\mathcal{Z}}$-stable\nand has stable rank one.\nTracial ${\\mathcal{Z}}$-stability follows from Theorem~\\ref{ATraciallyZStable}.\nFor stable rank one,\n$E$ is exact,\nso has strict comparison of positive elements\nby Corollary 4.6 of~\\cite{Rdm7}.\nChoose any one point subset $Y \\S X$.\nThen $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)_Y$\n(see Definition~\\ref{D_4Y12_OrbSubalg})\nis ${\\mathcal{Z}}$-stable\nby Corollary~\\ref{EStableGivesZStable}.\nThis algebra\nis simple by Proposition~\\ref{P_4Y15_CPSimple},\nso by Theorem 6.7 of~\\cite{Rdm7} it has stable rank one.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)_Y$\nis centrally large\nin $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$\nby\nCorollary \\ref{AyCentLargeConds}(\\ref{9212_AyCentLargeConds_StrCmp}),\nso Theorem 6.3 of~\\cite{ArPh} implies\nthat $C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ has stable rank one.\n\\end{exa}\n\nIn Example~\\ref{E_9214_ShiftAndCStFI_2}, we don't know whether\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$ is ${\\mathcal{Z}}$-stable.\nWe also don't know whether tracial ${\\mathcal{Z}}$-stability implies\nstable rank one, although this is expected to be true.\n\nThere is nothing special about the specific formulas\nfor $x \\mapsto \\alpha_x$\nin Example~\\ref{E_9214_ShiftAndUHF}\nand Example~\\ref{E_9Z26_ShiftAndUHF_v2},\nand $x \\mapsto \\beta_x$ in Example~\\ref{E_9214_ShiftAndCStFI_2}.\nThey were chosen merely to show that interesting examples exist.\n\nExample~\\ref{E_9214_ShiftAndUHF},\nExample~\\ref{E_9Z26_ShiftAndUHF_v2},\nand Example~\\ref{E_9214_ShiftAndCStFI_2}\nwere constructed so that the homeomorphism~$h$\ndoes not have mean dimension zero.\nIf $X$ is finite dimensional,\nthen one can get all we do by using known results\nfor crossed products by actions with finite Rokhlin dimension\nwith commuting towers.\nWith finite dimensional~$X$, one even gets ${\\mathcal{Z}}$-stability in\nexamples like Example~\\ref{E_9214_ShiftAndUHF},\nExample~\\ref{E_9Z26_ShiftAndUHF_v2},\nand Example~\\ref{E_9214_ShiftAndCStFI_2},\nby Theorem~5.8 of~\\cite{HWZ}.\nHowever,\nwe know of no results which apply to examples of this type\nwhen $X$ is infinite dimensional{} and $h$ has mean dimension zero.\n\nThe next most obvious choice for a minimal homeomorphism{} of an infinite dimensional{} space~$X$\nseems to be as follows.\nTake $X = (S^1)^{{\\mathbb{Z}}_{> 0}}$,\nfix $\\theta = (\\theta_n)_{n \\in {\\mathbb{Z}}_{> 0}} \\in {\\mathbb{R}}^{{\\mathbb{Z}}_{> 0}}$,\nand define $h \\colon X \\to X$ by\n$h \\bigl( (\\zeta_{n})_{n \\in {\\mathbb{Z}}_{> 0}} \\bigr)\n = \\bigl( e^{2 \\pi i \\theta_n} \\zeta_{n} \\bigr)_{n \\in {\\mathbb{Z}}_{> 0}}$.\nIf $1, \\theta_1, \\theta_2, \\ldots$ are linearly independent over~${\\mathbb{Q}}$,\nthen $h$ is minimal\nby Proposition~\\ref{P_4X04_ProdRotation}.\nHowever, by considering the action in just one coordinate,\none sees that, regardless of $\\zeta \\mapsto \\alpha_{\\zeta}$,\nthe action has finite Rokhlin dimension with commuting towers.\nBy Theorem 6.2 of~\\cite{HWZ}, irrational rotations\nhave finite Rokhlin dimension with commuting towers.\nRemark 6.3 of~\\cite{HWZ}, according to which the result\nextends to any homeomorphism which has an irrational\nrotation as a factor,\napplies equally well to any automorphism of $C (S^1) \\otimes B$,\nfor any unital C*-algebra~$B$,\nwhich lies over an irrational rotation on~$S^1$\nin the sense of Definition~\\ref{D_4Y12_OverX}.\nIf we start with $C \\bigl( (S^1)^{{\\mathbb{Z}}_{> 0}}, \\, D \\bigr)$,\ntake $B$ above to be $C \\bigl( (S^1)^{{\\mathbb{Z}}_{> 0} \\setminus \\{ 1 \\}}, \\, D \\bigr)$.\nTherefore Theorem~5.8 of~\\cite{HWZ} applies,\nand our results give nothing new.\n\nWe now give some examples of the second type discussed\nin the introduction to this section.\nFor easy reference,\nwe recall a result on the stable rank of reduced free products.\nMany reduced free products have stable rank~$1$.\nThe following result is from~\\cite{DHR}.\n(It is not affected by the correction~\\cite{DHRc}.)\nReduced free products of unital C*-algebra{s}\nin~\\cite{DHR}\nare implicitly taken to be amalgamated over~${\\mathbb{C}}$;\nsee Section~2.2 of~\\cite{DHR}.\n\n\\begin{prp}[Corollary~3.9 of~\\cite{DHR}]\\label{P_5416_TsrFrGp}\nLet $G$ and $H$ be discrete groups\nwith ${\\mathrm{card}} (G) \\geq 2$\nand ${\\mathrm{card}} (H) \\geq 3$.\nThen $C^*_{\\mathrm{r}} (G \\star H)$ has stable rank~$1$.\n\\end{prp}\n\n\\begin{exa}\\label{E_5416_CSrFGp}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nLet $\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\nbe as in Notation~\\ref{N_9214_FreeShift}.\nLet\n$\\alpha \\in {\\mathrm{Aut}} \\big( C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}) \\big)$\nbe the tensor product of the automorphism\n$f \\mapsto f \\circ h^{-1}$ of $C (X)$ and~$\\sigma$.\nThen\n$C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}),\n    \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes C^*_{\\mathrm{r}} (F_{\\infty}),\n    \\, \\alpha \\big)$\nhas stable rank~$1$.\n\nTo prove the claim,\nwe apply Theorem~\\ref{P_9Z26_XisCantor} with\n$D = C^*_{\\mathrm{r}} (F_{\\infty})$ and\n$\\alpha$ as given.\nUse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nBy Proposition 6.3.2 of \\cite{Rbt},\n$D$ has strict comparison of positive elements.\nBy construction, $\\alpha$ lies over~$h$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_StrCmp}).\n\\end{exa}\n\nApparently no previously known results give\nanything about the crossed product in this example.\nIn particular, as discussed after Problem~\\ref{Pb_9922_RokhlinTsr},\nknowing that the action has the Rokhlin property\ndoesn't seem to help.\n\nThere are many other automorphisms\nof $C^*_{\\mathrm{r}} (F_{\\infty})$\nwhich could be used in place of~$\\sigma$.\nFor example,\none could take a quasifree automorphism,\nas in Notation~\\ref{N_9214_QFreeFn}.\nHere is a more interesting version.\n\n\\begin{exa}\\label{E_5416_DenjFst}\nFix any $n \\in \\set{ 2, 3, \\ldots, \\infty }$\nand take $D = C^*_{\\mathrm{r}} (F_{n})$.\nAdopt Notation~\\ref{N_9214_QFreeFn}.\nLet $h \\colon X \\to X$ be a restricted Denjoy homeomorphism{}\nwith rotation number $\\theta \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nLet $\\zeta \\colon X \\to S^1$ be the continuous{} surjective map\nwith $\\zeta (h (x)) = e^{2 \\pi i \\theta} \\zeta (x)$\nof Lemma~\\ref{L_9X04_DjSemiCj}\n(gotten from Corollary~3.2 and Proposition~3.4 of~\\cite{PtSmdSk}).\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof~$D$,\nfor $x \\in X$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( D )$\nbe determined\nby $\\alpha_x (u_k) = \\zeta (x) u_k$\nfor $k = 1, 2, \\ldots, n$ (or $k \\in {\\mathbb{Z}}_{> 0}$ if $n = \\infty$).\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} \\big( C (X, D ) \\big)$,\nwhich we can think of as\na kind of noncommutative Furstenberg transformation.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X, D ), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, D ), \\, \\alpha \\big)$\nhas stable rank one.\nTo prove the claim,\nuse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\n\\end{exa}\n\nAgain, apparently no previously known results give\nanything about the crossed product.\nSince $C^*_{\\mathrm{r}} (F_{n})$ is not ${\\mathcal{Z}}$-stable,\nknowing that the action has the Rokhlin property\ndoesn't seem to help.\n\nWe can generalize Example~\\ref{E_5416_DenjFst} as follows.\n\n\\begin{exa}\\label{E_9217_MultltDenj}\nFix $n \\in \\{ 2, 3, \\ldots, \\infty \\}$.\nFix $\\theta_1, \\theta_2, \\ldots, \\theta_n \\in {\\mathbb{R}}$\n(or $\\theta_1, \\theta_2, \\ldots \\in {\\mathbb{R}}$)\nsuch that $1, \\theta_1, \\theta_2, \\ldots, \\theta_n$\n(or $1, \\theta_1, \\theta_2, \\ldots$)\nare linearly independent over~${\\mathbb{Q}}$.\nFor $k = 1, 2, \\ldots, n$,\nlet $h \\colon X_k \\to X_k$ be a restricted Denjoy homeomorphism{}\nwith rotation number $\\theta_k \\in {\\mathbb{R}} \\setminus {\\mathbb{Q}}$,\nas in Definition~\\ref{D_9X04_RestrDj}.\nTake $X = \\prod_{k = 1}^n X_k$,\nand let $h \\colon X \\to X$\nact as $h_k$ on the $k$-th factor.\nThen~$h$ is minimal by Proposition~\\ref{P_4X04_ProdRstDj}.\nDefine $\\zeta_k \\colon X_k \\to S^1$\nanalogously to the definition of $\\zeta$\nin Example~\\ref{E_5416_DenjFst},\nand take $\\alpha_x (u_k) = \\zeta_k (x_k) u_k$.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon\n {\\mathbb{Z}} \\to {\\mathrm{Aut}} \\big( C (X, \\, C^*_{\\mathrm{r}} (F_{n}) ) \\big)$.\nThen\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, \\, C^*_{\\mathrm{r}} (F_{n}) ),\n    \\, \\alpha \\big)$\nis simple and has stable rank one\nfor the same reasons as in Example~\\ref{E_5416_DenjFst}.\n\\end{exa}\n\n\\begin{exa}\\label{R_5416_More}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nChoose any decomposition $X = X_1 \\amalg X_2$\nof $X$ as the disjoint union of two nonempty closed subsets.\nSet $D = C^*_{\\mathrm{r}} (F_{\\infty})$,\nwith the generators of $F_{\\infty}$ indexed by~${\\mathbb{Z}}$.\nFix any $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in (S^1)^{{\\mathbb{Z}}_{> 0}}$.\nFor $x \\in X_1$ take\n$\\alpha_{x}$ to be the automorphism\n$\\varphi_{\\zeta}$ of Notation~\\ref{N_9214_QFreeFn},\nexcept using ${\\mathbb{Z}}$ in place of ${\\mathbb{Z}}_{> 0}$,\nand for $x \\in X_2$ take\n$\\alpha_{x}$ to be the automorphism\n$\\sigma$ of Notation~\\ref{N_9214_FreeShift}.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} ( C (X, D) )$.\nThe algebra $C^* ({\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nThe algebra $D$ has strict comparison of positive elements by\nProposition 6.3.2 of~\\cite{Rbt},\nso $C^* ({\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha)$ has\nstable rank one\nby Theorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_StrCmp}).\n\\end{exa}\n\nExample~\\ref{R_5416_More} admits many variations.\nHere are several.\n\n\\begin{exa}\\label{E_9217_AltDenj}\nLet $h \\colon X \\to X$ be a restricted Denjoy homeomorphism,\nas in Example~\\ref{E_5416_DenjFst},\nand let $\\zeta \\colon X \\to S^1$ be as there.\nChoose any decomposition $X = X_1 \\amalg X_2$\nof $X$ as the disjoint union of two nonempty closed subsets.\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof $C^*_{\\mathrm{r}} (F_{2})$,\nfor $x \\in X_1$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe determined\nby $\\alpha_x (u_1) = \\zeta (x) u_1$ and $\\alpha_x (u_2) = u_2$,\nand for $x \\in X_2$\nlet $\\alpha_x \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe determined\nby $\\alpha_x (u_1) = u_1$ and $\\alpha_x (u_2) = \\zeta (x) u_2$.\nThe algebra\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n  \\otimes C^*_{\\mathrm{r}} (F_{2}),  \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\n\nWe claim that\nthis algebra\nhas stable rank one.\nTo prove the claim,\nuse Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes}\nto see that $D$ is simple and has a tracial state.\nBy Proposition~\\ref{P_5416_TsrFrGp}, we have ${\\mathrm{tsr}} (D) = 1$.\nNow apply\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\n\\end{exa}\n\n\\begin{exa}\\label{E_9217_DenjAndSw}\nIn Example~\\ref{E_9217_AltDenj},\nreplace the definition of $\\alpha_x$\nfor $x \\in X_2$ with\n$\\alpha_x (u_1) = u_2$ and $\\alpha_x (u_2) = u_1$.\nProposition~\\ref{P_4Y15_CPSimple}\nand Theorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen})\nstill apply,\nso\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n  \\otimes C^*_{\\mathrm{r}} (F_{2}),  \\, \\alpha \\big)$\nagain is simple and has stable rank one.\n\\end{exa}\n\nWe now give examples in which we also get real rank zero.\nThe following lemma will be used for them.\n\n\\begin{lem}\\label{L_0916_MakeSimple}\nLet $M$ be factor of type $\\mathrm{II}_1$,\nlet $\\alpha \\colon G \\to {\\mathrm{Aut}} (M)$\nbe an action of a countable group~$G$ on~$M$,\nand let $P \\S M$ be a separable C*-subalgebra.\nThen there exists a simple separable unital C*-subalgebra\n$D \\subset M$ which contains~$P$,\nis invariant under ${\\overline{\\sigma}}$,\nhas a unique tracial state\n(the restriction to $D$ of the unique tracial state on $M$),\nhas real rank zero and stable rank one,\nand such that the order on projections over~$D$\nis determined by traces.\n\\end{lem}\n\n\\begin{proof}\nThe proof is the same as that of Proposition~3.1 of~\\cite{Ph57},\nexcept for $G$-invariance.\nWe construct by induction on $n \\in {\\mathbb{Z}}_{\\geq 0}$ separable unital subalgebras\n$A_n, B_n, C_n, D_n, E_n, F_n \\S M$ with\n\\[\nP \\S A_0 \\S B_0 \\S C_0 \\S D_0 \\S E_0 \\S F_0 \\S \\cdots\n  \\S A_n \\S B_n \\S C_n \\S D_n \\S E_n \\S F_n \\S \\cdots\n\\]\nsuch that $A_n$, $B_n$, $C_n$, $D_n$, and $E_n$ have the\nproperties in the proof of Proposition~3.1 of~\\cite{Ph57}\n($A_n$ is simple, etc.),\nand such that $F_n$ is $G$-invariant.\nIn the induction step, after constructing $E_n$,\nwe take $F_n$ to be the C*-subalgebra generated by\n$\\bigcup_{g \\in G} \\alpha_g (E_n)$.\nThen ${\\overline{\\bigcup_{n = 0}^{\\infty} F_n}}$ is $G$-invariant.\nThe proof in~\\cite{Ph57} now gives the desired conclusion,\nexcept that the conclusion is that $K_0 (D) \\to K_0 (M)$\nis an order isomorphism onto its range\ninstead of that the order on projections over~$D$\nis determined by traces.\nBut the conclusion we get implies the conclusion we want\nif $A$ has cancellation,\nand stable rank one implies cancellation\nby Proposition 6.4.1 and Proposition 6.5.1 of~\\cite{Blkd3}.\n\\end{proof}\n\n\\begin{exa}\\label{E_5416_DenjRR0}\nLet $X$ be the Cantor set\nand let $h$ be an arbitrary minimal\nhomeomorphism of~$X$.\nLet $\\sigma \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{\\infty}) )$\nbe as in Notation~\\ref{N_9214_FreeShift}.\n\nWe regard $C^*_{\\mathrm{r}} (F_{\\infty})$\nas a subalgebra of the group\nvon Neumann algebra $W^* (F_{\\infty})$\nin the usual way,\nand we let ${\\overline{\\sigma}} \\in {\\mathrm{Aut}} (W^* (F_{\\infty}))$\nbe the von Neumann algebra automorphism\nwhich shifts the generators of $F_{\\infty}$\nin the same way that $\\sigma$ does.\nThus ${\\overline{\\sigma}} |_{C^*_{\\mathrm{r}} (F_{\\infty})} = \\sigma$.\nChoose a ${\\overline{\\sigma}}$-invariant\nsubalgebra $D \\subset W^* (F_{\\infty})$\nwhich contains $C^*_{\\mathrm{r}} (F_{\\infty})$\nas in Lemma~\\ref{L_0916_MakeSimple},\nwith the properties there.\nSet $\\gamma = {\\overline{\\sigma}} |_D \\in {\\mathrm{Aut}} (D)$.\nThen define $\\alpha \\in {\\mathrm{Aut}} ( C (X) \\otimes D )$\nin the same was as in Example~\\ref{E_5416_CSrFGp},\nusing $\\gamma$ in place of~$\\sigma$.\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (X) \\otimes D, \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nIt has stable rank~$1$ and real rank zero\nby Theorem \\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_SP}).\n\nThe algebra $D$ is not nuclear\nbecause the Gelfand-Naimark-Segal representation\nfrom its tracial state gives a nonhyperfinite factor.\n\\end{exa}\n\nIt is perhaps interesting to point out that every quasitrace\non the algebra~$D$ in Example~\\ref{E_5416_DenjRR0} is a trace.\nThis is a consequence of the following lemma.\n\n\\begin{lem}\\label{L_9929_UniqQTr}\nLet $A$ be a unital C*-algebra{} with real rank zero.\nSuppose that $\\tau$ is a quasitrace on~$A$ and\nthat whenever $p, q \\in M_{\\infty} (A)$ are projection{s}\nwith $\\tau (p) < \\tau (q)$,\nthen $p \\precsim q$.\nThen $\\tau$ is the only quasitrace on~$A$.\n\\end{lem}\n\n\\begin{proof}\nSuppose the conclusion is false,\nand let $\\sigma$ be some other quasitrace on~$A$.\nBy definition, for any quasitrace~$\\rho$ on~$A$\nand any $a, b \\in A_{\\mathrm{sa}}$,\nwe have $\\rho (a + i b) = \\rho (a) + i \\rho (b)$.\nTherefore there is $a \\in A_{\\mathrm{sa}}$\nsuch that $\\sigma (a) \\neq \\tau (a)$.\nSince quasitraces are continuous,\nit follows from real rank zero that there is $a \\in A_{\\mathrm{sa}}$\nsuch that $a$ has finite spectrum and $\\sigma (a) \\neq \\tau (a)$.\nSince quasitraces are linear on commutative C*-subalgebras,\nthere is a projection{} $p \\in A$ such that $\\sigma (p) \\neq \\tau (p)$.\n\nSuppose $\\sigma (p) < \\tau (p)$.\nChoose $m, n \\in {\\mathbb{Z}}_{\\geq 0}$ such that $m \\sigma (p) < n < m \\tau (p)$.\nDefine projection{s} $e, f \\in M_{\\infty} (A)$ by $e = 1_{M_m} \\otimes p$ and\n$f = 1_{M_n} \\otimes 1_A$.\nThen $\\tau (f) < \\tau (e)$, so $f \\precsim e$.\nBut $\\sigma (f) > \\sigma (e)$, a contradiction.\nSimilarly, $\\sigma (p) > \\tau (p)$ is also impossible.\nThis contradiction shows that $\\sigma$ does not exist.\n\\end{proof}\n\n\\begin{exa}\\label{E_9217_BiggerCstFn}\nLet $n \\in \\set{ 2, 3, \\ldots }$.\nFollowing Notation~\\ref{N_9214_QFreeFn} for the generators\nof $C^*_{\\mathrm{r}} (F_{n})$,\nfor $\\rho$ in the symmetric group~$S_n$\nlet $\\psi_{\\rho} \\in {\\mathrm{Aut}} ( C^*_{\\mathrm{r}} (F_{n}) )$\nbe the automorphism determined\nby $\\psi_{\\rho} (u_k) = u_{\\rho^{-1} (k)}$ for $k = 1, 2, \\ldots, n$,\nand let ${\\overline{\\psi_{\\rho} }}$ be the corresponding automorphism\nof the group von Neumann algebra $W^* (F_{n})$.\n\nBy Theorems 9.2.6 and 9.2.7 of \\cite{PhNotes},\nthe algebra $C^*_{\\mathrm{r}} (F_{n})$ is simple\nand has a unique tracial state.\nUse Lemma~\\ref{L_0916_MakeSimple}\nto find a simple separable unital C*-algebra{}\n$D \\subset W^* (F_{n})$ which contains $C^*_{\\mathrm{r}} (F_{n})$,\nis invariant under all the automorphisms\n${\\overline{\\psi_{\\rho} }}$ for $\\rho \\in S_n$,\nand has the other properties given in Lemma~\\ref{L_0916_MakeSimple}.\n\nLet $X$ be the Cantor set, and let $h \\colon X \\to X$ be any minimal homeomorphism.\nChoose any decomposition $X = \\coprod_{\\rho \\in S_n} X_{\\rho}$\nof $X$ as the disjoint union of nonempty closed subsets.\nFor $x \\in X_{\\rho}$ let $\\alpha_x = {\\overline{\\psi_{\\rho} }} |_D$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\n\nThe algebra $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}.\nIt has stable rank one and real rank zero by\nTheorem~\\ref{P_9Z26_XisCantor}(\\ref{9217_GetTsr1_CptGen}).\nThe algebra $D$ is not nuclear\nfor the same reason as in Example~\\ref{E_5416_DenjRR0}.\n\\end{exa}\n\nThe next example is of the third type discussed\nin the introduction to this section.\n\n\\begin{exa}\\label{E_9217_GenPType}\nLet $(\\pi_1, \\pi_2, \\ldots)$\nbe a sequence of unital finite dimensional{} representations\nof $C^* (F_2)$\nsuch that, for every $n \\in {\\mathbb{Z}}_{> 0}$,\nthe representation\n$\\bigoplus_{k = n}^{\\infty} \\pi_k$ is faithful.\nFor $n \\in {\\mathbb{Z}}_{\\geq 0}$,\nlet $d (n)$ be the dimension of $\\pi_n$,\nand define\n$l (n) = d (n) + 4$\nand $r (n) = \\prod_{k = 1}^n l (k)$.\nLet $u_1, u_2 \\in C^* (F_2)$\nbe the ``standard'' unitaries in $C^* (F_2)$,\nobtained as the images of the standard generators of~$F_2$,\nas in Notation~\\ref{N_9214_FreeShift}\nexcept that we are now using the full C*-algebra{} instead\nof the reduced C*-algebra.\nLet $\\gamma_1, \\gamma_2, \\gamma_3 \\in {\\mathrm{Aut}} (C^* (F_2))$\nbe the automorphisms determined by\n\\[\n\\gamma_1 (u_1) = u_1^{-1}\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_1 (u_2) = u_2,\n\\]\n\\[\n\\gamma_2 (u_1) = u_1\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_2 (u_2) = u_2^{-1},\n\\]\nand\n\\[\n\\gamma_3 (u_1) = u_1^{-1}\n\\qquad {\\mbox{and}} \\qquad\n\\gamma_3 (u_2) = u_2^{-1}.\n\\]\nFor $n \\in {\\mathbb{Z}}_{> 0}$\ndefine\n$D_n = M_{r (n)} \\otimes C^* (F_2)$,\nwhich we identify as\n\\[\nM_{l (n)} \\otimes M_{l (n - 1)} \\otimes \\cdots\n    \\otimes M_{l (1)} \\otimes C^* (F_2),\n\\]\nand define\n$\\gamma_{n, n - 1} \\colon D_{n - 1} \\to D_n$\nby,\nfor $a \\in D_{n - 1} = M_{r (n - 1)} \\otimes C^* (F_2)$,\n\\begin{align*}\n\\gamma_{n, n - 1} (a)\n& = {\\mathrm{diag}} \\bigl( a, \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_1 ) (a),\n        \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_2 ) (a),\n\\\\\n& \\hspace*{3em} {\\mbox{}}\n        \\, ({\\mathrm{id}}_{M_{r (n - 1)}} \\otimes \\varphi_3 ) (a),\n        \\, (\\pi_n \\otimes {\\mathrm{id}}_{M_{r (n - 1)}} (a) \\otimes 1\n     \\bigr)\n\\\\\n& \\in M_{l (n)} \\otimes D_{n - 1} = D_n.\n\\end{align*}\nLet $D$ be the direct limit of the resulting direct system.\nIt follows by methods of~\\cite{Ddlt}\nthat $D$ is simple, separable, and has tracial rank zero.\nIn particular, $D$ has stable rank one\nand real rank zero by Theorem 3.4 of~\\cite{HLinTrAF},\nand the order on projections over $D$ is determined by traces\nby Theorem 6.8 of~\\cite{HLinTTR}.\nMoreover, using the known result for $K_* (C^* (F_n))$\n(see~\\cite{CuFree}),\none gets $K_1 (D) = 0$.\n\nFor $\\zeta \\in S^1$ define inductively\nunitaries $u_n (\\zeta) \\in D_n$ as follows.\nSet $u_0 (\\zeta) = 1$,\nand, given $u_{n - 1} (\\zeta) \\in D_{n - 1}$,\nset\n\\[\nu_n (\\zeta)\n = \\bigl( {\\mathrm{diag}} \\bigl( \\zeta, 1, 1, \\ldots, 1 \\bigr)\n             \\otimes 1_{D_{n - 1}}  \\bigr)\n         \\gamma_{n, n - 1} ( u_{n - 1} (\\zeta) )\n \\in M_{l (n)} \\otimes D_{n - 1} = D_n.\n\\]\nThen the actions\n$\\zeta \\mapsto {\\mathrm{Ad}} (u_n (\\zeta)) \\in {\\mathrm{Aut}} (D_n)$\nof $S^1$ on~$D_n$\nare compatible with the direct system,\nand so yield a continuous{} map\n(in fact, an action)\n$\\zeta \\mapsto \\alpha_{\\zeta} \\colon S^1 \\to {\\mathrm{Aut}} (D)$.\nLet $h \\colon S^1 \\to S^1$ be an irrational rotation.\nApply Lemma~\\ref{L_4Y12_GetAuto}\nto get an action\n$\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} ( C (S^1, D) )$.\n\nThe algebra $C^* \\big( {\\mathbb{Z}}, \\, C (S^1, D), \\, \\alpha \\big)$\nis simple by Proposition~\\ref{P_4Y15_CPSimple}\nand has stable rank one\nby\nTheorem \\ref{T_9222_Tsr1AndRR0_CrPrd}(\\ref{9927_AyCentLargeConds_SP}).\n\\end{exa}\n\nMethods of~\\cite{NiuWng} will probably show that\nthe algebra $D$ in Example~\\ref{E_9217_GenPType} is not ${\\mathcal{Z}}$-stable\n(see Question~\\ref{Q_9217_GenPType_IsZStab}),\nalthough it definitely is tracially ${\\mathcal{Z}}$-stable.\n\nFinally, we give purely infinite examples.\n\n\\begin{exa}\\label{E_9214_ShiftAndOI}\nLet $(X, h)$ be the minimal subshift\n(of the shift on $(S^1)^{{\\mathbb{Z}}}$) with nonzero mean dimension\nused in Example~\\ref{E_9214_ShiftAndUHF}.\nLet $D = {\\mathcal{O}}_{\\infty}$,\nbut with standard generators indexed by~${\\mathbb{Z}}$,\nsay $s_k$ for $k \\in {\\mathbb{Z}}$.\nLet $\\alpha_{\\zeta}$ be the gauge automorphism,\ngiven by $\\alpha_{\\zeta} (s_k) = \\zeta_k s_k$.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (C (X, D))$\nbe the corresponding action as in Lemma~\\ref{L_4Y12_GetAuto}.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase}.\nSince $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nis nuclear,\nit is necessarily ${\\mathcal{O}}_{\\infty}$-stable\nby Theorem 3.15 of \\cite{KiPh}.\n\\end{exa}\n\nIn Example~\\ref{E_9214_ShiftAndOI},\nTheorems 4.1 and~5.8 of~\\cite{HWZ} don't apply,\nsince there is no reason to think that $\\alpha$\nhas finite Rokhlin dimension with commuting towers.\n\n\\begin{exa}\\label{E_9214_OnPINotZStab}\nLet $D$ be the reduced free product\n$D = (M_2 \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$,\ntaken with respect to the Lebesgue measure state on $C ([0, 1])$\nand the state on $M_2 \\otimes M_2$\ngiven by tensor product of the usual tracial state ${\\mathrm{tr}}$\nwith the state\n$\\rho (x) = {\\operatorname{tr}} \\big(\n {\\mathrm{diag}} \\big( \\frac{1}{3}, \\frac{2}{3} \\big) x \\big)$\non $M_2$.\nIt is shown in Example 5.8 of~\\cite{AmGlJmPh}\nthat $D$ is purely infinite and simple\nbut not ${\\mathcal{Z}}$-stable.\n\nTake $X = (S^1)^4$,\nand for $\\zeta = (\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4) \\in X$\ntake $\\alpha_{\\zeta}$\nto be the free product automorphism\nwhich is given by\n\\[\n{\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       \\zeta_1 & 0 \\\\\n       0     & \\zeta_2\n     \\end{matrix} \\right) \\right)\n\\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       \\zeta_3 & 0 \\\\\n       0     & \\zeta_4\n     \\end{matrix} \\right) \\right)\n\\]\non $M_2 \\otimes M_2$\nand is trivial on $C ([0, 1])$.\nChoose $\\theta_1, \\theta_2, \\theta_3, \\theta_4 \\in {\\mathbb{R}}$\nsuch that $1, \\theta_1, \\theta_2, \\theta_3, \\theta_4$\nare linearly independent over~${\\mathbb{Q}}$.\nTake $h \\colon X \\to X$\nto be\n\\[\n(\\zeta_1, \\zeta_2, \\zeta_3, \\zeta_4)\n  \\mapsto \\bigl( e^{2 \\pi i \\theta_1} \\zeta_1, \\,\n       e^{2 \\pi i \\theta_2} \\zeta_2, \\,\n       e^{2 \\pi i \\theta_3} \\zeta_3, \\,\n       e^{2 \\pi i \\theta_4} \\zeta_4  \\bigr).\n\\]\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase}.\n\\end{exa}\n\nThe action in Example~\\ref{E_9214_OnPINotZStab}\nhas finite Rokhlin dimension with commuting towers.\nHowever, we don't know any theorem on pure infiniteness\nfor crossed products by such actions\nwhen the original algebra is not ${\\mathcal{O}}_{\\infty}$-stable.\nWe address this in Question~\\ref{Pb_0111_FinRDim_PI} below.\nOur result also does not imply that the crossed product\nis ${\\mathcal{O}}_{\\infty}$-stable, or even ${\\mathcal{Z}}$-stable,\nalthough it seems plausible that it might be.\n\nIn the next two examples, $D$ is again not ${\\mathcal{Z}}$-stable.\nAlso, $X$ isn't finite dimensional, and $h$ doesn't even have mean dimension zero.\nSo a positive answer to Question~\\ref{Pb_0111_FinRDim_PI}\npresumably would not help.\n\n\\begin{exa}\\label{E_9214_OnPINotZStab_2}\nLet $D$ be as in Example~\\ref{E_9214_OnPINotZStab},\nand let $(X, h)$ be as in Example~\\ref{E_9214_ShiftAndUHF}\n(and reused in Example~\\ref{E_9214_ShiftAndOI}).\nFor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X$\ntake $\\alpha_{\\zeta}$ to be the free product automorphism\nwhich is given by\n\\[\n{\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       \\zeta_1 & 0 \\\\\n       0     & \\zeta_2\n     \\end{matrix} \\right) \\right)\n\\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       \\zeta_3 & 0 \\\\\n       0     & \\zeta_4\n     \\end{matrix} \\right) \\right)\n\\]\non $M_2 \\otimes M_2$\nand is trivial on $C ([0, 1])$.\n(We are only using the coordinates with indexes $1, 2, 3, 4$.)\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase},\nbut may well not be ${\\mathcal{Z}}$-stable.\n\\end{exa}\n\nWith a small modification,\nwe can give an example of this type in which the\nunderlying action of $(S^1)^{{\\mathbb{Z}}}$\nis effective.\n\n\\begin{exa}\\label{E_9Z26_OnPINotZStab_v2}\nLet $B$ be the $2^{\\infty}$~UHF algebra,\nand let $\\tau$ be its (unique) tracial state.\nLet ${\\mathrm{tr}}$ be the tracial state{} on~$M_2$,\nand let $\\rho$ be the state on $M_2$ given by\n$\\rho (x)\n = {\\mathrm{tr}} \\big( {\\mathrm{diag}} \\big( \\frac{1}{3}, \\frac{2}{3} \\big) x \\big)$.\nLet $D$ be\nthe reduced free product\n$D = (B \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$,\ntaken with respect to the state $\\tau \\otimes \\rho$ on $B \\otimes M_2$\nand the Lebesgue measure state on $C ([0, 1])$.\nWe claim that $A$ is purely infinite and simple\nbut not ${\\mathcal{Z}}$-stable.\n\nTo prove pure infiniteness,\nin Examples 3.9(iii) of~\\cite{Dkm2}\ntake $A_1 = B$\nwith the state~$\\tau$,\ntake $F = M_2$\nwith the state~$\\rho$,\nand take $B = C ([0, 1])$\nwith the state given by Lebesgue measure.\nThese choices satisfy the hypotheses there.\nSo $A$ is is purely infinite and simple.\n\nTo prove that $A$ is not ${\\mathcal{Z}}$-stable,\nlet $\\pi$ be the Gelfand-Naimark-Segal representation of~$B$\nassociated with~$\\tau$,\nset $N = \\tau (B)''$,\nand also write $\\tau$ for the corresponding tracial state{} on~$N$.\nIn Proposition 5.6 of~\\cite{AmGlJmPh},\ntake $P_1 = N \\otimes M_2$\nwith the state $\\tau \\otimes \\rho$,\nand take $P_2 = L^{\\infty} ([0, 1])$\nwith the state given by Lebesgue measure.\nDefine\n\\[\na = \\left[ \\left( \\begin{matrix} 1 & 0 \\\\ 0 & -1 \\end{matrix} \\right)\n        \\otimes 1 \\otimes 1 \\otimes \\cdots \\right]\n      \\otimes 1\n    \\in \\left( \\bigotimes_{n = 1}^{\\infty} M_2 \\right) \\otimes M_2\n    = B \\otimes M_2\n    \\S P_1.\n\\]\nTake $G_1 = \\{ 1,  a\\} \\S P_1$,\nand take $G_2 \\S P_2$ to be the set of functions\n$\\lambda \\mapsto e^{2 \\pi i n \\lambda}$ for $n \\in {\\mathbb{Z}}$.\nThese choices\nsatisfy the hypotheses there.\nMoreover, $G_1 \\S B \\otimes M_2$ and $G_2 \\S C ([0, 1])$.\nTherefore the reduced free product\n$A = (B \\otimes M_2) \\star_{\\mathrm{r}} C ([0, 1])$\nis a subalgebra of the algebra~$P$\nin Proposition 5.6 of~\\cite{AmGlJmPh}\nwhich contains $a$, $b$, and~$c$,\nso is not ${\\mathcal{Z}}$-stable by Proposition 5.6 of~\\cite{AmGlJmPh}.\nThe claim is proved.\n\nLet $(X, h)$ be\nas in Example~\\ref{E_9214_ShiftAndUHF}\n(and reused in Example~\\ref{E_9214_ShiftAndOI}).\nLet $\\sigma \\colon {\\mathbb{Z}}_{> 0} \\to {\\mathbb{Z}}$ be a bijection\n(as in Example~\\ref{E_9214_ShiftAndUHF}).\nFor $\\zeta = (\\zeta_n)_{n \\in {\\mathbb{Z}}} \\in X$\ntake $\\alpha_{\\zeta}$\nto be the free product automorphism\nwhich is the identity on $C ([0, 1])$\nand which is the infinite tensor product\n\\[\n\\left[ \\bigotimes_{n \\in {\\mathbb{Z}}_{> 0}}\n  {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       1 & 0 \\\\\n       0 & \\zeta_{\\sigma (n)}\n     \\end{matrix} \\right) \\right) \\right]\n     \\otimes {\\mathrm{Ad}} \\left( \\left( \\begin{matrix}\n       \\zeta_{\\sigma (1)} & 0 \\\\\n       0             & \\zeta_{\\sigma (2)}\n     \\end{matrix} \\right) \\right)\n  \\in {\\mathrm{Aut}} (B \\otimes M_2)\n\\]\non~$B \\otimes M_2$.\nThen $C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$ is\npurely infinite and simple by Theorem~\\ref{T_0101_PICase},\nbut may well not be ${\\mathcal{Z}}$-stable.\n\\end{exa}\n\n\\section{Open problems}\\label{Sec_Qs}\n\nIn this section,\nwe collect some open questions\nsuggested by the examples and results in this paper.\n\n\\begin{pbm}\\label{Pb_9922_RokhlinTsr}\nLet $A$ be a unital C*-algebra,\nand let $\\alpha$ be an action of ${\\mathbb{Z}}$ on~$A$\nwhich has finite Rokhlin dimension with commuting towers.\nSuppose that $A$ has stable rank one and $C^* ({\\mathbb{Z}}, A, \\alpha)$ is simple.\nDoes it follow that $C^* ({\\mathbb{Z}}, A, \\alpha)$\nhas stable rank one?\n\\end{pbm}\n\nThis seems to be unknown even if $A$ is simple\n(in which case simplicity of $C^* ({\\mathbb{Z}}, A, \\alpha)$ is automatic)\nand $\\alpha$ has the Rokhlin property.\nWithout assuming simplicity of $C^* ({\\mathbb{Z}}, A, \\alpha)$,\nthe answer is definitely no.\nFor aperiodic homeomorphism{s} of the Cantor set\nwhose transformation group \\ca{s} don't have stable rank one,\nsee Theorem~3.1 of~\\cite{Pt1}\n(it is easy to construct examples there\nwhich have more than one minimal set)\nor Example~8.8 of~\\cite{Phl10}.\nCorollary~2.6 of~\\cite{Szb}\nimplies that the corresponding actions of~${\\mathbb{Z}}$\nhave Rokhlin dimension with commuting towers at most~$1$.\nIn fact, though,\nat least in Example~8.8 of~\\cite{Phl10} we get the Rokhlin property.\n\n\\begin{lem}\\label{L_9922_HasRokhlin}\nLet $h \\colon X \\to X$ be the aperiodic homeomorphism{}\nof the Cantor set in Example~8.8 of~\\cite{Phl10}.\nThen the induced automorphism of $C (X)$ has the Rokhlin property.\n\\end{lem}\n\n\\begin{proof}\nRecall from~\\cite{Phl10} that $X_1 = {\\mathbb{Z}} \\cup \\{\\pm \\infty \\}$,\n$h_1 \\colon X_1 \\to X_1$ is $n \\mapsto n + 1$,\n$X_2$ is the Cantor set,\n$h_2 \\colon X_2 \\to X_2$ is minimal,\n$X = X_1 \\times X_2$,\nand $h = h_1 \\times h_2$.\n\nLet $N \\in {\\mathbb{Z}}_{> 0}$.\nThe standard first return time construction provides\n\\[\nn, r (1), r (2), \\ldots, r (n) \\in {\\mathbb{Z}}_{> 0}\n\\]\nwith $N < r (1) < r (2) < \\cdots < r (n)$,\nand compact open subsets $Z_1, Z_2, \\ldots, Z_n \\S X_2$,\nsuch that\n\\[\nX_2 = \\coprod_{k = 1}^{n} \\coprod_{j = 0}^{r (k) - 1} h_{2}^j (Z_k).\n\\]\nThen the sets\n\\[\nX_1 \\times Z_1, \\quad X_1 \\times Z_2,\n \\quad \\ldots, \\quad X_1 \\times Z_n\n\\]\nform a system of Rokhlin towers for~$h$,\nwith heights $r (1), r (2), \\ldots, r (n)$,\nall of which exceed~$N$.\n\\end{proof}\n\nIn fact,\nit seems to be known that any aperiodic homeomorphism{}\nof the Cantor set~$X$\ninduces an automorphism of $C (X)$ with the Rokhlin property.\nWe have not found a reference,\nand we do not prove this here.\n\nThe following question asks for a plausible generalization\nof Lemma~\\ref{L_4Y11_Comp}.\n\n\\begin{qst}\\label{Qst1_20190103D}\nLet $D$ be a simple unital C*-algebra.\nLet $X$ be a compact metric space, let $G$ be a discrete group,\nand let $(g,x) \\mapsto gx$ be a minimal\nand essentially free action of $G$ on $X$.\nLet $\\alpha \\colon G \\to {\\mathrm{Aut}} (C (X, D))$ be an action of $G$\nwhich lies over the action of $G$ on $X$\nand which is pseudoperiodically generated.\nSet $A = C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$.\n\nDoes it follow that for every $a \\in C (X, D)_{+} \\setminus \\{ 0 \\}$\nthere exists $f \\in C (X)_{+} \\setminus \\{ 0 \\} \\subset A$\nsuch that $f \\precsim_A a$?\n\\end{qst}\n\nWe expect the techniques in the proof of  Lemma \\ref{L_4Y11_Comp}\n(taking $Y$ to be the empty set) can be used to answer\nthis question in the affirmative.\n\n\\begin{qst}\\label{Q_9212_ExistsNotPseudoper}\nIs there a simple unital C*-algebra~$D$\nsuch that ${\\mathrm{Aut}} (D)$\nis not pseudoperiodic in the sense of Definition~\\ref{D_4Y12_PsPer}?\n\\end{qst}\n\nPresumably such examples exist,\nbut we don't know of any.\nIndeed,\nwe don't see any reason why there should not be $\\alpha \\in {\\mathrm{Aut}} (D)$\nsuch that $\\set{\\alpha^n \\colon n \\in {\\mathbb{Z}}}$ is not pseudoperiodic.\n\n\\begin{qst}\\label{Q_9214_IsZStab}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (X, E), \\, \\beta \\big)$\nin Example~\\ref{E_9214_ShiftAndCStFI_2}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nWhat about crossed products by similarly constructed actions?\n\\end{qst}\n\n\\begin{qst}\\label{Q_9214_IsZStab_2}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (X)\n  \\otimes C^*_{\\mathrm{r}} (F_{\\infty}),  \\, \\alpha \\big)$\nin Example~\\ref{E_5416_CSrFGp}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nWhat about crossed products by similarly constructed actions?\nWhat about Example~\\ref{E_5416_DenjFst}?\n\\end{qst}\n\n\\begin{qst}\\label{Q_9217_GenPType_IsZStab}\nConsider the crossed product\n$C^* \\big( {\\mathbb{Z}}, \\, C (S^1, D), \\, \\alpha \\big)$\nin Example~\\ref{E_9217_GenPType}.\nIs this algebra ${\\mathcal{Z}}$-stable?\n\\end{qst}\n\nThe following question\nis motivated by Example~\\ref{E_9214_OnPINotZStab}.\n\n\\begin{qst}\\label{Pb_0111_FinRDim_PI}\nLet $A$ be a nonsimple unital C*-algebra{} which is purely infinite\nin the sense of Definition~4.1 of~\\cite{KiRo2000}.\nLet $\\alpha \\colon {\\mathbb{Z}} \\to {\\mathrm{Aut}} (A)$\nbe an action with finite Rokhlin dimension with commuting towers.\nDoes it follow that $C^* ({\\mathbb{Z}}, A, \\alpha)$\nis purely infinite?\n\\end{qst}\n\nIf $A$ is ${\\mathcal{O}}_{\\infty}$-stable,\nthen $C^* ({\\mathbb{Z}}, A, \\alpha)$ is at least ${\\mathcal{Z}}$-stable\nby Theorem~5.8 of~\\cite{HWZ}.\nIf $A$ is also exact,\nthen so is $C^* ({\\mathbb{Z}}, A, \\alpha)$\n(by Proposition 7.1(v) of~\\cite{Krh2}),\nand $C^* ({\\mathbb{Z}}, A, \\alpha)$ is traceless\n(as at the beginning of Section~5 of~\\cite{Rdm7})\nbecause $A$ is,\nso $C^* ({\\mathbb{Z}}, A, \\alpha)$ is purely infinite\nby Corollary~5.1 of~\\cite{Rdm7}.\n(In this case, one obviously wants ${\\mathcal{O}}_{\\infty}$-stability.\nSee Question~\\ref{Pb_0111_PI_ZStab_to_OIStab} below.)\nBut the question as stated seems to be open,\neven if one assumes that $C^* ({\\mathbb{Z}}, A, \\alpha)$ is simple.\nIf $A$ itself is simple,\nthen $C^* ({\\mathbb{Z}}, A, \\alpha)$\nis purely infinite even just assuming that $\\alpha$ is pointwise outer,\nby Corollary~4.4 of~\\cite{JngOsk}.\nProvided one uses the reduced crossed product,\nthis remains true if ${\\mathbb{Z}}$ is replaced by any discrete group.\n\n\\begin{qst}\\label{Pb_0111_OIStabQ}\nConsider the crossed product\n$C^* \\bigl( {\\mathbb{Z}}, \\, C (X, D), \\, \\alpha \\bigr)$\nin Example~\\ref{E_9214_OnPINotZStab_2}.\nIs this algebra ${\\mathcal{Z}}$-stable?\nIs it ${\\mathcal{O}}_{\\infty}$-stable?\n\\end{qst}\n\nWe hope that ${\\mathcal{Z}}$-stability should come from the\naction in the ``$(S^1)^4$~direction''.\nWe suppose that if a purely infinite simple C*-algebra{}\nis ${\\mathcal{Z}}$-stable,\nthen it is probably ${\\mathcal{O}}_{\\infty}$-stable,\nbut this seems to be open in general.\n\n\\begin{qst}\\label{Pb_0111_PI_ZStab_to_OIStab}\nLet $B$ be a ${\\mathcal{Z}}$-stable purely infinite simple C*-algebra.\nDoes it follow that $B$ is ${\\mathcal{O}}_{\\infty}$-stable?\n\\end{qst}\n\nIf $B$ is separable and nuclear,\nthe answer is yes, by Theorem~5.2 of~\\cite{Rdm7}.\nBut this result doesn't help with Example~\\ref{E_9214_OnPINotZStab_2},\neven if we prove ${\\mathcal{Z}}$-stability there.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n                                    \\label{sec:Introduction}\n\nThe analysis of gravitational instability in the expanding universe\nmodel was first presented by Lifshitz in 1946 in a general relativistic \ncontext \\cite{Lifshitz}.\nHistorically, the much simpler, and in hindsight, more intuitive \nNewtonian study followed later \\cite{Bonnor}.\nThe pioneering study by Lifshitz is based on a gauge choice which is\ncommonly called the synchronous gauge. \nAs the later studies have shown, the synchronous gauge is only \none way of fixing the gauge freedom out of several available\nfundamental gauge conditions\n\\cite{Harrison,Nariai,Bardeen-1980,KS,Reviews,PRW}.\nAs will be summarized in the following, out of the several gauge conditions\nonly the synchronous gauge fails to fix the gauge mode completely and \nthus often needs more involved algebra.\nAs long as one is careful of the algebra this gauge choice does \nnot cause any kind of intrinsic problem; there exist, however, some persisting\nalgebraic errors in the literature, see Sec. \\ref{sec:Discussion}.\nThe gauge condition which turns out to be especially suitable for\nhandling the perturbed density is the comoving gauge used first by\nNariai in 1969 \\cite{Nariai}.\nSince the comoving gauge condition completely fixes the gauge transformation\nproperty, the variables in this gauge can be equivalently considered as\ngauge-invariant ones.\nAs mentioned, there exist several such fundamental gauge conditions\neach of which completely fixes the gauge transformation properties.\nOne of such gauge conditions suitable for handling the gravitational\npotential and the velocity perturbations is the zero-shear gauge\nused first by Harrison in 1967 \\cite{Harrison}.\nThe variables in such gauge conditions are equivalently gauge-invariant.\nUsing the gauge freedom as an advantage for handling the problem\nwas emphasized by Bardeen in 1988 \\cite{Bardeen-1988}.\nIn order to use the gauge choice as an advantage a {\\it gauge ready method}\nwas proposed in \\cite{PRW} which will be adopted in the \nfollowing, see Sec. \\ref{sec:Relativistic}.\n\nThe variables which characterize the self gravitating Newtonian fluid flow are\nthe density, the velocity and the gravitational potential \n(the pressure is specified by an equation of state),\nwhereas, the relativistic flow may be characterized by various components\nof the metric (and consequent curvatures) and the energy-momentum tensor.\nSince the relativistic gravity theory is a constrained system we have the \nfreedom of imposing certain conditions on the metric or the energy-momentum \ntensor as coordinate conditions.  \nIn the perturbation analyses the freedom arises because we need to introduce\na fictitious background system in order to describe the physical perturbed\nsystem.\nThe correspondence of a given spacetime point between the perturbed spacetime\nand the ficticious background one could be made with certain degrees \nof freedom.\nThis freedom can be fixed by suitable conditions (the gauge conditions)\nbased on the spacetime coordinate transformation.\nStudies in the literature show that a certain variable in a certain gauge\ncondition correctly reproduces the corresponding Newtonian behavior.\nAlthough the perturbed density in the comoving gauge shows the Newtonian\nbehavior, the perturbed potential and the perturbed velocity in the\nsame gauge do not behave like the Newtonian ones; for example, in the comoving\ngauge the perturbed velocity vanishes by the coordinate (gauge) condition.\nIt is known that both the perturbed potential and the perturbed velocity\nin the zero-shear gauge correctly behave like the corresponding\nNewtonian variables \\cite{Bardeen-1980}.\n\nIn the first part (Sec. \\ref{sec:Six-gauges}) we will elaborate establishing \nsuch correspondences between the relativistic and Newtonian perturbed \nvariables.\nOur previous work on this subject is presented in \\cite{MDE,Hyun};\ncompared with \\cite{MDE} in this work we will explicitly compare \nthe relativistic equations for the perturbed density, potential and velocity \nvariables in several available gauge conditions with the\nones in the Newtonian system.\nWe will include both the background spatial curvature ($K$) \nand the cosmological constant ($\\Lambda$).\nIn the second part (Sec. \\ref{sec:Hydrodynamics}), using the variables \nwith correct Newtonian correspondences,\nwe will extend our relativistic results to the situations\nwith general pressures in the background and perturbations.\nWe will present the relativistic equations satisfied by the gauge-invariant\nvariables and will derive the {\\it general solutions} valid in the \nsuper-sound-horizon scale (i.e., larger than Jeans scale)\nconsidering both $K$ and $\\Lambda$, and the generally evolving\nideal fluid equation of state $p = p(\\mu)$.\n\nSection \\ref{sec:review} is a review of our previous work\ndisplaying the basic equations in both Newtonian and relativistic contexts\nand summarizing our strategy for handling the equations. \nIn Sec. \\ref{sec:Six-gauges} we consider a pressureless limit of the\nrelativistic equations.\nWe derive the equations for the perturbed density, the perturbed potential \nand the perturbed velocity in several different fundamental gauge conditions.\nBy comparing these relativistic equations in several gauges with\nthe Newtonian ones we {\\it identify} the gauge conditions which reproduce \nthe correct Newtonian behavior for certain variables in general scales.\nIn Sec. \\ref{sec:Hydrodynamics} we present the fully relativistic equations \nfor the identified gauge-invariant variables with correct Newtonian limits, \nnow, considering the general pressures in the background and perturbations.\nWe derive the general large-scale solutions valid for general $K$ and $\\Lambda$\nin an ideal fluid medium, which are thus valid for general equation of \nstate of the \nform $p = p(\\mu)$, but with negligible entropic and anisotropic pressures.\nThe solutions are valid in the super-sound-horizon scale, thus are valid \neffectively in all scales in the matter dominated era where the\nsound-horizon (Jeans scale) is negligible.\nWe discuss several quantities which are conserved in the large-scale\nunder the changing background equation of state.\nThese are the perturbed three-space curvature in several gauge conditions,\nand in particular, we find the three-space curvature perturbation\nin the comoving gauge shows a distinguished behavior:\nfor $K = 0$ (but for general $\\Lambda$), it is conserved in a\nsuper-sound-horizon scale independently of the changing equation of state.\n{}For completeness we also summarize the case with multiple fluid\ncomponents in Sec. \\ref{sec:Multi}, and cases of the rotation and\nthe gravitational wave in Sec. \\ref{sec:rot-GW}.\nSec. \\ref{sec:Hydrodynamics} is the highlight of the present work.\nSec. \\ref{sec:Discussion} is a discussion.\nWe set $c \\equiv 1$.\n\n\n\\section{Basic Equations and Strategy} \n                                    \\label{sec:review}\n\n\n\\subsection{Newtonian Cosmological Perturbations} \n                                    \\label{sec:Newtonian}\n\nSince the Newtonian perturbation analysis in the expanding medium is well known\nwe begin by summarizing the basic equations \\cite{MDE,Weinberg,Peebles-1980}.\nThe background evolution is governed by\n\\begin{eqnarray}\n   & & H^2 = {8 \\pi G \\over 3} \\varrho - {K \\over a^2} + {\\Lambda \\over 3}, \n       \\quad \\varrho \\propto a^{-3},\n   \\label{Newt-BG}\n\\end{eqnarray}\nwhere we allowed the general curvature (total energy) \nand the cosmological constant; $a(t)$ is a cosmic scale factor,\n$H (t) \\equiv \\dot a \/ a$, and $\\varrho (t)$ is the mass density. \nIn Newtonian theory the cosmological constant can be introduced in the\nPoisson equation {\\it by hand} as \n$\\nabla^2 \\Phi = 4 \\pi G \\varrho - \\Lambda$.\nPerturbed parts of the mass conservation, the momentum conservation\nand the Poisson's equations are [see Eqs. (43,46) in \\cite{MDE}]: \n\\begin{eqnarray}\n   & & \\delta \\dot \\varrho + 3 H \\delta \\varrho = - {k \\over a} \\varrho\n       \\delta v, \\quad\n       \\delta \\dot v + H \\delta v = {k \\over a} \\delta \\Phi, \\quad\n       - {k^2 \\over a^2} \\delta \\Phi = 4 \\pi G \\delta \\varrho,\n   \\label{Newt-eqs} \n\\end{eqnarray}\nwhere $\\delta \\varrho ({\\bf k}, t)$, $\\delta v ({\\bf k}, t)$ and\n$\\delta \\Phi ({\\bf k},t)$ are the Fourier modes of the perturbed mass density, \nvelocity, and gravitational potential, respectively \n[see Eqs. (44,45) in \\cite{MDE}];\n$k$ is a comoving wave number with $\\nabla^2 \\rightarrow - k^2\/a^2$.\n[For linear perturbations the same forms of equations are valid in the \nconfiguration and the Fourier spaces.\nThus, without causing any confusion, we often ignore distinguishing the\nFourier space from the configuration space by an additional subindex.]\nEquation (\\ref{Newt-eqs}) can be arranged into the closed form equations\nfor $\\delta$ ($\\equiv \\delta \\varrho \/ \\varrho$), $\\delta v$ and\n$\\delta \\Phi$ as:\n\\begin{eqnarray}\n   & & \\ddot \\delta + 2 H \\dot \\delta - 4 \\pi G \\varrho \\delta \n       = {1 \\over a^2 H} \\left[ a^2 H^2 \\left( {\\delta \\over H} \\right)^\\cdot\n       \\right]^\\cdot = 0,\n   \\label{Newt-delta-eq} \\\\\n   & & \\delta \\ddot v + 3 H \\delta \\dot v + \\left( \\dot H + 2 H^2 \n       - 4 \\pi G \\varrho \\right) \\delta v = 0,\n   \\label{Newt-v-eq} \\\\\n   & & \\delta \\ddot \\Phi + 4 H \\delta \\dot \\Phi\n       + \\left( \\dot H + 3 H^2 - 4 \\pi G \\varrho \\right) \\delta \\Phi \n       = {1 \\over a^3 H} \\left[ a^2 H^2 \n       \\left( {a \\over H} \\delta \\Phi \\right)^\\cdot \\right]^\\cdot = 0.\n   \\label{Newt-Phi-eq} \n\\end{eqnarray}\nWe note that Eqs. (\\ref{Newt-eqs}-\\ref{Newt-Phi-eq}) are\nvalid for general $K$ and $\\Lambda$.\nAlthough redundant, we presented these equations for later comparison\nwith the relativistic results.\nThe general solutions for $\\delta$, $\\delta v$ and $\\delta \\Phi$\nimmediately follow as (see also Table 1 in \\cite{MDE}):\n\\begin{eqnarray}\n   & & \\delta ({\\bf k},t) = k^2 \\left[ \n       H C ({\\bf k}) \\int^t_0 {dt \\over \\dot a^2} \n       + {H \\over 4 \\pi G \\varrho a^3} d ({\\bf k}) \\right], \n   \\label{delta-N} \\\\\n   & & \\delta v ({\\bf k}, t) = - \\left[ {k \\over aH} C ({\\bf k})\n       \\left( 1 + a^2 H \\dot H \\int^t_0 {dt \\over \\dot a^2} \\right)\n       + {k \\dot H \\over 4 \\pi G \\varrho a^2} d ({\\bf k}) \\right],\n   \\label{v-N} \\\\\n   & & \\delta \\Phi ({\\bf k}, t) = - \\left[ 4 \\pi G \\varrho a^2 H C ({\\bf k}) \n       \\int^t_0 {dt \\over \\dot a^2} + {H \\over a} d ({\\bf k}) \\right].\n   \\label{Phi-N}\n\\end{eqnarray}\nThe coefficients $C ({\\bf k})$ and $d ({\\bf k})$ are two integration constants, \nand indicate the relatively growing and decaying solutions, respectively;\nthe coefficients are matched in accordance with the solutions with a general\npressure in Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) using \nEq. (\\ref{Correspondences-2})\n(different dummy variables for $t$ in the integrands are assumed).\nIn the literature we can find various analytic form expressions of the \nabove solutions in diverse situations, \nsee \\cite{Lifshitz,Harrison,Nariai,Weinberg,Peebles-1980,MDE-sols}.\n\n\n\\subsection{Relativistic Cosmological Perturbations} \n                                    \\label{sec:Relativistic}\n\nIn the relativistic theory the fundamental variables are the metric and\nthe energy-momentum tensor.\nAs a way of summarizing our notation we present our convention of\nthe metric and the energy-momentum tensor.\nAs the metric we consider a spatially homogeneous and isotropic spacetime\nwith the most general perturbation\n\\begin{eqnarray}\n   d s^2 = - a^2 \\left( 1 + 2 \\alpha \\right) d \\eta^2\n       - a^2 \\left( \\beta_{,\\alpha} +B_\\alpha \\right) d \\eta d x^\\alpha\n       + a^2 \\left[ g_{\\alpha\\beta}^{(3)} \\left( 1 + 2 \\varphi \\right)\n       + 2 \\gamma_{|\\alpha\\beta} + 2 C_{(\\alpha|\\beta)} + 2 C_{\\alpha\\beta}\n       \\right] d x^\\alpha d x^\\beta,\n   \\label{metric} \n\\end{eqnarray}\nwhere $0 = \\eta$ with $dt \\equiv a d \\eta$,\nIndices $\\alpha$, $\\beta$ $\\dots$ are spatial ones and\n$g^{(3)}_{\\alpha\\beta}$ is the three-space metric of the homogeneous and\nisotropic space.\n$\\alpha ({\\bf x}, t)$, $\\beta ({\\bf x}, t)$, $\\gamma ({\\bf x}, t)$,\nand $\\varphi ({\\bf x}, t)$ indicate the scalar-type structure with\nfour degrees of freedom.\nThe transverse $B_\\alpha ({\\bf x}, t)$ and $C_\\alpha ({\\bf x}, t)$\nindicate the vector-type structure with four degrees of freedom.\nThe transverse-tracefree $C_{\\alpha\\beta} ({\\bf x}, t)$\nindicates the tensor-type structure with two degrees of freedom.\nThe energy-momentum tensor is decomposed as:\n\\begin{eqnarray}\n   & & T^0_0 \\equiv - \\bar \\mu - \\delta \\mu, \\quad\n       T^0_\\alpha \\equiv - {1 \\over k} \n       \\left( \\bar \\mu + \\bar p \\right) v_{,\\alpha}\n       + \\delta T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha},\n   \\nonumber \\\\\n   & & T^\\alpha_\\beta\n       \\equiv \\left( \\bar p + \\delta p \\right) \\delta^\\alpha_\\beta \n       + {1 \\over a^2} \\left( \\nabla^\\alpha \\nabla_\\beta\n       - {1 \\over 3} \\Delta \\delta^\\alpha_\\beta \\right) \\sigma\n       + \\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n       + \\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta},\n   \\label{Tab}\n\\end{eqnarray}\nwhere $\\mu ({\\bf k}, t) \\equiv \\bar \\mu (t) + \\delta \\mu ({\\bf k}, t)$ and \n$p \\equiv \\bar p + \\delta p$ are the energy density and the pressure, \nrespectively; an overbar indicates a background order quantity and will \nbe ignored unless necessary.\n$v$ is a frame-independent velocity related perturbed order variable\nand $\\sigma$ is an anisotropic pressure.\nSituation with multiple fluids will be considered in Sec. \\ref{sec:Multi}.\n$\\delta T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha}$, $\\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}$,\nand $\\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta}$ are vector and tensor-type\nperturbed order energy-momentum tensor.\nAll the spatial tensor indices and the vertical bar are based on \n$g^{(3)}_{\\alpha\\beta}$ as the metric.\nThe three types of structures are related to the density condensation,\nthe rotation, and the gravitational wave, respectively.\nDue to the high symmetry in the background the three types of perturbations\ndecouple from each other to the linear order.\nSince these three types of structures evolve independently, we can handle\nthem separately.\nEvolutions of the rotation and the gravitational wave will be discussed\nin Sec. \\ref{sec:rot-GW}.\n\n{}For the background equations we have [Eq. (6) in \\cite{Ideal}]:\n\\begin{eqnarray}\n   H^2 = {8 \\pi G \\over 3} \\mu - {K \\over a^2} + {\\Lambda \\over 3}, \\quad\n       \\dot H = - 4 \\pi G ( \\mu + p ) + {K \\over a^2}, \\quad\n       \\dot \\mu = - 3 H \\left( \\mu + p \\right). \n   \\label{BG-eqs} \n\\end{eqnarray}\nThese equations follow from $G^0_0$ and $G^\\alpha_\\alpha - 3 G^0_0$\ncomponents of Einstein equations and $T^b_{0;b} = 0$, respectively;\nthe third equation follows from the first two equations.\n{}For $p = 0$ and replacing $\\mu$ with $\\varrho$ Eq. (\\ref{BG-eqs}) \nreduces to Eq. (\\ref{Newt-BG}).\nThe relativistic cosmological perturbation equations without fixing\nthe temporal gauge condition, thus in a gauge ready form,\nwere derived in Eq. (22-28) of \\cite{PRW}\n[see also Eqs. (8-14) in \\cite{Ideal}; we use \n$\\delta \\equiv \\delta \\mu \/\\mu$ and $v \\equiv - (k\/a) \\Psi\/(\\mu + p)$]:\n\\begin{eqnarray}\n   & & \\kappa \\equiv 3 \\left( - \\dot \\varphi + H \\alpha \\right)\n       + {k^2 \\over a^2} \\chi,\n   \\label{eq1} \\\\\n   & & - {k^2 - 3K \\over a^2} \\varphi + H \\kappa = - 4 \\pi G \\mu \\delta,\n   \\label{eq2} \\\\\n   & & \\kappa - {k^2 - 3K \\over a^2} \\chi = 12 \\pi G \\left( \\mu + p \\right)\n       {a \\over k} v,\n   \\label{eq3} \\\\\n   & & \\dot \\chi + H \\chi - \\alpha - \\varphi = 8 \\pi G \\sigma,\n   \\label{eq4} \\\\\n   & & \\dot \\kappa + 2 H \\kappa = \\left( {k^2 \\over a^2} - 3 \\dot H \\right) \n       \\alpha + 4 \\pi G \\left( 1 + 3 c_s^2 \\right) \\mu \\delta \n       + 12 \\pi G e,\n   \\label{eq5} \\\\\n   & & \\dot \\delta + 3 H \\left( c_s^2 - {\\rm w} \\right) \\delta \n       + 3 H { e \\over \\mu } = \\left( 1 + {\\rm w} \\right)\n       \\left( \\kappa - 3 H \\alpha - {k \\over a} v \\right),\n   \\label{eq6} \\\\\n   & & \\dot v + \\left( 1 - 3 c_s^2 \\right) H v = {k \\over a} \\alpha\n       + {k \\over a \\left( 1 + {\\rm w} \\right) } \\left( c_s^2 \\delta\n       + {e \\over \\mu} - {2\\over 3} {k^2 - 3K \\over a^2} {\\sigma \\over \\mu}\n       \\right),\n   \\label{eq7} \n\\end{eqnarray}\nwhere we introduced a spatially gauge-invariant combination\n$\\chi \\equiv a ( \\beta + a \\dot \\gamma)$.\nAccording to their origins, Eqs. (\\ref{eq1}-\\ref{eq7}) can be called: \nthe definition of $\\kappa$, \nADM energy constraint ($G^0_0$ component of Einstein equations), \nmomentum constraint ($G^0_\\alpha$ component), \nADM propagation \n($G^\\alpha_\\beta - {1 \\over 3} \\delta^\\alpha_\\beta G^\\gamma_\\gamma$ component),\nRaychaudhuri ($G^\\alpha_\\alpha - G^0_0$ component), \nenergy conservation ($T^b_{0;b} = 0$),\nand momentum conservation ($T^b_{\\alpha;b} = 0$) equations, respectively.\nThe perturbed metric variables $\\varphi ({\\bf k}, t)$, $\\kappa ({\\bf k},t)$,\n$\\chi ({\\bf k}, t)$ and $\\alpha ({\\bf k}, t)$ are the perturbed part of\nthe three space curvature, expansion, shear, and the lapse function, \nrespectively; meanings for $\\varphi$, $\\kappa$, and $\\chi$ are based on\nthe normal frame vector field.\nThe isotropic pressure is decomposed as\n\\begin{eqnarray}\n   \\delta p ({\\bf k}, t) \\equiv c_s^2 (t) \\delta \\mu ({\\bf k}, t) \n       + e ({\\bf k}, t), \\quad\n       c_s^2 \\equiv {\\dot p \\over \\dot \\mu}, \\quad\n       {\\rm w} (t) \\equiv {p \\over \\mu}.\n   \\label{p-decomposition}\n\\end{eqnarray}\n\nUnder the gauge transformation $\\tilde x^a = x^a + \\xi^a$, the variables\ntransform as (see Sec. 2.2 in \\cite{PRW}):\n\\begin{eqnarray}\n   & & \\tilde \\alpha = \\alpha - \\dot \\xi^t, \\quad\n       \\tilde \\varphi = \\varphi - H \\xi^t, \\quad\n       \\tilde \\chi = \\chi - \\xi^t, \\quad\n       \\tilde \\kappa = \\kappa \n       + \\left( 3 \\dot H - {k^2 \\over a^2} \\right) \\xi^t,\n   \\nonumber \\\\\n   & & \\tilde v = v - {k \\over a} \\xi^t, \\quad\n       \\tilde \\delta = \\delta + 3 \\left( 1 + {\\rm w} \\right) H \\xi^t,\n   \\label{GT}\n\\end{eqnarray}\nwhere $\\xi^t \\equiv e^s \\xi^\\eta$, and $e$ and $\\sigma$ are \ngauge-invariant.\n[Due to the spatial homogeneity of the background, the effect \n from the spatial gauge transformation has been trivially handled; \n $\\chi$ and $v$ are the spatially gauge-invariant combinations of \n the variables, and the other metric and fluid variables \n are naturally spatially gauge-invariant \\cite{Bardeen-1988}.]\nThus, the perturbed variables in Eqs. (\\ref{eq1}-\\ref{eq7})\nare {\\it designed} so that any one of the following conditions\ncan be used to fix the freedom based on the temporal gauge transformation:\n$\\alpha \\equiv 0$ (the synchronous gauge), \n$\\varphi \\equiv 0$ (the uniform-curvature gauge),\n$\\chi \\equiv 0$ (the zero-shear gauge), \n$\\kappa \\equiv 0$ (the uniform-expansion gauge), \n$v\/k \\equiv 0$ (the comoving gauge), \nand $\\delta \\equiv 0$ (the uniform-density gauge).\nThese gauge conditions contain most of the gauge conditions used in the\nliterature concerning the cosmological hydrodynamic perturbation in a single\ncomponent situation; for the multiple component case see Eq. (\\ref{Gauges-2}).\nOf course, we can make an infinite number of different linear combinations \nof these gauge conditions each of which is also suitable for the temporal \ngauge fixing condition.\nOur reason for choosing these conditions as fundamental is partly based \non conventional use, but apparently, as the names of the conditions indicate,\nthe gauge conditions also have reasonable meanings based on the metric and \nthe energy-momentum tensors.\n\nEach one of these gauge conditions, except for the synchronous gauge, \nfixes the temporal gauge transformation property completely.\nThus, each variable in these five gauge conditions \nis equivalent to a corresponding gauge-invariant combination\n(of the variable concerned and the variable used in the gauge condition).\nWe proposed a convenient way of writing the gauge-invariant variables \n\\cite{PRW}.\n{}For example, we let\n\\begin{eqnarray}\n   & & \\delta_v \\equiv \\delta + 3 (1 + {\\rm w}) {aH \\over k} v\n       \\equiv 3 (1 + {\\rm w}) {a H \\over k} v_\\delta, \\quad\n       \\varphi_\\chi \\equiv \\varphi - H \\chi \\equiv - H \\chi_\\varphi, \\quad\n       v_\\chi \\equiv v - {k \\over a} \\chi \\equiv - {k \\over a} \\chi_v.\n   \\label{GI-combinations}\n\\end{eqnarray}\nThe variables $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ are gauge-invariant \ncombinations; $\\delta_v$ becomes $\\delta$ in the comoving gauge which takes\n$v\/k=0$ as the gauge condition, etc.\nIn this manner we can systematically construct the corresponding \ngauge-invariant combination for any variable based on a gauge condition which \nfixes the temporal gauge transformation property completely.\nOne variable evaluated in different gauges can be considered as different \nvariables, and they show different behaviors in general.\nIn this sense, except for the synchronous gauge, the variables in the \nrest of five gauges can be considered as the gauge-invariant variables.\n[The variables with a subindex $\\alpha$ are not gauge-invariant, \nbecause those are equivalent to variables in the synchronous gauge.]\nAlthough $\\delta_v$ is a gauge-invariant variable which becomes\n$\\delta$ in the comoving gauge, $\\delta_v$ itself \ncan be evaluated in any gauge with the same value.\n\nEquations (\\ref{eq1}-\\ref{eq7}) are the basic set of gauge-ready form\nequations for hydrodynamic perturbations proposed in \\cite{PRW}: \n``The moral is that one should work in the gauge that is mathematically\nmost convenient for the problem at hand'', \\cite{Bardeen-1988}.\nAccordingly, in order to use the available gauge conditions as the\nadvantage, Eqs. (\\ref{eq1}-\\ref{eq7}) are designed \nfor easy implementation of the fundamental gauge conditions.\nUsing this {\\it gauge-ready formulation}, complete sets of solutions for the \nsix different gauge conditions are presented in an ideal fluid case \n\\cite{Ideal}, and in a pressureless medium \\cite{MDE}.\nEquations (\\ref{BG-eqs}-\\ref{p-decomposition}) include general pressures which \nmay account for the nonequilibrium or dissipative effects in hydrodynamic \nflows in the cosmological context with general $K$ and $\\Lambda$.\nThe equations are expressed in general forms so that the fluid quantities \ncan represent successfully the effects of the scalar field and \nclasses of generalized gravity theories \\cite{PRW}.\nThe applications of the gauge-ready formalism to the minimally coupled scalar\nfield and to the generalized gravity theories have been made in \\cite{MSF-GGT}.\n\n\n\\section{Newtonian Correspondence of the Relativistic Analyses} \n                                    \\label{sec:Six-gauges}\n\nIn \\cite{MDE,Hyun} we developed arguments on the correspondences\nbetween the Newtonian and the relativistic analyses.\nIn order to reinforce the Newtonian correspondences of certain\n(gauge-invariant) variables in certain gauges, in the following we\nwill present the closed form differential equations for \n$\\delta$, $v$ and $\\varphi$ in the six different gauge conditions\nand compare them with the Newtonian ones in \nEqs. (\\ref{Newt-delta-eq}-\\ref{Newt-Phi-eq}).\n\nThere are reasons why we should know about possible Newtonian behaviors\nof the relativistic variables.\nVariables in the relativitistic gravity are only parameters appearing in\nthe metric and energy-momentum tensor.\n{}For example, later we will find that only in the zero-shear gauge does\nthe perturbed curvature variable behave as the perturbed Newtonian potential,\nwhich we have some experience of in Newtonian physics.\nThe same variable evaluated in other gauge conditions is simply other\nvariable and we cannot regard it as the perturbed potential.\nAs we have introduced several different fundamental gauge conditions\nit became necessary to identify the correct gauge where a variable shows\nthe corresponding Newtonian behavior.\nAs will be shown in this section, we rarely find Newtonian correspondences.\nIn fact, there exists an almost unique gauge condition for each\nrelativistic perturbation variable which shows Newtonian behavior.\nThe results will be summarized in Sec. \\ref{sec:Correspondences}.\n\nIn this section we consider a {\\it pressureless medium},\n$p = 0$ (thus ${\\rm w} = 0 = c_s^2$) and $e = 0 = \\sigma$.\nHowever, we consider general $K$ and $\\Lambda$.\n\n\n\\subsection{Comoving Gauge} \n                                    \\label{sec:CG}\n\nAs the gauge condition we set $v\/k \\equiv 0$.\nEquivalently, we can set $v\/k = 0$ and leave the other variables as\nthe gauge-invariant combinations with subindices $v\/k$ or simply $v$.\n{}From Eq. (\\ref{eq7}) we have $\\alpha_v = 0$.\nThus the comoving gauge is a case of the synchronous gauge; this is\ntrue {\\it only} in a pressureless situation.\n{}From Eqs. (\\ref{eq1}-\\ref{eq7}) we can derive:\n\\begin{eqnarray}\n   & & \\ddot \\delta_v + 2 H \\dot \\delta_v - 4 \\pi G \\mu \\delta_v = 0,\n   \\label{CG-delta-eq} \\\\\n   & & \\ddot \\varphi_v + 3 H \\dot \\varphi_v - {K \\over a^2} \\varphi_v = 0.\n   \\label{CG-varphi-eq} \n\\end{eqnarray}\nThus, Eq. (\\ref{CG-delta-eq}) has a form identical to Eq. \n(\\ref{Newt-delta-eq}).\nHowever, Eq. (\\ref{CG-varphi-eq}) differs from Eq. (\\ref{Newt-Phi-eq}),\nand apparently we do not have an equation for $v$.\n\n{}For $K = 0$ Eq. (\\ref{CG-varphi-eq}) leads to\ntwo solutions which are $\\varphi_v \\propto {\\rm constant}$ and \n$\\int_0^t a^{-3} dt$. \nSince we have $\\alpha_v = 0$, for $K = 0$, from Eqs. (\\ref{eq1},\\ref{eq3})\nwe have $\\dot \\varphi_v = 0$.\nThis implies that, for $K = 0$, the second solution, \n$\\varphi_v \\propto \\int_0^t a^{-3} dt$, should have the vanishing coefficient,\nsee Eq. (\\ref{varphi_v-sol3}).\nThe solution of Eq. (\\ref{CG-varphi-eq}) for general $K$ will be\npresented later in Eq. (\\ref{varphi_v-sol}).\n\n\n\\subsection{Synchronous Gauge} \n                                    \\label{sec:SG}\n\nWe let $\\alpha \\equiv 0$.\nThis gauge condition does not fix the temporal gauge transformation property\ncompletely.\nThus, although we can still indicate the variables in this gauge condition\nusing subindices $\\alpha$ without ambiguity, these variables are not \ngauge-invariant; see the discussion after Eq. (\\ref{varphi-GI}).\nEquation (\\ref{eq7}) leads to\n\\begin{eqnarray}\n   v_\\alpha = c_g {k \\over a},\n\\end{eqnarray}\nwhich is a pure gauge mode.\nThus, fixing $c_g \\equiv 0$ exactly corresponds to the comoving gauge.\nWe can show that the following two equations are not affected by the\nremaining gauge mode.\n\\begin{eqnarray}\n   & & \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha \n       - 4 \\pi G \\mu \\delta_\\alpha = 0,\n   \\label{SG-delta-eq} \\\\\n   & & \\ddot \\varphi_\\alpha + 3 H \\dot \\varphi_\\alpha \n       - {K \\over a^2} \\varphi_\\alpha = 0.\n   \\label{SG-varphi-eq} \n\\end{eqnarray}\nEquation (\\ref{SG-delta-eq}) is identical to Eq. (\\ref{Newt-delta-eq}).\nThis is because in a pressureless medium the behavior of the gauge modes \nhappen to coincide with the behavior of the decaying physical solutions\nfor $\\delta_\\alpha$ and $\\varphi_\\alpha$.\nHowever, for the variables $\\kappa_\\alpha$ and $\\chi_\\alpha$ the gauge mode \ncontribution appears explicitly.\n\nThus, in a pressureless medium, variables\nin the synchronous gauge behave the same as the ones in the comoving gauge, \nexcept for the additional gauge modes which appear in the synchronous gauge\nfor some variables.\nIn a pressureless medium, we can simultaneously impose both the comoving gauge\nand the synchronous gauge conditions.\nHowever, this is possible only in a pressureless medium, see the Appendix.\nIn Sec. \\ref{sec:Discussion} we indicate some common errors in the literature\nbased on the synchronous gauge.\n\n\n\\subsection{Zero-shear Gauge} \n                                    \\label{sec:ZSG}\n\nWe let $\\chi \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\chi$.\nWe can derive:\n\\begin{eqnarray}\n   & & \\ddot \\delta_\\chi + { 2 k^2 \/ a^2 - 36 \\pi G \\mu \\left[ 1 + ( H^2 \n       - 2 \\dot H ) a^2 \/ \\left( k^2 - 3 K \\right) \\right] \\over\n       k^2 \/ a^2 - 12 \\pi G \\mu \\left[ 1 + 3 H^2 a^2\/\\left( k^2 - 3K \\right) \n       \\right] }\n       H \\left( \\dot \\delta_\\chi \n       - 12 \\pi G \\mu {a^2 \\over k^2 - 3 K} H \\delta_\\chi \\right)\n   \\nonumber \\\\\n   & & \\qquad \n       - 4 \\pi G \\mu \\left[ 1 - 3 ( H^2 - 2 \\dot H )\n       {a^2 \\over k^2 - 3 K} \\right] \\delta_\\chi = 0,\n   \\label{ZSG-delta-eq} \\\\\n   & & \\ddot v_\\chi + 3 H \\dot v_\\chi \n       + \\left( \\dot H + 2 H^2 - 4 \\pi G \\mu \\right) v_\\chi = 0,\n   \\label{ZSG-v-eq} \\\\\n   & & \\ddot \\varphi_\\chi + 4 H \\dot \\varphi_\\chi + \\left( \\dot H + 3 H^2 \n       - 4 \\pi G \\mu \\right) \\varphi_\\chi = 0.\n   \\label{ZSG-varphi-eq} \n\\end{eqnarray}\nThus, Eqs. (\\ref{ZSG-v-eq},\\ref{ZSG-varphi-eq}) have forms identical \nto Eqs. (\\ref{Newt-v-eq},\\ref{Newt-Phi-eq}), respectively.\nHowever, only in the small-scale limit is the behavior of $\\delta_\\chi$\nthe same as the Newtonian one.\n\n\n\\subsection{Uniform-expansion Gauge} \n                                    \\label{sec:UEG}\n\nWe let $\\kappa \\equiv 0$, and substitute the remaining variables into\nthe gauge-invariant combinations with subindices $\\kappa$.\nWe can derive:\n\\begin{eqnarray}\n   & & \\ddot \\delta_\\kappa \n       + \\left( 2 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n       \\right) \\dot \\delta_\\kappa - \\left[ 4 \\pi G \\mu {k^2\/a^2 + 6 H^2 \\over\n       k^2\/a^2 - 3 \\dot H} + \\left( {12 \\pi G \\mu H \\over\n       k^2\/a^2 - 3 \\dot H} \\right)^\\cdot \\right] \\delta_\\kappa = 0,\n   \\label{UEG-delta-eq} \\\\ \n   & & \\ddot v_\\kappa + \\left[ 2 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n       + {4 \\pi G \\mu \\over k^2\/a^2 - 3 \\dot H}\n       \\left( {k^2\/a^2 - 3 \\dot H \\over 4 \\pi G \\mu} \\right)^\\cdot\n       \\right] \\dot v_\\kappa\n   \\nonumber \\\\ \n   & & \\quad \n       + \\left[ \\dot H + H^2 - 4 \\pi G \\mu {k^2\/a^2 + 3 H^2 \\over\n       k^2\/a^2 - 3 \\dot H} + H {4 \\pi G \\mu \\over k^2\/a^2 - 3 \\dot H}\n       \\left( { k^2\/a^2 - 3 \\dot H \\over 4 \\pi G \\mu} \\right)^\\cdot\n       \\right] v_\\kappa = 0,\n   \\label{UEG-v-eq} \\\\ \n   & & \\ddot \\varphi_\\kappa \n       + \\left( 4 H - {12 \\pi G \\mu H \\over k^2\/a^2 - 3 \\dot H}\n       \\right) \\dot \\varphi_\\kappa\n       + \\left[ \\dot H + 3 H^2 - 4 \\pi G \\mu {k^2\/a^2 + 9 H^2 \\over\n       k^2\/a^2 - 3 \\dot H} - \\left( {12 \\pi G \\mu H \\over k^2\/a^2\n       - 3 \\dot H} \\right)^\\cdot \\right] \\varphi_\\kappa = 0.\n   \\label{UEG-varphi-eq}\n\\end{eqnarray}\nIn the small-scale limit we can show that\nEqs. (\\ref{UEG-delta-eq}-\\ref{UEG-varphi-eq}) \nreduce to Eqs. (\\ref{Newt-delta-eq}-\\ref{Newt-Phi-eq}), respectively.\nThus, {\\it in the small-scale limit}, all three variables $\\delta_\\kappa$, \n$v_\\kappa$ and $\\varphi_\\kappa$ correctly reproduce the Newtonian behavior.\nHowever, outside or near the (visual) horizon scale, the behaviors of all these\nvariables {\\it strongly deviate} from the Newtonian ones.\n\nIn Sec. 84 of \\cite{Peebles-1980} we find that in order to get the usual \nNewtonian equations a coordinate transformation is made so that\nwe have $\\dot h \\equiv 0$ in the new coordinate.\nWe can show that $\\dot h = 2 \\kappa$ in our notation.\nThus, the new coordinate used in \\cite{Peebles-1980} is the uniform-expansion \ngauge\\footnote{This was incorrectly pointed out after Eq. (49) in \\cite{MDE}:\n         In \\cite{MDE} it was wishfully mentioned that in Sec. 84 of \n         \\cite{Peebles-1980} the gauge transformations were made into the \n         comoving gauge for $\\delta$ and into the zero-shear gauge for \n         $v$ and $\\varphi$, respectively.\n     }.\n\n\n\\subsection{Uniform-curvature Gauge} \n                                    \\label{sec:UCG}\n\nWe let $\\varphi \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\varphi$.\nWe have\n\\begin{eqnarray}\n   & & \\ddot \\delta_\\varphi + 2 H { (k^2 - 3K)\/a^2 + 18 \\pi G \\mu \\over \n       (k^2 - 3K)\/a^2 + 12 \\pi G \\mu } \\dot \\delta_\\varphi\n       - { 4 \\pi G \\mu k^2\/a^2 \\over \n       (k^2 - 3K)\/a^2 + 12 \\pi G \\mu } \\delta_\\varphi = 0,\n   \\label{UCG-delta-eq} \\\\\n   & & \\ddot v_\\varphi + \\left( 5 H + 2 {\\dot H \\over H} \\right) \\dot v_\\varphi\n       + \\left( 3 \\dot H + 4 H^2 \\right) v_\\varphi = 0. \n   \\label{UCG-v-eq}\n\\end{eqnarray}\nIn the small-scale limit Eq. (\\ref{UCG-delta-eq}) reduces to\nEq. (\\ref{Newt-delta-eq}).\nIn the uniform-curvature gauge, the perturbed potential is set to be equal\nto zero by the gauge condition.\nThe uniform-curvature gauge condition is known to have \ndistinct properties in handling the scalar field or the dilaton field \nwhich appears in a broad class of the generalized gravity theories\n\\cite{MSF-GGT,GRG-MSF-GGT}.\n\n\n\\subsection{Uniform-density Gauge} \n                                    \\label{sec:UDG}\n\nWe let $\\delta \\equiv 0$, and substitute the other variables into the \ngauge-invariant combinations with subindices $\\delta$.\nWe have\n\\begin{eqnarray}\n   & & \\ddot v_\\delta + 2 \\left( 2 H + {\\dot H \\over H} \\right) \\dot v_\\delta\n       + 3 \\left( H^2 + \\dot H \\right) v_\\delta = 0,\n   \\label{UDG-v-eq} \\\\\n   & & \\ddot \\varphi_\\delta + 2 H { (k^2 - 3 K)\/a^2 + 18 \\pi G \\mu\n       \\over (k^2- 3K)\/a^2 + 12 \\pi G \\mu} \\dot \\varphi_\\delta\n       - {4 \\pi G \\mu k^2\/a^2\n       \\over (k^2- 3K)\/a^2 + 12 \\pi G \\mu} \\varphi_\\delta = 0.\n   \\label{UDG-varphi-eq} \n\\end{eqnarray}\nThese equations differ from Eqs. (\\ref{Newt-v-eq},\\ref{Newt-Phi-eq}).\nOf course, we have no equation for $\\delta$ which is set to be equal to zero \nby our choice of the gauge condition.\n\n\n\\subsection{Newtonian Correspondences} \n                                    \\label{sec:Correspondences}\n\nAfter a thorough comparison made in this section we found that \nequations for $\\delta$ in the comoving gauge ($\\delta_v$), and\nfor $v$ and $\\varphi$ in the zero-shear gauge ($v_\\chi$ and $\\varphi_\\chi$)\nshow the same forms as the corresponding Newtonian equations \n{\\it in general scales} and considering general $K$ and $\\Lambda$.\nUsing these gauge-invariant combinations in Eq. (\\ref{GI-combinations}),\nEqs. (\\ref{eq1}-\\ref{eq7}) can be combined to give:\n\\begin{eqnarray}\n   \\dot \\delta_v = - {k^2 - 3K \\over a k} v_\\chi, \\quad\n       \\dot v_\\chi + H v_\\chi = - {k \\over a} \\varphi_\\chi, \\quad\n       {k^2 - 3K \\over a^2} \\varphi_\\chi = 4 \\pi G \\mu \\delta_v.\n   \\label{GI-eqs}\n\\end{eqnarray}\nComparing Eq. (\\ref{GI-eqs}) with Eq. (\\ref{Newt-eqs}) \nwe can identify either one of the following correspondences:\n\\begin{eqnarray}\n   & & \\delta_v \\leftrightarrow \\delta |_{\\rm Newtonian}, \\quad\n       {k^2 - 3 K \\over k^2} v_\\chi \n       \\leftrightarrow \\delta v |_{\\rm Newtonian}, \\quad\n       - {k^2 - 3 K \\over k^2} \\varphi_\\chi \\leftrightarrow \n       \\delta \\Phi |_{\\rm Newtonian},\n   \\label{Correspondences-1} \\\\\n   & & {k^2 \\over k^2 - 3 K} \\delta_v \\leftrightarrow \n       \\delta |_{\\rm Newtonian}, \\quad\n       v_\\chi \\leftrightarrow \\delta v |_{\\rm Newtonian}, \\quad\n       - \\varphi_\\chi \\leftrightarrow \\delta \\Phi |_{\\rm Newtonian}.\n   \\label{Correspondences-2}\n\\end{eqnarray}\nIn \\cite{MDE} we proposed the correspondences in Eq. (\\ref{Correspondences-1}),\nbut the ones in Eq. (\\ref{Correspondences-2}) also work well.\nIn fact, the gravitational potential identified in \nEq. (\\ref{Correspondences-2}) is the one often found in the Newtonian limit \nof the general relativity, e.g. in \\cite{Weinberg}.\nUsing Eqs. (\\ref{GI-combinations},\\ref{eq3},\\ref{eq4},\\ref{GI-eqs}) \nwe can also identify the following relations:\n\\begin{eqnarray}\n   & & \\delta_v = 3 H {a \\over k} v_\\delta, \\quad\n       v_\\chi = - {k \\over a} \\chi_v = - {ak \\over k^2 - 3 K} \\kappa_v, \\quad\n       \\varphi_\\chi = -\\alpha_\\chi = - H \\chi_\\varphi.\n\\end{eqnarray}\n\n{}From a given solution known in one gauge we can derive all the rest of the \nsolutions even in other gauge conditions.\nThis can be done either by using the gauge-invariant combination of variables \nor directly through gauge transformations.\nThe complete set of exact solutions in a pressureless medium is presented \nin \\cite{MDE}.\n{}From our study in this section and using the complete solutions presented \nin Tables 1 and 2 of \\cite{MDE} we can identify variables in certain gauges \nwhich correspond to the Newtonian ones.  These are summarized in Table 1.\n\n\\centerline{ [[{\\bf TABLE 1.}]] }\n\n\\noindent\nAs mentioned earlier, all three variables in the uniform-expansion gauge show \nNewtonian correspondence in the small-scale; however all of them change \nthe behaviors near and above the horizon scale. \nIn the small-scale limit, except for the  \nuniform-density gauge where $\\delta \\equiv 0$, \n$\\delta$ in all gauge conditions behaves in the same way \\cite{Bardeen-1980}.\nMeanwhile, since only $\\delta_v$ ($\\delta$ in the comoving gauge)\nshows the Newtonian behavior in a general scale, it may be natural to identify \n$\\delta_v$ as the one most closely resembling the Newtonian one.\n\nNotice that, although we have horizons in the relativistic analysis\nthe equations for $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$\nkeep the same form as the corresponding Newtonian equations\n{\\it in the general scale}.\nConsidering this as an additional point we regard\n$\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ as the one most closely corresponding\nto the Newtonian variables.\nThis argument will become stronger as we consider the case with a general\npressure in the next section.\n\n\n\\section{Relativistic Cosmological Hydrodynamics}\n                                \\label{sec:Hydrodynamics}\n\n\n\\subsection{General Equations and Large-scale Solutions}\n                                \\label{sec:Eqs}\n\nIn the previous sections we have shown that the gauge-invariant\ncombinations $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ behave most similarly\nto the Newtonian $\\delta \\equiv \\delta \\varrho\/\\varrho$, \n$\\delta v$ and $\\delta \\Phi$, respectively.\nThe equations remain the same in a general scale which includes the\nsuperhorizon scales in the relativistic situation considering \ngeneral $K$ and $\\Lambda$.\nIn this section, we will present the fully general relativistic equations \nfor $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ including the effects of the \ngeneral pressure terms.\nEquations (\\ref{eq1}-\\ref{eq7}) are the basic set of \nperturbation equations in a gauge-ready form.\n\n{}From Eqs. (\\ref{eq3},\\ref{eq6},\\ref{eq7}),\nEqs. (\\ref{eq4},\\ref{eq7}), Eqs. (\\ref{eq2},\\ref{eq3}), and\nEqs. (\\ref{eq1},\\ref{eq3},\\ref{eq4}), we have, respectively:\n\\begin{eqnarray}\n   & & \\dot \\delta_v - 3 H {\\rm w} \\delta_v\n       = - \\left( 1 + {\\rm w} \\right) {k \\over a} {k^2 - 3 K \\over k^2} v_\\chi\n       - 2 H {k^2 - 3K \\over a^2} {\\sigma \\over \\mu},\n   \\label{P-1} \\\\\n   & & \\dot v_\\chi + H v_\\chi = - {k \\over a} \\varphi_\\chi\n       + {k \\over a \\left( 1 + {\\rm w} \\right)} \n       \\left[ c_s^2 \\delta_v + {e \\over \\mu}\n       - 8 \\pi G \\left( 1 + {\\rm w} \\right) \\sigma\n       - {2 \\over 3} {k^2 - 3 K \\over a^2} {\\sigma \\over \\mu} \\right],\n   \\label{P-2} \\\\\n   & & {k^2 - 3 K \\over a^2} \\varphi_\\chi = 4 \\pi G \\mu \\delta_v,\n   \\label{P-3} \\\\\n   & & \\dot \\varphi_\\chi + H \\varphi_\\chi\n       = - 4 \\pi G \\left( \\mu + p \\right) {a \\over k} v_\\chi \n       - 8 \\pi G H \\sigma.\n   \\label{P-4}\n\\end{eqnarray}\nConsidering either one of the correspondences in \nEqs. (\\ref{Correspondences-1},\\ref{Correspondences-2})\nwe immediately see that Eqs. (\\ref{P-1}-\\ref{P-4}) have the \ncorrect Newtonian limit expressed in Eqs. (\\ref{Newt-eqs},\\ref{GI-eqs}).\nEquations (\\ref{P-1}-\\ref{P-4}) were presented in \\cite{Bardeen-1980}.\n\nCombining Eqs. (\\ref{P-1}-\\ref{P-4}) we can derive closed\nform expressions for the $\\delta_v$ and $\\varphi_\\chi$\nwhich are the relativistic counterpart of Eqs. \n(\\ref{Newt-delta-eq},\\ref{Newt-Phi-eq}).\nWe have:\n\\begin{eqnarray}\n   & & \\ddot \\delta_v + \\left( 2 + 3 c_s^2 - 6 {\\rm w} \\right) H \\dot \\delta_v\n       + \\Bigg[ c_s^2 {k^2 \\over a^2} \n       - 4 \\pi G \\mu \\left( 1 - 6 c_s^2 + 8 {\\rm w} - 3 {\\rm w}^2 \\right)\n       + 12 \\left( {\\rm w} - c_s^2 \\right) {K \\over a^2}\n       + \\left( 3 c_s^2 - 5 {\\rm w} \\right) \\Lambda \\Bigg] \\delta_v\n   \\nonumber \\\\\n   & & \\qquad \\qquad\n       = {1 + {\\rm w} \\over a^2 H} \\left[ {H^2 \\over a (\\mu + p)} \n       \\left( {a^3 \\mu \\over H} \\delta_v \\right)^\\cdot \\right]^\\cdot\n       + c_s^2 {k^2 \\over a^2} \\delta_v\n   \\nonumber \\\\\n   & & \\qquad \\qquad\n       = - {k^2 - 3 K \\over a^2} { 1 \\over \\mu }\n       \\left\\{ e + 2 H \\dot \\sigma\n       + 2 \\left[ - {1 \\over 3} {k^2 \\over a^2} + 2 \\dot H\n       + 3 \\left( 1 + c_s^2 \\right) H^2 \\right] \\sigma \\right\\},\n   \\label{delta-eq}\\\\\n   & & \\ddot \\varphi_\\chi + \\left( 4 + 3 c_s^2 \\right) H \\dot \\varphi_\\chi\n       + \\left[ c_s^2 {k^2 \\over a^2} \n       + 8 \\pi G \\mu \\left( c_s^2 - {\\rm w} \\right)\n       - 2 \\left( 1 + 3 c_s^2 \\right) {K \\over a^2}\n       + \\left( 1 + c_s^2 \\right) \\Lambda \\right] \\varphi_\\chi\n   \\nonumber \\\\\n   & & \\qquad \\qquad\n       = {\\mu + p \\over H} \\left[ {H^2 \\over a (\\mu + p)} \n       \\left( {a \\over H} \\varphi_\\chi \\right)^\\cdot \\right]^\\cdot\n       + c_s^2 {k^2 \\over a^2} \\varphi_\\chi\n       = {\\rm stresses}.\n   \\label{varphi-eq}\n\\end{eqnarray}\nThese two equations are related by Eq. (\\ref{P-3}) which resembles the \nPoisson equation.\nNotice the remarkably compact forms presented in the second steps of\nEqs. (\\ref{delta-eq},\\ref{varphi-eq}).\nIt may be an interesting exercise to show that the above equations\nare indeed valid for general $K$ and $\\Lambda$, and\nfor the general equation of state $p = p(\\mu)$;\nuse $\\dot {\\rm w} = - 3 H (1 + {\\rm w}) ( c_s^2 - {\\rm w} )$.\nEquation (\\ref{delta-eq}) became widely known through Bardeen's seminal paper \nin \\cite{Bardeen-1980}.\nIn a less general context but originally \nEqs. (\\ref{delta-eq},\\ref{varphi-eq}) were derived by\nNariai \\cite{Nariai} and Harrison \\cite{Harrison}, respectively;\nhowever, the compact expressions are new results in this paper.\n\nIf we ignore the entropic and anisotropic pressures ($e = 0 = \\sigma$) \non scales larger than Jeans scale\nEq. (\\ref{varphi-eq}) immediately leads to a general integral form \nsolution as\n\\begin{eqnarray}\n   \\varphi_\\chi ({\\bf k}, t)\n   &=& 4 \\pi G C ({\\bf k}) {H \\over a} \\int_0^t {a (\\mu + p) \\over H^2} dt\n       + {H \\over a} d ({\\bf k})\n   \\nonumber \\\\\n   &=& C ({\\bf k}) \\left[ 1 - {H \\over a} \\int_0^t\n       a \\left( 1 - {K \\over \\dot a^2} \\right) dt \\right]\n       + {H \\over a} d ({\\bf k}).\n   \\label{varphi_chi-sol}\n\\end{eqnarray}\nThis solution was first derived in Eq. (108) of \n\\cite{HV}\\footnote{We correct a typographical \n         error in Eq. (129) of \\cite{HV}: $c_0\/a$ should be replaced by $c_0$.\n         },\nsee also Eq. (55) in \\cite{PRW}.\nSolutions for $\\delta_v$ and $v_\\chi$ follow from Eqs.\n(\\ref{P-3},\\ref{P-4}), respectively, as\n\\begin{eqnarray}\n   & & \\delta_v ({\\bf k}, t) = {k^2 - 3 K \\over 4 \\pi G \\mu a^2} \n       \\varphi_\\chi ({\\bf k}, t), \n   \\label{delta_v-sol} \\\\\n   & & v_\\chi ({\\bf k}, t)= - {k \\over 4 \\pi G ( \\mu + p) a^2}\n       \\left\\{ C ({\\bf k}) \\left[ {K \\over \\dot a} \n       - \\dot H \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt \\right]\n       + \\dot H d ({\\bf k}) \\right\\}.\n   \\label{v_chi-sol}\n\\end{eqnarray}\nWe stress that these large-scale asymptotic solutions are\n{\\it valid for the general $K$, $\\Lambda$, and $p = p(\\mu)$};\n$C({\\bf k})$ and $d({\\bf k})$ are integration constants considering \nthe general evolution of $p = p(\\mu)$.\nWe also emphasize that {\\it the large-scale criterion is the sound-horizon},\nand thus in the matter dominated era the solutions are valid even far \ninside the (visual) horizon as long as the scales are larger than Jeans scale.\nIn the pressureless limit, considering Eq. (\\ref{Correspondences-2}),\nEqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol})\ncorrectly reproduce Eqs. (\\ref{delta-N}-\\ref{Phi-N}).\n\nUsing the set of gauge-ready form equations in\nEqs. (\\ref{eq1}-\\ref{eq7}) the complete set of corresponding general\nsolutions for all variables in all different gauges can be easily derived;\nfor such sets of solutions with less general assumptions see \\cite{MDE,Ideal}.\n{}For $K = 0 = \\Lambda$ and ${\\rm w} = {\\rm constant}$ we have\n$a \\propto t^{2\/[3(1+{\\rm w})]}$ and Eqs. \n(\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) become:\n\\begin{eqnarray}\n   \\varphi_\\chi ({\\bf k}, t) \n   &=& {3 (1 + {\\rm w}) \\over 5 + 3 {\\rm w}} C ({\\bf k})\n       + {2 \\over 3 (1 + {\\rm w})} {1 \\over at} d ({\\bf k})\n   \\nonumber \\\\\n   &\\propto& {\\rm constant}, \n       \\quad t^{- {5 + 3 {\\rm w} \\over 3 ( 1 + {\\rm w} )} }\n       \\quad \\propto \\quad {\\rm constant}, \n       \\quad a^{- {5 + 3 {\\rm w} \\over 2} },\n   \\nonumber \\\\\n   \\delta_v ({\\bf k}, t) \n   &\\propto& t^{2(1 + 3 {\\rm w}) \\over 3 (1 + {\\rm w})}, \n       \\quad t^{- {1 - {\\rm w} \\over 1 + {\\rm w} } }\n       \\quad \\propto \\quad a^{1 + 3 {\\rm w}}, \n       \\quad a^{- {3 \\over 2} (1 - {\\rm w}) },\n   \\nonumber \\\\\n   v_\\chi ({\\bf k}, t)\n   &\\propto& t^{1 + 3 {\\rm w} \\over 3 (1 + {\\rm w})}, \n       \\quad t^{- {4 \\over 3(1 + {\\rm w}) } }\n       \\quad \\propto \\quad a^{1 + 3 {\\rm w} \\over 2}, \n       \\quad a^{- 2}.\n   \\label{Sol-w=const}\n\\end{eqnarray}\nThese solutions were presented in \\cite{Bardeen-1980} and\nin less general contexts but originally in \\cite{Harrison,Nariai}.\n{}For ${\\rm w} =0$ we recover the well known Newtonian behaviors.\n      \n\n\\subsection{Conserved Quantities}\n                                        \\label{sec:Conservation}\n\nIn an ideal fluid, the curvature fluctuations in several gauge conditions \nare known to be conserved in the {\\it superhorizon scale} independently\nof the changes in the background equation of state.\n{}From Eqs. (41,73) in \\cite{Ideal} we find [for $K = 0 = \\Lambda$,\nbut for general $p(\\mu)$]:\n\\begin{eqnarray}\n   \\varphi_v = \\varphi_\\delta = \\varphi_\\kappa = \\varphi_\\alpha = C ({\\bf k}),\n   \\label{varphi-conserv}\n\\end{eqnarray}\nand the dominating decaying solutions vanish (or, are cancelled).\nThe combinations follow from Eq. (\\ref{GT}) as:\n\\begin{eqnarray}\n   \\varphi_v = \\varphi - {aH \\over k} v, \\quad\n       \\varphi_\\delta \\equiv \\varphi +{\\delta \\over 3 (1 + {\\rm w})} \n       \\equiv \\zeta, \\quad\n       \\varphi_\\kappa \\equiv \\varphi + {H \\over 3 \\dot H - k^2\/a^2 } \\kappa, \n       \\quad\n       \\varphi_\\alpha \\equiv \\varphi - H \\int^t \\alpha dt.\n   \\label{varphi-GI}\n\\end{eqnarray}\nThe $\\varphi_\\alpha$ combination which is $\\varphi$ in the synchronous gauge\nis not gauge-invariant; the lower bound of integration gives the behavior\nof the gauge mode; in Eq. (\\ref{varphi-conserv}) we ignored the gauge mode.\nThe large-scale conserved quantity, $\\zeta$, proposed in\n\\cite{Bardeen-1988,BST} is $\\varphi_\\delta$.\nIn \\cite{Ideal} the above conservation properties are shown assuming \n$K = 0 = \\Lambda$; in such a case the integral form solutions for a complete\nset of variables are presented in Table 8 in \\cite{Ideal}.\nIn Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}) we have the large-scale \nintegral form solutions in the case of general $K$ and $\\Lambda$.\nThus, now we can see the behavior of these variables considering the \ngeneral $K$ and $\\Lambda$.\n\n{}Evaluating $\\varphi_v$ in Eq. (\\ref{varphi-GI}) in the zero-shear gauge,\nthus $\\varphi_v = \\varphi_\\chi - (aH \/ k) v_\\chi$, and using\nthe solutions in Eqs. (\\ref{varphi_chi-sol},\\ref{v_chi-sol}) we \ncan derive\n\\begin{eqnarray}\n   \\varphi_v ({\\bf k}, t) \n   &=& C ({\\bf k}) \\left\\{ 1 + {K \\over a^2}\n       {1 \\over 4 \\pi G (\\mu + p)} \\left[ 1 \n       - {H \\over a} \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt \n       \\right] \\right\\}\n       + {K \\over a^2} { H\/a \\over 4 \\pi G ( \\mu + p)} d ({\\bf k})\n   \\nonumber \\\\\n   &=& C ({\\bf k}) \\left[ 1 + {K \\over a^2}\n       {H\/a \\over \\mu + p} \\int_0^t {a (\\mu + p) \\over H^2} dt \\right]\n       + {K \\over a^2} { H\/a \\over 4 \\pi G ( \\mu + p)} d ({\\bf k}).\n   \\label{varphi_v-sol}\n\\end{eqnarray}\nThus, for $K = 0$ (but for general $\\Lambda$) we have\n\\begin{eqnarray}\n   \\varphi_v ({\\bf k}, t) = C ({\\bf k}),\n   \\label{varphi_v-sol2}\n\\end{eqnarray}\nwith the vanishing decaying solution. \nThe disappearance of the decaying \nsolution in Eq. (\\ref{varphi_v-sol2}) implies that the dominating decaying \nsolutions in Eqs. (\\ref{varphi_chi-sol},\\ref{v_chi-sol}) cancel out \nfor $K = 0$.\nIn fact, for $K = 0$, from Eqs. \n(\\ref{eq1},\\ref{eq2},\\ref{eq3},\\ref{eq6},\\ref{eq7}) we can derive\n\\begin{eqnarray}\n    & & {c_s^2 H^2 \\over a^3 ( \\mu + p)} \\left[ {a^3 ( \\mu + p) \\over c_s^2 H^2}\n        \\dot \\varphi_v \\right]^\\cdot\n        + c_s^2 {k^2 \\over a^2} \\varphi_v = {\\rm stresses}.\n    \\label{varphi_v-fluid-eq}\n\\end{eqnarray}\n{}For a pressureless case ($c_s^2 = 0$), instead of\nEq. (\\ref{varphi_v-fluid-eq}) we have $\\dot \\varphi_v = 0$;\nsee after Eq. (\\ref{CG-varphi-eq}).\nIn the large-scale limit, and ignoring the stresses, we have an integral\nform solution\n\\begin{eqnarray}\n   & & \\varphi_v\n       = C ({\\bf x}) - \\tilde D ({\\bf x})\n       \\int^t_0 {c_s^2 H^2 \\over a^3 ( \\mu + p ) } dt.\n   \\label{varphi_v-sol3}\n\\end{eqnarray}\nCompared with the solution in Eq. (\\ref{varphi_v-sol2}), the $\\tilde D$ \nterm in Eq. (\\ref{varphi_v-sol3}) is $c_s^2(k\/aH)^2$ order higher\nthan $d$ terms in the other gauge.\nTherefore, for $K = 0$, $\\varphi_v$ is conserved for the generally varying\nbackground equation of state, i.e., for general $p = p(\\mu)$.\nSince the solutions in \nEqs. (\\ref{varphi_chi-sol},\\ref{delta_v-sol},\\ref{v_chi-sol}) are valid \nin super-sound-horizon scale, the conservation property in \nEq. (\\ref{varphi_v-sol2}) is valid in all scales in the matter \ndominated era.\n\n{}For $\\varphi_\\delta$, by evaluating it in the comoving gauge we have\n\\begin{eqnarray}\n   \\varphi_\\delta \\equiv \\varphi_v + {\\delta_v \\over 3 (1 + {\\rm w})}.\n\\end{eqnarray}\nUsing Eqs. (\\ref{varphi_v-sol},\\ref{delta_v-sol}) we have\n\\begin{eqnarray}\n   \\varphi_\\delta ({\\bf k}, t)\n   &=& C ({\\bf k}) \\left\\{ 1 + {k^2 \\over a^2}\n       {1 \\over 12 \\pi G (\\mu + p)} \\left[ 1\n       - {H \\over a} \\int_0^t a \\left( 1 - {K \\over \\dot a^2} \\right) dt\n       \\right] \\right\\}\n       + {k^2 \\over a^2} { H\/a \\over 12 \\pi G ( \\mu + p)} d ({\\bf k}).\n   \\label{varphi_delta-sol}\n\\end{eqnarray}\nSince the higher order term ignored in Eq. (\\ref{varphi_v-sol}) \nis $c_s^2 (k\/aH)^2$ order higher, in the medium with $c_s \\sim 1 (\\equiv c)$,\nthe terms involving $(k\/a)^2\/(\\mu + p)$ are not necessarily valid.\nThus, to the leading order in the superhorizon scale we have\n\\begin{eqnarray}\n   \\varphi_\\delta ({\\bf k}, t) = C ({\\bf k}).\n\\end{eqnarray}\nThus, in the superhorizon scale $\\varphi_\\delta$ is conserved \napparently  considering $K$ and $\\Lambda$.\n[In the case of nonvanishing $K$ we may need to be \n       careful in defining the superhorizon scale. \n       Here, as the superhorizon condition we simply {\\it took} $k^2$ term \n       to be negligible.\n       In a hyperbolic (negatively curved) space with $K = -1$, \n       since $R^{(3)} = {6 K \/ a^2}$, the distance \n       $a\/\\sqrt{|K|}$ introduces a curvature scale: on smaller scales the \n       space is near flat and on larger scales the curvature term dominates.\n       In the unit of the curvature scale $k^2$ less than one ($\\sim |K|$)\n       corresponds to the super-curvature scale, and $k^2$ larger than \n       one corresponds to the sub-curvature scale.  \n       Thus, when we ignored the $k^2$ term compared with $|K|$ in \n       a negatively curved space, we are considering the super-curvature scale.\n       In other words, for the hyperbolic backgrounds $k^2$ takes continuous \n       value with $k^2 \\ge 0$ where $k^2 \\ge 1$ corresponds to subcurvature \n       mode whereas $0\\le k^2 <1 $ corresponds to supercurvature mode, and \n       our large-scale limit took $k^2 \\rightarrow 0$.\n       A useful study in the hyperbolic situation can be found \n       in \\cite{Lyth-etal}.]\nHowever, in a medium with the negligible sound speed (like the matter dominated \nera), Eq. (\\ref{varphi_delta-sol}) is valid considering the $k^2$ terms.\nThus, as the scale enters the horizon in the matter dominated era \n$\\varphi_\\delta$ wildly changes its behavior and is {\\it no longer conserved \ninside the horizon};\ninside the horizon $\\varphi_\\delta$ is dominated by the $\\delta$ part\nwhich shows Newtonian behavior of $\\delta$ in most of the gauge conditions, \nsee Table 1.\n\n{}For $\\varphi_\\kappa$, by evaluating it in the uniform-density gauge\nand using Eq. (\\ref{eq2}), which gives \n$\\kappa_\\delta = (k^2 - 3K)\/(a^2 H) \\varphi_\\delta$, we have\n\\begin{eqnarray}\n   \\varphi_\\kappa = \\varphi_\\delta\n       \\left[ 1 + {k^2 - 3K \\over 12 \\pi G (\\mu + p) a^2} \\right]^{-1}.\n   \\label{varphi_kappa-sol}\n\\end{eqnarray}\nThus, in the superhorizon scale and for $K = 0$, $\\varphi_\\kappa$ is conserved.\nAs in $\\varphi_\\delta$, in a medium with the negligible sound speed, \nthe solution in Eq. (\\ref{varphi_kappa-sol}) with Eq. (\\ref{varphi_delta-sol}) \nis valid considering the $k^2$ terms.\nThus, as the scale enters the horizon in the matter dominated era \n$\\varphi_\\kappa$\nalso changes its behavior and is {\\it no longer conserved inside the horizon}.\n\nNow we summarize:\nIn the superhorizon scale $\\varphi_v$ and $\\varphi_\\kappa$ are conserved for \n$K = 0$, while $\\varphi_\\delta$ is conserved considering the general $K$.\nIn the matter dominated era, for $K = 0$, $\\varphi_v$ is still conserved in the\nsuper-sound-horizon scale which virtually covers all scales,\nwhereas $\\varphi_\\delta$ and $\\varphi_\\kappa$ change their behaviors\nnear the horizon crossing epoch and are no longer conserved inside the horizon.\nIn this regard, we may say, $\\varphi_v$ is the {\\it best conserved quantity}.\nCuriously, although $\\varphi_\\chi$ is the variable most closely resembling\nthe Newtonian gravitational potential, it is not conserved for changing \nequation of state, see Eq. (\\ref{varphi_chi-sol});\nhowever, it is conserved independently of the horizon crossing in the\nmatter dominated era for $K = 0 = \\Lambda$, see Eq. (\\ref{Sol-w=const}).\n\nSimilar conservation properties of the curvature variable in certain gauge\nconditions remain true for models based on a minimally coupled scalar \nfield and even on classes of generalized gravity theories.\nIn the generalized gravity the uniform-field gauge is more suitable\nfor handling the conservation property, and the uniform-field gauge\ncoincides with the comoving gauge in the minimally coupled scalar field.\n{}Thorough studies of the minimally coupled scalar field\nand the generalized gravity, assuming $K = 0 = \\Lambda$, are made in \n\\cite{MSF-GGT,MSF-GGT-2} and summarized in \\cite{GRG-MSF-GGT}.\n\n\n\\subsection{Multi-component situation}\n                               \\label{sec:Multi}\n\nWe consider the energy-momentum tensor composed of multiple components as\n\\begin{eqnarray}\n   T_{ab} = \\sum_{(l)} T_{(l)ab}, \\quad\n       T^{\\;\\;\\;\\; b}_{(i)a;b} \\equiv Q_{(i)a}, \\quad\n       \\sum_{(l)} Q_{(l)a} = 0,\n   \\label{Tab-multi}\n\\end{eqnarray}\nwhere $i,l (= 1,2, \\dots, n)$ indicates $n$ fluid components, and\n$Q_{(i)a}$ considers possible mutual interactions among fluids.\nSince we are considering the scalar-type perturbation we decompose \nthe $Q_{(i)a}$ in the following way:\n\\begin{eqnarray}\n   Q_{(i)0} \\equiv - a (1 + \\alpha) Q_{(i)}, \\quad\n       Q_{(i)} \\equiv \\bar Q_{(i)} + \\delta Q_{(i)}, \\quad\n       Q_{(i)\\alpha} \\equiv J_{(i),\\alpha}.\n\\end{eqnarray}\nThe scalar-type fluid quantities in Eq. (\\ref{Tab}) can be considered \nas the collective ones which are related to the individual component as:\n\\begin{eqnarray}\n   & & \\bar \\mu = \\sum_{(l)} \\bar \\mu_{(l)}, \\quad\n       \\bar p = \\sum_{(l)} \\bar p_{(l)}; \\quad\n       \\delta \\mu = \\sum_{(l)} \\delta \\mu_{(l)}, \\quad\n       \\delta p = \\sum_{(l)} \\delta p_{(l)}, \n   \\nonumber \\\\\n   & & (\\mu + p) v = \\sum_{(l)} (\\mu_{(l)} + p_{(l)} ) v_{(l)}, \\quad\n       \\sigma = \\sum_{(l)} \\sigma_{(l)}.\n   \\label{fluid-multi}\n\\end{eqnarray}\n\n{}For the background Eq. (\\ref{BG-eqs}) remains valid for the collective fluid\nquantities.\nAn additional equation follows from the individual\nenergy-momentum conservation in Eq. (\\ref{Tab-multi}) as\n\\begin{eqnarray}\n   \\dot \\mu_{(i)} + 3 H \\left( \\mu_{(i)} + p_{(i)} \\right) = Q_{(i)}.\n   \\label{BG-multi}\n\\end{eqnarray}\n{}For the perturbations Eqs. (\\ref{eq1}-\\ref{eq7}) remain valid for the \ncollective fluid quantities. \nThe additional equations we need follow from the individual\nenergy-momentum conservation in Eq. (\\ref{Tab-multi}).\n{}From $T^{\\;\\;\\;\\; b}_{(i)0;b} = Q_{(i)0}$ and using Eq. (\\ref{eq1})\nwe have the energy conservation of the fluid components \n\\begin{eqnarray}\n   \\delta \\dot \\mu_{(i)} + 3 H \\left( \\delta \\mu_{(i)}\n       + \\delta p_{(i)} \\right)\n       = - {k \\over a} \\left( \\mu_{(i)} + p_{(i)} \\right) v_{(i)}\n       + \\dot \\mu_{(i)} \\alpha\n       + \\left( \\mu_{(i)} + p_{(i)} \\right) \\kappa + \\delta Q_{(i)}.\n   \\label{eq8}\n\\end{eqnarray}\n{}From $T^{\\;\\;\\;\\; b}_{(i)\\alpha;b} = Q_{(i)\\alpha}$ we have\nthe momentum conservation of the fluid components\n\\begin{eqnarray}\n   \\dot v_{(i)} + \\left[ H \\left( 1 - 3 c_{(i)}^2 \\right)\n       + { 1 + c_{(i)}^2 \\over \\mu_{(i)} + p_{(i)} } Q_{(i)} \\right] v_{(i)}\n       = {k \\over a} \\left[ \\alpha\n       + {1 \\over \\mu_{(i)} + p_{(i)}} \\left( \\delta p_{(i)}\n       - {2 \\over 3} {k^2 - 3 K \\over a^2} \\sigma_{(i)}\n       - J_{(i)} \\right) \\right].\n   \\label{eq9}\n\\end{eqnarray}\nBy adding Eqs. (\\ref{eq8},\\ref{eq9}) over the components we have\nEqs. (\\ref{eq6},\\ref{eq7}).\nIn the multi-component situation \nEqs. (\\ref{eq1}-\\ref{eq7},\\ref{eq8},\\ref{eq9}) are the complete set \nexpressed in a gauge-ready form.\nIf we decompose the pressure as\n$\\delta p_{(i)} = c_{(i)}^2 \\delta \\mu_{(i)} + e_{(i)}$ with\n$c_{(i)}^2 \\equiv \\dot p_{(i)}\/\\dot \\mu_{(i)}$, comparing with\nEq. (\\ref{p-decomposition}) we have\n\\begin{eqnarray}\n   e = \\delta p - c_s^2 \\delta \\mu\n       = \\sum_{(l)} \\left[ \\left( c_{(l)}^2 - c_s^2 \\right) \\delta \\mu_{(l)}\n       + e_{(l)} \\right].\n   \\label{e}\n\\end{eqnarray}\n\nUnder the gauge transformation, from Eq. (20) in \\cite{PRW}, we have:\n\\begin{eqnarray}\n   & & \\delta \\tilde \\mu_{(i)} = \\delta \\mu_{(i)} - \\dot \\mu_{(i)} \\xi^t, \\quad\n       \\delta \\tilde p_{(i)} = \\delta p_{(i)} - \\dot p_{(i)} \\xi^t, \\quad\n       \\tilde v_{(i)} = v_{(i)} - {k \\over a} \\xi^t, \n   \\nonumber \\\\\n   & & \\tilde J_{(i)} = J_{(i)} + Q_{(i)} \\xi^t, \\quad\n       \\delta \\tilde Q_{(i)} = \\delta Q_{(i)} - \\dot Q_{(i)} \\xi^t.\n   \\label{GT-2}\n\\end{eqnarray}\nThus, we have the following additional gauge conditions:\n\\begin{eqnarray}\n   \\delta \\mu_{(i)} \\equiv 0, \\quad\n       \\delta p_{(i)} \\equiv 0, \\quad\n       v_{(i)}\/k \\equiv 0, \\quad\n       {\\rm etc}.\n   \\label{Gauges-2}\n\\end{eqnarray}\nAny one of these gauge conditions also fixes the temporal gauge condition\ncompletely.\nStudies of the multi-component situations\ncan be found in \\cite{KS,PRW,KS-2}.\n\n\n\\subsection{Rotation and Gravitational Wave}\n                                \\label{sec:rot-GW}\n\n{}For the vector-type perturbation, using Eqs. (\\ref{metric},\\ref{Tab}),\nwe have:\n\\begin{eqnarray}\n   & & 8 \\pi G T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha}\n       = - {\\Delta + 2 K \\over 2 a^2} \\left( B_\\alpha + a \\dot C_\\alpha \\right),\n   \\label{rot-1} \\\\\n   & & 8 \\pi G \\delta T^{(v)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n       = {1 \\over 2 a^3} \\left\\{ a^2 \\left[ B^\\alpha_{\\;\\;|\\beta}\n       + B_\\beta^{\\;\\;|\\alpha} + a \\left( C^\\alpha_{\\;\\;|\\beta}\n       + C_\\beta^{\\;\\;|\\alpha} \\right)^\\cdot \\right] \\right\\}^\\cdot,\n   \\label{rot-2} \\\\\n   & & {1 \\over a^3} \\left( a^4 T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha} \\right)^\\cdot\n       = - \\delta T^{(v)\\beta}_{\\;\\;\\;\\;\\;\\alpha|\\beta},\n   \\label{rot-3}\n\\end{eqnarray}\nwhere Eq. (\\ref{rot-3}) follows from $T^b_{\\alpha ; b} = 0$, and \napparently Eqs. (\\ref{rot-1},\\ref{rot-2}) follow from the Einstein equations.\nWe can show that Eq. (\\ref{rot-3}) follows from Eqs. (\\ref{rot-1},\\ref{rot-2}).\nWe introduce a gauge-invariant velocity related variable \n$V^{(\\omega)}_\\alpha ({\\bf x}, t)$ as \n$T^{(v)0}_{\\;\\;\\;\\;\\;\\alpha} \\equiv (\\mu + p) V^{(\\omega)}_\\alpha$.\nWe can show $V^{(\\omega)}_\\alpha \\propto a \\omega$ where $\\omega$ \nis the amplitude of vorticity vector, see Sec. 5 of \\cite{PRW}.\nThus, ignoring the anisotropic stress in the RHS of Eq. (\\ref{rot-3})\nwe have the conservation of angular momentum as\n\\begin{eqnarray}\n   {\\rm Angular \\; Momentum} \\sim a^3 \\left( \\mu + p \\right)\n       \\times a \\times V^{(\\omega)}_\\alpha ({\\bf x}, t)\n       = {\\rm constant \\; in \\; time},\n\\end{eqnarray}\nwhich is valid for general $K$, $\\Lambda$, and $p(\\mu)$ in general scale.\n\n{}For the tensor-type perturbation, using Eqs. (\\ref{metric},\\ref{Tab})\nin the Einstein equations, we have\n\\begin{eqnarray}\n   8 \\pi G \\delta T^{(t)\\alpha}_{\\;\\;\\;\\;\\;\\beta}\n       = \\ddot C^\\alpha_\\beta + 3 H \\dot C^\\alpha_\\beta\n       - {\\Delta - 2 K \\over a^2} C^\\alpha_\\beta.\n   \\label{GW-eq}\n\\end{eqnarray}\nIn the large-scale limit, assuming $K = 0$ and ignoring the anisotropic\npressure Eq. (\\ref{GW-eq}) has a general integral form solution\n\\begin{eqnarray}\n   C^\\alpha_\\beta ({\\bf x}, t)\n       = C^{\\;\\alpha}_{1\\beta} ({\\bf x}) - D^\\alpha_\\beta ({\\bf x})\n       \\int^t {1 \\over a^3} dt,\n\\end{eqnarray}\nwhere $C^{\\;\\alpha}_{1\\beta}$ and $D^\\alpha_\\beta$ are integration constants\nfor relatively growing and decaying solutions, respectively.\nThus, ignoring the transient term the amplitude of the gravitational\nwave is conserved in the super-horizon scale considering general\nevolution of the background equation of state.\n\nThe conservation of angular momentum and the equation for the\ngravitational wave were first derived in \\cite{Lifshitz}.\nEvolutions in the multi-component situation were considered in\nSec. 5 of \\cite{PRW}.\n\n\n\\section{Discussions}\n                                \\label{sec:Discussion}\n\nWe would like to make comments on related works in several {\\it textbooks}:\nEq. (15.10.57) in \\cite{Weinberg}, Eq. (10.118) in \\cite{Peebles-2},\nEq. (11.5.2) in \\cite{Coles}, Eq. (8.52) in \\cite{Moss},\nand problem 6.10 in \\cite{Padmanabhan} are in error.\nAll these errors are essentially about the same point involved with\na fallible algebraic mistake in the synchronous gauge.\nThe correction in the case of \\cite{Weinberg} was made in \\cite{GRG}:\nin a medium with a nonvanishing pressure the equation for the density \nfluctuation in the synchronous gauge becomes third order because of \nthe presence of a gauge mode in addition to the physical growing \nand decaying solutions which is true even in the large-scale limit.\nThe truncated second order equation in \\cite{Weinberg,Moss} \npicks up a gauge mode instead of the physical decaying solution in the \nsynchronous gauge.\nThe errors in \\cite{Peebles-2,Padmanabhan} are based on imposing \nthe synchronous gauge and the comoving gauge simultaneously, and thus \nhappen to end up with the same truncated second order equation \nas in \\cite{Weinberg}.\nIn a medium with nonvanishing pressure one cannot impose the two gauge \nconditions simultaneously (even in the large-scale limit).  \nIn Sec. 11.5 of \\cite{Coles} the authors proposed a simple modification\nof the Newtonian theory which again happens to end up with the same incorrect \nequation as in the other books.\nIn the Appendix we elaborate our point.\nFor comments on other related errors in the literature, see Sec. 3.10 in \n\\cite{Ideal} and \\cite{GRG}.\n\nWe would like to remark that these errors found in the synchronous\ngauge are not due to any esoteric aspect of the gauge choice in the \nrelativistic perturbation analyses.\nAlthough not fundamental, the number of errors found in the literature\nconcerning the synchronous gauge seem to indicate the importance of using \nthe proper gauge condition in handling the problems. \n{}For our own choice of the preferred gauge-invariant variables \nsuitable for handling the hydrodynamic perturbation and consequent analyses, \nsee Sec. \\ref{sec:Hydrodynamics}; the reasons for such choices are made in \nSec. \\ref{sec:Six-gauges}.\nWe wish to recall, however, that although one may need to do more algebra \nin tracing the remnant gauge mode, even the synchronous gauge condition\nis adequate for handling many problems as was carefully done in the \noriginal study by Lifshitz in 1946 \\cite{Lifshitz}.\n\nIn this paper we have tried to identify the variables in the relativistic\nperturbation analysis which reproduce the correct Newtonian behavior\nin the pressureless limit.\nIn the first part we have shown that\n$\\delta$ in the comoving gauge ($\\delta_v$) and $v$ and $\\varphi$ \nin the zero-shear gauge ($v_\\chi$ and $\\varphi_\\chi$) \nshow the same behavior as the corresponding Newtonian \nvariables {\\it in general scales}.\nIn fact, these results have already been presented in \\cite{MDE}. \nAlso, various general and asymptotic solutions for every variable\nin the pressureless medium are presented in the Tables of \\cite{MDE}.\nCompared with \\cite{MDE}, in Sec. \\ref{sec:Six-gauges} we tried to reinforce \nthe correspondence by explicitly showing the second order differential \nequations in several gauge conditions.\nWe have also added some additional insights gained after publishing \\cite{MDE}.\nThe second part contains some original results.\nUsing the gauge-invariant variables $\\delta_v$, $v_\\chi$ and $\\varphi_\\chi$ \nwe write the relativistic hydrodynamic cosmological perturbation equations in \nEqs. (\\ref{P-1}-\\ref{P-4}); actually, these equations \nare also known in \\cite{Bardeen-1980}.\nThe new results in this paper are the compact way of deriving the\ngeneral large-scale solutions in Eqs. (\\ref{varphi_chi-sol}-\\ref{v_chi-sol}), \nand the clarification of the general large-scale conservation property of\n$\\varphi_v$ in Eq. (\\ref{varphi_v-sol}).\n\nThe underlying mathematical or physical reasons for \nthe variables $\\delta_v$, $\\varphi_\\chi$, $v_\\chi$, and $\\varphi_v$ \nhaving the distinguished behaviors, compared with many other available\ngauge-invariant combinations, may still deserve further investigation.\n\n\n\\subsection*{Acknowledgments}\n\nWe thank Profs.  P. J. E. Peebles and A. M\\'esz\\'aros for useful discussions\nand comments, and Prof. R. Brandenberger for interesting correspondences.\nJH was supported by the KOSEF, Grant No. 95-0702-04-01-3 and\nthrough the SRC program of SNU-CTP.\nHN was supported by the DFG (Germany) and the KOSEF (Korea).\n\n\n\\section*{Appendix: Common errors in the synchronous gauge}\n\nIn the following we correct a minor confusion in the literature\nconcerning perturbation analyses in the synchronous gauge.\nThe argument is based on \\cite{GRG}.\n{}For simplicity, we consider a situation with $K = 0 = \\Lambda$,\n$e = 0 = \\sigma$ and ${\\rm w} = {\\rm constant}$ (thus $c_s^2 = {\\rm w}$).\n\nThe equation for the density perturbation in the {\\it comoving gauge}\nis given in Eq. (\\ref{delta-eq}).\nIn our case we have\n$$\n   \\ddot \\delta_v + ( 2- 3 {\\rm w}) H \\dot \\delta_v\n      + \\left[ c_s^2 {k^2 \\over a^2}\n      - 4 \\pi G \\mu (1 - {\\rm w}) ( 1 + 3 {\\rm w} ) \\right] \\delta_v = 0.\n   \\eqno{(A1)}\n$$\nThe solution in the super-sound-horizon scale is presented in\nEq. (\\ref{Sol-w=const}).\n\nIn the {\\it synchronous gauge}, from Eqs. (\\ref{eq5}-\\ref{eq7}) we have\n(the subindex $\\alpha$ indicates the synchronous gauge variable)\n$$\n   \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha\n       + \\left[ c_s^2 {k^2 \\over a^2}\n       - 4 \\pi G \\mu ( 1+ {\\rm w} ) ( 1 + 3 {\\rm w} ) \\right] \\delta_\\alpha\n       + 3 {\\rm w} ( 1 + {\\rm w}) {k \\over a} H v_\\alpha = 0,\n   \\eqno{(A2)}\n$$\n$$\n   \\dot v_\\alpha + ( 1- 3 {\\rm w}) H v_\\alpha\n       - {k \\over a} {{\\rm w} \\over 1 + {\\rm w}} \\delta_\\alpha = 0.\n   \\eqno{(A3)}\n$$\nNotice that for ${\\rm w} \\neq 0$ these two equations are generally coupled\neven in the large-scale limit.\n{}From these two equations we can derive a third order differential equation\nfor $\\delta_\\alpha$ as\n$$\n   \\delta^{\\cdot\\cdot\\cdot}_\\alpha\n       + {11 - 3 {\\rm w} \\over 2} H \\ddot \\delta_\\alpha\n       + \\left( {5 - 24 {\\rm w} - 9 {\\rm w}^2 \\over 2} H^2\n       + c_s^2 {k^2 \\over a^2} \\right) \\dot \\delta_\\alpha\n$$\n$$\n       + \\left[ {3 \\over 4} \\left( 1 + {\\rm w} \\right)\n       \\left( 1 + 3 {\\rm w} \\right) \\left( -1 + 9 {\\rm w} \\right) H^3\n       + {3 \\over 2} \\left( 1 + {\\rm w} \\right) H c_s^2 {k^2 \\over a^2}\n       \\right] \\delta_\\alpha = 0.\n   \\eqno{(A4)}\n$$\nIn the large-scale limit the solutions are\n$$\n   \\delta_\\alpha \\quad \\propto \\quad\n       t^{{ 2(1+ 3{\\rm w}) \\over 3(1 + {\\rm w})}}, \\quad\n       t^{{9{\\rm w} - 1 \\over 3( 1+ {\\rm w})}}, \\quad\n       t^{-1}.\n   \\eqno{(A5)}\n$$\nIn Appendix B of \\cite{GRG} we showed that\nthe third solution with $\\delta_\\alpha \\propto t^{-1}$\nis nothing but a gauge mode for a medium with ${\\rm w} \\neq 0$.\n[As mentioned before, the combination $\\delta_\\alpha$ is not gauge-invariant.\n {}From Eq. (\\ref{GT}) we have\n $\\delta_\\alpha \\equiv \\delta + 3 H ( 1 + {\\rm w}) \\int^t \\alpha dt$\n and the lower bound of the integration gives rise to the gauge mode \n which is proportional to $t^{-1}$.]\n{}For a pressureless case the physical decaying solution also behaves \nas $t^{-1}$ and the second solution in Eq. (A5) is invalid,\nsee Sec. 4 in \\cite{GRG}.\nIn Eq. (B5) of \\cite{GRG} we derived the relation between solutions\nin the two gauges explicitly; the growing solutions are the same in both gauges\nwhereas the decaying solutions differ by a factor\n$(k \/ aH)^2 \\propto t^{2(1 + 3 {\\rm w}) \\over 3 (1 + {\\rm w})}$.\n\nNow, we would like to point out that, in a medium with ${\\rm w} \\neq 0$,\none cannot ignore the last term in Eq. (A2)\neven in the large-scale limit.\nIf we {\\it incorrectly} ignore the last term in Eq. (A2) we recover\nthe {\\it wrong equation} in the textbooks mentioned in \nSec. \\ref{sec:Discussion} which is\n$$\n   \\ddot \\delta_\\alpha + 2 H \\dot \\delta_\\alpha\n       + \\left[ c_s^2 {k^2 \\over a^2}\n       - 4 \\pi G \\mu ( 1 + {\\rm w} ) ( 1 + 3 {\\rm w} ) \\right] \\delta_\\alpha\n       = 0.\n   \\eqno{(A6)}\n$$\nIn the large-scale we have solutions\n$$\n   \\delta_\\alpha \\quad \\propto \\quad\n       t^{{ 2(1+ 3{\\rm w}) \\over 3(1 + {\\rm w})}}, \\quad\n       t^{-1}.\n   \\eqno{(A7)}\n$$\nBy ignoring the last term in Eq. (A2), in the large-scale\nlimit we happen to recover the fictitious gauge mode under the price of losing\nthe physical decaying solution [the second solution in Eq. (A5)].\nThus, in the large-scale limit one cannot ignore the last term in Eq. (A2)\nin a medium with a general pressure; the reason is obvious if we see\nEq. (A4).\nAlso, one cannot impose both the synchronous gauge condition and the\ncomoving gauge condition simultaneously.\nIf we simultaneously impose such two conditions,\nthus setting $v \\equiv 0 \\equiv \\alpha$, from Eq. (\\ref{eq7}) we have\n$$\n   0 = {k \\over a \\left( 1 + {\\rm w} \\right) } \\left( c_s^2 \\delta\n       + {e \\over \\mu} - {2\\over 3} {k^2 - 3K \\over a^2} {\\sigma \\over \\mu}\n       \\right).\n   \\eqno{(A8)}\n$$\nThus, for $e = 0 = \\sigma$ and medium with the nonvanishing pressure\nwe have $\\delta = 0$ which is a meaningless system;\nthis argument remains valid even in the large-scale limit.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{ Introduction }\nIt has long been a problem that the $\\Delta I=3\/2$ \n$K^+\\to\\pi^+\\pi^0$ amplitude calculated in quenched lattice QCD is \nabout a factor two too large compared to the experimental value\\cite{BSReview}.\nIn this article we report a new study of this problem,\nincorporating various theoretical and technical advances \nin recent years for analysis.\nIn particular we discuss in detail how one-loop corrections of \nchiral perturbation theory (CHPT) recently calculated\\cite{ONECPTH} affect\nphysical predictions for the decay amplitude from lattice QCD simulations. \n\nOur simulation is carried out in quenched lattice QCD employing \nthe standard plaquette action for gluons at $\\beta=6.1$ and the Wilson action for quarks. \nTwo lattice sizes, $24^3\\times 64$ and $32^3\\times 64$, are employed.\nWe take up, down and strange quarks to be degenerate,\nand make measurements at four values of the common hopping parameter,\n$\\kappa = 0.1520$, $0.1530$, $0.1540$ and $0.1543$, which correspond to\n$m_\\pi\/m_\\rho = 0.797$, $0.734$, $0.586$ and $0.515$.\n\\section{ Extraction of decay amplitude }\nThe 4-quark operator most relevant for the \n$\\Delta I=3\/2$ $K\\to\\pi\\pi$ decay is\n$Q_+ = [ ( \\bar{s} d )_L\n         ( \\bar{u} u )_L\n    +    ( \\bar{s} u )_L\n         ( \\bar{u} d )_L ]  \/ 2$.\nWe extract the decay amplitude from the 4-point correlation function defined by\n$M_Q  = \\langle 0 | W_+  W_0 Q_{+}(t) W_K | 0 \\rangle$ \nwhere $W_{0,+,K}$ are wall sources for $\\pi^0$, $\\pi^+$ and $K^+$. \nIn our calculations on a lattice of a temporal size $T=64$,   \nthe walls are placed at the time slices $t_{K^+}=4$, $t_{\\pi^+}=59$ and $t_{\\pi^0}=60$.\nThe mesons are all created at rest, and the 4-quark operator $Q_+$ is \nprojected to zero spatial momentum.\n\nWe can use two types of factors to remove the normalization factors in $M_Q$.\nIf we define\n\\begin{eqnarray}\n\\lefteqn{ M_W  =\n                   \\langle 0 | W_0  \\pi^{0} ( t ) | 0 \\rangle\n                   \\langle 0 | W_+  \\pi^{+} ( t ) | 0 \\rangle\n                   \\langle 0 | K(t)   W_K         | 0 \\rangle ,} && \\nonumber \\\\\n\\lefteqn{ M_P  = \n    \\langle 0 | K(t) W_K      | 0 \\rangle \n    \\langle 0 | W_+ W_0  \\pi^{+} ( t ) \\pi^{0} ( t ) | 0 \\rangle , }  && \n\\end{eqnarray}\nwe find\n$R_W \\equiv M_Q \/ M_W$\n$\\sim \\langle \\pi^+\\pi^0 | Q_+ | K^+ \\rangle$\n$   \/ \\langle \\pi | \\pi | 0 \\rangle^3 \\cdot {\\rm exp}(t-t_+)\\Delta$\nand\n$R_P \\equiv M_Q \/ M_P$ \n$\\sim \\langle \\pi^+\\pi^0 | Q_+ | K^+ \\rangle \/ \\langle \\pi | \\pi | 0 \\rangle^3$\nfor $t_K\\ll t\\ll t_+, t_0$, where $\\Delta=m_{\\pi\\pi}-2m_\\pi$ is the \nmass shift of the 2-pion state due to finite lattice size \neffects~\\cite{LUSHER}.\n \nIn Fig.~\\ref{RWFRPF} we plot $\\langle \\pi | \\pi | 0 \\rangle^3 \\cdot R_W$ and \n$\\langle \\pi | \\pi | 0\\rangle^3 \\cdot R_P$ at \n$\\kappa=0.1530$ as a function of time $t$ of the weak operator.\nWe clearly observe a non-vanishing slope for $R_W$, while $R_P$ exhibits a \nplateau as expected.\nThe decay amplitude can be obtained by fitting $R_W$ to a single exponential \nor $R_P$ to a constant. We find the results to be mutually consistent \nwithin the statistical error.\n\\begin{figure}[t]\n\\vspace*{-0.2cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{L24L32all.1.eps}}\n\\vspace*{-1cm}\n\\caption{\\label{RWFRPF}\n$R_W \\cdot \\langle \\pi | \\pi | 0 \\rangle^3$ and $R_P\\langle \\pi | \\pi | 0 \\rangle^3$\nat $\\kappa=0.153$.\nOpen and filled circles refer to data for $24^3$ and $32^3$ lattices.\n}\n\\vspace*{-0.7cm}\n\\end{figure}\n\\begin{figure}[t]\n\\vspace{-0.6cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{o00210o11210.00.eps}}\n\\vspace{-1.0cm}\n\\caption{ \\label{COMPARISON}\nComparison of our results normalized by the experimental value obtained \nwith tree-level CHPT relation for $q^*=\\pi\/a$ at $\\beta=6.1$ \nwith those of previous work{\\protect\\cite{BSKpipi}} at $\\beta=6.0$ (crosses).\nOpen symbols refer to results with the traditional normalization \n$\\protect\\sqrt{2\\kappa}$ and no tadpole improvement, \nand filled ones are those with the KLM normalization and tadpole improvement.\n}\n\\vspace{-0.7cm}\n\\end{figure}\n\nThe one-loop renormalization factor relating the operator $Q_+$ on the lattice \nand that in the continuum was obtained in Refs.~\\cite{Z-M,Z-BS}. \nWe set $q^*=1\/a$ or $\\pi\/a$ as the matching point and employ tadpole-improved\nperturbation theory with \nthe mean-field improved $\\overline{\\rm MS}$ coupling constant\\cite{TPI}.  \nQuark fields are normalized with the KLM factor~\\cite{KLM}, \n$\\sqrt{1-3\\kappa\/4\\kappa_c}$.\n\\section{ Result }\nIn earlier calculations tree-level chiral perturbation theory (CHPT) \nformula was used to obtain the physical amplitude $A^{phys}$ from the \nlattice counterpart $A^{lat}$.  The formula takes the form \n$ A^{phys} = ( m_K^2 - m_\\pi^2 )\/( 2 M_\\pi^2 ) \\cdot A^{lat}$\nwhere $m_K=497 {\\rm MeV}$ and $m_\\pi=136 {\\rm MeV}$ are physical masses, \nand $M_\\pi$ is the degenerate mass of $K$ and $\\pi$ mesons on the lattice.\nClearly the physical amplitude should be independent of the \nlattice mass $M_\\pi$ if the matching formula is exact.\n\nIn Fig.~\\ref{COMPARISON} we compare results for the physical decay amplitude \nfrom a previous work\\cite{BSKpipi} carried out at $\\beta=6.0$ on a $24^3$ \nspatial lattice with those of our simulation at $\\beta=6.1$. \nOur values obtained with the conventional quark normalization $\\sqrt{2\\kappa}$\n(crosses and open squares) are consistent with their results,\nbut are larger than the experimental result roughly by a factor two.\n\nLet us also note with our results (circles and squares) \nthat (i) the KLM normalization and tadpole improvement of \nthe operator have a significant effect on the amplitude, (ii) there is \na clear dependence of the amplitude on the lattice meson mass $M_\\pi$, \nand (iii) a significant finite-size effect is observed between the two \nlattice sizes.\nThese features show that the tree-level CHPT is inadequate to extract \nthe physical amplitude from lattice results.\n\nIn Fig.~\\ref{FRHO} we show how predictions for the physical amplitude \nchange if we apply the one-loop formula of CHPT~\\cite{ONECPTH} to our results.\nHere $\\Lambda^q$ and $\\Lambda^{cont}$ are the cutoffs of CHPT for quenched and full theory.\nIn converting to physical values we ignore effects of $O(p^4)$ contact terms \nof the CHPT Lagrangian since their values are not well known.\nAn interesting point is that a size dependence seen with \nthe tree-level analysis in Fig.~\\ref{COMPARISON} is absent.\nAnother important point is that the magnitude of the amplitude decreases \nby $30-40$\\% over the range of meson mass covered in our simulation.\nThe amplitude depends significantly on the quenched lattice cutoff \n$\\Lambda^q$, in particular toward larger values of $M_\\pi$.\n\n\\begin{figure}\n\\vspace*{-0.6cm}\n\\centerline{\\epsfxsize=7.0cm \\epsfbox{o11210.1113.eps}}\n\\vspace*{-1cm}\n\\caption{ \\label{FRHO} \nDecay amplitude normalized by experimental value for $q^*=\\pi\/a$ \nobtained with one-loop CHPT for $\\Lambda^{cont} = 770{\\rm MeV}$.\nCircles and squares  refer to data for $24^3$ and $32^3$ spatial sizes. \nOpen symbols are for $\\Lambda^{q}=770{\\rm MeV}$ and filled ones for \n$1\\ {\\rm GeV}$.\n}\n\\vspace*{-7mm}\n\\end{figure}\nOur results in Fig.~\\ref{FRHO} show that a sizable meson mass dependence still remains.\nThis may be attributed to $O(p^4)$ contact terms which were not taken into account.\nIf we assume that the contact terms of the full theory are as small \nas suggested by the phenomenological analysis~\\cite{BK},\nwe can estimate the physical amplitude by a linear extrapolation of $M_\\pi^2$ to the chiral limit.\nIn Table~\\ref{FINALFIT} we list results for several choices of the cutoff\nand the operator matching point $q^*$.\nAs is expected from Fig.~\\ref{FRHO}, variation of results on the \nchoice of the cutoff parameters is small, being of the order of 10\\%.\nAllowing for this uncertainty, the values in Table~\\ref{FINALFIT}\nare consistent with the experimental value of $10.4\\times 10^{-3} {\\rm GeV}^3$.\n\\section{ Conclusions }\nThe present study has shown that the quenched result for the \n$K^+\\to\\pi^+ \\pi^0 $ decay amplitude agrees with experiment much better \nthan previously thought, especially if the chiral limit is taken \nfor the lattice meson mass.\nAway from the chiral limit, the amplitude depends significantly on the cutoff scales of CHPT, however.  \nThis problem is largely ascribed to a mismatch of chiral logarithms \nbetween the quenched and full theories\\cite{ONECPTH}.  \nIn order to obtain $K^+\\to\\pi^+ \\pi^0 $ decay amplitude free from this uncertainty,\nwe need simulations in full QCD.\n\n\\begin{table}[t]\n\\setlength{\\tabcolsep}{0.2pc}\n\\begin{tabular}{llllll}\n\\hline\n\\hline\n&&\\multicolumn{4}{c}{$C_+\\langle \\pi\\pi |Q_+| K \\rangle (\\times 10^{-3} {\\rm GeV}^3)$}\\\\\n$\\Lambda^{cont}$   &  $\\Lambda^{q}$   &  \\multicolumn{2}{c}{$24^3$}&\\multicolumn{2}{c}{$32^3 $}\\\\\n({\\small GeV}) &  ({\\small GeV}) &$q^*=\\frac{1}{a}$&$q^*=\\frac{\\pi}{a}$&$q^*=\\frac{1}{a}$&$q^*=\\frac{\\pi}{a}$\\\\[1mm]\n\\hline\n0.77  &  0.77  &  $ 9.3(19)$&$10.2(21)$  & $8.9(17)$&$ 9.7(19)$ \\\\\n0.77  &  1.0   &  $ 9.4(13)$&$10.3(14)$  & $8.8(11)$&$9.6(12)$ \\\\\n1.0   &  0.77  &  $10.3(21)$&$11.3(23)$  & $9.8(19)$&$10.7(21)$ \\\\\n1.0   &  1.0   &  $10.4(14)$&$11.4(15)$  & $9.7(12)$&$10.6(13)$ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\vspace*{3mm}\n\\caption{\\label{FINALFIT}\nResults of linear extrapolation of the decay amplitude to $M_\\pi^2=0$. \nStatistical and extrapolation errors are combined.\nThe experimental value is $10.4\\times 10^{-3} {\\rm GeV}^3$.\n}\n\\vspace*{-7mm}\n\\end{table}\n\\hfill\\break\nThis work is supported by the Supercomputer \nProject (No.~97-15) of High Energy Accelerator Research Organization (KEK),\nand also in part by the Grants-in-Aid of \nthe Ministry of Education (Nos. 08640349, 08640350, 08640404,\n09246206, 09304029, 09740226).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn 1932, the physicist P. Jordan proposed a program to discover a new algebraic setting for quantum mechanics, so Jordan algebras were created in this proceeding. Moreover, Jordan algebras were truned out to have illuminating connections with many areas of mathematics. A. A. Albert renamed them Jordan algebras and developed a successful structure theory for Jordan algebras over any field of characteristic zero in \\cite{A1}.\n\nHom-type algebras have been studied by many authors in \\cite{A2}, \\cite{A3}, \\cite{L1}, \\cite{D1}. In general, Hom-type algebras are a kind of algebras which are obtained by twisting the identity defining the structure with one or several linear maps(called the twisted maps) on the basis of the original algebras. The concept of Hom-Jordan algebras was first introduced by A. Makhlouf in his paper \\cite{M2}. He introduced a Hom-Jordan algebra and showed that it fits with the Hom-associative structure, that is a Hom-associative algebra leads to a Hom-Jordan algebra by considering a plus algebra.\n\nAs is well known, derivations and generalized derivations play an important role in the research of structure and property in algebraic system. The research on generalized derivations was started by Leger and Luks in \\cite{L2}. They gave many properties about generalized derivations of Lie algebras. From then on, many authors generalized the results in \\cite{L2} to other algebras. R. X. Zhang and Y. Z. Zhang generalized it to Lie superalgebras in \\cite{Z1}. And L. Y. Chen, Y. Ma and L. Ni generalized the results to Lie color algebras successfully in \\cite{C1}. J. Zhou, Y. J. Niu and L. Y. chen generalized the above results to Hom-Lie algebras in \\cite{Z2}. As for Jordan algebras, A. I. Shestakov generalized the results into Jordan superalgebras in \\cite{S1}. He showed that for semi-simple Jordan algebras over any field of characteristic $0$ or simple Jordan algebras over any field of characteristic other than $2$, the generalized derivations are the sum of the derivations and the centroids with some exceptions.\n\nThe purpose of this paper is to generalized some beautiful results in \\cite{Z2} and \\cite{M3} to Hom-Jordan algebras. We proceed as follow. In Section \\ref{se:2}, we recall some basic definitions and propositions which will be used in what follows. In Section \\ref{se:3}, we'll give some properties about generalized derivations of Hom-Jordan algebras and their Hom-subalgebras. In Section \\ref{se:4}, we show that quasiderivations of a Hom-Jordan algebra can be embedded as derivations into a larger Hom-Jordan algebra and obtain a direct sum decomposition of $Der(\\breve{V})$ when the centralizer of $V$ equals to zero. In Section \\ref{se:5}, we give some propositions with respect to the centroids of Hom-Jordan algebras.\n\n\\section{Preliminaries}\\label{se:2}\n\n\\begin{defn}\\cite{M1}\nAn algebra $J$ over a field $\\rm{F}$ is a Jordan algebra satisfying for any $x, y \\in J$,\n\\begin{enumerate}[(1)]\n\\item $x \\circ y = y \\circ x$;\n\\item $(x^{2} \\circ y) \\circ x = x^{2} \\circ (y \\circ x)$.\n\\end{enumerate}\n\\end{defn}\n\\begin{defn}\\cite{M2}\nA Hom-Jordan algebra over a field $\\rm{F}$ is a triple $(V, \\mu, \\alpha)$ consisting of a linear space $V$, a bilinear map $\\mu : V \\times V \\rightarrow V$ which is commutative and a linear map $\\alpha : V \\rightarrow V$ satisfying for any $x, y \\in V$,\n\\[\\mu(\\alpha^{2}(x), \\mu(y, \\mu(x, x))) = \\mu(\\mu(\\alpha(x), y), \\alpha(\\mu(x, x))),\\]\nwhere $\\alpha^{2} = \\alpha \\circ \\alpha$.\n\\end{defn}\n\\begin{defn}\nA Hom-Jordan algebra $(V, \\mu, \\alpha)$ is called multiplicative if for any $x, y \\in V$, $\\alpha(\\mu(x, y)) = \\mu(\\alpha(x), \\alpha(y))$.\n\\end{defn}\n\\begin{defn}\\cite{B1}\nA subspace $W \\subseteq V$ is a Hom-subalgebra of $(V, \\mu, \\alpha)$ if $\\alpha(W) \\subseteq W$ and\n\\[\\mu(x, y) \\in W,\\quad\\forall x, y \\in W.\\]\n\\end{defn}\n\\begin{defn}\\cite{B1}\nA subspace $W \\subseteq V$ is a Hom-ideal of $(V, \\mu, \\alpha)$ if $\\alpha(W) \\subseteq W$ and\n\\[\\mu(x, y) \\in W,\\quad\\forall x \\in W, y \\in V.\\]\n\\end{defn}\n\\begin{defn}\\cite{B1}\nLet $(V, \\mu, \\alpha)$ and $(V^{'}, \\mu^{'}, \\beta)$ be two Hom-Jordan algebras. A linear map $\\phi : V \\rightarrow V^{'}$ is said to be a homomorphism of Hom-Jordan algebras if\n\\begin{enumerate}[(1)]\n\\item $\\phi(\\mu(x, y)) = \\mu^{'}(\\phi(x), \\phi(y))$;\n\\item $\\phi \\circ \\alpha = \\beta \\circ \\phi$.\n\\end{enumerate}\n\\end{defn}\n\\begin{lem}\\cite{B1}\\label{le:2.7}\nLet $(V, \\mu, \\alpha)$ be a Hom-Jordan algebra over a field $\\rm{F}$, we define a subspace $\\mathcal{W}$ of $End(V)$ where $\\mathcal{W} = \\{w \\in End(V) | w \\circ \\alpha = \\alpha \\circ w\\}$, $\\sigma : \\mathcal{W} \\rightarrow \\mathcal{W}$ is a linear map satisfying $\\sigma(w) = \\alpha \\circ w$.\n\\begin{enumerate}[(1)]\n\\item A triple $(\\mathcal{W}, \\nu, \\sigma)$, where the multiplication $\\nu : \\mathcal{W} \\times \\mathcal{W} \\rightarrow \\mathcal{W}$ is defined for $w_{1}, w_{2} \\in \\mathcal{W}$ by\n    \\[\\nu(w_{1}, w_{2}) = w_{1} \\circ w_{2} + w_{2} \\circ w_{1},\\]\n    is a Hom-Jordan algebra over $\\rm{F}$.\n\\item A triple $(\\mathcal{W}, \\nu^{'}, \\sigma)$, where the multiplication $\\nu^{'} : \\mathcal{W} \\times \\mathcal{W} \\rightarrow \\mathcal{W}$ is defined for $w_{1}, w_{2} \\in \\mathcal{W}$ by\n    \\[\\nu^{'}(w_{1}, w_{2}) = w_{1} \\circ w_{2} - w_{2} \\circ w_{1},\\]\n    is a Hom-Lie algebra over $\\rm{F}$.\n\\end{enumerate}\n\\end{lem}\n\\begin{defn}\\cite{B1}\nFor any nonnegative integer $k$, a linear map $D : V \\rightarrow V$ is called an $\\alpha^{k}$-derivation of the Hom-Jordan algebra $(V, \\mu, \\alpha)$, if\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D$;\n\\item $D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)),\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\n\\begin{lem}\\cite{B1}\\label{le:2.9}\nFor any $D \\in Der_{\\alpha^{k}}(V)$ and $D^{'} \\in Der_{\\alpha^{s}}(V)$, define their commutator $\\nu^{'}(D, D^{'})$ as usual:\n\\[\\nu^{'}(D, D^{'}) = D \\circ D^{'} - D^{'} \\circ D.\\]\nThen $\\nu^{'}(D, D^{'}) \\in Der_{\\alpha^{k + s}}(V)$.\n\\end{lem}\n\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra. We denote the set of all $\\alpha^{k}$-derivations by $Der_{\\alpha^{k}}(V)$, then $Der(V) = \\dotplus_{k \\geq 0}Der_{\\alpha^{k}}(V)$ provided with the commutator and the following map\n\\[\\sigma : Der(V) \\rightarrow Der(V);\\quad \\sigma(D) = \\alpha \\circ D\\]\nis a Hom-subalgebra of $(\\mathcal{W}, \\nu^{'}, \\sigma)$ according to Lemma \\ref{le:2.9} and is called the derivation algebra of $(V, \\mu, \\alpha)$.\n\\begin{defn}\nFor any nonnegative integer $k$, a linear map $D \\in End(V)$ is said to be a generalized $\\alpha^{k}$-derivation of $(V, \\mu, \\alpha)$, if there exist two linear maps $D^{'}, D^{''} \\in End(V)$ such that\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D,\\; D^{'} \\circ \\alpha = \\alpha \\circ D^{'},\\; D^{''} \\circ \\alpha = \\alpha \\circ D^{''}$;\n\\item $\\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D^{'}(y)) = D^{''}(\\mu(x, y)),\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\n\\begin{defn}\\label{defn:2.11}\nFor any nonnegative integer $k$, a linear map $D \\in End(V)$ is said to be an $\\alpha^{k}$-quasiderivation of $(V, \\mu, \\alpha)$, if there exist a linear map $D^{'}\\in End(V)$ such that\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D,\\; D^{'} \\circ \\alpha = \\alpha \\circ D^{'}$;\n\\item $\\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)) = D^{'}(\\mu(x, y)),\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\n\nLet $GDer_{\\alpha^{k}}(V)$ and $QDer_{\\alpha^{k}}(V)$ be the sets of generalized $\\alpha^{k}$-derivations and of $\\alpha^{k}$-quasiderivations, respectively. That is\n\\[GDer(V) = \\dotplus_{k \\geq 0}GDer_{\\alpha^{k}}(V),\\quad QDer(V) = \\dotplus_{k \\geq 0}QDer_{\\alpha^{k}}(V).\\]\nIt's easy to verify that both $GDer(V)$ and $QDer(V)$ are Hom-subalgebras of $(\\mathcal{W}, \\nu^{'}, \\sigma)$(See Proposition \\ref{prop:3.1}).\n\\begin{defn}\nFor any nonnegative integer $k$, a linear map $D \\in End(V)$ is said to be an $\\alpha^{k}$-centroid of $(V, \\mu, \\alpha)$, if\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D$;\n\\item $\\mu(D(x), \\alpha^{k}(y)) = \\mu(\\alpha^{k}(x), D(y)) = D(\\mu(x, y)),\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\n\\begin{defn}\nFor any nonnegative integer $k$, a linear map $D \\in End(V)$ is said to be an $\\alpha^{k}$-quasicentroid of $(V, \\mu, \\alpha)$, if\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D$;\n\\item $\\mu(D(x), \\alpha^{k}(y)) = \\mu(\\alpha^{k}(x), D(y)),\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\nLet $C_{\\alpha^{k}}(V)$ and $QC_{\\alpha^{k}}(V)$ be the sets of $\\alpha^{k}$-centroids and of $\\alpha^{k}$-quasicentroids, respectively. That is\n\\[C(V) = \\dotplus_{k \\geq 0}C_{\\alpha^{k}}(V),\\quad QC(V) = \\dotplus_{k \\geq 0}QC_{\\alpha^{k}}(V).\\]\n\\begin{defn}\nFor any nonnegative integer $k$, a linear map $D \\in End(V)$ is said to be an $\\alpha^{k}$-central derivation of $(V, \\mu, \\alpha)$, if\n\\begin{enumerate}[(1)]\n\\item $D \\circ \\alpha = \\alpha \\circ D$;\n\\item $\\mu(D(x), \\alpha^{k}(y)) = D(\\mu(x, y)) = 0,\\quad\\forall x, y \\in V$.\n\\end{enumerate}\n\\end{defn}\n\nWe denote the set of all $\\alpha^{k}$-central derivations by $ZDer_{\\alpha^{k}}(V)$, then $ZDer(V) = \\dotplus_{k \\geq 0}ZDer_{\\alpha^{k}}(V)$.\n\nAccording to the definitions, it's easy to show that $ZDer(V) \\subseteq Der(V) \\subseteq QDer(V) \\subseteq GDer(V) \\subseteq End(V)$.\n\\begin{defn}\nLet $(V, \\mu, \\alpha)$ be a Hom-Jordan algebra. If $Z(V) = \\{x \\in V | \\mu(x, y) = 0,\\quad\\forall y \\in V\\}$, then $Z(V)$ is called the centralizer of $(V, \\mu, \\alpha)$.\n\\end{defn}\n\n\\section{Generalized derivation algebras and their subalgebras}\\label{se:3}\n\nAt first, we give some basic properties of center derivation algebras, quasiderivation algebras and generalized derivation algebras of a Hom-Jordan algebra.\n\\begin{prop}\\label{prop:3.1}\nSuppose that $(V, \\mu, \\alpha)$ is a multiplicative Hom-Jordan algebra. Then the following statements hold:\n\\begin{enumerate}[(1)]\n\\item $GDer(V)$, $QDer(V)$ and $C(V)$ are Hom-subalgebras of $(\\mathcal{W}, \\nu^{'}, \\sigma)$;\n\\item $ZDer(V)$ is a Hom-ideal of $Der(V)$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). Suppose that $D_{1} \\in GDer_{\\alpha^{k}}(V)$, $D_{2} \\in GDer_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$,\n\\begin{align*}\n&\\mu(\\sigma(D_{1})(x), \\alpha^{k + 1}(y)) = \\mu(\\alpha \\circ D_{1}(x), \\alpha^{k + 1}(y)) = \\alpha(\\mu(D_{1}(x), \\alpha^{k}(y)))\\\\\n&= \\alpha(D_{1}^{''}(\\mu(x, y)) - \\mu(\\alpha^{k}(x), D_{1}^{'}(y))) = \\sigma(D_{1}^{''})(\\mu(x, y)) - \\mu(\\alpha^{k + 1}(x), \\sigma(D_{1}^{'})(y)).\n\\end{align*}\nSince $\\sigma(D_{1}^{''}), \\sigma(D_{1}^{'}) \\in End(V)$, we have $\\sigma(D_{1}) \\in GDer_{\\alpha^{k + 1}}(V)$.\n\\begin{align*}\n&\\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y)) - \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y))\\\\\n&= D_{1}^{''}(\\mu(D_{2}(x), \\alpha^{s}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}^{'}(\\alpha^{s}(y))) - D_{2}^{''}(\\mu(D_{1}(x), \\alpha^{k}(y))) +\\\\ &\\mu(\\alpha^{s}(D_{1}(x)), D_{2}^{'}(\\alpha^{k}(y)))\\\\\n&= D_{1}^{''}(D_{2}^{''}(\\mu(x, y)) - \\mu(\\alpha^{s}(x), D_{2}^{'}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}^{'}(\\alpha^{s}(y)))\\\\\n&- D_{2}^{''}(D_{1}^{''}(\\mu(x, y)) - \\mu(\\alpha^{k}(x), D_{1}^{'}(y))) + \\mu(\\alpha^{s}(D_{1}(x)), D_{2}^{'}(\\alpha^{k}(y)))\\\\\n&= D_{1}^{''} \\circ D_{2}^{''}(\\mu(x, y)) - D_{1}^{''}(\\mu(\\alpha^{s}(x), D_{2}^{'}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}^{'}(\\alpha^{s}(y)))\\\\\n&- D_{2}^{''} \\circ D_{1}^{''}(\\mu(x, y)) + D_{2}^{''}(\\mu(\\alpha^{k}(x), D_{1}^{'}(y))) + \\mu(\\alpha^{s}(D_{1}(x)), D_{2}^{'}(\\alpha^{k}(y)))\\\\\n&= D_{1}^{''} \\circ D_{2}^{''}(\\mu(x, y)) - \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}^{'}(y))) - \\mu(\\alpha^{k + s}(x), D_{1}^{'}(D_{2}^{'}(y)))\\\\\n&- \\mu(\\alpha^{k}(D_{2}(x)), D_{1}^{'}(\\alpha^{s}(y))) - D_{2}^{''} \\circ D_{1}^{''}(\\mu(x, y)) + \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}^{'}(y)))\\\\\n&+ \\mu(\\alpha^{k + s}(x), D_{2}^{'}(D_{1}^{'}(y))) + \\mu(\\alpha^{s}(D_{1}(x)), D_{2}^{'}(\\alpha^{k}(y)))\\\\\n&= D_{1}^{''} \\circ D_{2}^{''}(\\mu(x, y)) - D_{2}^{''} \\circ D_{1}^{''}(\\mu(x, y)) - \\mu(\\alpha^{k + s}(x), D_{1}^{'}(D_{2}^{'}(y)))\\\\\n&+ \\mu(\\alpha^{k + s}(x), D_{2}^{'}(D_{1}^{'}(y)))\\\\\n&= \\nu^{'}(D_{1}^{''}, D_{2}^{''})(\\mu(x, y)) - \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}^{'}, D_{2}^{'})(y)).\n\\end{align*}\nSince $\\nu^{'}(D_{1}^{''}, D_{2}^{''}), \\nu^{'}(D_{1}^{'}, D_{2}^{'}) \\in End(V)$, we have $\\nu^{'}(D_{1}, D_{2}) \\in GDer_{\\alpha^{k + s}}(V)$.\n\nTherefore, $GDer(V)$ is a Hom-subalgebra of $(\\mathcal{W}, \\nu^{'}, \\sigma)$.\n\nSimilarly, we have $QDer(V)$ is a Hom-subalgebra of $(\\mathcal{W}, \\nu^{'}, \\sigma)$.\n\nSuppose that $D_{1} \\in C_{\\alpha^{k}}(V)$, $D_{2} \\in C_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$,\n\\[\\sigma(D_{1})(\\mu(x, y)) = \\alpha \\circ D_{1}(\\mu(x, y)) = \\alpha(\\mu(D_{1}(x), \\alpha^{k}(y))) = \\mu(\\sigma(D_{1})(x), \\alpha^{k + 1}(y)).\\]\nSimilarly, we have $\\sigma(D_{1})(\\mu(x, y)) = \\mu(\\alpha^{k + 1}(x), \\sigma(D_{1})(y))$. Hence, we have $\\sigma(D_{1}) \\in C_{\\alpha^{k + 1}}(V)$.\n\\begin{align*}\n&\\nu^{'}(D_{1}, D_{2})(\\mu(x, y))\\\\\n&= D_{1} \\circ D_{2}(\\mu(x, y)) - D_{2} \\circ D_{1}(\\mu(x, y))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y))) - D_{2}(\\mu(D_{1}(x), \\alpha^{k}(y)))\\\\\n&= \\mu(D_{1}(D_{2}(x)), \\alpha^{k + s}(y)) - \\mu(D_{2}(D_{1}(x)), \\alpha^{k + s}(y))\\\\\n&= \\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y)).\n\\end{align*}\nSimilarly, we have $\\nu^{'}(D_{1}, D_{2})(\\mu(x, y)) = \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y))$.\n\nHence, we have $\\nu^{'}(D_{1}, D_{2}) \\in C_{\\alpha^{k + s}}(V)$.\n\nTherefore, we have $C(V)$ is a Hom-subalgebras of $(\\mathcal{W}, \\nu^{'}, \\sigma)$.\n\n(2). Suppose that $D_{1} \\in ZDer_{\\alpha^{k}}(V)$, $D_{2} \\in Der_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$,\n\\[\\sigma(D_{1})(\\mu(x, y)) = \\alpha \\circ D_{1}(\\mu(x, y)) = 0.\\]\n\\[\\sigma(D_{1})(\\mu(x, y)) = \\alpha \\circ D_{1}(\\mu(x, y)) = \\alpha(\\mu(D_{1}(x), \\alpha^{k}(y))) = \\mu(\\sigma(D_{1})(x), \\alpha^{k + 1}(y)).\\]\nHence, $\\sigma(D_{1}) \\in ZDer_{\\alpha^{k + 1}}(V)$.\n\\begin{align*}\n&\\nu^{'}(D_{1}, D_{2})(\\mu(x, y))\\\\\n&= D_{1} \\circ D_{2}(\\mu(x, y)) - D_{2} \\circ D_{1}(\\mu(x, y))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y)) + \\mu(\\alpha^{s}(x), D_{2}(y))) = 0.\n\\end{align*}\n\\begin{align*}\n&\\nu^{'}(D_{1}, D_{2})(\\mu(x, y))\\\\\n&= D_{1} \\circ D_{2}(\\mu(x, y)) - D_{2} \\circ D_{1}(\\mu(x, y))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y)) + \\mu(\\alpha^{s}(x), D_{2}(y))) - D_{2}(\\mu(D_{1}(x), \\alpha^{k}(y)))\\\\\n&= \\mu(D_{1}(D_{2}(x)), \\alpha^{k + s}(y)) + \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y))) - \\mu(D_{2}(D_{1}(x)), \\alpha^{k + s}(y))\\\\\n&- \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= \\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y)).\n\\end{align*}\nHence, we have $\\nu^{'}(D_{1}, D_{2}) \\in ZDer_{\\alpha^{k + s}}(V)$.\n\nTherefore, $ZDer(V)$ is a Hom-ideal of $Der(V)$.\n\\end{proof}\n\\begin{prop}\\label{prop:3.2}\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra, then\n\\begin{enumerate}[(1)]\n\\item $\\nu^{'}(Der(V), C(V)) \\subseteq C(V)$;\n\\item $\\nu^{'}(QDer(V), QC(V)) \\subseteq QC(V)$;\n\\item $\\nu^{'}(QC(V), QC(V)) \\subseteq QDer(V)$;\n\\item $C(V) \\subseteq QDer(V)$;\n\\item $QDer(V) + QC(V) \\subseteq GDer(V)$;\n\\item $C(V) \\circ Der(V) \\subseteq Der(V)$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). Suppose that $D_{1} \\in Der_{\\alpha^{k}}(V)$, $D_{2} \\in C_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&\\nu^{'}(D_{1}, D_{2})(\\mu(x, y))\\\\\n&= D_{1} \\circ D_{2}(\\mu(x, y)) - D_{2} \\circ D_{1}(\\mu(x, y))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y))) - D_{2}(\\mu(D_{1}(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D_{1}(y)))\\\\\n&= \\mu(D_{1}(D_{2}(x)), \\alpha^{k + s}(y)) + \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y))) - \\mu(D_{2}(D_{1}(x)), \\alpha^{k + s}(y))\\\\\n&- \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}(y)))\\\\\n&= \\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y)).\n\\end{align*}\nSimilarly, we have $\\nu^{'}(D_{1}, D_{2})(\\mu(x, y)) = \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y))$. Hence, $\\nu^{'}(D_{1}, D_{2}) \\in C_{\\alpha^{k + s}}(V)$. Therefore $\\nu^{'}(Der(V), C(V)) \\subseteq C(V)$.\n\n(2). Suppose that $D_{1} \\in QDer_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&\\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y)) - \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y))\\\\\n&= D_{1}^{'}(\\mu(D_{2}(x), \\alpha^{s}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y))) - \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= D_{1}^{'}(\\mu(\\alpha^{s}(x), D_{2}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y))) - \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y))) + \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y)))\\\\\n&- \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y))) - \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y))) - \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y))) - \\mu(\\alpha^{k + s}(x), D_{2}(D_{1}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y)).\n\\end{align*}\nHence, we have $\\nu^{'}(D_{1}, D_{2}) \\in QC_{\\alpha^{k + s}}(V)$. So $\\nu^{'}(QDer(V), QC(V)) \\subseteq QC(V)$.\n\n(3). Suppose that $D_{1} \\in QC_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&\\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y)) - \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y))) - \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}(y))) - \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{2}(D_{1}(y))) - \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y)))\\\\\n&= -\\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y)),\n\\end{align*}\ni.e., $\\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y)) + \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y)) = 0$.\n\nHence, we have $\\nu^{'}(D_{1}, D_{2}) \\in QDer_{\\alpha^{k + s}}(V)$, which implies that $\\nu^{'}(QC(V), QC(V)) \\subseteq QDer(V)$.\n\n(4). Suppose that $D \\in C_{\\alpha^{k}}(V)$. Then for any $x, y \\in V$, we have\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) = \\mu(\\alpha^{k}(x), D(y)).\\]\n\nHence, we have\n\\[\\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)) = 2D(\\mu(x, y)),\\]\nwhich implies that $D \\in QDer_{\\alpha^{k}}(V)$. So $C(V) \\subseteq QDer(V)$.\n\n(5). Suppose that $D_{1} \\in QDer_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{k}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&\\mu((D_{1} + D_{2})(x), \\alpha^{k}(y))\\\\\n&= \\mu(D_{1}(x), \\alpha^{k}(y)) + \\mu(D_{2}(x), \\alpha^{k}(y))\\\\\n&= D_{1}^{'}(\\mu(x, y)) - \\mu(\\alpha^{k}(x), D_{1}(y)) + \\mu(\\alpha^{k}(x), D_{2}(y))\\\\\n&= D_{1}^{'}(\\mu(x, y)) - \\mu(\\alpha^{k}(x), (D_{1} - D_{2})(y)).\n\\end{align*}\nSince $D_{1}^{'}$, $D_{1} - D_{2} \\in End(V)$, we have $D_{1} + D_{2} \\in GDer_{\\alpha^{k}}(V)$. So $QDer(V) + QC(V) \\subseteq GDer(V)$.\n\n(6). Suppose that $D_{1} \\in C_{\\alpha^{k}}(V)$, $D_{2} \\in Der_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&D_{1} \\circ D_{2}(\\mu(x, y))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y)) + \\mu(\\alpha^{s}(x), D_{2}(y)))\\\\\n&= \\mu(D_{1}(D_{2}(x)), \\alpha^{k + s}(y)) + \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y))),\n\\end{align*}\nwhich implies that $D_{1} \\circ D_{2} \\in Der_{\\alpha^{k + s}}(V)$. So $C(V) \\circ Der(V) \\subseteq Der(V)$.\n\\end{proof}\n\\begin{thm}\nSuppose that $(V, \\mu, \\alpha)$ is a multiplicative Hom-Jordan algebra. Then $GDer(V) = QDer(V) + QC(V)$.\n\\end{thm}\n\\begin{proof}\nAccording to Proposition \\ref{prop:3.2} (5), we need only to show that $GDer(V) \\subseteq QDer(V) + QC(V)$.\n\nSuppose that $D \\in GDer_{\\alpha^{k}}(V)$. Then there exist $D^{'},\\; D^{''} \\in End(V)$ such that\n\\[D^{''}(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D^{'}(y)),\\quad\\forall x, y \\in V.\\]\n\nSince $\\mu$ is commutative, we have\n\\[D^{''}(\\mu(y, x)) = \\mu(D^{'}(y), \\alpha^{k}(x)) + \\mu(\\alpha^{k}(y), D(x)),\\quad\\forall x, y \\in V,\\]\nwhich implies that $D^{'} \\in GDer_{\\alpha^{k}}(V)$.\n\nMoreover, we have\n\\begin{align*}\n&\\mu\\left(\\frac{D + D^{'}}{2}(x), \\alpha^{k}(y)\\right) + \\mu\\left(\\alpha^{k}(x), \\frac{D + D^{'}}{2}(y)\\right)\\\\\n&= \\frac{1}{2}(\\mu(D(x), \\alpha^{k}(y)) + \\mu(D^{'}(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)) + \\mu(\\alpha^{k}(x), D^{'}(y)))\\\\\n&= D^{''}(\\mu(x, y)),\n\\end{align*}\nwhich implies $\\frac{D + D^{'}}{2} \\in QDer_{\\alpha^{k}}(V)$.\n\\begin{align*}\n&\\mu\\left(\\frac{D - D^{'}}{2}(x), \\alpha^{k}(y)\\right)\\\\\n&= \\frac{1}{2}(\\mu(D(x), \\alpha^{k}(y)) - \\mu(D^{'}(x), \\alpha^{k}(y)))\\\\\n&= \\frac{1}{2}(D^{''}(\\mu(x, y)) - \\mu(\\alpha^{k}(x), D^{'}(y)) - D^{''}(\\mu(x, y)) + \\mu(\\alpha^{k}(x), D(y)))\\\\\n&= \\mu\\left(\\alpha^{k}(x), \\frac{D - D^{'}}{2}(y)\\right),\n\\end{align*}\nwhich implies that $\\frac{D - D^{'}}{2} \\in QC_{\\alpha^{k}}(V)$.\n\nHence, $D = \\frac{D + D^{'}}{2} + \\frac{D - D^{'}}{2} \\in QDer(V) + QC(V)$, i.e., $GDer(V) \\subseteq QDer(V) + QC(V)$.\n\nTherefore, $GDer(V) = QDer(V) + QC(V)$.\n\\end{proof}\n\\begin{prop}\nSuppose that $(V, \\mu, \\alpha)$ is a multiplicative Hom-Jordan algebra where $V$ can be decomposed into the direct sum of two Hom-ideals, i.e., $V = V_{1} \\oplus V_{2}$ and $\\alpha$ is surjective. Then we have\n\\begin{enumerate}[(1)]\n\\item $Z(V) = Z(V_{1}) \\oplus Z(V_{2})$;\n\\item If\\; $Z(V) = \\{0\\}$, then we have\n      \\begin{enumerate}[(a)]\n      \\item $Der(V) = Der(V_{1}) \\oplus Der(V_{2})$;\n      \\item $GDer(V) = GDer(V_{1}) \\oplus GDer(V_{2})$;\n      \\item $QDer(V) = QDer(V_{1}) \\oplus QDer(V_{2})$;\n      \\item $C(V) = C(V_{1}) \\oplus C(V_{2})$.\n      \\end{enumerate}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). Obviously, $Z(V_{1}) \\cap Z(V_{2}) = \\{0\\}$. And it's easy to show that $Z(V_{i})(i = 1, 2)$ are Hom-ideals of $Z(V)$. For all $z_{i} \\in Z(V_{i})(i = 1, 2)$, take $x = x_{1} + x_{2}$ where $x_{1} \\in J_{1}$, $x_{2} \\in J_{2}$. We have\n\\[\\mu(z_{1} + z_{2}, x) = \\mu(z_{1} + z_{2}, x_{1} + x_{2}) = \\mu(z_{1}, x_{1}) + \\mu(z_{2}, x_{2}),\\]\nsince $z_{i} \\in Z(V_{i})(i = 1, 2)$, we have\n\\[\\mu(z_{i}, x_{i}) = 0,\\; i = 1, 2.\\]\nHence,\n\\[\\mu(z_{1} + z_{2}, x) = 0,\\]\nwhich implies that $z_{1} + z_{2} \\in Z(V)$, i.e., $Z(V_{1}) \\oplus Z(V_{2}) \\subseteq Z(V)$.\n\nOn the other hand, for all $a \\in Z(V)$, suppose that $a = a_{1} + a_{2}$ where $a_{i} \\in V_{i}(i = 1, 2)$. Then for all $x_{1} \\in V_{1}$,\n\\[\\mu(a_{1}, x_{1}) = \\mu(a - a_{2}, x_{1}) = \\mu(a, x_{1})\\]\nsince $a \\in Z(V)$, we have\n\\[\\mu(a, x_{1}) = 0.\\]\nHence,\n\\[\\mu(a_{1}, x_{1}) = 0,\\]\nwhich implies that $a_{1} \\in Z(V_{1})$. Similarly, we have $a_{2} \\in Z(V_{2})$. Hence, $Z(V) \\subseteq Z(V_{1}) \\oplus Z(V_{2})$.\n\nTherefore, we have $Z(V) = Z(V_{1}) \\oplus Z(V_{2})$.\n\n(2). $\\bf{Step 1}$. We'll show that $\\forall i = 1, 2$, $D(V_{i}) \\subseteq V_{i},\\quad\\forall D \\in Der_{\\alpha^{k}}(V)(k \\geq 0)$.\n\nSuppose that $x_{i} \\in V_{i}$, then\n\\[\\mu(D(x_{1}), \\alpha^{k}(x_{2})) = D(\\mu(x_{1}, x_{2})) - \\mu(\\alpha^{k}(x_{1}), D(x_{2})) \\in V_{1} \\cap V_{2} = 0,\\]\nsince $V_{1}$, $V_{2}$ are Hom-ideals of $(V, \\mu, \\alpha)$.\n\nSuppose that $D(x_{1}) = u_{1} + u_{2}$ where $u_{1} \\in V_{1}$, $u_{2} \\in V_{2}$. Then\n\\[\\mu(u_{2}, \\alpha^{k}(x_{2})) = \\mu(u_{1} + u_{2}, \\alpha^{k}(x_{2})) = \\mu(D(x_{1}), \\alpha^{k}(x_{2})) = 0,\\]\nwhich implies that $u_{2} \\in Z(V_{2})$ since $\\alpha$ is surjective. Note that $Z(V) = \\{0\\}$, we have $Z(V_{i}) = \\{0\\}(i = 1, 2)$. Hence, $u_{2} = 0$. That is to say $D(x_{1}) \\in V_{1}$. Similarly, we have $D(x_{2}) \\in V_{2}$.\n\n$\\bf{Step 2}$. We'll show that $Der_{\\alpha^{k}}(V_{1}) \\dotplus Der_{\\alpha^{k}}(V_{2}) \\subseteq Der_{\\alpha^{k}}(V)(k \\geq 0)$.\n\nFor any $D \\in Der_{\\alpha^{k}}(V_{1})$, we extend it to a linear map on $V$ as follow\n\\[D(x_{1} + x_{2}) = D(x_{1}),\\quad\\forall x_{1} \\in V_{1}, x_{2} \\in V_{2}.\\]\n\nThen for any $x, y \\in V$, suppose that $x = x_{1} + x_{2}$, $y = y_{1} + y_{2} \\in V$, where $x_{1}, y_{1} \\in V_{1}$, $x_{2}, y_{2} \\in V_{2}$, we have\n\\[D(\\mu(x, y)) = D(\\mu(x_{1} + x_{2}, y_{1} + y_{2})) = D(\\mu(x_{1}, y_{1}) + \\mu(x_{2}, y_{2})) = D(\\mu(x_{1}, y_{1})),\\]\n\\begin{align*}\n&\\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y))\\\\\n&= \\mu(D(x_{1} + x_{2}), \\alpha^{k}(y_{1} + y_{2})) + \\mu(\\alpha^{k}(x_{1} + x_{2}), D(y_{1} + y_{2}))\\\\\n&= \\mu(D(x_{1}), \\alpha^{k}(y_{1})) + \\mu(\\alpha^{k}(x_{1}), D(y_{1})),\n\\end{align*}\nsince $D \\in Der_{\\alpha^{k}}(V_{1})$,\n\\[D(\\mu(x_{1}, y_{1})) = \\mu(D(x_{1}), \\alpha^{k}(y_{1})) + \\mu(\\alpha^{k}(x_{1}), D(y_{1})),\\]\nwe have\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)),\\]\nwhich implies that $D \\in Der_{\\alpha^{k}}(V)$, i.e., $Der_{\\alpha^{k}}(V_{1}) \\subseteq Der_{\\alpha^{k}}(V)$. Moreover, $D \\in Der_{\\alpha^{k}}(V_{1})$ if and only if $D(x_{2}) = 0,\\; \\forall x_{2} \\in V_{2}$.\n\nSimilarly, we have $Der_{\\alpha^{k}}(V_{2}) \\subseteq Der_{\\alpha^{k}}(V)$ and $D \\in Der_{\\alpha^{k}}(V_{2})$ if and only if $D(x_{1}) = 0,\\; \\forall x_{1} \\in V_{1}$.\n\nThen we have $Der_{\\alpha^{k}}(V_{1}) + Der_{\\alpha^{k}}(V_{2}) \\subseteq Der_{\\alpha^{k}}(V)$ and $Der_{\\alpha^{k}}(V_{1}) \\cap Der_{\\alpha^{k}}(V_{2}) = \\{0\\}$. Hence, $Der_{\\alpha^{k}}(V_{1}) \\dotplus Der_{\\alpha^{k}}(V_{2}) \\subseteq Der_{\\alpha^{k}}(V)$.\n\n$\\bf{Step 3}$. We'll prove that $Der_{\\alpha^{k}}(V_{1}) \\dotplus Der_{\\alpha^{k}}(V_{2}) = Der_{\\alpha^{k}}(V)$.\n\nSuppose that $D \\in Der_{\\alpha^{k}}(V)$. Set $x = x_{1} + x_{2}, x_{i} \\in V_{i}$. Define $D_{1}, D_{2}$ as follows\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\nD_{1}(x_{1} + x_{2}) = D(x_{1}),\\\\\nD_{2}(x_{1} + x_{2}) = D(x_{2}).\n\\end{aligned}\n\\right.\n\\end{equation}\nObviously, $D = D_{1} + D_{2}$.\n\nFor any $u_{1}, v_{1} \\in V_{1}$,\n\\begin{align*}\n&D_{1}(\\mu(u_{1}, v_{1})) = D(\\mu(u_{1}, v_{1})) = \\mu(D(u_{1}), \\alpha^{k}(v_{1})) + \\mu(\\alpha^{k}(u_{1}), D(v_{1}))\\\\\n&= \\mu(D_{1}(u_{1}), \\alpha^{k}(v_{1})) + \\mu(\\alpha^{k}(u_{1}), D_{1}(v_{1})).\n\\end{align*}\nHence, $D_{1} \\in Der_{\\alpha^{k}}(V_{1})$. Similarly, $D_{2} \\in Der_{\\alpha^{k}}(V_{2})$.\n\nTherefore, $Der_{\\alpha^{k}}(V_{1}) \\dotplus Der_{\\alpha^{k}}(V_{2}) = Der_{\\alpha^{k}}(V)$ as a vector space.\n\nHence, we have\n\\begin{align*}\n&Der(V) = \\dotplus_{k \\geq 0}Der_{\\alpha^{k}}(V) = \\dotplus_{k \\geq 0}(Der_{\\alpha^{k}}(V_{1}) \\dotplus Der_{\\alpha^{k}}(V_{2}))\\\\\n&= (\\dotplus_{k \\geq 0}Der_{\\alpha^{k}}(V_{1})) \\dotplus (\\dotplus_{k \\geq 0}Der_{\\alpha^{k}}(V_{2})) = Der(V_{1}) \\dotplus Der(V_{2}).\n\\end{align*}\n\n$\\bf{Step 4}$. We'll show that $Der(V_{i})(i = 1, 2)$ are Hom-ideals of $Der(V)$.\nSuppose that $D_{1} \\in Der_{\\alpha^{k}}(V_{1})$, $D_{2} \\in Der_{\\alpha^{s}}(V)$. Then for any $x_{1}, y_{1} \\in V_{1}$, we have\n\\begin{align*}\n&\\sigma(D_{1})(\\mu(x_{1}, y_{1})) = \\alpha \\circ D_{1}(\\mu(x_{1}, y_{1})) = \\alpha(\\mu(D_{1}(x_{1}), \\alpha^{k}(y_{1})) + \\mu(\\alpha^{k}(x_{1}), D_{1}(y_{1})))\\\\\n&= \\mu(\\sigma(D_{1})(x_{1}), \\alpha^{k + 1}(y_{1})) + \\mu(\\alpha^{k + 1}(x_{1}), \\sigma(D_{1})(y_{1})),\n\\end{align*}\nwhich implies that $\\sigma(D_{1}) \\in Der_{\\alpha^{k + 1}}(V_{1})$.\n\\begin{align*}\n&\\nu^{'}(D_{1}, D_{2})(\\mu(x_{1}, y_{1}))\\\\\n&= D_{1} \\circ D_{2}(\\mu(x_{1}, y_{1})) - D_{2} \\circ D_{1}(\\mu(x_{1}, y_{1}))\\\\\n&= D_{1}(\\mu(D_{2}(x_{1}), \\alpha^{s}(y_{1})) + \\mu(\\alpha^{s}(x_{1}), D_{2}(y_{1}))) - D_{2}(\\mu(D_{1}(x_{1}), \\alpha^{k}(y_{1})) + \\mu(\\alpha^{k}(x_{1}), D_{1}(y_{1})))\\\\\n&= \\mu(D_{1}(D_{2}(x_{1})), \\alpha^{k + s}(y_{1})) + \\mu(\\alpha^{k}(D_{2}(x_{1})), D_{1}(\\alpha^{s}(y_{1}))) + \\mu(D_{1}(\\alpha^{s}(x_{1})), \\alpha^{k}(D_{2}(y_{1})))\\\\\n&+ \\mu(\\alpha^{k + s}(x_{1}), D_{1}(D_{2}(y_{1}))) - \\mu(D_{2}(D_{1}(x_{1})), \\alpha^{k + s}(y_{1})) - \\mu(\\alpha^{s}(D_{1}(x_{1})), D_{2}(\\alpha^{k}(y_{1})))\\\\\n&- \\mu(D_{2}(\\alpha^{k}(x_{1})), \\alpha^{s}(D_{1}(y_{1}))) - \\mu(\\alpha^{k + s}(x_{1}), D_{2}(D_{1}(y_{1})))\\\\\n&= \\mu(\\nu^{'}(D_{1}, D_{2})(x_{1}), \\alpha^{k + s}(y_{1})) + \\mu(\\alpha^{k + s}(x_{1}), \\nu^{'}(D_{1}, D_{2})(y_{1})),\n\\end{align*}\nwhich implies that $\\nu^{'}(D_{1}, D_{2}) \\in Der_{\\alpha^{k + s}}(V_{1})$. So $Der(V_{1})$ is a Hom-ideal of $Der(V)$.\n\nSimilarly, we have $Der(V_{2})$ is a Hom-ideal of $Der(V)$.\n\nTherefore, $Der(V) = Der(V_{1}) \\oplus Der(V_{2})$.\n\n(3), (4), (5) similar to the proof of (2).\n\\end{proof}\n\\begin{thm}\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra, $\\alpha$ a surjection and $Z(V)$ the centralizer of $(V, \\mu, \\alpha)$. Then $\\nu^{'}(C(V), QC(V)) \\subseteq End(V, Z(V))$. Moreover, if $Z(V) = \\{0\\}$, then $\\nu^{'}(C(V), QC(V)) = \\{0\\}$.\n\\end{thm}\n\\begin{proof}\nSuppose that $D_{1} \\in C_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{s}}(V)$ and $x \\in V$. Since $\\alpha$ is surjective, there exists $y^{'} \\in V$ such that $y = \\alpha^{k + s}(y^{'})$ for any $y \\in V$.\n\\begin{align*}\n&\\mu(\\nu^{'}(D_{1}, D_{2})(x), y)\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y^{'})) - \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y^{'}))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y^{'}))) - \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y^{'})))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y^{'}))) - \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y^{'})))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y^{'}))) - D_{1}(\\mu(\\alpha^{s}(x), D_{2}(y^{'})))\\\\\n&= D_{1}(\\mu(D_{2}(x), \\alpha^{s}(y^{'})) - \\mu(\\alpha^{s}(x), D_{2}(y^{'})))\\\\\n&= 0,\n\\end{align*}\nwhich implies that $\\nu^{'}(D_{1}, D_{2})(x) \\in Z(V)$, i.e., $\\nu^{'}(C(V), QC(V)) \\subseteq End(V, Z(V))$.\n\nFurthermore, if $Z(V) = \\{0\\}$, it's clearly that $\\nu^{'}(C(V), QC(V)) = \\{0\\}$.\n\\end{proof}\n\\begin{thm}\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra. Then $ZDer(V) = C(V) \\cap Der(V)$.\n\\end{thm}\n\\begin{proof}\nAssume that $D \\in C(V)_{\\alpha^{k}} \\cap Der_{\\alpha^{k}}(V)$. Then for any $x, y \\in V$, we have\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y))\\]\nsince $D \\in Der_{\\alpha^{k}}(V)$.\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) = \\mu(\\alpha^{k}(x), D(y))\\]\nsince $D \\in C_{\\alpha^{k}}(V)$.\n\nHence we have $D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) = 0$, which implies that $D \\in ZDer_{\\alpha^{k}}(V)$. Therefore, $C(V) \\cap Der(V) \\subseteq ZDer(V)$.\n\nOn the other hand, assume that $D \\in ZDer_{\\alpha^{k}}(V)$. Then for any $x, y \\in V$, we have\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) = 0,\\]\nhence we have\n\\[\\mu(\\alpha^{k}(x), D(y)) = \\mu(D(y), \\alpha^{k}(x)) = 0.\\]\nTherefore\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y))\\]\nwhich implies that $D \\in Der_{\\alpha^{k}}(V)$.\n\nAnd\n\\[D(\\mu(x, y)) = \\mu(D(x), \\alpha^{k}(y)) = \\mu(\\alpha^{k}(x), D(y))\\]\nwhich implies that $D \\in C_{\\alpha^{k}}(V)$.\n\nTherefore $D \\in C(V)_{\\alpha^{k}} \\cap Der_{\\alpha^{k}}(V)$, i.e., $ZDer(V) \\subseteq C(V) \\cap Der(V)$.\n\nHence, we have $ZDer(V) = C(V) \\cap Der(V)$.\n\\end{proof}\n\\begin{thm}\\label{thm:3.7}\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra. Then $(QC(V), \\nu, \\sigma)$ is a Hom-Jordan algebra.\n\\end{thm}\n\\begin{proof}\nAccording to Lemma \\ref{le:2.7} (1), we need only to show that $QC(V)$ is a Hom-subalgebra of $(\\mathcal{W}, \\nu, \\sigma)$.\n\nSuppose that $D_{1} \\in QC_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{s}}(V)$. Then for any $x, y \\in V$, we have\n\\begin{align*}\n&\\mu(\\sigma(D_{1})(x), \\alpha^{k + 1}(y)) = \\mu(\\alpha \\circ D_{1}(x), \\alpha^{k + 1}(y)) = \\alpha(\\mu(D_{1}(x), \\alpha^{k}(y)))\\\\\n&= \\alpha(\\mu(\\alpha^{k}(x), D_{1}(y))) = \\mu(\\alpha^{k + 1}(x), \\sigma(D_{1})(y)),\n\\end{align*}\nwhich implies that $\\sigma(D_{1}) \\in QC_{\\alpha^{k + 1}}(V)$.\n\\begin{align*}\n&\\mu(\\nu(D_{1}, D_{2})(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y)) + \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y))\\\\\n&= \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y))) + \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y)))\\\\\n&= \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}(y))) + \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{2}(D_{1}(y))) + \\mu(\\alpha^{k + s}(x), D_{1}(D_{2}(y)))\\\\\n&= \\mu(\\alpha^{k + s}(x), \\nu(D_{1}, D_{2})(y)),\n\\end{align*}\nwhich implies that $\\nu(D_{1}, D_{2}) \\in QC_{\\alpha^{k + s}}(V)$. Therefore, $QC(V)$ is a Hom-subalgebra of $(\\mathcal{W}, \\nu, \\sigma)$, i.e., $(QC(V), \\nu, \\sigma)$ is a Hom-Jordan algebra.\n\\end{proof}\n\\begin{thm}\nSuppose that $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra over a field $\\rm{F}$ of characteristic other than $2$.\n\\begin{enumerate}[(1)]\n\\item $(QC(V), \\nu^{'}, \\sigma)$ is a Hom-Lie algebra if and only if $(QC(V), \\iota, \\sigma)$ is a Hom-associative algebra where $\\iota(D_{1}, D_{2}) = D_{1} \\circ D_{2},\\quad\\forall D_{1}, D_{2} \\in QC(V)$.\n\\item If $\\alpha$ is a surjection and $Z(V) = \\{0\\}$, then $(QC(V), \\nu^{'}, \\sigma)$ is a Hom-Lie algebra if and only if $\\nu^{'}(QC(V), QC(V)) = \\{0\\}$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n(1). $(\\Leftarrow)$. For all $D_{1}, D_{2} \\in QC(V)$, we have $D_{1} \\circ D_{2} \\in QC(V)$, $D_{2} \\circ D_{1} \\in QC(V)$. So $\\nu^{'}(D_{1}, D_{2}) \\in QC(V)$. Moreover, $\\sigma(D_{1}) \\in QC(V)$. Therefore, $(QC(V), \\nu^{'}, \\sigma)$ is a Hom-Lie algebra.\n\n$(\\Rightarrow)$. For all $D_{1}, D_{2} \\in QC(V)$, we have $\\iota(D_{1}, D_{2}) = \\frac{1}{2}(\\nu^{'}(D_{1}, D_{2}) + \\nu(D_{1}, D_{2}))$. According to Theorem \\ref{thm:3.7}, we have $\\nu(D_{1}, D_{2}) \\in QC(V)$. Note that $\\nu^{'}(D_{1}, D_{2}) \\in QC(V)$, we have $\\iota(D_{1}, D_{2}) \\in QC(V)$. Hence, $(QC(V), \\iota, \\sigma)$ is a Hom-associative algebra.\n\n(2). $(\\Rightarrow)$. Suppose that $D_{1} \\in QC_{\\alpha^{k}}(V)$, $D_{2} \\in QC_{\\alpha^{s}}(V)$ and $x \\in V$. Since $\\alpha$ is a surjection, there exists $y^{'} \\in V$ such that $y = \\alpha^{k + s}(y^{'})$ for any $y \\in V$. Note that $(QC(V), \\nu^{'}, \\sigma)$ is a Hom-Lie algebra, then $\\nu^{'}(D_{1}, D_{2}) \\in QC_{\\alpha^{k + s}}(V)$. Then\n\\[\\mu(\\nu^{'}(D_{1}, D_{2})(x), y) = \\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y^{'})) = \\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y^{'})).\\]\n\nNote that\n\\begin{align*}\n&\\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y^{'}))\\\\\n&= \\mu(D_{1} \\circ D_{2}(x), \\alpha^{k + s}(y^{'})) - \\mu(D_{2} \\circ D_{1}(x), \\alpha^{k + s}(y^{'}))\\\\\n&= \\mu(\\alpha^{k}(D_{2}(x)), D_{1}(\\alpha^{s}(y^{'}))) - \\mu(\\alpha^{s}(D_{1}(x)), D_{2}(\\alpha^{k}(y^{'})))\\\\\n&= \\mu(D_{2}(\\alpha^{k}(x)), \\alpha^{s}(D_{1}(y^{'}))) - \\mu(D_{1}(\\alpha^{s}(x)), \\alpha^{k}(D_{2}(y^{'})))\\\\\n&= \\mu(\\alpha^{k + s}(x), D_{2} \\circ D_{1}(y^{'})) - \\mu(\\alpha^{k + s}(x), D_{1} \\circ D_{2}(y^{'}))\\\\\n&= -\\mu(\\alpha^{k + s}(x), \\nu^{'}(D_{1}, D_{2})(y^{'})).\n\\end{align*}\nHence, we have $\\mu(\\nu^{'}(D_{1}, D_{2})(x), y) = \\mu(\\nu^{'}(D_{1}, D_{2})(x), \\alpha^{k + s}(y^{'})) = 0$, which implies that $\\nu^{'}(D_{1}, D_{2})(x) \\in Z(V)$. Note that $Z(V) = \\{0\\}$,  we have $\\nu^{'}(D_{1}, D_{2})(x) = 0$, i.e., $\\nu^{'}(D_{1}, D_{2}) = 0$. Therefore, $\\nu^{'}(QC(V), QC(V)) = \\{0\\}$.\n\n$(\\Leftarrow)$. Obviously.\n\\end{proof}\n\n\\section{The quasiderivations of Hom-Jordan algebras}\\label{se:4}\n\nIn this section, we will prove that the quasiderivations of $(V, \\mu, \\alpha)$ can be embedded as derivations in a larger Hom-Jordan agebra and obtain a direct sum decomposition of $Der(\\breve{V})$ when the centralizer of $(V, \\mu, \\alpha)$ is equal to zero.\n\\begin{prop}\nSuppose that $(V, \\mu, \\alpha)$ is a Hom-Jordan algebra over $\\rm{F}$ and $t$ is an indeterminate. We define $\\breve{V} := \\{\\sum(x \\otimes t + y \\otimes t^{2}) | x, y \\in V\\}$, $\\breve{\\alpha}(\\breve{V}) := \\{\\sum(\\alpha(x) \\otimes t + \\alpha(y) \\otimes t^{2}) | x, y \\in V\\}$ and $\\breve{\\mu}(x \\otimes t^{i},  y \\otimes t^{j}):= \\mu(x,y) \\otimes t^{i + j}$ where $x, y \\in V, i, j \\in \\{1, 2\\}$. Then $(\\breve{V}, \\breve{\\mu}, \\breve{\\alpha})$ is a Hom-Jordan algebra.\n\\end{prop}\n\\begin{proof}\nIt's obvious that $\\breve{\\mu}$ is a bilinear map and commutative since $\\mu$ is a bilinear map and commutative.\n\nFor any $x \\otimes t^{i},  y \\otimes t^{j} \\in \\breve{V}$, we have\n\\begin{align*}\n&\\breve{\\mu}(\\breve{\\alpha}^{2}(x \\otimes t^{i}), \\breve{\\mu}(y \\otimes t^{j}, \\breve{\\mu}(x \\otimes t^{i}, x \\otimes t^{i})))\\\\\n&= \\mu(\\alpha^{2}(x), \\mu(y, \\mu(x, x))) \\otimes t^{3i + j}\\\\\n&= \\mu(\\mu(\\alpha(x), y), \\alpha(\\mu(x, x))) \\otimes t^{3i + j}\\\\\n&= \\breve{\\mu}(\\breve{\\mu}(\\breve{\\alpha}(x \\otimes t^{i}), y \\otimes t^{j}), \\breve{\\alpha}(\\breve{\\mu}(x \\otimes t^{i}, x \\otimes t^{i}))).\n\\end{align*}\nTherefore, $(\\breve{V}, \\breve{\\mu}, \\breve{\\alpha})$ is a Hom-Jordan algebra.\n\\end{proof}\n\nFor convenience, we write $xt(xt^{2})$ in place of $x \\otimes t(x \\otimes t^{2})$.\n\nIf $U$ is a subspace of $V$ such that $V = U \\dotplus \\mu(V, V)$, then\n\\[\\breve{V} = Vt + Vt^{2} = Vt + Ut^{2} + \\mu(V, V)t^{2}.\\]\nNow we define a map $\\varphi : QDer(V) \\rightarrow End(\\breve{V})$ satisfying\n\\[\\varphi(D)(at + ut^{2} + bt^{2}) = D(a)t + D^{'}(b)t^{2},\\]\nwhere $D \\in QDer_{\\alpha^{k}}(V)$, and $D^{'}$ is in Definition \\ref{defn:2.11} (2), $a \\in V, u \\in U, b \\in \\mu(V, V)$.\n\\begin{prop}\\label{prop:4.2}\n$V, \\breve{V}, \\varphi$ are defined as above. Then\n\\begin{enumerate}[(1)]\n\\item $\\varphi$ is injective and $\\varphi(D)$ does not depend on the choice of $D^{'}$.\n\\item $\\varphi(QDer(V)) \\subseteq Der(\\breve{V})$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). If $\\varphi(D_{1}) = \\varphi(D_{2})$, then for all $a \\in V, u \\in U, b \\in \\mu(V, V)$, we have\n\\[\\varphi(D_{1})(at + ut^{2} + bt^{2}) = \\varphi(D_{2})(at + ut^{2} + bt^{2}),\\]\nwhich implies that\n\\[D_{1}(a)t + D_{1}^{'}(b)t^{2} = D_{2}(a)t + D_{2}^{'}(b)t^{2}.\\]\nHence we have\n\\[D_{1}(a) = D_{2}(a),\\]\nwhich implies that $D_{1} = D_{2}$. Therefore, $\\varphi$ is injective.\n\nSuppose that there exists $D^{''}$ such that $\\varphi(D)(at + ut^{2} + bt^{2}) = D(a)t + D^{''}(b)t^{2}$ and $\\mu(D(x), \\alpha^{k}(y)) + \\mu(\\alpha^{k}(x), D(y)) = D^{''}(\\mu(x, y))$, then we have\n\\[D^{''}(\\mu(x, y)) = D^{'}(\\mu(x, y)),\\]\nwhich implies that $D^{''}(b) = D^{'}(b)$.\n\nHence, we have\n\\[\\varphi(D)(at + ut^{2} + bt^{2}) = D(a)t + D^{''}(b)t^{2} = D(a)t + D^{'}(b)t^{2}.\\]\nThat is to say $\\varphi(D)$ does not depend on the choice of $D^{'}$.\n\n(2). $\\forall i + j \\geq 3$, we have $\\breve{\\mu}(x \\otimes t^{i}, y \\otimes t^{j}) = \\mu(x, y) \\otimes t^{i + j} = 0$. Hence, we need only to show that\n\\[\\varphi(D)(\\breve{\\mu}(xt, yt)) = \\breve{\\mu}(\\varphi(D)(xt), \\breve{\\alpha}^{k}(yt)) + \\breve{\\mu}(\\breve{\\alpha}^{k}(xt), \\varphi(D)(yt)).\\]\n\nFor all $x, y \\in V$, we have\n\\begin{align*}\n&\\varphi(D)(\\breve{\\mu}(xt, yt))\\\\\n&= \\varphi(D)(\\mu(x, y)t^{2}) = D^{'}(\\mu(x, y))t^{2}\\\\\n&= \\mu(D(x), \\alpha^{k}(y))t^{2} + \\mu(\\alpha^{k}(x), D(y))t^{2}\\\\\n&= \\breve{\\mu}(D(x)t, \\alpha^{k}(y)t) + \\breve{\\mu}(\\alpha^{k}(x)t, D(y)t)\\\\\n&= \\breve{\\mu}(\\varphi(D)(xt), \\breve{\\alpha}^{k}(yt)) + \\breve{\\mu}(\\breve{\\alpha}^{k}(xt), \\varphi(D)(yt)).\n\\end{align*}\nHence, $\\varphi(D) \\in Der_{\\breve{\\alpha}^{k}}(\\breve{V})$. Therefore, $\\varphi(QDer(V)) \\subseteq Der(\\breve{V})$.\n\\end{proof}\n\\begin{prop}\nLet $(V, \\mu, \\alpha)$ be a Hom-Jordan algebra. $Z(V) = \\{0\\}$ and $\\breve{V}, \\varphi$ are defined as above. Then $Der(\\breve{V}) = \\varphi(QDer(V)) \\dotplus ZDer(\\breve{V})$.\n\\end{prop}\n\\begin{proof}\nAssume that $xt + yt^{2} \\in Z(\\breve{V})$, then for any $x^{'}t + y^{'}t^{2} \\in \\breve{V}$, we have\n\\[0 = \\breve{\\mu}(xt + yt^{2}, x^{'}t + y^{'}t^{2}) = \\mu(x, x^{'})t^{2},\\]\nwhich implies that $\\mu(x, x^{'}) = 0$. Note that $Z(V) = \\{0\\}$, we have $x = 0$. Hence, $Z(\\breve{V}) \\subseteq Vt^{2}$. Obviously, $Vt^{2} \\subseteq Z(\\breve{V})$. Therefore, $Z(\\breve{V}) = Vt^{2}$.\n\nSuppose that $g \\in Der_{\\breve{\\alpha}^{k}}(\\breve{V})$, $at^{2} \\in Z(\\breve{V})$. Since $\\alpha$ is surjective, $\\breve{\\alpha}$ is also surjective. For any $xt + yt^{2} \\in \\breve{V}$, there exists $x^{'}t + y^{'}t^{2} \\in \\breve{V}$ such that $xt + yt^{2} = \\breve{\\alpha}^{k}(x^{'}t + y^{'}t^{2})$.\n\\begin{align*}\n&\\breve{\\mu}(g(at^{2}), xt + yt^{2}) = \\breve{\\mu}(g(at^{2}), \\breve{\\alpha}^{k}(x^{'}t + y^{'}t^{2}))\\\\\n&= g(\\breve{\\mu}(at^{2}, x^{'}t + y^{'}t^{2})) - \\breve{\\mu}(\\breve{\\alpha}^{k}(at^{2}), g(x^{'}t + y^{'}t^{2}))\\\\\n&= 0,\n\\end{align*}\nwhich implies that $g(at^{2}) \\in Z(\\breve{V})$. Therefore, $g(Z(\\breve{V})) \\subseteq Z(\\breve{V})$.\n\nHence, we have $g(Ut^{2}) \\subseteq g(Z(\\breve{V})) \\subseteq Z(\\breve{V}) = Vt^{2}$.\n\nNow we define a map $f : Vt + Ut^{2} + \\mu(V, V)t^{2} \\rightarrow Vt^{2}$ by\n$$f(x) =\n\\left\\{\n\\begin{aligned}\ng(x) \\cap Vt^{2},\\quad x \\in Vt\\\\\ng(x),\\quad x \\in Ut^{2}\\\\\n0,\\quad x \\in \\mu(V, V)t^{2}\n\\end{aligned}\n\\right.\n$$\nIt's clear that $f$ is linear.\n\nNote that\n\\[f(\\breve{\\mu}(\\breve{V}, \\breve{V})) = f(\\mu(V, V)t^{2}) = 0,\\]\n\\[\\breve{\\mu}(f(\\breve{V}), \\breve{\\alpha}^{k}(\\breve{V})) \\subseteq \\breve{\\mu}(Vt^{2}, \\alpha^{k}(V)t + \\alpha^{k}(V)t^{2}) = 0,\\]\nwe have $f \\in ZDer_{\\breve{\\alpha}^{k}}(\\breve{V})$.\n\nSince\n\\[(g - f)(Vt) = g(Vt) - f(Vt) = g(Vt) - g(Vt) \\cap Vt^{2} = g(Vt) - Vt^{2} \\subseteq Vt,\\]\n\\[(g - f)(Ut^{2}) = 0,\\]\n\\[(g - f)(\\mu(V, V)t^{2}) = g(\\breve{\\mu}(\\breve{V}, \\breve{V})) \\subseteq \\breve{\\mu}(\\breve{V}, \\breve{V}) = \\mu(V, V)t^{2},\\]\nhence there exist $D, D^{'} \\in End (V)$ such that $\\forall a \\in V, b \\in \\mu(V, V)$,\n\\[(g - f)(at) = D(a)t, (g - f)(bt^{2}) = D^{'}(b)t^{2}.\\]\n\nSince $g - f \\in Der_{\\breve{\\alpha}^{k}}(\\breve{V})$, we have\n\\[(g - f)(\\breve{\\mu}(a_{1}t, a_{2}t)) = \\breve{\\mu}((g - f)(a_{1}t), \\breve{\\alpha}^{k}(a_{2}t)) + \\breve{\\mu}(\\breve{\\alpha}^{k}(a_{1}t), (g - f)(a_{2}t)),\\]\nwhich implies that\n\\[D^{'}(\\mu(a_{1}, a_{2}))t^{2} = \\mu(D(a_{1}), \\alpha^{k}(a_{2}))t^{2} + \\mu(\\alpha^{k}(a_{1}), D(a_{2}))t^{2}.\\]\nHence, we have $D \\in QDer_{\\alpha^{k}}(V) \\subseteq QDer(V)$.\n\nTherefore, $g - f = \\varphi(D) \\in \\varphi(QDer(V))$, which implies that $Der(\\breve{V}) \\subseteq \\varphi(QDer(V)) + ZDer(\\breve{V})$. According to Proposition \\ref{prop:4.2} (2), we have $Der(\\breve{V}) = \\varphi(QDer(V)) + ZDer(\\breve{V})$.\n\n$\\forall f \\in \\varphi(QDer(V)) \\cap ZDer(\\breve{V})$, there exists $D \\in QDer(V)$ such that $f = \\varphi(D)$. For all $a \\in V, b \\in \\mu(V, V), u \\in U$,\n\\[f(at + bt^{2} + ut^{2}) = \\varphi(D)(at + bt^{2} + ut^{2}) = D(a)t + D^{'}(b)t^{2}.\\]\n\nOn the other hand, for any $xt + yt^{2} \\in \\breve{V}$, there exists $x^{'}t + y^{'}t^{2} \\in \\breve{V}$ such that $xt + yt^{2} = \\breve{\\alpha}^{k}(x^{'}t + y^{'}t^{2})$ since $\\breve{\\alpha}$ is surjective. Then we have\n\\begin{align*}\n&\\breve{\\mu}(f(at + bt^{2} + ut^{2}), xt + yt^{2}) = \\breve{\\mu}(f(at + bt^{2} + ut^{2}), \\breve{\\alpha}^{k}(x^{'}t + y^{'}t^{2}))\\\\\n&= f(\\breve{\\mu}(at + bt^{2} + ut^{2}, x^{'}t + y^{'}t^{2}))\\\\\n&= 0\n\\end{align*}\nsince $f \\in ZDer(\\breve{V})$. Hence, $f(at + bt^{2} + ut^{2}) \\in Z(\\breve{V}) = Vt^{2}$.\n\nTherefore, $D(a) = 0$, i.e., $D = 0$. Hence, $f = 0$.\n\nTherefore, $Der(\\breve{V}) = \\varphi(QDer(V)) \\dotplus ZDer(\\breve{V})$.\n\\end{proof}\n\n\\section{The centroids of Hom-Jordan algebras}\\label{se:5}\n\n\\begin{prop}\nSuppose that $(V, \\mu, \\alpha)$ is a simple multiplicative Hom-Jordan algebra over an algebraically closed field $\\rm{\\mathbb{F}}$ of characteristic $0$. If $C(V) = \\rm{\\mathbb{F}}id$, then $\\alpha = id$.\n\\end{prop}\n\\begin{proof}\nFor all $k \\in \\mathbb{N}^{+}$, $\\forall\\; 0 \\neq \\psi \\in C_{\\alpha^{k}}(V)$, we have $\\psi = p\\;id,\\quad p \\in \\rm{\\mathbb{F}},\\;p \\neq 0$. So $\\forall x, y \\in V$,\n\\[p\\mu(x, y) = \\psi(\\mu(x, y)) = \\mu(\\psi(x), \\alpha^{k}(y)) = \\mu(px, \\alpha^{k}(y)) = p\\mu(x, \\alpha^{k}(y)),\\]\nwhich implies that $\\mu(x, y) = \\mu(x, \\alpha^{k}(y))$ for all $k \\in \\mathbb{N}^{+}$.\n\nSince $\\rm{\\mathbb{F}}$ is algebraically closed, $\\alpha$ has an eigenvalue $\\lambda$. We denote the corresponding eigenspace by $E_{\\lambda}(\\alpha)$. So $E_{\\lambda}(\\alpha) \\neq 0$. Let $k = 1$. For any $x \\in E_{\\lambda}(\\alpha)$, $y \\in V$, we have\n\\[\\alpha(\\mu(x, y)) = \\mu(\\alpha(x), \\alpha(y)) = \\mu(\\lambda x, \\alpha(y)) = \\lambda\\mu(x, \\alpha(y)) = \\lambda\\mu(x, y),\\]\nwhich implies that $\\mu(x, y) \\in E_{\\lambda}(\\alpha)$.\n\nMoreover, for any $x \\in E_{\\lambda}(\\alpha)$, we have\n\\[\\alpha(\\alpha(x)) = \\lambda^{2}x = \\lambda\\alpha(x),\\]\nwhich implies that $\\alpha(x) \\in E_{\\lambda}(\\alpha)$, i.e., $\\alpha(E_{\\lambda}(\\alpha)) \\subseteq E_{\\lambda}(\\alpha)$.\n\nSo $E_{\\lambda}(\\alpha)$ is a Hom-ideal of $(V, \\mu, \\alpha)$. Since $(V, \\mu, \\alpha)$ is simple, we have $E_{\\lambda}(\\alpha) = V$, i.e., $\\alpha = \\lambda id$.\n\nThen for any $x, y \\in V, k = 1$, we have\n\\[\\mu(x, y) = \\mu(x, \\alpha(y)) = \\mu(x, \\lambda y) = \\lambda\\mu(x, y),\\]\nwhich implies that $\\lambda = 1$, i.e., $\\alpha = id$.\n\\end{proof}\n\\begin{prop}\nLet $(V, \\mu, \\alpha)$ be a multiplicative Hom-Jordan algebra.\n\\begin{enumerate}[(1)]\n\\item If $\\alpha$ is a surjection, then $V$ is indecomposable if and only if $C(V)$ does not contain idempotents except $0$ and $id$.\n\\item If $(V, \\mu, \\alpha)$ is perfect(i.e., $V = \\mu(V, V)$), then every $\\psi \\in C_{\\alpha^{k}}(V)$ is $\\alpha^{k}$-symmetric with respect to any invariant form on $V$.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\n(1). $(\\Rightarrow)$. If there exists $\\psi \\in C_{\\alpha^{k}}(V)$ is an idempotent and satisfies that $\\psi \\neq 0,\\;id$, then $\\psi^{2}(x) = \\psi(x),\\quad x \\in V$.\n\nFor any $x \\in ker(\\psi), y \\in V$, we have\n\\[\\psi(\\mu(x, y)) = \\mu(\\psi(x), \\alpha^{k}(y)) = \\mu(0, \\alpha^{k}(y)) = 0,\\]\nwhich implies that $\\mu(x, y) \\in ker(\\psi)$.\nMoreover, $\\psi(\\alpha(x)) = \\alpha(\\psi(x)) = \\alpha(0) = 0$, so $\\alpha(x) \\in ker(\\psi)$, i.e., $\\alpha(ker(\\psi)) \\subseteq ker(\\psi)$.\n\nHence, $ker(\\psi)$ is a Hom-ideal of $(V, \\mu, \\alpha)$.\n\nFor any $y \\in Im(\\psi)$, there exists $y^{'} \\in V$ such that $y = \\psi(y^{'})$. And for any $z \\in V$, there exists $z^{'} \\in V$ such that $z = \\alpha^{k}(z^{'})$ since $\\alpha$ is surjective. Then\n\\[\\mu(y, z) = \\mu(\\psi(y^{'}), \\alpha^{k}(z^{'})) = \\psi(\\mu(y^{'}, z^{'})),\\]\nwhich implies that $\\mu(y, z) \\in Im(\\psi)$.\nMoreover, $\\alpha(y) = \\alpha(\\psi(y^{'})) = \\psi(\\alpha(y^{'}))$, so $\\alpha(y) \\in Im(\\psi)$, i.e., $\\alpha(Im(\\psi)) \\subseteq Im(\\psi)$.\n\nHence, $Im(\\psi)$ is a Hom-ideal of $(V, \\mu, \\alpha)$.\n\n$\\forall x \\in ker(\\psi) \\cap Im(\\psi)$, $\\exists x^{'} \\in V$ such that $x = \\psi(x^{'})$. So $0 = \\psi(x) = \\psi^{2}(y) = \\psi(y)$, which implies that $x = 0$. Hence, $ker(\\psi) \\cap Im(\\psi) = \\{0\\}$.\n\nWe have a decomposition $x = (x - \\psi(x)) + \\psi(x)$, $\\forall x \\in V$. So we have $V = ker(\\psi) \\oplus Im(\\psi)$. Contradiction.\n\n$(\\Leftarrow)$. Suppose that $V = V_{1} \\oplus V_{2}$ where $V_{i}$ are Hom-ideals of $(V, \\mu, \\alpha)$. Then for any $x \\in V$, $\\exists x_{i} \\in V_{i}$ such that $x = x_{1} + x_{2}$.\n\nDefine $\\psi : V \\rightarrow V$ by $\\psi(x) = \\psi(x_{1} + x_{2}) = x_{1} - x_{2}$. It's obvious that $\\psi^{2}(x) = \\psi(x)$.\n\nFor any $x, y \\in V$, suppose that $x = x_{1} + x_{2},\\;y = y_{1} + y_{2}$,\n\\[\\psi(\\mu(x, y)) = \\psi(\\mu(x_{1} + x_{2}, y_{1} + y_{2})) = \\psi(\\mu(x_{1}, y_{1}) + \\mu(x_{2}, y_{2})) = \\mu(x_{1}, y_{1}) - \\mu(x_{2}, y_{2}),\\]\n\\[\\mu(\\psi(x), y) = \\mu(\\psi(x_{1} + x_{2}), y_{1} + y_{2}) = \\mu(x_{1} - x_{2}, y_{1} + y_{2}) = \\mu(x_{1}, y_{1}) - \\mu(x_{2}, y_{2}),\\]\nwe have\n\\[\\psi(\\mu(x, y)) = \\mu(\\psi(x), y).\\]\nSimilarly, we have $\\psi(\\mu(x, y)) = \\mu(x, \\psi(y))$. So $\\psi \\in C_{\\alpha^{0}}(V) \\subseteq C(V)$. Contradiction.\n\n(2). Let $f$ be an invariant $\\rm{F}$-bilinear form on $V$. Then we have $f(\\mu(x, y), z) = f(x, \\mu(y, z))$ for any $x, y, z \\in V$.\n\nSince $(V, \\mu, \\alpha)$ is perfect,let $\\psi \\in C_{\\alpha^{k}}(V)$, then for any $a, b, c \\in V$ we have\n\\begin{align*}\n&f(\\psi(\\mu(a, b)), \\alpha^{k}(c)) = f(\\mu(\\alpha^{k}(a), \\psi(b)), \\alpha^{k}(c)) = f(\\alpha^{k}(a), \\mu(\\psi(b), \\alpha^{k}(c)))\\\\\n&= f(\\alpha^{k}(a), \\mu(\\alpha^{k}(b), \\psi(c))) = f(\\mu(\\alpha^{k}(a), \\alpha^{k}(b)), \\psi(c)) = f(\\alpha^{k}(\\mu(a, b)), \\psi(c)).\n\\end{align*}\n\\end{proof}\n\\begin{prop}\nLet $(V, \\mu, \\alpha)$ be a Hom-Jordan algebra and $I$ an $\\alpha$-invariant subspace of $V$ where $\\alpha|_{I}$ is surjective. Then $Z_{V}(I) = \\{x \\in V | \\mu(x, y) = 0,\\;\\forall y \\in I\\}$ is invariant under $C(V)$. So is any perfect Hom-ideal of $(V, \\mu, \\alpha)$.\n\\end{prop}\n\\begin{proof}\nFor any $\\psi \\in C_{\\alpha^{k}}(V)$ and $x \\in Z_{V}(I)$, $\\forall y \\in I$, $\\exists y^{'} \\in I$ such that $y = \\alpha^{k}(y^{'})$ since $\\alpha|_{I}$ is surjective, we have\n\\[\\mu(\\psi(x), y) = \\mu(\\psi(x), \\alpha^{k}(y^{'})) = \\psi(\\mu(x, y^{'})) = \\psi(0) = 0,\\]\nwhich implies that $\\psi(x) \\in Z_{V}(I)$. So $Z_{V}(I)$ is invariant under $C(V)$.\n\nSuppose that $J$ is a perfect Hom-ideal of $(V, \\mu, \\alpha)$. Then $J = \\mu(J, J)$. For any $y \\in J$, there exist $a, b \\in J$ such that $y = \\mu(a, b)$, then we have\n\\[\\psi(y) = \\psi(\\mu(a, b)) = \\mu(\\psi(a), \\alpha^{k}(b)) \\in \\mu(J, J) = J.\\]\nSo $J$ is invariant under $C(V)$.\n\\end{proof}\n\\begin{thm}\nSuppose that $(V_{1}, \\mu_{1}, \\alpha_{1})$ and $(V_{2}, \\mu_{2}, \\alpha_{2})$ are two Hom-Jordan algebras over field $\\rm{F}$ with $\\alpha_{1}$ is surjective. Let $\\pi : V_{1} \\rightarrow V_{2}$ be an epimorphism of Hom-Jordan algebras. Then for any $f \\in End_{\\rm{F}}(V_{1}, ker(\\pi)) := \\{g \\in \\mathcal{W}_{1} | g(ker(\\pi)) \\subseteq ker(\\pi)\\}$, there exists a unique $\\bar{f} \\in \\mathcal{W}_{2}$ satisfying $\\pi \\circ f = \\bar{f} \\circ \\pi$ where $\\mathcal{W}_{i}(i = 1, 2)$ are defined as Lemma \\ref{le:2.7}. Moreover, the following results hold:\n\\begin{enumerate}[(1)]\n\\item The map $\\pi_{End} : End_{\\rm{F}}(V_{1}, ker(\\pi)) \\rightarrow \\mathcal{W}_{2}$, $f \\mapsto \\bar{f}$ is a Hom-algebra homomorphism with the following proporties, where $(End_{\\rm{F}}(V_{1}, ker(\\pi)), \\circ, \\sigma_{1})$ and $(\\mathcal{W}_{2}, \\circ, \\sigma_{2})$ are two Hom-associative algebras and $\\sigma_{1} : End_{\\rm{F}}(V_{1}, ker(\\pi)) \\rightarrow End_{\\rm{F}}(V_{1}, ker(\\pi)),\\;f \\mapsto \\alpha_{1} \\circ f$ and $\\sigma_{2} : \\mathcal{W}_{2} \\rightarrow \\mathcal{W}_{2},\\;g \\mapsto \\alpha_{2} \\circ g$.\n    \\begin{enumerate}[(a)]\n    \\item $\\pi_{End}(Mult(V_{1})) = Mult(V_{2})$, $\\pi_{End}(C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi))) \\subseteq C(V_{2})$.\n    \\item By restriction, there is a Hom-algebra homomorphism\n         \\[\\pi_{C} : C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi)) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}.\\]\n    \\item If $ker(\\pi) = Z(V_{1})$, then every $\\varphi \\in C(V_{1})$ leaves $ker(\\pi)$ invariant.\n    \\end{enumerate}\n\\item Suppose that $V_{1}$ is perfect and $ker(\\pi) \\subseteq Z(V_{1})$. Then $\\pi_{C} : C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi)) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}$ is injective.\n\\item If $V_{1}$ is perfect, $Z(V_{2}) = \\{0\\}$ and $ker(\\pi) \\subseteq Z(V_{1})$, then $\\pi_{C} : C(V_{1}) \\rightarrow C(V_{2})$ is a Hom-algebra homomorphism.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\nFor any $y \\in V_{2}$, there exists $x \\in V_{1}$ such that $y = \\pi(x)$ since $\\pi$ is an epimorphism. Then for any $f \\in End_{\\rm{F}}(V_{1}, ker(\\pi))$, define $\\bar{f} : V_{2} \\rightarrow V_{2}$ by $\\bar{f}(y) = \\pi(f(x))$ where $x$ satisfies that $y = \\pi(x)$. It's obvious that $\\pi \\circ f = \\bar{f} \\circ \\pi$.\n\\begin{align*}\n&(\\alpha_{2} \\circ \\bar{f}) \\circ \\pi = \\alpha_{2} \\circ (\\bar{f} \\circ \\pi) = \\alpha_{2} \\circ (\\pi \\circ f) = (\\alpha_{2} \\circ \\pi) \\circ f = (\\pi \\circ \\alpha_{1}) \\circ f = \\pi \\circ (\\alpha_{1} \\circ f)\\\\\n&= \\pi \\circ (f \\circ \\alpha_{1}) = (\\pi \\circ f) \\circ \\alpha_{1} = (\\bar{f} \\circ \\pi) \\circ \\alpha_{1} = \\bar{f} \\circ (\\pi \\circ \\alpha_{1}) = \\bar{f} \\circ (\\alpha_{2} \\circ \\pi) = (\\bar{f} \\circ \\alpha_{2}) \\circ \\pi.\n\\end{align*}\nSince $\\pi$ is surjective, we have $\\alpha_{2} \\circ \\bar{f} = \\bar{f} \\circ \\alpha_{2}$, i.e., $\\bar{f} \\in \\mathcal{W}_{2}$.\n\nIf there exist $\\bar{f}$ and $\\bar{f}^{'}$ such that $\\pi \\circ f = \\bar{f} \\circ \\pi$ and $\\pi \\circ f = \\bar{f}^{'} \\circ \\pi$, then we have $\\bar{f} \\circ \\pi = \\bar{f}^{'} \\circ \\pi$. For any $y \\in V_{2}$, there exists $x \\in V_{1}$ such that $y = \\pi(x)$. Hence, $\\bar{f}(y) = \\bar{f}(\\pi(x)) = \\bar{f}^{'}(\\pi(x)) = \\bar{f}^{'}(y)$. So $\\bar{f} = \\bar{f}^{'}$.\n\n(1). For all $f, g \\in End_{\\rm{F}}(V_{1}, ker(\\pi))$, we have\n\\[\\pi \\circ (f \\circ g) = (\\pi \\circ f) \\circ g = (\\bar{f} \\circ \\pi) \\circ g = \\bar{f} \\circ (\\pi \\circ g) = \\bar{f} \\circ (\\bar{g} \\circ \\pi) = (\\bar{f} \\circ \\bar{g}) \\circ \\pi,\\]\nwhich implies that $\\pi_{End}(f \\circ g) = \\bar{f} \\circ \\bar{g} = \\pi_{End}(f) \\circ \\pi_{End}(g)$.\n\\[\\pi \\circ (\\alpha_{1} \\circ f) = (\\pi \\circ \\alpha_{1}) \\circ f = (\\alpha_{2} \\circ \\pi) \\circ f = \\alpha_{2} \\circ (\\pi \\circ f) = \\alpha_{2} \\circ (\\bar{f} \\circ \\pi) = (\\alpha_{2} \\circ \\bar{f}) \\circ \\pi,\\]\nwhich implies that $\\pi_{End}(\\sigma_{1}(f)) = \\alpha_{2} \\circ \\bar{f} = \\sigma_{2}(\\pi_{End}(f))$, i.e., $\\pi_{End} \\circ \\sigma_{1} = \\sigma_{2} \\circ \\pi_{End}$. Therefore, $\\pi_{End}$ is a Hom-algebra homomorphism.\n\n(a). For all $x \\in V_{1}$, $z \\in ker(\\pi)$, we have\n\\[\\pi(L_{x}(z)) = \\pi(\\mu_{1}(x, z)) = \\mu_{2}(\\pi(x), \\pi(z)) = \\mu_{2}(\\pi(x), 0) = 0,\\]\nwhich implies that $L_{x}(z) \\in ker(\\pi)$, i.e., $L_{x} \\in End_{\\rm{F}}(V_{1}, ker(\\pi))$.\n\nMoreover, we have $\\pi \\circ L_{x} = L_{\\pi(x)} \\circ \\pi$. So $\\pi_{End}(L_{x}) = L_{\\pi(x)} \\in Mult(V_{2})$. Hence, $\\pi_{End}(Mult(V_{1})) \\subseteq Mult(V_{2})$.\n\nOn the other hand, for any $L_{y} \\in Mult(V_{2})$, $\\exists x \\in V_{1}$ such that $y = \\pi(x)$. So\n\\[L_{y} = L_{\\pi(x)} = \\pi_{End}(L_{x}) \\in \\pi_{End}(Mult(V_{1})),\\]\nwhich implies that $Mult(V_{2}) \\subseteq \\pi_{End}(Mult(V_{1}))$. Therefore, $\\pi_{End}(Mult(V_{1})) = Mult(V_{2})$.\n\nFor any $\\varphi \\in C_{\\alpha_{1}^{k}}(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi))$, $\\forall x^{'}, y^{'}\\in V_{2}$, $\\exists x, y \\in V_{1}$ such that $x^{'} = \\pi(x),\\; y^{'} = \\pi(y)$, then we have\n\\begin{align*}\n&\\bar{\\varphi}(\\mu_{2}(x^{'}, y^{'})) = \\bar{\\varphi}(\\mu_{2}(\\pi(x), \\pi(y))) = \\bar{\\varphi}(\\pi(\\mu_{1}(x, y))) = \\pi(\\varphi(\\mu_{1}(x, y))) = \\pi(\\mu_{1}(\\varphi(x), \\alpha_{1}^{k}(y)))\\\\\n&= \\mu_{2}(\\pi(\\varphi(x)), \\pi(\\alpha_{1}^{k}(y))) = \\mu_{2}(\\bar{\\varphi}(\\pi(x)), \\alpha_{2}^{k}(\\pi(y))) = \\mu_{2}(\\bar{\\varphi}(x^{'}), \\alpha_{2}^{k}(y^{'})).\n\\end{align*}\nSimilarly, we have $\\bar{\\varphi}(\\mu_{2}(x^{'}, y^{'})) = \\mu_{2}(\\alpha_{2}^{k}(x^{'}), \\bar{\\varphi}(y^{'}))$. Hence, $\\bar{\\varphi} \\in C_{\\alpha_{2}^{k}}(V_{2})$. Therefore, $\\pi_{End}(C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi))) \\subseteq C(V_{2})$.\n\n(b). $\\forall f, g \\in C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi))$, we have\n\\[\\pi \\circ (f \\circ g) = (\\pi \\circ f) \\circ g = (\\bar{f} \\circ \\pi) \\circ g = \\bar{f} \\circ (\\pi \\circ g) = \\bar{f} \\circ (\\bar{g} \\circ \\pi) = (\\bar{f} \\circ \\bar{g}) \\circ \\pi,\\]\nwhich implies that $\\pi_{C}(f \\circ g) = \\bar{f} \\circ \\bar{g} = \\pi_{C}(f) \\circ \\pi_{C}(g)$.\n\\[\\pi \\circ (\\alpha_{1} \\circ f) = (\\pi \\circ \\alpha_{1}) \\circ f = (\\alpha_{2} \\circ \\pi) \\circ f = \\alpha_{2} \\circ (\\pi \\circ f) = \\alpha_{2} \\circ (\\bar{f} \\circ \\pi) = (\\alpha_{2} \\circ \\bar{f}) \\circ \\pi,\\]\nwhich implies that $\\pi_{C}(\\sigma_{1}(f)) = \\alpha_{2} \\circ \\bar{f} = \\sigma_{2}(\\pi_{C}(f))$, i.e., $\\pi_{C} \\circ \\sigma_{1} = \\sigma_{2} \\circ \\pi_{C}$. Therefore, $\\pi_{C}$ is a Hom-algebra homomorphism.\n\n(c). If $ker(\\pi) = Z(V_{1})$, $\\forall x \\in ker(\\pi), \\varphi \\in C_{\\alpha_{1}^{k}}(V_{1})$, for any $y \\in V_{1}$ there exists $y^{'} \\in V_{1}$ such that $y = \\alpha_{1}^{k}(y^{'})$ since $\\alpha_{1}$ is surjective. We have\n\\[\\mu_{1}(\\varphi(x), y) = \\mu_{1}(\\varphi(x), \\alpha_{1}^{k}(y^{'})) = \\varphi(\\mu_{1}(x, y^{'})) = \\varphi(0) = 0,\\]\nwhich implies that $\\varphi(x) \\in Z(V_{1}) = ker(\\pi)$. Hence, every $\\varphi \\in C(V_{1})$ leaves $ker(\\pi)$ invariant.\n\n(2). If $\\bar{\\varphi} = 0$ for $\\varphi \\in C_{\\alpha_{1}^{k}}(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi))$, then $\\pi(\\varphi(V_{1})) = \\bar{\\varphi}(\\pi(V_{1})) = 0$. Hence, $\\varphi(V_{1}) \\subseteq ker(\\pi) \\subseteq Z(V_{1})$.\n\nHence, $\\varphi(\\mu_{1}(x, y)) = \\mu_{1}(\\varphi(x), \\alpha_{1}^{k}(y)) = 0$. Note that $V_{1}$ is perfect, we have $\\varphi = 0$. Therefore, $\\pi_{C} : C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi)) \\rightarrow C(V_{2}),\\quad f \\mapsto \\bar{f}$ is injective.\n\n(3). $\\forall y \\in V_{2}, \\exists y^{'} \\in V_{1}$ such that $y = \\pi(y^{'})$. For all $x \\in Z(V_{1})$,\n\\[\\mu_{2}(\\pi(x), y) = \\mu_{2}(\\pi(x), \\pi(y^{'})) = \\pi(\\mu_{1}(x, y^{'})) = \\pi(0) = 0,\\]\nwhich implies that $\\pi(x) \\in Z(V_{2})$. So $\\pi(Z(V_{1})) \\subseteq Z(V_{2}) = \\{0\\}$. Therefore, $Z(V_{1}) \\subseteq ker(\\pi)$. Note that $ker(\\pi) \\subseteq Z(V_{1})$, we have $Z(V_{1}) = ker(\\pi)$.\n\nFor all $\\varphi \\in C_{\\alpha_{1}^{k}}(V_{1}), x \\in Z(V_{1}), y \\in V_{1}$, $\\exists y^{'} \\in V_{1}$ such that $y = \\alpha_{1}^{k}(y^{'})$,\n\\[\\mu_{1}(\\varphi(x), y) = \\mu_{1}(\\varphi(x), \\alpha_{1}^{k}(y^{'})) = \\varphi(\\mu_{1}(x, y^{'})) = \\varphi(0) = 0,\\]\nwhich implies that $\\varphi(x) \\in Z(V_{1}) = ker(\\pi)$. So $\\varphi(ker(\\pi)) \\subseteq ker(\\pi)$, i.e., $\\varphi \\in End_{\\rm{F}}(V_{1}, ker(\\pi))$. So $C(V_{1}) \\subseteq End_{\\rm{F}}(V_{1}, ker(\\pi))$. Therefore, $C(V_{1}) \\cap End_{\\rm{F}}(V_{1}, ker(\\pi)) = C(V_{1})$. According to (1) (b), we have $\\pi_{C} : C(V_{1}) \\rightarrow C(V_{2})$ is a Hom-algebra homomorphism.\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\\label{intro}\n\nMeasurements of single-top-quark production via the electroweak interaction, a process first observed in proton--antiproton (\\ensuremath{{p}\\bar{p}}\\xspace) collisions at the Tevatron~\\cite{Aaltonen:2009jj, Abazov:2009ii}, have entered the precision era at the Large Hadron Collider (LHC). It has become possible to measure top-quark properties using single-top-quark events~\\cite{Giammanco:2017xyn}. Single-top-quark production is sensitive to new physics mechanisms~\\cite{Tait:2000sh} that either modify the \\ensuremath{tWb}\\xspace coupling~\\cite{Cao:2007ea,Godbole:2011vw,Zhang:2010dr,AguilarSaavedra:2008gt,Dror:2015nkp,Berger:2015yha} or introduce new particles and interactions~\\cite{He:1999vp,Aguilar-Saavedra:2013qpa,Durieux:2014xla,Nutter:2012an,Hashemi:2013raa,Drueke:2014pla}. The production rate of single top quarks is proportional to the square of the left-handed coupling at the \\ensuremath{tWb}\\xspace production vertex, assuming that there are no significant \\ensuremath{tWd}\\xspace or \\ensuremath{tWs}\\xspace contributions. In the Standard Model (SM), this coupling is given by the Cabibbo--Kobayashi--Maskawa (CKM)~\\cite{CKM1,CKM2} matrix element $\\ensuremath{V_{\\myt\\myb}}\\xspace$. Indirect measurements of $|\\ensuremath{V_{\\myt\\myb}}\\xspace|$, from precision measurements of $B$-meson decays~\\cite{Tanabashi:2018oca} and from top-quark decays~\\cite{Abazov:2011zk,Aaltonen:2013luz,Aaltonen:2014yua,CMS-TOP-12-035}, rely on the SM assumptions that the CKM matrix is unitary\nand that there are three quark generations. The most stringent indirect determination comes from a global fit to all available $B$-physics measurements, resulting in $|\\ensuremath{V_{\\myt\\myb}}\\xspace| = 0.999105 \\pm 0.000032$~\\cite{Tanabashi:2018oca}. This fit also assumes the absence of any new physics mechanisms that might affect \\bquarks. The most precise indirect measurement using top-quark events was performed by the CMS Collaboration in proton--proton (\\ensuremath{pp}\\xspace) collisions at a centre-of-mass energy of \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, resulting in $|\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.007\\pm0.016$~\\cite{CMS-TOP-12-035}.\n\nA direct estimate of the coupling at the \\ensuremath{tWb}\\xspace production vertex, $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$, is obtained from the measured single-top-quark cross-section $\\ensuremath{\\sigma_{\\rm meas.}}\\xspace$ and its corresponding theoretical expectation $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$,\n\\begin{eqnarray}\n\\label{eq:vtb}\n\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace =  \\sqrt{\\frac{\\ensuremath{\\sigma_{\\rm meas.}}\\xspace}{\\ensuremath{\\sigma_{\\rm theo.}}\\xspace \\mathsmaller{\\;(\\ensuremath{V_{\\myt\\myb}}\\xspace=1)}}}.\n\\end{eqnarray}\nThe $\\ensuremath{f_{\\rm LV}}\\xspace$ term is a form factor, assumed to be real, that parameterises the possible presence of anomalous left-handed vector couplings~\\cite{AguilarSaavedra:2008zc}. By construction, this form factor is exactly one in the SM, while it can be different from one in models of new physics processes. The direct estimation assumes that $|\\ensuremath{V_{td}}\\xspace|,|\\ensuremath{V_{ts}}\\xspace| \\ll |\\ensuremath{V_{\\myt\\myb}}\\xspace|$~\\cite{Abazov:2012uga,Alwall:2006bx}, and that the \\ensuremath{tWb}\\xspace interaction involves a left-handed weak coupling, like that in the SM. The $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ determination via single-top-quark production is independent of assumptions about the number of quark generations and the unitarity of the CKM matrix~\\cite{Tait:1999cf,Belyaev:2000me,Tait:2000sh,Cao:2015qta}. Since the indirect determination of $|\\ensuremath{V_{\\myt\\myb}}\\xspace|$ gives a value close to unity, $\\ensuremath{V_{\\myt\\myb}}\\xspace$ is considered equal to one in theoretical calculations of the single-top-quark cross-section. The combination of single-top-quark measurements from the Tevatron gives $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace=1.02_{-0.05}^{+0.06}$~\\cite{Aaltonen:2015cra}.\n\nSingle-top-quark production at a hadron collider mostly proceeds, according to the SM prediction, via three modes that can be defined at leading order (LO) in perturbative quantum chromodynamics (QCD): the exchange of a virtual \\ensuremath{W}\\xspace boson in the \\myt-channel\\xspace\\ or in the \\mys-channel\\xspace, and the associated production of a top quark and a \\ensuremath{W}\\xspace boson (\\ensuremath{tW}\\xspace). Representative Feynman diagrams for these processes at LO are shown in Figure~\\ref{fig:feyn}.\n\n\\begin{figure}[!h!tbp]\n  \\begin{center}\n    \\subfloat[]{ \\includegraphics[height=0.12\\textheight]{intro\/tchanFeyn} }\n    \\subfloat[]{ \\includegraphics[height=0.12\\textheight]{intro\/WtFeyn} }\n    \\subfloat[]{ \\includegraphics[height=0.12\\textheight]{intro\/schanFeyn} }\n    \\caption{Representative Feynman diagrams at LO in QCD and in the five-flavour-number scheme for single-top-quark production in (a) the \\myt-channel\\xspace, (b) \\ensuremath{tW}\\xspace production, and (c) the \\mys-channel\\xspace.}\n    \\label{fig:feyn}\n  \\end{center}\n\\end{figure}\n\nIn \\ensuremath{pp}\\xspace collisions at the LHC, the process with the largest single-top-quark production cross-section is the \\myt-channel\\xspace, where a light-flavour quark \\ensuremath{q}\\xspace from one of the colliding protons interacts with a \\bquark by exchanging a space-like virtual \\ensuremath{W}\\xspace boson, producing a top quark (\\ensuremath{t}\\xspace-quark) and a recoiling light-flavour quark $\\ensuremath{q}\\xspace'$, called the spectator quark.\nFor \\myt-channel\\xspace production at LO, the \\bquark can be considered as directly emitted from the other proton (five-flavour-number scheme or 5FS) or it can come from gluon splitting (four-flavour-number scheme or 4FS)~\\cite{Frederix:2012dh}. The kinematic properties of the spectator quark provide distinctive features for this process~\\cite{TOPQ-2015-05,CMS-TOP-12-038}. The associated production of a \\ensuremath{W}\\xspace boson and a top quark has the second-largest production cross-section. In a representative process of \\ensuremath{tW}\\xspace production, a gluon interacts with an initial \\bquark by exchanging a virtual \\bquark, producing a \\ensuremath{t}\\xspace-quark and a \\ensuremath{W}\\xspace boson. The measurement of this process suffers from a large background from top-quark pair (\\ensuremath{t \\bar{t}}\\xspace) production~\\cite{TOPQ-2012-20,CMS-TOP-12-040}. The \\mys-channel\\xspace\\ cross-section is the smallest at the LHC. In this process, a quark--antiquark pair annihilates to produce a time-like virtual \\ensuremath{W}\\xspace boson, which decays to a \\ensuremath{t}\\xspace-quark and a \\ensuremath{\\bar{b}}\\xspace-quark. This process was observed in \\ensuremath{{p}\\bar{p}}\\xspace collisions at the Tevatron~\\cite{CDF:2014uma} \nand evidence of it was reported by the ATLAS Collaboration in \\ensuremath{pp}\\xspace collisions at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2015-01}.\n\nIn this paper, the \\myt-channel\\xspace, \\ensuremath{tW}\\xspace, and \\mys-channel\\xspace\\ single-top-quark cross-section measurements by the ATLAS and CMS experiments are combined for each production mode, separately at $pp$ centre-of-mass energies of \\ensuremath{ 7 \\text{ and } 8~\\TeV}. A combined determination of $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ is also presented, using as inputs the values of $\\ensuremath{\\absvtb^2}\\xspace$ calculated from the measured and predicted single-top-quark cross-sections in the three production modes at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}. Using the same approach, results are also shown for $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ combinations for each production mode.\n\nThe theoretical cross-section calculations are described in Section~\\ref{sec:theory}. Section~\\ref{sec:atlascmsxs} presents the cross-section measurements. The combination methodology is briefly described in Section~\\ref{sec:method}. Section~\\ref{sec:systcat} is devoted to a discussion of systematic uncertainties in the cross-section measurements as well as theoretical calculations, where the latter affect the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ extraction in particular.\nThe assumptions made about the correlation of uncertainties between the two experiments, as well as between theoretical calculations, are also discussed. \nSection~\\ref{sec:xscomb} presents the combination of cross-sections for each production mode at the same centre-of-mass energy.\nIn Section~\\ref{sec:vtbcomb}, determinations of $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ are performed using all single-top-quark cross-section measurements together or by production mode. Stability tests are also shown and discussed. In Section~\\ref{sec:sum}, the results are summarised.\n\n\n\\section{Theoretical cross-section calculations}\n\\label{sec:theory}\n\nThe theoretical predictions for the single-top-quark production cross-sections are calculated at next-to-leading order (NLO) in the strong coupling constant $\\alpha_{\\rm s}$, at NLO with next-to-next-to-leading-logarithm (NNLL) resummation (named NLO+NNLL), and at next-to-next-to-leading order (NNLO). The difference between 4FS and 5FS is small~\\cite{Campbell:2009ss, Campbell:2009gj}, and the calculations use the 5FS.\nThe NLO prediction is used in the \\ensuremath{V_{\\myt\\myb}}\\xspace\\ combination for the  \\myt-channel\\xspace\\ and \\mys-channel\\xspace, while the NLO+NNLL prediction is used for \\ensuremath{tW}\\xspace, as explained below.\nThe NLO prediction is calculated with \\textsc{HatHor}\\xspace (v2.1)~\\cite{Aliev:2010zk,Kant:2014oha}. Uncertainties comprise the scale uncertainty, the $\\alpha_{\\rm s}$ uncertainty, and the parton distribution function (PDF) uncertainty. The scale uncertainty is evaluated by varying the renormalisation and factorisation scales up and down together by a factor of two. The combination of the PDF+$\\alpha_{\\rm s}$ uncertainty is calculated according to the PDF4LHC prescription~\\cite{Botje:2011sn} from the envelope of the uncertainties at 68\\% confidence level (CL) in the MSTW2008 NLO, CT10 NLO~\\cite{Lai:2010vv}, and NNPDF2.3~\\cite{Ball:2012cx} PDF sets. \n\nThe NLO+NNLL predictions~\\cite{Kidonakis:2012rm} are available for all single-top-quark production modes~\\cite{Kidonakis:2011wy, Kidonakis:2010ux, Kidonakis:2010tc}. Uncertainties in these calculations are estimated by varying the renormalisation and factorisation scales between $m_{\\myt}\\xspace\/2$ and $2 m_{\\myt}\\xspace$, where $m_{\\myt}\\xspace$ is the top-quark mass, and from the 90\\% CL uncertainties in the MSTW2008 NNLO~\\cite{MSTW, MSTW2008} PDF set. The evaluation of the PDF uncertainties is provided by the author of Refs.~\\cite{Kidonakis:2011wy, Kidonakis:2010ux, Kidonakis:2010tc} and is not fully compatible with the PDF4LHC prescription.\nThe \\myt-channel\\xspace\\ cross-sections at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are also computed at NNLO in $\\alpha_{\\rm s}$~\\cite{Brucherseifer:2014ama}, with the renormalisation and factorisation scales set to $m_{\\myt}\\xspace$. This results in cross-sections which are about 0.3\\% and 0.6\\% lower than the NLO values at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ respectively. However, only a limited number of scale variations are evaluated~\\cite{Brucherseifer:2014ama}.\n\nA summary of all the available theoretical cross-section predictions for \\myt-channel\\xspace, \\ensuremath{tW}\\xspace, and \\mys-channel\\xspace production, $\\ensuremath{\\sigma^{\\myt\\textrm{-chan.}}_{\\mathrm{\\textrm{theo.}}}}\\xspace$, $\\ensuremath{\\sigma^{\\mytW}_{\\mathrm{\\textrm{theo.}}}}\\xspace$, and $\\ensuremath{\\sigma^{\\mys\\textrm{-chan.}}_{\\mathrm{\\textrm{theo.}}}}\\xspace$ respectively, with their uncertainties is shown in Table~\\ref{tab:xs}.\n\n\\begin{table}[!htb]\n  \\centering\n  \\caption{Predicted cross-sections for single-top-quark production at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ at the LHC. Uncertainties include scale and PDF+$\\alpha_{\\rm s}$ variations, except for the NNLO predictions, which only contain the scale variation. The PDF+$\\alpha_{\\rm s}$ uncertainties are evaluated according to the PDF4LHC prescription only for the NLO predictions. The uncertainties associated with the top-quark mass $m_{\\myt}\\xspace$ and beam energy $E_{\\rm beam}$ are also given for the NLO predictions for the $t$- and $s$-channels, and for the NLO+NNLL prediction for \\ensuremath{tW}\\xspace\\ production. The value of $m_{\\myt}\\xspace$ is set to $172.5~\\GeV$ in all predictions. The cross-sections marked with $^\\dagger$ are those used in the \\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace\\ combination.}\n  \\IfPackagesLoaded{adjustbox}{\\begin{adjustbox}{max width=1.0\\textwidth}}{}\n    \\begin{tabular}{l|l|l|l}\n      \\hline\n      \\hline\n      \\T\\B\n      $\\sqrt{s}$ & Process & Accuracy & $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace{}$\\,[pb] \\\\\n      \\hline\n      \\T\\B\n                 & & NLO${}^{\\dagger}$ & $63.9_{-1.3}^{+1.9}~\\text{(scale)} \\pm 2.2~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.7~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.1~\\text{($E_{\\rm beam}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & \\myt-channel\\xspace & NLO+NNLL & $64.6^{+2.6}_{-1.7}~\\text{(scale+PDF+$\\alpha_{\\rm s}$)}$\\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NNLO & $63.7^{+0.5}_{-0.3}~\\text{(scale)}$\\\\\n      \\cline{2-4}\n      \\T\\B\n      \\ensuremath{ 7~\\TeV} & \\multirow{2}{*}{\\ensuremath{tW}\\xspace} & NLO & $13.2_{-0.6}^{+0.5}~\\text{(scale)} \\pm 1.3~\\text{(PDF+$\\alpha_{\\rm s}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NLO+NNLL${}^{\\dagger}$ & $15.74 \\pm 0.40~\\text{(scale)} {}^{+1.10}_{-1.14}~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.28~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.04~\\text{($E_{\\rm beam}$)}$\\\\\n      \\cline{3-4}\n      \\cline{2-4}\n      \\T\\B\n                 & \\multirow{2}{*}{\\mys-channel\\xspace} & NLO${}^{\\dagger}$ & $\\;\\,4.29_{-0.10}^{+0.12}~\\text{(scale)} \\pm 0.14~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.10~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.01~\\text{($E_{\\rm beam}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NLO+NNLL & $\\;\\,4.63^{+0.20}_{-0.18}~\\text{(scale+PDF+$\\alpha_{\\rm s}$)}$\\\\\n      \\cline{3-4}\n      \\hline\n      \\hline\n      \\T\\B\n                 & & NLO${}^{\\dagger}$ & $84.7_{-1.7}^{+2.6}~\\text{(scale)} \\pm 2.8~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.8~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.2~\\text{($E_{\\rm beam}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & \\myt-channel\\xspace & NLO+NNLL & $87.8^{+3.4}_{-1.9}~\\text{(scale+PDF+$\\alpha_{\\rm s}$)}$\\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NNLO & $84.2^{+0.3}_{-0.2}~\\text{(scale)}$\\\\\n      \\cline{2-4}\n      \\T\\B\n                 \\ensuremath{ 8~\\TeV} & \\multirow{2}{*}{\\ensuremath{tW}\\xspace} & NLO & $18.77_{-0.82}^{+0.77}~\\text{(scale)} \\pm 1.70~\\text{(PDF+$\\alpha_{\\rm s}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NLO+NNLL${}^{\\dagger}$ & $22.37 \\pm 0.60~\\text{(scale)} \\pm1.40~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.38~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.06~\\text{($E_{\\rm beam}$)}$\\\\\n      \\cline{3-4}\n      \\cline{2-4}\n      \\T\\B\n                 & \\multirow{2}{*}{\\mys-channel\\xspace} & NLO${}^{\\dagger}$ & $\\;\\,5.24_{-0.12}^{+0.15}~\\text{(scale)} \\pm 0.16~\\text{(PDF+$\\alpha_{\\rm s}$)} \\pm 0.12~\\text{($m_\\ensuremath{t}\\xspace$)} \\pm 0.01~\\text{($E_{\\rm beam}$)}$ \\\\\n      \\cline{3-4}\n      \\T\\B\n                 & & NLO+NNLL & $\\;\\,5.61\\pm0.22~\\text{(scale+PDF+$\\alpha_{\\rm s}$)}$\\\\\n      \\cline{3-4}\n      \\hline\n      \\hline\n    \\end{tabular}\n    \\IfPackagesLoaded{adjustbox}{\\end{adjustbox}}{}\n  \\label{tab:xs}\n\\end{table}\n\nIn this paper, NLO predictions serve as the reference for the $t$- and $s$-channel processes, following the prescriptions presented above, because higher-order calculations and their uncertainties are not fully computed and available for the parameter values of choice. The advantage of the NLO cross-section calculations is that the configurable parameters in \\textsc{HatHor}\\xspace can be set according to those used to generate the ATLAS and CMS simulation samples. The $t$- and \\mys-channel\\xspace\\ processes do not interfere at NLO~\\cite{Willenbrock:1986cr}. For these two processes, the entire phase space is included in the integration in order to obtain the total cross-section. The \\ensuremath{tW}\\xspace cross-section prediction, $\\ensuremath{\\sigma^{\\mytW}_{\\mathrm{\\textrm{theo.}}}}\\xspace$, is available at NLO ~\\cite{Kant:2014oha} and NLO+NNLL~\\cite{Kidonakis:2013zqa,Kidonakis:2010ux}. The \\ensuremath{tW}\\xspace process at NLO interferes with the \\ensuremath{t \\bar{t}}\\xspace process at LO with the subsequent decay $\\ensuremath{\\bar{t}}\\xspace \\rightarrow W\\ensuremath{\\bar{b}}\\xspace$. In the NLO prediction for \\ensuremath{tW}\\xspace production provided in Ref.~\\cite{Kant:2014oha}, a kinematic cut-off is imposed on the transverse momentum (\\pt) of the outgoing \\bquark, suppressing the contribution from \\ttbar\\ production. Since the treatment of this interference in \\textsc{HatHor}\\xspace is still being developed~\\cite{Demartin:2016axk, Campbell:2005bb}, the NLO+NNLL calculation is used as reference for \\ensuremath{tW}\\xspace production. For the reference cross-section predictions, uncertainties corresponding to the dependence on $m_{\\myt}\\xspace$ and on the LHC beam energy, $E_{\\rm beam}$, are evaluated. The $m_{\\myt}\\xspace$ dependence is estimated by varying its central value of $172.5~\\GeV$ (the value used in the simulation samples used to measure the single-top-quark cross-sections) by $\\pm$1~\\GeV, using the functional form proposed in Ref.~\\cite{Czakon:2013goa}. The theoretical calculations are performed at a given centre-of-mass energy while the energy of the LHC beam is measured with an uncertainty. The single-top-quark cross-sections are assumed to depend on $E_{\\rm beam}$ according to the model given in Ref.~\\cite{Langenfeld:2009tc}, with a relative uncertainty $\\delta E_{\\rm beam}\/E_{\\rm beam}$ of 0.1\\%~\\cite{Todesco:2017nnk}. \nThe theoretical cross-sections that are used as reference are marked with a $^\\dagger$ in Table~\\ref{tab:xs}.\n\n\n\n \\section{Single-top-quark cross-section measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}}\n\\label{sec:atlascmsxs}\n\nThe \\myt-channel\\xspace single-top-quark production cross-sections, $\\ensuremath{\\sigma_{{\\myt\\textrm{-chan.}}}}\\xspace$, were measured by the\nATLAS and CMS Collaborations at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}~\\cite{TOPQ-2012-21,CMS-TOP-11-021}\nand 8\\,\\TeV~\\cite{TOPQ-2015-05,CMS-TOP-12-038}. Evidence of \\ensuremath{tW}\\xspace production was reported at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}~by \nATLAS~\\cite{TOPQ-2011-17} and CMS~\\cite{CMS-TOP-11-022}, \nwhile at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ its cross-section, $\\ensuremath{\\sigma_{\\mytW}}\\xspace$, was\nmeasured by both\nexperiments~\\cite{TOPQ-2012-20,CMS-TOP-12-040}. Evidence of \\mys-channel\\xspace\\\nproduction was reported by ATLAS, with a measured cross-section,\n$\\ensuremath{\\sigma_{\\mys\\textrm{-chan.}}}\\xspace$, at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2015-01}, whereas CMS set upper limits on\nthe \\mys-channel\\xspace\\ production cross-section at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}. The observed\n(expected) significance of the CMS measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ is $2.3$ ($0.8$) standard\ndeviations~\\cite{CMS-TOP-13-009}.\n\nThe ATLAS and CMS analyses use similar approaches to measure the single-top-quark production cross-sections. Both experiments select events containing at least one prompt isolated lepton (electron or muon) and at least one high-\\pt\\ jet. The analyses use various multivariate analysis (MVA) techniques, such as boosted decision trees~\\cite{FREUND1997119,Friedman2006,Hocker:2007ht}, neural networks~\\cite{Feindt:2006pm}, or the matrix element method (MEM)~\\cite{Kondo:1988yd, Kondo:1991dw}, to separate the signal from background. To measure the cross-section, analyses perform a binned maximum-likelihood fit to data using the distribution of the corresponding MVA discriminator. \nExceptions are the ATLAS \\mys-channel\\xspace\\ and CMS \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. In the ATLAS \\mys-channel\\xspace\\ analysis, the fit is performed simultaneously to the MEM discriminant in the signal region and the lepton-charge distribution in the \\ensuremath{W}+jets\\xspace control region. The CMS \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ is based on a simultaneous fit to the absolute pseudorapidity ($\\eta$) distributions of the recoiling light-flavour jet in events with negative and with positive lepton charge. \nThe analyses measuring different single-top-quark production modes within the same experiment and at the same centre-of-mass energy have disjoint signal regions. \nBoth experiments simulate the single-top-quark processes using the NLO \\POWHEGBOX\\ generator~\\cite{Nason:2004rx,Frixione:2007vw,Alioli:2009je,Alioli:2010xd,Re:2010bp} for the matrix-element (ME) calculations. ATLAS also uses the \\POWHEGBOX\\ generator to simulate top-quark-pair background events, while CMS uses the LO \\MADGRAPH generator~\\cite{Alwall:2011uj}. The \\PYTHIA~\\cite{Sjostrand:2006za} event generator is used for modelling the parton shower (PS), hadronisation and the underlying event in both the single-top-quark and \\ttbar\\ processes. The cross-sections are measured assuming a value of $172.5$~\\GeV\\ for $m_{\\myt}\\xspace$ for all top-quark processes\nand all centre-of-mass energies. \nA summary of the uncertainties in each measurement is shown in Table~\\ref{tab:xsec_meas}, with details given in Appendix~\\ref{app:allUncs}.\n\n\\begin{table}[!htbp]\n  \\centering\n  \\caption{Summary of the single-top-quark cross-section measurements published by the ATLAS and\n    CMS Collaborations at\n    \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}. Total uncertainties are shown. Small\n    differences between the integrated luminosity \n    values in different analyses within the same experiment and\n    centre-of-mass energy are due to different luminosity\n    calibrations at the time of publication.}\n  \\IfPackagesLoaded{adjustbox}{\\begin{adjustbox}{max width=1.0\\textwidth}}{}\n    \\begin{tabular}{l|l|r@{\\hskip 0cm}lc|r@{\\hskip 0cm}lc}\n      \\hline\n      \\hline\n      \\multicolumn{2}{c|}{} & \\multicolumn{3}{c|}{ATLAS} & \\multicolumn{3}{c}{CMS} \\\\\n      \\hline\n      $\\sqrt{s}$ & Process & \\multicolumn{2}{c}{$\\sigma{}$\\,[pb]} & Lumi.\\ [$\\ensuremath{\\rm fb^{-1}}\\xspace$] & \\multicolumn{2}{c}{$\\sigma{}$\\,[pb]} & Lumi.\\ [$\\ensuremath{\\rm fb^{-1}}\\xspace$] \\\\\n      \\hline\n                            & \\myt-channel\\xspace & $68$&$~\\pm~8$ & 4.59 & $67.2$&$~\\pm~6.1$ & 1.17--1.56 \\\\ \n      \\cline{2-8}\n      \\T\\B\n      \\ensuremath{ 7~\\TeV} & \\ensuremath{tW}\\xspace & $16.8$&$~\\pm~5.7$ & 2.05 & \\multicolumn{2}{c}{$16^{+5}_{-4}$} & 4.9 \\\\\n      \\cline{2-8}\n      \\T\\B\n                            & \\mys-channel\\xspace & \\multicolumn{2}{c}{---} & --- & $7.1$&$~\\pm~8.1$ & 5.1 \\\\\n      \\hline\n      \\hline\n      \\T\\B\n                            & \\myt-channel\\xspace & \\multicolumn{2}{c}{$89.6^{+7.1}_{-6.3}$} & 20.2 & $83.6$&$~\\pm~7.8$ & 19.7 \\\\\n      \\cline{2-8}\n      \\T\\B\n      \\ensuremath{ 8~\\TeV} & \\ensuremath{tW}\\xspace & \\multicolumn{2}{c}{$23.0^{+3.6}_{-3.9}$} & 20.3 & $23.4$&$~\\pm~5.4$ & 12.2 \\\\\n      \\cline{2-8}\n      \\T\\B\n                            & \\mys-channel\\xspace & \\multicolumn{2}{c}{$4.8^{+1.8}_{-1.5}$} & 20.3 & $13.4$&$~\\pm~7.3$ & 19.7 \\\\\n      \\hline\n      \\hline\n    \\end{tabular}\n    \\renewcommand{\\arraystretch}{1}\n    \\IfPackagesLoaded{adjustbox}{\\end{adjustbox}}{}\n  \\label{tab:xsec_meas}\n\\end{table}\n  \\section{Combination methodology}\n\\label{sec:method}\nThe ATLAS and CMS single-top-quark production cross-section measurements shown in Table~\\ref{tab:xsec_meas} are combined, and the combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ value determined, using the best linear unbiased estimator (BLUE) method~\\cite{Lyons:1988rp, Valassi:2003mu,Nisius:2014wua}. The BLUE method is applied iteratively in order to reduce a possible bias arising from the dependence of systematic uncertainties on the central value of the cross-section~\\cite{Lista:2014qia}. Convergence is reached when the central value changes by less than 0.01\\% compared with the previous iteration. \nIn each iteration, the BLUE method minimises the global $\\chi^2$ by adjusting the weight for each input measurement~\\cite{,Nisius:2014wua}. The global $\\chi^2$ is calculated taking correlations\ninto account. The sum of weights is required to be equal to one. Negative weights are allowed; these indicate strong correlations~\\cite{Valassi:2013bga}. The number of degrees of freedom is $n-1$, where $n$ is number of measurements in the combination. The $\\chi^2$ and $n$ are then used to calculate a corresponding probability~\\cite{Nisius:2014wua}. The systematic uncertainties are scaled with the cross-section in each iteration, i.e.\\ they are treated as relative uncertainties. The data and simulation statistical uncertainties are not scaled~\\cite{Lista:2014qia}. The systematic uncertainties in the \\mys-channel\\xspace\\ cross-section combination are also not scaled because the \\mys-channel\\xspace\\ measurements have large backgrounds.\n\nFollowing the same strategy as in the input measurements by the ATLAS and CMS Collaborations, the combined cross-sections are reported at $m_{\\myt}\\xspace = 172.5~\\GeV$, not including the uncertainty associated with the $m_{\\myt}\\xspace$ variation. The shift in the combined cross-section due to a variation of $\\pm 1~\\GeV$ in the top-quark mass is given where this information is available. For the determination of the combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ value, the uncertainty in the measured cross-sections due to a variation of $\\pm 1~\\GeV$ in the mass is considered.\nUncertainties in the measurements are symmetrised, before combination, by averaging the magnitude of the downward and upward variations. More details are given in Sections~\\ref{sec:systcat} and \\ref{sec:xscomb}.\n \\section{Systematic uncertainties and correlation assumptions}\n\\label{sec:systcat}\nIn order to combine single-top-quark cross-section measurements and $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ values, the sources of uncertainty are grouped into categories. While the categorisation and evaluation of uncertainties varies somewhat between experiments and between measurements, each individual measurement considers a complete set of uncertainties. Assumptions are made about correlations between similar sources of uncertainty in different measurements, as explained in Section~\\ref{sec:systcat_exp}. Uncertainties associated with theoretical predictions are taken into account in the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ combination. The correlations between similar uncertainties in different theoretical predictions are discussed in Section~\\ref{sec:systcat_theo}.\n\n\\subsection{Systematic uncertainties in measured cross-sections}\n\\label{sec:systcat_exp}\n\nSystematic uncertainties in the ATLAS \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are evaluated using pseudoexperiments, except the background normalisation uncertainties, which are constrained in the fit to data. In the ATLAS \\ensuremath{tW}\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ and the \\mys-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, systematic uncertainties are included as nuisance parameters in profile-likelihood fits. Systematic uncertainties in the CMS \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are included as nuisance parameters in fits to data, except the theory modelling uncertainties in signal and backgrounds, described below, which are evaluated using pseudoexperiments. All systematic uncertainties in the CMS \\mys-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are obtained through pseudoexperiments, except the background normalisation uncertainties, which are constrained in the fit to data. In the analyses where systematic uncertainties are included as nuisance parameters, the total uncertainty presented in Table~\\ref{tab:xsec_meas} is evaluated by varying all the nuisance parameters in the fit simultaneously. To extract the impact of each source of this type of uncertainty, these analyses use approximate procedures which neglect the correlations between sources of uncertainty introduced by the fits. Throughout this paper, individual uncertainties are taken as reported by the input analyses, regardless of the method used to determine them. The total uncertainties are evaluated as the sum in quadrature of individual contributions.\n\nAlthough the sources of systematic uncertainty and the procedures used to\nestimate their impact on the measured cross-section are partially different in the individual analyses, it is still possible to identify contributions that describe similar physical effects. These contributions are listed below; they are grouped together, and only the resulting categories are used in the combination. Categories are treated as uncorrelated among each other. \nFor each source of uncertainty, correlations between different measurements are assumed to be positive, unless explicitly mentioned otherwise. \nThe stability of the cross-section and $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ combinations is studied by varying the correlation assumptions for the dominant uncertainties, as discussed in Section~\\ref{sec:tests}.\n\nThe uncertainties in each category are listed below, with the correlation assumptions across experiments given in parentheses. These correlations correspond to those used in the cross-section combinations. They are also valid for the combination of the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ extractions, unless explicitly mentioned otherwise. The symbol ``---'' means that the uncertainty is either considered only in the ATLAS or the CMS measurement, or is not considered at all. A summary of uncertainties in the cross-section measurements together with the corresponding correlation assumptions between experiments is provided in Appendix~\\ref{app:allUncs}.\n\n\\noindent {\\bf Data statistical} (Correlation 0)\\\\ This statistical uncertainty arises from the limited size of the data sample. It is uncorrelated between ATLAS and CMS, between production modes, and between centre-of-mass energies.\n\n\\noindent {\\bf Simulation statistical} (Correlation 0 and --- for CMS \\ensuremath{tW}\\xspace\\ at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and \\mys-channel\\xspace\\ at \\ensuremath{\\sqrt{s}=\\lhcenergyEight})\\\\ This statistical uncertainty comes from the limited size of simulated event samples. It is uncorrelated between ATLAS and CMS, between production modes, and between centre-of-mass energies. For the CMS \\ensuremath{tW}\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and \\mys-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, this uncertainty is evaluated as part of the total statistical uncertainty, which is also considered uncorrelated, as discussed above. \nMore details are given in Appendices~\\ref{app:tw} and \\ref{app:sch}.\n\n\\noindent {\\bf Integrated luminosity} (Correlation 0.3)\\\\ This uncertainty originates from the systematic uncertainty in the integrated luminosity, as determined by the individual experiments using the methods described in Refs.~\\cite{DAPR-2011-01,DAPR-2013-01,CMS-PAS-SMP-12-008,CMS-PAS-LUM-13-001}. It affects the determination of both the signal and background yields. The integrated-luminosity uncertainty has a component that is correlated between ATLAS and CMS, arising from imperfect knowledge of the beam currents during van der Meer scans in the LHC accelerator~\\cite{vanderMeer:1968zz}, and an uncorrelated component from the long-term luminosity monitoring that is experiment-specific.\nAt \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, these components are 0.5\\% and 1.7\\% respectively for ATLAS and 0.5\\% and 2.1\\% respectively for CMS. At \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, they are 0.6\\% and 1.8\\% respectively for ATLAS and 0.7\\% and 2.5\\% respectively for CMS.\nAt both centre-of-mass energies, the correlation coefficient between the integrated-luminosity uncertainty in ATLAS and CMS at the same centre-of-mass energy is $\\rho=0.3$. Within the same experiment, the integrated-luminosity uncertainty is assumed to be correlated between production modes and uncorrelated between centre-of-mass energies. In Section~\\ref{sec:tests}, it is shown that the combined $\\ensuremath{\\absvtb^2}\\xspace$ result does not depend significantly on the correlation assumptions.\n\n\\noindent {\\bf Theory modelling}\\\\ This category contains the uncertainties in the modelling of the simulated single-top-quark processes, as well as smaller contributions from the modelling of the \\ensuremath{t \\bar{t}}\\xspace and \\ensuremath{W}+jets\\xspace background processes. Both signal and background modelling are included because the uncertainties in all top-quark processes are closely related.\n These include initial- and final-state radiation (ISR\/FSR), renormalisation and factorisation scales, NLO matching method, PS and hadronisation modelling, and PDF uncertainties. For the \\ensuremath{tW}\\xspace process, the uncertainty due to the treatment of interference between \\ensuremath{tW}\\xspace and \\ensuremath{t \\bar{t}}\\xspace final states is also included, as discussed below. These modelling uncertainties in signal and background processes are summed in quadrature in each input measurement.\n\n\\begin{itemize}\n\n\\item {\\it Scales and radiation modelling} (Correlation 1)\\\\ The renormalisation and factorisation scales and ISR\/FSR uncertainties account for missing higher-order corrections in the perturbative expansion and the amount of initial- and final-state radiation in simulated signal and background processes. In the ATLAS measurements of all three production modes, these uncertainties are estimated using dedicated single-top-quark and \\ensuremath{t \\bar{t}}\\xspace simulated event samples, by consistently varying the renormalisation and factorisation scales and the amount of ISR\/FSR in accordance with a measurement of additional jet activity in \\ensuremath{t \\bar{t}}\\xspace events at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}~\\cite{TOPQ-2011-21,ATL-PHYS-PUB-2014-005}.\nIn the ATLAS \\myt-channel\\xspace\\ measurements, they are also estimated in \\ensuremath{W}+jets\\xspace\\ simulated event samples, by varying the scale and matching parameters in the \\ALPGEN\\ LO multileg generator~\\cite{Mangano:2002ea} at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and by varying the parameters controlling the scale in the \\SHERPA\\ LO multileg generator~\\cite{Gleisberg:2008ta} at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. In the CMS measurements, these uncertainties are estimated by varying the renormalisation and factorisation scales, and ISR\/FSR, consistently in the simulated event samples. In the CMS \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, this uncertainty applies only to the signal modelling since the modelling of the dominant \\ensuremath{t \\bar{t}}\\xspace and \\ensuremath{W}+jets\\xspace background processes is obtained from data. However, for the \\myt-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the scales are varied in the simulated signal, \\ensuremath{t \\bar{t}}\\xspace, \\ensuremath{W}+jets\\xspace and other single-top-quark processes. The same approach is followed in the CMS \\mys-channel\\xspace\\ measurements at both centre-of-mass energies. The \\ensuremath{tW}\\xspace cross-section measurements of CMS account for this uncertainty only in the \\ensuremath{tW}\\xspace signal and \\ensuremath{t \\bar{t}}\\xspace background, given the negligible contributions from the \\ensuremath{W}+jets\\xspace and other single-top-quark processes in the dilepton final state.\n\nAlthough the methods are apparently different, they mostly address the same uncertainty, hence this uncertainty is considered correlated between ATLAS and CMS. It is also considered correlated between production modes and centre-of-mass energies. The combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ result does not depend significantly on this correlation assumption,\nas discussed in Section~\\ref{sec:tests}.\n\n\n\\item {\\it NLO matching} (Correlation 1 for \\myt-channel\\xspace\\ and --- for \\ensuremath{tW}\\xspace and \\mys-channel\\xspace)\\\\ The ATLAS measurements include an uncertainty to account for different NLO matching methods implemented in different NLO event generators. This is evaluated in single-top-quark and \\ensuremath{t \\bar{t}}\\xspace simulations by comparing the \\POWHEGBOX, \\MCatNLO~\\cite{Frixione:2002ik,Skands:2010ak}, and \\MGMCatNLO~\\cite{Alwall:2014hca} event generators, all interfaced to \\HERWIG~\\cite{Corcella:2000bw} (with \\JIMMY~\\cite{Butterworth:1996zw} for the underlying-event modelling). In the CMS \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the NLO matching uncertainty is evaluated by comparing \\POWHEGBOX\\ with \\Comphep~\\cite{Boos:2004kh,Pukhov:1999gg}. In the CMS \\myt-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, this uncertainty accounts for different NLO matching methods in the \\myt-channel\\xspace\\ signal event generator, as well as for differences between event generation in the 4FS and 5FS, by comparing \\POWHEGBOX\\ with \\MADGRAPH. The NLO matching uncertainty is considered correlated between ATLAS and CMS, between production modes, and between centre-of-mass energies. In the CMS \\ensuremath{tW}\\xspace and \\mys-channel\\xspace\\ analyses at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, this uncertainty is not considered, since the modelling uncertainties in the scheme to remove overlap with \\ensuremath{t \\bar{t}}\\xspace are dominant in the \\ensuremath{tW}\\xspace analysis and the renormalisation\/factorisation scale is dominant in the \\mys-channel\\xspace\\ analysis. The results of the stability test for this uncertainty are shown in Section~\\ref{sec:tests}.\n\n\\item {\\it Parton shower and hadronisation} (Correlation 1)\\\\ In both experiments, the difference between the \\PYTHIA and \\HERWIG showering programs is considered in the jet energy scale (JES)~\\cite{PERF-2011-03,PERF-2012-01,CMS-JME-10-011,CMS-JME-13-004} and \\btag calibration~\\cite{PERF-2012-04,ATLAS-CONF-2014-004,ATLAS-CONF-2014-046,CMS-BTV-12-001,CMS-PAS-BTV-13-001}. \nThe ATLAS analyses additionally  include an uncertainty in the PS and hadronisation modelling in simulated single-top-quark and \\ensuremath{t \\bar{t}}\\xspace events, evaluated by comparing the \\POWHEGBOX\\ event generator interfaced to \\PYTHIA or to \\HERWIG. \nThe CMS analyses additionally include an uncertainty in the \\ensuremath{t \\bar{t}}\\xspace and \\ensuremath{W}+jets\\xspace backgrounds estimated with the \\MADGRAPH event generator interfaced to \\PYTHIA. It is evaluated in simulated event samples where the value of the ME\/PS matching threshold in the MLM method~\\cite{Mangano:2006rw} is doubled or halved from its initial value. \nThe CMS \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ does not consider this uncertainty in the \\ensuremath{t \\bar{t}}\\xspace and \\ensuremath{W}+jets\\xspace backgrounds since the distribution and normalisation of the \\ensuremath{t \\bar{t}}\\xspace\\ and \\ensuremath{W}+jets\\xspace\\ processes are derived mostly from data.\nIn the CMS \\ensuremath{tW}\\xspace analyses at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, the contributions of the \\ensuremath{W}+jets\\xspace and other single-top-quark processes in the dilepton final state are negligible.\n\nThis uncertainty is considered correlated between ATLAS and CMS, between different production modes, and between different centre-of-mass energies. The combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ result does not depend significantly on this correlation assumption, as shown in Section~\\ref{sec:tests}.\n\n\\item {\\it Parton distribution functions} (Correlation 1)\\\\ The PDF uncertainty is evaluated following the PDF4LHC procedures~\\cite{Alekhin:2011sk,Botje:2011sn,Butterworth:2015oua} and is \nconsidered correlated between ATLAS and CMS, between different production modes, and between different centre-of-mass energies.\n\n\\item {\\it \\ensuremath{tW}\\xspace and \\ensuremath{t \\bar{t}}\\xspace interference} (Correlation 1 for \\ensuremath{tW}\\xspace\\ and --- for $t$- and $s$-channels)\\\\ The \\ensuremath{tW}\\xspace process interferes with \\ensuremath{t \\bar{t}}\\xspace production at NLO~\\cite{Frixione:2008yi, tWth2, tWth3}. In both ATLAS and CMS, two simulation approaches are compared: diagram removal (DR)~\\cite{Frixione:2008yi} and diagram subtraction (DS)~\\cite{Frixione:2008yi,Tait:1999cf}.\nIn the DR approach, all NLO diagrams that overlap with the doubly resonant \\ensuremath{t \\bar{t}}\\xspace\\ contributions are removed from the calculation of the \\ensuremath{tW}\\xspace\\ amplitude. This approach accounts for the interference term, but it is not gauge invariant (though the effect is numerically negligible)~\\cite{Frixione:2008yi}. In the DS approach, a subtraction term is built into the amplitude to cancel out the \\ensuremath{t \\bar{t}}\\xspace\\ component close to the top-quark resonance while respecting gauge invariance.\n\n\n The DR approach is the default, and the comparison with the DS approach is used to assess this systematic uncertainty. For the \\ensuremath{tW}\\xspace analyses, this uncertainty is considered correlated between the two experiments and between different centre-of-mass energies.\n\n\\item {\\it Modelling of  the top-quark \\ensuremath{p_{\\mathrm{T}}}\\xspace\\ spectrum} (Correlation ---)\\\\ In the CMS \\ensuremath{tW}\\xspace and \\mys-channel\\xspace\\ analyses at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, the simulated \\ensuremath{t \\bar{t}}\\xspace events are reweighted to correct the $\\ensuremath{p_{\\mathrm{T}}}\\xspace$\\ spectrum of the generated top quarks, which was found to be significantly harder than the spectrum observed in data in differential cross-section measurements~\\cite{CMS-TOP-12-028,Czakon:2016ckf}. To estimate the uncertainty related to this mismodelling, the \\ensuremath{tW}\\xspace\\ measurement is repeated without the reweighting, and the change relative to the default result is taken as the uncertainty. In the CMS \\mys-channel\\xspace\\ analysis, the measurement is repeated with the effect of the weights removed and doubled. The resulting variation in the cross-section is symmetrised. This uncertainty is not considered in the CMS \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ where the modelling of the \\ensuremath{t \\bar{t}}\\xspace\\ background is extracted from data. In the ATLAS measurements, modelling uncertainties in the top-quark \\ensuremath{p_{\\mathrm{T}}}\\xspace spectrum in \\ensuremath{t \\bar{t}}\\xspace events~\\cite{TOPQ-2015-06} are covered by the PS and hadronisation uncertainty and they are found to be small in comparison with other systematic uncertainties. This uncertainty is considered correlated between the CMS \\ensuremath{tW}\\xspace and \\mys-channel\\xspace\\ analyses at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}.\n\n\\item {\\it Dependence on the top-quark mass} (Correlation 1)\\\\ The measured single-top-quark cross-sections shown in Table~\\ref{tab:xsec_meas} assume a nominal $m_{\\myt}\\xspace$ value of 172.5~\\GeV. The dependence of the measured cross-section on $m_{\\myt}\\xspace$ is estimated for the ATLAS \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ and for the ATLAS \\ensuremath{tW}\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. It is determined using dedicated simulations of single-top-quark and \\ttbar\\ samples with different $m_t$ values. The cross-section measurements assuming the different $m_t$ values are interpolated using a first- or a second-order polynomial, for which the constant term is given by the central value of $m_{\\myt}\\xspace = 172.5$~\\GeV. \n  The CMS measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ provide information for a variation of $\\pm 2~\\GeV$ in the top-quark mass, which is scaled to a $\\pm 1~\\GeV$ shift assuming a linear dependence. For the CMS \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, the changes in cross-sections are symmetrised and reported as uncertainties.  In the CMS \\mys-channel\\xspace\\ analysis, the change in the cross-section is determined for the up and down variation of $m_{\\myt}\\xspace$.\n  No estimates are available for the CMS \\myt-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the ATLAS and CMS \\ensuremath{tW}\\xspace\\ analyses at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ or the ATLAS \\mys-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. The top-quark-mass uncertainty is small for each measurement, thus the impact of not evaluating it for these measurements is negligible.\n\n In this paper, a symmetrised uncertainty in the measured cross-section due to a variation of $\\pm 1~\\GeV$ in the top-quark mass is considered. When the full cross-section dependence on the top-quark mass is available for a given production mode at a given centre-of-mass energy, the sign of the dependence of the uncertainty per unit of mass is taken into account in the correlations.\nIn the case of the CMS \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, where the sign of the dependence is not available, it is assumed that the sign is the same as for the ATLAS measurement, since the phase space and background composition are comparable between CMS and ATLAS. Given that the uncertainty in the measured cross-section is considered for the same $m_t$ variation and considering the sign of the dependence when available, this uncertainty is considered correlated between ATLAS and CMS and between different centre-of-mass energies and uncorrelated between the \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ production modes. \n\n\\end{itemize}\n\n\\noindent {\\bf Background normalisation} (Correlation 0)\\\\\nThree background uncertainties are considered: in top-quark background (\\ttbar\\ and other single-top-quark processes), in other background determined from simulation (\\ensuremath{W\/Z}+jets\\xspace, diboson, and other smaller background channels), and in background estimated from data (multijet background from misidentified and non-prompt leptons). The exceptions are the \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, where the background from simulation includes top-quark background, as shown in Tables~\\ref{tab:tChan7XS}$-$\\ref{tab:sChan8XS} in Appendix~\\ref{app:allUncs}. \nThe normalisation of the main background processes is determined from data, either by inclusion of normalisation uncertainties as nuisance parameters in the fit used to extract the signal, or through dedicated techniques based on data. In the \\myt-channel\\xspace\\ and \\mys-channel\\xspace\\ measurements, the uncertainties in the theoretical cross-section predictions for the top-quark, \\ensuremath{W\/Z}+jets\\xspace, and diboson processes are included. In the \\ensuremath{tW}\\xspace\\ measurements, the uncertainties in the theoretical cross-section predictions for the top-quark and diboson processes are taken into account. In the ATLAS measurements of the \\myt-channel\\xspace\\ process at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, the uncertainty in the multijet background is estimated by comparing background estimates made using different techniques based on simulation and data samples.\nIn the ATLAS \\ensuremath{tW}\\xspace\\ analyses at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, the normalisation uncertainty in the background from misidentified and non-prompt leptons is obtained from variations in the data-based estimate.\nIn the ATLAS \\mys-channel\\xspace\\ analysis, the uncertainty assigned to the normalisation of the multijet background is based on control samples.\nFor all CMS measurements, background normalisations are constrained in the fits to data. In the CMS measurements of the \\myt-channel\\xspace\\ and \\mys-channel\\xspace\\ processes, the uncertainties in the multijet background are assessed by comparing the results of alternative background estimation methods based on data. \nHence, the associated uncertainties are considered uncorrelated between ATLAS and CMS, between different production modes, and between different centre-of-mass energies. \n\n\n\\noindent {\\bf Jets} \\\\ In the analyses, the uncertainties related to the reconstruction and energy calibration of jets are propagated through variations in the modelling of the detector response. These uncertainties, classified in categories as JES, jet identification (JetID), and jet energy resolution (JER), are discussed below.\n\n\\begin{itemize}\n\\item {\\it Jet energy scale} (Correlation 0 and --- for JES flavour)\\\\ \nThe JES is derived using information from data and simulation. Its uncertainty increases with increasing $|\\eta|$ and decreases with increasing $\\pT$ of the reconstructed jet.\n\nFor all of the ATLAS measurements, except the $\\ensuremath{tW}\\xspace$ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the JES uncertainty is split into  components originating from the jet calibration procedure; most of them are derived from in situ techniques based on data~\\cite{PERF-2011-03,PERF-2012-01}. These components are categorised as modelling, detector, calibration method, and statistical components, which are grouped into the ``JES common'' uncertainty, as well as a flavour-dependence component (``JES flavour''),  which accounts for the flavour composition of the jets and the calorimeter response to jets of different flavours. The modelling of additional $pp$ collisions in each bunch-crossing (pile-up) is considered separately, as discussed below. The  $\\eta$-dependent component is dominant for the \\myt-channel\\xspace\\ production mode. Thus, the JES common uncertainty is considered uncorrelated between the \\myt-channel\\xspace\\ and the other single-top-quark production modes. For the \\ensuremath{tW}\\xspace analysis at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, the modelling component, which is constrained in the fit to data, is dominant. The uncertainty in the flavour composition of the jets is dominant for the \\mys-channel\\xspace. \n\nFor the CMS measurements, sources contributing to the JES uncertainty are combined together into the ``JES common'' uncertainty, and the effect is propagated to the cross-section measurements through $\\eta$- and $\\ensuremath{p_{\\mathrm{T}}}\\xspace$-dependent JES uncertainties~\\cite{CMS-JME-10-011,CMS-JME-13-004}. The jet energy corrections and their corresponding uncertainties are extracted from data. The  JES uncertainty is estimated from its effect on the normalisation and shape of the discriminant in each analysis. The JES uncertainty is considered uncorrelated between the \\myt-channel\\xspace\\ and the other single-top-quark production modes because it is dominated by the forward jet in the \\myt-channel\\xspace.\n\nThe correlation between the JES common uncertainty (or the JES uncertainty for the $\\ensuremath{tW}\\xspace$ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}) in ATLAS and the JES uncertainty in CMS follows the prescription in Refs.~\\cite{ATL-PHYS-PUB-2014-020, ATL-PHYS-PUB-2015-049}, with the slight differences for the \\myt-channel\\xspace\\ described above. The JES common (or JES) uncertainty is considered uncorrelated between ATLAS and CMS, between centre-of-mass energies, and between production modes. Within the ATLAS experiment, the JES common uncertainty is considered correlated between \\ensuremath{tW}\\xspace\\ and \\mys-channel\\xspace\\ and uncorrelated between \\myt-channel\\xspace\\ and the other production modes. For the ATLAS \\myt-channel\\xspace\\ analyses, a correlation of 0.75 is assumed between \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, since these analyses are mainly affected by the same uncertainty components. This correlation value is estimated by comparing variations of the JES uncertainty components in these two measurements.\n\nIn all CMS measurements and in the ATLAS $\\ensuremath{tW}\\xspace$ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the JES uncertainty is not split and therefore the JES flavour uncertainty is included in the overall JES uncertainty. For the ATLAS measurements where this component is available, the JES flavour uncertainty is considered correlated between different production modes and uncorrelated between centre-of-mass energies.\n\n\nThe JES uncertainty is one of the dominant contributions in most of the single-top-quark measurements. To ensure the robustness of the results against the correlation assumptions for this large uncertainty, the combination is performed with alternative correlation values, as discussed in Section~\\ref{sec:tests}.\n\n\\item {\\it Jet identification} (Correlation ---)\\\\ In the ATLAS measurements, the JetID uncertainty includes the jet and vertex reconstruction efficiency uncertainties. In the CMS measurements, this uncertainty is included in the JES uncertainty.\n  For ATLAS, it is considered correlated between the different production modes at the same centre-of-mass energy and uncorrelated for the other cases.\n  \n\\item  {\\it Jet energy resolution} (Correlation 0)\\\\ The uncertainty in the JER, which is not split into components, is extracted from data. Generally, the JER uncertainty is propagated via a nuisance parameter in the signal extraction fit, except for the ATLAS \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, and the CMS \\mys-channel\\xspace\\ measurement, where this uncertainty is determined using pseudoexperiments. The JER uncertainty is considered uncorrelated between ATLAS and CMS, and between centre-of-mass energies. It is considered correlated between different production modes.\n\\end{itemize}\n\n\\noindent {\\bf Detector modelling}  \\\\ This category includes the uncertainty in the modelling of leptons, magnitude of the missing transverse momentum (\\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace), and identification of jets from $b$-quarks (\\btag).\n\n\\begin{itemize}\n\\item {\\it Lepton modelling} (Correlation 0)\\\\ \nThe lepton modelling uncertainty includes components associated with the lepton energy scale and resolution, reconstruction and trigger efficiencies. This uncertainty is considered uncorrelated between ATLAS~\\cite{PERF-2014-05,PERF-2013-05,PERF-2013-03,PERF-2016-01} and CMS~\\cite{Sirunyan:2017ulk} and between different centre-of-mass energies, since it is determined from data. It is considered correlated between different production modes.\n\n\\item {\\it Hadronic part of the high-level trigger} (Correlation ---)\\\\ \nIn the CMS \\myt-channel\\xspace\\ cross-section measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the high-level trigger (HLT) criteria for the electron channel are based on the presence of an electron together with a \\btagged jet. In this analysis, the uncertainty in the modelling of the hadronic part of the HLT requirement is determined from data. This uncertainty is only evaluated in this one measurement.\n\n\\item {\\it \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace modelling} (Correlation 0)\\\\ \nThe ATLAS measurements include separate components for the uncertainties in the energy scale and resolution of the \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace~\\cite{PERF-2014-04}. The CMS measurements account for a combined \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace scale and resolution uncertainty~\\cite{CMS-JME-10-011,CMS-JME-13-003}, arising from the jet-energy uncertainties.\nAdditionally, CMS accounts for an uncertainty in \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace arising from energy deposits in the detector that are not included in the reconstruction of leptons, photons, and jets.\nThe \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace uncertainty is considered uncorrelated between ATLAS and CMS, and between different centre-of-mass energies. It is considered correlated between production modes, except for the ATLAS and CMS \\ensuremath{tW}\\xspace analyses at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, where it is considered  uncorrelated with the other production modes because the \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace uncertainty is constrained in the fit to data. In the ATLAS \\ensuremath{tW}\\xspace analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, this uncertainty is included in the pile-up modelling uncertainty.\n\n\\item {\\it \\btag} (Correlation 0)\\\\ \nIn the ATLAS analyses, \\btag modelling uncertainties are split into components associated with \\bquark, \\ensuremath{c}\\xspace-quark, and light-flavour quark and gluon jets~\\cite{PERF-2012-04,ATLAS-CONF-2014-004,ATLAS-CONF-2014-046}. They are evaluated by varying the $\\pT$-dependence ($\\eta$-dependence in the case of light-flavour jets) of the flavour-dependent scale factors applied to each jet in simulation within a range that reflects the systematic uncertainty in the measured tagging efficiency and misidentification rates. This uncertainty is not considered in the ATLAS \\ensuremath{tW}\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ because no \\btag criterion is applied in the event selection. \nIn the CMS measurements, the uncertainties in \\btag efficiency and misidentification rates of jets initiated by light-flavour quarks and gluons are derived from data, using control samples~\\cite{CMS-BTV-12-001,CMS-PAS-BTV-13-001}. The CMS uncertainties are propagated to the cross-section measurements using pseudoexperiments. Exceptions are the \\myt-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and the \\ensuremath{tW}\\xspace measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, where these uncertainties are constrained in the fit to data.\n\nThe two collaborations split up the different sources of systematic uncertainties related to \\btag in a different way. However, the different sources are combined by adding their contributions in quadrature to obtain a single \\btag uncertainty per analysis. This means that the \\btag uncertainty also contains the uncertainties associated with the misidentification rates of jets initiated by charm quarks, light-flavour quarks and gluons. The resulting uncertainty is considered uncorrelated between ATLAS and CMS, and between different centre-of-mass energies. It is considered correlated between different production modes.\n\n\\item {\\it Pile-up modelling} (Correlation 0)\\\\ \nIn both ATLAS and CMS, simulated events are reweighted to match the distribution of the average number of interactions per bunch-crossing in data. The corresponding uncertainty is obtained from in situ techniques based on data and simulated event samples. In the ATLAS analyses at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the uncertainty due to pile-up is derived from the impact of the reweighting on \\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace. In the ATLAS analyses at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, this uncertainty is evaluated as a component of the JES, separated into four terms (number of primary vertices, average number of collisions per bunch-crossing, average pile-up energy density in the calorimeter, and \\pT\\ dependence) since the pile-up calibration (assuming average conditions during 8~\\TeV\\ data-taking) is applied to both data and simulation before selecting and calibrating the jets~\\cite{ATL-PHYS-PUB-2015-049}. \nIn CMS, the reweighting uses a model with a free parameter that can be interpreted as an effective cross-section for inelastic $pp$ interactions. This uncertainty is obtained from a fit to the number of additional primary vertices in simulation. In the CMS analyses, this uncertainty is introduced as a nuisance parameter in the fit. The only exception is the \\mys-channel\\xspace\\ measurement, where the pile-up uncertainty is estimated from pseudoexperiments. In all cases, the effects of pile-up on the jet energy and the isolation of leptons are taken into account in the jet and lepton uncertainties respectively. The pile-up uncertainty is considered uncorrelated between ATLAS and CMS and between different centre-of-mass energies. It is considered correlated between different production modes~\\cite{ATL-PHYS-PUB-2014-020, ATL-PHYS-PUB-2015-049}.\n\n\\end{itemize}\n\n\\subsection{Systematic uncertainties in theoretical cross-section predictions}\n\\label{sec:systcat_theo}\n\nThe systematic uncertainties in the combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ value are evaluated from uncertainties in the individual cross-section measurements $\\ensuremath{\\sigma_{\\rm meas.}}\\xspace$ and the theoretical predictions $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$. The uncertainties associated with $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$ are discussed in Section~\\ref{sec:theory}; they are summarised in Table~\\ref{tab:xs}. The correlation assumptions for the systematic uncertainties related to the theoretical cross-section are explained below. In Section~\\ref{sec:tests}, the stability of the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ combination against variations in the correlations is examined. For clarity, the  correlations are given in parentheses next to the systematic-uncertainty name. These correlations are used in the combination of the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ extractions.\\\\\n\n\\noindent {\\bf PDF+$\\alpha_{\\rm s}$} (Correlation 1 for centre-of-mass energies and 0.5 for production modes) \\\\ The PDF uncertainty is considered correlated between centre-of-mass energies and 50\\% correlated between production modes, since different production modes have one initial-state particle in common (a quark or a gluon), but not both.\n\n\\noindent {\\bf Renormalisation and factorisation scales} (Correlation 1 for \\myt-channel\\xspace\\ and \\mys-channel\\xspace\\ and 0 for \\ensuremath{tW}\\xspace) \\\\ The renormalisation and factorisation scale uncertainties in $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$ are\nconsidered correlated between production modes and centre-of-mass energies, except between the \\ensuremath{tW}\\xspace production mode and the other production modes, where they are considered uncorrelated because the \\ensuremath{tW}\\xspace prediction is computed at a different order in perturbation theory.\n\n\\noindent {\\bf Top-quark mass} (Correlation 1)\\\\ \nThe uncertainty due to $m_{\\myt}\\xspace$ is evaluated by varying $m_{\\myt}\\xspace$ from its central value of 172.5~\\GeV\\ by $\\pm1$~\\GeV\\ and evaluating the corresponding change in cross-section using the parameterisation given in Ref.~\\cite{Czakon:2013goa}, as discussed in Section~\\ref{sec:theory}. This uncertainty is considered correlated between centre-of-mass energies and production modes.\n\n\\noindent {\\bf $E_{\\rm beam}$} (Correlation 1)\\\\ The uncertainty in the cross-section due to the uncertainty in $E_{\\rm beam}$\nis estimated by computing the cross-section variation corresponding to a $\\pm 1$ standard deviation shift in the beam-energy uncertainty.\nIt is considered correlated between centre-of-mass energies and production modes.\n\n\n \\section{Combinations of cross-section measurements}\n\\label{sec:xscomb}\nThe cross-section measurements described in\nSection~\\ref{sec:atlascmsxs} are combined at each centre-of-mass\nenergy for each production mode. Systematic uncertainties are\ncategorised and correlation assumptions are employed according to\nSection~\\ref{sec:systcat}. The combinations are performed using the\niterative BLUE method, as described in Section~\\ref{sec:method}. \n\nAs discussed in Section~\\ref{sec:method}, the uncertainty in the\nmeasured cross-section associated with the $m_{\\myt}\\xspace$ variation is\nnot considered in the combination of cross-sections. However, the\nshift in the combined cross-section resulting from a variation of $\\pm\n1~\\GeV$ in the top-quark mass is provided where this information is\navailable. This is calculated by repeating the combination with the\nup-shifted and down-shifted input cross-sections. In measurements\nwhere only the magnitude of the shift is available for one experiment,\nthe sign of the shift is assumed to be the same for both experiments,\nas discussed in Section~\\ref{sec:systcat_exp}. If the uncertainty associated with the $m_{\\myt}\\xspace$ variation is not\navailable for one or both of the input measurements, then no shift in the combined cross-section is given.\n\n\nAdditional information about the uncertainties considered in the combination of cross-section measurements is provided in Appendix~\\ref{app:allUncs}.\n\n\\subsection{Combinations of \\myt-channel\\xspace\\ cross-section measurements}\n\\label{sec:xscomb_tch}\nThe combination of the ATLAS and CMS \\myt-channel\\xspace\\ cross-section measurements\nat \\ensuremath{\\sqrt{s}=\\lhcenergySeven}~\\cite{TOPQ-2012-21,CMS-TOP-11-021} results, after\none iteration, in \n\\begin{eqnarray*}\n\\ensuremath{\\sigmatch = 67.5\\pm2.4\\;(\\text{stat.})\\pm5.0\\;(\\text{syst.})\\pm1.1\\;(\\text{lumi.})\\,\\,\\text{pb} = 67.5\\pm5.7\\,\\,\\text{pb}}.\n\\end{eqnarray*}\nThe relative uncertainty is $\\ensuremath{8.4\\%}$, which improves\non the uncertainty of 9.1\\% in the most precise individual\nmeasurement from CMS~\\cite{CMS-TOP-11-021}. The $\\chi^2$ for the combination is\n$\\ensuremath{0.01}$, corresponding to a probability of\n$\\ensuremath{93}\\%$. The CMS weight in the combination is\n$\\ensuremath{0.58}$, while the ATLAS weight is\n$\\ensuremath{0.42}$. The overall correlation between the\ntwo measurements is \\ensuremath{20}\\%. The contribution from\neach uncertainty category to the total uncertainty in the combined \\myt-channel\\xspace\\ cross-section \nmeasurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ is shown in Table~\\ref{tab:tChanRes}(a).\n\nThe combination of the ATLAS and CMS \\myt-channel\\xspace\\ cross-section measurements\nat \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2015-05,CMS-TOP-12-038} results, after\ntwo iterations, in a cross-section of\n\\begin{eqnarray*}\n\\ensuremath{\\sigmatch = 87.7\\pm1.1\\;(\\text{stat.})\\pm5.5\\;(\\text{syst.})\\pm1.5\\;(\\text{lumi.})\\,\\,\\text{pb} = 87.7\\pm5.8\\,\\,\\text{pb}}.\n\\end{eqnarray*}\nThe relative uncertainty is $\\ensuremath{6.7\\%}$, which improves\non the uncertainty of 7.5\\% in the most precise individual\nmeasurement from ATLAS~\\cite{TOPQ-2015-05}. The $\\chi^2$ for the combination is\n$\\ensuremath{0.59}$, corresponding to a probability of\n$\\ensuremath{44}\\%$. This probability is lower than the\nprobability of the combination at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ because of the differences between the ATLAS and CMS measured cross-sections and their small uncertainties. The ATLAS weight in the combination is\n$\\ensuremath{0.68}$, while the CMS weight is\n$\\ensuremath{0.32}$. The overall correlation between the\ntwo measurements is \\ensuremath{42}\\%. This is larger\nthan the correlation between the measurements at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ because the statistical and detector uncertainties are lower, thus increasing the importance of the theory modelling uncertainty (which is correlated between the two experiments), as shown in Appendix~\\ref{app:tch}. The contribution from each uncertainty category to the\ntotal uncertainty in the combined \\myt-channel\\xspace\\ cross-section measurement at \n\\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ is shown in Table~\\ref{tab:tChanRes}(b).\n\n\\begin{table}[!htbp]\n  \\caption{Contribution from each uncertainty category to the combined\n    \\myt-channel\\xspace\\ cross-section ($\\ensuremath{\\sigma_{{\\myt\\textrm{-chan.}}}}\\xspace$) uncertainty at\n    (a) \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and (b) \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. The total\n    uncertainty is computed by adding in quadrature all the individual systematic\n    uncertainties (including the uncertainty in the integrated luminosity) and the statistical uncertainty in data. Correlations of systematic uncertainties between\n    experiments are presented in Appendix~\\ref{app:tch}.}\n  \\begin{center}\n    \\begin{minipage}[b]{0.46\\hsize}\\centering\n      \\captionof{subfigure}{}\n      \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n \\multicolumn{3}{c}{ $\\ensuremath{\\sigma_{{\\myt\\textrm{-chan.}}}}\\xspace$, $\\sqrt{s} = $ 7~\\TeV } \\\\\n\\hline\n\\hline\n{\\bf Combined cross-section} & \\multicolumn{2}{c}{67.5~pb} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  [pb] \\\\\n\\hline\nData statistical & 3.5 & 2.4\\\\\n\\hline\nSimulation statistical & 1.4 & 0.9\\\\\nIntegrated luminosity & 1.7 & 1.1\\\\\nTheory modelling & 5.1 & 3.5\\\\\nBackground normalisation & 1.9 & 1.3\\\\\nJets & 3.4 & 2.3\\\\\nDetector modelling & 3.4 & 2.3\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 7.5 & 5.0\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 7.6 & 5.2\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 8.4 & 5.7\\\\\n\\hline\n \\hline\n\\end{tabular}\n     \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.46\\hsize}\\centering\n      \\captionof{subfigure}{}\n      \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n \\multicolumn{3}{c}{ $\\ensuremath{\\sigma_{{\\myt\\textrm{-chan.}}}}\\xspace$, $\\sqrt{s} = $ 8~\\TeV } \\\\\n\\hline\n\\hline\n{\\bf Combined cross-section} & \\multicolumn{2}{c}{87.7~pb} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  [pb] \\\\\n\\hline\nData statistical & 1.3 & 1.1\\\\\n\\hline\nSimulation statistical & 0.6 & 0.5\\\\\nIntegrated luminosity & 1.7 & 1.5\\\\\nTheory modelling & 5.3 & 4.7\\\\\nBackground normalisation & 1.2 & 1.1\\\\\nJets & 2.6 & 2.3\\\\\nDetector modelling & 1.8 & 1.6\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 6.3 & 5.5\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 6.5 & 5.7\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 6.7 & 5.8\\\\\n\\hline\n \\hline\n\\end{tabular}\n     \\end{minipage}\n  \\end{center}\n  \\label{tab:tChanRes}\n\\end{table}\n\nAt both centre-of-mass energies, the uncertainties from theory\nmodelling are found to be dominant. Details of the central values, the impact of\nindividual sources of uncertainties, and their correlations between experiments at\n\\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ can be found in Appendix~\\ref{app:tch}. \n\n\n\nThe shift in the combined cross-section at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ from a variation of $\\pm 1~\\GeV$ in the top-quark mass is $\\mp 0.8$~pb, which is similar to the shifts in the input measurements for the same $m_{\\myt}\\xspace$ variation. The shift in the combined cross-section at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ is not evaluated since no estimate is available for the CMS input measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}.\n\n\\subsection{Combinations of {$\\boldmath \\ensuremath{tW}\\xspace$} cross-section measurements}\n\\label{sec:xscomb_tW}\nThe combination of the ATLAS and CMS $\\ensuremath{tW}\\xspace$ cross-section measurements\nat \\ensuremath{\\sqrt{s}=\\lhcenergySeven}~\\cite{TOPQ-2011-17, CMS-TOP-11-022} yields, after\ntwo iterations, a cross-section of \n\\begin{eqnarray*}\n\\ensuremath{\\sigmatW = 16.3\\pm2.3\\;(\\text{stat.})\\pm3.3\\;(\\text{syst.})\\pm0.7\\;(\\text{lumi.})\\,\\,\\text{pb} = 16.3\\pm4.1\\,\\,\\text{pb}}.\n\\end{eqnarray*}\nThe relative uncertainty is $\\ensuremath{25\\%}$, which improves on the uncertainty of 28\\% in the most precise individual\nmeasurement from CMS~\\cite{CMS-TOP-11-022}. The $\\chi^2$ for the combination is\n$\\ensuremath{0.01}$, corresponding to a probability of\n$\\ensuremath{91}\\%$. The CMS weight in the combination is\n$\\ensuremath{0.59}$, while the ATLAS weight is\n$\\ensuremath{0.41}$. The overall correlation between the\ntwo measurements is \\ensuremath{17}\\%. The contribution from\neach uncertainty category to the total uncertainty in the combined  \\ensuremath{tW}\\xspace\\ \ncross-section measurement at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ is shown in\nTable~\\ref{tab:tWRes}(a).\n\nThe combination of the ATLAS and CMS $\\ensuremath{tW}\\xspace$ cross-section measurements\nat \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2012-20,CMS-TOP-12-040} results, after\ntwo iterations, in\n\\begin{eqnarray*}\n\\ensuremath{\\sigmatW = 23.1\\pm1.1\\;(\\text{stat.})\\pm3.3\\;(\\text{syst.})\\pm0.8\\;(\\text{lumi.})\\,\\,\\text{pb} = 23.1\\pm3.6\\,\\,\\text{pb}}.\n\\end{eqnarray*}\nThe relative uncertainty is $\\ensuremath{15.6\\%}$, which improves on the uncertainty of 16.5\\% in the most precise individual\nmeasurement from ATLAS~\\cite{TOPQ-2012-20}. The $\\chi^2$ for the combination is\n$\\ensuremath{0.01}$, corresponding to a probability of\n$\\ensuremath{94}\\%$. The ATLAS weight in the combination is\n$\\ensuremath{0.70}$, while the CMS weight is\n$\\ensuremath{0.30}$. The overall correlation between the\ntwo measurements is \\ensuremath{40}\\%.\nSimilar to the \\myt-channel\\xspace, this is larger than the correlation between the\nmeasurements at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ due to the increased importance of the\ntheory modelling uncertainties.\nThe contribution from\neach uncertainty category to the total uncertainty in the combined \\ensuremath{tW}\\xspace\\ \ncross-section measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ is shown in\nTable~\\ref{tab:tWRes}(b).\n\n\\begin{table}[!htbp]\n  \\caption{Contribution from each uncertainty category to the combined\n    \\ensuremath{tW}\\xspace cross-section ($\\ensuremath{\\sigma_{\\mytW}}\\xspace$) uncertainty at\n    (a) \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ and (b) \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. The total\n    uncertainty is computed by adding in quadrature all the individual systematic\n    uncertainties (including the uncertainty in the integrated luminosity) and the statistical uncertainty in data.\n    Correlations of systematic uncertainties between\n    experiments are presented in Appendix~\\ref{app:tw}.}\n  \\begin{center}\n    \\begin{minipage}[b]{0.46\\hsize}\\centering\n      \\captionof{subfigure}{}\n      \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n \\multicolumn{3}{c}{ $\\ensuremath{\\sigma_{\\mytW}}\\xspace$, $\\sqrt{s} = $ 7~\\TeV } \\\\\n\\hline\n\\hline\n{\\bf Combined cross-section} & \\multicolumn{2}{c}{16.3~pb} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  [pb] \\\\\n\\hline\nData statistical & 14.0 & 2.3\\\\\n\\hline\nSimulation statistical & 0.8 & 0.1\\\\\nIntegrated luminosity & 4.4 & 0.7\\\\\nTheory modelling & 13.9 & 2.3\\\\\nBackground normalisation & 6.0 & 1.0\\\\\nJets & 11.5 & 1.9\\\\\nDetector modelling & 6.2 & 1.0\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 20.0 & 3.3\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 20.5 & 3.3\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 24.8 & 4.1\\\\\n\\hline\n \\hline\n\\end{tabular}\n     \\end{minipage}\n    \\hfill\n    \\begin{minipage}[b]{0.46\\hsize}\\centering\n      \\captionof{subfigure}{}\n      \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n \\multicolumn{3}{c}{ $\\ensuremath{\\sigma_{\\mytW}}\\xspace$, $\\sqrt{s} = $ 8~\\TeV } \\\\\n\\hline\n\\hline\n{\\bf Combined cross-section} & \\multicolumn{2}{c}{23.1~pb} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  [pb] \\\\\n\\hline\nData statistical & 4.7 & 1.1\\\\\n\\hline\nSimulation statistical & 0.8 & 0.2\\\\\nIntegrated luminosity & 3.6 & 0.8\\\\\nTheory modelling & 11.8 & 2.7\\\\\nBackground normalisation & 2.2 & 0.5\\\\\nJets & 6.2 & 1.4\\\\\nDetector modelling & 4.9 & 1.1\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 14.4 & 3.3\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 14.8 & 3.4\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 15.6 & 3.6\\\\\n\\hline\n \\hline\n\\end{tabular}\n     \\end{minipage}\n  \\end{center}\n  \\label{tab:tWRes}\n\\end{table}\n\nAt both centre-of-mass energies, the uncertainties in the theory\nmodelling are found to be dominant. The jet uncertainties are also\nimportant. Details of the central values, the impact of\nindividual sources of uncertainties, and their correlations between experiments at\n\\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are presented in Appendix~\\ref{app:tw}.\n\nThe shift in the combined cross-section at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ from a variation of $\\pm\n1~\\GeV$ in the top-quark mass is $\\pm 1.1$~pb, which is similar in magnitude to that in the input measurements for the same $m_{\\myt}\\xspace$ variation. The shift in the combined cross-section at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}\\ is not evaluated since no estimates are available for the input measurements at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}.\n\n\\subsection{Combination of \\mys-channel\\xspace\\ cross-section measurements}\n\\label{sec:xscomb_sch}\n\nThe ATLAS and CMS \\mys-channel\\xspace\\ cross-section measurements suffer from large backgrounds, and the cross-section measurements have large uncertainties. Since the systematic uncertainties mainly affect the background prediction, they are not scaled in the iterative BLUE procedure. Only the luminosity uncertainty is scaled with the central value. The combination of the ATLAS and CMS \\mys-channel\\xspace\\ cross-section measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2015-01, CMS-TOP-13-009} results, after\ntwo iterations, in a cross-section of\n\n\\begin{eqnarray*}\n\\ensuremath{\\sigmasch = 4.9\\pm0.8\\;(\\text{stat.})\\pm1.2\\;(\\text{syst.})\\pm0.2\\;(\\text{lumi.})\\,\\,\\text{pb} = 4.9\\pm1.4\\,\\,\\text{pb}}.\n\\end{eqnarray*}\nThe relative uncertainty is $\\ensuremath{30\\%}$, very similar to the most precise individual measurement from ATLAS~\\cite{TOPQ-2015-01}. The $\\chi^2$ for the combination is $\\ensuremath{1.45}$, corresponding to a probability of $\\ensuremath{23}\\%$. \nThe ATLAS weight in the combination is $\\ensuremath{0.99}$, while the CMS weight is $\\ensuremath{0.01}$. The overall correlation between the two measurements is \\ensuremath{15}\\%. The contribution from each uncertainty category to the total uncertainty in the combined \\mys-channel\\xspace\\ cross-section measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ is shown in Table~\\ref{tab:sChanRes}.\n\n\\begin{table}[!htbp]\n  \\caption{Contribution from each uncertainty category to the combined\n    \\mys-channel\\xspace\\ cross-section ($\\ensuremath{\\sigma_{\\mys\\textrm{-chan.}}}\\xspace$) uncertainty at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. The total\n    uncertainty is computed by adding in quadrature all the individual systematic\n    uncertainties (including the uncertainty in the integrated luminosity) and the statistical uncertainty in data.\n    Correlations of systematic uncertainties between\n    experiments are presented in Appendix~\\ref{app:sch}.}\n  \\begin{center}\n    \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n \\multicolumn{3}{c}{ $\\ensuremath{\\sigma_{\\mys\\textrm{-chan.}}}\\xspace$, $\\sqrt{s} = $ 8~\\TeV } \\\\\n\\hline\n\\hline\n{\\bf Combined cross-section} & \\multicolumn{2}{c}{4.9~pb} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  [pb] \\\\\n\\hline\nData statistical & 16 & 0.8\\\\\n\\hline\nSimulation statistical & 12 & 0.6\\\\\nIntegrated luminosity & 5 & 0.2\\\\\nTheory modelling & 14 & 0.7\\\\\nBackground normalisation & 8 & 0.4\\\\\nJets & 13 & 0.6\\\\\nDetector modelling & 8 & 0.4\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 25 & 1.2\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 25 & 1.2\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 30 & 1.4\\\\\n\\hline\n \\hline\n\\end{tabular}\n   \\end{center}\n  \\label{tab:sChanRes}\n\\end{table}\n\nSince the ATLAS measurement has a large weight in the combination, the importance of each uncertainty in the combination is similar to that in the ATLAS measurement, as presented in Appendix~\\ref{app:sch}.\n\nThe shift in the combined cross-section at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ from a variation in the top-quark mass is not evaluated since no estimate is available for the ATLAS input measurement.\n\n\\subsection{Summary of cross-section combinations}\nA summary of the cross-sections measured by ATLAS and CMS\nand their combinations in all single-top-quark production modes at each\ncentre-of-mass energy is shown in\nFigure~\\ref{fig:xssum}. The measurements are compared with the theoretical predictions shown in Table~\\ref{tab:xs}: NNLO for \\myt-channel\\xspace\\ only, NLO and NLO+NNLL for all three production modes. For the NLO calculation, the renormalisation- and factorisation-scale uncertainties and the sum in quadrature of the contributions from scale, PDF, and $\\alpha_{\\rm s}$ are shown separately. Only the scale uncertainty is shown for the NNLO calculation. \nFor the NLO+NNLL calculation, the sum in quadrature of the contributions from scale, PDF, and $\\alpha_{\\rm s}$ is shown. All measurements are in good agreement with their corresponding theoretical predictions within their total uncertainties.\n\n\n\\begin{figure}[!ht]\n  \\begin{center}\n    \\includegraphics[width=1.0\\textwidth]{xs\/singletop_xsec_combinations}\n  \\end{center}\n  \\caption{Single-top-quark cross-section measurements performed by ATLAS and CMS, together with the combined results shown in Sections~\\ref{sec:xscomb_tch}$-$\\ref{sec:xscomb_sch}. These measurements are compared with the theoretical predictions at NLO and NLO+NNLL for all three production modes and the prediction at NNLO for \\myt-channel\\xspace\\ only. The corresponding theoretical uncertainties are also presented. The scale uncertainty for the NNLO prediction is small and is presented as a narrow band under the dashed line.}\n  \\label{fig:xssum}\n\\end{figure}\n\nThe stability of the combinations of the cross-section measurements to variations in\nthe correlation assumptions, discussed in Section~\\ref{sec:systcat}, is  checked for the theory modelling, JES, the most important contributions to the theoretical cross-section predictions\n(i.e.\\ PDF+$\\alpha_{\\rm s}$ and scale) and the integrated luminosity. The results\nof these tests show that their impacts on the cross-section\ncombinations are very small, similar to the stability tests\nfor the combination of the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ values discussed in Section~\\ref{sec:tests}.\n\n\n\n\n\n \\section{Combinations of {\\boldmath $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$} determinations}\n\\label{sec:vtbcomb}\n\\newcommand{\\ensuremath{0.03}}{\\ensuremath{0.03}}\n\\newcommand{\\ensuremath{1.02}}{\\ensuremath{1.02}}\n\\newcommand{\\ensuremath{1.04}}{\\ensuremath{1.04}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.06\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.04\\pm0.07}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.06\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.04\\pm0.07}}\n\\newcommand{\\ensuremath{6.5\\%}}{\\ensuremath{6.5\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.02\\pm0.03}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.02\\pm0.03}}\n\\newcommand{\\ensuremath{1.02\\pm0.03}}{\\ensuremath{1.02\\pm0.03}}\n\\newcommand{\\ensuremath{3.2\\%}}{\\ensuremath{3.2\\%}}\n\\newcommand{\\ensuremath{0.63}}{\\ensuremath{0.63}}\n\\newcommand{\\ensuremath{89}}{\\ensuremath{89}}\n\\newcommand{\\ensuremath{0.09}}{\\ensuremath{0.09}}\n\\newcommand{\\ensuremath{0.25}}{\\ensuremath{0.25}}\n\\newcommand{\\ensuremath{0.48}}{\\ensuremath{0.48}}\n\\newcommand{\\ensuremath{0.53}}{\\ensuremath{0.53}}\n\\newcommand{\\ensuremath{0.16}}{\\ensuremath{0.16}}\n\\newcommand{\\ensuremath{0.16}}{\\ensuremath{0.16}}\n\\newcommand{\\ensuremath{0.27}}{\\ensuremath{0.27}}\n\\newcommand{\\ensuremath{-0.80}}{\\ensuremath{-0.80}}\n\\newcommand{\\ensuremath{0.42}}{\\ensuremath{0.42}}\n\\newcommand{\\ensuremath{0.46}}{\\ensuremath{0.46}}\n\\newcommand{\\ensuremath{0.40}}{\\ensuremath{0.40}}\n\\newcommand{\\ensuremath{0.21}}{\\ensuremath{0.21}}\n\\newcommand{\\ensuremath{0.33}}{\\ensuremath{0.33}}\n\\newcommand{\\ensuremath{0.20}}{\\ensuremath{0.20}}\n\\newcommand{\\ensuremath{1.56}}{\\ensuremath{1.56}}\n \\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{1.02}}{\\ensuremath{1.02}}\n\\newcommand{\\ensuremath{1.03}}{\\ensuremath{1.03}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.14\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.}) = 1.03\\pm0.16}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.14\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.}) = 1.03\\pm0.16}}\n\\newcommand{\\ensuremath{15.2\\%}}{\\ensuremath{15.2\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 1.02\\pm0.08}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 1.02\\pm0.08}}\n\\newcommand{\\ensuremath{1.02\\pm0.08}}{\\ensuremath{1.02\\pm0.08}}\n\\newcommand{\\ensuremath{7.6\\%}}{\\ensuremath{7.6\\%}}\n\\newcommand{\\ensuremath{0.02}}{\\ensuremath{0.02}}\n\\newcommand{\\ensuremath{100}}{\\ensuremath{100}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.11}}{\\ensuremath{0.11}}\n\\newcommand{\\ensuremath{0.61}}{\\ensuremath{0.61}}\n\\newcommand{\\ensuremath{-0.05}}{\\ensuremath{-0.05}}\n\\newcommand{\\ensuremath{0.16}}{\\ensuremath{0.16}}\n\\newcommand{\\ensuremath{-0.06}}{\\ensuremath{-0.06}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.43}}{\\ensuremath{0.43}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{0.12}}{\\ensuremath{0.12}}\n\\newcommand{\\ensuremath{0.35}}{\\ensuremath{0.35}}\n\\newcommand{\\ensuremath{0.29}}{\\ensuremath{0.29}}\n\\newcommand{\\ensuremath{0.17}}{\\ensuremath{0.17}}\n\\newcommand{\\ensuremath{1.55}}{\\ensuremath{1.55}}\n \\newcommand{\\ensuremath{0.14}}{\\ensuremath{0.14}}\n\\newcommand{\\ensuremath{0.97}}{\\ensuremath{0.97}}\n\\newcommand{\\ensuremath{0.93}}{\\ensuremath{0.93}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.}) = 0.93\\pm0.28}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.}) = 0.93\\pm0.28}}\n\\newcommand{\\ensuremath{29.7\\%}}{\\ensuremath{29.7\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 0.97\\pm0.14}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 0.97\\pm0.14}}\n\\newcommand{\\ensuremath{0.97\\pm0.14}}{\\ensuremath{0.97\\pm0.14}}\n\\newcommand{\\ensuremath{14.8\\%}}{\\ensuremath{14.8\\%}}\n\\newcommand{\\ensuremath{1.42}}{\\ensuremath{1.42}}\n\\newcommand{\\ensuremath{23}}{\\ensuremath{23}}\n\\newcommand{\\ensuremath{0.99}}{\\ensuremath{0.99}}\n\\newcommand{\\ensuremath{-1.19}}{\\ensuremath{-1.19}}\n\\newcommand{\\ensuremath{0.01}}{\\ensuremath{0.01}}\n\\newcommand{\\ensuremath{1.19}}{\\ensuremath{1.19}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{3.60}}{\\ensuremath{3.60}}\n \\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.04\\pm0.03\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.04\\pm0.05}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.04\\pm0.03\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.04\\pm0.05}}\n\\newcommand{\\ensuremath{1.04}}{\\ensuremath{1.04}}\n\\newcommand{\\ensuremath{0.03}}{\\ensuremath{0.03}}\n\\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{0.05}}{\\ensuremath{0.05}}\n\\newcommand{\\ensuremath{0.02}}{\\ensuremath{0.02}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.02\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.02\\pm0.02}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.02\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.02\\pm0.02}}\n\\newcommand{\\ensuremath{1.02\\pm0.02}}{\\ensuremath{1.02\\pm0.02}}\n\\newcommand{\\ensuremath{2.4\\%}}{\\ensuremath{2.4\\%}}\n\\newcommand{\\ensuremath{1.54}}{\\ensuremath{1.54}}\n\\newcommand{\\ensuremath{67}}{\\ensuremath{67}}\n\\newcommand{\\ensuremath{0.23}}{\\ensuremath{0.23}}\n\\newcommand{\\ensuremath{0.58}}{\\ensuremath{0.58}}\n\\newcommand{\\ensuremath{0.26}}{\\ensuremath{0.26}}\n\\newcommand{\\ensuremath{0.46}}{\\ensuremath{0.46}}\n\\newcommand{\\ensuremath{0.23}}{\\ensuremath{0.23}}\n\\newcommand{\\ensuremath{0.31}}{\\ensuremath{0.31}}\n\\newcommand{\\ensuremath{0.28}}{\\ensuremath{0.28}}\n\\newcommand{\\ensuremath{-1.23}}{\\ensuremath{-1.23}}\n\\newcommand{\\ensuremath{0.43}}{\\ensuremath{0.43}}\n\\newcommand{\\ensuremath{0.44}}{\\ensuremath{0.44}}\n\\newcommand{\\ensuremath{0.44}}{\\ensuremath{0.44}}\n\\newcommand{\\ensuremath{0.44}}{\\ensuremath{0.44}}\n\\newcommand{\\ensuremath{0.44}}{\\ensuremath{0.44}}\n\\newcommand{\\ensuremath{0.45}}{\\ensuremath{0.45}}\n\\newcommand{\\ensuremath{1.78}}{\\ensuremath{1.78}}\n \\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.04\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.04\\pm0.08}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.04\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.04\\pm0.08}}\n\\newcommand{\\ensuremath{1.04}}{\\ensuremath{1.04}}\n\\newcommand{\\ensuremath{0.03}}{\\ensuremath{0.03}}\n\\newcommand{\\ensuremath{0.07}}{\\ensuremath{0.07}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.02\\pm0.04}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 1.02\\pm0.04}}\n\\newcommand{\\ensuremath{1.02\\pm0.04}}{\\ensuremath{1.02\\pm0.04}}\n\\newcommand{\\ensuremath{3.8\\%}}{\\ensuremath{3.8\\%}}\n\\newcommand{\\ensuremath{0.57}}{\\ensuremath{0.57}}\n\\newcommand{\\ensuremath{90}}{\\ensuremath{90}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.58}}{\\ensuremath{0.58}}\n\\newcommand{\\ensuremath{0.38}}{\\ensuremath{0.38}}\n\\newcommand{\\ensuremath{-0.20}}{\\ensuremath{-0.20}}\n\\newcommand{\\ensuremath{0.18}}{\\ensuremath{0.18}}\n\\newcommand{\\ensuremath{-0.38}}{\\ensuremath{-0.38}}\n\\newcommand{\\ensuremath{0.36}}{\\ensuremath{0.36}}\n\\newcommand{\\ensuremath{0.27}}{\\ensuremath{0.27}}\n\\newcommand{\\ensuremath{0.68}}{\\ensuremath{0.68}}\n\\newcommand{\\ensuremath{0.70}}{\\ensuremath{0.70}}\n\\newcommand{\\ensuremath{0.69}}{\\ensuremath{0.69}}\n\\newcommand{\\ensuremath{0.70}}{\\ensuremath{0.70}}\n\\newcommand{\\ensuremath{0.70}}{\\ensuremath{0.70}}\n\\newcommand{\\ensuremath{0.72}}{\\ensuremath{0.72}}\n\\newcommand{\\ensuremath{1.70}}{\\ensuremath{1.70}}\n \\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 0.97\\pm0.05\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 0.97\\pm0.07}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 0.97\\pm0.05\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 0.97\\pm0.07}}\n\\newcommand{\\ensuremath{0.97}}{\\ensuremath{0.97}}\n\\newcommand{\\ensuremath{0.05}}{\\ensuremath{0.05}}\n\\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{0.07}}{\\ensuremath{0.07}}\n\\newcommand{\\ensuremath{0.03}}{\\ensuremath{0.03}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 0.98\\pm0.02\\;(\\text{stat.})\\pm0.02\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 0.98\\pm0.03}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 0.98\\pm0.02\\;(\\text{stat.})\\pm0.02\\;(\\text{syst.})\\pm0.00\\;(\\text{lumi.}) = 0.98\\pm0.03}}\n\\newcommand{\\ensuremath{0.98\\pm0.03}}{\\ensuremath{0.98\\pm0.03}}\n\\newcommand{\\ensuremath{3.4\\%}}{\\ensuremath{3.4\\%}}\n\\newcommand{\\ensuremath{300.77}}{\\ensuremath{300.77}}\n\\newcommand{\\ensuremath{0}}{\\ensuremath{0}}\n\\newcommand{\\ensuremath{0.97}}{\\ensuremath{0.97}}\n\\newcommand{\\ensuremath{-17.34}}{\\ensuremath{-17.34}}\n\\newcommand{\\ensuremath{0.03}}{\\ensuremath{0.03}}\n\\newcommand{\\ensuremath{17.34}}{\\ensuremath{17.34}}\n\\newcommand{\\ensuremath{0.55}}{\\ensuremath{0.55}}\n\\newcommand{\\ensuremath{18.56}}{\\ensuremath{18.56}}\n \\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{0.02}}{\\ensuremath{0.02}}\n\\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{1.02}}{\\ensuremath{1.02}}\n\\newcommand{\\ensuremath{1.04}}{\\ensuremath{1.04}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.08\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.04\\pm0.08}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.08\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.04\\pm0.08}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.06\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) = 1.04\\pm0.08}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.04\\pm0.01\\;(\\text{stat.})\\pm0.06\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) = 1.04\\pm0.08}}\n\\newcommand{\\ensuremath{4.2\\%}}{\\ensuremath{4.2\\%}}\n\\newcommand{\\ensuremath{2.1\\%}}{\\ensuremath{2.1\\%}}\n\\newcommand{\\ensuremath{7.7\\%}}{\\ensuremath{7.7\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.02\\pm0.04}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.01\\;(\\text{stat.})\\pm0.04\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.}) = 1.02\\pm0.04}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|=~&~1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 1.02\\pm0.04}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|=~&~1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 1.02\\pm0.04}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.})}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.})}}\n\\newcommand{\\ensuremath{1.02\\pm0.04}}{\\ensuremath{1.02\\pm0.04}}\n\\newcommand{\\ensuremath{3.9\\%}}{\\ensuremath{3.9\\%}}\n\\newcommand{\\ensuremath{0.63}}{\\ensuremath{0.63}}\n\\newcommand{\\ensuremath{89}}{\\ensuremath{89}}\n\\newcommand{\\ensuremath{0.09}}{\\ensuremath{0.09}}\n\\newcommand{\\ensuremath{0.25}}{\\ensuremath{0.25}}\n\\newcommand{\\ensuremath{0.49}}{\\ensuremath{0.49}}\n\\newcommand{\\ensuremath{0.54}}{\\ensuremath{0.54}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{0.16}}{\\ensuremath{0.16}}\n\\newcommand{\\ensuremath{0.27}}{\\ensuremath{0.27}}\n\\newcommand{\\ensuremath{-0.80}}{\\ensuremath{-0.80}}\n\\newcommand{\\ensuremath{0.54}}{\\ensuremath{0.54}}\n\\newcommand{\\ensuremath{0.55}}{\\ensuremath{0.55}}\n\\newcommand{\\ensuremath{0.52}}{\\ensuremath{0.52}}\n\\newcommand{\\ensuremath{0.33}}{\\ensuremath{0.33}}\n\\newcommand{\\ensuremath{0.44}}{\\ensuremath{0.44}}\n\\newcommand{\\ensuremath{0.31}}{\\ensuremath{0.31}}\n\\newcommand{\\ensuremath{1.45}}{\\ensuremath{1.45}}\n \\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.04}}{\\ensuremath{0.04}}\n\\newcommand{\\ensuremath{0.09}}{\\ensuremath{0.09}}\n\\newcommand{\\ensuremath{1.02}}{\\ensuremath{1.02}}\n\\newcommand{\\ensuremath{1.03}}{\\ensuremath{1.03}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.16\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.}) = 1.03\\pm0.17}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.16\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.}) = 1.03\\pm0.17}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.14\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.})\\pm0.07\\;(\\text{theo.}) = 1.03\\pm0.17}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 1.03\\pm0.05\\;(\\text{stat.})\\pm0.14\\;(\\text{syst.})\\pm0.03\\;(\\text{lumi.})\\pm0.07\\;(\\text{theo.}) = 1.03\\pm0.17}}\n\\newcommand{\\ensuremath{7.2\\%}}{\\ensuremath{7.2\\%}}\n\\newcommand{\\ensuremath{3.6\\%}}{\\ensuremath{3.6\\%}}\n\\newcommand{\\ensuremath{16.8\\%}}{\\ensuremath{16.8\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.03\\;(\\text{stat.})\\pm0.08\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 1.02\\pm0.09}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.03\\;(\\text{stat.})\\pm0.08\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 1.02\\pm0.09}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|=~&~1.02\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.09\\;(\\text{meas.})\\pm0.04\\;(\\text{theo.}) = 1.02\\pm0.09}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|=~&~1.02\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.09\\;(\\text{meas.})\\pm0.04\\;(\\text{theo.}) = 1.02\\pm0.09}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 1.02\\pm0.09\\;(\\text{meas.})\\pm0.04\\;(\\text{theo.})}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 1.02\\pm0.09\\;(\\text{meas.})\\pm0.04\\;(\\text{theo.})}}\n\\newcommand{\\ensuremath{1.02\\pm0.09}}{\\ensuremath{1.02\\pm0.09}}\n\\newcommand{\\ensuremath{8.4\\%}}{\\ensuremath{8.4\\%}}\n\\newcommand{\\ensuremath{0.02}}{\\ensuremath{0.02}}\n\\newcommand{\\ensuremath{100}}{\\ensuremath{100}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.11}}{\\ensuremath{0.11}}\n\\newcommand{\\ensuremath{0.61}}{\\ensuremath{0.61}}\n\\newcommand{\\ensuremath{-0.05}}{\\ensuremath{-0.05}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{-0.05}}{\\ensuremath{-0.05}}\n\\newcommand{\\ensuremath{0.16}}{\\ensuremath{0.16}}\n\\newcommand{\\ensuremath{0.08}}{\\ensuremath{0.08}}\n\\newcommand{\\ensuremath{0.49}}{\\ensuremath{0.49}}\n\\newcommand{\\ensuremath{0.22}}{\\ensuremath{0.22}}\n\\newcommand{\\ensuremath{0.20}}{\\ensuremath{0.20}}\n\\newcommand{\\ensuremath{0.39}}{\\ensuremath{0.39}}\n\\newcommand{\\ensuremath{0.34}}{\\ensuremath{0.34}}\n\\newcommand{\\ensuremath{0.21}}{\\ensuremath{0.21}}\n\\newcommand{\\ensuremath{1.54}}{\\ensuremath{1.54}}\n \\newcommand{\\ensuremath{0.12}}{\\ensuremath{0.12}}\n\\newcommand{\\ensuremath{0.02}}{\\ensuremath{0.02}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{0.97}}{\\ensuremath{0.97}}\n\\newcommand{\\ensuremath{0.93}}{\\ensuremath{0.93}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.}) = 0.93\\pm0.28}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.}) = 0.93\\pm0.28}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) = 0.93\\pm0.28}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|^2 = 0.93\\pm0.15\\;(\\text{stat.})\\pm0.23\\;(\\text{syst.})\\pm0.05\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) = 0.93\\pm0.28}}\n\\newcommand{\\ensuremath{4.6\\%}}{\\ensuremath{4.6\\%}}\n\\newcommand{\\ensuremath{2.3\\%}}{\\ensuremath{2.3\\%}}\n\\newcommand{\\ensuremath{30.1\\%}}{\\ensuremath{30.1\\%}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 0.97\\pm0.15}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.}) = 0.97\\pm0.15}}\n\\newcommand{\\ensuremath{|\\flv\\vtb|=~&~0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~0.97\\pm0.15\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 0.97\\pm0.15}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace|=~&~0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~0.97\\pm0.15\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 0.97\\pm0.15}}\n\\newcommand{\\ensuremath{|\\flv\\vtb| = 0.97\\pm0.15\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.})}}{\\ensuremath{|\\ensuremath{f_{\\rm LV}}\\xspace\\ensuremath{V_{\\myt\\myb}}\\xspace| = 0.97\\pm0.15\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.})}}\n\\newcommand{\\ensuremath{0.97\\pm0.15}}{\\ensuremath{0.97\\pm0.15}}\n\\newcommand{\\ensuremath{15.0\\%}}{\\ensuremath{15.0\\%}}\n\\newcommand{\\ensuremath{1.42}}{\\ensuremath{1.42}}\n\\newcommand{\\ensuremath{23}}{\\ensuremath{23}}\n\\newcommand{\\ensuremath{0.99}}{\\ensuremath{0.99}}\n\\newcommand{\\ensuremath{-1.19}}{\\ensuremath{-1.19}}\n\\newcommand{\\ensuremath{0.01}}{\\ensuremath{0.01}}\n\\newcommand{\\ensuremath{1.19}}{\\ensuremath{1.19}}\n\\newcommand{\\ensuremath{0.15}}{\\ensuremath{0.15}}\n\\newcommand{\\ensuremath{3.88}}{\\ensuremath{3.88}}\n The measured cross-section for a given single-top-quark production mode, $\\sigma_{\\rm meas.}$, has\na linear dependence on $\\ensuremath{\\absvtb^2}\\xspace$ as defined in Eq.~(\\ref{eq:vtb}). Thus, a value of $\\ensuremath{\\absvtb^2}\\xspace$ is extracted from each cross-section measurement and the corresponding theoretical prediction (presented in\nSections~\\ref{sec:atlascmsxs} and \\ref{sec:theory}\nrespectively). These values are then combined per channel, and in an\noverall $\\ensuremath{\\absvtb^2}\\xspace$ combination. In the overall combination, the value from the CMS measurement of $\\ensuremath{\\sigma_{\\mys\\textrm{-chan.}}}\\xspace$ is excluded. The reason for excluding the CMS \\mys-channel\\xspace\\ analysis from the overall $\\ensuremath{\\absvtb^2}\\xspace$ combination is that, at the same centre-of-mass energy, the CMS \\myt-channel\\xspace\\ determination has strong correlations with the \\mys-channel\\xspace\\ determination, which contains relatively large uncertainties. The strong\ncorrelation between these two measurements makes the combined $\\ensuremath{\\absvtb^2}\\xspace$ value strongly\ndependent on the correlation assumptions for the dominant uncertainties.\nThis results in a large variation of the combined $\\ensuremath{\\absvtb^2}\\xspace$ value for different correlation assumptions.\n\n\n\nAll uncertainties in  $\\ensuremath{\\sigma_{\\rm meas.}}\\xspace$ and $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$ are propagated to the $\\ensuremath{\\absvtb^2}\\xspace$ values,\ntaking into account the correlations described\nin Section~\\ref{sec:systcat}. The combined value of $\\ensuremath{\\absvtb^2}\\xspace$ is evaluated using the reference theoretical\ncross-section central values marked with a $^\\dagger$ in\nTable~\\ref{tab:xs}, where it can also be seen that the $E_{\\rm\n  beam}$ uncertainty is negligible compared to other\nuncertainties. For the most precise measurements (i.e.\\ for\n$\\ensuremath{\\sigma_{{\\myt\\textrm{-chan.}}}}\\xspace$ cross-section measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}), which have a large expected impact on the combination,\nthe other theoretical calculations from Table~\\ref{tab:xs} are used as\ncross-checks.\n\nTable~\\ref{tab:uncVtb} contains a summary of the individual $\\ensuremath{\\absvtb^2}\\xspace$\ndeterminations that are the inputs to the overall $\\ensuremath{\\absvtb^2}\\xspace$ combination, together with their experimental and theoretical uncertainties using the reference theoretical cross-sections and uncertainties. For the same processes and at the same centre-of-mass\nenergies, there are some important differences between uncertainty\ncategories. In analyses based on \\myt-channel\\xspace\\ events at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the data\nstatistical uncertainty is larger in CMS than in ATLAS because the two experiments use data samples of different size. Differences in the category of jet uncertainties are due to the evaluation of the JES uncertainty in ATLAS using pseudoexperiments, while this uncertainty is introduced as a nuisance parameter in the fit in CMS. At \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, the difference between ATLAS and CMS in the background-normalisation category is due to the different techniques used to estimate each background\nuncertainty. Additional details are discussed in Appendix~\\ref{app:tch}. In the CMS\n\\ensuremath{tW}\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, the uncertainty\nassociated with the size of the simulated samples is evaluated as part\nof the total statistical uncertainty. The large difference in the\npile-up uncertainty between ATLAS and CMS is due to the different\nmethods used to assess this uncertainty, as discussed in\nSection~\\ref{sec:systcat_exp}.\nAt \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, the sizes of the data and simulated samples used in the CMS \\ensuremath{tW}\\xspace\\ analysis are smaller than in the ATLAS analysis, resulting in larger data and simulation statistical uncertainties. The large difference between the two experiments in the category of jet uncertainties arises because the JES uncertainty in ATLAS is evaluated in different categories mostly using pseudoexperiments, while in CMS the JES uncertainty is introduced as a nuisance parameter in the fit. Further details are discussed in Appendix~\\ref{app:tw}.\nIn the CMS \\mys-channel\\xspace\\ analysis, the uncertainty associated with the size of the simulated samples is evaluated as part of the total statistical uncertainty. More details are discussed in Appendix~\\ref{app:sch}.\n\n\n\\begin{sidewaystable}[!htbp]\n  \\centering\n  \\caption{Results of the ATLAS and CMS individual $\\ensuremath{\\absvtb^2}\\xspace$ determinations that are the inputs to the overall $\\ensuremath{\\absvtb^2}\\xspace$\n     combination together with their experimental uncertainties. The values  of $\\ensuremath{\\absvtb^2}\\xspace$ may\n    slightly differ from those published for the different analyses since in this paper the theoretical cross-sections used are those marked with $^\\dagger$ in Table~\\ref{tab:xs}. Experimental uncertainties contributing\n    less than 1\\% are denoted by $<$0.01. Entries with $-$ mean\n    that this uncertainty was not evaluated for this\n    analysis. Descriptions of the background categories and of the\n    correlations of systematic uncertainties between experiments are presented in\n    Appendix~\\ref{app:allUncs}.}\n  \\IfPackagesLoaded{adjustbox}{\\begin{adjustbox}{max width=0.80\\textwidth}}{}\n    \\begin{tabular}{l|rrrrrrrrr}\n\\hline\n\\hline\n& $t$-channel& $t$-channel& $t$-channel& $t$-channel& $tW$& $tW$& $tW$& $tW$& $s$-channel\\\\\n& ATLAS& CMS& ATLAS& CMS& ATLAS& CMS& ATLAS& CMS& ATLAS\\\\\n& 8 TeV& 8 TeV& 7 TeV& 7 TeV& 8 TeV& 8 TeV& 7 TeV& 7 TeV& 8 TeV\\\\\n\\hline\\hline\n\\T\\B{\\boldmath $\\ensuremath{\\absvtb^2}\\xspace$}&1.06&0.99&1.06&1.05&1.03&1.05&1.07&1.02&0.92\\\\\n\\hline\n\\hline\n\\textbf{Uncertainties:}  & & & & & & & & &\\\\\n\\hline\n~~\\textbf{Data statistical}& 0.01& 0.03& 0.03& 0.06& 0.06& 0.09& 0.18& 0.21& 0.15\\\\\n\\hline\n~~\\textbf{Simulation statistical}& 0.01& 0.01& 0.02& 0.02& 0.01& 0.03& 0.02& $-$& 0.11\\\\\n\\hline\n~~\\textbf{Integrated luminosity}& 0.02& 0.03& 0.02& 0.02& 0.05& 0.03& 0.07& 0.04& 0.05\\\\\n\\hline\n~~\\textbf{Theory modelling}& & & & & & & & & \\\\\n~~~~ISR\/FSR, ren.\/fact.\\ scale& 0.04& 0.02& 0.03& 0.04& 0.09& 0.13& 0.05& 0.03& 0.06\\\\\n~~~~NLO match., generator& 0.03& 0.05& 0.02& 0.04& 0.03& $-$& 0.11& $-$& 0.10\\\\\n~~~~Parton shower& 0.02& $-$& $-$& 0.01& 0.02& 0.15& 0.16& 0.10& 0.02\\\\\n~~~~PDF& 0.01& 0.02& 0.03& 0.01& 0.01& 0.02& 0.02& 0.02& 0.03\\\\\n~~~~DS\/DR scheme& $-$& $-$& $-$& $-$& 0.04& 0.02& $-$& 0.06& $-$\\\\\n~~~~Top-quark $p_{\\rm T}$ rew.& $-$& $-$& $-$& $-$& $-$& $<$0.01& $-$& $-$& $-$\\\\\n\\hline\n~~\\textbf{Background normalisation}& & & & & & & & & \\\\\n~~~~Top-quark bkg.& $<$0.01& 0.02& 0.02& 0.01& 0.02& 0.02& 0.06& 0.06& 0.05\\\\\n~~~~Other bkg.\\ from sim.& 0.01& $<$0.01& $<$0.01& 0.03& 0.02& 0.03& 0.09& 0.04& 0.05\\\\\n~~~~Bkg.\\ from data& $<$0.01& 0.02& 0.01& 0.01& $<$0.01& $-$& 0.02& $-$& 0.01\\\\\n\\hline\n~~\\textbf{Jets}& & & & & & & & & \\\\\n~~~~JES common& 0.03& 0.04& 0.08& 0.01& 0.05& 0.04& 0.17& 0.15& 0.05\\\\\n~~~~JES flavour& $<$0.01& $-$& 0.02& $-$& 0.02& $-$& $-$& $-$& 0.01\\\\\n~~~~JetID& $<$0.01& $-$& 0.01& $-$& $<$0.01& $-$& 0.05& $-$& 0.01\\\\\n~~~~JER& $<$0.01& 0.01& 0.02& $<$0.01& 0.07& 0.01& 0.02& 0.04& 0.11\\\\\n\\hline\n~~\\textbf{Detector modelling}& & & & & & & & & \\\\\n~~~~Leptons& 0.02& 0.01& 0.03& 0.04& 0.03& 0.02& 0.07& 0.05& 0.02\\\\\n~~~~HLT (had.\\ part)& $-$& $-$& $-$& 0.02& $-$& $-$& $-$& $-$& $-$\\\\\n~~~~\\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace scale& $<$0.01& $<$0.01& 0.03& $<$0.01& 0.06& $<$0.01& $-$& 0.03& 0.01\\\\\n~~~~\\ensuremath{E_{\\mathrm{T}}^{\\mathrm{miss}}}\\xspace res.& $<$0.01& $-$& $-$& $-$& $<$0.01& $-$& $-$& $-$& 0.01\\\\\n~~~~\\btag& 0.01& 0.02& 0.04& 0.02& 0.01& 0.01& $-$& 0.02& 0.07\\\\\n~~~~Pile-up& $<$0.01& 0.01& $<$0.01& 0.01& 0.03& $<$0.01& 0.11& 0.01& 0.01\\\\\n\\hline\n~~\\textbf{Top-quark mass}& 0.01& $<$0.01& 0.01& $-$& 0.05& 0.05& $-$& $-$& $-$\\\\\n\\hline\n~~\\textbf{Theoretical cross-section}& & & & & & & & & \\\\\n~~~~PDF+$\\alpha_{\\rm s}$& 0.03& 0.03& 0.04& 0.04& 0.06& 0.07& 0.08& 0.07& 0.03\\\\\n~~~~Ren.\/fact.\\ scale& 0.03& 0.03& 0.03& 0.03& 0.03& 0.03& 0.03& 0.03& 0.02\\\\\n~~~~Top-quark mass& 0.01& 0.01& 0.01& 0.01& 0.02& 0.02& 0.02& 0.02& 0.02\\\\\n~~~~$E_{\\text{beam}}$& $<$0.01& $<$0.01& $<$0.01& $<$0.01& $<$0.01& $<$0.01& $<$0.01& $<$0.01& $<$0.01\\\\\n\\hline\n\\hline\n\\textbf{Total systematic uncertainty}&0.09&0.09&0.13&0.10&0.18&0.23&0.34&0.24&0.24\\\\\n\\hline\n\\textbf{Total uncertainty}&0.09&0.10&0.13&0.12&0.19&0.24&0.38&0.32&0.28\\\\\n\\hline\n\\hline\n\\end{tabular}\n     \\IfPackagesLoaded{adjustbox}{\\end{adjustbox}}{}\n  \\label{tab:uncVtb}\n\\end{sidewaystable}\n\n\\subsection{Results} \n\\label{sec:vtbresult}\nThe combination of $\\ensuremath{\\absvtb^2}\\xspace$ is performed using the inputs from\nall three single-top-quark production modes. Using the same method, the combination of $\\ensuremath{\\absvtb^2}\\xspace$ is also performed separately for each production mode for comparison.\n\nCombining the $\\ensuremath{\\absvtb^2}\\xspace$ values extracted from the \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ cross-section measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ from ATLAS and CMS, as well as the ATLAS \\mys-channel\\xspace\\ measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, results in\n\\begin{linenomath}\n  \\begin{equation*}\n    \\ensuremath{|\\flv\\vtb|^2 = 1.05\\pm0.02\\;(\\text{stat.})\\pm0.06\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) = 1.05\\pm0.08},\n    \\label{eq:resultSqd}\n  \\end{equation*}\n\\end{linenomath}\nwith a relative uncertainty of $\\ensuremath{7.4\\%}$. The contribution from each experimental uncertainty category to the\ntotal uncertainty in the combined $\\ensuremath{\\absvtb^2}\\xspace$ value is shown in Table~\\ref{tab:vtbRes}. The\ntheory modelling uncertainties in signal and background processes, discussed in\nSection~\\ref{sec:systcat_exp}, dominate the experimental\nuncertainty and the total uncertainty. The theoretical cross-section uncertainty is the\nsecond-largest contribution to the total uncertainty in the combined $\\ensuremath{\\absvtb^2}\\xspace$ value. Changes in the combined $\\ensuremath{\\absvtb^2}\\xspace$ value from using alternative NNLO and NLO+NNLL theoretical predictions for the \\myt-channel\\xspace\\ are less than 1\\%.\n\n\\begin{table}[htbp]\n  \\begin{center}\n    \\caption{Contributions from each experimental and theoretical\n      uncertainty category to the overall $\\ensuremath{\\absvtb^2}\\xspace$ combination. The total uncertainty is\n      computed by adding in quadrature all of the individual systematic\n      uncertainties (including the integrated luminosity and theoretical\n      cross-section) and the statistical uncertainty in data.}\n    \\begin{tabular}{l | r | r}\n\\hline\n \\hline\n\\T\\B\n{\\bf Combined} {\\boldmath $\\ensuremath{\\absvtb^2}\\xspace$} & \\multicolumn{2}{c}{1.05} \\\\\n\\hline\n\\hline\n\\multirow{2}{*}{Uncertainty category} & \\multicolumn{2}{c}{Uncertainty} \\\\\\cline{2-3}\n &  ~~[\\%] &  $\\Delta\\ensuremath{\\absvtb^2}\\xspace$ \\\\\n\\hline\nData statistical & 1.8 & 0.02\\\\\n\\hline\nSimulation statistical & 0.9 & 0.01\\\\\nIntegrated luminosity & 1.3 & 0.01\\\\\nTheory modelling & 4.5 & 0.05\\\\\nBackground normalisation & 1.3 & 0.01\\\\\nJets & 2.6 & 0.03\\\\\nDetector modelling & 1.6 & 0.02\\\\\nTop-quark mass & 0.7 & 0.01\\\\\nTheoretical cross-section & 4.3 & 0.04\\\\\n\\hline\nTotal syst.\\ unc.\\ (excl.\\ lumi.) & 7.1 & 0.07\\\\\nTotal syst.\\ unc.\\ (incl.\\ lumi.) & 7.2 & 0.08\\\\\n\\hline\\hline\n{\\bf Total uncertainty} & 7.4 & 0.08\\\\\n\\hline\n \\hline\n\\end{tabular}\n     \\label{tab:vtbRes}\n  \\end{center}\n\\end{table}\n\nFigure~\\ref{fig:rhosVtb} illustrates the correlations between the input measurements in the combination. The correlations are all below 0.6. The largest correlations are generally between the measurements in the same experiment at the same centre-of-mass energy, and for those that have large contributions from the same theory modelling components, such as the ATLAS \\mys-channel\\xspace\\ measurement, which has a correlation of over 0.5 with each of the \\ensuremath{tW}\\xspace\\ measurements.\n\nThe BLUE weights for each of the contributing measurements are shown in\nTable~\\ref{tab:wgtVtb}. The \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ have\nthe largest weight in the combination, followed by the \\myt-channel\\xspace\\ measurements at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}.\nThe $\\ensuremath{tW}\\xspace$ measurements have smaller cross-section uncertainties than the \\mys-channel\\xspace\\ measurements, but, in addition to the correlation between $\\ensuremath{tW}\\xspace$ and \\mys-channel\\xspace\\ measurements, the $\\ensuremath{tW}\\xspace$ measurements are also\nmore correlated with the \\myt-channel\\xspace\\ measurements in each experiment. The negative weights indicate the presence of large correlations between the corresponding measurement and some of the other measurements~\\cite{Valassi:2013bga}.\n\n\\begin{figure}[htbp]\n  \\begin{center}\n    \\includegraphics[width=0.85\\textwidth]{overallCorrComb}\n    \\caption{Correlation matrix of the overall \\ensuremath{\\absvtb^2}\\xspace\\ combination. Each bin\n      corresponds to a measurement in a given production mode,\n      experiment, and at a given centre-of-mass energy.}\n    \\label{fig:rhosVtb}\n  \\end{center}\n\\end{figure}\n\n\\begin{table}[htbp]\n  \\begin{center}\n    \\caption{BLUE weights for the overall $\\ensuremath{\\absvtb^2}\\xspace$ combination.}\n    \\begin{tabular}{c|c|c|L{.}}\n\\hline\n\\hline\nProcess & $\\sqrt{s}$ & Experiment & \\multicolumn{1}{c}{BLUE weight} \\\\\n\\hline\n\\multirow{4}{*}{$t$-channel} & \\multirow{2}{*}{8 TeV} & ATLAS&0.56\\\\\n\\cline{3-4}\n& & CMS&0.27\\\\\n\\cline{3-4}\n\\cline{2-3}\n & \\multirow{2}{*}{7 TeV} & ATLAS&0.07\\\\\n\\cline{3-4}\n& & CMS&0.15\\\\\n\\hline\n\\multirow{4}{*}{$tW$} & \\multirow{2}{*}{8 TeV} & ATLAS&0.05\\\\\n\\cline{3-4}\n& & CMS&-0.04\\\\\n\\cline{3-4}\n\\cline{2-3}\n & \\multirow{2}{*}{7 TeV} & ATLAS&-0.02\\\\\n\\cline{3-4}\n& & CMS&0.02\\\\\n\\hline\n$s$-channel & 8 TeV & ATLAS&-0.07\\\\\n\\hline\\hline\n\\end{tabular}\n     \\label{tab:wgtVtb}\n  \\end{center}\n\\end{table}\n\nThe combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ value from the cross-section measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, including uncertainties in $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$ for each\nproduction mode, is\n\\begin{linenomath}\n  \\begin{equation*}\n    \\begin{aligned}\n    \\ensuremath{|\\flv\\vtb|=~&~1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 1.02\\pm0.04},\n    \\end{aligned}\n    \\label{eq:result}\n  \\end{equation*}\n\\end{linenomath}\nwith a relative uncertainty of \\ensuremath{3.7\\%}, which improves on the precision of 4.7\\% of the most precise individual $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ extraction, which comes from the ATLAS \\myt-channel\\xspace\\ analysis at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}~\\cite{TOPQ-2015-05}. This is a 30\\% improvement over the Tevatron combination~\\cite{Aaltonen:2015cra}.\n\nThe $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ values are also combined for each production mode, combining across experiments and centre-of-mass energies. For the \\mys-channel\\xspace, the ATLAS and CMS measurements at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}\\ are combined. The results are\n\\begin{linenomath}\n  \\begin{equation*}\n    \\begin{aligned}\n      {t{\\text{-channel}}}: \\ensuremath{|\\flv\\vtb|=~&~1.02\\pm0.01\\;(\\text{stat.})\\pm0.03\\;(\\text{syst.})\\pm0.01\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 1.02\\pm0.04}, \\\\\n      {\\ensuremath{tW}\\xspace}: \\ensuremath{|\\flv\\vtb|=~&~1.02\\pm0.03\\;(\\text{stat.})\\pm0.07\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.04\\;(\\text{theo.}) \\\\=~&~1.02\\pm0.09\\;(\\text{meas.})\\pm0.04\\;(\\text{theo.}) = 1.02\\pm0.09}, \\\\\n      {s{\\text{-channel}}}: \\ensuremath{|\\flv\\vtb|=~&~0.97\\pm0.08\\;(\\text{stat.})\\pm0.12\\;(\\text{syst.})\\pm0.02\\;(\\text{lumi.})\\pm0.02\\;(\\text{theo.}) \\\\=~&~0.97\\pm0.15\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.}) = 0.97\\pm0.15}. \\\\\n    \\end{aligned}\n  \\end{equation*}\n\\end{linenomath}\nThe relative uncertainties are \\ensuremath{3.9\\%}, \\ensuremath{8.4\\%}\\ and \\ensuremath{15.0\\%}\\ respectively.  In all cases, these results\nare more precise than the best individual determinations of $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$, which have uncertainties of\n4.7\\%, 9.9\\% and 20.8\\% for the \\myt-channel\\xspace~\\cite{TOPQ-2015-05},\n\\ensuremath{tW}\\xspace~\\cite{TOPQ-2012-20} and \\mys-channel\\xspace~\\cite{TOPQ-2015-01} analyses respectively.\n\nFigure~\\ref{fig:vtbsum} shows a summary of the $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ combinations. The combination is dominated by the \\myt-channel\\xspace\\ analyses.\n\n\\begin{figure}[!h!tbp]\n  \\begin{center}\n    \\includegraphics[width=0.82\\textwidth]{figures\/vtb\/vtb_summary.pdf}\n    \\caption{The combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ value extracted  from the \\myt-channel\\xspace\\ and \\ensuremath{tW}\\xspace\\ cross-section measurements at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ from ATLAS and CMS, as well as the ATLAS \\mys-channel\\xspace measurement at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}, is shown together with the combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ values for each production mode.\n      The theoretical predictions for \\myt-channel\\xspace\\ and \\mys-channel\\xspace\\ production are computed at NLO accuracy, while the theoretical predictions for $\\ensuremath{tW}\\xspace$  are calculated at NLO+NNLL accuracy. The $\\ensuremath{\\sigma_{\\rm theo.}}\\xspace$ uncertainties used to compute $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ include scale, PDF+$\\alpha_{\\rm s}$, $m_{\\ensuremath{t}\\xspace}$, and $E_{\\rm beam}$ variations.\n    }\n    \\label{fig:vtbsum}\n\\end{center}\n\\end{figure}\n\n\\subsection{Stability tests}\n\\label{sec:tests}\n\nThe stability of the combination of the $\\ensuremath{\\absvtb^2}\\xspace$ values to variations in the correlation assumptions, discussed in Section~\\ref{sec:systcat}, is checked for the dominant uncertainty contributions. The correlation values are varied for the theory modelling, JES, and the most important contributions to the theoretical cross-section predictions (i.e.\\ PDF+$\\alpha_{\\rm s}$ and scale). Because of the scheme that is used for the correlations, stability tests are also performed for the uncertainties associated with the integrated luminosity. Figure~\\ref{fig:vtbstabilities} summarises the results of these stability tests, where the correlations between ATLAS and CMS (and also between centre-of-mass energies for the integrated luminosity) are varied.\n\n\\begin{figure}[!h!tbp]\n  \\begin{center}\n    \\includegraphics[width=1.0\\textwidth]{stability\/stability_style2.pdf}\n    \\caption{Results of the stability tests performed by varying\n      of the correlation assumptions in different uncertainty\n      categories: theory modelling (scales and radiation modelling,\n      NLO matching, and PS and hadronisation), JES, dominant\n      theoretical cross-section predictions (i.e.\\ PDF+$\\alpha_{\\rm s}$\n      and scale) and integrated luminosity. Two or three variations are considered depending on the\n      uncertainty category. The corresponding relative shifts\n      (with shift = varied $-$ nominal) in the central value, $\\Delta\\ensuremath{\\absvtb^2}\\xspace\/\\ensuremath{\\absvtb^2}\\xspace$, and in its uncertainty,\n      $\\Delta(\\delta\\ensuremath{\\absvtb^2}\\xspace)\/(\\delta\\ensuremath{\\absvtb^2}\\xspace)$, are shown.}\n    \\label{fig:vtbstabilities}\n  \\end{center}\n\\end{figure}\n\nThe uncertainties in the theory modelling category (i.e.\\ scales and radiation modelling, NLO matching, and PS and\nhadronisation) are varied from their default value of fully\ncorrelated to half correlated and to the more extreme tests of uncorrelated and half anti-correlated. The JES\ncategory is varied from its default value of uncorrelated to half correlated and half anti-correlated and the more extreme variation of fully correlated. The theoretical cross-section uncertainties, PDF+$\\alpha_{\\rm s}$ and\nscale, are varied from their default values of fully correlated to\nhalf correlated, uncorrelated and half anti-correlated. For the integrated luminosity, the correlation between ATLAS and CMS is varied from its default value of 30\\% correlated to half correlated and uncorrelated. The correlation between different centre-of-mass energies for each experiment is varied from the default of uncorrelated to half and fully\ncorrelated. The correlation of the theoretical scale uncertainty between different processes is also tested. For all variations, the relative changes in the central value of the combined $\\ensuremath{|f_{\\rm LV}\\vtb|}\\xspace$ are significantly smaller ($<$$0.5\\%$) than the relative total uncertainty of \\ensuremath{3.7\\%}. Additionally, the relative changes in the total uncertainty are below 0.004, i.e., less than $10\\%$ of the total uncertainty of \\ensuremath{0.04}. \nThese tests show that the result of the combination is robust and does not critically depend on any of the correlation assumptions. The cross-section combinations similarly do not depend significantly on any of the correlation assumptions.\n \n\\IfPackagesLoaded{adjustbox}{\\FloatBarrier}{}\n\\section{Summary}\n\\label{sec:sum}\nThe combinations of single-top-quark production cross-section measurements in the \\myt-channel\\xspace, \\ensuremath{tW}\\xspace, and \\mys-channel\\xspace\\ production modes are presented, using data from LHC $pp$ collisions collected by the ATLAS and CMS Collaborations. The combinations for each production mode are performed at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}, using data corresponding to integrated luminosities of 1.17 to 5.1~\\ensuremath{\\rm fb^{-1}}\\xspace at \\ensuremath{\\sqrt{s}=\\lhcenergySeven}, and of 12.2 to 20.3~\\ensuremath{\\rm fb^{-1}}\\xspace at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. \nThe combined \\myt-channel\\xspace\\ cross-sections are found to be $\\ensuremath{67.5\\pm5.7\\,\\,\\text{pb}}$ and $\\ensuremath{87.7\\pm5.8\\,\\,\\text{pb}}$ at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ respectively. The values of the combined \\ensuremath{tW}\\xspace\\ cross-sections at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ are $\\ensuremath{16.3\\pm4.1\\,\\,\\text{pb}}$ and $\\ensuremath{23.1\\pm3.6\\,\\,\\text{pb}}$ respectively. For the \\mys-channel\\xspace\\ cross-section, the combination yields $\\ensuremath{4.9\\pm1.4\\,\\,\\text{pb}}$ at \\ensuremath{\\sqrt{s}=\\lhcenergyEight}. \nThe square of the magnitude of the CKM matrix element $\\ensuremath{V_{\\myt\\myb}}\\xspace$ multiplied by a form factor accounting for possible contributions from physics beyond the SM, \\ensuremath{f_{\\rm LV}}\\xspace, is determined from each production mode at each centre-of-mass energy, using the ratio of the measured cross-section to its theoretical prediction, and assuming that the top-quark-related CKM matrix elements obey the relation $|\\ensuremath{V_{td}}\\xspace|,|\\ensuremath{V_{ts}}\\xspace| \\ll |\\ensuremath{V_{\\myt\\myb}}\\xspace|$. The values of $\\ensuremath{\\absvtb^2}\\xspace$ extracted from individual ratios at \\ensuremath{\\sqrt{s}=7 \\text{ and } \\lhcenergyEight}\\ yield a combined value of \\ensuremath{|\\flv\\vtb| = 1.02\\pm0.04\\;(\\text{meas.})\\pm0.02\\;(\\text{theo.})}. All combined measurements are consistent with their corresponding SM predictions.\n  \n\n\n\\clearpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nElectronic transport in mesoscopic systems is usually affected by the\nCoulomb interactions between the electrons. The interactions manifest\nthemselves most dramatically in small tunnel junctions. If a small\nmetallic grain is connected to a lead by a tunnel junction, an electron\ntunneling into such a grain charges it by the elementary charge $e$ and\nincreases the energy of the system by $e^2\/2C$. For a small grain, the\ntypical capacitance of the system can be very small, so that the\ncharging energy $e^2\/2C$ can significantly exceed the temperature of the\nsystem $T$. In this regime one observes the Coulomb blockade: strong\nsuppression of the tunneling into the grain at low temperatures. Various\naspects of the Coulomb blockade have been intensively studied in recent\nyears, both experimentally and theoretically.\\cite{Averin,Grabert}\n\nRecently it has become possible\\cite{Kastner} to observe the Coulomb\nblockade in semiconductor heterostructures. Instead of a metallic grain,\nin such experiments one creates a small region of the two-dimensional\nelectron gas (2DEG)---a quantum dot---separated from the rest of the\n2DEG by potential barriers created by applying negative voltage to the\ngates, Fig.~\\ref{fig:1}.  Experimentally, one measures the linear\nconductance through the quantum dot as a function of the voltage $V_g$\napplied to the central gates. The role of the gate voltage is to change\nthe electrostatic energy of the system,\n\\begin{equation}\nE = \\frac{(Q-eN)^2}{2C},\n  \\label{electrostatic_energy}\n\\end{equation}\nwhere $Q$ is the charge of the quantum dot, and $N$ is a dimensionless\nparameter proportional to the gate voltage and the capacitance $C_g$\nbetween the dot and the gate, $N=C_g V_g\/e$. If the potential barriers\nseparating the quantum dot from the leads are high, the charge $Q$ is\ninteger in units of $e$. Thus, the ground state of the system corresponds\nto some integer value of charge, and the states with different values of\n$Q$ are separated from it by the electrostatic gap $\\Delta\\sim e^2\/2C$. It\nis important to note that at the values of the gate voltage corresponding\nto half-integer values of $N$ the gap vanishes, i.e., the ground state is\ndegenerate in charge $Q$. For example, at $N=\\frac{1}{2}$ the ground state\ncan have either charge $Q=0$ or $e$. Therefore at half-integer $N$ the\ntunneling of an electron into or out of the quantum dot does not lead to\nthe increase of the electrostatic energy of the system, and the Coulomb\nblockade is lifted. As a result the conductance of the structure in\nFig.~\\ref{fig:1} shows periodic peaks as a function of $N$. The Coulomb\nblockade peaks in linear conductance as a function of the gate voltage are\nreadily observed in the experiments (see, e.g.,\nRef.~\\onlinecite{Kastner}).\n\n\n\\vfill\n\\begin{figure}\n\\narrowtext\n\\epsfxsize=3.1in\n\\hspace{.2em}\n\\epsffile{4chan1.ps}\n\\vspace{1.5\\baselineskip}\n\\caption{Schematic view of a quantum dot connected to two bulk 2D\n  electrodes. The dot is formed by applying negative voltage to the gates\n  (shaded). Solid line shows the boundary of the 2D electron gas (2DEG).\n  Electrostatic conditions in the dot are controlled by the voltage\n  applied to the central gates. Voltage $V_{L,R}$ applied to the auxiliary\n  gates controls the transmission probability through the left and right\n  constrictions.}\n\\label{fig:1}\n\\end{figure}\n\\vspace{\\baselineskip}\n\n\n\n\nThe shapes and heights of the peaks in conductance are usually described\nwithin the framework of the orthodox theory of the Coulomb\nblockade.\\cite{Averin} In this approach the conductance is found from the\nbalance of the rates of tunneling of electrons between the leads and the\ndot, with the rates being calculated in the lowest order perturbation\ntheory in the tunneling matrix elements. The resulting linear conductance\nhas the form:\\cite{Glazman}\n\\begin{equation}\n  G = \\frac{1}{2}\\frac{G_LG_R}{G_L + G_R}\n      \\frac{\\Delta\/T}{\\sinh(\\Delta\/T)},\n  \\quad \\Delta=\\frac{e^2}{C}\\delta N.\n  \\label{orthodox}\n\\end{equation}\nHere $G_L$ and $G_R$ are the conductances of the left and right barriers,\n$\\delta N$ is the deviation of $N$ from the nearest half-integer value.\nAway from the center of the peak, the tunneling of an electron into the\nquantum dot is suppressed, because only a small fraction of electrons has\nthe energy sufficient to overcome the electrostatic gap $\\Delta$. Thus the\nconductance (\\ref{orthodox}) decays exponentially at $T\\to0$. At very low\ntemperatures, however, another mechanism---usually referred to as {\\it\n  inelastic co-tunneling}---dominates the conductance.  Instead of a\nreal process of tunneling from a lead to the dot, electron tunnels from\nthe left lead to a virtual state in the dot, and then another electron\ntunnels from the dot to the right lead. As a result of such a two-step\nprocess, one electron is transferred from the left lead to the right one,\nand the charge of the quantum dot is unchanged. The corresponding\nconductance has only a power-law dependence on temperature,\\cite{AN}\n\\begin{equation}\nG=\\frac{\\pi\\hbar G_LG_R}{3e^2}\\left(\\frac{T}{\\Delta}\\right)^2,\n\\quad\nT\\ll\\Delta.\n\\label{cotunneling}\n\\end{equation}\nIn the case of weak tunneling, $G_L, G_R\\ll e^2\/\\hbar$, the results\n(\\ref{orthodox}) and (\\ref{cotunneling}) give the full description of the\nCoulomb blockade peaks in linear conductance. Near the center of the peak\nthe conductance is given by Eq.~(\\ref{orthodox}), whereas the tails of the\npeaks are dominated by the co-tunneling contribution (\\ref{cotunneling}).\n\nIn a number of recent experiments\\cite{van,Pasquier,Waugh,Molenkamp} the\ntunneling through quantum dots was studied in the regime when one or more\nquantum point contacts were tuned to the regime of strong tunneling,\ncorresponding to conductance $G\\to e^2\/\\pi\\hbar$. In particular, it was\ndemonstrated by van der Vaart {\\it et al.\\\/}\\cite{van} that the Coulomb\nblockade oscillations of linear conductance persist as long as the\nconductances of both barriers are below the conductance quantum\n$e^2\/\\pi\\hbar$.  Most of the existing theoretical works on the\nstrong-tunneling regime of the Coulomb\nblockade\\cite{GS,PZ,Falci,Schoeller} discuss the case of metallic\nsystems, where one can achieve large conductance of a tunnel junction\nbetween the grain and a lead by increasing the number of transverse modes\nwhile keeping the transmission coefficient ${\\cal T}$ small. Such theories\ncannot be applied to the experiments in semiconductor heterostructures,\nwhere the point contacts typically allow only a single transverse mode,\nbut the transmission coefficient can be tuned to ${\\cal T} \\to 1$. The\ntheoretical works for this case either do not concentrate on the Coulomb\nblockade oscillations\\cite{Flensberg} or study the oscillations of\nequilibrium properties of the system,\\cite{Matveev1} rather than its\nconductance.\n\nIn this paper we develop a theory of the Coulomb blockade oscillations of\nconductance in the regime of strong tunneling. It was shown in\nRef.~\\onlinecite{Matveev1} that the average charge of a quantum dot\nconnected to a lead by a single-mode quantum point contact shows small\nperiodic oscillations as a function of the gate voltage as long as the\ntransmission coefficient ${\\cal T}$ is below unity. Similarly, we will\nshow that the conductance through a quantum dot shows small periodic\noscillations at ${\\cal T} < 1$. One should note, however, that unlike the\naverage charge,\\cite{Matveev1} the amplitude of the oscillations of conductance\ndepends\non temperature and grows as $T$ is lowered. In the limit $T\\to 0$, small\noscillations develop into sharp periodic peaks, and the off-peak\nconductance vanishes as $T^2$, thus reproducing the characteristic\nquadratic temperature dependence for inelastic co-tunneling\n(\\ref{cotunneling}).\n\nIn section \\ref{weak} we use the analogy\\cite{GM,Matveev2} between the\nCoulomb blockade and multichannel Kondo problem to discuss the peak value\nof conductance at $T\\to0$. This discussion is done for the weak-tunneling\ncase, but the conclusion that the peak conductance is of the order of\n$e^2\/\\hbar$ is valid for the strong tunneling as well. The theory of\nconductance oscillations in the strong-tunneling regime is developed in\nsections \\ref{spinless} and \\ref{withspin}. In section \\ref{scaling} we\ndiscuss the scaling properties of our problem. In particular, we show that\nthe quadratic temperature dependence of the off-peak conductance at\n$T\\to0$ is universal and holds for arbitrary transmission coefficients.\n\n\\section{Conductance peaks in the weak-tunneling case}\n\\label{weak}\n\nLet us first consider the weak-tunneling case. The linear conductance is\ngiven by Eqs.~(\\ref{orthodox}) and (\\ref{cotunneling}). One should note\nthat in the derivation of Eqs.~(\\ref{orthodox}) and (\\ref{cotunneling})\nonly the lowest-order terms of the perturbation theory in tunneling\namplitudes were taken into account. This is usually justified at $G_L, G_R\n\\ll e^2\/\\hbar$. However, it was shown in Ref.~\\onlinecite{GM} that near a\nhalf-integer $N$ and at low temperatures the higher-order corrections lead\nto large logarithmic renormalizations of the tunneling amplitudes. The\nnature of these renormalizations can be understood in terms of the\nanalogy\\cite{Matveev2} between the Coulomb blockade and the Kondo problem.\nWe will now discuss this analogy and correct the results (\\ref{orthodox})\nand (\\ref{cotunneling}) to take into account the renormalizations of the\ntunneling amplitudes.\n\nFor simplicity we start with the quantum dot weakly coupled to a {\\it\n  single\\\/} lead. Such a system can be described by the following\nHamiltonian:\n\\begin{eqnarray}\nH &=& \\sum_{k} \\epsilon_k a_k^\\dagger a_k^{}\n      + \\sum_{p} \\epsilon_p a_p^\\dagger a_p^{}\n      + E_C (\\hat n - N)^2\n\\nonumber\\\\\n  & & +\\sum_{kp} (v_t^{} a_k^\\dagger a_p^{} + v_t^* a_p^\\dagger a_k^{}).\n  \\label{tunnel_hamiltonian}\n\\end{eqnarray}\nHere $a_k$ and $a_p$ are the annihilation operators for the electrons in\nthe lead and in the dot respectively, and $E_C$ is the characteristic\ncharging energy, $E_C = e^2\/2C$; coupling between the dot and the lead is\ndescribed by tunneling Hamiltonian with the matrix element $v_t^{}$.\nFinally, $\\hat n$ is the operator of the number of electrons in the dot,\ncounted from that at the ground state of the system without tunneling,\n$\\hat n = \\sum (a_p^\\dagger a_p^{} - \\langle a_p^\\dagger a_p^{}\n\\rangle_0^{})$.\n\nDue to the last term in Eq.~(\\ref{tunnel_hamiltonian}) electrons can\ntunnel from the lead to a virtual state in the dot. The energy of such a\nvirtual state is of the order of $E_C$, and to complete this scattering\nprocess an electron has to tunnel from the dot to the lead. It is\ninstructive to calculate the second-order amplitude of scattering between\ntwo states in the lead, $k$ and $k'$, near the Fermi level:\n\\begin{eqnarray}\n  T_{k\\to k'} = \\sum_p && \\left[\n                  \\frac{|v_t^{}|^2 \\theta(\\epsilon_p)}\n                       {E_C N^2 - E_C(1-N)^2 - \\epsilon_p}\n\\right.\\nonumber\\\\\n              && \\left.\n                  - \\frac{|v_t^{}|^2 \\theta(-\\epsilon_p)}\n                       {E_C N^2 - E_C(1+N)^2 + \\epsilon_p}\n                       \\right].\n  \\label{2_order_sum}\n\\end{eqnarray}\nHere the second term corresponds to the process in which first an electron\nfrom the filled state $p$ in the dot tunnels to the state $k'$ in the\nlead, and then the electron $k$ fills the hole in the dot. Clearly, the\ntotal amplitude diverges logarithmically at $N\\to\\pm\\frac12$,\n\\begin{equation}\n  T_{k\\to k'} = -\\nu_d |v_t^{}|^2 \\ln\\frac{1+2N}{1-2N},\n  \\label{2_order_log}\n\\end{equation}\nwhere $\\nu_d$ is the density of states in the dot. The logarithmic\nsingularity at $N\\to\\frac12$ originates from the degeneracy of the\nelectrostatic energy for states with $n=0$ and $n=1$ extra electrons in\nthe dot.\n\nLet us now concentrate on the case of the dimensionless gate voltage $N$\nvery close to $\\frac{1}{2}$. In this case the logarithmic renormalizations\nare large, and to find the scaling properties of the scattering\namplitudes, we can restrict our consideration to the energy scales smaller\nthan $E_C$. Thus when constructing the perturbation series for a tunneling\namplitude we will neglect all the processes in which the system visits\nvirtual states with the charge $n$ different from 0 or 1.  Another\nimportant property of the Hamiltonian (\\ref{tunnel_hamiltonian}) is that\nthe perturbation---the tunneling term---always changes the charge of the\ndot by 1. Suppose that at the first step of a high-order tunneling process\nan electron tunnels from the lead to the dot thus changing $n$ from 0 to\n1. Then at the next step an electron will have to tunnel from the dot to\nthe lead (changing $n=1\\to0$), because otherwise a virtual state with\n$n=2$ and correspondingly large energy $\\sim E_C$ would have been created.\nAt the third step an electron tunnels from the lead to the dot, changing\n$n=0\\to1$, and so on.\n\nOne can easily see that exactly the same perturbation series for\nscattering amplitudes is obtained from the Hamiltonian of the anisotropic\nKondo model,\n\\begin{eqnarray}\nH &=& \\sum_{k\\alpha} \\epsilon_k a_{k\\alpha}^\\dagger a_{k\\alpha}^{}\n      + h S^z\n\\nonumber\\\\\n  & & +J_\\perp\\! \\sum_{kk'\\alpha\\alpha'}\\!\n     (\\sigma^{+}_{\\alpha\\alpha'} S^{-} + \\sigma^{-}_{\\alpha\\alpha'} S^{+})\n     \\,a_{k\\alpha}^\\dagger a_{k'\\alpha'}^{},\n  \\label{Kondo}\n\\end{eqnarray}\nwhere $\\sigma^{\\pm}$ are the linear combinations of Pauli matrices,\n$\\sigma^{\\pm}= \\sigma^x \\pm \\sigma^y$, and {\\boldmath $S$} is the impurity\nspin, $S^{\\pm}=S^x\\pm S^y$.  A formal discussion of this mapping can be\nfound in Ref.~\\onlinecite{Matveev2}; here we only mention the relations\nbetween the parameters of the Hamiltonians (\\ref{tunnel_hamiltonian}) and\n(\\ref{Kondo}). The values of the electron spin $\\alpha=\\ \\uparrow$ and\n$\\downarrow$ correspond to the electron being in the lead and the dot\nrespectively. Similarly, the values of the impurity spin $S^z=\\\n\\downarrow$ and $\\uparrow$ correspond to the states with the number of\nparticles in the dot $n=0$ and 1. Magnetic field $h$ is identified with\nthe energy splitting of the $n=0$ and $n=1$ states, $h=2E_C(\\frac12-N)$.\nFinally, the tunneling matrix element $v_t^{}$ is mapped to the exchange\nconstant $J_\\perp$.\n\nIt is well known that at low temperatures and magnetic fields the exchange\nconstant $J$ of the Kondo model renormalizes logarithmically. As a result\n$J$ grows, and the system approaches the strong-coupling fixed point.  In\nthe context of the Coulomb blockade problem, this means that at\n$N\\to\\frac12$ and $T\\to0$ the tunneling amplitudes grow until the\nconductance of the barrier becomes of order $e^2\/\\hbar$.\n\nUsing the analogy with the Kondo problem, one can actually find the\nrenormalizations of the tunneling amplitudes. In the leading logarithm\napproximation one can use the poor man's scaling technique.\\cite{Anderson}\nThe result\\cite{GM} can be written down in terms of the renormalization of\nthe transmission coefficient ${\\cal T}$ of the barrier between the lead\nand the dot,\n\\begin{equation}\n{\\cal T} = \\frac{{\\cal T}_0}\n           {\\cos^2\\bigl(\\frac{1}{\\pi}\\sqrt{{\\cal T}_0}\\xi\\bigr)},\n\\quad\n\\xi = \\ln\\frac{E_C}{\\max\\{T,\\Delta\\}}.\n  \\label{poor_man}\n\\end{equation}\nHere ${\\cal T}_0$ is the bare value of the transmission coefficient, and\n$\\Delta = E_C|1-2N|$.  When ${\\cal T}$ becomes of the order of unity, the\nleading logarithm approximation fails. Thus the result (\\ref{poor_man}) is\nvalid as long as the renormalized transmission coefficient is small,\n${\\cal T}\\ll 1$.\n\nIn this regime one can still use the weak-tunneling results\n(\\ref{orthodox}) and (\\ref{cotunneling}) for the conductance through a dot\nconnected to two leads, taking into account the renormalizations of the\nconductances of the barriers. This can be done by noting the relation\nbetween the conductance of a barrier $G_b$ and its transmission\ncoefficient,\n\\begin{equation}\nG_b=\\frac{e^2}{2\\pi\\hbar}\\times\n    \\begin{cases}{\n        {\\cal T}, & without spins,\\cr\n        2{\\cal T}, & with spins.\n     }\\end{cases}\n  \\label{barrier_conductance}\n\\end{equation}\nIt is worth mentioning that within the leading logarithm approximation the\ntransmission coefficients of two barriers renormalize independently, so\none can use Eq.~(\\ref{poor_man}) for both left and right barriers.\n\nLet us find the temperature dependence of the peak value of the\nconductance in the case of symmetric barriers, i.e., assume $N=\\frac12$\nand $G_L = G_R = G_b = (e^2\/\\pi\\hbar) {\\cal T}$. According to\nEq.~(\\ref{orthodox}), the peak conductance is $\\frac14 G_b$. Taking into\naccount the renormalization (\\ref{poor_man}) of the transmission\ncoefficient, we find\n\\begin{equation}\nG = \\frac{e^2}{4\\pi\\hbar}\n    \\frac{{\\cal T}_0}\n           {\\cos^2\\left[\\frac{1}{\\pi}\\sqrt{{\\cal T}_0}\\ln(E_C\/T)\\right]}.\n  \\label{peak_conductance}\n\\end{equation}\nAs expected, when the temperature is lowered, the conductance grows.  This\nresult is valid only at temperatures exceeding the characteristic Kondo\ntemperature of this problem, $T_K\\simeq E_C\\exp(-\\pi^2\/2\\sqrt{{\\cal\n    T}_0})$. At too low temperatures, $T\\lesssim T_K$, the conductance is\nof the order of $e^2\/\\hbar$, transmission coefficient ${\\cal T} \\sim 1$,\nand the result of the leading logarithm approximation\n(\\ref{peak_conductance}) is no longer applicable.\n\nIt is also interesting to study the tails of the conductance peaks. If the\ngate voltage $N$ is not precisely half-integer, then at low temperatures,\n$T\\ll\\Delta=2E_C|N-\\frac12|$, one can still use the inelastic co-tunneling\nresult (\\ref{cotunneling}) with the values of conductances of the barriers\nrenormalized according to Eqs.~(\\ref{poor_man}) and\n(\\ref{barrier_conductance}). The resulting conductance has the form:\n\\begin{equation}\nG=\\frac{e^2{\\cal T}_L{\\cal T}_R}{3\\pi\\hbar}\n  \\frac{T^2}\n   {\\Delta^2\n    \\cos^2\\left[\\frac{1}{\\pi}\\sqrt{{\\cal T}_L}\\ln\\frac{E_C}{\\Delta}\\right]\n    \\cos^2\\left[\\frac{1}{\\pi}\\sqrt{{\\cal T}_R}\\ln\\frac{E_C}{\\Delta}\\right]}.\n  \\label{tails}\n\\end{equation}\nSimilarly to Eq.~(\\ref{peak_conductance}), the result (\\ref{tails}) is\nvalid only as long as ${\\cal T}\\ll 1$. In this case the condition allows\nonly the values of the gate voltage not too close to the resonance,\n$\\Delta\\gg T_K$.\n\nAs we have seen, if one starts with the weak-tunneling case, in the\nvicinity of the resonances in $G(V_g)$ the tunneling amplitude grows, and\nthe system approaches the strong-coupling regime. Therefore one can use\nthe results of the weak-tunneling approach to find the universal features\nof the peaks in conductance which should also persist in the\nstrong-tunneling case. Such features are: (i) the $T\\to0$ value of the\npeak conductance (\\ref{peak_conductance}) for symmetric barriers is of the\norder of $e^2\/\\hbar$, and (ii) away from the centers of peaks the\ntemperature dependence of the conductance at $T\\to0$ is $G\\propto T^2$. It\nis worth noting that unlike the temperature dependence of the co-tunneling\nconductance (\\ref{cotunneling}), its $1\/\\Delta^2$ dependence of the gate\nvoltage is not universal, see Eq.~(\\ref{tails}).\n\n\n\\section{Strong co-tunneling of spinless electrons}\n\\label{spinless}\n\nWe will start our treatment of the strong-tunneling case with a simplified\nmodel of spinless electrons. As we will see, in this case a complete\nsolution of the problem is possible. Experimentally, the spinless case can\nbe achieved by applying an in-plane magnetic field, sufficiently strong to\npolarize the electron gas in the constrictions.\n\n\\subsection{Theoretical model}\n\nWe assume that each of the quantum point contacts in Fig.~\\ref{fig:1}\nallows the transmission of electrons in only one mode. The transmission\ncoefficients of the two contacts ${\\cal T}_L$ and ${\\cal T}_R$ are assumed\nto be close to unity. This enables us to start with the Hamiltonian\ncorresponding to perfect transmission, ${\\cal T}_L = {\\cal T}_R = 1$, and\nthen construct the perturbation theory in small reflection amplitudes,\n$r_L$ and $r_R$. Following Ref.~\\onlinecite{Flensberg}, we will describe\neach of the quantum point contacts by a system of one-dimensional (1D)\nelectrons. The formal derivation of this model\\cite{Matveev1} is based on\nthe fact that if the boundaries of the channel are very smooth, the\ntransport though it can be described in terms of adiabatic\nwavefunctions\\cite{Lesovik} having essentially a 1D form.\n\nAnother simplification of the problem is due to the fact that the typical\nenergies involved in the Coulomb blockade phenomenon are of the order of\nthe charging energy $E_C$, which is much smaller than the Fermi energy.\nThis allows us to linearize the spectra of the 1D electrons near the\nFermi points and to formulate the model in terms of the left- and\nright-moving fermions:\n\\begin{eqnarray}\nH_0\\! =\\! i\\hbar v_F\\!\\int\\!\\! &&\n        \\left[\n         \\psi^\\dagger_{L1}(x)\\nabla\\psi_{L1}(x)\n         \\!-\\!\\psi^\\dagger_{L2}(x)\\nabla\\psi_{L2}(x)\n        \\right.\n\\nonumber\\\\\n      &&\\left.\\!\\!\\!\n         +\\psi^\\dagger_{R1}(x)\\nabla\\psi_{R1}(x)\n         \\!-\\!\\psi^\\dagger_{R2}(x)\\nabla\\psi_{R2}(x)\n        \\right] \\!dx.\n  \\label{H_0_fermionic}\n\\end{eqnarray}\nHere $\\psi_{L1}$ and $\\psi_{L2}$ are the annihilation operators for the\nelectrons of the left contact moving to the left and right respectively;\n$\\psi_{R1}$ and $\\psi_{R2}$ describe the left- and right-moving electrons\nof the right contact; $\\nabla\\equiv\\frac{\\partial}{\\partial x}$. We\nassociate the centers of the constrictions with points $x=0$, meaning that\nelectrons in the dot are described by $\\psi_{L1(2)}$ at $x>0$ and by\n$\\psi_{R1(2)}$ at $x<0$. Thus the charging energy can be presented as\n\\begin{equation}\n  H_C = E_C(\\hat n - N)^2,\n\\label{H_C}\n\\end{equation}\nwhere\n\\[\n\\hat n = \\int_{0}^{\\infty}\\!\\sum_{j=1,2}\n         \\!\\left[:\\!\\psi^\\dagger_{Lj}(x)\\psi_{Lj}^{}(x)\\!\\!:\n         +:\\!\\psi^\\dagger_{Rj}(-x)\\psi_{Rj}^{}(-x)\\!\\!:\\right]\\!dx.\n\\]\nHere $:\\!\\hat X\\!\\!:$ denotes the normal-ordered operator $\\hat X$.\nFinally, the presence of weak backscattering in the contacts, described by\nsmall reflection amplitudes $|r_L|, |r_R| \\ll 1$, gives rise to the term\nmixing the left- and right-moving particles,\n\\begin{equation}\nH' = \\hbar v_F\\!\\left[\n    |r_L|\\psi^\\dagger_{L1}(0)\\psi_{L2}^{}(0)\n   +|r_R|\\psi^\\dagger_{R1}(0)\\psi_{R2}^{}(0)\n   +\\mbox{h.c.}\\right]\\!.\n  \\label{backscattering_fermionic}\n\\end{equation}\n\nThe Hamiltonian $H=H_0+H_C+H'$ gives the complete description of the\ntransport through a quantum dot connected to two leads. In our model the\ntwo contacts are described by two independent 1D systems. Thus we neglect\nthe possibility of coherent transport of electrons from one quantum point\ncontact to the other. Such processes lead to the additional mechanism of\ntransport through the dot called {\\it elastic co-tunneling.\\\/} It is\nknown,\\cite{AN} however, that at temperatures exceeding the level spacing\n$\\varepsilon$ in the dot the elastic co-tunneling contribution to the\ntotal conductance is much smaller than that of inelastic co-tunneling.\nThus we will concentrate on the case $T\\gg\\varepsilon$ which is easy to\nsatisfy in the experiments, and in the following calculations we will\nassume $\\varepsilon=0$. At $T\\lesssim\\varepsilon$ the conductance becomes\ntemperature-independent and can be estimated by replacing $T$ by\n$\\varepsilon$ in our results.\n\nThe solution of the problem with the Hamiltonian given by\nEqs.~(\\ref{H_0_fermionic})--(\\ref{backscattering_fermionic}) is not\ntrivial because the interaction term $H_C$ is not quadratic in fermion\noperators. To overcome this difficulty, we will use the bosonization\ntechnique\\cite{Sol} which enables one to treat the interaction in 1D\nsystems exactly. Following Ref.~\\onlinecite{Sol}, we present the fermion\noperators as\n\\begin{equation}\n  \\begin{array}{rcl}\n     \\psi_{L1}(x) &=& \\left(\\frac{D}{2\\pi\\hbar v_F}\\right)^{1\/2}\n                 \\eta_{L1}^{} e^{i\\varphi_{L1}^{}(x)},\\\\\n     \\psi_{L2}(x) &=& \\left(\\frac{D}{2\\pi\\hbar v_F}\\right)^{1\/2}\n                 \\eta_{L2}^{} e^{i\\varphi_{L2}^{}(x)}.\n  \\end{array}\n  \\label{bosonization}\n\\end{equation}\nHere $D$ is the bandwidth, and the two independent bosonic fields\n$\\varphi_{L1}^{}$ and $\\varphi_{L2}^{}$ satisfy the following commutation\nrelations:\n\\begin{equation}\n[\\varphi_{L1}^{}(x), \\varphi_{L1}^{}(y)]\n = -[\\varphi_{L2}^{}(x), \\varphi_{L2}^{}(y)]\n = -i\\pi{\\rm sgn}(x-y).\n  \\label{commutators}\n\\end{equation}\nGiven (\\ref{commutators}), the operators (\\ref{bosonization}) satisfy\nusual anticommutation relations. To ensure proper anticommutation\nrelations for fermions at different branches, we have added\nthe local Majorana fermions $\\eta_{L1}^{}$ and $\\eta_{L2}^{}$.\n\n\nThe electrons of the right contact are\nbosonized in a similar way. One can then rewrite the Hamiltonian\n(\\ref{H_0_fermionic})--(\\ref{backscattering_fermionic}) in terms of\n$\\varphi_L$ and $\\varphi_R$. It will be more convenient, however, to\npresent the Hamiltonian in terms of slightly different bosonic fields:\n\\begin{equation}\n\\phi_L = \\frac{1}{2}(\\varphi_{L2} - \\varphi_{L1}),\n\\quad\n\\Pi_L = -\\frac{1}{2\\pi} \\nabla (\\varphi_{L1} + \\varphi_{L2}).\n  \\label{good_fields}\n\\end{equation}\nThe new variables satisfy the commutation relations typical for the\ndisplacement of a 1D elastic medium and its momentum density,\n\\begin{equation}\n[\\phi_L(x), \\Pi_L(y)] = i \\delta(x-y).\n  \\label{good_commutator}\n\\end{equation}\nThe form of the free-electron Hamiltonian (\\ref{H_0_fermionic}) supports\nthis observation,\n\\begin{eqnarray}\nH_0 = \\frac{\\hbar v_F}{2}\\int\n      &&\\left\\{\\frac{1}{\\pi}[\\nabla \\phi_L(x)]^2 + \\pi \\Pi_L^2(x)\\right.\n\\nonumber\\\\\n      &&\\hspace{.7em}\n        \\left. +\\frac{1}{\\pi}[\\nabla \\phi_R(x)]^2 + \\pi \\Pi_R^2(x)\n        \\right\\}dx.\n  \\label{H_0_bosonized}\n\\end{eqnarray}\n\nThis qualitative interpretation allows one to predict the correct\nbosonized form of the charging energy (\\ref{H_C}). One should expect that\nthe charge of the dot is proportional to the difference of the charge\n$Q_L\\sim e\\phi_L(0)$ brought into the dot through the left contact and the\ncharge $Q_R\\sim e\\phi_R(0)$ carried away through the right contact.\nExplicit calculation gives\n\\begin{equation}\nH_C = \\frac{E_C}{\\pi^2}\n      \\left[\\phi_R(0) - \\phi_L(0) -\\pi N\\right]^2.\n  \\label{H_C_bosonized}\n\\end{equation}\nIt is important to stress that the interaction term (\\ref{H_C_bosonized})\nis now quadratic in $\\phi_L$ and $\\phi_R$. Therefore the bosonization\napproach enables one to treat the charging energy exactly.\n\nFinally, the backscattering term (\\ref{backscattering_fermionic}) takes\nthe form\n\\begin{equation}\nH' = \\frac{D |r_L|}{\\pi} \\cos[2\\phi_L(0)]\n     +\\frac{D |r_R|}{\\pi} \\cos[2\\phi_R(0)].\n  \\label{backscattering_bosonized}\n\\end{equation}\nIn Eq.~(\\ref{backscattering_bosonized}) we neglected the products of\nMajorana fermions $\\eta_{L1}^{}\\eta_{L2}^{}$ and\n$\\eta_{R1}^{}\\eta_{R2}^{}$, as they commute with each other.  Unlike the\nfirst two terms, $H_0$ and $H_C$, the backscattering term is not quadratic\nin bosonic variables and cannot be treated exactly. Below we assume weak\nbackscattering and develop the perturbation theory in small parameters\n$|r_L|$ and $|r_R|$.\n\n\n\\subsection{Perturbation theory for the conductance}\n\nIn this paper we are only interested in the linear conductance through the\nquantum dot. To find it, we will use a Kubo formula, which in the\nzero-frequency limit can be written as\n\\begin{equation}\nG = \\frac{i}{2\\hbar}\n    \\int_{-\\infty}^{\\infty} t\n    \\langle [ I(t), I(0) ] \\rangle dt,\n  \\label{Kubo}\n\\end{equation}\nwhere $I$ is the operator of current through the dot.\n\nWe define $I$ as the average of the current $I_L$ flowing into the dot\nthrough the left contact and the current $I_R$ flowing out of the dot\nthrough the right contact.  Clearly, the current $I_L$ is proportional to\nthe time derivative of the displacement of the effective elastic medium\ndescribing the left contact, $I_L\\sim e\\dot\\phi_L$, and, similarly,\n$I_R\\sim e\\dot\\phi_R$. From the equations of motion corresponding to the\nHamiltonian $H_0 + H_C + H'$, one concludes that $I\\sim\nev_F(\\Pi_L+\\Pi_R)$. A careful calculation gives\n\\begin{equation}\nI = \\frac{1}{2}ev_F\n      \\left.\\left(\\Pi_L + \\Pi_R\\right)\\right|_{x=0}.\n  \\label{current_operator}\n\\end{equation}\n\nAs the first step, we will find the conductance in the absence of the\nbarriers. Since the current operator (\\ref{current_operator}) commutes\nwith $H_C$, one should expect the conductance to be insensitive to the\ncharging energy. Clearly, at $E_C = 0$ the system is equivalent to two\nresistances $2\\pi\\hbar\/e^2$ of the two quantum point contacts, connected\nin series. One then expects the conductance to be\n\\begin{equation}\nG_0 = \\frac{e^2}{4\\pi\\hbar}.\n  \\label{G_0}\n\\end{equation}\nThis result is easily derived from the Kubo formula (\\ref{Kubo}), because\nwith $H'=0$ the current-current correlator must be found for a quadratic\nHamiltonian.  As a result, one finds $\\langle [ I(t), I(0) ] \\rangle =\ni(e^2\/2\\pi)\\delta'(t)$, and after the substitution into (\\ref{Kubo}) one\nreproduces (\\ref{G_0}).\n\nThe result (\\ref{G_0}) is independent of temperature. Since we used the\nHamiltonian (\\ref{H_0_fermionic}) with linearized spectra of electrons,\nthe temperature was assumed to be much smaller than the Fermi energy, but\ncould still be of the order of the charging energy. In the presence of the\nbarriers the conductance must acquire some temperature dependence, which\nis obvious from the limit of very high barriers (\\ref{cotunneling}).\nIndeed, the lowest non-vanishing (2nd-order) correction in $r_L$ and $r_R$\nto the conductance is already temperature dependent. At low temperatures\n$T\\ll E_C$, we get\n\\begin{equation}\nG=\\frac{e^2}{4\\pi\\hbar}\\!\\left(1-\\frac{\\pi\\Gamma_0(N)}{4T}\\right)\\!,\n  \\label{2nd_order_spinless}\n\\end{equation}\nwhere $\\Gamma_0$ is defined as follows:\n\\begin{equation}\n\\Gamma_0(N)=\\frac{2\\gamma E_C}{\\pi^2}\n              \\left[|r_L|^2+|r_R|^2+2|r_L||r_R|\\cos(2\\pi N)\\right]\\!.\n\\label{Gamma_0}\n\\end{equation}\nHere $\\gamma=e^{\\bf C}$, with ${\\bf C}\\approx 0.5772\\ldots$ being the\nEuler's constant.  In Eq.~(\\ref{2nd_order_spinless}) we have neglected the\nterms small in $T\/E_C$.  The result (\\ref{2nd_order_spinless}) and\n(\\ref{Gamma_0}) is analogous to that for the conductance of a 1D wire with\ntwo defects\\cite{Furusaki,remark3} and can be obtained in a similar way,\nsee Appendix~\\ref{A1}.\n\n\nAs expected, the presence of weak scattering in the contacts gives rise to\na small periodic correction to the conductance. The same behavior was\npredicted\\cite{Matveev1} for the average charge of the dot. However,\nunlike the average charge, the correction to conductance strongly depends\non temperature. As the temperature is lowered, the conductance decreases.\nThe only case when conductance is temperature-independent is when the\nbarriers are symmetric, $|r_L|=|r_R|$, and the dimensionless gate voltage\n$N$ is half-integer, which corresponds to the center of a peak in the\nhigh-barrier case. In this regime we expect the conductance to be of the\norder of $e^2\/\\hbar$, cf.\\ Sec.~\\ref{weak}, which is confirmed by the\nperturbation theory (\\ref{2nd_order_spinless}) and (\\ref{Gamma_0}).\n\n\n\\subsection{Non-perturbative calculation}\n\nAt low temperatures $T\\lesssim \\Gamma_0$ the second-order correction\nbecomes large, and the perturbation theory fails. To find the conductance\nin this case, one has to sum up infinite number of terms of the\nperturbation theory. To perform such a summation, we will use the\ntechnique developed in Ref.~\\onlinecite{Matveev1}.\n\nAt the first step we perform a linear transformation of the bosonic\nfields,\n\\begin{equation}\n\\phi_L = \\frac{\\phi_I - \\phi_C}{\\sqrt{2}},\n\\quad\n\\phi_R = \\frac{\\phi_I + \\phi_C}{\\sqrt{2}},\n  \\label{transformation}\n\\end{equation}\nand similarly for $\\Pi_L$ and $\\Pi_R$. The advantage of using these new\nvariables is that the Coulomb energy (\\ref{H_C_bosonized}) is now\nexpressed in terms of the charging mode $\\phi_C$ only,\n\\begin{equation}\nH_C = \\frac{2E_C}{\\pi^2}\n      \\left[\\phi_C(0)-\\frac{\\pi N}{\\sqrt{2}}\\right]^2.\n  \\label{H_C_diagonalized}\n\\end{equation}\nThe charging energy (\\ref{H_C_diagonalized}) suppresses the fluctuations\nof $\\phi_C(0)$ at energy scales below $E_C$. Thus at $T\\ll E_C$ one can\nintegrate out the fluctuations of the charging mode by replacing the\nscattering term (\\ref{backscattering_bosonized}) with its value averaged\nover the fluctuations of $\\phi_C$. The resulting Hamiltonian is expressed\nin terms of the field $\\phi_I$ only and has the form\n\\begin{eqnarray}\nH_0 &=& \\frac{\\hbar v_F}{2}\\int\n        \\left\\{\\frac{1}{\\pi}[\\nabla \\phi_I(x)]^2 + \\pi \\Pi_I^2(x)\n        \\right\\}dx,\n  \\label{H_0_simplified}\\\\\nH' &=& \\left(\\frac{\\gamma E_C D}{2\\pi^3}\\right)^{1\/2}\n       \\left[re^{i\\sqrt{2}\\phi_I(0)}\n       + r^{*}e^{-i\\sqrt{2}\\phi_I(0)}\\right],\n  \\label{H'_simplified}\n\\end{eqnarray}\nwhere the dependence on the gate voltage is incorporated into the\ncomplex parameter $r = |r_L|e^{-i\\pi N} + |r_R|e^{i\\pi N}$.\n\nThe Hamiltonian (\\ref{H_0_simplified}) and (\\ref{H'_simplified}) can be\ntransformed to that of an impurity in a $g=\\frac12$ Luttinger\nliquid\\cite{Kane} by a simple transformation $\\phi=\\sqrt{2}\\phi_I$ and\n$\\Pi=\\Pi_I\/\\sqrt{2}$. The latter can be solved exactly\\cite{Kane} using\nthe technique developed in Ref.~\\onlinecite{Guinea}. For our purposes it\nwill be more convenient to use an alternative exact\nsolution.\\cite{Matveev1} The idea is to interpret $e^{i\\sqrt{2}\\phi_I}$ as\na bosonized form of some fermion annihilation operator.\\cite{Luther} This\ncan be done by rewriting the Hamiltonian in terms of two decoupled chiral\nbosonic fields defined as\n\\begin{equation}\n\\varphi_{\\pm}(x) = \\frac{\\phi_I(x) \\pm \\phi_I(-x)}{\\sqrt{2}}\n                   +\\pi\\int_{0}^{x}\n                    \\frac{\\Pi_I(x') \\pm \\Pi_I(-x')}{\\sqrt{2}} dx'.\n  \\label{chiralization}\n\\end{equation}\nIn these variables the Hamiltonian takes the form\n\\begin{eqnarray}\nH_0 &=& \\frac{\\hbar v_F}{4\\pi}\\int\n        \\left\\{[\\nabla \\varphi_{+}(x)]^2 + [\\nabla \\varphi_{-}(x)]^2\n        \\right\\}dx,\n  \\label{H_0_chiralized}\\\\\nH' &=& \\left(\\frac{\\gamma E_C D}{2\\pi^3}\\right)^{1\/2}\n       \\left[re^{i\\varphi_{+}(0)}\n       + r^{*}e^{-i\\varphi_{+}(0)}\\right].\n  \\label{H'_chiralized}\n\\end{eqnarray}\nOne can easily check that the fields $\\varphi_+$ and $\\varphi_-$ commute.\nThus the $\\varphi_-$ part of the Hamiltonian completely decouples.  Since\nthe field $\\varphi_+(x)$ satisfies the same commutation relations\n(\\ref{commutators}) as $\\varphi_{L1}$, one can in complete analogy with\nEq.~(\\ref{bosonization}) interpret $e^{i\\varphi_+}$ as a fermion\noperator,\\cite{remark}\n\n\\begin{figure}\n\\narrowtext\n\\epsfxsize=3in\n\\hspace{.2em}\n\\epsffile{4chan2.ps}\n\\vspace{1.5\\baselineskip}\n\\caption{Conductance (\\protect\\ref{conductance_spinless}) as a function\n  of the dimensionless gate voltage $N$ for the symmetric case,\n  $|r_L|=|r_R|=0.2$.  The two curves are calculated for $E_C\/T=25$ (a) and\n  $200$ (b). In this calculation we included the correction proportional\n  to $T\/E_C$ in Eq.~(\\protect\\ref{finally_obtain}). The dashed line\n  represents the conductance $G=G_LG_R\/(G_L + G_R)$ in the\n  high-temperature limit, $T\\gg E_C$.}\n\\label{fig:2}\n\\end{figure}\n\\vspace{\\baselineskip}\n\n\\begin{equation}\ne^{i\\varphi_+(x)} = \\left(\\frac{2\\pi\\hbar v_F}{D}\\right)^{1\/2}\n                    \\eta_{+}\\psi_{+}(x).\n  \\label{fermionization}\n\\end{equation}\n\nThe resulting Hamiltonian is quadratic in fermion operators,\n\\begin{eqnarray}\nH &=& i\\hbar v_F\\int\\psi_{+}^{\\dagger}(x)\\nabla\\psi_{+}(x)dx\n\\nonumber\\\\\n  & & + \\left(\\frac{\\gamma \\hbar v_F E_C}{\\pi^2}\\right)^{1\/2}\n      \\left[r\\eta_{+}^{}\\psi_{+}(0)\n            + r^{*}\\psi_{+}^{\\dagger}(0)\\eta_{+}^{}\n      \\right],\n  \\label{H_fermionized}\n\\end{eqnarray}\nand can be easily diagonalized.\\cite{Matveev1} This greatly simplifies the\ncalculation of the conductance. One can again use the Kubo formula\n(\\ref{Kubo}), with the properly transformed current operator\n(\\ref{current_operator}). In the new fermionic variables the current\noperator has the form\n\\begin{equation}\nI = -e v_F \\left. :\\!\\psi_{+}^{\\dagger}\\psi_{+}\\!\\!:\n           \\right|_{x=0}.\n  \\label{current_operator_fermionized}\n\\end{equation}\n\nThe actual calculation of the linear conductance is rather long, but\nstraightforward. First we find the current-current correlation function\nto be\n\\begin{eqnarray}\n& &\\langle [I(t),I(0)]\\rangle =\n   i\\frac{e^2}{2\\pi}\\delta'(t)\n\\nonumber\\\\\n & &\\hspace{3em}\n  - i\\frac{e^2}{2\\pi^2\\hbar^2}\\int\\!\\!\\!\\int\\!\n  \\frac{\\Gamma_0^2\\sin[(\\epsilon+\\epsilon')t\/\\hbar]d\\epsilon d\\epsilon'}\n       {(\\epsilon^2+\\Gamma_0^2)(e^{\\epsilon\/T}+1)(e^{\\epsilon'\/T}+1)}.\n  \\label{current-current}\n\\end{eqnarray}\nThe substitution of Eq.~(\\ref{current-current}) into the Kubo formula\n(\\ref{Kubo}) gives the following expression for the linear conductance:\n\\begin{equation}\nG=\\frac{e^2}{4\\pi\\hbar}\n  \\left[1-\\int^\\infty_{-\\infty}dE\n        \\frac{1}{4T\\cosh^2\\frac{E}{2T}}\n        \\frac{\\Gamma^2_0(N)}{E^2+\\Gamma^2_0(N)}\n  \\right].\n  \\label{conductance_spinless}\n\\end{equation}\nEquation (\\ref{conductance_spinless}) is the central result of this\nsection.  It is valid for any temperature below $E_C$, and, unlike the\nperturbative result (\\ref{2nd_order_spinless}), it is not restricted to\nonly sufficiently high temperatures $T\\gg\\Gamma_0$.  Of course, at\n$T\\gg\\Gamma_0$ the result (\\ref{2nd_order_spinless}) is reproduced.\n\nIn the opposite limit, $T\\ll\\Gamma_0$, the conductance is quadratic in\ntemperature,\n\\begin{equation}\nG = \\frac{\\pi e^2}{12\\hbar}\\left[\\frac{T}{\\Gamma_0(N)}\\right]^2.\n  \\label{low_T_spinless}\n\\end{equation}\nIt is important to note that the same temperature dependence was found for\nthe inelastic co-tunneling (\\ref{cotunneling}) in the weak-tunneling\nregime. As expected, the quadratic temperature dependence is a universal\nproperty of inelastic co-tunneling and is not limited to the\nweak-tunneling limit, cf.\\ Eq.~(\\ref{tails}).\n\nIn the special case of symmetric barriers, $|r_L| = |r_R| = r_0$, the\nlow-temperature limit (\\ref{low_T_spinless}) is never achieved on\nresonance. Indeed, we have $\\Gamma_0= (8\\gamma E_C\/\\pi^2)r_0^2\\cos^2 \\pi\nN$, i.e., $\\Gamma_0$ vanishes at half-integer values of $N$. Therefore in\nthe low-temperature limit the resonant value of conductance is\n$e^2\/4\\pi\\hbar$, whereas away from the resonance $\\Gamma_0\\neq 0$, and\n$G\\propto T^2$. It is clear then that at low temperatures the weak Coulomb\nblockade oscillation in conductance (\\ref{2nd_order_spinless}) transforms\ninto sharp periodic peaks, Fig.~\\ref{fig:2}. Unlike the weak-tunneling\ncase (\\ref{cotunneling}), the tails of the peak at $N=\\frac12$ are not\nLorentzian: $G\\propto1\/(N-\\frac12)^4$.\n\nFinally, it is worth noting that the conductance\n(\\ref{conductance_spinless}) coincides with that of a weak impurity in the\n$g=\\frac12$ Luttinger liquid.\\cite{Kane} This is not surprising,\nconsidering the aforementioned identity of the Hamiltonian\n(\\ref{H_0_simplified}), (\\ref{H'_simplified}) with the one for the quantum\nimpurity problem.\\cite{Kane} In the latter problem the condition\n$g=\\frac12$ requires special fine-tuning of the strength of interaction\nbetween electrons. In our case the condition $g=\\frac12$ is satisfied\nautomatically when we integrate out one of the two bosonic\nmodes.\\cite{Furusaki}\n\n\n\\section{Co-tunneling of electrons with spins}\n\\label{withspin}\n\nIn the previous section we considered the case of spinless electrons,\nwhich can be realized by applying strong in-plane magnetic field. In the\nabsence of the field, each of the contacts has two identical modes,\ncorresponding to two possible values of electron spin. The number of modes\nis an important parameter in our problem. This can be easily seen from the\nanalogy to the Kondo problem discussed in Sec.~\\ref{weak}. In the\nweak-tunneling case, the model of spinless electrons maps to the 2-channel\nKondo problem, because the fictitious spin of the impurity---i.e., the\ncharge of the dot---can be changed by tunneling of an electron in two\nindependent channels, corresponding to the left and right contacts. In the\npresence of real spins of electrons, the number of channels in the\neffective Kondo problem is 4. It is well known that the number of channels\nin the Kondo problem strongly affects its properties.\\cite{NB} Thus one\nshould expect that the presence of spins should have a dramatic effect\nupon the transport through a quantum dot.\n\n\n\\subsection{Strong-tunneling case}\n\nWe will start with the discussion of the strong-tunneling limit, ${\\cal\n  T}_L, {\\cal T}_R \\to 1$. To find the conductance, one can use the same\ntechnique as in Sec.~\\ref{spinless}. The Hamiltonian of the problem can be\nwritten in the bosonized form, analogous to\n(\\ref{H_0_bosonized})--(\\ref{backscattering_bosonized}),\n\\begin{eqnarray}\nH_0 &=& \\frac{\\hbar v_F}{2}\\!\\sum_{\\alpha}\\int\\!\n      \\left\\{\\frac{1}{\\pi}[\\nabla \\phi_{L\\alpha}(x)]^2\n                           + \\pi \\Pi_{L\\alpha}^2(x)\\right.\n\\nonumber\\\\\n     &&\\hspace{5.3em}\\left. +\\frac{1}{\\pi}[\\nabla \\phi_{R\\alpha}(x)]^2\n                                         + \\pi \\Pi_{R\\alpha}^2(x)\n        \\right\\}\\!dx,\n\\label{H_0_with_spins}\\\\\nH_{C} &=& \\frac{E_C}{\\pi^2}\\left\\{\\sum_{\\alpha}\n         [\\phi_{R\\alpha}(0) - \\phi_{L\\alpha}(0)] -\\pi N\\right\\}^2,\n\\label{H_C_with_spins}\\\\\nH' &=& \\frac{D}{\\pi} \\sum_{\\alpha}\n       \\{|r_L| \\cos[2\\phi_{L\\alpha}(0)]\\!\n              +\\! |r_R| \\cos[2\\phi_{R\\alpha}(0)]\n       \\}.\n  \\label{backscattering_with_spins}\n\\end{eqnarray}\nHere the fields with $\\alpha=\\ \\uparrow$ and $\\downarrow$ describe the\nelectrons with spins up and down respectively.\n\nOne can again find the linear conductance using the Kubo formula\n(\\ref{Kubo}), but the current operator (\\ref{current_operator}) must be\nmodified to account for the spins,\n\\begin{equation}\nI = \\frac{1}{2}ev_F\\!\\sum_{\\alpha=\\uparrow,\\downarrow}\\!\n      \\left(\\Pi_{L\\alpha} + \\Pi_{R\\alpha}\\right)\\Bigl|_{x=0}.\n  \\label{current_operator_with_spins}\n\\end{equation}\n\nIn the absence of barriers, $|r_L|=|r_R|=0$, the calculation is trivial,\nas the Hamiltonian (\\ref{H_0_with_spins}) and (\\ref{H_C_with_spins})\nis quadratic. Due to the doubling of the number of modes in each contact,\nin the presence of spins the conductance doubles compared to\nEq.~(\\ref{G_0}),\n\\begin{equation}\nG_0 = \\frac{e^2}{2\\pi\\hbar}.\n  \\label{G_0_with_spins}\n\\end{equation}\n\nWeak backscattering, $|r_L|,|r_R|\\ll1$, can be treated in perturbation\ntheory in $H'$, see Appendix \\ref{A2}. Up to the second order we get\n\\begin{equation}\n  G = \\frac{e^2}{2\\pi\\hbar}\n     \\left[1-\\frac{2\\Gamma(\\frac{3}{4})}{\\Gamma(\\frac{1}{4})}\n             \\sqrt{\\frac{\\gamma E_C}{\\pi T}}\n             \\left(|r_L|^2+|r_R|^2\\right)\n     \\right].\n  \\label{2nd_order_with_spins}\n\\end{equation}\nIt is instructive to compare this result with its analog for the spinless\ncase, namely Eq.~(\\ref{2nd_order_spinless}). In both cases the small\ncorrection to the conductance has the tendency to grow as the temperature\nis decreased. An important difference is that the second-order correction\nin (\\ref{2nd_order_with_spins}) does not depend on the gate voltage $N$,\nand as a result it grows even at half-integer $N$, i.e., on resonance.\nThus, in contrast to the spinless case, one should expect the resonance\nvalue to be smaller than the unperturbed conductance $G_0=e^2\/2\\pi\\hbar$.\n\nThis difference has a natural interpretation in terms of the analogy with\nthe Kondo model. As we already mentioned, the spinless case corresponds to\nthe 2-channel Kondo model. In the 2-channel case the strong-coupling fixed\npoint is stable at zero magnetic field, or, in terms of the Coulomb\nblockade problem, at half-integer $N$. Therefore, precisely on resonance a\nweak reflection must be an irrelevant perturbation, and the peak value of\nthe conductance should be given by Eq.~(\\ref{G_0}). On the contrary, the\n4-channel Kondo problem corresponding to the case with spins has a stable\nfixed point at an intermediate value of coupling, and both weak- and\nstrong-coupling fixed points are unstable.\\cite{NB} Thus even on resonance\nthe conductance is not given by its value in the strong-tunneling limit\n(\\ref{G_0_with_spins}), as confirmed by the perturbation theory\n(\\ref{2nd_order_with_spins}).\n\n\\subsection{Conductance in the case of asymmetric barriers}\n\\label{asymmetric}\n\nUnfortunately, it is not easy to calculate the conductance near the\nintermediate-coupling fixed point. Some conclusions about it can be drawn\non the basis of the scaling approach, Sec.~\\ref{scaling}. One should note,\nhowever, that this fixed point is stable only if the coupling in all the 4\nchannels is identical. In the case of the Coulomb blockade problem, the\ncoupling in the pairs of channels corresponding to different spin\norientations in the same contact must be identical (in the absence of\nmagnetic field). On the other hand, the coupling constants for the modes\nin different contacts are determined by the corresponding conductances and\nare not necessarily equal. In fact, in the structure shown in\nFig.~\\ref{fig:1}, the two conductances are controlled separately by the\ngate voltages $V_L$ and $V_R$. Therefore in experiments the coupling\nconstants corresponding to the left and right modes are not equal, unless\na special attempt to make them equal is made.\n\nIn this section we study the limit of very asymmetric barriers, namely we\nassume that one of the barriers is very high, whereas the other one is\nlow. In terms of the transmission coefficients of the barriers this limit\ncorresponds to ${\\cal T}_L \\ll 1$ and $1-{\\cal T}_R \\ll 1$. Such a regime\ncan be easily realized in an experiment by proper tuning of the gate\nvoltages $V_L$ and $V_R$. In addition, we will see in Sec.~\\ref{scaling}\nthat at $T\\to 0$ arbitrarily weak asymmetry grows and the system scales\ntowards the strongly asymmetric limit. Therefore at low temperatures the\nresults of this section can be applied to any asymmetric system.\n\nIn Sec.~\\ref{weak} we described the contact in the regime of weak\ntunneling by the tunnel Hamiltonian (\\ref{tunnel_hamiltonian}) written in\nterms of the original electrons. On the other hand, in Sec.~\\ref{spinless}\nit was more convenient to use the bosonized form of the Hamiltonian for\nthe electron gas in the contact with transmission coefficient close to\nunity. Therefore when considering the asymmetric case ${\\cal T}_L\\to0$ and\n${\\cal T}_R\\to1$ we shall bosonize only the electrons in the right\ncontact. Then the unperturbed fixed-point Hamiltonian $H_0 + H_C$ is given\nby\n\\begin{eqnarray}\nH_0 &=& \\sum_{k\\alpha} \\epsilon_k a_{k\\alpha}^\\dagger a_{k\\alpha}^{}\n        + \\sum_{p\\alpha} \\epsilon_p a_{p\\alpha}^\\dagger a_{p\\alpha}^{}\n\\nonumber\\\\\n    & & +\\frac{\\hbar v_F}{2}\\!\\sum_{\\alpha}\\!\\int\\!\n         \\left\\{\\frac{1}{\\pi}[\\nabla \\phi_{R\\alpha}(x)]^2\n                                         + \\pi \\Pi_{R\\alpha}^2(x)\n        \\right\\}dx,\n  \\label{H_0_asymmetric}\\\\\nH_C &=& E_C\\!\n        \\left[\\hat n+\\frac{1}{\\pi}\\sum_\\alpha\\phi_{R\\alpha}(0)\n              - N\\right]^2.\n  \\label{H_C_asymmetric}\n\\end{eqnarray}\nHere the operators $a_{k\\alpha}$ and $a_{p\\alpha}$ correspond to the\nleft-contact electrons in the lead and in the dot, respectively; the\nbosonic fields $\\phi_{R\\alpha}$ and $\\Pi_{R\\alpha}$ model the electrons in\nthe right contact. An integer-valued operator $\\hat n$ describes the number\nof electrons transferred to the dot through the left barrier.\n\nThe perturbation consists of two parts: weak tunneling through the high\nbarrier in the left contact, and weak scattering on the small barrier in\nthe right contact,\n\\begin{eqnarray}\n  H_L' &=& \\sum_{kp\\alpha}\n           (v_t^{} a_{k\\alpha}^\\dagger a_{p\\alpha}^{}F\n            + v_t^* a_{p\\alpha}^\\dagger a_{k\\alpha}^{}F^\\dagger),\n\\label{H_L'}\\\\\n  H_R' &=& \\frac{1}{\\pi}D|r_R| \\sum_{\\alpha}\n           \\cos[2\\phi_{R\\alpha}(0)].\n\\label{H_R'}\n\\end{eqnarray}\nWe have chosen not to write the operator $\\hat n$ of the number of\nelectrons transferred into the dot through the barrier in terms of the\nfermionic operators as $\\hat n=\\sum a_{p\\alpha}^\\dagger a_{p\\alpha}^{}$,\nbut to treat it as an independent variable. Thus the tunneling Hamiltonian\n(\\ref{H_L'}) acquires the charge-lowering and raising operators $F$ and\n$F^\\dagger$ defined as\n\\begin{equation}\n[F,\\hat n] = F.\n  \\label{lowering_operator}\n\\end{equation}\nOne possible representation\\cite{Ingold} of these operators is given by\n$F=e^{ix}$ and $\\hat n = i\\frac{\\partial}{\\partial x}$, where $x$ has the\nmeaning of a phase operator conjugated to $\\hat n$.\n\nOur goal in this section is to find the conductance of the system,\nassuming that the perturbations $H_L'$ and $H_R'$ are small. At the first\nstep we will utilize the smallness of the amplitude $v_t^{}$ of tunneling\nthrough the left barrier, and find the expression for the current in the\nsecond order in $v_t^{}$. The current can be defined as the time\nderivative of the number of electrons that have tunneled through the\nbarrier, $I = -e \\frac{\\partial \\hat n}{\\partial t} =\n\\frac{ie}{\\hbar}[\\hat n,H]$. Using the commutation relation\n(\\ref{lowering_operator}), we get\n\\begin{equation}\nI = i\\frac{e}{\\hbar}\n    \\sum_{kp\\alpha} \\left(\n    v_t^* a_{p\\alpha}^\\dagger a_{k\\alpha}^{} F^\\dagger -\n    v_t a_{k\\alpha}^\\dagger a_{p\\alpha}^{} F\\right).\n  \\label{current_asymmetric}\n\\end{equation}\nTo find the conductance through the dot, we will substitute the above\ncurrent operator into the Kubo formula (\\ref{Kubo}). This requires the\ncalculation of the correlator $\\langle[I(t),I(0)]\\rangle$. Since $I\\propto\n|v_t|$, up to the second order in tunneling matrix elements we can average\n$[I(t),I(0)]$ over the equilibrium thermal distribution of the system\nwithout tunneling, $H_L'=0$. Then the averaging over the fermionic degrees\nof freedom is straightforward, and we get\n\\begin{equation}\nG = -i G_L \\frac{\\pi T^2}{\\hbar^2}\n      \\int_{-\\infty}^{\\infty}\n      \\frac{tK(t)dt}{\\sinh^2[\\pi T(t-i\\delta)\/\\hbar]}.\n  \\label{conductance_asymmetric}\n\\end{equation}\nHere we have introduced the conductance of the left tunnel barrier,\n$G_L=(4\\pi e^2\/\\hbar)\n\\sum_{kp}|v_t^{}|^2\\delta(\\epsilon_k)\\delta(\\epsilon_p)$, and the\ncorrelator\n\\begin{equation}\nK(t) = \\langle F(t) F^\\dagger(0)\\rangle\n        = \\langle F^\\dagger(t) F(0)\\rangle.\n  \\label{K}\n\\end{equation}\nThe equality of the two correlators in Eq.~(\\ref{K}) follows from the fact\nthat the symmetry transformation $\\hat n \\to -\\hat n$, $\\phi_{R\\alpha}\\to\n-\\phi_{R\\alpha}$, and $N\\to -N$ does not affect the Hamiltonian\n(\\ref{H_0_asymmetric}), (\\ref{H_C_asymmetric}), and (\\ref{H_R'}), but\nchanges $F^\\dagger\\leftrightarrow F$.\n\nIt is instructive to apply the formula (\\ref{conductance_asymmetric}) to\nthe non-interacting case, $E_C = 0$. In this limit the operators $F$ and\n$F^\\dagger$ commute with the Hamiltonian (\\ref{H_0_asymmetric}) and\n(\\ref{H_R'}). Thus the correlator (\\ref{K}) is unity, and we get $G=G_L$.\nThis is the expected result, since we assumed $G_L\\to0$, and the\nconductance of the system must be determined by that of the weakest link.\n\nLet us now find the conductance at non-zero charging energy $E_C$, but in\nthe absence of the scattering in the right contact, $r_R=0$. In this\ncase the correlator (\\ref{K}) is no longer trivial, because the operators\n$F$ and $F^\\dagger$ do not commute with the interaction term\n(\\ref{H_C_asymmetric}). When an electron tunnels into the dot, $n\\to n+1$,\nthe system is shifted from its equilibrium state.  To accommodate the\nadditional electron brought by the tunneling process and to return the\nsystem to the equilibrium state, one electron must leave the dot\nthrough the right contact. Thus the time evolution of the operators $F$\nand $F^\\dagger$ becomes non-trivial.\n\nTo find the correlator $K(t)$, we will perform the following unitary\ntransformation of the Hamiltonian:\n\\begin{eqnarray}\n&U = e^{i\\hat n \\Theta},&\n  \\label{unitary_transformation}\\\\\n&{\\displaystyle\n\\Theta = \\frac{\\pi}{2} \\int_{-\\infty}^{\\infty}\n            \\left[\\Pi_{R\\uparrow}(y) + \\Pi_{R\\downarrow}(y)\\right]dy}.&\n  \\label{phase}\n\\end{eqnarray}\nThis transformation shifts the bosonic fields $\\phi_{R\\alpha}$ and adds a\nphase factor to the charge-lowering operator $F$,\n\\begin{eqnarray}\n \\phi_{R\\alpha}(x)&\\to&  \\phi_{R\\alpha}(x) - \\frac{\\pi}{2}\\hat n,\n\\label{transformed_phi}\\\\\n F&\\to& F e^{i\\Theta}.\n\\label{transformed_F}\n\\end{eqnarray}\nOne can easily see that the kinetic-energy term (\\ref{H_0_asymmetric}) is\ninvariant under the transformation (\\ref{unitary_transformation}), whereas\nthe charging term (\\ref{H_C_asymmetric}) transforms to a $\\hat\nn$-independent form,\n\\begin{equation}\n\\tilde H_C = U^\\dagger H_C U =\nE_C\\left[\\frac{1}{\\pi}\\sum_\\alpha\\phi_{R\\alpha}(0) - N\\right]^2.\n\\label{transformed_H_C}\n\\end{equation}\nUpon this transformation, the operators $F$ and $F^\\dagger$ commute with\nthe Hamiltonian (\\ref{H_0_asymmetric}) and (\\ref{transformed_H_C}), but\nthey acquire non-trivial phase factors, see Eq.~(\\ref{transformed_F}).\nThis enables us to find $K(t)$ as the correlator of the phase factors,\n$K(t) = K_{\\Theta}(t)\\equiv \\langle e^{i\\Theta(t)}\ne^{-i\\Theta(0)}\\rangle$.  Since the phase $\\Theta$ is linear in bosonic\nvariables, the calculation of the correlator $K_{\\Theta}$ is\nstraightforward:\n\\begin{eqnarray}\nK_{\\Theta}(t)\n   &=& \\exp\\{-\\langle[\\Theta(0)-\\Theta(t)]\\Theta(0)\\rangle\\}\n\\nonumber\\\\\n   &\\simeq& \\frac{\\pi^2 T}{2i\\gamma E_C}\n         \\frac{1}{\\sinh[\\pi T(t-i\\delta)\/\\hbar]}.\n  \\label{Debye_Waller}\n\\end{eqnarray}\nThe last equality is valid asymptotically at low energies, $T,\n\\hbar\/t\\ll E_C$. The substitution of $K=K_\\Theta$ into the\nexpression for the conductance (\\ref{conductance_asymmetric}) leads to a\nlinear temperature dependence of the conductance:\n\\begin{equation}\nG = G_L \\frac{\\pi^3 T}{8\\gamma E_C}.\n  \\label{peak_conductance_asymmetric}\n\\end{equation}\n\nPhysically the origin of the linear suppression of the conductance at low\ntemperatures (\\ref{peak_conductance_asymmetric}) is the orthogonality\ncatastrophe.\\cite{Anderson2} Unlike the non-interacting case $E_C = 0$\ndiscussed above, the charge fluctuations in the dot are now suppressed at\nenergies below $E_C$. Effectively one can interpret the effect of the\ncharging energy as a hard-wall boundary condition for the wavefunctions of\nthe right-contact electrons with energies within the band of width $W\\sim\nE_C$ near the Fermi level. When an electron tunnels through the left\nbarrier, the system must lower its charging energy by moving one electron\nthrough the right contact. Since the two spin channels are completely\nsymmetric, each mode transfers the charge $q=\\frac{e}{2}$. According to\nthe Friedel sum rule, this leads to a change in the boundary condition\ncorresponding to an additional scattering phase shift $\\delta=\\pm\\frac{\\pi\n  q}{e}=\\pm\\frac{\\pi}{2}$.  A sudden change of the boundary conditions\ncreates a state with a large number of electron-hole excitations, which is\nnearly orthogonal\\cite{Anderson2} to the ground state of the system. This\northogonality leads to a power-law suppression of the tunneling density of\nstates $\\nu(\\varepsilon)\\propto\\varepsilon^\\chi$, with the\nexponent\\cite{Schotte,ND,Mahan} determined by the phase shifts in all the\nelectronic channels, $\\chi=\\sum(\\delta\/\\pi)^2$. In our system, each spin\nmode gives two channels---in the dot and in the right lead---leading to\nthe total of 4 channels. Thus the exponent $\\chi$ is unity, and the\ndensity of states $\\nu(\\varepsilon)\\propto\\varepsilon$. The linear\nsuppression of the tunneling density of states results in the $T$-linear\nconductance (\\ref{peak_conductance_asymmetric}).\n\nThe linear temperature dependence of conductance\n(\\ref{peak_conductance_asymmetric}) appears to contradict to the $T^2$ law\nfor the inelastic co-tunneling, Eqs.~(\\ref{cotunneling}), (\\ref{tails})\nand (\\ref{low_T_spinless}). The reason is the absence of the barrier in\nthe right junction assumed in the derivation of\nEq.~(\\ref{peak_conductance_asymmetric}). We will now show that the\npresence of arbitrarily weak barrier in the right contact is crucial (see\nalso Appendix~\\ref{A3}) and leads to the quadratic temperature dependence\nof the conductance at $T\\to0$.\n\nThe weak backscattering in the right contact is described by the term\n(\\ref{H_R'}) in the Hamiltonian.  As it follows from\nEq.~(\\ref{transformed_phi}), under the unitary transformation\n(\\ref{unitary_transformation}) the scattering term (\\ref{H_R'}) acquires\nadditional sign:\n\\begin{equation}\n\\tilde H_R' = (-1)^{\\hat n}\n              \\frac{1}{\\pi}D|r_R| \\sum_{\\alpha}\n              \\cos[2\\phi_{R\\alpha}(0)].\n  \\label{transformed_H_R'}\n\\end{equation}\nThe operators $F$ and $F^\\dagger$ no longer commute with the Hamiltonian:\nthey commute with $H_0 + \\tilde H_C$ and anti-commute with $\\tilde H_R'$.\nTherefore the correlator (\\ref{K}) is no longer equal to $K_{\\Theta}$, but\nrather $K(t)=K_{\\Theta}(t)K_{F}(t)$, where\n\\begin{equation}\nK_F(t) = \\langle F(t) F^\\dagger(0)\\rangle\n  \\label{K_F}.\n\\end{equation}\nHere the averaging $\\langle\\ldots\\rangle$ is over the equilibrium thermal\ndistribution of the transformed Hamiltonian, $H=H_0+\\tilde H_C + \\tilde\nH_R'$.\n\nIn the absence of the backscattering in the right contact, $K_{\\Theta}$ is\ngiven by Eq.~(\\ref{Debye_Waller}), and $K_F(t)\\equiv 1$. When a weak\nbackscattering is added, both correlators are affected. It is clear,\nhowever, that $K_\\Theta$ is not modified significantly. Indeed, it follows\nfrom the above mentioned analogy with the orthogonality catastrophe that\nthe time dependence (\\ref{Debye_Waller}) is determined solely by the\ncharge transferred through the right junction, which in our case is always\n$e$---the charge of the electron brought into the dot through the left\nbarrier---and is not affected by the presence of the right barrier.\nTherefore the only change in $K_\\Theta(t)$ caused by the weak barrier can\nbe a small modification of the prefactor in Eq.~(\\ref{Debye_Waller}). On\nthe other hand, we will show that $K_F(t)$ is modified significantly,\nwhich leads to the quadratic temperature dependence of the conductance.\n\nTo find $K_F(t)$ we first note that the Hamiltonian $H=H_0+\\tilde H_C +\n\\tilde H_R'$ is identical to the one considered in Sec.~\\ref{spinless} and\nin Ref.~\\onlinecite{Matveev1}. The difference with the problem considered\nin Sec.~\\ref{spinless} is that instead of two modes corresponding to two\ndifferent contacts we now have two spin modes in the same (right) contact.\nThus the two scattering amplitudes are equal and given by $|r_R|$.\nAs we have seen in Sec.~\\ref{spinless}, the presence of the backscattering\nshows up at low-energy scales $\\epsilon\\sim |r_R|^2 E_C\\ll E_C$, where one\ncan effectively integrate out the mode related to the charge fluctuations\nin the dot, $\\phi_{R\\uparrow} + \\phi_{R\\downarrow}$. The resulting\nHamiltonian is expressed in terms of the spin field\n$\\phi_{Rs}=(\\phi_{R\\uparrow} - \\phi_{R\\downarrow})\/\\sqrt{2}$ and has the\nform:\n\\begin{eqnarray}\nH &=& \\frac{\\hbar v_F}{2}\\!\\int\\!\n         \\left\\{\\frac{1}{\\pi}[\\nabla \\phi_{Rs}(x)]^2\n                                         + \\pi \\Pi_{Rs}^2(x)\n        \\right\\}dx\n\\nonumber\\\\\n  & & + (-1)^{\\hat n}\n        \\sqrt{\\frac{8\\gamma E_C D}{\\pi^3}}\\,|r_R|\\cos\\pi N\n        \\cos\\!\\left[\\sqrt{2}\\phi_{Rs}(0)\\right]\\!.\n  \\label{spin_Hamiltonian}\n\\end{eqnarray}\nWe can now fermionize the Hamiltonian using a representation of the\noperators $e^{\\pm i\\sqrt{2}\\phi_{Rs}}$ in terms of the fermion creation\nand annihilation operators in a way identical to the one leading from the\nHamiltonian (\\ref{H_0_simplified}) and (\\ref{H'_simplified}) to\nEq.~(\\ref{H_fermionized}). The final Hamiltonian is again quadratic in\nfermion operators, but also depends on the operator $\\hat n$ of the number\nof particles transferred through the left barrier,\n\\begin{eqnarray}\nH &=& i\\hbar v_F\\int\\psi^{\\dagger}(x)\\nabla\\psi(x)dx\n\\nonumber\\\\\n  & & + (-1)^{\\hat n}\n      \\frac{2}{\\pi}\\sqrt{\\gamma \\hbar v_F E_C}\\,\n      |r_R|\\cos\\pi N\n\\nonumber\\\\\n  & &\\hspace{1em}\\times\n     (c+c^\\dagger)[\\psi(0) - \\psi^{\\dagger}(0)].\n  \\label{H_fermionized_asymmetric}\n\\end{eqnarray}\nHere we presented a Majorana fermion operator similar to $\\eta_+$ in\nEq.~(\\ref{H_fermionized}) as a sum of fermion creation and annihilation\noperators, $\\eta=c+c^\\dagger$. This representation is more convenient,\nbecause we can now replace the operators $F$ and $F^\\dagger$ in the\ndefinition (\\ref{K_F}) of the correlator $K_F$ by\n$F_s=F_s^\\dagger=1-2c^\\dagger c$. Indeed, these new operators commute with\nthe first term in the Hamiltonian (\\ref{H_fermionized_asymmetric}),\nanti-commute with the second one, and possess the property\n$F_s^{}F_s^\\dagger = FF^\\dagger=1$. Therefore one can find the correlator\n$K_F$ as\\cite{remark2}\n\\begin{equation}\nK_F(t) = \\langle[1-2c^\\dagger(t)c(t)]\n              [1-2c^\\dagger(0)c(0)]\\rangle.\n  \\label{K_F_fermionized}\n\\end{equation}\n\nThe actual calculation of the correlator (\\ref{K_F_fermionized}) is now\nstraightforward, as the Hamiltonian (\\ref{H_fermionized_asymmetric}) is\nquadratic in the fermion operators. The result for the correlator $K_F$\nhas the form\n\\begin{equation}\nK_F(t) = \\frac{2\\Gamma_R}{\\pi}\\int_{-\\infty}^{\\infty}\n            \\frac{e^{iEt}dE}\n                 {(E^2+\\Gamma_R^2)(e^{E\/T}+1)}.\n  \\label{K_F_final}\n\\end{equation}\nHere we have introduced the low energy scale\n\\begin{equation}\n\\Gamma_R = \\frac{8\\gamma}{\\pi^2}E_C |r_R|^2 \\cos^2 \\pi N,\n  \\label{Gamma_R}\n\\end{equation}\noriginating from the presence of a weak scatterer in the right contact. In\nthe weak-scattering limit, $\\Gamma_R\\to0$, Eq.~(\\ref{K_F_final})\nreproduces the result $K_F(t)=1$ mentioned above. However, in\nthe opposite limit $T,\\hbar\/t\\ll\\Gamma_R$, the correlator $K_F$ has the\nsame non-trivial time dependence as the correlator $K_{\\Theta}(t)$\ngiven by Eq.~(\\ref{Debye_Waller}),\n\\[\nK_F(t) \\simeq \\frac{2T}{i\\Gamma_R}\n                 \\frac{1}{\\sinh[\\pi T (t -i\\delta)\/\\hbar]}.\n\\]\nThis additional time-dependent contribution to $K(t)$ leads to the\nquadratic temperature dependence of the linear conductance at\n$T\\ll\\Gamma_R$.\n\nTo find the conductance at temperatures $T\\sim\\Gamma_R$, we can now\nsubstitute the correlator $K(t)=K_\\Theta(t)K_F(t)$ into\nEq.~(\\ref{conductance_asymmetric}), which leads to the following\nexpression:\n\\begin{equation}\nG = \\frac{TG_L}{8\\gamma E_C}\n    \\int^\\infty_{-\\infty}\n    \\frac{\\Gamma_R}{E^2+\\Gamma^2_R}\n    \\frac{\\pi^2+(E\/T)^2}{\\cosh^2(E\/2T)}dE.\n  \\label{conductance_asymmetric_final}\n\\end{equation}\nEquation (\\ref{conductance_asymmetric_final}) is the central result of\nthis section. It is valid for any $T\/\\Gamma_R$, if both temperature and\n$\\Gamma_R$ are much smaller than the charging energy $E_C$. The condition\n$\\Gamma_R\\ll E_C$ means $|r_R|\\ll1$, i.e., the scattering in the right\ncontact must indeed be weak.\n\n\n\n\\begin{figure}\n\\narrowtext\n\\epsfxsize=2.8in\n\\hspace{.4em}\n\\epsffile{4chan3.ps}\n\\vspace{1.5\\baselineskip}\n\\caption{Conductance (\\protect\\ref{conductance_asymmetric_final})\n  as a function of the dimensionless gate voltage $N$ for the asymmetric\n  case, $|r_R|=0.4$.  The three curves are calculated for\n  $E_C\/T=10$ (a), $40$ (b), and $160$ (c).}\n\\label{fig:3}\n\\end{figure}\n\\vspace{\\baselineskip}\n\nIn the limit $T\\gg\\Gamma_R$, Eq.~(\\ref{conductance_asymmetric_final})\nreproduces the previously obtained linear temperature dependence\n(\\ref{peak_conductance_asymmetric}) in the absence of the scattering in\nthe right contact. It is important to note, however, that the linear\ntemperature dependence is obtained not only at $r_R=0$, but also in the\npresence of the barrier if $N$ is half-integer, i.e., at the centers of\nconductance peaks.\n\nAway from the centers of the peaks, the conductance has a stronger\ntemperature dependence, $G\\propto T^2$. Indeed, at $T\\ll\\Gamma_R$ the\nexpression (\\ref{conductance_asymmetric_final}) reduces to\n\\begin{equation}\nG = \\frac{2\\pi^2}{3\\gamma}\\frac{T^2}{E_C\\Gamma_R} G_L.\n  \\label{off-peak_conductance}\n\\end{equation}\nThus the quadratic temperature dependence of the linear conductance is\nrestored for the off-peak values of the gate voltage. The evolution of the\nconductance peaks is illustrated in Fig.~\\ref{fig:3}.\n\n\n\n\\section{Scaling approach to inelastic co-tunneling}\n\\label{scaling}\n\nIn the sections \\ref{weak}--\\ref{withspin} we have considered several\ncases when the inelastic co-tunneling can be studied analytically even\nthough one or both contacts are in the strong-tunneling regime. A common\nproperty of all these results is that at $T\\to0$ the conductance behaves\nas $T^2$ away from the resonance values of the gate voltage, i.e., at\n$N\\neq\\pm\\frac12, \\pm\\frac32,\\ldots$. The same temperature dependence was\nobtained earlier\\cite{AN} for the off-resonance conductance in the\nweak-tunneling case, see Eq.~(\\ref{cotunneling}). In this section we study\nthe general scaling properties of our system and argue that the quadratic\ntemperature dependence of linear conductance is universal and should hold\nfor arbitrary barriers.\n\nWe will describe our system by the bosonized Hamiltonian\n(\\ref{H_0_bosonized})--(\\ref{backscattering_bosonized}) in the spinless\ncase and (\\ref{H_0_with_spins})--(\\ref{backscattering_with_spins}) in the\npresence of spins. In the absence of the backscattering the Hamiltonian is\nquadratic, i.e., $|r_L|=|r_R|=0$ is a fixed point of our Hamiltonian. In\nanalogy with the Kondo problem we will call it the strong-coupling fixed\npoint. To investigate the stability of this fixed point, one should find\nthe scaling dimension of the backscattering term\n(\\ref{backscattering_bosonized}) or (\\ref{backscattering_with_spins}).\n\nWe shall start with the spinless case, where at $T=0$ and $t\\gg\\hbar\/E_C$\na simple calculation gives\n\\begin{eqnarray*}\n\\langle H'\\!(t)H'\\!(0)\\rangle &=& \\frac{\\hbar\\Gamma_0(N)}{2\\pi i t}\n\\\\\n   &&+ \\frac{\\gamma\\hbar^3(|r_L|^2 + |r_R|^2 - 2|r_L||r_R|\\cos2\\pi N)}\n          {4\\pi i E_Ct^3},\n\\end{eqnarray*}\nwith $\\Gamma_0(N)$ defined by Eq.~(\\ref{Gamma_0}). The long-time\nasymptotics of the correlator $\\langle H'(t)H'(0)\\rangle$ is given by the\nfirst term, which is proportional to $t^{-1}$, unless $\\Gamma_0=0$. Thus\nthe dimension of the perturbation $H'$ is $\\frac12<1$, and the\nstrong-coupling fixed point is unstable. At a half-integer $N$ and in a\nperfectly symmetric case, $|r_L|=|r_R|$, the parameter $\\Gamma_0$\nvanishes, and the dimension of the operator $H'$ becomes $\\frac32>1$. In\nthis case the strong-coupling fixed point is stable. In the language of\nthe corresponding 2-channel Kondo problem this means that the\nstrong-coupling fixed point is stable only in the absence of magnetic\nfield and channel anisotropy. The leading irrelevant operator with\ndimension $\\frac32$ gives rise to the weak temperature dependence of the\npeak heights in the symmetric spinless case, see Appendix~\\ref{A1}.\n\nA similar analysis can be performed in the case of electrons with spins,\ndescribed by the Hamiltonian\n(\\ref{H_0_with_spins})--(\\ref{backscattering_with_spins}). The correlator\nof the perturbation operators now has the form\n\\[\n\\langle H'(t)H'(0)\\rangle =\n    \\bigl(|r_L|^2+|r_R|^2\\bigr)\n    \\sqrt{\\frac{4\\gamma E_C}{\\pi^5}}\\,\n    \\left(\\frac{\\hbar}{it}\\right)^{3\/2},\n\\]\nmeaning that $H'$ has dimension $\\frac34<1$ at any $N$ and any barrier\nstrengths. Thus as expected the strong-coupling fixed point in the\n4-channel case is unstable. At a half-integer $N$ this statement has a\nwell-known analog in the theory\\cite{NB} of the multichannel Kondo\nproblem, where the strong-coupling fixed point is unstable even in\nthe absence of the magnetic field.\n\nThe fact that the strong-coupling fixed point is unstable means that as\nthe temperature is lowered, the weak backscattering in the contacts\nbecomes stronger. It is natural to assume that the effective barriers grow\nindefinitely, and the system approaches the weak-coupling fixed point. To\ntest this hypothesis we will consider a system where the two contacts are\nin the weak-tunneling regime and study the stability of the corresponding\nfixed point.\n\nAs we saw in Sec.~\\ref{weak} and \\ref{withspin}, in the weak-tunneling\ncase it is convenient to describe the system in terms of the original\nfermionic variables. Generalizing the Hamiltonian of Sec.~\\ref{asymmetric}\nto the case of both barriers being in the weak-tunneling regime, we\nintroduce the Hamiltonian $H=H_0 + H_C + H_L'+H_R'$, where\n\\begin{eqnarray}\nH_0 &=& \\sum_{k\\alpha} \\epsilon_k a_{k\\alpha}^\\dagger a_{k\\alpha}^{}\n        + \\sum_{p\\alpha} \\epsilon_p a_{p\\alpha}^\\dagger a_{p\\alpha}^{}\n\\nonumber\\\\\n    & & + \\sum_{q\\alpha} \\epsilon_q a_{q\\alpha}^\\dagger a_{q\\alpha}^{}\n        + \\sum_{s\\alpha} \\epsilon_s a_{s\\alpha}^\\dagger a_{s\\alpha}^{} ,\n\\label{H_0_both_weak}\\\\\nH_C &=& E_C(\\hat n_L + \\hat n_R - N)^2,\n\\label{H_C_both_weak}\\\\\n  H_L' &=& \\sum_{kp\\alpha}\n           (v_L^{} a_{k\\alpha}^\\dagger a_{p\\alpha}^{}F_L^{}\n            + v_L^* a_{p\\alpha}^\\dagger a_{k\\alpha}^{}F_L^\\dagger),\n\\label{H_L'_both_weak}\\\\\n  H_R' &=& \\sum_{qs\\alpha}\n           (v_R^{} a_{s\\alpha}^\\dagger a_{q\\alpha}^{}F_R^{}\n            + v_R^* a_{q\\alpha}^\\dagger a_{s\\alpha}^{}F_R^\\dagger).\n\\label{H_R'_both_weak}\n\\end{eqnarray}\nHere $a_{k\\alpha}$ and $a_{s\\alpha}$ describe the electrons in the left\nand right leads, respectively; $a_{p\\alpha}$ and $a_{q\\alpha}$ correspond\nto the electrons in the dot coupled to the left and right contacts.\nSimilarly to the formalism of Sec.~\\ref{asymmetric}, we have introduced\nthe operators $\\hat n_L$ and $\\hat n_R$ of the number of electrons\ntunneled through the left and right contacts and the corresponding raising\nand lowering operators $F_L^{}$, $F_L^\\dagger$, $F_R^{}$, and\n$F_R^\\dagger$ in the tunnel Hamiltonians (\\ref{H_L'_both_weak}) and\n(\\ref{H_R'_both_weak}).\n\nAway from the resonance values of the gate voltage, $N\\neq \\pm\\frac12,\n\\pm\\frac32,\\ldots$, the states with different $n_L$ and $n_R$ have different\nenergies because of the Coulomb term (\\ref{H_C_both_weak}). Since\nthe tunnel Hamiltonian given by (\\ref{H_L'_both_weak}) and\n(\\ref{H_R'_both_weak}) changes $n_L$ or $n_R$, it does not describe the\ntransitions between the low-energy states of the Hamiltonian. Thus when\nconsidering the low-energy properties of the system, one should use the\ntunneling terms (\\ref{H_L'_both_weak}) and (\\ref{H_R'_both_weak}) to\nconstruct the perturbation which is not accompanied by the change of the\ntotal charge of the dot. In the simplest case, this can be done by\nselecting the terms containing equal number of operators $F_i$ and\n$F_i^\\dagger$ in the second-order perturbation\n\\begin{equation}\n H_2' = (H_L' + H_R')\\frac{1}{E_0 - H_0 - H_C}(H_L' + H_R').\n\\label{H_2'}\n\\end{equation}\nHere the denominator accounts for the energy of the virtual state created\nby adding (or removing) an electron to the dot. At very low energies the\nonly important contribution to the denominator is the change of the\nelectrostatic energy of the system.\n\nAt $N$ in the interval $\\bigl(-\\frac12, \\frac12\\bigr)$ a typical\nrepresentative of the variety of terms contained in Eq.~(\\ref{H_2'}) has\nthe form\n\\begin{eqnarray}\nA &=& -v_L^* v_R^{}\\left[\\frac{1}{E_C(1-2N)}+\\frac{1}{E_C(1+2N)}\\right]\n\\nonumber\\\\\n  & & \\times\n      F_L^\\dagger F_R^{}\n      \\sum_{kp\\alpha}  a_{p\\alpha}^\\dagger a_{k\\alpha}^{}\n      \\sum_{qs\\alpha'}  a_{s\\alpha'}^\\dagger a_{q\\alpha'}^{}.\n  \\label{A}\n\\end{eqnarray}\nThe terms included in Eq.~(\\ref{A}) correspond to the processes where one\nelectron tunnels into the dot through the left junction and another one\nescapes from the dot through the right junction. Depending on which of the\ntwo tunneling events is executed first, we get different energies of the\nvirtual states, corresponding to the two denominators in Eq.~(\\ref{A}).\n\nTo find out whether or not the weak-coupling fixed point is stable, one\nshould evaluate the scaling dimension of operator $A$ and other similar\ncontributions in Eq.~(\\ref{H_2'}). A straightforward calculation gives\n\\begin{eqnarray}\n\\langle A(t) A^\\dagger(0)\\rangle &=&\n   \\frac{\\hbar^2G_L G_R}{(2\\pi e^2)^2}\\!\n   \\left[\\frac{1}{E_C(1-2N)}+\\frac{1}{E_C(1+2N)}\\right]^2\n\\nonumber\\\\\n  & &\\times\n   \\left\\{\\frac{\\pi T}{\\sinh[\\pi T(t-i\\delta)\/\\hbar]}\\right\\}^4.\n  \\label{A_correlator}\n\\end{eqnarray}\nAt $T\\to0$ the correlator decays as $1\/t^4$, and therefore\nthe dimension of the operator $A$ is 2. One can also check that the other\ncontributions to Eq.~(\\ref{H_2'}) have the same dimension, whereas the\nhigher-order terms have even higher dimensions. Thus the weak-coupling\nfixed point is stable.\n\nSo far we have shown that away from the resonance values of the gate\nvoltage, $N\\neq \\pm\\frac12, \\pm\\frac32, \\ldots$, the strong-coupling fixed\npoint is unstable whereas the weak-coupling fixed point is stable. We now\nconjecture that there are no fixed points at intermediate coupling. Then\nat low energies the system always evolves towards the stable\n(weak-coupling) fixed point. One should also expect that near this fixed\npoint the temperature dependence of conductance is determined by the\ndimension of the leading irrelevant perturbation, which is a universal\ncharacteristic of the problem. Indeed, one can easily show that the\nleading contribution to the current operator $I=e\\dot n_L$ is given by\n\\begin{equation}\nI = -\\frac{ie}{\\hbar}(A-A^\\dagger).\n  \\label{current_in_terms_of_A}\n\\end{equation}\nWe now substitute this current operator into the Kubo formula\n(\\ref{Kubo}), and use (\\ref{A_correlator}) to find the following\nexpression for the conductance:\n\\begin{equation}\nG = \\frac{\\pi\\hbar G_LG_R}{3e^2}\n   \\left[\\frac{1}{E_C(1-2N)}+\\frac{1}{E_C(1+2N)}\\right]^2 T^2.\n  \\label{cotunneling_conductance}\n\\end{equation}\nThis expression for the conductance in the weak-tunneling limit coincides\nwith the well-known result for the inelastic co-tunneling,\\cite{AN} and\nreproduces Eq.~(\\ref{cotunneling}) near the resonances,\n\\mbox{$\\frac12\\pm N \\ll 1$}.\n\nIt is important to note that in the derivation of\nEq.~(\\ref{cotunneling_conductance}) we neglected all the contributions\nto the current operator with dimensions $d>2$. Such terms would lead to\nthe additional terms in the correlator (\\ref{A_correlator}), which are\nproportional to $\\{\\pi T\/\\sinh[\\pi T(t-i\\delta)\/\\hbar]\\}^m$ with $m>4$.\nClearly, the corresponding corrections to the conductance would behave as\n$\\delta G \\propto T^{m-2}\\ll T^2$. Thus at $T\\to 0$ the temperature\ndependence of the conductance is given by the leading irrelevant\nperturbation. As we have seen, the universal scaling dimension $d=2$ of\nthis perturbation translates into the universal temperature dependence of\nthe conductance $G\\propto T^2$. Therefore this temperature dependence is\nspecific not only for the cases of weak or strong tunneling, but should\nhold for any barrier strengths.\n\nThe above treatment of the stability of the weak-coupling fixed point was\nlimited to the off-resonance values of the gate voltage. We will now\ndemonstrate that the properties of the system are completely different at\nhalf-integer $N$. Consider, for example, the case of $N=\\frac12$. Then the\nstates with the dot charge equal to 0 and $e$ have the same energy. As a\nresult the terms in the tunnel Hamiltonians (\\ref{H_L'_both_weak}) and\n(\\ref{H_R'_both_weak}) responsible for the transitions between these two\nstates no longer lead to the large increase in the energy of the system\nand should not be discounted. It is easy to check that the dimension of\nsuch operators is $d=1$, i.e., the tunnel Hamiltonian is a marginal\noperator.  This conclusion is actually obvious, because the same\ncalculation of scaling dimension must be valid in the absence of the\ncharging effects, $E_C=0$, where we do not expect the tunneling amplitudes\nto be renormalized as the temperature is lowered.  However, the simple\ncalculation of the scaling dimension is not sufficient to determine the\nstability of the weak-coupling fixed point. To make such a determination,\none has to study the scaling equations for the tunneling amplitudes\nincluding the higher-order terms in $v_L$ and $v_R$.  We have performed\nthis analysis in Sec.~\\ref{weak}, where we saw from the mapping to the\nmultichannel Kondo problem that the tunneling amplitude is in fact a\nmarginally relevant perturbation, i.e., it grows logarithmically at low\ntemperatures.  Therefore on resonance the weak-coupling fixed point is\nunstable.\n\nThe on-resonance scaling properties of our system are strongly affected by\nthe presence of electron spins and by the symmetry of the two barriers.\nLet us first consider the case of symmetric barriers.  In the spinless\ncase we saw that on resonance the strong-coupling fixed point is stable if\nthe barriers are symmetric. Thus one should expect the system to always\nrenormalize towards this fixed point. Then the conductance must be on the\norder of $e^2\/\\hbar$. This is confirmed by the explicit calculation in the\nstrong-tunneling case, Eq.~(\\ref{conductance_spinless}).\n\nIn the presence of spins, both weak- and strong-coupling fixed points are\nunstable on resonance. Thus one should expect that there must be a stable\nfixed point at some intermediate coupling strength. This hypothesis is\nalso confirmed by the analogy with the 4-channel Kondo problem discussed\nin Sec.~\\ref{weak}. Indeed, it is well-known that the low-energy\nproperties of the Kondo problem with the number of channels exceeding 2 is\ngoverned by an intermediate-coupling fixed point, which gives rise to a\nspecific non-Fermi-liquid behavior of the susceptibility and specific\nheat.\\cite{NB,bethe,Affleck,Fabrizio} In Appendix~\\ref{4-channel} we show\nhow one can study the properties of the intermediate-coupling fixed point\nstarting from the vicinity of the strong-coupling point. We discover that\non resonance the thermodynamics of our problem is indeed identical to that\nof the 4-channel Kondo problem.\n\nThe technique presented in Appendix~\\ref{4-channel}, however, does not\nenable us to find the low-temperature conductance $G$. Nevertheless one\ncan estimate the order of magnitude of $G$ at the intermediate-coupling\nfixed point in the following way. From the fact that the strong-coupling\nfixed point is unstable and from the result (\\ref{2nd_order_with_spins})\nof the perturbation theory near this point we know that the conductance\nmust be smaller than $e^2\/2\\pi\\hbar$. On the other hand, at $G\\ll\ne^2\/\\hbar$ the weak-coupling treatment of Sec.~\\ref{weak} is applicable\nand predicts the growth of conductance at $T\\to0$. Therefore the limiting\nvalue of conductance must be of the order, but smaller than\n$e^2\/2\\pi\\hbar$. We further conjecture that the peak conductance is a\nuniversal characteristic of the intermediate-coupling fixed point, see\nAppendix~\\ref{4-channel}, meaning that in the limit $T\\to0$ the peak\nconductance does not depend on the heights of the barriers.\n\nThe scaling analysis in the spirit of Ref.~\\onlinecite{Kane} enables us to\nmake some conclusions about the shape of the peaks in $G(N)$. In the\nspinless case we see from Eq.~(\\ref{conductance_spinless}) that at the\ntails of the peak at $N=\\frac12$ the conductance decays as $G\\propto\n1\/\\bigl(N-\\frac12\\bigr)^4$.  This result can be interpreted within the\nscaling picture as follows. A small deviation of $N$ from $\\frac12$ is\nidentified as a magnetic field $h$ in the appropriate multichannel Kondo\nproblem, see Eq.~(\\ref{Kondo}).  Therefore such a deviation is a relevant\nperturbation with scaling dimension $d=2\/(2+k)$, where $k$ is the number\nof channels.\\cite{Affleck} In the spinless case we have $k=2$ and\n$d=\\frac12$. This means that as the temperature is lowered, the magnetic\nfield grows as $h(T)\\propto \\bigl(N-\\frac12\\bigr)T^{-1\/2}$. We now\nconjecture that near the fixed point, at $T\\to0$, the conductance is a\nuniversal function of $h$ only, $G=G(h(T))$.  As we discussed above, away\nfrom the center of the peak $G\\propto T^2$ at $T\\to0$. Therefore the\nmagnetic field dependence of the conductance must be given by $G\\propto\n1\/h^4\\propto T^2\/\\bigl(N-\\frac12\\bigr)^4$, in agreement with\nEq.~(\\ref{conductance_spinless}). We can now apply the same argument to\nthe case of electrons with spins, where the number of channels $k=4$ and\nthe scaling dimension of the magnetic field is $d=\\frac13$. The\ntemperature dependence of the renormalized magnetic field is consequently\n$h(T) \\propto \\bigl(N-\\frac12\\bigr)T^{-2\/3}$.  Therefore near the\nintermediate-coupling fixed point the condition $G\\propto T^2$ at $T\\to 0$\ndemands $G\\propto h^{-3}\\propto T^2\/\\bigl(N-\\frac12\\bigr)^3$. Thus we\nexpect the tails of the conductance peaks to decay as\n$1\/\\bigl(N-\\frac12\\bigr)^3$.\n\n\nSo far we discussed the case of symmetric barriers. The on-resonance\nbehavior of the system is very different if the two barriers are not\nidentical. This can be readily seem from the analogy to the Kondo problem.\n{}From the theory of the multichannel Kondo model\\cite{NB,Melnikov} it is\nknown that the asymmetry between the channels is a relevant perturbation.\nIn terms of the Coulomb blockade problem this means that on resonance a\nsmall asymmetry will grow and eventually the smaller transmission\ncoefficient, say ${\\cal T}_L$, will scale to zero, whereas the larger one,\n${\\cal T}_R$, must approach the fixed point of the problem with half the\nnumber of channels of the original one. Thus for both 2- and 4-channel\ncases ${\\cal T}_R \\to 1$.  For the most realistic case of electrons with\nspins, near this asymmetric fixed point the system is identical to the one\nconsidered in Sec.~\\ref{asymmetric}. Thus we conclude that for asymmetric\nsystems (i) the on-resonance conductance vanishes linearly in $T$, and (ii)\nthe shape of the resonances in $G(N)$ is universal and given by\nEq.~(\\ref{conductance_asymmetric_final}).\n\nThe on-resonance scaling properties of our problem are summarized in\nFig.~\\ref{fig:4}.\n\n\\vfill\n\\begin{figure}\n\\narrowtext\n\\epsfxsize=2.6in\n\\hspace{.1em}\n\\epsffile{4chan4.ps}\n\\vspace{1.5\\baselineskip}\n\\caption{Scaling diagram of the two-barrier system in the cases of (a)\n  spinless electrons and (b) electrons with spins. The gate voltage\n  corresponds to a resonance value, $N=\\pm\\frac12,\\pm\\frac32,\\ldots$. In\n  the symmetric case, ${\\cal T}_L={\\cal T}_R$, the system scales to either\n  strong- or intermediate-coupling fixed point, depending on the number of\n  channels. In an asymmetric case the system approaches the limit where\n  the weaker barrier disappears, while the stronger one becomes very\n  high.}\n\\label{fig:4}\n\\end{figure}\n\n\n\\section{Conclusion}\n\nIn this paper we have studied the transport through a quantum dot\nconnected to two leads by quantum point contacts, Fig.~\\ref{fig:1}. Unlike\nin conventional theories of Coulomb blockade,\\cite{Averin,Grabert} we\nconcentrate on the regime of strong tunneling, where one or both contacts\nare close to perfect transmission.  We find the linear conductance in a\nwide interval of temperatures, limited by the level spacing in the dot\nfrom below and by the charging energy $E_C$ from above.\n\nWe have shown that as long as there is any backscattering in the contacts,\ni.e., both transmission coefficients ${\\cal T}_{L,R}<1$, the Coulomb\nblockade peaks in the conductance as a function of the gate voltage should\nbe observed. Between the Coulomb blockade peaks the temperature dependence\nof the conductance at $T\\to0$ is determined by the weak-coupling fixed\npoint. As a result we get the $T^2$ dependence of the conductance, which\ncoincides with the known result\\cite{AN} for the inelastic co-tunneling\nmechanism.\n\nOur results show that the behavior of the conductance near the peaks\ndepends qualitatively on the presence of electron spins. In the spinless\ncase the strong-tunneling fixed point is stable if the barriers are\nsymmetric, and thus the technique developed in this paper allows a\ncomplete solution of the problem. The resulting peaks in conductance have\nthe height $e^2\/4\\pi\\hbar$, with the shape given by\nEq.~(\\ref{conductance_spinless}). In the presence of the spins, the\nstrong-tunneling fixed point is no longer stable, and our technique does\nnot allow a complete solution of the problem. Using a useful analogy with\nthe multichannel Kondo problem, we have concluded that the peak\nconductance must still be of the order of $e^2\/\\hbar$. In addition, we\nstudied the peaks in conductance in the case of asymmetric\nbarriers. The peak conductance in this case is linear in temperature at\n$T\\to0$, and its shape is given by\nEq.~(\\ref{conductance_asymmetric_final}).\n\nThe strong-tunneling regime of Coulomb blockade was explored to some\nextent in several recent experiments.\\cite{van,Pasquier,Waugh,Molenkamp}\nOur results are in qualitative agreement with the experiment.\\cite{van} We\nbelieve that our predictions can be easily tested in similar experiments.\n\n\n\n\n\n\\acknowledgements\n\nWe are grateful to L.~I.~Glazman and P.~A.~Lee for helpful discussions.\nThe work was sponsored by Joint Services Electronics Program Contract\nDAAH04-95-1-0038.\n\n\\end{multicols}\n\\widetext\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nLet $G \\subset GL_n(\\mathbb{C})$ be a finite complex reflection group and denote by ${\\mathcal A}(G)$ the union of all the reflecting hyperplerplanes of $G$, i.e. the hyperplanes in $\\mathbb{C}^n$ fixed by some $g \\in G$, $g \\ne Id$. For a general reference on complex reflection groups, see \\cite{LT} and \\cite{OT}.\nSince ${\\mathcal A}(G)$ is a central hyperplane arrangement, it has a defining equation $f=0$ in $\\mathbb{C}^n$, where $f$ \nis a homogeneous polynomial of some degree $d$. One can associate to this setting the Milnor fiber $F(G)$ of the arrangement  ${\\mathcal A}(G)$. This is a smooth hypersurface in $\\mathbb{C}^n$, defined by $f=1$, and it is  endowed with a monodromy morphism $h: F(G) \\to F(G)$, given by $h(x_1,...,x_n)=\\exp(2\\pi i\/d)\\cdot (x_1,...,x_n)$, see \\cite{BDY, BaSe, BDS, CS, DHA, DL1, DL2, L, MP, Se1, Se2, S2} for related results and to get a feeling of the problems in this very active area.\nThe study of the induced monodromy operator\n$$h^1(G): H^1(F(G),\\mathbb{C}) \\to H^1(F(G),\\mathbb{C})$$\nis the object of the papers \\cite{MPP} and \\cite{Dreflection}, while in the special case of real reflection groups $G$, there are some additional results on the higher degree monodromy operators  \n$$h^j(G): H^j(F(G),\\mathbb{C}) \\to H^j(F(G),\\mathbb{C})$$\nwhere $j>1$, see  \\cite{Se1, Se2, DL2}. In general, not only these monodromy operators are not known, but even the Betti numbers $b_j(F(G))$ are known only in a limited number of cases.\n\nThe complex reflection groups have been classified by Shephard and Todd, see \\cite{ST}, who showed that there is an infinite series $G(m,p,n)$ of such groups, plus 34 exceptional cases.\nThe exceptional complex reflection groups in this classification are usually denoted by $G_j$, with $4 \\leq j \\leq 37$.   \nConsider the following polynomial of degree 60 in $S=\\mathbb{C}[x,y,z,t]$ \n\\begin{equation}\n\\label{e6}\nf=xyzt(x^4-y^4)(x^4-z^4)(x^4-t^4)(y^4-z^4)(y^4-t^4)(z^4-t^4)\n\\end{equation}\n$$((x-y)^2-(z+t)^2) ((x-y)^2-(z-t)^2) ((x+y)^2-(z+t)^2)((x+y)^2-(z-t)^2)$$\n$$((x-y)^2+(z+t)^2)((x-y)^2+(z-t)^2)((x+y)^2+(z+t)^2)((x+y)^2+(z-t)^2)$$\n$$((x-z)^2+(y+t)^2) ((x-z)^2+(y-t)^2)((x+z)^2+(y+t)^2)((x+z)^2+(y-t)^2)$$\n$$((x-t)^2+(y+z)^2)((x-t)^2+(y-z)^2)((x+t)^2+(y+z)^2)((x+t)^2+(y-z)^2).$$\nThen the reflection hyperplane arrangement ${\\mathcal A}(G_{31})$ consists of 60 hyperplanes in $\\mathbb{C}^4$ and is given by $f=0$, see \\cite{HR}.\nIn this note we prove the following result.\n\n\\begin{thm}\n\\label{thmmain}\nThe  monodromy operator\n$$h^1:H^1(F(G_{31}),\\mathbb{C}) \\to H^1(F(G_{31}),\\mathbb{C})$$\nfor the exceptional complex reflection group $G_{31}$ is the identity. In particular, the first Betti number $b_1(F(G_{31}))$ is $59$.\n\\end{thm}\nThe description of the monodromy operator\n$h^1$ for all the other complex reflection groups is given in \\cite{Dreflection}, using a  method which cannot be applied to the exceptional group $G=G_{31}$ as explained in \n\\cite[Remark 6.2]{Dreflection}. The proof for the case $G=G_{31}$ involves a completely different approach, and this explains why it is written down here as a separate note. Indeed, this proof is close in spirit to our paper \\cite{DStCompM}, with computer aided computations playing a key role at several stages. Moreover, the rank of the group $G_{31}$ being $n=4$, we have to deal with higher dimensional singularities and hence with a more complicated spectral sequence than in \\cite{DStCompM}, where the case of plane curves corresponding to $n=3$ is discussed, see also Remark \\ref{rkdim} (i).\n\nIn the second section we recall a spectral sequence approach for the computation of the monodromy of the Milnor fibers of homogeneous polynomials with arbitrary singularities introduced in \\cite{Dcomp}, and developed in \\cite[Chapter 6]{D1}, \\cite{DStCompM}, with several key additions in the joint work of Morihiko Saito and the first author, see \\cite{DS1}.\nFor simplicity, we describe the results only in the case $n=4$, the only case needed in the sequel. However, the approach presented here is very general, and can be applied at least to all free hyperplane arrangements to get valuable information on their first Milnor cohomology.\nOn the other hand, the success of our method is based on the special properties enjoyed by the mixed Hodge structure on the first cohomology group $H^1(F(G),\\mathbb{Q})$, see the proof of Theorem \\ref{thmEV}, and hence this simple approach does not give complete results on\nthe higher cohomology groups $H^j(F(G),\\mathbb{Q})$, where $j>1$.\n\n\nIn the third section we recall basic facts on the exceptional complex reflection group $G_{31}$, in particular the construction of the basic invariants going back to Maschke \\cite{Mas} and the construction of a basis of Jacobian syzygies as described by Orlik and Terao in \\cite[Appendix B, pp. 280-281 and p. 285]{OT}.\n\nIn the fourth section we describe the algorithm used to determine the monodromy operator\n$h^1:H^1(F(G_{31}),\\mathbb{C}) \\to H^1(F(G_{31}),\\mathbb{C})$.  For this we need a careful study of the second cohomology group of the Koszul complex of the partial derivatives $f_x$, $f_y$, $f_z$ and $f_t$ of the polynomial $f$ of degree 60 given in \\eqref{e6}. At several points, this study is done by\nusing the software SINGULAR \\cite{Sing}. To get an idea why computations via SINGULAR are necessary, let us remark that each of the sequences of three homogeneous polynomials in $S$ which are shown to be regular sequences in Proposition \\ref{prop1} consists of polynomials of degrees\n$28, 12$ and $16$ respectively, and having $136,24$ and respectively $45$ monomials occurring with non zero (and, in fact, quite complicated rational) coefficients. And the final step in the proof is Lemma \\ref{lemfinal} where a homogeneous system of 19600 linear equations in 1424 indeterminates is shown to admit only the trivial solution.\nThe corresponding codes are available at \\url{http:\/\/math.unice.fr\/~dimca\/singular.html}.\n\n\\medskip\n\nWe would like to thank the referee for his useful suggestions.\n\n\\section{A spectral sequence for the Milnor monodromy} \nLet $S=\\mathbb{C}[x,y,z,t]$ and let $f \\in S$ be a homogeneous polynomial of degree $d$ without multiple factors. Consider the global Milnor fiber $F$ of the polynomial $f$ defined by $f(x,y,z,t)=1$ in $\\mathbb{C}^4$, with monodromy action $h:F \\to F$, \n$$h(x,y,z,t)=\\exp(2\\pi i\/d)\\cdot (x,y,z,t).$$\n\nLet $\\Omega^j$ denote the graded $S$-module of (polynomial) differential $j$-forms on $\\mathbb{C}^4$, for $0 \\leq j \\leq 4$. The complex $K^*_f=(\\Omega^*, d f \\wedge)$ is nothing else but the Koszul complex in $S$ of the partial derivatives $f_x$, $f_y$, $f_z$ and $f_t$ of the polynomial $f$. The general theory says that there is a spectral sequence $E_*(f)$, whose first term $E_1(f)$ is computable in terms of the homogeneous components of the cohomology of the Koszul complex $K^*_f$ and whose limit\n$E_{\\infty}(f)$ gives us the action of monodromy operator  on the graded pieces of $H^*(F,\\mathbb{C})$ with respect to a certain  pole order filtration $P$, see for details \\cite{Dcomp}, \\cite[Chapter 6]{D1}, \\cite{DS1}.\n\n More precisely,  for any integer $k \\in [1,d]$, there is a spectral sequence \n\\begin{equation} \n\\label{newspsq}\nE_1^{s,t}(f)_k=H^{s+t+1}(K^*_f)_{td+k},\n\\end{equation} \nconverging to\n\\begin{equation} \n\\label{limit}\nE_{\\infty}^{s,t}(f)_k=Gr_P^sH^{s+t}(F,\\mathbb{C})_{\\lambda },\n\\end{equation} \nwhere $\\lambda=\\exp (-2 \\pi ik\/d)$  and $H^{s+t}(F,\\mathbb{C})_{\\lambda }$ denotes the associated eigenspace of the monodromy operator.\nMoreover, the differential $d_1$ in this spectral sequence is induced by the exterior differentiation of forms, i.e. $d_1 :[\\omega] \\mapsto [d (\\omega)]$. We have the following.\n\n\\begin{thm}\n\\label{thmEV}\nLet $D:f=0$ be a reduced degree $d$ surface in $\\mathbb{P}^3$, and let $$\\lambda=\\exp (-2\\pi i k\/d)\\ne 1,$$  with $k \\in (0,d)$ an integer. \nThen $H^1(F,\\mathbb{C})_{\\lambda}=0$  if and only if\n$$E_2^{1,0}(f)_k =E_2^{1,0}(f)_{k'} = 0,$$\nwhere $k'=d-k$. \n\\end{thm}\n\\proof\nNote that on the cohomology group $H^1(F,\\mathbb{C})_{\\ne 1}$, which is by definition the direct sum of eigenspaces of $h^1$ corresponding to eigenvalues $\\ne 1$, one has $P^2=0$, $F^1 \\subset P^1$\nand $F^0 = P^0=H^1(F,\\mathbb{C})_{\\ne 1}$, where $F^1$ denotes the Hodge filtration, see \\cite{DS1}. Let now $\\lambda$ be an eigenvalue of the monodromy operator.\nThe fact that $H^1(F,\\mathbb{C})_{\\ne 1}$ is a pure Hodge structure of weight 1, see  \\cite{BDS}, \\cite{DP}, implies that either\n$H^{1,0}(F,\\mathbb{C})_{\\lambda } \\ne 0$ or $H^{0,1}(F,\\mathbb{C})_{\\lambda } \\ne 0$. In the first case we get\n$Gr_P^{1}H^{1}(F,\\mathbb{C})_{\\lambda }\\ne 0$, while in the second case we get $H^{1,0}(F,\\mathbb{C})_{\\overline \\lambda } \\ne 0$ and hence $Gr_P^{1}H^{1}(F,\\mathbb{C})_{\\overline \\lambda }\\ne 0$.\nThe result follows now from the fact that $H^1(K^*_f)=0$ since $f$ is a reduced polynomial.\nIndeed, it is known that the smallest $j$ with $H^j(K^*_f) \\ne 0$ is precisely the codimension of the singular locus $D_{sing}$ of $D$ in $\\mathbb{P}^3$, see \\cite[Theorems A.2.38 and A.2.48]{Eis}. This implies that $E_1^{s,t}(f)_k=0$ for $s+t=0$, and hence \n$$E_2^{1,0}(f)_k =E_{\\infty}^{1,0}(f)_k$$\nfor any $k \\in (0,d)$.\n\\endproof\nTo check the condition $E_2^{1,0}(f)_k =0$ in practice, we proceed as follows. Consider a 2-form\n \\begin{equation}\n\\label{e9}\n\\omega= \\sum_{1\\leq i<j \\leq 4}a_{ij}dx_i \\wedge dx_j \\in \\Omega^2,\n\\end{equation}\nwhere $a_{ij} \\in S_{k-2}$ and we use the convention $x=x_1$, $y=x_2$, $z=x_3$, $t=x_4$. \n\nWe have  the following two obvious results.\n \\begin{lem}\n\\label{dfomega=0}\nA differential 2-form $\\omega \\in \\Omega^2$ given by \\eqref{e9} satisfies $df \\wedge \\omega=0$ if and only if the following 4 equations in $S$ hold.\n$$\n(R_1) \\ \\ a_{34}f_y-a_{24}f_z+a_{23}f_t=0\n$$\n$$\n(R_2) \\ \\ a_{34}f_x-a_{14}f_z+a_{13}f_t=0\n$$\n$$\n(R_3) \\ \\ a_{24}f_x-a_{14}f_y+a_{12}f_t=0\n$$\n$$\n(R_4) \\ \\ a_{23}f_x-a_{13}f_y+a_{12}f_z=0\n$$\n\\end{lem}\n\n \\begin{lem}\n\\label{lemd=0}\nA differential 2-form $\\omega \\in \\Omega^2$ given by \\eqref{e9} satisfies $d \\omega=0$ if and only if the following 4 equations in $S$ hold.\n$$ (E1) \\  \\ a_{{23}_t}-a_{{24}_z}+a_{{34}_y}=0.$$\n$$ (E2) \\  \\ a_{{13}_t}-a_{{14}_z}+a_{{34}_x}=0.$$\n$$ (E3) \\  \\ a_{{12}_t}-a_{{14}_y}+a_{{24}_x}=0.$$\n$$ (E4) \\  \\ a_{{12}_z}-a_{{13}_y}+a_{{23}_x}=0.$$\n\\end{lem}\n\n\\section{On the exceptional complex reflection group $G_{31}$} \nThe exceptional complex reflection group $G_{31}$ has order 46080, and rank 4, hence it acts naturally on $\\mathbb{C}^4$. Its basic invariants $f_1$, $f_2$, $f_3$ and $f_4$, of degree $8,12,20$ and $24$ respectively, can be constructed as follows, see \\cite{Mas}.\nLet \n\\begin{equation}\n\\label{e1}\na=x^4+y^4+z^4+t^4, \u00a0\\ b=x^2y^2+z^2t^2, \\ c= x^2z^2+y^2t^2, \\ d=x^2t^2+y^2z^2, \\ e=xyzt. \n\\end{equation}\nWe define\n\\begin{equation}\n\\label{e2}\nA_1=a+6(-b-c-d), \\ A_2= a+6(-b+c+d),  \\  A_3= a+6(b-c+d), \n\\end{equation}\n$$ \\ A_4=a+6(b+c-d), \\ A_5=-2a-24e, \\ A_6=-2a+24e.$$\nLet $s_i$ be the $i$-th symmetric function in the variables $A_i$, such that\n$$\\prod_{i=1,6} (u+A_i)=\\sum _{j=0,6}s_ju^{6-j},$$\nin the polynomial ring $S[u]$, with $s_0=1$. Define the following homogeneous polynomials in $S$\n\\begin{equation}\n\\label{e3}\nF_8=-\\frac{1}{6}s_2, \\ F_{12}=-\\frac{1}{4}s_3, \\ F_{20}=\\frac{1}{12}s_5,\n\\end{equation}\nand note that $s_4=9F_8^2$. Define new polynomials as in \\cite{OT}, p. 285, by the following\n\\begin{equation}\n\\label{e4}\nf_1=F_8, \\ f_2=F_{12}, \\ f_3=  F_{20}, \\ f_4=\\frac{1}{265531392}Hess(f_1),\n\\end{equation}\nwhere $Hess(f_1)$ is the determinant of the Hessian matrix $H(f_1)$ of $f_1$.\n\n Moreover, as any reflection hyperplane arrangement, ${\\mathcal A}(G_{31})$ is a free arrangement, see \\cite[Theorem 6.60]{OT}, \\cite{Te}. This means that the graded $S$-module of Jacobian syzygies for $f$, namely\n$$AR(f)=\\{ r=(r_1,r_2,r_3,r_4) \\in S^4 \\ : \\ r_1f_x+r_2f_y+r_3f_z+r_4f_t=0 \\},$$\nwhich is the same as the $S$-module of derivations killing $f$, is free of rank $3$. A basis for this module can be computed starting from the basic invariants $f_1,f_2, f_3,f_4$ as follows, see \\cite[Theorem 6.53 and Appendix B, pp. 280-281 and p. 285]{OT}. Let $A(f_1)$ be the adjoint of the Hessian matrix  $H(f_1)$, namely a matrix such that\n$$A(f_1)H(f_1)=H(f_1)A(f_1)=Hess(f_1)I_4,$$\nwith $I_4$ the unit matrix of order 4. This matrix is, up to the factor $Hess(f_1)$, the inverse matrix of $H(f_1)$.\nLet $J(f)$ be the $4 \\times 4$ matrix having on the $j$-th column the first order partial derivatives of $f_j$ with respect to $x,y,z$ and $t$. Define a new $4 \\times 4$ matrix $C$, as described in  \\cite[Appendix B, p. 284]{OT} for the last six exceptional groups. For the group $G_{31}$, the matrix $C$ has the first, the third and the fourth rows the same as the unit matrix $I_4$, and the second row is given by \n\\begin{equation}\n\\label{e5}\n0, \\ f_4, \\ -\\frac{1}{5}f_1 , -\\frac{1}{1620}f_2,\n\\end{equation}\nsee \\cite[Appendix B, p. 285]{OT}.\nThen we set $B=A(f_1)J(f)C$ and it follows from \\cite[Theorem 6.53]{OT} that the first column in $B$ is a multiple of the Euler derivation. Let $b_{11}$ be the entry of this matrix $B$ situated on the first row and the first column, set $g=b_{11}\/x$ and let $D=g^{-1}B$, a new matrix with entries in the polynomial ring $S$. \nThe first column $D[1]$ in the matrix $D$, which is a normalized version of $B$, is now given by $x,y,z,t$, and the corresponding derivation of the polynomial ring $S$, denoted again by $D[1]$,  is precisely the Euler derivation.\nIn particular, one has $D[1]f=60f$. Note that the other columns $D[j]$  of the matrix $D$, for $j=2,3,4$, satisfy similar relations $D[j]f=g_jf$ for some homogeneous polynomials $g_j \\in S$, for $j=2,3,4$.\nDefine a new matrix $E$, whose columns $E[j]$ are given by the following relations.\n \\begin{equation}\n\\label{e7}\nE[j]=D[j]-\\frac{g_j}{60}D[1],\n\\end{equation}\nfor $j=1,2,3,4$, where we set $g_1=0$. It follows that $E[j]f=0$ for $j=2,3,4$, hence we have obtained 3 syzygies for $f$. Moreover, a direct computation using Singular shows that\n \\begin{equation}\n\\label{e8}\n\\det E= -486 f\n\\end{equation}\nand hence, in view of Saito's Criterion \\cite[Theorem 4.19]{OT}, $f$ is free with exponents $1,13,17,29$. In other words, the columns $E[j]$ for $j=2,3,4$ give a basis for the free $S$-module $AR(f)$. Note that $\\deg E[2]=29$, $\\deg E[3]=13$ and $\\deg E[4]=17$. The set of these degrees, to which we must add 1, the degree of the Euler derivation, correspond to the coexponents of $G_{31}$ as listed in \\cite[Table B.1., p. 287]{OT}. The entries of the matix $E$ are too complicated to display here, see Remark \\ref{rkcoeffsyz}.\n\nFinally we discuss the monodromy operator $h^1:H^1(F(G_{31}),\\mathbb{C}) \\to H^1(F(G_{31}),\\mathbb{C})$. To compute this operator, we can take a general hyperplane section $V(G_{31}): g=0$ of the projective surface $D:f=0$.\nIndeed, the first Milnor monodromy for $g$ and $f$ coincide, see \\cite[Theorem 4.1.24]{D1}. Table C.12 in \\cite{OT} shows that the corresponding curve $V(G_{31})$ in $\\mathbb{P}^2$ has degree $d=60$, 360 double points, 320 triple points and 30 points of multiplicity 6, corresponding to the isotropy group $G(4,2,2)$. It follows that $h^1$ can have only eigenvalues of order $1,2,3$ and $6$, since any eigenvalue of $h^1$ has to be a root of at least one of the local Alexander polynomials associated to the singularities of $V(G_{31})$, see \\cite{L1}, \\cite[Corollary 6.3.29]{D1}. The eigenvalues of order $2$ and $3$ are excluded by the results in \\cite{MPP}.\n\nTo prove Theorem \\ref{thmmain}, it remains to exclude the eigenvalues of order 6, and to do this, we apply  Theorem \\ref{thmEV}, for $d=60$ and $k=10$ or $k=50$.\nNote that for $k=10$ we have \n$E^{1,0}_1(f)_{10}=H^2(K^*_f)_{10}=0.$\nIndeed, by the above computations one has \n$AR(f)_8=0$ as all the generating syzygies $E[j]$ have degree $\\geq 13$, and then Lemma \\ref{dfomega=0}  implies that $H^2(K^*_f)_{10}=0$. \nTherefore it remains to show that $E^{1,0}_2(f)_{50}=0$, and this is done in the next section.\n \\begin{rk}\n\\label{rkdim}\n(i) In principle one can try to compute $h^1$ by applying the approach described in \\cite{DStCompM} to the curve $V(G_{31}): g=0$ defined above. However, this curve is no longer free, and it seems to us better from the computational point of view to work with the free surface $D$, having non isolated singularities, than to work with the non free curve $V(G_{31})$ with isolated singularities. This is due to the existence of the nice procedure to compute a basis for $AR(f)$ described above.\n\n\\noindent (ii) A computation of $h^1$ using the curve $V(G_{31}): g=0$ defined above can be found in \\cite{BDY}. The main techniques there are some subtle vanishing results for the cohomology of perverse sheaves on affine varieties, coupled again with computer aided\ncomputation via Singular needed to show that some surfaces are affine.\n\n\\noindent (iii) One can avoid using the results in \\cite{MPP} and exclude the eigenvalues of order 2 and 3 by the same approach as we use in the case of eigenvalues of order 6. These computations correspond to $k \\in \\{20,30,40\\}$. The cases  $k \\in \\{20,30\\}$ are trivial, since by an obvious modification of Theorem \\ref{thm1} below, one has  $H^2(K^*_f)_k=0$ in such situations. The case $k=40$ can be settled by  following the algorithm described below: with the notation of Theorem \\ref{thm1} we have in this case $\\deg A'_1=8$ and $A'_2=A'_3=0$, i.e. much simpler computations give the result. The corresponding Singular code $g31cubic$ is available at \\url{http:\/\/math.unice.fr\/~dimca\/singular.html}.\n\\end{rk}\n\n\n\n\\section{The algorithm}\n\nIn this section we assume that the coefficients $a_{ij}$ of the  differential 2-form $\\omega \\in \\Omega^2$ given by \\eqref{e9} are homogeneous polynomials in $S$ of degree 48, which corresponds to $\\deg \\omega=50$.\n\nWe denote the columns of the matrix $E$ above as follows:\n$E[2]=(m_1,n_1,p_1,q_1)$, $E[3]=(m_2,n_2,p_2,q_2)$, $E[4]=(m_3,n_3,p_3,q_3)$.\nAll the polynomials $m_i$ are divisible by $x$ (since $f_y$, $f_z$ and $f_t$ are clearly divisible by $x$ and $f_x$ is not) and we set $m_i=xm'_i$ for $i=1,2,3$.\nSimilarly one has $n_i=yn'_i$, $p_i=zp'_i$ and $q_i=tq'_i$ for $i=1,2,3$.\n \\begin{prop}\n\\label{prop1}\nEach of the four sequences $(m'_1,m'_2,m'_3)$, $(n'_1,n'_2,n'_3)$, $(p'_1,p'_2,p'_3)$ and $(q'_1,q'_2,q'_3)$ is a regular sequence in $S$.\n\\end{prop}\n\\proof\nTo show that $(m'_1,m'_2,m'_3)$ is a regular sequence, it is enough to check that the dimension of the corresponding zero set in $\\mathbb{C}^4$ is the expected one, i.e.\n$$\\dim V(m'_1,m'_2,m'_3)=1.$$\n And this can be checked by Singular. \n\\endproof\n \\begin{rk}\n\\label{rkcoeffsyz}\nThe four sequences $(m'_1,m'_2,m'_3)$, $(n'_1,n'_2,n'_3)$, $(p'_1,p'_2,p'_3)$ and $(q'_1,q'_2,q'_3)$ of polynomials in $S$ are listed in the print out of a Singular code here.\\\\\n\\url{http:\/\/math.unice.fr\/~dimca\/singular.html}\n\\end{rk}\n\n\nNote that the relation $R_1$ in Lemma \\ref{dfomega=0} is a syzygy in $AR(f)$ having the first component zero. Hence there are polynomials $P,Q,R$ in $S$ such that\n$$R_1=PE[2]+QE[3]+RE[4]$$\nand $Pm_1+Qm_2+Rm_3=0$. By dividing the last relation by $x$, we get\n\\begin{equation}\n\\label{r5}\nPm'_1+Qm'_2+Rm'_3=0.\n\\end{equation}\nBy Proposition \\ref{prop1}, it follows that $(P,Q,R)$ must be an $S$-linear combination of Koszul relations\nfor the regular sequence $(m'_1,m'_2,m'_3)$. Consider the following syzygies in $AR(f)$.\n\\begin{equation}\n\\label{r6}\nT_1=m'_2E[4]-m'_3E[3], \\ T_2=m'_1E[4]-m'_3E[2], \\ T_3=m'_1E[3]-m'_2E[2].\n\\end{equation}\nMore precisely, one has\n$$T_1=(0, y(m'_2n'_3-m'_3n'_2),z(m'_2p'_3-m'_3p'_2),t(m'_2q'_3-m'_3q'_2))$$\nand similar formulas for $T_2$ and $T_3$. One has $\\deg T_1=29$, $\\deg T_2=45$, $\\deg T_3=41$.\nIt follows that there are homogeneous polynomials $A_1,A_2,A_3$ in $S$ with $\\deg A_1= 19$, $\\deg A_2= 3$, $\\deg A_3= 7$\nsuch that\n\\begin{equation}\n\\label{r7}\nR_1=A_1T_1+A_2 T_2 +A_3T_3.\n\\end{equation}\nTo express $R_2$ as a combination of simpler syzygies, we have to consider for the same reason as above the syzygies in $AR(f)$ given by\n\\begin{equation}\n\\label{r8}\nU_1=n'_2E[4]-n'_3E[3], \\ U_2=n'_1E[4]-n'_3E[2], \\ U_3=n'_1E[3]-n'_2E[2].\n\\end{equation}\nMore precisely, one has\n$$U_1=(-x(m'_2n'_3-m'_3n'_2),0, z(n'_2p'_3-n'_3p'_2),t(n'_2q'_3-n'_3q'_2))$$\nand similar formulas for $U_2$ and $U_3$. One has $\\deg U_1=29$, $\\deg U_2=45$, $\\deg U_3=41$.\nIt follows that there are homogeneous polynomials $B_1,B_2,B_3$ with $\\deg A_j=\\deg B_j$, such that\n\\begin{equation}\n\\label{r9}\nR_2=B_1U_1+B_2 U_2 +B_3U_3.\n\\end{equation}\nHowever, the coefficient of $f_y$ in the relation $R_1$ coincides with the coefficient of $f_x$ in the relation $R_2$, hence we have\n$$yA_1(m'_2n'_3-m'_3n'_2)+yA_2(m'_1n'_3-m'_3n'_1)+yA_3(m'_1n'_2-m'_2n'_1)=$$\n$$-xB_1(m'_2n'_3-m'_3n'_2)-xB_2(m'_1n'_3-m'_3n'_1)-xB_3(m'_1n'_2-m'_2n'_1).$$\nIn other words,\n\\begin{equation}\n\\label{r10}\n(yA_1+xB_1)(m'_2n'_3-m'_3n'_2)+(yA_2+xB_2)(m'_1n'_3-m'_3n'_1)+(yA_3+xB_3)(m'_1n'_2-m'_2n'_1)=0,\n\\end{equation}\ni.e. a syzygy for the $2$-minors $MN_{ij}$ in the $2 \\times 3$ matrix $MN$ having the first row $(m'_1,m'_2,m'_3)$ and the second row $(n'_1,n'_2,n'_3)$.\n\\begin{rk}\n\\label{rk1}\nThe relation $(R_1)$ implies that $a_{34}$ is divisible by $g=xy(x^4-y^4)$, since both $f_z$ and $f_t$ are divisible by $g$, but not $f_x$ or $f_y$ as they have no multiple factors.\nA direct computation by Singular shows that all the minors $MN_{ij}$ are also divisible by $g_0=x^4-y^4$.\nTherefore, when we want to determine $a_{34}$ as an $S$-linear combination of $MN_{ij}$, we can simplify by $g_0$.\n\\end{rk}\nBy the above remark, we can consider new polynomials $MN'_{ij}=(m'_in'_j-m'_jn'_i)\/g_0$\nfor $1 \\leq i <j \\leq 3$ and hence \\eqref{r10} can be written as\n\\begin{equation}\n\\label{r11}\n(yA_1+xB_1)MN'_{23}+(yA_2+xB_2)MN'_{13} +(yA_3+xB_3)MN'_{12}=0.\n\\end{equation}\n\\begin{lem}\n\\label{lem12}\nThe minimal degree syzygy  \n$r_1MN'_{23}+r_2MN'_{13}+r_3MN'_{12}=0$ has multidegree $(\\deg r_1, \\deg r_2, \\deg r_3)=(24,8,12)$.\n\\end{lem}\n\\proof\nDirect computation by Singular.\n\\endproof\nIt follows that $yA_1+xB_1=yA_2+xB_2=yA_3+xB_3=0$, in other words, there are homogeneous polynomials\n$A'_1,A'_2,A'_3$ such that $A_i=xA'_i$ and $B_i=-yA'_i$ for $i=1,2,3$.\nConsider now the third syzygy $R_3$, having the third coordinate trivial. We define  new syzygies in $AR(f)$ by \n\\begin{equation}\n\\label{r12}\nV_1=p'_2E[4]-p'_3E[3], \\ V_2=p'_1E[4]-p'_3E[2], \\ V_3=p'_1E[3]-p'_2E[2].\n\\end{equation}\nMore precisely, one has\n$$V_1=(-x(m'_2p'_3-m'_3p'_2),-y(n'_2p'_3-n'_3p'_2), 0, t(p'_2q'_3-p'_3q'_2))$$\nand similar formulas for $V_2$ and $V_3$. One has $\\deg V_1=29$, $\\deg V_2=45$, $\\deg V_3=41$.\nIt follows that there are homogeneous polynomials $C_1,C_2,C_3$ with $\\deg A_j=\\deg C_j$, such that\n\\begin{equation}\n\\label{r13}\nR_3=C_1V_1+C_2 V_2 +C_3V_3.\n\\end{equation}\nHowever, the coefficient of $f_z$ in $R_1$ coincides with the opposite of the coefficient of $f_x$ in $R_3$, hence we have\n$$zA_1(m'_2p'_3-m'_3p'_2)+zA_2(m'_1p'_3-m'_3p'_1)+zA_3(m'_1p'_2-m'_2p'_1)=$$\n$$xC_1(m'_2p'_3-m'_3p'_2)+xC_2(m'_1p'_3-m'_3p'_1)+xC_3(m'_1p'_2-m'_2p'_1).$$\nIn other words,\n\\begin{equation}\n\\label{r14}\n(zA_1-xC_1)(m'_2p'_3-m'_3p'_2)+(zA_2-xC_2)(m'_1p'_3-m'_3p'_1)+(zA_3-xC_3)(m'_1p'_2-m'_2p'_1)=0,\n\\end{equation}\ni.e. a syzygy for the $2$-minors $MP_{ij}$ in the matrix $MP$ having the first row $(m'_1,m'_2,m'_3)$ and the second row $(p'_1,p'_2,p'_3)$.\n\n\\begin{lem}\n\\label{lem13}\n\\begin{enumerate}\n\n\\item The minors $(MP_{23},MP_{13},MP_{12})$ are all divisible by $x^4-z^4$.\n\n\\item Set $MP'_{ij}=MP_{ij}\/(x^4-z^4)$. Then the  minimal degree syzygy involving $(MP'_{23},MP'_{13},MP'_{12})$ has multidegree $(24,8,12)$.\n\n\\end{enumerate}\n\\end{lem}\n\\proof\nDirect computation by Singular.\n\\endproof\nIt follows as above that one has $C_i=zA'_i$ for $i=1,2,3$.\nConsider now the last syzygy $R_4$, having the fourth coordinate trivial. We define the syzygies\n\\begin{equation}\n\\label{r20}\nW_1=q'_2E[4]-q'_3E[3], \\ W_2=q'_1E[4]-q'_3E[2], \\ W_3=q'_1E[3]-q'_2E[2].\n\\end{equation}\nMore precisely, one has\n$$W_1=(-x(m'_2q'_3-m'_3q'_2),-y(n'_2q'_3-n'_3q'_2), -z(p'_2q'_3-p'_3q'_2),0)$$\nand similar formulas for $W_2$ and $W_3$. One has $\\deg W_1=29$, $\\deg W_2=45$, $\\deg W_3=41$.\nIt follows that there are homogeneous polynomials $D_1,D_2,D_3$ with $\\deg A_j=\\deg D_j$, such that\n\\begin{equation}\n\\label{r21}\nR_4=D_1W_1+D_2 W_2 +D_3W_3.\n\\end{equation}\nMoreover, the coefficient of $f_t$ in $R_1$ coincides with the coefficient of $f_x$ in $R_3$, hence we have\n$$tA_1(m'_2q'_3-m'_3q'_2)+tA_2(m'_1q'_3-m'_3q'_1)+tA_3(m'_1q'_2-m'_2q'_1)=$$\n$$-xD_1(m'_2q'_3-m'_3q'_2)-xD_2(m'_1q'_3-m'_3q'_1)-xD_3(m'_1q'_2-m'_2q'_1).$$\nIn other words,\n\\begin{equation}\n\\label{r22}\n(tA_1+xD_1)(m'_2q'_3-m'_3q'_2)+(tA_2+xD_2)(m'_1q'_3-m'_3q'_1)+(tA_3+xD_3)(m'_1q'_2-m'_2q'_1)=0,\n\\end{equation}\ni.e. a syzygy for the $2$-minors $MQ_{ij}$ in the matrix $MQ$ having the first row $(m'_1,m'_2,m'_3)$ and the second row $(q'_1,q'_2,q'_3)$.\n\n\\begin{lem}\n\\label{lem23}\n\\begin{enumerate}\n\n\\item The minors $(MQ_{23},MQ_{13},MQ_{12})$ are all divisible by $x^4-t^4$.\n\n\\item Set $MQ'_{ij}=MQ_{ij}\/(x^4-t^4)$. Then the  minimal degree syzygy involving $(MQ'_{23},MQ'_{13},MQ'_{12})$ has multidegree $(24,8,12)$.\n\n\\end{enumerate}\n\\end{lem}\n\\proof\nDirect computation by Singular.\n\\endproof\nIt follows as above that one has $D_i=-tA'_i$ for $i=1,2,3$. This proves the following result.\n \\begin{thm}\n\\label{thm1}\nA differential 2-form $\\omega$ of degree 50 given by \\eqref{e9} satisfies $df \\wedge \\omega=0$ if and only if there are homogeneous polynomials $A'_1 \\in S_{18}$, $A'_2 \\in S_2$ and $A'_3 \\in S_6$ such that\n$$R_1=x(A'_1T_1+A'_2T_2+A'_3T_3), \\ R_2=-y(A'_1U_1+A'_2U_2+A'_3U_3),$$\n$$R_3=z(A'_1V_1+A'_2V_2+A'_3V_3), \\ R_4=-t(A'_1W_1+A'_2W_2+A'_3W_3).$$\n\\end{thm}\n\\proof\nOne should check that any $a_{ij}$ which occurs in two distinct relations $R_k$ and $R_{k'}$\ngets the same values by this construction. As an illustration, let's check this for $a_{12}$.\nIn the syzygy $R_3$, the polynomial $a_{12}$ occurs on the last coordinate, hence its value is\n$$zA'_1[t(p'_2q'_3-p'_3q'_2)]+zA'_2[t(p'_1q'_3-p'_3q'_1)]+zA'_3[-t(p'_2q'_1-p'_1q'_2)].$$\nIn the syzygy $R_4$, the polynomial $a_{12}$ occurs on the third coordinate, hence its value is\n$$-tA'_1[-z(p'_2q'_3-p'_3q'_2)]-tA'_2[-z(p'_1q'_3-p'_3q'_1)]-tA'_3[z(p'_2q'_1-p'_1q'_2)],$$\nhence exactly the same value as above.\n\\endproof\nThe above formulas show that we have the equalities\n\t\\begin{itemize}\n\t\t\\item[i)] $a_{12}=zt(A'_1PQ'_{23}+A'_2PQ'_{13}+A'_3PQ'_{12}),$\n\t\t\\item[ii)] $a_{23}=xt(A'_1MQ'_{23}+A'_2MQ'_{13}+A'_3MQ'_{12}),$\n\t\t\\item[iii)] $a_{13}=-yt(A'_1NQ'_{23}+A'_2NQ'_{13}+A'_3NQ'_{12}).$\n\t\\end{itemize}\t\n\nThe polynomials $A'_i$ are unique, since the  minimal degree syzygy involving say  $(PQ'_{23},PQ'_{13},PQ'_{12})$ has multidegree $(24,8,12)$ by Lemma \\ref{lem23}. It follows that\n$$\\dim H^2(K^*_f)_{50}=\\dim S_{18}+\\dim S_2+\\dim S_6=1330+10+84=1424.$$\nA differential form $\\omega \\in  H^2(K^*_f)_{50}$ will survive to give an element in $E^{1,0}_2(f)_{50}$ if and only if it satisfies the equations\n$(E_i)$ for $i=1,2,3,4$ in Lemma \\ref{lemd=0}.\nAny such equation $(E_i)$ is in fact a system of $19600=\\dim S_{47}$ linear equations in the $1424$  indeterminates given by the coefficients of the polynomials $A'_1$, $A'_2$ and $A'_3$.\nOur proof is completed by the following result.\n\\begin{lem}\n\\label{lemfinal}\nThe system of linear equations corresponding to the equation $(E_4)$\nin Lemma \\ref{lemd=0} has only the trivial solution. In particular one has\n$E^{1,0}_2(f)_{50}=0$.\n\\end{lem}\n\\proof\nDirect, rather lengthy, computation by Singular. The corresponding code $g31$ is available at \n\\url{http:\/\/math.unice.fr\/~dimca\/singular.html}\n\\endproof\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and main results}\n\nIn this paper we prove the existence and  stability of Cantor families of  quasi-periodic solutions of  Hamiltonian \n{\\it quasi-linear} (also called  ``{\\it strongly nonlinear}'', e.g. in \\cite{k1}) perturbations of the KdV equation \n\\begin{equation}\\label{kdv quadratica}\nu_t + u_{xxx} - 6 u u_x + {\\mathcal N}_4 (x, u, u_x, u_{xx}, u_{xxx}) = 0\\,, \n\\end{equation}\nunder periodic boundary conditions $ x \\in \\mathbb T := \\mathbb R \/ 2 \\pi \\mathbb Z$, \nwhere \n\\begin{equation}\\label{qlpert}\n{\\mathcal N}_4 (x, u, u_x, u_{xx}, u_{xxx}) := - \\partial_x \\big[ (\\partial_u f)(x, u,u_x)  - \\partial_{x} ((\\partial_{u_x} f)(x, u,u_x)) \\big]   \n\\end{equation}\nis the most general quasi-linear Hamiltonian (local) nonlinearity. \nNote that $ {\\cal N}_4 $ contains as many derivatives as the linear part $ \\partial_{xxx} $.\nThe equation \\eqref{kdv quadratica}  is the Hamiltonian PDE\n$ u_t = \\partial_x  \\nabla H(u)  $\nwhere $ \\nabla H $ denotes the $L^2(\\mathbb T_x)$  gradient of the  Hamiltonian \n\\begin{equation} \\label{Ham in intro}\nH (u) = \\int_\\mathbb T  \\frac{u_x^2}{2} + u^3 + f(x, u,u_x) \\, dx \n\\end{equation}\non the real phase space   \n\\begin{equation}\\label{def phase space}\nH^1_0 (\\mathbb T_x) := \\Big\\{ u(x ) \\in H^1 (\\mathbb T, \\mathbb R) \\ : \\ \\int_{\\mathbb T} u(x) dx = 0  \\Big\\} \\, . \n\\end{equation}\nWe assume  that  the ``Hamiltonian density\" \n$ f \\in C^q (\\mathbb T \\times \\mathbb R \\times \\mathbb R; \\mathbb R ) $  \nfor some $ q $ large enough, and that \n\\begin{equation}\\label{order5}\nf = f_5(u,u_x) + f_{\\geq 6}(x,u,u_x) \\,,\n\\end{equation}\nwhere $ f_5 (u,  u_x)  $ denotes the homogeneous component of $ f $ of degree 5  and \n$ f_{\\geq 6} $ collects all the higher order terms. \nBy \\eqref{order5} the nonlinearity $ {\\mathcal N}_4 $ vanishes of order $ 4 $ at $ u = 0 $ and \n \\eqref{kdv quadratica}\nmay be seen, close to the origin, as  a ``small\" perturbation of the KdV equation\n\\begin{equation}\\label{KdVmKdV} \nu_t + u_{xxx} - 6 u u_x = 0 \\, , \n\\end{equation}\nwhich is completely integrable. \nActually, the KdV equation \\eqref{KdVmKdV} may be described by global analytic action-angle variables, see \\cite{KaP} and the references therein.\n\n\nA natural question  is  to know whether the  periodic, quasi-periodic or almost periodic solutions of \\eqref{KdVmKdV}\npersist under small perturbations.  This is the content of KAM theory. \n\n\\smallskip\n\nThe first KAM results for PDEs \nhave been obtained \nfor   $1$-d  semilinear Schr\\\"odinger and wave equations \nby Kuksin \\cite{Ku}, Wayne \\cite{W1}, \nCraig-Wayne \\cite{CW}, P\\\"oschel \\cite{Po2}, see \\cite{C}, \\cite{k1} and references therein. \nFor PDEs in higher space dimension the theory has been more recently extended \nby Bourgain \\cite{B5}, Eliasson-Kuksin \\cite{EK}, and Berti-Bolle\n\\cite{BB13JEMS}, Geng-Xu-You \\cite{GXY}, Procesi-Procesi \\cite{PP}-\\cite{PP1}, Wang \\cite{Wang}. \n\n\\smallskip\nFor {\\it unbounded} perturbations the first KAM results \nhave been proved by Kuksin \\cite{K2} and Kappeler-P\\\"oschel \\cite{KaP}\nfor KdV (see also Bourgain \\cite{B96}), and more recently by  Liu-Yuan \\cite{LY}, Zhang-Gao-Yuan \\cite{ZGY} for derivative NLS, and by Berti-Biasco-Procesi \\cite{BBiP1}-\\cite{BBiP2} for derivative NLW.\nFor a recent survey of known results for \nKdV, we refer to \\cite{K13}. \n\nThe KAM theorems in \\cite{K2}, \\cite{KaP} prove the persistence \nof the finite-gap solutions of the integrable KdV \n\\eqref{KdVmKdV} under semilinear \nHamiltonian perturbations $ \\varepsilon \\partial_{x} (\\partial_u f) (x, u) $, \nnamely when the density  $ f $ is independent of\n$ u_x $, so that \\eqref{qlpert} is a differential operator of order $ 1 $  (note that in \\cite{k1} such \nnonlinearities are called ``quasi-linear\" and \\eqref{qlpert} ``strongly nonlinear\"). \nThe key point is that the frequencies of KdV grow as $ \\sim j^3 $ and\nthe difference $ |j^3  - i^3| \\geq (j^2 + i^2)\/2 $, $i \\neq j $, so that  KdV gains (outside the diagonal) two \nderivatives.\nThis approach also works  for Hamiltonian pseudo-differential perturbations of order $ 2 $ (in space),\nusing the improved Kuksin's lemma in \\cite{LY}. \nHowever it does {\\it not} work for a general quasi-linear \n perturbation as in \\eqref{qlpert},\nwhich is a nonlinear differential operator of the {\\it same} order (i.e.\\ 3) as the constant coefficient linear operator $ \\partial_{xxx}$.\nSuch a strongly nonlinear perturbation term makes the KAM question quite delicate \nbecause of the possible phenomenon of formation of singularities in finite time, see \nLax \\cite{Lax}, Klainerman-Majda \\cite{KM} for quasi-linear wave equations,\nsee also section 1.4 of \\cite{k1}. \nFor example, Kappeler-P\\\"oschel \\cite{KaP} (Remark 3, page 19) wrote: \n``{\\it It would be interesting to obtain perturbation results which also include terms of higher order, at least in the region where the KdV approximation is valid. However, results of this type are still out of reach, if true at all}''.\n\n\\smallskip\n\nIn this paper we give the first positive answer to this problem, proving the existence of \nsmall amplitude, linearly stable, quasi-periodic solutions of \\eqref{kdv quadratica}, see Theorem \\ref{thm:KdV}.  \nNote that  \\eqref{kdv quadratica} does not depend on external parameters.\nMoreover the KdV equation \\eqref{kdv quadratica} is a {\\it completely resonant} PDE, \nnamely the linearized equation at the origin is the linear Airy equation $ u_t + u_{xxx} = 0 $, \nwhich possesses only the $ 2 \\pi $-periodic in time solutions\n\\begin{equation}\\label{Airyper}\nu(t,x) = {\\mathop \\sum}_{j \\in \\mathbb Z \\setminus \\{0\\} } u_j {\\rm e}^{{\\mathrm i}  j^3 t} e^{{\\mathrm i}  jx } \\, .\n\\end{equation}\nThus the existence of quasi-periodic solutions of \\eqref{kdv quadratica} is a purely nonlinear phenomenon \n(the diophantine frequencies in \\eqref{solution u} are $ O(|\\xi|) $-close to integers with $ \\xi \\to 0 $)\nand a perturbation theory is more difficult. \n\nThe solutions that we find are localized in Fourier space close to  finitely many {``tangential sites''}  \n\\begin{equation}  \\label{tang sites}\nS^+ := \\{ \\bar\\jmath_1, \\ldots, \\bar\\jmath_\\nu \\}\\,, \\quad \nS := S^+ \\cup (- S^+) = \\{ \\pm j : j \\in S^+ \\}\\,, \n\\quad {\\bar \\jmath}_i \\in \\mathbb N \\setminus \\{0\\}  \\, , \\quad \\forall i =1, \\ldots, \\nu \\,.\n\\end{equation}\nThe set $ S $ is required to be even because the solutions $ u $ of  \\eqref{kdv quadratica} have to be real valued. \nMoreover, we also assume the following explicit hypotheses on $ S $: \n\\begin{itemize}\n\\item {\\sc  $ ({\\mathtt S}1) $} \n$j_1 + j_2 + j_3 \\neq 0$ for all $j_1, j_2, j_3 \\in S$. \n\\item {\\sc  $ ({\\mathtt S}2) $}  $ \\nexists j_1, \\ldots, j_4 \\in S $ such that   \n$ j_1 + j_2 + j_3 + j_4 \\neq 0 $, $ j_1^3 + j_2^3 + j_3^3 + j_4^3 - (j_1 + j_2 + j_3 + j_4)^3 = 0 $. \n\\end{itemize}\n\\begin{theorem}  \\label{thm:KdV}\nGiven $ \\nu \\in \\mathbb N $,  \nlet $ f  \\in C^q $ (with $ q := q(\\nu) $ large enough) satisfy \\eqref{order5}.\nThen, for all the  tangential sites $ S $ as in \\eqref{tang sites} satisfying\n$ ({\\mathtt S}1) $-$ ({\\mathtt S}2) $,   \nthe KdV equation \\eqref{kdv quadratica} possesses small amplitude quasi-periodic solutions \nwith diophantine frequency vector $\\omega := \\omega(\\xi) = (\\omega_j)_{j \\in S^+} \\in \\mathbb R^\\nu$, \nof the form \n\\begin{equation}\\label{solution u}\nu(t,x) = {\\mathop\\sum}_{j \\in S^+} 2 \\sqrt{\\xi_j} \\, \\cos( \\omega_j t + j x) + o( \\sqrt{|\\xi|} ), \\quad \n\\omega_j := j^3 - 6 \\xi_j j^{-1} \\,,\n\\end{equation}\nfor a ``Cantor-like\" set of small amplitudes $ \\xi \\in \\mathbb R^\\nu_+ $ with density $ 1 $ at $ \\xi = 0 $.\nThe term $ o( \\sqrt{|\\xi|} ) $ is small in some $ H^s $-Sobolev norm, $ s < q $. \nThese quasi-periodic solutions are {\\sl linearly stable}.  \n\\end{theorem}\n\nThis result is deduced from Theorem \\ref{main theorem}. Let us make some comments. \n\\begin{enumerate}\n\\item \nThe set of tangential sites $ S $  satisfying  $({\\mathtt S}1)$-$({\\mathtt S}2) $ \ncan be  iteratively constructed in an explicit way, see the end of section \\ref{sec:NM}. \nAfter fixing  $  \\{\\bar\\jmath_1, \\ldots, \\bar\\jmath_n \\} $, \nin the choice of $\\bar\\jmath_{n+1}$ there are only finitely many forbidden values,\nwhile all the other infinitely many values are good choices for $\\bar\\jmath_{n+1}$. \nIn this precise sense the set $ S $ is ``generic''.  \n\\item \nThe linear stability of the quasi-periodic solutions is discussed after \\eqref{sistema lineare dopo}.\nIn a suitable set of symplectic coordinates $ (\\psi, \\eta, w) $, $ \\psi \\in \\mathbb T^\\nu $,  near the invariant torus,\n the linearized equations at the quasi-periodic solutions assume the form \n\\eqref{sistema lineare dopo}, \\eqref{vjmuj}. Actually there is a complete KAM normal form near the \ninvariant torus (remark \\ref{rem:KAM normal form}), see also \\cite{BB13}. \n\\item \nA similar result holds for perturbed (focusing\/defocusing) mKdV equations\n\\begin{equation}\\label{mKdV}\nu_t + u_{xxx} \\pm \\partial_x u^3 + {\\mathcal N}_4 (x, u, u_x, u_{xx}, u_{xxx})  = 0 \n\\end{equation}\nfor  tangential sites $ S $  which satisfy \n$ \\frac{2}{2\\nu - 1} \\sum_{i = 1}^\\nu \\bar \\jmath_i^{\\,2} \\notin \\mathbb Z $.\nIf the density $  f(u, u_x) $ is independent on $ x $, the result holds for {\\it all} the choices of the tangential sites.  \nThe KdV equation \\eqref{kdv quadratica} is more difficult  than \n \\eqref{mKdV} because the nonlinearity is quadratic and not cubic.\n\nAn important point is that the fourth order Birkhoff normal form of KdV and mKdV is completely integrable. \nThe present strategy of proof --- that we describe in detail below --- is a rather general \napproach for constructing small amplitude quasi-periodic solutions of quasi-linear perturbed KdV equations. For example it could be applied to generalized KdV equations with \nleading nonlinearity $ u^p $, $ p \\geq 4 $, by using \nthe normal form techniques of Procesi-Procesi \\cite{PP1}-\\cite{PP}. \nA further interesting open question concerns perturbations of the finite gap solutions of KdV.\n\\end{enumerate}\n\nLet us describe the strategy of proof of Theorem \\ref{thm:KdV}, which involves many different arguments. \n\\\\[2mm]\n{\\it Weak Birkhoff normal form.} \nOnce the finite set of tangential sites $ S $ has been fixed, \nthe first step is to perform  \na ``weak\" Birkhoff normal form (weak BNF), \nwhose goal is to find an invariant manifold of solutions of the third order approximate KdV equation \\eqref{kdv quadratica}, \non which the dynamics is completely integrable,  see section \\ref{sec:WBNF}.  Since the  KdV  nonlinearity is quadratic, two steps of weak BNF are required. \nThe present Birkhoff map is close to the  identity up to  {\\it finite dimensional} operators, see Proposition \\ref{prop:weak BNF}. \nThe key advantage is that it modifies  $ {\\mathcal N}_4 $ very mildly, only up to finite dimensional operators \n(see for example Lemma \\ref{lemma astratto potente}), and thus the spectral analysis of the linearized equations \n(that we shall perform in section \\ref{operatore linearizzato sui siti normali}) is essentially the\nsame as if we were in the original coordinates.\n\n\nThe weak normal form \\eqref{widetilde cal H} does not remove (or normalize) the monomials  $ O(z^2) $. \nThis could be done. However,\nwe do not perform such stronger normal form (called  ``partial BNF\" in P\\\"oschel \\cite{Po3})\nbecause the corresponding \nBirkhoff map is close to the identity only up to an operator of order $ O(\\partial_x^{-1}) $, and so it would \nproduce, in the transformed vector field \n$ {\\mathcal N}_4 $,  terms of order $ \\partial_{xx} $ and  $ \\partial_x $.\nA fortiori, we cannot either use the full Birkhoff normal form computed\nin \\cite{KaP} for KdV, which completely diagonalizes the fourth order terms, because \nsuch Birkhoff map is only close to the identity up to a bounded operator. \nFor the same reason, we do not use the global nonlinear Fourier transform in \\cite{KaP} (Birkhoff coordinates), which is close to the Fourier transform up to smoothing operators of order $ O(\\partial_x^{-1}) $.  \n\n\nThe weak BNF procedure of section \\ref{sec:WBNF}  is sufficient \nto find the first nonlinear (integrable) approximation of the solutions and \nto extract the  ``frequency-to-amplitude'' modulation  \\eqref{mappa freq amp}.  \n\nIn Proposition \\ref{prop:weak BNF}\nwe also remove  the terms   $O(v^5)$, $O(v^4 z)$ in order to have  sufficiently good approximate solutions  \nso that  the Nash-Moser iteration of section \\ref{sec:NM} will converge. This is necessary for KdV\nwhose nonlinearity is quadratic at the origin. \nThese further steps of Birkhoff normal form are not required if the nonlinearity  \nis yet cubic as for mKdV, see Remark \\ref{remark:cubic}. \nTo this aim, we choose the tangential sites $S$ such that $({\\mathtt S}2)$ holds. \nWe also note that we assume  \\eqref{order5} because \nwe use the conservation of momentum up to the homogeneity order 5, see \\eqref{cons mom}.\n \\\\[1mm]  \n {\\it Action-angle and rescaling.}\nAt this point we introduce action-angle variables on the tangential sites (section \\ref{sec:4})\nand, after the rescaling \\eqref{rescaling kdv quadratica}, \nwe  look for quasi-periodic solutions of the \nHamiltonian \\eqref{formaHep}.\nNote that the coefficients of the  normal form $ {\\cal N } $ in \\eqref{Hamiltoniana Heps KdV}\ndepend on the angles $ \\theta $, unlike the usual KAM theorems \\cite{Po3}, \\cite{Ku},\nwhere the whole normal form is reduced to constant coefficients.\nThis is because the weak BNF of section \\ref{sec:WBNF} did {\\it not} normalize the quadratic terms $ O(z^2) $. \nThese terms are dealt with the ``linear Birkhoff normal form'' (linear BNF)\nin sections \\ref{BNF:step1}, \\ref{BNF:step2}.\nIn some sense here the ``partial\" Birkhoff normal form of \\cite{Po3} is split into the weak BNF of section \\ref{sec:WBNF} and the linear BNF of sections \\ref{BNF:step1}, \\ref{BNF:step2}.\n \nThe action-angle variables are convenient for \nproving the stability of the solutions. \n\\\\[1mm]  \n {\\it The nonlinear functional setting.}\nWe look for a zero of the nonlinear operator \\eqref{operatorF}, whose unknown is the\nembedded torus and the frequency $ \\omega $ is seen as an ``external\" parameter.\nThe solution is obtained  by a Nash-Moser iterative scheme in Sobolev scales. \nThe key step  is to  construct  (for $ \\omega  $ restricted to a suitable Cantor-like set) \n an approximate inverse ({\\it \\`a la} Zehnder \\cite{Z1}) of the linearized operator\nat any approximate solution. Roughly, this means to find a linear operator which \nis an inverse  at an exact solution. A major difficulty is that the tangential and the normal dynamics near an invariant torus are \nstrongly coupled. \n\nThis difficulty is overcome by implementing the abstract procedure in Berti-Bolle \\cite{BB13}-\\cite{BB14}\n developed in order to\nprove existence of quasi-periodic solutions for autonomous NLW (and NLS) with a multiplicative\npotential. This approach reduces the search of an approximate inverse for \\eqref{operatorF}\nto \nthe invertibility of a quasi-periodically forced PDE \nrestricted on the normal directions. \nThis  method  approximately decouples the ``tangential\" and the ``normal\" dynamics around  \nan approximate invariant torus, introducing a suitable set of symplectic  variables\n$ (\\psi, \\eta, w) $ near the torus, see \\eqref{trasformazione modificata simplettica}.\nNote that, in the first line of \\eqref{trasformazione modificata simplettica}, $ \\psi $ is the ``natural\" angle variable which coordinates  the torus, and, in the third line, the normal variable $ z $ is \nonly translated by the component $ z_0 (\\psi )$ of the torus.\nThe second line \ncompletes this transformation\nto a symplectic one. The canonicity of this map  is proved in  \\cite{BB13} using the isotropy of\nthe approximate invariant torus $ i_\\d $, see Lemma \\ref{toro isotropico modificato}.\nThe change of variable \n\\eqref{trasformazione modificata simplettica} brings the  torus $ i_\\d $ ``at the origin\". The advantage is that\n the second equation in \\eqref{operatore inverso approssimato} (which corresponds to the action variables of the torus) \n can be immediately solved, see \n\\eqref{soleta}. Then it remains to solve the third equation \\eqref{cal L omega}, i.e. to invert\nthe linear operator  $ {\\cal L}_\\omega $. \nThis is, up to finite dimensional remainders,  \na quasi-periodic Hamiltonian linear  Airy equation perturbed by a \nvariable coefficients differential operator of order $ O(\\partial_{xxx} ) $.\nThe exact form of $ {\\cal L}_\\omega $ is obtained in Proposition \\ref{prop:lin}. \n\\\\[1mm]\n{\\it Reduction of the linearized operator in the normal directions.} In section \n\\ref{operatore linearizzato sui siti normali}\nwe conjugate the variable coefficients operator $ {\\cal L}_\\omega $ in \\eqref{Lom KdVnew} \nto a diagonal operator with constant coefficients\nwhich describes infinitely many harmonic oscillators\n\\begin{equation}\\label{linearosc}\n{\\dot v}_j + \\mu_j^\\infty v_j  = 0 \\,  , \\quad \\mu_j^\\infty := {\\mathrm i}  (-m_3 j^3 + m_1 j) + r_j^\\infty \\in {\\mathrm i}  \\mathbb R \\, , \\quad j \\notin S \\, , \n\\end{equation}\nwhere the constants $  m_3 -1 $, $ m_1 \\in \\mathbb R $ and  $ \\sup_j |r_j^\\infty | $ are small, see Theorem \\ref{teoremadiriducibilita}.\nThe main perturbative effect to the spectrum (and the eigenfunctions) of $ {\\cal L}_\\omega $\nis clearly due to  the term $ a_1 (\\omega t, x ) \\partial_{xxx} $ (see \\eqref{Lom KdVnew}), \nand it is too strong for the usual \n reducibility KAM techniques to work directly. \nThe conjugacy of $ {\\cal L}_\\omega $ with \\eqref{linearosc} is obtained in several steps. \nThe first task (obtained in sections \\ref{step1}-\\ref{step5}) is to conjugate \n$ {\\cal L}_\\omega $ to another \nHamiltonian operator of $ H_S^\\bot $ with constant coefficients \n\\begin{equation}\\label{L6 qualitativo}\n{\\cal L}_6 := \\omega \\cdot \\partial_\\varphi + m_3 \\partial_{xxx}  + m_1 \\partial_x   + R_6  \\, , \\quad m_1, m_3 \\in \\mathbb R \\, , \n\\end{equation}\nup to a  small bounded remainder $ R_6 = O(\\partial_x^0 ) $, see \\eqref{def L6}. \nThis expansion of $ {\\cal L}_\\omega $ in ``decreasing symbols\" with constant coefficients \nis similar to \\cite{BBM}, and  it is somehow in the spirit of the works of Iooss, Plotnikov and Toland \\cite{Ioo-Plo-Tol}-\\cite{IP11} \nin water waves theory, and Baldi \\cite{Baldi Benj-Ono} for Benjamin-Ono.\nIt is obtained by transformations which are very different from the usual KAM changes of variables. \nThere are several differences  with respect to \\cite{BBM}: \n\n\\begin {enumerate}\n\\item \nThe first step is to eliminate the $ x $-dependence from the  coefficient $ a_1 (\\omega t, x ) \\partial_{xxx} $ of the Hamiltonian operator $ {\\cal L}_\\omega $. \nWe cannot use the symplectic transformation $ {\\cal A } $ defined in \\eqref{primo cambio di variabile modi normali}, used in \\cite{BBM},  \nbecause  $ {\\cal L}_\\omega $ acts on the normal subspace $ H_S^\\bot $ only,\n and not on the whole Sobolev space as in \\cite{BBM}.\nWe can not use the restricted map $ {\\cal A}_\\bot :=  \\Pi_S^\\bot  {\\cal A} \\Pi_S^\\bot $\nwhich is {\\it not} symplectic. \nIn order to find a symplectic diffeomorphism of $ H_S^\\bot $ near  $ {\\cal A}_\\bot $, \nthe first observation is to realize $ {\\cal A } $  as the  flow map of the time dependent \nHamiltonian transport linear PDE \\eqref{transport-free}. \nThus we conjugate $ {\\cal L}_\\omega $ with the \nflow map of the projected Hamiltonian equation \\eqref{problemi di cauchy}. In \nLemma \\ref{modifica simplettica cambio di variabile} we prove that it differs from $ {\\cal A}_\\bot $\nup to finite dimensional operators.  \nA technical, but important, fact is that the remainders produced after this conjugation of ${\\cal L}_\\omega $ remain of the finite dimensional form \\eqref{forma buona con gli integrali}, see  Lemma  \\ref{cal R3}. \n\n\n\nThis step may be seen as a quantitative application of the Egorov theorem, see \\cite{Taylor},\nwhich describes how the principal symbol of a pseudo-differential operator (here $ a_1 (\\omega t, x) \\partial_{xxx} $)\ntransforms \nunder the flow of a  linear hyperbolic PDE (here \\eqref{problemi di cauchy}). \n \\item \nSince the weak BNF procedure of section \\ref{sec:WBNF} did not touch the quadratic terms $ O(z^2 ) $, \nthe operator $ {\\cal L}_\\omega $  has variable coefficients\nalso at the orders $ O(\\varepsilon)$ and $ O(\\varepsilon^2 )$, see \\eqref{Lom KdVnew}-\\eqref{a1p1p2}. \nThese terms cannot be reduced to constants by the \nperturbative scheme in \\cite{BBM},  which applies to terms $ R $  such that \n$ R \\gamma^{ -1} \\ll 1 $ where $ \\gamma $ is the diophantine constant of the frequency vector $ \\omega $.\nHere, since KdV is completely resonant,  \nsuch \n$ \\gamma = o(\\varepsilon^2 ) $, see \\eqref{omdio}.\nThese terms are reduced to constant coefficients  in sections \n\\ref{BNF:step1}-\\ref{BNF:step2} \nby means of purely algebraic arguments (linear BNF), which, ultimately,\nstem from the complete integrability of the fourth order BNF of the \nKdV equation \\eqref{KdVmKdV}, see \\cite{KaP}. \n\\end{enumerate}\n\nThe order of the  transformations of sections \\ref{step1}-\\ref{subsec:mL0 mL5}\nused to reduce $ {\\cal L}_\\omega $ is not accidental. The first two steps in sections \\ref{step1}, \\ref{step2}\nreduce to constant coefficients the quasi-linear  \nterm  $ O(\\partial_{xxx}) $ and eliminate the term $ O(\\partial_{xx})$, see \\eqref{L2 Kdv} (the second transformation is a\ntime quasi-periodic reparametrization of time).  \nThen, \nin section \\ref{step3}, we  apply the transformation ${\\cal T}$ \\eqref{gran tau} in such a way \nthat the space average of the coefficient $ d_1 (\\varphi, \\cdot ) $  in \n\\eqref{L3 KdV} is constant. This is done in view\nof the applicability of the descent method in section \\ref{step5}. \nAll these transformations are composition operators induced by diffeomorphisms of the torus. \nTherefore they are well-defined operators of a Sobolev space into itself, but their decay norm is infinite! \nWe perform the transformation $ {\\cal T } $  \\emph{before} \nthe linear Birkhoff normal form steps of sections \\ref{BNF:step1}-\\ref{BNF:step2}, because $\\mathcal{T}$ is a change of variable that preserves the form \\eqref{forma buona con gli integrali} \nof the remainders (it is not evident after the Birkhoff normal form). \nThe Birkhoff transformations are symplectic maps of the form  $ I + \\varepsilon O( \\partial_x^{-1}) $.  \nThanks to this property the coefficient $ d_1 ( \\varphi,x) $\nobtained in step \\ref{step3} is {\\it not} changed by these Birkhoff maps. \nThe transformation in section  \\ref{step5} is one step of ``descent method'' which transforms \n$ d_1 ( \\varphi,x) \\partial_x $ into a constant  $ m_1 \\partial_x  $. \nIt is at this point of the regularization procedure that the assumption $({\\mathtt S}1)$ on the tangential sites is used,\nso that the space average of the function $ q_{>2} $ is zero, see Lemma \\ref{p3 zero average}. \nActually we only need that the average of \nthe function  in \\eqref{unico pezzo} is zero. If \n $ f_5 = 0 $ \n (see \\eqref{order5})  then $({\\mathtt S}1)$  is not required. \nThis completes the task of conjugating ${\\cal L}_\\omega $ to $ {\\cal L}_6 $ in \\eqref{L6 qualitativo}.\n\nFinally, in section \\ref{subsec:mL0 mL5} we apply the abstract reducibility Theorem 4.2 in \\cite{BBM}, \nbased on a quadratic KAM scheme, which \ncompletely diagonalizes the linearized operator, obtaining  \\eqref{linearosc}. \nThe required smallness condition  \\eqref{R6resto} for $ R_6 $  holds.  Indeed\nthe biggest term in $ R_6 $ comes from the conjugation of \n$  \\varepsilon \\partial_x v_\\varepsilon (\\theta_0 (\\varphi), y_\\d (\\varphi)) $ in   \\eqref{a1p1p2}.\nThe linear BNF procedure of section \\ref{BNF:step1} had eliminated \nits main contribution $  \\varepsilon \\partial_x v_\\varepsilon ( \\varphi,0) $. \nIt remains  $ \\varepsilon \\partial_x \\big( v_\\varepsilon (\\theta_0 (\\varphi), y_\\d (\\varphi) ) - v_\\varepsilon ( \\varphi,0)  \\big) $ \nwhich has size $ O( \\varepsilon^{7-2b} \\gamma^{-1} ) $ due to the estimate \\eqref{ansatz 0} of the approximate solution.\nThis term enters in the variable coefficients of $ d_1 (\\varphi,x)  \\partial_x $\nand $ d_0 (\\varphi, x) \\partial_x^0 $. The first one had been reduced to the constant \noperator $ m_1 \\partial_x $ by the descent method of section \\ref{step5}.\nThe latter term is an operator of order $O(\\partial_x^0 )$ which satisfies \\eqref{R6resto}.\nThus $ {\\cal L}_6 $ may be diagonalized \nby the iterative scheme of Theorem 4.2 in \\cite{BBM} \nwhich requires the smallness condition $ O( \\varepsilon^{7-2b} \\gamma^{-2}) \\ll 1  $. This is the content of section \\ref{subsec:mL0 mL5}.  \n\\\\[1mm]\n{\\it The Nash-Moser iteration.}\nIn section \\ref{sec:NM} we perform the nonlinear Nash-Moser iteration which \nfinally proves Theorem \\ref{main theorem} and, therefore, Theorem \\ref{thm:KdV}.\nThe optimal smallness condition required for the convergence of the scheme is \n$ \\varepsilon \\| {\\cal F}(\\varphi, 0, 0 ) \\|_{s_0+ \\mu} \\gamma^{-2} \\ll 1 $, see  \\eqref{nash moser smallness condition}.\nIt is verified because $ \\| X_P(\\varphi, 0 , 0 ) \\|_s \\leq_s \\varepsilon^{6 - 2b} $ \n(see \\eqref{stima XP}), which, in turn, is a consequence of having eliminated  the terms\n$ O(v^5), O( v^4 z)$ from the original Hamiltonian \\eqref{H iniziale KdV}, \nsee \\eqref{widetilde cal H}. This requires the condition ($ {\\mathtt S}2$). \n\n\\smallskip\n\n\\noindent\n{\\it Acknowledgements}. We  thank M. Procesi, P. Bolle and T. Kappeler for many useful discussions. \nThis research was supported by the European Research Council under FP7, \nand partially by the grants STAR 2013 and \nPRIN 2012 ``Variational and perturbative aspects of nonlinear differential problems\".\n\n\\section{Preliminaries}\n\n\\subsection{Hamiltonian formalism of KdV}\\label{sec:Ham For}\n\nThe Hamiltonian vector field $X_H$ generated by a Hamiltonian \n$ H : H^1_0(\\mathbb T_x) \\to \\mathbb R$ is $ X_H (u) := \\partial_x \\nabla H (u) $, because\n$$\nd H (u) [ h] \n= ( \\nabla H(u), h )_{L^2(\\mathbb T_x)} = \\Omega ( X_H (u), h ) \\, , \\quad \\forall u, h \\in H^1_0(\\mathbb T_x) \\, , \n$$\nwhere $\\Omega$ is the non-degenerate symplectic form\n\\begin{equation}\\label{2form KdV}\n\\Omega (u, v) := \\int_{\\mathbb T} (\\partial_x^{-1 } u) \\, v \\, dx \\, ,   \\quad \n\\forall u, v \\in H^1_0 (\\mathbb T_x) \\, , \n\\end{equation}\nand $ \\partial_x^{-1} u $ is the periodic primitive of $ u $ with zero average. \nNote  that \n\\begin{equation} \\label{def pi 0} \n\\partial_x \\partial_x^{-1} =  \\partial_x^{-1} \\partial_x = \\pi_0 \\, , \\quad \n\\pi_0 (u) :=  u - \\frac{1}{2\\pi} \\int_{\\mathbb T} u(x) \\, dx \\,. \n\\end{equation}\nA map \nis symplectic if  it preserves \nthe 2-form $ \\Omega $.\n\nWe also remind that the Poisson bracket between two functions $ F $, $ G : H^1_0(\\mathbb T_x) \\to \\mathbb R$ is\n\\begin{equation}\\label{Poisson bracket}\n\\{ F (u), G(u) \\} := \\Omega (X_F, X_G ) = \\int_{\\mathbb T} \\nabla F(u) \\partial_x \\nabla G (u) dx \\, . \n\\end{equation}\nThe linearized KdV equation at $ u $ is  \n$$\nh_t = \\partial_x \\, (\\partial_u \\nabla H)(u) [h ] = X_{K}(h) \\, ,  \n$$\nwhere $ X_K $ is the KdV Hamiltonian vector field with quadratic Hamiltonian \n$ K = \\frac12  ((\\partial_u \\nabla H)(u)[h], h)_{L^2(\\mathbb T_x)} $ \n$= \\frac12 (\\partial_{uu} H)(u)[h, h]  $. By the Schwartz theorem, the Hessian operator $ A := (\\partial_u \\nabla H)(u) $ is symmetric, namely\n$ A^T = A $, with respect to the $ L^2$-scalar product. \n\n\\smallskip\n\n\\noindent\n{\\bf Dynamical systems formulation.} \nIt is convenient to regard the KdV equation also in the Fourier representation \n\\begin{equation}\\label{Fourier}\nu(x) = {\\mathop \\sum}_{j \\in \\mathbb Z \\setminus \\{0\\} } u_j e^{{\\mathrm i}  j x} \\, , \\qquad \nu(x) \\longleftrightarrow u := (u_j)_{j \\in \\mathbb Z \\setminus \\{0\\} } \\, , \\quad u_{-j} = \\overline{u}_j  \\, ,\n\\end{equation}\nwhere the Fourier indices $ j \\in \\mathbb Z \\setminus \\{ 0 \\}$ by the definition \\eqref{def phase space} of the phase space\nand $u_{-j} = \\overline{u}_j$ because $u(x)$ is real-valued.\nThe symplectic structure writes\n\\begin{equation}\\label{2form0}\n\\Omega = \\frac12 \\sum_{j \\neq 0}  \\frac{1}{{\\mathrm i}  j} du_j \\wedge d u_{-j} = \\sum_{j \\geq 1} \n\\frac{1}{{\\mathrm i}  j} du_j \\wedge d u_{-j}\n\\, , \\qquad \n\\Omega ( u, v ) = \\sum_{j \\neq 0} \\frac{1}{{\\mathrm i}  j} u_j v_{-j} = \\sum_{j \\neq 0} \\frac{1}{{\\mathrm i}  j} u_j { \\overline v}_{j} \\, , \n\\end{equation}\nthe Hamiltonian vector field $X_H$ and the Poisson bracket $\\{ F, G \\}$ are\n\\begin{equation}\\label{PoissonBr}\n[X_H (u)]_j = {\\mathrm i}  j (\\partial_{u_{-j}} H) (u) \\, , \\  \\forall j \\neq 0 \\, ,\n\\quad \n\\{ F (u), G(u) \\} = - {\\mathop \\sum}_{j \\neq 0} {\\mathrm i}  j   (\\partial_{u_{-j}} F) (u)  (\\partial_{u_j} G) (u) \\, .  \\end{equation}\n\\noindent\n\\noindent\n{\\bf Conservation of momentum.} \nA Hamiltonian \n\\begin{equation}   \\label{cons mom}\nH(u) = \\sum_{j_1, \\ldots, j_n \\in \\mathbb Z \\setminus \\{ 0 \\} } H_{j_1, \\ldots, j_n} u_{j_1} \\ldots u_{j_n}, \n\\quad \nu(x) = \\sum_{j \\in \\mathbb Z \\setminus \\{ 0 \\} } u_j e^{{\\mathrm i}  jx},\n\\end{equation}\nhomogeneous of degree $n$, \\emph{preserves the momentum} if the coefficients \n$H_{j_1, \\ldots, j_n}$ are zero for $j_1 + \\ldots + j_n \\neq 0$, \nso that the sum in \\eqref{cons mom} is restricted to integers \n such that $j_1 + \\ldots + j_n = 0$. \nEquivalently, $H$ preserves the momentum if $\\{ H, M \\} = 0$, where $M$ is the  momentum \n$ M(u) := \\int_{\\mathbb T} u^2 dx = $ $  \\sum_{j \\in \\mathbb Z \\setminus \\{0\\}} u_j u_{-j} $. \nThe homogeneous components of degree $ \\leq 5 $ \nof the KdV Hamiltonian $ H $ in \\eqref{Ham in intro} \npreserve the momentum because, by \\eqref{order5}, \nthe homogeneous component $f_5$ of degree 5 does not depend on the space variable $x$. \n\n\\smallskip\n\n\\noindent\n{\\bf Tangential and normal variables.} \nLet $\\bar\\jmath_1, \\ldots, \\bar\\jmath_\\nu \\geq 1$ be $\\nu$ distinct integers, \nand $S^+ := \\{ \\bar\\jmath_1, \\ldots, \\bar\\jmath_\\nu \\}$. \nLet $S$ be the symmetric set in \\eqref{tang sites}, \nand $S^c := \\{ j \\in \\mathbb Z \\setminus \\{ 0 \\} : j \\notin S \\}$ its complementary set in $\\mathbb Z \\setminus \\{ 0 \\}$. \nWe decompose the phase space as \n\\begin{equation}\\label{splitting S-S-bot}\nH^1_0 (\\mathbb T_x) := H_S \\oplus H_S^\\bot \\,, \n\\quad \nH_S := \\mathrm{span}\\{ e^{{\\mathrm i}  jx} : \\, j \\in S \\} , \\quad \nH_S^\\bot := \\big\\{ u = \\sum_{j \\in S^c} u_j e^{{\\mathrm i}  jx} \\in H^1_0(\\mathbb T_x)  \\big\\} ,\n\\end{equation}\nand we denote by $\\Pi_S $, \n$\\Pi_S^\\bot $ the corresponding orthogonal projectors. \nAccordingly we decompose\n\\begin{equation}  \\label{u = v + z}\nu = v + z, \\qquad  \nv = \\Pi_S u := {\\mathop \\sum}_{j \\in S} u_j \\, e^{{\\mathrm i}  jx}, \\quad  \nz = \\Pi_S^\\bot u := {\\mathop \\sum}_{j \\in S^c} u_j \\, e^{{\\mathrm i}  jx} \\, ,  \n\\end{equation}\nwhere $  v $ is called the {\\it tangential} variable and $ z $ the {\\it normal} one.\nWe shall sometimes identify $ v \\equiv (v_j)_{j \\in S } $ and  $ z \\equiv (z_j)_{j \\in S^c } $. \nThe subspaces  $ H_S $ and $ H_S^\\bot $ are {\\it symplectic}. \nThe dynamics of these two components is quite different. \nOn $ H_S $ we shall introduce the action-angle variables, see \\eqref{coordinate azione angolo}.\nThe linear frequencies of oscillations on the tangential  sites are \n\\begin{equation}\\label{bar omega}\n\\bar\\omega := (\\bar\\jmath_1^3, \\ldots, \\bar\\jmath_\\nu^3) \\in \\mathbb N^\\nu.\n\\end{equation}\n\n\\subsection{Functional setting}\n\n\n{\\bf Norms.} Along the paper we shall use the notation \n\\begin{equation}\\label{Sobolev coppia}\n\\| u \\|_s := \\| u \\|_{H^s( \\mathbb T^{\\nu + 1})} := \\| u \\|_{H^s_{\\varphi,x} }\n\\end{equation}\nto denote the Sobolev norm of functions $ u = u(\\varphi,x) $ in the Sobolev space $ H^{s} (\\mathbb T^{\\nu + 1} ) $. \nWe shall denote by $ \\| \\ \\|_{H^s_x} $ the Sobolev norm in the phase space of functions $ u :=  u(x) \\in H^{s} (\\mathbb T ) $.\nMoreover $ \\| \\ \\|_{H^s_\\varphi} $ will denote the Sobolev norm of scalar functions, like \nthe Fourier components $ u_j (\\varphi)  $.  \n\nWe fix $ s_0 := (\\nu+2) \\slash 2 $ so that   $ H^{s_0} (\\mathbb T^{\\nu + 1} ) \\hookrightarrow L^{\\infty} (\\mathbb T^{\\nu + 1} ) $ and the spaces \n$ H^s (\\mathbb T^{\\nu + 1} ) $, $ s > s_0  $, are an algebra.  \nAt the end of this section \nwe report interpolation properties of the Sobolev norm that will be currently used along the paper.\nWe shall also denote \n\\begin{align} \\label{HSbot nu}\nH^s_{S^\\bot} (\\mathbb T^{\\nu+1}) \n& := \\big\\{  u \\in H^s(\\mathbb T^{\\nu + 1} )  \\, : \\, u (\\varphi, \\cdot ) \\in H_S^\\bot \\  \n\\forall \\varphi \\in \\mathbb T^\\nu \\big\\} \\,,\n\\\\\n\\label{HS nu}\nH^s_{S} (\\mathbb T^{\\nu+1}) \n& := \\big\\{  u \\in H^s(\\mathbb T^{\\nu + 1} )  \\, : \\, u (\\varphi, \\cdot ) \\in H_{S} \\   \n\\forall \\varphi \\in \\mathbb T^\\nu \\big\\} \\,. \n\\end{align}\nFor a function $u : \\Omega_o \\to E$, $\\omega \\mapsto u(\\omega)$, where $(E, \\| \\ \\|_E)$ is a Banach space and \n$ \\Omega_o $ is a subset of $\\mathbb R^\\nu $, we define the sup-norm and the Lipschitz semi-norm\n\\begin{equation} \\label{def norma sup lip}\n\\| u \\|^{\\sup}_E \n:= \\| u \\|^{\\sup}_{E,\\Omega_o} \n:= \\sup_{ \\omega \\in \\Omega_o } \\| u(\\omega ) \\|_E \\, , \n\\quad\n\\| u \\|^{\\mathrm{lip}}_E \n:= \\| u \\|^{\\mathrm{lip}}_{E,\\Omega_o}  \n:= \\sup_{\\omega_1 \\neq \\omega_2 } \n\\frac{ \\| u(\\omega_1) - u(\\omega_2) \\|_E }{ | \\omega_1 - \\omega_2 | }\\,,\n\\end{equation}\nand, for $ \\gamma > 0 $, the Lipschitz norm\n\\begin{equation} \\label{def norma Lipg}\n\\| u \\|^{{\\mathrm{Lip}(\\g)}}_E  \n:= \\| u \\|^{{\\mathrm{Lip}(\\g)}}_{E,\\Omega_o}\n:= \\| u \\|^{\\sup}_E + \\gamma \\| u \\|^{\\mathrm{lip}}_E  \\, . \n\\end{equation}\nIf $ E = H^s $ we simply denote $ \\| u \\|^{{\\mathrm{Lip}(\\g)}}_{H^s} := \\| u \\|^{{\\mathrm{Lip}(\\g)}}_s $.  We shall use the notation \n$$\na \\leq_s b \\quad \\ \\Longleftrightarrow \\quad a \\leq C(s) b \\quad\n\\text{for  some  constant  } C(s) > 0  \\, . \n$$ \n\\noindent\n{\\bf Matrices with off-diagonal decay.}\nA linear operator can be identified, as usual, with its matrix representation. \nWe recall the definition  \nof the $ s $-decay norm (introduced in \\cite{BB13JEMS})  of an infinite dimensional matrix. \nThis norm is  used in \\cite{BBM} for the KAM reducibility scheme of the linearized operators. \n\\begin{definition}\\label{def:norms}\nThe $s$-decay norm of an infinite dimensional matrix $ A := (A_{i_1}^{i_2} )_{i_1, i_2 \\in \\mathbb Z^b } $, $b \\geq 1$, is  \n\\begin{equation} \\label{matrix decay norm}\n\\left| A \\right|_{s}^2 := \n\\sum_{i \\in \\mathbb Z^b} \\left\\langle i \\right\\rangle^{2s} \n\\Big( \\sup_{ \\begin{subarray}{c} i_{1} - i_{2} = i \n\\end{subarray}}\n| A^{i_2}_{i_1}| \\Big)^{2} \\, .\n\\end{equation}\nFor parameter dependent matrices $ A := A(\\omega) $, $\\omega  \\in \\Omega_o \\subseteq \\mathbb R^\\nu $, \nthe definitions \\eqref{def norma sup lip} and \\eqref{def norma Lipg} become \n\\begin{equation} \\label{matrix decay norm Lip}\n| A |^{\\sup}_s  := \\sup_{ \\omega \\in \\Omega_o } | A(\\omega ) |_s \\, , \n\\quad\n| A |^{\\mathrm{lip}}_s := \\sup_{\\omega_1 \\neq \\omega_2} \n\\frac{ | A(\\omega_1) - A(\\omega_2) |_s }{ | \\omega_1 - \\omega_2 | }\\,,\n\\quad\n| A |^{{\\mathrm{Lip}(\\g)}}_s := | A |^{\\sup}_s + \\gamma | A |^{\\mathrm{lip}}_s  \\,. \n\\end{equation}\n\\end{definition}\n\nSuch a norm is modeled on the behavior of matrices representing the multiplication\noperator by a function.\nActually, given a function $ p \\in H^s(\\mathbb T^b) $, the multiplication operator $ h \\mapsto p h $ is represented by the T\\\"oplitz matrix \n$ T_i^{i'} = p_{i - i'} $ and  $ |T|_s = \\| p \\|_s $. \nIf $p = p(\\omega )$ is a Lipschitz family of functions, then \n\\begin{equation}\\label{multiplication Lip}\n|T|_s^{\\mathrm{Lip}(\\g)} = \\| p \\|_s^{\\mathrm{Lip}(\\g)}\\,.\n\\end{equation}\nThe $s$-norm  satisfies classical algebra and interpolation inequalities, see \\cite{BBM}. \n\n \n\\begin{lemma} \\label{prodest}\nLet  $A = A(\\omega)$ and $B = B(\\omega)$ be matrices depending in a Lipschitz way on the parameter $\\omega \\in \n\\Omega_o \\subset \\mathbb R^\\nu $. \nThen for all $s \\geq s_0 > b\/2 $ there are $ C(s) \\geq C(s_0) \\geq 1 $ such that\n\\begin{align} \\label{algebra Lip}\n|A B |_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq  C(s) |A|_s^{{\\mathrm{Lip}(\\g)}} |B|_s^{{\\mathrm{Lip}(\\g)}} \\, , \n\\\\\n\\label{interpm Lip}\n|A B|_{s}^{{\\mathrm{Lip}(\\g)}} \n& \\leq C(s) |A|_{s}^{{\\mathrm{Lip}(\\g)}} |B|_{s_0}^{{\\mathrm{Lip}(\\g)}} \n+ C(s_0) |A|_{s_0}^{{\\mathrm{Lip}(\\g)}} |B|_{s}^{{\\mathrm{Lip}(\\g)}} .\n\\end{align}\n\\end{lemma}\nThe $ s $-decay norm controls the Sobolev norm, namely \n\\begin{equation}\\label{interpolazione norme miste}\n\\| A h \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq C(s) \\big(|A|_{s_0}^{\\mathrm{Lip}(\\g)} \\| h \\|_s^{\\mathrm{Lip}(\\g)} + |A|_{s}^{\\mathrm{Lip}(\\g)} \\| h \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\big).\n\\end{equation}\nLet now $ b := \\nu + 1 $. \nAn important sub-algebra  is formed by the {\\it T\\\"oplitz in time matrices} defined by\n\\begin{equation}\\label{Topliz matrix}\n A^{(l_2, j_2)}_{(l_1, j_1)}  := A^{j_2}_{j_1}(l_1 - l_2 )\\,  ,\n\\end{equation}\nwhose  decay norm \\eqref{matrix decay norm} is\n\\begin{equation}\\label{decayTop}\n|A|_s^2 =  \\sum_{j \\in \\mathbb Z, l \\in \\mathbb Z^\\nu} \\big( \\sup_{j_1 - j_2 = j} |A_{j_1}^{j_2}(l)| \\big)^2  \\langle l,j \\rangle^{2 s} \\, . \n\\end{equation}\nThese matrices are identified with the $ \\varphi $-dependent family\nof operators\n\\begin{equation}\\label{Aphi}\nA(\\varphi) := \\big( A_{j_1}^{j_2} (\\varphi)\\big)_{j_1, j_2 \\in \\mathbb Z} \\, , \\quad \nA_{j_1}^{j_2} (\\varphi) := {\\mathop\\sum}_{l \\in \\mathbb Z^\\nu} A_{j_1}^{j_2}(l) e^{{\\mathrm i}  l \\cdot \\varphi}\n\\end{equation}\nwhich act on functions of the $x$-variable as\n\\begin{equation}\\label{notationA}\nA(\\varphi) : h(x) = \\sum_{j \\in \\mathbb Z} h_j e^{{\\mathrm i}  jx} \\mapsto  \nA(\\varphi) h(x) = \\sum_{j_1, j_2 \\in \\mathbb Z}  A_{j_1}^{j_2} (\\varphi) h_{j_2} e^{{\\mathrm i}  j_1 x} \\, .  \n\\end{equation}\nWe still denote by $ | A(\\varphi) |_s $ the $ s $-decay norm of the matrix in \\eqref{Aphi}.\nAs in \\cite{BBM}, all the transformations that  we shall construct in this paper are of this type (with $ j, j_1, j_2 \\neq 0 $ because\nthey act on the phase space $ H^1_0 (\\mathbb T_x) $).\nThis observation allows to interpret the conjugacy procedure from a dynamical point of view, see \\cite{BBM}-section 2.2. \nLet us fix some terminology.\n\n\\begin{definition}\\label{operatore Hamiltoniano}\nWe say that:\n\nthe operator \n$(A h)(\\varphi, x) := A(\\varphi) h(\\varphi, x)$ is  {\\sc symplectic} if each \n$ A (\\varphi ) $, $ \\varphi \\in \\mathbb T^\\nu $,  is  a symplectic map of the phase space\n(or of a symplectic subspace like $ H_S^\\bot $);\n\n\nthe operator\n$\n\\omega \\cdot \\partial_{\\varphi} - \\partial_x G( \\varphi )  $ is  {\\sc Hamiltonian} \nif each $ G (\\varphi) $, $  \\varphi \\in \\mathbb T^\\nu $, is  symmetric; \n\nan operator is {\\sc real} if it maps real-valued functions into real-valued functions.\n\\end{definition}\n\n\nAs well known, a Hamiltonian operator $  \\omega \\cdot \\partial_{\\varphi} - \\partial_x G( \\varphi ) $ is transformed,\nunder a symplectic map $ {\\cal A} $, \ninto another  Hamiltonian operator $ \\omega \\cdot \\partial_{\\varphi} - \\partial_x E( \\varphi ) $, \nsee e.g. \\cite{BBM}-section 2.3.\n\n\\smallskip\n\nWe conclude this preliminary section recalling the following well known lemmata, see Appendix of \\cite{BBM}.\n\n\\begin{lemma} {\\bf (Composition)}\n\\label{lemma:composition of functions, Moser}\nAssume $ f \\in C^s (\\mathbb T^d \\times B_1)$, $B_1 := \\{ y \\in \\mathbb R^m  : \n|y| \\leq 1 \\}$. Then \n$ \\forall u \\in H^{s}(\\mathbb T^d, \\mathbb R^m) $ such that $ \\| u \\|_{L^\\infty} < 1  $, \nthe composition operator $\\tilde{f}(u)(x) := f(x, u(x))$ satisfies\n$ \\| \\tilde f(u) \\|_s \\leq C \\| f \\|_{C^s} (\\|u\\|_{s} + 1)  $\nwhere the constant $C $ depends on $ s ,d $.  If $ f \\in C^{s+2} $  \nand $ \\| u + h \\|_{L^\\infty} < 1$, then    \n\\begin{align*}\n\\big\\| \\tilde f(u+h) - {\\mathop\\sum}_{i = 0}^k \\frac{\\tilde{f}^{(i)}(u)}{i !} [h^i] \\big\\|_s\n& \\leq C \\| f \\|_{C^{s+ 2}} \\, \\| h \\|_{L^\\infty}^k ( \\| h \\|_{s} + \\| h \\|_{L^\\infty} \\| u \\|_{s}) \\, , \\quad k = 0, 1 \\, . \n\\end{align*}\nThe previous statement also holds replacing \n$\\| \\ \\|_s$  \nwith the norms $| \\ |_{s, \\infty} $.  \n\\end{lemma}\n\n\\begin{lemma} \\label{lemma:standard Sobolev norms properties}\n{\\bf (Tame product).} \nFor $s \\geq s_0 > d\/2  $, \n$$\n\\| uv \\|_s \\leq C(s_0) \\|u\\|_s \\|v\\|_{s_0} + C(s) \\|u\\|_{s_0} \\| v \\|_s \\, , \n\\quad \\forall u,v \\in H^s(\\mathbb T^d) \\, .\n$$\nFor $s \\geq 0$, $s \\in \\mathbb N$,\n$$\n\\| uv \\|_s \\leq \\tfrac32 \\,  \\| u \\|_ {L^\\infty}  \\| v \\|_s + C(s) \\| u \\|_ {W^{s, \\infty}} \\| v \\|_0  \\, , \n\\quad \\forall u \\in W^{s,\\infty}(\\mathbb T^d) \\, , \\ v \\in H^s(\\mathbb T^d)   \\, .\n$$\nThe above inequalities also  hold for \nthe norms $\\Vert \\  \\Vert_s^{{\\rm Lip}(\\gamma)}$.\n\\end{lemma} \n\n\n\\begin{lemma} {\\bf (Change of variable)}  \\label{lemma:utile} \nLet $p \\in W^{s,\\infty} (\\mathbb T^d,\\mathbb R^d) $,  $ s \\geq 1$, with \n$ \\| p \\|_{W^{1, \\infty}} \\leq 1\/2 $.  Then the function  $f(x) = x + p(x)$\nis invertible, with inverse  $ f^{-1}(y)  = y + q(y)$ where\n $q \\in W^{s,\\infty}(\\mathbb T^d,\\mathbb R^d)$, and \n $ \\| q \\|_{W^{s, \\infty}}  \\leq C \\| p \\|_{ W^{s, \\infty}} $.\nIf, moreover,  $p = p_\\omega $ depends in a Lipschitz way on \na parameter $\\omega \\in \\Omega \\subset \\mathbb R^\\nu $, \nand \n$ \\| D_x p_\\omega \\|_ {L^\\infty}  \\leq 1\/2 $, $ \\forall \\omega $, \nthen\n$ \\| q \\|_{W^{s, \\infty}}^{{\\rm Lip}(\\gamma)} \\leq  C  \\| p \\|_{W^{s+1, \\infty}}^{{\\rm Lip}(\\gamma)} $.\nThe constant $C := C (d, s) $ is independent of $\\gamma$.\n\nIf $u \\in H^s (\\mathbb T^d,\\mathbb C)$, then $ (u\\circ f)(x) := u(x+p(x))$ satisfies \n\\begin{align*}\n\\| u \\circ f \\|_s \n& \\leq  C (\\|u\\|_s + \\| p \\|_{W^{s, \\infty}} \\|u\\|_1),\n\\quad\n\\| u \\circ f - u \\|_s \n\\leq C ( \\| p \\|_{L^\\infty} \\| u \\|_{s + 1}  + \\| p \\|_{W^{s, \\infty}} \\| u \\|_{2} ) ,\n\\\\\n\\| u \\circ f \\|_{s}^{{{\\rm Lip}(\\gamma)}}\n& \\leq C  \\, \n\\big( \\| u \\|_{s+1}^{{{\\rm Lip}(\\gamma)}} + \\| p \\|_{W^{s, \\infty}}^{{\\rm Lip}(\\gamma)}\\| u \\|_2^{{\\rm Lip}(\\gamma)} \\big). \\nonumber\n\\end{align*}\nThe function  $u \\circ f^{-1} $  satisfies the same bounds.\n\\end{lemma}\n\n\n\n\\section{Weak Birkhoff normal form}\\label{sec:WBNF}\n\nThe Hamiltonian\nof the perturbed KdV equation \\eqref{kdv quadratica} is  $ H = H_2 + H_3 + H_{\\geq 5} $ (see \\eqref{Ham in intro})  where \n\\begin{equation} \\label{H iniziale KdV}\nH_2 (u):=\\frac{1}{2} \\int_{\\mathbb T} u_x^{2} \\, dx \\, , \\quad \nH_3(u) := \\int_\\mathbb T u^3 dx \\, , \\quad \nH_{\\geq 5}(u) := \\int_\\mathbb T f(x, u,u_x) dx \\,,\n\\end{equation}\nand $f$ satisfies \\eqref{order5}. \nAccording to the splitting \\eqref{u = v + z} $ u = v + z $, $ v \\in H_S $, $ z \\in H_S^\\bot $, \nwe have\n\\begin{equation}\\label{prima v z}\nH_2(u) = \\int_\\mathbb T \\frac{v_x^2}{2}\\, dx + \\int_\\mathbb T \\frac{z_x^2}{2} \\, dx, \n\\quad \nH_3 (u) =  \\int_{\\mathbb T}  v^3 dx + 3 \\int_{\\mathbb T}  v^2 z dx \n+ 3\\int_{\\mathbb T}  v z^2  dx  + \\int_{\\mathbb T}  z^3 dx  \\, . \n\\end{equation}\nFor a finite-dimensional space\n\\begin{equation} \\label{def E finito}\nE := E_{C} :=  \\mathrm{span} \\{ e^{{\\mathrm i}  jx} :  0 < |j| \\leq C \\}, \\quad C > 0,\n\\end{equation}\nlet $\\Pi_E $ denote the corresponding $ L^2 $-projector on $E$.\n\nThe notation $R(v^{k-q} z^q)$ indicates a homogeneous polynomial of degree $k$ in $(v,z)$ of the form \n$$\nR(v^{k-q} z^q) = M[\\underbrace{v, \\ldots, v}_{(k-q)\\, \\text{times}}, \n\\underbrace{z, \\ldots, z}_{q \\, \\text{times}} \\,], \\qquad \nM = k\\text{-linear} \\, .\n$$\n\\begin{proposition} \\label{prop:weak BNF}  \n{\\bf (Weak Birkhoff normal form)} Assume Hypothesis $ ({\\mathtt S}2) $. \nThen there exists an  analytic invertible symplectic transformation \nof the phase space $ \\Phi_B : H^1_0 (\\mathbb T_x) \\to H^1_0 (\\mathbb T_x) $ \nof the form \n\\begin{equation} \\label{finito finito}\n\\Phi_B(u) = u + \\Psi(u), \n\\quad\n\\Psi(u) = \\Pi_E \\Psi(\\Pi_E u),\n\\end{equation}\nwhere $ E $ is a finite-dimensional space as in \\eqref{def E finito}, such that the transformed Hamiltonian is\n\\begin{equation} \\label{widetilde cal H}\n{\\cal H} := H \\circ \\Phi_B  \n= H_2 + \\mathcal{H}_3 + \\mathcal{H}_4 + {\\cal H}_5 + {\\cal H}_{\\geq 6} \\,,\n\\end{equation}\nwhere $H_2$ is defined in \\eqref{H iniziale KdV}, \n\\begin{equation} \\label{H3tilde} \n\\mathcal{H}_3 := \\int_{\\mathbb T} z^3\\,dx + 3 \\int_{\\mathbb T} v z^2 \\,dx \\,, \\quad \n\\mathcal{H}_4 := - \\frac32 \\sum_{j \\in S} \\frac{|u_j|^4}{j^2} + \\mathcal{H}_{4,2} + \\mathcal{H}_{4,3} \\,, \\quad \n{\\cal H}_5 := \\sum_{q=2}^5 R(v^{5-q} z^q)\\,,\n\\end{equation} \n\\begin{equation} \\label{mH3 mH4}\n\\mathcal{H}_{4,2} := 6 \\int_\\mathbb T  v z \\Pi_S \\big((\\partial_x^{-1} v)(\\partial_x^{-1} z) \\big)\\,dx + 3 \\int_\\mathbb T z^2 \\pi_0 (\\partial_x^{-1} v)^2 \\,dx\\,,  \\quad \n\\mathcal{H}_{4, 3} := R(v z^3) \\,,  \n\\end{equation}\nand ${\\cal H}_{\\geq 6}$ collects all the terms of order at least six in $(v,z)$. \n\\end{proposition}\n\nThe rest of this section is devoted to the proof of Proposition \\ref{prop:weak BNF}.\n\n\n\nFirst, we remove the cubic terms $ \\int_{\\mathbb T} v^3 + 3 \\int_{\\mathbb T} v^2 z $ from the Hamiltonian $ H_3 $ defined in \\eqref{prima v z}.\nIn the Fourier coordinates \\eqref{Fourier}, we have\n\\begin{equation}\\label{H3 Fourier}\nH_2  = \\frac12 \\sum_{j \\neq 0} j^2 |u_j|^2, \n\\quad H_3  = \\sum_{j_1 + j_2 + j_3 = 0} u_{j_1} u_{j_2} u_{j_3} \\, .\n\\end{equation}\nWe look for  a symplectic transformation $ \\Phi^{(3)}  $ of the phase space \nwhich eliminates the monomials $ u_{j_1} u_{j_2} u_{j_3} $ of  $ H_3 $ with at most {\\it one} index\noutside $ S $. Note that, by the relation $ j_1 + j_2 + j_3 = 0 $, they are {\\it finitely} many.\nWe look for $\\Phi^{(3)} := (\\Phi^t_{F^{(3)}})_{|t=1}$ as the time-1 flow map generated by the Hamiltonian vector field $X_{F^{(3)}}$, with an auxiliary Hamiltonian of the form\n$$\nF^{(3)}(u) := \\sum_{j_1 + j_2 + j_3 = 0} F^{(3)}_{j_1 j_2 j_3} u_{j_1} u_{j_2} u_{j_3} \\,.\n$$\nThe transformed Hamiltonian is \n\\begin{align}\nH^{(3)} & := H \\circ \\Phi^{(3)} \n=  H_2 + H_3^{(3)}  +  H_4^{(3)}  +  H_{\\geq 5}^{(3)}  \\,,\n\\notag \\\\ \n\\label{H tilde 234}\nH_3^{(3)} & = H_3 + \\{ H_2, F^{(3)} \\}, \\quad \n H_4^{(3)}  = \\frac12 \\{ \\{ H_2, F^{(3)}\\}, F^{(3)}\\} + \\{H_3, F^{(3)} \\} , \n\\end{align}\nwhere $ H_{\\geq 5}^{(3)} $ collects all the terms of order at least five in $(u,u_x)$.\nBy \\eqref{H3 Fourier} and \\eqref{PoissonBr}  we calculate \n$$\nH_3^{(3)}  = \n\\sum_{j_1 + j_2 + j_3 = 0} \n\\big\\{ 1 - {\\mathrm i}  (j_1^3 + j_2^3 + j_3^3) F^{(3)}_{j_1 j_2 j_3} \\big\\} \\, u_{j_1} u_{j_2} u_{j_3} \\,.\n$$\nHence, in order to eliminate the monomials with at most one index outside $ S $, \nwe choose\n\\begin{equation}\\label{F3q}\nF^{(3)}_{j_1 j_2 j_3} := \\begin{cases}\n \\dfrac{1}{{\\mathrm i}  (j_1^3 + j_2^3 + j_3^3)} & \\text{if} \\,\\,(j_1,j_2,j_3) \\in {\\cal A}\\,, \n\\\\\n0 & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere ${\\cal A} := \\big\\{ (j_1 , j_2 , j_3) \\in (\\mathbb Z \\setminus \\{ 0 \\})^3$ : $j_1 + j_2 + j_3 = 0$, \n$j_1^{3} + j_2^{3} + j_3^{3} \\neq 0$, and at least 2 among $j_1 , j_2 , j_3$ belong to $S \\big\\}$.  \nNote that \n\\begin{equation} \\label{def calA quadratica}\n{\\cal A} = \\big\\{ (j_1 , j_2 , j_3) \\in (\\mathbb Z \\setminus \\{ 0 \\})^3 : \nj_1 + j_2 + j_3 = 0, \\, \\text{and at least 2 among}\\,\\, j_1 , j_2 , j_3 \\,\\, \n\\text{belong to} \\, S  \\big\\} \n\\end{equation}\nbecause of the elementary relation\n\\begin{equation}\\label{prodottino}\nj_1 + j_2 + j_3 = 0 \\quad \\Rightarrow \\quad j_1^3 + j_2^3 + j_3^3 = 3 j_1 j_2 j_3 \\neq 0 \n\\end{equation}\nbeing $ j_1, j_2, j_3 \\in \\mathbb Z \\setminus \\{ 0 \\}$. \nAlso note that $ \\mathcal{A} $ is a finite set, actually $ \\mathcal{A} \\subseteq [- 2 C_S, 2 C_S]^{3} $ where \nthe tangential sites $ S \\subseteq [- C_S, C_S ]$.\nAs a consequence, the Hamiltonian vector field $ X_{F^{(3)}} $  has finite rank and vanishes outside  \nthe finite dimensional subspace $ E := E_{2 C_S}$ (see \\eqref{def E finito}), namely\n$$\n X_{ F^{(3)}}(u)  = \\Pi_E X_{ F^{(3)}} ( \\Pi_E u ) \\, . \n$$\nHence its flow $ \\Phi^{(3)} : H^1_0 (\\mathbb T_x) \\to H^1_0 (\\mathbb T_x) $ has the form  \\eqref{finito finito} and it is analytic. \n\nBy construction, all the monomials of $ H_3 $   \nwith at least two indices outside $ S $ \nare not modified by the transformation $ \\Phi^{(3)}  $. Hence  (see \\eqref{prima v z}) we have  \n\\begin{equation}\\label{lem:H3tilde}\nH_3^{(3)}  = \\int_{\\mathbb T} z^3\\,dx + 3 \\int_{\\mathbb T} v z^2 \\,dx \\, .\n\\end{equation}\nWe now compute the fourth order term\n$ H_4^{(3)}  =  \\sum_{i=0}^4 H_{4,i}^{(3)} $   in \\eqref{H tilde 234}, \nwhere $ H_{4,i}^{(3)} $ is of type $ R( v^{4-i} z^i )$.\n\n\\begin{lemma}\nOne has (recall the definition \\eqref{def pi 0} of $ \\pi_0 $) \n\\begin{equation}\\label{Htilde41} \n{H}_{4,0} ^{(3)}   := \\frac32 \\int_\\mathbb T v^2 \\pi_0[(\\partial_x^{-1} v)^2] dx \\, ,  \\quad\nH_{4,2}^{(3)} := 6 \\int_\\mathbb T v z \\Pi_S \\big((\\partial_x^{-1} v)(\\partial_x^{-1} z) \\big)\\,dx \n+ 3 \\int_\\mathbb T z^2 \\pi_0 [(\\partial_x^{-1} v)^2] dx \\, . \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe write $ H_3 = H_{3, \\leq 1} +  H_3^{(3)}  $ where $ H_{3, \\leq 1}(u)  := \\int_\\mathbb T v^3 dx  + 3 \\int_\\mathbb T  v^2 z \\,  dx  $. \nThen, by \\eqref{H tilde 234}, we get\n\\begin{equation} \\label{grado 4 *}\nH_4^{(3)} \n=  \\frac12 \t\\big\\{ H_{3, \\leq 1} \\, , F^{(3)} \\big\\} + \\{ H_3^{(3)} , F^{(3)} \\} \\,.\n\\end{equation} \nBy \\eqref{F3q},  \\eqref{prodottino}, the auxiliary Hamiltonian may be written as\n$$\nF^{(3)} (u) \n= - \\frac{1}{3}\\sum_{(j_1, j_2, j_3) \\in {\\cal A}}  \n\\frac{u_{j_1} u_{j_2} u_{j_3}}{ ({\\mathrm i}  j_1) ( {\\mathrm i}  j_2) ( {\\mathrm i}  j_3)}  \n= - \\frac{1}{3}\\int_\\mathbb T (\\partial_x^{-1} v)^3 dx  - \\int_\\mathbb T (\\partial_x^{-1} v)^2 (\\partial_x^{-1} z) dx \\, . \n$$\nHence, using that the projectors $\\Pi_S$, $\\Pi_S^\\bot $ are self-adjoint and $\\partial_x^{-1}$ is skew-selfadjoint, \n\\begin{equation} \\label{grad F3 formula}\n\\nabla F^{(3)}(u) \n= \\partial_x^{-1}\\big\\{ (\\partial_x^{-1} v)^2 + 2 \\Pi_S \\big[ (\\partial_x^{-1} v)(\\partial_x^{-1} z)\\big] \\big\\}\n\\end{equation}\n(we have used that $\\partial_x^{-1} \\pi_0 = \\partial_x^{-1}$ be the definition of $\\partial_x^{-1}$). \nRecalling the Poisson bracket definition \\eqref{Poisson bracket}, \nusing  that $ \\nabla H_{3, \\leq 1}(u) = 3 v^2 + 6 \\Pi_S(v z) $ and   \\eqref{grad F3 formula},\nwe get \n\\begin{align}\n\\{ H_{3, \\leq1}, F^{(3)} \\} \n& = \\int_\\mathbb T \\big\\{ 3 v^2 + 6 \\Pi_S(v z) \\big\\} \\pi_0 \\big\\{ (\\partial_x^{-1} v)^2 + \n2 \\Pi_S \\big[ (\\partial_x^{-1} v)(\\partial_x^{-1} z)\\big]  \\big\\}\\,dx  \n\\nonumber\n\\\\\n& = 3 \\int_\\mathbb T v^2 \\pi_0 (\\partial_x^{-1} v)^2\\,dx  +\n 12 \\int_\\mathbb T \\Pi_S(v z) \\Pi_S [ (\\partial_x^{-1} v)(\\partial_x^{-1} z) ]\\,dx + R(v^3 z) \\, .  \\label{H31F}\n\\end{align}\nSimilarly, since $ \\nabla  H_3^{(3)}(u) = 3 z^2 + 6 \\Pi_S^\\bot (v z) $, \n\\begin{equation}\n\\{ H_3^{(3)}, F^{(3)} \\}  \n=  \n3 \\int_\\mathbb T z^2 \\pi_0 (\\partial_x^{-1} v)^2\\,dx  \n+ R(v^3 z) + R(v z^3) \\,.  \\label{H3tildeF}\n\\end{equation}\nThe lemma follows by \\eqref{grado 4 *}, \\eqref{H31F}, \\eqref{H3tildeF}.\n\\end{proof}\n\nWe now construct a symplectic map $ \\Phi^{(4)}  $ \nsuch that  the Hamiltonian system obtained transforming  $ H_2 + H_3^{(3)} +  H_4^{(3)} $ \npossesses the invariant subspace $ H_S $  (see \\eqref{splitting S-S-bot}) \nand its dynamics on $ H_S $ is integrable and non-isocronous.  \nHence we have to eliminate the term $ H_{4,1}^{(3)} $ \n(which is linear in $ z $),\nand to normalize  $ H_{4,0}^{(3)} $ (which is independent of $ z $). \nWe need the following elementary lemma (Lemma 13.4 in \\cite{KaP}).\n\n\\begin{lemma}  \\label{lemma:interi} \nLet $j_1, j_2, j_3, j_4 \\in \\mathbb Z $ such that $ j_1 + j_2 + j_3 + j_4 = 0 $. \nThen \n$$\nj_1^3 + j_2^3 + j_3^3 + j_4^3 = -3 (j_1 + j_2) (j_1 + j_3) (j_2 + j_3).\n$$\n\\end{lemma}\n\n\n\\begin{lemma}\nThere exists a symplectic transformation $ \\Phi^{(4)}  $ of the form \\eqref{finito finito}\nsuch that\n\\begin{equation}\\label{Ham4quadratica}\nH^{(4)} := H^{(3)} \\circ \\Phi^{(4)} = \nH_2 + H_3^{(3)} + H_4^{(4)} + H^{(4)}_{\\geq 5} \\,, \n\\qquad \nH^{(4)}_4 := - \\frac32 \\sum_{j \\in S} \\frac{|u_j|^4}{j^2}   +  H_{4,2}^{(3)} + H_{4,3}^{(3)} \\,,\n\\end{equation}\nwhere  $ H_3^{(3)} $ is defined in \\eqref{lem:H3tilde},\n$ H_{4,2}^{(3)} $ in \\eqref{Htilde41}, $ H_{4,3}^{(3)} = R( v z^3) $ \nand  $ H_{\\geq 5}^{(4)} $ collects all the terms of degree at least five in $(u,u_x)$.\n\\end{lemma}\n\n\\begin{proof}\nWe look for a map $\\Phi^{(4)} := (\\Phi_{F^{(4)}}^t)_{|t=1}$ which is the time $ 1$-flow map \nof an auxiliary Hamiltonian\n$$\nF^{(4)}(u) := \\sum_{\\begin{subarray}{c}\nj_1 + j_2 + j_3 + j_4 = 0 \\\\\n\\text{at least}\\,\\,3\\,\\,\\text{indices are in}\\,\\,S\n\\end{subarray}} F^{(4)}_{j_1 j_2 j_3 j_4} u_{j_1} u_{j_2} u_{j_3} u_{j_4} \n$$\nwith the same form of the Hamiltonian $ H_{4,0}^{(3)} + H_{4,1}^{(3)} $.\nThe transformed Hamiltonian is\n\\begin{equation}\\label{callH}\nH^{(4)} :=  H^{(3)} \\circ \\Phi^{(4)} \n= H_2 + H_3^{(3)} + H_4^{(4)} +  H_{\\geq 5}^{(4)}, \\quad \nH_4^{(4)} = \\{ H_2, F^{(4)} \\} + H_4^{(3)} , \n\\end{equation}\nwhere $ H_{\\geq 5}^{(4)} $ collects all the terms of order at least five. \nWe write $ H_4^{(4)} = \\sum_{i=0}^4 H_{4,i}^{(4)} $ where each $ H_{4,i}^{(4)} $ if of type $ R( v^{4-i} z^i ) $. \nWe choose the coefficients \n\\begin{equation}\\label{def F4}\nF^{(4)}_{j_1 j_2 j_3 j_4} := \\begin{cases}\n \\dfrac{ H^{(3)}_{j_1j_2j_3j_4}   }{{\\mathrm i}  (j_1^3 + j_2^3 + j_3^3+ j_4^3)} & \\text{if} \\,\\,(j_1,j_2,j_3, j_4) \\in {\\cal A}_4\\,, \n\\\\\n0 & \\text{otherwise},\n\\end{cases}\n\\end{equation}\nwhere \n\\begin{align*} \n{\\cal A}_4  := \\big\\{ (j_1 , j_2 , j_3, j_4) \\in (\\mathbb Z \\setminus \\{ 0 \\})^4 & : \nj_1 + j_2 + j_3 + j_4 = 0, \\,  j_1^{3} + j_2^{3} + j_3^{3} + j_4^{3} \\neq 0, \\\\\n& \\quad  \\text{and at most one among}\\,\\, j_1 , j_2 , j_3, j_4 \\,\\, \\text{outside} \\, S \\big\\}  \\, .  \n\\end{align*}\nBy this definition  $ H_{4,1}^{(4)}=  0 $\nbecause there exist no integers $ j_1, j_2 , j_3 \\in S$, $ j_4 \\in S^c $ satisfying \n$ j_1 + j_2 + j_3 + j_4 = 0 $, $ j_1^3 + j_2^3 + j_3^3 + j_4^3 = 0 $, by Lemma \\ref{lemma:interi} and the\nfact that $ S $ is symmetric. \nBy construction, the terms $ H_{4,i}^{(4)}=  H_{4,i}^{(3)} $, $ i = 2, 3, 4$, are not changed by $ \\Phi^{(4)} $.\nFinally, by \\eqref{Htilde41} \n\\begin{equation} \\label{H2F1}\nH_{4,0}^{(4)}=  \\frac32 \n\\sum_{\\begin{subarray}{c}\nj_1, j_2, j_3, j_4  \\in S \\\\\nj_1 + j_2 + j_3 + j_4 = 0 \\\\\nj_1^3 + j_2^3 + j_3^3 + j_4^3 = 0 \\\\\nj_1 + j_2 \\,,\\,j_3 + j_4 \\neq 0\n\\end{subarray}} \\frac{1}{({\\mathrm i}  j_3) ({\\mathrm i}  j_4)}u_{j_1} u_{j_2} u_{j_3} u_{j_4} \\, . \n\\end{equation}\nIf $ j_1 + j_2 + j_3 + j_4 = 0 $ and $ j_1^3 + j_2^3 + j_3^3 + j_4^3 = 0$, then $(j_1 + j_2)(j_1 + j_3)(j_2 + j_3) = 0$ by Lemma \\ref{lemma:interi}. \nWe develop the  sum in \\eqref{H2F1} with respect to the first index $j_1$.  Since $ j_1 + j_2 \\neq 0 $\nthe possible cases are:\n\\[\n(i) \\ \\big\\{ \nj_2 \\neq - j_1, \\ \nj_3 = - j_1, \\\nj_4 = - j_2 \\big\\}   \n\\qquad \\text{or} \\qquad\n(ii) \\ \\big\\{\nj_2 \\neq - j_1, \\ \nj_3 \\neq - j_1, \\ \nj_3 = - j_2, \\ \nj_4 = - j_1 \\big\\} . \n\\]\nHence, using $ u_{-j} = \\bar{u}_j $ (recall \\eqref{Fourier}), and since $S$ is symmetric, we have\n\\begin{equation} \\label{case ii}\n\\sum_{(i)}  \\frac{1}{j_3 j_4} u_{j_1} u_{j_2} u_{j_3} u_{j_4} \n= \\sum_{j_1, j_2 \\in S, j_2 \\neq - j_1}  \\frac{|u_{j_1}|^2 |u_{j_2}|^2 }{j_1 j_2} \n= \\sum_{j,j' \\in S}  \\frac{|u_j|^2 |u_{j'}|^2}{j j'}  + \\sum_{j \\in S}  \\frac{|u_j|^4}{j^2}   = \\sum_{j \\in S}  \\frac{|u_j|^4}{j^2}\\,,    \n\\end{equation}\nand in the second case ($ii$) \n\\begin{equation} \\label{case iii}\n\\sum_{(ii)} \\frac{1}{j_3 j_4}  u_{j_1} u_{j_2} u_{j_3} u_{j_4} \n = \\sum_{j_1, j_2, j_2 \\neq \\pm j_1} \\frac{1}{j_1 j_2}  u_{j_1} u_{j_2} u_{-j_2} u_{-j_1}  =\n  \\sum_{j \\in S} \\frac{1}{j}  |u_{j}|^2 \\Big( \\sum_{j_2 \\neq \\pm j} \\frac{1}{j_2}  |u_{j_2}|^2 \\Big) = 0\\,.   \n\\end{equation}\nThen \\eqref{Ham4quadratica} follows by \\eqref{H2F1}, \\eqref{case ii}, \\eqref{case iii}. \n\\end{proof}\n\nNote that the Hamiltonian $ H_2 + H_3^{(3)} + H_4^{(4)} $ (see \\eqref{Ham4quadratica}) possesses\nthe invariant subspace $ \\{ z =  0 \\} $ and the system restricted to $ \\{ z =  0 \\} $ is\ncompletely integrable and non-isochronous (actually it is formed by $ \\nu $ decoupled rotators). \nWe shall construct  quasi-periodic solutions which bifurcate from this invariant manifold. \n\nIn order to enter in a perturbative regime, \nwe have to eliminate further monomials of $ H^{(4)} $ in  \\eqref{Ham4quadratica}.   \nThe minimal requirement for  the convergence of the nonlinear  Nash-Moser iteration is to \neliminate the monomials $ R(v^5) $ and $ R(v^4 z)$.\nHere we need the choice of the sites of Hypothesis $ ({\\mathtt S}2) $. \n\n\\begin{remark}\\label{remark:cubic}\nIn the KAM theorems  \\cite{k1}, \\cite{Po3} (and \\cite{PP}, \\cite{Wang}),\nas well as for the perturbed mKdV equations \\eqref{mKdV}, \nthese further steps of Birkhoff normal form are not required because \nthe nonlinearity of the original PDE is yet cubic. A difficulty of KdV\nis that the nonlinearity is quadratic.  \n\\end{remark}\n\nWe spell out Hypothesis $( {\\mathtt S}2)$ as follows:\n\\begin{itemize}\n\\item {\\sc $( {\\mathtt S}2_0)$.} \nThere is no choice of $ 5 $ integers $ j_1, \\ldots, j_5 \\in  S $ such that \n\\begin{equation}\\label{scelta dei siti grado 7}\nj_1 + \\ldots + j_5 = 0\\,,\\quad \nj_1^3 + \\ldots + j_5^3 = 0 \\,.\n\\end{equation} \n\\item {\\sc  $({\\mathtt S}2_1)$.} \nThere is no choice of $4 $ integers $j_1, \\ldots, j_{4} $ in  $ S $ \nand an integer in the complementary set $ j_5 \\in S^c :=  (\\mathbb Z \\setminus \\{0\\}) \\setminus S   $ such that  \\eqref{scelta dei siti grado 7} holds. \n\\end{itemize}\n\nThe homogeneous component of degree $ 5 $ of $H^{(4)}$ is\n$$\nH^{(4)}_5 (u) = \\sum_{j_1+ \\ldots + j_5 = 0} H^{(4)}_{j_1, \\ldots, j_5}  u_{j_1} \\ldots u_{j_5} \\,.\n$$\nWe want to remove from $H^{(4)}_5$ the terms with at most one index among $ j_1, \\ldots , j_5 $ outside $ S $.\nWe consider the auxiliary Hamiltonian \n\\begin{equation} \\label{Fn odd}\nF^{(5)} = \\sum_{\\begin{subarray}{c} j_1 + \\ldots + j_5 = 0 \\\\ \n\\text{at most one index outside $ S $}\n\\end{subarray} } \nF_{j_1 \\ldots j_5}^{(5)} u_{j_1} \\ldots u_{j_5} \\, , \\quad \n F_{j_1 \\ldots j_5}^{(5)} := \\frac{H_{j_1 \\ldots j_5}^{(5)}}{ {\\mathrm i}  (j_1^3 + \\ldots + j_5^3)} \\,.\n\\end{equation}\nBy Hypotheses $ ({\\mathtt S}2_0), ({\\mathtt S}2_1) $, if $ j_1 + \\ldots + j_5 = 0 $ with at most one index outside $ S $ then \n$ j_1^3 + \\ldots + j_5^3 \\neq 0 $ and  \n$ F^{(5)}$ is well defined. \nLet $ \\Phi^{(5)} $ be the time $ 1 $-flow generated by \n$ X_{F^{(5)}} $.  The new Hamiltonian is\n\\begin{equation}\\label{callHn}\nH^{(5)} :=  H^{(4)} \\circ \\Phi^{(5)} \n= H_2 + H_3^{(3)} + H_4^{(4)} +  \\{ H_2, F^{(5)} \\} + H_5^{(4)} + H^{(5)}_{\\geq 6}  \n\\end{equation}\nwhere, by  \\eqref{Fn odd}, \n$$\nH_5^{(5)} := \\{ H_2, F^{(5)} \\} + H_5^{(4)} = {\\mathop\\sum}_{q = 2}^5 R(v^{5 - q} z^q) \\, .\n$$\n\n\nRenaming $ {\\cal H} := H^{(5)} $, namely $ {\\cal H}_n := H^{(n)}_n $, $ n =3, 4, 5 $, and \nsetting $ \\Phi_B := \\Phi^{(3)} \\circ \\Phi^{(4)}  \\circ \\Phi^{(5)} $,  formula \\eqref{widetilde cal H} follows. \n\nThe homogeneous component $ H^{(4)}_{5} $ preserves the momentum, see section \\ref{sec:Ham For}.  \nHence $F^{(5)} $ also preserves the momentum. \nAs a consequence, also $ H^{(5)}_k $, $ k \\leq 5 $, preserve the momentum. \n\n\nFinally, since $F^{(5)} $  is Fourier-supported on a finite set, \nthe transformation $\\Phi^{(5)}$ is of type \\eqref{finito finito} (and analytic), \nand therefore also the composition $ \\Phi_B $ is of type \\eqref{finito finito} (and analytic). \n\n\\section{Action-angle variables}\\label{sec:4}\n\n\nWe now introduce action-angle variables on the tangential directions by the change of coordinates\n\\begin{equation}\\label{coordinate azione angolo}\n\\begin{cases}\nu_j  := \\sqrt{\\xi_j + |j| y_j} \\, e^{{\\mathrm i}  \\theta_j}, \\qquad & \\text{if} \\  j \\in S\\,,\\\\\nu_j  := z_j, \\qquad  & \\text{if} \\  j \\in S^c \\, , \n\\end{cases}\n\\end{equation}\nwhere (recall $ u_{-j} = {\\overline u}_j $)\n\\begin{equation}\\label{simmeS}\n\\xi_{-j} = \\xi_j  \\, , \\quad \n\\xi_j > 0 \\, , \\quad \ny_{-j} = y_j \\, , \\quad \n\\theta_{-j} = - \\theta_j \\, , \\quad \n\\theta_j, \\, y_j \\in \\mathbb R \\, , \\quad  \n\\forall j \\in S \\,.\n\\end{equation}\nFor the tangential sites $ S^+ := \\{ {\\bar \\jmath_1}, \\ldots, {\\bar \\jmath_\\nu} \\} $ we shall also denote \n$ \\theta_{\\bar \\jmath_i} := \\theta_i $, $ y_{\\bar \\jmath_i} := y_i $, \n$ \\xi_{\\bar \\jmath_i} := \\xi_i $,  $ i =1, \\ldots \\, \\nu $.\n\nThe symplectic 2-form $ \\Omega $ in \\eqref{2form0} (i.e.  \\eqref{2form KdV}) becomes \n\\begin{equation}\\label{2form}\n{\\cal W} := \\sum_{i=1}^\\nu  d \\theta_i \\wedge d y_i  \n+ \\frac12 \\sum_{j \\in S^c \\setminus \\{ 0 \\} } \\frac{1}{{\\mathrm i}  j} \\, d z_j \\wedge d z_{-j} =  \n\\big( \\sum_{i=1}^\\nu  d \\theta_i \\wedge d y_i \\big)  \\oplus \\Omega_{S^\\bot} = d \\Lambda \n\\end{equation}\nwhere $ \\Omega_{S^\\bot} $ denotes the restriction of $ \\Omega $ to $ H_S^\\bot $ (see \\eqref{splitting S-S-bot}) and \n$ \\Lambda $ is the contact $ 1 $-form on $ \\mathbb T^\\nu \\times \\mathbb R^\\nu \\times H_S^\\bot $ defined by\n$ \\Lambda_{(\\theta, y, z)} : \\mathbb R^\\nu \\times \\mathbb R^\\nu \\times H_S^\\bot \\to \\mathbb R $, \n\\begin{equation}\\label{Lambda 1 form}\n\\Lambda_{(\\theta, y, z)}[\\widehat \\theta, \\widehat y, \\widehat z] := \n-  y \\cdot \\widehat \\theta + \\frac12 ( \\partial_x^{-1} z, \\widehat z )_{L^2 (\\mathbb T)} \\, .    \n\\end{equation}\nInstead of working in a shrinking neighborhood of the origin, it is a convenient devise to rescale the ``unperturbed actions\" $ \\xi $\nand the action-angle variables as\n\\begin{equation}\\label{rescaling kdv quadratica}\n \\xi \\mapsto \\varepsilon^2 \\xi \\, , \\quad  y \\mapsto \\varepsilon^{2b} y \\, , \\quad z \\mapsto \\varepsilon^b z  \\, . \n \\end{equation}\nThen the symplectic $ 2 $-form in \\eqref{2form} transforms into $  \\varepsilon^{2b} {\\cal W } $. Hence\nthe Hamiltonian system generated by $ {\\cal H} $ in \\eqref{widetilde cal H} \ntransforms into  the new Hamiltonian system \n\\begin{equation}  \\label{def H eps}\n \\dot \\theta = \\partial_y H_\\varepsilon (\\theta, y, z)   \\, , \\ \n \\dot y   = -  \\partial_\\theta H_\\varepsilon  (\\theta, y, z) \\, , \\  \n z_t  =  \\partial_x \\nabla_z H_\\varepsilon  (\\theta, y, z)  \\, , \\quad H_\\varepsilon := \\varepsilon^{-2b} \\mathcal{H} \\circ A_\\varepsilon\n\\end{equation}\nwhere \n\\begin{equation}  \\label{def A eps}\nA_\\varepsilon (\\theta, y, z) := \\varepsilon v_\\varepsilon(\\theta, y) + \\varepsilon^b z \n:= \\varepsilon {\\mathop \\sum}_{j \\in S} \\sqrt{\\xi_j + \\varepsilon^{2(b-1)} |j| y_j} \\, e^{{\\mathrm i}  \\theta_j} e^{{\\mathrm i}  j x} + \\varepsilon^b z \\, .\n\\end{equation}\nWe shall still denote by $ X_{H_\\varepsilon} = (\\partial_y H_\\varepsilon, - \\partial_\\theta H_\\varepsilon, \\partial_x \\nabla_z H_\\varepsilon) $ the Hamiltonian vector field in the variables $ (\\theta, y, z ) \\in \\mathbb T^\\nu \\times \\mathbb R^\\nu \\times H_S^\\bot $.\n\n\nWe now write explicitly the Hamiltonian $ H_\\varepsilon (\\theta, y, z) $ in  \\eqref{def H eps}. \nThe quadratic Hamiltonian $ H_2 $ in \\eqref{H iniziale KdV} transforms into\n\\begin{equation}\\label{shape H2}\n\\varepsilon^{-2b}H_2 \\circ A_\\varepsilon = const + \n{\\mathop \\sum}_{j \\in S^+} j^3 y_j + \\frac{1}{2} \\int_{\\mathbb T} z_x^2 \\, dx \\, , \n\\end{equation}\nand, recalling \\eqref{H3tilde}, \\eqref{mH3 mH4}, \nthe Hamiltonian $ {\\cal H }$ in \\eqref{widetilde cal H} \ntransforms into\n(shortly writing $ v_\\varepsilon := v_\\varepsilon (\\theta, y) $)  \n\\begin{align}\nH_\\varepsilon (\\theta, y, z) &  \n=  e(\\xi) + \\a (\\xi) \\cdot y + \\frac12 \\int_\\mathbb T z_x^2 dx + \\varepsilon^b\\int_\\mathbb T z^3 dx  + 3 \\varepsilon \\int_\\mathbb T v_\\varepsilon  z^2 dx \\label{formaHep}\n\\\\ & \\quad \n+ \\varepsilon^2 \\Big\\{6 \\int_\\mathbb T  v_\\varepsilon z\n \\Pi_S \\big((\\partial_x^{-1} v_\\varepsilon)(\\partial_x^{-1} z) \\big)\\,dx + 3 \\int_\\mathbb T z^2 \\pi_0 (\\partial_x^{-1} v_\\varepsilon)^2\\,dx \\Big\\} \n - \\frac32 \\varepsilon^{2b} {\\mathop \\sum}_{j \\in S} y_j^2   \\nonumber \n\\\\ & \\quad\n+ \\varepsilon^{b + 1} R(v_\\varepsilon z^3) \n+ \\varepsilon^3 R(v_\\varepsilon^3 z^2) \n+ \\varepsilon^{2+b} \\sum_{q=3}^5 \\varepsilon^{(q-3)(b-1)} R(v_\\varepsilon^{5-q} z^q) \n+ \\varepsilon^{-2b} {\\cal H}_{\\geq 6} (\\varepsilon v_\\varepsilon + \\varepsilon^b z )   \\nonumber \n\\end{align}\nwhere $e(\\xi)$ is a constant, and the frequency-amplitude map is\n\\begin{equation} \\label{mappa freq amp}\n\\a(\\xi) := \\bar\\omega + \\varepsilon^2 {\\mathbb A} \\xi  \\, , \\quad \n{\\mathbb A} := - 6 \\, \\text{diag} \\{ 1\/j \\}_{j \\in S^+} \\, .\n\\end{equation} \nWe write the Hamiltonian in \\eqref{formaHep} as \n\\begin{equation}  \\label{Hamiltoniana Heps KdV}\nH_\\varepsilon = {\\cal N} +  P \\,,   \\quad \n{\\cal N}(\\theta, y, z) \n= \\a (\\xi) \\cdot y + \\frac12 \\big(N(\\theta) z , z \\big)_{L^2(\\mathbb T)} \\,,\n\\end{equation}\nwhere  \n\\begin{align}\n\\frac12 \\big(N(\\theta) z, z \\big)_{L^2(\\mathbb T)}  & \n:= \\frac12 \\big( (\\partial_{z} \\nabla H_\\varepsilon)(\\theta, 0, 0)[z], z \\big)_{L^2(\\mathbb T)}  \\label{Nshape}  \n=  \\frac12 \\int_\\mathbb T z_x^2 dx  + 3 \\varepsilon \\int_\\mathbb T v_\\varepsilon(\\theta, 0)  z^2 dx  \\\\\n& + \\varepsilon^2 \\Big\\{6 \\int_\\mathbb T  v_\\varepsilon(\\theta, 0) z \\Pi_S \n\\big((\\partial_x^{-1} v_\\varepsilon(\\theta,0))(\\partial_x^{-1} z) \\big) dx \n+ 3 \\int_\\mathbb T z^2 \\pi_0 (\\partial_x^{-1} v_\\varepsilon(\\theta, 0))^2 dx \\Big\\} + \\ldots \\nonumber\n\\end{align}\nand $P := H_\\varepsilon - {\\cal N} $.\n\n\\section{The nonlinear functional setting}\\label{sec:functional}\n\nWe look for an embedded invariant torus \n\\begin{equation} \\label{embedded torus i}\ni : \\mathbb T^\\nu \\to \\mathbb T^\\nu \\times \\mathbb R^\\nu \\times H_S^\\bot, \\quad  \n\\varphi \\mapsto i (\\varphi) := ( \\theta (\\varphi), y (\\varphi), z (\\varphi)) \n\\end{equation}\nof the Hamiltonian vector field $ X_{H_\\varepsilon}  $ \nfilled by quasi-periodic solutions with diophantine frequency $ \\omega $. We require that $ \\omega $ belongs to the set \n\\begin{equation}\\label{Omega epsilon}\n\\Omega_\\varepsilon := \\a ( [1,2]^\\nu ) = \n\\{ \\a(\\xi) : \\xi \\in [1,2]^\\nu \\}  \n\\end{equation}\nwhere $ \\a $ is the diffeomorphism \n\\eqref{mappa freq amp}, and, in the Hamiltonian $ H_\\varepsilon $ in \\eqref{Hamiltoniana Heps KdV}, we choose \n\\begin{equation}\\label{linkxiomega}\n\\xi = \\a^{-1}(\\omega) = \\varepsilon^{-2} {\\mathbb A}^{-1} (\\omega - \\bar\\omega) \\, . \n\\end{equation}\nSince any $ \\omega \\in \\Omega_\\varepsilon$ is $ \\varepsilon^2 $-close to the integer vector\n$ \\bar \\omega $ (see \\eqref{bar omega}), we require that the constant $\\gamma$ in the diophantine inequality \n\\begin{equation}\\label{omdio}\n |\\omega \\cdot l | \\geq \\gamma \\langle l \\rangle^{-\\tau} \\, , \\ \\  \\forall l \\in \\mathbb Z^\\nu \\setminus \\{0\\}   \\, , \n\\quad \\text{satisfies} \\  \\  \\gamma = \\varepsilon^{2+a} \\quad \\text{for some} \\ a > 0 \\,.  \n\\end{equation}\nWe remark that the definition of $\\gamma$ in \\eqref{omdio} is slightly stronger than the minimal condition, which is $ \\gamma \\leq c \\varepsilon^2 $ with $ c $ small enough.\nIn addition to \\eqref{omdio} we shall also require that $ \\omega $ satisfies the first and second order Melnikov-non-resonance conditions \\eqref{Omegainfty}. \n\nWe\nlook for an embedded invariant torus \nof the modified Hamiltonian vector field $ X_{H_{\\varepsilon, \\zeta}} = X_{H_\\varepsilon} + (0, \\zeta, 0)$ which is generated by \nthe Hamiltonian \n\\begin{equation}\\label{hamiltoniana modificata}\nH_{\\varepsilon, \\zeta} (\\theta, y, z) :=  H_\\varepsilon (\\theta, y, z) + \\zeta \\cdot \\theta\\,,\\quad \\zeta \\in \\mathbb R^\\nu\\,.\n\\end{equation}\nNote that $ X_{H_{\\varepsilon, \\zeta}} $ is  periodic in $\\theta $\n(unlike $ H_{\\varepsilon, \\zeta}  $). \nIt turns out that an invariant torus for $ X_{H_{\\varepsilon, \\zeta}} $ is actually \ninvariant  for $ X_{H_\\varepsilon} $, see Lemma \\ref{zeta = 0}. \nWe introduce the parameter $\\zeta \\in \\mathbb R^\\nu $ in order to control the average in the \n$ y$-component\nof the linearized equations.\nThus we look for zeros of the nonlinear operator\n\\begin{align}\n {\\cal F} (i, \\zeta ) \n& :=   {\\cal F} (i, \\zeta, \\omega, \\varepsilon )  := {\\cal D}_\\omega i (\\varphi) - X_{H_{\\varepsilon, \\zeta}}  (i(\\varphi)) =\n{\\cal D}_\\omega i (\\varphi) - X_{\\cal N}  (i(\\varphi))  - X_P  (i(\\varphi)) + (0, \\zeta, 0 )  \\label{operatorF}  \\\\\n& \\nonumber  :=  \\left(\n\\begin{array}{c}\n{\\cal D}_\\omega \\theta (\\varphi) - \\partial_y H_\\varepsilon ( i(\\varphi)  )   \\\\\n{\\cal D}_\\omega y (\\varphi)  +  \\partial_\\theta H_\\varepsilon ( i(\\varphi)  ) + \\zeta  \\\\\n{\\cal D}_\\omega z (\\varphi) -  \\partial_x \\nabla_z H_\\varepsilon  ( i(\\varphi)) \n\\end{array}\n\\right) \\!\\! \n=  \\!\\!\n\\left(\n\\begin{array}{c} \n{\\cal D}_\\omega   \\Theta (\\varphi)  -  \\partial_y P (i(\\varphi) )   \\\\\n\\!\\! \\! {\\cal D}_\\omega  y (\\varphi)  + \\frac12  \\partial_\\theta ( N(\\theta (\\varphi)) z(\\varphi), z(\\varphi) )_{L^2(\\mathbb T)} \n\t+  \\partial_\\theta P  ( i(\\varphi) ) + \\zeta  \\!\\!\\!\\!  \\\\\n{\\cal D}_\\omega  z (\\varphi) - \\partial_x N ( \\theta  (\\varphi )) z (\\varphi)  \n-  \\partial_x \\nabla_z P  ( i(\\varphi) ) \n\\end{array} \n\\right) \n\\end{align}\nwhere $ \\Theta(\\varphi) := \\theta (\\varphi) - \\varphi $ is $ (2 \\pi)^\\nu $-periodic \nand we use the short notation \n\\begin{equation}\\label{Domega}\n{\\cal D}_\\omega := \\omega \\cdot \\partial_\\varphi \\, . \n\\end{equation}\nThe Sobolev norm of the periodic component of the embedded torus \n\\begin{equation}\\label{componente periodica}\n{\\mathfrak I}(\\varphi)  := i (\\varphi) - (\\varphi,0,0) := ( {\\Theta} (\\varphi), y(\\varphi), z(\\varphi))\\,, \\quad \\Theta(\\varphi) := \\theta (\\varphi) - \\varphi \\, , \n\\end{equation}\nis \n\\begin{equation}\\label{norma fracchia}\n\\|  {\\mathfrak I}  \\|_s := \\| \\Theta \\|_{H^s_\\varphi} +  \\| y  \\|_{H^s_\\varphi} +  \\| z \\|_s \n\\end{equation}\nwhere $ \\| z \\|_s := \\| z \\|_{H^s_{\\varphi,x}}  $ is defined in \\eqref{Sobolev coppia}.\nWe link the rescaling \\eqref{rescaling kdv quadratica}\nwith the diophantine constant $ \\gamma = \\varepsilon^{2+a} $ by choosing\n\\begin{equation}\\label{link gamma b}\n\\gamma = \\varepsilon^{2b}\\,, \\qquad\nb = 1 + ( a \\slash 2 ) \\, .\n\\end{equation}\nOther choices are possible, \nsee Remark \\ref{comm3}.\n\n\\begin{theorem}\\label{main theorem}\nLet the tangential sites $ S $ in \\eqref{tang sites} \nsatisfy\n$({\\mathtt S}1), ({\\mathtt S}2)$. \nThen, for all $ \\varepsilon \\in (0, \\varepsilon_0 ) $, where $ \\varepsilon_0 $ is small enough,   \nthere exists a Cantor-like set $ {\\cal C}_\\varepsilon \\subset \\Omega_\\varepsilon $,\nwith  asympotically full measure as $ \\varepsilon \\to 0 $, namely\n\\begin{equation}\\label{stima in misura main theorem}\n\\lim_{\\varepsilon\\to 0} \\, \\frac{|{\\cal C}_\\varepsilon|}{|\\Omega_\\varepsilon|} = 1 \\, , \n\\end{equation}\nsuch that, for all $ \\omega \\in {\\cal C}_\\varepsilon $, there exists a solution $ i_\\infty (\\varphi) := i_\\infty (\\omega, \\varepsilon)(\\varphi) $ \nof  ${\\cal D}_\\omega i_\\infty(\\varphi) - X_{H_\\varepsilon}(i_\\infty(\\varphi)) = 0 $.\nHence the embedded torus \n$ \\varphi \\mapsto i_\\infty (\\varphi) $ is invariant for the Hamiltonian vector field $ X_{H_\\varepsilon (\\cdot, \\xi)} $  \nwith $ \\xi $ as in \\eqref{linkxiomega}, and it is filled by quasi-periodic solutions with frequency $ \\omega $.\nThe torus $i_\\infty$ satisfies\n\\begin{equation}\\label{stima toro finale}\n\\|  i_\\infty (\\varphi) -  (\\varphi,0,0) \\|_{s_0 + \\mu}^{\\mathrm{Lip}(\\g)} = O(\\varepsilon^{6 - 2 b} \\gamma^{-1} ) \n\\end{equation}\nfor some $ \\mu := \\mu (\\nu) > 0 $. Moreover, the torus $ i_\\infty $ is {\\sc linearly stable}. \n\\end{theorem}\n\n\nTheorem \\ref{main theorem} is  proved in sections \\ref{costruzione dell'inverso approssimato}-\\ref{sec:NM}. It implies Theorem \\ref{thm:KdV}\nwhere the $ \\xi_j $ in \\eqref{solution u} are $ \\varepsilon^2 \\xi_j  $, $ \\xi_j \\in [1,2 ] $, in \\eqref{linkxiomega}.\nBy \\eqref{stima toro finale}, \ngoing back to the variables before the rescaling \\eqref{rescaling kdv quadratica}, \nwe get\n$ \\Theta_\\infty = O( \\varepsilon^{6-2b} \\gamma^{-1}) $, $ y_\\infty = O( \\varepsilon^6 \\gamma^{-1} ) $, $ z_\\infty = O( \\varepsilon^{6-b} \\gamma^{-1} ) $, which, \nas $ b \\to  1^+ $, tend to the expected optimal estimates. \n\n\\begin{remark} \\label{comm3}\nThere are other possible ways to link the rescaling \\eqref{rescaling kdv quadratica}\nwith the diophantine constant $ \\gamma = \\varepsilon^{2+a} $.\nThe choice $ \\gamma > \\varepsilon^{2b} $\nreduces to study perturbations of an isochronous system  (as in \\cite{Ku}, \\cite{k1}, \\cite{Po3}), and it is convenient to \n introduce $ \\xi (\\omega) $ as a variable. \nThe case  $ \\varepsilon^{2b} > \\gamma $, in particular $ b = 1 $, \nhas to be dealt with a perturbation approach  of a non-isochronous system {\\`a la} Arnold-Kolmogorov. \n\\end{remark}\n\n\nWe now give the tame estimates for the composition operator induced by the Hamiltonian vector fields $ X_{\\cal N} $ and  $ X_P $ in \\eqref{operatorF}, that we shall use in the next sections. \n\nWe first estimate the composition operator induced by $ v_\\varepsilon (\\theta, y) $  defined in \\eqref{def A eps}.\nSince the functions $ y \\mapsto \\sqrt{\\xi + \\varepsilon^{2(b - 1)}|j| y} $, $\\theta \\mapsto e^{{\\mathrm i}  \\theta}$ \nare analytic  for $\\varepsilon$ small enough and $|y| \\leq C$, \nthe composition Lemma \\ref{lemma:composition of functions, Moser} implies that, for all  $ \\Theta, y \\in H^s(\\mathbb T^\\nu, \\mathbb R^\\nu )$,  \n$   \\| \\Theta \\|_{s_0},  \\| y \\|_{s_0} \\leq 1 $, setting $\\theta(\\varphi) := \\varphi + \\Theta (\\varphi)$, \n$ \\|  v_\\varepsilon (\\theta (\\varphi) ,y(\\varphi) ) \\|_s \n\\leq_s 1 + \\| \\Theta  \\|_s + \\| y \\|_s  $. \nHence, using also \n\\eqref{linkxiomega},   the map $ A_\\varepsilon $ in \\eqref{def A eps} satisfies, for all \n$   \\| {\\mathfrak I} \\|_{s_0}^{\\mathrm{Lip}(\\g)}  \\leq 1 $ (see \\eqref{componente periodica})\n\\begin{equation} \\label{stima Aep}\n \\| A_\\varepsilon (\\theta (\\varphi),y(\\varphi),z(\\varphi)) \\|_s^{{\\mathrm{Lip}(\\g)}} \\leq_s \\varepsilon (1 + \\| {\\mathfrak I} \\|_s^{\\mathrm{Lip}(\\g)}) \\, . \n\\end{equation} \nWe now give tame estimates for the Hamiltonian vector fields $ X_{\\cal N} $, $ X_P $,  $ X_{H_\\varepsilon} $, \nsee \\eqref{Hamiltoniana Heps KdV}-\\eqref{Nshape}.\n\n\\begin{lemma}\\label{lemma quantitativo forma normale}\nLet $ {\\mathfrak{I}}(\\varphi) $ in \\eqref{componente periodica} satisfy\n$ \\| {\\mathfrak I} \\|_{s_0 + 3}^{\\mathrm{Lip}(\\g)}  \\leq C\\varepsilon^{6 - 2b} \\gamma^{-1} $. \nThen  \n\\begin{alignat}{2} \n\\| \\partial_y P(i) \\|_s^{\\mathrm{Lip}(\\g)}   & \\leq_s \\varepsilon^4 + \\varepsilon^{2b} \\| {\\mathfrak I}\\|_{s+1}^{\\mathrm{Lip}(\\g)} \\label{D y P} \\, ,  \n& \n\\| \\partial_\\theta P(i) \\|_s^{\\mathrm{Lip}(\\g)}\n& \\leq_s \\varepsilon^{6 - 2b} (1 + \\| {\\mathfrak I}  \\|_{s + 1}^{\\mathrm{Lip}(\\g)}) \n\\\\ \n\\| \\nabla_z P(i) \\|_s^{\\mathrm{Lip}(\\g)}\n& \\leq_s \\varepsilon^{5 - b} + \\varepsilon^{6- b} \\gamma^{-1} \\| {\\mathfrak I}  \\|_{s + 1}^{\\mathrm{Lip}(\\g)} \\, ,\n&\n\\| X_P(i)\\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^{6 - 2b} + \\varepsilon^{2b} \\| {\\mathfrak I}\\|_{s + 3}^{\\mathrm{Lip}(\\g)}  \\label{stima XP}\n\\\\\n\\label{stime linearizzato campo hamiltoniano}\n\\| \\partial_{\\theta} \\partial_y P(i)\\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^{4} + \\varepsilon^{5} \\gamma^{-1} \\| {\\mathfrak I}\\|_{s + 2}^{\\mathrm{Lip}(\\g)}  \\, ,\n& \n\\| \\partial_y \\nabla_z P(i)\\|_s^{\\mathrm{Lip}(\\g)}  \n& \\leq_s \\varepsilon^{b+3} +  \\varepsilon^{2b - 1} \\| {\\mathfrak I} \\|_{s+2}^{\\mathrm{Lip}(\\g)}   \n\\end{alignat}\n\\begin{equation}\n\\label{D yy P}\n\\| \\partial_{yy} P(i) + 3 \\varepsilon^{2b} I_{\\mathbb R^\\nu} \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\varepsilon^{2+ 2b} + \\varepsilon^{2b+3} \\gamma^{-1} \\| {\\mathfrak{I}} \\|_{s+2}^{\\mathrm{Lip}(\\g)}\n\\end{equation}\nand, for all  $ \\widehat \\imath :=  (\\widehat \\Theta, \\widehat y, \\widehat z) $, \n\\begin{align}\n\\label{D yii}\n\\| \\partial_y d_{i} X_P(i)[\\widehat \\imath \\,]\\|_s^{\\mathrm{Lip}(\\g)} &\\leq_s \\varepsilon^{2 b - 1} \\big( \\| \\widehat \\imath \\|_{s + 3}^{\\mathrm{Lip}(\\g)} + \\| {\\mathfrak I}\\|_{s + 3}^{\\mathrm{Lip}(\\g)} \\| \\widehat \\imath \\|_{s_0 + 3}^{\\mathrm{Lip}(\\g)}\\big)  \\\\\n\\label{tame commutatori} \\| d_i X_{H_\\varepsilon}(i) [\\widehat \\imath \\, ] + (0,0, \\partial_{xxx} \\hat z)\\|_s^{\\mathrm{Lip}(\\g)}  \n& \\leq_s \\varepsilon \\big( \\| \\widehat \\imath\\|_{s + 3}^{\\mathrm{Lip}(\\g)} \n+ \\|{\\mathfrak I} \\|_{s + 3}^{\\mathrm{Lip}(\\g)} \\| \\widehat \\imath\\|_{s_0 + 3}^{\\mathrm{Lip}(\\g)} \\big) \\\\\n\\label{parte quadratica da P}\n\\| d_i^2 X_{H_\\varepsilon}(i) [\\widehat \\imath, \\widehat \\imath \\,]\\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon\\Big( \\| \\widehat \\imath \\|_{s + 3}^{\\mathrm{Lip}(\\g)} \\| \n\\widehat \\imath\\|_{s_0 + 3}^{\\mathrm{Lip}(\\g)} + \\| {\\mathfrak I}\\|_{s + 3}^{\\mathrm{Lip}(\\g)} (\\| \\widehat \\imath\\|_{s_0 + 3}^{\\mathrm{Lip}(\\g)})^2\\Big).\n\\end{align}\n\\end{lemma}\n\nIn the sequel we will also use that, by the diophantine condition \\eqref{omdio}, the operator $ {\\cal D}_\\omega^{-1} $  (see \\eqref{Domega})\nis defined for all functions $ u  $ with zero $ \\varphi $-average, and satisfies \n\\begin{equation}\\label{Dom inverso}\n \\| {\\cal D}_\\omega^{-1} u \\|_s \\leq C \\gamma^{-1} \\| u \\|_{s+ \\tau} \\, , \\quad  \\| {\\cal D}_\\omega^{-1} u \\|_s^{{\\mathrm{Lip}(\\g)}} \\leq C \\gamma^{-1} \\| u \\|_{s+ 2 \\tau+1}^{{\\mathrm{Lip}(\\g)}} \\, . \n\\end{equation}\n\n\\section{Approximate inverse}\\label{costruzione dell'inverso approssimato}\n\nIn order to implement a convergent Nash-Moser scheme that leads to a solution of \n$ \\mathcal{F}(i, \\zeta) = 0 $ \n our aim is to construct an \\emph{approximate right inverse} (which satisfies tame estimates) of the linearized operator \n\\begin{equation}\\label{operatore linearizzato}\nd_{i, \\zeta} {\\cal F}(i_0, \\zeta_0 )[\\widehat \\imath \\,, \\widehat \\zeta ] =\nd_{i, \\zeta} {\\cal F}(i_0 )[\\widehat \\imath \\,, \\widehat \\zeta ]   = \n{\\cal D}_\\omega \\widehat \\imath - d_i X_{H_\\varepsilon} ( i_0 (\\varphi) ) [\\widehat \\imath ] + (0, \\widehat \\zeta, 0 )\n \\,,  \n\\end{equation}\nsee Theorem \\ref{thm:stima inverso approssimato}. Note that \n$ d_{i, \\zeta} {\\cal F}(i_0, \\zeta_0 ) = d_{i, \\zeta} {\\cal F}(i_0 ) $ is independent of $ \\zeta_0 $ (see \\eqref{operatorF}).\n\nThe notion of approximate right inverse is introduced in \\cite{Z1}. It denotes a linear operator \nwhich is an \\emph{exact} right inverse  at a solution $ (i_0, \\zeta_0) $  of $ {\\cal F}(i_0, \\zeta_0) = 0 $.\nWe want to implement the general strategy in \\cite{BB13}-\\cite{BB14}\nwhich reduces the search of an approximate right inverse of \\eqref{operatore linearizzato} \nto the search of an approximate inverse on the normal directions only. \n\n\nIt is well known that an invariant torus $ i_0 $ with diophantine flow \nis isotropic (see e.g. \\cite{BB13}), namely the pull-back $ 1$-form $ i_0^* \\Lambda $ is closed, \nwhere $ \\Lambda $ is the contact 1-form in \\eqref{Lambda 1 form}. \nThis is tantamount to say that the 2-form $ \\cal W $ (see \\eqref{2form}) vanishes on the torus $ i_0 (\\mathbb T^\\nu )$ \n(i.e. $\\cal W$ vanishes on the tangent space at each point $i_0(\\varphi)$ of the manifold \n$i_0(\\mathbb T^\\nu)$), because \n$ i_0^* {\\cal W} =  i_0^* d \\Lambda  = d i_0^* \\Lambda $. \nFor an ``approximately invariant\" torus $ i_0 $ the 1-form $ i_0^* \\Lambda$ is only  ``approximately closed\".\nIn order to make this statement quantitative we consider\n\\begin{equation}\\label{coefficienti pull back di Lambda}\ni_0^* \\Lambda = {\\mathop \\sum}_{k = 1}^\\nu a_k (\\varphi) d \\varphi_k \\,,\\quad \na_k(\\varphi) := - \\big( [\\partial_\\varphi \\theta_0 (\\varphi)]^T y_0 (\\varphi)  \\big)_k \n+ \\frac12 ( \\partial_{\\varphi_k} z_0(\\varphi),  \\partial_{x}^{-1} z_0(\\varphi) )_{L^2(\\mathbb T)}\n\\end{equation}\nand we quantify how small is \n\\begin{equation} \\label{def Akj} \ni_0^* {\\cal W} = d \\, i_0^* \\Lambda = {\\mathop\\sum}_{1 \\leq k < j \\leq \\nu} A_{k j}(\\varphi) d \\varphi_k \\wedge d \\varphi_j\\,,\\quad A_{k j} (\\varphi) := \n\\partial_{\\varphi_k} a_j(\\varphi) - \\partial_{\\varphi_j} a_k(\\varphi) \\, .\n\\end{equation}\nAlong this section we will always assume the following hypothesis \n(which will be verified at each step of the Nash-Moser iteration):\n\n\\begin{itemize}\n\\item {\\sc Assumption.} \nThe map $\\omega\\mapsto i_0(\\omega)$ is a Lipschitz function defined on some subset $\\Omega_o \\subset \\Omega_\\varepsilon$, where $\\Omega_\\varepsilon$ is defined in \\eqref{Omega epsilon}, and, for some $ \\mu := \\mu (\\t, \\nu) >  0 $,   \n\\begin{equation}\\label{ansatz 0}\n\\| {\\mathfrak I}_0  \\|_{s_0+\\mu}^{{\\mathrm{Lip}(\\g)}} \\leq C\\varepsilon^{6 - 2b} \\gamma^{-1}, \\quad \n\\| Z \\|_{s_0 + \\mu}^{{\\mathrm{Lip}(\\g)}} \\leq C \\varepsilon^{6 - 2b}, \\quad \n\\gamma = \\varepsilon^{2 + a}, \n\\quad \n b := 1 + (a\/2) \\,,\n \\quad a \\in (0, 1 \/ 6), \n\\end{equation}\nwhere ${\\mathfrak{I}}_0(\\varphi) := i_0(\\varphi) - (\\varphi,0,0)$, and \n\\begin{equation} \\label{def Zetone}\nZ(\\varphi) :=  (Z_1, Z_2, Z_3) (\\varphi) := {\\cal F}(i_0, \\zeta_0) (\\varphi) =\n\\omega \\cdot \\partial_\\varphi i_0(\\varphi) - X_{H_{\\varepsilon, \\zeta_0}}(i_0(\\varphi)) \\, .\n\\end{equation}\n\\end{itemize}\n\\begin{lemma}  \\label{zeta = 0}\n$ |\\zeta_0|^{{\\mathrm{Lip}(\\g)}} \\leq C \\| Z \\|_{s_0}^{{\\mathrm{Lip}(\\g)}}$ \\!\\!\\!. If $ {\\cal F}(i_0, \\zeta_0) = 0 $ then $ \\zeta_0 = 0  $, \n namely the torus $i_0 $ is invariant  for $X_{H_\\varepsilon}$.\n \\end{lemma}\n\n\\begin{proof}\nIt is proved in \\cite{BB13} the formula\n$$\n\\zeta_0 = \n\\int_{\\mathbb T^\\nu} - [\\partial_\\varphi y_0 (\\varphi)]^T  Z_1 (\\varphi) \n+ [\\partial_\\varphi \\theta_0 (\\varphi)]^T Z_2 (\\varphi) \n- [\\partial_\\varphi z_0 (\\varphi) ]^T \\partial_x^{-1} Z_3 (\\varphi) \\, d \\varphi \\, .\n$$\nHence the lemma follows by \\eqref{ansatz 0} and usual algebra estimate. \n\\end{proof}\n\nWe now quantify the size of $ i_0^* {\\cal W} $ in terms of $ Z $.\n\n\\begin{lemma}\nThe coefficients $A_{kj} (\\varphi) $ in \\eqref{def Akj} satisfy \n\\begin{equation}\\label{stima A ij}\n\\| A_{k j} \\|_s^{{\\mathrm{Lip}(\\g)}} \\leq_s \\gamma^{-1} \\big(\\| Z \\|_{s+2\\t+2}^{{\\mathrm{Lip}(\\g)}} \\|  {\\mathfrak I}_0 \\|_{s_0+ 1}^{{\\mathrm{Lip}(\\g)}}  \n+ \\| Z \\|_{s_0+1}^{{\\mathrm{Lip}(\\g)}} \\|  {\\mathfrak I}_0 \\|_{s+ 2 \\tau + 2}^{{\\mathrm{Lip}(\\g)}} \\big)\\,.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nWe estimate the coefficients of the Lie derivative\n$ L_\\omega (i_0^* {\\cal W}) := \\sum_{k < j} {\\cal D}_\\omega A_{k j}(\\varphi) d \\varphi_k \\wedge d \\varphi_j $. \nDenoting  by $ \\underline{e}_k  $ the $ k $-th versor of $ \\mathbb R^\\nu $ we have  \n$$\n {\\cal D}_\\omega A_{k j} \n=  L_\\omega( i_0^* {\\cal W})(\\varphi)[ \\underline{e}_k , \\underline{e}_j ] \n= {\\cal W}\\big( \\partial_\\varphi Z(\\varphi) \\underline{e}_k ,  \\partial_\\varphi i_0(\\varphi)  \\underline{e}_j \\big) \n+ {\\cal W} \\big(\\partial_\\varphi i_0(\\varphi) \\underline{e}_k , \\partial_\\varphi Z(\\varphi) \\underline{e}_j \\big) \n$$\n(see \\cite{BB13}). \nHence\n\\begin{equation} \\label{bella trovata}\n\\| {\\cal D}_\\omega A_{k j} \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\| Z \\|_{s+1}^{\\mathrm{Lip}(\\g)} \\| {\\mathfrak I}_0 \\|_{s_0 + 1}^{\\mathrm{Lip}(\\g)} \n+ \\| Z \\|_{s_0 + 1}^{\\mathrm{Lip}(\\g)} \\| {\\mathfrak I}_0 \\|_{s + 1}^{\\mathrm{Lip}(\\g)}  \\,. \n\\end{equation}\nThe bound \\eqref{stima A ij} follows applying $ {\\cal D}_\\omega^{-1}$ and using \n\\eqref{def Akj}, \\eqref{Dom inverso}.  \n\\end{proof}\n\nAs in \\cite{BB13} we first modify the approximate torus $ i_0 $ to obtain an isotropic torus $ i_\\d $ which is \nstill approximately invariant. We denote the Laplacian $ \\Delta_\\varphi := \\sum_{k=1}^\\nu \\partial_{\\varphi_k}^2 $ .  \n\n\\begin{lemma}\\label{toro isotropico modificato} {\\bf (Isotropic torus)} \nThe torus $ i_\\delta(\\varphi) := (\\theta_0(\\varphi), y_\\delta(\\varphi), z_0(\\varphi) ) $ defined by \n\\begin{equation}\\label{y 0 - y delta}\ny_\\d := y_0 +  [\\partial_\\varphi \\theta_0(\\varphi)]^{- T}  \\rho(\\varphi) \\, , \\qquad \n\\rho_j(\\varphi) := \\Delta_\\varphi^{-1} {\\mathop\\sum}_{ k = 1}^\\nu \\partial_{\\varphi_j} A_{k j}(\\varphi) \n\\end{equation}\nis isotropic. \nIf \\eqref{ansatz 0} holds, then, for some $ \\sigma := \\sigma(\\nu,\\t) $,   \n\\begin{align} \\label{stima y - y delta}\n\\| y_\\delta - y_0 \\|_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq_s  \\gamma^{-1} \\big(\\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\|  {\\mathfrak I}_0 \\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}} + \n\\| Z \\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\|  {\\mathfrak I}_0 \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\big) \\,,\n\\\\\n\\label{stima toro modificato}\n\\| {\\cal F}(i_\\delta, \\zeta_0) \\|_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq_s  \\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}}  +  \\| Z \\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\|  {\\mathfrak I}_0 \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\\\\n\\label{derivata i delta}\n\\| \\partial_i [ i_\\d][ \\widehat \\imath ] \\|_s & \\leq_s \\| \\widehat \\imath \\|_s +  \\| {\\mathfrak I}_0\\|_{s + \\sigma} \\| \\widehat \\imath  \\|_s \\, .\n\\end{align}\n\\end{lemma}\nIn the paper we denote equivalently the differential by $ \\partial_i $ or  $ d_i $. Moreover we denote \nby $ \\sigma := \\sigma(\\nu, \\tau ) $ possibly different (larger) ``loss of derivatives\"  constants. \n\n\\begin{proof}\nIn this proof we write $\\| \\ \\|_s$ to denote $\\| \\ \\|_s^{{\\mathrm{Lip}(\\g)}}$.\nThe proof of the isotropy of $i_\\d$ is in \\cite{BB13}. \nThe estimate \\eqref{stima y - y delta} follows by \\eqref{y 0 - y delta}, \\eqref{stima A ij}, \\eqref{ansatz 0}\nand  the tame bound for the inverse \n$ \\Vert [\\partial_\\varphi \\theta_0 ]^{-T}\\Vert_{s} \\leq_s 1 +  \\|  {\\mathfrak I}_0 \\|_{s + 1} $. \nIt remains to estimate the difference (see \\eqref{operatorF} and note that $ X_{\\cal N} $ does not depend on $ y $)\n\\begin{equation}\\label{F diff}\n{\\cal F}(i_\\delta, \\zeta_0) - {\\cal F}(i_0, \\zeta_0) =  \n\\begin{pmatrix} 0 \\\\ {\\cal D}_\\omega (y_\\delta  - y_0 )  \\\\ 0 \\end{pmatrix}\\, \n+ X_P(i_\\delta ) - X_P(i_0).\n\\end{equation}\nUsing \\eqref{stime linearizzato campo hamiltoniano}, \\eqref{D yy P}, we get \n$ \\| \\partial_y X_P (i) \\|_s \\leq_s \\varepsilon^{2b} + \\varepsilon^{2b - 1} \\| {\\mathfrak I}\\|_{s + 3} $.\nHence \\eqref{stima y - y delta}, \n\\eqref{ansatz 0}  imply\n\\begin{equation}\\label{XP delta - XP}\n\\|X_ P(i_\\delta ) - X_P(i_0 )\\|_s \n\\leq_s    \\|  {\\mathfrak I}_0 \\|_{s_0 + \\sigma} \\|Z \\|_{s + \\sigma} + \\|  {\\mathfrak I}_0 \\|_{s + \\sigma} \\|Z \\|_{s_0 + \\sigma}   \\, . \n\\end{equation}\nDifferentiating  \\eqref{y 0 - y delta} we have  \n\\begin{equation}\\label{D omega y 0 - y delta}\n{\\cal D}_\\omega (y_\\delta - y_0 ) \n= [\\partial_\\varphi \\theta_0(\\varphi)]^{-T} {\\cal D}_\\omega \\rho(\\varphi) \n+ ( {\\cal D}_\\omega [\\partial_\\varphi \\theta_0(\\varphi)]^{-T} ) \\rho(\\varphi)  \n\\end{equation}\nand \n$ {\\cal D}_\\omega  \\rho_j(\\varphi) = \\Delta^{-1}_\\varphi \\sum_{k = 1}^\\nu \\partial_{\\varphi_j} {\\cal D}_\\omega  A_{kj}(\\varphi) $. \nUsing \\eqref{bella trovata},  we deduce that \n\\begin{equation}\\label{primo pezzo D omega y 0 - y delta}\n\\|  [\\partial_\\varphi \\theta_0 ]^{-T} {\\cal D}_\\omega \\rho \\|_s \n\\leq_s \\| Z\\|_{s + 1} \\| {\\mathfrak I}_0\\|_{s_0 + 1} + \\| Z \\|_{s_0 + 1}  \\| {\\mathfrak I}_0\\|_{s + 1}\\,.\n\\end{equation}\nTo estimate the second term in \\eqref{D omega y 0 - y delta}, \nwe differentiate \n$ Z_1(\\varphi) = {\\cal D}_\\omega \\theta_0(\\varphi) - \\omega -  (\\partial_y P)(i_0(\\varphi)) $\n(which is the first component in \\eqref{operatorF}) with respect to  $\\varphi$. We get\n$ {\\cal D}_\\omega \\partial_\\varphi \\theta_0(\\varphi) \n= \\partial_\\varphi (\\partial_y P)(i_0(\\varphi)) + \\partial_\\varphi Z_1(\\varphi) $. \nThen, by \\eqref{D y P},\n\\begin{equation}\\label{bella trovata 2}\n\\|{\\cal D}_\\omega [\\partial_\\varphi \\theta_0]^T \\|_s \n\\leq_s  \\varepsilon^4 + \\varepsilon^{2b} \\| {\\mathfrak I}_0 \\|_{s + 2}   + \\| Z \\|_{s + 1}\\,.\n\\end{equation}\nSince \n$ {\\cal D}_\\omega [ \\partial_\\varphi \\theta_0(\\varphi)]^{-T}  \n= - [ \\partial_\\varphi \\theta_0(\\varphi)]^{-T} \n\\big( {\\cal D}_\\omega [\\partial_\\varphi \\theta_0(\\varphi)]^T \\big) [\\partial_\\varphi \\theta_0(\\varphi)]^{-T} $, \nthe bounds \\eqref{bella trovata 2}, \\eqref{stima A ij},\n\\eqref{ansatz 0} imply\n\\begin{equation}\\label{secondo pezzo D omega y 0 - y delta}\n\\| ({\\cal D}_\\omega [ \\partial_\\varphi \\theta_0]^{-T} )  \\rho \\|_s \n\\leq_s \\varepsilon^{6-2b} \\gamma^{-1} \n\\big( \\| Z\\|_{s + \\sigma}  \\| {\\mathfrak I}_0\\|_{s_0 + \\sigma} + \\| Z \\|_{s_0 + \\sigma}  \\| {\\mathfrak I}_0\\|_{s + \\sigma}\\big) \\, . \n\\end{equation}\nIn conclusion \\eqref{F diff}, \\eqref{XP delta - XP}, \\eqref{D omega y 0 - y delta},\n\\eqref{primo pezzo D omega y 0 - y delta}, \\eqref{secondo pezzo D omega y 0 - y delta} imply \n\\eqref{stima toro modificato}. The bound  \\eqref{derivata i delta} follows by \n\\eqref{y 0 - y delta}, \\eqref{def Akj}, \\eqref{coefficienti pull back di Lambda}, \\eqref{ansatz 0}.\n\\end{proof}\n\nNote that there is no $ \\gamma^{- 1} $ in the right hand side of \\eqref{stima toro modificato}.\nIt turns out that an approximate inverse of $d_{i, \\zeta} {\\cal F}(i_\\delta )$\n is an approximate inverse of $d_{i, \\zeta} {\\cal F}(i_0 )$ as well.  \nIn order to find an approximate inverse of the linearized operator $d_{i, \\zeta} {\\cal F}(i_\\delta )$ \nwe introduce a suitable set of symplectic coordinates nearby the isotropic torus $ i_\\d $. \nWe consider the map\n$ G_\\delta : (\\psi, \\eta, w) \\to (\\theta, y, z)$ of the phase space $\\mathbb T^\\nu \\times \\mathbb R^\\nu \\times H_S^\\bot$ defined by\n\\begin{equation}\\label{trasformazione modificata simplettica}\n\\begin{pmatrix}\n\\theta \\\\\ny \\\\\nz\n\\end{pmatrix} := G_\\delta \\begin{pmatrix}\n\\psi \\\\\n\\eta \\\\\nw\n\\end{pmatrix} := \n\\begin{pmatrix}\n\\theta_0(\\psi) \\\\\ny_\\delta (\\psi) + [\\partial_\\psi \\theta_0(\\psi)]^{-T} \\eta + \\big[ (\\partial_\\theta \\tilde{z}_0) (\\theta_0(\\psi)) \\big]^T \\partial_x^{-1} w \\\\\nz_0(\\psi) + w\n\n\\end{pmatrix} \n\\end{equation}\nwhere $\\tilde{z}_0 (\\theta) := z_0 (\\theta_0^{-1} (\\theta))$. \nIt is proved in \\cite{BB13} that $ G_\\delta $ is symplectic, using that the torus $ i_\\d $ is isotropic \n(Lemma \\ref{toro isotropico modificato}).\nIn the new coordinates,  $ i_\\delta $ is the trivial embedded torus\n$ (\\psi , \\eta , w ) = (\\psi , 0, 0 ) $.  \nThe transformed Hamiltonian $ K := K(\\psi, \\eta, w, \\zeta_0) $\nis (recall \\eqref{hamiltoniana modificata})\n\\begin{align} \nK := H_{\\varepsilon, \\zeta_0} \\circ G_\\d  \n& = \\theta_0(\\psi) \\cdot \\zeta_0 + K_{00}(\\psi) + K_{10}(\\psi) \\cdot \\eta + (K_{0 1}(\\psi), w)_{L^2(\\mathbb T)} + \n\\frac12 K_{2 0}(\\psi)\\eta \\cdot \\eta \n\\nonumber \\\\ & \n\\quad +  \\big( K_{11}(\\psi) \\eta , w \\big)_{L^2(\\mathbb T)} \n+ \\frac12 \\big(K_{02}(\\psi) w , w \\big)_{L^2(\\mathbb T)} + K_{\\geq 3}(\\psi, \\eta, w)  \n\\label{KHG}\n\\end{align}\nwhere $ K_{\\geq 3} $ collects the terms at least cubic in the variables $ (\\eta, w )$.\nAt any fixed $\\psi $, \nthe Taylor coefficient $K_{00}(\\psi) \\in \\mathbb R $,  \n$K_{10}(\\psi) \\in \\mathbb R^\\nu $,  \n$K_{01}(\\psi) \\in H_S^\\bot$ (it is a function of $ x  \\in \\mathbb T $),\n$K_{20}(\\psi) $ is a $\\nu \\times \\nu$ real matrix, \n$K_{02}(\\psi)$ is a linear self-adjoint operator of $ H_S^\\bot $ and \n$K_{11}(\\psi) : \\mathbb R^\\nu \\to H_S^\\bot$. Note that the above Taylor coefficients do not\ndepend on the parameter $ \\zeta_0 $.\n\nThe Hamilton equations associated to \\eqref{KHG}  are \n\\begin{equation}\\label{sistema dopo trasformazione inverso approssimato}\n\\begin{cases}\n\\dot \\psi \\hspace{-30pt} & = K_{10}(\\psi) +  K_{20}(\\psi) \\eta + \nK_{11}^T (\\psi) w + \\partial_{\\eta} K_{\\geq 3}(\\psi, \\eta, w)\n\\\\\n\\dot \\eta \\hspace{-30pt} & =- [\\partial_\\psi \\theta_0(\\psi)]^T \\zeta_0 - \n\\partial_\\psi K_{00}(\\psi) - [\\partial_{\\psi}K_{10}(\\psi)]^T  \\eta - \n[\\partial_{\\psi} K_{01}(\\psi)]^T  w  \n\\\\\n& \\quad -\n\\partial_\\psi \\big( \\frac12 K_{2 0}(\\psi)\\eta \\cdot \\eta + ( K_{11}(\\psi) \\eta , w )_{L^2(\\mathbb T)} + \n\\frac12 ( K_{02}(\\psi) w , w )_{L^2(\\mathbb T)} + K_{\\geq 3}(\\psi, \\eta, w) \\big)\n\\\\\n\\dot w \\hspace{-30pt} & = \\partial_x \\big( K_{01}(\\psi) + \nK_{11}(\\psi) \\eta +  K_{0 2}(\\psi) w + \\nabla_w K_{\\geq 3}(\\psi, \\eta, w) \\big) \n\\end{cases} \n\\end{equation}\nwhere $ [\\partial_{\\psi}K_{10}(\\psi)]^T $ is the $ \\nu \\times \\nu $ transposed matrix and \n$ [\\partial_{\\psi}K_{01}(\\psi)]^T $,  $ K_{11}^T(\\psi) : {H_S^\\bot \\to \\mathbb R^\\nu} $ are defined by the \nduality relation $ ( \\partial_{\\psi} K_{01}(\\psi) [\\hat \\psi ],  w)_{L^2}  = \\hat \\psi \\cdot [\\partial_{\\psi}K_{01}(\\psi)]^T w  $,\n$ \\forall \\hat \\psi \\in \\mathbb R^\\nu, w \\in H_S^\\bot $, \nand similarly for $ K_{11} $. \nExplicitly, for all  $ w \\in H_S^\\bot $, \nand denoting $\\underline{e}_k$ the $k$-th versor of $\\mathbb R^\\nu$, \n\\begin{equation} \\label{K11 tras}\nK_{11}^T(\\psi) w =  {\\mathop \\sum}_{k=1}^\\nu \\big(K_{11}^T(\\psi) w \\cdot \\underline{e}_k\\big) \\underline{e}_k   =\n{\\mathop \\sum}_{k=1}^\\nu  \n\\big( w, K_{11}(\\psi) \\underline{e}_k  \\big)_{L^2(\\mathbb T)}  \\underline{e}_k  \\, \\in \\mathbb R^\\nu \\, .  \n\\end{equation}\n\n\nIn the next lemma we estimate the coefficients $ K_{00} $, $ K_{10} $, $K_{01} $ \nin the Taylor expansion \\eqref{KHG}.\nNote that on an exact solution we have $ Z  = 0 $ and therefore\n$ K_{00} (\\psi) = {\\rm const} $, $ K_{10}  = \\omega $ and $ K_{01}  = 0 $. \n\n\\begin{lemma} \\label{coefficienti nuovi}\nAssume \\eqref{ansatz 0}. Then there is $ \\sigma := \\sigma(\\tau, \\nu)$  such that \n\\[ \n\\|  \\partial_\\psi K_{00} \\|_s^{{\\mathrm{Lip}(\\g)}} \n+ \\| K_{10} - \\omega  \\|_s^{{\\mathrm{Lip}(\\g)}} +  \\| K_{0 1} \\|_s^{{\\mathrm{Lip}(\\g)}} \n\\leq_s  \\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}}  +  \\| Z \\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\| {\\mathfrak I}_0 \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}}\\,.\n\\]\n\\end{lemma}\n\n\n\\begin{proof}\nLet $ {\\cal F}(i_\\d, \\zeta_0) := Z_\\d := (Z_{1,\\d}, Z_{2,\\d}, Z_{3,\\d}) $. \nBy a direct calculation as in \\cite{BB13} (using \\eqref{KHG}, \\eqref{operatorF})\n\\begin{align*}\n\\partial_\\psi K_{00}(\\psi) & =  \n- [ \\partial_\\psi \\theta_0 (\\psi) ]^T \\big( - Z_{2, \\d} - \n[ \\partial_\\psi y_\\d] [ \\partial_\\psi \\theta_0]^{-1} Z_{1, \\d}   \n+ [ (\\partial_\\theta {\\tilde z}_0)( \\theta_0 (\\psi)) ]^T \\partial_x^{-1} Z_{3,\\d} \n\\\\ \n&  \\quad + [ (\\partial_\\theta {\\tilde z}_0)(\\theta_0 (\\psi)) ]^T \\partial_x^{-1} \\partial_\\psi z_0 (\\psi) [ \\partial_\\psi \\theta_0 (\\psi)]^{-1} Z_{1,\\d} \\big) \\, ,  \n\\\\\nK_{10}(\\psi) &\n=  \\omega -   [ \\partial_\\psi \\theta_0(\\psi)]^{-1} Z_{1,\\d}(\\psi) \\,, \n\\\\\nK_{01}(\\psi) \n& = - \\partial_x^{-1} Z_{3,\\d} + \\partial_x^{-1} \\partial_\\psi z_0(\\psi) [\\partial_\\psi \\theta_0(\\psi)]^{-1} Z_{1,\\d}(\\psi)\\,.   \n\\end{align*}\nThen \\eqref{ansatz 0},  \\eqref{stima y - y delta}, \\eqref{stima toro modificato} (using Lemma \\ref{lemma:utile}) imply  the lemma.\n\\end{proof}\n\n\\begin{remark} \\label{rem:KAM normal form}  \nIf $ {\\cal F} (i_0, \\zeta_0) = 0 $ then $\\zeta_0 = 0$  by \nLemma \\ref{zeta = 0}, and Lemma \\ref{coefficienti nuovi} implies that\n\\eqref{KHG} simplifies to \n$ K  = const + \\omega \\cdot \\eta + \\frac12 K_{2 0}(\\psi)\\eta \\cdot \\eta \n+ \\big( K_{11}(\\psi) \\eta , w \\big)_{L^2(\\mathbb T)} \n+ \\frac12 \\big(K_{02}(\\psi) w , w \\big)_{L^2(\\mathbb T)} + K_{\\geq 3} $. \n\\end{remark}\n\nWe now estimate  $ K_{20}, K_{11}$ in \\eqref{KHG}. \nThe norm of $K_{20}$ is the sum of the norms of its matrix entries.  \n\n\\begin{lemma} \\label{lemma:Kapponi vari}\nAssume \\eqref{ansatz 0}. Then \n\\begin{align}\\label{stime coefficienti K 20 11 bassa}\n\\|K_{20} + 3 \\varepsilon^{2 b} I \\|_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq_s \\varepsilon^{2b+2} + \\varepsilon^{2b} \\| {\\mathfrak I}_0\\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}}  +  \\varepsilon^{3} \\gamma^{-1} \\| {\\mathfrak I}_0\\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}}  \\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} \n\\\\ \n\\label{stime coefficienti K 11 alta}  \n\\| K_{11} \\eta \\|_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq_s \\varepsilon^{5} \\gamma^{-1} \\| \\eta \\|_s^{{\\mathrm{Lip}(\\g)}} \n+ \\varepsilon^{2 b - 1} ( \\| {\\mathfrak I}_0\\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} + \\gamma^{-1} \\| {\\mathfrak I}_0\\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}}  \\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} ) \n\\| \\eta \\|_{s_0}^{{\\mathrm{Lip}(\\g)}} \n\\\\\n\\label{stime coefficienti K 11 alta trasposto}  \n\\| K_{11}^T w \\|_s^{{\\mathrm{Lip}(\\g)}} \n& \\leq_s \\varepsilon^{5} \\gamma^{-1} \\| w \\|_{s + 2}^{{\\mathrm{Lip}(\\g)}} \n+ \\varepsilon^{2 b - 1} ( \\| {\\mathfrak I}_0\\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} + \\gamma^{-1} \\| {\\mathfrak I}_0\\|_{s_0 + \\sigma}^{{\\mathrm{Lip}(\\g)}}  \\| Z \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} ) \n\\| w \\|_{s_0 + 2}^{{\\mathrm{Lip}(\\g)}} \\, . \n\\end{align}\nIn particular \n$ \\| K_{20} + 3 \\varepsilon^{2 b} I  \\|_{s_0 }^{{\\mathrm{Lip}(\\g)}}  \\leq C \\varepsilon^{6} \\gamma^{-1} $, and \n$$\n \\| K_{11} \\eta \\|_{s_0 }^{{\\mathrm{Lip}(\\g)}}  \\leq C \\varepsilon^{5} \\gamma^{-1}\\| \\eta \\|_{s_0}^{{\\mathrm{Lip}(\\g)}} , \n \\quad  \\| K_{11}^T w \\|_{s_0 }^{{\\mathrm{Lip}(\\g)}}  \\leq C \\varepsilon^{5} \\gamma^{-1} \\| w \\|_{s_0}^{{\\mathrm{Lip}(\\g)}} \\, .\n$$ \n\\end{lemma}\n\n\\begin{proof}\nTo shorten the notation, in this proof we write $\\| \\ \\|_s$ for $\\| \\ \\|_s^{{\\mathrm{Lip}(\\g)}}$.\nWe have\n$$\nK_{2 0}(\\varphi) = [\\partial_\\varphi \\theta_0(\\varphi)]^{-1} \\partial_{yy} H_\\varepsilon(i_\\delta(\\varphi)) \n[\\partial_\\varphi \\theta_0(\\varphi)]^{-T} \n= [\\partial_\\varphi \\theta_0(\\varphi)]^{-1} \\partial_{yy} P(i_\\delta(\\varphi)) \n[\\partial_\\varphi \\theta_0(\\varphi)]^{-T}.\n$$\nThen \\eqref{D yy P}, \\eqref{ansatz 0}, \\eqref{stima y - y delta} \nimply \\eqref{stime coefficienti K 20 11 bassa}.\nNow (see also \\cite{BB13})\n\\begin{align*}\nK_{11}(\\varphi) & = \n\\partial_{y} \\nabla_z H_\\varepsilon (i_\\delta(\\varphi)) [\\partial_\\varphi \\theta_0 (\\varphi)]^{-T} \n- \\partial_x^{-1} (\\partial_\\theta {\\tilde z}_0) (\\theta_0(\\varphi)) (\\partial_{yy} H_\\varepsilon) (i_\\delta(\\varphi)) [\\partial_\\varphi \\theta_0 (\\varphi)]^{-T}   \\nonumber\\\\\n& \\stackrel{\\eqref{Hamiltoniana Heps KdV}} = \n \\partial_{y} \\nabla_z P(i_\\delta(\\varphi)) [\\partial_\\varphi \\theta_0 (\\varphi)]^{-T} \n- \\partial_x^{-1} (\\partial_\\theta {\\tilde z}_0) (\\theta_0(\\varphi)) (\\partial_{yy} P) (i_\\delta(\\varphi)) [\\partial_\\varphi \\theta_0 (\\varphi)]^{-T}\\,,\n\\end{align*}\ntherefore, using \\eqref{stime linearizzato campo hamiltoniano}, \\eqref{D yy P}, \\eqref{ansatz 0},\nwe deduce  \\eqref{stime coefficienti K 11 alta}. \nThe bound \\eqref{stime coefficienti K 11 alta trasposto} for $K_{11}^T$ follows by \\eqref{K11 tras}.\n\\end{proof}\n\nUnder the linear change of variables \n\\begin{equation}\\label{DGdelta}\nD G_\\delta(\\varphi, 0, 0) \n\\begin{pmatrix}\n\\widehat \\psi \\, \\\\\n\\widehat \\eta \\\\\n\\widehat w\n\\end{pmatrix} \n:= \n\\begin{pmatrix}\n\\partial_\\psi \\theta_0(\\varphi) & 0 & 0 \\\\\n\\partial_\\psi y_\\delta(\\varphi) & [\\partial_\\psi \\theta_0(\\varphi)]^{-T} & \n- [(\\partial_\\theta \\tilde{z}_0)(\\theta_0(\\varphi))]^T \\partial_x^{-1} \\\\\n\\partial_\\psi z_0(\\varphi) & 0 & I\n\\end{pmatrix}\n\\begin{pmatrix}\n\\widehat \\psi \\, \\\\\n\\widehat \\eta \\\\\n\\widehat w\n\\end{pmatrix} \n\\end{equation}\nthe linearized operator  $d_{i, \\zeta}{\\cal F}(i_\\delta )$ \ntransforms (approximately, see \\eqref{verona 2}) into the operator obtained linearizing \n\\eqref{sistema dopo trasformazione inverso approssimato} at $(\\psi, \\eta , w, \\zeta ) = (\\varphi, 0, 0, \\zeta_0 )$\n(with $ \\partial_t \\rightsquigarrow {\\cal D}_\\omega $), namely \n\\begin{equation}\\label{lin idelta}\n\\hspace{-5pt}\n\\begin{pmatrix}\n{\\cal D}_\\omega \\widehat \\psi - \\partial_\\psi K_{10}(\\varphi)[\\widehat \\psi \\, ] - \nK_{2 0}(\\varphi)\\widehat \\eta - K_{11}^T (\\varphi) \\widehat w \\\\\n {\\cal D}_\\omega  \\widehat \\eta + [\\partial_\\psi \\theta_0(\\varphi)]^T \\widehat \\zeta + \n\\partial_\\psi [\\partial_\\psi \\theta_0(\\varphi)]^T [ \\widehat \\psi, \\zeta_0] + \\partial_{\\psi\\psi} K_{00}(\\varphi)[\\widehat \\psi]  + \n[\\partial_\\psi K_{10}(\\varphi)]^T \\widehat \\eta + \n[\\partial_\\psi  K_{01}(\\varphi)]^T \\widehat w   \\\\ \n{\\cal D}_\\omega  \\widehat w - \\partial_x \n\\{ \\partial_\\psi K_{01}(\\varphi)[\\widehat \\psi] + K_{11}(\\varphi) \\widehat \\eta + K_{02}(\\varphi) \\widehat w \\}\n\\end{pmatrix} \\! .  \\hspace{-5pt}\n\\end{equation}\nWe now estimate the induced composition operator. \n\\begin{lemma} \\label{lemma:DG}\nAssume \\eqref{ansatz 0} and let \n$ \\widehat \\imath := (\\widehat \\psi, \\widehat \\eta, \\widehat w)$. \nThen \n\\begin{gather} \\label{DG delta}\n\\|DG_\\delta(\\varphi,0,0) [\\widehat \\imath] \\|_s + \\|DG_\\delta(\\varphi,0,0)^{-1} [\\widehat \\imath] \\|_s \n\\leq_s \\| \\widehat \\imath \\|_{s} + ( \\| {\\mathfrak I}_0 \\|_{s + \\sigma} + \n\\gamma^{-1}  \\| {\\mathfrak I}_0 \\|_{s_0 + \\sigma} \\|Z \\|_{s + \\sigma} ) \\| \\widehat \\imath \\|_{s_0}\\,,\n\\\\ \n\\| D^2 G_\\delta(\\varphi,0,0)[\\widehat \\imath_1, \\widehat \\imath_2] \\|_s \n\\leq_s  \\| \\widehat \\imath_1\\|_s \\| \\widehat \\imath_2 \\|_{s_0} \n+ \\| \\widehat \\imath_1\\|_{s_0} \\| \\widehat \\imath_2 \\|_{s} \n+ ( \\| {\\mathfrak I}_0  \\|_{s + \\sigma} + \\gamma^{-1}  \\| {\\mathfrak I}_0 \\|_{s_0 + \\sigma} \\| Z\\|_{s + \\sigma} ) \\|\\widehat \\imath_1 \\|_{s_0} \\| \\widehat \\imath_2\\|_{s_0}\n\\notag \n\\end{gather}\nfor some $\\sigma := \\sigma(\\nu,\\t)$. \nMoreover the same estimates hold if we replace the norm $\\| \\ \\|_s$ with $\\| \\ \\|_s^{{\\mathrm{Lip}(\\g)}}$.\n\\end{lemma}\n\n\n\\begin{proof}\nThe estimate \\eqref{DG delta} for $D G_\\delta(\\varphi,0,0)$ follows by \\eqref{DGdelta} and  \\eqref{stima y - y delta}.\nBy \\eqref{ansatz 0},\n$ \\| (DG_\\delta(\\varphi,0,0) - I) \\widehat \\imath \\|_{s_0} \n\\leq $ $ C \\varepsilon^{6 - 2b} \\gamma^{-1} \\| \\widehat \\imath \\|_{s_0} \n\\leq \\| \\widehat \\imath \\|_{s_0} \/ 2 $.\nTherefore $DG_\\delta(\\varphi,0,0)$ is invertible and, by Neumann series, the inverse \nsatisfies \\eqref{DG delta}. \nThe bound for $D^2 G_\\delta$ follows by differentiating $DG_\\d$. \n\\end{proof}\n\n\nIn order to construct an approximate inverse of \\eqref{lin idelta} it is sufficient to solve the equation\n\\begin{equation}\\label{operatore inverso approssimato} \n{\\mathbb D} [\\widehat \\psi, \\widehat \\eta, \\widehat w, \\widehat \\zeta ] := \n  \\begin{pmatrix}\n{\\cal D}_\\omega \\widehat \\psi - \nK_{20}(\\varphi) \\widehat \\eta - K_{11}^T(\\varphi) \\widehat w\\\\\n{\\cal D}_\\omega  \\widehat \\eta + [\\partial_\\psi \\theta_0(\\varphi)]^T \\widehat \\zeta  \\\\\n{\\cal D}_\\omega \\widehat w  - \\partial_x K_{11}(\\varphi)\\widehat \\eta -  \\partial_x K_{0 2}(\\varphi) \\widehat w \n\\end{pmatrix} =\n\\begin{pmatrix}\ng_1 \\\\ g_2 \\\\ g_3 \n\\end{pmatrix}\n\\end{equation}\nwhich is obtained by neglecting in \\eqref{lin idelta} \nthe terms $ \\partial_\\psi K_{10} $, $ \\partial_{\\psi \\psi} K_{00} $, $ \\partial_\\psi K_{00} $, $ \\partial_\\psi K_{01} $ and\n$ \\partial_\\psi [\\partial_\\psi \\theta_0(\\varphi)]^T [ \\cdot , \\zeta_0] $\n(which are naught at a solution by Lemmata \\ref{coefficienti nuovi} and \\ref{zeta = 0}). \n\nFirst we solve the second equation in \\eqref{operatore inverso approssimato}, namely \n$ {\\cal D}_\\omega  \\widehat \\eta  = g_2  -  [\\partial_\\psi \\theta_0(\\varphi)]^T \\widehat \\zeta $. \nWe choose $ \\widehat \\zeta $ so that the $\\varphi$-average of the right hand side \nis zero, namely \n\\begin{equation}\\label{fisso valore di widehat zeta}\n\\widehat \\zeta = \\langle g_2 \\rangle \n\\end{equation}\n(we denote $ \\langle g \\rangle := (2 \\pi)^{- \\nu} \\int_{\\mathbb T^\\nu} g (\\varphi) d \\varphi $). \nNote that the $\\varphi$-averaged matrix  $ \\langle [\\partial_\\psi \\theta_0 ]^T  \\rangle = \n\\langle I + [\\partial_\\psi \\Theta_0]^T \\rangle = I $ because $\\theta_0(\\varphi) = \\varphi + \\Theta_0(\\varphi)$ and \n$\\Theta_0(\\varphi)$ is a periodic function.\nTherefore \n\\begin{equation}\\label{soleta}\n\\widehat \\eta := {\\cal D}_\\omega^{-1} \\big(\ng_2 - [\\partial_\\psi \\theta_0(\\varphi) ]^T \\langle g_2 \\rangle \\big) + \n\\langle \\widehat \\eta \\rangle \\, , \\quad \\langle \\widehat \\eta \\rangle \\in \\mathbb R^\\nu \\, , \n\\end{equation}\nwhere the average $\\langle \\widehat \\eta \\rangle$ will be fixed below. \nThen we consider the third equation\n\\begin{equation}\\label{cal L omega}\n{\\cal L}_\\omega \\widehat w = g_3 + \\partial_x K_{11}(\\varphi) \\widehat \\eta\\,,  \\\n\\quad  {\\cal L}_\\omega := \\omega \\cdot \\partial_\\varphi -  \\partial_x K_{0 2}(\\varphi) \\, .\n\\end{equation}\n\n\\begin{itemize}\n\\item {\\sc Inversion assumption.} \n{\\it There exists a set $ \\Omega_\\infty \\subset \\Omega_o$ such that  \nfor all $ \\omega \\in \\Omega_\\infty $, for every function\n$ g \\in H^{s+\\mu}_{S^\\bot} (\\mathbb T^{\\nu+1}) $  there \nexists a solution $ h :=  {\\cal L}_\\omega^{- 1} g  \\in H^{s}_{S^\\bot} (\\mathbb T^{\\nu+1})  $ \nof the linear equation $ {\\cal L}_\\omega h = g $\nwhich satisfies}\n\\begin{equation}\\label{tame inverse}\n\\| {\\cal L}_\\omega^{- 1} g \\|_s^{{\\mathrm{Lip}(\\g)}} \\leq C(s) \\gamma^{-1} \n\\big(  \\| g \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} + \\varepsilon \\gamma^{-1} \n\\big\\{ \\| {\\mathfrak I}_0 \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} + \\gamma^{-1} \\| {\\mathfrak I}_0 \\|_{s_0 + \\mu}^{{\\mathrm{Lip}(\\g)}} \\| Z \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} \\big\\} \\|g \\|_{s_0}^{{\\mathrm{Lip}(\\g)}}  \\big) \n\\end{equation}\n\\emph{for some $ \\mu := \\mu (\\tau, \\nu) >  0 $}. \n\\end{itemize}\n\n\n\\begin{remark}\nThe term $ \\varepsilon \\gamma^{-1} \n\\{ \\| {\\mathfrak I}_0 \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} + \\gamma^{-1} \\| {\\mathfrak I}_0 \\|_{s_0 + \\mu}^{{\\mathrm{Lip}(\\g)}} \\| Z \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} \\} $ arises because\nthe remainder $ R_6 $ in section \\ref{step5} contains the term \n$ \\varepsilon ( \\| \\Theta_0 \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} + \\| y_\\d \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}}) $ \n$\\leq \\varepsilon  \\| {\\mathfrak I}_\\d \\|_{s + \\mu}^{{\\mathrm{Lip}(\\g)}} $, see Lemma \\ref{lemma L6}. \n\\end{remark}\n\nBy the above assumption  there exists a solution \n\\begin{equation}\\label{normalw}\n\\widehat w := {\\cal L}_\\omega^{-1} [ g_3 + \\partial_x K_{11}(\\varphi) \\widehat \\eta \\, ] \n\\end{equation}\nof \\eqref{cal L omega}. \nFinally, we solve the first equation in \\eqref{operatore inverso approssimato}, \nwhich, substituting \\eqref{soleta}, \\eqref{normalw}, becomes\n\\begin{equation}\\label{equazione psi hat}\n{\\cal D}_\\omega \\widehat \\psi  = \ng_1 +  M_1(\\varphi) \\langle \\widehat \\eta \\rangle + M_2(\\varphi) g_2 + M_3(\\varphi) g_3 - \nM_2(\\varphi)[\\partial_\\psi \\theta_0]^T \\langle g_2 \\rangle \\,,\n\\end{equation}\nwhere\n\\begin{equation} \\label{cal M2}\nM_1(\\varphi) := K_{2 0}(\\varphi) + K_{11}^T(\\varphi) {\\cal L}_\\omega^{-1} \\partial_x K_{11}(\\varphi)\\,, \\quad\nM_2(\\varphi) :=  M_1 (\\varphi)  {\\cal D}_\\omega^{-1} \\, , \\quad \nM_3(\\varphi) :=  K_{11}^T (\\varphi) {\\cal L}_\\omega^{-1} \\, .  \n\\end{equation}\nIn order to solve the equation \\eqref{equazione psi hat} we have \nto choose $\\langle \\widehat \\eta \\rangle$ such that the right hand side in \\eqref{equazione psi hat} has zero average.  \nBy Lemma \\ref{lemma:Kapponi vari} and \\eqref{ansatz 0}, the $\\varphi$-averaged matrix \n$ \\langle M_1 \\rangle =- 3 \\varepsilon^{2 b} I + O( \\varepsilon^{10} \\gamma^{-3}) $.  \nTherefore, for $ \\varepsilon $ small,  $\\langle M_1 \\rangle$ is invertible and $\\langle M_1 \\rangle^{-1} = O(\\varepsilon^{-2 b}) = O(\\gamma^{- 1})$ \n(recall \\eqref{link gamma b}). Thus we define \n\\begin{equation}\\label{sol alpha}\n\\langle \\widehat \\eta \\rangle  := - \\langle M_1 \\rangle^{-1} \n[ \\langle g_1 \\rangle + \\langle M_2 g_2 \\rangle + \\langle M_3 g_3 \\rangle - \n\\langle M_2 [\\partial_\\psi \\theta_0]^T   \\rangle  \\langle g_2 \\rangle ].\n\\end{equation}\nWith this choice of $\\langle \\widehat \\eta \\rangle$ the equation \\eqref{equazione psi hat} has the solution\n\\begin{equation}\\label{sol psi}\n\\widehat \\psi :=\n{\\cal D}_\\omega^{-1} [ g_1 + M_1(\\varphi) \\langle \\widehat \\eta \\rangle + M_2(\\varphi) g_2 + M_3(\\varphi) g_3 -\nM_2(\\varphi)[\\partial_\\psi \\theta_0]^T \\langle g_2 \\rangle ].\n\\end{equation}\nIn conclusion, we have constructed \na solution  $(\\widehat \\psi, \\widehat \\eta, \\widehat w, \\widehat \\zeta)$ of the linear system \\eqref{operatore inverso approssimato}. \n\n\\begin{proposition}\\label{prop: ai}\nAssume \\eqref{ansatz 0} and \\eqref{tame inverse}. \nThen, $\\forall \\omega \\in \\Omega_\\infty $, $ \\forall g := (g_1, g_2, g_3) $,\n the system \\eqref{operatore inverso approssimato} has a solution \n$ {\\mathbb D}^{-1} g := (\\widehat \\psi, \\widehat \\eta, \\widehat w, \\widehat \\zeta ) $\nwhere $(\\widehat \\psi, \\widehat \\eta, \\widehat w, \\widehat \\zeta)$ are defined in \n\\eqref{sol psi}, \\eqref{soleta}, \\eqref{sol alpha}, \\eqref{normalw}, \\eqref{fisso valore di widehat zeta}  satisfying\n\\begin{equation} \\label{stima T 0 b}\n\\| {\\mathbb D}^{-1} g \\|_s^{{\\rm Lip}(\\gamma)}  \n\\leq_s \\gamma^{-1} \\big( \\| g \\|_{s + \\mu}^{{\\rm Lip}(\\gamma)}  \n+ \\varepsilon \\gamma^{-1} \\big\\{ \\| {\\mathfrak I}_0  \\|_{s + \\mu}^{{\\rm Lip}(\\gamma)}  \n+  \\gamma^{-1}  \\| {\\mathfrak I}_0  \\|_{s_0 + \\mu}^{{\\rm Lip}(\\gamma)} \\|{\\cal F}(i_0, \\zeta_0) \\|^{{\\rm Lip}(\\gamma)}_{s + \\mu} \\big\\} \\| g \\|_{s_0 + \\mu}^{{\\rm Lip}(\\gamma)}  \\big).\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n\nRecalling \\eqref{cal M2}, by  Lemma \\ref{lemma:Kapponi vari}, \\eqref{tame inverse}, \\eqref{ansatz 0}  we get $ \\| M_2 h \\|_{s_0} + \\| M_3 h \\|_{s_0}  \\leq C \\| h \\|_{s_0 + \\sigma} $. \nThen, by \\eqref{sol alpha} and $\\langle M_1 \\rangle^{-1} = O(\\varepsilon^{-2 b}) = O(\\gamma^{-1}) $, \nwe deduce \n$ |\\langle \\widehat \\eta\\rangle|^{{\\mathrm{Lip}(\\g)}} \\leq C\\gamma^{-1} \\| g \\|_{s_0+ \\sigma}^{{\\mathrm{Lip}(\\g)}} $\nand  \\eqref{soleta}, \\eqref{Dom inverso} imply \n$ \\| \\widehat \\eta \\|_s^{{\\mathrm{Lip}(\\g)}} \\leq_s \\gamma^{-1} \\big( \\| g \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} + \\| {\\mathfrak{I}}_0 \\|_{s + \\sigma } \\| g \\|_{s_0}^{\\mathrm{Lip}(\\g)}  \\big)$.\nThe bound \\eqref{stima T 0 b} is sharp for $ \\widehat w $ because $ {\\cal L}_\\omega^{-1} g_3 $ in  \\eqref{normalw}\nis estimated using \\eqref{tame inverse}. Finally $  \\widehat \\psi $ \nsatisfies \\eqref{stima T 0 b} using\n\\eqref{sol psi},  \\eqref{cal M2}, \\eqref{tame inverse}, \\eqref{Dom inverso} and Lemma \\ref{lemma:Kapponi vari}.\n\\end{proof}\nFinally we prove that the operator \n\\begin{equation}\\label{definizione T} \n{\\bf T}_0 := (D { \\widetilde G}_\\delta)(\\varphi,0,0) \\circ {\\mathbb D}^{-1} \\circ (D G_\\delta) (\\varphi,0,0)^{-1}\n\\end{equation}\nis an approximate right  inverse for $d_{i,\\zeta} {\\cal F}(i_0 )$ where\n$ \\widetilde{G}_\\delta (\\psi, \\eta, w, \\zeta) := $  $ \\big( G_\\delta (\\psi, \\eta, w), \\zeta \\big) $ \n is the identity on the $ \\zeta $-component. \nWe denote the norm $ \\| (\\psi, \\eta, w, \\zeta) \\|_s^{\\mathrm{Lip}(\\g)} := $ $  \\max \\{  \\| (\\psi, \\eta, w) \\|_s^{\\mathrm{Lip}(\\g)}, $ $ | \\zeta |^{\\mathrm{Lip}(\\g)}  \\} $.\n\n\n\\begin{theorem} {\\bf (Approximate inverse)} \\label{thm:stima inverso approssimato}\nAssume \\eqref{ansatz 0} and the inversion assumption \\eqref{tame inverse}. \nThen there exists $ \\mu := \\mu (\\tau, \\nu) >  0 $ such that, for all $ \\omega \\in \\Omega_\\infty $, \nfor all $ g := (g_1, g_2, g_3) $,  \nthe operator $ {\\bf T}_0 $ defined in \\eqref{definizione T} satisfies  \n\\begin{equation}\\label{stima inverso approssimato 1}\n\\| {\\bf T}_0 g \\|_{s}^{{\\rm Lip}(\\gamma)}  \n\\leq_s  \\gamma^{-1} \\big(\\| g \\|_{s + \\mu}^{{\\rm Lip}(\\gamma)}  \n+ \\varepsilon \\gamma^{-1} \\big\\{ \\| {\\mathfrak I}_0 \\|_{s + \\mu}^{{\\rm Lip}(\\gamma)}  \n+\\gamma^{-1} \\| {\\mathfrak{I}}_0 \\|_{s_0 + \\mu}^{\\mathrm{Lip}(\\g)} \n\\|{\\cal F}(i_0, \\zeta_0) \\|_{s + \\mu}^{{\\rm Lip}(\\gamma)}  \\big\\} \n\\| g \\|_{s_0 + \\mu}^{{\\rm Lip}(\\gamma)}  \\big). \n\\end{equation}\nIt is an approximate inverse of $d_{i, \\zeta} {\\cal F}(i_0 )$, namely \n\\begin{align}\n& \\| ( d_{i, \\zeta} {\\cal F}(i_0) \\circ {\\bf T}_0 - I ) g \\|_s^{{\\rm Lip}(\\gamma)}  \n\\label{stima inverso approssimato 2} \n\\\\ \n& \\leq_s \\gamma^{-1} \\Big( \\| {\\cal F}(i_0, \\zeta_0) \\|_{s_0 + \\mu}^{\\mathrm{Lip}(\\g)} \\| g \\|_{s + \\mu}^{\\mathrm{Lip}(\\g)}  \n+ \\big\\{ \\| {\\cal F}(i_0, \\zeta_0) \\|_{s + \\mu}^{\\mathrm{Lip}(\\g)} \n+ \\varepsilon \\gamma^{-1} \\| {\\cal F}(i_0, \\zeta_0) \\|_{s_0 + \\mu}^{\\mathrm{Lip}(\\g)} \\| {\\mathfrak I}_0 \\|_{s + \\mu}^{\\mathrm{Lip}(\\g)} \\big\\} \\| g \\|_{s_0 + \\mu}^{\\mathrm{Lip}(\\g)} \\Big).\n\\nonumber\n\\end{align}\n\\end{theorem}\n\n\n\\begin{proof}\nWe denote $\\| \\ \\|_s$ instead of $\\| \\ \\|_s^{{\\mathrm{Lip}(\\g)}}$. \nThe bound \\eqref{stima inverso approssimato 1} \nfollows from \\eqref{definizione T}, \\eqref{stima T 0 b},  \\eqref{DG delta}.\nBy \\eqref{operatorF}, since $ X_\\mathcal{N} $ does not depend on $ y $,  \nand $ i_\\d $ differs from $ i_0 $ only for the $ y$ component,  \nwe have \n\\begin{align} \\label{verona 0}\nd_{i, \\zeta} {\\cal F}(i_0 )[\\, \\widehat \\imath, \\widehat \\zeta \\, ]  - d_{i, \\zeta} {\\cal F}(i_\\delta ) [\\, \\widehat \\imath, \\widehat \\zeta \\, ]\n&  =   d_i X_P (i_\\delta)  [\\, \\widehat \\imath \\, ]  - d_i X_P (i_0) [\\, \\widehat \\imath \\, ]\n\\\\ & \n=  \\int_0^1 \\partial_y d_i X_P (\\theta_0, y_0 + s (y_\\delta - y_0), z_0) [y_\\d - y_0,   \\widehat \\imath  \\,  ] ds \n=: {\\cal E}_0 [\\, \\widehat \\imath, \\widehat \\zeta \\, ]  \\,. \\nonumber\n\\end{align}\nBy \\eqref{D yii}, \\eqref{stima y - y delta}, \\eqref{ansatz 0},  we estimate\n\\begin{equation}\\label{stima parte trascurata 1}\n\\| {\\cal E}_0 [\\, \\widehat \\imath, \\widehat \\zeta \\, ] \\|_s \\leq_s \n\\| Z \\|_{s_0 + \\sigma} \\| \\widehat \\imath \\|_{s + \\sigma} +  \n\\| Z \\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} +  \\varepsilon^{2b-1}\\gamma^{-1}\n\\| Z \\|_{s_0 + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} \\| {\\mathfrak{I}}_0 \\|_{s+\\sigma} \n\\end{equation}\nwhere $Z := \\mathcal{F}(i_0, \\zeta_0)$ (recall \\eqref{def Zetone}). \n Note that $\\mathcal{E}_0[\\widehat \\imath, \\widehat \\zeta]$ is, in fact, independent of $\\widehat \\zeta$.\nDenote the set of variables $  (\\psi, \\eta, w) =: {\\mathtt u} $. \nUnder the transformation $G_\\delta $, the nonlinear operator ${\\cal F}$ in \\eqref{operatorF} transforms into \n\\begin{equation} \\label{trasfo imp}\n{\\cal F}(G_\\delta(  {\\mathtt u} (\\varphi) ), \\zeta ) \n= D G_\\delta( {\\mathtt u}  (\\varphi) ) \\big(  {\\cal D}_\\omega {\\mathtt u} (\\varphi) - X_K ( {\\mathtt u} (\\varphi), \\zeta)  \\big) \\, , \\quad \nK = H_{\\varepsilon, \\zeta} \\circ G_\\delta   \\, ,\n\\end{equation}\nsee \\eqref{sistema dopo trasformazione inverso approssimato}. \nDifferentiating  \\eqref{trasfo imp} at the trivial torus \n$ {\\mathtt u}_\\delta (\\varphi) = G_\\delta^{-1}(i_\\delta) (\\varphi) = (\\varphi, 0 , 0 ) $, \nat  $ \\zeta = \\zeta_0 $, \nin the directions $(\\widehat {\\mathtt u}, \\widehat \\zeta)\n= (D G_\\d ({\\mathtt u}_\\d)^{-1} [\\, \\widehat \\imath \\, ], \\widehat \\zeta) \n= D {\\widetilde G}_\\d ({\\mathtt u}_\\d)^{-1} [\\, \\widehat \\imath , \\widehat \\zeta \\, ] $, \nwe get\n\\begin{align} \\label{verona 2}\nd_{i , \\zeta} {\\cal F}(i_\\delta ) [\\, \\widehat \\imath, \\widehat \\zeta \\, ]\n=  & D G_\\delta( {\\mathtt u}_\\delta) \n\\big( {\\cal D}_\\omega \\widehat {\\mathtt u} \n- d_{\\mathtt u, \\zeta} X_K( {\\mathtt u}_\\delta, \\zeta_0) [\\widehat {\\mathtt u}, \\widehat \\zeta \\, ] \n\\big) \n+ {\\cal E}_1 [ \\, \\widehat \\imath , \\widehat \\zeta \\, ]\\,,\n\\\\\n\\label{E1}\n{\\cal E}_1 [\\, \\widehat \\imath , \\widehat \\zeta \\, ] \n:=  & \nD^2 G_\\delta( {\\mathtt u}_\\delta) \\big[ D G_\\delta( {\\mathtt u}_\\delta)^{-1} {\\cal F}(i_\\delta, \\zeta_0), \\,  D G_\\d({\\mathtt u}_\\d)^{-1} \n[ \\, \\widehat \\imath \\,  ] \\big] \\,, \n\\end{align}\nwhere  $ d_{\\mathtt u, \\zeta} X_K( {\\mathtt u}_\\delta, \\zeta_0) $ is expanded in \\eqref{lin idelta}.\nIn fact, ${\\cal E}_1$ is independent of $\\widehat \\zeta$. \nWe split  \n\\[ \n{\\cal D}_\\omega \\widehat {\\mathtt u} \n- d_{\\mathtt u, \\zeta} X_K( {\\mathtt u}_\\delta, \\zeta_0) [\\widehat {\\mathtt u}, \\widehat \\zeta] \n= \\mathbb{D} [\\widehat {\\mathtt u}, \\widehat \\zeta \\, ] + R_Z [ \\widehat {\\mathtt u}, \\widehat \\zeta \\, ], \n\\]\nwhere $ {\\mathbb D} [\\widehat {\\mathtt u}, \\widehat \\zeta] $ is defined in \\eqref{operatore inverso approssimato} and \n\\begin{equation}\\label{R0}\nR_Z [  \\widehat \\psi, \\widehat \\eta, \\widehat w, \\widehat \\zeta]\n:= \\begin{pmatrix}\n - \\partial_\\psi K_{10}(\\varphi) [\\widehat \\psi ] \\\\\n\\partial_\\psi [\\partial_\\psi \\theta_0(\\varphi)]^T [  \\widehat \\psi, \\zeta_0] + \\partial_{\\psi \\psi} K_{00} (\\varphi) [ \\widehat \\psi ] + \n [\\partial_\\psi K_{10}(\\varphi)]^T \\widehat \\eta + \n [\\partial_\\psi K_{01}(\\varphi)]^T \\widehat w  \\\\\n - \\partial_x \\{ \\partial_{\\psi} K_{01}(\\varphi)[ \\widehat \\psi ] \\}\n \\end{pmatrix}\n\\end{equation}\n($R_Z$ is independent of $\\widehat \\zeta$). \nBy \\eqref{verona 0} and \\eqref{verona 2}, \n\\begin{equation} \\label{E2}\nd_{i, \\zeta} {\\cal F}(i_0 ) \n= D G_\\delta({\\mathtt u}_\\delta) \\circ {\\mathbb D} \\circ D {\\widetilde G}_\\delta ({\\mathtt u}_\\delta)^{-1} \n+ {\\cal E}_0 + {\\cal E}_1 + \\mathcal{E}_2 \\,,\n\\quad\n\\mathcal{E}_2 := D G_\\delta( {\\mathtt u}_\\delta) \\circ R_Z \\circ D {\\widetilde G}_\\delta ({\\mathtt u}_\\delta)^{-1}  \\, .\n\\end{equation}\nBy Lemmata \\ref{coefficienti nuovi}, \\ref{lemma:DG},    \\ref{zeta = 0}, and \\eqref{stima toro modificato}, \\eqref{ansatz 0},\nthe terms $\\mathcal{E}_1, \\mathcal{E}_2 $ (see \\eqref{E1}, \\eqref{E2}, \\eqref{R0})  satisfy \nthe same bound \\eqref{stima parte trascurata 1} as $\\mathcal{E}_0$\n(in fact even better).\nThus the sum $\\mathcal{E} := \\mathcal{E}_0 + \\mathcal{E}_1 + \\mathcal{E}_2$ satisfies \\eqref{stima parte trascurata 1}.\nApplying $ {\\bf T}_0 $ defined in \\eqref{definizione T} to the right in \\eqref{E2}, \nsince $ {\\mathbb D} \\circ  {\\mathbb D}^{-1} = I $ (see Proposition \\ref{prop: ai}), \nwe get $d_{i, \\zeta} {\\cal F}(i_0 ) \\circ {\\bf T}_0  - I \n= \\mathcal{E} \\circ {\\bf T}_0$. \nThen \\eqref{stima inverso approssimato 2} follows from \n\\eqref{stima inverso approssimato 1} and the bound \\eqref{stima parte trascurata 1} for $\\mathcal{E}$. \n\\end{proof}\n\n\n\n\\section{The linearized operator in the normal directions}\\label{linearizzato siti normali}\n\nThe goal of this section is to write an explicit  expression of the linearized operator \n$\\mathcal{L}_\\omega$ defined in \\eqref{cal L omega}, see Proposition \\ref{prop:lin}.\nTo this aim, we compute $ \\frac12 ( K_{02}(\\psi) w, w )_{L^2(\\mathbb T)} $, $ w \\in H_S^\\bot$, \nwhich collects all the components of $(H_\\varepsilon \\circ G_\\d)(\\psi, 0, w)$ that are quadratic in $w$, see \\eqref{KHG}.\n\n\\smallskip\n\nWe first prove some preliminary lemmata. \n\n\\begin{lemma}\\label{lemma astratto potente}\nLet $ H $ be a Hamiltonian of class $C^2 (  H^1_0(\\mathbb T_x), \\mathbb R )$ and consider a map\n$ \\Phi(u) := u + \\Psi(u) $ satisfying $\\Psi (u) = \\Pi_E \\Psi(\\Pi_E u)$, for all $ u $, \nwhere $E$ is a finite dimensional subspace as in \\eqref{def E finito}. Then \n\\begin{equation}\\label{lint2}\n\\partial_u \\big[\\nabla ( H \\circ \\Phi)\\big] (u)  [h] = (\\partial_u  \\nabla H )(\\Phi(u)) [h] + {\\cal R}(u)[h]\\,,\n\\end{equation}\nwhere  $ {\\cal R}(u) $ \nhas the ``finite dimensional\" form \n\\begin{equation}\\label{forma buona resto}\n{\\cal R}(u)[h] =  {\\mathop\\sum}_{|j| \\leq C} \\big( h , g_j(u) \\big)_{L^2(\\mathbb T)} \\chi_j(u) \n\\end{equation}\nwith $ \\chi_j (u) = e^{{\\mathrm i}  j x} $ or  $ g_j(u) = e^{{\\mathrm i}  j x} $. The remainder  \n$ {\\cal R} (u)  =  {\\cal R}_0 (u)  + {\\cal R}_1 (u) + {\\cal R}_2 (u)  $ with\n\\begin{align}\\label{resti012} \n{\\cal R}_0 (u) & :=  (\\partial_u \\nabla H)(\\Phi(u)) \\partial_u \\Psi (u), \\qquad  \n{\\cal R}_1 (u) :=  [\\partial_{u }\\{ \\Psi'(u)^T\\}] [ \\cdot , \\nabla H(\\Phi(u)) ], \\nonumber \\\\\n\\,  {\\cal R}_2 (u)  & :=  [\\partial_u \\Psi (u)]^T (\\partial_u \\nabla H)(\\Phi(u)) \\partial_u \\Phi(u). \n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy a direct calculation, \n\\begin{equation}\\label{nabla composto} \n\\nabla (H \\circ \\Phi)(u) = [\\Phi'(u)]^T \\nabla H(\\Phi(u)) = \\nabla H(\\Phi(u)) + [\\Psi'(u)]^T \\nabla H(\\Phi(u))\n\\end{equation}\nwhere $ \\Phi' (u) := ( \\partial_u \\Phi) (u) $ and $ [ \\ ]^T $ denotes the transpose with respect to the $ L^2 $ scalar product.\nDifferentiating \\eqref{nabla composto}, we get \\eqref{lint2} and \\eqref{resti012}.\n\nLet us show that each $ {\\cal R}_m $ has the form \\eqref{forma buona resto}. \nWe have \n\\begin{equation}\\label{Psiuh}\n\\Psi'(u) = \\Pi_E \\Psi'(\\Pi_E u) \\Pi_E  \\, \\, , \\quad  [\\Psi'(u)]^T = \\Pi_E   [\\Psi'( \\Pi_E u)]^T \\Pi_E \\, . \n\\end{equation}\nHence, setting $ A  := (\\partial_u \\nabla H)(\\Phi(u)) \\Pi_E \\Psi' ( \\Pi_E u)  $, we get \n\\[\n{\\cal R}_0(u)[h] \n= A [ \\Pi_E h ] \n=  {\\mathop\\sum}_{|j| \\leq C } h_j A ( e^{{\\mathrm i}  j x} ) \n=  {\\mathop\\sum}_{|j| \\leq C} (h, g_j )_{L^2(\\mathbb T)} \\chi_j \n\\]\nwith $ g_j := e^{{\\mathrm i}  j x} $, $ \\chi_j :=  A( e^{{\\mathrm i}  jx } )$.\nSimilarly, using \\eqref{Psiuh},  and setting  \n$ A  := [ \\Psi' (\\Pi_E u) ]^T \\Pi_E (\\partial_u \\nabla H)(\\Phi(u)) \\Phi'(u) $,  \nwe get\n$$\n{\\cal R}_2 (u)[h] =  \\Pi_E [ A h ] =  {\\mathop\\sum}_{|j | \\leq C } (A h ,  e^{{\\mathrm i}  j x} )_{L^2(\\mathbb T)} e^{{\\mathrm i}  jx}  \n =  {\\mathop\\sum}_{|j| \\leq C} (h, A^T   e^{{\\mathrm i}  j x}  )_{L^2(\\mathbb T)}  e^{{\\mathrm i}  j x} \\,,  \n$$\nwhich has the form \\eqref{forma buona resto} with $ g_j :=  A^T( e^{{\\mathrm i}  jx } )$ and  $ \\chi_j := e^{{\\mathrm i}  j x} $.\nDifferentiating the second equality in \\eqref{Psiuh}, we see that  \n$$\n{\\cal R}_1 (u)[h] =  \\Pi_E [ A h ] \\, , \\quad A h := \\partial_u \\{ \\Psi' ( \\Pi_E u)^T \\} [\\Pi_E h, \\Pi_E  \n(\\nabla H)(\\Phi(u)) ] \\,,\n$$\nwhich has the same form of $ {\\cal R}_2 $ and so \\eqref{forma buona resto}. \n\\end{proof}\n\n\n\\begin{lemma} \\label{sifulo}\nLet  $ H(u ) := \\int_\\mathbb T f(u) X(u) d x $\nwhere $ X(u) = \\Pi_{E} X ( \\Pi_E u) $ and $  f(u)(x) := f( u(x)) $ is the composition operator for a function of class\n$  C^2 $. Then\n\\begin{equation}\\label{lint1}\n(\\partial_u \\nabla H) (u) [h] = f''(u)  X(u) \\, h + {\\cal R} (u) [h] \n\\end{equation}\nwhere $ {\\cal R} (u) $ has  the form \\eqref{forma buona resto}\nwith $ \\chi_j (u) = e^{{\\mathrm i}  j x} $ or  $ g_j(u) = e^{{\\mathrm i}  j x} $.\n\\end{lemma}\n\n\\begin{proof}\nA direct calculation proves  that $\\nabla H(u) = f'(u) X(u) + X'(u)^T [f(u)]$, \nand \\eqref{lint1} follows with\n$ {\\cal R} (u) [h] = $ $  f'(u) X'(u)[h] + $ $  \\partial_u \\{  X'(u)^T\\} [h, f(u)] + $ $  X'(u)^T [ f'(u) h ]$, \nwhich has the form \\eqref{forma buona resto}. \n\\end{proof}\n\nWe conclude this section with a technical lemma  used from the end of section \\ref{step3} about the decay norms of ``finite dimensional operators\".\nNote that operators of the form \\eqref{forma buona con gli integrali} (that will appear in section \\ref{step1}) \nreduce to those in \\eqref{forma buona resto} when the functions $ g_j(\\tau) $, $ \\chi_j (\\tau)$ are independent of $ \\tau $\n\n\n\\begin{lemma}\\label{remark : decay forma buona resto}\nLet $ {\\cal R} $ be an operator of the form\n\\begin{equation}\\label{forma buona con gli integrali}\n{\\cal R} h = \\sum_{|j| \\leq C } \\int_0^1 \\big(h\\,,\\,g_j(\\tau) \\big)_{L^2(\\mathbb T)} \\chi_j (\\tau)\\,d \\tau\\,,\n\\end{equation}\nwhere the functions $g_j(\\tau),\\,\\chi_j(\\tau) \\in H^s$, $\\tau \\in [0, 1]$ depend in a Lipschitz way on the parameter $\\omega$. Then its matrix $ s$-decay norm (see \\eqref{matrix decay norm}-\\eqref{matrix decay norm Lip}) satisfies \n$$ \n| {\\cal R} |_s^{\\mathrm{Lip}(\\g)} \\leq_s {\\mathop \\sum}_{|j| \\leq C} {\\rm sup}_{\\tau \\in [0,1]} \\big\\{ \\| \\chi_j(\\tau) \\|_s^{\\mathrm{Lip}(\\g)} \\| g_j(\\tau) \\|_{s_0}^{\\mathrm{Lip}(\\g)} \n+ \\| \\chi_j(\\tau) \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\| g_j(\\tau) \\|_s^{\\mathrm{Lip}(\\g)} \\big\\} \\, . \n$$ \n\\end{lemma}\n\n\\begin{proof}\nFor each $\\tau \\in [0, 1]$, the operator $ h \\mapsto (h,g_j(\\tau)) \\chi_j(\\tau) $ is the \ncomposition $ \\chi_j(\\tau) \\circ \\Pi_0 \\circ g_j(\\tau) $ of the multiplication operators for $ g_j(\\tau), \\chi_j(\\tau) $ and \n$ h \\mapsto \\Pi_0 h := \\int_{\\mathbb T} h dx $. Hence\nthe lemma follows by the interpolation estimate \\eqref{interpm Lip} \nand \\eqref{multiplication Lip}. \n\\end{proof}\n\n\\subsection{Composition with the map $G_\\d$} \\label{section:appr}\n\nIn the sequel we shall use that \n$   {\\mathfrak{I}}_\\d := {\\mathfrak{I}}_\\d (\\varphi ; \\omega) := i_\\d (\\varphi; \\omega ) - (\\varphi,0,0) $ satisfies, \nby Lemma \\ref{toro isotropico modificato} and \\eqref{ansatz 0},  \n\\begin{equation}\\label{ansatz delta}\n\\| {\\mathfrak I}_\\d \\|_{s_0+\\mu}^{{\\mathrm{Lip}(\\g)}} \n\\leq C\\varepsilon^{6 - 2b} \\gamma^{-1}\\, .   \n\\end{equation}\nWe now study the Hamiltonian \n$ K := H_\\varepsilon \\circ G_\\d = \\varepsilon^{-2b} \\mathcal{H} \\circ A_\\varepsilon \\circ G_\\d $ defined in \\eqref{KHG}, \\eqref{def H eps}. \n\nRecalling  \\eqref{def A eps} and \\eqref{trasformazione modificata simplettica} the map $A_\\varepsilon \\circ G_\\d$ has the form  \n\\begin{equation}  \\label{A eps G delta}\nA_\\varepsilon \\circ G_\\d(\\psi, \\eta, w) \n= \\varepsilon \\sum_{j \\in S} \\sqrt{\\xi_j + \\varepsilon^{2(b-1)} |j| [ y_\\d(\\psi) + L_1(\\psi) \\eta + L_2(\\psi) w ]_j } \\, e^{{\\mathrm i}  [\\theta_0(\\psi)]_j} e^{{\\mathrm i}  jx} + \\varepsilon^b (z_0(\\psi) + w)\n\\end{equation}\nwhere \n\\begin{equation}\\label{L1 L2}\nL_1(\\psi) := [\\partial_\\psi \\theta_0(\\psi)]^{-T} \\, , \\quad\nL_2(\\psi) := \\big[ (\\partial_\\theta \\tilde{z}_0) (\\theta_0(\\psi)) \\big]^T \\partial_x^{-1} \\, . \n\\end{equation}\n By Taylor's formula, we develop \\eqref{A eps G delta} in $w$ at $\\eta=0$, $w=0$, and we get\n$ A_\\varepsilon \\circ G_\\d(\\psi, 0, w)  = $ $ T_\\delta(\\psi) + T_1(\\psi) w + T_2(\\psi)[w,w] + $ $  T_{\\geq 3}(\\psi, w) $, \nwhere \n\\begin{equation}\\label{T0}\nT_\\delta(\\psi) := (A_\\varepsilon \\circ G_\\d)(\\psi, 0, 0)\n= \\varepsilon v_\\d(\\psi) + \\varepsilon^b z_0(\\psi) \\, , \\ \\  v_\\d (\\psi):= \\sum_{j \\in S} \\sqrt{\\xi_j + \\varepsilon^{2(b-1)} |j| [ y_\\d(\\psi) ]_j } \\, e^{{\\mathrm i}  [\\theta_0(\\psi)]_j} e^{{\\mathrm i}  jx}\n\\end{equation}\nis the approximate isotropic torus in  phase space (it corresponds to $ i_\\d $ in Lemma \\ref{toro isotropico modificato}), \n\\begin{align}\nT_1(\\psi) w & = \\varepsilon \\sum_{j \\in S} \n\\frac{\\varepsilon^{2(b-1)} |j| [ L_2(\\psi) w ]_j \\, e^{{\\mathrm i}  [\\theta_0(\\psi)]_j}} \n{2 \\sqrt{ \\xi_j + \\varepsilon^{2(b-1)} |j| [ y_\\d(\\psi) ]_j }} \\,  e^{{\\mathrm i}  jx} \n+ \\varepsilon^b w =: \\varepsilon^{2b-1} U_1 (\\psi) w + \\varepsilon^b w \\, \\label{T1} \\\\\nT_2(\\psi)[w,w] &  \n= - \\varepsilon \\sum_{j \\in S} \n\\frac{\\varepsilon^{4(b-1)} j^2 [ L_2(\\psi) w ]_j^2 \\, e^{{\\mathrm i}  [\\theta_0(\\psi)]_j}} \n{8 \\{ \\xi_j + \\varepsilon^{2(b-1)} |j| [ y_\\d(\\psi) ]_j \\}^{3\/2} } \\,  \ne^{{\\mathrm i}  jx} =:  \\varepsilon^{4b - 3} U_2(\\psi)[w,w] \\label{T2}\n\\end{align}\nand $T_{\\geq 3}(\\psi, w)$ collects all the terms of order at least cubic in $w$. In the notation of  \\eqref{def A eps},\nthe function $v_\\d(\\psi) $ in \\eqref{T0} is \n$v_\\d(\\psi) = v_\\varepsilon( \\theta_0(\\psi), y_\\d(\\psi))$.\nThe terms  $U_1, U_2 = O(1)$ in $\\varepsilon$.\nMoreover, using that $ L_2 (\\psi) $ in \\eqref{L1 L2} vanishes as $ z_0 = 0 $,  \nthey satisfy\n\\begin{equation}\\label{extra piccolezza}\n\\| U_1 w \\|_s \\leq \\| {\\mathfrak{I}}_\\d \\|_s \\| w \\|_{s_0}   +  \\| {\\mathfrak{I}}_\\d \\|_{s_0}  \\| w \\|_s  \\, , \\quad\n\\| U_2 [w,w] \\|_s \\leq  \\| {\\mathfrak{I}}_\\d \\|_s \\| {\\mathfrak{I}}_\\d \\|_{s_0} \\| w \\|_{s_0}^2   +  \n\\| {\\mathfrak{I}}_\\d \\|_{s_0}^2 \\| w \\|_{s_0} \\| w \\|_s  \n\\end{equation}\nand also in the  $ \\|  \\ \\|_s^{\\mathrm{Lip}(\\g)} $-norm.\n\nBy Taylor's formula \n$ \\mathcal{H}(u+h) \n= \\mathcal{H}(u) + ( (\\nabla \\mathcal{H})(u), h )_{L^2(\\mathbb T)} \n+ \\frac12 ( (\\partial_u \\nabla \\mathcal{H})(u) [h], h )_{L^2(\\mathbb T)} \n+ O(h^3) $. \nSpecifying at $u = T_\\delta(\\psi)$ and $ h = T_1(\\psi) w + T_2(\\psi)[w,w] + T_{\\geq 3}(\\psi,w)$, \nwe obtain that the sum of all the components of $ K = \\varepsilon^{-2b} (\\mathcal{H} \\circ A_\\varepsilon \\circ G_\\d)(\\psi, 0, w) $ \nthat are quadratic in $w$ is  \n$$\n\\frac12 ( K_{02}w, w )_{L^2(\\mathbb T)} \n= \\varepsilon^{-2b} ( (\\nabla \\mathcal{H})(T_\\delta ), T_2 [w,w] )_{L^2(\\mathbb T)} \n+ \\varepsilon^{-2b} \\frac12 ( (\\partial_u \\nabla \\mathcal{H})(T_\\delta ) [T_1 w], \nT_1 w )_{L^2(\\mathbb T)} \\, . \n$$\nInserting the expressions \\eqref{T1}, \\eqref{T2} we get\n\\begin{align}\nK_{02}(\\psi) w \n& = (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [w]  \n+ 2 \\varepsilon^{b-1} (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [U_1 w] \n+ \\varepsilon^{2(b-1)} U_1^T (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [U_1 w]  \\nonumber\n\\\\ & \\quad \n+ 2 \\varepsilon^{2b- 3} U_2[w, \\cdot]^T (\\nabla \\mathcal{H})(T_\\delta). \\label{K02}\n\\end{align}\n\n\\begin{lemma}\\label{dopo l'approximate inverse}\n\\begin{equation}\\label{piccolezza resti}\n ( K_{02}(\\psi) w, w )_{L^2(\\mathbb T)} \n=  ( (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [w], w )_{L^2(\\mathbb T)} \n+  ( R(\\psi) w, w )_{L^2(\\mathbb T)}\n\\end{equation}\nwhere $R(\\psi)w $ has the ``finite dimensional\" form \n\\begin{equation}\\label{forma buona resto con psi}\nR(\\psi) w  =  {\\mathop\\sum}_{|j| \\leq C} \\big( w , g_j(\\psi) \\big)_{L^2(\\mathbb T)} \\chi_j(\\psi) \n\\end{equation}\nwhere, for some $\\sigma := \\sigma (\\nu, \\tau) > 0$, \n\\begin{align}\n\\label{piccolo FBR}\n \\| g_j \\|_s^{\\mathrm{Lip}(\\g)} \\| \\chi_j \\|_{s_0}^{\\mathrm{Lip}(\\g)} + \\| g_j \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\| \\chi_j \\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^{b+1} \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} \\\\\n\\| \\partial_i g_j [\\widehat \\imath ]\\|_s \\| \\chi_j \\|_{s_0} \n+ \\| \\partial_i g_j [\\widehat \\imath ]\\|_{s_0}  \\| \\chi_j \\|_{s} + \\| g_j \\|_{s_0} \\| \\partial_i \\chi_j [\\widehat \\imath ] \\|_s \n+ \\| g_j \\|_{s} \\| \\partial_i \\chi_j [\\widehat \\imath ]\\|_{s_0}  \n& \\leq_s \\varepsilon^{b + 1} \\| \\widehat \\imath \\|_{s + \\sigma}\\label{derivata piccolo FBR} \\\\ \n& + \\varepsilon^{2b-1} \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}  \\|\\widehat \\imath \\|_{s_0 + \\sigma} \\,,  \\nonumber\n\\end{align}\nand, as usual, $i = (\\theta, y, z)$ (see \\eqref{embedded torus i}), \n$\\widehat \\imath = (\\widehat \\theta, \\widehat y, \\widehat z)$.\n\\end{lemma}\n\n\\begin{proof}\nSince $ U_1 = \\Pi_S U_1 $ and  $ U_2 = \\Pi_S U_2 $, \nthe last three terms in \\eqref{K02} have all the form \\eqref{forma buona resto con psi}\n(argue as in Lemma \\ref{lemma astratto potente}). \nWe now prove that they are also small in size.  \n\nThe contributions in \\eqref{K02}  from $ H_2 $ are better analyzed by the expression\n$$\n\\varepsilon^{-2b} H_2 \\circ A_\\varepsilon \\circ G_\\delta (\\psi, \\eta, w) =   const + \\sum_{j \\in S^+} j^3 \n\\big[  y_\\delta (\\psi) + L_1(\\psi)\\eta + L_2 (\\psi) w  \\big]_j  + \\frac{1}{2} \\int_{\\mathbb T} ( z_0 (\\psi)  + w)_x^2 \\, dx \n$$\nwhich follows by \\eqref{shape H2}, \\eqref{trasformazione modificata simplettica}, \\eqref{L1 L2}.\n Hence the only contribution to \n$ (K_{02} w,w)  $  is $  \\int_{\\mathbb T} w_x^2 \\, dx  $. \nNow we consider the cubic term $ \\mathcal{H}_3 $ in \\eqref{H3tilde}.\nA direct calculation shows that for $ u = v + z $, \n$ \\nabla \\mathcal{H}_3 ( u )  = 3 z^2 + 6  \\Pi_S^\\bot (v z) $, and\n$ \\partial_u \\nabla \\mathcal{H}_3 ( u ) [U_1  w] = 6 \\Pi_S^\\bot ( z U_1 w) $ (since $ U_1 w \\in H_S $).\nTherefore\n\\begin{equation}\\label{mH 3 T delta}\n\\nabla \\mathcal{H}_3(T_\\delta) = 3 \\varepsilon^{2 b} z_0^2 + 6 \\varepsilon^{b + 1} \\Pi_S^\\bot(v_\\delta z_0 ) \n\\,,\\quad \\partial_u \\nabla \\mathcal{H}_3 ( T_\\delta ) [U_1 w] = 6 \\varepsilon^b \\Pi_S^\\bot ( z_0 \\, U_1 w) \\, . \n\\end{equation}\nBy \\eqref{mH 3 T delta} one has \n$  ( (\\partial_u \\nabla \\mathcal{H}_3)(T_\\delta) [U_1 w ], U_1 w )_{L^2(\\mathbb T)}  = 0 $, \nand since also $U_2 = \\Pi_S U_2$,  \n\\begin{equation}\\label{contributi H3}\n\\varepsilon^{b - 1}\\partial_u \\nabla {\\cal H}_3(T_\\delta)[U_1 w] + \\varepsilon^{2 b - 3} U_2[w, \\cdot]^T \\nabla {\\cal H}_3(T_\\delta) = 6 \\varepsilon^{2 b - 1} \\Pi_S^\\bot(z_0  U_1 w) + 3 \\varepsilon^{4 b - 3} U_2[w, \\cdot]^T  z_0^2 \\,.\n\\end{equation}\nThese terms have  the form  \\eqref{forma buona resto con psi} and, using  \\eqref{extra piccolezza}, \\eqref{ansatz 0}, \nthey satisfy \\eqref{piccolo FBR}. \n\nFinally we consider all the terms which arise from ${\\cal H}_{\\geq 4} = O(u^4)$.  \nThe operators\n$ \\varepsilon^{b - 1} \\partial_u \\nabla {\\cal H}_{\\geq 4}(T_\\delta) U_1  $, \n$ \\varepsilon^{2(b - 1)} U_1^T (\\partial_u \\nabla {\\cal H}_{\\geq 4})(T_\\delta) U_1$, \n$ \\varepsilon^{2 b - 3} U_2^T \\nabla {\\cal H}_{\\geq 4}(T_\\delta) $\nhave the form  \\eqref{forma buona resto con psi} and, using $ \\| T_\\delta \\|_s^{\\mathrm{Lip}(\\g)} \\leq \\varepsilon (1 + \\| {\\mathfrak{I}}_\\d \\|_s^{\\mathrm{Lip}(\\g)}) $,\n\\eqref{extra piccolezza}, \\eqref{ansatz 0}, \nthe bound  \\eqref{piccolo FBR} holds. Notice that the biggest term is  $ \\varepsilon^{b - 1} \\partial_u \\nabla {\\cal H}_{\\geq 4}(T_\\delta) U_1  $. \n\nBy \\eqref{derivata i delta} and using explicit formulae \\eqref{L1 L2}-\\eqref{T2} we get estimate \\eqref{derivata piccolo FBR}.\n\\end{proof}\n\nThe conclusion of this section is that,  after the composition with \nthe action-angle variables, the rescaling \\eqref{rescaling kdv quadratica}, \nand the transformation $ G_\\delta $, the linearized operator \nto analyze is $  H_S^\\bot \\ni w \\mapsto (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [w] $,  up to finite dimensional operators which have the form \\eqref{forma buona resto con psi} and size \\eqref{piccolo FBR}. \n\n\n\n\\subsection{The linearized operator in the normal directions}\n\nIn view of \\eqref{piccolezza resti} we now compute  \n$  ( (\\partial_u \\nabla \\mathcal{H})(T_\\delta) [w], w )_{L^2(\\mathbb T)} $, $ w \\in H_S^\\bot $, where\n$ \\mathcal{H}  = H \\circ \\Phi_B $ and $\\Phi_B $ is the Birkhoff map of Proposition \\ref{prop:weak BNF}. \nIt is convenient to estimate separately the terms in \n\\begin{equation}\\label{mH H2H3H5}\n\\mathcal{H}  = H \\circ \\Phi_B = (H_2 + H_3) \\circ \\Phi_B + H_{\\geq 5} \\circ \\Phi_B \n\\end{equation}\nwhere $ H_2, H_3, H_{\\geq 5}$ are defined in \\eqref{H iniziale KdV}.\n\nWe first consider $ H_{\\geq 5} \\circ \\Phi_B $. \nBy \\eqref{H iniziale KdV} we get\n$ \\nabla H_{\\geq 5}(u) = \\pi_0[ (\\partial_u f)(x, u, u_x) ] - \\partial_x \\{ (\\partial_{u_x} f)(x, u,u_x) \\} $, see \\eqref{def pi 0}. \nSince the Birkhoff transformation $ \\Phi_B $ has the form \\eqref{finito finito}, \nLemma \\ref{lemma astratto potente} (at $ u = T_\\delta $, see \\eqref{T0}) implies that\n\\begin{align} \n\\partial_u \\nabla ( H_{\\geq 5} \\circ \\Phi_B ) (T_\\delta)  [h] \n& = \n(\\partial_u \\nabla H_{\\geq 5})(\\Phi_B(T_\\delta)) [h] + {\\cal R}_{H_{\\geq 5}}(T_\\delta)[h] \n\\notag \\\\ \n& = \\partial_x (r_1(T_\\delta) \\partial_x h ) + r_0(T_\\delta) h + {\\cal R}_{H_{\\geq 5}}(T_\\delta)[h] \n\\label{der grad struttura separata5}\n\\end{align}\nwhere the multiplicative functions $r_0(T_\\d)$, $r_1(T_\\d)$ are\n\\begin{alignat}{2} \\label{r0r1 def}\nr_0 (T_\\delta) & := \\sigma_0(\\Phi_B(T_\\delta)), \\qquad & \n\\sigma_0(u) & :=  (\\partial_{uu} f)(x, u, u_x)  -  \\partial_x \\{ (\\partial_{u u_x} f)(x, u, u_x) \\}, \n\\\\\n\\label{sigma0sigma1 def}\nr_1 (T_\\delta) & := \\sigma_1(\\Phi_B(T_\\delta)), \\quad &\n\\sigma_1(u) & := -  (\\partial_{u_x u_x} f)(x, u, u_x) , \n\\end{alignat}\nthe remainder $ {\\cal R}_{H_{\\geq 5}}(u) $ has the form \\eqref{forma buona resto}\nwith $\\chi_j = e^{{\\mathrm i}  jx}$ or $g_j = e^{{\\mathrm i}  jx}$ and, using \\eqref{resti012}, \nit satisfies,  \nfor some $ \\sigma := \\sigma (\\nu, \\tau) > 0$, \n\\begin{align*}\n \\| g_j \\|_s^{\\mathrm{Lip}(\\g)} \\| \\chi_j \\|_{s_0}^{\\mathrm{Lip}(\\g)} + \\| g_j \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\| \\chi_j \\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^4 (1 + \\| {\\mathfrak{I}}_\\d \\|_{s+2}^{\\mathrm{Lip}(\\g)} )  \\\\\n\\| \\partial_i g_j [\\widehat \\imath ]\\|_s \\| \\chi_j \\|_{s_0} \n+ \\| \\partial_i g_j [\\widehat \\imath ]\\|_{s_0}  \\| \\chi_j \\|_{s} \n+ \\| g_j \\|_{s_0} \\| \\partial_i \\chi_j [\\widehat \\imath ] \\|_s \n+ \\| g_j \\|_{s} \\| \\partial_i \\chi_j [\\widehat \\imath ]\\|_{s_0}  \n& \\leq_s  \\varepsilon^4 ( \\| \\widehat \\imath \\|_{s+\\sigma} \n+ \\| {\\mathfrak{I}}_\\d \\|_{s+2} \\| \\widehat \\imath \\|_{s_0 + 2} ).  \n\\end{align*}\nNow we consider the contributions from $ (H_2 + H_3) \\circ \\Phi_B $.\nBy Lemma \\ref{lemma astratto potente} and the expressions of $ H_2, H_3 $ in \\eqref{H iniziale KdV} we deduce that\n$$\n\\partial_u \\nabla ( H_2 \\circ \\Phi_B) (T_\\delta) [h] \n= - \\partial_{xx} h + {\\cal R}_{H_2}(T_\\delta)[h] \\,, \\quad  \n\\partial_u \\nabla ( H_3 \\circ \\Phi_B) (T_\\delta) [h] \n= 6 \\Phi_B (T_\\delta)   h + {\\cal R}_{H_3}(T_\\delta)[h] \\,,\n$$\nwhere $  \\Phi_B (T_\\delta) $ is a function with zero space average, \nbecause $ \\Phi_B: H^1_0 (\\mathbb T_x) \\to H^1_0 (\\mathbb T_x)$ (Proposition \\ref{prop:weak BNF}) \nand  $ {\\cal R}_{H_2}(u) $, $ {\\cal R}_{H_3}(u) $ have the form \\eqref{forma buona resto}. \nBy \\eqref{resti012}, the size $ ( {\\cal R}_{H_2} + {\\cal R}_{H_3}) (T_\\delta )  = O( \\varepsilon ) $. \nWe expand\n$$\n( {\\cal R}_{H_2} + {\\cal R}_{H_3}) (T_\\delta ) = \\varepsilon {\\cal R}_1 + \\varepsilon^2 {\\cal R}_2 +  {\\tilde {\\cal R}}_{> 2} \\,, \n$$  \nwhere $\\tilde \\mathcal{R}_{>2}$ has size $o(\\varepsilon^2)$, \nand we get, $ \\forall h \\in H_S^\\bot $, \n\\begin{equation}\\label{useful repr}\n\\Pi_S^\\bot  \\partial_u \\nabla ((H_2 + H_3) \\circ \\Phi_B) (T_\\delta)  [h] \n= - \\partial_{xx} h +  \\Pi_S^\\bot (6 \\Phi_B (T_\\delta) h ) + \n\\Pi_S^\\bot  ( \\varepsilon {\\cal R}_1 + \\varepsilon^2 {\\cal R}_2 + {\\tilde {\\cal R}}_{> 2} ) [h] \\, . \n\\end{equation}\nWe also develop the function $ \\Phi_B (T_\\delta) $ is powers of $ \\varepsilon $. \nExpand $\\Phi_B (u) = u + \\Psi_2 (u) + \\Psi_{\\geq 3}(u) $, \nwhere $ \\Psi_2 (u) $ is quadratic, $ \\Psi_{\\geq 3} (u) = O(u^3)$, and both map  \n$ H_0^1(\\mathbb T_x)  \\to H_0^1(\\mathbb T_x) $.\nAt $ u = T_\\d = \\varepsilon v_\\d + \\varepsilon^b z_0 $ \nwe get\n\\begin{align}\n\\Phi_B ( T_\\delta ) & = T_\\delta + \\Psi_2 (T_\\delta) + \\Psi_{\\geq 3}(T_\\delta) \n= \\varepsilon v_\\d + \\varepsilon^2 \\Psi_2 ( v_\\d ) + \\tilde q \n\\label{funzione moltiplicativa}\n\\end{align}\nwhere $ \\tilde q := \\varepsilon^b z_0 + \\Psi_2 ( T_\\d ) - \\varepsilon^2 \\Psi_2 (v_\\d) \n+ \\Psi_{\\geq 3} (T_\\d) $ \nhas zero space average and it satisfies  \n$$\n\\| \\tilde q \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\varepsilon^3 + \\varepsilon^b \\| {\\mathfrak{I}}_\\delta \\|_s^{\\mathrm{Lip}(\\g)}\\,,\\quad \n\\| \\partial_i \\tilde q [\\widehat \\imath ] \\|_s \n\\leq_s \\varepsilon^b \\big( \\| \\widehat \\imath \\|_s + \\| {\\mathfrak I}_\\delta \\|_s \\| \\widehat \\imath \\|_{s_0} \\big)\\,.\n$$\nIn particular, its low norm $\\| \\tilde q \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\leq_{s_0} \\varepsilon^{6-b} \\gamma^{-1} = o(\\varepsilon^2)$.  \n\nWe need an exact expression of the terms of order $ \\varepsilon $ and $ \\varepsilon^2 $ in  \\eqref{useful repr}. \nWe compare the Hamiltonian \\eqref{widetilde cal H} with \\eqref{mH H2H3H5},  \nnoting that $ (H_{\\geq 5} \\circ \\Phi_B)(u) = O(u^5) $ \nbecause $ f $ satisfies \\eqref{order5} and $ \\Phi_B (u) = O(u) $. Therefore \n$$\n(H_2 + H_3) \\circ \\Phi_B =  H_2 + \\mathcal{H}_3 + \\mathcal{H}_4  + O(u^5) \\,,\n$$\nand the homogeneous terms of $ (H_2 + H_3) \\circ \\Phi_B $ of degree $ 2, 3, 4 $ in $ u $ are $ H_2 $, $\\mathcal{H}_3 $, $ \\mathcal{H}_4 $ respectively. \nAs a consequence, the terms of order $ \\varepsilon $ and $ \\varepsilon^2 $ in \\eqref{useful repr} (both in the function \n$ \\Phi_B (T_\\d ) $ and in the remainders $ \\mathcal{R}_1, \\mathcal{R}_2 $) come only from $ H_2 + \\mathcal{H}_3 + \\mathcal{H}_4 $. \nActually they come from $ H_2 $, $ \\mathcal{H}_3 $ and $ \\mathcal{H}_{4,2} $  (see \\eqref{H3tilde}, \\eqref{mH3 mH4})\nbecause, at $ u = T_\\d = \\varepsilon v_\\d + \\varepsilon^b z_0 $, \nfor all $ h \\in H_S^\\bot $,  \n$$\n\\Pi_S^\\bot (\\partial_u \\nabla \\mathcal{H}_4)(T_\\delta) [h] = \n\\Pi_S^\\bot (\\partial_u \\nabla \\mathcal{H}_{4,2})(T_\\delta) [h] + o(\\varepsilon^2) \\,. \n$$\nA direct calculation based on the expressions \\eqref{H3tilde}, \\eqref{mH3 mH4}\nshows that, for all $ h \\in H_S^\\bot $, \n\\begin{align}\n\\Pi_S^\\bot (\\partial_u \\nabla ( H_2 + \\mathcal{H}_3 + \\mathcal{H}_4)) (T_\\delta)[h] \n& =  - \\partial_{xx} h +   6 \\varepsilon \\Pi_S^\\bot ( v_\\d  h )  + 6 \\varepsilon^b \\Pi_S^\\bot ( z_0  h ) \n+ \\varepsilon^2 \\Pi_S^\\bot \\big\\{ 6 \\pi_0 [ (\\partial_x^{-1} v_\\d)^2  ]   h  \\nonumber \\\\\n& \n+ 6 v_\\d  \\Pi_S [ (\\partial_x^{-1} v_\\d) (\\partial_x^{-1} h) ] -\n 6  \\partial_x^{-1}  \\{  (\\partial_x^{-1} v_\\d) \\Pi_S [v_\\d  h] \\} \\big\\}  + \no(\\varepsilon^2). \\label{lin basso}\n\\end{align}\nThus, comparing the terms of order $ \\varepsilon, \\varepsilon^2  $ \nin \\eqref{useful repr} (using \\eqref{funzione moltiplicativa}) with those in \\eqref{lin basso} we deduce that the operators $\\mathcal{R}_1, \\mathcal{R}_2$ and the function $\\Psi_2 (v_\\d)$ are\n\\begin{equation}\\label{R nullo1R2}\n{\\cal R}_1 = 0 ,  \\quad \n{\\cal R}_2 [h ] = 6 v_\\d  \\Pi_S \\big[ (\\partial_x^{-1} v_\\d) (\\partial_x^{-1} h)  \\big] \n- 6 \\partial_x^{-1} \\{ (\\partial_x^{-1} v_\\d) \\Pi_S [v_\\d  h] \\} \\,, \\quad \n\\Psi_2 (v_\\d) =  \\pi_0 [ (\\partial_x^{-1} v_\\d)^2  ] . \n\\end{equation}\nIn conclusion,  by \\eqref{mH H2H3H5}, \n\\eqref{useful repr}, \\eqref{der grad struttura separata5},\n\\eqref{funzione moltiplicativa}, \\eqref{R nullo1R2}, we get, \nfor all $ h \\in H_{S^\\bot } $, \n\\begin{align}\n\\Pi_S^\\bot \\partial_u \\nabla \\mathcal{H} (T_\\delta)[h]  \n& =  - \\partial_{xx} h +  \\Pi_S^\\bot \n\\big[ \\big( \\varepsilon 6 v_\\d + \n\\varepsilon^2 6 \\pi_0 [ (\\partial_x^{-1} v_\\d)^2  ] +   q_{>2} + p_{\\geq 4 } \n\\big) h \\big] \\nonumber \\\\\n& \\quad +  \n \\Pi_S^\\bot  \\partial_x (r_1(T_\\delta) \\partial_x h ) + \n \\varepsilon^2   \\Pi_S^\\bot {\\cal R}_2[h] + \\Pi_S^\\bot  {\\cal R}_{> 2} [h] \\label{lafinalina}\n\\end{align}\nwhere  $ r_1 $ is defined in \\eqref{r0r1 def}, $ {\\cal R}_2 $ in \\eqref{R nullo1R2}, the remainder \n$  {\\cal R}_{> 2} :=     {\\tilde {\\cal R}}_{> 2} + {\\cal R}_{H_{\\geq 5}} (T_\\d)  $ \nand the functions\n(using also \\eqref{r0r1 def}, \\eqref{sigma0sigma1 def},  \\eqref{order5}), \n\\begin{align}\\label{def p3}\nq_{>2} & :=   6 \\tilde q \n + \\varepsilon^3 \\big( (\\partial_{uu} f_5)(v_\\d, (v_\\d)_x) -  \\partial_x \\{ (\\partial_{u u_x} f_5)(v_\\d, (v_\\d)_x) \\}  \\big)  \\\\\np_{\\geq 4 } & :=    \nr_0 (T_\\d) - \\varepsilon^3 \\big[ (\\partial_{uu} f_5)(v_\\d, (v_\\d)_x) -  \\partial_x \\{ (\\partial_{u u_x} f_5)(v_\\d, (v_\\d)_x) \\} \\big] \\, . \n \\label{def p>4} \n\\end{align}\n\n\\begin{lemma}\\label{p3 zero average}\n$ \\int_{\\mathbb T} q_{>2} dx = 0 $. \n\\end{lemma}\n\n\\begin{proof}\nWe already\nobserved that $ \\tilde q $ has zero $x$-average as well as\nthe derivative $ \\partial_x \\{ (\\partial_{u u_x} f_5)(v, v_x) \\}  $.\nFinally \n\\begin{equation}\\label{unico pezzo}\n(\\partial_{uu} f_5)(v, v_x) = \\sum_{j_1, j_2, j_3 \\in S } c_{j_1j_2j_3} v_{j_1} v_{j_2} v_{j_3} e^{{\\mathrm i}  (j_1+ j_2 + j_3) x } \\, , \n\\quad v := \\sum_{j \\in S} v_j e^{{\\mathrm i}  j x}\n\\end{equation}\nfor some coefficient  $ c_{j_1j_2j_3} $, and therefore it\nhas zero average by hypothesis (${\\mathtt S}1$). \n\\end{proof}\n\nBy Lemma \\ref{dopo l'approximate inverse}\nand the results of this section (in particular  \\eqref{lafinalina}) we deduce: \n\n\\begin{proposition}\\label{prop:lin}\nAssume \\eqref{ansatz delta}. Then  the Hamiltonian operator $ {\\cal L}_\\omega $ has the form, \n$  \\forall h \\in H_{S^\\bot}^s ( \\mathbb T^{\\nu+1}) $,\n\\begin{equation}\\label{Lom KdVnew}\n{\\cal L}_\\omega h  :=   \\omega \\!\\cdot \\!\\partial_{\\varphi} h - \\partial_x K_{02} h   \n=  \\Pi_S^\\bot \\big( \\omega \\! \\cdot \\! \\partial_\\varphi h + \\partial_{xx} (a_1 \\partial_x h)  +\n\\partial_x ( a_0 h ) \n- \\varepsilon^2 \\partial_x {\\cal R}_2 h - \\partial_x \\mathcal{R}_* h    \\big)  \n\\end{equation}\nwhere $ {\\cal R}_2 $ is defined in \\eqref{R nullo1R2}, \n$ {\\mathcal{R}}_* := {\\cal R}_{> 2} + R(\\psi) $ (with $R(\\psi)$  defined in Lemma \\ref{dopo l'approximate inverse}), the functions \n\\begin{equation}\\label{a1p1p2}\na_1  := 1 - r_1 ( T_\\delta ) \\, , \\quad \na_0 :=  - ( \\varepsilon p_1  + \\varepsilon^2 p_2 + q_{>2} +  p_{\\geq 4} ) \\, , \\quad \np_1 :=  6 v_\\d  \\, , \\quad p_2 :=   6 \\pi_0 [ (\\partial_x^{-1} v_\\d)^2  ]\\,, \n\\end{equation}\nthe function $ q_{>2} $ is defined in \\eqref{def p3} and satisfies $ \\int_{\\mathbb T} q_{>2} dx = 0 $, the function $ p_{ \\geq 4} $ is defined in \\eqref{def p>4},  \n$ r_1 $  in \\eqref{sigma0sigma1 def},  \n$ T_\\delta $ and $ v_\\d $ in \\eqref{T0}.\nFor $ p_k = p_1, p_2 $, \n\\begin{alignat}{2} \\label{stime pk}\n\\| p_k \\|_s^{\\mathrm{Lip}(\\g)}  \n& \\leq_s 1 + \\| {\\mathfrak I}_\\d \\|_s^{\\mathrm{Lip}(\\g)}, \n& \\quad \\qquad \n\\| \\partial_i p_k [ \\widehat \\imath ] \\|_s   \n& \\leq_s \\| \\widehat \\imath \\|_{s+1} +  \\| {\\mathfrak I}_\\delta \\|_{s+1} \\| \\widehat \\imath \\|_{s_0+1}, \n\\\\\n\\label{stima q>2}\n\\| q_{>2} \\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^3 + \\varepsilon^b \\| {\\mathfrak{I}}_\\delta \\|_{s}^{\\mathrm{Lip}(\\g)}\\,,\n& \\quad  \n\\| \\partial_i q_{>2} [\\widehat \\imath ] \\|_s \n& \\leq_s \\varepsilon^b \\big( \\| \\widehat \\imath \\|_{s+1} + \\| {\\mathfrak I}_\\delta \\|_{s+1} \\| \\widehat \\imath \\|_{s_0+1} \\big),\n\\\\\n\\| a_1 -1 \\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s \\varepsilon^3 \\big(  1 + \\| {\\mathfrak I}_\\d \\|_{s+1}^{\\mathrm{Lip}(\\g)} \\big) \\,,  \n& \\quad  \n\\| \\partial_i a_1[\\widehat \\imath ] \\|_s  \n& \\leq_s \\varepsilon^3 \\big( \\| \\widehat \\imath \\|_{s + 1} + \\| {\\mathfrak I}_\\delta \\|_{s + 1} \\| \\widehat \\imath \\|_{s_0 + 1} \\big) \\label{stima a1}  \\\\\n\\| p_{\\geq 4} \\|_s^{\\mathrm{Lip}(\\g)} & \\leq_s \\varepsilon^4 + \\varepsilon^{b + 2} \\| {\\mathfrak{I}}_\\delta \\|_{s + 2}^{\\mathrm{Lip}(\\g)}\\,,& \\quad \\| \n\\partial_i p_{\\geq 4}[\\widehat \\imath ] \\|_s & \\leq_s \\varepsilon^{b + 2} \\big( \\| \\widehat \\imath \\|_{s + 2} + \\| {\\mathfrak{I}}_\\delta\\|_{s + 2} \\| \\widehat \\imath  \\|_{s_0 + 2} \\big) \\label{p geq 4}\n\\end{alignat}\nwhere  $ {\\mathfrak{I}}_\\d (\\varphi) := (\\theta_0(\\varphi) - \\varphi, y_\\d(\\varphi), z_0(\\varphi)) $ corresponds to $T_\\delta$. \nThe remainder $ {\\cal R}_{2} $ has the form \\eqref{forma buona resto} with \n\\begin{align} \\label{stime resto 1}\n\\| g_j \\|_s^{\\mathrm{Lip}(\\g)} + \\| \\chi_j \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s  1+ \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} \\, , \\quad \n\\| \\partial_i g_j [\\widehat \\imath ]\\|_s  + \\| \\partial_i \\chi_j [\\widehat \\imath ]\\|_s\n  \\leq_s   \\| \\widehat \\imath \\|_{s + \\sigma} +   \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}  \\|\\widehat \\imath \\|_{s_0 + \\sigma} \n\\end{align}\nand also $ {\\cal R}_* $ has the form \\eqref{forma buona resto} with\n\\begin{align} \\label{stima cal R*}\n \\| g_j^* \\|_s^{\\mathrm{Lip}(\\g)} \\| \\chi_j^* \\|_{s_0}^{\\mathrm{Lip}(\\g)} + \\| g_j^* \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\| \\chi_j^* \\|_s^{\\mathrm{Lip}(\\g)} \n& \\leq_s  \\varepsilon^3 + \\varepsilon^{b+1} \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}\n \\\\\n\\| \\partial_i g_j^* [\\widehat \\imath ]\\|_s \\| \\chi_j^* \\|_{s_0} \n+ \\| \\partial_i g_j^* [\\widehat \\imath ]\\|_{s_0}  \\| \\chi_j^* \\|_{s} + \\| g_j^* \\|_{s_0} \\| \\partial_i \\chi_j^* [\\widehat \\imath ] \\|_s \n+ \\| g_j^* \\|_{s} \\| \\partial_i \\chi_j^* [\\widehat \\imath ]\\|_{s_0}  \n& \\leq_s  \\varepsilon^{b + 1} \\| \\widehat \\imath \\|_{s + \\sigma}\n\\label{derivate stima cal R*} \\\\ \n& +  \\varepsilon^{2b-1} \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}  \\|\\widehat \\imath \\|_{s_0 + \\sigma} \\nonumber  \\, . \n\\end{align} \n\\end{proposition}\n\nThe bounds \\eqref{stime resto 1}, \\eqref{stima cal R*} imply, by \nLemma \\ref{remark : decay forma buona resto}, estimates \nfor the $ s $-decay norms of $ {\\cal R}_2 $ and \n$  {\\cal R}_* $.   The linearized operator $ {\\cal L}_\\omega := {\\cal L}_\\omega (\\omega, i_\\d (\\omega))$ \ndepends on the parameter $ \\omega $ both directly  and also through the dependence on the \ntorus $   i_\\d (\\omega )  $.\nWe have estimated also the partial derivative $ \\partial_i $ with respect to  the variables $ i $ (see \\eqref{embedded torus i})\nin order to \ncontrol,  along the nonlinear Nash-Moser iteration, \nthe Lipschitz variation of the eigenvalues of $ {\\cal L}_\\omega $ with respect to  $ \\omega $\nand  the approximate solution $ i_\\d $. \n\n\n\n\\section{Reduction of the linearized operator in the normal directions}\\label{operatore linearizzato sui siti normali}\n\nThe goal of this section is to conjugate the Hamiltonian operator $ {\\cal L}_\\omega $ in \\eqref{Lom KdVnew} to the \ndiagonal  operator  $ {\\cal L}_\\infty $ defined in \\eqref{Lfinale}. \nThe proof is obtained applying different kind of symplectic transformations. We shall always assume \\eqref{ansatz delta}. \n\n\n\\subsection{Change of the space variable }\\label{step1}\n\nThe first task is to conjugate $ {\\cal L}_\\omega $ in \\eqref{Lom KdVnew} to  $ {\\cal L}_1 $ in\n \\eqref{cal L1 Kdv},  which has \nthe coefficient of $\\partial_{xxx}$ independent on the space variable. \nWe look for a  $ \\varphi $-dependent family of {\\it symplectic} diffeomorphisms $\\Phi (\\varphi) $ of $ H_S^\\bot $ \nwhich differ from \n\\begin{equation}\\label{primo cambio di variabile modi normali}\n{\\cal A}_{\\bot} := \\Pi_S^\\bot {\\cal A} \\Pi_S^\\bot  \\, ,  \\quad  \n({\\cal A} h)(\\varphi,x) := (1 + \\beta_x(\\varphi,x)) h(\\varphi,x + \\beta(\\varphi,x)) \\, , \n\\end{equation}\nup to a small ``finite dimensional\" remainder, see  \\eqref{forma buona resto cambio di variabile hamiltoniano}.\nEach \n$ {\\cal A}(\\varphi) $ is a  symplectic  map\nof the phase space, see  \\cite{BBM}-Remark 3.3. \nIf  $ \\| \\b \\|_{W^{1,\\infty}} < 1 \/ 2 $ then \n $ {\\cal A} $ is invertible, see Lemma \\ref{lemma:utile}, and \nits inverse and  adjoint maps are \n\\begin{equation}\\label{cambio di variabile inverso}\n({\\cal A}^{-1} h)(\\varphi,y) := (1 + \\tilde{\\beta}_y(\\varphi,y)) h(\\varphi, y + \\tilde{\\beta}(\\varphi,y)) \\,, \n\\quad \n({\\cal A}^T h) (\\varphi,y) = h(\\varphi, y + \\tilde{\\beta}(\\varphi,y)) \n\\end{equation}\nwhere\n$ x = y + \\tilde{\\b} (\\varphi, y) $ is the inverse diffeomorphism (of $\\mathbb T$) of $ y = x + \\b (\\varphi, x) $. \n\nThe restricted maps $ {\\cal A}_\\bot (\\varphi): H_S^\\bot \\to H_S^\\bot $ are not symplectic.  \nIn order to find a symplectic diffeomorphism near  $ {\\cal A}_\\bot (\\varphi) $, \nthe first observation is \nthat each $ {\\cal A }(\\varphi ) $ can be seen as the time $1$-flow of a time dependent \nHamiltonian PDE. Indeed $ {\\cal A }(\\varphi ) $ (for simplicity we skip the dependence on $ \\varphi $) \nis homotopic to the identity \nvia the path of symplectic diffeomorphisms \n$$ \nu \\mapsto  \n(1+ \\tau \\b_x ) u ( x+ \\tau \\b(x) ), \\quad  \\tau \\in [0,1 ] \\, , \n$$\nwhich is the trajectory solution of the time dependent, linear Hamiltonian PDE\n\\begin{equation} \\label{transport-free}\n\\partial_\\tau u = \\partial_x (b(\\tau, x) u) \\, , \\quad b (\\tau, x) := \\frac{\\beta(x)}{1 + \\tau \\beta_x(x)}\\, , \n\\end{equation}\nwith value $ u (x) $ at $ \\tau = 0 $ and $ {\\cal A}u  = (1+ \\b_x(x) ) u ( x+  \\b(x) ) $ at $ \\t = 1 $. \nThe equation \\eqref{transport-free} is a {\\it transport} equation. Its  \nassociated\ncharactheristic ODE is \n \\begin{equation}\\label{equazione delle caratteristiche}\n \\frac{d}{d\\tau} x  = - b(\\tau, x ) \\, .\n \\end{equation}\nWe  denote  its flow by $ \\gamma^{\\tau_0, \\t}  $, namely $  \\gamma^{\\tau_0, \\t} (y) $ is the \n solution of \\eqref{equazione delle caratteristiche} with $  \\gamma^{\\tau_0, \\tau_0} (y) = y $. \nEach  $ \\gamma^{\\tau_0, \\t} $ is a diffeomorphism of the torus $ \\mathbb T_x $. \n\n\\begin{remark}\\label{rem:ca}\nLet $ y \\mapsto y + \\tilde \\beta(\\tau, y)$ be the inverse diffeomorpshim of \n$x \\mapsto x + \\tau \\beta(x)$. \nDifferentiating the identity $\\tilde \\beta(\\tau, y) + \\tau \\beta(y + \\tilde \\beta(\\tau, y)) = 0$ with respect to $\\tau$ it results that \n $ \\gamma^\\t (y) := \\gamma^{0,\\tau} (y) =  y + \\tilde \\beta(\\tau, y) $. \n\\end{remark}\n\nThen we define a symplectic map $\\Phi $ of $ H_S^\\bot $  as the time-1 flow of the Hamiltonian PDE\n \\begin{equation}\\label{problemi di cauchy} \n \\partial_\\tau u = \\Pi_S^\\bot \\partial_x (b(\\tau, x) u) = \\partial_x (b(\\tau, x) u) - \\Pi_S \\partial_x (b(\\tau, x) u)  \\, ,\n \\quad u \\in H_S^\\bot  \\, .\n \\end{equation}\nNote that $ \\Pi_S^\\bot \\partial_x (b(\\tau, x) u) $ is the Hamiltonian vector field generated by \n$ \\frac12 \\int_{\\mathbb T} b(\\tau, x) u^2 dx  $ restricted to $ H_S^\\bot $. \nWe denote by $ \\Phi^{\\tau_0,\\tau} $ the flow of \n \\eqref{problemi di cauchy}, namely $ \\Phi^{\\tau_0, \\tau} (u_0 ) $ is the solution of  \\eqref{problemi di cauchy} \n with initial condition $  \\Phi^{\\tau_0, \\tau_0} (u_0 ) = u_0 $.\nThe flow is   well defined in Sobolev spaces \n$ H^s_{S^\\bot} (\\mathbb T_x) $  for $ b(\\tau, x)  $ is smooth enough\n(standard theory of linear hyperbolic PDEs, see e.g. \nsection 0.8 in \\cite{Taylor}). \n It is natural to expect that the \ndifference between the flow map $ \\Phi := \\Phi^{0,1} $ and $ {\\cal A}_\\bot $ is a  ``finite-dimensional\" remainder\nof the size of $  \\b $. \n\n\\begin{lemma}\\label{modifica simplettica cambio di variabile}\nFor  $ \\| \\beta \\|_{W^{s_0 + 1,\\infty}} $ small,  there exists\nan invertible symplectic transformation $  \\Phi = {\\cal A}_\\bot + {\\cal R}_\\Phi  $ of $ H_{S^\\bot}^s $, \n where $ {\\cal A}_\\bot $ is defined in \\eqref{primo cambio di variabile modi normali} \nand $ {\\cal R}_\\Phi $ is a ``finite-dimensional\" remainder \n\\begin{equation}\\label{forma buona resto cambio di variabile hamiltoniano}\n{\\cal R}_\\Phi h= \\sum_{j \\in S} \\int_0^1 (h, g_j (\\tau)  )_{L^2(\\mathbb T)} \\chi_j (\\tau) d \\tau + \\sum_{j \\in S} \\big(h, \\psi_j \\big)_{L^2(\\mathbb T)} e^{{\\mathrm i}  j x}\n\\end{equation}\nfor some functions $ \\chi_j (\\tau), g_j (\\tau) , \\psi_j \\in H^s $ satisfying \n\\begin{equation}\\label{stime forma buona resto cambio di variabile hamiltoniano}\n\\| \\psi_j\\|_s\\,,\\, \\| g_j(\\tau)\\|_s \\leq_s \\| \\beta\\|_{W^{s + 2, \\infty}}\\,,\n\\quad \\| \\chi_j(\\tau)\\|_s \\leq_s  1 + \\| \\beta \\|_{W^{s + 1, \\infty}} \\,,\\quad \\forall \\tau \\in [0, 1]\\, .\n\\end{equation}\nFurthermore,  \nthe following tame estimates holds\n\\begin{equation}\\label{stime Phi Phi -1}\n\\| \\Phi^{\\pm 1}h\\|_s \\leq_s \\| h \\|_s + \\| \\beta \\|_{W^{s + 2, \\infty}} \\| h \\|_{s_0} \\, , \\quad \\forall h \\in H^s_{S^\\bot} \\,  . \n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nLet $ w (\\tau, x ) := (\\Phi^\\t u_0)(x) $ denote  the solution  of \\eqref{problemi di cauchy} with initial condition \n$ \\Phi^0 (w) = u_0 \\in H_S^\\bot $.  \nThe difference \n\\begin{equation}\\label{differenza voluta}\n({\\cal A}_\\bot - \\Phi) u_0  = \\Pi_S^\\bot {\\cal A} u_0  - w(1, \\cdot) \n= {\\cal A}u_0  - w(1, \\cdot ) - \\Pi_S {\\cal A} u_0  \\, , \\quad \\forall u_0 \\in H_S^\\bot \\, , \n\\end{equation}\nand \n\\begin{equation}\\label{pezzo2}\n\\Pi_S {\\cal A} u_0  = \\Pi_S ({\\cal A} - I) \\Pi_S^\\bot u_0  = \\sum_{j \\in S} \\big(u_0\\,,\\,\\psi_j \\big)_{L^2(\\mathbb T)} e^{{\\mathrm i}  j x}\\,,\n\\quad \\psi_j := ({\\cal A}^T- I) e^{{\\mathrm i}  j x} \\, .\n\\end{equation}\nWe claim that the difference\n\\begin{equation}\\label{differenza}\n{\\cal A} u_0  - w(1, x) =  (1 +  \\beta_x (x) )\\int_0^1 (1 + \\t \\b_x (x) )^{-1}\n\\big[ \\Pi_S \\partial_x (b (\\tau ) w(\\tau) ) \\big] ( \\gamma^\\t ( x + \\b (x) )) \\,d \\tau \n\\end{equation}\nwhere  $ \\gamma^\\t (y)  := \\gamma^{0,\\t} (y) $ is the flow of \\eqref{equazione delle caratteristiche}.\nIndeed the  solution $ w(\\tau, x) $ of \\eqref{problemi di cauchy} satisfies \n$$\n\\partial_\\tau \\{ w(\\tau, \\gamma^\\t (y) ) \\} = b_{x}(\\tau, \\gamma^\\t (y)) w(\\tau, \\gamma^\\t (y)) - \\big[ \\Pi_S \\partial_x (b (\\tau ) w(\\tau) ) \\big] ( \\gamma^\\t (y))\\, .\n$$\nThen, by the variation of constant formula, we find\n$$\nw(\\tau, \\gamma^\\t (y)) =  e^{\\int_0^\\tau b_x (s, \\gamma^s (y))\\,d s} \\Big( u_0(y) - \\int_0^\\tau  e^{- \\int_0^s b_x(\\zeta, \n\\gamma^\\zeta (y) )\\,d\\zeta} \\big[ \\Pi_S \\partial_x (b ( s ) w(s) ) \\big] ( \\gamma^s (y)) \\, d s  \\Big) \\, . \n$$\nSince $ \\partial_y \\gamma^\\t (y) $ solves the variational equation \n$ \\partial_\\tau (\\partial_y \\gamma^\\t (y)) = - b_x (\\t, \\gamma^\\t (y) ) (\\partial_y \\gamma^\\t (y))  $\nwith  $ \\partial_y \\gamma^0 (y)  = 1 $ we have that \n\\begin{equation}\\label{identita equazione variazionale}\ne^{\\int_0^\\tau b_x(s , \\gamma^s (y) )d s} = \\big( {\\partial_{y} \\gamma^\\t (y) } \\big)^{-1} = 1 + \\tau   \\beta_x (x)\n \\end{equation}\n by remark \\ref{rem:ca}, and so we derive the expression \n$$\nw(\\tau, x) = (1 + \\tau \\beta_x (x)) \\Big\\{ u_0(x +  \\tau \\beta(x))  - \\int_0^\\tau \n( 1+ s \\b_x (x) )^{-1} \n \\big[ \\Pi_S \\partial_x (b ( s ) w(s) ) \\big] ( \\gamma^s ( x + \\t \\b (x) )) \\,d s \\Big\\} \\, .\n$$\nEvaluating at $ \\t = 1 $, formula  \\eqref{differenza} follows. \nNext, we develop (recall $ w(\\t) = \\Phi^\\tau (u_0 ) $) \n\\begin{equation}\\label{espressione esplicita gj}\n [\\Pi_S \\partial_x (b(\\tau) w(\\tau))] (x)  \n =  \\sum_{j \\in S} \\big(u_0, g_j(\\tau) \\big)_{L^2(\\mathbb T)} e^{{\\mathrm i}  j x} \\, , \\quad\n g_j(\\tau) := - (\\Phi^\\tau)^T[b(\\tau) \\partial_x e^{{\\mathrm i}  j x}]\\,, \n\\end{equation}\nand  \\eqref{differenza} becomes  \n\\begin{equation}\\label{formula utile 4}\n{\\cal A} u_0 - w(1, \\cdot) = - \\int_0^1 \\sum_{j \\in S} \\big(u_0\\,,\\, g_j(\\tau) \\big)_{L^2(\\mathbb T)} \\chi_j(\\tau, \\cdot )\\,d\\tau\\,,\n\\end{equation}\nwhere \n\\begin{equation}\\label{espressione chi j}\n\\chi_j(\\tau, x) :=  - ( 1 + \\b_x (x) ) (1 + \\t \\b_x (x))^{-1} e^{{\\mathrm i}  j \\gamma^\\t (x + \\b(x) )} \\, .  \n\\end{equation}\n\nBy \\eqref{differenza voluta}, \\eqref{pezzo2}, \\eqref{differenza}, \\eqref{formula utile 4} we deduce that \n$ \\Phi = {\\cal A}_\\bot + {\\cal R}_\\Phi $ as in \\eqref{forma buona resto cambio di variabile hamiltoniano}.\n\n\\smallskip\n\nWe now prove the estimates \\eqref{stime forma buona resto cambio di variabile hamiltoniano}. \nEach function $ \\psi_j  $ in \\eqref{pezzo2} satisfies $  \\|\\psi_j \\|_s \\leq_s \\| \\beta \\|_{W^{s, \\infty}} $, \nsee \\eqref{cambio di variabile inverso}.\nThe bound $ \\| \\chi_j(\\tau)\\|_s \\leq_s  1 + \\| \\beta \\|_{W^{s + 1, \\infty}} $ follows by \\eqref{espressione chi j}.\nThe  tame estimates for $ g_j(\\tau) $ defined in \\eqref{espressione esplicita gj} are more difficult because \nrequire tame estimates  for the adjoint $(\\Phi^\\tau)^T$,  $ \\forall \\tau \\in [0, 1] $.\nThe adjoint of the flow map can be represented \nas  the flow map of the ``adjoint'' PDE \n\\begin{equation}\\label{equazione aggiunta}\n\\partial_\\tau z =  \\Pi_S^\\bot \\{ b(\\tau, x) \\partial_x \\Pi_S^\\bot z \\}  =  \nb(\\tau, x) \\partial_x z - \\Pi_S (b(\\tau, x)  \\partial_x  z ) \\, , \\quad  z \\in H_S^\\bot \\, , \n\\end{equation}\nwhere  $ - \\Pi_S^\\bot b(\\tau,x) \\partial_x $ is the $ L^2 $-adjoint of the Hamiltonian \nvector field in \\eqref{problemi di cauchy}.\nWe denote by $ \\Psi^{\\tau_0, \\tau} $ \nthe flow of  \\eqref{equazione aggiunta}, namely $ \\Psi^{\\tau_0, \\tau} (v) $ \nis the solution of \n\\eqref{equazione aggiunta} with $ \\Psi^{\\tau_0, \\tau_0} (v)  = v  $. \nSince the derivative $  \\partial_\\t (\\Phi^\\t (u_0)  , \\Psi^{\\tau_0,\\t} (v) )_{L^2(\\mathbb T)}  = 0  $, $ \\forall \\t  $,  we deduce that \n$  ( \\Phi^{\\tau_0} (u_0)  , \\Psi^{\\tau_0,\\tau_0} (v) )_{L^2(\\mathbb T)} =  ( \\Phi^0 (u_0)  , \\Psi^{\\tau_0,0} (v) )_{L^2(\\mathbb T)} $, namely\n(recall that $ \\Psi^{\\tau_0,\\tau_0} (v) = v $) the adjoint \n\\begin{equation}\\label{adjoint flow}\n(\\Phi^{\\tau_0})^T = \\Psi^{\\tau_0, 0}  \\,, \\quad \\forall \\tau_0 \\in [0,1] \\, . \n\\end{equation}\nThus it is sufficient to prove tame estimates for the flow $ \\Psi^{\\t_0, \\tau} $.\nWe first provide a useful expression for the solution  $ z (\\tau, x) := \\Psi^{\\tau_0, \\tau} (v) $ \nof  \\eqref{equazione aggiunta}, obtained by the methods of characteristics. \nLet $ \\gamma^{\\t_0,\\tau} (y) $ be the  flow of \\eqref{equazione delle caratteristiche}.\nSince $ \\partial_\\tau z (\\tau, \\gamma^{\\t_0, \\t} (y)) = - [\\Pi_S (b(\\t) \\partial_x z (\\t) ] ( \\gamma^{\\t_0,\\t} (y))  $ we get \n$$\nz(\\tau,  \\gamma^{\\t_0,\\tau} (y)) = v(y) + \\int_\\tau^{\\tau_0} \n[\\Pi_S (b(s) \\partial_x z (s) ] ( \\gamma^{\\t_0,s} (y)) \\,d s\\, , \\quad \\forall \\t \\in [0,1] \\, . \n$$\nDenoting by $ y = x + \\sigma(\\tau, x)$ \nthe inverse diffeomorphism of $ x = \\gamma^{\\tau_0, \\tau} (y) = y +  {\\tilde  \\sigma}(\\tau, y) $, we get\n\\begin{align}\n\\Psi^{\\tau_0, \\tau}(v) = z ( \\tau, x)  & =  \nv(x + \\sigma(\\tau, x)) + \\int_\\tau^{\\tau_0}  \n[\\Pi_S (b(s) \\partial_x z (s) ] ( \\gamma^{\\t_0,s} (x + \\sigma(\\tau, x))) \\, d s \\nonumber \\\\\n& = v(x + \\sigma(\\tau, x)) +\\int_\\tau^{\\tau_0} \\sum_{j \\in S} (z(s), p_j(s)) \\kappa_j(s, x)\\,d s = v( x +   \\sigma(\\tau,x) ) +  {\\cal R}_\\tau v\\,,   \\label{Psi-expression} \n\\end{align}\nwhere \n$  p_j (s) := - \\partial_x (b(s) e^{{\\mathrm i}  j x}) $, \n$ \\kappa_j(s, x) :=  e^{{\\mathrm i}  j \\gamma^{\\tau_0, s} (x + \\sigma(\\tau, x))} $ and \n$$\n({\\cal R}_\\tau v)(x) :=  \\int_\\tau^{\\tau_0} \\sum_{j \\in S} (\\Psi^{\\tau_0,s}(v), p_j(s))_{L^2(\\mathbb T)} \\kappa_j(s, x)\\,d s\\,.\n$$\nSince \n$\\| \\sigma(\\tau, \\cdot) \\|_{W^{s,\\infty}} $, $ \\| \\tilde \\sigma(\\tau, \\cdot)\\|_{W^{s,\\infty}} \n\\leq_s \\| \\beta \\|_{W^{s + 1,\\infty}}$ (recall also \\eqref{transport-free}), we derive   \n$ \\| p_j \\|_s \\leq_s \\| \\beta \\|_{W^{s + 2,\\infty}} $,  \n$ \\| \\kappa_j\\|_s \\leq_s 1 + \\| \\beta\\|_{W^{s + 1,\\infty}} $\nand \n$ \\| v(x +  \\sigma(\\tau, x)) \\|_s \\leq_s \\| v\\|_s + \\| \\beta\\|_{W^{s + 1,\\infty}} \\| v\\|_{s_0} $, $ \\forall \\tau \\in [0,1] $. \nMoreover\n$$\n\\| {\\cal R}_\\tau v\\|_s \\leq_s {\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0,\\tau}(v) \\|_s \n\\| \\beta\\|_{W^{s_0 + 2,\\infty}} +{\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0, \\tau}(v) \\|_{s_0} \\| \\beta\\|_{W^{s + 2,\\infty}}\\, .\n$$\nTherefore, for all $ \\tau \\in [0, 1] $, \n\\begin{equation}\\label{tame aggiunto flusso parziale}\n\\| \\Psi^{\\tau_0,\\tau} v\\|_s \\leq_s  \\| v\\|_s + \\| \\beta\\|_{W^{s + 1,\\infty}} \n\\| v\\|_{s_0} + {\\rm sup}_{\\tau \\in [0, 1]} \\big\\{ \\| \\Psi^{\\tau_0, \\tau} v\\|_s  \\| \\beta\\|_{W^{s_0 + 2,\\infty}}  +\n\\| \\Psi^{\\tau_0, \\tau} v\\|_{s_0} \\| \\beta\\|_{W^{s + 2,\\infty}} \\big\\}\\,. \n\\end{equation}\nFor $s = s_0$ it implies \n$$\n{\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0, \\tau}(v)\\|_{s_0} \\leq_{s_0} \\| v \\|_{s_0}(1  + \n\\| \\beta \\|_{W^{s_0 + 1,\\infty}}) + {\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0, \\tau}(v)\\|_{s_0} \\| \\beta\\|_{W^{s_0 + 2,\\infty}} \n$$\nand so, for  $\\| \\beta\\|_{W^{s_0 + 2,\\infty} } \\leq c(s_0) $  small enough, \n\\begin{equation}\\label{tame norma bassa aggiunto flusso}\n{\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0, \\tau}(v)\\|_{s_0} \\leq_{s_0} \\| v \\|_{s_0} \\, .\n\\end{equation}\nFinally  \\eqref{tame aggiunto flusso parziale}, \\eqref{tame norma bassa aggiunto flusso} imply \nthe tame estimate\n\\begin{equation}\\label{tame aggiunto flusso}\n{\\rm sup}_{\\tau \\in [0, 1]} \\| \\Psi^{\\tau_0, \\tau} (v)\\|_s \\leq_s \\| v \\|_s + \\| \\beta\\|_{W^{s + 2,\\infty}} \\| v \\|_{s_0}\\,.\n\\end{equation}\nBy \\eqref{adjoint flow} and \\eqref{tame aggiunto flusso} \nwe deduce the bound \\eqref{stime forma buona resto cambio di variabile hamiltoniano} \nfor $ g_j $  defined in \\eqref{espressione esplicita gj}.\nThe tame estimate \\eqref{stime Phi Phi -1} for $\\Phi $ follows by  that of $ {\\cal A} $ and \n\\eqref{stime forma buona resto cambio di variabile hamiltoniano}\n (use Lemma \\ref{lemma:utile}). \nThe estimate for $\\Phi^{- 1}$ follows in the same way because \n$\\Phi^{-1 } = \\Phi^{1,0} $ is the backward flow.\n\\end{proof}\n\nWe  conjugate  $ {\\cal L}_\\omega $  in  \\eqref{Lom KdVnew} \nvia the symplectic map $ \\Phi = {\\cal A}_\\bot + {\\cal R}_\\Phi $\nof Lemma \\ref{modifica simplettica cambio di variabile}. We compute \n (split $ \\Pi_S^{\\bot} = I - \\Pi_S $)\n\\begin{equation}\\label{LAbot}\n{\\cal L}_\\omega \\Phi =  \\Phi {\\cal D}_\\omega + \n\\Pi_S^\\bot {\\cal A} \\big( \nb_3 \\partial_{yyy}  + b_2 \\partial_{yy}   + b_1 \\partial_{y}  + b_0   \\big) \\Pi_S^\\bot +  {\\cal R}_I \\,,\n\\end{equation}\nwhere the coefficients are\n\\begin{align}\n& b_3 (\\varphi,y) := {\\cal A}^T [ a_1 ( 1 + \\b_x)^3  ] \\label{step1: b3KdV} \n\\qquad \\qquad\nb_2 (\\varphi,y) := {\\cal A}^T \\big[ 2 (a_1)_x (1 + \\beta_x )^2 + \n6 a_1  \\beta_{xx} (1 + \\beta_x )\\big]  \\\\ \n& b_1 (\\varphi,y) := \n {\\cal A}^T \\Big[ ({\\cal D}_\\omega \\beta) + 3 a_1 \\frac{\\beta_{xx}^2 }{1 + \\beta_x} + \n 4 a_1 \\beta_{xxx} + 6 (a_1)_x \\beta_{xx}  + \n (a_1)_{xx} (1 + \\beta_x) + a_0 (1 + \\b_x) \\Big] \\label{tilde b1KdV} \n\\\\ \n& b_0 (\\varphi,y) := \n{\\cal A}^T \\Big[ \\frac{({\\cal D}_\\omega \\beta_x)}{1 + \\b_x} + \n a_1 \\frac{\\beta_{xxxx}}{1+ \\b_x} + \n 2 ( a_{1})_{x} \\frac{\\beta_{xxx}}{1+ \\b_x} +  ( a_{1})_{xx} \\frac{\\beta_{xx}}{1+ \\b_x} +\na_0  \\frac{\\beta_{xx}}{1+ \\b_x}  + (a_0)_x   \\Big] \\label{tilde b0KdV}\n\\end{align}\n and the remainder\n\\begin{align}\n {\\cal R}_I &:= -\n\\Pi_S^\\bot \\partial_x (   \\varepsilon^2 {\\cal R}_2 + \\mathcal{R}_{*}    ) {\\cal A}_\\bot  - \\Pi_S^\\bot\n \\big( a_1 \\partial_{xxx}  + 2 \n(a_1)_x \\partial_{xx}  + ( (a_{1})_{xx} + a_0)\\partial_x  \n+ (a_0)_x \\big)  \\Pi_{S} {\\cal A}  \\Pi_S^\\bot \\,  \\nonumber\\\\\n & \\quad +[{\\cal D}_\\omega, {\\cal R}_\\Phi] + ({\\cal L}_\\omega - {\\cal D}_\\omega) {\\cal R}_\\Phi\\,. \\label{calR1KdV}\n\\end{align}\nThe commutator $[{\\cal D}_\\omega, {\\cal R}_\\Phi] $ has the form \\eqref{forma buona resto cambio di variabile hamiltoniano} with ${\\cal D}_\\omega g_j$ or ${\\cal D}_\\omega \\chi_j$, ${\\cal D}_\\omega \\psi_j$ instead of $\\chi_j$, $g_j$, $\\psi_j$ respectively. Also the last term $({\\cal L}_\\omega - {\\cal D}_\\omega) {\\cal R}_\\Phi$ in \\eqref{calR1KdV} has the form \\eqref{forma buona resto cambio di variabile hamiltoniano} (note that ${\\cal L}_\\omega - {\\cal D}_\\omega$ does not contain derivatives with respect to $\\varphi$). By \\eqref{LAbot}, and decomposing $ I = \\Pi_S + \\Pi_S^\\bot $, we get \n\\begin{align} \\label{L A bot finaleKdV}\n{\\cal L}_\\omega \\Phi \n= {} & \\Phi  ( {\\cal D}_\\omega  +  b_3 \\partial_{yyy}  + b_2 \\partial_{yy}   + b_1 \\partial_{y}  + b_0 ) \\Pi_S^\\bot\n+ {\\cal R}_ {II} \\,,\n\\\\\n\\label{calR2KdV}\n{\\cal R}_{II}  \n:= {} &  \\big\\{\\Pi_S^\\bot ({\\cal A} - I) \\Pi_{S} - {\\cal R}_\\Phi  \\big\\} \n( b_3 \\partial_{yyy}  + b_2 \\partial_{yy}   + b_1 \\partial_{y}  + b_0 ) \\Pi_S^\\bot   + {\\cal R}_I \\,. \n\\end{align}\nNow we choose the function $ \\beta = \\b (\\varphi, x)  $  such that \n\\begin{equation}\\label{choice beta}\na_1(\\varphi, x) (1 + \\b_x (\\varphi, x))^3 = b_3 (\\varphi) \n\\end{equation}\nso that the coefficient $ b_3 $ in \\eqref{step1: b3KdV} depends only on $ \\varphi $\n(note that $ {\\cal A}^T [b_3 (\\varphi)]  = b_3 (\\varphi) $). \nThe only solution of \\eqref{choice beta} with zero space average is (see e.g. \\cite{BBM}-section 3.1)\n\\begin{equation}\\label{sol beta}\n\\b  := \\partial_x^{-1} \\rho_0 , \\quad \n\\rho_0 := b_3 (\\varphi)^{1\/3} (a_1 (\\varphi, x))^{-1\/3} - 1,  \\quad\nb_3 (\\varphi) := \\Big( \\frac{1}{2 \\pi} \\int_{\\mathbb T} (a_1 (\\varphi, x))^{-1\/3} dx \\Big)^{-3}. \n\\end{equation}\nApplying  the symplectic map  $ \\Phi^{-1} $ in \\eqref{L A bot finaleKdV} \nwe  obtain the Hamiltonian operator (see Definition \\ref{operatore Hamiltoniano})\n\\begin{equation}\\label{cal L1 Kdv}\n{\\cal L}_1 := \\Phi^{-1} {\\cal L}_\\omega \\Phi\n= \\Pi_S^\\bot \\big( \\omega \\cdot \\partial_\\varphi + b_3(\\varphi) \\partial_{yyy} \n+ b_1 \\partial_y + b_0 \\big) \\Pi_S^\\bot +  {\\mathfrak R}_1 \n\\end{equation}\nwhere $ {\\mathfrak R}_1 := \\Phi^{-1} {\\cal R}_{II} $. We used that, \nby the Hamiltonian nature of $ {\\cal L}_1 $, the coefficient \n$ b_2 = 2 (b_3)_y $ (see \\cite{BBM}-Remark 3.5) and so, by the choice \\eqref{sol beta}, we have\n$ b_2  =  2 (b_3)_y = 0 $.\nIn the next Lemma we analyse the structure of the remainder ${\\mathfrak R}_1$.\n\\begin{lemma} \\label{cal R3}\nThe operator $ {\\mathfrak R}_1 $ has the form \\eqref{forma buona con gli integrali}.\n\\end{lemma}  \n\n\\begin{proof}\nThe remainders $ {\\cal R}_I $ and $ {\\cal R}_{II} $ have the form \\eqref{forma buona con gli integrali}. Indeed\n $ {\\cal R}_2,  {\\cal R}_* $ in \\eqref{calR1KdV} have the form \\eqref{forma buona resto}\n (see Proposition \\ref{prop:lin}) and the term\n$ \\Pi_S {\\cal A} w = \\sum_{j \\in S } ( {\\cal A}^T e^{{\\mathrm i}  j x}, w)_{L^2(\\mathbb T) } e^{{\\mathrm i}  jx} $\nhas the same form. By \\eqref{forma buona resto cambio di variabile hamiltoniano}, the terms of ${\\cal R}_I$, ${\\cal R}_{II}$ which involves the operator ${\\cal R}_\\Phi$ have the form \\eqref{forma buona con gli integrali}.\nAll the operations involved preserve this structure: \nif $R_\\tau w = \\chi(\\tau) (w, g(\\tau) )_{L^2(\\mathbb T)} $, $\\tau \\in [0, 1]$, then \n\\begin{alignat*}{3}\nR_\\tau \\Pi_S^\\bot w & = \\chi(\\tau) (\\Pi_S^\\bot g(\\tau) ,  w)_{L^2(\\mathbb T)}\\,, \\  & \nR_\\tau {\\cal A} w & = \\chi(\\tau) ({\\cal A}^T g(\\tau) ,  w)_{L^2(\\mathbb T)} \\,, \\ &\n\\partial_x R_\\tau w & = \\chi_x(\\tau) (g(\\tau) , w)_{L^2(\\mathbb T)} \\,, \n\\\\\n\\Pi_S^\\bot R_\\tau w & = (\\Pi_S^\\bot \\chi(\\tau)) (g(\\tau) , w)_{L^2(\\mathbb T)} \\,, \\   & \n{\\cal A} R_\\tau w & = ({\\cal A} \\chi(\\tau)) (g(\\tau) ,  w)_{L^2(\\mathbb T)} \\,, \\  &\n\\Phi^{-1} R_\\tau w & = (\\Phi^{-1} \\chi(\\tau)) (g(\\tau), w)_{L^2(\\mathbb T)} \n\\end{alignat*}\n(the last equality holds because $ \\Phi^{-1} ( f (\\varphi) w ) = f (\\varphi) \\Phi^{-1} (  w ) $ for all function $f(\\varphi)$).  \n Hence $ {\\mathfrak R}_1 $ has the form \\eqref{forma buona con gli integrali} \n  where $ \\chi_j(\\tau)  \\in H_S^\\bot $ for all $\\tau \\in [0, 1]$.\n\\end{proof}\n\nWe now put in evidence the terms of order $ \\varepsilon, \\varepsilon^2, \\ldots $, in $ b_1 $, $ b_0  $, $ \\mathfrak R_1 $, recalling that $ a_1 -1 = O(\\varepsilon^3 )$ (see \\eqref{stima a1}), $ a_0 = O( \\varepsilon) $ (see \\eqref{a1p1p2}-\\eqref{p geq 4}),\nand  $ \\beta  = O( \\varepsilon^3)  $  (proved below in \\eqref{stima beta}).\nWe expand  $ b_1$ in \\eqref{tilde b1KdV} as\n\\begin{equation}\\label{b1 Kdv}\nb_1   = - \\varepsilon p_1 -\\varepsilon^2 p_2 - q_{>2} +  {\\cal D}_\\omega \\beta +  4  \\b_{xxx} + (a_1)_{xx} \n  + b_{1, \\geq 4}\n\\end{equation}\nwhere $ b_{1, \\geq 4} = O(\\varepsilon^4) $ is defined by difference (the precise estimate is in Lemma \\ref{lemma:stime coeff mL1}). \n \n\\begin{remark}\\label{media beta}\nThe function \n$ {\\cal D}_\\omega \\beta $ has zero average in $ x $ by \\eqref{sol beta} as well as $ (a_1)_{xx}, \\b_{xxx} $. \n\\end{remark}\n\nSimilarly, we expand $ b_0 $ in \\eqref{tilde b0KdV} as \n\\begin{equation} \\label{b0 Kdv} \nb_0 = - \\varepsilon (p_1)_x - \\varepsilon^2 (p_2)_x - (q_{>2})_x  +  {\\cal D}_\\omega \\beta_x + \\b_{xxxx} + b_{0, \\geq 4} \n\\end{equation}\nwhere $ b_{0, \\geq 4} = O(\\varepsilon^4) $ is defined by difference.\n\nUsing the equalities \\eqref{calR2KdV},  \\eqref{calR1KdV} and $ \\Pi_S {\\cal A} \\Pi_S^\\bot = \\Pi_S ({\\cal A} - I) \\Pi_S^\\bot $ we get\n\\begin{equation}\\label{resto1 Kdv}\n{\\mathfrak R}_1 := \\Phi^{-1} {\\cal R}_{II} = \n-  \\varepsilon^2 \\Pi_S^\\bot \\partial_x  {\\cal R}_2 + {\\cal R}_{*} \n\\end{equation}\nwhere $ {\\cal R}_2 $ is defined in \\eqref{R nullo1R2} and we have renamed $\\mathcal{R}_*$ \nthe term of order $ o(\\varepsilon^2) $ in $\\mathfrak{R}_1$. \nThe remainder $ {\\cal R}_{*} $ in \\eqref{resto1 Kdv} has the form \\eqref{forma buona con gli integrali}. \n\n\\begin{lemma} \\label{lemma:stime coeff mL1}\nThere is $\\sigma = \\sigma(\\t,\\nu) > 0$ such that \n\\begin{align} \\label{stima beta}\n\\| \\b \\|_s^{\\mathrm{Lip}(\\g)}   \\leq_s \\varepsilon^3 (1 + \\| {\\mathfrak{I}}_\\d \\|_{s + 1}^{\\mathrm{Lip}(\\g)} ), \\qquad \n\\| \\partial_i \\b [\\widehat \\imath ] \\|_s \n& \\leq_s \\varepsilon^3 \\big( \\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}  \\| \\widehat \\imath  \\|_{s_0 + \\sigma}\\big)\\,, \\\\\n\\label{stima b1 b0}\n \\| b_3 - 1 \\|_s^{\\mathrm{Lip}(\\g)}  \n \\leq_s \\varepsilon^4 + \\varepsilon^{b + 2} \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}, \\qquad  \\| \\partial_i b_3[\\widehat \\imath ]\\|_s & \\leq_s \\varepsilon^{b + 2} \\big( \\| \\widehat \n \\imath \\|_{s + \\sigma} + \\| {\\mathfrak{I}}_\\delta\\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\big)\n\\\\\n\\label{stima Di b1 b0}\n\\| b_{1, \\geq 4} \\|_s^{\\mathrm{Lip}(\\g)} + \\| b_{0, \\geq 4} \\|_s^{\\mathrm{Lip}(\\g)} & \\leq_s \\varepsilon^4 + \\varepsilon^{b + 2} \\| {\\mathfrak{I}}_\\delta\\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}  \\\\\n\\label{stima Di b geq 4}\n  \\| \\partial_i b_{1, \\geq 4}[\\widehat \\imath ] \\|_s + \\| \\partial_i b_{0, \\geq 4}[\\widehat  \\imath ]\\|_s  & \\leq_s \\varepsilon^{b + 2}\\big( \\| \\widehat \\imath \\|_{s + \\sigma} + \\| {\\mathfrak{I}}_\\delta\\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\big) . \n\\end{align}\nThe transformations $\\Phi$, $\\Phi^{-1}$ satisfy\n\\begin{align}\n\\label{stima cal A bot}\n\\|\\Phi^{\\pm 1} h \\|_s^{{\\mathrm{Lip}(\\g)}} & \\leq_s \\| h \\|_{s + 1}^{{\\mathrm{Lip}(\\g)}} + \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{{\\mathrm{Lip}(\\g)}} \\| h \\|_{s_0 + 1}^{{\\mathrm{Lip}(\\g)}} \\\\\n\\label{stima derivata cal A bot}\n\\| \\partial_i (\\Phi^{\\pm 1}h) [\\widehat \\imath] \\|_s & \\leq_s \n\\| h \\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} + \n\\| h\\|_{s_0 + \\sigma} \\| \\widehat \\imath \\|_{s + \\sigma} + \n\\| {\\mathfrak I}_\\delta\\|_{s + \\sigma} \\| h\\|_{s_0 + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\,.\n\\end{align}\nMoreover the remainder ${\\cal R}_{*}$ has the form \\eqref{forma buona con gli integrali},\nwhere the functions $\\chi_j(\\tau)$, $g_j(\\tau)$ satisfy the estimates \n\\eqref{stima cal R*}-\\eqref{derivate stima cal R*} uniformly in $\\tau \\in [0, 1]$.\n\\end{lemma}\n\n\n\\begin{proof}\nThe estimates \\eqref{stima beta} follow by  \\eqref{sol beta}, \\eqref{stima a1}, \nand the usual interpolation and tame estimates in Lemmata \n\\ref{lemma:composition of functions, Moser}-\\ref{lemma:utile} (and Lemma \\ref{stima Aep}) and \\eqref{ansatz delta}.  \nFor the estimates of $ b_3 $, by \\eqref{sol beta} and  \\eqref{a1p1p2} we consider \nthe function $ r_1 $ defined in \\eqref{sigma0sigma1 def}. Recalling also\n\\eqref{finito finito} and \\eqref{T0}, the  function \n$$ \nr_1 (T_\\delta ) =  \\varepsilon^3 (\\partial_{u_x u_x} f_5) (v_\\d, (v_\\d)_x ) + r_{1, \\geq 4} \\, , \\quad \nr_{1, \\geq 4} := r_1 (T_\\delta ) -  \\varepsilon^3 (\\partial_{u_x u_x} f_5) (v_\\d, (v_\\d)_x ) \\, . \n$$\nHypothesis (${\\mathtt S}1$) implies, as in the proof of Lemma \\ref{p3 zero average}, that the space average\n$ \\int_{\\mathbb T} (\\partial_{u_x u_x} f_5) (v_\\d, (v_\\d)_x ) dx = 0 $. Hence the bound \n\\eqref{stima b1 b0} for $ b_3 - 1 $ follows.\nFor the estimates on $\\Phi$, $\\Phi^{-1} $ we apply Lemma \\ref{modifica simplettica cambio di variabile} and the estimate \\eqref{stima beta} for $\\beta$ .\nWe estimate the remainder ${\\cal R}_*$ in \\eqref{resto1 Kdv}, using \\eqref{calR1KdV}, \\eqref{calR2KdV} and\n \\eqref{stima cal R*}-\\eqref{derivate stima cal R*}.\n\\end{proof}\n\n\n\n\\subsection{Reparametrization of time}\\label{step2}\n\nThe goal of this section is to make constant the coefficient of the highest order \nspatial derivative operator $ \\partial_{yyy} $, by a quasi-periodic reparametrization of time. \nWe consider the change of variable \n$$\n(B w)(\\varphi, y) := w(\\varphi + \\omega \\alpha(\\varphi), y), \n\\qquad  \n( B^{-1} h)(\\vartheta  , y ) := h(\\vartheta + \\omega \\tilde{\\alpha}(\\vartheta), y)\\,,\n$$\nwhere $ \\varphi = \\vartheta + \\omega \\tilde{\\alpha}(\\vartheta )$ \nis the inverse diffeomorphism of $ \\vartheta =  \\varphi + \\omega \\alpha(\\varphi) $ in $\\mathbb T^\\nu$.\nBy conjugation, the differential operators become \n\\begin{equation}  \\label{anche def rho}\nB^{-1} \\omega \\cdot \\partial_\\varphi B = \\rho(\\vartheta)\\,   \\omega \\cdot \\partial_{\\vartheta} ,\n\\quad \nB^{-1}  \\partial_y B = \\partial_y, \n\\quad \n\\rho := B^{-1} (1 + \\omega \\cdot \\partial_{\\varphi} \\a).\n\\end{equation}\nBy \\eqref{cal L1 Kdv}, using also that   $  B $ and $ B^{-1} $ commute with $ \\Pi_S^\\bot $, we get \n\\begin{equation} \\label{secondo coniugio siti normali Kdv}\n B^{-1} {\\cal L}_1 B \n= \\Pi_S^\\bot [ \\rho  \\omega \\cdot \\partial_{\\vartheta} \n + (B^{-1} b_3)  \\partial_{yyy}  \n + ( B^{-1}  b_1 ) \\partial_y + ( B^{-1} b_0 ) ] \\Pi_S^\\bot \n + B^{-1} {\\mathfrak R}_1 B.\n\\end{equation}\nWe choose $ \\alpha $ such that \n\\begin{equation}\\label{B3solu}\n(B^{-1}b_3 )(\\vartheta ) = m_3 \\rho (\\vartheta) \\,, \\quad m_3 \\in \\mathbb R \\, , \\quad \\text{namely} \\ \\ \\  \n b_3 (\\varphi) = m_3 ( 1 + \\omega \\cdot \\partial_\\varphi \\a (\\varphi) ) \n\\end{equation}\n(recall  \\eqref{anche def rho}). \nThe unique solution with zero average of \\eqref{B3solu} is\n\\begin{equation}  \\label{def alpha m3}\n\\a (\\varphi) := \\frac{1}{m_3} ( \\omega \\cdot \\partial_\\varphi )^{-1} ( b_3 - m_3 ) (\\varphi) , \n\\qquad \nm_3 := \\frac{1}{(2 \\pi)^\\nu} \\int_{\\mathbb T^\\nu} b_3 (\\varphi) d \\varphi  \\,. \n\\end{equation}\nHence, by \\eqref{secondo coniugio siti normali Kdv}, \n\\begin{alignat}{2} \\label{L2 Kdv}\n& B^{-1} {\\cal L}_1 B = \\rho {\\cal L}_2 \\, , \\qquad &\n& {\\cal L}_2 := \\Pi_S^\\bot ( \\omega \\cdot \\partial_{\\vartheta} + m_3 \\partial_{yyy} \n+ c_1 \\partial_y + c_0 ) \\Pi_S^\\bot +  {\\mathfrak R}_2\n\\\\\n\\label{tilde c1 Kdv}\n& c_1 := \\rho^{-1} (B^{-1} b_1 ) \\,, \\qquad &\n& c_0 :=  \\rho^{-1} (B^{-1}  b_0 ) \\, , \\qquad \n{\\mathfrak R}_2 := \\rho^{-1} B^{-1}{\\mathfrak R}_1 B \\, .\n\\end{alignat}\nThe transformed operator ${\\cal L}_2$ in \\eqref{L2 Kdv} is still Hamiltonian, since the reparametrization of time preserves the Hamiltonian structure (see Section 2.2 and Remark 3.7 in \\cite{BBM}).\n\nWe now put in evidence the terms of order $ \\varepsilon, \\varepsilon^2, \\ldots $ in  $ c_1, c_0 $.\nTo this aim, we anticipate the following estimates: \n$ \\rho (\\vartheta) = 1 + O(\\varepsilon^4) $,  \n$ \\alpha = O( \\varepsilon^4 \\gamma^{-1}) $, \n$ m_3 = 1 + O(\\varepsilon^4) $, $ B^{-1} - I = O( \\alpha ) $ (in low norm), \nwhich are proved in Lemma \\ref{lemma:stime coeff mL2} below. \nThen, by \\eqref{b1 Kdv}-\\eqref{b0 Kdv}, we expand the functions $ c_1, c_0 $ in \\eqref{tilde c1 Kdv} as\n\\begin{equation} \\label{tilde c1 KdV}\nc_1 =  - \\varepsilon p_1 - \\varepsilon^2 p_2 - B^{-1} q_{>2} + \\varepsilon ( p_1 - B^{-1} p_1)  + \\varepsilon^2 ( p_2 - B^{-1} p_2 )    + {\\cal D}_\\omega \\beta +  4 \\b_{xxx} +  (a_1)_{xx}   + c_{1, \\geq 4}  \\, ,  \n\\end{equation}\n\\begin{equation} \\label{tilde c0 KdV}\nc_0 = - \\varepsilon (p_1)_x  - \\varepsilon^2 (p_2)_x - (B^{-1} q_{>2})_x \n+ \\varepsilon ( p_1 - B^{-1} p_1)_x + \\varepsilon^2 ( p_2 - B^{-1} p_2 )_x + ({\\cal D}_\\omega \\beta)_x +  \\b_{xxxx}  + c_{0, \\geq 4}\\,, \n\\end{equation}\nwhere  $ c_{1, \\geq 4}, c_{0, \\geq 4} = O( \\varepsilon^4 ) $ are defined by difference.\n\n\n\n\n\\begin{remark}\nThe functions $\\varepsilon ( p_1 - B^{-1} p_1) = O( \\varepsilon^5 \\gamma^{-1} )$  and \n$\\varepsilon^2 ( p_2 - B^{-1} p_2) = O( \\varepsilon^6 \\gamma^{-1} )$, see \\eqref{commu B p1 p2}. \nFor the reducibility scheme,  the terms of order $ \\partial_x^0 $\nwith size  $  O(\\varepsilon^5 \\gamma^{-1}) $ are perturbative, since\n$ \\varepsilon^5 \\gamma^{-2} \\ll 1 $.\n\\end{remark}\n\n\nThe remainder $ {\\mathfrak R}_2 $ in \\eqref{tilde c1 Kdv} has still the form \\eqref{forma buona con gli integrali} and, by \\eqref{resto1 Kdv}, \n\\begin{equation}\\label{mathfrakR2}\n{\\mathfrak R}_2 :=  - \\rho^{-1} B^{-1} {\\mathfrak  R}_1 B   = \n-  \\varepsilon^2 \\Pi_S^\\bot \\partial_x  {\\cal R}_2 +   {\\cal R}_{*} \n\\end{equation}\nwhere $ {\\cal R}_2 $ is defined in \\eqref{R nullo1R2} and we have renamed $  {\\cal R}_{*}  $ the  term of order $ o( \\varepsilon^2 ) $ in $\\mathfrak{R}_2$.   \n\n\n\\begin{lemma} \\label{lemma:stime coeff mL2}\nThere is $ \\sigma = \\sigma(\\nu,\\t) > 0 $ (possibly larger than $ \\sigma $ in Lemma \\ref{lemma:stime coeff mL1}) such that \n\\begin{align} \\label{stima m3}\n| m_3 - 1 |^{\\mathrm{Lip}(\\g)}    \\leq  C \\varepsilon^4 , \\qquad  \n| \\partial_i m_3 [\\widehat \\imath ]| & \\leq C \n\\varepsilon^{b+2} \\| \\widehat \\imath  \\|_{s_0 + \\sigma}  \n\\\\\n\\label{stima Di b3} \n\\| \\a \\|_s^{\\mathrm{Lip}(\\g)}  \\leq_s \\varepsilon^4 \\gamma^{-1} + \\varepsilon^{b + 2} \\gamma^{- 1}\\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}, \\qquad \n \\| \\partial_i \\a [\\widehat \\imath] \\|_s\n& \\leq_s \\varepsilon^{b + 2} \\gamma^{-1} \\big( \\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}  \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\big)\\,, \n\\\\\n\\label{stima c1 c0}\n \\| \\rho -1 \\|_s^{\\mathrm{Lip}(\\g)} \n \\leq_s \\varepsilon^4 + \\varepsilon^{b + 2} \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} ,\\qquad    \\| \\partial_i \\rho[\\widehat \\imath ]\\|_s &  \\leq_s   \\varepsilon^{b + 2} \\big(\\| \\widehat \n \\imath \\|_{s + \\sigma} + \\| {\\mathfrak{I}}_\\delta\\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} \\big)      \\\\\n\\label{commu B p1 p2}\n\\| p_k - B^{-1} p_k \\|_s^{\\mathrm{Lip}(\\g)}  \n& \\leq_s \\varepsilon^4 \\gamma^{-1}  + \\varepsilon^{b + 2} \\gamma^{- 1} \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} , \\quad k = 1, 2 \n\\\\\n\\label{commu B p1 p2 der}\n\\| \\partial_i  (p_k - B^{-1} p_k) [\\widehat \\imath] \\|_s \n& \\leq_s \\varepsilon^{b + 2} \\gamma^{-1} \\big( \\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}  \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\big)\\, \n\\\\\n\\label{stima z0 - B-1 z0}\n\\| B^{-1} q_{>2} \\|_s^{\\mathrm{Lip}(\\g)}  \n& \\leq_s  \\varepsilon^3 + \\varepsilon^b \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} ,\n\\\\\n\\label{stima derivata z0 - B-1 z0}\n\\|\\partial_i (B^{-1} q_{>2}) [\\widehat \\imath] \\|_s \n& \\leq_s  \\varepsilon^b \\big( \\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}  \\| \\widehat \\imath \\|_{s_0 + \\sigma} \\big) \\,.\n\\end{align}\nThe terms  \n$ c_{1, \\geq 4},  c_{0, \\geq 4} $ satisfy the bounds \\eqref{stima Di b1 b0}-\\eqref{stima Di b geq 4}. \nThe transformations $B$, $B^{-1}$ satisfy the estimates \\eqref{stima cal A bot}, \\eqref{stima derivata cal A bot}.\nThe remainder $ {\\cal R}_{*} $ has the form \\eqref{forma buona con gli integrali}, and the functions $g_j(\\tau)$, $\\chi_j(\\tau)$ satisfy \nthe estimates \\eqref{stima cal R*}-\\eqref{derivate stima cal R*} for all $\\tau \\in [0,1]$.\n\\end{lemma}\n\n\\begin{proof}\n\\eqref{stima m3} follows from \\eqref{def alpha m3},\\eqref{stima b1 b0}. \nThe estimate  $ \\| \\a \\|_s \\leq_s \\varepsilon^4 \\gamma^{-1}  + \\varepsilon^{b + 2} \\gamma^{- 1} \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma} $ and the inequality for \n$ \\partial_i \\alpha $ in \\eqref{stima Di b3} follow \nby \\eqref{def alpha m3},\\eqref{stima b1 b0},\\eqref{stima m3}. \nFor the first bound in \\eqref{stima Di b3} we also differentiate \n\\eqref{def alpha m3} \nwith respect to the parameter $ \\omega $. The estimates for $\\rho$ follow from $\\rho - 1 = B^{-1}(b_3 - m_3) \/ m_3$.\n\\end{proof}\n\n\\subsection{Translation of the space variable}\\label{step3}\n\nIn view of the next linear Birkhoff normal form steps (whose goal is to eliminate the terms of size $ \\varepsilon $ and $ \\varepsilon^2 $), \nin the expressions \\eqref{tilde c1 KdV}, \\eqref{tilde c0 KdV} we split \n$ p_1 = {\\bar p}_1 + ( p_1 - {\\bar p}_1)$, $p_2 = {\\bar p}_2 + ( p_2 - {\\bar p}_2)$ (see \\eqref{a1p1p2}), where\n\\begin{equation}\\label{def bar pi}\n{\\bar p}_1 :=  6 {\\bar v},  \\qquad \n{\\bar p}_2 :=   6 \\pi_0 [ (\\partial_x^{-1} {\\bar v})^2  ], \\qquad \n{\\bar v} (\\varphi, x) := {\\mathop \\sum}_{j \\in S} \\sqrt{\\xi_j} e^{{\\mathrm i}  \\ell (j) \\cdot \\varphi } e^{{\\mathrm i}  j x}, \n\\end{equation}\nand $\\ell : S \\to \\mathbb Z^\\nu$ is the odd injective map (see \\eqref{tang sites})\n\\begin{equation}\\label{del ell}\n\\ell : S \\to \\mathbb Z^\\nu, \\quad \n\\ell(\\bar \\jmath_i) := e_i, \\quad \n\\ell(-\\bar \\jmath_i) := - \\ell(\\bar \\jmath_i) = - e_i, \\quad \ni = 1,\\ldots,\\nu,\n\\end{equation}\ndenoting by $e_i = (0,\\ldots,1, \\ldots,0)$  the $i$-th vector of the canonical basis of $\\mathbb R^\\nu$.\n\n\\begin{remark}\\label{remarkp1p2}\nAll  the functions $ {\\bar p}_1  $, $ {\\bar p}_2 $,  $ p_1 - {\\bar p}_1  $,  $ p_2 - {\\bar p}_2 $ \nhave zero average in $ x $.\n\\end{remark}\n\nWe write the variable coefficients $c_1, c_0$ of the operator $\\mathcal{L}_2$ in \\eqref{L2 Kdv} \n(see \\eqref{tilde c1 KdV}, \\eqref{tilde c0 KdV}) as\n\\begin{equation}\\label{tilde c1 KdVbis} \nc_1 = - \\varepsilon {\\bar p}_1 - \\varepsilon^2 {\\bar p}_2 +  q_{c_1} + c_{1, \\geq 4}\\, , \n\\qquad \nc_0 = - \\varepsilon ({\\bar p}_1)_x - \\varepsilon^2 ({\\bar p}_2)_x + q_{c_0} + c_{0, \\geq 4}\\, , \n\\end{equation}\nwhere we define \n\\begin{gather} \\label{def qc1 qc0}\nq_{c_1} := q + 4 \\b_{xxx} + (a_1)_{xx} \\,, \\quad \nq_{c_0} := q_x + \\b_{xxxx}, \n\\\\\n\\label{def q}\nq := \\varepsilon ( p_1 - B^{-1} p_1) + \\varepsilon ( {\\bar p}_1 - p_1)\n+ \\varepsilon^2 ( p_2 - B^{-1} p_2 )  + \\varepsilon^2 ( {\\bar p}_2 - p_2)\n- B^{-1} q_{>2} + {\\cal D}_\\omega \\beta \\,.\n\\end{gather}\n\n\\begin{remark}  \\label{rem:qc1 qc0}\nThe functions $ q_{c_1}, q_{c_0} $ have zero average in $ x $ (see Remarks \\ref{remarkp1p2}, \\ref{media beta} and Lemma \\ref{p3 zero average}). \n\\end{remark}\n\n\n\\begin{lemma} \\label{lemma:stime scorporo}\n The functions $\\bar p_k - p_k$, $k=1,2$ and $q_{c_m}$, \n$m = 0,1, $  satisfy\n\\begin{alignat}{2} \n\\label{stima bar pk - pk}\n& \\| {\\bar p}_k - p_k \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\| {\\mathfrak{I}}_\\d \\|_{s}^{\\mathrm{Lip}(\\g)} , \n\\quad & \n& \\| \\partial_i ({\\bar p}_k - p_k)[ \\widehat \\imath] \\|_s \n\\leq_s \\| \\widehat \\imath \\|_{s} + \\| {\\mathfrak{I}}_\\d \\|_{s} \\| \\widehat \\imath \\|_{s_0}\\,, \n\\\\\n\\label{stima q}\n& \\| q_{c_m} \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\varepsilon^{5}\\gamma^{-1} + \\varepsilon \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}\\,, \n\\quad & \n& \\| \\partial_i q_{c_m}[\\widehat \\imath ] \\|_s^{\\mathrm{Lip}(\\g)}\n\\leq_s \\varepsilon \\big( \\| \\widehat \\imath \\|_{s + \\sigma} + \\|{\\mathfrak I}_\\delta \\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} \\big)\\,. \n\\end{alignat}\n\\end{lemma}\n\n\\begin{proof}\nThe bound \\eqref{stima bar pk - pk} follows from \n\\eqref{def bar pi}, \\eqref{a1p1p2}, \\eqref{T0}, \\eqref{ansatz delta}. Then use \\eqref{stima bar pk - pk},\n\\eqref{commu B p1 p2}-\\eqref{stima derivata z0 - B-1 z0},\n\\eqref{stima beta},\n\\eqref{stima a1} to prove \\eqref{stima q}. The biggest term comes from $ \\varepsilon ({\\bar p}_1 - p_1 ) $. \n\\end{proof}\n\nWe now apply the transformation  $\\mathcal{T}$ defined in \\eqref{gran tau} \nwhose goal  is to remove the space average from the coefficient in front of $ \\partial_y $. \n\nConsider the change of the space variable $ z = y + p(\\vartheta ) $ which induces on \n$ H^s_{S^\\bot} (\\mathbb T^{\\nu+1}) $ the operators\n\\begin{equation}\\label{gran tau}\n ({\\cal T} w)(\\vartheta, y ) := w(\\vartheta , y + p(\\vartheta)) \\, , \\quad \n ({\\cal T}^{-1} h) (\\vartheta ,z ) = h(\\vartheta, z - p(\\vartheta))\n \\end{equation}\n(which are a particular case of those used in section \\ref{step1}). \nThe differential operator becomes \n$ {\\cal T}^{-1} \\omega \\cdot \\partial_{\\vartheta} {\\cal T} $ \n$ =  \\omega \\cdot \\partial_{\\vartheta} \n+ \\{ \\omega \\cdot \\partial_{\\vartheta}p (\\vartheta) \\} \\partial_z $, \n$ {\\cal T}^{-1} \\partial_{y} {\\cal T}  =  \\partial_{z} $. \nSince $\\mathcal{T}, \\mathcal{T}^{-1}$ commute with $ \\Pi_S^\\bot $, we get\n\\begin{align} \\label{L3 KdV}\n\\mathcal{L}_3 & := {\\cal T}^{-1}{\\cal L}_2 {\\cal T} = \\Pi_S^\\bot \\big(\\omega \\cdot \\partial_{\\vartheta} \n + m_3 \\partial_{zzz} + d_1 \\partial_z + d_0 \\big) \\Pi_S^\\bot \n + {\\mathfrak R}_3 \\,,\n\\\\\n\\label{d1d0R3}\nd_1 & := ({\\cal T}^{-1} c_1)   + \\omega \\cdot \\partial_{\\vartheta} p  \\, , \n\\qquad \nd_0 :=  {\\cal T}^{-1}  c_0   \\, , \n\\qquad \n{\\mathfrak R}_3 :=   {\\cal T}^{-1} {\\mathfrak R}_2 {\\cal T}.\n\\end{align}\nWe choose \n\\begin{equation} \\label{def m1 p Kdv}\nm_1 :=  \\frac{1}{(2\\pi)^{\\nu + 1}} \\int_{\\mathbb T^{\\nu + 1}}   c_1 d\\vartheta dy \\, , \\quad \np := (\\omega \\cdot \\partial_\\vartheta)^{-1} \n\\Big( m_1 -  \\frac{1}{2 \\pi} \\int_{\\mathbb T}  c_1  d y \\Big) \\, ,\n\\end{equation}\nso that $\\frac{1}{2 \\pi} \\int_{\\mathbb T} d_1 (\\vartheta, z) \\, dz = m_1$ for all $\\vartheta \\in \\mathbb T^\\nu$.\nNote that, by \\eqref{tilde c1 KdVbis}, \n\\begin{equation} \\label{media migliore}\n\\int_\\mathbb T c_1(\\th,y) \\, dy = \\int_\\mathbb T c_{1, \\geq 4}(\\th,y) \\, dy\\,,\n\\quad \n\\omega \\cdot \\partial_{\\vartheta} p(\\th) = m_1 - \\frac{1}{2\\pi} \\int_\\mathbb T c_{1, \\geq 4} (\\th,y)\\, dy \n\\end{equation}\nbecause $\\bar p_1, \\bar p_2, q_{c_1}$ have all zero space-average.  \nAlso note that $\\mathfrak R_3$ has the form \\eqref{forma buona con gli integrali}. \nSince ${\\cal T}$ is symplectic, the operator ${\\cal L}_3$ in \\eqref{L3 KdV} is Hamiltonian.\n\\begin{remark}\nWe require  Hypothesis (${\\mathtt S}1$) so that the function $ q_{>2}  $ has zero space average (see Lemma \\ref{p3 zero average}). \nIf  $ q_{>2} $ did not have zero average,  \nthen $ p $ in \\eqref{def m1 p Kdv} would have size $O(\\varepsilon^3 \\gamma^{-1})$ \n(see \\eqref{def p3}) and, since  $\\mathcal{T}^{-1} - I = O(\\varepsilon^3 \\gamma^{-1}) $, \nthe function $ \\tilde d_0 $ in \\eqref{d0 tilde KdV} would satisfy $  \\tilde d_0  =  O(\\varepsilon^4  \\gamma^{-1}) $.\nTherefore it would remain a term of order $ \\partial_x^0 $ which is  not \nperturbative for the reducibility scheme of section \\ref{subsec:mL0 mL5}.\n\\end{remark}\n\nWe put in evidence the terms of size $ \\varepsilon, \\varepsilon^2 $ in $ d_0 $, $ d_1 $,  $ {\\mathfrak R}_3 $. \nRecalling \\eqref{d1d0R3}, \\eqref{tilde c1 KdVbis}, we split \n\\begin{equation} \\label{d1 d0 KdV}\nd_1  = - \\varepsilon {\\bar p}_1 - \\varepsilon^2 {\\bar p}_2 +  \\tilde d_{1}  \\, , \\quad  \nd_0  = - \\varepsilon ({\\bar p}_1)_x - \\varepsilon^2 ({\\bar p}_2)_x + \\tilde d_0  \\, , \\quad  \n{\\mathfrak R}_3 =  \n-  \\varepsilon^2 \\Pi_S^\\bot \\partial_x \\bar {\\cal R}_2 +  \\widetilde \\mathcal{R}_{*}\n\\end{equation}\nwhere $ \\bar {\\cal R}_2 $ is obtained replacing \n$ v_\\d $ with ${\\bar v}$ in $  {\\cal R}_2 $  (see \\eqref{R nullo1R2}), and \n\\begin{align} \n\\label{d1 tilde KdV}\n\\tilde d_1 \n& := \\varepsilon (\\bar p_1 - {\\cal T}^{-1}{\\bar p}_1 ) \n+ \\varepsilon^2 (\\bar p_2 - {\\cal T}^{-1}{\\bar p}_2 ) \n+ {\\cal T}^{-1} ( q_{c_1} + c_{1, \\geq 4}) \n+ \\omega \\cdot \\partial_{\\vartheta} p , \n\\\\\n\\label{d0 tilde KdV}\n\\tilde d_0 \n& := \\varepsilon ( \\bar p_1 - {\\cal T}^{-1}{\\bar p}_1 )_x \n+ \\varepsilon^2 ( \\bar p_2 - {\\cal T}^{-1} {\\bar p}_2 )_x\n+ {\\cal T}^{-1} ( q_{c_0} + c_{0, \\geq 4} ),\n\\\\\n\\label{tilde mR 3}\n\\widetilde \\mathcal{R}_{*} \n& := \\mathcal{T}^{-1} \\mathcal{R}_{*} \\mathcal{T} \n+ \\varepsilon^2 \\Pi_S^\\bot \\partial_x (\\mathcal{R}_2 - \\mathcal{T}^{-1} \\mathcal{R}_2 \\mathcal{T}) + \\varepsilon^2  \\Pi_S^\\bot \\partial_x  (  \\bar {\\cal R}_2 - {\\cal R}_2 ),\n\\end{align}\nand $ \\mathcal{R}_{*} $ is defined in \\eqref{mathfrakR2}. \nWe have also used that ${\\cal T}^{-1}$ commutes with $\\partial_x$ and with $\\Pi_S^\\bot$. \n\n\\begin{remark}\\label{d1 media}\nThe space average \n$ \\frac{1}{2 \\pi} \\int_{\\mathbb T} \\tilde d_1 (\\vartheta, z) \\, dz =  \\frac{1}{2 \\pi} \\int_{\\mathbb T} \nd_1 (\\vartheta, z) \\, dz = m_1$ for all $\\vartheta \\in \\mathbb T^\\nu$.\n\\end{remark}\n\n\n\\begin{lemma} \\label{lemma:stime coeff mL3}\n There is $ \\sigma := \\sigma(\\nu,\\t) > 0 $ (possibly larger than in Lemma \\ref{lemma:stime coeff mL2}) such that \n\\begin{alignat}{2} \\label{stima m1}\n| m_1 |^{\\mathrm{Lip}(\\g)}   & \\leq C \\varepsilon^4 , \n\\quad & | \\partial_i m_1 [\\widehat \\imath ]|  & \\leq C \\varepsilon^{b+2}  \\| \\widehat{\\imath} \\|_{s_0 + \\sigma} \n\\\\ \\label{stima p} \n\\| p \\|_s^{\\mathrm{Lip}(\\g)}   & \\leq_s \\varepsilon^4 \\gamma^{-1} + \\varepsilon^{b + 2} \\gamma^{- 1} \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}\\,, \n\\quad & \\| \\partial_i p [\\widehat \\imath] \\|_s\n& \\leq_s \\varepsilon^{b + 2} \\gamma^{-1} \\big( \\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}  \\| \\widehat \\imath \\|_{s_0 + \\sigma}\\big)\\,, \n\\\\ \\label{tilde d1 d0 KdV}\n\\| \\tilde d_k  \\|_s^{\\mathrm{Lip}(\\g)}   & \\leq_s \\varepsilon^{5} \\gamma^{-1} + \\varepsilon \\| {\\mathfrak{I}}_\\d \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} \\,, \n\\quad & \\| \\partial_i \\tilde d_k [\\widehat \\imath] \\|_s \n& \\leq_s \\varepsilon \\big(\\| \\widehat \\imath \\|_{s + \\sigma} +  \\| {\\mathfrak I}_\\d \\|_{s + \\sigma} \n\\| \\widehat \\imath \\|_{s_0 + \\sigma}  \\big)\n\\end{alignat}\nfor $k = 0,1$. Moreover the matrix $ s $-decay norm (see \\eqref{matrix decay norm}) \n\\begin{align}\\label{nuove R3}\n|\\widetilde{\\cal R}_{*}|_s^{{\\mathrm{Lip}(\\g)}} & \\leq_s   \\varepsilon^{3} +  \\varepsilon^{2}\\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}  \\, , \\quad \n|\\partial_i \\widetilde{\\cal R}_{*} [\\widehat \\imath ]|_s  \\leq_s \\varepsilon^{2} \\| \\widehat \\imath \\|_{s + \\sigma} + \\varepsilon^{2 b - 1} \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma}  \\, .\n\\end{align}\nThe transformations ${\\cal T}$, ${\\cal T}^{-1}$ satisfy \\eqref{stima cal A bot}, \\eqref{stima derivata cal A bot}.\n\\end{lemma}\n\n\n\\begin{proof}\nThe estimates \\eqref{stima m1}, \\eqref{stima p} follow by \\eqref{def m1 p Kdv},\\eqref{tilde c1 KdVbis},\\eqref{media migliore}, and the bounds for $ c_{1, \\geq 4}, c_{0, \\geq 4} $ in Lemma \\ref{lemma:stime coeff mL2}. \nThe estimates \\eqref{tilde d1 d0 KdV} follow similarly  by \n\\eqref{stima q},\n\\eqref{media migliore}, \\eqref{stima p}.\nThe estimates \\eqref{nuove R3} follow because\n$ \\mathcal{T}^{-1} \\mathcal{R}_{*} \\mathcal{T} $  satisfies the bounds \\eqref{stima cal R*} \nlike $ \\mathcal{R}_{*} $ does (use Lemma \\ref{remark : decay forma buona resto} and \\eqref{stima p}) \nand \n$ |\\varepsilon^2  \\Pi_S^\\bot \\partial_x  (  \\bar {\\cal R}_2 - {\\cal R}_2 )|_s^{\\mathrm{Lip}(\\g)} \\leq_s \\varepsilon^2 \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} $. \n\\end{proof}\nIt is sufficient to estimate  $ \\widetilde \\mathcal{R}_{*} $ (which has the form \\eqref{forma buona con gli integrali})\nonly in the $ s $-decay norm (see \\eqref{nuove R3}) because the next transformations will preserve it. Such norms are used \nin the reducibility scheme of section \\ref{subsec:mL0 mL5}. \n\n\\subsection{Linear Birkhoff normal form. Step 1} \\label{BNF:step1}\n\nNow we  eliminate the terms of order $ \\varepsilon$ and $ \\varepsilon^2  $ of $ {\\cal L}_3 $.\nThis step is different from the \nreducibility steps that we shall perform in section \\ref{subsec:mL0 mL5}, \nbecause the diophantine constant $ \\gamma = o(\\varepsilon^2 ) $ (see \\eqref{omdio}) and so \nterms $ O( \\varepsilon), O(\\varepsilon^2) $ are not perturbative. \nThis reduction is possible thanks to the special form of the terms \n$ \\varepsilon {\\cal B}_1$, $ \\varepsilon^2 {\\cal B}_2 $ defined in \\eqref{def cal B1 B2}: \nthe harmonics of $ \\varepsilon {\\cal B}_1$, and $ \\varepsilon^2 T $ in \\eqref{mL4 T R4}, which \ncorrespond to a possible small divisor\nare naught, see  Corollary \\ref{B1 zero KdV}, and Lemma \\ref{pezzo epsilon 2 A}. \nIn this section we eliminate the term  $ \\varepsilon {\\cal B}_1 $. In section \\ref{BNF:step2}\nwe eliminate the terms of order $\\varepsilon^2$.  \n\nNote that, since the previous transformations $ \\Phi $, $ B $, $ {\\cal T} $ are \n$ O(\\varepsilon^4 \\gamma^{-1} ) $-close to the identity, the terms of order $\\varepsilon $ and $ \\varepsilon^2 $ in $ {\\cal L}_3 $ \nare the same as in the original linearized operator.\n\n\\smallskip\n\nWe first collect all the terms of order $ \\varepsilon $ and $ \\varepsilon^2 $ in the operator $ {\\cal L}_3 $ defined in \\eqref{L3 KdV}. \nBy  \\eqref{d1 d0 KdV},  \\eqref{R nullo1R2}, \\eqref{def bar pi}\nwe have, renaming $ \\vartheta = \\varphi $, $ z = x $, \n$$\n \\mathcal{L}_3 = \\Pi_S^\\bot \\big( \\omega  \\cdot \\partial_\\varphi\n + m_3 \\partial_{xxx}+ \\varepsilon {\\cal B}_1 + \\varepsilon^2 {\\cal B}_2 +\n {\\tilde d}_1 \\partial_x +  {\\tilde d}_0 \\big) \\Pi_S^\\bot \n +  {\\widetilde {\\cal R}}_{*} \n$$\nwhere $  {\\tilde d}_1 $, $  {\\tilde d}_0 $, $  {\\widetilde {\\cal R}}_{*}  $ are defined in \\eqref{d1 tilde KdV}-\\eqref{tilde mR 3} and \n(recall also \\eqref{def pi 0})\n\\begin{equation} \\label{def cal B1 B2}\n{\\cal B}_1 h := - 6  \\partial_x ( {\\bar v}  h), \n\\quad\n{\\cal B}_2  h := - 6  \\partial_x \\{  {\\bar v}  \n\\Pi_S [ (\\partial_x^{-1} {\\bar v} )\\,\\partial_x^{-1} h ] \n+ h \\pi_0 [ ( \\partial_x^{-1} {\\bar v} )^2 ] \\}  +\n 6 \\pi_0 \\{ (\\partial_x^{-1} {\\bar v}) \\Pi_S [ {\\bar v} h ] \\}.\n\\end{equation}\nNote that ${\\cal B}_1$ and ${\\cal B}_2$ are the linear Hamiltonian vector fields of $ H_{S}^\\bot $\ngenerated, respectively, by the   Hamiltonian $ z \\mapsto 3 \\int_\\mathbb T v z^2 $ in \\eqref{H3tilde}, and \nthe fourth order Birkhoff Hamiltonian $  {\\cal H}_{4,2}$  in \\eqref{mH3 mH4} at $ v = \\bar v $.\n\nWe transform $ {\\cal L}_3 $ by a symplectic operator \n$ \\Phi_1 : H_{S^\\bot}^s(\\mathbb T^{\\nu + 1}) \\to H_{S^\\bot}^s(\\mathbb T^{\\nu + 1}) $ of the form \n\\begin{equation}\\label{Phi_1}\n\\Phi_1 := {\\rm exp}(\\varepsilon A_1) = I_{H_S^\\bot} + \\varepsilon A_1 + \\varepsilon^2 \\frac{A_1^2}{2} +\n\\varepsilon^3 \\widehat A_1, \\quad \n\\widehat A_1 := {\\mathop \\sum}_{k \\geq 3} \\frac{\\varepsilon^{k - 3}}{k !} A_1^k  \\, ,\n\\end{equation}\nwhere $ A_1(\\varphi) h = {\\mathop \\sum}_{j,j' \\in S^c} ( A_1)_j^{j'}(\\varphi) h_{j'} e^{{\\mathrm i}  j x} $\nis a Hamiltonian vector field.\nThe map  $ \\Phi_1 $ is symplectic, because it is the time-1 flow of a Hamiltonian vector field. \nTherefore\n\\begin{align}\\label{L3 new KdV}\n& {\\cal L}_3 \\Phi_1 - \\Phi_1 \\Pi_S^\\bot ( {\\cal D}_\\omega + m_3 \\partial_{xxx} ) \\Pi_S^\\bot \\\\\n& = \\Pi_S^\\bot ( \\varepsilon \\{ {\\cal D}_\\omega A_1 + m_3 [\\partial_{xxx}, A_1] +  {\\cal B}_1 \\} \n+ \\varepsilon^2 \\{ {\\cal B}_1 A_1 + {\\cal B}_2 + \\frac12 m_3 [\\partial_{xxx}, A_1^2 ] + \n\\frac12 ({\\cal D}_\\omega A_1^2)\\} + {\\tilde d}_1 \\partial_x + R_3 ) \\Pi_S^\\bot\n\\nonumber\n\\end{align}\nwhere \n\\begin{align}\\label{def R3}\nR_3 & :=   {\\tilde d}_1 \\partial_x (\\Phi_1 - I) \\! +\\! {\\tilde d}_0 \\Phi_1\\! +\\! \\widetilde {\\cal R}_{*} \\Phi_1\\! + \\! \\varepsilon^2 {\\cal B}_2 (\\Phi_1 - I) \\! + \\!   \\varepsilon^3 \\big\\{ {\\cal D}_\\omega \\widehat A_1 \\! + \\!  m_3 [\\partial_{xxx}, \\widehat A_1]  \\! + \\! \\frac12 {\\cal B}_1 A_1^2 \\!+ \\! \\varepsilon {\\cal B}_1 \\widehat A_1 \\big\\}\\, . \n\\end{align}\n\n\\begin{remark}\n$ R_3 $ has no longer the form \\eqref{forma buona con gli integrali}. \nHowever $ R_3 = O( \\partial_x^0 ) $ because \n$ A_1 =  O(  \\partial_x^{-1} ) $  (see Lemma \\ref{lemma:Dx A bounded}), and therefore $\\Phi_1 - I_{H_S^\\bot} = O(\\partial_x^{-1})$. \nMoreover the matrix decay norm of $ R_3 $ is $ o(\\varepsilon^2) $. \n\\end{remark}\n\nIn order to eliminate the order $\\varepsilon$ from \\eqref{L3 new KdV}, we choose \n\\begin{equation}\\label{cal A 1}\n( A_1)_j^{j'}(l) := \\begin{cases}\n- \\dfrac{( {\\cal B}_1)_{j}^{j'}(l)}{{\\mathrm i}  (\\omega \\cdot l + m_3( j'^3 - j^3))} & \\text{if} \\  \\bar{\\omega} \\cdot l + j'^3 - j^3 \\neq 0 \\, ,  \\\\\n0 & \\text{otherwise},\n\\end{cases}\n\\qquad j, j' \\in S^c, \\ l \\in \\mathbb Z^\\nu.\n\\end{equation}\nThis definition is well posed. \nIndeed, by \\eqref{def cal B1 B2} and \\eqref{def bar pi}, \n\\begin{equation}\\label{def cal B1 b}\n( {\\cal B}_1)_{j}^{j'}(l):=\n\\begin{cases}\n-6 {\\mathrm i}  j \\sqrt{\\xi_{j - j'}}   & \\text{if} \\ j - j' \\in S\\,, \\  \\  l = \\ell(j - j') \\\\\n0 & \\text{otherwise}.\n\\end{cases}\n\\end{equation}\nIn particular $ ( {\\cal B}_1)_{j}^{j'}(l) = 0 $ unless $ | l | \\leq 1 $. Thus, for \n$ \\bar{\\omega} \\cdot l + j'^3 - j^3 \\neq 0 $, \nthe denominators in \\eqref{cal A 1} satisfy \n\\begin{align}\n|\\omega \\cdot l +m_3( j'^3 - j^3)| \n& = | m_3 (\\bar \\omega \\cdot l + j'^3 - j^3) + ( \\omega - m_3 \\bar \\omega ) \\cdot l | \n\\nonumber \\\\ & \n\\geq  |m_3| | \\bar \\omega \\cdot l + j'^3 - j^3 |\n- | \\omega - m_3 \\bar \\omega | |l| \\geq  1\/2  \\, , \\quad \\forall |l| \\leq  1 \\, ,  \n\\label{BNFdeno}\n\\end{align}\nfor $ \\varepsilon $ small, because the non zero integer \n$ | \\bar{\\omega} \\cdot l + j'^3 - j^3| \\geq 1 $, \\eqref{stima m3}, and $ \\omega = \\bar \\omega + O(\\varepsilon^2) $. \n\n$ A_1$ defined in \\eqref{cal A 1} is a Hamiltonian vector field as $\\mathcal{B}_1$.\n\n\\begin{remark} \n\\label{rem:Ham solving homolog}\nThis is a general fact: the denominators $ \\d_{l,j,k} := {\\mathrm i}  (\\omega \\cdot l + m_3( k^3 - j^3)) $ \nsatisfy $ \\overline{ \\d_{l,j,k} } = \\d_{-l,k,j} $ and \nan operator $G(\\varphi)$ is self-adjoint if and only if its matrix elements satisfy \n$ \\overline{ G_j^k(l) } = G_k^j(-l) $, see \\cite{BBM}-Remark 4.5. In a more intrinsic way, \nwe could solve \nthe  homological equation of this Birkhoff step directly for the Hamiltonian function whose flow generates $ \\Phi_1 $.\n\\end{remark}\n\n\n\n\\begin{lemma} \\label{lem:cubetto}  If $j,j' \\in S^c$, $j - j' \\in S$, $l =  \\ell(j - j')$, then  \n$ \\bar{\\omega} \\cdot l + j'^3 - j^3  = 3 j j' (j' - j) \\neq 0 $.\n\\end{lemma}\n\n\\begin{proof}\nWe have $ \\bar \\omega \\cdot l =  \\bar \\omega \\cdot \\ell( j - j' ) = (j-j')^3$ because $ j - j' \\in S$ (see \\eqref{bar omega} and \\eqref{del ell}).\nNote that $ j, j' \\neq 0 $ because $ j, j' \\in S^c $, and $ j - j' \\neq 0 $ because $ j - j' \\in S $.\n\\end{proof}\n\n\\begin{corollary}\\label{B1 zero KdV}\nLet $j, j' \\in S^c$. \nIf $\\bar{\\omega} \\cdot l + j'^3 - j^3 = 0$ then $( {\\cal B}_1)_{j}^{j'}(l) = 0$. \n\\end{corollary}\n\n\\begin{proof}\nIf $({\\cal B}_1)_{j}^{j'}(l) \\neq 0$ then $j - j' \\! \\in \\! S, l =  \\ell(j - j')$ \nby \\eqref{def cal B1 b}. \nHence $\\bar{\\omega} \\cdot l + j'^3 - j^3 \\! \\neq 0$ by Lemma \\ref{lem:cubetto}.\n\\end{proof}\n\nBy \\eqref{cal A 1} and the previous corollary, the term of order $\\varepsilon$ in \\eqref{L3 new KdV} is\n\\begin{equation}\\label{primo termine BNF1}\n\\Pi_S^\\bot \\big( {\\cal D}_\\omega A_1 + m_3 [\\partial_{xxx}, A_1]  +  {\\cal B}_1 \\big) \\Pi_S^\\bot = 0 \\, . \n\\end{equation}\nWe now estimate the transformation $ A_1 $.\n\n\\begin{lemma}\\label{lem: A decay} \n$(i)$ For all $l \\in \\mathbb Z^\\nu$, $j,j' \\in S^c$, \n\\begin{equation}\\label{Acoef}\n| (A_1)_{j}^{j'}(l)| \\leq C (| j | +  | j'  |)^{-1}\\, , \\quad \n| (A_1)_{j}^{j'}(l)|^{\\rm lip} \\leq \\varepsilon^{-2} (|j| + |j'|)^{-1} \\,.\n\\end{equation}\n$(ii)$ $ (A_1)_j^{j'}(l) = 0$ for all $l \\in \\mathbb Z^\\nu$, $j,j' \\in S^c$ such that $|j - j'| >  C_S $, where $C_S := \\max\\{ |j| : j \\in S \\}$. \n\\end{lemma}\n \n\n\\begin{proof}\n$(i)$ We already noted that $ (A_1)_j^{j'}(l) = 0$, $ \\forall |l| > 1$. \nSince  $ | \\omega | \\leq |\\bar \\omega | + 1 $, one has, for $ |l| \\leq 1 $, $ j \\neq j' $,\n\\[\n|\\omega \\cdot l + m_3 (j'^3 - j^3)| \n\\geq |m_3||j'^3 - j^3| - |\\omega \\cdot l| \n\\geq \\frac14 (j'^2 + j^2) - |\\omega| \n\\geq \\frac18  (j'^2 + j^2) \\, ,  \\quad \\forall (j'^2 + j^2) \\geq C, \n\\]\nfor some constant $C > 0$. \nMoreover, recalling that also \\eqref{BNFdeno} holds, we deduce that for $  j \\neq j' $, \n\\begin{equation} \\label{lower bound}\n (A_1)_j^{j'}(l) \\neq 0 \n\\quad \\Rightarrow \\quad\n|\\omega \\cdot l + m_3 (j'^3 - j^3)| \\geq c ( | j | +  | j' | )^2 \\, . \n\\end{equation}\nOn the other hand, if $ j = j' $, $ j \\in S^c$,  the matrix $ (A_1)_j^j (l) = 0 $, $ \\forall l \\in \\mathbb Z^\\nu$,  \nbecause $ ({\\cal B}_1)_j^j (l) = 0 $ by \\eqref{def cal B1 b} \n(recall that $ 0 \\notin S $).  Hence \\eqref{lower bound} holds for all $ j, j'  $.\nBy  \\eqref{cal A 1}, \\eqref{lower bound},  \\eqref{def cal B1 b}  we deduce the first \nbound in \\eqref{Acoef}. \nThe Lipschitz bound follows similarly (use also $ |j - j'| \\leq C_S $).  \n$(ii)$ follows by \\eqref{cal A 1}-\\eqref{def cal B1 b}. \n\\end{proof}\n\nThe previous lemma  means that $ A = O(| \\partial_x|^{-1})$. \nMore precisely we deduce that\n\n\\begin{lemma} \\label{lemma:Dx A bounded}\n$ | A_1 \\partial_x  |_s^{\\mathrm{Lip}(\\g)} + | \\partial_x A_1 |_s^{\\mathrm{Lip}(\\g)} \\leq C(s) $.\n\\end{lemma}\n\n\\begin{proof}\nRecalling the definition of the (space-time) matrix norm in \\eqref{decayTop},  \nsince $(A_1)_{j_1}^{j_2}(l) = 0$ outside the set of indices $|l| \\leq 1, |j_1 - j_2| \\leq C_S$, \nwe have\n\\begin{align*}\n| \\partial_x A_1 |_s^2 \n& = \\sum_{|l| \\leq 1, \\, |j| \\leq C_S} \n\\Big( \\sup_{j_1 - j_2 = j} |j_1| | (A_1)_{j_1}^{j_2}(l)| \\Big)^2 \n\\langle l,j \\rangle^{2s}  \n\\leq C(s) \n\\end{align*}\nby Lemma \\ref{lem: A decay}. The estimates for $ |A_1 \\partial_x |_s $ and the Lipschitz bounds \nfollow similarly. \n\\end{proof}\n\n\nIt follows that the symplectic map $ \\Phi_1 $ in \\eqref{Phi_1} is invertible for $ \\varepsilon $ small, \nwith inverse\n\\begin{equation}\\label{A1 check}\n \\Phi_1^{-1} = {\\rm exp}(-\\varepsilon A_1) =  I_{H_S^\\bot} + \\varepsilon {\\check A}_1 \\, , \\ \n  {\\check A}_1 := {\\mathop \\sum}_{n \\geq 1} \\frac{\\varepsilon^{n-1}}{n !} (-A_1)^n \\, , \\ \n| {\\check A}_1 \\partial_x  |_s^{\\mathrm{Lip}(\\g)} + | \\partial_x {\\check A}_1 |_s^{\\mathrm{Lip}(\\g)} \\leq C(s) \\, . \n\\end{equation}\nSince $ A_1 $ solves the homological equation \\eqref{primo termine BNF1}, the $ \\varepsilon  $-term in\n\\eqref{L3 new KdV} is zero, and, \nwith a straightforward calculation, \nthe $ \\varepsilon^2 $-term simplifies to $ {\\cal B}_2  + \\frac12 [{\\cal B}_1, A_1] $. \nWe obtain the Hamiltonian operator\n\\begin{align} \\label{bernardino1}\n{\\cal L}_4 &  :=    \\Phi_1^{-1} {\\cal L}_3 \\Phi_1 \n= \\Pi_S^\\bot ( {\\cal D}_\\omega + m_3 \\partial_{xxx} +  {\\tilde d}_1 \\partial_x \n+ \\varepsilon^2  \\{  {\\cal B}_2  + \\tfrac12 [{\\cal B}_1, A_1]  \\}  +     \\tilde{R}_4 ) \\Pi_S^\\bot \n\\\\\n\\label{bernardino 2}\n{\\tilde R}_4 & := (\\Phi_1^{-1} - I) \\Pi_S^\\bot [ \\varepsilon^2  ( {\\cal B}_2 + \\tfrac12 [{\\cal B}_1, A_1] ) + {\\tilde d}_1 \\partial_x ]  +  \\Phi_1^{-1} \\Pi_S^\\bot R_3 \\, .\n\\end{align}\nWe split $ A_1 $ defined in  \\eqref{cal A 1}, \\eqref{def cal B1 b} into $ A_1 = \\bar{A}_1 + \\widetilde{A}_1$ where, for all $ j, j' \\in S^c $, $l \\in \\mathbb Z^\\nu$, \n\\begin{equation}\\label{bar A1} \n( {\\bar A}_1)_j^{j'}(l) := \n\\dfrac{6 j \\sqrt{\\xi_{j - j'}}}{ \\bar\\omega \\cdot l + j'^3 - j^3} \n\\qquad \\text{if } \\  \\bar{\\omega} \\cdot l + j'^3 - j^3 \\neq 0, \\ \\  j-j' \\in S , \\ \\   \nl = \\ell(j - j'), \n\\end{equation}\nand $( {\\bar A}_1)_j^{j'}(l) := 0$ otherwise. \nBy Lemma \\ref{lem:cubetto}, for all $ j , j' \\in S^c $, $l \\in \\mathbb Z^\\nu$,  \n$ ( {\\bar A}_1)_j^{j'}(l) = \n\\frac{2  \\sqrt{\\xi_{j - j'}}}{ j' ( j' - j )} \n$ if $  j-j' \\in S $, $  l = \\ell(j - j') $, and    \n$ ( {\\bar A}_1)_j^{j'}(l) = 0 $  otherwise, \nnamely (recall the definition of $ \\bar v $ in \\eqref{def bar pi}) \n\\begin{equation}\\label{bar A1 esplicita}\n{\\bar A}_1h = 2 \\Pi_S^\\bot [( \\partial_x^{-1} {\\bar v}  )(\\partial_x^{-1} h)]   \\, , \n\\quad \\forall h \\in H_{S^\\bot}^s(\\mathbb T^{\\nu+1}) \\,.\n\\end{equation}\nThe difference is\n\\begin{equation}\\label{widetilde A1}\n(\\widetilde{A}_1)_j^{j'}(l) = (A_1 - \\bar{A}_1)_j^{j'}(l) = - \n\\frac{6 j \\sqrt{\\xi_{j - j'}}\\big\\{ ( \\omega - \\bar \\omega) \\cdot l + (m_3 - 1)(j'^3 - j^3) \\big\\}}{\\big( \\omega \\cdot l + m_3(j'^3 - j^3) \\big) \\big(\\bar\\omega \\cdot l + j'^3 - j^3 \\big)} \n\\end{equation}\nfor $ j, j' \\in S^c $, $  j - j' \\in S $,  $ l = \\ell(j - j') $, \nand $ (\\widetilde{A}_1)_j^{j'}(l) = 0 $ otherwise. \nThen, by \\eqref{bernardino1}, \n\\begin{equation}\\label{mL4 T R4}\n{\\cal L}_4 = \\Pi_S^\\bot \\big( {\\cal D}_\\omega + m_3 \\partial_{xxx} \n+  {\\tilde d}_1 \\partial_x \n+ \\varepsilon^2 T +  R_4 \\big) \\Pi_S^\\bot \\,, \n\\end{equation}\nwhere\n\\begin{equation}\\label{T hamiltoniano}\nT := {\\cal B}_2 +  \\frac12 [{\\cal B}_1, {\\bar A}_1]   \\,, \\qquad R_4 := \\frac{\\varepsilon^2}{2} [{\\cal B}_1, \\widetilde A_1] + \\tilde R_4\\,.\n\\end{equation}\nThe operator $ T $ is Hamiltonian as $ {\\cal B}_2 $,  $ {\\cal B}_1 $, $ {\\bar A}_1 $ (the commutator of two Hamiltonian vector fields is Hamiltonian). \n\n\\begin{lemma} \\label{lemma:R4}\n There is $ \\sigma = \\sigma(\\nu,\\t) > 0 $ (possibly larger than in Lemma \\ref{lemma:stime coeff mL3}) such that \n\\begin{align} \\label{stima Lip R4} \n| R_4 |_s^{{\\rm Lip}(\\gamma)} \n& \\leq_s \\varepsilon^{5} \\gamma^{-1} + \\varepsilon \\| {\\mathfrak I}_\\d \\|_{s + \\sigma}^{{\\rm Lip}(\\gamma)}  \\,, \\quad \n| \\partial_i R_4 [\\widehat \\imath] |_s \n\\leq_s \\varepsilon \\big( \\| \\widehat \\imath \\|_{s + \\sigma} + \\| {\\mathfrak I}_\\d \\|_{s + \\sigma} \n \\| \\widehat \\imath \\|_{ s_0 + \\sigma} \\big)  \\, . \n \\end{align}\n\\end{lemma}\n\n\\begin{proof} \nWe first estimate $ [{\\cal B}_1, \\widetilde A_1] = \n({\\cal B}_1 \\partial_x^{-1}) (\\partial_x {\\widetilde A}_1)- ({\\widetilde A}_1 \\partial_x)( \\partial_x^{-1} {\\cal B}_1) $.\nBy  \\eqref{widetilde A1}, $ | \\omega - \\bar \\omega | \\leq C \\varepsilon^2 $ \n(as $ \\omega \\in \\Omega_\\varepsilon $ in \\eqref{Omega epsilon}) and \\eqref{stima m3}, \narguing as in Lemmata \\ref{lem: A decay}, \\ref{lemma:Dx A bounded}, we deduce that \n$ | {\\widetilde A}_1 \\partial_x  |_s^{\\mathrm{Lip}(\\g)} +  $ $ | \\partial_x {\\widetilde A}_1 |_s^{\\mathrm{Lip}(\\g)} \\leq_s \\varepsilon^2 $.\nBy  \\eqref{def cal B1 B2} \nthe norm $ |{\\cal B}_1  \\partial_x^{-1}|_s^{\\mathrm{Lip}(\\g)} + |\\partial_x^{-1} {\\cal B}_1|^{\\mathrm{Lip}(\\g)} \\leq  C(s) $.  \nHence $ \\varepsilon^2 |[{\\cal B}_1,  {\\widetilde A}_1]|_s^{\\mathrm{Lip}(\\g)} \\leq_s \\varepsilon^4 $. \nFinally \\eqref{T hamiltoniano},\n\\eqref{bernardino 2}, \\eqref{A1 check}, \\eqref{def R3}, \\eqref{tilde d1 d0 KdV}, \\eqref{nuove R3}, \n and the interpolation estimate \\eqref{interpm Lip} imply \\eqref{stima Lip R4}.\n\\end{proof}\n\n\n\n\n\\subsection{Linear Birkhoff normal form. Step 2}\\label{BNF:step2} \n\n\nThe goal of this section is to remove the term $ \\varepsilon^2 T$\nfrom the operator $\\mathcal{L}_4$ defined in   \\eqref{mL4 T R4}.\nWe conjugate the Hamiltonian operator $\\mathcal{L}_4$  via a symplectic map\n\\begin{equation}\\label{def Phi2}\n\\Phi_2 := {\\rm exp}(\\varepsilon^2 A_2) = I_{H_S^\\bot} + \\varepsilon^2 A_2 + \\varepsilon^4 \\widehat A_2\\,,\\quad \n\\widehat A_2 := {\\mathop \\sum}_{k \\geq 2} \\frac{\\varepsilon^{2(k - 2)}}{k !} A_2^k\n\\end{equation}\nwhere $A_2(\\varphi) = {\\mathop \\sum}_{j,j' \\in S^c} ( A_2)_j^{j'}(\\varphi) h_{j'} e^{{\\mathrm i}  j x} $ \nis a Hamiltonian vector field. We compute \n\\begin{gather} \\label{L4 diff}\n{\\cal L}_4 \\Phi_2 - \\Phi_2 \\Pi_S^\\bot \\big( {\\cal D}_\\omega + m_3 \\partial_{xxx} \\big) \\Pi_S^\\bot \n= \\, \\Pi_S^\\bot ( \\varepsilon^2 \\{ {\\cal D}_\\omega  A_2 + m_3 [\\partial_{xxx}, A_2 ] + T \\}  \n+ \\tilde{d}_1 \\partial_x + \\tilde R_5 ) \\Pi_S^\\bot \\,,\n\\\\\n\\label{def tilde R5}\n\\tilde R_5  := \\Pi_S^\\bot \\{ \n\\varepsilon^4 ( ({\\cal D}_\\omega \\widehat A_2) + m_3 [\\partial_{xxx}, \\widehat A_2] )\n+ (\\tilde{d}_1 \\partial_x  + \\varepsilon^2 T)(\\Phi_2 - I) + R_4 \\Phi_2 \\} \\Pi_S^\\bot \\, . \n\\end{gather}\nWe define\n\\begin{equation}\\label{A2}\n( A_2)_j^{j'}(l) := \n- \\dfrac{T_j^{j'}(l)}{ {\\mathrm i}  (\\omega \\cdot l + m_3 (j'^3 - j^3) )} \n\\quad \\text{if } \\  \\bar\\omega \\cdot l + j'^3 - j^3 \\neq 0; \n\\qquad \n( A_2)_j^{j'}(l) := 0 \\quad \\text{otherwise.}\n\\end{equation}\nThis definition is well posed. \nIndeed, by \\eqref{T hamiltoniano}, \\eqref{def cal B1 b}, \\eqref{bar A1}, \\eqref{def cal B1 B2}, \nthe matrix entries $ T_j^{j'} (l) = 0 $ for all $ | j - j' | > 2 C_S $, $l \\in \\mathbb Z^\\nu $, \nwhere $C_S := \\max \\{ |j| \\, , j \\in S \\} $. \nAlso $ T_j^{j'} (l) = 0 $ for all $ j, j'  \\in S^c $, $ | l | > 2 $ (see also \n\\eqref{forma funzionale B1 A1}, \\eqref{B1B2 Fourier}, \\eqref{B3 Fourier} below).\nThus, arguing as in \\eqref{BNFdeno}, if $ \\bar\\omega \\cdot l + j'^3 - j^3 \\neq 0 $, then \n$ |\\omega \\cdot l + m_3 (j'^3 - j^3)| \\geq 1 \/ 2 $.\nThe operator $ A_2 $ is a Hamiltonian vector field because $ T $ is Hamiltonian and by Remark \\ref{rem:Ham solving homolog}.\n\nNow we prove that the Birkhoff map \n$\\Phi_2$  removes completely the term $\\varepsilon^2 T$. \n\n\\begin{lemma} \\label{pezzo epsilon 2 A}\n Let $j, j' \\in S^c$. If $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$, then $T_j^{j'}(l) = 0$.\n\\end{lemma}\n\n\\begin{proof}\nBy  \\eqref{def cal B1 B2}, \\eqref{bar A1 esplicita} we get\n$ {\\cal B}_1 {\\bar A}_1  h \n= - 12 \\partial_x \\{ \\bar v \\Pi_S^\\bot[( \\partial_x^{-1} \\bar v )(\\partial_x^{-1} h)] \\}\n$, ${\\bar A}_1{\\cal B}_1 h = $ $ - 12 \\Pi_S^\\bot [ (\\partial_x^{-1} \\bar v) \\Pi_S^\\bot (\\bar v h) ] $ for all $h \\in H_{S^\\bot}^s $, \nwhence, recalling \\eqref{def bar pi}, for all $ j, j' \\in S^c $, $ l \\in \\mathbb Z^\\nu $, \n\\begin{equation}\\label{forma funzionale B1 A1}\n( [{\\cal B}_1,  {\\bar A}_1])_{j}^{j'}(l) = 12 {\\mathrm i}   \\!\\!\\! \\sum_{\\begin{subarray}{c}\nj_1, j_2 \\in S, \\, j_1 + j_2 = j - j' \\\\\nj' + j_2 \\in S^c, \\, \\ell(j_1) + \\ell(j_2) = l \n\\end{subarray}}   \\!\\!\\!  \\frac{ j j_1 -j' j_2 }{j' j_1 j_2} \\sqrt{\\xi_{j_1} \\xi_{j_2}}\\,, \n\\end{equation}\nIf $([{\\cal B}_1,  {\\bar A}_1])_j^{j'}(l) \\neq 0 $\nthere are $ j_1, j_2 \\in S $ such that $ j_1 + j_2 = j - j' $, $j' + j_2 \\in S^c$, $ \\ell(j_1) + \\ell(j_2) = l  $.\nThen\n\\begin{equation} \\label{con lo zero}\n\\bar \\omega \\cdot l + j'^3 - j^3 \n = \\bar \\omega \\cdot \\ell(j_1 ) + \\bar \\omega \\cdot \\ell(j_2) + j'^3 - j^3 \n  \\stackrel{\\eqref{del ell}} \n = j_1^3 + j_2^3 + j'^3 - j^3\\, .\n\\end{equation}\nThus, if $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$, \nLemma \\ref{lemma:interi} implies $ (j_1 + j_2 )( j_1 + j') ( j_2 + j' ) = 0 $. Now $ j_1 + j' $, $  j_2 + j'  \\neq 0 $\nbecause $ j_1, j_2 \\in S $, $ j' \\in S^c $ and $ S $ is symmetric. \nHence $ j_1 + j_2 = 0 $, which implies $ j = j' $ and $ l = 0 $ (the map $ \\ell $ in \\eqref{del ell} is odd).\nIn conclusion, if  $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$, the only nonzero matrix entry \n$ ([{\\cal B}_1, {\\bar A}_1])_j^{j'}(l) $   is\n\\begin{equation}\\label{parte diagonale zero}\n([{\\cal B}_1, \\bar A_1])_j^j(0) \n\\stackrel{\\eqref{forma funzionale B1 A1}} = 24 {\\mathrm i}  \\sum_{j_2 \\in S, \\, j_2 + j \\in S^c } \n\\xi_{j_2} {j_2^{-1}}.\n\\end{equation}\nNow we consider $ {\\cal B}_2 $ in \\eqref{def cal B1 B2}. Split $ {\\cal B}_2 =  B_1 + B_2 + B_3 $, \nwhere \n$B_1 h := - 6 \\partial_x \\{ {\\bar v} \\Pi_S [ (\\partial_x^{-1} {\\bar v}) \\partial_{x}^{-1} h ] \\}$,  \n$B_2 h := - 6 \\partial_x \\{ h \\pi_0 [(\\partial_x^{-1} {\\bar v} )^2] \\}$,  \n$B_3 h := 6 \\pi_0 \\{ \\Pi_S ({\\bar v} h) \\partial_x^{-1} {\\bar v} \\}$.\nTheir Fourier matrix representation is\n\\begin{gather}\\label{B1B2 Fourier}\n(B_1)_j^{j'}(l) =  6 {\\mathrm i}  j \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\sum_{\\begin{subarray}{c}\nj_1, j_2 \\in S, \\, j_1 + j' \\in S \\\\\nj_1 + j_2 = j - j', \\, \\ell(j_1) + \\ell(j_2) = l \n\\end{subarray}} \\!\\!\\!\\! \\!\\!\\!\\!\\!\\!\\! \\frac{\\sqrt{\\xi_{j_1} \\xi_{j_2}}}{ j_1 j'} \\, ,  \n\\qquad \n(B_2)_{j}^{j'}(l) = 6 {\\mathrm i}  j  \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\sum_{\\begin{subarray}{c}\nj_1 , j_2 \\in S, \\, j_1 + j_2 \\neq 0 \\\\\nj_1 + j_2 = j - j', \\, \\ell(j_1) + \\ell(j_2) = l\n\\end{subarray}}   \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\frac{\\sqrt{\\xi_{j_1} \\xi_{j_2}}}{j_1 j_2} \\,, \n\\\\\n\\label{B3 Fourier}\n(B_3)_j^{j'}(l) =  6 \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\sum_{\\begin{subarray}{c}\nj_1 , j_2 \\in S,  \\, j_1 + j' \\in S \\\\\nj_1 + j_2 = j - j', \\,  \\ell(j_1) + \\ell(j_2) = l\n\\end{subarray}} \\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\! \\frac{\\sqrt{\\xi_{j_1} \\xi_{j_2}}}{{\\mathrm i}  j_2}\\,,\n\\qquad j,j' \\in S^c, \\ l \\in \\mathbb Z^\\nu.\n\\end{gather}\nWe study the terms $ B_1 $, $ B_2 $, $ B_3$ separately.\nIf $(B_1)_j^{j'}(l) \\neq 0$, there are $j_1 , j_2 \\in S$ such that \n$ j_1 + j_2 = j - j' $, $j_1 + j' \\in S$, $ l = \\ell(j_1) + \\ell(j_2) $ and\n\\eqref{con lo zero} holds. Thus, if $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$,  \nLemma \\ref{lemma:interi} implies \n$ (j_1 + j_2) (j_1 + j')(j_2 + j') = 0 $, \nand, since $j' \\in S^c $ and $ S $ is symmetric, the only possibility is $j_1 + j_2 = 0$. Hence $j = j'$, $l = 0$.\nIn conclusion, if  $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$, the only nonzero matrix element \n$ ( B_1 )_j^{j'}(l) $   is \n\\begin{equation} \\label{diag R0}\n(B_1)_j^j(0) = 6 {\\mathrm i}  \\sum_{ j_1 \\in S, \\, j_1 + j \\in S } \\xi_{j_1} j_1^{-1} \\,.\n\\end{equation}\nBy the same arguments, if  $ (B_2)_j^{j'} (l) \\neq 0 $ and $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$ we find $ (j_1 + j_2) (j_1 + j')(j_2 + j') = 0 $, \nwhich is impossible because also $ j_1 + j_2 \\neq 0$.\nFinally, arguing as for $ B_1 $, if  $\\bar \\omega \\cdot l + j'^3 - j^3 = 0$, then the only nonzero matrix element $ ( B_3 )_j^{j'}(l) $ is\n\\begin{equation} \\label{diag R1 R2}\n(B_3)_j^j(0) = 6 {\\mathrm i}  \\sum_{j_1 \\in S, \\, j_1 + j \\in S } \\xi_{j_1}  j_1^{-1} \\,.\n\\end{equation}\nFrom \\eqref{parte diagonale zero}, \\eqref{diag R0}, \\eqref{diag R1 R2} we deduce that, if \n$ \\bar \\omega \\cdot l + j'^3 - j^3 = 0 $, then the only non zero elements  $ (\\frac12 [\\mathcal{B}_1, \\bar A_1] + B_1 + B_3)_j^{j'} (l) $ must be for $ (l, j, j') = (0, j, j) $. \nIn this case, we get\n\\begin{equation}\\label{Tjj0}\n\\frac12( [\\mathcal{B}_1, \\bar A_1])_j^j(0) + (B_1)_j^j(0) + (B_3)_j^j(0) \n= 12 {\\mathrm i}  \\sum_{\\begin{subarray}{c} j_1 \\in S \\\\ j_1 + j \\in S^c \\end{subarray}} \\frac{\\xi_{j_1}}{ j_1}\n+ 12 {\\mathrm i}  \\sum_{\\begin{subarray}{c} j_1 \\in S \\\\ j_1 + j \\in S \\end{subarray}} \\frac{\\xi_{j_1}}{ j_1}\n= 12 {\\mathrm i}  \\sum_{j_1 \\in S} \\frac{\\xi_{j_1}}{ j_1} \n= 0\n\\end{equation}\nbecause \nthe case $j_1 + j = 0$ is impossible ($j_1 \\in S$, $j' \\in S^c$ and $S$ is symmetric), \nand the function $S \\ni j_1 \\to \\xi_{j_1} \/ j_1 \\in \\mathbb R$ is odd. \nThe lemma follows by \\eqref{T hamiltoniano}, \\eqref{Tjj0}.\n\\end{proof}\n\nThe choice of $ A_2 $ in \\eqref{A2} and Lemma \\ref{pezzo epsilon 2 A} imply that \n\\begin{equation}\\label{pezzo zero}\n \\Pi_S^\\bot \\big( {\\cal D}_\\omega A_2  + m_3 [\\partial_{xxx}, A_2] + T \\big) \\Pi_S^\\bot = 0 \\, .\n\\end{equation}\n\n\\begin{lemma}\\label{A2 decay}\n$ |\\partial_x A_2|_s^{\\mathrm{Lip}(\\g)} + |  A_2 \\partial_x|_s^{\\mathrm{Lip}(\\g)} \\leq C(s) $. \n\\end{lemma}\n\n\\begin{proof}\nFirst we prove that the diagonal elements $ T_j^j (l) = 0 $ for all $ l \\in \\mathbb Z^\\nu $. \nFor $l = 0$, we have already proved that $T_j^j(0) = 0$ (apply Lemma \\ref{pezzo epsilon 2 A} with $j = j'$, $l=0$).\nMoreover, in each term $[\\mathcal{B}_1, \\bar A_1]$, $B_1$, $B_2$, $B_3$ (see \\eqref{forma funzionale B1 A1}, \\eqref{B1B2 Fourier}, \\eqref{B3 Fourier}) \nthe sum is over $j_1 + j_2 = j - j'$, $l = \\ell(j_1) + \\ell(j_2)$. \nIf $j = j'$, then $j_1 + j_2 = 0$, and $l = 0$. \nThus $ T_j^j (l) =  T_j^j (0)  = 0 $. \nFor the off-diagonal terms $ j \\neq j' $ we argue as in Lemmata \\ref{lem: A decay},  \\ref{lemma:Dx A bounded},\nusing that all the denominators $ |\\omega \\cdot l + m_3 ( j'^3 - j^3 ) | \\geq c ( |j| + |j'|)^2 $.  \n\\end{proof}\n\nFor $ \\varepsilon $ small, the map $ \\Phi_2 $ in \\eqref{def Phi2} is invertible and $ \\Phi_2 = \\exp(-\\varepsilon^2 A_2 ) $. \nTherefore \\eqref{L4 diff}, \\eqref{pezzo zero} imply \n\\begin{align} \\label{L5 KdV}\n{\\cal L}_5 \n& := \\Phi_2^{-1} {\\cal L}_4 \\Phi_2 \n= \\Pi_S^\\bot ( {\\cal D}_\\omega + m_3 \\partial_{xxx} +  \\tilde{d}_1 \\partial_x  + R_5 ) \\Pi_S^\\bot \\,,\n\\\\\n\\label{R5}\nR_5 \n& := ( \\Phi_2^{-1} - I) \\Pi_S^\\bot \\tilde{d}_1 \\partial_x \n+ \\Phi_2^{-1}  \\Pi_S^\\bot {\\tilde R}_5 \\,. \n\\end{align}\nSince $ A_2 $ is a Hamiltonian vector field, the map $\\Phi_2$ is symplectic and so  ${\\cal L}_5$ is Hamiltonian.\n\\begin{lemma} \\label{lemma:R5}\n$R_5$ satisfies the same estimates \\eqref{stima Lip R4} as $R_4$ (with a possibly larger $\\sigma$).\n\\end{lemma}\n\n\\begin{proof} \nUse \\eqref{R5}, Lemma \\ref{A2 decay}, \\eqref{tilde d1 d0 KdV}, \\eqref{def tilde R5}, \n\\eqref{stima Lip R4} and the interpolation inequalities \\eqref{multiplication Lip}, \\eqref{interpm Lip}. \n\\end{proof} \n\n\n\\subsection{Descent method}\\label{step5}\n\nThe goal of this section is to transform $ {\\cal L}_5 $ in \\eqref{L5 KdV} so that the coefficient of $ \\partial_x $ becomes constant.\nWe conjugate $ {\\cal L}_5 $ via  a symplectic map of the form \n\\begin{equation}\\label{def descent}\n{\\cal S} := {\\rm exp}(\\Pi_S^\\bot (w \\partial_x^{-1}))\\Pi_S^\\bot = \\Pi_S^\\bot \\big( I  + w   \\partial_x^{-1} \\big) \\Pi_S^\\bot + \\widehat {\\cal S}\\,,\\quad \\widehat {\\cal S} := {\\mathop \\sum}_{k \\geq 2} \n\\frac{1}{k!} [\\Pi_S^\\bot (w \\partial_x^{-1})]^k \\Pi_S^\\bot\\,,\n\\end{equation}\nwhere $ w : \\mathbb T^{\\nu+1} \\to \\mathbb R $ is a function. \nNote that $\\Pi_S^\\bot (w \\partial_x^{-1}) \\Pi_S^\\bot$ is the  Hamiltonian vector field  generated by \n $ - \\frac12 \\int_\\mathbb T w (\\partial_x^{-1} h)^2\\,dx$, $h \\in H_S^\\bot$.\nRecalling \\eqref{def pi 0}, we calculate\n\\begin{align} \\label{L5 diff}\n& {\\cal L}_5 {\\cal S} - {\\cal S} \\Pi_S^\\bot\n( {\\cal D}_\\omega + m_3 \\partial_{xxx}  + m_1 \\partial_x ) \\Pi_S^\\bot \n= \\Pi_S^\\bot ( 3 m_3 w_x + \\tilde{d}_1 - m_1 ) \\partial_x \\Pi_S^\\bot + \\tilde R_6 \\,,\n\\\\\n& \\tilde R_6\n:= \\Pi_S^\\bot \\{ ( 3 m_3 w_{xx} + \\tilde d_1 \\Pi_S^\\bot w - m_1 w ) \\pi_0 \n+ ( ({\\cal D}_\\omega w) + m_3 w_{xxx} + \\tilde d_1 \\Pi_S^\\bot w_x ) \\partial_x^{-1}\n+ ({\\cal D}_\\omega \\widehat{\\cal S}) \n\\notag \\\\ & \\qquad \\ \\ \n+ m_3 [\\partial_{xxx}, \\widehat{\\cal S}] \n+ \\tilde{d}_1 \\partial_x \\widehat{\\cal S} \n- m_1 \\widehat{\\cal S} \\partial_x \n+ R_5 {\\cal S} \\} \\Pi_S^\\bot\n\\notag \n\\end{align}\nwhere $\\tilde R_6$ collects all the terms of order at most $\\partial_x^0$. \nBy Remark \\ref{d1 media}, we solve $ 3 m_3 w_x + \\tilde{d}_1 - m_1 = 0 $  \nby choosing $w := - (3 m_3)^{-1} \\partial_x^{-1} ( \\tilde{d}_1 - m_1 )$.\nFor $ \\varepsilon $ small, the operator $ {\\cal S} $ is invertible and, by \\eqref{L5 diff}, \n\\begin{equation}\\label{def L6}\n\\mathcal{L}_6 := \\mathcal{S}^{-1} \\mathcal{L}_5 \\mathcal{S} \n= \\Pi_S^\\bot ( {\\cal D}_\\omega + m_3 \\partial_{xxx} + m_1 \\partial_x ) \\Pi_S^\\bot + R_6 \\,, \n\\qquad R_6 :=  {\\cal S}^{-1} \\tilde R_6 \\, .\n\\end{equation}\nSince $ {\\cal S} $ is symplectic, ${\\cal L}_6$ is Hamiltonian (recall Definition \\ref{operatore Hamiltoniano}).\n\n\n\\begin{lemma}\\label{lemma L6}\n There is $ \\sigma = \\sigma(\\nu,\\t) > 0 $ (possibly larger than in Lemma \\ref{lemma:R5}) such that \n$$\n|{\\cal S}^{\\pm 1} - I|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\varepsilon^{5} \\gamma^{-1} + \\varepsilon \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)}\\,, \\quad \n|\\partial_i {\\cal S}^{\\pm 1} [\\widehat \\imath ]|_s  \n\\leq_s \\varepsilon ( \\| \\widehat \\imath\\|_{s + \\sigma} + \\|{\\mathfrak I}_\\delta \\|_{s + \\sigma} \\| \\widehat \\imath \\|_{s_0 + \\sigma} ).\n$$\nThe remainder $R_6$ satisfies the same estimates \\eqref{stima Lip R4} as $R_4$. \n\\end{lemma}\n\n\\begin{proof}\nBy \\eqref{tilde d1 d0 KdV},\\eqref{stima m1},\\eqref{stima m3}, $\\| w \\|_s^{\\mathrm{Lip}(\\g)} \\leq_s  \\varepsilon^{5} \\gamma^{-1} + \\varepsilon \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} $, and the lemma follows by \\eqref{def descent}. Since $ \\widehat { \\cal S } = O(\\partial_x^{-2})$ the commutator \n$ [ \\partial_{xxx}, \\widehat { \\cal S }] = O(\\partial_x^0 )$ and\n$ | [ \\partial_{xxx}, \\widehat { \\cal S }] |_s^{\\mathrm{Lip}(\\g)} \\leq_s \\| w \\|_{s_0+3}^{\\mathrm{Lip}(\\g)} \\| w \\|_{s+3}^{\\mathrm{Lip}(\\g)} $. \n\\end{proof}\n\n\\subsection{KAM reducibility and inversion of $ {\\cal L}_{\\omega} $} \\label{subsec:mL0 mL5}\n\nThe coefficients $ m_3, m_1 $  of the operator $ {\\cal L}_6 $ in \\eqref{def L6} are constants, and the remainder $ R_6 $ is a bounded operator of order $ \\partial_x^0 $ with  small matrix decay \nnorm, see \\eqref{decay R6}.\nThen we  can diagonalize $ {\\cal L}_6 $ \nby applying the iterative KAM reducibility Theorem 4.2  in \\cite{BBM}  along the \nsequence of scales  \n\\begin{equation}\\label{defN}\nN_n := N_{0}^{\\chi^n}, \\quad n = 0,1,2,\\ldots,  \n\\quad  \\chi := 3\/2, \\quad N_0  > 0 \\, .\n\\end{equation}\nIn section \\ref{sec:NM}, the initial $ N_0 $ will (slightly) increase to infinity \nas $ \\varepsilon \\to 0 $, see \\eqref{nash moser smallness condition}. \nThe required smallness condition (see (4.14) in \\cite{BBM}) is (written in the present notations)\n\\begin{equation}\\label{R6resto}\nN_0^{C_0} | R_6 |_{s_0 + \\beta}^{{\\mathrm{Lip}(\\g)}} \\gamma^{-1} \\leq 1 \n\\end{equation}\nwhere $ \\b := 7 \\tau + 6  $ (see (4.1) in \\cite{BBM}), \n$ \\tau $ is the diophantine exponent in \\eqref{omdio} and \\eqref{Omegainfty}, \nand the constant $ C_0 := C_0 (\\t, \\nu ) > 0 $  is fixed in\nTheorem 4.2 in \\cite{BBM}.\nBy Lemma \\ref{lemma L6}, the remainder $ R_6 $ satisfies the bound \\eqref{stima Lip R4}, \nand using \\eqref{ansatz delta} we get (recall \\eqref{link gamma b})\n\\begin{equation} \\label{decay R6}\n| R_6|_{s_0 + \\beta}^{{\\mathrm{Lip}(\\g)}} \\leq C \\varepsilon^{7 - 2 b} \\gamma^{-1} = C \\varepsilon^{3 - 2 a}, \\qquad\n| R_6 |_{s_0 + \\beta}^{{\\mathrm{Lip}(\\g)}} \\gamma^{-1} \\leq C \\varepsilon^{1 - 3 a}  \\, .\n\\end{equation}\nWe use that  $ \\mu $ in \\eqref{ansatz delta}  is assumed to satisfy $ \\mu  \\geq \\sigma + \\b $ where $ \\sigma := \\sigma (\\tau, \\nu ) $ \nis given in Lemma \\ref{lemma L6}.  \n\n\\begin{theorem} \\label{teoremadiriducibilita} \n{\\bf (Reducibility)}\nAssume that  $\\omega \\mapsto i_\\d (\\omega)\n$ is a Lipschitz function defined on some subset $\\Omega_o \\subset \\Omega_\\varepsilon $ (recall  \\eqref{Omega epsilon}), satisfying \n\\eqref{ansatz delta} with  $ \\mu \\geq \\sigma + \\b $ where $ \\sigma := \\sigma (\\tau, \\nu) $ is given in Lemma \\ref{lemma L6} and $ \\b  := 7 \\tau + 6 $. \nThen there exists $ \\delta_{0} \\in (0,1) $  such that, if\n\\begin{equation}\\label{condizione-kam}\nN_0^{C_0}  \\varepsilon^{7 - 2 b} \\gamma^{-2} = N_0^{C_0}  \\varepsilon^{1 - 3 a}\n\\leq \\delta_{0} \\, , \\quad \\gamma := \\varepsilon^{2 + a} \\, , \\quad a \\in (0,1\/6) \\, , \n\\end{equation}\nthen:\n\n$(i)$ {\\bf (Eigenvalues)}.\nFor all $ \\omega \\in \\Omega_\\varepsilon $ there exists a sequence \n\\begin{equation} \\label{espressione autovalori}\n\\mu_j^\\infty(\\omega) := \\mu_j^\\infty(\\omega, i_\\d (\\omega)) \n:=  {\\mathrm i}  \\big( - {\\tilde m}_3 (\\omega) j^3 +  {\\tilde m}_1(\\omega)  j \\big)\n+ r_j^\\infty(\\omega), \\quad j \\in  S^c \\, ,  \n\\end{equation}\nwhere $ {\\tilde m}_3, {\\tilde m}_1$  coincide with the coefficients $m_3, m_1$ of $ {\\cal L}_6 $ in \\eqref{def L6} for all $ \\omega \\in \\Omega_o $,\nand \n\\begin{align}\n\\label{autofinali}\n| {\\tilde m}_3 - 1 |^{{\\rm Lip}(\\gamma)} + | {\\tilde m}_1 |^{{\\rm Lip}(\\gamma)} \\leq C \\varepsilon^4 \\, , \\quad \n| r^{\\infty}_j |^{{\\rm Lip}(\\gamma)} & \\leq C \\varepsilon^{3 - 2 a} \\, , \n\\quad \\ \\forall j \\in  S^c \\,  , \n\\end{align}\nfor some  $ C > 0 $.\nAll the eigenvalues $\\mu_j^{\\infty}$ are purely imaginary. We define, for convenience,  $  \\mu_0^\\infty (\\omega)  := 0 $. \n\n\\smallskip\n\n$(ii)$ {\\bf (Conjugacy)}.\nFor all $\\omega$ in the set\n\\begin{equation}  \\label{Omegainfty}\n\\Omega_\\infty^{2\\gamma} := \\Omega_\\infty^{2\\gamma} (i_\\d) \n:= \\Big\\{ \\omega \\in \\Omega_o : \\,\n| {\\mathrm i} \\omega \\cdot l + \\mu^{\\infty}_j (\\omega) - \\mu^{\\infty}_{k} (\\omega) |\n\\geq \\frac{2 \\gamma | j^{3} - k^{3} |}{ \\langle l \\rangle^{\\tau}}, \n\\, \\forall l \\in \\mathbb Z^{\\nu}, \\, j ,k \\in S^c \\cup \\{0\\}\\Big\\} \n\\end{equation}\nthere is a real, bounded, invertible linear operator $\\Phi_\\infty(\\omega) : H^s_{S^\\bot} (\\mathbb T^{\\nu+1}) \\to \nH^s_{S^\\bot} (\\mathbb T^{\\nu+1}) $, with bounded inverse \n$\\Phi_\\infty^{-1}(\\omega)$, that conjugates $\\mathcal{L}_6$ in \\eqref{def L6} to constant coefficients, namely\n\\begin{equation}\\label{Lfinale}\n{\\cal L}_{\\infty}(\\omega)\n:= \\Phi_{\\infty}^{-1}(\\omega) \\circ \\mathcal{L}_6(\\omega) \\circ  \\Phi_{\\infty}(\\omega)\n=  \\omega \\cdot \\partial_{\\varphi} + {\\cal D}_{\\infty}(\\omega),  \\quad\n{\\cal D}_{\\infty}(\\omega)\n:= {\\rm diag}_{j \\in S^c} \\{ \\mu^{\\infty}_{j}(\\omega) \\} \\, .\n\\end{equation}\nThe transformations $\\Phi_\\infty, \\Phi_\\infty^{-1}$ are close to the identity in matrix decay norm,\nwith\n\\begin{equation} \\label{stima Phi infty}\n| \\Phi_{\\infty} - I |_{s,\\Omega_\\infty^{2\\gamma}}^{{\\rm Lip}(\\gamma)}\n+ | \\Phi_{\\infty}^{- 1} - I |_{s,\\Omega_\\infty^{2\\gamma}}^{\\mathrm{Lip}(\\g)}\n\\leq_s \\varepsilon^{5} \\gamma^{-2} + \\varepsilon \\gamma^{-1} \\| {\\mathfrak I}_\\delta \\|_{s + \\sigma}^{\\mathrm{Lip}(\\g)} .\n\\end{equation}\nMoreover $\\Phi_{\\infty}, \\Phi_{\\infty}^{-1}$  are symplectic, and \n$\\mathcal{L}_\\infty $ is a Hamiltonian operator.\n\\end{theorem}\n\n\n\\begin{proof}\nThe proof is the same as the one of Theorem 4.1 in \\cite{BBM}, \nwhich is based on Theorem 4.2, Corollaries 4.1, 4.2 and Lemmata 4.1, 4.2 of \\cite{BBM}.\nA difference is that here $\\omega \\in \\mathbb R^\\nu$, while in \\cite{BBM} the parameter $\\lambda \\in \\mathbb R$ is one-dimensional. \nThe proof is the same because Kirszbraun's Theorem on Lipschitz extension of functions also holds in $\\mathbb R^\\nu$ \n(see, e.g., Lemma A.2 in \\cite{Po2}). \nThe bound \\eqref{stima Phi infty} follows by Corollary 4.1 of \\cite{BBM} and the estimate of \n$ R_6 $ in Lemma \\ref{lemma L6}.\nWe also use the estimates \\eqref{stima m3}, \\eqref{stima m1} for $ \\partial_i m_3 $, \n$ \\partial_i m_1 $ which correspond to (3.64) in \\cite{BBM}.  \nAnother difference is that here the sites $ j \\in S^c \\subset \\mathbb Z \\setminus \\{0\\} $ unlike in \\cite{BBM} where $ j \\in \\mathbb Z $. \nWe have defined $ \\mu_0^\\infty := 0 $ \nso that also the first  Melnikov conditions \\eqref{prime di melnikov} are included  in the definition of $ \\Omega^{2 \\gamma}_{\\infty} $.\n\\end{proof}\n\n\\begin{remark}\nTheorem 4.2 in \\cite{BBM}\nalso provides the Lipschitz dependence of the (approximate) \neigenvalues $ \\mu_j^n $ with respect to the unknown $ i_0 (\\varphi) $,  \nwhich is used for the measure estimate Lemma \\ref{matteo 10}.\n \\end{remark}\n\nAll the parameters $ \\omega \\in \\Omega_\\infty^{2 \\gamma} $ satisfy  (specialize \\eqref{Omegainfty} for $ k = 0 $) \n\\begin{equation}\\label{prime di melnikov}\n|{\\mathrm i}  \\omega \\cdot l + \\mu_j^\\infty(\\omega)| \\geq \n2 \\gamma | j |^3 \\langle l \\rangle^{-\\tau} \\, , \\quad \\forall l \\in \\mathbb Z^\\nu , \\ j \\in S^c, \n\\end{equation}\nand the diagonal operator $ {\\cal L}_\\infty $ is invertible.  \n\nIn the following theorem we finally verify the inversion assumption \\eqref{tame inverse} for ${\\cal L}_\\omega $. \n\n\\begin{theorem}\\label{inversione linearized normale} {\\bf (Inversion of $ {\\cal L}_\\omega $)}\nAssume the hypotheses of Theorem \\ref{teoremadiriducibilita} and \\eqref{condizione-kam}. \nThen there exists $ \\sigma_1 := \\sigma_1 ( \\t, \\nu ) >  0 $ such that, \n$ \\forall \\omega \\in \\Omega^{2 \\gamma}_\\infty(i_\\d )$ (see \\eqref{Omegainfty}),\nfor any function $ g \\in H^{s+\\sigma_1}_{S^\\bot} (\\mathbb T^{\\nu+1})  $ \nthe equation  ${\\cal L}_\\omega h = g$ \nhas a solution $h = {\\cal L}_\\omega^{-1} g \\in H^s_{S^\\bot} (\\mathbb T^{\\nu+1})$, satisfying\n\\begin{align}\\label{stima inverso linearizzato normale}\n\\| {\\cal L}_\\omega^{-1} g \\|_s^{{\\rm Lip}(\\gamma)} \n& \\leq_s \\gamma^{-1} \\big( \\| g \\|_{s +\\sigma_1}^{{\\rm Lip}(\\gamma)} \n+ \\varepsilon \\gamma^{-1} \\| {\\mathfrak I}_\\delta\\|_{s + \\sigma_1}^{\\mathrm{Lip}(\\g)}\n\\| g \\|_{s_0}^{{\\rm Lip}(\\gamma)} \\big) \n\\\\\n& \\leq_s \\gamma^{-1} \\big( \\| g \\|_{s +\\sigma_1}^{{\\rm Lip}(\\gamma)} \n+ \\varepsilon \\gamma^{-1} \\big\\{  \\| {\\mathfrak{I}}_0 \\|_{s + \\sigma_1 + \\sigma}^{\\mathrm{Lip}(\\g)} + \\gamma^{-1} \n \\| {\\mathfrak{I}}_0 \\|_{s_0 +  \\sigma }^{\\mathrm{Lip}(\\g)}\n\\| Z \\|_{s + \\sigma_1 + \\sigma}^{\\mathrm{Lip}(\\g)} \\big\\}\n\\| g \\|_{s_0}^{{\\rm Lip}(\\gamma)} \\big)\\, . \\nonumber \n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nCollecting Theorem \\ref{teoremadiriducibilita} with the results of \nsections \\ref{step1}-\\ref{step5}, we have obtained the (semi)-conjugation of the operator \n$ \\mathcal{L}_\\omega $ (defined in \\eqref{Lom KdVnew}) \nto $\\mathcal{L}_\\infty $ (defined in \\eqref{Lfinale}), namely  \n\\begin{equation} \\label{def cal M 12}\n\\mathcal{L}_\\omega = {\\cal M}_1 \\mathcal{L}_\\infty {\\cal M}_2^{-1}, \\qquad \n{\\cal M}_1 := \\Phi B \\rho {\\cal T}  \\Phi_1 \\Phi_2  {\\cal S} \\Phi_{\\infty}, \\quad \n{\\cal M}_2 := \\Phi B {\\cal T}  \\Phi_1 \\Phi_2 {\\cal S} \\Phi_{\\infty} \\,,\n\\end{equation}\nwhere $ \\rho $ means the multiplication operator by the function $ \\rho $ defined in \\eqref{anche def rho}. \nBy \\eqref{prime di melnikov} and Lemma 4.2 of \\cite{BBM} we deduce that \n$ \\| {\\cal L}_\\infty^{-1} g \\|_s^{\\mathrm{Lip}(\\g)} \\leq_s \\gamma^{-1} \\| g \\|_{s+ 2 \\tau + 1}^{\\mathrm{Lip}(\\g)} $.\nIn order to estimate $\\mathcal{M}_2, \\mathcal{M}_1^{-1}$, we recall that the composition of tame maps is tame, see Lemma 6.5 in \\cite{BBM}. \nNow,  \n$ \\Phi ,  \\Phi^{-1}$ are estimated in Lemma \\ref{lemma:stime coeff mL1}, \n$ B, B^{-1} $ and $ \\rho $ in Lemma \\ref{lemma:stime coeff mL2}, \n$ {\\cal T}, {\\cal T}^{-1} $ in Lemma \\ref{lemma:stime coeff mL3}. \nThe decay norms $ |\\Phi_1|_s^{\\mathrm{Lip}(\\g)}$, $ |\\Phi_1^{-1}|_s^{\\mathrm{Lip}(\\g)} $, $ |\\Phi_2|_s^{\\mathrm{Lip}(\\g)} $, $ |\\Phi_2^{-1}|_s^{\\mathrm{Lip}(\\g)} \\leq C(s) $ by  Lemmata \\ref{lemma:Dx A bounded}, \\ref{A2 decay}. \nThe decay norm of $ {\\cal S}, {\\cal S}^{-1} $ is estimated in Lemma \\ref{lemma L6}, \nand $ \\Phi_\\infty, \\Phi_\\infty^{-1} $ in \\eqref{stima Phi infty}. \nThe decay norm controls the Sobolev norm by \\eqref{interpolazione norme miste}. \nThus, by \\eqref{def cal M 12}, \n\\[\n\\| \\mathcal{M}_2 h \\|_s^{\\mathrm{Lip}(\\g)} + \\| \\mathcal{M}_1^{-1} h \\|_s^{\\mathrm{Lip}(\\g)} \n\\leq_s \\| h \\|_{s+3}^{\\mathrm{Lip}(\\g)} + \\varepsilon \\gamma^{-1} \\| {\\mathfrak{I}}_\\d \\|_{s+\\sigma+3}^{\\mathrm{Lip}(\\g)} \\| h \\|_{s_0}^{\\mathrm{Lip}(\\g)} \\,,\n\\]\nand  \\eqref{stima inverso linearizzato normale}  follows. \nThe last inequality in \\eqref{stima inverso linearizzato normale} follows by\n\\eqref{stima y - y delta} and \\eqref{ansatz 0}.\n\\end{proof}\n\n\n\\section{The Nash-Moser nonlinear iteration}\\label{sec:NM}\n \nIn this section we prove Theorem \\ref{main theorem}. It will be a consequence of the Nash-Moser Theorem \\ref{iterazione-non-lineare} below.\n\nConsider the finite-dimensional subspaces\n\\[\nE_n := \\big\\{ {\\mathfrak{I}} (\\varphi) = ( \\Theta, y, z )(\\varphi)  : \\, \\Theta = \\Pi_n \\Theta, \\ y = \\Pi_n y, \\ z = \\Pi_n z \\big\\}\n\\]\nwhere $ N_n := N_0^{\\chi^n} $ are introduced  in \\eqref{defN}, and $ \\Pi_n $ are the projectors \n(which, with a small abuse of notation, we denote with the same symbol)\n\\begin{align}\n\\Pi_n \\Theta (\\varphi) :=  \\sum_{|l| < N_n} \\Theta_l e^{{\\mathrm i}  l \\cdot \\varphi}, \\quad \n\\Pi_n y (\\varphi) :=  \\sum_{|l| < N_n} y_l e^{{\\mathrm i}  l \\cdot \\varphi}, \\quad \n& \\text{where} \\ \\Theta (\\varphi) = \\sum_{l \\in \\mathbb Z^\\nu} \\Theta_l e^{{\\mathrm i}  l \\cdot \\varphi}, \\quad \ny(\\varphi) = \\sum_{l \\in \\mathbb Z^\\nu} y_l e^{{\\mathrm i}  l \\cdot \\varphi}, \\nonumber\n\\\\\n\\Pi_n z(\\varphi,x) := \\sum_{|(l,j)| < N_n} z_{lj} e^{{\\mathrm i}  (l \\cdot \\varphi + jx)}, \\quad  \n& \\text{where} \\ z(\\varphi,x) = \\sum_{l \\in \\mathbb Z^\\nu,  j \\in S^c} z_{lj} e^{{\\mathrm i}  (l \\cdot \\varphi + jx)}. \\label{Pin def}\n\\end{align}\nWe define $ \\Pi_n^\\bot := I - \\Pi_n $.  \nThe classical smoothing properties hold: for all $\\alpha , s \\geq 0$, \n\\begin{equation}\\label{smoothing-u1}\n\\|\\Pi_{n} {\\mathfrak{I}} \\|_{s + \\alpha}^{\\mathrm{Lip}(\\g)} \n\\leq N_{n}^{\\alpha} \\| {\\mathfrak{I}} \\|_{s}^{\\mathrm{Lip}(\\g)} \\, , \n\\  \\forall {\\mathfrak{I}} (\\omega) \\in H^{s} \\,,  \n\\quad\n\\|\\Pi_{n}^\\bot {\\mathfrak{I}} \\|_{s}^{\\mathrm{Lip}(\\g)} \n\\leq N_{n}^{-\\alpha} \\| {\\mathfrak{I}} \\|_{s + \\alpha}^{\\mathrm{Lip}(\\g)} \\, , \n\\  \\forall {\\mathfrak{I}} (\\omega) \\in H^{s + \\alpha} \\, .\n\\end{equation}\nWe define the constants\n\\begin{alignat}{3} \\label{costanti nash moser}\n& \\mu_1 := 3 \\mu + 9\\,,\\quad &\n& \\alpha := 3 \\mu_1 + 1\\,,\\quad &\n& \\alpha_1 := (\\alpha - 3 \\mu)\/2 \\,, \n \\\\\n& \\kappa := 3 \\big(\\mu_1 + \\rho^{-1} \\big)+ 1\\,,\\qquad &\n& \\beta_1 := 6 \\mu_1+  3 \\rho^{-1} + 3 \\, ,  \\qquad &\n& 0 < \\rho < \\frac{1 - 3 a}{C_1(1 + a)}\\,, \\label{def rho}\n\\end{alignat}\nwhere $ \\mu := \\mu (\\tau, \\nu) $ is the ``loss of regularity\" defined in Theorem \\ref{thm:stima inverso approssimato} \n(see \\eqref{stima inverso approssimato 1}) and $ C_1 $ is fixed below. \n\n\\begin{theorem}\\label{iterazione-non-lineare} \n{\\bf (Nash-Moser)} Assume that $ f \\in C^q $ with $ q > S := s_0 + \\b_1 + \\mu + 3 $. \nLet $ \\tau \\geq \\nu + 2 $. Then there exist $ C_1 > \\max \\{ \\mu_1 + \\a, C_0 \\} $\n(where $ C_0 := C_0 (\\tau, \\nu) $  is the one in Theorem \\ref{teoremadiriducibilita}), \n$ \\delta_0 := \\d_0 (\\tau, \\nu) > 0 $ such that, if\n\\begin{equation}\\label{nash moser smallness condition}  \nN_0^{C_1}  \\varepsilon^{b_* + 1} \\gamma^{-2}< \\d_0\\,,\\quad \\gamma:= \\varepsilon^{2 + a} = \\varepsilon^{2b} \\,,\\quad \nN_0 := (\\varepsilon \\gamma^{-1})^\\rho\\,,\\quad b_* := 6 - 2 b \\, , \n\\end{equation}\nthen, for all $ n \\geq 0 $: \n\n\\begin{itemize}\n\\item[$({\\cal P}1)_{n}$] \nthere exists a function \n$({\\mathfrak{I}}_n, \\zeta_n) : {\\cal G}_n \\subseteq \\Omega_\\varepsilon \\to E_{n-1} \\times \\mathbb R^\\nu$, \n$\\omega \\mapsto ({\\mathfrak{I}}_n(\\omega), \\zeta_n(\\omega))$,  \n$ ({\\mathfrak{I}}_0, \\zeta_0) := 0 $, $ E_{-1} := \\{ 0 \\} $,   \nsatisfying $ | \\zeta_n |^{\\mathrm{Lip}(\\g)} \\leq C \\|{\\cal F}(U_n) \\|_{s_0}^{\\mathrm{Lip}(\\g)} $, \n\\begin{equation}\\label{ansatz induttivi nell'iterazione}\n\\| {\\mathfrak{I}}_n \\|_{s_0 + \\mu }^{{\\rm Lip}(\\gamma)} \n\\leq C_* \\varepsilon^{b_*} \\gamma^{-1}\\,, \\quad \n\\| {\\cal F}(U_n)\\|_{s_0 + \\mu + 3}^{{\\rm Lip}(\\gamma)} \\leq C_*\\varepsilon^{b_*} \\,, \n\\end{equation}\nwhere $U_n := (i_n, \\zeta_n)$ with $i_n(\\varphi) = (\\varphi,0,0) + {\\mathfrak{I}}_n(\\varphi)$.\nThe sets ${\\cal G}_{n} $ are defined inductively by: \n$$\n{\\cal G}_{0} := \\big\\{\\omega \\in \\Omega_\\varepsilon \\, :\\, |\\omega \\cdot l| \\geq  2 \\gamma \\langle l \\rangle^{-\\tau}, \\,  \\forall l \\in \\mathbb Z^\\nu \\setminus \\{0\\} \\big\\}\\,, \n$$\n\\begin{equation}\\label{def:Gn+1}\n{\\cal G}_{n+1} :=  \n\\Big\\{ \\omega  \\in {\\cal G}_{n} \\, : \\, |{\\mathrm i}  \\omega \\cdot l + \\mu_j^\\infty ( i_n) -\n\\mu_k^\\infty ( i_n )| \\geq \\frac{2\\gamma_{n} |j^{3}-k^{3}|}{\\left\\langle l\\right\\rangle^{\\tau}}, \\, \\forall j , k \\in S^c \\cup \\{0\\}, \n\\, l \\in \\mathbb Z^{\\nu} \\Big\\}\\,,\n\\end{equation}\nwhere $ \\gamma_{n}:=\\gamma (1 + 2^{-n}) $ and $\\mu_j^\\infty(\\omega) := \\mu_j^\\infty(\\omega, i_n(\\omega)) $ \nare defined in \\eqref{espressione autovalori} (and  $ \\mu_0^\\infty(\\omega) = 0 $).\n\nThe differences $\n\\widehat {\\mathfrak I}_n\n:=  {\\mathfrak I}_n - {\\mathfrak I}_{n - 1} $\n(where we set $ \\widehat {\\mathfrak{I}}_0 := 0 $) is defined on $\\mathcal{G}_n$, and satisfy\n\\begin{equation}  \\label{Hn}\n\\| \\widehat {\\mathfrak I}_1 \\|_{ s_0 + \\mu}^{{\\mathrm{Lip}(\\g)}} \\leq C_* \\varepsilon^{b_*} \\gamma^{-1} \\, , \\quad \n\\| \\widehat {\\mathfrak I}_n \\|_{ s_0 + \\mu}^{{\\mathrm{Lip}(\\g)}} \\leq C_* \\varepsilon^{b_*} \\gamma^{-1} N_{n - 1}^{-\\alpha_1} \\, , \\quad \\forall n > 1 \\, .\n\\end{equation}\n\\item[$({\\cal P}2)_{n}$]   $ \\| {\\cal F}(U_n) \\|_{ s_{0}}^{{\\rm Lip}(\\gamma)} \\leq C_* \\varepsilon^{b_*} N_{n - 1}^{- \\alpha}$ \nwhere we set $N_{-1} := 1$.\n\\item[$({\\cal P}3)_{n}$] \\emph{(High norms).} \n\\ $  \\| {\\mathfrak{I}}_n \\|_{ s_{0}+ \\beta_1}^{{\\rm Lip}(\\gamma)} \\leq C_* \\varepsilon^{b_*} \\gamma^{-1}  N_{n - 1}^{\\kappa} $ and  \n$ \\|{\\cal F}(U_n ) \\|_{ s_{0}+\\beta_1}^{{\\rm Lip}(\\gamma)} \\leq C_* \\varepsilon^{b_*}   N_{n - 1}^{\\kappa} $.   \n\n\n\\item[$({\\cal P}4)_{n}$] \\emph{(Measure).} \nThe measure of the ``Cantor-like\" sets $ {\\cal G}_n $ satisfies\n\\begin{equation}\\label{Gmeasure}\n | \\Omega_\\varepsilon \\setminus {\\cal G}_0 | \\leq C_* \\varepsilon^{2(\\nu - 1)} \\gamma \\, , \\quad  \n\\big| {\\cal G}_n \\setminus {\\cal G}_{n+1} \\big|  \\leq  C_* \\varepsilon^{2(\\nu - 1)} \\gamma N_{n - 1}^{-1}  \\, . \n\\end{equation}\n\\end{itemize}\nAll the Lip norms are defined on $ {\\cal G}_{n} $, namely  $\\| \\ \\|_s^{{\\rm Lip}(\\gamma)} = \\| \\ \\|_{s,\\mathcal{G}_n}^{{\\rm Lip}(\\gamma)} $.\\end{theorem}\n\n\n\\begin{proof}\nTo simplify notations, in this proof we denote $\\| \\, \\|^{{\\rm Lip}(\\gamma)}$ by $\\| \\, \\|$. We first prove $({\\cal P}1, 2, 3)_n$.\n\n\\smallskip\n\n{\\sc Step 1:} \\emph{Proof of} $({\\cal P}1, 2, 3)_0$. \nRecalling \\eqref{operatorF} we have $ \\| {\\cal F}( U_0 ) \\|_s = $ \n$ \\| {\\cal F}(\\varphi, 0 , 0, 0 ) \\|_s = \\| X_P(\\varphi, 0 , 0 ) \\|_s \\leq_s \\varepsilon^{6 - 2b} $ by\n\\eqref{stima XP}. Hence (recall that $ b_* = 6 - 2 b $) the smallness conditions in\n$({\\cal P}1)_0$-$({\\cal P}3)_0$ hold taking $ C_* := C_* (s_0 + \\b_1) $ large enough.\n\n\\smallskip\n\n{\\sc Step 2:} \\emph{Assume that $({\\cal P}1,2,3)_n$ hold for some $n \\geq 0$, and prove $({\\cal P}1,2,3)_{n+1}$.}\nBy \\eqref{nash moser smallness condition} and \\eqref{def rho},\n$$\nN_0^{C_1} \\varepsilon^{b_* + 1} \\gamma^{-2} = N_0^{C_1} \\varepsilon^{1-3a} = \\varepsilon^{1 - 3 a - \\rho C_1(1 + a)}  < \\d_0 \n$$\nfor $\\varepsilon$ small enough, and the smallness condition \\eqref{condizione-kam} holds. \nMoreover \\eqref{ansatz induttivi nell'iterazione} imply  \\eqref{ansatz 0}\n(and so \\eqref{ansatz delta}) and  Theorem \\ref{inversione linearized normale} applies. \nHence the operator $ {\\cal L}_\\omega := {\\cal L}_\\omega(\\omega, i_n(\\omega))$ \ndefined in \\eqref{cal L omega} is invertible for all $\\omega \\in {\\cal G}_{n + 1}$ and \nthe last estimate in \\eqref{stima inverso linearizzato normale} holds. \nThis means that the assumption \\eqref{tame inverse} of Theorem \\ref{thm:stima inverso approssimato} \nis verified with $\\Omega_\\infty = {\\cal G}_{n + 1}$. \nBy Theorem \\ref{thm:stima inverso approssimato} there exists an approximate inverse \n${\\bf T}_n(\\omega) := {\\bf T}_0 (\\omega, i_n(\\omega))$ of the linearized operator \n$L_n(\\omega) := d_{i, \\zeta} {\\cal F}(\\omega, i_n(\\omega)) $, satisfying \\eqref{stima inverso approssimato 1}. \nThus, using also \\eqref{nash moser smallness condition}, \\eqref{smoothing-u1}, \\eqref{ansatz induttivi nell'iterazione},\n\\begin{align}\n\\| {\\bf T}_n g  \\|_s \n& \\leq_s \\gamma^{-1} \\big( \\| g \\|_{s + \\mu} \n+ \\varepsilon \\gamma^{-1}\\{\\| {\\mathfrak{I}}_n \\|_{s + \\mu} + \n\\gamma^{-1} \\|{\\mathfrak I}_n \\|_{s_0 + \\mu}  \\|{\\cal F}(U_n) \\|_{s + \\mu} \\} \\| g \\|_{s_0+ \\mu}\\big) \\label{stima Tn} \\\\\n\\| {\\bf T}_n  g \\|_{s_0} \n& \n\\leq_{s_0} \\gamma^{-1} \\| g \\|_{s_0 + \\mu}   \\label{stima Tn norma bassa}\n\\end{align}\nand, by \\eqref{stima inverso approssimato 2}, \nusing also \\eqref{ansatz induttivi nell'iterazione}, \n\\eqref{nash moser smallness condition}, \\eqref{smoothing-u1}, \n\\begin{align}\n\\| \\big(L_n \\circ  {\\bf T}_n -I \\big)  g \\|_s \n& \\leq_s \\gamma^{-1}  \\big( \\| {\\cal F}(U_n) \\|_{s_0 + \\mu} \\| g\\|_{s + \\mu} + \n\\| {\\cal F}(U_n) \\|_{s + \\mu} \\|  g \\|_{s_0 + \\mu}  \\nonumber\n\\\\\n& \\qquad + \\varepsilon \\gamma^{-1} \\| {\\mathfrak{I}}_n \\|_{s + \\mu} \\| {\\cal F}(U_n) \\|_{s_0 + \\mu} \\| g \\|_{s_0 + \\mu} \\big)\\, , \n \\label{stima Tn inverso approssimato} \\\\\n\\label{stima Tn inverso approssimato norma bassa}\n\\| \\big(L_n \\circ  {\\bf T}_n -I \\big) g  \\|_{s_0} \n& \\leq_{s_0} \\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\mu} \\| g \\|_{s_0 + \\mu}  \\nonumber\n\\\\\n& \\leq_{s_0}  \\gamma^{-1} \\big( \\|\\Pi_n  {\\cal F}(U_n)\\|_{s_0 + \\mu}+  \\|\\Pi_n^\\bot  {\\cal F}(U_n)\\|_{s_0 + \\mu} \\big) \\|  g  \\|_{s_0 + \\mu}  \\nonumber \n\\\\\n&  \\leq_{s_0} N_{n }^{\\mu} \\gamma^{-1} \\big( \\| {\\cal F}(U_n)\\|_{s_0} + N_{n}^{-\\beta_1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} \\big) \\|  g  \\|_{s_0 + \\mu}\\, .\n\\end{align}\nThen, for all $ \\omega \\in {\\cal G}_{n+1} $, $ n \\geq 0 $, we define \n\\begin{equation}\\label{soluzioni approssimate}\nU_{n + 1} := U_n + H_{n + 1}\\,, \\quad \nH_{n + 1} :=\n( \\widehat {\\mathfrak{I}}_{n+1}, \\widehat \\zeta_{n+1}) :=  - {\\widetilde \\Pi}_{n } {\\bf T}_n \\Pi_{n } {\\cal F}(U_n) \n\\in E_n \\times \\mathbb R^\\nu\\, ,  \n\\end{equation}\nwhere  $ {\\widetilde \\Pi}_n ( {\\mathfrak{I}} , \\zeta ) :=   ( \\Pi_n {\\mathfrak{I}} , \\zeta ) $ \nwith $ \\Pi_n $ in \\eqref{Pin def}.\nSince $ L_n := d_{i,\\zeta} {\\cal F}(i_n) $,  we write  \n$ {\\cal F}(U_{n + 1}) =  {\\cal F}(U_n) + L_n H_{n + 1} + Q_n $, where\n\\begin{equation}\\label{def:Qn}\nQ_n := Q(U_n, H_{n + 1}) \\, , \\quad \nQ (U_n, H)  :=  {\\cal F}(U_n + H ) - {\\cal F}(U_n) - L_n H \\,, \\quad \nH \\in E_{n} \\times \\mathbb R^\\nu . \n\\end{equation}\nThen, by the definition of $ H_{n+1} $ in \\eqref{soluzioni approssimate}, and writing $ {\\widetilde \\Pi}_n^\\bot ({\\mathfrak{I}}, \\zeta ) :=\n(\\Pi_n^\\bot {\\mathfrak{I}}, 0) $, we have \n\\begin{align}\n{\\cal F}(U_{n + 1}) & = \n {\\cal F}(U_n) - L_n {\\widetilde \\Pi}_{n } {\\bf T}_n \\Pi_{n } {\\cal F}(U_n) + Q_n = \n {\\cal F}(U_n) - L_n  {\\bf T}_n \\Pi_{n } {\\cal F}(U_n) + L_n  {\\widetilde \\Pi}_n^\\bot  {\\bf T}_n \\Pi_{n } {\\cal F}(U_n)\n + Q_n \\nonumber\\\\ \n& =  {\\cal F}(U_n)  - \\Pi_{n } L_n {\\bf T}_n \\Pi_{n }{\\cal F}(U_n) \n+ ( L_n  {\\widetilde \\Pi}_n^\\bot -  \\Pi_n^\\bot L_n ) {\\bf T}_n \\Pi_{n }{\\cal F}(U_n) + Q_n \\nonumber\\\\\n & = \\Pi_{n }^\\bot {\\cal F}(U_n) + R_n + Q_n + Q_n'  \n\\label{relazione algebrica induttiva}\n\\end{align}\nwhere \n\\begin{equation}\\label{Rn Q tilde n}\nR_n := (L_n  {\\widetilde \\Pi}_n^\\bot -  \\Pi_n^\\bot L_n) {\\bf T}_n \\Pi_{n }{\\cal F}(U_n) \\,,\n\\qquad \nQ_n' := - \\Pi_{n } ( L_n {\\bf T}_n - I) \\Pi_{n } {\\cal F}(U_n)\\,.\n\\end{equation}\n\n\n\\begin{lemma}\\label{lemma convergence}\nDefine \n\\begin{equation}\\label{riscalamenti nash moser}\nw_n := \\varepsilon \\gamma^{-2}  \\|{\\cal F}(U_n) \\|_{s_0}\\,,\\quad B_n := \\varepsilon \\gamma^{-1}\\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1} + \\varepsilon \\gamma^{-2}  \\|{\\cal F}(U_n) \\|_{s_0 + \\beta_1} \\,.\n\\end{equation}\nThen there exists $ K :=  K( s_0, \\b_1 ) > 0 $ such that,  for all $n \\geq 0$, \nsetting $ \\mu_1 := 3 \\mu + 9$ (see \\eqref{costanti nash moser}), \n\\begin{equation}\\label{relazioni induttive}\nw_{n + 1} \\leq K N_{n }^{\\mu_1 + \\frac{1}{\\rho} - \\beta_1} B_n +  K  N_n^{\\mu_1} w_n^2\\,,\\qquad \nB_{n + 1} \\leq K N_{n }^{\\mu_1 + \\frac{1}{\\rho}}  B_n\\, . \n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe  estimate separately the terms $ Q_n $ in \\eqref{def:Qn} and $ Q_n' , R_n $ in \\eqref{Rn Q tilde n}. \n\\\\[1mm]\n{\\it Estimate of $ Q_n $.}\nBy  \\eqref{def:Qn}, \\eqref{operatorF}, \\eqref{parte quadratica da P}\nand \\eqref{ansatz induttivi nell'iterazione}, \\eqref{smoothing-u1}, \nwe have the quadratic estimates\n\\begin{align}\n\\label{stima parte quadratica norma alta}\n\\| Q(U_n, H)\\|_s & \\leq_s \\varepsilon \\big(\\| \\widehat {\\mathfrak{I}} \\|_{s + 3} \\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + 3} + \\| {\\mathfrak{I}}_n \\|_{s + 3} \n\\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + 3}^2 \\big) \\\\\n\\label{stima parte quadratica norma bassa}\n\\| Q(U_n, H) \\|_{s_0} & \\leq_{s_0}\\varepsilon N_n^6 \\|  \\widehat {\\mathfrak{I}} \\|_{s_0}^2\\,, \\quad \\forall  \\widehat {\\mathfrak{I}} \\in E_n   \\, . \n\\end{align}\nNow by the definition of $H_{n + 1}$ in \\eqref{soluzioni approssimate} and \\eqref{smoothing-u1}, \n\\eqref{stima Tn}, \\eqref{stima Tn norma bassa}, \\eqref{ansatz induttivi nell'iterazione},   we get \n\\begin{align}\n\\|  \\widehat {\\mathfrak{I}}_{n + 1} \\|_{s_0 + \\beta_1} \n& \\leq_{s_0 + \\beta_1} N_n^{\\mu} \\big( \\gamma^{-1} \\|{\\cal F}(U_n) \\|_{s_0 + \\beta_1} + \\varepsilon \\gamma^{-2} \\| {\\cal F}(U_n)\\|_{s_0 + \\mu} \\{\\| {\\mathfrak{I}}_n\\|_{s_0 + \\beta_1} + \\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} \\} \\big)\n\\nonumber \\\\\n& \\leq_{s_0 + \\beta} N_n^{\\mu} \\big( \\gamma^{-1}\\| {\\cal F}(U_n) \\|_{s_0 + \\beta_1}  \n+ \\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1}\\big)\\, ,  \\label{H n+1 alta} \n\\\\\n\\label{H n+1 bassa}\n\\|  \\widehat {\\mathfrak{I}}_{n + 1}\\|_{s_0} \n& \\leq_{s_0} \\gamma^{-1}N_{n}^\\mu \\| {\\cal F}(U_n)\\|_{s_0} \\, .\n\\end{align}\nThen the term $ Q_n $ in \\eqref{def:Qn} satisfies, \nby \\eqref{stima parte quadratica norma alta}, \\eqref{stima parte quadratica norma bassa},  \n\\eqref{H n+1 alta}, \\eqref{H n+1 bassa},  \n\\eqref{nash moser smallness condition},  \\eqref{ansatz induttivi nell'iterazione}, \n$ ({\\cal P}2)_n $, \\eqref{costanti nash moser},   \n\\begin{align} \n\\| Q_n \\|_{s_0 + \\beta_1} \n& \\leq_{s_0 + \\beta_1}\n N_n^{2 \\mu + 9}  \\gamma \\big( \\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1}  + \n\\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1} \\big) \\, ,  \\label{Qn norma alta} \\\\\n\\| Q_n \\|_{s_0} \n&  \\leq_{s_0}  N_n^{2 \\mu + 6 } \\varepsilon \\gamma^{-2} \\| {\\cal F}(U_n) \\|_{s_0}^2\\, . \\label{Qn norma bassa}\n\\end{align}\n{\\it Estimate of $ Q_n' $.}\nThe bounds \\eqref{stima Tn inverso approssimato}, \\eqref{stima Tn inverso approssimato norma bassa},\n\\eqref{smoothing-u1}, \\eqref{costanti nash moser}, \\eqref{ansatz induttivi nell'iterazione} imply\n\\begin{align}\n\\label{Qn' norma alta}\n\\| Q_n' \\|_{s_0 + \\beta_1} &  \\leq_{s_0 + \\beta_1} N_n^{2 \\mu} \\big(\n\\| {\\cal F}(U_n) \\|_{s_0 + \\beta_1} + \\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1} \\| {\\cal F}(U_n)\\|_{s_0} \\big) \\, , \\\\\n\\label{Qn' norma bassa}\n\\| Q_n'\\|_{s_0} & \\leq_{s_0} \\gamma^{-1} N_{n }^{2\\mu }\\big(\\| {\\cal F}(U_n) \\|_{s_0} + N_{n}^{- \\beta_1} \\|{\\cal F}(U_n)\\|_{s_0 + \\beta_1} \\big) \\| {\\cal F}(U_n) \\|_{s_0}\\,.\n\\end{align}\n{\\it Estimate of $ R_n $.} For  $ H := (\\widehat {\\mathfrak{I}}, \\widehat \\zeta ) $ we have \n$ (L_n  {\\widetilde \\Pi}_n^\\bot -  \\Pi_n^\\bot L_n) H = $ $ [{\\bar D}_n, \\Pi_n^\\bot ] \\widehat {\\mathfrak{I}} = $\n$ [\\Pi_n  , {\\bar D}_n] \\widehat {\\mathfrak{I}} $ where $ {\\bar D}_n := d_i X_{H_\\varepsilon}(i_n) + (0,0, \\partial_{xxx} ) $. \nThus Lemma \\ref{lemma quantitativo forma normale},\n\\eqref{ansatz induttivi nell'iterazione}, \\eqref{smoothing-u1} and \\eqref{tame commutatori}  imply \n\\begin{align}\\label{stima commutatore modi alti norma bassa}\n\\| (L_n  {\\widetilde \\Pi}_n^\\bot -  \\Pi_n^\\bot L_n) H \\|_{s_0} & \\leq_{s_0+ \\b_1} \n\\varepsilon N_{n }^{- \\beta_1 + \\mu + 3} \\big(\\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + \\beta_1 - \\mu} + \n\\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1 - \\mu} \\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + 3}\\big)\\,, \\\\\n\\label{stima commutatore modi alti norma alta}\n \\| (L_n  {\\widetilde \\Pi}_n^\\bot -  \\Pi_n^\\bot L_n) H \\|_{s_0 + \\beta_1} & \n\\leq_s \n\\varepsilon N_n^{\\mu + 3} \\big(\\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + \\beta_1 - \\mu } + \\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1 - \\mu } \n\\|  \\widehat {\\mathfrak{I}} \\|_{s_0 + 3} \\big)\\,.\n\\end{align}\nHence, applying \n\\eqref{stima Tn}, \n\\eqref{stima commutatore modi alti norma bassa}, \\eqref{stima commutatore modi alti norma alta},  \n\\eqref{nash moser smallness condition}, \n\\eqref{ansatz induttivi nell'iterazione}, \n\\eqref{smoothing-u1}, \nthe term $R_n$ defined in \\eqref{Rn Q tilde n} satisfies\n\\begin{align} \\label{stima Rn norma bassa}\n\\| R_n\\|_{s_0} \n& \\leq_{s_0 + \\beta_1}  N_n^{ \\mu + 6 - \\beta_1} ( \\varepsilon \\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} + \\varepsilon \\| {\\mathfrak{I}}_n  \\|_{s_0 + \\beta_1} )\\,, \n\\\\\n\\label{stima Rn norma alta}\n\\| R_n \\|_{s_0 + \\beta_1} \n& \\leq_{s_0 + \\beta_1} N_n^{ \\mu + 6} ( \\varepsilon \\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} + \\varepsilon \\| {\\mathfrak{I}}_n  \\|_{s_0 + \\beta_1} )\\,.\n\\end{align}\n{\\it Estimate of $ {\\cal F}(U_{n + 1}) $.} \nBy \\eqref{relazione algebrica induttiva} and \\eqref{Qn norma alta}, \n \\eqref{Qn norma bassa}, \\eqref{Qn' norma alta}, \\eqref{Qn' norma bassa}, \n\\eqref{stima Rn norma bassa}, \\eqref{stima Rn norma alta},  \\eqref{nash moser smallness condition},  \n\\eqref{ansatz induttivi nell'iterazione}, \nwe get\n\\begin{align}\\label{F(U n+1) norma bassa}\n& \\| {\\cal F}(U_{n + 1})\\|_{s_0} \\leq_{s_0 + \\beta_1}   N_{n }^{\\mu_1 - \\beta_1} (  \\varepsilon \\gamma^{-1}\\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} +  \\varepsilon \\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1} ) \n+ N_n^{\\mu_1} \\varepsilon \\gamma^{-2} \\| {\\cal F}(U_n)\\|_{s_0}^2\\,, \n\\\\\n\\label{F(U n+1) norma alta}\n& \\| {\\cal F}(U_{n + 1}) \\|_{s_0 + \\beta_1} \n\\leq_{s_0 + \\beta_1}  N_n^{\\mu_1} ( \\varepsilon \\gamma^{-1}\\| {\\cal F}(U_n) \\|_{s_0 + \\beta_1} + \\varepsilon \\| {\\mathfrak{I}}_n \\|_{s_0 + \\beta_1} ) \\,,\n\\end{align}\nwhere $\\mu_1 := 3 \\mu + 9$.\n\n\\noindent \n{\\it Estimate of $ {\\mathfrak{I}}_{n+1} $.} \nUsing  \\eqref{H n+1 alta}  the term $ {\\mathfrak{I}}_{n+1} = {\\mathfrak{I}}_n + \\widehat {\\mathfrak I}_{n+1} $ is bounded by\n\\begin{equation}\\label{U n+1 alta}\n\\| {\\mathfrak{I}}_{n + 1}\\|_{s_0 + \\beta_1} \\leq_{s_0 + \\beta_1} \nN_n^\\mu ( \\| {\\mathfrak{I}}_n\\|_{s_0 + \\beta_1} +\\gamma^{-1} \\| {\\cal F}(U_n)\\|_{s_0 + \\beta_1} )\\, . \n\\end{equation}\nFinally, recalling  \\eqref{riscalamenti nash moser}, the  inequalities \\eqref{relazioni induttive} follow by \n\\eqref{F(U n+1) norma bassa}-\\eqref{U n+1 alta}, \\eqref{ansatz induttivi nell'iterazione} and \n$ \\varepsilon \\gamma^{-1} = N_0^{1\/\\rho} \\leq N_n^{1\/\\rho}$.\n\\end{proof} \n\n\\emph{Proof of $({\\cal P}3)_{n + 1}$}. \nBy \\eqref{relazioni induttive} and $ ({\\cal P}3)_n $, \n\\begin{equation}\nB_{n + 1} \n\\leq K N_{n}^{\\mu_1 + \\frac{1}{\\rho}} B_n \n\\leq 2 C_* K \\varepsilon^{b_* + 1} \\gamma^{-2} N_{n}^{\\mu_1+ \\frac{1}{\\rho}} N_{n-1}^\\kappa \\leq C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_n^{\\kappa} \\,,  \n\\label{stima B n+1} \n\\end{equation}\nprovided $ 2 K N_n^{\\mu_1 + \\frac{1}{\\rho} - \\kappa} N_{n-1}^\\kappa \\leq 1 $, $  \\forall n \\geq 0 $. \nThis inequality  holds by \\eqref{def rho}, taking $N_0$ large enough (i.e $\\varepsilon$ small enough). By \\eqref{riscalamenti nash moser}, the bound $B_{n + 1} \\leq C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_n^{\\kappa}$ implies $({\\cal P}3)_{n + 1}$.\n\n\\smallskip\n\n\\emph{Proof of $({\\cal P}2)_{n + 1}$}. \nUsing \\eqref{relazioni induttive}, \\eqref{riscalamenti nash moser} and $  ({\\cal P}2)_n, ({\\cal P}3)_n $, we get\n\\begin{align*}\nw_{n + 1}  \n& \\leq K N_n^{\\mu_1+ \\frac{1}{\\rho} - \\beta_1} B_n +  K N_{n }^{\\mu_1} w_n^2   \n\\leq K N_n^{\\mu_1 + \\frac{1}{\\rho} - \\beta_1} 2 C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_{n - 1}^{\\kappa} \n+ K N_{n}^{\\mu_1} ( C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_{n - 1}^{- \\alpha} )^2 \n\\end{align*}\nwhich is $ \\leq C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_n^{- \\alpha} $ provided that \n\\begin{equation} \\label{provided 2}\n4 K N_n^{\\mu_1 + \\frac{1}{\\rho} - \\b_1 + \\a} N_{n-1}^\\kappa \\leq 1, \n\\quad\n2 K C_* \\varepsilon^{b_*+1} \\gamma^{-2} N_n^{\\mu_1 + \\a} N_{n-1}^{-2\\a} \\leq 1 \\, , \n\\quad \\forall n \\geq 0.\n\\end{equation}\nThe inequalities in \\eqref{provided 2} hold by \\eqref{costanti nash moser}-\\eqref{def rho}, \n\\eqref{nash moser smallness condition}, $C_1 > \\mu_1 + \\alpha$, taking $\\d_0$ in \\eqref{nash moser smallness condition} small enough. \nBy \\eqref{riscalamenti nash moser}, the inequality $w_{n + 1} \\leq C_* \\varepsilon^{b_* + 1} \\gamma^{-2} N_n^{- \\alpha}$ implies $({\\cal P}2)_{n + 1}$.\n\n\\smallskip\n\n\\emph{Proof of $({\\cal P}1)_{n + 1}$}.\nThe bound \\eqref{Hn} for $  \\widehat {\\mathfrak{I}}_1 $ \nfollows by \\eqref{soluzioni approssimate}, \\eqref{stima Tn} (for $ s = s_0 + \\mu $) and \n$  \\| {\\cal F} ( U_0 ) \\|_{s_0 + 2 \\mu} = $ $  \\| {\\cal F}(\\varphi, 0, 0, 0) \\|_{s_0 + 2 \\mu} \\leq_{s_0+2\\mu } \\varepsilon^{b_*} $.\nThe bound \\eqref{Hn} for $ \\widehat {\\mathfrak{I}}_{n + 1} $ follows by \\eqref{smoothing-u1}, \n\\eqref{H n+1 bassa}, $({\\cal P}2)_n $, \\eqref{costanti nash moser}. \nIt remains to prove that \\eqref{ansatz induttivi nell'iterazione} holds at the step $n + 1$.\nWe have\n\\begin{equation}\\label{W n+1 norma bassa}\n\\| {\\mathfrak{I}}_{n + 1} \\|_{s_0 + \\mu} \n\\leq {\\mathop \\sum}_{k = 1}^{n + 1} \\| \\widehat {\\mathfrak I}_k \\|_{s_0 + \\mu} \n\\leq C_* \\varepsilon^{b_*}\\gamma^{-1} {\\mathop \\sum}_{k \\geq 1} N_{k - 1}^{- \\alpha_1} \n\\leq C_*  \\varepsilon^{b_*} \\gamma^{-1}\n\\end{equation}\nfor $ N_0 $ large enough, i.e. $ \\varepsilon $ small. \nMoreover, using  \\eqref{smoothing-u1}, (${\\cal P}2)_{n +1}$, (${\\cal P}3)_{n + 1}$, \\eqref{costanti nash moser}, we get \n\\begin{align*}\n\\| {\\cal F}(U_{n + 1}) \\|_{s_0 + \\mu+ 3} \n& \\leq  N_n^{\\mu + 3} \\|{\\cal F}(U_{n + 1}) \\|_{s_0} + N_{n}^{\\mu + 3 - \\beta_1} \\| {\\cal F}(U_{n + 1}) \\|_{s_0 + \\beta_1} \n\\\\\n& \\leq C_* \\varepsilon^{b_*} N_n^{\\mu + 3- \\alpha} + C_* \\varepsilon^{b_*} N_n^{\\mu + 3 - \\beta_1 + \\kappa}\n\\leq  C_* \\varepsilon^{b_*}\\,,\n\\end{align*}\nwhich is the second inequality in \\eqref{ansatz induttivi nell'iterazione} at the step $n + 1$.\nThe bound $ | \\zeta_{n+1} |^{\\mathrm{Lip}(\\g)} \\leq C \\|{\\cal F}(U_{n+1}) \\|_{s_0}^{\\mathrm{Lip}(\\g)} $ is a consequence of Lemma \n\\ref{zeta = 0} (it is not inductive).\n\n\\smallskip\n\n\n{\\sc Step 3:} \\emph{Prove $({\\cal P}4)_n$ for all $n \\geq 0$.} \nFor all $n \\geq 0$, \n\\begin{equation}\\label{Gn inclusioni}\n{\\cal G}_n \\setminus {\\cal G}_{n + 1} = \\!\\! \\!\\!\\!  \\bigcup_{\\begin{subarray}{c}\nl \\in \\mathbb Z^\\nu, \\, j , k \\in S^c \\cup\\{ 0 \\}\n\\end{subarray}} \\!\\! \\! \\!\\!  R_{ljk}(i_n)\n\\end{equation}\nwhere \n\\begin{equation}\\label{resonant sets}\nR_{ljk}(i_n) := \\big\\{ \\omega \\in {\\cal G}_n \\, : \\, |{\\mathrm i}  \\omega \\cdot l + \\mu_j^\\infty (i_{n}) -\n\\mu_k^\\infty (i_{n})| < 2\\gamma_{n} |j^{3}-k^{3}|\\left\\langle l\\right\\rangle^{- \\tau}\\big\\}\\,.\n\\end{equation}\nNotice that $R_{ljk}(i_n) = \\emptyset$ if $j = k$, so that we suppose in the sequel that $j \\neq k$.\n\n\\begin{lemma} \\label{matteo 10}\nFor all $n \\geq 1$, $|l| \\leq N_{n - 1}$, the set $R_{ljk}(i_n) \\subseteq R_{ljk}(i_{n - 1}) $. \n\\end{lemma}\n\n\\begin{proof}\nLike Lemma 5.2 in \\cite{BBM} \n(with $\\omega$ in the role of $\\lambda \\bar\\omega$,  and $N_{n-1}$ instead of $N_n$).\n\\end{proof}\n\nBy definition, $ R_{ljk} (i_n) \\subseteq {\\cal G}_n $ (see \\eqref{resonant sets})\nand Lemma \\ref{matteo 10} implies that, for all $ n \\geq 1$, $ |l| \\leq N_{n-1} $, the set\n$R_{ljk}(i_n) \\subseteq R_{ljk}(i_{n - 1}) $. On the other hand \n$ R_{ljk}(i_{n - 1})  \\cap {\\cal G}_{n} = \\emptyset $ (see \\eqref{def:Gn+1}). As a consequence,\nfor all $ |l| \\leq N_{n-1} $, $ R_{ljk} (i_n) = \\emptyset $ and, by \\eqref{Gn inclusioni}, \n\\begin{equation}\\label{Gn Gn+1}\n{\\cal G}_n \\setminus {\\cal G}_{n+1}  \\subseteq \\!\\!  \\bigcup_{|l| > N_{n-1}, \\, j, k \\in S^c \\cup \\{0\\}} \n\\!\\!\\! \\!\\!\\! R_{ljk} ( i_n)  \\qquad \\forall n \\geq 1. \n\\end{equation}\n\n\\begin{lemma}\\label{matteo 4}\nLet  $n \\geq 0$. If $R_{ljk}(i_n) \\neq \\emptyset$  \nthen $|l| \\geq C |j^3 - k^3| \\geq \\frac12 C (j^2 + k^2) $ for some $C > 0$.\n\\end{lemma}\n\n\\begin{proof}\nLike Lemma 5.3 in \\cite{BBM}. The only difference is that  $\\omega$ is not constrained to a fixed  direction. \nNote also that $ | j^3- k^3 | \\geq (j^2 + k^2) \/ 2  $, $ \\forall j \\neq k $. \n\\end{proof}\n\nBy usual arguments (e.g. see Lemma 5.4 in \\cite{BBM}), using Lemma \\ref{matteo 4} and \\eqref{autofinali} we \nhave:\n\n\\begin{lemma} \\label{lemma:risonanti}\nFor all $n \\geq 0$, the measure $ |R_{ljk}(i_n)| \\leq C \\varepsilon^{2(\\nu - 1)}  \\gamma \\langle l \\rangle^{- \\tau} $.\n\\end{lemma}\n\n\nBy \\eqref{Gn inclusioni} and Lemmata \\ref{matteo 4}, \\ref{lemma:risonanti} we get\n$$\n|{\\cal G}_0 \\setminus {\\cal G}_1 | \\leq \\sum_{l \\in \\mathbb Z^\\nu, |j|, |k| \\leq C |l|^{1\/2}} \n| R_{ljk} ( i_0)| \\leq \\sum_{l \\in \\mathbb Z^\\nu} \\frac{C  \\varepsilon^{2(\\nu-1)} \\gamma }{\\langle l \\rangle^{\\tau-1}} \\leq C' \\varepsilon^{2(\\nu-1)} \\gamma \\,.\n$$\nFor $ n \\geq 1$, by \\eqref{Gn Gn+1}, \n$$\n|{\\cal G}_n \\setminus {\\cal G}_{n+1} | \\leq \\sum_{|l| > N_{n-1}, |j|, |k| \\leq C |l|^{1\/2}} \n| R_{ljk} ( i_n)| \\leq \\sum_{|l| > N_{n-1}} \\frac{C  \\varepsilon^{2(\\nu-1)} \\gamma }{\\langle l \\rangle^{\\tau-1}} \\leq C' \\varepsilon^{2(\\nu-1)} \\gamma N_{n-1}^{-1} \n$$\nbecause $\\tau \\geq \\nu + 2$.\nThe estimate $ | \\Omega_\\varepsilon \\setminus {\\cal G}_0| \\leq C \\varepsilon^{2(\\nu-1)} \\gamma $ is elementary. \nThus \\eqref{Gmeasure} is proved. \n\\end{proof}\n\n\n\\noindent\n{\\bf Proof of Theorem \\ref{main theorem} concluded.}  \nTheorem \\ref{iterazione-non-lineare} implies that \nthe sequence $ ({\\mathfrak{I}}_n, \\zeta_n) $ \nis well defined for $ \\omega \\in {\\cal G}_\\infty := \\cap_{n \\geq 0} {\\cal G}_n $,  that \n$ {\\mathfrak{I}}_n $  \nis a Cauchy sequence in  $\\| \\ \\|_{s_0 + \\mu, {\\cal G}_\\infty}^{{\\mathrm{Lip}(\\g)}}$, see \n\\eqref{Hn}, and  $ |\\zeta_n |^{\\mathrm{Lip}(\\g)}  \\to 0 $. Therefore $ {\\mathfrak{I}}_n $ converges to a limit $ {\\mathfrak{I}}_\\infty $ \nin norm $\\| \\ \\|_{s_0 + \\mu, {\\cal G}_\\infty}^{{\\mathrm{Lip}(\\g)}}$ and, \nby $ ({\\cal P}2)_n $,  for all $\\omega \\in {\\cal G}_\\infty$, \n$ i_\\infty (\\varphi) :=  (\\varphi,0,0) + {\\mathfrak{I}}_\\infty(\\varphi)$,  \nis a solution of \n$$\n{\\cal F}(i_\\infty, 0 )= 0\\, \\quad \\text{with} \\quad \n \\| {\\mathfrak{I}}_\\infty \\|^{\\mathrm{Lip}(\\g)}_{s_0 + \\mu,  {\\cal G}_\\infty} \n \\leq C \\varepsilon^{6 - 2b} \\gamma^{-1} \n$$\nby \\eqref{ansatz induttivi nell'iterazione} (recall that $b_* := 6 - 2b$). \nTherefore $\\varphi \\mapsto i_\\infty(\\varphi)$ is an invariant torus for the \nHamiltonian vector field $ X_{H_\\varepsilon} $ (see \\eqref{hamiltoniana modificata}).\nBy  \\eqref{Gmeasure}, \n$$\n|\\Omega_\\varepsilon \\setminus {\\cal G}_\\infty| \n\\leq |\\Omega_\\varepsilon \\setminus {\\cal G}_0| + \\sum_{n \\geq 0} |{\\cal G}_n \\setminus {\\cal G}_{n + 1}| \n\\leq 2 C_* \\varepsilon^{2(\\nu - 1)} \\gamma + C_* \\varepsilon^{2(\\nu - 1)} \\gamma \\sum_{n \\geq 1} N_{n - 1}^{-1} \n\\leq C \\varepsilon^{2(\\nu - 1)} \\gamma \\,. \n$$\nThe set $\\Omega_\\varepsilon$ in \\eqref{Omega epsilon} has measure $|\\Omega_\\varepsilon | = O( \\varepsilon^{2 \\nu} ) $.\nHence $|\\Omega_\\varepsilon \\setminus \\mathcal{G}_\\infty| \/ |\\Omega_\\varepsilon| \\to 0$ as $\\varepsilon \\to 0$ \nbecause $ \\gamma = o(\\varepsilon^2)$, and therefore the measure of  $\\mathcal{C}_\\varepsilon := \\mathcal{G}_{\\infty}$\nsatisfies \\eqref{stima in misura main theorem}.\n\nIn order to complete the proof of Theorem \\ref{main theorem} we show the linear \nstability of the solution $ i_\\infty (\\omega t ) $. \nBy section \\ref{costruzione dell'inverso approssimato} \nthe system obtained linearizing the Hamiltonian vector field $ X_{H_\\varepsilon }  $ \nat a quasi-periodic solution $ i_\\infty (\\omega t ) $\nis conjugated to the linear Hamiltonian system\n\\begin{equation}\\label{sistema lineare dopo}\n\\begin{cases}\n\\dot \\psi & \\hspace{-6pt} = K_{20}(\\omega t) \\eta + \nK_{11}^T (\\omega t ) w  \n\\\\\n\\dot \\eta & \\hspace{-6pt} = 0 \n\\\\\n\\dot w - \\partial_x K_{0 2}(\\omega t ) w & \\hspace{-6pt} = \\partial_x  K_{11}(\\omega t) \\eta   \n\\end{cases} \n\\end{equation}\n(recall that the torus $ i_\\infty $ is isotropic and the transformed nonlinear Hamiltonian system is \\eqref{sistema dopo trasformazione inverso approssimato} where  $ K_{00}, K_{10}, K_{01} = 0 $, \nsee Remark \\ref{rem:KAM normal form}). \nIn section \\ref{operatore linearizzato sui siti normali} we have proved the reducibility of the linear system\n$ \\dot w - \\partial_x K_{0 2}(\\omega t ) w $, conjugating the last equation in \\eqref{sistema lineare dopo}\nto a diagonal system\n\\begin{equation}\\label{vjmuj}\n{\\dot v}_j +  \\mu_j^\\infty v_j = f_j (\\omega t)  \\, , \\quad j \\in S^c \\, , \\quad \\mu_j^\\infty \\in {\\mathrm i}  \\mathbb R \\, , \n\\end{equation}\nsee \\eqref{Lfinale}, and $ f (\\varphi, x) = \\sum_{j \\in S^c} f_j (\\varphi) e^{{\\mathrm i}  j x } \\in H^s_{S^\\bot}\n(\\mathbb T^{\\nu+1}) $. \nThus \\eqref{sistema lineare dopo} is stable.\nIndeed the actions $ \\eta (t) = \\eta_0 \\in \\mathbb R $, $ \\forall t \\in \\mathbb R $. \nMoreover the solutions of the  non-homogeneous equation \\eqref{vjmuj} are\n$$\nv_j (t) = c_j e^{ \\mu_j^\\infty t} + {\\tilde v}_j (t) \\,, \\quad \\text{where} \\quad \n{\\tilde v}_j (t)  := \\sum_{l \\in \\mathbb Z^\\nu} \\frac{f_{jl} \\, e^{{\\mathrm i}  \\omega \\cdot l t } }{ {\\mathrm i}  \\omega \\cdot l + \\mu_j^\\infty } \n$$ \nis a quasi-periodic solution (recall that the first Melnikov conditions \\eqref{prime di melnikov} hold at a solution).\nAs a consequence  (recall also $ \\mu_j^\\infty \\in {\\mathrm i}  \\mathbb R  $)\n the Sobolev norm of the solution of \\eqref{vjmuj} with initial condition $ v(0) = \\sum_{j \\in S^c} v_j (0) e^{{\\mathrm i}  j x } \n \\in H^{s_0} (\\mathbb T_x) $, $ s_0 < s $, does not increase in time.  \\qed\n\\\\[1mm]\n{\\bf Construction of the set $S$ of tangential sites.} \nWe finally prove that, for any $\\nu \\geq 1$, the set $S$ in \\eqref{tang sites} satisfying $({\\mathtt S}1)$-$({\\mathtt S}2)$ can be constructed inductively with only a \\emph{finite} number of restriction at any step of the induction.\n\nFirst, fix any integer $\\bar\\jmath_1 \\geq 1$. \nThen the set $J_1 := \\{ \\pm \\bar\\jmath_1 \\}$ trivially satisfies \n$({\\mathtt S}1) $-$({\\mathtt S}2)$.\nThen, assume that we have fixed $n$ distinct positive integers $\\bar\\jmath_1, \\ldots, \\bar\\jmath_n$, $n \\geq 1$, such that the set $J_n := \\{ \\pm \\bar\\jmath_1, \\ldots, \\pm \\bar\\jmath_n \\}$ satisfies $({\\mathtt S}1)$-$({\\mathtt S}2)$.\nWe describe how to choose another positive integer $\\bar\\jmath_{n+1}$, which is different from all $j \\in J_n$, such that $J_{n+1} := J_n \\cup \\{ \\pm \\bar\\jmath_{n+1} \\}$ also \nsatisfies $({\\mathtt S}1), ({\\mathtt S}2)$.\n\nLet us begin with analyzing $({\\mathtt S}1)$. \nA set of 3 elements $j_1, j_2, j_3 \\in J_{n+1}$ can be of these types: \n$(i)$ all ``old'' elements $j_1, j_2, j_3 \\in J_n$; \n$(ii)$ two ``old'' elements $j_1, j_2 \\in J_n$ and one ``new'' element $j_3 = \\sigma_3 \\bar\\jmath_{n+1}$, $\\sigma_3 = \\pm 1$;\n$(iii)$ one ``old'' element $j_1 \\in J_n$ and two ``new'' elements $j_2 = \\sigma_2 \\bar\\jmath_{n+1}$,  $j_3 = \\sigma_3 \\bar\\jmath_{n+1}$, with $\\sigma_2, \\sigma_3 = \\pm 1$;\n$(iv)$ all ``new'' elements $j_i = \\sigma_i \\bar\\jmath_{n+1}$, $\\sigma_i = \\pm 1$, $i = 1,2,3$. \n\n\nIn case $(i)$, the sum $j_1 + j_2 + j_3$ is nonzero by inductive assumption. \nIn case $(ii)$, $j_1 + j_2 + j_3$ is nonzero provided \n$\\bar\\jmath_{n+1} \\notin \\{ j_1 + j_2 : j_1, j_2 \\in J_n \\}$, which is a finite set. \nIn case $(iii)$, for $\\sigma_2 + \\sigma_3 = 0$ the sum $j_1 + j_2 + j_3 = j_1$ is trivially nonzero because $0 \\notin J_n$, while, for $\\sigma_2 + \\sigma_3 \\neq 0$, the sum $j_1 + j_2 + j_3 = j_1 + (\\sigma_2 + \\sigma_3) \\bar\\jmath_{n+1} \\neq 0$ if $\\bar\\jmath_{n+1} \\notin \\{ \\tfrac12 j : j \\in J_n \\}$,\nwhich is a finite set. \nIn case $(iv)$, the sum $j_1 + j_2 + j_3 = (\\sigma_1 + \\sigma_2 + \\sigma_3) \\bar\\jmath_{n+1} \\neq 0$ because \n$\\bar\\jmath_{n+1} \\geq 1$ and $\\sigma_1 + \\sigma_2 + \\sigma_3 \\in \\{ \\pm 1, \\pm 3\\}$. \n\n\\smallskip\n\nNow we study $({\\mathtt S}2)$ for the set $J_{n+1}$. \nDenote, in short, $b := j_1^3 + j_2^3 + j_3^3 + j_4^3 - (j_1 + j_2 + j_3 + j_4)^3$.  \n\nA set of 4 elements $j_1, j_2, j_3, j_4 \\in J_{n+1}$ can be of 5 types: \n$(i)$ all ``old'' elements $j_1, j_2, j_3, j_4 \\in J_n$; \n$(ii)$ three ``old'' elements $j_1, j_2, j_3 \\in J_n$ and one ``new'' element $j_4 = \\sigma_4 \\bar\\jmath_{n+1}$, $\\sigma_4 = \\pm 1$;\n$(iii)$ two ``old'' element $j_1, j_2 \\in J_n$ and two ``new'' elements $j_3 = \\sigma_3 \\bar\\jmath_{n+1}$,  $j_4 = \\sigma_4 \\bar\\jmath_{n+1}$, with $\\sigma_3, \\sigma_4 = \\pm 1$;\n$(iv)$ one ``old'' element $j_1 \\in J_n$ and three ``new'' elements $j_i = \\sigma_i \\bar\\jmath_{n+1}$, \n$\\sigma_i = \\pm 1$, $i=2,3,4$;\n$(v)$ all ``new'' elements $j_i = \\sigma_i \\bar\\jmath_{n+1}$, $\\sigma_i = \\pm 1$, $i = 1,2,3,4$. \n\nIn case $(i)$, $b \\neq 0$\nby inductive assumption. \n\nIn case $(ii)$, assume that $j_1 + j_2 + j_3 + j_4 \\neq 0$, \nand calculate\n\\begin{align*}\nb & = - 3 (j_1 + j_2 + j_3) \\bar\\jmath_{n+1}^2 \n- 3 (j_1 + j_2 + j_3)^2 \\sigma_4 \\bar\\jmath_{n+1} \n+ [j_1^3 + j_2^3 + j_3^3 - (j_1 + j_2 + j_3)^3] \\ \n=: p_{j_1,j_2,j_3, \\sigma_4}(\\bar\\jmath_{n+1}). \n\\end{align*}\nThis is nonzero provided $p_{j_1,j_2,j_3, \\sigma_4}(\\bar\\jmath_{n+1}) \\neq 0$ for all $j_1, j_2, j_3 \\in J_n$, $\\sigma_4 = \\pm 1$. \nThe polynomial $p_{j_1,j_2,j_3, \\sigma_4}$ is never identically zero because either the leading coefficient $-3(j_1 + j_2 + j_3) \\neq 0$ (and, if one uses $(\\mathtt{S}_3)$, this is always the case), or, if $j_1 + j_2 + j_3 = 0$, then $j_1^3 + j_2^3 + j_3^3 \\neq 0$ \nby \\eqref{prodottino} (using also that $0 \\notin J_n$). \n\n\nIn case $(iii)$, assume that $ j_1 + \\ldots + j_4  = j_1 + j_2 + (\\sigma_3 + \\sigma_4) \\bar\\jmath_{n+1} \\neq 0$, and calculate\n\\begin{align*}\nb & = - 3 \\a \\bar\\jmath_{n+1}^3 \n- 3 \\a^2 (j_1 + j_2)  \\bar\\jmath_{n+1}^2\n- 3 (j_1 + j_2)^2 \\a \\bar\\jmath_{n+1} \n- j_1 j_2 (j_1 + j_2) \n=: q_{j_1,j_2,\\a}(\\bar\\jmath_{n+1}), \n\\end{align*}\nwhere $\\a := \\sigma_3 + \\sigma_4$. \nWe impose that $q_{j_1,j_2,\\a}(\\bar\\jmath_{n+1}) \\neq 0$ for all $j_1, j_2  \\in J_n$, \n$\\a \\in\\{ \\pm 2, 0 \\}$. \nThe polynomial $q_{j_1,j_2,\\a}$ is never identically zero because either the leading coefficient $-3\\a \\neq 0$, or, for $\\a = 0$, the constant term $- j_1 j_2 (j_1 + j_2) \\neq 0$ (recall that $0 \\notin J_n$ and $j_1 + j_2 + \\a \\bar\\jmath_{n+1} \\neq 0$). \n\nIn case $(iv)$, assume that $j_1 + \\ldots + j_4 = j_1 + \\a \\bar\\jmath_{n+1} \\neq 0$, \nwhere $\\a := \\sigma_2 + \\sigma_3 + \\sigma_4 \\in \\{ \\pm 1, \\pm 3 \\}$, and calculate\n\\[\nb = \\a \\bar\\jmath_{n+1} r_{j_1,\\a}(\\bar\\jmath_{n+1}), \n\\quad\nr_{j_1, \\a}(x) := (1-\\a^2) x^2 - 3 \\a j_1 x - 3 j_1^2.\n\\]\nThe polynomial $r_{j_1, \\a}$ is never identically zero because $j_1 \\neq 0$. \nWe impose $r_{j_1, \\a}(\\bar\\jmath_{n+1}) \\neq 0$ for all $j_1 \\in J_n$, $\\a \\in \\{ \\pm 1, \\pm 3\\}$.  \n\nIn case $(v)$, assume that $j_1 + \\ldots + j_4 = \\a \\bar\\jmath_{n+1} \\neq 0$, with $\\a := \\sigma_1 + \\ldots + \\sigma_4 \\neq 0$, and calculate \n$b = \\a (1 - \\a^2) \\bar\\jmath_{n+1}^3$.\nThis is nonzero because $\\bar\\jmath_{n+1} \\geq 1$ and $\\a \\in \\{\\pm 2, \\pm 4\\}$. \n\nWe have proved that, in choosing $\\bar\\jmath_{n+1}$, there are only  finitely many integers to avoid.   \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\n\n\n\\section{Introduction}\n\\label{Sec:Introduction}\n\nAs numerical simulations of black-hole binaries improve, the criterion\nfor success moves past the ability of a code to merely persist through\nmany orbits of inspiral, merger, and ringdown.  Accuracy becomes the\ngoal, as related work in astrophysics and analysis of data from\ngravitational-wave detectors begins to rely more heavily on results\nfrom numerical relativity.  One of the most important challenges in\nthe field today is to find and eliminate systematic errors that could\npollute results built on numerics.  Though there are many possible\nsources of such error, one stands out as being particularly easy to\nmanage and---as we show---a particularly large effect: the error made\nby extracting gravitational waveforms from a simulation at finite\nradius, and treating these waveforms as though they were the\nasymptotic form of the radiation.\n\nThe desired waveform is the one to which post-Newtonian approximations\naspire, and the one sought by gravitational-wave observatories: the\nasymptotic waveform.  This is the waveform as it is at distances of\nover $10^{14}\\,M$ from the system generating the waves.  In typical\nnumerical simulations, data extraction takes place at a distance of\norder $100\\,M$ from the black holes.  At this radius, the waves are\nstill rapidly changing because of real physical effects.  Near-field\neffects~\\cite{Teukolsky1982, Boyle2008, Boyle2009} are plainly\nevident, scaling with powers of the ratio of the reduced wavelength to\nthe radius, $(\\lambdabar\/r)^k$.\\footnote{We use the standard notation\n  $\\lambdabar \\equiv \\lambda \/ 2\\pi$.} %\nExtraction methods aiming to eliminate the influence of gauge effects\nalone (\\foreign{e.g.}\\xspace, improved Regge--Wheeler--Zerilli or quasi-Kinnersley\ntechniques) will not be able to account for these physical changes.\n\nEven using a rather naive, gauge-dependent extraction method,\nnear-field effects dominate the error in extracted waves throughout\nthe inspiral for the data presented in this paper~\\cite{Boyle2008}.\nFor extraction at $r=50\\,M$, these effects can account for a\ncumulative error of roughly $50\\%$ in amplitude or a phase difference\nof more than one radian, from beginning to end of a 16-orbit\nequal-mass binary merger.  Note that near-field effects should be\nproportional to---at leading order---the ratio of $\\lambdabar\/r$ in\nphase and $(\\lambdabar\/r)^2$ in amplitude, as has been observed\npreviously \\cite{HannamEtAl2008, Boyle2008}.  Crucially, because the\nwavelength changes most rapidly during the merger, the amplitude and\nphase differences due to near-field effects also change most rapidly\nduring merger.  This means that coherence is lost between the inspiral\nand merger\/ringdown segments of the waveform.\n\nWe can see the importance of this decoherence by looking at its effect\non the matched-filtering technique frequently used to analyze data\nfrom gravitational-wave detectors.  Matched filtering~\\cite{Finn1992,\n  FinnChernoff1993, BoyleEtAl2009a} compares two signals, $s_{1}(t)$\nand $s_{2}(t)$.  It does this by Fourier transforming each into the\nfrequency domain, taking the product of the signals, weighting each\ninversely by the noise---which is a function of frequency---and\nintegrating over all frequencies.  This match is optimized over the\ntime and phase offsets of the input waveforms.  For appropriately\nnormalized waveforms, the result is a number between 0 and 1, denoted\n$\\Braket{ s_{1} | s_{2}}$, with 0 representing no match, and 1\nrepresenting a perfect match.  If we take the extrapolated waveform as\n$s_{1}$ and the waveform extracted at finite radius as $s_{2}$, we can\nevaluate the match between them.  If the extrapolated waveform\naccurately represents the ``true'' physical waveform, the mismatch\n(defined as $1-\\Braket{s_{1}|s_{2}}$) shows us the loss of signal in\ndata analysis if we were to use the finite-radius waveforms to search\nfor physical waveforms in detector data.\n\nThe waveforms have a simple scaling with the total mass of the system,\nwhich sets their frequency scale relative to the noise present in the\ndetector.  In Figs.~\\ref{fig:MismatchInitial}\nand~\\ref{fig:MismatchAdvanced}, we show mismatches between\nfinite-radius and extrapolated data from the Caltech--Cornell group\nfor a range of masses of interest to LIGO data analysis, using the\nInitial- and Advanced-LIGO noise curves, respectively, to weight the\nmatches.  The value of $R$ denotes the coordinate radius of extraction\nfor the finite-radius waveform.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig1}\n  \\caption{\\CapName{Data-analysis mismatch between finite-radius\n      waveforms and the extrapolated waveform for Initial LIGO} This\n    plot shows the mismatch between extrapolated waveforms and\n    waveforms extracted at several finite radii, scaled to various\n    values of the total mass of the binary system, using the\n    Initial-LIGO noise curve.  The waveforms are shifted in time and\n    phase to find the optimal match.  Note that the data used here is\n    solely numerical, with no direct post-Newtonian contribution.\n    Thus, for masses below $40\\,\\Sun$, this data represents only a\n    portion of the physical waveform.}\n  \\label{fig:MismatchInitial}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig2}\n  \\caption{\\CapName{Data-analysis mismatch between finite-radius\n      waveforms and the extrapolated waveform for Advanced LIGO} This\n    plot shows the mismatch between extrapolated waveforms and\n    waveforms extracted at several finite radii, scaled to various\n    values of the total mass of the binary system, using the\n    Advanced-LIGO noise curve.  The waveforms are shifted in time and\n    phase to find the optimal match.  Note that the data used here is\n    solely numerical, with no direct post-Newtonian contribution.\n    Thus, for masses below $110\\,\\Sun$, this data represents only a\n    portion of the physical waveform.}\n  \\label{fig:MismatchAdvanced}\n\\end{figure}\n\nThe data in these figures is exclusively numerical data from the\nsimulation used throughout this paper, with no direct contributions\nfrom post-Newtonian (PN) waveforms.  However, to reduce ``turn-on''\nartifacts in the Fourier transforms, we have simply attached PN\nwaveforms to the earliest parts of the time-domain waveforms,\nperformed the Fourier transform, and set to zero all data at\nfrequencies for which PN data are used.  The match integrals are\nperformed over the intersection of the frequencies present in each\nwaveform, as in Ref.~\\cite{HannamEtAl2009a}.\n\nThis means that the data used here is not truly complete for masses\nbelow $40\\,\\Sun$ in Initial LIGO and $110\\,\\Sun$ in Advanced LIGO,\nand that a detected signal would actually be dominated by data at\nlower frequencies than are present in this data for masses below about\n$10\\,\\Sun$.  These masses are correspondingly larger for shorter\nwaveforms, which begin at higher frequencies.  It is important to\nremember that this type of comparison can only show that a given\nwaveform (of a given length) is as good as it needs to be for a\ndetector.  If the waveform does not cover the sensitive band of the\ndetector, the detection signal-to-noise ratio would presumably improve\ngiven a comparably accurate waveform of greater duration.  Thus, the\nbar is raised for longer waveforms, and for lower masses.\n\nThese figures demonstrate that the mismatch can be of order $1\\%$ when\nextracting at a radius of $R=50\\,M$.  For extraction at $R=225\\,M$,\nthe mismatch is never more than about $0.2\\%$.  The loss in event rate\nwould be---assuming homogeneous distribution of events in\nspace---roughly 3 times the mismatch when using a template bank based\non imperfect waveforms~\\cite{Brown2004}.  Lindblom\n\\foreign{et~al.}\\xspace~\\cite{LindblomEtAl2008} cite a target mismatch of less than\n$0.5\\%$ between the physical waveform and a class of model templates\nto be used for \\emph{detection} of events in current LIGO detector\ndata.\\footnote{ This number of 0.5\\% results from assumptions about\n  typical event magnitude, template bank parameters, and requirements\n  on the maximum frequency of missed events.  The parameters used to\n  arrive at this number are typical for Initial LIGO.}  Thus, for\nexample, if these numerical waveforms were to be used in construction\nof template banks,\\footnote{ We emphasize that these waveforms do not\n  cover the sensitive band of current detectors, and thus would not\n  likely be used to construct template banks without the aid of\n  post-Newtonian extensions of the data.  Longer templates effectively\n  have more stringent accuracy requirements, so the suitability of\n  these extraction radii would change for waveforms of different\n  lengths.  In particular, our results are consistent with those of\n  Ref.~\\cite{HannamEtAl2009a}, which included non-extrapolated data of\n  shorter duration.} the waveform extracted at $R=50\\,M$ would not be\nentirely sufficient, in the sense that a template bank built on\nwaveforms with this level of inaccuracy would lead to an unacceptably\nhigh reduction of event rate.  The waveforms extracted at $R=100\\,M$\nand $225\\,M$, on the other hand, may be acceptable for Initial LIGO.\nFor the loudest signals expected to be seen by Advanced LIGO, the\nrequired mismatch may be roughly $10^{-4}$~\\cite{LindblomEtAl2008}.\nIn this case, even extraction at $R=225\\,M$ would be insufficient;\nsome method must be used to obtain the asymptotic waveform.  For both\nInitial and Advanced LIGO, estimating the parameters of the\nwaveform---masses and spins of the black holes, for\ninstance---requires still greater accuracy.\n\n\n\nExtrapolation of certain quantities has been used for some time in\nnumerical relativity.  Even papers announcing the first successful\nblack-hole binary evolutions \\cite{BuonannoEtAl2007,\n  CampanelliEtAl2006, BakerEtAl2006} showed radial extrapolation of\nscalar physical quantities---radiated energy and angular momentum.\nBut waveforms reported in the literature have not always been\nextrapolated.  For certain purposes, this is\nacceptable---extrapolation simply removes one of many errors.  If the\nprecision required for a given purpose allows it, extrapolation is\nunnecessary.  However, for the purposes of LIGO data analysis, we see\nthat extrapolation of the waveform may be very important.\n\nWe can identify three main obstacles to obtaining the asymptotic form\nof gravitational-wave data from numerical simulations:\n\\begin{enumerate}\n \\item Getting the ``right'' data at any given point, independent of\n  gauge effects (\\foreign{e.g.}\\xspace, using quasi-Kinnersley techniques and improved\n  Regge--Wheeler--Zerilli techniques);\n \\item Removing near-field effects;\n \\item Extracting data along a physically relevant path.\n\\end{enumerate}\nMany groups have attempted to deal with the first of these\nproblems.\\footnote{See~\\cite{NerozziEtAl2005}\n  and~\\cite{SarbachTiglio2001} and references therein for descriptions\n  of quasi-Kinnersley and RWZ methods, respectively.  Also, an\n  interesting discussion of RWZ methods, and the possibility of\n  finding the ``exact'' waveform at finite distances is found\n  in~\\cite{PazosEtAl2007}.}  While this is, no doubt, an important\nobjective, even the best extraction technique to date is imperfect at\nfinite radii.  Moreover, at finite distances from the source,\ngravitational waves continue to undergo real physical changes as they\nmove away from the system~\\cite{Thorne1980}, which are frequently\nignored in the literature.  Some extraction techniques have been\nintroduced that attempt to incorporate corrections for these physical\nnear-field effects~\\cite{AbrahamsEvans1988, AbrahamsEvans1990,\n  DeadmanStewart2009}.  However, these require assumptions about the\nform of those corrections, which we prefer not to impose.  Finally,\neven if we have the optimal data at each point in our spacetime, it is\neasy to see that extraction along an arbitrary (timelike) path through\nthat spacetime could produce a nearly arbitrary waveform, bearing no\nresemblance to a waveform that could be observed in a nearly inertial\ndetector.  In particular, if our extraction point is chosen at a\nspecific coordinate location, gauge effects could make that extraction\npoint correspond to a physical path which would not represent any real\ndetector's motion.  It is not clear how to estimate the uncertainty\nthis effect would introduce to the waveforms, except by removing the\neffect entirely.\n\nWe propose a simple method using existing data-extraction techniques\nwhich should be able to overcome each of these three obstacles, given\ncertain very basic assumptions.  The data are to be extracted at a\nseries of radii---either on a series of concentric spheres, or at\nvarious radii along an outgoing null ray.  These data can then be\nexpressed as functions of extraction radius and retarded time using\neither of two simple methods we describe.  For each value of retarded\ntime, the waveforms can then be fit to a polynomial in inverse powers\nof the extraction radius.  The asymptotic waveform is simply the first\nnonzero term in the polynomial.  Though this method also incorporates\ncertain assumptions, they amount to assuming that the data behave as\nradially propagating waves, and that the metric itself is\nasymptotically Minkowski in the coordinates chosen for the simulation.\n\nExtrapolation is, by its very nature, a dangerous procedure.  The\nfinal result may be numerically unstable, in the sense that it will\nfail to converge as the order of the extrapolating polynomial is\nincreased.  This is to be expected, as the size of the effects to be\nremoved eventually falls below the size of noise in the waveform data.\nThere are likely better methods of determining the asymptotic form of\ngravitational waves produced by numerical simulations.  For example,\ncharacteristic evolution is a promising technique that may become\ncommon in the near future~\\cite{BishopEtAl1996, Huebner2001,\n  BabiucEtAl2005, BabiucEtAl2008}.  Nonetheless, extrapolation does\nprovide a rough and ready technique which can easily be implemented by\nnumerical-relativity groups using existing frameworks.\n\nThis paper presents a simple method for implementing the extrapolation\nof gravitational-wave data from numerical simulations, and the\nmotivation for doing so.  In Sec.~\\ref{sec:TortoiseExtrapolation}, we\nbegin by introducing an extrapolation method that uses approximate\ntortoise coordinates, which is the basic method used to extrapolate\ndata in various papers~\\cite{BoyleEtAl2007b, BoyleEtAl2008a,\n  ScheelEtAl2009, BoyleEtAl2009a, BuonannoEtAl2009a} by the\nCaltech--Cornell collaboration.  The method is tested on the inspiral,\nmerger, and ringdown waveform data of the equal mass, nonspinning,\nquasicircular 15-orbit binary simulation of the Caltech--Cornell\ncollaboration.  We present the convergence of the wave phase and\namplitude as the extrapolation order increases, and we also compare\ndata extrapolated using various extraction radii.  In\nSec.~\\ref{sec:PhaseExtrapolation}, we propose a different\nextrapolation method using the wave phase---similar to the method\nintroduced in Ref.~\\cite{HannamEtAl2008}---to independently check our\nresults, again demonstrating the convergence properties of the method.\nIn Sec.~\\ref{sec:Comparison}, we compare the extrapolated waveforms of\nboth methods at various extrapolation orders, showing that they agree\nto well within the error estimates of the two methods.  A brief\ndiscussion of the pitfalls and future of extrapolation is found in\nSec.~\\ref{sec:Conclusions}.  Finally, we include a brief appendix on\ntechniques for filtering noisy data, which is particularly relevant\nhere because extrapolation amplifies noise.\n\n\n\\section{Extrapolation using approximate tortoise coordinates}\n\\label{sec:TortoiseExtrapolation} %\n\nThere are many types of data that can be extracted from a numerical\nsimulation of an isolated source of gravitational waves.  The two most\ncommon methods of extracting gravitational waveforms involve using the\nNewman--Penrose $\\Psi_{4}$ quantity, or the metric perturbation $\\ensuremath{h}$\nextracted using Regge--Wheeler--Zerilli techniques.  Even if we focus\non a particular type of waveform, the data can be extracted at a\nseries of points along the $z$ axis, for example, or decomposed into\nmultipole components and extracted on a series of spheres around the\nsource.  To simplify this introductory discussion of extrapolation, we\nignore the variety of particular types of waveform data.  Rather, we\ngeneralize to some abstract quantity $f$, which encapsulates the\nquantity to be extrapolated and behaves roughly as a radially outgoing\nwave.\n\nWe assume that $f$ travels along outgoing null cones, which we\nparametrize by a retarded time $\\ensuremath{t_{\\text{ret}}}$.  Along each of these null cones,\nwe further assume that $f$ can be expressed as a convergent (or at\nleast asymptotic) series in $1\/r$---where $r$ is some radial\ncoordinate---for all radii of interest.  That is, we assume\n\\begin{equation}\n  \\label{eq:FormOfExtrapolatedFunction}\n  f(\\ensuremath{t_{\\text{ret}}}, r) = \\sum_{k=0}^{\\infty}\\, \\frac{f_{(k)}(\\ensuremath{t_{\\text{ret}}})} {r^{k}}\\ ,\n\\end{equation}\nfor some functions $f_{(k)}$.  The asymptotic behavior of $f$ is given\nby the lowest nonzero $f_{(k)}$.\\footnote{For example, if\n  $f=r\\Psi_{4}$, then $f_{(0)}$ gives the asymptotic behavior; if\n  $f=\\Psi_{4}$, then $f_{(1)}$ gives the asymptotic behavior.}\n\nGiven data for such an $f$ at a set of retarded times, and a set of\nradii $\\{r_{i}\\}$, it is a simple matter to fit the data for each\nvalue of $\\ensuremath{t_{\\text{ret}}}$ to a polynomial in $1\/r$.  That is, for each value of\n$\\ensuremath{t_{\\text{ret}}}$, we take the set of data $\\left\\{ f(\\ensuremath{t_{\\text{ret}}}, r_{i}) \\right\\}$ and\nfit it to a finite polynomial so that\n\\begin{equation}\n  \\label{eq:FittingPolynomial}\n  f(\\ensuremath{t_{\\text{ret}}}, r_{i}) \\simeq \\sum_{k=0}^{N}\\, \\frac{f_{(k)}(\\ensuremath{t_{\\text{ret}}})}\n  {r^{k}_{i}}\\ .\n\\end{equation}\nStandard algorithms~\\cite{PressEtAl2007} can be used to accomplish\nthis fitting; here we use the least-squares method.  Of course,\nbecause we are truncating the series of\nEq.~\\eqref{eq:FormOfExtrapolatedFunction} at $k=N$, some of the\neffects from $k>N$ terms will appear at lower orders.  We will need to\nchoose $N$ appropriately, checking that the extrapolated quantity has\nconverged sufficiently with respect to this order.\n\n\\subsection{Radial parameter}\n\\label{sec:ChoiceOfR}\nOne subtlety to be considered is the choice of $r$ parameter to be\nused in the extraction and fitting.  For numerical simulation of an\nisolated system, one simple and obvious choice is the coordinate\nradius $\\ensuremath{r_{\\text{coord}}}$ used in the simulation.  Alternatively, if the data is\nmeasured on some spheroidal surface, it is possible to define an areal\nradius $\\ensuremath{r_{\\text{areal}}}$ by measuring the area of the sphere along with $f$, and\nsetting $\\ensuremath{r_{\\text{areal}}} \\equiv \\sqrt{\\text{area}\/4\\pi}$.  Still other choices are\ncertainly possible.\n\nOne objective in choosing a particular $r$ parameter is to ensure the\nphysical relevance of the final extrapolated quantity.  If we try to\ndetect the wave, for example, we may want to think of the detector as\nbeing located at some constant value of $r$.  Or, we may want $r$ to\nasymptotically represent the luminosity distance.  These conditions\nmay be checked by inspecting the asymptotic behavior of the metric\ncomponents in the given coordinates.  For example, if the metric\ncomponents in a coordinate system including $r$ asymptotically\napproach those of the standard Minkowski metric, it is not hard to see\nthat an inertial detector could follow a path of constant $r$\nparameter.\n\nSuppose we have two different parameters $r$ and $\\tilde{r}$ which can\nbe related by a series expansion\n\\begin{equation}\n  \\label{eq:rRelation}\n  r = \\tilde{r}\\, \\left[ 1 + a\/\\tilde{r} + \\ldots \\right]\\ .\n\\end{equation}\nFor the data presented in this paper, we can show that the coordinate\nradius $\\ensuremath{r_{\\text{coord}}}$ and areal radius $\\ensuremath{r_{\\text{areal}}}$ are related in this way.\nIntroducing the expansion coefficients $\\tilde{f}_{(k)}$, we can write\n\\begin{equation}\n  \\label{eq:FormOfExtrapolatedFunctionExpanded}\n  f(\\ensuremath{t_{\\text{ret}}}, r) = \\sum_{k=0}^{\\infty}\\, \\frac{f_{(k)}(\\ensuremath{t_{\\text{ret}}})} {r^{k}} =\n  \\sum_{k=0}^{\\infty}\\, \\frac{\\tilde{f}_{(k)}(\\ensuremath{t_{\\text{ret}}})} {\\tilde{r}^{k}}\\ .\n\\end{equation}\nInserting Eq.~\\eqref{eq:rRelation} into this formula, Taylor\nexpanding, and equating terms of equal order $k$, shows that $f_{(0)}\n= \\tilde{f}_{(0)}$ and $f_{(1)} = \\tilde{f}_{(1)}$.  Thus, if the\nasymptotic behavior of $f$ is given by $f_{(0)}$ or $f_{(1)}$, the\nfinal extrapolated data should not depend on whether $r$ or\n$\\tilde{r}$ is used.  On the other hand, in practice we truncate these\nseries at finite order.  This means that higher-order terms could\n``pollute'' $f_{(0)}$ or $f_{(1)}$.  The second objective in choosing\nan $r$ parameter, then, is to ensure fast convergence of the series in\nEq.~\\eqref{eq:FittingPolynomial}.  If the extrapolated quantity does\nnot converge quickly as the order of the extrapolating polynomial $N$\nis increased, it may be due to a poor choice of $r$ parameter.\n\nThe coordinate radius used in a simulation may be subject to large\ngauge variations that are physically irrelevant, and hence are not\nreflected in the wave's behavior.  That is, the wave may not fall off\nnicely in inverse powers of that coordinate radius.  For the data\ndiscussed later in this paper, we find that using the coordinate\nradius of extraction spheres is indeed a poor choice, while using the\nareal radius of those extraction spheres improves the convergence of\nthe extrapolation.\n\n\\subsection{Retarded-time parameter}\n\\label{sec:ChoiceOfRetardedTime} %\n\nSimilar considerations must be made for the choice of retarded-time\nparameter $\\ensuremath{t_{\\text{ret}}}$ to be used in extrapolation.  It may be possible to\nevolve null geodesics in numerical simulations, and use these to\ndefine the null curves on which data is to be extracted.  While this\nis an interesting possibility that deserves investigation, we propose\ntwo simpler methods here based on an approximate retarded time\nconstructed using the coordinates of the numerical simulation and the\nphase of the waves measured in that coordinate system.\n\nAgain, we have two criteria for choosing a retarded-time parameter.\nFirst is the physical suitability in the asymptotic limit.  For\nexample, we might want the asymptotic $\\ensuremath{t_{\\text{ret}}}$ to be (up to an additive\nterm constant in time) the proper time along the path of a detector\nlocated at constant $r$.  Again, checking the asymptotic behavior of\nthe metric components with respect to $\\ensuremath{t_{\\text{ret}}}$ and $r$ should be a\nsufficient test of the physical relevance of the parameters.  Second,\nwe wish to have rapid convergence of the extrapolation series using\nthe chosen parameter, which also needs to be checked.\n\nAs before, we can also show the equivalence of different choices for\nthe $\\ensuremath{t_{\\text{ret}}}$ parameter.  Suppose we have two different approximations\n$\\ensuremath{t_{\\text{ret}}}$ and $\\ensuremath{\\accentset{\\smile}{t}_{\\text{ret}}}$ that can be related by a series expansion\n\\begin{equation}\n  \\label{eq:trRelation}\n  \\ensuremath{t_{\\text{ret}}} = \\ensuremath{\\accentset{\\smile}{t}_{\\text{ret}}}\\, \\left[ 1 + b\/r + \\ldots \\right]\\ .\n\\end{equation}\nUsing the new expansion coefficients $\\accentset{\\smile}{f}_{(k)}$, we\ncan write\n\\begin{equation}\n  \\label{eq:FormOfExtrapolatedFunctionExpandedForTR}\n  f(\\ensuremath{t_{\\text{ret}}}, r) = \\sum_{k=0}^{\\infty}\\, \\frac{f_{(k)}(\\ensuremath{t_{\\text{ret}}})} {r^{k}} =\n  \\sum_{k=0}^{\\infty}\\, \\frac{\\accentset{\\smile}{f}_{(k)}(\\ensuremath{\\accentset{\\smile}{t}_{\\text{ret}}})}\n  {r^{k}}\\ .\n\\end{equation}\nNow, however, we need to assume that the functions $f_{(k)}$ can be\nwell-approximated by Taylor series.  If this is true, we can again\nshow that $f_{(0)} = \\accentset{\\smile}{f}_{(0)}$ or, \\emph{if} we\nhave $f_{(0)}=\\accentset{\\smile}{f}_{(0)}=0$, that $f_{(1)} =\n\\accentset{\\smile}{f}_{(1)}$.  The condition that $f$ be\nwell-approximated by a Taylor series is nontrivial, and can help to\ninform the choice of $f$.  Similarly, the speed of convergence of the\nextrapolation can help to inform the choice of a particular $\\ensuremath{t_{\\text{ret}}}$\nparameter.  While it has been shown \\cite{DamourEtAl2008} that a\nretarded-time parameter as simple as $\\ensuremath{t_{\\text{ret}}}=T-R$ is sufficient for some\npurposes, we find that convergence during and after merger is\ndrastically improved when using a somewhat more careful choice.\n\nSince we will be considering radiation from an isolated compact\nsource, our basic model for $\\ensuremath{t_{\\text{ret}}}$ comes from the Schwarzschild\nspacetime; we assume that the system in question approaches this\nspacetime at increasing distance.  In analogy with the\ntime-retardation effect on outgoing null rays in a Schwarzschild\nspacetime~\\cite{Chandrasekhar1992}, we define a ``tortoise\ncoordinate'' $\\ensuremath{r_{\\ast}}$ by:\n\\begin{equation}\n  \\label{eq:TortoiseCoordinate}\n  \\ensuremath{r_{\\ast}} \\equiv r + 2 \\Eadm \\ln\\left(\\frac{r}{2\\Eadm} - 1 \\right)\\\n  ,\n\\end{equation}\nwhere $\\Eadm$ is the ADM mass of the initial data.\\footnote{ Kocsis\n  and Loeb~\\cite{KocsisLoeb2007} pointed out that the propagation of a\n  roughly spherical gravitational wave should be affected primarily by\n  the amount of mass \\emph{interior to} the wave.  Because the waves\n  from a merging binary can carry off a significant fraction\n  (typically a few percent) of the binary's mass, this suggests that\n  we should allow the mass in this formula to vary in time, falling by\n  perhaps a few percent over the duration of the waveform.  However,\n  this is a small correction of a small correction; we have not found\n  it necessary.  Perhaps with more refined methods, this additional\n  correction would be relevant.} %\nIn standard Schwarzschild coordinates, the appropriate retarded time\nwould be given by $\\ensuremath{t_{\\text{ret}}} = t - \\ensuremath{r_{\\ast}}$.  It is not hard to see that the\nexterior derivative $\\d \\ensuremath{t_{\\text{ret}}}$ is null with respect to the Schwarzschild\nmetric.\n\nTaking inspiration from this, we can attempt to account for certain\ndifferences from a Schwarzschild background.  Let $T$ and $R$ denote\nthe simulation's coordinates, and suppose that we extract the metric\ncomponents $g^{TT}$, $g^{TR}$, and $g^{RR}$ from the simulation.  We\nseek a $\\ensuremath{t_{\\text{ret}}}(T,R)$ such that\n\\begin{equation}\n  \\label{eq:dTRetarded}\n  \\d\\ensuremath{t_{\\text{ret}}} = \\frac{\\partial \\ensuremath{t_{\\text{ret}}}}{\\partial T}\\, \\d T +\\frac{\\partial\n    \\ensuremath{t_{\\text{ret}}}}{\\partial R}\\, \\d R\n\\end{equation}\nis null with respect to these metric components.  That is, we seek a\n$\\ensuremath{t_{\\text{ret}}}$ such that\n\\begin{multline}\n  \\label{eq:NullCondition}\n  g^{TT}\\, \\left( \\frac{\\partial \\ensuremath{t_{\\text{ret}}}}{\\partial T} \\right)^{2} + 2\n  g^{TR}\\, \\left( \\frac{\\partial \\ensuremath{t_{\\text{ret}}}}{\\partial T} \\right)\\, \\left(\n    \\frac{\\partial \\ensuremath{t_{\\text{ret}}}}{\\partial R} \\right) \\\\ + g^{RR}\\, \\left(\n    \\frac{\\partial \\ensuremath{t_{\\text{ret}}}}{\\partial R} \\right)^{2} = 0\\ .\n\\end{multline}\nWe introduce the ansatz $\\ensuremath{t_{\\text{ret}}} = t - \\ensuremath{r_{\\ast}}$, where $t$ is assumed to be a\nslowly varying function of $R$,\\footnote{More specifically, we need\n  $\\lvert \\partial t\/\\partial R \\rvert \\ll \\lvert \\partial \\ensuremath{r_{\\ast}}\n  \/ \\partial R \\rvert$.  This condition needs to be checked for all\n  radii used, at all times in the simulation.  For the data presented\n  below, we have checked this, and shown it to be a valid assumption,\n  at the radii used for extrapolation.} %\nand $\\ensuremath{r_{\\ast}}$ is given by Eq.~\\eqref{eq:TortoiseCoordinate} with $R$ in\nplace of $r$ on the right side.  If we ignore $\\partial t \/ \\partial\nR$ and insert our ansatz into Eq.~\\eqref{eq:NullCondition}, we have\n\\begin{multline}\n  \\label{eq:NullConditionB}\n  g^{TT}\\, \\left( \\frac{\\partial t}{\\partial T} \\right)^{2} - 2\n  g^{TR}\\, \\left( \\frac{\\partial t}{\\partial T} \\right)\\, \\left(\n    \\frac{1}{1-2\\Eadm\/R} \\right) \\\\ + g^{RR}\\, \\left(\n    \\frac{1}{1-2\\Eadm\/R} \\right)^{2} = 0\\ .\n\\end{multline}\nWe can solve this for $\\partial t \/ \\partial T$:\n\\begin{equation}\n  \\label{eq:RetardedTimeSolutionA}\n  \\frac{\\partial t} {\\partial T} = \\frac{1}{1-2\\Eadm\/R}\\,\n  \\frac{g^{TR} \\pm \\sqrt{(g^{TR})^{2} - \\, g^{TT}\\, g^{RR}}} {g^{TT}}\n  \\ .\n\\end{equation}\nSubstituting the Schwarzschild metric components shows that we should\nchoose the negative sign in the numerator of the second factor.\nFinally, we can integrate (numerically) to find\n\\begin{equation}\n  \\label{eq:FullRetardedTimeSolution}\n  t = \\int_{0}^{T}\\, \\frac{1}{g^{TT}}\\, \\frac{g^{TR} -\n    \\sqrt{(g^{TR})^{2} - g^{TT}\\, g^{RR}}} {1-2\\Eadm\/R}\\, \\d T'\\ .\n\\end{equation}\nNow, in the case where $g^{TR}$ is small compared to 1, we may wish to\nignore it, in which case we have\n\\begin{equation}\n  \\label{eq:RetardedTimeSolutionB}\n  t = \\int_{0}^{T}\\, \\frac{\\sqrt{-g^{RR} \/ g^{TT}}} {1-2\\Eadm\/R}\\,\n  \\d T'\\ .\n\\end{equation}\nIt is not hard to see that this correctly reduces to $t=T$ in the\nSchwarzschild case.\n\nFor the data discussed later in this paper, we make further\nassumptions that $g^{RR} = 1-2\\Eadm\/R$, and that $R=\\ensuremath{r_{\\text{areal}}}$.  That is,\nwe define the corrected time\n\\begin{subequations}\n  \\label{eq:DynamicLapseCorrection}\n  \\begin{gather}\n    \\ensuremath{t_{\\text{corr}}} \\equiv \\int_{0}^{T}\\, \\sqrt{\\frac{-1\/g^{TT}}{1 -\n        2\\Eadm\/\\ensuremath{r_{\\text{areal}}}}} \\, \\d T' \\intertext{and the retarded time} \\ensuremath{t_{\\text{ret}}}\n    \\equiv \\ensuremath{t_{\\text{corr}}} - \\ensuremath{r_{\\ast}}\\ .\n  \\end{gather}\n\\end{subequations}\nWe find that this corrected time leads to a significant improvement\nover the naive choice of $t(T)=T$, while no improvement results from\nusing Eq.~\\eqref{eq:FullRetardedTimeSolution}.\n\n\\subsection{Application to a binary inspiral}\n\\label{sec:Application}\nTo begin the extrapolation procedure, we extract the (spin-weight\n$s=-2$) $(l,m)=(2,2)$ component of $\\Psi_{4}$ data on a set of spheres\nat constant coordinate radius in the simulation.\\footnote{See\n  Ref.~\\cite{ScheelEtAl2009} for details of the extraction procedure.\n  We use $\\Psi_{4}$ data here, rather than Regge--Wheeler--Zerilli\n  data because the $\\Psi_{4}$ data from this simulation is of higher\n  quality; it appears that the RWZ data is more sensitive to changes\n  in gauge conditions after the merger.  This problem is still under\n  investigation.} %\nIn the black-hole binary simulations used here (the same as those\ndiscussed in Refs.~\\cite{BoyleEtAl2007b, Boyle2008, BoyleEtAl2008a,\n  ScheelEtAl2009}), these spheres are located roughly\\footnote{\n  Explicitly, the extraction spheres are at radii $\\ensuremath{r_{\\text{coord}}}\/\\ensuremath{M_{\\text{irr}}} = \\{75,\n  85, 100, 110, 120, \\ldots, 190, 200, 210, 225\\}$, though we find\n  that the final result is not sensitive to the exact placement of the\n  extraction spheres.} %\nevery $\\Delta\\ensuremath{r_{\\text{coord}}} \\approx 10\\Mirr$ (where $\\Mirr$ is the sum of\nthe irreducible masses of the black holes in the initial data) from an\ninner radius of $\\ensuremath{r_{\\text{coord}}}=75\\Mirr$ to an outer radius of\n$\\ensuremath{r_{\\text{coord}}}=225\\Mirr$, where $\\Mirr$ denotes the total apparent-horizon\nmass of the two holes at the beginning of the simulation.  This\nextraction occurs at time steps of $\\Delta \\ensuremath{t_{\\text{coord}}} \\approx 0.5\\Mirr$\nthroughout the simulation.  We also measure the areal radius, $\\ensuremath{r_{\\text{areal}}}$,\nof these spheres by integrating the induced area element over the\nsphere to find the area, and defining $\\ensuremath{r_{\\text{areal}}} \\equiv\n\\sqrt{\\text{area}\/4\\pi}$.  This typically differs from the coordinate\nradius $\\ensuremath{r_{\\text{coord}}}$ by roughly $\\Mirr\/\\ensuremath{r_{\\text{coord}}}$.  Because of gauge effects, the\nareal radius of a coordinate sphere changes as a function of time, so\nwe measure this as a function of time.  Finally, we measure the\naverage value of $g^{TT}$ as a function of coordinate time on the\nextraction spheres to correct for the dynamic lapse function.  The\nareal radius and $g^{TT}$ are then used to compute the retarded time\n$\\ensuremath{t_{\\text{ret}}}$ defined in Eq.~\\eqref{eq:DynamicLapseCorrection}.\n\nThe gravitational-wave data $\\Psi_{4}$, the areal radius $\\ensuremath{r_{\\text{areal}}}$, and\nthe lapse $N$ are all measured as functions of the code coordinates\n$\\ensuremath{t_{\\text{coord}}}$ and $\\ensuremath{r_{\\text{coord}}}$.  We can use these to construct the retarded time\ndefined in Eq.~\\eqref{eq:DynamicLapseCorrection}, using $\\ensuremath{r_{\\text{areal}}}$ in place\nof $r$.  This, then, will also be a function of the code coordinates.\nThe mapping between $(\\ensuremath{t_{\\text{ret}}},\\ensuremath{r_{\\text{areal}}})$ and $(\\ensuremath{t_{\\text{coord}}},\\ensuremath{r_{\\text{coord}}})$ is invertible, so we\ncan rewrite $\\Psi_{4}$ as a function of $\\ensuremath{t_{\\text{ret}}}$ and $\\ensuremath{r_{\\text{areal}}}$.\n\nAs noted in Sec.~\\ref{sec:ChoiceOfRetardedTime}, we need to assume\nthat the extrapolated functions are well approximated by Taylor\nseries.  Because the real and imaginary parts of $\\Psi_{4}$ are\nrapidly oscillating in the data presented here, we prefer to use the\nsame data in smoother form.  We define the complex amplitude $A$ and\nphase $\\phi$ of the wave:\n\\begin{equation}\n  \\label{eq:AmplitudeAndPhaseDefinition}\n \n  \\ensuremath{r_{\\text{areal}}}\\, \\Mirr\\, \\Psi_4 \\equiv A\\, \\ensuremath{\\mathrm{e}}^{\\i \\phi}\\ ,\n\\end{equation}\nwhere $A$ and $\\phi$ are functions of $\\ensuremath{t_{\\text{ret}}}$ and $\\ensuremath{r_{\\text{areal}}}$.  Note that this\ndefinition factors out the dominant $1\/r$ behavior of the amplitude.\nThis equation defines the phase with an ambiguity of multiples of\n$2\\pi$.  In practice, we ensure that the phase is continuous as a\nfunction of time by adding suitable multiples of $2\\pi$.  The\ncontinuous phase is easier to work with for practical reasons, and is\ncertainly much better approximated by a Taylor series, as required by\nthe argument surrounding\nEq.~\\eqref{eq:FormOfExtrapolatedFunctionExpandedForTR}.\n\nA slight complication arises in the relative phase offset between\nsuccessive radii.  Noise in the early parts of the waveform makes the\noverall phase offset go through multiples of $2\\pi$ essentially\nrandomly.  We choose some fairly noise-free (retarded) time and ensure\nthat phases corresponding to successive extraction spheres are matched\nat that time, by simply adding multiples of $2\\pi$ to the phase of the\nentire waveform---that is, we add a multiple of $2\\pi$ to the phase at\nall times.\n\nExtrapolation of the waveform, then, basically consists of finding the\nasymptotic forms of these functions, $A$ and $\\phi$ as functions of\ntime.  We apply the general technique discussed above to $A$ and\n$\\phi$.  Explicitly, we fit the data to polynomials in $1\/\\ensuremath{r_{\\text{areal}}}$ for\neach value of retarded time:\n\\begin{subequations}\n  \\label{eq:ExtrapolationFormula}\n  \\begin{align}\n    \\label{eq:AxmplitudeExtrapolation}\n    A(\\ensuremath{t_{\\text{ret}}},\\ensuremath{r_{\\text{areal}}}) &\\simeq \\sum_{k=0}^N\\,\\frac{A_{(k)}(\\ensuremath{t_{\\text{ret}}})}{\\ensuremath{r_{\\text{areal}}}^k}\\ , \\\\\n    \\label{eq:PhaseExtrapolation}\n    \\phi(\\ensuremath{t_{\\text{ret}}},\\ensuremath{r_{\\text{areal}}}) &\\simeq \\sum_{k=0}^N\\,\\frac{\\phi_{(k)}(\\ensuremath{t_{\\text{ret}}})}{\\ensuremath{r_{\\text{areal}}}^k}\\\n    .\n  \\end{align}\n\\end{subequations}\nThe asymptotic waveform is fully described by $A_{(0)}$ and\n$\\phi_{(0)}$.  When the order of the approximating polynomials is\nimportant, we will denote by $A_{N}$ and $\\phi_{N}$ the asymptotic\nwaveforms resulting from approximations using polynomials of order\n$N$.\n\nWe show the results of these extrapolations in the figures below.\nFigs.~\\ref{fig:LapseCorrectionComparison_Corr_Amp}\nthrough~\\ref{fig:LapseCorrectionComparison_NoCorrMatched} show\nconvergence plots for extrapolations using orders $N=1$--$5$.  The\nfirst two figures show the relative amplitude and phase difference\nbetween successive orders of extrapolation, using the corrected time\nof Eq.~\\eqref{eq:DynamicLapseCorrection}.  Here, we define\n\\begin{subequations}\n  \\begin{gather}\n    \\label{eq:RelativeAmplitudeDifferenceDefinition}\n    \\frac{\\delta A}{A} \\equiv \\frac{A_{N_{a}} - A_{N_{b}}} {A_{N_{b}}}\n    \\\\ \\intertext{and}\n    \\label{eq:PhaseDifferenceDefinition}\n    \\delta \\phi \\equiv \\phi_{N_{a}} - \\phi_{N_{b}}\\ .\n  \\end{gather}\n\\end{subequations}\nWhen comparing waveforms extrapolated by polynomials of different\norders, we use $N_{b}=N_{a}+1$.  Note that the broad trend is toward\nconvergence, though high-frequency noise is more evident as the order\nincreases, as we discuss further in the next subsection.  The peak\namplitude of the waves occurs at time $\\ensuremath{t_{\\text{ret}}}\/\\ensuremath{M_{\\text{irr}}} \\approx 3954$.  Note\nthat the scale of the horizontal axis changes just before this time to\nbetter show the merger\/ringdown portion.  We see that the\nextrapolation is no longer convergent, with differences increasing\nslightly as the order of the extrapolating polynomial is increased.\nThe oscillations we see in these convergence plots have a frequency\nequal to the frequency of the waves themselves.  Their origin is not\nclear, but may be due to numerics, gauge, or other effects that\nviolate our assumptions about the outgoing-wave nature of the data.\nIt is also possible that there are simply no higher-order effects to\nbe extrapolated, so low-order extrapolation suffices.\n\nFigure~\\ref{fig:LapseCorrectionComparison_NoCorrMatched} shows the\nsame data as in Fig.~\\ref{fig:LapseCorrectionComparison_Corr}, except\nthat no correction is used for dynamic lapse.  That is, for this\nfigure (and only this figure), we use $\\ensuremath{t_{\\text{ret}}} \\equiv T - \\ensuremath{r_{\\ast}}$, where $T$\nis simply the coordinate time.  This demonstrates the need for\nimproved time-retardation methods after merger.  Note that the\nextrapolated data during the long inspiral is virtually unchanged\n(note the different vertical axes).  After the merger---occurring at\nroughly $\\ensuremath{t_{\\text{ret}}}\/\\ensuremath{M_{\\text{irr}}} = 3954$---there is no convergence when no\ncorrection is made for dynamic lapse.  It is precisely the merger and\nringdown segment during which extreme gauge changes are present in the\ndata used here~\\cite{ScheelEtAl2009}.  On the other hand, the fair\nconvergence of the corrected waveforms indicates that it is possible\nto successfully remove these gauge effects.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig3}\n  \\caption{\\CapName{Convergence of the amplitude of the extrapolated\n      $\\Psi_{4}$, with increasing order of the extrapolating\n      polynomial, $N$} This figure shows the convergence of the\n    relative amplitude of the extrapolated Newman--Penrose waveform,\n    as the order $N$ of the extrapolating polynomial is increased.\n    (See Eq.~\\eqref{eq:ExtrapolationFormula}.)  That is, we subtract\n    the amplitudes of the two waveforms, and normalize at each time by\n    the amplitude of the second waveform.  We see that increasing the\n    order tends to amplify the apparent noise during the early and\n    late parts of the waveform.  Nonetheless, the broad\n    (low-frequency) trend is towards convergence.  Note that the\n    differences decrease as the system nears merger; this is a first\n    indication that the extrapolated effects are due to near-field\n    influences.  Also note that the horizontal axis changes in the\n    right part of the figure, which shows the point of merger, and the\n    ringdown portion of the waveform.  After the merger, the\n    extrapolation is nonconvergent, though the differences grow slowly\n    with the order of extrapolation.}\n  \\label{fig:LapseCorrectionComparison_Corr_Amp}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig4}\n  \\caption{\\CapName{Convergence of the phase of the extrapolated\n      $\\Psi_{4}$, with increasing order of the extrapolating\n      polynomial, $N$} This figure is the same as\n    Fig.~\\ref{fig:LapseCorrectionComparison_Corr_Amp}, except that it\n    shows the convergence of phase.  Again, increasing the\n    extrapolation order tends to amplify the noise during the early\n    and late parts of the waveform, though the broad (low-frequency)\n    trend is towards convergence.  The horizontal-axis scale changes\n    just before merger.}\n  \\label{fig:LapseCorrectionComparison_Corr}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig5}\n  \\caption{\\CapName{Convergence of the phase of $\\Psi_{4}$,\n      extrapolated with no correction for the dynamic lapse} This\n    figure is the same as\n    Fig.~\\ref{fig:LapseCorrectionComparison_Corr}, except that no\n    correction is made to account for the dynamic lapse.  (See\n    Eq.~\\eqref{eq:DynamicLapseCorrection} and surrounding discussion.)\n    Observe that the convergence is very poor after merger (at roughly\n    $\\ensuremath{t_{\\text{ret}}}\/\\ensuremath{M_{\\text{irr}}} = 3954$).  This corresponds to the time after which\n    sharp features in the lapse are observed.  We conclude from this\n    graph and comparison with the previous graph that the correction\n    is crucial to convergence of $\\Psi_{4}$ extrapolation through\n    merger and ringdown.}\n  \\label{fig:LapseCorrectionComparison_NoCorrMatched}\n\\end{figure}\n\n\n\\subsection{Choosing the order of extrapolation}\n\\label{sec:ExtrapolationOrder} %\nDeciding on an appropriate order of extrapolation to be used for a\ngiven purpose requires balancing competing effects.  As we see in\nFig.~\\ref{fig:LapseCorrectionComparison_Corr_Amp}, for example, there\nis evidently some benefit to be gained from using higher-order\nextrapolation during the inspiral; there is clearly some convergence\nduring inspiral for each of the orders shown.  On the other hand,\nhigher-order methods amplify the apparent noise in the\nwaveform.\\footnote{So-called ``junk radiation'' is a ubiquitous\n  feature of initial data for current numerical simulations of binary\n  black-hole systems.  It is clearly evident in simulations as\n  large-amplitude, high-frequency waves that die out as the simulation\n  progresses.  While it is astrophysically extraneous, it is\n  nevertheless a correct result of evolution from the initial data.\n  Better initial data would, presumably, decrease its magnitude.  This\n  is the source of what looks like noise in the waveforms at early\n  times.  It is less apparent in $\\ensuremath{h}$ data than in $\\Psi_{4}$ data\n  because $\\Psi_{4}$ effectively amplifies high-frequency components,\n  because of the relation $\\Psi_{4} \\approx -\\ddot{h}$.} %\nMoreover, late in the inspiral, and on into the merger and ringdown,\nthe effects being extrapolated may be present only at low orders;\nincreasing the extrapolation order would be useless as higher-order\nterms would simply be fitting to noise.\n\nThe optimal order depends on the accuracy needed, and on the size of\neffects that need to be eliminated from the data.  For some\napplications, little accuracy is needed, so a low-order extrapolation\n(or even no extrapolation) is preferable.\\footnote{ We note that---as\n  expected from investigations of near-field effects\n  \\cite{Teukolsky1982, Boyle2008, Boyle2009}---the second-order\n  behavior of the amplitude greatly dominates its first-order behavior\n  \\cite{HannamEtAl2008}.  Thus, there is no improvement to the\n  accuracy \\emph{of the amplitude} when extrapolating with $N=1$; it\n  would be better to simply use the data from the largest extraction\n  radius.}  If high-frequency noise is not considered a problem, then\nsimple high-order extrapolation should suffice.  Of course, if both\nhigh accuracy and low noise are required, data may easily be filtered,\nmitigating the problem of noise amplification.  (See the appendix for\nmore discussion.)  There is some concern that this may introduce\nsubtle inaccuracies: filtering is more art than science, and it is\ndifficult to establish precise error bars for filtered data.\n\n\n\\subsection{Choosing extraction radii}\n\\label{sec:ExtractionRadii} %\n\nAnother decision needs to be made regarding the number and location of\nextraction surfaces.  Choosing the number of surfaces is fairly easy,\nbecause there is typically little cost in increasing the number of\nextraction radii (especially relative to the cost of---say---running a\nsimulation).  The only restriction is that the number of data points\nneeds to be significantly larger than the order of the extrapolating\npolynomial; more can hardly hurt.  More careful consideration needs to\nbe given to the \\emph{location} of the extraction surfaces.\n\nFor the extrapolations shown in\nFigs.~\\ref{fig:LapseCorrectionComparison_Corr_Amp}\nand~\\ref{fig:LapseCorrectionComparison_Corr}, data was extracted on\nspheres spaced by $10$ to $15\\Mirr$, from $R=75\\Mirr$ to\n$R=225\\Mirr$.  The outer radius of $225\\Mirr$ was chosen simply\nbecause this is the largest radius at which data exists throughout the\nsimulation; presumably, we always want the outermost radii at which\nthe data are resolved.  In choosing the inner radius, there are two\ncompeting considerations.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig6}\n  \\caption{\\CapName{Comparison of extrapolation of $\\Psi_{4}$ using\n      different sets of extraction radii} This figure compares the\n    phase of waveforms extrapolated with various sets of radii.  All\n    comparisons are with respect to the data set used elsewhere in\n    this paper, which uses extraction radii $R\/\\Mirr = \\{75, 85,\n    100, 110, 120, \\ldots, 200, 210, 225\\}$.  The order of the\n    extrapolating polynomial is $N=3$ in all cases.}\n  \\label{fig:RadiiComparison}\n\\end{figure}\n\nOn one hand, we want the largest spread possible between the inner and\nouter extraction radii to stabilize the extrapolation.  A very rough\nrule of thumb seems to be that the distance to be extrapolated should\nbe no greater than the distance covered by the data.  Because the\nextrapolating polynomial is a function of $1\/R$, the distance to be\nextrapolated is $1\/R_{\\text{outer}} - 1\/\\infty = 1\/R_{\\text{outer}}$.\nThe distance covered by the data is $1\/R_{\\text{inner}} -\n1\/R_{\\text{outer}}$, so if the rule of thumb is to be satisfied, the\ninner extraction radius should be no more than half of the outer\nextraction radius, $R_{\\text{inner}} \\lesssim R_{\\text{outer}}\/2$\n(noting, of course, that this is a \\emph{very} rough rule of thumb).\n\nOn the other hand, we would like the inner extraction radius to be as\nfar out as possible.  Extracting data near the violent center of the\nsimulation is a bad idea for many reasons.  Coordinate ambiguity,\ntetrad errors, near-field effects---all are more severe near the\ncenter of the simulation.  The larger these errors are, the more work\nthe extrapolation needs to do.  This effectively means that\nhigher-order extrapolation is needed if data are extracted at small\nradii.  The exact inner radius needed for extrapolation depends on the\ndesired accuracy and, again, the portion of the simulation from which\nthe waveform is needed.\n\nWe can compare data extrapolated using different sets of radii.\nFigure~\\ref{fig:RadiiComparison} shows a variety, compared to the data\nused elsewhere in this paper.  The extrapolation order is $N=3$ in all\ncases.  Note that the waveforms labeled $R\/\\Mirr = \\{50, \\ldots,\n100\\}$ and $R\/\\Mirr = \\{100, \\ldots, 225\\}$ both satisfy the rule\nof thumb that the inner radius should be at most half of the outer\nradius, while the other two waveforms do not; it appears that\nviolation of the rule of thumb leads to greater sensitivity to noise.\nOne waveform is extrapolated using only data from small radii,\n$R\/\\Mirr = \\{50, \\ldots, 100\\}$.  It is clearly not converged, and\nwould require higher-order extrapolation if greater accuracy is\nneeded.  The source of the difference is presumably the near-field\neffect~\\cite{Boyle2008}, which is proportionally larger at small\nradii.\n\nClearly, there is a nontrivial interplay between the radii used for\nextraction and the order of extrapolation.  Indeed, because of the\ntime-dependence of the various elements of these choices, it may be\nadvisable to use different radii and orders of extrapolation for\ndifferent time portions of the waveform.  The different portions could\nthen be joined together using any of various\nmethods~\\cite{AjithEtAl2008, BoyleEtAl2009a}.\n\n\n\\section{Extrapolation using the phase of the waveform}\n\\label{sec:PhaseExtrapolation}\n\nWhile the tortoise-coordinate method just described attempts to\ncompensate for nontrivial gauge perturbations, it is possible that it\ndoes not take account of all effects adequately. As an independent\ncheck, we discuss what is essentially a second---very\ndifferent---formulation of the retarded-time parameter, similar to one\nfirst introduced in Ref.~\\cite{HannamEtAl2008}. If waves extrapolated\nwith the two different methods agree, then we can be reasonably\nconfident that unmodeled gauge effects are not diminishing the\naccuracy of the final result.  As we will explain below, the method in\nthis section cannot be used naively with general data (\\foreign{e.g.}\\xspace, data on\nthe equatorial plane).  In particular, we must assume that the data to\nbe extrapolated consists of a strictly monotonic phase.  It is,\nhowever, frequently possible to employ a simple technique to make\npurely real, oscillating data into complex data with strictly\nmonotonic phase, as we describe below.  The results of this technique\nagree with those of the tortoise-coordinate extrapolation as we show\nin Sec.~\\ref{sec:Comparison}.\n\nInstead of extrapolating the wave phase $\\phi$ and amplitude $A$ as\nfunctions of time and radius, we extrapolate the time $\\ensuremath{t_{\\text{ret}}}$ and the\namplitude $A$ as functions of wave phase $\\phi$ and radius $\\ensuremath{r_{\\text{areal}}}$.  In\nother words, we measure the amplitude and the arrival time to some\nradius $\\ensuremath{r_{\\text{areal}}}$ of a fixed phase point in the waveform.  This is the\norigin of the requirement that the data to be extrapolated consist of\na strictly monotonic phase $\\phi(\\ensuremath{t_{\\text{ret}}}, \\ensuremath{r_{\\text{areal}}})$ (\\foreign{i.e.}\\xspace, it must be\ninvertible).  For the data presented here, the presence of radiation\nin the initial data---junk radiation---and numerical noise cause the\nextracted waveforms to fail to satisfy this requirement at early\ntimes.  In this case, the extrapolation is performed separately for\neach invertible portion of the data.  That is, the data are divided\ninto invertible segments, each segment is extrapolated separately, and\nthe final products are joined together as a single waveform.\n\n\n\\subsection{Description of the method}\n\\label{sec:PhaseDescription} %\nThis extrapolation technique consists of extrapolating the retarded\ntime and the amplitude as functions of the wave phase $\\phi$ and the\nradius $\\ensuremath{r_{\\text{areal}}}$.  In other words, when extrapolating the waveform, we are\nestimating the amplitude and the arrival time of a fixed phase point\nat infinity.  Here, we extract the same $\\Psi_{4}$, $g^{TT}$, and\nareal-radius data used in the previous section.  As in the previous\nmethod, we first shift each waveform in time using $\\ensuremath{t_{\\text{ret}}} = \\ensuremath{t_{\\text{corr}}} -\n\\ensuremath{r_{\\ast}}$, where $\\ensuremath{t_{\\text{corr}}}$ is defined in\nEq.~\\eqref{eq:DynamicLapseCorrection} and the basic tortoise\ncoordinate $\\ensuremath{r_{\\ast}}$ is defined in Eq.~\\eqref{eq:TortoiseCoordinate} with\nareal radius as the radial parameter.  The amplitude and wave phase\nare again defined using Eq.~\\eqref{eq:AmplitudeAndPhaseDefinition},\nand the phase is made continuous as in Sec.~\\ref{sec:Application}.\nThus, we begin with the same data, shifted as with the\ntortoise-coordinate method.\n\nNow, however, we change the method, in an attempt to allow for\nunmodeled effects.  Instead of extrapolating $\\phi(\\ensuremath{t_{\\text{ret}}}, \\ensuremath{r_{\\text{areal}}})$ and\n$A(\\ensuremath{t_{\\text{ret}}}, \\ensuremath{r_{\\text{areal}}})$, as with the previous method, we invert these functions\nto get $\\ensuremath{t_{\\text{ret}}}(\\phi,\\ensuremath{r_{\\text{areal}}})$ and $A(\\phi,\\ensuremath{r_{\\text{areal}}})$ as functions of the wave\nphase $\\phi$.  In other words, we extrapolate the arrival time and the\namplitude of a signal to a coordinate radius $R$ for each wave phase\nvalue.  This is done by fitting the retarded time $\\ensuremath{t_{\\text{ret}}}$ and the\namplitude $A$ data to polynomials in $1\/\\ensuremath{r_{\\text{areal}}}$ for each value of the\nwave phase:\n\\begin{subequations}\n  \\label{eq:ExtrapolationFormulaMethod2}\n  \\begin{align}\n    \\label{eq:AmplitudeExtrapolationMethod2}\n    A(\\ensuremath{r_{\\text{areal}}}, \\phi) &\\simeq \\sum^N_{k=0}\\frac{A_{(k)}(\\phi)}{\\ensuremath{r_{\\text{areal}}}^{k}}\n    \\ , \\\\\n    \\label{eq:PhaseExtrapolationMethod2}\n    t(\\ensuremath{r_{\\text{areal}}}, \\phi) &\\simeq \\ensuremath{r_{\\ast}} + \\sum^N_{k=0}\\frac{t_{(k)}(\\phi)}{\\ensuremath{r_{\\text{areal}}}^k}\n    \\ ,\n  \\end{align}\n\\end{subequations}\nwhere the asymptotic waveform is fully described by $A_{(0)}(\\phi)$\nand $t_{(0)}(\\phi)$.\n\nWith this data in hand, we can produce the asymptotic amplitude and\nphase as functions of time by plotting curves in the \\mbox{$t$--$A$}\nand \\mbox{$t$--$\\phi$} planes parametrized by the phase.  In order to\nbe true, single-valued functions, we again need monotonicity of the\n$t_{(0)}(\\phi)$ data, which may be violated by extrapolation.  The\nusable data can be obtained simply by removing data from times before\nwhich this condition holds.\n\nChoosing the extraction radii and extrapolation order for this method\nfollows the same rough recommendations described in\nSecs.~\\ref{sec:ExtrapolationOrder} and~\\ref{sec:ExtractionRadii}.\nNote also that the restriction that the data have an invertible phase\nas a function of time is not insurmountable.  For example, data for\n$\\Psi_{4}$ in the equatorial plane is purely real, hence has a phase\nthat simply jumps from $0$ to $\\pi$ discontinuously.  However, we can\ndefine a new quantity\n\\begin{equation}\n  w(t) \\equiv \\Psi_{4}(t) + \\i \\dot{\\Psi}_{4}(t)\\ .\n\\end{equation}\nThis is simply an auxiliary quantity used for the extrapolation, with\na smoothly varying, invertible phase.  The imaginary part is discarded\nafter extrapolation.\n\n\n\\subsection{Results}\n\\label{sec:PhaseResults} %\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig7}\n  \\caption{\\CapName{Convergence of the amplitude of $\\Psi_{4}$\n      extrapolated using the wave phase, with increasing order $N$ of\n      the extrapolating polynomial} This figure shows the convergence\n    of the relative amplitude of the extrapolated Newman--Penrose\n    waveform extrapolated using the wave phase, as the order $N$ of\n    the extrapolating polynomial is increased.  (See\n    Eq.~\\eqref{eq:ExtrapolationFormulaMethod2}.)  Increasing the\n    extrapolation order tends to amplify the apparent noise during the\n    early and late parts of the waveform, but it improves convergence.\n    The vertical axis at $\\ensuremath{t_{\\text{ret}}}\/\\ensuremath{M_{\\text{irr}}}\\approx 3950$ denotes merger.}\n  \\label{fig:AmplitudeComparisonMethod2}\n\\end{figure}\n\nIn Figs.~\\ref{fig:AmplitudeComparisonMethod2}\nand~\\ref{fig:PhaseComparisonMethod2} we plot the convergence of the\nrelative amplitude and phase of the extrapolated $(l,m) = (2,2)$ mode\nof the $\\Psi_4$ waveform for extrapolation orders $N=1, \\ldots, 5$.  A\ncommon feature of both plots is that during the inspiral, the higher\nthe extrapolation order, the better the convergence.  However, the\nnoise is amplified significantly for large orders of extrapolation.\nThis method of extrapolation amplifies high-frequency noise\nsignificantly, compared to the tortoise-coordinate method.\n\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig8}\n  \\caption{\\CapName{Convergence of the phase of $\\Psi_{4}$ as a\n      function of time extrapolated using the wave phase, with\n      increasing order $N$ of the extrapolating polynomial} Again,\n    increasing the extrapolation order tends to amplify the apparent\n    noise during the early and late parts of the waveform, though\n    convergence is improved significantly.}\n  \\label{fig:PhaseComparisonMethod2}\n\\end{figure}\n\nIn the inspiral portion, we have a decreasing error in the\nextrapolation of the phase and the amplitude as the wavelength of the\ngravitational waves decreases.  In the merger\/ringdown portion, a more\ncareful choice of the radii and order of extrapolation needs to be\nmade.  Since near-field effects are less significant in the data\nextracted at larger radii, extrapolation at low order ($N=2,3$) seems\nsufficient.  Data extrapolated at large order ($N=4,5$) has a larger\nerror in the phase and amplitude after merger than data extrapolated\nat order $N=2$ or $3$.  Moreover, the outermost extraction radius\ncould be reduced, say to $R_{\\text{outer}} \/ \\ensuremath{M_{\\text{irr}}} = 165$ instead of\n$R_{\\text{outer}} \/ \\ensuremath{M_{\\text{irr}}} = 225$, without having large extrapolation\nerror at late times.  Using the radius range $R\/\\ensuremath{M_{\\text{irr}}}={75,\\ldots,160}$\ninstead of the range $R\/ \\ensuremath{M_{\\text{irr}}} = {75,\\ldots,225}$ would leave the\nextrapolation error during the merger\/ringdown almost unchanged, while\nthe extrapolation error during the inspiral would increase by about\n70\\%.\n\nWe note that this method allows easy extrapolation of various portions\nof the waveform using different extraction radii and orders since---by\nconstruction---the wave phase is an independent variable. For example,\nsolve for the phase value of the merger $\\phi_{\\text{merger}}$\n(defined as the phase at which the amplitude is a maximum), then use\nthe radius range $R \/ \\ensuremath{M_{\\text{irr}}} = {75,\\ldots,225}$ for all phase values\nless than $\\phi_{\\text{merger}}$ and the range\n$R\/\\ensuremath{M_{\\text{irr}}}={75,\\ldots,160}$ for all larger phase values.\n\nThis method has been tested also using the coordinate radius $R$ and\nthe naive time coordinate $T$, in place of areal radius and corrected\ntime.  We found results similar to those discussed in\nSec.~\\ref{sec:TortoiseExtrapolation}.  Using the new time coordinate\n$\\ensuremath{t_{\\text{corr}}}$ instead of the naive time coordinate $T$ improved the\nextrapolation during the merger\/ringdown, as found in\nSec.~\\ref{sec:TortoiseExtrapolation}.\n\nAs with the previous extrapolation method, increasing the\nextrapolation order gives a faster convergence rate of waveform phase\nand amplitude, but it amplifies noise in the extrapolated waveform. To\nimprove convergence without increasing the noise, we need a good\nfiltering technique for the inspiral data. The junk-radiation noise\ndecreases significantly as a function of time, disappearing several\norbits before merger.  However, this noise could be reduced by using\nmore extraction radii in the extrapolation process, or by running the\ndata through a low-pass filter.  See the appendix for further\ndiscussion of filtering.\n\n\n\n\\section{Comparison of the two methods}\n\\label{sec:Comparison}\n\nBoth methods showed good convergence of the amplitude and the phase of\nthe waveform as $N$ increased in the inspiral portion.  (See\nFigs.~\\ref{fig:LapseCorrectionComparison_Corr_Amp}\nand~\\ref{fig:AmplitudeComparisonMethod2} for the amplitude, and\nFigs.~\\ref{fig:LapseCorrectionComparison_Corr}\nand~\\ref{fig:PhaseComparisonMethod2} for the phase.) The wave-phase\nextrapolation method was more sensitive to noise.  In the\nmerger\/ringdown portion, both methods have similar convergence as $N$\nincreases, especially when the correction is taken to account for the\ndynamic lapse.  The use of the time parameter $\\ensuremath{t_{\\text{corr}}}$ improved the\nagreement between the methods significantly in the merger\/ringdown\nportion for all extrapolation orders.  Extrapolating at order $N=2$ or\n$3$ seems the best choice as the noise and phase differences are\nsmallest for these values.\n\nIn Fig.~\\ref{fig:MroueBoyleRelativeAmplitudeDifferenceByOrder}, we\nshow the relative amplitude difference between data extrapolated at\nvarious orders ($N=1,\\ldots,5$).  There is no additional time or phase\noffset used in the comparison.  Ignoring high-frequency components,\nthe difference in the relative amplitude is always less than 0.3\\% for\ndifferent extrapolation orders.  Even including high-frequency\ncomponents, the differences between the two methods are always smaller\nthan the error in each method, as judged by convergence plots.  In\nFig.~\\ref{fig:MroueBoylePhaseDifferenceByOrder}, we show the\n\\emph{phase} difference between the data extrapolated using both\nmethods.  As in the relative amplitude-difference plots, the best\nagreement is achieved during the inspiral portion.  Ignoring\nhigh-frequency components, the difference is less than 0.02 radians\nfor all orders.  In the merger\/ringdown portion, the best agreement\nbetween extrapolated waveforms is at order $N=2$ or $3$ where the\nphase difference is less than 0.01 radians.\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig9}\n  \\caption{\\CapName{Relative difference in the amplitude of the two\n      extrapolation methods as we increase the order of extrapolation}\n    The best agreement between both methods is at low orders of\n    extrapolation, for which the relative difference in the amplitude\n    is less than 0.1\\% during most of the evolution.}\n  \\label{fig:MroueBoyleRelativeAmplitudeDifferenceByOrder}\n\\end{figure}\n\n\\begin{figure}\n \n  \\includegraphics[width=\\linewidth]{Fig10}\n  \\caption{\\CapName{Phase difference of the two extrapolation methods\n      as we increase the order of extrapolation} This figure shows the\n    phase difference between waveforms extrapolated using each of the\n    two methods. The best agreement between the methods is at orders\n    $N=2$ and $3$.  The relative difference in the phase is less than\n    0.02 radians during most of the evolution.}\n  \\label{fig:MroueBoylePhaseDifferenceByOrder}\n\\end{figure}\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:Conclusions}\n\nWe have demonstrated two simple techniques for extrapolating\ngravitational-wave data from numerical-relativity simulations.  We\ntook certain basic gauge information into account to improve\nconvergence of the extrapolation during times of particularly dynamic\ngauge, and showed that the two methods agree to within rough error\nestimates.  We have determined that the first method presented here is\nless sensitive to noise, and more immediately applies to arbitrary\nwavelike data; this method has become the basic standard in use by the\nCaltech--Cornell collaboration.  In both cases, there were problems\nwith convergence after merger.  The source of these problems is still\nunclear, but will be a topic for further investigation.\n\nAs with any type of extrapolation, a note of caution is in order.  It\nis entirely possible that the ``true'' function being extrapolated\nbears little resemblance to the approximating function we choose,\noutside of the domain on which we have data.  We may, however, have\nreason to believe that the true function takes a certain form.  If the\ndata in question are generated by a homogeneous wave equation, for\ninstance, we know that well-behaved solutions fall off in powers of\n$1\/r$.  In any case, there is a certain element of faith that\nextrapolation is a reasonable thing to do.  While that faith may be\nmisplaced, there are methods of checking whether or not it is:\ngoodness-of-fit statistics, error estimates, and convergence tests.\nTo be of greatest use, goodness-of-fit statistics and error estimates\nfor the output waveform require error estimates for the input\nwaveforms.  We leave this for future work.\n\nWe still do not know the correct answers to the questions numerical\nrelativity considers.  We have no analytic solutions to deliver the\nwaveform that Einstein's equations---solved perfectly---predict would\ncome from a black-hole binary merger; or the precise amount of energy\nradiated from any given binary; or the exact kick or spin of the final\nblack hole.  Without being able to compare numerical relativity to\nexact solutions, we may be leaving large systematic errors hidden in\nplain view.  To eliminate them, we need to use multiple, independent\nmethods for our calculations.  For example, we might extract $\\Psi_4$\ndirectly by calculating the Riemann tensor and contracting\nappropriately with our naive coordinate tetrad, and extract the metric\nperturbation using the formalism of Regge--Wheeler--Zerilli and\nMoncrief.  By differentiating the latter result twice and comparing to\n$\\Psi_4$, we could verify that details of the extraction methods are\nnot producing systematic errors.  (Just such a comparison was done in\nRef.~\\cite{BuonannoEtAl2009a} for waveforms extrapolated using the\ntechnique in this paper.)  Nonetheless, it is possible that\ninfrastructure used to find both could be leading to errors.\n\nIn the same way, simulations need to be performed using different\ngauge conditions, numerical techniques, code infrastructures, boundary\nconditions, and even different extrapolation methods.  Only when\nmultiple schemes arrive at the same result can we be truly confident\nin any of them.  But to arrive at the same result, the waveforms from\neach scheme need to be processed as carefully as possible.  We have\nshown that extrapolation is crucial for highly accurate gravitational\nwaveforms, and for optimized detection of mergers in detector data.\n\n\n\n\\begin{acknowledgments}\n  We thank Emanuele Berti, Duncan Brown, Luisa Buchman, Alessandra\n  Buonanno, Yanbei Chen, \\'{E}anna Flanagan, Mark Hannam, Sascha Husa,\n  Luis Lehner, Geoffrey Lovelace, Andrea Nerozzi, Rob Owen, Larne\n  Pekowsky, Harald Pfeiffer, Oliver Rinne, Uli Sperhake, B\\'{e}la\n  Szil\\'{a}gyi, Kip Thorne, Manuel Tiglio, and Alan Weinstein for\n  helpful discussions.  We especially thank Larry Kidder, Lee\n  Lindblom, Mark Scheel, and Saul Teukolsky for careful readings of\n  this paper in various draft forms and helpful comments.  This\n  research has been supported in part by a grant from the Sherman\n  Fairchild Foundation to Caltech and Cornell; by NSF Grants\n  PHY-0652952, PHY-0652929 and DMS-0553677 and NASA Grant NNX09AF96G\n  to Cornell; and by NSF grants PHY-0601459, PHY-0652995, DMS-0553302\n  and NASA grant NNG05GG52G to Caltech.\n\\end{acknowledgments}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{introduction}\nWhen we perform a quantum measurement and extract information from a system, the system is disturbed due to the backaction of the measurement.\nThe more information we extract by quantum measurement, the more strongly the system is disturbed.\nSuch a tradeoff relation has been recognized since Heisenberg pointed out the uncertainty relation~\\cite{He27} between error and disturbance.\nSince 1990's, with the tremendous development of the quantum information theory, various methods of quantifying the information and the disturbance has been proposed, and the tradeoff relations has been shown based on their definitions\\cite{Fu96,Da03,Sa06,BS06,BHH08,CL12}.\nIn most of the studies, the information and the disturbance are defined in the information theoretic settings, i.e., decoding the message encoded in quantum states by a measurement.\n\nIn this paper, we  formulate the information and the disturbance in the setting of estimating an unknown quantum state by quantum measurements, with an emphasis on the estimation accuracy.\nWe derive an inequality that shows the tradeoff relation between them, and give a sufficient condition to achieve the equality.\nWe also discuss the condition for divergences, which measure the distinguishability between two quantum states, to satisfy a similar inequality.\n\n\\section{information-disturbance relation based on estimation theory}\nSuppose that we estimate an unknown quantum state corresponding to a density operator $\\h\\rho_{\\bm\\theta}$ by performing a quantum measurement.\nHere, $\\bm\\theta\\in\\Theta\\subset\\mathbb R^m$ represents $m$ real parameters that characterize the unknown state, so that estimating the state is equivalent to estimating the parameters.\nSuch a parameterized family of states $\\{\\h\\rho_{\\bm\\theta}\\}_{\\bm\\theta\\in\\Theta}$ is called a quantum statistical model.\nA quantum measurement is characterized by a mapping from quantum states to the probability distribution of outcomes and the post-measurement state corresponding to each outcome.\nIf the measurement outcome is discrete, the probability $p_{\\bm\\theta,i}$ of obtaining the outcome $i\\in I$ and the post-measurement state $\\h\\rho_{\\bm\\theta,i}$ are respectively given by \n\\e{\np_{\\bm\\theta,i}&=\\sum_j \\tr{\\h K_{ij}\\h\\rho_{\\bm\\theta}\\h K_{ij}^\\dagger}\\\\\n\\h\\rho_{\\bm\\theta,i}&=\\frac{1}{p_{\\bm\\theta,i}}\\sum_j \\h K_{ij}\\h\\rho_{\\bm\\theta}\\h K_{ij}^\\dagger,\n}\nwhere measurement operators $\\{\\h K_{ij}\\}$ satisfy the normalization condition $\\sum_{i,j}\\h K_{ij}^\\dagger\\h K_{ij}=\\h I$.\n\nWhat is the natural quantification of the information in the setting of estimating an unknown state from the outcome of the quantum measurement?\nThe estimating process is characterized by a function $\\bm\\theta^{\\rm est}:I\\rightarrow \\Theta$ which is called an estimator.\nSince the outcome $i\\in I$ is a random variable, the estimator, which is calculated from the outcome, is also a random variable, and should be distributed around the true parameter $\\bm\\theta$.\nAccording to the classical Cram\\'er-Rao inequality, the variance-covariance matrix of the locally unbiased estimator has a lower bound which is determined only by a family of the probability distributions $\\{p_{\\bm\\theta}\\}_{\\bm\\theta\\in\\Theta}$:\n\\e{\n{\\rm Var}_{\\bm\\theta}[\\bm\\theta_{\\rm est}]\\ge (J^C_{\\bm\\theta})^{-1}.\n}\nHere, $J^C_{\\bm\\theta}$ is an $m\\times m$ matrix called the classical Fisher information whose elements are defined as\n\\e{\n[J^C_{\\bm\\theta}]_{ab}=\\sum_i  p_{\\bm\\theta,i} \\pd{1}{\\log p_{\\bm\\theta,i}}{\\theta_a} \\pd{1}{\\log p_{\\bm\\theta,i}}{\\theta_b}. \\label{info}\n}\nTherefore, a measurement with a larger classical Fisher information allows us to estimate the state more accurately. \nWe define the classical Fisher information as the information obtained by the quantum measurement.\n\nThe disturbance can be evaluated as the loss of information on the parameter $\\bm\\theta$ that can be extracted from the quantum state.\nThe information on the parameter $\\bm\\theta$ that the quantum state potentially possesses can be quantified by the quantum Fisher information, which is defined as\n\\e{\n[J^Q_{\\bm\\theta}]_{ab}:=\\tr{\\pd{1}{\\h\\rho_{\\bm\\theta}}{\\theta_a}\\bm K^{-1}_{\\h\\rho_{\\bm\\theta}} \\pd{1}{\\h\\rho_{\\bm\\theta}}{\\theta_b}}.\n}\nHere, the superoperator $\\h K_{\\h\\rho}$ is defined as\n\\e{\n\\h K_{\\h\\rho}=\\bm R_{\\h\\rho}f(\\bm L_{\\h\\rho}\\bm R_{\\h\\rho}^{-1}),\n}\nwhere $f:(0,\\infty)\\rightarrow (0,\\infty)$ is an operator monotone function satisfying $f(1)=1$ and $\\bm R_{\\h\\rho}  (\\bm L_{\\h\\rho})$ is the right (left)-multiplication of $\\h\\rho$:\n\\e{\n\\bm R_{\\h\\rho}(\\h A)=\\h A\\h\\rho, \\ \\bm L_{\\h\\rho}(\\h A)=\\h\\rho\\h A.\n}\nThe quantum Fisher information is the only metrics on the parameter space that monotonically decrease under an arbitrary completely positive and trace-preserving (CPTP) mapping $\\calE$~\\cite{Pe96}:\n\\e{\nJ^Q_{\\bm\\theta}(\\{\\h\\rho_{\\bm\\theta}\\})\\ge J^Q_{\\bm\\theta}(\\{\\calE(\\h\\rho_{\\bm\\theta})\\}).\n}\nFrom the monotonicity, the quantum Fisher information gives an upper bound of the classical Fisher information obtained by all the possible measurements, and the symmetric logarithmic derivative (SLD) Fisher information~\\cite{He68}, which corresponds to $f(x)=\\frac{1+x}{2}$, is known to be the least upper bound.\nTherefore, the quantum Fisher information, especially SLD Fisher information, can be interpreted as the information on the parameter $\\bm\\theta$ that can be extracted from the quantum state.\nWe define the disturbance of the measurement as \n\\e{\n\\Delta J^Q_{\\bm \\theta}:=J_{\\bm \\theta}^{Q}-\\sum_{i}p_ {\\bm\\theta,i} J_{i,\\bm \\theta}'^{Q},\\label{dist}\n}\nwhere $J_{\\bm \\theta}^{Q}$ and $J_{i,\\bm \\theta}'^{Q}$ are the quantum Fisher information of quantum statistical models $\\{\\h\\rho_{\\bm\\theta}\\}$ and $\\{\\h\\rho_{\\bm\\theta,i}\\}$, respectively.\nFor the sake of generality, we consider the disturbance using a general quantum Fisher information.\n\nThe information~\\eqref{info} and the disturbance~\\eqref{dist} satisfy the following inequality that shows a tradeoff relation between them:\n\\e{\nJ^C_{\\bm\\theta} \\le  \\Delta J^Q_{\\bm\\theta}.  \\label{i-d}\n}\nThis inequality means that if we perform a quantum measurement on an unknown state and extract information on the state, the state loses more intrinsic information.\nSince the inequality~\\eqref{i-d} is valid independently of the choice of the quantum Fisher information to define the disturbance, we obtain\n\\e{\nJ^C_{\\bm\\theta} \\le \\inf_Q \\Delta J^Q_{\\bm\\theta},\n}\nwhere the infimum is taken over all kinds of the quantum Fisher information to define the disturbance.\nWe note that it is nontrivial which quantum Fisher information gives the minimum disturbance, though the minimum and the maximum of the quantum Fisher information are known to be the SLD Fisher information and the real right logarithmic derivative Fisher information (real RLD), which corresponds to $f(x)=\\frac{2x}{x+1}$, respectively.\n\n\nThe proof of the inequality~\\eqref{i-d} is based on the monotonicity and the separating property of the quantum Fisher information.\nFor a given measurement $\\{\\h K_{ij}\\}$, we define a CPTP mapping $\\calE^{\\rm meas}$ as \n\\e{\n\\calE^{\\rm meas}(\\h\\rho)=\\bigoplus_i \\left( \\sum_j \\h K_{ij}\\h\\rho\\h K_{ij}^\\dagger \\right).\n}\nThen the quantum Fisher information of the quantum statistical model $\\{\\calE(\\h\\rho_{\\bm\\theta})\\}$ is equal to the sum of the classical Fisher information obtained by the measurement and the average quantum Fisher information of the post-measurement states as \n\\e{\nJ^Q_{\\bm\\theta}(\\{\\calE^{\\rm meas}(\\h\\rho_{\\bm\\theta})\\})=J^C_{\\bm\\theta}(\\{p_{\\bm\\theta}\\})+\\sum_{i}p_ {\\bm\\theta,i} J_{i,\\bm \\theta}'^{Q},  \\label{separating}\n}\nwhich we call the separating property (see Appendix~\\ref{proofA} for proof).\nBy applying the monotonicity under the CPTP mapping $\\calE^{\\rm meas}$, we obtain\n\\e{\nJ^Q_{\\bm\\theta}(\\{\\h\\rho_{\\bm\\theta}\\})&\\ge J^Q_{\\bm\\theta}(\\{\\calE^{\\rm meas}(\\h\\rho_{\\bm\\theta})\\}) \\nonumber \\\\\n&=J^C_{\\bm\\theta}(\\{p_{\\bm\\theta}\\})+\\sum_{i}p_ {\\bm\\theta,i} J_{i,\\bm \\theta}'^{Q},\n}\nwhich proves the inequality~\\eqref{i-d}.\n\n\\section{condition for equality}\nMeasurements that achieves the equality of the inequality~\\eqref{i-d} are efficient in a sense that they cause the minimum disturbance among those that extract a given amount of information.\nWhen we adopt the right logarithmic derivative (RLD) Fisher information~\\cite{YL73}, which corresponds to $f(x)=x$, to define the disturbance, a class of measurements called pure and reversible measurement achieves the equality of the inequality~\\eqref{i-d} (see Appendix~\\ref{proofB} for proof):\n\\e{\nJ^C_{\\bm\\theta}=\\Delta J^{\\rm RLD}_{\\bm\\theta} \\label{RLDequality}\n}\nHere, a measurement is called pure if the number of measurement operators is one for each measurement outcome so that the measurement operators can be written as  $\\{\\h K_i\\}$, and is called reversible if each $\\h K_i$ has the inverse operator $\\h K_i^{-1}$~\\cite{UIN96}.\n\nAs an example, a measurement on a spin-1\/2 system proposed by Royer~\\cite{Ro94} is pure, whose measurement operators are \n\\e{\n\\h K_1&=\n\\begin{pmatrix}\n\\cos(\\theta\/2-\\sigma\/4)&0\\\\\n0&\\cos(\\theta\/2+\\sigma\/4)\n\\end{pmatrix},\\\\\n\\h K_2&=\n\\begin{pmatrix}\n\\sin(\\theta\/2-\\sigma\/4)&0\\\\\n0&\\sin(\\theta\/2+\\sigma\/4)\n\\end{pmatrix}.\n}\nThis measurement is reversible if $\\theta\/2\\pm\\sigma\/4\\ne n\\pi\/2$.\n\nIn fact, pure measurements are the least disturbing measurements in the following sense.\nSuppose that two measurements $\\{\\h K'_{ij}\\}$ and $\\{\\h K_i\\}$ give the same positive operator-valued measure (POVM)\n\\e{\n\\sum_j \\h K_{ij}'^\\dagger\\h K'_{ij}=\\h K_i^\\dagger \\h K_i, \\quad\\forall i\\in I,\n}\nand hence give the same probability distribution.\nThen, the pure measurement causes less disturbance, and gives the same amount of information:\n\\e{\n\\Delta J^Q_{\\bm\\theta}(\\{\\h K_i\\})&\\le \\Delta J^Q_{\\bm\\theta} (\\{\\h K'_{ij}\\}),\\\\\nJ^C_{\\bm\\theta} (\\{\\h K_i\\})&=J^C_{\\bm\\theta}(\\{\\h K'_{ij}\\}).\n}\n\n\n\\section{information-disturbance relation based on distinguishability}\nIn Ref.~\\cite{BL05}, by using the classical and quantum relative entropies\n\\e{\nS^C(p\\|q)&=\\sum_i p_i\\log\\left( \\frac{p_i}{q_i} \\right),\\\\\nS^Q(\\h\\rho\\|\\h\\sigma)&=\\tr{\\h\\rho(\\log\\h\\rho-\\log\\h\\sigma)},\\label{q.rel.}\n}\na similar inequality to~\\eqref{i-d} was derived:\n\\e{\nS^C(p\\|q)\\le S^Q(\\h\\rho\\|\\h\\sigma)-\\sum_i p_i S^Q(\\h\\rho_i\\|\\h\\sigma_i),\\label{rel.ent}\n}\nwhere $p,q$ and $\\h\\rho_i,\\h\\sigma_i$ are the probability distributions and the post-measurement states of a quantum measurement performed on the quantum states $\\h\\rho, \\h\\sigma$, respectively.\nSince the quantum relative entropy is a measure of the distinguishability of two quantum states~\\cite{HP91,ON02}, Eq.~\\eqref{rel.ent} can also be interpreted as a tradeoff relation between information and disturbance.\nIn particular, if we choose two similar states $\\h\\rho_{\\bm\\theta}$ and $\\h\\rho_{{\\bm\\theta}+{\\rm d}{\\bm\\theta}}$ as the arguments of the relative entropy, Eq.~\\eqref{rel.ent} reproduces the inequality~\\eqref{i-d} with the disturbance defined by the Bogoliubov-Kubo-Mori (BKM) Fisher information, which corresponds to $f(x)=\\frac{x-1}{\\log x}$.\n\nIn the following, we discuss the extension of the inequality~\\eqref{rel.ent} to general divergences.\nLet $D^C(\\cdot\\|\\cdot)$ be a divergence between two probability distributions, and $D^Q(\\cdot\\|\\cdot)$ be its quantum extension, i.e.,\nif two quantum states $\\h\\rho,\\h\\sigma$ commute and therefore are simultaneously diagonalizable as $\\h\\rho=\\sum_ip_i\\ket i\\bra i, \\h\\sigma=\\sum_iq_i\\ket i\\bra i$, we obtain\n\\e{\nD^Q(\\h\\rho\\|\\h\\sigma)=D^C(p\\|q).\n}\nWe note that quantum extensions of a divergence is not unique in general.\n\nLet us consider a condition for divergences $D^C(\\cdot\\|\\cdot),D^Q(\\cdot\\|\\cdot)$ to satisfy the information-disturbance tradeoff relation\n\\e{\nD^C(p\\|q)\\le D^Q(\\h\\rho\\|\\h\\sigma)-\\sum_i p_i D^Q(\\h\\rho_i\\|\\h\\sigma_i).\\label{divergence}\n}\nThe essential properties for the proof of the inequality~\\eqref{i-d} are the monotonicity and the separating property of the quantum Fisher information.\nIn the same way, we require these two properties for divergences:\n\\e{\nD^Q(\\h\\rho\\|\\h\\sigma)&\\ge D^Q(\\calE(\\h\\rho)\\|\\calE(\\h\\sigma)),\\\\\nD^Q(\\calE^{\\rm meas}(\\h\\rho)\\|\\calE^{\\rm meas}(\\h\\sigma))&=D^C(p\\|q)+\\sum_i p_iD^Q(\\h\\rho_i\\|\\h\\sigma_i).\n}\nIf we also require a continuity of $D^C(p\\| q)$ with respect to $p,q$, then it satisfies Hobson's five conditions that axiomatically characterize the classical relative entropy~\\cite{Ho69}.\nTherefore, the divergence with the monotonicity, the separating property, and the continuity must be consistent with the relative entropy at least for classical probability distributions:\n\\e{\nD^C(p\\|q)=S^C(p\\|q).\n}\nAs is shown in~\\cite{BL05}, the well-known quantum relative entropy satisfies the tradeoff relation~\\eqref{divergence} because it has the monotonicity and the separating property.\nAnother quantum extension of relative entropy proposed by Belavkin and Staszewski~\\cite{BS82},\n\\e{\nS^{\\rm BS}(\\h\\rho\\|\\h\\sigma)=\\tr{\\h\\rho\\log(\\h\\rho^{1\/2}\\h\\sigma^{-1}\\h\\rho^{1\/2})}, \\label{BS.rel.}\n}\nalso satisfies the inequality~\\eqref{divergence}.\nHere, $S^{\\rm BS}(\\cdot\\|\\cdot)$ is known to be maximal among all the possible quantum extensions of the classical relative entropy~\\cite{Ma13}.\nBy substituting $\\h\\rho_{\\bm\\theta}$ and $\\h\\rho_{{\\bm\\theta}+{\\rm d}{\\bm\\theta}}$ to the inequality~\\eqref{divergence}, we again obtain the inequality~\\eqref{i-d} with the disturbance defined by the real RLD Fisher information.\n\n\n\\section{conclusion}\nIn this paper, we have formulated the tradeoff relation between information and disturbance in quantum measurement in view of estimating parameters that characterize an unknown quantum state.\nThe information is defined as the classical Fisher information of the probability distributions of measurement outcomes, and the disturbance is defined as the average loss of the quantum Fisher information due to the backaction of the measurement.\nWe have shown the tradeoff relation~\\eqref{i-d} between them.\nWhen we use the RLD Fisher information, the equality of the inequality~\\eqref{i-d} is achieved by pure and reversible measurements.\nIn fact, pure measurements are the least disturbing among those that provide us with a given amount of information.\n\nWe have also discussed the necessary condition for divergences between two quantum states to satisfy a similar tradeoff relation~\\eqref{divergence}.\nIt is necessary for divergences to coincide with the relative entropy at least for classical probability distributions.\nIn addition to the well-known relative entropy, the maximum relative entropy also satisfies the tradeoff relation~\\eqref{divergence}, which reproduces the inequality~\\eqref{i-d} for the disturbance defined by the real RLD Fisher information.\nIf there are quantum extensions of the relative entropy that give an arbitrary quantum Fisher information, another systematic derivation of the inequality~\\eqref{i-d} should be possible.\n\\section*{acknowledgement}\nThis work was supported by\nKAKENHI Grant No. 26287088 from the Japan Society for the Promotion of Science, \na Grant-in-Aid for Scientific Research on Innovation Areas ``Topological Quantum Phenomena'' (KAKENHI Grant No. 22103005),\nthe Photon Frontier Network Program from MEXT of Japan,\nand the Mitsubishi Foundation.\nT. S. was supported by the Japan Society for the Promotion of Science through Program for Leading Graduate Schools (ALPS).\nY. K. acknowledges supports by Japan Society for the Promotion of Science (KAKENHI Grant No. 269905).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nHealth anxiety is a significant problem in our modern medical system \\cite{asmundson, taylor}. The belief that one's common and benign symptoms are explained by serious conditions may have several adverse effects such as quality-of-life reduction, incorrect medical treatment and inefficient allocation of medical resources. The web has been shown to be a significant factor in fostering such attitudes \\cite{whitehorvitzcyber, whitehorvitzhypo}. A recent study found that 35\\% of U.S. adults had used the Web to perform diagnosis of medical conditions either for themselves or on behalf of another person, and many ($>$50\\%) pursued professional medical attention concerning their online diagnosis \\cite{foxduggan}. Motivated by the popularity of online health search, we investigated how search engines might improve their health information offerings. We hypothesize that searchers are less likely to develop unrealistic beliefs when they are given unbiased and well-presented information about their medical state. Thus, we believe an ideal search engine should use queries, and available search histories, to extract medically relevant information about the individual as well as detect any health anxieties.\n\nLet us give a (negative) example of a possible medical search session. A user experiencing anxiety about a headache might first spend some time searching for information on a serious condition such as ``brain tumor'' and then switch to a symptom query about headache (a type of transition that prior work shows occurs frequently \\cite{cartright}). The user might choose the query wording ``severe headache explanations'' because of the subjective concern they experience at query time. The engine, registering the words ``severe'' and ``explanations'' as well as the phrase ``brain tumor\" present in the user's search history might compile a search engine result page (SERP) that is biased towards serious conditions. The user, viewing the SERP through the lens of their current health anxiety, may be attracted towards serious conditions in captions \\cite{whitehorvitzbias} and hence select a concerning page, heightening their anxiety further.\n\nIn this paper, we highlight a range of challenges and opportunities in working towards improving exploratory health search and thus hope to outline an agenda that frames this problem. Achieving this requires an understanding of both user and search engine behavior. Users play a role in the search process in two ways: their choice of query formulation as well their subjective consumption of information on the SERP and pages clicked on. This naturally led us to formulate three research questions:\n\\begin{itemize*}\n\\item [{\\bf Q1}] How does a user's subjective medical concern shape his or her choice of wording for medical queries?\n\\item [{\\bf Q2}] How does the search engine interpret the subjective medical concern and objective medical information expressed in the query as well as other measurable characteristics (such as medical search history) when compiling the SERP?\n\\item [{\\bf Q3}] How do users respond to the SERP, both in terms of the consumption of the information on the SERP as well as changes in future behavior caused by viewing the SERP and pages clicked on?\n\\end{itemize*}\n\nResults pertaining to these questions are found in Sections 4, 5, and 6 respectively. To answer them, we studied the search logs of 190 thousand consenting users of a major commercial Web search engine. Search logs are a valuable resource for studying information seeking in a naturalistic setting, and such data has been used by several studies to explore how searchers obtain medical information \\cite{whitehorvitzcyber, cartright}. The pipeline used to process this data and the features extracted for analysis are further described in Section 3. \n\nThe main contributions from our analysis are:\n\n\\begin{itemize*}\n\\item Revealing how certain users have specific preferences for certain query formulations (e.g. ``i have muscle pain'' vs. ``muscle pain'') which also has a significant effect on search results, and potentially health outcomes. \n\\item Finding evidence that users might not be swayed by concerning content appearing on SERPs as we might expect based on prior studies.\n\\item Quantifying the extent to which users with prior medical concern receive more concerning SERPs in response to health queries and choose the most concerning pages to click on, potentially leading to a vicious cycle of concern reinforcement. \n\\item Determining to how users directly and indirectly influence the level of concern expressed in the SERP they receive, both through query choice and other factors (e.g., personalization).\n\\end{itemize*}\n\nWe discuss these findings and their implications in Section 8 and conclude in Section 9.\n\n\\section{RELATED WORK}\n\nRelated research in this area falls into three main categories: health search behavior; the quality of online health content; and health anxiety and the impact of reviewing health content on the Web.\n\nThere continues to be interest in search and retrieval studies on expert and consumer populations in a variety of domains, typically conducted as laboratory studies of search behavior \\cite{bhavnani, hershhickam}. Benigeri and Pluye \\cite{benigeripluye} showed that exposing novices to complex medical terminology puts them at risk of harm from self-diagnosis and self-treatment. It is such consumer searching (rather than expert searching) that we focus on in the remainder of this section.\n\nSearch engine log data can complement laboratory studies, allowing search behavior to be analyzed at scale in a naturalistic setting and mined for a variety of purposes. Logs have been used to study how people search \\cite{jansen}, predict future search and browsing activity \\cite{lauhorvitz}, model future interests \\cite{dupretpiwowarski}, improve search engine quality \\cite{joachims}, and learn about the world \\cite{richardson}. Focusing on how people perform exploratory health searching, Cartright et al. \\cite{cartright} studied differences in search behaviors associated with diagnosis versus more general health-related information seeking. Ayers and Kronenfeld \\cite{ayerskronenfeld} explored changes in health behavior associated with Web usage, and found a positive correlation between it and the likelihood that a user will change their health behavior based on the content viewed. \n\nThe reliability of the information in search results is important in our study; unreliable information can drive anxiety. The quality of online healthcare information has been subject to recent scrutiny. Lewis \\cite{lewis} discussed the trend toward accessing information about health matters online and showed that young people are often skeptical consumers of Web-based health content. Eysenbach and Kohler \\cite{eysenbachkohler} studied users engaged in assigned Web search tasks. They found that the credibility of Web sites was important in the focus group setting, but that in practice, participants largely ignored the information source. Sillence and colleagues \\cite{sillence} studied the influence of design and content on the trust and mistrust of health sites. They found that aspects of design engendered mistrust, whereas the credibility of information and personalization of content engendered trust. \n\nThe medical community has studied the effects of health anxiety, including hypochondriasis \\cite{asmundson}, but not in Web search. Health anxiety is often maladaptive (i.e., out of proportion with the degree of medical risk) and amplified by a lack of attention to the source of their medical information \\cite{taylor, kring}. Such anxiety usually persists even after an evaluation by a physician and reassurance that concerns lack medical basis. A recent study showed that those whom self-identified as hypochondriacs searched more often for health information than average Web searchers \\cite{whitehorvitzhypo}. By estimating the level of health concern via long-term modeling of online behavior, search engines can better account for the effect that results may have and help mitigate health concerns. Our research makes progress in this area.\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{>{\\centering\\arraybackslash}p{\\textwidth}} \nheadache, headaches, severe headache what do I do, which remedy for headache, headache top of head with back pain \\\\ \\hline\nheadache do I have a {\\it tumor}, {\\it headache rack}, my {\\it job} gives me a headache, headache {\\it national parks of california} \\\\ \\hline \na, low, be, he, sick, she, helps, standing, black, speech, male, between, acute, shaking, sensitive, bending, an, testing \\\\ \\hline\n\\end{tabular}\n\\captionsetup{justification=centering}\n\\caption{Examples of landmark queries (top), of non-landmark queries (middle; secondary topics and medical conditions are italic) and of admissible words we used to find potential landmark queries (bottom)}\n\\end{table*}\n\nSearchers may feel too overwhelmed by the information online to make an informed decision about their care\\cite{hart}. Performing self-diagnosis using search engines may expose users to potentially alarming content that can unduly raise their levels of health concern. White and Horvitz \\cite{whitehorvitzcyber} employed a log-based methodology to study escalations in medical concerns, a behavior they termed cyberchondria. Their work highlighted the potential influence of several biases of judgment demonstrated by people and search engines themselves, including base-rate neglect and availability. In a follow up study \\cite{whitehorvitzesc}, the same authors showed a link between the nature and structure of Web page content and the likelihood that users' concerns would escalate. They built a classifier to predict  escalations associated with the review of content on Web pages (and we obtained that classifier for the research described in this paper). Others have also examined the effect of health search on user's affective state, showing that the frequency and placement of serious illnesses in captions for symptom searches increases the likelihood of negative emotional outcomes \\cite{lauckner}. Other research has shown that health-related Web usage has been linked with increased depression \\cite{bessiere}.\n\nMoving beyond the psychological impact of health search, researchers have also explored the connection between health concerns and healthcare utilization. In one study \\cite{whitehorvitzhui}, the authors estimated that users sought medical attention by identifying queries containing healthcare utilization intentions (HUIs) (e.g., [physicians in san jose 95113]). Eastin and Guinsler \\cite{eastinguinsler} showed that health anxiety moderated the relationship between health seeking and healthcare utilization. Baker and colleagues \\cite{baker} examined the prevalence of Web use for healthcare, and found that the influence of the Web on the utilization of healthcare is uncertain. The role of the Web in informing decisions about professional treatment needs to be better understood. One of our contributions in this paper is to demonstrate the potential effect of health-related result pages on future healthcare utilization.\n\nOur research extends previous work in a number of ways. By focusing on the first query pertaining to a particular symptom observed in a user history, we show that small differences in query formulation can reflect significant differences amongst health searchers and their health-related search outcomes. To date, no research has demonstrated the impact and insight afforded from analyzing such landmark queries and the behavior around them. To our knowledge, we are also the first to use the search logs to devise statistical experiments which allow us to quantify effects such as user response to medical search results amongst real user populations and provide evidence for causal relationships where possible. We believe that such an approach is necessary to formulate definitive implications for search engine design as well as measuring search engine performance. \n\n\\section{STUDY}\nWe describe various aspects of the study that we performed, including the data, the features extracted, and the statistical methods used for analysis.\n\n\\subsection{Log Data}\n\nTo perform our study we used the anonymized search logs of a popular search engine. Users of this search engine give consent to having information about their queries stored via the terms of use of the engine. During this study, we focused on medical queries related to headache, as it is among the most common health concerns \\cite{nielsen}. We use the phrase {\\it headache query} to refer to queries that contain the substring ``headache'' and occurred during the six month period from September 2012 to February 2013. Amongst those, we call a {\\it landmark query} a query that shows an intent to explore the symptom ``headache'', that does not already contain a possible explanation for headache (e.g., migraine, tumor) and that is not otherwise off-topic. We found these landmark queries by manually assessing frequent headache queries and creating a list of 682 ``admissible'' terms that we believed could occur in landmark queries. We then compiled all queries that exclusively contain terms from that list into a dataset. Examples of landmark queries, non-landmark queries, and admissible words can be found in Table 1. From manual inspection, we concluded with confidence that the dataset captured over 50\\% of all landmark queries present in the logs and that over 95\\% of captured queries are proper landmark queries, the rest being headache queries which contain a significant secondary topic. We then excluded all but the first landmark query instance for each user, ensuring that each user only appears once in the dataset. Overall, our dataset contains over 50,000 unique queries and over 190,000 query instances \/ users.\n\nWe focus on headache since it a common medical symptom (e.g., over 95\\% of adults report experiencing headaches in their lifetime \\cite{rasmussen}) and there are a variety of serious and benign explanations (from caffeine withdrawal to cerebral aneurism), facilitating a rich analysis of content, behavior, and concern. While we believe that headache searching is sufficiently rich and frequent to warrant its own study, investigating queries related to symptoms other than headache could solidify our findings. A large-scale analysis similar to that reported here, but focused on multiple symptoms, is an interesting and important area for future work.\n\n\\subsection{Features}\n\nFor each landmark query in our dataset, we generated features. To frame our three research questions, we modeled the search process around a landmark query as five separate stages: (i) The user's search behavior prior to the landmark query, (ii) the user's choice of wording of the landmark query, (iii) the SERP returned to the user by the search engine, (iv) the user's decision which pages to click on (if any), and (v) the user's search behavior after the landmark query. Our research questions ask about the relationship of these 5 stages. Hence, the features we extracted come in five groups.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lm{2.3in}cc} \nName&Description  \\\\ \\hline\npasttopserious&Number of medical queries containing the most frequent serious condition amongst all serious conditions present \\\\ \\hline\npasttopbenign&Number of medical queries containing the most frequent benign condition amongst all benign conditions present \\\\ \\hline\npastdiffserious&Number of distinct serious conditions in medical queries \\\\ \\hline\npastdiffbenign&Number of distinct benign conditions in medical queries\\\\ \\hline\npastmedical&Number of medical queries \\\\ \\hline\npastheadache&Number of medical queries containing ``headache'' or a condition that is an explanation for headache \\\\ \\hline\npasthui&Number of HUI queries \\\\ \\hline\npasthuiclicked&Number of HUI queries where at least one page on the SERP was clicked \\\\ \\hline\n\\end{tabular}\n\\caption{User Search Behavior Features}\n\\end{table}\n\n\\begin{table*}\n\\centering\n\\begin{tabular}{lllc} \nName & Description (query ..) & Example & Frequency\\\\ \\hline\naudience & .. is about specific population &``headache in adults'' & 3.6\\%\\\\ \nfiller & .. contains a filler such as 'a' or 'and' &``headache and cough'' & 3.6\\%\\\\ \ngoal & .. contains a specific search goal & ``definition of headache'' & 26.3\\%\\\\ \ngoal:condition & .. indicates the goal of diagnosis & ``reasons for headache'' & 8.2\\%\\\\ \ngoal:symptom & .. indicates the goal of related symptoms & ``headache symptoms'' & 2.1\\%\\\\ \ngoal:treatment & .. indicates the goal of treatment & ``headache cure'' & 14.7\\%\\\\ \ngoal:medication & .. indicates the goal of treatment through medication & ``headache pills'' & 2.4\\%\\\\ \ngoal:alternative & .. indicates the goal of alternative treatment & ``natural headache remedy'' & 5.4\\%\\\\ \nsymptomhypothesis & .. states another symptom as cause & ``headache caused by back pain'' & 1.7\\%\\\\ \neventhypothesis & .. relates the headache to a life event & ``headache after hitting head'' & 3.3\\%\\\\ \nduration & .. specifies a duration for the headache & ``chronic headache'' & 10.9\\%\\\\ \nintensity & .. specifies that the headache is strong & ``severe headache'' & 5.0\\%\\\\ \nlocation & .. specifies the location of the headache & ``headache left side of head'' & 17.3\\%\\\\ \npronoun & .. contains a pronoun & ``i have headache'' & 5.8\\%\\\\ \nkindofheadache & .. specifies the kind of pain & ``stabbing headache'' & 2.5\\%\\\\ \nothersymptom & .. specifies an additional symptom & ``headache reflux'' & 27.9\\%\\\\ \ntriggered & .. indicates a headache trigger & ``headache when bending over'' & 1.4\\%\\\\ \ntimeofday & .. indicates a daily pattern & ``headache in afternoon'' & 3.2\\%\\\\ \nopenquestion & .. is phrased as an open question & ``what to do about headache'' & 11.1\\%\\\\ \n\\end{tabular}\n\\caption{High-level {\\it QueryFormulation} features. Note that queries may have multiple features, such as ``severe headache on top of head and cough'', so the percentages in the far-right column do not sum to 100\\%.}\n\\end{table*}\n\n{\\it BeforeSearching}. This group of features describes the level of medical searching before the landmark query (stage (i)). For each query in the user's search log, we first extracted whether that query was of a medical nature. For this, we used a proprietary classifier. From manual inspection, we concluded with confidence that its Type I and II errors are $< 0.1$. Queries so classified as medical will be called {\\it medical queries}. Secondly, we extracted phrases present in the query that describe medical conditions, such as ``common cold'' or ``cerebral aneurism''. The list of phrases we considered was based on the International Classification of Diseases 10th Edition (ICD-10) published by the World Health Organization as well as the U.S. National Library of Medicine's PubMed service and other Web-based medical resources. We also used manually curated synonyms from online dictionaries and standard grammatical inflections to increase coverage. For more information see the approach used by \\cite{whitehorvitzcyber}. The list was also separated into benign and serious medical conditions and we determined which conditions are possible explanations for headache. Thirdly, we extracted phrases that indicate the query is linked to an intention of real-world healthcare utilization (HUI), such as ``emergency care'' or ``hepatologist''. We call those queries {\\it HUI queries}. Finally, the individual-query-level features were aggregated over a time window just before the landmark query to form 8 distinct features, which are shown in Table 2. In our experiments, we considered five different aggregation windows: 1 hour, 1 day, 1 week, 30 days and 90 days. All experiments involving {\\it BeforeSearching} features were replicated for each aggregation window. \n\nWe believe that intense medical searching may be a sign of health concerns or anxieties. White and Horvitz \\cite{whitehorvitzhypo} extensively demonstrated that users believing their symptoms may be explained by a serious condition conduct longer medical search sessions and do so more frequently. Of course, there are many potential reasons for increased medical search such as different web search habits, random noise or even a recent visit to a physician \\cite{whitehorvitzhui}. Overall, we believe that it makes sense to view {\\it BeforeSearching} in light of possible medical concern experienced. However, even if there are significant other factors at play, we believe that the phenomenon of health search intensity remains interesting.\n\n{\\it QueryFormulation}. The choice of wording for the query (stage (ii)) was modeled, firstly, using 19 high-level features. We arrived at these by manually inspecting admissible words and landmark queries (such as in Table 1) and noting the most important high-level ideas expressed through them. We believe that those 19 features capture a significant portion of the semantic variation within queries. The features are shown in Table 3. They offer a useful characterization of the broad range of different types of search intent associated with headache-related queries. Four of these features ({\\it othersymptom}, {\\it duration}, {\\it intensity} and {\\it location}) were further divided by which key phrase was used to express this feature, yielding an additional 117 low-level features (examples are shown in the graph annotation of Figure 1.2-1.5). All feature extraction functions are based on substring matches joined by logical operators. From manual inspection, we concluded with confidence that the Type I and II errors of the extractors of all these features was low. All features in this category are binary.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=1.8]{F1p1.pdf}\n\\includegraphics[scale=1.8]{F1p2.pdf}\n\\includegraphics[scale=1.8]{F1p3.pdf}\n\\caption{Association between query formulation and search behavior features. From left to right: {\\bf {\\color{red} Red}}: {\\it pastmedical} (aggregation window: 90 days); {\\bf {\\color{blue} Blue}}: {\\it pastmedical} (window: 1 hour); {\\bf {\\color{Brown} Brown}}: {\\it futuremedical} (window: 1 hour); {\\bf Black}: {\\it futuremedical} (window: 90 days). All bars show the relative change of medical search intensity of users whose landmark query has a certain formulation relative to the global mean. Error bars are 95\\% confidence intervals.}\n\\end{figure*}\n\n{\\it SERPConcern}. We scored the level of medical concern expressed in each of the Top 8 pages on the SERP (stage (iii)) using a logistic regression classifier designed to predict searching for serious conditions and shown in \\cite{whitehorvitzesc} to have significant predictive power. The classifier was graciously provided to us by the authors for use in our study. Note that during the time period analyzed in this study, the search engine only returned eight results on a large number of SERPs so we disregarded possible further results. It has been shown that users rarely click below position 8, including for health queries \\cite{whitehorvitzbias}. The classifier is based on page features from the URL and HTML content similar to those shown in Table 2. It also includes features that attempt to measure the ``respectability'' of the page (e.g., expressed through the {\\it Health on the Net Foundation} certificate \\cite{hon}) to capture the effect of this on the user. It is then trained to discriminate between pages that lead to serious condition searches (concerning) and pages that lead to benign condition searches (non-concerning). The concern score of an arbitrary page is then the inner product between its feature vector and the learnt weight vector. Finally, we take the weighted sum of the 8 scores for the individual pages on any SERP to produce a single feature value for the full SERP. Note that even though we consider pages leading to benign condition searches less concerning than those leading to serious condition searches, throughout this study, we still consider benign condition searching as an indicator for medical concern experienced by the user, albeit less strong than serious condition searching.\n\n{\\it ClickFeatures}.  To measure the users click behavior (stage (iv)), we record whether the user clicked a page on the SERP. If the user did, we also recorded the concern score of the page clicked (as described in the last paragraph) as well as the position of the clicked page on the SERP. We call these three features {\\it hasclicked}, {\\it clickconcern} and {\\it clickposition}. (Users that did not click are excluded from analysis involving features {\\it clickconcern} and {\\it clickposition}.)\n\n{\\it AfterSearching}. The same 8 features as {\\it BeforeSearching}, but aggregated over a window just after the landmark query (stage (v)). In the name, we replace ``past'' with ``future'' (e.g. one feature is called {\\it futuremedical}).\n\n\\subsection{Statistical Methodology}\n\n\nLet $X$ be a dataset where each entry corresponds to a user \/ landmark query (either our full dataset of 190.000 entries or a subset of it.). Write $x_1, .., x_N$ for the data points and $x^1_n, .., x^d_n$ for the components \/ feature values of data point $x_n$. Throughout our analysis, we wish to measure whether there is a significant association between two feature values $i$ and $j$, say between {\\it pastmedical} and {\\it SERPConcern}. By this we mean that the p-value of a suitable independence test on the two variables is low. To measure this simply and robustly, we will choose a threshold $t^i$ and split the data set into two subpopulations $X_<$ and $X_>$ such that for all $x_n$, $n \\in \\{1, .., N\\}$, we have $x_n \\in X_< \\iff x_n^i < t^i$. We also call the two subpopulations the {\\it lower bucket} and {\\it upper bucket} respectively. Then, we either perform a two-sample t-test on the population means of $X_<$ and $X_>$ (Figure 3.3) or we consider a 95\\%-confidence interval around the mean of the smaller bucket if it is significantly smaller than the other bucket (Figures 1 and 2). (In practice, all our subpopulations are large enough to warrant the use of gaussian confidence intervals \/ tests.) A statistical association between features $i$ and $j$ is a symmetric relation, we may choose to split on either feature and compare the means of the other, based on convenience of presentation.\n\nSeveral times, we will encounter a more challenging case where we want to answer the question whether two feature values $i$ and $j$ are associated while controlling for a third feature value $k$. We do this by first dividing the dataset into many subpopulations according to the exact value of feature $k$ to obtain $X^{v^k_1}$, .., $X^{v^k_{N^k}}$, where $v^k_1, .., v^k_{N^k}$ are the values $x_n^k$ can take. Then, we split each of these subpopulations further according to thresholds on feature $i$ as before. Hence, we obtain $X_<^{v^k_1}$, .., $X_<^{v^k_{N^k}}$ and $X_>^{v^k_1}$, .., $X_>^{v^k_{N^k}}$. Because it is nontrivial to jointly compare this potentially large number of buckets, we adopt the following 2-step procedure. \n\nFirst, we take the union of all lower and upper buckets respectively and perform a two-sample t-test as before. For this to control for feature $k$, each lower bucket must be of the same size as its corresponding upper bucket. If, for some $X^{v^k_n}$ there is no threshold that achieves this, we randomly remove data points whose feature value is equal to the median across $X^{v^k_n}$ until this is possible. While this significantly reduces the number of data points entering the analysis, we do not believe this threatens validity or generalizability.\n\nIn the second step, we first individually compare each lower bucket to its corresponding upper bucket. Because many of these buckets are small (e.g. of size 1), we use the Mann-Whitney U statistic to obtain a p-value for each pair of buckets. Then, we aggregate all of those p-values by Stouffer's Z-score method where each p-value is weighted according to the size of its respective buckets. \n\nFinally, we wish to combine p-values obtained in both steps to either accept or reject the null-hypothesis at a given significance level. This is achieved by taking the minimum of both values and multiplying by 2. The cumulative distribution function of that combined quantity is below that of the uniform distribution under the null hypothesis and is thus at least as conservative in rejecting the null as each individual p-value, while gaining a lot of the statistical power of both tests. We quote this value in Figure 3.1-3.2 and 3.4-3.7.\n\n\\section{QUERY FORMULATION}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=1.8]{F2v2.pdf}\n\\caption{Association between query formulation features and {\\it SERPConcern}. All bars show the difference in {\\it SERPConcern} of users whose landmark query has a certain formulation relative to the global mean, measured in standard deviations of the global distribution.}\n\\end{figure*}\n\nWe began our analysis by investigating how users with different levels of {\\it BeforeSearching} phrase their queries. Figure 1.1 shows the mean value of {\\it pastmedical} for each high-level {\\it QueryFormulation} feature relative to the population mean, both aggregated over 1 hour and 90 days. Error bars indicate 95\\% confidence intervals. We chose the feature {\\it pastmedical} to represent {\\it BeforeSearching} because it is the least sparse and hence has the smallest confidence interval.\n\nTo our surprise, we found that most {\\it QueryFormulation} features are highly discriminative, i.e. users choosing queries with certain features have conducted significantly more medical searching than users choosing queries that have different features. In fact, every one of these 19 features is associated with a significant change in prior medical search activity during the hour before the landmark query. This effect could be explained by users near the beginning of their medical search process choosing different query patterns compared to users near the end of their medical search process. However, most {\\it QueryFormulation} features are still discriminative when aggregated over a 90 day window (red bars), and we found that the majority of medical searches in that window do not occur immediately prior to the landmark query. Figure 1.1 also shows the feature {\\it futuremedical}. We find that most {\\it QueryFormulation} features have the same association with past and future search activity, even over a 90 day window. This is evidence that these {\\it QueryFormulation} features characterize users and that heavy medical searchers prefer certain formulations compared to other users in a consistent fashion over the long term.\n \nOver the 1 hour window, users entering an additional symptom in their landmark query show the highest level of search activity. This might be because experiencing a larger number of symptoms causes the user to want to find information about each individual symptom, thus increasing the search need. Surprisingly, {\\it intensity} is not one of the strongest predictors of increased searching even though it appears to be the most intuitive indicator of increased concern. Also, the features {\\it openquestion} and {\\it pronoun} indicate less prior searching. Hence, the presence of sentence-like structures in queries is potentially associated with lower user health concern. It is difficult to intuitively interpret most of the {\\it QueryFormulation} features. Figure 1.2-1.5 shows low-level {\\it QueryFormulation} features relating to high-level features {\\it duration}, {\\it intensity} and {\\it othersymptom}. Unfortunately, the sparseness of {\\it pastmedical} leads to large margins of error, the exact size of which are also difficult to determine. Nonetheless, certain features which are more common and thus have lower margins or error such as ``constant'' or ``terrible'' are as discriminative as high-level features, suggesting they are useful for making inferences about the user. For example, our analysis suggests the very counterintuitive conclusion that users searching for ``headache everyday'' experience a different level of concern than users searching for ``daily headache'' (see difference in feature {\\it pastmedical} between {\\it daily} and {\\it everyday} (blue bar) in Figure 1.2). Similarly, different symptoms are associated with their own level of past search activity. We chose to display 10 relatively concerning symptoms (Figure 1.4) and 10 relatively benign symptoms (Figure 1.5). The choice was made using the findings of a separate crowdsourcing study that we omit from the paper due to space reasons. In that study, many participants were asked to estimate the level of medical concern associated with a set of symptoms. Even though we do not present this study, the difference between the two groups of symptoms is intuitively clear. Surprisingly, we do not see a clear trend that users under Figure 1.4 have searched more, which weakens the hypothesis that objective medical state is linked to search activity.\n\nIn summary, we find that both high-level {\\it QueryFormulation} features as well as individual word choices reveal information about the searcher, which is not necessarily expected. While some {\\it QueryFormulation} features are interpretable, more work is necessary to understand their precise meaning. Also, more data is needed to better study their effect on rarer search events such as HUI queries. \n\n\\section{EFFECT ON SERP}\n\n\n\nWhen personalization is employed by search engines a ``filter bubble\" can be created whereby only supporting information is retrieved \\cite{pariser}. As highlighted in the earlier example, this can be problematic in the case of health searching. In this section, we investigate in what ways the user influences the content of the SERP and the medical concern expressed therein. This question can be separated into two aspects: How do SERPs vary for a given query and how do SERPs vary between queries. \n\n\\subsection{Same Query, Different SERPs}\nTo determine the impact of the user on SERPs within each query, we first investigated how diverse those SERPs were to begin with. We analyzed the composition of SERPs of the 29 most frequent queries from our data set. (Each of these occurred at least 500 times.) We found that on average, 92\\% of SERPs contain the same top result. For example, the page ``www.thefreedictionary.com\/headache'' appears as the top result for the query ``headache definition'' for almost every user. The three most common top results together covered over 99\\% of SERPs. If we consider the Top 8 results on the SERP, we find that eight specific pages are enough to account for 61\\% of all results. Hence, most SERPs returned for a given query are highly similar. We do see considerably more diversity when we consider the ordering of pages on the SERP. Hence, we conclude that factors such as time of day, user location and user personalization may shuffle the ordering of pages, especially in the lower half of the SERP, but do not have the power to promote completely different pages to the top the majority of the time (one notable exception to this is personal navigation \\cite{teevan}, which agressively promotes pages that an individual visits multiple times, but the coverage of this approach is small). Since user outcomes are driven chiefly by top results, we thus expect the impact of non-query factors on user outcomes to be limited.\n\n\n\nNonetheless, we investigated the impact of prior search activity on the SERP by measuring the association between {\\it BeforeSearching} features and {\\it SERPConcern} while controlling for query choice. We measure significance as described in section 3.3. We split on {\\it SERPConcern} and compared the empirical means of the {\\it BeforeSearching} features. Figure 3.1 shows the percentage difference between the mean of the union of the upper buckets and the mean of the union of the lower buckets, aggregated over a 24 hour window before the landmark query (the largest window where significance was obtained). The p-value is shown above the bars. We only show {\\it BeforeSearching} features that were significant. \n\nWe see that users who received concerning SERPs relative to other users entering the same query searched for 3\\% more serious conditions and 2\\% more for the most frequent serious conditions in the 24 hours before the landmark query than than those receiving less concerning SERPs. So, there is an impact of prior searching on the SERP, albeit, as expected, a small one. The difference between the buckets becomes much larger when we consider only users who receive the 10\\% most and least concerning SERPs within each query. Figure 3.2 shows that users receiving especially concerning SERPs search for 12\\% more serious conditions, search 18\\% more for their most frequent serious condition and conduct 10\\% more medical searches overall in the 24 hours before the landmark query. If it was the case that prior searching indicates heightened concern, then search engines present significantly more concerning search results to already concerned users, which may be undesirable.\n\n\\subsection{Different Queries, Different SERPs}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=1.8]{F3p1.pdf}\n\\includegraphics[scale=1.8]{F3p2.pdf}\n\\caption{Association between user search features and features related to the landmark query. Analysis was conducted as described in section 3.3. In each case, we show the ratio between the search behavior feature value across all upper buckets and across all lower buckets. Hence, positive values imply a positive association between features and vice versa. The p-value is noted next to the bar. We do not show results that were not significant, except in Figure 3.7, were only {\\it futurehuiclicked} was significant. The effective sample size refers to the total number of individuals \/ datapoints included in the analysis. This varies for three reasons. (1) Some features are unavailable for some users. For example, in Figure 3.6, we cannot include users that did not click on the SERP. (2) In some Figures, we only include a specific subset of users. For example, in Figure 3.2, we only include users receiving the 10\\% most \/ least concerning SERPs for each query. (3) We have to subsample for statistical reasons (see section 3.3).}\n\\end{figure*}\n\nNow, we turn our attention to how SERPs vary across queries. As a starting point, we tested the association between {\\it BeforeSearching} and {\\it SERPConcern} (without controlling for query choice). Results are shown in Figure 3.3. {\\it BeforeSearching} values shown were aggregated over a one hour window. This window size yielded the most significant results, but significance was preserved over all window sizes up to 90 days. Interestingly, users receiving concerning SERPs are now significantly less likely to engage in medical searching before the landmark query. They search for 14\\% less benign conditions and 17\\% less for their most frequent benign condition. Since these values (and p-values) are much larger than in Figure 3.1, we conclude that the choice of query is the cause for the majority of this effect. At face value, this effect may be either desirable or counterproductive depending on whether {\\it BeforeSearching} is an indicator of subjective concern or objective health state.\n\nIn Figure 2.1, we break down {\\it SERPConcern} by high-level {\\it QueryFormulation} features. Error bars are much smaller than those for {\\it pastmedical} because the SERP scoring function is not sparse. We see that including an additional symptom in the landmark query (feature: {\\it othersymptom}) significantly lowers the level of concern in the SERP. This might be because most of these symptoms are not indicative of a serious brain condition (e.g. cough, stomach pain, hot flashes etc.), thus providing evidence to the search engine that such a condition might not be the underlying reason of the user's health state. We note that this association is the opposite of the association between {\\it othersymptom} and {\\it pastmedical}. We hypothesize that this effect might be responsible for the negative association in Figure 3.3. Indeed, if we exclude searches that include additional symptoms, all negative associations in Figure 3.3 disappear and we obtain significant positive associations between {\\it SERPConcern} and prior serious condition searching. \n\nIn contrast, searching specifically for conditions that explain headache (feature: {\\it goal:condition}) yields the most concerning pages. This is logical given what makes a page most concerning is content about (serious) conditions. Again, it is difficult to interpret the meaning of most of the {\\it QueryFormulation} features intuitively, but each feature is highly discriminative with respect to {\\it SerpConcern} and is thus warranting further study.\n\nFigure 2.2-2.5 breaks down SERPConcern by low-level {\\it QueryFormulation} feature. It appears that search engines do a decent job at ranking different symptoms based on how medically serious they are. {\\it SERPConcern} is overall much higher amongst relatively concerning symptoms versus relatively benign symptoms, showing that the search engine does respond to this factor. Furthermore, we see a form of consistency in the search engine in that queries with additional symptoms consistently receive below average concern scores. We compared this against different key phrases describing the feature {\\it location} (see Table 3) such as ``above eye'', ``temporal'' and ``base of skull''. If we model the {\\it SERPConcern} value associated with each additional symptom as well as each location descriptor as normally distributed, then the difference in mean between the two normals is highly significant ($p < 10^{-6}$).\n\nNonetheless, we still see that there is a considerable amount of unexplained variation in {\\it SERPConcern} values. For example, a user entering ``bad headache'' will receive a less concerning SERP than a user entering ``terrible headache''. We hypothesize that it is unlikely that this difference is always due to hidden semantics, but may often be attributable to random noise. This suggests that there is still significant scope for search engines to better reflect the medical content of queries and become more consistent. One caveat is that some of this noise might be caused by our classifier that is used to assign {\\it SERPConcern} values. The presence of this noise implies that different user populations which have a preference for certain query formulations may be inadvertently led down completely different page trails to different health outcomes.\n\n\\subsection{Summary}\n\nIn summary, we find that search engines do seem to show the ability to return less concerning SERPs for benign symptoms in queries as opposed to more serious symptoms. However, there is significant work to be done to achieve a state where search results reflect the medical information given in the query while being robust to irrelevant nuances. Also, we find that past user searching does have a direct impact in making SERPs more concerning, which might not be desirable. Further analysis might discover the exact cause.\n\n\\section{RESPONSE OF USER TO SERP}\n\nIn this section, we investigate how users respond to the SERP returned in response to the landmark query. We phrase this as two sub-problems: How do users with different levels of prior searching interpret the SERP by making click choices and what impact does the SERP have in altering the behavior of the user. \n\n\\subsection{Impact on Click Behavior}\n\nFirst, we looked at the impact of {\\it BeforeSearching} on click decisions. For this, we measured the association between {\\it BeforeSearching} and {\\it ClickFeatures}. However, to capture the impact of user predisposition, we can only compare users against each other who not only entered the same query, but also received identical SERPs, to control for those two confounders. Unfortunately, this means we cannot include users in our study that received a SERP none else received, which significantly reduces the size of our effective dataset to lie between 20,000 and 50,000 and users. We split each subpopulation based on high \/ low values of {\\it hasclicked}, {\\it clickconcern} and {\\it clickposition} and compared the means of {\\it BeforeSearching} features. Results are shown in Figures 3.4 (window: 30 days), 3.5 (window: 30 days) and 3.6 (window: 24 hours) respectively. \n\nWe found that users who select more concerning pages on any given SERP are significantly more likely to have conducted medical searches in the 30 days before the landmark query. They have searched for 19\\% more serious conditions over this time period. This is quite large given the size of the aggregation window and obfuscating factors such as the limited amount of information available about a page in a SERP caption, the ad hoc nature of a click decision in general and the overall preference for pages ranked near the top. This suggests that people are selectively seeking information and concerned users reinforce their opinion by focusing on concerning content on the SERP. Selective exposure to information has been studied in detail in the psychology community \\cite{frey}. The fact that the significance of these results is highest for a large aggregation window illustrates a user's page preference is formed over the medium term, suggesting it is an attitude rather than a momentary state. Furthermore, this result points to past medical search activity as a good proxy for level of health concern, making our previous analyses more meaningful.\n\nWe found that users with more prior health searching are more likely to click lower positions on the SERP and are less likely to click overall. This might be because users who have searched about similar topics before are likely to look for specific kinds of information and are thus more likely to reject pages as unsuitable, leading them further down the SERP and ultimately to abandon more of their queries. Connections between topic familiarity and search behavior have been noted in previous work \\cite{kelly}. Interestingly, {\\it huiclicked} breaks that trend and is positively associated with clicking on the SERP. This might be because {\\it huiclicked} is by definition associated with a user's general disposition to click on a SERP, which in turn affects the probability of the user clicking in response to the landmark query. \n\n\\subsection{Impact on Future Behavior}\n\nNow we turn to measuring the impact that the SERP has on the state of the user as measured by {\\it AfterSearching}. Previous research (e.g., survey responses in \\cite{whitehorvitzcyber,foxduggan}) showed that online content can have a direct impact on people's healthcare utilization decisions. It also has been shown that the content of pages viewed can be used to predict future serious condition searches \\cite{whitehorvitzesc}. We believe that finding evidence in logs that different SERPs cause users to respond differently would strengthen the motivation for improving search engine performance during health searches. \n\nBefore proceeding, we must point out that this task is quite difficult. We have already shown that users have a myriad of significant predisposition which shape the SERP. Hence, it is impossible to tell whether any change in user behavior after the landmark query was really caused by the SERP or is the result of a predisposition.\n\nOne way to mitigate this is, again, control for query choice. Unfortunately, the group of users receiving concerning SERPs within each query is very different from the group of users receiving concerning SERPs overall, as within-query variation of {\\it SERPConcern} is much smaller than between-query variation (see Section 5). Hence, the true impact of the SERP on users is likely much larger than is measurable in this experiment.\n\nSplitting on {\\it SerpConcern} and comparing the means of {\\it AfterSearching} features yielded no significant differences over any aggregation window. However, looking at the top 10\\% vs. bottom 10\\% of SERPs within each query yielded surprising results. They are shown in Figure 3.7 (window: 1 hour). Users receiving especially concerning SERPs perform 20\\% less HUI queries in the hour following the landmark query when compared to their peers who entered the same query but received an especially unconcerning SERP. Additionally, these users had a larger empirical probability of expressing HUI in the preceding hour (non-significant), amplifying the drop. The ratio of HUI queries between users receiving concerning vs. non-concerning SERPs changes from +20\\% to -20\\% from past to future. To solidify this result, we reran our experiment under exclusion of users who had performed any medical searching over the last 1 hour, 24 hours, etc. We found that the negative association of {\\it futurehui} and {\\it futurehuiclicked} with {\\it SERPConcern} remained significant even when excluding users who had performed no medical searching in the preceding 30 days. (Note that the more users we exclude, the less data we have and the more difficult it is to achieve significance.) This is a surprising result that warrants further investigation.\n\nEven though there is no causal argument to be made, we were still interested in the outright association of {\\it SERPConcern} and {\\it AfterSearching}. We made the interesting observation that the association of {\\it topserious} and {\\it diffserous} actually decreased from past to future. For example, users receiving concerning SERPs issued 2\\% more searches for their most frequent serious condition during the hour before the landmark query, but 1\\% less afterwards (both non-significant). On the contrary, users receiving concerning SERPs searched 17\\% less for their most frequent benign conditions in the hour before the landmark query and only 8\\% less in the hour after. These results are difficult to interpret. However, they do provide some evidence that the impact of the SERP on the user's concern might be more subtle than expected.\n\n\\subsection{Summary}\nIn summary, we found that concerned users are significantly more likely to click concerning pages which may be a serious problem. Additionally, we found only weak evidence that concerning SERPs cause an increase in user's health concerns. To the contrary, users receiving concerning SERPs may be less likely to seek HUI in the short term. Both of these effects need to be investigated further.\n\n\\section{DISCUSSION AND IMPLICATIONS}\n\nIn this section, we summarize our findings and discuss implications, limitations and opportunities for future research. Our main findings are as follows:\n\n\\begin{itemize*}\n\\item Innocuous details in query formulation can hold characterizing information about the search engine user.\n\\item While the search engine does respond to general trends in query formulation sensibly, there is a lot of variation in SERPs for different but similar word choices. \n\\item While users with a history of medical searching are significantly more likely to pick out especially concerning content, the search engine also serves those users more concerning SERPs to begin with.\n\\item There was only weak evidence for the effect of intensification of health concerns through concerning SERPs, but some evidence that concerning SERPs might reduce HUI queries and hence real-world health seeking in the short term.\n\\end{itemize*}\n\nConsidering this, we believe that the most important implication for improving medical search results from this study is clearly highlighting the need for a further rigorous quantification of the extent to which concern in SERPs influences users' levels of health concern. Utlimately, through methodologies to study consenting user cohorts in detail, we can move beyond our focus on health concerns to target health anxieties directly.\n\nWe also believe there is significant  potential in refining our understanding of the meaning of both landmark query formulation and general search behavior for extracting hidden information about the user. In this paper, we have shown how these might be used to better infer levels of health concern. In this way, we might be able to improve search engine performance, for example by promoting more medically trusted pages that discuss a wide variety of possible causes objectively if we suspect health anxiety in the searcher. However, a lot more can be done. For example, word choices might reveal the age of the user \\cite{torres} or their level of domain expertise \\cite{hembrooke}, which in turn would have implications for the medical meaning of symptoms. We also have not yet considered formulation nuances of queries other than the landmark query, which might hold much richer information. The need for such analysis to adjust for anxiety is exemplified by our finding that users who conduct more medical searching have a preference for concerning content, which might indicate a cycle of self-reinforcing concern.\n\nOther questions for future research include: In how far is frequent medical searching indicative of the user's actual state of health vs. subjective concern? How can we effectively integrate the results of a possible algorithm that learns searchers' true medical situation into a standard retrieval model without ``overmanaging'' or hurting the overall robustness of the engine? How is it possible to find the right balance between preventing health anxiety and the potential delay of important medical treatment when users are confronted with only benign explanations when they are actually sick?\n\nAlthough we studied the behavior many searchers, one  limitation is that we used a single search engine, meaning our findings pertaining to user behavior might not generalize. Another limitation is the imperfect measurement of {\\it SERPConcern}. Due to the fact that some queries occur very frequently (e.g. ``headache''), any error incurred by our scoring function on the top results for those queries might have a big impact on the outcome. As mentioned earlier, another limitation is that we only used data on queries that were related to headache. Finally, query logs offer only a limited view on health concerns, and we need to work with searchers directly to understand the motivations behind their search behaviors.\n\n\\section{CONCLUSIONS}\n\nWe presented a log-based longitudinal study of medical search behavior on the web. We investigated the use of medical query formulation as a lens onto searchers health concerns and found that those features were predictive for this task. We evaluated how a major search engine responds to changes in medical query formulation and saw that there were some trends and a lot of variance, which might have adverse affects by way of misinformation. We showed that a significant tendency for medically concerned users to view concerning content makes it important for engines to manage this effect (e.g., by considering estimated level of health concern as a personalization feature). Finally, we raised the need for detailed study of the impact of SERP composition on users' future behavior. We believe that our results can function as an initial guide for developing practical tools for better health search as well as to inform deeper investigations into health concerns and anxieties on the Web and in general.\n\n\n\n\n\\bibliographystyle{abbrv}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{sloppypar}\nMarkov chain Monte Carlo (MCMC) methods enable importance sampling from complicated, concentrated probability distributions.\nThe Metropolis-Hastings (MH) algorithm~\\cite{Metropolis1953Equation, Hastings1970Monte} is the prototype for a class of MCMC algorithms where transition proposals are accepted or rejected to generate each sample.\nIts applicability to distributions where probability ratios, but not absolute probabilities, are easily computed motivates its broad usage.\nSince the complexity and scale of these applications routinely push against computational resource limits, practical techniques for minimizing storage requirements and execution time are important aspects of implementations.\nIdeally, the full sequence of samples is recorded in a compact format that facilitates analysis and interpretation.\n\\end{sloppypar}\n\nCurrent implementations of Metropolis-Hastings-class algorithms typically reduce the information content of the full sample sequence in two ways before it is recorded.\nIn the first, a small fraction of samples are recorded in complete detail, but the rest are discarded.\nThis enables arbitrary post-analysis because all degrees of freedom are available for the retained samples.\nHowever, when sampling systems with many degrees of freedom, retaining even a small fraction of samples requires considerable storage capacity.\nEven if capacity is abundant, disk throughput would limit the fraction of samples that could be recorded without having disk write operations dominate the execution time.\nIn the second pattern, averages, moments, histograms, or other quantities are accumulated during simulation and recorded upon completion.\nThis is parsimonious with respect to demands on storage bandwidth and capacity, but decisions about which quantities to accumulate, their frequencies of accumulation, and the number of samples to discard due to starting configuration bias must be made in advance.\nAltering these choices requires repeating the entire simulation.\n\nThese tradeoffs and losses of information complicate the usage of MH-class algorithms in real applications.\nA typical experience from our lab provides an illustrative example in the area of biomolecular simulation.\nA study of conformational and dimerization equilibria~\\cite{Williamson2010Modulation} involved a set of $\\sim$$10$ proteins, each consisting of $\\sim$$10^3$ interacting atoms and modelled at $\\sim$$10$ different temperatures either individually or in pairs.\nAt least three independent replicate simulations were performed for each condition, with each replicate generating $\\sim$$10^8$ samples using MH.\nWhile some quantities of interest had their expectation values accumulated during the initial simulations, only 1 in $\\sim$$10^4$ samples had their full set of atomic coordinates recorded.\nDespite using Gromacs XTC compression with limited precision, these sparse recordings consumed a total of $\\sim$$10^{11}$ bytes of storage.\nSubsequent analyses could only be performed by reanalyzing these recorded samples or repeating the entire set of simulations while accumulating expectations for the new quantities of interest.\nGiven that the simulations consumed $\\sim$$10^5$ total hours of cpu time, both options were suboptimal in terms of efficient usage of computational resources.\n\nThese problems can be completely avoided.\nHere, we describe a simple method for recording the full sample sequence from a MH-class simulation that stores one bit per sample.\nThe method operates independently from any details of the system and the transition proposals, and is therefore generally applicable.\n\n\\section{Description of the method}\n\nRecording a bit string of 1's for acceptance and 0's for rejection, with one bit per transition, suffices to preserve the complete information content of all samples generated during one run of a MH-class algorithm.\nThis method follows naturally from an information theoretic perspective: since all influence from the underlying distribution is reduced to a binary accept-or-reject decision, MH-class algorithms can be viewed as communication channels with capacity of one bit per sample~\\cite[chapter 30.5]{MacKay2002Information}.\nAn existing implementation of a MH-class algorithm can be easily modified to perform this recording operation during each iteration.\nBuffered output is necessary because file systems do not allow writing of individual bits, and also helps to reduce the frequency of disk writes.\n\nRegenerating the full sequence of samples for subsequent reanalysis requires the recorded bit string, the starting state, and any pseudorandom number generator seed(s) from the original simulation.\nThe original simulation code should be reused to guarantee that the original sequence of transition proposals is recapitulated, but modified to use the recorded bit string for deciding whether to accept or reject proposals.\nSince the acceptance criterion no longer needs to be evaluated, all calculations involved in computing ratios of sample weights or proposal probabilities can be skipped during reanalysis.\nHowever, as with the original simulation run, all degrees of freedom for every sample are available for computing and accumulating distributions of any quantity of interest.\n\nCare must be taken to avoid corrupting the sequence of samples during reanalysis.\nThe stream of pseudorandom numbers generated during reanalysis must be identical to the original simulation's stream.\nIn particular, if a single pseudorandom number stream is used for generating transition proposals as well as evaluating the acceptance criterion, the pseudorandom number that would have been used for the acceptance criterion must be generated and discarded during each iteration.\nIf analysis routines themselves make use of pseudorandom numbers, they must generate their own independent streams.\nInsidious platform dependencies are another source of potential corruption when recordings generated on one computer are reanalyzed on another.\nFor example, the implementation of floating point arithmetic differs between processor architectures and compilers.\nThis source of error should be eliminated by adhering to best practices in the coding of floating point operations~\\cite{Goldberg1991What} or writing unit tests that assert platform-specific assumptions are valid during both original simulation and reanalysis runs.\n\n\\section{Demonstration of the method}\n\nTo create a nontrivial demonstration of this recording method, we implemented a MH simulation of a colloidal suspension of charged spherical particles.\nThis three-dimensional off-lattice system consists of $10^3$ particles confined within a spherical droplet.\nEach particle has two charge sites that freely diffuse on the particle's surface.\nThe potential energy is the sum of a pairwise Lennard-Jones potential between particle centers and a pairwise Debye-H\\\"{u}ckel screened electrostatic potential between charge sites.\nIntra-particle and inter-particle electrostatic interactions are screened using different dielectric constants.\nTransition proposals consist of local or full randomization of one particle's center coordinates or charge site positions.\nAn executable Java Archive file containing source code and compiled class files for this demonstration is available as supplementary material.\nTo run it on any computer where Java Runtime Environment version 6 is installed, execute \\texttt{java~-jar~RecordingDemonstration.jar} at a command line.\n\nA comparison with typical alternatives highlights the efficiency of the proposed recording method.\nA straightforward format would be a recording of all seven degrees of freedom $(x, y, z, \\theta_1, \\phi_1, \\theta_2, \\phi_2)$ for all particles.\nWhile this is obviously inefficient, it is simple and easy to parse, facilitating interoperability with other software.\nStoring the updated values, if any, for only the changed degrees of freedom during a transition would be significantly more efficient.\nHowever, this method would tie the recording format to the choice of transition proposals; modifying the simulation to use more sophisticated transitions that simultaneously perturb multiple particles would require redesigning the format.\nAn even more efficient method would record the potential energy (or more generally, the weight) of each sample.\nAs with the proposed method, this would require preserving the simulation code, but would enable full reconstruction of the original samples without calculating any weights.\nTable~\\ref{table:storagetable} compares the storage efficiency of these recording schemes to the proposed one.\n\\texttt{RecordingDemonstration.jar} can derive any of the other recordings starting from the bit string recording, proving that the bit string (along with simulation code and starting state) retains all information about the sample sequence.\n\n\\begin{table}[htbp]\n\\input{storagetable}\n\\label{table:storagetable}\n\\end{table}\n\n\\begin{sloppypar}\nSince the bit string recording obviates all sample weight and proposal probability ratio calculations, iterating over the sample sequence takes significantly less time than generating it.\nTo enable benchmarking, \\texttt{RecordingDemonstration.jar} provides two analyses for the demonstration system.\nThe first is a histogram of the central angle between intra-particle charge sites and the second is a pairwise distance histogram between particle centers.\nThe histograms are accumulated once every 100 and 1000 steps, respectively.\nTable~\\ref{table:timingtable} compares the execution time for performing both analyses during the original simulation versus using the bit string recording.\nNote that energy evaluations are efficiently implemented such that only changing terms are computed; each iteration after the first of the original simulation performs a number of computations that is linear, rather than quadratic, in the number of particles.\nHowever, once the bit string is recorded, energy evaluations are skipped; each iteration only needs to update the degrees of freedom for a single particle and therefore executes in constant time.\n\\end{sloppypar}\n\n\\begin{table}[htbp]\n\\input{timingtable}\n\\label{table:timingtable}\n\\end{table}\n\n\\section{Discussion}\n\nThis method enables efficient reanalysis as long as the time spent generating and executing transition proposals is small compared to the time spent computing ratios of sample weights and proposal probabilities.\nFortunately, this condition is naturally satisfied because sample weights are generally determined by an energy function or other interaction between degrees of freedom that is more expensive to update than the degrees of freedom themselves.\nBy recording the results of the most computationally expensive component of MH-class algorithm implementations, the method proposed here approaches the minimum achievable limits on both storage consumption and execution time.\n\nA closely related alternative method would be to record the full sequence of sample weights in addition to the accept \/ reject decision.\nIn some applications, the sample weights themselves are important subjects of analysis, and rederiving them from the samples during reanalysis would be computationally expensive.\nFor instance, in a simulation of a physical system in the canonical ensemble, one may wish to calculate the heat capacity and other properties of the energy distribution.\nAssuming that weights are represented as IEEE 754 double precision floating point numbers, as in Table~\\ref{table:storagetable}, each sample would require an additional 64 bits of storage.\nWhile much less compact than the bit string recording alone, it would still be far more efficient than most alternatives.\n\nNote that the size of the recording might be further reduced by compressing it using a lossless algorithm.\nOne bit per sample is an \\emph{upper} bound on the entropy rate~\\cite[chapter 30.5]{MacKay2002Information}; if the acceptance rate of the simulation is not exactly 1\/2, the recorded bit string will contain biases that a compression algorithm can exploit.\n\nThe compactness of this recording method makes it susceptible to corruption: during reanalysis, a single incorrectly recorded bit contaminates all subsequent samples.\nAn effective solution would be to encode the raw bit string using error correcting codes (ECC) that provide robustness at the cost of increased storage consumption~\\cite[chapter 11]{MacKay2002Information}.\nIn fact, employment of Reed-Solomon~\\cite{Reed1960Polynomial} and low-density parity-check (LDPC)~\\cite{Gallager1962Lowdensity} codes at the hardware level is already widespread among designers of random access memory, persistent storage devices, and networking interfaces.\nTherefore, ECC at the software level would constitute a redundant layer of protection against error.\nNon-redundant detection of error can be achieved by recording all degrees of freedom for a small number of checkpoint samples, and verifying equality for the corresponding samples generated during reanalysis.\nAlternatively, the common practice of comparing cryptographic hash function outputs can be applied to verifying the integrity of recorded bit strings.\n\nThis method is compatible with descendants of the MH algorithm that retain its general accept-or-reject architecture.\nExamples include expanded ensemble techniques~\\cite{Lyubartsev1992New}, replica exchange Monte Carlo~\\cite{Mitsutake2001Generalizedensemble}, multiple-try Metropolis~\\cite{Liu2000MultipleTry}, simulated annealing~\\cite{Kirkpatrick1983Optimization}, and simulated tempering~\\cite{Marinari1992Simulated}.\nFor some algorithms, such as Wang-Landau sampling~\\cite{Wang2001Efficient}, efficient post-analysis would require recording the system energy at every iteration as described above.\n\n\\section{Conclusion}\n\nThe method described here is a simple and effective approach to data storage and representation in the design of MH-class algorithm implementations.\nIt facilitates decoupling of simulation design from the constraints of data analysis.\nThe information theoretic perspective through which this method was conceived deserves broader appreciation in the development of Monte Carlo algorithms.\n\n\\section{Acknowledgements}\n\nThis work was supported by National Science Foundation MCB 0718924 and National Institutes of Health \u2013 National Institute of General Medical Sciences 5T32GM008802.\n\n\n\n\n\n\n\\bibliographystyle{model1-num-names}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nObservations of high-energy radiation from a variety of astrophysical\nsources imply the presence of significant populations of nonthermal\n(often relativistic) particles. Much of the observational data is\ncharacterized by strong variability on very short timescales. Nonthermal\ndistributions are naturally produced via the Fermi process, in which\nparticles interact with scattering centers moving systematically and\/or\nrandomly. The first-order Fermi mechanism treats particle acceleration\nin converging flows, such as shocks, and is thought to be important at\nthe Earth's bow shock, in certain classes of solar flares, for cosmic-ray\nacceleration by supernova remnants, and in sources with relativistic\noutflows, including blazars and $\\gamma$-ray bursts \\citep[for reviews,\nsee][]{be87,kir94}. The second-order process, as it was originally\nconceived by Fermi, involved the stochastic acceleration of particles\nscattering with randomly moving magnetized clouds \\citep{fer49}. Later\nrefinements of this idea replaced the magnetized clouds with\nmagnetohydrodynamic (MHD) waves \\citep[e.g.,][]{mel80}. The\nsecond-order, stochastic Fermi process now finds application in a wide\nrange of astrophysical settings, including solar flares\n\\citep{mgr90,lpm04a}, clusters of galaxies \\citep{sst87,pet01,bru04},\nthe Galactic center \\citep{lpm04b,ad04}, and $\\gamma$-ray bursts\n\\citep{wax95,dh01}.\n\nThe standard approach to modeling the acceleration of nonthermal\nparticles via interactions with MHD waves involves obtaining the\nsolution to a steady-state transport equation that incorporates\ntreatments of systematic and stochastic Fermi acceleration, radiative\nlosses, and particle escape \\citep[e.g.,][]{sch84,ss89,lmp06}. However,\nthe prevalence of variability on short timescales in many sources calls\ninto question the validity of the steady-state interpretation.\n\\citet{pp95} provided a comprehensive review of the various\ntime-dependent solutions that have been derived in the past 30 years. In\nmost cases, it is assumed that the momentum diffusion coefficient,\n$D(p)$, and the mean particle escape timescale, $t_{\\rm esc}(p)$, have\npower-law dependences on the particle momentum $p$. Although the set of\nexisting solutions covers a broad range of possible values for the\nassociated power-law indices, there are certain physically interesting\ncases for which no analytical solution is currently available. For\nexample, \\citet{dml96} have shown that for the stochastic acceleration\nof relativistic particles due to resonant interactions with plasma waves\nin black-hole magnetospheres, one obtains $D(p) \\propto p^q$ and $t_{\\rm\nesc}(p) \\propto p^{q-2}$, where $q$ is the index of the wavenumber\ndistribution (see \\S~2). The analytical solution for the time-dependent\nGreen's function in this situation is of particular physical interest,\nbut it has not appeared in the previous literature. This has motivated\nus to reexamine the associated Fokker-Planck transport equation for this\ncase, and to obtain a new family of closed-form solutions for the\nsecular Green's function. The resulting expression, describing the\nevolution of an initially monoenergetic particle distribution,\ncomplements the set of solutions discussed by \\citet{pp95}. Our primary\ngoal in this paper is to present a detailed derivation of the exact\nanalytical solution and to demonstrate some of its key properties.\nDetailed applications of our results to the modeling of particle\nacceleration in black-hole accretion coronae and other astrophysical\nenvironments will be presented in a separate paper.\n\nThe remainder of the paper is organized as follows. In \\S~2 we review\nthe fundamental equations governing the acceleration of charged\nparticles interacting with plasma waves. The transport equation is\nsolved in \\S~3 to obtain the time-dependent Green's function, and\nillustrative results are presented in \\S~4. The astrophysical implications\nof our work are summarized in \\S~5, and additional mathematical details are\nprovided in the Appendix.\n\n\\section{FUNDAMENTAL EQUATIONS}\n\nCharged particles in turbulent astrophysical plasmas are expected to be\naccelerated via interactions with whistler, fast-mode, and Alfv\\'en\nwaves propagating in the magnetized gas. Here we consider a simplified\nisotropic description of the wave energy distribution, denoted by\n$W(k)$, where $W(k) dk$ represents the energy density due to waves with\nwavenumber between $k$ and $k+dk$. The transport formalism we consider\nassumes a power-law distribution for the wave-turbulence spectrum, which\nimplies definite relations between the momentum diffusion coefficient\nand the momentum-dependent escape timescale\n\\citep[e.g.,][]{mel74,sch89a,sch89b,dml96}. MHD waves injected over a\nnarrow range of wavenumber cascade to larger wavenumbers, forming a\nKolmogorov or Kraichnan power spectrum over the inertial range with\n$W(k) \\propto k^{-q}$, where $q=5\/3$ and $q=3\/2$ for the Kolmogorov and\nKraichnan cases, respectively \\citep[e.g.,][]{zm90}. The specific forms\nwe will adopt for the transport coefficients in \\S~2.2 are based on the\nphysics of the resonant scattering processes governing the wave-particle\ninteractions \\citep[see][]{dml96}. We assume that the nonthermal\nparticle distribution is isotropic, and focus on a detailed treatment of\nthe propagation of particles in momentum space due to wave-particle\ninteractions. The spatial aspects of the transport (i.e., the\nconfinement of the particles in the acceleration region) will be treated\nin an approximate manner using a momentum-dependent escape term.\n\n\\subsection{Transport Equation}\n\nThe fundamental transport equation describing the propagation of\nparticles in the momentum space can be written in the flux-conservation\nform \\citep[e.g.,][]{bec92,sch89a,sch89b}\n\\begin{equation}\n{\\partial f \\over \\partial t}\n= - {1 \\over p^2} {\\partial \\over \\partial p}\\left\\{\np^2 \\left[\nA(p) \\, f - D(p) {\\partial f \\over \\partial p}\\right]\n\\right\\}\n- {f \\over t_{\\rm esc}(p)}\n+ {S(p,t) \\over 4 \\pi p^2}\n\\ ,\n\\label{eq1}\n\\end{equation}\nwhere $p$ is the particle momentum, $f(p,t)$ is the particle\ndistribution function, $D(p)$ denotes the momentum diffusion\ncoefficient, $t_{\\rm esc}(p)$ is the mean escape time, $S(p,t)$\nrepresents particle sources, and $A(p)$ describes any additional,\nsystematic acceleration or loss processes, such as shock acceleration or\nsynchrotron\/inverse-Compton emission. The quantity in square brackets in\nequation~(\\ref{eq1}) describes the flux of particles through the\nmomentum space \\citep{tnj71}, and the source term is defined so that\n$S(p,t) \\, dp \\, dt$ gives the number of particles injected into the\nplasma per unit volume between times $t$ and $t + dt$ with momenta\nbetween $p$ and $p + dp$. The total particle number density $n(t)$ and\nenergy density $U(t)$ of the distribution $f(p,t)$ are computed using\n\\begin{equation}\nn(t) = \\int_0^\\infty 4 \\pi \\, p^2 \\, f(p,t) \\, dp\n\\ , \\ \\ \\ \\ \\ \nU(t) = \\int_0^\\infty 4 \\pi \\, \\epsilon \\, p^2 \\, f(p,t)\n\\, dp \\ ,\n\\label{eq2}\n\\end{equation}\nwhere the particle kinetic energy $\\epsilon$ is related to the Lorentz factor\n$\\gamma$ and the particle momentum $p$ by\n\\begin{equation}\n\\epsilon = (\\gamma-1) \\, m c^2 \\ , \\ \\ \\ \\ \\ \n\\gamma=\\left({p^2 \\over m^2 c^2} + 1\\right)^{1\/2} \\ ,\n\\label{eq3}\n\\end{equation}\nand $m$ and $c$ denote the particle rest mass and the speed of light,\nrespectively.\n\nRather than solve equation~(\\ref{eq1}) directly to determine $f(p,t)$\nfor a given source term $S(p,t)$, it is more instructive to first solve\nfor the Green's function, $f_{_{\\rm G}}(p_0,p,t_0,t)$, which satisfies the\nequation\n\\begin{equation}\n{\\partial f_{_{\\rm G}} \\over \\partial t}\n= - {1 \\over p^2} {\\partial \\over \\partial p}\\left\\{\np^2 \\left[A(p) \\, f_{_{\\rm G}}\n- D(p) {\\partial f_{_{\\rm G}} \\over \\partial p}\n\\right]\\right\\}\n- {f_{_{\\rm G}} \\over t_{\\rm esc}(p)}\n+ {\\delta(p-p_0) \\, \\delta(t-t_0) \\over 4 \\pi p_0^2}\n\\ ,\n\\label{eq4}\n\\end{equation}\nwhere $p_0$ is the initial momentum and $t_0$ is the initial time.\nThe source term in this equation corresponds to the injection of a single\nparticle per unit volume with momentum $p_0$ at time $t_0$. The particle\nnumber and energy densities associated with the Green's function are\ngiven by\n\\begin{equation}\nn_{_{\\rm G}}(t) = \\int_0^\\infty 4 \\pi \\, p^2 \\,\nf_{_{\\rm G}}(p_0,p,t_0,t) \\, dp\n\\ , \\ \\ \\ \\ \\ \nU_{_{\\rm G}}(t) = \\int_0^\\infty 4 \\pi \\, \\epsilon \\, p^2 \\,\nf_{_{\\rm G}}(p_0,p,t_0,t) \\, dp\n\\ .\n\\label{eq5}\n\\end{equation}\nOnce the Green's function solution has been determined, the {\\it\nparticular solution} associated with an arbitrary source distribution\n$S(p,t)$ can be computed using the integral convolution\n\\citep[e.g.,][]{bec03}\n\\begin{equation}\nf(p,t) = \\int_0^t \\int_0^\\infty\nf_{_{\\rm G}}(p_0,p,t_0,t) \\,\nS(p_0,t_0) \\, dp_0 \\, dt_0\n\\ ,\n\\label{eq6}\n\\end{equation}\nwhere we have assumed that the particle injection begins at time $t=0$\nand no particles are present in the plasma for $t < 0$.\n\n\\subsection{Transport Coefficients}\n\nIn Appendix~A.1, we demonstrate that for\narbitrary particle energies, the mean rate of change of the\nparticle momentum due to stochastic acceleration is related to the\nmomentum diffusion coefficient $D(p)$ via\n\\begin{equation}\n\\Big\\langle {dp \\over dt} \\Big\\rangle \\Big|_{\\rm stoch} = {1 \\over p^2}\n{d \\over dp}\\left(p^2 D\\right)\n\\ .\n\\label{eq8}\n\\end{equation}\nThe corresponding result for the mean rate of change of the kinetic energy\nis \\citep[see Appendix~A.2 and][]{mr89}\n\\begin{equation}\n\\Big\\langle {d\\epsilon \\over dt} \\Big\\rangle \\Big|_{\\rm stoch}\n= {1 \\over p^2} {d \\over dp} \\left(p^2 v D \\right) \\ ,\n\\label{eq8b}\n\\end{equation}\nwhere $v$ is the particle speed. If the MHD wave spectrum has the\npower-law form $W \\propto k^{-q}$ associated with Alfv\\'en and fast-mode\nwaves, then the momentum dependences of the diffusion coefficient $D(p)$\nand the mean escape time $t_{\\rm esc}(p)$ describing the resonant\npitch-angle scattering of relativistic particles are given by\n\\citep[e.g.,][]{dml96,mr89}\n\\begin{equation}\nD(p) = D_* \\, m^2 c^2 \\left({p \\over m c}\n\\right)^q \\ , \\ \\ \\ \\ \\ \\ \\\nt_{\\rm esc}(p) = t_* \\left({p \\over m c}\n\\right)^{q-2} \\ ,\n\\label{eq7}\n\\end{equation}\nwhere $D_* \\propto {\\rm s}^{-1}$ and $t_* \\propto {\\rm s}$ are\nconstants. We shall focus on transport scenarios with $q \\le 2$, so that\n$t_{\\rm esc}$ is either a decreasing or constant function of the\nmomentum $p$. In order to maintain the physical validity of the\nescape-timescale approach used here, we must require that $t_{\\rm esc}$\nexceed the light-crossing time $L\/c$ for a source with size $L$. This\nimplies a fundamental upper limit to the particle momentum when $q < 2$.\n\nBy combining equations~(\\ref{eq8}) and (\\ref{eq7}), we find that the\nmean rate of change of the momentum for relativistic particles\naccelerated stochastically by MHD waves is given by\n\\begin{equation}\n\\Big\\langle {dp \\over dt} \\Big\\rangle\\Big|_{\\rm stoch} = (q+2) \\, D_*\n\\, m c \\left({p \\over m c}\\right)^{q-1}\n\\ .\n\\label{eq9}\n\\end{equation}\nFor simplicity, we shall assume that the momentum dependence of the\nadditional, systematic loss\/acceleration processes appearing in the transport\nequation (\\ref{eq1}), described by the coefficient $A(p)$, mimics that of the\nstochastic acceleration (eq.~[\\ref{eq9}]). We therefore write\n\\begin{equation}\nA(p) = A_* \\, m c \\left({p \\over m c}\\right)^{q-1}\n\\ ,\n\\label{eq10}\n\\end{equation}\nwhere the constant $A_* \\propto {\\rm s}^{-1}$ determines the positive\n(negative) rate of systematic acceleration (losses). Note that this\nformulation precludes the treatment of loss processes with a quadratic\nenergy dependence (e.g., inverse-Compton or synchrotron) since that\nwould imply $q=3$, which is outside the range considered here. However,\nfirst-order Fermi acceleration at a shock or energy losses due to\nCoulomb collisions can be treated by setting $q=2$ with $A_*$ either\npositive or negative, respectively. This suggests that the results\nobtained here are relevant primarily for the transport of energetic\nions. However, even in this application one needs to bear in mind that\nsynchrotron and inverse-Compton losses will become dominant at\nsufficiently high energies \\citep[e.g.,][]{sch84,ss89,lmp06}.\n\nIt is convenient to transform to the dimensionless momentum and time variables\n$x$ and $y$, defined by\n\\begin{equation}\nx \\equiv {p \\over m c} \\ , \\ \\ \\ \\ \\ \\ \\ \ny \\equiv D_* \\, t\n\\ ,\n\\label{eq11}\n\\end{equation}\nin which case the transport equation~(\\ref{eq4}) for the Green's function becomes\n\\begin{equation}\n{\\partial f_{_{\\rm G}} \\over \\partial y}\n= {1 \\over x^2} {\\partial \\over \\partial x}\\left(\nx^{2+q} \\, {\\partial f_{_{\\rm G}} \\over \\partial x}\\right)\n- {a \\over x^2} {\\partial \\over \\partial x} \\left(\nx^{1+q} \\, f_{_{\\rm G}}\\right) - \\theta \\, x^{2-q} \\, f_{_{\\rm G}}\n+ {\\delta(x-x_0) \\, \\delta(y-y_0) \\over 4 \\pi m^3 c^3 x_0^2}\n\\ ,\n\\label{eq12}\n\\end{equation}\nwhere\n\\begin{equation}\nx_0 \\equiv {p_0 \\over m c} \\ , \\ \\ \\ \\ \\ \\ \\ \ny_0 \\equiv D_* \\, t_0\n\\ ,\n\\label{eq13}\n\\end{equation}\nand we have introduced the dimensionless constants\n\\begin{equation}\na \\equiv {A_* \\over D_*} \\ , \\ \\ \\ \\ \\ \\ \\ \n\\theta \\equiv {1 \\over D_* \\, t_*}\n\\ .\n\\label{eq14}\n\\end{equation}\nNote that $x$ equals the particle Lorentz factor in applications involving\nultrarelativistic particles. The constant $a$ describes the relative\nimportance of systematic gains or losses compared with the stochastic\nprocess.\n\n\\subsection{Fokker-Planck Equation}\n\nAdditional physical insight can be obtained by reorganizing equation~(\\ref{eq12})\nin the Fokker-Planck form\n\\begin{equation}\n{\\partial N_{_{\\rm G}} \\over \\partial y}\n= {\\partial^2 \\over \\partial x^2} \\left(\nx^q \\, N_{_{\\rm G}}\\right)\n- {\\partial \\over \\partial x} \\left[(q+2+a) \\, x^{q-1}\n\\, N_{_{\\rm G}}\\right] - \\theta \\, x^{2-q} \\, N_{_{\\rm G}}\n+ {\\delta(x-x_0) \\, \\delta(y-y_0)}\n\\ ,\n\\label{eq15}\n\\end{equation}\nwhere we have defined the Green's function number distribution $N_{_{\\rm G}}$\nusing\n\\begin{equation}\nN_{_{\\rm G}}(x_0,x,y_0,y) \\equiv 4 \\pi \\, m^3 c^3 \\, x^2 \\, f_{_{\\rm G}}(x_0,x,y_0,y)\n\\ .\n\\label{eq16}\n\\end{equation}\nThe Fokker-Planck coefficients appearing in equation~(\\ref{eq15}), which\ndescribe the evolution of the particle distribution due to the influence\nof stochastic and systematic processes, are given by \\citep{rei65}\n\\begin{equation}\n{1 \\over 2}{d\\sigma^2 \\over dy} = x^q \\ , \\ \\ \\ \\ \\ \n\\Big\\langle{dx\\over dy}\\Big\\rangle = (q+2+a) \\, x^{q-1} \\ ,\n\\label{eq17}\n\\end{equation}\nwhere the first coefficient describes the ``broadening'' of the distribution\ndue to momentum space diffusion, and the second represents the mean ``drift''\nof the particles (i.e., the average acceleration rate).\n\nEquation~(\\ref{eq15}) is equivalent to equation~(49) from \\citet{pp95}\nif we set their parameters $a_{\\rm pp} = 2+a$ and $s_{\\rm pp} = q-2$,\nwhere $s_{\\rm pp}$ denotes the power-law index describing the momentum\ndependence of the escape timescale. Our particular choice for $s_{\\rm\npp}$ reflects the physics of the resonant wave-particle interactions, as\nrepresented by equations~(\\ref{eq7}). The Fokker-Planck form of\nequation~(\\ref{eq15}) clearly reveals the fundamental nature of the\ntransport process. In particular, we note that in the limit $a \\to 0$,\nthe drift coefficient $\\langle dx\/dy\\rangle$ reduces to the purely\nstochastic result (eq.~[\\ref{eq9}]), as expected when systematic\ngains\/losses are excluded from the problem. The total number and energy\ndensities are computed in terms of $N_{_{\\rm G}}$ using (cf. eq.~[\\ref{eq5}])\n\\begin{equation}\nn_{_{\\rm G}}(y) = \\int_0^\\infty N_{_{\\rm G}}(x_0,x,y_0,y) \\, dx\n\\ , \\ \\ \\ \\ \\ \nU_{_{\\rm G}}(y) = \\int_0^\\infty \\epsilon \\, x \\, N_{_{\\rm G}}(x_0,x,y_0,y) \\, dx\n\\ ,\n\\label{eq18}\n\\end{equation}\nwhere (see eq.~[\\ref{eq3}])\n\\begin{equation}\n\\epsilon = m c^2 \\left[\\left(x^2 + 1\\right)^{1\/2}-1\\right]\n\\ .\n\\label{eq19}\n\\end{equation}\nSince the physical situation considered here corresponds to the injection\nof a single particle per unit volume at ``time'' $y=y_0$, it follows that\n$n_{_{\\rm G}}(y_0)=1$.\n\n\\section{SOLUTION FOR THE GREEN'S FUNCTION}\n\nThe result obtained for the Green's function number distribution\n$N_{_{\\rm G}}$ has two important special cases depending on whether $q = 2$\nor $q < 2$. \\citet{pp95} obtained the exact solution to\nequation~(\\ref{eq15}) for the hard sphere case \\citep[][]{ram79} with\n$q=2$ and their parameter $s_{\\rm pp}=0$, corresponding to a\nmomentum-independent escape timescale. In this section we derive the\nexact solution to the time-dependent problem with $q < 2$ and $s_{\\rm\npp}=q-2$, which describes the physics of the wave-particle interactions\n(see eqs.~[\\ref{eq7}]).\n\n\\subsection{Laplace Transformation}\n\nWe define the Laplace transformation of $N_{_{\\rm G}}$ using\n\\begin{equation}\nL(x_0,x,s) \\equiv \\int_0^\\infty e^{-s y} \\, N_{_{\\rm G}}(x_0,x,y_0,y) \\, dy\n\\ .\n\\label{eq20}\n\\end{equation}\nBy operating on the Fokker-Planck equation~(\\ref{eq15}) with\n$\\int_0^\\infty e^{-s y} \\, dy$, we obtain\n\\begin{equation}\n{d^2 \\over d x^2} \\left(x^q \\, L \\right)\n- {d \\over d x} \\left[(q+2+a) \\, x^{q-1}\n\\, L \\right] - \\theta \\, x^{2-q} \\, L - s L\n= - e^{-s y_0} \\, \\delta(x-x_0)\n\\ ,\n\\label{eq21}\n\\end{equation}\nor, equivalently,\n\\begin{equation}\n{d^2 L \\over d x^2} + \\left({q - 2 - a \\over x} \\right)\n{d L \\over d x}\n+ \\left[{(1-q)(2+a) \\over x^2} - {\\theta \\over x^{2 q-2}}\n- {s \\over x^q} \\right] \\, L = - {e^{-s y_0} \\, \\delta(x-x_0) \\over x^q}\n\\ .\n\\label{eq22}\n\\end{equation}\nThis equation can be transformed into standard form by introducing the\nnew momentum variables\n\\begin{equation}\nz(x) \\equiv {2 \\sqrt{\\theta} \\over 2 - q} \\ x^{2-q} \\ , \\ \\ \\ \\ \\ \nz_0(x_0) \\equiv {2 \\sqrt{\\theta} \\over 2 - q} \\ x_0^{2-q} \\ .\n\\label{eq23}\n\\end{equation}\nAfter some algebra, we find that the equation for $L$ now becomes\n\\begin{equation}\nz^2 \\, {d^2 L \\over d z^2} + {a+1 \\over q-2} \\, z \\,\n{d L \\over d z} + \\left[{(1-q)(2+a) \\over (2-q)^2}\n- {z^2 \\over 4} - {s z \\over c_0 (2-q)^2}\\right] \\, L\n= - {c_0 \\, e^{-s y_0} \\, \\delta(z-z_0) \\over 2-q} \\, \\left({z \\over c_0}\n\\right)^{(3-2q)\/(2-q)}\n\\ ,\n\\label{eq24}\n\\end{equation}\nwhere\n\\begin{equation}\nc_0 \\equiv {2 \\sqrt{\\theta} \\over 2-q} \\ .\n\\label{eq25}\n\\end{equation}\nThe solutions to equation~(\\ref{eq24}) obtained for $z \\ne z_0$ that satisfy\nthe high- and low-energy boundary conditions are given by\n\\begin{equation}\nL(z_0,z,s) = e^{-z\/2} \\, z^{(a+2)\/(2-q)} \\,\n\\cases{\nA \\, U(\\alpha,\\beta,z) \\ , & $z \\ge z_0$ \\ , \\cr\nB \\, M(\\alpha,\\beta,z) \\ , & $z \\le z_0$ \\ , \\cr\n}\n\\label{eq26}\n\\end{equation}\nwhere $M$ and $U$ denote the confluent hypergeometric functions\n\\citep{as70}, and\n\\begin{equation}\n\\alpha \\equiv {s + (a+3) \\sqrt{\\theta} \\over 2 (2-q) \\sqrt{\\theta}}\n\\ , \\ \\ \\ \\ \\ \\ \\\n\\beta \\equiv {a+3 \\over 2-q} \\ .\n\\label{eq27}\n\\end{equation}\nThe constants $A$ and $B$ appearing in equation~(\\ref{eq26}) are determined\nby ensuring that the function $L$ is continuous at $z=z_0$, and that it also\nsatisfies the derivative jump condition\n\\begin{equation}\n\\lim_{\\varepsilon \\to 0} {dL \\over dz} \\bigg|_{z_0-\\varepsilon}\n^{z_0+\\varepsilon} = - {c_0 \\, e^{-s y_0} \\over (2-q) \\, z_0^2} \\,\n\\left({z_0 \\over c_0} \\right)^{(3-2q)\/(2-q)}\n= {- e^{-s y_0} \\over 2 \\, x_0 \\sqrt{\\theta}}\n\\ ,\n\\label{eq28}\n\\end{equation}\nobtained by integrating the transport equation~(\\ref{eq24}) with respect\nto $z$ in a small range around the source momentum $z_0$.\n\nThe constant $B$ can be eliminated by combining the continuity and derivative\njump conditions. After some algebra, the solution obtained for $A$ is\n\\begin{equation}\nA = - \\, {e^{-s y_0} \\, e^{z_0\/2} \\, z_0^{(a+2)\/(q-2)} \\, M(\\alpha,\\beta,z_0)\n\\over 2 \\, x_0 \\sqrt{\\theta} \\, W(z_0)}\n\\ ,\n\\label{eq29}\n\\end{equation}\nwhere $W(z)$ denotes the Wronskian, defined by\n\\begin{equation}\nW(z) \\equiv\nM(\\alpha,\\beta,z) {d \\over dz} U(\\alpha,\\beta,z)\n- U(\\alpha,\\beta,z) {d \\over dz} M(\\alpha,\\beta,z)\n\\ .\n\\label{eq30}\n\\end{equation}\nUsing equation~(13.1.22) from \\citet{as70}, we find that $W$ is given\nby the exact expression\n\\begin{equation}\nW(z) = - \\, {\\Gamma(\\beta) \\, z^{-\\beta} \\, e^z \\over\n\\Gamma(\\alpha)}\n\\ ,\n\\label{eq31}\n\\end{equation}\nwhich can be combined with equation~(\\ref{eq29}) and the continuity\nrelation to obtain\n\\begin{equation}\nA = {\\Gamma(\\alpha) \\, z_0^{\\beta+(a+2)\/(q-2)} \\, e^{-s y_0} \\, e^{-z_0\/2}\n\\, M(\\alpha,\\beta,z_0) \\over 2 \\, x_0 \\sqrt{\\theta} \\, \\Gamma(\\beta)}\n\\ ,\n\\label{eq32}\n\\end{equation}\n\\begin{equation}\nB = {\\Gamma(\\alpha) \\, z_0^{\\beta+(a+2)\/(q-2)} \\, e^{-s y_0} \\, e^{-z_0\/2}\n\\, U(\\alpha,\\beta,z_0) \\over 2 \\, x_0 \\sqrt{\\theta} \\, \\Gamma(\\beta)}\n\\ .\n\\label{eq33}\n\\end{equation}\nThe final solution for the Laplace transformation $L$ can therefore\nbe written as\n\\begin{equation}\nL(z_0,z,s) = {\\Gamma(\\alpha) \\, z_0^\\beta \\over \\Gamma(\\beta) \\,\n2 \\, x_0 \\sqrt{\\theta}} \\, \\left({z \\over z_0} \\right)^{(a+2)\/(2-q)}\n\\, e^{-s y_0} \\, e^{-(z+z_0)\/2} \\,\nM(\\alpha,\\beta,z_{\\rm min}) \\, U(\\alpha,\\beta,z_{\\rm max})\n\\ ,\n\\label{eq34}\n\\end{equation}\nwhere\n\\begin{equation}\nz_{\\rm min} \\equiv \\min(z,z_0) \\ , \\ \\ \\ \\ \\ \\ \\\nz_{\\rm max} \\equiv \\max(z,z_0)\n\\ .\n\\label{eq35}\n\\end{equation}\n\n\\subsection{Inverse Transformation}\n\n\\begin{figure}[t]\n\\hspace{35mm}\n\\includegraphics[width=85mm]{f1.eps}\n\\caption{Integration contour $C$ used to evaluate equation~(\\ref{eq36}).}\n\\label{fig1}\n\\end{figure}\n\nThe solution for the Green's function $N_{_{\\rm G}}$ can be found using the\ncomplex Mellin inversion integral, which states that\n\\begin{equation}\nN_{_{\\rm G}}(z_0,z,y_0,y) = {1 \\over 2 \\pi i} \\int_{\\gamma-i \\infty}\n^{\\gamma + i \\infty} e^{s y} \\, L(z_0,z,s) \\, ds\n\\ ,\n\\label{eq36}\n\\end{equation}\nwhere $\\gamma$ is chosen so that the line ${\\rm Re} \\, s = \\gamma$ lies\nto the right of any singularities in the integrand. The singularities\nare simple poles located where $|\\Gamma(\\alpha)| \\to \\infty$, which\ncorresponds to $\\alpha = -n$, with $n = 0,1,2,\\ldots$\nEquation~(\\ref{eq27}) therefore implies that there are an infinite\nnumber of poles situated along the real axis at $s = s_n$, where\n\\begin{equation}\ns_n = \\left[2 \\, (q-2) \\, n - a - 3\\right] \\sqrt{\\theta}\n\\ , \\ \\ \\ n = 0,1,2,\\ldots\n\\label{eq37}\n\\end{equation}\nWe can therefore employ the residue theorem to write\n\\begin{equation}\n\\oint e^{s y} \\, L(z_0,z,s) \\, ds\n= 2 \\pi i \\sum_{n=0}^\\infty \\ {\\rm Res}(s_n)\n\\ ,\n\\label{eq38}\n\\end{equation}\nwhere $C$ is the closed integration contour indicated in Figure~1 and\n${\\rm Res}(s_n)$ denotes the residue associated with the pole at $s=s_n$.\nBased on asymptotic analysis, we conclude that the contribution to the integral\ndue to the curved portion of the contour vanishes in the limit $R \\to \\infty$,\nand consequently we can combine equations~(\\ref{eq36}) and (\\ref{eq38}) to obtain\n\\begin{equation}\nN_{_{\\rm G}}(z_0,z,y_0,y) = \\sum_{n=0}^\\infty \\ {\\rm Res}(s_n)\n\\ .\n\\label{eq39}\n\\end{equation}\nHence we need only evaluate the residues in order to determine the solution\nfor the Green's function.\n\n\\subsection{Evaluation of the Residues}\n\nThe residues associated with the simple poles at $s=s_n$ are evaluated\nusing the formula \\citep[e.g.,][]{but68}\n\\begin{equation}\n{\\rm Res}(s_n) = \\lim_{s \\to s_n}\n(s-s_n) \\ e^{s y} \\, L(z_0,z,s)\n\\ .\n\\label{eq40}\n\\end{equation}\nSince the poles are associated with the function $\\Gamma(\\alpha)$\nin equation~(\\ref{eq34}), we will need to make use of the identity\n\\begin{equation}\n\\lim_{s \\to s_n} (s-s_n) \\ \\Gamma(\\alpha)\n= {(-1)^n \\over n!} \\, 2 \\, (2-q) \\, \\sqrt{\\theta}\n\\ ,\n\\label{eq41}\n\\end{equation}\nwhich follows from equations~(\\ref{eq27}) and (\\ref{eq37}). Combining\nequations~(\\ref{eq34}), (\\ref{eq40}), and (\\ref{eq41}), we find that the\nresidues are given by\n\\begin{equation}\n{\\rm Res}(s_n) = {(-1)^n \\, (2-q) \\, e^{s_n (y-y_0)} \\, z_0^\\beta \\over\nn! \\, \\Gamma(\\beta) \\, x_0} \\, \\left({z \\over z_0}\\right)^{(a+2)\/(2-q)}\n\\, e^{-(z+z_0)\/2} \\, M(-n,\\beta,z_{\\rm min}) \\, U(-n,\\beta,z_{\\rm max})\n\\ .\n\\label{eq42}\n\\end{equation}\nBased on equations~(13.6.9) and (13.6.27) from Abramowitz \\& Stegun\n(1970), we note that the confluent hypergeometric functions appearing in\nthis expression reduce to Laguerre polynomials, and therefore our result\nfor the residue can be rewritten after some simplification as\n\\begin{equation}\n{\\rm Res}(s_n) = {n! \\, e^{s_n (y-y_0)} \\, z_0^\\beta \\, (2-q)\n\\over \\Gamma(\\beta+n) \\, x_0} \\, \\left({z \\over z_0}\\right)^{(a+2)\/(2-q)}\n\\, e^{-(z+z_0)\/2} \\, P_n^{(\\beta-1)}(z) \\, P_n^{(\\beta-1)}(z_0)\n\\ ,\n\\label{eq43}\n\\end{equation}\nwhere $P_n^{(\\beta-1)}(z)$ denotes the Laguerre polynomial.\n\n\\subsection{Closed-Form Expression for the Green's Function}\n\nThe result for the Green's function number distribution $N_{_{\\rm G}}$ is obtained\nby summing the residues (see eq.~[\\ref{eq39}]), which yields\n\\begin{equation}\nN_{_{\\rm G}}(z_0,z,y_0,y) = \\sum_{n=0}^\\infty\n\\ {n! \\, e^{s_n (y-y_0)} \\, z_0^\\beta \\, (2-q)\n\\over \\Gamma(\\beta+n) \\, x_0} \\, \\left({z \\over z_0}\\right)^{(a+2)\/(2-q)}\n\\, e^{-(z+z_0)\/2} \\, P_n^{(\\beta-1)}(z) \\, P_n^{(\\beta-1)}(z_0)\n\\ .\n\\label{eq44}\n\\end{equation}\nThis convergent sum is a useful expression for the Green's function.\nHowever, further progress can be made by employing the bilinear\ngenerating function for the Laguerre polynomials, given by\nequation~(8.976) from \\citet{gr80}. After some algebra, we find that the\ngeneral closed-form solution can be written in the form\n\\begin{equation}\nN_{_{\\rm G}}(x_0,x,y_0,y) = {2-q \\over x_0} \\left({x \\over x_0}\\right)^{(a+1)\/2}\n\\, {\\sqrt{z z_0 \\xi} \\over 1 - \\xi} \\ \\exp\\left[-{(z + z_0)(1+\\xi) \\over 2\n\\, (1-\\xi)} \\right] \\, I_{\\beta-1}\\left({2 \\sqrt{z z_0 \\xi} \\over 1 - \\xi}\\right)\n\\ ,\n\\label{eq45}\n\\end{equation}\nwhere\n\\begin{equation}\nz(x) \\equiv {2 \\sqrt{\\theta} \\over 2 - q} \\ x^{2-q} \\ , \\ \\ \\ \\ \\ \nz_0(x_0) \\equiv {2 \\sqrt{\\theta} \\over 2 - q} \\ x_0^{2-q} \\ , \\ \\ \\ \\ \\ \n\\beta \\equiv {a+3 \\over 2 - q} \\ , \\ \\ \\ \\ \\ \n\\xi(y,y_0) \\equiv e^{2(q-2)(y-y_0)\\sqrt{\\theta}}\n\\ .\n\\label{eq46}\n\\end{equation}\nNote that the solution for $N_{_{\\rm G}}$ depends on the time parameters $y$\nand $y_0$ only through the ``age'' of the injected particles, $y-y_0$, as\nindicated by the form for $\\xi$. In the limit $\\theta \\to 0$,\ncorresponding to infinite escape time, the Green's function number\ndistribution reduces to\n\\begin{equation}\nN_{_{\\rm G}}(x_0,x,y_0,y) \\Big|_{\\theta=0} \\! = {(x x_0)^{(3-q)\/2} \\over (2-q)\n\\, x_0^2 \\, (y-y_0)}\n\\left(x \\over x_0 \\right)^{a\/2} \\!\\! \\exp\\left[-{(x^{2-q} + x_0^{2-q}) \\over\n(2-q)^2 (y-y_0)}\\right]\nI_{\\beta-1}\\!\\left[{2 (x x_0)^{(2-q)\/2} \\over (2-q)^2 (y-y_0)}\\right]\n\\ .\n\\label{eq47}\n\\end{equation}\n\nThe exact solution for the time-dependent Green's function describing the\nevolution of a monoenergetic initial spectrum with $q < 2$ given\nby equation~(\\ref{eq45}) represents an interesting new contribution to\nparticle transport theory. The corresponding solution for the hard-sphere\ncase with $q = 2$, given by equation~(43) of \\citet{pp95}, can be stated\nin our notation as\n\\begin{eqnarray}\nN_{_{\\rm G}}(x_0,x,y_0,y) \\Big|_{q=2} = {e^{-\\lambda (y-y_0)} \\over 2 x_0\n\\sqrt{\\pi (y-y_0)}} \\left({x \\over x_0}\\right)^{(a+1)\/2} \\, \\exp\n\\left[{-(\\ln x - \\ln x_0)^2 \\over 4 \\, (y-y_0)}\\right]\n\\ ,\n\\label{eq48}\n\\end{eqnarray}\nwhere\n\\begin{equation}\n\\lambda \\equiv {(a+1)^2 \\over 4} + 2 + a + \\theta\n\\ .\n\\label{eq49}\n\\end{equation}\nWe note that the general solution for $N_{_{\\rm G}}$ given by\nequation~(\\ref{eq45}) agrees with equation~(\\ref{eq48}) in the\nlimit $q \\to 2$ as required. Furthermore, the equation~(\\ref{eq45})\nallows $q$ to take on {\\it negative} values if desired, and it is also\napplicable over a broad range of both positive and negative values for\n$a$. Recall that when $a=0$, there are no systematic acceleration or\nloss processes included in the model. Positive (negative) values for $a$\nimply additional systematic acceleration (losses).\n\n\\subsection{Transition to the Stationary Solution}\n\nThe analytical solutions we have obtained for the Green's function\nprovide a complete description of the response of the system to the\nimpulsive injection of monoenergetic particles at any desired time. The\ngenerality of these expressions allows one to obtain the particular\nsolution for the distribution function $f$ associated with any\narbitrary momentum-time source function $S$ using the convolution\ngiven by equation~(\\ref{eq6}). One case of special interest is the\nspectrum resulting from the {\\it continual injection} of monoenergetic\nparticles beginning at time $t=0$, described by the source term\n\\begin{equation}\nS(p,t) = \\cases{\n\\dot N_0 \\, \\delta(p-p_0) \\ , & $t \\ge 0$ \\ , \\cr\n0 \\ , & $t < 0$ \\ , \\cr\n}\n\\label{eq50}\n\\end{equation}\nwhere $\\dot N_0$ denotes the rate of injection of particles with\nmomentum $p_0$ per unit volume per unit time. We assume that no\nparticles are present for $ t < 0$. Combining equations~(\\ref{eq6})\nand (\\ref{eq50}), we find that the time-dependent distribution function\nresulting from monoenergetic particle injection is given by\n\\begin{equation}\nf(p,t) = \\dot N_0 \\int_0^t f_{_{\\rm G}}(p_0,p,t_0,t) \\, dt_0 \\ .\n\\label{eq51}\n\\end{equation}\nBy transforming to the dimensionless variables $x$ and $y$ and employing\nequation~(\\ref{eq16}), we conclude that the particular solution for the\nnumber distribution associated with continual monoenergetic particle injection\ncan be written as\n\\begin{equation}\nN(x,y) \\equiv 4 \\pi \\, m^3 c^3 \\, x^2 \\, f(x,y)\n= {\\dot N_0 \\over D_*} \\int_0^y N_{_{\\rm G}}(x_0,x,y_0,y) \\, dy_0 \\ ,\n\\label{eq52}\n\\end{equation}\nwhere we have used equation~(\\ref{eq13}) to make the substitution $dt_0\n= dy_0 \/ D_*$. Since $N_{_{\\rm G}}$ depends on the time parameters $y$ and\n$y_0$ only through the combination $y-y_0$ (see eqs.~[\\ref{eq45}] and\n[\\ref{eq48}]), it follows that\n\\begin{equation}\nN(x,y) = {\\dot N_0 \\over D_*} \\int_0^y N_{_{\\rm G}}(x_0,x,0,y') \\, dy' \\ .\n\\label{eq53}\n\\end{equation}\nThis is a more convenient form for $N(x,y)$ because $y$ now appears only\nin the upper integration bound.\n\nFor general, finite values of $y$, the time-dependent particular\nsolution for $N(x,y)$ given by equation~(\\ref{eq53}) must be computed\nnumerically by substituting for $N_{_{\\rm G}}$ using either\nequation~(\\ref{eq45}) or (\\ref{eq48}), depending on the value of $q$.\nHowever, as $y \\to \\infty$, the solution rapidly approaches a stationary\nresult representing a balance between injection, acceleration, and\nparticle escape. The form of the stationary solution, called the {\\it\nsteady-state Green's function}, $N^{\\rm G}_{\\rm ss}$, can be obtained by\ndirectly solving the transport equation~(\\ref{eq1}) with $\\partial\nf\/\\partial t=0$ and $S(p,t) = \\dot N_0 \\, \\delta(p-p_0)$. Alternatively,\nthe steady-state Green's function can also be computed by taking the\nlimit of the time-dependent solution, which yields\n\\begin{equation}\nN^{\\rm G}_{\\rm ss}(x) \\equiv \\lim_{y \\to \\infty}\nN(x,y) = {\\dot N_0 \\over D_*} \\int_0^\\infty N_{_{\\rm G}}(x_0,x,0,y')\n\\, dy' \\ .\n\\label{eq54}\n\\end{equation}\nIn the $q < 2$ case, we can substitute for $N_{_{\\rm G}}$ using equation~(\\ref{eq45})\nand evaluate the resulting integral using equation~(6.669.4) from \\citet{gr80}.\nAfter some algebra, we obtain\n\\begin{equation}\nN^{\\rm G}_{\\rm ss}(x) = {\\dot N_0 \\over (2-q) \\, x_0 D_*}\n\\left(x \\over x_0\\right)^{(a+1)\/2} (x x_0)^{(2-q)\/2}\n\\, I_{\\beta-1 \\over 2} \\left(\\sqrt{\\theta} \\, x_{\\rm min}^{2-q} \\over 2-q\\right)\n\\, K_{\\beta-1 \\over 2} \\left(\\sqrt{\\theta} \\, x_{\\rm max}^{2-q} \\over 2-q\\right)\n\\ ,\n\\label{eq55}\n\\end{equation}\nwhere $\\beta = (a+3)\/(2-q)$, and\n\\begin{equation}\nx_{\\rm min} \\equiv \\min(x,x_0) \\ , \\ \\ \\ \\ \\ \\ \\\nx_{\\rm max} \\equiv \\max(x,x_0)\n\\ .\n\\label{eq56}\n\\end{equation}\nLikewise, for the case with $q=2$, we can substitute for $N_{_{\\rm G}}$ using\nequation~(\\ref{eq48}) and then utilize equation~(3.471.9) from \\citet{gr80}\nto conclude that the steady-state solution is given by\n\\begin{equation}\nN^{\\rm G}_{\\rm ss}(x) \\Big|_{q=2} = {\\dot N_0 \\over 2 \\, x_0 D_* \\sqrt{\\lambda}}\n\\left(x \\over x_0\\right)^{(a+1)\/2} \\left(x_{\\rm max} \\over x_{\\rm min}\\right)^{-\\sqrt{\\lambda}}\n\\ ,\n\\label{eq57}\n\\end{equation}\nwhere $\\lambda$ is defined by equation~(\\ref{eq49}). The steady-state\nsolutions given by equations~(\\ref{eq55}) and (\\ref{eq57}) agree with\nthe results obtained by directly solving the transport equation. Due to\nthe asymptotic behavior of the Bessel $K_\\nu(z)$ function for large $z$,\nequation~(\\ref{eq55}) indicates that $N^{\\rm G}_{\\rm ss}$ exhibits an\nexponential cutoff at high energies when $q < 2$ \\citep{as70}. This\ncorresponds physically to the fact that the escape timescale decreases\nwith increasing particle momentum in this case. However, when $q=2$\n(eq.~[\\ref{eq57}]), the spectrum displays a pure power-law behavior at\nhigh energies because the escape timescale is independent of the\nparticle momentum. Specific examples illustrating these behaviors will be\npresented in \\S~4.\n\n\\subsection{Escaping Particle Distribution}\n\nThe various expressions we have obtained for the Green's function\n$N_{_{\\rm G}}$ describe the momentum distribution of the particles remaining\nin the plasma at time $t$ after injection occurring at time $t_0$.\nHowever, since our model incorporates a physically realistic,\nmomentum-dependent escape timescale $t_{\\rm esc}(p)$ given by\nequation~(\\ref{eq7}), it is also quite interesting to compute the\nspectrum of the {\\it escaping} particles, which may form an energetic\noutflow capable of producing observable radiation. In general, the\nnumber distribution of the escaping particles, $\\dot N^{\\rm esc}(x,y)$,\nis related to the current distribution of particles in the plasma, $N(x,y)$,\nvia\n\\begin{equation}\n\\dot N^{\\rm esc}(x,y) \\equiv t_{\\rm esc}^{-1} \\, N(x,y)\n= \\theta D_* \\ x^{2-q} \\, N(x,y)\n\\ ,\n\\label{eq58}\n\\end{equation}\nwhere we have used equations~(\\ref{eq7}) and (\\ref{eq14}) to obtain the\nfinal result. The quantity $\\dot N^{\\rm esc}(x,y)\\,dx$ represents the number\nof particles escaping per unit volume per unit time with dimensionless\nmomenta between $x$ and $x+dx$.\n\nAn important special case is the evolution of the escaping particle\nspectrum resulting from {\\it impulsive monoenergetic injection} at\ndimensionless time $y=y_0$. In this application, equation~(\\ref{eq58})\ngives the Green's function number spectrum for the escaping particles,\ndefined by\n\\begin{equation}\n\\dotN_{_{\\rm G}}^{\\rm esc}(x_0,x,y_0,y)\n\\equiv \\theta D_* \\ x^{2-q} \\, N_{_{\\rm G}}(x_0,x,y_0,y)\n\\ ,\n\\label{eq59}\n\\end{equation}\nwhere $N_{_{\\rm G}}$ is evaluated using either equation~(\\ref{eq45}) or\n(\\ref{eq48}) depending on the value of $q$. We can use this expression\nfor $\\dotN_{_{\\rm G}}^{\\rm esc}$ to compute the {\\it total} escaping number\ndistribution resulting from an impulsive flare occurring at $t=0$ by\nintegrating the escaping distribution with respect to time, which yields\n\\begin{equation}\n\\DeltaN_{_{\\rm G}}^{\\rm esc}(x) \\equiv \\int_0^\\infty \\dotN_{_{\\rm G}}^{\\rm esc} \\, dt\n= \\theta \\, x^{2-q} \\int_0^\\infty N_{_{\\rm G}}(x_0,x,0,y') \\, dy'\n\\ ,\n\\label{eq60}\n\\end{equation}\nwhere we have used equation~(\\ref{eq59}) and taken advantage of the fact\nthat $N_{_{\\rm G}}$ depends on $y$ and $y_0$ only through the difference $y-y_0$\n(cf. eq.~[\\ref{eq53}]).\nBy comparing equations~(\\ref{eq54}) and\n(\\ref{eq60}), we deduce that\n\\begin{equation}\n\\DeltaN_{_{\\rm G}}^{\\rm esc}(x) = {\\theta D_* \\over \\dot N_0}\n\\ x^{2-q} \\, N^{\\rm G}_{\\rm ss}(x)\n\\ ,\n\\label{eq61}\n\\end{equation}\nso that the total escaping spectrum is simply proportional to the steady-state\nGreen's function resulting from continual injection, as expected. The process\nconsidered here corresponds to the injection of a single particle at time $t=0$,\nand therefore we find that the normalization of $\\DeltaN_{_{\\rm G}}^{\\rm esc}$ is\ngiven by\n\\begin{equation}\n\\int_0^\\infty \\DeltaN_{_{\\rm G}}^{\\rm esc}(x) \\, dx = 1\n\\ ,\n\\label{eq62}\n\\end{equation}\nwhich provides a useful check on the numerical results.\n\nAnother interesting example is the case of continual monoenergetic\nparticle injection commencing at time $t=0$, described by the source\nterm given by equation~(\\ref{eq50}). The time-dependent buildup of the\nescaping particle spectrum $\\dot N^{\\rm esc}(x,y)$ in this scenario can\nbe analyzed by using equation~(\\ref{eq53}) to substitute for $N(x,y)$ in\nequation~(\\ref{eq58}), which yields\n\\begin{equation}\n\\dot N^{\\rm esc}(x,y)\n= \\dot N_0 \\, \\theta \\ x^{2-q} \\int_0^y N_{_{\\rm G}}(x_0,x,0,y') \\, dy'\n\\ .\n\\label{eq63}\n\\end{equation}\nIn the limit $y \\to \\infty$, the escaping particle spectrum approaches the\nsteady-state result\n\\begin{equation}\n\\dot N_{\\rm ss}^{\\rm esc}(x) \\equiv \\lim_{y \\to \\infty} \\dot N^{\\rm esc}(x,y)\n= \\dot N_0 \\, \\theta \\ x^{2-q} \\int_0^\\infty N_{_{\\rm G}}(x_0,x,0,y') \\, dy'\n\\ .\n\\label{eq64}\n\\end{equation}\nBy comparing equations~(\\ref{eq54}), (\\ref{eq61}), and (\\ref{eq64}), we\nconclude that\n\\begin{equation}\n\\dot N_{\\rm ss}^{\\rm esc}(x)\n= \\theta D_* \\ x^{2-q} \\, N^{\\rm G}_{\\rm ss}(x)\n= \\dot N_0 \\ \\DeltaN_{_{\\rm G}}^{\\rm esc}(x)\n\\ .\n\\label{eq65}\n\\end{equation}\nThe final result can be combined with equation~(\\ref{eq62}) to show that\nthe escaping spectrum satisfies the normalization relation\n\\begin{equation}\n\\int_0^\\infty \\dot N_{\\rm ss}^{\\rm esc}(x) \\, dx = \\dot N_0 \\ ,\n\\label{eq66}\n\\end{equation}\nas expected for the case of continual steady-state injection.\n\nNote that analytical expressions for the steady-state escaping spectrum $\\dot\nN_{\\rm ss}^{\\rm esc}(x)$ can be obtained by substituting for $N^{\\rm G}_{\\rm\nss}(x)$ in equation~(\\ref{eq65}) using either equation~(\\ref{eq55}) or\n(\\ref{eq57}), depending on the value of $q$. We obtain\n\\begin{equation}\n\\dot N^{\\rm esc}_{\\rm ss}(x) = {\\dot N_0 \\, \\theta \\, x^{2-q} \\over (2-q) \\, x_0}\n\\left(x \\over x_0\\right)^{(a+1)\/2} (x x_0)^{(2-q)\/2}\n\\, I_{\\beta-1 \\over 2} \\left(\\sqrt{\\theta} \\, x_{\\rm min}^{2-q} \\over 2-q\\right)\n\\, K_{\\beta-1 \\over 2} \\left(\\sqrt{\\theta} \\, x_{\\rm max}^{2-q} \\over 2-q\\right)\n\\ ,\n\\label{eq67}\n\\end{equation}\nfor $q < 2$, and\n\\begin{equation}\n\\dot N^{\\rm esc}_{\\rm ss}(x) = {\\dot N_0 \\, \\theta \\, x^{2-q} \\over\n2 \\, x_0 \\sqrt{\\lambda}}\n\\left(x \\over x_0\\right)^{(a+1)\/2} \\left(x_{\\rm max} \\over x_{\\rm min}\\right)^{-\\sqrt{\\lambda}}\n\\ ,\n\\label{eq68}\n\\end{equation}\nfor $q = 2$.\n\n\\section{RESULTS}\n\nThe new result we have derived for the secular Green's function\n(eq.~[\\ref{eq45}]) displays a rich behavior through its complex\ndependence on momentum, time, and the dimensionless parameters $q$, $a$,\nand $\\theta$, which are related to the fundamental physical transport\ncoefficients $D_*$, $A_*$, and $t_*$ via equations~(\\ref{eq14}). Here we\npresent several example calculations in order to illustrate the utility\nof the new solution. Detailed applications to astrophysical situations,\nincluding active galaxies and $\\gamma$-ray bursts, will be presented in\nsubsequent papers.\n\n\\begin{figure}[t]\n\\includegraphics[width=6in]{f2.eps}\n\\vskip-0.6in\n\\caption{Green's function solutions to the time-dependent stochastic\nparticle acceleration equation for $x_0=1$. The left panels depict the\nimpulsive-injection solution, $N_{_{\\rm G}}$ (eqs.~[\\ref{eq45}] and\n[\\ref{eq48}]), and the right panels illustrate the response to uniform,\ncontinuous injection, $N$ (eq.~[\\ref{eq53}]), with $\\dot\nN_0=D_*=1$. Note that $N$ approaches the corresponding steady-state\nsolution (eqs.~[\\ref{eq55}] and [\\ref{eq57}]) as $y$ increases. The\nindices of the wave turbulence spectra are indicated, with $q = 2$ for\nhard-sphere scattering, $q = 5\/3$ for a Kolmogorov cascade, and $q =\n3\/2$ for a Kraichnan cascade.}\n\\label{fig2}\n\\end{figure}\n\nThe panels on the left-hand side of Figure~2 depict the time-dependent\nGreen's function solution, $N_{_{\\rm G}}$, describing the evolution of the\nparticle distribution in the plasma resulting from {\\it impulsive}\nmonoenergetic injection at $y=0$. Results are presented for the\nhard-sphere scattering case ($q = 2$), computed using\nequation~(\\ref{eq48}), and for the Kolmogorov ($q = 5\/3$) and Kraichnan\n($q = 3\/2$) cases, evaluated using equation~(\\ref{eq45}). The only\nacceleration mechanism considered here is the stochastic acceleration\nassociated with the second-order Fermi process, and therefore we set $a\n= 0$. The escape parameter $\\theta$ is set equal to unity, so that the\ntimescale for escape is comparable to the diffusion timescale. As the\nwave turbulence spectrum becomes steeper (i.e., as $q$ increases), a\nlarger fraction of the turbulence energy is contained in waves with long\nwavelengths, which interact resonantly with higher energy particles.\nSteeper turbulence spectra therefore produce harder particle\ndistributions, as can be confirmed in the plots. Consequently we\nconclude that an ensemble of hard sphere scattering centers is more\neffective at accelerating nonthermal particles compared with a\nKolmogorov wave spectrum, which in turn is more effective than a\nKraichnan spectrum.\n\nThe panels on the right-hand side of Figure~2 illustrate the buildup of\nthe particle spectrum in the plasma, $N(x,y)$, due to {\\it continual}\nmonoenergetic injection beginning at $y=0$, computed by evaluating\nnumerically the integral in equation~(\\ref{eq53}). We have set $a=0$ and\ntherefore the acceleration is purely stochastic. As $y \\to \\infty$, the\nspectrum approaches the steady-state form given by equation~(\\ref{eq55})\nfor $q<2$ or by equation~(\\ref{eq57}) for $q=2$. In the hard-sphere case\n($q=2$), the particle spectrum displays a power-law shape at high\nenergies in agreement with equation~(\\ref{eq57}). However, when $q < 2$,\nparticle escape dominates over acceleration at high energies, and\ntherefore the steady-state distribution is truncated, even in the\nabsence of radiative losses. This effect produces the quasi-exponential\nturnovers exhibited by the stationary spectra when $q = 5\/3$ and $q =\n3\/2$. Particle escape in these cases mimics energy losses due to, for\nexample, synchrotron emission from electrons.\n\n\\begin{figure}[t]\n\\hskip0.2in\\plottwo{f3a.eps}{f3b.eps}\n\\hskip0.5truein\n\\caption{Evolution of the particle distribution in the plasma resulting\nfrom monoenergetic injection with $q = 5\/3$, $a = 0$, $\\theta = 0.1$,\nand $x_0=1$. Panel~({\\it a}) depicts the Green's function, $N_{_{\\rm G}}$,\nresulting from impulsive monoenergetic injection (eq.~[\\ref{eq45}]), and\npanel ({\\it b}) illustrates the response to continual monoenergetic\ninjection (eq.~[\\ref{eq53}]) and the corresponding steady-state solution\n(eq.~[\\ref{eq55}]) for $\\dot N_0=D_*=1$.}\n\\label{fig3}\n\\end{figure}\n\nFigures~3 and 4 illustrate the effects of varying the values of the\nescape parameter $\\theta$ and the systematic acceleration\/loss parameter\n$a$ for the $q = 5\/3$ case. In Figure~3, we set $a=0$ and $\\theta =\n0.1$, which represents a particle escape timescale that is an order of\nmagnitude larger than the corresponding case depicted in Figure~2. The\nlonger escape time allows the particles to be accelerated to higher mean\nenergies before diffusing out of the plasma, and the decay of the\nparticle density at late times takes place much more slowly than for\nlarger values of $\\theta$. The hardening of the spectra causes the\nquasi-exponential cutoffs to move to higher energies, and the same\neffect can also be noted in the corresponding stationary solutions. The\ncalculations represented in Figure~4 include additional systematic\nacceleration processes that are modeled by setting $a = 2$. The enhanced\nparticle acceleration further hardens the spectrum and shifts the cutoff\nto even higher energies. Although the slope of the low-energy particle\ndistribution ($x < x_0$) is not altered much when only $\\theta$ is\nvaried (see Figs.~2 and 3), this slope becomes significantly steeper\nwhen $a$ is increased, as indicated in Figure~4.\n\nIn the left-hand panels of Figures~5 and 6 we plot the time-dependent\nsolution for the Green's function describing the escaping particles,\n$\\dotN_{_{\\rm G}}^{\\rm esc}$, resulting from {\\it impulsive} monoenergetic\nparticle injection (eq.~[\\ref{eq59}]) when $q = 5\/3$. The right-hand\npanels illustrate the buildup of the escaping spectrum, $\\dot N^{\\rm\nesc}$ (eq.~[\\ref{eq63}]), resulting from {\\it continual} injection\nbeginning at $y=0$. Note the transition to the steady-state solution,\n$\\dot N_{\\rm ss}^{\\rm esc}$ (eq.~[\\ref{eq64}]), as $y \\to \\infty$. In\nFigure~5 we set $\\theta=0.1$ and $a=0$, and in Figure~6 we set\n$\\theta=0.1$ and $a=2$. Included for comparison are the corresponding\nspectra describing the particle distributions in the plasma at the same\nvalues of $y$. We point out that the escaping particle spectra are\nsignificantly harder than the in situ distributions, which reflects the\npreferential escape of the high-energy particles resulting from the\nmomentum dependence of the escape timescale when $q<2$ (see\neq.~[\\ref{eq7}]).\n\n\\begin{figure}[t]\n\\hskip0.2in\\plottwo{f4a.eps}{f4b.eps}\n\\caption{Same as Fig.~3, except $a = 2$.}\n\\label{fig4}\n\\end{figure}\n\n\\section{DISCUSSION AND SUMMARY}\n\nWe have derived new, closed-form solutions (eqs.~[\\ref{eq45}] and\n[{\\ref{eq67}]}) for the time-dependent Green's function representing the\nstochastic acceleration of relativistic ions interacting with MHD waves.\nThe analytical results we have obtained describe the time-dependent\ndistributions for both the accelerated (in situ) and the escaping\nparticles. The Fokker-Planck transport equation considered here includes\nmomentum diffusion with coefficient $D(p) \\propto p^q$, particle escape\nwith mean timescale $t_{\\rm esc} \\propto p^{q-2}$, and additional\nsystematic acceleration\/losses with a rate proportional to $p^{q-1}$,\nwhere $p$ is the particle momentum and $q$ is the index of the wave\nturbulence spectrum. This specific scenario describes the resonant\ninteraction of particles with fast-mode and Alfv\\'en waves, which is one\nof the fundamental acceleration processes in high-energy astrophysics\n\\citep[e.g.,][]{dml96}.\n\nThe new analytical result complements the work of \\citet{pp95} since it\nis applicable for any $q < 2$ provided $s_{\\rm pp} = q-2$, where $s_{\\rm\npp}$ is the power-law index used by these authors to describe the\nmomentum dependence of the escape timescale. The most closely related\nprevious result is given by their equation~(59), which treats the case\nwith $q \\ne 2$ and $s_{\\rm pp}=0$, corresponding to a\nmomentum-independent escape timescale. Our analytical solution\n(eq.~[\\ref{eq45}]) agrees with theirs in the limit $q \\to 2$, as\nexpected. The general features of our new solution were discussed in\n\\S~4, where it was demonstrated that increasingly hard particle spectra\nresult from larger values of the wave index $q$, smaller values of the\nescape parameter $\\theta$, and larger values of the systematic\nacceleration parameter $a$. The rich behavior of the solution as a\nfunction of momentum, time, and the parameters $q$, $\\theta$, and $a$\nprovides useful physical insight into the nature of the coupled\nenergetic\/spatial particle transport in astrophysical plasmas.\n\n\\begin{figure}[t]\n\\hskip0.2in\\plottwo{f5a.eps}{f5b.eps}\n\\caption{Evolution of the particle distribution resulting from\nmonoenergetic injection with $q = 5\/3$, $a = 0$, $\\theta = 0.1$, $x_0 =\n1$, $\\dot N_0=1$, and $D_*=1$. Panel~({\\it a}) treats the case of\nimpulsive injection, with the thin lines representing the particle\ndistribution in the plasma (eq.~[\\ref{eq45}]) and the heavy lines denoting\nthe escaping particle spectrum (eq.~[\\ref{eq59}]) in units of $\\theta$.\nPanel~({\\it b}) illustrates the response to continual monoenergetic\ninjection for the particle distribution in the plasma (eqs.~[\\ref{eq53}] and\n[\\ref{eq55}]; thin lines) and for the escaping particle spectrum\n(eqs.~[\\ref{eq63}] and [\\ref{eq65}]; heavy lines).\nSee the discussion in the text.}\n\\label{fig5}\n\\end{figure}\n\n\\begin{figure}[t]\n\\hskip0.2in\\plottwo{f6a.eps}{f6b.eps}\n\\caption{Same as Fig.~5, except $a = 2$.}\n\\label{fig6}\n\\end{figure}\n\nThe solutions presented here can be used to describe the acceleration\nand transport of relativistic ions in astrophysical environments in\nwhich the turbulence spectrum is very poorly known and can be\napproximated by a power law, such as $\\gamma$-ray bursts, active\ngalaxies, magnetized coronae around black holes, and the intergalactic\nmedium in clusters of galaxies. For example, the hard X-ray emission\nfrom black-hole jet sources such as Cygnus X-1 \\citep{mal06} and the\nmicroquasar LS 5039 \\citep{aha05} could be powered by the stochastic\nacceleration of ions in a black-hole accretion disk corona that\nsubsequently escape and interact with surrounding material. In this\nscenario, persistent acceleration of monoenergetic particles injected\ninto the corona, or flaring episodes averaged over a sufficiently long\ntime, would produce a time-averaged escaping distribution of\nrelativistic protons given by equation~(\\ref{eq67}) for $q < 2$.\nAssuming only stochastic acceleration, so that $a = 0$, the number\ndistribution of escaping particles with $x \\geq x_0$ takes the form\n\\begin{equation}\n\\dot N_{\\rm ss}^{\\rm esc}(x) \n\\propto x^{(7-3q)\/2} K_{{\\beta - 1\\over 2}}\\left( {\\sqrt{\\theta} \nx^{2-q} \\over 2-q}\\right) \\propto \\cases{\nx^{3-2q}  \\ , & $x \\ll \\left( {2-q\\over \\sqrt{\\theta}}\\right)^{1\/(2-q)}$ \\  \\cr\nx^{(5-2q)\/2}\\exp\\left(-{\\sqrt{\\theta}x^{2-q}\\over 2-q}\\right)  \\ , & \n$x \\gg \\left({2-q\\over \\sqrt{\\theta}}\\right)^{1\/(2-q)}$ \\  \\cr\n}.\n\\label{eq69}\n\\end{equation}\nWhen the escaping hadrons collide with ambient gas or stellar wind\nmaterial, they would generate X-rays and $\\gamma$-rays via a pion\nproduction cascade with a very hard spectrum leading up to a\nquasi-exponential cutoff. The {\\it Gamma ray Large Area Space Telescope}\n(GLAST) telescope will be able to provide detailed spectra from Galactic\nblack-hole sources and unidentified EGRET $\\gamma$-ray sources to test\nfor the existence of this hard component. Our new analytical solution\ncan also be used to model the variability of radiation from ions\naccelerated in the accretion-disk coronae of Seyfert galaxies by\nchanging the level of turbulence. Flaring $\\sim 100\\,$MeV -- GeV\nradiation could be weakly detected by GLAST as a consequence of this\nprocess. Additional applications of our work include studies of the\nstochastic acceleration of relativistic cosmic rays in $\\gamma$-ray\nbursts \\citep{dh01}.\n\nThe results we have obtained describe both the momentum distribution of\nthe particles in the plasma, and the momentum distribution of the\nparticles that escape to form energetic outflows. Since the solutions do\nnot include inverse-Compton or synchrotron losses, which are usually\nimportant for energetic electrons, they are primarily applicable to\ncases involving ion acceleration. In order to treat the acceleration of\nrelativistic electrons, a generalized calculation including both\nstochastic particle acceleration and radiative losses is desirable, and\nwe are currently working to extend the analytical model presented here\nto incorporate these effects. Beyond the direct utility of the new\nanalytical solutions for probing the nature of particle acceleration in\nastrophysical sources, we note that they are also useful for\nbenchmarking numerical simulations.\n\n\\acknowledgements\nT.~L. is funded through NASA {\\it GLAST} Science Investigation No.~DPR-S-1563-Y.\nThe work of C.~D.~D. and P.~A.~B. is supported by the Office of Naval Research.\nThe authors also acknowledge the useful comments provided by the anonymous\nreferee.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n    Phase retrieval aims to reconstruct a complex-valued signal from its intensity-only measurements. It is a crucial problem in crystallography, optical imaging, astronomy, X-ray, electronic imaging, etc., because most existing measurement systems can only detect the relative low-frequency magnitude information of the wave fields. The study of the phase retrieval problems originated in the $1970$s. The researchers of the optics community developed many reconstruction algorithms \\cite{Gerchberg1972APA, Fienup:82, rodenburg2008ptychography}. \n    \n    Mathematically, the phase retrieval problem can be described by the following equation:\n    \\begin{equation}\n    \t\\label{CDP}\n    \t\\begin{aligned}\n    \t\t\\text { Find } \\mathbf{x} \\in \\mathbb{C}^{N} \\quad \\text { s.t. } \\ \\mathbf{y}_{i}=\\left|\\mathcal{F}\\left(\\mathbf{I}_{i} \\circ \\mathbf{x}\\right)\\right|^{2}, i=1, \\ldots, M,\n    \t\\end{aligned}\n    \\end{equation}\n    where $\\mathbf{x}\\in \\mathbb{C}^N$ is the complex-valued signal of interest, and $ \\mathbf{y}$ is its Fourier intensity measurements; $\\circ $ and $\\mathcal{F}$ refer to elementwise multiplication and Fourier transform operator, respectively. The pre-defined optical masks $ \\mathbf{I} $ provide the constraints to reduce the ill-posedness of the problem. It is implemented in many different ways. For instance, early-stage phase retrieval algorithms treat the support of the signal as the optical mask \\cite{Fienup:82}. Specifically, $|\\mathbf{x}_i| \\neq 0$ when $i \\in \\mathcal{S}$, where $\\mathcal{S}$ denotes the support of the signal. However, these early-stage algorithms cannot guarantee the convergence of the optimization process nor provide globally optimal solutions. Recently, pre-defined random optical masks have been used as the constraints to improve the reconstruction performance \\cite{Anand07, Horisaki2014SingleshotPI, candes2015phase, YE2022106808}. The random masks can be implemented using a spatial light modulator (SLM) or digital micromirror device (DMD) \\cite{Horisaki2014SingleshotPI, Zheng2017DigitalMD}. Although the usage of the random masks can lead to better reconstruction performance, there are several disadvantages. The cost of DMD and SLM is one concern, the error due to the global gray-scale phase mismatch and spatial non-uniformity from the devices is another. Besides, it was empirically shown that about $ 4 - 6 $ measurements are required for exact recovery with random binary masks \\cite{candes2015code, candes2015phase}. It increases the measurement time and can affect the quality of the reconstructed image, particularly for dynamic objects.\n    \n    Recently, data-driven methods, like deep neural networks, have been widely applied to explore the data distributions through extensive training samples \\cite{LeCun_2015}. Among them, the so-called Convolutional Neural Network (CNN) that exploits the features based on convolution kernels has been successfully adopted in image processing tasks, like image restoration \\cite{He_2016_CVPR, zhang2017beyond, wang2020lightening}. Besides, an important technique named attention mechanism that mimics cognitive attention became popular in the CNN structure to enhance some parts of the feature maps and thus improve the performance \\cite{attention_17}. Since CNN can learn discriminative representations from large training datasets, researchers have spent effort in using CNN for solving phase retrieval problems \\cite{deng_li_goy_kang_barbastathis_2020, White:19, Shi:19, Sinha17}. For instance, Chen et al. reconstructed signals from Fresnel diffraction patterns with CNN and the contrast transfer function \\cite{Bai:19}. Kumar et al. proposed the deep unrolling networks that solved the iterative phase retrieval problem with CNN \\cite{Kumar:21}. Besides, the conditional generative adversarial network (GAN), a kind of data-driven generative method based on specific conditions \\cite{cond_gan_14}, was adopted for reconstructing the phase information from Fourier intensity measurements \\cite{uelwer2021phase}. Wu et al. also proposed a CNN to reconstruct the crystal structures with coherent X-ray diffraction patterns \\cite{Wu_cw5029, Wu_2021}. The above  methods adopt supervised learning that trains the networks with the paired ground-truth data. In recent years, training with unpaired or unsupervised (no ground-truth data) datasets has been a popular topic. For example, Fus et al. proposed the unsupervised learning for in-line holography phase retrieval \\cite{Fus_18}, while Zhang et al. developed the GAN-based phase retrieval with unpaired datasets \\cite{Zhang_21}. While the above approaches have achieved some success, phase retrieval with Fraunhofer diffraction patterns (i.e., Fourier intensity) for general applications is still a challenge. It is partly because of the large domain discrepancy between the Fourier plane and image plane; the large variation in data distributions of general applications also introduces much difficulty in designing a model structure that is optimal for different applications.  In this paper, we propose a single-shot maskless phase retrieval method named \\textit{SiPRNet} that is benefited from the advanced deep learning technology. Compared with the traditional algorithms that need multiple intensity measurements, the proposed method only collects one Fourier intensity measurement for reconstruction. Besides, no optical mask is needed in the proposed method; thus, the cost  for expensive SLM\/DMD devices is saved. The proposed \\textit{SiPRNet} is designed to work with general image datasets without assuming a specific application domain. It has demonstrated its generality by achieving state-of-the-art performance on three different datasets, as shown in our experimental results. It significantly outperforms the existing optimization-based and learning-based phase retrieval methods.\n    \n    The contributions of our method are: (1) We propose a novel CNN structure - \\textit{SiPRNet} that utilizes advanced deep learning techniques for single-shot maskless phase retrieval. It can reliably reconstruct the original signal based on only one Fourier intensity measurement without using optical masks.  (2) We propose a Multi-Layer Perception (MLP) block to extract global representations from the intensity measurement. In addition, we propose to embed a dropout layer into the MLP block to constrain the behavior of the fully connected layers, which promotes the MLP to extract global representation. (3) We propose to use residual learning and self-attention mechanism in the Up-sampling and Reconstruction blocks, which improves the quality of the reconstructed signals. (4) We verify the proposed method on an optical platform to demonstrate its practicality. The source codes of the proposed method have been released on Github: \\url{https:\/\/github.com\/Qiustander\/SiPRNet}.\n    \n    The rest of this paper is organized as follows. Section \\ref{Sec:Method} introduces the defocused phase retrieval system we developed in this study and the proposed end-to-end deep-learning phase retrieval network \\textit{SiPRNet}. Section \\ref{Sec:simulation} presents the model analysis and simulation results. Section \\ref{Sec:experiment} provides the experimental results on an optical system for validating the performance of the proposed method. \n      \n\\section{Proposed Method} \n\\label{Sec:Method}\n\n\\subsection{Phase Retrieval System} \\label{Sec:PRsystem}\n\nAs indicated in Eq. \\eqref{CDP}, a phase retrieval system aims to reconstruct a complex-valued signal $ \\mathbf{x} $ from its Fourier intensity measurements. The proposed method does not require multiple measurements or optical masks, so $ M=1 $ and all the values of the mask $ \\mathbf{I} $ are equal to one in Eq. \\eqref{CDP}. In this case, the original complex-valued signal $ \\mathbf{x} $ is reconstructed only from its Fourier intensity. This seemingly impossible task is known to have a solution, although with ambiguities. From \\cite{hayes1982reconstruction}, it is known that $ \\mathbf{x} $ can be uniquely defined, except for trivial ambiguities, by its Fourier intensity, with an oversampling factor over $ 2 $ in each dimension, if $ \\mathbf{x} $ has a finite support and is non-symmetric. The above has an important implication to the optical system required for the proposed phase retrieval method. Fig. \\ref{fig:opticalpath} shows the system we have constructed to implement the proposed method. In the system, we increase the resolution of the CMOS camera to make sure that the captured Fourier intensity image has a sampling rate at least two times in each dimension higher than that of the object image $ \\mathbf{x} $. As a result, the number of samples in the zero diffraction order of the Fourier intensity image is $ 762\\times762 $, while the number of samples of $ \\mathbf{x} $ is $ 128\\times128 $. We then extract the central $ 128\\times128 $ pixels of the Fourier intensity measurements for feeding to the proposed CNN model for phase retrieval. \n\n\\begin{figure}[htb]               \n\t\\centering\\includegraphics[width=0.8\\linewidth]{Figures\/Fig_Optical_Path.pdf}\n\t\\caption{Optical path of the defocus phase retrieval system. $\\mathbf{x}$ represents the complex-valued object, and $\\mathbf{y}$ is its Fourier intensity measurement; $\\mathbf{X} = \\mathcal{F}\\left(\\mathbf{I}_{i} \\circ \\mathbf{x}\\right)$,  $\\mathbf{H}$ and $\\ast$ denote the Fourier plane in complex-valued form, the defocus function generated via Fresnel diffraction and the convolution operation, respectively. }\n\t\\label{fig:opticalpath}\n\t\n\\end{figure}\n\nWhen capturing Fourier intensity images, it is common to have the saturation problem. It is because most structured signals have the energy concentrated in the low-frequency areas, in particular the zero frequency. Due to the limited analog-to-digital resolution of standard scientific cameras ($14$ - $16$ bits), the central regions of the Fourier intensity measurement are often seriously saturated. To lessen the effect of the saturation problem, we convolve the Fourier intensity measurement with a defocus function $ \\mathbf{H} $ to reduce the dynamic range \\cite{goodman2017introduction}. It can be implemented by moving the camera beyond the Fourier plane. Fig. \\ref{fig:opticalpath} shows a typical optical path for implementing the defocus phase retrieval system. Fig. \\ref{fig:compvisual} shows some examples of the Fourier intensity measurements obtained in our experiments. \n\nBesides the lens-based system that we have implemented in this work, we can also find lensless phase retrieval systems \\cite{Sinha17}. Different from the lens-based system that reconstructs images from their Fraunhofer diffraction patterns, the lensless system reconstructs images based on their Fresnel diffraction patterns \\cite{Sinha17}. Compared with the lens-based system, the lensless system requires a much longer distance between the sensors and objects. For instance, in \\cite{Sinha17}, the distance between the sensor and object is reported as $37.5 cm$  or longer. It would lead to severe aberrations and energy loss. By using a lens, the lens-based system can generate high-quality Fraunhofer diffraction patterns at a short propagation distance between the lens and sensor ($3 cm$  or less), thus avoiding the drawbacks stated above.\n\n\n\\subsection{Deep Learning Model: \\textit{SiPRNet}}\n\nWith the oversampling constraint as mentioned above, it is possible to reconstruct a complex-valued signal (with trivial ambiguities) from only its Fourier intensity without other constraints (such as optical masks). However, such a reconstruction problem is highly non-convex that is difficult to solve with traditional optimization methods. Due to their non-linear characteristic, deep neural networks have been successfully applied to many non-convex optimization problems. In this paper, we propose to regard the single-shot maskless phase retrieval challenge as an end-to-end learning task, and formulate the retrieval process directly by a deep neural network model (denoted as $P(\\mathbf{y};\\theta)$) as:\n\n\\begin{dmath}\n\t\\hat{\\mathbf{x}} = P(\\mathbf{y};\\theta),\n\\end{dmath}\nwhere $\\theta$ represents the learnable parameters of the deep learning model. The training target is to minimize the error between the predicted $P(\\mathbf{y};\\theta)$ and the ground-truth complex-valued image $ \\mathbf{x} $ regularized with the total variation (TV) norm $ TV(\\hat{\\mathbf{x}}) $. The training process can be written as:\n\n\\begin{equation}\n\tP(y;\\theta) = \\mathop{argmin}\\limits_{\\hat{\\mathbf{x}}} \\left( \\left\\| \\mathbf{x}-\\hat{\\mathbf{x}} \\right\\|_1 +\\lambda TV\\left(\\hat{\\mathbf{x}}\\right)\\right), \n\\end{equation}\nwhere $\\left\\| \\cdot \\right\\|_1$ denotes the $\\ell_1$-norm distance  and $ \\lambda $ is a constant determined empirically.\n\nFig. \\ref{fig:structure} shows the overall structure of the proposed phase retrieval network: \\textit{SiPRNet}. It mainly consists of two parts: feature extraction and phase reconstruction. It takes the Fourier intensity measurement $ \\mathbf{y} $ as input and reconstruct the complex-valued image $ \\mathbf{x} $ as its output.\n\n\\begin{figure} [htb]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{Figures\/Fig_Network_Structure.pdf}\n\t\\caption{Structure of the proposed network that mainly consists of two parts: Feature Extraction and Phase Reconstruction.}\n\t\\label{fig:structure}\n\\end{figure}\n\n\\subsubsection{\\textbf{Feature Extraction}}\nTraditional learning-based approaches often use a CNN as the front end to extract the features of the Fourier intensity measurement. However, the convolution operation of CNN shares the parameters (e.g., weights and bias) for all data of the measurement but ignores their different characteristics. Therefore, we propose to use a new Multi-Layer Perceptron (MLP) block to fuse the information of different frequency data as shown in Fig. \\ref{fig:structure}(a). The MLP block consists of three Fully Connected (FC) and one Dropout layer. The input Fourier measurement is firstly reshaped from the size of $128\\times 128\\times 1$ to $1\\times1\\times16,384$, and then fed to the first FC layer of the MLP block. Each FC layer has $1,024$ neurons for extracting the features required for reconstructing the original complex-valued images. We propose to embed a Dropout layer after the first FC layer. The Dropout layer randomly deactivates a small number (empirically $20\\%$) of neurons at the training stage. At the inference stage, all neurons are activated that forms an ensemble to reduce the risk of overfitting \\cite{srivastava2014dropout}. Our design with embedding a Dropout layer after the first FC layer constrains the behavior of the FC layers so that the features extracted by the MLP block do not depend on a particular input frequency measurement. The effectiveness of the dropout layer has been evaluated and shown in Section \\ref{Sec:ablationstudy}. Fig. \\ref{fig:structure} shows an example of the original complex-valued image (real and imaginary), the Fourier intensity measurement, and the features extracted by the MLP block. It can be seen that the MLP block has performed the domain conversion. The extracted feature image contains the features of the original complex-valued image. A decoder is needed to reconstruct the real and imaginary images from the extracted features. \n\n\n\\subsubsection{\\textbf{Phase Reconstruction}}\nThe phase reconstruction part contains two Up-sampling and Reconstruction (UR) blocks and one Post-processing block, as shown in Fig. \\ref{fig:structure}(b) and (c). The UR block is responsible for reconstructing the complex-valued image from the extracted features and progressively upsampling it back to the original size. The basic processing unit of the UR block is the convolution block that has three components: convolutional, instance normalization, and Parametric Rectified Linear Unit (PReLU) layers. The convolutional layer contains $3\\times 3$ learnable filters that aim to implicitly approximate the nonlinear backward mapping from the extracted features to the desired complex-valued image $ \\mathbf{x} $. The instance normalization and PReLU layers are non-linear functions that increase the non-linear capacity and learning ability of the deep learning model. \n\nA self-attention unit is followed after each convolutional block. Recently, the self-attention technique has been popularly used in image reconstruction tasks, e.g., the scattering problem \\cite{non-local, Wang:21}. Traditional CNN using small convolution kernels can only extract the local correlation of an image. However, natural images are often structural that exhibit global correlation. Self-attention is an effective way to exploit the global correlation in images. The structure of the self-attention layer used in the UR block is similar to that in \\cite{non-local}, as shown in Fig. \\ref{fig:attn}. At the first stage, the input features are fed to three $1\\times1$ convolutional layers ($\\theta, \\phi$ and $g$). The outputs are denoted as key, query, and value, respectively, all with the same dimensions. Next, the cross-correlation matrix is computed between the key and query. It represents the dependency between any two pixels of the input features. The softmax function is then performed to normalize the correlation scores and build an attention map. Large values in the attention map denote strong correlations between the features concerned. Since the correlation is computed among all features, the attention is made not only to those with local correlations but also the global ones. The result is finally multiplied with the value to generate the self-attention feature maps. They are then added back to the original features to strengthen the features of interest. Such an attention mechanism is important to the training of the UR block. It allows the UR block to reconstruct images with correlated features. For instance, for the extracted features as shown in Fig. \\ref{fig:structure}, the UR block will try to reconstruct both the left and right eyes together since they have a high correlation as indicated in the training face images in the dataset. Since the correlation of the features may change at different resolutions, another UR block is used after the first one on the features of higher resolution.\n\n\\begin{figure} [htb]\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{Figures\/Fig_Atten_Structure.pdf}\n\t\\caption{Structure of the attention mechanism.}\n\t\\label{fig:attn}\n\\end{figure}\n\nDirect stacking covolutional and attention layers would lead to the obvious performance drop because of weak information flow and gradient vanishing \\cite{He_2016_CVPR}. Therefore, we add a shortcut connection after the convolution block and attention mechanism that forms a residual structure (shown in Fig. \\ref{fig:structure}(b)), which is widely adopted in image processing tasks \\cite{Wang_2017_CVPR, zhang2021explainable, park2021dynamic}. The residual setting avoids the gradient vanishing problem and therefore reduces the learning difficulty. As shown in Fig. \\ref{fig:structure}, the self-attention layers effectively estimate the residue between the input feature and desired one. Finally, we utilize the unshuffle layer to reshape the feature map to enlarge the spatial sizes. The effectiveness of the residual self-attention structure is further discussed in Section \\ref{Sec:ablationstudy}.\n\n\n\\subsubsection{\\textbf{Post-processing block}}\nThe Post-processing block contains two convolutional layers (shown in Fig. \\ref{fig:structure}(c)). It estimates the real $\\hat{\\mathbf{x}}_{Re}$ and imaginary $\\hat{\\mathbf{x}}_{Im}$ parts of the image from the up-sampled features. Then, the phase part $\\hat{\\mathbf{x}}$ of the image can be calculated by:\n\\begin{equation}\n\t\\label{Eq:converttophase}\n\t\\hat{\\mathbf{x}}=tan^{-1}\\left( \\frac{\\hat{\\mathbf{x}}_{Im}}{\\hat{\\mathbf{x}}_{Re}} \\right).\n\\end{equation}\n\n\n\\subsubsection{\\textbf{Loss Function}}\nWe regard the phase retrieval problem as a supervised learning task. At the training stage, we prepare a set of paired training data. Each Fourier intensity measurement has a paired complex-valued image as the ground-truth target. Then, the deep learning model can be optimized by minimizing the $\\ell_1$-norm distance between the estimation and the ground-truth images plus the TV-norm to improve the image edges. The loss function $\\mathcal{L}$ is defined as: \n\\begin{equation}\n\t\\mathcal{L} = \\left\\| \\mathbf{x}_{Re}-\\hat{\\mathbf{x}}_{Re} \\right\\|_1 + \\left\\| \\mathbf{x}_{Im}-\\hat{\\mathbf{x}}_{Im} \\right\\|_1 + \\lambda TV\\left(\\hat{\\mathbf{x}}\\right).\n\\end{equation}\nThe symbols $\\mathbf{x}_{Re}$ and $\\mathbf{x}_{Im}$ represent the ground-truth real and imaginary parts of the image, respectively. \n\n    \\section{Simulation Results} \\label{Sec:simulation}\n\n\\subsection{Simulation Setup}\n\nWe conducted the simulation on two different datasets: the Real-world Affective Faces (RAF) dataset \\cite{li2017reliable} and Fashion-MNIST dateset \\cite{xiao2017\/online}. Specifically, the training data was converted to gray-scale image and resized to $128 \\times 128$. Then the pre-processed images $\\mathbf{x} \\in [0, 1]$ were used to generate complex-valued images of two forms: phase-only and magnitude-phase. For generating the phase-only images, we mapped $\\mathbf{x}$ to the $2\\pi$ phase domain via an exponential function: $\\exp(2\\pi i \\mathbf{x})$, with the magnitude kept as $1$. On the other hand, we used the images of Fashion-MNIST dataset to generate the magnitude-phase images. To be specific, we directly used the pre-processed images $\\mathbf{x} \\in [0, 1]$ as the magnitudes of the images, i.e., $\\mathbf{x}_{mag} = \\mathbf{x} \\in [0, 1]$. Then we set the phase parts through $\\mathbf{x}_{phase} = \\exp(2\\pi i \\mathbf{x}_{mag})$ and combined them by $\\mathbf{x} = \\mathbf{x}_{mag} \\circ \\mathbf{x}_{phase}$ as the magnitude-phase training and testing images, where $\\circ$ denotes the elementwise multiplication.\n\nAs mentioned in Section \\ref{Sec:Method}, we suggest to reduce the saturation problem by convolving the Fourier intensity measurement with a defocus function. Based on the convolution property of the Fourier transform, we can simulate it by multiplying the testing image with a defocus kernel $\\mathbf{h}$ in the spatial domain. It can be generated through the Fresnel propagation \\cite{goodman2017introduction}. Denote the original image as $x(p, q)$ and $X(u, v)$ as its  Fourier transform, then the optical field $X_L(u', v')$ in the defocus plane is expressed as:\n\\begin{dmath}\n\t\\label{Eq:defocuskernel}\n\tX_L(u', v') \n\t=  C \\iint X(u, v) H(u - \\frac{u'}{\\lambda L}, v - \\frac{v'}{\\lambda L}) du\\ dv,\n\\end{dmath}\nwhere $\\lambda$, $L$ and $H$ denotes the wavelength, distance between the lens after the object and the defocus plane, and the Fourier transform of the defocus kernel $\\mathbf{h}$, respectively. $C$ is the constant and $h(p, q)$ is given as $e^{\\frac{j\\pi}{\\lambda L}c(p^{2}+q^{2})}$. Based on the convolution theorem, Eq. \\eqref{Eq:defocuskernel} is equivalent to the elementwise multiplication between $h$ and $x$ in the spatial domain with the scaling factor $\\lambda L$. \n\nThen, oversampled discrete Fourier transform (DFT) with size $ 762\\times762 $ (the same size as in real experiments) was performed to generate the intensity measurements. For each image, the Fourier intensity measurement was capped at the value $ 4095 $ ($12$-bit) to simulate the dynamic range of the practical scientific cameras. The central $128\\times 128$ patch was cropped as the input of the \\textit{SiPRNet} (the performance of different input sizes is discussed in Section \\ref{Sec:ablainputsize}). The total epoch and mini-batch size were set as $1,000$ and $32$. The Adam optimizer \\cite{kingma2017adam} with the learning rate of $10^{-4}$ was implemented to update the gradients in each training step. Note that the learning rate was updated through \\textit{StepLR} scheduler in PyTorch every $100$ epoches with $\\gamma = 0.9$.\n\nWe adopted Peak Signal to Noise Ratio (PSNR, higher the better), Structural Similarity (SSIM, higher the better) and Mean Absolute Error (MAE, lower the better) as the performance metrics to measure the difference between the reconstructed images $\\hat{\\mathbf{x}}$ and ground truth images $\\mathbf{x}$. For each dataset, we selected the first $1,000$ images in the test set as the testing samples for evaluation. We converted the reconstructed real and imaginary parts to phase-only images via Eq. \\eqref{Eq:converttophase}. The phase values of both images were shifted by $\\pi$ to ensure non-negative values for the computation of PSNR. The average PSNR, SSIM and MAE were acquired among all $1,000$ testing images for evaluation. \n\n\\subsection{Ablation Study} \\label{Sec:ablationstudy}\n\nTo gain a deeper insight into the improvements obtained by \\textit{SiPRNet}, we conducted additional ablation studies on different components of our framework and analyzed their impact on the speed and accuracy of the network.\n\n\n\\subsubsection{\\textbf{Impact of Input Size}} \\label{Sec:ablainputsize}\n\nFor the two-dimensional Fourier spectrum, the central regions represent the low-frequency information while outer areas contain the high-frequency components. Structured signals (e.g., natural images) usually have their energy concentrated in the low-frequency regions; hence the high-frequency components have relatively lower intensity and are dominated by noise. An ablation analysis was performed to determine the size of the central regions to be used. In our experiments, the intensity measurements have the size of $762 \\times 762$. We fed the central $[32 \\times 32, 64 \\times 64, 128 \\times 128, 256 \\times 256, 512 \\times 512]$ of the intensity measurement to the same network and study the performance, as shown in Fig. \\ref{fig:Inputsize}. The average testing PSNR from $30000th$ to $35000th$ training steps was chosen for comparison. The box-plot of the performance, the floating point operations per second (FLOPs) and the numbers of trainable parameters with respect to the sizes are shown in Fig. \\ref{fig:Inputsize}. Although the model performance keeps increasing along with the increase of input size, the number of the trainable parameters also boosts quickly, which brings a huge computational burden. For example, it can be seen that the number of parameters dramatically increases from $ 19.3M $ to $ 69.6M $ when the input size changes from $128 \\times 128$ to $256 \\times 256$. While the growth speed of the performance is linear when the input size is smaller than $256 \\times 256$, it saturates as the input size increases to about $128 \\times 128$. Therefore, we adopt $128 \\times 128$ as the input size to balance performance and efficiency. \n\n\\begin{figure}[htb]\t\t\n\t\\centering\n\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_Input_Size.pdf}\n\t\\caption{Ablation study on the input sizes. The results are obtained with the RAF dataset \\cite{li2017reliable}.}\n\t\\label{fig:Inputsize}\n\\end{figure}\n\n\\subsubsection{\\textbf{Impact of Defocus Distance}} \\label{Sec:abladefocusdist}\n\nThe effect of the defocus distance $L$ is discussed in this section. We investigated $6$ situations of the defocus distance: $0mm$ (no defocusing), $15mm$, $20mm$, $30mm$, $45mm$, and $75mm$. The defocus kernel is a pure-phase object generated by Holoeye built-in function.  The default defocus distance is $30mm$ in our proposed \\textit{SiPRNet}. The size of the input intensity measurements is $128 \\times 128$. The box-plot for average PSNR is presented in Fig. \\ref{fig:Defocusdist}. As shown in the figure, the reconstruction performance significantly degrades when there is no defocusing ($L = 0 mm$). It is because some pixels are severely saturated and give wrong information to the network. On the other hand, the performances are similar when $L = 15$ to $45 mm$ although the PSNR is slightly higher when $L = 30 mm$. It shows that the performance of the system is not sensitive to the choice of $L$ as long as it is not too small or too big.  When the defocus distance is large, i.e. $75 mm$, the PSNR significantly drops. It is because a portion of the Fourier intensity measurements (high-frequency areas) become very small due to the decrease in light intensity with the long defocus distance.  To balance the illumination efficiency and the reconstruction performances, the defocus distance $L$ was selected as $30 mm$ for all experiments.\n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_DefocusDist.pdf}\n\t\\caption{Ablation study on the defocus distance. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:Defocusdist}\n\\end{figure}\n\n\n\\subsubsection{\\textbf{Impact of Dropout Layer}} \\label{Sec:abladropout}\n\nWe investigated the effect of the number of Dropout layers in feature extraction part. We set the dropout rate as $0.2$ in every Dropout layer. Since $3$ FC layers are used in the MLP blocks, we investigated $4$ situations when introducing the Dropout layers: $0$ Dropout layer, $1$ Dropout layer after the first FC layer, $2$ Dropout layers after each of the first two FC layers, and $3$ Dropout layers after each of the FC layers. One Dropout layer is the default setting in our proposed \\textit{SiPRNet}. The size of the input intensity measurements is $128 \\times 128$. The box-plot for average PSNR is presented in Fig. \\ref{fig:Dropout}. As shown in the figure, if there is no Dropout layer, the average PSNR of the reconstructed images is the lowest because of the over-fitting problem. The network is lazy to investigate the hidden frequency representations, and the reconstruction relies on a small number of inputs. The Dropout layer randomly deactivates a part of the inputs at the training stage, which inspires the network to find more global and robust representations. At the testing stage, all input signals are used to form an ensemble learning structure that increases the generalization power of the whole network. After implementing the Dropout layer, the model performance boosts as expected. Interestingly, with more dropout layers applied, there is too much information ignored in the training stage that hurts the learning progress of the network and decreases the performance. Therefore, we adopt $30 mm$ as the defocus distance to balance performance and illumination efficiency. \n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_Dropout.pdf}\n\t\\caption{Ablation study on the number of Dropout layers in feature extraction. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:Dropout}\n\\end{figure}\n\n\n\\subsubsection{\\textbf{Impact of Residual Units}} \\label{Sec:ablaresidual}\n\nWe also investigated the effects of the number of residual units in the phase reconstruction part. As shown in Fig. \\ref{fig:structure}, each residual unit contains two layers of Conv Blocks interlaced with two self-attention layers plus the PReLU at the end. The size of the input images to the network is still $128 \\times 128$. With everything remaining the same but just changing the number of residual units in each UR block, the average PSNRs of the estimated images are obtained and shown in Fig. \\ref{fig:ResLayer}. It can be seen that using the residual units brings in better PSNR performance. The model with one residual unit has similar or even better performance than the models with more residual units. To pursue better PSNR and efficiency, we choose to use one residual unit in the UR block.\n\n\\begin{figure}[htb]\t\t\n\t\\centering\t\\includegraphics[width=0.7\\linewidth]{Figures\/Fig_Ablation_ResLayer.pdf}\n\t\\caption{Ablation study on the number of residual units. The results are obtained with the RAF dataset \\cite{li2017reliable}. }\n\t\\label{fig:ResLayer}\n\\end{figure}\n\n\\subsubsection{\\textbf{Model Robustness}} \\label{Sec:CrossValidation}\nBesides the generalization ability, the robustness is also a crucial metric to evaluate neural networks. The more robust the network, the less probable the overfitting problem would happen in the network. In order to demonstrate the robustness of the proposed \\textit{SiPRNet}, we utilized the K-Fold cross validation, a statistical method based on the resampling procedure \\cite{Stone_1974}. The procedure of the K-Fold cross validation can be briefly described as follows: \\textit{1.} Randomly shuffle the dataset. \\textit{2.} Split the dataset into $K$ groups.  \\textit{3.} In each fold, take a group as the testing dataset and the remaining groups as the training dataset. Then, fit the deep learning model on the training dataset and do the evaluation on the testing dataset. \\textit{4.} Reset the model parameters and repeat step \\textit{3} until $K$ folds are accomplished. \\textit{5.} Summarize the model evaluation scores obtained in each fold. To be specific, we adopted $K=5$ folds for evaluation and we ran $100$ epochs in each fold. In each fold, $80\\%$ of the samples were treated as the training data and the rest $20\\%$ were utilized for testing. We used the training (for the training dataset) and testing (for the testing dataset) losses as the evaluation scores. The training loss was recorded every $20$ training batches and the testing loss was collected every epoch. The simulated results on the RAF and Fashion-MNIST datasets are shown in Fig.\\ref{Fig:KFold}. As can be seen, both losses decrease quickly with the growth of the training epoch for both datasets. The curves of the losses are similar in each fold. Although there are small variations at the beginning of the training stage ($5-15$ epochs in Fig.\\ref{Fig:KFold}(a) and  $10-25$ epochs in Fig.\\ref{Fig:KFold}(b)), all folds converge to the optimal direction and thus are close to the average fold curve. In conclusion, the K-fold cross validation results demonstrate the robustness and effectiveness of \\textit{SiPRNet}. \n\n\\begin{figure}[htb]\n\t\\centering\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\begin{tabular}{*{2}{c@{\\extracolsep{0em}}} }\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_RAF.pdf}\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_RAF_test.pdf}\n\t\t\\end{tabular}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}[t]{\\textwidth}\n\t\t\\begin{tabular}{*{2}{c@{\\extracolsep{0em}}} }\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_Fashion.pdf}\n\t\t\t\\includegraphics[width=0.5\\textwidth]{Figures\/Fig_Ablation_KFold_Fashion_test.pdf}\n\t\t\\end{tabular}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\caption{Training \\& Testing errors of $5$-Fold cross validation on (a) RAF dataset, and (b) Fashion-MNIST dataset, respectively. Fold Avg. (colored in red) denotes the mean of $5$ folds.}\n\t\\label{Fig:KFold}\n\\end{figure}\n\n\n\\subsection{Simulation Results}\n\nIn this paper, we compare our \\textit{SiPRNet} with the traditional algorithms and learning-based methods. Compared methods include the traditional optimization-based algorithms such as Gerchberg\u2013Saxton (GS) \\cite{Gerchberg1972APA}, Hybrid Input-Output (HIO) \\cite{Fienup:82}, and total variation ADMM (ADMM-TV) \\cite{chang2018total}. In addition, we compare the proposed \\textit{SiPRNet} with other CNN-based methods including: Plain Network (denoted as PlainNet) \\cite{He_2016_CVPR}: the basic CNN structure that stacks convolutional layers one by one; Residual Network (denoted as ResNet) \\cite{He_2016_CVPR, Nishizaki_2020}: a popular deep-learning structure and has several skip connections among the convolutional layers; residual dense network (denoted as ResDenseNet) \\cite{Zhang_2018_CVPR}: a network that adds more skip connections to achieve better efficiency; Lensless imaging network (denoted as LenlessNet) \\cite{Sinha17}: a residual UNet to recover the phase-only images in a lensless imaging system; Conditional Generative Adversarial Phase Retrieval Network (denoted as PRCGAN) \\cite{uelwer2021phase}: a network that utilizes the conditional adversarial learning strategy  for phase retrieval task; and NNPhase \\cite{Wu_cw5029}: a network to recover the complex-valued images from Fraunhofer diffraction patterns. \n\n\\textbf{\\textit{Phase-only datasets:}} The phase retrieval results with the phase-only datasets are shown in Table \\ref{table:RAFresult}, Table \\ref{table:Fashionresult}, and Fig. \\ref{fig:compvisual}. To save space, the qualitative results of PlainNet and ResDenseNet are not included in Fig. \\ref{fig:compvisual} due to their inferior performances. All the CNN-based methods were trained with the same settings on the RAF and Fashion-MNIST datasets. Note that the  hyper-parameters of all traditional methods were optimally fine-tuned and fixed when the methods were evaluated with all testing images. For each traditional optimization-based method, we did three different trials of evaluation for each testing image, where each trial had $ 400 $ iterations. We manually selected the reconstructed image with the highest PSNR as the final result for comparison. For the CNN-based methods, we compared the average PSNR of reconstructed images given by the trained models at the last epoch.\n\nAs can be seen in Table \\ref{table:RAFresult} and Table \\ref{table:Fashionresult}, the proposed \\textit{SiPRNet} achieves much better performance than all compared methods. Compared with the traditional optimization-based methods, \\textit{SiPRNet} has average PSNR and SSIM gains of at least $6.544dB$ and $0.457$, respectively, on the RAF dataset (Table \\ref{table:RAFresult}). The PSNR and SSIM gains can even reach $19.58dB$ and $0.816$, respectively, on the Fashion-MNIST dataset (Table \\ref{table:Fashionresult}). This is because traditional optimization-based methods cannot converge to satisfactory results with only one intensity measurement. As shown in Fig. \\ref{fig:compvisual}, only ADMM-TV method can recover the contour of the facial images ($7$th column). In addition, \\textit{SiPRNet} can achieve PSNR and SSIM gains of at least $1.631dB$ and $0.065$, respectively, compared with other state-of-the-art CNN-based methods on the two datasets. It can be seen in Fig. \\ref{fig:compvisual} that the reconstructed images of the proposed \\textit{SiPRNet} outperform other CNN-based methods qualitatively. In addition, the error maps ($4$th column) also show that the estimated images are close to the ground-truth images. As can be seen, the reconstructed images only differ from the original ones in some local areas with minor errors, while most regions are close to $0$.\n\nNote that we train all the networks with our quantized defocused Fraunhofer diffraction patterns datasets. Although the reconstructed images via PRCGAN \\cite{uelwer2021phase} seem to have better high-frequency information (facial details), the reconstructed image does not share similarities with the original image (last second column).  In fact, the performance of PRCGAN in our simulation is poorer than that reported in the original paper of PRCGAN. It is because the dataset used in the original simulation of PRCGAN does not consider the saturation problem, which is commonly found when imaging the Fourier intensity. Besides, the authors assume the object images are magnitude-only (no phase information), which is far from real situations.  As for NNPhase \\cite{Wu_2021}, it aims to deal with the crystal data that the application is quite specific. Besides, LenslessNet \\cite{Sinha17} is initially designed for the phase retrieval task based on Fresnel diffraction patterns with lensless imaging systems. When testing with the more general and realistic datasets used in our simulations, the performances of these networks drop significantly compared with those reported in their original papers. We will show in the next section that these networks also suffer from the same problems in the real experiments. The above simulation results show that the proposed \\textit{SiPRNet} outperforms the traditional optimization-based and CNN-based methods both quantitatively and qualitatively. More simulation results can be found in \\nameref{Sec:Moreresults}. \n\nWe also evaluate the inference speed of different methods by collecting the mean execution time of one hundred samples. The results are presented in Table \\ref{table:RAFresult} (last column). Since our approach is designed in an end-to-end manner that can effectively utilize the huge computational power of the current advanced GPU devices, it can be seen in Table \\ref{table:RAFresult} that the proposed the \\textit{SiPRNet} is very fast and can realize real-time performance ($3.569$ ms\/img). It is comparable with other learning-based approaches. In contrast, the inference speed of the traditional algorithms is quite slow (in the order of seconds) since lots of iterations are required, and they can only be implemented with the CPU.\n\n\\textbf{\\textit{Magnitude-phase dataset:}} The simulation results are shown in Fig. \\ref{fig:ampphase}. We select HIO, NNPhase, and PRCGAN for comparison since they all perform Fourier phase retrieval. It can be seen that the proposed \\textit{SiPRNet} outperforms all compared approaches. In particular, the results show that PRCGAN cannot deal with the complex-valued phase retrieval tasks, and it is difficult for HIO to reconstruct complex-valued images with single-shot intensity measurement. The magnitude and phase images given by these approaches have poor quality. As is presented in Fig. \\ref{fig:ampphase}(b), the proposed \\textit{SiPRNet} achieves much better performance than all compared methods evaluated by different metrics. For the magnitude part, the proposed \\textit{SiPRNet} achieves average PSNR and SSIM gains of at least $2.145dB$ and $0.156$, respectively. For the phase part, the increases in PSNR and SSIM gains can reach $3.214 dB$ and $0.181$, respectively. The above simulation results show that the proposed \\textit{SiPRNet} can be generalized to complex-valued image datasets (containing both magnitude and phase parts) and outperform the existing methods.\n\n\\begin{table}[ht]\n\t\\centering\n\n\n\t\\caption{Quantitative comparison (average MAE\/PSNR\/SSIM\/Inference Time) with the state-of-the-art methods for phase retrieval on the RAF dataset. Best and second best performances are in \\textcolor{red}{red} and \\textcolor{blue}{blue} colors, respectively.}\n\t\\begin{adjustbox}{width=0.8\\columnwidth, center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} & \\multirow{2}{*}{\\parbox{1.2cm}{\\centering Inference Time (ms\/img) }}\\\\\n\t\t\t\\cline { 2 - 7 } & Sim. & Exp. & Sim. & Exp. & Sim. & Exp. & \\\\\n\t\t\t\\hline\n\t\t\t\\hline GS \\cite{Gerchberg1972APA} & $0.969$ & $1.147$ & $15.099$ & $12.770$ & $0.250$ & $0.129$ & $1.233\\times 10^{3}$\\\\\n\t\t\tHIO \\cite{Fienup:82} & $1.007$ & $1.185$ & $14.854$ & $12.554$ & $0.239$ & $0.127$ & $1.226\\times 10^{3}$\\\\\n\t\t\tADMM-TV \\cite{chang2018total} &  $1.119$ & $1.600$ & $12.660$ & $9.902$ & $0.206$ & $0.025$ & $1.324\\times 10^{4}$\\\\\n\t\t\tPlainNet \\cite{He_2016_CVPR} & $0.975$ & $1.180$ & $13.841$ & $12.479$ & $0.425$ & $0.377$ & $1.583$\\\\\n\t\t\tResNet \\cite{He_2016_CVPR, Nishizaki_2020} & $0.894$ & $0.812$ & $15.178$ & $15.831$ & $0.456$ & $0.471$ & $2.438$\\\\\n\t\t\tResDenseNet \\cite{Zhang_2018_CVPR} & $1.084$ & $1.010$ & $13.344$ & $13.800$ & $0.380$ & $0.387$ & $1.732$\\\\\n\t\t\tLenslessNet \\cite{Sinha17} & $\\textcolor{blue}{0.671}$ & $\\textcolor{blue}{0.704}$ & $\\textcolor{blue}{17.573}$ & $\\textcolor{blue}{17.132}$ & $\\textcolor{blue}{0.598}$ & $\\textcolor{blue}{0.596}$ & $7.396$\\\\\n\t\t\tPRCGAN \\cite{uelwer2021phase} & $0.951$ & $0.945$ & $14.738$ & $14.833$ & $0.511$ & $0.508$ & $0.983$\\\\\n\t\t\tNNPhase \\cite{Wu_cw5029} & $0.755$ & $0.931$ & $16.753$ & $15.416$ & $0.558$ & $0.544$ & $5.834$\\\\\n\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\textcolor{red}{0.570}$ & $\\textcolor{red}{0.582}$ & $\\textcolor{red}{19.204}$ & $\\textcolor{red}{18.905}$ & $\\textcolor{red}{0.663}$ & $\\textcolor{red}{0.663}$ & $3.569$\\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\label{table:RAFresult}\n\\end{table}\n\n\\begin{table}[ht]\n\t\\centering\n\n\n\n\t\\caption{Quantitative comparison (average MAE\/PSNR\/SSIM) with the state-of-the-art methods for phase retrieval on the Fashion-MNIST dataset. Best and second best performances are in \\textcolor{red}{red} and \\textcolor{blue}{blue} colors, respectively.}\n\t\\begin{adjustbox}{width=0.7\\columnwidth,center}\n\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} \\\\\n\t\t\t\\cline { 2 - 7 } & Sim. & Exp. & Sim. & Exp. & Sim. & Exp. \\\\\n\t\t\t\\hline\n\t\t\t\\hline GS \\cite{Gerchberg1972APA} & $1.571$ & $1.681$ & $9.852$ & $9.176$ & $0.088$ & $0.053$ \\\\\n\t\t\tHIO \\cite{Fienup:82} & $1.594$ & $1.709$ & $9.629$ & $9.082$ & $0.068$ & $0.052$ \\\\\n\t\t\tADMM-TV \\cite{chang2018total} & $0.935$ & $1.366$ & $13.772$ & $9.973$ & $0.226$ & $0.132$ \\\\\n\t\t\tPlainNet \\cite{He_2016_CVPR} & $0.589$ & $0.601$ & $16.916$ & $16.672$ & $0.629$ & $0.623$  \\\\\n\t\t\tResNet \\cite{He_2016_CVPR, Nishizaki_2020} & $0.289$ & $0.304$ & $21.394$ & $20.806$ & $0.739$ & $0.726$  \\\\\n\t\t\tResDenseNet \\cite{Zhang_2018_CVPR} & $0.381$ & $0.419$ & $19.406$ & $18.252$ & $0.675$ & $0.657$  \\\\\n\t\t\tLenslessNet \\cite{Sinha17} & $\\textcolor{blue}{0.179}$ & $\\textcolor{blue}{0.221}$ & $\\textcolor{blue}{26.784}$ & $\\textcolor{blue}{23.833}$ & $\\textcolor{blue}{0.842}$ & $\\textcolor{blue}{0.802}$\\\\\n\t\t\tPRCGAN \\cite{uelwer2021phase} & $1.598$ & $1.602$ &  $10.102$ & $10.126$ & $0.422$ & $0.417$ \\\\\n\t\t\tNNPhase \\cite{Wu_cw5029} & $0.404$ & $0426$ & $23.956$ & $22.778$ & $0.739$ & $0.443$ \\\\\n\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\textcolor{red}{0.127}$ & $\\textcolor{red}{0.144}$ & $\\textcolor{red}{29.209}$ & $\\textcolor{red}{28.132}$ & $\\textcolor{red}{0.884}$ & $\\textcolor{red}{0.872}$   \\\\\n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\label{table:Fashionresult}\n\\end{table}\n\n   \n\n\\begin{figure}[htbp]\n\n\n\n\t\\begin{adjustbox}{width=1\\textwidth,center}\n\t\t\\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_12_sim.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0011_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/GS_RAF_simulation_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/HIO_RAF_simulation_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/ADMM_sim.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/\/Res18_simulation.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_sim_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0011_angle_sim.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_12_exp.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_gt_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0011_angle_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0011_residual_exp.png}~\t\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_RAF_exp_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_RAF_exp_12.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/Res18_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0011_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0011_angle_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0011_angle.png}\n\t\t\t\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_304_exp.jpg}~\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0303_angle_gt.png}~\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0303_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_RAF_exp_304.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_RAF_exp_304.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/Res18_exp2.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0303_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0303_angle.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_0847.png}\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0847_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/SiPRNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/img_0847_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/GS_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/HIO_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/ADMM_RAF_0847.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/ResNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/LenslessNet_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/PRCGAN_img_0847_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t, valign=t]{Figures\/main_result\/NNPhase_img_0847_angle.png}\n\t\t\t\\\\    \t\t\t\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_646_exp_fashion.jpg}\n\t\t\t&\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0645_angle_gt.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0645_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_Fashion_exp_646.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_Fashion_exp_646.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_exp_fashion.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ResNet_exp_fashion.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/lensless_exp_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0645_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0645_angle.png}\n\t\t\t\\\\\n\t\t\t\n\t\t\t\\includegraphics[height=0.09\\textwidth, valign=t]{Figures\/main_result\/measurement_0007.png}\n\t\t\t&\\includegraphics[height=0.105\\textwidth, width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0007_angle_gt.pdf}~         &\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/SiPRNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/img_0007_residual.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/GS_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/HIO_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ADMM_Fashion_0007.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/ResNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/LenslessNet_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/PRCGAN_img_0007_angle.png}~\n\t\t\t&\\includegraphics[width=0.09\\textwidth, valign=t]{Figures\/main_result\/NNPhase_img_0007_angle.png}\n\t\t\t\\\\    \n\t\t\t\n\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\\\\\n\t\t\\end{tabular}\n\t\\end{adjustbox}\n\t\\vspace{-0.2cm}\n\t\\caption{Simulation and experimental results of different phase retrieval methods on the RAF \\cite{li2017reliable} and the Fashion-MNIST datasets \\cite{xiao2017\/online}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars (for experimental results) at the bottom left corners. The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. The first and second \\textbf{rows} are the simulation and experimental results of the same image from the RAF dataset. The third and the fourth \\textbf{rows} show the experimental performances of other images in the RAF dataset. The fifth and the sixth \\textbf{rows} indicate the experimental results of images from the Fashion-MNIST dataset. Please zoom in for better view. More results can be found in \\nameref{Sec:Moreresults}. }\n\t\\label{fig:compvisual}\n\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\t\n\t\\begin{subfigure}{\\textwidth}\t\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=1\\textwidth, center}        \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0093.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_magnitude_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0093_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0093_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0093_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0550.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_magnitude_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0550_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0550_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0550_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/measurement_0756.png}~\n\t\t\t\t&\\includegraphics[height = 0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_magnitude_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0756_magnitude.png}~\n\t\t\t\t&\\includegraphics[height = 0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/SiPRNet_img_0756_angle.png}~\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/HIO_img_0756_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/NNPhase_img_0756_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/mag_phase\/PRCGAN_img_0756_angle.png}~\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet} }  \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{HIO \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{NNPhase \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{PRCGAN \\\\ (mag.)}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}  } \\\\ (phase)}} & {\\scriptsize \\makecell[c]{HIO \\\\ (phase)}} & {\\scriptsize \\makecell[c]{NNPhase \\\\ (phase)}} & {\\scriptsize \\makecell[c]{PRCGAN \\\\ (phase)}} \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace*{-2mm}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\begin{subfigure}{\\textwidth}\t\n\t\t\\centering\n\t\t\\begin{adjustbox}{width=0.7\\columnwidth, center}\n\t\t\t\\begin{tabular}{|c|c|c|c|c|c|c|}\n\t\t\t\t\\hline \\multirow{2}{*}{ Methods } & \\multicolumn{2}{c|}{ MAE $\\downarrow$} & \\multicolumn{2}{c|}{ PSNR $\\uparrow$} & \\multicolumn{2}{c|}{ SSIM $\\uparrow$} \\\\\n\t\t\t\t\\cline { 2 - 7 } & Mag. & Phase & Mag. & Phase & Mag. & Phase \\\\\n\t\t\t\t\\hline\n\t\t\t\t\\hline HIO \\cite{Fienup:82} & $0.298$ & $1.898$ &  $7.709$ & $8.317$ & $0.330$ & $0.043$ \\\\\n\t\t\t\tPRCGAN \\cite{uelwer2021phase} & $0.073$ & $1.654$ &  $18.672$ & $9.924$ & $0.588$ & $0.387$ \\\\\n\t\t\t\tNNPhase \\cite{Wu_cw5029} & $0.109$ & $0.687$ & $18.169$ & $18.752$ & $0.395$ & $0.46$ \\\\\n\t\t\t\t\\textbf{\\textit{SiPRNet} (Ours)} & $\\mathbf{0.053}$ & $\\mathbf{0.359}$ & $\\mathbf{20.817}$ & $\\mathbf{21.966}$ & $\\mathbf{0.744}$ & $\\mathbf{0.594}$   \\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\caption{}\n\t\\end{subfigure}\n\t\n\t\\vspace{-0.2cm}\n\t\\caption{(a) Qualitative simulation results of different phase retrieval methods on the magnitude-phase Fashion-MNIST datasets \\cite{xiao2017\/online}. The pixel values of the Fourier intensity measurements (first column) range from $0$ to $4095$. The colormap of the magnitude parts (second to the sixth columns) ranges from $0$ to $1$, while the colormap of the phase parts (seventh to the last columns) ranges from $0$ to $2\\pi$. Please zoom in for better view. (b) Quantitative comparisons (average MAE\/PSNR\/SSIM) on the same dataset. Best performances are denoted in \\textbf{bold} font.}\n\t\\label{fig:ampphase}\n\t\n\\end{figure}\n\n    \\section{Experimental Results} \\label{Sec:experiment}\n\n\\subsection{Experimental Setup}\n\nWe also evaluated our proposed \\textit{SiPRNet} with a practical optical system. A picture of the experimental setup is shown in Fig. \\ref{fig:setup}.  It comprises a Thorlabs $ 10 mW$ HeNe laser with wavelength $ \\lambda = 632.8 nm$ ; and a $ 12 $-bit $ 1920\\times1200 $ Kiralux CMOS camera with pixel pitch $ 5.86 \\mu m$ . Besides, the optical system includes one $75 mm$, three $100 mm$, and one $150 mm$ lenses to form a standard imaging system. The choice of the lenses is for generating Fourier intensity images of appropriate size to match the size of the CMOS sensor. In practice, these lenses can be compactly arranged such that the whole imaging system can be made very small. However, the lens design is out of the scope of this study. A $ 1920\\times1080 $ Holoeye Pluto phase-only SLM with pixel pitch $ \\delta_{SLM}=8\\mu m $ was used to generate the testing objects in the experiments. Specifically, the images in the two datasets mentioned above were loaded to the SLM to impose multiple-level phase changes. The images were pre-multiplied with a defocus kernel; thus, there was no need to move the camera beyond the focal plane. It is to ensure that the defocusing process, which is not our main focus, will be perfectly implemented so that we only need to focus on the reconstruction performances of different methods. The defocus kernel is a pure-phase object generated by Holoeye built-in function corresponding to $L = 30 mm$. The reason for choosing $L = 30mm$ has been explained in Section \\ref{Sec:abladefocusdist}. We utilized the central $ 128\\times 128 $ SLM pixels in all experiments. The resulting Fourier intensity measurements were captured by the camera and formed the training and testing datasets for \\textit{SiPRNet}. The size of the acquired measurement is $\\frac{\\lambda L}{d}$ where $d$ denotes the pixel pitch of the SLM. \n\n\\begin{figure}[ht]               \n\t\\centering\\includegraphics[width=0.8\\linewidth]{Figures\/Fig_Setup.pdf}\n\t\\caption{Experimental setup of the defocus phase retrieval system.}\n\t\\label{fig:setup}\n\t\n\\end{figure}\n\n\\subsection{Experimental Results}\n\nWith the training dataset, we fine-tuned all the pre-trained models acquired in the simulation with the experimental measurements. That is, we re-trained the models with the trained parameters from the simulation and used the experimental measurements as the new training inputs. The quantitative and qualitative experimental results are presented in Table \\ref{table:RAFresult}, Table \\ref{table:Fashionresult}, and Fig. \\ref{fig:compvisual}, respectively. \n\nIt is clearly shown in Table \\ref{table:RAFresult} and Table \\ref{table:Fashionresult} that the experimental results of \\textit{SiPRNet} only decline a little compared with the simulation results. It proves the robustness of the proposed method. As shown in Fig. \\ref{fig:compvisual}, all traditional optimization-based methods fail to reconstruct the original images because it is insufficient to converge to satisfactory solutions with only one intensity measurement for those methods. Compared with other state-of-the-art CNN-based methods, \\textit{SiPRNet} can achieve at least $1.773dB$ and $0.067$ gains in PSNR and SSIM, respectively, on the two datasets. As can be clearly seen in Fig. \\ref{fig:compvisual}, the proposed \\textit{SiPRNet} can reconstruct the most similar contours as the ground truth and preserve more high-frequency details compared with other methods.  The above experimental results show that the proposed \\textit{SiPRNet} outperforms all traditional optimization-based and CNN-based methods quantitatively and qualitatively when using in practical phase retrieval systems. However, there is always room for improvement. It is shown in the results of all learning-based approaches that the reconstructed images are a bit blurry compared with the ground truths. We note that for the captured intensity images, the values of the high-frequency components are much smaller than those in the low-frequency areas. The noise in the images further makes the high-frequency components difficult to identify. It introduces much difficulty to the neural networks to extract the features of the high-frequency components. Further investigation is needed to improve the sharpness of the reconstructed images. Extra denoising and attention modules may be applied to better recover the high-frequency components.\n\n\\section{Conclusion} \n\tIn conclusion, this paper proposed an end-to-end CNN structure for single-shot maskless phase retrieval. We have shown how the optical system should be constructed to ensure the feasibility of the solution and reduce the saturation problem due to the high dynamic range of Fourier intensity measurements. We proposed a novel CNN named \\textit{SiPRNet} to reconstruct phase-only images with the central areas of the intensity measurements. To fully extract the representative features, we proposed an MLP block embedded with a Dropout layer to fuse the information of the frequency components. A self-attention structure was included to enhance the global correlation in the reconstructed phase images. The simulation and experimental results demonstrate that the proposed \\textit{SiPRNet} performs much better than the traditional phase retrieval algorithms and other CNN structures, making it a promising solution to practical phase retrieval applications. At the moment, we only conduct simulations and experiments on synthesized objects. Further research is underway to enhance the network structure for working with more complex image datasets and realistic objects. \n\n    \\section*{Appendix}  \\label{Sec:Moreresults}\n\n\tIn this Appendix, we show additional qualitative comparison results.  Fig. \\ref{fig:firstimage}, \\ref{fig:secondimage}, \\ref{fig:thirdimage} present the simulation and experimental results. The detailed explanations are attached with each figure.\n\t\n\t   \n\t\\begin{figure}[htb]\n\t\t\\begin{adjustbox}{width=1\\textwidth,center}        \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0287_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0287_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0287_angle.png}\n\t\t\t\t\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0544_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0544_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0544_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/measurement_722.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0721_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/SiPRNet_img_0721_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/img_0721_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/GS_RAF_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/HIO_RAF_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/ADMM_sim_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/Res18_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/lensless_sim_img_0721_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/CGAN_simulation_722.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/first\/NNPhase_img_0721_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Simulation results of different phase retrieval methods on the RAF dataset \\cite{li2017reliable}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images. The fourth \\textbf{column} denotes the error maps of the \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods.  Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. Each \\textbf{row} presents the simulation results of a testing image.  Please zoom in for better view.}\n\t\t\\label{fig:firstimage}\n\t\t\n\t\\end{figure}\n\t\n\t\n\t\n\n\t\\begin{figure}[htb]\n\t\t\\begin{adjustbox}{width=1\\textwidth,center}          \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0177_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0177_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0177_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0177_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_178.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0177_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0287_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0287_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0287_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_288.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0287_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/measurement_545.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth, width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0544_angle_gt.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/SiPRNet_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/img_0544_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/GS_RAF_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/HIO_RAF_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/ADMM_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/Res18_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/lensless_exp_img_0544_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/CGAN_exp_545.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/second\/NNPhase_img_0544_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Experimental results of different phase retrieval methods on the RAF dataset \\cite{li2017reliable}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars at the bottom left corners.  The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstructed images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. Each \\textbf{row} presents the experimental results of a testing image. Please zoom in for better view.}\n\t\t\\label{fig:secondimage}\n\t\t\n\t\\end{figure}     \n\t\n\t\n\n\t\\begin{figure}[H]\n\t\t\\begin{adjustbox}{width=1\\textwidth, center}          \\begin{tabular}{c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}c@{\\extracolsep{0em}}}\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_sim_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0106_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_sim_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_sim_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_simulation_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0106_angle_sim.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0106_angle_gt_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0106_residual_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_exp_img_0106_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_exp_107.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0106_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_sim_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_gt.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0926_residual.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_sim_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_sim_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_simulation_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0926_angle_sim.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/measurement_exp_927.png}~\n\t\t\t\t&\\includegraphics[height=0.118\\textwidth,width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_gt_exp.pdf}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/SiPRNet_img_0926_angle_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/img_0926_residual_exp.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/GS_Fashion_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/HIO_Fashion_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ADMM_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/ResNet_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/lensless_exp_img_0926_angle.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/CGAN_exp_927.png}~\n\t\t\t\t&\\includegraphics[width=0.1\\textwidth, valign=t]{Figures\/third\/NNPhase_img_0926_angle.png}\n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\t{\\scriptsize \\makecell[c]{Fourier\\\\measurement}} & {\\scriptsize \\makecell[c]{Ground-truth \\\\ (phase)}} & {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}}\\\\(ours)} }& {\\scriptsize \\makecell[c]{\\textbf{\\textit{SiPRNet}} \\\\(Error Map)} }  & {\\scriptsize GS } & {\\scriptsize HIO} & {\\scriptsize ADMM-TV} & {\\scriptsize ResNet}\n\t\t\t\t& {\\scriptsize LenslessNet}  & {\\scriptsize PRCGAN}  & {\\scriptsize NNPhase}  \n\t\t\t\t\\\\\n\t\t\t\t\n\t\t\t\\end{tabular}\n\t\t\\end{adjustbox}\n\t\t\\vspace{-0.2cm}\n\t\t\\caption{Simulation and experimental results of different phase retrieval methods on the Fashion-MNIST dataset \\cite{xiao2017\/online}. The first \\textbf{column} denotes the Fourier intensity measurements (pixel values: $0 \\rightarrow 4095$). The second \\textbf{column} shows the corresponding phase parts of the complex-valued images with scale bars (for experimental results) at the bottom left corners. The fourth \\textbf{column} denotes the error maps of \\textit{SiPRNet}. The other \\textbf{columns} present the reconstruction images through different methods. Except for the Fourier intensity measurements (the first \\textbf{column}), the colormap of the rest \\textbf{columns} ranges from $0$ to $2\\pi$. The first and second \\textbf{rows} present the simulation and experimental results of the same image. The third and the fourth \\textbf{rows} show the simulation and experimental results of another image. Please zoom in for better view.}\n\t\t\\label{fig:thirdimage}\n\t\t\n\t\\end{figure}   \n\n    \\section*{Data availability} The datasets used in this paper are available in  \\cite{li2017reliable, xiao2017\/online}. Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.\n   \t\n   \t\\section*{Funding} General Research Fund (PolyU 15225321), Centre for Advances in Reliability and Safety (CAiRS)\n    \n    \\section*{Acknowledgments} This work was supported by the Hong Kong Research Grant Council under General Research Fund no. PolyU 15225321, and the Centre for Advances in Reliability and Safety (CAiRS) admitted under AIR@InnoHK Research Cluster.\n    \n    \\section*{Disclosures} The authors declare no conflicts of interest.\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCertainly the existence of black holes was one of the most interesting predictions of General Relativity. However, the discovery that these objects, due to quantum fluctuations, emit as black bodies with temperatures dictated by the surface gravity \\cite{Bekenstein:1972tm,Bekenstein:1973ur,Hawking:1974rv,Hawking:1974sw}, a purely geometric magnitude, simultaneously exposed  black holes as  thermodynamic objects and showed that black holes are scenarios where geometry and thermodynamics intertwine.\n\nThe first law of the black hole thermodynamics (see for instance Ref.\\cite{Wald:1984rg}) is given by\n\\begin{equation}\\label{1aleySinPdV}\ndM=TdS+\\Omega dJ+ \\phi dQ,\n\\end{equation}\nand represents the balance of energy through the modification of the macroscopic parameters of the black hole. Here $M$ corresponds to the mass parameter and was considered the internal energy of the system, namely the (ADM) mass. Finally, in Eq.(\\ref{1aleySinPdV}) $T$ is the temperature computed as the $(4\\pi)^{-1} \\kappa$ with $\\kappa$ is the surface gravity of the black hole horizon, $S$ is the entropy, $\\Omega$ is the angular velocity, $J$ is the angular momentum, $\\phi$ is the electrostatic potential and $Q$ is the electric charge.\n\nBy comparing Eq.(\\ref{1aleySinPdV}) with the first law of thermodynamics one can notice the absence of the pressure\/volume term, namely $-pdV$, which would stand for \\textit{the macroscopic work} done by the system. In a matter of speaking, this was due to the lack of the concepts of volume and pressure for a black hole in the original derivation, based on accretion processes. Several works have studied this issue in the last 20 years \\cite{Padmanabhan:2002sha,Tian:2010gn,Cvetic:2010jb,Dolan:2011xt}. To address this problem let us consider the first law of thermodynamics (see for example \\cite{Dolan:2011xt}):\n\\begin{equation}\\label{1aley1}\ndU = TdS - pdV +\\Omega dJ+ \\phi dQ,\n\\end{equation}\nwhere $U$ stands for the internal energy. To extend Eq.(\\ref{1aleySinPdV}) to match Eq.(\\ref{1aley1}), one can think of assigning a volume to the black hole by considering the \\textit{volume}  defined its radius, for example $V=\\frac{4}{3}\\pi r_+^3$ for $d=4$ case. Unfortunately, since entropy is a function of the horizon radius as well, then Eq.(\\ref{1aley1}) would be inconsistent due to $dS$ and $dV$ would not be independent directions. To address this problem, in Ref.\\cite{Kastor1}, was proposed to reinterprete the mass parameter as the Enthalpy of the black hole, instead of internal energy $U$. Moreover, the cosmological constant was connected with the thermodynamic pressure. The law obtained is called the first law of (black hole) thermodynamics in the {\\it extended phase space}. In Ref.\\cite{MotorTermico1}, on the other hand, the promotion of the mass parameter to the enthalpy is based on the fact that to form a black hole would require to cut off a region of the space, and therefore an initial energy equal to $E_0 = -\\rho V$, with $\\rho$ the energy density of the system, is needed. In four dimensions the thermodynamics volume corresponds to $V=\\frac{4}{3}\\pi r_+^3$. Moreover, the presence of the cosmological constant defines $\\rho = -P$ and therefore the mass parameter can be considered equivalent to\n\\begin{equation}\nM=U-\\rho V = U+pV,\n\\end{equation}\nHere $M$ is to be recognized as the enthalpy $H$ of the system. With this in mind, the first law, in this  extended phase space, yields\n\\begin{equation} \\label{1aley2}\ndM=dH=TdS+Vdp+\\phi dQ+\\Omega dJ.\n\\end{equation}\nThis extended first law (\\ref{1aley2}) also was derived in reference \\cite{Mann1} by using Hamiltonian formalism. It is worth to mention that the definition of the extended phase space have allowed to construct a {\\it heat engine} in terms of a black hole, see some examples in references \\cite{MotorTermico1,MotorTermico5,MotorTermico7,MotorTermico13,MotorTermico14,MotorTermico15}, adding a new layer to our understanding of the black hole thermodynamics. Another interesting applications is the Joule Thompson expansion for black holes studied in \\cite{JT1,JT2,JT3,JT4}.\n\n\\subsection{Phase Transitions}\nThe study of phase transitions in black hole physics has called a renewed attention in the last years due to the AdS\/CFT conjecture. For instance, it is well known that the {\\it Hawking-Page phase transition} \\cite{Hawking:1982dh}, in the context of the AdS\/CFT correspondence, has been re-interpreted as the plasma gluon confinement\/deconfinement phase transition in the would-be dual (conformal) field theory. Similarly, the AdS Reissner Nordstr\\\"om's transitions in the $(\\phi-q)$ diagram have been interpreted as liquid\/gas phase transitions of Van der Waals fluids \\cite{Chamblin:1999tk,Chamblin:1999hg}.\n\nRecently the analysis of the $p-V$ critical behaviors (in the extended phase space) have been under studied extensively\\cite{RegularBHquimica, PV1,PV2,PV3,PV4,PV5,PV6,PV7,PV8,PV9,PV10,PV11,PV12,Hendi:2017fxp,Hendi:2015cqz,Hendi:2015hoa,Hendi:2014kha}. For instance, in \\cite{Mann2} was studied in the context of the charged $4D$ AdS black holes how the phase transitions between small\/large black hole are analogous to liquid\/gas transitions in a Van der Waals fluid. Moreover, it was also shown that the critical exponents, near the critical points, recovers those of Van der Waals fluid with the same \\textit{compressibility factor} $Z=3\/8$. In reference \\cite{Mann1} was introduced a \\textit{new interpretation} of the Hawking Page phase transition \\cite{Hawking:1982dh} mentioned above, but in the context of $p-V$ critical behavior. \n\n\\subsection{Higher Dimensions, Lovelock and thermodynamics}\n\nDuring the last years 50 years several branches of theoretical physics have noticed that considering higher dimensions is plausible. Now, considering higher dimension in gravity opens up a range of new possibilities that retain the core of the Einstein gravity in four dimensions. Lovelock is a one of these possibilities as, although includes higher powers of curvature corrections, its equations of motion are of second order and thus causality is still insured. Generic Lovelock theory is the sum of the Euler densities $L_n$ in lower dimensions ($2n \\leq d$) multiplied by coupling constants $\\alpha_n$ \\footnote{In even dimensions the maximum order, $n=d\/2$, is in turn the corresponding Euler density of the dimension and therefore does not contribute to the equations of motion,}. The Lagrangian is\n\\begin{equation}\n \\displaystyle L  =  \\sum_{n=0}^{[d\/2]} \\alpha_n L_n,\n\\end{equation}\nwhere $L_n=\\frac{1}{2^{n}}\\,\\sqrt{-g}\\ \\delta_{\\nu_{1}...\\nu_{2n}}^{\\mu_{1}...\\mu_{2n}}\\ R_{\\mu_{1}\\mu_{2}}^{\\nu_{1}\\nu_{2}}\\cdots R_{\\mu_{2n-1}\\mu_{2n}}^{\\nu_{2n-1}\\nu_{2n}}$. The $L _0 \\propto 1$ term is related by the cosmological constant, $L_1 \\propto R $ is related by the Ricci scalar, and $L_2 \\propto R^{\\alpha\\beta}_{\\hspace{2ex}\\mu\\nu}R^{\\mu\\nu}_{\\hspace{2ex}\\alpha\\beta}-4 R^{\\alpha\\nu}_{\\hspace{2ex}\\beta\\nu} R^{\\beta\\mu}_{\\hspace{2ex}\\alpha\\mu} + R^{\\alpha\\beta}_{\\hspace{2ex}\\alpha\\beta}R^{\\mu\\nu}_{\\hspace{2ex}\\mu\\nu}$ is the Gauss Bonnet density. The EOM are:\n\\begin{align}\\label{LovelockGenerico}\n\\frac{1}{\\sqrt{g}}\\frac{\\delta }{\\delta g_{\\mu\\nu}} (L\\sqrt{g}) &= G^{\\mu}_{ \\hspace{1ex} \\nu} \\nonumber \\\\\n&= \\sum_{n} \\alpha_{n} \\frac{1}{2^n} \\delta^{\\mu_1 \\ldots \\mu_{2n} \\mu}_{\\nu_1 \\ldots \\nu_{2n} \\nu} R^{\\nu_{1} \\nu_{2}}_{\\hspace{2ex}\\mu_{1} \\mu_{2}}\\ldots R^{\\nu_{2n-1} \\nu_{2n}}_{\\hspace{2ex}\\mu_{2n-1} \\mu_{2n}}\n\\end{align}\nwhere $G^{\\mu \\nu}_{(LL)}$   which, by definition, satisfies $\\nabla_\\mu G^{\\mu\\nu}_{(LL)} \\equiv 0$.\n\nIn higher curvature the $p-V$ criticality have been studied for example in reference \\cite{Mann3} for generic Lovelock black holes with conformal scalar hair arising from coupling of a real scalar field to the dimensionally extended Euler densities and in reference \\cite{Nam:2019clw} for a regular charged AdS black hole in Einstein Gauss Bonnet gravity. See also \\cite{Hendi:2015soe,Xu:2014tja}.\n\nOne drawback of a generic Lovelock gravity is the existence of more than single ground state, namely more than a single constant curvature spaces solution, or equivalently more than a single potential \\textit{effective cosmological constants} \\cite{Camanho:2011rj}. However, there are two families of Lovelock gravities that have indeed a single ground state. The first family has been originally studied in \\cite{Banados:1994ur} and has a unique but $k$ degenerated ground state. \n\nThe second case is usually called {\\it Pure Lovelock gravity}. In this case the Lagrangian is a just the $n$-single term of Lagrangian plus a cosmological constant, \\textit{i.e.}, $L = \\alpha_n L_n + \\alpha_0 L_0$. This has a single and non-degenerated ground state of negative curvature $-1\/l^2$ for $n$ odd. Several solutions of Pure Lovelock theories are known. For instance vacuum black hole solutions can be found in references \\cite{Cai:2006pq,DadhichBH,Dadhich3,PureLovelock1,PureLovelock2,Aranguiz:2015voa} and regular black holes in references \\cite{milko1,milko2}. See other applications in references \\cite{Dadhich1,Dadhich2,Dadhich4,Dadhich5}.\n\nIn the next sections the thermodynamic of the Pure Lovelock solutions will be carried out in details. For this, the cosmological constant will be promoted to the intensive thermal pressure of the system. Under this assumption, it will be shown that the system behaves as Van der Waals fluid. Finally, the critical coefficient near the critical points will be computed.\n\n\\subsubsection{{\\bf Vacuum black hole in Pure Lovelock gravity: Uncharged Asymptotically AdS Case}}\n\nLet us consider a static geometry, represented in Schwarzschild coordinates by\n\\begin{equation}\\label{linelement}\nds^2 =-f(r) dt^2+ \\frac{dr^2}{f(r)} + r^2 d \\Omega^2_{d-2}.\n\\end{equation}\nFor simplicity the cosmological constant is fixed such that\n\\begin{equation} \\label{Lambda1}\n    \\Lambda = - \\frac{(d-1)(d-2)}{2 l^{2n}}.\n\\end{equation}\nThis is similar to the definition in \\cite{Aranguiz:2015voa} but a numerical factor. The equations of motion reduce to merely\n\\begin{equation}\n\\frac{d}{dr} \\left ( r^{d-2n-1}\\big (1-f(r) \\big)^n +  \\dfrac{ r^{d-1}}{{l}^{2n}} \\right ) =0.\n\\end{equation}\nSolutions of these equations have been studied in references \\cite{Cai:2006pq,Dadhich1}. In terms of Eq.(\\ref{linelement}) the solution is defined by\n\\begin{equation} \\label{funcionPLvacio}\nf(r) =1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{r^{2n}}{l^{2n}} \\right )^{1\/n} .\n\\end{equation}\nOne can notice that $f(r)$ can take complex values for even $n$. Because of this, in this work will be considered only odd $n$. The thermodynamic pressure can be read off this definition as\n\\begin{equation} \\label{p1}\n    p=-\\frac{\\Lambda}{(d-2)\\Omega_{d-2}},\n\\end{equation}\nwhere $\\Omega_{d-2}$ is the unitary area of a $d-2$ sphere. It is worth to stress that Eqs.(\\ref{Lambda1},\\ref{p1}) coincide with the definitions in Ref.\\cite{Mann1,Mann2} for $n=1$ and $d=4$. Now, replacing Eqs.(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{funcionPLvacio}) yields\n\\begin{equation} \\label{funcionPLvacio1}\n    f(r)=1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{2 \\Omega_{d-2}r^{2n}}{d-1}p  \\right )^{1\/n},\n\\end{equation}\nwhere the dependence of $f(r)$ on $p,M$ have been made explicit.\n\n\\subsubsection{{\\bf Charged Pure Lovelock solution}}\nThe charged case is slightly difference since it is necessary to solve the Maxwell equations and to include an energy momentum tensor into the gravitational equations. Let us start by defining $A_\\mu = A_t(r) \\delta_\\mu^t$, which defines only the non-vanishing component of the Maxwell tensor \n\\begin{equation} \\label{PotencialVectorial}\n    F_{tr}=-\\partial_rA_t(r), \n\\end{equation}\nand the single non vanishing component of the Maxwell equations, \n\\begin{equation} \\label{TensorElectromagnetico}\n  \\nabla_{\\mu}F^{\\mu\\nu} =0 \\rightarrow  \\frac{d}{dr} \\left ( r^{d-2} F_{tr} \\right) =0.\n\\end{equation}\nLikewise, the gravitational equations are reduced to the single independent component\n\\begin{equation}\\label{GravitationalEMEq}\n \\frac{d}{dr} \\left ( r^{d-2n-1}\\big (1-f(r) \\big)^n +  \\dfrac{ r^{d-1}}{{l}^{2n}} \\right ) = (F_{tr})^2 r^{d-2}.\n\\end{equation}\nIt is straightforward to integrate Eqs.(\\ref{TensorElectromagnetico},\\ref{GravitationalEMEq}) yielding \\cite{Aranguiz:2015voa}\n\\begin{equation} \\label{funcionPLvacioQ}\nf(r) =1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{r^{2n}}{l^{2n}} - \\frac{Q^2}{(d-3) r^{2d-2n-4}} \\right )^{1\/n},\n\\end{equation}\nand\n\\begin{equation}\n    F_{tr}=\\frac{Q}{r^{d-2}} \\rightarrow A_t(r)= \\phi_\\infty + \\frac{Q}{(d-3)r^{d-3}},\n\\end{equation}\nAs usual $\\phi_{\\infty}=0$ is fixed such that $\\displaystyle \\lim_{r\\rightarrow \\infty} A_t(r) =0$. \n\nNow,  by direct observation, one can notice that for even $n$ the existence of certain ranges of $r$ where $f(r)$ can take complex values. To avoid this only odd $n$ will be considered from now on.\n\nAs done previously, by replacing Eqs(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{funcionPLvacioQ}), $f(r)$ can be written in terms of the thermodynamics variables as\n\\begin{equation} \\label{funcionPLvacio1Q}\n    f(r)=1 - \\left (\\frac{2M}{r^{d-2n-1}} - \\frac{2 \\Omega_{d-2}r^{2n}}{d-1}p  - \\frac{Q^2}{(d-3) r^{2d-2n-4}}   \\right )^{1\/n}.\n\\end{equation}\n \n\n\\section{Extended phase space in Vacuum Pure Lovelock Gravity}\n\nIn this section the extended phase space formalism, mentioned above, will be introduced to analyze the solutions. One can notice that the fist law of the thermodynamics, Eq.(\\ref{1aley2}), for the non rotating case takes the form\n\\begin{equation} \\label{1aleyconcarga}\n   dH= dM=TdS+Vdp + \\phi dQ.\n\\end{equation}\nNow, in order to construct a thermodynamic interpretation one must notice that under any transformation of the parameters the function $f(r_+,M,Q,l)$  must still vanishes, otherwise the transformation would not be mapping black holes into black holes in the space of solutions. Indeed, $\\delta f(r_+,M,Q,l) = 0$ and $f(r_+,M,Q,l)=0$ are to be understood as constraints on the evolution along the space of parameters. However, there is another approach by recalling that the mass parameter, $M$, is also to be understood as a function of the parameters $M(r_+,l,Q)$ as well.\n\nSince the thermodynamic parameters are $S,p$ and $Q$, therefore it is convenient to reshape $M=M(S,p,Q)$ in order to explicitly obtain \n\\begin{equation} \\label{1aley3}\n    dM= \\left ( \\frac{\\partial M}{\\partial S}    \\right)_{p,Q}dS + \\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q}dp + \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{p,S}dQ .\n\\end{equation}\nThis corresponds to the definitions of the component of the tangent vector in the space of parameters, but also correspond to the definitions of the temperature, thermodynamic volume and electric potential in the form of\n\\begin{eqnarray}\nT &=&\\left ( \\frac{\\partial M}{\\partial S}    \\right)_{p,Q},\\label{temperatura} \\\\\nV &=& \\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q} \\textrm{ and }\\label{volumen} \\\\\n\\phi &=& \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{S,p}. \\label{potencial}\n\\end{eqnarray}\n\nOn the other hand, the variation along the space of parameters of the condition defined by $f(r_+,M,p,Q)=0$,\n\\begin{eqnarray}\n  df(r_+,M,p,Q) &=& 0\\label{variacion}\\\\\n   &=& \\frac{\\partial f}{\\partial r_+}  dr_+ + \\frac{\\partial f}{\\partial M} dM + \\frac{\\partial f}{\\partial p} dp + \\frac{\\partial f}{\\partial Q} dQ,\\nonumber\n\\end{eqnarray}\nyields a second expression for $dM$ given by\n\\begin{align} \\label{variacion1}\n    dM &=  \\left (\\frac{1}{4\\pi} \\frac{\\partial f}{\\partial r_+} \\right ) \\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+ \\nonumber \\\\\n    &+ \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right ) dp +\\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial Q} \\right ) dQ,\n\\end{align}\nwhich must coincide with equation (\\ref{1aley3}). In Eq.(\\ref{variacion1}) one can recognize presence of the temperature, which geometrically is defined as\n\\begin{equation} \\label{temperaturaconocida}\n    T= \\frac{1}{4\\pi} \\frac{\\partial f}{\\partial r_+},\n\\end{equation}\n yielding \n\\begin{equation} \\label{dS1}\n\\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+= dS,\n\\end{equation}\nwhich is a very known result. Now, by the same token, the thermodynamic volume and electric potential are given by\n\\begin{equation} \\label{volumen1}\n    V=\\left ( \\frac{\\partial M}{\\partial p}    \\right)_{S,Q}= \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right ) \n\\end{equation}\nand\n\\begin{equation} \\label{potencial1}\n    \\phi = \\left ( \\frac{\\partial M}{\\partial Q}    \\right)_{S,p}= \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial Q} \\right ),\n\\end{equation}\nrespectively. These expressions will be discussed in the next sections.\n\n\\subsection{Smarr Expression}\nConsidering Plank units, one can notice that $p$, the pressure, has units of $[p]=\\ell^{-2n}$. See Eqs.(\\ref{Lambda1},\\ref{p1}, \\ref{presionPL1}). Likewise, one can check that $[M] = \\ell^{d-2n-1}$, $[Q] = \\ell^{d-n-2}$ and $[S] = \\ell^{d-2n}$. \n\nFollowing {\\it Euler's theorem} \\cite{Altamirano:2014tva}, with $M(S,p,Q)$, one can construct the {\\it Smarr formula} for Pure Lovelock gravity given by\n\\begin{equation}\n    \\frac{d-2n-1}{d-2n}M=TS+\\frac{d-n-2}{d-2n} \\phi Q - \\frac{2n}{d-2n}VP.\n\\end{equation}\nThis coincides with the definitions discussed in \\cite{Altamirano:2014tva,Mann2,Mann1} for $n=1$ . Derivations of the Smarr formula for generic Lovelock theory are discussed in \\cite{Dolan:2014vba,Kastor:2010gq}. \n\n\\subsection{ Uncharged asymptotically AdS}\nReplacing the solution of Eq.(\\ref{funcionPLvacio1}) into Eq.(\\ref{dS1}) yields\n\\begin{align}\n\\left( -\\frac{1}{4 \\pi} \\frac{\\partial f}{\\partial M} \\right)^{-1} dr_+&=2\\pi n r_+^{d-2n-1} \\nonumber \\\\\n&=d \\left ( \\frac{2}{d-2n}n \\pi r_+^{d-2n}  \\right ) = dS,\n\\end{align}\nor equivalently\n\\begin{equation} \\label{entropia1}\n    S= \\frac{2}{d-2n}n \\pi r_+^{d-2n},\n\\end{equation}\nwhich coincides with reference \\cite{Cai:2006pq}. Eq.(\\ref{volumen1}), on the other hand,  yields\n\\begin{equation} \\label{volumen11}\n    \\left( - \\frac{\\partial f}{\\partial M} \\right)^{-1} \\left ( \\frac{\\partial f}{\\partial p} \\right )  = V = \\frac{\\Omega_{d-2}}{d-1}r_+^{d-1},\n\\end{equation}\nwhich corresponds to the volume of a $(d-2)$ sphere of radius $r_+$ . This coincides with the definition in Refs.\\cite{Mann1,Mann2}.\n\n\\subsubsection{{ \\bf Fluid equation of state}}\nIt is worth to notice at this point that the temperature, see Ref.\\cite{milko1}, can be expressed as \n\\begin{equation} \\label{TemperaturaPL1}\n    4\\pi n T= \\frac{d-2n-1}{r_+} + (d-1)\\frac{ r_+^{2n-1}}{l^{2n}},\n\\end{equation}\nwhere, the temperature has units of $\\ell^{-1}$. One can make explicit the dependence on the pressure, by inserting Eqs.(\\ref{Lambda1},\\ref{p1}) into Eq.(\\ref{TemperaturaPL1}). Therefore, \n\\begin{equation} \\label{presionPL1}\n    2 \\Omega_{d-2}p = 4 \\pi n \\frac{T}{r_+^{2n-1}} - \\frac{d-2n-1}{r_+^{2n}}.\n\\end{equation}\n\n\\subsubsection{\\bf Physical Pressure}\nBefore to proceed a digression is necessary. As mentioned above $[p] = \\ell^{-2n}$, however the physical pressure, $p_G$, must be satisfied [Force\/Area]$=\\ell_p^{-2}$, since the area has units of [area]$=\\ell_p^{d-2}$. In the literature $p_g$ is called the geometrical pressure. Since $p$ and $p_G$ must be connected by $p_G \\sim \\alpha_n p$, namely $[p_{G}] = \\left [\\alpha_n p \\right]=\\ell_p^{-d}$, therefore\n \\begin{itemize}\n    \\item the {\\it coupling constants} must satisfy $[\\alpha_n]=\\ell^{2n-d}$ \\cite{Oliva1}. For $n=1$ this coincides with the inverse of the higher dimensional Newton constant $G_d^{-1}$ \\cite{Maartens:2003tw} and\n    \\item  there is still room for a dimensionless constant which can be used to adjust the definition. \n\\end{itemize}\nWith this in mind,  $p_G$ can be taken as \n\\begin{equation}\n    p_{G} =2 \\Omega_{d-2} \\ell_p^{2n-d} p = 4 \\pi n \\frac{T}{\\ell_p^{d-2n}r_+^{2n-1}} + \\ldots,\n\\end{equation}\nwhich, in turns, defines the \\textit{specific volume} of the system as\n\\begin{equation} \\label{volumenespecificoPL}\n    v=\\frac{\\Omega_{d-2}}{2\\pi} \\ell_p^{d-2n}r_+^{2n-1}.\n\\end{equation}\n\nNotice that $v$ indeed has units of volume, namely $[v]= \\ell_p^{d-1}$ \\cite{Hennigar:2018cnh}. Finally, after some replacements, \n\\begin{equation} \\label{presionPL2}\n   p = \\frac{n T}{v} - \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{2n\/(2n-1)}}\\frac{d-2n-1}{v^{2n\/(2n-1)}},\n\\end{equation}\nwhich can be recognized as the Van der Walls equation for $n=1$, namely $P=T\/(v-b)-a\/v^2$, with $b=0$ \\cite{Mann1}.  \n\n\\subsubsection{{\\bf Hawking-Page phase transition}}\n\nIn reference \\cite{Hawking:1982dh} was analyzed the thermodynamic behavior of Schwarzschild AdS space, which differs from the Schwarzschild case due to the presence of the AdS gravitational potential,  namely the presence of $\\sim r^2\/l^2$ in $f(r)$ \\cite{Czinner:2015eyk}. In this case there is a phase transition between black hole and AdS radiation at a critical temperature $T_{HP}$ where the Gibbs free energy, $G=M-TS$ vanishes \\cite{Mann2,Czinner:2015eyk,Wang:2019vgz}. \n\nFig.\\ref{HP} displays the numerical behavior of the Gibbs free energy v\/s temperature for $n=3$ and $d=10$ for Pure Lovelock. The upper curve represents the unstable small black hole (namely with negative heat capacity) and the lower curve represents the stable large black hole. It is direct to show that this behavior is similar for any other set of values of $n$ and $d$. \n\nThe generic behavior is displayed in Fig.\\ref{HPgenerico}. The upper green curve represents the unstable small black hole, the lower blue curve represents the stable large black hole, and the orange curve represents the thermal AdS radiation whose Gibbs free energy vanishes \\cite{Mann1,Czinner:2015eyk,Wang:2019vgz,Costa:2015gol}. One can notice that for $[T_{min},T_{HP}[$ the large stable black hole has positive Gibbs free energy,  therefore, the preferred state corresponds to the thermal AdS radiation. On the other hand, for $T>T_{HP}$, the preferred state is the large stable black hole whose Gibbs free energy is negative. This hints the existence of a Hawking Page phase transition between radiation and the large black hole states at $T=T_{HP}$.  \n\nFinally, it is worth to notice, from Fig.\\ref{HP}, that the value of $T_{HP}$ increases as the pressure increases. This behavior is similar to the HP phase transition for the Schwarzschild AdS black holes \\cite{Mann1,Belhaj:2015hha} or the polarized AdS black holes \\cite{Costa:2015gol}.\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{HP1.eps}}\n\\caption{HP phase transitions for $n=3$, $d=10$ and $Q=1$, $p=0,0000045$(blue), $p=0,0000050$(red), $p=0,0000055$ (green). T (horizontal axis) v\/s G (vertical axis). }\n\\label{HP}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{HPgenerico.eps}}\n\\caption{Generic behavior of Gibbs free energy (vertical axis) v\/s T (horizontal axis). HP phase transition at $T_{HP}$}\n\\label{HPgenerico}\n\\end{figure}\n\n\\subsection{ Charged Pure Lovelock solution}\nBy replacing Eq.(\\ref{funcionPLvacio1Q}) into Eqs.(\\ref{dS1},\\ref{volumen1}) the expressions for the entropy (\\ref{entropia1}) and the volume (\\ref{volumen11}) are obtained . On the other hand, replacing Eq.(\\ref{funcionPLvacio1Q}) into Eq.(\\ref{potencial1}) yields \n\\begin{equation} \\label{potencial2}\n    \\phi= \\frac{Q}{(d-3)r^{d-3}}.\n\\end{equation}\nThis thermodynamics quantities, obtained by mean of variation of parameters, are consistent with the values presented in the literature. \n\n\\subsubsection{\\bf Fluid equation of state}\nIn this case the temperature can be written as\n\\begin{equation} \\label{TemperaturaPL1Q}\n    4\\pi n T= \\frac{d-2n-1}{r_+} + (d-1)\\frac{ r_+^{2n-1}}{l^{2n}}- \\frac{Q^2}{r_+^{2d-2n-3}} .\n\\end{equation}\nBy inserting equations (\\ref{Lambda1},\\ref{p1}) into equation (\\ref{TemperaturaPL1Q}) is obtained\n\\begin{equation} \\label{presionPL2Q}\n    2 \\Omega_{d-2}p = 4 \\pi n \\frac{T}{r_+^{2n-1}} - \\frac{d-2n-1}{r_+^{2n}} + \\frac{Q^2}{r_+^{2d-4}}.\n\\end{equation}\nNow, by using the definition of the specific volume, defined in Eq.(\\ref{volumenespecificoPL}) with $\\ell_p=1$,\n\\begin{eqnarray}\n p &=& \\frac{n T}{v} - \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{2n\/(2n-1)}}\\frac{d-2n-1}{v^{2n\/(2n-1)}} \\nonumber \\\\\n   &+& \\frac{1}{2\\Omega_{d-2}(2\\pi\/\\Omega_{d-2})^{(2d-4)\/(2n-1)}}\\frac{Q^2}{v^{(2d-4)\/(2n-1)}}. \\label{presionPL3}\n\\end{eqnarray}\nOne can notice the similarity of this relation with the corresponding one for a Van der Waals fluid for $n=1$ but with the presence of an addition third term.  \n\n\\subsubsection{\\bf Critical points and compressibility factor}\nTo compare Eq.(\\ref{presionPL3}) with the behavior of a Van der Waals fluid it is necessary to determine the critical points of the system. The second order critical points are defined by the conditions,  \n\\begin{equation}\n    \\frac{\\partial p}{\\partial v}=0 \\textrm{ and } \\frac{\\partial^2 p}{\\partial v^2}=0.\n\\end{equation}\nThese determine the critical values\n\\begin{equation}\n    v_c = \\frac{\\Omega_{d-2}}{2\\pi}\\left ( \\frac{2d^2-2dn-7d+4n+6}{n(d-2n-1)}Q^2   \\right )^{(2n-1)\/(2d-2n-4)} ,\n\\end{equation}\nand\n\\begin{align}\n    T_c &= \\frac{n(d-2n-1)}{2\\pi n (2n-1)(2\\pi\/\\Omega_{d-2})^{1\/(2n-1)}v^{1\/(2n-1)}} \\nonumber \\\\\n        &-\\frac{(d-2)Q^2(2\\pi\/\\Omega_{d-2})^{(2n-2d+4)\/(2n-1)}v^{(2n-2d+4)\/(2n-1)}}{2\\pi n (2n-1)(2\\pi\/\\Omega_{d-2})^{1\/(2n-1)}v^{1\/(2n-1)}}\n\\end{align}\nNotice that $p_c=p(v_c,T_c)$ can be determined from equation (\\ref{presionPL3}) by evaluation on the critical values $T_c$ and $v_c$. \n\nIn the table (\\ref{tabla1}) critical values $v_c$, $T_c$ and  $p_c$ and the {\\it compressibility factor} $Z=p_cv_v\/T_c$ are displayed for different values of $n$ and $d$ . For $n=1$ and $d=4$ the compressibility factor has the exact value $Z=3\/8$ which coincides with the value of the compressibility for a Van der Waals fluid. In general, compressibility factor can be written as\n\\begin{equation} \\label{factorcompresibilidad}\n    Z=\\frac{2d-2n-3}{4(d-2)},\n\\end{equation}\nwhich implies that $Z<1$ extrictly. This implies that this can be interpreted as a low pressure gas. \n\\begin{table*} \n\\caption{Critical values and Compressibility Factor ($Z$) .}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}lccccc@{}}\n\\hline\n\\multicolumn{1}{c}{$n$   } & \\multicolumn{1}{c}{$d$ } & \\multicolumn{1}{c}{$v_c$}& \\multicolumn{1}{c}{$T_c$}&\\multicolumn{1}{c}{$p_c$}&\\multicolumn{1}{c}{$Z=\\dfrac{p_cv_c}{T_c}$} \\\\\n \\hline\n$1$ & $4$  & $\\Omega_{d-2}\/(2\\pi)Q\\sqrt{6}$ & $1\/(18\\pi Q) \\sqrt{6}$ &$1\/(24\\Omega_{d-2}Q^2)$ & $3\/8$   \\\\\n\\hline \n$1$ & $5$  & $\\Omega_{d-2}\/(4\\pi)(120Q^2)^{1\/4}$ & $4\/(75\\pi)(6750\/Q^2)^{1\/4}$ &$2\/(45\\Omega_{d-2}Q)\\sqrt{30}$ & $5\/12$   \\\\\n\\hline\n$1$ & $6$  & $\\Omega_{d-2}\/(6\\pi)(6804Q^2)^{1\/6}$ & $9\/(98\\pi)(201.684\/Q^2)^{1\/6}$ &$9\/(112\\Omega_{d-2})(294\/Q^2)^{1\/3}$ & $7\/16$   \\\\\n\\hline\n$3$ & $8$  & $\\Omega_{d-2}\/(2\\pi)(14Q^2)^{5\/6}$ & $3\/(490\\pi) (14^5\/Q^2)^{1\/6}$ &$1\/(280\\Omega_{d-2}Q^2)$ & $7\/24$   \\\\\n\\hline\n$3$ & $9$  & $\\Omega_{d-2}\/(4\\pi)(21^5\\cdot 2^3Q^{10})^{1\/8}$ & $8\/(945\\pi) (21^7\\cdot 2\/Q^2)^{1\/8}$ &$4\/(735\\Omega_{d-2})(21\\cdot 2^3\/Q^6)^{1\/4}$ & $9\/28$   \\\\\n\\hline\n$3$ & $10$  & $\\Omega_{d-2}\/(18\\pi)(792)^{1\/2}Q$ & $3\/(242\\pi) (2^7\\cdot 11^9 \\cdot 9\/Q^2)^{1\/10}$ &$9\/(704\\Omega_{d-2})(726\/Q^6)^{1\/5}$ & $11\/32$   \\\\\n\\hline\n$5$ & $12$  & $\\Omega_{d-2}\/(2\\pi)(22Q^2)^{9\/10}$ & $5\/(2178\\pi)(22^9\/Q^2)^{1\/10}$ &$1\/(792\\Omega_{d-2}Q^2)$ & $11\/40$   \\\\\n\\hline\n\\end{tabular*} \\label{tabla1}\n\\end{table*}\n\n\\subsubsection{\\bf $P-v$ curve.}\n\nIn Fig.\\ref{p-v} is displayed for $n=3$ and $d=10$ the behavior of the curve $p-v$ defined by equation (\\ref{presionPL3}). Although this is an example still this behavior is generic for any values of $n$ and $d$. \n\nFor values of temperature $T>T_{c}$, the second and third factors of Eq.(\\ref{presionPL3}) are negligible in comparison with the first one, and therefore the curve approximates the form  $p \\cdot v \\propto T$ and thus mimicking the behavior of an ideal gas. Conversely, for $T<T_{c}$  as $v$ increases the $p$, which diverges for $v=0$, decreases until reach a local minimum. Next, $p$ increases until reaching a local maximum. Finally $p$ decreases asymptotically until reaching $p=0$. Thus for $T<T_{c}$ ( or for $p<p_{c}$ due that $T \\propto p$) the behavior is analogue to the Van der Waals fluid.  \n\nIn standard vapor-liquid theory is well established that an increase in pressure must be correlated with a decrease in volume and vice versa. Therefore, see Fig.\\ref{p-v}, one can notice the existence of a range of the specific volume, says $]v_{min},v_{max}[$, which must be considered nonphysical due to both pressure and volume increase simultaneously. On the other hand, it can be also noticed that for a single value of the specific pressure, there might exist up to three possible values of specific volume $v$ with one of them always within the nonphysical region. Therefore, for analysis one must only consider the two proper solutions that satisfy either $v_1<v_{min}$ or $v_2>v_{max}$ with $p(v_1)=p(v_2)$. In fluid theory these two solutions are known as the {\\it Van der Waals loop} and physically this corresponds to a {\\it vapor liquid equilibrium}, highlighting that phase transitions take place. \n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{p-v.eps}}\n\\caption{$p-v$ curve for $T_1<T_2<T_3<T_{c}<T_4<T_5$}\n\\label{p-v}\n\\end{figure}\n\n\\subsubsection{\\bf Temperature}\nThe behavior of the temperature is displayed in figure \\ref{FigTemperatura}(a) for $p>p_{c}$, in figure \\ref{FigTemperatura}(b) for $p=p_{c}$ and in figure \\ref{FigTemperatura}(c) for $p<p_{c}$ . Although for these figures $n=3$ and $d=10$ it is straightforward to show that this enfolds the generic behavior for any $n$ and $d$. One can notice the presence of an extreme black hole case for small $r_+$ where the temperature vanishes. For $p>p_{c}$ the temperature is an increasing function of $r_+$. For $p=p_{c}$ the temperature has one inflexion point at $r_+=r_{infl}$. More relevant for this discussion is the case for $p<p_{c}$, where the fluid is analogous to the Van der Waals, and where the temperature has a local minimum and a local maximum at $r_+=r_{min}$ and $r_+=r_{max}$, respectively.\n\n\\begin{figure}\n\\centering\n\\subfigure[Temperature for $p=1>p_{c}$ .]{\\includegraphics[width=75mm]{TemperaturaMayor.eps}} \n\\subfigure[Temperature for $p=p_{c} \\approx 0,001608$ ]{\\includegraphics[width=75mm]{TemperaturaCritica.eps}}\n\\subfigure[Temperature for $p=0,00001<p_{c}$ ]{\\includegraphics[width=75mm]{TemperaturaMenor.eps}}\n\\caption{Temperature behavior for $n=3$ and $d=10$ with $Q=1$ .}\n\\label{FigTemperatura}\n\\end{figure}\n\n\\subsubsection{\\bf Heat Capacity}\nThe heat capacity is displayed in Fig.\\ref{CalorEspecificoMayoroigual}(a) for $p>p_c$, in figure \\ref{CalorEspecificoMayoroigual}(b) for $p=p_c$ and in figure \\ref{CalorEspecificoMenor} for $p<p_c$. As previously, although $n=3$ and $d=10$ it is straightforward to show that Fig.\\ref{CalorEspecificoMayoroigual} enfold the generic behavior for any $n$ and $d$. One can notice that \n\n\\begin{itemize}\n    \\item For $p>p_c$ the heat capacity is a positive increase function of $r_+$, and there is no phase transition, thus black hole is always stable.\n    \\item For $p=p_c$ small and large black hole coexist at the inflexion point $r_+=r_{infl}$, where the heat capacity $C \\to \\infty$.\n    \\item For the $p<p_c$ case, the derivative $(dT\/dr_+)$ can vanish for two values of  $r_+=r_{min}$ and $r_+=r_{max}$, as observed in the figure \\ref{FigTemperatura}(c). This implies, due to $C= (dS\/dr_+)\/(dT\/dr_+)|_{p,Q}$, that the heat capacity becomes ill-defined at those values of $r_+$. In this way $r_{min}$ and $r_{max}$ define three regions. For $r_+<r_{max}$ one can notice that $C>0$ defining a small stable black hole. Next, there is small unstable ($C<0$) region for $r \\in ]r_{max},r_{min}[$. Finally there is a third region $r_+>r_{min}$ where the system is a large stable black hole $(C>0)$. This hints the existence of phase transitions but, by means of the following analysis of the Gibbs free energy, one can check that only the \\textit{small stable bh\/large stable bh} transition is allowed. \n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\subfigure[Heat capacity for $p=1>p_c$.]{\\includegraphics[width=75mm]{Cmayor.eps}}\n\\subfigure[Heat capacity for $p=p_c \\approx 0,001608 $]{\\includegraphics[width=75mm]{Ccritica.eps}}\n\\caption{Behavior of Heat Capacity for $n=3$ and $d=10$ with $Q=1$  .}\n\\label{CalorEspecificoMayoroigual}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\subfigure[Heat capacity for $r_+ \\in (0.896,1.6)$.]{\\includegraphics[width=75mm]{CMenor1.eps}}\n\\subfigure[Heat capacity for $r_+ \\in (2.7,3.5)$]{\\includegraphics[width=75mm]{CMenor2.eps}}\n\\caption{Behavior of Heat Capacity for $n=3$ and $d=10$ with $Q=1$, $p=0,00001<p_c$ .}\n\\label{CalorEspecificoMenor}\n\\end{figure}\n\n\\subsubsection{\\bf Gibbs Free energy}\nThe Gibbs free energy, defined as $G=M-TS$ \\cite{Mann1,Hennigar:2018cnh,Czinner:2015eyk,Wang:2019vgz}, is displayed for $n=3$ and $d=10$ in figure \\ref{GibbsT}(a) for $p>p_c$, in figure \\ref{GibbsT}(b) for $p=p_c$ and in \\ref{GibbsT}(c) for $p<p_c$. It is direct to check that this behavior is similar for other values of $n$ and $d$. As well known \\cite{RegularBHquimica} the Gibbs free energy describes the global stability of the system. It is worth to recall that the global minimum represents the most likely state, while the preferred state at fixed temperature corresponds to the minimal value of the Gibbs free energy. \n\\begin{itemize}\n    \\item For pressure larger than the critical pressure, the Gibbs free energy is a single valued function.\n    \\item For $p=p_c$ there is a cusp at $T=T_c$, which coincides with the radius $r_+=r_{infl}$, thus, since at this point $C=T dS\/dT=-T (\\partial^2G\/\\partial T^2) \\to \\infty$, the discontinuity on the second derivative of Gibbs function implies the presence of second order phase transition between small stable\/large stable black holes.\n    \\item The generic behavior of the Gibbs free energy for $p<p_c$ is displayed in figure \\ref{GibbsT1}. We see three possibles black hole states: small stable, small unstable and large stable. The preferred state is such that the Gibbs free energy has the minimum value. Thus, for $]T_0,T_{max}]$ the preferred state correspond to stable large black hole. However, for $T<T_0$ the preferred state corresponds to the small stable black hole. Thus, at $T=T_0$ there is a first order phase transition between large\/small stable black hole.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\subfigure[Gibbs for $p=1>p_{c}$ .]{\\includegraphics[width=63mm]{GibbsMayor1.eps}} \\hspace{15mm}\n\\subfigure[ Gibbs for $p=p_{c} \\approx 0,001608$ ]{\\includegraphics[width=63mm]{GibbsCritico1.eps}}\n\\subfigure[Gibbs for $p=0,00001<p_{c}$ ]{\\includegraphics[width=63mm]{GibbsMenor1.eps}}\n\\caption{Gibbs free energy (vertical axis) v\/s T (horizontal axis) for $n=3$ and $d=10$ with $Q=1$.}\n\\label{GibbsT}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n{\\includegraphics[width=3in]{GibbsT1.eps}}\n\\caption{Generic behavior of Gibbs free energy (vertical axis) v\/s T (horizontal axis)}\n\\label{GibbsT1}\n\\end{figure}\n\n\\subsubsection{\\bf Behavior near critical points}\nLet's define the following dimensionless variables \n\\begin{equation} \\label{omega}\n    \\omega =\\left ( \\frac{V}{V_c} \\right)^{2n-1}-1=\\left ( \\frac{v}{v_c} \\right)^{d-1}-1\n\\end{equation}\nand \n\\begin{equation} \\label{t}\n    t = \\frac{T}{T_c}-1,\n\\end{equation}\nto analyze the behavior nearby the critical points. It is direct to check that near the critical points, \\textit{i.e.} $V \\to V_c$ (or $v \\to v_c$) and $T \\to T_c$, the variables $\\omega \\to 0$, and $t \\to 0$, respectively. The pressure in Eq.(\\ref{presionPL3}) is displayed in table \\ref{tabla2} at $O(t\\omega^2, \\omega^4)$. The truncation will be justified below . \n\\begin{table*} \n\\caption{Behavior of pressure near critical points .}\n\\begin{tabular*}{\\textwidth}{@{\\extracolsep{\\fill}}llc@{}}\n\\hline\n\\multicolumn{1}{l}{$n$   } & \\multicolumn{1}{l}{$d$ } & \\multicolumn{1}{c}{$p$} \\\\\n \\hline\n$1$ & $4$  & $p \\approx 1 + 8\/3 t - 8\/9t \\omega -4\/81 \\omega^3$    \\\\\n\\hline \n$1$ & $5$  & $p \\approx 1 + 12\/5t - 3\/5t \\omega -1\/32 \\omega^3$  \\\\\n\\hline\n$1$ & $6$  & $p \\approx 1 + 16\/7t - 16\/35t \\omega -8\/375 \\omega^3$   \\\\\n\\hline\n$3$ & $8$  & $p \\approx 1 + 72\/7t - 72\/49t \\omega -12\/8575 \\omega^3$   \\\\\n\\hline\n$3$ & $9$  & $p \\approx 1 + 28\/3t - 7\/6t \\omega -7\/6400 \\omega^3$   \\\\\n\\hline\n$3$ & $10$  & $p \\approx 1 + 96\/11t - 32\/33t \\omega -16\/18225 \\omega^3$   \\\\\n\\hline\n$5$ & $12$  & $p \\approx 1 + 200\/11t - 200\/121t \\omega -100\/323433 \\omega^3$    \\\\\n\\hline\n\\end{tabular*} \\label{tabla2}\n\\end{table*}\n\nIn general it is possible to approximate the pressure as\n\\begin{equation} \\label{presiongenerica}\n    p \\approx 1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3,\n\\end{equation}\nwhereas \n\\begin{equation}\n    dP= -\\frac{P_c n}{Z (d-1)} \\left ( t+ \\frac{(2d-2n-3)}{2(d-1)^2(2n-1)^2} \\omega^2 \\right ) d \\omega.\n\\end{equation}\nTo compute the \\textit{ critical exponents} one can follow \\cite{Mann2}. \n\\begin{itemize}\n    \\item The {\\it $\\alpha$ exponent} describes the behavior of the heat capacity at constant volume defined as\n    \\begin{equation}\n        C_v = T \\frac{\\partial S}{\\partial T} \\propto |t|^{-\\alpha}.\n    \\end{equation}\n    In the case at hand, since entropy and volume are both functions of horizon radius, see Eqs.(\\ref{entropia1},\\ref{volumen11}), then a constant volume implies a constant entropy as well. Therefore it is satisfied that $C_v=0$ which implies that there is no dependence on $|t|$. In turn this implies that\n    \\begin{equation}\n        \\alpha=0.\n    \\end{equation}\n  \\item The \\textit{exponent} $\\beta$ describes the behavior of the \\textit{order parameter} defined as\n  \\begin{equation} \\label{eta}\n      \\eta = V_l-V_s \\propto |t|^\\beta.\n  \\end{equation}\n  To compute this order parameter one can use the \\textit{Maxwell's area law}\n\\begin{equation} \\label{integral1}\n    \\oint VdP=0,\n\\end{equation}\nwhere, the volume (\\ref{omega}) is approximated such that $VdP$ is truncated to $O(t\\omega^3, \\omega^5)$, in other words, \n\\begin{equation} \\label{aproxVolumen}\n    V \\approx V_c \\left (  1 + \\frac{1}{2n-1} \\omega   \\right ).\n\\end{equation}\nTherefore, closed integral (\\ref{integral1}) becomes\n\\begin{equation} \\label{integral2}\n  V_c  \\oint  dP + \\frac{V_c}{2n-1} \\oint \\omega dP =0.\n\\end{equation}\nThe second integral of the left side of (\\ref{integral2}) yields\n\\begin{equation}\n \\int^{\\omega_s}_{\\omega_l}\\omega dP= \\int^{\\omega_s}_{\\omega_l} \\omega   \\left ( t+ \\frac{(2d-2n-3)}{2(d-1)^2(2n-1)^2} \\omega^2 \\right ) d \\omega =0.\n\\end{equation}\nwhich has the non trivial solution given by\n\\begin{equation} \\label{condicion}\n    \\omega_s = -\\omega_l.\n\\end{equation}\nOn the other hand, the first integral of the left side of (\\ref{integral2}) yields\n\\begin{align}\n    &\\int^{\\omega_s}_{\\omega_l}dP = 0 \\nonumber \\\\\n&1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega_l - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3_l = \\nonumber \\\\\n&1 + \\frac{n}{Z}t - \\frac{n t}{(d-1)Z} \\omega_s - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3_s,\n\\end{align}  \nThis, by condition (\\ref{condicion}), yields\n\\begin{equation} \\label{omegal}\n    \\omega_l=(d-1)(2n-1) \\sqrt{- \\frac{6}{2d-2n-3}t},\n\\end{equation}\n with $t<0$. Replacing Eqs. (\\ref{aproxVolumen}) and (\\ref{omegal}) into Eq.(\\ref{eta}) one can obtain \n \\begin{equation}\n     \\eta = \\frac{2V_c}{2n-1} \\omega_l = 2(d-1)V_c\\sqrt{- \\frac{6}{2d-2n-3}t},\n \\end{equation}\n Comparing this result with Eq.(\\ref{eta}) one can uncover that \n \\begin{equation}\n     \\beta = \\frac{1}{2}.\n \\end{equation}\n\\item Now one can compute the \\textit{exponent} $\\gamma$ which describes the behavior  under \\textit{isothermal compressibility}, $\\kappa_T$, defined by\n\\begin{equation}\n\\kappa_T= - \\frac{1}{V}\\frac{\\partial V}{\\partial P} \\bigg |_T \\propto |t|^{-\\gamma}.\n\\end{equation}\n\nBy using Eqs.(\\ref{presiongenerica},\\ref{aproxVolumen}) one can prove that\n\\begin{equation}\n    \\frac{\\partial P}{\\partial V}=P_c \\frac{\\partial p}{\\partial \\omega} \\frac{\\partial \\omega}{\\partial V} \\propto -(2n-1)\\frac{P_c}{V_c} \\frac{n}{(d-1)Z}t,\n\\end{equation}\nand therefore\n\\begin{equation}\n    \\kappa_T \\propto \\frac{(d-1) Z}{P_c n(2n-1)t},\n\\end{equation}\nfrom which is direct to read that\n\\begin{equation}\n    \\gamma=1.\n\\end{equation}\n\n\\item Finally, one can compute \\textit{exponent} $\\delta$ which describes the behavior on the critical isotherm $T=T_c$, and therefore for $t = 0$. In this case this is defined by\n\\begin{equation}\n    |P-P_c| \\propto |V-V_c|^\\delta.\n\\end{equation}\nFrom Eq.(\\ref{presiongenerica}), at $t=0$, it is possible to notice that\n\\begin{equation} \\label{presiongenerica1}\n    p -1 \\approx - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\omega^3.\n\\end{equation}\nUsing the approximation of Eq.(\\ref{aproxVolumen}) one can show that\n\\begin{equation}\n    \\frac{P-P_c}{P_c} \\approx - \\frac{(2d-2n-3)n}{6Z(d-1)^3(2n-1)^2} \\left ( (2n-1) \\frac{V-V_c}{V_c}   \\right )^3,\n\\end{equation}\nand therefore\n\\begin{equation}\n    \\delta=3. \n\\end{equation}\n\\end{itemize}\n\nThis critical exponents just computed are similar to those of Van der Waals gas. Although the presence of extra dimensions and the value of $n$ modify the value of the compressibility factor respect to the well known value $Z=3\/8$, they do not affect the value of the critical exponents, and thus, the behavior is still similar to that of Van der Waals fluid near the critical exponents.\n\n\\section{Conclusion and discussion} \nIn this article it has been analyzed the thermodynamics of the Pure Lovelock solutions in $d$ dimensions, in an extended phase space including the introduction of pressure and volume as dual thermodynamic variables and the mass parameter standing for the enthalphy of the system. A linear relation between the cosmological constant and the thermodynamics pressure, valid for all value of $n$ (odd) and $d$, has been established. The thermodynamic volume obtained corresponds to a generalization of the black hole \\textit{volume}. The Smarr formula for Pure Lovelock gravity was constructed as well.\n\nIn addition, it was shown that the variation of parameters are constrained to follow a first law of thermodynamics in the extended phase space. The temperature is defined by surface gravity, as usual, the entropy coincides with the value computed {\\it a la Wald} \\cite{Cai:2006pq, milko1} and the electric potential matches the usual known definition. \n\nThe equation state for the uncharged and charged cases was obtained. For the uncharged case, a Hawking-Page-Like phase transitions between thermal radiation and large stable black hole has been found. For the charged case, it was found that the compressibility factor, $Z$, is a generic function of $d$ and $n$ given by equation (\\ref{factorcompresibilidad}). Remarkably, $Z<1$ estrictly and therefore the behavior always corresponds to a real gas. \n\nIt was found the existence of a critical temperature, $T_c$, where phase transitions occur. The mapping of the $p-v$ curves indicate that for Pure Lovelock gravity the behavior is similar to an ideal gas for $T>T_c$. For $T<T_c$ the behavior is analogous to a Van der Walls fluid. Furthermore, there are a first order phase transition between small stable\/large stable black hole, which are analogous to liquid\/gas phase transitions.\n\nFinally, we have computed the critical exponents whose values are similar to those of the Van der Waals gas. Furthermore, it is possible to write the pressure as a generic function of $n$ and $d$ given by equation (\\ref{presiongenerica}). Thus, the presence of extra dimensions and the value of $n$ does not modify the known values of the critical points, therefore the general behavior can be considered similar to a  Van der Waals fluid near the critical exponents.\n\n\\section*{Acknowledgments}\nThis work was partially funded by grant DI-08-19 UNAB.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{References}}\n\\bibliographystyle{elsarticle-num}\n\n\\begin{document}\n\t\\begin{frontmatter}\n\t\t\\title{Minimum $2$-vertex strongly biconnected spanning directed subgraph problem}\n\t\t\\author{Raed Jaberi}\n\t\t\n\t\t\\begin{abstract}  \n\t    A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $2$-vertex-strongly biconnected if $|V|\\geq 3$ and the induced subgraph on $V\\setminus\\left\\lbrace w\\right\\rbrace $ is strongly biconnected for every vertex $w\\in V$. In this paper we study the following problem.\t\t\n\t\tGiven a $2$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{2sb} \\subseteq E$ of minimum size such that the subgraph $(V,E^{2sb})$ is $2$-vertex-strongly biconnected.\n\t\t\\end{abstract} \n\t\t\\begin{keyword}\n\t\t\tDirected graphs \\sep Approximation algorithms  \\sep Graph algorithms \\sep strongly connected graphs \\sep Strongly biconnected directed graphs\n\t\t\\end{keyword}\n\t\\end{frontmatter}\n\t\\section{Introduction}\n\t  A directed graph $G=(V,E)$ is strongly biconnected if $G$ is strongly connected and its underlying graph is biconnected. A strongly biconnected directed graph $G=(V,E)$ is called $k$-vertex-strongly biconnected if $|V|> k$ and for each $L\\subset V$ with $|L|<k$, the induced subgraph on $V\\setminus L$ is strongly biconnected. The minimum $k$-vertex-strongly biconnected spanning subgraph problem (denoted by MKVSBSS) is formulated as follows.\t\t\n\tGiven a $k$-vertex-strongly biconnected directed graph $G=(V,E)$, compute an edge subset $E^{ksb} \\subseteq E$ of minimum size such that the subgraph $(V,E^{ksb})$ is $k$-vertex-strongly biconnected. In this paper we consider the MKVSBSS problem for $k=2$.\n Each $2$-vertex-strongly-biconnected directed graph is $2$-vertex-connected, but the converse is not necessarily true. Thus, optimal solutions for minimum $2$-vertex-connected spanning subgraph (M2VCSS) problem are not necessarily feasible solutions for the $2$-vertex strongly biconnnected spanning subgraph problem, as shown in Figure \\ref{figure:sbiconnectedexampleoptimalsolutions}.\n\n\\begin{figure}[htp]\n\t\\centering\n\t\n\t\\subfigure[]{\n\t\n\\scalebox{0.79}{\n\n\t\t\\begin{tikzpicture}[xscale=2]\n\t\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\t\\node (v1) at (-1.6,3.1) {$1$};\n\t\t\\node (v2) at (-2.4,0) {$2$};\n\t\t\\node (v3) at (-0.5, -2.5) {$3$};\n\t\t\\node (v4) at (3.6,-2.1) {$4$};\n\t\t\\node (v5) at (0.5,1.6) {$5$};\n\t\t\\node (v6) at (3.2,0.5) {$6$};\n\t\t\\node (v7) at (3.7,3.2) {$7$};\n\t\n\t\t\\begin{scope}   \n\t\t\\tikzstyle{every node}=[auto=right]   \n\t\t\\draw [-triangle 45] (v2) to (v5);\n\t\t\t\\draw [-triangle 45] (v2) to (v6);\n\t\t\t\t\\draw [-triangle 45] (v3) to (v5);\n\t\\draw [-triangle 45] (v2) to (v1);\n\t\\draw [-triangle 45] (v1) to[bend right] (v2);\n\t\\draw [-triangle 45] (v1) to (v5);\n\t\\draw [-triangle 45] (v5) to[bend right] (v1);\n\t\\draw [-triangle 45] (v5) to (v7);\n\t\\draw [-triangle 45] (v7) to[bend right] (v5);\n\t\\draw [-triangle 45] (v7) to (v6);\n\t\\draw [-triangle 45] (v6) to[bend right] (v7);\n\t\\draw [-triangle 45] (v6) to (v4);\n\t\\draw [-triangle 45] (v4) to[bend right] (v6);\n\t\\draw [-triangle 45] (v4) to (v3);\n\t\\draw [-triangle 45] (v3) to[bend right] (v4);\n\t\\draw [-triangle 45] (v3) to (v2);\n\t\\draw [-triangle 45] (v2) to[bend right] (v3);\n\t\\draw [-triangle 45] (v1) to[bend left] (v7);\n\t\\draw [-triangle 45] (v5) to (v6);\n\t\\draw [-triangle 45] (v6) to (v3);\n\t\\draw [-triangle 45] (v4) to (v2);\n\t\\draw [-triangle 45] (v3) to (v1);\n\t\\draw [-triangle 45] (v4) to (v5);\n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v7) to [bend left](v4);\n\t\t\\end{scope}\n\t\t\\end{tikzpicture}\n\t}\n\t}\n\t\\subfigure[]{\n\t\t\\scalebox{0.79}{\n\t\t\\begin{tikzpicture}[xscale=2]\n\t\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\\node (v1) at (-1.6,3.1) {$1$};\n\\node (v2) at (-2.4,0) {$2$};\n\\node (v3) at (-0.5, -2.5) {$3$};\n\\node (v4) at (3.6,-2.1) {$4$};\n\\node (v5) at (0.5,1.6) {$5$};\n\\node (v6) at (3.2,0.5) {$6$};\n\\node (v7) at (3.7,3.2) {$7$};\n\t\t\n\t\t\\begin{scope}   \n\t\t\\tikzstyle{every node}=[auto=right]   \n\t\t\n\t\t\\draw [-triangle 45] (v2) to (v1);\n\t\t\\draw [-triangle 45] (v1) to[bend right] (v2);\n\t\t\\draw [-triangle 45] (v1) to (v5);\n\t\t\\draw [-triangle 45] (v5) to[bend right] (v1);\n\t\t\\draw [-triangle 45] (v5) to (v7);\n\t\t\\draw [-triangle 45] (v7) to[bend right] (v5);\n\t\t\\draw [-triangle 45] (v7) to (v6);\n\t\t\\draw [-triangle 45] (v6) to[bend right] (v7);\n\t\t\\draw [-triangle 45] (v6) to (v4);\n\t\t\\draw [-triangle 45] (v4) to[bend right] (v6);\n\t\t\\draw [-triangle 45] (v4) to (v3);\n\t\t\\draw [-triangle 45] (v3) to[bend right] (v4);\n\t\t\\draw [-triangle 45] (v3) to (v2);\n\t\t\\draw [-triangle 45] (v2) to[bend right] (v3);\n\t\n\t\t\\end{scope}\n\t\t\\end{tikzpicture}}}\n\\subfigure[]{\n\t\\scalebox{0.79}{\n\t\\begin{tikzpicture}[xscale=2]\n\t\\tikzstyle{every node}=[color=black,draw,circle,minimum size=0.9cm]\n\t\\node (v1) at (-1.6,3.1) {$1$};\n\\node (v2) at (-2.4,0) {$2$};\n\\node (v3) at (-0.5, -2.5) {$3$};\n\\node (v4) at (3.6,-2.1) {$4$};\n\\node (v5) at (0.5,1.6) {$5$};\n\\node (v6) at (3.2,0.5) {$6$};\n\\node (v7) at (3.7,3.2) {$7$};\n\t\n\t\\begin{scope}   \n\t\\tikzstyle{every node}=[auto=right]   \n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v2) to (v1);\n\n\t\\draw [-triangle 45] (v1) to (v5);\n\t\n\t\\draw [-triangle 45] (v5) to (v7);\n\t\n\t\\draw [-triangle 45] (v7) to (v6);\n\n\t\\draw [-triangle 45] (v6) to (v4);\n\n\t\\draw [-triangle 45] (v4) to (v3);\n\n\t\\draw [-triangle 45] (v3) to (v2);\n\t\n\t\\draw [-triangle 45] (v1) to[bend left] (v7);\n\t\\draw [-triangle 45] (v5) to (v6);\n\t\\draw [-triangle 45] (v6) to (v3);\n\t\\draw [-triangle 45] (v4) to (v2);\n\t\\draw [-triangle 45] (v3) to (v1);\n\t\\draw [-triangle 45] (v2) to (v5);\n\t\\draw [-triangle 45] (v7) to [bend left](v4);\n\t\\end{scope}\n\t\t\\end{tikzpicture}}}\n\\caption{(a) A $2$-vertex strongly biconnected graph. (b) An optimal solution for the minimum $2$-vertex-connected spanning subgraph problem. But note that this subgraph is not $2$-vertex strongly biconnected.(c) An optimal solution for the minimum $2$-vertex strongly biconnected spanning subgraph problem}\n\\label{figure:sbiconnectedexampleoptimalsolutions}\n\\end{figure}\n\nThe problem of finding a $k$-vertex-connected spanning subgraph of a $k$-vertex-connected directed graph is NP-hard for $k\\geq 1$ \\cite{G79}. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that the number of edges in each minimal $k$-vertex-connected directed graph is at most $2kn$ \\cite{CT00}. Cheriyan and Thurimella \\cite{CT00} gave a $(1+1\/k)$-approximation algorithm for the minimum $k$-vertex-connected spanning subgraph problem. Georgiadis \\cite{Georgiadis11} improved the running time of this algorithm for the M2VCSS problem and presented a linear time approximation algorithm that achieves an approximation factor of $3$ for the M2VCSS problem. Georgiadis et al. \\cite{GIK20} provided linear time $3$-and $2$- approximation algorithms based on the results of \\cite{GT16,G10,ILS12,FILOS12} for the M2VCSS problem. Furthermore, Georgiadis et al. \\cite{GIK20} improved the algorithm of Cheriyan and Thurimella when $k=2$. Strongly connected components of a directed graph and blocks of an undirected graphs can be found in linear time using Tarjan's algorithm \\cite{TAARJAN72}.\n Wu and Grumbach \\cite{WG2010} introduced the concept of strongly biconnected directed graph and strongly connected components.  The MKVSBSS problem is NP-hard for $k\\geq 1$. In this paper we study the MKVSBSS problem when $k=2$ (denoted by M2VSBSS). \n\t\n\\section{Approximation algorithm for the M2VSBSS problem} \nIn this section we present an approximation algorithm (Algorithm \\ref{algo:approximationalgorithmfor2sb}) for the M2VSBSS Problem. This algorithm is based on b-articulation points, minimal 2-vertex-connected subgraphs, and Lemma \\ref{def:addingedgestoreducesbcs}. A vertex $w$ in a strongly biconnected directed graph $G$ is a b-articulation points if $G\\backslash\\left\\lbrace w \\right\\rbrace $ is not strongly biconnected \\cite{Jaberi20}. \n\n\n\\begin{lemma} \\label{def:addingedgestoreducesbcs}\n\tLet $G_s=(V,E_s)$ be a subgraph of a strongly biconnected directed graph $G=(V,E)$ such that $G_s$ is strongly connected and $G_s$ has $t>0$ strongly biconnected components. Let $(u,w)$ be an edge in $E\\setminus E_s$  such that $u,w $ are in not in the same strongly\n\tbiconnected component of $G_s$. Then the directed subgraph $(V,E\\cup \\left\\lbrace (u,w) \\right\\rbrace )$ contains at most $t-1$ strongly biconnected components.\n\\end{lemma}\n\\begin{proof}\n\tSince $G_s$ is strongly connected, there exists a simple path $p$ from $w$ to $u$ in $G_s$. Path $p$ and edge $(u,w)$ form a simple cycle. Consequently, the vertices $u,w$ are in the same strongly biconnected component of the subgraph $(V,E\\cup \\left\\lbrace (u,w) \\right\\rbrace )$.\n\\end{proof}\n\n\\begin{figure}[htbp]\n\t\\begin{myalgorithm}\\label{algo:approximationalgorithmfor2sb}\\rm\\quad\\\\[-5ex]\n\t\t\\begin{tabbing}\n\t\t\t\\quad\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\quad\\=\\kill\n\t\t\t\\textbf{Input:} A $2$-vertex strongly biconnected directed graph $G=(V,E)$ \\\\\n\t\t\t\n\t\t\t\\textbf{Output:} a $2$-vertex strongly biconnected subgraph $G_{2s}=(V,E_{2s})$\\\\\n\t\t\t{\\small 1}\\> find a minimal $2$-vertex-connected subgraph $G_{1}=(V,E_{1})$ of $G$.\\\\\n\t\t\t{\\small 2}\\> \\textbf{if} $G_{1}$ is $2$-vertex strongly biconnected \\textbf{then} \\\\\n\t\t\t{\\small 3}\\>\\>output  $G_{1}$\\\\\n\t\t\t{\\small 4}\\> \\textbf{else}\\\\\n\t\t\t{\\small 5}\\>\\>$E_{2s} \\leftarrow E_{1}$\\\\\n\t\t\t{\\small 6}\\>\\>  $G_{2s}\\leftarrow (V,E_{2s})$ \\\\\n\t\t\t{\\small 7}\\>\\>  identify the b-articulation points of $G_{1}$.\\\\\n\t\t\t{\\small 8}\\>\\>  \\textbf{for} evry b-articulation point $b\\in V$ \\textbf{do} \\\\\n\t\t\t{\\small 9}\\>\\>\\>  \\textbf{while} $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $ is not  strongly biconnected \\textbf{do}\\\\ \n\t\t{\\small 10}\\>\\>\\>\\>  calculate the  strongly biconnected components of $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $\\\\\n\t\t{\\small 11}\\>\\>\\>\\> find an edge $(u,w) \\in E\\setminus E_{2s}$ such that $u,w $ are not in \\\\\n\t\t{\\small 12}\\>\\>\\>\\>the same strongly biconnected components of $G_{2s}\\setminus\\left\\lbrace b \\right\\rbrace $.\\\\\n\t\t{\\small 13}\\>\\>\\>\\> $E_{2s} \\leftarrow E_{2s} \\cup\\left\\lbrace  (u,w)\\right\\rbrace $ \\\\\n\t\t{\\small 14}\\>\\>\\>output $G_{2s}$\n\t\n\t\t\\end{tabbing}\n\t\\end{myalgorithm}\n\\end{figure}\n\n\\begin{lemma} \nAlgorithm \\ref{algo:approximationalgorithmfor2sb} returns a $2$-vertex strongly biconnected directed subgraph.\n\\end{lemma}\t\n\\begin{proof}\nIt follows from Lemma \\ref{def:addingedgestoreducesbcs}.\n\\end{proof}\n\nThe following lemma shows that each optimal solution for the M22VSBSS problem has at least $2n$ edges.\n\\begin{lemma} \\label{def:optsolutionatlesttwon}\n\tLet $G=(V,E)$ be a $2$-vertex-strongly biconnected directed graph. Let $O\\subseteq E$ be an optimal solution for the M2VSBSS problem. Then $|O|\\geq 2n$. \n\t\n\\end{lemma}\n\\begin{proof}\n\tfor any vertex $x\\in V$, the removal of $x$ from the subgraph $(V,O)$ leaves a strongly biconnected directed subgraph. Since each strongly biconnected directed graph is stronly connected,\n\tthe subgraph $(V,O)$ has no strong articulation points. Therefore, the directed subgraph $(V,O)$ is $2$-vertex-connected.\n\\end{proof}\nLet $l$ be the number of b-articulation points in  $G_{1}$. The following lemma shows that Algorithm \\ref{algo:approximationalgorithmfor2sb} has an approximation factor of $(2+l\/2)$.\n\\begin{theorem}\n\tLet $l$ be the number of b-articulation points in  $G_{1}$. Then, $|E_{2s}|\\leq l(n-1)+4n$.\n\\end{theorem}\n\\begin{proof}\n\tResults of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that $|E_1|\\leq 4n$  \\cite{CT00,Georgiadis11}. Moreover, by Lemma \\ref{def:optsolutionatlesttwon}, every optimal solution for the M22VSBSS problem has size at least $2n$. For every b-articulation point in line $8$,  Algorithm \\ref{algo:approximationalgorithmfor2sb} adds at most $n-1$ edge to $E_{2s}$ in while loop. Therefore, $|E_{2s}|\\leq l(n-1)+4n$\n\\end{proof}\n\\begin{Theorem}\n\tThe running time of Algorithm \\ref{algo:approximationalgorithmfor2sb} is $O(n^{2}m)$.\n\\end{Theorem}\n\\begin{proof}\n\tA minimal $2$-vertex-connected subgraph can be found in time $O(n^2)$ \\cite{Georgiadis11,GIK20}. B-articulation points can be computed in $O(nm)$ time.\n\tThe strongly biconnected components of a directed graph can be identified in linear time \\cite{WG2010}. \n\tFurthermore, by Lemma \\ref{def:addingedgestoreducesbcs}, lines $9$--$13$ take $O(nm)$ time.\n\\end{proof}\n\n\\section{Open Problems}\n Results of Mader \\cite{Mader71,Mader72} imply that the number of edges in each minimal $k$-vertex-connected undirected graph is at most $kn$ \\cite{CT00}. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that the number of edges in each minimal $k$-vertex-connected directed graph is at most $2kn$ \\cite{CT00}. These results imply a $2$-approximation algorithm \\cite{CT00} for minimum $k$-vertex-connected spanning subgraph problem for undirected and directed graphs \\cite{CT00} because every vertex in a $k$-vertex-connected undirected graphs has degree at least $k$ and every vertex in a $k$-vertex-connected directed graph has outdegree at least $k$ \\cite{CT00}. Note that these results imply a $7\/2$ approximation algorithm for the M2VSBSS problem by calculating a minimal $2$-vertex-connected directed subgraph of a $2$-vertex strongly biconnected directed graph $G=(V,E)$ and a minimal $3$-vertex connected undirected subgraph of the underlying graph of $G$.\n \n \\begin{lemma}\nLet $G=(V,E)$ be a $2$-vertex strongly biconnected directed graph. Let $G_1=(V,L)$ be a minimal $2$-vertex-connected subgraph of $G$ and let $G_2=(V,U)$ be a minimal $3$-vertex-connected subgraph of the underlying graph of $G$. Then the directed subgraph $G_s=(V,L\\cup A )$ is $2$-vertex strongly connected, where $A=\\left\\lbrace (v,w) \\in E \\text{ and } (v,w) \\in U \\right\\rbrace $. Moreover, $|L\\cup A|\\leq 7n$\n \\end{lemma}\n\\begin{proof}\n\tLet $w$ be any vertex of the subgraph $G_s$. Since the $G_1=(V,L)$ is $2$-vertex-connected, subgraph $G_s$ has no strong articulation points. Therefore, $G_s\\setminus\\left\\lbrace w\\right\\rbrace $ is strongly connected. Moreover, the underlying graph of  $G_s\\setminus\\left\\lbrace w\\right\\rbrace $ is biconnected because the underlying graph of $G_s$ is $3$-vertex-connected. Results of Edmonds \\cite{Edmonds72} and Mader \\cite{Mader85} imply that $|L|<4n$. Results of Mader \\cite{Mader71,Mader72} imply that $|U|\\leq 3n$.\n\\end{proof}\n \n An open problem is whether each minimal $2$-vertex strongly biconnected directed graph has at most $4n$ edges.\n\n Cheriyan and Thurimella \\cite{CT00} presented a $(1+1\/k)$-approximation algorithm for the minimum $k$-vertex-connected spanning subgraph problem for directed and undirected graphs. The algorithm of Cheriyan and Thurimella \\cite{CT00} has an approximation factor of $3\/2$ for the minimum $2$-vertex-connected directed subgraph problem. Let $G=(V,E)$ be a $2$-vertex strongly biconnected directed graph and let $E^{CT}$ be the output of the algorithm of Cheriyan and Thurimella \\cite{CT00}. The directed subgraph $(V,E^{CT})$ is not necessarily $2$-vertex strongly biconnected. But a $2$-vertex strongly biconnected subgraph can be obtained by performing the following third phase. For each edge $e\\in E\\setminus E^{CT}$, if the underlying graph of $G\\setminus \\left\\lbrace e\\right\\rbrace $ is $3$-vertex-connected, delete $e$ from $G$. We leave as open problem whether this algorithm has an approximation factor of $3\/2$ for the M2VSBSS problem.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\chapter{Acknowledgement}\\label{Acknowledgement}\n\n\nIt seems incredible but the Ph.D. really comes to the end! \n\nFor sure this has been the most intense period of my life both by the scientific point of view, with {\\em my} Di-Bosoni, the Higgs, tons of new things learned and discovered, and also by the personal point of view,\nwith thousands of miles across the World and several houses and cities where I spent my time. \n\nThe real risk was to get lost along the way but, luckily, I have been continuously surrounded, helped and supported by so so many nice people that it is really hard to thank everybody in one page.\n\nFor sure I couldn't had made it without the support of Prof. Giorgio Chiarelli (who now knows me by the far 2007); the work wouldn't had been possible without him but neither without the continuous exchange of ideas with Dr. Sandra Leone and within all the CDF Pisa Group.\n\nFermilab has been the other indispensable ingredient: the best place in the Chicago suburbs to do Physics and also all the rest. I met there a great research team (all the WHAM'ers), wonderful scientists and researchers from whom I learned so much.\n\nHowever there have not just been Physics in these years\\footnote{Although often friendship, study, work, spare time and Physics all overlaps in a indistinguishable way...}, my family has been always with me also from the other side of the ocean or farther away. No possibility to get anywhere without their help. Then there is Celine, ``la Piccolina'', that I met once, lost, and luckily I met again and now we are together!\n\nAnd finally there are all the friends from {\\em Sancasciani}, {\\em Cottonwood}, {\\em Siena}, {\\em Chicago}, {\\em Pisa}, the {\\em old} and the {\\em new Summer Students}, the {\\em ControTV} group and others I am surely missing. I spend with all of you a very good time and first or later I will meet again everybody scattered all around the World.\n\nThis travel lasted three years (well almost four), a pretty long period in the lifespan of somebody that is not even thirty but it flowed rapidly like a mountain torrent.\n\n\n\n\\chapter[First Evidence of Diboson in $\\ell\\nu+ HF$ ($7.5~fb^{-1}$)]{First Evidence of Diboson in Lepton plus Heavy Flavor Channel ($7.5~\\textrm{fb}^{-1}$)}\\label{App:7.5}\n\nThe first evidence of diboson production in the  $\\ell \\nu + HF$ final state was already obtained in the Summer of 2011 with a preliminary version of this analysis performed on a smaller dataset, $7.5$~fb$^{-1}$ of CDF data. \n\nThe differences with respect to the work described in the main parts of the thesis are the followings:\n\\begin{itemize}\n\\item a dataset of $7.5$~fb$^{-1}$ of CDF data.\n\\item The forward electrons category (PHX) was not used.\n\\item A previous version~\\cite{CHEP_svm} of the SVM multi-jet rejection algorithm was employed. The algorithm was optimized only for binary classification and it was not possible to use the shape of the output distribution. Therefore the \\mbox{${\\not}{E_T}$}  distribution was used to in the normalization of the $W + $ jets and the multi-jet backgrounds.\n\\item  No flavor separator ($KIT-NN$) information was used for single-tag events. Therefore only the di-jet invariant mass distribution was used as the final signal to background discriminator both for the single and double tagged signal regions.\n\\item The statistical analysis of the significance of the observation was performed in a different way. The likelihood ratio~\\cite{mclimit} of the signal and {\\em test} hypothesis, after the fit over the nuisance parameter, was used\\footnote{This method can not be easily applied to the significance estimate of two signals (for example $WW$ {\\em vs} $WZ\/ZZ$) therefore, in the main part of the thesis, we moved to the method described in Section~\\ref{sec:sigma_eval} for the statistical analysis}. \n\\item We also estimated $95$\\% Confidence Level (CL) limits, both in the case of diboson signal and no diboson signal. The first case can be used to constrain new physics models which produces an increase of the TGC couplings.\n\\end{itemize}\nA summary of the event selection, the background estimate and the statistical analysis is reported in the following.\n\n\n\n\\section{Event Selection and Background Estimate}\\label{sec:Data}\n\nWe select events consistent with the $\\ell\\nu+HF$ signature. \n\nThe charged lepton candidate online and offline identification is described in Chapter~\\ref{chap:sel} but we consider only the tight central lepton candidates (CEM, CMUP, CMX), the loose lepton candidates (BMU, CMU, CMP, CMIO, SCMIO, CMXNT) and the isolated track candidates (ISOTRK). Loose leptons and ISOTRK are classified together in the EMC category. The $W\\to \\ell \\nu$ selection is completed by a cut on the \\mbox{${\\not}{E_T}$}  variable, corrected for the presence of muons and jets. We require a \\mbox{\\mbox{${\\not}{E_T}$} $>20$~GeV} for CEM and EMC leptons while we relax the cut down to \\mbox{\\mbox{${\\not}{E_T}$} $>10$~GeV} for the tight muons categories. \n\nThe $HF$ jets selection is also identical to the one reported in Chapter~\\ref{chap:sel}: two central ($|\\eta_{Det}| < 2.0$) jets with E$_T^{cor} > 20$~GeV (energy corrected for detector effects) \non which we require the identification of one or two secondary vertices with the \\texttt{SecVtx} $b$-tagging algorithm. The $b$-tagging requirement divides the selected sample in a {\\em pretag} control region (no $b$-tag requirement) and two signal regions characterized by exactly one or two $b$-tagged jets.\n\nA relevant difference in the event selection comes from the multi-jet background rejection strategy. We used a previous version of the SVM algorithm~\\cite{CHEP_svm} based on the following six variables (described in Section~\\ref{sec:var_sel}):\n\\begin{itemize}\n\\item Lepton $p_T$, \\mbox{${\\not}{E_T}$} , $MetSig$, $\\Delta\\phi(lep,\\mbox{${\\not}{E_T^{raw}}$})$,  $E_T^{raw,jet2}$ and $E_T^{cor,jet2}$.\n\\end{itemize}\nThe discriminant was optimized to work in the central region of the detector and, as binary classifier, the performances were similar to the present algorithm described in Appendix~\\ref{chap:AppSvm}.\n\nThe background estimate was done with the same methodology described in Chapter~\\ref{chap:bkg}. The only difference is that the fits, used to estimate $W + $ jets and multi-jet normalizations in the different lepton categories and tag regions, were performed on the \\mbox{${\\not}{E_T}$}  distributions. Figure~\\ref{fig:qcd_fit_met} shows the result of the pretag maximum likelihood fit on the \\mbox{${\\not}{E_T}$} , pretag sample, distribution of the different lepton categories: CEM, CMUP, CMX, EMC.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CEM}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CMUP}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_CMX}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/PretafFit_ISOTRK}\n\\caption[\\mbox{${\\not}{E_T}$}  Fit Used for $W +$ Jets Estimates in the Pretag Sample]{$W + $ jets  and non-$W$ fraction estimates, on the pretag sample, with a maximum likelihood fit on the \\mbox{${\\not}{E_T}$}  distributions. The non-$W$ background is shown in pink while the $W + $ jets component is in green. The dashed line is the sum of all the backgrounds and the points are the data. The figures show (left to right and top to bottom) the CEM, CMUP, CMX and EMC \n  charged lepton categories.} \\label{fig:qcd_fit_met}\n\\end{center}\n\\end{figure}\n\nThe total background estimate is reported in Tables~\\ref{tbl:1tag} and~\\ref{tbl:2tags}. The final statistical analysis of the selected events is performed on the $M_{Inv}(jet1, jet2)$ distribution of four channels: single and double tagged candidates for the central tight leptons (CEM+CMUP+CMX) and EMC leptons. Figure~\\ref{fig:mjj_1tag} shows the high-statistics single-tag  $M_{Inv}(jet1, jet2)$ distribution of central tight leptons and of the EMC leptons.\n\n\\begin{table}[htb]\n  \\begin{center}\n    \\begin{tabular}{lcccc}\n      \\toprule\n      \\multicolumn{5}{c}{\\bf Single-tag Event Selection}\\\\\n      Lepton ID & CEM                       &             CMUP             & CMX    & EMC    \\\\  \n      \\midrule    \n\n     Pretag Data       & 61596  & 29036 & 18878  &  27946 \\\\\\midrule\n      $Z+$jets            &27.9 $\\pm$ 3.5     & 43.0 $\\pm$ 5.5    & 27.3 $\\pm$ 4.2     & 65.0 $\\pm$ 8.4     \\\\  \n      $t\\bar{t}$        &201.3 $\\pm$ 19.6   &109.8 $\\pm$ 10.7   & 55.0 $\\pm$ 7.1     & 171.9 $\\pm$ 16.9    \\\\  \n      Single Top $s$    &52.9 $\\pm$ 4.8     &28.2 $\\pm$ 2.6     & 14.0 $\\pm$ 1.8     & 38.0 $\\pm$ 3.5       \\\\  \n      Single Top $t$    &71.4 $\\pm$ 8.4     &37.4 $\\pm$ 4.4     & 19.8 $\\pm$ 2.9     & 49.5 $\\pm$ 5.8       \\\\  \n      $WW$                &68.0 $\\pm$ 9.4     &33.3 $\\pm$ 4.6     & 20.3 $\\pm$ 3.3     & 38.4 $\\pm$ 5.3       \\\\  \n      $WZ$                &21.8 $\\pm$ 2.3     & 11.5 $\\pm$ 1.25   & 7.4 $\\pm$ 1.0      & 14.1 $\\pm$ 1.6     \\\\ \n      $ZZ$                &0.44 $\\pm$ 0.04    & 0.65 $\\pm$ 0.06   & 0.42 $\\pm$ 0.05    & 0.86 $\\pm$ 0.08    \\\\    \n      $W + b\\bar{b}$    &632.9 $\\pm$ 254.2  & 309.8$\\pm$ 124.1  & 192.3 $\\pm$ 77.1   & 308.9 $\\pm$ 124.3  \\\\\n      $W + c\\bar{c}$    &331.0 $\\pm$ 133.7  & 155.1 $\\pm$ 62.5  & 96.2 $\\pm$ 38.8    & 164.2 $\\pm$ 66.4   \\\\\n      $W + cj$            &259.9 $\\pm$ 105.0  & 127.8 $\\pm$ 51.5  & 75.3 $\\pm$ 30.4    & 106.4 $\\pm$ 43.0   \\\\\n      $W+LF$            &605.2 $\\pm$ 71.3   & 283.8 $\\pm$ 31.7   & 181.0 $\\pm$ 20.6  & 346.2 $\\pm$ 39.2   \\\\   \n      Non-$W$             &173.9 $\\pm$ 69.6   & 45.8 $\\pm$ 18.3  & 2.8 $\\pm$ 1.1       & 100.9 $\\pm$ 40.4   \\\\   \n      \\midrule                                                                                               \n       {\\bf Prediction}        &{\\bf 2446 $\\pm$ 503} & {\\bf 1186 $\\pm$ 242 } &{\\bf  692 $\\pm$ 149}  & {\\bf 1404 $\\pm$ 243} \\\\\n       {\\bf Observed}          &{\\bf 2332}               & {\\bf 1137}              & {\\bf 699}                & {\\bf 1318}               \\\\\n      \\midrule                                                                                            \n      {\\bf Dibosons}             &{\\bf 89.7 $\\pm$ 10.2}    & {\\bf 44.8 $\\pm$ 5.05}   & {\\bf  27.7 $\\pm$ 3.9}   &  {\\bf 52.5 $\\pm$ 5.9}     \\\\\n      \\bottomrule                                                   \n    \\end{tabular}\n\n    \\caption[Observed and Expected Events with One \\texttt{SecVtx} Tags]{Summary of  observed and expected events with one \\texttt{SecVtx} tag, in the $W+2$ jets sample, in \n      7.5 fb$^{-1}$ of data.}\\label{tbl:1tag}\n  \\end{center}\n\\end{table}\n\n\\begin{table}[htb]\n\\begin{center}\n    \\begin{tabular}{lcccc}\n    \\toprule\n    \\multicolumn{5}{c}{\\bf Double-tag Event Selection}\\\\\n    Lepton ID & CEM                       &             CMUP             & CMX    & EMC    \\\\  \n    \\midrule    \n      \n       Pretag Data       & 61596   &  29036  &  18878  &  27946 \\\\\\midrule\n       $Z+$jets            & 0.9 $\\pm$ 0.1   &2.0 $\\pm$ 0.3    & 1.2 $\\pm$ 0.2   & 3.1 $\\pm$ 0.4  \\\\    \n       $t\\bar{t}$        & 42.2 $\\pm$ 6.1  &22.2 $\\pm$ 3.2   & 11.1 $\\pm$ 1.9  & 34.4 $\\pm$ 5.0  \\\\\n       Single Top $s$    & 14.1 $\\pm$ 2.0  &7.6 $\\pm$ 1.1    & 3.7 $\\pm$ 0.6   & 10.2 $\\pm$ 1.4  \\\\ \n       Single Top $t$    & 4.2 $\\pm$ 0.7   &2.3 $\\pm$ 0.4    & 1.2 $\\pm$ 0.2   & 3.1 $\\pm$ 0.5   \\\\ \n       $WW$                & 0.6 $\\pm$ 0.1   & 0.26 $\\pm$ 0.07 & 0.16 $\\pm$ 0.04 & 0.33 $\\pm$ 0.08\\\\  \n       $WZ$                & 4.0 $\\pm$ 0.6   & 1.9 $\\pm$ 0.3   & 1.4 $\\pm$ 0.2   & 2.4 $\\pm$ 0.4  \\\\  \n       $ZZ$                & 0.06 $\\pm$ 0.01 & 0.12 $\\pm$ 0.02 & 0.09 $\\pm$ 0.01 & 0.16 $\\pm$ 0.02  \\\\\n       $W + b\\bar{b}$    & 81.9 $\\pm$ 33.2 &42.2 $\\pm$ 17.1  & 23.4 $\\pm$ 9.5  & 44.9 $\\pm$ 18.2\\\\\n       $W + c\\bar{c}$    & 4.7 $\\pm$ 1.9   &2.3 $\\pm$ 1.0    & 1.3 $\\pm$ 0.5   & 2.8 $\\pm$ 1.1   \\\\ \n       $W + cj$            & 3.7 $\\pm$ 1.5   &1.9 $\\pm$ 0.8    & 1.0 $\\pm$ 0.4   & 1.8 $\\pm$ 0.7  \\\\    \n       $W+LF$            & 3.2 $\\pm$ 0.7   &1.6 $\\pm$ 0.3    & 0.9 $\\pm$ 0.2   & 2.2 $\\pm$ 0.4  \\\\  \n       Non-$W$             & 7.9 $\\pm$ 3.2   &4.8 $\\pm$ 1.9    & 0.1 $\\pm$ 0.5   & 0.0 $\\pm$ 0.5   \\\\   \n       \\midrule                                                                               \n       {\\bf Prediction}        & {\\bf 167.3 $\\pm$ 38.0} &{\\bf 88.9 $\\pm$ 19.6}  & {\\bf 45.4 $\\pm$ 10.9} & {\\bf 105.3 $\\pm$ 21.5}\\\\\n       {\\bf Observed}          & {\\bf 147}             &{\\bf 74}               &{\\bf  39}              & {\\bf 106}             \\\\  \n       \\midrule\n       {\\bf Dibosons}             &{\\bf  4.6 $\\pm$ 0.6}   &{\\bf  2.1 $\\pm$ 0.3}   & {\\bf 1.5 $\\pm$ 0.2}   &{\\bf  2.7 $\\pm$ 0.4}   \\\\\n       \\bottomrule\n       \\end{tabular}  \n\n       \\caption[Observed and Expected Events with Two \\texttt{SecVtx} Tags]{Summary of observed and expected events with two \\texttt{SecVtx} tags, in the $W+2$ jets sample, in $7.5$~fb$^{-1}$ of data.}\\label{tbl:2tags}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure} [!ht]\n\\centering\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/1svt_cen}\n\\includegraphics[width=0.495\\textwidth]{App7.5\/Figs\/1svt_emc} \n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for Single \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for the single \\texttt{SecVtx} tagged events. Tight leptons (CEM+CMUP+CMX combined) on the left and EMC leptons on the right. The best fit values for the rate and shape of the backgrounds are used in the figures.}  \n\\label{fig:mjj_1tag}\n\\end{figure}\n\n\n\n\\section{Statistical Analysis}\n\nAt the time of this analysis, the process $WZ\/WW \\rightarrow \\ell \\nu $ + Heavy Flavors was not yet observed at hadron colliders. Therefore we started by evaluating a 95\\% exclusion CL on a potential signal and then we evaluated the cross section of the process and its significance. \n\nMost of the statistical analysis procedure has been described in Chapter~\\ref{chap:StatRes}. We build a likelihood function (Equation~\\ref{eq:likelihood_full}) with templates derived from the selected data and backgrounds, both shape and rate systematics are taken into account as described in Section~\\ref{sec:sys_desc}. The only relevant change concerns the $b$-tag SF for $c$-marched jets where we applied the same prescriptions of the $WH$ CDF search, it does not double the uncertainty for $c$-matched quarks. \n\nThe last relevant difference is the evaluation of the 95\\% CLs for the diboson signal. The CLs are evaluated by integrating the likelihood distribution over the unknown parameter $\\alpha$ (i.e. the diboson cross section) up to cover 95\\% of the total possible outcomes. In formulas:\n\\begin{equation}\n\\int^{\\bar{\\alpha}}_{0} \\mathscr{L}(\\alpha)\\mathrm{d}\\alpha.  \n\\end{equation}\nwhere $\\mathscr{L}(\\alpha)$ is derived from Equation~\\ref{eq:likelihood_full} after the integration over the nuisance parameters, $\\vec{\\alpha}$ is reduced to just one dimension because we perform CLs on only one signal and the limit on the integration, $\\bar{\\alpha}$, is given by the condition of 95\\% coverage.\n\nA first set of expected CLs are obtained assuming no SM diboson production and generating Pseudo Experiments (PEs) on the base of the expected background yields varied within the assigned systematics. \nCombining single--tagged and double--tagged results for all lepton categories, we find an expected limit of:\n\\begin{equation}\n\\left( 0.575^{+0.33}_{-0.31} \\right) \\times \\textrm{SM prediction.}\n\\end{equation}\n\nWe also calculated a second set of expected CLs assuming the predicted SM diboson yield in the PEs generation. In this case an excess in the observed CLs would indicate the presence of {\\em new physics}.\nCombining single--tagged and double--tagged results for all lepton categories, we find an expected limit of:\n\\begin{equation}\n\\left( 1.505 ^{+0.46}_{-0.61}\\right) \\times \\textrm{SM prediction.}\n\\end{equation}\n\nThe observed limit of $1.46$ times the SM prediction is thereby consistent with the existence of a signal and no presence of new physics.\n\nTable~\\ref{tab:limits} summarizes the expected (and observed) 95\\% production limits in units of the SM prediction. \n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccc}\n\\toprule\n Category     & Expected 95\\% CL & Expected 95\\% CL & Observed \\\\ \n&  (No Diboson) &  (With Diboson)  & 95\\% CL \\\\\\midrule\nSingle Tag  &&&\\\\\nCEM+CMUP+CMX          & $0.715 ^{+0.42}_{-0.38}$ & $1.625 ^{+0.54}_{-0.69}$ & 2.07 \\\\\nEMC                   & $1.215 ^{+0.57}_{-0.46}$ & $2.125 ^{+0.76}_{-0.6}$ & 1.70 \\\\ \\midrule\nDouble Tag &&&\\\\\nCEM+CMUP+CMX         & $4.015 ^{+1.94}_{-1.50}$ & $4.875 ^{+2.15}_{-1.7}$ & 3.03 \\\\ \nEMC                  & $6.075 ^{+2.92}_{-2.20}$ & $7.015 ^{+3.15}_{-2.42}$ & 8.59 \\\\ \\midrule\nAll combined                & $0.575 ^{+0.33}_{-0.31}$ & $1.505 ^{+0.46}_{-0.61}$ & 1.46 \\\\\n\\bottomrule                                                                                                       \n\\end{tabular}\n\\caption[Expected and Observed 95\\% CL for Diboson Production]{Expected and observed 95\\% exclusion confidence levels for each lepton category, single and double tagged events, in units of the SM cross section for diboson production. Expected CLs are produced also including the diboson signal generated PEs, thus probing the contribution of new physics processes.}\\label{tab:limits}\n\\end{center}\n\\end{table}\n\n\n\n\\subsection{Significance and Cross Section Measurement}\n\nTo compute the significance of the signal, we performed a hypothesis test comparing the data to the likelihood ratio of the {\\em null} and {\\em test} hypotheses. \n\nThe {\\em null} hypothesis, $H_0$, assumes all the predicted processes except diboson production. The {\\em test} hypothesis, $H_1$, assumes that the diboson production cross section and the branching ratio into $HF$ are the ones predicted by the SM. The likelihood ratio is defined as:\n\\begin{equation}\\label{eq:l_ratio}\n-2 \\ln Q = -2 \\ln \\frac{ \\mathscr{L}(data|H_1,\\hat{\\theta})}{ \\mathscr{L}(data|H_0,\\hat{\\hat{\\theta}})}\n\\end{equation}\nwhere $\\mathscr{L}$ is defined by Equation~\\ref{eq:likelihood_full}, $\\theta$ represents the nuisance parameters describing the uncertain values of the quantities studied for systematic error, $\\hat{\\theta}$ the best fit values of $\\theta$  under $H_1$ and $\\hat{\\hat{\\theta}}$ are the best fit values of the nuisance parameters under $H_0$.\n\nTo perform the hypothesis test we generated two sets of PEs, one assuming $H_0$ and a second one assuming $H_1$ and we evaluated Equation~\\ref{eq:l_ratio} for each pseudo-data outcome. The distributions of the  $- 2 \\ln Q$ values are shown in Figure~\\ref{fig:gaussians}. The integral of the $H_0$ $- 2 \\ln Q$ distribution below the $- 2 \\ln Q$ value of the real data gives the observed $p-$value of the signal.\n\nWe obtained an observed $p-$value of 0.00120, corresponding to a 3.03$\\sigma$ excess and producing the {\\em evidence} of the diboson signal.  The result is compatible with the expectation as the {\\em test} hypothesis $p-$value, obtained from the median of the $H_1$ distribution of $ - 2 \\ln Q$, is $0.00126$.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.95\\textwidth]{App7.5\/Figs\/combined_significance}\n\\caption[Significance Estimate of Diboson Signal with $ - 2 \\ln Q$]{Distributions  of $ - 2 \\ln Q$  for the {\\em test} hypothesis $H_1$, which assumes the estimated backgrounds plus SM diboson production and decay into $HF$ (blue histogram), and for the {\\em null} hypothesis, $H_0$, which assumes no diboson  (red histogram). The observed value of $ - 2 \\ln Q$  is indicated with a solid, vertical line and the $p$-value is the fraction of the $H_0$ curve integral to the left of the data.}\\label{fig:gaussians}\n\\end{figure}\n\nIn order to measure the diboson production cross section, a Bayesian marginalization technique is applied as described in Section~\\ref{sec:likelihood_meas}. The nuisance parameters are integrated, the maximum of the posterior distribution returns the cross section measurement while the smallest interval containing 68\\% of the integral gives the 1-$\\sigma$ confidence interval. The resulting cross section measurement is:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(1.08^{+0.26}_{-0.40} \\right)\\times \\sigma_{Diboson}^{SM} =  \\left(19.9^{+4.8}_{-7.4}\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainties and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section derived from Table~\\ref{tab:mc_dib}\n\n\\chapter[SVM Multi-Jet Rejection]{Support Vector Machines Multi-Jet Rejection}\\label{chap:AppSvm}\n\nAn innovative multivariate method, based on the Support Vector Machines algorithm (SVM), is used in this thesis to drastically reduce the multi-jet background. \n\nOne of the crucial points in the search for diboson production in the $\\ell\\nu + HF$ final state is the maximization of the signal acceptance while keeping the background under control. This is a challenge because, in hadronic collider environment, jets are produced with a rate several order of magnitude larger than $W\\to \\ell \\nu$ events, therefore, as a jet can fake the lepton identification with not negligible probability (especially for electrons identification algorithms), multi-jet events are introduced in the sample.\n \nThe multi-jet background, a mixture of detector and physics processes, is challenging to parametrize and, usually, approximate data-driven models are obtained by appropriate fake-enriched selections (see Section~\\ref{sec:qcd}). These models are often statistically limited and the use of a different selection can produce unexpected biases in the simulated variables. It is obviously not trivial the use of multivariate techniques to tackle such a problem.\n\nThe SVM algorithm, described in Section~\\ref{sec:SVM}, is considered to perform well in this case as it offers good non-linear separation and stable solutions also on low statistical training samples~\\cite{svmbook, BShop_machine_learning}. It was never used before and we had to develop original solutions to address the major challenges: evaluate the robustness of the SVM against biases in the training set and establish the best, minimal set of input variables providing optimal performances. Section~\\ref{sec:method} describes how we solved the first problem while Section~\\ref{sec:var_sel} describes the input variable selection criteria.\n\nThe results, reported in Section~\\ref{sec:svm_res}, in terms of signal efficiency and background rejection, are superior to any other cut based or multi-variate method previously applied at CDF.\n\n\n\n\\section{Support Vector Machines}\\label{sec:SVM}\n\nThe SVM is a supervised learning binary classifier whose basic concept is the identification of the {\\em best separating hyper-plane} between two classes of $n$-dimension vectors. \n\nIn the case of linear separation the algorithm produces, given a training set of the vectors of the two classes, an {\\em unique} solution where the plane is defined by the minimum amount of vectors, called {\\em support vectors}, at the boundary of the two classes. In the case of non-linear separation, the plane is found in an abstract space, defined by a transformation of the input vectors. However it is not necessary to know the exact transformation, but just its effect on the scalar product between the vectors, named {\\em Kernel}, thus allowing a feasible solution. Finally the cases of not perfect separability of the two samples are included by introducing a penalty parameter accounting for the contamination. \n\nThe main advantages of the SVM with respect to other machine learning algorithms are the unique convergence of the problem, a small number of free tunable parameters (usually related to the Kernel choice) and good performances for low statistics training sets because only a small number of training vectors (the support vectors) are important  for the final solution.\n\nIt is possible to find more details in~\\cite{svmbook, BShop_machine_learning}, but a short overview of the algorithm is also given in the following. For the actual, numerical, implementation of the SVM algorithm we relied on the \\texttt{LIBSVM} open source library~\\cite{libsvm}.\n\n\n\n\\subsection{The Linear Case}\nFigure~\\ref{fig:svm_linear} shows a basic example of the SVM linear classification separating two classes of bi-dimensional training vectors with a maximum margin hyperplane (a line for this simple case)\n\n\\begin{figure}\n  \\begin{center}  \n    \\includegraphics[width=0.8\\textwidth]{AppSVM\/Figs\/vgv4}\n    \\caption[Example of SVM Linear Separation]{An example of SVM: two linearly separable classes of data are represented with red and blue dots. The hyperplane (in this case a simple line) leading to a maximum margin separation is defined by the weight vector $w$ and the bias vector $b$.}\\label{fig:svm_linear}\n  \\end{center}\n\\end{figure}\n\nThe problem can be formalized in a general way as the minimization of $|\\vec{w}|^2$ (with $\\vec{w}$ = vector normal to the plane) with the constraint:\n\\begin{equation}\\label{eq:constraint}\n  y_i(\\vec{x_i}\\cdot\\vec{w} + b)-1\\geq 0\\qquad\n  \\left\\{\\begin{array}{ll}\n  y_i=+1; & i \\in \\textrm{signal} \\\\\n  y_i=-1; & i \\in \\textrm{background}\\\\\n  \\end{array} \\right.\n\\end{equation}\nThe problem has an unique solution obtained by the maximization of:\n\\begin{equation}\\label{eq:lagrange_1}\n  L=\\sum_i \\alpha_i - \\frac{1}{2}\\sum_{i,j}\\alpha_i\\alpha_j y_i y_j\\vec{x_i}\\cdot\\vec{x_j} ,\n\\end{equation}\nobtained with the application of the Lagrange multipliers to Equation~\\ref{eq:constraint}.\nThe solution identifies, for some $i$, $\\alpha_i>0$. The associate vectors are the \\textit{support vectors}, i.e. a subset of the training sample that define the best hyper-plane (see Figure~\\ref{fig:svm_linear}).\n\nTo solve the case of not completely separable classes of vectors, a {\\em penalty parameter}, $C$, is added into the target function to account for the contamination. So that we have a new minimization condition:\n\\begin{equation}\n  |\\vec{w}|^{2} + C \\sum_{i}{\\xi_{i}};\n\\end {equation}\nand a new constraint (derived again from Equation~\\ref{eq:constraint}):\n\\begin{equation}\\label{eq:lagrange_2}\n  y_i(\\vec{x_i}\\cdot\\vec{w} + b) \\geq 1-\\xi_{i}\\quad\\textrm{with}\\quad \\xi \\geq 0\\textrm{.}\n\\end{equation}\n\nThe parameter $C$ defines, before the training, the SVM implementation therefore it represents as one of the {\\em hyper parameter} of the SVM.\n\nA newly seen vector, $\\vec{X}$, is classified according the position with respect to the plane defined by the support vectors $\\vec{x_i}$ and the parameters $\\alpha_i$:\n\\begin{equation}\\label{eq:distance}\n  D(\\vec{X})= \\sum_i\\alpha_i y_i \\vec{x_i}\\cdot \\vec{X_i} - b,\n\\end{equation}\nwhere $b$ is a bias term of the solution. The sign of $D(\\vec{X})$ defines the classification but the value itself can be seen as the distance of a test vector $\\vec{X}$ from the classification plane. However, as we will see in the next paragraph, a non-linear classification is possible only thanks to a not-explicit transformation in a different vector space, where $D$ looses it immediate geometrical meaning. \n\nCommonly SVMs are used as binary classifiers but, here, we add a large degree of flexibility by exploiting the full information of the variable $D$. We see it as a dimensionality reducer that summarizes all the information obtained during the training and classification process. \n\n\n\n\\subsection{Kernel Methods}\n\nNon-linearly separable classes of vectors can be classified by transforming them into linearly separable\nclasses. An opportune function, $\\Phi(\\vec{x})$, can be used to map the elements into another\nspace, usually with higher dimension where the separation is possible. \n\nHowever the  identification of $\\Phi(\\vec{x})$ is {\\em non trivial} and the, so called, {\\em Kernel trick} is often used: a Kernel function, $\\mathbf{K}(x_i,x_j)$, generalizes the scalar product appearing in Equation~\\ref{eq:lagrange_1} (or Equation~\\ref{eq:lagrange_2}) without the need of explicitly know $\\Phi(\\vec{x})$. Or in equations, we compose the mapping $\\Phi(\\vec{x})$ with the inner product:\n\\begin{equation}\n  \\mathbf{K}(x_i,x_j)= \\Phi(x_i)\\cdot\\Phi(x_j)\\quad\\mathrm{with}\\quad \\Phi : \\Re^n\\mapsto \\mathcal{H}.\n\\end{equation}\n\nThe function $\\mathbf{K}$ should satisfy to a general set of roles to be a Kernel, but we want only to briefly describe the {\\em Gaussian} Kernel we used in this work. It is expressed as:\n\\begin{equation}\\label{eq:gauss_k}\nK(x_{i},x_{j}) = e^{-\\gamma |\\vec{x}_{i}-\\vec{x}_{j}|^2}\n\\end{equation}\nThe corresponding $\\Phi(x)$ maps to an infinite dimension space and it is not known. The Kernel is defined only by one hyper-parameter, $\\gamma$, that should be defined before the training.\n\n\n\n\n\n\\section{SVM Training in a Partially Biased Sample}\\label{sec:method}\n\nThe assumption behind the supervised learning is that the labelled samples, used for the classifier training, are drawn from the same probability distribution of the unclassified events. \nHowever in our case of study, where only an approximate and statistically limited model of the background processes is available (see Sections~\\ref{sec:cdf_set} and~\\ref{sec:qcd} for the multi-jet background description), we do not expected the previous assumption to hold for every region of the phase space. To cope with this problem, we developed an original methodology to evaluate the SVM training performances.\n\nSection~\\ref{sec:SVM} shows that, for each choice of hyper-parameters and training vectors, only one optimal SVM solution exists and, for it, we need to evaluate the performances. \n\nAs a performances estimator we use the {\\em confusion matrix} of the classifier: the element $(i,j)$ of the matrix is the fraction of the class $i$ classified as member of class $j$. Figure~\\ref{fig:confusion} shows a representation of it in the two classes case, where one class is considered the {\\em background} and the other the {\\em signal}. We obtain a reliable estimate of the classifier quality by filling the confusion matrix in two independent ways and combining all the available information.\n\n\\begin{figure}[h!]\n  \\begin{center}   \n    \\begin{tabular}{|c|c|}\n      \\toprule\n      {\\em Sgn} classified as {\\em Sgn} & {\\em Bkg} classified as {\\em Sgn}\\\\\n      \\midrule\n      {\\em Sgn} classified as {\\em Bkg} & {\\em Bkg} classified as {\\em Bkg}\\\\\n      \\bottomrule\n    \\end{tabular}   \n    \\caption[Definition of Confusion Matrix]{Definition of confusion matrix for a two classes ($Sgn$ and $Bkg$) classification problem. This reproduces the case of an algorithm used to discriminate signal {\\em vs} background: the elements of the matrix are the signal and background classification performances and the cross contamination.}\\label{fig:confusion}\n  \\end{center}\n\\end{figure}\n\nThe first performance evaluation method is the {\\em $k$-fold cross-validation}: the training set is divided into $k$ sub-samples of which one is used as a validation set and the remaining $k - 1$ are used in the training; the confusion matrix is then evaluated applying the trained discriminant to the validation set. The cross-validation process is repeated $k$ times, the {\\em folds}, and the final performance is given by the average on all the folds. This method is solid against over fitting but it has no protection against biases on the complete training sample.\n\nThe second method, a {\\em key feature} of this work, is based on a bi-component fit that uses signal and background templates and it is performed on a significant distribution of the unclassified events, the {\\em data}. While the signal and background templates are derived in the same way of the training samples, the unclassified data events are, by definition composed by an unknown mixture of the {\\em true} signal and background events. The fit is performed by maximizing a binned likelihood function, $\\lambda$, where the Poisson statistic of the templates is used and the fractions of the signal and of background templates, from which we can derive the elements of the confusion matrix, are free parameters. The fitting function is implemented in the ROOT~\\cite{rootr} analysis package and it is derived from~\\cite{Barlow_1993}. \nFigure~\\ref{fig:toysamplefit} shows an example of the fit used on the toy model described in the next Section.\n\nIf the variable considered in the fit is not well reproduced in the simulation then we expect that the fitted fractions will differ greatly from the results obtained with the $k$-fold cross-validation. At the same time we can evaluate quantitatively the agreement between the data shape and the fitted templates because the quantity:\n\\begin{equation}\n  \\chi^2 = -2\\ln(\\lambda),\n\\end{equation}\nfollows a $\\chi^2$ probability distribution (under general assumptions).\n\nThe last critical point is the identification of a {\\em sensitive} variable to be used in the fit. In a previous work~\\cite{CHEP_svm} we exploited the \\mbox{${\\not}{E_T}$}  distribution as it is sensitive to the multi-jet contamination. A much more general approach, by the machine learning point of view, is the use of the SVM distance value, $D$, defined in Equation~\\ref{eq:distance}. If the SVM training performances are optimal, also the variable $d$ offers an optimal discrimination, furthermore the cross check on the $\\chi^2$ of the template fit ensure a good shape agreement between the data and signal and background templates. We verified the validity of the fit procedure with a toy example reported in the following.\n\n\\begin{figure}[!h]\n  \\begin{center}\n  \\includegraphics[width=0.9\\textwidth]{AppSVM\/Figs\/toysample}\n  \\caption[Bi-component Template Fit on Toy Data]{A bi-component fit is performed on the SVM distance, $D$ (Equation~\\ref{eq:distance}), of a signal (blue) and background (red) templates toy data.}\\label{fig:toysamplefit}    \n  \\end{center}\n\\end{figure}\n  \n\n\\subsection{A Toy Example}\n\nWe built a toy example in order to verify the robustness of the proposed method for an SVM performances evaluation when partially biased samples are available.\n\nThe toy is composed by three data-sets generated with known probability distributions:\n\\begin{description}\n\\item[signal model:] $10^5$ vectors generated from a $2-$Dim Gaussian distribution with the following mean, $\\vec{\\mu}_{Sgn}$ and standard deviation, $\\tilde{\\sigma}_{Sgn}$:\n  \\begin{equation}\\label{eq:sgn_gaus}\n    \\vec{\\mu}_{Sgn} = \\left[\n      \\begin{array}{c}\n        - 3  \\\\\n        0  \\end{array} \\right]\n    ,\\qquad \n     \\tilde{\\sigma}_{Sgn} = \\left[\n       \\begin{array}{cc}\n         8 & 0 \\\\\n       0 & 8  \\end{array} \\right].\n  \\end{equation}\n  \n \\item[Background model:] $10^5$ vectors generated from a $2-$Dim Gaussian distribution with the following mean, $\\vec{\\mu}_{Bkg}$ and standard deviation, $\\tilde{\\sigma}_{Bkg}$:\n   \n   \\begin{equation}\\label{eq:bkg_gaus}\n     \\vec{\\mu}_{Bkg} = \\left[ \n     \\begin{array}{c}\n       3  \\\\\n       0  \\end{array} \\right]\n ,\\qquad \n     \\tilde{\\sigma}_{Bkg} = \\left[\n     \\begin{array}{cc}\n       8 & 0 \\\\\n       0 & 8  \\end{array} \\right].\n   \\end{equation}\n \\item[Data:] a mixture of $5 \\cdot 10^4$ vectors generated from the same distribution of the signal model (Equation~\\ref{eq:sgn_gaus}) and $5\\cdot 10^4$ vectors generated from a {\\em true} background distribution similar to the background model (Equation~\\ref{eq:bkg_gaus}) but with $\\tilde{\\sigma}_{Bkg}$ increased by 20\\% in one direction to simulate a mismatch between the real background and the model.\n \\end{description}\n %\n We tested several combinations of the hyper-parameters $C$ and $\\gamma$ (over a grid) using the signal and background model in the training. For obtained SVM we evaluated the $k$-fold cross validation and we performed the template fit on the SVM distance, $D$, evaluated on the data sample. The result is reported in Figure~\\ref{fig:ToyResult} with the real performances reported on the $x$ axis of the diagram (we know the true label of the data vectors). The evaluation of the performances obtained with the fit is on the diagonal of the plane, therefore it gives a much more realistic estimate of the true performances of the classifier.\n\n \\begin{figure}[!h]\n   \\begin{center}\n   \\includegraphics[width=0.75\\textwidth,height=0.5\\textwidth]{AppSVM\/Figs\/ToyResult}\n   \\caption[Toy Model SVM Performances Estimate with Fit and Cross-Validation]{SVM performances estimate with a $k$-fold cross-validation (green circles) and with a bi-component signal and background template fit on the SVM distance (red triangle) of toy data of known composition. The {\\em true} performances of the SVM classifier are reported on the $x$ axis. The fit evaluation appears on the diagonal of the plane, signaling a more realistic estimate of the true performances of the classifier. }\\label{fig:ToyResult}     \n   \\end{center}\n \\end{figure}\n\n\n\n\\section{Performances on the CDF Dataset}\\label{sec:cdf_set}\n\nThe final goal of the SVM discriminant we discussed is the realization of a tool able to reject the multi-jet background in a wide range of searches performed in the lepton plus jets channel in a hadron collider environment. \n\nWe performed the SVM training on $W\\to e\\nu + $jets candidate, as the electron identification is more tamed by the multi-jet background. Furthermore  we performed the training process two times, one in the central ($|\\eta|<1.1$) and one in the forward ($1.2<|\\eta|<2.0$) region of the detector, as both the electron identification algorithm (see Section~\\ref{sec:lep_id}) and the kinematic of the events are different. \n\n\n\\subsection{Training Sample Description}\\label{sec:training}\n\nWe defined both a central and a forward training set using $7000$ $W\\to e\\nu +$jets signal events and $3500$ multi-jet background events:\n\\begin{description}\n  \\item[Signal:] $W + 2,3$ partons ALPGEN~\\cite{alpgen} MC, where the $W$ is forced to decay into electron and neutrino. We have about $10^5$ generated events and we keep approximately  $9\\times 10^4$ events as a control sample (i.e. not used for training). The CEM and PHX electron identification algorithms are used for the central and forward sample selection. \n\\item[Background:] we obtain a suitable background sample with a data-driven approach. The anti-electrons selection described in Section~\\ref{sec:qcd} is used for the central background training while for the forward training we had study a improved multi-jet model. \n\nIn particular, we noticed that the anti-PHX sample produced large over estimates in the detector region of $1.2<|\\eta|<1.4$ where two different calorimeter sub-systems are connected. These events are clear fake leptons and they are usually rejected by very loose kinematic requirements, nevertheless the quantity of them produced a too large bias in the SVM training efficiency estimate. Non-Isolated PHX electrons (with $IsoRel>0.1$) were found to give a reliable training set after a correction to the lepton $E_T$ equal to the amount of energy in the outer isolation cone ($E_T^{0.1<\\Delta R<0.4}$).\n\\item[Data:] The data sample used for the bi-component fit validation corresponds to the data periods (see Table~\\ref{tab:datasets}) from p18 to p24. They represent only a small fraction of the total CDF data with intermediate luminosity profile and run conditions.\n\\end{description}\nA close to final lepton plus jet selection with loose kinematic cuts is applied to all the samples. We require two or more jets reconstructed (see Section~\\ref{sec:jetCone}) in the central region of the detector ($\\eta<2.0$), with energy corrected at L5 and $E_T> 18$~GeV, also a minimal amount of \\mbox{${\\not}{E_T}$} $>15$~GeV is used as signature of the escaping neutrino. We also removed events with not understood behaviour like electron $E_T>300$~GeV, $M_T^W>200$~~GeV$\/c^{2}$ and, only for the forward sample, \\mbox{${\\not}{E_T^{raw}}$}$<20$~GeV (this last cut was needed to avoid trigger turn-on effects).\n\n\n\n\\subsection{Variable Selection}\\label{sec:var_sel}\n\nA multivariate algorithm relies on a given set of input variables. The {\\em feature selection} problem is fundamental in machine-learning and, if possible, even more in the present case where the background sample does not guaranteed a perfectly model of all the variables. \n\nWe started from a large set of twenty-four variables chosen according to two basic criteria: no correlation with respect to the lepton identification variables and exploit of the kinematic difference between real $W+$ jets events and multi-jet fakes. These requirements allowed the development of a flexible multi-jet rejection algorithm, applied with very good performances also to muon and isolated track lepton selections.\n\nTable~\\ref{tab:input_vars} shows all the input variables that we used in the optimization process. Many of them were introduced in Chapter~\\ref{chap:objects} but the following are new:\n\\begin{itemize}\n\\item \\mbox{${\\not}{p_T}$} is the missing momentum defined as the momentum imbalance on the transverse plane. It is computed adding all the reconstructed charged tracks transverse momenta, $\\vec{p_i}$:\n  \\begin{equation}\n    \\vec{{\\not}{p_T}}\\equiv - \\sum_i{\\vec{p^i}_T}\\quad\\textrm{with }|\\vec{p^i}_T|>0.5\\textrm{GeV}\/c\\textrm{;}\n  \\end{equation}\n\\item $M_T^W$ is the {\\em transverse mass} of the reconstructed $W$ boson:\n  \\begin{equation}\n    M_T^W=\\sqrt{2(E_T^{lep}\\cancel{E}_T - E_x^{lep}\\cancel{E}_x - E_y^{lep}\\cancel{E}_y)}\\mathrm{.}\n  \\end{equation}\n\\item $MetSig$ is the \\mbox{${\\not}{E_T}$}  {\\em significance}, a variable that relates the reconstructed \\mbox{${\\not}{E_T}$}  with the detector activity (jets and unclustered energy):\n  \\begin{equation}\n    MetSig =\\frac{\\cancel{E}_T}{\\sqrt{ \\Delta E^{jets}+ \\Delta E^{uncl}}}\\mathrm{,}\n  \\end{equation}\n  where:\n  \\begin{eqnarray}\n    \\Delta E^{jets} =\\sum_{j}^{jets}(cor_{j}^2\\cos^2\\Big(\\Delta\\phi\\big(\\vec{p_j},\\cancel{\\vec{E}}_T\\big)\\Big)E_{T}^{raw, j}\\mathrm{,}\\\\\n    \\Delta E^{uncl} = \\cos^2\\Big(\\Delta\\phi\\big(\\vec{E}_T^{uncl}\\cancel{\\vec{E}}_T\\big)\\Big)E_T^{uncl}\\mathrm{,}\n  \\end{eqnarray}\n  $uncl$ refers to the calorimeter energy not clustered into electrons or jets and $cor_j$ is the total correction applied to each jet.\n\\item $\\nu^{Min},\\quad \\nu^{Max}$ are the two possible reconstruction of the neutrino momenta. As the $p_z^{\\nu}$ component is not directly measurable we infer it from the $W$ boson mass and the lepton momentum. The constraints lead to a quadratic equation which may have two real solutions, one real solution, or two complex solutions\\footnote{The real part is chosen in this case}. The reconstructed $\\nu^{Min}$, $\\nu^{Max}$ derive from the distinction of $p_z^{\\nu,Max}$ and $p_z^{\\nu,Min}$.\n\\end{itemize}\n\n\\begin{table}\n  \\begin{tabular}{cl| cl|  cl| cl }\n    \\toprule\n    \\multicolumn{8}{c}{Possible Input Variables }\\\\\n    \\midrule\n    1 & $p_T^{lep}$           &  7 & $E_T^{raw, jet1} $   & 13 &  $\\Delta\\phi( {\\not}{p_T},$ $lep)$ &   19 & $\\Delta R( lep,$ $jet 2)$ \\\\\n    2 & \\mbox{${\\not}{E_T}$}                  &  8 & $E_T^{raw, jet2} $   & 14 &  $\\Delta\\phi( {\\not}{p_T},$ \\mbox{${\\not}{E_T}$} $)$ & 20 & $\\Delta R( \\nu^{min},$ $jet 1)$ \\\\ \n    3 & \\mbox{${\\not}{E_T^{raw}}$}                &  9 & $E_T^{cor,jet1} $       & 15 & $\\Delta\\phi( {\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$}$)$ &  21 & $\\Delta R( \\nu^{min},$ $jet 2)$ \\\\ \n    4 & \\mbox{${\\not}{p_T}$} & 10 & $E_T^{cor,jet2} $       &  16 & $\\Delta\\phi( lep,$ \\mbox{${\\not}{E_T}$} $)$ &    22 & $\\Delta R( \\nu^{min},$ $lep)$ \\\\ \n    5 & $M_T^W$              & 11 & $\\Delta\\phi( jet 1,$ \\mbox{${\\not}{E_T}$} $)$ & 17 & $\\Delta\\phi( lep,$ \\mbox{${\\not}{E_T^{raw}}$}$)$ &  23  & $\\Delta R( \\nu^{max},$ $jet 1)$ \\\\ \n    6 & $MetSig$             & 12 & $\\Delta\\phi( jet 2,$ \\mbox{${\\not}{E_T}$} $)$ &  18 & $\\Delta R( lep,$ $jet 1)$ &  24 & $\\Delta R( \\nu^{max},$ $jet 1)$ \\\\ \n\\bottomrule\n  \\end{tabular}\n  \\caption[Possible SVM Input Variables]{All the possible input variables used for the SVM training and optimization. See Section~\\ref{sec:var_sel} for a detailed description.}\\label{tab:input_vars}  \n\\end{table}\n\nUnluckily the extensive research over all the possible combinations of variables across all the $C, \\gamma$ phase space of a given SVM training, is computationally unfeasible. To scan the most relevant sectors of the phase space we applied factorized and incremental optimization:\n\\begin{itemize}\n\\item for all the configuration of {\\em three} variables and the given training set, we evaluate a grid of  $C, \\gamma$ values in the intervals\\footnote{The use of a logarithmic scale allows to scan the parameters across different orders of magnitude.}:\n  \\begin{equation}\n   \\log_2 C\\in [-3,8] \\quad\\textrm{and}\\quad \\log_2\\gamma\\in[-4,5].\n  \\end{equation}\nWe select only the best training configuration according to the confusion matrix evaluation.\n\\item For each {\\em best} SVM of a given variable configuration we perform a bi-component fit on the SVM distance $D$. We evaluate the  $\\chi^2$ of the fit, reduced by the Number of Degrees of Freedom ($NDoF$), and we compare the fitted background contamination, $f_{Bkg}^{Fit}$, against the one obtained from the $n$-fold cross-validation, $f_{Bkg}^{n-fold}$. The SVM under exam is rejected if:\n  \\begin{equation}\n    \\frac{\\chi^2}{NDoF}>3\\quad  or \\quad\\frac{f_{Bkg}^{Fit}}{f_{Bkg}^{n-fold}}>2\n  \\end{equation}\n\n\\item The remaining SVMs are displayed on a {\\em signal-efficiency vs background-contamination} scatter plot like the one in Figure~\\ref{fig:discriminants}. The 5 best variable combinations are \nselected for further processing.\n\\item We add other 2 or 3 variables to the best variables combinations obtained in the previous step and we iterate the chain.\n\\end{itemize}\nAfter a couple of iterations the best variable combination and $C,\\gamma$ hyper-parameters choice remains stable within $1\\div 2\\%$.\n\n\\begin{figure}[!h]\n  \\begin{center}    \n    \\includegraphics[width=0.9\\textwidth]{AppSVM\/Figs\/discriminants_bests_trainCEM3}\n  \\caption[Signal Efficiency {\\em vs} Background Contamination for Multiple SVM]{Different SVM configurations (i.e. with different input variables), obtained for the central region training, are displayed on a {\\em signal-efficiency vs background-contamination} scatter plot. The signal efficiency is directly estimated from simulation while the background contamination is obtained from the bi-component template fit of the SVM distance, $D$, described in Section~\\ref{sec:method}. The blue starts represent the performances obtained from a three (out of twenty-four) input variables training. The best five configurations w.r.t. the Euclidean distance from the optimal point ($\\epsilon_{Sig}=1$, $f_{Bkg}=0$) are circled in yellow. The Euclidean distance of the fifth best SVM is represented by a dotted line. Three iterations, with {\\em three, six and nine} input variables are represented by dotted lines, after that no more appreciable improvement occurs.}\\label{fig:discriminants}\n  \\end{center}\n\\end{figure}\n\n\n\\section{Final SVM Results}\\label{sec:svm_res}\nThe SVM configurations obtained by the process described in this appendix are finally used in the multi-jet rejection phase of the analysis (Section~\\ref{sec:svm_sel}) and in the background normalization estimates\\footnote{By construction a fit on the SVM distance, $D$, offers a reliable estimate of the multi-jet background normalization.} (see Sections~\\ref{sec:wjets_pretag} and~\\ref{sec:qcd}).\n\nThe optimal hyper-parameter configurations that we obtain for the central (superscript $c$) and forward (superscript $f$) SVMs are:\n\\begin{equation}\nC^c = 7,\\quad  \\gamma^c = \u2212- 1;\n\\end{equation}\n\\begin{equation}\nC^f = 8,\\quad  \\gamma^f = \u2212- 1.\n\\end{equation}\nTable~\\ref{tab:svm_fin_vars} reports the final eight input variables used for the central SVM and the six ones used for the forward SVM.\n\\begin{table}\n  \\begin{center}   \n  \\begin{tabular}{lccc}\n    \\toprule\n    \\multicolumn{4}{c}{ Final SVM Input Variables}\\\\\n    \\midrule\n    Central SVM: &  $M^W_T$ & \\mbox{${\\not}{E_T^{raw}}$} &  ${\\not}{p_T}$\\\\ \n                 & $MetSig$ & $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ) & $\\Delta \\phi(lep,$ \\mbox{${\\not}{E_T}$} $)$ \\\\\n                 & $\\Delta R(\\nu^{Min},$ $lep)$  & $\\Delta \\phi(Jet1,$ \\mbox{${\\not}{E_T}$} )  & \\\\\n    \\midrule\n    Forward SVM: & $M^W_T$ & \\mbox{${\\not}{E_T^{raw}}$} & ${\\not}{p_T}$ \\\\\n                 &  $MetSig$ &  $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ) & $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$})  \\\\\n    \\bottomrule\n  \\end{tabular}\n  \\caption[Final SVM Input Variables]{Input variables used for the configuration of the {\\em central} and {\\em forward} SVM multi-jet discriminants.}\\label{tab:svm_fin_vars}\n  \\end{center}\n\\end{table}\nFigure~\\ref{fig:final_svm} shows the complete shape of the two discriminants for the multi-jet background models and the $W+2$ partons signal. A quantitative measurement of the performances can be seen in Table~\\ref{tab:svm_eff}.\n\n\\begin{figure}[!h]\n  \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{AppSVM\/Figs\/central_svm}\n    \\includegraphics[width=0.495\\textwidth]{AppSVM\/Figs\/forward_svm}\n  \\caption[Final SVM Discriminants for Central and Forward Detector Regions]{Distribution of the SVM distance, $D$, described in Equation~\\ref{eq:distance} for the central (left) and the forward (right) SVM discriminants obtained from the optimization process. Multi-jet background models are shown in red, $W+2$ partons MC signal is shown in blue.}\\label{fig:final_svm}\n  \\end{center}\n\\end{figure}\n\nWe can conclude that we successfully built a multi-jet rejection tool based of the SVM algorithm, nevertheless the challenges of a multi-variate approach to this problem. The CDF II dataset was a perfect test-bench for this problem with very good performances, however the procedure can be exported also to any of the LHC experiments.\n\n\n\\chapter{Rate Systematics Summary}\\label{app:rate_sys}\n\nTables from~\\ref{tab:1tag_cem} to~\\ref{tab:2tag_emc}: rate systematics variations for each channel entering in the statistical analysis (see Chapter~\\ref{chap:StatRes}), same name systematics are fully correlated.\n\n\\renewcommand{\\arraystretch}{0.97}\n\\renewcommand{\\tabcolsep}{1.5pt}\n\n\\begin{table}[!h]\n  \\begin{scriptsize}\n    \\begin{tabular}{ c  c c c c c c c c c c c } \\toprule\n      & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_CEM}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_CEM}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-12\/12$  & $-3\/2$  & $-2\/1$  & $1\/6$  & $1\/-1$  & $-6\/6$  & $-3\/6$  & $-5\/6$  & $-3\/-4$  & $2\/-2$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-2\/-1$  & $4\/11$  & $5\/8$  & $3\/9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/17$  & $12\/12$  & $13\/13$  & $45\/46$  & $11\/11$  & $31\/33$  & $31\/32$  & $30\/32$  & $14\/14$  & $9\/11$  \\\\\n\\hline\n    \\end{tabular}\n  \\end{scriptsize}\n  \\caption[Rate Uncertainties: Single-Tag, CEM Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, CEM  channel.}\\label{tab:1tag_cem}\n  \\end{table}\n\n  \\begin{table}[!h]\n    \\begin{scriptsize}\n  \\begin{tabular}{ c c c c c c c c c c c c} \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_PHX}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_PHX}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-8\/12$  & $-1\/2$  & $-3\/3$  & $2\/-6$  & $-1\/-1$  & $-4\/4$  & $-5\/4$  & $-8\/6$  & $-7\/-5$  & $-10\/-3$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-1\/-2$  & $0\/15$  & $0\/18$  & $2\/10$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $14\/17$  & $12\/12$  & $13\/13$  & $45\/46$  & $11\/11$  & $30\/34$  & $30\/35$  & $31\/32$  & $15\/14$  & $14\/10$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n  \\caption[Rate Uncertainties: Single-Tag, PHX Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, PHX channel.}\\label{tab:1tag_phx}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_MU}$ & $0\/0$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $1\/-1$  & $1\/-1$  \\\\\n$\\texttt{ QCD\\_MU}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-11\/12$  & $-4\/3$  & $-1\/1$  & $9\/-6$  & $1\/1$  & $-8\/6$  & $-5\/5$  & $-6\/7$  & $-1\/-4$  & $1\/-2$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-4$  & $2\/16$  & $6\/12$  & $2\/15$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/17$  & $13\/12$  & $13\/13$  & $46\/46$  & $11\/12$  & $31\/35$  & $31\/33$  & $31\/34$  & $13\/14$  & $9\/16$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Single-Tag, CMUP $+$ CMX Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, CMUP $+$ CMX channel.}\\label{tab:1tag_emc}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $3\/-3$  & $3\/-3$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $5\/-5$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_EMC}$ & $0\/0$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/-5$  & $5\/-5$  \\\\\n$\\texttt{ QCD\\_EMC}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $11\/-11$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-10\/9$  & $0\/-2$  & $0\/-3$  & $11\/-8$  & $0\/-0$  & $-5\/6$  & $-8\/4$  & $-3\/2$  & $9\/-6$  & $8\/-6$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-2$  & $11\/13$  & $12\/11$  & $11\/16$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/16$  & $13\/13$  & $14\/14$  & $47\/46$  & $11\/11$  & $32\/33$  & $33\/32$  & $32\/34$  & $17\/15$  & $14\/19$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Single-Tag, EMC Channel]{Per-sample rate uncertainty (\\% up\/down). Single-tag, EMC channel.}\\label{tab:1tag_mu}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_CEM}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_CEM}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-4\/10$  & $-5\/3$  & $-4\/2$  & $-9\/-1$  & $3\/-3$  & $-7\/8$  & $-15\/2$  & $-11\/9$  & $-18\/6$  & $-4\/-5$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/1$  & $18\/9$  & $6\/17$  & $5\/-9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/19$  & $17\/16$  & $17\/17$  & $46\/45$  & $21\/21$  & $35\/32$  & $34\/35$  & $32\/32$  & $31\/26$  & $14\/14$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, CEM Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, CEM channel.}\\label{tab:2tag_cem}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_PHX}$ & $0\/0$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $2\/-2$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $2\/-2$  & $2\/-2$  \\\\\n$\\texttt{ QCD\\_PHX}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/11$  & $-5\/-2$  & $-9\/0$  & $1\/-5$  & $3\/-2$  & $-6\/5$  & $-17\/8$  & $-35\/32$  & $-50\/-50$  & $-4\/-4$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/0$  & $6\/-7$  & $-0\/35$  & $-41\/9$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $17\/20$  & $17\/16$  & $19\/17$  & $45\/46$  & $22\/21$  & $31\/31$  & $34\/47$  & $62\/45$  & $56\/56$  & $14\/14$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, PHX Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, PHX channel.}\\label{tab:2tag_phx}\n  \\end{table}\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c c c c c c c c c c c c } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_MU}$ & $0\/0$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $1\/-1$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $1\/-1$  & $1\/-1$  \\\\\n$\\texttt{ QCD\\_MU}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/9$  & $-3\/2$  & $-2\/0$  & $7\/2$  & $-0\/0$  & $-10\/9$  & $-6\/10$  & $-14\/7$  & $-1\/-6$  & $-2\/-3$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-2\/-5$  & $14\/10$  & $-6\/-8$  & $-8\/-11$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $18\/19$  & $16\/16$  & $17\/17$  & $46\/45$  & $21\/22$  & $35\/33$  & $31\/33$  & $34\/33$  & $25\/26$  & $13\/13$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, CMUP $+$ CMX Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, CMUP $+$ CMX channel.}\\label{tab:2tag_mu}\n  \\end{table}\n\n\n  \\begin{table}\n    \\begin{scriptsize}\n\\begin{tabular}{ c  c c c c c c c c c c c  } \\toprule\n    & nonW & $t\\bar{t}$ & s-top $s$ & s-top $t$ & Zjets & W+LF & Wbb & Wcc & Wcj & WW & WZ\/ZZ \\\\\\midrule\n$\\texttt{ XS\\_ttbar}$ & $0\/0$  & $10\/-10$  & $10\/-10$  & $10\/-10$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFBTAGSF}$ & $0\/0$  & $11\/-11$  & $11\/-11$  & $12\/-12$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $24\/-24$  & $11\/-11$  \\\\\n$\\texttt{ CDFLUMI}$ & $0\/0$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $6\/-6$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $6\/-6$  & $6\/-6$  \\\\\n$\\texttt{ LEPACC\\_EMC}$ & $0\/0$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $5\/-5$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $5\/-5$  & $5\/-5$  \\\\\n$\\texttt{ QCD\\_EMC}$ & $40\/-40$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFVHF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $30\/-30$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFWC}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $30\/-30$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ CDFMISTAG}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $21\/-21$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ZJETS}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $45\/-45$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  \\\\\n$\\texttt{ ISRFSRPDF}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $4\/-4$  & $4\/-4$  \\\\\n$\\texttt{ CDFJES}$ & $0\/0$  & $-7\/9$  & $-0\/-0$  & $1\/-5$  & $5\/-3$  & $-1\/1$  & $-10\/11$  & $-5\/19$  & $-7\/6$  & $-10\/-14$  & $8\/-8$  \\\\\n$\\texttt{ CDFQ2}$ & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $0\/0$  & $-3\/-1$  & $17\/5$  & $15\/0$  & $27\/20$  & $0\/0$  & $0\/0$  \\\\\n\\hline\n$\\texttt{TOT}$ & $40\/40$  & $18\/19$  & $17\/17$  & $18\/18$  & $46\/46$  & $21\/21$  & $36\/32$  & $34\/36$  & $41\/37$  & $27\/29$  & $16\/16$  \\\\\n\\hline\n\\end{tabular}\n    \\end{scriptsize}\n\\caption[Rate Uncertainties: Double-Tag, EMC Channel]{Per-sample rate uncertainty (\\% up\/down). Double-tag, EMC channel.}\\label{tab:2tag_emc}\n  \\end{table}\n\n\\renewcommand{\\tabcolsep}{6pt}\n\n\n\\chapter{Kinematic Distribution of the Signal Regions}\n\nFigures from~\\ref{fig:1svt_jet} to~\\ref{fig:2svt_dphi} show several kinematic variables for the single-tag and double-tag signal regions after the composition of all the background and all the lepton categories. The normalization of each background has been changed to match the {\\em best fit} values returned by statistical analysis of the likelihood described in by Equation~\\ref{eq:likelihood_full}, in Chapter~\\ref{chap:StatRes}. For each variable we show:\n\\begin{itemize}\n\\item the total background prediction overlaid to the selected data.\n\\item The reduced $\\chi^2$ of the prediction against the data distribution. As we use the best-fit normalization is used for the backgrounds, no rate uncertainty is applied in the $\\chi^2$ evaluation, however MC statistical uncertainty is accounted.\n\\item The probability of the Kolmogorov-Smirnov test derived from the predicted and observed shapes.\n\\item Background subtracted data histogram.\n\\item JES and $Q^2$ shape variations added in quadrature.\n\\end{itemize}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J2Et}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n  \\caption[Single Tag Region Jet Kinematic Variables]{Variables relative to the jet kinematic for all the lepton categories combined in the single-tag signal region. Jet 1 $E_T$ (top left), jet 2 $E_T$ (top right), jet 1 $\\eta$ (bottom left), jet 2 $\\eta$ (bottom left).}\\label{fig:1svt_jet}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_LepPt}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_LepEta}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_WMt}\n  \\caption[Single Tag Region Lepton Kinematic Variables]{Variables relative to the lepton kinematic for all the lepton categories combined in the single-tag signal region. Lepton $P_T$ (top left), lepton $\\eta$ (top right), \\mbox{${\\not}{E_T}$}  (bottom left), $M_T^W$ (bottom left).}\\label{fig:1svt_lep}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVT_CEMCMUPCMXISOTRKPHX_wPt}\\\\\n\\caption[Single Tag Region Angular Kinematic Variables]{Angular variables for all the lepton categories combined in the single-tag signal region. $\\Delta\\phi($\\mbox{${\\not}{E_T}$} $,jet1)$ (top left), $\\Delta R(Lep,jet1)$ (top right), $\\Delta R(jet1,jet2)$, (bottom left), $P_T^W$ (bottom left).}\\label{fig:1svt_dphi}\n\\end{sidewaysfigure}\n\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J2Et}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n  \\caption[Double Tag Region Jet Kinematic Variables]{Variables relative to the jet kinematic for all the lepton categories combined in the double tag-signal region. Jet 1 $E_T$ (top left), jet 2 $E_T$ (top right), jet 1 $\\eta$ (bottom left), jet 2 $\\eta$ (bottom left).}\\label{fig:2svt_jet}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_LepPt}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_LepEta}\\\\\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_WMt}\n  \\caption[Double Tag Region Lepton Kinematic Variables]{Variables relative to the lepton kinematic for all the lepton categories combined in the double tag-signal region. Lepton $P_T$ (top left), lepton $\\eta$ (top right), \\mbox{${\\not}{E_T}$}  (bottom left), $M_T^W$ (bottom left).}\\label{fig:2svt_lep}\n\\end{sidewaysfigure}\n\n\\begin{sidewaysfigure}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}\n\\includegraphics[width=0.49\\textwidth]{AppTagKin\/Figs\/WZ_2JET_SVTSVT_CEMCMUPCMXISOTRKPHX_wPt}\\\\\n\\caption[Double Tag Region Angular Kinematic Variables]{Angular variables for all the lepton categories combined in the double-tag signal region. $\\Delta\\phi($\\mbox{${\\not}{E_T}$} $,jet1)$ (top left), $\\Delta R(Lep,jet1)$ (top right), $\\Delta R(jet1,jet2)$, (bottom left), $P_T^W$ (bottom left).}\\label{fig:2svt_dphi}\n\\end{sidewaysfigure}\n\n\n\\chapter{WHAM: WH Analysis Modules}\\label{chap:AppWHAM}\n\nHigh energy physics data analysis is done with the help of complex software frameworks which allow to manage the huge amount of information collected by the detector. \nThe analysis framework which was used for this analysis is named $WH$ {\\em Analysis Modules} or WHAM and I was one of the main developers of the package.  \n\nThe aim of the software is a reliable event selection and background estimate in the $\\ell\\nu+HF$ channel, with the possibility to easily implement new features and studies that can improve the Higgs search.\nWHAM plays a relevant role in the CDF low-mass Higgs boson search, especially in the $WH\\to \\ell\\nu+b\\bar{b}$ channel but also in other contexts, like $t\\bar{t}H$ and lately also $ZH$ searches. \n\nThe analysis package tries to incorporate the CDF knowledge about the $\\ell\\nu+HF$ channel, most of it coming from the top~\\cite{top_lj2005} and single-top~\\cite{single_top} analyses. An effort was also made in the direction of code modularization and analysis customization with option loading at run-time.\n\nA detailed explanation of the package is beyond the goal of this thesis, however a general overview of the package structure and functionality is given. More information is available in the CDF internal pages~\\cite{wham_twiki} although a comprehensive documentation is not yet available.\n\n\n\\section{Package Structure}\n\nThe package is organized in a folder structure organized according to the purpose of each of the sub-elements. The {\\em first level} directories are:\n\\begin{description}\n\\item[Setup:] contains the scripts needed to setup the analysis environment, both first installation and every day use, the references to all the external tools and any patche that needs to be applied.\n\\item[Documentation:] contains all the internal and public documentation. It is easily accessible and customizable by all the analysis group collaborators.\n\\item[Inputs:] contains the database files of the MC samples, the parametrization of the triggers and $SF$s, the re-weighting templates, the option configuration files. Basically every input to the analysis elaboration is here, except the data and the MC ntuples themselves.\n\\item[Commands:] this is more an utilities repository, it contains scripts to run the analysis on the CDF Grid for parallel computing (named CAF~\\cite{caf_Sfiligoi2007}), plus a wide set of macros and scripts used for single-sample studies, text file processing or small data-handling tasks. \n\\item[Results:] contains all the information elaborated by the rest of the analysis packages. This includes pre-processed data and MC samples as well as the final templates obtained after the complete elaboration. Several commands expect to find the input files here.\n\\item[Modules:] this is the core of the analysis package. It contains the C++ code used for the selection, the background estimate and the final production of the templates used for the statistical analysis and the validation of the kinematic distributions. Next section will describe it in more details.\n\\end{description}\n\n\\subsection{WHAM Modules}\n\nThe core of the WHAM analysis framework is the Modules directory. Here each functional step of the analysis is classified in a {\\em module}, i.e. a self consistent C++ class built with standardized structure to allow straightforward compilation and testing. \n\nThe modules are of two kinds: {\\em functional modules} and {\\em construction modules}. The formers are in charge of actually perform an operation, for example the event selection or the drawing of stacked histograms. The second kind of modules are the sub-components used by the first, for example the selection code needs to known the format of the input and output data as well as the definition of the lepton-object or jet-object.\n\nThe level of abstraction offered by the building modules proved to be extremely powerful. Two minimal examples (on which I contributed) that revealed to be extremely useful are: the handling of the configuration options and of the $b$-tag efficiency estimate. \nFor the first, I implemented, using the \\verb'libconfig' library~\\cite{libconfig}, a text file reading utility that allows a single location definition and the run-time loading of all the options needed by the functional modules. For the second, it was necessary to identify the minimal amount of information needed to define a $b$-tagging algorithm. The only two values needed\\footnote{Functional dependencies and correlations should be already taken into account.} are: $SF_{Tag}$ and $p_{Mistag}^j$ (see Section~\\ref{sec:tag-sel}). With this information, it was possible to develop a single algorithm for the combination of any number of different $b$-tagging algorithms, for any required tag and jet multiplicity. The code works iteratively on the jets of the event requiring the definition of a $b$-tag in priority order, defined by the user.\n\n\nThe functional modules are four and, in the directory structure, are identified by the \\verb'process' prefix:\n\\begin{description}\n\\item[Sample Selection:] access to the production ntuples (see Section~\\ref{sec:data-struc}), as well as lepton and jet selections, are performed here. The result is a small size ntuple, the \\texttt{EvTree}, containing the 4-vectors of the identified particles plus all the relevant information needed in the next steps. The \\texttt{EvTree} is defined by a class with complex {\\em methods} working on the simple stored variables: this allows a huge saving of disk space and computing time at selection stage. Furthermore the portability of the ntuple class ease the reproducibility of the same algorithms and allows faster checks.\n\\item[Sample Pre-processing:] here the complete selection is applied to the \\texttt{EvTree}'s and the pre-processing with more analysis-specific algorithms is performed. For example, MC samples are scaled to the expected yield with the application of the latest available $SF$'s and trigger efficiencies. Also, the multivariate discriminants are evaluated here. The pre-processing has the possibility to be interfaced to other ntuples than the \\texttt{EvTree}.\n\\item[Background Estimate:] this is the last step in the analysis of the $\\ell\\nu+HF$ channel. The background estimate described in Chapter~\\ref{chap:bkg} is applied here and the templates of the different signals, backgrounds and data samples are stored in a ROOT file with the derived normalizations. In general any other background estimate method can be plugged at the end of the analysis chain but for the moment only the one described in Chapter~\\ref{chap:bkg} is available.\n\\item[Result Display:] this step completes the analysis in the sense that it allows the comparison of the final estimate with the observed data distributions. Histograms are produced together with statistical indicators of the shape agreement: $\\chi^2$, Kolmogorov-Smirnov tests, systematic overlay, background subtracted plots. Tables and histograms can appear in several formats, from html pages to simple eps files.\n\\end{description}\nThe very last step in each analysis, the statistical interpretation of the results, is implemented in a different software tool~\\cite{mclimit} that is interfaced with the templates produced by the WHAM background estimate.\n\n\n\\section{Relevant Results}\n\nBeyond the analysis presented here, a wide range of other analyses exploit the WHAM package. Between the most relevant: the new single-top cross section measuremens~\\cite{single_top_benwu_10793}, the $t\\bar{t}H$~\\cite{tth_10801} search in the lepton plus jets channel and, lately, also top-properties~\\cite{topR_10887} and SUSY searches are exploiting the package.\n\nFigure~\\ref{fig:WH_improve}, probably, shows the most striking result: the improvement of the $WH$ search sensitivity (7.5~fb$^{-1}$ and 9.4~fb$^{-1}$ versions~\\cite{wh75_note10596, wh94_note10796}). The several improvements produced for the Higgs search were readily implemented and tested thanks to the backbone of a reilable framework. \n\nNevertheless the shrinking of the CDF collaboration, the final sensitivity to the Higgs boson exceeded the best expectations.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth,height=0.75\\textwidth]{AppWHAM\/Figs\/WH_Projection_new}\n\\caption[Improvements to the CDF $WH$ Search Sensitivity]{Improvements to the CDF $WH\\to \\ell\\nu+ b\\bar{b}$ search sensitivity for a Higgs boson of $m_H=115$~GeV$\/c^{2}$. Both the 2011~\\cite{wh75_note10596} and 2012~\\cite{wh94_note10796} results are mainly obtained within the WHAM analysis framework. The sensitivity reached improves more than 125\\% over the first analysis of the same channel.}\\label{fig:WH_improve}\n\\end{center}\n\\end{figure}\n\n\n\n\\chapter{Background Estimate}\\label{chap:bkg}\n\nIn this analysis an accurate background evaluation is crucial since a low signal over background ratio is expected and several components contribute to the final $M_{Inv}$ line shape used for the signal extraction. \n\nBackground estimate should be as independent as possible from the signal regions, the single and double-tagged $\\ell\\nu+2 $~jets samples. One possible approach is the simulation of background events with Monte Carlo generators, another approach is the extrapolation of background components from data control regions. Both methods present advantages and disadvantages, for example a MC simulation is completely under control but can be limited by the knowledge of the underlying physics phenomena parametrization. Data-driven techniques, on the other hand, are based on the assumption that a background in the control region can be simply extrapolated to the signal region. This brings some approximations, eventually covered by systematic uncertainties. \n\nIn our case we use a mixture of MC and data-driven estimates depending on the different background component being addressed.\n\nThe complete machinery used for the $\\ell\\nu+HF$ jets background estimate is named {\\em Method II}\\footnote{The name derives from a methodology developed for the lepton plus jets $t\\bar{t}$ decay channel. Method II appeared after the {\\em Method I}, the data-driven background estimate used in the top discovery~\\cite{cdf_top_evidence}.} and it was extensively studied at CDF, first on $t\\bar{t}$~\\cite{top_lj2005} analyses, and later in single-top~\\cite{single_top} and $WH$~\\cite{wh75_note10596, wh94_note10796} analyses. Together with the MC, two data control samples play a major role in this kind of background estimate: the {\\em pretag} control sample, without any $b$-tag requirement, and the fake-$W$ enriched sample, without any multi-jet rejection requirement.\n\nThe background evaluation can be divided in four categories: \n\\begin{description}\n\\item[Electroweak and Top:] sometimes dubbed $EWK$, these are backgrounds due to contributions of well known physical processes:  $t\\bar{t}$, single-top and $Z+ $~jets  production. For these processes, described in detail in Section~\\ref{sec:mc_bkg}, we completely rely on MC simulation for the line-shape and theoretical prediction or previous measurements for the normalization. \n\\item[$W+$ Heavy Flavors:] also named $W+HF$, is the part of the $W+$~jets sample where the $W$ is produced in association with $HF$ quarks. It is classified as {\\em irreducible} because it has the same final state ($W+c\/b$ quarks) of the $diboson\\to \\ell\\nu+HF$ signal. The line shape is derived by a LO $W+n$~partons MC while the normalization is obtained from the {\\em pretag} data control region and from MC derived $W+HF$ fractions. Section~\\ref{sec:whf} describes in detail the procedure.\n\\item[$W+$ Light Flavors:] also named {\\em mistags} or $W+LF$. These are events where the $W$ is produced in association with one or more Light Flavor ($LF$) jets. This component is drastically reduced by the requirement of one or more $\\mathtt{SecVtx}$ tagged jets. Remaining events are due to mis-measured jet tracks and long living $LF$ hadrons. The background evaluation, explained in Section~\\ref{sec:mistag}, is obtained using LO $W+ n$~partons MC where appropriate reweighting and normalization are derived from a data control sample of multi-jet events.\n\\item[Fake-$W$:] sometimes also named multi-jet or non-$W$ background, is composed by events without a real $W$ passing the selection. Those are due to multi-jet QCD production where the jets fake the charged lepton and the \\mbox{${\\not}{E_T}$}  of the neutrino. Section~\\ref{sec:qcd} describes the normalization and line-shape evaluation procedure, both completely data-driven.\n\\end{description}\nThe normalization of the four categories proceeds through sequential steps: the $EWK$ contribution is evaluated first, then $W+$~jets normalization is extracted from the pretag control region so that the $W+HF$ fraction can be scaled to the single and double tag signal regions.\n\nNext step is the estimate, from data, of $W+LF$ contribution and, finally, the {\\em tagged} non-$W$ contribution is evaluated before the application of the SVM requirement.\n\n\n\\section{Electroweak and Top Processes}\\label{sec:mc_bkg}\n\nThis category contains all the processes whose shape is evaluated directly from MC samples and whose normalization is obtained from theory prediction or experimental measurements. Figure~\\ref{fig:fey_ewk} shows some of the tree-level Feynman diagrams contributing to this background set. \n\nThe components of $EWK$ backgrounds are: \n\\begin{description}\n\\item[$Z\/\\gamma$ + jets:] Drell-Yan and $Z\\to \\ell\\ell$ events can be produced in association with jets emitted by QCD radiation. If one lepton is misidentified, they enter as a background in our selection. The simulation of  $Z\/\\gamma+n$ partons is available only at LO level (with \\texttt{ALPGEN}~\\cite{alpgen} Matrix Element generator) therefore the process is evaluated with the same procedure of $W+$~jets background (see Section~\\ref{sec:whf}). Luckily, the small acceptance and the possibility to use a cross section measured at CDF~\\cite{z_jets_cx} allows us to evaluate the $Z\/\\gamma+$jets background directly from MC. The $Z+$jets cross section measurement refers only to the on-shell $Z$ contribution, therefore a further scaling factor is derived by comparing the \\texttt{ALPGEN} LO predictions for the on-shell and the inclusive $Z\/\\gamma+$ jets MC sample. \n\n\\item[Top pair:] $t\\bar{t}$ production and decay in the lepton plus jets ($\\ell\\nu+b\\bar{b}+2$ jets) and di-lepton ($2\\ell+2\\nu+b\\bar{b}$) channels can enter in our selection if one or more of the final state objects is misidentified. Although the probability is small, the background becomes sizable in the $HF$-tagged sample because of the presence of two $b$-quarks.  \n\\item[Single-top:] $t$ and $s$ channel production have cross sections comparable to the diboson one and the semi-leptonic decay channel produces the same final state we are interested in. On the other hand, the signal to background discrimination is performed by the $M_{Inv}(jet1,jet2)$ where the single-top has no peak.\n\\end{description}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.25\\textwidth]{Bkg\/Figs\/zjet}\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/ttbar}\\\\\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/single-top_s_t}\n\\caption[Some Tree-Level Diagrams Contributing to the $EWK$ Backgrounds]{Examples of tree-level Feynman diagrams contributing to the $EWK$ background set. They are $Z+$jets production (top left in association with a $b\\bar{b}$ quark pair), $t\\bar{t}$ production and decay in semi-leptonic (top center) and di-leptonic channels (top right), single-top production in the $s$ (bottom left) and $t$ (bottom right) channels and semi-leptonic decay.}\\label{fig:fey_ewk}\n\\end{center}\n\\end{figure}\n\nThe normalization, $N_{l,k}^{EKW}$, of each $EWK$ process is obtained for each lepton category, $l$, in the pretag and $1,2$-Tag regions, $k$, by using Equation~\\ref{eq:mc_estimate}.  A summary of the production cross sections and the MC generator information are reported in Table~\\ref{tab:mc_ewk}.\n\n  \\begin{table}[h]\n    \\begin{center}\n    \\begin{tabular}{cccc}\n      \\toprule\n      Sample & $\\sigma$ (pb)& Source & Initial Events \\\\\n      \\midrule\n      $Z\/\\gamma+$jets    & 787.4 $\\pm$ 85    & Measured~\\cite{z_jets_cx} & Several MCs \\\\ \n      $t\\bar{t}$         & 7.04  $\\pm$ 0.49  & NNLO~\\cite{ttbar_cx} & 6.7 M\\\\  \n       single-top, $s$   & 1.017 $\\pm$ 0.065 & NNLO~\\cite{stops_kidonakis} & 3.5 M\\\\ \n       single-top, $t$   & 2.04  $\\pm$ 0.18  & NNLO~\\cite{stopt_kidonakis} & 3.5 M\\\\ \n      \\bottomrule\n    \\end{tabular}\n    \\caption[MC Information for the $EWK$ Backgrounds]{MC information for backgrounds of the $EWK$ category, i.e. the ones estimated from simulation. $Z+$ jets cross section~\\cite{z_jets_cx} refers to the on-shell $Z$ contribution and \\texttt{ALPGEN} LO MC prediction is used to derive a correction factor. \\texttt{HERWIG} plus \\texttt{PYTHIA} PS are used for $t\\bar{t}$, \\texttt{POWHEG} plus \\texttt{PYTHIA} PS are used for single-top $s$ and $t$ channels. The generated top mass is $m_t=172.5$~GeV$\/c^{2}$.}\\label{tab:mc_ewk}\n    \\end{center}\n  \\end{table}\n\nAs the three MC derived background processes and the diboson signal are estimated in the same way, the label $EWK$ will be used to indicate generically all of them:\n\\begin{equation}\n  \\sum\\limits_{EWK} N_{l,k}^{EWK} = N_{l,k}^{Zjets} + N_{l,k}^{t\\bar{t}} + N_{l,k}^{s-top,s} + N_{l,k}^{s-top,t} +  \\sum\\limits_{diboson} N_{l,k}^{diboson}\\mathrm{.}\n\\end{equation}\n\n\n\n\\section{$W$ plus Heavy Flavor Background}\\label{sec:whf}\n\nThe associated $W+HF$ production (Figure~\\ref{fig:fey_wjets} shows few examples of tree-level diagram contributions) is the most significant contribution to the total background in the single and double tag signal samples. For its estimate we rely partially on MC and partially on data. \n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Bkg\/Figs\/wjet}\n\\caption[Some Tree Level Diagrams Contributing to $W+HF$ and $W+LF$ Backgrounds]{Examples of tree-level Feynman diagrams contributing to the $W+HF$ and the $W+LF$ backgrounds. NLO diagrams also play a relevant role in the $W+$~QCD radiation emission.}\\label{fig:fey_wjets}\n\\end{center}\n\\end{figure}\n\nNormalization and kinematic distributions of the $W+$~jets sample, and of the $W+HF$ sub-sample, present several theoretical difficulties.  NLO MC simulations became available lately for $W+b\\bar{b}$ processes~\\cite{wbb_powheg} but large higher order contributions enhance the theoretical uncertainty. Furthermore the prediction of the complete $HF$ spectrum ($W+c\\bar{c}$, $W+b\\bar{b}$, $W+c$) is beyond the current capabilities. \n\nThe two aspects of the problem can be solved by factorizing the evaluation of the kinematic properties from the estimate of the normalization:\n\\begin{itemize}\n\\item a solution to estimate the kinematic properties and the composition of $W+$~jets QCD emission was proposed in~\\cite{first_lo_match} in 2001: the appropriate matching scheme of a LO calculation with a properly tuned Parton Shower (PS) simulation can reproduce the experimental results. Section~\\ref{sec:alpgen_comp} describes in detail our use of the \\texttt{ALPGEN} LO generator, interfaced with \\texttt{PHYTIA} PS. $W+HF$ relative components or $HF$ fractions, $f^{HF}$, can be extracted with this method.\n\\item the only free parameter remaining is, now, the $W+$~jets normalization. We derive it completely from data using a maximum likelihood fit on the SVM output distribution: this variable, described in Appendix~\\ref{chap:AppSvm}, has a high discriminating power between non-$W$ and real $W$ events. Therefore, as all the other $EWK$ components are known, both $N^{nonW}_{pretag}$ and $N^{Wjets}_{pretag}$  can be extracted from the fit. Section~\\ref{sec:wjets_pretag} describes the procedure.\n\\end{itemize}\n\n\n\n\\subsection{$W +$ Jets Spectrum Composition}\\label{sec:alpgen_comp}\n\n$W+$~jets composition and kinematic properties are evaluated from MC. We employ a large set of $W+n$~partons \\texttt{ALPGEN}~\\cite{alpgen} LO MC that includes $HF$ production in the ME calculation and is interfaced with \\verb'PYTHIA'~\\cite{pythia} PS:\n\\begin{itemize}\n\\item[-]$W$ plus $0$, $1$, $2$, $3$, $\\ge 4$ partons (for $W+ n$~generic jets description);\n\\item[-]$W+b\\bar{b}$ plus $0$, $1$, $2$ partons;\n\\item[-]$W+c\\bar{c}$ plus $0$, $1$, $2$ partons;\n\\item[-]$W+c$ (or $W+\\bar{c}$) plus $0$, $1$, $2$, $3$ partons.\n\\end{itemize}\nIn total we used ninety samples: fifteen to be multiplied by the three leptonic $W$ decay modes ($W\\to e,\\mu,\\tau +\\nu_e,\\nu_{\\mu}\\nu_{\\tau}$) and by another factor two for {\\em low luminosity} and {\\em high luminosity} running periods. Table~\\ref{tab:samples} summarizes the total of ninety samples used along with the respective LO cross sections and the approximate number of events initially generated.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\nSample& $\\sigma$ (pb)& Initial Events\\\\\n\\midrule\n$W+0p$& 1810& 7 M\\\\\n$W+1p$& 225 & 7 M\\\\\n$W+2p$& 35.3& 1.4 M\\\\\n$W+3p$& 5.59& 1.4 M\\\\\n$W+4p$& 1.03& 0.5 M\\\\\n$W+b\\bar{b}+0p$& 2.98& 2.1 M\\\\\n$W+b\\bar{b}+1p$& 0.888& 2.1 M\\\\\n$W+b\\bar{b}+2p$& 0.287& 2.1 M\\\\\n$W+c\\bar{c}+0p$& 5& 2.8 M\\\\\n$W+c\\bar{c}+1p$& 1.79& 2.8 M\\\\\n$W+c\\bar{c}+2p$& 0.628& 2.8 M\\\\\n$W+c+0p$& 17.1& 2.8 M\\\\\n$W+c+1p$& 3.39& 2.8 M\\\\\n$W+c+2p$& 0.507& 2.8 M\\\\\n$W+c+3p$& 0.083& 2.8 M\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[ALPGEN $W + $ Partons MC Samples]{\\texttt{ALPGEN} LO plus \\texttt{PYTHIA} (PS) MC samples used to estimate the various contributions to the $W+HF$ and $W+LF$ backgrounds. The second column lists the LO cross section of the samples, the third column shows the approximate number of generated events. Different MC samples are used for the three lepton flavors ($e, \\mu, \\tau$) of the $W$ decay.}\\label{tab:samples}\n\\end{center}\\end{table}\n\nThe basic composition algorithm is simple: the acceptance of each sample, for a given jet multiplicity and for each lepton category, is weighted by its own LO production cross section, then a normalization factor equal to the sum of all the samples is applied. The key idea is that, if the PS matching is well tuned, the higher order corrections to the LO $W+n$~partons simulation, should simplify when the normalization factor is applied. \n\nThe procedure is more complex in our case because we want an accurate estimate of the $W+HF$ spectrum: we need to derive the $HF$ fractions, $f^{HF}$, from the $W+n$~partons samples and relate them to the physical observables of $W+b\\bar{b}\/c\\bar{c}\/c$ jets with one or two \\texttt{SecVtx} $HF$-tags.\n\nThe $W+HF$ production is evaluated explicitly at ME level in some of the $W+n$ partons samples, however a certain amount of $HF$ quarks is also produced by the PS algorithm in all the samples. Some care must be used when combining all the samples because the double counting of a process will degrades the relative weights.\n\nThe {\\em overlap removal} technique used in this work was previously developed by the CDF collaboration and it is documented in~\\cite{hf_8765}. It consists in the selection of $HF$ production from the PS {\\em or} from the ME on the base of detector level reconstructed jets. We explicitly veto events where the ME heavy flavor quarks wind up in the same jet, and we also remove events where the $HF$ quark pair from the shower is divided in two jets. In both cases, the distinction is made with simple kinematic cuts that define a parton to be {\\em in a jet} if:\n\\begin{equation}\n  \\Delta R(parton, jet)<0.4\\quad\\textrm{and}\\quad E_T^{jet,cor}> 12~\\textrm{GeV}\n\\end{equation}\nThe reason for this choice is that PS and ME predict different $\\Delta R$ (separation) between heavy quarks: PS model is tuned on the more collinear gluon splitting quark pairs but fails in the limit of large opening angles, while ME generation works better in latter regime. Figure~\\ref{fig:overlap_rem_spectrum} shows the generator level $\\Delta R$ distribution of $b\\bar{b}$ and $c\\bar{c}$ pair production after the composition of the $W+n$~partons samples and the overlap removal procedure. The $\\Delta R $ from PS-only is also reported in Figure~\\ref{fig:overlap_rem_spectrum} showing disagreement at large $\\Delta R $ with respect to the $HF$ generated with ME. The smooth transition between the different  $W+n$~partons contributions indicates that the overlap removal scheme is working properly.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/cc_dr}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/bb_dr}\n\\caption[Generator Level $\\Delta R$ Distributions of $b\\bar{b}$ and $c\\bar{c}$ ]{Generator level $\\Delta R$ distributions of $b\\bar{b}$ (left) and $c\\bar{c}$ (right) pair production after the composition of the $W+n$~partons samples and the overlap removal procedure~\\cite{hf_8765}. The $\\Delta R $ from PS-only is also reported showing disagreement at large $\\Delta R $ w.r.t. the $HF$ generated with ME.}\\label{fig:overlap_rem_spectrum}\n\\end{center}\n\\end{figure}\n\nThe complete procedure for the determination of the $f^{HF}$'s is the following:\n\\begin{itemize}\n\\item each MC sample is assigned a weight:\n  \\begin{equation}\\label{eq:weight}\n    w^{m}=\\frac{\\sigma^m}{N_{z_0}^m} \\mathrm{,}\n  \\end{equation}\n  where $m$ varies on all the different samples of Table~\\ref{tab:samples}, $\\sigma^m$ is the cross section and $N_{z_0}^m$ is the total number of generated events with $|z_0^{vtx}|<60$~cm.\n\\item After pretag selection, the denominators of the $HF$ fractions are calculated as:\n  \\begin{equation}\n    Den=\\sum_{m=1}^{samples} w^{m}N^m\\mathrm{,}\n  \\end{equation}\n  where $N^m$ are the {\\em unique} events (i.e. not vetoed by the overlap removal) falling in each jet-bin for sample $m$.\n\\item Each selected jet is classified as $HF$ jet or not: $HF$ jets have a $b$ or $c$ parton (at generation level) {\\em match}, i.e. laying inside the cone (\\mbox{$\\Delta R(jet,parton)<0.4$}). We distinguish four $HF$ categories depending on the number of matched jets and on the kind of $HF$ parton ($b$ or $c$). The categories are: $1B$, $2B$, $1C$ and $2C$ depending if we have 1 or $\\ge 2$, $b$ or $c$ quarks matched; no $b$ quarks should be matched in the case of $1,2C$ categories.\n\\item The numerators are defined by the sum of the events in each $HF$ category (with weights given by Equation~\\ref{eq:weight}) over all the samples, in each jet-bin:\n  \\begin{equation}\n    Num^{HF}=\\sum_{m=1}^{samples}w^m N^{m,HF}\\mathrm{.}\n  \\end{equation}\n\\end{itemize}\nThe computation of the $f^{HF}$'s is now straightforward. Table~\\ref{tab:hf_frac} summarizes the final $HF$ fractions averaged on all the lepton categories.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccc}\n\\toprule\n\\multicolumn{4}{c}{Heavy Flavor Fractions}\\\\\n\\midrule\n$f^{1B}$ & $f^{2B}$ & $f^{1C}$ & $f^{2C}$ \\\\\n$0.024$ & $ 0.014$  & $ 0.114$  & $ 0.022$ \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[$HF$ fractions, $f^{HF}$, for Two Jets Selection]{$HF$ fractions, $f^{HF}$, for the two jets selection derived from \\texttt{ALPGEN} MC composition. They are classified in the different categories. $1B$: events with $1$ $b$-jet. $2B$: events with $\\ge 2$ $b$-jet. $1C$: events with $1$ $c$-jet and no $b$-jet. $2C$: events with $\\ge 2$ $c$-jet and no $b$-jet.}\\label{tab:hf_frac}\n\\end{center}\\end{table}\n\nHowever $f^{HF}$'s  are not immediately usable to predict $W+HF$ background. They need an additional correction, $K$, to account for different $HF$ composition in data and in the LO ME simulation:\n\\begin{equation}\\label{eq:k}\nK^{HF}=\\frac{f^{HF}_{data}}{f^{HF}_{MC}}\\mathrm{,}\n\\end{equation}\nThe $HF$ calibration is carried out on a data sample not used in this measurement: we base our correction on the studies performed in~\\cite{single_top}. There, the $W+1$~jet data sample is used to derive a correction of \\mbox{$K^{c\\bar{c}\/b\\bar{b}}=1.4$} to the $W+c\\bar{c}\/b\\bar{b}$ processes, while no correction ($K^{c}=1.0$) is quoted for the $W+c$ process\\footnote{$W+c$ production is an electroweak production process measured at CDF in~\\cite{cdf_wc}.}. A large uncertainty of $30$\\% is associated to all these $K$ factors as the correction is extrapolated from the $W+1$~jet data sample to the $W\\ge 2$ jets data sample. A previous study~\\cite{k_8768}, on a multi-jet control sample, obtained a $K$ closer to $1$ and it is consistent within the uncertainty. Table~\\ref{tab:k-fac} shows a summary of the applied $K$ corrections.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\n\\multicolumn{3}{c}{Heavy Flavor $K-$Factor Corrections}\\\\\n\\midrule\n$K^{W+b\\bar{b}}$ & $K^{W+c\\bar{c}}$ & $K^{W+c}$ \\\\\n$1.4\\pm 0.42$ & $1.4\\pm 0.42$ & $1.0\\pm 0.33$ \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[$HF$ $K-$Factor Correction]{$K-$Factor corrections needed for the calibration of the $HF$ production in data. Studies performed in~\\cite{single_top}.}\\label{tab:k-fac}\n\\end{center}\\end{table}\n\nThe final estimate of $f^{HF}$ is:\n\\begin{equation}\nf^{HF}=K\\frac{Num^{HF}}{Den}\\mathrm{.}\n\\end{equation}\n\nThe $HF$ fractions can be used in the construction of the pretag sample shapes but are of marginal importance because of the overwhelming $LF$ contribution.\nOn the other hand $HF$ are fundamental in the tagged sample composition. To evaluate it we need the tagging efficiency of each $HF$ category ($\\epsilon^{HF}$): they are derived from the MC samples requiring a \\verb'SecVtx' tag, then we apply the $\\omega_{k-Tag}$ correction as described in Section~\\ref{sec:jetSel}. Tagging efficiencies for each $HF$ category are summarized in Table~\\ref{tab:hf_tag_ef}.\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccccc}\n\\toprule\n\\multicolumn{5}{c}{Heavy Flavor Tagging Efficiency}\\\\\n\\midrule\n& $\\epsilon^{1B}$ & $\\epsilon^{2B}$ & $\\epsilon^{1C}$ & $\\epsilon^{2C}$ \\\\\n1-Tag & $ 22$\\%  & $ 30$\\% & $6.7$\\%  & $ 9.1$\\% \\\\ \n2-Tag & $ 0.4$\\%  & $8.6$ & $ 0.06$\\%  & $ 0.4$\\% \\\\ \n\\bottomrule\n\\end{tabular}\n\\caption[Heavy Flavor Tagging Efficiency $\\epsilon^{HF}$ ]{Heavy Flavor tagging efficiency $\\epsilon^{HF}$ in each category. $1B$: events with $1$ $b$-jet. $2B$: events with $\\ge 2$ $b$-jet. $1C$: events with $1$ $c$-jet. $2C$: events with $\\ge 2$ $c$-jet.}\\label{tab:hf_tag_ef}\n  \\end{center}\n\\end{table}\n\n\nFinally we can combine all the pieces to obtain the {\\em final} $HF$ tagged contribution. For each lepton $l$ and tag category $k$, it is:\n\\begin{equation}\\label{eq:tag_hf}\nN^{HF}_{l,k}=N^{Wjets}_{l,k}\\cdot f^{HF} K^{HF}\\epsilon^{HF}_{k}\\mathrm{,}\n\\end{equation}\nWhere the $HF$ label distinguishes the number of jet-matched heavy flavor hadrons: $1,2B$, $1,2C$. The $1,2B$ fractions represent $W+b\\bar{b}$, $2C$ represents $W+c\\bar{c}$ prediction and $1C$ is related to $W+c$ production. The extraction of the $W+$~jets pretag normalization ($N^{Wjets}$) is discussed in the next Section.\n\n\n\n\\subsection{Pretag $W +$ Jets Normalization Estimate}\\label{sec:wjets_pretag}\n\nAccording to Equation~\\ref{eq:tag_hf}, the $W+$~jet normalization, $N^{Wjets}$,  is the only missing piece in the complete $W+HF$ estimate.  $N^{Wjets}$ is obtained solving:\n\\begin{equation}\\label{eq:pretag_comp}\nN^{Data}_{Pretag} = F^{nonW}_{Pretag} \\cdot N^{Data}_{Pretag} + F^{Wjets}_{Pretag} \\cdot N^{Data}_{Pretag} + \\sum_{EWK}N^{EWK}_{Pretag} \\mathrm{,}\n\\end{equation}\nwhere:\n\\begin{itemize}\n\\item $N^{EWK}$ is derived from Section~\\ref{sec:mc_bkg};\n\\item the fractions $F^{nonW}_{Pretag}$ and $F^{Wjets}_{Pretag}$ are unknown parameters obtained by a maximum likelihood fit in the pretag region. This provides a $W+$~jets estimate unbiased by the $b$-tag requirement;\n\\item the SVM output distribution is used in the fit before the application of the multi-jet rejection cut (Section~\\ref{sec:svm_sel}) so that the fake-$W$ fraction of the sample can be extrapolated from the sideband. \n\\end{itemize}\nWe build a binned likelihood function on the SVM distribution on the base of Equation~\\ref{eq:pretag_comp}. Events are Poisson distributed and a Gaussian constraint is imposed to the  $N^{EWK}$ normalizations while $F^{nonW}_{Pretag}$ and $F^{Wjets}_{Pretag}$ are left free to vary.\n\nTemplates used in the fit are derived in different ways according to the background source: $EWK$ and $diboson$ SVM distributions are derived from MC, $W+$~jets template is derived from the $W+n$~partons MC composition described in previous Section. Non-$W$ template is obtained in data by reversing appropriate lepton identification cuts, a detailed description is given in Section~\\ref{sec:qcd}.\nTemplates are built for each lepton category but we decided to combine all the EMC algorithms in one template to reduce statistical fluctuations exploiting the homogeneous behaviour of the different components of EMC.\n\nThe validation of the SVM output distribution is another important step of the fit procedure. Most of the previous analyses~\\cite{single_top, wh94_note10796} used a fit to the \\mbox{${\\not}{E_T}$}  distribution to derive the $W+$ jets fraction. Here a slightly different strategy is used as the SVM was trained to discriminate the $W+$~jets sample against multi-jet events. This physics interpretation was checked during the training process (see Appendix~\\ref{chap:AppSvm}). A simplified version of the fit (slightly different selection and composed only by non-$W$ and $W+$~jets samples) was performed at each training step to reject unreliable configurations. We based the rejection on two figures of merit: \n\\begin{itemize}\n\\item the $\\chi^2$ evaluation of post fit template\/data shapes; \n\\item the comparison of the non-$W$ and $W+$~jets expectation and fit estimate.\n\\end{itemize}\n\nThe final step is the actual fit evaluation for the CEM, PHX, CMUP, CMX and EMC lepton categories. Figure~\\ref{fig:pretag_fit} shows the result: the template composition with the fractions returned by the maximization likelihood fit. Table~\\ref{tab:wjet_pretag_qcd_fit} summarizes $F^{Wjets}_{Pretag}$ and $F^{nonW}_{Pretag}$ extracted from the fit. \n\nThe pretag non-$W$ fraction, $F^{nonW}_{Pretag}$, is not used any more in the background estimate but the total non-$W$ pretag normalization, $N^{nonW}_{Pretag}$, is needed to cross check the kinematic of the pretag control sample. We derive it with the following equation:\n\\begin{equation}\n  N^{nonW}_{Pretag} = F^{nonW}_{Pretag} \\cdot N^{Data}_{Pretag} + \\sigma^{EWK}\\textrm{;}\n\\end{equation}\nwhere $\\sigma^{EWK}$ is the change in the normalization of the $EWK$ backgrounds as returned by the Likelihood fit.\n\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccccc}\n\\toprule\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n$F^{nonW}_{Pretag}$ & $ 8.5 \\pm 0.2$\\% &  $ 13.6\\pm 0.1 $\\% &  $ 1.9\\pm 0.2 $\\% &  $2.2 \\pm 0.2 $\\% & $5.9\\pm 0.2$\\%\\\\\n$F^{Wjets}_{Pretag}$ &  $ 84.0 \\pm 0.4$\\% &  $ 82.5 \\pm 0.6$\\% &  $86.7 \\pm 0.7 $\\% &  $ 87.9 \\pm 0.9$\\% &  $ 78.7\\pm 0.9 $\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[$W +$Jets and Non-$W$ Pretag Sample Compositions]{$W+$ jets and non-$W$ fractional compositions of the pretag selection sample estimated by a maximum likelihood fit on the SVM output distributions reported in Figure~\\ref{fig:pretag_fit}. The statistical error of the fit is reported.}\\label{tab:wjet_pretag_qcd_fit}\n\\end{center}\\end{table}\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_PretagSVTSVT_ISOTRK}\n\\caption[$W +$ Jets and Non-$W$ Pretag Sample Estimate]{Background templates composition with proportions returned by the maximization likelihood fit on the SVM pretag distribution for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the selection cut value in each lepton category.}\\label{fig:pretag_fit}\n\\end{center}\n\\end{figure}\n\n\\subsection{$W +$ Jets Scale Uncertainty}\nThe simulation of the $W+$~jets sample should take into account one more effect, the uncertainty on the factorization and renormalization scale, $Q^2$, of the ALPGEN event generator. \n\nThe $Q^2$ of the generator can be seen as the momentum scale of the hard interaction. Technically this value is used in two contexts:\n\\begin{itemize}\n\\item in the evaluation of the PDFs of the hard interaction, the {\\em factorization scale};\n\\item in the calculation of the ME perturbative expansion of a given QCD process, the {\\em renormalization scale};\n\\end{itemize}\nA reasonable choice for it, in the $W+n$~partons MC, is given by the following equation: \n\\begin{equation}\\label{eq:scale}\n  Q^2 = M^2_W + \\sum^{partons} p_T^2  \n\\end{equation}\nwhere $M_W$ is the $W$ boson mass, the sum extends over all the partons and $p_T$ is their transverse energy.\n\nHowever the $Q^2$ is not a physical observable, as it is an artifact of the perturbative approximation needed to solve QCD problems, therefore an appropriate uncertainty should be taken into account. \nTable~\\ref{tab:q2samples} shows the characteristics of two new sets of $W+n$~partons MC samples used for systematic variation and generated with the $Q^2$ parameter doubled and halved.\n\\begin{table}\\begin{center}\n\\begin{tabular}{lcccc}\n\\toprule\n&\\multicolumn{2}{c}{\\bf $\\mathbf{Q^2=2.0}$ (Up)}& \\multicolumn{2}{c}{\\bf $\\mathbf{Q^2=0.5}$ Down}\\\\\nSample& $\\sigma$ (pb)& Initial Events &  $\\sigma$ (pb)& Initial Events\\\\\n\\midrule\n$W+0p$        & 1767  & 3.5 M & 1912  & 3.5 M\\\\\n$W+1p$        & 182.9 & 3.5 M & 303   & 3.5 M\\\\\n$W+2p$        & 24.6  & 0.7 M & 57.6  & 0.7 M\\\\\n$W+3p$        & 3.36  & 0.7 M & 10.8  & 0.7 M\\\\\n$W+4p$        & 0.54  & 0.7 M & 2.24  & 0.7 M\\\\\n$W+b\\bar{b}+0p$& 2.30  & 0.7 M & 4.14  & 0.7 M\\\\\n$W+b\\bar{b}+1p$& 0.550 & 0.7 M & 1.64  & 0.7 M\\\\\n$W+b\\bar{b}+2p$& 0.152 & 0.7 M & 0.615 & 0.7 M\\\\\n$W+c\\bar{c}+0p$& 3.89  & 1.4 M & 6.88  & 1.4 M\\\\\n$W+c\\bar{c}+1p$& 1.08  & 1.4 M & 3.18  & 1.4 M\\\\\n$W+c\\bar{c}+2p$& 0.323 & 1.4 M & 1.30  & 1.4 M\\\\\n$W+c+0p$       & 13.8  & 1.4 M & 23.3  & 1.4 M\\\\\n$W+c+1p$       & 2.30  & 1.4 M & 5.72  & 1.4 M\\\\\n$W+c+2p$       & 0.294 & 1.4 M & 1.03  & 1.4 M\\\\\n$W+c+3p$       & 0.042 & 1.4 M & 0.189 & 1.4 M\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[ALPGEN $W + $Partons MC Samples for $Q^2$ Systematics]{\\texttt{ALPGEN} LO plus \\texttt{PYTHIA} (PS) MC samples used to estimate the $Q^2$ systematic variation of the $W+HF$ and $W+LF$ backgrounds. The $Q^2$ is halved and doubled with respect to the default value given by Equation~\\ref{eq:scale}. Different MC samples are used for the three lepton flavors ($e, \\mu, \\tau$) of the $W$ decay.}\\label{tab:q2samples}\n\\end{center}\\end{table}\n\nThe $Q^2$ variation affects both the kinematic and the parton composition of the $W+$~jets sample, therefore, for each $Q^2$ systematic variation, the background estimate was redone as described Section~\\ref{sec:wjets_pretag}. The results are different shapes and rates for the $W+$~jets components. Figure~\\ref{fig:wjets_mjj_q2} shows the effect on the single-tag $M_{Inv}(jet1,jet2)$ distributions.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wlf_q2sys}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wc_q2sys}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wbb_q2sys}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/mjj_wcc_q2sys}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution and $Q^2$ Variations for $W + ~LF\/c\/c\\bar{c}\/b\\bar{b}$ Samples]{$M_{Inv}(jet1,jet2)$ distribution for single \\texttt{SecVtx} tagged events, after combination of all the lepton categories: the default estimate of the four $W+$~jets samples is marked by the continuous line while the $Q^2$ systematic variations are marked by the dashed lines. Top left: $W+LF$. Top right: $W+c$. Bottom left: $W+c\\bar{c}$. Bottom right: $W+b\\bar{b}$.}\\label{fig:wjets_mjj_q2}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{$W$ plus Light Flavors}\\label{sec:mistag}\n\nThe $W+$~jets events originating from a $LF$ quark can produce a secondary vertex for several reasons. Long living $LF$ hadrons produce a small amount of real \\verb'SecVtx' tags while false \\verb'SecVtx' tags are due to track reconstuction errors or interaction with the detector material and the beam pipe.\n\nThe $LF$ pretag fraction ($f_j^{LF}$) is derived from the $W+n$~partons MC composition, in the same way as the $HF$ fraction. However after requiring a $b$-tag, only a very small fraction of $LF$ jets remains in the sample. As the effects that generate mistags are not adequately simulated, appropriate parametrization is obtained studying the {\\em mistag} behaviour of the \\verb'SecVtx' algorithm. \n\nAs explained in Section~\\ref{sec:btag_sf}, a multi-jet control sample is used to parametrize a per-jet mistag probability,  $p^j_{Mistag}$, function of six variables specific of the event and the jet. The $p^j_{Mistag}$ evaluation can be applied to any data sample or to MC. This gives the possibility to divide the normalization and the shape evaluation issues:\n\\begin{itemize}\n\\item the normalization is derived directly from the selected data sample. For each event we calculate an {\\em event-mistag} estimate, $w_{Mistag}^{ev}$, by using  Equations~\\ref{eq:1tag} and~\\ref{eq:2tag} with the $p^j_{Mistag}$ of the jets as inputs. The sum of all the event-mistag gives a raw normalization of the total $LF$ contribution:\n  \\begin{equation}\n    N^{rawLF}=\\sum_{ev} w_{Mistag}^{ev}\\textrm{,}\n  \\end{equation}\nthat needs to be corrected for the contribution of the other backgrounds. The $W+LF$ only part is:\n\\begin{equation}\n  N^{W+LF} = N^{rawLF} \\frac{N^{Wjets} - N^{W+HF}- \\sum\\limits_{EWK}{N^{EWK}}}{N^{Wjets}+ \\sum\\limits_{EWK}N^{EWK}}\\textrm{.}\n\\end{equation}\n\n\\item Shape is derived from the $W+LF$ composition of ALPGEN MC (no $HF$ matched jets are used) where each simulated event is weighted by its own $w_{Mistag}$ to reproduce the kinematic beaviour of the mistagged jets. Figure~\\ref{fig:wlf_weight} shows the normalized ratios between the single-tagged $W+LF$ weighted MC and the original pretag $W+LF$ distribution: $LF$ jets have a higher mistag probability in the central $\\eta$ region of the detector (where also \\texttt{SecVtx} $b$-tagging efficiency is higher) and for large $E_T$. The effect on the $W+LF$ $M_{Inv}(jet1, jet2)$ spectrum after the $b$-tagging requirement is also shown.\n\n\\end{itemize}\n\nA rate uncertainty of $11$\\% and $21$\\% is included in the single and double-tagged mistag estimates respectively. It is derived from the mistag matrix parametrization and takes into account the statistical uncertainties and correlations between jets which fall in the same jet-bin of the mistag matrix. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/wlf_eta}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/wlf_et}\\\\\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/wlf_mjj}\n\\caption[Normalized Ratios of $W+LF$ Pretag and Tag Distributions]{Normalized ratios between the single-tagged $W+LF$ weighted MC and the original pretag $W+LF$ MC distribution: jet 1 $\\eta$ distribution (top left), jet 1 $E_T$ distribution (top right) and $M_{Inv}(jet1, jet2)$ (bottom).}\\label{fig:wlf_weight}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\section{Multi-jet Background}\\label{sec:qcd}\n\nAnother source of background comes from fake-$W$ events, where a QCD multi-jet event fakes the charged lepton and produces enough \\mbox{${\\not}{E_T}$}  to pass the event selection (\\mbox{${\\not}{E_T}$} $>15$~GeV). \n\nThe probability to fake the lepton identification both at trigger and at offline level selection is very small but, at hadron colliders, multi-jet events are produced at such a high rate that this background becomes important. The selection of a fake lepton can happen for a mixture of physics and detector effects: for example a jet with large EM fraction and low track multiplicity can easily fake an electron or a high $p_T$ hadron can reach the muon chambers leaving little energy in not well instrumented sections of the calorimeter. A simulation of these effects would need an extremely high statistics, a perfect simulation of the detector and an accurate QCD prediction at high orders. This is not feasible therefore we rely on data to model the multi-jet sample (both shape and normalization). \n\nA fake-$W$ sample is obtained by reversing one or more $W\\to \\ell \\nu$ identification criteria so that the final sample will be enriched in multi-jet events and orthogonal to the signal selection. However the correlation between the reversed identification variables and the related kinematic quantities are lost in the process. The fake-$W$ sample may present discrepancies with respect to the truly multi-jet events selected in the signal region. \n\nThe production mechanism is strictly related to the lepton identification algorithm, therefore we employ three different models:\n\\begin{description}\n\\item[Not isolated muons:] the $IsoRel <0.1$ cut is one of the most important $W$ identification requirements therefore we can employ real muons but in the sideband with $IsoRel>0.2$ to select fake-$W$ candidates for CMX, CMUP and EMC categories\\footnote{EMC is composed by 7 lepton identification algorithms but the non-isolated EMC category is build only by four of them: CMU, CMP, CMXNT and BMU. The multi-jet sample properties are reproduced well enough also with this simpler model.}.\n For a better simulation, the jet associated with the non-isolated muon is removed from the jet-multiplicity count and its energy (without the muon) accounted for the \\mbox{${\\not}{E_T}$}  correction (Section~\\ref{sec:metObj}).\n\\item[Fake-CEM:] CEM electron case is different because the isolation cut is correlated with the EM cluster energy, used both at trigger level selection and in the calorimeter \\mbox{${\\not}{E_T}$}  calculation. A fake-$W$ model that maintains $IsoRel<0.1$ is obtained reversing at least two out of five shower identification requirements: Table~\\ref{tab:antiEle} shows the reversed cuts. As in the muon case, jet multiplicity should be corrected for the jet associated with the lepton, however it is not straightforward to decide if any correction needs to be applied to the \\mbox{${\\not}{E_T}$}  calculation. \n\nWe studied the behaviour of the fake model on data, after full selection, but with no SVM cut and for \\mbox{\\mbox{${\\not}{E_T}$} $<15$~GeV}. The region is supposed to be dominated by multi-jet events (except for a small $Z$ contamination) therefore we compared data to the fake-$W$ model.  Figure~\\ref{fig:chi2} shows how the reduced $\\chi^2$ test changes when comparing $M^W_T$ shapes for data and fake-$W$ model with different \\mbox{${\\not}{E_T}$}  corrections. The best $\\chi^2$ value is obtained when the \\mbox{${\\not}{E_T}$}  is corrected as if the jet corresponding to the anti-CEM is present in the event but with an energy of $0.45\\cdot E_T^{jet,raw}$. We apply this correction.\n\n\\item[Fake-PHX:] also the fake-$W$ PHX model is obtained with an anti-PHX selection where 2 out of 5 shower identification cuts are reversed (listed in Table~\\ref{tab:antiEle}). In this case no special prescription to the \\mbox{${\\not}{E_T}$}  correction was found to drastically improve the kinematic model. Although some properties of the multi-jet sample are not perfectly reproduced, the high rejection power of the SVM discriminant reduces this to a minor problem.\n\\end{description}\nThe accurate description of the multi-jet background and its rejection by using the SVM discriminant played a major role in this analysis and in the latest $WH$ search results~\\cite{wh75_note10596,wh94_note10796}.\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{c|c}\n\\toprule\n\\multicolumn{2}{c}{\\bf Anti-Electron: $\\ge 2$ Failed Cuts }\\\\\nCEM  & PHX \\\\\\midrule\n$E^{Had}\/E^{EM}<0.055+0.0045 E^{EM}$& $E^{Had}\/E^{EM}<0.05$ \\\\        \n$L_{shr}<0.2$                      & PEM$3\\times 3$Fit $\\ne 0$ \\\\    \n$\\chi^2_{\\mathrm{{CES strip}}}<10$ & $\\chi^2_{PEM3\\times 3} <10$ \\\\  \n$\\Delta z(CES,trk)<3$~cm           & $PES5\\mathrm{by}9U>0.65$ \\\\     \n$-3.0<q \\Delta x(CES,trk)<1.5$~cm  & $PES5\\mathrm{by}9V>0.65$ \\\\     \n\\bottomrule\n\\end{tabular}\n\\caption[CEM and PHX Multi-jet Model Selection Cuts]{Multi-jet models for CEM and PHX electrons should fail at least 2 of the shower identification cuts listed here. The models built in this way are named {\\em anti-CEM} and {\\em anti-PHX} leptons.}\\label{tab:antiEle}\n\\end{center}\\end{table}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Bkg\/Figs\/cem_scan100}\n\\caption[Reduced $\\chi^2$ Used to Tune \\mbox{${\\not}{E_T}$}  Correction on Fake-$W$ Model]{Reduced $\\chi^2$ obtained comparing the normalized $M^W_T$ shapes of data and anti-CEM fake-$W$ model in the multi-jet enriched region  \\mbox{${\\not}{E_T}$} $<15$~GeV. The test changes as we correct the \\mbox{${\\not}{E_T}$}  for fractions of the energy associated with the jet corresponding to the anti-CEM.}\\label{fig:chi2}\n\\end{center}\n\\end{figure}\n\nThe fake-$W$ models can now be used in the background evaluation. Section~\\ref{sec:wjets_pretag} already gives the formula for the pretag $N^{nonW}$ estimate, however this result is not used in this analysis except for control purposes and to plot the background composition in the pretag region. On the other hand, the tagged region normalization of the fake-$W$ sample is the only missing piece in the full background evaluation. To estimate it we use again the multi-jet enriched sideband of the SVM output distribution.\n\nWe perform another maximum likelihood fit similar to the one described in Section~\\ref{sec:wjets_pretag} with two differences. The first is a modification of the template: we require events with at least $1$ ($2$) {\\em taggable} jets when evaluating the single (double) tag fake-$W$ templates, this allows an approximate simulation of the $b$-tag requirement retaining most of the statistics of the samples. The second difference is in the fit strategy: all the real-$W$ samples are added into one template whose normalization is left free to float in the fit together with $N^{nonW}_{k-Tag}$. This procedure and the use of the sidebands make the fake-$W$ determination as much uncorrelated as possible from the other backgrounds and from a possible unknown signal. Figures~\\ref{fig:nonw_tag1} and~\\ref{fig:nonw_tag2} show the results of the $N^{nonW}_{k-Tag}$ fit on the single and double tagged SVM distributions for the CEM, PHX, CMUP, CMX and EMC lepton categories.\nA conservative rate uncertainty of 40\\% is applied uniformly on all the lepton categories to account for low statistics in some of the fits and for the extrapolation from the sideband region. Fake-$W$ contamination is summarized in Table~\\ref{tab:wjet_tag_qcd_fit}, confirming the excellent rejection power of the SVM discriminant.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccccc}\n\\toprule\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n$F^{nonW}_{1-Tag}$ & $ 9.7 \\pm 0.5$\\% &  $ 19.1 \\pm 0.7  $\\% &  $ 3.7 \\pm 0.7  $\\% &  $ 3.3\\pm 1.0 $\\% & $ 6.2 \\pm 0.5$\\%\\\\\n $F^{nonW}_{2-Tag}$ &  $ 3.2 \\pm 1.8$\\% &  $ 10.8 \\pm 2.4$\\% &  $ 4.7 \\pm 3.6$\\% &  $ 0.0 \\pm 2.2$\\% &  $ 0.0\\pm 0.3$\\%\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Non-$W$ Background Contamination in Single and Double Tag Samples]{Estimate of the non-$W$ contamination in the single and double-tagged selection samples. Only the statistical error of the fit is reported, a systematic uncertainty of $40$\\% is used in the final estimate.}\\label{tab:wjet_tag_qcd_fit}\n\\end{center}\\end{table}\n\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVT_ISOTRK}\n\\caption[Single Tag SVM Distribution for Non-$W$ Background Estimate]{Result of the maximum likelihood fit performed on the single-tagged SVM distribution to estimate the non-$W$ background normalization: $N^{nonW}_{1-Tag} = F^{nonW}_{1-Tag}\\cdot N^{Data}_{k-Tag}$. Results are reported for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the used selection cut value in each lepton category.}\\label{fig:nonw_tag1}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CEM}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_PHX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CMUP}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_CMX}\\\\\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/NonWFit_2JET_SVTSVT_ISOTRK}\n\\caption[Double Tag SVM Distribution for Non-$W$ Background Estimate]{Result of the maximum likelihood fit performed on the double-tagged SVM distribution to estimate the non-$W$ background normalization: $N^{nonW}_{1-Tag} = F^{nonW}_{1-Tag}\\cdot N^{Data}_{k-Tag}$. Results are reported for the different lepton categories: CEM (top left), PHX (top right), CMUP (center left), CMX (center right), EMC (bottom). An arrow indicates the used selection cut value in each lepton category.}\\label{fig:nonw_tag2}\n\\end{center}\n\\end{figure}\n\n\\section{Correction of Luminosity Effects}\n\nThe effect of instantaneous luminosity should be corrected in the available MC samples before completing of the background estimate. \n\nThe instantaneous luminosity of the collisions influences several parameters like the occupancy of the detector, the number of vertices in the interactions and so on. Most of these effects are parametrized within the CDF simulation (see Section~\\ref{sec:mcGen}) and they are simulated in a run-dependent way (i.e. each MC section corresponds to a period of data taking with appropriate tuning). Unluckily the available MC samples reproduce only a first section (up to 2008) of the CDF dataset while the highest luminosities (easily above $\\mathscr{L} = 2\\times 10^{32}$~cm$^{-2}$s$^{-1}$) were delivered by the Tevatron in the latest years. \n\nAn approximate solution to the problem is the {\\em reweighting} of the MCs as a function of the observed number of good quality primary vertices\\footnote{Quality$\\ge 12$, see Table~\\ref{tab:pVtx}.} (a variable highly correlated with the instantaneous luminosity). The reweighting function is evaluated comparing, in the pretag control region, the number of vertices estimated in MC against the one observed in data. Figure~\\ref{fig:nvtx_rew} shows the distribution of the number of primary vertices withouth any correction and after correcting the background estimate with the reweighting function that we apply to the final evaluation.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_nQ12Vtx_noW}\n\\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_nQ12Vtx}\n\\caption[Number of \\mbox{{\\em Quality} $\\ge 12$} Before and After Reweighting]{Distribution (for all the lepton categories) of the number of \\mbox{{\\em Quality} $\\ge 12$} primary vertices for the events selected in the pretag control region before (left) and after (right) the application of a reweighting function to the MC events.}\\label{fig:nvtx_rew}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\n\\section{Final Background Evaluation}\n\nAs explained in the previous part of this Chapter, we derive the shape and normalization contributions of each background in an independent way. The composition of them should reproduce, within uncertainties, the observed number of selected events and the kinematic distributions of data in control and signal regions. If the background estimate is proven to be solid, then it is possible to investigate the presence of a signal with adequate tools. \n\nWe present here the final background estimate. Statistical analysis and the measured properties of the $diboson\\to \\ell \\nu + HF$ signal will be described in the next Chapter.\n\nThe first important validation criteria is the agreement of the kinematic distributions in the pretag control region. As the normalization of the region is constrained by data the most important effect comes from the evaluation of the shapes. Figures from~\\ref{fig:pret_jetEt} to~\\ref{fig:pret_mjj} show several kinematic variables after the composition of all the background for all the lepton categories. For each variable, we estimated a central pretag background value and four systematic variations: \n\\begin{itemize}\n\\item jet energy scale plus and minus one sigma (JES $+1\\sigma$, JES $-1\\sigma$);\n\\item $W+n$~partons renormalization scale variation: $Q^2$ is doubled ($Q^2_{Up}= 2.0 Q^2 $) and halved ($Q^2_{Down}= 0.5 Q^2$).\n\\end{itemize}\nThe positive (negative) systematic variations are added in quadrature and the result is shown together with the central background evaluation. The agreement is excellent within uncertainties. Few distributions that show mis-modeling, like the $\\eta$ of the second jet shown in Figure~\\ref{fig:pret_jetEta}, become irrelevant after the tagging requirement that enforces a more central selection of the jets.\n\n  \\begin{figure}\n    \\begin{center}    \n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J1Et}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J2Et}\n    \\end{center}\n  \\caption[Jets $E_T$ Distributions in Pretag Control Region]{Jets corrected $E_T$ distributions in the pretag control region, for all the lepton categories combined: jet 1 $E_T$ (left) and jet 2 $E_T$ (right) for all the lepton categories combined, in the pretag control region.}\\label{fig:pret_jetEt}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J1Eta}\n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_J2Eta}\n    \\end{center}\n    \\caption[Jets $\\eta$ Distributions in Pretag Control Region]{Jets $\\eta$ distributions in the pretag control region, for all the lepton categories combined: jet 1 $\\eta$ (left) and jet 2 $\\eta$ (right).}\\label{fig:pret_jetEta}\n  \\end{figure}\n  \n  \\begin{figure}\n    \\begin{center}    \n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_LepPt}\n    \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_LepEta}\n    \\end{center}\n    \\caption[Lepton $p_T$ and $\\eta$ Distributions in Pretag Control Region]{Two lepton related kinematic distributions in the pretag control region, for all the lepton categories combined: $p_T$ (left) and $\\eta$ (right).}\\label{fig:pret_lepKin}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_met}\n  \\includegraphics[width=0.495\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dPhiMetJet1}\n    \\end{center}\n  \\caption[$\\nu$ Kinematic Distributions in Pretag Control Region]{Two $\\nu$ related kinematic distributions in the pretag control region, for all the lepton categories combined: \\mbox{${\\not}{E_T}$}  (left) and $\\Delta\\phi(jet1, $\\mbox{${\\not}{E_T}$} $)$ (right).}\\label{fig:pret_metKin}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}      \n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_wPt}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_WMt}\n      \\caption[$W$ Kinematic Distributions in Pretag Control Region]{Two $W$ related kinematic distributions in the pretag control region, for all the lepton categories combined: $p_T^W$ (left) and $M_T^W$ (right).}\\label{fig:pret_wkin}\n    \\end{center}\n  \\end{figure}\n\n  \\begin{figure}\n    \\begin{center}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dRLepJ1}\n      \\includegraphics[width=0.49\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_dRJ1J2}     \n    \\end{center}\n    \\caption[Angular Separation Distributions in Pretag Control Region]{Two angular separation distributions in the pretag control region, for all the lepton categories combined: $\\Delta R(lep,jet1)$ (left), $\\Delta R(jet1,jet2)$, (right).}\\label{fig:pret_dphi}\n  \\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Bkg\/Figs\/WZ_2JET_PretagSVT_CEMCMUPCMXISOTRKPHX_Mj1j2}\n\\caption[$M_{Inv}(jet1, jet2)$ Distribution in Pretag Control Region]{Invariant mass distribution of the selected di-jet pair in the pretag control region, for all the lepton categories combined.}\\label{fig:pret_mjj}\n\\end{center}\n\\end{figure}\n\nThe other validation method is the actual comparison of the expected backgrounds in the signal region against the observed selection. Tables~\\ref{tab:1tag_final} and~\\ref{tab:2tag_final} report the complete background composition in each of the separate lepton categories for the single and double \\texttt{SecVtx}-tagged selections, neither JES nor $Q^2$ variations are reported in the tables. All the estimates are within the uncertainties, that are, however, rather large because of the $K^{HF}$ factor uncertainties listed in Table~\\ref{tab:k-fac}. \n\n\\begin{sidewaystable}\n  \\begin{center}\n    \\begin{tabular}{lccccc}\n  \\toprule\n        \\multicolumn{6}{c}{\\bf Single-tag Event Selection}\\\\\n        Lepton ID & CEM                  &  PHX         &             CMUP             & CMX    & EMC    \\\\  \n\\midrule                                                         \n   $Z+$jets           & 55.53 $\\pm$ 4.73     & 7.76 $\\pm$ 0.67       & 65.3 $\\pm$ 5.73        & 37.14 $\\pm$ 3.28    & 104.18 $\\pm$ 10.8  \\\\    \n   $t\\bar{t}$      & 237.55 $\\pm$ 23.3    & 46.93 $\\pm$ 4.59      & 139.68 $\\pm$ 13.8      & 62.22 $\\pm$ 6.14    & 228.58 $\\pm$ 25.6   \\\\   \n   Single-top $s$  & 64.23 $\\pm$ 5.88     & 11.08 $\\pm$ 1.02      & 36.42 $\\pm$ 3.35       & 16.11 $\\pm$ 1.49    & 51.68 $\\pm$ 5.48    \\\\   \n   Single-top $t$  & 84.95 $\\pm$ 9.99     & 16.86 $\\pm$ 1.98      & 47.75 $\\pm$ 5.64       & 22.5 $\\pm$ 2.66     & 66.22 $\\pm$ 8.56    \\\\   \n   $WW$              & 84.35 $\\pm$ 11.8          & 25.05 $\\pm$ 3.54       & 43.7 $\\pm$ 6.17        & 23.68 $\\pm$ 3.35      & 53.11 $\\pm$ 8     \\\\\n   $ZZ$              & 1.85 $\\pm$ 0.19           & 0.21 $\\pm$ 0.02        & 2.45 $\\pm$ 0.25        & 1.37 $\\pm$ 0.14       & 3.45 $\\pm$ 0.39  \\\\ \n   $WZ$              & 29.2 $\\pm$ 2.95           & 12.32 $\\pm$ 1.22       & 16 $\\pm$ 1.66          & 9.21 $\\pm$ 0.94       & 20.54 $\\pm$ 2.39  \\\\\n   $W+b\\bar{b}$    & 858.71 $\\pm$ 258        & 263.08 $\\pm$ 79.1       & 428.58 $\\pm$ 129       & 238.99 $\\pm$ 71.9    & 409.97 $\\pm$ 123   \\\\\n   $W+c\\bar{c}$    & 441.77 $\\pm$ 134        & 145.08 $\\pm$ 44.1       & 212.98 $\\pm$ 64.8      & 120.26 $\\pm$ 36.6    & 214 $\\pm$ 65       \\\\\n   $W+cj$            & 342.86 $\\pm$ 104        & 96.42 $\\pm$ 29.3       & 171.77 $\\pm$ 52.2      & 93.01 $\\pm$ 28.3     & 143.47 $\\pm$ 43.6  \\\\ \n   $W+LF$         & 809.01 $\\pm$ 87.1       & 302.78 $\\pm$ 32.2       & 408.74 $\\pm$ 43.3      & 230.75 $\\pm$ 24.7    & 463.65 $\\pm$ 53.5  \\\\\n   Non-$W$           & 302.69 $\\pm$ 121        & 205.06 $\\pm$ 82       & 58.75 $\\pm$ 23.5       & 27.96 $\\pm$ 11.2     & 106.35 $\\pm$ 42.5    \\\\\n\\midrule                                                                                                                                    \n   {\\bf Prediction}      & {\\bf 3312.68 $\\pm$ 521}       & {\\bf 1132.63 $\\pm$ 176 }      & {\\bf 1632.11 $\\pm$ 253 }     & {\\bf 883.21 $\\pm$ 140}     & {\\bf 1865.2 $\\pm$ 248}   \\\\\n   {\\bf Observed }       & {\\bf 3115}                    & {\\bf 1073}                  & {\\bf 1577 }                  & {\\bf 830}                   & {\\bf 1705}               \\\\\n\\midrule                                                                                                                                \n   {\\bf Dibosons}     & {\\bf 115.39 $\\pm$ 13}           & {\\bf 37.57 $\\pm$ 4.05}      & {\\bf 62.15 $\\pm$ 6.87}       & {\\bf 34.26 $\\pm$ 3.75}    & {\\bf 77.09 $\\pm$ 9.35}    \\\\\n\\bottomrule\n    \\end{tabular}  \n  \\end{center}\n    \\caption[Observed and Expected Events with One \\texttt{SecVtx} Tags]{Summary of observed and expected events with one secondary vertex tag (\\texttt{SecVtx}), in the $W+2$~jets sample, in \n      $9.4$~fb$^{-1}$ of data in the different lepton categories. Statistical and systematics rate uncertainties are included in the table except for the contribution of the JES and the $Q^2$  variations. Diboson expected yield is added in the prediction.}\\label{tab:1tag_final}\n\\end{sidewaystable}\n\n\\begin{sidewaystable}\n  \\begin{center}\n\n  \\begin{tabular}{lccccc}\n        \\toprule \n        \\multicolumn{6}{c}{\\bf Double-tag Event Selection}\\\\\n        Lepton ID         & CEM                     & PHX                    &      CMUP                   & CMX                     & EMC               \\\\ \n        \\midrule                                                                                                                             \n      \n      \n       $Z+$jets                  & 1.43 $\\pm$ 0.13      & 0.27 $\\pm$ 0.02      & 2.84 $\\pm$ 0.27        & 1.41 $\\pm$ 0.13          & 4.6 $\\pm$ 0.5           \\\\\n       $t\\bar{t}$             & 48.21 $\\pm$ 6.99     & 9.91 $\\pm$ 1.44      & 27.31 $\\pm$ 3.98       & 12.37 $\\pm$ 1.8          & 44.89 $\\pm$ 6.95        \\\\\n       Single-top $s$         & 16.89 $\\pm$ 2.36     & 2.87 $\\pm$ 0.4       & 9.68 $\\pm$ 1.36        & 4.17 $\\pm$ 0.58          & 13.72 $\\pm$ 2.05        \\\\\n       Single-top $t$         & 5.07 $\\pm$ 0.81      & 1.13 $\\pm$ 0.18      & 2.85 $\\pm$ 0.46        & 1.34 $\\pm$ 0.22          & 4.16 $\\pm$ 0.71         \\\\\n       $WW$                     & 0.72 $\\pm$ 0.19      & 0.18 $\\pm$ 0.05      & 0.35 $\\pm$ 0.09        & 0.2 $\\pm$ 0.05           & 0.49 $\\pm$ 0.13         \\\\\n       $ZZ$                     & 0.26 $\\pm$ 0.04      & 0.03 $\\pm$ 0         & 0.46 $\\pm$ 0.06        & 0.29 $\\pm$ 0.04          & 0.63 $\\pm$ 0.1          \\\\  \n      $WZ$                     & 5.28 $\\pm$ 0.75      & 2.6 $\\pm$ 0.37       & 2.52 $\\pm$ 0.36        & 1.67 $\\pm$ 0.24          & 3.52 $\\pm$ 0.54         \\\\\n       $W+b\\bar{b}$           & 114.7 $\\pm$ 35.1     & 33.92 $\\pm$ 10.4     & 59.06 $\\pm$ 18.1       & 29.49 $\\pm$ 9.04         & 60.71 $\\pm$ 18.6        \\\\\n       $W+c\\bar{c}$           & 6.68 $\\pm$ 2.1       & 2.16 $\\pm$ 0.68      & 3.41 $\\pm$ 1.08        & 1.63 $\\pm$ 0.51          & 4 $\\pm$ 1.25            \\\\     \n       $W+cj$                   & 5.18 $\\pm$ 1.63      & 1.43 $\\pm$ 0.45      & 2.75 $\\pm$ 0.87        & 1.26 $\\pm$ 0.4           & 2.69 $\\pm$ 0.84         \\\\   \n       $W+LF$           & 4.53 $\\pm$ 0.94      & 1.7 $\\pm$ 0.36       & 2.35 $\\pm$ 0.48        & 1.28 $\\pm$ 0.27          & 2.98 $\\pm$ 0.66         \\\\    \n       Non-$W$                  & 5.58 $\\pm$ 2.23      & 6.79 $\\pm$ 2.72      & 4.31 $\\pm$ 1.73        & 0 $\\pm$ 0.5              & 0 $\\pm$ 0.5             \\\\ \n       \\midrule                                                          \n      {\\bf  Prediction}            & {\\bf 214.53 $\\pm$ 40.5}    & {\\bf 62.99 $\\pm$ 12.1}     & {\\bf 117.92 $\\pm$ 21.1}      & {\\bf 55.11 $\\pm$ 10.4}         & {\\bf 142.38 $\\pm$ 23.3}       \\\\\n       {\\bf Observed}                 & {\\bf 175}                 & {\\bf 62 }                   & {\\bf 92}                        & {\\bf 49 }                        & {\\bf 126}              \\\\     \n       \\midrule                                                          \n       {\\bf Dibosons }          & {\\bf 6.26 $\\pm$ 0.79}           & {\\bf 2.8 $\\pm$ 0.38}       & {\\bf 3.34 $\\pm$ 0.4}         & {\\bf 2.15 $\\pm$ 0.26}          & {\\bf 4.64 $\\pm$ 0.61 }        \\\\\n       \\bottomrule\n      \\end{tabular}        \n  \\end{center}\n  \\caption[Observed and Expected Events with Two \\texttt{SecVtx} Tags]{Summary of observed and expected events with two secondary vertex tags (\\texttt{SecVtx}), in the $W+2$~jets sample, in \n      $9.4$~fb$^{-1}$ of data in the different lepton categories. Statistical and systematics rate uncertainties are included in the table except for the contribution of the JES and the $Q^2$  variations. Diboson expected yield is added in the prediction.}\\label{tab:2tag_final}\n\\end{sidewaystable}\n\nBackground is still large although the diboson signal is sizable. An analysis of the $diboson\\to \\ell\\nu+HF$ properties with a counting experiment is not feasible. Therefore, in the next Chapter, we exploit the separation power of two variables: \n\\begin{itemize}\n\\item the di-jet invariant mass, $M_{Inv}(jet1,jet2)$, improves the separation between the non-resonant $W+HF$ production and the diboson.\n\\item KIT-NN improves the $b$-jets $vs$ $c$-jets separation.\n\\end{itemize}\nThe shape analysis and the fitting procedure also constraint the large normalization uncertainties, thereby increasing the significance of the measurement.\n\n\n\n\n\\chapter{Conclusions}\\label{chap:conclusions}\n\nThe result of this thesis is the evidence, at $3.08\\sigma$, for the associate production of massive vector bosons ($W$ and $Z$) detected at the CDF II experiment in a final state with one lepton, \\mbox{${\\not}{E_T}$}  and $HF$-tagged jets. \n\nSuch a result, obtained on the complete CDF II dataset ($9.4$~fb$^{-1}$ of data), was possible thanks to a simultaneous effort in several directions. The signal acceptance was extended both at online and offline selection level; the multi-jet background was strongly suppressed using a SVM-based multi-variate algorithm; second vertex $HF$-tagging was used in conjunction with a flavor-separator NN (KIT-NN). \n\nThe signals, both inclusive diboson production and $WW$ {\\em vs} $WZ\/ZZ$ separately, were extracted from the invariant mass distribution, $M_{Inv}(jet1,jet2)$, of single and double $HF$-tagged jet pairs. For single-tagged  events the $b$ quark {\\em vs} $c$ quark discrimination was obtained by using a bi-dimensional distribution $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN.\n\nWe measure the total diboson production cross section, fixing the $WW$ and $WZ\/ZZ$ relative contribution to the SM prediction. We obtain:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainty and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section.\n\nThen, after removing the constraint on the $WW$ and $WZ\/ZZ$ relative contribution, we leave them free to float independently. We perform a simultaneous measurement of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ where all correlations are included. We can also perform a separate measurement of each contribution one at the time by considering the other as a background. In this case we obtain:\n\\begin{equation}\n\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb,}    \n\\end{equation}\nwhere the errors include statistical and systematic uncertainties.\nThe significance of the measurements is $1.78\\sigma$ for $WW$ signal and $2.54\\sigma$ for $WZ\/ZZ$ signal.\n\nAll the results are consistent with the SM prediction and they confirm the CDF capability of identify a small signal in this challenging final state. In particular the previous version of this analysis, performed with a dataset of $7.5$~fb$^{-1}$ and reported in Appendix~\\ref{App:7.5}, was the first evidence of diboson production in $\\ell\\nu+HF$ final state at a hadron collider.\n\nBeyond the pure testing of SM predicted processes, several of the techniques developed for this thesis were also applied to the $WH$ search at CDF, with a relevant improvement of the sensitivity to this process.\n\n\\chapter{The CDF II Experiment}\\label{chap:detector}\nIn this Chapter the accelerator facility and the detector apparatus are described in their main features. \n\nThe analyzed dataset presented in this thesis corresponds to $9.4$~fb$^{-1}$ of data collected by the CDF II (Collider Detector at Fermilab for Run II) experiment along its entire operation time\\footnote{Most of the presented work was performed during the data taking period therefore some experimental features are presented as if the operations are not yet concluded.}, from February 2002 to September $30^{th}$, 2011.\nThe CDF multi-purpose detector was located at one of the two instrumented interaction points along the Tevatron accelerator ring where $p\\bar{p}$ beams collided at an energy of $\\sqrt{s}=1.96$~TeV. \n\n\\section{The Tevatron}\n\nThe Tevatron collider was a proton-antiproton storage ring and circular accelerator located at the Fermi National Accelerator Laboratory (FNAL or Fermilab), $50$~Km west from Chicago (Illinois, U.S.A.). With a center-of-mass energy of $\\sqrt{s}=1.96$~TeV, it was the world highest energy accelerator~\\cite{tevatronweb} before the beginning of the Large Hadron Collider (LHC) era\\footnote{The Tevatron collision energy record was exceed by the LHC on March $30^{th}$, 2010, when the first $pp$ collisions at $\\sqrt{s}=7$~TeV took place.} and the largest anti-matter source in the world. The decommissioning of the accelerator started at the end of 2011 with the last collision and the stop of the operations on September $30^{th}$, 2011.\n\nThe history of the Tevatron is marked by impressive technology achievements and physics results. For example, starting the operations in 1983, it was the fist super-conducting magnet accelerator ring, in 1995, the {\\em top} quark was discovery here~\\cite{CDF_top_obs, d0_top_obs} or in 2006, $B_s$ mixing was observed~\\cite{cdf_bsmix_obs}.\n\nOne of most striking achievement of the Tevatron was the whole process of the proton-antiproton production and acceleration that involved the simultaneous operation of a chain of accelerator machines. Figure~\\ref{accel_sec} shows a view of Tevatron complex and of its sections~\\cite{tevatronweb,rookie_book}, next paragraphs summarize their operation and performances.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/accel_sec}\n\\caption[CDF Detector Schema]{Schematic view of Tevatron accelerator complex at Fermilab, different colors mark different accelerator sections.}\\label{accel_sec}\n\\end{center}\n\\end{figure}\n\n\\subsection{Proton and Antiproton Production}\n\nThe first stage, proton extraction and initial acceleration, takes place in the Pre-Accelerator (\\emph{PreAc}). Hot hydrogen gas molecules ($H_{2}$) are \nsplit by an intense local electrostatic field and charged with two electrons; $H^{-}$ ions are accelerated up to $750$~KeV by a Cockcroft-Walton accelerator every $66$~ms.\n\nPreAc ion source\\footnote{The ion sources are actually two, named H- and \nI-, and working alternatively.} constantly produces beams at $15$~Hz rate and \nsend them to the \\emph{Linac}: a linear accelerator that increases the ions \nenergy $750$~KeV to $400$~MeV. It is made of two sections: a low energy \ndrift tube and a high energy coupled cavity at the end.\n\nNext acceleration stage is performed by a circular accelerator (synchrotron)\nof $75$~m radius called \\emph{Booster}. The insertion of a thin carbon foil \nstrips off electrons from the $400$~MeV ions and a sweep, from $\\sim38$ to \n$\\sim53$~MHz in radio-frequency (RF), carries resulting protons to an energy \nof $8$~GeV. The use of negative ions permits injection of more particles from \nthe Linac, otherwise the magnetic field needed to catch the protons would also\nkick away protons already inside the Booster. Bunches are extracted when about\n$8\\cdot 10^{12}$ protons are collected.\n\nResulting bunches have the correct energy to be sent to the \\emph{Main Injector}, a larger synchrotron with radius of $\\approx 0.5$~Km): here conventional magnets and 18 accelerating cavities are employed to accelerate protons up to $120$~GeV or $150$~GeV, depending upon their use. The $120$~GeV protons are used to produce the antiprotons, while $150$~GeV protons are further accelerated into the Tevatron main ring. \n\nAntiproton production takes place in the \\emph{Antiproton Source}. This machine is composed by several parts (see Figure~\\ref{accel_sec}): first \nthere is a target station where the $120$~GeV protons, extracted from Main Injector, collide with a Nickel target and $8$~GeV $\\bar{p}$ are selected from all the resulting particles. Typically, $10\\div 20$ $\\bar{p}$ are collected for each $10^{6}$ protons on target. After production, antiprotons have a large spatial and momentum spread while acceleration into Main Injector requires narrow $8$~GeV packets. Therefore they are sent to the \\emph{Debuncher}: a triangular shape synchrotron, with a mean radius of $90$~m. The bunch signal is picked up and analyzed at one side of the ring and then it is corrected on the other side in a process named \\emph{stochastic cooling} and \\emph{bunch rotation}~\\cite{Marriner:2003mn_cooling}.\nThe final step of the $\\bar{p}$ production is the accumulation: the beam is sent to a smaller synchrotron (with a mean radius of $75$~m) inside Debuncher ring called \\emph{Accumulator}. Other cooling methods are applied here.\n\nFrom the Accumulator, $8$~GeV antiprotons can be transferred either to the Main\nInjector or to the Recycler ring. The latter is a $3.3$~Km long ring of permanent magnets, located in the Main Injector enclosure, which is used to gather antiprotons as a final storage\nbefore the injection into the Tevatron, thus allowing the Accumulator to operate at its optimal efficiency.\n\nDuring the last year of running, 2011, a store could start with up to $3.5\\times 10^{12}$ antiprotons, collected in $10\\div 20$ hours of production.\n\n \n\\subsection{Collision and Performance}\n\nLast acceleration stage takes place into the \\emph{Tevatron} Main Ring: with a radius of one kilometer this is the largest of the Fermilab accelerators, and, thanks to superconducting magnets, it can store and \naccelerate beams from an energy of $150$~GeV (Main Injector result) to $980$~GeV.\nTable~\\ref{tab:FermilabAcceleratorComplex} summarizes the acceleration characteristics of the different stages of the Fermilab $p\\bar{p}$ Accelerator Complex.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{small}\n\\begin{tabular}{cccccccc}\\toprule\nAcc. & H & $H^{-}$ & C-W & L & B & M & T\\\\\n\\midrule\nE &0.04 eV & 25 KeV &750 KeV & 400 MeV&8 GeV& 150 GeV&0.98 TeV\\\\\n$\\beta$ & $9.1\\cdot10^{-8}$ & $0.01$& $0.04$ & $0.71$ & $0.99$ & $1$ &$1$\\\\\n$\\gamma$ & $1$ & $1$ & $1$ & $1.43$ & $9.53$ & $161$ & $1067$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{small}\n\\caption[Performance of Fermilab Accelerator Complex]\n{Performances of the Fermilab Accelerator Complex. The steps along the accelerator chain (Acc.), with the corresponding labelling, are: Cockwroft-Walton (C-W), Linac (L), Booster (B), Debuncher and Recycler, Main Injector (M), Tevatron (T). The energy reached at the end of the step is E, $\\beta=\\frac{v}{c}$ expresses\nthe speed of the particle as a fraction of the speed of light in vacuum and \\mbox{$\\gamma=\\frac{E}{pc}=\\frac{1}{\\sqrt{1-(\\frac{v}{c})^{2}}}$} is the relativistic factor.}\\label{tab:FermilabAcceleratorComplex}\n\\end{center}\n\\end{table}\n\n\nWhen beams production and acceleration is complete, a Tevatron \\emph{store} is started: 36 protons and 36 antiprotons bunches, containing respectively \n$\\sim 10^{13}$ and $\\sim 10^{12}$ particles, are injected into the Main Ring at location\\footnote{The Tevatron is divided into six sections (see \nFigure~\\ref{accel_sec}) and each junction zone, named form A to F, has a different function: most important areas are B0, D0 and F0, the first two are experimental areas where CDF and DO~detectors are placed, while F0 contains RF cavity for beam acceleration and switch areas to connect Main Injector and the Tevatron.} F0 to be collided. The $0.5$~mm thin proton and antiproton bunches share the same beam pipe, magnets and vacuum system and they follows two non intersecting orbits kept $5$~mm away from each other. Beam control is obtained through nearly 1000 superconducting magnets $6$~m long, cooled to $4.3$~K and capable of $4.2$~T fields. \n\\\\\n\nBeside energy, the other fundamental parameter of an accelerator is the \ninstantaneous luminosity ($\\mathscr{L}$), as the rate of a physical process\nwith cross section $sigma$ is:\n\\begin{equation}\\label{ist_ev}\n\\frac{dN}{dt}[\\textrm{events s}^{-1}]=\\mathscr{L}[\\textrm{cm}^{-2}\\textrm{s}^{-1}]\\sigma[\\textrm{cm}^{2}]\\mathrm{.}\n\\end{equation}\nHigh energy permits an insight to incredibly small scale physics but only very \nhigh instantaneous luminosity and very large integrated (in time) luminosity\nallow to see rare events. Figure~\\ref{processes} shows the production\ncross section of different physical processes\\footnote{Due to the tiny cross sections we deal with, through most of this work we will be using {\\it picobarns} (pb) where $1$~pb$= 10^{-36}\\mathrm{~cm}^{2}$.}. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/processes_new}\n\\caption[Production Cross Section of Physical Processes at CDF]{Predicted production cross section of physical processes at CDF along with their measured values. Higgs cross section varies with the Higgs mass.}\\label{processes}\n\\end{center}\n\\end{figure}\n\nThe instantaneous luminosity of an accelerator, usually measured in cm$^{-2}$s$^{-1}$, is given by:\n\\begin{equation}\\label{ist_lum}\n\\mathscr{L}=\\frac{N_{p}N_{\\bar{p}}B f}{2\\pi(\\sigma^2_{p}+\\sigma^2_{\\bar{p}})} F\\Big(\\frac{\\sigma_l}{\\beta^\\ast}\\Big),\n\\end{equation}\nwhere $N_{p}$ ($N_{\\bar{p}}$) are the number of protons (antiprotons) per bunch, $B$ is the number of bunches inside accelerator, $f$ is the bunch \ncrossing frequency, $\\sigma_{p(\\bar{p})}$ is the r.m.s. of the proton (antiproton) beam at the interaction point and $F$ is a beam shape form factor depending on the ratio between the the longitudinal r.m.s. of the bunch, $\\sigma_l$,  and the beta function, $\\beta^\\ast$, a measure of the beam extension in the $x,y$ phase space.\nSeveral of these parameters are related to the accelerator structure so they are (almost) fixed inside the Tevatron:\n36 $p\\bar{p}$ bunches crossed with frequency of $396$~ns and, at interaction points, quadrupole magnets focus beams in $\\approx 30$~$\\mu$m spots with $\\beta^\\ast\\simeq 35$~cm and $\\sigma_l \\simeq 60$~cm.\n\nHigher luminosities were achieved in Run II thanks to the increased antiproton stack rate. Figures~\\ref{initLum} and~\\ref{lumi} show the delivered luminosity and the constant progress in the performances of the machine with an instantaneous luminosity record of $4.3107\\times 10^{32}$~cm$^{-2}$s$^{-1}$ on May, 3$^{rd}$ 2011~\\cite{acc_records}. \n\nThe total integrated luminosity produced by the Tevatron is more than $12$~fb$^{-1}$ and CDF wrote on tape, on average, about $85\\%$ of it, with small inefficiencies due\nto detector calibration, stores not used to collect data for physics or dead time during the start-up of the data taking. This analysis uses all the data collected by CDF in Run II, corresponding, after data quality requirements, to a integrated luminosity of about $9.4$~fb$^{-1}$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/initLum}\n\\caption[Initial Instantaneous Luminosity Delivered by the Tevatron to CDF]{Initial instantaneous luminosity delivered by the Tevatron accelerator to the CDF II detector. Performances grown by two order of magnitude from the beginning of the operation (2002) to the final collisions (2011)~\\cite{lum}.}\\label{initLum}\n\\end{center}\n\\end{figure}\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/lum}\n\\caption[Integrated Luminosity Delivered by the Tevatron for each Run]{Integrated luminosity delivered by the Tevatron for each physics run (blue) and averaged on a week by week basis (green bars)~\\cite{lum}.}\\label{lumi}\n\\end{center}\n\\end{figure}\n\n\n\n\\section{The CDF Detector}\n\nCDF II is a multi-purpose solenoidal detector situated at the B0 interaction point along the Tevatron main accelerator ring. Thanks to accurate charged \nparticle tracking, fast projective calorimetry and fine grained muon detection, the CDF II detector can measure energy, momentum and charge of most particles resulting from $\\sqrt{s}=1.96$~TeV $p\\bar{p}$ collisions. \n\nThe first original design goes back to 1981 but CDF underwent many upgrades \nduring the past twenty years. The last and most extensive one began in \n1996 and ended in 2001 when Tevatron \\emph{Run II} started. At present the \nCDF II experiment is operated by an international collaboration\nthat embraces more than 60 institutions from 13 different countries, for a total\nof about 600 researchers.\n\n\n\\subsection{Overview and Coordinate system}\n\nCDF is composed by many parts (sub-detectors) for a total of about \n5000 tons of metal and electronics, a length of $\\sim 16$~m and a diameter of \n$\\sim 12$~m. It is approximately cylindrical in shape with axial and \nforward-backward symmetry about the B0 interaction point. Before going further\nwe describe the coordinate system used at CDF and through this thesis.\n\nB0 is taken as the origin of CDF right-handed coordinate system: $x$-axis\nis horizontal pointing North\\footnote{Outward with respect to the center of \nTevatron.}, $y$-axis is vertical pointing upward and $z$-axis is along beam \nline pointing along proton direction, it identifies {\\em forward} and \n{\\em backward} regions, respectively at $z>0$, East, and $z<0$, West. Sometimes \nit is convenient to work in cylindrical ($r$, $z$, $\\phi$) coordinates where \nthe azimuthal angle $\\phi$ is on the $xy$-plane and is measured from the \n$x$-axis. \nThe $xy$-plane is called \\emph{transverse}, quantities projected on it are \nnoted with a T subscript. Two useful variables are the transverse momentum, $p_{T}$, and energy, $E_{T}$, of a particle:\n\\begin{equation}\n\\vec{p}_{T}\\equiv p\\sin(\\theta)\\mathrm{,}\\qquad E_{T}\\equiv E \\sin(\\theta)\\mathrm{,}\n\\end{equation}\nin collider physics another widely used variable, used in place of $\\theta$, is the \\emph{pseudorapidity}: \n\\begin{equation}\n\\eta\\equiv-\\ln\\big(\\tan (\\theta\/2)\\big).\n\\end{equation}\nIf $(E,\\vec{p})$ is the 4-momentum of a particle, the  pseudorapidity is the \nhigh energy approximation ($p\\gg m$) of the \\emph{rapidity}:\n\\begin{equation}\ny=\\frac{1}{2}\\ln\\frac{E+p\\cos(\\theta)}{E-p\\cos(\\theta)}\\stackrel{p \\gg m}{\\rightarrow} \\frac{1}{2}\\ln\\frac{p+p\\cos(\\theta)}{p-p\\cos(\\theta)}= -\\ln\\big( \\tan (\\theta\/2)\\big) \\equiv \\eta.\n\\end{equation}\nA Lorentz boost along the $\\hat{z}$ direction adds a constant $\\ln(\\gamma + \\gamma\\beta)$ to $y$, therefore rapidity differences are invariant.\nThe statistical distribution of final state particles is roughly flat in $y$ because, in hadronic colliders, the interactions between the (anti)proton \nconstituents, which carry only a fraction of the nucleon energy, may have large momentum imbalances along $\\hat{z}$.\n\nFigure~\\ref{CDF_schema} shows an isometric view of the CDF detector and of \nits various sub-detectors. The part inside the $1.4$~T superconducting \nsolenoid contains the integrated tracking system: three silicon sub-detectors \n(the \\emph{Layer00}, the \\emph{Silicon Vertex detector II} and the \n\\emph{Intermediate Silicon Layers}) are the inner core of CDF II. The high \nresolution capability of silicon microstrips is necessary to have good track \nresolution near the interaction point, where particle density is higher. Afterward \nan open cell drift chamber (the \\emph{Central Outer Tracker}) covers until \n$r\\simeq 130$~cm, in the region $|\\eta|<1.0$, the extended lever arm provides \nvery good momentum measurement (\\mbox{$\\Delta p_T\/p^2_T \\simeq 10^{-3}~$GeV$\/c^{-1}$}).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{Detector\/Figs\/CDF_schema2}\n\\caption[Isometric View of the CDF II Detector]{Isometric view of the CDF II detector, the various sub-detectors are highlighted in different colors and listed.}\\label{CDF_schema}\n\\end{center}\n\\end{figure}\n\nCalorimeter systems are located outside the superconducting solenoid. They are\nbased on \\emph{shower sampling calorimeters} made of sequential layers of \nhigh-Z passive absorbers and active signal generator plastic scintillators. \nThe system is composed by towers with $\\eta - \\phi$ segmentation, each one \ndivided in electromagnetic and hadronic part, they cover the region up to \n$|\\eta|\\simeq 3.6$ ($\\theta\\simeq3^{\\circ}$) and are organized in two main\nsections: the \\emph{Central Calorimeter} covering the region \n$|\\eta|\\lesssim 1.1$ and the \\emph{Plug Calorimeter} \nextending the coverage up to $|\\eta|\\simeq 3.6$. While the central calorimeter\nis unchanged since 1985, the plug calorimeter active part was completely \nrebuilt  for Run II, replacing gas chambers with plastic scintillator tiles to\nbetter cope with the higher luminosity.\n\nThe outermost part of CDF detector, outside calorimeters, is occupied by the\nmuon detectors. They are multiple layers of drift chambers arranged in various \nsubsections which cover the region $|\\eta| \\lesssim 1.5$. Only high penetrating\ncharged particles, such as muons, can go across the entire detector.\n\nOther detectors are used for a better particle identification, calibration or \nmonitoring. However a detailed description of the entire CDF detector is far \nfrom the scope of this work. The next paragraphs will focus on tracking and \ncalorimeter systems which play a significant role in the analysis. A complete \ndescription of CDF II detector can be found in~\\cite{TDR}.\n\n\n\n\n\\subsection{Integrated Tracking System}\\label{track_par}\n\nThe trajectory of a charged particle in a uniform \nmagnetic field in vacuum is a helix. A tracking detector identifies some points along particle path so \nthat it is possible to obtain momentum measurements by reconstructing the \nhelix parameters\\footnote{See Section~\\ref{sec:track} for track reconstruction \ndetails.}. A schematic view of CDF tracking volume can be seen in \nFigure~\\ref{CDF_track}: the three main components are the superconducting \nmagnet, the silicon sub-detectors and the central drift chamber.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/CDF_track}\n\\caption[CDF II Tracking Volume and Calorimeter]{View of CDF II tracking volume and calorimeter location.}\\label{CDF_track}\n\\end{center}\n\\end{figure}\n\nThe solenoidal magnet, made  by NbTi\/Cu superconducting coils, maintains a \nbending magnetic field with a central value of $1.4116$~Tesla, oriented along \nthe positive $\\hat{z}$ direction and nearly uniform in all the tracking volume \n($r\\lesssim 150$~cm and $|z|\\lesssim 250$~cm). \nThe momentum threshold for a particle to radially escape the magnetic field is \n$p_{T}\\gtrsim 0.3$~GeV\/$c$ and the radial thickness of the coil is $0.85$ \nradiation lengths ($X_{0}$).\n\n\n\n\\subsection*{Silicon System}\n\nThe silicon system is the first tracking sub-detector encountered by particles\nexiting from the primary interaction vertex. Semiconductor detectors offer\nexcellent spatial resolution and fast response time.\nTherefore it permits the reconstruction of secondary vertices displaced from the\nprimary, produced in the decay of long lived $b$-hadrons\\footnote{Correct \nidentification of $b$-hadrons is fundamental in many analyses e.g. $b$-hadrons \nare one of the decay products of $top$ quark and also Higgs boson has a high \nbranching ratio to $b$ quarks for $m_H\\lesssim 140$~GeV$\/c^{2}$.}.\n\nCDF employs $\\sim7$~m$^2$ silicon active-surface for a total \nof 722,432 different channels read by about 5500 integrated custom chips. The\ncomplete silicon tracking detector is displayed in Figure~\\ref{silicons}.\nOf the three subsystems composing the core of CDF, the \n\\emph{Layer00 }~\\cite{l00} (L00 ) is the innermost.\nIt consists of a single layer of single-sided silicon sensors directly mounted \non the beam pipe at radii, alternating in $\\phi$, of $1.35$~cm or $1.62$~cm, \ncovering the region $|z|\\lesssim 47$~cm. During the construction of the SVX II \nmicrovertex (see below) CDF realized that the multiple scattering due to the \npresence of read-out electronics and cooling systems installed \ninside tracking volume was going to degrade the impact parameter resolution.\nL\\O\\O~ was designed to recover it thanks to its proximity to the beam. \nFurthermore, being made of state-of-the-art radiation-tolerant sensors, it will\nensure a longer operating lifetime to the entire system.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.49\\textwidth]{Detector\/Figs\/cdfiisi_rz}\n\\includegraphics[width=0.49\\textwidth]{Detector\/Figs\/cdf_silicon_endview}\n\\caption[Side and Front View of Silicon System]{Side and front view of silicon tracking system at CDF.}\\label{silicons}\n\\end{center}\n\\end{figure}\n\nThe main component of the silicon system is SVX II~\\cite{svx}, the \n\\emph{Silicon VerteX detector}\nis made of three cylindrical barrels for a total length of about $96$~cm along\n$z$, covering the luminosity region until $\\simeq 2.5~\\sigma_{l}$, and with a \npseudo-rapidity\nrange $|\\eta|\\lesssim 2$. Each barrel is divided in twelve identical wedges in\n$\\phi$, arranged in five concentric layers between radii \n$2.4$~cm and $10.7$~cm. Each layer is divided into independent longitudinal \nread-out units, called {\\em ladders}. Each ladder consists of a low-mass support\nfor a double-sided silicon microstrip detector. Three out of five layers \ncombine an $r-\\phi$ measurement on one side with $90^{\\circ}$ stereo measurement\non the other, the remaining two layers combine an $r-\\phi$ measure with a small\nangle $r-z$ stereo measurement (with tilt angle of $1.2^{\\circ}$). \nThe highly parallel fiber based data acquisition \nsystem reads out the entire sub-detector in approximately $10~\\mu$s.\n\nThe \\emph{Intermediate Silicon Layers} detector~\\cite{isl} (ISL) is the outermost of the three silicon sub-detectors, radially\nlocated between SVX II and the drift chamber covering the region $|\\eta|\\lesssim 2$. It is divided in three barrels segmented into $\\phi$ \nwedges. The central barrel ($|\\eta|\\lesssim 1$) is made of one layer of silicon sensors at radius of $22$~cm, instead the two outer barrels \n($1\\lesssim|\\eta|\\lesssim 2$) are made of two layers at radii of $20$~cm and \n$28$~cm. Its purpose is to strengthen the CDF tracking in the central region and to add precision hits in a region not fully covered by the \ndrift chamber. Track reconstruction can be extended to the whole region $|\\eta|<2$ using the silicon detector.\n\nThe complete silicon sub-detector (L00, SVX II and ISL) has an asymptotic \nresolution of $40~\\mu$m in impact parameter and of $70~\\mu$m along $z$ \ndirection. The total amount of material varies roughly as:\n\\begin{equation}\n\\frac{0.1X_{0}}{\\sin(\\theta)}\n\\end{equation}\nin the central region and doubles in the forward region because of the presence of read-out electronics, cooling system and support frames~\\cite{sil_performance}.\n\n\n\n\\subsection{Central Outer Tracker}\n\nThe \\emph{Central Outer Tracker}~\\cite{cot} (COT) is an open-cell drift \nchamber used for particles tracking at large radii. It has an \nhollow-cylindrical geometry and covers $43.3<r<132.3$~cm, $|z|\\lesssim 155$~cm.\nFigure~\\ref{CDF_track} shows that COT fully covers the central region \n($|\\eta|\\lesssim 1$) with some residual capability up to $|\\eta| \\approx 1.8$\n\nThe COT (see Figure~\\ref{cot1}) is structured into eight {\\em super-layers} each \ndivided into $\\phi$ cells; each cell contains twelve sampling wires, spaced \n$0.583$~cm, to collect the ions produced by passing charged particles. The \narrangement of the cells has a $\\chi=35^{\\circ}$ tilt with respect to the \nchamber radius to partially compensate the Lorentz angle of the electrons \ndrifting in the magnetic field and obtain the best \nresolution\\footnote{Electrons drifting in a gas within an electromagnetic \nfield $(\\vec{E},\\vec{B})$ move with an angle \n$\\chi\\simeq\\arctan\\big(\\frac{v(E,B=0)B}{kE}\\big)$, where $k$ is empirical \nparameter of gas and electric field and $v(E,B=0)$ is the velocity without the\nmagnetic field. The angle $\\chi$ is also known as Lorentz angle.}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Detector\/Figs\/cot1}\n\\caption[A $1\/6$ Section of the COT End-plate]{A $1\/6$ section of the COT end-plate with the eight super-layers structure and the location of cell slots.}\\label{cot1}\n\\end{center}\n\\end{figure}\n\nThe final structure has $8 \\times 12$ sampling planes alternated with \nplanes of potential wires (see Figure~\\ref{cot2}), $96$ hits are measured for a \nparticle crossing the entire COT ($|\\eta|<1$). Four super-layers employ \nsense-wires parallel\nto the beam axis for measurements in $r-\\phi$ plane, the other four interspacing\nsuper-layers are named {\\em stereo} super-layers because their wires are \nalternately canted at angles of $+2^{\\circ}$ and $-2^{\\circ}$ with respect to \nthe beam line and are used to measure $r-z$ coordinates. The electric \ndrift field (see Figure~\\ref{cot2}) is $1.9$~kV\/cm. A $50:50$ gas admixture of \nargon and ethane bubbled through isopropyl alcohol ($1.7$\\%) constantly flows \nin the chamber volume. The drift velocity is about $100~\\mu$m\/cm for a\nmaximum drift space of $0.88$~cm. The material of the COT is about \n$0.017X_{0}$, mostly concentrated in the inner and outer shell.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.98\\textwidth]{Detector\/Figs\/cot2}\n\\caption[Equipotential Line and Wire Layout Inside one of the COT Cells]{Left: equipotential line inside one of the COT super-layer cell. \nRight: layout of sense-wires, field-wires and shaper-wires inside one COT cell.}\\label{cot2}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Calorimeter System}\n\nLocated immediately outside the solenoid, the calorimeter system covers\na solid angle of nearly $4\\pi$ around $p\\bar{p}$ interaction point and has the\nfundamental role to measure energies of electrons, photons, particle clusters \n(\\emph{jets}) and the imbalance in transverse energy flow (signature of \n\\emph{neutrinos}). \nThe location of calorimeter sections is visible in Figure~\\ref{CDF_track}.\nBoth plug and central are sampling calorimeters divided into an \nelectromagnetic section (lead\/scintillator), optimized to collect all the \nenergy of electrons and photons, and a subsequent hadronic section \n(iron\/scintillator), thicker and optimized for hadron energy measurement. \nCalorimeters have an in-depth segmentation, finer near the collision point and\ncoarser outward. The $\\eta-\\phi$ plane is mapped in rectangular cells, each one\ncorresponding to the independent read-out of a projective electromagnetic or \nhadronic tower. Thanks to the fast response of scintillators, signals from\ncalorimeters are quickly processed and used at various trigger levels. \nFollowing paragraphs explains in more detail the composition \nof the different subsections and Table~\\ref{cal_tab} summarizes their main \ncharacteristics.\n\\begin{table}\n  \\begin{center}\n    \\begin{small}\n\n      \\begin{tabular}{ccccc}\n        \\toprule\n        & En. Resolution & $\\eta$ Coverage & Absorber & Longitudinal Depth\\\\\n        \\midrule\n        CEM & $13.5\\%\/\\sqrt{E} \\oplus 2\\% $ &  $|\\eta|<1.1$  & lead  & $19X_{0}$,  $1\\lambda$\\\\\n        CHA & $50\\%\/\\sqrt{E}\\oplus 3\\%$ &  $|\\eta|<0.9$      & iron  & $4.5\\lambda$\\\\\n        WHA & $75\\%\/\\sqrt{E}\\oplus 4\\%$ &  $0.7<|\\eta|<1.3$  & iron  & $4.5\\lambda$\\\\\n        PEM & $16\\%\/\\sqrt{E}\\oplus 1\\%$ &  $1.1<|\\eta|<3.6$  & lead  & $21X_{0}$, $1\\lambda$ \\\\\n        PHA & $74\\%\/\\sqrt{E}\\oplus 4\\%$ & $1.3<|\\eta|<3.6$   & iron  & $7\\lambda$ \\\\\n        \\bottomrule\n      \\end{tabular}      \n    \\end{small}\n    \\caption[CDF II Calorimeter System]{Main characteristics of CDF II calorimeter system.}\\label{cal_tab}\n  \\end{center}\n  \\end{table}\n\n\n\\subsection*{Central Calorimeter}\n\nThe central region of the detector is covered by the \\emph{Central Electromagnetic} (CEM) and the \\emph{Central HAdronic} (CHA) calorimeters~\\cite{cen_cal}, corresponding to the pseudo-rapidity region $|\\eta|<1.1$ and $|\\eta|<0.9$ respectively. \n\nThe CEM is a hollow cylinder located at $173<r<208$~cm, divided in four $180^{\\circ}$ arches each composed by 12 azimuthal sections ($\\Delta\\phi=15^{\\circ}$) and 10 pseudo-rapidity sections ($\\Delta\\eta\\simeq0.11$) for a total of 478 instrumented towers\\footnote{Two towers are missing to permit access to the solenoid, the\nso-called {\\em chimney}.}. \nThe CHA covers region $|\\eta|<0.9$ and it is divided into 9x12 $\\eta-\\phi$ \ntowers corresponding to CEM segmentation for a total of 384 towers. Central \nhadronic calorimeter covering is extended up to $|\\eta|\\simeq 1.3$ thanks to \nthe \\emph{Wall HAdron Calorimeter}~\\cite{wha} (WHA). It has same $\\phi$ \nsegmentation and six additional $\\eta$ towers: the first three overlap\nCHA and the last three extend $\\eta$ coverage.\n\nFigure~\\ref{cen_sector} shows a wedge of the central calorimeter system. Each\nCEM sector is a sampling device made of 31 layers of polystyrene scintillator \n($5$~mm thick) radially alternated with layers of aluminum-clad lead \n($3.18$~mm thick). Some of the 30 lead layers are replaced by acrylic \n(Plexiglas) as a function of $\\theta$ to maintain a uniform thickness in \n$X_{0}$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/cen_sector}\n\\caption{Structure of a wedge of CDF central calorimeter.}\\label{cen_sector}\n\\end{center}\n\\end{figure}\nAs particles loose energy into the absorber medium, the blue light emitted by \nactive scintillator medium is collected by thin bars of blue-to-green \nwave-length shifter acrylic material placed on the sides of each tower that \nbring the light to two photomultiplier tubes (PMT) outside CHA.\nCEM contains also the \\emph{Central Electron Strip chambers} (CES) and the \n\\emph{Central PReshower detector} (CPR). CES is a multi-wire proportional \nchamber placed at a radial depth of $\\sim6X_{0}$ and is used to determine shower \nposition and transverse shower development with an accuracy of $\\sim 0.2$~cm. \nCPR is a layer of finely segmented scintillators located immediately outside \nthe solenoid and is used to monitor photon conversion started in the tracking \nregion.\n\nThe structure of hadronic calorimeters is similar to electromagnetic ones but\nabsorber materials are 32 steel, $2.5$~cm-thick, layers in CHA and 15 steel, \n$5.1$~cm thick, layers in WHA both alternated with acrylic scintillator, \n$1.0$~cm thick.\n\nThe total thickness of electromagnetic section is nearly uniform and \ncorresponds to $19$ radiation lengths ($X_{0}$) or $1$ interaction length \n($\\lambda_{int}$). Based on test beam data, the CEM energy resolution \nfor an electron going through the center of a tower is found to be:\n\\begin{equation}\\label{cem_res}\n\\frac{\\sigma_{E}}{E}=\\frac{13.5\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 2\\%.\n\\end{equation}\nThe total thickness of hadronic section is $\\sim4.5\\lambda_{int}$ and the \nenergy resolution is:\n\\begin{equation}\\label{cha_res}\n\\frac{\\sigma_{E}}{E}=\\frac{50\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 3\\%, \\qquad \\frac{\\sigma_{E}}{E}=\\frac{75\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 4\\%.\n\\end{equation}\nrespectively for CHA and WHA.\n\n\n\\subsection*{Forward Calorimeter}\\label{forward_cal_sec}\n\nPlug calorimeters~\\cite{plug} are two identical structures, East and \nWest, covering region $1.1\\lesssim|\\eta|\\lesssim 3.6$. Figure~\\ref{plug1} \nshows the structure\nof plug calorimeters, in a way similar to the central device: there is a \n\\emph{Plug ElectroMagnetic calorimeter} section (PEM), a \n\\emph{Plug PReshower} (PPR) detector before the calorimeter, a \n\\emph{Plug Electromagnetic Shower-maximum} detector (PES) embedded (at \n$6X_{0}$)  and a subsequent \\emph{Plug HAdronic calorimeter} section (PHA).\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/plug1}\n\\caption{Elevation view of one quarter of the CDF plug calorimeter.}\\label{plug1}\n\\end{center}\n\\end{figure}\n\nElectromagnetic section is $21X_{0}$ thick and is composed by 23 annular \nplates, of $2.77$~m outer diameter and an inner hole for the beam pipe made of\n$4.5$~mm thick lead absorber. Towers have \na segmentation with varying $\\Delta\\eta$ and $\\Delta\\phi$ as \nTable~\\ref{cal_segment} shows, with an azimuthal-angle covering of \n$7.5^{\\circ}$ down to $\\eta=2.11$ and of $15^{\\circ}$ further. Active elements\nare $4$~mm thick scintillator tiles read-out by embedded wavelength shifters \nconnected to PMT. All is assembled in triangular shape \\emph{pizza-pans} that\nenclose a slice of a $\\Delta\\phi=15^{\\circ}$ sector. Two layers are different:\nthe first scintillator layer is $10$~mm thick and is used as a preshower detector,\nand another layer, at about $5$~cm from the surface, is used as the Plug Electron Shower-max (PES) detector, it is made by two strips of scintillators that provide shower maximum \nposition measurement with $\\sim1$~mm accuracy. \n\n\\begin{table}\\begin{center}\n\\begin{tabular}{ccc}\n\\toprule\n$|\\eta|$ Range & $\\Delta\\phi$ & $\\Delta\\eta$ \\\\\n\\midrule\n$0.-1.1 (1.3 H)$ & $15^{\\circ}$ & $\\sim 0.1$ \\\\\n$1.1(1.3 H)-1.8$ & $7.5^{\\circ}$ & $\\sim 0.1$ \\\\\n$1.8-2.1$ & $7.5^{\\circ}$ & $\\sim 0.16$ \\\\\n$2.1-3.64$ & $15^{\\circ}$ & $0.2 - 0.6$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CDF II Calorimeter Segmentation]{CDF II calorimeter segmentation, $H$ stands for the hadronic section.}\\label{cal_segment}\n\\end{center}\n\\end{table}\n\nHadronic section is about $7\\lambda_{int}$ thick and segmented in \n$\\Delta\\phi=30^{\\circ}$ for a total of 12 sections of 23 iron $5$~cm-thick \nlayers alternated with $6$~mm scintillator active material layers. The \ncharacteristic {\\em plug} shape is due to the growing radii of the layers far \nfrom interaction point to match WHA coverage. Energy resolution is:\n\\begin{equation}\\label{pem_res}\n\\frac{\\sigma_{E}}{E}=\\frac{16\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 1\\%, \\qquad \\frac{\\sigma_{E}}{E}=\\frac{74\\%}{\\sqrt{E(\\mathrm{GeV})}}\\oplus 4\\%.\n\\end{equation}\nrespectively for PEM and PHA. Figure~\\ref{plug2} shows the segmentation of a\n$\\Delta\\phi=15^{\\circ}$ sector and describes the distribution of trigger \ntowers.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/plug2}\n\\caption[Plug Calorimeter Segmentation]{Segmentation of the plug calorimeter and tower location inside one wedge.}\\label{plug2}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Muon Detectors}\\label{sec:mu_cham}\nAlthough nearly all particles are absorbed by the calorimeter system, muons pass through the calorimeters as minimum ionizing particles and can exit the calorimeter system\\footnote{Muons from $Z^0$ decays, for instance, deposit on average about $0.4$~GeV in the electromagnetic portion of the calorimeter and $4$~GeV in the hadronic one.}, therefore the outermost sub-detector of CDF is the muon detection system~\\cite{muon}. It is made out of single wire drift chambers and scintillator counters for fast timing, located radially just outside the calorimeter system.\n\nThere are various muon subsystems with slightly different characteristics and named according to their locations: the Central Muon Detector (CMU), the Central Muon uPgrade Detector (CMP), the Central Scintillator uPgrade (CSP), the Central Muon eXtension Detector (CMX), the Central Scintillator eXtension (CSX), the Toroid Scintillator Upgrade (TSU), the Barrel Muon Upgrade (BMU) and the Barrel Scintillator Upgrade (BSU). \nThe CMU, CMP and CSP systems cover an $\\eta$ range of $|\\eta|<0.6$, the CMX and CSX systems cover an $\\eta$ range of $0.6<|\\eta|<1.0$ and the TSU, BMU and BSU subsystems cover an $\\eta$ range of $1.0<|\\eta|<2.0$. A diagram of the muon subsystems coverage can be seen in Figure~\\ref{figure:MuonCoverage} .\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[angle=0,width=0.75\\textwidth,clip=]{.\/Detector\/Figs\/mu_coverage}\n\\caption[CDF II Muon System $\\eta$-$\\phi$ Diagram]\n{Diagram in the $\\eta$-$\\phi$ plane of the muon systems at CDF: CMU, CMP, CMX and BMU muon detectors. The BMU detector is referred in this diagram as IMU.}\\label{figure:MuonCoverage}\n\\end{center}\n\\end{figure} \n\nThe innermost muon system is CMU, it was built for CDF I and is located just outside the CHA calorimeter, at a radius of $350$~cm and arranged in $12.6$$^\\circ$ wedges in $\\phi$. Each wedge is made of three layers ({\\em stacks}) composed by four rectangular drift tubes. Each drift tube operates in proportional mode, with an Argon-Ethane mixture gas and a single 50~$\\mu$m sense wire in the middle of the cell, parallel to the $z$ axis: absolute differences of drift arrival time between two tubes provide a measurement of the azimuthal coordinate, while the charge division at each end of a wire can be used to determine the $z$ coordinate. The CMU is followed by another muon system of similar structure, the CMP, installed beyond a $60$~cm thick layer of steel. The minimal $p_T$ threshold for the CMU (CMP) is $1.4$ ($2.2$)~GeV$\/c$. \n\nOutside the CMP we find the CSP: a fast response detector used for triggering and formed by a single scintillator layer connected to a light guide and a PMT. \n\nThe CMX muon system is located at each edge between the CDF barrel and forward regions. It has a conical geometry with drift chambers similar to the CMP. Also, it has a scintillating system called the CSX, similar to the CSP. The CMX system covers $360$$^\\circ$ with 15 wedges in $\\phi$. Each wedge is formed of eight layers of drift chambers in the radial direction. \nVarious properties of the CMU\/CMP\/CMX subsystems are summarized in Table~\\ref{table:MuonSubsystems}.\n\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{cccc}\\toprule\nGeneral Parameters & CMU & CMP & CMX\\\\\n\\midrule\n$\\eta$ coverage & 0-0.6 & 0-0.6   & 0.6-1.0\\\\\n$p_T$ Threshold [GeV$\/c$] & 1.4 & 2.2 & 1.4\\\\\n\\midrule \nDrift Tubes & CMU &  CMP & CMX\\\\\n\\midrule \nThickness [cm] & 2.68 & 2.5 & 2.5 \\\\\nWidth [cm] & 6.35 & 15 & 15 \\\\\nLength [cm] & 226 & 640 & 180 \\\\\nMax. drift time [$\\mu$s] & 0.8 & 1.4 & 1.4 \\\\\n\\midrule \nScintillators & N\/A & CSP & CSX\\\\\n\\midrule \nThickness [cm] & N\/A & 2.5 & 1.5 \\\\\nWidth [cm] & N\/A & 30 & 30-40 \\\\\nLength [cm] & N\/A & 320 & 180 \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CDF II Summary of Muon Subsystem Properties]{Summary of the properties of the muon subsystems at CDF. \\label{table:MuonSubsystems}}\n\\end{center}\n\\end{table}\n\n\nMuon identification proceed on the base of short ionization tracks left in the drift chambers (called {\\em stubs}) and reconstructed thanks to the timing information provided by the individual drift chambers. Then a COT track is matched to the stubs to confirm the muon candidate providing an accurate measurement of the the muon momentum.\n\n\\subsection{Cherenkov Luminosity Counters}\\label{clc_par}\n\nThe \\emph{Cherenkov Luminosity Counters}~\\cite{clc_performances} (CLC) are two symmetrical detector modules designed to measure the instantaneous luminosity through the rate of $p\\bar{p}$ interactions in the forward region.\n\nEach counter is made of 48 conical, isobuthane gas filled, Cherenkov counters pointing to the nominal interaction region and located inside \neach plug calorimeter in a forward pseudo-rapidity region (\\mbox{$3.7<|\\eta|<4.7$}). Cones are disposed in a concentric way, with smaller counters at the center (length $110$~cm, initial diameter $2$~cm) and larger ones outward (length $180$~cm, initial diameter $6$~cm). The narrow shape and orientation is optimal to collect particles outgoing from the \ninteraction point that produce an important Cherenkov light yield. On the other hand particles from beam halo or from secondary interactions have larger crossing angle, hence they produce a much smaller signal. The excellent time resolution (less than $100$~ps) allows the analysis of the coincidence between the two modules (East and West) and it is an additional tool to remove \nbackground interactions. Figure~\\ref{clc_coincidence} shows the time distribution of the hits on the two modules. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Detector\/Figs\/clc_coincidence}\n\\caption[Time Distribution of CLC Modules Signal]{Time distribution of East and West CLC modules signals. A $p\\bar{p}$ collision deposits a coincidence signal in the two modules.}\\label{clc_coincidence}\n\\end{center}\n\\end{figure}\n\nThe CLC signal shape is used to measure the average number of inelastic interactions per bunch crossing ($\\bar{N}$), the instantaneous luminosity $\\mathscr{L}$ is inferred from the relation:\n\\begin{equation}\\label{lum1}\n\\bar{N} f_{b.c.}=\\mathscr{L} \\sigma_{in} \\epsilon ,\n\\end{equation}\nwhere the bunch crossing frequency ($f_{b.c.}$) is precisely known from the Tevatron RF, $\\epsilon$ is the CLC acceptance for inelastic scattering and $\\sigma_{in}$ is the inelastic $p\\bar{p}$ cross section. The last parameter comes from the averaged CDF and E811 luminosity independent measurements at $\\sqrt{s}=1.80$~TeV~\\cite{cx_ppbar_cdf, cx_ppbar_e811}, extrapolated to $\\sqrt{s}=1.96$~TeV:\n\\begin{equation}\\label{lum2}\n\\sigma_{in}(1.80\\mathrm{~TeV})=60.4\\pm 2.3\\mathrm{~mb}\\rightarrow\\sigma_{in}(1.96\\mathrm{~TeV})=61.7\\pm 2.4\\mathrm{~mb}.\n\\end{equation}\nThe combined systematic uncertainty on the luminosity measurement~\\cite{clc_performance} is $6$\\%: a\n4\\% due to the extrapolation applied in Equation~\\ref{lum2} and about 4\\% due to the uncertainty of the CLC acceptance. \n\n\n\n\n\n\n\\section{Trigger and Data Handling}\\label{sec:trigger}\n\nThe purpose  of the trigger system is the on-line selection of useful physics events from the background of uninteresting processes produced at much higher rate. The online selection step is rather important because only a fraction of data can be stored for offline physics analysis.\n\nAt the B0 interaction point, with a bunch crossing frequency of $2.5$~MHz, an inelastic $p\\bar{p}$ cross section of $\\sigma_{in}\\simeq 60$~mb and an instantaneous luminosity of \n\\mbox{$\\mathscr{L}\\simeq 10^{32}$~cm$^{-2}$s$^{-1}$}, there are about $1\\div 2$ inelastic collision in each bunch crossing. It is clearly impossible to store the entire detector information for each collision, as the maximum recording rate is $50\\div 100$~Hz, and it would also be useless, because interesting processes have much smaller cross section than generic inelastic interactions (diboson production cross sections are $\\mathscr{O}(10^{-9})$ w.r.t. generic jet production, see Figure~\\ref{processes}). The CDF trigger system is designed for the efficient selection of the interesting events.\n\nThe system is composed by three levels, L1, L2 and L3 (see \nFigure~\\ref{trigger_lev}); each one provides a sufficient rate reduction to allow\nthe feeding and processing by the next level with, virtually, no dead-time\\footnote{\\emph{Dead-time} occurs when events must be rejected because trigger system is occupied processing a preceding event.}.\n\nEach level filters the events using a set of programmable conditions, step by \nstep more complex as the detector read-out completes and more elaboration time becomes available.\nA, so-called, \\emph{trigger path} is the logic combination of criteria from different levels. \n\nA peculiar requirement that needs to be described is the {\\em PreScale} (PS) condition: a known fraction of the events selected by L1 or L2 are immediately discarded before the elaboration of the following trigger level. PS can be fixed or {\\em Dynamic} (DPS): the first is applied to auxiliary trigger paths, used for efficiency estimates or data quality control, the second is used for physics trigger paths that would require excessive computing time, thus their scaling factor is optimized during the data taking according to the available bandwidth at each trigger level.\n\nThe several trigger paths used this analysis are described in Section~\\ref{sec:onlineSel}, however the CDF experiment collects about 150 trigger paths. They are arranged in a \\emph{trigger table} aimed to maximise the acceptance of interesting events allowing a maximum acquisition dead time of $5\\%$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.525\\textwidth]{Detector\/Figs\/trigger_lev}\n\\includegraphics[width=0.465\\textwidth]{Detector\/Figs\/trigger}\n\\caption[CDF II Trigger System]{The CDF II trigger system. Left: block diagram of three-levels trigger\n and DAQ system. Right: L1 and L2 trigger streams.}\\label{trigger_lev}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Level-1}\n\nThe first trigger level is a custom designed hardware system composed by three\nparallel streams which feed inputs to Global Level-1 decision unit (see Figure~\\ref{trigger_lev}). One stream, \\verb'L1_CAL', collects prompt colorimetric response: it is divided into {\\em object triggers}, i.e. single hadronic or electromagnetic deposits above threshold, and {\\em global triggers}, i.e. total event transverse energy ($\\sum{E_{T}}$) or the presence of raw missing transverse energy\\footnote{Missing transverse energy is defined as $\\vec{\\mathrm{\\cancel{E}}}_{T}\\equiv - \\sum_i{\\vec{E}_T^i}$, with $i$ index of the calorimeter towers. See Section~\\ref{sec:metObj} for more details.} (MET or \\cancel{E}$_T$ ).\nA second stream, \\verb'L1_MUON', collects information from muon chambers thus identifying muon stubs. The last stream, \\verb'L1_TRAK', comes from the eXtreme Fast Tracker (XFT), a powerful\nparallel pattern recognition algorithm used to feed COT raw tracking information to L1 trigger. \nThe collected information already allows a coarse but efficient reconstruction of candidate physics objects: a track plus matched EM deposit is an electron candidate, a track plus a matched stub is a muon candidate, MET can identify a neutrino, a hadronic cluster in the calorimeter can be a jet and so on. \n\nWhile the L1 trigger takes a decision, the events are stored in forty-two pipelined buffers synchronized with Tevatron clock cycles ($132$~ns). In a latency time of $132\\cdot42\\simeq 5.5~\\mu$s, L1 drastically reduces the rate of accepted events from $2.5$~MHz to about $50$~KHz, the feed rate of L2.\n\n\n\\subsection{Level-2}\n\nAt the second trigger level there is enough time to readout the information of slower sub-detectors and perform more complex physic object identification algorithms. The main components of L2 are:\nthe readout of the shower-max trigger boards, the 3D reconstruction of the XFT tracks~\\cite{xft_3d}, the \\verb'L2CAL' hardware~\\cite{l2cal_up} clustering of calorimeter towers and the Silicon Vertex Trigger~\\cite{svt_up} (SVT) for the reconstruction of displaced secondary vertices. \n\nThe shower-maximum detector information ensures a better electron or photon recognition with azimuthal information and better spatial resolution. Electron identification is also improved by the XFT 3D track matching. The requirement of a track matched to a muon chamber stub improves also the muon fake rejection.\n\n\\verb'L2CAL' is a custom hardware setup able to perform raw energy clustering. Adjacent towers above a predefined threshold (usually few GeV), are merged iteratively to build refined trigger objects like jets or EM clusters. The use of physics objects allows to define advanced selection criteria on the base of  detector $\\eta-\\phi$ position, multiplicity or $E_T$ threshold for the full objects.\n\nThe last fundamental piece of the L2 is SVT: the SVX II detector $r-\\phi$ side is readout and the silicon hits, together with XFT information, are compared on on the fly with a large batch of Associative Memories (AM) where all the possible track configurations (of a certain resolution) are simulate and stored. The comparison with pre-processed simulation is the key for an extremely fast reconstruction of the track parameters and it allows the identification of displaced secondary vertices, a possible signature of beauty hadrons present in the event.\n\nStarting from the L1 input rate of $\\sim 50$~KHz, L2 must bring the accept rate to $300$~Hz exploiting four memory buffers and a short latency time of \\mbox{$5.5\\times 4 = 22~\\mu$s}. The high luminosity delivered by the Tevatron after 2005 produced events with lager detector occupancy that required more time for L2 elaboration. To cope with this, an extensive upgrade of the L2 hardware  took place between 2005 and 2006. My first experience with CDF happened in this context and, in particular, I was part of the SVT~\\cite{svt_up} upgrade team.\n\n\n\\subsection{Level-3} \n\nThe last stage of the trigger system, L3, is composed by a farm \nof several hundreds processors exploiting LINUX OS and C++ based software for the full event reconstruction. The complete detector information is assembled by the \\emph{EVent Builder}~\\cite{evb}\n(EVB) and processed by a simplified version of the offline reconstruction code.\n\nIf the L3 trigger requirements are satisfied, the {\\em event record} corresponding to the given bunch crossing is transferred to the Consumer Server\/Data Logger (CSL) that streams the data to disk, while a fraction of the output is also used for real time monitoring. The L3 accept rate suitable for disk storage is $100$~Hz.\n\n\n\n\\subsection{Data Structure}\\label{sec:data-struc}\n\nWhen an event record is saved on disk, it is labelled with a progressive number, grouped and classified.\n\nAll the events belonging to a continuous data taking period  are expected to have  very similar detector conditions (i.e. active sub-detecors, calibration parameters, trigger table, etc) so they define a {\\em run}. A large set of runs is grouped into a {\\em data period}, that usually corresponds to an integrated luminosity of a few hundreds of pb$^{-\u22121}$. The thirty-eight data periods that compose the complete CDF dataset are summarized in Table~\\ref{tab:datasets} together with the corresponding integrated luminosities and run ranges.\n\nEvents are also classified into {\\em data-streams}, a four-character label, describing similar trigger properties or common physical interest. This analysis uses four of the several data-streams available at CDF: \n\\begin{description}\n  \\item[\\texttt{bhel}:] high $E_T$ central electron stream;\n  \\item[\\texttt{bpel}:] high $E_T$ forward electron stream;\n  \\item[\\texttt{bhmu}:] high $p_T$ muon stream;\n  \\item[\\texttt{emet}:] high \\mbox{${\\not}{E_T}$}  stream.\n\\end{description}\n\n\\renewcommand{\\arraystretch}{0.95}\n\\begin{table}\n  \\begin{center}\n    \\begin{small}\n    \\begin{tabular}{cccc}\\toprule\n      Data Period & Run range & Period $\\int\\mathscr{L}dt$ (pb$^{-1}$) & Total $\\int\\mathscr{L}dt$ (pb$^{-1}$)\\\\\n\n\\midrule\np0\t&138425-186598\t&550\t&550   \\\\\np1\t&190697-195408\t&130\t&680   \\\\\np2\t&195409-198379\t&130\t&810   \\\\\np3\t&198380-201349\t&100\t&910   \\\\\np4\t&201350-203799\t& 95\t&1005  \\\\\np5\t&203819-206989\t&135 \t&1140 \\\\\np6\t&206990-210011\t& 110 \t&1250 \\\\\np7\t&210012-212133\t& 50 \t&1300 \\\\\np8\t&217990-222426\t& 210 \t&1510 \\\\ \np9\t&222529-228596\t&180 \t&1690 \\\\\np10\t&228664-233111\t&280 \t&1970  \\\\\np11\t&233133-237795\t&264 \t&2234  \\\\\np12\t&237845-241664\t&185 \t&2419  \\\\\np13\t&241665-246231\t&317 \t&2736  \\\\\np14\t&252836-254683\t&44.5 \t&2780  \\\\\np15\t&254800-256824\t&159 \t&2939  \\\\\np16\t&256840-258787\t&142 \t&3081  \\\\\np17\t&258880-261005\t&188 \t&3269  \\\\\np18\t&261119-264071\t&407 \t&3676  \\\\\np19\t&264101-266513\t&287 \t&3963  \\\\\np20\t&266528-267718\t&256 \t&4219  \\\\\np21\t&268155-271047\t&520 \t&4739  \\\\\np22\t&271072-272214\t&292 \t&5031  \\\\\np23\t&272470-274055\t&232 \t&5263  \\\\\np24\t&274123-275848\t&283 \t&5546  \\\\\np25\t&275873-277511\t&236 \t&5782  \\\\\np26\t&282976-284843\t&189 \t&5971  \\\\\np27\t&284858-287261\t&422 \t&6393  \\\\\np28\t&287294-289197\t&333  \t&6726  \\\\\np29\t&289273-291025\t&360  \t&7086  \\\\\np30\t&291294-293800\t&460 \t&7546  \\\\\np31\t&293826-294777\t&172 \t&7718  \\\\\np32\t&294778-299367\t&435 \t&8153  \\\\\np33\t&299368-301303\t&357 \t&8510  \\\\\np34\t&301952-303854\t&359 \t&8869  \\\\\np35\t&304266-306762\t&364 \t&9233  \\\\\np36\t&306791-308554\t&462 \t&9695 \\\\\np37\t&308570-310441\t&174 \t&9869 \\\\\np38\t&310472-312510\t&252 \t&10121 \\\\\n\\bottomrule\n    \\end{tabular}\n    \\end{small}\n    \\caption[Data Period and Luminosity]{Summary of data taking periods for the complete CDF dataset. Corresponding run range and collected integrated luminosity are reported. \\label{tab:datasets}}\n  \\end{center}\n\\end{table}  \n\\renewcommand{\\arraystretch}{1.}\n\nAfter that the event information is saved on disk and properly classified, it is possible to start the offline processing, also named {\\em production}. At this stage, the low-level detector data is extracted, corrected with the calibration constants and appropriate algorithms are used to reconstruct high-level physics objects (tracks, electrons, muons and jets described in Chapter~\\ref{chap:objects}).\n\nThe software is an object oriented framework where all the algorithms are defined by a self consistent \\texttt{C++} module~\\cite{data_proc}; this allows an independent testing and development of the separate algorithms and improves flexibility. For example the same {\\em track module} can be used both to reconstruct a single track information or as an element of a more complex algorithm, like the electron identification module. The final format of the physics data is a large array ({\\em n-tuple}) that can be analyzed with commercial software\\footnote{The open source \\texttt{ROOT}~\\cite{rootr} analysis software is widely used in the high energy physics community.}.  \n\nAlthough each of the \\texttt{AC++} module can be improved or modified, the analyzed data should be as much stable and uniform as possible therefore there is always a {\\em recommended} analysis prescription for the production code version to use in each data period: Table~\\ref{tab:prod_prescription} reports the one used in this thesis and approved by the Higgs Discovery  Group (HDG).\n\n\\begin{table}\n  \\begin{center}\n    \\begin{tabular}{cccc}\\toprule\n      Data Period & p0 & p1-p17 & p18-p38 \\\\\\midrule\n      Production Version & 5.3.1 & 6.1.1 & 6.1.6p+ \\\\      \n      \\bottomrule \n    \\end{tabular}\n    \\caption[Data Code Production Version]{Production code versions used the different data periods.}\\label{tab:prod_prescription}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\section{Monte Carlo Simulation}\\label{sec:mcGen}\n\nAn accurate Monte Carlo (MC) simulation of the physics processes of interest and of the detector response are a fundamental tool for most of the high energy physics experiments. A wide variety of MC samples is available to the CDF users and, as explained in Chapters~\\ref{chap:sel} and~\\ref{chap:bkg}, this analysis relies on MC both for signal evaluation and for part of the background estimate. \n\nThe simulation of an high energy hadron collision event proceeds through four independent phases:\n\\begin{description}\n\\item[Parton Density Function Application:] quarks and gluons are the initial states of any interaction. Unluckily they are {\\em confined} by the strong interaction within the $p$ and the $\\bar{p}$ making impossible a perfect knowledge of the initial parameters. The Particle Density Functions (PDFs) overcome this problem by giving a parametrization of the interaction probability as a function of the momentum transfer ($Q^2$) during the $p\\bar{p}$ collision. When a MC event generator starts the evaluation of a process, the first step is the extraction of two partons of given $Q^2$ from the PDFs. Several parametrization of the PDFs exist but all the MCs used in this analysis employ the CTEQ5L~\\cite{cteq5l} PDFs. Systematic variations are obtained by using different  PDFs sets and varying the prediction within the theoretical uncertainties.\n\n\\item[Event Generation:] once that two initial partons are extracted from the PDFs functions, a hard interaction between them is calculated to obtain the simulation of the desired final state.\nWithin the perturbative approximation~\\cite{qftbook_weinberg}, the Matrix Element (ME) equation is derived with numerical integration of Leading-Order (LO) or Next-Leading Order (NLO) equations. The softwares performing such calculations are named {\\em generators}. Here we exploit a variety of them depending of the different final states: \\verb'ALPGEN'~\\cite{alpgen}, v2.1, for $W+$jets and $Z+$Jets prediction, \\verb'PYHTIA'~\\cite{pythia} v6.216 for $WW$, $WZ$, $ZZ$ and $t\\bar{t}$ prediction and, finally, \\verb'POWHEG'~\\cite{powheg1,powheg2, powheg3} v6.510, for single-top $s$ and $t$ channel NLO prediction.\n\n\\item[Parton Shower:] away from the hard interaction, higher order QCD processes are needed. The perturbative approach breaks and simulation is based on analytical parametrization and QCD models. Parton Shower (PS) programs like \\verb'PYHTIA' and \\verb'HERWIG'~\\cite{herwig} are used to simulate quark hadronization, soft gluon emission or underlying-event processes (i.e. secondary soft interactions and spectator quark interactions).\nPartons are evolved until they form real final state particles that can be undergo physical interaction in the detector. \n\n\\item[Detector Simulation:] the MC receives, as input, the positions, the four-momenta, and the identities of all particles and it reproduces the response of the different sub-detectors, including resolution effects, passage through passive material (such as cables or support structures) and secondary decays. CDF uses the \\verb'GEANT3'~\\cite{geant}, V.3.15, program to model the tracking volume of the detector. A mathematical model is used with full  simulation of charged particles passage, showering and secondary or tertiary particle production. The calorimeter section is not completely simulated because it would be too much time and CPU consuming. A much faster parametric response program, called \\verb'GFLASH'~\\cite{geant4}, tuned on test beam data, is used.\n\\end{description}\nWhen all the simulation is completed, the MC events are saved into a data structure identical to the one used for collision data, thus allowing reconstruction algorithms to work in the same way on data and MC events.\n\n\\chapter{Introduction}\n\nThe topic of this thesis is the measurement of $W$ and $Z$ bosons associate production in the lepton plus neutrino plus Heavy Flavor ($HF$) quarks final state:\n\\begin{equation}\n  p\\bar{p}\\to WW\/WZ \\to \\ell\\nu + HF,\n\\end{equation}\nidentified by the CDF II experiment at the Tevatron collider at $\\sqrt{s}=1.96$~TeV.\n\nThe associate production of the massive vector bosons $W$ and $Z$, as well as the different final states, are predicted by the Standard Model of the elementary particles (SM). The SM, briefly described in Chapter~\\ref{chap:physics}, is an extremely successful theory in which a minimal set of equations explains most of the known interactions. However the mechanism responsible for the mass of the particles still needs to be fully proved by the discovery of the (predicted) Higgs boson.\n\nThe CDF II experiment, described in Chapter~\\ref{chap:detector}, set tight constraints on the existence of the Higgs boson. Most of the sensitivity for the low mass Higgs boson production comes from:\n\\begin{equation}\n  p\\bar{p}\\to WH\\to \\ell\\nu+b\\bar{b}.\n\\end{equation}\nThis happens to be extremely similar to the process studied in this thesis\\footnote{I personally contributed to the $p\\bar{p}\\to WH\\to \\ell\\nu+b\\bar{b}$ CDF result as a developer of the common analysis framework described in Appendix~\\ref{chap:AppWHAM}.}, thus it allows to test, on a well known physics process, the correctness of the analysis procedure used in the Higgs search. \n\nHowever, the observation of diboson production in the  $\\ell\\nu + HF$ final state is not a simple task and, before this work (in particular its preliminary version described in Appendix~\\ref{App:7.5} and presented in 2011), no evidence was observed at a hadron collider experiment.\n\nThe actual analysis procedure is divided in four steps: object identification, event selection, background estimate and statistical analysis. \n\nFirst, a set of advanced identification algorithms, described in Chapter~\\ref{chap:objects}, is exploited for the recognition of the final state objects: one charged lepton ($\\ell$), a neutrino ($\\nu$), and two high energy jets, of which at least one tagged by the identification of the secondary decay vertex produced by a $HF$ hadron. Key element of the selection, the $HF$-tagging allows an efficient identification of \\mbox{$Z\\to c\\bar{c}\/b\\bar{b}$} signal candidates together with $W\\to c\\bar{s}$ candidates.\n\n\nFurther specific event selection criteria, described in Chapter~\\ref{chap:sel}, are imposed to maximise the accepted signal events while keeping the background under control. An original technique, based on the support vector machine algorithm and described in Appendix~\\ref{chap:AppSvm}, was developed to suppress the multi-jet events, a background, difficult to model, due to events in which no real $W\\to \\ell\\nu$ decay is present.\n\n\nSuccessively, as described in Chapter~\\ref{chap:bkg}, the total background is estimated. A variety of methods are used for the different background sources. We exploit Monte Carlo information for several backgrounds (e.g. for top quark production), a completely data-driven approach for the multi-jet contamination and a combination of data and Monte Carlo information for the $W+$ jets background estimate.\n\nAfter the full selection a large irreducible background fraction (i.e. with the same final state signature) remains. In particular, the total $W+HF$ non-resonant production is estimated to be more than a factor twenty larger than the expected signal. The shape analysis of the di-jet invariant mass distribution, $M_{Inv}(jet1, jet2)$, allows the extraction of the combined diboson signal, but the separation between $WW$ and $WZ$ contributions is still not feasible due to the close mass of the $W$ and the $Z$ bosons. To overcome this last issue, we exploited the discriminative power of a flavor separation neural network (KIT-NN described in Section~\\ref{sec:kitnn}) to classify the single-tagged events according to their $c$ quark or $b$ quark origin. This was not necessary for the double-tagged selection as it is dominated only by events with two $b$ quarks in the final state.\n\nChapter~\\ref{chap:StatRes} describes the statistical analysis of the bi-dimensional distribution $M_{Inv}(jet1, jet2)$ $vs$ KIT-NN, of the single-tagged events, together with the simple $M_{Inv}(jet1, jet2)$ distribution, of the double-tagged events. This allowed both the measurement of the cross sections for the total diboson associate production processes ($WW+ WZ\/ZZ$) and to separate the $WW$ and $WZ\/ZZ$ contributions.\n\nA summary of the results and the conclusions are reported in Chapter~\\ref{chap:conclusions}.\n\n\\chapter{Physics Objects Identification}\\label{chap:objects}\n\nThe final state topology of this analysis presents four different high-$p_T$ physics objects reconstructed combining the data of several sub-detectors. The identification and reconstruction methodologies are introduced here and discussed more in depth in the rest of the Chapter.\n\nEvents are selected online by the three-level trigger system described in Section~\\ref{sec:trigger} with two trigger strategies (described more in detail in next Chapter): {\\em single high-$p_T$ lepton} triggers and {\\em multiple-objects} triggers. After this, the digitalized electrical pulses recorded by the CDF sub-detectors are analyzed to reconstruct the physics objects of interest: the primary vertex of the interaction, one charged lepton, missing transverse energy (signaling the escape of an undetected neutrino) and two high-$E_T$ jets containing a reconstructed secondary vertex that {\\em tags} the presence of a Heavy Flavor ($HF$) hadron decay.\n\nGiven the small expected signal yield, the efficient identification of charged lepton candidates is a key feature of the analysis. We identify three main lepton categories: electrons, muons and isolated tracks. Electrons are defined as electromagnetic energy clusters matched to a charged track, muons are defined by ionization deposits in the muon chambers ({\\em stubs}) matched to a charged track and, finally, isolated tracks are just high-quality charged tracks isolated from other detector activity. The variety of identification criteria, with a total of {\\em eleven} different selection algorithms, allows to identify a large fraction of the $W\\to\\ell\\nu$ decays: factorizing the trigger efficiency and the jet and neutrino selection, we estimate an approximate lepton acceptance of $35$\\% for electrons, $45$\\% for muons and $4$\\% for taus\\footnote{$\\tau$ identification is not enforced but leptonic and one prong $\\tau$ decays contributes to the selected lepton categories.}.\n\n$W$ selection is completed by the neutrino identification (Section~\\ref{sec:metObj}). A neutrinos is the only particle that leaves the experimental apparatus completely undetected, however its presence is revealed by a large imbalance in the total transverse energy of the event (MET or \\mbox{${\\not}{E_T}$} ), since the total transverse energy of the $p\\bar{p}$ interaction is expected to be zero.\n\nAnother key feature of the analysis is the identification of $HF$ jets (Section~\\ref{sec:jetObj}). Jets are the experimental signature of high momentum quark and gluon production that hadronizes in a narrow shower of particles. They appear as a energy deposit (clustered) in both electromagnetic and hadronic calorimeters.\nHeavy flavor quarks hadronize to meta-stable particles that can travel a distance away from the primary interaction vertex before they decay in other particles. The identification of a secondary decay vertex, displaced from the primary, is used to {\\em tag} the jet as coming from a $HF$, $b$ or $c$ quark, hadronization. The selection efficiency for an event containing a $b\\bar{b}$ quark pair is approximately $50$\\%, going down to about $12$\\% for $c\\bar{c}$ pairs; a residual contamination of $1\\div 2$\\% Light-Flavor ($LF$) quarks selection is also present. A Neural-Network flavor separator (named KIT-NN and described in Section~\\ref{sec:kitnn}) is also applied on each tagged jet to separate $b$ quarks from $c$ and $LF$ quark components, thus giving further discrimination between $W\\to cs$ and \\mbox{$Z\\to b\\bar{b}$} signals.\n\n\n\\section{Charged Tracks Reconstruction}\\label{sec:track}\n\nThe ability to detect and reconstruct charged particle trajectories is essential for particle identification and momentum reconstruction. Precise, high efficiency tracking is the first step in the lepton identification, moreover track reconstruction allows the measurement of the track impact parameter, thus the identification of secondary vertices.\n\nA charged particle moving in a uniform magnetic field ($\\vec{B}$ with $|\\vec{B}|=1.4$~T for CDF) produces an helicoidal trajectory that can be uniquely described by five parameters. At CDF we use (see Figure~\\ref{coordinate}):\n\\begin{itemize}\n\\item $C$: the half-curvature of the trajectory, $C\\equiv 1\/2qr$ with $r$ equal to the helix radius and $q$ the measured charge of the particle. It has\n  the same sign of the particle charge and it is related to the transverse momentum of the track:\n  \\begin{equation}\n    p_T=\\frac{cB}{2 |C|}\n  \\end{equation}\n  %\n\\item $d_0$: the impact parameter, i.e. the distance of closest approach in the transverse plane between the helix and the origin. It is defined as:\n  \\begin{equation}\n    d_0=q\\Big(\\sqrt{x_0^2+y_0^2}-r\\Big)\\mathrm{,}\n  \\end{equation}\n  where $x_0$ and $y_0$ are the coordinates of the center obtained by the projection of the helix on the transverse plane and $r=1\/2C$. The quality of $d_0$ measurement is often parametrized by the {\\em impact parameter significance} defined as $|d_0\/\\sigma_{d_0}|$.\n  %\n\\item $\\lambda$: the helix pitch, i.e. the cotangent of the polar angle between the track and the $z$-axis ($\\cot \\theta_0$). The longitudinal component of the momentum is given by:\n \\begin{equation}\n   p_z=p_T\\cot \\theta_0\\mathrm{.}\n \\end{equation}\n  %\n\\item $z_0$: the $z$ position of the track vertex.\n  %\n\\item $\\phi_0$: the azimuthal angle of the track at its vertex. \n\\end{itemize}\nThe helix is completely described by these five parameters. Indeed every point along the trajectory satisfies the following equations~\\cite{track_eq}:\n\\begin{eqnarray}\nx&=& r\\sin \\phi -(r-d_0)\\sin \\phi_0\\mathrm{,}\\\\\ny&=& -r\\cos \\phi +(r-d_0)\\cos \\phi_0\\mathrm{,}\\\\\nz&=& z_0 +s\\lambda\\mathrm{,}\n\\end{eqnarray}\nwhere $s$ is the length projected along the track, and $\\phi= 2Cs+\\phi_0$.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{PhysObj\/Figs\/coordinate}\n\\caption[Track Reconstruction Parameters]{CDF track parameters and coordinate system.}\\label{coordinate}\n\\end{center}\n\\end{figure}\n\nCDF exploits several tracking algorithms~\\cite{cdf_trk}, optimized for different detector regions, to derive the previously defined parameters.\nThe main ones, described in the following paragraphs, are: the Outside-In algorithm (OI), the Silicon-Stand-Alone (SiSA) algorithm and the Inside-Out (IO) algorithm.\n\n\\paragraph{Outside-In Algorithm}\n\nThe Outside-In is the most reliable of CDF tracking algorithms as it is based on COT coverage, efficient up to $|\\eta|\\simeq 1$.\n\nTrack pattern recognition starts in the COT outer layers (lower hit density) and proceeds through four steps: first\neach super-layer is searched for groups of three aligned hits that are fit to a straight line with the least squares method. Then the \ntracks are reconstructed from the information of the {\\em axial} super-layers that are linked by two different algorithms ({\\em segment linking} and \n{\\em histogram linking} algorithms~\\cite{hist_alg}). During the third step, the information of the stereo layers is added and the algorithm searches for the vertex of the track. As final step, a global refit of the track is performed taking into account corrections for the non-uniformity of the magnetic field and for the modeling of the electrons drift.\n\nAt second stage, the track found in the COT is propagated into the silicon system. A road around a track is defined using the uncertainties on the COT track parameters and silicon hits are added if they lie inside this predefined road. When a hit is added, the track parameters are recalculated and the search is performed again. The impact parameter resolution of COT + SVX tracks is found to be $\\sigma_{d_0}\\simeq 20~\\mu$m.\n\n\\paragraph{Silicon-Stand-Alone Algorithm}\n\nThe hits in silicon sub-detectors not used by OI tracking are available to the Silicon-Stand-Alone algorithm~\\cite{sisa}, it covers the region $|\\eta|<2$ with a small residual capability up to $|\\eta|\\simeq 2.8$. \n\nThe SiSA algorithm starts from a collection of at least four hits in the SVX II detector in the $r-\\phi$ plane (SVX has five axial layers, three $90^{\\circ}$ layers and two small angle layers) and fits the $C$, $d_0$ and $\\lambda$ parameters to obtain a projection of the helix on the transverse plane. Then the algorithm creates a 3-D seed track adding small angle hits and the primary vertex information. At this point the $90^{\\circ}$ stereo hits are added and a global refit is performed. \n\nSiSA tracks reconstructed using only SVX II have a poor resolution for high $p_T$ tracks so hits are searched in LOO and ISL with the SVX II track as seed. The track is refit if other layers can be added. However, the performances on momentum and impact parameter resolution are limited and indeed SiSA tracks are not used for secondary vertexing.\n\n\\paragraph{Inside-Out Algorithm}\n\nThe third tracking algorithm, the Inside-Out~\\cite{io}, tries to recover efficiency and $p_T$ resolution in the region $1.2<|\\eta|<1.8$ where the COT coverage is limited. SiSA tracks are used as seeds which are extrapolated to the COT inner cylinder. Matching hits in the COT are added, the track is refitted and all duplicates are removed.\n\n\n\n\\section{Primary Vertex Identification}\\label{sec:primVtx}\n\nPrecise identification of the primary interaction vertex (PV) is the very first step in the event reconstruction process. Due to the relatively long $\\sigma_Z$ of the beam ($\\sigma_z\\simeq 28$~cm), important correction to $E_T$ and \\cancel{E}$_T$ may be needed. Furthermore, PV position allows the individuation of displaced secondary vertices in an event, the signature of long living $HF$ hadrons.\n\nThe algorithm used to reconstruct primary vertices is \\verb'PrimVtx'~\\cite{primVtx}: a seed vertex is calculated as the average $z$ position measured during \ncollisions and is provided as input, then all tracks with $|z_{0}-z_{PV}|< 1$~cm, $|d_0|<1$~cm and $|d_0\/\\sigma_{d_0}|<3$ are collected and ordered in decreasing\n$p_T$. They are fitted to a new $3D$ vertex and the tracks with $\\chi^2 > 10$ are removed. The procedure is iterated until all accepted tracks have $\\chi^2< 10$.\nA quality index (see Table~\\ref{tab:pVtx}) is assigned to the primary vertex depending on track multiplicity and type: a quality $\\ge 12$ is required for primary interaction vertex reconstruction. \n\nThe PV position is defined by $\\left(x_{PV},y_{PV},z_{PV}\\right)$. Typical $x_{PV}$ and $y_{PV}$ are of the order of tens of microns while a cut of $|z_{PV}|\\leq 60$~cm (luminosity region fiducial selection) is applied to constrain the collisions in the geometrical region where the detector provides optimal coverage.\n\n\n  \\begin{table}[h]\n\\begin{center}\n    \\begin{tabular}{cc}\\toprule\n       Criterion & Quality Value\\\\\\midrule\n       Number Si -tracks$\\ge$3 & 1\\\\\n       Number Si -tracks$\\ge$6 & 3\\\\\n       Number COT-tracks$\\ge$1 & 4\\\\\n       Number COT-tracks$\\ge$2 & 12\\\\\n       Number COT-tracks$\\ge$4 & 28\\\\\n       Number COT-tracks$\\ge$6 & 60\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[Primary Vertex Quality Criteria]{Primary Vertex quality criteria: a quality $\\ge 12$ is required for primary interaction vertex reconstruction.}\\label{tab:pVtx}\n\\end{center}\n  \\end{table}\n\n\n\n\n\\section{Lepton Identification Algorithms}\\label{sec:lep_id}\n\nIn this section the specific lepton identification algorithms\\footnote{A lepton is, by the experimental point of view, an electron or a $\\mu$ with no distinction between a particle and its anti-particle. Also $\\tau$ leptonic decays can enter in the lepton sample but the algorithms are not optimized for them.} are discussed. We distinguish two {\\em tight electron} identification algorithms (CEM, PHX), two {\\em tight muon} identification algorithms (CMUP, CMX), six {\\em loose muon} identification algorithms (BMU, CMU, CMP, SCMIO, CMIO, CMX-NotTrig) and one isolated track identification algorithm (ISOTRK). \nFigure~\\ref{fig:split_alg} shows, for a $WZ$ MC, the detector $\\eta-\\phi$ coverage separately for each lepton identification category. Figure~\\ref{fig:all_alg} shows them for all the categories together. The eleven lepton identification algorithms are described in the following sections.\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet1}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet2}\\\\\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet3}\n\\includegraphics[width=0.495\\textwidth]{PhysObj\/Figs\/lepSet4}\n\\caption[Coverage, in the $\\eta - \\phi$ Plane, for the Lepton Identification Algorithms]\n        {Coverage of the detector $\\eta - \\phi$ plane for the different lepton identification algorithms evaluated on a $WZ$ Monte Carlo: CEM and PHX tight electrons (top left); CMUP and CMX tight muons (top right); BMU, CMU, CMP, SCMIO, CMIO, CMX-NotTrig loose muons (bottom left); ISOTRK isolated tracks (bottom right).}\\label{fig:split_alg}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/lepSet0}\n\\caption[Combined coverage, in the $\\eta - \\phi$ Plane, for the Lepton Identification Algorithms Combined]\n{Coverage of the detector $\\eta - \\phi$ plane for all the combined lepton identification algorithms evaluated on a $WZ$ Monte Carlo.}\\label{fig:all_alg}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{Electron Identification}\\label{sec:eleSel}\n\nA candidate electron is ideally defined as an energy deposit ({\\em cluster}) in the EM section of the calorimeters and a charged track matched to it. The track requirements removes most of the ambiguity due to photon EM showers. \n\nThe two electron identification algorithms used in this analysis are divided according to the CDF calorimeter segmentation in CEM (Central ElectroMagnetic object) and PHX (Plug electromagnetic object with pHoeniX tracking), for $|\\eta|<1.1$ and $1.2<|\\eta|<3.6$ respectively.\n\nThe CDF EM clustering algorithm~\\cite{ele_cal} works in a simple but efficient way. The physical space corresponding to the calorimeter towers is mapped in the $\\eta-\\phi$ plane and the algorithm creates two lists of the calorimeter towers ordered by decreasing energy measurement: the {\\em usable list} (working towers with energy $>100$~MeV) and the {\\em seed list} (towers with energy $>2$~GeV). Then, it takes the first seed tower and creates an $\\eta-\\phi$ cluster by adding the neighboring towers to form a $2\\times 2$ or $3\\times 3$ $\\eta-\\phi$ area. An EM cluster is found if:\n\\begin{equation}\n  E^{Had}\/E^{EM}<0.125\\textrm{,}\n\\end{equation}\nwhere $E^{Had}$ is the energy deposited in the hadronic calorimeter section and $E^{EM}$ is the corresponding quantity for the EM section. As final step, the $\\eta-\\phi$ centroid of the cluster is calculated and the used towers are removed from the list. The algorithm selects the next seed tower and iterates the process until there are no more seed towers available. \n\nThe $3\\times 3$ clustering is used for the CEM algorithm while the $2\\times 2$ clusters are used in the Plug region. A cluster is not allowed to cross the boundary between different sub-detectors. \n\nSeveral corrections are applied to reconstruct the final energy: lateral leakage, location inside the physical tower, on-line calibration and response curve drawn by the test beam data. Also the energy measured in the shower-max (CES, PES) and preshower (CPR, PPR) detectors is added to the final reconstructed energy. The shower-max profile is also compared to the calibration profiles of electrons or photons and, last but not least, it is used to measure the position of the EM shower centroid. \n\nBeyond the EM energy measurement, the calorimeter information is further exploited for a better particle identification. The following variables are used:\n\\begin{itemize}\n\\item $E^{Had}\/E^{EM}$ ratio: studies performed with candidate $Z^0\\to e^+e^-$ events~\\cite{w_z_prl} show that electrons detected in the central or in the plug region have a little deposit in the hadronic part of the calorimeter (Figure~\\ref{ele_deposit}). \n\\item {\\em Lateral shower sharing} variable, $L_{shr}$, compares the sharing of energy deposition between the towers in the CEM to the expected in true electromagnetic showers taken with test beam data:\n\\begin{equation}\n  L_{shr}=0.14\\sum_i \\frac{E_i^{adj}- E_i^{expect}}{\\sqrt{(0.14\\sqrt{E_i^{adj}})^2+(\\Delta E_i^{expect})^2}}\\mathrm{,}\n\\end{equation}\nwhere the sum is over the towers adjacent (\\emph{adj}) to the seed tower of the cluster, $0.14\\sqrt{E_i^{adj}}$ is the error on the energy measure and $\\Delta E_i^{expect}$ is the error on the energy estimate.\n\\item The $\\chi^2$ of the fit between the energy deposit and the one obtained from test beam data ($\\chi^2_{strip}$ for CEM and $\\chi^2_{towers}$ for PHX).\n\\end{itemize}\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.92\\textwidth]{PhysObj\/Figs\/ele_deposit_cen}\\\\\n\\includegraphics[width=0.445\\textwidth]{PhysObj\/Figs\/ele_hadem_plug}\n\\includegraphics[width=0.445\\textwidth]{PhysObj\/Figs\/ele_iso_plug}\n\\caption[Electron Identification Criteria for $Z^0\\to e^+e^-$ Candidate Events]{$E^{Had}\/E^{EM}$ (left) and isolation (right) distribution of central (top) and plug (bottom) calorimeter electron selection from unbiased, second legs of $Z^0\\to e^+e^-$ candidate events in data~\\cite{w_z_prl}.}\\label{ele_deposit}\n\\end{center}\n\\end{figure}\n\nFinally, a reconstructed track matched to the EM cluster is used to suppress photons in the central region. Due to poor tracking in the forward region, PHX electrons are defined by a different strategy that relies on the {\\em Phoenix} matching scheme~\\cite{phx_alg}. \n\nTracks candidate electrons in the central region must satisfy the following requirements:\n\\begin{itemize}\n\\item track quality: $\\ge 3$ ($\\ge 2$) COT Axial (Stereo) segments with at least five hits each associated with the track.\n\\item $E^{cluster}\/p^{trk}<2$: bremsstrahlung photons are emitted colinearly to the electron and energy is radiated in the same EM cluster, therefore the measured track momentum is lower than the original $p_T$ of the electron.\n\\item Track-shower matching: \n  \\begin{equation}\n   3.0<q \\Delta x<1.5\\textrm{~cm}\\quad\\textrm{and}\\quad\\Delta z<3\\textrm{~cm,}\n  \\end{equation}\n  where $\\Delta x=|x^{CES}-x^{trk}|$, $\\Delta z=|z^{CES}-z^{trk}|$ and $q$ is the charge. \n\\end{itemize}\n\nElectron candidates in the PEM region do not have any track matching requirement due to the limited coverage of the tracking system for $|\\eta|>1.1$. However to provide some fake-electron rejection a road is built starting from the $x, y$ position in the PES detector, the PV position and the curvature provided by the $E_T$ cluster. The Phoenix matching succeeds if at at last three hits in the silicon detectors are found, thus allowing the selection of a PHX electron candidate. This is not considered a {\\em real} track matching because only four points are used in the fit and it is impossible to reconstruct a five-parameter helix, moreover some events have a high density of silicon hits, increasing the ambiguity of the matching. \n\nThe other important identification requirement is the {\\em Isolation} or $IsoRel$, a variable describing how much calorimeter activity surrounds the lepton. It is defined as: \n\\begin{equation}\n  IsoRel\\equiv E_T^{iso}\/E_T^{cluster}<0.1,\\quad E_T^{iso}=E_T^{0.4}-E_T^{cluster},\n\\end{equation}\n$E_T^{0.4}$ is the energy collected by the calorimeters within a radius $\\Delta R=0.4$ from the centroid of the EM cluster. Isolation is used in analyses involving a $W^{\\pm}$ or $Z^0$ boson because the kinematic region allowed to leptons coming from the bosons decay is usually far from jets or other particles (see Figure~\\ref{ele_deposit}).\n\nTable~\\ref{tab:ele_alg} summarizes all the CEM and PHX identification criteria while the top left part of Figure~\\ref{fig:split_alg} shows the detector coverage of the two algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nElectrons & Identification Cuts\\\\\n\\midrule\n& EM fiduciality \\\\\n& $E_T^{EM}>20$~GeV \\\\\n& $E^{Had}\/E^{EM}<0.055+0.0045 E^{EM}$ \\\\\n& $L_{shr}<0.2$ \\\\\n& $\\chi^2_{\\mathrm{{CES strip}}}<10$ \\\\\nCEM &$\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n& $\\Delta z(CES,trk)<3$~cm \\\\\n& $-3.0<q \\Delta x(CES,trk)<1.5$~cm \\\\\n&$E\/p^{trk}<2.0$ or $E_T^{\\mathrm{cluster}}>100$~GeV \\\\\n&$|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{trk}|<60$~cm \\\\\n&$IsoRel<0.1$ \\\\\n\\midrule\n& $1.2<|\\eta^{PES}|<2.8$ \\\\\n& $E_T^{EM}>20$~GeV \\\\\n& $E^{Had}\/E^{EM}<0.05$ \\\\\n& PEM$3\\times 3$Fit $\\ne 0$ \\\\\n& $\\chi^2_{PEM3\\times 3} <10$ \\\\\nPHX & $PES5\\mathrm{by}9U>0.65$ \\\\\n& $PES5\\mathrm{by}9V>0.65$ \\\\\n& $\\Delta R(PES,\\mathrm{cluster})<3$~cm \\\\\n& N Si Hits $\\ge 3$ \\\\ \n& $\\mathrm{Phoenix~Track}$\\\\\n&$|z_0^{PHX}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{PHX}|<60$~cm \\\\\n&$IsoRel<0.1$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CEM and PHX Electron Selection Requirements]{Summary of CEM and PHX \\emph{tight electron} selection requirements.}\\label{tab:ele_alg}\n\\end{center}\\end{table}\n\n\n\n\\subsection{Muon Identification}\n\nA muon behaves like a minimum ionizing particle due to its rest mass, which is about 200 times larger than the electron one. Therefore muons deposit very little energy in the calorimeter systems and can leave a signal in the outer layer of the CDF detector which is instrumented with arrays of gas detector and scintillators (muon chambers). Ionization deposits from a muon candidate in a given muon detector constitute a {\\em stub} and the candidate naming depends by the coverage of the muon detector that records them (see Section~\\ref{sec:mu_cham}). \n\nThe basic selection for a muon candidate~\\cite{muon_id} is a high quality COT track pointing to a Minimum Ionizing Particle (MIP) energy deposit in the EM and HAD calorimeters and matched to a stub in the muon chambers. The precise requirements are the following:\n\\begin{itemize}\n\\item track quality: the reconstructed COT track must have a minimal amount of axial ($\\ge 3$) and stereo ($\\ge 2$) COT super-layers;\n\\item the $\\chi^2_{trk}$ returned by the track fitting algorithm should be less than $2.3$;\n\\item track\/stub matching in the $\\phi$-plane is required by an appropriate $\\Delta x(trk,\\mathrm{stub}) = | x^{trk} - x^{\\mathrm{stub}}|$ with cut values different for each specific muon sub-detector\n\\end{itemize}\n\nIf we are interested to a muon coming from $Z$ or $W$ decay, we expect the muon to be isolated from other detector activity (as in the electron case). The isolation is defined exploiting the muon candidate momentum and MIP energy deposit in the calorimeter:\n\\begin{equation}\n  IsoRel\\equiv E_T^{iso}\/p_T<0.1;\\quad E_T^{iso}=E_T^{0.4}-E_T^{MIP},\n\\end{equation}\nwhere the $p_T$ is the COT track momentum and $E_T^{MIP}$ is the transverse energy deposited in the towers crossed by the track.\n\nThe two {\\em tight} muon reconstruction algorithms are the CMUP and CMX. The first covers the region $|\\eta|<0.6$ where a track is required to match stubs in both CMU and CMP muon detectors. The second covers the region $0.65<|\\eta|<1.0$: it requires a stub in the CMX muon detector and a minimum curvature of the COT track, $\\rho_{COT}$, of $140$~cm\\footnote{This last requirement  ensures appropriate efficiency of the CMX trigger.}. Table~\\ref{tab:central_mu_alg} summarizes all the CMUP and CMX identification criteria while the top right section of Figure~\\ref{fig:split_alg} shows the detector coverage of the two algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nMuons & Identification Cuts\\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n&$E_T^{EM}<2+\\mathrm{max}\\big(0, ((p^{trk}-100)0.0115)\\big)$~GeV \\\\\n&$E_T^{Had}<6+\\mathrm{max}\\big(0, ((p^{trk}-100)0.028)\\big)$~GeV \\\\\n& $\\ge 3$ axial $\\ge2$ stereo COT Layers \\\\\nCMUP & $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\nCMX & $|d_0^{trk}|<0.02$~cm (track with silicon) \\\\\n& $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\n& $|z_0^{trk}|<60$~cm \\\\\n& $\\chi^2_{trk}<2.3$\\\\\n&$IsoRel<0.1$ \\\\ \\midrule\n& CMU Fiduciality \\\\ \nCMUP & CMP Fiduciality \\\\\n& $\\Delta X_{CMU}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\Delta X_{CMP}(trk, \\mathrm{stub})<5$~cm \\\\\\hline\n& CMX Fiduciality \\\\ \nCMX & $\\rho_{COT}>140$~cm \\\\\n& $\\Delta X_{CMX}(trk, \\mathrm{stub})<6$~cm \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[CMUP and CMX Muon Selection Requirements]{Summary of CMUP and CMX \\emph{tight muons} selection requirements.}\\label{tab:central_mu_alg}\n\\end{center}\\end{table}\n\n\n\n\n\\subsection{Loose Muons Identification}\\label{sec:LooseMuId}\n\nMuons can be faked by cosmic rays or hadrons showering deep inside the calorimeters or not showering at all, however the muon candidates usually have a very clean signature because of the many detector layers used to identify them. On the other hand the signal acceptance for the tight muon categories is geometrically limited, therefore a set of lower quality muon identification criteria, {\\em loose muons}, was developed. Loose muons were used in other analysis~\\cite{single_top} to increase the signal yield. These are the main requirements of the loose muons algorithms:\n\\begin{itemize}\n\\item BMU: forward isolated muons ($1.0<|\\eta|<1.5$) with hits in {\\em Barrel} muon chambers.\n\\item CMU: central isolated muons with hits {\\em only} in the CMU chambers and not in CMP.\n\\item CMP: central isolated muons with hits {\\em only} in the CMP chambers and not in CMU.\n\\item SCMIO: a good quality track matched to an isolated MIP deposit and a {\\em non-fiducial} stub in the muon detector (Stubbed Central Minimum Ionizing particle).\n\\item CMIO: a good quality track {\\em only} matched to an isolated MIP deposit and failing any other muon identification criteria (Central Minimum Ionizing particle).\n\\item CMX-NotTrigger: also named CMXNT, are isolated muon detected in the CMX chamber but which are {\\em not} triggered by the CMX specific triggers because of the geometrical limits of the COT ($\\rho_{COT}<140$~cm).\n\\end{itemize}\nTable~\\ref{tab:loose_alg} summarizes all the Loose muons identification criteria while the bottom left section of Figure~\\ref{fig:split_alg} shows the detector coverage of the six algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nMuons & Identification Cuts \\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n&$E_T^{EM}<2+\\mathrm{max}\\big(0, ((p^{trk}-100)0.0115)\\big)$~GeV \\\\\nBMU &$E_T^{Had}<6+\\mathrm{max}\\big(0, ((p^{trk}-100)0.028)\\big)$~GeV \\\\\nCMU & $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\nCMP & $|d_0^{trk}|<0.02$~cm (track with silicon) \\\\\nSCMIO& $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\nCMIO& $|z_0^{trk}|<60$~cm \\\\\nCMXNT& $\\chi^2_{trk}<2.3$\\\\\n&$IsoRel<0.1$ \\\\ \\midrule\n\n& BMU Fiduciality \\\\ \nBMU & $\\Delta X_{BMU}(trk, \\mathrm{stub})<9$~cm \\\\\n& COT Hits Frac. $>0.6$ \\\\\\midrule\n\n& CMU Fiduciality \\\\ \nCMU & $\\Delta X_{CMU}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\\midrule\n\n& CMP Fiduciality \\\\ \nCMP & $\\Delta X_{CMP}(trk, \\mathrm{stub})<7$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\\midrule\n\n& Stub Not Fiducial  \\\\ \nSCMIO & $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n&$E_T^{EM} + E_T^{Had} > 0.1$  \\\\\\midrule\n\n& No Stub  \\\\ \nCMIO & $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n&$E_T^{EM} + E_T^{Had} > 0.1$  \\\\\\midrule\n\n\n& CMX Fiduciality \\\\ \nCMXNT & $\\Delta X_{CMX}(trk, \\mathrm{stub})<6$~cm \\\\\n& $\\ge 3$ axial $\\ge 2$ stereo COT Layers \\\\\n& $\\rho_{COT}<140$~cm, Not trigger CMX \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Loose Muons Selection Requirements]{Summary of the BMU, CMU, CMP, SCMIO, CMIO, CMX-Not-Trigger (CMXNT) {\\em Loose muons} selection requirements.}\\label{tab:loose_alg}\n\\end{center}\\end{table}\n\n\n\n\n\n\n\\subsection{Isolated Track Identification}\\label{sec:IsoTrk}\n\nThe isolated tracks are last lepton category used in this analysis. They are defined to be high-$p_T$ good quality tracks isolated from energy deposits in the tracking systems. The track isolation is defined as:\n\\begin{equation}\\label{TrackIsolation}\n\\rm{TrkIso}=\\frac{p_T({\\rm{trk\\ candidate}})}{p_T({\\rm{trk\\ candidate}})+\\sum p_T({\\rm{other\\ trk)}}}\\,\\rm{,}\n\\end{equation}\nwhere $p_T({\\rm{trk\\ candidate}}$ is the transverse momentum of the specific track we analyze (candidate) and $\\sum p_T({\\rm{other\\ trk)}}$ is the sum of the transverse momenta of all good quality tracks within a cone radius of 0.4 of the candidate track. The isolation requirement is necessary in order to ensure that the track corresponds to a charged lepton produced in a decay of a $W$ boson and it is not part of an hadronic jet. A track is fully isolated if $TrkIso = 1.0$, thus a cut value of $0.9$ is used in the analysis.\nA selection requirement of $\\Delta R>0.4$ between the track and any tight jet\\footnote{Central, high $E_T$ jets are classified as {\\em tight}: $|\\eta|<2.0$, $E_T^{cor}>20$ (see Section~\\ref{sec:jetObj}).} is also applied to remove jets with low track multiplicity.\n\nThe fact that the isolated track is not required to match a calorimeter cluster or a muon stub allows to recover real charged leptons that arrive in non-instrumented regions of the calorimeter or muon detectors, as seen in Figure~\\ref{fig:split_alg}. \n\nTable~\\ref{iso_tab} summarizes the criteria used to select good quality isolated tracks while the bottom right section of Figure~\\ref{fig:split_alg} shows the detector coverage of the algorithms in the $\\eta - \\phi$ plane for a $WZ$ Monte Carlo.\n\n\\begin{table}\\begin{center}\n\\begin{tabular}{cc}\n\\toprule\nTrack Leptons & Identification Cuts \\\\\n\\midrule\n&$p_T^{trk}>20$~GeV$\/c$ \\\\\n& $\\ge 24$ axial $\\ge 20$ stereo COT Hits \\\\\n& $|d_0^{trk}|<0.2$~cm (track w\/o silicon) \\\\\n& $|d_0^{trk}|<0.02$~cm (track with silicon)\\\\\nIsolated & $|z_0^{trk}-z_0^{PV}|<5$~cm \\\\\nTracks & $|z_0^{trk}|<60$~cm \\\\\n& $\\chi^2_{trk}$ probability $<10^{-8}$\\\\\n& N Si Hits $\\ge 3$ \\\\ \n&$TrkIso>0.9$ \\\\\n& $\\Delta R(trk, \\mathrm{tight~jets})>0.4$\\\\\n& $\\Delta \\phi(trk, \\mathrm{jet 1})>0.4$\\\\\n\\bottomrule\n\\end{tabular}\n\\caption[Isolated Tracks Selection Requirements]{Summary of ISOTRK \\emph{isolated track} leptons selection requirements.}\\label{iso_tab}\n\\end{center}\\end{table}\n\n\n\n\\section{Jet Identification}\\label{sec:jetObj}\n\nThe QCD theory tells us that the partons composing the (anti)proton can be treated perturbatively as free particles if they are stuck by an external \nprobe\\footnote{I.e. a lepton, a photon or a parton from a different hadron.} with sufficient high energy (so called {\\em hard scattering}). However partons resulting from the \ninteraction can not exist as free particles because at longer distances (i.e. lower energies) the \\emph{strong potential} can not be treated perturbatively and \npartons must form colorless hadrons. This process, called {\\em hadronization} or {\\em showering}, produces a {\\em jet}, i.e. a narrow spray of stable particles that retains the information of the initial parton (for a pictorial representation see Figure~\\ref{jet1}). \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{PhysObj\/Figs\/jet1}\n\\caption[Schematic Representation of a Hadronic Jet]{A parton originating from a hard scattering hadronizes and generates a\na narrow spray of particles identified as a {\\em jet}.}\\label{jet1}\n\\end{center}\n\\end{figure}\n\nFrom an experimenter's point of view a jet is defined as a large energy deposit in a localized area of the detector (see Figure~\\ref{lego_jet}). The challenge of a physics analysis\nis to recover from detector information the initial energy, momentum and, possibly, the kind of the parton produced in the original interaction. \n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/lego_jet}\n\\caption[Event Display with Jet Deposits]{Calorimetric deposit in the $\\eta-\\phi$ plane as represented in the CDF event display. EM deposits are red while HAD deposits are blue.}\\label{lego_jet}\n\\end{center}\n\\end{figure}\nA {\\em jet identification algorithm} is a tool to reconstruct such information and it should satisfy at best the following requirements~\\cite{jet_req}:\n\\begin{itemize}\n\\item\\emph{Infrared safety}: the presence of soft radiation between two jets may cause a merging of the two jets. This should not occur to avoid an incorrect parton attribution.\n\\item\\emph{Collinear safety}: the jet reconstruction should be independent from any collinear radiation in the event, i.e. different energy distribution of particles inside calorimetric towers.\n\\item\\emph{Invariance under boost}: the same jets should be found independently from boosts in longitudinal direction.\n\\item\\emph{Boundary stability}: kinematic variables should be independent from the details of the final state.\n\\item\\emph{Order independence}: the same reconstructed quantities should appear looking at parton, particle and detector levels.\n\\item\\emph{Straightforward implementation}: algorithm should be easy to implement in perturbative calculations.\n\\end{itemize}\nBeyond this theoretical aspects a jet algorithm should be experimentally efficient with a high reconstruction efficiency, good resolution and robust at high instantaneous luminosity.\n\n\n\\subsection{CDF Cone Algorithm}\\label{sec:jetCone}\n\nCDF uses several algorithms, none of them completely satisfying all the above requirements. The most common one, that is also the one used in this ana\\-lysis,\nis \\verb'JETCLU'~\\cite{jetclu}, an iterative fixed cone jet reconstruction algorithm based only on calorimetric information.\n\nThe algorithm starts by creating a list of the seed towers from all the calorimeter towers with transverse energy above the threshold of $1$~GeV. \nStarting with the highest-$E_T$ seed tower, a precluster is formed by combining\ntogether all adjacent seed towers within a cone of given radius $R$ in the $\\eta-\\phi$ space\\footnote{In this analysis we use $R=0.4$.}. This procedure is repeated, starting with the next unused seed tower, until the list is exhausted. The $E_T$-weighted centroid is then formed from the towers in the precluster and a new cone of radius $R$ is formed around this centroid. All towers with energy above the lower threshold of $100$~MeV within this new cone are added to the cluster. Then, a new centroid is calculated from the set of towers within the cluster and a new cone drawn. This process is iterated until the centroid of the energy deposition within the cone is aligned with the geometric axis of the cone (stable solution).\n\nThe initial clusters found can overlap so the next step is to merge or separate overlapping clusters, since each tower may belong only to one jet, each particle should not be assigned to more than one jet. Two clusters are merged if the total energy of the overlapping towers is greater than $75\\%$ of the energy of the smallest cluster. If the shared energy is below this cut, the shared towers are assigned to the cluster that is closer in $\\eta-\\phi$ space. The process is iterated again until the list of clusters remains fixed. \n\nThe final step of the jet identification happens on a second stage, after electron candidate identification and it is called {\\em reclustering}. If an EM calorimeter cluster is found to be compatible with a tight electron identification (CEM, PHX both isolated or not), the EM calorimeter towers are removed and the jet clustering algorithm is iterated.\n\nMassless four-vector momenta are assigned to the towers in the clusters for EM and HAD components with a magnitude equal to the energy deposited in the tower and the direction defined by a unit vector pointing from the event vertex to the center of the calorimeter tower at the depth that corresponds to the shower maximum. A cluster four-vector is then defined summing over the towers in the cluster:\n\\begin{eqnarray}\n&& E=\\sum_{i=1}^N(E_i^{EM}+E_i^{HAD})\\mathrm{,}\\\\\n&& p_x=\\sum_{i=1}^N(E_i^{EM}\\sin\\theta_i^{EM}\\cos\\phi_i^{EM}+E_i^{HAD}\\sin\\theta_i^{HAD}\\cos\\phi_i^{HAD})\\mathrm{,}\\\\\n&& p_y=\\sum_{i=1}^N(E_i^{EM}\\sin\\theta_i^{EM}\\sin\\phi_i^{EM}+E_i^{HAD}\\sin\\theta_i^{HAD}\\sin\\phi_i^{HAD})\\mathrm{,}\\\\\n&& p_z=\\sum_{i=1}^N(E_i^{EM}\\cos\\theta_i^{EM}+E_i^{HAD}\\cos\\theta_i^{HAD})\\mathrm{.}\n\\end{eqnarray}\nwhere the index $i$ runs over the towers in the cluster.\n\nIn order to study jet characteristics, other variables (number of tracks, energy deposited in the HAD and EM calorimeters, etc.) are reconstructed and associated to the final jet analysis-object.\n\n\n\n\\subsection{Jet Energy Corrections}\n\nThe ultimate goal of the jet reconstruction algorithm is the determination of the exact energy of the outgoing partons coming from the hard interaction, i.e. the Jet Energy Scale (JES). Clearly many factors produce a mismatch between the raw energy measured by the algorithm and the one of the parton before the hadronization. \n\nCDF developed a set of jet energy corrections depending of $\\eta$, $E_T^{raw}$ and $R$ of the jet reconstructed by \\verb'JETCLU' algorithm. The corrections are divided into five levels\\footnote{The actual naming skips $L2$, because it is absorbed in $L1$, and $L3$, as it was introduced as a temporary MC calibration in Run II.} ($L$-levels) that can be applied in a standard way to different analyses~\\cite{jet_corr1, jet_corr2}: $\\eta$-dependent response($L1$), effect of multiple interactions ($L4$), absolute energy scale ($L5$), underlying event ($L6$) and out-of-cone ($L7$) corrections. The correction $L1$ and $L5$ are multiplicative factors ($f_{L1}$ and $f_{L5}$) on the raw $E_T$ of the jet, the others are additive constants ($A_{L4}$, $A_{L6}$ and $A_{L7}$). The equation for the complete correction is:\n\\begin{equation}\\label{eq:fullCor}\nE_T^{FullCor}(\\eta, E_T^{raw}, R)=(E_T^{raw} f_{L1}-A_{L4})f_{L5}-A_{L6}+A_{L7}.\n\\end{equation}\nA description of each term is given in the following paragraphs while Figure~\\ref{totsys} shows the separate systematic uncertainties ($\\sigma_{JES}$) associated to each term of Equation~\\ref{eq:fullCor}.\n\nRecent studies~\\cite{gluonJES_10829} have shown that the simulated detector response to jets originating from high-$p_T$ gluons is improved by lowering the reconstructed $E_T$ by two times the uncertainty used in Equation~\\ref{eq:fullCor} ($-2\\sigma_{JES}$). More accurate studies are ongoing within the CDF collaboration to fully understand the effect of a {\\em parton dependent} jet energy correction. The $-2\\sigma_{JES}$ gluon-jets prescription  has been applied also to this analysis, however the result is  negligible due to the small fraction of gluon-jets present in the $HF$-enriched signal region.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/totsys2}\n\\caption[Jet Energy Scale Systematic Uncertainty]{Jet $p_T$ dependence of the systematic uncertainties ($\\sigma_{JES}$) relative to the different terms of Equation~\\ref{eq:fullCor} for the \\texttt{JETCLU} algorithm with $R=0.4$ cone size.}\\label{totsys}\n\\end{center}\n\\end{figure}\n\nIn this analysis, like in many others~\\cite{single_top, wh94_note10796}, we choose to use jet corrections only up to L5. Therefore we define the Level-5 only correction:\n\\begin{equation}\nE_T^{cor}(\\eta, E_T^{raw}, R)=(E_T^{raw} f_{L1}-A_{L4})f_{L5}.\n\\end{equation}\n\nL6 and L7 corrections are fundamental for the measurement of the unknown mass of a particle in the hadronic final state (for example top quark mass measurements) but they are much less relevant in a cross section measurement where only the relative data\/MC energy scale matters  and not its absolute value. Furthermore, Figure~\\ref{totsys} shows that a large uncertainty is associated to the out-of-cone correction for $R=0.4$ jets, which would decrease the di-jet invariant mass resolution, lowering the sensitivity of the present measurement.\n\nDepending on, L5 corrected, $E_T^{cor}$ and the jet centroid position in the detector, jets are classified as {\\em tight} or {\\em loose}:\n\\begin{itemize}\n\\item{\\bf Tight:} $|\\eta^{det}|<2.0$, $E_T^{cor}>20$~GeV.\n\\item{\\bf Loose:} $|\\eta^{det}|<2.4$, $E_T^{cor}>12$~GeV and the jet is not tight.\n\\end{itemize}\nEvent selection (see Section~\\ref{sec:jetSel}) is based on the number of tight jets while both tight and loose jets are used in the \\mbox{${\\not}{E_T}$}  correction. Jet clusters of lower energy are not analyzed, it is assumed that they produce a negligible noise on the top of the unclustered energy of the event.\n\n\n\n\n\n\\paragraph{Level-1: $\\eta$ Dependent Corrections}\n\n$L1$ correction is applied to raw jet energy measured in the calorimeters to \nmake the detector response uniform in $\\eta$, it takes into account aging of\nthe sub-detectors\\footnote{This was the $L2$ correction during Run I} and other \nhardware non-uniformities (for example the presence of cracks).\nThis correction is obtained using a large di-jet sample:\nevents with one jet (\\emph{trigger jet}) in the central region of the \ncalorimeter ($0.2<|\\eta|<0.6$), where the detector response is well known and \nflat in $\\eta$, and a second jet (\\emph{probe jet}), allowed to range anywhere\nin the calorimeter ($|\\eta|<3.6$). In a perfect detector the jets should be\nbalanced in $p_T$, a balancing fraction is formed:\n\\begin{equation}\nf_b\\equiv \\frac{\\Delta p_T}{p_T^{ave}}=\\frac{p_T^{probe}-p_T^{trigger}}{(p_T^{probe}+p_T^{trigger})\/2}\\mathrm{,}\n\\end{equation}\nthe average of $f_b$ in the analyzed $\\eta$ bin is used to define the $\\beta$\nfactor\\footnote{The definition of Equation~\\ref{beta_eq} has a average value equal\nto $p_T^{probe}\/p_T^{trig}$ but is less sensitive to presence of non-Gaussian \ntails in the usual $p_T^{probe}\/p_T^{trig}$ ratio.} (Figure~\\ref{eta_correction} \nshows the $\\beta$ distribution for different cone radii):\n\\begin{equation}\\label{beta_eq}\n\\beta\\equiv\\frac{2+<f_b>}{2-<f_b>}\\mathrm{.}\n\\end{equation}\nThe final $L1$ correction is defined as $f_{L1}(\\eta, E_T^{raw}, R)=1\/\\beta$ \nand reproduces an approximately flat response in $\\eta$ with an error varying \nfrom $0.5\\%$ to $7.5\\%$.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/eta_correction}\n\\caption[Jet Energy Correction as a Function of $\\eta$]{$\\eta$-dependence of $\\beta$ factor for cone radii $R=0.4$, $0.7$ and \n$1.0$, measured in the di-jet component of $\\mathtt{jet20}$\nsample.}\\label{eta_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-4: Multiple Interactions Corrections}\n\nJet energy measurement is also degraded by the presence of minimum-bias events\nthat come from multiple $p\\bar{p}$ interactions. This correction becomes more \nrelevant at high luminosity, indeed the number of $p\\bar{p}$ interactions is \nPoisson distributed with mean value approximately linear with instantaneous luminosity:\n\\begin{equation} \n\\langle N(\\mathscr{L}\\simeq 10^{32}~\\mathrm{cm}^{-2}\\mathrm{s}^{-1})\\rangle\\simeq3\\mathrm{,}\\quad \\langle N(\\mathscr{L}\\simeq 3\\cdot 10^{32}~\\mathrm{cm}^{-2}\\mathrm{s}^{-1})\\rangle\\simeq 8\\mathrm{.}\n\\end{equation}\nThe energy of particles coming from those processes is estimated from \nminimum-bias events drawing a cone in a random\nposition in the region $0.1<\\eta<0.7$. Figure~\\ref{mp_correction} shows that\nthe measured minimum-bias $E_T$ grows linearly with the number of primary \nvertices\\footnote{Good quality primary vertices are reconstructed through at \nleast 2 COT tracks.}, such quantity, $A_{L4}$, must be subtracted by jet raw \nenergy. The total uncertainty is about $15\\%$, it mostly depends on luminosity \nand event topology.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/mp_correction}\n\\caption[$E_T$ Correction due to Multiple Interactions]{$E_T$ correction deriving from multiple interactions as a function of primary vertex number for cones with $R=0.4$ (right) and $R=0.7$ (left).}\\label{mp_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-5: Absolute Energy Scale Corrections}\n\nWhile $L1$ and $L4$ make jet reconstruction uniform over the whole detector and\nover the global behavior of $p\\bar{p}$ beam interaction, $L5$ correction \n($f_{L5}$) aims to derive, from the detector jet energy measurement, the $E_T$ of particles originating the jet.\n\nThe study is MC driven: first jet events are generated with full CDF detector \nsimulation, then jets are reconstructed both at calorimeter and hadron\ngeneration levels (HEPG) with the use of same clustering algorithm. A \ncalorimeter jet (C) is associated to the corresponding hadron jet (H) if \n$\\Delta R <0.1$. For both HEPG and detector jets the transverse momentum, \n$p_T^C$ and $p_T^H$, is calculated. \nThe absolute jet energy is defined as $\\mathcal{P}(p_T^C|p_T^H)$, the probability\\footnote{Different \n$p_T^H$ can give the same $p_T^C$, in this case the maximum is taken.} to measure $p_T^C$ with a given $p_T^{H}$.\n\nFigure~\\ref{abs_correction} shows the correction factor $f_{L5}$ for different \ncone sizes as function of the different jet transverse energies. The total \nuncertainty is about $3\\%$ and it mainly arises from the determination of\ncalorimetric response to single particles and MC fragmentation modeling.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{PhysObj\/Figs\/abs_correction2}\n\\caption[Absolute Jet Energy Correction as a Function of $p_T$]{Absolute jet energy scale correction and systematic uncertainty ($f_{L5}$) for \\texttt{JETCLU} algorithm with $R=0.4$ cone size.}\\label{abs_correction}\n\\end{center}\n\\end{figure}\n\n\\paragraph{Level-6 \\& Level-7: Underlying Event and Out-of-cone Corrections}\n\nAlthough we do not use L6 and L7 corrections, their description is reported for completeness. They are the last two corrections needed to infer the initial energy of the parton originating the jet. \n\nThe underlying event correction ($L6$) takes into account the interaction \nprocesses which can occur between spectator partons or that originates from initial \nstate radiation (usually soft gluon radiation) while the out-of-cone correction\n($L7$) considers the fraction of particles coming from the original parton that fall outside the jet cone. \n\nThe underlying event energy ($A_{L6}$) must be subtracted to the total jet \nenergy. It was measured studying minimum-bias events during Run I and is \nparametrized with a constant value that scale with the cone radius. Out of cone\nenergy ($A_{L7}$) must be added to the total jet energy, studies are carried \nout with the same jet-parton matching method of $L5$. \n\n\n\n\\section{Neutrino Reconstruction}\\label{sec:metObj}\n\nNeutrinos are the only subatomic particles that leaves the detector completely undetected, therefore their signature is {\\em missing energy}. \n\nSince the longitudinal energies of the colliding partons are unknown and not necessarily equal, we can only say that the total transverse energy of the $p \\bar{p}$ collision is zero. Therefore the total amount of missing transverse energy \\cancel{E}$_T$ (or MET) gives a measurement of the neutrino transverse momentum\\footnote{For a massless neutrino $p_{T}=E_{T}$.} and it is defined as:\n\\begin{equation}\\label{rawmet}\n\\vec{\\mathrm{\\cancel{E}}}_T\\equiv - \\sum_i{\\vec{E}_T^i}\n\\end{equation}\nwhere $\\vec{E}_T^i$ is a vector with magnitude equal to the transverse energy \ncollected by the i-th calorimeter tower and pointing from the interaction\nvertex to the center of the tower. The sum involves all the towers with total \nenergy above $0.1$~GeV in the region $|\\eta|<3.6$.\n\nThe \\mbox{${\\not}{E_T}$}  obtained from Equation~\\ref{rawmet}, close to the online reconstructed missing energy, is often referred to as {\\em raw} \\cancel{E}$_T$ (or \\mbox{${\\not}{E_T^{raw}}$}). The fully reconstructed \\mbox{${\\not}{E_T}$}  is corrected for the true $z$ vertex position, for the difference between the raw and corrected $E_T$ of the tight and loose jets, for the presence of muons in the event, by subtracting the momenta of minimum ionizing high-$p_T$ muons and adding back the transverse energy of the MIP deposit in the calorimeter towers. These corrections can be summarized in the following equation:\n\\begin{equation}\n\\mbox{${\\not}{E_T}$}  = \\mbox{${\\not}{E_T^{raw}}$} - \\sum_{\\text{muon}} p_T + \\sum_{\\text{muon}} (E_T^{EM} + E_T^{HAD}) - \\sum_{jet}(E_T - E_T^{cor}).\n\\end{equation}\n\n\n\n\\section{Secondary Vertex Tagging}\\label{sec:secvtx}\n\nThe algorithms able to select a jet coming from a Heavy-Flavor ($HF$) quark hadronization process are called {\\em $b$-taggers} or {\\em heavy-flavor taggers} and they are of fundamental importance in this and in many other analyses. For example both the top quark and the SM Higgs boson (for $m_H\\lesssim 135$~GeV$\/c^{2}$) have large branching fraction in $b$-quark, therefore an efficient $b$-tagging can dramatically reduce the background of uninteresting physical processes which contain only light-flavor ($LF$) hadrons in their final state.\n\nWe employ the Secondary Vertex Tagger\\footnote{Historically it was the most important \ncomponent in top discovery in 1995.} algorithm (\\verb'SecVtx') to select a $HF$ enriched sample by requiring a $b$-tag on one or both the selected jets. In this way, it is possible to discriminate the $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$ decays against the generic $W\/Z\\to u,d, s$ decays. \n\nIn a successive phase of the analysis a flavor separator Neural Network~\\cite{Richter_stop2007,kit_note7816}, developed by the Karsrhue Institute of Technology (also named KIT Flavor Separator), is used to separate jets originating from $c$ and $b$ quarks, allowing a separate measurement of $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$.\n\n\n\\subsection{The SecVtx Algorithm}\\label{sec:secvtx_alg}\n\nThe \\verb'SecVtx' algorithm takes advantage of the long life time of $b$-hadrons: a $c\\tau$ value of about $450~\\mu$m together with a \nrelativistic boost due to a momentum of several GeV$\/c$ permits to a $b$-hadron to fly several millimeters\\footnote{The average transverse momentum of a $b$-hadron coming from a $WH$ events is about $40$~GeV$\/c$ for a Higgs boson mass of $120$~GeV$\/c^2$; in that condition a neutral $B^0$ meson of mass $5.28$~GeV$\/c^2$ undergoes a boost $\\beta\\gamma=7.6$ and the average decay length is $3.5$~mm.} away from the primary interaction \nvertex. The relevant quantity is the $c\\tau$ which is approximately the average\nimpact parameter of the outgoing debris of $b$-hadron decays. \nThe decay produces small sub-jets composed by tracks with large impact \nparameter ($d_0$). The silicon detectors (see \nsection~\\ref{track_par}) are able to reconstruct $d_0$ with adequate \nprecision to separate displaced tracks from the prompt tracks coming from the \nprimary interaction. Figure~\\ref{svx_example} shows as a $W+$ jets candidate\nevent with two displaced secondary vertices is identified by \\verb'SecVtx' and \nreconstructed by the CDF event display.\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{PhysObj\/Figs\/svx_example}\n\\caption[$W +$ Jets Candidate Event with Two Reconstructed Secondary Vertices]{$W+$ jets candidate event with two secondary vertices tagged by \n$\\mathtt{SecVtx}$ (run 166063, event 279746). The \\cancel{E}$_T$ direction, a \nmuon track, a prompt track and tracks from the secondary vertices are shown.}\\label{svx_example}\n\\end{center}\n\\end{figure}\n\nTagging is performed for all the jets with $|\\eta|<2.4$ in an event. The algorithm searches for secondary vertices using the tracks within the jet cone of radius \\mbox{$\\Delta R=0.4$} . The usable tracks must satisfy the following requirements:\n\\begin{itemize}\n\\item $p_T>0.5$~GeV$\/c$;\n\\item $|d_0|<0.15$~cm and $|d_0\/\\sigma_0|>2.0$;\n\\item $|z_0-z_{PV}|<2.0$~cm;\n\\item have a minimum number (depending on track reconstruction quality and position) of hits in the silicon detector;\n\\item be seeded or confirmed in the COT;\n\\end{itemize}\na {\\em taggable} jet is defined as a jet containing at least two usable tracks.\n\nThe algorithm works on a two step basis and has two main operation modes\\footnote{An {\\em ultra-tight} operation mode exists but it is rarely used.}, {\\em tight} (the standard one) and {\\em loose}. The operating modes are defined by track and vertex quality criteria~\\cite{secvtx_opt7578} but the two-step selection algorithm remains identical.\n\n\nIn the \\emph{Pass 1} at least three tracks are required to pass loose selection\ncriteria. At least one of the tracks used is required to have $p_T > 1.0$~GeV$\/c$. The selected tracks are \ncombined two by two until a seed secondary vertex is reconstructed, then all \nthe others are added one by one and a quality $\\chi^2$ is computed. Tracks are \nadded or removed depending of their contribute to the $\\chi^2$.\n\nThe \\emph{Pass 2} begins if \\emph{Pass 1} does not find a secondary vertex. Now only \ntwo tracks are required to form a secondary vertex but they must pass tighter requirements: $p_t>1.0$~GeV$\/c$, $|d_0\/\\sigma_0|>3.5$ and one of the tracks must have $p_T > 1.5$~GeV$\/c$.\n\nIf a secondary vertex is identified in a jet, the jet is {\\em tagged}. The bi-dimensional decay length $L_{xy}$ is calculated as the projection on the jet axis, in the $r-\\phi$ \nplane, of the \\verb'SecVtx' vector, i.e. the one pointing from the primary vertex \nto the secondary. The sign of $L_{xy}$ is defined by the angle $\\alpha$ between\nthe jet axis and the \\verb'SecVtx' vector. Figure~\\ref{fig:svx} explains the \ngeometry.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/svx}\n\\caption[Reconstructed Secondary Vertex: True and Fake $\\mathtt{SecVtx}$]{Schematic representation of $\\mathtt{SecVtx}$ variables and geometry. Left: true reconstructed secondary vertex. Right: negative $\\mathtt{SecVtx}$ tag, falsely reconstructed secondary vertex.}\\label{fig:svx}\n\\end{center}\n\\end{figure}\n\nA secondary vertex coming from a $HF$ hadron is expected to have \nlarge $L_{xy}$. To reduce background due to mismeasured tracks \n\\mbox{$|L_{xy}\/\\sigma_{L_{xy}}|>7.5$} is required\\footnote{Negative $L_{xy}$ has\nno physical meaning but it is important to estimate the mistag probability due \nto resolution effects.}. Other requirements are applied on the \ninvariant mass of the pair of tracks, to avoid $K$ and $\\lambda$ decays, and on\nvertex multiplicity and impact parameter to reject secondary vertices due to \ninteraction with material inside the tracking volume.\n\n\n\\subsection{Tagging Performances and Scale Factors}\\label{sec:btag_sf}\n\nThe performances of a $b$-tagger are evaluated on its efficiency, i.e the rate \nof correctly identified $b$-hadrons over all the produced $b$-hadrons, and on \nits purity, i.e the rate of falsely identified $b$-hadrons in a sample with no \ntrue $b$-hadrons. CDF uses $b\\bar{b}$ QCD MC to evaluate \\verb'SecVtx' efficiency \nrelying on detector and physical processes simulation. Figure~\\ref{fig:tag_eff} \nshows the $b$-tagging efficiency as a function of jet $\\eta$ and $E_T$ for the \ntight and loose \\verb'SecVtx' operating modes. Tagging efficiency drops at large $|\\eta|$ because of tracking acceptance.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_eff}\n\\caption[\\texttt{SecVtx} Efficiency]{Efficiency of the  \\texttt{SecVtx} $b$-tagging algorithm for the tight and loose operating modes.}\\label{fig:tag_eff}\n\\end{center}\n\\end{figure}\n\nAs MC does not reproduce the exact $b$-tagging efficiency of \\verb'SecVtx' a {\\em Scale Factor} ($SF_{Tag}$) is introduced to account for data\/MC difference in the form:\n\\begin{equation}\nSF_{Tag}\\equiv\\frac{\\epsilon_{data}}{\\epsilon_{MC}}.\n\\end{equation}\n\n\nCDF uses two methods to calculate $SF_{Tag}$: both use di-jet samples where one of the two jets contains a low-$p_T$ lepton, electron or muon, which increases the presence of a $HF$ hadron decay, while the other jet can be $b$-tagged by \\verb'SecVtx' or not. \n\nThe method exploiting the electrons~\\cite{eleSF_10178} finds algebraically the relative number of  $b$-tagged $HF$-jets with respect to the number of $b$-jets with a semileptonic electron decay. To extract the fraction, it also needs the number of jets with a low-$p_T$ electron but not containing $HF$ hadrons: this is obtained from a jet sample where the $\\gamma\\to e^+e^-$ conversion process is identified, and therefore, with depleted  $HF$ content. \n\nThe second method, used as a validation of the first one, exploits the muon semileptonic decay of the $HF$ hadrons~\\cite{muSF_8640}. The algorithm is similar to the electron based one, but the fraction of  low-$p_T$ muons not coming from $HF$ hadrons is extracted from a fit of the muon $p_T$ relative to the jet axis (named $p_T^{rel}$) that has a peculiar distribution for $HF$ hadron decays. \n\nThe efficiency on data are therefore compared to the MC results and the $SF_{Tag}$ is extracted.\nFigure~\\ref{fig:tag_sf} shows the $SF_{Tag}$ dependency with respect to the jet $E_T$ as determined by the electron method.\nTable~\\ref{tab:tag_sf} reports the $SF_{Tag}$ for the loose and tight \\verb'SecVtx' operation modes integrated over the variables of interest for the $SF_{Tag}$ parametrization (no strong dependency is seen in any of them).\n The total per-jet tagging efficiency, deconvoluted from tracking effects is about $40$\\% for $b$-jets and $6$\\% for $c$-jets.\n\\begin{table}\\begin{center}\n\\begin{tabular}{cccc}\n\\toprule\n mode & $SF_{Tag}$ & stat. err. & sys. err.\\\\\n\\midrule\nTight &  $0.96$ & $0.01$ & $0.05$ \\\\\nLoose & $0.98$ & $0.01$ & $0.05$ \\\\\n\\bottomrule\n\\end{tabular}\n\\caption[\\texttt{SecVtx} Efficiency Scale Factors]{$\\mathtt{SecVtx}$ Scale Factors ($SF_{Tag}$) for the tight and loose $\\mathtt{SecVtx}$ operation modes.}\\label{tab:tag_sf}\n\\end{center}\\end{table}\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_sf}\n\\caption[\\texttt{SecVtx} Scale Factor]{Evaluation of the \\texttt{SecVtx} $SF_{Tag}$ with the electron method~\\cite{eleSF_10178} and its dependence over the $E_T$ of the jet. The linear fit shows almost no dependency from the $E_T$ of the jet. Table~\\ref{tab:tag_sf} reports the evaluation of $SF_{Tag}$ integrated over the variables of interest for the $HF-$tagger parametrization.}\\label{fig:tag_sf}\n\\end{center}\n\\end{figure}\n\nThe number of falsely \\verb'SecVtx' tagged jets is dubbed \\emph{mistags}.\nMistags can be due to track resolution, long living $LF$ hadrons or \nsecondary interactions with detector material.\n\nThe rate of $W$ plus mistag jets is derived from a sample of events collected with \nan inclusive jet-based trigger with no $HF$ requirement\\footnote{The presence of a \nsmall $HF$ contamination in the sample is a source of systematic uncertainty.}. \nThe mistag parametrization~\\cite{mistag} is obtained from an inclusive jet sample using {\\em negative tags} (see Figure~\\ref{fig:svx}), i.e. $b$-jets which appear to travel back toward the \nprimary vertex. Resolution and material effects are expected to produce false \ntags in a symmetric pattern around the primary interaction vertex. The mistag \nrate is then corrected for the effects of long-lived $LF$ hadrons to take\ninto account the LF contamination giving real secondary vertices. \nA {\\em per-jet} mistag probability, $p^j_{Mistag}$ is parametrized in bins of:\n\\begin{itemize}\n\\item total number of jets in the event, \n\\item jet $E_T$, \n\\item jet $\\eta$, \n\\item track multiplicity within the jet, \n\\item total $E_T$ of the event, \n\\item number of interaction vertices,\n\\item the $z$ vertex position.\n\\end{itemize}\nThis defines the, so called, {\\em Mistag Matrix}. Figure~\\ref{fig:tag_mistag} shows the mistag rate as function of $E_T$ and $\\eta$ of the tagged jets for the tight and loose \\verb'SecVtx'\noperation modes. The approximate per-jet fake rate is about $1$\\% for the tight operation mode and $2$\\% for the loose one.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/tag_mistag}\n\\caption[$E_T$ and $\\eta$ Mistag Rate]{Rate of wrongly $\\mathtt{SecVtx}$ tagged jets (\\emph{mistags}) as a function of $E_T$ and $\\eta$ of the jets for the tight and loose $\\mathtt{SecVtx}$ operation modes. The rate is derived from an inclusive jet sample.}\\label{fig:tag_mistag}\n\\end{center}\n\\end{figure}\n\n\\section{Neural Network Flavor Separator (KIT-NN)}\\label{sec:kitnn}\n\nThe $b$-tag requirement drastically reduces the contamination of $LF$ jets, however also a per-jet fake rate of $1$\\% can produce enough mistag background to decrease the sensitivity to rare processes. On the other hand a more stringent $b$-tagging requirement would reduce the signal yield. The KIT Flavor Separator Neural Network~\\cite{Richter_stop2007,kit_note7816} (KIT-NN) offers a possible solution to these problems: after a \\verb'SevVtx' tag, the Neural-Network (NN) exploits a broad range of $b$ quark discriminative variables to obtain a continuous distribution with a good separation power between real $b$-jets and $LF$ jets (see Figure~\\ref{fig::kit_nn_out}). \n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/kit_yield}\n\\caption[KIT-NN Output Distribution and $b$-jet Efficiency\/Purity Curve]{Left: Output value of the KIT-NN function for $b$, $c$ and $LF$ jets. Right: curve of the $b$-jet efficiency {\\em vs} $b$-jet purity for different selection thresholds of the KIT-NN output.}\\label{fig::kit_nn_out}\n\\end{center}\n\\end{figure}\nBeyond the $LF$ separation power, in this analysis we exploit the KIT-NN distribution in an original way. Thanks to the substantial shape difference between $c$-jets and $b$-jets KIT-NN values, we managed to measure separately $W\\to cs$ and $Z\\to c\\bar{c}\/b\\bar{b}$ contributions (as shown in Chapeter~\\ref{chap:StatRes}).\n\nA short description of the KIT-NN implementation, developed for the single-top search at CDF~\\cite{single_top}, can be useful to understand the physical meaning of such variable. \n\nA Bayesian NN is a supervised learning algorithm~\\cite{BShop_machine_learning} that associates a score to an {\\em event}, according to its likeliness to be {\\em signal} or {\\em background}. An event is defined by an array of $K$ input variables, $\\alpha_i$, and the ouput score, $o \\in [-1,1]$, is obtained from the following function:\n\\begin{equation}\n  o =S  \\Big( \\sum^H_{j=0} \\omega_j S \\Big( \\sum_{i=0}^{K}\\omega_{i,j} \\alpha_i +\\mu_{0,j} \\Big) \\Big),\n\\end{equation}\nwhere $\\omega_i$, $\\omega_{i,j}$ and $\\mu_{0,j}$ are tunable parameters and $S$ is a sigmoid, or activation, function:\n\\begin{equation}\n  S(x) =\\frac{2}{1+e^{-x}} -1,\n\\end{equation}\nThe parameters are derived from a target function optimized over the training set of signal and background labeled events. \n\nThe final result is a per-event signal-or-background {\\em posterior probability} that takes into account non-linear correlations between the input variables. \n\nIn our case, after that a jet has been tagged by \\verb'SevVtx', twenty-five input variables, relative to the secondary vertex and to the jet, are fed into a NeuroBayes\\textsuperscript{\\textregistered} NN~\\cite{neurobayes}. The training signal sample is composed by $t\\bar{t}$, single-top and $W+b\\bar{b}$ MC samples, while the background is built with $W+c\\bar{c}$ and $W+LF$ MC samples. The variables exploit lifetime, mass, and decay multiplicity of the $b$ hadrons using several characteristics of the identified secondary vertex, the properties of the tracks inside the jet and information from the $b$-tag algorithm, for example if {\\em Pass 1} or {\\em Pass 2} reconstruction is used (see Section~\\ref{sec:secvtx_alg}).\nFigure~\\ref{fig::kit_nn_out} shows the obtained classification values for  $b$, $c$ and $LF$ jets.\n\nAlthough a careful validation of the input variables was performed, the use of a MC-driven $LF$ training sample introduced a relevant discrepancy in the KIT-NN evaluation of {\\em real} mistagged jets. Therefore a  $LF$ correction function was derived from a fake-enriched data sample selected by negative $\\mathtt{SecVtx}$ tags. The final $LF$ KIT-NN distribution is obtained by assigning, to each $LF$-jet, a random value extracted from the corrected $LF$ distribution. The uncorrected template distribution is used as an {\\em optimistic} systematic variation. A similar effect has been hypothized also for $c$-jets but it appears in a less relevan way. For $c$-jets the KIT-NN output is evaluated per-jet each jet and the $LF$-like variation is used as a {\\em pessimistic} systematic variation. Figure~\\ref{fig:kit_sys}\nshows the default (central) KIT-NN distribution and systematic variations for $LF$ and $c$ jets.\n\n\\begin{figure}[!h]\n\\begin{center}\n\\includegraphics[width=0.99\\textwidth]{PhysObj\/Figs\/kit_sys}\n\\caption[KIT-NN Distribution and Systematic Variation for $LF$ and $c$ Jets]{KIT-NN default distribution (continuous lines) and systematic variations (dashed lines) for $LF$-jets (left) and $c$-jets (right).}\\label{fig:kit_sys}\n\\end{center}\n\\end{figure}\n\n\\chapter{Events Selection}\\label{chap:sel}\n\nIn the following we describe the selection requirements applied to produce samples of data enriched in $diboson\\to \\ell\\nu+ HF$ candidates.\n\nEvery physics analysis starts with some kind of candidate signal selection. In the case of collider experiments, it is possible to see three ingredients in it: online (trigger) selection (Section~\\ref{sec:onlineSel}), offline selection (Sections from~\\ref{sec:w_sel} to~\\ref{sec:svm_sel}) and the efficiency evaluation of each of the steps.\n\nThe variety of different physics object that compose the diboson decay channel under investigation is a characteristic of this analysis: charged leptons (electrons, muons and isolated tracks), missing energy, jets and $HF$ jets. These represent a good fraction of all the objects that can be reconstructed at a hadron collider experiment. Furthermore the need to relax the selection requirements in order to increase the statistics and the background-rich $W+2$ jets sample increase the challenges. It worths noticing that the same issues arise also in other primary analyses at CDF: for example single-top or $WH$ searchs.\n\nIn order to deal with these difficulties, one of the major task of this thesis work was the development of a robust analysis framework that could exploit the $\\ell\\nu+HF$ channel in an efficient and reliable way. The package, named $WH$ Analysis Modules (WHAM), is described in Appendix~\\ref{chap:AppWHAM}. Its strengths are modularity, a wide set of configuration options and the use, with improvements, of CDF most advanced tools. The modular and object-oriented approach of the software allowed the handling of the complex selection formed by eleven different lepton types and four data-streams. At the same time it was also possible to develop a set of tools useful also in other contexts: for example a new multivariate multi-jet background rejection tool (see Appendix~\\ref{chap:AppSvm}) or a versatile $b$-tag efficiency evaluation algorithm.\n\nThe actual selection process is divided into two main steps. First we select candidates that have the $\\ell\\nu+2$ jets signature, this defines the {\\em pretag control} sample. Successively we require one or both the jets to be $b$-tagged, obtaining two signal samples ({\\em single} and {\\em double} tagged) enriched in $HF$. Section~\\ref{sec:data_sig} shows the final result of the selection on the CDF dataset and on the diboson signal.\n\n\n\\section{Online Event Selection}\\label{sec:onlineSel}\nThe trigger system, described in Section~\\ref{sec:trigger}, is in charge of the online selection.\n\nIn order to maximize the signal acceptance we use a complex trigger strategy based on several trigger paths collected in four data-streams:\n\\begin{itemize}\n\\item high energy central electrons or \\verb'bhel';\n\\item high energy forward electrons or \\verb'bpel';\n\\item high momentum central muons or \\verb'bhmu';\n\\item high missing transverse energy or \\verb'emet'.\n\\end{itemize}\nAfter online trigger selection but before offline analysis, the quality of the recorded data is crosschecked and the final luminosity is calculated (see Section~\\ref{sec:data-struc}).\n\nA detailed description of the trigger selection, the trigger efficiency ($\\epsilon_{trig}$) evaluation and of the data quality requirements follows in the Section. \n\n\n\\subsection{High Energy Central Electron Trigger}\n\nThe trigger path used for the selection of tight central electron candidate events (CEM) is named \\verb'ELECTRON_CENTRAL_18' and it is stored in the \\verb'bhel' data-stream together with other auxiliary triggers. Single high energy electron identification is based on a calorimeter electromagnetic cluster matched with a reconstructed track.\n\nThe selection proceeds through the three trigger levels, first with minimal requirements and then with more and more sophisticated object reconstruction. L1 requirements are a track with $p_T > 8$~GeV\/c, a central\n($|\\eta| < 1.2$) calorimeter tower with $E_T > 8$~GeV and the ratio between the\nenergy deposited in the hadronic calorimeter to that in the electromagnetic\ncalorimeter ($E^{HAD}\/E^{EM}$) less than $0.125$. At L2, it is required a calorimeter cluster with\n$E_T > 16$~GeV matched to a track of $p_T > 8$~GeV$\/c$. At L3, it requires an\nelectron candidate with $E_T > 18$~GeV matched to a track of $p_T > 9$~GeV$\/c$.\n\nThe \\verb'ELECTRON_CENTRAL_18' trigger path was extensively studied in precision CDF measurements involving $W$ and $Z$ bosons~\\cite{w_z_prl} and now, after each data taking period, the efficiency is evaluated with a standard set of tools~\\cite{perfidia}. A data sample of $W$ boson collected with a \\verb'NO_TRACK' trigger is used to measure the tracking efficiency while a backup trigger is used to measure the efficiency of the calorimeter clustering. The efficiency is \\mbox{$\\epsilon^{CEM}_{trig} =0.982\\pm 0.003$} across the complete dataset with a small $E_T$ and $\\eta$ dependency (see Figure~\\ref{fig:cem_trig}). The trigger turn-on $E_T$ dependency is parametrized with the following function:\n\\begin{equation}\\label{eq:cem_et}\n  \\epsilon_{trig}(x) = A - Be^{-c x};\n\\end{equation}\nwhere $x \\equiv E_T$ and $A$, $B$, $c$ are the parameters of the fit. The following equation is used to model the small effect across $\\eta$:\n\\begin{equation}\\label{eq:cem_eta}\n  \\epsilon_{trig}(x) = A - \\frac{C}{2\\pi\\sigma} e^{-\\frac{x^2}{2\\sigma^2}};\n\\end{equation}\nwhere $x \\equiv  \\eta$ and $A, C, \\sigma$ are the parameters of the fit. Figure~\\ref{fig:cem_trig} shows the parametrization, derived from the full CDF dataset, and applied to MC events.\n\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth, height=0.36\\textwidth]{Sel\/Figs\/cem_et_trig}\n\\includegraphics[width=0.495\\textwidth, height=0.36\\textwidth]{Sel\/Figs\/cem_eta_trig}\n\\caption[\\texttt{ELECTRON\\_CENTRAL\\_18} Trigger Efficiency Parametrization]{\\texttt{ELECTRON\\_CENTRAL\\_18} trigger efficiency parametrization as a function of $E_T$ (left) and $\\eta$ (right) of the candidate electrons. $E_T$ dependence is parametrized by Equation~\\ref{eq:cem_et}, $\\eta$ dependence by Equation~\\ref{eq:cem_eta}. The fit parameters are reported in the legends.}\\label{fig:cem_trig}\n\\end{center}\n\\end{figure}\n\n\n\n\\subsection{High Energy Forward Electron Trigger}\n\nThe candidate high energy forward electrons (in $1.2 < |\\eta_{Det}| < 2.0$ or Plug region) are stored in the \\verb'bpel' data-stream. \n\nElectron triggering in the plug region is particularly challenging because it lacks of the COT coverage and because a large number of soft interactions are boosted in the region closer to the\nbeam-line. A partial solution to these problems is the use of a multiple-object trigger, named \\verb'MET_PEM', dedicated to $W\\to e\\nu$ selection instead of an inclusive electron trigger: there is no track selection but large \\mbox{${\\not}{E_T}$}  is required together with a high energy cluster in the forward electromagnetic calorimeter. The L1 requirements are a EM tower of $E_T > 8$~GeV in the forward region and \\mbox{${\\not}{E_T}$} $ > 15$~GeV. L2 confirms \\mbox{${\\not}{E_T}$}  requirements but a reconstructed EM cluster of $E_T > 20$~GeV is required and, finally, L3 confirms again the same quantities (\\mbox{\\mbox{${\\not}{E_T}$} $> 15$}~GeV and EM Cluster \\mbox{$E_T > 20$}~GeV) with L3 reconstruction algorithms. \n\nThe trigger shows a turn-on for both the $E_T$ of the cluster and the \\mbox{${\\not}{E_T^{raw}}$}, i.e. corrected only for the position of the primary vertex. The following equation is used for the parametrization of the turn-on:\n\\begin{equation}\\label{eq:phx_trig}\n  \\epsilon_{trig}(x) = \\frac{1}{1+ e^{-A(x-B)}};\n\\end{equation}\nwhere $x\\equiv E_T$ or $x\\equiv$\\mbox{${\\not}{E_T^{raw}}$} and $A$ and $B$ are the parameters of the fit, reported in Figure~\\ref{fig:phx_trig}.\n\nThe \\verb'MET_PEM' trigger parametrization was part of my initial work in\nthe CDF collaboration~\\cite{met_pem_trig9493} and is now included in the CDF analysis\ntools~\\cite{perfidia}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/phx_et_trig}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/phx_rawmet_trig}\n\\caption[\\texttt{MET\\_PEM} Trigger Efficiency Parametrization]{\\texttt{MET\\_PEM} trigger efficiency parametrization as a function of $E_T$ (left) of the electron candidate and \\mbox{${\\not}{E_T^{raw}}$} (right) of the event. The \\mbox{${\\not}{E_T^{raw}}$} is corrected only for the position of the primary vertex and therefore it is closer to the trigger level \\mbox{${\\not}{E_T}$} . The $E_T$ and \\mbox{${\\not}{E_T^{raw}}$} dependence is parametrized by Equation~\\ref{eq:phx_trig} and the fit parameters are reported in the legends.}\\label{fig:phx_trig}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\subsection{High Momentum Central Muon Trigger}\n\nCentral tight muon candidate events (CMUP and CMX) are collected by the \\verb'MUON_CMUP18' and the \\verb'MUON_CMX18' triggers and saved into the \\verb'bhmu' data-stream. \n\nFor \\verb'MUON_CMUP18' the main trigger requirements are: at L1, a track (\\mbox{$p_T > 4$}~GeV$\/c$) matched to a stub in both the CMU and CMP detectors. At L2, a confirmed track ($p_T > 15$~GeV$\/c$) and a calorimeter deposit consistent with a MIP in the direction of the track and muon stubs. Finally at L3, a fully reconstructed COT track ($p_T > 18$~GeV$\/c$) whose extrapolation matches the CMU (CMP) hits within $\\Delta x_{CMU} < 10$~cm ($\\Delta x_{CMP} < 20$~cm). \nSimilar requirements are made also for \\verb'MUON_CMX18' but the sub-detector of interest is the CMX (covering $0.6 < |\\eta| < 1.1$) and the L1 requirement is a track with $p_T > 8$~GeV$\/c$ matched to a stub in the CMX chamber.  \n\nAs for tight electron candidates, also the central muon triggers were extensively studied~\\cite{w_z_prl} and the efficiency is\nmeasured with a standard procedure~\\cite{perfidia}. Using a data sample of $Z\\to \\mu^+\\mu^-$, a muon passes the trigger requirements and the second muon is checked to pass the trigger selection or not. \n\nAs the CMUP sample represents a relevant fraction of the good quality muon data, an additional effort was done, in the context of the $WH$ analysis~\\cite{hao_trig10747}, to improve the CMUP trigger collection efficiency. Basically we exploited all the CMUP trigger candidates present in the \\verb'bhmu' stream and collected by secondary and auxiliary triggers. The purity of those events was found to be the same of the \\verb'MUON_CMUP18' selected, and we evaluated, with the bootstrapping method, a recover of about $12$\\% efficiency on CMUP online selection. \n\nThe trigger efficiencies are \\mbox{$\\epsilon^{CMUP}_{trig} = 0.976\\pm 0.001$} and \\mbox{$\\epsilon^{CMX}_{trig}=0.873\\pm0.001$}, flat in the kinematic variables. \n\n\n\n\\subsection{High Missing Transverse Energy Triggers}\n\nIn order to recover lepton acceptance lost because of the limited geometrical coverage of the single-lepton triggers, we rely on data collected by the \\texttt{emet} data-stream. \n\nThe \\texttt{emet} stream contains samples with large online \\mbox{${\\not}{E_T}$}  or \\mbox{${\\not}{E_T}$}  plus reconstructed jets. No charged lepton information is used, therefore the acceptance of $\\ell\\nu + 2$~jets events can be recovered. In particular we use the \\verb'emet' stream to collect Loose Muons (Section~\\ref{sec:LooseMuId}) and Isolated Tracks (Section~\\ref{sec:IsoTrk}) candidates, altogether dubbed Extended Muon Categories (EMC).\n\nThe trigger optimization~\\cite{higgs_ttf8875} was performed with the aim to increase the sensitivity to Higgs boson production, but, as diboson and $WH$ production share the same $HF$ final state, an identical strategy was employed for this analysis. The three main triggers used are: \\verb'MET45', \\verb'MET2J' and \\verb'METDI'. The exact naming and requirements of the trigger paths changed many times along the data taking period due to different instantaneous luminosity conditions but the main classification and characteristics are the following:\n\\begin{itemize}\n\\item \\verb'MET45' requires large \\mbox{${\\not}{E_T}$}  in the event, first at L1 and, with increased quality and cut levels, at L2 and L3. The trigger was present along all the data taking with slightly different specifications, however in the last 2\/3 of the data taking the L3 \\mbox{${\\not}{E_T}$}  cut was $40$~GeV.\n\\item \\verb'MET2J' requires high \\mbox{${\\not}{E_T}$}  in the event at L1 (\\mbox{${\\not}{E_T}$} $ > 28$~GeV), two jets at\nL2 with one of them reconstructed in the central region of the detector ($|\\eta| < 0.9$) and, finally, L3 object confirmation and \\mbox{${\\not}{E_T}$} $ > 35$~GeV. The trigger was Dynamically Pre-Scaled (DPS) to cope with the increasing luminosity in the second half of the data taking.\n\\item \\verb'METDI' requires two jets and \\mbox{${\\not}{E_T}$}  in the event, very similarly to \\verb'MET2J', but the L1 requirement, with \\mbox{${\\not}{E_T}$} $ > 28$~GeV and at least one jet, is different.\nThis trigger was introduced after the first $2.4$~fb$^{-\u22121}$ of integrated luminosity.\n\\end{itemize}\n\nThe overlap and composite structure of the \\verb'emet' triggers makes their use less straightforward than the inclusive lepton ones. Furthermore the use of multiple-objects requires to model, on signal and background MC, the possible selection efficiency correlations (e.g. \\mbox{${\\not}{E_T}$}  and jets correlations).\nEfficiency studies were performed in~\\cite{Nagai:2010zza, Buzatu:2011zz}. The main results are the definition of appropriate kinematic regions where $\\epsilon_{trig}$ is flat (plateau region) with respect to the jet variables and can be parametrized as a function of the \\mbox{${\\not}{E_T}$}  only. \\mbox{${\\not}{E_T}$}  is corrected for primary vertex and jet energy but not for the presence of muons. In this way, tight muon candidates, collected from\nother triggers (e.g. \\verb'MUON_CMUP18'), can be used to evaluate the turn-on curves as their contribution to the \\mbox{${\\not}{E_T}$}  is not accounted at trigger level. \n\nAn optimal trigger combination strategy for \\texttt{emet} triggers was evaluated in~\\cite{Buzatu:2011zz}. The trigger selection maximizes the $WH$ acceptance after that the \\mbox{$\\epsilon_{trig}(\\mbox{${\\not}{E_T}$} )$} function is evaluated for each data period and kinematic configuration of the three triggers. \n\nThe final combined trigger efficiency reaches $\\epsilon_{trig}\\simeq 0.5$ for $WZ$ simulated events.\n\n\n\n\n\n\\subsection{Data Quality Requirements and Luminosity Estimate}\\label{sec:lumi}\n\nThe last step before offline event selection is the application of strict data quality requirements. These are ensured by the Good Run List (GRL) selection: i.e. it is possible to use only the run periods where the performances of the sub-detectors are optimal and well parametrized.\n\nA conservative approach, used in many of the CDF analyses, is to consider a run as good only if all the sub-detectors used to define the physics objects of the analysis are fully functional. However here, as in the latest $WH$ search results~\\cite{wh94_note10796}, an ad-hoc approach is used~\\cite{ext_grl}. We require a full functionality of the silicon detector (used for jet $b$-tagging) and calorimeters, while, as we are interested in the one-lepton final state, we de-correlate inefficiencies due to shower-max detectors functionality (CES and PES used for CEM and PHX identification) from muon chambers (CMU, CMP, CMX problematic runs). In this way it was possible to recover 8\\% of the collected luminosity.\n\nOnce that the trigger paths and the appropriate GRLs are defined, it is possible to calculate the final integrated luminosity used for the various lepton types:\n\\begin{itemize}\n\\item CEM and PHX electrons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.446$~fb$^{\u2212-1}$.\n\\item CMUP muons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.494$~fb$^{-\u22121}$. \n\\item CMX muons use an integrated luminosity of $\\int \\mathcal{L} \\mathrm{dt}= 9.396$~fb$^{-1}$.\n\\item EMC charged leptons use an integrated luminosity of $\\int \\mathcal{L}\\mathrm{dt} = 9.288$~fb$^{-\u22121}$.\n\\end{itemize}\nThe calculated luminosity has systematic uncertainty of $6$\\% (see Section~\\ref{clc_par}).\n\n\n\\section{Offline Event Selection}\n\nNow, on good data, we apply the object identification algorithms described in Chapter~\\ref{chap:objects} to select the desired final state of $\\ell\\nu + HF$ jets. \n\nTable~\\ref{tab:cuts} summarizes the complete selection {\\em cut flow} and the following Sections (from \\ref{sec:w_sel} to~\\ref{sec:svm_sel}) describe in detail the selection criteria. Part of the cut flow is common to many other CDF analyses while the applied multi-jet rejection algorithm, as well as some other specific choices, are completely original.\n\nBecause we rely on simulation to evaluate the signal acceptance, the data\/MC agreement is cross checked for each one of the employed algorithms with data driven methods. Well known physics processes are selected in control samples and the efficiency on data ($\\epsilon^{Data}_{alg}$) and MC ($\\epsilon^{MC}_{alg}$) is estimated. The ratio between them, with appropriate correlations and systematic errors evaluation, is a correction Scale Factor ($SF_{alg}$) to be applied on the simulation:\n\\begin{equation}\n  SF_{alg} =\\frac{ \\epsilon^{MC}_{alg}}{\\epsilon^{Data}_{alg}}\n\\end{equation}\nThe $SF_{alg}$ associated to the lepton identification ($SF_{l}$) and to the $b$-tagging ($SF_{Tag}$) are particularly important in this analysis because of the large number of different lepton reconstruction algorithms and because of the $HF$ final state of the signal. The $SF$s are discussed along with the selection criteria in the following Sections\n\n\n\\begin{table}[h]\n  \\begin{center}\n   \\begin{tabular}{c}\n        \\toprule\n  {\\bf Selection Cuts}\\\\\\midrule\n\n   Trigger Fired \\\\\n   Good Run Requirement\\\\\n  $|z_0^{PV}| \\le 60$~cm \\\\\n  Charged Lepton Selection \\\\\n  $|z_0^{lep} \u2212- z_0^{PV}| < 5$~cm\\\\\n  Di-Lepton Veto\\\\\n  $Z\\to \\ell\\ell$ Veto\\\\\n  Cosmic Ray Veto \\\\\n  $\\gamma\\to e^+e^-$ Conversion Veto\\\\\n  \\mbox{${\\not}{E_T}$} $ > 15$~GeV\\\\\n  2 Tight Jets Selection \\\\\n  $M_{Inv}(jet1, jet2)>20$~GeV$\/c^2$\\\\\n  1 or 2 \\texttt{SecVtx} Tags\\\\\n  {\\bf Multi-jet Rejection:} \\\\\n {\\em SVM}$_{CEM, EMC}>0$, {\\em SVM}$_{CMUP, CMX}>-0.5$, {\\em SVM}$_{PHX}>1$\\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[Selection Summary]{Summary of all the selection requirements applied in the analysis.}\\label{tab:cuts}\n  \\end{center}\n\\end{table}\n\n\n\n\n\\subsection{$W\\to\\ell\\nu$ Offline Selection}\\label{sec:w_sel}\n\nThe leptonic decay of the $W$ boson is selected by requiring exactly one high energy charged lepton and missing transverse energy signaling the presence of the neutrino.\n\nThe correct application of these two basic requirements exploits several of the identification algorithms described in the previous Chapter and other selection criteria in order to purify the lepton sample.\nThe summary of all the selection steps, applied on data and MC, is the following:\n\\begin{itemize}\n\\item identification of the primary interaction vertex (Section~\\ref{sec:primVtx}) within the interaction fiducial region of 60 cm from the detector center ($|z_0^{PV}| \\le 60$~cm). Minimum bias events are used to evaluate on data the efficiency of this cut. The result, averaged over all the data taking periods, is\\footnote{The initial number of MC events, used to normalize all the simulated acceptances, is derived after this cut so that $\\epsilon_{z_0}$ should be applied to obtain the final yields.}:\n  \\begin{equation}\n    \\epsilon_{z_0} = 0.9712\\pm 0.0006.\n  \\end{equation}\n\\item Identification of one charged and isolated lepton candidate originating\nfrom the primary vertex ($|z_0^{lep} \u2212- z_0^{PV}| < 5$~cm). A charged lepton candidate is one of the eleven lepton identification algorithms described in Sections from~\\ref{sec:eleSel} to~\\ref{sec:IsoTrk}: CEM, PHX (tight electrons), CMUP, CMX (tight muons), BMU, CMU, CMP, CMIO, SCMIO, CMXNT (loose muons) and ISOTRK (isolated tracks). This requirement does not produce any appreciable signal loss.\n\\item Veto of di-leptonic candidate events: if an event contains two tight or loose electrons or muons both isolated or not, it is rejected as a Drell-Yan event or $t\\bar{t}$ dileptonic candidate. Events with an ISOTRK candidate are vetoed if reconstructed together with another ISOTRK or a tight lepton. An efficiency of approximately 90\\% is estimated from signal MC.\n\\item Further rejection of $Z\\to\\ell\\ell$ candidates is obtained for CEM, PHX, CMUP and CMX leptons by vetoing events in which the tight lepton can be paired with a track or a EM calorimeter cluster that form an invariant mass within the range $[76, 116]$~GeV$\/c^2$.  An efficiency of approximately 90\\% is estimated from signal MC.\n\\item A cosmic tagger was implemented within the CDF offline code~\\cite{cosmic_tag}\nto reject high-$p_T$ cosmic muons which interact with the detector simultaneously with a bunch crossing. The algorithm exploits muon chambers, COT and TOF, calorimeter energy and timing information, providing\nalmost 100\\% rejection of cosmic ray events, with a negligible loss of signal efficiency.\n\\item Rejection of electrons originating from photon conversions, $\\gamma\\to e^+e^{-\u2212}$,\nis obtained by applying the CDF conversion tagger~\\cite{conversion_tag}. The algorithm looks for two opposite-sign tracks (one of them belonging to the identified electron) and requires $\\Delta \\cot(\\theta) \\le 0.02$ and the distance at the closest approach between them $D_{xy} \\le 0.1$~ cm. The rejection efficiency of the conversion veto is about $65$\\% with a $1\\div 2$\\% signal loss.\n\\item Finally, \\mbox{${\\not}{E_T}$} $ > 15$~GeV is required to signal the presence of a neutrino. The \\mbox{${\\not}{E_T}$}  is fully corrected for position of the primary interaction vertex, presence of jets and muons in the event (see Section~\\ref{sec:metObj}). Approximatly 7\\% of the signal is lost with this requirement.\n\\end{itemize}\n\nFor each run period and lepton identification algorithm, the agreement\nbetween data and MC is measured comparing the $Z\\to\\ell\\ell$ data sample against the simulation. Table~\\ref{tab:lepId} reports, averaged on all the run range, the yield correction, $SF_{l}$ that covers the\ndata\/MC differences. The main differences arise in the simulation of the isolation and of the muon chamber response.\n\n   \\begin{table}[h]\n \\begin{center}\n     \\begin{tabular}{cccc}\n       \\toprule\n\n       CEM & PHX & CMUP & CMX \\\\\n       $0.973 \\pm 0.005$ &   $0.908 \\pm 0.009$ & $0.868\\pm 0.008$ & $ 0.940\\pm 0.009$\\\\\n     \\end{tabular}\n  \n      \\begin{tabular}{cccccc}\n        \\midrule\n        BMU & CMU & CMP & SCMIO & CMIO & ISOTRK \\\\\n\n        $ 1.06\\pm 0.02$ & $ 0.88\\pm 0.02$ & $ 0.86\\pm0.01$ & $ 1.02\\pm 0.01$ & $ 0.97 \\pm 0.02$  &  $ 0.94 \\pm 0.04$ \\\\\n        \\bottomrule \n      \\end{tabular}\n\n     \\caption[Lepton Algorithm Data\/MC Scale Factors ($SF_l$)]{Data\/MC Scale Factors ($SF_l$) for each of the single lepton identification algorithms used in the analysis, tight charged leptons in the first row and EMC categories in the lower row.}\\label{tab:lepId}\n \\end{center}\n   \\end{table}\n\nIf several lepton identification algorithms cover the same detector region they can not be considered independent and a further correction\\footnote{This correction was introduced for the first time in this analysis and in the $WH\\to \\ell\\nu+ b\\bar{b}$ search~\\cite{wh94_note10796}.} to the simulation is needed. As shown in Figure~\\ref{fig:split_alg}, loose muon and ISOTRK largely overlap with the tight lepton selection and, often, a MC lepton candidate is reconstructed by more than one lepton selection algorithm. Because of this and to avoid ambiguity, we apply a prioritized selection, i.e. we label a lepton candidate with the highest priority algorithm according to:\n  \\begin{equation}\n\\begin{gathered}\n CEM > PHX > CMUP > CMX > BMU > CMU >\\\\\n CMP > SCMIO > CMIO > CMXNT >  ISOTRK\n\\end{gathered}\n  \\end{equation}  \n\n\nIn most of the cases, selection cuts are orthogonal and only one lepton reconstruction is possible, however, if this is not true {\\em and} $SF_{l}\\ne 1$, the same simulated lepton has a probability greater than zero of not being identified by the highest priority algorithm\\footnote{Quality criteria and historical reasons are at the base of the prioritization of the algorithms. Prioritization is indirecly present also in the $SF_{l}$ measurements: for example in the ISOTRK $SF_{l}$ measurement the lepton candidates are checked not to be reconstructed as tight leptons, while the check is not performed in the opposite case, i.e. no ISOTRK identification is performed during the tight $SF_{l}$ measurement.}. In this case the MC event under consideration is classified under more lepton categories with different weights, according to:\n\\begin{equation}\\label{eq:composite_sf}\n  \\omega_{l} = SF_{l} + SF_{l}\\prod_{l'>l} \\big(1- SF_{l'} \\big)\n\\end{equation}\nwhere $\\omega_{l}$ is a composite $SF_{l}$. Equation~\\ref{eq:composite_sf} reduces to $\\omega_{l} = SF_{l}$ when a lepton candidate is reconstructed only by one identification algorithm, otherwise it takes into account the probability of {\\em not} identification from algorithms with higher priority ($l'>l$).\n\nFigure~\\ref{fig:eta_multisf} shows the corrected  $\\eta$  distribution for the EMC lepton category (combination of loose muon and ISOTRK algorithms) while Figure~\\ref{fig:eta_pre_post} shows the complete background estimate distribution for the combined $\\eta$ of the EMC category before and after the correction of Equation~\\ref{eq:composite_sf}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Sel\/Figs\/emc_eta_sf}\n\\caption[$\\eta$ Distribution of the $\\omega_{l}$ Correction on EMC]{Effect of the $\\omega_{l}$ correction described in Equation~\\ref{eq:composite_sf} across the $\\eta$ distribution of the EMC lepton category selected on $WZ$ MC. The correction appears as a factor $f_{\\omega_l}\\ne 1$ for each event where the reconstructed lepton is identified by more than one algorithm.}\\label{fig:eta_multisf}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/WZ_Pretag_ISOTRK_LepEta_noSFCorr}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/WZ_Pretag_ISOTRK_LepEta_wSFCorr}\n\\caption[Effect of $\\omega_{l}$ on EMC $\\eta$ Distribution after Background Estimate]{$\\eta$ distribution of the EMC lepton category with full background evaluation before (left) and after (right) the application of the multiple $SF_{l}$ correction defined by Equation~\\ref{eq:composite_sf}.}\\label{fig:eta_pre_post}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\subsection{Offline Jets Selection}\\label{sec:jetSel}\n\nThe final aim of this analysis is the study of the invariant mass spectrum of a two jet, $HF$ enriched, final state. \n\nJet selection is based on the \\verb'JETCLU' $R = 0.4$ algorithm, described in Section~\\ref{sec:jetObj}, and the \\texttt{SecVtx} $HF$-tagger algorithm, described in Section~\\ref{sec:secvtx}.\nOne of the goals of this analysis is to demonstrate the ability of the CDF experiment to identify a resonance in the background rich sample of $W+HF$ jets. Therefore the jet selection was kept simple with the aim to understand the sample itself. The specific requirements are the following:\n\\begin{itemize}\n\\item  selection of exactly two tight jets. Jets central in the detector and with large $E_T$ (after L5 energy correction) are named tight jets if:\n\n\\begin{equation}\n  E^{jet,cor}_T > 20\\mathrm{~GeV}\\quad\\textrm{and}\\quad |\\eta^{jet}_{Det}| < 2.0;\n\\end{equation}\n\n\\item a minimal lower bound on the invariant mass distribution of the jet pair: $M_{Inv}(jet1, jet2) > 20$~GeV$\/c^2$;\n\\item one or both of the selected jets $HF$-tagged by the \\texttt{SecVtx} algorithm;\n\\item jets are ordered according to their $E_T^{jet,cor}$ and classified as: $jet1$ and $jet2$. \n\\end{itemize}\nNo other requirements are applied on the jets and, in particular, we do not apply any criteria to the loose jets present in the event. \n\nThe $HF$-tag requirement divides the $\\ell\\nu+ 2$ jets sample in three analysis regions:\n\\begin{description}\n\\item[Pretag:] no $HF$-tag requirement is applied and, due to the small $HF$ contamination, the sample is used as a control region (see Section~\\ref{sec:wjets_pretag}).\n\\item[Single-tag:] events with exactly one jet tagged as $HF$ compose this signal region. The statistics of the selected sample is quite large therefore it is important for the sensitivity of this analysis to  $WW\\to\\ell\\nu + c\\bar{s}$ decay.\n\\item[Double-tag:] events with both jets tagged as $HF$. The statistics of this sample is small because of the strict double-tag requirement, however a large fraction of the sensitivity for the $WZ\\to\\ell\\nu + b\\bar{b}$ signal comes from this region as the background contamination is also low.\n\\end{description}\n\nAccording to Section~\\ref{sec:btag_sf}, the $HF$ selection efficiency of the \\texttt{SecVtx} algorithm must be corrected with an appropriate data\/MC $SF_{Tag}$. Furthermore a contamination due to $LF$ quarks producing secondary vertices will contribute to the signal sample. The $HF$ and $LF$ components for $1$\u2212-Tag and $2$--Tag selection are accounted in the signal MC with a per-event tagging probability, $\\omega_{k-\u2212Tag}$ obtained by the combination of per-jet tagging probabilities:\n\\begin{equation}\\label{eq:1tag}\n  \\omega_{1-Tag}=\\sum_{i=1}^{n}p^i_{Tag}\\cdot\\Big(\\prod^n_{j=1,j\\ne i}\\big( 1 - p^j_{Tag}\\big)\\Big),\n\\end{equation}\n\\begin{equation}\\label{eq:2tag}\n  \\omega_{2-Tag}=\\sum_{i=1}^{n-1}p^i_{Tag}\\cdot\\Big( \\sum_{j>i}^{n}p^i_{Tag} \\cdot \\Big( \\prod^n_{h=1,h\\ne i,h\\ne j}\\big( 1 - p^k_{Tag}\\big)\\Big)\\Big),\n\\end{equation}\nwhere the per-jet tagging probability is defined by $p^j_{Tag} \\equiv SF_{Tag}$, if the jet is matched\\footnote{The matching is satisfied if $\\Delta R(jet, had) < 0.4$ where we test the fully reconstructed jet 4-vector against all the final state hadrons 4-vectors.}  to a $HF$ hadron, or $p^j_{Tag} \\equiv p^j_{Mistag}$, if the event is matched to a $LF$ hadron.\n\nEquations~\\ref{eq:1tag} and~\\ref{eq:2tag} are applied to obtain \\texttt{SecVtx} 1-Tag or 2-Tags probabilities.  A further generalization of the previous equations was implemented in the WHAM analysis framework (see Appendix~\\ref{chap:AppWHAM}) so that efficiencies can be calculated for any $b$-tagging algorithm (and combinations of them), any tag and jet multiplicity.\n\n\n\\subsection{Multi-Jet Background Rejection}\\label{sec:svm_sel}\n\nMulti-jet events can fake the $W$ boson signature when one jet passes the high $p_T$ lepton selection criteria and fake \\mbox{${\\not}{E_T}$}  is generated through jets mis-measurement. Although the probability for such an event is small, the high rate of multi-jet events, combined with the small cross section of the processes of interest, make this an important background. Additional difficulties arise in the simulation of the  mixture of physics and detector processes contributing to this background. Therefore it is desirable to reduce the multi-jet events as much as possible.\n\nFor this purpose, an original and efficient multi-variate method based on the Support Vector Machine (SVM) algorithm was developed for this analysis.\n\nThe basic concept behind a SVM classifier is quite simple: given two sets of $n$-dimensional vectors (the training sets) the algorithm finds the {\\em best separating hyper-plane} between the two. The plane is then used again to classify a newly presented vector in one of the two sets. Non linear separation can be achieved with the use of appropriate transformations called {\\em Kernels}. SVM classification performs better than other multivariate algorithms, like Neural Networks, in the case of statistically limited training sets and shows more stability with respect to over-training~\\cite{svmbook}.\n\nThe work described here improves a previously developed software package\\footnote{The previously created SVM based multi-jet rejection~\\cite{CHEP_svm} was already tested and employed in previous analyses~\\cite{wh75_note10596, wh94_note10796, cdfDiblvHF_Mjj2011}.}, based on the \\texttt{LIBSVM}~\\cite{libsvm} library, able to perform algorithm training, variable ranking, signal discrimination and robustness test. Here we report only a summary of the results while a detailed description of optimization, training procedure and variable definition is given in Appendix~\\ref{chap:AppSvm}. \n\nWe trained two specific SVMs, one for the {\\em central} and one for the {\\em forward} region of the detector. \nThe SVMs aimed to improve the purity of the $W\\to e\\nu+2$ jets sample, however we avoided the use of input variables related to the electron identification (i.e. cluster or track related) while we focused on the kinematic variables. This gave an additional result as the multi-jet rejection proved to be optimal also for other lepton identification algorithms. For example the central SVM was used on all the lepton identification algorithms in the region $|\\eta|<1.2$. \n\nAppendix A describes in details all the input variables used in the trainings.\nThe SVM for the {\\em central} region exploits the following eight input variables:\n\\begin{itemize}\n\\item $W$ related variables: $M^W_T$, \\mbox{${\\not}{E_T^{raw}}$}, ${\\not}{p_T}$, $\\Delta \\phi(lep,$ \\mbox{${\\not}{E_T}$} $)$, $\\Delta R(\\nu^{min},$ $lep)$;\n\\item global variables:  $MetSig$, $\\Delta \\phi(jet1,$ \\mbox{${\\not}{E_T}$} ), $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ).\n\\end{itemize} \nThe specific SVM for the {\\em forward} region of the detector (used only for the PHX electron selection) exploits six input variables:\n\\begin{itemize}\n\\item $W$ related variables: $M^W_T$, \\mbox{${\\not}{E_T^{raw}}$}, ${\\not}{p_T}$; \n\\item global variables:  $MetSig$, $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T^{raw}}$}), $\\Delta \\phi({\\not}{p_T},$ \\mbox{${\\not}{E_T}$} ).\n\\end{itemize}\n\nTable~\\ref{tab:cuts} shows different SVM selection values for the different lepton types:\n\\begin{equation}\n\\textrm{{\\em SVM}}_{CEM, EMC}>0\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  \\textrm{{\\em SVM}}_{CMUP, CMX}>-0.5\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  \\textrm{{\\em SVM}}_{PHX}>1\\textrm{.}\n\\end{equation}\nWe choose different thresholds due to the different probabilities to fake a $W$ signature: lower for the tight muon selection (CMUP, CMX), higher for CEM electrons and EMC lepton category, and way higher for PHX electrons, identified in a detector region with low track reconstruction efficiency and large calorimeter occupancies. \n\nFigure~\\ref{fig:svm_templates} shows the SVM output distribution for CEM and PHX leptons for the signal and background MC samples as well as for the multi-jet background models (see Section~\\ref{sec:qcd} for their description).\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/svm_cem} \\includegraphics[width=0.495\\textwidth]{Sel\/Figs\/svm_phx}\n\\caption[SVM Output Distribution for CEM and PHX Leptons]{SVM output distribution for CEM (left) and PHX (right) leptons for the signal and background MC samples as well as for the multi-jet background model (see Section~\\ref{sec:qcd} for the description).}\\label{fig:svm_templates}\n\\end{center}\n\\end{figure}\n\nTable~\\ref{tab:svm_eff} summarizes the signal selection efficiencies for the SVM cuts in the different lepton categories; all the analysis selection cuts are applied except for the tagging requirement. The evaluation of the final multi-jet background contamination is described in Section~\\ref{sec:qcd} and a summary of the results is reported in Tables~\\ref{tab:wjet_pretag_qcd_fit} and~\\ref{tab:wjet_tag_qcd_fit}.\n\nThe training procedure requires the SVM to pass several quality criteria about data\/MC agreement. We also checked the stability of the selection efficiency. We found it stable against a variation of up to 5\\% of the SVM input variables. Therefore we did not find necessary to add a specific SF for the SVM selection.\n\n  \\begin{table}\n\\begin{center}\n    \\begin{tabular}{cccccc}\n      \\toprule\n      Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n      $\\epsilon_{SVM}^{WW}$ & $0.95 \\pm 0.02$ & $0.82 \\pm 0.03$ & $0.97 \\pm 0.03$ & $0.97 \\pm 0.04$ &  $0.94 \\pm 0.03$ \\\\\n      $\\epsilon_{SVM}^{WZ}$ & $0.96 \\pm 0.03$ & $0.83 \\pm 0.04$ & $0.98 \\pm 0.04$  &  $0.99 \\pm 0.05$ & $0.95 \\pm 0.03$ \\\\\n      \\bottomrule\n    \\end{tabular}\n    \\caption[$WW$ and $WZ$ SVM Selection Efficiency]{$WW$ and $WZ$ signal selection efficiencies for the SVM cuts used in the lepton selection. Reported errors are statistical only.}\\label{tab:svm_eff}\n\\end{center}\n  \\end{table}\n\n\n\n\\section{Final Signal Estimate and Data Selection}\\label{sec:data_sig}\n\nThe complete event selection can now be  applied to both the full data sample and the signal MC. \n\nThe final signal yield is derived from the following equation, for each signal process ($Sig=WW$, $WZ$, $ZZ$), each lepton identification algorithm ($l$) and for single and double tagged ($k=1$, $2$) selections:\n\\begin{equation}\\label{eq:mc_estimate}\n  N^{Sig} = \\epsilon_{l, k}^{Sig}\\cdot\\sigma_{p\\bar{p}\\to Sig}\\cdot\\int\\mathcal{L}_{l}\\mathrm{dt};\n\\end{equation}\nthe factors are:\n\\begin{itemize}\n\\item the cross sections, $\\sigma_{p\\bar{p}\\to Sig}$, for inclusive diboson production, reported in Table~\\ref{tab:mc_dib}, are evaluated at NLO~\\cite{mcfm_prd,mcfm_url};\n  \\begin{table}\n    \\begin{center}\n    \\begin{tabular}{ccc}\n      \\toprule\n      Signal & $\\sigma$ (pb), NLO &  Initial Events \\\\\n      \\midrule\n      $WW$    & 11.34 $\\pm$ 0.66  & 4.6 M\\\\ \n      $WZ$    & 3.47  $\\pm$ 0.21  & 4.6 M\\\\  \n      $ZZ$    & 3.62  $\\pm$ 0.22  & 4.8 M\\\\ \n      \\bottomrule\n    \\end{tabular}\n    \\caption[Diboson MC Information and NLO Cross Section]{MC information about the diboson signal. The reported cross sections are evaluated at NLO precision~\\cite{mcfm_prd,mcfm_url}.  \\texttt{PYTHIA}~\\cite{pythia} v6.216 generator and parton-showering is used for the MC simulation.}\\label{tab:mc_dib}\n    \\end{center}\n  \\end{table}\n\n\\item the integrated luminosity, $\\int\\mathcal{L}_{l}dt$, is calculated in Section~\\ref{sec:lumi} for the different leptons\/data-streams;\n\\item $\\epsilon_{l, k}^{Sig}$ is the simulated selection acceptance accounting for trigger and $|z_0|$ cut efficiencies ($\\epsilon_{trig}$, $\\epsilon_{z_0}$), lepton and $b$-tag efficiency corrections ($\\omega_l$, $\\omega_{k-Tag}$):\n  \\begin{equation}\\label{eq:all_w_tag}\n    \\epsilon_{l, k}^{Sig} = \\epsilon_{trig}\\cdot\\epsilon_{z_0}\\cdot\\omega_{l}\\cdot\\omega_{k-Tag}\\cdot\\frac{N^{Sig}_{l,k}}{N^{Sig}_{z_0}}\n  \\end{equation}\nwhere $N^{Sig}_{l,k}$ is the number of MC events passing the lepton and tag selection and $N^{Sig}_{z_0}$ is the number of events generated in the fiduciality region of the detector. Equation~\\ref{eq:all_w_tag} simplifies slightly in case we are evaluating the pretag region yield:\n\\begin{equation}\\label{eq:all_w_pretag}\n    \\epsilon_{l, pretag}^{Sig} = \\epsilon_{trig}\\cdot\\epsilon_{z_0}\\cdot\\omega_{l}\\cdot\\frac{N^{Sig}_{l,pretag}}{N^{Sig}_{z_0}}\n\\end{equation}\nwhere $N^{Sig}_{l,pretag}$ is the number of MC events passing the lepton selection at pretag level. \n\\end{itemize}\nThe final result of the event selection is reported in Table~\\ref{tab:data_sig} for the CDF dataset and the signal expectation; also the pretag control region is shown for comparison. \nThe selection suppresses the $Z\\to\\ell\\ell$ events making the total $ZZ$ contribution almost negligible ($\\approx 3$\\% of the total diboson yield), however this component is included in the signal because no distinction is possible between the $Z\\to HF$ decay originating from $WZ$ or $ZZ$ production.\n\n\\begin{table}[h]\n  \\begin{small}\n    \\begin{center}\n    \\begin{tabular}{cccccc}\n      \\toprule\n \\multicolumn{6}{c}{\\bf Pretag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n    {\\bf Data} & {\\bf 80263}  &{\\bf 27759}  &    {\\bf 39045}  & {\\bf 22465}      & {\\bf 35810} \\\\                      \n  $WW$            & 2012.4 $\\pm$ 168.7    &  705.5 $\\pm$ 59.3     & 1047.7 $\\pm$ 88.5   & 537.0 $\\pm$ 45.5  &1157.7 $\\pm$ 115.3 \\\\\n  $ZZ$            & 22.4 $\\pm$ 1.9        &  6.1 $\\pm$ 0.5        & 25.2 $\\pm$ 2.2      & 13.2 $\\pm$ 1.1    &37.5 $\\pm$ 3.8\\\\\n  $WZ$            &  307.0 $\\pm$ 26.2     & 135.7 $\\pm$ 11.6      & 168.6 $\\pm$ 14.5    & 95.3 $\\pm$ 8.2    &225.7 $\\pm$ 22.8\\\\\n    \\midrule\n   \\multicolumn{6}{c}{\\bf Single-Tag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\ \\midrule\n    {\\bf Data}        & {\\bf 3115}           &{\\bf 1073 }     & {\\bf 1577}          & {\\bf 830}          & {\\bf 1705}          \\\\                           \n\n   $WW$            & 84.35 $\\pm$ 11.8 & 25.05 $\\pm$ 3.54          & 43.7 $\\pm$ 6.17        & 23.68 $\\pm$ 3.35      & 53.11 $\\pm$ 8           \\\\\n   $ZZ$            & 1.85 $\\pm$ 0.19  & 0.21 $\\pm$ 0.02           & 2.45 $\\pm$ 0.25        & 1.37 $\\pm$ 0.14       & 3.45 $\\pm$ 0.39        \\\\ \n   $WZ$            & 29.2 $\\pm$ 2.95  & 12.32 $\\pm$ 1.22          & 16 $\\pm$ 1.66          & 9.21 $\\pm$ 0.94       & 20.54 $\\pm$ 2.39        \\\\\n\n\\midrule                                                                                        \n  \\multicolumn{6}{c}{\\bf Double-Tag Selection}\\\\\n Lepton & CEM & PHX & CMUP & CMX &  EMC \\\\\\midrule\n  {\\bf Data}               &{\\bf 175 }       & {\\bf 62 }     & {\\bf 92 }            & {\\bf 49 }             & {\\bf 126}              \\\\     \n       $WW$           & 0.72 $\\pm$ 0.19   & 0.18 $\\pm$ 0.05        & 0.35 $\\pm$ 0.09        & 0.2 $\\pm$ 0.05           & 0.49 $\\pm$ 0.13      \\\\\n       $ZZ$           & 0.26 $\\pm$ 0.04   & 0.03 $\\pm$ 0.01           & 0.46 $\\pm$ 0.06        & 0.29 $\\pm$ 0.04          & 0.63 $\\pm$ 0.10     \\\\  \n       $WZ$           & 5.28 $\\pm$ 0.75   & 2.6 $\\pm$ 0.37         & 2.52 $\\pm$ 0.36        & 1.67 $\\pm$ 0.24          & 3.52 $\\pm$ 0.54      \\\\\n       \\bottomrule\n\n    \\end{tabular}\n    \\caption[Final Signal Estimate and Data Selection]{Final result of the event selection for the $W+2$ jets dataset and signal expectation in the different lepton categories. The pretag control region is shown together with single and double-tagged signal regions. The uncertainties from MC statistics, productionn cross section, luminosity, lepton identification, trigger and $HF$-tagging are considered in the table.}\\label{tab:data_sig}\n    \\end{center}\n  \\end{small}\n  \\end{table}\n\n\\chapter{Statistical Analysis and Results}\\label{chap:StatRes}\n\nThe measurement of a potential signal (and its properties) over a predicted background requires a statistical analysis of the selected events.\n\nThe detection of diboson events in the $\\ell\\nu+HF$ final state is a challenging problem as it combines a small expected signal yield, a sizable irreducible background and large systematic uncertainties typical of the hadronic environment.\n\nAfter the full event selection (Chapter~\\ref{chap:sel}), the remaining background processes are mainly of irreducible nature ($W+c\/c\\bar{c}\/b\\bar{b}$) and still overwhelm the signal by more than a factor twenty. \n\nAs the diboson production is a resonant process while $W+HF$ is not, a feasible and optimal signal extraction strategy is the shape analysis of the di-jet invariant mass distribution, $M_{Inv}(jet1,jet2)$.\n\nThe use of the di-jet mass has two effects. Firstly, we can not distinguish $WZ\\to\\ell\\nu+b\\bar{b}\/c\\bar{c}$ decay from $ZZ\\to\\ell\\not{\\ell}+b\\bar{b}\/c\\bar{c}$ decay where one lepton is lost. Therefore, from now on, the $WZ$ diboson production is considered together with the small amount of selected $ZZ$ events\\footnote{Table~\\ref{tab:data_sig} shows that $ZZ$ is about $1\/10$ of the $WZ$ expected events and 1\/30 of the total diboson sample.}. A more important effect is that, due to the low mass resolution, typical of the hadronic final states, we can not separate contribution from $WW\\to \\ell\\nu+c\\bar{s}$ from the $WZ\\to \\ell\\nu+b\\bar{b}\/c\\bar{c}$, in the single-tagged event selection\\footnote{In the double-tagged event selection the only signal contribution comes from $WZ\\to \\ell\\nu+b\\bar{b}$.}. We solve this problem with the use of a second variable, the NN flavor-separator (KIT-NN) described in Section~\\ref{sec:kitnn}. We use it on single-tagged events to build a bi-dimensional distribution:\n\\begin{equation}\n  M_{Inv}(jet1,jet2)\\textrm{ {\\em vs }KIT-NN}  \\textrm{.}\n\\end{equation}\nThis, together with the simple $M_{Inv}(jet1,jet2)$ distribution of the double-tagged events, allows to measure $WW$ and $WZ\/ZZ$ contributions separately.\n\nThe statistical analysis is performed with a methodology and a software tool, named {\\em mclimit}~\\cite{mclimit, single_top}, also used in the CDF and Tevatron Higgs searches~\\cite{cdf_cmb, tev_cmb} and adapted to the present analysis.\n\nThe technique, described in Section~\\ref{sec:likelihood}, is based on the integration, over data, of a binned likelihood function where the diboson production cross section is a free parameter. The systematic uncertainties, listed in Section~\\ref{sec:sys_desc}, are included with a Bayesian approach and marginalized to increase the sensitivity. \n\nThe likelihood is built by dividing the selected data in eight orthogonal channels depending on the kinematic properties and on the background composition:\n\\begin{itemize}\n\\item {\\bf 4 Lepton categories} are derived from the data-streams used to collect the events (see Section~\\ref{sec:onlineSel}): central electrons (CEM), forward electrons (PHX), central muons (CMUP+CMX) and extended muons (EMC). The kinematic is homogeneous within each sample.\n\\item {\\bf 2 Tag categories} are derived from the single or double \\texttt{SecVtx} tagged event selection for each lepton category. The sample composition drastically changes in the two regions. Section~\\ref{sec:mjj_kit_distrib} shows the $M_{Inv}(jet1,jet2)$ distribution in the double-tagged categories and slices of the bi-dimensional distribution, $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN, used for the single-tagged categories. Figure~\\ref{fig:2d_templates} shows four examples (for the CEM lepton selection) of the bi-dimensional templates used for the single-tagged channels evaluation: $WW$, $WZ\/ZZ$ signals and $W+c\\bar{c}$, $W+b\\bar{b}$ backgrounds.\n\\end{itemize}\n\\begin{figure}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/ww_templ}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wzzz_templ}\\\\\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wcc_templ}\n  \\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/wbb_templ}\n  \\caption[Example of $M_{Inv}(jet1,jet2)$ {\\em vs } KIT-NN Templates]{Four examples of the bi-dimensional templates, $M_{Inv}(jet1,jet2)$ {\\em vs } KIT-NN, used for the statistical analysis of the  single-tagged channels in the CEM lepton selection: $WW$ signal (top left), $WZ\/ZZ$ signals (top right), $W+c\\bar{c}$ background (bottom left), $W+b\\bar{b}$ background (bottom right). The contribution of $c$ and $b$ quarks cluster at opposite sides of the KIT-NN distribution, the diboson production cluster around an invariant mass resonant peak while the generic $HF$ production spreads along the $M_{Inv}(jet1,jet2)$ direction.}\\label{fig:2d_templates}\n\\end{figure}\n\nThe measured diboson cross sections, $\\sigma_{WW}$ and  $\\sigma_{WZ\/ZZ}$, are obtained in Section~\\ref{sec:cx_measure} from the evaluation of the Bayesian posterior distribution of the combined likelihood. \n\nAs a first result, we measure the total diboson production cross section, fixing the $WW$ and the $WZ\/ZZ$ relative contributions to the SM prediction. Then we remove the constraint on the $WW$ and the $WZ\/ZZ$ relative fractions. We let them free to float independently so to obtain a measurement where the full correlation between $\\sigma_{WW}$ and  $\\sigma_{WZ\/ZZ}$ is accounted. Finally we also obtain two individual measurements integrating out $\\sigma_{WW}$ or  $\\sigma_{WZ\/ZZ}$ one at the time from the Bayesian posterior distribution.\n\nIn Section~\\ref{sec:sigma_eval}, we discuss the significance of the different measurements and we show the evidence of the total diboson production. The previous version of this analysis~\\cite{cdfDiblvHF_Mjj2011}, performed with a dataset of $7.5$~fb$^{-1}$ and reported in Appendix~\\ref{App:7.5}, was the first evidence of this process at a hadron collider.\n\nIn general the result confirms the SM prediction for diboson production also when they are detected in the $HF$ final state. This provides additional confidence on the capability and the methods used by the CDF collaboration for the Higgs search in the $H\\to b\\bar{b}$ channel. \n\n\n\\section{Construction of the Likelihood Function}\\label{sec:likelihood}\n\nThe comparison of observed data with the templates derived from signal and background is possible with the use of a likelihood function in which we use a Bayesian approach to include the prior probabilities of the background and systematic effects.\n\nWe assume that the outcome of the data observation $n_{i,j}$, in the $j^{th}$ bin of the $i^{th}$ input histogram follows the Poisson statistics and the expectation value, $\\mu_{i,j}$, depends by the estimated backgrounds and signals. Therefore the complete likelihood function, $\\mathscr{L}$, of the bin by bin outcome is:\n\\begin{equation}\\label{eq:likelihood}\n  \\mathscr{L}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{n}, \\vec{\\beta}) =\\prod^{Hists}_{i} \\prod^{Bins}_{j} \\frac{e^{-\\mu_{i,j}}\\mu_{i,j}^{n_{i,j}}}{n_{i,j}!}\n\\end{equation}\nwhere the first product runs over the total number of input channels, $Hist$ (i.e. the different histograms provided for them), and the second runs on the bins of each distribution. The bin expectation value contains the signal and background dependence: \n\\begin{equation}\\label{eq:mu_likelihood}\n  \\mu_{i,j}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{\\beta}) = \\sum_p^{Sgn} \\alpha_{p} s_{i,j,p}(\\vec{\\beta}) + \\sum_q^{Bkg} b_{i,j,q}(\\vec{\\beta})\\textrm{,}\n\\end{equation}\nwhere $s_{i,j,p}$ represents the $p^{th}$ unknown signal, $b_{i,j,q}$ is the $q^{th}$ estimated background process and $\\alpha_{p}$ is the {\\em scaling parameter} used to measure the amount of the different signals. The unknown expectation of the signal is parametrized by flat, uniform, positive prior distribution in $\\vec{\\alpha}$. The last element of Equations~\\ref{eq:likelihood} and~\\ref{eq:mu_likelihood} is $\\vec{\\beta}$, the vector of the nuisance parameters: it incorporates in the likelihood the systematic uncertainties. They are parametrized as fractional variations on the $s_{i,j,p}$ and $b_{i,j,q}$ rates: \n\\begin{equation}\n   s_{i,j,p}(\\vec{\\beta})= s_{i,j,p}^{central}\\prod_{k}^{Sys}(1+\\frac{\\sigma_{i,j,k}}{s_{i,j,p}^{central}}\\beta_k)\n\\end{equation}\n\\begin{equation}\n   b_{i,j,p}(\\vec{\\beta})= b_{i,j,p}^{central}\\prod_{k}^{Sys}(1+\\frac{\\sigma_{i,j,k}}{b_{i,j,p}^{central}}\\beta_k)  \n\\end{equation}\nwhere $\\frac{\\sigma_{i,j,k}}{s_{i,j,p}^{central}}$, and $\\frac{\\sigma_{i,j,k}}{b_{i,j,p}^{central}}$ are the relative uncertainties related to the systematic effect $k$ and the $\\beta_k$ variables. It is important to notice that we distinguish:\n\\begin{itemize}\n\\item {\\em total rate uncertainties}, like the luminosity dependence of the signal, where there is no bin by bin variation of the systematic effect: $\\sigma_{i,j,k}\\equiv \\sigma_{i,k}$.\n\\item {\\em shape uncertainties} where the rate variation can change on a bin by bin basis, evaluated by the ratios between the central and the varied histograms.\n\\end{itemize}\nTo account for the nuisance parameters effect and correlate them across different channels we introduce in Equation~\\ref{eq:likelihood}, for each systematic, a Gaussian probability constraint centered in zero and with unitary variance. The final result is:\n\\begin{equation}\\label{eq:likelihood_full}\n  \\mathscr{L}(\\vec{\\alpha}, \\vec{s}, \\vec{b}| \\vec{n}, \\vec{\\beta}) =\\prod^{Hists}_{i} \\prod^{Bins}_{j} \\frac{e^{-\\mu_{i,j}}\\mu_{i,j}^{n_{i,j}}}{n_{i,j}!}\\prod_{k}^{Sys}e^{-\\beta_k^{2}\/2}\\textrm{,}\n\\end{equation}\nthe previous equation is the full likelihood used in the analysis of the results.\n\n\\subsection{Parameter Measurement and Likelihood Integration}\\label{sec:likelihood_meas}\n\nBy maximizing the likelihood, we obtain the {\\em measurement} of the unknown $\\vec{\\alpha}$.\n\nThe maximum of Equation~\\ref{eq:likelihood_full} can be found in different ways: with a fit in the multidimensional space of $\\vec{\\alpha}$ and $\\vec{\\beta}$ or with the integration ({\\em marginalization}) of the nuisance parameters over their prior probability distributions. \n\nThe second technique, exploited here for the measurements, returns a {\\em Bayesian posterior distribution} of $\\vec{\\alpha}$: the minimal extension that covers $68$\\% of the distribution around the maximum gives the one-standard-deviation confidence band and the uncertainty on the measurement.\n\nSection~\\ref{sec:cx_measure} reports the results of the marginalization, where the integration of the nuisance parameters is performed numerically with a Markov-chain adaptive integration~\\cite{mk-chain_book}.\n\nHowever we perform also a fit of the likelihood function because it gives the {\\em best} outcome for all the nuisance parameters. Those values, the result of the fit and the reduced $\\chi^2$, should be consistent with the results obtained by the marginalization.\n\n\n\n\\section{Sources of Systematic Uncertainties}\\label{sec:sys_desc}\n\nAlong the preceding Chapters, we discussed several sources of rate and shape systematics and we need to include them in the likelihood Equation~\\ref{eq:likelihood_full}. \n\nThe {\\em rate-only} systematic effects that we include are summarized in the following points:\n\\begin{description}\n %\n\\item{\\em Initial and Final State Radiation} uncertainties are estimated by changing (halving and doubling) the parameters related to ISR and FSR emission on the signal MC. Half of difference between the two shifted samples is taken as the systematic uncertainty on the signal samples. The total effect on the signal acceptance is about $4\\%$.\n\\item{\\em Parton Distribution Functions} uncertainties are evaluated by reweighting each event of the signal MC according to several PDFs parametrisation and to the generator level information of the event. Then the acceptance is evaluated again giving a rate variation with respect to the original PDFs. The exact procedure is described in~\\cite{pdf_note7051}. The effect on the signal acceptance was evaluated in previous similar analyses to be around $1\\div 2$\\%. It is added in quadrature with the ISR\/FSR systematic as they both influence only the signal acceptance.\n\\item{\\em $b$-tag Scale Factor} uncertainty  comes from the measured \\texttt{SecVtx} $b$-tag efficiency variation. The $SF_{b-Tag}$ is $0.96\\pm0.5$ for a single $b$-matched and \\texttt{SecVtx} tagged jet. The uncertainty, propagated through the per-event tagging probability, is assumed to be double in the case of $c$-jets. The uncertainty is applied on the signal and on all the EWK backgrounds described in Section~\\ref{sec:mc_bkg}.\n\\item{\\em Luminosity} measurement uncertainty contributes for an overall 6\\% rate uncertainty on the signal and on all the EWK backgrounds.\n\\item{ \\em Lepton Acceptance} uncertainty derives from the quadrature sum of trigger efficiency measurements and lepton ID Scale Factors. They range from $1 \\div 2$\\% for tight lepton categories (CEM, CMUP+CMX, PHX) to $6$\\% for the EMC leptons collected by the \\mbox{${\\not}{E_T}$}  plus jets triggers. These uncertainties are applied on signal and EWK samples.\n\\item {\\em Top Production} uncertainty  is a $10$\\% rate uncertainty applied on all the top-related processes ($t\\bar{t}$, single-top $s$ and $t$ channels). It covers the cross section theoretical prediction uncertainty and the acceptance differences due to systematic top quark mass variation. \n\\item {\\em QCD} normalization, independent for each lepton channel, is constrained to be within 40\\% of the value extracted in Section~\\ref{sec:qcd}.\n\\item{\\em Mistag} uncertainty is derived from the mistag matrix $\\pm 1\\sigma$ variation and propagated to the final $W+LF$ sample. The rate variations are $11$\\% and 21\\% for single and double-tagged channels respectively.\n\\item{\\em K-Factor} uncertainty is of 30\\% on $W+HF$ rate estimate. It also includes the $b$-tag SF uncertainty used to derive this correction. We consider an uncorrelated uncertainty for $W+c\\bar{c}\/W+b\\bar{b}$  and  $W+c$ backgrounds as the first process is produced by strong interaction while the latter is of electroweak nature.\n\\item {\\em $Z+$Jets} normalization uncertainty is, conservatively, set to $45$\\% as it includes the uncertainty on $Z+HF$ production.\n\\end{description}\nAppendix~\\ref{app:rate_sys} summarizes the systematic rate variation on the different templates.\n\nWe also account for {\\em shape} systematic variations from three sources: JES, $Q^2$ and KIT modeling.\n\\begin{itemize}\n\\item {\\em Jet Energy Scale} shape uncertainty (JES) is estimated  by shifting the JES of the input templates by $\\pm 1\\sigma$ from the nominal value. The acceptance of the process is allowed to change, therefore a new background estimate is performed with the JES varied templates. This produces a simultaneous rate and shape uncertainty. All the templates are affected except the fake-$W$ sample that is derived from data.\n\\item {\\em $W+$ Jets $Q^2$ Scale} uncertainty is obtained by halving and doubling the nominal generation $Q^{2}$ (defined in Equation~\\ref{eq:scale}) of all the $W+$ jets samples ($W+b\\bar{b}$, $W+c\\bar{c}$, $W+c$, $W+LF$). Again, we have a shape and rate uncertainty as a new background estimate is performed with the  varied templates.\n\\item {\\em KIT-NN} shape uncertainty is the last systematic that we take into account. We apply two kind of independent variations: one on the fake-$W$ templates and the other on $c$-quarks and mistag templates. \nFor the multi-jet template, a variation of the flavor composition is applied allowing more or less $b$-quark like events in the fake-$W$ KIT-NN template. For templates with relevant $c$-quark component  ($WW$, $WZ$, $ZZ$, $W+c$ and $W+c\\bar{c}$) and $W+LF$ a correction to the KIT-NN output is derived from negative tags in data. This systematic variation influences only the shape and not the rate of the final discriminant because it leaves untouched the background composition.\n\n\\end{itemize}\n\nIn the case of templates with few events passing all requirements\\footnote{The MC events can be spread across a large number of bins, especially in the bi-dimensional case where each template is composed by $50\\times 4$ bins.} large statistical fluctuations can introduce a bias in the evaluation of the shape systematics. Therefore, shape variations are filtered to reduce the statistical noise. Filters are widely employed in modern image processing for different purposes. We choose a {\\em median filter smoothing}~\\cite{median_filter} because it maintains long range correlations among the histogram bins. We use a 5-bin filter along the $M_{Inv}(jet1,jet2)$ direction for the bi-dimensional, single-tagged, templates while a 3-bin filter is used for double-tagged templates\\footnote{A double bin width is used for double-tagged $M_{Inv}(jet1,jet2)$ templates w.r.t. single-tagged ones as a 5-bin long range correlation would spoil the systematic variation.}. We also apply a low and high boundary to the possible template variations to reduce them in $[0.5,2.0]$ range.\nFigure~\\ref{fig:smoothing} shows an example of the double-tagged CEM lepton $WZ$ template before and after the smoothing filtering.\n\n\n\\begin{figure} [!h]\n\\centering\n\\includegraphics[width=0.4\\textwidth,height=0.35\\textwidth]{StatRes\/Figs\/noSmooth_SVTSVT_CEM}\\hspace{0.7cm}\n\\includegraphics[width=0.4\\textwidth,height=0.35\\textwidth]{StatRes\/Figs\/smooth_SVTSVT_CEM}\n\\caption[Example of $M_{Inv}(jet1,jet2)$ Template Before and After Smoothing]{Example of the double-tagged CEM lepton $WZ$ $M_{Inv}(jet1,jet2)$ template rate variation due to JES systematic before (left) and after (right) a 3-bin median filter smoothing~\\cite{median_filter}. The red (blue) line indicates the one sigma down (up) JES variation.}\\label{fig:smoothing}\n\\end{figure}\n\n\n\n\n\n\\section{$M_{Inv}(jet1,jet2)$ and KIT-NN Distributions}\\label{sec:mjj_kit_distrib}\n\nThe final templates, including all the systematic effects, are then used to build the likelihood of Equation~\\ref{eq:likelihood_full}. \n\nWe combine a total of eight different channels: {\\em four} lepton sub-samples (CEM, PHX, Tight Muons, EMC) times {\\em two} $b$-tag prescriptions (single and double \\texttt{SecVtx} tags). For the double \\texttt{SecVtx} tagged events, the signal discrimination is based only on the di-jet invariant mass, $M_{Inv}(jet1,jet2)$, while for the single-tagged events we exploit the bi-dimensional distribution of $M_{Inv}(jet1,jet2)$ {\\em vs} KIT-NN flavor separator. The KIT-NN output ranges from $-1$ to $1$ and it is divided in four equal size bins: the rightmost is highly enriched in $b$-like jets, while the others have variable composition of $b$-like, $c$-like and $LF$-like jets. \n\nAs noted in Section~\\ref{sec:likelihood_meas}, we first obtain the maximum value of the likelihood with a fit. The  best fit result allows to check the agreement of the predicted distribution with data. The reduced $\\chi^2$ of the fit is:\n \\begin{equation}\n   \\frac{\\chi^2}{ NDoF}=\\frac{744.4}{664} = 1.12\\textrm{,}\n \\end{equation}\ncorresponding to a probability $P=0.984$. To further investigate the agreement of data and prediction, several post-fit distributions are shown from Figure~\\ref{fig:mjj_2tag_nokit} to~\\ref{fig:mjj_1tag_allkit}. We evaluate on them both the $\\chi^2$ the Kolmogorov-Smirnov tests.\n\\begin{itemize}\n\\item Figure~\\ref{fig:mjj_2tag_nokit} shows the  $M_{Inv}(jet1,jet2)$ distribution for double-tagged events;\n\\item Figure~\\ref{fig:kit_1tag} shows the KIT-NN distribution for single-tagged events after the integration of all the $M_{Inv}(jet1,jet2)$ values used in the bi-dimensional distribution;\n\\item Figure~\\ref{fig:mjj_1tag_allkit} shows the $M_{Inv}(jet1,jet2)$ distribution  for the single-tagged channel, added for all the lepton categories and integrated across all the KIT-NN values as well as the two most interesting KIT-NN regions: KIT-NN$>0.5$, $b$-enriched, and KIT-NN$<0.5$ with contribution from $c$ and $LF$ quarks.\n\\end{itemize}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_DT_TIG_EMC_PHX}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for 2 \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for candidate events with two \\texttt{SecVtx} tags, all the lepton categories have been added together. The best fit values for the rate and shape of the backgrounds are used in the figure. }  \n\\label{fig:mjj_2tag_nokit}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_ST_MjjKitProj_TIG_EMC_PHX}\n\\caption[KIT-NN Distribution for 1 \\texttt{SecVtx} Tag Events]{KIT-NN distribution for candidate events with a single \\texttt{SecVtx} tag, all the lepton categories have been added together. The best fit values for the rate and shape of the backgrounds are used in the figure.}  \n\\label{fig:kit_1tag}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitAll_TIG_EMC_PHX}\\\\\\vspace{0.5cm}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitLow_TIG_EMC_PHX}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/AddSelected_ST_KitHigh_TIG_EMC_PHX}\n\\caption[$M_{Inv}(jet1,jet2)$ Distribution for 1 \\texttt{SecVtx} Tag Events]{$M_{Inv}(jet1,jet2)$ distribution for candidate events with a single \\texttt{SecVtx} tag, all the lepton category have been added together. Top distribution shows the results for all KIT-NN values while the bottom distributions are separated for KIT-NN$<0.5$ (bottom left) and KIT-NN$>0.5$ (bottom right). The best fit values for the rate and shape of the backgrounds are used in the figures.}\n\\label{fig:mjj_1tag_allkit}\n\\end{figure}\n\n\n\n\n\n\n\\section{Cross Section Measurement}\\label{sec:cx_measure}\n\nThe actual cross section measurement is performed by marginalizing the likelihood with respect to the nuisance parameters and studying the resulting Bayesian posterior as a function of the diboson signal cross section.\n\nFirst we measure the total diboson cross section, $\\sigma_{Diboson}^{Obs}$, constraining the relative  $WW$ and $WZ\/ZZ$ cross sections to the SM prediction. The resulting Bayesian posterior distribution is shown in Figure~\\ref{fig:posterior_dib} together with the 68\\% and 95\\% confidence intervals. The measured cross section is:\n\\begin{equation}\n \\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the errors include statistical and systematic uncertainties and $\\sigma^{SM}_{Diboson}$ is the SM predicted cross section derived from Table~\\ref{tab:mc_dib}:\n\\begin{equation}\n  \\sigma_{Diboson}^{SM}= \\sigma^{SM}_{WW}+ \\sigma^{SM}_{WZ\/ZZ} = \\left( 18.43\\pm0.73\\right)\\textrm{~pb.}\n\\end{equation}\n\nIn order to separate the different components, we exploit the $c$ versus $b$ classification power of KIT-NN and the different sample composition of single and double-tagged events to obtain a separate measurement of $WW$ and $WZ\/ZZ$. \n\nWe iterate the cross section measurement procedure but, this time, $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ are left free to float independently (i.e. not constrained to the SM ratio). Figure~\\ref{fig:posterior_2d} shows the resulting Bayesian posterior distribution. The maximum of the posterior distribution gives the value of the measured cross sections:\n\\begin{equation}\n\\sigma^{Obs,2D}_{WW}=\\left(0.50^{+0.51}_{-0.46}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.7^{+5.8}_{-5.2}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\sigma^{Obs,2D}_{WZ\/ZZ}=\\left(1.56^{+1.22}_{-0.73}\\right)\\times\\sigma^{SM}_{WZ\/ZZ}= \\left(11.1^{+8.7}_{-6.3}\\right)\\textrm{~pb,}\n\\end{equation}\nwhere the SM predictions are $\\sigma^{SM}_{WW} = 11.34\\pm0.66$~pb, $\\sigma^{SM}_{WZ\/ZZ} =7.09\\pm0.30$~pb and the errors are evaluated by the intersection of the $x$ and $y$ position of the maximum with the boundary of the smallest area enclosing the 68\\% of the posterior distribution. The smallest areas enclosing 68\\%, 95\\% and 99\\% of the posterior integrals give the contours of one, two and three standard deviations and are explicitly shown in Figure~\\ref{fig:posterior_2d} with correlation between $\\sigma^{Obs,2D}_{WW}$ and  $\\sigma^{Obs,2D}_{WZ\/ZZ}$.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{StatRes\/Figs\/1d_posterior_blessSyle}\n\\caption[Bayesian Posterior Distribution of $\\sigma_{Diboson}$ Measurement]{The Bayesian posterior, marginalized over nuisance parameters, is shown. The maximum value is the central value of the cross-section, the blue and azure areas represent the smallest intervals enclosing 68\\%, 95\\% of the posterior integrals, respectively. The final cross section measurement is\n\\mbox{$\\sigma_{Diboson}^{Obs}=\\left(0.79\\pm 0.28\\right)\\times \\sigma_{Diboson}^{SM} = \\left(14.6\\pm 5.2\\right)\\textrm{~pb}$}.}\n\\label{fig:posterior_dib}\n\\end{figure}\n\nThe $WW$ and $WZ\/ZZ$ channels are also analyzed separately by projecting the two-dimensional Bayesian posterior on the $\\sigma_{WW}$ and the $\\sigma_{WZ\/ZZ}$ axes. In this way, the two processes are considered as background one at the time. For both $WW$ and $WZ\/ZZ$ we re-computed the maximum values and confidence intervals. Figure~\\ref{fig:posterior_2d_proj} shows the results, the measured cross sections are:\n\\begin{equation}\n\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}\n\\end{equation}\nand\n\\begin{equation}\n  \\quad \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb,}    \n\\end{equation}\nwhere the errors include statistical and systematic uncertainties.\n\n\\begin{figure}[h!]\n  \\centering\n\\includegraphics[width=0.99\\textwidth]{StatRes\/Figs\/2d_posterior}\n\\caption[2-Dim Bayesian Posterior Distribution of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ Combined Measurement]{The Bayesian posterior, marginalized over nuisance parameters (scaled to SM expectation), is shown in the plane $\\sigma_{WW}$ {\\em vs} $\\sigma_{WZ\/ZZ}$. The measured cross sections correspond to the maximum value of $\\sigma^{Obs,2D}_{WW}=0.50\\times\\sigma^{SM}_{WW} = 5.7\\textrm{~pb}$ and $\\sigma^{Obs,2D}_{WZ\/ZZ}=1.56\\times\\sigma^{SM}_{WZ\/ZZ}= 11.1\\textrm{~pb}$. The red, blue and azure areas represent smallest areas enclosing 68\\%, 95\\% and 99\\% of the posterior integrals, respectively.}\n\\label{fig:posterior_2d}\n\\end{figure}\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/1d_WW_posterior_blessSyle}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/1d_WZZZ_posterior_blessSyle}\n\\caption[Bayesian Posterior Distributions of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ Separate Measurements]{The Bayesian posterior, function of $\\sigma_{WW}$ and $\\sigma_{WZ\/ZZ}$ marginalized over nuisance parameters, is shown after projection on the $\\sigma_{WW}$ (left) and $\\sigma_{WZ\/ZZ}$ (right) axes.\nThe maximum value is the central value of the cross-section, the blue and azure areas represent the smallest intervals enclosing 68\\%, 95\\% of the posterior integrals, respectively. The final cross section measurements are: \\mbox{$\\sigma^{Obs}_{WW}=\\left(0.45^{ +0.35}_{ -0.32}\\right)\\times\\sigma^{SM}_{WW} = \\left(5.1^{+4.0}_{-3.6}\\right)\\textrm{~pb}$} and \\mbox{$\\quad \\sigma^{Obs}_{WZ\/ZZ}=\\left(1.64^{+0.83}_{-0.78}\\right)\\times\\sigma^{SM}_{WZ\/ZZ} = \\left(11.6^{+5.9}_{-5.5}\\right)\\mathrm{~pb}$}.}\\label{fig:posterior_2d_proj}\n\\end{figure}\n\n\n\\section{Evaluation of the Statistical Significance}\\label{sec:sigma_eval}\n\nTo compute the significance of the measurements we perform a {\\em hypothesis test} comparing data observation to the {\\em null hypothesis} ($H_0$).\n\nRandom generated Pseudo Experiments (PEs) are extracted from the predicted background processes distribution in the $H_0$ hypothesis (i.e. excluding  the diboson production): this is straightforward once we know the probability distribution of the background and of the nuisance parameters. Then we repeat the cross section measurements with the complete marginalization of the likelihood. We expect a distribution peaking at $\\sigma_{Diboson}^{PEs}\/\\sigma_{Diboson}^{SM} =0$ and we compare it to the measured cross section.\n\nFigure~\\ref{fig:significance_dib} shows the possible outcomes of many cross section PEs in a back\\-ground-only and in a background-plus-signal hypothesis. The number of times that a background fluctuation produces a cross section measurement greater than \\mbox{$\\sigma^{Obs}_{Diboson}=0.79\\times\\sigma^{SM}_{Diboson}$} has a $p-$value of $0.00209$. \n\nThe result is an evidence for the diboson production in $\\ell\\nu +HF$ final state with a significance\\footnote{A two sided significance estimate is used because both upper and lower fluctuation of the cross section measurement are considered in the integral of the {\\em null} hypothesis cross section distribution above the measured value.} of $3.08\\sigma$.\n\nThen, we evaluate the single $WW$ and $WZ\/ZZ$ significances in a similar way: PEs are generated with null hypothesis for both $WW$ and $WZ\/ZZ$ signals. Then, the cross section PEs measurements are projected along the $\\sigma_{WW}$ vs $\\sigma_{WZ\/ZZ}$ axes and compared with $\\sigma^{Obs}_{WZ}$ and $\\sigma^{Obs}_{WZ\/ZZ}$. The result of the $p-$value estimates are shown in Figure~\\ref{fig:significance_2d_proj}. \n\nWe obtain: $p-$value$_{WW}=0.074565$ and $p-$value$_{WZ\/ZZ}=0.011145$. They correspond to a significance of $1.78\\sigma$ and $2.54\\sigma$ for $WW$ and $WZ\/ZZ$ respectively.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{StatRes\/Figs\/significance_diboson}\n\\caption[Diboson Significance Evaluation]{Possible outcomes of many diboson cross section measurements from Pseudo Experiments (PEs) generated in a background-only and in a background-plus-signal hypothesis. The $p-$value for $\\sigma^{Obs}_{Diboson}=0.79\\times\\sigma^{SM}_{Diboson}$ is $0.00209$, corresponding to a significance of $3.08\\sigma$.}\n\\label{fig:significance_dib}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/significance_WW}\n\\includegraphics[width=0.495\\textwidth]{StatRes\/Figs\/significance_WZZZ}\n\\caption[$WW$ and $WZ$ Separate Significance Evaluation]{Possible outcomes of many diboson cross section measurements from Pseudo Experiments (PEs) generated in a background-only and in a background-plus-signal hypothesis in the $\\sigma_{WW}$ vs $\\sigma_{WZ\/ZZ}$ plane and then projected on the  $\\sigma_{WW}$ (left) and $\\sigma_{WZ\/ZZ}$ (axis). The $p-$values of $0.074565$ and $0.011145$ correspond to a significance of $1.78\\sigma$ and $2.54\\sigma$ for $WW$ and $WZ\/ZZ$ respectively.}\n\\label{fig:significance_2d_proj}\n\\end{figure}\n\n\n\n\\chapter{Theoretical and Experimental Overview}\\label{chap:physics}\n\n\nThe goal of particle physics is the understanding of the principles of Nature.\n\nThe quest is pursued through the scientific method: the observation of a phenomenon is explained by a hypothesis that must be, successively, verified or rejected by experimental evidences.\n\nIn this prospect, the observed phenomenon is the existence itself of the atomic and sub-atomic structure of matter, the hypothesis is the Standard Model theory of Elementary Particles and Fundamental Interaction (SM) while the experimental tools are the high energy physics colliders and detectors, available nowadays.  \n\nThe main infrastructure of the SM~\\cite{pdg2010, glasgow, weinberg_lep} was developed in the 70's and, since then, it showed to be a very successful theory. One of its main success was the prediction of new elementary particles, later observed at hadron collider experiments. The discovery of the $W$ and $Z$ force carrier vector bosons~\\cite{ua1_w_obs, ua2_w_obs, ua1_z_obs,ua2_z_obs} and of the $top$ quark~\\cite{CDF_top_obs,d0_top_obs} shed light on the fundamental structure of the matter. \n\nStrengthened by these results, SM describes the {\\em electromagnetic}, {\\em weak} and {\\em strong} interactions, three of the four fundamental forces that compose the physics description of Nature. The fourth force, gravitational interaction, is left out but it is negligible at atomic and subatomic scale. \n\nDespite the great success of the SM, one predicted particle has not yet been observed: the \\emph{Higgs boson}, an essential element for the inclusion of the mass of the particles in the equation of motion~\\cite{higgs1964,englert1964}. Because of this, a considerable effort is ongoing to prove or disprove the existence of the Higgs boson\\footnote{Some hints of its existence are confirmed by the present experiments~\\cite{tev_cmb, atlas_cmb, cms_cmb}.} and the analysis of the data collected by the CDF II experiment, situated at the Tevatron $p\\bar{p}$ collider, plays a relevant role in it~\\cite{cdf_cmb}.\n\nIn the scenario~\\cite{Tev4LHC} of a low-mass Higgs ($m_H\\lesssim 135$~GeV$\/c^{2}$), one of the most relevant CDF search channels is the $WH$ associate production with a $\\ell\\nu+b\\bar{b}$ final state. In light of this, the  diboson decay channel considered for the presented analysis becomes a perfect benchmark for the Higgs boson search. The accurate SM prediction for the diboson production and decay can be used as a standard comparison for an unknown process. \n\nThis Chapter introduces the relevant aspects of the SM theory (Section\\ref{sec:sm_teo}), some of the latest results of the Higgs boson search (Section\\ref{sec:higgs_search}) and several, diboson related, experimental confirmation of the SM validity (Section\\ref{sec:dib_res}).\n\n\n\n\n\\section{The Standard Model of Elementary Particles}\\label{sec:sm_teo}\n\nThe SM is defined by the language of mathematics and theoretical physics so that it can be used to produce accurate predictions that have to be verified by the experiments. \nIn this language a particle is defined by a local quantum field. If no interaction is present, the free field is described by only two quantum numbers, the spin and the mass; if interactions are presents, the {\\em Gauge symmetries} can elegantly describe them: new quantum numbers classify the type and the strength of force while new particles, force-mediator vector bosons, are used to propagate the interaction.\n\nThe fundamental building blocks of matter, observed up to now, are the spin-$1\/2$ fields (\\emph{fermionic}), named \\emph{quarks} and \\emph{leptons}, and the spin-$1$ ({\\em vector}) fields, named \\emph{gauge bosons}. The leptons are divided into three \\emph{generations}, or \\emph{families}, and are\ngrouped in a left weak isospin doublet\\footnote{See Section~\\ref{sec:SM} for \nthe explanation of the weak isospin quantum number.} and a right weak isospin singlet. Also quarks are divided into three \\emph{flavor families} but weak \nisospin classification mixes quark doublets of different families. Quarks are also subject to the strong interaction, described by the \\emph{color} quantum number. Finally the {\\em charge} quantum number is used, for both quarks and leptons, to describe the electromagnetic interaction. \n\nThe force mediators are $W^\\pm$, $Z^0$, $\\gamma$, that carry electroweak \nforce, and $g$ (\\emph{gluons}), which mediate strong interaction. A short \nsummary of the SM fundamental particles is reported in Figure~\\ref{fig:particles}.\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Theory\/Figs\/SM_Particles}\n\\caption[SM Particles and Their Properties]{Quarks, leptons and gauge bosons in Standard Model and some of their \ncharacteristics~\\cite{pdg2010}, for each particle the corresponding antiparticle exists.}\\label{fig:particles}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Gauge Theory Example: QED}\n\nThe Quantum Electro-Dynamic (QED) is a perfect example to explain the importance of gauge invariance.\n\nThe equation that describes free fermionic fields is the {\\em Dirac} Lagrangian:\n\\begin{equation}\\label{eq:dirac}\n\\mathscr{L}(x)=\\bar{\\psi}(x)(i\\gamma^{\\mu}\\partial_{\\mu}- m)\\psi(x),\n\\end{equation}\nwhere $\\psi$ is the Dirac field of mass $m$ and $\\gamma^{\\mu}$ are the Dirac's \nmatrices. Equation~\\ref{eq:dirac} satisfies the \\emph{global} $U(1)$ symmetry\ntransformation: \n\\begin{equation}\n\\psi(x)\\to e^{iQ\\alpha}\\psi(x),\n\\end{equation}\nwith the electric charge $Q$ and the space independent parameter $\\alpha$ ($x$ is a space-time 4-vector).\nThe \\emph{Noether theorem}~\\cite{qftbook_weinberg} states that when a symmetry \nappears in a Lagrangian there is a corresponding conserved current. In the case of the Dirac field:\n\\begin{equation}\n\\partial_{\\mu}j^{\\mu} = 0\\quad\\mathrm{with}\\quad j^{\\mu} = -Q\\bar{\\psi}\\gamma^{\\mu}\\psi,\n\\end{equation}\ndescribes the conservation of charge, i.e the time component of the current 4-vector $j^{\\mu}$, integrated over the space, is a constant.\n\nAn elegant way to introduce interaction in the free Lagrangian is to shift from\nthe global, i.e. space independent, $U(1)$ transformation to a \\emph{local} \n$U(1)$ transformation, i.e. with a space dependent parameter $\\alpha(x)$:\n\\begin{equation}\n\\psi(x)\\to e^{iQ\\alpha(x)}\\psi(x).\n\\end{equation}\nTo maintain the gauge invariance condition in the Lagrangian~\\ref{eq:dirac}, a \ncovariant derivative $D_{\\mu}$ is introduced:\n\\begin{equation}\\label{eq:em_derivative}\n\\partial_{\\mu}\\to D_{\\mu}=\\partial_{\\mu}+iQA_{\\mu}, \\qquad\nD_{\\mu}\\psi(x)\\to e^{iQ\\alpha(x)}D_{\\mu}\\psi(x),\n\\end{equation}\nwhere the new vector field $A_{\\mu}$ is defined to transform in the following manner:\n\\begin{equation}\nA_{\\mu}\\to A_{\\mu}-\\frac{1}{Q}\\partial_{\\mu}\\alpha(x)\\mathrm{.}\n\\end{equation}\nEquations~\\ref{eq:em_derivative} and~\\ref{eq:dirac} can be composed to give the final QED Lagrangian:\n\\begin{equation}\\label{eq:L_qed}\n\\mathscr{L}_{QED}=\\bar{\\psi}(x)(i\\gamma^{\\mu}D_{\\mu}- m)\\psi(x) - \\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu},\n\\end{equation}\nwhere $F_{\\mu\\nu}\\equiv\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$ is the covariant kinetic term of $A_{\\mu}$.\nThe Dirac equation of motion for a field $\\psi$ undergoing electromagnetic interaction is obtained by applying the Euler-Lagrange equation~\\cite{qftbook_weinberg} to the QED Lagrangian:\n\\begin{equation}\n(i\\gamma^{\\mu}\\partial_{\\mu}- m)\\psi(x) = Q\\gamma^{\\mu}A_{\\mu}\\psi(x)\\mathrm{,}\n\\end{equation}\nthe force is mediated by the massless vector field $A_{\\mu}$. A mass term in \nthe form $\\frac{1}{2}m^{2}A_{\\mu}A^{\\mu}$ would break apart gauge invariance of\nEquation~\\ref{eq:L_qed}, indeed this is consistent with zero mass of the photon.\n\n\n\n\\subsection{Standard Model Theory}\\label{sec:SM}\n\nThe leptonic sector of the SM\\footnote{Only electroweak interaction on the leptons is considered here to simplify the discussion.} is based on the gauge group:\n\\begin{equation}\n  SU(2)\\otimes U(1)\\textrm{,}  \n\\end{equation}\nwhere $SU(2)$ is the non-Abelian group used in the spin algebra, and $U(1)$ is the Abelian group equivalent to the one used in QED. \nThe quantum number arising from $SU(2)$ is the \\emph{weak isospin}, $\\vec{T}$, and the one arising from $U(1)$ is {\\em hypercharge}, $Y$. They are related to the observed charge of real particles, $Q$, by the \nthe Gell-Mann-Nishijima equation:\n\\begin{equation}\nQ = T_{3}+\\frac{Y}{2}\\mathrm{.}\n\\end{equation}\nwhere $T_{3}$ is the third component of weak isospin.\n\nElectroweak interaction can be explained with a simplified model containing only two spin $1\/2$, elementary, massless, fermions, $f$ and $f'$, such that \n$Q_{f}=Q_{f'}+1$ ($Q$ is the electric charge). Weak interaction is built from V-A currents, i.e. left and right components are \ndefined and collected into a left doublet field and into two right singlet fields:\n\\begin{equation}\\label{eq:smL}\n\\psi_{1} \\equiv \\left( \\begin{array}{c} f_{L}(x) \\\\ f'_{L}(x) \\end{array}\\right), \\qquad \\psi_{2} \\equiv f_{R}(x) \n\\qquad \\psi_{3} \\equiv f'_{R}(x),\n\\end{equation}\nwith:\n\\begin{equation}\nf_{L,R}(x)=\\frac{1}{2}(1\\pm\\gamma_{5})f(x), \\qquad \n\\bar{f}_{L,R}(x)=\\frac{1}{2}\\bar{f}(x)(1\\pm\\gamma_{5}),\n\\end{equation}\n\\begin{equation}\nf'_{L,R}(x)=\\frac{1}{2}(1 \\pm \\gamma_{5})f'(x), \\qquad \n\\bar{f'}_{L,R}(x)=\\frac{1}{2}\\bar{f'}(x)(1 \\pm \\gamma_{5}).\n\\end{equation}\nAll the leptonic sector of the SM is explained by such pattern: the left doublet with $T_{3}=\\pm 1\/2$, $Y = 1$ is the charged lepton $f$\nplus the corresponding neutrino $f'$, while the right singlet with $T_{3}=0$, $Y = -2$ is only the charged lepton.\n\nThe electroweak interaction is introduced through $SU(2)\\otimes U(1)$ gauge transformation:\n\\begin{equation}\n\\psi_{j}(x)\\to\\psi'_{j}(x)= e^{i\\frac{\\tau}{2}\\cdot\\vec{\\alpha}(x) + iY_{j}\\beta(x)}\\psi_{j}(x),\n\\end{equation}\nof the free field Lagrangian:\n\\begin{equation}\\label{eq:freeL}\n\\mathscr{L_{0}}(x)=\\sum_{j=1}^{3}i\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\partial_{\\mu}\\psi_{j}(x)\\textrm{,}\n\\end{equation}\nwhere a covariant derivative is also introduced to maintain gauge invariance. The result is:\n\\begin{equation}\\label{eq:SML}\n\\mathscr{L}_{I}(x)=\\sum_{j=1}^{3}i\\bar{\\psi}_{j}(x)\\gamma^{\\mu}D_{\\mu}^{j}\\psi(x)_{j} - \\frac{1}{4}\\vec{W}_{\\mu,\\nu} \\vec{W}^{\\mu,\\nu} - \\frac{1}{4} B_{\\mu,\\nu}B^{\\mu,\\nu},\n\\end{equation}\n\\begin{equation}\\label{eq:bosons}\n\\mathrm{with}\\qquad D_{\\mu}^{j}= \\partial_{\\mu}-ig\\frac{\\tau}{2}\\cdot\\vec{W}_{\\mu}(x)-ig'Y_{j}B_{\\mu}(x)\\mathrm{,}\n\\end{equation}\n\\begin{equation}\\label{eq:tensor}\n\\mathrm{and}\\quad \\vec{W}_{\\mu,\\nu}=\\partial_{\\mu}\\vec{W}_{\\nu} - \\partial_{\\nu} \\vec{W}_{\\mu} + g \\vec{W}_{\\mu}\\times\\vec{W}_{\\nu}\\textrm{,}\\quad B_{\\mu,\\nu}= \\partial_{\\mu}B_{\\nu} - \\partial_{\\nu} B_{\\mu}.\n\\end{equation}\nEquation~\\ref{eq:bosons} contains three vector bosons ($\\vec{W}_{\\mu}$) from the $SU(2)$ \ngenerators, one vector boson ($B_{\\mu}$) from the $U(1)$ generator and four coupling constants: \n\\begin{equation}\n  g,\\quad g'Y_{j} \\quad \\mathrm{with}\\quad j=1,2,3\\mathrm{.}\n\\end{equation}\nAfter some algebra the Lagrangian~\\ref{eq:SML} can be written in the form:\n\\begin{equation}\n\\mathscr{L}_{I}(x)= \\mathscr{L}_{CC}(x) + \\mathscr{L}_{NC}(x)\\mathrm{,}\n\\end{equation}\nwith a {\\em charged current} contribution ($\\mathscr{L}_{CC}$) and a {\\em neutral\ncurrent contribution} ($\\mathscr{L}_{NC}$). The charged current contribution is seen only by left doublet fields:\n\\begin{equation}\\label{eq:W}\n\\mathscr{L}_{CC}(x)=\\frac{g}{2\\sqrt{2}}\\Big\\{\\bar{f}(x)\\gamma^{\\mu}(1-\\gamma_{5})f'(x)\\frac{1}{\\sqrt{2}}W_{\\mu}^{+}(x) + h.c.\\Big\\},\n\\end{equation}\nwith $W_{\\mu}^{+}(x)$ defined by a linear combination of $W_{\\mu}^{1}(x)$ and $W_{\\mu}^{2}(x)$. Equation~\\ref{eq:W} defines the Lagrangian for charged current \ninteractions mediated by the $W$ boson.\n\nThe fermion coupling to $Z^{0}$ field and photon ($A$) field is produced in a similar way, by an appropriate orthogonal linear combination of neutral vector fields $B_{\\mu}(x)$\nand $W_{\\mu}^{0}(x)$:\n\\begin{equation}\n\\mathscr{L}_{NC}(x)= \\mathscr{L}_{NC}^{A}(x) + \\mathscr{L}_{NC}^{Z}(x)\\textrm{,}\n\\end{equation}\nwhere:\n\\begin{eqnarray}\n\\mathscr{L}_{NC}^{A}(x)=\\sum^{3}_{j=1}\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\big[g\\frac{\\tau_{3}}{2}\\sin \\theta_{W}\n+ g'Y_{j}\\cos \\theta_{W}\\big]\\psi_{j}(x)A_{\\mu}(x),\\\\\n\\mathscr{L}_{NC}^{Z}(x)=\\sum^{3}_{j=1}\\bar{\\psi}_{j}(x)\\gamma^{\\mu}\\big[g\\frac{\\tau_{3}}{2}\\cos \\theta_{W}\n+ g'Y_{j}\\sin \\theta_{W}\\big]\\psi_{j}(x)Z_{\\mu}(x),\n\\end{eqnarray}\nthe parameter $\\theta_{W}$ is named Weinberg angle and the generic four coupling constants, arising from the SM group structure, have now a physical meaning:\n\\begin{equation}\ng\\sin \\theta_{W} = e,\n\\end{equation}\n\\begin{equation}\ng'\\cos \\theta_{W}Y_{1}=e(Q_{f}-1\/2), \\qquad g'\\cos \\theta_{W}Y_{2}=eQ_{f},\n\\end{equation}\n\\begin{equation}\ng'\\cos \\theta_{W}Y_{3}=eQ_{f'}\\textrm{.}\n\\end{equation}\nPrevious equations are the core of the Standard Model. However one problem remains as no mass term appears for any of the fields: the spontaneous symmetry breaking and Higgs mechanism can generate the mass term without breaking the gauge invariance.\n\n\n\\subsection{Spontaneous Symmetry Breaking}\n\nSpontaneous symmetry breaking can be applied to Equation~\\ref{eq:smL} to give mass to $W^{\\pm}$ and $Z^{0}$ bosons. The actual application procedure is named {\\em Higgs mechanism}: two complex scalar fields are introduced such that they form an iso-doublet with respect to $SU(2)$:\n\\begin{equation}\\label{eq:phi_doublet}\n\\phi(x)\\equiv\\left(\\begin{array}{c} \\phi^{+}(x) \\\\\\phi^{0}(x) \\end{array}\\right),\n\\end{equation}\nthe field $\\phi^{+}(x)$ is the charged component of the doublet and $\\phi^{0}(x)$ is neutral component. The Higgs potential, $ V_{H}(x)$, is then defined as:\n\\begin{equation}\\label{eq:h_pot}\n V_{H}(x)\\equiv -\\mu^{2}\\phi^{\\dagger}(x)\\phi(x)-h\\big[\\phi^{\\dagger}(x)\\phi(x)\\big] ^{2},\n\\end{equation}\nwith $h>0$ and $\\mu^{2}<0$. The neutral scalar field $\\phi^{0}(x)$ has an unconstrained (i.e. to be obtained from measurements) vacuum expectation value of $\\frac{\\lambda}{\\sqrt{2}}$, so that (at first order)\nthe field  $\\phi(x)$ is:\n\\begin{equation}\\label{eq:doublet}\n\\phi(x)=e^{\\frac{i}{\\lambda}\\vec{\\tau}\\cdot\\vec{\\theta(x)}}\\left(\\begin{array}{c} 0 \\\\\\frac{1}{\\sqrt{2}}\\big(\\lambda + \\chi(x)\\big) \\end{array}\\right),\n\\end{equation}\nwhere the $SU(2)$ gauge freedom is explicit. This permits to \\emph{gauge away} three of the four components of field $\\phi(x)$ leaving only one real scalar field: \n\\begin{equation}\\label{eq:}\n  \\phi^{0}(x)= \\frac{1}{\\sqrt{2}}\\big(\\lambda + \\chi(x)\\big).\n\\end{equation}\nThe explicit evaluation of Equation~\\ref{eq:h_pot} and the coupling of $\\phi^{0}(x)$ with the electroweak force carriers ($W^{\\pm}$, $Z^0$) gives the last piece of the SM Lagrangian:\n\\begin{eqnarray}\\label{eq:finalL}\n\\mathscr{L}(x)&=& \\frac{1}{4}g^{2}\\lambda^{2}W_{\\mu}^{\\dagger}(x)W^{\\mu}(x) + \\frac{1}{1}(g^{2}+g'^{2})\\lambda^{2}Z_{\\mu}(x)Z^{\\mu}\\\\\n&&+\\frac{1}{2}g^{2}\\lambda W^{\\dagger}_{\\mu}(x)W^{\\mu}(x)\\chi(x)+\\frac{1}{4}g^{2}W^{\\dagger}_{\\mu}W^{\\mu}\\chi^{2}(x)\\nonumber\\\\\n&&+\\frac{1}{4}(g^{2}+g'^{2})\\lambda Z_{\\mu}(x)Z^{\\mu}(x)\\chi(x)+\\frac{1}{8}g^{2}Z_{\\mu}(x)Z^{\\mu}(x)\\chi^{2}(x)\\nonumber\\\\\n&&+\\frac{1}{2}\\big[\\partial^{\\mu}\\chi(x)\\partial_{\\mu}\\chi(x) + 2\\mu^{2}\\chi^{2}(x)\\big]\\nonumber\\\\\n&&+\\frac{\\mu^{2}}{\\lambda}\\chi^{3}(x)+\\frac{\\mu^{2}}{4\\lambda^{2}}\\chi^{4}(x)-\\frac{1}{4}\\lambda^{2}\\mu^{2}\\nonumber\\textrm{.}\n\\end{eqnarray}\nWe conclude that the $Z^{0}$ and $W^{\\pm}$ bosons have acquired mass:\n\\begin{equation}\n  M_{W}=\\frac{1}{2}\\lambda g,  \n\\end{equation}\n\\begin{equation}\n  M_{Z}=\\frac{1}{2}\\lambda \\sqrt{g^{x}+g'^{x}}=\\frac{1}{2}\\frac{\\lambda g}{\\cos \\theta_{w} },  \n\\end{equation}\nsome parameters are now constrained, for example:\n\\begin{equation}\n  M_{Z}=\\frac{M_{W}}{cos\\theta{w}}\\geqslant M_{W},\n\\end{equation}\n\\begin{equation}\n  \\frac{G_{F}}{\\sqrt{2}}=\\frac{g^{2}}{8M^{2}_{W}},\n\\end{equation}\nwhile the Higgs mass, $M_{\\chi}=\\sqrt{-2\\mu^{2}}$ ($m_{H}$ is also used), remains a free parameter to be measured by the experiments. \nThe Higgs mechanism can generate also fermion masses if a Yukawa coupling is added:\n\\begin{eqnarray}\\label{eq:h_yukawa}\n\\mathscr{L_{f}}(x)&=&c_{f'}\\Bigg[(\\bar{f}(x),\\bar{f}'(x))_{L} \\left(\\begin{array}{c} \\phi^{+}(x) \\\\\\phi^{0}(x) \\end{array}\\right)\\Bigg]f_{R}'(x)\\\\\n&&+ c_{f}\\Bigg[(\\bar{f}(x),\\bar{f}'(x))_{L} \\left(\\begin{array}{c} -\\bar{\\phi}^{0}(x) \\\\\\phi^{-}(x) \\end{array}\\right)\\Bigg]f_{R}(x) + h.c.\\mathrm{,}\\nonumber\n\\end{eqnarray}\ntherefore, after symmetry breaking, fermion masses have the form:\n\\begin{equation}\nm_{f}=-c_{f}\\frac{\\lambda}{\\sqrt{2}}\\mathrm{,}\\qquad m_{f'}=-c_{f'}\\frac{\\lambda}{\\sqrt{2}}\\mathrm{,}\n\\end{equation}\nwhere the constants $c_{f}$ and $c_{f'}$ can be derived by the measurements of the fermion masses.\n\n\n\n\n\\section{Higgs Boson Search and Results}\\label{sec:higgs_search}\n\nThe mechanism that generates the mass of all the SM particles is a key element for the understanding of Nature, therefore it is not a surprise that the Higgs boson search is considered, by the High Energy Physics community, one of the most interesting research topics. \n\nAlthough the existence of the Higgs particle is unknown, its hypothetical couplings and decay properties are important for the interpretation of the experimental results: Figure~\\ref{fig:higgs_cx} shows the Higgs production cross section~\\cite{Tev4LHC}, at $\\sqrt{s}=1.96$~TeV and $\\sqrt{s}=7$~TeV, and Figure~\\ref{fig:higgs_br} shows the Higgs decay Branching Ratios~\\cite{LHCHiggs_cx_wg:2011ti} (BR) for a mass range $100< m_H<200$~GeV$\/c^{2}$. \n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[angle=-90,width=0.495\\textwidth]{Theory\/Figs\/tev_higgs_cx}\n\\includegraphics[angle=-90,width=0.495\\textwidth]{Theory\/Figs\/lhc_higgs_cx}\n\\caption[Higgs Boson Production Cross Sections]{Higgs boson production cross sections in different modes~\\cite{Tev4LHC}. At the Tevatron $p\\bar{p}$, $\\sqrt{s}=1.96$~TeV (left) and at the LHC $pp$, $\\sqrt{s}=7$~TeV (right). In 2012 the LHC raised the collision energy to  $\\sqrt{s}=8$~TeV increasing still more the Higgs production cross section.}\\label{fig:higgs_cx}\n\\end{center}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/YRHXS2_BR_Fig1-2}\n\\caption[Higgs Boson Decay Branching Ratios]{Higgs boson decay branching ratios~\\cite{LHCHiggs_cx_wg:2011ti}. The Higgs boson couples to the mass of the particles therefore the decay to the $b\\bar{b}$ quark pair is favored for $m_H\\lesssim 135$~GeV$\/c^{2}$ while $H\\to W^+W^-$ decay dominates for larger masses.}\\label{fig:higgs_br}\n\\end{center}\n\\end{figure}\n\nThe LEP experiments were the first to test the existence of  Higgs boson for masses larger that $100$~GeV$\/c^{2}$, but, as no signal evidence was found~\\cite{lep}, all the searches were combined to provide a lower mass limit of $m_H>114.4$~GeV$\/c^{2}$, at 95\\% Confidence Level (CL). In the latest years also the experiments situated at the Tevatron and LHC colliders provided several mass exclusion limits~\\cite{tev_cmb, cms_cmb, atlas_cmb}. Figure~\\ref{fig:tev_cmb} gives a summary of the 95\\% CLs of all the three colliders, overlaid to the Tevatron result in the mass range $100< m_H<200$~GeV$\/c^{2}$, only a tiny fraction of the phase space is still available to the Higgs presence and, interestingly, a broad excess appears in mass range $110\\lesssim m_H \\lesssim 140$~GeV$\/c^{2}$. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Theory\/Figs\/tev_lhx_excl}\n\\caption[Tevatron Combined 95\\% Exclusion CL]{Mass exclusion limits (95\\% CL) obtained from the combination of all the Tevatron searches for a SM Higgs boson~\\cite{tev_cmb}. Exclusion limits obtained  from the CMS~\\cite{cms_cmb} and Atlas~\\cite{atlas_cmb} Collaboration are overlaid.}\\label{fig:tev_cmb}\n\\end{center}\n\\end{figure}\n\nIn a short time, as the LHC continues the data taking, a conclusive statement about the Higgs existence will be possible.\n\nHowever, the Tevatron and LHC results are also complementary because they investigate different couplings of the Higgs boson.\nThe LHC experiments base most of the low-mass ($m_H\\lesssim 135$~GeV$\/c^{2}$) sensitivity on the $H\\to\\gamma\\gamma$ final state. This channel offers an excellent mass resolution and background rejection although at the price of a very low BR (see Figure~\\ref{fig:higgs_br}). This is optimal for the higher background rate and Higgs production cross sections availables at a the LHC $pp$ collisions of energy $\\sqrt{s}=7$~TeV and $\\sqrt{s}=8$~TeV.\n\nThe Tevatron experiments rely more on the $H\\to b\\bar{b}$ final state, where the Higgs is produced in association with a vector boson ($WH$ and $ZH$ production). The lower production cross section is compensated by the larger BR (see Figures~\\ref{fig:higgs_cx} and~\\ref{fig:higgs_br}) while the presence a leptonic decay of the $W$ or $Z$ boson allows to keep the background under control. Figure~\\ref{fig:tev_bb} shows the  $H\\to b\\bar{b}$ only Tevatron combined search result. Furthermore the investigation of the $H\\to b\\bar{b}$ BR is important to understand the coupling of the Higgs with the fermion masses and for the confirmation of the SM assumption coming from Equation~\\ref{eq:h_yukawa}.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Theory\/Figs\/tev_bb_cmb}\n\\caption[Tevatron $H\\to b\\bar{b}$ Combined 95\\% Exclusion CLs]{Mass exclusion limits (95\\% CL) obtained from the combination of the Tevatron searches~\\cite{tev_cmb} exploiting the $H\\to b\\bar{b}$ final state.}\\label{fig:tev_bb}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\section{Status of the Diboson Measurements}\\label{sec:dib_res}\n\nIn the context of the Higgs searches at the Tevatron, the diboson observation in  $\\ell\\nu + b\\bar{b}$  final state is particularly relevant as it is a direct check of the  $p\\bar{p}\\to WH\\to \\ell\\nu + b\\bar{b}$ analyses.\n\nThe relevant tree-level diagrams involved in dibosons production are shown in Figure~\\ref{fig:wz_prod}. \n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/wz_prod}\n\\caption[$WW$ and $WZ$ Tree-Level Diagrams]{$WW$ and $WZ$ production Feynman diagrams at tree-level, $t$-channel\n(left) and $s$-channel (right).}\\label{fig:wz_prod}\n\\end{center}\n\\end{figure}\nThe simultaneous emission of a $W$ and the $Z$ vector bosons can happen in the $t$-channel (left of Figure~\\ref{fig:wz_prod}), with the exchange of a virtual quark, or in the $s$-channel (right of Figure~\\ref{fig:wz_prod}) with the exchange of a virtual force carrier. The second case is due to the {\\em non-Abelian} characterisctics of the $SU(2)$ group that origins a {\\em Triple Gauge Coupling} (TGC) in the kinetic term (Equation~\\ref{eq:tensor}) of the SM Lagragnian. The cross sections for the $WW$ and $WZ$ production\\footnote{In this analysis we consider also $ZZ\\to \\ell\\ell + HF$ as a signal when a lepton is misidentified, however this contributes to less than 3\\% of the total diboson signal yield.} calculated at NLO~\\cite{mcfm_prd,mcfm_url}, for $p\\bar{p}$ collision at $\\sqrt{s}=1.96$~TeV, are:\n\n\\begin{equation}\n  \\sigma_{p\\bar{p}\\to WW} = 11.34 \\pm 0.66\\mathrm{~pb};\\quad\\sigma_{p\\bar{p}\\to WZ} = 3.47 \\pm 0.21\\mathrm{~pb};  \n\\end{equation}\nAn increase in the TGC, $s$-channel, production cross section would point to a possible contribution from New Physics (NP) processes. However, the precision that we can obtain in the $\\ell\\nu+HF$ final state is not comparable to the one achievable in other channels with higher leptonic multiplicity (see Table~\\ref{tab:dib_measurements}).\n\nFigure~\\ref{fig:torte} shows the small dibosons BR in $b$ or $c$ quarks\\footnote{The experimental identification of $HF$ quarks, described in Section~\\ref{sec:secvtx} has also a low efficiency.}, furthermore \nthe hadronic final state is background rich and has a low invariant mass resolution.  The search is challenging but it is a perfect standard candle to confirm the understanding of the $\\ell\\nu+HF$ dataset on a well known SM process.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.75\\textwidth]{Theory\/Figs\/wzpie}\\vspace{0.2cm}\n\\caption[Diboson Final States BR]{Branching ratios into the different final states allowed to $WW$ and $WZ$ production. $\\ell\\nu+ HF$ final state is highlighted in red.}\\label{fig:torte}\n\\end{center}\n\\end{figure}\n\n\nDiboson related experimental results are widely present in literature. LEP~\\cite{lep_diboson} performed the first measurements exploiting all the decay channels of the $WW$ and $ZZ$ processes: their cross sections were measured with good precision. The use of an $e^+e^-$ machine allowed also the observation of hadronically decaying $W$'s, in addition to the semi-leptonic $WW$ decays. However $WZ$ production was not allowed at LEP, since it is forbidden by charge conservation.\n\nHadron colliders, both Tevatron and LHC, observed $WW$, $WZ$ and $ZZ$ production in their fully leptonic decay modes, obtaining excellent agreement with the SM prediction~\\cite{cdfWZlllv_2012, d0WZZZlep_2012, cmsDib_2011, cdfZZllll_2012, atlasZZ_2012, cdfWWllvv_2012, d0WWllvv_2009, cmsWW_2012, atlasWW_2012, cdfDibMetJet_2009, cdfDiblvJet_Mjj2010, cdfDiblvJet_ME2010, d0DiblvJet_2011, cdfDiblvHF_Mjj2011, cdfWZZZCombHF_2012, d0WZZZCombHF_2011,  tevWZZZCombHF_2012}.\n\nThe semi-leptonic final states, more difficoult to isolate due to the background rich hadronic environment, were observed at the Tevatron, both at CDF and D0 experiments. In this thesis we present the measurement in the channel $p\\bar{p}\\to WW\/WZ\\to \\ell\\nu +HF$ with an update and improvement of the analysis described in Appendix~\\ref{App:7.5}~\\cite{cdfDiblvHF_Mjj2011}, performed on a smaller dataset in 2011 ($7.5$~fb$^{-1}$). \n\nRecently~\\cite{cdfWZZZCombHF_2012, d0WZZZCombHF_2011, tevWZZZCombHF_2012}, CDF and D0 produced the evidence for $WZ\/ZZ$ production with $HF$ jets in the final state and $WW$ production considered as background. Both experiments produced single results in the three semileptonic diboson decay modes:\n\\begin{equation}\n  WZ\/ZZ \\to \\ell\\ell+ HF\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  WZ\/ZZ \\to \\ell\\nu + HF\\textrm{;}\n\\end{equation}\n\\begin{equation}\n  WZ\/ZZ \\to \\nu\\nu+ HF\\textrm{;}\n\\end{equation}\nwhere the $\\nu$ may indicate a lepton failing the identification. The analyses were performed as an exact replica of corresponding Higgs searches in those channels, with the final signal discriminants re-optimized for $WZ\/ZZ$ extraction.  Heavy usage of multivariate tecniques, as for example in the $HF$ selection strategy~\\cite{HOBIT, BDT_D0} and in the final signal--background discrimination, is the key of the impressive sensitivity of these analyses. The final combined cross section measurement, with a significance of $4.6\\sigma$, is: \n\\begin{equation}\n  \\sigma_{WZ+ZZ}  =  4.47\\pm 0.67^{+0.73}_{-0.72}\\textrm{~pb,}\n\\end{equation}\nwhere the SM ratio between $WZ$ and $ZZ$ is imposed. This confirms the SM production prediction\\footnote{Both $\\gamma$ and $Z$ components are assumed in the neutral current exchange and corresponding production of dilepton final states for $75\\le m_{\\ell^+\\ell^{-}} \\le 105$~~GeV$\/c^{2}$.} of  $\\sigma_{WZ+ZZ}=4.4\\pm0.3$~pb~\\cite{mcfm_prd,mcfm_url}. The evidence of the $WZ\/ZZ$ signal, obtained independently by each experiment and their combination, strongly supports the Tevatron $H\\to b\\bar{b}$ search results.\n\nA summary of all the present diboson measuremens is reported in Table~\\ref{tab:dib_measurements}.\n\n\\renewcommand{\\tabcolsep}{3pt}\n\\begin{table}[h]\n\\begin{center}\n  \\begin{small}\n\n\\begin{tabular}{ccccc}\\toprule\nChannel & Experiment &  $\\mathscr{L}$ (fb$^{-1}$) & Measured $\\sigma$(pb) & Theory $\\sigma$(pb) \\\\\\midrule\n\n$WZ\\to \\ell\\ell\\ell\\nu$ & CDF II\\cite{cdfWZlllv_2012} & $7.1$ & $3.9\\pm 0.8$    &  $3.47 \\pm 0.21$ \\\\\n                                    & D0\\cite{d0WZZZlep_2012} & $8.6$ & $4.5^{+0.6}_{-0.7} $  &  $3.47 \\pm 0.21$ \\\\\n                                    & CMS\\cite{cmsDib_2011} & $1.1$ & $17.0 \\pm 2.4\\pm 1.5$    &  $ 17.3^{+1.3}_{-0.8}$ \\\\\n                                    & Atlas\\cite{cdfWZlllv_2012} & $$ & $20.5 \\pm ^{+3.2+1.7}_{-2.8-1.5}$ & $ 17.3^{+1.3}_{-0.8}$ \\\\\n\\midrule\n\n$ ZZ \\to \\ell\\ell\n+\\ell\\ell\/\\nu\\nu$ \n & CDF II\\cite{cdfZZllll_2012} & $6.1$ & $ 1.64^{+0.44}_ {-0.38}$ & $1.4\\pm 0.1$\\\\\n&  D0\\cite{d0WZZZlep_2012} & $8.6$ & $ 1.44^{+0.35}_{-0.34}$ & $1.4\\pm 0.1$\\\\\n\\midrule\n\n  $ZZ \\to \\ell\\ell\\ell\\ell$ & Atlas\\cite{atlasZZ_2012} & $1.02$ & $ 8.5^{+2.7+0.5}_ {-2.3-0.4}$ & $6.5^{+0.3}_{-0.2}$\\\\\n                            & CMS\\cite{cmsDib_2011} & $1.1$ & $ 3.8^{+1.5}_{-1.2}\\pm 0.3 $ & $6.5^{+0.3}_{-0.2}$\\\\\n\n\\midrule\n\n$WW\\to \\ell\\ell\\nu\\nu$ & CDF II\\cite{cdfWWllvv_2012} & $3.6$ & $12.1\\pm 0.9 ^{+1.6}_{-1.4}$  &  $11.3 \\pm 0.7$ \\\\\n                                    & D0\\cite{d0WWllvv_2009} & $1.0$ & $11.5\\pm 2.2$   &  $11.3 \\pm 0.7$ \\\\\n                                    & CMS\\cite{cmsWW_2012} & $4.92$ & $52.4\\pm 2.0\\pm 4.7$    &  $ 47.0\\pm 2.0 $ \\\\\n                                    & Atlas\\cite{atlasWW_2012} & $1.02$ & $54.4\\pm 4.0 \\pm 4.4$  &  $ 47.0\\pm 2.0$ \\\\\n\n\\midrule\n$ Diboson\\to \\nu\\nu + jets$ & CDF II\\cite{cdfDibMetJet_2009} & $3.5$ & $18.0\\pm 2.8 \\pm 2.6$  &  $16.8 \\pm 0.5$ \\\\\n\n\\midrule\n$ WW\/WZ\\to \\ell\\nu + jets$ & CDF II\\cite{cdfDiblvJet_Mjj2010} & $4.3$ & $18.1\\pm 3.3\\pm 2.5$  &  $16.8 \\pm 0.5$ \\\\\n                                      & CDF II\\cite{cdfDiblvJet_ME2010} & $4.6$ & $16.5^{+3.3}_{-3.0}$  & \\\\\n\n  & D0\\cite{d0DiblvJet_2011} & $4.3$ & $19.6^{+3.2}_{-3.0}$  &  $16.8 \\pm 0.5$ \\\\\n \n\\midrule\n\n$ WW\/WZ\\to \\ell\\nu + HF$ & CDF II\\cite{cdfDiblvHF_Mjj2011} & $7.5$ & $18.1^{+3.3}_{-6.7}$  &  $16.8 \\pm 0.5$ \\\\\n\n\\midrule\n\n & CDF II\\cite{cdfWZZZCombHF_2012} & $9.45$ & $4.1^{+1.8}_{-1.3}$  &  $4.4 \\pm 0.3$ \\\\\n$  WZ\/ZZ \\to \\ell\\ell\/\\ell\\nu\/\\nu\\nu+ HF$ & D0\\cite{d0WZZZCombHF_2011} & $8.4$ & $5.0 \\pm 1.0 ^{+1.3}_{-1.2}$  &  $4.4 \\pm 0.3$ \\\\\n & D0,CDF II\\cite{ tevWZZZCombHF_2012} & $7.5 - 9.5$ & $4.47\\pm 0.67^{+0.73}_{-0.72}$  &  $4.4 \\pm 0.3$ \\\\\n\n\\bottomrule\n\\end{tabular}\n\\caption[Recent Diboson Measurements]{Summary of the recent measurements of the diboson production cross section in leptonic and semi-leptonic final states. The reference to the individual measurements are reported in the table as well as the used integrated luminosity, the statistical uncertainty appears before the systematic uncertainties (when both are available), the theoretical predictions are calculated with NLO precision~\\cite{mcfm_prd,mcfm_url}.}\\label{tab:dib_measurements}\n    \n  \\end{small}\n\\end{center}\n\\end{table}\n\\renewcommand{\\tabcolsep}{6pt}\n\n\\chapter*{\\centering \\begin{large}Abstract\\end{large}}\n\n\n\\clearpage\n\\include{Introduction\/Introduction}\n\n\n\\mainmatter\n\\clearpage\n\\include{Theory\/Theory}\n\n\\clearpage\n\\include{Detector\/Detector}\n\n\\clearpage\n\\include{PhysObj\/PhysObj}\n\n\\clearpage\n\\include{Sel\/Sel}\n\n\\clearpage\n\\include{Bkg\/Bkg}\n\n\\clearpage\n\\include{StatRes\/StatRes}\n\n\\clearpage\n\\include{Conclusions\/Conclusions}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe current generation of short pulse lasers reach intensities beyond $\\mathrm{I} > 10^{20}\\ \\mathrm{W\/cm^2}$, and hold great potential for many applications in high energy density science such as high energy x-ray backlighters and particle beams\\cite{Note1}\\cite{Note2}. While lower intensity lasers couple their energy into the transverse motion of electrons, relativistic electrons can couple energy into longitudinal motion via the laser's magnetic field. However, as intensity increases on relativistic laser platforms, the long ns pre-pulse pedestal that arrives prior to the main interaction intensifies as well. This pre-pulse, generated by amplified spontaneous emission processes in the laser, typically has an intensity contrast ratio $\\mathrm{(I_{Main Beam}\/I_{Pre-Pulse})}$ of $10^{6}$. In high intensity interactions this results in a pre-pulse of high enough intensity to ionize material, creating an underdense pre-plasma that extends for hundreds of\\SI{}{\\ \\micro\\meter}. \n\nWhile pre-plasma has been seen as detrimental to electron acceleration for applications such as fast ignition due to filamentation and self focusing instabilities\\cite{MacPhee}\\cite{Sunahara}, it actually can be beneficial in other ways, enabling a number of mechanisms that can generate energetic electrons. Accelerating electrons is the primary way to couple the energy in a laser to a plasma and provides the basis for a wide range of phenomena and applications. The generation of large quantities of x-ray and energetic secondary particles, such as ions \\cite{Pomerantz}, neutrons\\cite{Schollmeier} and positrons\\cite{Chen} highly depend on relativistic electrons. The electron acceleration in these experiments occur in a regime where the plasma response time is short compared to the laser pulse. In these experiments the density profile evolves slowly relative to the laser and develops into a quasi-steady state, opposite of the regime necessary for processes such as wakefield acceleration. Therefore, understanding the parameters that control the quantity, energy and trajectory of these electrons is critical to properly understanding the production of other energetic particles from secondary interactions.\n\nA common element for electron acceleration mechanisms is the important role played by quasi-static transverse and longitudinal pre-plasma electric fields in enhancing the energy transfer from the laser pulse to pre-plasma electrons \\cite{Booshan1}-\\cite{Robinson}. These fields are relatively weak compared to the field of the laser pulse and are unable to directly transfer considerable energy to the electrons. However, they do change the phase between the oscillating electric field of the laser and the electron velocity, which can result in a net energy gain with each laser period. This is the essence of the mechanism called direct laser acceleration (DLA) that leads to acceleration of the so-called super-ponderomotive electrons in an extended pre-plasma. These electrons have a corresponding relativistic $\\gamma$-factor greatly exceeding the conventional estimate of $\\gamma \\approx a_0^2$, where $a_0$ is the normalized laser amplitude. Indeed it has been shown that these super-ponderomotive electrons can enhance the energy of target normal sheath accelerated (TNSA) protons for radiography purposes but requires carefully controlled conditions for the rise time and trajectory of the electron beam \\cite{Kim}. \n\nPrior experiments using pulses with durations of 150 fs \\cite{Peebles},400 fs \\cite{Tanaka}, 500 fs \\cite{Toshi} and 700 fs \\cite{Ma} suggest that appreciable quantities of super-ponderomotive electrons should be accelerated for pulse lengths 400 fs or longer. The goal of the experiment presented here was to fill a gap between theory and experiment by accelerating large quantities of super-ponderomotive electrons using a very high intensity Texas Petawatt (TPW) laser while varying the interaction pulse length. These super-ponderomotive electrons were observed with energy up to $150$ MeV or more. However, contrary to prior experiments and simulations they were only detected at the longest pulse length tested (600 fs). In this paper we report the experimental measurement of super-ponderomotive electrons accelerated by DLA and the significance that a quasi static magnetic field has on their trajectory.\n\n\\section{Experimental Setup and Initial Simulations}\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Exp-Setup3}\n\t\\caption{Setup for the experiment on the TPW laser. A long pulse beam (green) generates an underdense plasma on the surface of the target while the high intensity short pulse (red) accelerates electrons measured by the primary EPPS diagnostics.}\n\\end{figure}\n\nSimilar to previous experiments \\cite{Peebles}-\\cite{Willingale} a low intensity, 2 ns beam with a large focal spot was used to generate a controllable, uniform, underdense plasma. This beam was injected 4 ns prior to the arrival of a short pulse beam, which had a variable pulse length (150, 450, 600 fs) and energy (30-105 J). Intensity was kept nominally to $2-3 \\times 10^{20}\\ \\mathrm{W\/cm^2}$ for all pulse lengths, and the beam was focused to a \\SI{10}{\\ \\micro\\meter} diameter spot, incident on the target at a $21.8^{\\circ}$ angle. The intrinsic pre-pulse from amplified spontaneous emission of the high intensity beam was nearly negligible after recent upgrades to the TPW, which improved the pre-pulse intensity contrast to over $3 \\times 10^{10}$ \\cite{GaulContrast}. Both beams had a wavelength of 1057 nm. The high-contrast laser was incident on a 1 $\\mathrm{mm^2}$ \\SI{100}{\\ \\micro\\meter} thick planar foil target composed primarily of aluminum, with a buried \\SI{20}{\\ \\micro\\meter} thick copper layer for diagnostic purposes. This foil was attached to a bulk plastic block used to inhibit electron refluxing, preventing double counting of electrons in our diagnostics. The experimental setup with the positions of diagnostics are shown in Figure 1.\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Sims1-a-revised3}\n\t\\caption{Simulation results of a 150, 450 and 600 fs beam (traveling left to right) incident on a target with underdense plasma. Electrons with energy $>$ 100 MeV traveling in the forward direction are color coded by angle relative to the target normal. The deep purple and green sections represent areas with significant electromotive force in the transverse direction on a 100 MeV electron.}\n\\end{figure}\n\nDiagnostics were fielded to characterize hot electron temperature and trajectory. While bremsstrahlung spectrometers (BMXS) and copper K$\\alpha$ imaging (SCI) were used, their signals are more sensitive to lower temperature electrons. High energy electrons were measured with EPPSes, a set of  calibrated magnetic electron spectrometers \\cite{EPPS}. The spectrometers had 1 x 2 mm entrance pinholes, a strong magnet to deflect charged particles and differentiate them by energy and charge and two image plate detectors. The energy range was 5-100 MeV with energy resolution scaling with energy, from 5 keV at low energies to 1 MeV at higher energies. Two spectrometers were placed in the chamber, the first 43 cm away from the interaction facing the rear surface normal of the target and the second was placed 55.5 cm away from the interaction, 3 degrees off the front surface normal of the target. Spectra taken from the EPPS were characterized by the Half Maximum Integrated Energy (HMIE) value, which represents the energy value where 50 percent of the total energy in the spectrum is contained, e.g. a HMIE of 10 MeV means 50\\% of the energy in the spectrum is contained in electrons with less than 10 MeV energy. Hence a higher HMIE value will result from a spectrum with a greater proportion of high energy electrons.\n\nInitial hydrodynamic simulations with 2D FLASH \\cite{FLASH} were conducted to estimate the density profile of the underdense plasma using the parameters of the long pulse beam in the experiment. The resulting density profile was approximated as a sum of exponential decays with a density of around $10^{19}\\ \\mathrm{cm}^{-3}$ \\SI{150}{\\ \\micro\\meter} away from the solid surface. This density profile was input into the 2D particle-in-cell (PIC) code EPOCH \\cite{EPOCH} and truncated \\SI{160}{\\ \\micro\\meter} away from the target surface. The simulation box size was \\SI{250}{\\ \\micro\\meter} long, \\SI{60}{\\ \\micro\\meter} wide, with 10 electron and 5 proton macro particles per cell. The resolution was 30 cells\/\\SI{}{\\\\\\micro\\meter} in the x direction and 15 cells\/\\SI{}{\\\\\\micro\\meter} in the y direction. A laser was injected into the box with an intensity of $3 \\times 10^{20}\\ \\mathrm{W\/cm^2}$, a 70 fs rise time and a \\SI{10}{\\ \\micro\\meter} diameter spot.\n\nIn Figure 2 we present initial snapshots simulations for pulse lengths of 150, 450 and 600 fs taken just after the center of the pulse impacts the target in time. With the 150 fs pulse no significant super-ponderomotive electrons were accelerated. For 450 fs super-ponderomotive electrons are accelerated and travel primarily along the laser trajectory, while at 600 fs these electrons are dispersed by fields that develop near the critical density. Based on these simulations we anticipated measurable quantities of super-ponderomotive electrons ($\\mathcal{E} >$  60 MeV, $a_0$ = 15.6) using pulse lengths greater than 400 fs.\n\n\\section{Results and Discussion}\n\nOver 35 experimental shots were performed over the course of the campaign with time split evenly between the 3 pulse length settings (150, 450 and 600 fs); half of the total shots did not use the long pulse beam to serve as the no pre-plasma cases. We verified that no super-ponderomotive electrons were measured with the shortest pulse length. When comparing the spectra for the 150 fs pulse length case there was no difference in the HMIE values when the long pulse beam was included. Upon increasing the pulse length to 450 fs the results were largely the same and no significant departure was seen between shots with and without pre-plasma. This was a surprising result since prior experiments \\cite{Tanaka}\\cite{Toshi} and our simulations with similar pulse lengths exhibited super-ponderomotive electrons and a change in spectrum shape.\n\n\\begin{figure}[b]\n\t\\includegraphics[width=\\columnwidth]{Exp-EPPS1-d2}\n\t\\caption{(a): Raw data from EPPS1 comparing cases with and without underdense plasma. Background subtracted lineouts are taken of the data and convolved with the spectrometer's calibrated dispersion to produce electron spectra. \\\\\\hspace{\\textwidth} (b): Electron spectra leaving the rear of the target measured by EPPS1 in the calibrated region of the diagnostic. Two 600 fs shots with pre-plasma shown in black and red produced substantial quantities of super-ponderomotive electrons $>$ 60 MeV resulting in a dramatic change to the electron spectrum compared to the no pre-plasma case (blue). \\\\\\hspace{\\textwidth}(c): HMIE values taken for all shots at 600 fs. Shots from the above spectra are circled with their respective color. At 600 fs, 3 out of 9 shots with underdense plasma measured a significant quantity of super-ponderomotive electrons}\n\\end{figure}\n\nExtending the pulse length further to 600 fs yielded significant changes to the electron spectrum shown in Figure 3; 3 out of the 9 shots with the long pulse beam had differently shaped spectra than their counterparts. These spectra contained electrons so energetic that they were detected at the end of the spectrometer (beyond its known calibration) and had energy exceeding 150 MeV. These results contradict the initial expectations from simulations shown in Figure 2. At 450 fs in simulations copious quantities of high energy electrons are accelerated nearly \\SI{50}{\\ \\micro\\meter} from the target surface and move forward into the target, while in the experiment no super-ponderomotive electrons were measured. At 600 fs super-ponderomotive electrons were measured on experiment while in simulations most electrons were severely deflected and few super-ponderomotive electrons propagated directly forward toward the diagnostic.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{Sims2-revised2}\n\t\\caption{Identical simulation setup to Figure 2 with the laser at a 20 degree incidence angle on target. The electrons are color coded by angle bin with respect to the EPPS diagnostic placed in the target normal direction. At 450 fs electrons travel in the direction of the laser similar to those in Figure 2 (b), while at 600 fs the electrons are deflected upwards in the target normal direction. This is due to the development of a large asymmetry in the electromotive force.}\n\\end{figure}\n\nWhat could cause this large discrepancy between the simulations and experimental data? The cause can be partly attributed to the EPPS diagnostic itself. The EPPS collects electrons only near the target rear normal and the solid angle ($\\sim 10^{-5}$ steradian) is an extremely small fraction of the solid angle where electrons are seen to travel in the simulations. Another difference between simulation and experiment lies in the inclusion of the laser incidence angle. In our experiment the laser was incident at roughly a $\\mathrm{20^o}$ angle and the EPPS diagnostic was placed facing the target normal, while our simulations used a normal incidence beam. When conducting another simulation with the same parameters but with the laser incidence angle and diagnostic angles included, the trajectories of super-ponderomotive electrons were altered significantly.\n\nFigure 4 shows the results from the angled simulation taken at the same times as those in Figure 2. Initially the $>$ 100 MeV electrons propagate primarily in the laser direction, not in the direction of the diagnostic. Measuring the angle of the electrons with respect to the viewing angle of the EPPS shows that the EPPS would miss the majority of super-ponderomotive electrons at 450 fs. However, later in time, electrons were scattered weakly toward the target normal rather than in all directions. At 600 fs the bulk of high energy electrons were deflected in the direction of the target normal and EPPS diagnostic, resulting in a greater chance of detection, which is directly supported by the experimental results.\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\columnwidth]{Exp-EPPS2-b}\n\t\\caption{HMIE values for electrons measured from the front surface of the target. Higher energy electrons were measured when an underdense plasma was present, regardless of pulse length suggesting that electrons in the pre-plasma were accelerated by the reflected beam.}\n\\end{figure}\n\\begin{figure}[]\n\t\\includegraphics[width=\\columnwidth]{Sims3-4-revised2}\n\t\\caption{(a): A current density plot for the 20 degree laser incidence angle case taken at the same time as Figure 4 (c). The effect of the asymmetric magnetic field on incident hot electrons is gradual causing a smaller deflection. Arrows indicate the location and direction of the majority of current flow.\\\\\\hspace{\\textwidth} (b): A plot of a simulation with a 10 degree laser incidence angle taken earlier in time. \\\\\\hspace{\\textwidth} (c): Plot of the 10 degree incidence simulation at the same time as the top 20 degree image. The current accelerated by the reflected beam expands and encounters the incident current more directly than in the 20 degree case. The super-ponderomotive electrons see a sudden, larger increase in the deflecting magnetic field as the fields join together.}\n\\end{figure}\n\nUpon closer examination of electron deflection at these later times it is found that the electrons encounter a counter-propagating current of colder electrons. The effect of this reverse current can be seen by mapping the electromagnetic force on a forward going electron (seen in purple and green in Figs 2 and 4). The forward going electrons generate a self confining magnetic field based on their current. Extending from the target surface, a counter-propagating current generates an opposite field that causes the electrons to split upwards or downwards. As pulse length increases this opposing field gains strength and causes more severe deflections. This effect has the largest impact on super-ponderomotive electrons, as they are accelerated via DLA far from the target surface and travel through this field. Lower energy, ponderomotive scaling electrons are less affected since they are accelerated near the critical density, closer to the target surface and do not interact with these fields over an extended distance.\n\nThe development of counter-propagating current can be traced to electrons in the pre-plasma near the target surface that are accelerated by reflected laser light away from the critical surface. This effect has been characterized in 3D PIC simulations conducted by F. Perez \\textit{et al.}\\cite{Perez}, which show the development of strong magnetic fields due to counter-propagating current after 473 fs. In the normal incidence case the two currents directly oppose each other and incoming electrons are deflected in all directions; in the angled case the opposing fields develop at an angle with respect to the target surface. This angle causes the fields from the counter-propagating current to build upon the fields from the incident current rather than oppose them, leading to a large asymmetry and significant deflection toward target normal.\n\nExperimental evidence of pre-plasma electrons accelerated by reflected light was found in the data of the spectrometer facing the front surface of the target (EPPS2). Measuring the HMIE values for electrons traveling in this direction shows distinctly higher energies for nearly all shots with the long pulse beam, regardless of pulse length (Fig 5). However, as our simulations show, the field from this reflected current requires time to develop and only has enough strength to deflect significantly electrons after 500 fs.\n\n\\section{Summary and Future Considerations}\n\nIn summary, we found that in experiments no super-ponderomotive electrons were detected in the forward direction for $\\tau_L$ = 150 fs, nor for $\\tau_L$ = 450 fs, in contradiction to earlier normal-incidence simulations, and only sporadically for $\\tau_L=$ 600 fs. Simulations including the experimental incidence angle showed that the super-ponderomotive electrons in fact were generated as early as $450$ fs, but were likely traveling in the direction of the laser direction and not towards the diagnostic placed in the target normal direction. As fields from an opposing current increased later in time these electrons were eventually deflected preferentially in the direction of the diagnostic. These results raise the question, how sensitive are super-ponderomotive electron trajectories to changes in laser incidence angle? To address this, another simulation with identical parameters was conducted, which used a 10 degree incidence angle rather than 20. As seen in figure 6 (c), electrons in the small angle case were deflected \\textit{more} significantly than in either of the previous two cases (0 and 20 degree), with a large portion deflected away from the target entirely late in time. Due to the smaller incidence angle, the region where laser light reflects off the critical density surface becomes narrower causing the opposing current to become more concentrated. More importantly, halving the incidence angle causes the relative angle between the reflected and incident light to be quartered. The opposing current therefore has a more head on trajectory with the incident electrons leading to greater deflection angle unlike the ``glancing blow'' seen in the large angle case.\n\nThese results point to a few considerations for future experiments and simulations. Understanding the effect of laser incidence angle is crucial for setting up an experiment. Many high intensity laser facilities do not allow for direct normal laser incidence due to the potential damage caused by reflected light, which means that incidence angle effects are a factor that must be considered for experiments that have been or will be performed at these facilities. Even for experiments that use perfectly normal incidence, the effect of counter-propagating current can result in deflecting electrons away from the laser direction as shown in Figure 2. Electron trajectories change significantly when incidence angle is varied and must be taken into consideration when placing diagnostics and when accelerating protons via TNSA using non-relativistic beams \\cite{Kim}. Due to the nature of high energy electrons, diagnostics capable of measuring super-ponderomotive electrons use very small solid angle apertures. Measuring these directional, high energy electrons poses a challenge to the diagnostics community to develop new tools capable of providing spatial information for higher energy electrons.\n\n\n\\section*{Acknowledgments}\n\nExperimental time and resources were supported by the United States Department of Energy under contract DE-NA0001858 (DOE). Simulations were performed using the EPOCH code (developed under UK EPSRC Grants No. EP\/G054940\/1, No. EP\/G055165\/1, and No. EP\/G056803\/1) using HPC resources provided by the TACC at the University of Texas. The work by A. Arefiev ,CHEDS researchers, and the TPW itself were supported by NNSA Cooperative Agreement DE-NA0002008.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nQuantum Key Distribution (QKD) allows two remote parties, Alice and Bob, to generate a secret shared string of bits, that can be used for symmetric cryptography or other cryptographic protocols.\nUnlike classical key distribution, whose security is based on the computational difficulty of solving certain classes of problems, the security of quantum key distribution comes from the impossibility for an eavesdropper (Eve) to acquire information about an exchanged state without perturbing it.\nThis goal is obtained by using non-orthogonal states, as in the first QKD protocol, introduced by Bennett and Brassard in 1984 (BB84), that uses four different states of two mutually unbiased bases \\cite{Bennett1984}.\nIn 1992, Bennett showed that two non-orthogonal states are sufficient for QKD \\cite{Bennett1992}.\nThis protocol, however, has the drawback that the secure key rate is strongly affected by losses: indeed, Eve can extract information by increasing the losses and performing the so called unambiguous state discrimination (USD) attack~ \\cite{Tamaki2003,Tamaki2004}.\nThe addition of a third state is sufficient to make the B92 protocol unconditionally secure independently from the noise in the quantum channel \\cite{Fung2006,luca09pra}: however, rates comparable to the BB84 protocols can be obtained only when four states are detected at the receiver.\n\nThe optimal three-state QKD protocol, introduced in 2000 by Phoenix-Barnett-Chefles (PBC00)~\\cite{Phoenix2000}, uses states that form an equilateral triangle in the X-Z plane of the Bloch sphere.\nThe symmetry of this protocol can indeed be exploited to obtain rates comparable with the BB84 protocol but requiring only three detectors instead of four.\nThis protocol, however, still requires the public exchange of part of the sifted key in order to estimate the QBER.\nAn improvement of this protocol, introduced by Renes in 2004 (R04)~\\cite{Renes2004}, estimates the error rate from the number of inconclusive events, thus allowing to use all conclusive ones for key extraction.\nThe unconditional security of the PBC00 has been demonstrated, in the asymptotic case, in 2005 \\cite{Boileau2005}, for a bit error rate of up to $\\unit[9.81]{\\%}$.\nThe security of the PBC00 has been demonstrated also in the case of finite key \\cite{Mafu2014}, using the framework introduced by Scarani and Renner \\cite{Scarani2008} and the postselection technique \\cite{Christandl2009}.\nTheir work, however, does not consider the security of the R04 protocol, therefore excluding one of the most interesting features of this class of symmetrical codes.\nDespite all these theoretical results, no experimental implementation of equiangular three state QKD protocols has been reported so far.\nThe measurement apparatus proposed in the original work implements the trine measurement using an interferometric setup at the receiver~\\cite{Phoenix2000}.\nThis scheme requires careful alignment \\cite{Clarke2001,Clarke2001-1} and is not assured to have the long term stability required by Quantum Key Distribution (a stability of about half an hour is reported in \\cite{Clarke2001-1}).\nA new experimental scheme implementing the trine measurement using only passive optical elements has been proposed in \\cite{Saunders2012}.\nUsing this apparatus, we demonstrate the feasibility of equiangular three state QKD and assess its key generation rate, both in the asymptotic limit of infinite key and taking into account finite key effects.\n\n\\section{Results}\n\\subsection{Protocol}\nThe R04 protocol uses three quantum states, $\\{ \\ket{\\psi_1}, \\ket{\\psi_2}, \\ket{\\psi_3} \\}$, placed in an equilateral triangle in the X-Z plane of the Bloch sphere, as shown in Figure \\ref{fig:bloch}.\n\\begin{figure}[t]\n   \\centering\n      \\includegraphics[width=0.7\\linewidth]{images\/bloch.pdf}\n      \\includegraphics[width=0.7\\linewidth]{images\/povm.pdf}\n      \\caption{States used in the R04 protocol (above) and POVM $\\{ \\Pi_i \\}$ used for the measurement (below). The states lie in the X-Z plane of the Bloch sphere. They are grouped into the sets $S_1 = \\{ \\ket{\\psi_1}, \\ket{\\psi_2} \\}$, $S_2 = \\{ \\ket{\\psi_2}, \\ket{\\psi_3} \\}$, and $S_3 = \\{ \\ket{\\psi_3}, \\ket{\\psi_1} \\}$, where the first element of each set corresponds to bit 0 and the second to bit 1. The POVM is implemented by using a partially polarizing beam-splitter (pPBS), a half-wave plate (HWP) at $22.5^\\circ$ and a polarizing beam-splitter (PBS).}\n      \\label{fig:bloch}\n\\end{figure}\nThe states are grouped into three different sets, $S_1 = \\{ \\ket{\\psi_1}, \\ket{\\psi_2} \\}$, $S_2 = \\{ \\ket{\\psi_2}, \\ket{\\psi_3} \\}$, and $S_3 = \\{ \\ket{\\psi_3}, \\ket{\\psi_1} \\}$. \nIn each set, the first state is associated with the bit $0$ and the second with the bit $1$.\nDifferently from other QKD protocols, each state brings no information about its associated bit before the information about the used set is disclosed. \\\\\n\nWe implemented an entanglement-based version of the protocol, using polarization-entangled photon pairs in the singlet state\n\\begin{equation}\n\t\t\t\t\\ket{\\Psi^{-}} = \\frac{\\ket{H}_A \\ket{V}_B - \\ket{V}_A \\ket{H}_B}{\\sqrt{2}},\n\t\t\t\t\\label{eq:psi-}\n\\end{equation}\nwhere the subscripts $A$ and $B$ indicate the photons going to, respectively, Alice's and Bob's detection apparatus.\nIn this state, photons A and B are anti-correlated in any measurement basis.\nAlice measures photon $A$ using the POVM $\\{\\Pi_i\\equiv\\frac23\\ket{\\psi_i^\\perp}\\bra{\\psi_i^\\perp}\\}$, with $\\ket{\\psi_1^\\perp} = \\ket{V}$, $\\ket{\\psi_2^\\perp} = \\frac{\\sqrt{3}}{2}\\ket{H} - \\frac{1}{2}\\ket{V}$, $\\ket{\\psi_3^\\perp} = \\frac{\\sqrt{3}}{2}\\ket{H} + \\frac{1}{2}\\ket{V}$. In our notation the state $\\ket{\\psi_i^\\perp}$ is orthogonal to $\\ket{\\psi_i}$.\nWhen Alice obtains a detection in the state $\\ket{\\psi_i^\\perp}$ (each with probability $\\frac{1}{3}$), she is sending to Bob the state $\\ket{\\psi_i}$ where  $\\ket{\\psi_1} = \\ket{H}$, $\\ket{\\psi_2} = \\frac{1}{2}\\ket{H} + \\frac{\\sqrt{3}}{2}\\ket{V}$, and $\\ket{\\psi_3} = \\frac{1}{2}\\ket{H} - \\frac{\\sqrt{3}}{2}\\ket{V}$.\nThis operation corresponds to the random preparation, with equal probability, of one of the three states $\\{\\ket{\\psi_1}, \\ket{\\psi_2}, \\ket{\\psi_3}\\}$, as in the prepare-and-measurement scheme of the R04 described in \\cite{Boileau2005,Scarani2009}.\nBob performs his measurements in the same POVM as Alice $\\{ \\Pi_i \\}$.\nAfter all measurements, Bob and Alice compare the\ninstants of their events, \nkeeping only those where both have a detection within a fixed coincidence window. \n\nEven if they have already exchanged all symbols, Alice and Bob do not share any bit string yet, because each state can mean both 0 or 1.\nAlice uses a QRNG to choose the bit value for each symbol.\nThe combination of the state and the bit value unambiguously determines the set $S_i$ used for that event (for example, if Alice sends $\\ket{\\psi_2}$ and the QRNG gives $1$, the set used for that event is $S_1$).\nFor each event, Alice tells Bob the corresponding set by sending him the value of the index $i$.\nBob uses $i$ to associate $\\ket{\\psi_2^\\perp}$ (for $i = 1$), $\\ket{\\psi_3^\\perp}$ (for $i = 2$), and $\\ket{\\psi_1^\\perp}$ (for $i = 3$) with bit 0, and $\\ket{\\psi_1^\\perp}$ (for $i = 1$), $\\ket{\\psi_2^\\perp}$ (for $i = 2$), and $\\ket{\\psi_3^\\perp}$ (for $i = 3$) with bit 1.\nAll other combinations are marked as inconclusive, since Bob is not able to determine the state sent by Alice.\nBob tells Alice which events are inconclusive and they both discard them.\nThey then estimate the quantum bit error rate (QBER) from the fraction of inconclusive events \\cite{Renes2004,Boileau2005}, and use this information to distill the key using error correction and privacy amplification \\cite{Scarani2008-1}. \n\n\\subsection{Setup}\nThe experimental setup is shown in Figure \\ref{fig:setup}.\n\\begin{figure}[t]\n   \\centering\n      \\includegraphics[width=\\linewidth]{images\/Sagnac_noPOVM.pdf}\n      \\caption{Experimental setup used for the experiment. The light issued by a laser diode (LD) at $\\unit[404.5]{nm}$ is collected into a polarization maintaining single mode fiber (PM-SMF) for spatial mode filtering. The polarization is adjusted by a PBS followed by a HWP and the beam is focused by a $\\unit[200]{mm}$ lens (L) into the center of the interferometer. The interferometer is formed by a PBS, a dual-wavelength HWP (d-HWP) and a PPKTP crystal, where the down-conversion takes place. Photons exiting the interferometer are collected into single mode fibers (SMF), preceded by a long-pass filter (LPF) to filter out residual pump intensity. Fiber birefringence is compensated by two quarter-wave plates (QWP) and a half-wave plate (HWP). The receiver consists in an implementation of the POVM $\\{ \\Pi_i \\}$, with output Ai and Bi corresponding to $\\ket{\\psi_i^\\perp}\\bra{\\psi_i^\\perp}$ for, respectively, Alice and Bob. All the output signals are timed by a common timetagger.} \n          \n          \n\\label{fig:setup}\n\\end{figure}\nEntangled photon pairs are produced by using a $\\unit[30]{mm}$ periodically poled KTP crystal in a polarization-based Sagnac interferometer \\cite{Kim2006,Fedrizzi2007}.\nThe source is pumped with a continuous wave (CW) laser at $\\unit[404.5]{nm}$, with a power of $\\unit[3.5]{mW}$.\nThe down-converted photons have a central wavelength of $\\unit[809]{nm}$, with $\\unit[0.2]{nm}$ full width at half maximum (FWHM), and are collected into single-mode fibers.\nIn this configuration, the setup has a mean coincidence rate of $\\unit[29]{kHz}$, with a 5\\% heralding ratio.\nThe fraction of multi-pair over one pair events, measured by putting one output of the source into a Hanbury-Brown-Twiss interferometer, is $\\sim 3 \\cdot 10^{-3}$.\nAmong these events, only those which are partially correlated in the polarization degree of freedom are exploitable by Eve through the photon number splitting (PNS) attack \\cite{Scarani2008}.\nThe ratio of the number of these events to all multi-pair ones is $\\zeta \\simeq \\frac{\\tau_c}{\\Delta t} = 5 \\cdot 10^{-3}$, where $\\tau_c = \\unit[8]{ps}$ is the coherence time of down-converted photons and $\\Delta t = \\unit[1.5]{ns}$ is the coincidence window.\nThe fraction of correlated multi-photon events over the total number of detection events is $\\sim 1.5 \\cdot 10^{-5}$, thus the information leaked to Eve is negligible.\nA set of two quarter-wave plates (QWP) and one half-wave plate (HWP) is placed at the exit of the fiber at Bob's side in order to compensate polarization rotations induced by fiber birefringence. \n\nThe receiving apparatus implementing the POVM $\\{\\Pi_i\\}$ consists of a partially polarizing beam-splitter (pPBS), that completely transmits the horizontal polarization and has a reflectivity of $\\unit[66.7]{\\%}$ for the vertical polarization, followed by a HWP at $\\theta = 22.5^\\circ$ and a polarizing beam-splitter (PBS).\nGiven an arbitrary input state $\\ket{\\phi} = \\alpha \\ket{H} + \\beta \\ket{V}$, with $|\\alpha|^2 + |\\beta|^2 = 1$, the pPBS routes it to detector $1$ with probability $P_1 = \\frac{2}{3}|\\beta|^2$.\nThe state at the transmitting output of the pPBS, $\\alpha \\ket{H} + \\frac{1}{\\sqrt{3}} \\beta \\ket{V}$, is transformed into $\\alpha \\ket{-} + \\frac{1}{\\sqrt{3}} \\beta \\ket{+} = \\frac{1}{\\sqrt{2}} ( \\alpha + \\frac{1}{\\sqrt{3}} \\beta) \\ket{H} - \\frac{1}{\\sqrt{2}} ( \\alpha - \\frac{1}{\\sqrt{3}} \\beta) \\ket{V}$ by the\nHWP.\nThus, the probabilities of a click at detectors $2$ and $3$ are, respectively, $P_2 = \\frac{1}{2} |\\alpha - \\frac{1}{\\sqrt{3}} \\beta |^2$  and $P_3 = \\frac{1}{2} | \\alpha + \\frac{1}{\\sqrt{3}} |^2$.\nIt is easy to check that the above\nprobabilities can be also written as $P_i = \\bra{\\phi} \\Pi_i \\ket{\\phi}$, with $\\Pi_i = \\frac{2}{3} \\ket{\\psi_i^\\perp} \\bra{\\psi_i^\\perp}$: then, the above described apparatus implements the POVM $\\{\\Pi_i\\}$.\nPhotons are detected using silicon single photon counting modules (SPCM), characterized by a dead time of $\\unit[21]{ns}$ and a jitter of $\\sim800$ ps FWHM. Detection events are time-tagged with a resolution of $\\unit[81]{ps}$. \n\n\\subsection{Data acquisition}\nA two hour continuous run of the apparatus has led to the exchange of about $10^{9}$ symbols within a coincidence window of $\\unit[1.5]{ns}$.\nThe events can be described as pairs (Ai,Bj), with $i$ the number of the detector clicking at Alice's side and $j$ the one at Bob's side.\nEvent distribution is shown in table \\ref{tab:data} and Figure \\ref{fig:histogram}.\n\\begin{table}[h]\n   \\centering\n   \\begin{tabular}{|c|c|c|c|}\n      \\hline\n      & A1 & A2 & A3 \\\\\n      \\hline\n      B1 & 0.6 & 35.8 & 33.6 \\\\\n      B2 & 35.1 & 0.6 & 32.8 \\\\\n      B3 & 33.4 & 33.2 & 0.4 \\\\\n      \\hline\n   \\end{tabular}   \n   \\caption{\\label{tab:data} Total number of coincidences at the different detectors (million events). The cell (Ai,Bj) corresponds to a coincidence of Alice's detector $i$ and Bob's detector $j$.}\n\\end{table}\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{images\/histogram.pdf}\n   \\caption{Total number of coincidences at the different detectors. Full (red) bars correspond to detected events and (blue) contours represent the expected number of detection events. }\n   \\label{fig:histogram}\n\\end{figure}\nAfter the collection of all data, a QRNG \\cite{Vallone2014} is used to generate the bit value for each symbol.\nCoincidence events are then analyzed using the sifting procedure summarized in Table \\ref{tab:sifting}.\n\\begin{table}[h]\n   \\centering\n   \\begin{tabular}{|c|c|c|c||c|c|c|}\n      \\hline\n      & \\multicolumn{3}{c||}{Bit = 0} & \\multicolumn{3}{c|}{Bit = 1} \\\\\n      \\hline\n      & A1 & A2 & A3 & A1 & A2 & A3\\\\\n      \\hline\n      B1 & 1 & Inc & 0 & 0 & 1 & Inc \\\\\n      B2 & 0 & 1 & Inc & Inc & 0 & 1 \\\\\n      B3 & Inc & 0 & 1 & 1 & Inc & 0\\\\\n      \\hline\n   \\end{tabular}\n   \\caption{\\label{tab:sifting} Sifting procedure, according to the random choice of the bit at Alice's side (on the left for 0, on the right for 1). The cell (Ai,Bj) stands for a coincidence between Alice's detector $i$ and Bob's detector $j$. Inconclusive events are marked as ``Inc''. The events in the diagonal (Ai,Bi) give an error independently from the bit choice. The other combinations (Ai,Bj), with $i \\neq j$, are either a ``good'' conclusive or an inconclusive event, according to Alice's choice.}\n\\end{table}\nThe events of the form (Ai,Bi) are bit errors, while the others are either a ``good'' conclusive or an inconclusive result according to Alice's choice.\nThe string of conclusive results gives the sifted key, from which a secret key can be distilled using classical post-processing. \n\n\\subsection{Secret key rate}\nPost-processing consists of a series of passages that transform a partially correlated, partially secret key into a new one Eve has negligible information of \\cite{Scarani2008-1,Scarani2009}.\nThe effect of these tasks is a reduction of the number of bits and can be quantified using the secret fraction $r$, defined as the ratio between secure and conclusive bits \\cite{Scarani2009}.\nIn the asymptotic limit of infinitely long key, the key fraction of the R04 is given by \\cite{Boileau2005}\n\\begin{equation}\n   r = 1 - f_{EC} h(Q) - h\\left({\\frac{5}{4}Q}\\right),\n   \\label{eq:asymp_r}\n\\end{equation}\nwhere $f_{EC} = 1.1$ is the efficiency of the error correction protocol \\cite{Mateo2015}.\nThe number of secure bits is given by $N_{conc} r$ and, dividing it by the exposure time, the secret key rate is obtained. \n\nFigure \\ref{fig:keyrate} shows the behavior of the QBER and of the key rate during a two hour acquisition.\nThe slight reduction in the sifted key rate is probably due to a misalignment in the fiber coupling of the entangled source.\nThe losses can be estimated from the ratio of coincidences over single counts.\nThe measured $5\\%$ heralding efficiency corresponds to a total loss level of $\\unit[13]{dB}$, with a contribution of $\\unit[1.5]{dB}$ due to the POVM.\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=\\linewidth]{images\/keyrate.pdf}\n\t \\caption{Result obtained during 2 hour of continue acquisition. The time has been divided into $90$ blocks of about $\\unit[80]{s}$ each, with a mean number of $1.1 \\cdot 10^{6}$ sifted bits. The QBER is estimated from inconclusive events, with Poissonian error bars. The secret key rate is estimated for the case of infinitely long key, using equation (\\ref{eq:asymp_r}).}\n   \\label{fig:keyrate}\n\\end{figure}\nThe QBER is estimated as $Q = \\frac{1-2I}{1-I}$, where $I$ is the fraction of inconclusive results \\cite{Boileau2005}.\nThe QBER remains almost constant at a level below $2\\%$ for all the acquisition, thus confirming the stability of both the source and the POVM $\\{ \\Pi_i \\}$ during the acquisition.\nThe visibility of the source in two mutually unbiased bases at the exit of the Sagnac interferometer, before fiber injection, has been measured to be between $97\\%$ and $98\\%$: then the measured QBER level can be attributed almost completely to the source.\nA small contribution to the QBER is due to the small imbalances between the different channels of the POVM $\\{ \\Pi_i \\}$.\nThese values, estimated by computing the ratio between the raw counts at the different detectors, vary between $0.95$ and $1.05$, in line with what observed in the previous implementation of the POVM \\cite{Saunders2012}.\n\nIn a real scenario, the number of exchanged signals is always finite and the security analysis must take this fact into account.\nThe finite key analysis of the R04 protocol is very similar to the one of the PBC00 \\cite{Mafu2014}, the only substantial difference lying in the estimation of the bit error rate.\n\\begin{figure}[t]\n\\includegraphics[width=\\linewidth]{images\/finite.pdf}\n\\caption{Finite key analysis of the R04 protocol.\nThe y-axis represents the fraction of exchanged symbols giving a secure bit. Each point is calculated using the first N collected data for the estimation of the QBER. The security parameter for both collective and general attacks is fixed to $4 \\cdot 10^{-10}$.}\n\\label{fig:finite}\n\\end{figure}\nThe PBC00 estimates it by comparing part of the sifted key through the public channel, while the R04 use the fraction of inconclusive results.\nThe method is based on the fact that the choice between ``good'' conclusive and inconclusive results is given by a random event at Alice's side after the exchange of all the qubits, therefore Eve has no way of differentiate between the two and their number is approximately equal \\cite{Boileau2005}.\nDefining $I$ the fraction of inconclusive results, the fraction of ``good'' conclusive ones can be written as $(1-Q)(1-I)$.\nUsing the Hoeffding bound \\cite{Hoeffding1963}, the inequality\n\\begin{equation}\n\t\t\t\t\\left| (1-Q)(1-I) - I \\right| \\leq \\xi(\\epsilon,N) := \\sqrt{\\frac{2}{N}\\log\\frac{2}{\\epsilon}}\n\t\t\t\t\\label{eq:hoeffding}\n\\end{equation}\nis valid with probability at least $1-\\epsilon$.\nThis implies that, with the same probability, the QBER $Q$ is less than $\\widetilde{Q} := \\frac{1-2I+\\xi(\\epsilon,N)}{1-I}$.\nThe secret key fraction, in the case of collective attacks, is \\cite{Scarani2008,Mafu2014}\n\\begin{align}\n\t\t\t\tr_{col} = & \\left[ 1 - h\\left( \\frac{5}{4} \\widetilde{Q} \\right) \\right] - 7 \\sqrt{\\frac{1}{N} \\log_2{\\frac{2}{\\bar{\\epsilon}}}} \\nonumber \\\\\n\t\t\t\t    & - \\frac{1}{N}\\log_2{\\frac{1}{\\epsilon_{EC}}} - \\frac{1}{N}\\log_2{\\frac{2}{\\epsilon_{PA}}} - f_{EC} h(Q)\n\t\t\t\t\\label{eq:r_finite}\n\\end{align}\nfor $N$ exchanged signals, with security parameter $\\epsilon_{col} := \\bar{\\epsilon} + \\epsilon_{EC} + \\epsilon_{PA} + \\epsilon_{PE}$.\nThis result can be extended to the case of general attacks by exploiting the postselection technique \\cite{Christandl2009}, giving \\cite{Mafu2014} \n\\begin{equation}\nr_{gen} = r_{col} - \\frac{6 \\log_2{(N+1)}}{N},\n\\label{eq:r_finite_gen}\n\\end{equation}\nwith security parameter $\\epsilon_{gen} = (N+1)^{3} \\epsilon_{col}$. \n\nFigure \\ref{fig:finite} shows the secret key fraction of the R04 protocol in the finite key scenario.\nFor each point, both the QBER and the number of conclusive events is evaluated on the first $N$ exchanged symbols.\nThe secret key fraction is calculated by using equations (\\ref{eq:r_finite}) and (\\ref{eq:r_finite_gen}).\nFor collective attacks, the chosen security parameter is $\\epsilon_{col} = 4 \\cdot 10^{-10}$, with $\\bar{\\epsilon} = \\epsilon_{EC} = \\epsilon_{PA} = \\epsilon_{PE} = 10^{-10}$.\nThe same value has been chosen for $\\epsilon_{gen}$, therefore the term $r_{col}$ of equation (\\ref{eq:r_finite_gen}) is calculated using $\\bar{\\epsilon} = \\epsilon_{EC} = \\epsilon_{PA} = \\epsilon_{PE} = \\frac{10^{-10}}{(N+1)^3}$.\nThe plots show that at least $10^{4}$ - $10^{5}$ signals are necessary to exchange a key, while already $N = 10^{6}$ (slightly more than half a minute at $\\unit[29]{kHz}$) gives a reasonable key fraction.\nThe difference between the key fraction for collective and general attacks is more marked for lower values of $N$ and tends to disappear for large number of exchanged symbols, where both approach the asymptotic key fraction. \n\n\\section{Discussion}\nIn this work, we demonstrated the experimental feasibility of the equiangular three state QKD protocol R04.\nWe showed that the scheme proposed by \\cite{Saunders2012} for the POVM is suitable for applications in Quantum Key Distribution.\nState preparation was simplified by using an entanglement-based version of the protocol, with the same POVM at both Alice's and Bob's side.\nWe also showed that the estimation of the bit error rate from inconclusive results is feasible in the finite key scenario.\nThe implemented scheme was demonstrated to be stable and highly reliable, allowing a two hour data acquisition without any significative change in the QBER value.\nThe performance of the protocol is comparable with the BB84, despite the less efficient parameter estimation, both in the asymptotic limit and for finite key.\nIts simpler receiving apparatus, requiring only three single photon detectors, makes it a valid alternative to current implementations of QKD based on the BB84 protocol.\nFinally, this work extends the experimental investigation of equiangular spherical codes to a still uncovered area of quantum information: quantum key distribution.\n\n\\acknowledgements\nWe thank Davide G. Marangon and Nicola Laurenti for helpful discussions.\nThis work has been carried out within the Strategic-Research-Project QUANTUMFUTURE of the University of Padova and the Strategic-Research-Project QUINTET of the Department of Information Engineering, University of Padova.\nM.S. acknowledges the Ministry of Research and the Center of Studies and Activities for Space (CISAS) ``Giuseppe Colombo'' for financial support.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nEleven-dimensional, \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}-based Chern--Simons (CS) Supergravity was put forward by\nR.~Troncoso and J.~Zanelli (TZ) in 1997~\\cite{Tro97,Tro98}.\nThe appearance of this eleven-dimen\\-sional theory followed previous efforts in lower dimensions by\nS.~Deser, R.~Jackiw and S.~Templeton~\\cite{Des82b,Des83},\nA.~Ach\u00facarro and P.~K.~Townsend~\\cite{Ach86}, and\nA.~H.~Chamseddine~\\cite{Cha90}.\n\nThe ``supergravity'' name has proven a bit controversial.\nOn one hand, the TZ theory includes gravity and fermionic matter and sports an exact, off-shell\n\\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\ symmetry which mixes bosons and fermions.\nOn the other, it differs quite radically%\n\\footnote{For instance, the TZ theory lacks equality between bosonic and fermionic degrees of freedom (at least in the off-shell counting).\nThis matching, which has been taken as a watermark of supersymmetry in the literature, lies also at the foundations of the MSSM.\nThe recent negative results of the search for supersymmetric particles at the LHC~\\cite{CMS11a,Str11,ATL11a,ATL11b,Buc11},\nalthough clearly still at an early stage,\nshould not then be taken as an irrefutable sign that supersymmetry as a principle is wrong.}\nfrom the ``standard'' 1978 theory of Cremmer, Julia and Scherk (CJS)~\\cite{Cre78},\nwhich is widely regarded as unique~\\cite{Des97}.\n\nThe purpose of this paper is to shed new light on the relation between the TZ and the CJS theories.\n\nIn this regard, we would like to highlight here the work of\nM.~Ba\u00f1ados~\\cite{Banh01}, who showed that the linearized perturbations of the TZ theory are the same as the ones of CJS supergravity\n(see also Ref.~\\cite{Hor97}).\nThat result, however, is based on the identification of the CJS three-form $A_{3}$ with $e^{a} T_{a}$\n(where $e^{a}$ and $T^{a}$ are the elfbein and torsion, respectively),\nwhich might lead to inconsistencies~\\cite{Ede08}.\nOur own approach\nsuggests that\n$A_{3}$ should to be related to the CS fields through a much more complicated ansatz\n[see eq.~(\\ref{eq:A3})].\n\nRather than focusing on the dynamics of the TZ theory, in this paper we deal with its Lagrangian structure.\nThis proves to be a more tractable approach, since CS theories possess highly nonlinear dynamics and a complicated phase space structure~\\cite{Mis03}.\n\nWe first show that the TZ Lagrangian may be cast as a polynomial of degree~11 in $1\/l$, where $l$ is a length.\nWhen the CS five-index one-form $b^{abcde}$ vanishes, the leading term is just a cosmological constant,\nwhile the sub-leading term includes both torsion and a mass term for the fermions.\nOur main results follows:\nthe next term (proportional to $1\/l^{9}$) is essentially the CJS Lagrangian (see section~\\ref{sec:main}).\nThis discovery shows the existence of a previously unknown relation between the TZ and the CJS theories.\n\nThat this result may be new owes much to the great complexity of the TZ Lagrangian (see section~\\ref{sec:TZ}).\nWhile a compact closed formula exists [see eq.~(\\ref{eq:TZLag})],\na sufficiently explicit version would amount to roughly a thousand terms and has never been attempted.\n\nTo achieve our result we have first split the TZ Lagrangian into several meaningful terms\nby means of the CS subspace separation method introduced in Ref.~\\cite{Iza06a}.\nA careful dimensional analysis has then allowed us to extract only those terms proportional to the required powers of $l$.\nFor the sake of simplicity, we have restricted ourselves to the case when $b^{abcde} = 0$, which is roughly equivalent to setting $A_{3} = 0$.\nThe details of the calculation are summarized in~\\ref{sec:CJSTZ}.\n\nFurther comments and an outlook for future work are given in section~\\ref{sec:final}.\n\n\n\\section{A brief look at the TZ theory}\n\\label{sec:TZ}\n\nThe TZ theory is a pure gauge theory in eleven-dimen\\-sional spacetime\nwhose Lagrangian $\\mathcal{L}_{\\text{TZ}}$ is the CS eleven-form for the \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\ superalgebra.\nA compact closed formula for $\\mathcal{L}_{\\text{TZ}}$ reads%\n\\footnote{Wedge product between differential forms is assumed throughout, e.g., $\\bm{A}^{2} = \\bm{A} \\wedge \\bm{A}$.}\n\\begin{equation}\n \\mathcal{L}_{\\text{TZ}} =\n 6 \\int_{0}^{1} dt\n \\left\\langle\n  \\bm{A}\n  \\left(\n   t \\mathrm{d} \\bm{A} +\n   t^{2} \\bm{A}^{2}\n  \\right)^{5}\n \\right\\rangle,\n \\label{eq:TZLag}\n\\end{equation}\nwhere $\\bm{A}$ is an \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}-valued one-form and\n$\\left\\langle \\cdots \\right\\rangle$\nstands for an \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}-invariant supersymmetric polynomial of rank six.%\n\\footnote{This is essentially a higher-order version of the Killing metric, which may include a number of dimensionless coupling constants.}\n\nThe general properties of CS forms~\\cite{Che74,deAz95} ensure that $\\mathcal{L}_{\\text{TZ}}$\nchanges at most by a closed form under \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\ gauge transformations, which close off-shell\nwithout introducing auxiliary fields.\n\nChoosing a convenient basis for \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\ one may write $\\bm{A}$ as\n\\begin{equation}\n \\bm{A} =\n \\frac{1}{l} e^{a} \\bm{P}_{a} +\n \\frac{1}{2} \\omega^{ab} \\bm{J}_{ab} +\n \\frac{1}{5!l} b^{abcde} \\bm{Z}_{abcde} +\n \\frac{1}{\\sqrt{l}} \\bar{\\bm{Q}} \\psi,\n \\label{eq:A}\n\\end{equation}\nwhere\n$\\bm{P}_{a}$ are (AdS nonabelian) translations,\n$\\bm{J}_{ab}$ are Lorentz rotations,\n$\\bm{Z}_{abcde}$ are extra bosonic generators (not central charges---$\\bm{Z}_{abcde}$ behaves as a five-index, fully antisymmetric Lorentz tensor), and\n$\\bm{Q}$ is a fermionic generator.\nThis basis, with its associated commutation relations,\nallows the interpretation of\n$e^{a}$ as the orthonormal frame of one-forms%\n\\footnote{The spacetime metric $g_{\\mu \\nu}$ is not a fundamental field in the TZ theory,\nbeing derived from $e^{a}$ by means of the relation\n$g_{\\mu \\nu} = \\eta_{ab} \\nwse{e}{a}{\\mu} \\nwse{e}{b}{\\nu}$.}\n(the \\textit{elfbein} one-form) and\n$\\omega^{ab}$ as the spin connection,\nwhich shows that gravity (in its first-order formulation) is included in the TZ theory.\n\nDimensional consistency demands that we introduce a length parameter $l$ in eq.~(\\ref{eq:A}).\nThis length scale allows the one-form $\\bm{A}$ to remain dimensionless while giving the right dimensions to the component fields\n$e^{a}$, $\\omega^{ab}$, $b^{abcde}$ and $\\psi$.\n\nThe definition of $\\mathcal{L}_{\\text{TZ}}$\nis only complete after we specify the precise meaning of\n$\\left\\langle \\cdots \\right\\rangle$.\nPerhaps the simplest way to build this invariant polynomial\nis to use the trace of the supersymmetrized product of six generators in a suitable supermatrix representation.%\n\\footnote{This is not the most general choice, though, and more general invariant polynomials may prove useful. We return to this in section~\\ref{sec:final}.}\nOur choice for such supermatrix representation is as follows:\n\\begin{align}\n\\bm{P}_{a} & = \\left[\n\\begin{array}[c]{cc}\n\\frac{1}{2} \\nwse{\\left( \\Gamma_{a} \\right)}{\\alpha}{\\beta} & 0 \\\\\n0 & 0\n\\end{array}\n\\right], \\\\\n\\bm{J}_{ab}  &  =\\left[\n\\begin{array}[c]{cc}\n\\frac{1}{2} \\nwse{\\left( \\Gamma_{ab} \\right)}{\\alpha}{\\beta} & 0 \\\\\n0 & 0\n\\end{array}\n\\right], \\\\\n\\bm{Z}_{abcde}  &  =\\left[\n\\begin{array}[c]{cc}\n\\frac{1}{2} \\nwse{\\left( \\Gamma_{abcde} \\right)}{\\alpha}{\\beta} & 0 \\\\\n0 & 0\n\\end{array}\n\\right], \\\\\n\\bm{Q}^{\\rho} & = \\left[\n\\begin{array}[c]{cc}\n0 & C^{\\alpha \\rho} \\\\\n\\delta_{\\beta}^{\\rho} & 0\n\\end{array}\n\\right],\n\\end{align}\nwhere $\\Gamma_{a}$, $\\Gamma_{ab}$ and $\\Gamma_{abcde}$ are Dirac Gamma matrices in $d=11$,\nand $C_{\\alpha \\beta} = -C_{\\beta \\alpha}$ is the charge conjugation matrix.\n\nThe computation of\n$\\left\\langle \\cdots \\right\\rangle$\nis thus reduced to the calculation of traces of products of Gamma matrices,\nwhich is conceptually straightforward but may be quite challenging in practice.\nIn fact, the invariant polynomial so-defined has yet to be fully computed explicitly,\nwhich means that the detailed structure of the TZ theory is not well known.\n\n\n\\section{CJS Supergravity and the TZ Theory}\n\\label{sec:main}\n\nThe inclusion of the length parameter $l$ in eq.~(\\ref{eq:A}) and the algebraic structure of CS Lagrangians\n[see eq.~(\\ref{eq:TZLag})]\nimply that the TZ Lagrangian can be written as a polynomial of degree~11 in $1\/l$, namely\n\\begin{equation}\n\\mathcal{L}_{\\text{TZ}} =\n\\sum_{p=0}^{11}\nl^{-p}\n\\mathcal{L}^{\\left( p \\right)}.\n\\label{eq:LTZpol}\n\\end{equation}\nWe expect $l$ to be\na small quantity.\nThe most relevant terms in eq.~(\\ref{eq:LTZpol}) are then\n\\begin{equation}\n\\mathcal{L}_{\\text{TZ}} =\nl^{-11} \\mathcal{L}^{\\left( 11 \\right)} +\nl^{-10} \\mathcal{L}^{\\left( 10 \\right)} +\nl^{-9} \\mathcal{L}^{\\left( 9 \\right)} +\n\\cdots.\n\\label{eq:LTZ3}\n\\end{equation}\nWithout yet evaluating the invariant polynomial\n$\\left\\langle \\cdots \\right\\rangle$,\nthe first two terms are given by\n(see~\\ref{sec:CJSTZ})\n\\begin{align}\n\\mathcal{L}^{\\left( 11 \\right)} & =\n \\frac{6}{11}\n \\left\\langle\n   \\bm{e}\n   \\left(\n     \\bm{e}^{2}\n   \\right)^{5}\n \\right\\rangle, \\label{eq:L11sim} \\\\\n\\mathcal{L}^{\\left( 10 \\right)} & =\n 15\n \\left\\langle\n  \\bm{\\psi}\n  \\left[ \\bm{e} , \\bm{\\psi} \\right]\n  \\left(\n   \\bm{e}^{2}\n  \\right)^{4}\n \\right\\rangle +\n 3\n \\left\\langle\n  \\bm{eT}\n  \\left( \\bm{e}^{2} \\right)^{4}\n \\right\\rangle, \\label{eq:L10sim}\n\\end{align}\nwhere, for simplicity, we have set $\\bm{b} = 0$.%\n\\footnote{Here we use the abbreviations\n$\\bm{e} = e^{a} \\bm{P}_{a}$,\n$\\bm{\\omega} = \\frac{1}{2} \\omega^{ab} \\bm{J}_{ab}$,\n$\\bm{b} = \\frac{1}{5!} b^{abcde} \\bm{Z}_{abcde}$, and\n$\\bm{\\psi} = \\bar{\\bm{Q}} \\psi$.\nAll powers of $l$ are displayed explicitly.}\nAs advertised, the leading term is a cosmological constant,\nwhile the sub-leading term includes both torsion and a mass term for the fermions.\n\nThe evaluation of the next term\n(the one proportional to $1\/l^{9}$)\nis significantly more involved,\nsince it requires the computation of many components of\n$\\left\\langle \\cdots \\right\\rangle$.\nWe have left all details concerning this calculation for~\\ref{sec:CJSTZ},\nwhere we also explain the notation and conventions used below.\nThe result reads\n\\begin{align}\n\\mathcal{L}^{\\left( 9 \\right)} & =\n- \\frac{1}{4 \\times 9! \\kappa_{2}^{2}} \\varepsilon_{a_{1} \\cdots a_{11}} R^{a_{1} a_{2}} e^{a_{3}} \\cdots e^{a_{11}}\n- \\frac{7i}{4} \\bar{\\psi} \\Gamma_{\\left( 8 \\right)} \\mathrm{D} \\psi\n- \\frac{1}{2} ab\n+ \\nonumber \\\\ &\n+ \\frac{i}{8} \\left( T^{a} - \\frac{i \\kappa_{1}^{2}}{4} \\bar{\\psi} \\Gamma^{a} \\psi \\right) e_{a} \\bar{\\psi} \\Gamma_{\\left( 6 \\right)} \\psi\n+ \\frac{7}{2^{4} \\times 5} a \\left. \\ast a \\right.\n- \\frac{1}{2^{10} \\times 3^{2}} \\frac{\\left( 4! \\right)^{5}}{6!} b \\left. \\ast b \\right.,\n\\label{eq:L9}\n\\end{align}\nwhere we have also set $\\bm{b} = 0$.\n\nCompare now eq.~(\\ref{eq:L9}) with the CJS Lagrangian~\\cite{Jul99},\n\\begin{align}\n\\mathcal{L}_{\\text{CJS}} & =\n- \\frac{1}{4 \\times 9! \\kappa^{2}} \\varepsilon_{a_{1} \\cdots a_{11}} R^{a_{1} a_{2}} e^{a_{3}} \\cdots e^{a_{11}}\n+ \\frac{i}{2} \\bar{\\psi} \\Gamma_{(8)} \\mathrm{D} \\psi\n- \\frac{1}{2} ab\n+ \\nonumber \\\\ &\n+ \\frac{i}{8} \\left( T^{a} - \\frac{i \\kappa^{2}}{4} \\bar{\\psi} \\Gamma^{a} \\psi \\right) e_{a} \\bar{\\psi} \\Gamma_{\\left( 6 \\right)} \\psi\n- \\frac{1}{2} \\left( a + F \\right) \\left. \\ast F \\right.,\n\\label{eq:LCJS}\n\\end{align}\nwhere we have set $A_{3} = 0$.\n\nThe similarity between eqs.~(\\ref{eq:L9}) and~(\\ref{eq:LCJS}) is, of course, striking.\nApart from some slightly different numerical coefficients,\nthe main difference is to be found among the last few terms,\nwhich are precisely those more sensitive to the truncation made by setting\n$\\bm{b} = 0$ and $A_{3} = 0$.\n\nIt could be argued that this similarity is just accidental,\nand that, given the highly nonlinear structure of the TZ Lagrangian,\nalmost any imaginable kind of term will appear in it.\nInterestingly enough, this is not the case:\nthere are many possible combinations which could in\nprinciple appear in the $l^{-9}$ sector of the TZ Lagrangian,\nand which do not show up in the CJS supergravity Lagrangian.\nNone of these appear.\nSome examples are\n\\begin{align}\n\\left[ EB \\right]^{abcde} & = \\nwse{E}{abcd}{f} B^{fe}, \\label{noshowfirst} \\\\\n\\left[ EBBB \\right]^{abcde} & = \\nwse{E}{ab}{fgh} B^{fc} B^{gd} B^{he}, \\\\\n\\left[ EBB^{2} \\right]^{abcde} & = \\nwse{E}{abc}{fg} B^{fd} \\nwse{B}{g}{h} B^{he}, \\\\\n\\left[ EB^{3} \\right]^{abcde} & = \\nwse{E}{abcd}{f} \\nwse{B}{f}{g} \\nwse{B}{g}{h} B^{he}, \\\\\n\\left[ BEB \\right]^{abc} & = B_{de} \\nwse{E}{deab}{f} B^{fc}, \\\\\n\\left[ BBEB \\right]^{a} & = B_{bc} B_{de} \\nwse{E}{bcde}{f} B^{fa}, \\label{noshowlast}\n\\end{align}\nwhere $B^{ab}$ is any antisymmetric two-index tensor constructed from the physical fields\n(e.g., $e^{a} e^{b}$, $R^{ab}$, etc.).\nSimilarly, $E^{abcde}$ stands for any antisymmetric five-index combination, for which,\neven with $b^{abcde}=0$, there are several possibilities:\n$\\bar{\\psi} \\Gamma^{abcde} \\psi$, $e^{a} e^{b} e^{c} e^{d} e^{e} $, etc.\n\n\n\n\\section{Discussion}\n\\label{sec:final}\n\nWe have shown that the TZ Lagrangian may be cast as a polynomial of degree 11 in $1\/l$,\nwhere $l$ is a (small) length parameter,\nand that the term proportional to $1\/l^{9}$\nis essentially identical to the CJS Lagrangian.\n\nOne significance of our result lies on its improbability:\nthe TZ theory and CJS supergravity emerge from entirely different backgrounds,\nyet both seem to be related at a deep level.\nWe would like to emphasize this:\nto the best of our knowledge,\nthere is a priori no reason whatsoever for the TZ theory to be related in this way to CJS supergravity,\nyet the relation appears.\n\nFor instance, the CJS Lagrangian includes the Hodge $\\ast$-operator,\nwhereas the TZ Lagrangian, being formulated with no a priori background metric, does not.\nThis sole fact could be na\\\"{\\i}vely argued to provide a reason for the absence of a relation between both theories.\nHowever, terms such as $a \\left. \\ast a \\right.$ do end up appearing in the TZ Lagrangian.\nThis can be traced back to the fact that a duality relation exists between different Dirac Gamma matrices in eleven-dimensional spacetime.\n\nA further interesting aspect deals with the \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\ symmetry:\nour results suggest that the CJS Lagrangian fails to be invariant under \\ensuremath{\\mathfrak{osp} \\left( 32 \\middle\\vert 1 \\right)}\\\nbecause it lacks all the other terms that ensure the invariance of the TZ Lagrangian.%\n\\footnote{It has been known for some time, however, that CJS supergravity has on-shell symmetry under a one-parameter family of superalgebras~\\cite{DAu82,Ban04a,Ban04b}.}\n\nThe cosmological constant issue is seen here under a new light.\nWhile it is well known that CJS supergravity forbids a cosmological constant,\nthe TZ theory includes it on a different level, namely, in the term proportional to $1\/l^{11}$\nrather than in the one proportional to $1\/l^{9}$ (where the CJS terms reside).\nThe implications of this fact have yet to be fully assessed.\n\nThese observations lead us to propose the conjecture that CJS supergravity might actually be a truncation of a more general theory, such as the TZ theory.\nIn fact, the possibility that the CJS theory may be included within the TZ theory was already put forward by its authors~\\cite{Tro97}.\n\nOur result also highlights the usefulness of the CS subspace separation method~\\cite{Iza06a}\nfor manipulating CS Lagrangians in higher-dimensional spacetime.\nGiven the complexities involved, such powerful methods become essential tools\nto extract any meaningful information from CS theories (see~\\ref{sec:CJSTZ}).\n\nThere are several ways in which this work could be extended.\nThe inclusion of a nonzero $b^{abcde}$ is obviously one of them,\nbut the technical hurdles involved may be hard to overcome.\nSince only one-forms are allowed in CS theories,\nthe $A_{3}$ three-form must be first decomposed in terms of the TZ fields before a comparison is attempted.\nThe idea of decomposing $A_{3}$ in terms of one-forms has been already analyzed in the literature\n(see, e.g., Refs.~\\cite{Zan05,Ban04b}).\nOur construction has, however, an additional constraint;\n$A_{3}$ must be of order $l^{-3}$,\nand therefore, there are forbidden combinations.\nFor instance, the gravitino cannot be part of $A_{3}$,\nbecause it will lead to orders $l^{-1}$ and $l^{-2}$.\nDerivatives of the fields are also forbidden,\nsince it would lead at most to order $l^{-2}$ and order $l^{-3}$ would never be reached.\nTherefore, the identification of $A_{3}$ with a term such as $e^{a} T_{a}$ \\cite{Banh01}\nis ruled out from the outset.\nA moment's thought shows that the only consistent way of decomposing $A_{3}$\nin terms of the TZ fields is through the ansatz\n\\begin{align}\nA_{3} & =\nC_{abc} e^{a} e^{b} e^{c} +\n\\frac{1}{5!} C_{abc_{1} \\cdots c_{5}} e^{a} e^{b} b^{c_{1} \\cdots c_{5}}\n+ \\frac{1}{\\left( 5! \\right)^{2}} C_{ab_{1} \\cdots b_{5} c_{1} \\cdots c_{5}} e^{a} b^{b_{1} \\cdots b_{5}} b^{c_{1} \\cdots c_{5}} +\n\\nonumber \\\\ & +\n\\frac{1}{\\left( 5! \\right)^{3}} C_{a_{1} \\cdots a_{5} b_{1} \\cdots b_{5} c_{1} \\cdots c_{5}} b^{a_{1} \\cdots a_{5}} b^{b_{1} \\cdots b_{5}} b^{c_{1} \\cdots c_{5}},\n\\label{eq:A3}\n\\end{align}\nwhere the various $C$-tensors are fully anti-symmetric zero-forms to be determined.\n\nAs mentioned in section~\\ref{sec:TZ}, the invariant polynomial we have used is but the simplest choice.\nMore general invariant polynomials may prove useful for eliminating the slight differences that still remain between the CJS Lagrangian and the $1\/l^{9}$ sector of the TZ theory.\n\nAnother interesting issue is the exploration of the solution space of the TZ theory.\nWhile the full theory is difficult to analyze due to the nonlinearities of the Lagrangian~\\cite{Banh95,Banh96b,Cha99,Mis03},\nthe truncated version (at order $1\/l^{9}$) should prove to be more tractable.\n\n\n\\section*{Acknowledgments}\n\nThe authors wish to thank\nJos\u00e9~A.\\ de~Azc\u00e1rraga,\nAlfredo P\u00e9rez,\nPatricio Salgado and\nJorge Zanelli\nfor many fruitful discussions regarding the topics covered in the present work.\nWe are also indebted to\nRicardo Troncoso\nfor helpful comments on the first draft of this manuscript.\nF.~I.\\ extends his thanks to\nJos\u00e9~A.\\ de~Azc\u00e1rraga\nfor his kind hospitality at the Department of Theoretical Physics of the Universitat de Val\\`{e}ncia, and to\nMicho {\\DJ}ur{\\dj}evich\\\nfor his kind hospitality at the Institute of Mathematics of the Universidad Nacional Aut\u00f3noma de M\u00e9xico, where part of the present work was done.\nE.~R.\\ is grateful to\nJorge Zanelli\nfor his kind hospitality at the Centro de Estudios Cient\u00edficos (CECS), Valdivia, where part of this work was done.\nThe authors also thank\nKasper Peeters\nfor his computer algebra system ``Cadabra'' \\cite{Pee06,Pee07}, which was of invaluable help when performing lengthy Dirac-matrix algebra computations, and\nRicardo Ram\u00edrez,\nfor helping to create the necessary code to evaluate some of the components of the invariant polynomial.\nF.~I.\\ and E.~R.\\ were supported by the\nNational Commission for Scientific and Technological Research, Chile, through Fondecyt research grants 11080200 and 11080156, respectively.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t        \nThe majority of stars do not form in isolation but in areas of clustered star formation \\citep[e.g.][]{lada2003}. In some cases, the densest areas of these large regions result in gravitationally bound stellar systems, commonly referred to as star clusters. These bound systems can survive for hundreds of Myr. To distinguish them from ancient stellar objects like the globular clusters (GCs), we refer to them as young stellar clusters (YSCs). They usually populate star-forming galaxies in the local universe (e.g. \\citealp{larsen2006b}) and their physical properties (ages, masses) can in principle be used to determined star formation histories (SFHs) of the hosting galaxies \\citep[e.g.][]{miller1997,goudfrooij2004,konstantopoulos2009,glatt2010}.\n\nOver the past 20 years, studies of the distributions of YSC luminosities and masses in local galaxies have shown that they are well-described by a power law function of the form $Ndm \\propto M^\\alpha dm$, with a slope $\\alpha \\sim-2$, observed both for low-mass clusters in the Milky Way \\citep{piskunov2006}, in the Magellanic Clouds \\citep{baumgardt2013,degrijs2006} and in M31 \\citep{fouesneau2014}, and for sources up to masses of $\\sim10^5-10^6$ M$_\\odot$\\ in nearby spirals and starburst galaxies \\citep{chandar2010,konstantopoulos2013,whitmore2010}. This result is expected if star formation happens in a hierarchical manner, dominated by interstellar medium (ISM) turbulence, and the clusters occupy the densest regions (e.g. \\citealp{elmegreen2006}; see also \\citealp{elmegreen2010b} for a review).\n\nDespite observational and theoretical progress over the past few decades, many questions concerning the properties of YSC populations remain open. Among these: is cluster formation only driven in space and time by size--of--sample effects \\citep[e.g.][]{hunter03}, with an increasing number of clusters found in galaxies with higher star formation rate (SFR)? Will the galactic environment (ISM conditions, gas fraction, galaxy type) where clusters form leave an imprint on the final properties of the YSC populations? When we look at YSC populations in local spirals \\citep[e.g.][]{larsen2004}, merger systems \\citep[e.g.][]{whitmore1999} and dwarf galaxies \\citep[e.g.][]{billett2002} it is challenging to discern the role played by statistical sampling \\citep[e.g.][]{fumagalli2011} and environment.\n\nEven the exact shape of the mass function is still debated, in particular concerning its high mass end. Some early studies \\citep[e.g.][]{larsen2006} have pointed out the dearth of massive YSCs if a single power law fit of slope $-2$ describes the upper-part of the YSC mass function. \\cite{gieles2006} have proposed a Schechter function as a better description of the YSC mass function in local galaxies, due to a mass truncation at a characteristic mass above which the likelihood of forming massive clusters goes rapidly to zero. \n\n\nIn order to be able to characterise how star clusters form and evolve, \nit is important to study a statistically meaningful sample. The Legacy Extragalactic UV Survey (LEGUS) is a Cycle 21 HST Treasury program which observed 50 nearby galaxies from the UV to NIR bands, with the goal of deriving high quality star cluster catalogues, and, more in general, of studying star formation at intermediate scales, linking the smallest (stellar) scales to the larger (galactic) ones \\citep[see][]{legus1}. \nIn general, the large number of galaxies and galaxy properties available in LEGUS will enable us to statistically study YSC populations over a wide range of galactic environments \\citep{legus2}. \n\nAmong the most interesting galaxies in the LEGUS catalogue is NGC 5194 (also known as M51a or the Whirlpool Galaxy), because of its proximity and the number of star clusters that it hosts.\nIt is a spiral galaxy, catalogued as SAbc\\footnote{According to the Nasa Extra-galactic Database (NED)}, almost face-on (inclination angle $i\\approx22^\\circ$, \\citealp{colombo2014b}) at a distance of 7.66 Mpc \\citep{distancem51}. M51a is interacting with the (smaller) companion galaxy NGC 5195 and it is probably this interaction that is the cause of a marked spiral geometry and a high star formation process (a SFR value of 2.9 M$_\\odot$\/yr is derived from published total fluxes in the far-$UV$ and 24$\\mu$m, combined using the recipe by \\citealp{hao2011}) sustained over time \\citep[e.g.][]{dobbs2010_m51}. The two galaxies together form the M51 system. In the remainder of this paper we will use the name M51 mainly referring to the main spiral galaxy M51a.\nThis galaxy hosts numerous star formation complexes \\citep{bastian2005b}, HII regions \\citep{thilker2000,lee2011} and YSCs and it has been a benchmark in the study of extragalactic star and cluster formation. \n\nHigh-brightness blue sources in M51 have been studied already by \\citet{georgiev1990}.\nIn more recent years, broadband and narrowband imaging with the Hubble Space Telescope (HST) Wide-Field Planetary Camera 2 (WFPC2) in various bands from $UV$ to $NIR$ were used for initial studies of the cluster population in small parts of the galaxy \\citep{bik2003,bastian2005,gieles2005,lee2005}. Later optical observations with the higher resolution and more sensitive ACS camera were obtained in the $BVI$ bands and covered uniformly the entire galaxy allowing to extend the investigation of the YSC population to the whole galaxy \\citep{scheepmaker2007,hwang_lee2008, chandar2011}. More recently, the coverage by the WFPC2 F336W filter ($U$-band) has been expanded, with 5 more pointings, along with H$\\alpha$\\, data, allowing improved age determination for a significant fraction of the cluster population\\footnote{The $U$-band filter (or bluer filters) is fundamental to break the age-extinction degeneracy when SEDs are compared to stellar population synthesis models, see \\citet{anders2004}} \\citep{scheepmaker2009,hwang_lee2010,chandar2011,chandar16}.\n\nAll this effort led to the consensus that the star cluster population in M51 can be described by a standard mass distribution, i.e. a simple power law with slope -2. However, whether the single-power law function is also a good representation of the upper-mass end of the cluster mass function, in terms of the eventual presence of a truncation at high masses, is still under debate (compare e.g. \\citealp{gieles2006} and \\citealp{chandar2011}). \nThe analyses of the cluster mass function evolving in time, and, more in general, of the cluster number densities evolving with time, reach  different conclusions on the disruption properties of the clusters in M51. Some studies observe a mass function evolution consistent with a disruption time dependent on the mass of the clusters \\citep[e.g. the mass-dependent disruption -MDD- model by][]{gieles2009}, while in others the study of the mass function (MF) evolution seems to exclude this model, and to favour a constant disruption time of clusters \\citep[e.g. mass independent disruption -MID- model by][]{chandar16}. \n\nThe interaction of M51 has been studied using simulations in order to describe the current geometrical and dynamical properties of the star formation  \\citep{salo2000,dobbs2010_m51}. \nCluster properties have then been compared with the expectations based on simulations in order to test the models for the formation of the spiral structure (e.g. \\citealp{chandar2011} ruled out the possibility of self-gravity as the cause of the generation of the spiral structure). \n\nStar formation in M51 has also been studied from the point of view of molecular gas via radio observations (\\citealp{schuster2007,koda2009,koda2011} and \\citealp{schinnerer2010,schinnerer13} among the most recent). High resolution interferometric data have been used to study in detail the properties of giant molecular clouds (GMCs) \\citep{koda2012,colombo2014a}.\nThe possibility of studying the galaxy at high resolution at different wavelengths allows studying star formation at different ages, in particular to compare the properties of the progenitors (GMCs) and the final products (stars and star clusters). \n\nOne of the goals of the present work is to conduct a statistically driven study of the YSC population of M51 using the new data and cluster catalogue produced by the LEGUS team. The new LEGUS dataset of M51 provides 5 new pointings in the $NUV$ (F275W and F336W) with the Wide Field Camera 3 (WFC3). The improved spatial resolution of the WFC3 and sensitivity in the $NUV$ give a better leverage on the physical determinations of the YSC properties \\citep{legus1}.\nIn order to compare our new catalogue with previously published works we investigate, in this paper, YSC mass and luminosity functions for the whole galaxy. With the help of simulated Monte Carlo cluster populations we build a comprehensive picture of the cluster formation and evolution in the galaxy as a whole. In a forthcoming paper (Messa et al., in prep, hereafter Paper II) we test whether YSC properties change across the galaxy as a function of star formation rate (SFR) density ($\\Sigma_{\\textrm{SFR}}$) and gas surface density.  These results can shed light on a possible environmental dependences in the properties of the cluster population and whether studies of YSC populations can be used to characterise the galactic environment.\n\nThe paper is divided as follows: a short description of the data is given in Section~\\ref{sec:data} and the steps necessary to produce the final cluster catalogue are described in Section~\\ref{sec:ccp}. In Section~\\ref{sec:global} the global properties of the sample (luminosity, mass and age functions) are studied, while in Section~\\ref{sec:montecarlo} the same properties are analysed using simulated Monte Carlo populations. The fraction of star formation happening in a clustered fashion is studied in Section~\\ref{sec:cfe}.\nFinally, the conclusions are summarised in Section~\\ref{sec:conclusions}. \n\n\n\\section {Data}\n\\label{sec:data}\nA detailed description of the LEGUS general dataset and the standard data reduction used for LEGUS imaging is given in \\citet{legus1} and we refer the reader to that paper for details on the data reduction steps.\n\nHere we summarize the properties of the data used in this study. The M51 system (NGC 5194 and NGC 5195) spans $\\sim7 \\times 10$ arcmin on the sky at optical wavelengths (at an assumed distance of 7.66 Mpc, from \\citealp{distancem51}) and several pointings are therefore necessary to cover their entire angular size. \nThe LEGUS dataset includes multi-band data spanning the wavelength range from near-$UV$ to near-$IR$; data for M51 cover the $UV$ (F275W), $U$ (F336W), $B$ (F435W), $V$ (F555W) and $I$ (F814W) bands. Even though no conversion is applied to the Cousins-Johnson filter system, we keep the same nomenclature, due to the similarity of the central wavelength between that system and our data. Concerning the $B$, $V$ and $I$ filters, ACS WFC archival data\navailable from the Mikulski Archive for Space Telescopes (MAST) have been re-processed. The data in these bands include 6 pointings that cover the entire galaxy and the companion galaxy NGC 5195 (GO-10452, PI: S. Beckwith).\n\nWithin the LEGUS project, the coverage has been extended to the $U$ and $UV$ bands. The new $UV$ data consist of 4 pointings covering the arms and outskirts of the galaxy combined with a deep central exposure (GO-13340, PI: S. Van Dyk) covering the nuclear region of the galaxy. Exposure times for all filters are summarized in Tab. \\ref{tab:data} while the footprints of the pointings are illustrated in Figure \\ref{fig:footprints}.\n\n\\begin{table}\n\\centering\n\\caption{Exposure times for the different filters and number of pointings (the exposure times refer to each single pointing). As can be noted also in Fig.~\\ref{fig:footprints} the ACS data cover the entire galaxy with 6 pointings, while for the $UV\/U$ band the observations consist of 5 pointings only.}\n\\begin{tabular}{clccl}\n\\hline\nInstr. & Filter & Expt. & \\# & Project Nr. \\& PI \\\\\n\\hline\n\\hline\nWFC3\t& F275W($UV$)& 2500 s\t& 4\t& GO-13364\\ \\ \t D. Calzetti \\\\\n\\ \t\t & \\\t\t\t& 7147 s\t& 1\t& GO-13340\\ \\ \t S. Van Dyk \\\\\nWFC3\t& F336W($U$)\t& 2400 s\t& 4\t& GO-13364\\ \\\t D. Calzetti \\\\\n\\\t\t & \\\t\t\t& 4360 s\t& 1\t& GO-13340\\ \\ \t S. Van Dyk \\\\\nACS \t\t& F435W($B$)\t& 2720 s\t& 6\t& GO-10452\\ \\   S. Beckwith \\\\\nACS\t\t& F555W($V$)\t& 1360 s\t& 6\t& GO-10452\\ \\ \t S. Beckwith \\\\\nACS\t\t& F814W($I$)\t& 1360 s\t& 6\t& GO-10452\\ \\ \t S. Beckwith \\\\\n\\hline\n\\end{tabular}\n\\label{tab:data}\n\\end{table}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{01_ngc5194_footprints.png}\n\\caption{UVIS (red) and ACS (orange) footprints on a DSS image of the NGC5194 and NGC5195 system. The UVIS (white) footprint corresponds to proposal 13340 (PI: S. Van Dyk). See also Tab. \\ref{tab:data} for more info on the observations.}\n\\label{fig:footprints}\n\\end{figure}\n\n\n\\section{Cluster Catalogue production}\n\\label{sec:ccp}\n\\subsection{Cluster Extraction}\n\\label{sec:cluster_extraction}\nIn order to produce a cluster catalogue of the M51 galaxy we follow the procedures described in \\cite{legus2}, where a detailed description of the standard reduction steps can be found. Hereafter we describe these steps along with the specific parameters used for the M51 dataset. The catalog\nproduction is divided into 2 main parts, the cluster extraction and the cluster classification.\n\nThe cluster extraction is executed through a semi-automatic custom pipeline available inside the LEGUS collaboration. As the first step we extracted the source position of the cluster candidates in the $V$ band (used as reference frame in our analysis) with \\texttt{SExtractor} \\citep{sextractor}. The parameters of \\texttt{SExtractor} were chosen to extract sources with at least a 10$\\sigma$ detection in a minimum of 10 contiguous pixels.\nIn the same band, we measured the concentration index (CI) on each of the extracted sources.\nWe use the definition for the CI as the magnitude difference between the fluxes in circular regions of 1 and 3 pixels radius, centred on the source position. It measures how much the light is concentrated in the centre of the source and can also be used as also a tracer of the cluster size \\citep[see][]{ryon2017}. \n\n\\begin{figure*}\n\\centering\n\\subfigure{\\includegraphics[width=0.9\\columnwidth]{02a_plot_histogram_ci_fullcat.pdf}}\n\\hspace{1mm}\n\\subfigure{\\includegraphics[width=0.9\\columnwidth]{02b_plot_histogram_ci.pdf}}\n\\subfigure{ \\includegraphics[width=0.18\\textwidth]{02c_plot_apcor_275_f275w.png}}\n\\subfigure{ \\includegraphics[width=0.18\\textwidth]{02d_plot_apcor_336_f336w.png}}\n\\subfigure{ \\includegraphics[width=0.18\\textwidth]{02e_plot_apcor_435_f435w.png}}\n\\subfigure{ \\includegraphics[width=0.18\\textwidth]{02f_plot_apcor_f555w.png}}\n\\subfigure{ \\includegraphics[width=0.18\\textwidth]{02g_plot_apcor_814_f814w.png}}\n\\caption{\\textit{Top left:} Distribution of the CI values for all the sources extracted with \\texttt{SExtractor}. The red solid line indicates the value of 1.35 used to select the cluster catalogue. \\textit{Top right:} CI distribution for visually selected stars (in red) and clusters (in blue). \\textit{Bottom:} Distributions of the aperture corrections given by visually selected clusters. Black dashed lines set the interval within which values are considered to calculate the average value (blue bin). \n}\n\\label{fig:histo_CI}\n\\end{figure*}\nThe distribution of the CI values for the extracted sources looks like a continuous distribution, peaked around a value of 1.3, as Fig.~\\ref{fig:histo_CI} (top left) shows, but is in fact the sum of two sub-distributions, one for stars and one for clusters. To understand how the distributions of CI values changes between stars and clusters we select via visual inspection a sample of stars and clusters that are used as training samples for our analysis. In Fig.~\\ref{fig:histo_CI}, top right, we show the CI distributions of stars and clusters. It becomes clear that the distributions of CIs in the two cases are different. Stars, being point sources, have CI values that do not exceed values of $\\sim$1.4, while clusters have on average higher CI values and a broader distribution. The distributions also suggest that considering sources with a CI bigger than 1.35 strongly decreases the chances of erroneously including stars in the catalogue, thus facilitating the selection of most of the clusters.  Following the CI versus effective radius relation showed in Fig. 4 of \\citet{legus2}, we estimate that a CI cut at 1.35 mag corresponds to a cluster effective radius of 0.5 pc. Because the distribution star cluster radii peak at $\\sim$3 pc \\citep{ryon2017}, placing a CI cut at 1.35 mag will not negatively impact the recovery of clusters in this system.\n\nAperture photometry was performed on the CI-selected sample, using fixed apertures of 4 pixels radius, and local sky is subtracted from an annulus with 7 pixels (px) interior radius and 1 px width. \nA fixed aperture correction was estimated using the photometric data of the visually selected sample of clusters. The sample was adjusted in each filter in order to consider only isolated bright clusters with well defined PSF wings. The number of visually selected sources used in each filter are listed in Tab~\\ref{tab:corrections}. During the visual selection, sources were chosen to span different cluster sizes and to also include compact clusters. In this way the resulting aperture correction is not biased towards the very large clusters which are more easily detectable. For each source the single aperture correction was calculated subtracting the standard photometry (aperture: 4 px and sky at 7 px) with the total photometry within a 20 px radius (with a 1 px wide sky annulus at a radius of 21 px). The final correction in each filter was calculated taking the average of the values within an allowed range of values. The single aperture correction values of the selected sample along with the final mean value in each filter are plotted in Fig.~\\ref{fig:histo_CI} (bottom) and the values are also reported in Tab.~\\ref{tab:corrections}. The standard deviation ($\\sigma_{\\lambda}$) is added in quadrature to the photometric error of each source. \n\\begin{table}\n\\centering\n\\caption{Corrections applied to the photometry of all the sources in different filters. With reddening we refer to the Galactic extinction in magnitudes (in each filter). The uncertainty on the average aperture correction has been used to estimate the final error on the magnitude.}\\begin{tabular}{ccccc}\n\\hline\nFilter\t\t& Reddening\t& \\# Clusters \t& Avg ap.corr. \t& $\\sigma_{\\textrm{ap.corr.}}$\t\\\\\n\\\t\t& [mag]\t\t&  \t\t\t& [mag]\t \t& [mag]\t\\\\\n\\hline\n\\hline\nF275W\t& 0.192\t\t& 36\t\t\t\t& -0.628  \t\t& 0.034 \\\\\nF336W\t& 0.156\t\t& 66\t\t\t\t& -0.668  \t\t& 0.030 \\\\\nF435W\t& 0.127\t\t& 62\t\t\t\t& -0.665  \t\t& 0.026 \\\\\nF555W\t& 0.098\t\t& 71\t\t\t\t& -0.663\t\t& 0.023 \\\\\nF814W\t& 0.054\t\t& 56\t\t\t\t& -0.830\t\t& 0.031 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:corrections}\n\\end{table}\n\nA final cut was made excluding sources which are not detected in at least 2 contiguous bands (the reference $V$ band and either the $B$ or $I$ band) with a photometric error smaller than 0.3 mag. The positions of the 30176 sources satisfying the CI cut of 1.35 mag and this last selection criterion are collected, along with their photometric data, in a catalogue named \\textit{``automatic\\_catalog\\_avgapcor\\_ngc5194.tab''}, following the LEGUS naming convention. Note that, being automatically-selected, this catalogue probably includes contaminant sources (e.g. foreground stars, background galaxies, stars in the field of M51).\n\nTo remove the contamination of non-clusters in the automatic catalog, we created a high-fidelity sub-catalog including all sources detected in at least 4 bands with a photometric error below 0.30 mag and having an absolute $V$ band magnitude brighter than -6 mag. Selecting only bright sources reduces considerably the number of stars in the catalogue, while the constraint on the number of detected bands allows a reliable process for the SED fitting analysis (see Section~\\ref{sec:sed}). Note that, differently from the standard LEGUS procedure, we applied the -6 mag cut to the \naverage--aperture--corrected magnitudes and not to the CI--based--corrected ones (see \\citealp{legus2} for a description of the CI--based correction).\nThis choice is motivated mainly by the use of the average--aperture--corrected catalogue as the reference one: when testing the completeness level of our catalogue (see Section~\\ref{sec:completeness}) we noticed that applying the cut on the average--aperture--corrected magnitudes improves the completeness, being more conservative (i.e. allowing the inclusion of more sources). This high-fidelity catalogue counts 10925 sources, which have been all morphologically classified (see Section~\\ref{sec:classification}).\n\n\\subsection{Morphological classification of the cluster candidates, human versus machine learning inspection}\n\\label{sec:classification}\n\nSources in the high fidelity catalogue were visually inspected, in order to morphologically classify the cluster candidates and exclude stars and interlopers that passed the automatic selection.\nLike for the other galaxies of the LEGUS sample, visually inspected sources were divided into 4 classes, described and illustrated in \\citet{legus2}. Briefly, class 1 contains compact (but more extended than stars) and spherically symmetric sources while class 2 contains similarly compact sources but with a less symmetric light distribution. Both these classes include sources with a uniform colour. Class 3 sources show multi-peaked profiles with underlying diffuse wings, which can trace the presence of (small and compact) associations of stars. Sources in class 3 can have colour gradients. Contaminants like single stars, or multiple stars that lie on nearby pixels even if not part of a single structure, and background galaxies are all stored in class 4 and excluded from the study of the cluster population of the galaxy.\n\n\\begin{table}\n\\centering\n\\caption{Number of sources in each class for the human-classified (2nd column) and machine-learning classified (3rd column) sources. In parentheses are the percentage of sources over the number of the visually inspected sources. The fourth column lists the percentage of the sources in each class for which the ML has assigned the same class as the humans. In the fifth column the number of sources per class classified by the machine learning algorithm is given. In brackets we include the recovered fraction with respect to the total number of sources in the catalogue (i.e. 10925 sources).}\n\\begin{tabular}{cllll}\n\\hline\nClass \t& Human \t\t\t& ML\t\t\t\t&HvsML\t\t& ML\t (tot cat.) \\\\\n\\hline\n\\hline\ntot\t\t& 2487\t\t\t& 2487\t\t\t&\\ \t\t\t& 10925\t\t\\\\\n\\hline\n1\t\t& 360 ($14.5\\%$)  \t& 377 ($15.2\\%$)\t& $95.3\\%\t$\t& 1324 ($12.1\\%$)\\\\\n2\t\t& 500 ($20.1\\%$) \t& 516 ($20.7\\%$)\t& $93.4\\%$\t& 1665 ($15.2\\%$)\\\\\n3\t\t& 365 ($14.7\\%$)  \t& 338 ($13.6\\%$)\t& $92.1\\%$\t& 385 ($3.5\\%$)\\\\\n4\t\t& 1262 ($50.7\\%$)\t& 1256 ($50.5\\%$)\t& $95.6\\%$\t& 7551 ($69.1\\%$)\\\\\n\\hline\n\\end{tabular}\n\\label{tab:classification}\n\\end{table}\nDue to the large number of sources entering the automatic catalogue we have implemented the use of a machine learning (ML) optimised classification. We have visually inspected only $\\sim$1\/4 of the 10925 sources, located in different regions of the galaxy. This visually inspected subsample has been used as a training set for the ML algorithm to classify the entire catalogue (details of the algorithm are discussed in a forthcoming paper by Grasha et al., in prep). \nThe ML code is run on the entire sample of 10925 sources, including the already humanly--classified ones. In this way we can use the sources having a classification with both methods to estimate the goodness of the ML classification for M51. Table~\\ref{tab:classification} gives the number of sources classified in each class by human and ML, as well as the comparison between the two classification (forth column). We recover a 95\\% of agreement between the two different classifiers, within the areas used as training sets. To assess the goodness of the ML classification on the entire sample, we list in Table~\\ref{tab:classification} between brackets the relative fraction of each class with respect to the total number of sources classified with different methods. We observe that the relative number of class 1 and 2 sources  with respect to the total number of sources (10925) classified by the ML approach is only a few per cent smaller than the fraction obtained with the control sample (2487 sources).  However, there is a striking difference in the recovery fractions of  class 3 and 4 sources. When considering the entire catalogue, the relative number of class 3 objects is much smaller (and on the contrary the one of class 4 is significantly more numerous). We consider very unlikely that there are so few class 3 objects in the total sample. So far the ML algorithm fails in recognising the most variate class of our classification scheme that contains sources with irregular morphology, multi-peaked, and some degree of colour gradient. From the absolute numbers of sources per class it is easy to conclude that the ML code is able to reclassify correctly almost all the class 3 objects given as a training sample, but is unable to recognize new class 3 sources, considering many of them as class 4 objects.\nFuture improvements for the classification will be produced by the use of different ML recognition algorithms. For our current analysis we will focus on the properties of class 1 and 2 cluster candidates and exclude class 3 sources, as explained in Sec \\ref{sec:finalcat}.\n\n\nWe can summarise the photometric properties of the M51 cluster population using a two colour diagram (Fig~\\ref{fig:ccd}).\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{03_M51_CCD_optical_bestm_cl12.png}\n\\caption{Colour--colour diagram with $V-I$ on the x axis and $U-B$ on the y axis. Black points are all the sources in the catalogue that passed the automatic selection, while the orange shaded area gives the density of only the class 1 ad 2 sources. The SSP evolutionary track from Padova-AGB models is overplotted. The ages covered by the track goes from 1 Myr to 14 Gyr. The arrow indicates how an object would move if corrected for a reddening E($B-V$)=0.2. The cross at the top-right shows the average error in colours. }\n\\label{fig:ccd}\n\\end{figure}\nContours based on number densities of clusters show the regions occupied by the class 1 and class 2 M51 cluster population with respect to the location of the 10925 sources included by the automatic selection. A simple stellar population (SSP) track showing the cluster colour evolution as a function of age is also included. Sources are mainly situated along the tracks, implying the high quality of our morphological classification. Extinction spreads the observed colours towards the right side of the evolutionary tracks. Correcting for extinction, in the direction indicated by the black arrow, would move the sources back on the track, at the position corresponding to the best--fitted age.\nThe diagram shows a broad peak between 50 and 100 Myr. The contours are quite shallow towards younger ages and show a pronounced decline around $\\sim1$ Gyr suggesting that most of the sources detected are younger than 1 Gyr. We use SED fitting analysis to derive cluster physical properties (see Section~\\ref{sec:sed}), including the age distribution (Fig~\\ref{fig:agemass}b). \n\n\n\n\\subsection{Completeness}\n\\label{sec:completeness}\n\nTo investigate the completeness limit of the final catalogue, we use the custom pipeline available within the LEGUS collaboration as described in \\citet{legus2}.\nThe pipeline follows closely the selection criteria adopted to produce the final automatic catalogues.\nFor each filter we produce frames containing simulated clusters of different luminosities and sizes which are added to the original scientific frames. Effective radii ($R_\\textrm{eff}$) between 1 and 5 pc have been used, as studies of cluster sizes in similar galaxies suggest that most of the sources fall in this range \\citep{ryon2015,ryon2017}. The synthetic clusters span an apparent magnitude range between 19 and 26 mag and are created using the software \\texttt{BAOlab}, freely available online\\footnote{\\hyperref[http:\/\/baolab.astroduo.org\/]{http:\/\/baolab.astroduo.org\/}}. All clusters are simulated as symmetric sources with a MOFFAT15 profile \\citep[see][]{baolab} considered a standard assumption for the YSC light profiles \\citep{eff1987}.\nCluster extraction (via \\texttt{SExtractor}) and  photometry are repeated on the resulting coadded scientific and synthetic frames. \nA signal of 10$\\sigma$ in at least 10 contiguous pixel is requested to extract sources in the $B$, $V$, and $I$ bands, while a value of 5$\\sigma$ over at least 10 contiguous pixels is chosen for the $UV$ and $U$ bands.\nThe recovery rate of sources as a function of luminosity yields the completeness.\nThe magnitude limits above which the relative number of the recovered sources falls below 90\\% level is summarized in Tab.~\\ref{tab:completeness} for each filter. \n\\begin{table}\n\\centering\n\\caption{Completeness limits in each filter. The second and third column give the 90\\% completeness limit calculated with the completeness test code (described in the text). The completeness was computed in the disk (area outside a 35'' radius, second column) and in the central region (area inside 35'' radius, third column). The last column gives the peak magnitude of the luminosity distribution, as plotted in Fig~\\ref{fig:checkmaglimits}.}\n\\begin{tabular}{lccc}\n\\hline\nFilter\t\t& compl. (disk) & compl. (centre)\t& Lum$_\\textrm{peak}$\t\\\\\n\\\t\t& [mag] \t\t& [mag]\t\t\t& [mag]\t\\\\\n\\hline\n\\hline\nF275W\t& 22.17 \t& 21.75 \t& 21.75\t\\\\\nF336W\t& 22.75\t& 21.79\t& 22.00\t\\\\\nF435W\t& 24.17 \t& 22.65 \t& 23.25\t\\\\\nF555W\t& 23.70\t& 22.31\t& 23.25\t\\\\\nF814W\t& 22.70\t& 21.61\t& 22.25\t\\\\\n\\hline\n\\end{tabular}\n\\label{tab:completeness}\n\\end{table}\n\nThe completeness test code only gives a completeness limit estimated for each filter independently. These values should be considered as completeness limits resulting from the depth of the data. However, our cluster candidate catalogue is the result of several selection criteria that cross correlate the detection of sources among the 5 LEGUS bands. This effect can be visualized in Fig~\\ref{fig:checkmaglimits}, where we show the recovered luminosity distributions of sources at different stages of the data reduction. The requirement of detecting the sources in 4 filters with a photometric error smaller than 0.30 mag diminishes the number of recovered objects, mainly due to the smaller area covered by UVIS compared to ACS.\nThe cut at $M_V = -6$ mag not only excludes the sources which are faint in $V$ band, but also modifies the luminosity distributions in the other filters (see differences between red solid line distributions and blue dashed ones). Both these conditions affect the resulting catalogue and modify the shape of the luminosity distributions in each filter in a complicated way at the faint limits, therefore modifying also the completeness limits with respect to our approach of treating each filter independently. \nWe use the observed luminosity distributions to understand how the completeness limits change as a function of waveband and adopted selection criteria. \nWe observe in each filter an increase going from bright to low luminosities and we know that incompleteness starts to affect the catalogue where we see the luminosity distribution turning over (see e.g. \\citealp{larsen2002}). \n\\begin{figure*}\n \\centering\n \\includegraphics[width=\\textwidth]{04_plot_mag_lim_a.pdf}\n \\caption{Luminosity distributions showing how the completeness changes when different cuts are applied in the selection process. Each distribution represents the catalogue at a different stage:  the black dotted line shows the distribution of all the sources with photometric errors smaller than 0.3 mag, the blue dashed line represents the sources left after requiring that they have a detection in at least 4 filters and the red solid line is the distribution of the sources after the -6 mag cut in the $V$ band. Vertical lines are plotted corresponding to the peak of the distribution. The peak values after the magnitude cut is applied in $V$ band are listed in Table~\\ref{tab:completeness}. The sources are grouped in bins of 0.25 mag width. The black (orange) arrows show the value of the 90\\% completeness limit evaluated with the completeness test code in the disk (centre) of the galaxy (see Tab.~\\ref{tab:completeness}).\n }\n\\label{fig:checkmaglimits}\n\\end{figure*}\n\nWe draw the following conclusions from the analysis conducted in Fig~\\ref{fig:checkmaglimits}. First, the $-6$ mag cut in the $V$ band strongly reduces the number of selected sources in all the bands. At the distance of M51, this brightness corresponds to an apparent magnitude in $V$ of 23.4 mag, thus, it is brighter than the 90\\% completeness limit recovered in the $V$ band (23.84 mag, see Tab.~\\ref{tab:completeness}). Secondly, the 90\\% completeness limits fall rightwards of the peak in the luminosity distributions with the only exception for the F275W filter. For this reason we prefer to apply a more conservative approach and use the peak of the luminosity distribution as a limit for the cluster luminosity function analysis. Only in the case of F275W the 90\\% completeness limit is brighter than the peak magnitude, therefore the latter is adopted as the completeness limit value.\nWe stress that the part of the luminosity distribution leftwards of the peak remains almost identical after the selection cut (check Fig~\\ref{fig:checkmaglimits}) suggesting that the distributions are not affected by our selection criteria and completeness recovery at magnitudes brighter than the peak of the distributions.\n\nWe also tested completeness variations on sub-galactic scales. Our analysis shows that the completeness is worse towards the centre of the galaxy. Outside the inner region (radius larger than 35''), the $V$ band 90\\% completeness value is fainter than the cut at 23.4 mag (see Tab.~\\ref{tab:completeness} and Fig.~\\ref{fig:checkmaglimits}). Similar results are observed in the other filters. Because of the completeness drop at radii smaller than 35'' (1.3 kpc), we have excluded from our analysis this inner region.\n \n\\subsection{SED fitting}\n\\label{sec:sed}\nSources with detection in at least four filters were analyzed via SED fitting algorithms in order to derive physical properties of the clusters. We use uniformly sampled IMF to derive SSP models that include a treatment for nebular continuum and emission lines as described in \\citet{legus2}.  \nPutting together the different aperture correction methods, different stellar libraries and different extinction curves, 12 final catalogues are produced with the deterministic fitting method (and will be available online on the LEGUS website\\footnote{\\hyperref[https:\/\/archive.stsci.edu\/prepds\/legus\/]{https:\/\/archive.stsci.edu\/prepds\/legus\/}}). All catalogues use models with solar metallicity for both stars and gas and an average gas covering factor of $50\\%$.\n\nThe analyses and results presented hereafter are obtained using the final catalogue with:\n\\begin{itemize}\n\\item photometry corrected by average aperture corrections;\n\\item Padova evolutionary tracks produced with solar metallicity stellar libraries. \n\\item Milky Way extinction curve \\citep{cardelli1989}.\n\\item  \\citet{kroupa2001} stellar initial mass function.\n\\end{itemize}\n\n\\section{Global Properties of the cluster sample}\n\\label{sec:global}\n\\subsection{Final Catalogue}\n\\label{sec:finalcat}\nIn Fig.~\\ref{fig:agemass} we show the ages and masses of the sources classified as class 1, 2 and 3. In the same plot, the completeness limit of 23.4 mag in the $V$ band discussed in the previous section is converted into an estimate of the minimum mass detectable for each age. The line representing this limit follows quite accurately the detected sources with minimum mass at all ages. \nUncertainties on the age and mass values are on average within 0.2 dex. Uncertainties can reach 0.3 dex close to the red supergiant phase (visible as a loop at ages $\\sim10-50$ in Fig.~\\ref{fig:ccd}).\nIn order to study only the cluster population of the grand design spiral and avoid the clusters of NGC 5195, we have neglected the clusters with y coordinate bigger than 11600 (in the pixel coordinates of the LEGUS mosaic) both from Fig.~\\ref{fig:agemass} and from the rest of the analysis. This cut is similar to removing the UVIS pointing centred on NGC 5195. \n\\begin{figure*}\n \\centering\n \\subfigure[]\n   {\\includegraphics[width=0.49\\textwidth]{05a_plot_agemass_123.pdf}}\n \\hspace{0mm}\n \\subfigure[]\n   {\\includegraphics[width=0.49\\textwidth]{05b_plot_agemass_123_hist.pdf}}\n \\hspace{0mm}\n \\caption{(a) Age-mass diagram for the sources in class 1 and 2 (blue circles) and class 3 (red triangles). The solid black line represents the mass limit as function of ages estimated from the evolutionary tracks assuming a completeness limit of 23.4 mag in V band. The dotted horizontal line shows a mass of 5000 M$_\\odot$. The sources in each age bin were furnished with a small amount of artificial scatter around their respective bin to make the plot easier to read. (b) Histogram of the age distribution of the class 1+2 (blue) and class 3 (red) sources: the total height of each bin gives the total number of sources of the classes 1, 2 and 3. The number of class 3 sources drops fast in the first 10 Myr, such that in the range Log(age) = 7 - 8.5 their number is very small even if the age spanned is more that 100 Myr.}\n\\label{fig:agemass}\n\\end{figure*}\n\nThe different classes are not distributed in the same regions of the plot, with class 3 sources having on average smaller masses and younger ages. Previous studies \\citep{grasha2015,ryon2017,legus2} have shown that our morphological cluster classification is with good approximation also a dynamical classification. \nCompact associations (i.e. class 3) in NGC 628 are on average younger and less massive than class 1 and 2 clusters \\citep{grasha2015}. In addition, the age distributions of class 3 systems suggest that they are more easily and quickly disrupted (see \\citealp{legus2} for details), probably because they are not bound. \\citet{grasha2015,grasha2017}, focusing on the clustering function of clusters in the LEGUS galaxies, have also shown that class 3 clusters behave differently than class 1 and class 2 clusters. These results contribute to the idea that the morphological classification chosen has also some dynamical implication: class 3 sources seem to be short-lived systems (see Fig.~\\ref{fig:agemass}b), possibly already unbound at the time of formation. \n\nFor the remainder of this work we will only analyse class 1 and 2 objects, which we consider to be the candidate stellar clusters, i.e. the  gravitationally bound stellar systems that form the cluster population of M51. In total we have 2834 systems classified as class 1 and 2 out of the 10925 sources with 3 sigma detection in at least 4 LEGUS bands. \n\nIn Fig~\\ref{fig:ebv} we show how the recovered extinction changes as a function of cluster age. We see that on average the internal extinction of YSCs changes from $E(B-V)\\sim$0.4 mag at very young ages to 0.2 mag for clusters older than 10 Myr, and is even lower ($E(B-V)\\sim$0.1 mag) for clusters older than 100 Myr. We observe also a large scatter at each age bin, suggesting that the extinction may not only be related to the evolutionary phase of the clusters but also to the region where the cluster is located within the galaxy. \n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{06_ageVSebv.pdf}\n\\caption{Age-extinction diagram for the class 1 and 2 sources in the catalogue (blue circles). The red squares are the median values of E(B-V) in age bins of 0.5 dex. The red lines marks the range between the first and the third quartiles in the same age bins. }\n\\label{fig:ebv}\n\\end{figure}\n\n\n\\subsection{Comparison with previous catalogues}\n\\label{sec:comparison}\nNumerous studies of the cluster population in M51 are available in the literature. We compare our catalogue with recently published ones, testing both our cluster selection and agreement in age and mass estimates. These comparisons will be used when we compare the results of our analyses to values reported in the literature.\nAmong the works that studied the entire galaxy, \\citet{scheepmaker2007}, \\citet{hwang_lee2008} and \\citet{chandar16} used the same $BVI$ data as our work. However, we decided to focus our comparison only on two catalogues for which we have access to estimates of ages and masses.\nThe first catalogue is the one compiled by \\citet{chandar16} (hereafter CH16), using the same $BVI$ observations, plus F658N (H$\\alpha$) filter observations from the same program (GO-10452, PI: S. Beckwith) and WFPC2\/F336W filter ($U$ band) observations (from GO-10501, PI: R.Chandar, GO-5652, PI: R.Kirshner and GO-7375, PI: N.Scoville). The cluster candidates catalogue used in their analysis includes 3812 sources in total (of which 2771 lies in the same area covered by our UVIS observations) and has been made publicly available \\citep{chandar16cat}. \nThe second catalogue is taken from \\citet{bastian2005} (hereafter BA05), covers only the central part of the galaxy and is mainly based on HST observations with the WFPC2 camera. It contains 1150 clusters, 1130 of which are in an area in common with our UVIS pointings.\nThese two catalogues are very different both in terms of coverage and instruments used. For this reason we compare the catalogue produced with LEGUS with each of them separately.\n\n\\subsubsection{Cluster Selection}\nWe first compare the fraction of clusters candidates in common between the catalogs. When doing so, we include class 3 sources in the LEGUS sample, as this class of sources is considered in BA05 and CH16 catalogues. Tab.~\\ref{tab:comparison} collects the results of the comparison. Among the 2771 candidates detected in the same area of the galaxy by CH16 and LEGUS, 1619 (60\\%) systems are in common. We have repeated the comparison in the regions covered by human visual classification in LEGUS, finding a better agreement ($\\sim75\\%$). We take this last value as reference for the common fraction of candidates and justify the drop observed when considering the entire catalogue as given by the ML misplacing class 3 objects into class 4 (as discussed in Section~\\ref{sec:classification}).\nFig.~\\ref{fig:comparison} shows a blow-up of the galaxy with the cluster positions of both catalogues. We selected two different regions, one where the sources have been classified via visual classification and one where only ML is available. We notice that some of the CH16 candidates which do not appear in LEGUS catalogue of classes 1, 2 and 3, have been assigned a class 4. This is true for both regions. Those sources were extracted by the LEGUS analysis but were discarded based on their morphological appearance. The differences between the two catalogues are therefore mostly due to source classification and not by the extraction process.\n\nThe comparison with the BA05 catalogue indicates a poorer agreement, with less than $40\\%$ of sources in common. This discrepancy, observable in Fig.~\\ref{fig:comparison}, may be caused by the difference in the data and in the approach used to extract the clusters. BA05 analysis is based on WFPC2 data, whose resolution is a factor of $\\sim$2 worse than ACS. In addition they do not apply any CI cut, increasing the contamination from stars.\n\n\\begin{table*}\n\\centering\n\\caption{Comparison between our cluster catalogue and the catalogues by \\citet{chandar16} (CH16) and \\citet{bastian2005} (BA05). The number of clusters reported in the columns are: (1) number of cluster candidates detected in the CH16 and B05 catalogues within the same region covered by LEGUS; (2) number of cluster candidates in common between CH16, B05 and LEGUS (class 1,2,3), respectively; (3) number of cluster candidates in the Ch16 and B05 catalogues restricted to the area of the galaxy that has been visually inspected by humans; (4) number of cluster candidates of CH16 and B05 in common with LEGUS class 1, 2 and 3 sources that have been classified by human.\n$^1$ These fractions would increase to 95\\% if we include cluster candidates classified as 4 within the LEGUS catalogue. $^2$ This fraction would increase to 71\\% if class 4 cluster candidates are included. $^3$ The agreement would increase to 67\\% if class 4 objects are considered. \nThe drop in the common fraction of cluster candidates from column (4) to column (2) is likely due to the ML classification misplacing class 3 sources into class 4 (see Section~\\ref{sec:classification}). \nThe low fraction of clusters in common with BA05 is most likely due to a different approach for the cluster extraction analysis and low resolution imaging data. BA05 is based on WFPC2 data (a factor of 2.5 worse spatial resolution with respect to WFC3) and does not include a CI cut. This causes a higher contaminations from unresolved sources, which in our analysis were excluded by the CI selection. \n}\n\\begin{tabular}{lcccc}\n\\hline\nCatalogue\t\t\t&\t\\# clusters\t&  \\# clusters\t\t&  \\# clusters\t\t& \\# clusters\t\t\\\\\n\\\t\t\t\t& \t(1) \t\t& (2)& (3) & (4)\\\\\n\\hline\n\\hline\nCH16\t\t\t\t\t\t& 2711\t\t\t& 1619 (60\\%)$^1$\t\t\t& 732\t\t\t& 535 (73\\%)$^1$\t\t\\\\\nLEGUS (1,2,3)$_{CH16\\ area}$\t \t& 3240\t\t\t& $-$\t\t\t\t\t& 1294\t\t\t& $-$\t\t\t\\\\\n\\hline\nBA05\t\t\t\t\t\t& 1130\t\t\t& 388 (35\\%)$^2$\t\t\t& 214\t\t\t& 83 (39\\%)$^3$\t\t\\\\\nLEGUS (1,2,3)$_{BA05\\ area}$ \t& 1238\t\t\t& $-$\t\t\t\t\t& 267\t\t\t& $-$\t\t\t\\\\\n\\hline\n\\end{tabular}\n\\label{tab:comparison}\n\\end{table*}\n\n\\begin{figure*}\n \\centering\n\\subfigure{\\includegraphics[width=\\textwidth]{07a_comparison2.png}}\n\\subfigure{\\includegraphics[width=\\textwidth]{07b_comparison3.png}}\n \\caption{Comparison between different cluster catalogues in a spiral arm in the galaxy covered by human visual classification (top) and by machine learning classification (bottom) in LEGUS. Green circles are the clusters candidates in \\citet{chandar16} catalogue. Yellow circles are the cluster candidates in \\citet{bastian2005} catalogue. Red circles are cluster candidates of class 1 and 2 in LEGUS, blue circles are class 3 sources in LEGUS and white dashed circles are sources which have been assigned class 4 in LEGUS. In the bottom right corner an inset shows the position of the zoomed region inside M51.}\n\\label{fig:comparison}\n\\end{figure*}\n\n\\subsubsection{Comparison of Ages and Masses}\n\\begin{figure*}\n \\centering\n \\subfigure{\\includegraphics[width=0.43\\textwidth]{08a_plot_ages_ch16.pdf}}\n \\subfigure{\\includegraphics[width=0.43\\textwidth]{08b_plot_ages_ba05.pdf}}\n \\subfigure{\\includegraphics[width=0.43\\textwidth]{08c_plot_1to1_ch16.pdf}}\n  \\subfigure{\\includegraphics[width=0.43\\textwidth]{08d_plot_1to1_ba05.pdf}}\n \\subfigure{\\includegraphics[width=0.43\\textwidth]{08e_plot_masses_ch16.pdf}}\n \\subfigure{\\includegraphics[width=0.43\\textwidth]{08f_plot_masses_ba05.pdf}}\n \\caption{Comparison of ages and masses retrieved from the broadband SED fitting in the LEGUS analysis and in the works of \\citealp{chandar16} (CH16, red, left column) and \\citet{bastian2005} (BA05, green, right column). Only the sources in common between LEGUS and CH16 or BA05 are plotted. First row: ages distributions. Second row: 1--to--1 comparison between ages. The 1--to--1 agreement line is shown in solid blue, the 0.3 dex scatter lines in dashed blue. Third row: masses distributions.}\n\\label{fig:comparison_am}\n\\end{figure*}\nThe comparison of age distributions for the sources in common between LEGUS and CH16 is plotted in Fig.~\\ref{fig:comparison_am} (top left). The age distribution of CH16 has a strong peak for sources younger than log(age\/yr)$=7$ and a subsequent drop in the range $7-7.5$, both of which are not observed in our catalogue. The one--to--one comparison between age estimates in Fig.~\\ref{fig:comparison_am} (middle left) shows a large fraction of clusters with young ages in CH16 which have a wide age spread in the LEGUS catalogue. More in general, the differences in the age estimates are mostly caused by the different broadband combinations used in fitting the data, as already noticed in CH16. In addition to the ``standard'' $UBVI$ filter set used for SED fitting of both CH16 and our LEGUS catalogues, we use an extra $UV$ broadband while CH16 use the flux of the narrow band filter centred on the H$\\alpha$ emission line, from an aperture of the same size as the broadband ones. Both approaches aim at breaking the age-extinction degeneracy weighting different information. The LEGUS standard approach is to use two datapoints below the Balmer break ($\\lambda < 4000$) which give a stronger constraint on the slope of the spectrum, and thus, extinction. The approach used by CH16 is to use the detection of H$\\alpha$ emission from gas ionised by massive stars to determine the presence of a very young stellar population in the cluster. From Fig.~\\ref{fig:comparison_am} (middle left) we observe that the two methods agree within 0.3 dex in $\\sim50$\\% of the cases. \nThe correlation between the ages derived in the two methods is confirmed by a Spearman's rank correlation coefficient $r_s=0.7$ with a p-value: $10^{-234}$.\nWe will address in a future work (Chandar et al. in prep.) the systematics and differences in the two methods. In this work we will take into account the differences observed in the age distributions when discussing and comparing our results to those available in the literature.\n\nThe mass distributions (Fig.~\\ref{fig:comparison_am}, bottom left) show a more similar behaviour, with a broad distribution and a decrease at low masses caused by incompleteness. Note that CH16 retrieve higher mass values at the high-mass end of the distribution. This difference can be important in the study of the mass function shape (Section~\\ref{sec:massfunc}).\n\nThe comparison of age and mass distributions for the sources in common between LEGUS and BA05 is shown in the right column of Fig.~\\ref{fig:comparison_am}.\nSince the youngest age assigned by BA05 is log(age\/yr)=6.6, for the sake of the comparison in Fig.~\\ref{fig:comparison} (top right) we have assigned log(age\/yr)=6.6 age to all the sources that in our catalogue are younger. The general trends of the distributions looks similar, but the one--to--one comparison in Fig.~\\ref{fig:comparison} (middle right) reveals that the two methods agree within 0.3 dex in $\\sim50\\%$ of the cases. \nThe correlation found with a Spearman's rank test is $r_s=0.5$. A p-value of $10^{-24}$ confirms that this correlation is not random, but the moderate value of $r_s$ is caused by the difference in the age distribution observed in Fig.~\\ref{fig:comparison} (middle right). \nBA05 use very different data from our own and allow fits with $BVI$ bands only, with large uncertainties on the recovered properties. For example, the large cloud of systems that sit in the upper left part of the plot has been assigned younger ages in our catalogue. Also in this case we can conclude that the differences in age determinations are mostly caused by the different fitting approach, with our catalogue having more information to break the age-extinction degeneracy. The mass distributions (Fig.~\\ref{fig:comparison_am}, bottom right) show the same overall shape, with the BA05 distribution shifted by 0.2 dex to higher values of masses.\n\nIn general, for both catalogues, we notice that differences in the derived properties can be also caused by differences in the stellar templates adopted, which are different for all catalogues (CH16 uses \\citealp{bruzual_charlot2003} models, while BA05 uses updated GALEV simple stellar population models from \\citealp{schulz2002} and \\citealp{anders2003}). We will use the differences outlined among these previously  published catalogues and ours when we will discuss the results of our analyses.\n\n\n\\subsection{Cluster Position as a function of age}\n\\label{sec:position}\nIn order to understand where clusters form and how they move in the dynamically active spiral arm system of M51 we plot the position of the clusters inside the galaxy in Fig.~\\ref{fig:positions}. The sample is divided in age bins ($1-10$, $10-100$, $100-200$ and $200-500$ Myr). \nClusters in our sample are mostly concentrated along the spiral arms. This trend is particularly clear for the very young clusters (age $<10$ Myr) but can also be spotted in the ranges $10-100$ and $100-200$ Myr. In general we observed that young sources are clustered. Moving to older sources the spatial distribution becomes more spread, but it can still be recognized that sources are more concentrated along the spiral arms. In the last age bin, probing clusters older than 200 Myr, the number of available sources is much smaller and is therefore hard to define a distribution, although the sources appear to be evenly spread across the area covered by observations. The strength and age-dependency of the clustering will be further investigated in a future paper (Grasha et al, in prep).\n\\begin{figure*}\n \\centering\n \\subfigure\n   {\\includegraphics[width=0.4\\textwidth]{09a_positions1.pdf}}\n \\subfigure\n   {\\includegraphics[width=0.4\\textwidth]{09b_positions2.pdf}}\n \\subfigure\n   {\\includegraphics[width=0.4\\textwidth]{09c_positions3.pdf}}\n \\subfigure\n   {\\includegraphics[width=0.4\\textwidth]{09d_positions4.pdf}}\n \\caption{Position of the clusters divided in age bins in the ranges 1-10 Myr (left), 10-100 Myr (middle-left), 100-200 Myr (middle-right) and 200-500 Myr (right). The central circle of radius 1.3 kpc encloses the region where detection is poorer (see text). The UVIS footprint, restricted to y values below 11600, is overplotted as a grey solid line.}\n\\label{fig:positions}\n\\end{figure*}\n\nThe lack of age gradient as a function of distance from the spiral arm observed in Fig. 8 is in agreement with the detailed study of azimuthal distances of clusters as a function of their ages collected in a forthcoming paper (Shabani et al., in prep.) where the origin of spiral arm and dynamical evolution is investigated.\nThe observed trend has been predicted by \\citet{dobbs2010} which, modeling a spiral structure induced by tidal interactions find that clusters of different ages tend to be found in the same spiral arm without a defined age gradient. \nIn a more recent numerical work, \\citet{dobbs2017} analyse the evolution of stellar particles in clustered regions, i.e. simulated star clusters within spiral fields. They observe that up to the age range they are able to follow (e.g. 200 Myr) their simulated clusters are mainly distributed along the spiral arms. The trend observed in M51 is thus compatible with that found in \\citet{dobbs2017} simulations. Simulations on the evolution of M51 \\citep[e.g][]{dobbs2010_m51} suggest that the interaction with the companion galaxy, started $\\sim300$ Myr ago, is responsible for creating or strengthening the spiral arms and may have helped keep the old clusters we see now fairly close to the arms.\n\nFrom Fig.~\\ref{fig:positions} we clearly see that our detection is very poor in the centre of the galaxy where the bright background light of the diffuse stellar population is much stronger than in the rest of the galaxy. This effect could explain why we do not detect sources older than 10 Myr (i.e. when cluster light starts to fade), causing a drop in the completeness limit, as already pointed out in Section~\\ref{sec:completeness}. For this reason we ignore the clusters within 35'' (1.3 kpc) from the centre of the galaxy from the following analyses. \n\n\n\n\n\\subsection{Luminosity function}\n\\label{sec:lumfunc}\n\nThe luminosity function is intrinsically related to the mass function (luminosity is proportional to mass, with a dependence also on the age) but it is an observed quantity, and therefore, like the colour-colour diagrams, available without any assumption of stellar models and without any SED fitting. \nThe luminosity function of YSCs is usually described by a power law function $dN\/dL\\propto L^{-\\alpha}$, with an almost universal index close to $\\alpha\\approx -2$ as observed in local spiral galaxies (e.g. \\citealp{larsen2002,degrijs2003}, see also the reviews by \\citealp{whitmore2003} and \\citealp{larsen2006b}).\n\nWe analyse the cluster luminosity function by building a binned distribution with the same number of objects per bin, as described in \\citet{maiz2005} and performing a least-$\\chi^2$ fitting.\nThe errors on the data are statistical errors given by $\\sigma_{\\textrm{bin}}=\\sqrt{\\frac{n_{\\textrm{bin}}(n_\\textrm{tot}-n_\\textrm{bin})}{n_\\textrm{tot}}}$, where $n_\\textrm{bin}$ is the number of sources in each bin and $n_\\textrm{tot}$ is the total number of sources. \nThe results of the fits are listed in Tab.~\\ref{tab:lumfit} and plotted in Fig.~\\ref{fig:lumfit}. \nWe have fitted the data up to the completeness limits described in Section~\\ref{sec:completeness}.\nThe function is fitted with both a single and a double power law (PL). The single PL fit gives slopes close to a value of $\\alpha=-2$,\nhowever, for all filters, the double power law results in a better fit, as the $\\chi^2_{red}$ in this second case is always lower. \n\\begin{table*}\n\\centering\n\\caption{Results of the fit of the binned luminosity functions.}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c}\n\\hline\n\\multicolumn{1}{|c|}{Filter} \t\t& \\multicolumn{1}{c|}{$\\textrm{Mag}_{\\textrm{cut}}$} \t& \\multicolumn{2}{l|}{Single PL fit} \t& \\multicolumn{4}{l|}{Double PL fit} & \\multicolumn{1}{l|}{Cumulative fit}\\\\\n\\multicolumn{1}{|c|}{} \t\t& \\multicolumn{1}{c|}{} \t\t& \\multicolumn{1}{c|}{$\\alpha$} & \\multicolumn{1}{c|}{$\\chi^2_{red.}$}\t\t& \\multicolumn{1}{c|}{$\\alpha_1$}\t& \\multicolumn{1}{c|}{$\\textrm{Mag}_{\\textrm{break}}$} \t& \\multicolumn{1}{c|}{$\\alpha_2$} & \\multicolumn{1}{c|}{$\\chi^2_{red.}$} & \\multicolumn{1}{c|}{$\\alpha$}\t\\\\\n\\hline\n\\hline\nF275W  &\t21.71\t& $1.84\\ _{\\pm 0.04}$\t& 2.17 &\t$1.57\\ _{\\pm 0.06}$ &\t$19.39\\ _{\\pm  0.25}$ & \t$2.24\\ _{\\pm 0.10}$  &   0.85 & $2.08\\ _{\\pm 0.01}$ \\\\\nF336W  &\t22.00\t& $1.89\\ _{\\pm 0.04}$\t& 1.80 &\t$1.67\\ _{\\pm 0.05}$ &\t$19.49\\ _{\\pm  0.26}$ & \t$2.32\\ _{\\pm 0.12}$  &   0.92 & $2.10\\ _{\\pm 0.01}$ \\\\\nF435W  &\t23.25\t& $1.99\\ _{\\pm 0.03}$\t& 1.72 &\t$1.74\\ _{\\pm 0.04}$ &\t$20.97\\ _{\\pm  0.17}$ & \t$2.41\\ _{\\pm 0.09}$  &   0.80 & $2.17\\ _{\\pm 0.01}$ \\\\\nF555W  &\t23.25\t& $2.02\\ _{\\pm 0.03}$\t& 1.70 &\t$1.79\\ _{\\pm 0.03}$ &\t$20.83\\ _{\\pm  0.17}$ & \t$2.48\\ _{\\pm 0.10}$  &   0.86 & $2.18\\ _{\\pm 0.01}$ \\\\\nF814W  &\t22.25\t& $2.04\\ _{\\pm 0.04}$\t& 2.31 &\t$1.60\\ _{\\pm 0.07}$ &\t$20.73\\ _{\\pm  0.13}$ & \t$2.40\\ _{\\pm 0.08}$  &   1.03 & $2.28\\ _{\\pm 0.01}$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:lumfit}\n\\end{table*}\n\\begin{figure*}\n \\centering\n \\subfigure[binned]\n   {\\includegraphics[width=0.48\\textwidth]{10a_plot_lumfunc_DP_new.pdf}}\n \\hspace{0mm}\n \\subfigure[cumulative]\n   {\\includegraphics[width=0.48\\textwidth]{10b_plot_lumcumfit_DP_new.pdf}}\n \\caption{Binned (a) and cumulative (b) luminosity functions. Fit results are reported in Tab. \\ref{tab:lumfit}. The curves are for different filters, from UV band at the top to I band at the bottom. Errorbars in (a) are of the same size of the markers. In both panels the vertical lines mark the completeness limit in each filter.}\n\\label{fig:lumfit}\n\\end{figure*}\n\nSimilar results were found by \\citet{haas2008} using a cluster catalogue based only on $BVI$ photometry.\nThey found that the low-luminosity part of the function could be fitted by a shallow power-law, with slopes in the range $\\sim1.7-1.9$, while the high-luminosity end was steeper, with slopes $\\sim2.3-2.6$. We similarly found that the low luminosity part of the function is shallower ($\\alpha\\sim1.6-1.8$) than the high luminosity part ($\\alpha\\sim2.4-2.5$). \nIn both analyses a double power law is a better fit of the luminosity function in all filters. As suggested by \\citet{gieles2006}, a broken power law luminosity function suggests that also the underlying mass function has a break.\nThe possibility that the underlying mass function is truncated is further explored with the study of the mass function in Section~\\ref{sec:massfunc} and via Monte Carlo simulations in Section~\\ref{sec:montecarlo}.\n\nWe compared the binning fitting method with the one presented in \\citet{bastian2012}, involving the use of cumulative functions. In the case of a single power law behaviour, the two functions are expected to show the same shape. The cumulative function is given by $y_{\\textrm{c}}(m)=\\left(1 - \\frac{k}{n_{\\textrm{data}}}\\right)$, where $k$ is the index of the object of magnitude m in the sorted array containing the magnitudes and $n_{\\textrm{data}}$ is the length of the array. In case of a simple power law, it has a slope $\\alpha_c=\\alpha_b-1$ which can  be directly compared to the slope $\\alpha_b$ of the binned function. Also in this case a least $\\chi^2$ fit is performed. In the cumulative distribution no error is associated with the data, therefore the fit is made with a linear function in the logarithmic space, assigning the same uncertainty to all points. The errors on the fitted parameters have been estimated via a bootstrapping technique: 1000 Monte Carlo realisations of the distribution are simulated, where the luminosity of each cluster is changed using uncertainties normally distributed around $\\sigma_{mag}=0.35$ mag. Each realisation was then fitted in the same way as the original one. The standard deviation of the 1000 values recovered for each parameter gave the final uncertainty associated with the recovered slopes.\nResults and slopes are collected in Fig.~\\ref{fig:lumfit}(b) and Tab.~\\ref{tab:lumfit}. We converted $\\alpha_c$ into $\\alpha_b$ in Tab.~\\ref{tab:lumfit}, for an easier comparison with the binned function.\nThe trends observed in the analysis of the binned distributions are traceable in the cumulative function as well. In particular, we observe that the single PL fit is not a good description of the bright end of the cumulative distributions in all the filters (Fig.~\\ref{fig:lumfit}b). \nAlso with the cumulative function, the brightest sources fall below the expected curve of a single power law distribution, sign of a break in the luminosity function and therefore also in the underlying mass function.\nWe note that the fit of the cumulative functions results in steeper slopes than the ones recovered with the binned distributions.  This discrepancy caused by the differences in the two techniques is discussed at length in \\citet{legus2}.\n \n\n\n\n\\subsection{Mass Function}\n\\label{sec:massfunc}\nThe results obtained with the analysis of the luminosity function can be further explored with the study of the properties derived from the SED fit, i.e. the mass and the age distributions.\nIn the following analyses we use a mass-complete sample, by selecting only clusters above $5000\\ \\textrm{M}_{\\odot}$. This value has been chosen in order to avoid low mass sources, affected by inaccuracies in the SED fitting and by stochastical sampling of the stellar IMF (see e.g. the comparison between deterministic and Bayesian fitting of cluster SEDs in Figure 15 of \\citealp{slug}).\nThe age-mass plot of Fig. \\ref{fig:agemass} suggests that we are complete in recovering sources more massive than $5000\\ \\textrm{M}_{\\odot}$ only up to 200 Myr. At older ages, even sources more massive than 5000 M$_\\odot$ can fall below our magnitude detection limit. \nOur mass-limited complete sample therefore contains sources with M $>5000\\ \\textrm{M}_{\\odot}$ and  ages $< 200$ Myr.\n\nThe cluster mass function is expected to evolve from a cluster initial mass function (CIMF), usually assumed as a power law $dN\/dM \\propto M^{\\beta}$ with a $\\beta=-2$ slope. This slope is interpreted as the sign of the formation of clusters from a turbulent hierarchical medium  \\citep{elmegreen2010b}. The initial function is then expected to evolve due to cluster evolution and disruption. \n\nThe cluster mass function of our sample is plotted in Fig.~\\ref{fig:massfit}, where bins of equal number of sources were used.\nWe recover a shape which is well fitted with a single power law of slope $-2.01 \\pm0.04$ ($\\chi^2_{red.}$  of 1.6), even if a double power law with a  steeper high mass slope fits better the function ($\\chi^2_{red.}$ of 1.1, see Tab.~\\ref{tab:massfit}). \n\\citet{gieles2009} and CH16 found similar slopes, $\\beta=-2.09 \\pm0.09$ and $\\beta=-1.97 \\pm0.09$ respectively, considering only clusters in the age range from 10 to 100 Myr. Restricting to the same age range, we find a consistent value of $\\beta=-2.03 \\pm0.04$ ($\\chi^2_{red} = 0.77$, see Fig.~\\ref{fig:massfunc_ages}).\n\nAs done in the analysis of the cluster luminosity function, we also plot the mass function in a cumulative form (Fig.~\\ref{fig:massfit}) and fit it with a pure power law. As already observed for the luminosity functions, the cumulative mass distributions show a steepening at the high-mass end. As observed in \\citet{bastian2012} and \\citet{legus2}, while the equal number of object binning technique is statistically more robust, it is insensitive to small scales variations, like the dearth of very massive clusters. The cumulative form is therefore more appropriate to study the high-mass end of the mass (and luminosity) function.\n\\begin{table}\n\\centering\n\\caption{Values derived by the fit of the mass function with a least-$\\chi^2$ fitting of the binned function. Fits have been performed considering a low mass cut of either 5000 or $10^4$ M$_\\odot$, as indicated in the second column.}\n\\begin{tabular}{lccccc}\n\\hline\nMethod\t& \tM cut \t&  $-\\beta_1$\t\t\t& $M_{break}$\t& $-\\beta_2$ \t& $\\chi^2_{red.}$\\\\\n\\\t\t&\t[M$_\\odot$]\t& \\\t\t\t\t\t& [M$_\\odot$]\t\t& \\\t\t\t& \\\t\t\t\\\\\n\\hline\n\\hline\nSingle PL\t&\t5000\t\t& $2.01_{\\pm0.04}$\t\t& $-$\t\t\t& $-$\t\t\t& 1.6\\\\\nDouble PL\t&\t5000\t\t& $1.52_{\\pm0.12}$\t\t& $1.5\\times10^4$\t& $2.31_{\\pm0.09}$\t& 1.1\\\\\n\\hline\nSingle PL\t&\t$10^4$\t& $2.19_{\\pm0.07}$\t\t& $-$\t\t\t& $-$\t\t\t& 1.6\\\\\nDouble PL\t&\t$10^4$\t& $1.71_{\\pm0.39}$\t\t& $1.8\\times10^4$\t& $2.36_{\\pm0.13}$\t& 1.5\\\\\n\\hline\n\\end{tabular}\n\\label{tab:massfit}\n\\end{table}\n\\begin{table}\n\\centering\n\\caption{Values derived by the fit of the cumulative mass function with a maximum-likelihood fit. For a description of the values $N_0$ and $M_0$ see Eq.~\\ref{eq:cumulative}. In the last row the result of the fit of the GMC population is reported.}\n\\begin{tabular}{lccccc}\n\\hline\nMethod\t\t& M cut \t& Age \t&$-\\beta$\t\t\t\t&$N_0$\t\t\t& $M_0$  \\\\\n\\\t\t\t& [M$_\\odot$]\t& [Myr]\t& \\\t\t\t\t\t& \\\t\t\t\t&  [$10^5$ M$_\\odot$] \\\\\n\\hline\n\\hline\nSingle PL\t\t& 5000\t& 1-200\t& $2.30_{\\pm0.03}$\t\t& $-$ \t\t\t& $-$ \\\\ \nTruncated \t& 5000\t& 1-200\t& $2.01_{\\pm0.02}$\t\t& $66_{\\pm6}$\t\t& $1.00_{\\pm0.12}$\\\\\nTruncated\t\t& 5000\t& 1-10\t& $2.12_{\\pm0.22}$\t\t& $10_{\\pm7}$\t\t& $0.56_{\\pm0.08}$\\\\\nTruncated\t\t& 5000\t& 10-100\t& $1.97_{\\pm0.06}$\t\t& $43_{\\pm15}$\t& $0.91_{\\pm0.16}$\\\\\nTruncated\t\t& 5000\t& 100-200\t& $2.01_{\\pm0.05}$\t\t& $28_{\\pm4}$\t\t& $1.15_{\\pm0.27}$\\\\\n\\hline\nSingle PL\t\t& $10^4$\t& 1-200\t\t& $2.67_{\\pm0.03}$\t\t& \\ \t\t\t\t& \\ \\\\ \nTruncated\t\t& $10^4$\t& 1-200\t\t& $2.34_{\\pm0.03}$\t\t& $22_{\\pm10}$\t& $1.34_{\\pm0.24}$\\\\\n\\hline\nGMC pop \t\t& $-$\t& $-$\t& $2.36_{\\pm0.16}$     \t& $12_{\\pm5}$\t\t& $160_{\\pm32}$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:massfit2}\n\\end{table}\n\nIn order to test the hypothesis of a mass truncation, we have fitted the cumulative distribution with the \\texttt{IDL} code \\texttt{mspecfit.pro}, implementing the maximum-likelihood fitting technique described in \\citet{rosolowsky2007}, commonly used for studying the mass functions of GMCs \\citep[e.g.][]{colombo2014a}. The code implements the possibility of having a truncated power law mass function, i.e.\n\\begin{equation}\n\\label{eq:cumulative}\nN(M'>M)=N_0\\left[\\left(\\frac{M}{M_0}\\right)^{\\beta+1}-1\\right],\n\\end{equation}\nwhere $M_0$ is the maximum mass in the distribution and $N_0$ is the number of sources more massive than $2^{1\/(\\beta+1)}M_0$, the point where the distribution shows a significant deviation from a power law (for the formalism, see \\citealp{rosolowsky2005}).\nA value of $N_0$ bigger than $\\sim1$ would indicate that a truncated PL is preferred over a simple one. On the other hand, $N_0<1$ would mean that the truncation mass is not constrained and that a single power law is a good description of the distribution. The resulting parameters of the fit for our sample, considering normally distributed 0.1 dex errors on the masses, are collected in Tab.~\\ref{tab:massfit2}. \nThe resulting $N_0 = 66 \\pm6$ suggests that the fit with a truncated function,with $M_0=10^5$ M$_\\odot$, is preferred over the simple PL. The best fit for the slope is $\\beta=-2.01 \\pm0.02$.\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{11_plots_massfit_cat_bestm_all.pdf}\n\\caption{Binned (top) and cumulative (bottom) mass function. The solid lines are the best fits with a single power law, the dashed line in the case of the cumulative function shows the slope $-2$ for comparison. The binned function is steeper because the slope of the cumulative function in a simple power law case is $B=\\beta+1$, where $\\beta$ is the slope of the original function (see Eq.~\\ref{eq:cumulative}).}\n\\label{fig:massfit}\n\\end{figure}\n\nIn order to test for possible incompleteness at masses close to 5000 M$_\\odot$, we repeated the analysis of the mass function using a mass cut of $10^4$ M$_\\odot$.\nResults are collected in Tab.~\\ref{tab:massfit} and \\ref{tab:massfit2}. Different lower limits at the low mass yield steeper than $-2$ power laws but consistent truncation masses. \nThe binned function is well fitted with a single power law with $\\beta=-2.19 \\pm0.07$ ($\\chi^2_{red.}=1.55$). The maximum likelihood fit of the cumulative function gives $\\beta=-2.34 \\pm0.03$, $M_0= (1.34 \\pm 0.24) \\times10^5$ M$_\\odot$\\ and $N_0=22 \\pm 10$, thus a truncation is still statistically significant. \n\nIt has been reported in the literature that the YSC mass function is probably better described by a Schechter function with a $\\beta=-2$ slope and a truncation $M_c$ at the high-mass end. In the case of M51, \\citet{gieles2006} found that a Schechter function with $M_c = 10^5$ M$_\\odot$\\ would reproduce closely the luminosity function observed. With a very different approach, \\citet{gieles2009} derived $M_c = (1.86 \\pm0.52) \\times 10^5$ M$_\\odot$\\ from the analysis of an evolving mass function. Both results are consistent with our results. \n\n\\citet{chandar2011,chandar16} find that a simple PL  is a good description of the YSC mass function in M51, however they only considered a binned MF. Different mass estimates for high-mass clusters, as noted in Fig.~\\ref{fig:comparison_am}, could produce differences in the mass function slopes. Nevertheless, we retrieve the same results of CH16 if a binned function is used. We notice that the bin containing the most massive clusters encompasses the whole range in masses where the truncation mass is found (see Fig.~\\ref{fig:massfit}). Thus binning techniques that use equal number of objects are therefore unable to put a constraint on the truncation. \n\nThe truncation mass we recover is smaller but similar to what was found in other spirals, like M 83 ($M_c=(1.60 \\pm0.30) \\times10^5$ M$_\\odot$, \\citealp{adamo2015}), NGC 1566 ($M_c=2.5 \\times10^5$ M$_\\odot$, \\citealp{hollyhead2016}) and NGC 628 ($M_c=(2.03 \\pm0.81) \\times 10^5$ M$_\\odot$, \\citealp{legus2}). On the other hand, some galaxies still exhibit a truncated mass function but with very different truncation masses. In M31, for example, \\citet{johnson2017} found a remarkably small truncation mass of $\\sim10^4$ M$_\\odot$, while the Antennae have a MF that exhibits a PL shape that extends up to masses larger than $10^6$ M$_\\odot$\\ \\citep{whitmore2010}. These differences spanning orders of magnitudes suggest that the maximum cluster mass in galaxies must be determined by the internal (gas) properties of the galaxies themselves. \\citet{johnson2017} suggested that $M_c$ should scale with the $\\Sigma_{\\textrm{SFR}}$.\nDifferences in the recovered truncation mass have also been found within the same galaxy (e.g. \\citealp{adamo2015}). We will investigate possible environmental dependencies of the mass function properties of M51 in a follow up work (Paper II).\n\n\n\n\\subsubsection{Comparison With GMC Masses}\n\\label{sec:GMC}\nWe compare our cluster mass function with the mass function of the GMCs in M51 from the catalogue compiled and studied in \\citet{colombo2014a}. Clusters are expected to form out of GMCs, via gravitational collapse and fragmentation, and therefore the mass distribution of the latter can in principle leave an imprint on the mass distribution of YMCs.\n\nThe mass function of GMCs in M51 steepens continuously going from low to high masses, and cannot be described by a single power law \\citep{colombo2014a}, as is instead the case for other galaxies like LMC, M33, M31 and the Milky Way \\citep{wong2011,gratier2012,rosolowsky2005}.  We perform a fit of GMC masses with the same code described in the previous section, up to a lower limiting mass of $10^6$ M$_\\odot$\\ (discussed in Section 7.2 and 7.3 on the mass functions in \\citealp{colombo2014a}). The resulting best value for the slope and the maximum mass are $\\beta=-2.36 \\pm0.16$ and $M_0=(1.6 \\pm0.3) \\times10^7$ M$_\\odot$. \nThe value of $M_0$ implies a truncation mass which is $\\sim100$ times bigger in the case of GMCs similar to what has been observed in M83 by \\citet{freeman2017}. \nThe mass function of GMCs looks steeper than the one of the clusters, within a 3$\\sigma$ difference. Analysing the mass function of simulated GMCs and clusters, \\citet{dobbs2017} found the opposite trend of a steeper function in the case of clusters. Part of the difference between the simulated and observed trends can be due to the different regions covered within the two surveys: the PAWS survey from which the GMC data are derived covers only the central part of the galaxy, while our clusters also occupy more distant regions from the centre.\nThe study of the mass function in different regions of M51 in Paper II will enable us to compare the CMF with the GMC one on local scales, testing closely the link between GMC and cluster properties.\n \n\\subsubsection{Evolution of the Mass Function}\n\\label{sec:massage}\nCluster disruption affects the mass function and could, in principle, modify its shape: for this reason we study the function in different age bins. \nIn order to be able to see how significant the disruption is, we look at the evolution of the CMF normalized by the age range (i.e. $dN\/dMdt$).\nIn case of constant star formation and no disruption the mass functions should overlap. \nCluster disruption can in principle affect the mass function in different ways according to the disruption model considered. \n\nTwo main empirical disruption scenarios have been proposed in the literature and they differ in the dependence with the cluster mass. A first model, firstly developed to explain the age distribution of clusters in the Antennae galaxies (see \\citealp{fall2005} and \\citealp{whitmore2007}), proposes that all clusters lose the same fraction of their mass in a given time. This implies that the disruption time of clusters is independent on the cluster mass and we therefore call this model mass-independent disruption (MID). It is characterized by a power-law mass decline and therefore by a disruption rate which depends linearly on the mass \\citep{fall2009}, i.e.\n\\begin{equation}\nM(t)\\propto t^\\lambda, \\ \\ \\ \\ \\frac{dM}{dt}\\propto M\n\\end{equation}\nOn the other hand, the mass dependent disruption (MDD) time scenario assumes that the lifetime of a cluster depends on its initial mass, with a relation $t_{dis}\\propto M^k$ (with $k=0.65$, i.e. less massive clusters have shorter lifetimes).\nInitially suggested by \\citet{boutloukos2003} considering only instantaneous disruption to explain the properties of the cluster populations in the SMC, M33 and M51, this model has been updated to account also for gradual mass-loss in \\citet{lamers2005}. This model is characterized by a disruption rate which depends sub-linearly on the mass as:\n\\begin{equation}\n\\frac{dM}{dt}\\propto M^{1-k}\n\\end{equation}\nThe two scenarios predict different evolutions for the cluster mass function \\citep[e.g.][]{fall2009}. In the MID model, the mass function shape is constant in time, it only shifts to lower masses due to all clusters losing the same fraction of mass. In the MDD model, instead, low-mass clusters have shorter lifetimes and this results in a dearth of clusters at the low-mass end of the function, as the time evolves.\n\nIn Figure~\\ref{fig:massfunc_ages} we observe that the normalised CMF at ages below 10 Myr is detached from the CMFs of the other two age bins, suggesting a stronger drop in the number of sources. In the age range $10-100$ Myr, compared to the range $100-200$ Myr, the main difference between the two CMF is seen at low masses as a bend in the CMF of the oldest clusters, i.e. $100 - 200$ Myr. This trend would suggest a shorter disruption time for low mass clusters, however as can be seen in Fig~\\ref{fig:agemass}, at these ages also incompleteness could start affecting the data. So the flattening could be the result of both mass dependent disruption time and incompleteness. On the other hand, at high masses the functions seem to follow each other quite well. \n\\begin{figure}\n\\centering\n\\includegraphics[width=1.\\columnwidth]{12_plots_massfit_cat_bestm_all_ages_new.pdf}\n\\caption{Mass function divided in age bins and normalized by the age range in each bin. The black dotted line is the low-mass limit of 5000 M$_\\odot$. The shift in normalisation between the young function (blue) and the others suggest that cluster disruption is already happening between 10 and 100 Myr. The old function (red) flattens at low masses, but it is difficult to separate the effects of disruption and incompleteness.}\n\\label{fig:massfunc_ages}\n\\end{figure}\n\nEach function is fitted with a least-$\\chi^2$ approach. Single power laws are fitted and the resulting slopes for the age bins $1-10$, $10-100$ and $100-200$ Myr are $\\beta=-2.02 \\pm0.11$,  $-2.03 \\pm0.04$ and $-1.85 \\pm0.06$, respectively. We can compare these values with the results of CH16 and \\citet{gieles2009} as both of them studied the mass function in age bins. CH16  found a slope  $\\beta=-2.06 \\pm 0.05$ for sources younger than 10 Myr and $\\beta=-1.97 \\pm 0.09$ for sources in the range 10-100 Myr. For older sources they consider a bin with ages in the range 100-400 Myr finding a slope of $\\beta=-2.19 \\pm 0.06$. \nUsing age bins of $4-10$ Myr, $10-100$ Myr and $100-600$ Myr, \\citet{gieles2009} found slopes of $\\beta=-2.08 \\pm 0.08$, $-2.09 \\pm 0.09$ and $-2.76 \\pm 0.28$ respectively, using the cluster catalogue of B05. While up to 100 Myr those values are comparable with what we find, at old ages their results seem to strongly deviate from our own. A first reason for this deviation may be the smaller size of our last age bin, which extends only to 200 Myr and therefore neglects older sources. \nHowever, a more likely explanation can be found in the definition of the minimum mass considered in each age range: because of our completeness limit, we always consider sources more massive than 5000 M$_\\odot$, while the cut of the older bin in \\citet{gieles2009} is 6$\\times10^4$ M$_\\odot$\\ and in CH16 is $\\approx 10^4$ M$_\\odot$. In both cases the low-mass part of the function is not considered in the fit, thus their fit may be more sensitive to the presence of a bend in the form of a truncation. As seen in Tab.~\\ref{tab:massfit}, fitting only the high mass part of the CMF results in stepper slopes also in our catalogue, even if a shorter age range is used.\n\nFitting cumulative instead of binned distributions with the maximum-likelihood fit with the \\texttt{mspecfit.pro} code yields slopes and truncation masses collected in Tab.~\\ref{tab:massfit2}. Results for age bins $10-100$ Myr and $100-200$ Myr are very similar to the results found for the whole population. In both age ranges the presence of a truncation ($N_0 >>1$) is statistically significant. The CMF in the age range $1-10$ Myr has a fitted $M_0$ which is a factor 2 smaller. The statistical significancy $N_0$ of the latter fit is, within uncertainties, not much larger than 1 ($N_0=10 \\pm7$). This result seems driven by size--of--sample effects. Uncertainties in this last case are larger because the sample is small, counting only 140 clusters, compared to the other two age bins hosting more than 500 clusters each. These uncertainties prevent to statistically test the truncation for the mass function in the bin $1-10$ Myr.\n\n\n\n\\subsection{Age Function}\n\\label{sec:agefunc}\nWe can investigate the cluster evolution analysing the age distributions of the clusters. The YSC age function is determined by the star (and cluster) formation history (SFH and CFH) convolved with cluster disruption.\n\nIn first approximation, the SFH of spiral galaxies can be assumed constant  for extended periods, unless external perturbations (like interactions, minor, or major mergers) change the condition of the gas in the galaxy. \nYSC disruption is usually inferred by changes in the number of clusters as a function of time, assuming that the SFH has been constant. In the presence of enhancement in SF, the change brought by the increasing SFR can be misinterpreted as disruption. Thus it is of fundamental importance to know the recent  SFH of the host galaxy. \nThe easiest assumption of a constant star formation rate allows a very straightforward interpretation of the age function which it is not necessarily true. In the case of M51, we know it is an interacting system and that tidal interactions can enhance the star formation \\citep{pettitt2017}. Many simulations of the M51 evolution have suggested a double close passage of the companion galaxy, the older approximately $400-500$ Myr ago and a more recent one $50-100$ Myr ago \\citep[see][]{salo2000,dobbs2010_m51}. Whether the enhancement of star formation during the close passages with the interacting galaxy has a visible impact on the age function is difficult to assess, without an accurate star formation history. Our analysis is limited to an age range $<200$ Myr, hence we expect our analysis to be only partially affected.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\columnwidth]{13_plots_agefunc.pdf}\n\\caption{Age function of the cluster catalogue, comparing the effect of different bin widths, namely widths of 0.5 dex (blue squares), 0.6 dex (orange triangles) and 0.7 dex (red circles). The grey-shaded area marks the ages at which incompleteness causes a steepening of the slope, preventing the study of the function.}\n\\label{fig:agefit}\n\\end{figure}\nWe build an age function dividing the sample in age bins of 0.5 dex width and taking the number of sources in each bin. This number is normalised by the age range spanned by each bin (Fig.~\\ref{fig:agefit}). Points have been fitted with a simple power law $dN\/dt \\propto t^{-\\gamma}$ up to $\\textrm{log(age\/yr)}=8.5$. After that the incompleteness strongly affects the shape of the function, which starts declining steeply. The resulting slope is $\\gamma=0.38 \\pm0.06$,  smaller than reported  by CH16 who found $\\gamma$ in the range $0.6-0.7$ using a set of mass selections and age intervals. \nNote that, in their cluster selection, CH16 do not remove clusters in the internal part of the galaxy, which happen to have a much steeper slope. However the main difference can probably to be attributed to the different age modeling of the two catalogues. As can be seen from Fig. \\ref{fig:comparison} the ages of the CH16 catalogue have a prominent peak between logarithmic ages of 6.5 and 7. The resulting cluster ages are therefore younger on average and result in a steeper decline.\n\nA source of uncertainty in the recovered slope of the age function is related to the binning of the data. The distribution of the ages is discrete, and therefore binning is necessary, but the choice of the bins can affect the recovered slopes. To test possible variations we repeat our analysis changing the bin size. Results are shown in Fig.~\\ref{fig:agefit}. Differences on the recovered slopes are within the errors and therefore in this case the age function is statistically not sensitive to the choice of binning.\n\nCaution must be taken when considering the age function at young ages: in the literature it has been proposed that an ``infant mortality'' (introduced by \\citealp{lada2003}), caused by the expulsion of leftover gas from star formation, could in principle cause a rapid decline in the number of clusters surviving after $\\sim10$ Myr. However numerical simulations show that gas expulsion does not have strong impact of the dynamical status of the stars within a gravitationally bound cluster (see \\citealp{longmore2014} review). As already discussed in \\citet{legus2}, at young ages it could be easier to include in the sample sources that are unbound at the origin. We are not considering the sizes of clusters, or their internal dynamics, therefore we are unable to assess the boundness of clusters. However, given a typical size of a few parsecs for the cluster radius \\citep{ryon2015,ryon2017}, we know that clusters older than 10 Myr have ages larger than their crossing time, and we can consider them bound systems. \nThe inclusion of unbound sources would cause a strong decline in the age function because they contaminate our sample at ages younger than 10 Myr. If we exclude the youngest age bins below 10 Myr, we find that the fit results in a shallower slope, $\\gamma=0.30 \\pm0.06$ (compare the resulting slopes in Fig.~\\ref{fig:agefit} and Fig~\\ref{fig:agefit_mass}). In the hypothesis of constant SFR over the last 200 Myr, we conclude that disruption is not very significant in the age range $10-200$ Myr. \n\nIn order to test if the disruption time is mass dependent, we divided the catalogue in three subsamples of increasing mass, in the ranges: $5000-10^4$ M$_\\odot$, $10^4-3\\times10^4$ M$_\\odot$\\ and masses $>3\\times10^4$ M$_\\odot$. The slopes of the resulting age functions (Fig.~\\ref{fig:agefit_mass}) present small differences, with more massive sources having flatter slopes. Incompleteness at ages $\\approx200$ Myr may start affecting the less massive sources ($\\sim$5000 M$_\\odot$) which seem to have significantly more disruption. The differences between the two more massive bins are within 1 $\\sigma$, thus very similar.\nCH16 found slopes compatible within 1 $\\sigma$ in different mass bins, i.e. $\\gamma=-0.71 \\pm0.03$, $-0.64 \\pm0.20$ and $-0.62 \\pm0.07$ for mass ranges log(M\/M$_\\odot$)$=3.8-4.0$ , $4.0-4.5$ and $4.5-5.0$ respectively. These slopes are systematically steeper than what we find, with differences close to $\\sim3 \\sigma$ from our values.\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{14_plots_agefunc_mass.pdf}\n\\caption{Age function divided in mass bins. Dashed lines represent the best fitting curve for the bins in the range Log(age\/yr)$=7-8.5$. The grey-shaded areas mark the part of the functions excluded from the analysis due to incompleteness (old ages) and possible contamination by unbound sources (young ages).}\n\\label{fig:agefit_mass}\n\\end{figure}\n\nThe difficulty in retrieving the correct model for mass disruption can be also due to the simultaneous action of different processes dispersing the clusters mass.\n\\citet{elmegreen2010} propose a model in which clusters are put into an hierarchical environment in both space and time and show that, under reasonable assumptions, many different processes of mass disruption (or the combination of them) can reproduce an age function with a power law decline, as generally observed.\n\nUnder the assumption of a MDD time, we can derive a typical value for cluster disruption. We consider $t_4$, i.e. the time necessary to disrupt a cluster of $10^4$ M$_\\odot$, as it is an indicative physical value. We use a maximum-likelihood fitting technique, introduced by \\citet{gieles2009}, where we assume an ICMF described by a $-2$ power law with a possible exponential truncation at $M_*$ (which is left as a free parameter) and a disruption process which is mass dependent in time, with a timescale-mass relation given by $t_{dis}\\propto M^{0.65}$. In this analysis we considered sources with ages up to $10^9$ yr, limiting the sample to $M>5000$ M$_\\odot$\\ and $V_{mag}<23.4$ mag (see Fig.~\\ref{fig:agefit_maxlk}). The mass cut, as previously pointed out, allows us the study of a mass-complete sample (up to 200 Myr). The cut in magnitude instead allows us to consider sources older than $200$ Myr, accounting for the fading of old sources below the magnitude completeness limit.\n\nThe results of this maximum-likelihood analysis are $M_*=(8.6\\ \\pm\\ 0.5)\\times10^4$ M$_\\odot$\\ and $t_4=230\\ \\pm\\ 20$ Myr (probability distributions are given in Fig.~\\ref{fig:agefit_maxlk}). \nThe Schechter truncation mass is compatible to what we found in the previous section within the uncertainties. The disruption timescale is of the same order of $t_4\\simeq130$ Myr found in the analysis of \\citet{gieles2009}, slightly above the interval $100\\lesssim t_4\\lesssim200$ Myr retrieved for different assumptions for the cluster formation history in \\citet{gieles2005}. However, the analysis of the age function suggests that some disruption, possibly also with mass independent time, may have been effective from the beginning, as we see a mild decrease in $dN\/dt$ already at young ages.\n\\begin{figure}\n \\centering\n\\subfigure{ \\includegraphics[width=0.95\\columnwidth]{15a_maxlk_age6to9_mcut37_magl23_4_tot_2ndpart.png}}\n\\subfigure{ \\includegraphics[width=1.\\columnwidth]{15b_maxlk_age6to9_mcut37_magl23_4_tot_1stpart.png}}\n\\caption{Probability distributions for the Schechter mass $M_*$ and for the typical disruption time of a $10^4$ M$_\\odot$\\ cluster, $t_4$, in the maximum-likelihood test (top). The red dot indicates the best fit. The plots on the bottom collect the age-mass distributions, with a dashed line indicating the limit (both due to the imposed mass cut and the limiting magnitude) above which we have selected the sources for the analysis. Black points are the selected sources, while blue points are the total sample. \n}\n\\label{fig:agefit_maxlk}\n\\end{figure}\n\n\n\n\\section{Simulated Montecarlo Populations}\n\\label{sec:montecarlo}\nIn order to better understand the results of the previous sections, we perform simulations, producing Monte Carlo populations of synthetic clusters, with properties similar to what we observe for the cluster population of M51. We use different initial sets of parameters for the mass distribution and the disruption and compare the synthetic populations with the observed one. This is particularly useful for the luminosity function, which, while easy to obtain, is the result of many generations of YSCs formed and evolved within the galaxy.\nAnalytical and semi-analytical models have tried to derive an expected LF shape from basic assumptions on the CIMF and on the age distribution and have been applied to the studies of cluster populations (e.g. \\citealp{haas2008} in M51, \\citealp{fall2006} in the Antennae, \\citealp{hollyhead2016} in NGC 1566). We will also refer to those studies in order to compare expectations, simulations and observations.\n\n\\subsection{Simulated Luminosity Functions}\nWe simulate populations of clusters with masses larger than 200 M$_\\odot$. The number of clusters per simulation is set such that we have the same number of clusters with $M>5000$ M$_\\odot$\\ as the observed population, i.e. $\\sim1200$. The star (and cluster) formation history is assumed constant for 200 Myr, which is also the maximum age we assign to the simulated clusters. After simulating age and mass for each of the synthetic clusters we assign a magnitude to them in the $B$, $V$ and $I$ bands, using the same models adopted for the SED fitting and described in Section~\\ref{sec:sed}. We can then build the luminosity functions, both in the binned and in the cumulative way (see Sec~\\ref{sec:lumfunc} for ref.), and fit them with a power law. The models used for the initial mass function and for the disruption are:\n\\begin{description}\n\\item[\\textbf{PL-2:}] pure $-2$ power law, no disruption.\n\\item[\\textbf{PL-2\\_MDD:}] pure $-2$ power law, mass dependent disruption time model with $k=0.65$ and $t_4=230$ Myr (as the results of the maximum-likelihood fit of Sec~\\ref{sec:agefunc} suggest, see the same Section also for the formalism).\n\\item[\\textbf{SCH:}] Schechter function\\footnote{The Schechter mass function, when not specified, is assumed to have a slope $-2$ in the power law part, i.e. $dN\\propto M^{-2}\\ e^{-M\/M_*} dM$} with $M_*=10^5$ M$_\\odot$, no disruption.\n\\item[\\textbf{SCH\\_MDD:}] Schechter function with $M_*=10^5$ M$_\\odot$, mass dependent disruption time model with $k=0.65$ and $t_4=230$ Myr.\n\\end{description}\nWe have not included MID in the models because it will only  change the normalisation of the LF, not the shape. For this reason the two models without disruption (PL-2 and SCH) can be used to also study the expected LF in the case of mass independent disruption time model.\n\nThe results of the analysis are collected in Tab~\\ref{tab:mc_lum} and plotted in Fig~\\ref{fig:lumfit_simulated}.  \nAs expected, a $-2$ power law mass function has a luminosity function with the same shape, when no disruption is considered. \nSlopes close to $-2$ are retrieved in all filters with both a binned and a cumulative function fit. The values for the cumulative function are slightly steeper, but both methods gives comparable results.\n\\begin{table}\n\\centering\n\\caption{Results of the fit of the luminosity functions derived from the simulated cluster populations with a pure power law. The models are described in the text. The fit of the observed luminosity function (with only sources younger than 200 Myr) is given for comparison. The errors, not reported in the table, are on the order of $\\pm0.03$ mag. The magnitude limits used are the same of the real data, listed in Tab.~\\ref{tab:lumfit}.}\n\\begin{tabular}{llllllll}\n\\hline\n\\multicolumn{1}{l}{MF}\t&  \\multicolumn{3}{l}{binned function} \t& \\multicolumn{1}{c}{\\ }\t& \\multicolumn{3}{c}{cumulative function} \\\\\n\\\t\t\t\t\t& B \t\t& V \t\t& I\t\t\t\t& \\ \t\t\t\t\t& B \t\t& V \t\t& I \t\t\t\t\\\\\n\\hline\n\\hline\nOBS.\t\t\t\t& 1.94\t& 1.97\t& 1.99\t\t\t& \\\t\t\t\t\t& 2.14\t& 2.14\t& 2.25\t\t\t\\\\\n\\hline\nPL-2\t\t\t\t\t& 1.90\t& 2.00\t& 1.96\t\t\t& \\\t\t\t\t\t& 2.02\t& 2.03\t& 2.06\t\t\t\\\\\nPL-2\\_MDD\t\t\t& 1.65\t& 1.70\t& 1.76\t\t\t& \\\t\t\t\t\t& 1.78\t& 1.79\t& 1.83\t\t\t\\\\\nSCH\t\t\t\t\t& 1.94\t& 2.06\t& 2.13\t\t\t& \\\t\t\t\t\t& 2.19\t& 2.20\t& 2.38\t\t\t\\\\\nSCH\\_MDD\t\t\t& 1.81\t& 1.90\t& 2.02\t\t\t& \\\t\t\t\t\t& 2.08\t& 2.09\t& 2.29\t\t\t\\\\\n\\hline\n\\end{tabular}\n\\label{tab:mc_lum}\n\\end{table}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{16_lumfunc_simulated.pdf}\n\\caption{Luminosity functions of the simulated populations in the binned (left) and cumulative (right) form. $BVI$ bands are plotted in both cases. The models plotted are $-2$ power law (PL-2, solid line), $-2$ power law including MDD (PL-2\\_MDD, dashed) and Schechter (SCH, dashed-dotted) functions. Even if the underlying mass functions are very different, the luminosity functions are graphically quite similar in the binned form. When plotted cumulatively, instead, the differences are clear. In this second case the SCH model has the most similar LF shape to the one observed in the data.}\n\\label{fig:lumfit_simulated}\n\\end{figure*}\n\nConsidering disruption, MID would not change the shape of the mass or luminosity function, as discussed in Section~\\ref{sec:massage}. On the other hand a MDD would remove more quickly low-mass sources, modifying the luminosity function. The recovered slopes in this case are shallower, indicating that many low-luminosity sources have been removed (or have fallen below the completeness limit). \nThe effect of MDD is producing shallower LF in all filters. \nWe also observe that the slope is steeper in redder filters, as was also observed in the real data. This trend is mainly due to the difference in the the magnitude range fitted. The $I$ band, having a brighter completeness limit, is fitted only up to a magnitude of 22.25 mag, which is less affected by disruption than less bright magnitudes. If all filters will be fitted up to the same limiting magnitude the trend would not appear.  \n\nWhen considering a Schechter mass function, the corresponding LF also has a steeper end. This is reflected in the recovered slopes, which in many cases have values more negative than $-2$. Including MDD still produces the effect of bending the low-luminosity end of the LF, but in this case, with slopes shallower than $-2$ at low luminosity and steeper than $-2$ at high luminosities, the resulting slope with a single power law can still be more negative than $-2$, as observed in the real data.\n\nWe can conclude that in order to produce luminosity functions with slopes steeper than $-2$ we need the underlying mass function to be truncated, or at least steeper than $-2$. MDD can affect the luminosity function, but only producing shallower functions. \n\n\n\n\\subsection{Age-Luminosity relation}\nAnother characteristic of the luminosity function that can be tested is the relative contribution of sources of different ages to the total luminosity in each magnitude bin. It has been proposed in the literature that, for a truncated mass function, the median age of clusters vary as function of luminosity, with the brightest clusters being on average younger than the faintest \\citep[see e.g. ][]{adamobastian2015}. This expectation has been confirmed with semi-analytical models \\citep[e.g.][]{larsen2009,gieles2010} and compared successfully with observations \\citep{larsen2009,bastian2012b}. \n\nTo verify that a similar trend is visible also with our catalogue we use the simulated cluster population, considering the PL-2 and SCH runs. \nFor both cases we divide the sources in magnitude bins of 1 mag and we take the median age of the sources inside each bin. We repeated this process 100 times and in Fig.~\\ref{fig:lumfit_ages} we plot the area covered by the distribution of the central 50\\% median ages per magnitude bin. In the case of PL-2 the median ages are more or less constant at all magnitude, even if towards the bright end the spread between the percentiles increases, due to the lower number of sources there. On the other hand, for SCH, a trend with younger ages towards brighter bins is clear. \nWe also include the median and 25th and 75th percent intervals obtained from the observed luminosities of the clusters. They show the same decreasing trend as the SCH model, bringing additional support to it. Similar conclusions are reached studying the age-luminosity relation for $\\sim$50 of the brightest clusters of M51 with spectroscopically-derived ages in a forthcoming paper (Cabrera-Ziri et al., in prep).\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{17_plot_lumages_cut.pdf}\n\\caption{Median ages of the clusters in magnitude bins of 1 mag. The observed data (black points, with bars extending from the 1st to the 3rd quartiles) are plotted over the expectations from the models. The shaded areas encompass the central 50\\% distributions of medians from the simulations. The light areas (with an almost uniform behavior) are from the PL-2 model, the (declining) darker areas are from the SCH model. The data show a trend for younger ages associated in brighter clusters.  This is in agreement with expectations from a Schechter mass function with a mass truncation at $10^5$ M$_\\odot$.}\n\\label{fig:lumfit_ages}\n\\end{figure*}\n\nWe have not considered disruption in this simple comparison. Anyway we do not expect the disruption to change drastically the results: MID is unable to produce the observed trend, and MDD could in principle only produce an opposite trend, with brighter sources being on average older \\citep{larsen2009}. We must conclude that the trend we see between ages and luminosities is another sign of an underlying truncated mass function.\n\n\n\\subsection{Simulated Mass Function}\n\\label{sec:mcmass}\nWe compare the simulated mass functions with the observed one. For each model we set the number of simulated clusters in order to be the same of the observed ones. In doing so we are able to test for the effect of random sampling from the mass function, which could produce a ``truncation-like'' effect (see \\citealp{dasilva2012}). We repeat the simulations 1000 times and compare the observed mass function with the median, the 50\\% and the 90\\% limits of the simulated functions in Fig.~\\ref{fig:massfit_simulated}. \nWe plot the mass functions in the cumulative form as we have seen that in this way the differences are graphically easier to spot. The models of the mass function considered for the simulations are a simple $-2$ power law, the single power law best fit of the cumulative function and a Schechter function with truncation mass $1.0\\times10^5$ M$_\\odot$\\ (best fit found in Tab.~\\ref{tab:massfit2}).\nWhen comparing the mass functions behaviours in Fig.~\\ref{fig:massfit_simulated} we immediately notice that a simple $-2$ power law (left panel) overestimates the number of cluster at high masses. A steeper power law (middle panel) follows better the observed data on average, but it underestimates the low mass clusters and overestimates the high mass ones. \nThe Schechter case (right panel) instead follows quite nicely the observed mass function at all masses.\n\\begin{figure*}\n\\centering\n\\subfigure{\\includegraphics[width=0.325\\textwidth]{18a_montecarlo_mass_pl2.pdf}}\n\\subfigure{\\includegraphics[width=0.325\\textwidth]{18b_montecarlo_mass_pl230.pdf}}\n\\subfigure{\\includegraphics[width=0.325\\textwidth]{18c_montecarlo_mass_sch.pdf}}\n\\caption{Monte Carlo simulations (black lines) compared to the observed mass function (blue triangles). 1000 Monte Carlo populations were simulated. The median mass distributions (solid lines) and the limits within 50\\% (dashed) and 90\\% (dotted) of the simulations are plotted. The resulting probability of the AD test is also reported (see text for description). The models for the simulated clusters are pure power laws with different slopes (left and center) and Schechter mass functions (right).}\n\\label{fig:massfit_simulated}\n\\end{figure*}\n\nWe test the null hypothesis that the observed masses are described by our models. We are mainly interested in the upper part of the mass function, which is the only part that possibly deviates from a simple power law description. We run the Anderson-Darling (AD) test comparing the observed masses larger than $1.0\\times10^4$ M$_\\odot$\\ with the ones produced in the Monte Carlo simulations. The AD test returns the probability that the null hypothesis (the two tested samples are drawn from the same distribution) is true and a typical value for rejecting the null hypothesis is $p\\sim10^{-4}$. The resulting probabilities of our test are collected in the plots of Fig~\\ref{fig:massfit_simulated}. \nThey confirm that $-2$ power law is a poor description for the massive part of the function (p $\\approx10^{-5}$) while the other cases perform better (p $=0.070$ for the steeper power law and p $=0.334$ for the Schechter function). Both for the steeper power law and for the Schechter function the null hypothesis is not rejected. The test gives a better agreement with the Schechter function.\n\nWe can conclude that the analysis of the mass function suggests that there is a mechanism that inhibits the formation of clusters at very high masses. As pointed out in Section~\\ref{sec:GMC}, the same is seen for GMCs, and this mass cut could therefore come from the progenitor structures.\n\n\n\n\\section{Cluster Formation Efficiency}\n\\label{sec:cfe}\nThe Cluster Formation Efficiency, CFE (also called $\\Gamma$), is the fraction of star formation happening within bound clusters \\citep[see][]{bastian2012}. Previous studies have found that $\\Gamma$ varies positively with the SFR density, $\\Sigma_{\\textrm{SFR}}$. Galaxies with higher $\\Sigma_{\\textrm{SFR}}$\\ also have on average a larger $\\Gamma$ (see \\citealp{goddard2010,adamo2011,ryon2014} and Fig. \\ref{fig:gamma}). Variations of CFE values have been observed also inside single galaxies, e.g. in the study of M83 \\citep{silvavilla2013,adamo2015}, suggesting that the main dependence of the efficiency is on the $\\Sigma_{\\textrm{SFR}}$.\n\nIn order to calculate $\\Gamma$, we compared the SFR with the cluster formation rate (CFR). We calculated the SFR from FUV, correcting for dust using 24$\\mu$m emission according to the recipe from \\citet{hao2011} and assuming a $0.1-100$ M$_\\odot$\\ Kroupa IMF. The resulting value is collected in Tab.~\\ref{tab:gamma}.\n\nThe CFR has been calculated in the age ranges, $1-10$, $1-100$, $1-200$ and $10-100$ Myr.\nThe sum of the masses of clusters in our catalogue with $M>5000$ M$_\\odot$\\ in the selected age range, divided by the age spanned, gives the CFR of the mass limited sample of sources. In order to correct for the missing mass of clusters less massive than 5000 M$_\\odot$, we have assumed a model for the ICMF and derived the contribution of the low-mass clusters down to 100 M$_\\odot$.\nThe assumed ICMF is a Schechter function with a truncation mass of $1.0\\times10^5$ M$_\\odot$, as resulting from the fit in Section~\\ref{sec:massfunc}.\n\nValues for measured masses, CFRs and $\\Gamma$ are collected in Tab.~\\ref{tab:gamma}. \nErrors on $\\Gamma$ have been estimated considering uncertainties of 0.1 dex on age and mass values, a Poisson error \non the number of clusters used for calculating the total mass, the errors on the mass function given in Tab.~\\ref{tab:massfit2} and assuming an uncertainty of $10\\%$ on the SFR value. \n$\\Gamma_{1-10}$ has been derived using two different tracers for the SFR. In one case H$\\alpha$+24$\\mu$m has been used following the recipe by \\citet{kennicutt2009}. This recipe assumes that SFR has been constant for $\\sim100$ Myr. In the second case H$\\alpha$\\ has been used without correction for obscured SF. This second case traces only the SF younger than 10 Myr but a 50\\% uncertainty is associated to the SFR value.\n\\begin{table}\n\\centering\n\\caption{CFR and CFE ($\\Gamma$) values for M51. $^a$ SFR derived from FUV+24$\\mu$m, associated uncertainty of $\\sim10\\%$. $^b$ SFR derived from H$\\alpha$+24$\\mu$m, associated uncertainty of $\\sim10\\%$. $^c$ SFR derived from H$\\alpha$\\ only, not corrected for obscured SF, associated uncertainty of $\\sim50\\%$.}\n\\begin{tabular}{ccccc}\n\\hline\nAge range\t\t& CFR\t\t& SFR\t\t& $\\langle\\Sigma_{\\textrm{SFR}}\\rangle$ \t\t\t\t& $\\Gamma$\t\\\\ \n\\ [Myr]\t\t& [M$_\\odot$\/yr]\t& [M$_\\odot$\/yr]\t& $\\left[\\textrm{M}_{\\odot}\/\\textrm{yr kpc}^{-2}\\right]$\t\t& [\\%]\t\\\\\n\\hline\n\\hline\n$10-100$\t\t& 0.305\t\t& 1.636$^a$\t\t\t& 0.0139\t\t& 18.6$\\ _{\\pm2.4}$\t\t\\\\\n\\hline\n$1-10$\t\t& 0.465\t\t& 1.437$^b$\t\t\t& 0.0132\t\t& 32.4$\\ _{\\pm12.1}$\t\\\\\n$1-10$\t\t& 0.465\t\t& 0.734$^c$\t\t\t& 0.0062\t\t& 63.3$\\ _{\\pm39.0}$\t\\\\\n$1-100$\t\t& 0.321\t\t& 1.636$^a$\t\t\t& 0.0139\t\t& 19.6$\\ _{\\pm2.5}$\t\t\\\\\n$1-200$\t\t& 0.264\t\t& 1.636$^a$\t\t\t& 0.0139\t\t& 16.2$\\ _{\\pm1.9}$\t\t\\\\\n\\hline\n\\end{tabular}\n\\label{tab:gamma}\n\\end{table}\n\n$\\Gamma_{10-100}=(18.6 \\pm2.4)\\%$ is probably the value that is less affected by systematics. The absence of clusters younger than 10 Myr effectively avoid the possible inclusion of unbound sources in the sample, while the restriction to ages younger than 100 Myr lowers the effects of the cluster disruption on the $\\Gamma$ derivation \\citep[see][]{kruijssen2016}.\nHowever, Tab.~\\ref{tab:gamma} shows that all $\\Gamma$ are consistent within 2$\\sigma$ with a 20\\% value. $\\Gamma_{1-10}$ and $\\Gamma_{1-100}$ are bigger due to the possible contamination by unbound sources already discussed in Section~\\ref{sec:agefunc}. $\\Gamma_{1-200}$ is slightly lower, but we know from the age function analysis (Section~\\ref{sec:agefunc}) that disruption is affecting the cluster population.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\columnwidth]{19_gamma_m51_yesmodel.pdf}\n\\caption{Cluster Formation Efficiency $\\Gamma$ in function of the average $\\Sigma_{\\textrm{SFR}}$ derived in the age range 10-100 Myr (blue square). Values for other galaxies \\citep[taken from][]{goddard2010,adamo2011,annibali2011,ryon2014,adamo2015,lim2015,johnson2016} are shown for comparison. The black solid line is the $\\Sigma_{\\textrm{SFR}}-\\Gamma$ model presented in \\citet{kruijssen2012} with a 3$\\sigma$ uncertainty enclosed by the dotted lines.}\n\\label{fig:gamma}\n\\end{center}\n\\end{figure}\n\n$\\Gamma_{10-100}$ is compared with CFEs from other galaxies in Fig.~\\ref{fig:gamma}. Our value for M51 is similar to the $\\Gamma$ values found for other local galaxies, in particular for similar spiral galaxies like M83 \\citep{adamo2015}. More in general, it fits well into the $\\Sigma_{\\textrm{SFR}}$$-\\Gamma$ relation modeled by \\citet{kruijssen2012}, which predicts the amount of star formation happening in the cluster to increase with increasing surface density of star formation. \nIn a recent work on M 31, \\citet{johnson2016} suggested a comparison of the $\\Gamma$ values with the surface density of molecular gas. Their results show that the cluster formation efficiency scales with $\\Sigma_{\\textrm{gas}}$, with the mid-plane pressure of the galactic disk and with the fraction of molecular over atomic gas. These finding suggests that environments with higher pressure can form denser gas clouds which in turn result in a higher fraction of star formation happening in a clustered form. The environmental analysis of Paper II, based on regions with different $\\Sigma_{\\textrm{SFR}}$\\ and $\\Sigma_{\\textrm{gas}}$ inside M51, will help testing the cluster formation efficiency scenario.\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nUsing LEGUS \\citep{legus1} broadband observations of M51 we have built a new catalogue of young stellar clusters. The new WFC3 coverage allows very accurate photometry in the $UV$ and $U$ bands, necessary for deriving precise ages of the young sources. \nThe cluster catalogue is automatically extracted using the steps described in \\citet{legus2}. A critical parameter of the extraction is the minimum concentration index considered (1.35 in our analysis). The SEDs of the extracted sources are fitted with \\textit{Yggdrasil} SSP models \\citep{yggdrasil} in a deterministic approach. \n\nSources with detection in at least 4 filters are initially visually classified by humans. This subsample has been used as a training set for a ML algorithm that has classified the entire catalogue.\nWe focus our analyses on 2834 sources which are compact and uniform in color, neglecting multi-peaked sources.\n\nClusters are spatially associated with the spiral arms of the galaxy up to ages of 200 Myr, as recently shown in a simulation of spiral galaxies by \\citet{dobbs2017}.\nThe luminosity, mass and age functions of the cluster sample are analyzed and compared to the results of simulated Monte Carlo cluster populations, in order to be able to better interpret the observed features. \n\nWe list hereafter the main results of this work.\n\\begin{itemize}\n\\item A double power law provides the best fit for the luminosity function, suggesting a truncation at bright magnitudes which is better observed when the function is plotted in a cumulative way. A trend of steeper slopes with redder filter is observed, as already pointed out by \\citet{haas2008}, and, in each filter, brighter sources have on average younger ages. \n\\item The mass function has been directly studied with a maximum--likelihood fit, supporting the hypothesis of a truncated function: the cluster population of M51 is consistent with a Schechter function with slope $-2$ and a mass truncation at $M_{*} = 10^5$ M$_\\odot$. \nThe analysis is repeated considering only high mass clusters (M$>10^4$ M$_\\odot$). A steeper slope is retrieved but the fit still gives preference to a truncated function with $M_{*}\\sim10^5$. \n\\item The age function indicates the presence of a moderate disruption over the range $10-200$ Myr.\nThe study of disruption in the first 10 Myr is precluded by the contamination of possibly unbound sources. Under the assumption of a mass-dependent disruption in time, a typical timescale for the disruption of $10^4$ M$_\\odot$\\ clusters is derived, $t_4=230$ Myr.  \n\\item Simulated Monte Carlo populations are used to test the luminosity and mass function analyses. The trends observed in the luminosity function are recovered when a cluster population with an underlying Schechter mass function is analysed. A cluster population simulated with a pure power law mass function fails instead to produce the observed trends. \nWe notice that a cumulative function is the preferred way to study the bright end of the luminosity function and to investigate a possible deviation from a simple power law. The inclusion of cluster disruption in the simulations have also an impact on the luminosity function, but only at the faint end.\nMonte Carlo population are also used for a direct study of the mass function. A careful study of the high mass end of the function in the comparison with the simulated populations rejects the hypothesis of the function following a simple $-2$ power law, with a probability of $\\simeq0.001\\%$ given by an Anderson-Darling statistics. \n\\item We derive the fraction of star formation happening in clusters, which for M51 is $\\simeq20\\%$. This value is in line with the model of \\citet{kruijssen2012} linking the cluster formation efficiency with the SFR density, as well as consistent with observations for $\\gamma$ within local star-forming galaxies. \n\\end{itemize}\n\nIn a forthcoming paper (Paper II) clusters will be analyzed as a function of environments inside M51, enabling us to link cluster properties with the interstellar medium and GMC properties. \n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nBased on observations made with the NASA\/ESA Hubble Space Telescope, obtained  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. \nThese observations are associated with program \\# 13364.\nSupport for program \\# 13364 was provided by NASA through a grant from the Space Telescope Science Institute.\nWe acknowledge the help and suggestions provided by M. Gieles and N. Bastian.\nCLD acknowledges funding from the European Research Council for the FP7 ERC starting grant project LOCALSTAR. \nDAG kindly acknowledges financial support by the German Research Foundation (DFG) through program GO 1659\/3-2.\nMF acknowledges support by the Science and Technology Facilities Council \n[grant number  ST\/P000541\/1].\n\n\n\n\n\n\n\\bibliographystyle{mnras.bst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe puzzling nature of the hidden order (HO) phase of URu$_2$Si$_2$\\ is still not understood. The central interest of this enigma is that it reflects the duality between the local and itinerant characters of the $5f$ electrons. These different facets often play a major role in the field of strongly correlated electronic systems. In spite of more than two decades of intense search\\cite{Palstra:1985}, there is still no direct access to the order parameter (OP) as it occurs for the sublattice magnetization of the high pressure antiferromagnetic (AF) ground state with the wave-vector \\textbf{Q$_{AF}$}=(0,0,1)\\cite{Amitsuka:1999,Bourdarot:2005} which appears above $P_x\\simeq0.5$ GPa via a first order transition switching from HO to AF phases\\cite{Motoyama:2003,Amitsuka:2007,Hassinger:2008}. However, recently it was noticed that an unambiguous signature of the HO phase is the sharp resonance at $E_0\\simeq1.8$ meV for the commensurate wave-vector \\textbf{Q$_0$}=(1,0,0) (equivalent wave-vector to \\textbf{Q$_{AF}$}) as this resonance mode collapses through $P_x$ while the other resonance at $E_1\\simeq4.1$ meV for the incommensurate wave-vector \\textbf{Q$_1$}=(1.4,0,0) persists through $P_x$\\cite{villaume:2008}. Furthermore, above $P_x$, a magnetic field leads to the ''resurrection'' of the resonance at $E_0$, when the HO phase is restored \\cite{aoki:2009}. As the strong inelastic signal at \\textbf{Q$_0$} is replaced above $P_x$ by a large elastic signal, fingerprint of the AF ground state with \\textbf{Q$_{AF}$}=(0,0,1), it was proposed that, in both HO and AF phases, a lattice doubling along the \\textbf{c} axis occurs at the transition from paramagnetic (PM) to either HO or AF ground states\\cite{aoki:2009}. The nice feedback is that the change in the class of tetragonal symmetry via development of a new Brillouin zone generates a drastic decrease in the carrier number as pointed out by a large number of theories\\cite{Yamagami:2000,Harima:2009,Elgazzar:2009} and experiments\\cite{Hassinger:2008a}. Furthermore, as the HO-AF line touches the PM-HO and PM-AF lines at a critical pressure $P_c\\simeq 1.4$ GPa and critical temperature $T_c\\simeq19.5$ K, a supplementary symmetry breaking must occur between the HO and AF phases. Two recent theoretical proposals for the OP of the HO phase were a hexadecapole - from Dynamical Mean Field Theory (DMFT) calculations\\cite{Haule:2009};  or a $O_{xy}$ type antiferroquadrupole - from group theory analysis \\cite{Harima::2010}. For both models, the pressure-switch from HO to AF will add a supplementary time reversal breaking in the AF phase. For these models as well as for recent band structure calculations \\cite{Elgazzar:2009}, partial gapping of the Fermi surface may happen at $T_0=17.8$ K with a characteristic gap $\\Delta_G$. It was even proposed in the last model, that in the HO phase the gapping is produced by a spontaneous symmetry breaking occurring through collective AF moment excitations.\n\nThis article presents a careful revisit of the inelastic neutron response at \\textbf{Q$_0$} \\cite{Broholm:1991,Mason:1995,Wiebe:2007} using recent progress in polarized inelastic neutron configuration of the spectrometer IN22 and in the performance of the cold-neutron three-axis spectrometer IN12, both installed at the high flux reactor of the Institute Laue Langevin (ILL). Thus, this new generation of experiments provide a careful basis on the temperature and pressure evolution of $E_0$. They give a new insight on previous neutron studies thanks to our recent proof\\cite{villaume:2008} that the resonance at $E_0$ is up to now the major signature of the HO phase. Comparison will be made with the case of charge ordering observed in the skutterudite PrRu$_4$P$_{12}$ as well as with PrFe$_4$P$_{12}$ where a sequence of HO-AF phases are observed \\cite{JPSJ77SA:2008}.\n\n\\section{Experimental set up}\n\nHigh quality single crystals of URu$_2$Si$_2$\\ were grown by the Czochralski method in tetra-arc furnace. The details are described elsewhere \\cite{aoki:2010}. The sample already used for inelastic neutron scattering in the superconducting state \\cite{Hassinger:2010}, was installed in an ILL-type orange cryostat with $\\bf{c}$ axis oriented vertically on the thermal triple-axis IN22 in its polarized configuration and on the cold triple-axis IN12 in its standard configuration (both CEA-CRG spectrometer at ILL).\n\nThe IN22 experiment was performed at 1.5 K, with two fixed final energies : 14.7 meV ($k_f$=2.662 \\AA$^{-1}$) and 30.3 meV ($k_f$=3.84 \\AA$^{-1}$). The beam was polarized by a Heusler monochromator vertically focusing and analyzed in energy and polarization by a Heusler analyzer vertically (fixed) and horizontally focusing. The flipping ratio was around 17 and the energy resolutions were 0.95 meV and 2.4 meV for both energies, respectively. No collimation was installed. The background was optimized by an optical calculation of the dimension openings of the slits placed before and after the sample. The measurements were performed in the non-spin-flip channel which provides all the necessary information. The inelastic scans were performed with $\\bf{Q}$ parallel to the $\\bf{a}$ axis. When the polarization is along the $\\bf{a}$ axis, the intensity measured (I$^a_{NSF}$) corresponds to the nuclear and background contributions; when the polarization is along the $\\bf{b}$ or $\\bf{c}$ axis, the intensities measured (I$^b_{NSF}$) and (I$^c_{NSF}$) correspond to the same contributions plus the (imaginary part of the) susceptibility along the $\\bf{b}$ or $\\bf{c}$ directions. In the following, the data shown are the subtraction of inelastic scans performed in the non-spin-flip channel with polarization along the $\\bf{b}$ or $\\bf{c}$ axis, with an inelastic scan performed in the same conditions with polarization along the $\\bf{a}$ axis. The subtractions correspond to the imaginary part of the dynamical susceptibilities with for the $\\bf{b}$ axis $\\chi''_y\\propto (I^b_{NSF}-I^a_{NSF})$ the transverse susceptibility and for the $\\bf{c}$ axis $\\chi''_z\\propto (I^c_{NSF}-I^a_{NSF})$ the longitudinal susceptibility.\n\nThe  IN12 experiment was performed with a fixed final energy $E_f=4.7$ meV ($k_f$=1.5 \\AA$^{-1}$) that gives a good compromise between intensity of the excitation and energy resolution $l=0.11$ meV. The incident neutrons were selected by a (0,0,2) graphite vertically focusing monochromator with vertically focusing and analyzed in energy by a (0,0,2) graphite analyzer with horizontal focusing. No collimation was installed. The temperature was measured by a calibrated carbon thermometer and it was checked before each scan that the temperature was stable. As for IN22, the background was optimized by an optical calculation of the dimension openings of the slits placed before and after the sample. The raw scans were corrected for the electronic background (29 counts per hour) and for the $\\lambda\/2$ contamination of the monitor.\n\n\\section{Description of the model used for fitting the inelastic spectrum}\nIn a neutron scattering experiment, the neutron intensity $I(q,\\omega)$ in the detector is proportional to the convolution of the scattering function $S(q,\\omega)$ with the instrumental resolution function. $S(q,\\omega)$  is related to the imaginary part of the dynamical spin susceptibility $\\chi''(q,\\omega)$ ($\\chi(q,\\omega)= \\chi'(q,\\omega)+\\imath\\chi''(q,\\omega)$) via the fluctuation-dissipation theorem : $S(q,\\omega)=n(\\omega,T)\\chi''(q,\\omega)$  where $n(\\omega,T)= 1\/(1-e^{-\\hbar\\omega\/{k_BT}})$ is the detailed balance factor. \n Below $T_{0}$, the excitation is well-defined with an asymmetrical shape related to the finite extent of the resolution function that integrates the dispersion of the excitation in the vicinity of the nominal $q$ wave-vector at which the spectrometer is set-up. To analyze the data, firstly a harmonic oscillator function is taken for $\\chi\"(q,\\omega$) ; this corresponds to the difference of two normalized Lorentzian functions multiplied by $\\frac{\\chi_o\\,\\Omega^2_0}{\\omega_0\\gamma}$, where $\\chi_0=\\chi(q,\\omega=0)$ is the static susceptibility, $\\Omega_0$ is the oscillator frequency, $\\gamma$ its damping, and $\\omega_0$ is given by the equation $\\Omega_0=\\sqrt{\\omega^2_0+(\\gamma\/2)^2}$. Secondly, a simplified convolution with the resolution function is made : to this aim the resolution function is approximated by a 4D parallelepiped (instead of an ellipsoid) and the $q$ dispersion is taken as linear in all directions. This linear dispersion simplified the calculation and gives a better description of the dispersion than a usual quadratic law in the case of URu$_2$Si$_2$. This description is called the \\bm{$\\gamma$} \\textbf{model} when $\\gamma \\gg l$ and the \\bm{$l$} \\textbf{model} when $l \\gg \\gamma$, $l$ being the energy resolution. It leads to a simple analytic expression for $I(q,\\omega)$ at the minimums of the dispersion ($q_0$):\n\n\\begin{equation}\nI(q_0,\\omega)=n(\\omega,T)\\left(F(q_0,\\omega,)-F(q_0,-\\omega)\\right)\\\\\n\\label{m3}\n\\end{equation}\nwith\n\\begin{eqnarray}\n\\begin{split}\nF(q_0,\\omega)&=L\\,K(q_0,\\omega)\\left(\\sum_{i=x,y,z}\\sqrt{1+(2\\alpha_i\\,l\/\\omega_0(q_0))^2}\\right.\\nonumber\\\\\n&\\left.e^{-\\left(4\\ln{2}\\left(\\alpha_i\\frac{\\omega-\\omega_0(q_0)}{\\omega_0(q_0)}\\right)^2\\right)} \\right)\\\\\nK(q_0,\\omega)&=\\frac{1}{2}\\left(1+\\frac{2}{\\pi}arctan\\left(\\frac{2\\,(\\omega-\\omega_0(q_0))}{\\beta}\\right)\\right)\\nonumber\\\\\n\\end{split}\n\\end{eqnarray}\n\\\\\n\n\\noindent\nwhere $\\beta=\\gamma(q_0)$ for the \\bm{$\\gamma$} \\textbf{model} and $\\beta=l$ for the \\bm{$l$} \\textbf{model} \n$\\alpha_i=\\alpha_{i0}(\\omega_0\/\\Omega_0)^2$, where $\\alpha_{i0}$ is the ratio between the slope of the dispersion and the $q$-width of the resolution function. The L parameter, which depends of the magnetic form factor, the incident  and final energies ($k_i$ and $k_f$) is supposed to stay constant at first approximation.\n\nFinally the susceptibility $\\chi_{0}$ for $T<T_0$ is determined using the integrated intensity by applying the formula:\n\n\\begin{equation}\n\\begin{split}\nI_{\\Omega_0}=\\int^\\infty_0\\chi''(q_0,\\omega)d\\omega=\\\\\n\\frac{\\chi_0 \\Omega^2_0(q_0)}{\\omega_0(q_0)}\\arctan\\left(\\frac{\\omega_0(q_0)}{\\gamma(q_0)\/2}\\right)\n\\end{split}\n\\label{m4}\n\\end{equation}\n\nAbove $T_{0}$, the signal is much broader than the resolution and no convolution is needed (at least on IN12). For $\\chi\"(q,\\omega)$, we use the magnetic quasi-elastic model, which corresponds to a Lorentzian function of susceptibility $\\chi_{L0}$ and the full-width at half-maximum $\\Gamma_L$, multiplied by $\\omega$:\n\n\\begin{equation}\n\\chi''(q_0,\\omega)=\\frac{\\chi_{L0}\\,\\omega\\,\\Gamma_L}{\\omega^2+\\Gamma^2_L}\n\\label{lor}\n\\end{equation}\n\n\n Between \\bm{$\\gamma$} \\textbf{model} and the quasi-elastic model, the widths and the susceptibilities are linked by the relations: $\\Gamma_L=\\gamma\/2$ and $\\chi_0=\\chi_{L0}$.\n\n\\section{Results}\n\n\\subsection{Inelastic Polarized neutron scattering}\n\nAt first this experiment was to unambiguously determine the polarization of the resonances at \\textbf{Q$_0$} = (1,0,0) and \\textbf{Q$_1$} = (1.4,0,0) and to determine the nature of the signal occurring at much higher energy than the resonant modes. The origin of such a signal is often referred to as coming from multi-phonons. However, it may come from magnetic process as was hinted by C. Broholm \\cite{Broholm:1991}.\n\nFigure \\ref{fig5} shows the longitudinal and transverse magnetic response at \\textbf{Q$_0$} measured on IN22 with a final energy $E_f=14.7$ meV. No transverse magnetic response is detected for a range of energy transfer going from -3 meV to  27 meV. The longitudinal magnetic response shows two contributions: the well-known and well-defined resonance with a gap value $E_0$ fitted to 1.86(5) meV, where $E_0$ is the harmonic oscillator energy $\\Omega_{0}$  (the gap value $E_0$ measured with the cold triple-axis spectrometer IN12 is around 1.70(5) meV) and a broad magnetic contribution. This broad magnetic contribution, which looks like to a magnetic continuum persists at least up to 27 meV as seen in the inset of Figure \\ref{fig5} for measurements performed with a final energy of 30.3 meV (27\\ meV was the maximum energy transfer we could reach in this configuration). In this paper the intensity of this continuum is described by a Lorentzian function (eq.\\ref{lor}) of width $\\Gamma_{c}$. An extra elastic signal (at $\\omega=0$) corresponding to the small antiferromagnetic moment is detected. This well known signal is currently believed to be a parasitic contribution due to the survival of AF droplets generated near defects \\cite{Matsuda:2001,Amato:2004}. \n\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=80mm]{F5.eps}\n\\end{center}\n\\caption{(color online) Transverse (open circles) and longitudinal (filled circles) magnetic response of URu$_2$Si$_2$\\ at \\textbf{Q$_0$} and $T=1.5$ K with a final energy $E_f=14.7$ meV.  The full and dashed curves correspond to a \\bm{$l$}\\textbf{-model} of eq.(\\ref{m3}) and quasi-elastic function for the magnetic continuum, respectively. The vertical black arrow indicates the gap position of the resonance (1.86(5) meV) for the \\bm{$l$}\\textbf{-model}. The inset shows the same magnetic response but with  a final energy $E_f=30.3$ meV.}\n\\label{fig5}\n\\end{figure}\n\nFigure \\ref{fig6} shows the longitudinal and transverse magnetic response at \\textbf{Q$_1$} measured with a final energy $E_f=14.7$ meV.  As previously at \\textbf{Q$_0$}, no transverse magnetic response is detected in the range of 0 meV to 27 meV. Again the longitudinal response shows two inelastic magnetic contributions: the well-known and well-defined resonance with a gap $E_1$ of 4.06(6) meV, and the broad continuum of linewidth $\\Gamma_c$. Note that the gap $E_1=4.06(6)$ meV is slightly lower than the usual gap value $E_1\\simeq4.5$ meV found in previous experiment \\cite{Bourdarot:2003}. Let us point out that it is the first time that the resolution and the dispersion are taken into account to analyse this excitation. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F6.eps}\n\\end{center}\n\\caption{Transverse (open circles) and longitudinal (filled circles) response of URu$_2$Si$_2$\\ at \\textbf{Q$_1$} and $T=1.5$ K. The full and dashed curves correspond to a \\bm{$l$}\\textbf{-model} of eq.(\\ref{m3}) and quasi-elastic function for the magnetic continuum, respectively. The vertical black arrow indicates the gap position of the resonance (4.06(6) meV) for the \\bm{$l$}\\textbf{-model}.}\n\\label{fig6}\n\\end{figure}\n\nFigure \\ref{fig8} shows the longitudinal and transverse magnetic response for a \\textbf{Q}-scan performed with an energy transfer of 15 meV. A constant signal corresponding to the continuum is measured from \\textbf{Q}=(1,0,0) to \\textbf{Q}=(1.7,0,0), then the magnetic signal decreases approaching to nuclear zone center \\textbf{Q}=(2,0,0). The vanishing of the continuum has to be verified by new \\textbf{Q}-scans at different energy transfer in the futur.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F8.eps}\n\\end{center}\n\\caption{Transverse (open circles) and longitudinal (filled circles) response of URu$_2$Si$_2$\\ at $T=1.5$ K for a \\textbf{Q}-scan (Q$_x$,0,0) at 15 meV. The hatched area corresponds to the magnetic continuum.}\n\\label{fig8}\n\\end{figure}\n\nTo summarize, the inelastic polarized neutron scattering experiments performed at \\textbf{Q$_0$} and \\textbf{Q$_1$} confirm without any ambiguity that, at low temperature (below $T_0=17.8$ K), the magnetic response is exclusively longitudinal. We also evidence a broad magnetic continuum that may be fitted at least for the two main \\textbf{Q} positions (\\textbf{Q$_0$} and \\textbf{Q$_1$}) by exactly the same quasi-elastic function with a half-width $\\Gamma_c=7.8$ meV. This magnetic contribution was never taken into account in the previous studies of temperature dependence of the gaps at \\textbf{Q$_0$} and at \\textbf{Q$_1$}. This motivates us to reinvestigate the temperature dependence of the excitation at \\textbf{Q$_0$}.\n\n\\subsection{Low energy study at \\textbf{Q$_0$}}\n\nThe aim of this study is a precise determination of the temperature dependence of the magnetic resonance at the position \\textbf{Q$_0$}. This experiment was already performed by T.E. Mason \\cite{Mason:1995b} but without a large precision and more important without taking into account the magnetic continuum described by a quasi-elastic function and revealed by our recent inelastic polarized neutron scattering.  From their study they concluded that the gap follows a singlet ground state model but moreover that the evolution of the susceptibility $\\chi(\\mathbf{Q}_0,\\omega=0)$ shows a large enhancement at $T_0$. We will discuss these points below.\n\nFigures \\ref{fig11} and \\ref{fig12} show some representative inelastic spectrums obtained at \\textbf{Q$_0$} just below and above the transition temperature $T_0=17.8$ K respectively. At $T$=1.5 K, the signal measured at energies above 5-6 meV is derived from the continuum previously detected by inelastic polarized neutron scattering. As the half-width $\\Gamma_c=7.8$ meV of the quasi-elastic function which fit the continuum is already known at low temperature, and assuming that for temperatures lower than $\\Gamma_c\/k_B \\approx 88$ K, this width does not change, only the amplitude can depend on temperature. A good fit is obtained by taking a constant amplitude for all temperatures: the continuum being, at $T=27.1$ K, the unique contribution (as seen in Fig. \\ref{fig12}).\n\nFor $T<T_0$ (Fig. \\ref{fig11}), in parallel to the continuum, the well-defined resonance is detected as for $T=1.5$ K. The inelastic spectrums show clearly that the width $\\gamma_0$ and the gap $E_0$ change substantially with temperature only close to $T_0$. The gap $E_0$ and the width $\\gamma_0$ determined using the \\bm{$\\gamma$} \\textbf{model} are shown in Fig. \\ref{fig10} and \\ref{fig14}. As expected from the spectrum, the gap $E_0$ decreases only close to $T_0$ but more abruptly as the width $\\gamma_0$ increases. At $T_0$, $E_0\\approx\\gamma_0\/2$, that confirms that we enter into an over-damped regime for temperatures larger than $T_0$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F11.eps}\n\\end{center}\n\\caption{Energy scans measured at \\textbf{Q$_0$} for temperatures below $T_0=17.8$ K. Fits are performed with the \\bm{$\\gamma$} \\textbf{model} plus the continuum (quasi-elastic with $\\Gamma_c=7.8$ meV).}\n\\label{fig11}\n\\end{figure}\n\nFor $T>T_0$ (Fig. \\ref{fig12}), the magnetic response is over-damped, and the spectrum is treated with a quasi-elastic magnetic function of width $\\Gamma_L$ added to the continuum ($\\Gamma_c=7.8$ meV). As expected, $\\Gamma_L=\\gamma_0\/2$ at $T_0$, which validates the \\bm{$\\gamma$} \\textbf{model} and indicates that the gap $E_0$ drops to zero in the paramagnetic state (above $T_0$).  The half-width of this excitation is plotted versus temperature in Figure \\ref{fig14} (open circles). $\\Gamma_L$ increases rapidly (maybe linearly) with temperature. It was not possible to follow this signal for temperatures approaching $T=27.5$ K: above this temperature only the large tail of the continuum can be detected.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F12.eps}\n\\end{center}\n\\caption{Energy scans measured at \\textbf{Q$_0$} for temperatures just around the transition temperature $T_0=17.8$ K (and at 1.5 K for reference). Fits are described in the text. At $T=27. 1$ K (triangles), the scan is only fit by the continuum (quasi-elastic signal with $\\Gamma_c=7.8$ meV).}\n\\label{fig12}\n\\end{figure}\n\nAs the temperature evolution of the signal is not usual, the Figure \\ref{fig20} gives the variation of $\\chi''(\\mathbf{Q}_0,\\omega)\/\\omega$ as a function of $\\omega$ at different temperatures. Thus, this plot shows that this signal saturates at low temperature and the abrupt drop of E$_0$ when approaching $T_0$. Clearly the integration of $\\chi''(\\mathbf{Q}_0,\\omega)\/\\omega$ increases on cooling below $T_0$ as discussed latter as a consequence of Fermi Surface reconstruction.\n\nFigure \\ref{fig15} shows the magnetic susceptibility at $Q_0$, $\\chi(\\mathbf{Q}_0,\\omega=0)$. The susceptibility for $T<T_0$ is determined using eq. (\\ref{m4}) (filled circles), then $\\chi_{L0}$ is a fitted parameter of the quasi-elastic expression for $T>T_0$ (open circles). Of course, in addition, the susceptibility coming from the magnetic continuum (with $\\Gamma_c=7.8$ meV) has to be added (constant low contribution in Fig \\ref{fig15}). Without any scaling-factor, the susceptibilities from the \\bm{$\\gamma$} \\textbf{model} and the second magnetic quasi-elastic contribution ($\\Gamma_L$) are equal at $T_0$. The total susceptibility versus temperature shows a saturation at low temperature, then decreases  but with a bump around $T_0$. The susceptibility stays almost constant above $T = 30$ K. In contrast to Mason's analysis no marked divergence of $\\chi(\\mathbf{Q}_0,\\omega=0)$ is observed at $T_0$. Furthermore, in our data below 16 K, we found that $\\chi(\\mathbf{Q}_0,\\omega=0)$ increases on cooling before saturating at low temperature.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F10.eps}\n\\end{center}\n\\caption{Temperature dependence of the gap $E_0$. The curves are guide for the eyes.}\n\\label{fig10}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F14.eps}\n\\end{center}\n\\caption{Temperature dependence of the half-width $\\gamma_0$ of the resonance at \\textbf{$Q_0$} below $T_0$ (filled circle) and of the quasi-elastic $\\Gamma_L$ above $T_0$ (open circle). The curve below $T_0$ corresponds to a fit with a Korringa model.}\n\\label{fig14}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F20.eps}\n\\end{center}\n\\caption{Imaginary part of the dynamical spin susceptibility without the magnetic continuum divided by energy $(\\chi''(\\mathbf{Q}_0,\\omega)\/\\omega$) for different temperatures. The curves are deduced from the $\\gamma$ model corrected by the thermal factor and divided by $\\omega$. The integration of these curves gives $\\chi(\\textbf{Q$_0$},\\omega=0)$.}\n\\label{fig20}\n\\end{figure}\n\nThe integration $\\int^{6.3 meV}_0\\chi''(\\mathbf{Q}_0,\\omega)d\\omega$ ($\\simeq I_{E_0}$)  of the magnetic resonance at \\textbf{$Q_0$} without the continuum contribution ($\\Gamma_c$) is plotted in Fig. \\ref{fig13}. The filled circles correspond to the contribution with the oscillator model and the open circles correspond to the integration of the quasi-elastic contribution of width $\\Gamma_L$. This integration, below $T_0$, seems to mimic the temperature variation of an OP vanishing at $T_0$, while above $T_0$ it behaves like the contribution of a critical regime. Furthermore, the temperature variation of $I_{E_0}(T)$ is well described below $T_0$ by BCS formula used for the temperature variation of the superconducting gap \\cite{Bardeen:1957}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F15.eps}\n\\end{center}\n\\caption{Temperature dependence of the static susceptibility $\\chi(\\textbf{Q}_0,\\omega=0)$. In filled circles are data coming from the \\bm{$\\gamma$} \\textbf{model}, the open circles from the quasi-elastic model $\\Gamma_L$, and the hatched area coming from the magnetic continuum ($\\Gamma_c=7.8$ meV).}\n\\label{fig15}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=80mm]{F13.eps}\n\\end{center}\n\\caption{Temperature dependence of the integrated imaginary part of the dynamical spin susceptibility. In filled circles are data coming from the \\bm{$\\gamma$} \\textbf{model}, the open circles from the quasi-elastic model. The intensity coming from the magnetic continuum ($\\Gamma_c=7.8$ meV) was not taken into account. The line is the BCS-type gap below $T_0$, and a guide for the eyes above $T_0$.}\n\\label{fig13}\n\\end{figure}\n\n\\section {Discussion}\n\nThe occurrence of a magnetic continuum ($\\Gamma_c = 7.8$ meV) for energies $\\omega$ far higher than the energy $k_BT_0$ of the hidden order phase is a general phenomenon: large energy transfer leads to excite energy states not only from the ground state.\n\nAs the study is focused on the hidden order phase, we have concentrated our studies on the frequency response at \\textbf{Q$_0$}. Below $T_0$, the resonance at $E_0$ represents a collective mode with a half-width $\\gamma_0\/2$ much smaller than the gap energy $E_0$: the collective mode is long lifetime as shown by the large ratio $\\frac{E_0}{\\gamma_0\/2}\\approx35$. As shown on Fig. \\ref{fig10}, in decreasing temperature $E_0$ very rapidly reaches its final value $\\approx1.7$ meV. On crossing $T_0$, the resonance collapses and the magnetic response appears quasi-elastic with a half-width $\\Gamma_L$ becoming rapidly larger than the resonance energy $E_0$ at $T = 0$ K.\n\nThe susceptibility $\\chi($\\textbf{Q$_0$},$\\,\\omega$=0) reported in Fig. \\ref{fig15} is calculated for $T<T_0$ using eq. (\\ref{m4}). Due to the weakness of the resonance signal close to $T_0$ the uncertainty in the determination of  $\\chi($\\textbf{Q$_0$},$\\,\\omega$=0) is large. However it is obvious that no divergence  of  $\\chi($\\textbf{Q$_0$},$\\,\\omega$=0) occurs at $T_0$ as it is observed at the onset of an antiferromagnetic ordering. In heavy fermion compounds close to AF-PM critical point, as in Ce$_{1-x}$La$_x$Ru$_2$Si$_2$ series, a sharp maximum of $\\chi(\\textbf{Q$_0$},\\omega=0$) occurs at $T_N$ (for x$<$0.075) which is accompanied on cooling , below $T_N$ by a decrease of $\\chi(\\textbf{Q$_0$},\\omega=0)$ \\cite{Knafo:2009}. Here on cooling below $T_0$, by contrast, $\\chi(\\textbf{Q$_0$},\\omega=0)$ increases at it is observed for critical concentration x$_c=0.075$ in Ce$_{0.925}$La$_{0.075}$Ru$_2$Si$_2$. However, the major difference is that for URu$_2$Si$_2$\\ the saturation regime is achieved very steeply in temperature as the key energy is the gap energy $\\Delta_G$ (see Figure \\ref{fig10}). Also a drastic difference between URu$_2$Si$_2$\\ and CeRu$_2$Si$_2$ cases is that the resonance at E$_0$ collapses suddenly above P$_x$ at the benefit of the establishment of a large sublattice magnetization. The occurrence of a weak maximum of $\\chi(\\textbf{Q$_0$},\\omega=0)$ at $T_0$ may be consequence of the Fermi surface reconstruction with the particularity that the U ions through $T_0$ will go from an intermediate valence behavior above $T_0$ to a quasi-tetravalent dressing below $T_0$  \\cite{Hassinger:2008a}. This image, based on thermodynamical considerations, is supported by the recent tunneling microscope experiments which appear during the revision of our paper \\cite{aynajan:2010,schmidt:2010}. Thus this observation confirms the lack of antiferromagnetism at \\textbf{Q$_0$} in the hidden order phase. Thanks to our previous measurements\\cite{villaume:2008}, we have specified the emergence of the resonance at \\textbf{Q$_0$}=(1,0,0) as the signature of the hidden order (see also \\cite{Ohkawa:1999,Miyake:2010,Miyake:2010b}), we interpret the temperature dependence $I_{E_0}(T)$ plotted in Fig. \\ref{fig13} as the temperature dependence of the OP\\cite{villaume:2008}.\nIt was suggested by P.M. Oppeneer according the model developed in reference \\cite{Elgazzar:2009,Oppeneer:2009} that $I_{E_0}(T)$ may be related with the hidden order parameter via an even function of the magnetization amplitude which will not vanish in time even in the hidden order phase as the sublattice magnetization amplitude for an AF.\n\nAs the quasi-elastic contribution with the characteristic energy half-width $\\Gamma_L$ above $T_0$ seems to vanish above $T\\approx30$ K, a simple picture would be that only the continuum with $\\Gamma_c=7.8$ meV persists above $T=30$ K. According to the Kondo impurity model which predicts \\cite{1}\n$\\Gamma\\gamma=750$ mJ mol$^{-1}$K$^{-2}$, the linear term of the specific heat of $\\gamma$ can be approximated to $\\approx 96$ mJ mol$^{-1}$K$^{-2}$ with the continuum $\\Gamma_c$. This agrees with the observed magnitude of the linear term of the specific heat \\cite{vanDijk:1997b}. Of course an open problem is the modification of this continuum at low energy when crossing $T_0$ since transport measurements, as well as NMR and specific heat measurements, indicate clearly a Fermi surface reconstruction with a carrier drop by a factor 3 to 10 \\cite{Schoenes:1987,Kasahara:2007,Behnia:2005,Bel:2004,Kohara:1986,Kohara:1987,Maple:1986}, which favors localized magnetism.\n\nA key point is the rapid temperature evolution of the energy gap $E_0$ which is much faster than the temperature evolution of the BCS-type gap. The emerging image is that, at $T_0$, the change in lattice symmetry associated with the paramagnetic to hidden order phase transition (from $body-centered\\ tetragonal$ to $simple\\ tetragonal$) induced presumably by a multipolar ordering leads to large gapping of the Fermi surface with characteristic gap energy $\\Delta_G$ much larger than $E_0$. The origin of $E_0$ may come from crystal-field splitting with dispersion coming from dipolar and quadrupolar interactions (see for example \\cite{JPSJ77SA:2008,Kuramoto:2009,Balatsky:2009}). Opening a gap $\\Delta_G$ at $T_0$ leads to a drop of the carrier number which allows the observation and development of the resonance: the resonance at \\textbf{Q$_0$} is over-damped far above $T_0$ become a well-defined excitation below $T_0$. Of course the change of carrier number acts as well on the over-damped mode observed above $T_0$ for \\textbf{Q}$_1$=(1.4,0,0), allowing again the appearance of a sharp resonance below $T_0$ at $E_1\\approx4$ meV \\cite{Broholm:1991,Wiebe:2007}. \nSince $E_1$ is quite comparable to $\\Gamma_c$ below $T_0$, this explains why the inelastic contribution at \\textbf{Q$_1$} is still observed in the over-damped regime, above $T_0$. It was suggested that the hidden order phase may be an incommensurate antiferromagnet at \\textbf{Q}$_1$ \\cite{Balatsky:2009}. However, inelastic magnetic response at \\textbf{Q}$_1$ does not give evidence of a crossing through a phase transition: no divergence of the static susceptibility $\\chi(\\mathbf{Q}_1,\\omega=0)$ has been reported \\cite{Broholm:1991}. Even the shallow maxima of $\\chi(\\mathbf{Q}_1,\\omega=0)$ at $T_0$ may be an artifact of the fitting, or it can also be, as discussed for $\\chi(\\textbf{Q$_0$},\\omega=0)$ a consequence of the Fermi Surface reconstruction.\n\nAs recently proposed in two different approaches, DMFT \\cite{Haule:2009} and group theory analysis \\cite{Harima::2010}, the promising explanation is that the hidden order phase would be a multipolar phase: a hexadecapolar order in DMFT studies, a quadrupolar order in group theory. In this last scenario, the hidden order phase is still hidden as it corresponds to a second order phase transition from the space group $I4\/mmm$ (No.139) to the space group $P4_2\/mnm$ (No.136) with no lattice distortion and invariance of the Ru-site at the crossover transition from hidden order to antiferromagnetic phase. Switching from the hidden order to the antiferromagnetic phase will preserve the $P4_2\/mnm$ (No.136) symmetry of the lattice but will add of course the time-reversal symmetry operator. Let us emphazise that even if the resonance at \\textbf{Q}$_0$=(1,0,0) is not a direct proof of the hidden order parameter, there is no doubt that it is a key signature which supports strongly the change from $I4\/mmm$ (No.139) to $P4_2\/mnm$ (No.136) symmetry at the paramagnetic-hidden order border. If no quadrupolar signature will be detected, a possibility is that the HO phase of URu$_2$Si$_2$\\ may be regarded as an electronic spin Peierls transition with only tiny displacement of the atoms ($\\delta d\/d <10^{-6}$) \\cite{fomin}.\n\nFurthermore the collection of previous data with fine pressure tuning \\cite{Bourdarot:2004b,Bourdarot:2005,villaume:2008} and of recent data at a fixed pressure $P$ between $P_x$ and $P_c$ with supplementary magnetic field $H$ scans\\cite{aoki:2009}, allow to extract the $P$ dependence of $E_0$ and $E_1$ and to compare with the $P$ dependence of the gap $\\Delta_G$ derived from resistivity measurements (figure \\ref{fig16}) \\cite{Hassinger:2010,Hassinger:2010b,Jeffries:2007}. This gap $\\Delta_G$ is directly related to the partial gap opening at the Fermi surface which occurs at $T_0$.\nUsing a simple formula $\\rho = \\rho_0+A\\,T^2 + B\\,e^{-\\Delta_G\/T}$ and not the currently used $\\rho = \\rho_0+A\\,T^2 + B\\,T\/\\Delta_G\\,(1+2\\,T\/\\Delta_G)\\,e^{-\\Delta_G\/T}$, which is suitable only if the scattering process is due to spin waves, $\\Delta_G$ is quite close to $E_1$ by comparison to $E_0$ and comparable to $\\Gamma_c$. In the vicinity of $T_0$, as pointed out for example by the entropy drop, the contribution of the $E_1$ resonance may play a major role. However on cooling, the \\textbf{Q}$_1$ role is defeated by the \\textbf{Q}$_0$ wave-vector response which occurs at lower energy than the $E_1$ one.\n\nUnder pressure, as shown in Fig.\\ref{fig16}, $E_0(P)$ decreases from 1.8 meV at $P=0$ to 0.8 meV just below  $P_x$ and collapses above  $P_x$ as the response at \\textbf{Q}$_0$ is dominated by the onset of a large static sublattice magnetization. On the contrary, for the wave-vector \\textbf{Q}$_1$=(1.4,0,0), the resonance at $E_1$ persists through $P_x$; $E_1$ increases smoothly under pressure and exhibits a jump at $P_x$. As for the gap $E_0$, the resonance at \\textbf{Q}$_1$ becomes sharp below $T_0$, where the number of carriers drops, but broadens, with a width comparable to $\\Gamma_c$ in the over-damped regime. The thermal dependence of the width $\\gamma_0$ can be fitted using the Korringa model \\cite{Iwasa:2005}: $\\gamma_0 \\sim \\gamma_{0,T=0K} + a\\ n(T)^2\\ k_BT$ where $n(T)=n_0\\,e^{-\\Delta(T)\/k_BT}$ is the number of carriers which reduced when the gap $\\Delta(T)$ opens at the Fermi surface. A gap (temperature independent) $\\Delta=7.7$ meV is found below $T_0$ surprisingly very close to our derivation of $\\Delta_G$ (70 K) or to the gap value deduced from specific heat measurements ($\\Delta_{cp}=73$ K)\\cite{vanDijk:1997}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=72mm]{F16.eps}\n\\end{center}\n\\caption{Behavior of the resonance energies $E_0$ and $E_1$ versus pressure, above and below $P_x\\simeq5$ kbar and of the the gap $\\Delta_G$ reflecting the gap opening below $T_0$.}\n\\label{fig16}\n\\end{figure}\n\nThus our data give sound basis for further theoretical developments. They confirm the dual character of the phase transition with $\\Delta_G$ directly linked to the itinerant nature of the $5f$ electrons and $E_0$ and $E_1$ collective modes associated to the local character of the $5f$ electrons.\nThe description of quantities such as $I_{E_0}$ and presumably $I_{E_1}$ as well as the temperature variation of the specific heat below $T_0$ \\cite{Balatsky:2009}, by BCS-type formula, may reflect the feedback between the local and itinerant properties. Of course, the possibility that the strong resonance ($E_1$) at \\textbf{Q}$_1$ is a mark of an incommensurate spin density wave cannot completely be ruled out.  Our support for the choice of \\textbf{Q}$_{AF}$ as the wave-vector of the HO phase are; its occurrence only in the HO phase \\cite{villaume:2008}, the quasi-invariance of the frequencies detected in the de Haas-van Alphen effect\\cite{Nakashima:2003}  and Shubnikov-de Haas effect through $P_x$ \\cite{Hassinger:2010b} that indicate no change of the wave-vector between HO and AF phases, the lower value of $E_0$ by comparison to $E_1$ with furthermore a field convergence of $E_0$ towards $E_1$ in high magnetic field when the intersite dipolar and quadrupolar interaction  are smeared out on entering in the paramagnetic polarized phase where the magnetic response will be $q$ independent \\cite{Bourdarot:2003,Levallois:2009}. \n\nLet us compare the results on URu$_2$Si$_2$\\ with \ntwo Pr skutterudite systems PrRu$_4$P$_{12}$ and PrFe$_4$P$_{12}$ where strong feedbacks occur between Fermi sea and multipole dynamics (see references in \\cite{JPSJ77SA:2008}). The interest in the last reference is that the U ions in their tetravalent configuration will have two electrons in the $5f$ shell as do Pr ions in their trivalent configuration in the $4f$ shell.\n\n\n\n\n\nThese are systems where a strong feedback may occur between band structure, charge and multipolar ordering \\cite{JPSJ77SA:2008}. In PrRu$_4$P$_{12}$, it is now well established that at low pressure a charge order phase transition at $T_0=63$ K occurs through a switch from $body\\ centered\\ cubic$ to $simple\\ cubic$ lattice with clear evidence of a formation of two sublattice leading here to an unambiguous detection of $Ru-ion$ displacement \\cite{Lee:2001,Iwasa:2005}. The strong similarity of PrRu$_4$P$_{12}$ with URu$_2$Si$_2$\\ in inelastic neutron scattering experiment is the smearing of the inelastic response above $T_0$ and the appearance of a sharp feature below $T_0$ with nuclear Bragg displacement following BCS-type dependence leading to the claim that the crystal-field level variation through $T_0$ is coupled to the carrier change itself \\cite{Iwasa:2005}(as for URu$_2$Si$_2$, the temperature variation of integrated intensity of $E_0$). Another interesting case is PrFe$_4$P$_{12}$ where for $P_x=2$ GPa, the system switches from HO semi-metallic phase to AF insulator phase \\cite{Sato:2000}. NMR experiments on P sites \\cite{Kikuchi:2007} have recently lead to the conclusion that the HO phase has a scalar OP\\cite{Sakai:2007,Kiss:2008}. The difference between URu$_2$Si$_2$\\ and the last two skutterudite system appears that for the first case in the paramagnetic regime the system is clearly in a mixed valence state for the U ions (valence v $\\sim$ 3.5)\\cite{Hassinger:2008a,Barzykin:1995}. As discussed for systems like TmSe \\cite{Derr:2006}, it is the crossing to a well ordered phase at $T_0$, which makes that the uranium centers look to be tetravalent and leads consequently to the idea that a ThRu$_2$Si$_2$ description for the electronic bands may be a good starting point \\cite{Haule:2009,Harima::2010}.\n\n\\section {Conclusion}\nThe present work leads to a precise study of the resonance $E_0$ at \\textbf{Q$_0$} which is up to now the main signature of the OP of the HO phase. The key results are; the control of the temperature dependence of the resonance energy $E_0$ by the partial gap opening at the Fermi surface ($\\Delta_G$), the temperature like BCS dependence of the integrated inelastic intensity of the resonance. It was suggested that this variation may reflect the temperature evolution of the order parameter. Clearly, the itinerant and local character of the $5f$ electrons must be treated in equal footing. These new data will certainly push to theoretical developments.\n\n\\section{Acknowledgements}\n\nWe thank H. Harima, K. Miyake, G. Knebel, L. Malone, M. Zhitomirsky, V Mineev, J.P. Sanchez and J.P. Brison for useful and fruitful comments. \nThis work is supported by the Agence Nationale de la Recherche through the ANR contracts Delice, Sinus, and Cormat.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nNeutral hydrogen (HI) is a fundamental element in the Universe and its late-time distribution traces the underlying matter field, making it an innovative key probe of the large scale structure (LSS) \\citep[e.g.][]{review,Ansari_2012,Santos:2015}. \nDespite the success of Cosmic Microwave Background (CMB) experiments~\\citep{WMAP,planck:2018,ACT} and the LSS measurement via galaxy surveys \\citep[e.g.][]{BOSS:2016} in constraining the cosmological parameters of the $\\Lambda$CDM model, we still lack an understanding of the nature of dark energy and dark matter and an explanation for some of the tensions among different observables~\\citep[e.g.][]{riess:2019,wong:2019,Verde:2019}. The measurement of the large scale distribution of HI and its evolution with time can thus play an important role in the upcoming years, providing a complementary probe to traditional galaxy surveys~\\citep[e.g.][]{bull:2015}. \n\nThe 21cm signal, originating from the spin-flip transition in the hyperfine structure of the hydrogen ground state (e.g.~\\citealt{Furlanetto:2006}), is redshifted by the expansion of the Universe, and, thus, it is detectable on Earth at radio frequencies. Several planned and ongoing experiments, either purpose-built compact interferometers (such as CHIME~\\citep{Bandura:2014gwa,CHIMEdetection}, CHORD or HIRAX~\\citep{Newburgh:2016mwi}) or single-dish telescopes (such as GBT~\\citep{Masui2013,Wolz2022} or FAST~\\citep{Hu:2019okh}) aim to measure it with intensity mapping (IM) techniques~\\citep{Bharadwaj:2000,Battye:2004,McQuinn:2005,Chang:2007,Seo:2009fq,Kovetz:2017,Villaescusa-Navarro:2018} and some of them have achieved the detection of the HI signal in cross correlation with galaxy surveys \\citep{Chang2010,Masui2013,Anderson2018,Wolz2022}.\n\nRadio cosmology is also one of the main science goals of the SKA Observatory (SKAO)\\footnote{\\url{https:\/\/www.skao.int\/}}, that will be composed by SKA-Low and\nSKA-Mid telescopes located in Australia and South Africa, respectively. Using the SKA-Mid telescope array as a collection of single-dishes~\\citep{Battye:2013,Santos:2015} it will be possible to perform 21cm intensity mapping observations at the large scale important for cosmology up to redshift 3~\\citep{Bacon:2018}.  \nThe SKAO is currently under construction, and MeerKAT, the SKA-Mid precursor, has been conducting IM survey for cosmology~\\citep[MeerKLASS]{Santos:2017}.\nPreliminary data analysis have provided promising results~\\citep{Wang:2021,irfan2022} and a first detection of the HI signal in cross correlation with the WiggleZ galaxies~\\citep{Cunnington:2022uzo}. However, the level of foreground residuals is preventing a direct detection and this issue has triggered an extensive simulation work on foreground cleaning performances~\\citep{Alonso2015,Wolz2016,Carucci2020,Matshawule:2021,Irfan2021, Cunnington2021,Soares:2020,Soares2021GPR,Spinelli:2021emp}.\nIn parallel with the effort in improving the data analysis and the foreground separation, it is of key importance to refine the forecast for the constraining power of the 21cm IM alone and in combination with other probes in order to make a better case for radio cosmology with the SKAO or optimize the survey design.\n\n\\medskip\nIn this work, we focus on the parameters of the $\\Lambda$CDM model taking into account the redshift-space nature of the 21cm power spectrum. In this way we can exploit the tomographic, i.e. refined in redshift, potential of the observations. We build on the formalism of~\\citet{Blake:2019,Cunnington:2020, Soares:2020} and study redshift-space power spectrum monopole and quadrupole. Following \\citet{Bacon:2018}, \nwe construct mock 21cm power spectrum measurements in six redshift bins, in the redshift range $z=0.25 - 3$.\nWe expand the code \\texttt{CosmoMC}~\\citep{Lewis:2002,Lewis:2013} to include a new likelihood module to compute constraints through Markov-Chain Monte-Carlo (MCMC) techniques and we assess the constraining power of our mock 21cm data set alone and combined with CMB data. \n\nWe note that forecasts for future IM observations based on the Fisher Matrix formalism, thereby using a complementary approach to the one described here, have been presented in \\citet{obuljen18}.\n \nThe novelty aspects of this study with respect to \\citet{Soares:2020}, are: $i)$ we produce constraints on the full set of cosmological parameters; $ii)$ we combine the forecasted 21cm power spectrum multipoles with Planck 2018 CMB data~\\citep{planck:2018} ; $iii)$ we employ the multipole formalism to construct and study the constraining power of a tomographic data set. \nWe neglect the effect of foregrounds but we include uncertainties on the astrophysical quantities that connect the measured 21cm power spectrum to the underlying matter field, as the atomic hydrogen bias and the brightness temperature (see also \\citealt{Berti:2021}).\n\n\\medskip\nThe structure of the paper is the following. The adopted methodology, including the modelling of the 21cm power spectrum and multipoles, the construction of the forecasted data set and the likelihood implementation, is discussed in \\autoref{sec:methods}. Results are presented in \\secref{sec:results}. The constraining power of the mock 21cm power spectrum multipoles is evaluated in \\secref{sec:res_P21}. We investigate how 21cm observations affect the constraining power of other probes, i.e. CMB measurements, in \\secref{sec:res_P21_P21+Planck}. A discussion on the impact of opening the parameter space to the brightness temperature, the HI bias and the growth rate is given in \\secref{sec:res:nuisances}. We also investigate the extension to non-linear scale of our mock data in \\secref{sec:res_non_linear}. A summary of the results and our conclusions are outlined in \\secref{sec:conclusions}.\n\n\\medskip\nThrough all this work we assume a $\\Lambda$CDM universe described by a Planck 2018~\\citep{planck:2018}, fiducial cosmology i.e. $\\{\\Omega_b h^2 = 0.022383,\\, \\Omega_c h^2 = 0.12011,\\, n_s= 0.96605 ,\\, \\ln (10^{10} A_s) =3.0448 ,\\, \\tau= 0.0543,\\, h = 0.6732,\\, \\Sigma m_\\nu = 0.06 \\mathrm{eV} \\}$.\n\n\\section{Methods}\n\\label{sec:methods}\n\nIn this section we outline the formalism and the methods used \nthroughout this work. \nWe describe the 21cm linear power spectrum from \\secref{sec:P21_model} to~\\secref{sec:mult_exp}. In \\secref{sec:errors} and~\\secref{sec:tomographic_data_set} we construct the mock tomographic data set of SKA-Mid observations. The description of the likelihood function and the parameter estimation method are reported in \\secref{sec:bayes}.\n\n\\subsection{The theoretical 21cm signal linear power spectrum}\n\\label{sec:P21_model}\n\nWe model the 21cm linear power spectrum as~\\citep{kaiser1987,Villaescusa-Navarro:2018, Bacon:2018}\n\\begin{equation}\\label{eq:P21}\n     P_{21}(z,\\,k,\\, \\mu) = \\Bar{T}_{\\rm b}^2(z) \\left[ b_{\\mathrm{HI}}(z) + f(z)\\, \\mu^2\\right]^2 P_{\\rm m}(z,k),\n\\end{equation}\nwhere $\\bar{T}_{\\rm b}$ is the HI mean brightness temperature, $b_{\\rm HI}\\xspace$ is the HI bias, $f$ is the growth rate, $\\mu= \\hat{k} \\cdot \\hat{z}$ is the cosine of the angle between the wave number and the line-of-sight, and $P_{\\rm m}(z,k)$ is the linear matter power spectrum. \nWe neglect in \\autoref{eq:P21} the shot noise term, which is believed to be negligible at linear scales~\\citep{villa14,Pourtsidou:2016,Villaescusa-Navarro:2018,Spinelli:2019}. We refer to \\secref{sec:res_non_linear} for a discussion on non-linear scales and the shot noise term.\n\nWe use the parametrization of the brightness temperature from~\\cite{Battye:2013} \n\\begin{equation}\\label{eq:Tb}\n    \\bar{T}_{\\mathrm{b}} (z) = 180 \\, \\,  \\Omega_{\\mathrm{HI}}(z) \\frac{h\\, H_0}{ H(z)} (1+z)^2 \\text{mK},\n\\end{equation}\nwhere we consider the HI density parameter to evolve mildly in redshift as $\\Omega_{\\mathrm{HI}}(z) = 4. \\times 10^{-4} (1+z)^{0.6} $~\\citep[see][]{Crighton:2015}. Given that we lack an analytical model, $b_{\\rm HI}\\xspace (z)$ at given redshift is computed by interpolating numerical results from hydro-dynamical simulations~\\citep{Villaescusa-Navarro:2018,villa15}.\n\nThe growth rate $f(z)$ and the linear matter power spectrum $P_{\\rm m}(z,k)$ are, instead, computed numerically by means of the Einstein-Boltzmann solver \\texttt{CAMB}\\footnote{See \\url{https:\/\/camb.info\/}.}~\\citep{Lewis:2000}.\n\n\\subsection{Modeling the observed 21cm signal power spectrum as measured by SKA-Mid}\n\\label{sec:beam_model}\n\\begin{table}\n\t\\centering\n\t\\caption{Assumed specifications for SKA-Mid survey~\\citep{Bacon:2018}.}\n\t\\label{tab:SKA_specifics}\n\t\\begin{tabular*}{\\columnwidth}{l@{\\hspace*{40pt}}l@{\\hspace*{40pt}}c} \n\t\t\\hline\n\t\tParameter & & Value\\\\\n\t\t\\hline\n\t\t$D_{\\rm dish}\\, [\\rm m]$ & SKAO dish diameter & 15 \\\\\n\t\t$N_{\\rm dish}$ & SKAO dishes & 133 \\\\\n\t\t$t_{\\rm obs}\\, [\\rm h]$ & observing time & 10000 \\\\\n\t\t$T_{\\rm sys}\\, [\\rm K]$ & system temperature & 25 \\\\\n\t\t$\\delta \\nu\\, [\\rm MHz]$ & frequency range & 1 \\\\\n\t\t$A_2 \\, [\\rm deg^2]$ & survey area (Band 2) & 5000 \\\\\n\t\t$\\Omega_{\\rm sur,1} \\, [\\rm sr] $ & survey area (Band 2) & 1.5 \\\\\n\t\t$A_1 \\, [\\rm deg^2]$ & survey area (Band 1) & 20000 \\\\\n\t\t$\\Omega_{\\rm sur,2} \\, [\\rm sr] $ & survey area (Band 2) & 6.1 \\\\\n\t\t$f_{\\rm sky,2}$ & covered sky area (Band 2) & 0.12\\\\\n\t\t$f_{\\rm sky,1}$ & covered sky area (Band 1) & 0.48\\\\\n\t\t$\\Delta z$ & width of the redshift bins& 0.5 \\\\\n\t\t\\hline\n\t\\end{tabular*}\n\\end{table}\n\nIn this work, we construct mock single-dish 21cm power spectrum observations of the SKA-Mid telescope, modeling the 21cm IM survey as in \\citet{Bacon:2018}\n\nIn the following, we discuss the effect of the telescope beam and of the instrumental noise on the power spectrum. The telescope specifications relevant for our work are reported in \\autoref{tab:SKA_specifics}. \n\n\\subsubsection{The effect of the telescope beam}\nOne of the main instrumental contamination is the effect of a Gaussian telescope beam, which suppresses the power spectrum on scales smaller than the beam full width at half maximum \\citep{Cunnington:2022,Soares:2020,Cunnington:2020,Villaescusa-Navarro:2016, Battye:2013}.\n\nIts effect can be written in terms of $R_{\\rm beam}$, the beam physical dimension\n\\begin{align}\n\\begin{split}\n R_{\\rm beam} (z) &= \\sigma_\\theta r(z) \\\\\n &= \\frac{\\theta_{\\rm FWHM}}{2 \\sqrt{2\\ln 2}} r(z),\n \\end{split}\n\\end{align}\nwhere $r(z)$ is the comoving distance, $\\theta_{\\rm FWHM} = \\frac{1.22 \\lambda_{21}}{D_{\\rm dish}} (1+z) $ is the full width at half maximum, and $D_{\\rm dish}$ is the diameter of the dish. \n\nThe beam damping factor in Fourier space  $\\tilde{B} (z,k,\\mu)$ can thus be written as\n\\begin{equation}\n    \\tilde{B} (z,k,\\mu) = \\exp \\left[ \\frac{-k^2 R_{\\rm beam}^2(z) (1-\\mu^2)}{2}\\right].\n\\end{equation} \n\nNote that the factor $(1-\\mu)$ model the smoothing only along the transverse direction since the damping along the radial direction is negligible due to the high frequency resolution of 21cm observation~\\citep{Villaescusa-Navarro:2016}. \n\nThe beam convolved 21cm power spectrum is then\n\\begin{equation}\n    \\label{eq:P_21_full}\n     \\Hat{P}_{21}(z,\\,k,\\, \\mu) = \\tilde{B}^2 (z,k,\\mu) P_{21}(z,k,\\mu),\n\\end{equation}\nwhere $P_{21}(z,k,\\mu)$ is defined in~\\autoref{eq:P21}.\n\n\\subsubsection{Instrumental noise}\n\\label{sec:instrumental_noise}\nFor a single-dish intensity mapping SKAO-like experiment, the noise power spectrum can be modeled as~\\citep{Santos:2015,Bernal:2019}\n\\begin{equation}\\label{eq:noise}\n    P_{\\rm N}(z) = \\frac{T_{\\rm sys}^2 4\\pi f_{\\rm sky}}{N_{\\rm dish} t_{\\rm obs} \\delta \\nu} \\dfrac{V_{\\rm bin} (z)}{\\Omega_{\\rm sur}}.\n\\end{equation}\nHere, given a redshift bin centered at $z$ and of width $\\Delta z$, the volume of the redshift bin $V_{\\rm bin}(z)$ can be computed as\n\\begin{align}\\label{eq:volume_bin}\n\\begin{split}\n       V_{\\rm bin}(z) &=  \\Omega_{\\rm sur} \\int_{z - \\Delta z \/2}^{z + \\Delta z \/2} \\mathrm{d}z' \\, \\frac{\\mathrm{d} V}{\\mathrm{d} z'\\mathrm{d}\\Omega} \\\\\n       &=  \\Omega_{\\rm sur}  \\int_{z - \\Delta z \/2}^{z + \\Delta z \/2} \\mathrm{d}z' \\, \\frac{c r(z')^2}{H(z')}.\n\\end{split}\n\\end{align}\nwith $r(z)$ being the comoving distance.\nA description of all the other parameters that appear in \\autoref{eq:noise} and their assumed values can be found in \\autoref{tab:SKA_specifics}. \n\n\\subsubsection{Variance}\nTo construct mock observation we need an estimation of the errors on the power spectrum. Following \\citet[e.g.][]{Bernal:2019} we write the variance per $k$ and $\\mu$ bin $\\sigma(z,k,\\mu)$ as \n\\begin{equation}\\label{eq:variance}\n    \\sigma^2(z,k,\\mu) = \\frac{\\Big(\\Hat{P}_{21}(z,k,\\mu) + P_{\\rm N}(z)\\Big)^2}{N_{\\rm modes}(z,k,\\mu)},\n\\end{equation}\nwhere $\\Hat{P}_{21}(z,k,\\mu)$ is the 21cm signal power spectrum, defined in \\autoref{eq:P_21_full}, and $P_{\\rm N} (z)$ is the noise power spectrum of \\autoref{eq:noise}.  $N_{\\rm modes}(z,k,\\mu)$ is the number of modes per $k$ and $\\mu$ bins in the observed sky volume. We can compute it as\n\\begin{equation}\n\\label{eq:nmodes_mu}\n    N_{\\rm modes}(z,k,\\mu) = \\frac{k^2 \\Delta k(z) \\Delta \\mu(z)}{8\\pi^2} V_{\\rm bin}(z).\n\\end{equation}\nHere, $V_{\\rm bin}(z)$ is the volume of the redshift bin centered at $z$, while $\\Delta k(z)$ and $\\Delta \\mu(z)$ are the $k$ and $\\mu$ bin width respectively. In our analysis, however, we integrate over all the possible values of $\\mu$ in the interval $\\mu \\in (-1,1)$ (as we will discuss in more detail in the next section). Thus, computing the number of $\\mu$ modes, \\autoref{eq:nmodes_mu} reduces to\n\\begin{equation}\n     N_{\\rm modes}(z,k) = \\frac{k^2 \\Delta k(z)}{4\\pi^2} V_{\\rm bin}(z).\n\\end{equation}\n\n\\subsection{Multipole expansion}\n\\label{sec:mult_exp}\nThe non isotropic redshift space 21cm power spectrum\ncan be decomposed using Legendre polynomials $\\mathcal{L}_\\ell(\\mu)$ as\n\\begin{equation}\n    \\Hat{P}_{21} (z,k,\\mu) = \\sum_\\ell \\Hat{P}_\\ell (z,k) \\mathcal{L}_\\ell(\\mu).\n\\end{equation}\nThe first Legendre polynomials are the following functions of $\\mu$\n\\begin{equation}\n    \\mathcal{L}_0 (\\mu) = 1, \\qquad \\mathcal{L}_2 (\\mu) = \\frac{3\\mu^2}{2} - \\frac{1}{2}.\n\\end{equation}\nThe coefficients of the expansion, i.e. the multipoles of the 21cm power spectrum, are then given by\n\\begin{equation}\n\\label{eq:ell_P}\n    \\Hat{P}_\\ell (z,k) = \\frac{(2\\ell + 1)}{2} \\int_{-1}^{1} {\\rm d}\\mu\\, \\mathcal{L}_\\ell(\\mu) \\Hat{P}_{21}(z,k,\\mu),\n\\end{equation}\nwhere the expression for $\\Hat{P}_{21}(z,k,\\mu)$ can be found in \\autoref{eq:P_21_full}.\n\nIn our analysis, we construct mock observations for the monopole $\\Hat{P}_0(z,k)$ and the quadrupole $\\Hat{P}_2(z,k)$, i.e. $\\ell = 0$ and $\\ell = 2$ respectively. We refer to \\appref{sec:app_equations} for the explicit analytical expression of these quantities. \n\n\\subsection{Errors and covariance matrix}\n\\label{sec:errors}\nTo compute the errors on the monopole and the quadrupole, we begin by defining the covariance between the multipoles $\\ell$ and $\\ell'$ as a function of $k$ and $z$ \\citep[see][]{Bernal:2019,Chung:2019}\n\\begin{equation}\n\\label{eq:Cell}\n    C_{\\ell\\ell'}(z,k) = \\frac{(2\\ell + 1)(2\\ell' + 1)}{2} \\int_{-1}^1  {\\rm d}\\mu\\, \\mathcal{L}_\\ell(\\mu)\\, \\mathcal{L}_{\\ell'}(\\mu)\\,\\,\\sigma^2(z,k,\\mu),\n\\end{equation}\nwhere we neglect mode coupling. Here, $\\sigma^2(z,k,\\mu)$ is the variance per $k$ and $\\mu$ bin at redshift $z$, as defined in~\\autoref{eq:variance}.\n\n\\autoref{eq:Cell} can be used both to estimate the error on a single data point and to compute the covariance matrix for the multipoles. Given a set of $N$ measurements of the 21cm multipole $\\Hat{P}_\\ell$ at scales $\\{k_1, \\dots, k_N\\}$, the computed error on each point of the data set is \n\\begin{align}\\label{eq:errors}\n\\begin{split}\n        \\sigma_{\\Hat{P}_\\ell}(z,k_i) &=  \\sqrt{C_{\\ell\\ell}(z,k_i)} \\\\\n        &=\\sqrt{\\frac{(2\\ell + 1)^2}{2} \\int_{-1}^1  {\\rm d}\\mu\\, \\mathcal{L}^2_\\ell(\\mu)\\,\\sigma^2(z,k_i,\\mu)},\n\\end{split}\n\\end{align}\nfor each $k_i$ in $\\{k_1,\\dots, k_N\\}$. \nIn our analysis we focus only on the monopole $\\Hat{P}_0$ and the quadrupole $\\Hat{P}_2$.\n\nThe multipole covariance defined in~\\autoref{eq:Cell} allows to estimate both the covariance for a given multipole and between different multipoles. Thus, the most general covariance matrix for $\\Hat{P}_0$ and $\\Hat{P}_2$ combined is a $2N \\times 2N$ symmetric block matrix. At fixed redshift, it is constructed as\n\\begin{equation}\\label{eq:full_covariance}\n    \\bm{\\mathsf{C}}(z) = \\begin{pmatrix}\n\\bm{\\mathsf{C}}_{00}(z) & \\bm{\\mathsf{C}}_{02}(z)\\\\\n\\bm{\\mathsf{C}}_{02}(z) & \\bm{\\mathsf{C}}_{22}(z)\n\\end{pmatrix}.\n\\end{equation}\nEach block $\\bm{\\mathsf{C}}_{\\ell\\ell'}$ is a diagonal matrix of dimensions $N\\times N$, defined as $\\bm{\\mathsf{C}}_{\\ell\\ell'}(z) = \\diag(C_{\\ell\\ell'}(z,k_1), \\dots, C_{\\ell\\ell'}(z,k_N))$, where the elements $C_{\\ell\\ell'}(z,k_i)$ are computed as in~\\autoref{eq:Cell} at each $\\{k_1,\\dots, k_N\\}$.\n\nThe blocks along the diagonal, i.e. $\\bm{\\mathsf{C}}_{00}(z)$ and $\\bm{\\mathsf{C}}_{22}(z)$, are the covariance matrices for the monopole and the quadrupole alone. The off-diagonal block $\\bm{\\mathsf{C}}_{02}(z)$, instead, describes the covariance between $\\Hat{P}_0$ and $\\Hat{P}_2$. \n\nIn the simplified case where the monopole $\\Hat{P}_0$ and the quadrupole $\\Hat{P}_2$ are uncorrelated, we can neglect the off-diagonal terms in $\\bm{\\mathsf{C}}(z)$ and assume the block-diagonal covariance matrix\n\\begin{equation}\\label{eq:covariance_diagonal}\n    \\bm{\\mathsf{C}}_{\\rm diag}(z) = \\begin{pmatrix}\n\\bm{\\mathsf{C}}_{00}(z) & 0\\\\\n0 & \\bm{\\mathsf{C}}_{22}(z)\n\\end{pmatrix}.\n\\end{equation}\nIn our work, we compute the covariance matrices $\\bm{\\mathsf{C}}_{00}(z)$, $\\bm{\\mathsf{C}}_{02}(z)$, $\\bm{\\mathsf{C}}_{22}(z)$, $\\bm{\\mathsf{C}}_{\\rm diag}(z)$, and $\\bm{\\mathsf{C}}(z)$ numerically, from the analytical expression for the monopole and the quadrupole (see \\appref{sec:app_equations}). \n\n\n\\subsection{Tomographic data set}\n\\label{sec:tomographic_data_set}\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{gfx\/Fig1.pdf}\n    \\caption{Tomographic mock data set for 21cm linear power spectrum monopole (upper panel) and quadrupole (lower panel) observations. The considered six redshift bins are centered at redshifts $\\{0.25,\\, 0.75,\\, 1.25,\\, 1.75,\\, 2.25,\\, 2.75 \\}$. For the first redshift bin (black dashed line) we assume a SKA-Mid Band 2 survey. The data sets for the other five bins, instead, assume a SKA-Mid Band 1 survey (see \\autoref{tab:SKA_specifics}). We refer to \\autoref{sec:tomographic_data_set} for further details on how the signal and the errors are computed.}\n    \\label{fig:P_data_set}\n\\end{figure}\nWe follow \\citet{Bacon:2018} to construct the mock data set and assume a combination of two surveys: a Medium-Deep Band 2 survey that covers a sky area of 5,000 deg$^2$ in the frequency range $0.95-1.75$ GHz (i.e. the redshift range $0-0.5$); a Wide Band 1 survey that covers a sky area of 20,000 deg$^2$ in the frequency range $0.35-1.05$ GHz (i.e. the redshift range $0.35-3$). \nWe forecast observations for six equi-spaced, non-overlapping redshift bins, in the range $z=0-3$ with $\\Delta z = 0.5$. The six bins are centered at redshifts $z_c = \\{0.25,\\, 0.75,\\, 1.25,\\, 1.75,\\, 2.25,\\, 2.75 \\}.$ We assume the Band 2 survey specification for the mock 21cm power spectrum at redshift $0.25$ and Band 1 survey specification for all the others. Note that in our analysis each bin is regarded as independent.\n\n\\medskip\nThe survey sky coverage and the redshift range define the volume for the mock observations thus fixing the range of accessible scales for our analysis. \nIn Fourier space, the largest scale available at each central redshift is $k_{\\rm min}(z_c) = 2\\pi \/ \\sqrt[3]{V_{\\rm bin}(z_c)}$, where $V_{\\rm bin}(z_c)$ is the volume of each redshift bin, computed using \\autoref{eq:volume_bin}. \n\nThe smallest scale is, instead, imposed by the size of the telescope beam and it can be estimated as $k_{\\rm max}(z_c) = 2\\pi\/R_{\\rm beam}(z_c)$. At scales smaller than $k_{\\rm max}$, the signal is dominated by the beam providing no relevant information on cosmology. Although we do not show results here, we tested pushing the $k_{\\rm max}$ limit beyond the beam scale. We found no significant impact on the cosmological parameters constraints. To avoid entering the non-linear regime for the power spectrum we impose a cut-off at $k_{\\rm max} = 0.2\\, h \\rm Mpc^{-1}$. \n\nFinally, we choose a fixed k-bin width as function of redshift $\\Delta k(z_c)$ in order for to be large enough for modes to be independent, assuming $\\Delta k(z_c)\\sim 2k_{\\rm min}(z_c)$.\n\n\\medskip\nAt each central redshift $z_c$ and data point $k$ we compute the monopole $\\Hat{P}_0(z_c,k)$, the quadrupole $\\Hat{P}_2(z_c,k)$ and the errors, as discussed in \\secref{sec:mult_exp} and~\\secref{sec:errors} respectively. In table~\\autoref{tab:values_redshift}, we gather some of these redshift dependent quantities for the interested reader. The resulting forecasted data sets for the monopole and the quadrupole are shown in \\autoref{fig:P_data_set}.\n\n\\begin{table*}\n \\caption{Computed values of redshift dependent quantities at each central redshift $z_c$. For the first redshift bin ($z=0.25$) we assume SKA-Mid Band 2 specifications, while we use SKA-Mid Band 1 parameters for the other five bins. We refer to \\autoref{tab:SKA_specifics} for a list of used SKA-Mid specifications.}\n \\label{tab:values_redshift}\n \\begin{tabular*}{2\\columnwidth}{llccccccl} \n  \\hline\n   $z_{c}$ & central redshift & {0.25} & $0.75$ & $1.25$ & $1.75$ & $2.25$ & $2.75$\\\\\n  \\hline\n    $\\bar{T}_{\\rm b} $  & mean brightness temperature & ${0.78\\times 10^{-1}}$ & $1.3$ $\\times 10^{-1}$& $1.9$ $\\times 10^{-1}$& $2.5$ $\\times 10^{-1}$& $3.0$ $\\times 10^{-1}$ &  $3.6$ $\\times 10^{-1}$ &[mK]\\\\\n    $b_{\\rm HI}$ & HI bias & {1.03}& $1.33$& $1.62$& $1.89$& $2.17$& $2.45$ \\\\\n    $V_{\\rm bin} $ & volume of the bin & $ {0.11\\times 10^{10}}$ & 1.98 $ \\times 10^{10}$& 3.14 $ \\times 10^{10}$& 3.71 $ \\times 10^{10}$&\n       3.90 $ \\times 10^{10}$ & 3.88 $ \\times 10^{10}$ &$ [ h^{-3}\\mathrm{Mpc}^3]$\\\\\n    $R_{\\rm beam}$ &  beam size  &  {6.38}&  23.36&  43.74&  66.13&\n        89.83 & 114.5 & $ [h^{-1}\\,\\mathrm{Mpc}]$\\\\\n    $P_{\\rm N} $ & noise power spectrum  & {0.6} & 10 & 16 & 19 & 20 & 20 & $[\\mathrm{mK^2}\\, h^{-3}\\mathrm{ Mpc^3 }]$\\\\\n    $k_{\\rm max}$  & minimum scale & {0.2}      & 0.2    & 0.14 & 0.09 & 0.07&\n       0.05 &  $[h \\rm Mpc^{-1}]$\\\\\n    $k_{\\rm min} $ & maximum scale& ${6.00\\times 10^{-3}}$& 2.32$\\times 10^{-3}$ & 1.99$\\times 10^{-3}$& 1.88 $\\times 10^{-3}$& 1.85 $\\times 10^{-3}$& 1.85  $\\times 10^{-3}$ & $[ h\\, \\rm Mpc^{-1}]$\\\\\n    $\\Delta k $ & $k$-bin width & ${12\\times 10^{-3}}$ & 5 $\\times 10^{-3}$& 4 $\\times 10^{-3}$ & 4 $\\times 10^{-3}$& 4 $\\times 10^{-3}$& 4  $\\times 10^{-3}$&$ [ h \\rm Mpc^{-1}]$\\\\\n    $N$ & number of data points & {16} & 43 & 36 & 25 & 18 & 14\\\\\n  \\hline\n \\end{tabular*}\n\\end{table*}\n\\subsection{Constraining the cosmological parameters}\n\\label{sec:bayes}\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{gfx\/Fig2.pdf}\n\t\t\\includegraphics[width=\\columnwidth]{gfx\/Fig2_2.pdf}\n    \\caption{Signal-to-noise ratio as a function of $k$ (see \\autoref{eq:signal_to_noise}). In the upper panel, we show the signal-to-noise at given redshift $z=0.75$, for different combinations of multipoles: the monopole alone (blue dashed-dotted line), the quadrupole alone (pink dotted line), the two combined (green dashed line) and the two combined considering the full non-diagonal covariance matrix (yellow solid line). In the lower panel, we show the signal-to-noise redshift dependence for the monopole and the quadrupole combined, considering a diagonal covariance matrix (dashed lines) and the full non-diagonal one (solid lines).}\n    \\label{fig:signal_to_noise}\n\\end{figure}\n\nIn order to exploit the constraining power of the mock data set presented in \\secref{sec:tomographic_data_set}, we define a likelihood function (\\secref{sec:likelihood}) and then set up the framework to constrain the cosmological parameters adopting a Bayesian approach (\\secref{sec:numerical_analysis}). Given a set of observations and a theory that depends on set of parameters, the Bayes theorem links the posterior distribution to the likelihood function. The high-dimensional posterior can then be sampled using Monte-Carlo Markov-Chain (MCMC) methods~\\citep[e.g.][]{gilks:1995,gamerman2006markov}.\n\n\\subsubsection{Likelihood function and signal-to-noise}\n\\label{sec:likelihood}\nGiven a set of measurements at scales $\\{k_1, \\dots, k_N\\}$ and redshift $z_c$, to compute the likelihood function we define the vector \n\\begin{equation}\n    \\bm{\\Theta}(z_c) = \\Big( \\bm{\\Hat{P}}_0(z_c), \\bm{\\Hat{P}}_2(z_c) \\Big),\n\\end{equation}\nwith $\\bm{\\Hat{P}}_\\ell(z_c) = (\\Hat{P}_\\ell(z_c,k_1), \\dots, \\Hat{P}_\\ell(z_c,k_N))$.\nWhen we use both the monopole and the quadrupole, the logarithmic likelihood is computed as\n\\begin{equation}\n    \\label{eq:likelihood}\n    -\\ln\\big[ \\mathcal{L}\\big] = \\sum_{z_c} \\,\\frac{1}{2} \\,\\Delta \\bm{\\Theta}(z_c)^{\\rm T}\\, \\bm{\\mathsf{C}}^{-1}(z_c)\\,  \\Delta \\bm{\\Theta}(z_c),\n\\end{equation}\nwhere we define $\\Delta \\bm{\\Theta}(z_c) = \\bm{\\Theta}^{\\rm th}(z_c) - \\bm{\\Theta}^{\\rm obs}(z_c)$, the difference between the values of $\\bm{\\Theta}(z_c)$ predicted from theory and observed. Here, $\\bm{\\mathsf{C}}(z_c)$ is the covariance matrix defined in \\autoref{eq:full_covariance}, which becomes $\\bm{\\mathsf{C}}_{\\rm diag}(z_c)$, i.e. \\autoref{eq:covariance_diagonal}, when we neglect multipole covariance. We consider independent redshift bins, i.e. we simply sum over the contribution from each central redshift. When studying $\\Hat{P}_0$ and $\\Hat{P}_2$ separately we use only the relevant blocks of the covariance matrix $\\bm{\\mathsf{C}}(z_c)$, thus using a simplified version of \\autoref{eq:likelihood}.\n\n\\medskip\nUsing a similar formalism, we can compute the signal-to-noise at a specific $k$ for each central redshift as\n\\begin{equation}\n\\label{eq:signal_to_noise}\n    [\\mathrm{S\/N}]^2(z_c,k) = \\bm{\\Theta}_k(z_c)^{\\rm T}\\, \\bm{\\mathsf{C}}_k^{-1}(z_c)\\, \\bm{\\Theta}_k(z_c)\n\\end{equation}\nwhere $\\bm{\\Theta}_k(z_c) = (\\Hat{P}_0(z_c,k), \\Hat{P}_2(z_c,k))$ and $\\bm{\\mathsf{C}}_k(z_c)$ is a matrix defined as \n\\begin{equation}\n \\bm{\\mathsf{C}}_k(z_c) =   \\begin{pmatrix}\n    C_{00}(z_c,k) & C_{02}(z_c,k) \\\\\n     C_{02}(z_c,k) & C_{22}(z_c,k)\n    \\end{pmatrix}.\n\\end{equation}\nThe expression of the signal-to-noise when we neglect the multipole covariance or when we use only $\\Hat{P}_0$ or $\\Hat{P}_2$ is modified accordingly, as described for the likelihood function above. \n\n\\medskip\nThe resulting signal-to-noise as a function of $k$ is shown in \\autoref{fig:signal_to_noise}. We recall that that the maximum scales explored is the minimum scale between the maximum scale imposed by the beam width and the linear regime cut-off of $k_{\\rm max} = 0.2\\, h \\rm Mpc^{-1}$ (see \\secref{sec:tomographic_data_set}).\n\nAt fixed redshift (upper panel of \\autoref{fig:signal_to_noise}), we observe that, when the monopole and the quadrupole are used together ($\\ell =0,2$), we get a higher signal-to-noise with respect to the monopole ($\\ell =0$) and the quadrupole ($\\ell=2$) alone. For the $\\ell =0,2$ case, we observe that when we consider multipole covariance (yellow solid line) we get an enhancement of the signal-to-noise at higher scales and a suppression at lower ones. As discussed in~\\cite{Soares:2020}, this effect is caused by the beam smoothing factor in the model for $\\Hat{P}_{21}$ (see \\secref{sec:beam_model}). \nWe examine the implications of using different combinations of multipoles on the parameters constraints in \\secref{sec:res_P21_P21+Planck}. \n\nWe show how the signal-to-noise decreases as a function of redshift in the lower panel of \\autoref{fig:signal_to_noise}. Its shape is consistent at all redshifts (the full signal-to-noise for the mock measurements in the Band 2 ($z=0.25$), can be found in~\\autoref{fig:non_linear}). In our analysis we will always consider the cumulative contribution from all the redshift bins.\n\n\\subsubsection{Numerical analysis}\n\\label{sec:numerical_analysis}\nTo perform the MCMC analysis we use an expanded version of the MCMC sampler \\texttt{CosmoMC}\\footnote{See \\url{https:\/\/cosmologist.info\/cosmomc}.}~\\citep{Lewis:2002,Lewis:2013}. We modify it in order to include the computation of the theoretical expectations for the monopole and the quadrupole (see \\appref{sec:app_equations}) and of the likelihood function defined above (see \\autoref{eq:likelihood}). \n\nThe analysis of the MCMC samples to compute the marginalised constraints is performed with the Python package \\texttt{GetDist}\\footnote{See \\url{https:\/\/getdist.readthedocs.io}.}~\\citep{Lewis:2019xzd}.\n\nWe conduct an MCMC analysis varying the six parameters describing the $\\Lambda$CDM model, i.e. we vary $\\{\\Omega_b h^2,\\, \\Omega_c h^2, n_s,\\, \\ln (10^{10} A_s),\\, \\tau,\\, 100\\theta_{\\rm MC} \\}$. Results on other parameters, such as $H_0$ and $\\sigma_8$, are derived from results on this set. The assumed fiducial cosmology is Planck 2018~\\citep{planck:2018}. We list the fiducial values and the flat prior we use in \\autoref{tab:fiducial_cosmology}.\n\n\\subsubsection{CMB data sets}\n\\label{sec:Planck_data_set}\n\nIn this study, we combine our mock 21cm IM data set with Planck 2018~\\citep{planck:2018}. The CMB likelihood includes the high-$\\ell$ TT, TE, EE lite likelihood in the interval of multipoles $30\\leq\\ell\\leq 2508$ for TT and $30\\leq\\ell\\leq 19696$ for TE, EE. Lite likelihoods are  calculated with the \\texttt{Plik lite} likelihood~\\citep{planck:2018like}. Instead for the low-$\\ell$ TT power spectrum we use data from the \\texttt{Commander} component-separation algorithm in the range $2 \\leq \\ell \\leq 29$. We adopt also the Planck CMB lensing likelihood and the low EE polarization power spectrum, referred to as lowE, in the range $2 \\leq \\ell\\leq 29$, calculated from the likelihood code \\texttt{SimAll}~\\citep{planck:2018_maps}. In the rest of the paper with the label \"Planck 2018\" we refer to the combination TT, TE, EE + low-$\\ell$ + lowE + lensing.\n\n\\section{Results}\n\\label{sec:results}\n\\begin{table}\n\t\\centering\n\t\\caption{Assumed fiducial cosmology~\\citep{planck:2018} and used flat priors.}\n\t\\label{tab:fiducial_cosmology}\n\t\\begin{tabular*}{\\columnwidth}{l@{\\hspace*{0.8in}}c@{\\hspace*{0.7in}}c@{\\hspace*{0.7in}}} \n\t\\hline \n\tParameter & Fiducial value & Prior\\\\\n\t\t\\hline\n        $\\Omega_b h^2$ & 0.022383 & $[0.005, 0.1]$\\\\\n        $\\Omega_c h^2$ & 0.12011 & $[0.001, 0.99]$\\\\\n        $n_s$ & 0.96605 & $[0.8, 1.2]$\\\\\n        $\\ln (10^{10} A_s)$ & 3.0448 & $[1.61, 3.91]$\\\\\n        $\\tau$ & 0.0543 & $[0.01, 0.8]$\\\\\n        $100\\theta_{\\rm MC}$ & 1.040909 & $[0.5, 10]$\\\\\n\t\t\\hline\n\t\\end{tabular*}\n\\end{table}\nWe present in this section the results of our analysis. We first explore the constraining power of the mock 21cm data set, using different combinations of multipoles (\\secref{sec:res_P21}); we then combine the mock data set with Planck CMB data (\\secref{sec:res_P21_P21+Planck}); we study\nthe effect of nuisance parameters describing the neutral hydrogen astrophysics in \\secref{sec:res:nuisances}; finally, we discuss the impact of non-linear scales in \\secref{sec:res_non_linear}.\n\nWe show the marginalized 1D and 2D posteriors for the studied set of parameters. Note that 68\\% confidence level constraints are presented as percentages with respect to the marginalized mean value. \n\n\\subsection{Probing the constraining power of 21cm signal observations}\n\\label{sec:res_P21}\nIn \\autoref{fig:results_P21} and \\autoref{tab:constraints_P21_alone}, we show the marginalized contours and constraints\\footnote{We specify that, when dealing with asymmetrical posterior distributions, we estimate the percentage constraint using the mean value between the left and right marginalized error. Given that this is a forecasts analysis, this approximation is enough for the purpose of this work.} that we obtain using our SKA-Mid tomographic data set alone, for different combinations of multipoles. Note that we show only some of the model parameters for brevity.\n\nAs one could expect from the signal-to-noise prediction of  \\autoref{fig:signal_to_noise}, using only the quadrupole leads to the broadest constraints, while the most constraining results are obtained for the monopole and the quadrupole combined. \nThe off-diagonal terms of the multiple covariance do not affect much the constraints. The marginalized percentage constraints for the baseline case ($\\Hat{P}_0 + \\Hat{P}_2$ considering the full covariance) for the complete set of cosmological parameters can be found in \\autoref{fig:full_P21} and \\autoref{tab:constraints_P21}.\n\n\\begin{figure}\n\t\\includegraphics[width=\\columnwidth]{gfx\/Fig3.pdf}\n    \\caption{Joint constraints (68\\% and 95\\% confidence regions) and marginalized posterior distributions on a subset of the cosmological parameters. We show the constraints obtained using the mock tomographic data set for the monopole only (\"$\\Hat{P}_0$\"), the quadrupole only (\"$\\Hat{P}_2$\"), the two combined with (\"$\\Hat{P}_0 + \\Hat{P}_2$ non diagonal\") and without (\"$\\Hat{P}_0 + \\Hat{P}_2$ diagonal\") considering multipole covariance. The relative constraints are listed in \\autoref{tab:constraints_P21_alone}. }\n    \\label{fig:results_P21}\n\\end{figure}\n\\begin{table}\n\t\\centering\n\t\\caption{Marginalized percentage constraints on cosmological parameters at the 68\\% confidence level. We show the results obtained using the mock tomographic data set for the monopole only (\"$\\Hat{P}_0$\"), the quadrupole only (\"$\\Hat{P}_2$\"), the two combined with (\"$\\Hat{P}_0 + \\Hat{P}_2$ full covariance\") and without (\"$\\Hat{P}_0 + \\Hat{P}_2$ diagonal\") considering multipole covariance. Confidence regions for the same set of results are shown in \\autoref{fig:results_P21}.}\n\t\\label{tab:constraints_P21_alone}\n\t\\begin{tabular}{lcccc} \n\t\\hline \n\t\\multirow{2}{*}{Parameter} & \\multirow{2}{*}{$\\Hat{P}_0$} & \\multirow{2}{*}{$\\Hat{P}_0$}& $\\Hat{P}_2 + \\Hat{P}_2$ & $\\Hat{P}_0 + \\Hat{P}_2$ \\\\\n    \t& & & diagonal & full covariance \\\\\n\t\t\\hline\n$\\Omega_c h^2$ & 16.71\\%& 21.57\\%& 12.71\\%& 13.36\\% \\\\\n$n_s$ & 4.59\\%& 5.59\\%& 3.55\\%& 3.44\\% \\\\\n${\\rm{ln}}(10^{10} A_s)$ & 10.94\\%& 15.26\\%& 8.26\\%& 8.83\\% \\\\\n$H_0$ & 9.09\\%& 12.01\\%& 6.91\\%& 7.39\\% \\\\\n$\\sigma_8$ & 9.56\\%& 11.92\\%& 7.11\\%& 7.64\\% \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nOur results show, as expected, that the constraining power on the cosmological parameters of the SKA-Mid mock data set is greater than what one could obtain with MeerKAT alone~\\citep{Berti:2021}. The data set we constructed is enough to constrain five out of six of the cosmological parameters. This is because 21cm observations are not sensitive to variations on $\\tau$, that remains unconstrained. \nThe marginalised confidence levels are broad with respect to Planck constraints~\\citep{planck:2018}, but comparable with other probes. E.g., with tomographic observations of the monopole and the quadrupole combined we constrain $H_0$ with a relative error of $\\sigma_{H_0} =7.4\\%$,\n\\begin{equation}\n\\label{eq:H0_linear}\n   H_0 =  71.6^{+3.8}_{-6.8}\\,\\, \\text{ km s}^{-1}\\text{Mpc}^{-1} \\quad {\\scriptstyle(68\\%,\\ \\Hat{P}_0 + \\Hat{P}_2 \\text{ - full covariance})}.\n\\end{equation}\nWe stress that when we state a constraint on a single parameter obtained using our mock 21cm data set, the central value does not have any physical meaning and it is driven by the input fiducial cosmology value.\n\nAlthough not competitive with Planck, our tomographic measurements with six redshift bins and at linear scales provide an estimate of $H_0$ with an uncertainty comparable with others late Universe measurements~\\citep{Verde:2019}, and, as we further discuss in \\secref{sec:res_non_linear}, constraints on $H_0$ are improved if we extend our data set to non-linear scales. SKA-Mid 21cm observations will have thus the potential to provide new information for the discussion on the $H_0$ value~\\citep{Schoneberg:2021qvd}.\n\nLooking at the 2D contours, we observe that there is a marked degeneracy between the cosmological parameters. As already found for mock MeerKAT observations, 21cm measurements show a strong degeneracy in the $H_0$ - $\\Omega_ch^2$ plane~\\citep{Berti:2021}. This feature is pivotal when combining intensity mapping data with CMB measurements, as we discuss in the following section.\n\n\\subsection{21cm signal observations combined with CMB data}\n\\label{sec:res_P21_P21+Planck}\nIn the above section we studied the constraining power of the 21cm multipoles. Here, we combine our baseline data set (the monopole $\\Hat{P}_0$ plus the quadrupole $\\Hat{P}_2$ considering the full covariance) with Planck CMB measurements. The rationale behind this is to investigate if and how 21cm observations can complement the detailed information on the cosmological parameters carried by the CMB. \n\n\\medskip\nWe refer to \\secref{sec:Planck_data_set} for a description of the used Planck 2018~\\citep{planck:2018} data sets and likelihoods. For consistency, \nwe first run the Planck likelihood in our framework and reproduce constraints in agreement with the Planck 2018 results\n\\begin{equation}\n    \\begin{rcases}\n    \\Omega_c h^2 = 0.1201\\pm 0.0012 \\\\\n   H_0 = 67.32 \\pm 0.53\\ \\text{ km s}^{-1}\\text{Mpc}^{-1}\\\\\n      {\\rm{ln}}(10^{10} A_s) = 3.045\\pm 0.014 \\\\\n   \\sigma_8 = 0.8115\\pm 0.0060\n    \\end{rcases}\n    \\ {\\scriptstyle(68\\%,\\ \\text{Planck 2018}).}\n\\end{equation}\n\n\\medskip\nOur results for the combination of the CMB data and our mock 21cm observation are presented in \\autoref{tab:constraints_P21_Planck} and in \\autoref{fig:full_results}.\nAdding the 21cm power spectrum multipoles to the CMB, significantly improves the constraining power on the majority of the cosmological parameters. The effect is particularly pronounced on $\\Omega_c h^2$ and $H_0$, for which the error is reduced by approximately a fourth. This gain in constraining power is due to the combination of opposite degeneracy directions between the CMB and the 21cm power spectrum on these cosmological parameters. \nThis effect is particularly strong in the $H_0$ - $\\Omega_ch^2$ plane, where the degeneracy is completely removed. In $A_s$ - $\\sigma_8$ plane, the effect is milder, but still significant. \n\n\\medskip\nIn more detail, we find\n\\begin{equation}\n    \\begin{rcases}\n    \\Omega_c h^2 = 0.12014\\pm 0.00030 \\\\\n   H_0 = 67.28\\pm 0.11\\ \\text{ km s}^{-1}\\text{Mpc}^{-1}\\\\\n       {\\rm{ln}}(10^{10} A_s) = 3.0463\\pm 0.0052 \\\\\n   \\sigma_8 = 0.8125\\pm 0.0021\n    \\end{rcases}\n    \\ {\\scriptstyle(68\\%,\\ \\text{Planck 2018} + \\Hat{P}_0 + \\Hat{P}_2).}\n\\end{equation}\nWe recall that the central value of the obtained constraints does not have a physical meaning and it is driven by the input fiducial cosmology we use for our mock 21cm observations. However, these values are useful to properly visualize the constraining power of our mock observations. \n\nMore important from a quantitative point of view is, instead, the relative error. \nWe find $\\sigma_{\\Omega_ch^2} = 0.25\\%$ and $\\sigma_{H_0} = 0.16\\%$, to be compared with the Planck only estimates of $\\sigma_{\\Omega_ch^2} = 0.99\\%$ and $\\sigma_{H_0} = 0.79\\%$. \nThe estimate of the error on $H_0$ we obtain combining 21cm power spectrum multipoles with CMB is competitive with other LSS probes, e.g. with Euclid\\footnote{\\url{https:\/\/sci.esa.int\/web\/euclid}} forecasts~\\citep{Euclid:2019clj}. The errors on $A_s$ and $\\sigma_8$ are significantly reduced too, by more than a factor two: the relative errors are $\\sigma_{\\ln (10^{10} A_s)} = 0.17\\%$ and $\\sigma_{\\sigma_8} = 0.26\\%$ to be compared with $\\sigma_{\\ln (10^{10} A_s)} = 0.46\\%$ and $\\sigma_{\\sigma_8} = 0.73\\%$ of the Planck only result.\nMoreover, it is interesting to see how the improvement on the other cosmological  parameters induces a better estimate of $\\tau$, although the 21cm observable alone has not a significant constraining power on it. \n\nWe conclude that 21cm observations provide complementary information to the CMB, allowing for a significantly improved estimation of the cosmological parameters. \n\nNote that the improvement is stronger than the effect of adding BAO measurements to the CMB~\\citep{planck:2018}. Although we do not show results here, we tested also the effect of using BAO~\\citep{Beutler:2011,Ross:2014,BOSS:2016} along with the multipoles and the CMB, finding no significant repercussion on the constraints.\n\n\n\\begin{figure*}\n\t\\includegraphics[width=2\\columnwidth]{gfx\/Fig4.pdf}\n    \\caption{Joint constraints (68\\% and 95\\% confidence regions) and marginalized posterior distributions on cosmological parameters. The label \"Planck 2018\" stands for TT, TE, EE + lowE + lensing, while the label \"$\\Hat{P}_0 + \\Hat{P}_2$\" stands for the baseline tomographic data set for the monopole and the quadrupole combined and with multipole covariance taken into account. The label \"nuisances\" (dashed line) indicates that we vary the nuisances parameters along with the cosmological ones. The relative constraints are listed in \\autoref{tab:constraints_P21_Planck}.}\n    \\label{fig:full_results}\n\\end{figure*}\n\n\\begin{figure}\n\t\\includegraphics[width=1\\columnwidth]{gfx\/Fig6.pdf}\n    \\caption{Redshift evolution for the nuisance parameters $\\bar{T_{\\rm b}} b_{\\rm HI} \\sigma_8(z) $ and $\\bar{T_{\\rm b}} f \\sigma_8 (z)$. We show the theory predicted values for the six redshift bins we consider (circles and squares) and the best-fit redshift evolution (solid lines) modelled as a $3^{\\rm rd}$-degree polynomial (see \\autoref{eq:fitting_nuisances}).}\n    \\label{fig:nuisances_redshift}\n\\end{figure}\n\n\n\\begin{figure*}\n\t\\includegraphics[width=2\\columnwidth]{gfx\/Fig5.pdf}\n    \\caption{Joint constraints (68\\% and 95\\% confidence regions) and marginalized posterior distributions on cosmological parameters. Here, the label \"$\\Hat{P}_0 + \\Hat{P}_2$\" stands for the baseline tomographic data set for the monopole and the quadrupole combined and with multipole covariance taken into account. The label \"$\\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL}$\" indicates that the full non-linear data set has been used. The label \"nuisances\" (dashed lines) indicates that we vary the nuisances parameters along with the cosmological ones. The relative constraints are listed in \\autoref{tab:constraints_P21}.}\n    \\label{fig:full_P21}\n\\end{figure*}\n\n\\begin{table}\n\t\\centering\n\t\\caption{Marginalized percentage constraints on cosmological parameters at the 68\\% confidence level. Here, the label \"Planck 2018\" stands for TT, TE, EE + lowE + lensing, while the label \"$\\Hat{P}_0 + \\Hat{P}_2$\" stands for the baseline tomographic data set for the monopole and the quadrupole combined and with multipole covariance taken into account. The label \"nuisances\" indicates that we vary the nuisances parameters along with the cosmological ones. Confidence regions for the same set of results are shown in \\autoref{fig:full_results}.}\n\t\\label{tab:constraints_P21_Planck}\n\t\\begin{tabular}{lccc} \n\t\\hline \n    Parameter & Planck 2018 & $+ \\Hat{P}_0 + \\Hat{P}_2$& $+$ nuisances \\\\\n\t\t\\hline\n$\\Omega_b h^2$ & 0.64\\%& 0.49\\%& 0.49\\% \\\\\n$\\Omega_c h^2$ & 0.99\\%& 0.25\\%& 0.27\\% \\\\\n$n_s$ & 0.42\\%& 0.27\\%& 0.31\\% \\\\\n${\\rm{ln}}(10^{10} A_s)$ & 0.46\\%& 0.17\\%& 0.45\\% \\\\\n$\\tau$ & 13.44\\%& 6.09\\%& 12.19\\% \\\\\n$100\\theta_{MC}$ & 0.03\\%& 0.03\\%& 0.03\\% \\\\\n$H_0$ & 0.79\\%& 0.16\\%& 0.20\\% \\\\\n$\\sigma_8$ & 0.73\\%& 0.26\\%& 0.70\\% \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Introducing astrophysical uncertainties}\n\\label{sec:res:nuisances}\nIn the analysis discussed above, we assumed a perfect knowledge of the astrophysics involved in the estimate of 21cm signal observations. In particular, we assumed to know the total HI density $\\Omega_{\\rm HI}$ (that enters in \\autoref{eq:Tb}) and the HI bias $b_{\\rm HI}$ as a function of redshift. However, these quantities depend on the detailed baryon physics at play and their connection with dark matter is not completely understood \\citep[e.g.][]{Zoldan2017,Guo2017,Villaescusa-Navarro:2018,Spinelli:2019}. \nTo take into account our ignorance on these parameters in our analysis, we follow \\citet{Bernal:2019} and rewrite the power spectrum of \\autoref{eq:P_21_full} as\n\\begin{align}\n    \\begin{split}\n        \\Hat{P}_{21}(z,\\,k,\\, \\mu) &= \\tilde{B}^2 (z,k,\\mu) \\Big[ \\Bar{T}_{\\rm b}(z)b_{\\mathrm{HI}}(z)\\sigma_8(z) + \\\\\n        & \\qquad +\\Bar{T}_{\\rm b}(z)f(z)\\sigma_8(z)\\, \\mu^2\\Big]^2 \\frac{P_{\\rm m}(z,k)}{\\sigma_8(z)}.\n    \\end{split}\n\\end{align}\nThe redshift dependent combinations of functions $\\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8(z)$ and $\\Bar{T}_{\\rm b}f\\sigma_8(z)$ can be added to the set of estimated parameters as nuisances. \n\nThe most general parametrization for these nuisance parameters does not impose any specific redshift evolution. Given that we have six redshift bins, we need twelve new parameters: six $[\\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8]_i$ and six $[\\Bar{T}_{\\rm b}f\\sigma_8]_i$, one for each redshift, with $i$ being $i=\\{1,\\dots,6\\}$. However, the high dimensionality of this configuration impact significantly the required computational time for the convergence of the MCMC procedure for the exploration of the posterior.\n\nAlternatively, one can lower the number of nuisance parameters by assuming a parametrization for the redshift evolution of $\\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8(z)$ and $\\Bar{T}_{\\rm b}f\\sigma_8(z)$ in agreement with their theoretical prediction. We use a $3^{\\rm rd}$-degree polynomial model \n\\begin{equation}\n    \\label{eq:fitting_nuisances}\n    \\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8(z), \\: \\Bar{T}_{\\rm b}f\\sigma_8(z) = az^3 + bz^2 + cz + d,\n\\end{equation}\nand reduce the nuisances from twelve to eight:  four coefficients $[\\Bar{T}_{\\rm b}f\\sigma_8]_q$, with $q=\\{a,b,c,d\\}$, and other four $[\\Bar{T}_{\\rm b}f\\sigma_8]_q$. We find that assuming this redshift evolution gives the same results with respect to the twelve nuisances case while we achieve a better and faster convergence. Thus, we choose to work with this latter parametrization of the nuisances.\n\nIn \\autoref{fig:nuisances_redshift} we show the theoretical redshift evolution (under the assumption discussed in \\secref{sec:P21_model}) and the fitted one for $\\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8$ and $\\Bar{T}_{\\rm b}f\\sigma_8$. In the following we show the results we obtain varying the eight nuisances parameters.\nNote that we assume a very wide flat prior, centered at the theoretical expected value for each nuisance parameter.\n\n\nWe present the results for the multipoles alone in \\autoref{tab:constraints_P21} and \\autoref{fig:full_P21}. \n\n\\medskip\nWhen openning the parameter space to the nuisances, we see a complete loss of constraining power on $A_s$. This is expected, due to the fact that varying the nuisances we loose information on the amplitude of the power spectrum. The deterioration of the constraint on $A_s$ \ntranslates in a weakening of the constraining power on $\\sigma_8$. \n\nNevertheless, the impact of nuisance parameters is limited to these two parameters. The constraints on the other cosmological parameters remain unaffected showing the power of tomography: using the six redshift bins allows to include the evolution of the 21cm power spectrum multipoles and, thus, preserves their constraining power in particular on $\\Omega_ch^2$ and $H_0$.\n\n\\medskip\nThe same discussion applies when we combine the multipoles with CMB, as in \\secref{sec:res_P21_P21+Planck} but also varying the nuisance parameters. Results are shown in \\autoref{fig:full_results} and \\autoref{tab:constraints_P21_Planck} from which it can be seen that the constraints on $\\Omega_ch^2$ and $H_0$ remain essentially unvaried. Note, however, that the constrain on $A_s$ and  $\\tau$ and, consequently, on $\\sigma_8$ are driven just by the Planck data.\n\nFor completeness, 2D contours and the marginalized posteriors for the nuisance parameters themselves are shown in \\autoref{fig:full_nuisances} and discussed in \\appref{sec:app_nuisances}.\n\n\n\\subsection{Extending to non-linear scales}\n\\label{sec:res_non_linear}\nUp to now, we investigated the constraining power on the cosmological parameters of 21cm observations at linear scales that are the once best sampled by the large beam of the single-dish intensity mapping. At the linear scales, it is also possible to explore beyond $\\Lambda$CDM models \\citep{Berti:2021}, for which we often lack non-linear scale predictions. In a $\\Lambda$CDM scenario and for the low-z bins, we can, however, push our analysis to larger $k$ and study their constraining power. \n\nFor all redshift bins but the first two, the $k_{\\rm max}$ cut-off due to the frequency dependent beam (see \\secref{sec:tomographic_data_set}) is much stronger than the linear-scale cut-off $k = 0.2 \\, h \\, \\text{Mpc}^{-1}$. The two lowest redshift bins ($z=0.25$ and $z=0.75$) can instead be extended to larger $k$ if we relax the linear-scale cut off. We acquire 15 and 67 new points and we are able to reach $k \\sim 0.27 \\, h \\, \\text{Mpc}^{-1}$ and \n$k \\sim 1 \\, h \\, \\text{Mpc}^{-1}$ at redshifts $z=0.75$ and $z=0.25$, respectively.\nIn this new k-range the shot noise is non-negligible and it needs to be considered in the modelling.\n\n\\medskip\nWe create the new mock non-linear data set as\n\\begin{align}\n\\begin{split}\n    \\label{eq:P_21_full_NL}\n     \\Hat{P}_{21}^{\\rm NL}(z,\\,k,\\, \\mu) &= \\tilde{B}^2 (z,k,\\mu) \\big[ {P}_{21}^{\\rm NL}(z,k) + {P}_{\\rm SN}(z)\\big],\n\\end{split}\n\\end{align}\nwhere ${P}_{\\rm SN}$ is the shot noise level estimated at different redshift interpolating results from hydro-dynamical simulations~\\citep{Villaescusa-Navarro:2018}.\nThe non-linear 21cm power spectrum $P_{21}^{\\rm NL}(z,k)$ is obtained as in~\\autoref{eq:P21}, but substituting the linear matter power spectrum with the non-linear one, computed numerically with \\texttt{CAMB}\\footnote{We use the \\texttt{HALOFIT}~\\citep{Smith:2002} version from~\\cite{Mead:2016}.}. The expressions for the 21cm multipoles are changed accordingly. \n\\begin{table}\n\t\\centering\n\t\\caption{Marginalized percentage constraints on cosmological parameters at the 68\\% confidence level. Here, the label \"$\\Hat{P}_0 + \\Hat{P}_2$\" stands for the baseline tomographic data set for the monopole and the quadrupole combined and with multipole covariance taken into account. The label \"$\\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL}$\" indicates that the full non-linear data set has been used. The label \"nuis.\" indicates that we vary the nuisances parameters along with the cosmological ones. The symbol \"---\" stands for unconstrained. Confidence regions for the same set of results are shown in \\autoref{fig:full_P21}.}\n\t\\label{tab:constraints_P21}\n\t\\begin{tabular}{lcccc} \n\t\\hline \n    Parameter & $\\Hat{P}_0 + \\Hat{P}_2$ & $+$ nuis. & $ \\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL}$ & $+$ nuis. ($\\Hat{P}_{\\rm SN}$) \\\\\n\t\t\\hline\n$\\Omega_b h^2$ & 21.04\\%& 22.81\\%& 3.02\\%& 17.30\\% \\\\\n$\\Omega_c h^2$ & 13.36\\%& 14.66\\%& 1.16\\%& 12.27\\% \\\\\n$n_s$ & 3.44\\%& 3.94\\%& 0.95\\%& 4.45\\% \\\\\n${\\rm{ln}}(10^{10} A_s)$ & 8.83\\%& --- & 0.49\\%& 3.00\\% \\\\\n$100\\theta_{MC}$ & 1.53\\%& 1.62\\%& 0.18\\%& 1.61\\% \\\\\n$H_0$ & 7.39\\%& 8.10\\%& 0.49\\%& 5.90\\% \\\\\n$\\sigma_8$ & 7.64\\%& --- & 0.37\\%& 2.41\\% \\\\\n\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\medskip\nIn \\autoref{fig:non_linear}, we show the signal-to-noise for the new non-linear tomographic data set obtained with the model of~\\autoref{eq:P_21_full_NL}. \nFor the scales larger than $k = 0.2 \\, h \\, \\text{Mpc}^{-1}$, the results for the various redshift are analogous to the ones of \\autoref{fig:signal_to_noise}.\n\\begin{table}\n\t\\centering\n\t\\caption{Shot noise values used in the computation of the non-linear 21cm power spectrum, at each central redshift $z_c$. For the first redshift bin ($z=0.25$) we assume SKA-Mid Band 2 specifications, while we use SKA-Mid Band 1 parameters for the other bin. We refer to \\autoref{tab:SKA_specifics} for more instrumental details.}\n\t\\label{tab:shot_noise}\n\t\\begin{tabular}{clccc} \n\t\\hline \n    $z_c$  & central redshift & 0.25 & 0.75 \\\\\\hline\n$P_{\\rm SN}$ & shot noise  &0.72 & 2.4 & $[\\mathrm{mK^2}\\, h^{-3}\\mathrm{ Mpc^3 }]$\\\\\n$N$ & number of data points & 83 & 58 \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nWith this non-linear data set we perform an analysis similar to the one discussed in the previous sections. We study the constraining power of non-linear 21cm observations alone and combined with CMB. We first assume perfect knowledge on the quantities $\\Omega_{\\rm HI}$ and $b_{\\rm HI}$ linked to baryon physics. Note that, for this ideal case without any nuisance parameters, we assume also that the level of the shot noise is known. \n\n\n\\begin{figure}\n\t\\includegraphics[width=1\\columnwidth]{gfx\/Fig7.pdf}\n    \\caption{Computed signal-to-noise ratio as a function of $k$ (see \\autoref{eq:signal_to_noise}). We show the signal-to-noise computed for the six redshift bins and for the monopole and the quadrupole combined, considering a diagonal covariance matrix (dashed lines) or the full non-diagonal one (solid lines). The shaded area highlights the new scales acquired extending the mock 21cm power spectrum to non-linear scales.}\n    \\label{fig:non_linear}\n\\end{figure}\n\nResults are presented in Tab.~\\ref{tab:constraints_P21_alone} and \\autoref{fig:full_P21}. \n\n\\medskip\nEven if only the first two bins are concerned,\nthe extension of the data set to non-linear scales significantly improves on the constraining power of 21cm observations. We find\n\\begin{equation}\nH_0 = 67.28 \\pm 0.33\\,\\, \\text{ km s}^{-1}\\text{Mpc}^{-1}\\qquad {\\scriptstyle(68\\%,\\ \\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL})}\n\\end{equation}\nthus a relative error of $\\sigma_{H_0} = 0.49\\%$, competitive with the one from Planck 2018 data alone (i.e. $\\sigma_{H_0} = 0.79\\%$).\n\nWe then test the more realistic case where we vary the nuisances parameters. Along with the eight nuisances for the redshift evolution of $\\Bar{T}_{\\rm b}b_{\\mathrm{HI}}\\sigma_8(z)$ and $\\Bar{T}_{\\rm b}f\\sigma_8(z)$ (see \\secref{sec:res:nuisances}), we include six additional parameters $P_{\\rm SN, i}$, with $i=\\{1,\\dots,6\\}$, to model the shot noise in each redshift bin.\n\nFor example, we get\n\\begin{equation}\n    H_0 = 68.3^{+3.4}_{-4.7}\\,\\, \\text{ km s}^{-1}\\text{Mpc}^{-1}  \\quad {\\scriptstyle(68\\%,\\ \\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL} + \\text{ nuis. } (P_{\\rm SN}))}.\n\\end{equation}\nThe relative error is $\\sigma_{H_0} = 5.9\\%$, better than the \ncorresponding result for the linear-scale case discussed in \\secref{sec:res:nuisances} (i.e. $\\sigma_{H_0} = 7.4\\%$). When we open to nuisances with the non-linear data set, the presence of the shot noise terms helps to fix the height of the power spectrum and we do not loose all the constraining power on $A_s$ and $\\sigma_8$. We can thus constrain all the cosmological parameters, with the exception of $\\tau$. The posteriors for the shot noise parameters $P_{\\rm SN, i}$ and the other nuisances are shown in \\autoref{fig:full_nuisances_shot} and commented in \\appref{sec:app_nuisances}.\n\nIn summary, extending the data set to non-linear has an essential role in increasing the constraining power of the 21cm multipoles alone. Our results suggest that competitive constraints independent from other probes could be obtained with 21cm intensity mapping observations at lower redshifts and non-linear scales. \n\n\\medskip \nWhen combining the non-linear data set $\\Hat{P}_0^{\\rm NL} + \\Hat{P}_2^{\\rm NL}$ $+$ nuisances with Planck data, instead, we do not observe substantial changes in the constraints with respect to the Planck + $\\Hat{P}_0 + \\Hat{P}_2$ $+$ nuisances case (shown in \\autoref{fig:full_results}).\nAs discussed in \\secref{sec:res_P21_P21+Planck}, the improvement in combining the two probes mainly comes from the interaction of opposite degeneracy direction for some of the cosmological parameters between the CMB and the 21cm power spectrum. These are unaffected by the extension to non-linear scales and thus,  when combined with the Planck data, this extended mock data set does not add much information with respect to the linear one.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\nIn this work, we forecast the constraints on the $\\Lambda$CDM cosmological parameters \nfor a neutral hydrogen intensity mapping survey with the SKAO telescope, assuming the measurement of the first multipoles of the redshift-space 21cm power spectrum. We construct and analyse this mock data set as an alternative large-scale structure probe alone and in combination with Planck CMB data. \nWe model monopole and quadrupole signal of the 21cm power spectrum at linear scales as in \\citet{Blake:2019,Cunnington:2020,Soares:2020} and we include in our analysis the full non-diagonal covariance matrix between the multipoles.\n\n\\medskip\nWe follow the SKAO Red Book~\\citep{Bacon:2018} proposal and simulate single-dish observations with the SKA-Mid telescope both in Band 2 (frequency range $0.95-1.75$ GHz) and in Band 1 (frequency range $0.35-1.05$ GHz). \nAssuming a Planck 2018 fiducial cosmology, we construct a tomographic data set of observations within six different redshift bins. \nTo test the constraining power on the cosmological parameters of the constructed data set, we implement the computation of the likelihood function for the monopole and the quadrupole, fully integrated with the MCMC sampler \\texttt{CosmoMC}.\nWe include a discussion on the impact of our lack of knowledge on the baryonic physics involved in the computation of the 21cm power spectrum, as nuisance parameters in the analysis.\n\n\\medskip\nWe first focus on the 21cm power spectrum measurements at linear scales, that are the preferred target of single-dish intensity mapping observations with SKA-Mid, due to the large beam on the sky. \nHowever, for the lowest redshifts, the telescope beam is small enough to allow to probe also the non-linear scales. We thus extend our mock data set to non-linearities and the shot noise contribution to check if this could improve on the constraining power.\nThe results of our analysis can be summarized as follows. \n\n\\medskip\nWe find that the mock SKA-Mid 21cm observations have a good constraining power on the cosmological parameters. The constraints we obtain are comparable with other probes. E.g., with the 21cm monopole and quadrupole combined, both $H_0$ and $\\sigma_8$ are constrained at the $\\sim 7\\%$ level. The 2D contours presents very marked degeneracies between the parameters, especially in the $\\Omega_ch^2$ - $H_0$ and $\\ln (10^{10} A_s)$ - $\\sigma_8$ planes. \n\n\\medskip\nAdding the mock 21cm observations to Planck 2018 CMB data, it is possible to significantly narrow the constraints, with respect to Planck alone.\nAlthough the effect is observable on all the parameters, we get the most significant improvement on $\\Omega_c h^2$ and $H_0$, for which the errors are lessen by a fourth. With 21cm multipoles $+$ Planck we estimate $\\Omega_c h^2$ and $H_0$ at the $0.25\\%$ and $0.16\\%$ levels respectively, to be compared with $0.99\\%$ and $0.79\\%$, obtained with Planck alone. For $\\ln (10^{10} A_s)$ and $\\sigma_8$ the errors are reduced by more then a factor two. We constrain $(10^{10} A_s)$ at the $0.17\\%$ and $\\sigma_8$ at the $0.26\\%$ level, to be compared with the $0.46\\%$ and $0.73\\%$ Planck estimates, respectively. Furthermore, we observe that combining the tomographic 21cm data set with CMB alleviates some of the degeneracies between the parameters, resulting in improved constraints. The strongest effect is visible in the $\\Omega_c h^2$ - $H_0$ plane. Although 21cm observations are not sensitive to $\\tau$, we find that with Planck the improvement on the other parameters is reflected also on $\\tau$, reducing the error by a factor two. \n\n\\medskip\nTo take into account the lack of knowledge on the brightness temperature $T_b$ (that depends on the total HI density $\\Omega_{\\rm HI}$) and the HI bias $b_{\\rm HI}$, \nwe repeat our analysis including nuisance parameters. In particular, we consider the combinations $\\bar{T}_{\\rm b} b_{\\rm HI} \\sigma_8$ and $\\bar{T}_{\\rm b} f \\sigma_8$, where $f$ is the growth factor. We find that, when we open the parameter space to these nuisances, the constraining power of 21cm multipoles on $A_s$, and consequently on $\\sigma_8$, is crucially reduced. However, the results obtained for $\\Omega_ch^2$ and $H_0$ remain unaffected, for both the 21cm data set alone and combined with Planck. This result confirms the strength of 21cm tomographic measurements and motivates even more the current observational effort in this field.\n\n\\medskip\nWhen we extend the 21cm data set to non-linear scales we find a tightening in the constraints.\nThe most noteworthy result is that the constraining power of 21cm multipoles observations on $A_s$ and $\\sigma_8$ is remarkably improved, even when we open up the parameter space to the nuisance parameters. This is due to fact the information at lower scales helps fixing the amplitude of the power spectrum. \n\n\\medskip\nWe conclude that 21cm SKAO observations will provide a competitive cosmological probe, complementary to CMB and, thus, pivotal for gaining statistical significance on the cosmological parameters constraints.\n\n\\medskip\nThe formalism presented in this work and the mock data set we construct can be straightforwardly adapted to forecast constraints on the neutrino mass and beyond $\\Lambda$CDM models. These extensions are currently under study.\nNote that our modelling does not include possible residual foregrounds contamination. A discussion on how the constraints on the cosmological parameters could be biased by this systematic is left for future work. \n\n\\section*{Acknowledgements}\n\nThe authors would like to thank Paula S. Soares, Steven Cunnington, and Alkistis Pourtsidou for useful discussion and feedback.\nMB and MV are supported by  the INFN INDARK PD51 grant. MV acknowledges contribution from the agreement ASI-INAF n.2017-14-H.0.\nMS acknowledges support from the AstroSignals Synergia grant CRSII5\\_193826 from the Swiss National Science Foundation. \n\n\n\\section*{Data Availability}\n\nAccess to the original code is available upon reasonable request to the corresponding author.\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\n\nISBN-13: 978-1-62205-003-1\n\nSWEET, SWEET POISON\n\nCopyright \u00a92012 InfinityBox Press\n\nAll rights reserved. Except for the use of brief quotations in critical articles and reviews, no part of this book may be used or reproduced in any form by any electronic, mechanical, or other means in any manner without the written permission of the publisher.\n\nAll characters, groups, places, and events portrayed in this novel are fictitious.\n\nCover: Richard Wilhelm\n\nInfinityBox Press\n\n7060 North Borthwick Avenue\n\nPortland, OR 97217\n\nwww.infinityboxpress.com\n\nwww.facebook.com\/InfinityBoxPress\n\nwww.facebook.com\/kate.wilhelm.52\n\nFor Geoffrey Simmons, with gratitude.\n\nThere are debts that cannot be repaid, but only acknowledged.\nSweet, Sweet poison\n\nKate Wilhelm\nCHAPTER 1\n\nAL ZUKAL DROVE INTO the driveway of the new house with caution. You never knew out here in the country, he thought; things happen, dead animals, trees down, sinkholes... .\n\n\"They ain't got here yet,\" Sylvie muttered. And, in fact, no one else was in sight.\n\nThere was the gravel drive close pressed by evergreen trees at the county road and dense undergrowth that neither recognized, with grass that already needed cutting and probably would need it on a weekly basis. The driveway curved a few times, and at those places neither the county road nor the house was visible; it was like being in a wilderness, they both thought uneasily. The yard had not been maintained for many years. Untrimmed roses sprawled with dead lower branches; lilacs grew as high as trees and made a thicket with too many spindly new sprouts. Spent blossoms hung in brown clusters. Sumac had crept in and threatened to claim the entire acreage but was contested by many young trees\u2014maples, pines, a few oaks\u2014all too crowded and weak-looking. There were fourteen acres altogether, reverting back to a natural state where nature would do the thinning.\n\nBut no moving van was in sight. Al pulled in behind the house, where he stopped and turned off the motor. There was a two-car garage, unusable until tons of junk were cleaned out. He looked at Sylvie.\n\nShe had been a red-haired girl when they married, thirty years ago, and she would die red-haired at the end, no matter how long-delayed it was. He was used to her red hair and liked it. He liked her pear-shape, too, and would have admitted readily that they made a good couple, each pear-shaped, with her rounder end down, and his up. It made for better sleeping that way, he liked to think.\n\n\"What if they don't make it today?\" she demanded, not yet moving to open her door and get out.\n\n\"They'll make it. They want to get paid, they get here. Simple arithmetic. Come on, gives us time to make sure where we want the bed.\"\n\nActually the only thing they were moving of any consequence from their Bronx apartment was the bed. \"That was my mother's bed!\" Sylvie had said early on, as if that settled that. And it had. Just about everything else in the van that had not yet arrived was new. The new station wagon had boxes of bedding, two feather beds, also inherited from her mother, God bless her soul, and a mishmash of dishes, no four alike. But Sylvie knew when and how they had acquired each and every one of them, and when the missing pieces had vanished, or had been broken. \"I want them all,\" she had said through tight lips, and there they were in the wagon waiting to be lugged inside. There were also boxes of new clothes; at least she hadn't insisted on keeping every rag they ever owned.\n\nHe said, \"Well, why don't you get on out and let's get at it?\"\n\nActually he said, \"why doncha,\" and when she responded she said, \"Well, aincha the impatient one!\" But they didn't hear it like that and presently they were both standing before their new home. It was a two-story building, the lower half finished with river stone, the upper section white clapboard. It had an attic that would sleep many grandchildren, and a basement that would probably hold more junk than they could accumulate in the years remaining to them\u2014they were both in their fifties. The house was large, five bedrooms, a den and breakfast room besides the kitchen, a big dining room, and even bigger living room. The biggest apartment they had ever lived in had had three bedrooms, and one of them was really a dining room or a parlor or something. But with four girls growing up in the house they had needed bed space more than eating space or sitting space. Now they looked at their new home with silent awe.\n\nLooming over the many trees behind the house the mill rose, and that was theirs also. The mill had been built in 1848 and operated until World War I, or thereabouts, and used as a school at one time, and a bootlegger's production site at another, but for the past twenty years it had not been used for anything.\n\nSylvie cleared her throat and said, \"Listen.\" It came out as a whisper.\n\n\"I don't hear nothing.\"\n\n\"That's what I mean.\"\n\nBefore, when they had come out to their house, they had been with the real estate man, and then with their daughter Flora and her husband Bobby and their two kids, and there had been noise. But now there was a breeze shaking the trees a little, and off beyond the mill the rustle of water, and a faint sound of the falls, not much, not enough.\n\n\"I don't know, Al,\" Sylvie said softly.\n\n\"Jaysus Crackers! Now you don't know! Come on, grab that first box and let's get at it. Wait and see. When Flora and Bobby are here, and the kids yelling, the radio and stereo blasting, it'll be just like home.\"\n\nThey began to carry the boxes inside and stack them on counters in the kitchen, and soon the movers came, and the sight of the four-poster with the pineapple knobs comforted Sylvie. She began to relax. She fussed at Al for getting in the way, and he cursed the movers for bumping the bed against the stair rails, and everything was normal again.\n\nThe movers also brought in a new sofa, new chairs and tables, dining table, kitchen table... a new stove, a Jenn Aire that had a grill and six burners, and a refrigerator with two doors. Al had stared at it in the store. \"Jaysus Crackers! You know what I grew up with? A little box this big, with a chunk of ice delivered every Monday, and by God it had to last until the next Monday.\"\n\nAfter the movers had finished, Al nudged Sylvie with his elbow. \"Well, watcha think?\"\n\nShe nodded. \"We need more furniture than I thought we would. It's a big house, Al.\"\n\n\"You don't have to fill it, you know.\"\n\nShe was surveying the living room with narrowed eyes and did not answer. The spanking-new and factory-clean furniture looked barren somehow, and that was because there was no clutter, no piles of newspapers and magazines, no sneakers in corners, no beer cans on the tables, no pretzel bags, nothing to make it look like home. It even smelled strange, Al realized, not liking the untouched, un-lived-in look and smell. Sylvie would fill it, he knew, and after the kids got here, they would help, and for now, well, there was work to do.\n\nA while later she was stashing things in cabinets in the kitchen when he decided to do the shopping they had agreed on ahead of time. \"You coming?\"\n\nShe shook her head. \"Too dirty. I thought they just had this many cabinets in them fancy magazines with houses that no one really ever lives in.\"\n\nShe'd fill the cabinets, too, he thought gloomily and left her there. Spender's Ferry was less than three miles away, one hour and twenty minutes out of New York City, with a commuter train station and everything. Goddamn suburbanites, he thought in surprise. That's what they had suddenly become. Al's father had come over from Czechoslovakia to strike gold and had set himself up in a butcher shop, which Al inherited in due time. Then the Bronx caught on fire. That was how he thought of it. The shop vanished, and Al got a job butchering for a supermarket. At least, he often thought when things got tough, they always had meat on their table. Stolen meat, meat carried home in pockets, in his lunch box, meat sometimes slipped to Sylvie when she came in to shop. When things got really bad Sylvie worked too, cleaning offices in the middle of the night. Somehow they had made out okay. Not great, but okay. Then, almost a year ago, Al Zukal had spent five dollars on the lottery and had won big. Just like his father always said, he thought: work hard all your life and you'll make it. Guaranteed income of two hundred twenty-five thou a year, he thought, as he had very, very often over the intervening months. Goddamn suburbanites!\n\nJill Ferris was leading the way across a swinging bridge below the mill at that moment. \"Dad's always maintained the bridge, just because he likes it,\" she said over her shoulder to a tall man behind her. Jill was thirty, a bit underweight, with long pale hair, no longer really blond, although she had been very blond as a child. Now it was the color of darkening golden oak; it gleamed with red highlights when she moved into sunlight. The man following her was Sebastian Pitkin, a few years older than Jill; he had a solemn expression, lank hair darker than hers, and large blue eyes that bulged a little. The bridge swayed and he clutched the handrail involuntarily; she seemed not to notice.\n\nJill stopped near the end of the bridge and pointed. \"Pretty, isn't it?\" Below them was Spender's Lake, seven acres in all, with a short strip of beach at the upper-right end. That was on the side of her father's land, several hundred acres, mostly woods, with an orchard that was invisible from here. On the other side of the lake was the university experimental farm. The narrow bridge, twenty feet across, swung a few inches above the dam; the waterfall had a drop of eight feet. The stream that wound away through a gorge was only two feet wide below the dam, although farther on it widened again to become Spender's Creek. The setting was incredibly beautiful, untouched-looking. The trees were massive in the forest, the lake shockingly blue, grasses edged the water in places, and even the picturesque swinging bridge was like an illustration in a book from the turn of the century.\n\n\"See, that's state forest land,\" Jill said, pointing again. \"Dad's land borders it on this end and the creek and mill property make up the boundary here. The experimental farm is on the other side of the mill property. No one will ever be able to encroach in any way. Perfect, isn't it?\"\n\nSebastian was nodding thoughtfully, eyeing the mill that towered over them. The waterwheel looked intact, although some of the buckets probably would need replacing. Hard to tell, though; there was too much moss on the lower portion.... No water flowed through the diversion trough, but of course it wouldn't, not with the mill shut down. \"You say the mill's in good shape?\"\n\n\"Absolutely. They built it out of cypress, cedar, and oak, and they built it to last forever. The upper floors would be great for dormitories and the entire lower part could be classrooms, mess hall, kitchens, meeting rooms. Wait until you see.\" She started to walk again; the bridge began to swing again. He caught the handrail and followed cautiously.\n\nThere were mammoth double doors that had admitted horse-drawn wagons loaded with grain at one time, but no longer opened. Weeds grew thick at the base of the doors, all around the mill. White Queen Anne's lace, blue forget-me-nots, pink and yellow sweet peas trailing everywhere. Almost too pretty, too planned. Jill led the way around to a normal-sized door that opened at her touch. The interior was dim; makeshift partitions had been added at one time or another, obviously temporary additions that never had been torn down again.\n\n\"See,\" Jill said, in the center of one of the small rooms, \"a private meditation room, or a sleeping cell, or even a meeting place for a group of four or five.\" The room was about eight by ten, stifling, bake-oven hot. The single window appeared to be painted shut.\n\nThey left the room and were in the central, open space where the mechanism for the grinding of the corn and wheat was all there, gears and wheels and chains and pulleys, but the stone grinding wheels had been removed. This section was pleasantly cool. They went up very sturdy stairs and looked around, came down a different staircase, and roamed the back part of the lower floor.\n\nThey were emerging from around one of the partitions when Jill gasped at the sight of a figure in the open doorway.\n\n\"Who are you? This is private property,\" Jill said, louder than she intended, startled.\n\n\"I could say the same,\" the woman snapped back at her. \"It's private property, all right, my private property, me and my husband's. What are you doing in here?\"\n\nJill could only stare at this awful woman, with her awful, thick Bronx accent and voice that sounded like a demented duck quacking. She was dressed in awful dirty, baggy pants, with a dirty shirt, and she had awful red-dyed frizzy hair poking out from under a grass-green kerchief.\n\n\"I'm sorry,\" Sebastian said. \"We certainly didn't mean to invade your privacy. I thought the property was for sale. Mrs. Ferris was showing it to me. Please forgive us.\" He smiled gently. \"This is Mrs. Ferris, and I'm Sebastian.\"\n\nFor a moment, with his light blue eyes fixed, his gaze on her so directly, she felt confusion. A kind man, she thought, understanding, but then the girl made a noise and Sylvie looked at her, the moment broken, gone, leaving a bewildering set of contradictions in its place. Sylvie crossed her arms over her chest and nodded. \"Right. Well, the mill and the house ain't for sale, not no more. Me and Al bought it all last week and we're moving in right now.\"\n\n\"Come along,\" Sebastian said to Jill then. \"It's a mistake, after all. No harm done.\"\n\nSylvie watched them narrowly as they left the mill and stepped onto the bridge. She waited until they had vanished among the trees on the other side before she went back to the house, muttering to herself. A politician, or a preacher, she thought. She shook her head. Preacher. Not just because no politician would ever own up to a name like Sebastian, but the way he looked at her. Making up to her, she thought with surprise. The kind who got on Sunday television and looked at you like that, like he'd be all over you in a flash if he could, like if he could just lay on a hand, you'd say yes no matter what he asked. Making up to her, at her age, she muttered angrily. Sylvie didn't trust religion that relied on sorrowful blue eyes and the laying on of hands, and the intimate little smiles that suggested he knew your secrets. Religion should be cooler than that, she thought, more dignified. And that skinny girl. Putty in the hands of a man like that. Her lips tightened at the thought of the girl. Who are you? Like she owned the world. Well, she showed them, Mr. Sebastian-let-me-save-your-soul, and Mrs. I-own-the-world Ferris.\n\nShe was still grumbling under her breath when David Levy turned up at the kitchen door of the house. He was a stick of a boy, way over six feet, with a great big pumpkin grin, and wanting a haircut, like always. Ever since she knew him, he had needed a haircut. Must get it done sometime, she thought, but it never showed. Great mop of black hair. Rosa, their youngest daughter, had told David they were looking for a place in the country, and David had told them about this, and now here he was to lend a hand. A good boy. And Rosa didn't know good when she saw it, off in California on a scholarship, studying to be some kind of ocean scientist, and some other girl would see David Levy and not be so blind... .\n\n\"What can I do?\" David asked. \"I wanted to come earlier but we've been pretty busy at the farm. Anyway, here I am.\" He had a backpack slung over his shoulder. He put it on the table and she began directing him exactly the same way she directed Al and the girls and their husbands and children.\n\nPretty soon Al came back with the station wagon loaded with groceries, and they got to work carrying them inside. \"I told my boss about you,\" David said, putting a box down on the table. \"I hope it's okay if she comes by to meet you.\"\n\n\"Grand Central,\" Sylvie muttered. \"Fine, David, fine. Al, put that on the table until I have a chance to see what all you've got. Did you buy some hot dogs like I told you?\"\n\n\"Sure, sure. And buns, what you didn't tell me. And catsup. And beer. And I'm for a beer right now. David?\"\n\nHe shook his head. Al opened two cans and handed one to Sylvie. David rummaged in his backpack and brought out a thermos. \"Lemonade,\" he said, and drank.\n\nHe put the thermos down and went to the door then. \"Here she comes,\" he said. In just a moment a woman appeared and he opened the door for her.\n\n\"Dr. Wharton, these are my friends, Mr. and Mrs. Zukal.\"\n\nShe was dressed in jeans and a T-shirt and sneakers. She had the kind of sturdy little body that would let her wear clothes like that for many more years. Her hair was short and straight, dark with streaks of gray at the temples. When she smiled, a dimple appeared in her cheek and her eyes smiled along with her mouth.\n\nSylvie nodded. \"You want a beer?\" she asked. And then, \"You're a doctor?\"\n\n\"Just a professor. And call me Lois,\" she said. \"I'd love a beer.\" She was looking around at the furniture, at the boxes, and cartons with undisguised interest. \"This has been a hell of a day.\"\n\nAl Zukal popped another can and handed it over to her and she drank thirstily. \"For us, too,\" he said. \"Moving's a bitch.\"\n\n\"I was really surprised when David said you had bought the place,\" Lois said. \"Not you, but anyone. This property has been on the market for years.\"\n\n\"Two looks,\" Al said. \"All it took. Two trips and me and Sylvie, we seen it was exactly what we're after. Believe you me, we looked at a lot of dumps, real dumps, and a lot of mansions, too.\"\n\nSylvie began to talk about some of the other places they had inspected, and Al was still talking, now about the realtors who would sell a doghouse and call it a palace. Lois finished her beer and caught the look of affection and amusement on David's face. She grinned again and although both Al and Sylvie were talking, she did, too.\n\n\"Anyway, welcome, and thanks for the beer. I have to go and scrub. I'm all over muck. Why don't you join us for a cookout? David, you know the way up the trail, don't you? You can be guide. Seven or thereabouts.\"\n\nSylvie and Al exchanged glances; hers said sure, why not, and he nodded.\n\n\"Good. No one should have to cook on moving day, and restaurants are just too much trouble if you've been working hard. Don't dress up or anything. It's a real cookout, first of the season, and with the weather cooling off the way it does as soon as the sun goes down, it's not likely to be a late-night affair.\" She reached out to put down her empty can, and somehow her arm brushed David's thermos and it teetered, then fell with the unmistakable sound of breaking glass.\n\n\"Goddamn it,\" Lois said with a sigh. \"David, I'm sorry. As I said, it's been that kind of day. I'll replace it, of course.\" She bent over to pick it up, but he was there first.\n\n\"It's all right,\" he said hurriedly. ^'Really. Don't worry about it.\"\n\nBut she would get him a new one, Sylvie thought. Lois left then, and Sylvie turned to the bags and boxes on the table. \"Now, let's see what you got and try to figure out where it should go.\"\n\nAl was looking thoughtfully out the door. \"David, she said up the trail, walk to her place. Where? There ain't no place out there that I know about.\"\n\nDavid squirmed uncomfortably. \"I didn't know she'd invite us all over,\" he said, not looking happy. \"I mean, she's my boss, a professor, Dr. Wharton, and all, but she's also Mrs. Wollander. She lives in the big house at the other side of the lake.\"\n\nAs Lois walked away she continued to hear Sylvie's raucous voice, and Al's equally raucous voice in some kind of weird harmony. She was still grinning.\n\n\"What's so funny?\"\n\nShe looked up, startled, but not really surprised to see her husband Warren approaching with both hands outstretched.\n\n\"Hi! Didn't see you there. Sun in my eyes maybe.\"\n\nHe drew her close and kissed her. Warren Wollander was six feet tall and large through the shoulders and chest, strong-looking at sixty-two. \"I didn't intend you should see me,\" he said, softly. \"That's just about where I was standing the first time I saw you. Remember? I like watching you when you don't know I'm there.\"\n\n\"Who could forget?\" she said, and put her arm about his waist. His arm was across her shoulders as they started up the trail.\n\nHe had taken the walk that day even though there was still snow in every shadowed cranny, thick on the north slopes, piled up behind stone walls. And across the lake he had seen a person who, he thought, was spiking the experimental trees. A short, shapeless figure in a down jacket, heavy pants, boots, with a red stocking cap that had a long tail that swung back and forth as she applied a drill to a tree, braced for purchase, and started to drill.\n\n\"I thought you were a yeti,\" she said gravely, and they both laughed. Actually, she had been drilling out core samples to measure for growth. He tightened his arm around her shoulders. \"I was smiling at our new neighbors,\" she said. \"Mr. and Mrs. Zukal. I invited them to the cookout. And David, too. It's time you met David, I think.\"\n\nThe pressure of his hand on her shoulder tightened, just for a moment, and perhaps not in response to what she had said, she thought. She did not look up at him, but watched the trail. It was well-groomed, with dense mayapple colonies on both sides.\n\n\"Well, well, the place finally got sold? That will surprise Jill, I suspect.\"\n\n\"I guess so,\" she agreed, and suddenly felt tired, and unhappy about the invitation that she had issued without thought. \"Do you mind that I asked them over?\"\n\nThis time she knew his hand tightened, but it was a reassuring pressure, and reassurance was in his voice when he replied. \"Lois, this is your house, not Jill's. Remember? You're the lady of the house. You want company, invite them. Tell me about our new neighbors.\"\n\nShe began to talk and presently he was chuckling, and then laughed louder. \"God, it will certainly surprise Jill.\"\n\nShe laughed, too, but she was bothered about his daughter Jill. What if Jill came on in one of her bitchy moods? She could do that. And this man, this guru, whatever he was, what if he was impossible? She knew that Jill's husband Stanley would be all right, and Warren would treat the Zukals exactly the same way he treated everyone. With a politeness that would be excessive if he didn't like them, and with warmth and humor if he did. She sighed and wished she could take back the spontaneous invitation. And even with the thought, she found herself denying it.\n\nNo, by God, she thought firmly; this was her house, she was the lady of the house, and Jill had to be reminded now and then. Jill had arrived for a visit three weeks ago and Lois was beginning to think of her as the Woman Who Came to Dinner. Yesterday Jill had announced plans for a cookout, even though the late April weather could be as fickle as any poet ever suggested. She had invited a stranger, Sebastian, and had taken it for granted that it would be fine with everyone else. Well, it was, only the party had suddenly increased by three.\n\nSometimes she wondered if Warren was fully aware of the cautious maneuvers she and Jill used with each other, and usually she knew the answer was that he was fully aware. She suspected that he knew exactly why she had invited their new neighbors, and she suddenly felt ashamed of herself, and immensely grateful for his support. His position was not enviable either, she knew. He was deeply in love with her, no doubt about that, and he loved his daughter without reservation, even though she could be... difficult was the kindest word Lois could think of. She gave his waist a little squeeze when they had to separate to walk single file. She appreciated these short periods together more than he could realize; she knew he was the wisest man she had ever met, and if he thought it would all work out, that made her think so too. Most of the time anyway.\nCHAPTER 2\n\nTHIS WAS ONE OF THE worst evenings of his life, David Levy thought as he and the Zukals approached the big house. They had come up the trail, as Dr. Wharton had suggested, and Mrs. Zukal had been watching for animals or snakes or bugs the entire way. Now they emerged onto a meticulously maintained lawn that stretched out forever. Already, some people on the patio behind the house had turned to watch them. David felt like a pinned butterfly. He swallowed hard and deliberately looked past the group of people to study the house.\n\nThe original structure had been tall and narrow, three stories up with a peaked roof, but wings had been added, and now it sprawled and the central part looked almost like a ship's mast to which the lower sections were tethered. The patio was flagstone; a long trellis covered with the first extravagant flush of red and white roses screened what lay beyond.\n\nAlthough he intended to keep focusing on the house, his gaze was drawn to Lois as she came to meet them. She was dressed in chino pants and a gray silk shirt that made Sylvie's blue polyester pantsuit look tawdry. She got between Sylvie and Al to take their arms and steer them to her husband. Her greeting was warm and cheerful; she made it seem natural for the three newcomers to be part of the group, and to David's surprise no one else seemed to think it strange either. He mumbled something to Mr. Wollander, and then to his daughter Jill who looked like a movie star in skintight jeans and a scoop-neck sweater that accentuated her sharp collarbones. He nodded to the other man whose name was simply Sebastian.\n\n\"You've certainly come to a beautiful setting,\" Sebastian said to Al as they shook hands. \"We were just marveling over the lake.\"\n\nIt was about twenty feet lower than the lawn where they were standing, and in the evening sunlight it gleamed like a sapphire without a ripple to mar it.\n\n\"Yeah, it's pretty, all right,\" Al said, and Sylvie said something at the same time; behind them David suppressed a groan. But Sylvie shut up and the moment passed.\n\n\"I have pictures of the mill that go back sixty years or more,\" Warren said then. \"I bet you'd like to see them.\"\n\nBefore they could respond, Jill said to Sylvie, \"This afternoon I acted like an idiot. You gave me such a start. We used to play in the mill, you know how children will get into a place like that. We told ghost stories and there you were suddenly, just like out of one of the stories. I'm sorry.\"\n\n\"You started me, too,\" Sylvie said. \"I thought you was squatters or something. No harm done.\"\n\n\"She was absolutely forbidden to go there, just as I was,\" Warren said, laughing. \"We did the same thing when I was a boy. Come on, let's look at those pictures.\"\n\nAs he led them inside, Lois began to relax and she wondered why she had been tense at all. Jill had behaved like a perfect lady, and had taken the news of the addition to her party with such good grace that Lois had felt ashamed. She went to one of the long tables where drinks had been prepared.\n\n\"David, Sebastian? What would you like?\"\n\nThey both took juice and she had gin and tonic. Jill helped herself to white wine. Sebastian was still regarding the lake.\n\n\"I had no idea this was back here in the hills,\" he said after a moment. \"I'm surprised that people aren't swarming to it. Isolation and peace are priceless commodities these days.\"\n\n\"Too small and too out of the way,\" Lois said. \"And hiking's better on the other side of the village, up into the mountains. But when hunting season opens, they swarm.\" She stood at his side and also gazed at the unlikely blue of the water. \"You realize that color comes from the fact that the lake is dead?\"\n\nSebastian looked at her with disapproval, she thought, surprised at his expression. \"Acid rains did it in,\" she explained. \"Unfit for fish or plant life. It's doing the same to the experimental farm,\" she added, and heard the bitterness in her voice.\n\n\"Perhaps,\" Sebastian said in his gentle voice, \"we are seeing a warning sign that must be acknowledged. A lake here, a farm there. The earth is speaking to us. But who is listening? Perhaps there has been too much experimenting already.\"\n\nLois felt a surge of anger that caused a trembling in her fingers. \"Not just a lake here and a farm there,\" she snapped. \"Whole forests are dying. In Bavaria, the whole damn Fichtelgebirge forest is at risk. More than fifty percent of the mature trees are dead. And it's happening in Vermont, on the West Coast. Not just a little patch here and there, I'm afraid.\"\n\n\"I understand the peril,\" he said gravely, \"and what I suggest is that if we can restore the harmony of humans and nature, of living with nature instead of pitting ourselves against it over and over, perhaps even this peril can be overcome.\"\n\nHe believed it, she realized, when she turned to study his face, to see if he was mocking. He looked as if he wanted to touch her, to soothe her, calm her; he looked as if he ached because of her pain and anger. Suddenly she remembered a high school teacher, more than twenty years ago, who had looked at her pityingly and said, \"Do you really believe we can pollute all the oceans of the world?\" Then, as a girl, she had been comforted by such knowledge, such serene wisdom, such absolute faith. This man, Sebastian, with his understanding, his sympathy, his empathy, his firm convictions could be just as soothing, just as comforting. He would dispense faith, trust, harmony with a word, a gentle smile. And meanwhile the trees were dying from causes so complex that few people even attempted to define them, and those who studied them walked in fear. Looking at him, she had the wild impression that he was following her thoughts, that he was telling her she did not have to live in fear, that she could recapture the same kind of tranquillity that softened his face. The tranquillity of ignorance, she said to herself sharply, and lifted her glass and forced herself to look away from him.\n\nShe drank, and then remembered that she had not yet asked David about his weekend plans. She turned toward him. \"Are you free this weekend? I thought we might start the lime before another rain comes.\"\n\nToo late she realized that Jill had been talking in a low voice with David; she bit her lip in exasperation. \"Sorry. I can't leave my work behind, it seems.\" But David had left Jill with an embarrassing swiftness and was already at her side.\n\n\"I wasn't going anywhere. The lime shipment came this afternoon. And a couple of the hands will be around.\"\n\nShe shook her head, and waved him away. \"No. No more work today, no more talk about lime or trees or acid rain.\" Then, as if to prove her point about no more work talk, she returned to the table with the decanters, and asked over her shoulder, \"Sebastian, can you leave your work when the day is over?\"\n\n\"I'm going to the village to pick up Stanley,\" Jill said, and walked stiffly off the patio to vanish into the house.\n\nLois sighed and refilled her glass.\n\n\"But how does one say a day ends?\" Sebastian asked as if bemused. \"One day flows into another, and then another endlessly. And my work... That, too, is endless. I am never not working, if that is even the right word.\"\n\nLois glanced toward the house, wishing Warren would come back and bring Al and Sylvie with him, let them do the talking. She said to David, \"Sebastian is a... what, counselor, preacher, teacher? What do you call yourself?\"\n\nFor a moment, when he had come to her side to talk about lime, David had looked almost at ease, now he looked wretched again. But Sebastian paid no attention to him, he kept his gaze fixed on Lois.\n\nHe smiled. \"None of the above, I think. A guide at times, when I have found a piece of the path myself. But most of the time I think I am a student, a searcher, and the few stretches of path that have opened to me have revealed that the path is as endless as the day, as my work... .\" He stopped speaking and regarded her for a moment. \"You have a headache?\"\n\nAfter a brief hesitation she said yes. \"Tension, I know. Stiff neck, stiff shoulders. Working too hard, racing the clock most of the time. It will pass.\"\n\n\"Let me demonstrate one of the techniques I found along the way,\" he said. \"It will help very fast, or not at all. But first you have to sit down.\"\n\nIt was a challenge, she thought, one she did not have to accept. But his expression was guileless and kindly, and with reluctance she seated herself at the table. At the other side of the table David swirled his ice cubes around and around in his glass.\n\nA few minutes later when Warren returned, laughing with Al and Sylvie, she was still seated at the table. \"Warren, let Sebastian show you how to get rid of a headache. It works like magic!\"\n\nSebastian demonstrated on them all. Sylvie was stiff all over as he approached her, confirming her worst suspicions, that he was a laying-on-of-hands minister, and Al was wishing they could go back inside and look at more pictures. What a house, he thought. Made theirs look like kid's stuff. David was wishing he had never been born. He had known this would be the worst evening of his life, but he had not known just how bad it would get. Seeing Dr. Wharton\u2014Lois\u2014with her husband, and then with Sebastian's hands on her neck and shoulders... . He had thought he was braced, but that was before he had realized just how bad it could get. Women were fussing with another long table and that gave him hope that they would eat soon, and the evening would end.\n\n\"See,\" Sebastian was saying to Sylvie. \"You are stiff with apprehension, and that causes the muscles to contract, and that is the source of pain.\"\n\n\"I ain't even got a headache,\" she said.\n\n\"But you are not relaxed, either.\" His voice was soothing and kind, and when he felt her neck, his fingers were gentle at first; they became probing, and hurting. He pressed at the base of her skull, and then at the top of her shoulders, and it seemed to her that she did loosen up a little. But there wasn't no miracle cure, she added. He went on to Al who steeled himself as if getting ready for hand-to-hand combat. Sebastian laughed.\n\nSoon after that Jill returned with Stanley. His arm was around her waist, her hand over his on her hip, as if to keep it anchored there, not let it roam. His face was round and smooth, his eyes pale blue. Like a little doll, Sylvie thought. A bright and smiling little doll. She guessed that Jill had told him about the company for supper; he acted like a proper little gentleman with Sebastian, didn't look him up and down or anything, just shook hands, and then turned to her and Al. At first, she expected him to cold-shoulder them; he looked like one of the magazine ads for the type who made a million before thirty and had no use, thank you, for anyone who didn't. But instead, he was like a school kid with her and Al.\n\n\"Wow! You guys won the big one! Hey, that's great! Do you have a financial advisor yet?\"\n\nWarren burst out laughing. \"He can turn carrot juice into oil fields, folks. If he gives out advice, listen.\"\n\nAl surveyed Stanley thoughtfully. \"Financial advisor? What's he do?\"\n\n\"You don't, do you? Oh, God! Let's take a little walk.\" When Al started to move toward the side of the patio where the trellises were heavy with roses, Stanley took his arm and drew him instead out onto the lawn. \"No flowers. Allergic. Mostly to bees, but it's been my experience that if you avoid flowers, you're more likely to avoid the little bastards. Now look, you need someone who makes it a career thinking about money, someone who really likes to see it grow, see....\"\n\nIt wasn't that he had snubbed Sebastian, Lois thought, but he had shown no enthusiasm either. She listened absently as Jill explained to Sebastian that with Stanley money was a vocation, and a hobby, which made him a happy man since there was so much of it to work and play with. And Sebastian murmured something to the effect that Stanley might be very good for Al and Sylvie. With a start Lois understood that Jill had been apologizing for her husband and that Sebastian had accepted the apology. Lois wondered if Jill had planned this whole cookout simply to bring Sebastian and Stanley together in a friendly manner, and she decided she simply did not care.\n\nDavid didn't know exactly how it happened, but Warren Wollander asked him a simple question or two, and the next thing he knew he was talking about the need to introduce new foods that would grow in marginal lands, the need to supply complete proteins in crops that could be maintained by peasants everywhere. The need to develop fast-growing trees that would supply fuel and food and fodder and hold the soil, restore the balance of atmospheric gases... .\n\n\"You can't do it,\" Jill said flatly when he paused for a breath. \"You can't keep up with human fertility through food. You have to educate people, reduce fertility first, and then maybe you will accomplish something.\"\n\n\"But you can't just let them starve if they're already here,\" Sylvie said.\n\nAl and Stanley returned, apparently having reached an agreement of some sort. Al looked at Sylvie fondly and said, \"First thing, we get that check, and first thing, ten percent right off the top, to charity. Her idea.\"\n\nStanley groaned.\n\n\"And where do you stand in this perennial problem of distributing the world's largess, Sebastian?\" Warren asked. Lois felt herself become very still at the tone of his voice, too polite.\n\n\"When enough people find the path, realize the human potential, experience the wholeness that is available to all, then there will be no problem,\" Sebastian said softly. \"It doesn't take everyone, just enough to lead the way. We all find different ways to help bring this about, some through the green revolutions that arise, some through other means.\"\n\n\"Well, the means I intend to apply this minute are at hand at the grill,\" Warren said. \"Steaks in five minutes.\"\n\nIt turned out that neither David nor Sebastian would eat meat; Lois felt, with a twinge of irritation, that she should have known, and she went inside to advise Mrs. Carlysle, the housekeeper, that they would also need a lot of cheese. Her headache had come back.\n\nThe steaks were superb. Al asked for the name of the butcher, the grocery, wherever they got them. The supermarket he had worked in never had meat like that. Mrs. Carlysle had found smoked salmon that David accepted although Sebastian did not. \"To hell with it,\" Lois said distinctly under her breath. Sebastian was not her problem. Let Jill see to his diet. Warren talked to Sylvie about growing up here, how he and his brother, who was killed in Iwo Jima, had played up and down the hills, in the mill. Stanley talked to Al about money matters and to Jill about the hideous mess the painters were making in their New York City condo that was being redecorated. Jill shrugged. \"I told you to move into a hotel until they're finished. You keep telling me how awful it is, but you don't do anything about it. You knew it would be a mess.\" She was quiet for most of the meal, her smile forced and mechanical. Sylvie caught her glancing from David back to Lois, and saw the little glint in her eyes as she reached some kind of conclusion. \"You swim every morning, don't you?\" Jill asked David, who blushed and nodded. Jill shivered. \"I couldn't stand the idea of leaving a warm bed and plunging into that cold water before breakfast even.\" It was innocent, Lois told herself. It was. David looked agonized and averted his gaze to his plate. Sebastian and Stanley launched into a tortured philosophical discussion about the ethics of feeding the poor. Sebastian's voice was melodious, his accent softened by a touch of the south. Now that Lois had disengaged herself from trying to entertain him, she listened to the rhythms of his voice instead of the actual words. It was almost as if his first language had been French perhaps, or Spanish, and he had learned English from a southerner who had lived many years in the north. Lois smiled to herself over her own tortured path in trying to identify his origins, and she wondered what Warren had seen in the man to cause his distrust. Aside from the fact that he was some kind of guru, she added.\n\n\"But don't you think it would be cruel to give them vitamins and adequate food for a limited time, knowing that tomorrow they will be starving again?\" Sebastian asked. His voice had become sorrowful.\n\n\"You have to work with it one day at a time,\" David said, sounding nearly desperate but unable not to join the discussion, his words too fast and sharp. His New York sound, Lois thought.\n\nSylvie had grown increasingly restless, certain that none of them was talking about real people, just ideas. \"What you're saying is just like don't go to the doctor today because maybe you'll get sick again or get hit by a truck tomorrow. Well, maybe if your belly's full and you're healthy tomorrow you can face whatever's around the corner better.\" Her voice rose in indignation. \"It's crazy to just give up. Might as well not be born in the first place. Or have a bat ready to bash every baby's head the minute it shows up. Is that what you mean?\"\n\n\"Of course not,\" Sebastian said. \"But I think people should accept what happens. Endless, futile struggling destroys whatever peace is available, and peace of the soul is more important than a full stomach.\"\n\n\"Yeah, but peace is easier with a full stomach than an empty one,\" Sylvie muttered. \"You don't know that, you ain't talked to the right people.\"\n\nWarren laughed. \"She's got you, I'm afraid.\"\n\nIt was the signal that that conversation had drifted too far into unpleasant matters to continue at the dinner table, Lois knew. Apparently Stanley understood the message also.\n\n\"Speaking of vitamins and such,\" Stanley said, \"I read that if you take bee pollen in increasing doses, you can build up a tolerance to stings.\"\n\nJill toyed with her food and looked terminally bored.\n\nDavid began to talk about the health food and vitamin catalog that he ordered his vitamins from and Sebastian promised to send him a price list from the company he used, less expensive, he said, than David's supplier. Stanley asked if they carried bee pollen, and Sebastian said coolly that anyone with the right thoughts, the right attitude, reverence for all living creatures never got stung to begin with.\n\n\"Right,\" Al Zukal said with a satisfied grunt. He pushed his plate back, finished. \"And them people can whistle snakes up out of baskets and climb ropes that ain't hooked on nothing, and their shit smells sweet. Yeah.\"\n\nThere was a long silence, broken when Warren rose from his chair. \"I'll see if coffee's about ready,\" he said, his voice strained.\n\nBelatedly Lois realized that he was nearly choking to keep from laughing. \"I'll help,\" she said and followed him inside where they collapsed holding each other in the hallway to the kitchen, laughing out of control.\n\nThat night in bed, Al said, \"Well, watcha think?\" And Sylvie said, \"Doncher get ideas.\"\n\n\"What's that supposed to mean?\"\n\n\"We're not their kind and they know it and so do we.\"\n\n\"Stanley's all right.\"\n\nStanley had insisted on driving them home, to Sylvie's great relief. She had been terrified of the dark trail, and the probable dangers it held.\n\n\"Sure he is. They all are. But don't get ideas. Listen. What's that?\" They both listened, and she said, \"Wolves.\"\n\n\"You crazy or something. There's no wolves anymore. Except on TV. Dogs.\"\n\nThey lay in rigid silence listening to howling that sounded more like wolves than dogs.\n\n\"Stanley gave me some good advice already,\" Al said after a moment. \"And he's coming over Sunday to talk before he goes back to town.\"\n\n\"What advice?\"\n\n\"He said... Well, not all that much, I guess.\"\n\n\"Spit it out,\" Sylvie said. \"You won't go to sleep and neither will I until you do.\"\n\n\"Yeah. Well, he said I should first thing, next check next month, take out insurance, a big policy.\"\n\nShe was silent for a long time, then said, \"You already got insurance.\"\n\n\"I mean big. Really big. You know that lottery ticket was mine, and if I kick, that's all she wrote.\"\n\nSylvie drew closer to him. Out here in the country it was cold at night, not like back home, she thought. \"Well, don't kick,\" she said finally. \"Why should you? You're as healthy as a horse. Listen. What's that?\"\n\nAfter a moment he said, \"Frogs or something. Crickets maybe.\"\n\nAfter another minute or two, she got up and stood at the window looking out. The blackness seemed to start right at the end of her nose. \"I can't see a thing, not a light, nothing. Black as a coal bin out there. I want a dog.\"\n\nShe groped her way to the bathroom where she switched on the light, and then returned to bed, leaving the door open a few inches. Now the dimly lighted bedroom seemed more natural, like the bedroom in their old apartment where streetlights had filtered in no matter what she did.\n\n\"Dog's a good idea,\" Al said, as they adjusted themselves to fit each other again. Her feet were like ice. \"Yeah, real good idea. Tomorrow, you and Flora should do that, go out and get a watch dog.\"\n\nThey both began to think about the visit of their daughter Flora and her husband Bobby and their two children, and the noise they would bring with them, and they thought that somehow light would come with them, too. Finally they could drift off to sleep.\n\nThe dog was a golden retriever, two years old, female. They named her Sadie. Al regarded her with great distrust. They had never owned a dog before\u2014who wants to clean up dog shit on the streets? Or take it out for a walk in the rain? But out here it seemed the dog took care of herself. And she slept on the porch so no one had to take her out. But his distrust was focused on the fact that she acted like a kid, just like Flora's two kids, running, romping, having a good time. Not like a watchdog should act, he decided. Mikey held her while Janie went off to hide, and Sadie tore out after her when he let go and found her in two shakes. They tossed a stick and she brought that back. She played Frisbee like a pro. So, Al brooded, good-time Sadie, what good was a dog like that, one that loved everybody? And she did, it seemed, love everybody. She laughed at the kids and smiled at Flora and Bobby, and treated Sylvie like she was a Marine sergeant and Sadie a recruit in love with being a Marine.\n\nOn Monday morning when Lois appeared on the swinging bridge Sadie stiffened and growled, although she couldn't see that far, or hear either, Al was sure. He and Sylvie went with the dog to the edge of the property where Lois was just about to step onto their land, and Sadie was shivering, growling. She began to bark and Lois stopped in her tracks. It was the first time Al had heard the dog bark, he realized with approval. Not that she should bark at Lois, but it was a good sign.\n\nSylvie had to introduce Sadie to Lois, real polite and formal like; the dog sniffed Lois all over, and then wagged her rear end in approval. It was embarrassing to Al the way Sadie stuck her nose in Lois's crotch, but she seemed to have to do that; she had done it to everyone she was introduced to so far.\n\n\"She's beautiful!\" Lois said when the inspection ended, and she had been accepted. \"Has she been in swimming yet? Retrievers usually love the water.\"\n\nAl hadn't even thought of letting the dog in the lake. But that afternoon he and Sylvie went down and he tossed in a stick, and Sadie tore out after it as if she were half fish. By then he was sold on her, anyway, but it was another good sign. If one of the kids ever fell in, or anything like that happened... he thought vaguely, not quite ready to admit aloud that he sort of liked the silly animal for herself. She had set up a hell of a ruckus when the carpenters came to work on the mill, and after they left she went over every inch of the ground they had walked on, everything they had touched. She repeated this the next morning, and the next; they had not introduced her to the carpenters whom she continued to treat as trespassers. But she met Lois at the bridge and smiled, waving her plumy tail, and escorted her to the break in the fence that separated this property from the experimental farmlands. She did not step through with Lois and she refused to go on the swinging bridge unless under direct order by Sylvie.\n\nNow, when Al and Sylvie went to bed, they didn't lie in rigid silence, holding their breath, listening. Sadie was on the job.\n\nOn Saturday Flora, Bobby, and the kids came back, and the kids wore themselves and Sadie out again, as they had done before. When the kids had gone on to bed, Al shook his head at the dog, lying in exhaustion on the porch. \"That's how it's gonna be,\" he said. She raised her head and smiled at him. She thumped her tail on the porch once, sighed, and made no further attempt to be polite.\n\nAl was the first one up the next morning. He went to the kitchen to start coffee, and glanced out for Sadie. She would take food from him now, and he liked to feed her, but she wasn't on the porch. He walked out and whistled, expecting her to come bounding in from the fringe of woods, or up from the mill, overjoyed to see one of her people, the way she was every morning. He had disturbed the early birds, silenced them; gradually their twitterings resumed, and there was a slight breeze in the treetops, but no other sound. He walked a few feet from the house and whistled once more, a little louder; when she did not appear, he started to walk toward the mill and the water, vaguely concerned that she might have fallen in, even though he had seen her swimming like a fish before.\n\nA lot of trees had grown up between the house and mill, not as thick as the real woods, but enough so that he could not see the mill until he stepped out from the thicket, and then he came to a full stop.\n\n\"Sadie? Sadie!\" He ran to the dog, lying on her side in high grasses. \"Jaysus!\" he whispered hoarsely and knelt on the ground near her, not close enough to touch her. \"Oh, Jaysus,\" he said again, and it sounded like a whimper.\nCHAPTER 3\n\nTHE PROBLEM WAS THAT when she was gone the house changed, Charlie Meiklejohn brooded, sitting at the kitchen table, scowling at the rain that continued without letup. It had rained yesterday, too. Too damn much. It was a comfortable house, if a bit untidy at the moment because he believed in carrying out papers and stuff only when they began to get in the way, and that point hadn't been reached yet. This was Wednesday, and Sunday's paper was still there; soon it would be time. But this little bit of clutter wasn't the real difference, he knew, even though when she was home, clutter seemed to disappear without anyone's doing anything about it. When she was around, he would see something and think vaguely that next time through he would pick it up, but next time through, more often than not, it was gone.\n\nFor one thing, he thought then, pursuing the cause of the difference in the house, it felt too big and too empty, and when she was home it never occurred to him that the house might be too big. Three rooms upstairs, one of them her office with her computer now nicely covered, everything in place, awaiting her return. Their bedroom, and their grown daughter's bedroom that was now their official guest room. His office on the first floor was a jumble of miscellaneous stuff. He kept the door closed, he said to keep the cats out. There was the living room where they spent most of their evenings, a dining room, and a tiny room that they never could figure out what to call. They had put the television in there. Earlier that day he had made a tour of the house wondering that he never had noticed before that there was too much room. And that was when the mystery had started. When she was home the house was exactly the right size, and with her gone, it was ten times too big.\n\nThere were dishes in the sink from last night, and from that morning, and eventually he would rouse himself and cram them all into the dishwasher, but not yet. There was this mystery to solve: What was it that vanished with her gone? How could her presence fill an entire house no matter where she was physically within it? The house was the same. The yard beyond the sliding glass door, with flowers in bloom, the patio awash at the moment, but not changed, everything tangible exactly the same as usual, but everything different anyway.\n\nEven the cats were different. Brutus was not speaking to him at all, disdainful of any attempt at friendliness, accepting food grudgingly, without appreciation, and spending far too much time simply staring at Charlie through slitted eyes that could turn into devil's eyes altogether too easily. That cat was too damn smart for his own good, Charlie brooded. He had come to them in New York, nearly full grown, and street smart; he had looked them over, looked over their apartment, examined the two other cats and found them , acceptable, and moved in. But he had never fully forgiven them for retiring out to the country, not yet, probably never would. And he blamed Charlie; that was obvious in the way he stared at him through devil eyes. Ashcan, the cowardly gray-suited one, kept prowling about, as if Constance might be found under the sofa, or behind the drapes. And Candy, the big-mouth, had gone into her invalid act. She did it very well, dragged herself from room to room, lay without motion for hours at a time, and looked at him with her butterscotch eyes in a way that suggested death was near, and it was his fault.\n\nIt wasn't perfume, he knew, returning to the mystery; she seldom wore any. But she had a fragrance that came from her shampoo, her own physical body, whatever. That was part of it. Not enough. Her soft steps throughout the house, the sound of the sliding door opening and closing, her low murmurs to the cats. Another part of it. Not enough yet. A few days ago Mrs. Grayson had come to do her weekly cleaning and she had talked to the cats, and had even elicited a response from them, and that hadn't been enough to fill the void. So it wasn't just another person, another voice that was lacking.\n\nHe missed her in bed, of course, the warmth of her body next to his, the rhythm of her breathing, a sigh now and then, and that wasn't enough either. After all, he didn't spend all day every day in bed with her. Not any more anyway. A grin softened his face, smoothed out deep lines, as he thought of the various times they had tried to do just that. Stop, he told himself. That made it worse.\n\nCharlie had been a New York City fireman, an inspector, a city detective, altogether more than twenty-five years, and he had seen enough, had done enough to put those lines on his face. It was as if, when Constance was home, a curtain was drawn somewhere in him that screened out all those lines, those many years, and with her gone, the curtain was removed, his past was written clearly on the landscape of his face. Although he had taken an early retirement, it became evident when he was alone that it had not been too early. He was not generally introspective, and never gazed on his own face except during the act of shaving; he had no way of realizing the change he was fretting about was internal, and he continued to try to find a cause for the palpable difference in the surroundings of the house.\n\nWhen the phone finally rang he scooped it up before the first peal ended. \"Do you believe in auras?\" he asked.\n\n\"Sometimes. Charlie, are you drunk?\"\n\n\"Nope. How is she? How's your dad? How are you?\"\n\n\"She's coming along fine. We get to spring her on Friday. Dad's going crazy. And I'm lonely. How are you?\"\n\n\"Swinging from the chandelier. The Mitchums are so concerned they've taken to sending the boys over to keep an eye on me.\"\n\nShe chuckled. Now that the crisis with her mother was over\u2014gall bladder surgery that had gone sour\u2014she could laugh again, but for several days the strain in her voice had made him feel as if he were choking. Another mystery. They chatted; he read her a letter from an associate of hers who wanted permission to reprint one of her papers in psychology; she reported on her father's eccentricities in restaurants\u2014nothing with sauce because you never knew what they were trying to hide\u2014and they both laughed. When he hung up, the house felt emptier than ever.\n\nCandy was regarding him with tragic eyes. \"All right,\" he snarled at her. \"She didn't ask to speak to you. So there.\"\n\nHe started to load the dishwasher then and was still at it when Pete Mitchum tapped on the back door. With him was another boy. Young man, Charlie corrected himself, as he crossed the kitchen to admit them. Pete was about twenty and his friend looked a year or so older. Dressed in dripping ponchos with hoods, rubber boots, they looked like Christopher Robin twice.\n\n\"Hi, Mr. Meiklejohn. This is David Levy, the guy I told you about.\"\n\nCharlie motioned toward a chair under the wide overhang out of the rain, and they deposited their wet gear and then came on inside.\n\nDavid Levy was very tall and angular, with a lot of thick black hair, and blue eyes. \"Thanks for seeing me, sir,\" he said as they shook hands. His grip was surprisingly strong.\n\nSir, Charlie thought with a feeling of resignation. He asked if they wanted a beer, a Coke, coffee, which he was at that very moment going to make, and motioned them toward the table. They both refused his hospitality, and he abandoned the coffee project and joined them.\n\n\"Before I ask you anything,\" David said awkwardly, \"I should find out your consultation fees, I guess.\"\n\nPete looked mortified and Charlie said seriously, \"Advice is cheap, so cheap that it's held to have no value, and no one takes it anyway. For advice, no fees.\"\n\nPete had already told him a long story about how he had met David, through a network of mutual college friends that seemed to stretch out over the country. His friend, he had said, needed some professional advice, would it be okay to bring him over? Since at the time Charlie had been at the Mitchums' dining room table eating chicken and dumplings it would have been churlish to refuse. Now he regarded the young man patiently and waited.\n\n\"What I need to know is how do you go about proving that someone poisoned a dog. I mean, if it's something you really know happened, then what?\"\n\n\"Your dog?\"\n\n\"No sir. -You see, I work at the university experimental farm over at Spender's Ferry, about forty-five minutes from here, I guess. And my friend Rosa's parents won the lottery last year and they were looking for a place in the country and I told them about the old mill over there...\"He took a deep breath.\n\n\"Take it easy,\" Charlie said. \"There's no rush. Rosa's folks won the lottery. She's your girlfriend?\"\n\nDavid blushed. \"That's what my dad asked. It's not like that, sir. I mean, she's female and she's my friend. I think men and women are friends more than they used to be maybe. Anyway, her parents won the lottery. And they decided to move out of the city. They've always lived in the Bronx, I guess. And because I'm doing graduate work at the farm, I knew about the mill. They were looking for a big place. Anyway, they liked it and bought it, and Mrs. Zukal bought this dog, a beautiful golden retriever, because she was afraid out in the country. And someone poisoned the dog. We all know it was poisoned. But now what?\"\n\n\"Did they send you? Mrs. Zukal and her husband? Why didn't they come themselves?\"\n\nDavid shook his head. \"No one sent me. They don't even know I came here. I guess I feel sort of responsible because I found the place and suggested they look at it, and all.\"\n\nCharlie grunted and stood up. \"I'm going to make that coffee. You guys sure you don't want something? Cold drinks are in the fridge. Help yourselves.\"\n\nPete helped himself. \"He just drinks juice or herb teas,\" he said, opening a can of Coke.\n\n\"Sorry,\" Charlie said. When she was home there was always freshly squeezed juice, but he never thought of it. He busied himself with coffee and then returned to the table. \"Okay,\" he said. \"There is a problem with proof. Dogs get into things. You said a mill. Maybe rat poison, something like that. Maybe a neighbor put out something that he ate and died from. Did a vet look at him?\"\n\nDavid shook his head. \"Her,\" he said. \"Her name was Sadie. They found her the next morning, I guess. She was good and dead. No point in the vet. Mr. Zukal buried her.\"\n\n\"Then you don't really know what she died from.\"\n\nDavid shook his head even harder, a stubborn look settling in on his face. \"One day she was playing, chasing a ball, in great shape, the next morning she's dead. We know. And the mill hasn't been used in half a century maybe. There aren't any rats there. Sadie was trained in a special school to be a watchdog. She wouldn't go chasing around the neighborhood. She had to be poisoned on her own property by someone. I just don't know what to do about it. How to go about finding out anything.\"\n\n\"Okay,\" Charlie said briskly. \"You haven't ruled out accident, you know. She could have caught a diseased animal, or a poisoned animal. A squirrel or a rabbit, even a rat. They're everywhere even if you don't see them. And no doubt dogs are susceptible to a host of things you also don't know anything about. So the first thing you'd have to do is find out what killed her. That means exhumation. You would take her remains to a veterinary pathologist and have tests run. Then you have something to start with. Right now all you have is suspicion. Is there any one person you suspect?\"\n\nDavid had turned very pale at the word exhumation. Charlie pretended not to notice.\n\n\"I can't think of anyone, or a reason,\" David said faintly.\n\n\"Right. So, after you learn the actual cause of death, if it was poison, you would call the county animal control office and ask if other poisonings have been reported. If there have been, they might take over at that point. If not, you're still on your own. But the first step, in any event, is to find out the cause of death.\"\n\n\"If I find out that she was really poisoned, will the sheriff investigate? Or the animal control office? Will anyone?\"\n\nCharlie felt a stab of pity for this earnest young man burdened with his load of guilt and responsibility. \"Maybe,\" he said slowly. \"But, David, I won't kid you, unless there have been other cases, and unless livestock gets involved, probably not much. Someone might ask the Zukals a few questions. But it's almost impossible to prove something like this unless you actually catch the person in the act, and even then, chances are good they'll get off with a slap on the wrist. And the animal pathologist will charge, you know. It will be expensive, and you'll run up against indifference. Just the way it goes.\"\n\n\"That's really shitty!\" Pete protested.\n\n\"Yep.\"\n\nDavid stood up then and held out his hand. \"Thanks, Mr. Meiklejohn. I appreciate your talking to me about this.\"\n\nCharlie wanted to pat him on the head, but he knew that would not be welcome, and besides, he would have had to get on a chair. Instead, he shook hands with David, and then stood in the doorway watching the boys walk across the backyard in the rain. So young they probably would splash in every puddle they came to, getting a sampling of how truly shitty the world could be. They would climb the fence, walk through the pasture that separated the Mitchum house from his house, and that would be that, he thought, almost wishing he had told Pete no, not to bring his young friend around. The advice he had given had left a bad taste in his mouth.\n\nSaturday morning Lois was surveying the greenhouse that held her cloned trees. Some of them were outgrowing the bench already and would have to be moved onto a lower one, and that was going to be a mess, she knew. It would take several hours to do them all, and meanwhile she had the metabolism studies to work on, and the measurements... She glanced at her watch again. At ten-thirty she walked outside and started for David's apartment, but then she saw Tom Hopewell emerging from his corn rows, and she slowed down.\n\nHe waved at her and she waved back, glanced toward the apartments, and waved again.\n\n\"Are you busy right now?\" she called.\n\n\"Sure. But I can stop. Why?\"\n\nSometimes she felt a bit of confusion with Tom, but she had no idea of what caused it. He was a year or two younger than she was, and he dressed like a ragged child, and like a child he could grin so disarmingly that people often forgot that he was a truly fine botanist with a Ph.D. There was something unfathomable about him, about the way he looked at her, the way he could stop what he was doing just like that. He looked and acted like a child playing at being a scientist, his hair too long, knees out of his jeans, dirty hands.\n\n\"It's probably nothing,\" she said, as he drew near. \"David's so late, and I need him. I was going to go wake him up, but it might embarrass him if I walk in. Would you mind?\"\n\n\"Oh, it might embarrass him indeed,\" he said gravely, but his eyes were dancing with amusement, and she didn't know what he found funny.\n\nThey walked on to the apartments together, and she knocked on the door. There was no response from inside. Tom entered and continued on through to the bedroom; Lois stood in the doorway uncomfortably, keeping her gaze averted.\n\n\"Oh my God!\" Tom said in a voice that was low but carried to where she was standing. \"Lois, there's been an accident or something.\"\n\nShe ran into the apartment, into the bedroom where he was standing ashen-faced, staring at the bed, at the figure that had covers drawn up to his chin. She reached past Tom and touched David's cheek, then jerked her hand back. She heard a moaning sound and did not realize she was making it until she found herself standing at the sink, hanging onto it as if to keep from falling, and behind her she could hear Tom on the telephone.\n\nThe day before, Friday, Charlie had gone to the city to give a deposition in an arson case he had worked on, and he got talked into a poker game that lasted too long, but made him feel filthy rich afterward. And because on Monday Constance was due, he stayed in town to meet her at Kennedy, and then they celebrated by staying the rest of the week and seeing two Broadway shows. By the time they returned home, the weather had changed dramatically, into summertime, and the house was filled with her presence and once more was exactly the right size.\n\nHe had forgotten David Levy and his problem and never would have thought of him again, except in a passing moment, if Constance had not come looking for him in the basement a few days later.\n\n\"Charlie, there's a man to see you. A Mr. Levy. He's pretty upset.\"\n\n\"David?\" Charlie put down the spinning reel he was trying to fix. It kept snarling the line, and for the life of him, he thought moodily, he couldn't see why. He wiped his hands and followed Constance back upstairs. It was a pleasure to watch her move. She was long and lithe and moved like a dancer. When they emerged from the dimness of the stairs into the sunlight in the kitchen, he got another jolt of pleasure in seeing her hair gleam when she moved through the brilliant light. Only God knew how much of that color was the original color, how much was because the original color had started to turn into spun platinum. His own black crinkly hair was unmistakably turning gray, but she was keeping her secrets.\n\nShe preceded him into the living room where a tall, gangly man stood. \"This is Mr. Levy,\" Constance said.\n\nCharlie felt bewilderment. Not David. He started to extend his hand, but the other man stepped backward, swaying. He looked very ill, cadaverous almost, with sunken eyes and a pallor that suggested confinement, hospital, sickbed. Deathbed.\n\n\"My boy came to you for help,\" Levy said heavily. \"He did what you told him and they killed him. I just wanted to look at you. See you. What kind of a man are you to turn him away like that? Send a boy off to get killed.\"\nCHAPTER 4\n\nCHARLIE WATCHED IN ICY shock while Constance took Mr. Levy by the arm and maneuvered him to the kitchen table with a cup of coffee before him. She said little as she managed him, and what she said made no difference; Levy allowed himself to be led and now he picked up the cup and sipped from it.\n\n\"Tell us what happened,\" Constance said gently, when he put the cup down again.\n\nIn his mind's eye Charlie was seeing the lanky boy unfold himself from that same chair; he remembered his own impulse to pat him on the head, remembered watching the two boys trudge out into the rain toward the Mitchum house.\n\n\"They called me,\" Levy said. \"They said there's been an accident. They said they had to do an autopsy. They released him to me and I took him home and buried him. Today they said the investigation is over, accidental death. They said he used drugs and overdosed.\"\n\nHe looked into Charlie's face across the table from him and shook his head. \"My boy didn't do drugs,\" he said with finality.\n\n\"No, he didn't,\" Charlie said. \"Mr. Levy, do you know why he came to see me? Did he tell you?\"\n\nConstance would have taken it slower, he knew; she would have gentled Levy some more, got him to relax some more, but Constance hadn't met that string bean of a boy. Maybe she didn't realize that Levy was a dying man. Constance didn't have the icy rage that filled Charlie at that moment.\n\n\"He told me in a letter,\" Levy said after a hesitation. He pushed his cup back.\n\n\"Hold it a minute, Mr. Levy,\" Charlie said, his voice suddenly harsh. \"Just hold it a minute. He came here for advice and I gave it to him. Did he tell you if he followed up on it, what he learned?\"\n\nLevy faced Constance, directed his words at her. \"He couldn't get anyone to help him, the sheriff, the animal agency, no one. Your husband. I told the sheriff about the dog, what Davy was doing, and he said Mr. Meiklejohn was high-priced and wouldn't get involved with a dead dog, that no one in his right mind would get involved with it. He said Davy took the drugs himself and that's that. He didn't even read the letters. Pretended to, that's all. He already had his mind fixed on what he thought he knew.\"\n\nConstance was totally bewildered by all this information. Charlie had not told her about David Levy or a dead dog. And this poor man, she thought, had not fully accepted the death of his son, no matter if there had been a funeral; he still had hope behind the desperation in his eyes. And he was so very ill. The skin was drawn tight over the bones of his face; his eyes were bloodshot, sunken, the rims red. There was a tremor in his hands.\n\n\"Mr. Levy,\" she said, \"did your son find out who killed the dog? Did he tell you?\"\n\nHe sighed deeply and shook his head. \"I think he knew, but he didn't say who it was in the letter. I'd better be going. It's a long drive.\"\n\nHis pain was so wrenching that Constance wanted to hold him and soothe him, and she knew that would be absolutely wrong. Not now. \"Mr. Levy,\" she said, and Charlie heard her professional voice, the crisp manner and tone of a trained psychologist, \"you made an accusation about Charlie that we have to air. When did your son die?\"\n\n\"Saturday, the twentieth, in the morning. Today they said the investigation was over. I came to get the stuff they were holding and to show them his letters. No use.\"\n\nCharlie's eyes were narrowed in thought. He shook his head. \"He couldn't have followed my advice already. Listen, let me tell you what we talked about the day he was here. That was on Wednesday, late afternoon. He said the Zukals' dog had been poisoned and I said he would need to exhume it and have tissue examination tests to find the cause of death. There's no way on earth he could have done that and got the results in two or three days, by Saturday.\"\n\n\"Just like the sheriff,\" Levy said with weary bitterness. \"You're all alike.\" He started to get up.\n\n\"Just sit still, will you?\" Charlie snapped at him. \"Maybe there's a connection and maybe not. I sure as hell don't know at this point and apparently neither do you. But, Mr. Levy, neither do I believe your son took drugs of any kind. Now let's talk about it.\"\n\n\"Why?\" Levy asked. \"I can't afford to pay you anything.\"\n\n\"For crying out loud! Do you have a buck? Just haul it out and put it on the table.\"\n\nLevy looked from him to Constance, and then withdrew a shabby wallet and fished out a dollar bill.\n\n\"You're paid in full. Now, just answer a few questions, will you? Honey, is there more coffee?\"\n\nTwo hours later they walked to his car with him, a twelve-year-old Ford. Home for him was a veterans' hospital in New Jersey, a long drive.\n\nAt the car he hesitated briefly, then held out his hand to Charlie. \"Let me know what you find out,\" he said, and got in and started the engine; he looked almost surprised when it turned over. \"All he wanted to do was help feed the world,\" he said softly, looking straight ahead, his cavernous face twisted. He shook himself and engaged the gears and left without speaking to either of them again.\n\n\"We'd better find out something pretty damn fast or he won't be around to hear it,\" Charlie said, as the old car backed out to the road. \"Let's go read those letters.\"\n\nIn the kitchen once more, Constance gazed at the dollar bill that had been pushed aside to make room for the letters that Levy had produced. He was dying, he had said candidly; he and his son had pretended that death could be put off until David was through with school. And now he could go ahead and die, Constance thought, but she knew that he would try to hold on until he knew the truth.\n\nShe began to arrange the letters chronologically. David had been a faithful correspondent, had written at least once a week; there were a lot of letters to go through. Levy had also left a box of personal effects that the sheriff had released to him\u2014snapshots, a few magazines, a calendar, catalogs, a grocery list, bills, a checkbook, and several bank statements. It was a pitiful assortment.\n\n\"Look at this,\" Charlie said a few minutes later in a grumpy voice. \"Three pages about the party at Wollander's place, a blow-by-blow account of a cookout! And half a page with one line about the goddamn dog, the last letter he wrote.\"\n\nConstance nodded.\n\nCharlie read it in disgust: Remember Sadie, the dog that got killed, I think I'm onto something about that. More next time.\n\nHe tossed the letter down on top of the others and stood up. \"Want to go for a ride in the country?\"\n\n\"Where to?\" she asked. \"Spender's Ferry?\"\n\nHe grinned at her. \"Why ask anything ever? Just say the answers. Works just the same.\"\n\n\"Charlie.\" She began to gather up the letters. Watching her hands, she said, \"If a boy came through that door right now and asked what to do about a possible dog murder, you'd tell him just about what you told David Levy, and you know it.\"\n\n\"Yeah,\" he said. \"I know that. Somehow it doesn't seem to help much. And, like the sheriff said, maybe there's no connection.\"\n\n\"The sheriff must be Greg Dolman,\" Constance said a few minutes later in the Volvo, drawing up a memory of the sheriff. Middle-aged, paunchy, amiable, good politician, and he did not want trouble. That was how he put it, he didn't want trouble, kept a clean county. At one time, it was said, he also had run a speed trap that had netted someone a lot of money that never seemed to show up on the books. The next election his vote had been bigger than ever; his constituents understood the value of free enterprise and approved. But someone had put a stop to that, without publicity, without scandal; it just stopped.\n\nHaving sorted him out in her mind, Constance began to regard the passing scenery with great pleasure. God must have been having a very fine day when he invented June, she thought. Today the sun was warm, without the cruel heat that could come later in the season; the leaves were at their brightest, most tender stage, flowers abounded, and even the traffic seemed manageable, civilized and polite.\n\nThey drove through Spender's Ferry slowly. Like most New York villages, it had been discovered by tourists, and it played up its picturesque appearance. A grocery store with a boardwalk in front, and a mammoth parking lot behind it; three antique shops, one with a sign that said: CUSTOM-MADE ANTIQUES. A gift shop. A church with a white steeple and a golden bell. Red-brick elementary school, a restaurant that probably was very good, housed in a beautifully maintained Victorian three-story house\u2014it had a sign that said only HAZELTINE'S.\n\n\"Hokey,\" Charlie said.\n\n\"But nice.\"\n\n\"You betcha. Pays to look nice.\"\n\nShe sighed as the fleeting romanticism disappeared. Beyond the village, working farms stretched out on the right, fields and fields of satiny corn six inches high, and on the left the experimental university farm began. It looked very little different from all the other farms they had passed. Then Charlie turned into the driveway of the Zukals' house.\n\n\"Now, this is nice,\" Charlie murmured when he drew up to the house and stopped. He opened his door and got out as Constance got out on her side, and they both stood very still.\n\nA woman's raucous voice was screeching; a man's equally raucous voice was yelling at the same time. Just then Sylvie and Al appeared from the trees behind the house. She had a green kerchief tied on her head; wisps of red hair had escaped and clung to her sweaty brow. She was dressed in baggy pants and a man's shirt. He was in jeans that rode too low on his ample belly, and sneakers with shoelaces flying. He had a fringe of pale hair and a bright red scalp.\n\n\"You want they should nail you to the floor? Why doncher just leave them to their work?\"\n\n\"You hear anyone saying I'm in the way? You hear that? I don't hear nothing like\u2014\"\n\nThey broke it off when they saw Charlie and Constance, and now the sound of hammering could be heard, and a loud radio.\n\n\"You another lousy inspector? Go inspect already! Watcha waiting for? Go inspect until your eyes fall out!\"\n\n\"Mr. Zukal? Mrs. Zukal?\"\n\n\"Yeah, who you think, Queen of Sheba or something? It's over there!\" Sylvie began to stamp toward the house, muttering over her shoulder to Al. \"And stay out of their way! They don't need help!\"\n\nMy God, Charlie realized suddenly, with amazement, he had missed this! He had known people like this all his life and he had missed them. Constance had come to his side; he took her hand and walked forward.\n\n\"Hold it, Mrs. Zukal. I want to talk to you about your dog.\"\n\nWell, he thought, satisfied, as they both froze, he had their attention. The stillness was short-lived, and one of them was demanding to know what for, and the other who did he think he was, and either or both, what did he know about their dog, and other things that got lost in the tumult. He held his hand up, shaking his head.\n\n\"Let's sit down and talk,\" he said firmly. \"I'm Charlie Meiklejohn, and this is my wife Constance, and we are not inspectors of any sort. We are private investigators. Can we talk?\"\n\nSylvie looked them both up and down, and then nodded toward the house. \"But don't you go to the living room,\" she snapped at Al. \"All that sawdust!\"\n\nThey went inside the house and stopped at the kitchen. \"This is fine,\" Charlie said; he pulled out a chair for Constance, another for himself, and sat down at the kitchen table.\n\nThe Zukals sat down after hesitating briefly, and Al demanded, \"Now, you. Who sent you and what you want here? And what do you know about our dog?\"\n\n\"Not enough by half,\" Charlie said, and went on to describe David's visit again. \"Today his father hired us to look into David's death,\" he finished. \"His father doesn't think he ever took drugs in his life.\"\n\n\"See!\" Sylvie cried. \"See! I told you! That David, he wouldn't even drink beer!\"\n\nBut Al was scowling fiercely at Charlie. \"You sure as hell didn't do David no good.\"\n\n\"I know that,\" Charlie said quietly. \"That's why I'm here now. Will you answer a few questions?\"\n\nSylvie yanked the kerchief from her head and ran her fingers through her red hair until it looked like the hair of the bride of Frankenstein. \"You axe, then we see,\" she said.\n\n\"Yeah, just axe. No promises,\" Al said.\n\n\"Right. First. Did David actually dig up the dog?\"\n\n\"Nope. Didn't even know he was thinking about it. Never said a word.\"\n\nSylvie contradicted him instantly. \"He nearly cried when I told him. He would go stand by the grave and just look. You knew what he was thinking, all right. It hurt him. He was such a good boy. Rosa should have seen it, that he was such a good boy.\"\n\nCharlie asked where the dog had come from, and was rewarded with an avalanche of information. He let them both talk without interruption.\n\n\"Brenda Ryan's kennels over by Albany. Me and Flora, she's our girl, me and Flora and the kids took the wagon and went over. I thought it might need lots of room in the car, you know, chains and stuff like that, but she was like a puppy, after Brenda introduced us, I mean. Introduced us like she was a person. Sadie, this is Sylvie, and this is Flora, and this is Mike and this is Janie. Just like that, and the dog smelled us, and then began to wag her whole rear end.\"\n\n\"And when they got her home, we had to take her all around the place and let her pee now and then, and smell the trees,\" Al chimed in. \"She had to do that, you know, to find out where she lived. And if we wanted her to be nice to people we had to do like Brenda done and go Sadie this is David, or Sadie this is Lois. Don't do no good to put it any other way. I mean, you can't just go meet the gang or nothing.\"\n\n\"Brenda Ryan weighs three hundred and forty pounds,\" Sylvie said suddenly, in awe.\n\n\"Jaysus Crackers! What difference does that make? You told me that a hundred times already. Who cares?\"\n\nSylvie turned to Constance. \"I just axed her right out. That's the way. People don't care if you just go ahead and axe instead of beating the bush like. She said lady dogs are even better than boy dogs to guard things. They take their responsibility seriouser or something. I thought it would be one of those great big ugly black dogs like on television all the time, you know, all big teeth and hair like painted on. But Sadie was real pretty, and she liked to play\u2014\"\n\n\"Sylvie, who cares if she was pretty! Let the man axe his questions, will you?\"\n\nConstance glanced at Charlie who looked absolutely at ease and relaxed, as well he might, she thought. This was going to take the rest of the day.\n\nSlowly he drew the story out of them. Sylvie had brought the dog home and for the next few days they had felt secure, well guarded, and then the dog died. They had it one week.\n\n\"But dogs don't just die any more than people do,\" Charlie murmured at that point. \"They die of something.\"\n\nAl and Sylvie exchanged uneasy glances.\n\n\"Who did you introduce the dog to during those days?\"\n\n\"David, of course. He was in and out all the time, just like he was at home like. And Lois. Mrs. Wollander. She cuts through the property walking to work and home.\"\n\n\"So on Sunday morning you found the dog dead?\"\n\n\"Yeah. Stiff already. Puked a lot first, then kicked.\"\n\n\"Were you here Saturday night?\"\n\n\"Sure. And Bobby and Flora and the kids. They come on weekends, but as soon as school's out, they're moving in with us. Anyways, they was here. Kids played with Sadie right up to bedtime. Crazy about her. Real smart, she was. They'd hide things and she'd go find them. Real smart.\"\n\n\"Where did Sadie sleep?\"\n\n\"On the porch most nights. But she'd be up prowling around all hours. Investigating noises, I guess.\"\n\n\"And did she eat outside, too?\"\n\n\"Yeah. On the porch...\" Al's voice faded out and Sylvie said, \"But we found her dish way over by the mill. Remember, Al?\"\n\n\"Yeah, just thinking about that. But the kids was hiding it on her. They must have left it there.\" The certainty that had been in his voice was gone, however.\n\nCharlie was feeling more and more unhappy as they talked, and when Al asked if he wanted a beer he nodded gratefully. Al brought one for Constance, too, and, bless her, Charlie thought with appreciation, she pretended she wanted it.\n\n\"We'll have to exhume the dog's remains,\" Charlie said a few minutes later. \"Can you tell me anything about David? He didn't tell you he was coming to see me, I know. I asked. But something made him decide to poke around himself. Any idea what it was?\"\n\nAl began to pick at the beer can as if the painted-on label might be peeled off, and Sylvie began to pat her hair down, suddenly aware that it was unkempt.\n\nCharlie waited. If there was anything people like the Zukals could not stand, he knew very well, it was silence.\n\n\"It was the insurance,\" Sylvie muttered. \"Set him right off.\"\n\n\"I'll be damned if it was. It was them letters. You had to go show him, didn't you?\"\n\nFrom then on Charlie listened. The Zukals had met David when he and their daughter had attended a special science-oriented high school in the Bronx. The kids had become good friends and David had spent a lot of time at their apartment. His father was ill even then, in and out of the hospital, and David was lonely. They had treated him like a son, and maybe that was the problem with him and Rosa, Sylvie added thoughtfully. Anyway, they didn't try to hide things from him. When they were broke he knew it, and when they struck gold, he knew that, and when the letters came, it had seemed natural for Sylvie to show him. Gently Charlie drew her back to the letters.\n\n\"Yeah,\" Sylvie said heavily. \"They don't want our kind here, you know. Don't much blame them, but here we are. And here we stay, like it or not. Anyways, Monday, after Sadie died a letter came, nasty like. Go back where you belong. That's all. Looked like a kid must have wrote it. And the next Monday another one came. What does it take? A bullet? Would that penetrate your thick skulls? David seen that one and I told him about the first one.\"\n\n\"And two days later he came to see me,\" Charlie said. \"Did you keep the letters?\" They both shook their heads. \"Have there been any more?\" They shook their heads again. \"Okay. What about insurance?\"\n\n\"Nothing to it,\" Al said. \"Sylvie got spooked, is all. See, Stanley, Wollander's son-in-law, he tells me I should get more insurance just as soon as the next check comes, seventh of July. He's talking a big policy, something no load, or front load, or something like that. I can borrow on it even, and Sylvie tells David, and he goes maybe it's not so safe out here for us. But, hell, wouldn't do nobody around here no good if I kicked.\"\n\n\"Someone might think Sylvie wouldn't hang around out here if you were gone,\" Charlie mused.\n\n\"Well, someone's got another think coming,\" Sylvie screeched.\n\nAnd now Al told in great detail how his two sons-in-law were geniuses. \"See, we get the big check and first thing Sylvie tops it right off, ten percent by Jaysus, to a school for girls.\"\n\n\"Well, there's boys' camps everywhere, homes for troubling boys, special schools, and girls get the short stick every time. Sure I did, and next time another ten percent.\"\n\n\"Yeah, yeah. We been all over that. Okay by me. See, we got these four girls\u2014\"\n\n\"And whose fault is that, Mister? You got the special doohickey that makes a boy baby, not me. Rosa told me all about that,\" she explained to Constance. \"She's a scientist. Or will be some day. Real smart, Rosa.\"\n\n\"Jaysus! Will you let me finish! Anyways, we was going to spread the dough around, you know, let the girls and their husbands have some, and they all got together and talked it over, and they go, not the way to do it. If we really want to help, we'd help Bobby and Harmon start a business, this year and next, and then help Les and Nola get a bait shop they been wanting, and like that. Seeds, they call it. So Bobby and Harmon\u2014Bobby can make things out of wood, really good stuff, chairs and tables, stuff like that, and Harmon can do business, sell stuff. They make a good pair. They're going to have the mill for a shop. Handmade furniture. Won't make a fortune right off, but it'll grow. Bobby comes out and looks it over and he goes, you know, Pop, that mill can make electricity, all we'll ever need, and more, and the electric company has to buy all the leftovers. We'll break even right off the bat, and that means profits faster than anyone thought at first.\" Al looked puzzled, then shrugged. \"You can have leftover electricity? I don't know about that. But...\"\n\nNow Sylvie chimed in and her voice overrode his. \"That's what we mean, you get it? We bought this place, and they can send inspectors and zoners and whatnot, but we're here. And we're here for good.\"\n\nCharlie asked a few more questions and got more answer than he wanted each time. They had not seen David after his visit to Charlie's house. What with the kids all over the place Saturday and Lois working David's butt off over at the farm, Al said, it hadn't struck them as unusual not to see him. And she's working her butt off, too, remember, Sylvie put in, and they went on from there.\n\nThey hadn't even known anything was wrong until Lois had come to tell them. When David didn't show up on time, even though it was Saturday, she had gone to his place looking for him, and she had found him dead. She was real shook up by it, Al said soberly. It hurt her real bad.\n\n\"So she came to tell you?\"\n\n\"Yeah. She thought it wasn't right for us to find out in the village, or something like that.\"\n\n\"Did you believe he overdosed himself?\"\n\nAl and Sylvie looked at each other, and she said in a low voice, so low it was hard to catch her words. \"You know, we're from the Bronx. We wanted out, want to get our kids and our grandkids out of there. Sure, we know it's not everyone shooting up, snorting, messing themselves up, but who's to say who is and who ain't anymore? I didn't believe it about David but I don't know. We seen too much. Been fooled too many times. You know what I mean?\"\n\nCharlie nodded. He knew.\n\nHe asked Al to show him where they had found the food dish, and where the dog was buried, and for permission to dig it up for tests. And when Al asked him with incredulity if he was going to dig it up himself, Charlie shuddered. He said he would send someone. They all went out to walk over the grounds, down to the bridge, around the mill, to the mound that was Sadie's grave, to the spot where Al had found her dead. Al and Sylvie talked over the noise of the carpenters in the mill with their radio turned on at maximum volume. Lois was going to help choose which trees had to come out. Al was going to put in a little beach at this end of the lake. Here was the place in the fence that Lois used every day to go to work. They planned to put in a real gate and fence off the bridge, because of the kids... .\n\nCharlie paused at the break in the fence and looked back at the mill, several hundred yards away; the loud music was dimmed by distance but still too clearly audible. A path led from this point, skirted the lake side of the mill and continued to the swinging bridge. No comparable path led to the house through the tangle that many years of neglect had produced. Beyond the break in the fence the path was even more sharply defined, three stone steps down, then a hard-packed trail.\n\nHe glanced at Constance and nodded toward the path. \"Let's see a bit more of it.\"\n\n\"It's like a sidewalk,\" Al said behind them as they started down.\n\nAnd it was, almost. It led them through a few feet of brambles and low sumac and suddenly they were in a grove. The ground still sloped, but more gently here, hardly noticeable as they walked slowly. On the left the trees were all about twenty-five feet tall, straight up to a canopy, but on the right the trees had been allowed to keep their lower branches and they made a thicket that would be all but impenetrable. They had been planted precisely, meticulously, and looked totally artificial in almost military ranks. Every tree was tagged.\n\nConstance looked at the experimental trees with a sense of unease. No grasses, no vines, nothing grew under them, but more, the trees themselves looked alien, foreign to her. Not like oaks or maples or conifers, these had smooth, pale tan trunks, leaves too shiny and thick, and many small round nuts no bigger yet than peas, and they also looked alien. Orchards were always arranged carefully, she reminded herself; she had planted their apple trees according to the book, so many feet from one to another, but every one was different from the next, gnarly, unique, with the differences more apparent year by year. And these were all the same.\n\nThey came to a different section, smaller trees, obviously younger, identical... . Abruptly Charlie stopped, shaking his head. The path continued but it wound here and there and the end was not in sight. Neither was the lake. Stippled sunlight, rows and rows of identical trees, and finally silence; they had gone far enough to lose the rock music.\n\n\"How far to the farm?\" he asked Al over his shoulder.\n\n\"Three, four hundred more feet of trees and the buildings start. It's all like this, all the way. Up at the fence you coulda turned toward the lake, and that's different, right down to the water, but let me tell you, I sure don't like this way.\"\n\n\"Yeah. See your point,\" Charlie said, and turned back. \"Is this the way David came, through the trees?\"\n\nAl nodded. \"Closest this way. Lois uses the path every day. Lots of people seem to, why it's all packed down.\"\n\nThey returned to the mill property and the noise of hammering and music. They walked to the bridge and surveyed it and the dam.\n\n\"That dog, Sadie? She wouldn't set foot on the bridge,\" Al said. \"Can't blame her none. Bridges ain't supposed to move when you do, way I see it.\"\n\nSilently Charlie agreed. Finally, he and Constance got in their car to leave and he felt that he had missed whatever it was he had been looking for.\n\n\"You think the dog was poisoned, don't you? Deliberately,\" Constance said as he backed and turned around, headed out to the road. \"But, Charlie, a dog trained the way that one apparently was wouldn't take food from a stranger, would it? And how could anyone have gone in that far? The dish was at least three hundred yards from the edge of the property.\"\n\nHe grunted. \"On my list,\" he said. \"Both items. Tomorrow we ask Brenda Ryan a thing or two. And not how much she weighs.\"\n\nAfter a moment, Constance murmured, \"Lois Wharton Wollander is the only one outside the family who was introduced to the dog, apparently. And David, of course. And David worked for her.\"\n\n\"It'll be a busy day tomorrow,\" Charlie said, slowing down at Spender's Ferry. \"Brenda Ryan, Lois W. W., Sheriff Greg Dolman. A busy day. Let's check out Hazeltine's.\" He drew up before the Victorian mansion that housed the restaurant and turned off the engine. \"You know who Wollander is?\"\n\n\"Vaguely familiar name. Tell me.\"\n\n\"What Al would call a crowner. A kingmaker. A man who knows which closets hold skeletons, and how to unlock the doors to get at them, and where every body is buried.\"\n\n\"Oh, dear,\" Constance said. \"Al said all those inspectors, zoning people.\"\n\n\"Yep. Someone must have sicced them onto him.\"\nCHAPTER 5\n\nTHAT NIGHT AFTER DINNER in the Wollander house, Warren Wollander said, \"Jill, would you mind if we all go to the study for a little talk?\"\n\nLois felt her muscles tighten at his tone of voice. Carefully she placed her napkin on the table and just as carefully she did not look across the table at Jill.\n\n\"Let's talk here,\" Jill said. \"What now? I feel like I'm twelve years old again.\"\n\n\"You're not a child,\" Warren said evenly, \"and that's part of the problem. People are talking in the village about your... involvement with Sebastian and his group.\"\n\n\"What I do is my own business,\" Jill said coldly. \"I don't give a damn what they're saying in the village. And you never used to.\"\n\n\"Now I do. Mrs. Carlysle tells me you've ordered a special diet for yourself and tonight you barely touched your dinner. Is that at Sebastian's suggestion?\"\n\nJill flung down her napkin and started to jump up, then decided not to, looking intently at her father. She took a deep breath. \"Yes. He understands nutrition thoroughly. I am to eat meat sparingly, drink a lot of milk and juice, no alcohol in any form, make sure I have plenty of fresh vegetables and fruits, many of them raw. I am to watch my weight and not gain more than sixteen pounds, and get plenty of exercise.\"\n\n\"My God,\" Lois said softly. \"Jill! That's wonderful!\"\n\nWarren looked confused for a moment, then a wide smile spread over his face and he blinked rapidly. He jumped to his feet and took Jill's hands, drew her from her chair, and wrapped his arms around her, held her hard against him. After a moment he pulled back and examined her face; his eyes were bright. \"Are you sure? I thought you couldn't... Your mother said\u2014\"\n\n\"She was wrong,\" Jill said. \"That's what the meditation has been all about, why I've practiced and practiced. And it worked. I had to learn to relax, to center myself, become receptive. It worked! That's what Sebastian was trying to explain out here: If you can master yourself, you can do anything.\"\n\n\"Let them talk,\" her father said then, and laughed and hugged her fiercely again. \"Let the bastards talk all they want. When? How are you? Have you seen a doctor yet? Will you stay here until it's time, afterward if you want?\" He looked at her closely again. \"Stanley doesn't know yet, does he?\"\n\nJill turned helplessly to Lois and shrugged. \"I wasn't going to tell anyone for a couple more weeks. It's too soon. But I know. Next weekend, when Stanley comes up, I'll tell him, now that it's out anyway. But let me be the one. Promise. Not a word yet.\"\n\nLois nodded, smiling. And Warren said, \"We should have a toast in champagne. A tiny preliminary celebration.\"\n\nJill shook her head. \"No alcohol, remember? Don't tempt me. You two go ahead. I want to walk a little bit and then read, and then sleep. I should get as much sleep as I can, too.\"\n\nShe kissed Warren's cheek, nodded to Lois, and started to leave the dining room. At the door she paused, her hand on the knob, her head bowed. She looked like a too-thin adolescent in her long pale skirt and tank top. Her arms were nearly fleshless, her hands sharply boned. She turned to look at them and said softly, \"Dad, I'm really sorry. The things I said when Mother died. I'm sorry. I was a little bit crazy. And, Lois, I'm sorry. You've been wonderful to me, and I've been a bitch. I know. I know. I'm sorry. Can we all start over, now, tonight?\"\n\nWarren started to move toward her; she waved him back again and smiled slightly. \"No emotional upsets, either. Let's just pretend this is day one and go on from here. Okay?\" She gazed at him steadily, then at Lois, and nodded. \"Thanks.\" Then she opened the door and left.\n\nLois tried to convince herself that the glint in Jill's eyes had been happiness, not triumph.\n\n\"Well, let's take a walk, too,\" Warren said then to Lois. He was beaming, his eyes glistening.\n\nThey walked slowly down to the lake, and then along the tiny beach to where there was an immense log. All around them the night sounds were like music not quite in harmony, stopping and starting, speeding and slowing. Cicadas, crickets, tree frogs, the whir of wings\u2014a night hawk or an owl. Overhead, the sky was clear, midnight dark with myriad pinpricks of stars. The air was warm, but cooling fast. It rarely stayed warm throughout the night, not until late July, or even August.\n\nThey sat on the log, his arm around her, her cheek on his shoulder. \"I'm so happy for her,\" Lois said. \"And for you.\"\n\nHe squeezed her slightly. \"She surprised the hell out of me. The last thing I expected her to say. She made me forget altogether the other matter I wanted to bring up with both of you.\"\n\nLois started to pull away a bit, but he held her firmly.\n\n\"I've been in the spotlight for more years than you've lived,\" he said after a moment. \"I've made enemies. And some damn good friends. But there are enemies. And some of them are nothing but slime.\"\n\nLois shivered, whether at the falling temperature or the harshness of his voice she could not have said. The chill entered her and stayed.\n\n\"I want you to promise me something,\" he went on.\n\n\"You know I will.\"\n\n\"Maybe you will and maybe you won't. We'll see. Promise me that if you get any poison-pen letters, you'll bring them to me. Will you promise that?\"\n\nShe jerked away from him with a suddenness he could not counter. She leaped up and backed away from him. \"You've had letters like that? About me? Someone's found out, hasn't he? I told you it would happen!\"\n\nHe stood up and caught her hands and held them tightly. \"Not about that. That's what I'm afraid of, that someone will send such a letter to you. I've had filth like this before in the past. Slime can't face a man directly, they act like this, underhanded, mean, nasty, hurting. Today I got a letter about Jill and Sebastian. Last week one about you. Both lies. Meant to destroy, to kill something fine and wonderful.\"\n\n\"About me? What about me?\" Her mouth had gone dry, and she was shivering uncontrollably.\n\n\"It doesn't matter what it said. Believe me, I know how this kind strikes out. They don't want to hurt you, just me. And they know the way to do it is through the two people on earth that I love. They always know how. If he doesn't see a reaction from these first two, I'm afraid he might send you a letter. If he does, will you just hand it to me?\" He pulled her to him and held her, and she stared, unresponsive, dry-eyed at the lake. Starlight reflected from its surface in an unquiet shimmer that appeared and vanished with the motion of a breeze. Dead lake, she thought, and felt as dead as she knew the waters were.\n\n\"It must be Coughlin,\" she said finally. \"He must have found out what you're doing.\"\n\n\"Shh. No jumping to conclusions. It could be a number of people for a number of reasons. Let's leave it at that for now.\"\n\nShe might not have heard. \"If it's Coughlin, he won't stop. He'll dig and dig until our grandparents' private lives are on the line.\"\n\nWarren sighed. \"Darling, this has no payoff. You can speculate all night, all week, for nothing. You'll just lose sleep. Either this slime will keep it up or he won't. Either he'll make demands or he won't. Either he'll send you a letter or he won't. I just wanted to prepare you, in case. Now we leave it alone.\" He released her and stretched. \"Speaking about grandparents, how about Jill? My God, I'll be a grandfather! And you a grandmother!\"\n\nThey began to stroll and he talked about his strategy for keeping his daughter here, at home, during the pregnancy. \"Can't pressure her,\" he admitted, thinking out loud. \"Through Stanley. He'll see that it's for her own good, fresh air, rest, no stress of the city, no muggers. That's the ticket. A little chat with Stanley. No pressure.\"\n\nShe said the appropriate things at the appropriate times, but her thoughts were on Bill Coughlin, who had told Warren to keep out of his way because he had every intention of becoming governor first, and then president when the time was ripe. Over my dead body, Warren had murmured, and Coughlin had nodded. If it takes that, he had said. Two weeks later Warren had started writing his memoirs. But no one could have learned what he was doing, she told herself firmly. No one knew yet except her, and his secretary, and, no doubt, Jill. His secretary, Carla Mercer, had been with Warren for over twenty years; she knew more of his secrets than he could remember, he sometimes said. When Warren began to chuckle, she realized she had heard nothing for a long time, and she realized that her thoughts and worries were spinning in all directions and had become too chaotic to track.\n\nLater that night, unable to sleep for thinking about his grandchild, grandson, he corrected firmly, Warren stood at his darkened window gazing at the land his grandfather had bought so many years ago. A moon sliver shed enough light to see the oak tree that he had built a treehouse in, and had fallen out of. But even without the moon, he knew every inch, where every shrub lay, every rise and fall of his land. And now another generation to maintain it, preserve it, love it.\n\nHe always had known that Jill would come home, he thought then. She had spent her first eight years right here, and after those good years, at least half her time had been spent here, absorbing the countryside even if she had not realized it then. She knew this piece of land as well as he did, it had reached out to claim her exactly the way it had claimed him, and her son would know and love it, too.\n\nSuddenly he stiffened. A figure was hurrying up the driveway toward the county road, a woman dressed in a long black coat with a hood, a raincoat from the downstairs closet, he knew. She was out of sight almost instantly, behind the clump of birch trees.\n\nHe felt leaden as he turned from the window. His room was too dark to see anything, but he crossed to the door, out into the hallway where dim lights were always on. Moving like a stone man he walked through the hall to the wide stairs. Halfway down he realized that he had no slippers on, but he kept going, to the first floor, to the den which was dark. Without turning on a light he went to the bar and poured a drink from a decanter there; he knew it contained scotch. He added no water. He took the glass across the room and sat in his favorite chair, his father's favorite chair before him, and he drank deeply.\n\nAt first he had thought to await her return, confront her, accuse her? Even that, he admitted to himself. Accuse her. Drive her away? Possibly, he admitted, and got up for another drink. The first half hour passed, then a second half hour. He was certain he had not dozed, certain he would hear the door open, would hear steps eventually, but the deep silence persisted. Another half hour. Then he cursed himself under his breath. She could have entered through the back door, or the side door. He wasn't even sure which door she had left by. He went to the front hall and looked inside the closet. The black coat was there, as usual. Heavily he returned to his room where he stood for a long time regarding the border of light outlining the connecting door to Lois's bedroom. He did not touch it.\n\nBehind the door Lois lay with her face pressed into her pillow. Trapped, she kept thinking. This was what it meant to be trapped. She couldn't stay, and neither could she leave. She was not weeping although her eyes felt afire with unshed tears. And the word, trapped, repeated like a drumbeat in her temples. She had told Warren in the beginning that it wouldn't work, and he had assumed she meant the difference in their ages. No, no, no! she had cried and had told him about her first husband. They had been in Hazeltine's, a corner table, she remembered distinctly. The vase had two irises in it, a velvety deep blue one, and one that was only faintly touched with blue.\n\n\"He got in trouble at school,\" she had said that night, speaking of her first marriage. \"He was my advisor, and then we married, and my name was on the paper along with his. We both were in trouble. He... he fudged some results and it was discovered. He was ruined, and I dropped out and went to California to start the degree process all over, using my maiden name, Wharton. When I applied here, in New York, I had to tell the committee, of course, but no one else knows. Not my colleagues at the farm, no one. The name on the paper was L. Malik, his name; L. Malik is dead and buried, and she has to stay dead and buried. She could ruin both of us if anyone ever makes the connection.\"\n\nWarren had looked blank. \"History,\" he said. \"What does it have to do with us now?\"\n\n\"You don't understand how serious something like that can be. If anyone brought it out, smeared me with it, I would have to go away. But you could be hurt, too.\"\n\nHe reached across the table and put his finger on her lips. \"One question,\" he said softly. \"Did you do anything wrong?\"\n\nHis finger pressed against her lips. She shook her head.\n\n\"I thought not. Will you marry me?\"\n\nLois turned over in the bed and stared at the ceiling. She should have known, she thought wearily. She had known. Secrets always surface again. Nothing stays buried. But now that Jill was pregnant, Warren would have his grandchild, his heir... She bit her lip. Six months ago she had brought home a new prescription for birth-control pills, had filled her little silver box, and then had stood holding it tightly for a long time, and finally had put it away without taking a pill. She had prayed, if only she could give him the heir he so desperately wanted. She had not told him, afraid that if nothing happened he would be too disappointed, and now she was relieved that she hadn't mentioned it, hadn't talked about it. And she realized that she had better start them again.\n\nShe did not move yet, but continued to stare straight up at the ceiling. And the pulse in her temple beat trapped, trapped over and over. When she finally rose to take a birth-control pill, she could not find the container, the little silver one-a-day dispenser that she had bought the day after Warren's proposal and her acceptance. She was so tired she could not remember where she had put it, when she had last seen it even. It was three in the morning and she was too tired to think anything. She dragged herself back to bed.\n\nThen, in that state of not yet sleep, but not awake either, she realized that Warren must have taken the pills. In her mind's eye his face shaped itself, smiling at her, coming closer and closer, and she smiled back and held out both hands to him, and they laughed together and made love.\n\n\"What they needed was a good watchdog, and a good dog for kids, and good around water. Sadie all the way,\" Brenda Ryan told Charlie and Constance the next morning. She was probably more than three hundred forty pounds and she was enraged. Her face was red-splotched, her jowls shook as she talked. She wore a tentlike garment with great pink blossoms of some improbable flower, and the garment shivered and rustled and shook also, as if her rage had communicated itself to her clothing. Only her hair, a mass of tight golden curls, did not move. They were in her office outside Albany.\n\nVery mildly Charlie said, \"You're sure the dog wouldn't have taken food from a stranger?\"\n\nBrenda snorted and everything about her shook harder. \"Look, I train those dogs myself. You don't want a dog that'll take poisoned meat from a burglar, now do you? I see to it they don't. We see those caper movies just like the burglars do, and we take care of that. Mr. and Mrs. Zukal, they could feed her, that's it. And I'm not too sure about him. Her, I know. I trained her, too. Mrs. Zukal, I mean. If she doesn't handle the dish, handle the food, Sadie doesn't eat. It's like that. I tell her, Mrs. Zukal, I mean, it's good to have a backup or two. I mean, she gets sick or something, dog doesn't eat from strangers, dog dies. I never saw it happen, but it could, that's how they're trained.\"\n\n\"You never even met Mr. Zukal, did you?\"\n\n\"No. I told her how to teach Sadie to take from him, too. They both handle it for a few days, and then he can do it alone. Not until two, three days. One of them kids could have dropped a steak under her nose and she'd leave it there. That's how they're trained. Poisoned? God, I don't see how.\"\n\nShe talked on about the dog, wanted to show them the kennels, demonstrate how she trained them, but Charlie and Constance declined, and left in morose silence.\n\nHe slouched in the passenger seat and she drove. \"Maybe Sadie wasn't poisoned,\" she said.\n\n\"Maybe. Maybe Sylvie did it herself.\"\n\n\"Hah!\"\n\n\"Right.\"\n\nMiles later she said, \"Maybe Al did it.\"\n\nThis time he said, \"Hah!\"\n\nIt was midmorning when she pulled to a stop in front of the administration building of the experimental farm. If there hadn't been a sign on the door they might have mistaken it for a farmhouse. There was a wide porch and a nice lawn out front with well-trimmed hedges and specimen shrubs but nothing to suggest that it was a university facility.\n\nCharlie had the feeling that if he rang the bell a plump little housewife in a starched apron would appear and the fragrance of apple pie would fill the air. He ignored the bell and tried the doorknob instead. The door opened onto a long hall. On one side there was a narrow table with metal in-and-out baskets, some of them nearly overflowing with mail, others empty. Above each basket was a hand-lettered nameplate'. He looked them over and spotted one that said Dr. Wharton; the basket below it was empty. There were several closed doors on both sides of the hall and a staircase. No carpeting, no other furniture, no other person.\n\nSomewhere a typewriter was going and there was the sound of an air conditioner that apparently was in serious trouble. A door closed somewhere out of sight. Charlie shrugged at Constance and they moved on down the hall looking for a person, anyone. A door opened and a young woman in jeans and a faded Pink Floyd T-shirt stepped out carrying papers. She regarded them uncertainly.\n\n\"Are you looking for someone?\" she asked.\n\nCharlie turned to Constance. \"See? I told you we weren't the last man and woman on earth.\" To the bewildered young woman he said, \"Dr. Wharton.\"\n\n\"Oh.\" Her relief was evident, she even smiled. \"Out back. You can go through here.\"\n\nShe indicated the hallway, possibly to a second door to the outside.\n\n\"And after we are in the open air again, then where?\" Charlie asked gravely. They all looked at the stairs as a man began to descend, speaking to someone still out of sight on the second floor.\n\n\"Just tell them if they can't be in and out again within two weeks we'll have to get someone else.\" He saw Charlie and Constance then. \"Hi. Can I help you?\"\n\n\"We're looking for Dr. Wharton.\"\n\n\"I'm going that way. Come along. I'm Tom Hopewell.\"\n\nCharlie regarded him with interest. One of the hand-lettered nameplates had been Dr. Hopewell, but this man looked more like a rag picker than a doctor of anything. His worn shirt was a faded, red-tinged mud color, and both elbows were out. His jeans were shabby, one knee out, never hemmed, and now ragged at the bottoms. He had what looked like a two-day-old stubble. Thirty-five at the most, slender with narrow shoulders, and very dirty hands.\n\n\"Lois is back in the tree section,\" Tom Hopewell said, setting a brisk pace through the hall, out to the back of the building. The appearance of a family farm stopped abruptly here. Rows of tomato plants came to the door almost, and they seemed to be all tagged. Beyond them were rows of immature corn that looked bedraggled and stunted. Some of the plants had plastic tents over them. Some had plastic bags over portions of them only. Small boxes spotted the ground here and there. Tom Hopewell ignored it all and maintained his fast pace. Suddenly he stopped and pointed. \"There she is. I've got to go.\" Although his words indicated a need to hurry, his body seemed of two minds about it. The lower part started to turn, the feet started to move in the other direction but his head did not move immediately as he gazed another moment at the woman he had pointed out. Then, abruptly, he turned all the way and trotted off.\n\nThe woman had just emerged from a greenhouse near the start of the tree plantings; she stood in the doorway talking over her shoulder to someone still inside. She nodded and walked toward a quonset hut. Another farmhand, Charlie thought, another worker dressed in jeans, T-shirt, and sneakers. Doctors were not what they used to be, he added to himself, as he and Constance went forward to meet Lois Wharton Wollander.\n\nThey caught up with her outside the quonset hut. \"Dr. Wharton,\" Charlie said, \"can we have a few minutes?\" At first she had appeared to be as young as the woman in the building\u2014thirty maybe, with very attractive gray streaks in her dark hair that were dramatic enough to have been added at a salon\u2014but close up it was obvious that she was a bit older than that. She wore no makeup and was very good-looking in a careless sort of way, as if she hadn't thought about other ways to fix her hair, or how beautiful her wide eyes actually were.\n\nShe glanced from him to Constance and back. \"Why?\"\n\nHe introduced himself and Constance. \"We've been hired to investigate the death of your student, David Levy.\"\n\nOne moment her expression had suggested preoccupation, a bit of impatience, and not a lot of interest in them, but now she simply looked startled and puzzled. \"But why? I thought... The sheriff said he overdosed.\"\n\n\"Yes. But his father isn't convinced, and, frankly, Dr. Wharton, neither am I. Can we talk?\"\n\nShe hesitated, then nodded. \"I have an office in the admin building. I guess we could go there.\"\n\n\"Where did David live?\" Constance asked suddenly. \"One of the units over there?\" Off behind the greenhouse, only partially visible, was a long, low building that looked like a motel of the fifties.\n\nLois looked at her more sharply and nodded again.\n\n\"I stayed in a place just like that years ago, a summer at Indiana State,\" Constance said. \"I bet they used the same plans. Can we talk in there?\"\n\nReluctantly Lois said the units were open and they started to walk. \"Is the whole operation here organic?\" Constance asked, and this time Charlie looked at her sharply. \"The bees,\" she said. \"All those little boxes. Hired bees. You can't bring in bees if you're using sprays. Well, some sprays, I guess, but not most.\"\n\n\"It's all organic,\" Lois said. \"That's why we're understaffed and overworked and underfunded. We have a continuing grant for the next four years. We're six years into this project.\"\n\nThe apartments they were approaching were very ugly, institutional ugly, with aluminum windows and doors, no porches or even stoops. There were eight doors.\n\n\"No one's staying in them now,\" Lois said, leading the way to the end door. \"They're all due for painting, refurbishing. Thorough cleaning. The usual summer maintenance. This one was David's.\"\n\nShe opened the door and they walked into a cramped, L-shaped living room with a kitchenette; one door to the right opened to a bath, and another straight ahead to the bedroom. There was only one narrow bed, but the room had been designed for two; two chests of drawers, two chairs, two desks were crowded into the space. Constance nodded. Exactly like the one she had shared with another student many years ago. By the end of summer they had been ready to throttle each other.\n\n\"Of course, nothing of David's is still here,\" Lois said, standing near the door as Charlie and Constance looked over the apartment. The spartan rooms were stripped to such essentials they looked uninhabitable. The apartment felt loveless and gray.\n\nThey returned to the living room where they all sat down on plastic-covered chairs with very little padding. Charlie regarded Lois steadily and asked, \"Do you believe David took drugs? That he took an overdose?\"\n\nMiserably she looked down at her hands. \"I don't know. I didn't. Then the sheriff said he must have done it. I just don't know.\"\n\n\"Okay. Tell us something about him. Good student?\"\n\nShe nodded. \"Good, but not brilliant. He was idealistic. His plan was to get his degree and then go to some third-world country and teach. He could have done that.\"\n\n\"Why don't you just talk about him. How he got along with the others here. Who his friends were. Anything that comes to mind.\"\n\nThe picture she painted of David Levy in halting words, with pauses and hesitations, was that of a shy young man with few friends among the other graduate students. He was not asocial but he didn't do the same kinds of things they did, and he was an outsider. He didn't dance, or like rock music, or drink anything alcoholic, or do drugs. He was a strict vegetarian, a health food advocate. He got along with everyone, as far as she had been able to tell but he wasn't intimate with anyone. When the school year ended only two other graduate students remained and they lived in an apartment in the village. They both had been away that weekend.\n\n\"So he was over here by himself?\"\n\n\"Yes.\"\n\n\"How did he appear to you the last few days before he died?\" Charlie asked.\n\nLois shrugged helplessly. \"I didn't notice anything out of ordinary. We were working very hard, seven days a week usually, and he was as busy as I was. It isn't as if we had much time for socializing.\"\n\n\"Okay. What about the day he died. You found him?\"\n\nShe took a breath and nodded. \"We have two hundred cloned trees that need daily tending. Saturday, Sunday, it doesn't matter. It was David's job to see to them early every morning, and that Saturday he didn't show up. Over the weekends it was just for a few hours, but it had to be done, and I had other work to do myself, and after a while, about ten or ten-thirty, I was becoming angry that he wasn't there, I'm afraid. I assumed it was the usual kind of thing; you know, the clock didn't go off, or too late a night, something of that sort. Anyway, I started to come over here.\" She paused. Her face was pinched looking, strained. Now she looked nearer forty than the thirty she had seemed at first glance. \"I saw one of the scientists here, Tom Hopewell. I asked him to come with me. It occurred to me that if David was in bed, it would be embarrassing for him if I showed up.\"\n\nCharlie nodded encouragingly at her.\n\n\"We both came to the door. I knocked, and called David, and tried the knob. It turned, and I opened the door, but Tom actually entered. I stood in the doorway and he went on into the bedroom and found David.\"\n\n\"Did you go into the bedroom at all?\"\n\n\"Yes. I couldn't believe... I thought... I don't know what I thought. I just ran across the room and to the bed. I... He was dead.\"\n\n\"Okay.\" Charlie studied her for a moment; she was pale and strained, too tight, but in control. Too controlled? \"Dr. Wharton,\" he said then, \"was anything out of the ordinary that you could tell?\"\n\n\"I don't know what would be ordinary,\" she said, shaking her head. \"I never had been in here before. It was very neat. David was fastidious, nothing out of order, nothing out of place. I didn't see a glass or a bottle or anything like that. The sheriff asked.\"\n\n\"What happened next? You saw him. Was he covered by the blanket or anything?\"\n\n\"Yes. I touched his cheek. Then...\" She paused again. \"I'm really not sure. I was standing at the sink, holding onto it while Tom was on the phone to the police. I kept thinking, he intended to make an omelette. I remember thinking that he wouldn't have his omelette. I was crying,\" she added in a low voice, her eyes downcast.\n\n\"What made you think of omelette?\" Charlie asked. \"What did you see at the sink?\"\n\nShe shook her head helplessly. \"I don't know. I was crying, and holding on, and hearing Tom on the phone saying he was dead. I must have been in shock for a minute or two.\"\n\nCharlie caught a signal from Constance, nothing she said, nothing she did, no motion she made, nothing that would have been discernible to anyone else on earth, he knew, but there it was, almost like a hand on his back, fingers digging slightly into his spine. He held the next question. Constance stood up and went to Lois, took her arm.\n\n\"Let's go to the sink, the way you were then,\" she said.\n\nFor a moment Lois resisted the pressure, but then she got to her feet and walked slowly across the room to stand stiffly at the sink. It was dull metal with little dents here and there. Water had discolored a trail to the drain. A few inches of the same dull metal made up drain boards on both sides. A three-burner stove was to the right, a tiny refrigerator to the left.\n\n\"I think,\" Constance said, \"you should move closer, hold it the way you did that morning.\"\n\nWith obvious reluctance Lois moved closer and grasped the edges of the sink and bowed her head.\n\n\"Now, close your eyes and see what you saw that morning,\" Constance said, as if this were the most natural thing in the world.\n\nLois obeyed, and after a second or two she said, \"There was a yellow rubber dish drainer on that side, with a small pot in it. Two eggs were on the other side, and a piece of cheese. And a glass. His vitamins were on that side, too. Several containers of vitamins.\" She lifted her head and looked startled. \"I smelled orange juice,\" she said. \"I forgot that before, and the eggs. I forgot them. There was an omelette pan on the burner.\" Her startlement increased. \"Dear God, there was a towel, wet.\" She spun around and looked at the room. \"A chair was right here,\" she said indicating the place in the tiny kitchenette. \"There was a towel draped over it. He must have been swimming. He swam every morning before breakfast. We had to walk around the chair to get to the bedroom.\"\n\nThey waited, but there was nothing else. She had not seen bathing trunks, but she had not gone into the bathroom. After Tom Hopewell called the sheriff's office they had gone outside to wait.\n\nWhen they were finished in the apartment, Charlie stood just outside the door looking around. The units had been placed close to the boundary of the grounds. Behind the structure the experimental trees started. From the door nothing of the greenhouses or the quonset hut was visible. Trees were in the way, and the trees continued down to the lake. He studied the layout unhappily. David's apartment could be entered without anyone on the farm seeing a thing. For these few seconds that he looked over the grounds, Lois stood silently also.\n\n\"It just won't work, will it?\" Lois said in a very quiet voice that seemed almost resigned. \"He wouldn't have gone swimming, had his juice, and then take anything like quaaludes. He wouldn't have taken phenobarbital before his chores were done.\"\n\n\"No,\" Charlie agreed. \"It won't work.\"\nCHAPTER 6\n\nTOM HOPEWELL GOT TO his feet and brushed dirt off his knees when Charlie and Constance approached him at the edge of one of the corn patches. A few feet beyond him there was a twelve-foot-high fence made of posts with netting. Beans were starting to spiral up the lines. Hopewell was just as ragged-looking as Charlie remembered.\n\n\"Dr. Hopewell,\" Charlie said, \"would you mind answering a couple of questions?\" He introduced himself and Constance, who was eyeing the bean trellis thoughtfully.\n\n\"About David Levy?\"\n\nCharlie raised his eyebrows.\n\n\"Well, what else?\" Tom Hopewell said with a shrug. \"First Lois, now me. I thought that whole matter was handled a bit fast.\"\n\n\"How will you pick them?\" Constance asked then, still studying the trellis.\n\nHopewell glanced at it, then back to her, and grinned. \"We hire very tall workers.\" When she grinned back, he said, \"Actually we want the biomass more than the beans. For compost. And we need the windbreak. Those are scarlet runner beans, not even a true bean. They'll cover the netting, with any luck, and pollen from the corn down past the peas won't blow this far and mess up this stand.\" He shrugged and added, \"We're working with very limited space here, I'm afraid.\"\n\nCharlie waited patiently as Constance asked another question or two about the farm, but when it appeared that Tom Hopewell intended to take her away on a tour, Charlie cleared his throat quite loudly. Constance smiled at him.\n\n\"Another time,\" Hopewell said to her. Then to Charlie he said, \"And you want to know about that morning, I guess.\"\n\n\"Yep. But first, how well did you know David?\"\n\n\"Hardly at all. I was down in Peru all summer, their summer that is, and came back in April with seeds. Corn seeds. I was on a collecting trip. We're looking for better strains, more resistant, hardier, faster to yield, more yield. And perennial strains, of course.\" He was like a little boy who wanted to show off a new kite. When Charlie cleared his throat again, he looked apologetic. \"Yes, David. Anyway, I met him in April, and we worked together a little, not as much as you might think, though. Lois had him pretty well tied up. Properly so, I might add.\" He grinned suddenly. His grin was infectious and made him look too young to be off in Peru collecting seeds, too young to be a doctor of anything. \"We're always trying to snare each other's grad students, of course. Free labor, and God knows we can all use more help than we have. Early on I tried to lure David, to get him to put in a few extra hours. No dice. That was probably the only time I ever really talked to him, more than passing the time of day, I mean. Anyway. I hardly knew him.\"\n\n\"Who's actually in charge here? Who was around that weekend? Besides you, I mean.\"\n\n\"In charge,\" Hopewell said. \"That's Dr. Clarence Bosch, the head of the farm here, our boss. His daughter got married that weekend. He took off on Friday and got back Monday. There were the hired helpers, locals for the most part, but not that early on Saturday. A couple of them were due at ten, for half a day. They were mine. The other two grad students were off for the weekend. All the rest had already left for the year. School's out, off they scoot, credits in their grimy little fists.\"\n\n\"Okay. So that morning, what happened?\"\n\nHis expression became sober. \"That morning. I got here after eight or eight-thirty, I guess. I don't wear a watch, you see, and I didn't check the time anywhere. After I'd been busy for a bit, Lois called me, and I met her near the apartments. She was sore and didn't want to go in herself.\" He glanced from Charlie to Constance and said ruefully, \"Levy had a crush on her, although I don't think she knew that. But I did, and I didn't blame her for not wanting to go in and catch him in his underwear or anything like that. Anyway, she knocked and tried the door, but I went in first. She followed me inside, but just inside the doorway. I found him in bed. Then she came on into the bedroom. We went back to the other room where I called the sheriff's office. Then we waited outside for them to get there.\"\n\nCharlie nodded. \"Let's back up a step. When you went into the bedroom, she was standing at the door. Was she still there when you called her?\"\n\nHopewell's face tightened. \"The sheriff asked me if she could have removed anything, and I told him no. The way the apartment is set up, it's a straight line from the door through the kitchen space and into the bedroom. I saw him, and looked back at her instantly. She hadn't moved. I said some dumb thing, like there's been an accident. Something in my voice made her run into the apartment, straight through to join me by the bed. She reached down and touched his face, that's all. Then I took her arm and moved her to the sink. Frankly, I was afraid she might upchuck or even pass out. She went dead white, and she was crying. I kept an eye on her while I made the call. She just stood there hanging onto the edge of the sink. Then I took her outside. I was watching her every second.\"\n\nHe had not seen a container for the pills, had not seen anything out of the ordinary, accepting that he had no way of knowing what was ordinary. \"I never had been in his apartment,\" he said. \"Like I said, I hardly knew him.\"\n\n\"So you have no way of knowing if he did drugs,\" Charlie said.\n\n\"Exactly. But I'd guess not. He just wasn't the type. Now, my grad student Preston Heywood, that's different. But he left in mid-May. And he was in Peru with me, so he knew Levy even less than I did, I guess.\"\n\n\"Maybe he could have sold something to Levy,\" Charlie suggested hopefully. \"You don't have to be bosom pals to deal.\"\n\n\"Not unless it was coke. I said he was in Peru with me.\"\n\nHe had started to fidget restlessly, and glanced at two workmen who were standing several hundred feet away under a tree, apparently waiting for him.\n\n\"One more thing,\" Charlie said, dissatisfied. \"Do you know the Zukals?\"\n\nHopewell looked surprised. \"Sure. To be neighbors with them is to know them. So?\"\n\n\"You ever go up to their place?\"\n\n\"What the devil does that have to do\u2014? As a matter of fact, when they bought that mill, they closed down one of the favorite lovers' lanes hereabout. But I haven't been up there for a spell, not since last summer, at least.\"\n\n\"They came down here?\"\n\n\"No. I saw them by the lake. Look, I really do have work to do.\" His impatience was turning into real irritation. He waved to the waiting men and started to edge away.\n\nCharlie might not have noticed his growing restlessness. \"Did they have the dog with them?\"\n\nHopewell looked at him angrily. \"No. I've got to go now.\"\n\nThey watched him take a couple of steps. Then Charlie called after him. \"What did you do when you found out that your student had smuggled coke into the country?\"\n\nHopewell was caught in midstride. He wheeled about, palefaced. \"I didn't say he did.\"\n\n\"What did you do?\" Charlie repeated softly.\n\n\"I fired him. Are you through now?\"\n\n\"Oh yes. See you around, Dr. Hopewell.\"\n\nThey watched him stride away quickly. He looked stiff.\n\n\"Well,\" Constance said. \"If you wanted to get his goat, you succeeded.\"\n\n\"I did, didn't I?\" He took her arm. \"Let's go see the sheriff.\"\n\nSheriff Greg Dolman was grayer than Constance remembered, and a bit stouter, but his smile was just as wide, just as false as it ever had been. His eyes did not know the rest of his face was smiling.\n\n\"Charlie! You're looking great. And, Constance, prettier than ever. Come on in and sit down. How are you? How's the little plantation? What can I do you for?\"\n\nHe ushered them into his private office in the county courthouse and stood beaming at them until they were seated in the two wooden chairs. Then he went around his desk and took his own seat.\n\n\"It's about David Levy, Greg,\" Charlie said. \"His father came to see us.\"\n\n\"Poor guy,\" Dolman said, shaking his head in sorrow. \"Walking dead man. Only son. You know how that goes.\"\n\n\"Uh-huh. Fill us in, will you? About the boy.\"\n\n\"Nothing there, Charlie. Like I told the old man. Just nothing there. Kid takes an overdose. What're you going to do? Can't supervise them all their lives.\"\n\n\"I'd like to see the reports,\" Charlie said, just as amiable as the sheriff, his eyes just as hard.\n\n\"Charlie, the case is closed. Did the old man hire you? What with? He doesn't have two dimes to rub together.\"\n\n\"The reports, Greg. Medical examiner's report, you know. He hired me. Got to earn my keep. You know how it goes.\"\n\nDolman regarded him flatly for a moment, then smiled his expansive smile again. \"Sure, Charlie. But I can wrap it all up in a couple of sentences. The kid took methaqualone and phenobarbital together. A real bad combination. Real bad. Took them with orange juice, on an empty stomach, the worst way. Took all he had. We did an analysis on everything in that dump, vitamins, aspirins, juice. Not a trace of anything. He took them all, probably flushed the paper they were in. Then he went back to bed. Just not a good morning for him, I guess.\" He smiled again. \"Frankly, Charlie, between you, me, and the woodwork, I don't give a shit if the dopeheads go out that way.\"\n\n\"That sounds exactly like what we've been hearing,\" Constance said. \"Poor Mr. Levy. It won't take more than a couple of minutes, I think, to make copies of the various reports. Then we can write our report for his father. Poor man.\"\n\nDolman looked at her suspiciously before his smile returned, but she was as innocent as dawn. \"Yeah,\" he said finally. \"Why not? There's just nothing there.\"\n\nFifteen minutes later they were back in their Volvo, Charlie driving. \"Pretty slick,\" he said. \"I take it you wanted out of there.\"\n\n\"Damn right,\" she said. \"He's worse than ever.\"\n\nCharlie chuckled and rested his hand on her thigh. \"Let's go home and do some reading.\"\n\nYears before, when they bought the house in the country, long before they could spend much time in it, Charlie had said in a burst of exuberance, \"When we retire, it's fifty-fifty, kiddo. We take turns with chores, okay?\"\n\n\"Like what chores? I say we hire some things done, like snow plowing.\"\n\n\"Maybe. I meant in the kitchen. You cook, I cook, tit for tat.\"\n\n\"Wonderful,\" she said, laughing. \"And, Charlie, I won't forget, you know.\"\n\nAnd she hadn't. If either of them was really crushed for time, that was different, but generally it was tit for tat. But with a new case on their hands, Charlie was thinking at the kitchen table, maybe they could discuss it again.\n\nConstance broke into his thoughts. \"Charlie, I'm glad we worked things out in advance, cooking, things like that. It could be awkward on a day-by-day basis, couldn't it?\"\n\nHe examined her carefully; her clear, pale blue eyes were guileless, her expression serene. He sighed. \"Fish on the grill,\" he said. \"Later.\"\n\n\"Wonderful, darling,\" she murmured. \"Look, here's the list of the stuff they examined in David's apartment. He was a real health-food addict.\" She continued to read as the orange cat, Candy, sidled up to her nonchalantly and sneaked onto her lap, as if hoping she would not be noticed and sent off to Siberia. Constance adjusted to the cat and stroked her absently. Candy began to purr. Constance was reading over the shopping list they had found, more of the same: tofu, yogurt, black beans, lemons, herb tea... .\n\nHe sat opposite her at the table and started on the medical examiner's report. For a long time neither spoke. Candy got bored and flowed out of Constance's lap like a sluggish mass of syrup. Brutus stalked in and glared at them, stalked out again twitching his tail. Constance was skimming the early letters from David Levy to his father. When she got to April she began to make a few notes. Charlie was making notes too.\n\nThey would both read it all, and then discuss everything. But Constance was having a harder and harder time with the boy's letters to his dying father.\n\nFinally she put one facedown and stared out the screen door to the backyard. The early bulbs were all finished and now irises were making a show, and baby's breath had come into bloom, making the irises appear to rise from white clouds. Yellow, orange, bronze marigolds, calendulas in lemony colors, some almost white. A cardinal flitted past, then two chickadees. All too pretty to introduce such ugliness into, she was thinking. Peaceful, bucolic even, the air alive with buzzes and hummings, the twitter of birds, a song now and then\u2014\n\n\"Tough going?\" Charlie asked. He reached across the table and covered her hand with his.\n\n\"He didn't kill himself,\" she said quietly. \"He didn't do drugs in any form. He was murdered.\"\n\n\"I know. And we'll find out who and why. I'm going to start the grill.\"\n\nShe set the table on the patio, and now the three cats prowled between her and Charlie, who was cursing softly. The grill never seemed to work exactly right without starter fluid, and he was opposed to using a chemical in his food, not even one that had long since burned off. He danced around the cats and cursed fluently and finally had a bed of coals, not ready yet, but coming along. When he backed away from the grill with a satisfied nod, she put a glass into his hand. He tasted and nodded more vigorously. Gin and tonic.\n\nThe fish was delicious, the salad superb, the wine chilled just right, potatoes crisp and brown and tender. They sat with coffee as the first stars began to appear in the postcard-blue sky.\n\n\"Okay,\" Charlie said then. \"Okay.\" He leaned back in his chair. \"'Star light, star bright, first star...' You do that when you were a kid? Make a wish on the first star?\"\n\n\"Absolutely. '... I see tonight. Wish I may, wish I might, have the wish I wish tonight.'\" She closed her eyes a moment and when she opened them it was to see him regarding her with a soft expression.\n\n\"Bet I know what you wished,\" he said.\n\nAfter a second she nodded. \"I think you do.\"\n\n\"Enough of the mushy stuff,\" he said. Then very briskly he went on. \"The way I read it, David Levy got up at six that morning and went down to the lake for his swim. He went back to the apartment and had orange juice and his vitamins. He got the eggs and cheese out but he wanted a shower before breakfast. He showered, shaved, shampooed, and dried off and put on his shorts, but by then he was feeling sleepy, or dopey, or something and he crawled back into bed and fell asleep and died. Death between eight and nine at the very latest.\"\n\nConstance poured more coffee. The tiger cat Brutus was stalking the grill, planning his attack. Charlie had given them fish already, but with cats enough was a meaningless word. \"I can see why Greg had a problem,\" she said after a minute.\n\n\"Yep. They opened the vitamins, all what they said they were\u2014A, B complex, E, C, mineral supplements, there's a long list. Not many of any one of them, but all okay. Containers okay, no trace of anything other than the legitimate contents, all as they should be. The juice was okay. All the food okay. Nothing in the aspirins but aspirin. One missing from a container of twenty-five, expiration date last summer. The kid just didn't take things. Period. And unless Lois Wharton Wollander or Tom Hopewell removed something, there was no container for any other drugs. But they had to come from somewhere. And if he did unwrap them and flush the paper, that means he did it himself. And there's no way we can prove or disprove a thing.\"\n\n\"He could have had company,\" Constance said after a moment. \"Someone else could have made the juice and fixed a glass for him with the dope in it, and then washed it out again.\"\n\n\"I really hate that a lot,\" Charlie said, frowning. \"It's the damn time element. That junk had to be in his system by six-thirty. He comes in from swimming and someone says, here's your juice, and he drinks it and goes to shower and comes back out in his shorts. That someone must have been a good friend, and he didn't have good friends at the farm. Okay, he could have had a robe on, and could have taken it off again to go to bed. He wasn't expecting company for breakfast\u2014two eggs only. I just hate that a bunch.\"\n\n\"Were any of the vitamin containers empty?\" she asked after another minute or so.\n\n\"Nope. Our friendly sheriff is an asshole, granted, but he's thorough. He thought of that, too. A few of this, a few of that, and so on, but something in each of them, and no container in the trash. He was running low, but the new supply must not have arrived yet. He ordered them from a mail-order company.\" He sighed. \"Your turn.\"\n\n\"Not much we don't already know, I guess. It's pathetic, how he wrote to his father pretending nothing was wrong, that his father wasn't a dying man. He must have told his dad everything there is to know about the work he was doing. There's a lot about Lois Wharton. He really did have a crush on her. He mentioned Tom Hopewell's arrival and Hopewell's fight with the other student, and he told his father when the bees were delivered. He told him about cloning the trees, and a fight they all had over space and priorities. Apparently they are really strained for space. He referred to a time when they all thought Lois might approach her husband to buy the mill and the grounds there for her work with the trees. The other scientists begrudge her the space that trees take. He didn't refer to it again. I guess it was a just a wish on their part. I haven't got much further. The Zukals are due, that's when I stopped reading.\"\n\n\"That damn cat,\" Charlie muttered, and then yelled wildly. Brutus was on his hind legs, straining to see the top of the grill. He dropped to all four feet and streaked away. Ashcan slunk off into the bed of irises, and Candy crouched near the door, her eyes enormous, her hair standing up like a razorback hog's, her ears flattened. Charlie went to the grill and closed the lid; when he turned toward the door, Candy bellied away from him. \"Let's get back to that stuff,\" Charlie said cheerfully. Any time he had all three cats buffaloed at once, he thought, he had earned his Brownie points for the day.\n\n\"You'll give them all traumas,\" Constance said.\n\n\"Good. I'll take the coffee, you bring the cups.\"\n\nIt was nine-thirty when Lois got home that night, so tired that she felt numb. Until she had a replacement for David it would be like this she had told Warren. The metabolism tests had to be done within a limited time period or they would be meaningless. Even a week made such a difference...\n\n\"He said you knew,\" Mrs. Carlysle said helplessly when Lois asked where Warren was. \"One of those committee meetings.\"\n\nMrs. Carlysle was a kindly woman in her sixties, white-haired and a bit overweight. She had been with Warren for twenty-eight years. Now she said, \"I'll make you something to eat. You must be starving.\"\n\nA few minutes ago Lois would have agreed that she was starving, but now she felt only her fatigue. \"Just a sandwich,\" she said. \"I can do it.\" She started to walk through the hall toward the kitchen, Mrs. Carlysle close behind her.\n\n\"We have ham, and there's some tunafish, of course. Cheese. You just tell me what you want and I'll make it. You look so tired.\"\n\nLois stopped at the wide staircase and felt her shoulders droop. \"Tunafish would be fine,\" she said. \"And milk. I'll go up and soak in the tub for a few minutes. Thanks.\"\n\nWhen she called to tell him she would be late, he had not mentioned a meeting, she felt certain. She would remember that. But she had forgotten that tonight was Jill's night to go to Sebastian's meeting. Every Wednesday night they did chants or something. She tried to relax in the warm water, tried to remember what Warren had said, but nothing came. Nothing. He hadn't said anything. All this week she had been taking her car to work because she no longer had time for the leisurely stroll over to the farm, and then back again. And this morning she had left before Warren was up. Had he been too abrupt on the phone? Or had she? It had been a short conversation. \"I'm sorry, but I'll be late. Go ahead without me and I'll have something when I get home.\" Had she said more than that? She shook her head irritably. What difference did it make? He had a meeting, and he had forgotten to mention it.\n\nBut something was wrong, she thought. Something was slipping away and she didn't even know what it was, or how it was happening. The poison-pen letter he had received had something to do with it. He had refused to tell her exactly what it said. Lies, all lies, was all he would say about it. And then he forgot to mention a meeting. Or she forgot that he was going out. Wearily she climbed out of the tub and toweled herself dry, put on her gown and robe and went to the bedroom, where Mrs. Carlysle had left a tray. A sandwich, salad, milk, coffee. Even a bowl of strawberries. Lois found that she could eat little of it. All she wanted was to lie down and stretch as far as she could and close her eyes.\n\nShe would hear him when he came upstairs, she thought. She always heard him in the next room, moving around, opening a drawer, opening his closet, his bathroom door. Water running. He would come in to tell her goodnight, kiss her goodnight. She turned onto her stomach.\n\nThree years ago when she tried to explain why they should not marry, she had said, \"I have this work to do. I don't mean that it's something I just would like to do if it's convenient. I have to do it. I can't explain it any other way.\"\n\n\"And you don't have to explain it any way at all. That's one of the things I love about you, your determination, your unswerving march to your own goal. I promise I'll never get in your way. I won't ever try to get between you and your work. I love you very much, Lois. More than I know how to tell you.\"\n\n\"People will say I married you for your money. Your own daughter will think that.\"\n\n\"Now you listen to my secrets,\" Warren had said soberly. \"Two years ago I had a heart attack. Not serious, just a warning, the doctors told me. Jill doesn't know. She was in Paris with her mother. I went on a cruise to think and rest, and I decided that all my life was a lie. I lied to myself and to the world, pretending I was doing public service. Public disservice is more like it. You've read how a person changes when death comes too close? Believe it. I changed. When I came back, I began to sever ties, to disengage myself. It's been slower than I'd like, but I'm still working on it. I don't know what I'll do with the rest of my life, but not what I've been doing. People who look up to me look up to an illusion, nothing more. You think I care what they say about you, about us? Jill's mother married me for money and it killed her. Addicted to everything that money can buy, she died of money. She was respected because she was from a good family, and she sold herself like a common whore. You would do me great honor if you will marry me. I know I'm rushing you, but I'm afraid of time. Maybe I see salvation through you. Don't say anything else now. Maybe we've both said too much for one evening. I'll take you home.\"\n\nLois rolled to her side and breathed deeply, remembering. Salvation. But now with a grandchild on the way, with reconciliation between him and Jill, everything had changed. It was like looking into a kaleidoscope that had been still for so long that she had forgotten how swiftly the pattern could change.\n\nThis was ridiculous, she told herself as she slipped into sleep. He would come in to kiss her goodnight. She would remember that he had mentioned a meeting. She would get a replacement for David and not have to work such long hours. And they would walk hand in hand in the garden again.\n\nThis way! This way! he called, and she ran toward the sound of his voice, up there, past a gate, running without effort. Then she stopped in terror. All around were ghost trees, pale and stripped of leaves, rising from drifting sand. The sand made a whispering sound as wind stirred it; nothing else moved. The trees had been denuded of smaller limbs, and those that remained were stiletto-like, gleaming in a light without a source. She backed away, but was mired in sand; when she dragged one foot loose, the sand whispered. This way! His call again. All the trees were gray, silver, unmoving, the trunks too thin to hide anyone, but she could not see him, could only try to follow the sound of his voice. Her legs ached with the effort and when she would have fallen, she found herself unable to touch one of the ghost trees for support, staggered instead to her knees, and had to work desperately to stand up once more.\n\nNow the sand started a slide, and she was caught in it, dragged along downward, twisting and turning to avoid the sharpened tree branches, breathless with fear. Below was the sapphire lake, unreal, painted-looking, but then churning with motion. Alive, she cried out in her dream, laughing and crying at once, eager to finish the slow-motion slide. As she watched the boiling water, a figure emerged, upright. David! His hair swirled about his face, then was plastered to his cheeks, across his open, staring eyes. He rose as far as his shoulders, and began to sink again as she gazed with horror.\n\nThis way! No! She tried to scream, and the sound was no more than a moan as her slide toward the lake continued. The water was subsiding, becoming still again, unreal again.\n\nStanding in the open doorway between the bedrooms Warren watched her toss and twist. The sounds she made were unintelligible; her thrashing and her moans reached a crescendo and for a moment he thought she would surely wake up, but instead she sighed deeply and then became still. In the dim light he could see a sheen of sweat on her forehead, her cheeks. Silently he backed into his own room and closed the door.\n\nWhen she woke up it was six in the morning.\nCHAPTER 7\n\n\"OKAY,\" CHARLIE SAID ON Friday afternoon. \"We're stymied until we get the report on the dog. Right?\" Constance nodded. They had learned so much in the past few days, she thought, mildly surprised again by how much of everyone's lives had been recorded, filed away, remembered by unlikely people. Clarence Bosch, in charge of the experimental farm, world-famous for his introductions of various vegetables, bigger and better tomatoes, better peppers, earlier melons... Jill Wollander, a wild girl in years gone by, now respectable, married to a millionaire... Lois Wharton, brilliant in silviculture circles, introducing a new tree that would grow twenty feet in a season in the north... Warren Wollander, power behind the visible power. The reason no one could construct any of them, Constance knew, was because there were great gaps in all of their lives. Jill had flitted back and forth between mother and father for many years. What had she done in those years in Europe with her mother, who apparently had been in and out of sanatoriums for the last ten years of her life, addicted to alcohol, drugs, who knew what all? Even Clarence Bosch. He had sued Warren Wollander for a million dollars fifteen years ago, charging malicious slander. It had been dropped, but the mystery persisted. What had that been about? Lois Wharton. She had appeared with a Ph.D. in hand and had been hired on at the farm out of nowhere. Six years of her life a blank. Nothing on record for those years. And Sebastian. Con man? Preacher? Enlightened? Just suddenly there in the picture, looking into buying the mill for a school, or possibly a temple. Or something. They had found nothing yet about Sebastian.\n\nThere was no point in calling Wilbur Palmer, the pathologist who had dug up the dog Sadie. He would simply snap at them and hang up until he was ready. There was little they could do until they had his report. There might be little they could do even after they had the report.\n\nCharlie's thoughts were equally gloomy. This was the kind of case he had hated back when he was a New York City detective. Pointless deaths. No direction to strike out in. No direction for questions except the obvious one: Did you kill that boy? He had dumped out the contents of the box Mr. Levy had left with them, and morosely moved stuff around. Such a pitiful little pile of stuff. A few bills, a checkbook, postcards.... He had lived on practically nothing, had no extravagant habits, damn little to show for having lived at all. He put it all back inside the box.\n\n\"When David came to talk to you,\" Constance said then, \"he didn't know who killed the dog. So, if there's a connection, something must have happened in the next day or two to make him suspect he knew. Or maybe he did know by then. But, Charlie, no one would commit murder to keep something like that hidden.\"\n\n\"A nut might.\"\n\nShe rolled her eyes. He had more faith in crazy people behaving in crazy ways than she did, he knew, but nuts did nutty things all the time; they earned the title. And killing David Levy seemed to be in the category of nuttiness.\n\n\"So, let's make a case,\" he said, tapping his fingers on the table. \"Someone wanted to buy the mill property, but the Zukals got there first.\" She made a noise; not an interruption, she never did that, but still.... He paused, waiting.\n\n\"Maybe it wasn't that someone wanted to buy it, just that someone didn't want the Zukals there,\" she said. \"For a lot of reasons. Starting a furniture factory, trucks maybe, traffic, noise. All those grandchildren at play, Sylvie and Al themselves yelling. Or maybe for a reason we can't even guess\u2014a treasure buried on the grounds.\" She spread her hands. \"There's so much we can't even guess about.\"\n\nCharlie nodded. \"I know, but let's assume a scenario, that someone wants them out, for whatever the reason. So someone decides to drive them away. First the dog. Give them a scare. Then David catches on, and he has to go.\" He looked at her and shrugged. \"Won't play in Baltimore, will it?\"\n\n\"I wouldn't buy it.\"\n\n\"Me neither. Let's go to Sebastian's service tomorrow.\"\n\nSebastian had private, invitation-only meetings every day for one thing and another, they had learned, but on Saturday he had open services.\n\n\"We'll have to sit on the floor,\" Constance said. Charlie made a face.\n\nLois watched Jill and Warren climb the bank from the lake that Friday evening. For days she had been unwilling to go near the brilliant water; even glancing at it filled her with dread. Jill and her father were talking and laughing. All this time, Lois was thinking, how she had wished they would be close again, father and daughter, friends, loving, confiding in each other. The kind of relationship she had yearned for with her own father, who had died too soon for it ever to happen. Warren's first wife Shelley, and then Jill, had hurt him so much; Lois had ached for him, and now he had his daughter back. She knew she should be rejoicing, but her stomach felt leaden and her head throbbed with a persistent pain. She sipped her wine and forced a smile when they drew near.\n\n\"Al Zukal has come up with a fantastic scheme!\" Jill said. \"He's going to put in a beach at his end of the lake. Limestone. He's planning to ship it in from Tennessee or Kentucky. It'll look like snow!\"\n\nLois looked past Jill and Warren. From up here she could see only the upper end of the lake, but now Al and Clarence came into sight. They were strolling back in the direction of the mill, their heads lowered. Limestone? She felt a rising excitement as she considered it. Every rain, every snow would dissolve some of it. How much would it take? Maybe if they did their beach over.... She looked at Warren; he averted his gaze.\n\n\"I've got to change and meet Stanley's train,\" Jill said as she hurried to the house. \"I made a reservation for us at Hazeltine's, but we won't be out very late.\"\n\nAs soon as she was gone, Lois said, \"Warren, what's wrong? What's happened?\"\n\nHe shook his head. \"I'm just a little tired. I'll lie down before dinner.\"\n\nHer hands were shaking. Carefully she put down her wineglass and took a step in his direction. \"Something's happened, hasn't it? Warren, what is it?\" Her voice rose and she stopped abruptly as he walked past her.\n\n\"We have to talk about it,\" she said, controlling her voice so much that it sounded strange to her own ears. \"Whatever it is, we have to talk about it or this weekend will be hell for all of us.\"\n\nHe paused and turned finally to regard her. \"I'll be here when you want to tell me what you're hiding.\" He entered the house then.\n\nShe sank into one of the lawn chairs and stared blindly at nothing. For a long time she could not track any single thought from start to finish; they merged and blended, and disconnected things joined one another haphazardly. Last night Warren had pressed Jill for an explanation of exactly what it was that Sebastian taught, exactly what she meant by enlightenment, by the rapture of nothingness.\n\n\"Nothingness,\" Jill had said hesitantly, \"has to be by choice. But if you choose, you are actively involved, and that can't be real nothingness. But it can't be simple mind-wipe fatigue, because that's negative, and it has to be positive.\" She had thrown up both hands. \"I can't explain. You have to ask Sebastian.\"\n\n\"It sounds exactly like my problems with algebra back in the good old high school days,\" Warren had said, smiling. \"I followed the teacher's explanations, watched the steps on the blackboard, and each time I thought I had it finally. It was so clear, so simple, logical. Then I would tackle the homework and it was all gone again.\"\n\nJill had set her mouth stubbornly. \"It's not like that.\"\n\nLois had left them to return to her quonset hut and the metabolism studies she was trying to finish. Now, thinking about that conversation, about nothingness, she felt that she had come back from nothingness herself, and it had not been enlightening or rapturous. The shadows had lengthened perceptibly and the only thoughts she could recall had to do with nothingness. Tired, she rose and walked into the house. She told Mrs. Carlysle she would not be home for dinner, collected her purse from her room, and left.\n\nMuch later she was still staining one slide after another, studying it under the microscope, making the cell count, and recording it. Her eyes burned and her headache was like the surf\u2014pounding, receding, pounding.\n\n\"I taught you well, didn't I?\"\n\nShe dropped the slide she was holding and spun her chair around. \"What are you doing here? Get out!\"\n\n\"Just wanted to see for myself.\" Earl Malik was slender, wiry, with black hair untouched by gray. His eyes were very dark, and bloodshot at the moment. His eyebrows nearly met and were bushy, much too heavy for his thin face; they gave him a made-up look, like a performer inept with the tools of his trade. He leaned against the door frame, looking about the lab with contempt. \"They're going cheapo, all right,\" he said.\n\nThere was her workbench with the microscope, the tray of slides, notebooks, a computer with figures in columns on the monitor. Behind her station were two more workbenches, sinks, shelves of flasks, burners... . Many high schools were better equipped.\n\nLois stood up and pushed her chair aside. \"Get out of here, Earl. Now. I don't have any more money. I don't have anything at all for you. Warren suspects you're around, and if he finds out for sure, he'll make a lot of trouble for you. Just get out and leave me alone.\"\n\n\"But, Lois, I'm your collaborator, remember? I taught you everything you know, and that's my work you're messing around with. Let's be reasonable. I don't want your money. I want us to work together like we used to.\"\n\n\"You've been drinking,\" she said with disgust. \"How did you get out here?\"\n\n\"Took a little walk, not that far, couple of miles. I waited for you last night, and the night before, and tonight I said to myself, old friend, she's not coming, so you'd better get your ass over there and see why not. So here I am.\"\n\n\"Earl, listen to me. I don't have any money left. I don't have access to his money, and even if I did, I wouldn't give any to you.\"\n\nHe shoved himself clear of the door frame and stood swaying. \"We'll work together, Lois. Like before. So you won't mess up again. I'll check things out this time.\"\n\nShe went to the computer and saved what was on the screen, turned it off, and began to stow away her materials on the workbench.\n\n\"Too early to stop now,\" he said. \"Couple more hours, finish up here. My work, Lois. Remember that, it's my work.\"\n\n\"It isn't!\" she cried, suddenly furious with him, with her self for not knowing how to get him out again, furious with the whole bloody mess, she thought. \"You botched it once, remember? You botched it. I started over from scratch, and this is my own work and it's damn good work. Now get out of here!\" She was screaming at him, trembling all over. She drew in a deep breath. \"If he does find out, I'll tell him you're blackmailing me, Earl. I will! He'll put you in jail!\" He smiled and took a step toward her and she rushed at him and shoved him backward, back through the open door to the outside. He staggered, caught himself, then fell into the hydrangea bush, cursing hoarsely. She grabbed her purse, flipped the light switch off, and pulled the door closed and locked it. He was scrabbling in the bushes, still cursing, when she ran to her car and got in and locked the door. Gravel flew as the wheels dug in and spun before the car jerked away and she raced up the drive to the road.\n\n\"What the hell are you doing?\" Al Zukal demanded Saturday morning. Sylvie was bustling around the kitchen as if preparing a holiday dinner, and it was just eight-thirty.\n\n\"Apple kuchen,\" she said with a withering look. Any idiot could tell that much if he just took the trouble to look.\n\n\"Yeah, yeah, I seen that. What for?\"\n\n\"Stanley might like a bite of something. You ever seen anyone turn down hot apple kuchen?\"\n\n\"He ain't coming for no breakfast. This is a business meeting, like I told you.\"\n\n\"He don't eat none, the kids'll polish it all off later. Won't go to waste.\"\n\n\"Yeah, yeah.\" Flora was due with the children by noon. This weekend Bobby was tied up at his job. With only two more weeks there he was trying to clean up things, not leave his boss in a lurch. Good guy, Bobby. Considerate like. Al looked at the table and groaned.\n\nHe had spread brochures and folders all over the kitchen table, but he knew he was out of his depth here. Stanley had sent the stuff ahead of time to give him a chance to look over the various proposals, and Stanley would be here before nine to review them with Al, but so far none of the stuff had made any sense at all. Insurance come-ons, certificates of deposit, trust deeds, treasury notes, equity loans. He understood none of it. Sylvie put two pans in the oven and a strong whiff of cinnamon drifted across the kitchen. He'd end up doing whatever Stanley thought best, Al had already decided. It was like a game with Stanley, Monopoly or something. See how much money you could make with money, watching it every second, keeping track, knowing when to move, when to sit still. And it was a headache bigger than the house for Al.\n\nWhen he heard a car in the drive, he went to the door to watch Jill come to a stop near the turnaround at the garage. The new station wagon was still parked outside the garage, but maybe this week he'd get to work hauling the junk out, hauling it away. Stanley got out of the car and crossed in front of it to stop at Jill's side. He leaned in and kissed her and then stood watching until she made the turn, headed out, and disappeared at the first curve in the driveway. When he turned so that Al could see his face, he was grinning like a kid with his own strawberry jam pot.\n\nHe started to walk toward the house, then veered in the direction of the station wagon and went to it instead. Mystified, Al watched him pull open the back and then scream hoarsely and throw both hands up over his face, backing up, screaming.\n\n\"Jaysus!\" Al ran out of the house, crossed the yard and came to a stop. Stanley had fallen down and was rolling over and over, and clustered all over him were bees, hundreds of bees. Stanley convulsed suddenly, his back arched; he shrieked and went limp.\nCHAPTER 8\n\nELLIS STREET WAS LINED with venerable maples that met overhead and made a tunnel with diffused lighting. Parked along the street that morning were two Cadillacs, one Continental, one Saab, several Volvos, and half a dozen other cars far less remarkable. Charlie found a parking space and pulled in. When he got out he noticed that their three-year-old Volvo needed a good washing. He took Constance's arm and they strolled back to number 1242, a neat, well-kept two-story frame house with a postage stamp-sized lawn closely sheared. On the porch they were greeted by a pretty, slender woman in white pants, a white silk shirt, and a blue sash at her waist.\n\n\"Welcome,\" she said softly and opened the door for them. They entered to find a closed door on one side of a foyer, and open double doors on the other. Many shoes were lined up in the foyer. They added theirs and went into the meeting room where a dozen other people had already assembled.\n\nAn assortment of cushions was at the end wall. They selected one each and looked around for a place to settle. The others in the room were in many attitudes, some lotus position, but not many. One woman was kneeling. Most of them were cross-legged or had their legs sprawled out. Charlie felt a touch of relief at that. He knew damn well that he could not sit cross-legged for more than a few minutes, and as for lotus, forget it. He indicated a place that would allow him to see the double doors, as well as a single door in the back of the room, and they sat on their cushions and waited.\n\nAt the far end of the room was the only furniture\u2014a tier of long tables covered with white cloths held many arrangements of flowers. Like a funeral, Charlie thought gloomily. On the floor in front of this was a red cushion on a white rug. A pagan funeral, he added. The windows were covered with white drapes that admitted light. On each side wall were two pairs of gold sconces with tapered lamps in them. The floor was covered with straw mats. Very simple, very soft sitar music drifted in from somewhere; there was no other sound except for the occasional rustling movement of one of the other attendees.\n\nAlready present, and seated, were eight women, four men. Most of them were simply dressed, in white or pastels. Most of them were waiting with their eyes closed, breathing evenly, some so deeply that it appeared they were in a trance state. Charlie recognized Jill from many newspaper photos when she entered. She was in white, a jumpsuit of some sort with a scarlet sash. She sat down cross-legged, bowed her head, cradled one hand in the other, and did not move again.\n\nSebastian's entrance was so underplayed that it would have been easy to miss it. He glided into the room and took up a lotus position on the red pillow; the sitar music ended abruptly. Most of those gathered bowed, some touched the floor with their foreheads, and then they resumed their silent, unmoving positions.\n\n\"The student came to the master and said, In my village is a very powerful man, a very rich landowner who tells me I must finish my studies quickly in order to return home and enlighten my fellow villagers and thus lead them to a happier life more quickly.'\" Sebastian's voice was pleasant, low-pitched, and conversational. He seemed to be making no effort whatever, but his voice carried throughout the room. He looked totally relaxed, his expression both serene and eager as he gazed at his listeners, first here, then there, including everyone in a very personal way that was engaging. He was dressed entirely in white, even his sash.\n\nWhen his gaze lingered on Charlie, the effect was strangely unsettling. He had the gift, Charlie thought then, the same gift that carried politicians into high office, that made evangelists draw thousands to their television sets, to stadiums where they preached. Myopic, Charlie told himself, that accounted for the peculiar staring quality of the man's eyes, but he knew it was more than that. Sebastian liked him personally; he liked everyone in that room personally. They all felt it and responded. That was the gift he possessed, the ability to project such warmth, such acceptance, liking, even loving the other, the stranger. Uh-huh, Charlie thought, and watched as Sebastian turned his myopic blue eyes here, then there.\n\n\"'What should I tell this powerful landowner?' the student asked. The master laughed delightedly and raised his staff and whacked the student across the head. 'Just this,' he cried. 'No more, no less.'\"\n\nSebastian smiled widely, as if he shared the master's delight. \"What a beautiful story,\" he said. \"What a beautiful man was this master, how fortunate this student. But let us examine this student. A student is one who is receptive, who comes empty, yearning to be filled, who comes with no thought of what he has left behind, but only of what is before him. The student must sever his ties to the past, because the past shackles him to his ignorance, to his preconceptions, his false thoughts... .\"\n\nCharlie's legs were going to sleep. He glanced at Constance who had taken her aikido position of sitting back on her heels, her hands loosely joined before her. She looked as comfortable as Sebastian; she looked as if she could stay that way for hours. He had tried that position once, only to find that after ten minutes he could not straighten his legs at all. He had hobbled for an hour, he remembered bitterly. He knew she was thinking, I told you we'd have to sit on the floor. He refused to look at her again, but began to ease one leg inward, thinking if he could just bend that knee a little, then work the other leg in a little. He caught a glance from one of the women, and stopped moving temporarily.\n\n\"... can have but one master. A student can have no thoughts for those he left in his village, whether family, friends, lovers, whoever... \"\n\nHe had not even got to the rich man yet, Charlie thought, and began to ease his leg again. Across the room a woman went from lotus to simple cross-legged. When he realized that no one gave her a glance, he drew in his right leg. Several people looked at him, their concentration broken.\n\n\"How long had this student been harboring this burdensome problem that he evidently brought with him to the ashram? Lost time, every second of it gone, never to be recaptured, and he would have to start...\"\n\nJill had not yet moved. How could she do that? Her head was turned so that he could see only the tip of her ear, a bit of her chin. Too thin, much too thin. Anorectic, like most young women, he decided, especially most rich young women. Why did the ones who could afford to eat anything so often choose to eat practically nothing?\n\n\"... did not say young man, or old man, a wise man or a fool, a good man or an evil man. No, he defined him in the simplest terms available, and the most untrustworthy: a powerful rich man and a landowner...\"\n\nCharlie looked at Sebastian with awe. He was still going on about the student. But it seemed he intended to slop over to the landowner any day now, any month now...\n\nAnd he, Charlie, was going to die right here, gangrene would set in, both legs would fall off.\n\nSuddenly Warren Wollander stood in the doorway looking at the group. Charlie recognized him, too. There was a flutter of stirred air, as if his presence affected the charge, and molecules were rushing back and forth in a dither searching for equilibrium again. Warren spotted Jill and went to her quietly, whispered into her ear, took her arm, drew her to her feet and took her away.\n\nSebastian's voice faltered less than a second, then he resumed his analysis, although most heads had turned to watch the exit of father and daughter.\n\nCharlie jumped to his feet as soon as Warren led Jill out of the room, but one of his legs really had fallen asleep, and he would have plunged headlong into his nearest neighbor if Constance had not caught and steadied him. Leaning on her heavily, he hopped from the room, dragging his useless leg.\n\nThey reached the porch in time to see Warren depositing his daughter in the passenger seat of a Buick that he then entered and sped away in.\n\n\"I'll be damned,\" Charlie grunted. \"Now what?\"\n\n\"I don't know. Wait here and I'll get the car. Flex that leg to get the blood running again. Be right back.\"\n\n\"I can make it,\" he protested.\n\nShe was already moving. \"You hold onto that rail and exercise. I'd rather get it alone than carry you down the street.\"\n\nShe hurried away and he bent his knee, straightened the leg, bent it again. He was now getting pins-and-needles, and they were all red-hot.\n\nShe drove to the curb moments later and he walked to the car, determined not to limp or hobble. \"Fine,\" she said, grinning. \"Where to?\"\n\n\"Let's just drive past the Wollander house, and Al's place. See if anything's stirring.\"\n\nActually they didn't get farther than Al Zukal's house, and it was clear that whatever had happened, had taken place there. The sheriff's men were on the road, waving traffic on, and a medic unit was pulling out when Constance drew near.\n\n\"Make a U-turn, back to the farm,\" Charlie said, craning to see something besides the traffic cops. The medic unit had left without its siren blaring, meaning no one inside it would benefit by being rushed to a hospital.\n\nHe slouched down in the seat. First the poor dumb mutt, then the boy next door, and now what? Al? Sylvie? He remembered Al's words: \"You sure didn't do him no good.\" Christ, he thought then, there was another possibility they had left unexamined: What if Al or Sylvie was really the target of the murderer? But who out here could want either of them dead? He scrunched down in the seat scowling.\n\nConstance made a turn and headed back. She pulled into the driveway of the farm, and there was another police car with an officer leaning against the door.\n\nThis officer's instructions had been simple: keep those people at the farm until the sheriff got through up at the mill. Nothing about keeping anyone out, or stopping anyone from talking to anyone else. He didn't interfere when Constance parked and she and Charlie got out of the Volvo and entered the building.\n\n\"What's going on?\" Charlie asked Tom Hopewell, who had opened a door to peer out when he heard them enter.\n\n\"Come on in,\" Hopewell said, and swung the door open farther to admit them. \"Clarence, Mr. and Mrs. Meiklejohn. Private investigators. Dr. Bosch,\" he finished, and closed the door again.\n\nThey were in a small office cluttered with too much furniture and too many file cabinets. There were also three computers and monitors, all working with blinking lights and ever-changing columns of figures. Clarence Bosch was standing at the window when they entered, a slender, pale man in his sixties, with thin gray hair that did not quite cover his scalp, and oversized glasses with dark frames. He seemed confused by their presence.\n\n\"What are you investigating?\"\n\n\"David Levy's murder,\" Charlie said.\n\nClarence Bosch sank into a chair behind the desk, staring at him. \"Good God!\"\n\n\"What's going on up at the mill?\" Charlie demanded. \"Why are the cops here?\"\n\nBosch shook his head helplessly and Tom Hopewell said, \"I was hoping you could tell us. All I know is that Lois went tearing out about forty-five minutes ago, and soon after that the cop outside arrived and said we should hang around and wait for the sheriff.\" He spread his hands and shrugged, then sat down at one of the computers and gnawed on his finger, watching the numbers scrolling past.\n\n\"Come on,\" Charlie said to Constance.\n\n\"But we're supposed to wait,\" Bosch said.\n\n\"Wait then. See you later.\"\n\nThey went through the hall, out the back, past the rows of tagged and tented vegetables and on to the quonset hut. Charlie stopped and looked at the gravel scattered on the path, on the single step that led inside.\n\n\"He said she left in a hurry,\" Constance said.\n\n\"Twice. Come on.\"\n\nThey continued to the path among the tagged trees, and a few minutes later they emerged onto the mill property. They had almost reached the house before one of the deputies spotted them and escorted them to Sheriff Greg Dolman, which was where Charlie wanted to be at the moment.\n\n\"How the hell did you get in here?\" Greg snapped.\n\n\"Just out walking in the woods and came to call. What's up?\"\n\nThey were near the driveway where the station wagon was parked at the side of the garage. Half a dozen deputies were standing well back from the station wagon. Squinting, Charlie could just make out the box inside the wagon. And bees seemed to be everywhere, zooming back and forth purposefully.\n\n\"We're waiting for the guy who owns the hives,\" Dolman said. \"He knows how to get that hive out of there. We're not about to touch it.\"\n\nCharlie nodded. \"Who opened the back?\" he asked grimly.\n\nDolman gave him a quick look and nodded. \"You read it right,\" he said with a sigh. \"Stanley Ferris. Dead. And old man Wollander's after someone's scalp for this.\"\n\nIt took several seconds for the name to register. Ferris, Jill's husband, Wollander's son-in-law. He exhaled softly. \"Boy oh boy, Greg. You've got a hot one this time.\"\n\n\"Yeah. Punk kids did it, trying to tie a can to Zukal's tail, get him out of here. Pothead friends of that Levy, I bet. Got high, thought, what a gag it'd be. And now Ferris is dead. I'll get them, Charlie. You better believe.\"\n\n\"How are the Zukals?\"\n\n\"Real shook up. Pretty bad. She told me about those letters. Guess they're getting a message all right. Goddamn punk potheads!\"\n\n\"I'll go see how they are,\" Constance said in a low voice.\n\nCharlie squeezed her arm slightly and she left them standing near each other regarding the hurrying bees. She went to the kitchen door and peered through the screen to see Al and Sylvie at the table, both voices going.\n\n\"Damn it, it's getting in my eyes.\"\n\n\"I can't help it. You're hot as a firecracker. Ice melts soon as it touches you. Hold still.\"\n\n\"Let me have that.\"\n\nConstance entered. Al had been stung too. His face was puffed on one side, and both hands were blotchy with red welts. Another one was on his neck.\n\n\"Did someone take the stingers out?\" she asked over their voices.\n\n\"Miz... Miz...\"\n\n\"Just Constance. Did the medics treat him?\"\n\n\"Yeah. They used tweezers and got the stings, and they said to keep ice on them, but he won't sit still.\"\n\n\"I'm sitting still, but it's running down my neck and in my eyes. Damn it, Sylvie, watch it.\"\n\nShe had turned to speak to Constance and was jabbing a cloth with ice cubes wrapped in it against his ear.\n\n\"What my mother used to do,\" Constance said, \"was use baking soda. You have any?\"\n\nSylvie nodded toward the cabinet, and presently Constance had a paste made and began to dab it on the stings. \"How about some vodka?\" she asked.\n\n\"You want a drink?\" Sylvie asked in wonder.\n\n\"No. Just another trick from the old country.\"\n\nSylvie went out and returned with a bottle of vodka. Constance put ice cubes in two glasses and added an inch of vodka to them and swirled them around a few seconds, then took them both to the table and handed one to Sylvie, one to Al. \"Aspirins?\" Al lifted his glass and downed the vodka in a gulp, and after a second Sylvie drank hers.\n\n\"Aspirin in the bathroom, down that hallway.\" She kept smearing the baking soda paste on Al.\n\nConstance found the bathroom, and aspirins in the cabinet, along with cotton balls, and brought both back to the kitchen. \"Another thing she did,\" she said, mashing a dozen of the tablets in another glass. \"The paste dries out and falls off. Makes a real mess, I'm afraid. But when it did, she would swab off the sting with this solution, clean it, and then put more paste on.\" She poured vodka on top of the aspirin granules and stirred it. \"Helped us. How are you doing?\" she asked Al kindly.\n\n\"I'll live,\" he muttered. Some of the paste on one of his hands was already flaking off. Constance dipped a cotton ball in the vodka-and-aspirin solution and cleaned the sting, and then reapplied the soda paste. Al's wrists were very hairy.\n\nOne of the deputies appeared at the door, wanting keys for the station wagon. The bee man had moved the hive, he reported, but he wanted to move the wagon away from the area because the bees kept going back inside it.\n\nA few minutes after that Charlie appeared at the door. \"I'm going down to the farm with Dolman. Won't be long. How are you, Al?\"\n\n\"Not bad. Not bad. Constance knows a trick or two them medicine men could use.\"\n\n\"I bet she does,\" Charlie said, grinning, and trotted off.\n\nSylvie cleared all the brochures and pamphlets from the table and brought the apple kuchen over along with the coffee.\n\n\"See,\" Al said, as she moved back and forth arranging things, \"Stanley, he was a genius with money, and he was coming to be my advisor. What he said, my advisor. Early, before the kids get here making a whoop and holler all over the place.\"\n\n\"They're coming pretty soon,\" Sylvie muttered. \"I don't know, Al. This just about does it for me. You know what I mean?\"\n\n\"Yeah, yeah. I know. So he's coming over, like I said, and Jill drives up and turns around, and he gets out and smooches her real big through the window and off she goes. And I'm thinking now he's coming on in, but he turns to the station wagon and pulls the back open and the bees are all over him. Just like that. He's yelling and throwing his hands over his face trying to get them off, and he falls down, and by then I'm going out there, and I'm trying to get them off him and pull him away. I tried artificial resitation. I took a class once, but it don't do no good. He's a goner even before I reach him. Bees on his face, even his eye, all over his arms.\"\n\n\"And I'm right behind Al and I seen him fall down and I run back in and call that emergency number wrote down on the phone book and the bitch goes you gotta stay on the line and I go, no way, I gotta call Wollander, and he shows up about the same time them near doctors get here in their truck and when he sees that Stanley is a goner, he goes tearing out for Jill.\"\n\nSylvie dabbed more paste on Al's cheek, and they all ate kuchen and drank coffee and Sylvie and Al talked about staying here, or going back to the city, or getting a hotel room for a few days, and about installing a real professional security system on Monday, and maybe trying another dog. And finally Charlie came back with Greg Dolman.\n\nThe sheriff's tone was solicitous. \"You doing better now?\" His eyes were as hard as ever. Al nodded, and he said, \"Look, if you get any more letters, give me a call, will you? And if you decide to go somewhere else for the next few days, let me know where I can find you. Wouldn't blame you for leaving for awhile. But we'll get them kids. We'll get them.\"\n\nHe started to leave, then paused. \"My guys have been holding the reporters off, but when the word gets out that it was Stanley Ferris, they'll be thicker than fleas out here. I'll leave someone to hold them off while I go talk to Wollander. Another hour, hour and a half.\" He nodded to them and went out the door.\n\nWordlessly Sylvie got up and poured coffee for Charlie. She examined the paste on Al's stings and cleaned off a couple of them. \"What'll we do, Al?\" she asked then, subdued. \"That sheriff ain't going to do us no good.\"\n\n\"Yeah. I don't know. Maybe we'd better just take off.\"\n\n\"Al,\" Charlie said gravely, \"your station wagon still has some bees hanging out in it. The beekeeper plans to come back after dark, after they're all back in the hive, and take the hive back down to the farm. Meanwhile some of the bees seem confused and keep homing in on the wagon.\"\n\nAl shuddered and Sylvie made a moaning sound so deep in her throat it sounded inhuman.\n\n\"Our car is down at the farm,\" Constance said. \"After your daughter comes and you send her back home, you can go down through the trees and take our car. You can go to our house and wait for us. Do you mind cats?\"\n\nCharlie blinked at her, then nodded slightly.\n\n\"Cats?\" Sylvie asked uncertainly. \"Your house? Where's that?\"\n\n\"We'll give you a map,\" Constance said. \"Three cats. They'll hate having you show up in our car, and they may even just take off and sulk. Don't worry about them. You can relax and try to decide what you want to do without any reporters bugging you. We'll wait until the beekeeper comes back and clears them all out and then join you.\"\n\nAl and Sylvie looked at each other, then looked more closely at Constance and Charlie, and in the end they nodded and began to talk about what they would take with them. \"Not too much to lug through them woods,\" Al said, and she said, \"Three, four days of clothes. Is the suitcase in the attic or the basement?\" When they began to talk together, Charlie leaned back in his chair and grinned faintly at Constance.\n\n\"Well?\" Constance asked later, when she and Charlie were alone in the house. The daughter and grandchildren had come and gone, the Zukals had left. The reporters had not landed yet.\n\n\"No one knows when the hive was taken. It could be one that was put under the trees in April when the trees were blooming. They're checking now. You can move a hive at night, no problem. Probably it was done overnight. The sun heated up the wagon and the bees got mad when they couldn't get out and tend to business. And that's about all. No fingerprints on the hive, or the wagon.\"\n\n\"It's so ugly,\" she said after a moment. \"What if one of those children had opened the station wagon first?\"\n\n\"Everyone's thought of that,\" he said quietly.\n\nShe told him the story Al had told her. He didn't know why Stanley Ferris had opened the station wagon. There wasn't any reason for him to go near it that Al knew. He and Sylvie had used it yesterday and there wasn't anything in it, nothing that should have been of any interest to Stanley.\n\n\"They know it was meant for them,\" she said matter-of-factly. \"They're taking it a lot better than I would be doing.\"\n\nCharlie nodded. He knew he would be out with a gun looking for someone to shoot along about now.\n\nThe reporters came to the porch and Charlie got rid of them. He didn't know where the Zukals had gone, he was house sitting for them, he didn't know anything about anything. That was the station wagon, and it still had bees in it; if they wanted to go get pictures, fine. They left. Constance straightened up the table and put the baking soda away again, put dishes in the dishwasher, and stood looking at a can of dog food left in one of the cabinets. Sylvie's green kerchief was on the counter, and the stack of brochures that Stanley had sent them. She sighed and turned to the door as someone knocked. More reporters, she thought.\n\nBut it was Lois Wharton Wollander. She was very pale and had been weeping; her eyes were inflamed and puffy.\n\n\"Oh,\" she said uncertainly when Charlie opened the door for her. \"I came to see how Al is.\"\n\n\"He'll survive,\" Charlie said. \"Come in. How's Mrs. Ferris?\"\n\nLois shook her head. \"It hit her pretty hard. She... she miscarried. The doctor has her sedated now.\"\n\n\"Hospital?\"\n\n\"No. She was only a few weeks pregnant, he said that physically it shouldn't be too serious. She just needed calming down for now.\" Lois sank down into one of the kitchen chairs. \"Have Sylvie and Al left for good?\"\n\n\"Probably not. They're tough, and once they get over the shock, they'll be ready to fight back.\" Charlie studied her thoughtfully, then asked, \"Were you interested in buying this property?\"\n\nStartled, she shook her head. \"What for?\"\n\n\"Your work, maybe. For the school, maybe. I don't know.\"\n\n\"That's a crazy idea. It never occurred to me.\"\n\n\"What's the cause of the hard feelings between Clarence Bosch and your husband?\"\n\nShe flushed. \"I don't know that there are hard feelings.\"\n\n\"Mrs. Wollander,\" Charlie said, drawing out a chair and sitting down, \"murder has been done here twice now. A lot of questions are going to be asked, and if they aren't answered, a lot of digging will be done anyway. There aren't many secrets that will stay under wraps from here on out. If there's something between Bosch and your husband, it'll come out.\"\n\n\"Ask them,\" she said sharply then. \"Why are you asking me things like that? Ask them.\"\n\n\"I will. And I'm asking you because you're here. There's been a death in the family, your stepdaughter's had a miscarriage, your husband must be devastated by such tragedies, and you're inquiring about a neighbor you just met recently. I'm surprised you're not over at your own house holding someone's hand, or making tea, or answering the phone. You know, rallying around.\"\n\nShe rose stiffly. \"I was on my way to lock up the lab. I won't be in it for days, of course. I just stopped in to see if Al was all right. I'll be going now.\" She went to door and paused. \"The sheriff is calling Stanley's death accidental. Some boys were playing a prank on Al Zukal and it got out of hand.\"\n\nCharlie nodded. \"I know what the sheriff is saying, but I'm still investigating murder, Mrs. Wollander.\"\n\nShe walked out, and he watched her return through the yard toward the break in the fence. She had just reached the area where trees and unruly undergrowth would hide her when Tom Hopewell and Clarence Bosch emerged from the tangle. The three scientists stood together talking for a few minutes. Lois turned and headed toward the mill and the swinging bridge. Hopewell watched her go out of sight, then he and Bosch continued to the house.\n\n\"Reporters,\" Tom Hopewell said with disgust when they entered. \"They're getting a human-interest story from some of the workers. We decided to duck out until they're done.\"\n\n\"Did Al and Sylvie get out before they arrived?\" Constance asked, ushering them both into the living room. The furniture was very good in here\u2014two sofas, comfortable chairs, and pleasant lamps; all subdued colors, ivories and tans with green and blue accents. One of Sylvie's scarfs was on the sofa, a beer can was on an end table, a child's toy vacuum cleaner stood in the corner. A window seat at a bow window was completely covered with children's coloring books and crayons.\n\nTom Hopewell nodded. He seemed about to say something, but held it back. All the friendliness that he had shown them earlier was gone now. He looked angry and upset.\n\n\"Al said you would be up here until dark, and I thought this would be a good time. I wanted a word with you,\" Bosch said. \"You made a serious statement earlier. You said David Levy was murdered. Why, Mr. Meiklejohn? On what basis?\"\n\nCharlie shook his head. \"Let's just say it's what I believe. Why were you feuding with Wollander?\"\n\nBosch groaned. \"This is what I was afraid of,\" he said. He was perspiring and took off his coat. He had on a short-sleeved shirt, open at the throat. His arms were tanned up to the edge of the shirt sleeves, and when he moved, the white above the sleeve appeared and vanished. \"Questions. The past dug out and aired all over again. That's old history. It has nothing to do with the present.\"\n\n\"If anyone knows, I'm bound to find out,\" Charlie said reasonably. \"Why not let me have the real story?\"\n\nWhen Bosch hesitated, Constance said, \"I can see the tabloid headlines now: attack of the killer bees, old feuds settled at last.\" She looked at him kindly. \"You know that's how it will be treated.\"\n\n\"I know. It's just a silly thing, sordid and unpleasant, but not relevant. And I know you're right, it will come out again. Back in the seventies, you remember all the turmoil on campuses, everywhere. It was here, too. Wild kids, rebellious, experimenting with LSD, magic mushrooms, whatever came down the road. They got out of hand here, too. I threatened them with expulsion\u2014there were eleven graduate students, supposedly young adults, too old to require supervision. Anyway, there were parties, and people from around here were upset because some of the local kids got involved. I had my students in and warned them that at the next party I'd call the sheriff and round them all up. And I did. One of the locals who got hauled in was Jill Wollander. A kid, fifteen, sixteen. And Warren threatened to have my head. He said things he shouldn't have said and in self-defense I got an attorney and slapped him with a suit. To shut him up, stop his threats. It worked, and he hasn't spoken to me since.\"\n\n\"What kind of threats?\"\n\n\"What you might expect. He would close down the farm here, or have me removed, things like that. Called me incompetent, said I provided the atmosphere, and maybe even the drugs, let things get out of hand. It was ugly.\"\n\nHe looked grim, his mouth set in a tight line. His soft, slender appearance had been deceptive; now with his coat off, his arms revealed wiry muscles when he moved, and his expression was obstinate. He would take on Wollander again if he had to, Charlie thought. A tough old bird who would fight city hall and all the king's men, whoever got in his way.\n\n\"And now your bees kill his son-in-law. Oh yes, it will be aired again, I'm afraid.\" Charlie regarded him with commiseration. \"Tell me something about the bees. Could anyone have gone in there and just picked up a hive and walked out again?\"\n\n\"Walked out, maybe. But our night watchman says no one drove in or out after Lois left at ten-thirty. The sheriff practically accused him of lying about it, falling asleep on the job, or something, but I believe him. He's a good man, and he's been with us ever since I've been there, twenty-one years. There's no reason for him to lie.\"\n\n\"How heavy are the hives?\"\n\nBosch glanced at Hopewell and shrugged. \"Seven, eight pounds. They're lightweight, meant to be carried out to fields or berry patches. And if it was the one left under the trees, it didn't have to be carried far.\"\n\n\"If you don't think it was neighborhood kids, who do you suspect?\" Charlie asked softly then.\n\n\"I didn't say that,\" Bosch protested. \"I'm just saying they didn't drive in with a truck and collect a hive. And if the hive was up in the trees, they wouldn't have had to drive in. Maybe they spotted it earlier and knew it was just a few feet from the fence. That fence is down here and there, has been for years. Harry wouldn't have seen them necessarily if they had gone in that way, through the trees, but he would have seen a truck. There wasn't one. That's all I'm saying.\"\n\n\"What happened to Jill after the cops collected her?\"\n\nBosch shook his head. \"I don't know. I think Warren packed her off to live with her mother. Maybe they put her in a convent. Where she belonged in those days. With a high fence topped with barbed wire. The next time I saw her, four, five years later, things were different here. That crazy phase had ended.\"\n\n\"Does she speak to you?\" Charlie asked.\n\n\"Sure. She's been over a time or two to bring Lois mail that the rural delivery woman leaves over at the house. Jill is as friendly as a pup. But Warren will never forgive or forget.\"\n\nTom Hopewell was getting restless, tapping his fingers on the arm of his chair, crossing and recrossing his legs. Charlie glanced at him. \"Anything you can add?\"\n\n\"Nope. I agree with Clarence about the truck. Actually, what I suggested we do is inspect the fence line and see just where the breaks are, and if there are traces someone might have left. And we decided that we shouldn't do it ourselves, alone, I mean. We thought maybe you would come along.\" He stood up and began to move about the room.\n\n\"You really don't want the sheriff to pursue the truck idea, do you?\" Charlie murmured. \"Afraid of a blot on the reputation of your watchman? What is it?\"\n\nBosch looked more obstinate, more irritated than before. \"That's partly it, of course. We don't want a blot, not now. I'm due to retire the month after our grant runs out, three years, six months, nine days. And I don't want to get into another fight with Warren. He could start a lot of trouble, talking about dangerous, uncontrolled bees, that sort of thing. He couldn't shut us down, but it would be a nuisance that I frankly just don't have time for. But more than that. You said that David Levy was murdered. Presumably you have your reasons. Lois said the dog was poisoned. I assume she has her reasons for believing that. And now, the bees. Someone obviously has launched a campaign of terror directed at the Zukals with the possible intention of driving them away. And it's just a matter of time until the rumors start flying that one of us at the farm may be behind it.\" His voice had gone very dry, his words precise and clipped. \"It is no secret that I tried to get our board to buy this property on two different occasions. The second time it seemed a possibility, but then funds vanished and there simply wasn't enough money to make it feasible. But that's the sort of thing people will be saying if this doesn't get cleared up quickly. And if the sheriff is off searching for an illusory truck, it won't be cleared up quickly or any other way. I told him about the fence being down and he sent an incompetent to have a look, but that was the extent of his interest. I thought you might be more interested, if you are seriously looking for a killer.\"\n\nCharlie nodded. \"I think we should go look at that fence. How I heard it was that when Lois Wharton married Wollander, it was assumed that she would add the mill property to the farm.\"\n\nBosch shook his head vehemently. \"Rubbish. Warren would be just as likely to cut off his arm and present it as a gift. Let's go inspect the fence.\"\n\nTom Hopewell had been standing at the bow window gazing out; abruptly he swung around. \"Wait a minute,\" he said. \"You know where rumors like that lead. If it wasn't bad kids playing bad jokes, it must have been an outsider, someone who tried to move in on the old families by marrying one of the big shots. They sure don't need you to suggest any such thing, Meiklejohn. They'll think of it all by themselves. Come on.\" He strode from the room, out of the house, and led them to the break in the fence. \"I found it on one of my rambles years ago,\" he said. \"There's another section down nearer the road, and of course the one at the lake end. Everyone knows about all of them.\"\n\n\"We talked about fixing it at a board meeting,\" Bosch said dryly. \"But it's mill property, not ours, and there's just no money for anything extra.\"\n\nOn the mill side of the fence there were fine old maple trees with a thicket of straggly new shoots that competed for light and room. Closer to the fence there were sumacs and a few blackberry brambles. The break wouldn't be noticeable to anyone not aware of it. Then the ground was too rocky to support even that underbrush, and here the fence had been bent over. The break could have been used recently, or not; it was impossible to say.\n\n\"Deer do it getting down to the lake,\" Bosch said, surveying it. \"Snow piles up, they walk over it. Comes a thaw, it's low enough they still walk over it, or maybe jump over it. Two, three seasons and it's all the way down.\"\n\nThey stepped over the fence and kept to the rocky ground for fifteen or twenty feet where more sparse undergrowth struggled, and then they were among the experimental trees.\n\n\"Where was the hive?\" Charlie asked.\n\n\"I don't know. Lois's chart will show where it was placed. But you can see that I was right, no one had to drive in. No one had to carry the hive more than a couple hundred feet.\"\n\nCharlie nodded slowly, and said even more slowly, \"I see. And you're right, it would have been easy for anyone who knew the hive was up in the trees here, and who knew there was a broken section of fence, and who knew his way around in the dark, and who didn't mind leaving a car or truck parked somewhere and walking in nearly half a mile. Doesn't sound a hell of a lot like your typical teenagers. How I hear it is that they won't walk to the mailbox if they can help it. And probably not your typical transient from Toledo.\"\n\nBosch returned his skeptical gaze steadily. \"Warren probably knew all those things. He's a good walker, and he knows every inch of all this area, dark or light.\"\nCHAPTER 9\n\n\"WHAT DO YOU THINK?\" Charlie asked a few minutes later. He and Constance were walking at the edge of the lake on the farm grounds. Across the lake the woods had ended at a grassy slope; the upper part of the Wollander house was visible, as if rising from an emerald carpet.\n\n\"I think that if our culprits are neighbor kids, the sheriff is in a much better position to find them than we are.\"\n\n\"Granted. What about Bosch?\"\n\nShe knew that Charlie had a lot more faith in instant analysis than she did, and that he believed implicitly that she had the ability to peer through all layers of masking to see to the core. She had disputed this too many times to try yet again. Instead, she said slowly, \"He's driven to complete his work, of course. I plant half a dozen vegetable varieties that he developed and introduced. Tomatoes, peas, some peppers. He's done very important work in horticulture. I can imagine the fights he's had just to get where he is, spending the last ten years of his professional life doing exactly what he wants. The bureaucracy probably put up roadblocks all the way, but he won. I wonder what really lies behind his hatred for Warren Wollander.\"\n\n\"You don't believe the story about the bust, about Jill being caught in the net?\"\n\n\"Oh, certainly. But I suspect he knew ahead of time that she would be. Maybe that's why he called the police,\" she added thoughtfully. \"It really isn't how the system works. You have to see the university as a clan; those who wear the tartan keep the problems at home, keep the family secrets hidden away, they don't call in the police and get their own into that kind of trouble. They just handle it in house, the way Tom Hopewell handled his student. Nothing on the record, no police, no bust, matter ended. And Clarence Bosch has been a clansman his entire life, as a student, then a teacher; it's the only world he's ever lived in. He wouldn't have done that unless there was something else behind it. And it seems to me that Warren must have overreacted to the incident if Jill just happened to be one of several. But if she was a target, then fifteen years of hostility isn't so excessive, is it?\"\n\nCharlie nodded, and gripped her hand tighter. That hadn't occurred to him, and he believed every word of it. See? he wanted to tell her. See what you can do without any effort at all. He said nothing like that because they had had that debate too many times, and it was one he couldn't prove or win. But he knew. He nodded toward the general view before them.\n\n\"Pretty, isn't it?\"\n\nOn this side the ground sloped gently to the lake, with knee-high grasses, waist-high in spots, and wildflowers in bloom. Butterflies and bees were everywhere, but here they seemed benign, a piece with the setting. On the other side the grass was clipped, and the slope was steeper and higher. They had drawn even with the small beach, a length of golden sand twenty feet wide, fifty or sixty feet long. It looked as artificial as it was, but it, too, was pretty, uncluttered and clean. How much value did Warren Wollander place on this serenity, this privacy, this quiet? Charlie tried to imagine the lake filled with loud-mouthed youngsters, and Sylvie and Al's voices raised in warnings, and radios blaring, bonfires blazing, hotdogs burning. Paradise lost? Maybe, he decided. Maybe.\n\nYet the thought was swiftly followed by another; he preferred the chaotic imaginary scenes he had conjured. Knowing the lake was beautiful and dead, that the water flowing into it brought only more death tainted the beauty; unbidden, unseen, unidentified evil had insinuated itself into the pristine valley and now owned the dead waters. Silver wind ripples played on the surface, a mockery. He forced his gaze from the lake.\n\n\"What I'd like to do is walk to the end, see how hard it might be to cross the stream up yonder. Game?\"\n\nShe nodded. \"I suspect what you'd really like to do is walk across the water and beard the lion in the den,\" she murmured, her expression calm, her eyes busy taking in the landscape.\n\nHe grunted, and they walked on toward the end of the lake.\n\nIt would be hard, he had to admit a few minutes later. The lake narrowed until it became a swift stream that ran ten to twelve feet down a rocky gorge. Not wadable, he decided, unless you didn't mind rock climbing, and getting soaked, and then what was the point? You might just as well swim across the lake itself.\n\n\"You know,\" he said, \"there might not be a connection, and that's why we can't see it. Like the little man on the stair.\"\n\n\"Three murders\u2014if you count Sadie, and I do\u2014and no connection? How likely do you suppose that is?\"\n\n\"Not very, but possible,\" he said grumpily. \"And I'll be damned if I can link them. A dog that's trained not to take food from anyone but the owner, poisoned. A boy who doesn't do drugs, overdosed. The wrong schmuck opens the car door, dead of bee venom.\"\n\n\"You agreed to look into the death of David Levy,\" she reminded him. \"Just that.\"\n\n\"Yeah, yeah. But the other two tag along like kids from the first marriage. What does David's death have to do with forcing the Zukals out of here? Why on earth would anyone here want to kill the Zukals? What kind of nut would kill someone just because he saw him toss bad hamburger to a dog?\"\n\n\"Maybe someone whose reputation is extremely important to him,\" she said after a moment.\n\n\"And we don't have anyone around here like that,\" he grumbled. \"Just a few world-famous scientists, a presidential advisor, and a guru.\"\n\nShe laughed softly. \"You're just hungry. Al and Sylvie will eat our food, we eat theirs. Let's go see what kind of food they provide.\"\n\nWhen they reached the break in the fence that Lois always used, they saw a woman waiting for them. She was middle-aged, with gray hair, wearing a simple skirt and blouse, and sandals; the sort of woman, Charlie thought, whom you saw in supermarkets and discount stores clutching coupons and comparing prices, and whom you forgot instantly.\n\n\"Mrs. Meiklejohn? Mr. Meiklejohn? I'm Carla Mercer, Mr. Wollander's secretary. He would like to talk to you this afternoon, if it's convenient.\"\n\nShe did not offer to shake hands, and got no closer than ten feet. Charlie imagined that she could type up a storm. He nodded. \"Fine. We were hoping to get to talk with him, too. When?\"\n\n\"He suggested three.\"\n\n\"We'll be there. Up through the woods okay? The station wagon has been invaded by bees and we're on foot.\"\n\n\"It's the best way,\" she said seriously. \"I'll tell him.\"\n\nThey watched her cross the swinging bridge, staying well in the middle, not looking down at the water at all. Holding her breath, Charlie thought; when she reached the other side, he heard Constance exhale softly. He chuckled and took her elbow. \"Food. And I don't care what it is as long as it's soon.\"\n\nLois sat in a small room they called the flower room, a solarium, sun-drenched in the winter, filled with greenery and blooming plants that got carried out to the patio at the start of summer each year. The room was white, with tall windows, and window seats; although it looked stark now without the greenery, in winter it was a cheerful haven from the often-bitter cold, and it was her favorite room. She held a book in her lap but she had not read any of it since entering the room; it was a shield, to be used only if someone came in. She did not expect anyone to come in.\n\nNow and then she heard herself say under her breath, \"Poor Jill.\" But it was mechanical, as if someone else were saying it. Dissociated, she thought, remembering an article she had read. One of the stages of schizophrenia was dissociation, in which the person saw herself as if she were an actress in a play. And that was exactly right. She kept seeing herself at different stages of her life, herself as a child, as Earl's bride, as a student, facing her committee. Warren's bride. It was strange that the figure she saw was so identifiable and yet so distant.\n\nHer mother screaming at her father: \"Don't you see that she doesn't want you pawing her like that?\" Her father drunk, trying to draw her close, his arm around her shoulders, swaying and leaning on her more and more heavily. \"You love the old man, don't you, sweetheart? You know I don't blame you, don't you?\" Until then she hadn't even considered that he might blame her for anything; until then she had loved him, although she tried to avoid him when he was drinking because at those times she was afraid of him. The fear was too general to have a real basis, she knew, but it was there, strong, pounding, making her shrink away from him then. At that moment she had realized that she was to blame for his failure; if she had not been born, then maybe he would have become the scientist he yearned to be, not the junior high school teacher he actually was. Her fault, she accepted, and she shrank away from him because he might have to punish her for destroying his life, and finally the fear of his drunkenness had a cause.\n\nTen years later, her father long since dead, her mother insurance-rich, she had been confronted by the ghost of the past. Earl, drunk, weaving back and forth, lurching against a door frame, clutching the table for support, cursing the committee that was persecuting him. And she shrank away from him because she was afraid of him. She could say nothing, because anything she did say would provoke a tirade aimed at her; he would lash out, once he even swiped at her and fell heavily onto the couch when he lost balance. And the next day he would forget, or pretend to forget, all of it.\n\nNow the ghost was walking again, had entered her life again, and it brought the same fear as before, the same helplessness. She could see the twelve-year-old child pleading with God to make it not happen, to make it stop. \"I'm sorry,\" the child whispered into her pillow. \"I'm sorry.\" She could see herself at twenty-two: \"I'm sorry. I should have checked everything closer. I should have double-checked. I shouldn't have been so trusting, so confident. So stupid.\" That time she whispered into the dark in their bedroom while Earl crashed around in the living room beyond the locked door.\n\nLast night, when he crashed into the bushes outside the quonset hut, her fear and helplessness had been exactly the same as the child's many years earlier. Nothing had changed.\n\nShe stared out the window at an oak tree unmoving in the sunlight and wished for a storm, a hurricane, a volcano to erupt, a tornado to wipe everything away, a flood, anything. Exactly the way a child wishes for a catastrophe, she thought wryly, and turned to see Warren standing in the doorway. How long he had been there she didn't know, but she had the feeling that he had been watching her.\n\n\"I have to tell you something,\" she said then, surprised at the steadiness in her voice. \"Earl Malik is hanging around. I tried to pay him to go away again, but I don't think he will.\"\n\nWarren shook his head, as if trying to recall who Earl Malik was, or why she was bringing this up now. She felt a wrench at the sight of him; for the first time she realized that at sixty-two he was starting to look old.\n\nBefore he could say anything she went on, \"I wasn't going to tell you. You were right, of course; I did have a secret. I didn't want to worry you with my old problems, but with so much other trouble now, there can't be anything like this getting between us. I'll handle him some way, Warren. I don't want you to get involved with him.\"\n\n\"Maybe he's behind all this trouble,\" Warren said. \"That would explain it.\"\n\nShe shook her head. \"Not Earl. He wouldn't do anything like that. He just wants money, and recognition, and... I don't know what all he wants, but he wouldn't deliberately harm anyone. He isn't dangerous that way.\"\n\n\"You're still protecting him,\" Warren said harshly, sending a shock through her. \"Why, Lois? What's that man to you now?\"\n\nShe put down the book she was holding and very carefully walked across the room, trying to quiet her shaking hands, trying to control her breathing. She felt as if there wasn't enough air to go around, that he could breathe, or she could, but not both of them. He moved aside as she drew near the door.\n\n\"I was trying to protect you,\" she said finally. \"And maybe myself. I don't care if Earl Malik is alive or dead.\" She walked past him stiffly, out into the broad hallway, and stood still, no longer knowing where she could go in this house, where she could feel safe, not like an intruder, an unwelcome guest who had lingered beyond decency.\n\nHalf an hour later she was pacing in her room, back and forth, to the window to gaze at the wide expanse of lawn that looked painted, back to the door where the wood grain revealed a multitude of faces, some in profile, some full view, one grinning, one leering, back to the window again. She watched Carla Mercer scurry up from the woods and cross the lawn, and envied her. A break from her work, then back to mind-numbing details. She yearned for her own work, her tiny laboratory and the endless slides to examine and catalog. She was at the window when Warren tapped on the connecting door between their rooms.\n\n\"It's open,\" she said, not turning around.\n\n\"Lois, I'm sorry.\"\n\nShe looked at him then. He did not move into the room. She waited.\n\n\"Too many shocks, too fast,\" he said hesitantly. \"And I'm a fool.\"\n\nSlowly she crossed to him and then reached out her hands. He took them and drew her to his chest and held her. After a moment he said, \"My first reaction was to call the sheriff and tell him to look for Malik. An easy out, I thought. But I didn't. Then I called Diedrick and asked him what he knows about Meiklejohn, and he called back to say we can trust him. He's discreet and he's thorough. I sent for him. We don't have to tell anyone else. Let him find some answers. If Malik isn't involved, that's that. And if he is... We'll go on from there. But, at least, there won't be publicity.\"\n\nIt didn't occur to her to wonder when he had found time to call Diedrick, his attorney, when Diedrick had found time to learn anything about the private detectives, when he could have sent for them. She felt only a great relief as she stood sheltered in his arms, secure again, safe. Not until they were in the study with Charlie and Constance did she realize that this must be why Warren had been searching for her in the solarium, to tell her that he was bringing in private investigators. And by then she had already told them about Earl Malik.\n\nThey were in Warren's study, with its good old cherry furniture, green-leather upholstered chairs and sofa, floor to ceiling shelves with neatly arranged books. Very nice, all of it. Charlie had glanced around with obvious approval, then said bluntly, \"Why us, Mr. Wollander?\" Wollander's retainer check was on the desk; Charlie had not touched it yet.\n\nWarren Wollander was equally blunt. \"I don't want anyone to make any connection between those unfortunate occurrences and this house. My... resources... would surely establish an interest on my part. I want someone to be very discreet. Since you are already on the scene asking questions, you seem the obvious choice.\"\n\n\"What is the link you want to keep under wraps?\" Charlie asked. His voice was bland, his eyes as hard as obsidian chips.\n\nWarren studied him for a moment, glanced at Lois, then said, \"My wife's former husband has been hanging around\u2014a stranger in these parts. And we all know a stranger is the likeliest suspect if there's trouble. I want that to stay unpublicized. Unless, of course, he is actually involved. But we doubt that he is.\"\n\nCharlie looked at Lois, who now had two patches of high color on her pale cheeks, as if rouge had been applied hastily; it gave her a look of vulnerability. She held her head up and sat with her back straight; only her hands, clutching one another, betrayed her nervousness.\n\nBriefly she told them about Earl Malik. When she was finished, Charlie shook his head at her. \"You know we want more than just stripped facts. Is he the one who knocked over bushes outside the quonset hut last night?\"\n\nThe color left her cheeks and her hands clutched spasmodically. She nodded. \"He came there and I ran out. I drove around a while to calm down and then came home. But he wouldn't have moved the bees. He couldn't have known anything about them or the station wagon, or anything else up there. Besides,\" she finished in a very low voice, as if she had cause to feel shame, \"he was very drunk. He wouldn't have been able to carry out anything. I... he used to go to sleep and not wake up for ten or twelve hours when he was that drunk.\"\n\nShe didn't know, she said, what time she had left him; after ten was as close as she could come. She didn't know what time she had arrived home; she had driven around to calm down, and had stopped at a diner somewhere for a sandwich first. This morning she had not seen him and the lock had not been disturbed on the quonset hut. When she finished, she leaned back, visibly relieved.\n\n\"What does he expect of you?\" Charlie asked. \"Did he mention buying the Zukal property, having you buy it?\"\n\n\"Of course not,\" she snapped. \"That's a ridiculous idea.\"\n\n\"I agree,\" Charlie said, nodding. \"But so is blackmail when you come down to it. You've paid that man over twenty thousand dollars, you say, just to get him away from here, and yet he hasn't gone. How much more does he want? How much more are you willing to pay? Is it going to be a monthly annuity for him? For how long? You say he showed up in April. Before anyone knew the Zukals were interested in the mill. Why not consider that he might have wanted it for his work, his and yours? You see, the problem is that you don't usually pay blackmail to someone simply because you don't like him. In that case, you sic the dog on him, or call the cops, or tell your husband, who takes steps. When did you tell your husband, Mrs. Wollander?\"\n\nAt that moment everything fell apart again, and the pieces shattered into dust, and the dust was in her mouth. She realized that Warren had sent for Charlie before she had confessed that Earl was in the area. With a startled look she turned to Warren, who was gazing out the window as if none of this concerned him, and she knew that the security she had felt in his arms, the safety she had found there was an illusion, that this large, important, self-assured man whom she had thought so wise was a stranger.\n\nShe swallowed painfully, and said in a steady voice, keeping her gaze fixed on Charlie Meiklejohn, \"I didn't tell him until today. I knew he would do something, just not what. And I knew that Earl would go to the press and betray me. If Clarence learns that I have been involved with Earl in the past, he will let me go. He made Tom fire his student a month or two ago. He won't keep me either if I become a liability. It's his project; he pushed it through the administration, through the committees, he wrote the grants and got the money. It's all his, and he has the power to hire or fire. He saw to that. No tinge of scandal is to be allowed, or the entire operation might be suspect, and it's important work. I don't blame him. I might do the same in his place. It is very important work.\"\n\nWhen Charlie spoke again his voice was gentle. \"The trouble with paying blackmail is that sooner or later you run out of money. Have you considered what you would do then?\"\n\nTiredly she shook her head. \"I told him I don't have any money, not the kind of money he's talking about. I have my salary. I gave most of it to him already. I told him that Warren and I signed a prenuptial agreement that I had insisted upon. I can't touch Warren's money. Earl chose not to believe me. And then, I don't know what I will do. My salary, as long as that satisfies him. I don't know. I haven't thought ahead that far.\"\n\nIt was interesting, Charlie was thinking, how at the beginning of this conversation she had glanced again and again at her husband, as if for reassurance, support, something. And now she was apparently oblivious of him. As she might as well be; he was as remote-looking as the sphinx.\n\n\"Why did you confess this now if you don't think there's a connection with the trouble at the Zukals' property?\" he asked then.\n\nShe felt her muscles go stiff in her effort not to look at Warren. \"There were some letters,\" she said, \"sent to Warren. One about me. I don't know what was in it, but obviously someone must have seen me with Earl, or something. And if someone knows, no doubt, there will be talk now. The sheriff may learn about it. I thought if there are questions, we should both be prepared.\"\n\nCharlie turned to Warren Wollander. \"May I see the letters?\"\n\n\"I burned them.\" He continued his fascinated study of the great outdoors.\n\n\"Uh-huh.\" Charlie got up from his comfortable chair and crossed the room to the desk, where he lifted the check made out to him. \"Mr. Wollander, presumably you want an investigator who won't ask embarrassing questions, and to whom you can lie. I'm not that man, sir. Sorry.\"\n\n\"What the devil are you talking about?\" Wollander swung around. \"I mean, what's the point of hiring me and lying to me? There are letters. There were other letters that the Zukals received. There are links and cross-links all through this crazy case, and you burned the letters you got. Fat chance, Mr. Wollander. I rather think you handed copies of them over to your lawyer, or those other resources you mentioned earlier, and at this very minute someone is trying to find the author. Or has someone already found the author? You think you can hand out a smidgeon of information here, another smidgeon there, another to the sheriff, and when we all bring in our crumbs you can piece them together and make a cake?\" He looked at the check regretfully and then tore it in half and let the pieces fall to the desk. \"I'm already hired, you know. So I'll just keep on asking questions here and there and do the best I can.\"\n\nWell, he thought, he hadn't whacked him over the head, but he might as well have. He certainly had Wollander's attention. Suddenly that attention was diverted and Wollander was on his feet, deep concern etching his handsome face.\n\n\"Jill! What are you doing up? You should be resting.\"\n\nShe entered the study to stand by the door. \"I called Sebastian,\" she said. \"He said he came and they stopped him at the gate. I want to see him. Call them and tell them to let him in.\"\n\n\"Jill, baby, you need to rest. I told Duane I'd call him when you woke up. He's standing by.\"\n\nShe shook her head. She was very pale, with the waxy unnatural look of a gardenia; the long dressing gown she was wearing was pastel green, so little tinged with color that as she moved it went from a suggestion of green to silver. The green reflected from her skin eerily. Her hair was mussed as if she had been tossing and turning quite a while. \"No more of your cronies. No more of the good old boys. No Duane or Richard or Herman. I wouldn't let Richard touch me with his fat pink fingers and I don't want Duane talking about his fat pink God. No more. His God took my mother and my baby and my husband. No more. I want to talk to Sebastian. Call them.\"\n\n\"Honey, you're hysterical. Just lie down and rest\u2014\" He moved to her side and she flinched away from him, shaking her head. Her eyes were wide and too bright, the pupils pinpricks.\n\n\"I wouldn't let him touch me but I took his pills and I dreamed. I was showering Mother with money, tons of money. Crisp, green, like lettuce leaves, so pretty in the sunlight. Up to her knees, her hips, and she was spinning around with her hands out, laughing, trying to catch some of it. Then I woke up. I know what it means. I have money now. More money than I ever dreamed of. I can go anywhere, stay anywhere. You can't stop me. I would give it to her if she were alive. God, to see her happy like that! I could do it. And I can see Sebastian. Here or at his place in town. You can't stop me, Father! Call them at the gate!\"\n\n\"For God's sake, Jill. This is shock, hysteria. You don't need company.\" His voice had become peremptory, harsh. \"I'm going to hire a nurse. You need nursing care, bed rest.\"\n\nNow Constance got up and went to Jill. \"She isn't hysterical,\" she said briskly. \"But she's doped to the eyes. I think you should call and tell them to admit Sebastian. She certainly can't go out this way.\"\n\nJill looked at her gratefully, and when Constance took her hand, she did not resist. Neither did she allow herself to be drawn from the room. She watched Warren Wollander, who finally went back to his desk and lifted the phone; he spoke to his security people. When he replaced the receiver, Jill turned and walked out with Constance.\n\n\"I'll stay with her until Sebastian arrives,\" Constance said over her shoulder.\n\nFor a time no one in the study moved. Lois looked frozen in her chair, and Warren stared at the open door blankly. Charlie had not left the desk. When he stirred, Warren glanced at him and then seemed to remember where they were, what they had been saying and doing.\n\n\"Sit down,\" he said. His voice was flat, tired. \"The letters are in my safe-deposit box. I'll get them on Monday. No, Tuesday. Funeral's on Monday. In Bridgeport.\"\n\nCharlie nodded and sat down. \"And the report on Sebastian.\"\n\nLois looked surprised, but Warren merely nodded, almost absently. \"That, too. I couldn't leave it around here because Jill would have found it and been angry.\" Again he seemed to make an effort to collect himself, and more briskly he opened the desk drawer and withdrew his checkbook.\n\nCharlie got the address where Earl Malik was staying. He asked a few more questions, but nothing of consequence, and when Warren pushed a new check across the desk, he took it without comment. \"I'd like to walk around the house and grounds a little,\" he said. \"And when Constance comes back, we'll leave. I'll see you on Tuesday, around two.\"\n\nWarren Wollander nodded, and now Lois was watching him with concern. Charlie was anxious to get out of the room, out of the house where it felt as if an impossibly heavy weight had descended. And Warren looked like a man shell-shocked, he thought, as he began his stroll. He corrected himself. Warren Wollander looked like a man who had only this minute realized how passionately his beloved daughter hated him.\n\nJill's room was spacious and all over the same pastel green that she was wearing, and gold and white. A white, fluffy rug was mashed down in a regular pattern, from window to bed to bathroom, as if she had paced miles around and around. The bed was a heap of sheets and a satin spread, a pillow on the floor. Jill stood in the center of the room and seemed to shrink, to become a child in front of Constance's eyes.\n\n\"He always does that,\" Jill said. \"Says do this, do that, stop doing this, stop doing that...\" She felt her hair, glanced down at her gown. \"Excuse me.\" She went into the bathroom and closed the door.\n\nOn a white-and-gold table flanked by two white and gold chairs there were magazines, fashions and decorating, one of them French, and a number of pamphlets and booklets, all spiritual material. One of them had a smiling Buddha on the cover. A large music system lined one wall; another had a portrait of a beautiful woman. Constance studied it. Jill's mother, she decided, dressed almost exactly the way Jill was dressed now, in a flowing gown of silvered green, a large emerald on one finger, her eyes almost as green as the gem.\n\nJill returned and stood at her side. \"Isn't she beautiful? How could he treat her the way he did? She loved light and gaiety, dancing, music, everything alive, and he wouldn't leave this mausoleum. Never. He treated her like a hired prostitute, made her sign an agreement never to leave him, never to even ask for a divorce, never to demand anything more than their contract allowed.\" Tears were streaming down her cheeks as she gazed at the portrait of her mother; she seemed unaware of them. She continued to talk rapidly about her mother, life here, life in France and Switzerland, as if all these words had been stored in a container too tight for them and, now that it had been opened, the words were not to be denied.\n\nConstance moved away from Jill and went to the bathroom where she wet a washcloth and wrung it out. She glanced swiftly over the counter at twin gold basins, hastily opened both cabinets and looked inside, then returned to Jill, who had not moved and was still talking in that out-of-control staccato. Constance didn't miss a word, and when she gently wiped the girl's cheeks with the cool cloth, Jill seemed as oblivious of her as of the tears.\n\nJill was talking about the parties, about the house they had had outside Paris. \"And she was so ill, so much of the time, but she never let anyone know. She was so brave. When she was too sick to stay home, I came back here and she went to a rest home in the country. And then she got better and called me and I flew back. I hardly even needed an airplane. She could have had any man in Europe, but he wouldn't release her, and she had nothing. Nothing. And so sick so much of the time. And she had to take care of me. I tried to make her leave him anyway, and she was afraid. What would become of us? Of me? Always it came back to that\u2014what would become of me?\"\n\nSuddenly she stopped talking and snatched the washcloth from Constance. \"Oh, God, look at me!\"\n\nShe dashed into the bathroom and turned on the water.\n\n\"Let me brush your hair,\" Constance said, following her. The brush was gold and very heavy. She began to pull it through Jill's hair. \"Are you bleeding very heavily? Perhaps you really should see a doctor.\"\n\nJill drew in a shuddering breath and held the washcloth to her eyes. \"It isn't that bad,\" she said dully. \"Lois talked to her gynecologist. He told us what to watch for, made an appointment for me, in a few weeks, I forget when. I'm all right.\"\n\nConstance finished with her hair; it was silky, with a slight wave. \"Well, come sit down for now, and keep the cloth on your eyes.\"\n\nA minute or two later, when there was a tap on the door, Jill was composed. Constance opened the door to see the housekeeper and Sebastian. He ignored her and went straight in, and before the door closed, she saw him kneeling at Jill's chair, taking both her hands in his.\nCHAPTER 10\n\nAS SOON AS THEY LEFT the lawn and walked behind enough trees to hide them from the Wollander house, Charlie stopped and caught Constance up in his arms. He nuzzled her hair, and kissed her, and then kissed her a second time.\n\n\"Well,\" she said. \"Well.\"\n\n\"Touching base with reality,\" he said, grinning, and took her arm, steered her toward the path back to the mill. \"What happened?\"\n\nDreamily she said, \"I think that could catch on. Sharing reality. Your reality or mine, baby? What do you think?\"\n\nHe pinched her bottom.\n\n\"Oh well,\" she said. \"Beautifully furnished room, lots of white and gold and pale green. If she sat still in that dressing gown she would melt right into it and be invisible. No books. A few magazines, some religious material, booklets. One on meditation, one titled Unwinding the Universe. Stanley was a hypochondriac, lots of prescription medications, more over-the-counter do-it-yourself nostrums. Stomach, bowels, muscle tone, hair loss, you name it, he was probably treating himself for it. Plus heavy-duty sleeping pills, chloral hydrate, a prescription. And high blood pressure medicine. And allergy medications, ointments, pills, Caladryl lotion and tablets, and a sting kit still sealed.\"\n\n\"Good Lord,\" Charlie said when she paused. \"You were gone ten minutes.\"\n\n\"I know,\" she said regretfully. \"If she hadn't talked so much I probably would have had time to snoop around a little. Let's see, what else did I intend to mention? Oh yes. A portrait of her mother. Remember that scene in Tales of Hoffmann where Beverly Sills plays the girl who sings a duet with her dead mother? I always forget the names of the various women, but you know the one I mean. The father either is or has been dealing with the devil, and it's time to pay up. Anyway, it reminded me of that. Jill looks very much like her mother, and of course the mother is highly idealized in the portrait, great tragic eyes, honey silk hair, boneless hand draped beautifully over her pale arm. Dressed in a pale green satiny thing with lots of highlights. I guess it's too late for me to have a portrait done, isn't it?\"\n\nCharlie dug in his heels on the path and brought her to a stop for a second time. \"The biggest mistake I ever made was in packing you off to learn self-defense. Otherwise I would threaten to toss you over the hill into the lake. What did she say?\"\n\n\"I was just getting to that,\" Constance explained. \"It was all about her mother. That's why I prefaced it with the portrait, you see, just so you would have a basis for underst...\"\n\nShe broke it off to say warningly, \"Now, Charlie. It seems that her father kept her mother on a very short leash for a very long time. He had her sign a prenuptial agreement, and if she insisted on it, Jill didn't mention that. The agreement stated that if she left him, divorced him, she would take only what she had brought to the marriage, an inheritance of ten or fifteen thousand. He would pay for the education of any children and their maintenance, but not a penny alimony. If he broke it off and got a divorce, an arbitrator would settle the financial terms. He had an obsession, Jill said, about no divorce, the need to keep a good public image of an intact family, just in case he decided to run for office himself instead of pulling wires from behind the screen. So they all lived here when she was a young child, and then the first Mrs. Wollander split for France and never came back. In and out of hospitals, boyfriends, drugs, alcohol, the whole Fellini bit. When she was too disabled, she sent Jill home, but she never stayed more than a few months at a time, and then back into the party with Mother. She died a little over four years ago, and Jill married Stanley a few months later.\"\n\n\"Wow. And Wollander has a clean image, electable by a landslide.\"\n\n\"I'd say so. It's quite an in thing to admit to dependency treatments, attempts at cures. And he stood by her through thick and thin.\" Her voice had become detached and too thoughtful, too professional, as if she had Warren Wollander on a slide and was interested the way a doctor might be interested in a bacterium. \"I wonder if the current Mrs. Wollander really believes the idea of a prenuptial agreement was altogether her own.\" She paused, then added, \"Something came up today that shook her. Remember? It had to do with when she told him that her ex was in the neighborhood. She changed completely after that.\"\n\nHe remembered clearly. Now they had reached the place in the path where they had to walk single file, and he stood aside for her to go first. He liked to watch her move, liked the way the low sun was filtered through the trees to touch her hair again and again, the way her hair changed from brilliant to pale. The path was a pleasant walk, never steep, and, except for this one stretch, comfortably wide. Here the trees had funneled the trail to a scant two feet, and even that was nice, the fact that they had let the trees have their way. She looked very nice here among the trees in the soft dappled sunlight. When they emerged from this section, the lake was visible again. Constance turned to reach for his hand, and paused at the look in his eyes.\n\n\"Another reality check?\" she murmured.\n\n\"You're a wicked woman. And it's not too late to have a portrait. By God, it's not!\"\n\nThis time she kissed him.\n\nThe beekeeper came and Charlie watched closely, but there was little to see. The man looked inside the station wagon and nodded in satisfaction. It was not altogether dark, but the bees had had it for the day, apparently. He simply picked up the box by handles on each side and put it on his truck, and that was the end of that. \"Italian strain,\" he said. \"They're real peaceful if you treat them right.\" He got in the truck and drove away.\n\nWhen they got home, after winding through several unfamiliar roads to make certain they were going home alone, they found Al and Sylvie in the television room playing gin. The television was turned on too loud, and they were arguing, and Al was holding Brutus in his lap. Brutus barely acknowledged Charlie and Constance.\n\n\"I'll be damned,\" Charlie muttered. \"Perverse beast.\" Brutus hated strangers, had always hated strangers, and wasn't too crazy about people he knew.\n\n\"This one's a pussy cat,\" Al said, stroking the cat. \"Aincha, tiger? The others ain't showed up.\"\n\n\"He thinks he's back prowling city streets,\" Charlie said, and Sylvie said at the same time, \"Betcha you're starving, aincha? Me and Al, we snacked some but we ain't ate. Thought it'd be polite like if we waited. We got Polish sausages and potato salad and a couple other things. We seen this little store on the way over here and stopped and we got plenty lost when we come out again, got real turned around, but then we got on the right track again.\"\n\nAl's stings were all but gone now, no welts, not even an itch, he said proudly. \"Some tricks,\" he said to Constance. \"What we thought,\" he said then, \"was that while Sylvie's finishing up supper, I'd show you the stuff we wrote down today. Everything we can think of that happened with David after Sadie kicked. Ain't much, but who knows?\"\n\n\"That's right,\" Sylvie said. \"Who knows?\"\n\nThey all went to the kitchen to keep her company and Charlie looked over the notes they had made while Constance set the kitchen table.\n\nSylvie talked disparagingly about her cooking, and Al talked about the security system he had in mind, the kind where if someone moves without you doing nothing the lights come on and an alarm, and she said that they would just shoot out the lights, and he said that would be a signal too, they could fix it like that.\n\nCharlie found it hard to read what they had written down. Sylvie capitalized every noun, and Al capitalized nothing at all. Neither seemed to have much use for periods.\n\nSylvie was a superb cook and dinner was too good to talk through; it was late by then and they were all very hungry. Sausages and potato salad with a sweet-and-sour creamy dressing like none that Charlie had ever had before, with bits of green onions and olives chopped up in it. She had made a green bean casserole with mushrooms, and a tossed salad that had shredded raw beets and carrots and even pine nuts. He sighed his contentment when he finally pushed his plate back half an inch, his signal that another bite would do him in.\n\nConstance made coffee and Charlie brought in brandy; they all cleared the table and sat down again, and this time Charlie actually read some of the items they had written down.\n\n\"Them with checks,\" Al said, \"that means we both seen him together.\"\n\nCharlie went to the beginning of Al's paper. David had come to the mill to see how things were going on Monday;\n\nAl began to talk about the work being done in the mill, and with great concentration Charlie managed to blank out his voice and read on. Thursday of that week David had dropped in briefly to visit with Al while Sylvie was shopping. Now she talked about Betsy, the lady with the new antiques, whose son made them and made them look old by torturing the wood. They don't torture it, Al said, they just torment it a little, make holes in it with a drill. They worry it, that's what they do. Charlie looked up helplessly and found Constance following their talk with very bright eyes and not a trace of a grin. Betsy might even show some of the furniture Bobby made, Sylvie continued, pointedly ignoring Al, but not pretend it was antique, and Bobby sure wouldn't think of torturing good wood.\n\nCharlie gritted his teeth. On Friday David had brought their mail in; that was confirmed by both of them. He stayed for a glass of milk and a couple of cookies. Sylvie talked about how thin he always had been. That night they agreed they had him for supper.\n\nSunday he brought her head scarf, found it on the beach or somewhere. Filthy. Past washing. She had got used to wearing them when she cleaned offices in the city. No one had any idea of how filthy some doctors and lawyers could be. Maybe out here in the country she would get out of the habit, but her head felt naked...\n\nMonday, Sylvie told David about the poison-pen letters, and he had got real upset, she said soberly. Al nodded. Thursday, the day after he had talked to Charlie, although they didn't know nothing about that, but it was just like him, anyways, he had asked Al exactly where the dog died, where they had found her. Sylvie said he stood at the grave and looked real troubled, and he walked around the mill, troubled real bad. Al nodded again.\n\n\"We figured that maybe he was trying to come up with his own answers,\" Al said. \"And maybe he did.\"\n\n\"Next day,\" Sylvie said, \"we both seen him tossing sticks in the water. He looked just like the little Robin boy with his teddy bear. You know how they stood on the bridge and threw sticks in the water? Only he wasn't playing no games. He looked worried and bothered, and then he walked down the side of the lake and we didn't see him no more.\"\n\n\"Sticks?\" Charlie said faintly, and put down the papers in defeat. He would read them tomorrow.\n\n\"Little sticks. Twigs, like.\" Al measured about three or four inches with his hands, then adjusted them a bit. \"Little sticks. Charlie,\" he said, leaning forward, \"something that we both seen. Everything bad that happened was always on a Saturday. You see that? Sadie dies. Then David dies, and now Stanley dies. Always on Saturday.\"\n\nCharlie nodded. \"I noticed.\"\n\nAl looked disappointed, then brightened again. He and Sylvie exchanged a meaningful glance and she nodded. Al said, \"Something else, Charlie. Me and Sylvie, we think you want to be more careful. I mean, if there's a high school kid out there with ants in his pants, that's one thing, but if there's a nut out there, you want to be careful. You're too trusting, Charlie.\"\n\n\"How's that?\"\n\n\"Well, like you let us come here and we coulda robbed you blind. You don't know if I'm a nut or a thief or what have you. See what I mean?\"\n\n\"It goes both ways, Al. We could have robbed your house.\"\n\n\"She wouldn'ta.\" Al nodded toward Constance, who nodded back emphatically.\n\n\"Well, she's the one who invited you to come here, if you'll recall.\" And she, he thought, a bit disgruntled, didn't pipe up to say he wouldn'ta, either.\n\n\"You wouldn'ta axed us?\"\n\n\"Not on my own, probably.\"\n\nAl nodded in satisfaction. \"That's okay then.\"\n\n\"I told you,\" Sylvie said.\n\n\"Something you might be able to tell us,\" Charlie said then. \"How did David get over here to see me that day? Do you know?\"\n\n\"Sure,\" Al said, and Sylvie said at the same time, \"In Lois's car. He borrowed it.\"\n\n\"Then probably everyone at the farm knew he was going somewhere, even if not where. And maybe at Lois's house, too. She could have mentioned it, I guess.\"\n\n\"Gonna buy us a gun or two,\" Al said suddenly, making an abrupt right turn in the conversation without warning.\n\nCharlie groaned. \"Do you know anything about guns? Ever do any shooting?\"\n\n\"Nope, but all them punk kids in California know, and just about everybody on television. Even the president's wife had a gun, you know? Can't be that much to learn, even if she did say it was just a little tiny one.\"\n\n\"You'll kill yourself, or Sylvie,\" Charlie muttered unhappily.\n\n\"You think we should go back there without no protection at all?\"\n\n\"Why don't you take a little trip? Just a few days, a week. Go to Atlantic City and play the slots or something for a few days.\"\n\nSylvie screeched something and Al said something, and altogether it added up to no. They weren't going to be driven out of their house after all these years. Sylvie began to talk about her grandparents who had come from Poland, and Al talked about his father setting up a butcher shop, only to be burned out, and now they were trying the same with him and Sylvie... .\n\n\"Okay, okay,\" Charlie said, shouting over both of them. \"Let me go over the security system when it's in place, will you? Before you go back, I mean.\"\n\n\"Sure,\" Al said agreeably.\n\n\"My grandmother had a stove like that,\" Sylvie said, pointing to the wood-burning stove that wouldn't be used again until fall. \"And one of the places we seen, they had a stove like that. Remember, Al? The place with the horses?\"\n\n\"Yeah. Horses. Never get them stables smelling like roses in a million years. You know horses, Charlie?\" he asked with a thoughtful look. \"Big. Real big.\"\n\nHelplessly Charlie nodded. \"Yes, Al. I know.\"\n\n\"We gotta go now,\" Sylvie said, standing up. \"We called a hotel we seen on our way here, and if you'll just point us in the right direction, maybe we can find it again.\" She sounded as if she doubted it. \"It sure is dark in the country, you ever notice?\" she asked Constance. \"Real dark. You'd think they could put in a street light or two, for appearances or something.\"\n\nThey both talked and Charlie went out with them and pointed directions and then rejoined Constance inside. \"Well,\" he said, \"watcha think?\"\n\n\"I think they're swell,\" she said judiciously. \"And I think we had better catch that nut who's keeping them from enjoying their new fortune.\"\n\nWhile Constance finished straightening up in the kitchen, Charlie went through the house turning off lights. He was carrying one of Sylvie's headscarfs when he came back; she had left it in the television room. All three cats were prowling around the kitchen now to see if any interesting tidbits had fallen, and Candy was complaining bitterly, her hoarse voice very like Sylvie's. Charlie watched the cats approach to investigate the scarf, which he had draped over a chair-back to be returned the next day. He was thinking of Sylvie cleaning offices, Al butchering for a supermarket, four girls growing up in a small apartment, all four of them getting educated, and he agreed with Constance. They should be allowed to enjoy their place in the sunshine, their place in the country. Candy kept getting closer to the scarf, then backing off suspiciously, and she kept complaining. He felt certain her raucous words were very obscene.\n\nConstance flicked off the light at the sink; he put his arm around her waist, and they went upstairs, leaving the cats to decide if the green scarf was friend or foe.\nCHAPTER 11\n\nSUNDAY WAS A BUST, Charlie decided in mid afternoon. They had read all of David's letters, and the papers Al and Sylvie had left with them. They had gone over David's other belongings and there was nothing new. He had been pitifully poor, saving for three months to buy a new pair of running shoes, doing without a new watch when his old one quit, making do. Just making do with what he could afford, and writing letters week after week to his father. His checking account had bottomed out each month while he waited for his scholarship money, but he never was overdrawn. He had done without instead of bouncing checks. It all made Charlie grouchy and on edge.\n\nConstance brought in peas and asparagus from the garden and he prowled around the house and yard, making the cats uneasy. They kept starting at noises he couldn't hear, and staring at objects he couldn't see, and casting swift, frightened glances over their shoulders at nothing at all. After dinner Charlie announced that he wanted to see a movie.\n\n\"What movie?\"\n\n\"Any movie. Preferably a western, with lots of shooting and frantic riding around on the prairies and deserts.\"\n\n\"Oh, that kind of movie.\" She found the weekly local newspaper and scanned the two ads for coming attractions. It really didn't matter which movie he saw when he was in this state, but if she had to sit through something with him, and if there was a decent choice, she preferred to make it herself. When Charlie said any movie, he meant it. He might see it, or maybe not, or maybe some of it. It was as if at times he had to have something taking place before his eyes in order to let his brain get on with its own business of making sense when no sense was apparent. Later, he might walk endlessly. Other times he might play solitaire, but now it was a movie that he needed.\n\nShe drove to the theater in Woodbury, ten miles away, and they entered to find a Woody Allen movie already started, full of New York scenes done in sepia. Charlie settled down contentedly with a bag of popcorn, and she watched the movie. When it was over, she had to nudge him with her elbow.\n\nThey went out and walked the half block to the Volvo and she got behind the wheel again. Halfway home, he grunted, and said, \"Where are the damn vitamins?\"\n\n\"Didn't the police lab destroy them when they were looking for drugs?\"\n\n\"I mean the new package,\" he said softly.\n\nShe thought, recalling David's belongings, and then said, \"Oh! Of course.\"\n\nIn just a few minutes they were back inside their house, with David's personal effects once again spread on the table. Charlie found the police report and the list of vitamins they had destroyed: A, E, B-complex, C... Constance found the other list of what the sheriff's men had taken, and later had returned to David's father. No new package of vitamins. She picked up the catalog that David had ordered from and began to look for a price list, an order form. It had been torn out. But there had been a check made out to the company, she remembered, and now looked for that in the register.\n\n\"He ordered a three-month supply on May twelfth,\" she said a few minutes later. \"And before that, back in February. Orders spaced just over three months, one hundred of this and that. It's an in-state company, wouldn't take more than a few days for the order to be processed and delivered.\"\n\nCharlie made a noise that wasn't really a response. He was comparing the catalog to the list from the sheriff's office of destroyed vitamins. \"Four of this and five of that,\" he murmured. \"You'd think with one-a-day type vitamins, the numbers would be the same. And he was almost out of them. As close as he was with money, if the new supply hadn't been delivered before he died, they should have been in the next week while the police were collecting his mail.\"\n\n\"Four of some, five of others? But, Charlie, if the drugs had been in vitamins, there'd be some trace, wouldn't there? I mean, Greg Dolman uses a good lab technician, surely. And who could count on David's taking the drugged ones? That's just too chancy.\"\n\nHe nodded. Then abruptly he read down the list again. \"I'll be damned,\" he muttered. \"The fives are all capsules, the fours tablets.\"\n\nConstance waited, trying to reason it out, drawing a blank.\n\n\"Try this,\" Charlie said at last. \"You see that there are a lot of small containers, each one with five vitamins, and you take away the ones with capsules. You replace them with identical containers of capsules that have been tampered with. It doesn't matter which ones he takes of the capsules, they're all poison, but they'd look normal, and still have the right smell. Then, after he's taken the doped capsules, you return the containers with the harmless vitamins, and there are still five in each of them, but only four in each container that held tablets. Tablets are too hard to duplicate, but capsules are a snap to empty and refill. No trace of drugs would be found, obviously, since none of these ever had drugs in them.\"\n\nSlowly Constance nodded. \"Someone could have taken the new package, his new supply, and tampered with them, thrown away everything afterward.\" She began to study the catalog list again. \"Some of those capsules would have been awfully big\u2014a thousand milligrams of C, with rose hips, for instance, in a time-release form. That means hundreds, even thousands of little individual bits, like beads, to dissolve over eight hours or so. That begins to get pretty large for a single dose. Some of these are in the mega-vitamin category, really big, but would there have been enough of the capsules? Enough space for a lethal dose?\"\n\n\"I don't know. We'll have to get some and see.\"\n\n\"Oh!\" she said then. \"Look, he ordered cod-liver oil. A lot of drugs could have been dissolved in it, and the taste and smell would hide just about anything.\"\n\nCharlie looked over the list of things the sheriff had tested; no cod liver oil was there.\n\n\"Maybe he was just going to start it,\" Constance said.\n\n\"We'll sit on it for now,\" Charlie said. \"I want to get the damn pathology report on the dog. But I think we've got this link. And it sounds a lot more plausible than a six-in-the-morning visitor.\"\n\n\"Just one more thing,\" Constance said then. \"This other price list. Did you notice it? I thought at first it was more of the same, but look.\" She held up a list and pointed to the top margin, where words had been scrawled: The list I promised. S.\n\nCharlie looked from it to her and shook his head. \"So?\"\n\n\"It's a different company altogether,\" she said. \"And he could have saved nine dollars by ordering from this one. And who do you suppose S is? And why didn't David order from this company instead of the higher-priced one? They seem to be offering nearly identical items.\"\n\n\"S,\" Charlie said then. \"Sebastian? Remember David's letter about the party at Wollander's? They talked about bees and about vitamins.\"\n\n\"I know.\"\n\n\"Well, well,\" Charlie said then. \"Interesting, isn't it? No matter where we toss our net, the same fish keep swimming into it. I told you I wanted to see a western movie.\"\n\nShe suppressed a groan and he started to put everything back into the box.\n\nEarly Monday they found the address Lois had provided for Earl Malik; it turned out to be a trailer court, Hunt Acres. The manager's name was Petey Wilson, and he looked like a wrestler, with great bulging muscles and a neck that was the same diameter as his head. His head was shaved and it was deeply tanned, as was the rest of him. He was wearing an undershirt, cut off jeans, and sandals. Even his toes appeared muscular. His eyes were bright blue and clear.\n\n\"Earl? Sure. Third trailer from the end, left. But he's not here.\"\n\n\"Say when he'd be back?\" Charlie asked.\n\n\"Man, he hasn't said anything for days. Haven't seen him in nearly a week. His tough luck. He's paid up in advance; I should worry. If he isn't back by the end of the week, I'll clean out his junk and rent the unit to someone else. I could care less.\"\n\nCharlie nodded and surveyed the trailer park. Ten trailers were visible and probably there were others beyond them in a grove of trees. A gravel drive wound around them all. At the moment the whole complex seemed almost empty, but in hunting season, he suspected, it filled to overflowing. A blond woman in pink short shorts was sweeping her two steps endlessly, watching them. She was heavily made up, as if for a performance.\n\n\"You live out here, too?\" Charlie asked Petey Wilson.\n\n\"Yep.\" Petey pointed toward the road and the first trailer in the place.\n\n\"So anyone coming or going would have to pass right by your place.\"\n\nPetey looked somewhat disgusted and shrugged. \"If they come in this way they do. There's a back entrance in from Huntaker Lane. Lots of them use that one. Closer to Spender's Ferry that way.\"\n\nCharlie was eyeing the third trailer from the end, not yet moving toward it, but evidently intending to do so. Constance said, \"Haven't I seen you on television? Wrestling? Or was it football?\"\n\nPetey peered at her suspiciously, then nodded. \"It's been a long time.\"\n\n\"But you've kept in such good shape! You look as if you're ready to go back to it right now, this minute, if you wanted to.\"\n\nHe flexed his biceps. \"Could. Pays to keep in shape. You watch wrestling?\"\n\n\"Pretty often in the past. It's not the same now, for some reason. I knew the minute I saw you that you were familiar. Did you ever get hurt much? It always looks so cruel.\"\n\n\"Naw. We learn to take it, that's all. Nobody watches it much anymore. All fake now. Not like it used to be.\"\n\nShe nodded sympathetically, then turned to Charlie. \"Why don't you take that stuff to Earl's unit? I'll wait for you here.\" Then she looked at Petey again. \"Mr. Wilson, is it really true that those matches are all decided in advance?\"\n\nPetey glanced from her to Charlie and back, then he shrugged slightly. \"Now maybe, but not in my day.\" Charlie left them talking.\n\nThe door was locked, but even a kid on his first solo burglary could have opened it. Charlie slipped his plastic card in and opened the door. The trailer was hot, and smelled foul\u2014too many spilled drinks, too many fried onions and hamburgers, too many cigarettes, never enough air. It was dirty and messy, with clothes scattered about and magazines and newspapers everywhere, and a thick coating of greasy dirt on the window frames and the back of the stove. There were many books, science books, biographies, books about trees, plants... The refrigerator smelled even worse than the apartment, but there were steaks and apples, milk, butter, eggs. A broken egg had dried on the bottom of the refrigerator. On the only table was a notebook opened to a page of numbers and cryptic notes. It looked like the same sort of thing that had been scrolling on the computer monitor back at the farm. A stack of computer printouts with more of the same was by the side of the notebook. Earl Malik was keeping up, apparently.\n\nThere were two unopened bottles of bourbon, one of gin, three bottles of beer, some Collins mix, and a trash can filled with empties. Charlie was hurrying, trying to get a picture of the man, and the picture was too uneven. A pig, but a smart pig was the best he could do. A tiny chest of drawers held a few shirts and socks; two drawers were empty, and a pair of slacks and a jacket were in a small closet. All the pockets were empty. He ran his hand under the mattress on the sagging bed, and he looked under the cushions on the sad-looking sofa. Nothing. He picked up a crumpled map near the table and opened it. Then he folded it and tucked it inside his pocket and stood surveying the place for another second before he left. A real search by experts might turn up something, he thought morosely, but he doubted it.\n\nConstance was still in an intent conversation with Petey when he rejoined them. The woman in the short shorts had stopped pretending to sweep; she was simply watching them now. Constance looked at Charlie brightly. \"You wouldn't believe some of the tricks they use now to fool the audience!\"\n\n\"I'd believe,\" he said. They all looked toward the road as a siren blared nearer and nearer, then passed. Medical rescue unit, Charlie thought distantly. The road was hidden by trees.\n\n\"Well\u2014\" Petey started; he stopped again as another siren wailed, approaching.\n\n\"Let's go see,\" Charlie said, taking Constance by the arm. \"When Earl comes, tell him we'll be back. Just say Charlie. Okay?\"\n\nPetey nodded, following the sound of the sirens. A third one was approaching.\n\nCharlie drove back up the gravel driveway to the county road, and waited there for a sheriff's deputy to pass with the siren screaming. His face was without expression as he watched it go by. He listened briefly, then pulled out after it.\n\n\"The farm!\" Constance exclaimed a few minutes later.\n\nThey were waved off the driveway to the front parking lot, but no one stopped them when they entered the admin building and hurried through and out the back door. Charlie was nearly trotting as he led the way past the rows of plants to a group of people near the quonset hut. They were farm workers and a few others that Charlie had not seen before. Tom Hopewell was with them, his hands jammed into his pockets, his shoulders hunched. Everyone was watching the activity at the apartment units, although deputies were keeping them all well back.\n\nTom Hopewell started, then nodded to Charlie and Constance when they reached him; a deep frown lined his face, aging him many years. \"My God,\" he said. \"My God!\"\n\nCharlie looked past him. Several of the apartment doors were wide open, one unit had windows broken out. Greg Dolman was near it, but no one, as far as Charlie could tell, had entered.\n\n\"What the hell?\" he muttered, and then he got a whiff of gas. He gave Constance's arm a slight squeeze. \"Wait here,\" he said, and strode toward Greg Dolman.\n\nA deputy moved toward him, to intercept him probably; he ignored the man. \"Greg, what the hell is going on?\" he called.\n\nDolman scowled at him and motioned the deputy away. \"Gas,\" he said unnecessarily. \"Place was full of it when the painters opened the door. Thank God, no one tossed a cigarette or match in.\"\n\nCharlie nodded. \"Someone's in there?\"\n\nDolman's expression turned suspicious, and Charlie said irritably, \"For Christ's sake! What are the medics waiting for if there isn't someone in there?\"\n\n\"Just to haul him away,\" Dolman muttered. \"He's been in there a long time apparently. Too long. Bosch took one of my guys up to the main building to try to find a fan. But that sucker in there, won't matter to him.\"\n\nCharlie's hands bunched up into fists, and he stared at the apartment, every muscle taut now. Another one, he kept thinking. Another body.\n\nThe fan was produced; someone had to go find an extension cord, and finally it was working to air out the apartment. Bosch was the color of wet newspaper, and said not a word. He stood near Charlie watching the apartment as if he expected to see fireworks. He looked as tightly wound as Charlie was.\n\nWhen Dolman approached the apartment, Charlie followed. No one seemed to notice. He entered, sniffing first just to make sure, and then stood well out of the way, keeping his arms tightly crossed, observing. A dead man with dark hair, dressed in jeans and short-sleeved shirt, running shoes. On his side in the kitchen area. Bedroom door closed, bathroom door closed. The technicians\u2014death technicians, he thought bleakly\u2014did the usual things; some took pictures, then more pictures, while others dusted for prints here and there. Dolman prowled around purposefully, and finally they turned the man over and there was blood-soaked hair matted on that side. Charlie left.\n\nBosch had joined Constance and Tom Hopewell. \"Who is it?\" Bosch asked. A little color had returned to his face, but he still looked bad. A tic jerked one eye again and again.\n\nCharlie shrugged. \"They didn't find any ID. It may take a while to identify him. I never saw him before.\" He looked at Tom Hopewell, included him in his question, \"What happened?\"\n\nBosch said quickly, \"I was in the admin building when I heard people yelling, running for a telephone.\"\n\nTom Hopewell was looking everywhere, and likely seeing nothing. His gaze flitted here, there, somewhere else too fast for anything to be registering. Charlie touched his arm and he jumped. Very quickly he said, \"I was in the field. One of the painters opened the door, and, thank God, he had enough sense to yell for me to turn off the gas. I looked in through the window, and saw that someone was in there. That's why we broke the windows. I covered my face with my shirt and went in, but there wasn't any point to it. So we waited for the sheriff to get here.\"\n\n\"Did you touch anything?\"\n\n\"No! Just the body, and only enough to make sure he was dead.\" He looked like a man who had only this minute finally come to believe in the irreversible fact of death. And he looked like a man lying in his teeth. Charlie wondered if he had become this pale, this ancient when he found David's body, if he had looked this guilty then. Would finding a third body finish him off entirely?\n\nGreg Dolman came to the small group. \"Why don't you folks go back to the main building. I'll want a statement when we're through here, and a list of everyone who's been around the last few days. You could get it together while we're finishing up.\"\n\nCharlie lingered as Bosch and Hopewell started toward the building. After a glance at him, Constance followed the two scientists. Tom Hopewell was moving so fast she would have had to run to keep up.\n\n\"Any ID?\" Charlie asked Dolman then.\n\n\"Nothing. Transient. I think this wraps it all up, Charlie.\"\n\n\"Is that how you're going to play it, Greg?\"\n\n\"That's how it is,\" Greg Dolman said flatly. \"A bum moved the bees. Who knows why? Then he went out in an accident. Finis.\"\n\n\"Was there a bottle in there? I didn't see any.\"\n\n\"No. Why?\"\n\n\"Wondered, that's all.\"\n\n\"Well, stop wondering. I don't give a shit about a bottle. I say it's complete as it is.\"\n\nCharlie shook his head. \"It won't work, and you damn well know it won't.\"\n\n\"Butt out, Charlie. Just butt the hell out.\" He turned away abruptly and strode back to his men still working in the apartment.\n\nWhen they reached the admin building, Clarence Bosch sent Tom Hopewell to talk to the rest of the staff, and get the list of employees. \"If those two people show up for interviews, tell Wanda to keep them in the front office. I'll get to them as soon as I can.\" He busied himself with making coffee. \"Something about coffee,\" he said absently. \"I think it's just something to do, keep the hands busy, that's why we always think of it.\"\n\nConstance agreed; she studied the room they were in. Apparently this had been the original kitchen of the house that had been turned into the administration building. There was a sink, some cabinets painted brown with the paint flaking off, and a refrigerator, but there were also file cabinets, a computer and work station, and an array of test tubes and flasks on two of the counters, also a coffee maker. A box of latex gloves was on the table, and a metric scale.\n\n\"Ever since the Zukals came,\" he said, stopping in the middle of measuring the grounds into the basket. \"It's as if they brought a bad wind with them. Nothing but trouble.\"\n\n\"That piece of ground is so small, just fourteen acres. It's hard to believe anyone could want it so desperately as to cause all this trouble.\"\n\n\"No one wanted it for more than ten years,\" he said, and finished measuring the coffee, plugged in the machine. \"Frankly, I was glad to see someone buy it, maybe put an end to the lovers' lane reputation it had. Ever since I can remember, since I was a boy, it's been that, a place to rendezvous.\"\n\n\"You grew up here?\"\n\n\"I, my wife, Warren, his first wife... all kids together.\" His gaze was distant, as if he were looking into the past. \"We used the mill, too, in those days,\" he said with a faint smile that was so fleeting it might not have appeared at all. \"Things were different, though. No drugs, a little beer, maybe; now and then a little booze. Then we all grew up and went our own ways. Warren to Harvard, I to Penn State. Warren got Shelley, and after a few years I got Barbara. And we were all grown up.\"\n\n\"I understand that Jill's mother was very beautiful.\"\n\n\"Oh yes. Every one of us wooed the beautiful Shelley, and Warren won. Or lost. Who's to say anymore?\"\n\nHe turned his back on her and began fishing around in cabinets, pulling out a cup here, another there. Constance knew she was pushing it, but she kept up the questions as if she simply wanted to pass the time with him, distract him from the gruesome business at the apartment units.\n\n\"Was she really as wild as rumor has it?\"\n\nHe laughed harshly. \"Rumor can't even touch the reality,\" he said, and counted the cups he had assembled, went back to rummage for more.\n\n\"And was Jill like that?\"\n\nHe stiffened and stood motionless for a second or two, then turned to regard her carefully. \"I've been rather naive, I'm afraid,\" he said. \"You're working, aren't you? I forgot.\"\n\n\"I'm working,\" she admitted. \"Dr. Bosch, the present doesn't exist independently of the past. The gun that was loaded thirty years ago sometimes is fired in the present, but the question remains: Why was it loaded at all?\" She had heard Charlie enter the building, but he was not coming into this room yet. She sensed that he was on the other side of the closed door, listening, letting her carry it for the time being.\n\n\"Our past has nothing to do with the troubles going on now,\" Bosch said almost primly. \"Believe me, it doesn't.\"\n\n\"Probably you're right, but there are such big blanks. You called the police and turned in Jill Wollander when she was an adolescent. There are many ways to interpret that action. Revenge. Hatred for the Wollanders. Hatred for Warren in particular, striking at him where he was most vulnerable.\" Clarence Bosch's face looked frozen, his eyes unfocused. Very gently Constance said, \"And another way to interpret it is to wonder if maybe you were trying to save Jill. You knew what her mother had been like, and you could have seen a chance to prevent the daughter's following that path.\"\n\nHe closed his eyes for a moment and shook himself slightly. He looked down at the cups on the table and moved one, then another aimlessly, keeping his gaze on them. \"I... loved Shelley very much. Most men did. And she taunted me when she became engaged to Warren. She said he would do anything she demanded, and that he had money and position, things I didn't have. I was going to be a teacher, a poor teacher. She laughed at me. And suddenly I hated her. Just like that. Later, I pitied her, I think. And then Jill was slipping across the lake at night, meeting the students in the mill, in the apartments. I don't know. She was so young, so ignorant. I told my wife, and she tried to talk to Warren, but he... he wouldn't hear a word against Jill. I called the police, knowing he would have to pay attention, be aware. I knew the price I would pay, that we all would pay, but she was so like Shelley in those days. Barbara, my wife, agreed with me, that we should at least try to help her, even if it meant scandal and hostility and an end to our relationship with Warren.\" He looked up finally and shrugged. \"As you see, it has nothing to do with what's been happening around here.\"\n\nConstance nodded. \"I think your wife must be a very fine person.\"\n\nHe blinked in surprise. \"Of course. The coffee smells done, don't you think?\"\n\nHe was pouring coffee when Charlie entered the kitchen. Constance watched Charlie make a swift survey of the room, and she knew that later, if necessary, he would be able to list proportions, contents, windows, everything about it. Training, he said; she called it magic.\n\nBosch looked at Charlie with an expression that was almost pleading. Charlie shook his head. \"The sheriff will be along in a couple of minutes. We'll take off before he gets in. Just a question or two first, if you don't mind.\"\n\nBosch ran his hand through his thin hair. \"Go ahead.\"\n\n\"Right. When will Dr. Wharton return to work?\"\n\n\"Tomorrow.\"\n\n\"And you're interviewing her applicants for her? Will you hire them, or one of them?\"\n\n\"No, of course not. I told Lois I would simply determine if they are qualified. It would have been best to contact them both and put off the interview until tomorrow, but with the weekend, and they were already on the way...\"\n\n\"I understand. Dr. Bosch, when this project is finished, you'll retire, won't you? And Tom Hopewell? What will he do?\"\n\n\"You'll have to ask him. I imagine he'll cast about for another position where organic methods will be approved. It's far enough in the future that we haven't really discussed it.\"\n\n\"And Dr. Wharton, she would have to start a job hunt, too?\"\n\nThis time Clarence Bosch shifted uncomfortably. He sipped the coffee he had been holding and paying no attention to. \"I really don't know. Her situation is different, of course. A commitment to trees is a commitment to time itself. A lifetime is really too short.\"\n\n\"You haven't discussed it with her?\"\n\nHis discomfort increased and a flush of color tinged his pale cheeks. \"We talked about it,\" he said. \"She said at the time that she didn't know what she would do.\"\n\n\"Did she discuss the possibility of continuing her work without the sponsorship of the university?\"\n\n\"She didn't, but I did. Not on the mill property, not on fourteen acres. But damn it!\" he said with his first show of real passion, \"Warren owns over four hundred acres, much of it in pasture, there's a big apple orchard, open land. Good land. I suggested that she could work at home, and she was adamant about not doing that. That was the end of the discussion.\"\n\n\"Do you know that the organic project will end at the ten-year mark?\"\n\n\"Absolutely. They'd like to phase us out now, if they could. The high-tech bioengineering people can't wait to get in here. That's already in the planning stage.\"\n\nCharlie cocked his head, listening, then said quickly, \"Just one more thing, Dr. Bosch. Who would know if a liquor bottle was found anywhere on the grounds over the past couple of days?\"\n\n\"I'd know,\" Bosch said, looking bewildered. \"The grounds keeper reports to me. If anything unusual had turned up after the bees were taken, I'd know. Believe me, everyone's on full alert, a bit late maybe, but they are now.\"\n\n\"Thanks,\" Charlie said. \"You've really helped, Dr. Bosch. We'll be on our way.\" He took Constance's arm and hurried her through the hallway out the front of the building as Greg Dolman entered the back.\n\n\"Are we on the lam?\" Constance asked, seated in the Volvo, with Charlie behind the wheel.\n\n\"Not yet,\" he said, engaging gears. He waved to the deputy, who waved back, and drove out the gravel driveway to the county road.\n\n\"Was it Earl Malik's body?\" she asked after a moment.\n\n\"Probably.\"\n\n\"And did you mention that to Greg?\"\n\n\"It didn't come up.\"\n\nShe nodded slowly. \"Eventually he'll find out that you suspected it was Earl Malik and that you deliberately withheld the information. How long do you suppose it will be before we are on the lam?\"\n\nCharlie chuckled and patted her thigh, and then let his hand rest there.\nCHAPTER 12\n\nTODAY NO CARS WERE parked along Ellis Street. The postage stamp-sized lawn was just as immaculate as it had been on Saturday, but when Charlie knocked on the door, the young woman who answered was wearing jeans and a T-shirt.\n\n\"We would like to see Sebastian,\" Charlie said.\n\n\"I'm sorry, do you have an appointment?\" She looked sorry to the point of tragedy. She was a pretty young woman with very long dark hair held off her face by a blue ribbon, no makeup, no jewelry. Her eyes were a lovely clear brown. She looked as if her sorrow might lead to tears momentarily.\n\n\"Just tell Sebastian that I am a private investigator employed by Mr. Wollander, and I would like to talk to him.\"\n\nHer eyes widened and her mouth made a little O; she withdrew and closed the door. Charlie glanced at Constance and winked.\n\n\"Ten seconds,\" she said gravely.\n\nIt was no longer than that before the door opened again, and this time Sebastian appeared. He motioned for them to enter.\n\n\"You were at the lecture Saturday,\" he said.\n\n\"Indeed we were,\" Charlie said agreeably. He introduced himself and Constance. \"This won't take long. I know you must be a very busy man, what with services and lectures and private counseling and house-hunting, and all.\"\n\nSebastian smiled and shrugged. \"My office,\" he said, and opened a door in the foyer. The office was small and had no desk. It was painted white, with a soft gold carpet, and soft gold draperies. There was a rack of books and pamphlets on one wall; Constance recognized titles that she had seen in Jill's room. There were many cushions piled along the opposite wall, and between the two walls were several upholstered chairs. Still smiling, Sebastian motioned them to chairs.\n\n\"I noticed that you found the seating arrangement not altogether satisfactory the last time,\" he said.\n\n\"Very observant,\" Charlie said, as he and Constance sat down in chairs. Sebastian stood near the window. \"Are you still looking for a place to buy?\"\n\n\"I have not looked at all for a place to buy, much less still.\" His eyes bulged slightly, and his lank hair fell continually into his face. He used both hands to push it back again and again. The slight smile did not waver.\n\n\"Ah. I understood that Mrs. Ferris was showing you the mill property a few weeks ago.\"\n\n\"Many people show me many things, Mr. Meiklejohn.\"\n\n\"I bet they do,\" Charlie murmured. \"Mr. Pitkin, there have been several deaths in and around the mill, as you well know. We can play games, or you can level with me and I'll go away.\"\n\n\"Please, just Sebastian. And believe me, I want very much to help in your investigation. I'm afraid I have a small problem, however. I don't quite understand exactly what it is you are investigating.\"\n\n\"The murder of David Levy. And the death of Stanley Ferris.\"\n\nHe sobered at the words, and now looked sorrowful. \"Murder? David? Surely not. I met him only one time, I'm afraid, so I have no knowledge of his usage of drugs, but I assume that when the authorities decide a death by self-administered drugs is accidental, they have sufficient grounds to make that assessment.\"\n\n\"Murder, Mr. Pitkin. You gave him a list of vitamins. When was that?\"\n\n\"Vitamins? Oh, yes. I remember. We talked about vitamins at that cookout. Weeks ago, the same day Jill escorted me through the mill. He was paying far too much. That company is without a conscience, I'm afraid. I did tell him he could buy less expensively through the company I recommend to my own people, and a day or two later I gave the list to Jill to pass along to him. She said she would give it to her stepmother who would actually give it to David. I heard nothing more of the matter.\"\n\nHe looked so self-satisfied, so superior, Charlie thought, that he wanted to give him a swift kick. \"And you said that evening that a person with the right attitude wouldn't get stung by bees, didn't you?\"\n\nSebastian shrugged, and his hair fell into his face; he pushed it back. \"It's possible I said that, of course, because it is true. But I confess that I don't recall saying it.\"\n\n\"Is it also true that if you have the right attitude you won't get bitten by a dog?\"\n\n\"Of course. You see, Mr. Meiklejohn, you are the bee, and you are the dog, and the tiger, and the snake.\" The cadence of the words, the rhythm suggested thinly disguised mockery. \"Naturally there are many quite self-destructive people who haven't grasped that yet, and they are at risk, but with understanding and acceptance, there is no longer any danger with anything of nature.\"\n\nConstance had been watching him intently, but now she left her chair, as if bored, or perhaps simply restless. She moved about the room and came to a stop at the books and pamphlets. At first Sebastian watched her movements with annoyance as she strolled farther and farther from Charlie, forcing Sebastian to turn his head from side to side to include both of them in his sweeping gazes. When she began to examine the various books, he seemed to give up and concentrated on Charlie, only now and then glancing at her as she browsed in the reading material as if unconcerned with the questions and answers.\n\n\"Uh-huh,\" Charlie said, ignoring Constance. \"If you aren't looking for property to buy, what were you doing at the mill with Jill Ferris?\"\n\n\"Looking.\" He held up his hand and shook his head. \"Don't be angry, Mr. Meiklejohn. Your questions are difficult because they carry so many presuppositions, so many assumptions. Why is it, I wonder, that you ask only questions that you already believe you have answers for? Anyway, at one of our discussion meetings here, the matter of a school came up. I did not bring it up myself, but it was the subject of discussion once it did arise. It was suggested by one of our people that if we had a school, a large enough building to accommodate sleeping space and lecture rooms, that then some people could use it for a retreat, immerse themselves more fully in study and meditation without the interference of pressures of the outside world. A cloistered retreat seemed idyllic to many of us, I admit. But I have no money and I did not pursue the idea. Since then, several people here have offered to show various properties that they had reason to believe might be suitable. Jill made such an offer and I accepted. Not with any intention of buying, however, because I can buy nothing.\"\n\nCharlie regarded him for a moment, and then said, \"And if one of your people takes you on a tour and you find the ideal place, and if that person then says, 'Hey, don't sweat it, I'll buy it for you,' then you're in like Flynn. Neat.\"\n\n\"There are many paths to enlightenment, Mr. Meiklejohn. That may well be one that one of our people will follow.\"\n\n\"How long has Jill Ferris been coming here?\"\n\nNow Sebastian shook his head, in sorrow as deep as his receptionist's had been. \"You know I won't talk about any of my people, Mr. Meiklejohn. I came here in March this year, and I met Jill Ferris soon after that. She never offered to buy a building for me, or to help me financially to purchase one, and that's all I intend to say on the subject.\"\n\n\"I see. Was she driving all the way out from the city, just to attend your meetings? When did she become a regular?\"\n\nSebastian shook his head again, harder, sending his hair flying out both sides. \"You will have to ask her.\"\n\n\"Oh, I will. Do any of your... what do you call them? Parishioners? Disciples? Students? Do any of them ever stay here in this house?\"\n\nSebastian folded his hands before him. \"Exactly what are you implying now, Mr. Meiklejohn?\"\n\n\"Just curious. You got here in March. Jill turned up for her prolonged visit in April, I understand, and yet she's become quite an adept in meditation. Makes me wonder how she had time to advance so fast.\"\n\n\"Some people are ready to progress rapidly. Others never are able to see beyond their own noses. I really must go now, Mr. Meiklejohn. Is there anything else?\"\n\n\"One thing,\" Charlie said, rising lazily. \"Why did you agree to see us at all? You had no intention of telling us anything, did you? And did you notice that I asked not a single question about where you're from, how you got into the soul business, your qualifications to counsel troubled people. Nary a question. No doubt, that sort of thing will be neatly wrapped up in Wollander's report on you. If I see that I need more, I'll come back.\"\n\nSebastian did not move, but his stance changed subtly, his hands became tighter in their grasp, one of the other. \"What report are you talking about?\"\n\n\"Did you really think you could set up camp in his backyard and not be investigated? Now that is really naive. Especially where his daughter is concerned. I haven't asked you about your movements over the past couple of weeks, last Friday night particularly, or if you've been over to the farm, communing with the bees, your alter egos. We'll see how much of your time is already accounted for.\"\n\n\"I have nothing to hide, and nothing to fear from any investigation,\" Sebastian said with tight lips. He strode across the room and yanked the door open, then stood aside for them.\n\n\"Thanks for your time,\" Charlie said good-humoredly as they went through the foyer to the front door. \"Be seeing you around, Mr. Pitkin.\"\n\nSebastian's lips tightened even more, and his eyes looked strange, more bulging than before, as if suppressed rage was pushing on them from behind. A high flush was on his cheeks now. He said nothing.\n\nCharlie opened the door and let Constance go out first, then pulled the door closed hard. He did not let go, however, and did not allow it to latch. Silently, with an amused expression, he counted to seven and then pushed the door open an inch or two in time to hear Sebastian's voice:\n\n\"Goddamn son of a bitch! If they come back tell them I'm in Poughkeepsie! That bastard Wollander!\"\n\nSilently Charlie pulled the door closed again, this time releasing the knob. Grinning, he took Constance by the hand and they went down the front steps.\n\n\"Made you mad, did he?\" Constance said softly.\n\n\"Damn right. But I made him madder. What was that with the books?\"\n\nThey reached the car and got inside. He turned the key and then waited for her answer.\n\n\"All his own work. I was just curious. Jill's reading matter seemed a bit odd\u2014high-fashion magazines and spiritual material. I guess he hands out the pamphlets and articles, or sells them, to his people.\"\n\n\"His people,\" Charlie snorted. He started the car, began to drive. \"I'm in the wrong racket. Nobody ever bought me a school or a church, or even a house.\"\n\n\"You could grow a beard and wear a serape, or a ruana, or even a simple blanket\u2014go barefoot, of course\u2014and talk about taking the right road, scorning the wrong road, not being distracted by enticing signs, like luncheonettes, or anything that... . Where are we going?\"\n\n\"You know damn well we're going to have lunch and mull things over.\"\n\nShe smiled slightly.\n\n\"What I want,\" Charlie said in the Spender's Ferry Coffee Shop ten minutes later, \"is a roast beef sandwich with lots of gravy.\" Constance cleared her throat, and he added, \"But what I'll have is the tuna salad with tomatoes and dressing on the side.\" He looked at her balefully. \"Okay?\"\n\n\"Very good,\" she said, and ordered the same. \"And coffee right away.\"\n\n\"We need a spy,\" he said, and fell silent until the waitress poured coffee for both of them. As soon as she was gone, he continued, \"I want to know the minute that crew gets home over at the Wollander house.\"\n\n\"Charlie,\" she said, a little shocked, \"they're at a funeral!\"\n\n\"I know. Jill can hightail it to her room if she wants. But you can count on it that Greg Dolman won't waste any time in telling them the whole thing's settled. They've got their bum, in the best possible condition as far as Greg's concerned. He sure as hell isn't going to deny anything. So, case closed with even better speed than he managed when he closed the case of David Levy. The county's clean and all's well with the world. I want to get there first, that's all.\"\n\nShe nodded thoughtfully. Not the Zukals, she decided. They were probably still at the hotel. Not anyone at the farm, too many unknowns concerning all of them. Not anyone. \"I could hang out at the mill. I wonder if you can see enough from there?\"\n\n\"No way. Not you.\"\n\nShe looked searchingly at him then. No trace of amusement was visible on his face.\n\n\"Honey, there's a real killer loose in that neck of the woods. Someone as opportunistic as hell and pretty damn smart. And desperate by now, I expect.\"\n\n\"Can Greg make a case for Earl Malik? Will it stick?\"\n\n\"Not if I can help it. They'll find a high alcohol level in the blood, no doubt, and Greg will think that cinches it, but he can't have it both ways. If Malik was drunk enough to fall down and bash in his head, he was too drunk to be lurching around the woods carrying a beehive. They didn't find a bottle, remember. He didn't sober up and then start drinking again. And there weren't any prints on the stove. Whatever was on the doorknob was messed up by Hopewell when he opened the door. No prints. So Malik goes in the apartment and turns on the gas, and wipes off his prints, and then obligingly falls down and knocks himself out. Won't work.\"\n\n\"Then he must have seen whoever got the bees, and that's the motive for killing him.\"\n\nCharlie nodded. Their salads arrived and neither said any more for the next several minutes. \"What they'll do,\" Charlie said then, \"is mill about for a while after the funeral, more than likely. Say they start for home again around three. Be back by seven. I guess our best bet is simply to be on hand and wait for them.\"\n\n\"We'd better go through the mill property,\" she said. \"The security people probably won't let us in the front gate.\"\n\n\"Right. So we go home and brood about that damn dog, and light a fire under Wilbur Palmer if he hasn't got his report ready yet.\"\n\n\"You brood. I intend to freeze peas and speak encouragingly to the tomatoes.\" She had left most of her tomato on the plate. It was picture-pretty and tasted like plastic.\n\nWhen Charlie called the animal pathologist that afternoon the report was ready. \"One of the organophosphates,\" Wilbur said. \"More than a dozen trade names for the stuff\u2014insect sprays, dusts, that sort of thing.\"\n\n\"A smart dog would eat something like that?\"\n\n\"Charlie, the word smart is relative, right? So the dog's smarter than a possum, what's that mean? You mix this stuff with some canned dog food, and watch.\"\n\n\"But if it was insect spray, she could have got into it accidentally.\"\n\n\"If there's a store of it around. There was a lot in the body. Still is. I doubt that even a pretty dumb dog would lap up that much unless it was mixed with something. Oh, maybe some fruit got sprayed heavily, but dogs sure don't eat many apples or pears, now do they?\"\n\nCharlie thanked him and hung up and then stood at the patio door watching Constance move about in the garden. She picked a pea pod and ate it and continued with the chore of picking enough for the freezer. Hers was an organic garden, no poisons. You could go out there and eat whatever struck your fancy, he thought, and shuddered at the idea of the meal that had been fed to that poor dumb dog. \"But how?\" he muttered to himself. How had anyone managed to get her to eat it?\n\nSomeone out there was an opportunistic and smart killer, he thought again. Too smart. Taking advantage of every opening to get whatever the hell he was after. Or she, he corrected himself. First the dog Sadie, feeding her poison when she was trained never to take food from anyone but her owners. Then David Levy with drugs when it was known that he did not do drugs. Then the attack on the Zukals with the bees. If Al were allergic, he would be as dead as Stanley. One of the grandchildren, both of them could have been killed. And now the man he was certain was Earl Malik. Each one killed, using materials at hand, leaving no trail, hitting and vanishing in good terrorist form. Kill and run. If Greg announced the end of the mystery, the closing of the case, then whoever was responsible could strike again, and again, until... until what? he thought bleakly. Who next?\n\nThe phone rang in the kitchen as he stood watching Constance. He took the call before the answering machine got its act together. It was Al Zukal. The sheriff had called, he said, and they could go back home. The crazy killer was dead.\nCHAPTER 13\n\nLOIS FELT LIKE AN INTERLOPER, an intrusive guest who was always in the way, no matter what she did. Coming home from Stanley's funeral, Warren had held Jill as if she were a small child, her face against his chest, his arm protectively around her shoulders. One of the security men, Robert something, had driven the Buick, and Lois had been superfluous. The Ferris family had virtually ignored her, and again she was superfluous. That family's grief was so consuming, so overwhelming that even Warren and Jill had felt excluded somewhat; no one had suggested they linger after the funeral services, after the obligatory wake that had been cut short when Mrs. Ferris left the room on the arm of her husband. \"We'll go home now,\" Warren had said, and they got in the Buick and Robert drove them home. No one had uttered a word during the drive.\n\nWorst of all, Lois thought, gazing at the landscape without recognition, she had gone blank. She tried to think of work and nothing came. Two people to be interviewed, she reminded herself, and forgot them again. Where had she stopped the metabolism studies Friday night? No answer came. Had she put everything away on Saturday? No answer. Would Clarence think to check the moisture level of the cloned trees in their pots in the greenhouse? Would Tom? When the Buick stopped at the front of the Wollander house, she seemed to come up from a trance state that left no memory.\n\nMrs. Carlysle admitted them to the house and Jill started up the stairs silently. She was deathly pale, and looked as if she had lost five pounds in the last day or two. Should she be made to see a doctor, be examined? When Lois had asked hesitantly if she was bleeding heavily, if she needed anything, Jill had simply shrugged. \"I'm all right.\" Lois saw again how she had looked Saturday in the brilliant white jumpsuit, with the blood-red sash, and blood running down the legs. Jill had looked mortally wounded. At the stricken look on Warren's face, she knew he was seeing that same image again. She wanted to take his hand, but their new awkwardness permitted no such intimacy now.\n\n\"Mr. Wollander,\" Mrs. Carlysle said in a soft voice that was almost fearful, \"there is a message from Mr. Meiklejohn.\"\n\n\"For God's sake! Not now!\"\n\n\"I'm sorry, sir. He said to tell you it's extremely urgent, that it can't wait. He left a number to call.\" She held out a slip of paper, but Warren pushed past her and went into the study. Lois took the message, and Mrs. Carlysle looked relieved and grateful. \"Thank you,\" she whispered, with a frightened glance toward the study. \"He said it's vital and even told me what to write down, or I wouldn't have brought it up now.\"\n\nLois watched her hurry away down the broad hall toward the kitchen, and then opened the folded paper. I have to talk to you both immediately. It's about Dr. Wharton's former colleague. There was a telephone number.\n\nShe watched the paper start to shake, and realized that she was trembling all over. She closed her eyes and took several deep breaths, and then went into the study.\n\nShe offered the note to Warren, who was standing at the window looking out toward the front of the property. He ignored the slip of paper.\n\n\"I'm going to call him,\" she said.\n\n\"Leave it alone until tomorrow. My God, that man knows we've just come from the funeral!\"\n\n\"I know he does. That's what makes it frightening.\" She went to the desk and dialed the number, watching Warren's back all the while. He was suffering, she knew, losing his son-in-law, his grandchild, maybe his daughter... being brought face to face with his own mortality again. Charlie's voice sounded in her ear.\n\n\"I got your message,\" she said. \"Can't it wait until tomorrow?\"\n\n\"No. Tell your security people to let us through. We're at the mill property. We'll be there in five minutes.\"\n\n\"But what's hap...\"\n\n\"Five minutes.\" He hung up.\n\n\"They're coming from the mill.\"\n\nWarren's shoulders seemed to hunch as if he were bracing against a cold wind.\n\n\"I'll go meet them,\" she said. She started to leave, but stopped again and said pleadingly, \"Warren...\"He didn't move. She left the room after a moment to wait on the patio.\n\nThey never had needed security before, she was thinking distantly. She had felt as secure as a child in her parents' home, protected, safe from the world and everything it might threaten her with. And now there were men on duty to keep out reporters, sightseers, curiosity seekers. Killers? Maybe even that. And the more security they hired, the less secure she felt. She felt as if the threat existed at her elbow, not out there at all. She started to walk across the expanse of manicured lawn, feeling more exposed with every step.\n\nShe waved away one of the guards when Charlie and Constance emerged from the woods. They were an odd couple, she thought, still with the distance that seemed to be her self-protective armor. He was so dark and bearlike, she so tall and slender and fair. Lois liked the way they looked at each other. One plus one made a bigger one, she thought then, and thought of them that way, bigger than one, but still one. She felt a rush of envy, and she had an impulse to warn them that the cleaver could fall, even on them. They wouldn't believe her, any more than she would have believed such a warning just a few weeks ago.\n\n\"What's happened?\" she asked, as soon as they were in range.\n\n\"Let's save it for you and your husband,\" Charlie said. The crinkles at the sides of his face were gone, and deeper lines etched his features now.\n\nHer fear returned, redoubled; she nodded and led them inside the house. They went straight through to the study where Warren was now seated in his favorite chair, with a drink in hand, an invisible wall all around him. It started at his eyes; they were cold and hard. He nodded curtly at Charlie, and ignored Constance altogether.\n\n\"This shouldn't take very long,\" Charlie said. \"Today another body was found in the apartments at the farm. The sheriff will be around either this evening or in the morning. I wanted to get here first.\"\n\nLois went paper-white and sank down into a chair, her eyes wide and staring. \"My God,\" she whispered. \"Who?\"\n\n\"Unidentified at the moment.\"\n\n\"Earl?\" she whispered. \"Was it Earl?\"\n\n\"I never met him or saw a picture of him. Hard to say for sure. But I think so.\"\n\nWarren's eyes had narrowed and he looked different; he looked dangerous. This must be his political face, Charlie thought. Briefly and succinctly he told them what he knew about the body in the apartment, watching them both closely. Neither moved until he finished.\n\nCharlie caught a motion from the corner of his eye and glanced at Constance who was moving toward the study door. She opened it. Jill stood there.\n\n\"Come on in,\" Charlie said. \"You heard enough to know what's happening, I take it.\" No name had been mentioned, he knew; Lois's whisper Earl could not have carried to Jill. What she had heard would be broadcast news, and it made no difference. The interesting part was that she had learned it this way, by eavesdropping.\n\nShe had changed her clothes and was wearing jeans, a black shirt, and sandals. She looked childish and ill. She went to the sideboard and poured gin, added tonic and ice cubes, and sat down without a word. She looked too young to be drinking anything but a Coke or juice.\n\nLois stirred finally and said, \"I'm sorry. Do you want a drink? Sit down please.\" She looked at Charlie and Constance in turn, but without real invitation behind the words.\n\n\"You were right to tell us first,\" Warren Wollander said then. \"A man without ID. Died the same night that the bees were taken. He probably did it himself, then. They may never find out who he was.\" He looked steadily at Charlie as he spoke in a very deliberate voice.\n\n\"They'll find out,\" Charlie said just as deliberately.\n\n\"Meiklejohn, I hired you to do a job for me. I find now that I made a mistake. Keep the retainer, but you're no longer in my employ. I think that's all we need say. Our business is concluded as of this moment.\" He lifted his glass and drank, and set it down on the table by the side of his chair.\n\n\"Dr. Wharton,\" Constance said then, \"we have to talk, you know.\"\n\n\"My wife is under no obligation to talk to either of you,\" Warren said coldly.\n\n\"Mrs. Wollander may be protected by your name and your position and your wealth, but Dr. Wharton is at grave risk,\" Constance said.\n\nLois was watching her, pale down to her lips now. She glanced at Warren, who returned her look, stony-faced. Suddenly Jill jumped up, knocking over her glass that she had put on the arm of her chair. Everyone ignored it.\n\n\"Look at you!\" she cried, facing Lois. \"Mrs. Wollander will obey! I always envied you, did you know that? I did! Your degrees, your brains, your work, your freedom. Your freedom! You could tell the world to fuck off if you wanted to. But you're just like the rest of us! One look from him and you're just like the rest of us. I thought someone finally would say no. That's all, just no!\" Tears were streaming down her face by now. She wiped at them with the back of her hand. \"Shit! I'm going down to the lake.\" She ran from the room and a moment later a door slammed, the sound muffled by distance.\n\n\"What do you want to know?\" Lois asked. Her voice was harsh, grating.\n\nOf them all, Charlie thought at that moment, only Constance appeared unruffled by the scene Jill had made. And Constance had a thoughtful expression\u2014her professional expression, was how he thought of it. And since it was her show now, he waited for her to go where she was heading.\n\nConstance nodded to Lois, but turned to Warren and asked, \"Why didn't you ever run for office?\"\n\nHe started at her question. Jill's outburst had affected him deeply, it was apparent; he looked very tired, and very old. After a second he said, \"I don't even know her. She's like a stranger over and over.\" He took a breath. \"I'll get you both a drink. Lois?\"\n\nHe busied himself at the sideboard, handed out gin and tonics, and then said to Constance, \"I thought I could be of more service in the background.\" He stood at the sideboard watching her.\n\nConstance sipped her drink, then put it down again. \"Thank you,\" she said. It wasn't clear if she was thanking him for the gin and tonic, or for responding. \"And in the background there would be no public scrutiny, no inquisitive press hounding your every movement.\"\n\nHe shrugged. \"That too, but I have nothing to hide, nothing to fear.\"\n\n\"Maybe not personally. But what about the first Mrs. Wollander? What would have come up about her?\"\n\nHis jaw muscles tightened noticeably, but he controlled his voice so much that it was flat, expressionless. \"It was never an issue.\"\n\n\"I want to tell you something about your daughter, Mr. Wollander, but first you must tell me something. Why didn't you divorce your first wife? What was her hold?\"\n\nCharlie tensed, then made himself relax again. Wollander looked murderous, but he was not moving, not threatening. And if he did, Charlie thought, Constance would flatten him like a pancake. He waited.\n\nWhen Wollander remained silent, Constance said gently, \"She was a lovely young woman, much sought-after. She could have had her pick, and she accepted you. She made demands that you were happy to concede to. And the demands were binding long after the time when you would have divorced her if there hadn't been a hold. Isn't that right, Mr. Wollander?\"\n\nHe was regarding her with disbelief. \"Who told you any of that?\" He rubbed his eyes suddenly and left the sideboard where he had been standing, took his chair again, and looked at nothing in particular. \"You have it pretty much as it was. I agreed to sign over the property here to her, in the event that I ever divorced her. I agreed that no illegitimate child could inherit the estate. We were drinking champagne, celebrating her acceptance, laughing, joking. I thought we were joking, playing a game. I agreed in advance to a divorce settlement that seemed as remote as Mars. Divorce was the last thing on my mind then. I was deeply in love with her, crazy in love, I think the expression was in those days. I doubt that she could have suggested anything I would have balked at.\"\n\n\"I thought it must have been something of that sort,\" Constance said, still gentle and thoughtful. \"You see, Mr. Wollander, your daughter believes that your prenuptial agreement bound your first wife financially in such a way that if she sued for divorce she would be penniless. Did Jill have any money of her own?\"\n\nHe shook his head, looking baffled now. \"Just an allowance. But that's a preposterous idea.\" His expression changed, became harder. \"Shelley did it deliberately, to turn her against me. She had an instinct for that, finding the sore spot, knowing how to turn people against people. She told Jill a half-truth; if she sued for divorce, she would have been comfortably fixed, but without power over me. She wanted me to start the proceedings and I wouldn't. That explains so much. I knew she lied to me about Jill, I just never considered that she also lied to her about me. I didn't think of that.\" He turned to Lois. \"That's what she meant, that she was envious of you. It didn't make sense. But if she thinks I held her mother in bondage\u2014\"\n\n\"And now me,\" Lois said. She looked at Constance in dismay. \"We had a talk with her and Stanley, before we announced .our engagement. We told her about our agreement, and I tried to make it clear that I wasn't interested in the money or the property, that I didn't want to get between father and daughter in any way. I thought it was important for her to hear that from me. And she must have assumed that we were repeating what she thought she knew about her mother and her father.\" Her voice trailed off and she shook her head. \"We'll have to talk to her again.\"\n\n\"It won't be easy to overcome a lifetime of believing something else,\" Constance said. \"You may need help.\"\n\nAnd now she was finished, Charlie realized, and could not have said how he knew that.\n\n\"I have copies of everything,\" Warren said. \"She'll have to believe.\"\n\nConstance shrugged and did not reply. She glanced at Charlie.\n\nHe turned to Lois. \"How drunk was Malik Friday night? Staggering, falling down, incoherent?\"\n\nLois started, and then closed her eyes hard for a moment, as if she had to clear out the present revelations in order to think back to Friday night. \"Coherent,\" she said then, faintly, \"but weaving back and forth. Unstable.\"\n\n\"How did you push him? Stand up, show me. I'm Malik and you're coming out.\"\n\nShe hesitated, and he said impatiently, \"Stand up and show me.\"\n\nWith reluctance she stood up, still uncertain. Charlie motioned. \"Here's the door. I'm blocking it, is that right?\"\n\nShe nodded, and now picked up her purse and put it over her shoulder. She glanced around, at Constance, at Warren, and took a step forward toward Charlie. When he did not move, she reached out with both hands and shoved him in the chest. He took a step backward and sat in a chair. \"He fell like this, straight back? Into the bush?\"\n\nShe moistened her lips, nodded.\n\n\"Then what?\"\n\n\"I ran to my car, it was only a few steps away, and got in and drove off.\"\n\n\"Did he get up?\"\n\n\"I don't know. I didn't look at him again.\"\n\n\"How did you pay him off? Cash?\"\n\nShe nodded.\n\n\"Where?\"\n\n\"I met him on the road, at the entrance of the trailer court. He got in my car and I drove about half a mile to a turnoff. I parked there a few minutes and he took the money. Then I drove back.\"\n\n\"Where did he live?\"\n\n\"I don't know. He said in the city, that's all.\"\n\n\"Was he using his own name?\"\n\n\"I don't know. Probably.\"\n\n\"Did you ever enter the trailer court?\"\n\n\"No.\"\n\n\"Did you give a vitamin list to David Levy?\"\n\nShe looked bewildered and shook her head.\n\n\"Did you give work printouts, computer printouts to Malik?\"\n\n\"Yes. He insisted on knowing what we were doing.\"\n\n\"Did you find David Levy's thermos at the farm?\"\n\n\"I don't know what this is all about! What difference does it make? No. He kept it in his backpack.\"\n\n\"Could Malik have moved a hive of bees in his condition?\"\n\n\"No. I don't know. He was staggering. Maybe if he slept a while and sobered up. But he didn't know anything about the bees or the mill property, or the Zukals. He didn't even know where I lived. He thought this house was miles away, that the only way to get here was by car. He came by train and stayed a couple of days and then left again. Then he came back. But he knew nothing about this area. No.\"\n\n\"Was he ever fingerprinted? In the service? Arrested?\"\n\nShe shook her head. \"I don't know. Not that I'm aware of.\"\n\n\"Did he have a family, close ties who would start a search if he stayed missing?\"\n\n\"No, not that I know.\" She looked exhausted. She bit her lower lip when it began to tremble visibly.\n\nCharlie looked at Warren Wollander, and without moderating his tone, he said, \"Did you have Malik investigated?\"\n\n\"I think I've had just about enough of this,\" Warren said coldly.\n\n\"It hasn't even started,\" Charlie said, in a voice even colder and harder. \"You had him investigated and that fact will come out. This is going to blow up in your faces and you know it. When did you have it started?\"\n\n\"Last week,\" Warren said finally. \"But my men don't talk about private work.\"\n\n\"Last week?\" Lois asked in a faint voice. \"You told me you didn't believe the letter.\"\n\n\"I didn't believe it,\" he said harshly. \"My God, I thought you were faithful! It never occurred to me to believe it until I saw you leave that night. The night Jill told us she was pregnant. I couldn't sleep for thinking of it and I saw you, someone, going up the driveway. I came down to wait, to speak to you, someone, I don't know what I intended, but no one came back in. When I went upstairs again, your lights were still on. What was I supposed to think? The next day I called Myers and had him look into it.\"\n\n\"And a few days later Malik is dead,\" Charlie said softly. \"Mr. Wollander, you know this isn't going to stay a secret.\"\n\nLois was staring at her husband with incomprehension. \"I didn't go out that night, or any other night, not to meet him, or anyone else.\"\n\nHe looked at her with a penetrating gaze and finally rubbed his eyes. \"It wouldn't have been Jill,\" he said slowly. \"Why would she sneak out in a black raincoat like that? She was with that man Sebastian for hours every day, no need to meet him like that. Who else? Mrs. Carlysle?\"\n\n\"Show me,\" Charlie said then. \"You were where?\"\n\nWarren went to the window and pointed. \"She vanished behind the copse of birch trees. I didn't see her again. I thought I'd hear her when she came back in but I didn't.\"\n\nCharlie dismissed it. \"Okay. Is it possible Sebastian believed that Jill had enough money of her own to buy the mill? Did she have money of her own?\"\n\n\"No, of course not. Stanley was more than generous, but it was his money. Now it's hers, of course. What Sebastian believed, who knows?\"\n\nLois seemed to have shrunk into herself; she sat huddled in a tight mass, her arms hard about her body. She looked from Constance to Charlie, and she asked quietly, \"You think they will find out that dead man is Earl, don't you? No matter what we do or say, they'll learn that?\"\n\n\"Routine police work usually uncovers an identity,\" Charlie said. \"It may take a little time. Maybe fairly soon. What will you do if they bring around a picture and ask if you know this man?\"\n\nBefore she could speak, Warren said, \"I'm sending my daughter and my wife away for a rest. My daughter has had two terrible shocks and needs complete rest, and she needs a companion.\"\n\n\"No,\" Lois said, still not looking at him, keeping her gaze on Charlie. \"I have my work to do. That won't help. Running away never helps.\" She moistened her lips. \"Tell us what to expect.\"\n\n\"Right. I expect the sheriff already knows I'm in here with you, and that's enough to make him come calling this evening. He'll be polite and insist that the case is neatly tied up, the killer is dead of gas inhalation, accidental death, and that's that. But the routine will take place anyway. They like names to attach to corpses, after all. And he'll find out. That will bring him back here with more questions. And he'll find out that you've withdrawn large sums of money from your bank account. He'll find out that Mr. Wollander has had a private investigation of Malik. He already knows about the campaign to get rid of the Zukals. Probably there will come a time when he'll say to himself what if this Malik character and Dr. Wharton decided they wanted the property themselves in order to continue the work they started together so many years ago? And he'll come back with even more questions. Somewhere along the line, they'll connect material found on Malik's clothing with the bushes outside the quonset hut, and someone will remember the stones that were thrown up by tires, and someone will remember two sets of skid marks, two fast departures. Someone will suggest a scuffle between Malik and his former wife, on the night the bees were moved. Now Greg Dolman doesn't like trouble, but the way this is shaping up, the Wollanders are really out of it. This is between a flat-land foreigner who is now dead, and a lady scientist who doesn't really belong here anyway. That will reassure him and he'll keep coming back with more and more questions.\"\n\nAbruptly Warren snapped, \"Stop this! This is nothing but speculation! It won't get that far!\"\n\nLois regarded him very steadily, as if he were a stranger. \"Why not?\" she asked quietly. \"You know me better than anyone else here, and you were ready to believe the worst. You still believe the worst.\" She turned to Constance. \"I was naive. But now I understand what you meant. High-priced lawyers may be able to save Mrs. Wollander, but Dr. Wharton is doomed, isn't she? Dr. Wharton cannot survive another scandal, especially of this magnitude. Suspicion, innuendo, accusations, even if it never gets as far as an arrest.\"\n\nThere was a soft knock on the door and Warren strode over to open it. \"What is it? We're busy in here.\"\n\nMrs. Carlysle nodded. \"Yes sir, but it's the sheriff. He says he has to see you.\"\n\n\"Our cue,\" Charlie said, when Warren looked over his shoulder into the room again. \"We'll leave by the back door.\"\n\n\"I'll show you,\" Lois said, rising. \"I doubt that he'll want to see me at this stage.\" She led them out without another glance at Warren.\n\nThey went through the wide hallway, into a narrower one. Good, old-money comfort, Charlie was thinking, glancing around with approval, not new-money chic; there wasn't a trace of an interior decorator more recent than the turn of the century. Good woods, cherry and walnut, chestnut paneling, stained-glass windows here and there. Very nice.\n\nThey came to a side door and Lois paused after opening it. This was the side of the building; a short brick path led to the wide lawn; beyond the lawn was the trail through the woods. The sun had set; violet, shadowless twilight yielded to darkness under the trees.\n\n\"Thank you for... for preparing us,\" Lois said formally then. \"Mr. Meiklejohn, you think that all this is connected, don't you? David's death, the bees, Earl.\"\n\n\"Oh yes. Starting with Sadie.\"\n\n\"Sadie?\" Momentarily she lost the tight control. She closed her eyes and drew in a long breath, and then said, \"Thank you. I'd better go back in now.\"\n\nCharlie and Constance walked across the lawn in silence. At the edge of the woods he stopped to look at the driveway that curved around the birch trees and vanished. The woods were close on the other side of it. It was a wonder Wollander had been able to see anyone moving there, he decided, and took Constance by the hand and entered the woods. It was getting dark fast and under the trees it was almost too dark already. Still, the trail was easy to follow.\n\n\"Charlie,\" she said in a low voice, \"that scenario you outlined, I think Greg Dolman would buy it in a second if things get that far.\"\n\n\"Oh, so do I.\"\n\n\"And it will come out that she was friendly with Sadie. She's the only one beside the Zukals who was.\"\n\n\"Yep. I know.\"\n\n\"And she certainly had access to David's apartment.\"\n\nCharlie took out a penlight and they walked single file, and then they were out of the woods at the end of the swinging bridge. They had taken only a step or two on it when bright lights came on at the corners of the mill. Charlie grunted. Al's new security system was up and running.\n\n\"If our killer decides to strike again,\" he muttered unhappily, \"those lights won't mean diddly.\"\nCHAPTER 14\n\nTHEY TURNED DOWN THE offer of dinner from Sylvie and inspected Al's security system for a second time. Smart lights that came on when anyone got within twenty-five feet, dead-bolts, sensors on doors and windows. All very good stuff, Charlie admitted, and knew that none of it would matter in this particular case with this particular opportunistic killer. He did not tell Al and Sylvie this. Instead, he merely cautioned them to be alert and not to put too much faith in the sheriff's pronouncement that the case was over and done with. Then he and Constance drove to El Gordo's where they ordered Mexican pizza with green chilies and authentic white cheese that was very like Feta, and a seafood platter for two, Vera Cruz-style.\n\n\"With a pitcher of margaritas,\" Charlie said firmly. \"Soon.\"\n\n\"A whole pitcher?\" Her protest was mild, without conviction. He pretended not to hear.\n\n\"Okay,\" he said, as soon as the waiter Rene appeared with the pitcher that was so cold it looked frosted. \"All right!\" He poured for them both. \"You were saying?\"\n\n\"Not a thing.\" She sipped, then took a real drink. \"Charlie, what about the thermos? Why bring that up with Lois Wollander?\"\n\n\"Funny that it's missing, that's all. Not listed among his things. Just wondered.\"\n\nShe drank again and settled back against the leather seat. The furnishings in El Gordo's were authentic\u2014massive black tables and chairs with dark leather. The service pieces were heavy pewter that retained heat no matter how long the diners lingered. A nice place, she thought contentedly, and the margarita was not the only reason, although it helped.\n\n\"So,\" she said after a moment, \"he used the thermos for tea or lemonade. Not the sort of thing you would leave in it overnight, especially if the next day you planned to work only a few hours, with a late starting time. I think he was out of lemons, at least none were mentioned in the inventory, and he had them on his grocery order. You especially wouldn't leave tea overnight,\" she added with a grimace. \"The dishes were all washed and put away. The thermos should have been put away too, or at least on the counter ready for use. Then what?\"\n\nThe way she did this, voiced the thread that was running through his mind, sometimes bothered him, sometimes amused him, most of the time startled him no matter how used to it he pretended to be. Once or twice, many years ago, he had asked her to explain how she did that, and she had looked puzzled and said simply, \"I don't have an idea in the world what you're blithering on about.\" The one time the question had taken the form of a demand, she had turned her pale blue eyes on him in a way that reduced him to a thieving boy caught dead to rights in the act. His demands had been very infrequent over the years.\n\nTonight he followed her reasoning with a grin and nodded when she finished. \"Then what, indeed,\" he said. \"I wish I knew. And who took a late-night walk in a black raincoat on a warm night without rain? Why? As Warren Wollander said, Jill spent hours every day with Sebastian, no need to meet him like that. And as Lois said, Earl Malik never went to the house. I guess he didn't. I found a map with the farm marked on it, but not the house. Remember, he was on foot. That's a long walk by road, seven or eight miles from the trailer court. Just over two miles to the farm from there. That road curves and recurves before it passes by the Wollander place.\"\n\nThe pizza arrived and they became silent as they ate the spicy food that was piquant just to the point of pain, and drank the tangy margaritas. A couple they knew, the Martinsons, passed by their table and stopped briefly to chat. Rene brought the seafood platter and lifted the pitcher in invitation. Charlie shook his head regretfully. Rene shrugged with artistry and went away.\n\n\"Tomorrow,\" Charlie said, \"I want to mosey over to the county board of zoning commissioners and bat the breeze a little. And I think I'd like a chat with the real-estate company that handled that piece of property. Routine stuff. No point in dragging you through it.\" Routine that he disliked, he thought, but did not add that. She already knew. This was the sort of thing he used to have underlings take care of. Sometimes he missed that privilege, of giving orders that would be followed without question, more or less the way he expected them to be.\n\n\"You know, your Aunt Maud's birthday is coming up fairly soon,\" she said thoughtfully. \"And there's an anniversary present for my parents to shop for. Maybe I'll browse the antique stores in Spender's Ferry.\"\n\n\"Aunt Maud is an Aries,\" he said coldly. \"And your parents aren't entitled to celebrate more than once a year, just like the rest of us. We sent them something back in March.\"\n\n\"But shopping takes so much time,\" she said. \"Never too early to begin.\"\n\n\"Christ,\" he muttered. \"Tell me something. Do you think any of those people are crazy?\" She rolled her eyes and sighed. \"All right, not crazy, just demented, psychotic.\"\n\n\"I see. But not crazy. Clarence Bosch is determined to try to keep the project up and running for the duration of his grant. How far would he go to do that? Pretty far. Lois Wharton Wollander is determined to continue her work; she is driven by her need to accomplish good work. How far would she go to get rid of anything in her way? Pretty far. Warren is determined to protect his daughter, his property, and his wife. Same question, same answer. Pretty far. The Zukals are determined to keep that land. And so on. Crazy people? Of course not, but obsessional people, yes.\"\n\n\"You left out a couple,\" he said after a moment.\n\n\"I know. Sebastian. Charlatan, simpleton, or believer? A bit of all, I suspect. Did Jill make him a promise to buy that land and the mill? How far would he go to get it? I think the answer depends on the strength of his faith in what he's doing, and I can't judge. I haven't seen enough of him to have an opinion. Do you think he's a complete charlatan?\"\n\nHe hesitated, then shook his head. Not really, he had to admit. He wanted to believe in Sebastian as one hundred percent charlatan, but not yet. He had that particular gift, and in Charlie's experience people with it usually came to believe in what they preached, to some extent, anyway. \"Jill?\"\n\nConstance looked remote. \"She's been twisted out of shape by too much hatred. Shelley must have been extremely clever, also twisted out of shape. I think Jill is going through a terrible time. Coming of age shouldn't be this hard, but she's waited a long time to start the process. I wouldn't like to be Warren.\"\n\n\"What about when he tells her the truth about the pre-nuptial agreement? What will that do to her?\"\n\nConstance shrugged. \"Probably not what he wants. She may take it as an attempt to whitewash him, and to turn her against her mother, whom she obviously loved. She could become even more furious with him for not telling her years ago, for allowing her to live the lie of her mother's invention. She could simply start running and never stop until she's back in Paris or wherever she lived with her mother. It takes a certain amount of readiness to begin the reconciliation process and I don't think Jill is ready just yet.\"\n\nCharlie thought about this for a time, and then asked softly, \"And what about Tom Hopewell?\" It pleased him that she looked surprised.\n\n\"You've noticed?\"\n\n\"Come on,\" he said. \"Think I'm that blind?\"\n\n\"I don't think he's even aware of it himself.\"\n\n\"Let me have a go at him,\" Charlie said. \"He sees Lois as a brilliant, very attractive woman, exactly as I see her. Well, maybe not exactly as I see her, but anyway, as a colleague that he respects and admires, and he thinks that she's wasted on an old man like Wollander.\"\n\n\"I doubt that he's thought through any of that.\"\n\n\"Yet.\"\n\n\"But what could he hope to accomplish by driving the Zukals away?\"\n\n\"I don't know. Maybe he thinks that Lois could have bought the property and they could have worked on it together.\"\n\nConstance shook her head. \"He's into corn, not trees.\"\n\n\"Right. But two other men who looked at Lois with lust in their hearts are dead, remember. Speaking of lust, let's hit the trail. I have this little proposition I've been meaning to bring up practically all day.\"\n\n\"The game plan,\" he said the next morning, slowing down in Spender's Ferry, \"is that we meet at Al's place.\n\nYou wait there if you're done first, or I wait. You're sure you want to hike out?\"\n\n\"It's only two miles or so,\" she said. \"Of course.\"\n\n\"And, honey, watch your step. I think that crazy or not, there's someone hanging around who is sure deadly.\"\n\n\"I do, too,\" she said.\n\nHe stopped at a corner and she opened her door, then leaned across the seat and kissed him. \"I wouldn't be Lois Wharton Wollander for anything,\" she murmured, withdrawing. \"See you later.\"\n\nWhat the devil was that all about, he wondered, as he pulled away, and watched her in the rearview mirror. She waved, smiling softly. And he thought he knew. He began to hum under his breath. And the people she grilled today wouldn't even suspect a fire had been lighted under them.\n\nIn the first antique shop Constance compared a milk-glass bowl to a depression-glass vase, and priced a set of iron sconces. She asked a few questions and listened, and asked a few more questions. In the next shop she examined a wicker chair and end table, and listened. And so it went for over an hour. She dropped in at the bank to cash a check and met the sister of Carla Mercer, who was Warren Wollander's secretary. She remarked on the resemblance and listened some more, and moved on. Midway down the street she saw the blond woman who had been at the trailer court; today the woman had on yellow slacks and a yellow tank top. The dark roots of her hair made her look like a sunflower.\n\nConstance nodded to her as they neared one another; the other woman paused, then stopped.\n\n\"You were out at the court, weren't you?\"\n\n\"Oh, that's where we met!\" Constance said in apparent relief.\n\n\"You're new around here, aren't you?\"\n\n\"I'm just waiting for my husband. He has some business to take care of and I'm burning up. I was thinking of a nice cool drink. Iced coffee, maybe. Is there a place around here for something like that?\"\n\n\"Sure. Coffee shop up in the next block.\"\n\n\"Want to keep me company? Have a coffee with me. Charlie said fifteen minutes but he's been hours already. And my feet hurt.\"\n\n\"Husbands!\" the blond woman said eloquently. \"Up this way.\"\n\nHer name was Ellen Thurmond and her husband was canvassing the county for a national survey company. He had said it would be fun for her to come along for the ride, but then he parked her in the trailer court and took off every morning. Husbands!\n\nIt was cool and dim in the coffee shop. They sat in a booth and Constance sighed. \"This feels good. That looked like a pretty nice trailer court, I thought. And that Petey! He's something else, isn't he?\"\n\n\"But what? That's the question. It's a good thing your husband got there yesterday. Would have been too late by now, I guess. That Earl, he's something else, too. Flits back and forth to New York like a commuter flight or something.\"\n\n\"I never got to know him, but Charlie seems to think he's pretty weird.\"\n\n\"Weird! That's putting it mildly. A nut is more like it. Your husband a process server?\"\n\nConstance looked around, then said conspiratorially. \"Something like that, why?\"\n\n\"Thought so. It figures. See, Earl comes to the court and he's there alone, without a car, and I'm there all day alone without a car, and I think, well he's someone to talk to, at least. Can't talk to Petey. All he wants to talk about is Petey. So I go over to Earl's place and look in, and the place is a mess. I mean a real mess. And there's all these computer printouts everywhere. And he acts like it's national security time suddenly.\" She fell silent as the waitress brought their coffee. Then she leaned forward and said in a harsh whisper, \"I think he's one of those hackers who breaks into the big computer systems and does them dirt or something.\" She sat back, satisfied with herself, and began to sip her coffee.\n\n\"Oh dear,\" Constance said. \"I wonder if Charlie suspects anything like that. Go on.\"\n\n\"Well, the way he goes back and forth into New York City, I mean, why doesn't he just stay there if it's that important? Anyway, this morning a guy comes and says he's Earl's buddy and he's supposed to pick up his stuff for him. He has a letter from Earl and some extra rent money for the inconvenience. He says he wants to keep renting the place for another couple of months, and even pay extra just to hold it, in case Earl decides to come back and finish his work. And he even signs the register for Earl.\" She glanced swiftly about the coffee shop, exactly as Constance had done, as if checking for eavesdroppers, and then added, \"But he spelled the name wrong. Earl Marker. That's how he signed it.\"\n\n\"But that's awful! I don't even know his last name. What is it?\"\n\n\"I don't remember. But not Marker. Maxwell, something like that. But it's Marker now.\"\n\n\"It really sounds as if there's something funny going on,\" Constance said with a frown.\n\n\"Yeah. Let me tell you. You know how when you want to make a phone call, you put in your money and get the other person, and you talk. Right? Not Earl. He puts in his money and says a word or two, and hangs up. Then in a couple of minutes the phone rings, and he's right there, and this time he talks. We don't even have phones in those dumps. Petey's afraid we're going to run out without paying up, I guess. It's a pay phone out by his apartment. In clear view from my place. I saw him with my own eyes, twice. Like in the movies or something.\" She finished her coffee. \"Anyways, I think he got wind that the process server was closing in or something like that. And he's skipped.\"\n\n\"I guess you're right,\" Constance said. \"But that's how it goes.\" She listened to Ellen go on for a little while longer, but when it became apparent that nothing of interest would be added, she looked at her watch and then asked, \"Would you like more coffee?\"\n\n\"No, thanks. I've got to get over the store before it gets much hotter. It's a long walk home. And he'll come in wanting supper on the table like always. I tell him it's a long walk, but he's out in the air-conditioned car all day, what does he care about that?\"\n\nA few minutes later they stood on the sidewalk and Constance waited to see in which direction Ellen was headed, and then walked the opposite way. She dropped in at the drugstore, and looked over the racks of magazines and books, then chatted with the pharmacist while she mulled over the numbers on the sunscreen lotions and finally chose a number twenty. He approved. Skin as fair as hers needed protecting, he said. Finally she decided she was ready to start the long walk to Al and Sylvie's house. She felt her morning had been well used.\n\nShe wished she had paused to make some notes, but that would have been too conspicuous; now, since she hadn't done that, she began to review the facts. Just the facts, Ma'am, she told herself, rubbing sunscreen on her nose as she walked, sometimes in the sunlight, then in a shady area. The trailer court was only half a mile from town, not at all the long walk that Ellen had complained about. She passed it. The houses were far apart now, farmhouses mostly. The road was narrow and curvy, bordered by fields of corn, a truck garden, a small apple orchard, and some abandoned cherry trees that were gnarly and beautiful, and alive with birds eating their fill and making a raucous, joyous cacophony.\n\nThe experimental farm started, fields of corn that were planted in blocks, not like the uninterrupted fields of commercial farmers. Although from the car the farm looked like the others, on foot it was easy enough to tell the difference; there were many labels here, rope dividers, and interplanted with the corn were other crops\u2014a block of amaranth with furry, red flowers already; squash vines sprawled; blocks of what looked vaguely like bush beans... .\n\nWhen she reached the entrance to the farm, she turned in. She walked around the admin building, nodded to workers here and there, and kept walking toward the quonset hut where Lois worked. When she drew near the second greenhouse, a plump young woman in shorts came out carrying a clipboard. She looked as if she had fallen asleep in the sun. Her nose was bright red; her cheeks, her forearms, and thighs all glowed.\n\n\"You looking for Dr. Wharton?\" she asked.\n\n\"Yes.\"\n\n\"In there,\" she said, pointing to the greenhouse she had just left. She hurried on to the next one and went inside.\n\n\"Thanks,\" Constance said after her, and stood in the open doorway peering in. Toward the far end she could see Lois Wharton and another young woman. Lois was watching this woman do something with an instrument that from here looked like a complicated hybrid of thermometer and syringe.\n\n\"Hi,\" Constance called.\n\nLois looked up, spoke to the other woman, and came toward Constance. She looked haggard, in jeans and a tank top and sneakers, her face moist, her hair stuck to her forehead here and there. There was a smudge of dirt on one cheek.\n\n\"It all comes back to me,\" Constance said with a smile when Lois got near. \"The reason I chose psychology instead of one of the hard sciences. Actually there are several reasons, I'm sure. I never could remember to keep both eyes open when I used a microscope, for instance. I always felt that if they wanted you to use both eyes, there should be two eyepieces.\"\n\n\"Mostly there are these days,\" Lois said. She was smiling slightly now. \"When your husband introduced you, he said Leidl, but I didn't make any connection with the name, I'm afraid. One of the grad students is awed, though. He's studied some of your work, he says. I'm not very good with people, associating names, remembering things that are important to others.\" There was hardly a hint of apology in her tone even now when it was clear that she intended to be apologizing.\n\n\"That's a strange relationship, isn't it?\" Constance said. \"Student and teacher, I mean. I've had students act as if they really believed I was born with a Ph.D. They sometimes seem to want to deny that we put in the hours, did the classwork, were disciples of teachers who were as difficult taskmasters as the ones they face in us. TA's, waiting for the acceptance notification, summer workshops, never quite enough money, the orals, they seem to think we skipped all that. Especially the women students,\" she added. Neither of them mentioned Jill.\n\nLois was nodding emphatically. If she noticed that Constance had started them strolling toward the trees that screened this part of the farm from the opposite side of the lake, she gave no indication. \"What I was always so terrible with was the politics in academia,\" Lois said, her gaze on the ground before her. \"Somehow I kept forgetting that everything ended up being political, no matter where or how it got started. I still do. I guess, coming out of psychology, you were better at that part.\"\n\nConstance laughed with a touch of bitterness. \"What I seemed to be good at in those days was butting my head against walls. Idealistic and all that.\"\n\nLois glanced at her with understanding. \"Me, too.\"\n\nThey had reached the trees where the shade and the breeze coming off the lake were both welcome. Constance lifted her face and breathed in deeply. \"I was pretty hot,\" she said. Before Lois could respond, she went on, \"I was thinking of you last night, the patience it must take to work with trees. I had a friend once who was going through a difficult time with a physical problem that was quite undecided. She planted half a dozen trees on their property during the period when she was stewing over the outcome of tests. I thought it was an act of supreme faith. The trees are very lovely now.\"\n\n\"And your friend?\"\n\n\"She succumbed after five or six years but she nourished the trees to the end. She said they were her legacy. A butternut for the squirrels; a red twig dogwood for beauty in winter and for the birds; a noble fir for protection of wild things when the weather was brutal; a sugar maple... . Each one was chosen for a particular purpose. It was, I think, a great act of love, to leave this legacy.\"\n\nFor a time they were both silent as they gazed at the lake that was such an unnatural, sapphire blue. The wind played on the surface in random patterns. It riffled a section that smoothed out again, then another, and another.\n\nSuddenly Lois sat on the ground and hugged her knees tightly. Constance sat down near her. They both kept watching the unending patterns rise and fade on the surface of the blue water.\n\nWhen Lois spoke again, Constance was taken aback by the harshness of her voice. \"It isn't patience,\" she said, \"it's desperation. So few people understand the situation we're in, how fast it could become critical. Tom Hopewell, Clarence, a few others. Earl knew at one time. You build a generating plant back in the Midwest, Gary, Indiana, or Chicago, somewhere, and the vapors rise and years later come down as poison, rains so acidic that lakes die, trees die, plants twist and writhe in their own brand of agony. Fish, frogs, aquatic plants, algae, all gone. And the lake looks like a reflection of heaven. Isn't that ironic, that it should be so beautiful and so deadly?\"\n\n\"But can't it be countered?\"\n\nLois continued to stare at the lake stirring in the soft breeze. In her mind's eye she saw the ghost trees, saw the figure rising from the waters, David's blind eyes wide open... . She shuddered. \"A seedling puts down a deep tap root,\" she said, almost as if lecturing. \"A zone of undifferentiated cells forms where the rootlet and soil merge, and then the acid releases the aluminum in the soil and that means the tree is doomed. The tree no longer absorbs the potassium it needs, the calcium, magnesium, manganese, whatever nutrients it relied on before; aluminum replaces them or seals the cells that would take them in. Desperately the tree puts out surface roots and they take up the nitrates that also rain down from the heavens. In this form, in this amount, more poison. Or else drought comes along and the shallow roots wither. Our need for electricity, heat, light, our need for crops, fertilizers, they doom the forests. Can we undo it? No one knows. No one.\"\n\nThe breeze that had felt so welcome only moments ago now was chilling; the lake that had been so incredibly beautiful now seemed a mockery. Constance hugged her arms about her. \"Why are you telling me this now?\"\n\n\"You said patience. But that's not it,\" Lois said, following her own line of thoughts, paying little attention to Constance. \"Patience is benign, maybe even saintly. Sebastian has patience. People with too much patience are dangerous. They're the ones the world listens to, wants to hear, wants to follow. Not us, the doomsayers. Do you have any idea what kind of hours we're putting in here at the farm? Clarence, Tom, I? We're not driven by patience!\"\n\n\"Yet you were willing to drop your work and come chat with me,\" Constance said. \"Why?\"\n\nLois pressed her forehead hard against her knees. Her voice became muffled and thick. \"You, your husband, you have to leave all this alone. It's over with. The sheriff's satisfied, everyone else is. I have to get back to my work. We all do, yet as long as you're still asking questions, saying it's still open, no one can.\"\n\n\"Why do you drive to the farm now? I thought you enjoyed the walk?\"\n\nLois looked up then and said sharply, \"It's late when I stop work, too late to walk home. I'm too tired.\"\n\n\"Or afraid to walk through the woods in the dark suddenly?\"\n\n\"That's absurd.\"\n\n\"I think it's very sensible not to be alone in the dark around here,\" Constance murmured.\n\nLois took a deep breath, started to say something, and turned to look at the poisoned lake instead. Very quietly she said, \"There's nothing in this for your husband. Warren paid him; he won't stop payment or anything. Just leave, let us all get on with the work we have to do.\"\n\n\"Your husband thinks he can delete Earl Malik from living memory?\" Constance asked.\n\n\"Last night,\" Lois said, speaking to the wind, the water, the air, \"Warren and I talked a long time. There's something he has to do, something very important to him that he must do. He had a heart attack a few years ago, before we were married. Coming that close made him reexamine his life, everything he's ever done, and he realized that the most important thing is... what he has to do now. Anyway, we have to get all this... this trouble behind us so we can both get on with these other things. We simply have to.\" Her words rushed out suddenly, and she was almost whispering. \"He knows how to manage things, make things work out. He always has known how to do that. For him there's no mystery to politics, how to make people do what he wants them to do. It will be weeks before Jill can touch Stanley's money, but he will lend her some so that she can go back to Paris where she wants to live. She has friends there. That poor girl never was here long enough to make any real friends, always back and forth, back and forth. He told me last night that she never quite finished high school. She went to a private school in France, and another one in Switzerland, another one in England, but altogether she never actually graduated, never went on to college. That explains her outburst about me, her jealousy of my work. She never had a chance.\"\n\n\"And Charlie's warning last night, that an identification will be made eventually? What about that?\"\n\n\"You don't understand!\" Lois said, her voice rising. \"Warren can manage all that. He can manage Charlie, too, if he has to. I believe him. He's always been able to manage things. Except for Shelley. How shrewd she must have been to get that agreement signed early. She understood him thoroughly. She used to taunt him about inheriting the land here, about selling it piecemeal, plowing up the lawn to put in tract houses or something. She was a devil! She taunted him about Jill, how the line had run out, how there'd be no heirs to keep the property in the family. She made him believe Jill could never bear a child, that the land would go to strangers and be destroyed. I knew how important it was to him, to leave the property to his own blood. I even stopped taking birth-control pills last year, thinking that maybe I would be the one to provide the heir.\" She laughed bitterly. \"I was so convinced of the Tightness of providing him with a child that I actually lost the pills, misplaced them beyond finding again. Is that a Freudian slip or something?\" She put her forehead on her arms again. Her words became nearly inaudible. \"I know the order of things now. I guess I always did but I let myself pretend to be stupid. His daughter, his grandchild, his land, and then me. That's the order of things. And it's all right. Jill is young enough. She'll marry again and present him with grandchildren. I have my work. He has his memoirs to complete. And we have to get on with it. We have to!\" When she raised her face, it was tear-streaked. She ignored the tears that were still flowing. \"We'll do it Warren's way. He'll take care of everything. Father knows best and all of that.\"\n\nConstance regarded her for several seconds, then turned to look at the lake again. \"When did your father die?\" she asked. \"You loved him very much, didn't you?\"\n\nLois scrambled to her feet. \"What do you mean?\"\n\n\"Oh, you married your mentor years ago, and you let him take you into dangerous if not illegal waters. Although he finally betrayed you, when he turned up again, you paid him handsomely, out of pity, out of remorse, out of fear. Only you know why. And now you are willing to let another father figure plan your future for you, take you into dangerous waters, perhaps betray you. And this time, through enough love, enough patience, enough forgiveness perhaps you can atone. Children feel such guilt, such betrayal when parents die prematurely that occasionally they spend a lifetime trying to rewrite that particular past. It never works, you know. Actually, it's one of the most egotistical fantasies of childhood, that you as a child could bring about the death of your parent through wishing. The more sinned against you feel, the more forgiveness you must offer.\"\n\nConstance had been gazing at the blue lake, but now glanced at Lois, who stood rigidly before her as if paralyzed. \"You and I both know that Clarence Bosch is well aware of your past. As director of this project he would have been informed. That's how the system works, and we know it. He values your work and your honesty or you wouldn't be here, and you know that, too. He'll keep you until the grant runs out, unless you are arrested. He has nothing to lose by it\u2014not at his age, with retirement coming. And he has much to gain through your work.\" She shrugged slightly, stood up, and began to brush traces of leaf mold from her slacks. \"No, you're not going along with this just for yourself and your work. But why? To protect your husband? I wonder why you think he needs your protection. A deep-seated psychological need of your own, or a concern of greater immediacy?\"\n\nLois turned ashen and swayed. She held onto a tree for a moment, then pushed herself off and ran staggeringly. She bumped into another tree a second later, steadied herself, and continued to run toward the quonset hut. Constance watched.\n\nWhen Lois was out of sight, Constance started up the path that led to the Zukal house. She was startled when Charlie stepped out from behind a tree and held out his hands to her. Wordlessly he drew her close and held her, stroking her back.\n\nWhen she pulled away after several minutes, she studied his face. \"You were watching?\"\n\n\"For a while. I got here first and decided to walk around the farm a bit. I didn't want to interrupt you. Are you okay?\"\n\n\"Fine,\" she said wryly. \"Doctor's fine, patient is a mess.\" They started to walk together, his arm around her shoulders, and she told him what Lois had said, that Warren would arrange things. \"Manage things, is how she put it,\" Constance said. \"Including you. He could be a very difficult enemy, Charlie. He has a lot of power around here.\"\n\nCharlie tightened his grasp of her shoulders and did not reply.\nCHAPTER 15\n\nWHEN THEY WERE WITHIN sight of the mill, Charlie stopped and said, \"Anything interesting in town?\" He was watching men stack lumber torn down from the interior of the mill; the little rooms were all gone now to make way for the large work space Bobby would need. Next week, he thought distantly, Bobby and Flora and their two children were due to move out from the city.\n\n\"It's a fascinating little village,\" Constance said. \"I think if a grasshopper sneezed in that town word would be out to everyone within ten minutes. And anything to do with the Wollanders or the farm is prime news, of course. And the people who work out here for the most part either live in town or have relatives who do. Every sneeze is reported.\"\n\n\"So tell me,\" he said in a growly sort of way.\n\n\"I am,\" she said in her equable sort of way. \"First, Clarence Bosch didn't drive to his daughter's wedding the weekend that David died, after all. His wife drove over to Long Island on Monday of that week and he flew out of Albany at noon on Saturday. They drove home together, arrived Sunday night late.\"\n\nCharlie grunted. \"What on earth made you go after that tidbit?\"\n\n\"Curiosity. Usually the wedding is in the hometown of the bride. I just wondered why not this time. It's because the groom is in the navy and had a very short shore leave, and his parents live on Long Island and could swing the affair, and he had to report for duty again on Sunday at noon. They all decided that on the whole it could be managed better without extra traveling for the bride and groom. And Clarence Bosch didn't have to leave home until after ten-thirty on Saturday to make his flight.\"\n\n\"Check. Next.\"\n\nShe told him about Ellen from the trailer court, and about someone's cleaning out Earl Malik's possessions. This time he cursed softly.\n\n\"That's Warren Wollander's doing. He's managing things, the bastard. Go on.\"\n\n\"Warren's secretary's sister is a teller in the bank where Lois banks.\" She said this rather flatly, not liking the implications. \"Everyone in town could have known that she was withdrawing large sums of money. On the other hand, Jill has charge accounts everywhere and Stanley wrote the checks. He was very generous, they say. She lost a rather expensive gold bracelet a few weeks ago and he just bought her a new one. And you never know where you're going to see her flashing past in that little foreign car of hers.\"\n\nHe chuckled as her voice took on the tones of disapproval she had registered when listening to the townspeople. She was a good mimic.\n\n\"And the mill was a regular den for years and years, but recently it's been quiet, for the last ten years anyway. It's because those kids don't have to sneak around to do things now the way they used to. They just live together openly and don't care who knows it.\" Again she was mimicking one of the people she had talked to that morning. It didn't matter which one, Charlie knew; if he ever talked to that person, he would recognize that voice instantly.\n\nConstance gave him a complete summary of her talk with Lois and they were both silent after this for several moments.\n\n\"You know,\" he said at last, \"the case I made for her being It could even work. She certainly had reason to kill Malik.\"\n\n\"And David?\"\n\n\"He could have tumbled to the fact that her ex was hanging out, that she was seeing him.\"\n\n\"And Sadie? And the campaign against the Zukals? I believe Clarence Bosch, that this little parcel of ground would be meaningless for the work she does.\"\n\n\"Yeah,\" he said. \"I know. But maybe Malik knew more than she told us about the setup here, the bees and all. Maybe he actually did it the way Greg thinks he did, and then she had to kill him, to keep him from striking again.\"\n\n\"Good heavens!\" she cried. \"What for? Malik, I mean?\"\n\n\"David because he caught on to his being here. Sadie because he hated dogs. The bees because he wanted to worry Warren to death so Lois would inherit the plantation and he could step in and run it.\"\n\nGravely Constance nodded. \"And then Lois told him about her prenuptial agreement, that she won't inherit, after all, and he hit himself in the head, turned on the gas, lay down and died out of remorse. I'd buy that.\"\n\nCharlie scowled. \"What else did you hear?\"\n\nShe told him a few more things she had heard in town, mostly confirming what they both had already known. Wollander had this county in his pocket; it was hinted that he owned the sheriff, that he was the one who had made Greg Dolman stop operating a speed trap years ago, but that he had allowed it to go on until Greg had made a killing first.\n\n\"Your turn,\" Constance said finally.\n\n\"Not as much as you turned up,\" he said. \"An anonymous phone call to the zoning commission started the whole investigation from that department. Everything's in apple-pie order. No one knows if it was a male or female who called. Period. And no one even looked at this place for the past four years until the Zukals showed up. There was a little interest earlier, but nothing serious.\"\n\n\"That reminds me,\" Constance said. \"Apparently Sebastian has at least half a dozen rather wealthy patrons showing him real estate all over the county.\"\n\nCharlie grimaced. \"And one of them will buy a place for him, want to bet?\"\n\n\"No way. You know very well that I never bet.\"\n\nHe laughed outright and took her hand. \"Let's pay our respects to the Zukals and be on our way.\"\n\n\"You're not going to Wollander's house?\"\n\n\"Nope. If what Lois said is right, there's no point. Let him stew about it. Next time we talk, let him come to me.\"\n\nSylvie had one of her green kerchiefs tied on her red hair. She had on baggy pants and an orange-and-blue plaid shirt with the shirttails out. She was standing with her fists on her hips, glaring at Al on the porch, when Charlie and Constance reached the house.\n\n\"Charlie, tell him what they do when you shoot a deer out of season. Just tell Mr. Thick Skull, will you?\"\n\n\"Dear God,\" Constance breathed. \"Did you buy a gun, Al?\"\n\n\"What's this, a third degree, all of a sudden? I ain't got a right to have a gun?\"\n\n\"Did the deer keep you up all night?\" Constance asked him.\n\nAl gave her a suspicious look. \"Yeah. Lights go on, lights go off, on, off, all night. And you look out and it's like they're getting ready for Christmas or something, getting ready for the long ride.\" He looked at Sylvie with a scowl. \"I told you, I ain't going to shoot at them, just shoot and make a noise, scare the bejaysus out of them so they'll go have a meeting somewheres else. What's the crime? A man can't shoot a gun?\"\n\n\"All night,\" Sylvie said. \"You wouldn't believe it. All night. The lights go on and you look out and it's this monster with horns like this.\" She spread her arms wide. \"Just looking back at you. Red eyes. Like the devil in the movies, you know what I mean? Real red eyes. And just standing looking back. Like they own the world or something.\" She frowned in puzzlement. \"I thought deers was supposed to be scared of people.\"\n\nWhile Constance talked to Sylvie about deer and the hypnotizing effects of light, Charlie went with Al to inspect the gun, which turned out to be a twelve-gauge shotgun. He gave Al the name and address of an expert who would teach him to load it and how to shoot it, and extracted a promise that Al would not even try until he had instruction.\n\n\"Well,\" Constance said in the car, driving home again. \"If he does try to shoot that thing, it will buck and he'll think he's broken his shoulder. A shotgun!\" She glanced over at Charlie who was scowling at nothing in particular. \"Don't you have any ideas? This is giving me the willies.\"\n\nHis scowl deepened and he didn't even bother to answer.\n\nIn his dream Charlie was hiding bitter pills in bleu cheese and feeding them to Brutus, who gulped them down as fast as Charlie could prepare them. He woke up enough to hear rain and reluctantly got up to take care of the windows. He closed the bedroom door without a sound and dodged all three cats as he made his way through the house, checking windows. The cats always became manic if anyone got up at night or if anything unusual took place before breakfast. Ashcan wove in and out between his feet, and Brutus sniffed his ankles, his knees, and swatted Candy, and made a nuisance of himself. Then Candy and Ashcan raced after each other, and Brutus ambushed them both at the door to the dining room, and all of them wanted to go outside. If he was up, they forgot about their own swinging door that he had installed to end the doorman's job he had inherited somewhere along the line. Doorman to a bunch of dumb cats, was how he had put it then, and he repeated the phrase now as he stood with the door open, and they sniffed the air suspiciously, and then cursed him for making it rain.\n\nHe poured himself a bit of bourbon, added water, and went to the refrigerator for a piece of cheese, and Brutus was back inside at his feet magically, watching for a dropped crumb. He could move faster than thought if cheese fell, and he could hear the refrigerator door from anywhere in the yard, especially in the middle of the night. He could hear the can opener from the next county, Charlie thought sourly, dancing around the cat. Cues, Constance had said. He had learned all those cues and they were indelible in his brain, new synapses had been formed, and probably never would be erased again. At least, not without extensive counter-training.\n\nCharlie sat at the table and found that he was no longer thinking of the insatiable cat, but of Sadie, who also had learned cues. The trainer had transferred the cue to Sylvie, he was thinking. A word, maybe, or something in the introduction that let the dog know it was all right to take food from her new mistress. And in turn Sylvie could transfer a cue to Al. He remembered what she had said, they both had to handle the dish and food for a few days, and then Al could do it alone. Scent, Charlie decided. That cue was scent. Sylvie's scent had to be on the dish while the dog was learning to accept Al, too.\n\nHe nodded to himself. The rain stopped and the sky was lightening, not in a proper dawn because there was still too much overcast; he realized he did not really want the bourbon. He got up to make coffee instead. A moment later he caught a flash of movement as Brutus jumped on his chair and snapped up the pieces of cheese Charlie had left on the table. The cat streaked away as Charlie watched in resignation.\n\nA little after seven Constance stood in the doorway as Charlie moved about the kitchen muttering. Muttering in their house could mean either that the person speaking needed reminding that what was being said was unintelligible, and at those times it was okay to interrupt, to join in, to ask questions. The other muttering was different: it was not to be interrupted. Muttering at seven in the morning, alone in the kitchen, was definitely the other kind. He was making pancakes, his voice a soft rumble; the aroma of coffee was rich and irresistible. She entered the kitchen without a sound, and he looked at her with an almost blank expression and said, \"But why?\"\n\n\"I give. Why? What why?\"\n\n\"I think I know how someone got rid of Sadie, but Jesus, why? What was the point?\" He glanced at the clock on the stove. At seven-thirty he would call Brenda Ryan, and it was taking hours for the time to creep to that number. Several hours had passed since the last time he looked at the clock, two minutes ago.\n\nConstance blinked, and he grinned at her. \"I'll bring coffee, you sit.\" He poured a cup for her and brought it to the table where she was yawning. \"I'll pretend you're awake. Ready?\"\n\n\"In a second.\" She drank some of the coffee first, then nodded.\n\nAt nine-thirty they drove up to the Ryan Kennels. The sun had broken through the clouds and the morning was brilliant and pleasantly cool, fresh-smelling, sparkling with water droplets everywhere. The trembling drops looked as pure, sweet and life-giving as they were meant to be.\n\nBrenda Ryan was waiting for them. This morning she was wearing a bright-pink-and-white print dress, and pink earrings that looked like bracelets. Three hundred eighty pounds, Charlie thought, shaking her hand, no matter what she had told Sylvie. She had a very firm grip. A large black Labrador retriever was at her side. Brenda's fingers brushed the dog's head, and it began to wag its tail languidly, as if out of duty rather than interest.\n\n\"This is Duchess,\" Brenda said. \"And if you can get her to eat, I better go into another line of work.\"\n\n\"You held back part of her food this morning, so she'll have an appetite?\" Charlie asked, suddenly not quite as certain as he had been only moments ago. He was carrying a shopping bag.\n\n\"Sure did, but it wouldn't make any difference. I mean, most dogs will eat whenever food's available, hungry or not, unless they're really overfed. And I don't overfeed my dogs.\"\n\nFor a moment she looked as if she dared him to comment on that, but he pretended no automatic comment had come to mind, and he and Constance moved into the office, which instantly became very crowded.\n\nBrenda watched as he began to rummage in the bag. First he pulled out a plastic-wrapped package of latex gloves. He pulled on the thin rubber gloves without a word and felt inside the bag again. This time he took out a package that held a scarf. He handed it to her unopened.\n\n\"Would you mind taking out the scarf and tying it on your head a couple of minutes?\"\n\nShe hesitated, watching him through narrowed eyes, then shrugged, sending her patterned dress into a flurry of motion. Her hair was bouffant, curled, sprayed, dyed, with smells Duchess would recognize instantly, Charlie thought, hoped. He watched her put the scarf over her hair and tie it loosely. Next he took out a can of dog food, and finally a can opener.\n\nHe held up the can for her inspection. \"Gourmet dog food,\" he said, unnecessarily; she was already nodding at it. \"Just tell me this,\" he said, opening the can. \"Is there anything in particular that you do or say when you feed her?\"\n\n\"Nope. She's trained to stand back and wait, and then eat. Don't have to do anything more than put it down for her.\"\n\n\"Good. I was afraid there might be something else.\" The top came off, stuck to the magnetized opener. He put it on the desk. \"If it's going to work, that's probably time enough for the scarf,\" he said. \"Let's finish. I'll take it now.\"\n\nHis uncertainty was increasing by the second. Duchess had not shown so much as a flicker of interest at the sound of the can opener, or the malodorous contents of the can. Brutus would be climbing his head by now, Charlie thought gloomily. Brenda handed him the scarf and he spread it out on the desk and began to spoon dog food onto it. He put a few spoonfuls down and cast a doubtful look at the beast who sat there looking as inscrutable as a damn Buddha. Sighing, he wrapped the food enough to move it, and lowered it to the floor, and stepped away from it.\n\n\"Now, I guess we just wait,\" he said. \"You didn't give her a signal to sit still or anything, did you?\"\n\nBrenda was looking very interested now. She shook her head. \"She's waiting to see if I'm going to give her a signal. See her look at me?\"\n\nCharlie didn't see the dog move a muscle.\n\n\"If this works,\" Brenda said soberly, \"I'll have to change my methods, take it into account.\" She became silent now as Duchess began to shift.\n\nThe dog stood up, still apparently waiting for a sign, and when none was forthcoming, she ambled over to the food on the scarf and sniffed it with interest, then sniffed all around it, and back to the scarf again. Then she gulped it down in two bites. She licked her chops and sat down again.\n\n\"Goddamn it!\" Brenda muttered. \"I never thought of that. Of course, you'd have to have an in, get something that belonged to the owner, but even so, it's gotta be trained out. Goddamn it!\"\n\nCharlie looked at Constance who was smiling, more at his relief than at his victory. \"So now we know how,\" she said. \"But why?\"\n\nShe drove back. He talked.\n\n\"We made such a fundamental mistake,\" he said. \"The dish was found at the mill and we assumed that food had been put in it over there. Let's rethink the whole thing. Someone doctored the dog food and put it at the edge of the property. Sadie wouldn't have barked unless someone actually stepped over the line, and that wasn't necessary, after all. Just reach in and lay down the scarf with the food on it, then stand back and wait. You saw how that mutt wolfed down the food. Two bites and done. Then the poisoner reached in and grabbed the scarf back and was finished. David found the scarf and returned it to Sylvie. I'd guess he didn't realize until later what it meant. There could have been traces of the food on it as well as grease, or he could have seen someone with it earlier. He saw something, that's for sure. So he was next.\"\n\nHe became silent, brooding about it. Finally Constance said, \"Charlie, it doesn't make much sense, does it?\"\n\n\"Why not?\"\n\n\"Well, killing David because of the dog poisoning, and then killing Earl Malik because of the bees. It just doesn't make sense. Neither of those things, killing a dog, or moving bees to the station wagon was a capital offense; a reprimand, a fine, that's all either warranted. Murder to prevent discovery is just too much. In each case the response to being found out was too great. It keeps getting muddled in my head. I mean, it seems to be what happened, but it's so senseless.\"\n\nShe was a good driver; when she needed to pay close attention she focused on the road and other traffic, then her thoughts were tightly contained. After the road was clear again, the other thoughts took precedence. It was an automatic process that she didn't question. Now traffic thickened a little and she passed a Grand Union truck, and a Honda, and a few minutes later she glanced at Charlie and started to speak. He was slouched in the passenger seat, his chin tucked in, a scowl on his face.\n\nShe held the words and looked straight ahead again. Beside her, Charlie stirred, and the words were in his head just as if they had been uttered. He grunted again, considering. They had made a second, more serious fundamental mistake.\n\nWarren Wollander was going over the several days' stack of accumulated mail with his secretary, Carla Mercer. Pleas for money; would he put in an appearance at a dinner for Representative Lorenzo; two committee meetings were coming up, an agenda was needed... a policy meeting in Washington... . Tiredly he rubbed his eyes when the phone rang. He motioned for her to take it, and he leaned back in his chair with his eyes closed.\n\n\"I'll ask,\" she said. \"It's Richard at the gate. He says Sebastian is out there and is asking permission to come in.\"\n\nWith a jerk Warren sat upright. \"Let him come in. And you can get started on that stuff. We'll continue this afternoon with the rest of it.\"\n\nShe spoke softly into the phone and hung up. She gathered up papers, checked around the desk, and left for her own office down the wide corridor, toward the rear of the house. Warren went to the window to watch for Sebastian.\n\nSebastian's car, a powder-blue Cadillac, came into sight at the exact spot where someone had vanished the night Jill had announced her pregnancy. He rubbed his eyes again, harder this time, as if to erase the memory that was filled with pain, and he walked through the study, through the hallway to the main house entrance, and admitted Sebastian before he had a chance to ring the bell.\n\n\"I would like to see Mrs. Ferris,\" Sebastian said.\n\n\"Come in. This way.\" Warren pushed the door shut and started back to the study. When Sebastian hesitated, he motioned with his hand. \"Come along.\"\n\nAt the study, he waited until Sebastian was inside, and then closed that door and stood at it. \"Sit down, if you'd like. This won't take long, but you may as well be comfortable.\"\n\n\"Mr. Wollander, I have no wish to take up your time. I really would like to see Jill.\" He remained standing in the middle of the room.\n\n\"I know. I'll call her in a minute.\" He studied Sebastian for another moment, then started walking slowly toward his desk. When he spoke again, his voice was machinelike, completely expressionless, as if he were reading from cue cards. \"Sebastian Pitkin. Boy evangelist. Petty thief. Trick knee that keeps you out of the army but permits you to sit in lotus. Married twice, divorced twice. One son, sixteen years old now. Expensive tastes and the ability to indulge them without any taxable income. Out of the country for seven years, never accounted for.\"\n\nNot a flicker of expression crossed Sebastian's face. He listened almost gravely, as if checking off the items against a list in his mind. When Warren stopped, Sebastian said, \"I would like to see Jill, if you please.\"\n\n\"Of course.\" Warren had come to a stop at his desk. He moved around it and reached for the phone and paused once more with his hand on it. \"I have very little influence outside this state, but for various reasons a good deal inside it. Did you know that many IRS investigations arise from information gathered first at the state level?\" He picked up the phone and touched a button. The wait was brief. \"Jill, Sebastian is here, in my study. Will you come down, please?\"\n\nSebastian's expression continued imperturbable. Warren sat down and leaned back in his chair and neither said anything more until Jill entered.\n\n\"I came to see how you are,\" Sebastian said, studying her face intently. \"I was concerned about you.\"\n\n\"Well, as you see, I'm fine. In mourning, but okay.\"\n\n\"Are you doing your meditations? Are they helping?\"\n\nShe shrugged. \"Not really. That served its purpose, and I appreciate all you did for me, but I think I'm through with that now.\"\n\n\"Jill, you are the best student I've ever had. You have enormous promise. Don't let it slip away because of this tragedy. Let this experience strengthen you; find its meaning and grow from it.\"\n\n\"See it as opportunity?\" she said bitterly. \"Look, Sebastian, for a time I believed I could follow your path, but no longer. I'm sorry.\"\n\n\"I have the name of a camp, up in the hills about an hour's drive, they tell me. It's a private camp that has come on the market. Go there with me, Jill. A ride up into the hills, a walk under the trees. We won't even have to talk if you prefer silence during this healing period. Silence brings its own power to heal the soul if you let it. And, yes, see it as opportunity to learn, to grow, to contemplate the absolute in a way that has never been available to you before, because you were not prepared before to face it.\"\n\nShe was moving restlessly about the room. She looked irritable and impatient when she took a breath and said, \"Leave me alone, Sebastian. I wish everyone would just leave me the hell alone.\"\n\n\"I know he had me investigated, and that he told you things that he learned, but, Jill, I already talked about those things in our group meetings. Think back. You know I already talked about those things. Each one, every mistake filled with pain and regret became an opportunity to grow. And we grow or we cease to exist. We all have things in our pasts that are better left behind. We talked about that, too, Jill. Remember?\"\n\nShe whirled around to face him. \"What are you saying?\"\n\n\"Only the truth.\"\n\n\"Well, listen to me, Sebastian! I don't have anything in my past that can hurt me now. Nothing! And I have no intention of helping you finance a school, or a church, or a camp, or even a barbecue! Now leave me alone!\" She ran from the room and slammed the door after her.\n\n\"You'd better go,\" Warren said in a flat voice.\n\nSebastian looked at him, then looked past him out the window. \"There is an image in my mind,\" he said softly, \"of a puppet master whose fingers drip venom that runs down the wires that control the puppets, and the puppets absorb the poison, more and more until they are sickened and die of it.\"\n\nHe turned and walked from the room. Warren swiveled in his chair to watch out the window until the blue Cadillac curved around the birch trees and disappeared. He felt chilled and there was a tightness in his chest that he hadn't felt since his heart attack five years ago. His hands were clenching the arms of his chair so hard that when he forced them away, they hurt.\n\nGradually the noises of the house and yard returned. The sound of a riding mower swelled and receded; he could hear the rustle of Mrs. Carlysle in the hall, and a door closing somewhere in the back of the house. Outside, birdsong that had been silenced by Sebastian's departure now resumed. Slowly he arose and walked out through the side hall, out into the yard where the overnight rain had refreshed everything. Two gardeners were at work in the flowerbeds where majestic delphiniums made a deep blue backdrop for golden irises, and masses of pink sweet Williams and carnations were intermingled with clouds of baby's breath. The lawn glistened with raindrops in the shade where it had not yet evaporated. So much beauty, he thought. So much beauty. He found himself standing under the oak tree that he thought of as his tree and he touched the bark almost tenderly. He wanted to wrap his arms around it as far as they would reach, press his cheek into the trunk, and weep.\nCHAPTER 16\n\nWHEN CONSTANCE CAME within sight of their house, she slowed down. A pale-blue Cadillac was in the drive and leaning against it was Sebastian.\n\n\"Well, well,\" Charlie murmured. \"Well, well.\"\n\nShe pulled into the driveway and stopped a short distance from the Caddy; Sebastian straightened up. He looked grim and very angry. \"Hi,\" she called pleasantly, getting out of the Volvo; she headed for the front door with her key in her hand.\n\n\"I want to talk to you,\" Sebastian said to Charlie.\n\n\"How'd you even find me?\" Charlie asked, following Constance.\n\n\"Jill said you lived around here somewhere. I asked at a gas station down the road. I said I want to talk to you. Do you mind?\"\n\n\"Well, of course not. Come on in.\" He looked at Sebastian with disbelief. \"You don't mean out here, do you?\"\n\n\"It will take just a second.\"\n\n\"In the house,\" Charlie said firmly and entered without glancing back again.\n\nConstance had gone on through to the sliding door to the patio which she opened, and then continued to the kitchen, moving around cats without paying any attention at all to them. Candy was complaining in a loud voice and Brutus was looking his slitty-eyed meanest. As soon as they saw a stranger, Candy streaked away, back through the cat door to make her escape, and Ashcan pretended he was a rug.\n\n\"Well, sit down,\" Charlie said to Sebastian, motioning toward a chair at the dinette table. \"Honey, you going to make coffee, or you want me to?\"\n\n\"I'm doing it. Or would you rather have tea, Sebastian?\"\n\n\"Neither one,\" he said. He did not sit down, but glared at Charlie, who did. \"I have a message for you to deliver to your boss, Wollander. Just tell him that if he, or anyone else, harasses me in any way, I have enough information to make life uncomfortable for him, and I'll not keep it to myself.\"\n\n\"You mean about that woman? What's-her-name?\" Constance asked from the kitchen end of the long room.\n\n\"That's what I mean. I'm sure he would prefer not to have everyone in the county looking at him with raised eyebrows the next time he presides at a committee meeting. Especially the one on ethics in government that he's so fond of.\"\n\n\"Gave you the bum's rush, did he?\" Charlie asked with interest. \"But Jill's still on your side, isn't she? I mean, she's of age and all.\"\n\nAbruptly Sebastian pulled out a chair and sat down. He looked uncomfortable, and for a moment Charlie wondered if he should offer him a cushion for the floor instead.\n\n\"Jill's completely dominated by him,\" Sebastian said. \"I didn't believe she would fall into that kind of trap again but he's controlling her; she's like a doll that he moves around here and there.\"\n\nThe fragrance of brewing coffee began to float around the room. Constance brought cups and saucers to the table, then cream and sugar. She talked as she moved about, almost as if absently, \"Of course, we can tell him that, but I don't see what difference it would make, do you, Charlie?\" Her back was turned when she paused, not really long enough for him to answer. She picked up the carafe and came back to the table with it. \"I mean if everyone already knows...\"\n\n\"Well, everyone doesn't know that he had a motive for killing David Levy,\" Sebastian said harshly. \"I can rake that over a few times and see how he likes it.\"\n\n\"Oh, really,\" Constance murmured, and now she sat down and poured coffee for them all. \"Do you take sugar? Cream?\"\n\nHe looked down at the cup before him in surprise, then back to her. She was smiling easily.\n\n\"Why don't you tell us about it?\" she said. She looked at Charlie with raised eyebrows and he nodded almost imperceptibly. \"You see, Sebastian, we aren't working for Mr. Wollander anymore. He fired us, I'm afraid. But we are working for Mr. Levy; we are investigating the death of David Levy.\"\n\nSebastian set his mouth in a tight line. He pushed back his coffee cup and seemed ready to leave.\n\n\"The sheriff is convinced that David's death was accidental overdose,\" Charlie said evenly. \"I'm convinced that it was murder. But the sheriff may have the last word. Again, I just don't see why Wollander would give a damn about what you think.\"\n\n\"The sheriff will say what the puppet master tells him to say. Wollander would go to any length necessary to prevent a word of scandal from touching his wife, and David had more than just a word. He had a whole history to talk about. That's motive enough.\" Now he stood up.\n\n\"Sit down,\" Charlie said sharply. \"Are you really so naive that you think you can get away with blackmail, especially with someone as powerful as Wollander?\" Sebastian didn't move. Charlie waved toward the chair, the coffee on the table. \"You'd better sit down,\" he said tiredly, \"and hear me out. You see, the natural response to your little blackmail threat is to say that you had even more motive. You wanted the property for yourself\u2014\"\n\n\"I didn't. I told you, Jill didn't have a cent. I wasn't expecting to get that property at any time.\"\n\n\"I know quite well what you told me. Now I tell you things. I said sit down, damn it! I'm getting a crick in my neck.\"\n\nSebastian's eyes were bulging more than ever, but he sat down, his face flushed, his hands clenched.\n\n\"Use your own method and relax,\" Charlie muttered. \"My God, you'd think I had a gun to your head. Relax, damn it. See, this is how Wollander could play it. David sees you and Jill together, maybe he even hears her telling you there's nothing she won't give you if you can help her get pregnant. And that means the mill property. Then the Zukals arrive and they have to be driven out again. First the dog, then David.\" He paused, then looked at Sebastian with a hopeful expression. \"By the way, you haven't come across a stray thermos, have you? David was missing one.\"\n\n\"No, I haven't. And you're stringing a line of BS and you know it.\"\n\n\"Oh well,\" Charlie said blandly, and continued. \"Anyway, the Zukals dig in, get their backs up, and another dirty little trick is called for, this time the bees in the station wagon. And this time a transient sees what's happening, and he has to get it, too. But now Jill has inherited a mint, and, no doubt, the Zukals will clear out, and the mill will be up for grabs again, but suddenly Jill's giving you the cold shoulder, telling you to get lost, and you want to retaliate by spreading dirt on the old man. That's how he'll play it, Sebastian. Believe me, I've seen the type many times, and that scenario will hold up. Yes indeed, it would work just fine. If anyone has to leave a tidbit here, another there to clinch it, I'd guess tidbits will be strewn around.\" He sighed and drank his coffee. \"And as you said, the sheriff will buy whatever he has to sell.\"\n\nSebastian's head began to shake from side to side, as if he had nothing to do with it, was not even aware of it. His lank hair swayed back and forth. His gaze remained fixed on Charlie, who had to turn away because it was too damn eerie, he decided.\n\n\"It wasn't like that,\" Sebastian said finally. \"It wasn't. She never even mentioned pregnancy until after Stanley was dead. I had no idea. She never promised me anything for anything. I was in the role of guide to meditation, nothing more than that. And she was a good student. That day at the mill was the only time we were ever alone together for more than a counseling session a few times at the house, where there were always others about. I have witnesses. He can't stick me with anything like that. I have too many witnesses.\"\n\n\"She seems to have found plenty of time to blab to you about the family, about David. Not in front of all those witnesses, I'd bet.\"\n\n\"No. Of course not. We had private counseling sessions. She was very disturbed in the beginning, very. She seemed to derive comfort, serenity even, from our sessions\u2014for a time, at least. She never had anyone to talk to before, not even her mother, certainly not her father or stepmother. She needed to talk to someone and I filled that role for her. She talked a lot about her father, how much she hated him, and the reasons. Good reasons, I can see now, but the emotion of hatred was harming her, truly harming her.\" He looked very earnest, very sincere. \"That emotion, hatred that fierce, does far more damage to the hater than to the one who is hated. I was teaching her that lesson.\"\n\nConstance got up and took away his coffee, which had become cold. She refilled the cup and brought it back and this time he lifted it and sipped.\n\n\"So Wollander found out that David knew something about Lois and might tell it out of turn,\" Charlie said. \"Is that it then? Just that? Will you keep adding more details as needed?\"\n\n\"What more?\" Sebastian took a deep breath. \"From what Jill has said about her father, that was more than enough, Mr. Meiklejohn.\"\n\n\"Just Charlie,\" he said, then added with a grin, \"Doesn't have quite the same ring as when you say 'Just Sebastian,' but it's the best I can do. And, no, that isn't enough, not even for a man like Wollander. What's the rest of it?\"\n\n\"Never mind,\" Sebastian said. \"I'll tell him myself.\"\n\n\"I doubt it. You had a chance and blew it. Don't be an ass now. You wanted us to do your dirty work for you. Pass the word that you're talking to people, no matter about what, just talking. He would then go to Jill and ask what you could possibly know that gives you any clout, and she would stew and steam over it, and maybe even come up with an item or two, and then the high and mighty Mr. Wollander would treat you with respect. Or at least leave you alone. But, you see, Sebastian, you're playing in a new league here, a new game, new players, and you don't know the rules they play by. You don't even know the name of the game. It's Put the Bugger in His Place.\" He stood up lazily and stretched. \"I'm afraid, Sebastian, old buddy, that you've been used, and now you're being sent packing. Just the way it goes.\" He smiled kindly at him. \"And you know very well that I won't help you in a blackmail or extortion scheme, now don't you?\"\n\nSebastian glared at him and pushed himself up from the table. \"Try this one, then,\" he said harshly. \"Who was the man Lois Wollander was keeping on the side in the trailer court? Does the whole world know about that little affair?\" He stalked from the room, with Charlie close behind. Constance began to clear the table.\n\n\"I wonder just how much about that he actually knows,\" she said when Charlie returned, grinning.\n\n\"Remember that question we asked about Sebastian back in the beginning?\" he said. \"I've got my answer. How about you?\"\n\nShe tilted her head, considering, and then said, \"I think we probably agree. Poor man,\" she murmured. \"He's trying so hard to keep his toe in the door. You really shook his tree for him, and you offered no comfort at all to soften it.\"\n\n\"Hah! That's my mission for today, shake trees, see which nuts come tumbling out. And as for him, he tried to put the bite on the wrong woman, that's his problem. And Daddy came to the rescue. I'll give Pete Mitchum a call now.\"\n\nThree times Charlie had had to tell Pete, nothing yet, still working on it. The boy's disappointment had been hard to take each time. Mr. Levy's disappointment had been much harder on the two occasions that Charlie had returned calls from him. Soon, he thought, soon.\n\n\"Nothing definite yet,\" Charlie said to Pete when he appeared a few minutes later. \"I could use some help, though.\"\n\nPete nodded with enthusiasm. \"Whatever I can do, Mr. Meiklejohn.\" Although he had been disappointed, his faith was undiminished.\n\nConstance brought a plate of cookies and a can of soda. Now and then she made Scandinavian delicacies that her grandmother had spoiled her and her sisters with, and she always gave most of them to the Mitchum boys. Too much butter, too much cholesterol for Charlie, who could stand to lose a few pounds, she thought, but she didn't nag about it. Instead, she made the special cookies that he loved and let him have a few, and gave the rest away. And the reason, she knew, was the way the boy was looking at Charlie at that moment. Charlie pretended not to notice that the Mitchum boys had a serious case of hero worship, and he harrumphed if she hinted at it, but there it was. Fortunately he had never said anything like give me a hand, will you, or one of them might well have run for a saw.\n\n\"What I want is four or five guys about your size,\" Charlie was saying, opening the map he had found in Malik's trailer. Pete Mitchum stood five inches taller than Charlie, and his brothers were just as tall and broad. He nodded with a serious expression now that they had got down to business.\n\n\"Here's Spender's Ferry, out the county road here to the farm,\" Charlie said, pointing. \"And the mill property. Now, as soon as you cross this bridge, all the land on the left is state forest.\" He traced the curvy road, and then stopped. \"And this is the Wollander driveway, in about ten feet from the fence here. State forest on the other side of the fence.\" He pulled out a felt-tip pen and drew a wavy line on the map, the boundary, and then drew in the Wollander driveway. \"Along about here, I think, is a clump of birch trees that will be your guide. The driveway turns just past the trees and goes on to the house. I want you guys to start at the county road, on the state forest side, and search the woods from the fence on in for about three hundred feet, from the county road on past the birch trees another thirty feet. It's a lot of ground to cover.\"\n\n\"Naw, it's not that much,\" Pete said. \"What are we hunting for, Mr. Meiklejohn?\"\n\nCharlie glanced at Constance. \"I'm looking for a thermos,\" he said. \"And you might not find anything at all, but I want a good show of a bunch of guys searching. Carry big bags, wear gloves, pick up anything and everything that doesn't belong in the woods, even beer cans. Now, someone will probably come over to see what you're up to, and just tell them you're working for me, and you're searching for a thermos. You might even find a thermos, but I don't want you to fix your mind on that. Chances are you won't find it. If you do find one, don't open it, just chuck it in the bag and keep on searching until you cover the area. Got it?\"\n\n\"Yes sir!\" He looked like a horse at the starting gate.\n\nCharlie thought for a second. \"It could be that the sheriff or a deputy will come along and order you to leave. Don't argue. Just tell them you're working for me, and that you're on state land and have a right to be there. Tell them where I am, at the Zukal place, if they want to talk about it. If they insist that you leave, do it. Bring your loot over to the Zukal place. I'll be there.\"\n\n\"What if they try to take the bags away from us?\"\n\n\"Well, don't let them,\" Charlie said mildly. \"One of the things you'll have to have is a whistle. Use it if they want to steal your loot. I'll come running. But I doubt it will come to that.\"\n\nPete looked ecstatic. \"They won't get it away from us, Mr. Meiklejohn. Don't worry about that.\"\n\n\"Okay, I won't. Now, about time. It will take you a while to gather your forces, round up the bags, the whistle, gloves. Do you think you can have it ail together by four?\"\n\n\"Yes sir!\"\n\n\"Good. When you finish, come over to the Zukals' and leave the bags, and you're done.\"\n\nCharlie watched Pete cross the yard at a run. When he reached the fence this time he didn't bother to climb it; he put one hand on top and vaulted over as easily as a deer.\n\n\"One more little chore, and then lunch, and then off to the wilderness to shake a few more trees.\" He grinned at Constance, who looked up from cutting dense wheat bread. She already had roast beef out, and lettuce, and other good things, he saw with approval.\n\nHe dialed the Wollander house. Carla Mercer, Wollander's secretary, answered on the second ring.\n\n\"Charlie Meiklejohn,\" he said. \"I'd like to speak with Mrs. Carlysle, please.\"\n\nThere was a pause on the other end, and then she said, just a moment; he nodded in satisfaction when he heard a click on the line. He put his hand over the mouthpiece and silently formed the words: \"They're taping it.\"\n\nHe waited almost two minutes before Mrs. Carlysle's tentative voice said hello.\n\n\"Mrs. Carlysle, this is Meiklejohn,\" he said in a brisk voice. \"I need some information that I think you can help me with. Would it be convenient if I drop over at about one, or one-thirty?\"\n\n\"Oh, no, I couldn't. I mean, I'm working. What is it?\"\n\n\"Not over the phone, please. It won't take more than a minute or two, I assure you. And it has nothing to do with you personally.\" She made a throat-clearing kind of sound which he ignored. \"I don't want to bother the family anymore and I believe you can tell me what I need to know. About one, then. I'll come around to the kitchen.\" He hung up, and looked over at Constance. \"Is it lunch yet?\"\n\nAt one-twenty-five he and Constance walked from the woods onto the Wollander lawn. Since he had planned to make them wait until almost one-thirty, he was satisfied with his timing. He was humming under his breath.\n\n\"I think we have a reception committee,\" Constance said softly.\n\n\"Pretend you don't notice.\"\n\nWarren Wollander and Lois were on the patio watching them all the way. Warren moved stiffly to the edge of the flagstones as they drew near.\n\n\"I want to talk to you,\" he said.\n\nIt sounded to Charlie much more like an order than an invitation. \"I have some business with your housekeeper, and then I'll be on my way,\" he said pleasantly.\n\n\"I don't think you have any business with anyone in this house. And I don't want to broadcast what I have to say. Will you come up here, please.\"\n\nStill not a real invitation, Charlie thought with regret. He shrugged, and he and Constance changed course and headed for the patio. \"All I really want to know is who hauls your trash away, and to where,\" he said, still on the lawn a few feet away. \"I thought the housekeeper was the logical one to tell me.\"\n\nLois was staring at him in bewilderment. \"Why?\" She looked as if she had not slept enough for a month, and the effects were ravaging. There were hollows under her eyes and she was very pale.\n\nWarren gave her a sharp look and she sat down abruptly in a chaise. Charlie could imagine the scene that must have taken place a short time ago: Warren in his good, gruff, masculine voice saying, \"I'll handle this. You just keep still.\"\n\n\"I'm still on the trail of that missing thermos,\" Charlie said with a shrug. \"The way I figure it, David had to take the pills somehow, and it could be that he dissolved them in lemonade and did it that way. But then, where is the thermos? See? If it turns up in a landfill somewhere with traces of quaalude and barbiturates, then we'll know, won't we?\"\n\n\"You're still harping on that?\" Warren said in disbelief. \"My God, everyone else is satisfied that it was an accidental overdose. What I want to tell you, Meiklejohn, is that I am convinced, as is the sheriff, that the culprit was that transient in the apartment. We're both satisfied that nothing more is to be done except get on with life now. I don't want you nosing around here any more. You are no longer in my employ, and if you enter this property again, you will be treated as a common trespasser. Is that clear?\"\n\n\"Mrs. Wollander,\" Charlie said, looking past Wollander, \"was that thermos you bought a regular, glass-lined one, or stainless steel?\"\n\n\"Just a regular one, glass lining, I guess.\" Her voice rose. \"Are you going to search the landfill? For a thermos?\"\n\n\"If it comes to that. First the woods. I thought of the lake, but an empty thermos would float, you see, so I gave that up. I thought that if David took it with him when he swam that morning, he could have left it at the edge of the water, but someone would have spotted it, either there, or floating around.\" He was pointedly ignoring Warren Wollander, who was livid with fury now. Charlie gazed across the lake. \"I'm hiring a few of the workers to put in some overtime, after-hours, to search the farm. Eventually I may have to tackle the landfill.\"\n\n\"There could be a dozen such thermos bottles in the landfill!\" Warren Wollander said harshly. \"You're wasting your time, and right now you're wasting mine. Get out, Meiklejohn.\"\n\n\"I know there could be others,\" Charlie said. \"And we'll haul every damn one in for laboratory examination. You realize how hard it is to hide something if people make a determined effort to find it? Very hard, Mr. Wollander. Very hard.\"\n\nLois had risen, and was gazing at the lake also. Hesitantly she said, \"It could be in there. At the bottom. If he didn't put the cap back on, it would fill with water and sink.\" She looked at her husband. \"I think we should call in someone to make a search of the lake. If they find it in there, that will pretty much settle it. Then we'll know.\"\n\n\"I already have someone lined up,\" Charlie said. \"But that's good reasoning. I don't think you leave the cap off myself, because you might lose it, but it's possible. If nothing turns up at the farm, or in the woods, bright and early tomorrow morning you'll see people in the lake. Even if it's in the water, filled with water, there probably will still be traces of the drugs. It doesn't take much to show up in a good lab with good technicians. I have them lined up, too.\"\n\n\"I won't tolerate this,\" Warren said. \"I won't have people here in the lake.\"\n\n\"But the Zukals will,\" Charlie said easily. \"I looked it up in the county records. They have as much right to the lake and what goes on with it as you do. And my people will enter through their property, not yours. Are you going to give me the name of your trash collector, or do I have to do that the hard way, too?\"\n\n\"Get out of here!\" Warren turned and stamped back inside the house. After a moment, Lois followed him.\n\n\"She looks dreadful,\" Constance said in a low voice as they retraced their steps across the lawn.\n\nCharlie shrugged. His face was set in hard lines, his eyes were flinty, like chips of coal. \"Let's go see Bosch,\" he said.\n\nConstance felt chilled. This was how he had become just before they both retired: hard, merciless, finally showing as little sympathy for the victim as for the criminal. The fact that he had become like this was what had made her declare an ultimatum: He had to quit the force, or she would leave him. It came back over him from time to time, that unswerving, unpitying coldness that once had threatened their marriage, threatened everything in her life that she cared about. When he turned into this kind of iceman, the chill extended out from him to enwrap her.\n\nIn fact, Charlie was not thinking of Lois at that moment, nor of Constance, either. That order, \"Get out of here!\" reverberated in his head. All his years on the force, as a fireman, an arson investigator, a city detective, he had been subordinate to men with power just like that, power to decide when to prosecute, when to plea bargain, when to persist, when to call off everything. The power to order, \"Get out of here!\" and never have to look back to see that they were obeyed. You stay with it long enough, something rubbed off on you, he had come to understand finally. Something that couldn't be washed away. Not as simple as hatred, or contempt, not necessarily corruption. But something rubbed off that lingered, that made you see them all as pieces moving in an insane game\u2014the politicians, prosecutors, judges, cops, those who lashed out and those who were lashed, victimizers, victims, all players in the same hellish game. Each and every one of them willing to do whatever it took to survive the game. Even him, Charlie knew, and the knowledge filled him with icy fury, because he wanted, needed, a world built on sanity and reason and law, and they, the ones who made the game rules, had destroyed it, and had forced him to participate in the destruction. By the time you were smart enough to realize that something had rubbed off, you were a bona fide member of the wrecking crew, and the stain, the something, had penetrated right down to the bone marrow.\n\nWith those few words, Get out of here! Wollander had established his place in the game, Charlie's place, Lois's, everyone's. Those few words had caused to surface a deeper bitterness than Charlie had experienced for a long time.\n\nHe felt Constance take his hand, and then squeeze it hard, and he looked at her with surprise.\n\n\"Nothing,\" she said. \"Onward. Let's go shake Clarence Bosch's tree.\"\n\nCharlie grinned at her, but it was superficial. It did not touch his eyes. His eyes looked like dull chips of stone.\n\n\"Look, Dr. Bosch,\" Charlie said a few minutes later, \"I don't give a shit if you like it or not. I want to hire your people. If I can't, then I'll bring in outsiders. I'll get a court order if I have to, but I want the search made today.\"\n\nThey were standing outside the admin building. Bosch was grayer than ever, more troubled than ever. \"Do you have any idea of what all this nonsense is doing to us? Do you? We're weeks behind schedule. Weeks! I told you my caretaker here found nothing. Nothing!\"\n\nFrom the corner of his eye Charlie could see Tom Hopewell approaching, not in a big rush, but covering the ground. \"Did you get a look at that man in the apartment?\"\n\nBosch shook his head. \"No. I looked in the window, but he was just a man on the floor.\"\n\n\"How long have you known about Lois Wharton, who she is?\"\n\nBosch looked startled. He glanced at Tom who was only a dozen yards away, and motioned him not to come closer. In a low voice he said, \"I won't have you dragging any of that into this mess, Meiklejohn! That's uncalled for. It has nothing to do with anything here and now. Lois Wharton is a brilliant scientist, doing brilliant work, and that's all I intend to say on that matter.\"\n\n\"Why didn't you mention that you left town for your daughter's wedding late the morning that David died?\"\n\n\"Oh, my God, what are you driving at? I didn't conceal it. It simply never came up.\"\n\n\"I bet.\" Charlie waved to Tom Hopewell to join them. Brusquely he said, \"I'm hiring some of your workers here to search this end of the farm when they knock off for the day. Any objections?\"\n\n\"What the hell for?\" Tom Hopewell demanded.\n\nWhy didn't he buy some new jeans? Charlie thought irritably. He was tired of these worn-out clothes, worn like a badge of merit, a symbol of his dedication to work, his un-awareness of the amenities.\n\n\"If David took pills dissolved in lemonade, he might have left his thermos at the edge of the lake, or he might have dropped it on his way back to his apartment. I want a thorough search made of the grass at the lakefront on this side, and through the undergrowth from the lake to the apartment units. Understood?''\n\nTom Hopewell blinked, and Clarence Bosch raised his eyes beseechingly toward the sky. Bosch said, \"You know how much area you're talking about? The whole side of the lake?\"\n\n\"Absolutely. Do you know precisely where he went in, where he came out? Do you know precisely where he might have sat down to take a drink? Or the exact path he followed back to his place? The whole damn side!\" Grimly he added, \"If we don't find it today, I'm sending in divers tomorrow to search the whole damn lake bottom!\"\n\n\"Jesus God,\" Bosch whispered in despair. \"Let's get them started and get it over with. For God's sake!\" He looked at Tom Hopewell. \"Tell Henry, let them do it now, for God's sake.\" He reentered the admin building.\n\nTom Hopewell looked at Charlie, then at Constance bitterly. \"This is a damn fucking mess. Come on, meet Henry Tremont.\"\n\nWhen they returned to the Zukals' house, Al and Sylvie were both yelling at a strange man, who was dressed in olive-drab shirt and trousers. A red slash insignia was on his shirt pocket. \"I don't care! I don't want no deers looking me in the eyes all night. You can't fix them lights so no deers set them off, what good are they?\" Sylvie was screaming. Al was saying, an octave lower, but more menacingly, \"I think youse guys loused it up so we'd buy more of your fancy systems. Systems! Systems to get dough!\"\n\n\"You're from Frederick's Security?\" Charlie asked the man who was very red in the face.\n\n\"What's the matter with these people? They nuts or something?\"\n\n\"Al, I've been thinking about your problem,\" Charlie said soothingly. \"Let me talk to the man, okay? I'm Meiklejohn,\" he said. \"You?\"\n\n\"Bergman. Carl.\"\n\nCharlie took his arm and steered him toward the mill.\n\nAl hesitated, started to follow, hesitated again. \"Leave them alone!\" Sylvie said. \"Let Charlie take care of it, like he said. You deaf or something?\"\n\n\"Sylvie, could I have a drink of water?\" Constance asked then.\n\n\"Water? Hell, we can do better than that for you,\" Al said in evident relief that he had an excuse for not following Charlie. \"A beer? Vodka? Even coffee. I say a beer would go all right. Come on in.\"\n\nA few minutes later, when Charlie joined them in the kitchen, Constance was pouring freshly brewed coffee over a tall glass of ice cubes. \"And that's all there is to it,\" she was saying. \"Once when I ordered iced coffee the waiter brought me a steaming cup of coffee with a single ice cube melting away in it.\"\n\nThey all looked at Charlie. \"Al, Sylvie, I guarantee that no deer will keep you awake tonight,\" he said. \"And, Al, if you have another beer, that sure would go down just right.\"\nCHAPTER 17\n\nA FEW MINUTES AFTER four Greg Dolman arrived. Charlie watched him leave his sheriff's car and walk toward the Zukal house, stiff with anger. The trouble with Greg, Charlie thought, was that he looked like a bookkeeper, not a lawman. Or even a preacher, he decided; Greg would look at home at the pulpit. Not a fire-and-brimstone type, but the soft-spoken, even mournful type who suffered over the sins of his congregation, and frequently told them so.\n\nCharlie leaned against the porch railing, his arms crossed, a slight smile on his face as he imagined Greg sadly exhorting the sinners to sin no more. \"Afternoon,\" he said.\n\n\"What the hell are you up to?\"\n\n\"Just leaning against a porch rail, taking life easy.\"\n\n\"You're messing with the wrong people, Meiklejohn. I told you\u2014\"\n\n\"Is that the sheriff?\" Sylvie yelled from inside the house. She, Al, and Constance trailed out to the porch. \"I want to know what's the good of having a sheriff who don't do no sheriffing,\" Sylvie said. And Al said, \"We think, Mr. Sheriff, you don't look too hard when it don't suit you. Charlie here, me, and Sylvie, we don't think that tramp carried no bees nowhere.\"\n\n\"I don't give a damn what Charlie thinks,\" Greg snapped.\n\nCharlie lifted an eyebrow and glanced at Constance. \"I think we should take a little walk, Greg. Constance, you want to come along?\"\n\nShe nodded.\n\n\"This won't take too long,\" Charlie said to Al and Sylvie. \"Greg? Let's wander down this way a bit.\"\n\nThey walked toward the mill. The sound of workmen had diminished, and the numbers were down now, but there were still several of them coming and going. Today, a Con Ed truck was parked near the mill between a van and a pickup.\n\n\"What the devil are they doing now?\" Greg muttered.\n\n\"Looking it over to see if the waterwheel is in good shape, looking over the dam, the sluice gate, things like that,\" Charlie said. \"This is going to be a working establishment very soon now, it seems.\"\n\nWhen they were past the mill, on the edge of the property overlooking the lake\u2014which appeared bluer than ever\u2014Charlie stopped. Briefly he wondered if the acid content of the recent rain had already affected the lake. How blue could it get? He glanced at the sheriff and said, \"Greg, there's a murderer running around doing murderous things, and I intend to get him. I take it the boys have shown up by now in the woods, and that Wollander sent you over to give me a warning. Right?\"\n\n\"Charlie, do you have a shred of proof? Do you have anything better than the scenario I have? If you do, unload it, and then let's talk.\"\n\nCharlie shook his head. \"The problem as I see it,\" he said in a meditative way, \"is that I'm not real sure who you're working for. I mean, is it the Zukals? Then why didn't you find out who killed their dog, and who's been making life hell for them? Is it the public at large? Then why didn't you go after David Levy's killer? Is it Wollander? If it is, then anything I say will end up on his table, and that would put a crimp in my plans.\"\n\nA vein throbbed in Greg Dolman's temple, and his face became redder. \"If you have information, it's your duty to tell me,\" he grated. \"And if you don't cooperate, I'll go after your license. And you know I can take it if you're playing games with me.\"\n\n\"And you haven't answered my question,\" Charlie murmured. \"You see, what I thought I might do is contact Captain See at the state police detective bureau. I've known him for more than twenty years; we used to work together back in the city now and then. Good man, See.\"\n\n\"You're trying to blackmail me!\" Greg Dolman cried in disbelief.\n\n\"Gentle persuasion,\" Charlie protested. \"I've had a lot of role models around here these past few days. Wollander's washed up, Greg. Did you know that?\"\n\nThe sheriff snorted and jammed his hands down into his pockets.\n\n\"Oh, he probably would deny it himself, but that's the way it is. He's bowing out voluntarily, he thinks, but actually I think his days are numbered by forces even bigger than he is. Writing his memoirs, I hear.\"\n\nA new intentness stilled Greg. He glanced from Charlie to Constance.\n\nShe said, \"You remember when Eisenhower retired, Greg? He gave this impassioned warning about the industrial-military complex. Rather like biting the hand, don't you think? And Admiral Rickover? He did about the same kind of thing when he retired, a dire warning about the dangers of nuclear submarines, nuclear weapons of all sorts. This seems so common, doesn't it? The insider who gets out so often has an overwhelming desire to tell all. Educators do it, preachers do it, politicians. Even actors.\"\n\nShe gazed thoughtfully at the quiet lake. \"Funny, isn't it? When they start to tell about whatever it is they are leaving, they don't seem to stop. They go all the way.\"\n\n\"I don't believe you. He wouldn't bow out, no matter what. The game he's playing is the only game he knows.\"\n\n\"But he's tired of it,\" Charlie said quietly. \"Too many upstarts. The New Age politicians probably turn him off. Managerial types with their clipboards, spin-control specialists, four-second sound bites. It's not his game any longer, and he knows it. And he knows a lot of other things, too. He's been around a long time.\"\n\nVery softly Constance said, \"Think about the little remarks you must have heard in the past few years, the never-confirmed rumors you must have heard. Probably you didn't want to have to think too much about them, but I suspect they're floating around. Aren't they?\"\n\nHe looked at her sourly. \"There are always rumors.\" He turned to Charlie. \"For Christ's sake, what are you after? Give me something.\"\n\n\"A killer. I told Wollander the same thing. I simply want him to leave me alone to do my work. If I ask his wife a question I don't want his answer. Or the housekeeper, or the daughter. And I want to know that if I tell you anything, it stays with you. I don't want him second-guessing my every movement. And I sure as hell don't want you getting his permission every step along the way.\"\n\nGreg Dolman looked meaner than ever, his eyes little more than slits. \"Not enough, Charlie. He says you're after a thermos, guys searching the woods over there, people searching the farm on this side. Hell, it's all over town, that you're after a thermos. Why the thermos? Tell me something I don't know about the thermos.\"\n\n\"It's the key to the mystery,\" Charlie said, and if Greg Dolman heard the undercurrent of mockery in his voice, he gave no sign of it. \"If we find it in the right place, it'll mean David Levy probably took the drugs himself. If it turns up in the wrong place, it'll prove murder. See what I mean? The key.\"\n\nGreg Dolman's indecision was painful to watch. He glared at Charlie with an expression of near-hatred. Abruptly he faced away. \"Beat it, both of you. I have to think.\"\n\n\"But not too long, Greg. It's going to be you or Danny See. I want an official lawman around if the thermos turns up. Wouldn't like to be accused of rigging the sails the wrong way. Know what I mean? We'll be up at the house.\" Charlie took Constance's hand, and they left Sheriff Greg Dolman standing near the swinging bridge staring out at nothing in particular.\n\n\"Charlie,\" Constance said as they walked back, \"did you ever really work with Danny See?\"\n\n\"We need definitions,\" he began.\n\n\"I thought not.\"\n\n\"Dolman doesn't know one way or the other and the very thought of bringing in the state police makes him ache in a place he can't reach with ointment. But Danny did come to town a couple of times, you know. He took some workshops in investigating arson. It just happens that I conducted a couple of them. So there.\"\n\nShe nodded. \"I wonder if Greg fully realizes that he's between the rock and the hard place.\"\n\n\"He does now. Bet he'll be back in half an hour, and then he'll hang around like an attached shadow. Bet?\"\n\n\"You know I won't. Besides, why do you suggest a bet only when you know you'll win?\"\n\n\"Because I hate to lose,\" he said with complete honesty.\n\nConstance looked through picture albums with Sylvie and made all the right comments about the pretty daughters and beautiful grandchildren. Her comments were true, it was a handsome family. And Al and Charlie played the \"Did you know X?\" game. It had turned out that Charlie was on a special investigative force during the period that the Bronx was burning most fiercely, the time that the butcher shop Al's father had started was burned out.\n\n\"Did you know Mort Wurstman? Over on Trinity?\" Al asked.\n\n\"Yep, and the son Dooley.\"\n\n\"A bad one, Charlie. Real bad. Got sent up.\"\n\n\"I know. I was part of the crew that sent him.\"\n\n\"Really? Good job, but it shoulda been done years before. Did you know Sal Maldane?\"\n\n\"The hock shop? Sure. He's okay.\"\n\n\"Yeah, for a hock-shop guy. He got burned out, too.\"\n\n\"Prettiest case of arson you could ever hope to see.\"\n\n\"Yeah? Thought you said he was okay.\"\n\n\"Doesn't mean he can't light a fire, now does it? He is okay.\"\n\nThen the sheriff came back, still glowering, but his mind made up. \"You hanging out here these days?\" he asked Charlie meanly.\n\n\"Until the boys get through and bring the stuff over. And the people at the farm check in. Want some dinner after that?\"\n\n\"Yeah. And talk. Especially talk.\"\n\nTom Hopewell arrived first with Henry Tremont, who looked a bit like a tree himself, brown and tough. No sign of a thermos on the farm grounds, he said truculently, and he'd swear to that.\n\nCharlie thanked him nicely, and asked Tom, \"Will people be around at the farm for a while?\"\n\n\"Are you kidding?\" Hopewell said fiercely. \"Most of the night is more like it. I tell you, we are running a little bit behind, like three weeks behind by my figuring.\"\n\nHenry Tremont seemed to plant his feet even more firmly, and cleared his throat. \"Dr. Bosch,\" he said deliberately, \"he don't like the idea of people coming and going all night, taking bees, getting in the apartments, and such. We're putting on a couple of extras at night for the next week or so.\" He said this as if it implied a warning to Charlie, who nodded gravely.\n\nThen several cars pulled into the driveway; Tom Hopewell and Henry Tremont stalked away, and Pete Mitchum and his friends spilled out of the cars and expanded like foam monsters released from too-small boxes. Pete shook his head in regret, no thermos, and Charlie said that was all right. He said hi to Pete's two brothers and was introduced to the three boys he did not know. He introduced them all around, and it was obvious that they didn't want to leave, but he thanked them nicely, too, and sent them away. They had deposited four plastic garbage bags on the lawn before the front porch.\n\n\"We had witnesses to it all,\" Pete said before he left. \"Sheriff deputies stayed right with us the whole time.\"\n\nCharlie grinned. \"I thought they might.\"\n\nPete grinned even broader. \"I thought that's what you thought,\" he said in triumph. He swung around and returned jauntily to his old Ford that he kept looking spiffy.\n\nCharlie regarded the bags, and shrugged. \"Might as well have a look,\" he said. \"I bummed a tarp from the carpenters.\" He spread the clear plastic tarp on the ground. Al helped position it and weight it down with rocks, and then Charlie emptied the first bag. About what he had expected: mostly beer cans, a few soda-pop cans, a few bottles. A beret. A leather glove that had started to mildew. Sodden newspapers. He used a stick to push stuff to one side as he went through it all. Nothing of interest. Greg Dolman held the garbage bag open and Charlie and Al lifted the tarp and dumped the stuff back in and tied it. The next bag had a syringe with a rusty needle, partly dirt-filled, and a broken Swiss Army knife in addition to miscellany much like that in the first bag. There were many spent shells; hunters made use of that land in season. The third bag was more of the same. Two condoms, and a string of dime-store beads, and a little silver-colored pillbox were among the junk. Charlie pushed the pillbox to one side with the stick, and Constance picked it up carefully by the edges, using a tissue. The fourth bag had nothing to interest Charlie.\n\n\"What's that?\" Greg motioned toward the silver box that Constance had put on the porch rail.\n\nCharlie was through and all the trash was again packaged up. He looked at the pillbox without touching it. \"You have any gear in your car?\" he asked. \"Might as well check for prints. Probably aren't any, but let's give it a go.\"\n\nGreg went to his car and returned with a fingerprint kit. There were only smudges, worthless. Charlie picked up the box then and turned it over thoughtfully. \"Real silver,\" he said. \"It's just beginning to tarnish.\"\n\n\"That's too small to hold dope,\" Greg said.\n\n\"I know.\" The box was only an inch and a half on the side, and about a quarter-inch deep. Charlie opened it, empty, and then slipped it into his pocket. \"It's a pretty little thing.\" He stretched. \"Let's leave these people in peace for now,\" he said. \"Al, Sylvie, thanks for the use of your place. See you later.\"\n\nGreg Dolman followed Charlie and Constance to the Volvo. \"Now where are you going? Is that it for today?\"\n\n\"Dinner,\" Charlie said firmly. \"Remember? We take you to dinner? I suggest you follow in the police car, or do you want to ride with us?\"\n\n\"I'll follow,\" he said. \"Where?\"\n\n\"I spotted a roadhouse about ten miles out of Spender's Ferry. Don't know a thing about it, but that's where we're headed.\"\n\n\"Now,\" Greg Dolman said in the roadhouse. They were seated in a booth with drinks before them.\n\n\"Now,\" Charlie agreed, studying Greg thoughtfully. He knew Constance didn't trust the sheriff any more than he did, but she had said sensibly, \"What choice does he have but to go along with it?\" Charlie wished he knew.\n\n\"First,\" he said finally, \"I'll want you to stick closer than a tick the rest of the night. Deal?\"\n\nGreg flushed a deep red, but nodded.\n\n\"I don't want any alarms going off, or any of those private security people nosing around. You, as a representative of the law, can handle them if they do turn up. Okay?\" Again the sullen nod. \"Okay. I've been all over the place today baiting a trap, and I'm pretty sure that it will be sprung tonight. Late tonight. I'll want you, or someone, to be there.\"\n\n\"Tell me what you're up to, Charlie. No games; no puzzles, just what you're up to.\" Greg took a swallow of his bourbon, straight up, and added, \"You know I cut loose today, damn you. Wollander suggested I talk to you and let him know what you were doing. Well, I haven't gone back since. So, I guess you could say I'm committed.\" He sounded very depressed.\n\n\"Right. All day I've been broadcasting my belief that David Levy was drugged by the contents of his thermos. And the thermos is missing. That you already know. Everyone in the damn county must know it by now. I made it plain that if the thermos doesn't turn up in the lake, or on the edge of it, or on his route back to his apartment, it will prove beyond a doubt that he was murdered, and the investigation will go forward. But if it does turn up, I'll have to accept that he probably did it to himself. I think everyone concerned with this mess wants it over with, wants me to accept that David killed himself accidentally.\"\n\nGreg scowled. \"Goddamn it, Charlie, what if the damn thermos is in the lake? Then what?\"\n\n\"It isn't. Not yet anyway. I think someone will make an attempt tonight to put it there.\"\n\n\"Oh shit,\" Greg said in a low voice, disgusted and discouraged. \"You're fishing. You don't have anything, do you? You've been around, Charlie. You know there's people you don't play mind games with.\"\n\nHe was not even questioning his own acceptance of the game rules, Charlie knew, and Charlie understood without a doubt the process Greg Dolman was going through; looking inward, searching for his own way out of this predicament, probably examining the words he would recite to Wollander. Brusquely Charlie said, \"When I walked away from NYPD, I walked away from those people who like to tell you what to do and when. I'm calling the plays tonight, Greg, and, by God, you're in! Shut your face and stop the bellyaching!\"\n\nHe couldn't stop Greg, if the sheriff decided to take a walk, Charlie knew, not short of pulling a gun anyway; and he didn't have a gun along. The moment hung there while Greg made up his mind. Finally he lifted his drink and finished it.\n\n\"He didn't take the thermos to the lake for a morning swim,\" Charlie said, exactly as if neither of them knew what had gone on during that brief interlude. \"He had a routine. Up early, a swimming workout, back to the apartment for breakfast, and then to work. You don't take a thermos for a workout when you're a minute away from home. No, the killer took it. Maybe to plant it on someone else, maybe to lace lemonade with dope to make it look like that's how he took the stuff, maybe for some other reason altogether. But the accidental death was accepted so easily there wasn't any real need to do anything with it. It's been stashed away somewhere all this time. If it was tossed, then our killer probably has a new one in hand. I'm betting on it.\"\n\n\"You're planning to stake out a whole damn lake in the middle of the night?\" Greg shook his head.\n\n\"Part of it, Greg. Just part of it.\" Charlie pulled out a notebook and began to sketch rapidly. \"Here's the lake.\" It looked like a teardrop. \"Over here, the farm, with extra guards patrolling, and lights on. Not a good way to approach. Next, the Zukal place. Security lights everywhere. Even worse. On around the state forest land, and to the Wollander property. Again, guards everywhere. On past, and there's the orchard. Dirt roads back among the trees. Short walk to the lake from there. So, just this end, near the Wollander beach. It's the best place, after all. Toss the thermos as far as possible. It lodges among the rocks on the bottom and looks like it rolled in when David swam that morning.\" He took a gadget out of his pocket and placed it on the table. It looked very much like an infrared television control. He pushed it across the table to Greg. \"I had the Frederick's Security Company put in a light, a portable unit. This controls it. It'll light up most of the lake. We'll catch our killer in the act of getting rid of the thermos, and then the real work can begin, the nitty-gritty detail work that cinches things. First you have to have a direction, then you can run with it. You know that's how it goes. I know it, too.\"\n\n\"You've got someone in mind?\" Greg asked, a new intentness making his face taut, his eyes narrowed.\n\n\"Oh yes. And a lot of little things to corroborate it, but we need something big now.\"\n\nTheir dinners were brought then and they were silent until the waiter had left again. They all had ordered steak because they didn't trust anything else in the menu. The steaks turned out to be very good.\n\n\"Start at the beginning,\" Greg said. \"Give.\"\n\n\"You won't like it,\" Charlie said; he stopped chewing to speak. \"You see, it starts with the dog Sadie.\" Greg made a low groaning sound and Charlie said, \"Sorry about that*, but that's the way it goes.\" He looked thoughtful then and said in a wondering sort of way, \"I just remembered Captain Bertlesman. For God's sake, good old Cap Bertlesman. He used to come in to deliver lectures when I was a green-as-grass rookie, back when the dinosaurs roamed. He used to say, 'Get the who and the how, and let the shrinks take care of the why.'\" Charlie looked apologetically at Constance who had made a noise that was not quite a snort. \"Didn't say he was a nice man,\" Charlie murmured. \"Anyway, that's what I have for you, Greg. But only the how for the time being. If there aren't any slip-ups, you'll get the who in a couple of hours. Why will take care of itself.\"\n\nFor a moment Greg Dolman simply looked puzzled, but then a dull flush spread across his face. \"You son of a bitch!\"\n\nCharlie nodded. If there were any slip-ups, he knew he needed that card in the hole; he might have to persuade Greg to tag along another time. Greg looked as if he wanted to reach across the table and choke him, and that was all right, as long as he didn't move in that direction. Charlie waited.\n\n\"Go on,\" Greg finally muttered.\n\nCharlie described how he and Constance had demonstrated how to get a trained dog to take food from a stranger. \"So, no problem,\" he said. \"The killer dosed dog food with insect spray and killed Sadie. But then David came across the head scarf. Maybe the killer saw him with it, or they even talked about it. Anyway, David had to be next.\"\n\nGreg was attacking his steak ferociously, as if this might be his last meal. Indigestion in the making, Charlie thought sadly.\n\n\"So, everyone knew about David and his vitamins, and his morning swim.\" When Greg made a growly sound, Charlie explained about the cookout and the meticulous report David had written to his father. \"So, as I said, everyone knew. Everyone involved in this mess, in any event. No one was questioning the death of the dog, and now if David's death could be made to look accidental, so much the better. And the vitamins showed up. Heaven sent,\" he added drily.\n\nGreg shook his head. \"How you going to prove that, Charlie? No way.\"\n\n\"Wrong, Greg. Two facts. One, the company David ordered his vitamins from shipped them on the fourteenth, from Rochester. They would have been delivered by the sixteenth, but they aren't listed among his effects. Two, examine the autopsy report, the contents of the stomach. You see, a few of those vitamins were time-release doses, the vitamin C, for example. No vitamin C is listed, and it would have been there in granular form for many hours, only gradually dissolving, being absorbed. Our killer entered the apartment, swiped the new shipment of vitamins, doped some of the capsules, and substituted them for the ones on the counter. Then, when David was dying, or even after he was dead, the killer returned and took away the doped containers, replaced the untouched ones, and it was done. There was one too many of each one that was in capsule form, but who would notice? Who did notice?\"\n\n\"And the thermos?\"\n\nCharlie shrugged. \"An ace in the hole, probably, but one that wasn't needed, not until right now, anyway.\" He looked with regret at his plate; his steak was nearly gone and he had no real recollection of eating it.\n\n\"So,\" he said. \"Two down. No hubbub. No real investigation. On to the next item, moving the bees. And the tramp. You know, Greg, if you had found a bottle, that would have been much harder; much, much harder. But either the guy was so drunk that he turned on the gas and forgot about it and then fell down and bashed his head in, or he was sober enough to walk through the woods carrying a hive of bees. But not both. Let's say he was sleeping off a drunk under a tree and he hears someone messing around with the hive, and he follows out of curiosity, or even calls out. Our killer has to act fast now. A witness is the last thing on earth needed at this moment. Probably he said something like, 'Let's duck inside one of the apartments and talk. The watchman will spot us out here.' Something like that. And he picks up a rock. Inside the apartment, a quick bash on the head, turn on the gas, close up the place, and get back to the work at hand. There must have been a wallet, and he lifts that, just to muddy the waters. You could see the outline in the hip pocket,\" he added. A distant, thoughtful look crossed his face. \"It might even turn up,\" he said. \"And the rock, of course, gets tossed into the lake, one among five million others.\"\n\n\"And not an iota of proof,\" Greg Dolman muttered. \"Not a single print anywhere.\"\n\n\"But that's proof,\" Charlie said. \"Why would the tramp wipe his prints from the gas stove? Now that wouldn't have made any sense at all.\"\n\nThe waiter cleared the table and they sat in silence until he returned with coffee, and then left again.\n\n\"I don't like it,\" Greg said. \"We need more people. Three's not enough. Too much territory to cover.\"\n\n\"Too many people and you start tripping over each other.\"\n\n\"How you plan to go in? Your car's a dead giveaway and so's mine.\"\n\n\"We'll park in the Zukals' driveway, about a third of the way to the house, where it curves, out of sight of the road and the house.\"\n\n\"And those damn lights they installed blast you. You said yourself, that's out.\"\n\n\"Urn. The lights won't come on after ten tonight. I fixed that with the security guy today. He's the only one who knows,\" he added. \"Not even the Zukals are in on it.\"\n\nGreg thought some more, and they ordered more coffee. \"Have you considered that the damn thermos might have been ditched someplace not accessible now?\" Greg asked finally.\n\nCharlie looked at him in surprise. Constance had asked the same question. \"Sure. But I think a thermos will end up in the lake tonight, if it gets that far. One's pretty much like another, and who would pipe up and say it's not the same one? And I'd put up money that it will have measurable amounts of methaqualone and a barbiturate,\" he added grimly.\n\nGreg was not happy about any of it, Charlie knew, and he also knew there wasn't a thing he could do about that. It was chancy; the killer might not have the thermos, or any thermos. He might be throwing a party that no one would show up for. Maybe the damn thermos was already in the lake. Maybe David had lost it himself days before his death. He understood that Greg was going along with it for now, this one night, and if it didn't work out, he'd be back in Wollander's pocket tomorrow. Case closed, again.\n\nWhat if he was wrong? He didn't believe it, but it was a possibility he began to consider. What if they had read all the signs wrong? What if... ? He felt Constance's hand on his thigh under the table and looked at her. She knew, he realized, and she was saying, it's okay. He felt some of the tension seep from him, and he covered her hand with his.\n\nConstance now said to Greg, \"We both have dark raincoats in the car. Do you have something to put on? That shirt will be like a beacon.\"\n\nIt was going on nine-thirty. Right or wrong, Charlie thought, it was time.\nCHAPTER 18\n\nCHARLIE HAD POSITIONED both Greg and Constance, and he no longer could see either of them. Greg had complained. \"Hey, it's been a long time since I pulled an all-nighter.\"\n\n\"So go to sleep. You'll wake up when the lights come on.\"\n\n\"If.\"\n\n\"When. But, Greg, if you start snoring, I'll come over and kick you in the butt.\"\n\nConstance was invisible, merging with a clump of bushes until not even a nighthawk would have been alerted to her presence. Charlie admired, and even envied, her ability to go into a meditative state\u2014alpha state, she called it\u2014effortlessly. She would be wide awake and alert and totally relaxed for as long as it took. That had been part of her aikido training and she had mastered the skill, but he had not been willing to learn it himself and he could not say why. She had explained it, rather too airily, he thought. \"You aren't willing to give up any control, even to yourself.\" Maybe that was it, maybe not, and now, he was wishing he did not have a cramp in one leg, wishing that time would pass faster, wishing this was over.\n\nGreg was against the bluff at the edge of the beach, his back against the rise, no doubt. And no doubt he was asleep. Now and then Charlie had heard his rustlings, but not for some time. Charlie was farther down where the bluff was not so high and he could see over the edge if he tried. Some large round rocks were on the top here, and he hoped his head appeared to be just another such rock when he stood up to look.\n\nAt first, he was thinking now, you didn't hear anything. Or only those noises that were expected, insects, cicadas, crickets, distant traffic sounds, a whippoorwill and a faraway echo of an answer. Then other sounds became noticeable\u2014a bird flapping its wings, a bat's high screech, a small animal moving in the grass. No fish jumping, he thought with regret; no frogs yelling here I am. Two does and a buck had come down to drink and had stood outlined against the sky briefly, but they were not fooled by human immobility; they had left fast, the sounds of their feet loud, and with a snorting sound almost horselike.\n\nNo new stars had appeared in a long time. At first, he had been able to pick out a section of sky and study it, count the stars in it, close his eyes for a count of ten, and when he looked again the number of stars in the sector had doubled, tripled, quadrupled. But the slow swing of the heavens was impossible to see without a reference point; he tried to align certain stars with a tree branch, only to lose them when he closed his eyes again. Too many of them now.\n\nHe cupped his hand over his watch face; it was twelve-forty. He had not expected any action until after twelve, after one even, but time was being contrary; it had slowed to an unbelievable crawl. He stood up to scan the area again, and now that his eyes had adapted so well to the night, he could make out individual trees, individual bushes. He hoped Constance was stretching from time to time, and he knew that Greg had done that earlier, but not for too long a time. He inched away from his position and eased himself to where Greg was propped up, sound asleep.\n\nHe shook him lightly, and put his fingers over Greg's lips. \"Get up and move back and forth a couple of minutes,\" he said softly. \"I'll keep an eye out.\"\n\nGreg made a mumbling noise and pulled himself upright jerkily. He had become stiff already. He bent and stretched a few times and then walked down the beach and back, then again.\n\n\"Okay,\" he whispered. \"Thanks.\"\n\n\"Yeah. It's nearly one. Thought you'd want to know.\"\n\nCharlie carefully made his way back to his position and studied the landscape again. Nothing. He wanted to go check on Constance but he couldn't do that. He would be silhouetted if anyone happened to be watching. But, he told himself, he had put her where he did because he knew he could trust her not to fall asleep and not to stiffen up, and he had put Greg where he did because he knew Greg would do both.\n\nThe doubts that had troubled him earlier kept swimming back into his mind. What if nothing happened? What if he was overestimating the lure of the bait he had spread around altogether too thick all day? He personally would suspect a trap, he told himself, and then told himself to shut up. If it didn't work, well, he'd think of something else. But the other part of his mind, the part that resisted the idea of achieving a meditative state, and that now was arguing with him over the futility of this night watch, that part would not shut up. Instead, it was saying, if it doesn't work there probably won't be another day for you, asshole. He would lose Greg's cooperation, and did he really think he had enough evidence to stir up any interest with the state cops? Dummy.\n\nHe shifted his weight. Now he was keeping a close watch over the top of the rise, but even this, that other voice was saying derisively, how long could he keep this up? He was damned uncomfortable. His was the worst possible position he could have claimed; nowhere to rest his weight, no way to lean against a tree for relief, just stand upright and watch, or sink back down on this side and be blind. He was cursing himself silently when a figure appeared.\n\nHis hand closed over the infrared control in his pocket and drew it out. The person had vanished again, blended into the shadows of trees and shrubs. He didn't move, barely breathed, and presently he heard a sound that was new, then another. Footsteps on grass, hardly audible but there, coming closer, apparently hurrying.\n\nCharlie caught an eclipse of a cluster of stars as the figure started down the slope to the lake, and now he counted silently, one, two, three, four. He pressed the button on the infrared control, and was blinded momentarily by the brilliance even though he had been expecting it.\n\nHe already had started to run the dozen steps to the woman in the black raincoat, when she lifted her face and screamed. Constance had materialized at her side, one hand on the thermos held in the air ready to be thrown, the other on the woman's wrist.\n\n\"Hello, Jill,\" Charlie said and took her other arm. Greg was right behind him.\n\nCharlie had to admit that Greg behaved admirably during those first few seconds. He blinked a lot, but that could have been from the blinding lights. He looked at Charlie with large questions in his eyes, and when Charlie nodded firmly, he managed a nod also.\n\nTwo security men were racing toward the beach, one with a gun drawn, both with flashlights that were no longer needed. The portable spotlight array had turned the entire end of the beach into noontime glare.\n\n\"Mrs. Ferris,\" Greg said soberly, \"I'll have to ask you a few questions. Let's go inside.\" He swallowed hard, and then recited her rights in a swift monotone.\n\nCharlie gave Constance a quick look; she nodded, very somber, tired. She carried the thermos; it was filled, the top on loosely, ready to slide off any second. Charlie kept a grasp of Jill's arm; the sheriff waved the private security men out of the way, and they all walked toward the Wollander house where lights were coming on upstairs and down.\n\nBy the time they reached the patio Warren was hurrying toward them, tying his robe belt. \"What the devil is going on?\"\n\n\"I couldn't sleep,\" Jill said shrilly. \"I was taking a walk by the lake and found the thermos in the water and then they all jumped me.\"\n\n\"Take your hands off her!\"\n\n\"Inside,\" Charlie said. \"Come on, inside.\"\n\nWollander grabbed at the thermos, and Constance easily fended off his hand and kept moving with Charlie and Jill. Warren looked past them and yelled at his security men, \"Get these people out of here! Right now!\"\n\n\"Mr. Wollander,\" Charlie said patiently, \"it's over. Now, let's just go inside and talk. The sheriff will have to use your phone to call his deputies, and while we're waiting, we can talk things over.\"\n\n\"You heard what my daughter said. She found that damn thermos.\"\n\n\"Filled with tap water. I heard. Greg, do you want to just go straight in with her? Constance can bring a car around if that's the only way to handle this.\"\n\nAt the door to the house Lois said, \"Come in.\" She held the door open as they all passed her, and then pulled it closed again. She was in a long gray robe and slippers, her hair disheveled.\n\nWarren glared at Greg. \"Do you know what you're doing?\"\n\n\"Yes, sir. I'm afraid so.\"\n\nLois led the way to the study and when they were inside closed that door also. \"What happened?\" she asked then.\n\nCharlie released Jill's arm. \"You'd better sit down,\" he said, not too unkindly. She sank down into one of the leather chairs and drew up her feet into it. She looked childish, like a sick little girl. He felt in his pocket for the silver pill box and handed it to Lois. \"Do you recognize this?\"\n\nShe turned it over and over, then looked at him sharply. \"Where did you find it? That's mine. Or just like mine.\"\n\n\"I think it's yours.\"\n\nWarren walked stiffly to his desk and sat down. He picked up the phone and began to punch numbers. \"I'm calling my attorney.\"\n\nCharlie shrugged and spoke to Jill. \"When they arrest you,\" he said, \"the first thing they do is search you thoroughly, and there will be a medical examination. By a gynecologist.\" He heard the phone clatter back to the receiver but did not turn to look at Warren. \"I'm not a doctor,\" he went on, \"but I have it on good authority that there are certain physical symptoms that are present in the event of pregnancy; even if a pregnancy has recently terminated in any way, the signs are still there. Will they find the right symptoms, Jill? The right hormones, the right physiological changes?\"\n\nShe was staring at him speechlessly, the color gone from her face, her lips.\n\n\"When they send the thermos to the lab, will they find tap water? And maybe methaqualone and phenobarbital? You know the sheriff will have to search your rooms here. You must have had access to those drugs recently if you put them in the thermos. Are they still up in your room? Will they find them?\"\n\n\"There's medicine,\" she said, keeping her gaze on him as if hypnotized. \"My medicine. Stanley's. Maybe what you're talking about\u2014\"\n\n\"Jill, for God's sake! Just shut up! You don't have to answer his questions. Just shut up until an attorney is here to advise you.\" Warren's hand was on the phone but he did not lift it again.\n\nShe turned her wide, staring gaze to him, and began to shake her head. \"Just shut up. Be still. Be good. Do what Daddy tells you. If I'm not good, you'll send me to Mother, and if she gets tired of me, she'll send me back to you. Back and forth. Back and forth. I saw the letter she wrote to you. She showed it to me, laughing and laughing. We both laughed. See another doctor. Get more opinions, and more, and more. Remember? It can't be true. I don't believe it. See this man, he's good. Remember? She dragged me to one doctor after another in their prissy white coats, with their prissy fat fingers, and they kept telling her the same thing, didn't they? All of them. Over and over. Then a new letter from you would come: Try this one, that one, this treatment, that treatment. And she wrote to you again and again, and still, try this, try that.\"\n\nWhen she became quiet, Charlie said in a low voice, \"You're barren, aren't you?\"\n\n\"Yes,\" she whispered.\n\n\"And he's known that for many years, hasn't he?\"\n\nShe nodded. \"Yes.\"\n\n\"Did you take your stepmother's birth-control pills and use them?\"\n\nShe nodded again.\n\n\"To make it appear that you had had a miscarriage?\"\n\n\"Yes! Why are you asking me all this? You already know the answers! Yes! Yes! Yes! To everything.\"\n\n\"And the only victim you ever really had in mind was your husband, Stanley.\"\n\n\"Yes. Of course. Yes!\"\n\nBehind him Charlie heard Warren Wollander make a choking, sobbing sound. He did not look at him. He was gazing instead at Lois, who was pasty, and looked as if she might faint. The fury in his low intense voice was startling when he said to Lois, \"He's known from the start about the fake miscarriage, about the fake pregnancy, about the killings. He was going to throw you to the wolves if necessary to save her skin.\"\n\nBehind him he could hear Warren Wollander using the phone. His voice was toneless. \"Get over here,\" Warren said. \"We have desperate trouble. Come now.\"\n\nGreg called his deputies then, and asked Warren if he would need a search warrant. Warren said yes, but Jill said no. Greg spoke into the phone again, ordered a search warrant drawn up, and began to say what it should include when Charlie interrupted him again. \"Tell the judge you're also looking for a solid gold bracelet carved with flowers.\" Greg looked mystified, but included the bracelet.\n\n\"We'll have to wait,\" he said uncomfortably when he was done.\n\n\"I'll make coffee,\" Lois said. She left without a glance at Warren.\n\nCharlie looked at Constance, almost expecting her to go after Lois, but she moved her head fractionally and did not get up from her chair.\n\nIn a very low voice Warren said, \"Greg, you don't have to do this. We go back a long ways.\"\n\nGreg Dolman looked more uncomfortable but Charlie noticed that Constance tightened her grasp on the thermos. Son of a bitch, he thought, she still didn't trust Greg Dolman worth a damn.\n\n\"Jill,\" Charlie said softly, \"one question. Why did you take the thermos?\"\n\n\"You mean there's something you don't already know?\" she asked bitterly. \"When I went to the apartment to switch the vitamins, I found that I had left the cod-liver oil here, and I couldn't replace it. I panicked. I was afraid someone might connect it to the dope, and I thought that I could mix the drugs in the thermos if anyone questioned anything.\"\n\nThe deputies finally arrived and they all watched as the thermos was decanted into a sterile jar that was sealed and labeled, and then the thermos was put into a bag and sealed and labeled. The attorney Warren had called turned up, looking mean and sleepy. Charlie and Constance began to edge toward the door. Greg followed them.\n\n\"I'll want more,\" he said tiredly. \"Tomorrow, sometime. What's that about a bracelet?\"\n\n\"Tomorrow,\" Charlie said, just as tired as Greg. \"I told Sylvie and Al we'd drop in around one. And before that, our phone will be off the hook and the doors sealed tight.\"\n\nGreg didn't argue. He nodded, called one of the deputies to drive them over to collect their car, and turned back to the routine that awaited him inside.\n\nCharlie looked past him for just a moment; the lawyer was in a chair drawn up close to Jill's. His face was inches from hers and she was still huddled with her feet under her, still looked like a sick little girl. He took Constance's arm and they left.\n\nWhen they were once again in their own car, heading for their own house, Charlie said with a groan, \"I am wiped out from here to Thursday and I ache from top to bottom.\"\n\n\"When we get home, soak first and then I'll give you a Swedish massage.\"\n\nHe squeezed her thigh. \"You're good.\"\n\n\"And after I do you, you can give me one,\" she added. He squeezed her thigh again and she covered his hand with hers.\nCHAPTER 19\n\nCHARLIE WOKE UP GRINNING, and was still grinning broadly when he entered the kitchen to find Constance making waffles. Fresh strawberries were marinating on the counter. She eyed him suspiciously.\n\n\"So you ate the canary, after all?\"\n\nHe laughed out loud. \"Dreaming. I was dreaming that I was married to Sylvie and I kept trying to dump her in the lake so I could run away with you. See, even in my dreams, I'm chasing you like a fourteen-year-old with two bucks running after the only whore in town.\"\n\nShe raised her eyebrows and looked like a schoolteacher with a recalcitrant eight-year-old. \"You're just after my body. I always suspected that.\"\n\n\"Damn right,\" he said, nuzzling her neck. \"And I would dump Sylvie in the lake, or run Sophia off a cliff, or ditch Elizabeth in a second to get at you.\"\n\nShe reached behind her and gave him an indecent tweak that was just short of vicious. He howled and danced away, tripped over Brutus and Candy, and ended up at the table, laughing and cursing.\n\n\"Are you ready for breakfast, by any chance?\" she asked demurely.\n\n\"If they only knew,\" he said as soon as he could speak. \"Oh, if only they knew, all those people who are ready to canonize you. Saint Constance. Hah!\"\n\nShe brought him a steaming waffle covered with strawberries, and a pitcher of cream on the side. \"You realize that this is your ration of high-caloric food for the day,\" she said severely. \"Steamed fish for dinner. Fruit and carrot sticks for lunch.\"\n\nHe caught her wrist and kissed the palm of her hand. She leaned over and kissed the top of his head.\n\nAl Zukal met them in his driveway. He had a look of total disbelief on his face. \"Charlie! Lois said Jill's the one! That skinny girl? What for?\"\n\n\"Is Lois here?\"\n\n\"Yeah. Syivie said you said she should call Lois and get her over here. She's been crying real bad, Charlie.\"\n\nThey turned to see the sheriff's car coming in the driveway.\n\n\"I didn't know we was going to have a party,\" Al muttered.\n\n\"Sorry,\" Charlie said. \"I thought this would be the best way to wrap things up. You and Syivie will want to hear where you stand now. And Lois needs to know a few things. Okay?\"\n\n\"And if it ain't? Yeah, yeah. Come on in.\"\n\nThey sat at the kitchen table where Syivie had coffee ready, and two kuchens, one apple, one cinnamon. Constance shook her head firmly at Charlie. Greg Dolman, he noticed, aggrieved, had a very large piece of both coffee cakes.\n\nLois was in jeans and a T-shirt and sneakers. Her working clothes. Her eyes were red-rimmed and swollen. She did not speak when they came in, but ducked her head and moved her cup around and around in the saucer.\n\n\"This won't take very long,\" Charlie said briskly. \"Constance and I came to the same conclusion at the same moment, I think. We had been misdirected royally, and as soon as we admitted that, the rest began falling into place. Actually, Constance is the one who said it must be Jill.\"\n\n\"Everything pointed to her,\" Constance said, \"but we were all so busy looking the other way we couldn't see her as a suspect.\"\n\nLois looked at her in bewilderment. \"Nothing seemed to point to her. I thought she was pregnant and I lived in the same house with her.\"\n\n\"I know. But when I went to her room there wasn't a single book about pregnancy, no books on babies, no books on how to achieve pregnancy. None of the things you'd expect an anxious mother-to-be to have around. The drugstore in town has a nice selection, so it wasn't that they aren't available locally. She simply wasn't interested. And I kept wondering why she was here. It's obvious that she is not fond of her father, and she has no friends here, so why was she here?\"\n\n\"They were redecorating their condo in the city,\" Lois said. \"The place was completely torn up.\"\n\nConstance shook her head. \"She could have gone on a trip, or to an apartment, or to a hotel, or to visit friends, a number of things. But she came here, and stayed. Anyway, once we decided we had made a fundamental mistake, we backtracked, and it kept coming around to Jill when we asked: What if Stanley had been the intended victim from the start? After that it was obvious.\" She lifted her coffee then and Charlie knew the rest would be up to him.\n\n\"We backtracked all the way back to the day you and Sylvie came here,\" Charlie said, nodding to Al. \"And that made us realize that Jill already had a plan in mind by then. I'll just take it step by step. She has no money of her own and is jealous of Lois because she is independent. She saw her mother as a kept woman all her life, and resented it, and she saw herself repeating the pattern with her husband. Out here he could be with her only on the weekends and it gave her time to think and make her plans. It's really very hard to arrange an accidental death in the city, without drawing suspicion, you know. But out here? She learned about the bees, no doubt, when they arrived.\" Lois looked up, startled, and nodded slightly. Charlie went on. \"Stanley was allergic to bees. It could be made to look accidental if he got multiple stings, and no suspicion would be attached to her, especially if she was a grieving widow who also had lost her child. But how to get him to where the bees would attack? That must have presented itself as a massive problem, until she found Sebastian. I think she brought Sebastian out only to verify that she had gone into the mill, into some of the little rooms with him. She had no hope of buying any property, or anything else. Stanley gave her gifts, and paid her charge accounts, but you can't buy a piece of property like this on your Visa card. So, why take Sebastian into the mill, unless she had a different purpose altogether? She claimed to have lost a heavy gold bracelet, remember?\" He glanced at Greg who nodded.\n\n\"We found it in her room.\"\n\n\"I thought you might. Anyway, she said it was lost, and Stanley filed an insurance claim, and got her a new one. But if she could get the bees into one of those little rooms, and close the door on them, and then tell Stanley that she remembered that she had worn it the day she showed the mill to Sebastian, she could get him to open the door to go look for it. The rooms would have heated up just fine by midmorning; the bees would have been swarming around in a rage, just as they were later. It would have worked. The mill used to have a reputation as a hangout; kids could have moved the bees as a prank, and poor Stanley. She would have turned her big eyes on you and said she had no idea why he went in there, just as she did later about the station wagon.\"\n\nHe turned to Sylvie, \"But you showed up. And then a dog arrived. And carpenters began taking the mill apart. Everything she had planned was going to hell all at once.\"\n\n\"She killed Sadie?\" Sylvie cried. \"But how? Sadie didn't even like her. And why?\"\n\nCharlie detailed how Sadie could have been poisoned. Then he asked, \"Sylvie, when David found one of your scarves and returned it, was there anything funny about it?\"\n\n\"Filthy. It was filthy and I threw it away. I didn't even want to wash it.\"\n\n\"Mud? Grass stains? Filthy how?\"\n\n\"All that and greasy.\" Her eyes widened.\n\n\"Greasy,\" Charlie said. \"David found it and later on he must have made the connection. How he connected it to Jill I don't know, but he did, and she had to get rid of him, too.\" He shrugged. \"As for why she killed Sadie, that's pretty obvious now. She still needed to be able to cross the bridge to collect the bees, and she needed a place to put the bees so she could send Stanley to them. Not on the Wollander property, too suspicious. If not the mill, then this house, or the station wagon, but not if a watchdog was barking its head off at her. It must have come to her, there was the perfect smokescreen; she could arrange things so that it looked like an attempt to get the newcomers out of the neighborhood; kids could take the blame.\"\n\nHis face tightened. \"But David saw something and found the scarf. Maybe he saw her toss it. Maybe he even asked her about it, accused her of killing the dog.\"\n\n\"I don't understand what the fuss was over the thermos,\" Lois said. \"What were you doing?\"\n\n\"Well, it was missing. So the killer must have taken it. I was sure she had tampered with the vitamins, but why take the thermos? And I thought, if she had taken it, she would see this as a chance to sew things up good and tight. Let someone find it in the lake and be done with it.\n\n\"So now everything is in place for Stanley's death. She has announced her pregnancy, everyone is jubilant, even though Warren Wollander actually knew she was barren. He kidded himself at first about believing her, I imagine. She had taken the birth-control pills in order to plan exactly when she would start bleeding, and, to a certain extent, how much.\"\n\nAl looked uncomfortable and shifted in his chair, and Sylvie's mouth pursed; Charlie was talking about things men didn't talk about, their attitudes clearly said.\n\nHe went on. \"That night Warren saw her walk on the driveway to the place where it comes very close to the state forest. Why the midnight walk? Why the raincoat? Not to meet Sebastian. The raincoat was to hide under. A shadow moving out among shadows. Lights were still on, people still awake and she didn't want to be seen. And the reason could have been to throw something away. Not in the house trash because she already had said she had lost a bracelet and people might be sifting through trash more carefully. She had enough pills for her purposes, the rest could be flushed down the John, but there was the silver pillbox, and she had to get rid of it. She could have returned it to your bathroom, but if you knew it was missing before, that would raise questions, and she didn't want any questions about birth control or pregnancy to come up.\"\n\n\"You were looking for that?\" Lois exclaimed.\n\n\"Not really. For something. I didn't know what she had tossed, or even if she had tossed anything, and it could have been the thermos, although I didn't think so. But that's what they found.\n\n\"So everything was ready, at last. Stanley came up for the weekend. He had a medicine cabinet full of medications, including some heavy-duty sleeping pills. If he didn't take one himself, she could see that he got it. And in the night she went over to move the bees to the station wagon. Then a witness showed up, a transient. He was drunk, but he saw what she was doing, and must have started to make a ruckus. She got him to enter the apartment and hit him on the head, turned on the gas, and left to complete her real mission of murdering her husband.\"\n\nAl made a grunting sound and Sylvie shifted in her chair, but no one spoke. Lois didn't move at all.\n\n\"Saturday morning Stanley has an appointment with Al, here, and she has her regular class, service, whatever it is, with Sebastian. What's more natural than that she would drive? Stanley doesn't like wandering in the woods; allergies, you know. She pulls up in the driveway to turn around, where she can see in the wagon, make sure the bees are still there. She says, 'Oh, I think I lost my bracelet in the station wagon. Seeing it has just reminded me. Be a darling and get it for me, will you? It's in the back. Do it before you start talking to Al, or you'll forget.'\"\n\nLois made a deep throaty sound and closed her eyes.\n\n\"Right. So now it's done. I suspect that she stopped in the driveway, out of sight, long enough to hear the yelling, and then went on to Sebastian's meeting. Daddy shows up and rushes her home, she gags herself in the bathroom, takes out the tampon, and becomes an ashen-faced widow who has just lost a baby. But she won't let a doctor near her. There's only one more thing she has to do and that's get Sebastian over and tell him she was pregnant. She said he knew, that he was already advising her, and she has to make sure that he actually knows, just in case someone mentions it. She makes a big scene about seeing her spiritual advisor, and finally she can relax, wait for her inheritance to work through the legalities, and take off for France.\"\n\nThere was a long silence, broken finally by Lois. \"And last night, it was all planned? All just a trap for her?\" she whispered.\n\n\"Sure. There was no proof yet. I was pretty sure that if a scene started with her father present, taking over, giving orders, she would confess.\" Get out of here! She had killed three men to escape those orders. Charlie had not simply been pretty sure; he had known beyond doubt that when she saw that trap closing yet again, she would escape in the only way remaining. He said no more now, but watched Lois, and wondered if she saw escape as a possibility.\n\nLois sat unmoving for several seconds, then she shook her head hard. \"He must have known, but he was so desperate to believe,\" she said, nearly inaudibly. \"And he was as happy as a child. She lied about her pregnancy exactly the same way that her mother lied, to hurt him, to make him know he had to choose.\" She closed her eyes very tightly for a moment. \"No man should be forced into that lifeboat with his wife and child,\" she whispered. Abruptly she turned to Greg Dolman. \"Sheriff, I'd like a word with you alone.\"\n\nConstance stood up. \"We should be on our way, too. Dr. Wharton, is Jill under a doctor's care now?\"\n\n\"Yes. They took her to jail and did the usual, I suppose, but then the lawyer got a psychiatrist in to talk to her. He will attend her. Why?\"\n\n\"I just thought that if that is the case, we may never hear any more details, when she did this or that, or how, or who that man was in the apartment.\" Her gaze was very steady as she regarded Lois. \"Between a doctor and a lawyer, she may never have a chance to make another statement. This is what she feared and dreaded most, isn't it? Never to be free, to be independent. She really is to be pitied, and so is her father. And you, of course. But you have your work. You may be the most fortunate of the three.\"\n\nLois moistened her lips and slowly she nodded.\n\n\"We can step outside, if that's all right with you,\" Greg Dolman said then, rising, pushing back his chair.\n\n\"It's nothing,\" Lois said, her voice low, but decisive. \"I should be getting back.\" She held out her hand to Charlie and then to Constance. \"Thank you both.\" She walked away, back toward the mill, toward the swinging bridge.\n\nGreg Dolman shrugged. \"We'll be tying up loose ends for the next few months, more than likely. We found the bracelet and the drugs, and the water in the thermos is tap water with something in it. They're analyzing it now. We'll have the doctor's report on Mrs. Ferris in a few days. I'll be in touch.\" He also held out his hand. Lois's hand had been like ice, his was sweaty.\n\n\"And I have to pick up that portable light unit and return it,\" Charlie said. \"And youse guys don't have to worry none about being drove outta here, get it?\"\n\n\"It don't work, Charlie,\" Al said. \"The words are okay, maybe, but the tune's wrong, know what I mean?\" And Sylvie said, \"Watcha acting like that for, Mr. Wiseguy? Aincha got no manners? He don't mean nothing by it, Charlie.\"\n\nIn a few minutes Charlie and Constance were walking hand in hand toward the farm lakefront where the portable spotlights had been set up.\n\n\"One more little thing to see to,\" Charlie said, and gave her hand a squeeze.\n\nShe tried to think of what they had omitted; nothing came to mind. When they emerged from the tree sections, Charlie turned toward the row crops and began to peer up the paths. He waved when Tom Hopewell showed up between two rows.\n\n\"I'd like a word,\" Charlie called.\n\nTom came to them, his hands covered with moist dirt. \"I heard,\" he said. \"Everyone's heard. It's really over?\"\n\n\"Yep. I suspect Lois Wharton will be back at work in a few days and gradually things will return to normal. There are a few missing pieces, but nothing we can't live with.\"\n\n\"Well,\" Tom said doubtfully, and eyed the corn row he had just left.\n\nCharlie helped him look down the row. \"You know, if I had gone into that apartment first, I would have been overcome with curiosity about the identity of the dead man. Especially since I rather like Lois Wharton and wouldn't want to see her injured in any way. But then, if I snitched a man's wallet, I might not know what to do with it, might even be tempted to keep it around, and I would resist that temptation the way Adam should have resisted the apple.\" Tom Hopewell's face turned crimson. Charlie patted him on the arm. \"Since I didn't find any such thing I'm not going to worry about it. Just wanted to say so long. And pick up those lights. See you around.\"\n\nAfter they retrieved the lights and were in their car again, Constance said, \"I can't believe you told him to get rid of that wallet.\"\n\n\"You've got a nerve! After you told Lois to clam up? And right in front of the sheriff, too.\"\n\n\"Well, that's different.\"\n\nCharlie began to chuckle and after a moment he was laughing out loud. His laughter died when he turned south to start the drive to New Jersey. In the back of the Volvo was the box containing David Levy's few possessions. They had to be delivered in person; this report had to be made in person. Constance rested her hand on his thigh as he drove.\n\nJust before Christmas that year Lois paid them a visit. At first she was awkward. \"I really just wanted to tell you,\" she said, \"Warren is very ill, did you know? Last week he had his lawyer in and destroyed our agreement and changed his will. He set up a trust fund to care for Jill as long as she lives. But she may never be able to leave the institution. Anyway, he also transferred the property, the whole parcel of ground over to me, to use as an experimental farm.\"\n\nConstance took her arm and led her to the kitchen and saw her seated, then began to make coffee. Lois looked almost dazed.\n\n\"There will be enough money to run it. I intend to hire Clarence when he retires, as a consultant. And Tom, if he ever stops chasing corn in Peru.\" She looked up at Constance and said in a tone of wonder, \"I told Warren the story about your friend who planted trees as an act of faith. Last summer when I was planning to leave him, I told him that, and he asked me to stay and help him through this crisis. He needed someone.\" She shook her head. \"No. He needed me. I am grateful that he's lived long enough to know I forgive him. We are at peace. I wanted to tell you.\" Before Constance could say anything, she went on, \"We're already working very hard to get things ready. Al Zukal put in his limestone beach and it looks exactly like snow. Maybe it isn't too late.\"\n\nShe didn't stay long. At the door, ready to leave, she hugged Constance hard, and kissed her cheek. \"Thank you,\" she whispered and was gone.\n\nThey watched her back out of the drive and then looked at each other. \"He'll stop chasing corn in Peru when Warren kicks,\" Charlie said. He glanced at the sky; it had started to snow. He closed the door.\n\nAnd won't she be surprised at what he'll have to say!\n\nCharlie didn't know if she had said the words, or if he had, or if either of them had uttered a sound. He cast a suspicious glance at Constance and then sighed. She was smiling that certain smile she sometimes had, and it didn't matter.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nSundown,  \nYellow  \nMoon\nCONTENTS\n\nTitle Page\n\nDedication\n\nEpigraph\n\nSundown, Yellow Moon\n\nAcknowledgments\n\nA Reader's Guide: Sundown, Yellow Moon\n\nA Conversation with Larry Watson\n\nQuestions and Topics for Discussion\n\nAbout the Author\n\nAlso by Larry Watson\n\nCopyright\n\nTo Susan\n\nSundown, yellow moon, I replay the past  \nI know every scene by heart, they all went by so fast\n\n\u2014BOB DYLAN, \"If You See Her, Say Hello\"\nSundown,  \nYellow  \nMoon\n\nAlthough I have devoted much of my life to writing stories, they are all, I have come to realize, part of a single story that has shifted and swelled over time but never strayed far from my core. I will call its beginning the January day when I was sixteen years old and walking home from school with my best friend, Gene Stoddard. We heard sirens, but we couldn't see where they were coming from. Their combined howl, however, was so close that we believed police cars or fire trucks, and an ambulance, had to be nearby. We ran toward the sound, hoping to catch a glimpse of the vehicles and perhaps discern their destination.\n\nWhen we came to the Will-Moore Elementary School playground, its new snow packed and rutted to bare ground in places from the boots of hundreds of children, we saw a police cruiser and an ambulance speeding up Fourth Street. Those were the last vehicles in the procession.\n\nIt was obvious we'd get no closer to them and their mystery, so we stopped, our heaving, slowing breaths fogging the frigid air. Gene and I both lived a few blocks away on Keogh Street. (Our street was in a fairly new section of Bismarck, North Dakota, and the developer's wife, a student of history, had arranged to have some of the street names in that part of town named for the officers\u2014Keogh, Yates, Cooke, Reno\u2014in Custer's command when Custer and the Seventh Cavalry left Fort Abraham Lincoln, just west of Bismarck, on their ill-fated campaign that ended at Little Bighorn.) For all of our school years Gene and I had walked to and from school in each other's company. Recently, however, that had begun to change. We had a couple friends with cars, and one of them usually picked Gene and me up in the morning. They were both on the basketball team, however, and practices meant they stayed later at school. Also, Gene had started going steady with a sophomore girl, Marie Ryan, and he was often with her after school, but on that day she was in rehearsal for a choir concert. So we were walking together again, and we'd planned to go to his house or mine. We had to memorize the dagger speech from _Macbeth_ for Miss Cordell's English class, and we needed to work on our recitations. For a moment we were tempted to take a detour in an attempt to find where all those sirens were going. Their wailing seemed to have stopped, and not far away. Could they be at one of the stores at the shopping center on Third Street? At the state capitol, only a few blocks north of us on Fourth Street? Or could they have been going to a private home on one of the residential streets or avenues that surrounded us?\n\nPerhaps if we had not had that homework assignment, or if it had not been so cold\u2014the snow underfoot squeaked when we shifted our weight\u2014or if one of us had been allowed the use of the family car, we might have followed the trail of undulating sound that still hung in the crystalline winter air. But we did not. We turned up the collars of our wool overcoats (although this fashion would reverse itself before the decade was out, in 1961 teenagers imitated the attire of adults), pulled our stocking caps down to our eyebrows, hunched our shoulders to our necks, and moved on. Would it have mattered if we had not trudged immediately homeward? It would not. Sirens sound when deeds are done.\n\nWe went to Gene's home because it was closer, by four houses, and when we entered, we were surprised to find Gene's father home. He was sitting at the kitchen table, smoking a cigarette, and wearing his hat and overcoat. Is it only hindsight that makes me want to say he looked gray, drawn? Cold of course leaches color from as many cheeks as it fills.\n\n\"Where's your mother?\" he asked.\n\n\"I'm pretty sure she works at the church library on Wednesdays.\"\n\nGene's father\u2014Raymond\u2014nodded gravely. \"Wednesday. That's right. It's Wednesday.\"\n\nMy discomfort grew. Raymond Stoddard was a somber, reserved man by nature, but he wasn't usually tensely grim, as he appeared to be at that moment. That he wasn't aware of Mrs. Stoddard's schedule, something that the women in the other houses up and down the block would certainly know, struck me as unusual, but no more so than the fact that a seemingly healthy male was home before five o'clock. He hadn't taken off his hat and coat, and he hadn't greeted me in any way. And I knew from hearing my parents talk that Mr. Stoddard had once lost a job because of his drinking. I'd be willing to concede that my entire sense of unease might not have been true of the moment but was added only with the aid of hindsight, except for this fact: I didn't remain at the Stoddards' that day. Before I removed my own cap and coat, I made up an excuse and told Gene I couldn't stay. Later that evening, we decided, we'd get together and practice the dagger speech on each other.\n\nAt our house nothing suggested anything but the ordinary. My mother was in the kitchen moving from refrigerator to sink to cupboard to stove, preparing the evening meal and pausing only to take a drag from the Viceroy burning in the ashtray on the kitchen table. My sister, younger than me by eight years, sat on the living room floor watching cartoons and eating some kind of sweet that had already made her heavy for her age and that would lead to lifelong unhappiness over her weight. All was as it should be. The furnace was running, its constant exhalation heating the house until the insides of the windows perspired and the kitchen's cooking smells drifted all the way to the bedrooms and back again. Warmth, food, family\u2014it was a scene of reassuring comfort, and although I might have felt it as such, nothing about it registered itself as rare, and that I took it for granted only testifies to how few interiors I knew well.\n\n\"Do you know what those sirens were about?\" I asked my mother.\n\n\"I was going to ask you the same thing.\"\n\n\"Gene and I saw an ambulance and a cop car going up Fourth Street, but there had to be more than just those two.\"\n\n\"Police car,\" she corrected. \"And two went past on Divide, too.\"\n\n\"Going east?\"\n\n\"Going east.\"\n\n\"There wasn't anything on the radio?\" My mother had a radio on in the kitchen from early morning until night.\n\n\"Haven't heard a thing.\"\n\n\"Well, if you hear anything from my room, it's me. We have to memorize this Shakespeare speech for English, and the only way I can get it is to keep saying it out loud.\"\n\n\"Let me know if you want me to test you.\"\n\nI went off to my room and behind its closed door paced back and forth and recited Macbeth's words:\n\nArt thou not, fatal vision, sensible  \nTo feeling as to sight? or art thou but  \nA dagger of the mind, a false creation,  \nProceeding from the heat-oppressed brain?  \nI see thee yet, in form as palpable  \nAs this which now I draw.\n\nTo this day I remember the soliloquy, although I've had no occasion to repeat those lines. That they're sealed in my memory makes me believe that disaster has the power to cast not only every attendant fact in the sharpest relief but collateral details as well.\n\nOver the sound of my own voice I soon heard sirens again, though faintly at first. Because of what we had witnessed earlier, I stopped speaking and listened carefully. Their wail traveled so easily to my ears it seemed as though I could map their movement in the air\u2014up Fourth Street again, west on Divide Avenue, closer, closer, until they were right _there._\n\nI jumped to my window in time to see one police car and then another slide around the corner and career down our street. Their lights sliced through the early evening dusk and tinged the snowbanks and the houses on the block a pale, pulsing red. That same light found its way into our home, its presence a violation. In other parts of the city did they hear the sirens and calculate that they came from our block? I ran to the living room, from whose windows I could see farther down the street.\n\nI didn't have to look far. Both cruisers pulled awkwardly to the curb in front of . . . the Stoddards'? Was that possible? Even more improbable was the car already parked there. In the driveway, right behind Raymond Stoddard's dark blue Ford Galaxie was an emerald-green Nash Rambler. If anyone in town owned a Nash in that same color other than my father, I didn't know about it.\n\nSurely there had been other moments in my life when I had been immobilized by conflicting impulses, but that was the first time I was aware of it. I was caught between the desire to flee back into the interior of our house, back to where those lights couldn't touch me or those sirens reach me, and a wish to run in the other direction\u2014out of the house, across the drifted lawns and the snow-packed street, and down the block to where the real invasion was taking place, to my best friend's house. But perhaps it was confusion that truly paralyzed me. _Dad's car?_\n\nIn recalling the events of that and subsequent days, I'm sometimes unsure about chronology, about what occurred before or after. In part, that uncertainty is a consequence of how I've learned about the events\u2014often in fragments\u2014and my own version of what happened is a patchwork (although no less accurate for that). Scenes and realizations have been stitched together, sometimes overlapping one another, as I have made, and occasionally revised, my discoveries over time. About the order of days and hours I'm generally confident, but occasionally the seconds or minutes jump ahead or behind one another. For example, I have the impression that my mother came to the window too, and as she was looking over my shoulder, I asked her why Dad's car was there, but before she could answer me, the phone rang and she ran to pick it up. After a brief conversation, she came back to the window and said, \"That was your father. He said neither you nor your sister is to go outside. Do you hear? You're to stay in the house.\" But in truth, the phone had already rung, and my mother had talked to my father for a few minutes, long enough for him to give her a brief account of why he was at the Stoddards' and why the police would be there soon. True, he told my mother to keep my sister and me in the house, but his real concern was that neither of us go anywhere near the Stoddards', no matter how hard the obligations of friendship or the temptations of curiosity tried to move us out the door. He had instructed my mother to stay home as well, or at least until he phoned again or until she saw Mrs. Stoddard return home, and then she was to come running.\n\nI could continue to withhold information, revealing it only in slivers and shards, and in that way try to duplicate in these pages the suspense that I lived through that night, and in the process induce in you the apprehension and uncertainty that I felt, but while that might be effective narrative strategy, it would not be entirely true to the situation. I don't recall exactly how or when I came to know what had happened in our city and our neighborhood, because to this day I am still augmenting the story, though more now with human understanding than with factual details.\n\nNevertheless, when I finally went to bed, it was with sufficient knowledge\u2014sufficient but incomplete\u2014to allow me to fall asleep without being totally bedeviled by questions. This much I knew:\n\nTrue to one of Gene and my guesses, those sirens had been heading toward the state capitol, only a few blocks from our homes on Keogh Street. I had been in and out of the capitol building so often over the years\u2014on my own or with school groups, with relatives and friends who came to visit\u2014that I knew its architecture and floor plan almost as well as I knew my school's, the public library's, or the civic auditorium's. I had no difficulty picturing the setting for the afternoon's incident.\n\nIn the capitol, both the senate and the house were in the legislative wing, an impressive high-ceilinged, wood-paneled, art-deco-inspired hall. Right outside the legislative chambers were upholstered banquettes, intended no doubt as places where the senators and representatives could meet, during session breaks, with lobbyists and constituents. In January 1961, the legislature was in session (worth noting because to this day North Dakota's senate and house convene only every other year), and during a recess Monty Burnham, a popular, charismatic senator from Wembley, in Cleave County in the north central part of the state, sat on one of those padded benches in the hall. At Senator Burnham's side was John Ritterbush, a Fargo attorney.\n\nWhile the two men conferred, a lean, swarthy man approached them, reached into his overcoat pocket, and brought out a pistol. Without discussion or warning, he fired at Senator Burnham. The bullet struck the senator in the chest. Burnham tried to rise from the bench, but failed and slid to the marble floor. While he lay there, the gunman shot him again. This bullet went through Monty Burnham's throat. Ritterbush was not a target, but he was injured slightly when a fragment of either lead or stone struck him in the cheek.\n\nThe senator's wounds were mortal, and while he lay dying on the floor of the capitol's Great Hall, the gunman calmly walked away, exiting the building through a revolving door on the south side of the building. The fact that a weapon was in his possession may have kept anyone from trying to detain him.\n\nThere were many witnesses to the shooting and its immediate aftermath, and in spite of the inevitable confusion and panic, it was not long before someone volunteered an identification of the man in the overcoat. He was recognized as an employee of the North Dakota Department of Accounts and Purchases, whose offices were on the sixth floor of the capitol. His name was Raymond Stoddard.\n\nConsidering the span between the time when the shooting occurred and when Gene and I saw his father, Mr. Stoddard must have driven directly home after leaving the capitol.\n\nAnd it couldn't have been long after I left my friend's house that Mr. Stoddard went out into the garage, secured a length of clothesline rope around a two-by-four beam, balanced himself on a stack of tires, and hanged himself. To this day I don't know what propelled Gene to the garage, where he discovered his father swinging from the rafter, but once he did, he went right back into the house and tried to telephone his sister, hoping she might know what he should do before their mother returned. The call was long distance since Marcia Stoddard was a student at the University of North Dakota in Grand Forks. When she didn't answer, Gene called the office of a Bismarck attorney, the only other person he could think of who might know what should be done. That person was my father.\n\nMy father arrived at the Stoddard residence shortly before the police, in just enough time to try to calm Gene and to get as much information from him as possible. There wasn't enough time for my father to cut down Raymond Stoddard's body or to open the envelope lying on the kitchen table, both of which he had intended to do. When the police opened the envelope, they found inside a note typed with these words: \"I murdered Senator Monty Burnham. I acted alone.\" Mr. Stoddard signed the note, and since it was on Accounts and Purchases letterhead, the assumption was that he had typed it in his office before going down to the legislative wing to commit his crime.\n\nAnd that was that. About who did the deed there was to be no doubt. But in spite of the popularity of mystery novels, is _who_ the question we really want answered? Isn't _why_ what we truly want to know? A name, an identity, won't fully satisfy us; in order to understand his motive we want to see inside the mind and heart of the murderer, the lover, the coward, or the hero. Barring that, we want to be presented with enough evidence that we can construct a reasonable explanation of what occurred and why. And in fairness I have to say, if you are one of those readers who must have all your questions answered, you must set this narrative aside immediately. If I could have warned you earlier, I would have, but you've spent only as much of your time as it takes to read a little more than ten pages. It can't have been more than minutes. All I can promise you, if you choose to continue reading, is enough information that you might be able to look into your own heart and mind and find an answer there for why humans behave as they do, even in moments most extreme. And although it's not likely to console you, your investment of time will be nothing compared to mine, almost a lifetime by some standards of measurement.\n\nI said I knew enough to allow me to fall asleep that night, but of course many questions still troubled me, some of them of the obvious, profound variety that unsettled everyone, and some close to trivial, if indeed anything related to those events could be regarded as insignificant.\n\nFor example, I was curious about the murder weapon. I had never seen firearms of any sort in the Stoddard household, so I wondered if Mr. Stoddard had had the gun well hidden somewhere or if he had recently acquired it in order to shoot Senator Burnham. And why had he hanged himself\u2014a method of suicide that struck me as torturous and uncertain\u2014rather than shot himself?\n\nThe mystery of the gun was eventually solved, and in a way that illustrated how information was added to the basic narrative over time, just as it is for readers of any mystery. Raymond Stoddard couldn't have driven home and killed himself with the same pistol with which he shot Senator Burnham because he no longer had it. When he walked out of the capitol building, he threw the gun away. A groundskeeper found it months later when temperatures rose, snow melted, and the drift the gun had landed in gave up its secret. The event was a literal example of an expression my mother often used. _Det som g\u00f6 i sn\u00f6 kommer fram i t\u00f6._ She learned it from her mother, who came to this country from Sweden. _That which is hidden in the snow reappears in the thaw._ One of the witnesses to the crime had said he believed the murder weapon was an Army issue .45 automatic, and the groundskeeper's discovery verified the accuracy of the witness's observation.\n\nA fact I had long been in possession of also troubled me, and troubled me especially because I had no way of determining its importance. Did it mean anything that Raymond and Alma Stoddard and Monty Burnham all came from Wembley, North Dakota, a small town 150 miles northeast of Bismarck? I hesitated to make too much of it because my father and I were from Wembley as well. Both of us were born there, but our family moved to Bismarck when I was five years old. Was that common place of origin nothing more than coincidence, a detail that teases us with its suggestion of significance when it possesses none beyond itself?\n\nThroughout the evening, neighbors knocked on our door, sometimes with casseroles and questions about whether they could be of any help. The Stoddards were closer, in physical distance and perhaps in friendship, to other neighbors, but these people came to our house. I flattered myself that my friendship with Gene had given us special status, but of course they were there because of my father. The news of Monty Burnham's murder had been officially made public, but somehow the unofficial reports of my father's involvement had gotten around as well.\n\nAnd that no doubt was why the police and the press also visited our home that night. They wanted what the neighbors wanted, but the reporter from _The Bismarck Tribune_ and the two detectives came right out and said they were seeking information and flipped open their notebooks.\n\nMy mother did her best to keep up with the number and variety of visitors. She brewed pots of coffee, she put out plates of cookies and cups of nuts, she emptied ashtrays, and she kept adjusting the thermostat to make up for the heat the house lost with the door so often open. Strangers crossed the threshold, lights burned that were seldom turned on, snowy shoes and galoshes melted puddles on carpets that were usually kept dry, yet my mother remained a model of smiling courtesy, equanimity, and efficiency. Which is to say that in crisis, she was who she always was.\n\nMy father, on the other hand, seemed utterly transformed by the day's events. He was, in ordinary circumstances, sociable, if somewhat aloof, a calm, capable man who gave the impression of being in control of everything in his life from his gutters and eaves to his emotions. He was devoted to his family, his profession, and his community, and in return for his steadfastness, respect and admiration flowed his way.\n\nBut on that evening, my father seemed on the verge of tipping into rage. Scowling and tight-lipped, he answered questions posed to him with answers so terse I wondered if the police had cautioned him not to speak of what he had seen and heard at the Stoddard home. He tried to assist my mother with her duties as host, but his heart was plainly not in it. People barely put down their coffee cups before my father swept them up and carried them back to the kitchen. And on one occasion he was openly rude. When he walked into the living room and saw Pastor Lundgren speaking to Dolores Lemke, my father said to the minister, \"Don't you think you could do more good at the house down the street?\" When he finally sent my sister and me to our rooms, it was with a command that made us seem no different from the curiosity seekers who had been in and out of the house all evening. \"All right,\" he said to us, \"off to bed. Show's over.\"\n\nWhere had his anger come from? That question may have perturbed me more than any other. Granted, what he had confronted when he'd stepped into the Stoddard garage must have been shocking, disturbing, sickening. He knew both of the day's dead men, although his relationship with the senator was little more than a superficial acquaintance. Those circumstances, however, should have left my father shaken, grief-stricken, bewildered (the emotion most in evidence that evening). But rage? Had it always been inside him, waiting for the tiniest crack to come bursting forth? Or did he know something that the sad, puzzled rest of us didn't?\n\nMy bedroom was in the northwest corner of the house, the corner that took the brunt of the season's arctic winds. Built in the city's housing boom of the 1950s, our home was not especially well insulated, and during the coldest months, frost formed on the wall next to my bed. As I lay there thinking about what had happened in our city, our neighborhood, our block, I reached out and scraped my fingernails through the rime. Because the world's calamities had come so close, our home no longer felt like a refuge, and its walls seemed as insubstantial as that thin layer of ice. Nowhere were we protected, and only fools believed otherwise.\n\nI didn't speak to or see Gene until the following evening when he and his mother came to our house for dinner. To say I was nervous about being in his presence would have been an understatement. Could I have come up with something appropriate to say to a friend whose father had just died? Perhaps. But to a friend whose father was a murderer and a suicide? Impossible. As it was, I reacted with something close to joy when my mother woke me that morning with the news that Gene wouldn't be attending school that day. _Today?_ I thought. He shouldn't go back to school _ever again._ I might have pretended that a wish to save him from the stares, whispers, silences, and insults he would have to endure brought on that thought, but my own discomfort was my first concern.\n\nEvening inevitably fell, however, and with the dark came Gene and his mother, treading carefully up the block as if the real alteration to life on Keogh Street was the dusting of snow that had fallen that day.\n\nWhen I saw them approach, I grabbed the newspaper with the intention of hiding it and its \"Senator Slain!\" headline and the accompanying photograph of the capitol's bloodstained marble floor. But as soon as I picked it up, I realized how futile it was to believe the surviving Stoddards could be spared anything. _The_ _Bismarck Tribune_ was delivered to their house, too. And what if it were not? Almost every newspaper in the land and many of its magazines were likely to carry stories about Monty Burnham's assassination, and even if those media could be kept out of their home, the news could still enter through the airwaves. They would have to keep their television and radio turned off as well. And if they ventured outside their door, they would have to see themselves reflected in others' eyes. Even if they remained behind the walls of their own home, they would still have to look at each other. They were Stoddards and forever after would carry the burden of that name. I was sixteen years old, and I hadn't yet learned of time's astonishing ability to diminish or erase almost anything, so it wasn't long before the iron logic of hopelessness led me to the conclusion that the only relief from their circumstances would be to do with their lives what Raymond Stoddard had done with his. I gave up and put the paper back down on the coffee table. In the black-and-white photograph it was next to impossible to tell whether a swirl on the capitol floor was from blood or variegation in the stone.\n\nAnd then they were in our home. Because we couldn't do anything else, Gene and I said hi, and just like that a realization overturned all the morbid conclusions I had reached. Nothing needed to be kept from him. The television reporters could say the worst things imaginable about Raymond Stoddard, and the gossips could spread the most malicious rumors about the family. The newspapers could publish the most lurid photographs they could find. Gene had been inoculated against their pain. He would never be exposed to anything worse than what he had seen when he'd opened that door leading to the garage.\n\nIt was difficult to determine what physical toll the events had taken on Alma Stoddard because she had frequently looked weary and slightly frail. She and my mother were the same age and both were pretty women, but placing these brunettes side by side only emphasized my mother's vitality. Mrs. Stoddard was pale and thin, while my mother was ruddy-cheeked and robust. Even in the best of times Mrs. Stoddard didn't smile often, but when my mother helped her with her coat and in the process squeezed Mrs. Stoddard's narrow shoulders, she gave my mother a quick small smile that carried a world's worth of gratitude.\n\nMy father, mother, and Mrs. Stoddard ate at the dining room table, while Gene, my sister, and I were allowed to eat on TV trays in the living room. The arrangement worked well for me. Gene and I didn't have to speak much\u2014that night's episode of _My Three Sons_ did that work for us\u2014and I was near enough to the three adults that I could listen in on their conversations.\n\nThrough the meal and dessert and the clearing of dishes, through coffee and cigarettes, I waited in vain. Would no one ever speak about what had to be on everyone's mind: _Why would_ _Raymond do such a thing?_ Instead the talk was of practical matters. The funeral would have to be in Bismarck. If Raymond had died under . . . other circumstances, the service could have been held in Wembley, and he would have been buried there, in the family plot next to the graves of his parents and a sister who had died in infancy. But Wembley was reserved for Monty Burnham. According to the newspaper's biography, he was the hometown boy who'd made good\u2014a three-sport letterman in high school, a decorated World War II veteran, a successful businessman (he owned Ford auto and John Deere farm implement dealerships), and a prominent Republican legislator. And martyr, though to what cause no one knew. His funeral would be a public occasion in Wembley, and his murderer would be welcome nowhere near the community, even in the local cemetery.\n\nOf the three local funeral parlors, Mrs. Stoddard had picked Metzger's\u2014did my father and mother think she had made the right choice? I was looking into the kitchen as she asked the question, and she leaned across the table as if she were desperate to hear that in this matter she had chosen wisely.\n\nMy mother reached out and patted Mrs. Stoddard's hand, while my father nodded sagely and said, \"Sam Metzger is a good man. And if you don't mind me putting in my two cents' worth: Services sooner would be better than later.\"\n\nAs if in relief, Mrs. Stoddard began to weep, and just at that moment Gene asked, \"Can I get the homework for English and history? I'm going to school tomorrow.\" Perhaps he asked just to stop me from staring at his crying mother. If so, his strategy worked. The two of us went to my bedroom and my schoolbooks.\n\nWhile I copied down the assignments for Gene, he said, \"Do you want a ride tomorrow morning? I'm going to have the car. In fact, from now on I'll have the car a lot.\"\n\nHe didn't say this ruefully or ironically but expressionlessly, and my initial reaction was, My God, what a monstrously self-interested thing to say! Then, in the next beat, I reminded myself that none of the usual standards of human behavior applied to my friend.\n\n\"Sure,\" I said. \"Do you want to pick me up or should I come down?\"\n\n\"I'll pick you up. By the way, don't try to call. My mom unplugged the phone.\"\n\nWhen we returned, Gene's mother was still weeping. And confessing, if it's possible to confess to ignorance.\n\n\"I don't know. I _just don't know._ \" Her sobs twisted her face. \"He never said anything to me. _Never._ \"\n\nMy mother was up out of her chair and standing behind Mrs. Stoddard, patting her back and comforting the sobbing woman as best she could. My father, meanwhile, remained seated, and his expression\u2014stern, still on the edge of anger\u2014made me wonder if it had been his questioning that had brought on Mrs. Stoddard's state.\n\n\"You think you know someone,\" Mrs. Stoddard said, \"and then . . . then something like this happens. I've racked my brain . . . for an explanation. A clue. I've looked _everywhere._ \"\n\nNow I saw what might have been an additional emotion cross my father's face. Was that skepticism? Did he know a place\u2014a hidden area, a dark corner\u2014where Mrs. Stoddard hadn't looked?\n\n\"Don't blame yourself,\" my mother said. \"Please, Alma. Don't.\"\n\nMy sister ran into the room, and at her entrance the three adults tried to give an impression of normalcy. My father reached for his cigarettes. My mother asked if anyone needed more coffee. Alma Stoddard sat up straight and wiped her cheeks with the heels of her hands.\n\nI turned away too, but Gene was right there, ready to answer the question I would never ask. \"I don't know why he did it either,\" he said. He could not have been more straightforward and unapologetic in delivering that line if he were in Mr. Bollinger's algebra class admitting that he could not solve an equation. Gene had never been a particularly effusive boy. Neither of us were, but growing up I had always envied his placidity. I might have affected the same demeanor (ours was an especially chilly Midwestern version of cool), but I knew I was faking it. Inwardly I was often nervous\u2014about grades, girls, sports, the proper clothes, the correct companions. That night, however, as I confronted Gene's eerie serenity, I felt myself permanently cured of some of my anxiety. I couldn't imagine a life less enviable than his.\n\nAs if the real reason for coming to our house had been to make the announcement that Raymond Stoddard's behavior mystified them as much as anyone, having discharged that duty, Gene and his mother soon left.\n\nIn their absence we were, of course, as bewildered as ever, and almost immediately after their departure, my father went to the telephone. \"I'm calling Burt. Find out what they're saying about this up there.\"\n\nMy father and his younger brother, Burt, left Wembley for college\u2014my father to the University of North Dakota and law school and Burt to the University of Minnesota and pharmacy school\u2014and after graduation both returned to their hometown to start their careers. But while my father eventually moved on, Burt remained, working in the drugstore that their father owned. After my grandfather's death, Burt took over the family business, and he continued to live alone in the house the boys grew up in.\n\nMy mother and I waited in the living room while the brothers talked. Their conversation lasted long enough that we knew something of substance was being discussed, and Burt was obviously doing most of the talking. My father sat at the kitchen table and smoked, nodding in understanding or affirmation, and only interrupting to inject an occasional question. When he hung up, he remained in the kitchen, untangling the twisted telephone cord. It was not in his nature to torment us deliberately, so I can only imagine he was wondering how he would convey the information he had gleaned from his brother.\n\nWhen he finally came into the living room, he said, \"Well, that was Burt.\" He looked at me and then to my mother. I understood his hesitation. He wasn't sure how much he should say in front of me.\n\n\"Go ahead,\" my mother said. \"He needs to hear. Gene is his friend.\" With those words she conferred upon me a new status. Did I want it? I believed I did. It meant I would no longer have to eavesdrop or conjecture. But I also knew the time might come when I'd wish I could step back down in rank.\n\n\"Was Burt drunk, by the way?\" my mother asked. She would never have asked that in my presence before, though her question merely acknowledged what was generally known about her brother-in-law.\n\n\"Not yet, but he's getting there.\"\n\nMy mother moved over on the sofa, inviting my father to join us. \"So what did Burt have to say?\"\n\nMy father sat down between us and for the first time that day loosened his tie. \"Well, he said something that Alma didn't see fit to mention.\"\n\nOnce again, he cast a glance in my direction, but with nothing more than a hand touched lightly to his shoulder, my mother urged him on. My father cleared his throat and began. \"It seems Alma and Senator Burnham used to go together. Were you aware of that?\"\n\nShock crossed her face, and she pushed back against the sofa's cushions. \"When was this?\"\n\n\"High school. Or thereabouts.\"\n\n\"Oh. Well. Teenagers.\" Her dismissive tone wounded me.\n\n\"I'm simply reporting what Burt said. Do you want to hear it or not?\"\n\n\"Go ahead. But I'll keep in mind where this is coming from, too.\"\n\nI knew what that meant, or thought I did. Uncle Burt was not an admirer of Monty Burnham or his politics. Burt was a Democrat, a fact that put him in the minority in conservative North Dakota (although we had just elected a Democratic governor), while Monty Burnham was not just a Republican but a leader in the party and someone who had often been talked about as a possible candidate for higher office: governor, United States representative, or senator. But as my father talked\u2014that night, and in the days, weeks, months, and even years to come\u2014I learned of other reasons for Burt's antipathy.\n\nIn most respects, however, Burt was a good source to consult. He was intelligent and observant, he had a good memory, and, unlike his older brother, he was not averse to gossip. Burt was a Wembley resident past and present, and he was from the same high school graduating class as Monty Burnham, Raymond Stoddard, and Alma Shumate. (Betty Donfils, Monty Burnham's wife, now widow, graduated two years behind them.) My father, six or seven years older than these people, was never their peer, and he knew them as little more than the children of Wembley families and the younger siblings of friends. The Shumates were poor, and Alma grew up one of six children in what today would be known as a single-parent home. When she was a child, her father died in a farm accident; he fell into a grain silo and suffocated. The Stoddards, on the other hand, were comfortably middle class, thanks to Raymond's father's job with the Soo Line Railroad. My ancestors were among Wembley's first settlers, and while they were a prominent, respected, and well-established family in the community, they never enjoyed the power or prosperity that the Burnhams had. Monty's father served two terms as town mayor, and his uncle was state's attorney. Over the years the Burnhams owned a real estate company, various downtown buildings, and the two successful dealerships that Monty eventually took over. \"Generations of glad-handers and wheeler-dealers,\" my father said, perhaps repeating his brother's phrase. Neither brother would have been intending the remark as a compliment.\n\nMonty Burnham and Raymond Stoddard were both World War II veterans, having served together in the Pacific theater. Senator Burnham left the military as a decorated Army tank commander. During the battle for one of the Mariana Islands, according to one of the obituaries, Monty Burnham \"resolutely carried on even when his tank was cut off from the rest of the platoon and battalion, and by taking the fight to the Japanese, he carried the day for the Allied Forces.\" Mr. Stoddard never rose above the rank of private, and as far as Burt knew, neither man came through the war with anything worse than Raymond's case of malaria. (My father, also an officer, fought in Europe, while Uncle Burt was in the Navy but saw no combat.) After the war, Raymond Stoddard worked for a county agency in Wembley for a few years before he and Alma moved to Bismarck. Monty Burnham capitalized on his distinguished war record as well as his family name to rise to prominence in Wembley and eventually the state.\n\n\"Maybe there's something in those war years,\" my mother suggested. \"Could Monty Burnham have been Ray's commanding officer? Maybe there was something unfair about Monty Burnham's promotion or\u2014\"\n\nShe hadn't finished before my father began shaking his head. \"Didn't work that way. Anger, jealousy, resentment\u2014sure, there was bad blood among the men. No shortage of it. But by the time we mustered out, we were ready to put it all behind us. Wartime grudges don't have any staying power, in my experience.\"\n\n\"Did you and Ray talk about the war?\"\n\n\"Never.\"\n\n\"Never? That's hard to believe.\"\n\nNot for me. Many of my friends had fathers who had been in the war, and some of them never got over it. Or wouldn't let themselves. My friend Stan Gronlund, for example, never stopped talking about his father's experiences as an Air Force bombardier, but in that, Stan was simply imitating his father, who was amazingly adept at relating almost everything to his years in the military. Jim Kieper's father had what amounted to a World War II shrine in his basement, consisting mostly of German military artifacts\u2014helmets; medals; binoculars; bayonets; a disarmed grenade; even a singed, bullet-punctured swastika flag. But my father and Raymond Stoddard brought home no souvenirs, and they passed no war stories on to their sons.\n\n\"Don't believe it, then,\" my father said. \"But it was so.\"\n\nMy mother shrugged. \"What about the job that Ray drank himself out of? Was that in Wembley? Or here?\"\n\n\"That was here. And to tell you the truth, I'm inclined to give Ray a pass on that one. When they first moved to Bismarck, he was doing construction work, and then during the winter when things were slow he took a job as night clerk at the Frontier. Construction picked up, but he didn't want to let go of the extra money. Anyway, he was burning the candle at both ends, and one night they had some sort of small scale emergency at the hotel\u2014a small fire back in the kitchen or something\u2014and Ray wasn't at the desk, where he was supposed to be. He was off catching forty winks somewhere. When the manager of the Frontier caught wind of it, he fired Ray.\"\n\n\"What did his drinking have to do with it?\"\n\nNow it was my father's turn to shrug. \"The manager smelled liquor on Ray's breath, and the bartender said that Ray had been in the bar that night, so the manager put one and one together and got three. He assumed Ray was drunk and sleeping it off.\"\n\n\"But Ray was drinking then?\"\n\nMy father leaned forward and with the heel of his hand rubbed the coffee table's glass top. If he had seen a smudge there, it wouldn't last long, not in my mother's home. \"What makes us think he ever quit? Maybe he just got it under control.\"\n\n\"I suppose I just assumed . . .\"\n\nMy father looked at me, the first time I had been included as more than a listener. \"Well? Did you see any evidence that Mr. Stoddard was still drinking?\"\n\n\"He might have been. I once saw a couple empties in their garbage can.\"\n\n\"Bourbon?\" my father asked.\n\nI didn't want to appear too knowledgeable. \"Old Crow? It was a brown bottle. A little one.\"\n\n\"Bourbon,\" my father concluded. \"Raymond once said he was too fond of bourbon.\"\n\n\"When was this?\" my mother asked.\n\n\"I can't remember exactly,\" I answered. \"Last summer, maybe? We were just goofing around and the garbage can tipped over.\"\n\n\"No, no,\" she said, nodding in my father's direction. \"I mean, when did Ray confess this fondness for bourbon?\"\n\nMy father sat back again. \"A summer evening a few years back. We'd both been mowing our lawns, and Ray just wandered up the street. This was a driveway conversation, nothing more.\"\n\n\"But how did it come out, about the bourbon?\"\n\n\"I don't know. Hot weather? Wouldn't a beer hit the spot? And Ray said bourbon had always been his drink of choice. You call it a confession. It was nothing like that. Small talk is what it was.\"\n\n\"Did Burt say anything about why Ray and Alma came to Bismarck in the first place?\"\n\n\"Burt didn't. But I have the answer to that one. He came to build houses. Alma's brother was a foreman with the construction company Ray went to work for. He still is, for that matter. Harbring Construction has Len Harbring's name over the door, but Alma's brother runs the outfit. I'm not sure why Ray quit them. I suppose because working for the state is steadier.\"\n\n\"But if he was still drinking. . . .\"\n\n\"Alcohol doesn't explain murder. A man in trouble with the bottle might take his own life, but not another's.\" He clapped his hands on his thighs and rocked back and forth. \"None of it makes any sense.\"\n\n\"Burt didn't have anything else to say?\"\n\n\"Nothing comes to mind. What he kept coming back to was the business about Alma once going out with Monty Burnham.\" He looked at my mother. \"Is that something you could ask her about?\"\n\n\"Alma and I don't have that kind of relationship.\" My mother's tone was icy. \"We never have.\"\n\n\"Could you ask Iris?\" The Friedrichs lived next door to the Stoddards, and Iris Friedrich was Alma Stoddard's closest friend. \"See if Alma ever said anything to her about Monty Burnham?\"\n\n\"And what would that make me?\" My mother answered her own question. \"A snoop and a gossip.\" There were no worse labels that could be hung on her.\n\n\"Then we're stuck,\" my father said. He reached for his cigarettes\u2014then stopped, balled up his fist, and pounded it into his other open hand. \"God _damn_ it! I keep thinking I should go down there and grab Ray by the shoulders and shake him and talk some sense into the man!\"\n\nMy mother reached out and, as if she were playing rock, scissors, paper with my father, covered his fist with her hand before he could strike his palm again. Her gesture told me that she had long known what I had just learned\u2014that in my father grief and rage could wrap themselves as tightly together as those fingers that were seeking something to punch.\n\n\"You don't have to take this on yourself, you know,\" she said gently. \"This isn't a mystery you have to solve.\" Only my mother could have spoken to him in this way and have her words calm rather than anger him further.\n\nIn our home the living room draperies were seldom drawn, but since the traffic on Keogh Street, usually limited to its few residents, had increased significantly in the past twenty-four hours, my mother had closed the curtains just after Gene and his mother came over. Cars traveled slowly up and down the street as they would have a few weeks earlier when Bismarck residents patrolled the city's neighborhoods to look at the Christmas lights and decorations. Now, of course, they were searching for the Stoddard residence, hoping perhaps that its blank rectangles of glass, stucco, and wood might give off a unique and lurid glow.\n\nBut even if our curtains had been open, they would have revealed at that moment a tableau so ordinary that not even the most curious would look twice. Under a framed reproduction of a Constable landscape, father, mother, and son sat together on a sofa covered in a gray-green fabric. The three of them might have gathered to have a talk about the dangers of smoking\u2014even as the parents lit their own cigarettes\u2014or about the son's slipping grades in geometry. Perhaps they were asking him if he understood how important it was that he apply himself to his studies. On the television across the room Jack Paar spoke and gestured with a flamboyance rare among Midwestern men. Nothing\u2014nothing whatsoever\u2014in the scene revealed that a bomb had exploded on the block or that these people would henceforth have to live in its wreckage. If their faces were uncontorted by anguish, it was only because they had all learned the consoling power of the prayer of selfishness _\u2014Thank God it's not us, thank God it's_ _not us, thank God it's not us._\n\nThere was something in my parents' conversation that night that I found especially puzzling. I couldn't understand why they\u2014why my mother, really\u2014had passed so swiftly over the fact that Monty Burnham and Alma Stoddard had once dated. In my view, nothing of greater significance had, to that point, been revealed that could approach a motive for Raymond Stoddard. Yet my mother seemed to think that because they had both been teenagers at the time, the relationship couldn't have been of consequence. Had she been standing where I had been only a few weeks earlier, next to the pool tables at Midway Bowling Lanes, and had she seen and heard Russell Batt, a classmate of mine, she might have realized how easily a ruined adolescent romance can turn someone's thoughts to murder.\n\nRussell Batt and Jennifer Oslund had been a couple since junior high school, but then suddenly\u2014no one quite knew what the exact sequence of events was\u2014just after Thanksgiving she dropped Russell and began to date Curt Forney. Since Russell still loved Jennifer, he blamed Curt for the breakup, and that night at the bowling alley Russell was waiting for Curt to walk through the door. When he did, Russell said, he'd kill him.\n\nA tall, raw-boned, rope-muscled kid, Russell Batt had grown up on a farm and had moved to Bismarck as a sixth grader. He was quick-tempered and belligerent, and almost from the moment he arrived in town, he had been getting into fistfights. But it wasn't just his history and pugnacious nature that convinced those of us who listened to him that night that his rants were more than talk.\n\nHe had opened his jacket to reveal, tucked into the waistband of his jeans, a stag-handled hunting knife. Lest anyone think it was just a prop, he pulled it from its leather sheath, allowing its curved blade to glint in the overhead light meant to illuminate the green felt of the billiard tables. \"I'm going to slip this between his ribs,\" Russell said, \"before he can say a goddamn thing. I don't give a shit what his side of it is.\" Russell was drunk, but to those of us gathered around him, that only enhanced his credibility.\n\nCurt Forney, however, didn't enter Midway Bowling Lanes that night, and whether that was because someone warned him in advance or headed him off in the parking lot, I never knew. Besides, there was no reason to believe that Russell's vow had any temporal boundaries. If he didn't kill Curt then, he'd do it another time.\n\nIf Curt Forney had appeared, would we\u2014those of us clustered around Russell Batt and basking in his menace\u2014have tried to intercede on Curt's behalf? I like to think we would have, although Russell's jealousy, anger, and murderous determination shrunk all of us; we were boys listening to a man's threats.\n\nWe were spared that awful responsibility not just by Curt's nonappearance but by an intercession. Also at the Midway lanes was the former Janice Robichaud. A coarsely pretty woman in her early twenties, Janice was married to Russell's older brother, Morris, but that night she was bowling with her girlfriends. Someone who had overheard Russell sought out Janice and told her what Russell was threatening to do. The bowling lanes and the pool tables were some distance from each other, but when Janice learned of her brother-in-law's agitated state, she headed toward him at a dead run.\n\nShe made no attempt to calm Russell with compassion or soft words. Neither did she try to mount a moral appeal. Instead, she lit furiously into him as soon as she was within shouting distance.\n\n\"You're talking about throwing your life away for what\u2014that two-timing little twat? And you think that will bring her back to you? She'll hate your fucking guts. Now get the hell out of here, and if I hear about any more of this bullshit, I'm telling your brother, and he'll kick your ass from here to Fargo.\"\n\nHer tirade awed and terrified us, and it obviously made an impression on Russell as well. No harm befell Curt Forney on that night or any other, but neither did his relationship with Jennifer Oslund last. Before the end of the school year she was dating Chuck Vogel, and after Chuck, Wes Lahr, and after Wes, someone else, but never again Russell Batt. Or Curt Forney. When Jennifer left for college, she seemed to leave Bismarck for good, while Russell married Nancy Lawler, and the two of them settled on the city's south side to raise their family.\n\nBut what if\u2014what if Raymond Stoddard were made of the same combustible material as Russell Batt, although of the slower-burning sort? What if the passion that almost instantly burst into flame in Russell smoldered for years in Raymond Stoddard until he too was ready to kill the one who . . . who . . . Here, of course, my speculations bumped against reality. If there were once a competition for Alma Shumate, Raymond Stoddard was the winner, not Monty Burnham. But because I was reluctant to admit, even to myself, that I had nothing to offer from my limited store of observations and experiences that could explain Raymond Stoddard, I tried to invent a set of circumstances that would still involve a romantic triangle.\n\n_Yes, Raymond and Alma ended up together, but Ray worried_ _that Monty Burnham remained Alma's real love, so when_ _Ray had an opportunity to leave Wembley, he jumped at it. The_ _Stoddards moved to Bismarck not for Ray's higher-paying job_ _but because Ray wanted to remove his wife from the town_ _where her former boyfriend was a constant presence. Ray's_ _strategy worked. Or so he believed, though he couldn't keep_ _from worrying that his former rival might someday reappear_ _and win Alma back. Ray's concerns increased when Monty_ _Burnham was elected to the state senate, which meant that as a_ _legislator he would come to Bismarck every other year when_ _congress convened._\n\n_And, finally, exactly what Ray feared came to pass. On a_ _blustery winter day, Alma Stoddard took the bus downtown._ _Her destination was Schreiber's Fabric Shop, where she hoped_ _to find some material with which to re-cover a chair. When she_ _left the store, the wind was gusting so hard\u2014little clouds of_ _fine-grained snow billowed down the street and even the traffic_ _light posts were swaying\u2014that she decided to duck into the_ _Coffee Cup Caf\u00e9 for a few minutes of warmth. Meanwhile,_ _Monty Burnham was also in downtown Bismarck. He was supposed_ _to meet with an oil abstractor, but since he was early, he_ _decided to wait across the street in the caf\u00e9 with the frosted_ _windows and the neon sign of a cup and saucer. When Monty,_ _now Senator, walked in, he spotted Alma\u2014he recognized her_ _immediately\u2014and without invitation sat down at her table._\n\n_Alma was not only shy but uncomfortable. No other customers_ _were in the establishment, and there they were\u2014sitting_ _together at a table in the corner. It might have looked as though_ _this had been an arranged meeting! Or maybe her discomfort_ _came not entirely from her concern over what other people_ _might think but from her own feelings. Over the years, she had_ _thought often of Monty Burnham and had wondered what her_ _life would have been if she had stayed with him. Which was_ _harder to imagine\u2014being married to a wealthy well-known_ _man or being married to a charming ebullient man who seemed_ _to smile his way through every day? Monty Burnham was both_ _of those._\n\n_Alma shrank back from Monty Burnham's presence, which_ _only made him strive harder to wrest a smile or laugh from her._ _Nostalgia was the tactic he chose. \"Remember the night of the_ _holiday party that ended with us marching through the streets_ _of Wembley at two o'clock in the morning singing Christmas_ _carols? Remember when we put sugar in the gas tank of old Mr._ _Pettinger's Reo? Remember when we drove to Devils Lake for_ _the county fair, and on the way back my car broke down and it_ _was close to dawn when we finally pulled up in front of your_ _house and your mother came striding down the driveway and_ _didn't say a word but damned near dragged you back inside?_ _Remember the night you told me you feared we were becoming_ _too serious and that we should stop seeing each other? That_ _practically destroyed me, and the very next day I left Wembley_ _and drove into Canada. I took a job on a cattle ranch outside_ _Calgary, and when I returned, you were engaged to Ray Stoddard._ _. . .\" No, no. On that day, their first reunion in years,_ _Monty Burnham would keep the tone light. \"I bent over in Mrs._ _Schmidt's class and split the seat of my pants\u2014remember?\" Finally,_ _finally, he got a smile from Alma, and once he did, successful_ _car salesman and skillful politician that he was, Monty_ _Burnham knew he was home free._\n\n_They arranged to meet again, but Alma didn't want it to be_ _in a place so public. When he suggested that he come to her_ _home, Alma could think of nothing but all the windows up and_ _down Keogh Street, and in each one she imagined a woman_ _peering out and wondering, who could that be entering the_ _Stoddard home? He looks like that politician. . . ._\n\n_\"What about my place,\" Monty Burnham suggested. \"During_ _the legislative session I rent a little basement apartment over_ _on Avenue B. You can park in the alley and enter through the_ _side door. . . .\"_\n\nThis plot required adjustment. Their renewed relationship moved too quickly. Perhaps they met in just this way, but years ago, and they never saw each other in the two years between legislative sessions. _But gradually over time, Alma Stoddard and_ _Monty Burnham fell in love again. They not only met clandestinely,_ _they also exchanged letters, and while Raymond Stoddard_ _was searching for . . . for a needle, yes, a needle\u2014he had a_ _sliver that could be extracted only with a needle\u2014he opened his_ _wife's sewing kit. He accidentally lifted the box's top tray, the_ _one containing all the spools of colorful thread_ (I was of course picturing my mother's sewing kit), _and when he did, he found_ _the letters Monty Burnham had been sending Alma. None of the_ _envelopes had a return address\u2014in case Alma's husband would_ _be the one to pick up the mail one day\u2014but as soon as Ray saw_ _the signature\u2014\"Monty\"\u2014he knew. In fact, he didn't have to_ _read more than that name and the closing that preceded it\u2014_ _\"Love\"\u2014to realize the worst: He had lost his wife._\n\n_But read on he did. He learned that the correspondence had_ _flowed in more than one direction; Alma wrote as often to_ _Monty Burnham as he wrote to her. Ray couldn't see the texts_ _of any of her letters or how she signed them, but he didn't have_ _to. It was obvious that she felt for Monty what he felt for her,_ _and their rekindled relationship blazed hotter than before because_ _now it was fed not only by the incendiary passions left_ _over from youth but also by the slower-burning fuel of adult_ _love. Ray deduced too that they had done more than adore each_ _other from a distance. Plainly, they had been physically intimate,_ _as Monty wrote that \"I live for the time when once again_ _I can hold your naked body in my arms, when I can. . . .\"_\n\nAs you can see, I was already practicing the craft of the novelist I would someday be, though less with language than with imagination\u2014speculating on what lives other than my own were like, and on what forces underlay people's behavior. But in spite of what educators would have us believe about the creative powers of the young, reality hems in their minds more than it does adults'. At sixteen I couldn't imagine what I could at twenty-six, thirty-six, forty-six, fifty-six . . . so I couldn't really envision a world in which Alma Stoddard, clothed or otherwise, would lie in anyone's arms or a world in which anyone would want her to. The problem was not just that she was the mother of my best friend. I could, after all, summon up a few lascivious thoughts about Mrs. Crisp, the flaxen-haired, tanned, curvaceous mother of my friend Jeff, whereas Alma Stoddard had the appearance of a woman who tried to de-emphasize not only her beauty but her entire physical being. She didn't wear much makeup. She wore her hair in a tightly bobby-pinned, unflattering, unvarying style. Her clothes were dark, shapeless, and severe. A time would come when I would realize that ardor and austerity can coexist, but I couldn't get there when I was sixteen, not with Alma Stoddard.\n\nNevertheless, though I had nothing in the way of evidence to support it, I wasn't about to relinquish the idea that the relationship that had existed in the past between Monty Burnham and Alma Stoddard must have had something to do with the present, and I couldn't figure out why my mother brushed aside the possibility so quickly. Who knows\u2014maybe if I had gone on to serve in the military someday I would have been similarly vexed by my father's unwillingness to consider Raymond Stoddard and Monty Burnham's war experiences as the early stimulus for Raymond's murderous deed.\n\nYears later, however, it was my mother's behavior that I mentioned to my sister. The occasion was Christmas, and in the past she had always traveled to Bismarck for the holiday. Our mother, however, had died months earlier, and since my unmarried sister had no other family, she spent the holiday with my wife, daughters, and me in Montana.\n\nIt was late on Christmas Eve, and she and I were the only ones awake in the house. We were sitting in the living room, in front of a fire's dying embers, and we each had a glass of calvados, her Christmas gift to me, purchased on a summer trip to Normandy. We were reminiscing, as brothers and sisters do, but unlike other siblings, we were working from a script. No one who lived in 1961 within the radius of that small circle that took in the capitol building on its eastern edge and Keogh Street on the western margin could talk about the past without eventually, inevitably, discussing the Stoddard-Burnham tragedy.\n\nI had just expressed my long-ago puzzlement about why our mother had found the onetime relationship between Alma Stoddard and Monty Burnham to be of so little interest, and before I finished the thought, my sister began to laugh softly.\n\nDidn't I know? Our mother was engaged to be married when she was still a teenager, and not to the man who would be our father. All I could do was shake my head.\n\nOur mother grew up in a small town in western Minnesota, not far from Fargo, North Dakota, and her boyfriend, a year older, lived on a farm. They dated throughout high school, and on the day of his graduation he asked her to marry him the following year when she received her diploma. Our mother certainly had opportunities aplenty to observe the harsh realities of farm life, but the notion of an isolated, rural existence held a kind of romantic, bucolic appeal for her. And she loved the young man.\n\n\"Well, what happened?\" I asked. \"What made us a lawyer's kids and not a farmer's?\"\n\n\"He was helping out on someone's farm, and he was in an accident. A tractor flipped over on him. His legs were crushed, and he couldn't stand the thought of being a cripple, so he broke off the engagement. Or tried to. Mom, being Mom, said that his condition didn't matter to her, that they'd find some way to make a life together. She could keep her job at the drugstore where she'd been working since she was fourteen. She'd support them.\"\n\n\"Sounds like love, all right.\"\n\n\"On Mom's part. But he ended up marrying another girl, a girl who lived on that farm where he'd been working when he had his accident. So I guess their relationship was a variant of the farmer's daughter joke. And since she was pregnant, apparently he wasn't completely disabled. But Mom really took it hard. She said she wasn't sure how she'd be able to go on.\"\n\nI'm sure my sister believes to this day that I set down my drink and walked from the room because I was upset by the news that our mother had once had a youthful romance that fractured her heart.\n\nBut that was not it, not at all.\n\nAkira Kurosawa said that to be an artist means never to avert one's gaze, advice I've tried to take to heart as a writer and as someone who wants to understand the human mysteries. Even without knowing any specifics, I had readily acknowledged that my parents had lives before they became spouses and parents, and I never held them to any standard that lay beyond the obligations of those roles. Besides, I once lived on Keogh Street\u2014where was there a better school to learn the lesson that the life of every man and woman was so much more than it appeared to be?\n\nNo, disclosures about my mother's and father's pasts weren't likely to distress me. The real reason I couldn't remain in my sister's company wasn't the story she told, but what in hearing it I learned about myself. How could I have been such a fool as to think that our mother dismissed the relationship between Alma Stoddard and Monty Burnham because it was a product of their teen years? She rushed past it _because_ she knew how deep and durable a long-ago love could be.\n\nThe next day\u2014Christmas day\u2014I told my sister I was sorry I'd walked out on her, and I let her think that the subject had simply been more than I could handle.\n\nShe waved away my apology. \"I knew you hadn't heard the story.\"\n\n\"Did Dad know about it?\"\n\n\"He knew she once went with a boy she was serious about. He didn't know they were engaged.\"\n\nI wanted to defend my father, to say that knowledge of his wife's past wouldn't have made a damn bit of difference in his life, but I couldn't, not with complete confidence. The events of January 1961 kept me from that certainty as well. \"So that was a secret best kept from the males in the family?\"\n\nMy sister shrugged. Every sibling wants to believe that he or she is favored in some way, and what could confer that status better than to be taken into a parent's confidence? Maybe my sister held on to that story because our mother had asked her to tell it to no one. Or perhaps my sister hoarded it for a small measure of revenge because years before I had been included in some of our parents' conversations while she was excluded. Whatever the reason, I was willing to let my sister have her satisfaction. Moreover, though what I learned about my mother forced a revision of who I had always believed she was, it did nothing to any theory I held about why Raymond Stoddard did what he did.\n\nI was worried about riding to school with Gene on that first day when he would be permitted to drive the family car, and my worry was quite specific. I feared that if people saw us getting out of the same car, it would only contribute to their confusion about us.\n\nBecause we grew up together on Keogh Street and were so often in each other's company and because we were similar in so many respects, Gene and I were frequently mistaken for the other. We were both brown-haired, slender, and slightly taller than average. For the second consecutive winter we dressed in identical dark green overcoats. We were both serious about our studies, reasonably well behaved, and responsible. We were both shy and comfortable with silence. We were among the youngest of our class, our birthdays one month apart to the day\u2014his October 2 and mine November 2. Indeed, if the world were intent on telling us apart, it had little to go on. Eye color\u2014his brown, mine blue and peering out from behind glasses. He played the clarinet; I ran on the cross-country team. He had a girlfriend; I did not. I had tonsils; he did not. I had a father; he did not.\n\nAs a teenager I was, like many adolescents, frustrated by my anonymity. I wanted, I thought, to be known, to have a reputation, an identity. I wasn't a hood, a rebel, a varsity athlete, a class officer; I wasn't popular with girls or watched by the police. I wasn't tough or talented. But suddenly I saw that fame\u2014or notoriety\u2014could attach itself to you and yet have nothing to do with you. Gene Stoddard had been, like me, simply one of the indistinguishable mass of nice guys until, overnight, he became Bismarck's most famous teenager. And he would have given anything to be obscure again. Or, put another way, to be mistaken for me. If I could have, I would have worn a sign around my neck: I AM NOT GENE STODDARD.\n\nAs it turned out, Gene didn't drive to school, not that morning. Overnight the temperature dropped to almost thirty below, cold even by North Dakota standards. The Stoddards, no doubt because they couldn't bring themselves to enter their garage, had left their Ford parked outside, and in the morning it wouldn't start. My father ended up taking Gene and me to school, and for some reason it seemed to me a sufficient differentiation of our identities that we climbed from a car belonging to our family.\n\nAfter delivering us to school, my father returned to Keogh Street and used his jumper cables to get the Stoddards' car running again. Then he told Alma Stoddard that while he could understand why she might have difficulty going into the garage, if that was the case, then she should consider having a headbolt heater installed in the Ford and plugging it in on winter nights.\n\nI'm not sure if it was precisely _during_ Raymond Stoddard's funeral that I fell in love with Marie Ryan, but by the end of the day it was a fait accompli.\n\nThe process began when she telephoned me the night before. No, no, it started before that. It started with her looks. Marie Ryan was a beautiful girl, and if her beauty occasionally passed unnoticed, it was because she herself paid so little attention to it. She had large, almond-shaped green eyes with a slight oriental tilt, high cheekbones, a strong jaw, creamy skin, deep dimples, and perfectly formed lips. Without the aid of orthodontia, her teeth were flawlessly straight. Although she was only fifteen, she had a woman's lush body on her compact frame. To save her from classical beauty's boring predictability, she had a chicken pox scar on her forehead, and her nose was a little wide, causing a series of diagonal wrinkles to break out along the bridge when she laughed. Consistent with her lack of vanity, she dressed plainly, almost carelessly, and wore her long lustrous reddish-brown hair in the simplest of styles.\n\nBut no amount of physical description\u2014at least none within my powers to offer\u2014will do justice to what Marie Ryan was. Today the term is \"hot.\" Then, she was a babe, a honey, sexy, stacked. In 1961 we may not have known about pheromones, but we still trailed helplessly after someone who secreted them.\n\nAs incomparable as Marie Ryan was in appearance, she was also unique in character. She was intelligent, outspoken, and unconventional. Indeed, it was this last quality that enabled her to pick up the telephone and dial my number. In 1961 teenage girls simply did not call boys. Furthermore, the reason for Marie Ryan's call was another example of her ungovernable nature. Her parents had forbidden her to attend Raymond Stoddard's funeral, and she wanted to know if she could go with me. Without hesitation I said yes, and as evidence of how pleased I was at the prospect of being in her company, I even allowed myself to think, fleetingly, pathetically, inappropriately, that it was almost as if she and I would be on a date.\n\nOn the day of the funeral my parents needed to arrive early at First Lutheran Church. My mother, with a few of the women who belonged to the same Women's League as Alma Stoddard, would be serving coffee, sandwiches, and cookies in the church basement after the funeral, so she had to begin preparations an hour before the service. My father wanted to get to the church early because Alma Stoddard had asked him to be a pallbearer, and he had agreed. I dropped my parents off at First Lutheran, so when it was time to go to the actual services, I had the car to myself, and, according to our arrangements, I picked Marie Ryan up on a street corner between the high school and the church.\n\nWhen she climbed into the car, I asked her if she had been excused from school.\n\nShe shook her head. \"I just left.\"\n\n\"So your afternoon classes\u2014those will be unexcused absences?\"\n\nMarie shrugged. \"I'll tell Mr. Fedder where I was. He'll understand.\" Vernon Fedder was the school's vice principal and in charge of attendance and discipline.\n\n\"What if they call your house?\"\n\n\"They almost never call if you miss in the afternoon,\" she said, and then turned to look at me. \"Don't you want me to go with you? Is that what this is all about? Because if you don't, you can let me out and I'll walk. But I'm going to that funeral. I told Gene I'd be there, and I will be.\" And she was loyal. How could I have forgotten to mention that? Loving and loyal.\n\n\"I just don't want you to get into trouble.\"\n\n\"Don't worry about it.\"\n\nI couldn't figure out why she didn't share my concerns about the possible consequences of her attending the funeral, especially since in doing so she was defying her parents and breaking school rules. (No one could be excused from school without a written note from a parent, and few parents, and certainly not Marie's, wanted their children at a service for a murderer.) Once we arrived at the church, however, I understood.\n\nTogether Marie and I walked up the steps and entered the church, but then she left my side and hurried up the aisle to where Gene sat with his mother and his sister. Marie slid into the pew right behind the Stoddards, leaned forward, and rested her hand on Gene's shoulder.\n\nI trailed after Marie and seated myself next to her. My mother, coming up from the basement, soon appeared at the side door at the front of the church, and she also entered our pew. Before my mother sat down next to Marie, she bent over Alma Stoddard, whispered something to her and, through the net of Alma's veil, kissed the new widow on the cheek. My father sat a few rows back with the other pallbearers.\n\nThe service was sparsely attended. I could count the mourners\u2014no more than twenty\u2014and most of them I could identify as well from our neighborhood. Mrs. Holan and Mrs. Gustafson (but without their husbands); Mr. and Mrs. Hendricks from the newer brick house on Divide Avenue across from us; Judy Neville, Marcia Stoddard's best friend in junior high and high school. There was also a tall balding man with a bulbous nose whom I took to be Raymond Stoddard's coworker or perhaps even his boss; Earl Shumate, Alma's brother, and her sister. Raymond Stoddard also had a brother and a sister but neither was present. Among the few faces I wasn't sure of were two men I believed had been at our house on the evening of the day of the murder. One I thought was a newspaper reporter, the other a police officer or sheriff's deputy, and unless they were both going to act as pallbearers, it looked as though Raymond Stoddard would come up short of the traditional six.\n\nThe police or the press couldn't have been present for any reason but to satisfy their own curiosity. There was no crime that required further investigation, and the media had quickly run out of things to write and say after the facts had been reported and Monty Burnham's life had been celebrated.\n\nIndeed, the assassination seemed especially troublesome to journalists. They wanted to continue to write and report on the incident\u2014interest remained high, people couldn't stop talking about it, and, best or worst of all, depending on your point of view, the murder had brought attention to Bismarck and North Dakota. But what were the media supposed to _say_? They had chronicled Burnham's life, emphasizing the small-town-boy-rises-to-prominence story line; offered testimonials from friends, colleagues, and constituents; and played over and over the film clip (or displayed an accompanying still photograph) of Senator Burnham heartily welcoming presidential hopeful Richard Nixon to the state in the campaign summer of 1960. Burnham personally placed a Sioux headdress on Nixon's head and handed the candidate a bag of dried buffalo chips \"so he'd have something to fling at the Democrats when they started throwing their bull . . . manure his way.\" Nixon was a good sport, but you could tell he wanted to shed the headdress at the earliest opportunity. Another media favorite was a photograph of Monty Burnham in midair. As a high school senior he took first place in the broad jump in the state track meet, and the picture caught him during one of his prize-winning leaps, arms and legs akimbo, and his mouth wide open as if he were shouting the commentary to accompany his own feat.\n\nWhat seemed to frustrate the media was that they could not close off Monty Burnham's life with a theme. If only they could have written or said that Monty Burnham had died for a cause, but that plainly wasn't so, and it wasn't possible to make someone a martyr to senselessness or enigmatic violence. (In their efforts to make Monty Burnham one of North Dakota's distinctive citizens, they might have pointed out how rare he truly was: In 1960 there had been only six murder victims out of the state's population of 640,000. I looked up that statistic quite recently.) Journalists were no different from the people for whom they printed their newspapers and magazines, or broadcast their reports and features. What else was to be done about an event obviously momentous but to which the response ultimately seemed to be little more than a throwing up of hands?\n\nRaymond Stoddard's life and behavior might have presented a potentially greater problem, but since he wasn't famous, it was much easier to summarize him with relatively few words. In this regard, Pastor Lundgren had the advantage. He could follow the standard funeral service, take refuge in the platitudes of religion and scripture, and never refer to Raymond Stoddard in any but the most general outlines. And I don't mean to mock what the minister said. Alma, Marcia, and Gene Stoddard and most of those in attendance probably took comfort in his words. I know I did. It would be a few years before I lost my religion, and I was relieved to hear that Raymond Stoddard, no matter that he had committed murder, was not disqualified from entry into heaven.\n\nYet it was as a murderer that I kept trying to think of him. Since we were confined to those hard pews for an hour, and assigned, so to speak, to reflect on Raymond Stoddard's life, I scavenged my memories for anything that, with the benefit of hindsight, I could now point to and say, yes, there it was, murder's earliest sign. Raymond Stoddard was a murderer for only an hour of his time on earth, but that deed's stain immediately seeped backward throughout his existence. He would be defined forever after as a murderer, but that identity must have shown up before his final day.\n\nYet for the life of me, I couldn't see him as anything but one of the standard fathers of my boyhood, less outgoing than some, more generous than others, a father who took his turn giving us rides to the movies or picking us up after the basketball games; who gave his son money to buy ice cream at the Dairy-O or a hamburger at Jack Lyon's; who taught Gene how to throw a spiral or oil his baseball glove; who borrowed money to send his daughter to college; who mowed his grass in summer, shoveled his walk in winter, put up his storm windows in autumn, and turned over his garden's soil in spring. I couldn't think of a single example of behavior tinged with cruelty, much less evil, or any instance of a violently insane act.\n\nThe best I could come up with was an incident that occurred a couple years earlier. For much of August, Mrs. Stoddard had been out of town, gone to Fargo to help an elderly aunt recovering from surgery. In her absence Mr. Stoddard was in charge of preparing the meals for himself and Gene, and for Marcia on the few occasions when she was home. One evening Gene invited me to join them for supper, and I accepted.\n\nOn the menu were tomatoes and corn on the cob, both products of the Stoddard garden, pork and beans, and fish\u2014walleye caught in a Canadian lake and given to Mr. Stoddard by a coworker. If the fare wasn't particularly notable, the method of preparation was.\n\nAt the edge of their lawn, Mr. Stoddard dug a shallow pit and bordered it with stones. From the garage he brought out scrap wood, a small stack of lath, and blocks of two-by-four. He arranged those in the pit with some newspaper for kindling and squirted on a little lighter fluid. He lit the pile, and over the blaze placed a wire rack. The Stoddards, like everyone else in the neighborhood, had a conventional charcoal grill for cooking outdoors, but Mr. Stoddard insisted on this open fire. It was, he said, exactly how his father used to fry freshly caught fish when the family spent their summers in a cabin on the bank of Lake Liana.\n\nMr. Stoddard seemed uncharacteristically lighthearted as he built the fire and fixed the meal, his mood perhaps attributable to what was in the glass he sipped from as he worked. He made a special effort to show us the steps of preparing the fish. He dipped the long white filets in egg and milk, and then dredged them with a mixture of crushed soda crackers, corn flakes, and bread crumbs. He had placed an iron skillet with a half inch of oil on the rack over the fire, and when the oil was heated to bubbling, he dropped in the slabs of walleye. While he cooked and moved back and forth from the kitchen to the makeshift grill, he kept up an almost steady stream of talk, behavior as unusual as his mood. He chattered about how his father used to time himself\u2014with his railroad watch\u2014to see how quickly he could have the freshly caught fish transferred from the lake to the frying pan, and how the entire Stoddard family would get in on the game\u2014calling out the time when their father pulled his boat up to the dock and then continuing to shout out the elapsed minutes.\n\nNext door to the Stoddards lived Bill and Mary McCutcheon and their three children, a family fairly new to Keogh Street. While we were watching the fish fry, Mr. McCutcheon walked over from his yard to the Stoddards'.\n\n\"Your grill rust through or something?\" Bill McCutcheon asked. \"I would've loaned you ours.\"\n\nFrom the way he glowered at the fire and hesitated before answering, it was apparent Mr. Stoddard didn't welcome the question or the offer. \"Just thought I'd try an open fire,\" he said.\n\n\"Using a frying pan isn't exactly the same, is it?\"\n\nMr. Stoddard poked at the fish but said nothing. The smoke from the fire and the frying fish mingled in a blue-gray cloud, and when the wind blew it in Mr. McCutcheon's direction, he waved it away with an exaggerated motion.\n\n\"That fish come with some guarantee of freshness?\" Mr. McCutcheon asked. \"Because I have to tell you, you can smell it up and down the block, and it sure as hell ain't doing anything for my appetite.\"\n\nAt this remark, Mr. Stoddard rose from his crouch next to the cooking fire and walked away without saying anything about where he was going or why. He soon came back with the spade he had used to dig the fire pit.\n\nStill without explanation, he scooped up a shovelful of the loose dark loam from the pile next to the fire, and he dumped the dirt on fish, pan, grill, and fire. The oil ceased its spattering with a final choking hiss, and the smoke plumed out to the sides. Raymond Stoddard kept shoveling dirt onto the fire until it was extinguished and only the frying pan's handle stuck up from the tiny burial mound.\n\n\"Jesus Christ, Ray,\" Bill McCutcheon said, but I'm not sure Mr. Stoddard heard. When he finished shoveling, he flung the spade into the garden and marched toward the house. Gene and I followed him, and once we were all in the kitchen, Mr. Stoddard, his usual dark and doleful demeanor returned, announced that he would boil wieners for our supper.\n\nTo his son's stricken face, Mr. Stoddard said, \"Don't worry. They're the skinless kind you like.\"\n\nThis was the episode I thought back to while Pastor Lundgren intoned his vague and uncertain eulogy. It was, however, exceedingly difficult for me to dwell on Raymond Stoddard's soul when the body of Marie Ryan was so solidly beside me.\n\nShe was sitting close enough\u2014the cotton sleeve of her white blouse brushed the wool sleeve of my suit coat\u2014that I could feel her heat and smell her hair spray and hear her sniffling attempts not to cry too loudly. Ah, but not a single sensual detail\u2014or a page full of them\u2014can adequately convey what it felt like to have her near me! I could only hope that if anyone noticed how I kept trying to take in great gulps of air, they would merely think that I was trying to compose myself. Like the speaker of Robert Frost's great poem \"To Earthward,\" when I was young it didn't take much to stimulate me, and for the span of Raymond Stoddard's funeral, it seemed as though I too \"lived on air \/ That crossed me from sweet things . . .\" That day I would have said that to breathe in the warm essence of Marie Ryan was enough. Forever. That's what I would have said that day. But read the poem for yourself.\n\nIn the car on the way to the cemetery we replicated the seating arrangement from the church. I drove, Marie sat next to me, and my mother sat by the door. My father rode in the hearse, two vehicles ahead of us in our abbreviated procession through the city.\n\nDuring the ride, my mother asked how Gene was holding up. Although her question may have been meant for both of us, Marie and I shared the assumption that Marie was the one who should answer.\n\n\"He hardly talks at all,\" Marie said. \"He just goes around with this look like he's not really there.\"\n\n\"Pastor Lundgren asked me if I thought he should call Gene in for a talk.\"\n\n\"I don't think that would do much.\"\n\n\"A psychologist? I wonder if it might not be better if Gene visited someone like that.\"\n\nMarie shook her head so vigorously that I could feel the movement.\n\n\"I _know_ he wouldn't like that,\" Marie said. \"Mr. Wallich\"\u2014he was a guidance counselor at our high school\u2014\"suggested Gene come see him, and Gene got mad. He said, 'I'm not the crazy one in the family. That was my dad, not me.' \"\n\nI wondered if Gene had said that only to Marie or to Mr. Wallich as well.\n\n\"Is that what he believes,\" my mother asked, \"that his father was insane?\"\n\nMarie turned toward my mother and asked, a note of incredulity in her voice, \"Doesn't _everyone_ think that?\"\n\nOut of the small group of mourners at the church, fewer still made the trip to the cemetery. North Dakotans will sometimes say that \"it's too cold to snow,\" but the day that Raymond Stoddard was buried provided one more example of the falsity of that belief. The sky had been clear that morning but soon turned leaden, and while we stood clenched and shivering around the grave, a light snow began to fall. The flakes were so dry they hardly had weight to find their way to the ground, but a few caught in Marie's hair and remained there until the end of the ceremony.\n\nHardly had Pastor Lundgren finished his final words regarding the certainty of resurrection when Marie fairly ran to Gene and threw her arms around him. When she tilted her head back to look up into his red-rimmed eyes, those same snowflakes that I had watched gather tumbled from her hair.\n\nJust as quickly as she had raced into the embrace, Marie broke free and walked away from the small gathering.\n\nI had never been to a graveside service before, and its sudden conclusion\u2014as well as Marie's departure\u2014caught me off guard. I was still dwelling on other matters of weight and weightlessness\u2014weren't we going to witness Raymond Stoddard's coffin lowered into the earth?\n\nBut I soon collected myself and ran after Marie.\n\n\"Where are you off to? Aren't you coming back to the church?\"\n\nShe shook her head and swiped the tears from her cheeks. \"I have to get home.\"\n\n\"I'll give you a ride.\"\n\n\"I don't mind walking.\"\n\n\"It's a long way. Just wait a minute. I'll tell my mother, and she can ride with someone else.\"\n\n\"I _want_ to walk.\" Her jaw was set determinedly. \"Thanks for bringing me today.\" And with that she was off.\n\nIt didn't take me long to decide what to do. I ran back to my mother, gave her the keys to our car, and told her I was going to walk Marie Ryan home. \"Wait,\" she said. \"How far is it? You're not dressed warm enough. . . .\"\n\nBut I was already on the move and making my own heat, sprinting after Marie.\n\nI caught up to her as she was exiting the cemetery gates, and she seemed unsurprised to see me. When I paused to catch my breath, she didn't stop but only slowed and, walking backward, said, as if we had been discussing this matter all along, \"Did you know Mrs. Stoddard had to pay extra to have the ground thawed?\"\n\nMy puzzlement must have shown.\n\n\"To bury him,\" Marie said. \"When the ground's frozen, it has to be thawed. Otherwise they store the body until spring. So of course people will pay. No one likes to think of their loved one stuck in a freezer.\"\n\n\"Is that what it is\u2014a freezer?\"\n\n\"I assume. I mean, it has to be someplace cold.\"\n\n\"You'd think it would be storage you'd pay extra for.\"\n\n\"Maybe you do. They probably charge you more no matter what. They're such crooks. Anything to get their money . . .\" Marie was walking so quickly I had to lengthen my stride to keep up. \"Did Gene tell you about the coffin they ended up buying?\"\n\n\"No.\"\n\n\"He was excited about it because it's almost the same color as their Ford. My God, if I hear another word about that car, I swear I'll scream. Gene keeps talking about how he's sure he'll get the car anytime he wants now.\"\n\n\"Yeah. He said the same thing to me.\"\n\n\"Did he? I mean, I know he's having a hard time, and he's just trying to find something good in all this, but if he says that around the wrong people, they're liable to misunderstand and think he's happy his dad died.\"\n\nHer remark pleased me because she apparently thought she and I belonged in the same category\u2014people who understood, and although I knew it was concern for Gene that was truly supposed to unite us, I didn't care. I would welcome anything\u2014a funeral, a friend\u2014that brought Marie Ryan and me closer.\n\nMy mother was right. I wasn't dressed warmly enough, and Marie's house was miles away. I could hardly complain, however, since Marie's winter coat was thinner than mine, and under it she had only a cotton blouse, while I had the additional layer of a suit coat.\n\nWe walked mostly in silence, and not only because I was tongue-tied in Marie's presence or because funerals push everyone deeper into their own thoughts. Theories abound on the reasons for Midwestern taciturnity, but anyone who has spent time outdoors on a sub-zero day knows that at some point speech becomes physically difficult. The hinges of the jaw stiffen, the nose becomes clotted with mucus, and real effort is required to form words and propel them toward another person. Even then every utterance comes out cloaked in its little cloud of steam, a visual reminder of how difficult clear communication can be.\n\nNevertheless, some signal must have been transmitted between us because when Marie and I came within a block of her home, we both began to run.\n\nWe entered through a garage door and then into the house, but I didn't get far. A short flight of stairs led up to the kitchen, but Marie stopped on the first step, turned, and in a whisper thanked me again. It was all so abrupt that I knew I was not being invited to stay, something I certainly wanted, not only because it would have given me more time with Marie but also because it would have afforded me a chance to warm up before the walk home, still another mile.\n\n\"Anytime,\" I said breezily. \"Feel free to call whenever there's a funeral you don't want to miss.\"\n\nShe scowled and put a finger to her lips. I had forgotten that her parents had forbidden her to attend the funeral.\n\nI turned to go, but Marie put her hand on my shoulder to stay me for a moment. \"Don't ask Gene what he saw when he went into their garage. Don't ever ask him. Trust me\u2014you don't want to know.\" She didn't say so, but her warning made it clear: Gene had told her what he saw.\n\nShe released me, and I left the darkened entryway for the dim garage and then the barely brighter day.\n\nI walked a long way before the sensation of being cold replaced the tingling in my shoulder where Marie had touched me. She had touched Gene's shoulder in the church . . . and when she stood on the step above me, she had been standing, I knew, right where Gene often kissed her good night. He was willing to tell me about every increment in the physicality of their relationship\u2014kisses, their variance in number and kind, where and when he touched her and for how long\u2014and he had said that Marie especially favored kissing just inside her door, where that step made them nearly equal in height. . . . I didn't want to be Gene Stoddard, I didn't\u2014no matter that she had put her hand on his shoulder and mine, had stood on that step. . . . But I wore glasses and he did not; I had tonsils and he did not; while he stood at his father's grave, I was walking at his girlfriend's side. His girlfriend. _His._\n\nIn time, I would be invited inside Marie's home, and I would learn that the reason she was reluctant to allow me\u2014to allow anyone\u2014inside was that her mother was an alcoholic. When Marie entered the house on the day of the funeral, she immediately recognized the signs\u2014no lamps burning on an overcast day, dolorous piano music coming from the living room\u2014and knew her mother was drunk.\n\nAnd eventually I would see for myself the process by which the earth was thawed for winter interment. When I, like so many other Americans, took up jogging, one of my preferred routes was through a cemetery in the city I lived in at the time. Oak Lawn Memorial Gardens had flat paved paths, little or no traffic, and enough shade to be slightly cooler in summer than the surrounding city streets. And when I ran there in winter, I occasionally saw placed on the ground a fire-blackened metal hood, shaped like a large coffin lid, and inside a hole on one end of the dome, something like a large blowtorch was inserted, its flame heating the ground until the backhoe could dig it to grave depth. Later still I had a friend who dug wells for a living, and he informed me that in some snowless winters the frost line can go so deep that every burial must wait for spring\u2014ice's triumph over fire.\n\nBoth my parents asked to be cremated, and I often wondered if their decisions originated on that January day when the earth had to be heated to receive Raymond Stoddard.\n\nAs I walked home from Marie's, I passed the Stoddards' and saw their Ford, lightly sprinkled with snow, in the driveway. Were the day's rituals now complete, and all the Stoddards returned to Keogh Street? Or had Gene or Marcia left early and returned home? I didn't speculate for long on why the car was there, because its color commanded my attention. Yes, I supposed its dark blue, a shade just short of black, was the same color as the coffin. And without question, Raymond Stoddard had loved that vehicle. How often I had seen him in the driveway, carefully washing, drying, and waxing the car, jobs that other fathers often entrusted to sons and daughters. You probably remember as well as I the media reports of a man who was buried in his beloved car. Raymond Stoddard was ahead of his time. I never rode in that Ford again without thinking, I'm climbing into the coffinmobile.\n\nThen I crossed the street and soon was in front of my own home with its warmth and normalcy. Yet in spite of the fact that I was so cold I could barely feel my toes, and my shoulders were hunched so tightly to my neck they felt as though they might never come down, I would have turned around gladly and walked back to Marie's and, risking frostbite, circled her block endlessly if I had thought there was a chance she might see me and ask me in. It's tempting to say that that's what it is to live in love\u2014the willingness to sacrifice even physical comfort just to be near the object of one's love\u2014but I was probably experiencing an even more accurate definition: Love is the willingness to turn away from one's home.\n\nMonty Burnham's funeral was held the next day, and according to my uncle, it was an event as grand as any in Wembley's history.\n\nThe town's florist could not handle all the orders that came in, and the hotel and two motels didn't have enough rooms for all those who traveled to Wembley to pay their last respects. Schools and businesses were closed on the afternoon of the funeral, and Good Shepherd Methodist Church was literally filled to overflowing. Once there was no longer any place to sit in the pews or even to stand in the back or along the side, folding chairs were set up in the church basement, and the service was piped down there over loudspeakers.\n\nAmong the congregants were three members of the United States House of Representatives, a U.S. senator, the under secretary of agriculture, and two former governors of North Dakota. Roger Maris, a Fargo native and the New York Yankee who would break Babe Ruth's single-season home-run record before the year was out, sent condolences, a brief message that the minister read during the service along with the announcement that a Wembley grade school would be renamed. Washington Elementary School would become Monty Burnham Elementary. A scholarship would be established in his name for a student majoring in political science at one of the state's universities, and if the funding came through for a new gym for the high school, it was agreed it would bear Monty Burnham's name.\n\nIf the dead could be embarrassed by excessive displays of veneration and grief, Uncle Burt said, almost any corpse would have blushed over how Wembley turned itself inside out over Monty Burnham. Then again, Monty might have been the only man who, could he have witnessed his own funeral, would have thought the whole affair fell a little short of what he deserved. Uncle Burt also advanced the theory that if Raymond Stoddard had not disrupted the natural course of things, Monty Burnham might have eventually revealed his true self, got caught in some political scandal, and died in disgrace. And Wembley children could have kept walking through the doors of Washington Elementary School.\n\nAll of this my father reported again to my mother and me after the nightly phone call with his brother, and I guess I had been brought in on enough of those conversations that I finally felt I could ask a question of my own.\n\n\"Did you dislike Senator Burnham as much as Uncle Burt did?\"\n\nMy father seemed willing to repeat all his brother's remarks, no matter how tasteless or mordant, but now that he had been asked to offer his own view, his characteristic caution returned. \"I had very few dealings with the man. Not back in Wembley and not in the years since.\"\n\n\"But did you _hate_ him?\"\n\n\"He wasn't our kind of people, let me just say that.\"\n\n_Our?_ Did that pronoun refer to him and his brother? Were my mother and I included? Could it have meant my father's and Raymond Stoddard's?\n\n\"Why not? What kind was he?\" I have often thought that I would have gotten much further had I asked, _whose_ kind was he?\n\nAs I recall, my father once again subjected me to a long gaze of assessment. He was uncertain about my ability to understand, yes, but I believe he was also unsure about whether he could truly articulate his thoughts and feelings about Monty Burnham.\n\n\"He was dishonest.\"\n\n\"You mean like a crook?\" It didn't take long for my imagination to fashion a fantasy\u2014yes, yes, Monty Burnham was a crooked politician; he was stealing money that should have gone to schools or hospitals or to the poor. Raymond Stoddard somehow discovered what Burnham was doing, yet he couldn't get the authorities to believe him, and the only way to stop the senator was to\u2014\n\n\"A crook?\" My father shook his head. \"I don't believe Monty Burnham was a crook. The man had all the money he needed and then some. He had no reason to steal. Besides, money wasn't what mattered to him. Fame\u2014that's what he was out for. No, that's not the sort of dishonesty I'm talking about. I mean, he was the kind of man who had no regard for the world as it is. He had his way of seeing the world, and he had no doubt about the rightness of his view. Monty Burnham, like most politicians, wasn't burdened with doubt. So he went right ahead and said, in so many words, 'This is the way things are.' \"\n\n\"So he was a phony?\" If there is anything teenagers understand, it's hypocrisy, and no failing is greater.\n\n\"A phony?\" My father ran his fingers through his hair in frustration. \"Not exactly. I mean, maybe that's part of it. But only a small part of it. Men like Monty Burnham\u2014if they have their way, we'll never get at the truth of things in this world. And they won't mind one damn bit.\"\n\nBy now I thought I understood what my father was talking about. \"He was a liar?\"\n\nBut that remark only brought home to my father how wide the gulf was between what he said and what his son comprehended. He shook his head and winced. It was a characteristic expression for him\u2014my father felt frustration as pain\u2014and I've seen that same look\u2014you have too\u2014on Humphrey Bogart's face in movies, and just as with my father, it was usually accompanied by a death-deep drag on a cigarette.\n\n\"I'm not explaining this very well. He wasn't necessarily lying. He just didn't give a damn that what he said might not be true. He probably believed that as long as his heart was in the right place, it would be okay.\"\n\n\"And his heart\u2014\"\n\nBy this time my mother took pity on both of us, but her interruption was also in the interest of keeping the record straight. \"We might not have cared for Monty Burnham's type, dear, but that doesn't mean we wished him ill. Besides\"\u2014now she addressed my father\u2014\"if you still lived in Wembley, you probably would have voted for him, wouldn't you?\"\n\nMy father might have disapproved of Monty Burnham's character, but that didn't mean he could bring himself to vote for a Democrat. \"Probably,\" he said, and walked from the room.\n\nMarie didn't warn me not to ask my father about what _he_ saw when he opened the door to the Stoddards' garage, but she didn't have to. I loved my father and he loved me, but nothing in our relationship furnished the kind of intimacy that would have to exist before I could ask, What did Raymond Stoddard look like when he was hanging from the crossbeam? I don't doubt that there are sons in the world who could pose such a question and fathers who could answer, but we weren't of that breed.\n\nNevertheless, my father did eventually volunteer information about Raymond Stoddard's appearance in death, though strictly speaking, my father told us\u2014my mother and me\u2014about something he _didn't_ see.\n\nPeople who hang themselves usually die slowly\u2014they choke to death\u2014and they have time to reconsider their action and attempt to undo it. Almost always their hands exhibit the signs of their self-doubt, the palms and fingers raw with friction burns from grasping at the rope, cord, string, line, or wire that is cutting off their oxygen supply. Raymond Stoddard's hands were unmarked, evidence of a mind undeterred from its purpose and able to overpower the self-preserving impulses of the body.\n\nMy father came into possession of this arcanum of the self-destructive through his friendship with the Bismarck police detective who had been at the funeral (my guess had been right) and at our house the night of the murder-suicide. Soon, in fact, this new friend, Lee Mauer, was a regular visitor in our home, and even without knowing that some men were our kind while others were not, I knew that Mr. Mauer was not. He was a big, beefy, barrel-chested man, vulgar, loud, and profane and about as unlike my father as another man could be. They had curiosity in common, however, and each believed the other might possess some information that could solve the mystery of Raymond Stoddard's motive.\n\nInitially, Lee Mauer came to our house mostly by invitation. My parents felt sorry for him because he lived alone\u2014he was divorced and his wife and two children lived in Casper, Wyoming\u2014and my mother believed that what was most lacking in his life was home cooking. He came often for Sunday dinner, our most formal, elaborate meal of the week, and he ate so heartily my mother had to increase significantly the size of portions to accommodate Mr. Mauer's appetite. Furthermore, she had to counsel her children to overlook Mr. Mauer's poor table manners and his inappropriate talk. Her warnings were futile; almost everything Lee Mauer said\u2014and how he said it\u2014I found fascinating.\n\nHe had difficulty getting through a sentence without a profanity, something he was aware of, because he told my sister and me that we should be careful that we didn't develop the \"bad speech habits\" that he couldn't rid himself of. Worse, in my mother's view, was his willingness to bring up topics unfit for the dinner table or for children. For that matter, some subjects were probably matters of police confidentiality and shouldn't have been discussed in anyone's presence.\n\nHe said, for example, that he suspected a Bismarck politician of driving down to South Dakota in order to visit prostitutes, and although he didn't give the man's name, Lee Mauer implied that he was hoarding the man's identity for possible future use. A name he _was_ willing to mention was a local doctor's, a man Lee Mauer said was nothing better than a drug pusher. Lee Mauer's ex-wife and her friends had become virtual drug addicts, hooked on the sedatives and painkillers that the doctor prescribed. Lee Mauer told us that the police were keeping their eye on a well-known local athlete, a recent graduate of Bismarck High School, because they suspected him of stealing cars. Before any of my friends knew that the son of a sheriff's deputy had been arrested for selling liquor to minors, I acquired that information courtesy of Lee Mauer.\n\nHis gossip favored the lurid, the gruesome, the dangerous, and the scandalous, and all those qualities were enhanced by his side-of-the-mouth confidential style. One could easily imagine him passing on a bit of intelligence about the hands of a hanged man. In fact, if there were ever any doubt about his willingness to discuss any disaster in detail, it was dispelled completely when he told about the death of two young people in a motorcycle accident the previous summer. It happened in Bismarck's Riverside Park, and Lee Mauer happened to be in the park that evening and so hurried to the crash scene to see if he could be of help. \"Split in two that young fellow was,\" Lee Mauer said, \"from crotch to neck. Like he was a goddamn wishbone for a couple of giants. I've never seen anything like it.\"\n\nTales like that, coupled with Mr. Mauer's general crudeness, finally alienated my mother completely, and she stopped inviting him for meals.\n\nBut he kept coming, usually in the late evening. No doubt he came out of loneliness, but since he couldn't admit that, he always had an excuse for stopping by. He was investigating the report of a break-in nearby or he had to check out a suspicious vehicle parked on a neighborhood street. Even when he said he had been driving past, saw a light, and decided to stop, he made it sound as though that too was official business. Because he had learned that we didn't have much in the way of liquor in the house, he often brought with him a six-pack of Grain Belt Premium beer or a pint of schnapps or blackberry brandy. My father would occasionally drink a beer, but he seldom indulged in anything stronger, and certainly none of Lee Mauer's sweet liquors.\n\nThrough Lee Mauer I first learned the lesson\u2014the one I had to learn and relearn over the years and which I still haven't mastered\u2014that even (or perhaps especially) the motives of others are often understood by reference back to the self.\n\nLee Mauer grumbled constantly about politicians, \"those greedy sonsabitches,\" with their unconcern for everything but their own advancement. \"They don't give a good goddamn who they step on.\" And Lee Mauer could voice those sentiments over and over and each time act as though the emotions were newly minted. His voice would rise to a brassy pitch that rang with anger. His features would bunch and his face would redden. He'd pound the table with his big square fist. There, one couldn't keep from thinking, was the kind of rage that would drive a man to murder.\n\nFurthermore, Lee Mauer fixed his political anger on an issue, and there, he believed, his emotions and Raymond Stoddard's coincided. The 1961 North Dakota legislature, as one of its first orders of business, voted itself both a raise and a cost of living increase, yet a 4 percent wage increase proposed for the state's classified and unclassified employees was voted down. This inequity was worsened, in some people's view, by public statements by legislators, such as Monty Burnham, who cheerfully professed not to have known that their salary hike had been attached to a particular bill. Lee Mauer had no trouble imagining that a disgruntled state worker\u2014like Raymond Stoddard\u2014could take out his anger and resentment on a legislator\u2014like Monty Burnham\u2014who had enriched himself while denying the clerks, secretaries, bureaucrats, and administrators their share of the state's resources.\n\nMy father didn't argue with Lee Mauer, but he didn't buy the policeman's theory and instead saw it stemming from Lee Mauer's own job dissatisfaction. Lee Mauer held the rank of lieutenant with the Bismarck police department, yet he felt continually slighted in the workplace and complained constantly of the \"ass-kissers and brownnosers\" who surpassed him in rank, salary, or privilege. My father knew all this, but merely shook his head and to my mother and me said, \"Lee just doesn't get it. It's exactly his griping that makes trouble for him. The police department is like any organization. They want the good team players, not someone who's poisoning the well.\"\n\nThis judgment my father expressed with more than a touch of disapproval. In his view men and women were supposed to work hard and keep their complaints to themselves. It's worth noting, however, that he always practiced law alone, in spite of offers to work for various firms, companies, and institutions. He must have known his nature well enough to realize that he would be a saner, if poorer, man if he kept to a minimum the number of people with power and authority over him. My father's son, I earn my living as a novelist, an occupation as solitary as any.\n\nBut you must decide for yourself. Have you had experiences that would lead you to believe a man's perceptions of injustice in the workplace could move him to murder? In 1961 the term \"going postal\" wasn't part of our cultural vocabulary, but language often lags behind deeds.\n\nWhen I recall my father and Lee Mauer sitting in our darkened kitchen nightly, their white shirts the only brightness in the room, a brimming ashtray between them, each man's hand curled lightly around a can of beer, it seems as if they turned the room into their private enclave, taking up almost all the space in the room and forcing anyone who wanted to enter to wedge themselves in and then slide along the walls. Perhaps I have that impression because my mother assiduously avoided the kitchen when the men were in there, or perhaps it comes from Lee Mauer sometimes obviously and deliberately falling silent when someone came in. But I hung around the edges as often as I could, and on one of those occasions I overheard a Lee Mauer remark that was so unsettling it haunted me for years. I finally exorcised its effect with a narrative of my own.\n\n_In December 1942, when she was nineteen years old, Alma_ _Stoddard had a series of experiences that made her feel as_ _though she were living in a world turned upside down. She left_ _Wembley, North Dakota, her little town near the top of the_ _country, for a visit to a city on the bottom\u2014Killeen, Texas\u2014and_ _when she left North Dakota in December the weather was uncharacteristically_ _mild. Temperatures were in the fifties, and_ _there was no snow. Texas, she expected, would be hot. After all,_ _the letters she received from her husband were often filled with_ _complaints about the heat, yet when she stepped off the bus in_ _Killeen, a good three inches of wet snow covered the ground,_ _and a cold northwest wind blew so hard that tears sprang to her_ _eyes. As it turned out, the weather was the least puzzling feature_ _of her visit._\n\n_She had traveled to Texas to see her husband, Raymond,_ _who was stationed at Camp Hood. He would soon be sent overseas,_ _and since they were both realistic people who were not automatically_ _given to optimistic assumptions about the future,_ _they knew they might not have another chance to be together_ _again for a long time, perhaps ever again. As it was, they had already_ _scaled back on their plans. For a while they had discussed_ _the possibility of Alma moving to Texas but had finally decided_ _that would be impractical. By the time she found a place to live_ _and work, Raymond might be relocated, and then Alma would_ _have given up her job at Hudson's Pharmacy back in Wembley_ _and her rent-free home with her mother._\n\n_There had been other compromises. For a time it seemed_ _possible that Raymond would be able to come home to North_ _Dakota for an extended leave. But then all furloughs in his unit_ _were canceled, and the best the Stoddards could arrange was a_ _three-day weekend in a Texas hotel whose lobby was decorated_ _with a dried-out pine tree sparsely decorated with a few glass_ _bulbs._\n\n_Alma was sitting on an overstuffed chair near that tree when_ _she saw something more surprising\u2014and more unsettling\u2014than_ _Texas snow. She was waiting for Raymond to meet her, but the_ _first man in uniform to walk through the glass door was not her_ _husband but Monty Burnham, a young man she had dated in_ _high school._\n\n_Then, as if to demonstrate that human relationships had_ _been turned as upside down as the country's climate, Monty_ _Burnham greeted her with the words \"Alma! There's my girl!\"_ _And he said this within earshot of the man who came through_ _the revolving door right behind him\u2014her husband._\n\n_She had known, from Raymond's letters, that Monty Burnham_ _was also stationed at Camp Hood, but she didn't think the_ _men spent much time in each other's company. Her husband_ _hadn't said anything more than that he \"saw Monty and Dusty_ _Boyd, and a few other guys from back home, around the_ _camp.\"_\n\n_The men took turns hugging her, and because she had spent_ _more time in Monty Burnham's arms over the years than in_ _Raymond's, both embraces felt familiar. Alma had been Monty's_ _steady girlfriend from the age of fifteen to the age of seventeen._ _She and Raymond had been together\u2014dating, engagement, and_ _marriage combined\u2014for only a little more than a year. Both_ _men smelled of tobacco, damp wool, and hair tonic, but she_ _gasped when she felt how thin and bony Raymond seemed. She_ _hoped that he would think her response came only from emotion._ _And perhaps her shock was merely a consequence, now as_ _in the past, of having first had her arms around Monty's more_ _substantial girth._\n\n_Even though Monty Burnham was standing right there,_ _Alma asked her husband, \"Do you want to see the room?\" In_ _one of their late-night telephone conversations Raymond had_ _been unashamedly, even shockingly, specific about what they_ _would do when they were together once again._\n\n_Now, however, Raymond said, \"Later's fine.\" Then he_ _asked, \"Did you have to show your marriage license?\"_\n\n_Raymond had repeatedly told her to bring it with her, that_ _she might need it to check into the hotel, but she had still forgotten_ _it at home. Fortunately, she had not been asked for it._ _She shook her head._\n\n_\"Because you look so respectable,\" Monty said. She_ _couldn't tell if there was any irony in his remark. \"No one_ _would ever expect anything untoward of a woman who looks_ _like you.\"_\n\n_Raymond said, \"Monty's going to join us tonight. He knows_ _his way around town, and he can keep us moving in the right_ _direction.\" He said this without enthusiasm, and for the first_ _time Alma considered the possibility that Monty Burnham had_ _somehow forced his company upon Raymond. She knew little_ _about Army rank, its designations and privileges, but the differences_ _between Monty's uniform and Raymond's were obvious._ _Perhaps Monty was there as the result of a military command_ _that Raymond had to obey. Her husband's sullen remoteness_ _could certainly be explained as the behavior of a man who had_ _been forced into his circumstances. Alma wondered if she_ _should say she wasn't feeling well enough to go out; perhaps_ _that would send Monty on his way and allow her and Raymond_ _to be alone. She kept quiet, however, because she felt there was_ _a chance that Raymond might then choose to spend the evening_ _out on the town with his comrade._\n\n_\"Monty's got a girl who's going to meet us later.\"_\n\n_To Alma, Monty explained, \"She's working now.\"_\n\n_Alma brightened at this news. \"Oh, where does she work?\"_\n\n_\"At a laundry. She's off at five, but she has to clean up and_ _change clothes. She works up such a sweat she has to sit under a_ _cold shower for damn near an hour to cool off.\"_\n\n_She wanted to encourage Monty to talk about his girl._ _\"Does this young woman have a name?\"_\n\n_\"Dinah. And she's older than us.\"_\n\n_\"That's a pretty name. Do I see signs you might be serious_ _about Dinah?\"_\n\n_Monty laughed. \"Not likely! She's what soldiers call a diversion.\"_\n\n_Whatever term soldiers had for a woman like Dinah, she_ _knew it wasn't \"diversion.\" For that matter, she was surprised_ _that the word was part of Monty Burnham's vocabulary. Perhaps_ _Raymond would tell her later what Dinah was more likely_ _to be called._\n\n_\"Anyway, you'll meet her later,\" Monty said. \"But don't get_ _your hopes up.\"_\n\n_\"Monty Burnham! What kind of thing is that to say?\"_\n\n_He merely shrugged._\n\n_Alma turned to Raymond for support, but he was busy_ _looking his wife up and down, an examination Alma thought_ _more appropriate for a stranger\u2014and a rude one at that\u2014to be_ _conducting than a husband. She knew he hadn't seen the dress_ _before; it was a royal-blue cotton print, and she'd bought it at_ _Whitestone's right after Thanksgiving, when it had been on sale_ _because it belonged to a different season. Perfect for Texas in_ _December, Alma had thought, and had justified the purchase on_ _that basis. It clung to her in a way that she thought was just this_ _side of immodest._\n\n_To cover her discomfort under his stare she stepped back_ _and tried to make a joke. \"Well, soldier, do I pass inspection?\"_\n\n_\"I just wondered if you were ready to go, or if you wanted_ _to change.\"_\n\n_\"Isn't this appropriate for . . . where we're going?\"_\n\n_Apparently Monty took it upon himself to answer because_ _he would be their guide for the evening. \"You look great. But_ _bring your coat.\"_\n\n_And then they were off, leaning into the cold Texas wind,_ _threading their way carefully down sidewalks clotted with soldiers_ _and shopgirls. Alma was glad to be in the company of_ _Raymond and Monty. She had never seen so many men publicly_ _intoxicated and behaving so badly. Even a woman walking arm-in-_ _arm with a man was likely to be subjected to crude remarks_ _and propositions._\n\n_Their first stop was a restaurant large enough to be a dining_ _hall, and it too was crowded with soldiers. Casa Robles was unlike_ _any eatery Alma had ever been in. Its walls were painted_ _the color of mustard, brown-skinned waiters scurried about carrying_ _enormous serving trays, and the general din made conversation_ _next to impossible. Alma wanted to suggest they find a_ _quieter place to eat, but before they were even seated at a table,_ _Raymond and Monty ordered beers. Alma rarely drank beer,_ _and they were served a brand she had never heard of. She said_ _she'd prefer Coke._\n\n_Monty shook his head. \"Not with Mexican food. Coke_ _won't put the fire out.\"_\n\n_Raymond agreed. \"Pretty spicy.\"_\n\n_Alma couldn't eat more than a few bites of her meal, but the_ _heat wasn't the problem. She had told the men that she didn't_ _like spicy food, so they had studied the menu carefully, trying to_ _find something mild for her. But the food's textures were all_ _wrong\u2014mealy and soft\u2014and the restaurant made no effort to_ _separate one item from another on her plate. Neither of the men_ _noticed how little she ate._\n\n_Raymond didn't finish his food either. He pushed aside his_ _half-eaten tamales and lit another cigarette. Alma still could not_ _get used to the sight of him smoking. He had taken up the practice_ _when he'd joined the service because, he said in a letter,_ _smokers were given extra breaks. He wrote about his \"new_ _habit\" in a joking way, but there was nothing lighthearted in the_ _grim, resolute way he pulled smoke deep into his lungs._\n\n_They had entered Casa Robles in the full sunlight of afternoon,_ _but when they exited, the streetlamps had all come on._ _This too shocked Alma. So many miles separated Texas from_ _North Dakota\u2014yet December dark came as early to one place_ _as another._\n\n_Monty announced their next destination, and its name_ _sounded more exotic and interesting than it turned out to be._ _The Alhambra was nothing but a bar, small, smoky, and dimly_ _lit to hide its dirt and disrepair. After a couple of drinks they_ _walked to another bar, Vic's Place, but since it was so similar to_ _the one they'd just left, Alma wondered why they had bothered_ _to move. She supposed it might have been to meet Monty's girlfriend,_ _yet he had made no effort to survey either establishment_ _for her presence. He didn't even bother turning to look when_ _the door rattled open with new customers and a breath of cold_ _wind._\n\n_After their second round of drinks was served\u2014the men had_ _switched from beer to rum and Coke\u2014Alma asked, \"What does_ _this girlfriend of yours look like, Monty?\"_\n\n_\"A Mexican. But she's not. And I told you, she's not my_ _girlfriend.\"_\n\n_\"Your date, then. Dinah.\"_\n\n_\"Why do you ask?\"_\n\n_\"That girl over there is up on her tiptoes looking around,_ _and I wondered if she could be Dinah.\"_\n\n_Monty shook his head._\n\n_With his index finger Raymond nudged Alma's drink, a_ _Schenley's and Seven, closer to her. \"You're not drinking,\" he_ _said. \"You want something else?\"_\n\n_\"It's fine.\" She was disappointed that he didn't seem to remember_ _how little taste she had for either liquor or its effects._\n\n_\"You got to drink up, Alma,\" Monty said. His ruddy_ _cheeks, part of his soft, youthful good looks, became even more_ _flushed as he drank. Even in the bar's gloom she could see his_ _ears glowing bright red. \"This is a celebration!\"_\n\n_\"What are we celebrating?\"_\n\n_\"Why, that you made the trip! Welcome to Texas!\"_\n\n_Alma glanced at Raymond to see if Monty spoke for both of_ _them._\n\n_But Monty wasn't finished. \"And I never had a chance to_ _drink to your marriage.\" He raised his glass again._\n\n_\"We didn't have any alcohol at the reception,\" Alma said._\n\n_Raymond raised his own glass but only inches off the table,_ _and he addressed Alma's earlier question. \"Because we haven't_ _shipped out yet,\" he said. \"Because we can still breathe the_ _open air, and we're not trapped in a tank somewhere waiting to_ _get blown all to hell. Not yet.\"_\n\n_\"Jesus Christ, Ray,\" Monty said._\n\n_Alma put a comforting hand on her husband's wrist, but he_ _misunderstood her gesture. He obviously thought she was trying_ _to keep him from bringing his drink to his lips, and he twisted_ _away from her touch._\n\n_\"I'm just saying what we're all thinking.\"_\n\n_\"That doesn't mean you have to say it. It's going to happen_ _or it isn't. No point in dwelling on it.\"_\n\n_Monty's view on this matter was Alma's, too. She dreaded_ _the possibility of spending these days with Raymond and having_ _him fill the hours with talk of all the horrible things that could_ _happen to men in war. She'd prefer they spend every moment_ _together in complete silence rather than have to listen to her_ _husband's hideous, hopeless thoughts. Did that make her a bad_ _wife? She didn't care if it did._\n\n_\"Can we go back to the hotel?\" Alma asked apologetically._ _\"I couldn't sleep much on the bus.\"_\n\n_She directed her request to Raymond, but Monty rose as_ _well. He, in fact, reached into the pocket of his khakis and_ _brought out a money clip with a tight packet of folded bills. He_ _tugged loose a dollar and dropped it carelessly onto the table._ _Alma had never seen a man keep his currency in anything but a_ _billfold, exactly where Monty had kept his during their time together._ _What strange practices men adopted when they left their_ _homes._\n\n_They did not walk directly to the hotel. Monty Burnham insisted_ _on stopping at a liquor store, and Alma and Raymond remained_ _outside on the sidewalk. Alma hoped the frigid air_ _would help sober her husband, who had slipped into a slow-blinking,_ _slack-jawed silence. He looked as though he could fall_ _asleep leaning against the liquor store's brick wall._\n\n_She tugged on the sleeve of his uniform. Without her gloves_ _she could feel how rough was the wool of his jacket. \"Ray._ _Have you ever met this Dinah?\"_\n\n_\"I dunno. Maybe.\"_\n\n_\"Well, have you or haven't you? She was supposed to meet_ _us, but there's no sign of her. I'm beginning to wonder if she's_ _just one of Monty's stories.\"_\n\n_Either her question or his effort to answer her seemed to_ _sober him momentarily. \"You know him better'n I do. He seem_ _like the sort who needs to make up a woman?\"_\n\n_There was no right way for Alma to respond. She stepped_ _away from her husband and looked up the street as if she were_ _the one expecting to meet someone. On a wire above the street_ _hung two slices of tin cut in the shape of Christmas bells. In the_ _wind the tin bells had slid out of place on the wire and were rattling_ _against each other. Why, Alma wondered, had the town_ _not had to donate those tin decorations to the war effort? Back_ _in Wembley the townspeople had already collected enough tin_ _cans to build a tower as tall as a house. She turned back to Raymond,_ _thinking she might put him in an improved humor with_ _this observation about their hometown, but he had already_ _drifted back into his drunken trance._\n\n_Then Monty burst out of the liquor store, his smile wide, his_ _cheeks as red as his ears had been earlier. \"Let's go,\" he said._ _\"Let's show these folks how North Dakotans celebrate!\"_\n\n_But his remark made no sense, since they proceeded directly_ _back to the hotel and to Raymond and Alma Stoddard's room,_ _where, once the door was closed behind them, no one would_ _know whether the people inside were raising their glasses in celebration_ _or hanging their heads in despair._\n\n_Monty pulled the cork on the newly purchased bottle of_ _rum, and using a bottle opener screwed into the wall next to the_ _bathroom sink, he levered open a bottle of Coke. The room had_ _only two glasses, so after mixing drinks for Ray and Alma, he_ _poured rum into the Coke bottle for his own drink. \"I didn't_ _make yours quite so strong,\" he said to Alma. \"As I recall,_ _you'd just as soon not taste the liquor.\"_\n\n_She didn't want another drink at all, but she accepted it_ _without comment or complaint. The day had been so filled with_ _the unexpected that she hadn't even been surprised when Monty_ _followed them to the room and walked right in behind them._\n\n_\"To the newlyweds!\" Monty said._\n\n_\"Hardly,\" Raymond said, and sat down heavily on the edge_ _of the bed, which gave out a stiff-springed rusty groan._\n\n_Alma didn't raise her glass either, but Monty was undeterred._ _\"I can't imagine a couple better than the two of you!\"_\n\n_In order not to hear the demurral that she feared Raymond_ _might make, Alma excused herself, went into the bathroom, and_ _closed the door behind her._\n\n_Alma took out her cosmetics from her overnight case and_ _lined them up on the back of the sink. Some of the makeup was_ _newly purchased, never opened, and she had brought them from_ _North Dakota with a specific plan in mind. In Wembley she_ _would never dare venture out in public wearing much more_ _makeup than a little rouge and lipstick. \"You're pretty enough_ _all on your own,\" her mother used to tell her. \"You don't need_ _that gunk.\" Alma had never doubted that she was pretty, but_ _what she had in mind wasn't simply to make herself more attractive._ _She wanted to transform herself into a woman who_ _would be regarded as\u2014she didn't even know what term men_ _used. Not \"pretty.\" Not \"a diversion.\" Some word or phrase_ _that had more than appreciation in it, that expressed not just a_ _woman's looks but a man's desire. All she could think of were_ _the words from movie magazines. \"Glamorous.\" \"Stunning.\"_ _\"Alluring.\" Not the language that a man would use when a_ _woman with a certain look passed, and he'd nudge his friend_ _and say\u2014Well. Even if Alma couldn't put a word to it, she_ _knew what the look was. And she set about giving it to herself._\n\n_With her rouge she made her cheeks look as if they were_ _flushed with passion. With lipstick she not only darkened her_ _lips, she swelled their outline as well. With her eyelash curler_ _and eyebrow pencil she gave her eyes what she considered a_ _cruel, dramatic look. She believed that no one but Raymond_ _would see her this way. Soon Monty would leave, and then she_ _would walk out of the bathroom, startling her husband with her_ _new glamorous look. She and Raymond would spend most of_ _the remainder of her visit in their hotel room\u2014_ alone _in their_ _hotel room\u2014and if he wanted her to use her makeup to recreate_ _this movie star mien, she would happily do so. Or if he preferred_ _her scrubbed as clean as the morning of their wedding_ _day, that would be fine, too._\n\n_Alma put her ear to the bathroom door. She couldn't hear_ _any drunken voices or clinking glasses or matches being struck._ _She supposed it was possible that Monty had left when she was_ _running water in the sink and therefore she hadn't heard the_ _door open and close, but then why wouldn't Ray call out to her_ _and tell her they were alone? Deciding that she'd simply wait a_ _little longer, she sat down on the edge of the bathtub._\n\n_During this visit Alma had hoped to discuss with Ray a matter_ _that had been weighing heavily on her. She was living in her_ _mother's house, and since Alma was married to a serviceman,_ _she thought it was perfectly legitimate to affix a gold star to her_ _mother's window. Then, just last week, when Raymond's parents_ _learned that Alma would be traveling to Texas, they invited_ _her over for supper and to give her a small package to carry to_ _Ray. As Alma stepped onto the Stoddards' front porch, she saw_ _a gold star shining in the corner of their living room window._\n\n_Was this all right, Alma wanted to ask Ray, two homes and two_ _stars yet only one serviceman? But somehow, in the miles between_ _North Dakota and Texas, the dilemma had lost its importance._ _In fact, if she were to tell Ray about it now, she was_ _certain he would laugh at the foolishness of her concerns._\n\n_At the bathroom door came two soft knocks and Alma_ _jumped up to answer. When she opened the door, however, it_ _was Monty Burnham's face that peered into the room._\n\n_\"Well, I got him tucked in,\" Monty said. \"And he's on his_ _stomach just in case\u2014don't want him gagging on his supper. I_ _put that wastebasket beside the bed, too.\"_\n\n_Monty announced all this as if it had been done according_ _to some plan worked out with Alma in advance of the evening._ _She didn't know what to say but thank you._\n\n_Because Monty was wearing his officer's cap, Alma assumed_ _he was about to leave, but he tilted the cap back on his head,_ _stepped into the bathroom, and reached for a cigarette. He_ _leaned back against the door until it shut softly. The room was_ _tiny, and Alma backed up until her legs touched the cool porcelain_ _of the bathtub._\n\n_Monty blew a cloud of smoke toward the ceiling, and in a_ _voice barely more than a whisper said, \"Old Ray. He's holding_ _on but barely.\"_\n\n_If this was to be a conversation about her husband's state of_ _being, Alma would welcome it. \"I believe he's lost weight.\"_\n\n_\"That ain't the half of it.\" He cocked his head as if the bathroom's_ _bright light allowed him to see Alma in a new way. \"But_ _you\u2014you're looking prettier than ever, Alma. And that's saying_ _something.\"_\n\n_\"Pretty,\" he had said, but she realized that her face was_ _made up in that other way, and she stepped quickly to the sink_ _with the intention of washing immediately._\n\n_Monty must have misunderstood her movement. He came_ _up behind her and put a hand on her shoulder. \"Alma? You_ _okay?\"_\n\n_She nodded but turned on the faucet and began to splash_ _water on her face._\n\n_\"Thought you were gonna be sick or something.\"_\n\n_\"I'm fine.\"_\n\n_Monty didn't move away but started to rub Alma's back, his_ _hand caressing a small circle high between her shoulder blades_ _and on her neck up under her hairline._\n\n_\"When I saw you today . . . ,\" he said, \"well, it just started_ _up all over again. I mean, not that they ever went away, my feelings_ _for you. But Jesus Christ, I was a fool to let you go.\"_\n\n_Let me go? Alma believed she had been the one to walk_ _away from their relationship, fed up with the number of times_ _he'd stood her up because he preferred hunting or fishing with_ _his friends or drinking beer and shaking dice at Sugar's Bar to_ _being with her. But there was no point arguing with him. He_ _obviously had his own version of their breakup._\n\n_\"What makes me so damn mad,\" Monty continued, \"is that_ _I'm a better man now. Stronger, more sure of myself. Now I_ _could put up a helluva fight for you, Alma. A helluva fight.\"_\n\n_He was fairly murmuring these words in her ear, bent over_ _her back, his hands braced on the sink on either side of her._\n\n_\"If I could make Ray straighten up and treat you right, I'd_ _cut the orders tomorrow.\"_\n\n_This remark she might have disputed, but by now Monty's_ _hand was at the front of her throat, softly stroking its length as_ _if he were trying to help her swallow his words._\n\n _It would not have been accurate or honest for Alma to pretend_ _that what happened in that hotel bathroom was completely_ _without her acquiescence. But neither would it have been far_ _from the truth._\n\n_She may have been mistaken in thinking that what was happening_ _was happening in discrete stages, that the process that_ _had been set in motion could easily be halted or altered at anytime,_ _one moment easily snipped away from the moment that_ _preceded or followed it. But Monty's seduction was, like so_ _many events in life, composed of units of action and time linked_ _inextricably to one another._\n\n_And then, of course, what was occurring was not entirely_ _unfamiliar to Alma. During their time together Monty had often_ _touched her here and here, like this and like this, and he had_ _tried to touch her there and there. What was different now was_ _that it was now . . . and Monty was soon doing what Alma_ _didn't know could be done._\n\n_But a woman doesn't allow herself to be fucked standing up_ _and holding on to a bathroom fixture just to satisfy her curiosity_ _or dispel her disbelief. She must have been willing. . . . She_ _must have been. . . ._\n\n_Because she certainly could have stopped it with a scream or_ _a violent twist of her hips, or she could have grabbed his hand_ _and wiped it across her face so he could feel her tears or maybe_ _it would have been enough to lift her head so he could see her_ _reflection in the mirror. . . . As his thrusts increased in force and_ _frequency, the sink's plug, a black rubber stopper hanging from_ _a little chain looped over the spigot, began to move back and_ _forth. Alma watched it closely, and as it swung from side to_ _side, she imagined it was swinging between the poles of her_ _life\u2014North Dakota, Texas, Texas, North Dakota, Raymond,_ _Monty, Monty, Raymond. . . . Even after Monty gasped and_ _staggered back from her, she kept her focus on the chain and the_ _plug. Soon it was motionless again, hanging above a drain circled_ _with a rusty stain from a faucet that must have dripped for_ _years with no one bothering to fix it._\n\n _The following morning, as the first light of a cold Texas dawn_ _found its way into the room, Alma stared up at a blotch on the_ _ceiling. It was about the size of a pumpkin, and its irregular_ _shape was outlined in the same rust-orange color as the stain in_ _the bathroom sink. Only moments before, Raymond had finally_ _roused from his rum-steeped stupor, and now, as she lay under_ _his weight, it was all Alma could do not to interrupt her husband's_ _grunting efforts to ask him if he thought water was leaking_ _in the room above them._\n\nIn the original version of this story, the characters were named Donald and Lois Culpepper and Nick Anschutz. Otherwise the story is exactly as it appears, under the title \"A Mild Winter,\" in my MFA thesis and later in _The Bozeman Review,_ a now discontinued literary magazine once published at Montana State University. I have never included it in any collection.\n\nReaders sometimes ask me where I get my ideas, and I'm seldom able to provide a satisfactory answer, so vague and various are the sites where most of my fiction is born. But if I were asked about \"A Mild Winter,\" I could provide a number of answers, all true and all inadequate, as explanations of creative work so often are.\n\nOn one of the few occasions when Gene, Marie, and I were together in the Stoddard house (after his father's death, Gene seldom invited anyone into his home), we looked through the Stoddard family photo albums. One black-and-white photograph, blown up to an eight-by-ten, depicted a small gathering of casually arranged laughing soldiers standing in fresh snow, and Raymond Stoddard, with his lopsided smile and eyes squinting against the sun, stood in the back of the group. The back of the photo carried the penciled caption \"761st Tank Battalion, Camp Hood, Texas, December '42.\" If Monty Burnham was in the picture, I didn't recognize him.\n\nIn addition to what the photo provided, I had, in the writing of \"A Mild Winter,\" my parents' recollections of life during the war (as I recall I cut a passage of dialogue between \"Donald\" and \"Lois\" about ration cards), a little research (also eliminated was a brief mention of mixing egg yolks with margarine to give it the color of butter), and of course the date and year of Marcia Stoddard's birth. And, more than anything, I had the stimulus\u2014or the irritant\u2014of that Lee Mauer remark I alluded to earlier.\n\nOn a warm summer evening I was sitting on our front porch watching for a group of friends to drive up and take me to an American Legion baseball game. My father and Lee Mauer were in the kitchen and, though they kept their voices low and the window screen sifted out parts of their conversation, I heard enough to know they were talking about, as usual, the why of Raymond Stoddard's last day on earth. Months had passed, and the event had exceeded most people's attention spans but not my father's and Lee Mauer's. Their interest was undiminished, and a new season simply brought new theories, as when Mr. Mauer said, \"That Miss Stoddard\u2014am I the only one who sees a helluva resemblance between her and a certain state senator?\"\n\nJay Garner's Plymouth pulled up to the curb, so I didn't hear my father's response, but I had eavesdropped on the two men often enough to know what pattern their talk would follow. Soon one or both of them would be speculating on what might have happened if it were determined that Marcia Stoddard was Monty Burnham's child.\n\n_And suppose Ray found out?_\n\n_But how could he? After all this time?_\n\n_Same as I did. He looks at the girl one day, and god_ damn _\u2014_ _he knows. He just_ knows.\n\n_A man makes a discovery like that\u2014he's sure as hell going to_ _get himself a gun._\n\n_Sure as hell._\n\nI never saw the likeness that seemed so obvious to Mr. Mauer, nor do I know if my father did. At one point, years after his remark, I went so far as to lay open two high school yearbooks, one with Monty Burnham's senior picture and the other with Marcia Stoddard's. I couldn't see any resemblance besides youth and the unlined optimism that so often attends it.\n\nWriting \"A Mild Winter\" didn't purge me completely of the idea that Monty Burnham was Marcia's father, but at least I was finally able to do something useful with one of the many notions that kept bubbling to the surface of my thoughts over the years. Far more plentiful are the false starts, the abortive attempts, the unrealized fragments that have accumulated in my notebooks and journals, fading and yellowing in filing cabinets and manila folders.\n\nMy father not only had a need to know what had happened with and to Raymond Stoddard, he also felt an obligation to Raymond's family. Perhaps that began when Gene discovered his father's body and decided my father was the person to call for help. From that day forward, he was ready to provide any support the Stoddards might need, whether it was jump-starting a car, providing legal advice (adding to the bureaucratic complications that can accompany a death was the fact that Raymond didn't leave a will), helping with lawn care or household repairs, or offering sympathy. My mother certainly helped Alma Stoddard too, but her aid was perhaps less obvious since it was consistent with the kind of comfort that women generally provide for each other. Lee Mauer was a handier man than my father, and eventually\u2014I think it began with an electrical wiring problem\u2014my father recruited Lee Mauer to join him on those missions to make Alma Stoddard's life easier. By the time summer came around\u2014and Marcia Stoddard returned to Bismarck\u2014the two men were as likely to be doing chores at the house down the block as they were to be sitting in our kitchen drinking beer.\n\nThe amount of time my father\u2014and Mr. Mauer\u2014spent at the Stoddards' finally led to a quarrel between my parents.\n\nMy father and mother had been to a hardware store and had returned with samples to help them decide what color to paint the house. They walked from back to front, holding those little rectangles of color against a sunlit wall and then a shaded one, but they were talking about much more than tints of blue.\n\nThey had obviously been arguing for a time when I heard them, their voices raised near my bedroom window.\n\n\"Are you jealous?\" my father asked. \"For Christ's sake, are you _jealous_ of Alma Stoddard? After what she's been through?\"\n\n\"Don't be ridiculous. Although I am curious to see whose storm windows get taken down first, mine or Alma Stoddard's. No, you know what's bothering me: Your reason for being down there. If I could be sure you wanted nothing more than to help someone whose life has turned into hell on earth, I'd say fine, wonderful. But if you're just down there to spy on that poor woman\u2014that I can't forgive.\"\n\n\"Spy . . . ?\"\n\n\"Oh, don't pretend. Please. You know what I'm talking about. You and your newfound friend would like nothing better than to discover some kind of secret down there. Playing detective. . . .\"\n\nI heard a faint scraping sound, as though one of them were flicking the siding with the cardboard paint sample.\n\n\" _Playing?_ Is that what you think we're doing? Jesus. . . .\"\n\n\"What I call it or what you call it isn't the issue. Alma Stoddard deserves some peace in her life. And the two of you intend to use her for your own\u2014\"\n\nMy mother stopped then, but if I could finish her sentence, my father certainly could as well. _Pleasure._ My father and Lee Mauer wanted to use Alma Stoddard for their own pleasure.\n\nAs it has been tiresomely noted, the early 1960s were a sexually repressed era, at least as far as public expressions were concerned. The teenage males of that time\u2014I include myself\u2014were starved not only for sexual experience but for sexual knowledge, but since so much was denied us, almost any phrase that had\u2014or could be twisted into having\u2014a sexual connotation was titillating. To get inside. To enter. To penetrate. And we had somehow learned that \"to know\" had the archaic meaning of \"to have sexual intercourse,\" so that verb too was an occasion for sly smiles or outright ribald laughter.\n\nYet when I overheard my mother scold my father, the denotative and connotative meanings of words suddenly collapsed onto themselves, tangled, and could not be teased apart.\n\nI knew, on an imaginative if not an experiential level, what it meant to use a woman for one's own pleasure\u2014it certainly involved _know_ ing her\u2014but my mother accused my father and Lee Mauer of using Alma Stoddard for knowledge of the epistemological rather than the sexual sort. They wanted to get inside, to enter, to penetrate the Stoddard house, the Stoddard psyche, in order to know.\n\nMoments after my mother's sentence fragment, the front screen door slammed. I peered out of my bedroom and saw that it was my father alone who had walked away from their argument and back into the house.\n\nAnd whether it was the result of my mother chastising him I couldn't be sure, but in the house was where my father was soon more often to be found. His fellow spy, however, the real detective, began to log more hours at the Stoddards'. In fact, instead of parking in front of our house and sauntering up our walk with a six-pack under his arm, Lee Mauer was now more likely to pull up to the Stoddards' curb, and while his presence there was often visible\u2014edging the sidewalk, sealing cracks in the driveway, cleaning the window wells\u2014on as many days and nights he couldn't be seen. But his car was there. Explanations that he was working out of sight in the backyard or in the basement or garage could last only so long, and with my new awareness of sex and secrets conflated, I soon concluded that Mr. Mauer was there for reasons other than being a Good Samaritan.\n\nIt wasn't quite the same as when I looked out the window on that January day and wondered why my father's car was parked at the Stoddards', but I was still curious about why Lee Mauer was so often there. And since my interest was prurient, I was reluctant to ask my parents or Gene Stoddard, the people who would best be able to satisfy my curiosity.\n\nI decided I'd ask Marie Ryan, and of course what I was really doing was using my concern as an excuse to call her.\n\nWhen I telephoned and told her that I wanted to talk about the Stoddard family, I tried to imply that the matter was serious enough that we shouldn't discuss it over the telephone. My strategy worked. She invited me to meet her at Elks Swimming Pool during the evening hours set aside for family swimming. She would be there with the two children she was babysitting that summer. Gene had a job with his uncle's construction crew, and when the weather was good he worked late.\n\nHere is how memory can deceive: I know perfectly well that in North Dakota the summer sun sets late, and for much of the season, darkness doesn't descend until close to ten o'clock. I showed up at the swimming pool at around six\u2014an hour when sunlight was unabated. Yet it seems to me that when I stepped out of the locker room and saw the lovely, voluptuous Miss Marie Ryan sitting on the concrete at the edge of the pool, she was washed in the golden light of the setting sun. Perhaps I misremember reality because of the way her tanned skin replied to the sun's glow\u2014and still more radiance caught in her hair, turned amber in that light. At the sight of her I must have gasped just as I had seconds earlier when I'd walked through the cold shower on my way out to the pool. So let's allow this trick of memory to stand\u2014it creates something truer than the inexpressive, unalterable fact of the hour and minute when the sun sinks below the horizon, and reminds us that we see with more than our eyes.\n\nMarie's swimsuit was the color of lime sherbet, and it didn't have straps but a loop that circled her neck and left her sweet shoulders bare and unmarked. She was toweling the blond hair of a chunky sunburned little girl who was shouting an accusation of some sort at her brother, who was still in the water. The flesh of Marie's back was as smooth and unblemished as the little girl's. Marie didn't see me approach, for which I was grateful. I wouldn't have wanted her to scrutinize my gawky, sharp-shinned, long-boned pale body the way I examined hers.\n\nAs I squatted down next to her, I felt as though I were there merely to present a contrast in geometry\u2014my abrupt angles alongside her luxuriant curves. I also noticed that though Marie was sitting at the pool's shallow end, not all the footprints around her were child-size. Men and boys must have made up any excuse they could in order to walk out of the water near where she sat.\n\nWe couldn't make anything but small talk with the little girl on her towel between us, but when the child fussed to go back into the water, I was only too glad to lead her by the hand and help her wade out to where her brother was trying to keep a beach ball submerged. Again and again it exploded back to the surface, and he acted shocked every time. His sister lent her weight to his, and while they worked on sinking the ball for good, I told Marie about what was on my mind.\n\n\"Has Gene said anything about that police detective hanging around all the time? His car is there almost every evening.\"\n\nShe kept her eyes on her charges in the water, so I couldn't glean much from her expression as she spoke.\n\n\"He thinks something is going on, all right. A few nights ago he came into the house through the side door, and his mother and Lee Mauer were sitting at the kitchen table. But side by side. And close. Gene thought maybe his mother rearranged her blouse when he walked in.\"\n\nI didn't have to feign shock. \"She's fooling around with Lee Mauer? Her husband hasn't been dead a year!\"\n\n\"Gene doesn't know that for sure. But he said that ever since then Mr. Mauer has been real buddy-buddy to him. He even asked Gene if he wanted to drive his police car.\"\n\n\"Mrs. Stoddard and Lee Mauer? I can't believe it!\"\n\nMarie obviously didn't find it so hard to believe. She merely shrugged.\n\nOur talk had turned frank enough that I could share with Marie my theory that Lee Mauer was trying to get close to Mrs. Stoddard because he thought that she, and perhaps the Stoddard house itself, was holding back a secret that would explain why Raymond had behaved as he had.\n\nThis line of speculation didn't particularly impress her either. She said matter-of-factly, \"Men are always trying to see what they can get.\" The knowing smile that Marie turned on me said that she knew this generalization could apply to me as well. I was flattered and chagrined to be one of her specimens, but she had me so neatly pinned I couldn't do anything but sit and squirm.\n\nWhen I finally found my tongue, I asked, \"What does Gene think about all this?\"\n\n\"He acts like he doesn't care. His mother has her life, and he has his. That's what he says, anyway. But he doesn't like it. The house feels strange enough without having a stranger in it.\"\n\nI nodded eagerly. \"That's sort of what I thought. I'm hardly ever around the house anymore, though.\"\n\nMarie held me in her gaze. \"Whose idea is that?\"\n\nWhen I didn't answer, she released me. \"He sure as hell doesn't care about driving a police car.\"\n\nTo cover my discomfort I stood and stepped into the pool, pretending as though the children's play needed correction. Someone needed to show them what to do with a beach ball.\n\nWith my instruction and encouragement the kids eventually took to batting the ball back and forth rather than trying to drown it.\n\nWhen I splashed out of the water, I found Marie was not alone. Standing over her, blocking her sun and probably trying to look down her swimsuit, was a lifeguard. I knew him, or at least knew his reputation. Tim Townley had just graduated from Bismarck High School and in the fall would be heading to St. Olaf to major in music and to compete on the college swim team. Slender, tanned, sun-bleached blond, and handsome, he had, so gossip had it, been successfully seducing the city's girls for years. He usually targeted females a few years younger than himself, no doubt believing that they would be especially susceptible to the attentions of an older boy.\n\nThat was probably his strategy with Marie, but when I came closer, I heard her sternly say, \"I'm going with someone.\"\n\nTim Townley turned in the direction of my wet feet slapping on the concrete. When he saw I was coming toward Marie, he looked questioningly at her. She said nothing to correct his impression, and Tim Townley shook his head and walked away. A profound gratitude instantly augmented my already considerable love for Marie Ryan.\n\nShe signaled the kids to come out of the pool and began to gather up their towels. I asked her if I could give them a ride home, but she told me that Mrs. Linstrom would be picking them up.\n\nBefore we parted, she said, \"You know, you could have asked Gene about all this yourself.\"\n\nHer remark sounded like a mild reproach, and I took it in that spirit. But I had never received a scolding that came accompanied with a smile like the one Marie Ryan shone on me.\n\nIf the purpose of Marie's observation had been to suggest that I could be a better friend to Gene, it was right on the mark. I know it now, and I knew it then, yet I couldn't seem to find the actions that corresponded to that knowledge. I knew how not to hurt but not how to help.\n\nI can think of only one occasion when I did something that, in a small way, might have aided Gene during that difficult time.\n\nHe and I were at a party at Jay Garner's house, and though Jay's parents were out of town, the gathering wasn't a large or wild one\u2014just six or seven of us, all males. The month was April, but in North Dakota that's often nothing but a calendar page. While the temperature had been in the seventies the week before, that night a wet snow was falling, fat heavy flakes filling up the window well right outside the basement rec room where we sat drinking beer, our empty cans carefully stacked atop Jay's father's bar. Just below the ceiling's acoustical tiles blue smoke pooled from our cigarettes and cigars (someone had stolen a box of White Owls). Most of our conversation I can't recall, but I'm still certain of what we talked about: athletic skills and how they could be improved; cars and how they could be altered to run faster; and girls and how they could be induced to put out. As predictably as the belches that accompanied our beer drinking, those subjects came up whenever our group gathered. On other occasions we might also have talked about why Raymond Stoddard had killed Monty Burnham, but since Gene was present that night, the topic had to be avoided, even if the sight of Gene was enough to bring the question to mind all over again.\n\nOn that evening, however, the taboo was violated, and while I was there to witness the occurrence, I'm still not quite sure how it came to pass. We were at Jay Garner's house, and perhaps he believed that privileged him. Or maybe he was too drunk to observe the usual restrictions. Whatever the cause, at some point in the evening, Jay asked Gene if he had a theory about why his father had done what he'd done. Jay asked the question politely enough, and he seemed motivated, at least initially, by nothing but curiosity. For his part, Gene simply tried to deflect the question. \"Sorry, I don't.\"\n\nWe were all relieved when Gene's shrugged apology seemed to close off the subject, and we quickly returned to our discussion of carburetors.\n\nBut Jay got drunker, and as he did, he resumed his interrogation. At least an hour had passed since his impertinent question, but Jay picked up right where he had left off. \"That's fucking hard to believe,\" Jay said to Gene. \"That you don't have a clue about why he did it. I mean, he was your dad, for Christ's sake.\"\n\n\"Hard to believe or not,\" Gene said, \"it's true. Sorry.\"\n\n\"I'm not saying you know for sure. I'm asking for a fucking hunch. A goddamn guess. A hypothesis. You sure as shit have one. Everybody does.\"\n\n\"But I don't.\"\n\nJay turned to the rest of the group, most of whom were staring uncomfortably into their beer cans. \"You know what we ought to do? We should torture him until he tells us what he knows. What do you say?\"\n\nSomeone, it might have been Bill Forston, said, \"Why don't you give it a rest.\"\n\nBut something\u2014or someone, and maybe it was Gene's passivity\u2014had opened a vein of meanness in Jay Garner. And, typical of a drunk, he was now taken with his own sadistic idea. He scraped the ash from his cigar, then blew on the tip until it glowed red-orange. \"If someone'll hold him down, I'll administer the pain.\"\n\nNo one moved, but Bill Forston said, \"I know you're just trying to be funny, but you're not. You're not fucking funny.\"\n\nJay continued to contemplate his cigar.\n\nI looked over at Gene, expecting him to tell Jay where he could stick his cigar, but Gene said nothing and didn't move. It seemed to me, however, and here perhaps I gave myself more credit than I deserved, that his eyes were taking on the first glisten that would eventually lead to tears.\n\n\"I have a better idea,\" I said. I stood and walked over behind Jay. \"How about Chinese water torture?\" And with that I poured beer on his cigar.\n\nThe cigar hissed, Jay yelped and jumped back from the beer streaming down on him, everyone laughed, and the moment's tension, along with all talk of torture and truth-telling, was extinguished.\n\nAs I was returning to my chair, Gene rose and walked silently past me.\n\nI couldn't give myself too much credit for bailing out my friend. After all, I hadn't directly confronted Jay on his stupidly cruel comments; instead I had tried to use a joke of my own to defuse the situation. And had it been compassion for Gene that finally prompted me to act, or a wish to save him, me, all of us, from the embarrassment that would have followed if he had begun to cry? This for a young man who should have been understood and forgiven if he broke down in tears every single day.\n\nTime passed and it became apparent Gene had not merely gone to the bathroom or to get another beer, so I drained the remainder of my beer, conspicuously shook the can to demonstrate that it was empty, and then left the room.\n\nI couldn't find him anywhere, not in the bathroom, not in the laundry room, where the beers chilled in a cooler, and not in any of the upstairs rooms, generally understood to be off-limits to us. I would have assumed he'd left the party altogether, but his coat was still draped over a kitchen chair.\n\nAs I turned to go back downstairs, I glanced out the back door and saw him.\n\nGene was standing on the small porch, unsheltered, the adhesive spring snow cloaking and capping him.\n\nI opened the door and stepped out, and he didn't even turn to see who had joined him. Maybe he assumed all along that I'd come after him.\n\n\"Hey, Garner's an asshole,\" I said. \"Everybody knows that.\"\n\nGene nodded. If I was looking for an expression of gratitude, it was not to come.\n\n\"I thought maybe you'd take that cigar and put it out on his forehead.\"\n\nWe both knew how out of character that would have been, and he didn't bother responding to my suggestion. Instead, he turned to me and through the veil of the falling snow said, \"I don't know anything. I _don't._ \" The tears that I believed I saw forming in his dark eyes earlier were now in his voice. \"If he doesn't stop, I'll . . . I'll . . .\"\n\nIf there was a threat behind his words it wasn't intended for Jay Garner but for Gene Stoddard himself.\n\n\"Okay,\" I said, putting my hand on his shoulder and then bringing it back cold and wet with snow. \"Okay. I'll say something. Now let's go back inside. I'm freezing my ass.\"\n\nWe both went back into the house, but Gene didn't follow me down the stairs. Behind me I heard the door open and close again. Gene was not driving that night, but I felt certain that if I checked on the direction of his footsteps in the snow, they wouldn't track north, toward his home and mine, but south, toward Marie Ryan's, only three blocks away and where he'd find comfort that he couldn't in the company of his oldest friend.\n\nAs a way of honoring my promise to Gene, I said to Jay, upon rejoining the party with a fresh beer, \"Lay off Gene, why don't you.\" I might have added but didn't\u2014Besides, what makes you think you can conceive of a torture more painful than his daily life?\n\nThis happened, as I said, in April, months before Marie delivered her mild reprimand at the swimming pool, and though I didn't bother trying to defend myself to her, I might have pointed out that the distance between Gene and me had been increasing for some time. He soon had an entirely new set of friends, and between those companions and Marie, he probably didn't have much room left in his life for me.\n\nAdolescents have always been adept at categorizing (even as they resent it being done to them), and nowhere are systems of classification as rigorously imposed as upon high school populations. Over the years there have been jocks, preps, greasers, and hoods, nerds, brains, foxes, and studs, gearheads, metalheads, losers, and geeks\u2014and that's without even venturing into the innumerable sub-genres. Yet Gene Stoddard's new friends were a kind of featureless mix that defied designation.\n\nToday they might be characterized as slackers, but back then we had no term for them. They were heavy drinkers, and if drugs had been available then, I'm sure they would have been among the first users. They were poor students yet did enough to get by. None was on an athletic team or belonged to an extracurricular organization. They lacked the dark-hearted malice of the hoods, yet they were in and out of trouble mostly because they were inattentive or not clever enough not to get caught. Most of them worked, but only so they'd have money for gas, booze, or cigarettes. They lived in their cars but did nothing to customize or personalize them. Their usual gathering spot was the gravel lot behind a Mobil station on the east end of the city, and from there they drove endless circuits of Main Avenue. A few girls, plain and anonymous, hung out with them, but I think Gene was the only one who had a steady girlfriend. Gene's cousin, Del Shumate, provided the entr\u00e9e to the group, and a similar family connection, Del's father and Gene's uncle, arranged for Gene's summer job with Harbring Construction.\n\nFrom my perspective they seemed an apathetic bunch, but perhaps that was precisely what made them attractive to Gene. Your father was a murderer?\u2014We don't care. You found him hanging in the garage?\u2014Who gives a damn. You caught your mother with another man's hands on her?\u2014It doesn't matter. Your past looms so hugely and darkly over your life that even the light of the future can't find its way in?\u2014Shut up and have another beer!\n\nI had never shunned my friend, but I'd be a liar if I didn't say that a certain relief flowed in to fill the widening gap between Gene and me. There'd be no further possibility of confusion between us, and no more uncomfortable questions that assumed I had some insider's knowledge of the Stoddard family.\n\nBut while the diminution of a friendship didn't concern me greatly, something else did. Without Gene I'd have fewer opportunities to get close to Marie Ryan. No more double dates (there had been only a few), no more tagging along with them to a school function or sporting event, and no more hanging out with them at a party. After my swimming pool meeting with Marie\u2014ostensibly about my unease over Lee Mauer\u2014I couldn't manufacture another reason to call. Gene, however, just as he had on the occasion of his father's funeral, unwittingly provided a means by which I could have access to Marie Ryan.\n\nIt was close to eleven o'clock on a hot summer evening when the telephone rang. My mother and I were the only ones up\u2014we were watching _Lost Horizon\u2014_ and after she answered the phone, she handed it to me with a disapproving twist of her eyebrows.\n\nIt was Marie, and her first words were, \"Is he with you?\"\n\n\"Who?\"\n\n\"Who else? Gene.\"\n\n\"He's not here. I haven't seen him today.\" Or this week and maybe not even this month, I might have added. I'm not sure why I couldn't admit that I seldom saw him. Hadn't Gene told Marie that our friendship had dwindled to almost nothing?\n\n\"That figures. I should have known. He told me that you two were doing something together and then he'd be here before ten. Well, guess what. He missed.\"\n\n\"I'm sorry.\"\n\n\"He didn't even bother talking to you first so you'd cover for him? Stupid bastard. Stupid, stupid, _stupid._ \"\n\n\"You called his house?\"\n\n\"Twice. Even though his mother gets pissed off when I do. Your mom probably isn't too happy I called either.\"\n\n\"No, it's okay.\" From the kitchen I could see into the living room, where my mother sat on the couch. Anyone else might have believed that all her attention was focused on _The Channel_ _Twelve Late Show,_ but I could see signs that indicated otherwise. She exhaled cigarette smoke with a little more force than usual, and one tanned leg bounced impatiently. \"Look, do you want me to go down to his house and see if he's there? Maybe he went in the side door, and his mom doesn't even know he came home.\"\n\n\"Don't bother. I know where he is. Out drinking with those low-life friends of his.\"\n\n\"Sorry.\"\n\n\"That's the second time you've said that. It's not your fault. Don't apologize for him.\"\n\n\"There must have been some kind of mix-up. . . .\"\n\nHer laughter came through the telephone line as if it traveled on its own impulse of energy. \"He doesn't _have to_ set up anything with you beforehand,\" Marie said. \"You make excuses for him all on your own.\"\n\nWhen I returned to the living room and Ronald Colman's dilemma, my mother said, \"Rather late for telephone conversations, isn't it?\"\n\n\"Sorry. That was Marie Ryan. Gene stood her up, and she wondered if he was here.\"\n\n\"Is she worried?\"\n\n\"Just hurt, I think. And mad.\"\n\n\"Are those two serious about each other?\"\n\n\"I think it's mostly one way.\" And then I added, as if clarification were necessary, \"Marie for Gene.\"\n\nSo much time passed before my mother spoke again that I thought she was entirely engrossed in the movie. Then she leaned forward to stub out her cigarette, and said, \"Plenty of that going around.\"\n\nThe next day Marie apologized for having called so late, but over the coming weeks we had a few more similar conversations about Gene's whereabouts, though they were usually held after the fact. If she saw me on a Saturday morning, for example, she might ask if Gene had been with me the night before. I couldn't lie to her, yet I still felt some residual loyalty to my friend. Or maybe the loyalty wasn't to Gene but to my gender, an allegiance still strong in adolescence. \"Was he supposed to be?\" I had learned to ask.\n\nShe would return my smile. \"You just answered me.\"\n\nBut on a night in August this matter took a turn that neither of us could joke about.\n\nI was sound asleep when I heard my name being called, but rather than wake me, the sound insinuated itself into my dream, and in that dream I strained to hear who was summoning me. It seemed as though the voice were faint because it had to travel over a distance not of space but of time, as though someone far in the past or in the future were trying to gain my attention. The voice's persistence finally cracked my slumber, and I woke with the realization that someone was outside my bedroom window.\n\nIt was Marie, hissing my name and scratching the screen.\n\nI slept in nothing but my briefs, so I dragged the sheet with me as I got out of bed and crouched down by my window.\n\nMore than her presence at that place and time, her somber expression and tear-spangled eyes told me that something was seriously wrong.\n\n\"What is it? What's happened?\"\n\n\"Gene . . .\"\n\n\"Is he\u2014? What happened?\"\n\nShe shook her head violently. \"I don't know. I don't know. He's . . . I don't know where he is!\"\n\nBehind her, every window in every house was curtain-drawn and dark, and the street down which police cars had once careered was empty and quiet. The night was hot and still, and Marie's urgency and distress seemed one with the steamy air.\n\nI glanced back at the clock. It was almost two o'clock. \"Can you tell me what happened?\"\n\nWe both leaned forward, and as she began to whisper to me, her lips were barely more than an inch from mine, but of course the screen was between us.\n\nGene and Marie had gone out that evening, one of their first dates in weeks, first to a movie and then to Teen Canteen, where they danced until the lights were turned up and the club closed. Her father and mother believed Marie was sleeping at Donna Petracca's house that night, and since Donna's parents wouldn't notice or care when\u2014or if\u2014Marie came in, she and Gene could stay out as late as they liked.\n\nThey drove around for a while and then parked behind the museum on the capitol grounds. She didn't tell me what they were doing there, but she didn't have to. I no longer had reports from Gene on how far he was getting with Marie, but I had no reason to think his sexual progress had slowed or halted. She was wearing a sleeveless white blouse that fastened up the back.\n\n_She shrugged her shoulders forward, the blouse fell onto the_ _floor of the car, and Gene tossed it into the backseat. He_ _reached behind her and unclasped her brassiere with a deftness_ _that surprised her. Usually he groped clumsily for so long that_ _they both had to make a joke\u2014Mr. Fumble Fingers!\u2014before_ _Marie went ahead and did the job for him. Was it possible that_ _being drunk made him more dexterous rather than less?_\n\n_So she was naked to the waist as she lay back across the_ _front seat, and that was often enough for him, but this time his_ _hands dove almost immediately for the waistband of her shorts,_ _and rather than try to squeeze his hand inside, he tried to tug_ _them down. She decided to allow it, though when she felt her_ _panties slipping off with her shorts, she reached down to grab_ _them and keep them on._\n\n_\"Those too,\" Gene said, his voice so husky the words almost_ _seemed croaked._\n\n_In the future when Marie Ryan thought or felt or said she_ _was ready for sex, she would almost always mean nothing more_ _than physiologically ready, and in the front seat of that Ford her_ _body was certainly as ready as it would ever be\u2014her own panting_ _breaths came echoing back to her from the little cavern_ _under the dashboard, and she was moist everywhere with her_ _own heat and the night's. But at sixteen she was not simply a_ _body._\n\n_Was it the smell of bourbon on his breath, distressingly similar_ _to her mother's when she would bend over Marie's bed, ostensibly_ _to wish her good night but really to scold Marie once_ _more for that day's minor misdeed? Was it that Gene, in this_ _moment of sexual exigency, had stopped kissing her and his caresses_ _had become unimaginatively monotonous, as if he were_ _counting repetitions before moving on to another part of her_ _body? Was it that her sweating skin was sticking to the upholstery_ _and made a vulgar sucking sound whenever she shifted positions?_ _Was there some symbolism in the location where they_ _were parked that made it impossible for her to relax into the experience?_ _They were secluded behind the museum, yet nearby_ _was the state capitol, at seventeen stories the tallest building for_ _hundreds and hundreds of miles, its rectangular height insistently_ _phallic, its rows of windows inscrutably watchful, and_ _when Marie lifted her hips to let Gene pull her shorts off, she_ _tilted her head back and saw\u2014or imagined she saw\u2014the building_ _and knew that what was happening was within its energy_ _field. Or did the building cause her to make connections that_ _could override even the oblivion-inducing ardor of sex? In that_ _structure a man was murdered. I am practically naked in the_ _murderer's car. The murderer's son's hand is between my legs\u2014_\n\n_Whatever the cause, Marie pushed Gene away\u2014_\n\nThey had a quarrel, about what, Marie didn't say, but just before Gene walked away, he said, \"Maybe my dad had the right idea.\"\n\n\"Jesus,\" I said. \"Had he been drinking?\"\n\nMarie nodded.\n\n\"Didn't you go after him?\"\n\n\"I couldn't. Not right away. When I did, I couldn't find him.\"\n\n\"Maybe he went home.\"\n\n\"I checked. I looked in his window. He's not there. I know it.\"\n\n\"Did you look\"\u2014I swallowed hard before asking this\u2014\"in the garage?\"\n\nShe nodded again.\n\n\"And he walked away from his car?\"\n\n\"I didn't know if I should drive it or leave it there. Then I decided that where I was going to look for him I couldn't drive. If he'd walked off into the trees on the capitol grounds, I'd have to be on foot to find him.\"\n\n\"The trees? Why would he\u2014\" Suddenly I realized what she was telling me: Gene had threatened to hang himself, and the first opportunity that would present itself would be from the branch of a tree on the capitol grounds.\n\n_Usually no matter how much Gene Stoddard drank, he_ _could keep the world in focus, but as he looked down at the_ _nearly naked Marie Ryan, her outline kept blurring._\n\nMy next question was a coldly practical one: \"Was he wearing a belt?\"\n\n_He had the most difficulty with the borders on her body_ _where her tanned flesh gave way to the pallor her swimsuit had_ _created. But sometimes she had the straps down, or they shifted,_ _or she wore longer shorts, or she borrowed a swimsuit from a_ _friend and that allowed the sun to shine where before it had_ _been forbidden, and the lines weren't as sharp as they used to_ _be. And the front seat of the car was dark, and Marie was moving_ _sinuously under his touch and sight, and all that made it still_ _harder to know where something left off and something else_ _began. He could eliminate the problem by closing his eyes, but_ _then even the darkness would shift and sway and he was liable_ _to get sick. Besides, he couldn't get enough of the sight of her\u2014_ _the sudden flash of white as her eyelids fluttered briefly open,_ _the dark circles of her nipples, the faint shadows as her belly_ _sank below her ribs, the brighter swell of her thighs. . . . He had_ _to stare, yet the harder he stared the more unfocused everything_ _became. It reminded him of being in first grade, when he was_ _trying to learn how to read. He knew being able to read would_ _come only from looking at those letters, yet he looked and_ _looked and they would not yield what he needed from them._\n\n_It occurred to him then that his difficulty might be arising_ _not from bourbon but from desire, swirling in him like those_ _powerful currents that were said to rule the Missouri just below_ _its murky surface. He unbuckled his belt._\n\nMarie nodded, and with her index finger began to rub her lower lip, a gesture that meant\u2014as I would have ample opportunity to learn in the future\u2014that she was trying not to cry.\n\n\"Just a minute. I'll get dressed and come right out.\"\n\nIn my youth I was as modest as they come, yet at that moment I heedlessly dropped the sheet that covered me and reached for the clothes I had thrown onto a chair.\n\n_They had done this often enough that Marie knew what he_ _wanted\u2014or thought she did\u2014and reached inside the waistband_ _of his briefs._\n\n_\"Huh-uh,\" Gene said. \"Not tonight. More. I want more.\"_ _He knew that wasn't the way he should say it\u2014it wasn't the_ _way he_ wanted _to say it\u2014yet when he tried to improve it, he_ _only made it worse. \"Not with your hand.\"_\n\n_She propped herself up on her elbows. \"What?\"_\n\n_He pushed her back down. \"You know.\"_\n\n_And when her eyes closed again and her head lolled to the_ _side, and when she lifted her hips off the seat and let him pull_ _down her shorts, Gene believed he would have what he had to_ _have. But then she grabbed the waistband of her panties to keep_ _them from slipping off with her shorts._\n\n_\"Those too,\" he said in a voice that he barely recognized as_ _his own, but before she could answer or act, he lay down on top_ _of her, hoping that his volume, his entire being, pressed down_ _upon her would convey to her the urgency and import that his_ _words couldn't carry._\n\n_She lay unmoving under him for so long that Gene wondered_ _if she had fallen asleep. Then it occurred to him that perhaps_ _her stillness was a kind of permission. But he hadn't done_ _anything more than move his hand a few scant inches down her_ _side when she spoke, and she obviously had no trouble locating_ _the language or voice she needed._\n\n_\"No,\" Marie Ryan said._\n\n_Maybe, he told himself, she was talking out of a dream, and_ _he kept her pinned under his weight._\n\n_But another single word issued from her\u2014\"Off!\"\u2014and this_ _time action accompanied it. Somehow she got her hands between_ _her chest and his and thrust him upward. He had been_ _working on a construction crew that summer, and his own_ _strength had increased significantly. Furthermore, he knew what_ _the repetition of each task\u2014lifting plywood sheets, carrying_ _bricks and buckets of mortar, hoisting lengths of lumber\u2014had_ _done to harden each group of muscles, yet Marie's strength astonished_ _him, and Gene wondered what she had done over and_ _over again in her life to give her the power to push him off her_ _with what seemed to be such ease._\n\n_But Gene wasn't about to give up. He knew, however, that if_ _he was going to get what he desired, it would have to be with_ _language. Up until that night Marie had let him have what he_ _wanted, and all he had had to do was reach for it. But now_ _what was he supposed to say? I want to fuck you, screw you,_ _sink my cock into you? That was the vocabulary of his friends,_ _of the men he worked with, and of his own mind, yet he knew_ _that those words would get him nowhere._\n\n_Then Gene glanced out the window, and when he did, a_ _better idea occurred to him. A short distance away was the_ _capitol, its height hovering over them as if it were a spaceship_ _slowly landing in that area. He usually tried not to look at the_ _building\u2014next to impossible since it loomed over the city and_ _could be seen from every direction\u2014because he couldn't stop_ _thinking of it as the place where his father's life ended. Gene_ _knew that wasn't literally true\u2014his father died in the garage\u2014_ _but if it weren't for what had happened in the capitol . . ._\n\n_He opened the car door and stepped out, then bent down_ _and leaned back in. He would have thought she'd cover herself,_ _but she sat up unashamed of her nakedness. Boys in the locker_ _room were more self-conscious about their bodies than Marie_ _Ryan._\n\n_\"Maybe,\" he said, \"just maybe my dad had the right idea.\"_\n\n_He turned then and began to walk away, knowing that she'd_ _call him back, knowing that by the time he climbed back into_ _the car, she'd be lying back down, her panties off, her legs_ _spread as wide as the front seat would allow\u2014_\n\nI stayed close to the wall as I crept past my parents' and sister's bedrooms, knowing that the floor could creak and give me away if I walked down the middle. The screen door opened noiselessly, and then I was free, ready to join Marie in her search.\n\nFor the first few blocks as we walked, we watched for motion\u2014Gene's form under the light of a streetlamp or his silhouette against a house's white wall or his shadow stretching toward us as he was backlit by the headlights of the occasional car\u2014but once we entered the dark aisles of trees on the capitol grounds, we were on the alert for stillness\u2014his lifeless body hanging from a low branch.\n\nIt was Marie's idea that we search separately so we could cover a larger area in less time, and while I didn't argue with her, I wasn't convinced that was the best strategy. I didn't want to be alone when I found him, and I didn't want her to be either. We agreed that we'd shout out for the other if we saw Gene, no matter what his state.\n\nAnyone walking through the North Dakota capitol grounds and its paved arboretum trail today would find the trees widely spaced and clearly labeled. Plaques set into the earth at the base of every bush or tree identify the Russian olive or the ponderosa pine or the Chinese or American elms or the green ash or the cottonwood. But on the night I hunted for my friend, the trees were more thickly planted, and they were simply trees\u2014black vertical shapes whose trunks could be mistaken time and again for the hanging figure of a young man. Although the summer had been dry, the earth was soft and loamy underfoot, and I stumbled more than once because it was impossible to see where I was stepping. I was further slowed by having to walk hunched over, the only posture that allowed me to look upward and distinguish dark forms by examining them against the lighter darkness of the night sky. Had I had any foresight at all, I would have brought a flashlight, but then someone driving past or gazing out a window across the street might have seen a beam of light moving through the trees and called the police. More than once I wondered if that was exactly what we should have done.\n\nOccasionally I could hear Marie, just a corridor or two away from me, and I could tell she was moving much faster than I, but then being in love equips us well for moving blindly through the dark.\n\nThose of you who have been with me for these many pages must possess by now a sense of the man behind these words and of the boy behind that man. It should come as no shock or surprise to learn that I was ambivalent about my friend's possible fate. I can't say that I hoped to find him dead\u2014to contemplate that prospect was not just horrifying and sorrowful but grisly\u2014but I certainly calculated the advantages that might accrue to me if Gene were no longer alive. Marie would require solace, and I would be there to provide it. Furthermore, Gene would no longer block my path to Marie. But I also knew that loss can preserve love forever in a present state\u2014seal it, as it were, in amber\u2014and turn the loved one into a timeless ideal with whom no living human can compete.\n\nAfter searching for close to two hours and not finding Gene, Marie and I finally gave up. But in a curious application of faith\u2014belief in things unseen, as it is defined in the Old Testament\u2014having not found him, we became convinced that he was dead. If not in the small night forest where we were searching, then hanging from another branch, beam, or rafter for someone else to find. We debated whether we should take his car back to his house, but decided that it would be too upsetting\u2014to say nothing of mysterious\u2014for Mrs. Stoddard to see the Ford parked in the driveway yet with her son nowhere to be found.\n\nMarie wanted to check his house one more time, so we walked together away from the capitol and toward Keogh Street. As we stepped out of the trees and into the light, I dared put my hand on her shoulder, a gesture that I hoped she would read as intended only to comfort.\n\n\"God, your hand's sweaty,\" she said, and I took it away.\n\nWe were crossing Keogh, still a couple blocks from Gene's house and mine, when a car approached. I've often wondered, since Marie and I turned toward the car simultaneously, what she saw and heard that caused her to begin running immediately toward its headlights. However, I too must have soon perceived the truth of the situation, because I never called out or ran after her, even when the car kept coming at her.\n\nIt was Gene, and by the time the car pulled alongside me, Marie was already inside. I looked in the open passenger window and saw that she was practically curled into his lap, and Gene had one hand on the steering wheel and the other arm casually draped around her. Marie's face was turned from me, but I guessed tears of joy and relief were staining his shirt.\n\n\"Hey, man,\" Gene said. \"You want a ride?\" It was difficult to imagine that anyone so cheerful might have been considered capable of suicide.\n\nI shook my head and waved them on their way. They didn't try to argue with me but sped off. I stood and watched the Ford's taillights until they disappeared around the dip and curve that Keogh Street makes where it intersects with Cooke Avenue. Then I walked home, but when I got there I didn't go inside.\n\nInstead I sat on our front porch step. From there I could see when Gene returned home, no matter which direction he came from, yet it was unlikely he would notice me keeping my vigil. Time passed slowly, and every hour Gene didn't appear made the next hour longer.\n\nDawn finally came, light suffusing the block so subtly and gradually that even the most attentive observer would have found it impossible to name the instant when darkness gave in to day. One moment the numbers on the house across the street couldn't be read, and the next moment there they were. Preceding any visible glow was the chirp and twitter of birds, but they were soon in full-throated song, gloating or sighing that they had made it through another night. Gene had still not driven past, and that fact alone provided me with all the information I needed. I gave up and returned to my bed.\n\nI tried many times to write a story about that night, but every effort frustrated me. Because there was so much backstory\u2014the film term that my writing students invariably use when referring to almost anything in the past\u2014I finally decided to limit the narrative by concentrating on what happened in the car between the two lovers, to write, in other words, a seduction story. Neither the male nor the female quite worked as the focus of narration, so I alternated points of view. Still the story wasn't what I hoped it would be. Perhaps, as so often happens in life, the present has too little meaning when it's not attached to the past, the past with its power to clarify and distill. And perhaps that story was simply not meant to be because it was willed into existence\u2014over and over again\u2014not by a desire to make something artful and true but by a compulsion to lacerate myself with my own imagination.\n\nBut while the Ford's absence that night and early morning took on such importance in my psyche, the car's actual appearance had, for another viewer, significance of quite a different sort. A friend of my father's believed it explained Monty Burnham's murder.\nAs law students at the University of North Dakota, Ross Wilk and my father roomed together, and while both eventually found themselves practicing law in Bismarck, Ross Wilk was a more financially successful attorney than my father. Mr. Wilk did a great deal of work for the oil companies that swarmed over the state in the 1950s, and though the boom didn't last long, by the time it was over, Ross Wilk was a wealthy man and a partner in one of the city's top law firms. He would gladly have brought my father into the practice, but my father always refused, citing as reason not his stubborn independence but that he didn't \"look good in a cowboy hat.\" The remark made sense only to those familiar with Ross Wilk's appearance and then only barely. Ross Wilk wore expensive, hand-tooled boots and wide-brimmed Stetsons, attire that wasn't\u2014and isn't\u2014especially unusual in Bismarck. (In his _Travels with Charley,_ John Steinbeck said that the West began at the Missouri River, and Bismarck was built on the river's eastern bank.) Ross Wilk, however, had no Western roots or cowboy experiences in his past. He came from a small town in northwestern Minnesota where his father had been a dentist. But he knew, in a way my father never did, how a persona could be shaped and projected for one's personal gain. An insider in state and local politics, Ross Wilk had considerable influence in both circles (largely through the Republican party). My father generally trusted his friend's take on issues, believing they were arrived at by Mr. Wilk's keen intellect and by his access to information that most citizens didn't have.\n\nOnce or twice a year my parents had the Wilks over for dinner, and that was the occasion that brought Ross Wilk to our living room on a warm October day in 1961, nine months after Monty Burnham's murder. Mr. Wilk looked down the street and saw the Stoddards' Ford outside their house, and at the sight of the car, Ross Wilk tapped his index finger against our window. Along with his hat and boots, Ross Wilk affected a kind of Gary Cooper laconism, and he turned to my father (the wives were in the kitchen) and simply said, \"There's your answer.\"\n\nThe men had not been talking about Raymond Stoddard or Monty Burnham, and Ross Wilk's comment was made without preamble, but in Bismarck during that period\u2014and nowhere was this more true than on Keogh Street\u2014that subject was close enough to the surface of every conversation that it needed no introduction. Neither did Ross Wilk need any encouragement to give voice to his theory.\n\n\"For some reason when I learned that Raymond Stoddard had recently purchased a new Ford, I kept thinking that had to be important. I just didn't know how. And then when I had a talk with the governor, who was mad as hell about the prices Bob Borglund was charging to make repairs on state vehicles, it all started to make sense.\"\n\nRobert Borglund owned a very successful Ford-Mercury dealership in Bismarck, and for years he had the contract for providing\u2014and servicing\u2014vehicles to the state. The arrangement seemed to work to everyone's satisfaction until North Dakota elected a Democrat as governor. Then George Bartell, the new governor, found out that Borglund Ford-Mercury was charging the state exorbitant rates for routine repairs to its vehicles. When the governor began to look into that irregularity, he discovered an even greater one: The state was not receiving a particularly favorable price when it purchased its new cars.\n\nNow what state agency, Ross Wilk wondered, would be able to find out what the bill was for a new car? Why, Accounts and Purchases, of course, the department that employed Raymond Stoddard. And who would notice that what the state paid for a new Ford was out of line? An employee who had recently bought a car for himself, that's who. And that employee might then point out this discrepancy to his superior, who in turn would report it to the governor.\n\nFurthermore, the governor learned that the connection between the state's purchase of new Fords and state politicians was even tighter than first suspected. Robert Borglund was not only a heavy contributor to the Republican party, he was often mentioned as a future candidate for political office, perhaps the governorship. He was ambitious and wealthy; he was comfortable in front of the cameras (he appeared in all his own television commercials for Borglund Ford-Mercury); and he came across as confident, cheerful, honest, and direct. He was, in other words, a man cut from the same cloth as Monty Burnham.\n\nAccording to Ross Wilk, the men were more than similar personality types. They shared a political philosophy built on thrift, self-reliance, patriotism, and tough attitudes toward criminals and foreign governments. In practical terms most of these lofty values translated into little more than attempts to keep taxes low, to balance budgets, and to reject social programs. Both Monty Burnham and Bob Borglund liked to pretend\u2014or perhaps they believed\u2014that they had risen to stature and success on nothing more than strength of character and hard work. In truth, both men stepped into prosperous, established family businesses, and the businesses were the same\u2014selling cars, specifically Fords.\n\nRoss Wilk believed that Monty Burnham had somehow influenced the system that awarded the contract for state vehicles\u2014sales and service\u2014to Borglund Ford-Mercury. Bob Borglund returned the favor by donating generously to Burnham's campaigns and by sharing a portion of the state contract with Burnham Motors (also a Ford dealership), all on an unofficial basis, of course. And once again, Ross Wilk pointed out, a man with a new car (purchased at Borglund Ford-Mercury), a man who worked in the state's Accounts and Purchases department, would be the one likely to discover and then unravel these strands of corruption.\n\nBut if this man had his own connection to Senator Burnham\u2014as Raymond Stoddard had\u2014then he could perhaps be brought in on the plot. So instead of an employee who informed his superior of the corruption, there's someone who, by virtue of his position in Accounts and Purchases, was able to contribute to the scheme. He could hide or alter documents, manipulate figures\u2014he'd be a perfect partner. And maybe it was no recent discovery. Maybe it had gone on for years, an operation successful because many people benefited\u2014maybe Raymond Stoddard's new Ford was a payment for his participation\u2014and so few people were harmed. Who would have thought this new governor would get so worked up over something as minor as what the state was being charged for an oil change? One discovery leads to another, and before long you have a man taking his own life to avoid the humiliation of being uncovered as a crook. And, Ross Wilk concluded, he takes down with him the fellow who talked him into being a part of the plot. Or maybe he wants to put a bullet in the good senator because he knows that the senator will sell him out to save his own hide.\n\n\"I'm not convinced,\" my father said.\n\n\"Because the man was your friend, and you don't want to believe it of him. But people don't commit murder for noble reasons. And they kill themselves out of desperation. Shame. Even embarrassment. You know that.\"\n\n\"It just doesn't feel right.\"\n\n\"Feel? What do feelings have to do with anything? What would old Professor Saint Clair say if he heard you talking like that? Evidence. Logic. Reasoning. That's how guilt and innocence are determined. If you like, I can keep digging. I have no doubt the evidence is out there.\"\n\n\"Not necessary,\" my father said, waving his hands at his old friend as if in surrender. \"Not necessary.\"\n\nLater that evening, after our guests had gone home, my father summoned my mother and me to the living room. My mother was in the midst of putting away the good china she had used for that day's meal, and ordinarily she would have preferred to finish that task, but something in my father's voice must have told her this matter couldn't wait.\n\nHe sat back in his overstuffed chair while my mother and I perched on the edge of the sofa's cushions. He related Ross Wilk's theory about the plot to overcharge the state, and with Raymond featured as a willing accomplice it was a convincing presentation. Nevertheless, as my father finished talking, he wore the same distressed look as when he had begun.\n\nMy mother, on the other hand, received this latest hypothesis with a mixture of exuberance and relief. \"Well, that's as plausible as anything I've heard,\" she said.\n\n\"I suppose.\"\n\n\"What's wrong? You act as if you're disappointed. Why\u2014because you didn't crack the case? You and your detective friend?\"\n\nHer remark struck me as cruel, yet my father didn't flinch or attempt to defend himself.\n\nIn the absence of any further comment from my father, my mother turned to me and volunteered her own explanation of where we had now arrived. \"This is exactly why your father gets in trouble with some of his cases. He identifies too much with his client, and then when the verdict doesn't come out the way they'd hoped, he takes it personally.\"\n\nShe rose and walked over to sit on the arm of my father's chair, leaning her weight seductively against him and running her fingers through his hair. He stared straight ahead. \"You probably thought,\" my mother continued, \"that your dad was only interested in finding out the truth of that awful day. What you didn't know, what none of us knew, was that he had taken on Raymond Stoddard as a client. An unwilling client, of course. Since Raymond was dead, he was in no position to retain counsel. But in your father Raymond had a devoted advocate, an attorney working pro bono. Not only without pay but without even being asked. And an attorney who wouldn't be satisfied with any verdict except one that found his client innocent.\"\n\nWithout saying another word, my father pushed himself to his feet and stalked out of the room.\n\nFor another moment my mother remained balanced on the arm of the chair, and the fingers that had been combing my father's thinning hair now scratched at the chair's nubby upholstery. \"I guess I got a little too close,\" she said, and I knew she wasn't talking about physical proximity. Then she rose too and left the living room, but while my father had gone down the hall toward the bedrooms, she went back to her kitchen.\n\nIf I hadn't known it before, that scene between my parents tried to teach me the lesson: Lies can unsettle us and send us angrily from the room, but the truth can do the same.\n\n_Because time seems to have speeded up\u2014and if it seems so, isn't_ _it so?\u2014you feel as though you must act swiftly. Were you not so_ _pressed for time, perhaps you would evaluate your options carefully,_ _and with that care you might be more realistic. Instead,_ _once you happen upon a workable strategy, you must commit to_ _it unequivocally._\n\n_For that reason you decide early on that it will be necessary_ _to take your own life. You contemplated life in prison only for a_ _moment, and perhaps not even for that long. You recalled a_ _Sunday drive in the country, a mild March day when uncharacteristic_ _warmth bumped against the cold of unmelted snow and_ _a fog rose that made mid-afternoon as dusky as, well, dusk. The_ _car swept around a curve and there it was\u2014the state penitentiary,_ _its walls burning their way blackly through the vapor, its_ _stone towers too solid for a mist to cover. That was enough._ _Though on that day guilt had not yet encumbered your life, you_ _stored away a realization unrealized until now\u2014you could_ _never handle life in prison. Not for a year and certainly not for_ _ten._\n\n_Besides, it might be that your suicide will be regarded not as_ _an admission of guilt, an attempt to escape the inescapable, but_ _as an expression of frustration and exhaustion\u2014_ because he couldn't convince the world of his innocence, he took the only way out available to him.\n\n_But you are not innocent. Oh, no. Even in those moments_ _when you treat yourself most gently, when you interpret your_ _actions as generously as possible, even in the depth of self-pity_ _or at the height of indignation and self-justification, you still_ _must admit: You are guilty._\n\n_When your brother-in-law\/old friend\/former classmate came_ _to you with the scheme to fix prices\/misappropriate funds\/embezzle\/_ _steal\/conspire, you knew what was being asked of you._ _Yes, yes; he talked you into it. Nothing was your idea, not initially._ _Only when you became necessary to a successful outcome,_ _only then did you contribute what only you could_ _contribute. But right from the start you knew. You knew and_ _you assented. No matter that you were a minor (but essential)_ _functionary in the plan; that role only adds pathos to your guilt;_ _it does nothing to absolve you._\n\n_Odd, how familiar it all is. Until recently, you've led a fairly_ _respectable life, you've committed no previous criminal acts, yet_ _this feeling of being ensnared, guilty, caught, trapped, culpable,_ _is one you know. You took fifty cents from your mother's purse._ _You stole a pack of your father's cigarettes. You lied and told_ _your girlfriend you wouldn't be there, but then you were and so_ _was she. The teacher compared your answers and your friend's_ _and found them the same. You denied, you dissembled, you_ _cheated, you pilfered, you pretended. . . . And you were caught._ _The scale so different, the potatoes so small, yet the feeling the_ _same, as if now you are fulfilling your entire life's promise, finally enacting what you have rehearsed so long. Though now it_ _may be the police\/congressional committee\/investigative journalist\/_ _government agent closing in on you, it feels no different from_ _trudging home from grade school when you knew the phone_ _call from the principal would arrive before you. It seemed then,_ _as now, that the world was about to end._\n\n_Now you know of course that it will be over only for you._ _Or . . . ?_\n\n_It's the only question that remains. Should you take your_ _brother-in-law\/old friend\/former classmate\/colleague out with_ _you? To do so would be murder, an act of rage, retribution, revenge,_ _or, less grandly, of petulance, but murder without question._ _Yet it would also contain an element of justice. It's quite_ _possible that your death will end the investigation, close the_ _case, stop the proceedings. Once, after all, a guilty party is identified_ _and found, no one feels quite the same need to uncover_ _another. So, who, if not you, can guarantee punishment to the_ _blameworthy? In this you may well have a mission, a meaning,_ _a reason for being that you never had when your life was innocently_ _your own. . . ._\n\nThat story, if it can be called that, was published in _Epiphanies,_ another now-defunct publication. It was one of the longest pieces ever to appear in the avant-garde magazine; many of the \"discoveries and revelations,\" as the editors said they featured in their pages, were no more than a paragraph. The story's McInerney-fashionable (at the time) second-person point of view probably gave it special appeal to the _Epiphanies_ staff, but its source, of course, was that conversation between Ross Wilk and my father when Mr. Wilk speculated that in the corruption of the workplace Raymond Stoddard's murderous intention was born.\n\nIt would not be accurate to say that I was able to observe an actual lightening of my mother's spirit after the Wilks' visit, but I think it's fair to say that Mr. Wilk's rendition of the murder-suicide and what led to it satisfied my mother. I never again heard her express any curiosity about those awful circumstances, and if she didn't broadcast the Wilk theory to others, it was only because she didn't believe it was hers to share.\n\nMy father, on the other hand, resisted his friend's interpretation and, to some extent, actively lobbied against it. I know because I was one of those he tried to dissuade from believing that Raymond Stoddard murdered Monty Burnham and then killed himself because he was involved in a political scandal.\n\nEvery Halloween my father tried to organize his family for an evening of jack-o'-lantern decorating, an activity for which only he had any aptitude or enthusiasm. That year, however, I, unlike my mother and sister, was unable to come up with an excuse in time and so found myself sitting at the kitchen table alongside my father, both of us up to our elbows in pumpkin innards.\n\nHe introduced the subject of Ross Wilk as abruptly as Mr. Wilk had begun talking about Raymond Stoddard just a few weeks earlier.\n\n\"Ross Wilk is a helluva bright man,\" my father said, \"but he's making the mistake a lot of people make. Just because something happens in a particular location doesn't necessarily mean the location has anything to do with it. They see Monty Burnham being murdered in the capitol as important, and since the capitol is a government building, they think the murder must have to do with politics. But for legislators and state employees it's the place they work, simple as that. Raymond knew Senator Burnham was in the building. Nothing more to it, as far as I'm concerned. If you want an explanation of why, you can't allow yourself to be distracted with where.\"\n\nMy father was not being disingenuous; he believed that. But another set of beliefs was also at work in him and leading him away from Ross Wilk's account. My father didn't want place to figure at all in what Raymond Stoddard had done because my father wanted to dissociate his city and state from the crime.\n\nAlthough he didn't belong to any service organizations\u2014no membership in the Kiwanis Club or the Rotary Club\u2014and though he found embarrassingly superficial and hollow the Babbittry of many of the community's and state's boosters, my father took second place to no one in his love for North Dakota. He cherished its homesteading heritage of hard work and stoicism, its paradoxical high-plains ethic of both self-reliance and neighborliness, its lack of pretense, and its unspoken credo of honesty and forthright plain speech. He was neither hunter nor fisherman, but he loved the state's empty, almost featureless spaces, its shadowed valleys and sunlit prairies. Even North Dakota's fabled extremes of weather were for my father nothing but the tests that made us all stronger for having passed them. He couldn't stand the thought that people might see a connection between the state he loved and its most famous crime. It was bad enough that he and Raymond Stoddard and Monty Burnham shared a hometown\u2014tainted forever by the murder\u2014but he hoped that attempts to determine cause would concentrate on the murderer's psyche and not the place where he pulled the trigger.\n\nI sympathized with my father's position, but I couldn't agree with it. I feebly believed at the time\u2014and time has only firmed my belief\u2014that what happens can't be pulled apart from where.\n\nIf we had been carving our pumpkins fifteen years in the future, my father's jack-o'-lantern might have been characterized as wearing a Jimmy Carter grin. As it was, I couldn't help but look at that horizontal, symmetrical smile and think of a Ford's small-toothed grille.\n\n_Every year the lawyer's wife suggested that he draw the face on_ _the pumpkin before he began to carve, but he always disregarded_ _her advice. Instead he proceeded freestyle, cutting into_ _the hollowed-out pumpkin with one of his tools of choice, a fish_ _fileting knife or an old paring knife, both of which could be_ _sharpened to a razor's edge. Once the jack-o'-lantern's first features_ _took shape\u2014a wolfish smile, an astonished eye, a flared_ _nostril\u2014he invented from that beginning, sometimes letting an_ _accidental slice dictate the entire design. It was as close as he_ _came to improvisation, to art, or perhaps even to spontaneity._ _Certainly nothing in his professional life\u2014his specialty was advising_ _oil companies on how to deal with western states' land_ _use policies and regulations\u2014allowed him to act without a well-defined_ _sense of outcome and consequence._\n\n_The telephone rang, and though it was only a few feet from_ _where he sat at the kitchen table, he called out to his wife to an_ _swer it. In explanation he held up his hands, still wet and slippery_ _from scooping out the pumpkin's stringy pulp._\n\n_She had been engrossed in an episode of_ Ben Casey, _but she_ _rose from her chair in the living room and soon held out the_ _phone to him, mouthing the words, \"It's Lee.\"_\n\n_It was not a call the lawyer wanted to take, but he felt he_ _had no choice. While he washed his hands, his wife patiently_ _dangled the phone from its cord as if it were an object she didn't_ _want to bring near her mouth or ear. Just before he took it from_ _her, she whispered, \"He wants to ask for a favor.\" Then she left_ _the kitchen and returned to her television program._\n\n_\"Lee. What can I do for you?\"_\n\n_\"I hate to take you out of your house on a night like this\"\u2014_ _the day had not been only cold, but once the sun had set, the_ _wind occasionally tore loose a shower of fine-grained snow\u2014_ _\"but I could sure use a hand. Can I get you down here?\"_\n\n_The lawyer sighed, hoping the sound traveled through the_ _wire. He knew he would say yes to the request, but he wanted_ _Lee Mauer to hear his reluctance. \"Can you give me a clue,_ _Lee?\"_\n\n_\"It's not something I'm eager to discuss over the phone.\"_\n\n_The lawyer might have been puzzled, even worried, if someone_ _else had made that statement, but Lee Mauer, a former police_ _lieutenant, never missed an opportunity to dramatize or add_ _intrigue to almost any situation. \"I'll be there in a few minutes.\"_\n\n_\"Just come in the back door. I'm in the basement.\"_\n\n_Since he was only crossing the street and walking a few_ _houses down the block, the lawyer didn't bother with a jacket._ _And he was familiar with the home he was visiting. His friend_ _Raymond Stoddard had lived there until he had taken his own_ _life the previous January, but it was still home to Raymond's_ _wife and son. Lee Mauer originally began visiting the house, in_ _the lawyer's company, to help the widow with yard work and_ _light household repairs, but the lawyer's wife believed that a_ _romantic\u2014and sexual\u2014relationship had now developed between_ _Mrs. Stoddard and the former police officer. The lawyer,_ _however, seldom committed himself to belief without substantial_ _evidence._\n\n_Although he was uncomfortable walking into the house_ _without knocking or ringing the bell, the lawyer did as Lee_ _Mauer requested. Just inside the door, however, the lawyer_ _called out, \"Hello?\"_\n\n_From the darkened stairway came a feeble \"Down here.\"_\n\n_The Stoddard basement was only partially finished. The_ _laundry and storage rooms, separated by only a wall of widely_ _spaced two-by-fours, had concrete walls and floors and exposed_ _ceiling joists. Before his death, however, Raymond Stoddard_ _had made progress on a recreation room. He had installed ceiling_ _acoustical tiles and recessed lights, and he had laid flooring_ _with the markings for a shuffleboard court. On one paneled_ _wall was a gun rack, though Raymond Stoddard owned no guns_ _himself. Protruding from another wall were the pipes that were_ _supposed to connect to the fixtures of a future wet bar._\n\n_In spite of Raymond's plans for the room, it had, as basements_ _inevitably do, gone over to storage. The lawyer was sure_ _that the cardboard boxes, the mildew-spotted suitcases, the_ _dresser missing a few drawer pulls, and the cedar chest were all_ _full. Similar items in his own basement certainly were. Positioned_ _almost in the center of the room was a space heater, its_ _bars glowing orange and giving off the odor of hot dust. The_ _appliance must have been running for hours. The basement was_ _so warm that the lawyer felt immediately the onset of sweat_ _prickling at his hairline._\n\n_And in a darkened corner, lying flat on his back on a rollaway_ _bed's bare mattress, was Lee Mauer. Mauer was wearing_ _nothing but briefs, yellowed with age, and above the waistband_ _rose the great mound of his belly, a shape like an overturned_ _bowl. His stomach, chest, and shoulders were thickly covered_ _with dark hair, and the pink nipples peeking through the thatch_ _were an incongruous sight, looking as though they belonged on_ _the body of a woman in an oil painting from another century._ _His round face was moist and flushed, and though he smiled at_ _the lawyer, pain lined his forehead._\n\n_\"Thanks for coming.\"_\n\n_\"What can I do for you, Lee?\"_\n\n_\"My back's gone out.\" He twisted his lips as though the_ _words themselves hurt._\n\n_\"I'm sorry to hear that.\"_\n\n_Mauer weakly waved away the lawyer's sympathy. \"It's happened_ _to me for years. There's a doctor who thinks he can fix it,_ _but that'd mean surgery. You know me\u2014no knives.\"_\n\n_In fact, the lawyer didn't know Mauer well at all, and the_ _revelations about both the man's back problems and his dread_ _of surgery came as news to the lawyer._\n\n_\"Usually,\" Mauer continued, \"if I just take it easy for a couple_ _days, the pain eases and then I'm okay until the next time it_ _happens. Not that I have any choice in the matter. I can't do_ _much more than lie flat.\"_\n\n_\"Okay.\" The lawyer was not a fastidious man\u2014he had,_ _after all, recently had his hands covered with pumpkin slime\u2014_ _but he guessed that Lee Mauer was going to ask for his help in_ _getting dressed and moving, perhaps up the stairs, and the_ _lawyer wasn't eager to put his arms around the fat man's sweaty_ _body._\n\n_\"So there's some things I just can't do for myself when I'm_ _in this condition. Goddamnit.\"_\n\n_All right, here it comes, thought the lawyer._\n\n_\"If you'd drive my car, I'd sure appreciate it.\"_\n\n_The lawyer wasn't prepared for that request, but then he_ _was thrown further off balance by something that hadn't occurred_ _to him earlier. If Lee Mauer couldn't get up, how had he_ _made the telephone call that brought the lawyer to where he_ _now stood?_\n\n_\"Your car, Lee?\"_\n\n_\"It's in the driveway.\"_\n\n_\"I saw it. . . .\" But the lawyer would have to use his own_ _car to take Mauer home, wouldn't he? Unless Lee wanted the_ _lawyer to keep Mauer's car until he was up and around. And_ _while the lawyer was trying to puzzle through this matter, the_ _answer to an earlier question was provided. Just under the rollaway_ _was the black shape of a telephone, and from under the_ _foot of the bed snaked the telephone cord. The long cord ran to_ _a wall where it was plugged into a phone jack, not secured but_ _hanging from the outlet by its wires. Raymond, Raymond, to_ _leave with so much undone . . ._\n\n_Mauer said, \"I wouldn't want the neighbors to see the car_ _here all night.\"_\n\n_Would the neighbors be likely to conclude something if they_ _saw the car parked there at seven_ _A_ _._ _M_ _. that they didn't conclude_ _now when they saw it late at night? The lawyer supposed they_ _would. After all, he had withheld his own judgment until, as he_ _put it, all the evidence was in._\n\n_Just at that moment, as if she had read his thoughts on matters_ _of proof, Alma Stoddard walked into the room. She was_ _wearing an old flannel bathrobe, and from the way her breasts_ _moved beneath the worn fabric, the lawyer guessed that she was_ _naked under the robe. He also surmised that she had been in the_ _room recently, and that its warmth was what had caused tendrils_ _of her hair to stick to her perspiring forehead._\n\n_\"His back . . . ,\" Alma said. She crossed her arms and held_ _her elbows. \"If you could just . . .\"_\n\n_The lawyer held up his hands. \"It's not a problem.\"_\n\n_She smiled gratefully and then, to save them both further_ _embarrassment, hurried from the room. Perhaps she had made_ _an appearance as a way of confessing to the nature of her relationship_ _with Lee Mauer. Your life is your own, the lawyer_ _would have said to her; you live it as you see fit. You don't owe_ _me any explanations._\n\n_As soon as she was gone, Lee Mauer said, \"I was thinking_ _you could just park the car a couple blocks away. In the lot over_ _at First Presbyterian, maybe. Then if I'm up and around tomorrow,_ _I'll go get it. You wouldn't even have to come back here._ _You could just leave the keys on top of the right rear passenger_ _tire. An old cop trick . . .\"_\n\n_The lawyer wondered what kind of trick that was supposed_ _to be, but he didn't ask. Just as he didn't ask why Alma couldn't_ _drive the car._\n\n_\"Where are your keys, Lee?\" The lawyer preferred to get on_ _with the mission and return to his home. He planned to describe_ _this scene to his wife and allow her the satisfaction of having_ _her suspicions confirmed._\n\n_\"Pants pocket.\" He pointed to the clothing draped over a_ _rocking chair with a torn cane seat._\n\n_The lawyer picked up the trousers and carried them to Lee_ _Mauer._\n\n_Before he turned the car keys over to the lawyer, Lee Mauer_ _separated them from the rest of the keys on the chain. Holding_ _out the keys, he said, \"Park it under the light, if you can.\"_\n\n_The lawyer was able to leave the car just where he had been_ _instructed to park it, but before he reached inside the wheel well_ _to put the keys on top of the tire, he surveyed the area. Most of_ _the nearby houses were dark, and where there was a lighted_ _window, he saw no human form looking out. When he was as_ _certain as he could possibly be that he was not observed, he left_ _the keys and walked away. Snow was falling continuously now,_ _and though the flakes were still widely spaced and airy, by_ _morning the car would be veiled in white._\n\n_There were two routes he could take back to his home and_ _his waiting pumpkins. He could walk north and then east, staying_ _on sidewalks all the way, until he came to his own block, or_ _he could plot a diagonal course, which would allow him to_ _come up through the Stoddards' backyard. That was the most_ _tempting because it would allow him to peer into the basement_ _where Lee Mauer lay and where, perhaps, he and Alma had resumed_ _their sexual frolic. After all, for what other reason would_ _Lee insist that the lawyer not bring the car keys back to the_ _house?_\n\n_But the lawyer had not traveled far in that direction when a_ _number of thoughts stopped him. And it was not just his twined_ _senses of pride and propriety that held him back. Enough snow_ _had fallen that his footsteps across the lawn would be revealed,_ _and he couldn't do anything to cover or conceal his tracks. Besides,_ _Lee Mauer's injury, Alma Stoddard's sweaty forehead\u2014_ _whatever the act they had been engaged in, they were doubtless_ _finished by now. And if not, Lee's back would prevent them_ _from resuming._\n\n_So the lawyer took the longer route, staying on the sidewalks_ _that still held enough of the day's heat that the snow_ _couldn't accumulate on the concrete. He wished now that he_ _had worn a coat. The web of dark hair covering Lee Mauer's_ _torso probably would have done as much to keep out the cold_ _as the lawyer's white shirt did._\n\n_Besides, even if the lawyer had indulged in that brief_ _voyeuristic impulse, even if he had looked in that window and_ _seen Lee and Alma naked again on that narrow bed, the sight_ _wouldn't have satisfied his real curiosity about those two. He_ _may not have been particularly knowledgeable or sophisticated_ _when it came to sexual practices, but neither was he na\u00efve; he_ _was aware of a range of carnal activities that men and women_ _could engage in, some more likely than others to cause a man's_ _back to go out. But the question that most tormented the_ _lawyer couldn't be answered by peeking through a window._ _What truly perplexed him had to do with the fully clothed Lee_ _Mauer and Alma Stoddard. How could a fat, bald cop, a loud,_ _vulgar man with two chins and no charm persuade an attractive_ _woman\u2014a woman less than a year into her grief\u2014to disrobe_ _and . . . and . . . what? To love him?_\n\n_Once the lawyer's house came into view, he picked up his_ _pace to hurry toward its warmth. On his block, from front_ _porches and kitchen windows, his neighbors' jack-o'-lanterns_ _grinned, leered, or stared in cross-eyed disbelief at him as he_ _passed. His own pumpkins were of course still without expression._ _He had gotten no further than cutting open their skulls_ _and scooping out the tangled mess inside._\n\nThat story eventually appeared in _The North Coast Fiction Review._ My father is, of course, the model for the lawyer protagonist, and Lee Mauer (Ken Crowder in the fiction) did injure his back and call on my father for assistance. That incident occurred not on Halloween but in the week before Christmas, and beyond the fact that my father returned home shaken, I never knew what happened when he went to Lee Mauer's aid. If he told my mother, she never told me.\n\nA little more than a year after I sat in an almost empty First Lutheran Church for Raymond Stoddard's funeral, I attended another ceremony in the same building. Once again, few were present, and again, a Stoddard was the reason we were gathered. This time, however, the sparse turnout was predictable. Alma Stoddard had not invited many people to witness her marriage to Lee Mauer. Furthermore she must have decided that since it wouldn't be possible to keep her dead husband out of people's thoughts, she might as well go ahead and recite her vows in front of the same altar where his coffin had been so recently stationed.\n\nAnd those of us there . . . I have no doubt we all spent our time in the pews in the same manner\u2014by trying to determine what had been in Alma Stoddard's heart and mind in the days, weeks, and months leading up to the wedding. What, we wondered, moved her to say yes to Lee Mauer's proposal? Among the countless varieties of human love was there one that explained what the widow of a murderer and a suicide might have felt for a fat policeman?\n\nFor her part, my mother didn't believe affection of any kind was involved. When she learned of the impending marriage, she had a simple explanation.\n\n\"She needs protection,\" my mother had said.\n\nMy father scoffed. \"In Bismarck? On Keogh Street? Protection from what? From whom?\"\n\n\"You're a man,\" my mother answered. \"I wouldn't expect you to understand.\"\n\n\"Enlighten me.\"\n\n\"If you're lying in bed alone at night, and you hear something, it's no help to tell yourself there's nothing to be afraid of. You feel what you feel. And perhaps for Alma, having Lee Mauer beside her is preferable to having that feeling.\"\n\nAs he had in his conversation with Ross Wilk, my father rejected someone's theory of human motivation. And again, he had no formulation of his own to put in its place, for when my mother pressed him\u2014\"All right, then. You tell me. What's behind this? Has your friend taken you into his confidence? You seem awfully incurious about the upcoming nuptials.\"\n\nMy father answered with a question of his own. \"Is that what you've felt, when I've been out of town? Afraid?\"\n\n\"I'm not Alma Stoddard,\" my mother said, thereby ending both lines of inquiry.\n\nI wasn't sure why my parents insisted that I attend the wedding. Perhaps it was a desperate attempt to swell the congregation. Perhaps they wanted to bring home to me, through the unmistakable parallelism of the two Stoddard rites, a lesson about the cycles of human life. If that was the case, it was wasted on me. I did observe that after the funeral and the wedding, similar fare\u2014ham sandwiches and date bars\u2014was served in the church basement, but I had no insight into the nature of the rituals that attended grief and joy.\n\nAfter only a few minutes of milling around at the reception, Gene nudged me and asked, \"Want to grab a smoke?\"\n\nI seldom smoked, and Gene and I had reached the point where we rarely said more to each other than a passing hello in the halls at school, but I followed him out of the fellowship room. How could I refuse him on the day when his mother had married a substitute for his father?\n\nIt was too cold to go outside, so we made our way over to another section of the church basement, the darkened wing where we had both once attended Sunday school classes.\n\nInside the boys' lavatory, with its undersize porcelain fixtures for its undersize patrons, Gene offered me a Camel. We lit up and before I had taken my first timid drag, he asked his question.\n\n\"Would you be my best man?\"\n\nSince my father had just served as Lee Mauer's best man, I assumed that Gene was asking me a hypothetical question, that the occasion had revived in him some of the affections of our once-close friendship, but I hadn't even formed a response on that basis when Gene added, \"Marie's late.\"\n\nI thought Gene meant that Marie was supposed to attend the wedding\u2014I had wondered about her absence\u2014but that her tardiness had caused her to miss the ceremony.\n\n\"And she's never late.\"\n\nStruck suddenly by the full force of what Gene was telling me, it was all I could do to stop myself from saying what a na\u00efve, self-deceiving, heartbroken parent might say _\u2014how could this_ _happen?\u2014_ as if the only consolation left was in learning that it was the result of a single, mindless, aberrant moment.\n\nBut Gene addressed the matter without my question, not that his response offered any satisfaction. \"I don't even know how the hell it happened. . . .\"\n\n\"Oh, come on.\"\n\n\"Really. We hardly even\u2014Oh, what the hell.\" As long as I had known him, Gene had had a nervous little habit of tugging at a lock of his hair right at the hairline. When he did that now with the same hand that held the cigarette, I expected to hear the sizzle of burning hair. But, deftly, he managed the maneuver by pinching his hair with his thumb and little finger. \"If that's the way it is,\" he said, in a phrase I had never heard him use before, \"that's the way it is.\"\n\nI knew I was supposed to commiserate with him and his predicament, but at that moment it was my own anguish I was concerned with. As coolly as I could, I said, \"So, when's the wedding?\"\n\nSmoke tumbled out of his mouth with his sardonic laugh. \"How about as soon as I get out of the hospital? Because that's probably where I'll end up if her old man finds out I knocked up his daughter. Unless he has a heart attack before he can kick my ass.\"\n\n\"Jesus, I don't envy you.\" In truth, I would have traded places with him in an instant. Even if a beating were a part of the exchange.\n\n\"Yeah, and then good old Lieutenant Mauer would probably want to get his licks in because I'd have upset his precious new bride.\"\n\nGene backed up and wedged himself into the space between the sink and the towel dispenser, a position that was doubtless a physical analogue for how cornered he felt in his life.\n\n\"How far along is she?\" It may have seemed the most natural thing to ask, but I was still desperate to hear something that would make this news easier to bear _\u2014I know exactly how pregnant_ _she is because I got her drunk and raped her on\u2014_\n\n\"Fuck if I know. She's supposed to keep shit like that figured out.\"\n\n\"You don't sound like you're ready to be a daddy.\"\n\n\"Ready? I haven't even . . . Yeah. Raymond Stoddard's kid has a kid. . . . What a fucking joke.\"\n\n\"And this is for sure? You're getting married for sure?\"\n\nHe scraped the ashes from his cigarette on the edge of the sink. \"If she told me she wanted to go to the Florence Crittendon Home, I sure as hell wouldn't argue. Or if she'd go live with her sister in Minneapolis and have the kid there. Or if she'd just . . . Hell, I don't know.\"\n\nThat did it. When Gene spoke of his wish to escape from a predicament I would have given anything to be in\u2014I became enraged.\n\nAnger, however, almost always renders me inarticulate, and I couldn't come up with anything more to say than, \"Find another goddamn best man.\" A part of me must have sensed how inadequate that line was, because I accompanied it with a gesture, and for once action and intention matched. I flicked my cigarette at Gene, and the butt struck his lapel, spraying sparks down the front of his suit.\n\nMaybe, I thought, as I walked out of the lavatory, the next time he puts on his suit\u2014on his and Marie's wedding day, perhaps\u2014he'll see a burn mark or two and remember this occasion.\n\nBut as I was walking home from the church in the glittering January sunlight, after having told my parents that I was leaving, I realized that Gene probably didn't have any idea why I was angry. Why would he? I'd certainly never told him how I felt about Marie, and nothing in our shared history would lead him to believe that I was the sort to become furious over his unchivalrous attitude. Then again, he was Raymond Stoddard's son, and no training in the world could have better prepared him for the irrational behavior of others.\n\nLee Mauer not only married Raymond Stoddard's wife and moved into Raymond's house, like Raymond he brought the sound of sirens to Keogh Street.\n\nThis happened a few years after I moved from my parents' home, but my sister remembered the incident vividly. Late on a Saturday night she was in the bathroom, scrubbing off her makeup and trying to stop singing to herself the songs from _Brigadoon,_ the fall musical put on by Bismarck High School and in which she had performed.\n\nThe water was running, and that may have prevented her from hearing the siren earlier, but once she did, she knew it was coming toward Keogh Street. I didn't disbelieve her account, but my sister sometimes tries to wring more drama from a moment than it actually holds. She is also prone to claiming powers of prescience that are nothing more than a storyteller's hindsight.\n\nBut whether my sister's premonition at the time was fictional or authentic, the siren was real and fast approaching. And just as it had years before, the ambulance pulled up to the Stoddard house. But this time, my sister noted, when it sped away, its siren was wailing as loudly as when it had arrived.\n\nLee Mauer had had a heart attack, and as so often happens, there had been clues that such an event could occur and they had been ignored. For weeks he had been experiencing bouts of nausea, excessive sweating, and pain in his chest, in his neck, and between his shoulder blades. Like many men, however, he rejected the seriousness of his symptoms and treated himself with Bromo-Seltzer. But that night his heart clenched for good, and Lee Mauer was dead by the time the ambulance arrived at the hospital.\n\nAlma Stoddard Mauer would once again sleep alone, but she was not the only woman on Keogh Street who had her bed to herself. When the siren came for Lee Mauer, the sound did not disturb my father's rest. He was sleeping 140 miles away in Wembley in the house that had once belonged to his parents and where he was then living with his brother.\n\nMy parents first separated, or so they both told it, when my father began traveling frequently to Wembley to assist his brother with a project that required a lawyer's skills. Uncle Burt had decided that the land that was his parents' homestead when they first came to North Dakota should be restored to the family. With my father's assistance, my uncle determined where the original house and acreage had been, and he negotiated to buy the property from its present owners. Before long, the brothers were working to remodel the dilapidated farmhouse where their parents had briefly lived before they'd moved into town.\n\nGradually more and more of my father's life was spent in the town he grew up in. Along with the small farm, he and his brother owned the house in Wembley, and after a time they were both living in the rooms that had been theirs as boys. Through that small real estate venture with the homestead, my father found other work in and around Wembley, and soon his services were more in demand there than in Bismarck. (I should add here that just as I was no longer living on Keogh Street when Lee Mauer died, neither did I make my home there when my father's relocation became permanent. His business trips to Wembley started when I was in college, and we never again dwelt under the same roof.) But had the advantages of my father relocating to his hometown been only financial, my parents would have found a different solution. My mother and sister would have moved to Wembley, if not immediately, then upon my sister's graduation from high school. Or my father would have said his Bismarck income was sufficient. No, something else kept him in Wembley and apart from his wife of twenty-plus years.\n\nBut neither of my parents ever talked much about the dissolution of their marriage, and when they did, it was only in the most general way. Their silence on the subject added more mystery to the way life was once lived on Keogh Street, that stretch of houses that had\u2014pre\u2013Raymond Stoddard\u2014seemed emblematic of ordinary, predictable human existence.\n\nAnd if I believe that Raymond Stoddard's fate played a role in the breakup of my parents' marriage, then I must also believe that their marriage began to crack very soon after that January day when the police cars skidded down Keogh Street. That meant I was there to observe the breakup in its first if not final stages.\n\nAdolescents, however, are notoriously limited in their ability to see anything that lies outside their immediate concerns, and in that regard I was not merely a typical representative of my age group, I was probably worse. As an adult, I realize how frequently my self-involvement impaired my vision. For example, during part of the period when my parents' marriage was no doubt diminishing, all my powers of observation were instead keen for signs of growth. Specifically, I was watching for evidence of Marie Ryan's pregnancy, and I finally saw what I was looking for on a Friday night in late March.\n\nJulie Benske's parents were out of town, and in their absence Julie decided to hold a party. She tried to keep the numbers down, but to no avail. By the time my friends and I arrived, cars practically circled the block, and the house pulsed with the sounds of a hi-fi turned to full volume and with the shouts of teenagers who were either drunk already or intent on getting there. The neighbors must have been very tolerant, or else they believed Dr. and Mrs. Benske had sanctioned the party.\n\nI hadn't been there long when I saw Marie. And I heard her before I saw her. As soon as I entered the kitchen, I caught her laughter. She was with a group standing around a makeshift bar, and her gaiety traveled across the room as easily as her laugh. Why, I wondered, would she be as happy as she seemed to be, given her circumstances? Perhaps she and Gene had settled on what they'd do about their predicament, and what I saw as happiness was merely relief, even if for Marie Ryan a solution meant taking on the name more notorious than any other in the state. The crowd made it impossible to get closer to her, and I wasn't sure that I dared to anyway. I hadn't spoken to her since Gene had told me she was pregnant, and who knew how he had represented my boys' room behavior to her? I couldn't see Gene anywhere at the party, and I wasn't eager to.\n\nWhile the hard liquor was upstairs in the kitchen, the beer was in the basement, and that was where my friends were headed. Without taking off my coat, I followed them.\n\nGene wasn't downstairs either, and then it was my turn to feel relieved. Which may be the reason I drank as much as I did. Or maybe it was because I was at the same party as Marie, though she might have dwelt in a different universe for all the good it did me.\n\nSo I huddled on the rec room floor and guzzled beer after beer, rising only when my bladder was full.\n\nAnd I was on my way back up to the first floor\u2014the downstairs bathroom was occupied\u2014when I again caught sight of Marie.\n\nIn Bismarck in 1962 the markers signifying the status of the community's wealthier members were few and subtle. Even the city's doctors, lawyers, and business owners lived in modest houses that didn't stand apart much from the homes of their less prosperous neighbors. But there were differences, and I was standing on one of them.\n\nThe Benskes' home was plushly carpeted, even the stairs, and just as I was ascending them, Marie Ryan was coming down, and I stopped a few steps below her. The softness underfoot was such an uncustomary sensation that I felt for a moment as though I might keep sinking until I dropped from view, something I might have hoped for, so flustered was I by our meeting.\n\nShe laughed and, spreading her arms and legs wide to block my passage, said, \"You can't get past unless you say the secret word.\"\n\nI've never been quick-witted, and, then as now, drink only increases my tongue's sluggishness. No reply, clever or otherwise, came to mind.\n\nBut perhaps my being tongue-tied had another cause. With the three steps between us, Marie's waist and abdomen were right at my eye level. I did some mental arithmetic, difficult in my condition and further complicated by my ignorance of some matters of reproduction and anatomy. Gene had asked me to be his best man in January, and Marie was what\u2014two or three months pregnant then? It was March now, which meant she could be at least four months along. . . . But she was wearing a white blouse tucked into a snug skirt, and if a pregnancy had swelled her tummy in the least, wouldn't it have been visible by now? Even clothed, however, wrapped tight in a girdle's spandex, a skirt's tight wool, a brassiere's hooks and straps, and a blouse's buttoned-up cotton, it was apparent that her body's sensual configuration was exactly what it had been the previous summer in a bathing suit.\n\nSuddenly I had a new theory for her high spirits. She wasn't pregnant\u2014she couldn't be, not looking like that\u2014and perhaps she had only recently learned of her true condition. She wasn't celebrating her upcoming marriage\u2014she was jubilant because it wasn't necessary.\n\nElated as I was to make this discovery, I kept my happiness tamped down. Not difficult, since in the essential way, her circumstances hadn't changed. \"If you're looking for Gene,\" I said, \"he's not down there.\"\n\nShe relaxed her stance, leaning against the stair railing. The pose was seductive, although I'm sure that was not how she intended it. \"I know where he is. And isn't. He chose not to come tonight.\"\n\n\"Not his kind of party?\" The question was legitimate. None of Gene's newer acquaintances was likely to be at a gathering at the Benskes', no matter how much alcohol was present.\n\n\"He's at home.\"\n\n\"Is he sick?\"\n\n\"He's fine. But considering the occasion, he didn't feel like going to a party.\"\n\n\"The occasion?\"\n\nShe tilted her head and gave me a sidelong flirtatious look, or so I perceived the gesture. She probably wasn't sending that message, but she couldn't help how I received it.\n\n\"Think about it,\" Marie said. \"It's Gene's dad's birthday.\"\n\n\"Oh, yeah. I forgot.\"\n\nThe date had never had any purchase on my memory, even though on one awkward occasion I was a part of Raymond Stoddard's birthday celebration.\n\nMaybe there were Bismarckers of that era who dined out often, but that certainly wasn't true of my parents\u2014we could count on fewer than four or five such experiences each year, and those always occurred spontaneously, when my mother decided at the last minute that she didn't care to prepare the family's meal\u2014and it was even less true of the Stoddards. But five years earlier, Gene had asked me if I'd like to come along with him and his family for a birthday supper at the Wagon Wheel, a restaurant on the city's east side.\n\nThe Wagon Wheel was not a supper club or steak house. It was a small diner, more likely to be patronized by truck drivers than Bismarck families. I wasn't sure why we went there, but it seemed to have something to do with Mr. Stoddard's preferences, some belief he held that the Wagon Wheel's chili or fried chicken was superior to any other restaurant's.\n\nYet his unhappiness at being there became obvious almost as soon as we slid into our booth. A cold rain had begun to fall, and Mr. Stoddard worried that it might freeze on the streets, making even the short drive back to Keogh Street treacherous. The food was not to his liking\u2014\"grease, nothing but grease,\" he complained\u2014and the waitress failed to bring the full order.\n\nMrs. Stoddard tried to cheer him up. \"Birthday boy,\" she persisted in calling him. \"Let's see a smile from the birthday boy.\" His mood only darkened, and before the meal was finished, he fell into a trancelike silence. Seated next to the window, he stared out at the puddles gathering in the parking lot's gravel. At one point he pressed the heel of his fist to the glass and rubbed small hard circles, as if the window were frosted or fogged and needed to be cleared. It did not, but we soon looked away, certain that we couldn't see what Raymond Stoddard saw, no matter how transparent the surface.\n\n\"But I just decided,\" Marie said, \"no matter what day it is\u2014I'm going out. I'm going out and I'm going to have a good time. Just because March thirtieth is a bad day for him doesn't mean everyone else has to suffer along with him.\"\n\n\"That's right. Your name isn't Stoddard.\" It was as close as I dared come. Now, if she wished, she could say, But soon it will be.\n\n\"Exactly.\"\n\n\"Did you say all this to Gene?\"\n\n\"More or less.\"\n\n\"How did he take it?\"\n\nSomething in her shrug told me they had probably quarreled over the matter.\n\n\"So,\" I said, \"you're taking the day off.\"\n\nShe smiled, apparently pleased with the notion. \"Sort of.\"\n\n\"How's your vacation going so far?\"\n\n\"Very well.\"\n\nWho knows what went into the mix that caused me to consider what I was about to say? Was it her good spirits? Our staircase rapport? Her disenchantment\u2014even if only for the night\u2014with Gene? The beers I'd drunk certainly contributed. But no matter what the reason, I decided I'd tell her about Gene's crude remarks about her condition\u2014at least what they thought her condition was at the time.\n\nBefore I could frame the words, however, Doug Bauer appeared suddenly at the top of the stairs. \"Cops!\" he said. \"In the driveway!\"\n\nI froze, but Marie acted instantly. She hurried down the stairs, sweeping me along with her.\n\nThe Benskes had a walk-out basement, and we scurried toward that door. Somehow, even without an explicit warning, everyone downstairs knew that the party was being raided, and there was much scrambling to get out. Together Marie and I pushed our way through the door and out into the backyard. People scattered in every direction, and if they were like us, they had no real destination in mind. We just knew we had to get as far from the Benske home as we could and in so doing stay away from the streets and streetlights.\n\nMarie had to be cold\u2014she had left her coat behind\u2014and her low heels kept slipping on and breaking through the crusted snow we had to run across, but I heard only laughter from her and not complaint.\n\nEventually we decided that it would be safe to return to the sidewalks, so we cut through a yard, traveled down an alley, and came out on a street blocks from the Benskes'. At a stop sign just ahead we saw Rick Withers's car, jammed with fugitives from the party, and we ran to join them.\n\nThey made room for us, which meant that I was squeezed against the back door. As tight as the fit was, I didn't mind because Marie was forced to sit on my lap. I felt her shivering subside as the car's heater and all the bodies\u2014there must have been nine or ten people in the car\u2014generated their warmth. I smelled the sweet musk of her perfume and the slightly medicinal scent of vodka on her breath. When she turned her head, strands of her hair brushed across my face. I would have gladly remained in the back of that old Chevrolet forever!\n\nBut Rick Withers wasn't interested in driving around. Like almost everyone else who had been at the party, he was still shaken over his close call. He wanted to know where he should drop off his passengers, and sadly, Marie's house was nearby.\n\nHe pulled into her driveway, and she jumped out immediately. \"Thanks for the ride,\" she said to Rick. And to the rest of the car, she added softly, \"I'll bet we wouldn't have been as tightly packed in jail!\"\n\n\"Wait!\" I said. \"I'll walk you.\"\n\nI was still a little drunk, but fear and cold had sobered me somewhat. Fear . . . I _had_ been afraid. If I had been arrested that night, my parents would have been angry and disappointed, and they certainly would have decreed a penalty of some kind. But it would have been relatively mild. They were not nearly as strict or severe as, for example, Marie's parents were. Yet throughout our getaway and flight, she had displayed no fear. In the car she hadn't gone on, as some of the partygoers had, about how narrow their escape had been or about how awful their punishment would be if they were caught.\n\nWas Marie Ryan brave because she had once stood loyally beside a Stoddard as that family went through their terrible ordeal, and in the process had she acquired a courage that would serve her forever after, no matter what the situation? Or had she always possessed the kind of bravery that allowed her to step forward and stay when almost anyone else would have hung back? But even if the source of Marie's courage might have been unknowable, the fact of it was without dispute.\n\nAnd perhaps I took a cue from her character and elevated my own courage that night, for I attempted something that was for me uncharacteristically bold.\n\nOnce again we were in her garage, darker now than it had been on the day I had walked her home from Raymond Stoddard's funeral, and once again she stood on the step above me just where she'd be when she'd kiss Gene good night.\n\n\"Do you mind if I kiss you?\" I asked.\n\nAt least I asked, though I leaned forward with the hope I'd get the answer I wanted.\n\nShe didn't say no. Neither did she laugh at me or slap me or turn away in disgust. She had too much grace for any of those actions. Instead, she gently said, \"Do you think that's a good idea? I'm still going with Gene.\"\n\nWhat could I say to that\u2014let me make my betrayal complete? She was offering me the opportunity to be better than I was.\n\nWithout saying another word I hurried from the garage. The night's cold couldn't touch me, burning as I was with my own shame.\n\n_Monty Burnham considered making his confession on the day_ _when the Sherman tank he commanded was mired in black volcanic_ _ash, and Japanese artillery fire was shrieking all around._ _He believed then that it was only a matter of time before one of_ _the armor-piercing 150-millimeter shells found its target and he_ _and his crew of five were incinerated in their vehicle. For good_ _reason the tanks had been nicknamed \"Ronsons.\" And with_ _death imminent wouldn't it have made sense to blurt out a soul-cleansing_ _statement right at that time, especially since the man_ _whose wife Burnham had fucked was right there in the tank_ _with him? And if the confession weighed down one soldier at_ _the same time that it unburdened another, so what? They were_ _all doomed anyway._\n\n_He tapped his friend, Corporal Raymond Stoddard, the_ _tank's loader, on the shoulder with the full intention of owning_ _up to what had happened between him and Raymond's wife,_ _Alma. Yet when Stoddard turned toward him, Burnham_ _couldn't speak. He and Stoddard had attended the same high_ _school in Wembley, North Dakota, and they had played football_ _together, and at that moment his former classmate, his features_ _almost erased by grease and dirt, looked, in his leather tank helmet,_ _just the way he had looked on the playing field. Monty_ _Burnham shook his head and waved away Raymond's inquiring_ _glance. He could have confessed to a fellow soldier, a subordinate,_ _but not a former teammate._\n\n _Two weeks later, at their island camp, Monty and Raymond_ _Stoddard were sitting outside on planks placed on top of_ _stumps, watching_ The Affairs of Susan _with a hundred other_ _men. For the hour before the movie began, Monty and Raymond_ _Stoddard had been drinking, passing back and forth a_ _pint of Four Roses with two other members of the tank crew. So_ _maybe it was the whiskey. Maybe it was something Joan_ _Fontaine said to George Brent up on the screen. Whatever the_ _cause, the impulse to confess returned, and Monty Burnham_ _nudged Raymond Stoddard. As casually as he might comment_ _on something happening in the movie, he said softly, \"Hey. I_ _fucked your wife, you know. Back in Texas.\"_\n\n_The rain that had been falling steadily for days had subsided_ _now to little more than a mist, just enough to keep the moviegoers_ _wet but doing nothing to cool them. Monty and Raymond_ _were sitting apart from everyone else, in a back row near the_ _projector, and in its flickering beam the drizzle separated into_ _individual droplets that looked like tiny bits of silver floating in_ _the light._\n\n_Raymond Stoddard didn't acknowledge Monty's remark in_ _any way, but when Monty leaned in to repeat his disclosure,_ _Raymond said, \"I ain't deaf. I heard you.\"_\n\n_Someone else obviously heard as well, and not just Raymond's_ _response. From one of the rows ahead a soldier said,_ _\"Christ. There's a buddy for you.\"_\n\n_Immediately Monty wanted to seek that man out, to explain_ _that he and Raymond's wife had a history, that they had dated_ _throughout high school, and that she had all but acknowledged_ _that marrying Raymond had been a mistake. First, however, he_ _had to make Raymond understand._\n\n_\"It's not like I planned it,\" Monty whispered._\n\n_\"If you say so.\" Raymond kept his eyes focused on the_ _screen. On a tropical night like this one, how you could tell_ _whether the moisture on a man's face was rain, sweat, or tears,_ _Monty had no idea._\n\n_\"It's not like it was even my idea. Not entirely.\"_\n\n_\"Uh-huh.\"_\n\n_\"It was just something we had to get out of our systems._ _Both of us.\" Monty had now returned comfortably to the mental_ _script he had prepared well in advance of this occasion._\n\n_Raymond Stoddard, however, didn't offer any of the lines_ _that Monty had rehearsed on Raymond's behalf. Without saying_ _another word, Raymond slid off the plank, as if it were one of_ _Dennis O'Keefe's remarks that upset him rather than Monty's_ _confession. Monty almost shouted out an order\u2014Soldier, get_ _back here!\u2014before he realized that would not be the appropriate_ _tone for the transaction he was trying to conduct with Raymond_ _Stoddard._\n\n_From boyhood Monty Burnham had known that in his nature_ _ran two parallel streams. One was the pleasure he took in_ _inflicting pain. Oh, not serious pain\u2014just teasing that sometimes_ _went on a little too long, roughhousing that might cross a_ _border but still didn't venture far into the territory of hurt, or_ _joking that had a jagged edge. The other current running_ _through him was the need to be liked and the need to overcome_ _any prejudice, animosity, or grudge someone might hold against_ _him. Occasionally these two streams could converge, and_ _Monty Burnham would find that the person whose goodwill_ _had to be restored was exactly the person he had insulted, injured,_ _or grieved. When this occurred, when he had to persuade_ _a girl he had stood up to let go of her anger or a soldier he had_ _ridden too hard to give up his resentment, Monty would attempt_ _to set things right with talk, with charming, candid talk._ _He would usually be able to accomplish this without explicitly_ _asking for forgiveness or apologizing, but he knew he always_ _had those in reserve. And Monty Burnham could not be contented_ _until approval was once again flowing his way._\n\n_This was why Monty Burnham left the movie to find Raymond_ _Stoddard. The theater, however, had been set up near a_ _small forest whose trees had provided the stumps for movie_ _viewing. If Raymond wandered into that dense, dark grove,_ _Monty might never track him down._\n\n_He was standing at the entrance to the trees, trying to look_ _down their lightless corridors, when he heard the clank of a_ _Zippo lighter off to his left. There was Raymond Stoddard,_ _lighting a cigarette and shielding it expertly to keep it burning in_ _the rain._\n\n_\"I had a few more things I wanted to say,\" Monty said._\n\n_\"And you think I want to hear them.\" Raymond stood next_ _to a knee-high pile of brass\u2014anything above .50 caliber in_ _size\u2014that only days before the infantry had been forced to pick_ _up from the battle lines. The rain brought a dull glint to the_ _shell casings._\n\n_\"I guess I figure you have a right to know.\"_\n\n_\"About you and Alma. I know all about it. She told me.\"_\n\n_\"What happened back in Killeen? She told you that?\"_ _Monty considered the possibility that Raymond now only pretended_ _to know about what had gone on in that hotel bathroom,_ _in order to save face. Monty decided to test him._ _\"Everything? She told you everything?\"_\n\n_\"Enough.\" The smoke Raymond Stoddard exhaled seemed_ _to hang in the air as if it couldn't make its way through the drizzle._ _From the nearby forest came the hisses, chirps, scrapings,_ _whines, and pipings of strange insects and even stranger birds._ _If Monty stepped even a few feet into the jungle, some of those_ _calls would cease immediately while others would become_ _louder and more rapid._\n\n_\"Then I suppose you know I had her like you never did.\"_ _The streams in Monty had diverged once again, and there was_ _no doubt which current was running strongest now. He did find_ _it odd that war hadn't satisfied that need to hurt. Then again,_ _maybe it had only stirred the desire._\n\n_Raymond Stoddard, however, was not outwardly perturbed._ _\"You mean I never had her like that_ before. _I sure as hell did_ _after.\"_\n\n_\"Glad I could show the way.\"_\n\n_To that Raymond Stoddard said nothing. He simply raised_ _his hand to his forehead in a parody of a salute. God_ damn, _would nothing rile this man?_\n\n_\"I don't know,\" Monty said, \"how a man can talk about his_ _wife like that.\"_\n\n_Again, Raymond said nothing, but this time he snorted_ _softly, and the sound coincided almost exactly with a burst of_ _laughter that came from the rows of soldiers staring up at the_ _screen. Was Raymond even listening to him, Monty wondered,_ _or was most of his attention focused on the movie?_\n\n_Monty took a step back and straightened his shoulders._ _\"Well. I just needed to get that off my chest. I can't imagine it_ _was too easy to hear. You probably feel like taking a poke at_ _me. Can't say I blame you. Officer or not, I'd probably slug a_ _man who told me a tale like this one.\"_\n\n_\"Take a poke at you? Take a_ poke _at you?\" This time Ray_ _mond's laugh came out as sharp as a dog's bark. He flicked_ _away his cigarette and stepped so close to Monty that Monty_ _could smell the rank, curdled odor of tobacco and whiskey on_ _Raymond's breath. \"I never climbed into a tank with you but_ _that I thought this would be the time I pulled the pin on a_ _grenade and blew the both of us to kingdom come.\"_\n\n_\"And every other soldier in there with us?\"_\n\n _In the summer of 1935 Monty Burnham was eleven years old_ _and returning from a morning spent fishing in Ripley's Creek,_ _just outside Wembley. He was less than a mile from his home_ _when he came across an excavation site near an abandoned_ _farm. Curious as to why a sizable hole had been dug out there_ _in the country, Monty climbed to the top of the pile of dirt and_ _sod in order to survey the entire scene._\n\n_He guessed someone was digging the foundation for a_ _house, but the hole was crudely, unevenly dug, and Monty saw_ _no sign that any work had been done recently. While he was_ _wondering how he might have missed this alteration to the_ _landscape on the other occasions when he'd walked out to Ripley's_ _Creek, three boys approached. Monty didn't notice them_ _until they were right behind him, and then they caught his attention_ _by throwing a dirt clod that barely missed his head._\n\n_Monty turned quickly. The three had spread themselves out_ _at the base of the mound as if they planned to attack him from_ _different directions. From their similar high foreheads, close-set_ _eyes, and dirty, ragged overalls Monty guessed they were brothers._ _The oldest was probably a year or two older than Monty,_ _but he was so skinny Monty figured he could handle him. The_ _other two might have been twins, and though they were smaller_ _and younger than Monty, they had a determinedly nasty look_ _that said they would have been trouble even in a fair fight. And_ _three against one wouldn't be fair. . . ._\n\n_\"Hey,\" Monty said in a voice as friendly as any he owned,_ _\"you live around here?\"_\n\n_\"Yeah,\" the oldest one said, \"and you're trespassing.\"_\n\n_Monty pretended to look over the surrounding countryside._ _\"Nothing says this land is posted.\"_\n\n_\"Well, now you know.\"_\n\n_One of the twins kicked at the fishing pole Monty had set_ _down before he'd scrambled up the dirt pile. \"And you ain't_ _fishing the creek no more without our say-so.\"_\n\n_\"All this is our property now,\" the other twin added._\n\n_\"Not the creek. Nobody can buy the creek.\" Monty gestured_ _toward the water, and as he did, as if on cue, a red-winged_ _blackbird whistled its three notes from that direction._\n\n_\"Like hell,\" the oldest said, and when he reached down to_ _pick up a dirt lump, it was plain he'd had enough of oral argument._\n\n_Monty tried once more to extricate himself from the moment_ _with words. \"So if I ask first from now on, you'll let me_ _fish in your creek?\"_\n\n_The twins were silently sorting through the dirt at their feet._ _Their feet were bare. They were searching for rocks._\n\n_Because of the hole on the other side, Monty couldn't take_ _flight down the hill away from the brothers, yet if he tried to_ _run in any other direction, he'd only charge right at them. And_ _maybe, he decided, that was exactly what he should do. Charge_ _them, swinging wildly all the way, and hope that he could get_ _past them onto the road and then outrun them. Even if he took_ _a few punches in the process, that would be better than standing_ _here and allowing himself to be a target for their rocks and dirt_ _lumps._\n\n_Before he could put his plan into action, however, someone_ _else appeared on the scene._\n\n_From out of a field of tall grass came a boy Monty's age. He_ _too was carrying a fishing pole, and as he walked, grasshoppers_ _leaped in arcs all around him. It was Monty's friend Raymond_ _Stoddard, and he must have been fishing upstream from Monty,_ _near the bend in the creek where the cottonwood trees shaded_ _the water._\n\n_Raymond said, \"Hey, Morris. What the hell are you and_ _your snotnose brothers up to?\"_\n\n_Morris dropped his dirt clod, but the twins continued to_ _gather stones._\n\n_\"This fella's trespassing. . . .\"_\n\n_Raymond also positioned himself at the bottom of the dirt_ _pile, but he stood apart from the other three. \"You're the ones_ _trespassing. Just because you moved into that old barn don't_ _mean you own the place.\"_\n\n_\"We're gonna build here. My dad says\u2014\"_\n\n_\"Your folks are nothing more than squatters. That's what_ _my old man said.\" He set his fishing pole down carefully. To_ _Monty he said, \"What do you say. You want to come down here_ _and help me beat the shit out of these assholes? Maybe that'll_ _send 'em back where they came from.\"_\n\n_Monty, however, could not call up a desire for brutality on_ _such short notice. Furthermore, he could not abandon so_ _quickly his earlier strategy, which was to escape through charm._\n\n_\"Where are you boys from?\" he asked his enemies._\n\n_Raymond answered for them. \"Iowa. Lost their farm down_ _there and now they're gonna mooch off their North Dakota relations.\"_ _To Monty he explained, \"Their mom and my mom are_ _sort of cousins.\"_\n\n_Monty believed that conversation would now ensue among_ _the five of them, that goodwill would be the rule all around,_ _and that if he ever needed the aid or service of the dull-witted_ _Morris or the malicious twins, they would be available._\n\n_But Raymond had other ideas. He lurched threateningly_ _toward Morris and the other two, stamping his shoe in the dirt_ _as if they could be frightened off like animals. \"Go on,\" he said._ _\"Get the hell out of here. Crawl back in your holes and stop_ _making trouble for folks.\"_\n\n_And that was all it took. The three of them backed away_ _slowly, gradually moving together like birds in flight resuming_ _their formation. Side by side they walked back down through_ _the field. Monty noticed, however, that the twins had never_ _dropped the stones they'd collected. Sure enough, once they_ _gained some distance, the twins turned and heaved them at_ _Monty and Raymond. They were out of range, however, and the_ _two older boys just laughed as the missiles did nothing but kick_ _up dust at their feet._\n\n _The boy who saved Monty on that day had grown into the man_ _who now made his own confession. Once Raymond Stoddard's_ _laughter subsided, he said, \"That's right. I didn't give a shit who_ _I took with us.\"_\n\n_\"So what stopped you?\"_\n\n_Raymond shrugged. \"Figured the Japs would do the job for_ _me.\"_\n\n_\"You know, I could have you court-martialed for what you_ _just said.\"_\n\n_\"But you won't.\"_\n\n_\"Don't be so sure, Soldier. Don't. Be. So. Sure.\"_\n\n_The two men were still standing so close together that Raymond_ _Stoddard had to raise his hand right in front of his own_ _face in order to give Monty the finger._\n\n_Now Monty understood. Raymond wanted Monty to swing_ _at_ him. _A man who was willing to take the consequences that_ _might come from leaving his wife alone with an old boyfriend_ _would certainly be willing to take a punch just to get his superior_ _officer in trouble._\n\n_\"You know what, Ray? I feel sorry for you. I surely do.\"_\n\n_Raymond Stoddard's only response was to dig into his_ _pocket for another cigarette. As Monty walked away, he heard_ _again the clink of Raymond's lighter. And was there a similarly_ _distinctive sound, Monty wondered, when the pin was pulled_ _from a grenade? And would he ever be able to stop listening for_ _either sound, whether in war-or peacetime?_\n\nOf my many fictions, and fictional efforts, that had their origins in the Stoddard-Burnham saga, the preceding narrative (\"Got a Light?\" as it was titled when it appeared in _Blue Parchment,_ a Seattle magazine) was the only one that featured Monty Burnham (Tony Kroll in the published story) as a protagonist and point-of-view character. While there was never a point of personal contact\u2014an observed gesture, an overheard remark\u2014that allowed me to conjure that character's inner life (or to imagine an entire series of fictional episodes in which he played a prominent part), and while there's little reason to believe \"my\" Monty Burnham bore any resemblance to the actual one, I've always felt that my early attempts to imagine my way into the real Raymond Stoddard's mind (and only after the man's death) inevitably led me to try to enter others. And disposed me to create fictions more concerned with the motives behind actions than with the actions themselves.\n\nDuring my senior year I signed up for Introduction to Psychology, a course that had never been offered before at Bismarck High School. Edith Ehrlich taught the class, and I suspect she was given the assignment as a reward for having taught for almost fifty years in the city's school system. Miss Ehrlich looked forward to the prospect of spending months with us exploring \"the mysteries of our mental processes,\" but the semester had barely begun when her health forced her out of the classroom and out of the profession altogether. Miss Ehrlich had a stroke that deprived her of the power of speech, probably the only infirmity that could have kept her from taking her place in front of rows of students.\n\nThe school was in a quandary. No one else was willing to step forward and take over Miss Ehrlich's class, yet something had to be done with the students who were enrolled. We couldn't simply be given credit and sent on our way, and the semester was too far along to place us in other courses. The administration took the unusual step\u2014for this they must have needed a special dispensation from the board of education\u2014and hired an outsider, a non-teacher, to take over Introduction to Psychology.\n\nThat was how I came to know Frances Fenzer, Ph.D. Dr. Fenzer was a Bismarck psychologist, and someone on the school board apparently thought he would be a perfect substitute for Miss Ehrlich. And it worked out exactly as hoped. On a Wednesday, Miss Ehrlich was taken to the hospital. On the following Thursday and Friday, our Introduction to Psychology class was turned into a study hall, but when we showed up on Monday, we were met by a pink, plump, smiling man in a rumpled brown suit.\n\nWe were immediately pleased with the new development. Rather than stand behind a lectern, Dr. Fenzer sat on top of the desk. He straightened and twisted paper clips while he talked, and he freely admitted that this nervous behavior was caused by having to go an entire hour without a cigarette. Best of all, Dr. Fenzer didn't so much lecture as gossip. He refused to use the textbook that Miss Ehrlich had ordered, and instead structured the class around case histories\u2014culled from his own experience. Furthermore, we knew that the patients\u2014the neurotics, the compulsives, the depressives\u2014were almost surely Bismarck residents. He never used real names, but hadn't his entire professional career been spent in North Dakota? From what other sources could he be lifting those examples? He even teased us occasionally with a remark like, \"Now, this is behavior you've all had a chance to observe, especially if you've spent any time at all in a certain local establishment.\" Talk like that probably violated the standards of his profession, but we felt fortunate to be in his class and privileged to be taken into his confidence.\n\nHe favored us\u2014a few of us\u2014further by inviting us to his home, a small stucco house not far from the high school and surrounded by gardens unlike any other in Bismarck. \"Flowers and imported cigarettes,\" Dr. Fenzer said, \"my indulgences.\" Some of the students were invited to visit him after school; others in the evening.\n\nI went, along with Mike LaPorte and Joe McDonald, on a rainy night in April. I remember the weather conditions so well because Dr. Fenzer unapologetically required us to remove our wet shoes before we stepped on any of his rugs.\n\nThe house was as elaborately furnished as any I had ever been in. I realize now how much time and money Dr. Fenzer put into his home\u2014the walls were covered with paintings and prints, objets d'art were everywhere, a grand piano filled a sunporch, and every piece of furniture was covered with rich fabrics or made of heavy, polished woods\u2014but to my adolescent sensibility it all seemed fussy, ostentatious, and uncomfortable.\n\nDr. Fenzer was a gracious host, providing us with soft drinks and potato chips and encouraging us to smoke if we were so inclined. I sank into the corner of a brocade couch and lit a cigarette, eager to hear what kinds of anecdotes the psychologist would tell outside the school's walls, since his narratives in the classroom could often be outrageous.\n\nSomehow we began talking about suicide, a subject that any psychologist, even one practicing in a community as stolid and stoic as Bismarck, could hold forth on for hours. Dr. Fenzer told us that, contrary to the belief that attempted suicides were cries for help, many of these people were quite determined to die. What did he do when he had a patient with that resolve? he asked rhetorically. After a dramatic pause, he said, \"I watch the obituaries.\" Then he moved to the case closest to all our hearts.\n\nDr. Fenzer said, \"Take for example that fellow who murdered the legislator and then took his own life. . . . I've believed all along that the crime was puzzling to so many people because they couldn't figure out why he'd want to kill the good senator. But perhaps they couldn't come up with an answer because they'd asked the wrong question. They should have looked into why the fellow wanted to take his own life.\n\n\"Suicides are often motivated by a sense of worthlessness, of smallness, and more than one man has taken his own life thinking that at last he is doing something large, something dramatic. This chap may have felt as though he'd always been anonymous, overlooked, misunderstood. So he decided he'd do something that would finally cause people to notice him. He'd perform some act, commit some deed, that couldn't be disregarded. As a result, he'd become famous. The pathogenesis is really quite remarkable. These people feel so insignificant, yet they grow these monstrously large egos. . . .\" Dr. Fenzer looked around at his rapt, slightly shocked audience. \"Well? Have you any thoughts?\"\n\nAfter Dr. Fenzer's brief monologue, the conversation, which had been lively in its back and forth until then, fell silent. The doctor didn't know that Raymond Stoddard's son was our classmate, much less that I had a special connection to the family. Mike and Joe looked to me in deference to my Keogh Street address. This was my moment to shine, to show off my insider status, and to impress Dr. Fenzer, all of which I thought I wanted.\n\nPresented with the opportunity, however, I merely shrugged and said, \"Makes sense, I guess.\"\n\nDr. Fenzer must have sensed something in my recalcitrance that I didn't wholly comprehend myself. He quickly changed the subject, and for the rest of the evening I had no responsibilities but to listen and keep my cigarette's ashes from drifting to the floor.\n\nTwo days later, however, at the end of the school day, Dr. Fenzer stopped me in the hall as I was on my way out of the building. \"Do you mind if I walk with you?\" he asked.\n\nThe day and its sudden changes were typical of spring\u2014warm when the sun was shining, chilly when clouds obscured the sun. Dr. Fenzer lit a cigarette as soon as we stepped outside the school, but we didn't walk far. We stopped at a fairly new white Lincoln Continental, easily the grandest car in the faculty parking lot. \"All right,\" he said, smiling sheepishly. \"Flowers and cigarettes aren't my _only_ indulgences.\"\n\n\"It's nice.\"\n\nSmall talk finished, he got down to business. \"The other night at my house, I might have caused you some discomfort with my remarks, and I wanted to apologize. After you left, Joe mentioned that you were a close friend to the Stoddard boy and a neighbor to the family. I had no idea. I'm sorry.\"\n\n\"That's okay.\"\n\n\"That friendship must have weighed you down at times.\"\n\nEven as a callow adolescent I knew that to accede to that point would represent a failure of proportion. No matter how much I had come to dislike Gene, I had to admit that I could never own troubles that remotely rivaled his. \"Not really.\"\n\nDr. Fenzer was a good six inches shorter than me, so he stood on his toes and leaned forward to force me to look him in the eye. \"Perhaps your experiences are something you'd like to talk about. Sometimes we're so focused on someone else's pain that we lose sight of the fact that we're hurting too. We somehow think that we're not entitled.\"\n\nI was not a tough kid, not physically or emotionally, but like most males of my era, I knew how to fake it in certain situations and with certain people. \"It wasn't that big a deal.\"\n\nWith that, Dr. Fenzer gave up. He stepped back, bowed, and made a sweeping motion, a gesture that indicated both my dismissal and his defeat.\n\nDr. Fenzer invited me neither to his home nor to a private conversation with him again. In Introduction to Psychology I received a B, in spite of what I was sure was a poor performance on the final exam.\n\nThe night I graduated from high school was warm for May, and for weeks we had had no rain. As a result, my life took a turn that would not have been possible had the temperature been ten degrees cooler or the Missouri River six inches deeper.\n\nThe river's sandbars used to be (for that matter, perhaps they still are) a favored location for parties. We gathered there day and night, playing football or softball in the sand, swimming in pools and potholes, building bonfires, and drinking beer no matter what the hour. Depending on the river's height, we either drove out to one of those stretches of sand or trekked through a channel of icy, muddy, fast-flowing water. The Missouri was notorious for its treacherous currents and invisible drop-offs, and every summer it seemed to claim at least one drowning victim. Only someone very drunk or unfamiliar with the river and its reputation tried to swim through the main waterway. And the Missouri didn't care who it swallowed.\n\nBut the sandbars' remoteness and inaccessibility were exactly what made them so popular with the area's young people. Only the sheriff and his deputies had jurisdiction on the river (and the matter of jurisdiction was complicated by the fact that the river separated Burleigh and Morton counties), and even on the rare occasion when the law did try to patrol the area, we could usually spot them coming from a long way off.\n\nSo of course the river was a logical place for Bismarck High School's class of 1962 to hold its graduation party, and since the spring had been so dry, the river was low, which meant we could easily drive out to a sandbar southeast of the city for the night's festivities.\n\nI'd had family obligations earlier in the evening that prevented me from appearing at the river until well after dark, and by that time, hundreds of cars were parked in a line the length of at least two football fields, and bonfires of various sizes burned along the water's edge. The stage for the party itself was the long, narrow strip of sand between the cars and the river. Kids were everywhere, saying hello and goodbye to their fellow graduates, offering one another a beer or a drink from a bottle of hard liquor, throwing chunks of driftwood onto one of the fires. Many people were already drunk by the time I arrived, and the jubilation and delirium that traditionally accompanied the occasion had been replaced in some cases by belligerence\u2014rumor had it that kids from Mandan were among our number, and a group of wrestlers and football players were patrolling for party crashers\u2014and lust: Couples had not only gone off into the backseats of cars, but were also making out in full view. Among this second group I saw Gene and Marie. They were leaning against his car, tightly, passionately, locked in each other's arms. Marie stood on a case of beer\u2014whether the cans were empty or full, I couldn't tell\u2014the additional height facilitating their deep kisses and allowing their bodies to mesh in a way no doubt special to them. It was probably a fair approximation of the difference in height provided by the step inside Marie's garage. I walked close to them, testing not only their oblivion (it was complete) but also my vulnerability.\n\nTo say I was unaffected by the sight of them would not be true, but neither was the pain as sharp as it might have been. The entire evening I had been feeling a bit above it all, and not just in a metaphoric sense. Before I joined the party, I drove around for a while, at one point crossing the river and returning over the Memorial Bridge, whose height allowed me to look down on the sandbar and my classmates, at the dark glimmer of their cars and the blaze of their fires.\n\nHigh school was over, and I had only a few months left in the city. In the fall I would be attending the University of North Dakota in Grand Forks, where, as far as I knew, nothing would mark me as a Bismarcker. I had done a fair job of convincing myself that I would enjoy living in a place where I wouldn't have an identity that I'd have to share with anyone or anything Stoddard. Of course I had also often wondered how far the notoriety of the Stoddard name had spread and what subtle methods might be available to me to let others know that I had dwelt on Bismarck's Keogh Street during that stretch of pavement's most infamous hour.\n\nEven though I was moving at thirty miles per hour, my position on the bridge looking down at my classmates below provided a fitting emblem for my character. Did he wish to join the party or stay above it? Did he wish to be involved or ignored?\n\nMetaphors and symbols aside, any thought I might have entertained about living a lofty (in both senses of the word), solitary, or isolated existence no longer had to be merely theoretical. I now had my own car.\n\nEarlier that day, at a private graduation party to which only family and neighbors were invited, my parents presented me with a 1947 Studebaker Commander Regal DeLuxe. It had a flat-head six and a balky manual transmission, and though its original color had been red, the finish had oxidized over the years until it was a dusty pink. My father bought the car (or more likely accepted it in lieu of a fee) from a client\u2014a literal little old lady\u2014who lived in Sterling, a small town east of Bismarck. It was, in short, a car so obviously uncool that I felt an instant, inexplicable affection for it.\n\nI was certainly not embarrassed to be driving it, though when I finally decided to unite with my classmates down on the river, I didn't park on the sandbar but in a stand of cottonwoods farther up on the bank. The walk down gave me more time to consider what, if anything, I wanted from the night.\n\nAs I said, the party was in full progress when I arrived, and for the first hour I did nothing but stroll the sand and observe. As a result of some odd, unthought-through notion that self-denial could uplift my character, I opted not to drink that night, which meant I had to say no to any number of freely offered cups, cans, and bottles. I had to stay away from my friends, since a simple refusal would not have been enough for them. They would have insisted on an explanation for my abstinence, and I knew I couldn't make them understand what I couldn't understand myself.\n\nEventually I settled into a conversation with Bob Mullen and Diane Burgie, boyfriend and girlfriend and both planning to attend Carleton College in Minnesota in the fall. We stood near the biggest bonfire, which had in turn attracted the largest crowd to its fiery border. Nearby a smaller fire failed in its attempt to compete for people's attention. Someone had thrown an inner tube onto that fire, and as it burned it not only gave off the stink of scorched rubber, but its smoke rose black, even against the night sky. Bob and Diane weren't drinking either\u2014in fact, I was surprised to see them at such a gathering; I don't think I had ever seen them at a party before that night\u2014and the three of us made sad, ironic comments about our classmates' drunken behavior and what it portended for their future. Right in front of us, for example, a shirtless Mickey Lawson was lying flat on his back and trying, without anyone's help, to bury himself in the sand. He had barely covered his torso when he heard\u2014we all heard\u2014the persistent bleat of a car horn. That sound was accompanied by the roar of an engine. Everyone looked to the right, and Mickey did more than look\u2014he scrambled to his feet, the dry sand showering from him while the wetter stuff fell in clumps. A car was coming right toward him, churning its way along the open strip between the water and the line of parked cars. It lurched and wallowed through the soft sand and finally stopped near the largest bonfire, very near the space where Mickey had lain only moments earlier.\n\nWell before the driver came into view, I knew who he was. I had recognized those headlights and that grille coming out of the dark on another occasion. But before I looked to confirm that Gene was driving, I checked for passengers. He appeared to be alone.\n\nOnce the car stopped, the crowd that had pulled back, unsure of the car's direction and intent, surged forward, so when Gene opened the door and stepped out, he was surrounded by many of his fellow graduates. He tugged at his T-shirt, stuck to him with sweat\u2014both his and Marie's, I couldn't help thinking\u2014and combed his fingers through his dark hair. Smiling widely, he raised his arms over his head in that gesture that's supposed to quiet an assemblage but just as often serves to excite it. With everyone's attention focused on him, Gene stepped up onto his car's back bumper and proceeded to clamber from there onto the trunk and then to the roof of his car. I wondered what Gene's father would have said if he'd found the finish of his car scratched from Gene's shoes.\n\nBacklit by the bonfire's flames, he raised his arms again and shouted, \"Stew-Dents! Stew-Dents!\" Just that word, pronounced with two distinct, equally accented syllables, was enough to bring laughter and applause from the crowd. One of Bismarck High School's most eccentric and unintentionally comic teachers was the large-bosomed, blue-haired Miss Bonner, and when she needed to restore order in her sophomore English classes, she waved her arms and cried out, \"Stew-Dents!\" The previous year we were afforded even more opportunities to hear her hail us in her singular way: She was promoted on an interim basis to vice principal, and she often delivered announcements over the PA system and settled the crowd at school assemblies. She always began with \"Stew-Dents,\" though the volume at which she spoke the word varied.\n\nGene continued with his imitation. \"I have an announcement to make . . .\" It was not just this extravagant, attention-seeking public display that told me Gene was drunk; if I'd had no other evidence to go on but the toothy width of his smile, I would have reached the same conclusion.\n\n\". . . to all the wonderful, important, special, _special_ members of the class of '62.\" Now his impression of Miss Bonner turned into an imitation of Mrs. Harway, a school counselor who used the same approach in trying to help every student who came into her office, no matter what his or her problem. Mrs. Harway praised them, fawned over them, and told them what wonderful, unique individuals they were and how, if they could just accept that about themselves, their troubles would soon fade away. Mrs. Harway was decades ahead of her time.\n\n\"And you know what makes you so special?\" Gene cooed to us. Had he somewhere along the way spent enough time in Mrs. Harway's presence to work on this impression? Was it possible that he'd asked for Mrs. Harway's help when he was worried that Marie was pregnant? That possibility I rejected quickly. After all, he'd completely scorned the suggestion, right after his father's suicide, that he should see the school counselor, and if he wouldn't visit Mr. Wallich's office, he was not likely to seek Mrs. Harway's counsel. Then again, there had been so many ways that my friend had astonished me over the course of the previous year. Why couldn't a gift for mimicry simply be another surprise?\n\nGene was drunk, of that there was no question, but everything\u2014his entrance, his impressions, his timing\u2014spoke of someone confidently at ease in front of that drunken throng, someone who knew he had everyone's attention and was now able to toy with it. Here was a Gene Stoddard I had never seen before.\n\nHe turned first in one direction and then quickly pivoted to face another, almost as if he were expecting an attack.\n\n\"Do you?\" Gene asked again of the young people clustered around his car. \"Do you know why you'll always be a special, special member of Bismarck High's class of '62?\"\n\nI attributed Gene's out-of-character behavior to his drunkenness. If he weren't drunk, I reasoned in my sober state, he wouldn't have been able to act like that. And his drunkenness, I had concluded long before that night, was due to his being Raymond Stoddard's son. Father had bequeathed to son not only a penchant for alcohol but also a reason for drinking it. But as I watched Gene sway and balance on the roof of his car, another thought came to me. What if alcohol played no part in the alteration of his personality? What if the change in him occurred entirely because he lived inside the walls of a stucco house where a man had hanged himself? Surely that life-transforming event sent out ripples well beyond the walls of that home, ripples that rocked the neighborhood, the city, the state. . . . But if the lives closest to Raymond Stoddard were most affected, wasn't it logical to suppose that those next closest were the next most changed? Could there be a way to calculate closeness other than physical proximity? Was there a method that would allow another Keogh Street resident to believe that he was who he was through the strength or weakness of his own will rather than the accident of his address? Was it the weight of realizations like those that Dr. Fenzer had wondered if I needed help carrying?\n\nBut though such thoughts were earth-unsettling enough to make the sand underfoot feel solid by comparison, I set them aside when I saw Marie running down the beach.\n\nShe appeared to have her bathing suit on, and over it she wore cutoff jeans and a man's shirt, which, unbuttoned, fluttered whitely about her as she ran. She pumped her arms strenuously and lifted her knees high to keep from bogging down in the sand, and by the time she reached Gene's car\u2014her obvious destination\u2014her chest was heaving for oxygen. But she was not about to let shortness of breath keep her from her mission. \"No,\" she cried out. \"Gene, _please_!\"\n\nUp until that moment, I had tried to appear as nonchalant as possible at the spectacle of my former friend on top of his car. I hung back from those crowding forward and, unlike them, did nothing to cheer him on. But Marie's presence always quickened my interest, and now I was torn. Did I take her side in wishing he'd stop whatever he was doing or did I want more than ever\u2014because someone wanted him not to go on\u2014to hear what he had to say? Was it possible that he intended to announce something about his father that had never before been revealed? With that thought I considered trying to pull Marie back from the car myself.\n\nHer pleas did nothing to deter him, however. He drew himself up to his full, unsteady height, and answered the question he himself had posed. \"Because of me, that's why. Me. Because you graduated in the same class as Gene Stoddard, son of Raymond and Alma Stoddard.\"\n\n\"Don't do this,\" Marie begged. \" _Please,_ Gene.\" She moved closer, and now she too was both shadowed and illuminated by the flames behind her. In their flickering light her reddish hair took on a coppery glow.\n\n\"I made you all famous,\" Gene said. \"No matter where you go, you can always say, 'I went to high school with\u2014' \"\n\nHer spoken appeal ineffectual, Marie took action. Following the same route that Gene had taken, she began to climb onto the car. She had barely stepped from the bumper to the trunk, however, when Gene turned to repel this encroachment on his territory.\n\n\"God _damn it_!\" he said. \"Let me do this\u2014this is mine!\"\n\nGene had stepped toward Marie, and she obviously believed he was coming to help her up to his post. She reached out to him, but rather than grab her open hand and pull her to him, Gene pushed her.\n\nHer balance on the car was precarious, so not much force was necessary to send her reeling backward. She fell in two stages. First she groped behind her with one foot, trying to find a stable step, and for an instant it seemed as though she might regain her balance as she teetered on the bumper. But then she half-slid, half-fell from that perch, toppling sideways to the sand. If her arms had not been reaching in Gene's direction, she might have been able to break her fall. As it was, she struck the earth with a thud as audible as Gene's shouted curse.\n\nBut her tumble brought only laughter from the throng. They were glad she'd failed. They were enjoying Gene's performance and wanted it to continue.\n\nHer fall\u2014her fall and Gene's cruelty\u2014must have knocked the breath from Marie, but she was soon on her feet and shoving her way through the crowd and away from her boyfriend's car.\n\nGene meanwhile barely missed a beat. \"As I was saying . . .\" Cheers now accompanied the laughter.\n\nAs fascinating as I found Gene's performance, I had to turn away to see where Marie was headed. As she wove her way through the rows of parked cars, I had a moment of panic when I lost sight of her, but her white shirt soon reappeared. She was obviously walking away from the party and by the most direct passage available.\n\n\"No matter where you go,\" Gene continued, \"you can say, 'Remember that guy who shot the senator\u2014' \"\n\nAhead of Marie I could see glinting in the darkness a channel of the river that cut into the sandbar. If she walked right through the water, she'd have a head start that I'd have difficulty overcoming. As determined as she was, however, she still circled around, and I took that to be a sign. I pushed back from the crowd, and ran after Marie. Before Gene's voice faded completely from earshot, I clearly heard, \". . . the assassin's son!\"\n\nUp ahead, Marie had a steep grassy bank to negotiate, and for some reason I believed I had to get to her before she struggled to the top. That meant I had to go through the water, and as I approached the channel, I looked for the furrowed shadows of car tracks\u2014their presence would mean that it was shallow enough to run through. I saw none, but by then I had made my commitment. I went in at a full run, and initially at least, the footing was firm.\n\nMy third stride, however, carried me to a point where the bottom dropped away sharply, and I splashed face-forward and went under so suddenly and completely it seemed for an instant as though the elements had become mixed up and it was the sky that had turned to river water and night that had closed over my head. To further the confusion the icy water tasted like dirt.\n\nI must not have truly believed I would drown because I was able to appreciate the irony of the moment: You are going to drown chasing after a girl who belongs to someone else. . . . Within that inlet the river didn't seem to have sufficient power to do anything but steal my breath with its cold. No current tried to pull me down or push me out into the main channel, and I soon adjusted to my situation enough to probe for the bottom. It was near and firm enough that I could thrust back to the surface. When I came up I was soaked, sputtering for air, and missing my shoes, but half-paddling and half-running, I continued after Marie.\n\nShe had heard the splashing behind her and had stopped, no doubt hoping it was Gene pursuing her. From the top of the bank she gazed down at me, and it seemed as if she were about to say something, but in the dark it was hard to tell. Before I could climb to her height, she turned and disappeared.\n\nEach step I took came with more difficulty since both sand and water weighted down my socks, but I kept on and my perseverance was rewarded. Up on level ground, not far from the cottonwoods where my car was parked, Marie had paused to empty sand from her tennis shoes.\n\n\"What's the matter,\" I said to her, panting my way through the line I had rehearsed in case I caught up to her, \"didn't you want to stay back there and be famous with Gene?\"\n\nFor a long moment she stared, unsmiling, at me. The sounds of the party could still reach us, but they were nothing compared to the belching, trilling songs of frogs coming out of the surrounding darkness.\n\nThen Marie turned to go, and I quickly called after her, \"Wait! I'm sorry. I shouldn't have said that.\"\n\nI hobbled after her and was soon at her side.\n\n\"Did he send you after me?\" she asked. \"That would be just like him. . . .\"\n\n\"No, no. I came . . . I came on my own.\" Considering what I felt for her, those words seemed like a declaration of love.\n\n\"That asshole,\" she said, assuming we were both interested in discussing Gene Stoddard's character. \"I told him not to do something like that. I told him he'd just make a fool of himself and say something that tomorrow he'd regret. But no. _Famous . . ._ That's exactly what he thinks he is. Or should be. What a stupid _shit._ \"\n\n\"He's drunk\u2014\"\n\n\"Don't make excuses for him.\"\n\n\"I meant it as more of a question.\"\n\n\"He's drunk so often it's not even worth asking.\"\n\nShe began to walk away again, and now I noticed that she seemed to be favoring her right side. \"Where are you going?\"\n\n\"I don't know. Away from here. Home, I suppose.\"\n\nThe city and her house were miles away. \"You can't walk. Let me give you a ride.\"\n\n\"I'm not going back down there.\" She nodded in the direction of the party and the rows of parked cars. From where we stood the largest bonfire was still clearly visible, its flames licking even higher than before. Someone must have found another source of fuel.\n\n\"You don't have to. My car's right over there.\"\n\nAs we moved through the trees, I watched Marie carefully. Yes, she was injured, of that there could be no doubt. She kept her right arm pinned tight to her side.\n\nI pointed to her shoulder, which was slumped down and forward. \"What happened? Are you hurt?\"\n\n\"When I fell off Gene's car, I landed kind of hard.\"\n\nI almost corrected her _\u2014you didn't fall; you were pushed\u2014_ but I let it pass.\n\nWe arrived at my car, and I announced, \"This is it. All mine. A graduation present.\"\n\nMarie squinted through the darkness. \"Pink? Is it _pink_?\"\n\n\"More like a pale red. When it's washed, it'll be easier to tell the real color.\"\n\nWe both laughed at my defense of the car. \"Okay. Have it your way. But it's a _very_ pale red,\" she said.\n\nI had another little clutch of fear when it occurred to me that the keys might have fallen out when I tumbled into the water, but they were still in the pocket of my jeans. I unlocked Marie's door and held it open for her.\n\n\"You'll get your car seat wet.\" She waggled her finger up and down in my direction to indicate my drenched condition.\n\n\"Oh well.\"\n\n\"You don't have anything to put on the seat? Gene always keeps a blanket in the car.\"\n\nHer statement nicked me, but I loved her all the more for it, and in the process learned one more lesson about the inextricability of pain and love. And another lesson about my own character. Some other boys\u2014most other boys?\u2014in my place would have followed up Marie's remark with a wink and salacious question of their own\u2014Yeah? What was the blanket for? And I have no doubt she would have answered.\n\n\"It's not like it's a new car or anything,\" I said.\n\nWhen she entered the car, I noticed again how carefully she moved, and now she was supporting her right arm with her left hand.\n\n\"Is it your arm? Or your shoulder?\"\n\nJust the question was enough to make her wince. \"Shoulder, I guess.\"\n\n\"Can you lift your arm?\"\n\nShe made an effort, but it brought a gasp from her. \"Not very well. It feels like something's grinding in there.\"\n\n\"Can I take a look?\"\n\nI stepped back, and she slowly, gingerly, extricated her right arm from the shirt. Once that was done, she sat back against the car seat to facilitate my examination.\n\nLeaning over her, I caught the faint but unmistakable odor of Old Spice. I knew it so well because it was my father's aftershave, and I seldom stepped into the bathroom in the morning without smelling it. But I hadn't thought Gene used it. Finally I concluded that the white shirt must have been her father's, and the scent imbedded in the collar his.\n\nMarie's breasts swelled above her swimming suit, the top of which had slipped down just enough to expose a line of paler flesh. . . . I forced myself, however, to attend to the duty I had volunteered for, and I concentrated on her right shoulder, which, even though she was sitting back, was still hunched forward.\n\nThe car's dome light was not very bright, and shadows extending from the hollow of her throat made it difficult to be certain, but it looked as though she had a distinct bump on her collarbone and perhaps the start of a bruise there as well.\n\n\"I should take you to the emergency room. In case something's broken.\"\n\nThat suggestion caused her to crane her neck in an attempt to look down at herself, and that movement brought only another grimace to her face. Then, in order to prove to me her injury wasn't severe, she quickly smiled and tried again to lift her arm. That failed too. \"Shit,\" she said softly but angrily.\n\n\"Here. Let me try something. Can you take your shirt off all the way?\"\n\nIf she was skeptical or suspicious, her expression didn't show it. Nevertheless, to demonstrate the purity of my motives, I stepped back again and looked away while she struggled out of the shirt. Once she was free of it, I took it from her, unrolled the sleeves, and knotted them together. I folded the fabric until it was in the approximate shape of a sling, which I then slipped over Marie's head. Now both the strap of her bathing suit and the sleeves of the shirt were looped around her neck.\n\n\"Try this,\" I said. \"If it's too long or short, I can adjust it.\"\n\n\"Much better,\" she said, smiling her gratitude.\n\n\"All that Boy Scout training . . . finally good for something.\"\n\n\"You're a Scout?\"\n\n\"Not really. Was. A Cub Scout. I quit after a couple years. The knot-tying always gave me trouble.\"\n\n\"I didn't think you were a Scout or an Explorer or whatever they call themselves. Troop 109 tried to make me their mascot or an honorary member or something. I think it was just a ruse to get me to go on their overnight camping trips with them.\"\n\n\"Can't blame guys for trying.\"\n\n\"Oh, no,\" Marie said, adjusting her sling slightly. \"I never blame them for trying. Guys wouldn't be guys if they didn't try.\"\n\nAs I backed the Studebaker out of the cottonwood grove, it was with a gladdened heart and a relieved conscience.\n\nWhen we arrived at the hospital, Marie insisted she didn't have to go in. Her shoulder felt better, she said, and she was sure her injury wasn't severe. To destroy her argument I only had to ask her to lift her arm. When she couldn't do that, I pulled into the emergency room bay.\n\nI helped Marie from the car, and then said, \"I'll park the car and be right in.\"\n\nMy father and mother wouldn't be waiting up for me\u2014staying out all graduation night was a tradition that many parents honored\u2014and Marie didn't have to worry about coming in late either. Her parents were out of town, and the older sister who was staying with Marie and her younger brother didn't care what time Marie came in. We were both free to spend as many hours at the hospital as might be required, hardly where either of us had believed we'd end up on that night.\n\nAs if in a concession to the lateness of the hour, the emergency room area was hushed and dimly lit. The only hospital employee I could see was a nurse behind a high counter, and Marie was speaking to her when I walked in. When I took my place alongside Marie, the nurse, a sour-faced older woman with tiny eyes, stared intently at me. I must have made quite a sight\u2014barefoot, clothes soaking wet, and wearing a smile probably rare in those surroundings. But then the other people who came into the hospital didn't make the trip in the company of Miss Marie Ryan. But I could clearly read the inquiry in the nurse's gaze: Who are you? Since I had no answer for the question, I simply grinned a little harder and tried to appear as though I was exactly where I belonged.\n\nThe nurse pointedly addressed Marie. \"You can wait over there. Someone will be with you shortly.\"\n\nWe had barely sat down when Marie said, \"His problem is he can't get out of his head the idea that something, _something,_ good should come out of what his dad did. This stupid show he put on tonight _\u2014God!\u2014_ that was just his latest crazy idea. Famous . . . He'll be famous, all right.\"\n\nSince we'd left the river, Marie had not once spoken Gene's name.\n\n\"Yeah, they'll be talking about that for a long time,\" I said.\n\n\"I've tried to tell him, you're never going to forget what happened. Never. But you have to stop pulling the past up to the present. Let it stay back there. Let the days keep putting more distance between _now_ and _then._ \" She shook her head and grimaced, and I couldn't be sure if her expression of pain came from her injury or this renewal of her frustration at trying to help Gene and failing. \"You can see how much success I've had,\" she said ruefully.\n\nShe looked up at me, and I had the sense that she was waiting for me to confess my own inability to aid my friend through his life's worst period. What could I say? I have stopped short of openly professing my love for his girlfriend. . . .\n\n\"He'd be worse off without you,\" I said. \"A hell of a lot worse off.\"\n\nThe height to which her eyebrows rose indicated that my words had failed to persuade her. I couldn't summon any enthusiasm, however, for extending the argument. Besides, the worsening pain in Marie's shoulder was commanding all her attention. Suddenly light-headed, she bent forward in her chair in order to lower her head and keep from fainting.\n\n\"Should I call the nurse?\" I asked.\n\nMarie shook her head, but just then the nurse came out from behind her desk to take Marie to an examining room. That didn't however, mean a diagnosis or treatment would be forthcoming. Hospitals today of course are staffed around the clock with emergency room physicians and support staff, but in 1962 Marie couldn't be attended to until a doctor on call was notified and an X-ray technician was wakened and summoned to the hospital. I saw both of them arrive. The tech, a sleepy, slow-moving, dark-haired, handsome young man not much older than Marie and I, showed up first, and not long after, Marie was ushered away, but more than an hour passed before the doctor appeared. I recognized him as Dr. DuFresne, an older physician famous in Bismarck for his elegance, his encyclopedic medical knowledge, and his vinegary disposition. He stopped just inside the door and took a last drag on the cigarette poised at the tip of his long fingers. When he thrust the butt into the ashtray's sand, it was a gesture performed with impatience. I worried that Marie would not be treated gently or compassionately, at least not in his hands.\n\nOnce the requisite medical personnel were in attendance, however, the wait for Marie was still long. And frustrating. Since we were both trapped in the hospital, why couldn't we be allowed to pass the time in each other's company?\n\nAt one point I leaned back in my chair and dozed off, an act that would likely be noted in fiction only because it would present an opportunity to reveal a character's dream. But in life we remember few of our dreams, and those we do recall are usually not tied to a time or a place. Dreams\u2014and more commonly nightmares\u2014create their own settings, and override our waking sense of our surroundings. But since so many of the moments of that time in my life have, as I recollect them, dreamlike qualities\u2014unpredictable, emotionally intense yet ambiguous, imagistically vivid\u2014I feel as though it's appropriate to make up a dream to accompany that scene in the hospital's waiting room.\n\n_I've been led into a ward with a long row of beds or gurneys,_ _each separated from the next by a gauzy cloth curtain. I suspect_ _that Marie is just on the other side of the partition, and soon that_ _suspicion hardens into certainty. I can see her, albeit only in silhouette,_ _but a shadow is enough for me to know it's her. She's_ _moving around, dancing perhaps, and occasionally a part of_ _her\u2014her forehead, a leg, an elbow, her entire torso, especially her_ _breasts\u2014presses against the fabric, strains against it, almost as_ _though she is trying to tear through. I don't understand whether_ _I'm being invited to touch her through the cloth or whether she_ _doesn't even know I'm there. My dilemma is complicated when I_ _hear her voice. The sound of it is so faint that I can't make out_ _any words, yet its murmur has a rhythm\u2014rising and falling,_ _louder and softer\u2014that tells me she's speaking in sentences. But_ _again, I don't know if she's talking to me. I'm paralyzed with the_ _fear that if I reach out or speak to her, only to learn that her_ _speech and movements are intended for someone else, someone_ _whose shadow I can't see, I'll not only embarrass myself\u2014something_ _I'm willing to risk\u2014but also frighten and alienate her when_ _I'm revealed as an eavesdropper. Suddenly from over the top of_ _Marie's side of the partition something is thrown\u2014again,_ _whether accidentally or purposely I can't know\u2014over to my side._ _I can't tell what the object is, but it's right overhead, floating at_ _first, then falling at great speed. I stretch out my hands to catch_ _it, and in so doing startle myself awake._\n\nSunrise was less than an hour away when Marie finally reappeared. She was wearing a sling, just as she had been when she went in, but not the jury-rigged version I had knotted from her father's shirt. This one was made for its purpose. Canvas, it cradled her arm completely, and its work was supplemented by Ace bandages elaborately swathed around her, immobilizing her arm and shoulder. The elastic extended diagonally from the base of her neck to below her breasts and wound around her chest. To accommodate the wrap, the top half of her swimming suit was pulled down and bunched around her waist, while her shirt was draped around her shoulders.\n\nShe looked tired, but the color that pain had siphoned away earlier had returned to her cheeks. She smiled at me and asked, \"Have you dried out yet?\"\n\n\"Almost.\" In truth, only a little dampness remained in the creases of my clothes. \"How about you? Can you leave or are they going to check you in?\"\n\n\"No, I can go.\" Her glance over her shoulder made me wonder if she was escaping rather than being released.\n\nJust at that moment Dr. DuFresne emerged from the small office behind the nurse's desk. \"Remember, young lady,\" he said sternly, \"you're to check in with your family doctor next week. He'll want to see how that bone's healing. I'm not convinced that surgery isn't called for.\"\n\nMarie nodded curtly at his admonition, and kept moving toward the door. I rose and followed her, catching a glimpse, on my way out, of Dr. DuFresne's look of disgust at my damp, disheveled, barefoot appearance.\n\nMarie's collarbone was broken, and if the combination of the wrap and the sling didn't work, then it would be necessary, in Dr. DuFresne's view, to knit the bone back together with a surgical screw. For the next six weeks she was to use her arm as little as possible and to keep that beautiful body tightly bound.\n\nThe hours she spent inside the hospital were not all devoted to treatment of her injury. The first delay occurred because her sister had to be telephoned and her permission given for Marie to be treated. Her sister was willing to give that but wondered if she should come to the hospital herself, something she didn't want to do because their younger brother couldn't be left alone. Marie assured her that wouldn't be necessary and that someone was there who could bring her home. Next, both the X-ray technician and the doctor were called away to attend to other patients. The first was an old woman already in the hospital who fell attempting to go to the bathroom on her own. The technician wasn't supposed to read the X-rays, much less reveal what he saw on them, but he told Marie that the old woman had broken her hip. Dr. DuFresne, on the other hand, said nothing about the patient he was called away to see, but he was even more curt with Marie when he returned.\n\n\"Grumpy old bastard,\" Marie said as we drove away from the hospital. \"Why would anyone who hates people as much as he does want to be a doctor in the first place?\"\n\n\"How did it feel getting all trussed up like that?\"\n\n\"It hurt like hell.\"\n\n\"Well, there's your answer.\"\n\nWe were both tired, Marie was in pain, and the eastern sky's night blue was already surrendering its darkest hue to morning's lighter tint, but Marie didn't want to go home. Not just yet. \"Take the long way,\" she said.\n\nI wasn't sure what that route would be. If I drove all around the city's perimeter, the trip probably wouldn't take more than fifteen minutes. Then it came to me. Marie wanted to be driven past Gene's house.\n\nWhile I drove, Marie worried out loud about how her life would accommodate a broken clavicle. She was supposed to start her summer job at J. C. Penney's in the next week\u2014would they want a clerk who could use only one arm and who would have to wear blouses with an empty sleeve? Who was going to change her Ace bandage and wrap her up again? Marie would prefer asking that favor of her sister rather than her mother, but her sister might attend summer school in Minneapolis. How often could she bathe? She wouldn't be able to wear a bra or shave her armpits for weeks!\n\nShe fell silent, however, when I turned up Keogh Street. Within the distance of a couple blocks\u2014there could be no doubt about the direction in which we were headed\u2014she raised the hand of her uninjured arm as if she were blocking traffic. \"I want to go home,\" she said. \"Now.\"\n\n\"Is something wrong?\"\n\n\"Just turn at the corner.\"\n\nI did as I was told.\n\nBy the time I parked in Marie's driveway and walked her to the door, doves somewhere overhead muttered their _ooh_ s and _aah_ s over the hour we were coming in, and there was sufficient light to read the address on the house across the street. That was the same test I had subjected dawn to on another occasion. This time I knew where Marie was.\n\nI held the door to the garage open for her. \"Will your sister be waiting up?\"\n\n\"I doubt it. When I talked to her from the hospital and convinced her I was okay, she just told me to come in quietly.\"\n\nOnce again we were standing at her back door. Once again she thanked me for escorting her through a difficult time. Once again I thought of what it must be like to stand in this place and be more than Marie Ryan's friend.\n\nThen, abruptly, she said, \"Okay. Now.\"\n\n\"Now?\"\n\n\" _Now_ you can kiss me.\" She laughed at how her statement sounded like a regal command. \"Said the queen.\"\n\n\"I'm not sure that would be such a good idea. Gene\u2014\"\n\n\"Forget about Gene. You must know that he and I are finished.\"\n\n\"He's gone through so much\u2014\"\n\nShe clapped her hand over my mouth. \"Stop it. Stop defending him.\"\n\nHer hand slipped away, but its sensation\u2014softness, warmth, a faintly antiseptic smell from the hospital\u2014lingered. Once I could form words again, I continued to try to talk my way out of the moment that I had desired more than any other. \"Down at the river tonight . . . that wasn't the real Gene. He would never\u2014\"\n\n\"Don't you think I know by now who and what the real Gene is? And better than you? Now, are you going to take me up on my offer? You don't want to make a girl feel rejected.\"\n\nBefore I could answer, she hooked her fingers inside the collar of my T-shirt, pulled me close, and kissed me.\n\nI had kissed a few girls before Marie, but her kiss, with its wide openmouthed intensity, its ability to at first be hard and then yield to pliancy, was astonishing. Because of her injury she had to withhold most of her body, yet her entire being still seemed behind her kiss.\n\nOur lips came apart, and immediately I went back for more, to confirm both my good fortune and my physical impressions.\n\nThe next kiss was amazingly deeper, firmer, and softer than the first, and eventually I had to step back, breathless, unbelieving, and a little frightened that I couldn't match her passion.\n\n\"Whew!\" was all I could manage, and with that, Marie laughed and sent me on my way, albeit with the admonition that I had best call her the following afternoon.\n\nI parked the Studebaker in front of the house and was surprised to see both my father and mother hurry down the front walk to meet me. It was still too early for them to have gotten up, especially on a Saturday, so obviously they had been waiting for my arrival. The night before, they had cheerfully given me their permission to stay out as long as I liked, but as they approached, their expressions were drawn tight with anything but approval. Just as sunrise's gradual light brings to view one feature of the landscape after another, I saw, dawning across my parents' faces, anger, worry, and relief, all within the span of seconds.\n\nMy father put his arm across my shoulders, an action that should have seemed an expression of affection; instead it seemed as if he were gently restraining me. On my other side, my mother also walked close, ready to block my movement in the other direction. Together they herded me toward the house, but neither said anything; that was left to Uncle Burt, who came out onto the porch as we approached.\n\nUncle Burt had driven down from Wembley for my graduation, and he had stayed the night. He too must have been waiting for me. His cigar had burned down to a stub, and the coffee cup in his hand was probably not his first of the morning.\n\nHe said, in that voice that was always a strange combination of delicacy and heartiness, \"Your folks have been worried about you. As you might know, your classmates had a party down on the river last night, and there was an accident, a drowning.\"\n\nI couldn't hear any more. I twisted away from my father's loose embrace and tore off across the lawn. I must have heard one of my parents\u2014both of them\u2014yell after me, \"It wasn't Gene! It wasn't Gene!\" just as I must have concluded, once I saw his car parked in front of the Stoddard house, that he could not have been the drowning victim, but I ran on and didn't stop until I came to that car and saw, faintly in the early morning light, the dusty outlines of footprints on the trunk, rear window, and roof, evidence of its having been used as Gene's stage. I ran my hand along the rear fin, not far from where Marie had fallen. I backed away from the Ford. In its wheel wells were traces of sand.\n\nMy mother had followed me down the street, and now she caught up to me. Gently she turned me back toward our house. \"Sshh. That's okay,\" she said, as if I had been crying and needed to be quieted. \"He's all right.\"\n\nI certainly didn't argue with her, but I might have said, He is not all right, he is definitely not all right, and soon he will be worse. . . .\n\nBack in the house, I allowed myself to be comforted and reassured. My parents were convinced that I had broken away out of fear and concern for my friend. In truth, what had panicked me _was_ fear, and a familiar one. I had felt it on the night when Marie and I searched the branches of the trees for Gene's hanging body, and my fear then was that his death would preserve Marie's love for him. How much worse would it have been if on the night when Marie and I had exchanged our breaths, he lost his forever. She would never have recovered.\n\nAs it turned out, neither Bob Mullen nor Diane Burgie attended Carleton College in the autumn of 1962. Diane enrolled at Bismarck Junior College, and her decision to remain in her hometown was largely determined by her fragile emotional state at the time. And Bob? Diane's boyfriend and fellow scholarship winner was the young man who drowned in the Missouri River on the night of our high school graduation.\n\nShe blamed herself for what happened, and Bob likely would not have had it any other way. After I left the party in pursuit of Marie, after Gene slid off his car at the end of his act (as it turned out, his performance was mostly complete at the time of my departure), Diane was persuaded to drink a can of beer. The person who talked her into consuming alcohol for the first time was: A football player from Mandan High School; A trumpet player from St. Mary's Central High School; A farm kid from a small town west of Bismarck. All three accounts made the rounds, but probably the important detail, present in every version, was that this young man was someone who had been Diane's playmate in childhood. He had moved to another town or another neighborhood.\n\nIt doesn't matter. He was, as narratives so often require such a character to be, an outsider, the other. Diane was pleased to see him. Or at least she pretended to be. She and Bob had quarreled. He apparently had wanted to leave the party, but she had wanted to stay. The attention she paid the new boy, as well as the beer she drank (and another and another followed the first), was calculated to anger Bob and to demonstrate to him that she didn't have to shape all the contours of her life to his.\n\nHer tactic worked. Or was it a tactic? Perhaps she allowed herself, under the influence of alcohol or not, to be attracted to someone other than Bob. Perhaps the occasion, an end and a beginning, prompted her to consider alternatives that she had previously barred from consideration.\n\nBob, desperate to win back her attention\u2014or to garner the attention of others and in so doing demonstrate that he didn't need Diane's\u2014began to do front and back flips on the sandbar. Bob was known for his conscientiousness, his seriousness, his intelligence; he was, in other words, someone who readily gained the respect and admiration of adults while he had few qualities that made him cool or popular or attractive among his peers. He wasn't an athlete, at least not in any of the sports that had audiences or followers. But Bob Mullen participated in gymnastics throughout high school, and it was that training that enabled him to impress the drunken crowd at the sandbar. In addition to his flips, he did a somersault off the roof of a car. But that was not his most impressive performance. He twisted through the air; he landed on the earth. But then he needed to bring the other elements into play. For fire, he launched himself over one of the bonfires, landing with a perfect roll on the other side of the flames.\n\nBob Mullen backed up from the river and then took a run at it. When his feet touched the waterline, he catapulted himself into the air, flipped, and came down right into the river's main current. Had he been heavier and less skilled at hurling his body, perhaps he would not have been able to propel himself so far from the sandbar. Had he been taller, perhaps he would have landed where he could touch bottom and still keep his head above water.\n\nAlmost immediately the laughter and shouts of encouragement that had accompanied Bob's attempt to entertain his classmates died when they saw him swept away\u2014from light, the fire's flickering orange reflection on the water, to darkness, the river's true night face\u2014the instant he splashed down.\n\nAnd yet something\u2014drunkenness, disbelief, all those years of parental warnings about the river's treacherousness, something\u2014paralyzed the witnesses. A long moment passed before anyone stepped into the river for a rescue attempt. By then it was already too late. Most people said he vanished immediately into the black water, pulled under by a current that wouldn't permit him to surface even for a second to wave or shout for help. At least that's what most people said. But Karen Conroy, who had been standing right at the river's edge, told me she caught one final glimpse of Bob Mullen. She said she could see his pale form tumbling through the water, as if he were still doing his somersaults and flips while the Missouri rushed him off.\n\nHis body was discovered two days later, tangled in the driftwood and brush at a bend in the river fifteen miles south of Bismarck.\n\n_Although Bob Mullen died with a mouth full of water, the_ _taste of dirt was on his tongue._\n\nThe previous sentence appears in one of my journals, but the line never found its way into a poem, story, or novel. For the insight\u2014if it qualifies as such\u2014on that sensation, I had my own tumble into a Missouri River pothole to thank.\n\nWhy have I bothered to relate the Bob Mullen story when it is a tributary that never joins the main narrative? Because while I know\u2014rationally, logically, sensibly, I _know_ that Raymond Stoddard's deeds had nothing to do with that young man's drowning, I can't separate them emotionally. Everything that occurred near that place and time seems somehow to owe its causality to Raymond Stoddard and his murder of Monty Burnham. Raymond Stoddard? He set the universe in motion.\n\n_The slain politician's son doesn't want to board the train._\n\n_His mother and his uncle, however, remind him that being_ _sent away is not a punishment but a privilege, a reward for_ _being a good student and a respectful, dutiful boy who has_ _borne up under difficult circumstances. His mother kisses his_ _forehead, and when she embraces him, he suddenly thinks he_ _has a legitimate excuse for not getting on the train. He's ill! In_ _his throat he feels that scratchy constriction that's often the first_ _sign of tonsillitis. But that thought is enough to carry him back_ _from the verge of tears, and with that retreat the sensation vanishes._\n\n_Just as he lifts his foot onto the step stool that will carry him_ _into the train car, his uncle hisses into his ear, \"Don't stay in the_ _bathroom longer than you have to.\" Another passenger comes_ _up behind him, so the slain politician's son doesn't have time to_ _ask his uncle the question that seems so obvious:_ Why would I?\n\n_He walks down the train's aisle, looking for a seat on the_ _side of the train that won't force him to look out on the platform_ _where his mother and uncle will be standing, and as he_ _does, his feeling of banishment is complete. He is being sent to a_ _strange country whose customs, population, even food, will be_ _alien to him. He is on his way to spend two weeks at Camp_ _Way-Tah-Ga in northern Wisconsin, and he knows that the congregation_ _of First Lutheran Church chose him to go, rather than_ _a boy from a poor family as in years past, because the congregation,_ _indeed the entire town, feels sorry for him. But why not_ _leave, he thinks, since he is obviously no longer understood in_ _his own land. Otherwise why would his mother, who has always_ _been perfectly attuned to his likes and dislikes, his enthusiasms_ _and fears, believe that living with boys he has never met_ _would be anything but punitive?_\n\n_Further evidence of this estrangement he comes upon two_ _hours into his journey when he unwraps the ham sandwich his_ _mother prepared for his lunch. The bread has been spread_ _thickly with butter. Mustard\u2014he likes mustard only. He dislikes_ _butter on everything but pancakes and then only the thinnest_ _film, which must melt completely. He folds the sandwich back_ _into its waxed-paper wrapper and then tucks it inside the copy_ _of_ Boys' Life _that he's already read. He lays both on the empty_ _cushion beside him and tries to act as though they belong to the_ _person who will soon return to his seat. If the sandwich has_ _done nothing else, it has replaced some of his despondency with_ _disgust. Adults\u2014they act as though they understand, but they_ _fall so far short you wonder why they even bother with the pretense._\n\n_Once the slain politician's son actually goes to the bathroom,_ _he believes he understands his uncle's warning. The toilet_ _flushes with such swirling force and the water pours into the_ _washbasin so swiftly that it's a temptation to linger, turning the_ _water off and on, filling the sink and watching it drain, listening_ _to the way the toilet's formless_ whoosh _contrasts with the train's_ _rapid but still rhythmical rush. He feels as though he's on the_ _edge of a discovery about the essential difference between liquid_ _and machine, but remembering his uncle's words, he returns to_ _his seat. By the time he next visits the restroom, his sorrow and_ _self-pity have returned, and he no longer has any interest in insights_ _or discoveries._\n\n _When he disembarks in Wisconsin, he is met by a tall, sunburned,_ _redheaded young man, a counselor from the camp who_ _already has in his company a boy who rode the very same train_ _as the slain politician's son. Jimmy Hogan boarded the train in_ _St. Paul, but unlike the slain politician's son, Jimmy is an _experienced_ traveler and he roamed from one car to another, searching_ _for boys he remembered from his previous years at camp._ _The slain politician's son saw the fat boy repeatedly waddle_ _down the train's aisle, but the slain politician's son was too shy_ _to speak to anyone._\n\n_Once they are in the station wagon that will take them the_ _thirty miles to Camp Way-Tah-Ga, Jimmy Hogan suggests to_ _the counselor that in the future the camp should send special_ _caps to campers, something that the boys could wear on their_ _way to Camp Way-Tah-Ga and in that way identify their common_ _destination to one another. The counselor, a high school_ _senior or college student, the slain politician's son guesses,_ _doesn't acknowledge Jimmy's suggestion. Instead he lights another_ _Camel, which he keeps bouncing between his lips with his_ _vigorous gum-chewing._\n\n_Into the silence, the slain politician's son cautiously says to_ _Jimmy, \"That's a good idea. The caps.\"_\n\n_And with that, the two boys become friends. For the rest of_ _the ride, Jimmy tries to educate the slain politician's son about_ _the camp's culture and conventions. He begins, of course, with_ _the food. The hamburgers might look good, but they're usually_ _dried out and barely warm. The macaroni and cheese is okay if_ _you get a serving from the middle of the tray. The best breakfast_ _is French toast. The best dessert is the butterscotch pudding. He_ _advises the slain politician's son to arrive at the archery range_ _early when they have target practice, or else he'll get one of the_ _bows that's so tight he'll hardly be able to pull the string back._\n\n_Don't get in a canoe with the thirteen-year-olds because they_ _won't even let you paddle. Don't sit on the side of the campfire where the big trees are because the wind always blows sparks_ _and smoke in that direction. \"Chapel\" means the same thing as_ _\"church.\" \"Hike\" and \"nature walk\" mean pretty much the_ _same thing. Don't wear shorts to play baseball or you won't be_ _able to slide. And whatever the slain politician's son does, he_ _should make certain he doesn't piss off Castle, the oldest counselor_ _at the camp. Castle is always in a foul temper, but no one_ _is quite sure why. Some say it's because he's a junior high_ _teacher during the year, and by the time summer rolls around,_ _he's sick of kids. Another theory has it that Castle is an alcoholic_ _who is kept from drink during the camp sessions. A more_ _forgiving, romantic explanation of Castle's dark moods suggests_ _that he once had a beautiful wife who was killed in an automobile_ _accident. As Jimmy finishes his list of admonitions, he_ _glances up at their gum-chewing driver, who says only, \"Yeah,_ _Castle's a prick.\" The obscenity delights the boys._\n\n _The slain politician's son soon learns, however, that because of_ _the way the camp is set up, he and the redoubtable Castle are_ _unlikely to have much contact. Castle works mostly with the_ _younger boys, and the slain politician's son belongs with the_ _Wolverines (the ten-and eleven-year-olds), a group with whom_ _he quickly grows quite comfortable. Thanks to Jimmy, the slain_ _politician's son is assigned a bunk in Jimmy's dormitory, and_ _Jimmy introduces the slain politician's son to the other boys_ _their age and makes sure he is always gathered into the group_ _before any activity\u2014flag races, hikes, swim instruction, candle_ _making, or woodworking\u2014begins. The slain politician's son becomes_ _so close to these boys that he wonders if he should reveal_ _to them how he is different from every one of them and always_ _will be. Finally he decides against it. He won't share with the_ _Wolverines his unique history, one that includes newspaper_ _headlines and reporters in the kitchen extinguishing cigarettes in_ _his mother's coffee cups. Indeed, though the slain politician's_ _son is the only Wolverine who has never before attended Camp_ _Way-Tah-Ga, his family owns a cabin on a small lake in North_ _Dakota, and in his mind the slain politician's son substitutes_ _those experiences for the not dissimilar ones the other boys have_ _had in previous summers. Ironically, it is Jimmy Hogan who believes_ _he has an identity unlike anyone else's; Jimmy is not only_ _fat, he is left-handed and he has had his appendix removed._\n\n _Most of the boys have been looking forward to the day when_ _they'll be allowed to fish in the swift-flowing waters of the_ _Goose River. Since they arrived in camp they've been allowed to_ _fish only in the lake, and even then only with bamboo poles,_ _tiny hooks, red worms, and bobbers. They never catch anything_ _but crappies, bluegills, and sunfish. Today, however, they're_ _being driven to a location where the river suddenly straightens_ _its course and runs fast and frothy over and around a series of_ _big rocks. There the boys will be handed real fishing rods with_ _artificial lures and treble hooks and given instruction on how to_ _cast their lines into the pools and eddies where the big fish\u2014_ _bass and walleye and northern and even muskie\u2014lie. But before_ _any of them can wet a line, they must sit through fishing lessons,_ _even if they've been doing this kind of fishing for years._\n\n_On that day, the Wolverines are divided and assigned to_ _other groups, and the slain politician's son is placed with five_ _boys who will have_ \u2014oh, shit!\u2014 _Castle as their teacher. They_ _gather near the pilings of a railroad bridge. The river there has_ _shrunk back from the tree line, and they have a wide beach on_ _which to line up and listen to their teacher's advice. The sand_ _the slain politician's son sits upon feels simultaneously warm,_ _soft, and damp, a new and peculiar sensation, and he adds it to_ _the collection of never-felt-before that he has been compiling for_ _the past seven months._\n\n_As it turns out, Castle, in spite of his profession in the non-summer_ _months\u2014it's been verified; he is a junior high school_ _teacher\u2014has no aptitude for or interest in instruction. He simply_ _tells the boys, \"Watch how I do it.\"_\n\n_Next he performs a series of movements so rapid that none_ _of the boys understands exactly what he's doing or why. He_ _clicks something on the reel\u2014the bale? Did he lock the bale_ _back? Did he do something with the drag? He brings the rod_ _back, then flings it forward, and the lure\u2014a spinner with a tuft_ _of fur attached to its end\u2014whirs out across the water. Did he_ _have a target in mind? How did he direct it there? Almost immediately_ _after the lure splashes down, Castle reels it in swiftly,_ _straightening the rod's tip from time to time. \"The action has to_ _be like this,\" he says, \"so the fish will want to hit it.\"_\n\n_The last thing the slain politician's son wants at that moment_ _is to be handed the rod and reel and told to duplicate the_ _instructor's act, but he doesn't have to worry. Castle has apparently_ _forgotten both his charge and his charges. He's just a man_ _fishing on a summer day, casting over and over again to that_ _boulder-blocked expanse of water where the river seems to back_ _up on itself._\n\n_Suddenly there is a small eruption right where Castle's lure_ _splashes down. It looks as if sunlight has not just spangled the_ _water's surface but burst up from beneath it. Castle jerks back_ _on his rod, and then it's plain\u2014he's hooked a fish and the fish is_ _leaping into the air._\n\n_Castle hasn't addressed the boys, but even without being_ _asked, they all rise from the sand and stand behind him. From the_ _expressions on their faces it's obvious that none of them believed_ _that today's lessons could truly lead to an outcome like this._ _That's a real fish, large enough to bend the rod and create its own_ _wake as it fights across the current, the line, and Castle's strength._\n\n_The fish must lose of course, and by the time Castle has_ _reeled it in close to the shore, it can no longer leap clear of the_ _river or even shake itself on the line. And why, the slain politician's_ _son wonders, would the fish try to escape in the first place_ _by leaving the water for the air, the element it can't survive in?_\n\n_One of the boys steps forward with the net, but Castle_ _brushes him aside. Holding the rod in one hand, he steps into_ _the river, bends down, and pulls the fish from the water. To the_ _slain politician's son it appears that Castle grabs the fish by the_ _lip, but that can't be. Fish don't have\u2014Then he sees. Castle has_ _hooked the fish again, this time with his finger, right under the_ _fish's mouth._\n\n_A boy more knowledgeable about fish than the slain politician's_ _son whistles softly and says, \"A walleye!\"_\n\n_\"What a pig!\" Castle says. \"That sonofabitch has gotta be_ _twelve pounds.\"_\n\n_The fish twists and wriggles so slowly in Castle's hand it_ _seems as if it might be moving in the breeze rather than by its_ _own muscle and will. Castle quickly and deftly unhooks the_ _walleye, sets down his rod and reel, and holds the fish aloft with_ _two hands. Its pale belly sags as if it were weighted with stones._ _Its eyes look as though they have been cut out of aluminum foil._ _\"You pig,\" Castle says again, this time almost affectionately._\n\n_Castle finally acknowledges the boys who have been gazing_ _at him as if he had been enacting a play for their entertainment._ _\"One of you want to put him in the catch bucket?\"_\n\n_Like the other boys, the slain politician's son raises his hand_ _and jumps into the air. He is sure that only he, however, has no_ _interest in the fish. Instead he wants to do what no other boy_ _has been able to do\u2014to win Castle's approval and esteem._\n\n_And it is that purity of motive, the slain politician's son believes,_ _that separates him from every other ten-year-old bouncing_ _up and down on the sand, and that causes Castle to reach_ _the fish out toward him and say, \"Hold him tight. And watch_ _out for the fins. They're sharp as razors.\"_\n\n_Once the slain politician's son holds the fish, with its heft_ _that doesn't match its light silver slickness, he can smell it, and_ _that odor brings back his past\u2014the family cabin, the heat that_ _seemed to rise every morning from the lake's algae-choked_ _shoreline, the grown-ups with their skinny pale legs bare the_ _way they never would be in town, and their loud laughter, the_ _hollow thunk of oar and boat, and the little panfish strung on a_ _line as if they were beads run through with a sewing needle, and_ _smelling like, like, like nothing else. . . . Is that past only a single_ _summer removed from the slain politician's son? Is that possible?_ _The memory seems as if it has to struggle an exhausting_ _distance back to him. The walleye's gill yawns open so slowly it_ _must be for the last time. For a moment the slain politician's son_ _glimpses the fish's blood-red interior, and the sight appalls him_ _so that by a laborious process\u2014he slides the fish slowly across_ _his shirt front\u2014he shifts the walleye to his other hand and now_ _holds the fish backward, its head and dead eyes and awful_ _starving gills facing the other way._\n\n_While he wades into the water and opens the lid of the catch_ _bucket, the slain politician's son has to squeeze the fish tightly_ _but not too tightly, because he can feel how with just a little_ _more pressure it could squirt from his hand._\n\n_The river water inside the bucket looks like weak tea, and_ _for an instant the slain politician's son wonders if he is making a_ _mistake. No, he is doing exactly what Castle asked of him, and_ _there is no other way to gain the man's favor. He lowers the fish_ _into the bucket._\n\n_It all happens so fast that afterward the slain politician's son_ _can't be sure of the precise sequence\u2014did Castle first cry out,_ _\"Not tail first! Not tail first!\" or did those shouted words come_ _after the fish, instantly revived when the cold water washed_ _over it, with one sudden sinuous effort, burst from the bucket_ _and escaped into the river's current?_\n\n_No matter what the chronology, the consequence is unchanged._ _The walleye, a flashing golden shimmer, swims away,_ _and Castle grabs the slain politician's son by the collar of his_ _T-shirt and pulls him backward with such force that the boy_ _lands heavily on the sand, his breath flying out of him so completely_ _that he might as well have been shoved underwater._\n\n_\"That was a trophy fish!\" Castle shouts. \"A goddamn trophy_ _fish!\"_\n\n_He looms over the slain politician's son, who believes that_ _the physical threat from Castle hasn't passed. The boy rolls_ _across the sand until he can be certain Castle can't reach him,_ _even with a kicking foot._\n\n_But Castle isn't interested in pursuit. He simply stares down_ _at the slain politician's son in disgust. \"Can't even put a fish in a_ _bucket,\" Castle says. \"Jesus. What a moron. What a little_ _moron.\"_\n\n_And then the slain politician's son makes the situation_ _worse. He laughs, though his lack of breath soon causes his_ _laughter to change into a fit of coughing._\n\n_It is the laughter that provokes Castle into grabbing the_ _slain politician's son by the hair, shaking him, and then throwing_ _him back toward the sand. It is the boy's helpless coughing_ _fit that convinces the camp authorities that Castle's harshness_ _can no longer be tolerated and that the man must be dismissed_ _from working at Camp Way-Tah-Ga._\n\n_After the incident, the slain politician's son explains to his_ _new friends why he couldn't keep from laughing. That was the_ _summer of the \"little moron\" jokes\u2014Why did the little moron_ _throw his alarm clock out the window? Because he wanted to_ _see time fly. Why did the little moron put his father in the refrigerator?_ _Because he wanted cold pop\u2014and when Castle insulted_ _him with that very phrase, laughter bubbled uncontrollably_ _from him._\n\n_In truth, the slain politician's son's laughter came from the_ _sudden realization, insofar as such a thing can be realized by a_ _ten-year-old boy whose bony ass is denting the sand of a river_ _beach, that he had no special standing on this earth and that he_ _was therefore subject to the same vagaries of reward and retribution_ _as every other human being. He was no different from_ _any other Wolverine._\n\nThis story came into being from an assignment of sorts. A friend, to whom I had been complaining of a personal publishing drought, suggested that maybe I needed to try writing fiction that was removed from my own life, times, and circumstances. \"The Slain Politician's Son,\" which appeared in _The Stopped Clock Review,_ was my attempt to follow that advice, and while it may not have seemed to most readers like such a radical divergence from my usual concerns, it was. I knew that Monty Burnham, like Raymond Stoddard, had a son, so I tried to invent a life for him. Believe me, any turn that took me away from the murderer's existence\u2014and Keogh Street\u2014and toward the victim's was a radical one.\n\nI had ample opportunity to observe Diane Burgie in the aftermath of Bob Mullen's death because I was in two of her classes, biology and psychology, at Bismarck Junior College. But while she chose to stay close to home because her guilt-ridden, grief-stricken state left her too weak to stray far, I attended a community college in order to be close to Marie Ryan. By the end of summer 1962 she and I were going steady, and since she had a year of high school left, I decided that enrolling at the university in Grand Forks could wait. I say \"I decided,\" but really, there was no decision to be made at all.\n\nIn fact I sometimes wondered why, since we were in classes together, Diane Burgie and I weren't used for an academic demonstration. The biology or psychology teacher\u2014the demonstration would have worked for either subject\u2014might have displayed both of us in front of the classroom and said to the other students, \"Look closely at these specimens. They illustrate what love can do to a human being.\" Pointing to Diane, the professor might have noted the hair, once blond, bright, and shining, now lusterless and lank; the complexion, pocked and sallow; the expression, dour and despairing; the entire being, enervated and drawn inward. I, on the other hand, could have been used as an example of love's power to vitalize, to fulfill, to bestow happiness and hope. In fact, had my state of bliss not been so complete, Diane Burgie's doleful presence might have made me a little embarrassed over my great good luck.\n\nMarie's broken collarbone did not require surgery, but it was slower in healing than originally projected. Not until the end of July was she able to remove for good the sling and elaborate webbing that had held her body tight to itself. By that time J. C. Penney's had assigned her to work in their credit department, so her temporary handicap didn't affect her ability to do her job. It did, however, have a strange effect on our burgeoning relationship.\n\nIt should go without saying that I could not get enough of Marie Ryan, yet her physical condition, coupled with my timidity, imposed a restraint on what we could do. While I could kiss her for hours, because of her Ace bandage binding, there was no possibility that we could go beyond kissing (or so I believed; I had a very clear notion of what should be the natural progression of sexual intimacies). Furthermore, she could put no more than one arm into an embrace, and my own passion had to operate with a governor on it\u2014squeeze too hard or press too close and she might flinch with pain.\n\nThe eventual removal of the wrap and the sling freed Marie in ways that went beyond the ability to raise her arm or twist her torso or draw an unencumbered deep breath. After those weeks of control, her ardor could suddenly match mine.\n\nThe night of the day the doctor pronounced her healed we were parked on a hill west of Bismarck. The sun had barely gone down when the moon, full and antique yellow, rose to claim the sky. If we looked out Marie's side of the Studebaker, we could see the dark reflection of the Missouri unspooling itself far below. If we looked out my side, we could see the random cluster of the city's lights. Over the entire town the capitol building towered, each of its four sides presiding over a compass point. But notice I say \" _If_ we looked out, we could see . . .\" Our eyes were fixed on nearer things. For the first time we were able to embrace with all the strength we owned, and we pressed ourselves together as if we were determined to seal our bodies to each other in spite of the resistances of clothing, skin, muscle, and bone.\n\nDuring one of the brief moments when our lips and tongues were not twinned and twined, I mentioned how strange it was to feel Marie's actual flesh beneath her blouse rather than a bandage. Laughing, I added, \"And you're wearing a bra for the first time!\"\n\nIn response to a request I did not make, Marie unbuttoned her blouse and wriggled herself free of it. She twisted around so her back was to me. \"I still have trouble reaching around,\" she said. \"Unhook me.\"\n\nMy trembling, inexperienced fingers eventually managed to accomplish the task. Then, with a movement so subtle and deft I didn't quite understand how she performed it, Marie allowed her brassiere to fall off.\n\nShe turned back to me, her crossed arms covering her breasts. She let her hands fall away and at the same time reclined across the Studebaker's front seat.\n\nIn these pages are examples aplenty of my inventive powers (or deficiencies). There is evidence as well of how my imagination is as likely to lead to misapprehension and mistake as to truth and understanding. Let me merely say at this point that I could more successfully imagine my way into a murderer's mind than anticipate the wonder of Marie Ryan's perfect breasts exposed to my sight and touch.\n\nMarie, however, had no patience with my awe. She extended her arms, beckoning me to lie down with her, and just in case I didn't understand the invitation, she whispered, \"Come here.\"\n\nThe sight of the almost-naked Marie Ryan was intoxicating, but even more thrilling\u2014and astonishing\u2014was the feel of her. Her skin was warm and amazingly soft to the touch\u2014silk right off the ironing board.\n\n_\"Now,\"_ she said.\n\nWhen my kisses became too long and too intense, Marie turned her head to the side. As if to expand her body and its openness to pleasure, she extended one arm out under the dashboard and with the other reached up and pressed against the seat.\n\nIf the months-later occasion when we finally had intercourse was somewhat less ecstatic, mark it down to my overeagerness and awkwardness. The act occurred in the unfinished attic of Marie's parents' home. The attic entrance was through the garage, and we were frequently able to climb up there undetected. Under the slanting roof and the cobwebbed timbers, lying on our makeshift pallet (flattened cardboard boxes covered with winter coats and old clothes), we explored our bodies and their secret delights, and finally, on a warm April day that was warmer still in the attic, we went all the way. That was the phrase in currency at the time (and perhaps it still is), and while I felt its accuracy\u2014I had plainly gone somewhere I had never been before\u2014I also sensed its inadequacy: I knew there were further realms to travel to, and one journey was barely finished before I wanted another to begin. And gradually, with Marie's help, I overcame the ineptness of that first time. In sexual matters, she knew my nature better than I, and she taught me that the greatest pleasures came from being controlled by the moment rather than trying to control it.\n\nAcross from Marie's parents' home was an elementary school, and I remember being in the attic once when the shouts and laughter of children at play rose to our height. We had just finished making love, and while Marie still lay on our \"bed,\" I rose and went to the high, small window to try to take advantage of what little ventilation it might provide.\n\nThe day, like so many on North Dakota's calendar, was breezy, and just enough air moved through the window to cool my sweating body. I couldn't see the children, but I could hear them, and the sound was enough for me to imagine their activity, their play. . . . What Marie and I had been doing was, of course, playful, yet just as certainly it was different. I didn't at that time have Frost's phrase\u2014\"play for mortal stakes\"\u2014but it applied. In our stifling garret Marie and I played together. Life and death depended on it. Whose, I couldn't be sure.\n\nI didn't keep from my parents the fact that I was going steady with Marie Ryan, but neither did I go out of the way to advertise the relationship. I was concerned, a worry probably without foundation, that they might display an unseemly interest in what Marie might know about Raymond Stoddard. And though I say \"they,\" I really mean my father. A year and a half after the murder-suicide he was still seeking the truth behind it. His quest was no longer very active, but he was not yet satisfied with any of the theories that might have settled the minds of others. My mother, as I said earlier, was content with the version that Ross Wilk had presented to my father.\n\nBut though I didn't want anyone else to try to take advantage of Marie for her inside information on the Stoddard family, I had no compunction myself about seeking to learn any secrets she might possess. I often questioned her about her memories of Mr. Stoddard and for any theories she might have had about his behavior. Marie, however, had little to offer, and not only because she had observed almost no examples of behavior that would lead one to conclude that the father of her then-boyfriend harbored murderous impulses.\n\nAs she made plain as far back as the day of Raymond Stoddard's funeral, she believed that the man was obviously deranged, if not legally insane. His acts were, ipso facto, evidence of that condition. The minds of such people couldn't be understood. Furthermore, pragmatic to her core, she thought it futile to try.\n\nAnd though she wouldn't hold up any memory to explain Raymond Stoddard, she did tell me about an incident that corroborated her conviction that he was mentally ill.\n\nMarie first met Raymond and Alma Stoddard when Gene invited her to a family picnic in September 1960. The day was windy, cold, and overcast, and to make matters worse the picnic site wasn't a Bismarck park but an unsheltered prairie hilltop north of the city. Mr. Stoddard, however, insisted that the spot, which he chose, was perfect. They spread their old wool blanket over buffalo grass so wiry and stubborn the stubble poked right through the fabric. Stones and the picnic basket weighted down the cloth at the corners and kept it from blowing away. The paper plates had to be heaped with food immediately, or the wind would tear them away like dry leaves.\n\nAnd the food was not standard picnic fare. It was leftovers, and not particularly impressive ones at that. Chunks of cold (though previously overcooked) pot roast, unheated potatoes and carrots, apples (for weight, if nothing else), and buttered bread. Two thermoses\u2014one filled with coffee and the other with milk.\n\nOdder still was Marie's introduction to Mr. Stoddard. Shortly after meeting her, he wanted to know where she lived. When she told him, Mr. Stoddard said he knew the neighborhood well. When he first came to Bismarck, before he found an apartment for his family, he stayed with Mrs. Hills, who lived on Marie's street but had once lived in Wembley and was a friend of his grandmother's. Did Marie know Mrs. Hills?\n\nYes, she knew the old woman.\n\n\"How unfortunate,\" Mr. Stoddard said, \"that she's no longer with us.\"\n\nMarie didn't understand. \"Not with us?\"\n\n\"Why, she had a stroke,\" Mr. Stoddard said. \"She passed away.\"\n\nShe wanted to make a good impression on her boyfriend's parents, but Marie Ryan couldn't allow this misinformation to stand. \"No, she's alive. She lives just down the street from us.\"\n\n\"You're mistaken. She's no longer with us.\" Mr. Stoddard kept using the same phrase. No longer with us.\n\nAnother teenage girl might have backed down in the face of an adult's certainty. \"She's alive,\" Marie reiterated. \"Alive. I saw her recently.\"\n\nMr. Stoddard never stopped smiling or insisting. Mrs. Hills had had a stroke. She had passed away. He was sorry he hadn't attended her funeral.\n\nEventually, like the relentless prairie wind, he wore her down. Marie didn't submit to his belief, but she stopped arguing.\n\nWhen Marie related this anecdote, I confessed that I didn't quite understand. So he had made a mistake; I didn't see how that was evidence of derangement.\n\nMarie shook her head strenuously. It wasn't just that he was wrong\u2014though that could have been a sign that he was delusional in his convictions\u2014but that he wouldn't admit to doubt. He was so certain of the rightness of his belief.\n\nThat sounds, I suggested, as though you're describing someone religious.\n\n\"Okay,\" Marie said. \"I don't have a problem with that. If you add 'zealot.' \"\n\nYet for the most part, Marie resisted my efforts to pry from her any inside information on the Stoddard family, especially as such information might relate to Raymond Stoddard's pathology. In fact, she was critical of my curiosity.\n\nI remember very well an evening when we sat in the darkened kitchen of her parents' house, facing each other across the table where the Ryan family ate their meals. I had been questioning her again about her memories of Mr. Stoddard and what Gene had told her about his father.\n\n\"Why,\" she wondered, \"is it so important for you to know why he did what he did?\"\n\nOne psychology class in high school and another in college provided me with a suitably high-minded and personal response. \"If we can understand people like Raymond Stoddard\u2014their, you know, their motivation\u2014then maybe they can be stopped before they kill.\"\n\n\"Do you really believe _knowledge_ can keep people from killing each other?\" Marie had a gift for phrasing matters in such a way that the ground under your argument began to erode even before it was built. \"Or that we can ever understand people like him?\"\n\n\"Maybe.\"\n\n\"But there was only one Raymond Stoddard. And he did what he did. And he'll never do it again. Why not just let it all go?\"\n\n\"Isn't it human to be curious? To want to know?\"\n\n\"It just seems so pointless. Even if Raymond Stoddard had lived and he could tell us why he did what he did, we'd still only have a madman's word to go on.\"\n\nI could never formulate an effective refutation to Marie's position, yet I never took that inadequacy to mean that my own beliefs were flawed. She was, I told myself, simply a better debater than I. So there was no reason for me to cease my interrogations, no matter how they might exhaust and exasperate her.\n\nLet me offer just one more of Marie's memories of Raymond Stoddard, and I present it not as evidence of anything but simply to share an image that was lodged in her mind (and therefore in mine).\n\nMarie never had that many occasions to observe Raymond Stoddard, and that was a consequence of the age as much as anything. In the early 1960s a boy might spend plenty of time at his girlfriend's parents' house, but she might never enter the interior of his. We all understood this, even if the _why_ of it escaped us.\n\nBut Marie was there a few times (and if anyone was likely to defy the more idiotic strictures of the era, it was she), and on one occasion she and Gene were sitting on the living room floor in front of the black-and-white console television that I myself spent so many hours watching.\n\nMrs. Stoddard sat on the sofa and Mr. Stoddard was in his easy chair, but he was certainly not at ease. At one point he swiveled suddenly toward the living room window and the darkened night beyond and said, \"Did you hear that?\"\n\n\"Hear what?\" Mrs. Stoddard calmly asked.\n\nSuddenly embarrassed, Mr. Stoddard said, \"Nothing. It's nothing.\"\n\nYet from where Marie sat, she had a clear view of him, and for the rest of the evening he seemed on alert, as if he were listening for that sound to repeat itself and its message that no one else could hear.\n\nIn the weeks and months after Marie and I started dating, not a word passed between Gene and me. From others I learned that for most of the summer and fall he was working out of town, continuing in the employ of his uncle's construction company. But Keogh Street was still his home, and occasionally I saw him drive by, now in a white Chevrolet convertible that he'd started driving about the time I acquired the Studebaker. His car was newer, faster, and cooler than mine, and he probably bought it with his own money. His job, according to all accounts, paid well.\n\nWas I worried about how he would react to Marie and me being a couple? Yes. Was that worry intensified because he was Raymond Stoddard's son? Yes. And the worry played out on a daily basis as suspense, so while I was happy with Marie, happier than I had ever been, I was continually looking over my shoulder, constantly on the lookout for an attack from a wounded, enraged, drunken Gene Stoddard.\n\nBut Gene troubled my life in another way. As odd as it might seem, I missed him. He had been a part of my almost daily existence for years, while for only months had our friendship changed, so I couldn't quite adjust to the idea that now we were\u2014what were we? Enemies? Rivals? Whatever term might have applied, the fact was we no longer had that regular, casual contact _\u2014Did you understand that algebra assignment? Are you_ _going to play touch football on Saturday? Do you want to get a_ _burger at Jack Lyon's?\u2014_ that had for so long been a feature of both our lives. Are you going to play poker at Billy's on Friday? Do you want to drive or should I? Got an extra cigarette? _Want_ _to know what Marie let me get away with last night?_\n\nPerhaps, then, it was both a wish for reconciliation and a desire to eliminate the tension that I was living with that made me do what I did.\n\nOn the Friday after Thanksgiving 1962, snow began falling shortly after sunrise. By evening, close to a foot had fallen on the city. The snow was the dry, downy sort, so its effect was fairly benign. It was relatively easy to shovel, too light to snap branches or power lines, and since the wind was calm, flakes didn't stray far from where they fell. Still, even in a land as accustomed to winter's challenges as North Dakota, twelve inches of snow has an effect, and as I drove home late from Marie's, the streets were largely deserted. Snowplows had not cleared the streets, and I had to steer through the ruts left by cars that had earlier passed that way.\n\nI had just turned onto Keogh Street when I saw Gene Stoddard trudging through one of those furrows. He had probably stepped out into the street because on that section of the block most of the sidewalks were unshoveled, the owners probably away for the holiday.\n\nCould I have simply driven past? Certainly. And that option would have been in keeping with what our relationship had become. But as I said, the suspense had gotten to me, and this seemed an opportunity to address it, if not eliminate it altogether. Besides, he would recognize my car.\n\nI stopped in the middle of the street and asked him if he wanted a ride.\n\nGene was hatless, gloveless, and wearing only a light jacket. His shoes must have been soaked through. Nevertheless, he hesitated before finally shrugging and climbing into the Studebaker.\n\n\"Remember Thanksgiving last year?\" I said. \"Sixty degrees.\" An innocuous remark about the weather seemed like a safe way to begin, even if it would allow him the opportunity to comment on how many other things were different a year ago.\n\n\"How about that time it snowed on Halloween? Who was the kid whose trick-or-treat bag got so wet it tore open and all his candy fell out? And then he went home crying, so we kept his candy?\" Gene asked.\n\n\"Jerry Blessum.\"\n\n\"Jerry Blessum. Yeah.\" Gene was wearing aftershave or cologne, but its aroma was faint next to the stale but still overpowering smell of cigarette smoke and liquor. Was he drunk? Probably. \"Fucking North Dakota. Snow on Halloween. . . . Why does anyone live here?\"\n\nWhat was I supposed to say\u2014in order to be close to Marie? I changed the subject. \"Where's your car?\"\n\n\"Sitting in the driveway at Vicky Morhoeffer's. The battery's so fucking dead it won't even turn over.\"\n\nFor a moment I considered offering to drive him back there and help him start his car. In my trunk I kept, at my father's insistence, a set of jumper cables. They were there not only so I could get myself out of trouble, but so I could help others, as my father had done with the Stoddard vehicle on the second morning after Raymond Stoddard had hanged himself. I lacked, however, my father's Samaritan spirit. Indeed, it was hardly even an impulse to goodness that had prompted me to invite Gene into my car in the first place.\n\nAnd in the general direction of that topic was where I ventured next, albeit with trepidation. \"Vicky Morhoeffer, huh?\"\n\n\"What can I say\u2014she's a fucking slut. But she'll do anything. _Anything._ And her folks are hardly ever home.\"\n\nI took that to mean that, unlike in a previous relationship, love was not involved. I understood as well that I was free to ask what Vicky Morhoeffer was willing to do. Instead, I headed for safer terrain. \"She's a junior?\"\n\n\"A junior? Yeah, I guess.\" We were on our block now, and he pointed toward his house. \"Don't pull in the driveway. You'll pack down the snow, and it'll be harder to shovel in the morning.\"\n\nI stopped under a streetlamp across from his house. I hadn't scraped the windshield very well when I left Marie's, and frost stars glittered on the glass under the light's glow. A few weightless flakes still floated in the air.\n\n\"Hey, you got any smokes?\" Gene asked.\n\nI reached into my shirt pocket for my pack of Chesterfields and handed it to him.\n\nHe shook out two cigarettes. \"A couple to get me through the night?\"\n\n\"Take all you need. Keep the pack, if you like.\"\n\nWithout warning, he grabbed my wrist.\n\nI stiffened and that might have prevented Gene from pulling me toward him. But maybe moving me was not his intention. Using my weight as an anchor, he drew himself closer to me. I could smell the rank, curdled odor of tobacco and whiskey on his breath.\n\nStrangely, having my wrist in Gene's grasp didn't so much feel like a physical threat as it reminded me of a time in our childhood when we had taken each other's measure in quite a different way.\n\nDuring one of the summers of our Little League play, Gene and I embarked on a strenuous program to improve as ballplayers. We practiced long hours, hitting fly balls and grounders to each other, pitching, batting (and sprinting after the batted ball), and with the help of instructional, inspirational articles in _Baseball_ _Digest_ and other publications, tried to strengthen our baseball muscles. Following the example of Hank Aaron's success, we tried to build stronger wrists by squeezing tennis balls and rolling weighted bars back and forth. To gauge our success, we were constantly calibrating each other's wrists, hoping to feel that increase in circumference that would indicate muscle growth.\n\nGene's grip on my wrist tightened, and when he spoke, the urgency of what he said seemed as much an appeal for validation as an expression of menace. \"I had her first, you know. I had her first and I can have her back anytime I want. Any fucking time.\"\n\nFear might have motivated my reply. I was\u2014I am\u2014easily intimidated, and while I'm not eager to confess to cowardice, my commitment to honesty in this narrative won't let me back away from it. Nevertheless, I believe that something else was working on me. I felt sorry for Gene. And I leave to others the question of whether I betrayed Marie with my reply. Since I never reported this conversation to her, I can't report on her reaction.\n\nTo Gene I said simply, \"I know.\"\n\nHe released me, and then he and my cigarettes were gone.\n\nLying in bed that night I felt none of the fear of Gene Stoddard that I had been experiencing for months. Instead what kept me awake was fear that flowed the other way. Would the day come when I, perhaps like Gene's father, could no longer live with the worry that someone was loose in the world who might at any time take my love from me? What was _I_ willing to do to keep her close?\n\nThose were my thoughts on a snowy night when almost nothing followed the path prepared for it.\n\nAs if to prove how fickle the climate of the northern plains could be, the next year's November gave us a day so fair it seemed a gift from the gods. The temperature was mild, the sky was a pale and limitless blue, and Marie and I drove right through the day's benevolent heart.\n\nWe had left Grand Forks, where by that time we were both students at the university, early on a Friday morning, and we were pointed toward Minneapolis. Marie's sister had extra tickets for a Brothers Four concert, and we planned to stay with her for the weekend. We not only had the concert to look forward to, we would have two days to take in the city's attractions, and if that weren't enough, there was also the possibility that Marie and I would be allowed to share a bed. For an entire night. The prospect was tantalizing, for while we were having sex often, we never had the chance to _sleep_ together. To literally sleep through the night in each other's arms, head to dreaming head, took on an importance comparable to those earlier sexual milestones that marked the advance of our relationship. We both lived in dormitories at that time, but we were already making plans for the next school year. I was going to rent an apartment, and though I'd need a roommate or two to share rent, the apartment would provide a haven for Marie and me. Maybe she'd occasionally be able to check out of the dorm for a night or even a weekend, and we could truly live together.\n\nAs we drove through central Minnesota's gentle undulations\u2014turn your hand palm-up and raise it to eye level and its contours should provide an analogue for that landscape, a relief after the pancake flatness of eastern North Dakota\u2014Marie and I felt as though we could pass for husband and wife. We had stopped at a diner and refilled our thermos, and we passed the single cup back and forth. Marie lit my cigarettes for me. Our conversation was as familiar, comforting, and unrestricted as the sky. No longer living under our parents' roofs, our lives seemed completely and happily our own. Had we passed a highway sign announcing that the road we were traveling on would go on forever, it would have only deepened my contentment.\n\nBut what we passed, at intervals just regular enough to be puzzling, were cars pulled over to the side of the road. Eventually, I slowed for one, and the sight of the family inside, with identical expressions of confusion and grief on the faces of father, mother, and three children, alerted us to the possibility that something might be happening in the world that had nothing to do with conditions along Highway 10.\n\nMarie snapped on the radio\u2014and it was never off again when we were in the car\u2014and we learned what in the instant of knowing it suddenly seemed strange not to have known: The president had been assassinated.\n\nDid I steer the Studebaker onto the shoulder of the road? I must have. Yet the force of the radio news seemed sufficient to propel the vehicle's tons without any help. That same force moved Marie and me into an embrace.\n\nFor a moment we debated whether we should go back to Grand Forks, or perhaps even drive to our hometown. Certainly Bismarck would find this news especially upsetting, considering its own history of assassination. But not every vehicle had pulled over. Cars and trucks continued to whiz past us, and Marie and I thought we had already learned the lesson. Deliveries had to be made. Appointments had to be kept. Rituals had to be observed. Leaves had to fall. Clocks had to tick.\n\nIn fact, America was still unsure, in the autumn of 1963, what it should do and what it should postpone. The Brothers Four concert was canceled, but the National Football League played its scheduled games. Of course, before the decade was over, the country would have its protocols for post-assassination behavior well established.\n\nMarie and I were allowed to sleep together that weekend, at least after a fashion. Her sister kept her bedroom and her bed for herself and gave Marie the couch. I unrolled my sleeping bag on the floor next to her.\n\nAs we lay in our makeshift beds, Marie reached her hand down to me. I took it, though I didn't pull myself up toward her or tug her down to me. Her sorrow, I assumed, canceled her desire to make love. I was well aware of the line that ran from sex to pleasure. I even knew of the connection between sex and pain. But of the power of sex to heal or affirm, I was ignorant and would remain so for years to come.\n\nDuring the drive back to Grand Forks on Sunday, Marie was quiet. She was no doubt thinking about what had happened in the nation in the days just passed and how unsettled and unsettling the future now seemed. And perhaps that uncertainty caused her to question other matters\u2014how many of us in those days of late November suddenly let go of assumptions and never picked them up again? So Marie may also have been given over to contemplation of what life would be like in the company of a boy\u2014a man\u2014who could not guess her need. And wouldn't ask.\n\nAfter the assassination, however, Marie and I were no different from most Americans: We resumed the lives we'd expected to lead and acted as though the future was once more in our control. We continued with our studies, Marie as an education major who hoped to teach elementary school, and I as an English major with the intent to teach as well. My ambition to write I kept to myself.\n\nJust as I had planned, at the start of my junior year I moved off campus. I found a dark, damp, two-bedroom basement apartment close enough to the university that I could walk to class. For the necessary roommate I recruited Rob Varley, an acquaintance from Bismarck. Rob was quiet, solitary, and dedicated to his studies. A chemical engineering major, he spent almost all his spare time either at the university's new computer center or in a laboratory. His focus on the future rendered the past of little interest, so we seldom reminisced about our hometown or our high school years. As I recall, Rob made only a single comment about Monty Burnham's murder. The entire episode in the city's history was, he said, \"stupid,\" an unsurprising assessment from someone who was interested only in problems that had solutions. When Rob was home, he was sleeping or studying, usually in his bedroom with the door closed.\n\nIn other words, Marie and I had plenty of opportunities for privacy, and though the university had a regulation prohibiting female students from visiting men in off-campus housing, the apartment's entrance was on an alley and was blocked from general view by a garage and a shed, allowing Marie to come and go undetected. She probably spent as much time at the apartment as Rob, and she was responsible for whatever touches the place had that made it feel like something other than an underground cell. She hung curtains on the tiny windows that looked not so much out as up. She found a bookcase at a yard sale, painted it white, and arranged Rob's and my books on the shelves. She threw a blue checked tablecloth over the lopsided, rusting table at which we ate our morning cereal. On the wall above the sagging couch she hung framed prints of Alpine scenes. As part of her dormitory contract she could eat in a campus dining hall, but she often prepared meals for us in the apartment, using the groceries we shopped for together at the local Piggly Wiggly. From those years two images dominate because of their frequency. The first\u2014Marie, walking away from my bed after lovemaking, heading for the bathroom with a sheet awkwardly wrapped around her, her lovely back and backside exposed. And Marie and me on the couch, both of us reading; she's lying down, her skirt is above her knees, her legs are across my lap, and I am absentmindedly rubbing her feet.\n\nWe were so comfortable and established as a couple that we could, like real husbands and wives, presume permanence in our relationship without discussing it. So it was only natural that Marie should walk into the apartment one October day and announce, as a wife would to a husband, \"I think I'm pregnant.\"\n\nHere is what I should have done. I should have turned off the radio (the Yankees were playing the Cardinals in the World Series), walked over to her (she was standing at the entrance to the tiny kitchen, one hand on each side of the door frame, as if she were bracing herself for my reception of the news), taken her in my arms, and said, Wonderful. Terrific. Let's go talk to Pastor Shoup over at Christus Rex right now and tell him we want to get married. You don't need your parents' permission, and I know I can get either my mother or my father to say yes without any trouble. And then we'll go on with our lives and nothing will change except we'll be married and that's what we're going to be anyway. No attempts to be clever or eloquent. Just simple, straightforward talk, without pose or artifice.\n\nBut though Marie's announcement should have come as no surprise\u2014we had always been inconsistently careful about birth control, and during the previous year we had become even more lax about the matter\u2014I was shocked. In my contentment I had become complacent to the point of intransigence\u2014I liked things just the way they were. Fittingly, I remained seated while Marie stood less than two yards away. In attempting not to demonstrate that I had been jolted, I tried for cool.\n\nI asked the question that men have asked for millennia, and, next to \"Is it mine?\" it must be the one that women hate most to hear. \"Are you sure?\"\n\nHer shrug said that she would try for a nonchalance to match mine.\n\nThere was still time for me to say or do something that would help close the distance between us. But there was another reason for my cool, and it came out in my next question. \"Is this any different from before?\"\n\nMarie cocked her head quizzically. \"Before?\"\n\n\"You know. With Gene. What happened that time?\"\n\nNow it was Marie who stepped closer, but while she sat down across from me, she kept her arms and hands off the table, perhaps to prevent me from reaching over and touching her.\n\n\"That time?\" she asked warily. \"I'm not sure I know what we're talking about.\"\n\n\"Hey, it's okay. I know all about it. He told me on his mom's wedding day. Asked me if I'd be his best man. He wasn't exactly overjoyed at the prospect of being a father, but he was ready to do the right thing.\"\n\n\"Was he?\"\n\n\"So when did you realize you . . . you know, that you weren't pregnant. That time.\"\n\nMarie pointed to my cigarettes. Instead of just taking one from the pack, she said, \"May I?\"\n\n\"Help yourself.\"\n\nFor the next few moments Marie did nothing but bring the cigarette to her lips and inhale and exhale, directing forceful plumes of smoke toward the low ceiling. I suppose I might have said something, but it seemed as though the right of next comment had been reserved in her name. Throughout this silent time she often stared right at me, and rather haughtily it seemed to me, something I was prepared to note if a quarrel broke out.\n\nShe eventually looked away\u2014into the cracked and chipped soup bowl we used for an ashtray\u2014and that was the moment when she began to speak.\n\n\"It took me a moment to remember when the wedding was . . . when you say Gene told you. But it's coming back to me. Yes, I was once late with my period, and that could have been the time. That was unusual for me. And I'd been having pains and problems with my period. Something that still happens from time to time, not that you've noticed. I had an appointment to see a gynecologist. So Gene must have let his fear override his brain. After all, he was accustomed to things going badly in his life. But he knew where babies came from, so he had to have known I couldn't have been pregnant. _Could not have been._ \" She leaned over across the table, and the urgency of her words seemed to lift her from her chair. \"Do you understand? _Not possible._ He should have known that . . . that we hadn't done anything that could result in pregnancy. But maybe he had some confused notions about human anatomy or reproduction. Maybe he believed in virgin birth. Maybe he was simply making up a story just for you, something to impress you. Or depress you. But I do recall that for a few days he wore this look like his world was about to end. Not unlike the look you have right now.\" She sat back down.\n\n\"The Yankees are losing. Maybe that's what you see.\"\n\n\"That's very funny.\"\n\nI lit a cigarette of my own.\n\n\"But all jokes and sports talk aside, I have to make sure: You understand what I'm telling you, don't you? I could not have been pregnant. _I could not have been._ Not then. Not anytime Gene and I were together. It was not possible. We did some things . . . and we would have done more if he had had his way. God knows he tried. But you were the first. Do you understand? The first. The _only._.\"\n\n\"All right.\"\n\n\"Maybe you need to know what Gene and I did? What happened that might have let him think\u2014wrongly, stupidly\u2014that I could have been pregnant?\"\n\n\"That's okay.\"\n\n\"No, that you wouldn't want to know, would you? Instead of the truth, you'd rather have your own version of things. Your own fantasies. After all, you could have asked me anytime. God knows you've asked me enough questions about the past. But you've always cared more about what was impossible to know than what you could know. You're as haunted by the past\u2014part of the past\u2014as Gene was. But _his_ ghosts were forced on him. He didn't choose them.\"\n\nIf it weren't for that remark, I might have allowed myself to be chastened, and, repentant, I might have done exactly what the moment called for\u2014apologized and asked for Marie's forgiveness. After all, I believed her. There was never a moment when I didn't believe her. Never. But when she compared me to Gene, I became angry.\n\n\"Okay. So now I have two versions of the past. Another chance to choose.\"\n\nShe crushed her cigarette out so forcefully the bowl jumped and skidded across the table. Long after the cigarette was extinguished she kept jabbing the butt into the ashes. When she finally looked up, her eyes were flaring with fury, and even her tears had no power to put out the blaze. \"You know what?\" she said. \"Fuck you. Just _fuck you._.\"\n\nBefore I could make a move to stop her, she pushed violently back from the table, and stood, toppling her chair in the process.\n\nEarlier I mentioned images that branded themselves on my memory because of their frequency. Here is one I carry because of its singularity: The apartment is tiny, but Marie manages to gain running speed as she heads toward the door. Although her progress is nothing but forward, away from me, her shoulder-length hair\u2014brown in her first few strides but closer to red as she dashes through a shaft of sunlight that has found its angled way through a ground-level window\u2014waves from side to side as she runs. Similarly, the motion of her plaid wool skirt is lateral as she runs.\n\nWounded pride and self-righteous anger kept me tethered to the apartment for an hour or two, and then I left in search of Marie.\n\n_Because of what happened at his tenth high school reunion,_ _Raymond Stoddard did not want to attend his twentieth. But in_ _the first week of January, the representatives of Wembley High_ _School's class of 1941 sent out invitations to their forty-six fellow_ _graduates, and Raymond's wife, Alma, immediately marked_ _off on their calendar the days in August when the reunion_ _would take place. Raymond knew he could do nothing to dissuade her from making plans for them to attend, or to make her_ _understand why he didn't want to go._\n\n_Right from the start everything about the ten-year reunion_ _had been wrong, wrong, wrong. First of all, the planners decided_ _to hold the event in conjunction with the Catholic high_ _school's reunion, and in the process ignored the fact that the_ _two schools had been rivals, not allies. Next, the band, such as_ _it was\u2014piano, bass, drums, and trumpet\u2014played music popular_ _during the war years, neglecting the obvious fact that in the_ _spring of 1941 the United States' entry into the war was seven_ _months away. Over the course of the evening, Raymond noted_ _renditions of, among others, \"Don't Sit Under the Apple Tree,\"_ _\"You'd Be So Nice to Come Home To,\" and \"They're Either_ _Too Young or Too Old.\" His fellow classmates, however, applauded_ _those songs as if the songs had actually furnished the_ _score to their high school years. Twice the band played \"Skylark.\"_ _Raymond knew that song had come out in 1942, because_ _he remembered exactly where he was when he first heard it. He_ _had left Wembley on the train with a small group of friends and_ _acquaintances, new recruits like himself, and all of them bound_ _for Kansas and basic training. The train had a stopover in St. Paul, and Raymond got off to stretch his legs. He had never ridden_ _a train for that distance before, and he had never been that_ _far east. He walked up and down the platform, never straying_ _more than three cars from his own, breathing in the odor of_ _diesel fumes, and listening to a song that came from somewhere_ _inside the station. Its melody was both melancholy and expectant,_ _and in that regard served as the perfect signature for his_ _state of being. He would hear the song often enough in the fu_ _ture that he would soon know its name and composer, and the_ _versions he favored. While he stood on the platform listening, a_ _thin, cold rain fell, and Raymond felt as if he were already in a_ _foreign country._\n\n_It was raining on the night of the ten-year reunion dance_ _too. The preceding week had been exceptionally warm and the_ _summer of 1951 exceedingly dry, so when the rain began, it was_ _treated as another cause for celebration on a night when most_ _people's spirits were already high and their moods festive. The_ _dance was held in the high school gymnasium, and not only the_ _gym's doors but the school's outer doors were thrown open to_ _take advantage of the rain-cooled air. In fact, couples almost_ _immediately began dancing their way out of the gym and into_ _the cooler, darker school corridor, often swaying in time to the_ _music right next to the double doors propped open to the night._\n\n_In Raymond's view the weather was to blame for what occurred_ _that night, though he was willing to concede that things_ _might have ended as they did even without the rain._\n\n_Throughout the evening, in dance after dance, partner-switching_ _had been customary. Not only men but women too_ _felt free to cut in on other couples, and the intent behind this activity_ _was often to restore couples who had dated in high_ _school. The husbands and wives who were not from Wembley,_ _and the unmarried men and women, good-naturedly went along_ _with the constant realignment of couples._\n\n_For most of their high school years, Alma had dated Monty_ _Burnham, the most popular boy in school. Their relationship_ _was so long and durable that their classmates assumed that_ _Monty and Alma would someday marry. But the romance_ _foundered, for the rumored reason that Monty \"got too serious.\"_ _And though they each went on to marry other people\u2014_ _Raymond and a girl from Grand Forks\u2014there still seemed_ _something fated about them\u2014the class president and the prettiest_ _girl\u2014being together._\n\n_That would certainly explain why, when Monty cut in on_ _Raymond, laughter, cheers, and applause spontaneously erupted_ _from their classmates._\n\n_Playing to the approving crowd, Monty danced off with_ _Alma, whirling, spinning, exaggeratedly swinging their arms,_ _moving them off together with speeded-up, elongated steps, as if_ _they were waltzing around a nineteenth-century ballroom,_ _though Monty paid no attention to the rhythm of the actual_ _music. The song in fact was, as Raymond remembered well,_ _\"Begin the Beguine,\" one of the few instances when the band_ _accurately matched the music to their high school years, though_ _not to the year of their graduation. Alma's expression\u2014her unrestrained_ _smile, her astonished laughter\u2014made it plain that she_ _enjoyed being part of the show._\n\n_Once they exited the gym, they kept right on going, into the_ _night, into the steady rain. Other couples followed their example,_ _and soon half the reunion crowd was dancing exuberantly_ _on the high school's soggy lawn. Out there they certainly_ _couldn't hear the music anymore, but it didn't matter. Their_ _laughter would have drowned out the band anyway._\n\n_Raymond stood in the doorway, dry and watching it all. Or_ _trying to. The rain was a heavy veil that made it difficult to tell_ _one dancing couple from another. He thought he was able to_ _keep track of Monty and Alma, but he couldn't be sure\u2014so_ _many of the men wore dark suits and white shirts, and Alma_ _was one of three women who came to the reunion in a light_ _blue dress, and that blue turned dark when it became wet._\n\n_Finally, Raymond turned away. He didn't want to do what_ _he felt like doing\u2014running out into the rain, grabbing Alma by_ _the arm, and hauling her away from Monty Burnham and the_ _entire reunion. Behavior like that would only reveal him to be_ _more or less what he was\u2014a humorless, jealous fool._\n_Instead, putting on an expression that he hoped would be_ _read as sophisticated nonchalance, he strolled back to the gym,_ _shaking his head in amusement over his classmates' antics._\n\n_He walked to the refreshments table. There, two identical_ _punch bowls were set up. One container of pink liquid, however,_ _had been spiked with Everclear, and Raymond assumed_ _this was the nearly empty bowl. He hadn't had anything to_ _drink all evening, but now he scooped out a full cup, lit a cigarette,_ _and waited under the backboard of one of the gym's two_ _baskets for his wife to reappear. The net had been removed_ _from the rim, and Raymond wondered if that had been done_ _just for this occasion or if it had been taken down for the summer._ _The cr\u00eape-paper decorations that had been hung the length_ _of the court had gone limp from humidity._\n\n_At the other end of the gym stood a cluster of men talking,_ _Raymond was sure, about the war and their roles in it. Raymond_ _was entitled to join the group, but he had tired of war_ _stories and their distortions before the war was even over. On_ _more than one occasion he had listened to accounts of battles_ _on the very day they'd been fought, battles that Raymond himself_ _had been in, and he invariably found those reports, like the_ _songs played at the reunion, false. Yet those storytellers weren't_ _necessarily lying; they were simply trying to make sense of the_ _senselessness they'd lived through, and in the process warped_ _the truth for the sake of their tales._\n\n_His cigarette burned down to the filter. Raymond had been_ _willing to flick his ashes onto the gymnasium's floor, but he_ _didn't want to crush out the butt on the varnished hardwood. He considered returning to the outside door. There he could_ _throw his cigarette out into the rain and hope that it might seem_ _his only reason for going there. Instead he walked over to the_ _water fountain outside the locker room door and deposited his_ _cigarette there. There must have been twenty butts already_ _sticking wetly to the porcelain._\n\n_Monty and Alma were by now the only couple who had not_ _yet returned from their dance in the rain. Raymond surveyed_ _the gym for Mrs. Burnham, wondering if she was as troubled_ _and curious over her spouse's absence as Raymond was. She,_ _however, was involved in a raucous conversation with four_ _other women, one of them so pregnant she looked as though_ _she might give birth at any moment. In what must have been a_ _joking reference to her condition, the woman opened her eyes_ _wide and puffed out her cheeks. The other women laughed uproariously._\n\n_The band began to play its rickety rendition of \"Moonlight_ _Serenade,\" another of the few examples of songs that really_ _were popular when they were in school, though Monty and_ _Alma would have been more likely to dance to it than Raymond_ _and Alma._\n\n_And just at that moment Alma entered the gym. Her dress_ _was soaked, and while its skirt was wrinkled and shapeless_ _below her waist, the fabric above clung shamelessly to her_ _torso. Her hair hung down in wet tendrils. Alma wore a bewildered_ _expression, and perhaps she'd been crying, but then again_ _the rain might have created that effect. In each hand she carried_ _a shoe. Raymond made no move to approach her or signal his_ _presence._\n\n_Eventually of course Alma saw him standing under the basket,_ _and she began to hurry toward his end of the gym. Alma_ _had not been a cheerleader in high school, but she ran across_ _the gym floor just as she might have after leading a cheer for the_ _home team. And Mrs. Burnham and the other women in her_ _group regarded Alma as coolly as high school girls might have_ _watched a cheerleader for a rival team._\n\n_Before Alma reached him, Raymond decided that she had_ _allowed Monty Burnham to fuck her somewhere out there in_ _the rain. When she came close to him and stood on her tiptoes_ _to kiss his cheek, Raymond looked over her shoulder, examining_ _the back of her dress for mud or grass stains, evidence that she_ _had been lying on her back on the wet ground. The fabric was_ _only wet, but Raymond told himself that didn't mean anything._ _She and Monty might have done the deed in the backseat of a_ _car or leaning up against the school's bricks. Not that it mattered._ _. . . In his mind Raymond had moved into the realm_ _where neither logic nor fact was necessary for belief to be sustained._\n\n_Because she might admit it, he didn't ask Alma if she had_ _just had sex with Monty Burnham. And he didn't ask her because_ _she might deny it. From that moment forward, Raymond_ _Stoddard would live as a man of faith._\n\n_A few minutes later, Monty entered the gym. He stood at_ _center court, gazed expectantly up at the ceiling, and turned up_ _both palms. It seemed to Raymond that men especially found_ _Monty's act humorous._\n\n _For ten years Raymond did nothing in thought or action that_ _might alter his conviction about what had happened between_ _his wife and Monty Burnham. Indeed, he learned how he could_ _make practical use of it, particularly in his sexual relations with_ _Alma. If he felt no desire for her he could justify his rejection of_ _her by reminding himself that she had been unfaithful to him. And when that same thought\u2014Alma being fucked by another_ _man\u2014oddly stirred Raymond, he could make love to his wife_ _with a passion that was equal parts passion and anger. Have sex_ _with him, will you\u2014take that, and that, and thatthatthat. . . ._\n\n_So of course Alma wanted to attend the twentieth reunion What other opportunity would she have, since they had left_ _Wembley and moved to Bismarck, to renew her relationship_ _with her former lover? And of course Raymond was determined_ _that they not go. He could not face the possibility that another_ _man would once again dance off with his wife. Yet without_ _speaking to Alma of his belief, what could he possibly say or do_ _that would explain his unwillingness to reunite with his classmates?_\n\n_For months he worried the problem. Then, one night while_ _he was watching an episode of_ Peter Gunn, _a solution came to_ _him. He had been drinking beer since he'd come home from_ _work, and alcohol in any form often allowed his thoughts to_ _travel into orbits they could never reach when he was sober. It_ _was so simple, really. He could get a gun, Raymond reasoned,_ _and kill Monty Burnham. The notion had barely formed itself_ _when he laughed out loud at its ridiculousness. Alma heard him_ _and came in to ask him what was so funny. Raymond gestured_ _vaguely in the direction of the television._\n\n_Barely a week later that coldly murderous thought revisited_ _Raymond. He was sober this time, and though he laughed once_ _again, after a few seconds his laughter halted as abruptly as if a_ _hand had clutched his throat. The next time, he didn't laugh at_ _all._\n\nThis story could be read as another chapter in the lives of the Monty Burnham and Raymond and Alma Stoddard that have been revealed through the other fictions in these pages. Indeed, those fictional characters perhaps have more substance than the real people. The stories allow access to their inner lives, something that life generally won't offer. But notice that \"Reunion\" (published in the magazine _Windsong_ ) says nothing about Killeen, Texas, and what occurred in the fictional bathroom of a fictional hotel between a fictional Monty Burnham and a fictional Alma Stoddard. It contains no reference to a wartime confession or a confrontation between two soldiers, and of course I have no evidence that either event occurred. In fact, the narrative has no basis in any reality I'm aware of except that Raymond, Monty, and Alma graduated from the same high school in the same year. But I suspect that the story had its origin in the emotions I experienced that October when Marie vanished so suddenly from my life.\n\nAfter Marie ran from my apartment, I looked everywhere but couldn't find her. Not in the library, not in the student union. Not in the registrar's office, where she had a work-study job. Not on the banks of the English Coulee, where she sometimes sat and watched the water's slow swirl and flow. I phoned Blackmore Hall, Marie's dorm, but was told she wasn't in her room. I called Jackie Rickinger, Marie's friend from Bismarck who lived in another dormitory. She hadn't seen Marie since their Educational Psychology class that morning. I went to Neville Hall, the home of the education department, and walked up and down the corridor where the faculty had its offices, believing that she might be conferring with a professor. After I made all those rounds, I went back and visited or called each site again. And then again.\n\nIn spite of my search's futility I was confident that I'd find Marie and sure that she would accept my apology. Because he had once meant something to both of us and because of his hard-luck history, I had always been careful to temper any remarks I made about Gene Stoddard. But no more. As part of my contrition I would gladly portray him as a lout and a liar. Further, I was certain that I could convince her, because this was now my belief, that I regarded the news of her pregnancy as cause for celebration. If I found her soon enough, I'd propose that we commemorate the occasion with a steak dinner at the Bronze Boot.\n\nBut when the curfew for women arrived at ten o'clock (men had no restriction on their hours), I was pacing the sidewalk in front of Marie's dormitory, and I still hadn't seen her or heard from her.\n\nThe receptionist at Blackmore Hall was in my Shakespeare class. Phyllis. Stout, stringy-haired, earnest Phyllis Orr . . . After flattering her with questions about _King Lear_ and the upcoming exam, I finally persuaded her to reveal that Marie had checked out of the dorm hours earlier. On the form where she signed out, she listed her parents' address and phone number as her location until Sunday afternoon at five o'clock.\n\nMarie had no car. No train ran between Grand Forks and Bismarck, and the bus left much earlier in the day. It seemed unlikely that anyone with whom Marie could ride would leave for Bismarck on a Thursday afternoon or evening. Those factors helped me decide: I wouldn't drive to Bismarck that night\u2014no matter how hard I pushed the Studebaker, I wouldn't arrive before three A.M.\u2014but if I didn't hear from Marie before morning, I would leave then.\n\nI slept little that night, so when dawn came, I didn't waste any time. I tossed a hastily packed suitcase into the trunk and hit the highway. Less than five hours later, I was in Bismarck, and before I went to my own home, I drove to Marie's.\n\nAs soon as Mrs. Ryan answered the door, I immediately knew my being there was a mistake. I had not only Marie's accounts but the evidence of my own eyes to know how Mrs. Ryan had struggled with alcohol over the years. A nervous, high-strung woman, she used her worries, real or imagined, as her justification for drinking, so when I asked if Marie was there and Mrs. Ryan's eyes widened with her answer\u2014\"Isn't she in Grand Forks? With you?\"\u2014I realized that I simultaneously had caused her to be apprehensive about her daughter and had given her a reason for pouring herself a stiff drink. Stammering, I tried to say something that would allay her concerns. \"I-I'm sure that's where she is. But when I told her I was coming to Bismarck for the weekend, she said maybe she'd follow me here. I was sure she was joking, but I thought I'd check just to be sure. I'm sorry I bothered you.\" Mrs. Ryan habitually chewed her fingernails to the quick, and before I backed away from the door, she had the tip of her little finger between her teeth.\n\nMy mother too was skeptical of my reason for coming home. \"I needed a little break,\" I told her. \"You know, sleep in my own bed. Eat a home-cooked meal.\"\n\nBy that time my parents were living apart, and my mother had perhaps developed the ability to read the signs of a troubled heart in the faces of her family's men. \"Are you and Marie having difficulties?\" she asked.\n\nI couldn't manage a response any more convincing than, \"Nothing serious. We'll work it out.\"\n\nThroughout the day I drove by Marie's house, hoping I'd see her in the yard, in a window. When I wasn't driving, I was back at my mother's house or in the phone booth next to the Mobil station, calling Grand Forks\u2014Marie's dorm, Jackie Rickinger's room, my apartment. I gave my roommate my mother's telephone number and ordered him to phone there immediately if Marie, if any female, should call. Finally, late in the afternoon, self-conscious of my many circuits of the Ryans' block, I parked a short distance away and continued my watch from the car.\n\nI'm not sure how long I sat there but it was with no sign of Marie. In the gutter in front of his house a neighbor of the Ryans' burned a pile of prematurely fallen leaves. The air was still, and since the block was canopied with trees, the leaf smoke didn't rise very high. It hovered over the street until evening dusk arrived, and then, mingling, the smoke and the dusk completed the task of bringing autumn darkness to the block. I drove away in defeat.\n\nSince it seemed as though it had lately become my habit to distress mothers and mothers-to-be, I decided to risk disturbing one more. After parking my car in front of my mother's home, I walked down to the house that would always be the Stoddards' to me.\n\nAlma Stoddard\u2014I could never get used to her as Mrs. Mauer\u2014answered the door, and I had barely uttered a word of greeting, much less offered a reason for my visit, before she embraced me and ushered me inside.\n\nThe living room was little changed from the room I had spent so many hours in as a child. The console television was now a color set _\u2014The Wild Wild West_ was on\u2014and on top of the TV, Gene's graduation portrait joined his sister's. In place of the wing-backed chair where Mr. Stoddard used to sit there was now a Naugahyde recliner. This was where Gene's mother insisted I sit. She pulled up a footstool and sat at my feet.\n\nShe was even thinner than when I had last seen her, and the vestiges of her beauty had vanished with those pounds. Her dark hair was shot through with gray, and her eyes were sunken and dim, like a votive candle's flame guttering in its own liquid before it goes out. Creases ran down from the corners of her mouth. The smile she struggled to shine on me, however, seemed genuine, and genuine too seemed her interest in my college career.\n\nAfter I told her about the courses I was taking and my possible plans for graduate school or law school (no longer a serious ambition but one I still occasionally expressed, especially on Keogh Street), she said, \"I hope Gene changes his mind and applies for college one day.\"\n\n\"He still has time. There's a woman in my Modern Drama class in her sixties.\"\n\n\"I remember when the two of you used to sit at the kitchen table and work on your arithmetic together. . . . He's certainly smart enough to attend college. . . .\"\n\n\"Sure, he is.\"\n\n\"But he doesn't have the discipline. And you need discipline, don't you? Brains and discipline?\" She bent toward me, eager for confirmation of her theories on educational success.\n\n\"I suppose. . . .\"\n\n\"A chemist. I always thought Gene would make a good chemist.\"\n\n\"He would. . . .\"\n\n\"But when he and the Ryan girl broke up, something went out of him. He just couldn't get himself back on track.\"\n\nHer sense of causality was off. Gene's derailment coincided with his father's suicide and subsequent notoriety, but I didn't correct her.\n\nMrs. Stoddard turned and gazed toward the kitchen, remembering perhaps the hours Gene and I had spent hunched over our homework there. Her choice of subject was appropriate; while math problems stubbornly resisted my efforts to solve them, numbers gave up their secrets quickly to Gene. Then she sat up straight on her stool and coolly said, \"And your life is on track, isn't it?\"\n\n\"I guess.\"\n\n\"Completely on track. Of course it is. College. Plans for the future. A girlfriend. She's still your girlfriend, isn't she?\"\n\nAlthough at the moment I was unsure of that fact, I nodded.\n\n\"He kept calling her. . . . My God, how long did he call her, trying to talk her into taking him back?\" She looked again in the direction of the kitchen. Their telephone, I recalled, hung on the wall next to the cupboard. \"It reached the point where I thought of calling her myself. Can you imagine that? I don't know what I would have said. . . . And I even suggested that he call you, his old friend.\" Alma Stoddard turned a mirthless smile in my direction. \"But he wouldn't do that.\"\n\nI couldn't think of anything to say in my defense.\n\n\"Only later did I realize how foolish that suggestion was. Because you weren't his old friend anymore, were you? You weren't about to help him with _that_ problem, were you?\"\n\nI stood abruptly. \"I have to go, Mrs. Stod\u2014Mrs. Mauer. I'm home for the weekend, so I thought I'd see if Gene might be here. Does he have his own place here in town?\"\n\nAlma Stoddard didn't rise. She cupped her chin in her hand as if she were bracing herself. \"He's living in Minot now. Working for his uncle's construction company up there.\" She quite deliberately did not look at me.\n\nI thanked her and without escort left the house. Keogh Street in those years was largely treeless, yet just as in Marie's neighborhood, someone in the vicinity was burning leaves. I could smell the fire, though I couldn't see smoke or flames. Why that smell is regarded so fondly is beyond me. It is, after all, the odor of endings.\n\nThe following day, Saturday, I was once again on the road soon after sunrise, driving up Highway 83 from Bismarck to Minot. The early start was necessary. I wasn't sure how I'd find Gene, but I was prepared to drive up and down every street in the city until I sighted that distinctive white convertible of his. And when\u2014or if\u2014I came upon it? Then I would seek out its owner, and determine if he had recently driven to Grand Forks in response to a call from a Miss Marie Ryan, who required rescue or refuge from the fool who had fathered her child.\n\nAs it turned out, Gene was not nearly as difficult to locate as I'd feared. I stopped at a gas station on the south end of the city, opened a telephone directory, and there it was: G. Stoddard, 17051\u00bd Arapaho, KL5-2232. The station attendant provided directions to an avenue on the city's hilly north side.\n\nWe were no longer friends, Gene and I, but our lives continued to parallel in odd ways: We both continued to live in North Dakota, not in our hometown but in cities in the state's northern half, and we had both rented basement apartments. His white Impala was parked in front of a stucco house that had a covered apartment entrance built into its side like a chute.\n\nThe convertible's top was down, and when I peered into its interior, I guessed that it had been left down all night. The day might have been warm for October, but the night had been cool and the car's dashboard and upholstery were, at that late morning hour, still beaded with dew. Gene must have had matters more important than his car's welfare on his mind when he pulled to the curb.\n\nWith my palm still wet from where it swiped a path through the moisture on the car door, I marched on. The leaves littering the stairwell clattered like castanets as I kicked through them on my way down the steps to the apartment. I had not quite reached the door when I tried to formulate a plan that good sense would have said I'd needed well before I got that far.\n\nIf Gene admitted that Marie was inside\u2014or if his denial was unconvincing\u2014I was prepared to push past him and . . . and what? Haul her bodily from the apartment? Carry her to my car and transport her back to Grand Forks against her will? I could as well put what love she might still have for me on a chopping block and butcher it before her eyes.\n\nBut I had gotten that far, and I heard in the apartment's interior someone moving toward the door in answer to the buzzer I had already pushed twice and was about to lean on again.\n\nIt was probably not my desperate, bloody thoughts that somehow showed in my features and frightened the young woman who came to the door. More likely it was simply the sight of a stranger.\n\n\"Yes?\" She peered around the door she was willing to open only inches. I guessed her to be close to Gene's and my age. Her blond hair was tousled, she wore no makeup, and her eyes were still heavy-lidded from sleep.\n\n\"I'm looking for Gene, Gene Stoddard?\"\n\nNothing in my manner apparently conveyed goodwill, because she didn't open the door any wider or alter her guarded expression. \"He's at work?\"\n\n\"I saw his car. . . .\"\n\n\"Len picked him up? Like always?\"\n\nShe made every sentence into a question, a mannerism I adopted as well. \"I'm a friend of his? From Bismarck?\" In my case, however, the tone matched my dubiety.\n\nEither the word \"friend\" or \"Bismarck\" convinced her of my harmlessness. She opened the door and stepped back, an invitation for me to enter.\n\nMy mission to Minot was completed. Marie and this young woman could not have both been at 1705\u00bd Arapaho. Nevertheless, I felt too foolish at that point simply to walk away. I stepped into Gene Stoddard's home.\n\nHis place was not furnished any better than mine, but without Marie Ryan's touch, it had nothing to raise it above the level of slovenly, threadbare utilitarianism. The couch was covered with newspapers. Empty Hamm's beer cans littered the top of the coffee table. A bent coat hanger served as an antenna on the television that sat on a chair. The room smelled of cigarette smoke and motor oil\u2014Gene must have been operating heavy equipment for his uncle's company.\n\nThe young woman reached out her hand. \"I'm Joy?\" After we shook hands, she stepped quickly back and folded her arms across her chest, self-conscious because nothing covered her ample breasts but her pink cotton pajama top. Her plaid pajama bottoms didn't match the top and barely reached her ankles. Joy and Gene slept together. In the same bed. All the night through. Head to dreaming head.\n\nJoy had a pretty round face and a plump, shapely figure. She was self-conscious about her hair, but whenever she made a tentative move to reach up and pull her fingers through the tangles, she had to quickly bring her arm back down to cover her breasts.\n\n\"Are you from Minot, Joy?\"\n\nShe nodded eagerly. \"I work at First National? Bank?\"\n\n\"And Gene's working for his uncle?\" Raymond Stoddard, I recalled, also once worked for the construction company owned by Alma's brother.\n\n\"He really likes it? He says it reminds him of when he was a kid? Digging in the dirt with his toy Caterpillar?\"\n\nHis play must have been solitary. I had no recollection of the two of us on our hands and knees pushing dirt around.\n\n\"What will he do when winter comes?\"\n\nHer shoulders rose as automatically as the ends of her sentences, an embarrassed shrug that made me think she and Gene didn't talk much about the future.\n\n\"Well. Winter. It's not here yet, is it?\" That mention of time, though it was to a season rather than the hour, provided me with the opportunity to make my exit. I pretended to look at the watch I didn't wear. \"I have to hit the road. I was just driving through Minot, and I thought I'd take a chance and see if I could catch Gene.\"\n\n\"He'll be sorry he missed you?\"\n\nYes, that was open to question.\n\nI was halfway out the door when I realized I had left something unsaid. \"I hope you and Gene have a very, very happy life together.\"\n\nThat sentence I had no difficulty ending on a dying note of conviction.\n\nUpon the new planks and between the freshly painted columns, my father stood, hands on hips, a happy man on the porch of the house he and his brother had worked so hard to restore. His smile was as wide as any I'd ever seen cross his face, and though it was turned in my direction, I would learn soon that I was not the reason my father was beaming as if lit from within.\n\nThe highway that led from Minot back to Grand Forks ran right through Wembley, so I'd phoned ahead and asked my father if I could stop there and spend the night with him and Uncle Burt. Of course, of course, he said; it would be wonderful to see me and hear firsthand how I was doing in college.\n\nWhile I carried my suitcase up the front walk, Uncle Burt came out too, and both brothers watched me approach. Whether it was Burt or my father who was in charge of preparing their meals, the food must have agreed with my father. Since he and my mother had separated, he had gained weight, and now the brothers truly looked brothers\u2014a matched pair of tall, smiling, balding big-bellied men. There it was, I thought, the body of my future.\n\nMy father and uncle were both eager to show me the work they had done on the house, but before we went inside, we had to walk around the outside of the house so they could show me what was _not_ there but had been in their youth\u2014the outhouse, the poles that first brought electricity and then telephone service to the house, the huge garden whose harvest fed them throughout the year, the pit and the barrel in which the family burned much of its trash, and the barn where their father stabled his horse and the small buggy it pulled through the streets of Wembley.\n\nUpon entering the house, I was immediately taken on another tour, and as we moved from room to room, the brothers continued to take turns telling me about the remodeling work. They had even tried to furnish the house according to their memories. My father, who had never shown the slightest interest in the way my mother decorated our home, now boasted about how the iron bed frame in the largest bedroom was just like the one that once held their parents' mattress, which had also been covered with a blue coverlet. The rocking chair in the parlor was a duplicate of the one their father had sat in while reading the _Wembley Daily_ _News_ after working all day at the drugstore.\n\nThen, once I had been shown the claw-foot bathtub, the wainscoting, the newel post, the coved ceiling, the brass doorknobs, and the leaded glass windows, the three of us sat down at the dining room table (its mahogany surface gleaming like a mirror). For the first time in my life, my father offered me a beer. I accepted, and then as we three men who shared a last name raised our bottles of Budweiser, my father revealed that he and his brother had at long last solved the mystery of Raymond Stoddard's motivation for murdering Monty Burnham, or so they believed.\n\nRaymond's father, I was reminded, had worked for the railroad, a job that didn't make him wealthy but provided a steady income and a comfortable life for his family. In fact, they were sufficiently well fixed that the senior Mr. Stoddard was able to purchase a vacation place, rare indeed among North Dakotans of that generation. Their little cabin was on a lake in northeastern North Dakota, not far from Wembley.\n\nI recalled that Gene often lamented having to spend two weeks of his summer at Lake Liana. It meant he had to miss part of the Little League season, and to make matters worse, there was nothing he particularly liked to do at the lake. Fishing didn't appeal to him, and that was the activity his grandfather and father engaged in from dawn to dark. Whenever Gene's grandmother saw him idle, she would put him to work, doing everything from hauling water to picking blueberries. His grandparents were strict and insisted that the lights be turned out early every evening and that everyone go to bed in order to rise early the following morning. For Gene, the time spent at the cabin was more punishment than vacation.\n\nGene's feelings about the place were not, however, the family's feelings. Everyone else loved the cabin, no matter how dilapidated it was or how primitive living conditions were there. After his retirement from the railroad, the senior Stoddard managed to spend as much time there as in Wembley.\n\nUntil, that is, his and his wife's health began to fail. She developed a heart condition that made even walking across a room a venture that taxed her to the limit. His bones became brittle; he fell and broke a hip. A coughing fit cracked a rib. He shrunk and stiffened; bending down or raising an arm overhead was next to impossible. Then, as if to make his diminution complete, his wife died, and his spirit shrunk to match his body.\n\nGiven those circumstances, his beloved Lake Liana might as well have been on the far side of the moon, so inaccessible was it to him. Others now spent vacations at the cabin that he had worked to make livable, and upon their return he had to listen to their reports on how many fish they'd caught, how many glorious sunsets they'd witnessed, how bracing was the morning air.\n\nHis health worsened. Pneumonia put him into the hospital, and as he lay in the room he believed he might never leave alive, a visitor came and presented Mr. Stoddard with a business proposition.\n\nSince the old man would never again cast a line into Lake Liana or spend a night under the cabin's roof, and since he could not divide the place equally among his sons and daughter, who were in any event liable to squabble over the property after their father was dead, wouldn't Mr. Stoddard prefer to sell it himself and thereby exercise some control over the cabin's future? If you sell it to me, the visitor argued\u2014pleaded\u2014you can be sure that the cabin will be as well cared for\u2014as loved\u2014as it has been with a Stoddard as owner. The visitor even shed a few tears as part of his petition.\n\nThe old man was sufficiently selfish and grudging that this appeal worked on him. From his hospital bed he signed the papers that deeded the cabin and the waterfront land on which it sat to his visitor. The salesman\u2014swindler, some might have it\u2014who talked Mr. Stoddard into this transaction paid the dying old man exactly what the original purchase price had been.\n\nWhen Raymond learned of the sale, he was furious, but though plenty of people told him he could have the deal nullified\u2014an enfeebled old man in a hospital, for God's sake!\u2014Raymond refused to do anything. If his father was going to be that petty, then he, then the entire family, would have to live with the consequences.\n\nLike his father, Raymond had loved the cabin, and as long as his father was alive, Raymond blamed him for what he had done. Once the senior Stoddard died, however, Raymond transferred his rage to the man who had talked the old man out of his property. That smooth-talking son of a bitch was none other than Monty Burnham.\n\nMy father sat back, and then it was left to my uncle to explain the sources and the evidence that they had uncovered and relied on in piecing together their story.\n\nTheir research started, strangely enough, on the occasion of my high school graduation. Burt had been at our home when the news came of Bob Mullen's drowning at the river, and the tragedy brought back to him his own graduation celebration. It too had been held by a body of water, though one much more benign than the Missouri River. The family of one of his classmates had a cabin on a lake about forty miles from Wembley, and they convened there for a night of bonfires and beer drinking. Burt could recall many of the specifics of that evening\u2014including an altercation between two classmates\u2014but for the life of him, he couldn't remember whose family owned the land that was the site of the celebration. Ordinarily he might have been able to live with his curiosity unsatisfied, but for some reason this little mystery nagged at him.\n\nHe went for long drives in the country around Wembley, searching for the cabin or the lake. Finally, on the shores of tiny Lake Liana, he found what might have been the site of his graduation night celebration. The cabin, however, looked much different from the way he remembered it, and he might have marked that down to another of memory's inaccuracies, except that memory usually tricks us by enlarging the buildings of our past. We return to our elementary school classroom or high school auditorium, and we're shocked to discover how small it is. Then Burt realized what had caused the disjunction between memory and reality. The cabin had recently been remodeled and was almost doubled in size because of an additional story having been added.\n\nDorling was the little town nearest Lake Liana, and a gas station attendant there was able to tell Burt who owned the cabin. It was the Burnham family. Of course\u2014that made sense. After all, Monty was in the same graduating class, so it would stand to reason that the Burnham cabin could have been used for the party. Burt had a vague memory of Monty on that occasion\u2014was he involved in that fight with another classmate? Burt also seemed to recall that it was Monty who departed early from the party. But he wouldn't have left his own family's cabin, would he?\n\nBut Burt had to live with these uncertainties and inconsistencies until quite recently when he had a chance encounter with another member of the Burnham family. Monty's cousin, Irv Schmitz, came into the pharmacy to have a prescription filled, and Burt asked him what had happened with the cabin. Oh, it was still in the family, Irv said; Monty had been so proud of the deal he'd made to acquire the cabin that no one would ever consider selling it. Monty's sister and her husband had completely remodeled the place, and all the branches of the family took their turns vacationing at Lake Liana. Not much questioning was needed to get Irv to divulge the particulars of Monty's negotiating coup. \"There was nothing,\" Irv said, \"that old Monty appreciated more than talking someone into or out of something.\"\n\nAt that point Burt pushed his chair back, just as my father had earlier when he'd finished his portion of the narrative. And there the brothers sat, two aging fat men finally contented that yesterday was fixed, its mysteries solved, its story complete. Of course any explanation for Raymond Stoddard's behavior that had him acting to avenge the loss of an ancestral property would be especially satisfying to those two. They were, after all, living in their own reclaimed, restored past.\n\nDarkness, which on dwindling autumn days hasn't much patience anyway, could wait no longer and now filled the room. As if any movement required enormous effort, my father partially lifted from his chair, reached up, and twisted on the light above the dining room table before sitting heavily back down. For a moment, we squinted at one another in the sudden illumination as if we were verifying the company we kept. The brothers smiled at me to see that I was one of them.\n\nWith the same abruptness but none of the violence with which Marie rose from my table only days before, I stood. \"I have to go,\" I said. \"Back to Grand Forks.\"\n\n\"You're spending the night,\" my father said. It was not a command but a gentle reminder. My suitcase already stood in the bedroom that had always been reserved for guests.\n\n\"I can't stay.\"\n\nThe brothers regarded me quizzically. I hadn't time to concoct an excuse because I had nothing more than a spontaneous impulse to flee. Yes, the more time that passed with no word from Marie, the more my fear and concern increased, but something additional was working on me\u2014in me\u2014and trying to push me toward the door. I hadn't sufficient time or maturity or wisdom to absorb the lesson Marie had tried to teach me, but perhaps without her recent angry words on the subject I wouldn't have sensed anything wrong in how those brothers lived.\n\nI repeated my words, \"I have to go,\" and both my father and my uncle looked at me strangely, an indication, I hoped, of what I was to them, a stranger, not their kind, not someone who dwelt in the past, whether refashioned or otherwise.\n\n\"We were thinking,\" Uncle Burt said, \"that we'd walk downtown for supper. Get some steaks at the Windmill.\" He spoke so calmly, so mildly, that he must have perceived me to be in an agitated state.\n\n\"I can't. There's . . . there's a . . .\" I started to explain, to _try_ to explain, that a girl was out there somewhere, a girl I had to find. But I stopped short when I realized that I was speaking to men unlikely to understand. My father had left the woman he loved\u2014didn't he? didn't he still love my mother?\u2014for this, this boy's life. And Burt? Well, in the previous few minutes I had suddenly moved much farther down the road toward comprehending my uncle's nature, something that I wouldn't have full understanding of for years to come.\n\n\"I can't stay.\"\n\nThe brothers looked at each other again, and maybe in that instant they both recalled something about youth that they had forgotten, no matter that they were living in the house where they must have first felt youth's passions. \"I'll get his suitcase,\" Burt said.\n\nOr perhaps Burt left us alone so that we could have a father and son talk, during which I would disclose what was troubling me. But I was not interested in anything that might delay my departure.\n\nMy father was the one to snap the silence stretching tauter between us, and he did it with the question that was always his easiest substitute for intimacy. \"Do you need any money?\" He stood and reached for his wallet.\n\nThat question might have been my entr\u00e9e to say, Money? I do indeed need money, and lots of it. I'm getting married and soon. But I had already determined that the distance I had closed that day between my father and me was for nothing more than traveling convenience. \"I'm okay,\" I said.\n\n\"Gas? I believe the only station open today is the Conoco you passed on the way into town.\"\n\n\"I filled up in Minot.\"\n\nHe nodded. \"The Studebaker still does well on mileage?\"\n\n\"Almost as good as your old Rambler.\"\n\nHis faint smile told me that I had pleased him by recalling one of his vehicles. His pleasure would have been diminished if I'd told him that I remembered that car, would _always_ remember that car, because it had been parked in the Stoddards' driveway on that day. I hadn't a single image in my memory cache of his emerald-green Rambler in front of our house.\n\nBurt clumped into the room with my suitcase, I took it from him, and my father and I walked out into the October evening's chill. I opened the trunk and lifted the suitcase in.\n\n\"That theory you have about Monty Burnham cheating old Mr. Stoddard out of his cabin . . . did you tell Mom about it?\"\n\n\"That subject wouldn't be . . .\" He paused, as though his concern for the precision of language was coming back to him. \"An area of interest for her. Not any longer.\"\n\nWhen I drove away, my father was gazing up into the bare branches of a maple tree that overhung the house and no doubt dropped a good many of its leaves onto the roof and into the rain gutters every year. If that tree had been there when he was a boy, it couldn't have been more than a sapling.\n\nBack in Grand Forks I resumed my search for Marie, phoning, driving by, and looking into all the places I had tried only a few days before. I even tried some new locations. Since it was Saturday night, I checked a few bars, though Marie disliked them and their atmosphere. There were undoubtedly parties throughout the city, and while Marie was not averse to attending them, I had no way of knowing where they might be held.\n\nIt was well past midnight when I gave up and returned to my apartment. My roommate was not there, but that was not unusual. Somewhere on campus there was a laboratory he could get into and study in, no matter what the hour. I checked the kitchen table and the stand the telephone sat on to see if he had left a note informing me that Marie had called. There was nothing.\n\nFor many of the hours since Marie had run out, I had been in motion, and that had enabled me to keep my anxieties under control. But once I stopped moving, with no destination or mission but to get up the next morning and begin again, my fears leaped to the fore, and though they had no specific form, that didn't mean they didn't have the power to rattle me right to the core. I didn't believe that Marie would try to harm herself. A quarrel like ours would never be enough to shake her love of and commitment to life. Similarly, she would never contemplate ending the life inside her, even if that had been possible, and, need I remind you, in 1964 abortions were not only illegal but generally unavailable. And while those beliefs might have had their birth in her Catholic background, it wasn't the church that sustained them. She and I had both forsaken the religions we grew up in, and besides, Marie's strongest faith had always been reserved for the values and certainties of her own character. But my fear didn't need sharp definition. I had only to imagine the permanence of what was presently true _\u2014Marie was lost to me\u2014_ to experience a dread unlike any other.\n\nDuring the days and hours and minutes of my futile hunting for Marie, I had begun increasingly to feel like a failure, not as a searcher but as a lover, a mate. I was sure of my love for her, yet I questioned its power\u2014shouldn't it have been strong enough to lead me to her, no matter where she might be?\n\nOn Sunday morning I added a new location to my search circuit. I drove past Saint Michael's Cathedral, making sure I passed when crowds of congregants were exiting. No luck. I had tried libraries, bars, and churches, hometowns and old boyfriends; I had staked out her family and her dormitory, and nothing, not the personal or the institutional, had yielded results. By mid-afternoon the October sky had darkened to a stony gray and a cold autumn wind had begun to blow. I decided to go back to my apartment to try the telephone for a few hours.\n\nAnd there she was. In my kitchen.\n\nMarie Ryan was sitting in the same chair that she had knocked over in her haste to get away from me. On the table in front of her was an educational psychology textbook. A cigarette was burning in the ashtray, and an open Coke was near at hand. She was wearing a floral print blouse and tan corduroy slacks. She had kicked off her shoes.\n\n\"Rob let me in,\" she said. \"And I helped myself to a Coke. I hope you don't mind.\"\n\nI had driven her away by failing to respond as I should have to the news she had brought me, and now I was failing again and knew it even while it was happening. She had brought herself to me\u2014back to me\u2014and rather than overturning the table\u2014or any other obstacle standing between Marie Ryan and me\u2014and wrapping her in my arms, I remained where I was across the room, noting her brand of cigarettes, the angle of her shoes under the table, the pallid green of the Coke bottle contrasting with the living green of her eyes. In my life there have been so many times when, rather than like a hand opening and reaching, I have been a fist, closed and tight. I have never told my writing students this, but I have long known there are two ways to make use of yourself in your fiction: You may stand back and notice, as Saul Bellow says is our human purpose, or you may live fully and store up experiences that may one day find their way into your art. The cigarettes were Pall Malls. The toes of her loafers were scuffed because when she sat in a chair, she usually folded her legs under her, with her toes touching the floor.\n\nAt least I had the good sense\u2014or was it so?\u2014not to immediately demand where she had been or to try to shame her with all the searching I had done.\n\n\"Look,\" I said, \"I'm sorry. I've had time to think about it. And really, it's good news, you being pregnant, and we\u2014\"\n\nShe waved off my little speech. \"Forget it. My mistake. False alarm. Aunt Bertha's back.\" It was a phrase Marie and her friends used as their euphemism for menstruating.\n\n\"You're sure?\"\n\n\"Please. Give me some credit. Yes, I'm sure.\" She crushed out her cigarette and closed her textbook. \"Relieved?\"\n\nAnd I knew enough not to answer that one. \"I was worried about you.\"\n\n\"I know, I should have called you. I was pissed off. What can I say? I've just been so on edge lately. I think it's that statistics class\u2014it's _killing_ me.\"\n\n\"So where were you?\"\n\n\"A girl down the hall from me was driving to Minneapolis, so on a whim I said I'd ride with her. I stayed with my sister.\"\n\nI hadn't thought to call there. \"I was worried. . . .\"\n\n\"I should have called. I _know_ that. I'm sorry.\"\n\nWe might have continued like that, mired in my guilt-inducing expressions of worry and her subsequent apologies, but Marie saw a way out. She walked over to me, looped her arms around my neck, and kissed me vehemently enough to loosen\u2014slightly\u2014my clenched self.\n\nIf you have never experienced simultaneous passion and relief, I recommend it; the concoction is powerful and obliterating. It mattered then only that I was holding her again.\n\nAfter a few moments of breathless kissing, I turned Marie around 180 degrees and pulled her tight to my chest. She reached up and back and again put her arms around my neck. This exposed her exactly as I wanted, allowing me to unbutton her blouse and caress those magnificent breasts. Gradually I ran my hands down the concavity of her stomach and dipped my fingers inside her waistband. Since her slacks rode low\u2014hip-huggers, in the parlance of the day\u2014that move brought me very close to where I wanted to be.\n\nWhen I reached down a little farther, however, Marie dipped and twisted her torso, and I had no choice but to pull my hands away. \"Huh-uh,\" she said, and between quickened breaths added, \"I told you. I have my period.\"\n\nShe spun around another 180 degrees, kissed me again, and asked, \"Or was that a test?\"\n\n\"I forgot. For a moment.\"\n\n\"You have a very short memory.\" She tugged at my belt. \"We could do something else. . . .\" Now it was her hand squeezing its way inside my waistband.\n\n\"That's okay.\"\n\n\"What's the matter\u2014afraid of accumulating debts you can't repay?\"\n\nOh, how well she knew me! \"I'm just happy to have you back.\"\n\n\"I can feel how happy you are,\" she said.\n\nSince early childhood, the existence that has held the most appeal for me is one of unruffled routine, and almost immediately after those sirens came wailing down Keogh Street, I wanted things to revert to what seemed to me the calm of what had been. Taking the wrong lesson from those tumultuous days, I wanted\u2014or so I believed\u2014a return to a life in which I could take certain things for granted. This desire for placidity is perhaps too strong in me, and as a consequence I have often convinced myself it's present when in fact something is still roiling below the surface.\n\nNevertheless, it seemed to me that once Marie came back from her brief trip to Minneapolis, our lives quickly returned to what they had been, and it seemed that that was what we both wanted. We soon resumed having sex, although for a few months we were a bit more cautious. We were both busy with work and our studies, but we were together as often and, if questioned I would have said, as happily as before. When the semester ended, we packed up the Studebaker and drove together back to Bismarck for the holidays.\n\nWhere I promptly fell ill with strep throat, and for a few days I lay in bed with a fever and a throat so sore I could barely swallow. By Christmas day I felt better, but I still used my sickness as an excuse not to attend church with my mother and sister. After church they would go to the Christmas luncheon held every year by the Burnett sisters, another ritual I was pleased I could pass on. For a few hours I would have the house to myself.\n\nI poured myself a cup of coffee, lit the first cigarette I could smoke in days, and, still in my pajamas, sat up in bed with the copy of _The Short Stories of Ernest Hemingway_ that Marie gave me for Christmas. (The book, the Scribner's paperback edition priced at $2.95, is still on my shelf.) I had barely begun to read when someone knocked on my bedroom door. I'd thought I was alone in the house, and I hadn't heard the doorbell, but I wasn't startled for long. The door eased open, and there was Marie's smiling face.\n\n\"Are you feeling up to company?\"\n\n\"Yours\u2014yes.\"\n\nShe was skipping out of church too, she said, and when she took off her coat, I could see she was dressed for mass in a wool skirt and a turtleneck.\n\n\"I think God will forgive me,\" she said, \"if I spend the morning tending to the sick.\"\n\n\"Not necessary. I'm feeling much better. Really.\"\n\nThrowing back my blankets, I started to get up, but Marie pushed me back onto the bed. \"I told you. I'm here to nurse you back to health.\" She took my cigarette from me, inhaled, and then put it out.\n\n\"The first thing I have to do,\" she said, pulling my T-shirt off, \"is make sure your lungs are clear.\" She put her ear to my chest, but any pretense of listening soon fell away as she covered my torso with kisses.\n\nShe sat up. \"Your heart _seems_ sound, but I wonder . . . can it stand up to a little excitement?\"\n\n\"I'm willing to risk it.\"\n\nWithout any more talk, Marie kicked off her shoes, pulled her sweater off over her head, unhooked her brassiere, wriggled out of the girdle that she needed only to hold up her nylons, and stepped out of her panties. She made no effort to fold or neatly stack her garments, but left them all lying in a small pile on the floor. Then, wearing nothing but her skirt, she climbed onto my narrow bed.\n\nI reached for her, but she brushed my hands away. \"We can't have you exerting yourself. Not in your weakened condition. You lie still and let me\"\u2014she pulled her skirt up to her hips and straddled me\u2014\"take care of you.\"\n\nWhat was happening in that bedroom that had been mine when I was a boy was of course a boy's fantasy\u2014a beautiful girl enters his room and gets into his bed. Inevitably I remembered when Marie had come to my room in quite a different way and for quite a different purpose\u2014the night when, scratching on my screen, she woke me, and we went out together to search for her boyfriend. And suddenly through that tiny fissure of memory Gene squeezed his way into the bedroom.\n\nBut only for an instant.\n\nWith Marie moving rhythmically above me, her eyes half-closed, her breasts rising and falling, how could I think for long of anything but her? She leaned forward, pinning down my wrists and reminding me that I wasn't to grab or touch. Then she sat back, putting her hands on her own buttocks. When she bent down again her hair fell forward over her face and mine, and within that tent every breath I took was full of her heat and essence. Then, arching back up, she tossed her head and shook her hair back over her shoulder, but in the room's furnace-dried electric air, a few strands still floated free. Having taught me what I must not do, Marie now caressed her own breasts. The winter wind gusted hard and rattled the storm window in its frame, and when I came, it was with the same kind of shudder and gasp I might have displayed had I been dropped naked into snow. Marie's orgasm was a less dramatic trio of softly uttered _oh_ s.\n\nMy memory of what happened is reliable, but occasionally my brain tries to put something _before_ when in fact it was _after._ The mind performs these reversals or alterations no doubt because it has a narrative sense that life lacks and wants its stories to be as dramatic, harmonic, emphatic, and orderly as possible.\n\nFor example, I keep wanting to remember that it was on that Christmas Day that Marie, lying in my arms, us sharing a post-coital cigarette, announced that she was transferring to the University of Minnesota. But I know that wasn't so. I know it. She received notification before Christmas that she had been accepted into a special, experimental program that would allow her to seamlessly combine the completion of her undergraduate degree with teacher certification and admission to graduate study. When I traced my finger along her naked clavicle, feeling for that tiny irregularity where the bone broke, I knew\u2014we both knew\u2014that a separation lay ahead for us. We had already had our discussions of the difficulties that would pose, but we always concluded those conversations quickly with assurances that our love would overcome any problems.\n\nAnd I know that when we made love in that narrow bed it was not for the last time. I know that. Just as I know that our relationship didn't end that day. But perhaps because the mind wants the symmetry of a story that would both begin, as this one did, and end on Keogh Street, it tries to force my memory to delete the remaining six months of our relationship. Maybe I misremember the end of our life together so I can make Marie's educational opportunity responsible rather than my own behavior on the day when she told me she might be pregnant. And maybe something in the way Marie and I made love that day\u2014made love and made memory\u2014tinged it with valediction, a farewell fuck for each to remember the other by. It may have been nothing more than what seemed a game that day _\u2014don't use your_ _hands!\u2014_ but perhaps it was in fact Marie's wish, still unconscious at that point, not to be contaminated by my touch.\n\nBut however false my memory tries to play some facts, it situates precisely in time and place\u2014Christmas Day 1964, my bedroom\u2014what Marie said after she rose from my bed and unself-consciously dressed in front of me. The little tale she told had probably been thrust to the forefront of her consciousness by the nearness of the Stoddard home, just across the street and down the block. Had she opened the curtains and looked out, she could have seen it from my window.\n\n\"Did you know,\" she asked me, \"that Gene didn't call your father immediately after discovering his father's body?\"\n\n\"Yeah. He tried to phone Marcia first. And then when he couldn't reach her\u2014\"\n\nClasping her brassiere, Marie shook her head. \"There was a little time between those calls. Did you know that? What he did before he called your dad?\"\n\n\"I don't believe I do.\"\n\nShe adjusted her breasts within the brassiere's cups. \"First, Gene got two pans from the cupboard, one he filled with hot water and laundry detergent and the other with hot water and Pine-Sol.\"\n\nWhen Marie pulled her sweater over her head, static electricity again gave her hair a will of its own, and the strands stuck out in a way that reminded me of the graduation night party when she tried to coax Gene down from the roof of his car, and the bonfire backlit her and gave her hair that same look of coppery filament. Once again she lifted her skirt, this time to pull on her panties and shimmy her girdle into place. \"Mr. Stoddard had . . . When he died he . . . wet and soiled himself, and Gene wanted to clean up his father and the garage before anyone else came on the scene.\"\n\nShe adjusted her nylons, smoothed down her skirt, and stepped into her shoes. \"He did a decent job on the garage floor, and the tires his father had stood on Gene just threw into the backyard. He couldn't clean his father's trousers, though, and for a minute Gene considered taking them off and finding a clean pair of pants. But in the end he just couldn't make himself do it. He couldn't undress his dad.\"\n\nMarie put on her coat, gold wool and double-breasted, and buttoned it to her throat. The heat our bodies had made had cooled. She pulled her collar as high as it would go. I reached for the T-shirt she had pulled off me. Christmas services were over. Marie had her own family dinner to attend.\n\n\"You're right,\" I said. \"Gene never told me that story.\"\n\n\"Why would he?\" she asked on her way out the door. \"He wasn't trying to seduce you.\"\n\nIn addition to the Hemingway collection, Marie also gave me a Rooster tie and a copy of _The Freewheelin' Bob Dylan,_ the album with the cover of Dylan and Suze Rotolo arm-in-arm in the middle of a slushy New York street. I have that album still, like the Hemingway, although I no longer own a turntable to play the record on. I hadn't asked for the album, the book, or the tie, but Marie was well enough attuned to my tastes and desires that it would have been reasonable for me to expect any of those gifts.\n\nAnd while I didn't expect the little anecdote she told about Gene, in a way it too was a gift. Here, she might have said, I know this is something you'll like. And in giving it to me perhaps she was trying to get rid of it\u2014Here, I don't want this and I know you do. I present it to you, confident that you'll never lose it and that one day you'll find a use for it.\n\nYes. How well she knew me and my desires. And perhaps since she could not face a life with a man who would never stop hungering for exactly what she wished to divest herself of, she had to choose a life apart from his. I've often wondered\u2014did Marie Ryan ever tell the man she eventually married anything about Raymond and Alma Stoddard? About their son, Gene? About both the boys who once lived on Keogh Street? It would not surprise me to learn that she didn't. Nor would I blame her.\n\n_Since Marie W. remembered the past with such clarity, she wondered_ _why its appearance in her dreams was so muddled and_ _confused. Even by the surrealistic standard of dreams, hers_ _seemed to operate on the far edge of unreason, especially in_ _matters of chronology._\n\n_For example, no matter how many decades had passed since_ _she'd lived under her parents' roof, her dreams never played_ _themselves out in the home she currently lived in\u2014nor in any_ _house or apartment she had once called home\u2014but in the Bismarck,_ _North Dakota, dwelling she grew up in. No matter how_ _old her four sons were, they were almost always infants or, at_ _most, grade-schoolers in her dreams. Her parents, of whom she_ _never dreamed when they were alive, once deceased appeared to_ _her regularly. They were alive and in the fullness of health in_ _those dreams, yet they commented often on the inconveniences_ _of being dead._\n\n_Marie's own age fluctuated in her dreams but within a narrow_ _range. She was always an adult, no matter the age of her_ _parents or children, but she couldn't find any means, not her appearance,_ _not external events, that enabled her to locate her age_ _precisely._\n\n_But the greatest confusion came from three males, a_ _boyfriend from high school, a boyfriend who followed the first_ _and with whom she had a relationship that began in high school_ _and went into her college years, and her husband. These three_ _were as different as could be. The first was a brash, opportunistic,_ _unpredictable boy. The second was sensitive but shy, brooding_ _and often remote. And her husband was a confident,_ _ebullient, unshakably cheerful man. Yet these three freely_ _changed identities in her dreams. She might be driving somewhere_ _with her husband and their children only to find, upon_ _arrival, that it was her first boyfriend who lifted her son out of_ _the backseat, and her second boyfriend who held open the door_ _to the house they were visiting. The oddity wasn't, however,_ _that all three males were present. There was only one man but_ _with an identity that wasn't fixed. One would suddenly become_ _another, and with no discernible transition, and no matter how_ _strange these shifts were to her waking mind, within the dream_ _they were unremarkable._\n\n_Of course the reason for this shape-shifting, this identity_ _swapping, was within Marie, yet she had no idea what curl or_ _twist in her brain or psyche might be causing it. Any explanation_ _she came up with seemed pedestrian and inadequate. Yes,_ _these were three males to whom she had professed love. Yes,_ _she'd had sexual relations (but not intercourse) with all three. Nothing in those facts, however, explained the phenomenon. She didn't pine after former boyfriends or even possess much_ _curiosity about them. She didn't feel she had made a mistake in_ _severing her relationships with them. She loved her husband and_ _lived a contented life with him. She had a forceful, well-defined_ _personality\u2014others said it of her, and she felt its truth as well\u2014_ _and she had never sublimated her own identity to any male, not_ _to her father, her husband, her sons, or to an old boyfriend. In_ _the end, no amount of self-analysis brought her an answer or_ _decreased the frequency of the dreams._\n\n_Then Marie received notice of her high school's twentieth reunion,_ _to be held in Bismarck in mid-August. Both those_ _boyfriends of her youth graduated in the same class as Marie, so_ _though it was by no means a certainty, it was possible that if_ _Marie and her husband attended the reunion, the three men_ _who exchanged identities within her dreams might be in the_ _same room at the same time. She'd had no interest in attending_ _her ten-year reunion, but now it seemed to her that this social_ _ritual had the potential of correcting the confusion of her dream_ _life._\n\n_As soon as her husband came home from the Minneapolis_ _clinic where he had his pediatric practice, she presented the letter_ _of invitation to him. \"What do you think?\" Marie asked._\n\n_\"Feel like driving to Bismarck in August?\"_\n\n_He laughed, but something in her demeanor must have told_ _him she was serious. \"Why not? Will I get to meet your old_ _boyfriends?\"_\n\n_Marie had told him nothing of her dreams, and though his_ _question, both eerie and prescient, jolted her, she tried not to_ _show it. Instead, she attempted to match his jaunty tone. \"Get_ _to? I'll insist on it.\"_\n\n_He rubbed his hands together in gleeful imitation of a_ _scheming silent movie villain. \"At last\u2014I'll find out if you always_ _made that little mewing sound when the back of your_ _neck is kissed, or if that's only for my benefit.\"_\n\n_She took the invitation from his hand. \"That's just for you,_ _baby. Just for you.\"_\n\n_In truth, Marie came into the marriage with little in the way_ _of sexual history, especially compared to her husband's, but_ _what she had, or hadn't, done before they met was of no consequence_ _to him. Had Marie told him about her dreams, she was_ _sure he would find them only amusing. He was not a jealous_ _man, nor was he given to angst or introspection, and until the_ _dreams began, Marie had always felt her spirit was a match for_ _his._\n\n _The first of the reunion weekend's events was a mixer at the_ _Sheraton hotel. Marie and her husband were staying at a Holiday_ _Inn on the city's opposite end, and once they settled their_ _sons in their adjoining room and reminded them they were limited_ _to a single Spectravision movie, the couple set out for the_ _evening._\n\n_In the parking lot, Marie's husband tossed her the car keys. \"You drive,\" he said. \"It's your town.\" And then he launched_ _into a falsetto rendition of the J. D. Souther song. \"It used to be_ _his town, it used to be her town. . . .\"_\n\n_Her town . . . yes. A town whose streets and avenues were_ _once as familiar to Marie as the halls and rooms of her own_ _home, yet that night she had trouble finding the hotel. Over the_ _years, the city's hotels had changed hands; what was once a_ _Sheraton was now a Radisson, and the new Sheraton was on a_ _street that had been changed to a one-way. Eventually, Marie_ _had to park two blocks away because she couldn't figure out_ _how to negotiate her way to the hotel's block._\n\n_As they walked toward the Sheraton, a solitary man kept_ _pace with them on the opposite side of the street. He was wearing_ _a white shirt, dark slacks, and cowboy boots, and though_ _Marie didn't recognize him, she assumed he was on his way to_ _the reunion too. Why couldn't he be one of her former_ _boyfriends? Twenty years was more than enough time for bodies_ _to shrink and expand, for hair to grow out or fall out, for_ _features to flesh out or seam in ways that would make people_ _look nothing like their yearbook portraits. Why did Marie believe_ _she would know either of those boyfriends on sight? Did_ _she think that the intimacies of love were a kind of imprinting,_ _rendering her able to identify them by instinct? Or was she_ _counting on them to recognize her and then announce themselves?_ _For that matter, what process had her mind followed to_ _make those boys appear as men in her dreams? Did her unconscious_ _have a talent, Marie wondered, like a police sketch artist,_ _to produce accurately aged portraits of someone she knew two_ _decades before?_\n\n_Once again, her husband seemed to know her thoughts. \"Now, those old boyfriends,\" he said, \"any chance there might_ _be some residual bad feelings? Should I be ready to defend your_ _honor or fight for the right to take you home?\" He bobbed and_ _feinted and threw phantom jabs into the night air._\n\n_Marie laughed in spite of herself. If nothing else, this little display of his demonstrated how different he was\u2014with his_ _buoyant spirit and his always-ready willingness to take himself and everyone else lightly\u2014from her previous too-serious boyfriends, and she wondered again how it was possible that they could be confused or conflated, even in a dream. She felt a_ _sudden rush of love for this man she'd married, and she stepped_ _inside his boxer's stance, raised up on her tiptoes, and kissed_ _him on the long, clean line of his freshly shaven jaw._\n\n_He dropped his hands, running them languorously down her_ _back. When he reached her buttocks, he squeezed and pulled_ _her tightly to him. \"Or should I\"\u2014he assumed his comic British_ _accent\u2014\"find other ways to demonstrate my rights of conjugal_ _primacy?\"_\n\n_\"You're doing a pretty good job of that right now.\"_\n\n_He squeezed harder, and she pivoted away from his embrace._ _\"Before we go in there,\" she said, \"maybe we should set_ _up signals. You know, if we need to be rescued from someone_ _boring, or if the whole evening is just too excruciating.\"_\n\n_\"Good idea,\" he said. \"How about an erection? If you see_ _that I've got a hard-on, you'll know that means we've got to get_ _back to the hotel right away.\"_\n\n_\"I was thinking more along the lines of a tug on the earlobe.\"_\n\n_\"A page from the Carol Burnett playbook. Okay. I like_ _mine better, but the earlobe tug will work, too.\" He pushed_ _open the door of the hotel. \"Let us go among them.\"_\n\n_The mixer was being held in a large banquet room, and_ _Marie hesitated before its open doors. The room was dark but_ _for candles on tables, and the flickering light gave everyone inside_ _a spectral presence as they moved in and out of the wavering_ _shadows. Marie had no more than glanced inside when her_ _heart instantly misgave her, and she was ready to give up on the entire enterprise. But what could she say? Minneapolis was 450_ _miles away, their night's accommodation had already been paid_ _for, and her husband was standing expectantly by her side._\n\n_A plump scowling woman who must have been from_ _Marie's class but whom Marie did not recognize sat at a long_ _table just outside the doors. With obvious consternation, she_ _said, \"They haven't provided me with any of the registration_ _materials, so just write your name on one of these labels and_ _stick it somewhere on yourself.\"_\n\n_Marie's husband, for whom any unsmiling countenance was_ _a challenge, said, \"Any place visible, you mean.\"_\n\n_She looked at him blankly._\n\n_\"Because I could stick it on my chest,\" he said. \"Under my_ _shirt.\" He reached inside the gap between his buttons and_ _tapped his fingers in imitation of a beating heart. He bestowed_ _upon her the irresistible smile that he swore he never practiced._\n\n_Grudgingly returning his smile, she said, \"Someplace_ visible.\"\n\n_He pretended to slap the label onto his forehead, and the_ _plump woman allowed her grin its full release. She asked him_ _the question she never asked Marie. \"Do I know you?\"_\n\n_He pointed to his wife. \"She's a senior. I already graduated.\"_\n\n_As they walked away from the registration table with their_ _names pasted onto their clothes over their hearts, Marie asked,_ _\"Must you charm everyone?\"_\n\n_The question's petulance surprised Marie\u2014it had leaped_ _from her before she could stop it\u2014but her husband seemed not_ _to take offense. \"I must! I will not cease from my labors until_ _every man, woman, and child is charmed!\"_\n\n_At the bar, Marie ordered a white wine and her husband_ _asked for a light beer, and while they stood with the sweating_ _containers in their hands, two couples approached them. Marie_ _wasn't sure how it happened, but almost immediately after the_ _introductions were made an invisible partition descended on the_ _group. The men drifted off, talking, as near as Marie could tell,_ _about the brown suits Ronald Reagan wore, and Marie was left_ _in the company of two women she only vaguely remembered_ _from high school but who now pretended as though they had all_ _been closer than they really were._\n\n_Neither of the women had aged well. Rhonda Veach, now_ _Rhonda Schneeberg, had been a raucous, overweight tag-along_ _in high school, and while she was still heavy, she apparently believed_ _her pounds now worked to her advantage. She had_ _squeezed herself into a tight, low-cut dress that revealed an embarrassing_ _abundance of cleavage. Janice Schmidt, n\u00e9e Kalsow,_ _with whom Marie had been closer in elementary school than in_ _later years, was also wearing a low-cut dress, but she was so_ _thin and darkly tanned that the skin of her neck and chest_ _looked, even in the poor light, like paper that had been crumpled_ _and then smoothed out._\n\n_While they were catching up\u2014an activity that largely consisted_ _of an enumeration of children's names and ages\u2014another_ _woman joined them. Peggy Pilquist and Marie_ had _been good_ _friends, and their embrace was spontaneous and genuine. Peggy_ _looked disturbingly similar to the way she'd looked in high_ _school\u2014pretty in a wide-eyed, well-scrubbed way. Her blond_ _hair was still carefully waved around her face just as it was in_ _her graduation portrait._\n\n_The group's talk soon became a recitation of notorious high_ _school incidents\u2014the cow smuggled into the gymnasium, the_ _bowling ball rolled down the stairs, the principal's car pelted_ _with eggs\u2014that challenged no one's memory. When they began_ _to speculate about certain individuals\u2014the boy arrested for_ _stealing tires, the girl pregnant at prom, the algebra teacher and_ _the phys ed instructor found out in an affair\u2014Marie felt herself_ _tested. But her memories seemed to coincide with everyone_ _else's. The conversation gave her no reason to believe that the_ _confused identities in her dreams had anything to do with a general_ _disorientation about the past. Perhaps if Marie could separate_ _Peggy from the others, Marie could ask her if she ever_ _dreamed about old boyfriends or if the present and the past ever_ _exchanged themselves in dreams._\n\n_And just at that moment an old boyfriend stealthily approached._ _For a good portion of high school Janice Schmidt had_ _dated Randy Oslund, and that was exactly who now loomed_ _behind her. Randy's index finger in front of his wide grin_ _warned Marie and Peggy not to give away his presence._\n\n_Randy was over six feet tall, and even thinner than he'd_ _been in high school. He had close-set eyes and fine blond hair_ _that looked windblown even indoors, and when he raised his_ _arms over Janice Schmidt, he looked like a predatory bird. His_ _white shirt and dark trousers gave him away as the man Marie_ _had seen earlier walking toward the hotel._\n\n_Janice must have sensed he was there, because she whirled_ _about. When she saw who it was, she shrieked, and instantly_ _they fell into each other's arms._\n\n_No, not a bird. A vampire, for Randy Oslund pronounced,_ _in his Transylvanian accent, \"I haff come for you, my darling.\"_\n\n_Their embrace went on so long that Marie imagined they_ _were giving their bodies a chance to remember the sensations of_ _the past. Even when they broke their clinch, they still kept their_ _arms around each other._\n\n_Janice pretended to punch Randy in the ribs. \"And where_ _were you planning to take me?\"_\n\n_\"The Haystacks, of course!\" That was an area north of_ _town whose dirt roads and isolated groves had once provided_ _teenagers a private place to park. Marie had spent her share of_ _time in cars with fogged windows out at the Haystacks, and on_ _one occasion her and her boyfriend's passions ran so hot that as_ _their bodies pressed and undulated against each other, they_ _didn't notice that her glasses had slipped from the armrest and_ _gotten trapped under them. In the car's muffled silence the snapping_ _of the plastic frame made a sound like a breaking bone. Marie had no trouble recalling the incident or the boyfriend in_ _whose arms she'd rolled around on the car seat._\n\n_Laughing, Janice said, \"I'm ready to go. But won't your_ _wife wonder where you are?\"_\n\n_\"Hell, we'll be back before she knows I'm gone.\"_\n\n_\"So, Speedy Gonzales, I guess that hasn't changed.\"_\n\n_Randy Oslund affected a stricken expression and clapped_ _his hands over his heart, but Marie could tell Janice's remark_ _caused him no real pain._\n\n_In spite of the jokes, Marie recognized that their feelings for_ _each other were genuine, though the emotion likely had more to_ _do with history and nostalgia than with still-fresh desire or affection._ _The intimacy of their display, however, no matter that it_ _was feigned or performed for an audience, shocked Marie. Was_ _there a door to the past that she had somehow left open, and_ _did that explain her dreams? Or had the dreams pushed open_ _the door?_\n\n_But while Randy and Janice shocked Marie, she also envied_ _them. She could easily imagine her husband acting like Randy_ _Oslund with an old girlfriend. And inviting Marie over to meet_ _the woman. Unlike Marie, he talked openly about women with_ _whom he once had relationships, even to the extent of describing_ _their sexual behavior. Marie had once accompanied her husband_ _to an Austin, Texas, medical conference, and he_ _introduced her there to a woman he had dated during his first_ _year of medical school. A neonatologist, she soon excused herself_ _to attend a seminar, and then Marie's husband told Marie_ _about how the woman had never seemed to warm to him during_ _their time together, and how she'd preferred to make love on_ _all fours, a position that no doubt suited her because it didn't_ _require her to touch him or look at him. And that anecdote reminded_ _him of another woman who, from that same position,_ _would reach back and grab his testicles, a move that once_ _brought a yelp of pain from him that she mistook for ecstasy._\n\n_Even when such frankness embarrassed Marie, she knew it_ _signaled his confidence with her, and she had to admit it was_ _preferable to a husband who hoarded and cherished his memories_ _of other women._\n\n_Her husband detached himself from the group of men and_ _returned to Marie's side. \"Babe, I'm going to join these gentlemen_ _up in their room. Someone needs a physician's second opinion.\"_ _He pressed the tips of his thumb and index finger together_ _and, bringing them to his lips, inhaled audibly. Although he_ _had, in deference to his status as a physician and a father,_ _stopped buying marijuana years ago, he could not turn down an_ _invitation to smoke someone else's._\n\n_\"Be careful,\" Marie said. \"Please.\"_\n\n_As he walked away grinning, he said, \"If you can't trust the_ _class of '63, who can you trust?\"_\n\n_Rhonda, Janice, and Randy also walked away, leaving_ _Marie with her old friend._\n\n_Peggy leaned close, bringing with her the scent of cigarette_ _smoke and hair spray. \"Guess who's here\u2014and he's looking for_ _you?\"_\n\n_Marie felt herself blush. \"How can anyone be looking for_ _me? How does anyone know who's here and who isn't?\"_\n\n_\"Okay, asking. He's_ asking _if you're here.\"_\n\n_\"I have no idea.\"_\n\n_\"Just think for a minute. I believe you know.\"_\n\n_\"Oh, please, Peggy. This isn't junior high. Just tell me.\"_\n\n_Peggy smiled coyly. \"I don't believe I will. Anyway, keep_ _standing right here. I'm sure he'll find you.\" As if she were enacting_ _an exit she had rehearsed, Peggy bowed and backed_ _away._\n\n_One of them was looking for her. . . . Marie tried to determine_ _which former boyfriend that was likely to be. She couldn't_ _believe that shy young man would seek her out. Instead, he_ _would stake out a position, just as he had on that winter night_ _when he'd stood on a street corner and watched the house\u2014was_ _it Peggy's?\u2014where Marie was attending a slumber party. It was_ _after midnight when someone saw him out there and told_ _Marie. She stepped into her shoes and ran out to him in her pajamas._ _While they both stood shivering under a streetlamp, he_ _explained that it wasn't that he hadn't trusted her when she'd_ _told him she'd be there, but his worry caused his faith to tremble._ _So perhaps he was watching Marie now from some dark_ _corner, waiting for her to notice him, then approach him, and_ _say, \"I've been dreaming about you.\"_\n\n_When Marie finally severed the relationship with him, both_ _he and she shed tears. Finally, however, his moodiness, his inaccessibility,_ _and his passivity wore her out. She felt she expended_ _so much energy in closing the distance between them, in sending_ _out signs of affection so he could learn through imitation to_ _match them with his own, in fulfilling the needs he could not or_ _would not articulate for himself, that she was losing too much_ _of herself. Simply put, he exhausted her._\n\n_Yet he was not the likeliest candidate to be searching for her._ _No, if someone was sorting through the assemblage, checking_ _name tags and staring into faces until he found Marie's, it was_ _probably her first boyfriend, that smiling, intense, determined-to-_ _get-what-he-wanted-at-any-cost young man. Why wouldn't_ _he cross identity boundaries in her dreams? He could seldom be_ _restrained in the time he and Marie were together._\n\n _When Marie was sixteen, she traveled with her family to a Minnesota_ _lake for their summer vacation. Marie hadn't wanted to_ _go. Although the summers of her adolescence seemed long, two_ _weeks apart from her boyfriend was a deprivation too much to_ _bear. She wrote him twice a day, and she promised him, almost_ _to the minute, when they would be reunited. Her father pulled_ _into the driveway, and Marie jumped from the car. She ran into_ _the house, heading for her basement bedroom from where she_ _would telephone her boyfriend and tell him she was back\u2014_ Come over right away!\n\n_But he was already there, waiting for her just inside her bedroom_ _door._\n\n_Once Marie's fright subsided, her joy at seeing him\u2014after_ _fourteen days and nights!\u2014took over and she launched herself_ _into his arms. In the busyness of unloading the car and unpacking_ _suitcases, Marie's parents didn't notice that her boyfriend_ _was already there when the family arrived; they assumed he had_ _hurried over in record time to see Marie._\n\n_Over the next few weeks, Marie learned that that had not_ _been the only day he'd taken the key from under the pot of_ _geraniums and let himself into the house. He confessed that he_ _had been in there almost every day, and as evidence he had hidden_ _little notes throughout her room. She put her hand under_ _her pillow and found a note; she opened a drawer and found_ _another. He put one in her jewelry box and tucked another inside_ _a high school yearbook. None of the notes said anything_ _more than \"Hi there!\" \"It's me!\" \"Surprise!\" and initially Marie_ _found them endearing. But when she came across one taped to_ _the back of a framed photograph of her parents, she suddenly_ _felt a chill. The site of that note\u2014\"It's me again!\"\u2014was certainly_ _more innocent than the drawer where she kept her pajamas,_ _nylons, brassieres, panties, and slips, but the note on her_ _bureau changed everything that surrounded it. Suddenly Marie's_ _room felt less like hers; it had been transformed into a space_ _anyone might occupy. The bedroom door fit poorly in its frame_ _and only with great effort could it be pushed or pulled to shut_ _tight, and even then the door had no lock. Not that it mattered._ _As those little scraps of paper\u2014\"Guess who!\"\u2014demonstrated,_ _Marie had no threshold she could cross where the world_ _wouldn't follow. When Marie broke up with him a few months_ _later, she told him it was because they were becoming too serious,_ _a phrase that back then almost always meant the girl no_ _longer knew how to restrain her boyfriend's sexual demands or_ _how to keep her own desire from duplicating his._\n\n _Marie hadn't finished her glass of wine, but she made her way_ _back to the bar. From there she thought it would be easier to_ _not only see someone approach but also to read the history in_ _any face's features._\n\n_In fact, Marie had barely reached out her glass to the young_ _bartender to ask for a refill when a man unexpectedly thrust his_ _face in front of her and demanded, \"Why'd you break my_ _heart?\"_\n\n_Startled, Marie pulled back, and when she did, she saw who_ _was confronting her. Fortunately, she caught herself before she_ _made matters worse and spoke the words that came first to her_ _tongue. Oh, Terry, it's only you._\n\n_Instead, Marie greeted that pop-eyed, mock-belligerent face_ _with, \"Hello, Terry. It's nice to see you.\"_\n\n_\"Why? So you can rip my heart out and stomp on it and finish_ _the job?\"_\n\n_They both knew what he was jokingly referring to. Throughout their grade-school years Terry Bart had had a crush_ _on Marie. He plagued her on the playground, following her_ _everywhere, teasing her, making her his special target in dodge-ball;_ _he went blocks out of his way just so he could walk past_ _Marie's house after school. Finally, when they reached junior_ _high, he became bold enough to ask Marie if she would go with_ _him to a school dance. \"I'm sorry,\" she told him, \"but I'm_ _going with Danny McCabe.\" Danny was an eighth grader and_ _someone with whom Terry knew he could not compete. In the_ _next few years Terry learned to make a joke of the rejection, but_ _Marie knew his sense of defeat and disappointment had been_ _real. In high school when she was briefly between boyfriends,_ _Marie consented to attend a party with Terry, and though she_ _tried to make it clear to him that he would never be more than a_ _friend, he still tried, drunkenly, roughly, persistently, to kiss her_ _at the party. Marie eventually pushed him away and left the_ _party, walking home through four inches of fresh snow in kitten_ _heels._\n\n_\"Do you still live in Bismarck, Terry?\"_\n\n_\"You mean you haven't kept track of my every move? Denver._ _We've lived in Denver for the last seven years.\"_\n\n_\"We?\"_\n\n_\"My wife. We met at Augustana. She's right over there.\" Terry pointed across the room, but in the dim light his direction_ _was useless. \"Can you wait right here? I'll go get her. She'd love_ _to meet the girl who broke my heart. God knows she's heard_ _enough about you.\"_\n\n_The idea of being introduced to Mrs. Bart in that context_ _held no appeal for Marie, and as soon as Terry left, she slid_ _away from the bar. She wanted to head toward the door and_ _then wait for her husband over by the elevators, but that meant_ _stepping into the light. Anyone looking for her, whether an old_ _boyfriend or an aspirant like Terry, would find her easier to spot_ _if she left the banquet room's shadows._\n\n_She circled behind the bar, back by the troughs of iced beer_ _and soda and the stacked boxes of inexpensive wines and_ _liquors. One of the bartenders, a young man not much older_ _than Marie's sons, looked at Marie and raised an eyebrow conspiratorially_ _in her direction. Just as Randy Oslund had done_ _earlier, Marie raised her finger to her lips, and the bartender_ _smiled and nodded in understanding. But what, Marie wondered,_ _did the young man believe he understood?_\n\n_What if Terry Bart, and not one of Marie's ex-boyfriends,_ _were the anonymous seeker Peggy had told her about? Wouldn't_ _that make sense? Terry was there; he had obviously been looking_ _for her. Perhaps Marie was fleeing from a rendezvous she_ _had already had. . . ._\n\n_Ceiling-to-floor draperies encircled the room, and Marie_ _moved close to the heavy dark fabric. She had no idea what the_ _curtains were for, but she was ready to adapt them for her own_ _purposes, to step inside a fold or opening and vanish from sight._ _As unobtrusively as possible she began to move around the_ _room, all the while glancing back at the entrance so she_ _wouldn't miss her husband when he returned._\n\n_As soon as he came into view, she planned to run to him_ _and tell him they must leave immediately, that it had been a mistake_ _to come there, that she wanted nothing to do with the past_ _and certainly wanted none of its scenes enacted ever again. It_ _wasn't feasible to leave Bismarck tonight, but she'd insist they_ _leave first thing in the morning. She couldn't bear the thought of_ _being in the city in the full exposure of daylight._\n\n_Just ahead a small knot of people stood in Marie's path. She_ _could try to go around them, a move that would necessitate_ _leaving the available cover and possible escape route of the_ _draperies; she could go back in the direction from whence she_ _had come; or she could stop right where she was and wait, hoping_ _she wouldn't be noticed and that they would soon disperse_ _and clear the way ahead. She stood still._\n\n_Someone in the group ahead must have said something_ _funny, because every one of them either threw their head back_ _or bent over with laughter. Almost as if they were in on the_ _joke, the people at the table nearest her also began to laugh uproariously._\n\n_Suddenly, through the sounds of all that merriment and beneath_ _the room's general din, came a voice, barely more than a_ _whisper, and from right behind her. A man said\u2014she_ thought _she heard him say\u2014\"Would you like to go to Paris with me?\"_\n\n_Marie didn't recognize the voice, but given the room's noise_ _and its poor acoustics, how could she? It could have been her_ _husband speaking to her and she wouldn't have been able to_ _identify him._\n\n_And perhaps the man wasn't even addressing her. If she_ _waited for just a moment before turning around, he would certainly_ _repeat his question and then she would know exactly who_ _was there and why._\n\n_The question never came again, but eight years later, when Marie stood on the Pont des Arts with the man she loved, she_ _was sure that, in spite of her dreams' efforts to confuse her, her_ _decisions in life had been the right ones. At that moment in_ _Paris, however, who was beside her seemed of less importance_ _than the fact of the May sun warming her back and lighting the_ _gilt of a building that looked to her like a wedding cake._\n\nAlthough the relationship-cancer that eventually put an end to the us that was Marie and me may well have been in place on that Christmas Day when she came to check on my health\u2014indeed, maybe it was present from the very beginning\u2014we continued to see each other after she transferred to the University of Minnesota. I drove to Minneapolis, she rode the bus to Grand Forks, or we met in Bismarck when we both went back there for one school vacation or another. But in none of our meetings, letters, or many telephone conversations did we ever make arrangements for permanence, and eventually the simple fact of the 315 miles between us may have been too much to overcome. When we each began to find excuses not to make the journey\u2014an exam to study for, a paper to write, a lecture to attend, a party not to be missed, a snowstorm predicted, a car with a fussy carburetor\u2014our life together was all but over. We both decided to attend summer school at our respective institutions (one of us, I don't remember who, must have been the first to make that choice, and the other no doubt followed suit out of spite). Our breakup, like my parents', was not the result of a single explosive incident but more a gradual loss of the energy relationships use for fuel. By autumn of 1965 we were finished.\n\nAt first I told myself that was all right. My pride had been wounded\u2014Marie didn't have to transfer to another school. Then another girl briefly captured my interest. A creative writing professor overpraised my poems and stories and told me I could be admitted to the University of Iowa's MFA program. He was wrong, but the damage was done; my head was turned. When I graduated, I, like Marie, left North Dakota for an adjoining state, but she headed east and I went west, to the University of Montana and its new graduate writing program. Once there, I was, I told myself, exactly where I wanted to be and ready to begin realizing my deepest literary aspirations. The mysteries of the human heart could be known, and I would devote my life to the careful search for words to convey that knowledge.\n\nThen on a winter morning, after three days of warm Chinook winds rolling down the Rockies, after a night of drinking in a Missoula, Montana, bar with a group of students who shared my na\u00efve ambitions, I woke with a profound hangover and a refrain running through my brain as if on a perpetual circuit: _I need_ _Marie, I need Marie, I need Marie, I need Marie, I need Marie . . ._ the vowels _\u2014eye ee ah ee eye ee ah ee eye ee ah ee\u2014_ sounding for all the world like the siren of a Parisian police car. The words may not have come to me previously in quite that form, but the emotion that underlay them, I realized, had never left me and never would. That feeling, as if it too had been covered in snow, surfaced in January's thaw.\n\nSunk though I was that morning in self-pity and despair, my sense of irony was still intact. When Marie and I were together, I used to worry that someday I would lose her to Gene because, while I could make and state the case that I _wanted_ her, he could make the stronger claim by asserting that he _needed_ her. Now, however, it was too late for both of us. I had lost touch _\u2014wonderful_ _phrase!\u2014_ with Marie, and though this is another sequence I can't be entirely sure of, she may have already found the man she would marry, a man who, unlike the boys of Keogh Street, would answer _her_ wants and needs.\n\nAs long as my mother was alive\u2014and it took many years before her two-packs-a-day habit caught up to her\u2014I returned to Bismarck every year. Once I married (more of that soon) and eventually started teaching, largely because I needed to find work that provided benefits and brought in a paycheck larger and steadier than the erratic royalties and option money that came my way, those annual returns to my hometown were usually in the summer. At decade-long intervals, however, I made _certain_ that those trips were in the summer. Specifically, I made it a point to be in Bismarck during the days when the class of 1963\u2014Marie's class\u2014held its reunion.\n\nBut as far as I knew, she never returned. And why would she? In the late 1960s Marie's father was transferred, and her family moved to Illinois, and even if Marie had had enough pleasant memories of the city and her time there\u2014enough, that is, to overpower her association with the lurid, tragic Raymond Stoddard affair\u2014she had never been attached to the past. A writer I revere, William Maxwell, once said, \"I don't think I have outlived any part of my life, it all seems to coexist,\" and while I could say the same thing about my life, that sentiment was never Marie's. Remember her advice to Gene? He had to \"stop pulling the past up to the present.\" Besides, had she come back to Bismarck, for a high school reunion or any other reason, what good would it have done me? She would have been in the company of the man she'd married, a doctor she met in Minneapolis when she was in graduate school and he was interning. To this day they live in Edina, a Minneapolis suburb, with their four sons.\n\n(Incidentally, my informant on matters Marie has usually been the wife of my college roommate, Rob Varley. After earning a master's degree from the University of North Dakota, Rob returned to Bismarck to work for a utility company, and he married Karen Holmes, a former friend of Marie's. Karen was the secretary of their high school class, and she's been in charge over the years of keeping track of classmates and issuing invitations for reunions.)\n\nFor many years that wife I mentioned earlier accompanied me on trips to my hometown. A native of Billings, Montana, she was a nursing student when we met in Missoula. A soft-spoken, sincere, pretty woman, it would never have occurred to her trusting nature that on any occasion when we were in Bismarck, I was constantly scanning the streets in the hope that Marie Ryan might also be back in town and I could catch a glimpse of her. A glimpse. I told myself that I wanted nothing more. Just something to satisfy my curiosity, to see if she was as lovely as ever (and as lovely as Karen, who visited Marie in Minneapolis, said she was). She was my first love, an adolescent obsession. Many such youthful infatuations stay with men and women. Sometimes for years. They're normal. Harmless. That was what I told myself. Then, just when I thought I had convinced myself that all those things were true, the refrain would begin again, its vowels wailing through my brain _\u2014eye ee ah ee eye ee ah ee eye ee ah ee._ _I need Marie, I need Marie, I need Marie, I need Marie, I need_ _Marie. . . ._\n\nDuring the years of our marriage my wife never suspected that I was in love with another woman. Why would she? I never lied to her about where I was or who I was with. How could I love someone I hadn't seen for five years . . . for ten . . . for twenty? When our marriage inevitably ended, it wasn't because I had been caught in an affair. But my emotional remoteness, my inaccessibility, finally exhausted my wife.\n\nI remember the occasion when the end came. We were in the car together, running errands or going out for lunch. The circumstances don't matter. I might have said something, but it's more likely that I didn't. My silences were more common, and more wearing, than any speech, even of the wounding sort. She suddenly let her head fall back against the headrest, and said, without a trace of anger or vitality in her voice, \"I've had it with you.\"\n\nIndeed she had. Caring for the sick or injured was more than her profession; it was her natural disposition. For months, for years, she had tried to revive our marriage, willing to do almost anything to bring it back to health. \"What do you want from me?\" she used to ask. I didn't take advantage of her invitation. What was I supposed to say _\u2014be someone else?_ And since I was unwilling to be anyone other than who I was\u2014joyless and remote\u2014our marriage expired.\n\nEven when I was no longer making those annual pilgrimages to Bismarck in the company of a wife, I was usually not traveling alone. We had twin daughters, and the divorce didn't diminish their willingness\u2014their eagerness\u2014to make the trip to Grandmother's house and to the city that had been their father's boyhood home.\n\nWhen the girls were nine or ten, I decided to take them on a tour of the state capitol and its grounds. One of our first stops was the building's legislative wing, the art deco Great Hall where Raymond Stoddard murdered Monty Burnham. I planned to show them the banquette where the senator had been sitting when the assassin approached, the floor that had once been stained with blood, its red swirling into the marble's pattern. I intended to give them a lesson in history. . . .\n\nI couldn't do it.\n\nWhile the scene and the story probably would have been to them nothing more than a bit of gruesome Bismarck trivia, I couldn't take the chance. What if a detail\u2014as slight as the bullet-pocked stone floor or as large as the connection to Keogh Street and their ancestry\u2014lodged in my daughters the way the details had made a home in me? Of some matters ignorance is preferable to knowledge.\n\nI've brought the matter of Raymond Stoddard and his motivation about as far as I can. If I haven't made the choices clear, let me present them one more time. You might choose to believe, as my mother did, that Raymond Stoddard killed to avoid a scandal involving his job with the state. You might side with my father and think that resentment and anger over the Stoddard family's being cheated of a dwelling and acreage was enough to make Raymond buy a gun and use it. It may be your conviction that murder always proceeds from a deranged mind whose purposes are never available to us. Your character, your experiences, your values, your worldview\u2014all these will cause you to favor one theory over another. Have you ever succumbed to an irrational impulse? Could jealousy corrupt your soul? Have you ever acted with nothing but bitterness behind the behavior? My own hypothesis doesn't require much beyond what's known of the principals' lives\u2014that and the knowledge that nothing has a power quite like love to tilt us toward murder and self-destruction.\n\nI imagine\u2014and it hardly seems appropriate to term this theory the product of imagination, so plausible does it seem to me\u2014that _Raymond Stoddard had always known that his wife loved_ _another man before she loved him. For years, this was a source_ _of satisfaction for him. He regarded himself as blessed that_ _Alma Shumate chose him; he had a special pride that he possessed_ _what others wanted. In economic terms, her value increased_ _according to how strongly she was desired by others. However, just as the rich man worries that he will lose his treasure,_ _so Raymond Stoddard's anxiety eventually outpaced his_ _happiness._\n\n_Maybe Raymond's doubts needed nothing to grow but his_ _own insecurities, and maybe they began with a casual remark, a_ _comparison his wife made between him and other men. It might_ _have been a perfectly harmless observation. Why, Alma asked_ _her husband when he reached for his money to pay the restaurant_ _bill, do you always carry your wallet in your jacket pocket?_ _Every other man I've known\u2014my father, my brother, Monty,_ _Pastor Lundgren, Bill McCutcheon\u2014carries his in the back_ _pocket of his trousers. Monty. Only a first name was necessary. They both knew of whom she was speaking. Or maybe it_ _started with Monty Burnham's rise to prominence. One evening_ _he was featured on the news, a brief segment showing him at_ _the dedication of the Lewis and Clark monument overlooking_ _the Missouri, and Alma leaned farther forward than she ever_ _did for an act she liked on_ The Ed Sullivan Show _or with more_ _interest than she showed in any episode of_ The Loretta Young Show, _her favorite program._\n\n_And that was all it took._\n\n_She still thought of Monty Burnham. She knew Monty_ _Burnham, knew him and his habits well enough to recollect_ _without effort how he carried his wallet. He wasn't only part of_ _her past; he was in her present. She might be looking out a window,_ _but she wasn't seeing the snow falling or the wind sweeping_ _away the fallen leaves, she was thinking about Monty_ _Burnham. He was more real to her than the sunset pinks and_ _lavenders of the horizon._\n\n_Who is to say where worry crosses over to fear and then_ _hardens into agonized belief? We either have the ability to talk_ _ourselves out of our fears or we don't. And Raymond Stoddard_ _reached the point where he suspected that anytime Alma left the_ _house it might be for the final time. Her suitcase was packed,_ _and around the corner her lover was waiting. Of course he forgot_ _that Wednesdays were her days for working at the church or_ _that she had told him that she was walking over to Mrs. Morton's_ _house for coffee or that she said she'd stop at Super Valu;_ _he believed that anytime she left the house it was for another_ _assignation._\n\n_Over the years Raymond Stoddard had tried various mea_ _sures of self-treatment to salve his perpetually wounded pride or_ _to alleviate his misery. He had always drunk, but at some point_ _the act become purposeful, determined. The benumbed man,_ _after all, feels no pain. Anger too can be a narcotic, and though_ _he was never able to use it as successfully as bourbon, the few_ _times his rage burst forth, Raymond found himself oddly released_ _from pain. For that matter, his own reveries could sometimes_ _offer him relief, even when the object of contemplation_ _was his wife in the arms of another man. He pictured so vividly_ _the scenes of Alma in bed with Monty Burnham that Raymond_ _was briefly taken outside himself; he was no longer a man in_ _agony, he was an impersonal creator, a visionary, a vehicle for_ _his own lurid imagination._\n\n_None of his strategies for relieving his pain lasted for long,_ _however. It always returned, and frequently redoubled. Finally_ _in January of 1961\u2014and how many residents of the northern_ _plains have come to a life-altering conclusion in deepest winter,_ _as though something in their brains could crystallize and take_ _form just as their breath does in the frigid air\u2014Raymond Stoddard_ _concluded it all had to stop, even if his own life was subsumed_ _within that \"all.\"_\n\n_Marcia Stoddard's semester grades arrived, and they were_ _not good. A C in Introduction to Drama. A C_ \\- _in biology, and_ _they went down from there. A D in Spanish, and an F in Educational_ _Psychology._\n\n_Her father was furious. \"She had to attend the university,\" he said, slapping the printed grade form down on the kitchen_ _counter. \"She had to graduate early from high school, and then_ _junior college wasn't good enough for her. Oh, no. It had to be_ _the university. And I'm shelling out good money for grades like_ _these?\"_\n\n_Alma tried to calm her husband and to keep his ranting_ _from being overheard by her son and his friend watching television_ _in the living room. \"Shhh. You know she was struggling. She said so at Thanksgiving.\"_\n\n_\"This isn't just struggling. See this. F. That's failing. Not_ _struggling. Failing.\"_\n\n_\"She promised she'd do better. It was a bad semester.\"_\n\n_\"She promised. And does she know how she'll do better? I'm going to have a talk with her.\"_\n\n_\"What do you think that will accomplish? She's let us down._ _She knows that. She's let herself down.\"_\n\n_\"If she doesn't have a plan for how she's going to improve, I_ _might have a few suggestions for her.\"_\n\n_\"I really don't think that's what she needs to hear right_ _now.\"_\n\n_\"What she needs . . . Maybe what she needs isn't the important_ _thing right now. Maybe\u2014just a minute. Are you telling me_ _that I shouldn't speak to my own daughter? Is that what you're_ _saying? Because\u2014\"_\n\n_\"Oh, for God's sake. Will you stop? You're not going to_ _solve this with a talk. You've never bothered talking to her before,_ _about her grades or anything else._ You _. . . you're barely a_ _father to her.\"_\n\n_Because Raymond couldn't quite comprehend what his wife_ _had said to him\u2014_ he wasn't Marcia's father? _\u2014he looked elsewhere_ _for assistance. The boys in the living room\u2014if they'd_ _heard Alma's statement, surely they would register its shock,_ _wouldn't they?_ What the hell? Dad isn't\u2014? Mr. Stoddard isn't Marcia's father? _And Raymond's son was rising to his feet\u2014was_ _he coming out to the kitchen to demand the truth of_ his _paternity?_ _But Gene merely walked to the television set and with a_ _half turn of the horizontal hold stopped the picture from_ _rolling. His friend never took his eyes off that week's episode of_ Gunsmoke. _Was it Raymond who had misunderstood? Had his_ _wife merely offered a comment on him as an insufficiently attentive,_ _barely engaged father, and in that regard not very different_ _from every other father? That interpretation was certainly_ _available._\n\n_Yet Raymond rejected it and chose instead to take Alma's_ _blurted remark as revelation, the confession he had dreaded and_ _longed to hear._\n\n_His wife had not just loved another before she loved him\u2014_ _that he could handle and indeed in some moments could even_ _gloat about\u2014but she had had sex with that man. That was_ _harder to bear. Of course she had never stopped thinking about,_ _caring about Monty Burnham\u2014the reminder of him, of them_ _and their love for each other, was constantly present in Marcia. Their child. And that was, quite literally, a fact he could not digest._ _He bolted from the kitchen on his way to the bathroom,_ _but when he saw his son and his son's friend, Raymond turned_ _around. He ran to the garage and there he lurched to the trash_ _can just in time: He vomited onto the sack of garbage that he_ _had put there only an hour earlier._\n\n_What was he to do? He thought he had known what his_ _darkest worry was\u2014that Alma constantly, secretly compared_ _him to another man and found her husband wanting\u2014but Ray_ _mond realized now that even in his bleakest, most pessimistic_ _self-pitying moods he had not imagined circumstances as black_ _as these._\n\n _With the new year, the legislators invaded the city for their biennial_ _session, and soon their plans to create or thwart new laws_ _became the news that dominated local and state media. Newspapers_ _and television news programs could be counted on to_ _carry articles and interviews in which senators and representatives_ _lobbied, through the press, for or against the projects and_ _issues closest to or furthest from their hearts. Monty Burnham_ _was a rising star in the Republican party, and whether he sought_ _the attention of the cameras or the cameras looked for him is irrelevant;_ _he was a reliable media presence, as close to famous as_ _any North Dakotan was likely to come during the first cold,_ _snowy days of 1961._\n\n_On its six o'clock nightly news broadcast, Bismarck station_ _KFYR televised a brief interview with Senator Burnham on the_ _contentious subject of the state's property tax, and as it happened,_ _Raymond Stoddard, who usually favored the news on_ _KXMB, the CBS affiliate, was watching KFYR during the Burnham_ _segment. Had Raymond's wife been in the room at the_ _time, Raymond might well have turned to her, pointed at the_ _screen, and issued his ultimatum: Him or me\u2014which is it? Right now, make your decision. No deliberation, no discussion,_ _no denials, no explanations, no lists of pros and cons. Right_ _now\u2014him or me._\n\n_But the only other person in the room was the teenage boy_ _who lived up the block, waiting for Raymond's son so the two of_ _them could attend that evening's basketball contest\u2014Bismarck_ _High School versus its in-town rival, St. Mary's. They wanted to_ _leave early enough to be in time for the junior varsity game as_ _well._\n\n_But what if Alma had been there? It wouldn't have mattered._ _Raymond knew what she'd say. He was foolish, imagining_ _things. The past was past. She might profess her love for_ _Raymond, depending on whether he'd angered or touched her_ _with his accusation. But Raymond Stoddard now lived in a_ _realm where neither protest nor proof, nor protestations of love_ _or innocence, could reach him. Nothing of the usual life of humans_ _could affect his faith. Alma loved another man, and Raymond_ _knew it. The only thing left was to leave this life with_ _nothing behind but this message to his wife:_ I know.\n\n_Raymond tried composing a suicide note to Alma, a letter_ _that would make it clear that he was aware that she loved someone_ _else and always had. But every attempt to write this communication_ _failed. Sheet after sheet he tore up or crumpled and_ _tossed aside. He couldn't get the tone right. He wanted to sound_ _noble, self-sacrificing, yet he also felt that he was entitled to be_ _angry and aggrieved\u2014her contempt for him finally became too_ _much to bear. He wanted to outline the relevant background to_ _his suicide, yet any attempt to go into the past enlarged the_ _scope of his note and soon he was writing page after page of_ _personal history. He didn't want to write an autobiography. Indeed,_ _any account of his life only served to emphasize how_ _undistinguished it was. Yet he had to find a way to leave a message_ _for Alma, something that would tell her that he knew_ _about her feelings for another man, and that he could no longer_ _go on living with this knowledge._\n\n_He purchased a .45 automatic, the make and model of_ _sidearm that he and Monty Burnham trained with during the_ _second World War. In the privacy of the basement laundry room_ _Raymond filled the clip to capacity with cartridges, slammed_ _the clip into place, and pulled back on the automatic's slide to_ _chamber a bullet. Then he clicked on the pistol's safety. Although_ _he hadn't picked up a gun since his years in the service,_ _the feel of the weapon in his hand was so familiar that he_ _thought its loading and operation might well belong in the same_ _category as riding a bicycle\u2014once learned, never forgotten. He_ _put the .45 into the pocket of his overcoat\u2014first wrapping it in_ _a dish towel so its shape wouldn't give itself away\u2014and each_ _day he carried it to work with him. That was the extent of his_ _plan; the rest would be improvised as chance presented itself. He couldn't be sure when he'd see Monty Burnham, but Raymond_ _knew when the senate was in session and where the legislators_ _were likely to be when they were not in chambers. Raymond made it a point to visit the legislative wing frequently._\n\n_The day soon came when he saw Monty Burnham sitting on_ _a red leather banquette outside the senate, which was in recess. Raymond wished the man had been alone, but he could hardly_ _afford to let the opportunity pass. Who knew when he'd have_ _another? He returned to his office and put on his overcoat with_ _its familiar weight on the right side. Into his left pocket he_ _placed the envelope containing the folded, signed confession_ _that he had typed days before. That communication he had had_ _no trouble composing; it came out exactly right the first time,_ _and he was quietly proud of its straightforward\u2014yet still_ _slightly enigmatic\u2014uncomplaining acknowledgment of guilt._\n\n_His decision to say nothing about his motive before shooting_ _Monty Burnham was governed by the same logic, if it can_ _be called that, that decided Raymond against leaving a note for_ _Alma. He didn't know what to say because he didn't know what_ not _to say. But if the act itself was going to tell her everything_ _that needed to be said, then the sight of the .45 pointed at_ _Monty Burnham would have to be sufficient explanation for the_ _senator._\n\n_Only the actual sound of the gun going off was Raymond_ _unprepared for. That long hall, those high ceilings, the walls and_ _floors of stone\u2014the .45 boomed and echoed like none ever had_ _during his years of training and combat. He knew the gun_ _would jump in his hand and send a jolt all the way up his arm\u2014_ _he was heedful to aim his second shot as carefully as the first\u2014_ _but the noise, the noise! For a moment his hearing was affected,_ _and shouting and screaming reached Raymond's ears as if they_ _came from a much greater distance than from the men and_ _women scattering around him. Then, as his hearing gradually_ _improved, the noise from those same people grew louder, as if_ _they were now coming closer, when in fact everyone wanted to_ _move away from the man with the gun._\n\n_After shooting the senator, Raymond walked away calmly\u2014_ _his calm was a pleasant surprise\u2014away from the scene, away_ _from the man twitching and jerking and clutching at his throat_ _as if his death couldn't come fast enough and he wanted to_ _choke off his last breath, away from the blood-smeared stone_ _floor, away from the panicked onlookers who weren't sure of the_ _direction in which to run. At an even pace, Raymond exited the_ _building through the south door, and not until he was negotiating_ _the icy steps leading down from the building was he even_ _aware that he was still holding the gun, and even then his awareness_ _wasn't of a weapon, an instrument with which he had just_ _committed murder, but of weight, and that awareness led to another:_ _He was a fugitive, fleeing a crime, and fugitives needed_ _neither the extra weight nor evidence connecting them to the_ _crime. No matter that Raymond Stoddard had no expectation of_ _escape from capture or incrimination, as he walked away from_ _the capitol, he instinctively lightened his load by flinging the gun_ _away, and when it landed in a snowdrift, it had not only its_ _weight to help it sink through the snow but also the heat it still_ _held from Raymond's hand and from being so recently fired._\n\n _Raymond Stoddard's Ford was not the only car leaving the capitol's_ _parking lot. As he waited at the stop sign for his turn to_ _pull out onto Fourth Street, he counted five cars ahead of him,_ _all employees who had probably heard\u2014or heard about\u2014the_ _shooting and were driving away from the grounds as quickly as_ _possible. Surprises kept multiplying. He was shocked and somewhat_ _indignant that he could make his escape\u2014although he hesitated_ _to think of it in that way; after all, he had no hope, no_ _desire, of a permanent getaway\u2014with so little difficulty. Shouldn't the entire capitol building and grounds be sealed as_ _quickly as possible to prevent the assassin's flight?_\n\n_He drove the few blocks to his home, and once he arrived_ _there, he pulled into the driveway and automatically got out to_ _open the garage door in order to park the car inside. Then he_ _stopped himself, remembering that he needed the garage's empty_ _space for what lay ahead. He entered his house through the_ _front door, walked unhesitatingly through the living room and_ _kitchen, and exited into the garage._\n\n_The tires were piled in a corner, waiting to be put on the_ _Ford once winter's snow and ice melted from the roads. Raymond_ _reached inside the dark well of the heaped tires to bring_ _out the coiled clothesline rope. He tested again the knot that he_ _had tied days before. It held tight, as he knew it would. The_ _knot was a variation of one he tied in fishing line when he was a_ _boy. Seven twists and then circle through\u2014it looked more like a_ _noose than the actual noose would. He threaded the other end_ _of the rope through the loop but left an open circle. When he_ _reached up to gauge whether it was large enough to fit over his_ _head, he knocked off his hat, which under the circumstances_ _struck him as hilarious, the setup to a joke without a punch_ _line: Did you hear the one about the man about to hang himself?_ _He forgot to take off his hat. . . ._\n\n_Continuing to move quickly yet deliberately, Raymond unfolded_ _the stepladder and climbed high enough to toss the rope_ _over a crossbeam, a process he repeated a few times. He wanted_ _to make certain the rope wouldn't slide from side to side. The_ _other end of the rope he anchored by tying it to one of the legs_ _of a built-in workbench. With the stepladder still in place he_ _brought the tires over and stacked them under the knotted rope,_ _the height of which had to be adjusted one more time. The ladder_ _would enable him to climb up to the top of the pile of tires. He'd loop the rope around his neck\u2014making sure to take off_ _his hat first!\u2014and then he could push the ladder and send it_ _crashing to the floor. He'd have to balance only long enough to_ _kick the tires out from under him, and only two would have to_ _fall away to guarantee that his decision would be irrevocable._\n\n_Once all these preparations were in order, Raymond went_ _back inside the house. He didn't have a lot of time, but he could_ _have one more cigarette and warm up a bit. Working in the_ _garage had been almost as cold as it would have been to work_ _outside. He sat down at the kitchen table. He took the typed_ _confession from his pocket and put it under the ashtray. No_ _need to remove his hat or overcoat. He wouldn't be here long._\n\n_One small matter was still plaguing him. Those failed attempts_ _to write a suicide letter\u2014had he thrown them all away? Raymond was sure he had. But had he only crumpled them up_ _before tossing them into the wastebasket, or had he torn them_ _to pieces? Had he emptied that basket into the garbage can in_ _the garage? He thought he remembered taking those precautions,_ _but if he hadn't, someone could retrieve those sheets\u2014_ _some of them containing no more than a few words, others_ _filled with clumsy crossed-out sentences that were supposed to_ _explain his life and why he chose to end it as he did\u2014smooth_ _out the paper, decipher his scrawled handwriting, and believe_ _they had the answer to Raymond Stoddard._\n\n_Well. Let them try. If the papers were there, they were there._ _Raymond didn't have the energy to walk down the stairs and_ _climb back up again. Not when he still had those ladder steps to_ _negotiate. And now he heard the sirens that he had been listening_ _for, though he couldn't determine if they were moving_ _toward him or if they were still speeding toward the capitol_ _building. But if they weren't coming his way yet, they would be_ _soon._\n\n_He wondered from how far away they could be heard. Or,_ _put another way, how far his fame had already traveled. If Raymond_ _could hear the sirens from his kitchen, certainly they_ _could be heard in the homes of his neighbors on Keogh Street. People in the older houses and apartments in the center of the_ _city could probably hear them. And farther away? Could they_ _hear the sirens in those expensive homes on the western hills? In_ _the run-down houses south of the tracks? In the offices of his_ _brother-in-law's construction company out on Airport Road\u2014_ _could they hear them there? In the Frontier bar on Main Avenue?_ _His wife and son\u2014could they hear them? And was_ _everyone within earshot asking, as people so often do at the_ _sound of a siren, what's happening? What's wrong? Raymond_ _hoped they were heard everywhere and that to every ear they_ _would inspire the same questions, not just today but every day_ _hereafter. What happened? What went wrong?_\n\n_He had just lit his second Old Gold when Raymond heard_ _the front door open. He had an impulse\u2014an absurd, ridiculous,_ _impossible impulse\u2014to rush to the garage to complete his mission_ _before Alma or Gene\u2014no one else, not even the police,_ _would enter without knocking, ringing the bell, or shouting_ _their presence\u2014discovered him, but then Raymond relaxed. He_ _still didn't have to hurry. Not yet. The garage was his domain;_ _he could walk out there at anytime and neither his wife nor his_ _son would have any curiosity about what he might be doing out_ _there on a winter's day._\n\n_It was his son, his son and his friend who lived up the_ _street, their faces red and their noses running from their walk_ _home from school. Before Gene could ask his father why he_ _was sitting at the kitchen table rather than at his desk in the_ _capitol, Raymond put a question to his son. \"Where's your_ _mother?\"_\n\n_\"I'm pretty sure she works at the church library on Wednesdays.\"_\n\n_Raymond nodded. \"Wednesday. That's right. It's Wednesday.\"_ _In his mind he added to the day a calendar notation, calculating_ _the date that would for years after bear his mark and_ _name more than his birthday ever had. \"What time will she\u2014\"_\n\n_But Gene was already turned to the refrigerator, reaching inside_ _for the milk he would swig from the bottle. The neighbor_ _kid, however, kept watching Raymond, and with the gaze of_ _someone who doesn't recognize what his gaze fixes upon._\n\n_And perhaps Raymond did look different. He was, after all,_ _someone he hadn't been the last time these two young men had_ _seen him. Why wouldn't his appearance be altered\u2014what identity_ _had more strength and power than \"murderer\"? For that_ _matter, Raymond hadn't, he realized, looked at himself since_ _he'd left the capitol. Perhaps before he went into the garage he_ _would seek out a mirror for a quick self-inspection, just to see if_ _he could see what this kid saw. Then again, why bother? He had_ _never known who he was better than at this moment._\n\n_He tilted his hat back and took a last drag on his cigarette. Look all you like, kid; Raymond Stoddard will never be anything_ _but a mystery to you._\n\nSince my mother's death, and instead of those regular visits to her and my hometown, I've been making a very different journey. As a result of my friendship with a French editor, I'm able to live for a few weeks in Paris. His apartment is in the Latin Quarter, my Montana home is near a number of first-class trout streams, and because he is a fanatic fly fisherman, he is more than happy to swap residences every summer.\n\nHis apartment, in a building dating to the seventeenth century, has much to recommend it. Rue Xavier Privas is a quiet street, but only steps from the Seine, the Saint-Michel Metro stop, Notre Dame, and from wonderful bridge views. Boulevard St.-Germain is close enough that I hear those distinctive French police sirens regularly. Shakespeare & Company is nearby, and I browse its shelves frequently. Two or three evenings a week I eat dinner at Chez Pento, a wonderful tiny restaurant that has been feeding diners for over a hundred years. But those are the attractions that Jerome might list in a brochure if he were interested in subletting to any American. Only to me might he mention the slanted ceilings with their exposed beams . . . which remind me of the attic where Marie and I first made love. I can stand at the window, and though I am looking down at a cobbled courtyard where children play and my neighbors greet one another, those sights are not as poignantly real as the remembered image behind me. After lovemaking, Marie lies unashamedly naked on our makeshift mattress, and the late afternoon sun slants through the window, lighting her just so and revealing the sheen of perspiration that covers her from head to foot. She glistens. She shimmers. She shines. She glistens. She shimmers. She shone, shone, shone. . . .\nACKNOWLEDGMENTS\n\nFor their advice and support I am deeply grateful to my agent, Ralph Vicinanza, and my editor, Bruce Tracy. I am fortunate indeed to work with these consummate professionals and to count them as friends. For the information and insights they provided about the past, heartfelt thanks to Bruce and Barb Evanson and to Mark Miller. Special thanks to Eben Weiss and Bara MacNeill. Thanks also to Jennifer Hershey, Laura Ford, Beth Pearson, Ryan Doherty, and everyone else at Random House for their expertise and enthusiasm. Finally, for her love and inspiration I owe a debt that can't be repaid to my wife, Susan, who is present on every page of this novel.\nSundown,  \nYellow  \nMoon\n\nLARRY WATSON\n\nA Reader's Guide\nA CONVERSATION WITH LARRY WATSON\n\n**Random House Reader's Circle:** You tell the story through a timid and unnamed narrator. Was he modeled after anyone? At times he is likeable, yet ultimately he betrays his best friend. What relationship did you have with the narrator while writing? How do you think the reader should react to his actions?\n\n **Larry Watson:** I'm strongly tempted to duck this question, because the narrator\u2014timid and traitorous, as the question suggests\u2014is based on me. We grew up in similar neighborhoods in Bismarck, North Dakota. We both lived within view of the state capitol building, and in childhood came to know that distinctive structure well. We both graduated from Bismarck High School. (He's from the class of 1963; I'm from 1965.) We both attended the University of North Dakota, and we both became fiction writers. Each of us in adolescence fell in love with a girl who had been dating our best friend. In fact, the narrator is one of two characters in the novel with a real-life counterpart. Anyone who knew my wife, Susan, when she was younger (or, for that matter, who knows her now) will almost certainly identify her as the model for Marie Ryan. I seldom base my characters so closely on people from life, but these two characters in _Sundown, Yellow_ _Moon_ are exceptions.\n\nI suppose I could console myself because the question also applies the word \"likeable\" to the narrator, but I've already read reviews, met with book clubs, and heard from readers who have said they find the narrator unlikeable. I can live with those opinions, and I understand why readers respond to the narrator in that way. He lives too much in his head and too much in the past. He is calculating and self-absorbed. But he is also punished for his failings. The man who looks back on his life and tells this story\u2014these stories\u2014is haunted and anguished. I, of course, escaped his unhappy condition. I never broke up with my Marie Ryan; my wife and I have been happily married for more than forty years. There are other essential differences. I never had a neighbor who was an assassin and committed suicide, or had a friend who was a murderer's son, and no matter the extent to which fictional characters might be based on real people, they can't be the same people when their experiences are different.\n\nThis is a long way of saying that the narrator is and isn't me. Even when a writer works from the actual, the very act of writing produces an aesthetic and an emotional distance that guarantees the written version will differ from its real-life analogue. The narrator and I also share a reverence for the writer William Maxwell who wrote in _So Long, See You Tomorrow_ that \"in talking about the past we lie with every breath we draw.\" That strikes me as a little strong, since \"lie\" to me suggests an intent to deceive. It would be more accurate, in my view, to say that our memory always lies to us. But then fiction is a lie to get at the truth.\n\n **RHRC:** The entire town of Bismarck, North Dakota, becomes obsessed with Raymond's murder and suicide and, of course, his motives. You grew up in Bismarck. Was it how you portrayed it in the book? Was there an instance in your youth that inspired this story? What is it about this landscape that you wanted to bring to the reader?\n\n **LW:** The correlation between the Bismarck of the novel and the \"real\" Bismarck is similar to the correlation between the narrator and me. I wrote about a place that resembled a real place but whose true existence was in my imagination. For the purposes of the novel I altered a few facts about the city, and I'm sure my memory altered others. Nevertheless, the real Bismarck of the early 1960s was, I believe, comparable to the Bismarck of the novel in that they were both fairly conservative, largely middle class, and somewhat repressed homogeneous communities, removed in many respects from the rest of the nation by climate and geography. One of the ironies of the novel is that Bismarck is brought out of its isolation and anonymity by an act of violence. There have been, unfortunately, too many real-life instances of communities made famous, however briefly, by the violence that has occurred within their borders. In that regard, it should be noted that _Sundown, Yellow Moon_ takes place not only in a small city on the northern plains but also in America. From the political assassinations of the 1960s to last year's insane slaughter on the Virginia Tech campus, Americans have had event after gruesome, grievous event that have forced us to ask again and again who we are as a people and why we live in such a culture of violence.\n\n **RHRC:** First loves, first lovers, childhood friendships, and people's past relationships are big themes in this book. How do you think each relationship affects the next? Which relationship in the story was primary for you? Which one did you feel contained the crux of the story?\n\n **LW:** I do believe in the power and durability of first love, but whether that early relationship lasts, as my wife's and mine has, or doesn't, it's still capable of forming a template that many people apply to subsequent relationships. These people spend a good part of their lives seeking to find a partner with whom they can recapture the passion, purity, and intensity of a youthful love affair. This is the predicament of the narrator of _Sundown, Yellow_ _Moon_. Because of that, I regard his relationship with Marie Ryan as the crux of the story. To the end of his days the narrator will long for something he once had but can never have again. And he knows it.\n\n **RHRC:** Raymond Stoddard's motives for the murder of Monty Burnham and his suicide are never revealed. Do you believe there is a single motive? Did you write it with a clear answer in mind? The protagonist seems to believe it was jealousy. Do you agree? Do crimes always need motives?\n\n **LW:** I didn't write _Sundown, Yellow Moon_ with an answer in mind as to Raymond Stoddard's motive. I never doubted that he would have one, or more than one, but the desire to know what is ultimately unknowable\u2014what was going on Raymond Stoddard's heart and mind\u2014obsesses not just the narrator but other characters in the novel.\n\nJealousy is plausible as a motive, but of course, the narrator is drawn to that as an explanation because he himself is prone to jealousy. I had hoped that one of the ways readers would participate in the novel would be through examining their own personalities and by speculating on how that examination would lead them toward certain explanations. But it would be a brave, introspective reader willing to do that.\n\nYes, I believe that crimes always have motives, no matter how irrational, crazed, twisted, or inadequate they may be.\n\n **RHRC:** _Sundown, Yellow Moon_ contains many short stories written by the protagonist, who is a writer himself. How did you come up with the idea?\n\n **LW:** Almost immediately after the murder and suicide, the narrator struggles to comprehend what has happened. Such acts are beyond puzzling; they're inexplicable. But that doesn't stop people from trying to understand and explain them. Reason won't provide answers, so the narrator tries to imagine his way into lives that are otherwise closed to him. At first this activity takes place only in his mind, but that signals the beginning of his life as a writer. It made sense to me that he would continue to do that throughout his life, though, as for any artist, at some point the aesthetic demands of making a good art object\u2014a short story, in this case\u2014takes precedence over the personal stimulus that initially moved him toward making that object in the first place. Artists are, in my opinion, people finally more interested in making than knowing. The narrator of _Sundown, Yellow Moon_ has a dramatic past that provides him with all the raw material he needs for a lifetime of stories. I thought that, by including his stories in the text, readers would be able to see how his imaginative mind works. Do we know him any better through reading those stories? I believe we do.\n\n **RHRC:** When you began writing this story, what were you hoping to accomplish? What did you want to find out or share with your readers? You also speak directly to the reader. What did you set out to accomplish with this technique?\n\n **LW:** I could never be classified as a postmodernist, fabulist, or metafictionist, but anyone who writes fiction as long as I have is bound to notice some self-consciousness about the practice creeping into his thoughts. And while I'm dug in too deeply in the realist trenches (and am too devoted to story) to write a novel that is entirely a self-reflexive riff on the nature of narrative, I am interested in conducting fictional experiments and in attempting to see whether I can challenge the borders of fiction\u2014without losing readers in the process, of course. In _Orchard_ I tried a nonchronological form. And in _Sundown, Yellow Moon_ I tried to address some fundamental questions about stories, such questions as Where do stories come from? Why do humans need them? What uses do we make of stories? How do fictional truths differ from other varieties? What is the relationship between memory and imagination? So in the writing of _Sundown, Yellow_ _Moon_ I was aware that I was a writer remembering and imagining his past and in the process writing about a writer remembering and imagining his past. But I didn't want this to be an exercise in academic theorizing or in navel-gazing; I wanted, as I always want, a story that would engage, entertain, and move readers, and in the process provoke some thought. Addressing readers directly was a way, I hoped, to indicate that they and I were in this together.\n\n **RHRC:** Do you have a writing routine or any rituals surrounding your work? How long was it after you first had the idea for this story before you started writing it?\n\n **LW:** I don't have routines or rituals that must be followed\u2014no necessity for twenty sharpened pencils or a pot of Earl Grey tea. I can work in any place and at any time as long as I have the materials at hand. But I do make sure that I work on a novel\u2014producing new material, not just reworking old\u2014every day. I will probably also make a journal entry, and I might work on a poem, an essay, or a short story. But I must work on a novel every day without fail. I began writing _Sundown, Yellow Moon_ almost as soon as the idea came to me.\n\n **RHRC:** What are you working on now?\n\n **LW:** I've been working on a novel set, once again, in the early 1960s, and for this novel, whose working title is _The Doctor's Boys_ , I've returned to Montana. A teenage boy has become infatuated with a woman a few years older than he. In pursuing her, he finds himself in competition with a charismatic, powerful man, a doctor in the community. \nQUESTIONS AND TOPICS FOR DISCUSSION\n\n1. What does _Sundown, Yellow Moon_ say about the nature, source, and durability of young love?\n\n2. Of the available possible explanations for Raymond Stoddard's actions, which do you favor and why?\n\n3. Each character seems to favor a particular explanation. What does that preference reveal about his or her character?\n\n4. Does the explanation you favor reveal something about your character and experience?\n\n5. The narrator writes stories to explain and understand what happened in his neighborhood. Is that a universal human response, or does it stem from his personal nature?\n\n6. Does _Sundown, Yellow Moon_ say that storytelling is a basic human impulse?\n\n7. The narrator doesn't emerge as an entirely likable character. Why? Is he made less than sympathetic because of what he says and does, or because of what he thinks and feels? Or because of what he writes?\n\n8. What does _Sundown, Yellow Moon_ say about the nature of memory? Of memory and imagination?\n\n9. In some respects, the narrator is stuck in the past. What prevents him from living in the present?\n\n10. How is the setting, both the time and the place, important to the action in the novel?\n\n11. Because of the many stories within stories, it's not always possible to determine what \"really happened\" in the narrative. How does that uncertainty figure in the novel's themes?\n\n12. If you knew the narrator based only on the stories he's written, would you characterize him in the same way you would based on his behavior, speech, thoughts, and emotions?\n\n13. Do you have a favorite character?\n\n14. There have been many assassinations and attempted assassinations of politicians in the United States. How does this novel comment on the social, psychological, and cultural response to such events?\n\n15. What does _Sundown, Yellow Moon_ say about violence in America?\n\nLARRY WATSON is the author of _In a Dark Time,_ _Montana 1948, Justice, White Crosses, Laura,_ and _Orchard._ He has won the Milkweed Fiction Prize, a National Endowment for the Arts fellowship, the Mountain and Plains Booksellers Association Regional Award, and numerous other literary prizes. He lives with his wife in Milwaukee, Wisconsin. Visit www.larry-watson.com.\n\nALSO BY LARRY WATSON\n\n_In a Dark Time_   \n_Montana 1948_   \n_Justice_  \n_White Crosses_   \n_Laura_  \n_Orchard_\n_Sundown, Yellow Moon_ is a work of fiction. Names, characters, places, and incidents are the products of the author's imagination or are used fictitiously. Any resemblance to actual events, locales, or persons, living or dead, is entirely coincidental.\n\n2008 Random House Trade Paperback Edition\n\nCopyright 2007 by Larry Watson Reading group guide copyright 2008 by Random House, Inc.\n\nAll rights reserved.\n\nPublished in the United States by Random House Trade Paperbacks, an imprint of The Random House Publishing Group, a division of Random House, Inc., New York.\n\nRANDOM HOUSE TRADE PAPERBACKS and colophon are trademarks of Random House, Inc.  \nRANDOM HOUSE READER'S CIRCLE and colophon are trademarks of Random House, Inc.\n\nOriginally published in hardcover in the United States by Random House, an imprint of The Random House Publishing Group, a division of Random House, Inc., in 2007.\n\nGrateful acknowledgment is made to Ram's Horn Music for permission to reprint an excerpt from \"If You See Her, Say Hello\" by Bob Dylan, copyright \u00a9 1974 by Ram's Horn Music. All rights reserved. International copyright secured. Reprinted by permission.\n\nLibrary of Congress Cataloging-in-Publication Data   \nWatson, Larry.  \nSundown, yellow moon: a novel \/ Larry Watson.  \np. cm.  \nLegislators\u2014Crimes against\u2014Fiction. 2. Suicide victims\u2014Fiction. 3. Children of suicide victims\u2014Fiction. 4. Teenage boys\u2014Fiction. 5. Bismarck (N.D.)\u2014Fiction. I. Title.  \nPS3573.A853S57 2007  \n813'.54\u2014dc22 2007000031\n\nwww.randomhousereaderscircle.com\n\neISBN: 978-1-58836-819-5\n\nv3.0\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTable of Contents\n\nTitle Page\n\nCopyright Page\n\nDedication\n\nPART I - Changing Courses\n\nCHAPTER 1 - We'll Always Have Paris\n\nCHAPTER 2 - What Would Julia Do?\n\nCHAPTER 3 - The Secret Language of Kitchens\n\nCHAPTER 4 - It's Not About the Knife\n\nCHAPTER 5 - A Matter of Taste\n\nCHAPTER 6 - Fowl Play\n\nCLASS BREAK - Cruise Control\n\nPART II - A Bit About Meats, Bread, Pasta, Salads, and Vinaigrettes\n\nCHAPTER 7 - The Bread Also Rises\n\nCHAPTER 8 - Tossed Salad & Scrambled Eggs\n\nCHAPTER 9 - Udder Confusion\n\nCLASS BREAK - The Red Velvet Dinners\n\nPART III - Seafood, Soup, and the Importance of Leftovers\n\nCHAPTER 10 - The Pleasures of the Fish\n\nCHAPTER 11 - What's in the Box?\n\nCHAPTER 12 - Waste Not, Want Not\n\nCHAPTER 13 - The Power of Soup\n\nCHAPTER 14 - Kitchens, Revisited\n\nExtra Recipes\n\nAcknowledgements\n\nSelected Bibliography\n\nRecommended Reading\n\nIndex of Recipes\n\nALSO BY KATHLEEN FLINN\n**ALSO BY KATHLEEN FLINN**\n\n_The Sharper Your Knife, the Less You Cry: Love, Laughter, and Tears at the World's Most Famous Cooking School_\nVIKING\n\nPublished by the Penguin Group\n\nPenguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014, U.S.A. \u2022 Penguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario, Canada M4P 2Y3 (a division of Pearson Penguin Canada Inc.) \u2022 Penguin Books Ltd, 80 Strand, London WC2R 0RL, England \u2022 Penguin Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd) \u2022 Penguin Books Australia Ltd, 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty Ltd) \u2022 Penguin Books India Pvt Ltd, 11 Community Centre, Panchsheel Park, New Delhi\u2013110 017, India \u2022 Penguin Group (NZ), 67 Apollo Drive, Rosedale, Auckland 0632, New Zealand (a division of Pearson New Zealand Ltd) \u2022 Penguin Books (South Africa) (Pty) Ltd, 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd, Registered Offices: 80 Strand, London WC2R 0RL, England\n\nFirst published in 2011 by Viking Penguin, a member of Penguin Group (USA) Inc.\n\nCopyright \u00a9 Kathleen Flinn, 2011\n\nAll rights reserved\n\n_Names and identifying characteristics have been changed to protect the privacy of certain individuals._\n\n_Photograph credits_\n\nPage 5: Maggie Savarino; 125: Kathleen Flinn; 183: Jeff Maness\n\nLIBRARY OF CONGRESS CATALOGING IN PUBLICATION DATA\n\nFlinn, Kathleen.\n\nThe kitchen counter cooking school : how a few simple lessons transformed nine culinary novices into fearless home cooks\/Kathleen Flinn.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nISBN : 978-1-101-54451-8\n\n1. Cooking\u2014Study and teaching\u2014Anecdotes. 2. Flinn, Kathleen\u2014Anecdotes. 3. Cook books. I. Title.\n\nTX661.F57 2011\n\n641.507\u2014dc23 2011016222\n\nWithout limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the above publisher of this book.\n\nThe scanning, uploading, and distribution of this book via the Internet or via any other means without the permission of the publisher is illegal and punishable by law. Please purchase only authorized electronic editions and do not participate in or encourage electronic piracy of copyrightable materials. Your support of the author's rights is appreciated.\n\n<http:\/\/us.penguingroup.com>\n_For my mother, Irene_\n\n_For encouraging me to find joy in everything, not just cooking_\n**Author's Note**\n\n In theory, the story presented herein represents a year of my life teaching one round of lessons to nine volunteer students. In reality, it's the result of numerous additional kitchen visits, scores of interviews with cooks, researchers, and cooking teachers, plus my own teaching experiences in the year since the project ended.\n\nI recorded every home interview and most of the original classes, resulting in about fifty hours of video and two hundred pages of handwritten notes. I've done my best to portray events accurately, but consolidated or shifted events and comments in some situations. In a few scenes, the dialogue in classes was built from memory. The names of some of the volunteers and their children have been changed to protect their privacy.\n\nWhen it comes to the techniques taught here, ardent foodies or culinary professionals may debate the \"right\" method to roast a chicken or some other technique. I might even look back on these lessons to find that I have changed the way I teach the same recipes. That's one of the great things about cooking; there's more than one way to skin a fish. As long as an approach yields good, nourishing food, it isn't \"wrong.\"\nPROLOGUE\n\n**The Woman with the Chicken**\n\n_\"You teach best what you most need to learn.\"_\n\n\u2014Richard Bach, _Illusions: The Adventures of a Reluctant Messiah_\n\n  Normally, I do not stalk people in grocery stores.\n\nI confess to the occasional practice of supermarket voyeurism. But who doesn't sometimes notice the curious collections of fellow shoppers, then contemplate what they may reveal about them?\n\nWhat goes on in the home of a hunched, graying woman with nineteen cans of cat food, iceberg lettuce, a family pack of steaks, and a copy of _In Style_ magazine? Or a young woman in full stage makeup oblivious to the world outside her headphones, a pack of tofu hot dogs among the contents of a hand basket nestled in the crook of her tattoo-littered arm? Or an elegant man with a perfect manicure who lingers over the imported cheese counter, his cart filled with organic greens, expensive olives, and four bottles of champagne? Every grocery cart tells a story.\n\nLate on an otherwise average Tuesday afternoon, a sight near the canned tuna stopped me dead in my tracks. The cart sat as if abandoned in the middle of the aisle. It contained two dozen haphazardly piled boxes of dehydrated mixes for pasta, casseroles, rice, and stuffing and dubious jars of gravy. Despite being half full, the cart contained no _real_ food. As I stood contemplating its contents, a heavyset woman in her late thirties, casually attired in an eggplant-colored fleece, claimed the cart. Her preteen daughter twirled impatiently around her, quietly singing a Lady Gaga song under her breath.\n\nWould it be wrong if I followed her to find out what else she might buy?\n\nSmall basket in hand, I trailed behind her to stealthily observe. I feigned interest in various items along the aisles as she stocked up on packaged waffles and pizza pretzel bites, a collection of frozen dinners, chicken potpies, and a family-style package of pot roast with mashed potatoes and gravy.\n\nBy the time we hit the meat department, I suspected she was onto me. Hugging my shoulders against the damp chill and trying to avoid inhaling the vague smell of chlorine, I clumsily relied upon my peripheral vision to spy her selecting a heavy family-sized pack of ground hamburger at the other end of the case. She pushed her cart in my direction. I pretended to peruse the plastic-wrapped turkey options. \"Can you believe how expensive chicken breasts are these days? Crazy,\" she said out loud, to no one in particular. She reluctantly tossed a package into her cart.\n\nI seized the opportunity to say something. \"Whole chickens are on sale,\" I said. \"Ninety-nine cents a pound, I think.\"\n\nShe chuckled. \"Thanks, but I would have _no_ idea what to do with a whole chicken.\"\n\nIt hit me. After a year deboning chickens and stuffing meat with other meats at a famous Paris cooking school, I had information this woman needed. For some reason, at that moment, I felt compelled to give it to her. \"Come with me. I'll get someone to show you how to cut up a chicken.\"\n\n\"Ah, no, thanks,\" she said. A reasonable response given that I was a complete stranger who had followed her for twenty minutes through the maze of grocery store aisles.\n\nSomehow I assured her that I was not trying to sell a time-share in front of the turkey kielbasa. She shrugged and said, \"Okay, why not?\" We headed over to the butcher.\n\n\"Sure, I'm happy to show you how to cut it up, no problem,\" the butcher said as I handed him the chicken. The woman peered over the glass case to his thick white cutting board as he sectioned the bird deliberately. He stopped to show her how each cut was done. As he finished, he crackled fresh butcher paper around the pieces.\n\n\"So just how much is that whole chicken?\" she asked.\n\nHe looked at the tag. \"Let's see, it's on sale, so $5.20.\"\n\n\"How much would that go for if you sold me the pieces all cut up and packaged, like those breasts over there?\" She waved toward the meat case.\n\nHe looked up, mumbled abstractly under his breath, counting on his fingers. \"Well, breasts are out at $5.99 a pound, the thighs at $2.29, so I'd say about $10 or so.\"\n\n\"Get out!\" she exclaimed. \"So I pay twice the price to buy it as separate pieces? Well, who knew that!\" She smiled broadly.\n\nHe winked and passed her the freshly wrapped chicken. It landed heavy in her hand. She looked thoughtful. \"What is it?\" I asked.\n\nShe looked around, leaned forward, and whispered in a conspiratorial tone, \"I don't know what to do with the other parts of the chicken. I only know how to cook the breasts.\" She shrugged, embarrassed. \"But thanks for your help.\"\n\nAs she pushed her cart away, her daughter in tow, I stopped her. I could not let this woman go without knowing what to do with the rest of her chicken. By chance, this supermarket happened to be carrying the paperback of my first book. I fetched a copy. I flipped to a recipe for braised chicken thighs with mustard and then to one for stock.\n\nAt first, she didn't believe it was my book. I showed her my driver's license. \"I'm not trying to sell you a book,\" I assured her. \"I'm happy to buy it for you. I can't explain it, but I just really want to help you.\"\n\nFor the next hour, I led her around the store, making notes in the margins and writing new recipes in the notepad that I always carry in my purse. We discussed why she bought so many boxes and cans, and as we did, I slowly convinced her to clear out most of them from her cart and replace them with the real food that the boxed versions attempted to replicate. A three-pound beef roast replaced four shelf-stable individual pot roast dinners. When rounded out with inexpensive vegetables, the roast would yield a dozen servings for the same price.\n\n\"You know, I can't thank you enough for all this,\" she said earnestly as we made our way to the checkout, where, as promised, I bought her the book. \"At first, I thought you were some crazy person. But this feels like Wonder Woman stopping to help fix a flat tire.\" She and her daughter waved an enthusiastic good-bye. I didn't even get her name.\n\nThat afternoon stayed with me. It awakened a curiosity that I hadn't realized I had. Somehow, I knew this chance encounter was going to change my life.\n**PART I**\n\n**Changing Courses**\n\n_\"For most people, the only real stumbling block is fear of failure. In cooking you've got to have a what-the-hell attitude.\"_\n\n\u2014Julia Child\n\n**Terri, Trish, Shannon, and Sabra practice knife skills**\nCHAPTER 1\n\n**We'll Always Have Paris**\n\n_\"They say that time changes things but you actually have to change them yourself.\"_\n\n\u2014Andy Warhol, pop art icon\n\n  Standing on the stage delivering the graduation speech at Le Cordon Bleu in Paris is not the optimal time for an existential crisis.\n\nYet there I stood in my simple black dress, speech in hand, dwarfed by the vaulted gilded ceilings of the opulent ballroom in Le Club du Cercle just off the Champs-\u00c9lys\u00e9es. Two years earlier, I had awaited my diploma sitting right where this year's graduates were now.\n\nThe school knew that I would be in Paris to lead a culinary tour and asked if I would speak. I'd said yes, flattered, before thinking the whole thing through. The applause at my introduction drifted away during the eternity it took to walk to the podium. I silently surveyed the rows of the immaculately dressed audience. In the front sat two of the many young Japanese women in full celebration kimonos, their small hands carefully folded in their laps. Behind me, a dozen instructor chefs in tall toques spanned the width of the stage in a single row, waiting. I hid my panic behind a manic smile.\n\nWhat was I doing here? What words of wisdom could I possibly offer when I wasn't sure what to do next with my _own_ life? At age thirty-seven, I had realized my lifelong dream to graduate from the world's most famous cooking school. But what was I supposed to do now?\n\nI kept hearing two questions again and again, often from the same people. \"You graduated from culinary school, where is your restaurant?\" and \"When are you two having kids?\" I had no answer for either.\n\nA lot had happened since I'd left school. I'd settled into married life in Seattle while my husband, Mike, had joined a start-up company. We'd experienced the usual stuff of life as it marched unrelentingly onward; friends and relatives got married, others divorced, a few had children, and a couple got sick. I dealt with the curious, feast-andfamine life of authorship. Two days after I read to a packed crowd at the James Beard House in New York, I stood before an audience of three at a bookstore event in San Francisco, and _that_ included two of the store employees. The next week in Milwaukee, a quirky woman wearing her sweater inside out asked me to sign a book\u2014to her cats. I wrote, \"Dear Mr. Hinkel and Winkie Pie, I hope you enjoy the book, especially the parts about the fish.\"\n\nAbout three weeks before this trip to Paris, Mike had a strange premonition. \"I'm going to join my dad on his vacation in Mexico after all. I can't explain it, but I feel that if I don't go, I'll never see him again.\"\n\nWithin minutes of Mike's arrival at his dad's time-share, something went wrong. His dad, Floyd, pawed awkwardly against the dresser and suddenly fell heavily forward. Mike caught him and they tumbled to the floor. A doctor arrived on the scene and said Floyd had suffered a small stroke. A standard procedure to break up the stroke-causing clot led to a massive brain hemorrhage, sending Floyd into a coma. Mike arranged for an airlift to a Miami hospital for possible surgery. After an agonizing thirty-six-hour wait, doctors declared that nothing could be done. Over the next week, we watched helplessly as Floyd slipped away.\n\nOn a clear May afternoon, Mike broke down as he delivered the eulogy at a packed church in Spokane, Washington. \"He was my best friend . . . the best man at my wedding and . . .\" Mike stopped, fighting back tears. His sister came up and took his hand. \"I always wanted to grow up to be just like him.\"\n\nWe left for Paris three days after the funeral. When the seat belt light dinged, we finally exhaled. We felt a strange sense of relief, as if leaving the grief and stress behind.\n\nStanding at the podium with all this fresh in my mind, I cleared my throat as I surveyed the audience sitting in quiet expectation. I unfolded my speech. The sharp crinkling of paper echoed loudly in the vast room. I looked at Mike just offstage, proudly gripping a video camera. He tipped his head forward in a nod of encouragement. I folded my speech back up.\n\n\"So, today you've earned your culinary degree. What are you going to do with it?\" I asked. \"How many of you know? Raise your hands.\" A small group held their arms aloft.\n\n\"Here's what some of my classmates did with theirs.\" I talked about the friends I'd met at Le Cordon Bleu. Sharon had labored at Internet start-up companies, but now resided as a chef and kosher caterer in her hometown of Tel Aviv. Lely had traded corporate marketing to run a cooking school in Jakarta. Jose had left his job selling shoes to attend Le Cordon Bleu and was now an apprentice in a Michelin-starred restaurant in Madrid. Isabella, a former psychologist, had worked as the private chef for a Russian paper baron and later moved to Los Angeles, where she taught cooking classes in French to Hollywood celebrities' kids. Chinese-born and British-bred LP had been an attorney in London, but she now managed a French wine import business in Shanghai. Another friend had returned to work as a management consultant, but did pro bono work analyzing food distribution issues to combat hunger in places such as Mozambique.\n\n\"People ask, 'What can you do with a culinary degree?' The reality is that you can go anywhere and do anything,\" I continued. \"You're only limited by your passion and your imagination. Be open to the possibilities, take chances. Your career doesn't have to be traditional. It doesn't even have to be in a restaurant or even in a kitchen.\n\n\"For some reason, your passion for cooking and your desire to nourish people led you to this moment and place in time. Consider for a moment what success looks like when passion enters the equation. Is it money? Is it fame? Or is it having the strength to follow that passion? To have the will to go down a path you never thought you'd venture?\"\n\nI looked at the students' faces. Most were impossibly young. \"No matter what, remember that life is short, shorter than you think.\" I looked at Mike again. I thought back to our final moment in the Miami hospital with his father, to the last time I saw my own dad when I was thirteen, and the tear-filled interviews that I conducted writing obits in my early twenties. \"It can be gone just like that,\" I continued, and snapped my fingers in front of the microphone. The sound echoed like a stone dropping into a well. \"You think you'll figure it out someday, only to find that someday never comes.\" My throat caught on the words. \"One thing that I hope you'll do is try not to focus on what other people expect from you. At one point in my life I was so concerned about the next rung on the corporate ladder, and only later realized I missed the entire point of the climb. Find something _you_ believe in. Then, just do it. That's what matters.\"\n\nThe crowd applauded, but the toques rustled behind me. _\"O\u00f9 est la traduction?\"_ Where is the translation? I heard a chef ask. Between the funeral and being unsure of what to say, I'd turned in the speech too late for an official translation. Everyone agreed that if I did it on the fly in my paltry French, we'd be there all night. For once, most of the chefs couldn't understand _me_.\n\nAt the reception, as I was clutching my champagne, a radiant young woman approached. Her silky yellow-orange robes and sequined headdress perched atop her jet-black hair reminded me of an extra from the Elizabeth Taylor version of _Cleopatra_. Hessa hailed from Bahrain, but she spoke English in an accent more befitting a midwestern college sorority sister with a twinge of Valley Girl. \"I, like, really identified with your speech. I'm such a question mark. How do people figure out what they want to do? What are you doing? Do you have a restaurant?\"\n\nBefore I could answer, the fabulously suave Chef Savard arrived at my side. He had changed from his chef's whites into a tan suede blazer and black turtleneck to striking effect. After kissing me on both cheeks, he asked, \"So, _ch\u00e9rie,_ tell me, what are _you_ doing with your culinary training? Are you and Mike going to have any children?\"\n\nI downed my champagne.\n\nDespite our emotionally fragile states, the trip started well. Fresh from our flight, Mike dropped our bags in the tidy rental apartment near Rue Moufftard in the 5th arrondissement and immediately went downstairs. He returned with a pair of warm, fresh baguettes and a small bag of goods from a nearby _alimentation,_ the Parisian equivalent of a convenience store. We ripped greedily at the still-warm bread, slathering it with sweet French butter and finishing each bite with a sip of cold, bracing Chablis.\n\nJet-lagged, we napped mightily until dinnertime. We awoke, disoriented, to the sticky summer heat and crimson twilight, the distinct wail of French ambulances drifting through the open shutters. Half awake, we slipped outside to experience the city as if in a dream. The wide sidewalks reflected the heat of the day as we reacquainted ourselves with the marvels of seventeenth-century architecture and the brazen yet sophisticated female nudity of modern French advertising. We saw Parisians sitting at small round tables, sipping tiny espressos or glasses of wine. We walked the streets to the Seine, drew in deep breaths of its earthy scent, and looked out over the late summer sunset. We felt strangely at home.\n\nThe speech occurred at the start of our trip, but our primary reason for being there was to lead a vacation tour organized by the American Automobile Association. It was billed as a \"culinary tour of Paris\"; Mike and I agreed to take a group of Americans through an abbreviated version of our life in the city. The week the tour started, we met Sabine, the tour operator in Paris. \"It is a small group, yes, but the people here have all read your book and are very excited to meet you both.\" The group was a diverse mix that ranged from a paralegal to a flight attendant to a wealthy Florida homemaker who wielded a diamond ring the size of a baby food jar.\n\nTo kick off the tour, Mike and I led a walking tour of our former neighborhood near Les Halles. Among a smattering of meals and sightseeing, the itinerary included a tour of the wholesale food market Rungis, followed by a lunch buffet of impeccably fresh fish delivered only hours earlier to the massive seafood hangar across from the restaurant. The next day, we headed north in minibuses to Rouen, best known as the place where Joan of Arc was imprisoned by the English, then burned at the stake. Our visit was for dinner at an equally important landmark, at least for me: La Couronne, the restaurant where Julia Child ate her first meal in France. Julia Child is my longtime hero. After earning her diploma, she threw herself into writing _Mastering the Art of French Cooking,_ an effort that turned into a decade-long project that defined the later part of her life. I met Julia at the edge of an earlier existential crisis when I was twenty-three years old, shortly after I first began longing to study at Le Cordon Bleu but kept my desire secret. She was among the first people to whom I revealed what seemed like an impossible dream then: to study cooking in Paris.\n\nEach evening after dinner, the weeklong agenda descended into the \"where will Mike and Kathleen take us drinking?\" tour. If there's one bright spot in my French-speaking skills, it's my ability to navigate the delicate rigors of ordering a round of beverages. One evening Mike led an impromptu \"Introduction to Calvados\" lesson at the hotel bar. Another night, we familiarized our group with French beers and martini blancos at a place near the Palais Royal.\n\nWe enjoyed the group so much that we organized a dinner with them on the one free evening on the itinerary. As we dined on the patio of a small bistro on Rue Moufftard, we began to notice workers carrying fake hams and rubber lamb shanks passing by our table. Our waiter told us that scenes from the film version of _Julie & Julia_ would be shot on the street the next day.\n\nBy the next morning, a small army of prop designers and carpenters finished transforming the corner into a street market from the 1950s. Workers installed brand-new signs designed to appear weathered above the doors. They brought in aging handcarts, dozens of barrels, antique cash registers, and even a 1940s-era delivery truck. A few extras walked around in 1950s garb, smoking and talking on their cell phones, careful to avoid the thick tangle of cables that led to a massive camera and lighting setup choking the small street's intersection.\n\nMike and I watched Meryl Streep, dressed as Julia, wandering the market during Julia's early days learning to cook in Paris as a student at Le Cordon Bleu. Even from a distance, we could see a radiant smile as \"Julia\" took it all in with delight.\n\nBy chance, the tour ended the next day with its advertised highlight: a private class at Le Cordon Bleu. As we entered the familiar glass doors, I felt pangs of stress, as if I were still a student, a living version of an anxiety dream from which I'll probably never awaken. Just as I began a brief tour, one of my toughest teachers, the Gray Chef, appeared from around the corner. He had once scolded me mercilessly for a too-sweet duck \u00e0 l'orange sauce, prompting me to flee the kitchen in tears. I introduced him, first in stuttering English and then in clumsy French. Recognizing his name from the book, the group descended as a hungry pack on the man\u2014who now seemed much shorter than I remembered\u2014circling for a photo. I offered to take it. Suddenly, the once-imposing Gray Chef looked vulnerable and intimidated. I felt overwhelming tenderness for him. _\"\u00cates-vous c\u00e9l\u00e8bre maintenant?\"_ he asked gingerly. Are you famous now?\n\n_\"Non, chef, vous \u00eates,\"_ I told him sweetly. No, you are. He smiled broadly and then spread his arms around the group, confident again. I clicked the photo.\n\nAfter the rest of the group departed the next morning, one person remained for an extra day. Holly was an affable silver-haired flight attendant from Orlando in her early sixties. Holly had come to Paris with a secret mission. As we stood on the historic Pont Neuf bridge, she confessed that she had been discreetly leaving bits of her late sister's ashes on the trip.\n\n\"She wanted to visit Paris all her life, but she always found reasons not to do it,\" Holly said. My mind went to Floyd. We'd invited Mike's dad to visit us when we lived in Paris and even offered to buy him a spot on the AAA tour. He always declined, although he promised he'd visit \"someday.\" Just then, as the Eiffel Tower erupted in a sparkling display of strobe lights, Holly reached into her pocket, pulled out a small bag, and emptied it into the Seine. \"Now she finally got to come here, and stay,\" Holly said. Just before her sister died, they watched the movie _Something's Gotta Give,_ starring Diane Keaton, whose character visits Paris every year on her birthday to dine on roast chicken at a bistro called Le Grand Colbert. \"She was going through chemo, and she kept saying, 'I'm going to get through this because I want to eat at that restaurant in Paris.' The idea of that meal and coming to Paris got her through all of it, at least mentally. But finally, her body just gave up.\"\n\nThat night, the three of us dined at Le Grand Colbert. As we bid farewell to the gilded decor, Holly deposited a bit of her sister in a potted fern near the bar.\n\nMike and I had arranged a week to ourselves in Paris after the tour group ended. We thought that we'd be relieved to have the time alone, but Mike and I missed the enthusiasm of the group. Spurred on by that dinner with Holly, to make up for their absence and to avoid thinking of Mike's dad, we ate.\n\nWe began by haunting our favorite bistros. Curiously, a disproportionate number of their names contain _cochon,_ the French word for \"pig.\" At Au Pied de Cochon at Les Halles, we ordered steaming bowls of arguably the best French onion soup in Paris. We made a game of digging into the hot crust of Gruy\u00e8re and pulling the oozing, salty cheese with our forks a foot off the table as the pungent-yet-sweet aroma of onion-scented beef stock wafted around the room like a heavy cloud. At Aux Trois Petits Cochons, we started dinner with thick slabs of foie gras, spreading it like the smoothest butter onto crisp toasts. Au Cochon de Lait in the 19th arrondissement, a nondescript yet welcoming place that once served the workers of the slaughtering trade that operated nearby, had the aroma of grilled steaks, campfire, and stale red wine. The only foreigners in a sea of regulars, we ordered their specialty, l'Onglet Villette, a wickedly tender hanger steak topped with a buttery heap of caramelized onions splashed with red wine and a side of knife-cut crisp fries. We both closed our eyes to chew, and opened them to find our waitress automatically refilling our _pichet_ of red wine with a knowing smile.\n\n\"I wish that my dad had come to Paris, at least once. He would have loved this,\" Mike said. Neither of us wanted to talk about the empty place in our hearts that would greet us when we left the distractions of Paris.\n\nOn our last night in the city, in lieu of a standard author reading at a bookstore, I presented a knife skills demonstration at WHSmith, a splendid English-language bookstore at Place de la Concorde, the intersection of the grand boulevards and the former site of the ghastly guillotines of the French Revolution. Being in a city most famous for its food, I worried that my demonstration on basic cuts would be too remedial, but the crowd stood transfixed.\n\n\"I've always thought that if I could hold a knife properly, it would change my whole life,\" a South African woman in the crowd commented. \"I feel like I just never really learned. I always assumed that I'm doing it wrong.\"\n\nI had four chef's knives with me. I gathered everyone into groups and taught the fundamentals of holding a knife and drilled them on the basics of dicing, slicing, and julienne. Simple stuff, but then I realized that I'd learned the same fundamentals only a couple of years earlier. As the crowd disbanded, the South African woman pumped my hand in thanks. \"I know this sounds totally stupid, but I don't cook at all and you've just inspired me to learn. I thought this whole knife thing was so much more complicated. I feel like I've had a complete epiphany! Have you ever thought of teaching?\"\nCHAPTER 2\n\n**What Would Julia Do?**\n\n_\"You have to give yourself that dream assignment. No one is going to give it to you.\"_\n\n\u2014Penny de los Santos, photographer\n\n  How quickly we shifted back into our regular lives in Seattle. Still, after that moment on the stage, what _was_ my regular life? The notion weighed heavily on me. So did my weight. In France, we ate as if training for an Olympic eating event. Yet I returned weighing a few pounds less. So did Mike. Less than a month back in the United States I gained nearly ten pounds. How? Sure, we walked more in Paris. But what was it about being back in the States that led us to gain weight?\n\nThe French eat less in general and lean toward more fresh food and few snacks. As in other European cities, Parisians shop more often for groceries. Some of it is cultural, but most of it is practical. When I first moved to London in 1999, I had to completely shift my thinking about grocery shopping thanks to the dorm-room-sized fridge in my minuscule kitchen. I couldn't \"stock up\"; I had no physical space. My freezer was only slightly larger than the size of a paperback novel. Plus, I knew that I had to carry home with me whatever I bought. Since I shopped frequently, I chose mostly fresh food and prepared it that night. In Paris, we did the same thanks to the easily accessible street markets. Even so, I saw a lot of shoppers buying frozen quiche Lorraine in Parisian supermarkets, not to mention the customers who flocked to American-style fast-food outlets. While it might once have been true that French women don't get fat, more recent surveys show that as the French adopt more American-inspired habits of eating, notably consuming long-shelf-life products, their national weight steadily increases.\n\nContemplating all this not long after we returned from Paris, I wandered over to my local supermarket, a vast sixty-thousandsquare-foot store that's open twenty-four hours a day in the urban Capitol Hill neighborhood of Seattle. When it comes to diversity and cart voyeurism, it's hard to beat the collection at my local grocery.\n\nThat was the day I met the woman and her daughter, the one with the cart full of boxed and frozen ultraprocessed food products. She was shopping in a store with a bakery, a full-service deli, a sushi bar, a large meat department with trained butchers, a seafood counter replete with water tanks featuring live crabs and lobsters, plus signs bragging that the store carried 129 different varieties of produce, a third of them organic. With all those options, why did she actively select mostly food in boxes and cans?\n\nWe started talking after the butcher demonstrated how to cut up a chicken. \"When I make stuff from a box, it always turns out right,\" she explained. \"I never really learned to cook. Mom made dinner when we were young, but by the time I was in high school, she worked a lot, so my brother and I ate a lot of frozen dinners.\"\n\nShe picked up one box of pasta, the kind that makes a side dish in a few minutes. \"I know that Alfredo sauce is made with cream, but I would have no idea how to make it.\"\n\nI spent a year in culinary school learning endless variations on cream sauce. I explained a simple technique\u2014boil cream until it reduces and then extend it with a bit of the cloudy water left over from cooking pasta. \"That's it? Oh, wow, I thought it was a lot more complicated.\"\n\nShe agreed that if I wrote down the recipe, she'd give it a try. Out went the nine boxes, and in went two packages of whole wheat pasta, a quart of cream, and a small wedge of Parmesan cheese\u2014for roughly the same amount of money yet enough to make twice as many servings.\n\nThis result made her curious about what else we could replace from her cart. Boxes of Hamburger Helper were swapped for ingredients to make simple skillet dishes\u2014onions, garlic, peppers, canned tomatoes, and a block of Cheddar cheese. We visited the bulk herbs and spices area and stocked up on several, including a Cajun blend, chili powder, mixed herbs, oregano, thyme, and red pepper flakes. Her daughter, initially bored by our conversation, took over filling and labeling the plastic bags of spices and herbs. \"This is fun, Mama,\" she said as she sealed one bag. \"We should do this every time we shop.\"\n\nReal potatoes picked out by her daughter (along with a pink peeler) replaced the dehydrated variety. As we stood in the produce section, the woman looked around feebly. \"I know that I don't make vegetables enough,\" she said. \"I'm not very good at figuring out what to do with them. I kind of avoid anything that requires a lot of cutting something up. I see those chefs on TV and it looks so easy. I guess I'm not very good at it, and it always feels like it takes a long time.\" We bought a few bags of a precut broccoli and cauliflower mixture on sale that day, a bottle of olive oil, and some lemons. I wrote down instructions for how to steam and roast the vegetables, then top them with some lemon or a bit of her new herbs and spices.\n\nI couldn't stop thinking about that afternoon. I was certain I had overwhelmed her with information. She seemed like a smart woman and a good mom, but when we talked about cooking, she was discouraged, frustrated, and convinced that shortcuts remained the only path she had the time or skill to navigate.\n\n\"I don't mind boxed mashed potatoes\" was not the sort of comment that crept into my usual conversations. As a food writer, I'd slipped into what I call \"the foodie bubble,\" a magical place where everyone talks about ramps, perfect local peaches, and smoked duck prosciutto. People casually name-drop obscure chefs and discuss how many recipes they've tried from _The French Laundry_ _Cookbook_. Don't get me wrong: Life is _great_ in the bubble. It's just that most people don't reside there. Normal people live and shop in the center aisles of the grocery store, just like the woman I met. That afternoon, I decided it was time to abandon the bubble.\n\nI stumbled upon a TV interview with English chef Jamie Oliver a few days later in which he talked about why the United Kingdom had been gripped by an onset of obesity. \"The thing, I think, is that as people stop cooking, they get less healthy, yeah? People are going by the chippie to pick up dinner, but chips aren't dinner.\" Without realizing it, Jamie got animated, even upset, his adorable East London accent growing even more pronounced. \"If there was one thing that I could do, one thing that I could change, it would be to get people to just realize that cooking isn't that tough. It's a walk.\"\n\nHis words rang in my head. The woman and her cart of boxed food . . . the idea that cooking is too difficult . . . the refrain from the woman in Paris about knife skills. Ever the reporter, I started to conduct research on how they all related. Several studies back up Jamie's assertions that the less a person cooks real food, the more they rely on processed or convenience foods, whether at home or in the fast-food line, and the more weight-related health problems they experience. To a large extent, the more you cook, the less you weigh.\n\nWhat intrigued me was that the woman I met felt that she _was_ cooking. To her, opening a box and doing something with it was creating a meal. I disagree. Yet neither of us is right or wrong. Researchers can't even seem to agree on the definition of _cooking._ While a lot of food writers bemoan the loss of home cooking, I found surprisingly little research into the matter. Sure, various studies examined the amount of time people spend cooking, such as one led by a Harvard team that found people spend about twenty-seven minutes a day preparing food, about half the time spent in the 1960s.\n\n\"There's this notion that there is some kind of decline in cooking and that people aren't doing it anymore. But that's not so clear. It's just that there are so many other choices on what they can do to get food,\" said Dr. Amy Trubek of the University of Vermont, a food historian and researcher who has spent more than a decade studying how people cook. \"People aspire to cook what they believe is good, healthy food. But then they find the food environment very complex. There's also a strong sense of 'time poverty' in the American culture, this sense that you don't have time. Cooking is a thing many people perceive they don't have time to do.\" She equated it with going to the gym. \"Everyone knows that you should exercise, so they say that they will go five days a week, but when it comes down to it, they don't.\"\n\nThe woman in the supermarket lacked confidence and skill when it came to cooking. She wasn't sure how to turn whole foods into dinner, and as a result, she found her choices were limited. But that's the issue. If you can't cook, you put yourself at the mercy of companies whose interests are overwhelmingly financial.\n\nFrances Short, the author of _Kitchen Secrets: The Meaning of Cooking in Everyday Life,_ says that while consumers may want to eat healthy and even actively seek out this information, it doesn't have much effect if they can't act on it. \"Advice to, say, grill or steam food can only be followed if you know how to grill or steam,\" Short noted.\n\nTrubek told me that what I probably gave to the woman in the supermarket was awareness, a good start. \"But awareness is no good unless you have repetition associated with it. That's how knowledge becomes practice and practice becomes habit.\"\n\nOne reason that Julia Child made such a formidable impact was her unique ability to inspire people to get off their couches and go into their kitchens. While viewers watched her make _potage parmentier,_ they often took a crucial step\u2014they made it themselves. They searched out leeks, chopped potatoes, and maybe even crafted their own chicken stock. But somewhere along the line, people stopped getting off the couch. Cooking turned into a spectator sport. While Julia demonstrated how to fillet a fish or wrestle the bones out of a roast, most modern cooking shows fall into what industry experts refer to as \"dump and stir\" shows or reality-based competitions such as _Iron Chef_. While some viewers may follow along, even executives for the network admit that they focus more on entertainment and less on instruction.\n\n\"I watch Wimbledon but it has no relationship to my ability as a tennis player,\" Trubek said. \"It's beautiful and aesthetic, but practice is the only thing that is going to help my backhand.\"\n\nBack in the 1960s, Julia battled the idea that adding some ingredients to a box or heating something up somehow constituted cooking. Mike's late mother, pressed for time while working the night shift as a telephone operator on a military base, relied heavily on convenience foods. In fact, when we first married, he waxed so fond about something called Noodles Romanoff that in my dizzy honeymoon state, I spent three full days trying to track down either a box or a recipe for it. It turns out Betty Crocker ceased selling Noodles Romanoff in May 1994. I'm sure it's a coincidence, but that's the same month the FDA required manufacturers to include nutritional labels on food products. (I ended my quest when I learned that faithful replication of Noodles Romanoff required a portion of an orange flavoring packet from Kraft macaroni and cheese and commercial powdered mayonnaise.)\n\nWe live in a world where experts and the government preach that we should all eat leafy green vegetables, but then we're bombarded with messages that sugar-laden cereals are part of a \"nutritious breakfast,\" and commercials present Subway sandwiches as the holy grail to weight loss. TV cooking show host Sandra Lee cheerfully suggests that \"gravy is too hard to make, so just buy jars of gravy\" and advises viewers to buy jars of garlic because \"mincing garlic just takes too long.\"\n\nTo be fair, I think her voice is just one in a cathedral-worthy chorus shilling the idea that convenience is the _most_ important asset when it comes to eating. No wonder we've forgotten that the most essential thing we do is to feed ourselves and the people we care about. When I saw the stuff the woman had in her basket, it struck me as _antinourishment_.\n\nConsider the ingredient list for a brand-name box of a pasta Parmesan side dish. The goal of the product, a company spokesperson told me, is to approximate the flavor of pasta tossed with a bit of Parmesan cheese and olive oil. In order to do it, they used the following:\n\nrefined bleached wheat flour, partially hydrogenated palm oil, salt, whey, reduced lactose whey, corn syrup, natural flavors, palm oil, monosodium glutamate, cultured nonfat buttermilk, Parmesan cheese (cow's milk, cheese cultures, salt, enzymes), sodium caseinate, modified corn starch, freeze-dried parsley, nonfat milk, onions, spices, lactic acid, ferrous sulfate, niacin, soy lecithin, yellow 5, yellow 6 lake, yellow 6, thiamin mononitrate, riboflavin, folic acid\n\nThat's twenty-seven ingredients, mostly chemicals, in place of three real items. To make this palatable, it's loaded with corn and so much monosodium glutamate and cholesterol-agitating palm oil that they had to list all three at least twice. All this to simulate the flavor of three ingredients: pasta, olive oil, and Parmesan cheese.\n\nI believe that everyone has it within them to boil pasta, add olive oil, and grate a little fresh Parmesan cheese over it. I believe that anyone can learn to chop up garlic in less than a minute. So how do companies get consumers to buy a box filled with chemicals, dehydrated milk, and pasta fused with deeply fattening oils so that it will cook two minutes faster? Or to buy products that are little more than dressed-up military rations? They do this by convincing people that making pasta tossed with some olive oil and cheese falls beyond their culinary grasp.\n\nDecades of savvy marketing conspired to make the woman I met at the supermarket believe that a simple cream sauce fell outside her abilities, and who could blame her?\n\n\"As a culture, there's a lack of balancing cost as opposed to actual value,\" Trubek said. \"We surrender our best interests for the sake of seeming convenience. By failing to understand what's involved in certain kinds of basic food preparations, American consumers have been duped.\"\n\nAll of this thinking converged when I stumbled on the show _What Not to Wear_ on cable TV one night. The immaculately coiffed host and hostess went through the weekly guest's wardrobe, critiqued her normal choices, and then tutored her on what to wear. That week, they transformed a frumpy housewife into a smartly turned-out woman with a chic haircut with particularly striking blond highlights. They visited her two months later to see if she'd stuck to her new look. She had, plus she'd lost her \"last ten pounds\" and radiated enthusiasm. \"You just gave me a push, the confidence I needed to make the changes that I knew I should make for myself. I feel proud of myself.\"\n\nI bolted upright on the couch.\n\nArmed with a yellow notepad, I interrupted Mike in the other room as he worked on a kitchen redesign for one of our rental apartments.\n\n\"I have this crazy idea . . .\" I started.\n\n\"Oh, no. Not another one,\" he joked. He put his pencil down. \"You hungry?\"\n\nWe talked as I sat on a stool while he made dinner, his specialty, chicken and vegetable Thai curry with brown rice. I gave him the rundown.\n\n\"I want to try to understand what could motivate people to cook more often. I want to give people different cooking lessons and see which of the things they learn might stick with them.\" But before we started, I told him, I needed to educate myself on what people had to work with at home and get a sense of the choices they had already made. So I would go into homes and look into fridges, freezers, and cupboards\u2014the culinary equivalent of auditing their closets. \"I'll have them make a dish they usually make, so I can see how they cook. Then I'll put together some lessons around the skills I think they're missing. Afterward, we'll follow up and see how they're doing.\" He poured the coconut milk into the curry. \"Well, what do you think?\"\n\n\"Exactly where are you going to find people willing to let a stranger come into their house?\" Mike asked.\n\n\"Well, I don't know. I haven't figured that part out yet. I thought of this just now.\"\n\nThe smells of coconut and curry filled the kitchen, now so quiet that I could hear the metal spoon dinging around the bottom of the wok as he stirred.\n\n\"Our kitchen is too small for lessons,\" he said, concentrating on his dish. \"What about the commercial kitchen your friend Ace is using?\"\n\n\"So you think it's a good idea?\" I asked, relieved.\n\n\"Well, it's an admirable notion, anyway. You can't force people to cook differently. It's like that old joke, How many therapists does it take to change a lightbulb?\"\n\n\"One,\" I answered. \"But the lightbulb has to _want_ to change.\"\n\n\"Exactly. If you approach this as a way to encourage people and give them a few skills, maybe it could motivate them. But remember that everyone's life is different and complicated, and people are smart enough to know if they are willing to change.\" I nodded. He handed me a bowl of red curry chicken and brown rice. \"Now eat.\"\n\nAs part of my author duties, I was scheduled to be a guest on a longrunning local radio show hosted by Seattle celebrity chefs Tom Douglas and Thierry Rautureau. I've known both since my days as a restaurant critic in the 1990s. The owner of a group of popular restaurants and an _Iron Chef America_ winner, Tom balances the titles of culinary icon and astute businessman, yet for all his success he's remarkably down-to-earth. Thierry reminds me of my chefs in France. He started as a classic apprentice in French restaurants at age fourteen and worked his way up. He now owns the impeccable French restaurants Rover's and Luc, plus hosted a radio segment on NPR called _What's in the Fridge?_ Listeners called in and described ingredients in their pantries and refrigerators, and he'd coach them on the possibilities. They both have won the James Beard award for Best Chef in the Northwest, plus seemingly every other culinary award possible. If that's not enough, they've both appeared on _Top Chef_.\n\nAs I pulled on headphones in the pleasant, cluttered studio, Chef Thierry offered me a glass of a nice French red he had brought in to sip during the show. I told them about the woman in the supermarket and what I'd been learning about home cooking. Tom and Thierry were intrigued.\n\n\"This is a person you just met out of the blue?\" Thierry asked.\n\n\"She went along with you, in this cold call in the grocery store?\" Tom asked incredulously.\n\nI threw in a few statistics that I'd picked up. Americans waste about a third of the food they purchase, for instance. I told them about the project and that I wanted to include some of their listeners.\n\n\"How long do you think it would take to find a person who can't boil an egg and change them into a person who can open up a fridge door and figure out how to make something from what they find inside?\" Thierry asked.\n\n\"I don't know, I guess we'll find out,\" I said.\n\nTom nodded behind the tangle of microphones. \"Also, maybe they can realize that cooking can be fun, and a good way to spend your time.\"\n\nThen we took a call. \"I used to listen to these radio shows and the old-time commercials were thrown in,\" the caller said. \"I remember there was one for what to make with evaporated milk. That woman must have rattled off various recipes in a minute or so. It struck me that no one could understand those recipes today. No one would even understand the vocabulary. We've moved into a place where people don't understand basics of cooking and food.\"\n\nTom thanked the caller. Then he looked at me. \"How do you plan to address this lack of knowledge?\"\n\n\"That's an interesting point,\" I said. \"You know, recipe writers don't use certain words anymore, like _braise._ Instead, they write, 'Cover and simmer in the oven,' because people don't know what _braise_ means. There's a loss of language specific to the kitchen, and it's evaporated just like that milk.\"\n\n\"But it's not just _braise,_ \" Tom said. \"I mean, people are uncomfortable with the idea of a pinch of salt. What's a pinch?\"\n\nWe talked about the demise of home economics in public schools and the notion that many Americans are two generations away from knowing how to cook.\n\n\"I think you can trace a lot of those skills back to Madison Avenue and the ad agencies,\" Tom said. \"They're the ones who sold us the idea that you didn't need to cook, you could simply buy products instead.\" He looked at the clock on the wall. \"Well, thanks for coming on the show. I'm excited to see how it goes. And you know what? In a funny way, I'm interested to see what you get out of it.\"\n\nWhen I got home from the show, I found two dozen messages waiting in my e-mail inbox. More streamed in over the weekend.\n\nAs I read through them, I realized that the conversation on the air had appeared to touch a nerve. I glimpsed moments of guilt, embarrassment, and downright melancholy.\n\n\"When I was nineteen, I was asked to make the salad at my boyfriend's parents' for Thanksgiving. I didn't know where to start, and they all made fun of me. It made me never want to try to cook anything ever again,\" wrote one woman.\n\n\"I rely on boxed products because it's easy, but I don't want to. I have NO idea how to butcher anything. I can't fillet a fish, cut up a chicken, etc. When I look at an artichoke . . . well, I just walk right on by,\" wrote Cheryl, a thirty-three-year-old mother of two young sons. \"I'd like to learn how to make 'real' food, more food from scratch and what I have on hand, rather than pulling out a frozen pizza for my family.\"\n\n\"My mother never let me in the kitchen; she thought I was in the way,\" wrote Shannon, thirty-two, also a mother of two. \"When I read about little girls cooking side by side with their mothers . . . it makes me so sad for what I missed.\"\n\nAnother wrote, \"I grew up with a grandmother that could make a meal out of nothing and make it taste as good as any five-star menu. I've lived for years on frozen dinners and anything that is easy or fast.\" She considered herself \"addicted\" to cooking shows. She watched them all\u2014 _Top Chef, Iron Chef, The Next Food Network Star,_ Alton Brown, Giada, Emeril. \"But then I am totally lost when it comes to knowing what to do when I try to fix anything. I've watched Gordon Ramsey while eating Tuna Helper. I'm so ashamed.\"\n\nIn the end, I selected ten people for what I began to refer to simply as \"The Project.\" They shared little in common in terms of background, except that they all identified themselves as a \"poor cook\" or an \"aspiring cook\" who relied regularly on processed or fast foods. I told them little about what to expect, other than a few dates and times. I wasn't trying to be mysterious. I didn't really have a plan.\n\nIn retrospect, I'm not sure what I was thinking. I'm not an academic type or a researcher; I had taught only a couple of informal cooking classes. My thoughts on what lessons I'd teach were murky. I would have to make it all up as I went along.\n\nThe first stop was a generic apartment building in the rustic former logging town of Granite Falls, Washington. After three hours, I climbed back into our Mini Cooper and leaned my head against the steering wheel. \"What did I get myself into?\"\nCHAPTER 3\n\n**The Secret Language of Kitchens**\n\n  **LESSON HIGHLIGHTS:**\n\n**What Really Lurks in Cupboards, Fridges, and Freezers**\n\n#  **SABRA**\n\n\"This is what you call White Trash Garlic Bread,\" announced Sabra, a lovely, fair-haired twenty-three-year-old young woman clad in a skin-tight blue shirt and strategically ripped jeans. \"This is one of the few things that I learned about cooking from my mother.\" She slathered one-half of a soft hamburger bun with Gold 'n Soft margarine, added a few hearty shakes of generic garlic salt, and topped it with dried Parmesan cheese from a can. After repeating the process with a half dozen buns, she slid the baking sheet into her immaculate white oven.\n\nSabra was the first volunteer I met. She shared an apartment in a basic but pleasant complex with her boyfriend in a sleepy former lumber town an hour north of Seattle. Her standard-issue kitchen was small but tidy. In the living room, a tiny kitten she had rescued the day before took halting steps on the black leather sectional, her occasional mewing competing with the big flat-screen TV tuned to a poker tournament that Sabra wasn't watching.\n\nWhile the buns lingered under the broiler, Sabra alternated between sips of Red Bull and peach schnapps mixed together in an orange plastic tumbler while waxing poetic about her favorite food. \"I _love_ Gold 'n Soft,\" she said, holding up the half-full one-pound plastic tub of margarine. \"If it had Gold 'n Soft in it, that's what my mother and I ate growing up. Now pretty much everything I eat has Gold 'n Soft in it. Anything else, especially butter, just doesn't taste right.\"\n\nThe \"garlic bread,\" combined with Stouffer's five-cheese lasagna, straight from thirteen minutes in the microwave, constituted her lunch. As the microwave counted down, she and I finished the inventory of her cabinets, freezer, and fridge. Among the goods: nine varieties of Stouffer's frozen dinners, six boxes of Hamburger Helper, a five-pack of mac and cheese, half a case of Cup Noodles, and the remainder of a case of Red Bull, all of it from a recent warehouse-store haul. Sabra and her boyfriend dedicated the rest of the pantry to an impressive liquor assortment that featured thirty-eight bottles, everything from bitters to vodka and five varieties of schnapps.\n\nIn the fridge, newly purchased bundles of broccoli and cauliflower filled the crisper drawers to capacity, a change for her. A recent pap test found precancerous cells, curable but a worrying sign that she inherited a familial propensity for cancer. A doctor suggested Sabra overhaul her diet to include more fruits and vegetables. \"We found a fruit stand near here, and it's cheap. We got three pounds of cauliflower for three dollars,\" she said. \"I couldn't get Cheetos that cheap.\"\n\nSabra was nominated for the project by her stepmother, my friend Lisa, a culinary school graduate, a part-time chef, and the kind of woman who routinely makes her own mayonnaise. Back in 2006, I decided to hire someone to help test recipes for my first book. I put an ad on Craigslist describing appalling pay and erratic hours. Within twenty-four hours, eighty-five people had applied. Many sent r\u00e9sum\u00e9s and long pleas for the position. Lisa had assumed that I was a man and responded with a snarky \"Is this a dream job or are you looking for a date?\" When we met, the intense, briny smell of cheese preceded her. She walked into my office and handed me a bag. \"My mom owns a cheese shop. If you hire me, there's more where this came from.\"\n\nLisa quickly became one of my best friends. She tested recipes, helped me organize various events, and Mike insisted she play Clinton Kelly from _What Not to Wear_ when I went shopping for clothes to wear on book tours. (\"You want jewel tones, no pencil skirts, and at least two pairs of Spanx.\") When I told her about the project she said, \"That sounds like a ton of work. If you're really going to do it, I'm all in.\"\n\nOver the years as we worked together, Lisa relayed stories of frustration in terms of food as it related to her relationship with Sabra. When she married her husband in her early twenties, he had primary custody of Sabra, then six years old. When Sabra was a child, Lisa would make dinner, such as roast chicken with vegetables and a salad. Sabra would sit back in her chair with her little arms crossed over her chest and refuse to touch it. After rejecting dinner, Sabra would run from the table to call her mother, who lived less than a mile away. Lisa would overhear the litany of complaints about the fare Sabra said she'd been served. Invariably, they'd hear a honk from the driveway minutes later\u2014it was Sabra's mother, there to take her to McDonald's.\n\nAfter a couple of years of this, Lisa bought a case of ramen noodles. If Sabra didn't want what she made for dinner, fine. But no more McDonald's, not for dinner. Sabra could eat ramen instead, which she liked, but she had to make it herself. After four nights of noodles, Sabra started eating dinner with them again.\n\nI asked Sabra about all of this, and she nodded, taking another long pull on her drink. \"Oh, yeah, I _love_ McDonald's. When I was a kid, that's how my parents showed me they loved me after they were divorced. Who took me to McDonald's the most? That's who loved me the most.\"\n\nI pondered this as Sabra checked on her garlic bread. How much of this was about the food? Or did that trip to McDonald's mean time alone with her parent, forced to focus only on her as she swung, slid, and ran around the colorful play area?\n\nAs part of the kitchen visit, I asked each volunteer to prepare a lunch or dinner, something that they routinely ate, so I could get a sense of their kitchen skills and eating habits. Could they hold a knife? Did they taste as they cooked? What kind of portions did they serve up? Frozen dinners were Sabra's go-to dinners. Breakfast involved toaster pastries and half a can of Red Bull. \"That's because I don't drink coffee,\" she explained. The Cup Noodles was her midmorning snack. She hit McDonald's for lunch. Throughout the day, she consumed a few soft drinks. \"And there's the constant grazing on a lot of chips and popcorn. There's always a bowl of something on my desk,\" she added.\n\nRed Bull for breakfast isn't as unusual as you'd think for someone her age. It has roughly 70 percent of the energy-drink market, a narrow but influential wedge of consumers between ages fifteen and thirty. At age sixteen, Mike's niece, Michelle, lived primarily on various energy drinks for three months. The caffeine gave her energy, while the high sugar content\u2014roughly equivalent to a glazed doughnut\u2014kept her from feeling hungry. She drank several every day. Her doctor commanded her to quit. \"You're not getting any nutrients,\" he said. \"You're starving yourself.\"\n\nSabra offered some bright spots. The guy at her liquor store introduced her to a woman who supplied fresh eggs from a nearby farm. As a downside, Sabra warned me that she didn't want to spend more than twenty minutes making dinner. \"If it takes longer than that, I'll just get fast food.\" Plus, there was her devotion to Gold 'n Soft.\n\nWe sat down for lunch. To me, the lasagna had a vaguely sweet tomato flavor. The \"garlic bread\" was an exercise in blandness. I asked her how she liked the lasagna. \"Well, this brand has really good deals, like, you can get four for ten bucks sometimes,\" she started.\n\n\"No, what I mean is, do you like its taste? The flavor, I mean.\"\n\nShe thought about it. \"I don't know. I like it better than some of the other frozen lasagnas.\"\n\n\"But how does it compare to homemade lasagna?\"\n\nShe tilted her head to one side. \"I don't know. I've never thought about it. I like it, but I mainly buy it because it's cheaper than the other ones.\"\n\nLater, I calculated that over the course of a typical day, Sabra consumed a cup and a half of sugar. Sabra's food choices were motivated by money, ease, and that refreshing live-forever mentality that grips so many of us in our twenties. Who has to worry about their diet when they're so young? Isn't youth enough to overcome a high-sugar, high-fat, high-salt diet? She represented the dark side of \"taste memory\" that comes from the early adoption of unhealthy foods such as fast foods. Everyone has their comfort food, that flavor that harkens back to a time in childhood that they felt safe and loved. This explained her fondness for Gold 'n Soft and McDonald's. But it was clear that she put more thought into creating cocktails than into developing dinner.\n\n#  **TRISH**\n\nOur next stop offered insight into a whole different generation. Trish was a sixty-one-year-old psychologist living in a modest condo on the edge of the affluent Madison Park neighborhood. She fretted when Mike and I showed up with a small video camera in hand to record the proceedings.\n\n\"I'm sure that I'm going to be doing everything wrong,\" she said, a tinge of defeat in her voice. She literally wrung her hands. Mike kept the camera low, out of her face, and joked with her to keep her talking as she gave us a tour of her comfortably modern apartment dominated by obviously inherited antique furniture, dark yet luminous with a patina developed through many decades of careful polishing. Trish wanted to feel like she belonged in the kitchen. \"I want to feel at ease. I want to feel like I can make meals for my friends who are good cooks.\" She pounded her fist into her palm. \"I want to like cooking!\"\n\nTrish grew up outside Washington, D.C., in what she describes as a \"fairly typical\" 1950s suburban family. Her grandmother played an important role in their lives. She taught Trish and her siblings to bake, read \"old-fashioned\" cookbooks, and lent a formal but friendly note to the family dinners and conversations. In Trish's house, dinner was served at six o'clock sharp on a table set with silver and china. Despite her grandmother's baking prowess, her mother wasn't much of a cook; she relied on _The Compleat I Hate to Cook Book_. Dinners starred a meat, supported by some kind of potato and at least one canned or frozen vegetable cooked to the point of gray. Her mother's star turn was fried chicken.\n\n\"I remember some truly awful stuff, such as spaghetti sauce made from tomato soup!\" Trish said, laughing. \"I guess I was kind of a picky eater. I loved the mahogany furniture, the chandelier, the silver and china, but I never cared about eating.\"\n\nTrish's cabinets and fridge offered a striking contrast to Sabra's, with baking supplies, cans of beans, tomatoes, tuna, salmon, and chicken, boxes of organic soup and broth, jars of pasta sauce, salsa, pickles, jams, salad dressings, and artichokes, oils and vinegars, dry pasta, spices and herbs. \"For a meal, we usually open a can, a box, or a jar,\" she said. But most of it was real food, just preserved. She had the same kinds of stuff in her cupboard that I had in mine at home.\n\nShe climbed on a stool and pulled down three flat silver boxes. As she opened one, a strong smell of curry mingled with cinnamon escaped. Inside she kept carefully marked small flat disks filled with spices and dried herbs. \"I got these storage things somewhere years ago,\" she said. She shrugged while looking at them. \"I wish that I knew how to use all these spices better.\" She closed the box and put it back up on the shelf.\n\n\"I guess I can cook, but I'm not very pleased with the results. I've never learned to cut or chop things properly. Lots of times I try a new recipe but the results are disappointing. It's hard to get enthusiastic about cooking a meal that's going to take more than thirty minutes to make and it doesn't usually turn out well.\"\n\nWhen I asked about recipes, Trish took us over to her well-ordered bookcase. She pulled out volume after volume of excruciatingly organized recipes, categorized and neatly maintained in a series of white binders, another marvel of organization. I gazed in wonder. A protective plastic sheath housed each clipping. As a Gemini and a creative type who can barely keep track of bills, I could never imagine having done such a thing.\n\nShe pulled out a binder labeled \"Entrees.\" She flipped through it. \"Yeah, this turned out sort of bland. And this one, I did something wrong, the chicken was tough.\" She flipped a few more pages. \"Oh, this one, it came out all right. I was kind of surprised.\"\n\nAs I looked through the recipes she'd clipped from magazines, I recognized the names of a couple of food writer friends, including the dish she referred to as bland. I later talked to the author about it. \"Oh, I know the recipe she's talking about,\" she said with a sigh. \"I was only allowed to include six ingredients in the list. So I cut out at least four from my original version. Plus, I had fresh basil in the dish, and that really made it work. But my story got bumped from August to November, and since basil isn't seasonal then, the food editor cut the basil. It's funny, home cooks think recipes turn out bland because it's their fault. The reality is that there is so much pressure to make recipes short that food writers have to cut out steps or ingredients to make them look simpler, or, in the case of the basil, less expensive.\"\n\nFor lunch, Trish made ratatouille. She chopped an eggplant in a curious moment of butchery with a blunt paring knife. \"Knife skills,\" I wrote on my notepad. Next she opened a couple of cans of tomatoes. As it cooked, she talked. Mike kept filming her discreetly. She and her husband avoided fast food, red meat, and pork for health reasons. Trader Joe's was their main stop for groceries. \"We don't have a budget for food, but, then, we don't have extravagant tastes either. We're coming close to retirement age, and, like most everyone else, there is less money now.\" She broiled the occasional fish fillet in the toaster oven or baked sweet potatoes. Her husband made the nightly salad. She bought a lot of bottled salad dressings. I made another note on my pad: \"Teach vinaigrette.\"\n\n\"In terms of seasonings, they're basic. I sprinkle a little balsamic vinegar on green vegetables, lemon on the fish, and butter and salt on potatoes. That's what I know how to do.\"\n\nAs we sat down for lunch, she fussed, setting the table with expensive antique dishes, embroidered white cloth napkins, and real silver. Her ratatouille was good, just a little undersalted, which made sense as Trish seemed absolutely terrified of salt.\n\nAll this made me curious as to how a smart, organized woman who possessed the basic building blocks of cooking had ended up so tentative. When asked a question, she'd invariably answer, \"I don't know, is that right?\" Yet she's a psychologist, a professional paid for thoughtful commentary and insight into others' lives. She had so much emotional baggage wrapped up in cooking that I kept stifling my urge to say, \"And Trish, how does that make you feel?\"\n\n#  **SHANNON**\n\nThe next day, Lisa and I pulled up to a classic 1960s-era ranch house with a white picket fence in a quiet working-class neighborhood. Toys in various states of repair languished on torn-up sod in the front yard. Shannon was a thirty-two-year-old stay-at-home mom with two kids. She had recently purchased a couple of chickens to keep in her backyard and grew a small patch of vegetables. Shannon subscribed to food magazines and combed Internet sites for recipes. None of it made her feel what she called \"kitchen confident.\"\n\nHer mother answered the door. \"I don't know what you're going to teach her,\" she said dismissively to Lisa and me, turning away as soon as we entered. \"She burns everything.\" Shannon appeared behind her, a trim, pretty brunette with an easy smile and a pixie haircut, a lean baby girl balanced on her hip. Visibly irritated by her mother's remark, she let it pass. Shannon handed her the baby and took us into her kitchen, a large sunny space with pale Formica countertops and a classic suite of nondescript white appliances. She kept a row of cookbooks on one shelf, something of a rarity in the kitchens we visited.\n\nShannon has an easygoing way about her, occasionally punctuating sentences with flinging arms or wide-eyed expressions or a curious rolling-winking of one eye, an endearing tic. \"I'd describe my cooking skills as pretty basic,\" she started. \"I can bake pretty well but cooking has always kind of escaped me. I can read a recipe and follow it, but most stuff turns out pretty bland. I make a lot of casserole-type dishes, which aren't very popular in my house.\" Cue eye-rolling tic.\n\nUnlike the other cooks I would meet in the project, Shannon did try to plan meals before she hit a grocery store. \"My problem is when I'm making up the menu for the week. I just run out of ideas and end up cheating by filling up two days with burritos and spaghetti.\" By \"spaghetti\" she meant a jar of sauce. \"Oh, yeah, that's it. I've never made it from scratch. Sauces, that kind of thing, it's all just kind of beyond me. I buy those or get a seasoning packet.\"\n\nOtherwise, a meal to her included a meat-plus-veggie-plus-starch combination. \"I'm not very good at cooking meat. I'm scared to death of uncooked chicken so it's usually super overdone.\" Pork chops are pretty easy, she said, but she usually did the same seasoning every single time, a mix of salt, garlic powder, and pepper. \"I try to fit in fish once a week, but it is always a boring night because I really don't know what to do with fish.\"\n\nShe budgeted about seven hundred dollars a month for groceries and meals out. Given they're a family of five, restaurant trips were a rarity. Her big splurge was a sushi place near her house. \"At the beginning of the month, things are good. By the end, the meals get increasingly basic as I try to stick to the budget.\"\n\nI asked about her mother's remark. \"Oh, my mom.\" Her cheeks flushed, a wince flashed across her face. \"My mother started every single meal with a can of soup. She cooked but she never really wanted me in the kitchen, so I didn't learn much. I want to teach my kids to cook when they're older, but since I don't really feel like I know what I'm doing, what am I going to pass on?\" Shannon would occasionally make variations on the stuff her mother had served her, such as Parmesan chicken, sweet and sour beef with egg noodles, and tuna casserole, even though she now considered it \"old lady food.\"\n\n\"I am so interested in cooking, but I find it frustrating. I can't look at a recipe and conceptualize how it will taste. I can't figure out what is necessary in a recipe and what can be left out. I wish that I were one of those people who could look at my cupboards and my fridge and just improvise something. Or go to a restaurant and eat something I like and then replicate it at home. I don't feel like I have the skills to do that, you know?\"\n\nShannon had the desire, motivation, and time to cook but felt she lacked the core competencies. Like so many people, she didn't learn to cook from her mother, nor did she learn any cooking skills in school. By contrast, women of her mother's generation had multiple opportunities to learn\u2014from their own mothers and in high school back in the days when home economics enjoyed a more robust place in the curriculum. She struck me as similar to a frustrated aspiring musician who just wanted to get the scales down so that one day she could riff.\n\n#  **DRI**\n\n\"Welcome to the hood,\" Dri said, spreading her arms wide in a welcome as we walked up the neat path to her well-tended apartment building in the city's Central District. In theory, this is the \"rough\" part of town, but in recent years much of it has been undergoing serious gentrification. When there's a Starbucks on a nearby corner, it's tough to think you're in a ghetto, even in Seattle.\n\nVivacious and good-natured, Dri had the air of a nervous comic about her. She kept a smile fixed on her face for almost the entire visit. Dri planned to move soon into a condo she had bought in another part of town. \"Good-bye, eight-by-six-foot kitchen!\"\n\nDri was dressed entirely in black, possibly an effort to hide the extra fifty-plus pounds she carried on her tall, sturdy frame. She had recently started hitting the gym. \"But the whole food piece is just kind of missing,\" she told Lisa and me as she started to empty the contents of her cupboard. Her kitchen was so small, only she could fit in it; I stood in the doorway and watched.\n\nShe was as delighted as a kid at Christmas by the first thing she dug out\u2014a caked packet of spices. \"Oh, here's the magic herb mix that they gave me [at the food co-op] to make pot roast one time. Once. I only did it once. How sad.\" She dropped it dramatically on the counter. Dri pulled down aging jars of paprika and curry, a bulk jar of peppercorns, catering-sized jugs of olive oil and balsamic vinegar. \"Every time I make a real meal I seem to buy new spices. I usually buy them in bulk, so, hello\"\u2014she pulled down another quart container. \"Give a hearty Dri's Kitchen\u2013style welcome to another stupidly large container of cinnamon!\"\n\nAlthough she lived alone, Dri bought a lot of her food at a warehouse store. It's a habit she adopted from growing up in a family with seven kids. Her kitchen seemed so small yet food kept coming out of the cupboards like clowns from a circus car. At one point, she produced industrial-sized boxes filled with granola bars and instant oatmeal. \"For breakfast,\" she said. \"Oh, here's five pounds of cheap not-very-good-for-the-world kind of rice.\"\n\nDri described her \"vicious circle,\" one that will likely resonate with a lot of people.\n\n\"Okay, so I go to the grocery store and I have all these great intentions. I think, Okay, I'm going to make my lunch every day this week! I see all these great greens and I stock up on them. But then it's late when I get home, I'm tired. I promise myself that I'll eat them the next day. By then, I'm back in my routine that doesn't involve a lot of food preparation.\" Eventually, she finds a big, wilted, green stain at the bottom of the drawer. \"So I throw away a LOT of food. It's tragic how much I waste.\" A statement she followed up by noting that she has a degree in environmental studies; she now works in urban planning. \"I try to buy organic because sustainability is important to me, as a life choice.\"\n\nHer fridge was nearly empty. I suspected that she'd cleaned it out for my visit, but she insisted that wasn't it; she was going out of town the next day. On first inspection, it appeared immaculate save a few organic condiments, a half dozen eggs, and a solitary green pepper.\n\nIn a dairy drawer, she had the remnants of a large package of feta cheese. She sniffed. \"Ewww. Okay, this cheese is probably bad. That's the problem with bulk, right? It's cheaper to get a big thing than it is to buy a small one at the co-op. So I get it, eat part of it, and then the rest goes bad.\" She closed the fridge door and shook her head. For a moment, her permanent smile faded. \"It makes me feel terrible.\"\n\n\"Hey, you know, food waste is really common,\" I said. \"They say that people waste a third of what they bring home.\"\n\nDri pulled her mouth to one side. \"It's not very sustainable behavior, is it?\" she said flatly. She spent a quiet moment in self-judgment. Then she brightened. \"Want to see the freezer?\"\n\nShe pointed out artichoke hearts purchased months ago for a recipe that she never made. \"I'm not sure what to do with them now,\" she mused. She had organic hamburger as well as individual chicken breasts wrapped in plastic sleeves and several bags of random food that she didn't recognize. As with other people we visited, the freezer often became The Land of Food That Time Forgot. \"I like leftovers for one day but I don't want them on day two. So I think, I'll freeze them. And then they just sit in there until I move or the next millennium rolls around.\"\n\nWe moved on to a cupboard featuring what she referred to as her meager baking supply area. \"I bake so infrequently that I forget that I have stuff and buy it again.\" As if to prove her point, she uncovered fifteen pounds of white sugar.\n\nFor dinner, she decided to make a staple meal: spaghetti with a jar of tomato sauce. That's what she ate at least a couple of nights each week. As she waited for the water to boil, she explained that while growing up, her mom's cooking repertoire had relied heavily on pot roast and taco salad. \"We ate a LOT of taco salad,\" Dri said. \"We often had fend-for-yourself kinds of nights. Dad was a contractor, so we had very irregular hours.\"\n\nIt wasn't until she moved away from home that Dri realized her mother's food was, well, boring. \"If she went crazy, she used a bit of Lawry's seasoning salt. She doesn't believe in garlic, which I now believe is critical to life. Depending on where our family was financially, she would use stuff like powdered milk. I remember there would be times when she canned fruit and cooked much more from scratch. But there were a lot of frozen things in our lives.\"\n\nDri put the pasta into a bowl, a portion large enough for at least three people. \"You know, I fall back on pasta because, well, it doesn't take long, I can't screw it up, and I don't have to plan for it. But I want to have more options. I want to feel like I'm an adult who can feed herself. Sometimes I still feel like a kid in the kitchen; I'm just not sure what I'm doing. I mean, what am I going to do with those artichoke hearts in my freezer?\"\n\nTo me, Dri represented someone who was relying on old behaviors (shopping at a warehouse store when she lived alone) and couldn't stick to new ones that she wanted to adopt (eating more vegetables, cooking more often). There's nothing wrong with either, but how could she find strategies to do both and still fit in with her personal philosophies around sustainable food? I wondered.\n\n#  **JODI**\n\nA cute Japanese American with a quick laugh and a dimpled smile, Jodi lived in a comfortable split-entry house on a quiet suburban cul de sac. Her house was among the more upscale I visited, with expensive overstuffed furniture and a kitchen equipped with marble countertops and stainless steel appliances. At five feet two inches, she weighed about 135 pounds. \"I could probably stand to lose fifteen or twenty pounds,\" she admitted. \"It's funny, in America, I look pretty normal. But I went to Japan to visit my relatives recently. They all say\"\u2014she did a fairly pathetic Japanese accent here\u2014\" 'You so fat! You just sooooo faaaat!' \" They took her shopping and nothing fit. The shopkeepers looked at her and shook their heads. \"She too fat.\"\n\nJodi purposely never learned to cook. \"My mother has spent her life basically being my father's slave. It's part of Asian culture. So I figured that if I couldn't cook, then I couldn't be forced into that role.\" Jodi grew up with wildly conflicting maternal advice. The expectation cycle went something like this: Study hard, get good grades, get a good job, then get married. Wait, now you're married, why are you still working? My grandson is in day care? You're a horrible mother! You should quit your job.\n\nJodi married a good-looking, strikingly tall Thai American man with a flair for cooking. From the very start of their relationship, he cooked most of their weekend meals and dinner once a week. The rest of the time, they ate out or ordered takeout, an arrangement that satisfied them both\u2014until she was laid off from her job with a high-tech firm. Suddenly, they had half the income, plus a fussy three-year-old who refused to eat much of anything. Newly unemployed, tight for cash, and at home all day, she felt that the least she could do was to cook her husband dinner and feed her kid. They had long shopped at warehouse stores for food, where the savings held huge appeal. As a result, they bought a second fridge for their garage.\n\n\"All this food takes up so much space!\" she lamented, as food items fell out of her freezer when she opened the door. We pulled it all out: a vertical stack of twenty-plus breaded chicken cutlets, steaks of multiple shapes and sizes, a vacuum-packed whole beef tenderloin, dozens of individually wrapped boneless chicken breasts, and catering-sized bags of frozen vegetables.\n\nA closer inspection told an interesting tale. This penchant for bulk buying was not a new habit; a few steaks were four years old, and the chicken breasts had a sell-by date of two years prior. As she dug farther, she found whole salmon fillets and ribs bought for a camping trip three years earlier. \"Oh, wow, I forgot we had these!\"\n\nJodi and her husband infrequently made anything from the freezer, yet they kept buying food to put into it. Only two items had a good rotation: fish sticks and breaded chicken cutlets, both items her son would eat, and she could cook them easily on a tray in the oven. Their fridge was similarly packed. They purchased most condiments in industrial-sized jars. The mayonnaise in a hearty gallon jug had a sell-by date of a year earlier. \"I hate to throw it out, though, there's so much left. I guess we don't really eat that much mayonnaise.\"\n\nThe crisper drawer had three browning heads of iceberg lettuce. \"Oh, I hate it when this happens,\" she said. \"We always throw out lettuce. It's hard to go through it. They come five to a pack where we shop.\" Then she led us to the pantry, a closet larger than the kitchen of either Dri or Sabra, and we found it stuffed floor to ceiling with enough boxes and cans to last through the start of a nuclear winter: cases of soup, canned vegetables, boxed pasta dishes, instant ramen noodles (\"kid food\"), microwavable cups of mac and cheese, and a twenty-pound sack of white rice. \"There's more in the garage, too,\" she said.\n\nMany of her evening meals consisted of the \"brown and stir\" variety, some kind of sauce that involved adding a protein, usually chicken. For lunch, she worked up her most ambitious dish: cooked strips of chicken and chopped-up onion and red pepper in a skillet, flavored by adding a cube of \"golden curry\" seasoning. \"This is a Japanese household standby,\" she said, showing off the shimmering brown, gelatinous cube before she dropped it into the pan. She added water and stirred. Like a miracle, it thickened and created a thick, shiny brown sauce. She boiled water for Minute rice. \"I'm such a bad Asian! But our rice cooker has been broken for ages,\" she confessed.\n\n\"So how often do you eat rice?\" I asked.\n\nJodi cocked her head to one side, stirring her dish. \"Pretty much every day,\" she said thoughtfully. Then she laughed. \"I guess we should fix it, huh?\"\n\nI looked at the golden curry package. One serving, a mere quarter cup, contained 41 percent of the recommended sodium intake. The cube's flavor came primarily from monosodium glutamate. I told her it affects many people like a toxin. \"It's funny, now that you mention it, I always get a headache after I eat this stuff,\" she said. Although the package claimed five servings, Jodi said that she and her husband usually split a package\u2014each eating two and a half servings. Or 120 percent of the recommended daily sodium intake and 60 percent of the recommended daily fat intake, which didn't include the oil needed to stir-fry the chicken and vegetables. As she sat at the table, she fell into her chair with a heavy sigh.\n\n\"I didn't know all that. I guess I shouldn't make it anymore.\" But what would happen to the six packages in her pantry? Plus the curry remained one of the few things that her son, Koji, would eat. \"He will eat a little of the chicken and sauce sometimes, but not the rice and definitely none of the vegetables,\" she said. Like many toddlers, Koji was a wildly fussy eater. He eschewed vegetables. Unpleasant scenarios occurred when they tried to force him to eat anything other than chicken nuggets, pizza, fish sticks, or mac and cheese. \"It all started with day care. That's what they feed him there. Now he won't eat anything else. I worry about him. I mean, that can't be healthy, right?\"\n\nResearch studies have found that an increasing number of American children may get enough food to eat yet remain undernourished due to overreliance on foods that are high in fat, salt, and sugar yet lack the fundamental nutrients. A 2004 study found that nearly a third of the calories in a typical American child's diet came from junk foods, defined as ultraprocessed foods with little nutrition.\n\nIt's hard to blame kids, according to Dr. David A. Kessler, the author of _The End of Overeating: Taking Control of the Insatiable American Appetite._ Many of the foods on the common kid-food list\u2014chicken nuggets, powder-based mac and cheese, fish sticks\u2014have been engineered to stimulate pleasure centers in the brain. Studies found that as a result, rats can become addicted to junk food in the same way that they do to cocaine or heroin. Just as with drug addictions, rats often reject their standard \"rat chow\" and starve to death when denied junk food. That may explain the difficulty\u2014or sometimes impossibility\u2014of trying to force broccoli into a four-year-old in place of dinosaur-shaped pizza bites.\n\nAn acquaintance of mine took her fussy, plump toddler to the doctor when she noticed he had become grumpy and started to gain weight. The doctor described his condition as \"a sort of starving.\" He was dehydrated, an unsurprising fact given that he shunned water and insisted on sugar-spiked fruit juices or flavored milk. When she tallied up his collective meals from day care and at home, she was horrified to realize that he was subsisting on juice boxes, chicken nuggets, cheeseburgers, French fries, and hot dogs. She couldn't place the last time she had been able to make him eat a vegetable. I told Jodi that story.\n\n\"It's not like she was a bad mother,\" I said. \"She started realizing that wherever they went, the children's menu invariably included mac and cheese, fries, pizza, hot dogs, and hamburgers. It gives people the message that that's how kids should eat.\"\n\nThe normally bubbly Jodi stared at the counter. \"That pretty much describes what Koji eats, too.\" She stared into her coffee. \"I'm the one who is supposed to take care of him. I know that I shouldn't give him that kind of stuff. Sometimes, though, I come home and I think, I will make him eat a healthy dinner. I look in the cupboards and the fridge and pull out some vegetables and think, I don't even know how to actually cook these. Do you boil them?\" She appeared suddenly defeated. The golden curry had seemed like a good option because at least it wasn't breaded or fried. \"I never even thought to look at the label. Is that something you could make without a cube?\"\n\n#  **TERRI**\n\nTerri was a soft-faced, strawberry-blond-haired forty-six-year-old who had ditched a law career in the wake of a crumbled marriage and battles with alcoholism a dozen years ago. She managed a small tourism business from her one-bedroom condo. Due to the sedentary nature of her work, plus a recent broken ankle, she figured she was forty pounds overweight. She was battling high blood pressure, among other health problems.\n\nAccumulated papers, brochures, magazines, newspapers, bills, and unopened junk mail rose like a small mountain off her dining room table, with a portion of the pile cascading like a glacier into a puddle on the floor. Her kitchen remained pristine. \"That's because I don't use it that much!\" she said, laughing nervously.\n\nTerri struggled to find the motivation to cook for herself. Dinner tended to be takeout oriented. \"I rely on Chinese food and baguette sandwiches from a local bakery,\" she said. \"I make far too many runs to McDonald's and Jack in the Box for burgers, shakes, that sort of thing.\" Most weeks, she ordered a large pizza and ate it over the course of two or three days.\n\n\"The thing is that I like vegetables, but I don't feel like I cook them very well. I also tend to go overboard at the farmers' market and then I find all this stuff dead in my fridge.\" The previous summer, she had signed up for a Community Supported Agriculture (CSA) basket filled with fresh produce direct from a farmer. \"Even though I got it every other week, it was still too much for one person. I felt even worse when that went bad.\"\n\nHer fridge was a graveyard of expired condiments, a heavy vine of aging grapes, and a container of Greek yogurt with an eightmonth-old expiration date. She pointed to a few jars with unidentifiable goo in them. \"Attempts at making vinaigrette,\" she said, nodding. \"I make too much so it stays in there forever.\"\n\nThe freezer was equally stark; one of its few contents was a four-year-old turkey dinner. \"That's from a Thanksgiving when I decided I couldn't deal with my family,\" she said, then laughed nervously again. On Thanksgiving, she went to McDonald's. She looked a bit sad at the memory as she shut the door on the frozen-food tour.\n\nHer shelves were nods to healthy eating and falls from grace of it. Next to the quinoa (unopened) there was a shelf-stable microwavable meatloaf dinner and cracked bulgur next to fried onions (\"for green bean casserole,\" she interjected).\n\nFor her meal, she boiled whole wheat pasta and tossed it with olive oil. \"I would have no idea how to make pasta sauce,\" she said, grinding sea salt onto her pasta. \"I'm feeling kind of virtuous today because I'm not using Hamburger Helper or a jarred sauce.\" She'd been relying on them less after she learned that she had high blood pressure. As she sat down with her pasta in a beige La-Z-Boy chair in her living room, she talked wistfully about the days when, as a newlywed, she tried her hand at cooking. She made holiday roasts and even hosted dinner parties.\n\n\"I lost interest in cooking after I stopped drinking,\" she said honestly. \"But I'm realizing now that by not cooking I'm hurting myself, probably more than I realize. I want to be excited by it again,\" she said. \"I don't want to be on a diet. I just want to change the way I eat. But I don't know where to start or how to sustain that, you know?\"\n\nTerri struck me as a tough case. For high blood pressure, the best step she could take would be to cook more often. The vast majority of average Americans' sodium intake\u2014nearly 80 percent\u2014comes from fast food or ultraprocessed fare. By comparison, only 5 percent of sodium comes from home cooking. She mentioned time as an issue, but then talked about a lot of trips to physically pick up food and eat out, time that she might use to make dinner instead. I feared that what she wanted wasn't cooking lessons, but a magic bullet.\n\n#  **DONNA**\n\nIn Tacoma, we met Donna, a shy, sweet-faced twenty-six-year-old newlywed who resided in a row of modest starter homes. Her hair reminded me of Shirley Temple's, dark and spiraling down just above her shoulders. In a tinny, little-girl voice she asked, \"Do you want any iced tea? I have some in the garage.\"\n\nDonna was gosh-darned adorable. She and her husband had purchased the house two years earlier the week before they got married, a symbol of their mutual gung-ho commitment. Her pleasant kitchen looked as if it came directly out of a scene from _Mad Men,_ replete with canary-colored retro appliances. The fridge sported an extra door, presumably child-sized, built into the front. \"Isn't this weird? It's kind of a midget door or something. We want to redo it,\" Donna trilled brightly. \"But it's so darned expensive. Sure you don't want any tea?\"\n\n\"Do you have any vodka?\" I asked, only partly in jest.\n\nDonna laughed. \"Of course not, silly!\"\n\nDonna worked long hours in communications for an international aid organization that helps families in Africa. \"We should have two people to do my job, but of course it's a nonprofit so they just work you to the bone. But it's very rewarding.\"\n\nWhen Donna and her husband married, they agreed that he would cook and do dishes and she would do all the laundry. \"But now I realize that I can't rely on my husband to cook, or that what he makes will be healthy,\" she said. \"He goes back and forth about eating healthy, but when it comes down to it, he just eats what he wants. He says that he'll be more interested in it as I lose more weight, but right now he's not much of a good sport.\"\n\nAlthough Donna dutifully attended Weight Watchers for months, she'd lost only five of the fifty pounds she wanted to lose. She can cite the program's \"points\" for anything. But she finds they're not helpful when eating out, and lately they've been eating out\u2014a lot.\n\n\"My husband and I fight to the death about this. He doesn't eat all day or all afternoon. So he's supposed to be the cook in the family, but then by the time we're driving home together, he's starving. He stops and gets fast food or he wants to go out. Then he snacks all night. He thinks this is supposed to help him lose weight?\"\n\nShe revealed that just before they got married two years ago, he'd lost ninety pounds. Since then, he'd gained it all back.\n\n\"I guess back then he had to find a girl to impress,\" she said wryly. \"One reason that I want to learn to cook is because I can cut down on calories all day, but I when I get home from work, it's like a freefor-all. I don't know how to cook; it's not my element, so I am kind of at the mercy of whatever he wants to do.\"\n\nShe recently learned she's allergic to soy, and has long had an unusual reaction to raw vegetables. In the first cupboard, we hit abandoned cans of Slim-Fast. \"Oh, right, I forgot about those,\" she said sheepishly. \"It's got soy in it, so it's going to have to go.\"\n\nFrom the next one, she pulled out boxes for a hamburger-based skillet casserole. \"I grew up eating it, so I bought those when we were first married, but my husband hates it because he didn't grow up on it,\" she said. \"These might even be expired. I don't know, does Hamburger Helper have an expiration date?\" She examined the box. \"November 2008. Funny, I wonder what's in it that could expire?\"\n\nIn the same cupboard she found dehydrated mashed potatoes, boxes of Jell-O, and a block of Velveeta cheese. \"This I use for a dip when guests come over. You mix it with a can of chili.\" She dug in and pulled out an assortment of spices, many of them duplicates. \"Oh, we have three or four of the same kinds of spices. When we find a recipe we buy all the ingredients, not realizing that we already have the same herb or spice until we get home.\"\n\nOn a higher shelf, she had multiple bags full of flour, sugar, and other baking goods. \"I'd like to think that I do a lot of baking, but I don't. I think most of the stuff in this cupboard we haven't touched in more than a year.\"\n\nShe moved on. Condensed soup, cans of black olives, kits to make Mexican food, tins of tuna, a jar of pineapple chunks. \"We bought these for shish kebabs for camping,\" she said. As she pulled out more cans, she made a discovery. \"Here's the pineapple chunks from last year.\" She found many cans of turkey chili. \"We put them on Fritos or things like that for a snack.\"\n\nFrom another cupboard, she set out eight boxes of cornbread mix. \"You just add water, it's easy.\" I asked why she had so many boxes of the same items.\n\n\"We spend so much at this one warehouse store that they told us to upgrade to the business level because we would save money.\" Another cupboard revealed cases of granola bars and microwavable brown rice, an echo to Dri's shopping habits. A low drawer had a cache of hundred-calorie snacks. She picked up a small puffy bag of Cheez-Its and handled it disdainfully. \"Yeah, I was into these but they don't work for me. I'm kind of an addictive personality so I can't have just one. I can't think of anything else until I have another bag.\" She dropped it back into the drawer and quickly slammed it shut.\n\nWithout a word, she moved on to the fridge. A gallon of some pink drink dominated the top shelf. \"Yeah, my husband bought this. I didn't want to throw it away, so I've been drinking it.\" She looked at the label. \"I mean, this looks bad. I know anything that ends in - _ose_ is probably bad, or if the additive has an _x_ in it.\" She put it back.\n\n\"This is why I want to be educated, so that I can make better decisions.\" She bought books on nutrition, but rarely made it past the first couple of chapters. \"I buy a lot of cookbooks but I never really use them.\" She's made only one recipe from a book, a chicken dish from _Fresh Food Fast_ by _Cooking Light_ magazine. \"I was disappointed that it was so bland. My chicken was too dry, and I don't know why.\"\n\nIn the rest of the fridge, we found bricks of butter and a drawer devoted to an array of low-fat cheeses. A food chemist once described the process of making fat-free cheese as a complicated puzzle, one that typically required a lot of chemicals, gums, sugars, and added salt to solve. \"Do you ever wonder what they do to make them low-fat?\" I asked.\n\nShe cocked her head to one side. \"No, I guess I've never thought about it.\"\n\nWe moved on. There was a lot of packaged fruit, including some in small clear plastic containers. \"Oh, yeah, I stopped eating the grapefruit since it doesn't expire until next year. That kind of scared me. Isn't fruit supposed to expire?\"\n\nWhen she opened a crisper drawer, a putrid, vinegar-like odor escaped. She'd purchased a dozen apples in bulk, and half of them sat rotting in the bin. \"I totally forgot about those.\" Polite Donna looked visibly mortified. \"My husband thinks that if you can get it cheaper in bulk, you should buy it even if you throw half of it away.\"\n\n\"But you look upset,\" Lisa said, training the small video camera on her.\n\n\"I don't have an opinion on it,\" she replied. But then the color rose in her cheeks. She continued with the fridge tour. We discussed expired condiments and a rainbow of diet sodas. Then she discovered two browning heads of lettuce in another drawer. She clenched her jaw. She picked one up and examined it, not unlike Hamlet contemplating Yorick's skull.\n\n\"You know, I grew up not having much food in our house,\" she said, talking more to the head of lettuce than to us. \"We went to a lot of food banks as a kid. For me to throw away food, that's kind of sacrilegious. And, well, I work with starving kids in Africa.\" Her voice ratcheted up an octave. She dropped the lettuce back into the bin. \"So, yeah, it does bug me that we're throwing food away as if it's not important. That we're wasting anything.\"\n\nLisa and I looked at each other. We'd tapped something deep. \"You know, Donna, earlier you said that you didn't have an opinion,\" I said gently.\n\nShe started to drum her fingers against the door, avoiding our gaze. \"Apparently, I do have an opinion.\" She banged the door shut. She paused and purposefully slowed her breathing. Then she opened the freezer door.\n\nBig bags of frozen fried snacks dominated the freezer. Among them was a family-sized bag of Any'tizers, differently shaped and stuffed fried chicken products from Tyson. \"Those are my husband's.\" I asked Donna how many they normally eat at once. \"Oh, he'll make half the package for us to eat as a snack after dinner while we watch television.\" That's six servings, or 1,280 calories and 66 grams of fat.\n\nShe moved quickly on to the many bags of frozen vegetables, but they never used them because they didn't cook with vegetables. \"I know that I don't get enough fruit and vegetables in my diet. I don't know how to make them taste good.\"\n\nPart of the appeal of frozen vegetables is that they come precut. \"My husband makes fun of me when I cut things. Whenever I cook, I screw everything up and then I lose my courage.\" Her brother used to tease her if she helped in the kitchen. \"He'd say, 'Oh, you're not going to cook, are you? You're going to do it wrong and we're all going to die.' It wasn't actually all that funny.\"\n\nShe went quiet for a bit, organizing the elements of the meal she planned to cook. When she spoke, it was almost as if she were simply saying her thoughts aloud. \"I think it could be fun to cook. When I watch people cook, I get inspired. When I do it myself, I just get really freaked out. I panic. I don't want to cook for anyone. I used to lack confidence in everything in my life, and now I think I have confidence about everything else except cooking.\"\n\nFor her meal, she decided on \"El Paso Casserole,\" a menu staple she learned from her mother that featured canned tomato soup, canned turkey chili, canned cream-style corn, and shredded Cheddar cheese. As she used the can opener to open the chili, I noticed her hands shaking. Sweet Donna. This must have been so hard for her, to have not one but two people come into her kitchen, ask nosy questions, and film her while asking about the one thing she felt she didn't do well.\n\n\"Hey, I screw things up and I went to culinary school,\" I said, approaching her. I gave her a quick hug around the shoulder. \"I burned toast this morning. I overcooked a steak the other night. I mean, it happens. Even Julia Child screwed up sauces and dropped potatoes, right on TV.\"\n\nShe smiled at me thankfully and offered a polite laugh, but then went quiet as she started to brown a pound of hamburger. I decided to shift the conversation to how she would remodel her kitchen. Her mood lightened. As she topped the layered casserole with Cheddar cheese, she offered some final thoughts.\n\n\"I have friends who say, 'Oh, cooking is so easy, let me show you.' But to me, it's so intimidating. My friend comes over, and she's an amazing cook, she makes it all look so fun. But when she comes over and wants me to cook, I say, Let's go out. I'm too self-conscious.\"\n\nShe thinks some of it stemmed from growing up in a household where cooking was looked on as a chore rather than a rewarding act of creation and sharing. \"We go to my in-laws' house and you can tell that the women in his family love to cook. Everything my mother-in-law makes is wonderful and tastes great. But at my house, my mom and my grandmother, they don't really cook. Everything is very bland and heavy. Kind of unhealthy and artificial.\" She looked down at the dish she had just put together. \"Well, just like this casserole.\" She shoved it into her canary-colored retro wall oven.\n\n\"More than anything, I wish that cooking could become natural to me. I've come to realize that it's important for me to find options around food that I don't feel I have right now. If I'm going to eat right, I have to finally learn to do it myself.\"\n\nCooking seemed like a minefield to Donna, in terms of both her own relationship to food and the power struggle she felt in her relationship with her husband. He could cook, she couldn't, and that unfortunate balance led her to a place where she felt uncomfortable and out of control in the heart of her own home. She described it as \"his kitchen\" and commented that she didn't even know where to find some utensils and pans. She worked fifty-hour weeks trying to help feed kids in Africa, and her husband's buying decisions meant that she ended up wasting a lot of food. Yet she had clearly hit a point in her young life where she realized that she needed to make a change. Of all the people I visited, I felt that Donna had both the most to gain and potentially the most to lose.\n\n#  **ANDRA**\n\n\"I'm sorry, it's going to be so hot!\" Andra apologized repeatedly as she walked us up the stairs to her apartment. Edging toward the nineties, it was unseasonably hot for early June. \"I have no AC, so I hope you don't mind, but it's just too hot to wear a bra.\"\n\nThe life of Andra, a forty-three-year-old paralegal, had clearly taken some curious turns. Raised in an affluent family, she lived in what she called \"a sort of a slum,\" a complex of low-slung apartments that offered subsidized housing adjacent to the airport. In the wake of the economic meltdown of 2008, her firm slashed both her hourly rate and her hours during a major round of cost cutting. On a good week she took home $180 after taxes. For the past four months she'd been using food stamps for the first time in her life, a fact she kept secret from her colleagues and from her parents living nearby in an upscale suburb. \"They have a personal chef now,\" she said, an edge of what could have been either jealously or dismay in her voice. She hadn't seen them in three weeks, not since she sold her ten-year-old Honda Civic to keep up with her rent and cover her daily living expenses.\n\nLarge furniture overwhelmed the small apartment. A curious collection of knickknacks gave the impression that she had moved here from much larger quarters, and that perhaps not all of her decor originated from the same place. Expensive handmade Venetian masks hung on one wall, while pink-cheeked sweet Hummel characters and Franklin Mint\u2013style woodland critters cluttered shelves not far from items emblazoned with Harley-Davidson logos. Crescent moons and astrological symbols decorated everything from pillows to a faux wall tapestry, and an inflatable celestial globe hovered in the corner of her dining room. On her coffee table, there was an inexpensive plaster cast of a bald eagle clutching a struggling fish in its claws. The beaded curtain separating the living room and kitchen somehow did not seem out of place.\n\nAndra apologized for not having much on hand. \"It's the end of the month, and my food stamps don't kick in until the first.\" She had visited the food bank in the past week, where they had given her a package of frozen boneless chicken thighs. \"I'm really not sure what to do with those,\" she said, thumping the hard pack with her wrists.\n\nIn stark contrast to Jodi, she had absolutely nothing in most of her cupboards. In one, she had just four items\u2014a bottle of dried Italian herbs, a small jar of mustard, a bottle of Wesson oil, and half a package of dried egg noodles. In another, she had just two cans, cream of celery soup and beef ravioli in sauce, both marked with bright orange \"Damaged\u2014Half Off!\" stickers. Equally lacking, her fridge had only a few items, mostly condiments and a head of iceberg lettuce. For dinner, she planned to make the only other item in her freezer: miniature frozen pizzas that she had purchased at a supermarket outlet store. \"They have good deals there, but it's sort of a long time to get there by bus,\" she said.\n\nAs she placed the hard disks on a cookie sheet, Andra said that she believed people can eat well inexpensively but she hadn't yet figured out how. \"I used to have enough money that I never had to think about what to eat. That's changed dramatically. My options are much more limited. It's especially true now that I don't have a car. There's nothing near here in terms of food shopping, really.\"\n\nShe evaded questions about her past, but alluded to having lived a much different lifestyle not that long ago. \"I used to have money, but no time. Now I have time, but no money. When you have the funds, it's a lot easier if you don't know how to cook. You can go out to dinner, or you can get good takeout. But when you're strapped for cash like me, you end up eating Domino's pizza or stuff like that. It's cheaper to eat badly, or at least it seems that way, especially if you're a poor cook like me.\"\n\nMost people don't realize that the average food stamp recipient receives about a dollar per meal, roughly twenty-seven dollars per week. Andra's assessment of her situation struck me as accurate. Knowing how to cook does stretch food dollars. For the price of pizza delivery (at least ten dollars), it's possible to make a whole roasted chicken with mashed potatoes and a side of vegetables. But it requires access to a grocery store, plus enough knowledge to improvise dishes, take advantage of sales, and avoid wasting leftovers.\n\nThe speed with which Andra's lifestyle had collapsed was striking. Economists say that many Americans are only two paychecks away from being homeless. Andra had a roof over her head, but little more. She was willing to do what it took to weather the economic storm and hope for a better horizon.\n\n#  **CHERYL**\n\nBy the time we met the final two volunteers, strong themes had come through. Most volunteers were terrified of raw chicken and bemoaned their inability to make palatable vegetables. Bulk shopping and overzealous trips to the produce section or farmers' markets led to a lot of wasted food when combined with a lack of insight or inspiration for what to do with leftovers.\n\nCheryl, thirty-two, was the mother of a four-year-old boy and an infant son. She lived in an upscale home on the edge of a bay with commanding water and mountain views in a town well north of Seattle. Cheryl was a quiet, rail-thin woman with large, expressive brown eyes who hailed from a remote town in Canada. She focused on buying organic food and made a point to drive forty-five minutes to a local co-op.\n\nYet on the day we met her, she made herself a can of soup in her large, well-appointed kitchen with wide granite countertops and stainless steel appliances. When I asked if she'd ever tried to make soup, she smirked. \"No. Well, okay, once, but it didn't go well so I never tried it again.\"\n\n#  **GENEVIEVE**\n\nThe final volunteer, Genevieve, was a pretty twenty-five-year-old brunette who lived with three roommates in a spacious house in the city. Among them was John, who had become her boyfriend. She made one of the few dishes she knew, a combination of store-bought Asian slaw combined with a bottle of teriyaki sauce. Gen wanted to learn to cook in advance of a big life transition\u2014she and John were planning to move into a place of their own. \"I have friends who cook. It's like they belong to a sorority and I somehow never got asked to pledge, and now they seem to be guarding their secrets, you know?\"\n\nBoth Cheryl and Gen came from homes in which their mothers cooked, yet neither learned much before leaving home. \"It's so easy not to cook,\" Cheryl said. \"You can always pull out a frozen pizza. But I don't want to do that. I want to feel like I'm nourishing my family, not just giving them food to subsist on.\"\n\nGen echoed her comments. \"I guess I never felt compelled to learn because it feels like you could go your whole life without learning to cook. But as I've gotten older, I've just started to realize that seems like the least healthy way to go through life. I want to be able to control what I'm eating. You really just don't know what's in stuff anymore.\"\n\nThe goal for both of them, and for all the volunteers, was something that felt remarkably elusive. Cheryl articulated it well. \"I want to be one of those people who can open up the fridge, look inside it, and come up with a meal. I simply cannot do that now. It's like some kind of magic to be able to do that.\"\n\nI thought about what I had learned in the course of the week. It wasn't what I had anticipated, although, admittedly, I hadn't been sure what to expect. But I could not have predicted the residue and damage that a lack of cooking skills had on people's daily lives. Among the boxes and cans, I found a larger story of perceived failure that left them struggling with guilt, frustration, and a stinging lack of confidence.\n\nThe themes that I found in these kitchens were echoed in the others we visited. The women were so different in terms of age, background, and socioeconomic status, but they struggled with similar issues. A lot of them struggled with their weight. Lack of planning led them to rely on processed food or to stop for fast food. Some purchased too much and the wasted food made them feel bad. All of them talked about their mother, spouse, or grandmother and their ability or inability to cook, and how it impacted them as adults. None of them could hold a knife properly; then neither did I before culinary school.\n\nFrom the moment I learned of Sabra's devotion to margarine, I worried that I'd gotten myself into something far too complicated. In Donna's home, meals were an emotional riptide that represented more than brown lettuce, but involed morals, money, and worldview. Jodi struggled with how to balance her fears of succumbing to Asian wife stereotypes and trying to feed her son something other than chicken nuggets. Andra represented those most affected by the economic breakdown in 2008, her comfortable life turned upside down by an unexpected downsizing that shifted her from a comfortable living to food stamps within less than six months through no apparent fault of her own. For all her good intentions, Dri ultimately lived on massive portions of starch while her greens suffered a slow death in her fridge.\n\nI had started this on a whim, and it wasn't until I actually visited them that I realized how much courage it took all these women to allow a stranger into their homes, to poke and pry and play voyeur, then to sit in judgment of one the most intimate human acts. If you don't think of eating as intimate, think of those quiet moments you've stood alone in a kitchen, the wedge of light leaking from an open refrigerator door, seeking to satisfy a craving.\n\nConsider a doughnut. Everyone knows that doughnuts aren't good for you. They're sugar and white flour fried in fat and traditionally topped with more sugar. But it's hard to deny the lure of the delicate aroma of a warm doughnut and the guilty pleasure of sinking your teeth into the mushy, powerful sweetness of a freshly glazed bear claw. Moments later, you feel the sugar rush, subconsciously knowing that it's too strong to last and that it's only a matter of time before the inevitable crash. How many of us have had relationships like that? The tug of something forbidden and giving in only to experience joy, disappointment, and, ultimately, regret.\n\nAs I looked over my notes and started to review the video of those visits, I wondered what my cupboards would reveal.\n\nUnable to sleep, I slipped out of bed at two A.M. and conducted an inventory of my own kitchen. I found boxes of partially used pasta, mostly whole wheat. Three bags of white sugar. Really? Stacks of canned tomatoes, artichokes, olives, imported tuna, and locally packed clams hid two expired cans of foie gras and p\u00e2t\u00e9 from France. Damn, how did I forget they were there? Plastic bags with various grains I'd experimented with once or twice that Mike didn't like, so they remained untouched. Half jars of protein-shake mix from an ambitious gym period. A messy drawer crammed with tired spices, some labeled in foreign languages. In the cupboard, dying vinegars, Wondra gravy thickener, green peppercorns, a box of Krusteaz crust mix, and a couple of boxes of macaroni and cheese.\n\nCondiments dominated the fridge, a curious collection that ranged from Baconnaise to fish sauce to four varieties of mustard. Two small takeout boxes of Thai food sat on the middle shelf next to plastic containers filled with duck fat, bacon fat, and expired yogurt. Despite efforts to diligently use produce, I found a moist green bag of what had been organic mesclun salad, and next to it, decomposing zucchini, and a handful of molding limes lurking in the crisper drawer under fresh bundles of Swiss chard. Tucked in the back was an abandoned year-old bag of decaf coffee bought after a fertility specialist suggested that I give up caffeine. In the freezer, bones for chicken stock mingled next to vodka-spiked ice pops, a bag of white pork fat (a gift from Lisa), ready-to-cook chicken dumplings from a place in Chinatown, half-used bags of spinach, frozen blueberries from two summers ago, and the remnants of an undated batch of chocolate chip cookie dough.\n\nI was in a battle with myself. It seemed that I had as much to learn as any of the people I'd just visited.\nCHAPTER 4\n\n**It's Not About the Knife**\n\n  **LESSON HIGHLIGHTS:**\n\n**Kitchen Tools, Basic Cuts, and Why You**\n\n**Really Need Only a Couple of Knives**\n\nOn a balmy June evening, the volunteers straggled into the kitchen one by one. To each we handed an apron, a notebook, and a cloth diaper.\n\n\"They make the best side towels,\" Lisa explained of the diapers to the puzzled volunteers. \"The middle is padded, so it's like an oven mitt. You use it to pick up hot pans and stuff.\" Neither of us said it, but they're also dirt cheap.\n\nShannon picked one up and laughed. \"I have these exact same diapers at home. I just don't picture them in a kitchen.\"\n\n\"I swear, these are new,\" I said.\n\n\"Oh, I believe you,\" Shannon said, worried that she'd offended. \"I just think it's funny.\"\n\nDiapers in hand, they wrote their names on pieces of masking tape because I had not thought to get name tags. They took it in stride and slapped the masking tape to their chests.\n\nAs Mike noted, our Seattle kitchen was too small to teach more than a couple of students at a time. If you've never tried, finding a kitchen that can accommodate a dozen students proves a bit more complicated than you might expect. Only a few days before the first class, I managed to secure the use of a kitchen owned by a catering company. I discovered the place via Anne-Catherine, one of my fellow classmates at Le Cordon Bleu in Paris, affectionately known as \"Ace.\" \"I can't wait to move to Seattle. It's where I want to spend the rest of my life,\" Ace told the group while sipping _vin rouge_ on a crisp fall Parisian evening out with our fellow students at a beaux arts bistro near Les Halles. My dream transported me to Paris, while hers was to settle in Seattle. Ace served as the executive chef of the catering company for a time and held a series of communal dinners in the kitchen's small dining room before she left to open her own restaurant.\n\nThe owner required a business license, insurance, and state health cards in order for us to rent her place. Lisa and I sat through a threehour kitchen sanitation and safety class with several hundred aspiring kitchen workers. The short version: Wash your hands often for at least twenty seconds, roughly the time it takes to sing \"Happy Birthday\" twice, and don't defrost meat in the trunk of your car.\n\nEverything secured, the owner handed me the first set of keys that I've ever had to a commercial kitchen. It wasn't _Top Chef_. The kitchen is housed in a 1950s storefront, most recently the site of a pizza joint. The extremely canted windows bring to mind the architectural works of Mike Brady. The trapezoid-shaped room feels as if it started life as a coin-operated laundry or perhaps a dry cleaner. The neighbors include a jumble of houses, a Chinese food joint, used-car dealerships, old-school mini-malls, and a Cuban place known for its mojitos.\n\nSomehow, the owner made this spot work. A rugged six-burner gas stove anchored one wall, flanked by a grill and a pair of well-worn pastry ovens. A matching set of handsome stainless steel refrigerators lined the other wall to complement a six-by-eight-foot walk-in cooler. Organization reigned in the chrome industrial racks around the room, piled with the stuff you'd expect to find in a catering kitchen: stacks of plates, serving pieces, industrial-sized jars of oils, vinegars, condiments, and spices, plastic bins, bottles, pitchers, boxes of glasses, cooking utensils, stacks of pans and bowls. It smelled vaguely of floor cleaner, residual cooking odors, and the lavender growing in the front planter box.\n\nThe owner left Post-it notes stuck throughout the kitchen. The tone varied from shrill to verge-of-hysteria warnings: \"For catering use ONLY!\"; \"DO _NOT_ FORGET TO TURN OFF LIGHTS!\" Despite its quirky charm, or maybe because of it, I felt lucky to have found this place.\n\nTheir hands washed after multiple renditions of \"Happy Birthday\" and their diapers tucked into their white aprons, the volunteers stared nervously at the pile of knives.\n\nEach volunteer hauled in her own knives from home as instructed. The cutlery cluttered the stainless steel tables pushed together in the center of the room. With that, the project officially began.\n\nI toured the selection. Sabra had purchased her serrated-knife block set with five knives for twenty dollars because \"they looked cool,\" she said. Trish had a set of Cutco knives custom made to fit into a shallow drawer in a house she no longer owned; she admitted that she used only a blunt vegetable knife for all tasks. Donna showed off a wedding present, an eleven-piece set from the Pampered Chef, kept pristine in the original protective covers. Cheryl brought in her knives, drawer and all, which featured everything from a few antique Ginsus to some classic all-carbon cleavers to her husband's assortment of hunting knives. Shannon had a set of expensive German knives, a wedding present from years earlier. Andra took three buses from Sea-Tac to get to the kitchen. \"Sorry, I left the bigger ones at home. I didn't want to carry a big bag of knives on the bus,\" she said.\n\n\"Okay, show me your favorite knife, the one you use most of the time. Hold them up.\" They raised their knives aloft, like tentative knights at a medieval feast fearing their swords would not pass muster. Most held a vegetable or a paring knife. Only one person lifted a chef's knife. I had them put their arms down. I asked why more people didn't use a chef's knife.\n\n\"That kind of knife scares me,\" Trish said. \"It's so big.\"\n\nSabra shrugged. \"What different does it make? A knife is a knife.\"\n\n\"Most of these knives look pretty expensive,\" Andra said, perusing the collection.\n\n\"Your mileage will vary,\" I replied. In an effort to present the spectrum, Lisa had raided a restaurant supply store and purchased several Mundial plastic-handled chef's knives, the default tool of restaurants everywhere. I had picked up a twenty-five-dollar OXO knife at Target and a ten-dollar knife at IKEA. I set them out next to my own collection of knives, a United Nations\u2013style parade of metallurgy made by W\u00fcsthof, Shun, Henckels, Global, and Sabatier and a ceramic number by Kyocera, among others. \"Since I wrote a book with the word _knife_ in it, people give me knives,\" I explained. \"Maybe my next book title should include _diamonds, cash,_ or _Learjet_.\"\n\nWe went through Knife Anatomy 101, starting with the obvious. \"As you can guess, the blade is the sharp part. The blunt back of the blade is a spine, just like a book. Many knives sport a raised lip edge near the handle known as a bolster.\n\n\"See this W\u00fcsthof?\" Like a hand model, I stroked the length of the knife. \"This is what we call a 'full tang' because the blade extends all the way to the butt of the knife. It's held together by rivets. A bolster is heavy, designed to help balance the knife.\" I put it down and picked up one of the inexpensive restaurant knives. \"See? I bet the tang extends about an inch into the handle. Second, no bolster. It's flat.\" That means that the knife has been stamped out of a sheet of metal, kind of like a cookie cutter. To make a knife with a bolster requires forging, a more complicated process that requires human interaction, and so it's always more expensive.\n\n\"Take two key considerations into account when buying a knife. The steel and 'the feel.' You want a knife with the kind of steel that can take an edge and hold it.\" Not all steel is created equal. Harder steel takes an edge better, resulting in a sharper knife. But particularly hard steel can be brittle and trickier to maintain.\n\nThere's a complicated measure of hardness, but for most retail knives the main concern is carbon content. \"Carbon makes steel stronger. If you look for the phrase 'high-carbon steel,' that's a start.\" For instance, the knives from Victorinox, of Swiss Army knife fame, are made from high-carbon steel, and their chef's knives start at around thirty dollars.\n\n\"Marketers don't talk about steel since most consumers don't care,\" Lisa chimed in. \"Instead, their job is to try to blind you with a lot of features. They know that people like value, so they will pack a set with all kinds of cheap knives, but the reality is that you don't need them all.\"\n\n\"Okay, here's what I use.\" I picked up an eight-inch extra-wide German knife. \"I used this one throughout my training at Le Cordon Bleu. It's heavy, but I have small hands, and for some reason, the extra weight just feels good to me. How a knife fits into your hand is the 'feel.' Go to a place that has a good selection of knives, such as a cookware store, cutlery shop, and some department stores or restaurant supply places. Feel the subtle differences in the weight and the grip of the handle. A comfortable knife is a highly personal thing.\n\n\"Next, buy only the knives you'll use. Start with a good chef's knife. Supplement that with a paring knife and a bread knife.\" I picked up my German-made bread knife. \"With care, good knives last for a long time. My mom bought me this for my birthday nearly twenty years ago. Other than my life, still arguably the best gift she ever gave me.\"\n\nTrish asked, \"What about a vegetable knife?\"\n\n\"Honestly, I don't know what you would do with a vegetable knife that you couldn't do with a chef's knife or a paring knife,\" I told her. \"I use a chef's knife at least ninety percent of the time. I use a paring knife once in a while. I have a boning knife and a cleaver from my set from Le Cordon Bleu that I break out a few times a year. I haven't used my fillet knife in months. My chef friend Ted has four knives: a chef's knife, a paring knife, a bread knife, and a fillet knife. That's it. He was a professional chef for twelve years.\"\n\nDri raised an eyebrow and nodded her head. \"All of this is good to know,\" she said. \"My budget can handle getting one good knife. I was looking at three hundred dollars for a block set and thinking, There's no way.\"\n\n\"You can get a decent knife for thirty to fifty dollars, and that's a better deal than buying a cheap block set,\" I said. \"But honestly, if you spend money on anything in your kitchen, invest it in the best knife that you can afford. If you take care of it, you'll have it for twenty or thirty years.\"\n\nI looked at the clock. \"Okay, no more looking. Find a chef's knife and pick it up. You can use any of mine, too.\" The women stirred, staking out a spot at the stainless steel table.\n\nCheryl reached for a Henckels knife close to her. She had her infant son, Liam, strapped to her bosom in a BabyBj\u00f6rn. \"I couldn't get a sitter,\" she had explained at the start of class. \"Don't worry. I cook with him like this all the time.\" It is not every day that you see a baby next to a pile of a hundred knives. Liam reached for the shiny, pretty things. Cheryl kissed his head as she pulled his small hand back, \"Oh, no, sweet pea, those are not for you.\"\n\n\"Show me how you hold a knife.\" All of them held it the way that I had before I went to culinary school, by making a fist around the handle.\n\n\"Don't strangle your knife,\" I said. \"You want to sort of shake hands with it. Place the handle across your palm. Wrap your hand around it. With your thumb and index finger, pinch the juncture where the blade meets the handle. This will be on the bolster, if your knife has one. This should leave your other three fingers tucked around the handle.\"\n\nI watched as they all tried this out, a study in collective awkwardness and furrowed brows. Lisa and I walked around the table, inspecting and adjusting holds here and there. It reminded me of my first day in the kitchen at Le Cordon Bleu, when the pleasant Chef Bruno Stihl walked through and stopped to correct my own grip on my brand-new knife. _\"Non,\"_ he said as he gently unwrapped my fist and carefully reworked my fingers into the correct position. He held my hand in his as he demonstrated the rocking motion to use when cutting. _\"Oui, comme \u00e7a. Un couteau doit \u00eatre une extension de votre main,\"_ he said, smiling, and then walked away. Loosely, the phrase translates to 'Your knife should be one with your hand.' Once I figured out that phrase in French, I never forgot it. I never thought a knife demo could make me so nostalgic.\n\n\"How does that feel?\"\n\n\"Weird,\" Sabra said. Other people nodded.\n\n\"It's a little bit like how you hold a golf club,\" Trish observed, looking at her knife as if it had somehow just become miraculously attached to her body.\n\n\"The reason you want to hold it like this is that you'll have more control over the blade,\" I said. \"Plus, your hand won't get as fatigued. Okay, now hand your knife to the person on your right, handle first. _Carefully_.\"\n\n\"This one is a lot heavier,\" Sabra said. She had been holding an inexpensive knife from her own block and shifted to a Japanese brand. \"This is, like, well, a real knife.\"\n\n\"This is why it's great if you can test out a knife before you buy it. A comfortable knife will prompt you to chop more, which will encourage you to cook, and that's the whole point,\" I said.\n\nSomeone asked where to store knives without a block, so I pulled out a simple black plastic sheath that clipped on to the blade. \"This is a knife cover. Slip it on like this\"\u2014I clicked it in place over a blade\u2014\"and voil\u00e0. It will protect your knives and your hands from the blade when fumbling around in a drawer.\" Magnetic strips on a wall work great. Mike ingeniously snapped powerful bar magnets from IKEA on the inside of our stainless steel stove hood. Knives snap solidly to the outside, up and out of the way. \"I wash them by hand as soon as I use them, and then bang! They snap right to the magnet, high and out of the way.\" That brought up an important point.\n\n\"Always wash knives by hand. Never, ever, put a knife in the dishwasher. Steel is tempered with heat. The high heat from your dishwasher will damage the steel, dull the edge, and probably not do much for your handle either. Knives should never go into the dishwasher. Ever. Say it, everyone.\"\n\n\"Knives should never go in the dishwasher,\" they said in unison. \"Ever.\"\n\n\"One more thing to remember: Knives are like dogs. They need occasional grooming. Get them sharpened at least once a year. Most cookware or cutlery stores will either offer this service or can tell you where to take them. It will set you back a few bucks per knife, but it's so worth it,\" I said.\n\n\"But doesn't this thing,\" asked Dri, holding up a honing steel, \"sharpen your knife? Can't you do that at home?\"\n\nLisa jumped in. \"No, this is called a steel. When you run a blade across this, it's more like a fine-tuning by taking away bits of metal that collect on the edge. It keeps it straighter. But the metal should be a little bit magnetic to work and a lot of the steels aren't magnetized. So they're just for the finishing touch, really.\" It's the difference between brushing your dog and having it professionally groomed.\n\nWith that, we got down to business. We distributed thick cutting boards and settled a wet paper towel under each to keep it from slipping, a trick I learned from the French chefs. Then Lisa and I made sure each person had a proper chef's knife.\n\n\"Proper cutting technique involves sticks and cubes. That's about it.\" I grabbed a zucchini from the table. I cut it in half lengthwise and then cut each half into four vertical strips. \"Remember that ad that said a Ronco-something-or-other could make julienne fries and you wondered what they were? Well, they're sticks. That's how you do it.\" Next, I took three sticks and cut them into cubes. \"This is diced. The thinner the stick, the smaller the dice.\"\n\nI diced another four sticks. \"Notice that I'm not chopping down like a guillotine. Instead, I'm starting with the tip of the knife on the cutting board and then bringing the blade down. It should feel like a rocking motion.\"\n\nNext, a stalk of celery. \"It's always the same,\" I said, slicing it into two vertically. Then I diced it. \"Always curl your fingers under to avoid cutting them. Then use the flat of your knuckles like a guide for the blade. If you get into that habit, you'll cut yourself a lot less often.\"\n\nWith a grunt, Lisa banged a five-gallon bucket containing thirty pounds of zucchini onto the table. The group stood back and stared in disbelief.\n\n\"Uh, that's a lot of zucchini,\" Jodi observed. \"What do you want us to do with that?\"\n\n\"I want you to practice,\" I said. \"A lot of practice. So let's go.\"\n\nEach person grabbed a zucchini and made her first tentative cuts. After a bit of chatter, the room settled into an air of quiet concentration. I saw Trish struggling to get a grip on her knife and went over to her. \"Is everything okay?\"\n\nShe said quietly, \"I cut myself really bad when I was eight years old and so I have always been afraid of the blade. I think that's why I have always felt so unsure with a knife.\"\n\nI placed the chef's knife back in her hand and held it as she made her first cuts. I could tell it was hard for her. \"You know, years ago I badly cut one of my fingers trying to saw through a frozen bagel. See my scar?\" I held out my finger. \"I completely understand. But if you hold your knife back and keep your fingers out of the way, you're going to be fine.\" She nodded and grabbed a zucchini. I watched her deliberately slice it and then slowly carve sticks from one of the halves. When she diced her first batch, she looked relieved.\n\nBy her second zucchini, Trish moved more quickly. Soon, the regular thump-thump of knives filled the room like a healthy heartbeat.\n\n\"After a while it gets so much easier,\" Shannon said as I came around her side of the table to collect some of her diced vegetables. \"I feel like I have more control over the knife.\"\n\nWith that, the mood shifted into one of cordial focus. The women chatted sporadically yet kept their eyes on their cutting boards. Halfway through her final zucchini, Jodi insisted that everyone look at the perfection of her dice. \"I can't believe that I am doing this. People, I want you to know that I have never so much as peeled an apple in my life.\"\n\nWithin a half hour, the pile of zucchini disappeared, having been diced and then swept away into a massive thirty-two-quart pot on the six-burner stove. Then we handed each person two large yellow onions.\n\n\"When you cut an onion the classic way, it's easier, faster, and you cry less. But it also impresses people,\" I said. \"First, cut the onion in half across the root so that a portion of the root stays intact on both pieces,\" I said. \"If you cut it correctly, you should see _le c\u0153ur d'un oignon,_ or the heart of the onion. It looks like a Georgia O'Keeffe painting. It's usually easier to peel onions after they've been cut.\" I pulled the papery skin away. \"Now cut a vertical slit down the middle, but don't cut through the root. Make two more slits on either side.\" I picked it up. \"It should look a little like a fan if you pick it up and spread the slits open. Now just slice it across at the end, the way you might normally slice an onion,\" and the cubes of onions tumbled onto the cutting board.\n\n\"Cool!\" Sabra said. The group looked impressed. A few even golfclapped.\n\nSabra, a quick study with a knife, tackled an onion first. I coached her on keeping her neon-green fingernails tucked back. \"Hey, I'm totally getting this!\" Sabra yelled to everyone victoriously, showing off her chopped onion. \"I'm actually _good_ with a knife! Who knew? This is _sooo_ cool.\"\n\nI watched Donna quietly study her onion at the corner of the table. She bought her vegetables frozen and precut to avoid having to chop in front of her husband, who mocked her. I walked around the table and picked up a bit of her chopped onion. \"This looks great; perfect, actually.\" She flashed a quick, proud smile. \"Nice job on your grip, too.\"\n\nMost people got it. Terri struggled and reverted back to a modified choke hold; she made a clumsy knife wielder.\n\nThe nutty smell of saut\u00e9ing zucchini drifted toward the worktable. Months earlier, Lisa had traveled to the south of Italy, where she had a remarkable pasta dish at a donkey farm. She went into the kitchen to pry the recipe from the woman who had cooked it, a professor of ancient studies at the university in Palermo. The woman spoke no English. Lisa's Italian is limited to what she refers to as \"Tarzan Italian.\" The discussion went something along the lines of \"Me want recipe. You give recipe?\"\n\nEventually, she learned that the dish consisted of zucchini slow-cooked in olive oil, then paired with al dente pasta and finished with a lot of salt and pepper. As an extra bonus for that night, Lisa caramelized the chopped onions and added them to the zucchini. At the end of class, we handed out takeout boxes of the Italian-inspired zucchini pasta. Everyone seemed enthused and the volunteers left in a group, chattering excitedly.\n\nWe had reached the end of the first official cooking class, and it had gone without major slip-ups. We'd both been on our feet for at least six straight hours by the end of it, and an hour of cleanup remained. I searched one of the fridges marked with a \"For Catering Use ONLY!\" Post-it and dug out a bottle of pinot grigio. \"You want some?\" I asked Lisa. We feared breaking one of the glasses in the catering inventory, marked with a Post-it screaming, \"Do NOT use any of the wineglasses except for DINNERS!\" So we found a couple of small water glasses and clinked. \"Do you think any of it sunk in?\" I asked her.\n\nShe shrugged. \"We'll find out.\"\n\nIt was time to wrap up. We divvied up the tasks. I pulled out the rolling mop and bucket. She resigned herself to the dish-cleaning area in the back. As part of my first job at age sixteen, I often helped mop the restaurant floors at night. Just as I started to sink into the memory, I noticed something on a side counter. Sabra had left behind her knife block set. I texted her.\n\n\"I won't be needing them,\" Sabra responded.\n\nI debated what that meant. Was she not coming back to the class next week?\n\nThe next day, Sabra sent another text: \"So excited! Got new chef's knife! Full tang, good steel, great feel for $45! See you Monday!\"\n\n#    **Rustic Italian Farmhouse Zucchini \"Sauce\" with Penne**\n\n_This recipe is based on one developed by my friend Lisa Simpson after a meal at a picturesque Sicilian donkey farm. The slow-cooked zucchini takes on a nutty, earthy flavor and the pasta water \"melts\" the vegetables in a sort of thick sauce. This dish is best prepared in a stainless steel or cast iron pan to effectively caramelize the vegetables. Cook the pasta to not quite al dente so that it will finish cooking in the zucchini and absorb more flavor. Brown rice works well in place of pasta. Season the final outcome with liberal doses of coarse salt and ground black pepper. I sometimes add cooked Italian sausage, fresh basil, or pine nuts at the end of cooking. This pasta pairs well with a creamy white wine, such as Chardonnay._\n\n**SERVES 2 TO 3 AS A HEARTY MAIN DISH, 4 AS A SIDE**\n\n3 tablespoons olive oil   \n3 pounds zucchini, chopped into   -inch dice   \n6 ounces dried whole wheat pasta, such as penne   \nCoarse salt and freshly ground black pepper   \n1 cup caramelized onions (optional) (See note below.)\n\n  In a large skillet, heat the olive oil over medium heat. Add the zucchini and toss to coat evenly. Stir frequently over medium heat. Depending on your pan, the heat level, and the size of the dice, it should take 12 to 25 minutes until the zucchini browns and starts to fall apart.\n\n  Meanwhile, add the pasta to boiling salted water. Cook 2 minutes less than the package instructs. Before draining, _carefully_ scoop out 3 cups of the hot pasta water. Add 1 cup to the browned zucchini. Bring to a soft boil, adjusting the heat if necessary, stirring to scrape up any browned bits. Add another half cup of water and repeat the process until the zucchini takes on a thick, almost creamy consistency.\n\n  Add the cooked pasta, at least a couple of pinches of coarse salt, and plenty of freshly ground black pepper to the zucchini. Cook for a couple of minutes, until the pasta softens. If using caramelized onions, stir them in along with the pasta. After removing the skillet from the heat, taste again to see if the dish needs salt or pepper before serving.\n\n_**Note: Caramelized Onions**_\n\nCaramelized onions are an easy way to add a lot of flavor to a dish. I make a big batch and freeze it in 1-cup portions. A large ten-ounce onion cooks down to about a half cup, or five ounces, of caramelized onion. _Basic technique:_ Thinly slice 2 onions. Add 2 tablespoons of olive oil to a large heavy-bottomed skillet over medium-high heat. Cook the onions until softened, then lower the heat to a simmer and gently brown them for about 25 minutes.\nCHAPTER 5\n\n**A Matter of Taste**\n\n  **LESSON HIGHLIGHTS:**\n\n**Why You Must Taste, Taste, Taste**\n\n\"This tastes weird, like a chemical,\" Jodi said, rolling her tongue around her mouth. The rest of the students nodded, murmuring agreement. The subject? Iodized table salt. \"Gosh, it never occurred to me that salt could taste different from one another.\"\n\nI planned to focus on vegetables for the second class, a major culinary blind spot for the volunteers. Yet I could not shake the ongoing references to bland recipes and the sense of inadequacy that I'd heard in regard to their palates. The simple phrase \"Season to taste\" was greeted with the enthusiasm of a tax audit.\n\n\"I hate it when a recipe says that,\" Shannon said when we visited her. \"What does that mean? Whose taste? Mine? How do I know what tastes right?\"\n\n\"I find it daunting,\" Trish said of the equally vague standard recipe phrase \"Check seasonings.\" \"Check for what? I don't know what I'm trying to make it taste like, you know? What if my palate is just way off?\"\n\nOne of the most crucial lessons at Le Cordon Bleu\u2014and for any cook\u2014is the concept of taste, taste, taste. The chefs admonished us to taste or sniff every ingredient _before_ we added it to a dish, and then to taste the dish during the cooking process, and then to taste it yet again before serving. \"How can you tell how your dish will turn out if you don't taste the ingredients you're putting into it?\" a chef would say. If you wait to taste a dish when it's done, it's often too late to fix anything wrong.\n\nOne reason the woman in the supermarket relied on boxed or frozen food was that \"it always turned out.\" Most of the volunteers reported that even careful attention to some recipes yielded bland results. Both stemmed from a fundamental lack of understanding of flavor. Gen had described a recipe for a simple steamed broccoli dish. When I asked if she'd considered adding something extra, say, some lemon or extra black pepper, she shook her head. The recipe didn't say to do it, and she feared \"messing it up.\"\n\n\"Slavish followers of recipes, who treat them as gospel instead of guidelines, make the mistake of putting more faith in someone else's instructions than they do in themselves,\" note Andrew Dornenburg and Karen Page, authors of _The Flavor Bible._ \"Many people would do better in the kitchen if they _didn't_ blindly follow recipes.\"\n\nOne key is to start with good ingredients, understand their affinity with other flavors, and go from there. Some flavors naturally go together, such as basil and tomato, fig and bleu cheese, or even chocolate and peanut butter. The key here would be to figure out a way to convince the volunteers that spending a bit extra for flavorful quality ingredients was worth it. Most said that when it came to food, cost trumped any consideration of flavor, echoing Sabra's comments that most frozen meals tasted the same to her, so she went for the cheapest options.\n\nThis focus on cost versus flavor was not limited to the group or even to this generation. In her 1937 story \"Pity the Blind in Palate,\" M. F. K. Fisher lamented that many Americans shovel in the same fare with dogged regularity, rarely stopping to think about what they are eating. \"We eat, collectively, with a glum urge for food to fill us,\" she wrote. \"We are ignorant of flavor. We are as a nation taste-blind.\" But Fisher acknowledged that if a person sampled a variety of things, stopping to ponder the sensations and subtle differences, then hope existed to awaken the palate.\n\nBut how would I get this point across? As I often do when I've got something to think about, I went to the supermarket.\n\nAt nearly 10:30 P.M., the place was relatively quiet, with few shoppers. The scent of floor wax drifted over to the canned-goods aisle. I started to spend more time in the center aisles, where I had met the woman with the chicken, hoping to run into her again.\n\n\"Mommy, which kind of canned tomatoes do you want?\" I turned, hoping it was her daughter. Instead, it was a preteen with braces in a pink fleece sweater over a pink dance outfit and pink sneakers.\n\nHer mother, staring down at her phone, answered without looking up. \"Whatever's the cheapest, sweetie. It doesn't matter. They're all the same.\"\n\nThe girl tossed two cans of diced store-brand tomatoes into their cart and the pair rolled away. I stood contemplating the wall of tomatoes featuring nineteen different brands that offered them whole, diced, peeled, fire-roasted, organic, imported, and packed with basil.\n\n_Do_ they taste the same? I thought of M. F. K. Fisher's argument and of the chefs who made us taste everything in order to develop what's often referred to as \"taste memory.\" I thought of the book _The Tasting Club_ by Dina Cheney. Essentially, the book suggests comparative tastings of olive oil, cheese, olives, coffees, teas, and the like as a centerpiece for social gatherings. Why not comparatively taste canned tomatoes? I selected nine different varieties of diced and took them in my arms to the checkout. \"Someone likes tomatoes,\" the checker said blithely as he ran them across the scanner.\n\n\"Taste is personal. We all taste very differently,\" Cheney told me later. \"We have different sensory thresholds and different taste memories. Ultimately, it's about finding out what we like and trusting our own palates. That's all that matters.\"\n\nA veteran of more than 150 comparative tastings across the country, Cheney discovered that people are frequently taken aback by how much variety exists within one category of food or drink. In her experience, she found that people remain loyal to one brand or dedicated to the notion of buying the cheapest option, but rarely consider tasting different brands or varieties back to back. \"When people do, they're surprised at how many differences there are between, say, various brands of seventy percent dark chocolate bars or canned tuna. They might realize the brand to which they've been so loyal is actually mediocre and they much prefer another one. That's why tastings often change our buying habits.\"\n\nFor the next week, I studied everything that I could find about the issue of taste and flavor. Taste is physical. Our tongues detect only five options: sweet, sour, bitter, salty, and umami, a savory sense of earthiness equated with soy, meat, and mushrooms. Flavor happens in our brains, determined by what we perceive from our senses. Not everyone tastes the same. Some \"supertasters\" experience certain tastes more intensely than others, particularly salt and bitter. At the other end of the spectrum fall the \"nontasters\" with a dulled ability to distinguish individual tastes. Everyone else falls somewhere in between. Most scientists believe genetics is a factor.\n\nSo our ability to distinguish taste is nature. But preference is nurture; what flavors we like is something we learn. None of this is new to the wine world, where it's long been accepted that you can \"train your palate\" to appreciate minute subtleties in wine. That in turn leads to the confidence of someone declaring he's detected flavors such as \"black currant,\" \"rotted wood,\" or \"gym socks\" after taking a cursory sip from a glass of Cabernet Franc.\n\nPrompted by Cheney's book and my cache of canned tomatoes, I constructed a series of blind tastings with everyday cooking ingredients: olive oil, salt, Parmesan cheese, the tomatoes, and chicken stock. Lisa and I raided our own pantries and then hit a couple of grocery stores. We walked through the aisles and tossed various cans and bottles into the cart.\n\nWe arrived at the teaching kitchen in time to meet up with Lauri Carter, a local chef who had heard about the project and offered to volunteer to teach a lesson on vegetables. Lauri is an energetic petite brunette with deep dimples and a cheerleader-like can-do spirit. I half expect her to break out into a cheer at times, and I mean that in a good way. That summer, Lauri faced her own existential challenge. After four years as the chef and owner of her popular bistro, Moxie, her landlord unexpectedly broke her lease. She found herself covering shifts at a fashionable Spanish wine bar downtown, trying to figure out what to do next. She's smart, a gifted cook, and a good explainer. If anyone could inspire people about vegetables, Lauri could do it. She took the sudden addition of a tasting to her planned vegetable lesson in stride.\n\nAs the students wandered in from one of the first sunny Seattle summer days, they looked around, puzzled. Clusters of small plates took up every open countertop. Lisa and I labeled each with a letter or number. As they put on their aprons and grabbed a diaper for their side towels, we handed each person a small yellow legal pad.\n\n\"We'll explain what these are for once we cover some vegetable basics,\" I said. I introduced Lauri and she took my spot at the center of the worktable. She started first with potatoes: russet, red, white Idaho, Yukon Gold, and even purple. The volunteers examined them raw. \"The purple ones are so pretty!\" Gen said.\n\nTo prepare the potatoes, Lauri went on to demystify a commonly misunderstood term\u2014 _saut\u00e9._ Based on the French word for \"jump,\" _saut\u00e9_ simply means to cook something quickly in a bit of oil at high heat. \"So, where the jump comes in is from the way professional chefs cook. I'll show you.\" The crew huddled around the big six-burner stove. The skillet looked comically large in her small hand. \"First, add some oil to a pan and let it get hot,\" she explained. \"When you saut\u00e9 food, it should sizzle when it hits the pan.\" Lauri tossed in a handful of sliced potatoes that hissed when they hit the hot skillet. She shook the pan as soon as they landed. \"Shake it right away, enough so that the food moves a bit, and then it won't stick during the rest of cooking.\" Lauri added a couple of pinches of coarse salt from a small ramekin by the stove.\n\n\"So notice the salt in the dish,\" she said, holding it overhead. \"Chefs have dishes with salt rather than shakers. You know why?\" The group collectively shook their heads. \"If you shake salt from a shaker, you can't see the salt. It's hard to tell how much you're putting in. But with salt in a little dish, you can literally grab a pinch of salt.\" An \"ahhhh\" went over the group.\n\nLauri tugged at the pan's handle and tossed the potatoes in the air, and they fell back into the pan.\n\n\"Jump. It makes sense now,\" Jodi said.\n\nLauri had each volunteer do at least one shake of the pan. \"You want to cook the vegetables until they start to brown. That's known as caramelizing, which just means the heat has drawn out some of the natural sugar in the vegetables. That's when they start to turn tasty.\"\n\nTo practice their knife skills from the week prior, we then dumped about ten pounds of potatoes on the counter. The students huddled around Sabra as she showed off her new knife, purchased from a restaurant supply store.\n\n\"Oh, that's a smart idea. I didn't think about shopping at a restaurant supply store,\" Jodi said. \"I thought they might card me at the door or something.\"\n\nThe students dived into the task, peeling and chopping the potatoes. Everyone except Terri held their knives correctly. \"Oh, hey, do you want me to show you how to hold a knife again?\" I asked her.\n\n\"No, I just prefer to hold it this way, but thanks,\" she said, with a choke hold on the end of the knife. Huh, I thought. I didn't have much experience teaching. Should I force her to do it my way? I decided to let it go.\n\nSoon a rhythmic chopping and light banter filled the room. Lauri demonstrated how to cut a leek, chopping off the hard dark-green ends and then the stubby root. She cut the light green and white portions lengthwise (sticks) and then diced them (cubes). \"Leeks sometimes trap dirt between their layers,\" she advised. \"So once they're chopped up, we're going to toss them in a big bowl of water to clean them off.\"\n\nThe potatoes and leeks were destined for a classic _potage parmentier,_ potato and leek soup. \"Basically, it's always the same,\" Lauri said. \"We're going to saut\u00e9 the rinsed leeks in some butter and olive oil. Once the leeks soften, we'll toss in potatoes and chicken stock. Once they're soft, we'll puree them and stir in some cream.\"\n\n\"That's it?\" Shannon asked.\n\nLauri nodded. \"That's it. You could do a variation of it with almost anything. You could use leeks and asparagus, or onions and broccoli, shallots and cauliflower, onions and carrots. It's just a simple formula. If you've got kids who think they don't like vegetables, I've found they often will eat them when they're made into soup. They'll also eat more vegetables in the form of a soup than they would raw. It's easier for them to eat and digest.\"\n\nWe went through the basics of other common ways to cook vegetables. The volunteers chopped up cauliflower and Brussels sprouts and shoved them into one big collective pan with some olive oil to roast. We set up four portable burners on the worktable and paired the students together to saut\u00e9 Swiss chard and diced potatoes. The room turned lively. Pans clanked, oil sizzled, people laughed or shrieked, as the smells of the vegetables erupted around the table.\n\nOn Gen and Sabra's corner, some potatoes escaped as they shook the pan. \"Watch out, we're flying potatoes over here!\"\n\nTrish fretted at first. \"Am I doing this right?\" she asked tentatively as Lauri checked her status saut\u00e9ing potatoes. \"I don't usually cook over high heat.\"\n\n\"Looks perfect to me,\" Lauri said. Trish seemed pleased with herself.\n\nI brought in my electric steamer. \"This sounds cheesy, but it's one of the few things I leave on my counter. Simply add some water, pile in some vegetables, and turn on the timer. It's hands-off; you can forget about them until the timer dings. They never overcook and pretty much any vegetable can be steamed.\" I threw in baby artichokes and small cobs of corn.\n\nThen Lauri explained what Julia Child called \"cooking vegetables the French way.\" The method aims to keep the color and bite of vegetables such as asparagus, broccoli, green beans, and peas. We brought a pot of water to a boil on one of the portable burners. I set a bowl of ice water next to it.\n\n\"Briefly cook the vegetables in the boiling water until they're softened but still a little firm,\" Lauri said, dropping a handful of green beans into the pot. After a couple minutes, she pulled them out with a slotted spoon. \"Then plunge them into the ice water. Actually, cold tap water is fine, too. It slows the cooking process, and it keeps the chlorophyll intact. When you keep boiling them, the chlorophyll evaporates and you get . . .\"\n\n\"The gray vegetables that my mother used to serve,\" Trish said.\n\n\"Exactly,\" Lauri replied. She offered a green bean to everyone around the table.\n\n\"These are nice, crisp, and so green,\" Trish said. \"So fresh tasting. If my mother would have served me these, I might have thought I liked vegetables growing up.\"\n\nDone with the vegetables for now, we shifted from cooking to tasting.\n\nI said, \"Okay, all this stuff you see laid out around the room? We're going to taste all of them. There is no right or wrong. Just taste each ingredient and write down your impressions. Compare all the olive oils to one another, then the chicken stock, and so on.\"\n\nAt first, people were tentative. A light dip here, a scribble there, but no talking. Then came a chorus of \"I don't know if this right\" and \"I don't know what I'm supposed to taste\" and \"This just tastes like salt.\"\n\nAfter fifteen minutes, they grew bolder. They started to compare notes. The room grew louder, busy with voices. \"Did you like number six in the olive oils? No? Me neither.\" \"Didn't that chicken stock taste odd?\" \"Yeah, I thought so, too.\"\n\nPeople dodged from the nine types of olive oil to the eight types of chicken stock to the twelve cans of tomatoes, comparing notes. The salt was the hardest, and in retrospect we overdid it by offering nine varieties: standard table salt, kosher salt, sea salt, Himalayan pink salt, French gray sea salt, _fleur de sel,_ black salt, and salt substitute, plus a handcrafted sea salt made from Puget Sound water. The salt tasting led to a lot of water drinking and spitting.\n\nAt one point, Shannon sat down. \"I'm just seriously overwhelmed,\" she said. \"I don't know if I can taste anymore.\"\n\nWe passed around an \"aroma wheel,\" a multicolored circle designed to help wine drinkers articulate what they experience when tasting wine. Soon the students were consulting it after various tastes. This changed the discussions. \"That stock was kind of yeasty, don't you think?\" \"I found those tomatoes astringent.\" \"I'm definitely getting something metallic in this salt.\"\n\nThen we unveiled what had been tasted and compared notes.\n\nWe started with the salt. The expensive artisan salt made from Puget Sound water won best flavor overall. In second, a gray _sel gris_ that a friend had brought back from France. The inexpensive coarse kosher salt was deemed \"exactly what I think salt should taste like,\" Andra said. Collectively, the group disliked two of them. First was the salt labeled \"E.\"\n\nEveryone thought it tasted odd. \"It's just got a strange taste,\" Jodi said. \"It's harsh and like a chemical.\"\n\n\"It tastes too salty but also kind of like metal,\" Sabra said.\n\nThe dreaded E turned out to be standard iodized table salt.\n\n\"Oh, that stuff is awful,\" Shannon said, shaking her head. \"I am not using that anymore. I had no idea that salts could taste so different.\"\n\nThe other loathed salt was option D, which most of them referred to as harsh and bitter. Lisa developed a chemical burn on her tongue after tasting it. \"What I think a car battery would taste like,\" someone wrote.\n\n\"The weird thing is that it doesn't really taste like salt at all,\" said Gen, the young woman with roommates. She'd been relatively quiet up to that point. \"It's like fake salt.\" The culprit: salt substitute.\n\n\"I would not have believed this if I had not seen it for myself, but let me read this line on the label,\" Lisa said. \"Consult a doctor before using this product.\"\n\nNext we looked at the winners in the canned-tomato taste test. Both Hunt's and Cento scored high. \"Those two actually taste like tomatoes,\" Cheryl said. \"They're kind of sweet, but not so salty.\"\n\n\"I'm so relieved,\" Lauri whispered into my ear. \"I used to use Cento tomatoes in my restaurant. I was worried they wouldn't be any good.\"\n\nThe biggest loser was also the most expensive, a brand of San Marzano tomatoes from Italy. I didn't like them either after a direct comparison to the others. \"It's like they had a slightly sour flavor that left a strange aftertaste,\" Shannon said.\n\nIn the olive oil category, the winner was the delicate flavor of an expensive Italian extra virgin olive oil. \"Fruity, subtle,\" Cheryl said, reading her notes.\n\n\"Very delicate, not overly oily,\" Trish said.\n\n\"It's like white grapes,\" Gen said. \"Almost a little bit like honey.\" Lauri and I looked at each other. We tasted it again. Now that she mentioned it, we picked those flavors up, too.\n\nOlive oil is made by pressing olives to extract their natural oil. The first time olives are processed for oil is known as \"extra virgin\" and accounts for about 10 percent of all the olive oil produced. Each subsequent pressing extracts less flavor, from \"extra\" to \"pure\" to just plain olive oil.\n\n\"Use extra-virgin olive oil in something uncooked, like a salad dressing,\" Lauri said. \"It's better to buy good oils in small quantities. People think that oil lasts forever, but it has a shelf life of six months. Keep it in a cool, dark place and _not_ next to or above your stove! Heat breaks it down.\"\n\n\"You can use a less expensive kind of olive or vegetable oil for everyday cooking. If you use that a lot, then you can get a bigger size.\"\n\nUnfortunately, the bulk olive oil that I had bought for class was unpopular. \"It just tastes oily and bland,\" Dri said. \"Funny, I have this kind at home and I never realized that I don't like it. I have a lot of it, too.\"\n\nThe Parmesan cheese comparison required a brief education. \"The real deal is called Parmigiano-Reggiano,\" explained Lisa, who had done this spiel a million times at her mom's cheese shop. \"It's kind of like champagne. You can't call it Parmigiano-Reggiano unless it's from one of five provinces in Italy. To get the name, the cheese makers have to follow very strict guidelines. So that's why everything else is Parmesan.\"\n\nThe difference extends beyond the name. Commercially produced Parmesan is aged a shorter time and due to its production methods can contain up to 70 percent more sodium than the Italian variety. Most of the Parmesan found in supermarkets is mechanically processed to wrest out extra moisture, which extends its shelf life.\n\nThe winner in the category was an aromatic wedge of Parmigiano-Reggiano from Lisa's mother's shop. Shavings of it were stacked up against the pale slices of an American brand and three versions of grated cheese, including the familiar canned variety. Results went as expected.\n\n\"This tastes like soap,\" Sabra said of the exact brand of cheese she had used on her White Trash Garlic Bread. \"But this stuff, it rocks,\" she said, picking up another slice of the Italian cheese.\n\n\"The thing about something like Parmigiano-Reggiano is that, yeah, it's more expensive to buy a piece of that,\" Lisa said. \"But it has so much more flavor that a little goes a long way. In the end, you'll use less, so it's a better value than you think.\"\n\nWe concluded by tasting the various chicken stocks.\n\n\"I always just thought chicken stock was chicken stock,\" Shannon said, verbalizing what several had murmured during the class. \"There's so much difference, it's amazing.\" Dri described one as a \"salt lick.\" Others were not \"chicken-y.\"\n\nOne had a strange flavor. \"It's like how orange juice is so orange-y,\" Gen said. \"It's almost as if it is chicken stock made from concentrate. It's also weirdly salty.\" That one turned out to be stock made from a bouillon cube.\n\nOnly one person favored my lovingly handcrafted homemade stock. \"It's nice and chicken-y, and it has a good body to it, but it's bland,\" Trish said.\n\nAs usual, I hadn't added any salt to it. They were comparing unsalted homemade stock against mostly heavily salted prepared stocks, some of which had more than a third of an adult's daily sodium intake in one cup. Swanson and Pacific got the thumbs-up, and after a dose of salt was added to the homemade stock, the group preferred that one, too.\n\n\"I guess the thing is that after tasting them all, I'd only buy ones with lower sodium,\" said Trish, her taste buds sensitive to salt because she avoided it. \"I use one of these brands and, my goodness! I never realized just how _salty_ it was until I actually tasted it here tonight.\"\n\nEveryone nodded. \"One thing to remember, you can always add salt, but you can't take it out,\" I said. \"This is also why you should taste things before you cook with them. Taste the olive oil you're using before making a dressing. Sample a bit of the cheese before you add it to pasta. Try a bit of that chicken stock before you put it into a soup.\"\n\nWith that, Lauri and Lisa unveiled the final test of the night. In the middle of the table sat five bowls of leek and potato soup. Three were the newly crafted _potage parmentier_ split into different bowls, one with no salt, one mightily oversalted, and the final bowl seasoned to Lauri's taste. The other two bowls also contained leek and potato soup, one from a condensed-soup can while the other was an expensive dehydrated \"gourmet\" version.\n\n\"Select the one you think tastes best,\" Lauri challenged. Everyone selected the moderately seasoned soup. \"What's wrong with soup A?\" she asked. Too much salt, everyone agreed. \"What's wrong with D?\" Not enough salt, the group said. \"So now when you see a recipe that says 'Salt to taste,' that's all it means. If it doesn't have enough salt, like D, then add some. Just make it taste good to you.\"\n\nThe group pondered the remaining two bowls.\n\n\"Ugh, I can taste the iodized salt in both of these,\" Jodi said.\n\n\"Yeah, how weird, me, too,\" Shannon replied.\n\nThe canned soup generated a range of intriguing descriptions: \"strange mouthfeel,\" \"feels fatty on the back of my tongue,\" \"strange aftertaste,\" \"not particularly leek-y or potato-y,\" and just plain \"yuck.\" The dehydrated version fared better, but was still described as \"too salty,\" \"tastes like chemicals,\" \"odd spices,\" and\u2014my favorite\u2014\"kind of like a bowl of liquid blah.\"\n\nAs she took off her apron, Jodi quietly admitted to Shannon that she had a case of the canned soup back at home. \"If I had known it was so easy to make, I wouldn't have bought it, you know?\"\n\n#    **Five Marvelous Ways to Cook Vegetables with \"Flavor Splashes\"**\n\n_You just need to know the meaning of four words:_ saut\u00e9, steam, roast, _and_ grill _. Vegetables are done when they are tender but still crisp. Err on the side of undercooking. You can also toss in a vinaigrette from chapter 8 or a flavor splash (below), or toss with a handful of grated cheese and broil for a minute or two. There's nothing wrong with plain frozen vegetables. Flash-frozen on harvest, they're often as fresh as or fresher than their produce-aisle cousins. Frozen vegetables do best saut\u00e9ed or steamed._\n\n_**Saut\u00e9 or Stir-fry**_\n\n  Virtually all vegetables lend themselves to being quickly cooked over high heat in a bit of fat. Cut them into   -inch pieces, get a skillet hot, and add 1 or 2 glugs of vegetable, olive, peanut, canola, or coconut oil. Add the vegetables and stir a few times. Most take less than 10 minutes. Firm vegetables, such as potatoes and carrots, take the longest, up to about 15 minutes. Cook greens such as Swiss chard and spinach with a bit of chopped garlic until wilted. Finish with salt and pepper. If you mix firm vegetables with softer ones, be sure to add the harder ones first.\n\n_**Roast**_\n\n  This is simple yet yields great flavor. Line an oven-safe pan with aluminum foil or parchment paper. Cut the vegetables into uniform 1-inch or smaller pieces. Toss them generously with oil or butter, and add salt, pepper, and maybe a sprig of thyme. Roast at high heat (475\u00b0F) and stir a couple of times while cooking. Once diced, sliced, or otherwise broken down into smaller pieces, the following vegetables will cook in less than 15 minutes: asparagus, beets, bell peppers, Brussels sprouts, carrots, cauliflower, eggplant, green beans, leeks, mushrooms, parsnips, white and sweet potatoes, diced squash (hard outer skin discarded), and zucchini.\n\n_**Steam**_\n\n  This is a no-fat way to cook utilizing a multitude of setups. Cut up larger vegetables into 1-inch or smaller cubes. Use an electric steamer, or add vegetables to a pan with a tight-fitting lid filled with an inch or two of water, or simply microwave a small batch of vegetables with a bit of water in a bowl covered with a plate. Cook until just tender. Add salt and pepper after steaming. This method is recommended for asparagus, green beans, broccoli, Brussels sprouts, cabbage, carrots, cauliflower, corn, kale, kohlrabi, onions, parsnips, peas, potatoes, spinach, diced squash, Swiss chard, cubed turnips, and zucchini.\n\n_**Grill**_\n\n  If you grill often, invest in one of those square grill pans with holes. Cut vegetables into 1-inch pieces, and toss them with generous doses of oil, salt, and pepper. Put in the basket over hot coals, cover, and stir regularly until they are tender.\n\n##  **FLAVOR SPLASHES**\n\nJust mix the ingredients together in a small saucepan, heat briefly, pour onto vegetables, and toss to coat. Each recipe coats enough for about four side servings of vegetables. (For more ideas, see the \"Cheat Sheet\" to Flavor Profiles in the Extra Recipes section at the back of the book.)\n\n_**Asian Ginger Lime**_\n\n  Warm 2 teaspoons of sesame oil, then add 1 teaspoon of fresh grated ginger or a couple of pinches of dried ginger, a few squeezes of fresh lime juice, and   teaspoon of soy sauce. Heat through for about 3 minutes.\n\n_**Cajun Oil**_\n\n  Warm 2 teaspoons of olive oil, add 3 finely chopped green onions, and cook until tender, then add 1 teaspoon of Cajun spice blend, a few squeezes of fresh lemon juice, and a couple of drops of hot sauce.\n\n_**Garlic Citrus Butter**_\n\n  Heat 2 tablespoons of butter, add 2 small cloves of minced garlic, a bit of fresh thyme or mixed dried herbs, and 1 teaspoon of lemon or orange juice, and saut\u00e9 for a couple of minutes, until the garlic softens.\n\n_**Herb-Lemon Oil**_\n\n  Warm 2 teaspoons of olive oil, then add the zest of   lemon and 1 tablespoon of chopped fresh herbs such as rosemary, oregano, tarragon, thyme, or basil. Heat through.\n\n_**Thai-Style**_\n\n  Heat 2 teaspoons of peanut oil, then add 2 finely chopped green onions, 1 tablespoon of crushed peanuts, a couple of squeezes from a fresh lime, and a bit of hot sauce. Gently heat through.\nCHAPTER 6\n\n**Fowl Play**\n\n  **LESSON HIGHLIGHTS:**\n\n**The Value of Learning How to Use a Whole Chicken**\n\n\"If there's one skill that I think people need to learn, it's how to cut up a whole chicken,\" said Rick Rodgers, the author of thirty-five cookbooks. \"Roast chicken is iconic, we all love it and that's a great skill to learn, but being able to do something with all the various parts of chicken is something even more people need to know in terms of how it fits in with their daily life.\"\n\nKnowing your way around a chicken is a valuable thing to learn. On average, a whole chicken costs about the same as a package of boneless skinless chicken breasts, whether it's a standard supermarket chicken or a more expensive organic variety. Used efficiently, a single chicken can provide the goods for two or three meals.\n\nThe breast can be left on the bone and baked, a boneless breast can be quickly saut\u00e9ed, the thigh meat can be cut up for a stir-fry, the legs oven-fried, and the wings collected and frozen for snacks. If you decide to roast the whole bird, the meat can be used in a seemingly endless parade of dishes: salads, pasta, burritos, casseroles, sandwiches, and so on\u2014virtually any recipe that calls for cooked chicken. As Rodgers notes, you get not only the eight chicken pieces but all the bones, the back, and the giblets, too. Given that popular brands of stock average about $2.50 per quart and remnant bones from just one chicken can yield up to two quarts, one chicken can provide about $5.00 worth of stock.\n\nIn culinary terms, it's hard to beat the importance of chicken. It's the number one search term on recipe sites such as Epicurious.com, Allrecipes.com, and Foodista.com. On Google, worldwide traffic for \"chicken recipes\" dwarfs requests for beef, fish, or vegetarian recipes. Americans consume an average of 60 pounds of chicken annually, edging out beef as the nation's preferred meat. American chicken growers process 38 million chickens daily\u2014one for every man, woman, and child in the state of California every single day. That's 43 billion pounds each year, says the American Meat Institute, equivalent to the weight of 860 luxury cruise ships.\n\nThe students arrived, donned their aprons, grabbed a diaper, and selected a chef's knife from the box before heading to the worktable. They were getting to know one another and chatted amiably. I overheard snippets of small talk.\n\n\"I threw out all of my iodized salt. I had two of them . . .\"\n\n\"Yeah, me, too.\"\n\n\"I went through my cupboards and found most of my oils were rancid . . .\"\n\n\"I made kale, and my boyfriend was like, 'This is so good! Where did you get the idea to make _kale_?' \"\n\nHelping out that day was my friend Maggie, a thirtyish streetwise Sicilian American beauty from Chicago with jet-black hair and dragonfly tattoos cascading down each arm. A kitchen veteran turned culinary consultant, Maggie agreed to assist with some of the more complicated classes. Just as the students were filing in, she arrived, aggravated from a day at her cupcake client. The small-baked-goods industry appears to be a surprisingly ruthless affair. Her client was locked in a battle for cupcake dominance with an arch competitor and the stress had prompted cutthroat internal politics. That day, she had had to snap at someone, \"Does this kind of attitude belong around cupcakes!? I don't think so.\"\n\nBuoyed by the apparent success of the initial tasting evening, I decided to start each class with one. Today we set out small plates featuring five varieties of Dijon mustard. Now they knew the drill. Each person daintily dabbed a bit of mustard onto a small appetizer plate with a tiny espresso spoon. As each jotted down notes on a small yellow pad, they made thoughtful faces. We talked about flavors and words to describe them. The most common: _sour, spicy, bitter, smooth_. No one liked the cheapest store brand. \"It's like chalk,\" Gen said.\n\nGrey Poupon was the easy favorite, even beating out Maille, a pricey classic French import. \"You can taste the white wine in both, though,\" Jodi said. An expensive organic brand was written off as bland. Shannon summed up the group's general thoughts. \"The more that I taste, the more it makes me want to pay attention to what I used to just take for granted.\"\n\nAlthough it was past seven P.M., the warmth of the late-June day remained trapped in the room, intensified by the heat of the commercial ovens. Cars buzzed by outside the door, which was propped open in a futile attempt to draw in fresh air. Before we moved on to taking apart the chickens, I rounded up the group around the worktable set with a patchwork of colored cutting boards. Maggie passed around glasses of ice water as we talked chicken.\n\n\"Okay, I have a few questions. Have you ever cut up a chicken?\" Universal shakes of the head.\n\n\"Do you know what _braise_ means?\" One person ventured a guess.\n\nI asked if anyone had ever roasted a whole chicken. Only Trish raised her hand. \"I have, but I've mostly bought them that way,\" she said.\n\nI lined up four chickens, \u00e0 la Julia Child in a famous episode of her 1960s TV show, _The French Chef_. In that scene, she described the differences between a broiler, fryer, roaster, capon, stewer, and \"Old Madam Hen.\" Instead of different ages, I presented differently raised chickens, all broilers weighing three to four pounds. Two were supermarket chickens, a commercially raised standard variety and a free-range bird that cost just less than a dollar more per pound. The third was a certified organic free-range chicken from a butcher. The last was a pasture-raised chicken purchased directly from a local farmer.\n\nThe supermarket chicken felt wet, and the breast was noticeably heavier than that of the rest of the chickens. The organic chicken from the butcher and the pasture-raised chicken, which didn't come prepackaged, felt dry. When I sat all four up as if they were lounging upright, the supermarket and the free-range chickens fell forward, propelled by the weight of their overbuilt breasts. I sliced a breast off each one. I held up the largest, from the least expensive supermarket chicken.\n\n\"How much do you think this weighs?\" They all ventured a guess: six ounces, eight.\n\n\"How many servings do you think this is?\" Everyone agreed that it was one serving.\n\nCheryl nuzzled Liam in his baby carrier. \"A half a serving in my house. Yeah, Daddy could eat two of those, couldn't he, sweet pea?\"\n\n\"How much do you think a serving should weigh?\"\n\nDonna, knower of all things Weight Watchers, piped up. \"About four ounces,\" she trilled brightly. \"A chicken breast is three Weight Watchers points.\"\n\nI put the breast on a scale. It tipped in at more than a pound. \"So this one breast ought to provide four servings.\" More than two-thirds of the weight of this chicken came from the breast alone. The breast from the pasture-raised bird from the butcher weighed the least, about nine ounces.\n\nAs Michael Pollan noted in his book _The Omnivore's Dilemma,_ commercial poultry growers have researched how to engineer chickens through breeding to grow significantly greater portions of breast meat. From a monetary point of view, it makes perfect fiscal sense. Boneless, skinless chicken breasts fetch as much as six dollars per pound\u2014roughly five times the retail price for whole chickens and at least twice as much as the other portions of the bird. The downside? To maximize profits chickens are often confined to massive barracks where they are provided an endless supply of feed around the clock but given little exercise. As depicted in the documentary _Food Inc.,_ some birds grow so big so fast, they ultimately cannot support themselves under the weight of their mighty breasts and fall down. Detractors of the industrial poultry process say that the system ratchets up the stress level for the chickens and breaks down their immune systems, thus requiring mass poultry growers to use a lot of antibiotics.\n\nBy comparison, organic farmers aren't supposed to give their chickens any antibiotics or feed containing pesticides.\n\n\"So what does it mean when the package says 'all natural,' like this guy here?\" Dri asked, pointing to a supermarket chicken.\n\n\"In theory, it means that the chicken doesn't have any artificial flavor, but in practice it's a little meaningless,\" I said. Chickens are natural products, just like beef or pork. They don't make synthetic chickens, at least not yet. \"With some companies, free-range is kind of similar, since the USDA requires only that the chicken have access to a door should it want to go outside, but it doesn't necessarily mean that it ranged anywhere, free or not.\" The only way you can know how chickens are raised is to know the poultry grower, or if your butcher knows the grower. \"Some free-range chicken growers really make an effort to treat the chickens humanely. The farm where this guy was raised,\" I said, patting the pasture-raised chicken, \"they make a point to physically transport the chickens to a grassy grazing area. But he's also triple the cost of a standard supermarket chicken.\"\n\nA small puddle had formed under the first chicken. Dri pointed. \"Um, did that one need the ladies' room?\"\n\n\"That's probably saline.\" Producers sometimes inject chickens with water or brine. It keeps the meat moist, plus it adds more weight, adding cost, even though it's water, not meat.\n\nA whisper went through the group. \"That's bullshit,\" Sabra said. \"They shot it up with water so I could pay more?\" She shifted from one foot to the other and crossed her arms. She looked ready to kick some commercial-chicken-growing butt.\n\n\"Listen, I am not going to say that you should never buy a supermarket chicken because that's just not realistic. But look up the poultry grower at the supermarket where you shop. Ask questions of the butcher in your market. If you don't want your chicken injected with water, tell them. If consumers care about something, it changes the kind of products offered. I try to buy from my butcher because I trust that they know their suppliers, but I don't always do that. Remember, if you're buying something cheap, but then you don't use it all, then it's not quite the good deal it seemed when you bought it.\"\n\nI didn't want this class to just be about _cooking_ chickens. One of the things that I had come across in our kitchen visits reinforced what researchers already know. Most people don't associate chicken, especially in the form of boneless, skinless chicken breasts\u2014with a physical animal. In three kitchens, we found packages of chicken meat well past their sell-by dates in the fridge. \"Oh, no, I forgot about those,\" one of the volunteers had said of a family-sized package of chicken breasts. She picked them up to look at the date and shrugged, then tossed it into the trash. \"I hate wasting money like that.\"\n\nChickens are living creatures, a lesson that I learned early in life growing up on a farm in Michigan that has never left me. I told the volunteers the story about it.\n\n\"On the farm where I grew up, we kept chickens,\" I started. A slightly askew chicken coop graced the far corner of the barnyard, and shortly after buying the house in late winter, my folks went to get some chicks from the local farm-supply store. The weathered old man who owned the place suggested that they purchase 125 chicks to start. \"And I'll tell you what\"\u2014he rolled a piece of hay around with his dark teeth. \"You buy that 125, I'll give you 125 for free.\"\n\nAn hour later, they packed five crates with 250 wildly cheeping chicks into the back of Dad's battered Chevy pickup. My brothers and sisters loved the chicks, but lost interest in them after about a month. About the same time, my parents began to realize that the chicks had been a great deal\u2014for the guy who sold them. He made his real money off chicken feed. My parents stopped by to purchase massive bags every few days to satisfy the voracious chickens. The adorable yellow balls of fluff grew at a startling pace. As adults, they turned into a massive and noisy gaggle of feathers that moved as a single wavering blanket of white.\n\nAfter a couple of months, panic set in. \"We were going broke feeding them,\" my mom told me later. They traded some chickens for seed to start the garden and gave others away. It didn't put a dent in the flock.\n\nThe day after school let out for the summer, Mom rounded up the kids. She set up the chickens into a production line in the barn. A teenage neighbor caught the birds, and then held them down as she whacked off their heads, gutted them. Then she handed the carcasses to my oldest brother, Milton, to plunge into her huge canning pot filled with boiling water to loosen the feathers. He was only eleven years old. The other kids sat around for days plucking, the feathers sticking to their sweaty legs and arms in the Michigan heat. In one day, they killed and gutted twenty chickens. They would take a three-day break, and then fell another twenty more days later. Within two weeks, they killed one hundred and twenty chickens.\n\nWhen you've got that many birds in the freezer, you make a lot of chicken. Roasted, braised, saut\u00e9ed, ground up for spaghetti in place of hamburger\u2014Mom did it all. But one thing she always did was avoid waste. \"Oh, no, not after all that work. And besides, I could remember when that chicken was still walking around. I would feel too guilty,\" she said.\n\n\"When you're starting with a whole chicken, it helps to remember something important\u2014that this was a live animal. For a lot of people, knowing that makes it harder to throw parts of it away.\" The group nodded somberly. That was a buzz kill, I thought.\n\nDiscussion over, we got to work. Lisa pulled a huge plastic bin heaped with chickens from the walk-in. She set one in front of each student.\n\n\"First, we'll break them down,\" I said, glancing at the clock. \"Do this once or twice and it gets easier.\" I could sense the resistance. \"Go ahead, touch it. Don't be afraid.\"\n\nThe volunteers stood in a semicircle around the worktable, apprehensively eyeing the whole chickens resting on their cutting boards. Sweet Donna pulled back and curled her hands against her chest. Dri tentatively poked hers with an index finger. Even Cheryl's eightmonth-old, Liam, stared at the chicken suspiciously from the safety of his Bj\u00f6rn carrier. \"C'mon, you're not going to get salmonella from touching it. If you're going to cut up a chicken, you're going to have to put your hands on it.\"\n\nSabra went first, caressing the pale skin of the breast with neon-green fingernails that matched her eye shadow du jour. \"It kind of reminds me of Thanksgiving, you know, when you've got the whole turkey,\" she said. \"But instead, it's a chicken. Like a little turkey. Kind of sweet, really. I've never felt a whole, raw chicken before.\"\n\nIt's not surprising that none of them had ever handled a whole chicken. When Julia Child debuted in the early 1960s, shoppers purchased more than half of all retail chickens whole. Now it's around 10 percent, and that's with an uptick in sales in the midst of a recession. As the students stared at their chickens without touching them and the clock ticked down, I worried that perhaps the night's agenda was a little too ambitious.\n\nI started by holding up one of the legs and pulling it away from the body. Then I turned it upside down, and the leg splayed away. Using my knife, I cut around the \"oyster,\" the soft, tender round of meat tucked at the top of the thigh joined at the back. As I cleaved the leg from the body, I explained that the French think the oyster is the best part of the chicken. \"They even have a phrase for this, _le sot-l'y-laisse,_ which means 'only a fool leaves it behind.' \" Then I slid the tip of the knife along the breast bone and carefully sliced against the rib cage until the breast fell free. I felt for the shoulder joint that pins the wing to the bird and located the soft spot to push the knife through.\n\nOn the other side, I put my blade atop the sternum and leaned with my weight on top to crack it open, leaving the bone attached to the breast meat. With pressure on my knife, I cut through the rib bones on the side of the bird around the breast until it detached. The wing came off the same way, and thunked solidly onto the cutting board.\n\nNow it was their turn. Brow furrowed, knife in hand, each volunteer gamely attempted to cut up her chicken. Donna held the wings ever so daintily, trying to minimize her contact. To my left, Gen cut the leg right off, perfectly. \"I did it! Look!\"\n\nElsewhere around the table there were varying levels of success. Maggie, Lisa, and I circled the table offering advice. Terri, the one most resistant to the ways of a knife, started to saw with vigor into the backbone. Trish tried to hack through the leg bone, missing the joint completely.\n\n\"Okay, wait. If you find a bone, stop. Take a deep breath. Pull your knife back. You shouldn't get that much resistance if you're doing it right.\"\n\nIt's hard to teach what's become second nature. How do you articulate the feeling of knowing you've hit the joint at the right angle, so it comes off easily? Then I had a morbid idea.\n\n\"Touch your knuckle,\" I commanded. Everyone touched their first knuckle with their other hand. \"Now feel down at the leg joint on your chicken. Can you find a similar knuckle? Feel around for it.\"\n\nTerri poked around the leg joint until she found it. \"Now take your knife and cut through it.\" She hit the joint and it fell away easily. I observed as Donna followed the curve of the leg and in one quick motion separated it. Without realizing it, she let out a quick squeal of delight and threw her hands overhead in a cheer. \"I did it!\" Then she looked around and pulled her arms back down, embarrassed.\n\nLegs separated, we shifted to the breast area. The sound of cracking filled the room as the volunteers cut through the bones. Lisa brought over a big bowl for each cut of meat and a plastic container for the mound of remnant bones.\n\nKnives down, Maggie whisked the cutting boards to the dish area and the students rotated through the hand-washing station, merrily belting out various renditions of \"Happy Birthday.\" I eyed the clock; half an hour behind already and we hadn't even started cooking. The goal for the night was to demonstrate roasting, baking, saut\u00e9ing, grilling, and braising.\n\n\"First, we'll roast a whole chicken and a few individual pieces. You'll want some kind of fat, maybe butter, olive oil, or sesame oil, plus some seasonings. A little bit of acid is nice, too, maybe some lemon, lime, vinegar, or white wine.\"\n\nTo demonstrate, I took the organic chicken and set it on my own cutting board. Starting with the breast area, I carefully stretched my fingers between the skin and the meat to loosen the membrane between them. In the cavity created, I rubbed in a mix of chopped garlic, fresh herbs, and olive oil, then topped the skin with salt, freshly ground black pepper, and a bit of cayenne pepper and rubbed it in. To finish, I wedged pieces of lemon into the center cavity along with two cloves of garlic.\n\n\"Forget the idea that you need a lot of complicated cooking equipment,\" I said. \"All you need is an ovenproof pan with sides at least one inch high that is not too much bigger than the chicken. You can even use a skillet, as long as it can go in the oven.\" A roasting rack can make life easier, but it's not necessary. In this case, we used a simple rectangular pan from the catering supplies. I scattered a few hunks of carrots, celery, and onions on the bottom and then perched the chicken on top. \"The vegetables should elevate the chicken a bit, allowing fat to drain off. Plus, they keep it from sticking.\"\n\nDri raised her hand. \"Okay, so it seems like Julia Child was always trussing a chicken. It was all truss, truss, truss.\"\n\nEveryone laughed. \"You know, at Le Cordon Bleu, we learned a complicated way to truss that stitched the whole thing up like a tight little football. But you don't need to do anything, really. I like to at least tie the feet together to help it keep its shape, but it's not mandatory.\"\n\nThere's heated debate among chefs around the \"best\" roasting technique. I keep it simple. \"Put it in at high heat, about 425 degrees. When it occurs to me, I turn it over after forty minutes to brown the other side, but if you're not so keen on getting tongs out to turn over a hot chicken, by all means leave it alone. Lisa doesn't turn hers, do you?\" Lisa shook her head. \"See? Some cookbooks advise roasting at 350 degrees, others at 400 degrees. This isn't science, it's more of an art. After an hour, check it with an instant-read thermometer to see if it registers close to 175 degrees, or if the juice that runs off when you pick it up looks clear. Pink? Put it back. Clear? Take it out. Let it rest for a few minutes, lightly covered with foil.\"\n\nMeat will continue to cook for several minutes after leaving the oven. It's called \"carryover cooking.\" Sitting also allows the juices to reabsorb into the meat.\n\n\"What if you undercook poultry?\" Shannon asked, bringing up something we had talked about in her kitchen. \"I am always worried about it, so I just cook it and cook it and then cook it some more. I think that's why mine always turns out so badly. I mean, you're supposed to cook chicken pretty much to death, right?\"\n\n\"Suggested cooking temperatures tend to be higher than what's really needed to kill off any latent bacteria,\" I said. \"Cooking it past that point isn't going to do anything. It's kind of like trying to get more pregnant. If you're really worried about undercooking chicken, just spend a few dollars and get a meat thermometer. It will take out the guesswork.\"\n\nA blast of hot air hit the worktable as I shoved the chicken into the upright oven to roast.\n\nNext, I grabbed a whole chicken breast with the rib bone still attached and set it on my cutting board. \"Again, it's what gets put under the skin, just like a whole roast chicken. You need a bit of fat and a bit of flavor.\" As a demonstration, we used a recipe from Ina Garten of _Barefoot Contesssa_ fame that involved smearing a bit of goat cheese under the skin and then sliding in a couple of whole basil leaves. I drizzled the top with olive oil, coarse salt, and pepper. Everyone did the same, and we put all the breasts onto a sheet pan topped with parchment. We put them into the kitchen's other oven at 375\u00b0F. \"These will stay in for about a half hour.\"\n\nWe moved on. \" _Braise_ simply means cooking something covered at a low heat with a bit of liquid.\" Lisa and I started with the big bowl of thighs and legs. We had settled on a classic mustard chicken dish, utilizing some of the Dijon from the tasting.\n\n\"Dark meat is great for braises because it can take long, slow cooking,\" I said. The volunteers gathered around the commercial stove as Lisa added oil to a large pot and waited for it to heat. \"First, we're going to brown the meat. Getting a nice, dark color makes a big difference. What you want is a hot pan with hot oil,\" I said as Lisa set up a large pan to demonstrate. \"Then add your food and give the pan a shake, just like Lauri demonstrated last week. That's how you keep food from sticking.\"\n\nLisa put a thigh into the hot oil with long tongs. As if in protest, the pan crackled with a loud persistent sizzle. I added a few more pieces.\n\n\"Do you hear that?\" I asked. This was an important lesson. \"Don't cook just with your eyes. Teach yourself to cook with your ears. Listen. It's loud, it's sizzling and sounds sort of angry. That's what you want. Most home cooks are afraid of high heat. Don't be. Smell this chicken right now.\" Lisa waved them closer. \"Seriously, get close and take a whiff. What does it smell like?\"\n\nDonna, Shannon, Terri, and Sabra leaned close to the pan and wrinkled their noses. In unison, they said, \"Hot oil.\"\n\n\"Exactly. When it starts to smell like chicken, then you're getting somewhere.\" We shifted back over to the worktable, where Maggie had replaced the cutting boards and set out big bowls of onions, carrots, and celery. Lisa stayed at the stove to tend to the chicken as we moved on to chopping the vegetables. Only two weeks after the knife class, the volunteers grabbed onions, sliced them in half, peeled, and chopped as if they had been working a restaurant line for ages. Only Terri struggled. Maggie tried to help her again with holding a knife. Terri nodded and thanked her. When I checked back after a few minutes, she had reverted again to her former hold. Was this willful resistance or a deep need of remedial instruction? I didn't know. It troubled me.\n\nI asked if anyone was practicing at home. \"Oh, yeah! Did I tell you that I got my knife sharpened? It's so great. It goes so much faster now,\" Shannon said.\n\n\"I haven't bought baby carrots since that class,\" Jodi said proudly, without even glancing up from her cutting board. \"My husband likes carrots in stir-fry and my son will eat them now, so I am going through about two or three pounds of carrots a week. So instead of paying two dollars a pound, I'm paying sixty cents or something. It saves a lot.\"\n\nI walked to the end of the table where Donna was chopping. During Donna's home visit, she had said that her husband often mocked her when she used a knife. In their relationship, food seemed like a feast of control issues. I noticed that she'd brought in her own knife today. I casually asked how it was going at home.\n\n\"My husband and I made dinner the other night and guess what? I cut up all the vegetables.\" She looked up and smiled, dimples flaring. \"Now that I've been coming here, I can say, Hey, I know what I'm doing. Leave me alone.\" She chopped silently for a minute. \"I told him that I was going to start shopping.\"\n\n\"How did that go?\" I ask.\n\nShe kept her high-pitched voice low. \"We're still discussing it.\" Her face flushed crimson. Something in Donna brought out my maternal instincts. I wanted to hug her, but I settled for patting her shoulder instead.\n\nAs we finished the last of the vegetables, a powerful smell of chicken hit the worktable. The group looked up, as if they could see the scent hovering overhead. \"The chicken . . .\" Shannon said, thinking out loud.\n\nWe gathered around the stove as Lisa pulled out a piece with the tongs. After ten minutes, it had taken on a dark brown hue, the skin a bit shrunken and crinkly.\n\nCheryl raised her hand. The baby looked up. \"So does that seal in the juices, when you brown it like that?\"\n\n\"A culinary myth,\" I explained. \"Heat actually releases juices inside the meat, not seals them in. You brown the meat to caramelize the exterior of the meat, drawing out sugars and extra flavor in the process.\"\n\n\"Or, in other words, browning equals delicious goodness,\" Lisa said, and turned to put the thigh back into the pot.\n\n\"Exactly,\" I said. \"Braising is always the same. First, you brown. Add some vegetables and enough liquid to cover the meat halfway, cover the pan, and simmer. That's it. Brown, vegetables, liquid, cover, and simmer. Learn this one technique, five words, and you can cook almost anything.\"\n\nI removed the chicken to a bowl and tossed the onions, celery, and carrot that we'd just chopped into the pot. While they softened over the heat, we returned to the worktable. Each volunteer brushed a browned chicken piece with some Dijon mustard. We put them back into the pan and added some chicken stock, a few herbs, a bit of white wine. I covered the pot and set it inside the oven to simmer.\n\n\"The biggest problem with cooking chicken breasts is that while the poultry industry strives for a bigger breast, it hasn't developed them into a uniform shape or thickness. So they tend to be much thicker in the middle and thin on the ends.\n\n\"Here's a trick that I do at home.\" I laid a boneless breast on my cutting board, then sliced it through the center horizontally, resulting in two thinner fillets. I held them up. \"Now the chicken breast is half as thick and more uniform, so it'll cook more evenly. Not to mention, it's a more realistic serving size.\"\n\n\"That's a great tip,\" Shannon said. \"Usually the center isn't cooked all the way through while the edges are overcooked and tough.\"\n\nTo give breasts a bit of flavor, I showed them a lesson that I learned years ago from an aging Italian grandmother during a cooking class outside Florence. \"She called it a _'bacio di sapore,'_ or a kiss of flavor.\" I combined a bit of olive oil, a squeeze of lemon juice, salt, pepper, and dried thyme in a bowl. I added my two slices of breast meat and mixed them together. \"Let this sit for a few minutes. Then it's onto the heat.\"\n\nWe circled around the stove. I grabbed a small saut\u00e9 pan and added oil. Once it was hot, I slapped the breasts into the pan with a pair of tongs, was greeted by a hearty sizzle, and gave the pan a shake. I turned them after three minutes. After three more minutes, I added a hearty splash of stock, turned down the heat, and covered the pan for a few more minutes. \"At a restaurant where I once worked, the chef added some liquid like stock or wine at the end of cooking and I do that all the time now,\" I said. \"It infuses it with both flavor and moisture.\"\n\nMaggie handed out forks for everyone to take a bite. \"This is great,\" Shannon said, using the edge of her fork to cut off another bite. \"I mean, it tastes good, it's not dry. Mine never turn out like this.\"\n\nTrish had also commented on tough chicken during our visit. \"Neither do mine. I wonder if it's because I usually cook them at lower heat and longer,\" she mused. \"Well, a lot longer.\"\n\n\"Now it's your turn,\" I said. While I'd been cooking, Maggie and Lisa had assembled a collection of oils, vinegars, garlic, herbs, and spices from around the kitchen. From the walk-in, they had gathered jars of pesto, sun-dried tomatoes, gingerroot, lemons, and limes. The selection now sat as a jumble in the center of the worktable. Each person had a small bowl.\n\n\"First, you'll slice a breast in half like I did. One part we'll saut\u00e9, the other we'll grill,\" I said. Maggie walked around the table distributing chicken breasts from the chickens we had broken down earlier. \"Then make your own kiss of flavor. Think of what chefs call 'flavor profiles.' These sound all chef-y, but they're easy. For instance, what flavors does Asian food have in common?\"\n\nJodi raised her hand. \"Sesame oil, soy sauce, maybe rice wine vinegar?\"\n\n\"Great,\" I said. \"Put those together. What does Italian food taste like?\"\n\nDonna ventured a guess. \"Olive oil, Italian herbs, maybe pesto?\"\n\nAnd so it went. Caribbean? Jerk seasoning, hot sauce, limes, a bit of coconut oil. Tex-Mex? Corn oil, chili powder, garlic, and cumin with a bit of lime. The group erupted briefly in unmanageable enthusiasm. \"This is actually fun!\" Jodi said. \"I mean, I can do this!\"\n\nOne key is to evaluate the flavor of each ingredient. \"Smell or taste the oil and the vinegar, try a bit of the spice. As you taste each one, think how it will interact with the others you're using.\"\n\nThe volunteers started to pick up various bottles and jars, sniffing and sometimes tasting with one of the dozens of small spoons set in a plastic shoebox on the counter. A couple looked tentative, watching the others for clues. The three of us wandered around the table offering suggestions until each volunteer came up with a flavor kiss in which to briefly bathe her two chicken breast halves. Maggie put small bits of masking tape with their names on dinner plates. As each person finished saut\u00e9ing one of her breast halves, it went on her plate. Then they moved to the gas grill adjacent to the stove.\n\n\"The thing to remember about grilling chicken breasts is to avoid placing them directly over a flame. The meat's too fragile. An easy trick for grilling chicken is to cover it with a metal bowl. It will surround the chicken with heat, making it cook more quickly and keeping it from drying out. Use tongs to put it in place and lift it up so you don't burn yourself.\"\n\nThe grilled chicken went onto their plates. The baked chicken came out of the oven, and everyone claimed their pieces. The roast chicken emerged deeply brown and highly fragrant.\n\nBack at the worktable, clutching knives and forks and giddy with anticipation, everyone took a bite of their chicken.\n\n\"This is so good. And it took what? Less than ten minutes?\" Shannon said. \"I don't know why I never thought of cutting it like that or just putting a bit of oil and seasoning on it first.\"\n\nSabra jumped up and down after tasting her saut\u00e9ed chicken. \"Seriously! This is awesome. Taste mine.\" She offered her plate to Gen. It gave me an idea.\n\n\"Hey, everyone shift to the right,\" I said. \"I want you to taste everyone else's.\"\n\nLike a continuous Chinese fire drill, the students shifted from plate to plate to plate. The consensus: Donna and Sabra had the besttasting saut\u00e9ed chicken of the bunch, while Jodi and Trish won the baked chicken round. Cheryl had not wanted to get near the stove with baby Liam in the carrier but had seasoned all of her food and Lisa had cooked it. Her spicy Tex-Mex combo won rave reviews.\n\nMaggie retrieved the mustard braise at nearly ten P.M., as everyone was preparing to leave. The dish had the classic elements of a good braise, deeply flavored and so tender it fell off the bone. We packed up the leftovers and a full meal's worth of braised chicken in takeout containers that Lisa had picked up at a local restaurant supply store.\n\nSabra could barely contain herself. \"This is the best class ever! I've done stuff tonight that I've never done before!\"\n\nAs she clutched her takeout box, Trish lingered, reluctant to leave until she was the final volunteer in the kitchen while Maggie, Lisa, and I started to clean up. \"What is it?\" I asked her.\n\nShe wiped her eyes. \"Tonight I learned things that I really ought to have known. Why didn't I ever just learn it?\"\n\n\"Julia Child said she didn't learn to cook until she was thirty-two. Until then, she just ate. So you see, it's never too late.\" I gave her a hug and then handed her a takeout box. \"See you next week.\"\n\n#    **Your Basic Roasted Chicken**\n\n_Chefs around the world continually debate the best possible way to roast a whole chicken. Don't overcomplicate it. You don't need a fancy roasting pan. Any kind of ovenproof pan, skillet, or saut\u00e9 pan with sides one inch or higher in which the chicken fits comfortably will work. A rack is nice, but you can just roughly chop up carrots, onions, and potatoes to spread them across the bottom and balance the chicken on top to allow the juices to drain. Include some kind of fat such as oil or butter, plus salt and pepper, and baste it at least once. I prefer to start the bird breast side up and then, using tongs, turn it over for the last twenty minutes of cooking to brown the other side, but that's optional._\n\n##  **BASIC TECHNIQUE FOR A WHOLE CHICKEN**\n\n  Preheat the oven to 425\u00b0F. Mix up some flavorings. Remove the giblets from inside the chicken cavity.\n\n  Gently ease your fingers under the chicken's skin to separate it from the bird, creating a cavity across the top of the breast and around the legs. Shove your flavoring under the skin. Smear a bit over the top and generously season the skin with coarse salt and ground pepper.\n\n  If you want to, tie the legs together with some string; this will help the bird keep its shape and cook evenly.\n\n  The larger the bird, the longer it will take to roast. Depending on your oven and your bird, a standard 3-pound chicken will take about an hour; allow about 10 minutes for each additional half pound. After half an hour, baste it by using a spoon, pastry brush, or bulb-style baster to collect the juices from the pan to moisten the skin.\n\n  If desired, turn the chicken over for the last 20 minutes of cooking. Baste again.\n\n  See if it's done. The best method is to insert an instant-read thermometer into the thigh meat and again into the breast, avoiding the bones. It should read close to 180\u00b0F, but double-check by pulling the thigh away. If the juices that ooze out are clear, it's done. If they are pink, baste the chicken again and put it back in the oven for another 10 minutes. Repeat as needed.\n\n  Let your chicken rest for about a few minutes before serving.\n\n##  **TEN WAYS TO FLAVOR ROAST CHICKEN**\n\n  Add coarse salt and pepper to each combination, and be generous with the oil, butter, or other type of fat; the excess will run off and it will help to keep the meat moist. Mix each into a paste before applying it to the bird. You can't go wrong shoving some fresh herbs, some lemon, a wedge of onion, or a couple of cloves of garlic inside the bird's cavity, along with some salt and pepper.\n\n_**Lemon-Herb Butter**_\n\n cup fresh lemon juice; put the leftover lemon inside the cavity 3 tablespoons butter, softened\n\nHandful of chopped fresh tarragon, thyme, or rosemary\n\n_**Italian Herb Oil**_\n\n3 tablespoons olive oil   \n2 teaspoons grated Parmesan cheese   \n2 garlic cloves, chopped   \n2 tablespoons finely chopped fresh herbs, such as thyme, oregano,   \nparsley, or basil (or about 2 to 3 teaspoons dried herbs)\n\n_**French Dijon**_\n\n3 tablespoons butter, softened, or olive oil   \n1 tablespoon red wine vinegar   \n3 tablespoons Dijon mustard   \n teaspoon red chili flakes   \n1 teaspoon dried thyme\n\n_**Soy Ginger Oil**_\n\n2 tablespoons peanut or vegetable oil   \n1 tablespoon sesame oil   \n2 teaspoons soy sauce   \n1 tablespoon rice wine vinegar   \n1 teaspoon freshly grated ginger\n\n_**Pesto-rama**_   \n  cup any kind of pesto (basil, sun-dried tomato, etc.)\n\n_**Goat Cheese with Prosciutto and Herbs**_\n\n1 ounce finely minced prosciutto or pancetta or crumbled cooked   \nbacon   \n2 tablespoons goat cheese   \n3 tablespoons olive oil   \nHandful of chopped basil or 1 teaspoon dried oregano\n\n_**Thai-Style**_\n\n2 tablespoons coconut oil   \n1 teaspoon sesame oil   \n1 tablespoon finely crushed peanuts   \n3 tablespoons fresh lime juice   \n2   teaspoons Thai curry paste   \n2 tablespoons finely chopped fresh basil or cilantro\n\n_**Greekesque**_\n\n  cup plain yogurt   \n4 garlic cloves, minced   \n1 tablespoon chopped dill   \n2 teaspoons fresh lemon juice\n\n_**China Spice**_\n\n3 tablespoons sesame oil   \n1 tablespoon orange juice   \n1 tablespoon Chinese 5-spice powder   \n4 green onions, minced   \n3 garlic cloves, minced   \n1 teaspoon soy sauce\n\n_**Tex-Mex**_\n\n3 tablespoons vegetable oil or butter   cup lime juice 1 teaspoon chili powder 2 garlic cloves, minced 1 teaspoon dried oregano   teaspoon ground cumin\n\nPinch or 2 of cayenne or a few drops of hot sauce\n\n##  **FOR INDIVIDUAL CHICKEN PIECES**\n\n  You can add any of these flavorings to individual pieces to roast separately. Again, press the flavorings under the skin. Put the pieces on a baking sheet atop aluminum foil or parchment or in an ovenproof baking dish in an oven preheated to 375\u00b0F. Thighs and legs will take about 45 minutes, bone-in breasts about 30 minutes, and boneless breasts about 15 minutes, depending on the thickness and your oven. Aim for a final internal temperature of around 165\u00b0F. If roasting mixed pieces, remove them as they're done and keep them warm by covering them with foil.\nCLASS BREAK\n\n**Cruise Control**\n\n_We Interrupt This Project for a_   \n_Cruise to the Mediterranean_\n\n  The morning after the chicken class, I received a frantic call. \"Can you be in Rome next Tuesday?\" a woman's voice said. \"A chef canceled. Ten-day cruise on the Mediterranean. It would _really_ help me out.\"\n\nSuch things never happened in my corporate life. Months earlier, Holland America had invited me to work for a week on one of its ships as a \"guest chef.\" When Mike and I joined the cruise on the _MV Amsterdam_ for its final leg from Honolulu to San Diego, we met Erika, the manager of \"culinary entertainment\" for the cruise line. Mike and I liked her instantly and bonded with her over the short trip. Back on land in Seattle, we kept in touch. Now she was in need of a replacement guest chef for one of the line's culinary-themed cruises\u2014stat. \"I'm racking my brain thinking about who could go on such short notice,\" she said. \"We'll pay your airfare.\" She called not realizing the trip would fall during our fifth wedding anniversary and Mike's forty-fifth birthday. We'd have to reschedule two weeks of classes, and I worried about losing momentum with the project, but then I looked at the itinerary: Rome, Monte Carlo, Barcelona, Palma de Mallorca, Tunis, Palermo, and Naples, with three days at sea during which I was expected to entertain cruisers with cooking demonstrations and hands-on classes.\n\nI immediately called Mike. \"I know you've been planning something for our anniversary, and this is so crazy last-minute. But what do you think?\"\n\n\"I don't think I can top that,\" he said. \"Let's do it.\"\n\nWithin four hours he'd sorted flights and scored a last-minute deal on a hotel room in Rome. Just three days later, swarms of buzzing Vespas escorted our death-defying taxi ride from the airport into the city center. We abruptly turned off a main road to creep along a narrow, ancient street to the Hotel de Ville nestled at the top of the Spanish Steps.\n\nIt felt decadent jetting into a foreign city with no itinerary on short notice. On our first night, we walked arm in arm to dinner at the restaurant Tulio. Mike ordered us each a glass of champagne to start. We eavesdropped on conversations around us and flirted across the table. I ordered the day's special, handmade fettuccine pasta dosed heavily with eggs. Mike splurged on a bottle of earthy Italian wine. As the immaculate waiter shaved slices of black truffle onto my plate of handcut noodles, I jumped up and down in my seat, giddy with anticipation.\n\nMike took my hand across the table and twisted his fingers in mine. \"I love that you can get this worked up over pasta,\" he said, and tenderly kissed the back of my hand. The scene was impossibly perfect: a romantic candlelit restaurant in Italy, Mike in a smart jacket, my complete disregard for carbohydrates.\n\n\"You want to hear something?\" I whispered. He leaned in across the table. \"I would marry you all over again just for this one moment with you.\"\n\nThe waiter cleared his throat. We had failed to notice his return. He stood over us with the bottle of red. We politely waited for him to pour while my heart pounded. I picked up my fork to start on my pasta. I stole a quick look up, assuming Mike would be tending to his plate of osso bucco. Instead, he was looking at me. Without a word, he picked up his glass of wine in a toast.\n\n\"To five more years,\" I offered.\n\n\"No, wait.\" He stopped me as I held my glass up in midtoast. \"I believe we agreed to forever.\" We clinked glasses.\n\nThe next morning, he lazily threw an arm around me as we started the slow ascent of waking from our jet lag. He nuzzled his chin against my hair. \"You still smell like truffles,\" he murmured softly.\n\nBeing a guest chef on a cruise ship is not quite work, but it's not exactly a vacation. To earn my passage, I had to perform two live cooking demonstrations before an audience, and then conduct two hands-on classes for a small group of guests. The demonstrations took place in the ship's Culinary Arts Center, a no-expense-spared TV studio\u2013style demonstration kitchen set atop a stage in front of about three hundred plush upholstered audience seats. Huge plasma screens showed the action live to the audience, while the audio-visual crew pumped the \"show\" into every single stateroom on the boat. So it was more like planning two forty-five-minute live cooking shows with an unfamiliar kitchen crew plus cooking a tasting for hundreds of audience members. Piece of cake.\n\nOn the _MV Noordam,_ guest chef activities were coordinated by party planner Linda. Fun spirited and high energy, she spoke in a throaty northern England accent and was a dead ringer for Patsy from _Absolutely Fabulous,_ although the similarity ended there since she's not a shallow drunk. Guest chef status provides access to parts of the ship most cruisers never see. Behind the gilded mirrors and velvet wallpaper exists a maze of kitchens and prep areas that span five decks. Linda led me to Chef Don, the executive chef in charge of the kitchens. He noticed my bare head and fished a paper chef's toque out of a drawer. I'd never worn a tall toque before. It didn't quite fit and kept falling off as I walked through the sea of Indonesian and Central American kitchen workers, all 337 of them male. Women kitchen workers, much less chefs, are a rarity. The sea of workers parted as they saw me. They smiled sweetly and nodded furiously. \"Oh, _hello_ , Lady Chef.\"\n\nChef Don and I zigzagged through the vast kitchens stacked on multiple floors. The scale of the operation overwhelmed. The _MV Noordam_ served eleven thousand meals a day from five separate restaurants, plus busy stateroom room service and special events. One area equipped with a floor mixer the size of an impressive Jet Ski turned out nothing but the morning pastries and evening desserts. Rows of tables stood at the ready in one prep room awaiting the hundreds of plates the crew assembled each day. Don took me through two baking kitchens and an entire floor dedicated to _garde manger_ that managed cold foods such as salads and refrigerated desserts. A full-time butcher worked in a walk-in cooler the size of my first apartment tucked deep belowdecks. During meals, the kitchen buzzed at a frantic yet practiced pace. At ten A.M., it might be quiet in the fine dining kitchens, but the crew of the Lido Deck buffet\u2014the hub of breakfast onboard\u2014was just clocking out. \"At every hour of the day, there's someone working somewhere,\" Chef Don said. \"We never really close, we just slow down production.\"\n\nOn our tour, Chef Don asked, \"You want to see something?\" We took an elevator down to the cargo hold in the belly of the ship. In front of us, pallets the size of double-wide trailers held tons of produce. \"We have a crew who spend their days getting the food off the pallets and moving it to the right parts of the kitchens,\" he said. \"It goes faster than you can imagine.\"\n\nChef Don explained that the complexities of the supply and thin profit margins of the operation required a delicate balancing act. \"Managing our food supply is one of the most critical things that we do. We can't go through it too quickly or we will run out before our next port. But we can't afford to let it go bad or waste too much food either,\" he explained. \"We try hard to figure out how to maximize everything yet never run out.\" My thoughts went to the volunteers and to my own kitchen, and to the larger issue of wasted food. Just how many wilted vegetables in our crisper awaited my return?\n\nIn the elevator heading back up to the kitchen, Chef Don glanced again at my demonstration list. \"Paella? I can't wait. You have much experience with it?\"\n\nAs it happens, for a girl from Michigan, paella has odd prominence in my life. It was the subject of the first food story I ever published back in the early 1990s. Days after we got engaged in Spain, Mike and I happened upon a paella-making competition in the heart of Valencia, home of the classic rice and seafood dish. Knowing that, our chef friend Ted built a custom-made box and shipped his three-foot-wide paella pan across the country from Seattle to Florida for our rehearsal dinner on Anna Maria Island. If that's not friendship, I don't know what qualifies. The pan was so massive that Ted rented a portable commercial four-burner stove to ensure adequate heat. Our seventyfive guests hovered, clutching cocktails in the sweltering Florida night, as he browned chicken, cooked sausage, saut\u00e9ed onions, and added the rice and stock, stirring with a spoon the size of an oar. To finish, he tossed in a plethora of Gulf-fresh seafood from Cortez, a historic fishing village nearby. As a tradition, we've made paella every year to commemorate our wedding; it's as much a ritual in our lives as turkey at Thanksgiving. By fate or sheer coincidence, my last demonstration was scheduled for the Fourth of July, our fifth anniversary\u2014right after we'd visit two ports in Spain. We both agreed that I had no choice but to make paella, and to pick up a pan along the way.\n\nHowever, we arrived in Barcelona on Sunday and most of the stores were closed. Instead, we hung out at La Sagrada Familia, the famous unfinished church designed by artist Antoni Gaudi. Before that trip, I'd seen the church only from a distance; the spires always remind me of tapered candles left out too long in the sun. It's impressive on many levels. But what I admire most is that Gaudi started a church that he knew he would never live to see finished, perhaps the ultimate definition of faith.\n\nThe next day, the ship docked at the Spanish island of Palma de Mallorca. We stepped up our hunt for a pan. Mike confirmed that standard paella pans are crafted from carbon steel and thus would not work on the induction burners in the ship's Culinary Arts Center. The best way to tell if a pan will work on an induction burner is to see whether a magnet will stick to it. Mike sent me on my way and headed into town on his hunt armed with a tourist magnet from the Sagrada Familia. I'd booked a day-long excursion to watch a Spanish chef make paella over a blazing fire in the countryside. A bus delivered me and two dozen cruisers to a restaurant set up in a stone farmhouse.\n\nUnder a small wooden pavilion, the chef's crew commenced with threading three whole baby pigs onto monstrous wrought-iron poles. The poles hooked into a medieval-style spit device over a row of white-hot fires. The distraught look on some of the cruisers' faces signaled that many had likely never seen a whole pig, much less witnessed an iron rod jammed up one's backside. Only a cruise passenger skewered and put onto the spit could have yielded more horrified stares.\n\nAs the crew tended to the pigs, the chef worked the paella. He started with hacked-up pork bones on a flat, shallow pan nestled above another bracing fire in the 102-degree heat. Chef tossed out points in Spanish to his young Catalan assistant, who caught and translated them for the group. \"Chef says that he likes to start with bones because the marrow will caramelize and add richness to the dish,\" she said. The pork crackled and snapped as the pool of fat slowly widened and the remaining meat on the bones darkened with its own sugar.\n\nChef stooped over the heat to place two dozen white chicken thighs into the pan. Each sputtered as it hit the oil. Then he tossed in bowls of chopped onions, tomatoes, peppers, and rice, one by one. Finally, he poured in a bucket of dark chicken and fish stock. The watching crowd sat mesmerized, most slightly drunk from drinking sparkling wine in high heat at midday. Chef handed a long paddle to a clean-shaven midwestern guy in a golf shirt. He eagerly took it. Everyone took a turn stirring, including an eighty-four-year-old man in Sansabelt trousers who teetered over his walker as he pushed the oar around in the rice.\n\n\"Dad, are you okay?\" his concerned daughter asked, approaching. \"It's pretty hot over these coals. Maybe you should sit down.\"\n\nHe waved the paddle at her, nearly falling off his walker. \"Back off! I'm having a good time,\" he said gruffly. \"That chef isn't much younger than me, you know. This is the first time I've cooked anything in years and I'm liking it.\" She slunk away.\n\nThe area took on a meaty smell that mingled with the aroma of charred onions and peppers, joined later by the sweet scent of massive, lobsterlike langoustines, the final addition by the chef near the end of cooking. When finished, the pan was substantial enough with the rice and other ingredients that it required four strapping Spaniards to haul it away. Armed with fireman-style gloves, they squatted in unison and with a collective grunt hauled the pan up and walked it inside the dining room like pallbearers.\n\nAs the food arrived, I noticed that even women who had been aghast at the sight of the skewered pigs now happily gnawed on the pieces of roughly cut pork. The room went silent as everyone concentrated on lunch. The chef's assistant emerged from the kitchen with one last tip. \"Try a bite of paella and then a bite of the raw green pepper slices you'll find on the table. It enhances the flavor.\" He was right. It drew out the earthy, soft, and sweet flavor of the saffron rice and the light sea-scented stock.\n\nThe bus dropped us back in Palma, a town filled with beautiful Spanish architecture and populated by women dressed in beach attire even in the middle of downtown. Mike alerted me to meet him in the kitchen section of the department store El Cortes Ingl\u00e9s. He'd located two paella pans that he thought would work on the ship's burners. After some debate, we purchased both. We then boarded a bus whose destination was \"Airport\/Harbor.\"\n\nMike asked, \"Are you going to the harbor?\"\n\n\"Yes,\" the driver replied.\n\nAfter a few stops, we realized that we were heading out of town to the airport. We nervously looked at our watches.\n\nAt the airport, Mike talked to the driver again. \"Are you going to the port next?\"\n\n\"Yes,\" the driver said.\n\n\"With the ships?\"\n\n\"Yes.\"\n\n\"Do you go to the moon?\"\n\n\"Yes,\" the driver answered, nodding.\n\nMike waved me off the bus. \"Let's take a taxi back to the ship.\"\n\nThe night before, we'd barely made it onto the ship in Barcelona before the crew pulled up the gangway. Once again, we ran through the port, this time each of us clutching a paella pan.\n\nThe next day, I told the story of our chef friend Ted and our wedding. \"Oh, it's not fair that you have to work on your anniversary,\" a woman from Asheville, North Carolina, said.\n\n\"Are you kidding? If I was at home, this is exactly what I'd be doing, making paella.\"\n\nWhen we recounted the trip back home, we both measured the days in meals and countries: lunch in Monte Carlo overlooking a white sand beach populated with beautiful topless French women (crisp salad Ni\u00e7oise thick with anchovies, roasted whole fish with fennel and tomatoes); in Barcelona, a late bite just off Las Ramblas (spitroasted rabbit, broiled mussels with garlic and parsley sauce); the hike to a hillside cliff for hot, sweet tea above Tunis (sugary dateheavy Tunisian pastries); a bite in an ancient square in Palermo, Italy, after being overcharged by a crafty seventeen-year-old Sicilian cartand-buggy driver (handcrafted pizza with fresh tomatoes, grilled squid with sweet fennel sausage); and, finally, the best _pasta vongole_ (pasta with clams) of my life high above the harbor on the island of Capri.\n\n#    **Paella Valenciana**\n\n_Paella sounds complicated, but it's just a casserole that originated as a poor fisherman's supper designed to use up scraps left in the nets. Make it once, and then adapt the technique to make a one-pot dish that's simple or extravagant based on what you've got on hand. Just keep intact the base aromatics known as the_ sofrito: _onions, garlic, and tomatoes. I've made paella with and without meat, with green beans, asparagus, fresh corn, rabbit, garbanzo beans, scallops, andouille sausage, littleneck clams, and even Dungeness crab._\n\n_Paella takes its name from the traditional pan, a wide, flat-bottomed carbon-steel double-handled skillet with flared sides. This recipe is designed for paella pans from twelve to fifteen inches wide. If you don't have one, a twelve-inch or larger shallow skillet or similar-sized pan will work. In Spain, paella is often cooked over an open fire, but it can be cooked on a stovetop and finished in a 450\u00b0F oven or over a large, round barbecue. Lovely with a rioja, either red or white._\n\n**SERVES 6 TO 8**\n\n_**Shrimp-Flavored Stock**_\n\n1 pound medium shrimp, deveined, shells reserved 1 quart chicken stock\n\n_**Seasoned Chicken**_\n\n6 chicken thighs and\/or legs   \nCoarse salt and freshly ground black pepper   \n1 teaspoon dried thyme   \nCouple of pinches of cayenne\n\n_**Basics**_\n\n2 tablespoons olive oil   \n8 ounces chorizo, cut into bite-sized pieces   \n4 garlic cloves, minced   \n1 large yellow onion, diced   \n1 green pepper, diced   \n2 large tomatoes, seeded and chopped   \n2 bay leaves   \n1   cups uncooked Bomba, Arborio, or other short-grain rice   \nPinch of saffron threads (about   teaspoon crushed)\n\n_**To Finish**_\n\nOne 14-ounce can artichokes   \n1 cup frozen peas, thawed   \n2 ounces diced pimento   \nAbout 1 pound mussels, bearded and scrubbed   \n2 or 3 lemons, cut into wedges   \nStrips of green pepper\n\n Combine the shrimp shells and chicken stock in a pan, and simmer until needed.\n\n Season the chicken pieces with generous doses of coarse salt and pepper, the thyme, and the cayenne.\n\n Heat the pan over medium-high heat or a hot grill. Add the oil and saut\u00e9 the chorizo until it is partially cooked, about 5 minutes. Add the chicken to the pan and brown it well, turning occasionally, for about 15 minutes. Remove the chorizo and chicken and set aside. Add the garlic, onion, and green pepper to the pan and cook until the vegetables are softened and starting to brown. Add the tomatoes and bay leaves and cook another 3 minutes. Meanwhile, strain the shrimp shells from the stock.\n\n Add the rice to the vegetables and cook, stirring, for a couple of minutes, until gently toasted. Add the strained stock, saffron, several cranks of black pepper, and 3 pinches of salt. Stir as the mixture comes back to a boil.\n\n Return the chicken and chorizo to the pan. Cover loosely with aluminum foil and keep at a bubbling simmer for about 25 minutes, until the rice is tender. If excess liquid remains, remove the foil and cook until it is absorbed. Scatter the artichokes, peas, and pimentos on top of the rice, then press the shrimp and mussels into the hot rice. Cover again until the seafood cooks through, about 8 to 10 minutes. Remove the pan from the heat and let it stand, covered, for a few minutes before serving. Discard any mussels that are not open after cooking. Remove the bay leaves before serving. Serve with the lemon wedges and strips of raw green pepper.\n**PART II**\n\n**A Bit About Meats, Bread, Pasta, Salads, and Vinaigrettes**\n\n_\"The smell of good bread baking, like the sound of lightly flowing water, is indescribable in its evocation of innocence and delight.\"_\n\n\u2014M. F. K. Fisher\n\n_\"How can a nation be great if its bread tastes like Kleenex?\"_\n\n\u2014Julia Child\n\n**Trish makes bread**\nCHAPTER 7\n\n**The Bread Also Rises**\n\n  **LESSON HIGHLIGHTS:**\n\n**No-Knead Artisan Bread, Simple Pasta Sauces**\n\nOn the farm, my parents made bread twice a week. In one of my earliest memories I perched atop a wooden chair in the hot farmhouse kitchen on a snowy winter's day. I drew hearts on the steamy window, my small fingers cold against the pane of glass as my father shaped and patted loaves of dough into four pans.\n\nDad moved a chair over to the table. \"Time for sugar!\" he said. I leaped up on the chair and pressed my tiny right hand on top of each petal-soft loaf and he drew an outline of my hand with a butter knife. My tiny handprint, Dad said, replaced any need for sweeteners. I helped him fold laundry warm from the dryer while the yeasty sour fragrance grew fierce as the bread finished baking. Dad turned the pans upside down and shook the still-warm loaves from them with a thump. He smeared half a stick of butter across the top of each one, the butter creeping into my handprint, distorted by the heat, like a melting clock from a Salvador Dal\u00ed painting.\n\nBy lunchtime at school, such pleasant memories faded fast. Taking sandwiches with thick slabs of homemade bread to school made kids of my era look both poor _and_ distinctly uncool. By first grade, I longed so desperately for soft white store-bought bread that I saved up for a month to buy a loaf with my allowance, assuming it would taste like candy. Compared to our bread, it had the flavor of what Julia Child once described as Kleenex. The bread did offer remarkable sculpting qualities when softened with water, so it wasn't a total loss. I molded tiny statues rather than eat it.\n\nWhen I was in first grade we moved from the farm into the town of Davison, a suburb of Flint, Michigan. An early sign of our improved social standing: store-bought bread. When Mom discovered a series of barely touched PB and J's in my Scooby-Doo lunchbox, she assumed I'd developed a sudden dislike of the entire concept of sandwiches. It wasn't that; I missed my parents' homemade bread. But they seemed too busy to make it anymore, and I didn't want to hurt her feelings. She abruptly started to shove quarters in my hand every morning for hot lunches. I would sit with my hard plastic tray eating soggy Tater Tots and stare at the kids in the bagged-lunch ghetto unwrapping their thickly sliced sandwiches. As they sometimes had to risk dislocating their jaws to open their mouths wide enough to take a bite, I turned away, distressed by the depth of my envy.\n\nDespite my inherent love for homemade bread, I never mastered it. Single life in a series of tiny kitchens, with a deficit in counter space, and dodgy ovens did not inspire many forays into bread making. The curriculum at Le Cordon Bleu merely glanced at the subject. In France, _boulangerie_ equates to both a high art and a culinary science that warrants its own focused study, like butchery or patisserie. My personal reference to homemade bread made the whole thing feel so intimidating\u2014until I came across no-knead artisan bread.\n\nMost people became aware of the no-knead bread phenomenon thanks to Jim Lahey of the Sullivan St Bakery in New York when food writer Mark Bittman documented his method in a 2006 story for _The New York Times_. As Bittman noted, breakthroughs are rare in something as fundamental as bread making, so developing a strategy to transform freshly made bread into almost a convenience food counted as a groundbreaking achievement. I tried Lahey's version, and later versions from various books on the subject. The result: a crusty, artisan-style loaf for about sixty cents. Mike loved the stuff, and he adopted the role of primary baker for our small household.\n\nFor the next class, we'd focus on pasta (since I'd just been in Italy) and no-knead bread. \"You sure? It's going to be crazy hot in there with burners and that oven going full blast,\" Lisa said over the phone. I could hear her starting to sweat at the thought. \"Also, something's up with Maggie. I don't know if she's going to be able to help out for a while.\"\n\nDuring my absence abroad, the ongoing cupcake battle consumed Maggie's schedule. \"You won't even believe what I'm going through,\" she said on the phone, pained. The company was on the brink of opening its third store in two weeks and had scheduled three events back to back that involved handing out thousands of samples. I had heard that a major film being shot in Seattle might feature cupcakes. Maggie said nothing, but if this was true, I imagined pursuit of cupcake placement in a film held the possibility of bloody intrigues worthy of film noir. \"I've got to go back to the samples now,\" she said with a weary voice. I felt bad for Maggie. The idea of a business consultant having to help frost thousands of mini-cupcakes struck me as one of Dante's circles of Hell.\n\nI hung up and wandered downstairs to find Jeff. In an odd confluence of events, my former upstairs neighbor in London arrived in Seattle ready to relocate, but in the brutal job market of 2008 had yet to find work. Mike and I live in a 1902 Dutch Colonial that had been rebuilt into a duplex thirty years ago by an architect and his son. We live upstairs and rent out the larger main level. Mike was in the middle of an extensive remodel of the kitchen in the downstairs unit. Jeff arrived as our renters moved out, so Mike set him up in the place for free but warned him that he would be without a kitchen for at least a couple months. That didn't bother Jeff.\n\n\"Sure, I'll help out with your project. You know, you should teach risotto. I made it in a hotpot in the bathroom last night,\" he told me as he ironed his trousers. \"It came out pretty well.\" He seemed genuinely pleased with his ingenuity. \"I think you should write a whole book about cooking in the bathroom with a hotpot. I bet _that's_ never been done before. Oh, hey, I've got pinot gris chilling in a bucket in the bathtub. Want some?\"\n\nBy lunchtime on the day of the bread-making class, the air refused to budge outside. It was even hotter inside the kitchen; the catering crew had the ovens and commercial stove at full blast for most of the day. When we arrived, the kitchen's thermometer registered 101\u00b0F. After we set up most of the gear, we needed a break. Mike and I headed for the walk-in cooler. Jeff followed with a cold bottle of sauvignon blanc and three glasses. We perched on produce boxes and sipped wine to cool down. \"This is really nice,\" Jeff said, sitting with his legs crossed atop a crate of cabbage. \"Maybe we can do the whole class in here.\"\n\nThe door flew open. \"What, there's a party in the walk-in, and nobody told me?\" exclaimed Lisa, who looked wilted from a hot, horrible commute. We got her a glass before the four of us returned to the hot vat of a kitchen.\n\nShannon and Cheryl were the first to arrive, the latter without baby Liam. \"I'm liberated!\" she said. \"I'm ready to get near the big stove, and of course it's too hot outside to want cook anything.\"\n\nWithout a word, the early arrivals put on their aprons, grabbed a diaper, and headed to the hand-washing area. Then they found a spot at the table to start chopping. As each person arrived she joined in, like a bystander jumping into a moving parade. \"Hey, are there any more vegetables?\" Terri asked. After her initial struggles, she seemed to finally get the hang of it and made quick work of a few zucchini. \"We've run out of things to chop.\"\n\nI scoured the table. \"Nope, that's it. Today we're going to make pasta and sauces as promised, and we're also going to make bread.\" Half fanned themselves limply with copies of the recipes we'd handed out at the start of class. As I said the word _bread,_ most instinctively looked at the commercial ovens. I could almost hear what they were thinking. _Bread? In this heat? You've got to be kidding._\n\nTraditional bread has just four ingredients: flour, salt, yeast, water. I started the class by revealing a plastic sleeve from a loaf of supermarket bread. I read the ingredients out loud:\n\nRefined white flour, water, high fructose corn syrup, contains 2% or less of: wheat gluten, soybean oil, salt, molasses, yeast, monoand diglycerides, exthoxylated mono- and diglycerides, dough conditioners (sodium stearoyl lactylate, calcium iodate, calcium dioxide), datem, calcium sulfate, vinegar, yeast nutrient (ammonium sulfate), extracts of malted barley and corn, dicalcium phosphate, diammonium phosphate, calcium propionate (to retain freshness)\n\nI stumbled through the polysyllabic words. I was familiar with high-fructose corn syrup (HFCS), the heavily processed sweetener made from starch-rich corn. HFCS is common in baked goods, as it helps stave off spoilage and keep breads soft. Recent studies have linked high intake of HFCS to childhood obesity and diabetes. But in research that I conducted for the class, I was surprised to learn that soybean oil is so ubiquitous, a leading researcher has stated that some nutritionists believe it makes up as much as 20 percent of all calories consumed in the United States. Diglycerides are simply fancy names for fatty, often hydrogenated oils or trans fats.\n\nBy the end of it, the sweaty crew stood with their mouths agape. \"Some bread is healthier than this one. Read labels. Consumers have a voice in a free market. If people buy bread without a lot of additives, you'll see more bread without them. Or you can make your own. You can control what's in it, and it's way cheaper.\"\n\nJeff settled a big plastic container with flour on the worktable. \"Okay, first we're going to talk about the right way to measure flour. Don't dig a measuring cup into it. Instead, use one measuring cup to fill another.\" I spooned flour into a stainless steel cup measure. \"When full, use a knife to scrape it level across the top.\"\n\nTrish raised her hand. \"So I've always heard that, but why do you do it?\"\n\nJeff fetched a kitchen scale from a side table. I set a metal bowl on top and zeroed out the scale. I measured a cup of flour as I'd just demonstrated and plopped it into the bowl. \"Okay, 150 grams.\" I poured it back into the flour bin, set the bowl atop the scale, and zeroed the scale again. This time, I dug the scoop directly into the flour and scraped the top. \"This is 190 grams. If you add 40 grams on each cup, you'll completely ruin a baking recipe. In baking, weight is more precise than volume. That's why most professional recipes are outlined in weight, not cups.\"\n\nDri slapped the table. \"I think this explains most of my baking failures.\"\n\nYeast went in next. \"Any yeast will do. You can buy fresh yeast in the supermarket, like this,\" I said, holding up a cube of fresh yeast. \"Or you can use instant yeast in the packets or jars. If you think you'll make a lot of bread, buy a block of yeast at a warehouse store. Then invite a couple of your friends over, show them how to make this bread, and divvy up the yeast with them.\"\n\n\"Yeast is a living thing, be nice to it,\" I continued. I reached for a pitcher of warm tap water on the table and measured out a cup and then stuck an instant-read thermometer into the water. It registered 100\u00b0F. I handed it to Cheryl. \"Feel this and pass it around. It should feel lukewarm. If it's too cold or too hot, you'll kill the yeast. Go for around body temperature.\" The little metal cup went around the table as each person dutifully stuck a finger into the water. As it got to Donna, she shook her head.\n\n\"And _this_ explains a few of _my_ baking failures. I think sometimes my water is too hot,\" she said, passing the cup back to Jeff, who drained it into the sink.\n\nI measured the water, yeast, and salt and added them to a large bowl, dumped in the flour, and stirred. The dough became a stiff, lumpy, sticky, and moist mass. \"That's it. Here, pass it around. Smell, touch it, so you can remember what it's like.\" Everyone peered into the bowl, sniffed, and tentatively poked a finger into it. As it made the rounds, I pulled out a five-quart clear plastic bin.\n\nEarlier, I'd made a double batch of the same dough recipe. We'd left it in the back of our car while we shopped for ingredients for class, a point that reminded me of the state health exam and the question about not leaving meat to defrost in the trunk of one's car. It had shifted from a dense, gloopy mass to a bubbling, airy beige blob three times the size. I took back the original bowl and compared them side by side. \"So this is 'The Blob.' It's what it looks like after it has risen.\" The volunteers looked suitably impressed. \"Now we're going to shape some loaves.\"\n\nEveryone grabbed a hunk of the risen dough with floured hands and slathered it in a cloak of flour. Mike, our resident baker, stopped videotaping the class for a moment. \"The point is to cover it lightly all over,\" he advised. \"I think of it like creating the shape of the top of a mushroom by smoothing and stretching the dough out and tucking it under the loaf.\" He and I went around the table helping everyone mold their little grapefruit-sized balls of dough into round _boules_. The volunteers smiled and patted their blobs tentatively at first, and then with a bit more gusto, rather like kids making mud pies. Jeff carefully set up trays lined with parchment paper and cast a generous handful of cornmeal on each to keep the loaves from sticking. I watched as Dri set her loaf on the sheet and wrote her name next to it on the parchment paper. She gazed at it wistfully. If no one had been watching, I wondered if she might have kissed it. I knew the feeling. Baking can do that to you. The loaves needed at least a half hour to rise, so we moved on to pasta.\n\nPasta earned its place as a household staple in the space of a generation. Italian immigrants started commercially producing the tender, pleasantly neutral-tasting white pasta known to most diners in the early twentieth century, back when it was considered an ethnic food. In the United States and Canada, pasta consumption doubled between 1975 and 1995. Even with the rise of low-carb diets and an increased appetite for gluten-free foods, dried pasta is still a nearly $2-billion-a-year business. Ever since the major food companies took notice, they've found big profits trotting out various forms of flavored, prepackaged pasta dishes. We may have a vision of a group of Italian grandmothers in some countryside locale turning out noodles, but the retail arena is controlled by a handful of multinationals. In the 1990s, Hershey's was the largest pasta manufacturer until it divested itself of a couple of brands to focus more on its main business, chocolate and confectionaries.\n\nOne of my longest conversations with the woman in the supermarket was around her presumed inability to make a simple sauce. Pasta was a staple in the diets of all the volunteers, more ubiquitous than even chicken. But not all pastas are made the same. For that day's tasting, I had compiled a wide selection, including traditional white spaghetti and a couple of gluten-free alternatives, but mostly focusing on whole wheat options, mixing expensive and inexpensive brands. Shifting from white semolina pasta to 100 percent whole wheat pasta can double the fiber and nearly triple the protein of any dish.\n\nWhile we were all playing with dough, Lisa tended six pots of boiling water, preparing to cook the twelve different types of pasta to the same exact level of al dente and keep track of them all. On its face, this seems a simple task. Some had to cook for six minutes, others fourteen. As the rest of us worked on the bread, I could hear her clanging around in pots to dig out pasta to settle into colanders and slapping down a dozen small plates on the stainless side table.\n\nBy the time we were ready to taste, Lisa looked ready to melt. The walls behind the stove dripped condensation. \"Okay!\" she said, blowing out a huge sigh. \"Ready?\" As everyone circled the plates, she disappeared into the walk-in cooler and shut the door behind her. I'm fairly sure that Jeff was already in there, possibly with the wine.\n\nEveryone nibbled. The results intrigued. Favorites included the store-brand whole wheat pasta from a warehouse store. \"It's got a nice nutty flavor to it,\" Terri noted. The least favorite was a national whole wheat brand that I had bought months earlier but left untouched after Mike insisted on referring to it as \"dirt pasta.\" I used to give him a hard time for not liking it, but then I discovered no one else did either.\n\n\"No, it really does taste like dirt,\" Shannon said earnestly. \"I mean, it's got a strange grit thing going on. My kids would never eat that.\" She set down her small plate and pushed it away.\n\nMike leaned out from behind the video camera. \"See?\" he said, vindicated.\n\nTrish pointed at some brightly colored rotini on her plate. \"What are these colored pastas? They're kind of odd tasting.\" They were pasta spirals flavored with vegetables, another pasta reject from my household. \"Oh, yeah, I bought these for our friend's kids who come over to visit a lot. Neither of the girls were crazy about it, though.\"\n\nTasting one of the two white pastas, Donna said that she detected something fishy. \"It's like tuna. I mean, that's probably not right. That's just me.\" Then we looked at the package. It contained omega-3. That would explain the fish flavor. No one else had picked it up, but on tasting it again, it was obvious.\n\nShannon seemed enthused. \"This is a great tasting to do,\" she said. \"I'm going to start buying a couple of different pastas and trying them out with my kids. In some ways, I think this was my favorite tasting so far since it's something that I cook all the time.\"\n\nWhen it comes to cooking pasta, a few tips make or break the whole experience. First, be sure that you use plenty of water. Second, the pasta will take on the flavor of the water that it's boiled in. One classic way to add flavor to pasta is by salting the water. \"The water for pasta should taste salty, kind of like the sea,\" said Lisa, who was back and looking refreshed from a turn in the walk-in. Finally, don't overcook it, especially whole wheat pasta.\n\n\"Don't rinse the noodles,\" I added. \"Sauce will stick better to drier pasta.\"\n\n\"Why do they say to rinse it, then?\" Terri asked.\n\n\"As long as it's hot, it will keep cooking,\" I replied. \"In fact, if you're going to put pasta into a sauce, which you will be doing in a couple of dishes, undercook it a bit. It will cook a little bit as it cools down and a bit more when it sits in hot sauce.\"\n\n\"Did any of you know that?\" Dri asked the group. They shook their heads and kept listening intently. \"Okay, good, I thought maybe it was just me.\"\n\n\"Pasta can be a fridge-clearing powerhouse,\" I explained. \"Clean out your crisper drawer, chop up the contents, saut\u00e9, steam, or boil them, and add them to a bowl of pasta. Add a little olive oil and some Parmigiano-Reggiano cheese or a classic tomato or Alfredo sauce, and voil\u00e0, dinner.\"\n\nWe began with a basic cream sauce. Having worked in a restaurant that served a popular Alfredo dish, Lisa grabbed a whisk and a pan and took over. \"Okay, I know it's hot, but everyone over to the stove,\" she commanded. \"You know that fifteen dollars you pay at the Olive Garden for Alfredo pasta? Here's what they start with,\" she said, and she poured two cups of cream into a pan atop the six-burner stove. \"It's just cream reduced with a little bit of pasta water. It's such a simple recipe that you'll wonder why you ever bought it in a box with a flavor packet.\"\n\nThe stove still radiated massive heat from the earlier pasta cooking. She pointed at the cream gurgling in the pan. \"See how it's starting to bubble? When this happens, add in half a teaspoon of salt.\" She kept stirring. Large glassy bubbles popped then died on the surface slowly and then quickly until it resembled slow and lazy foam. The smell changed from bland milk to a strong, intoxicating, almost cheeselike scent. \"Smell that? Now let it keep doing that until it thickens.\" After a minute, she took a spatula and pulled it across the bottom of the pan, painting a wide swath in the stainless steel. She grabbed a spoon from a holder near the stove and stuck it into the sauce, then held it up. The cream cleaved thickly to the back of the spoon, melting in a glacial slide of goo.\n\n\"When the sauce thickens enough to cover the back of a spoon or leaves a clean line in the bottom of the pan, it's time to add some pasta water.\" She dug a ladle into one of the still-hot pots of pasta water and splashed it into the pan with the cream. She simmered it over medium-high heat until bubbles erupted again. \"Now add a little bit more cream, heat it through, and then add some shredded cheese, maybe some garlic, and a few cranks of pepper. Jeff, can you hand me some of that cooked pasta?\" Jeff brought over a colander of cooked linguine. She took hearty grabs with tongs and swirled the strands into the sauce.\n\n\"And that's it. Cream, pasta water, some cheese, maybe some garlic. Easy, right?\" She poured it into a pasta plate. \"That's just a start. Add in some leftover chicken, some ham, peas, saut\u00e9ed mushrooms, steamed asparagus, cooked shrimp, and chopped basil, whatever. Just think of dishes you've seen on restaurant menus when you're looking in your fridge thinking, Huh, what could go into pasta? You want to make macaroni and cheese? After the cream reduces, just add shredded Cheddar cheese and macaroni.\"\n\nShe brought the warm pasta with the Alfredo sauce to the worktable. Jeff ladled out a serving to each person on a small plate. For comparison, I had asked Jeff to prepare a boxed version of fettuccine Alfredo. The fresh Alfredo was white with a velvety texture and had a faint scent of garlic in the warm cream. The boxed version had a thin sauce and an unnatural sheen; it smelled strongly of cheese.\n\n\"Don't make us eat that,\" Jodi said sternly, screwing up her face. I knew from the inventory that I had made of her overstuffed pantry that she had at least half a dozen boxes of this exact same stuff.\n\n\"Oh, come on,\" I said. \"You need to taste the difference.\"\n\nShe sniffed it. \"I can actually smell the iodized salt in it.\" She took a bit and put it on her plate, then lifted it to choke down a bite.\n\nThe rest of the group tentatively picked up a single ribbon of the boxed pasta. \"This doesn't taste anything like that,\" Donna said, motioning to the freshly made sauce. \"It's not terrible, but it's not good, especially compared directly to the real version we just had. I never thought to do that.\"\n\nI tried both, too. The cream sauce had a luxurious, fat mouthfeel with the slightest spikes of pepper and garlic. By comparison, the boxed version had an almost overpronounced sweet cheesy flavor at first bite that vanished into a salty aftertaste. It was an obvious knockoff, like an inexpensive purse attempting to stand in for an authentic Italian leather bag.\n\n\"Oh, wow, look at the bread!\" Dri said as we set the first of the baking trays onto the worktable. In the half hour that had passed, the raw loaves that they had left to rest had plumped and softened. Mike grabbed a chef's knife and slashed three diagonal slits lightly across the top of the loaves as I explained: \"Scoring the bread allows a bit of steam to escape during baking, resulting in a better crumb and a prettier loaf. You can do a pattern if you want.\"\n\nArmed with their knives, the women found their loaves and exchanged murmurs. \"It's so soft!\" and \"Look, it's starting to look like bread.\" Most carefully sliced slashes in theirs; Sabra branded hers with a star. Jeff waved everyone away from the oven and pulled open the doors. The heat blast felt almost like a physical punch. I slid the two sheets of bread into the oven. Earlier, I had set a small pan in the bottom. I poured water into the hot pan. As it hissed and steamed violently, I slammed the door shut. The steam helps the dough \"spring\" in the oven with a sudden burst of heat.\n\nAs it baked, we moved on to a basic tomato sauce. Many jars and cans of spaghetti sauce include significant amounts of added sodium and sugar. Also, given that a sauce is little more than some aromatics, seasonings, and tomatoes, it's often less expensive to make at home. During the Alfredo lesson, Jeff had set up five portable electric and propane burners from the kitchen's inventory around the table; it was simply too hot to keep cooking over the big stove.\n\n\"Who wants to demonstrate?\" Gen raised her hand. She skipped around the table and grabbed the spoon. \"Okay, Gen, warm some olive oil in a pan. Add some garlic, onions, and herbs until the onions are soft.\" While she stirred I opened a can of diced tomatoes, and when the vegetables were ready, Gen plopped them into the saucepan along with a couple of pinches of red pepper flakes and salt, some water, and a bay leaf. She brought it to a boil and then reduced the heat to a simmer. \"And let's have a big hand for Gen and her sauce!\" I said. Everyone clapped. Gen took a bow.\n\nAs the sauce bubbled lightly, we shifted to another burner for a quick _pomodoro,_ or fresh tomato sauce. I took off my watch and asked Cheryl to time the process. The sauce started with heating olive oil, then adding chopped garlic, sliced cherry tomatoes, and a splash of the pasta water. We let it simmer briefly and then added two handfuls of cooked linguine to the sauce along with a toss of chopped basil. Start to finish, the sauce clocked in at less than five minutes. By the time it was done, so was the spaghetti sauce.\n\n\"Now it's your turn.\" I waved my hand over the center of the worktable, where bowls sat holding the vegetables chopped at the start of class, along with some cream, various cheeses, cherry tomatoes, fresh basil, cilantro, parsley, and lemon. Lisa brought over bowls of cooked spaghetti, penne, and macaroni. Everyone shifted into action, grabbed a pan, found an empty burner on the table or stove, and settled in to work.\n\nI observed their selections. Each person made something completely different. Terri faithfully replicated the _pomodoro_ sauce. Sabra made creamy macaroni and cheese. Jodi saut\u00e9ed all the vegetables quickly, then added the penne, chopped basil, and olive oil and lovingly grated Parmigiano-Reggiano cheese over the top. With a graceful, seemingly practiced move, she dropped it into a pasta bowl, then she stood back to admire her work. \"Wow, look at this,\" she said to herself. Then she turned to Shannon. \"Can you believe that I made this? It's so pretty.\"\n\nDing! We retrieved the dark blond loaves from the oven. Although it's best to let bread rest and cool, the smell insisted that we try one loaf almost immediately. Steam escaped as Mike held a hot loaf in a diaper and broke open the crust. \"I like to rip bread,\" he said. \"It's not the same if you slice it.\" The kitchen went quiet as everyone lingered, dipping the still-hot bread into small bowls of olive oil laced with herbs and garlic. The crusts of the other loaves crackled as they cooled on a nearby counter. \"I could do this, I could really do this,\" mused Jodi. \"I bet my son would eat this, too.\" Jeff brought out a bottle of white wine and poured short glasses. Everyone sampled one another's pasta sauce, plus the _pomodoro_ and spaghetti sauces. The room erupted into a series of \"mmmm's\" and \"yumm's.\"\n\n\"It's all good, I mean really good,\" Dri said. \"It's kind of surprising, really, all the different dishes you can do with a few ingredients.\"\n\nThe group collected their pasta and still-warm loaves to take home. Sweating under my own apron, I wished everyone a good night. Mike took my hand and led me across the street. We sat on the curb cloaked in shadow in the cooling night air. Late on a Monday, the normally busy thoroughfare was quiet. A waning moon lit the sky. \"It's amazing,\" Mike said. \"Last week we were on the other side of the world looking up at the moon, remember?\" I leaned my head on his shoulder.\n\nThen we heard a voice, a familiar one. \"I made bread. Yes, me. Yes, really.\" Donna emerged, balancing her cell phone on her shoulder, holding a box of pasta and a loaf of bread in her arms. \"I have pasta, too.\" Pause. \"No, _I_ made it. Yes, _me_. Don't sound so surprised!\" She got into her car and drove off into the night.\n\n#   **No-Knead Artisan Bread for Busy People**\n\n_This recipe is adapted from the master recipe in the excellent book_ Artisan Bread in Five Minutes a Day _by Jeff Hertzberg and Zo\u00eb Fran\u00e7ois. It's simple to prepare and the dough keeps in the fridge for up to two weeks. I often add a tablespoon of dried thyme, rosemary, or herbes de Provence to the water to infuse the bread with extra flavor. The recipe was designed to work on a baking stone, but I get similar results with a shallow cast iron skillet. A cookie sheet will work, but your loaf may not get quite as brown and crusty. You can find the original recipe plus helpful photos and variations atwww.artisanbreadinfive.com._\n\n**YIELDS ABOUT FOUR ONE-POUND LOAVES**\n\n3 cups lukewarm water (about 100\u00b0F)   \n1   tablespoons yeast   \n1 tablespoon kosher salt   \n6   cups (32 ounces\/900 grams) unsifted unbleached all-purpose   \nwhite flour   \nAdditional flour to create loaves   \nCornmeal\n\n Combine the water, yeast, and salt in a 5-quart bowl or plastic food container with a lid. Stir to mix. Add all of the flour at once and mix with a wooden spoon until the dough is wet and sticky with no dry patches. Cover with a lid or plastic wrap, but do not seal airtight. Let it rise for about 2 hours at room temperature. If you are not using it immediately, refrigerate the dough, covered, for up to 2 weeks.\n\n To make a loaf, lightly sprinkle some flour onto the dough's surface. Scoop up a handful the size of a grapefruit, and cut or tear it away from the remainder. Rub the dough with a layer of flour while gently stretching the top around to tuck the sides into the bottom to form a round, smooth loaf. Put the loaf on a pizza peel or cutting board dusted with cornmeal to prevent sticking. Let it rise, uncovered, for at least a half hour or as long as 90 minutes. The loaf will plump but not change radically in size.\n\n About 20 minutes before baking, preheat the oven to 450\u00b0F. Place a broiler tray or other metal pan on the bottom rack of the oven. Put the baking stone or cast iron skillet on the middle rack.\n\n Dust the loaf liberally with flour. Slash the top with a cross or three lines with a sharp knife and slide it onto the preheated baking surface. Carefully pour about 1 cup of hot water into the broiler tray or metal pan and close the oven door to trap the steam. Bake for about 30 minutes, until the crust is browned and the loaf feels light and hollow. Cool to room temperature.\n\n_**Note**_\n\nFor a \"lazy sourdough,\" mix the next batch in the same container without cleaning it first. You can substitute 2 cups of whole wheat flour for the white flour if desired.\n\n#   **Basic Alfredo Sauce**\n\n_This is a great way to use up leftovers such as shredded chicken, cooked shrimp, grilled vegetables, and so on. Just toss them in near the end of cooking. Use cream, not milk. Start cooking the pasta before you begin the sauce. As an easy shortcut, toss chopped vegetables such as broccoli, asparagus, peas, artichokes, and the like into the pasta water to cook them briefly; frozen vegetables work well. Add cooked chicken, vegetables, diced ham, cooked shrimp, and whatever you have on hand with the pasta and heat through._\n\n**MAKES 3 TO 4 SERVINGS**\n\n8 ounces cooked pasta   \n2 cups heavy cream (2 tablespoons reserved)   \n  teaspoon salt   \n cup grated Parmesan or Parmigiano-Reggiano cheese   \n1 garlic clove, minced (optional)   \nFreshly ground black pepper\n\n Prepare the pasta according to package directions. Carefully reserve one cup of the pasta water to use in the sauce. Over medium-high heat, add all but 2 tablespoons of the cream to a saut\u00e9 pan or skillet. When it bubbles, add the salt. Small bubbles will erupt into larger bubbles. Stir. When the sauce thickens enough to cover the back of a spoon or leaves a clean line in the bottom of the pan when you pull a spatula across it, add the pasta water. Cook over medium-high heat for about 3 minutes, until it bubbles again and the sauce thickens. Add the reserved 2 tablespoons of cream, heat through, and then add the cheese, garlic (if using), and a few cranks of pepper. Taste, and add more salt if needed. Add the cooked pasta and any additional ingredients and stir well to coat.\nCHAPTER 8\n\n**Tossed Salad & Scrambled Eggs**\n\n  **LESSON HIGHLIGHTS:**\n\n**Salads and Omelets as Vehicles for Leftovers, and the Miracle of Vinaigrette**\n\nThe autumn before I started the project, I donated the first of several cooking-class dinners to charity. A sweet couple forked over seven hundred dollars to a worthy cause for my services, blissfully unaware that, at the time, I had never taught a cooking class.\n\nAs Lisa and I arrived at the door clad in chef's jackets and armed with cooking gear, an elegant woman with a bob of highlighted blond hair holding a white wine spritzer opened the door. \"Hi, we're all so excited!\" she greeted us enthusiastically. \"Yep, it's the chef. She's here!\" she yelled over her shoulder to her guests. She kept one hand on the doorknob and waved us inside with the other clasped firmly on a flute of champagne. She smelled vaguely of Chanel No. 5.\n\nHer guests gathered around the granite-topped kitchen island littered with wineglasses. The house was a well-decorated atomic-era ranch style with an expansive backyard blanketed with lush green grass and ringed by evergreens. The friends stood around in casual summer attire, the women in capris and the men in Dockers and button-down short-sleeved shirts. Two of the men put down their drinks and left to help Lisa lug in the rest of our cooking gear from her truck. As I exchanged pleasantries with the hosts, a dark-haired fellow holding a gin and tonic cheerfully asked, \"So, Chef, where's your restaurant?\"\n\nUgh, that question. \"I don't have one. I'm a food writer,\" I replied.\n\n\"Oh, so you're not a real chef,\" he said, a little confused.\n\nThe hostess interjected, \"Don't put her on the spot. Of course she's a real chef! She went to culinary school. In Paris, no less!\"\n\nBooze comes in handy for dodging questions. \"Oh, hey, I almost forgot! We brought champagne to make kir royales.\"\n\nAfter I made a round of cocktails, I opened the fridge door to put the remnant champagne inside. Three dozen bottles of vinaigrette clanked like wind chimes caught in a Category 1 hurricane. \"That's quite a lot of dressing you've got in there,\" I said to the hostess. \"Why don't you make your own?\"\n\n\"Oh, I'd have no idea how,\" she said, fingering her pearl earrings.\n\n\"Do you have a jar?\" I asked.\n\nWe ransacked the cupboards, pulling a motley collection of barely touched bottles of oil and vinegars of seemingly every shape, size, and flavor and set them on the kitchen island. \"Okay, everyone, gather around,\" I said. As I explained the impromptu tasting session of oils and vinegars, the guests exchanged dubious glances.\n\n\"But are those even edible alone?\" a woman asked.\n\n\"Of course,\" I said. \"You need to know what each tastes like so that you'll have an idea when you put them together.\"\n\nWe started with the oils, a variety of olive, canola, safflower, and sesame and a small aging bottle of walnut. We set aside a bottle of olive oil that had turned rancid. \"Oils don't have the shelf life most people think,\" I said. \"It's, like, six months, and less if you store them over the stove like you're doing here because the heat breaks down the oil.\" Lisa and I had everyone taste the good oils and then the bad oil. \"Rancid oil isn't bad for you, but the unpleasant flavor and acrid taste will ruin whatever you decide to do with it.\" We moved on to the vinegars, which ranged from cider to white wine to red wine to balsamic.\n\nThen Lisa explained that vinaigrette dressing is just a simple ratio. \"It's generally three to one,\" she said. \"Three parts oil, such as olive oil or sesame oil, to one part acid, such as vinegar or citrus juice.\" The hostess and I measured three tablespoons of olive oil and one tablespoon of balsamic vinegar into her empty jar. We added some salt and pepper. I gave it to the hostess to shake fiercely.\n\n\"Now what do we do?\" she asked excitedly.\n\n\"Congratulations. You just made vinaigrette,\" I said. She looked proud and held it above her head as if she'd won a trophy at Wimbledon. Everyone cheered. A month later, she sent an e-mail thanking me for the class, adding, \"I haven't bought a bottle of salad dressing since that night. This sounds crazy, but it changed my life. It made me start to wonder what else I have been buying that I could just be making myself.\"\n\nI printed out that e-mail and brought it to the next class. I told the volunteers the story, and then read the e-mail aloud to them as they gathered around the worktable. Only one short week after the numbing heat of the pasta and bread class, the kitchen temperature was nearly twenty-five degrees cooler.\n\n\"That woman had everything she needed to make vinaigrette,\" I said. \"She was missing only one ingredient: the know-how to make it.\"\n\nVinaigrette is one of the more expensive products to buy by weight at a grocery store. A sixteen-ounce bottle of dressing can run anywhere from $3.60 to more than seven dollars. At an average of thirty-two cents an ounce, that's nearly forty-one dollars per gallon. Yet it's one of the easiest and cheapest things to make at home.\n\nAfter the story, we circulated copies of a basic vinaigrette recipe. At the top of each page: \"oil + acid = yummy.\"\n\nLisa and I led the volunteers on a raid around the kitchen to collect oils, vinegars, soy sauce, jams, chutney, fig paste, cheeses, garlic, dried coconut, gingerroot, spices, red wine, lemons, limes, olives, and some of the Dijon mustard from the chicken lesson. We broke the class into two-person teams. \"If you can think of something you had eating out, consider the flavor profiles, the way we did in the chicken class. What flavors make something taste Italian? Name them.\" We went around the table: Parmesan cheese, oregano, red wine, tomatoes, basil.\n\n\"Thai?\" Basil, coconut, curry, lime, hot chilies.\n\n\"I usually buy a strawberry balsamic. How would I make that?\" Shannon asked.\n\n\"In summer, you can add mashed-up fresh strawberries. The rest of the time, you can just use a bit of strawberry jam,\" I said, an answer that generated a collective \"ah\" of enlightenment.\n\nThe group went to work tasting and talking, a cacophony of whisks in motion, stopping and tasting again. Jodi and Dri made an Asian version with sesame oil, grated gingerroot, soy, and lime. Another team crafted a balsamic with fig paste and bleu cheese. Sabra and Gen made classic mustard and white wine vinaigrette with fresh thyme thrown in.\n\nThen we stopped everyone. \"Okay, whisks down!\" Lisa commanded. She and I had bought a couple of packages of mixed greens. We set them on the table. \"Pick out different salad leaves and dip them into your vinaigrette. You want to see how the flavor pairs with your lettuce, since all lettuces taste different,\" she explained. \"Remember, there is no right or wrong answer. Whatever tastes good to you is all that matters.\"\n\nEveryone stopped and delicately selected a few leaves\u2014radicchio, butter lettuce, romaine, arugula, fris\u00e9e among them. The group chatted and murmured. \"This one's bitter.\" \"This one doesn't go with the ginger.\" \"Huh, it's great with the arugula but terrible with that red leaf.\"\n\nThen each pair rotated around the table, tasting the other teams' creations with all manner of lettuce. They were all good, but the fig and bleu cheese balsamic knocked it out of the park. I made a mental note to try it myself.\n\nEveryone had such a great time with the salad and vinaigrette lesson that it was tough to pull them away from it. But it was time to move on to the eggs.\n\n\"Omelets don't get the respect they deserve,\" says Jamie Oliver. He's so right. An omelet is inexpensive, easy, satisfying, and a good way to use up leftovers. Plus, who says that eggs are only for breakfast? A famous food writer, Elizabeth David, wrote a whole story about the beauty of an omelet and a glass of wine for dinner.\n\nOne of my chefs at Le Cordon Bleu used to say, _\"Apprendre \u00e0 faire une omelette et vous n'aurez jamais faim.\"_ If you learn to make an omelet, you'll never go hungry. He told my Intermediate Cuisine class a legendary story about Napoleon Bonaparte. His army was traveling in southern France and spent a night in a small town, Bessi\u00e8res. The owner of the inn where Napoleon stayed made him an omelet. He had never had one before and found it so wonderful that the next day he requested the townspeople to gather up all the eggs in the village to prepare an enormous omelet for his army. They won the battle, of course. If they hadn't, no one would make omelets.\n\n\"Omelets are great for leftovers because, well, you can put almost anything into them and serve them for breakfast, lunch, or dinner,\" I said. \"I've made omelets filled with every possible leftover including steak and bleu cheese. Otherwise, the method is always the same.\"\n\nI put a seven-inch nonstick skillet over one of the portable burners and dropped in a dollop of butter. As it warmed, I cracked two eggs into a bowl, added salt, pepper, and a bit of thyme, and whisked. The eggs made a \"shhhh\" sound as they hit the surface of the warmed pan. I turned the heat to low, then tilted the pan around to assure that the egg coalesced evenly. Once it had cooked through, I added a bit of cheese, folded it over with the edge of a spatula, and slid it from the pan. Voil\u00e0.\n\nEach volunteer made an omelet. There may be nothing more gratifying than watching someone make an omelet for the first time. Some came out better than others, but student after student had the wide smile of accomplishment as she watched hers fold onto her plate. Andra openly chuckled. \"This is so easy! I don't know why it never worked when I tried them. I'd start out with an omelet but end up with scrambled eggs.\"\n\n\"Funny you mention that\u2014making scrambled eggs is the last lesson of the day.\"\n\nThe trick to great scrambled eggs is to start with a cold pan. Add a couple pats of butter and the eggs, and then turn the heat up to medium-low and stir regularly. It's a longer process but yields a softer result than the classic high-heat version in which the eggs take on an elastic flavor and a rubbery consistency. Lisa whipped up some cream and added a bit of cayenne to it, then spooned it over the scrambled eggs, a variation on a recipe by famed chef Jean-Georges Vongerichten. \"This is so good; I could eat a mountain of these eggs,\" Andra said.\n\nI had been worried that everything we taught was too simple. Yet that day I was reminded that in an unsure world, few things are invariably as good and true as a humble omelet, a simply dressed salad, and a glass of wine. Later, Gen told me that shortly after the class, she found herself without a plan for dinner one night. \"But I realized that I had eggs, so I made an omelet. I chopped up some leftover asparagus and threw it in. My roommate was so impressed. She was like, 'Wow, I can't believe that you just made an omelet, just like that.' So I taught her how to make one, too.\"\n\n#   **DIY Vinaigrette**\n\n_Vinaigrette is among the simplest things to make yourself, yet one of the most additive-riddled items in the supermarket. Extra vinaigrette will keep for a few days in an airtight container in the fridge. Experiment. Use leftovers for inspiration. If you hate it, you've wasted thirty cents in ingredients. Some lettuces reduce an acid tang while others amplify it, so taste your concoction with a leaf or two to taste before serving._\n\n_Here's the basic formula:_\n\n1 part acid + 3 parts oil = fabulous stuff\n\n In a bowl, combine the one part acid, a pinch of coarse salt, and a few grinds of fresh pepper, then whisk in the three parts oil, or add them all to ajar and shake vigorously. Taste. Congratulations, you've made vinaigrette. If it's too acidic, add oil. If it's too sour, add a little sweetener. If it's too oily, add more acid. Just add very small amounts until it tastes balanced and good to you. If you have time, let it rest for at least 15 minutes, then taste again.\n\n Acid and oil don't like to stay mixed, so you can add an emulsifier, like an egg yolk or a dab of mayonnaise or mustard or honey, before adding the oil. Additional seasonings to consider include some minced shallots or garlic, herbs, or spices.\n\n Want fancy designer gourmet vinaigrette? Raspberry balsamic, for example? Add a handful of berries, fresh or thawed from frozen, or a teaspoon of berry-based jam. Asian vinaigrette? Try some combination of lime juice, ginger, garlic, miso, sesame oil, peanut oil, soy sauce. French? Dijon mustard, shallots, tarragon, white wine vinegar. For more help developing vinaigrette flavors, check out the \"Cheat Sheet\" to Flavor Profiles in the Extra Recipes section at the back of the book.\n\n#   **Your Basic Omelet**\n\n_Omelets cook quickly, so assemble any fillings that you want to add before you start. For a two-egg omelet, you can use a seven-inch nonstick skillet; bump it up a size for three eggs. Otherwise, all you need is a small bowl, a fork, and a spatula. I like to use a dab of olive oil and butter to give the omelet a little bit of flavor. If you want to use vegetables such as onions, peppers, or mushrooms, you may want to precook them briefly._\n\n2 or 3 eggs   \nSalt and freshly ground black pepper to taste   \nA pinch of dried thyme or mixed herbs (optional)   \nA splash of milk or water (optional)   \nAbout 1 teaspoon oil (such as canola or olive), cooking spray, and\/   \nor a pat of butter   \nIngredients to fill your omelet, such as cheese, ham, vegetables,   \netc.\n\n  Combine the eggs in a small bowl. Add a pinch of salt, a couple of grinds of pepper, and the dried herbs, if using. Add a bit of milk if you want a fluffier omelet. Whip using a fork or a whisk. The longer you whip, the fluffier the omelet.\n\n Heat a nonstick pan over medium heat. Add the oil, cooking spray, or butter. Add the egg mixture. Tilt the pan around so that the egg runs to the open areas of the pan to form a consistent layer. Let the egg cook through briefly and then carefully use a spatula to pull up one side and let the remaining uncooked egg go under the omelet to the pan's surface. Once the egg layer thickens, lower the heat and add the other ingredients, such as cheese, vegetables, and cooked meat or seafood. Heat the omelet just until the egg is cooked through, but not too browned on the bottom. Using the spatula, carefully tuck around the edges to ensure that the omelet will slide freely from the pan.\n\n Fold your omelet in one of two ways. Carefully fold it in the pan using your spatula, and then slide the finished omelet onto the plate. Or put a serving plate on the counter and carefully tilt the omelet toward it. Once the edge of the omelet hits the plate, use the spatula to guide the other edge over in a fold.\nCHAPTER 9\n\n**Udder Confusion**\n\n  **LESSON HIGHLIGHTS:**\n\n**When It Comes to Meat, It's Worth Knowing**   \n**Your Butt from Your Round**\n\nNot long after I moved to London in the autumn of 1999, I scanned newspaper headlines while waiting for a bus at Oxford Circus on a dreary winter's morning. TWO MORE DEATHS LINKED TO MAD COW, screamed one headline. TODDLER ORPHANED! blared another.\n\nLike a lot of people living in the United Kingdom in the wake of the BSE epidemic, I stopped eating red meat. The problem wasn't limited to British beef, although the issue had the greatest impact in the United Kingdom. I swore off poultry and pork, too. What tripped me up were the crispy, fragrant ducks hanging in the windows of the city's small Chinatown area that I passed by on my way home from work. I tried changing my route, but the ducks, shiny as if varnished with a chestnut-colored lacquer, called to me like an artful siren luring the unsuspecting to treacherous rocks.\n\nI relented the way I imagine alcoholics fall off the wagon. I went into Lee Ho Fook's alone late one night. I'll just get a few steamed pork dumplings, I told myself. I can stop anytime I want. Everyone knows that dumplings are a gateway food. Before I knew it, I'd inhaled half a duck. I sorted through the remnants, prying tiny bits of dark meat from the bones with the enthusiasm of someone who has just returned from a space mission and subsisted on nothing other than gel packs and Tang for months. I walked home through the busy environs of Chinatown with my head down. I swore that it would never happen again. The next night, I went back. Settled in a discreet booth and wearing dark glasses, I broke the land speed record for consuming a full order of moo shu pork. After that, the floodgates opened. I bought my first whole chicken in more than three years and I never looked back.\n\nAll this transpired not long before I started studying at Le Cordon Bleu in Paris. I felt particularly unprepared for the routine dismemberment of lambs, ducks, chickens, and beef that was de rigueur in the course of culinary school.\n\nEven more than chicken, beef comes with baggage. So many people buy chicken in nonoffensive white breast meat hunks that it's ubiquitous enough that the \"meat thing\" doesn't even factor into the discussion. It's kind of like that scene in _My Big Fat Greek Wedding_ when an aunt, on hearing that the main character's fianc\u00e9 is a vegetarian, cannot fathom that it's an issue. The aunt replies, stunned, \"He don't eat no meat?! What do you mean, he don't eat no meat? Oh, that's okay, I make lamb.\"\n\nDuring the kitchen visits, many of the volunteers expressed that the meat department in their local supermarket was often a place of confusion. How to tell what cuts work for different types of cooking? \"I look at packages of meat and it's just baffling,\" Shannon said. \"I know that a T-bone is a good steak, but I am not sure what should be roasted and what should be grilled. And what the heck is chuck?\"\n\nCheryl shopped at a local food co-op. The labels confused her. Choice? Prime? Natural? Pasture-raised? Grass-finished? She didn't know what they meant.\n\nThey're all good questions. I didn't know the answers to many of them either, especially around meat classification. So I did some research.\n\nUnsurprisingly, beef is big business. The United States produced nearly 27 billion pounds of it in 2008, the result of processing about 33 million head of cattle. On average, Americans consume about 54 pounds of beef per capita each year. That's nearly one Quarter Pounder from McDonald's for every man, woman, and child in the United States every day of the year. More than 40 percent of all cattle raised meet their destiny as hamburger.\n\nI wanted to break the lessons out of the grind, so to speak, so I felt it was time for another guest teacher. I'd recently met Robin Leventhal, the former chef\/owner of Crave, a minuscule wedge of a restaurant in my neighborhood that specialized in beautiful comfort food. For the past few years, Robin had been winning a battle against two forms of lymphoma, yet lost the lease on her popular restaurant after a protracted negotiation with her landlord.\n\n\"It was both devastating and liberating,\" she said on reflection. \"For once, I wasn't a slave to my phone twenty-four hours a day.\" Released from the pressures of a restaurant, on a whim she auditioned for _Top Chef_ and earned a place as a cheftestant in the sixth season of the series from a field of thousands of candidates. Over a couple of celebratory cocktails, we ended up chatting more about her background. She started life in the art world, earning a master's degree in ceramics. \"I didn't want to be a starving artist,\" she said. Instead, she went to work feeding people. In food, she found a creative outlet that paid the bills and that was as emotionally satisfying as art. \"For me, feeding people is primal and maternal. It's arguably the most satisfying experience in the world.\"\n\nA week later, thinking about her comment, I called to ask if she would be willing to teach a class. Among the best dishes that I'd had in years were her simple braised specials at Crave. She agreed. For her class, we'd focus on cuts of meat, how to prep them and get the most from inexpensive cuts. But I hoped some of the ethos that Robin felt about cooking and her passion to nourish people would come through in the lesson as much as anything she could tell the volunteers about shoulder roasts.\n\nWhen I told Lisa and Maggie that Robin was coming in to teach, they had identical reactions. \"No freakin' way,\" Maggie said. No cupcakes would keep her from that class. \"Seriously, whenever I would go to Crave, the waiter would start to rattle off specials, and once he said 'braised whatever,' I'd just say, 'I'll take that.' The server would say, 'Don't you want to know what it comes with it?' I'd say, 'Uh-unh. Don't know, don't need to know. Just bring it.'\"\n\nThe day of the class, Lisa and I raided a local supermarket to collect twenty-two packages of beef, lamb, and pork, essentially clearing out the \"manager's special\" discounted area. We had ribs, round steak, tri-tips, oxtails, London broil, T-bones, beef shanks, lamb shanks, lamb steaks, pork shoulders, and something called chuck Denver steak. We dragged nearly forty pounds of meat into the teaching kitchen. To witness that much meat in one place is both impressive and disturbing. The students filed in, and as each one approached the table she stopped dead in her tracks. \"Oh, wow, that's a lot of meat,\" Cheryl said, summing up the group's reaction.\n\nRobin has stores of quiet energy, yet a big personality. She's also physically impressive, a square-jawed, solid woman with dark, olive skin and black eyes. Her black short-sleeved shirt accentuated hard biceps. Whenever I saw her, I made a mental note to check out Pilates and never to cross her because, frankly, she could take my pale carcass in a fight without breaking a sweat. She stood demurely behind me as I introduced her. \"In two days, she'll be on the first episode of this season's _Top Chef,_ but tonight she's going to be teaching something very important\u2014meat.\"\n\nEveryone seemed suitably impressed. She was going to be on TV! We handed out diagrams of cows and pigs with dotted lines to demarcate various cuts. Even at the paper version Trish recoiled a bit. As Robin talked, Trish pulled me aside between two towering metal racks of dried goods and tableware as everyone took turns washing their hands. \"So, this is kind of awkward,\" she said, literally wringing her hands, the way she had when I visited her kitchen. She cast a glance back at the dense collection of red meat on the worktable. \"We don't eat much meat now, so I may not be able to stay for the full class.\" Her anxiety was palpable. I assured her that I understood and told her of my fling with vegetarianism.\n\nWe started with a basic lesson on recognizing cuts. The July heat crept into the kitchen. Two minutes into the explanation, half the class started fanning themselves with the meat diagrams. Their collective eyes started to glaze over as Robin and I began to discuss chuck and sirloin. I looked at Maggie and Lisa. We all sensed we were losing the audience already. Lisa bent over abruptly.\n\n\"So, everybody, look at me, I'm a cow,\" she said loudly. Possible comebacks rattled around my brain. The students exchanged glances and snickered. Lisa waved a finger good-naturedly at the group. \"No wisecracks!\" she warned. \"Okay, see my shoulder? It gets a lot of work. Feel your shoulders and arms. There's a lot of muscle there. It's tough. This is where chuck comes from,\" Lisa said. The group snapped out of its stupor. Each person started to feel her own shoulders.\n\nThen Lisa slapped her thighs. \"This is where shank and brisket come from, and you'll want to braise them the way we did with the legs and thighs in the chicken class. We'll do another one tonight.\" The class kept their eyes on her, feeling their upper legs. \"The meat near the ribs is tender. It isn't used as much. This is where you get your prime rib and rib-eye steaks.\" She put her hands on her lower back. \"Behind the ribs are the short loins and then sirloin. That's where the most expensive cuts come from. Think tenderloin, filet mignon, that kind of thing.\"\n\nAs they prodded their own backs and ribs, it struck me that while effective, there was something curiously macabre about demonstrating meat cuts using the human anatomy.\n\nLisa took one last fling as a cow and put her hands on her buttocks. \"Feel your butt. Round, right? That's where round comes from. It works a little harder than the back, so it's tougher.\" Then she bolted upright. \"Okay, I can't believe that I just did that,\" she said, laughing. Everyone clapped, and she took an embarrassed, quick bow.\n\nThen we quizzed them. \"Where is your short loin?\" The group tentatively reached to their lower backs.\n\n\"Round?\" Looking at one another and stifling giggles, they grabbed their buttocks.\n\n\"Your chuck?\" They pointed to their shoulders.\n\nWe then looked over the vast assortment of meat on the table. Maggie videotaped the class as Lisa, Robin, and I started to pick up various packages and explain how to prepare them. We grouped them by cuts. Robin took over. \"Most people know steaks or ribs, so we're just going to focus on the lesser-known cuts. Anything that says chuck is tough, so these are great for stews, pot roast, or braising.\n\n\"The other one is round,\" Robin continued. \"If it says round, just think of that as a cue to marinate it before you cook it. For instance, London broil is a cut from round, and normally you marinate that and then grill it.\"\n\nEvery so often, though, you may run into a cut you've never heard of. \"For instance, let's discuss this Denver chuck steak,\" Robin said as I picked up a vacuum-packed oblong cut of beef. It turned out to be one of the \"new\" cuts developed by the National Cattlemen's Beef Association. The good folks at the Cattlemen's Association dedicated $1.5 million and five years to finding new ways to market lesser cuts of meat. Cuts that might have normally found their way to hamburger had been salvaged as whole pieces, including the Denver, an inexpensive, well-marbled piece of chuck that could be cooked like a steak.\n\n\"This piece used to be ground into hamburger. You'd pay three or four dollars a pound or whatever for it. Now that they call it a steak, they can charge seven to twelve dollars a pound. I'm not saying that it's a bad cut of meat, it's just that meat producers are always coming up with new things, so don't get discouraged if you don't recognize it.\"\n\nRobin picked up two T-bones. \"But this is a good example of why you should learn a little about meat cuts, or get to know your butcher,\" she said. \"This should have meat on both sides of the bone. One side is a strip steak, and the smaller part is the tenderloin. But the butcher cut off the tenderloin and left just the strip steak. The difference?\" She selected another package of meat. \"Here's a New York strip. It's $8.99 a pound. This T-bone is $12.99 a pound. But you didn't get half the T-bone, or the most expensive part of the steak. It's hard to know if this was a mistake or if it was on purpose. But if you bought this, you'd have been robbed.\"\n\nEveryone had questions. Among them, what the difference is between organic and \"natural,\" two common distinctions, and the controversy over grain-fed beef.\n\nTo start, it's worth knowing what's in commercially produced meat. About 70 percent of all the antibiotics produced in the United States are fed to healthy livestock, including pigs, chickens, and cows. Most commercial beef breeders in the United States and Canada inject their cattle with hormones designed to promote rapid growth. The beef industry maintains that the hormones have been vigilantly tested for safety. Critics argue that hormones cause everything from early onset of puberty to increased predisposition for certain types of cancer.\n\nMost commercially produced cattle start on an individual farm but are \"finished\" in a feedlot by being fed copious amounts of corn and grain, foods that cows are not physically designed to eat. But much of that stems from consumer demand; many people prefer corn-fed beef and its more heavily marbled texture to the leaner flavor and mouthfeel of grass-fed beef.\n\nBy contrast, organic beef farmers are prohibited from giving their charges antibiotics, growth hormones, or anything other than organic feed. The \"natural\" description for beef allows more flexibility, but generally the beef must be free from growth hormones and antibiotics and minimally processed, although in some cases it simply means that it wasn't ground into hamburger.\n\n\"A lot of smaller farms that produce grass-fed beef don't meet organic requirements, and some organic beef comes from cattle raised in feedlots that are just fed organic corn,\" I said. \"I sound like such an old saw on this, but the only way you know the difference is to know the company that developed your beef. Look up the companies that supply your supermarket or butcher. Ask them questions.\"\n\nRobin chimed in. \"Meat isn't cheap. Look at your grocery bill for a month. How much are you spending on meat? It's probably a bigger investment than you think. So it's worth thinking about.\"\n\nBeyond cooking, I wanted to instill the idea of considering the provenance of meat, not just its cost. So I told them the story of Betsy, the one and only cow we ever had on the farm growing up in Michigan. Betsy was a gentle bovine with a lazy disposition and impossibly long eyelashes. When we named her, my mother was not pleased.\n\n\"Don't name the cow,\" Mom said.\n\nMy brother, a gifted artist, drew a loving portrait of her. \"Don't draw any more pictures of the cow,\" Mom said.\n\nThe kids spent hours taking turns walking her through our ample meadows blanketed with clovers and petting her rough brown coat. \"Don't pet the cow,\" Mom said.\n\nOne day in autumn, a man in faded gray overalls showed up at the farm with a trailer. Dad said that he was going to take Betsy to a nice farm for the winter. Betsy docilely filed past the kids, who stared, stunned, and then, as if on cue, started bawling in unison. My sister couldn't take it. She tore herself away from my dad's leg and ran over to her, dramatically throwing her small arms up toward the cow's neck. Sobbing, she kissed her good-bye. \"We'll miss you, Betsy! I will dream about you every night until you come back next spring.\" My father gently pried her away.\n\nAbout two weeks later, my sister discovered new paper-wrapped parcels in the chest freezer in the barn. The first night we had spaghetti made with beef, she looked up at my brother's charcoal sketch of Betsy and then stared back down at her plate. She ate it anyway.\n\nHalf the class looked mortified. \"You ate your cow,\" Donna said flatly.\n\n\"Yes, and all this\"\u2014I waved my hand over the remainder of the meat we'd moved to the side table, still covered in plastic wrap\u2014\"was once a Betsy,\" I said. \"But we knew what _she_ ate; she didn't get any antibiotics or hormones. She also didn't eat any corn. She just ate grass, which is what cows are supposed to eat. So the closer you can get to meat from a cow like Betsy, the better.\"\n\nAlthough we could have talked about beef all night, the clock was ticking. We set out bowls of onions, carrots, and celery. Everyone plowed through their mirepoix as if they'd been doing it for years. What would have taken a half hour in the first class took less than ten minutes. \"Wow, you guys are good,\" Robin observed as Lisa and I collected it all in bowls. I felt like a proud stage mother.\n\nThen we shifted to pork to practice on a cut no one in the class had ever purchased: pork shoulder. Robin, Lisa, and I handed a hunk of pork shoulder to each student. Robin guided them through breaking down the larger pieces, feeling for bones and removing unwanted pieces such as gristle, tough sinew, and hard tracts of white fat.\n\nTrish, the gentle sixty-one-year-old part-time vegetarian, gave it her best. I watched her from across the room. She looked anguished as she held her chef's knife poised above her pinkish meat. But then she calmly set it down, stood back from the table, and purposefully took off her apron. Trish walked over to where I stood near the door. \"I'm sorry, but this is just too much meat for me,\" she said apologetically. She picked up her purse and left.\n\nThe rest of the class, however, seemed transfixed. While they had seemed so squeamish around the chicken, they had no such inhibitions with beef or pork. Most had never tackled a chunk of meat, and who could blame them? Given this era of prepackaging, you can go your whole life never having to cut anything beyond the bite-sized portions required for mastication. No reason exists to explore the sinew, to feel the complexities of muscular development, or to feel the stiff line of a bone giving way to softer flesh unless you plan to break it down as we were doing that day. Dri, who had been so reluctant to touch her whole chicken, now maneuvered her knife with the laser focus of a passionate scientist. Jodi explored the flesh with her fingers, massaging it and trying to understand the contours.\n\nRobin wandered around the table inspecting, offering advice and quiet support. \"Exactly, let your blade follow the line of the meat,\" she told Shannon. Cheryl had seemed a bit intimidated to start. Robin literally held her hand. \"Just like that. You're doing really well. See? You can do it.\" Cheryl asked if she'd learned about meat in culinary school. \"Oh, no, I didn't go to school,\" Robin said. \"I kind of fell into cooking. A chef took me under her wing and I never looked back.\"\n\nStanding next to Cheryl, Shannon overheard the conversation. \"Was your mother a big cook?\" she asked without looking up as she tenderly cut her meat into cubes. I thought of Shannon's response on her initial questionnaire, that her own mother had invariably shooed her out of the kitchen.\n\nRobin let out a robust hoarse laugh. \"Ha, no! My mother is a terrible cook. Her ideal house would contain no kitchen. I think part of the reason I started to cook was to rebel against her.\"\n\nWe shifted to start the evening's braise. This time, we handed out a two-quart saut\u00e9 pan to each person and gave her the option of a single gas or electric burner at stations around the kitchen. \"We can also have at least two people at the big six-burner stove,\" I said. Like a shot, Dri moved into position before anyone could take it away from her. During the classes, some of the students had seemed intimidated by the high heat emitted from the commercial stove. Not Dri. She seemed almost ready to trade her urban planning job for a spot on a restaurant hot line.\n\nAs we had for the chicken, we started by searing the pork in hot oil to brown, nearly crispy. Then each volunteer added the mirepoix, some stock, wine, and some herbs. The air took on a heady, meaty fragrance. One by one, the volunteers covered their pans and shifted them into the oven, holding the hot handles with their diapers. \"Remember where you put it in the oven, so you can remember which one was yours,\" I advised as we slammed the oven doors shut.\n\n\"Great!\" Robin said. Aware of the time, she moved swiftly onward. She heated a large skillet on one of the electric burners and waved everyone to join her around the worktable.\n\n\"One simple way to add a lot of flavor to any cut of meat is to coat it in a spice rub,\" Robin began. Spices have a shelf life of about a year, less if exposed to light or heat. Spices left intact in their original form, such as allspice berries, cinnamon sticks, or whole nutmeg, last significantly longer. \"I'm as guilty as the next person. I've got ground oregano that I've had for twenty years in my cupboard. I'm sure it tastes like nothing. But one thing you can do, either with whole spices or those that you've kept around a little too long, is to bring out the flavor by toasting them.\"\n\nShe demonstrated by adding cardamom to the pan. A common ingredient in Indian curry, cardamom pods resemble sunflower seeds shrouded with a light green or dark brown papery exterior. \"Notice I'm not using any oil,\" she said. \"We're just dry-toasting them.\" She swirled them around in the pan. After a minute, everyone started to sniff at the air as the pods released their strong licoricelike scent.\n\n\"I can smell that, totally,\" Sabra said. Everyone nodded.\n\nRobin smiled. \"Mmmm, doesn't it smell great? After a last swirl, she dumped the pods onto a plate. \"Next, I'm going to do the same with cumin. A lot of people know it from making chili and usually buy it ground.\" She tossed the oblong, ridged, beige-hulled seeds into the pan. Again she swirled them over the heat. \"Just heat until you can smell them and dry them a bit further.\" Cumin has a distinct scent, at once reminiscent of a pot of chili and falafel. It's a common ingredient in both Tex-Mex and Middle Eastern cuisine, arguably one of the few things those two regions have in common.\n\nWhen the toasting was over, Robin grabbed the large gray marble mortar and pestle from the kitchen's spice rack. She combined a few pinches of both the toasted cardamom and the toasted cumin in the heavy bowl along with some coarse salt and whole peppercorns. \"Who owns a mortar and pestle?\" she asked. \"These are a great investment,\" she said as she put her weight into the pestle, crunching and crushing the whole seeds into the bottom until they crumbled into dust. \"You know, you can get one for about ten bucks. You can buy whole spices and then grind them up. They'll store longer and taste fresher. Plus it's therapeutic, a great way to take out the frustration of a bad day.\"\n\nRobin put a spoonful of the powder on a plate. She instructed them to smell or taste it and passed it around the table. \"It's really powerful, almost pungent,\" Shannon said. \"That's just something that I would never have thought to do.\"\n\n\"Now we want you to make your own,\" Robin said. \"Just think of flavors you like. Your taste is something to listen to. Trust your instincts,\" she said. In many ways, it was a reprise of the vinaigrette lesson, but this time without oil or vinegar. \"Be bold,\" Robin advised. \"Experiment. Strong flavors make good rubs.\"\n\nAs she talked, Lisa and I had gathered up all the various dried herbs and spices in the kitchen and put them into a jumble on the worktable. At Robin's invitation, the women exploded with activity, grabbing for jars and canisters, inhaling deeply, rejecting or selecting one here and there to spoon onto their plates. A few took turns at the mortar and pestle. Everyone seemed happy and talkative yet mindful of their own creative mission, rather like kindergartners at art time. \"Taste it,\" Robin said. \"If it's strong or spicy, just put a tiny bit on the end of your tongue.\"\n\nDri and Jodi looked at each other as they daintily tasted a dab and started laughing at the sight of their outstretched tongues. \"Tongues are _so_ not attractive,\" Jodi said, once she regained her composure. \"But this cumin tastes yummy.\"\n\nEach person coated a small piece of steak or a pork chop with her spice rub. One by one they rotated through the grill. \"The key to grilling something like this is to start it hot,\" Robin said, swirling her tongs around the preheated portion of the commercial grill. \"You want to sear it on one side until it's slightly caramelized with a good brown color, but then move it off the fire to a cooler spot and cover it. Do the same for the other side.\"\n\nGen shrugged. \"That explains why I always end up with charred meat. I must be leaving it on the actual fire for too long,\" she said, which led to a lot of discussion around grilling mishaps.\n\nOnce again, Maggie put a piece of tape with each person's name onto a plate. As the meat came off the grill, each of the volunteers took a bite of her own and then passed it on. The rubs varied from a fiery combination of cumin, cayenne, and oregano to a sweet one seasoned with cardamom, brown sugar, and cinnamon. \"The great thing about rubs is that you can use them on anything, even vegetables. They're good for last-minute seasoning and they can be simple or complex. Just trust yourself,\" Robin said.\n\nOne by one, Lisa and I pulled the hot pans with the braised pork from the oven. The heat from the center of the table felt as if we'd set a fire. Each person claimed hers and tasted.\n\n\"This is great,\" Jodi said of the pork. \"It tastes . . . I don't know, like _home_. Does that make any sense?\" Everyone went quiet, eating. She noticed. \"Wow, this must be good. No one is saying anything.\"\n\n\"This is better than food that I've had at a restaurant, and I made it. I mean, really made it, from scratch. Cutting up the meat, everything,\" Dri said. \"It makes me feel like I'm cooking _Top Chef_ food or something.\" It was an offhand reference, but particularly amusing given that we had a cheftestant teaching the class.\n\nEveryone thanked Robin profusely. As she packed up her knives and we cleaned, we tried to pry some gossip about the season from her. Robin wouldn't share anything other than that she was the only contestant who didn't have a culinary degree. In past seasons, that had always been an obstacle. No one without a culinary degree had ever won. Robin had never worked in fine dining, nor did her r\u00e9sum\u00e9 include a stint under a famous chef.\n\n\"All I can say is that my food is heartfelt and not pretentious because that's who I am, so that's the way that I cook. That's what you'll see.\"\n\nThat season I watched Robin struggle on _Top Chef_. Her lack of culinary training did put her at a disadvantage at times when challenges called for complex dishes with tactics such as deconstruction or refined haute cuisine techniques. She was criticized more harshly by her fellow contenders than she was by any of the celebrity judges. At one point, I cringed when a chef in his twenties referred to her as \"Grandma\"; Robin and I are around the same age. Another complained that she wasn't even \"a real chef.\"\n\nWhat defines a chef, anyway? What defines any of us? I had a culinary degree, but Robin had spent a dozen years cooking in the culinary industry and owned a successful restaurant for five of those. If lack of a culinary degree disqualifies, what about all the Michelinranked chefs who worked their way up the food chain as apprentices?\n\nIt takes guts to go on a national reality show, especially one essentially devoted to fetishizing the entire concept of cooking. It was that foodie bubble all over again. In my kitchen, she was generous with her knowledge, patient, and encouraging. It struck me that in a curious way Robin could be a stand-in for home cooks looked down on by culinary elitists.\n\nI thought back to the original questionnaires. Shannon's mother had snipped that she burned everything, but in the space of six weeks, I'd seen no evidence of anything other than someone with terrific curiosity and an unrealized natural talent in the kitchen. Another person had admitted that she ate Tuna Helper while watching Gordon Ramsay. When I pressed her for a reason why, she responded that cooking felt beyond her reach and not worth attempting. She was so discouraged by her cooking skills that she declined to be a part of the project\u2014and it was _free_.\n\n\"I think that one of the things that has ruined home cooking are the TV cooking shows,\" said veteran cookbook author Rick Rodgers. In his life, he's taught more than five hundred cooking classes and met his share of struggling cooks. \"First, the majority of the shows teach lots of stuff but cooking is not one of them.\" When he sees twentysomethings obsessing about foam or rushing around the kitchen in a competitive cooking challenge, \"it's kind of like watching pornography. I think I'd love to do that, but I'm afraid I'll throw out my back.\"\n\nI suppose that's what made Julia Child so endearing. Not long after the meat class, I looked up the episode of _The French Chef_ in which she attempts to flip a pan-sized disk of potatoes. \"When you flip anything, you just have to have the courage of your convictions,\" she stated, even though she looked less than convinced that the whole thing was going to work. She gave the pan a shake and a hearty flip, but rather than landing neatly back in the pan, half of its contents fell and splattered on the stovetop. As she investigated the fallen pieces, she explained what went wrong. \"When I flipped it, I didn't have the courage to do it the way I should have. You can always pick it up.\"\n\nThen came the moment that sealed her fans for a lifetime: \"Besides, if you're alone in the kitchen, who is going to see?\"\n\n\"Julia's little kitchen catastrophe was a liberation and a lesson,\" wrote Michael Pollan in a story on the decline of home cooking in _The New York Times_ in 2009. She offered a simple but obvious observation. \"The only way you learn to flip things is to just flip them! It was a kind of courage\u2014not only to cook but to cook the world's most glamorous and intimidating cuisine\u2014that Julia Child gave my mother and so many other women like her.\"\n\nPerhaps that was her greatest contribution. She didn't limit herself to offering instructions on how to execute a particular dish, but instead imparted a more generalized sense of courage for cooks willing to navigate unfamiliar culinary terrain.\n\nHow often are we rewarded for going off the map anymore? Or given the freedom to trust ourselves and the permission to make mistakes, cooking or otherwise? My mother has many phrases, two worth noting here. \"Who says you can't?\" with emphasis on the \"who\" was her invariable refrain whenever any of her kids alleged that something simply couldn't be done. The other: \"In a hundred years, no one will know the difference.\"\n\nSo _who_ says you can't cook? Not every meal has to be from scratch, nor does everything you consume have to be organic, locally sourced, and pasture raised. Try to find a comfortable place somewhere between Tuna Helper and _Top Chef_. If you burn, scorch, drop, overcook, undercook, underseason, or otherwise put a meal together that's less than a success, in the end it doesn't matter. It's just one meal. You'll make another one tomorrow. In a hundred years, no one will know the difference.\n\n#    **Your Basic Braise**\n\n_The term \"braise\" promotes much confusion, yet it simply means to cover and simmer something with a bit of liquid so that it cooks slowly with moist heat. Braising can yield comfort foods on many levels; a traditional pot roast is a simple braise; so is beef bourguignon._\n\n_There are plenty of reasons to add braising to your bag of culinary tricks. Long, slow cooking provides great meals from the most inexpensive cuts of meat by gently breaking down the connective tissue, yielding a tender, flavorful result. Second, once you put it in the oven, it's pretty much on autopilot, leaving time to devote to other tasks or pleasures. Braises provide a star turn for entertaining, as most can be made a day ahead and reheated. Leftover braised meats can be used in a variety of ways, from inclusion in tacos or burritos to folding into omelets or scrambling with eggs, or they can be added to soups, beans dishes, salads, pasta, or sandwiches. Braising infuses great flavor into vegetables for side dishes, too, notably carrots, asparagus, green beans, onions, leeks, and Brussels sprouts. In a skillet, lightly brown vegetables in butter or oil, add a bit of water, stock, or wine, cover, and simmer for about 20 minutes or until tender._\n\n_**The Basic Technique**_\n\n  Season the main ingredient with coarse salt and ground pepper.\n\n  Brown the meat thoroughly in oil over medium-high heat.\n\n  Saut\u00e9 a mix of aromatics, such as onions, garlic, shallots, leeks, carrot, or celery, or even fruit, such as apples or pears, in some oil.\n\n Add some liquid, such as water, stock, or wine. Stir to scrape up any browned bits on the bottom of the pan; this is known as \"deglazing.\"\n\n Return the meat to the pan and cook until thoroughly tender, from 2 to 5 hours.\n\n If desired, take out the meat, vegetables, or fruit from the pan and remove any extraneous fat. Thicken the sauce by boiling it until some evaporates; this is known as \"reducing.\"\n\n_**A Few Tips**_\n\n Use a heavy pan with a tight-fitting lid (see note on Dutch ovens on page 171).\n\n Don't use bouillon cubes in place of stock; the result will be too salty. If you don't have stock, use water instead.\n\n Tougher cuts of meat, such as shoulder roasts, short ribs, brisket, and shanks, work best for braising. Whole chickens or dark poultry braise beautifully; boneless chicken or turkey breasts tend to overcook and stiffen.\n\n You can simply focus on braising meat, or extend it with vegetables, such as potatoes, cabbage, carrots, leeks, parsnips, or squash. Vegetables cook faster than the meat, so add them during the last hour of the cooking process. Here's an example:\n\n#   **Braised Pork with Potatoes and Cabbage**\n\n_Not enough people braise pork, yet it's an inexpensive dish resulting in deeply useful leftovers. You can replace the pork with beef, chicken thighs, or even a couple of meaty lamb shanks. If the meat is on the bone, leave it. If you start with a larger piece, just try to cut the meat into chunks roughly the same size so that they cook evenly, and trim off excess fat. This can be served alone or pairs nicely with wide pasta noodles or mashed potatoes._\n\n**SERVES ABOUT 6 WITH GENEROUS LEFTOVERS**\n\n3 tablespoons vegetable oil   \nAbout a 3-pound piece of pork shoulder or picnic ham, cut into   \n1- to 2-inch cubes   \nCoarse salt and freshly ground black pepper   \n1 large onion, chopped (about 1   cups)   \n2 carrots, chopped (about 1 cup)   \n2 stems of celery, chopped (about 1 cup)   \n4 garlic cloves, peeled   \n1 bay leaf   \nSeveral sprigs fresh thyme, or 1 teaspoon dried thyme   \nAbout 1   cups chicken stock or water\n\n_**Vegetables**_\n\n  head cabbage, shredded or sliced thin (about 8 ounces)   \n2 to 3 large tomatoes, diced, or one 14-ounce can diced tomatoes   \n1 large potato, peeled and cubed (about 1   cups)\n\n  Preheat the oven to 325\u00b0F. Heat 2 tablespoons of oil in a 5-quart Dutch oven over medium-high heat. Season the meat with coarse salt and freshly ground black pepper. When the oil is hot, add the cubes of pork, working in batches if needed to avoid crowding, which will steam the meat rather than brown it. Brown well, about 3 to 6 minutes per side. Remove all the meat from the pan and set aside. Add or remove oil so that there are about 2 tablespoons in the pan. Reduce the heat to medium.\n\n Add the onion, carrots, celery, garlic, bay leaf, and thyme. Saut\u00e9 until the vegetables are soft. Add the browned pork and enough stock or water to come to the top of the meat. Stir and scrape up any browned bits on the bottom of the pan. Cover and put the pot into the oven to simmer for about 2 to 2   hours, or until the meat is tender enough to cut with a fork.\n\n Add the shredded or sliced cabbage, tomatoes, and potatoes to the pot, then return it to the oven for about 35 to 45 minutes, until the cabbage is tender. Remove the bay leaf before serving. Taste to see if it needs salt and pepper before serving.\n\n_**A Note on Dutch Ovens**_\n\nA Dutch oven is a squat, thick-walled cooking pot with a tight-fitting lid. You can buy an enamel-coated steel number for as little as fifteen dollars, but for the quality and longevity, I recommend cast iron. A standard black preseasoned five-quart version can be had for thirty dollars, while enamel-coated cast iron varieties start at sixty dollars. Both varieties will last a lifetime with proper care.\nCLASS BREAK\n\n**The Red Velvet Dinners**\n\n_To Pay for the Classes, I Invite You to a Series_   \n_of Unusual Communal Dinners_\n\n  By the end of July, the bills for the class project started to mount. The rent for the Post-it notes kitchen, the barely above minimum wages for Lisa and Maggie, and the costs for food, business insurance, and even linens such as cloth diapers added up to a couple of thousand dollars. Meanwhile, I'd donated more cooking-class dinners to charity; one fetched thirty-four hundred dollars. The dinners promised to combine elements of the project, including a brief cooking or tasting lesson and dinner. Of course, since they were donations, all those same costs would apply\u2014out of our pocket\u2014to the tune of a few hundred dollars per dinner. Mike and I had a discussion about the mounting costs.\n\n\"It's a nonprofit research endeavor,\" I said, making the case for the project. \"And those dinners are for good causes.\"\n\n\" _We're_ not a charity,\" Mike said. \"How are we going to cover all this?\"\n\nMy culinary-school friend who had led me to the teaching kitchen, Anne-Catherine, had offered communal dinners in the space and had made a small profit. Another friend had done the same on Vashon Island. A communal dinner simply means that everyone shares one big table. No tables for two or four, but rather tables for twelve or twenty or thirty-two, with one menu shared family style. I decided to try a few dinners to raise money for our little charity project. We'd ask people to bring their own wine and suggest a donation. The first was nothing short of a haphazard affair.\n\nMike arranged the tables and chairs the catering company used for events. Jeff ironed white linen tablecloths to an immaculate finish and artfully crafted bits of the lavender growing outside into small glasses and littered the table with tea lights. The result resembled a _Martha Stewart Living_ magazine spread.\n\nBy coincidence (or fate?), we held the dinner on what would have been Julia Child's ninety-seventh birthday. I expected fourteen guests, yet twenty-two showed up. I forgot to add salt to the dough, so even though Mike led a great lesson on fashioning lovely loaves, the final bread had little flavor. My chef friend Ted and I hustled to cook the food, but the scene took on a strange Keystone Kops aura. We bought six dozen oysters but forgot to pack an oyster knife. The pilot light went out on the stove and we struggled to figure out how to relight it. Once Ted and Mike got started, my side towel caught fire. I dropped a huge vat of cooked green beans on the floor. The sole meuni\u00e8re fell into pieces as we started to cook it. We served the final dishes around eleven P.M.\n\nDue to the steady stream of wine, few seemed to mind the lateness of the dinner as most everyone was hammered by the time the food hit the table. Due to clumsy execution, we collected less than we needed to cover the cost of the food, much less earn anything toward the project.\n\nJulia Child consoled chefs by reminding them that they were alone in the kitchen, but this was not true in ours. The small dining room was open to the entire kitchen, so there was no escape. Before the next dinner, Mike engineered ceiling-to-floor-length inexpensive red velvet curtains from IKEA along the length of the kitchen to separate the cooking area from the tables. As he stood back to evaluate whether he needed to improve the torque to keep the heavy curtains level, Jeff stood next to him. Mike asked him whether perhaps he needed a turnbuckle, a piece of hardware, to keep the line straight. Jeff responded that the drapes needed more pleats.\n\n\"Pleats?\" Mike's voice scaled up.\n\n\"Oh, yeah, I think they're tight enough, they just need more pleats. I can take care of it,\" he said. \"My sisters and I used to put on musicals when we were kids. My job was always fixing the sheets so they looked a bit fuller when we used them as curtains. If the pleats aren't even, they look cheap.\" He borrowed our car to hit a store for small clamps and then spent the next ninety minutes moving a stool along the length of the curtain, painstakingly adding perfect pleats. I have to admit, the result looked fabulous. With that, the Red Velvet Dinners were born.\n\nAfter the dubious start, I strategized the next dinners as if executing a battle plan. In earlier years, I'd been known for over-the-top dinner parties, and even then I planned the menus weeks in advance. Invariably, though, the party buzzed in the kitchen, drinks took prominence, and dinner arrived on the table late, often very late.\n\nWhen someone pays for a meal, it's different. An entr\u00e9e served hours late isn't charming; it simply shows you failed to plan. The French chefs would not be pleased. Learning as we went along, we became more proficient at performance. Maggie and Jeff, both experienced kitchen hands, agreed to help prep food and manage the tables. I made time lines, schedules, and multiple versions of the menu. I even started to litter the kitchen with my own Post-it notes. \"Do NOT touch dough. For RVD only!\" The menu for the first series included the following:\n\n _Salt tasting_   \n_Calvados and peach sorbet (palate cleanser)_   \n_Fresh sweet corn soup with chili oil and seared scallops_   \n_Pear and lemon verbena sorbet_   \n_Artisanal beef tasting_   \n_Seasonal green salad with local goat cheese and cherry vinaigrette_   \n_A cheese tasting with savory tart and fresh fruit_\n\nThe red curtain invited something . . . sexy. We decided to add an \"entertainment\" to the dinners. \"What about burlesque to go with the steaks? Kind of fits, doesn't it?\" asked my friend Deirdre. She'd recently directed a documentary on the neoburlesque movement, and she knew all the best performers in town. She helped me arrange for The Shanghai Pearl, a nationally known burlesque artist, to perform during a break in the dinner.\n\nFor the beef tasting, a friend organized a variety of grass-fed beef options. Ted set his own Weber grill on the sidewalk in front of the storefront, tasked with cooking twelve steaks from four different purveyors to exactly the same degree; it was so complicated, it would likely have been easier to dribble three basketballs at once.\n\nThe lovely Shanghai was no diva. The first night, she arrived at the back door in jeans and a simple T-shirt, an odd combination with her intense stage makeup, which included bright blue drag-queenlike eye shadow and glitter lipstick, and a complicated updo. \"I'm so sorry, I've lost my voice,\" she whispered hoarsely. She held up a bulky garment bag. \"Where can I change?\" I directed her to the bathroom. \"Oh, here's my music.\" She went to change. I turned to give the CD to Mike, who was managing the audio. Instead, I bumped into Ted, lurking in the rear of the kitchen with a postgrilling glass of Syrah in hand.\n\nHe looked awestruck. \"Uh, wow, she's, um . . . ravishing.\"\n\nMoments later, Shanghai emerged in a scarlet-sequined hip-slit burlesque costume strutting in five-inch heels with two large red plume fans balanced in one hand. A massive blood-red flowery ostrich plume sprouted from the side of her head. Ted tried to look casual as she readied herself behind the red velvet curtain. She turned to him. \"Do you think you could hold these fans and hand them to me on cue?\" she whispered hoarsely. \"When I've stripped, can I hand you my dress? Do you mind?\"\n\nTed stared at her and nodded without blinking. \"Yes, I think I can handle that.\"\n\nHe took the job seriously, watching transfixed as the dancer bumped, grinded, and charmed the tiny audience around the long table while she strutted through her set. As she discarded gloves, garters, and portions of her dress, Ted dutifully caught them and carefully smoothed them over his arms. She concluded by removing a tiny red silk bra to reveal bright silver pasties and bowed in time to finish with her music. The dinner guests exploded in cheers. Nothing completes a dinner like a beautiful naked woman. As she returned behind the curtains, Ted handed her back her dress and accoutrements. \"I can do this anytime you want,\" he assured her earnestly.\n\nMaggie, Jeff, and I watched from the sidelines, clutching tumblers of wine. As her act concluded, Maggie leaned in to me. \"I'm fairly sure that after The Shanghai Pearl, no one is going to remember any of the food,\" she said. \"I mean that in a really nice way.\"\n\nEmpowered by that success, we developed more dinners. Demand overwhelmed the seats available. At first, the dinners were open to friends only, then friends of friends. Then we invited people to get on \"the list.\" Within a week, we had 370 names.\n\nSix of those with reservations could opt to come early to help prepare dinner via a cooking lesson. After they chopped vegetables, plated the cheese course, or made pasta, they then sat with the rest of the patrons for the five- or seven-course dinner. All of them followed a variation of the first dinner: a main course that featured a comparative tasting, such as oysters, salmon, or differently raised types of poultry. Each featured an entertainment\u2014usually live music, but once we brought in a tarot card reader.\n\nFor Halloween, we took the Red Velvet Dinners on the road to the upper floor of the Richard Hugo House, a literary center set in a rambling 1902 Victorian that once served as the city morgue. The house has been featured in news outlets from ABC's _Nightline_ to National Public Radio for one simple attraction\u2014it's haunted.\n\nI did not know this minor fact when I rented a writing cubicle situated in the former morgue. I heard whispers past midnight. Doors slammed when no one was around. At two o'clock one morning, I caught the sound of what sounded like silverware rattling atop something metallic. It's such a clich\u00e9, but the hair on my neck did, in fact, stand straight up. Later, I understood the sound when I heard the clang of medical instruments atop a tray at my dentist's office. The next day, I put my name in to transfer to a different writing office.\n\nFor the masquerade dinner, we shifted the drab six-foot classroom tables into one comically long dining tabletop set with three-foot-high candelabras, immaculate linens, and the ramshackle mismatched thrift-store plates, forks, and glassware from the center's own kitchen. All twenty-eight guests arrived in costume and character for a mystery dinner, masks intact. Hugo House doesn't have much of a kitchen, so we stuck to an elegant yet simple menu:\n\n _Trays of figs, duck liver p\u00e2t\u00e9, olives, assorted soft cheeses_   \n_Cassoulet with duck confit, garlic sausage, and braised lamb_   \n_Autumn green salad with Riesling-soaked pears_   \n_Apple tarte Tatin with brandy sauce_\n\nThat night's cassoulet dinner by candlelight had an air of sensuality. Champagne flowed. People flirted behind their masks. As everyone dug into three steaming vats of the traditional French white bean casserole and started to eat, the room went eerily quiet. A door slammed somewhere unexpectedly. Everyone suddenly looked around, startled. \"I think I left a window open in the other room,\" I said. Everyone returned to their dinner.\n\nI never left a window open. But how do you tell people that it might be the sign of a ghost signaling his displeasure at failing to get an invite?\n\nThe last of the series featured an Italian theme. The evening included an olive oil and olive tasting, plus handmade pasta in a fragrant saffron broth with shrimp and tomatoes, a reprise of something I'd taught during the hands-on classes on the cruise earlier that summer. The scheduled entertainment was an opera singer performing excerpts from the works of Italian-born composer Giacomo Puccini.\n\nIt was a last supper, so I invited all the guests into the kitchen early to cook, and more than a dozen of them took me up on it. I'd developed the recipe as a lesson on tackling that issue so often mentioned by the volunteers: How do you keep a recipe from tasting bland?\n\nAfter everyone helped to make the pasta and left it to dry on trays stacked on the counter, I discussed some classic seasoning points from Italian cooking. \"Buy a basil plant and keep it in your kitchen window all year long,\" I started. \"You can just pull off a few leaves at a time and it will add freshness to the end of a dish, and it won't die in the bottom of your crisper. While you're at it, consider getting a thyme plant, too.\" Next tip, keep lemons or limes on hand; the acid brightens flavor. Garlic adds bite and hot chili adds punch.\n\nI made one very bland plate of pasta and let everyone taste it. Then we divvied the pasta into three bowls and let them experiment with shifting the flavor by adding lemon, fresh herbs, and hot chili. In another batch, this time dosed with garlic. \"Wow, this is pretty boring without garlic,\" one guy said.\n\n\" _Life_ is boring without garlic,\" replied another.\n\nAfterward, as the \"cooks\" took off their aprons to take their seats at the tables with the other guests, a woman approached me. \"I can't get over how the lemon and the garlic changed everything,\" she said. \"I can't tell you how often I make a recipe and it's kind of blah. It will need _something,_ but I can't figure out what, and, well, I've always been kind of afraid to add anything because I'm afraid I'll screw it up.\"\n\nAs a little girl, I used to play restaurant, and in some ways, this was simply the big-girl version. With each dinner, I gained more confidence in the kitchen. I managed a regular crew, and we bonded in our kitchen camaraderie, quietly plating entr\u00e9es behind the curtains. The dinners required developing menus, managing food costs and budget, and the full responsibility of what went right or wrong. For the first time, I felt like I was starting to earn the title of chef, whatever that meant. The odd thing was that just as I thought that I might warrant it, I decided it wasn't all that important anymore.\n\n#   **Spicy Shrimp in Saffron Tomato Sauce**\n\n_To appear short, many recipes for \"quick\" meals often leave off steps that would upgrade the flavor. This recipe utilizes four important concepts to awaken otherwise bland dishes: fresh herbs, acid, garlic, and spice. Frozen shrimp is a great freezer staple, but try to opt for wild-caught North American shrimp over Asian farmed varieties. If you can find only peeled shrimp, c'est la vie. This recipe cooks quickly, so be sure to prep and measure all the ingredients in advance. This is equally charming served over fettuccine pasta, basmati rice, or saffron rice. No shrimp? You can substitute bay scallops (take care not to overcook them), a solid white fish (such as cod), extra-firm tofu, or boneless chicken breasts cut into half-inch cubes. You can substitute turmeric for saffron and dill or cilantro in place of the basil._\n\n**SERVES 4**\n\n1 pound uncooked large shrimp   \n cup chicken stock or water   \n2 teaspoons plus 1   tablespoons olive oil   \nCoarse salt and freshly ground black pepper   \n teaspoon red pepper flakes   \nHalf a lemon   \n cup dry white wine or vermouth   \nPinch of saffron threads   \n2 or 3 large garlic cloves, minced   \n1 tablespoon tomato paste   \nOne 14   -ounce can diced peeled tomatoes in juice   \n1   teaspoons dried oregano or mixed Italian herbs   \nHandful of minced basil or flat-leaf parsley\n\n Peel the shrimp. Add the shrimp shells to the chicken stock or water and simmer until needed. Toss the shrimp with 2 teaspoons of the olive oil, a couple of pinches of coarse salt, a few grinds of black pepper, the red pepper flakes, and a squeeze of juice from the lemon half. Combine the wine and saffron.\n\n  Heat 1   tablespoons of olive oil in a large saut\u00e9 pan over medium-high heat. Cook the shrimp until they are opaque, about 2 to 3 minutes, then transfer them to a bowl with a slotted spoon or tongs. Add a bit more oil to the pan if necessary. Cook the garlic for about 1 minute, then add the tomato paste and the wine and saffron, and cook for about 3 minutes, until some of it evaporates.\n\n Remove the shells from the stock with a slotted spoon and add the liquid to the sauce. Next add the tomatoes and dried herbs. Continue to cook over medium-high heat, stirring frequently, about 8 minutes, until it is reduced to a saucelike consistency. Taste to see if it needs more salt or pepper.\n\n Add the cooked shrimp and simmer until they're heated through, about a minute or two. Remove from the heat, then stir in the minced basil or parsley and add a few squeezes of lemon over the top. Serve over hot pasta or rice.\n\n_**Easy Habits for Good Flavor**_\n\n_These \"tricks\" are the kinds of things chefs do to elevate what could be a bland dish into something much more satisfying:_\n\n  Preseason protein with a few simple seasonings before cooking, such as a bit of olive oil, coarse salt, pepper, simple herbs or seasonings, a squeeze of citrus, or a splash of vinegar.\n\n Use fresh garlic. It adds punch the way jarred garlic doesn't. Some markets offer cloves of peeled garlic as a time saver; they are commonly used in professional kitchens.\n\n Spice it up. Many cuisines add spicy ingredients as an inexpensive way to add a sassy, satisfying note to otherwise simple dishes.\n\n Finish dishes with a \"bright\" flavor at the end. Normally this is a bit of acid and\/or an herb, in this case lemon and fresh herbs. Small amounts of vinegar, especially flavored ones, can add a lot of flavor in a finished dish.\n**PART III**\n\n**Seafood, Soup, and the Importance of Leftovers**\n\n_\"So long, and thanks for all the fish.\"_\n\n\u2014Douglas Adams\n\n**Donna and Dri with their fish**\nCHAPTER 10\n\n**The Pleasures of the Fish**\n\n  **LESSON HIGHLIGHTS:**\n\n**How to Buy Fish, Plus Plenty of Ways to Cook It**\n\nOne of the first things I discovered about Mike was that he didn't like fish. Sure, he was a fan of fish and chips, but most people would probably eat a brick if it was deep-fried. In Paris, when I brought home a complicated dish from culinary school featuring monkfish stuffed with delicate crab meat and wrapped in prosciutto coupled with a beurre blanc sauce, he simply said, \"It's not bad. It tastes like chicken.\"\n\nIn the years since, he's warmed up to fish. But the reasons for his initial resistance were often expressed by many of the volunteers. They felt uncomfortable selecting seafood, and I suspected this led to the inadvertent purchase of lower-quality fish. They told tales of fish stinking up kitchens and relentlessly overcooked fillets that turned out too tough, too oily, or too bland.\n\nNorth Americans eat relatively little seafood, about sixteen pounds total per capita per year, a paltry showing compared to the sixty-plus pounds of chicken we consume each year. Just 7 percent of our diet includes fish and shellfish. By comparison, it makes up more than 25 percent of that in Asian countries. Most health-care experts suggest increasing fish consumption to ensure adequate levels of omega-3 fatty acids. A growing body of research suggests that omega-3 can help deflect everything from heart disease to stroke, Alzheimer's disease, cancer, and even clinical depression.\n\nI was gung ho on teaching a class about fish\u2014until I went to see a screening of the utterly depressing documentary _The End of the Line._ If you're unfamiliar with this, it focuses on the havoc wreaked on various fish species due to commercial overfishing and is often referred to as _An Inconvenient Truth_ for the oceans. Overfishing happens based on consumer whims. Take the blackened redfish craze that gripped the nation in the 1980s. To feed the extraordinary and sudden demand from restaurants and retailers, commercial netters caught redfish by the millions, devastating the species within the space of a few years. When Chilean sea bass ended up on every menu years later, the same thing happened. More than twenty species are included on various overfished lists.\n\nSo my initial enthusiasm shifted into a dilemma: Should I even do a class on fish? Most of the students commented that they might eat more if they knew how to make it.\n\n\"Are you kidding? You have to do a class on fish,\" Lisa said. \"This is Seattle. The airport has bronze salmon embedded in the floor. Why don't you bring in Ted as a guest teacher? He's a total Jedi master when it comes to fish.\"\n\nI mention my chef friend Ted often but fail to point out that he trained at the Culinary Institute of America and spent a dozen years in the industry. His early career during the 1980s involved catering for the PBS channel in Boston. As a result, two of the regular visitors to his kitchen were Paul and Julia Child. \"She'd come in and ask, 'What are you making, Chef?' But then she'd have to go out and mingle,\" he told me. \"You could tell she'd rather have been in the kitchen.\" Although he's now a financial consultant to the tech industry, Ted never lost his appetite for the kitchen.\n\nSeveral of the volunteers arrived early. I took it as a chance to catch up with them as Ted, Jeff, and Maggie set everything up.\n\n\"So we've started to buy vegetables regularly from this farm stand near our house,\" Gen said. \"We told them we were going to a Mexicanthemed party that night. Seriously, you should have seen the reaction from the owner!\" She seemed anxious to tell the story. \"He and his wife went back and forth, insisting that we had to make this amazing sangria from a recipe his grandmother brought from Peru.\"\n\nThe farm-stand owners gave Gen the recipe and helped the pair organize all the stuff to make the sangria. Just when they wondered if perhaps it was a grand marketing ploy to persuade them to purchase extra produce, the owner asked if they wanted any tamales. \"They were making tamales for this big party. We said sure, we'd try some. They gave us this huge bag of tamales for free.\"\n\nGen and her boyfriend rushed home. \"We were using this big wine jug for the sangria, trying to shove the fruit through the ittybitty hole at the top,\" she said, pantomiming the scene and laughing. \"It was kind of chaos. We had to cook all these tamales and make the sangria and rush to the party.\" But they were the hit of the night. People were impressed.\n\n\"It's funny, but that made me realize that there's a whole world beyond the supermarket,\" Gen said. \"That kind of interaction would not happen at a grocery store. I feel like I've made friends with them. We're part of this little community. It's really kind of cool, and I don't know, this sounds sort of dorky but it makes me feel like, wow, cooking can be fun in a way I never imagined.\"\n\nJodi had arrived in time to hear the end of Gen's story. She joined the discussion.\n\n\"That's so crazy you just used the word _fun,_ \" Jodi said. The past weekend, her friend had come over to cook to see what she was learning from the project. \"After an hour of cooking together, my friend turned to me and said, 'Hey. What's up with you? I can just feel the tension! You are like a madwoman. Cooking is supposed to be fun!' \"\n\nJodi said that she looked at her friend, sipping wine and relaxing. \"It just kind of hit me. I realized that cooking is not supposed to be so stressful. Why do I put myself through all this anxiety? There was always a part of me that worried about screwing something up. But at that moment, I don't know, I just had this moment of clarity.\" She paused a minute. \"God, I sound so confessional!\"\n\nJust then, Maggie clapped her hands. \"Ladies? We have some salmon for you to taste.\"\n\nTed had saut\u00e9ed five pieces and placed them in a line from least to most expensive: farm-raised Atlantic, keta, sockeye, coho, and king. Side by side, they looked remarkably different. The keta and Atlantic versions appeared pale, while the sockeye was a vivid orange-red, its denser meat marked by thick waves. The coho was almost magenta. Meanwhile, the king had the deep color of a ripe peach with a softer texture.\n\nThe group was torn on their favorite. Some preferred the \"classic\" salmon taste of the sockeye. Others thought the coho was more complex. \"The king tastes buttery, if you can say that about a fish,\" Dri said. The keta and the farm-raised Atlantic were judged \"meh\" by the group.\n\nAs a group, we contemplated the uncooked salmon, halibut, and black cod fillets set out on the table atop a bed of ice. \"The most important thing to remember is that fresh fish doesn't smell fishy,\" Ted said. \"Take a whiff. What do you smell?\" All the women leaned in. Among the responses: cucumbers, cantaloupe, sea air, the brine of an oyster. \"See? No one said 'fishy.' Those are all nice, clean fragrances. That's how fresh fish should smell,\" he said.\n\nWhether you buy fish at a grocery store or a dedicated market, you should ask to smell the fish first. If it's still whole, you should be able to touch it.\n\n\"If the guy at the counter doesn't grant letting you smell it without question, find another place to buy your fish,\" Ted said. \"Since smelling it is so important, try not to buy it wrapped in plastic. Ideally, you want to eat fish the day you buy it.\" If you must store whole fish, put it belly down in crushed ice, and remove the water as it melts to keep the fish relatively dry. \"If you don't live somewhere where you can get good fresh seafood, fish that's been flash-frozen on the boat is a great option. That's true of shrimp, too.\"\n\nAs consumers, we can be part of the problem or part of the solution based on what we opt to buy. I handed out copies of the pocket guide to buying fish from Seafood Watch. The card offers suggestions ranked as \"Best Choices,\" \"Good Alternatives,\" and \"Avoid.\" Everyone perused the cards. We reviewed the fish on the table in front of us. Good choices included the wild-caught Alaskan salmon and halibut and the trout. Bad choice? The farm-raised Atlantic salmon.\n\nI told the group that former _Gourmet_ editor Ruth Reichl made a comment at a conference I attended that stuck with me. \"You only get to vote for a president once every four years,\" Reichl said. \"But you get to vote three times a day, every day, with your dollar.\"\n\nTed picked up on that. \"Exactly. Ask questions. Ask them, 'Is this fresh? Is it frozen? Where is it from? How is it caught?' I personally try to avoid eating shrimp from Asia if I can help it. If that's the only kind of shrimp available, I change my meal plans.\"\n\nSomeone else asked about the constant refrain that eating too much fish leads to mercury poisoning. Longer-living large fish such as tuna tend to be more susceptible to mercury contamination. \"I recently heard that one good rule is to avoid eating any fish that's longer than your arm,\" I said. \"Smaller fish are good both for you and for the environment. Anchovies and sardines are wicked good for you.\" As with the beef discussion, we could have gone on all night. But it was time to cook. First, the crew dutifully spent a few minutes chopping up all the necessary vegetables for the class. As they did, they chatted.\n\n\"Oh, you'll find this funny,\" Jodi said, clearly fine with her confessional mode now. \"We were cleaning out the garage and we found one of those appliances to make roast chicken. I had forgotten about it. But now it seems strange to think that we once bought a _machine_ to make roast chicken knowing what I know now.\"\n\nSabra said that she and a friend had started to cook together once a week. \"Yeah, it's pretty cool. My friend comes over and we cook. This week we made beef stew and it was pretty awesome.\" Then she looked around. \"Oh, is that it? No more vegetables?\"\n\nWe started with a dish that I learned from an Italian friend I met in London. In a roasting pan, I added handfuls of freshly sliced red pepper, onion, asparagus stalks, garlic, and chopped black olives and covered them generously with olive oil, salt, and pepper.\n\n\"Basically, you just roast a whole bunch of vegetables, and then you nestle some fish into it,\" I said. \"I've done it with chicken breasts, too.\" I put them in a hot oven to roast. \"Okay, Ted, you're up.\"\n\nHe slapped his hands together. \"Now let's fire up some fish!\" He grabbed a saut\u00e9 pan and heated it over one of the portable burners on the worktable. As with chicken, it's key to start it on high heat and cook it quickly to get the center hot. Ted turned up the heat on the burner and added a small pool of oil to the skillet. He coated a small halibut fillet in some flour. \"This will help release it from the pan.\" He slapped the fillet into the skillet and it popped and hissed loudly as it hit the oil. Ted talked loudly over the sizzle. It sounded like incessant static. \"Always cook the presentation side first. If there's skin, then you put the side _without_ skin down first. It will get the most even heat and so it will be nicely browned.\"\n\nHe gave the pan a quick shake. \"I'm sure you've heard this before, but if you shake right after you put it in, then it's not going to stick.\" He watched it closely. After about five minutes, the bottom and edges of the translucent white fish shifted to opaque. He flipped the fillet over, covered it, and turned off the heat. \"For a fillet that's less than an inch thick, the heat of the pan will cook it through. Let it continue for about the same amount of time you did the other side.\"\n\n\"Now you need just one tool to see if the fish is done,\" he said, and held up his right index finger. Theatrically and with great purpose, he slowly brought it down to the top of the fish.\n\n\"It should feel firm in the center and hot to the touch.\" He took a fork and pulled off a section \"When it comes away in layers like this, that's called 'flaking.' That's what you want.\" He slid the fish onto a plate.\n\n\"Now we'll make a quick sauce in the same pan.\" He turned the heat back on. \"See, there's flour stuck to the bottom of the pan.\" He held it up for everyone to see. \"This is what we call ' _fond_.' It means foundation in French. This stuff has a lot of flavor.\"\n\nHe poured in a bit of white wine and it steamed immediately. \"When you put cold liquid into a hot pan to release the fond, it's called 'deglazing.' I'll tell you a trick, too. It's also a way to clean out a pan that's got a lot of gunk stuck to it.\" He added some sliced onions, zucchini, and red bell pepper, then seasoned them with salt and pepper. After a couple of minutes, he poured it over the fish. \"And bang! It's done.\" He clanged the pan down onto the counter. He topped the fish with some chopped basil.\n\n\"This is simple and fast. Learn to do this and you can cook a sauce from stuff in your fridge in fifteen minutes. If you don't drink wine or don't have it, a little stock, extra lemon juice, or lime juice will work, too.\"\n\nEveryone tasted his dish. \"Delicious!\" Trish said. \"I normally find halibut so boring. This is nice. And moist; the sauce really adds to it.\"\n\nDri asked if the flavor kisses we learned for chicken could work, too.\n\n\"Sure, it's the same concept,\" I said. \"You can try it out right now. Everyone, get a partner.\"\n\nWatching people cook week after week, I observed something interesting. On their own, the volunteers seemed tentative. When asked to pair in teams, the lesson went quickly and they experimented more freely. Perhaps it was the collaborative nature of cooking with someone else, or maybe it just felt like a safety net for any gaps in knowledge or confidence, or possibly both. It made me wonder if perhaps people ought to cook together more often, to spend some time socializing as they tackle a recipe. Sabra had mentioned that she had started to cook with a friend once a week. Perhaps we should all be meeting for a \"cooking break\" rather than coffee? Maybe mothers could cook together while their children have playdates.\n\nMy attention went back to the activity of the kitchen as each team grabbed a skillet. They carefully selected fillets as if they were either precious jewels or plastic explosives. Dri again took up residence at the big stove, with Donna working by her side. Dri felt confident with the high heat. When she saw Donna shrink back, Dri encouraged her to take over cooking the fish. They finished, each testing it with their index finger. Then they made a quick sauce with garlic, green onions, and sticks of red pepper and zucchini. As they poured it atop their fillet, they could not contain their enthusiasm.\n\n\"Check this out! We made this. It could be in a magazine.\" They were so proud, I got my camera and took their photo.\n\nSabra and Gen finished their fish so quickly that I never even saw them cook it at all. \"We're just pros, that's why,\" Sabra explained as they nibbled at their fish. Jodi and Andra took the endeavor seriously, carefully navigating each of the steps. Ted teamed up with Trish.\n\n\"But how will I know when it's done? I don't know what it's supposed to feel like,\" Trish asked him. He advised that touching it regularly as it cooked to feel how the fish firmed helps to \"train\" the ever-sensitive index finger. Like anything, the way to cook fish well is to just do it.\n\nAs everyone finished their saut\u00e9, I pulled the roasted vegetables from the oven. They'd been in for about ten minutes. \"So here's the deal. The vegetables are partially cooked. Now all you have to do is place a piece of fish on top and cover it with some hot vegetables.\" I laid a long piece of halibut in the center and scooped some of the vegetables over it. \"We'll let it roast for another fifteen or twenty minutes and then see how it's doing.\"\n\nNext, we moved onto cooking _en papillote,_ or cooking in paper, a technique wildly overlooked in American households as a simple and fast cooking method, not to mention that it requires no pans, and leaves no dishes to clean. The process is simple. Put a little oil on a large piece of parchment or foil, then add some salt and pepper. Lather a thin fillet of fish with oil or butter and salt and pepper, and then add finely sliced or diced vegetables, some herbs, a bit of wine, perhaps some citrus or vinegar. Fold the parchment or foil in two and crimp the edges tightly. Put it in the oven for fifteen minutes at 400\u00b0F. This also works for thin slices of chicken.\n\n\"You want to keep the vegetables small so they cook quickly and keep the fish or chicken sliced thin,\" Ted explained. Then he demonstrated how to close the sides by thoroughly pinching and squeezing the edges together with his fingers. \"Be sure to get a nice, tight seal so that the moisture stays trapped inside. It bakes and steams at the same time.\"\n\nEach team went to work. I wandered around the table to see what they assembled with their choice of thin fillets of black cod, salmon, or snapper. Jodi and Andra added lime, ginger, shallots, rice wine vinegar, a touch of fish sauce, and chopped basil. Jen and Sabra flavored theirs with diced cherry tomatoes, finely chopped zucchini, minced garlic, chopped dill, and lemon. Dri and Donna debated. \"Hmmm, would balsamic work or would it be too strong a flavor?\" Dri mused. Donna thought about it.\n\n\"Let's think about what we like. The olives look good. What would go with them?\"\n\n\"Tomatoes,\" Dri offered. From there, they each took some of the sliced vegetables to chop them even more finely. Their fish looked beautiful topped with diced olives, red peppers, onions, shallots, and basil and christened with wine and a touch of white balsamic vinegar.\n\nEach team wrote their names on their paper packets and slid them onto a baking sheet. I pulled out the vegetables with fish and set them on the side table. \"Wow, that looks amazing,\" Andra said. \"Smells good, too.\" I slid the trays with the parchment into the oven.\n\nEveryone chatted, tasting one another's food and the vegetable-roasted fish. \"It's all so good, and so easy,\" Gen said as she sampled her own saut\u00e9ed piece. \"I think that I am truly going to start tackling fish more often.\"\n\nAbout fifteen minutes later, Maggie pulled the paper-cooked fish from the oven. Each team carefully retrieved their hot portion and settled it onto a plate. On my signal, they opened their packets in unison by slashing them with a knife. Steam escaped from each and the collective smells burst around the room. Donna clapped her hands in delight.\n\n\"It's like opening a tasty present!\" she exclaimed. Then the room went quiet as everyone ate. \"I'm definitely doing this again!\"\n\n\"No kidding, I'm doing it, too,\" Cheryl said. \"My husband will be knocked out by this.\"\n\nThat evening made me think of that old saying \"Give a man a fish, he eats for a day. Teach him to fish and he'll eat for a lifetime.\" That always struck me as such a clich\u00e9, but as I cleaned up that night I realized that these days, catching the fish isn't the issue, so teaching someone to simply cook it might accomplish the same thing.\n\n#    **Fish en Papillote, or Baked in Paper**\n\n_Simple enough for weeknights and elegant enough for guests, the smell that escapes when opening the package is reason enough to try this. To ensure thorough cooking, use thin fish fillets or chicken breast slices; this works well for salmon and mild-flavored white fish such as snapper and cod. It works best in parchment paper, but you can also use aluminum foil. Use sheets at least eight by twelve inches for each individual packet. Change up the ingredients. For instance, use sesame oil in place of olive oil and add lime, cilantro, and ginger to the package for an Asian flavor. Consult the \"Cheat Sheet\" to Flavor Profiles in the Extra Recipes section at the back of the book for more ideas._\n\n**SERVES 2 WITH INDIVIDUAL PACKETS**\n\n1  tablespoons olive oil   \nSalt and freshly ground black pepper   \nTwo 4- to 6-ounce pieces of fish or thinly sliced chicken breast   \nFew sprigs of a fresh herb (dill, basil, thyme, rosemary, or cilantro)   \nFew thin lemon or lime slices or a dash of vinegar   \n  cup white wine, water, or stock\n\n_**Vegetables**_\n\nAbout   cup finely chopped or sliced vegetables (such as shallots, onion, garlic, zucchini, carrots, broccoli, fennel, mushrooms) for flavor and garnish\n\n  Preheat the oven to 400\u00b0F. Start with two pieces of parchment paper (or aluminum foil), about 10 by 12 inches each. Fold the pieces in half. On one side of the middle crease of each piece, drizzle the olive oil and add a pinch of salt and a couple grinds of pepper. Add the fish or chicken and turn over to coat. Place the herbs, lemon, wine, and vegetables on top of the fish. Fold the parchment or foil over like a book and crimp the edges securely to avoid allowing any liquid or steam to escape from the package during cooking. Place the package on a baking sheet and bake for 15 minutes. Allow to sit at least 2 minutes. Open carefully by unraveling the edges to ensure the fish or chicken is cooked through, then serve.\nCHAPTER 11\n\n**What's in the Box?**\n\n  **LESSON HIGHLIGHTS:**\n\n**Why It's Worth Cooking Outside the Box**\n\nA fancy KitchenAid stand mixer dominates the kitchens of most food writers I know. In an online tour of her kitchen, famous food author Amanda Hesser introduces her handsome gray model by saying, \"I sometimes think of this as my third child. I use it for everything.\"\n\nBy contrast, we have a simple yet heavy chrome 1960s-era Hamilton Beach stand mixer. No pasta hook, no paddle, and certainly no pasta-maker attachment. Ours is limited to a set of standard beaters; its capabilities peak at \"heavy mixing.\" The mixer is one of Mike's most cherished possessions and one of the few items inherited from his late mother. He fondly recalls scraping batter from the bowl forty years ago as the two of them made chocolate chip cookies or\u2014his favorite\u2014yellow cake with chocolate frosting. He cracked the eggs into the bowl. She gave him the spoon to lick. (Yes, with raw eggs. As a nation, we used to be less fastidious about such things.) It was their special time together and remains a sweet memory. Sure, the cake came from a box and the frosting came from a can, but the cake itself wasn't the point: His mother set aside time for him. His emotional connection outweighs any of the functionality that we might gain from fancy attachments. As a mixer, it works great. Although Hamilton Beach gave up on the model two decades ago, Mike lovingly maintains it, sourcing replacement parts from an obscure outpost in Ohio. So we won't be replacing the mixer, and that's absolutely fine by me.\n\nOne night after Mike got a little too worked up watching his alma mater's football team take a sound drumming, he snapped off the TV. He abruptly arose from the couch where I sat sifting through a stack of food magazines and announced, \"I'm going to make a cake.\"\n\nThis did not surprise me. Whenever he's upset, Mike needs something constructive to do with his hands. He might disappear to rebuild a carburetor, replace a light switch, or, as in this case, bake a cake. He started to ransack the cupboards. \"Hey, don't we have any cake mix?\"\n\n\"We have all the stuff for cake,\" I replied without looking up. \"Just look up a recipe.\"\n\n\"Really?\" Mike asked. \"You mean you can make a cake without a mix?\"\n\nA few minutes later, after looking up a recipe, Mike called out, \"So what's in the box?\"\n\n\"What are you talking about?\"\n\nHe brought a printout of a recipe for yellow cake into the living room. \"You've got to see this. So get this, it's just flour, eggs, baking soda, milk, sugar, and butter. But with a box you already add eggs, milk, and oil, so what's in the freakin' box?\" He was agitated. \"Just flour, sugar, and baking soda?\"\n\nA fundamental truth had hit him: You don't need a box to make a cake.\n\nFor the first time in his life, Mike made a cake from scratch with his mother's forty-five-year-old mixer. \"So that's it? This doesn't take any longer than doing it from a mix.\"\n\nAs the cake baked in the oven, he had another revelation. \"That's ALL that's in frosting? Seriously?\" Another first for Mike with the mixer: chocolate frosting from scratch, made with confectioners' sugar, butter, vanilla, cocoa, and milk.\n\nI'm among the minority of women who aren't keen on chocolate and cake isn't my thing, but I had to try the results. The flavor and texture of the cake were more interesting and varied than the onenote sugar sensation from a mix. \"It's good, don't you think?\" Mike asked as we stood at the counter contemplating the flavor. His voice had a spark of pride, and for good reason. His cake _was_ good.\n\nBut that led to a great question: What _is_ in the box? I went back to the supermarket.\n\nSugar, enriched bleached wheat flour (flour, niacin, reduced iron, thiamine mononitrate, riboflavin, folic acid), vegetable oil shortening (partially hydrogenated soybean oil, propylene glycol mono- and diesters of fats, mono- and diglycerides), leavening (sodium bicarbonate, dicalcium phosphate, sodium aluminum phosphate, monocalcium phosphate). Contains 2% or less of: wheat starch, salt, dextrose, polyglycerol esters of fatty acids, partially hydrogenated soybean oil, cellulose gum, artificial flavors, xanthan gum, maltodextrin, modified cornstarch, colored with yellow 5 lake, red 40 lake.\n\nCuriously, in the boxed version, sugar is the ingredient in the largest quantity. Compare that list to the ingredients from the recipe Mike used for his cake:\n\nUnbleached flour, sugar, milk, eggs, unsalted butter, vanilla, baking powder.\n\nThe label for the frosting was equally unsettling:\n\nSugar, water, vegetable oil shortening (partially hydrogenated soybean and cottonseed oils, mono- and diglycerides, polysorbate 60), cocoa powder processed with alkali, corn syrup. Contains 2% or less of: cornstarch, salt, invert sugar, natural and artificial flavors, caramelized sugar (sugar, water), caramel color, acetic acid, preservatives (potassium sorbate), sodium acid pyrophosphate, citric acid, sodium citrate.\n\nMike's frosting contained only five ingredients:\n\nConfectioners' sugar, cocoa powder, butter, evaporated milk, vanilla extract.\n\nAgain, there's no oil and no corn syrup in the recipe. The cost for the raw ingredients is roughly the same. Cake mixes aren't even big time savers. Multiple studies conducted from the 1950s onward comparing the time it takes to make a cake from scratch versus a boxed version have found that the average time savings ranges from one to six minutes. So why make cake from a box at all? As it happens, cake illustrates an interesting story about what we think about food and cooking. After World War II, food manufacturers had to figure out other ways to market all the food-science technology developed during the war for army rations. The end of the war meant the loss of a massive market. So instead they focused their sights on our mothers and grandmothers, starting a not-so-subtle decades-long campaign to convince people that at least some elements of cooking were not worth the effort, says Laura Shapiro, author of _Something from the Oven: Reinventing Dinner in 1950s America_.\n\n\"The food industry created a basic assumption about cooking generations ago, and it's now fully settled into place as reality. Cake mixes exist, therefore they are easier than real baking, therefore real baking is hard,\" Shapiro said. \"Another factor might be that we have frozen and boxed versions of things that really _are_ hard\u2014frozen croissants, for instance\u2014so perhaps the very fact of packaging something gives it that aura of being out of reach.\"\n\nThe difference between a real cake and a cake-mix cake was apparent to most women baking for their families in the fifties. \"But as generation followed generation, the number of home cooks recognizing that difference dwindled,\" Shapiro added. Bluntly put, consumers not only get used to but prefer the flavor of the artificial version, not the real thing.\n\nMany people believe that women started to use convenience foods in droves in the 1950s. But as Shapiro documents, women then as well as now were responsible for the vast majority of meals in homes, and they initially avoided many prepared foods for fear that they'd appear to be shirking their duty as housewives _._ Boxed cake mixes were a prime example. When they first hit the scene in the 1950s, a homemaker needed to add only water. Women dreaded the guilt of serving such a cake, not to mention the depressing faux egg flavor. So despite initial interest, sales remained stagnant. Researchers found that cooks wanted more involvement with their cakes in order to have the necessary pride of ownership that goes with baking. Food scientists never cared for the outcome of cakes using dried egg whites anyway, so they changed the formula so that home cooks could participate by adding eggs. How delightful! Women could have their boxed cake and contribute, too. With that subtle change, the sales of cake mixes skyrocketed.\n\nEven now, many boxed foods require the addition of eggs, milk, butter, oil, or margarine, even though it's unnecessary given the state of food science. But by requiring those ingredients, they give the perception of \"cooking\" without too much fuss.\n\nHowever, the \"shortcut\" of boxed cake mixes typically includes twenty-two to thirty-three ingredients, many of them polysyllabic chemicals. A friend of mine majored in chemistry in college and later went to work for a major food company. To this day, he refuses to eat ultraprocessed foods. One reason is that the method to approve food additives requires that individual ingredients be tested and weighed in isolation, and as a result no one has any idea how they all interact together.\n\n\"When it comes to food additives, we're the mice,\" he said.\n\nThat isn't all bad. Scientists love mice. Well, they make great subjects, anyway. In _The End of Overeating,_ author Dr. David A. Kessler documents the decades-long pursuit of scientists working for food manufacturers to fine-tune heavily processed and fast-food fare. Their goal: to hit a \"sweet spot\" of the holy trinity of fat, salt, and sugar, trying to turn just the right keys to unlock a dopamine response in your brain. Dopamine triggers neurotransmitters to provide an artificially enhanced pleasure response. Finding that combination for them is like the scene in old movies where a guy uses a stethoscope to listen for clicks in a huge bank vault. Suddenly, he hears the last click and, voil\u00e0! The safe opens.\n\nThe goal of food science isn't flavor but consumption. \"When scientists say a food is palatable, they are referring primarily to its capacity to stimulate the appetite and prompt us to eat more,\" Kessler noted.\n\nFor example, eat a handful of blueberries and you're easily satiated by the natural sugar. Not so with an ultraprocessed frozen blueberry waffle. Even the scent that escapes the toaster has been carefully orchestrated to heighten your anticipation. Perhaps you don't even think it tastes great, yet you still want another one . . . and perhaps another one. This odd response triggered in the brain led researchers to find that rats fed a steady diet of junk food quickly become addicted to it. Waiting for their dopamine fix from high-sugar, fatty, and salty treats, they rejected their normal \"rat chow.\" Some of them starved to death. Notably, serious habit-forming drugs such as cocaine or heroin trigger the same response in the rats.\n\nTargeting this reaction is the reason why more than three-quarters of the sodium that Americans consume comes from ultraprocessed convenience and fast foods. Does a cup of soup really need 38 percent of an adult's recommended daily salt intake to taste good? No. Does a piece of frozen lasagna need three teaspoons of sugar, the equivalent of the amount in a glazed doughnut? No. But both do if the manufacturers want you to buy more soup and more lasagna.\n\nThis explains cravings for potato chips or fast food. It's also why some children reject nonprocessed food. It takes very little time for tiny haywired brains, hungry for dopamine, to reject anything other than what might match the high-fat, high-salt, and high-sugar makeup of chicken nuggets, a frozen pizza, or a box of macaroni and cheese. Jodi's son came to mind. \"It's so hard,\" Jodi told me about trying to feed her son something other than his preferred kid's diet. \"I do my freakin' best to be patient, but then we have these scenes. So I admit it, a lot of the time I just give in and give him what he wants.\"\n\nMany nutritionists believe toddlers should consume no more than 1,000 milligrams of sodium a day; the USDA cites 1,500 milligrams. Yet one can of Campbell's condensed alphabet vegetable soup contains 2,100 milligrams. That cup of soup has a teaspoon and a half of sugar, the same amount that's in a slice of apple pie.\n\nMost packaged foods are engineered to mimic a pharmacopoeia of flavors, even if you're expected to consciously taste only one. The more complex the flavor, the more you'll eat. This explains complicated variations of fried chicken or dipping sauces for chicken nuggets. But research suggests that when one or two flavors dominate, people eat less.\n\n\"There's the obvious fact that the single most important thing about food is taste,\" says Dr. David Katz, author of _The Flavor Point Diet._ \"If we know that limiting food to simple flavors causes people to fill up faster, it really makes sense that having a wide variety of flavors engineered into foods would make people fill up slower and need to eat more. If you are choosing simpler foods . . . you will fill up faster on fewer calories.\"\n\nAll of these are great reasons to learn to pay attention to labels and focus on cooking simple foods. So for the next class, I brought in Beve Kindblade. She had been a nutritionist for nineteen-plus years and I liked her pragmatic approach. The crew wouldn't be cooking, so I lured them in with dinner. But what to serve for dinner when a nutritionist comes calling? We made lentil soup and organic greens topped with a quick strawberry vinaigrette and baked up some fresh whole wheat baguette that Mike was experimenting with at home.\n\nBeve started by casually asking the group some questions. \"So just what brings you to a class like this?\"\n\nGen went first. \"So it's kind of funny, but my mother was insulted when she found out that I was taking a cooking class,\" she said. Growing up, her mother had asked her to make dinners on Monday nights. \"She said, 'I taught you how to cook! Why do you need a class? What did I do wrong?' So I had to say, 'Oh, yes, Mom, you're right, you taught me. I just wasn't listening.'\"\n\n\"Did she teach you anything?\" Beve ask.\n\nGen thought about it. \"The only thing that I really remember was tuna curry. It's a can of tuna mixed with curry powder and sour cream. But I still don't know how to make that either.\"\n\n\"That sounds kind of disgusting,\" Terri said.\n\n\"It kind of was,\" Gen said. \"That's probably why I didn't learn to cook from her.\"\n\n\"I enjoy going out a lot more than I like cooking,\" Terri told Beve. \"I feel like the hassle of cooking for just me isn't worth it. But now that I've got high blood pressure, my doctor wants me to cut sodium, and that's hard to do if you eat a lot of fast food, which I do.\"\n\n\"I know what you mean,\" Dri said. \"I don't eat a lot of fast food, but I have to say that when I cook, I tend to make a lot. Maybe it's from growing up in a big family. Then I eat too much of it or I have leftovers forever.\"\n\nBeve listened to all their stories with genuine interest. Then she spoke about herself, her voice full of Southern twang. Her decision to go into the field came from her own \"What's in the box?\" moment. Growing up in North Carolina farming country, Crisco was a pantry staple used for everything. \"Heck, I even won the Crisco is Cooking Award and still have the trophy! So why would a _shortening_ get me interested in nutrition, anyway? Because I asked one simple question: How is it made?\" She learned that Crisco was a man-made product that shifted natural liquid plant oil into a solid that doesn't really exist in nature. \"My response was, If it doesn't exist in nature, how does my body know what to _do_ with it? But no one then could really answer my question. That's how my career got started.\"\n\nThe group paid rapt attention to the down-to-earth Beve. \"My goal is to bring people back to the joy of eating. The best medicine for you is good food. I have eight hundred patients, and when I see most of them, they're wiped out.\" Her youngest patient is three years old; her oldest, ninety-one. She orders labs to find nutritional issues such as those involving blood sugar, vitamin D, or potassium. A couple of people nodded. Terri noted that she'd heard about vitamin D on _Oprah_. \"It's funny, this kind of stuff has to get on _Oprah_ or _Dr. Oz_ before people start to believe it, doesn't it?\"\n\nSome of Beve's information was new to me. \"If you consume coffee, you should wait an hour before you eat anything because the caffeine will stall your body from getting nutrients from the food,\" she said. \"Increasing fiber is one of the best things you can do in your diet. If you eat pasta, make it one hundred percent whole grain. Fiber helps drop your sugar. I have one client whose weight can fluctuate by sixty pounds just based on how much rice he eats.\"\n\nMost people have cravings due to deficiencies. \"If you increase protein in your diet, cravings for carbohydrates and sweets go away. It doesn't matter if the protein comes from meat, fish, or beans.\" For people who eat meat, she suggests consuming half as much, but spending the same amount of money to buy better quality. Grass-fed beef doesn't hike up cholesterol or estrogen levels and includes more omega-3 fatty acids than traditional beef. \"If you normally use a pound of ground beef in spaghetti or eat an eight-ounce steak, halve the amount and buy grass-fed beef instead.\"\n\nShe advocated reading labels on everything. \"A 'health nut muffin' sold at a coffee shop near my office had more sugar than cake does,\" she said. But the name would lead one to believe that it's healthy. \"You want muffins? Make your own and cut down the sugar. Learn to make your own salad dressings, too.\"\n\n\"We did that! I make vinaigrette all the time now,\" Gen piped up. \"It's so easy.\"\n\nBeve nodded. \"Yes, it is. By learning to make some simple things you'd normally buy prepared, you'll be amazed at how easily you can avoid ingredients like high-fructose corn syrup, cornstarch, the hydrogenated oils, sodium, that sort of thing. We consume way too much soy in our diet, especially soybean oil.\"\n\nDri took a lot of notes. She had been practicing every lesson at home. Weight had been an issue for her for years, and part of her interest in cooking was to simply eat a healthier diet. \"I'm finding that as I make more of my own food, I don't want some of the things that I used to eat. I mean, I look at boxes of pasta or rice dishes and I think that doesn't even appeal to me anymore.\"\n\nTerri nodded. \"The thing with me, though, is that I would probably not be good at memorizing a lot of rules, or I don't want to change what I'm eating too much. I mean, after listening to all this, I'm thinking that fast food, for sure, simply has to go.\"\n\nBeve's basic rules were simple. \"My clients can eat almost anything as long as a serving exceeds three grams of fiber and has less than six grams of sugar and more than six grams of protein. That pretty much eliminates ninety percent of the prepared stuff you find in the supermarket.\"\n\nMost convenience foods fall short of the fiber requirement not only due to the nature of processing but also because the ideal shelf-stable foods lack fiber. Their softer state makes them easier to freeze and ship, as well as faster to consume. Plus fiber fills. That's not good for manufacturers who want you to eat a lot. That's their profit margin.\n\nI had seen a presentation called \"Sugar: The Bitter Truth\" by Robert H. Lustig. He argues that sugar in any form is bad news. \"When god made the poison, he packaged it with the antidote. Sugarcane is a stick. You can't even chew it,\" he said.\n\nWhen I mentioned that to Beve, she agreed that everyone consumes too much sugar, often without knowing it. \"Food labels don't help.\" Sugar is listed in grams on nutrition labels, and there's no \"daily requirement\" for sugar. \"Why not? You don't really need fructose. You can get all the sugar you need from carbohydrates, dairy, and fruit. The last thing you need is sugar added to food. But if you're going to eat something like cake, make it from scratch, not a box.\" Her reasoning, in part, is that you'll see the sugar you're creaming into the butter. \"Most sugar is so hidden that you never know it's there.\"\n\nLisa had been sitting next to me. She got up and started looking at the labels on various items in the kitchen. She quietly came back, sat down, and whispered, \"How much sugar is in a gram, anyway?\" I didn't know. I later checked. Four grams equals a teaspoon of sugar, so a tablespoon is twelve grams. As I began to routinely note the sugar on every can, jar, or bottle I picked up, I was surprised to find the sugars lurking in foods I didn't necessarily consider sweetened, such as granola bars, yogurt, or even ketchup; a tablespoon of the latter contains a full teaspoon of sugar.\n\nBeve fielded a hail of questions. Are frozen fruits and vegetables a good choice? \"Sure,\" Beve answered. \"Sometimes frozen vegetables are actually better than fresh vegetables, as they haven't been shipped a long distance and may have more nutrients still intact.\"\n\nWhat about fat? \"Fat isn't necessarily bad. Some fats are good. I think everyone could use eating half an avocado every day,\" she said. \"Trans fats, anything that's hydrogenated, that's not good. But you're better off eating a little fat than eating a lot of something that's nonfat. Often nonfat stuff has additional sugar in it.\"\n\nHer final conclusion: It's your life. No one will judge you for ordering in a pizza every so often or if you have the occasional hit of ramen noodles for lunch. \"But when it comes right down to it, Michael Pollan had it right. Don't eat anything that your grandmother wouldn't recognize as food, or at least don't eat it all the time,\" Beve said. \"Just eat simply. If you cook more and you think about it, even just get in the habit of reading labels, you're more than halfway there.\"\n\n#    **Mike's Yellow Cake**\n\n_Most off-the-shelf cake mixes are filled with excessive sugars, hydrogenated oils, and other additives. Considering that a boxed cake requires getting out a mixer or beater to add at least three ingredients, consider trying one from scratch. You can use a mesh or small-holed colander to sift the flour if you don't have a sifter. You just want to get rid of lumps and add some air. It's best to measure the f lour after you've sifted it. Cake flour will yield the best results, but all-purpose flour will work if you don't have it, or make your own as noted below. Just be aware that plain all-purpose flour will result in a denser, flatter, and pale-colored cake. This is based on a recipe from the folks at Wilton, the baking and cake decorating goods company._\n\n3 cups sifted cake flour (300 grams), or 2  cups sifted all-purpose   \nflour (300 grams)   \n2  teaspoons baking powder   \n  teaspoon salt   \n1  cups sugar   \n6 tablespoons unsalted, softened butter (  cup or 150 grams)   \n1  teaspoons vanilla   \n2 eggs   \n1  cups milk\n\n  Preheat the oven to 350\u00b0F. Grease the bottoms of two 8-inch round cake pans; line the bottoms with wax paper or parchment paper. Sift together the flour, baking powder, and salt; set aside.\n\n  With an electric mixer on medium-high speed, beat the sugar and butter together until they become light and fluffy, at least 5 minutes. Add the vanilla, then beat in the eggs one at a time until thoroughly incorporated, about 4 minutes.\n\n  Next, add one-third of the flour mixture, then half the milk, then more flour mixture, then the rest of the milk, and end with the rest of the flour mixture, beating well after each addition. When it's all combined, beat for one more minute.\n\n  Spread the batter evenly into the prepared pans. Bake for 25 to 35 minutes, until a wooden pick inserted in the center comes out clean. Cool for at least 10 minutes before removing from the pans. Cool completely before frosting.\n\n_**Note**_\n\nYou can make your own version of cake flour by sifting   cup all-purpose flour (84 grams) with 2 tablespoons cornstarch (15 grams) for each cup of cake flour called for in a recipe.\n\n#    **Creamy Chocolate Frosting**\n\n_Making your own frosting eliminates heavy doses of corn syrup and partially hydrogenated oils, which form the bulk of most canned frostings. If this gets too slippery for frosting a cake, toss it in the fridge for a few minutes to let it thicken. Don't skip the sifting of the sugar and cocoa as that would change the texture of the frosting. Don't bother using fancy cocoa powders or blends here; plain cocoa powder such as Hershey's will provide the best outcome._\n\n2  cup confectioners' sugar   \n6 tablespoons cocoa powder   \n6 tablespoons butter, softened   \n5 tablespoons evaporated or 2 percent milk   \n1 teaspoon vanilla extract\n\n  Sift together the confectioners' sugar and cocoa; set aside.\n\n  Using a mixer or a beater, cream the butter until smooth. Gradually shake in the sugar-cocoa mixture, alternating with the milk. When all is combined thoroughly, add the vanilla. Beat until light and fluffy. Add more milk or cocoa powder if needed to adjust consistency and to taste.\nCHAPTER 12\n\n**Waste Not, Want Not**\n\n  **LESSON HIGHLIGHTS:**\n\n**Using Your Leftovers Can Save Money and Help the World**\n\nFor the four years I lived in London, I held a traditional Thanksgiving dinner. Most people outside North America seem confused by the whole concept. An English friend once asked, \"So what does Thanksgiving have to do with the Fourth of July again?\" It may take the isolation of expatriation to appreciate the curious nature of Thanksgiving. We commemorate a holiday prompted by the most puritanical of Christians with one of the seven deadly sins, gluttony.\n\nThe reason for the original Thanksgiving, at least the one attributed to the Pilgrims, was to celebrate bounty in a time otherwise gripped by hunger, since the Puritans spent the bulk of their time and resources figuring out how to avoid starving to death.\n\nMany things interest me about Thanksgiving. It's one of the rare opportunities when most people give genuine thought to meal planning, cook a whole animal, make a lot of food from scratch, and celebrate the use of leftovers. Part of the experience of the holiday is the nature of the feast, each person assigned a role, from setting the table to mashing potatoes to crafting a pie.\n\nOf course, that's before everyone falls asleep in a tryptophan haze in front of the television. As my husband likes to point out, the occasional Cowboys and Redskins football games scheduled on the day demonstrate just how far we've drifted from the whole concept of that first Thanksgiving with our Native American hosts.\n\nEven as they bemoan food prices, American consumers are generally unaware that they spend less of their wages on food than any other country in the world; just under 10 percent of their paychecks. Compare that to 1900, when 40 percent of wages went toward food. Around 1960, the first time the amount spent on food was no longer the biggest expenditure, the figure was about 25 percent. The declining cost comes with the rise of industrialization of farming practices and the shift of everything we eat\u2014from pigs to cows to orange juice\u2014into mass-produced merchandise.\n\nPerhaps it's the lack of investment that leads to a cavalier attitude toward food. We may give thanks for our bounty once a year, but then as a country we collectively waste about 40 percent of the food produced for consumption the rest of the time. Anthropologist Timothy Jones spent more than a decade studying food waste. His research finds that some crops sit abandoned or unharvested in the fields where they're grown. Supermarkets or suppliers discard another few percent dismissed as too imperfect for retail. The rest\u2014about 25 to 30 percent\u2014we throw away at home. That food goes into landfills to rot, where it emits clouds of methane, a greenhouse gas more toxic and damaging than carbon monoxide.\n\n\"By treating edibles as a disposable commodity, we teach our children not to value food,\" says Jonathan Bloom, author of the book _American Wasteland: How America Throws Away Nearly Half of Its Food (and What We Can Do About It)_. He puts the figure on what we waste at more than $100 billion annually. This jived with what I found in the interviews with the volunteers and the kitchen visits and what I observed in my own house and in the homes of friends. A few of the volunteers agreed to keep a journal of what they bought, ate, and threw out for two weeks. The result? They reported less waste due to the guilt they felt knowing they had to write it down, but even then, an average of 18 percent of their grocery bills went into the trash.\n\nBut why do we waste so much? Both Jones and Bloom offer some interesting insights.\n\nFirst, people often shop for the life they aspire to, not their real one. Everyone knows that they're supposed to eat fruit and vegetables, so we stock up on perishables. Since most people don't plan meals for the week, those beets or greens that looked so great at the farmers' market sit untouched as we end up eating convenience foods. With proper planning, buying in bulk or loading up on twofor-one deals can be a genuine money saver; without a plan, it's just a recipe to double or triple the amount of food tossed away.\n\nDr. Trubek from the University of Vermont has studied the activities of home cooks for years. To her, the greatest lack of skill when it comes to cooking isn't the inability to wield a knife. \"Planning menus is the greatest skill that we've collectively lost,\" she said. \"That, and what to do with leftovers.\" Fortunately, I found two people to help out with both subjects.\n\nAs part of my search for volunteers, I had been a guest on a radio show with celebrity chefs Tom Douglas and Thierry Rautureau. A week after the show, Chef Thierry e-mailed me. \"I'm curious about your plan. Would you like some help?\" As I noted earlier, Thierry hosted a regular radio segment called _What's in the Fridge?_ Callers would dial in and discuss the contents of their kitchen, and Chef Thierry would offer suggestions. Thierry had walked callers through using everything from slivers of avocado to too much zucchini to half a turkey. If anyone could inspire someone to use leftovers, he was the man.\n\nChef Thierry walked into our kitchen in shorts, a tasteful tropical shirt, and flip-flops, his expensive chef's jacket flung over his shoulder.\n\n\"I am here!\" he announced with a flourish, smiling and extending his arm in a wide embrace. Thierry has the classic dark coloring of a Frenchman and immediately owns a room. \" _Mes ch\u00e9ris,_ do you have anything for a cocktail? Tonight is my night off.\"\n\nThat week, people were crazy from the heat. The temperature spiked up to 103 degrees\u2014a record. Lisa went to five places looking for ice and found a lone bag stranded at the bottom of an ice case at a gas station. We chiseled an area out of the packed stand-up freezer for our sacred bag. We presented him with a bowl of it and led him over to the fridge. Jeff and Mike moved around the table and chairs to set up an ersatz demo area as Thierry demonstrated his first use of leftovers, a pitcher of cocktails from remnant vodka, vermouth, and limes. He held up his glass and smacked his lips after tasting his creation. \"Ah! _Fantastique!_ Now I can talk leftovers!\"\n\nFor him, teaching people not to waste food is personal. He grew up on a small farm in the Muscadet region of France in an area where cows and chickens outnumbered people. His family cooked only what they grew, and as the oldest child, Thierry routinely took on the task of helping with dinner. His family ate meat only once a week, usually on Sundays. He remembers many lean times. At age fourteen, he started an apprenticeship with a local restaurant, training that eventually took him throughout France, to Chicago, and ultimately to Seattle. One of the best meals I've had in my entire life was at his restaurant Rover's. The simple roasted squab sat in a warm bath of seafood _nage,_ a kind of light broth. I qualify that meal as a near sexual experience; it was embarrassing to eat in front of the two people with me at the table. This kind of reaction may account for the popularity of his restaurants.\n\nWith a celebrity chef as a teacher, the volunteers turned up in force, a few bringing along friends or family members. As the crowd spilled in, Thierry put his chef's jacket over his shirt and downed his drink. \"Ah, the chef must stay hydrated,\" he said as he poured another.\n\nI welcomed everyone in. \"No diapers today, you are just going to watch.\" Everyone took a seat.\n\n\"But I want Chef Thierry to see my knife,\" Sabra said, disappointment in her voice, as she sat down next to Gen.\n\n\"Oh, speaking of that, I got a new knife,\" Gen said. \"And I just bought a whole chicken yesterday. It was so much cheaper than buying the chicken breasts already separated. My boyfriend was pretty confused about how I am going to turn this whole chicken into fajitas, but he shall soon see how it's done.\"\n\nDri piped in. \"Oh, I buy them all the time now. I'm getting so fast at it. I mean, they're not the most beautiful pieces of chicken, but they taste the same, right?\"\n\n\"I roasted a chicken, too,\" Shannon added. \"And I have to say that I've rarely done that because I was so scared of undercooking it. We have two meat thermometers and neither seemed to work. One said it was 130 degrees and the other one said it was 230 degrees. I was like, What's going on with this chicken? So I pulled the legs away toto check on the juices and pierced the thigh to tell if it was done. My husband was like, 'How did you know to do that?' It was like a Jedi trick or something.\"\n\n\"Hey, have any of you made the bread?\" Trish asked. Jodi, Shannon, Dri, and Cheryl said they had at least once. \"Mine didn't turn out. I wonder if my yeast was bad.\" As they started to discuss their various bread experiences, Chef Thierry clapped his hands at the demonstration table. Almost immediately, sweat beaded on his neck. It felt about a thousand degrees in the kitchen.\n\n\"I'm not going to turn on the stove since it's a record heat day here, yes?\" said Thierry, the seasoned charmer. \"So _What's in the Fridge?_ is based on a very simple idea that happens in every home, even mine.\" People buy ingredients for a recipe but don't use them all. Or they're left with bits of random food. It's easy to throw those out.\n\n\"In America, we have a full fridge and it's so full that we can't even close the door. People look inside at this stuffed fridge and think, There's nothing to eat! I always wonder, What are you waiting for? A hand to hold out a sandwich for you? But one of the most important things to learn in life is to nourish yourself and those around you. I've been doing it for thirty-five years, which is amazing considering that I am only thirty-nine years old!\"\n\nI mentioned to him that I had just taken part in a challenge posted by a food writer friend, Kim O'Donnel, as part of the national movement called \"Eating Down the Fridge.\" The idea is to avoid buying groceries for a week, and instead try to use the remnant food instead.\n\n\"I love that!\" Thierry approved. He suggested putting a favorite photo in the back of the fridge and freezer. \"You'll want to see it, and if you can't, there's something wrong. Your fridge shouldn't be that full.\"\n\nUsing up older products first is known as rotation in restaurants. Home cooks need to learn it, he said. \"Buy one pepper, not three. Buy three potatoes, not three pounds. You'll have less waste and it will help you as a cook.\n\n\"If you have less food in your fridge it will actually push you to cook better. You will have to make something different. It will force substitutions. You think, I don't have a green pepper but, oh, wait, I have a zucchini, so I will try that. It's a good thing. That's how you learn.\"\n\nPeople give up on food too easily. They throw out a whole apple due to one dent that could easily be cut away. \"That zucchini that looks a little soft today? It's like anything, like paying bills. The more you avoid it, the worse it gets. It will never look any better. It will just look worse in a week when you get around to throwing it out.\"\n\nThierry has some fundamental strategies. \"Soup is a gift for leftovers. In the summer, I love to make cold soups; in the winter, I make hot soups.\" He will unload all the unused vegetables from his fridge. \"Get half an onion, caramelize it, add your leftover vegetables, some water or stock; it does not take long to make.\" Salads are big with him. \"It is easy, and it's almost like not really cooking.\" With that, we went to the fridge.\n\nEveryone shifted off their seats to the set of handsome upright commercial fridges. By this point, we had accumulated leftovers from all the various classes. We had also asked people to bring in leftovers. Sabra brought in the remnants of a cheeseburger and fries; someone else brought in a hard-boiled egg; another, leftover chicken salad.\n\n\"Wow, this is so well stocked, we will eat like _cochon_ tonight,\" Thierry said, tossing out the French word for \"pig\" as he pawed through the bounty, which included mushrooms, basil, garlic, eggs, lemon halves, bell peppers, a red onion, and various cheeses. \"Wow, this is fun, but this is probably a lot more food than you have at home. Most of you have things more like that,\" he said, picking up the bit of chicken salad. \"Oh, look, here's some zucchini.\"\n\nSome volunteers traded glances. \"We've cut up a _lot_ of zucchini,\" Terri said.\n\nJeff set down a big bowl for Thierry to collect his choices from the fridge. He started dropping the produce into it. Then he came to some cream left over from last month. \"Oh, this doesn't go into dinner. This should go into a museum!\" he said.\n\nHe stumbled onto tomatoes. \"Oh, no, tomatoes in the fridge.\" Uh-oh, I put those in there. \"Never do this, do you know why?\" he asked the group.\n\nTrish raised her hand. \"Because it kills the flavor?\"\n\nThierry pointed at her and then slapped his hands together in affirmation. \"Yes! Lemons shouldn't really go in the fridge either.\"\n\nHe concentrated again on the fridge and took out a nearly empty jar of mustard. He turned to the group. \"Love it. This is great. You get to the bottom of a jar of mustard, you add some lemon\"\u2014he picked up half a lemon from the bowl\u2014\"and then some olive oil and shake it up. Voil\u00e0, you have vinaigrette. Or just add some vinegar and canola oil. Use what's in your house; that's the name of the game.\"\n\n\"We learned to make vinaigrette, but I love the tip about the mustard jar,\" Shannon said. \"You could do that with the end of the jam in a jar, too.\"\n\n\"Or the end of a bottle of olive oil, since there's always some on the sides,\" Dri added.\n\nI conjured up many options: the bottom of a jar of pesto, soy sauce, balsamic vinegar, or sesame oil. \"That's a great tip, Chef,\" I said.\n\nHe stumbled onto a bouquet of parsley. \"Greens, store them like flowers, the stems in some water. And a batch of parsley like this?\" He held up a bunch of the green stuff. \"It is a massive amount for a couple of people. Make a kind of pesto sauce with it. Chop it up in a food processor with olive oil, garlic, perhaps walnuts. Freeze it in ice cube trays. It will be a frozen cube of green. You can then take some hot, drained pasta and toss in a cube of the pesto, or you can add it to the top of steamed or roasted vegetables. You will look like Martha Stewart!\"\n\n\"That could be a great flavoring for chicken,\" Dri observed.\n\nHis inventory complete, he brought his choices back to the demonstration table and everyone sat down again. Lisa circulated more ice water. The traffic was loud outside, so we closed the door so that Thierry could be heard, but the heat became unbearable. Thierry waved for us to open the door and just talked louder.\n\n\"It's so tempting to overbuy, but discipline yourself,\" he said. \"Don't buy enough fruit for the next two months. Buy it just for the next two days. Just because you can buy something doesn't mean you should. That half flat of raspberries? Yes, it looks great. But that's six pints, it's a lot. You need a Plan A _and_ a Plan B _and_ a Plan C on what to do with that much. We've lost the art of preserving and of canning, the whole idea of harvest,\" he said. \"There is seasonality to food, but we don't feel it. There was a tradition to get as much as you could from the harvest and save it for the year.\"\n\nI thought about that point. We live in a time when you can get peaches in January in Cleveland thanks to an international transportation system that can ship food long distances. In America, food comes from sunnier states such as California and Florida, Mexico, or Central or South America. In Europe, the food comes from Spain, Turkey, or various African countries.\n\nThierry advised that if you buy too much, you can do home IQF, which stands for \"individually quick frozen.\" Spread berries or vegetables on a tray, freeze them, and when frozen, put them into a plastic bag. \"In January, you can pull out a handful onto your pancakes and it's summer again.\" He took a sip of his cold drink and mopped his face with a diaper.\n\nAll the volunteers had small legal pads. He noticed. \"I see you writing. Here's an assignment. Go home. Open your fridge. Take everything out. Toss out only the truly bad expired stuff. Then come up with a plan to use what's left.\"\n\nThis is especially true with spices, he said. \"Most people have a museum of spices. If you have spices that are more than two years old, toss them out. Find a place that sells spices in bulk and buy one ounce of all the spices you think you'll use. After a year, see how much you've used. When you have fresh spices, you'll notice. Everything will taste so much better. You will taste cinnamon, not dust.\"\n\nThen Chef made a salad. He started to discuss onions. \"I like to caramelize them; it adds so much flavor.\" He picked up a knife and got ready to chop. \"Do you all know how to chop an onion? Does anyone want to come up and chop with me?\"\n\nSabra bolted her hand in the air. \"We're professionals at that,\" she declared. She jumped up to cut with the chef. Thierry chopped his onion, a fluid motion. Sabra then chopped hers quickly and confidently. Thierry raised his eyebrows. \"Ah, you do know how to chop an onion. I am impressed.\"\n\nChef showed Sabra how to skin a bell pepper, a tricky technique that involved sliding the knife under the pepper. Sabra did it. She raised her hands over her head in victory. Gen and Shannon highfived her as Sabra returned to her seat. \"You all are very good. You must have a good teacher,\" he said, and winked at me.\n\nNot to be outshone by a twenty-three-year-old prodigy, Chef finely sliced a tomato into paper-thin slices and then curved them into a circle, a classic French technique. He briefly saut\u00e9ed the onions, adding strips of the peeled red pepper. He placed them artfully into the center of the sliced tomato and topped the dish with a drizzle of balsamic vinegar, chopped basil, and a sprinkle of grated Parmigiano-Reggiano. Finally, he topped it off with two anchovies.\n\n\"It is nice to really think about different ways to make salad. You do not necessarily need lettuce, see? It doesn't take too many components to make it work. It's funny, most people are afraid of messing up their kitchens by cooking. I want to ask, How do you feel about your bedroom?\" he said, raising an eyebrow.\n\nThen he presented his dish. It was beautiful, with the deep red of the tomatoes set against the green of the basil and zucchini and the velvety caramelized onions.\n\n\"Trust yourself. Yes, I am a chef, but I am like all of you. I just want something good to eat after a long day working,\" he said. \"I believe all of you can do it. Prove me right.\" With that, the chef took a bow, tipped his hat, and the crowd applauded. \"You know what I have for dinner some nights? Fresh bread, jam, and hot chocolate. You don't have to cook every night. Just eat real food.\"\n\nThen he looked at his glass. \"Oh, no, this is empty. Is there any more . . . water?\" With that, I understood why he wanted a _clear_ cocktail.\n\nI knew that Chef Thierry would be a hard act to follow. But the way to avoid a lot of leftovers in the first place is to plan meals. So the next week another guest speaker came to talk to the group. Jenny, the supermarket chef, waddled into the kitchen eight months pregnant. A fair-skinned blonde who was likely a waif in her less exaggerated state, she looked tired when she arrived. \"I'm so excited to teach tonight, but do you have a stool? I can't stand up for more than eight minutes.\"\n\nJenny started her culinary career working in restaurants (at one time for Chef Tom Douglas, in fact). But she decided that she wanted to make more of an impact on how people cook at home. Her present position was as a chef with a high-end grocery where she interacted daily with shoppers and home cooks. Most people acquire their food through supermarkets and Jenny knows how markets work.\n\n\"Strategists figure out how to use every single inch of a supermarket to get the most profit out of it,\" she said. \"Nothing is left to chance.\"\n\nFew of us realize that we shoppers are mice in a complex retail maze. Supermarkets spend a vast amount of money to figure out how shoppers behave. Every detail is purposeful, from the music they play to the size of the font declaring sales. For instance, you probably notice that it's chilly in a supermarket. I used to think that was because the chill helped to preserve the food. In fact, cold triggers hunger. If you're hungry, you'll buy more. The first thing that you run into in a supermarket is the produce section. The tactile experience of touching food and the bright colors get you in the mood for shopping. The milk, flour, and cereal are invariably spaced far apart. Why? Supermarkets are designed to slow you down. The longer you spend in the maze of a store trying to find staples, the more likely you'll buy something on impulse. Food manufacturers pay for premium shelf placement at eye level, or, in the cereal aisle, at the eye level of children.\n\n\"God, that explains why I hate the cereal aisle!\" Shannon said. \"I dread it. My kids will come up to me and say, 'Why can't we have this cereal, Mommy, it's got Cinderella on it? Most of the time, I will be like, Where did you get that? I didn't even see it.\"\n\n\"Manufacturers also pay for premium space at the ends of aisles, known as the end caps,\" Jenny said. \"Sometimes this stuff is on sale, sometimes not. A lot of times the best deal is not on the end cap but around the corner in the aisle.\"\n\nTerri nodded and took notes. She had said during her kitchen visit that she loathed shopping. \"I have never felt like a very savvy shopper,\" she said. \"I always want to get in and get out and I could never figure out why it took so long or why sometimes I would pick up stuff that I thought was on sale, like on the end stands, and then it wasn't.\"\n\nLike Thierry, Jenny pulled leftover items out of the fridge. While Thierry was classically trained via the rigors of the French apprentice system, Jenny had earned her degree from a well-respected program at a local community college. Yet much of their messages were the same.\n\n\"If you want to save money and eat well, worry less about buying in bulk or what's on sale,\" Jenny started. \"The number one way to save money on your grocery bill is to not waste food. You can buy in bulk, within reason, on nonperishables, but for the fresh stuff, just buy less and shop more often.\"\n\nSmart shoppers plan meals and use thorough lists. They also stock up on basic staples. The meal plan doesn't have to be a strict \"tuna casserole on Tuesday,\" but a looser structure that simply means planning five or six meals for the week.\n\n\"There's nothing wrong with eating the same things routinely. The goal is to feed yourself and the people around you with real food. Cook on the weekends and use leftovers during the week. If you'll eat the leftovers, cook twice as much as you'll eat and put the rest aside for lunches. Or cook twice the amount you'll eat and feed your neighbors once a week and have them do the same for you.\"\n\n\"For nights you have no plan, I tell shoppers to figure out a few simple strategies that are quick and use up the bits of food you've got in your fridge. Some require very little cooking. I have some strategies that I call 'Desperation Dinners.' \"\n\nThe first involved a whole wheat flatbread. \"You can do this with naan or tortillas, too.\" From our leftovers, she added a handful of mozzarella, cut-up tomatoes, some garbanzo beans, some chopped red onion, half a red pepper, and then she cracked an egg in the middle. \"I am a trained chef, but you know what I cook with most on weekdays? Our toaster oven.\"\n\nShe slid the flatbread and egg into the kitchen's toaster oven. She made three different versions using the leftovers: a bit of ham, sliced Parmesan, chopped leeks, zucchini, sliced mushrooms. On each she cracked an egg and then chucked it into the toaster oven. When they emerged, the egg whites and yolks cooked, she topped them with a handful of arugula lightly dressed in olive oil. \"It's an easy way to use up bits of greens, and kids like anything that looks like a pizza. Tortillas or flatbreads keep well, or you can always freeze them and then quickly thaw for a few seconds in the microwave.\"\n\nNext she demonstrated \"Desperation Pasta.\" She seasoned the water with salt and pepper. As the pasta started to cook, she cut a few florets of broccoli off the stem and crafted strips of carrots with a vegetable peeler. \"I like to do this with carrots. They cook quickly, and they add some nice color.\" She dropped both into the water as the pasta finished cooking. After two minutes, she drained the pasta and dropped it into a bowl and tossed it with a handful of greens and grated cheese. \"Now I'm just going to top it with some olive oil, some vinegar, and taste to see if it needs more salt and pepper. Vinegar is wildly overlooked. It's great to add flavor, and it has no calories, plus it has great shelf life. Really wakes up food.\"\n\nShe explained the concept of dinners as \"layers.\" Her daughter will eat the first layer, say, plain pasta with grated cheese and maybe some broccoli. For Jenny and her husband, she'll \"finish\" the dish with pine nuts, chopped chilies, greens, saut\u00e9ed shrimp, and so on. \"If your kids are fussy, you don't have to make a completely different dish. Just evolve it into something more suited for adult tastes.\"\n\nThe class asked about gadgets. She doesn't use many. \"A microplaner is great. You can use it to grate cheese, garlic, gingerroot, just a ton of stuff.\" We handed out slices of her Desperation Pizzas and Pasta. While everyone ate, she discussed more shopping tactics. Make a list of foods with strong flavors that store well in your pantry\u2014things like capers, artichoke hearts, beans, dried mushrooms, and olives, that sort of thing. Buy basics in bulk, and buy fresh sparingly until you routinely use _all_ of your produce, and then add more. \"For produce, I look for what's fresh, in season, and hopefully on sale. We buy a whole chicken every time we shop. Sometimes I break it down but usually I just roast it for dinner that night. I have learned endless uses for it. Salads, pasta, burritos, chicken potpie, chicken salad, risotto. The list goes on. Then I make chicken stock, which also has endless uses.\"\n\n\"Oh, and a basil plant is a great investment,\" she said. \"Any herbs that you buy regularly, consider keeping those plants in your kitchen window. You can get an herb plant at a nursery for the same price as a package of herbs at a grocery and just use what you need.\"\n\nLike Thierry, she advocated trying to force using the last elements in the fridge. \"Try this: Open your fridge. Take out three ingredients that sound like they go together. Put them into the search engine of a recipe site that you trust. If it turns out terrible, well, it's just one meal.\"\n\nJenny summed up the evening with one message. \"You have to define what value is to you in your food. Is it cheap? Or are you going to get a lot of enjoyment from it? Maybe I'll splurge on some great flank steak, but then we'll get three meals from it. But the key thing is to think of food as money. You wouldn't toss a five-dollar bill in the garbage can, would you? If you throw a head of lettuce and some dead cucumbers in the trash, it's exactly the same thing. It adds up.\"\n\nShannon took a lot of notes. She had told us that she spent about seven hundred dollars a month on food, which turned out to be average for a family of four, according to the USDA. But she was always looking for ways to extend her budget. \"Everything that we've been learning in class has been so helpful, but thinking about this whole leftover piece is a big thing for me. I've never been one of those people who could open the fridge and figure out stuff to make from it. I think that's how stuff goes bad, by not knowing what to do with it. This is all super helpful.\"\n\nTrish had been trying to plan more meals, but she had recently figured out the problem. \"My husband likes to be spontaneous. I'll have dinner halfway done, and then we'll go to yoga. Afterward, he'll say, 'Let's go out.' But I have dinner half-finished at home. So that's still a challenge for me.\" On the positive side, they rarely throw away food thanks to his fearless approach to sell-by dates. \"Oh, he'll eat everything, even stuff with mold on it!\"\n\nThe morning after Jenny's class, I took everything out of my fridge, from the condiments to the last remnants of vegetables from the crisper. I remembered the heavy use of Post-it notes throughout the commercial kitchen. I estimated the cost of every item and tagged each with a price on a Post-it. In the course of two weeks, if I had to throw an item away, I'd take the Post-it note and stick it to an area inside one of my cabinet doors.\n\nAlmost immediately, tossing something signaled defeat by surrender. My mind-set changed: Oh, no, I don't want to put that Post-it for this bell pepper on my door. Hmmm, what can I do with it? At the end of two weeks, I'd thrown out about sixteen dollars in food, less than usual but still nothing to make me proud. Among the culprits: remnants of hummus of unknown origin, the estimated cost of leftover bits from dinners, the end of a bag of red grapes, a smudge of mesclun salad left in the container, a nearly full package of sour cream, half a lime that turned brown, leftover chicken salad, half a sandwich brought home from a restaurant that got pushed behind our bread dough, and crumbled bleu cheese that had taken on a disturbing consistency. But the exercise forced me to use some items that I might have tossed: bits of bread pulverized into bread crumbs, the last of a jar of horseradish added to mayo for a spread for sandwiches, black bananas pur\u00e9ed into a kind of ice cream, limp carrots and green onions forced into duty in stock, wizened apples cut up and baked and topped with brown sugar on top of oatmeal, and half an avocado whipped with olive oil and sparkling water for a kind of dressing. One came directly from Jenny, her ersatz \"pizza,\" with the last half handful of just-starting-to-wilt spinach.\n\nDuring the process, I started to read _More-with-Less Cookbook_ by the late Doris Janzen Longacre, a classic cookbook developed in the early 1970s with the Mennonite Central Committee that's still in use twenty-five years later. Longacre preached avoiding heavily processed foods, eating simple meals and more whole grains, and relying less on meat. She advocated shortening the shopping list and developing a stable of recipes on which to rely rather than trying to reinvent the culinary wheel for every single meal. In an opening scene, she describes a four-color advertising pitch she received in the mail for a new recipe-card set that promised to \"make cooking easier and more exciting than ever before!\" The ad stunned her sensibilities. \"The pitch indicates again how we try to turn eating into a super-experience,\" she wrote.\n\nIt reminded me of an old issue of a food magazine that had offered a week of \"fast meals.\" The lineup included a shrimp curry, Moroccan spiced lamb chops, a Mediterranean fish saut\u00e9, a beef stir-fry, and seared scallops with braised cabbage. The shopping list next to it contained more than fifty ingredients. Sure, some people might have some of the ingredients on hand as staples, but I examined the recipes closer. If you made all those dishes, among the leftovers you'd be left with half a head of cabbage, half a can of coconut milk, the remainder of bunches of cilantro, parsley, and basil, among other things. So what exactly to do with all of those extra remnants of food?\n\nWhen I thought about Longacre's views, why would I take on a set of menus that would leave me with so much? Does anyone need lamb, scallops, beef, fish, and shrimp all in the space of five days?\n\nChef Thierry, Jenny, and Longacre all had a point. While exploring my crisper drawer, I found a nearly full two-pound bag of organic carrots tagged with a four-dollar Post-it. Half an onion beckoned. Rosemary sprigs sat waiting on my counter. Rosemary and carrots together? I would never have thought of that combination if I had not been forced to figure out how to use them up. The result was a savory yet sweet chilled soup. I didn't know what the volunteers had learned through these classes, but I had learned something important. We all can do more with less.\n\n#    **Velvety Chilled Rosemary Carrot Soup**\n\n_This savory and sweet soup can be served at any temperature, but it's excellent chilled. Immersion, or \"stick,\" blenders are great for soup because you can plunge them directly into the pot. Hot soup can create a vacuum in conventional blenders, so if you use one, let the soup chill slightly first, and then take the cap off and cover with a towel. Running soups through a food mill is a low-tech option. If you have none of the above, simply mash the softened vegetables with a fork or potato masher; it will lend a rustic feel to the finished product. Add the rosemary, branch and all, but be sure to remove it before pureeing._\n\n2 tablespoons olive oil   \n1 medium onion, chopped (about 1  cups)   \n2 leeks (white and light green parts), chopped   \n1 pound carrots, diced   \nSeveral fresh rosemary sprigs   \n1 bay leaf   \n2 quarts chicken or vegetable stock   \nCoarse salt and freshly ground black pepper   \nPinch of cayenne (optional)   \n  cup quality plain yogurt (optional)   \nCroutons (optional)\n\n  Heat the olive oil in a 4-quart or larger saucepan. Add the onion and leeks and saut\u00e9 until softened. Add the carrots, rosemary sprigs, bay leaf, stock, a couple of pinches of coarse salt, a few grinds of coarse pepper, and a pinch of cayenne if using. Bring to a boil, then cover and reduce the heat to simmer until the carrots soften, about 1 hour.\n\n  Remove from the heat. Discard the rosemary and the bay leaf. Puree until smooth. Add additional water if necessary. Return to the pot. Check the seasonings, adding salt, black pepper, and cayenne to taste. Serve warm or cooled. Garnish with a scoop of yogurt or croutons if desired.\nCHAPTER 13\n\n**The Power of Soup**\n\n  **LESSON HIGHLIGHTS:**\n\n**It Turns Out That** _**Supper**_ **and** _**Soup**_   \n**Come from the Same Place**\n\nMy mother always said that if you can boil water, you can make soup. Chef Thierry had remarked that it's a gift for leftovers. From the beginning, I planned a class on stocks and soups as the last class, knowing that we'd end with a fridge full of leftovers.\n\nAs the volunteers filed in to claim a diaper and an apron, they all fretted about the night being the last. \"It's going to be so strange not to see everyone each week,\" Gen said. I asked what people had been up to in their kitchens.\n\n\"You know, I've been thinking a lot about what the nutritionist said,\" Dri said. \"It's funny, we hear so much about how fats are bad, but then she talked about the difference between good fats and bad fats, and how olive and coconut oil are good, but, then, palm oil isn't so good. So I went through everything in my cupboards and I looked at all of the labels.\" She considered the impact of feeding bad fats not only to herself but to her nieces and nephews who visit regularly. She was surprised by what she found in her wares, already edited significantly as part of her recent move. \"Really, after the Alfredo versus Alfredo night, I just kind of decided to ditch most of that stuff anyway. I've just decided that I simply will not eat out of a box anymore.\"\n\nThe comment spurred a lot of conversation. \"I never thought I'd be like that but I'm getting there,\" Jodi said. \"I went to make pancakes the other day and I looked at the label. It was basically just flour, hydrogenated oil, corn syrup, and baking soda. I thought, Do I really want to feed this stuff to my son? I looked up a pancake recipe in a cookbook and thought, That's it?\" She had whipped up the batter and then mashed in an overripe banana. Her face took on an obvious look of pride when she reported that her son loved them. \"He was like, 'Mommy, these are the greatest pancakes!' \"\n\nCheryl was baby-free again this week. Of all the classes, she confessed that this was a big one for her. When we visited her kitchen, she had made a can of soup for lunch. \"I buy a ton of soup,\" she admitted. The organic kinds can be expensive and often still come packed with salt. \"I want to master soup so that I can cut down on how much I buy. After all these cooking classes, it feels a little lame to be opening a can.\"\n\nFew nutriments date back as far as soup, which likely debuted shortly after cavemen discovered the joys of boiling water. The word _soup_ stems from the same Germanic word that led to the English word for \"supper,\" writes Alan Davidson in _The Penguin Guide to Food_. \"From that came a noun, _suppa,_ which passed into the Old French as _soupe_. This meant 'piece of bread soaked in liquid' . . . which ultimately led to the word 'sop.' \" So the words for _supper, soup,_ and _sopping_ up the soup with bread all derived from the same source.\n\nBy the late 1600s, soup was so beloved, people didn't want to leave home without it, resulting in the development of \"portable soup,\" which was meat stock boiled so long it reduced to a thick paste that was dried and cut into strips. The strips were then reconstituted with hot water. Davidson quotes a portable soup enthusiast from 1736 who referred to it as \"veal glue.\" Mind you, those were words used by someone who _liked it_. The term _restaurant_ stems from the French verb for \"to restore,\" a reference to the shops that sold soup in the late nineteenth century. To complete the cycle in something of an ironic twist, modern restaurants make \"soup du jour\" from leftovers of their nonsoup menus items. I knew that Lisa had a story about the trauma of soup du jour and asked her share it.\n\n\"My very first job cooking, I spent a year making soup,\" Lisa started. She reported to work on the lunch shift with a chef who had worked in the industry for nearly forty years. She was impressed by his seemingly laissez-faire attitude toward soup du jour. Broccoli looked a little shaky? He'd whip up cream of broccoli soup. Too much cabbage? He'd ask Lisa to shred it and then add some ham, white beans, and carrots. The owners fired the chef without warning a few days after she started. In her first week out of culinary school, she found herself managing a lunch shift for a sixty-four-seat restaurant all by herself.\n\n\"It felt like being thrown out of a plane with a scribbled five-point list on how to complete a parachute jump,\" Lisa said. In addition to a hundred other tasks that she had to complete the morning she first worked alone, she had to come up with the soup du jour. In the walk-in, she found a Thai curry base used for a seafood dish. She thinned it with chicken stock. A revelation! Sauce is thick soup!\n\nAlthough she got a handle on the rest of the job after she figured out that there would be no replacement for the axed chef, she fretted about the soup. It kept her awake at night. \"The very nature of it, that it can be _anything,_ just freaked me out,\" she told the group. You could feel her tension in the room. Lisa studied culinary school textbooks and looked up recipes trying to combat her soup angst. \"But then I would get to the kitchen and find that I didn't have all the ingredients, so I'd panic.\"\n\nThen a friend gave her _The Daily Soup Cookbook_ by Leslie Kaul and Bob Spiegel. This simple book offers straightforward instructions on two hundred soup recipes organized by ingredient or theme, such as tomatoes, beans, or gumbo. She showed off her battered copy to the class.\n\n\"I would pile up my dying ingredients on a counter, and then flip through this book to find recipes that would fit. I could make something like minestrone or a tortilla lime soup, but not exactly. I would have hamburger and not sausage, or sausage but not chicken.\" At first, this struck her as the culinary equivalent of forcing a square set of ingredients into a round hole. But she discovered that no matter what she changed, as long as the flavors seemed to go together, the soup always turned out anyway.\n\nThe soup du jour changed her perspective as a cook. \"All those substitutions taught me that I do not have to be a slave to a recipe, or even to convention,\" she said. \"It also taught me something critical. You don't have to buy ingredients for soup.\"\n\nWith that, we all ransacked the fridge, pulling out vegetables and the remnants of a chicken I'd roasted the day before. \"Soups generally follow the same formula,\" I began. \"You saut\u00e9 some aromatics, usually chopped garlic or leeks, and then you add in vegetables, meat, or poultry that needs some time to cook. Add in stock or water. That's a good time to add a bit of salt, some herbs and spices. Simmer for at least an hour. Give your soup some time to develop.\" Foods that don't take much time to cook, such as shellfish or pasta, go in at the end. \"Then taste it. Add salt or whatever it might need to pep up the flavor. That could be lemon, vinegar, maybe minced garlic or fresh herbs.\" Garnishes such as croutons or grated cheese are great but unnecessary.\n\nWe started two pots, then split up the volunteers into teams and let them figure out a soup from the leftovers. One team settled on chicken noodle soup, the other on a variation of minestrone. Each built an initial layer of flavor by saut\u00e9ing onions and leeks. Team Chicken added carrots, celery, fresh corn, and a fistful of fresh thyme and parsley tied together, along with the remains of the roasted chicken. Team Minestrone added zucchini, green bell pepper, cauliflower, garlic, red pepper flakes, a can of tomatoes, and the rind of some Parmigiano-Reggiano.\n\n\"You can also just start any pot of soup with half a roast chicken, whether you've bought it or you've made it, and go from there. It's an easy shortcut,\" Lisa said. \"If you keep your pantry stocked with some basics it's super easy to pull together a soup with minimal effort. I usually have canned tomatoes on hand, coconut milk, curry paste, rice, some type of pasta, dried beans, bacon, fresh herbs, stock, onions, carrots, celery, some dried chili pods. That's about it, that's most of my pantry. Everything else is accessories.\"\n\nOften, the difference between boring soup and fabulous soup is just time. Soup almost always has to simmer for at least an hour, usually two. It takes time to draw all the flavors out of the components. \"Trying to boil it like mad for a half hour is not going to trick the laws of cooking into thinking it's simmered for two hours,\" Lisa said.\n\nWe left each pot to simmer as we turned our attention to the notion of stocks.\n\nAs if on cue, Ted sauntered into the kitchen. \"I heard there was some stock action going on in here,\" he said. Ted is a stock aficionado; he once penned a two-thousand-word missive on the subject. \"Thought I'd just drop by and have a look.\"\n\nI waved him in. \"So, stock is the extra bonus from a roast chicken,\" I started. \"You can just simmer the bones with some vegetables. One chicken can generate a couple of quarts of stock. Considering that you pay two or three dollars for a quart of chicken stock, it's worth it not to throw them away. When it comes to the vegetables for stock, some of them can be odds and ends or trimmings you might normally throw away, like the hard heel of celery, scraps of onion, and the tough green tops of leeks. Those can all go into stock.\"\n\n\"So it's basically free,\" Sabra commented. \"That's cool.\"\n\nThe roasted chicken version is an easy shortcut, but applies the same principles of all stock. \"Now we're going to start with some bones.\"\n\nOver the weekend, I had rounded up plastic bags marked \"For chicken stock\" and \"For beef stock\" from my freezer. I never let a bone go, whether it comes from a chicken I've broken down, hot wings that I made at home, or even leftovers from restaurants.\n\nNot long before the class, we ate at the Space Needle Restaurant with friends from out of town. Mike and his friend Bill ordered the day's special, a twenty-five-ounce steak that had a _Flintstone_ s-style bone jutting from the meat. The server looked at me like I was a crazy woman when I told him I wanted to take home the bones, but he good-naturedly wrapped them up in a takeout bag. As we waited for the elevator, the Italian ma\u00eetre d' asked what was in the bag. His face lit up. \"Ah, now, that's a smart cook! I never understand why people leave without them! My nana would kill for bones like that.\"\n\nI'd roasted all the bones and made most of them into stock. I kept a small pan of each set of bones to show off the result of roasting. The chicken bones had a crisp quality and a mahogany color in some places where the bone was exposed, and wept brown puddles of caramelized goodness onto the pan. The beef bones looked dried and nearly charred, like trees in a forest recently ravaged by fire.\n\nI had Ted take over the explanation. For chicken, beef, veal, or other meat-based stock, the method remains the same. \"You can just simmer chicken in water with the vegetables. That's known as white stock,\" Ted said. \"But you'll get more flavor if you roast the bones first. You want your oven nice and hot, around 400 degrees. The goal is to caramelize the bones a bit. About a half hour or forty-five minutes is usually enough. When you can really start to smell them, that's when you're getting somewhere.\"\n\nI waved everyone over to the big stove. \"Okay, here are the two pots of stock that I made from the rest of the bones. They've been simmering about two hours. I want you to smell it and taste it.\"\n\nLisa handed everyone a spoon. Each person dutifully sniffed the gurgling liquids, then dipped her spoon in and sipped a taste. They looked thoughtfully at one another. \"It reminds me of the stock tasting,\" Cheryl said. \"It's more meaty and chickeny.\"\n\n\"So you take the browned bones, put them in a pot, and cover them with cold water. It's great to add some water to the roasting pans while they're hot and scrape up any browned bits left in them,\" Ted said. \"It will clean your pans and boost the flavor of your stock. Add vegetables, typically onion, carrot, and celery. The usual ratio is one pound of vegetables for every three pounds of bones. Add a bay leaf, some thyme, parsley. Some people like to add whole peppercorns and some garlic.\"\n\nThe point is to simmer them for as long as it takes for \"the bones to give their all,\" as Julia Child once wrote. For chicken, that's usually two to four hours; for beef, about double that.\n\nTed pulled a ladle from among the coterie of utensils and demonstrated a key technique. First he skimmed off a slight oil slick on the top of the stock, and then tackled an island of bubbling foam. \"Skimming simply means to do a seek-and-destroy of any foam or fat on the surface,\" Ted said. \"It makes a world of difference to both stock and soup. It keeps it from being greasy, for one thing.\"\n\nTed had a few other points on meat-based stocks. \"Try not to boil it or it will turn cloudy. What you want to do is get it superhot and then reduce the heat until you get the occasional, or 'lazy,' bubble. Don't add salt. As the water evaporates, the salt flavor will concentrate and it can be too salty.\"\n\nSince Lisa had brought in her soup book for show-and-tell, I opened a copy of _Vegetarian Cooking for Everyone_ by Deborah Madison. During my fling with vegetarianism, I started to make vegetable stocks, a habit that I keep up. \"They're built the same way. Roast the vegetables and then simmer for about a half hour. This book has a great section on vegetable stocks.\" I flipped through the section. \"There's a whole breakdown of vegetables to use by season, stocks for stirfries and curries, mushroom stock, and even tomato-style stock. You just want to avoid strong flavors such as cabbage, beets, broccoli, or, as she puts it, 'funky or spoiled' vegetables that you wouldn't eat.\"\n\nNext, we moved on to fish stock. In French, it's known as \"fish fumet,\" a common foundation in chowders and seafood dishes. I had bought two pounds of fish bones for a dollar from my regular fishmonger, an assortment of fragile skeletons and thick pieces of white bones from larger fish.\n\n\"A lot of recipes will call for clam juice, but what the food writer really wants you to use is fish stock.\" Personally, I loathe clam juice. Most supermarket varieties are simply too brackish. \"Even the cheap stuff is almost three bucks for eight ounces, so you're paying twelve dollars for a quart. You want white fish without too strong a flavor. No mackerel, no salmon.\" I combined some onion, celery, half a lemon, a bay leaf, a few sprigs of parsley, and the fish bones in a pot with cold water.\n\n\"You can do the same thing with shrimp, crab, or lobster shells, too. I sometimes get Dungeness crab shells from my fish market. They're free and it's awesome in gumbo,\" I said.\n\nTogether, Ted and I strained the pot of gurgling chicken stock. \"If you've got a big pot, don't try to pour it out. Remove the bones with tongs, and then ladle the liquid out.\" He ladled the stock through a mesh sieve lined with cheesecloth over a massive bowl. \"This is traditional but a colander with a coffee filter works, too.\" After the chicken, he strained the fish fumet.\n\n\"I'm totally not digging that smell,\" Dri said of the fumet. \"I mean, I guess it's good for chowder, but it's pretty fishy.\" Sabra nodded. She wasn't into it either.\n\nA hot steaming pot of stock cools slowly and has a tendency to linger in what food safety types refer to as the \"danger zone,\" or temperatures between 40 and 140 degrees. \"You want to make sure your stock stays out of the danger zone by cooling it quickly,\" Ted said. \"You can do a few things. Take the stock and put it into an ice bath in the sink and keep stirring. You can pour it into a shallow pan, like the bottom of broiler. Or you can wait until the stock cools below 180 degrees and then plop plastic bags filled with ice into the stock. Whatever, once it cools to room temperature or below, put it into the fridge right away.\"\n\nThe other option is the remarkably low-tech \"cold porch\" method, Lisa added. \"In winter, I just put a big pan of stock outside uncovered and stir it every so often until it cools down.\"\n\nShannon signaled a time-out. \"Okay, I'm a little confused. What's the difference between stock and broth?\"\n\n\"Stock is made from bones, broth is not. Technically, there's no such thing as vegetable stock. But since people aren't sure, a lot of things are called stock interchangeably with broth.\"\n\n\"All these years, and I never knew that,\" Shannon said.\n\nWe went back to the soups. Team Chicken finished theirs by tossing in handfuls of pasta left over from the pasta class, shredded cooked chicken, and chopped fresh parsley and oregano. Team Minestrone added chopped tomatoes, a can of red beans, then garnished each bowl with grated Parmigiano-Reggiano and chopped basil.\n\nThe streetlights came on outside. Each person slurped the various soups. Lisa produced a bottle of cava from the fridge. With great drama, she launched the cork and we all toasted.\n\n\"Here's a toast to you,\" Dri said to Lisa, Ted, and me. We clinked glasses.\n\n\"Here's to all of you and your willingness to chop zucchini!\" I said. Clink.\n\n\"To zucchini!\" Sabra said. Clink.\n\n\"To diapers!\" Shannon said. Clink.\n\nThen the mood turned bittersweet as the volunteers reluctantly took off their aprons and dropped their diapers into the bag we hauled around for the dirty laundry.\n\nAfter they left, Lisa, Ted, and I did a final cleanup in the kitchen. I stayed behind to mop. Alone in the kitchen, I dragged the bright yellow roller and swished the mottled gray-yarn mop across the black-and-white tile. I was deep into the Zen of the rhythmic motion when my phone rang.\n\n\"It's Eddie,\" my mother said.\n\nMy stepfather, Eddie, had suffered from an impressive string of maladies in the past dozen years. A recent surgery had left him weak, and since then he'd been falling. That morning, he collapsed and broke a chair. My mother tried for forty-five minutes to get him up. An ER doctor diagnosed him with pneumonia, a serious illness for a seventy-eight-year-old who already had enough health problems.\n\nThis time I heard something different in my mother's voice: sheer exhaustion. He'd barely slept for a week. I knew that as his caretaker, she hadn't either. \"Mom, I'm coming home,\" I told her. She insisted that it wasn't necessary. I hung up and called Mike, then I finished mopping the floor. By the time I got home, Mike had booked me a ticket to leave the next afternoon.\n\nWe had set up a couple of makeup classes for the following week. I called Lisa. \"Oh, no, should we reschedule?\" she said.\n\n\"No, you're going to teach them,\" I said.\n\n\"I can't teach them by myself,\" she replied quickly. \"Let's reschedule.\"\n\n\"Are you kidding? _I've_ learned stuff from you. So of course you can. I have total faith in you.\"\n\nIn Florida, the relief on Mom's face said it all. She looked as if she'd aged years. Eddie's gaunt appearance threw me. A thin guy by nature, he'd lost twenty-five pounds. His cheeks had an unhealthy hollow. As I hugged him, I asked, \"What can I do for you, Eddie?\"\n\n\"Make me dinner?\" he asked. \"It's not like I travel now. The highlight of my day is hitting Walgreens for prescriptions. Food is all that I have to look forward to.\"\n\nThat week, my sister, Sandy, and I spent two full days making huge vats of food, from spaghetti with meatballs to beef stew to scalloped potatoes. Fresh from the soup class, I cleared out my mother's fridge to make three different varieties, including chicken noodle, Eddie's favorite. Ever competitive, Sandy made Eddie's all-time favorite meal, a classic full-on Thanksgiving spread replete with turkey, mashed potatoes with gravy, and homemade stuffing. We diligently labeled and froze all of it into portions sized for two\u2014enough for two months' worth of meals.\n\nDoing this gave my exhausted mom a break from both shopping and cooking. She rested, and we took long walks on the beach together. The stockpile of varied foods kept Eddie interested in eating. Each morning, she'd open the freezer and ask what he wanted. \"How about we have the turkey gumbo for lunch and the cassoulet for dinner?\" He gained back some weight and his condition improved. Before leaving for the airport, he gave me a lingering hug as he thanked me for the meals. \"I can tell with every bite that you love us.\"\n\nAs I sat on the plane back to Seattle, I thought about the power of cooking to nourish, to comfort, and to heal. It was the fourth time in one year that I had contributed meals to people's freezers in the midst of a crisis: Mike's sister had gone through chemotherapy for breast cancer, a friend's husband had had surgery to remove a brain tumor, and our friend Amy had suffered life-threatening complications after the birth of her son.\n\nIt's a simple act, but to bring someone chicken soup when they're sick is not just about a meal, it's a tangible and physical sign of caring. If you buy a chicken and make it from scratch, the message is completely different from bringing over a can. It says, \"You're important, and I care about you enough to take the time to help restore you.\" Like laughter, soup is not the same when it's canned.\n\n#    **Blissfully Simple Chicken Stock**\n\n_Gather up all the bones from a roast chicken after you've wrested all possible use from the meat. Depending on how much water you add and how long it simmers, the yield will be six cups to three quarts._\n\nBones from 1 roast chicken   \n  medium onion, quartered   \n1 celery stalk, roughly chopped   \n1 large carrot, chopped   \nFew sprigs of fresh thyme and\/or parsley   \n1 garlic clove   \n1 bay leaf   \n4 to 5 quarts cold water\n\n  Put the chicken bones, onion, celery, carrot, fresh herbs, garlic clove, and bay leaf into a 5-quart or larger pot. Add 4 to 5 quarts of cold water. Bring just to a boil and then turn the heat down until it simmers. Let it simmer for at least 1 hour and up to 3 hours. Skim any foam or fat from the top with a spoon. Drain it in a colander or mesh sieve lined with cheesecloth or a coffee filter into a large bowl. Cool, then refrigerate or freeze until needed.\nCHAPTER 14\n\n**Kitchens, Revisited**\n\n  **LESSON HIGHLIGHTS:**\n\n**What Effect Did the Classes Have on the Volunteers?**\n\nNot long after the last class in early September, an acquaintance shared a most discouraging quote. Her grandfather used to say that he'd \"rather try to change someone's religion than change the way they eat.\"\n\n\"I'm only telling you this so you're not discouraged if your project didn't work,\" she said. I may have defriended her on Facebook.\n\nAlthough the signs had seemed encouraging over the summer, that conversation dogged me. What if I had spent all this time and effort\u2014not to mention the work of everyone else\u2014yet it had no impact? I waited until after Thanksgiving to follow up. As we had the first time, we'd audit their cabinets and ask them to prepare a meal that they ate regularly. I wanted the distance of time to see what, if anything, had stuck.\n\n#  **SABRA**\n\nSix months after our first visit, a frigid November wind wailed outside as I stood in Sabra's warm kitchen with Lisa and Sabra's dad. He was just as curious as we were. Sabra had assembled a casserole in a new pan, one of many recent additions to her kitchen. \"I got this idea from Stouffer's. It's a Cheddar and potato casserole with broccoli and bacon,\" she said, sliding the prepared dish into oven. \"I can make twelve servings of this for the same price as a couple of dinners, it tastes way, way better, and I know what's in it.\"\n\nI asked what her biggest takeaway was from the course. \"Confidence,\" she replied quickly. \"I can look at any recipe and know that I can make it now. I never thought that before.\" She picked up her chef's knife from the counter and handled it with reverence. \"Learning knife skills changed everything. One reason I used to get intimidated by cooking was I'd see a bunch of stuff to cut up, but now I know that's not a big deal. I actually like that part.\"\n\nFrom the minute we arrived, Sabra seemed eager to show off her freezer. \"You ready?\" she asked. We all nodded. With a flourish, she flung it open. \"Notice there are _no_ frozen meals in here,\" she said proudly. Two whole frozen turkeys took up the space once occupied by stacks of frozen dinners. \"They're a Thanksgiving gift from my dad's business,\" she explained. \"I haven't made them yet because I don't have a pan big enough, but I've asked for one for Christmas,\" she said, eyeing her dad sitting at the kitchen counter.\n\nRounding out the freezer were bags of tortellini, vegetables, and individual portions of various leftovers. \"Those are lunch,\" she explained. \"Oh, look!\" She grabbed a plastic bag and shook it. \"Chicken bones! For stock! I made some, but I'm out.\"\n\nIf the project had succeeded in nothing else, Sabra, the young woman who had served us frozen lasagna and White Trash Garlic Bread a mere six months ago, no longer wandered the frozen-foods aisle looking for dinner.\n\n\"While they're cheap, I just realized they aren't such a good deal after all,\" she said. \"I find that if I cook a couple of times a week, then there's always something my boyfriend and I can eat, which is cool. It's just like convenience food, except you make it.\"\n\nAvoiding frozen dinners wasn't the only shift in Sabra's habits. She found herself shopping more often, hitting a farm stand nearby for vegetables. Whenever she buys food, she always does so with at least two meals in mind. \"I think we're spending about the same on groceries but less on food overall because we eat less takeout. We waste a lot less food, too.\"\n\nSabra used to eat Hamburger Helper every other week. She admitted that she hadn't made it since July, about midway through the project. Her remaining box of the stuff had been banished to a high shelf to make way for new additions to her modest pantry. \"It's funny, a lot of things expire that I never thought about, like oils, flour, and spices.\" She had used up the rest of her expired flour, had replaced her dead spices, and now buys smaller doses of everything. \"Now when I buy something like olive oil, I get the smaller size even though it's more expensive. But I know that it will stay good until I can use it all.\"\n\n\"Oh, and check it out!\" She reached into the dairy drawer of her fridge to retrieve a paper package. She unwrapped a bit to expose the tip of a hard wedge of Parmigiano-Reggiano cheese. She hadn't seemed that engaged in the tasting class, but she reported that it had had a powerful effect on her. \"The tastings made me really think about what I buy. People can say you should go and spend the extra money and get real cheese, but until you taste it for yourself, it's almost impossible to know that it's worth it.\"\n\nHer fascination with McDonald's had subsided, too. \"It's funny how if you eat well and then you go eat fast food, you can really feel it. It kind of sinks in my stomach, and I can hear my body saying _whoa whoa whoa,_ what the . . . ? I don't want this.\" She held her stomach and groaned.\n\nThat comment, it must be noted, came from the woman who equated love with McDonald's and who told me that if cooking dinner would take more than twenty minutes, she'd go out for fast food.\n\n\"Oh, yeah, I know I said that. But a big thing I got out of it was that cooking is worth my time,\" she said. \"Homemade stock is basically free and you can make so much stuff out of it. Although I cut myself twice in class when I first started learning to use a knife, it was so worth it,\" she said with a laugh. \"The chicken class was important because I was afraid of chicken, that whole salmonella thing. And I'm going to be honest, I was kind of freaked out by cutting up the whole chicken at first. But that made me rethink food in a lot of ways. You're right, it's easier to remember that it was once, well, a chicken.\"\n\nYet for all these strides, some things she cannot let go of, White Trash Garlic Bread among them. \"But that's more about the memory of that, since I ate it growing up with my mother. But I use the good cheese now, not the stuff from a can.\"\n\nThe other untouchable? Gold 'n Soft. \"I can't give that up because butter still tastes funny to me,\" she admitted. \"But I use less, probably because I use olive oil more now.\"\n\nIn a way, Sabra was railroaded into this project by her stepmother, Lisa. I would not have faulted her if she had attended, feigned polite enthusiasm, and resumed her life in the fast-food lane. Instead, she embraced it and made extraordinary changes. Given all that, how could anyone deny her a few tastes of her childhood? I certainly could not.\n\n#  **TRISH**\n\nWhen I first visited Trish, Mike filmed the proceedings. This time around, Lisa came along. Like me, she marveled at Trish's collection of immaculate white binders filled with recipes she'd been clipping for decades, each encased in its own plastic sheath.\n\nShe exuded a sense of calm on this visit. \"So I don't think what I've bought or cooked has changed as much as something has changed in _me,_ \" she started. \"I'm more relaxed and I'm not so hard on myself.\"\n\nShe gave the recipes in the binders as an example. She pulled out one that said \"appetizers\" and started to flip through it. \"I went to all this trouble to put these together, but then I really didn't make many of the recipes. I was so intimidated and worried that they wouldn't work out. So I used to just look at them. Now I'm actually _cooking_ out of them. I make notes, and the ones that I don't like or don't work, I throw them away.\"\n\n\"So did you keep recipes that didn't work out before?\" I asked. She nodded. \"Why?\"\n\n\"I thought there couldn't be anything wrong with the recipe and so whatever went wrong was my fault,\" she explained. This led her to a bigger realization. \"I have this psychological thing that if I can do it then it must not be special or good. Plus, my mother didn't like to cook. She thought she was terrible at it and I absorbed that,\" she said. \"Now I think I may not be the greatest cook, but I know that I _can_ do it. I am not afraid of it.\"\n\nHer mother was the one who owned the _I Hate to Cook Book,_ a title that sold three million copies. The premise of the book was that women should get the whole cooking thing over with, thus avoiding any more drudgery in the kitchen than was absolutely necessary. This advice was couched in humorous terms by the author, Peg Bracken, who also advised women not to throw too much angst into meals either. Trish's mother managed to pass down the dislike of cooking, and not Bracken's ultimate message, which was that, in the end, cooking isn't that big of a deal. The irony was that of all the initial kitchen visits, Trish's ratatouille was probably the best food I ate, certainly the healthiest, and the only one made with whole foods. She slipped the binder back onto the shelf and we went into the kitchen.\n\nShe started to organize the makings of a vegetable and bean stew from _Feeding the Whole Family_ by Cynthia Lair, which is, incidentally, one of my favorite cookbooks.\n\nTrish set up her cutting board on the counter, carefully placing a wet paper towel underneath. She showed off her two new knives, a seven-inch chef's knife and a santoku, a Japanese-style design characterized by its curved spine and small indents in the blade. She set out her metal bowl for scraps. \"There are little things, like using a bowl for scraps, that have made a big difference. My kitchen stays cleaner and I feel more organized.\"\n\nShe cut an onion perfectly. \"I had a big aha over the onion,\" she said. \"In the knife skills class, I thought, So that's the secret! She used to use a garlic press. \"I stopped using that because now I like cutting, and that just dirties the garlic press.\" She turned to a six-quart stainless steel pot\u2014another recent purchase\u2014and tossed the vegetables into hot oil.\n\n\"Oh, and the other thing, I made stock.\" She opened the freezer and showed off a collection of carefully marked blue-lidded plastic containers. She pulled one out to show us.\n\n\"Wow, Trish, check you out!\" I said.\n\nShe looked proud. After she tucked the container back in the freezer, she went to the stove and held up a small blue handmade glazed ceramic box with a lid. \"I had wondered for years what to do with this. My son made it. Now I keep my coarse salt in it so I can pinch salt. That's another little thing. It seems silly, but trying to figure out a pinch with a shaker is hard!\"\n\nShe ticked off items she'd tried since the summer. \"I'm still terrible at cutting up chicken, but I love to do it.\" She regretted leaving the meat class early. \"I should have stayed. I would have liked to learn more about braising.\" She had enjoyed learning to make the vinaigrette but rarely did it. \"I don't know, buying it in the bottle is just so easy.\"\n\nOf the other classes, she used the soup class the most. \"I now have a sense of how to make soup from just about anything, and we love soup.\"\n\nOnce again, she set the table with beautiful dinnerware, a sculpted silver breadbasket in the center of the table, and immaculately ironed cloth napkins. The stew was light yet savory and brightened with the flavor of cilantro. It was the perfect lunch for a cold, rainy day. The timer dinged for the pear tart she had made for dessert. \"When I made it the last time, I took it to my friend's house and she had a perfect silver oblong pan and it looked so pretty. Everyone said, 'Wow, Trish, you're so good in the kitchen.'\n\n\"And this may seem silly, but it was a big moment for me. I was proud of something that I cooked.\" Her eyes got a little misty at the memory. \"No one expected that from me. I didn't expect it from myself. It is remarkable that at my age I can still change, and that I can still surprise myself.\"\n\n#  **JODI**\n\nWhen I met Jodi, she had found herself suddenly a stay-at-home mother after being laid off from her high-tech job. She had gone back to work again that autumn, juggling her home life with a three-year-old and trying to keep up with expectations in her new job. When I learned that she had gone back to work, I worried that whatever momentum Jodi had gained in the project might have dissipated.\n\nAs soon as I walked into her kitchen, I realized I didn't need to worry. Jodi could barely contain her excitement when we arrived. It was as if she had a secret that she just couldn't wait to share. It was a remarkable shift from the air of defeat and tentativeness I had felt on the day that she made Japanese curry from a cube.\n\nBy contrast, she was smiling and relaxed as she told us she was planning to roast a chicken. She showed off the simple roasting pan that she had bought for twenty dollars at a restaurant supply store. \"Oh, yeah, I do this all the time now,\" she said about the chicken. \"I've learned that as long as the chicken isn't too big, it takes right around an hour.\" She has made a hobby of finding uses for the leftovers. \"I made chicken Alfredo and it was _so_ good,\" she gushed. \"I make the fresh tomato sauce so often that I think I'm going to have to take a break from it because my husband and son might be getting sick of it, but I still really like it.\"\n\nShe had Brussels sprouts that had somehow gotten overlooked at Thanksgiving. She'd never made them before, but she figured she'd give them a saut\u00e9. We could smell bread baking in the oven. Jodi opened her fridge door to reveal a plastic bin with dough nestled into the center. \"I've used more flour in the last three or four months than I've ever used in my entire life! I got a couple of big plastic containers for white and whole wheat flour, and I'm refilling them all the time.\"\n\n\"I make a version that's kind of half wheat and half white now. My son likes it and I like to make it. I find it rewarding to pull hot bread from the oven, and it's something that I can make after work at least a couple of times a week.\"\n\nShe proudly showed off her new copy of _Joy of Cooking._ \"This summer completely changed the way that I think about cooking and nutrition, especially around processed foods, you know? I have tried a ton of recipes, which I would have _never_ done before. Also, we buy way more organic now and I read the labels on everything. Don't get me wrong, I'm still not making mayonnaise from scratch or anything like that, but I've come a long way.\"\n\nOn my earlier visit, the small walk-in had been packed with cases of canned and boxed goods stacked in piles on the floor. Now it looked organized and functional, and notably absent were ultraprocessed foods, save for a few boxes of macaroni and cheese. \"I tried to eat through a lot of things like canned tomatoes, but some of the stuff I cleared out and took to a food bank.\"\n\nHer fridge was similarly less cluttered. In her fridge door was a jar of parsley pesto, a tip she had taken from Chef Thierry. She was still working on eating through the freezer.\n\nInventory over, she went back to her cutting board. She tenderly separated leaves from a few sprigs of rosemary, chopped garlic, and effortlessly seasoned her chicken as she talked. She seemed as if she had been doing it for years. Even so, she still feared falling into the gender role that had trapped her mother in what Jodi viewed as a life of voluntary servitude.\n\n\"I don't mind cooking, as long as my husband and I both agree that it's a partnership and it's not just _my_ job,\" Jodi said. \"I think that it should be about who gets home first rather than the assumption that I'm supposed to do it, even if, to be honest, I rather like to cook now.\"\n\nInterestingly, her newfound interest in the kitchen both gave her equal footing and threatened the delicate balance of the power that had existed when her husband was the main cook in the family.\n\n\"We've had a couple of arguments about cooking, but that's probably because I never used to have an opinion.\" She now rejected suggestions and debated technique. Recently, she made a dish that left a layer of food stuck to the bottom of a skillet. Her husband kept scrubbing and complaining. \"I said, 'Hey, it's no big deal. Just deglaze it.' He was all 'What?' I said, 'Get the pan hot and add some liquid. It will come right off.'\" He kept scrubbing and finally threw the pan in the sink in disgust. \"I went to the stove, got it hot, and added water, and it was clean like that,\" she said, snapping her fingers. \"Ha, so I knew that, and he didn't.\"\n\n\"So, this is funny,\" she said in an obvious move to change the conversation. \"I have been trying to find a recipe for Japanese curry that doesn't use a cube. But all the Japanese people I asked say they use the dang box! I'd say, 'Hey, don't you know what's in that?'\" She laughed. \"So I asked a food writer friend who is Asian, and she thought it was a great question. So she's going to research it and I'm going to help with recipe testing.\"\n\nSo while she still makes golden curry, she's thinking about it more and making it less. \"I know this sounds dumb, but I used to think that the stuff in a box was something you couldn't make. Now I know they all just mimic real foods, so if it's in a box, there's got to be a way to make it for real. If you go to a good restaurant in Japan, then they'll have this same curry and it's amazing. So I guess that's my holy grail now. One day I will learn how to make that same curry _without_ the darn cube!\"\n\n#  **DRI**\n\nDri had moved from the hood into a comfortable condo in a leafy urban neighborhood. \"The greatest thing? I'm just across the street from Mutual Seafood,\" she said, naming one of the city's most respected seafood outlets.\n\nDri seemed happy and relaxed. She also looked as if she'd lost a bit of weight. Compared to her former one-cook-is-too-many-sized kitchen, her new one seemed vast and luxurious with its open plan, dark maple wood cabinets, and stainless steel appliances. \"It's great. I sometimes wake up in the morning and think, Wow, I live here. I feel like such a grown-up.\"\n\nShe had a basket on her counter filled with the kinds of stuff that she had discussed abandoning in the final class: dehydrated soup mixes, pasta side dishes, broccoli soup mix, and corn muffins among the lot. She blushed, a bit embarrassed.\n\n\"So this is awkward,\" she said. \"I did a favor for my friend and as a thank-you gift she gave me this lovely basket with all this packaged stuff.\" She could not stifle her laughter. \"It was such a nice idea, but I thought, I just don't do that anymore. I guess for some people it's a treat, but now I know better.\"\n\nLike a lot of people who live on their own, Dri reported that she found it hard to get motivated to cook for one. \"The project amplified the importance of taking time to cook and actually to make it a priority,\" she said. \"Plus it changed the way that I plan and stock my kitchen by about 175 million percent. I know that's a really big number, but it's true.\"\n\nShe opened a cabinet to show off a collection of oils and vinegars. \"I got most of this stuff after our lesson in salad dressings.\" She'd stocked up on red wine, balsamic, and champagne vinegars, soy sauce, sesame oil, and a good extra-virgin olive oil. \"One big takeaway was that you don't need to go out and buy something specific, but use what you've got. I can make an almost endless variation of dressings. I used to eat plain steamed vegetables, which are pretty blah and hard to get excited about. Now I think, Huh, what can I put together to make those actually taste _good_?\"\n\nThe contents of Dri's French door\u2013style stainless fridge resembled those of other volunteers, with a bouquet of greens in water and a plastic container with bread dough. \"I don't shop for meals as much as I shop for basics, which I consider onions, carrots, celery, garlic, lemons, limes, and a few other ingredients. I always have those on hand now,\" she explained. \"I feel like I can make most anything from these staples.\"\n\nThis is radically different from the way she used to shop. \"I used to go to the grocery store or a farmers' market and I'd see all these various things and think, Oh, that looks good, this looks good. I'd buy a whole bunch of random stuff. But then I wouldn't really have anything in mind about what to do with it. That's where a lot of my food waste came from.\" Dri said she made many decisions with Thierry's voice in her head. \"Now when I look at something highly perishable, I think, Okay, if I buy that, then what am I going to do with it in the next two days?\"\n\nAs she chatted, Dri started dinner. She moved with calm purpose as she chopped vegetables for a curried cucumber salad with a new chef's knife and pulled a beautiful small loaf of bread from her oven. Where Dri had dubiously poked at her chicken only a few months earlier, that night she demonstrated how to \"spatchcock,\" a technique in which the backbone is removed to butterfly a whole chicken so that it lies flat to cook faster and more evenly. \"I learned this from _Cook's Illustrated_. The editors use poultry shears, which I don't own. It's a little tricky with a knife but it works.\"\n\nShe put her weight down on the backbone to break it. Using the tip of her chef's knife, she made a surgical-like outline around the length of the spine. With one big tug, she ripped out the backbone and set it aside. She turned the chicken over and placed her hands one over the other as if to perform CPR on the bird. Instead she put her weight down on the sternum to break it with a mighty crack. \"Funny, I remember not being terribly thrilled by the sight of a chicken in that class,\" she said. \"I guess it doesn't bother me now, huh?\" She placed the chicken on a cushion of thinly sliced potatoes lining the bottom of a roasting pan, \"to soak up the yummy sauce,\" she explained. She seasoned it simply with fresh thyme, olive oil, and salt and pepper before shoving it in the oven.\n\nDri cooks one big meal during the week and a pot of soup on Sunday. With leftovers, that covers meals for at least four days. Other nights, she makes a quick salad, sandwich, or pasta with staples she keeps on hand. \"I still suffer from this issue of cooking for one and trying to get portions right. I try not to have _too many_ leftovers. I don't mind them on day two, but on day four? I usually can't go there. Sometimes I just take all the leftovers out of my fridge and bring them to work. I'll say to somebody, 'Hey, you want some chicken? I brought you lunch.'\"\n\nI didn't realize in the first visit that behind her quirky, smiling exterior, Dri's innate desire to make a radical life shift ran deep. \"When it came to how I approached food, I knew that I needed to change course, but I had no idea where to start.\"\n\n#  **ANDRA**\n\nAndra seemed in good spirits when we met her in December at her apartment in Sea-Tac. She looked more rested and her face seemed thinner. When I mentioned that I thought she looked like she'd lost weight, she laughed. \"Oh, that's so nice of you to say! I can't say that I've lost a ton of weight but I definitely feel like I'm eating better.\"\n\nI'm endlessly fascinated with her apartment. This time, I noticed that the entire length of the fridge was plastered with a mosaic of different magnets. A dress covered up a magnetic statue of David in the top right-hand corner; magnetic poetry spelled out \"Yes, a shadow wakens.\"\n\nAfter the project, Andra made a concerted effort to add more fruits and vegetables to her diet and to cook more. \"As usual, money is tight, but it's getting better now because I'm working more. Of course, the holidays are always a bit of a drain financially.\" She relied on public transport instead of a car. \"Sometimes I'm on ten buses in one day. It's exhausting, but that's my choice. I've started to take the new light rail train, which is nice. But some nights, I get home exhausted from all the travel.\"\n\nDue to the maze of buses that makes up her daily commute, she now relied on sandwiches or other foods that are easy to eat en route. She avoided the inexpensive burger joints that she used to frequent for lunch. \"Fast food is not a good value. I am hungry not long afterward and I don't feel good after eating it.\" She timed both to confirm that it took longer to stand in line to order than it took to make a sandwich and pack an apple at home. \"It's less money, less time, and healthier. It just takes planning.\"\n\nAndra missed more classes than any of the other volunteers. A shame, because as we talked, it was apparent that she took away a lot from the ones she did attend. As the others agreed, the knife skills class proved to be the most useful. Shortly after the last class, she made her first roast chicken. \"It was just me and my cat, but she seemed to really like it!\" Roasted chicken thighs and vegetables had become dinner staples, and she regularly made artisan bread. She sometimes cooked salmon in paper. In a pinch, she relied on scrambled eggs. \"I've cooked for some of my friends and I have had no complaints.\"\n\nAt Thanksgiving, she made stuffing from scratch with her mother and helped cook the turkey. \"She noticed my new chopping skills, and that felt good. I even made the vinaigrette for the salad.\" She was impressed by tasting different types of lettuce. \"I confess that I buy bags of salad, but that's a big move for me. I wouldn't have bought greens before at all.\"\n\nHer cupboards were more filled out this time, with whole wheat pasta and cans of tomatoes, beans, and soups. She selected one of the small plastic bags of herbs she'd purchased in bulk to use in the eggs she was making for lunch. \"I think about what your chef said, that if you don't have anything else, you can make an omelet. I make scrambled eggs more often, but an omelet feels more like a meal.\" We watched her crack the eggs into a bowl, quickly whisk the yolks together with salt, pepper, and a bit of thyme, and then pour them into a pan. She added a bit of grated cheese to the center, and then slid the omelet to fold it onto a plate. It was perfect.\n\nIt struck me that her omelet was a world away from the hyperprocessed pizza bites that she had made the first time I visited. I asked her about the contrast as she cut the edge of her eggs with a fork. \"I don't know if I would buy those now,\" she said. \"I think more about value rather than just cost.\" A big bag of greens costs the same as a fast-food dinner, she observed. She searched out places to buy vegetables less expensively than in a conventional supermarket, such as a local farm stand. \"I've got limited money for my food, so I need to get the most out of it that I can.\n\n\"Now that I can cook better, I don't have to settle for crappy food. It's been ages since I ordered a pizza, and those guys used to know my voice on the phone.\"\n\n#  **TERRI**\n\nTerri had finally tossed her four-year-old frozen turkey dinner. \"I didn't want you to come back and find that it was still there!\" she said with a nervous laugh. Her fridge was still relatively bare. \"I've gotten rid of all the science experiments,\" she admitted. Little else had changed in her cabinets. \"I find that I'm going grocery shopping more often, which is funny since I actually hate grocery shopping,\" she said. \"But I also don't want to waste food, so I usually just buy enough for a day or two and that's why I don't have a big stock of it here.\"\n\nShe had signed up as a client with Beve, the nutritionist. Her counter was lined with brand-new bottles of vitamins she'd been assigned to take after Beve had Terri's medical doctor run various tests. Like most people who live above the thirty-seventh parallel, Terri was wildly deficient in vitamin D; the sun is too weak most of the year for most people to produce it naturally. She had shifted to eating shredded wheat for breakfast to up her fiber intake and relied on scrambled eggs at lunch to help increase her protein.\n\nOn my first visit, she had made whole wheat pasta with a bit of olive oil. She'd advanced to adding the fresh tomato sauce from class, something she made at least twice a week. \"I'll be honest. I don't mind eating the same thing all the time. I kind of like routines,\" she said. \"It's also something that I don't really have to plan for. I always keep whole wheat pasta, tomatoes, and garlic around.\"\n\nTerri was the only one who didn't chop an onion as demonstrated in class. Instead, she cut off the ends, quartered them and dropped them into a food processor, and gave them a quick whirl. \"I know, I know, I'm cheating,\" she apologized. \"I find myself reverting back to old habits.\"\n\nWell, not all of them. Terri used to eat fast food for dinner up to four times a week; now it was down to less than once every couple of weeks. She had cut back on another indulgence, regular visits to Starbucks for breakfast or lunch, although that had as much to do with money as with nutrition.\n\nOf all the people in the study, self-employed Terri had the most enduring sense of \"time poverty.\" I heard a lot about \"time\" in the first round of visits with the volunteers. In the second round, the issue was noticeably less significant. When I later reviewed the videos, Terri mentioned the word _time_ on twenty-two occasions in both her first and second kitchen visits. Like the White Rabbit, she seemed invariably late and rushed, even as she mentioned that her tour business more or less ground to a halt during the off-season.\n\n\"I'm glad to know about how to cut up a chicken, although I don't think I will do it myself,\" she said. \"It was just a little too . . .\" She struggled for a word. \"I guess that I'm just squeamish, and it doesn't help that I am not that crazy about chicken in the first place.\"\n\nOf all the classes, she liked the meat class with Robin best; it involved cutting up red meat and pork, but that bothered her less. \"Learning to make rubs was helpful because pork chops are one of my standards, although I haven't made a rub yet. I liked the soup class and I plan to try that one day. The vinaigrette lesson was useful in terms of thinking about flavor combinations, but I don't make salads at home, I only eat them when I go out. As you can see, I don't keep a lot of food in my fridge, so leftovers aren't that big of a deal for me.\"\n\nBut she explained that this didn't mean that for her the project wasn't successful.\n\n\"It did help me to realize that I should think more about what I eat, but not necessarily worry about what I cook.\" Terri had slashed her fast-food runs from twenty to two per month. By doing so, she had eliminated 195,000 mostly empty calories in the course of a year, the equivalent of 550 hours on a treadmill. She explained that while some people can change their habits overnight, she learned while overcoming alcoholism that time and routine are critical for her. \"When I try to change everything at once, that never works. I feel like I have started to develop a foundation and I have to be happy with where I'm going.\"\n\nGiven all that, so what if she used her food processor to cut onions to make something healthier at home? More power to her.\n\n#  **CHERYL**\n\nAt Cheryl's, the Christmas tree fell down on her four-year-old son. He had pulled on an ornament, sending the tree straight down. He started wailing.\n\n\"Oh, no!\" Lisa and I raced over to rescue him. As we righted the tree, Cheryl grabbed her son. She conducted a quick inventory for injury before clutching him to her chest tightly. Then she held him out at arm's length. \"I told you to stop playing with the tree!\" she rebuked. \"I want you to sit on the couch and read your books while Mommy finishes her visit.\" Her son grabbed two picture books off the coffee table and dutifully climbed up onto the brown leather sofa.\n\nOn our first visit, Cheryl had opened a can of organic soup for lunch. This time, she made a pot from scratch, a fragrant curryscented number with coconut milk and vegetables. \"It's really easy,\" she said. \"I make one pot and then my son and I eat it for a couple of days for lunch. My husband's in construction, so he takes it in his thermos. I freeze the rest in small portions for lunches later.\"\n\nWe watched her efficiently chop an onion, carrots, and celery for her soup. Baby Liam scooted around the kitchen floor at her feet. She had recently had all her kitchen knives sharpened. \"It makes a huge difference having a super sharp knife. I can cruise through chopping.\" She tossed the vegetables into her pot and started to simmer them.\n\nHer husband hunts and brings home a motley assortment of game. She used to struggle about how to prepare it. \"I used to think, What do I do with a pheasant?\" she said. \"Then I learned to braise! Now a lot of my cooking is braising, braising, braising. I have to say that my braised pheasant is a big hit.\"\n\nShe picked up Liam, who had grown noticeably bigger, and balanced him heavily on her slight hip. \"The class definitely changed some fundamental basics about how I cook,\" she said. \"I make more sauces, soups, and meals from scratch. I am more confident in the kitchen because I just have more faith in my skills. I think the class not only helped me learn new things but also taught me not to be afraid of cooking and trying new things on my own.\"\n\nCheryl had been fairly label conscious and the class had just made her even more so. \"For instance, I rarely buy bread.\" She opened her fridge to reveal two containers of bread dough, one white and one whole wheat. \"My husband is the official baker in the family. He experiments with all kinds of versions. You know, I've gotten to the point that I just want to know what's in everything that I feed my family now.\"\n\n#  **SHANNON**\n\nShannon is endlessly up for a challenge. \"It's funny, I made some applesauce. I had all these apples,\" she said, holding her arms out wide to indicate a massive crate of apples. \"Weirdly, it was really fun. I was totally cruising through chopping them.\" She smiled broadly. \"My hand was a little sore after doing it, but I totally dug the Zen of getting into it.\"\n\nBy this point, the volunteers had some consensus on the efficacy of classes. Knife skills, the chicken class, stock, soup, no-knead bread, and learning to cook fish in parchment came out as strong themes. Shannon was no different. But I was interested in her takeaways. She was the mother of two young children and someone who carefully watched her food budget.\n\nSome things she'd been less hip to. \"I've made my own stock and I really like it, but sometimes I just can't get there,\" she said. \"It's hard to compare my own stock against supermarket stock because it's just so different. But if I buy it, I totally check the sodium. I get it as close to unsalted as I can. I found a brand that has only three grams of sodium that I like.\"\n\nOne class that she doesn't remember fondly was the tasting class. In retrospect, eager to offer a lot of options, I may have presented _too_ many and pushed a couple of volunteers into sensory exhaustion. Nine types of salt were probably six too many. \"I remember leaving that class feeling that I had been physically beaten.\" She liked the later classes in which we did a single tasting at the beginning of class. \"Don't get me wrong, I liked comparing tastes of things that I actually cook with, such as pasta and canned tomatoes and chicken stock. It was just too much at one time.\" Even so, she was the first student to announce she had tossed her iodized salt. She had seen a notice online for the comparative beef tasting at a Red Velvet Dinner. \"That's something I'd be interested in, the whole grass-fed versus corn-fed, or the taste of organic chickens versus the kind from the meat department.\"\n\nAs far as the fish class, the emphasis on hitting a separate store didn't factor into the complexities of being a mother and a cook. \"While I'd love to go to a fish market, the reality is that at the end of the day I have only so many trips in me. I've got to guide a baby and a young child and sometimes it's all I can do to just get milk. So I buy fish from the higher-end supermarket where I know they have good fish and I buy some of my other stuff there, too. Honestly, the reason why I like the parchment is because it always turns out. I can prep it in advance while they're napping and there are no dishes.\"\n\nWho could argue with Shannon? She had the information and weighed her options based on a frank and realistic set of expectations for her life right now. For instance, nurturing a pot of stock for hours wasn't a priority, but she didn't want to add the liquid equivalent of a salt lick to her food. So her decision to seek out unsalted chicken stock seemed like a completely reasonable solution. I'm sure the list went on.\n\nThat has not kept her from exploring new culinary territory, in part based on the confidence and information she gained from the classes. \"I've always been kind of scared of my Crock-Pot. I don't really understand it,\" she said. But she had dragged it out of a closet and dropped a couple of lamb shanks into it recently. \"When I came back in the meat was separated from the bones and I thought, Yes, that's right, that's what Robin said was supposed to happen. It tasted really awesome.\"\n\nIn June, Shannon noted that she felt uncomfortable cooking without a recipe. So it was a surprise when she shared the provenance of her lunch. \"I had a lamb ragout in a restaurant when we were on vacation,\" she started. \"It was so yummy! Oh, my God, it was so good! So then I wondered how I could re-create it. I looked online and found some recipes for ragout, but they all used ground lamb, which wasn't what they used. But I realized, Oh, that's just a braise. So I got a lamb shoulder and I cut it up into pieces and braised it. For the polenta, I made it from cornmeal and threw some cheese into it to make it gooey and yummy and it all worked.\"\n\nHer family approved. \"We were like, 'Oh, yum!' The funny thing about that story is that before class, I just don't think I would have ever thought to tackle something like that. Now, I can. I feel like I deserve a pat on the back for that one.\"\n\nAt that moment, I realized that not only had Shannon passed Cooking Basics 101 with flying colors, she was probably ready for a 301 class.\n\n#  **GENEVIEVE**\n\nDue to moving schedules and life in general, it took us months to catch up with Genevieve. She had been living with three roommates in a comfortable rental house when we met her, but she had since moved into a new condo with her boyfriend, John. The sleek kitchen boasted dark cherrywood cabinets, gray slate floor tile, a black man-made quartz countertop, and stainless steel appliances. \"It's funny, part of what we liked about this place was the cabinets, but it turns out some are super shallow.\" She demonstrated by showing that a canister she uses for tea barely fits inside. \"Oh, well, they look nice.\"\n\nThe shallow cabinets included a healthy roundup: wheat pasta, cans of tomatoes, artichokes, and olives, brown rice, and a collection of oils and vinegars. \"It turns out that I really like vinegar,\" she said. \"I was kind of surprised. Now I'm all over them. My latest favorite is tarragon. It is super tasty.\"\n\nIn her fridge was a spectrum of colored vegetables, a wrapped packet of fish, some greens, organic milk, white wine, prepackaged Jell-O pudding, and leftover butternut squash soup from the weekend. She kept cilantro in a small glass of water like flowers, as Thierry had suggested. In her cheese drawer was a wedge of real Parmigiano-Reggiano. Among the items she kept in her freezer were frozen strawberries, pot stickers, and a couple of frozen pizzas, plus some butter because she doesn't use it often.\n\n\"One thing that's I've changed is my buying habits. I don't buy premade packages of stuff, like those frozen pasta dinners with the sauce. I used to get those a lot. But now I realize that it's no comparison to what I can make fresh, plus the list of ingredients kind of turned me off. I find it hard to order pasta in a restaurant. I think, Twenty dollars? I could totally make this for three.\"\n\nShe and John hit a farm stand a couple of times a week rather than stocking up at the grocery store and sought out a stand-alone butcher and fishmonger. \"I rely less on getting my food from one place.\" Shopping together, they both try to rotate the goods in their fridge to avoid food waste. \"A couple of times a week I do a sweep of the fridge for what needs to be used. Whether it's leftover chicken or vegetables or herbs, I'll just chop them up and add them to a salad and then make vinaigrette. Sometimes that's dinner.\"\n\nOn her counter rests a small bowl filled with avocados and tomatoes. Beve, the nutritionist, had made a comment that everyone could eat more avocados, a source of healthy fat. So Gen had learned to make guacamole. John chimed in. \"I have to say, her guac is great.\"\n\nDuring our first visit, Gen combined a bag of prepared cabbage slaw with a jar of teriyaki sauce. On the follow-up visit, she made wild-caught Alaskan salmon cooked in parchment paper and roasted asparagus. \"I make this pretty often. I've even taught the paper trick to a couple of people. I remember when you asked how many portions a big chicken breast made, and then you weighed it and it was more than a pound, so it was really four servings. I use that as a reason to buy better meat or fish but eat less of it.\"\n\nShe reminded me of something that I'd said offhand in class. \"You said, 'No one is going to make you pack your knives and go home if a dish doesn't turn out.' I think about that a lot. That kind of attitude makes me more confident, and that's why I enjoy it more. It allows me to try things without getting worked up about it.\" A week ago, she had been saut\u00e9ing a piece of fish and decided to make a sauce with diced apple, rosemary, and white wine. \"I thought, What the heck? I'll give that a try. You know what? It was really good.\"\n\n#  **DONNA**\n\nOf all the volunteers, I was most intrigued by Donna. I genuinely liked her and was sympathetic to the power struggle over food and cooking in her new marriage. We never managed a date for a second home visit. After a few months, I sensed that the project had had no lasting effect on her and she was simply too polite to let me down.\n\nFinally, a year after the project ended, we caught up on the phone. \"Honestly, I didn't change my lifestyle much right after the classes,\" she started, my initial suspicions confirmed. \"But there's one phrase that repeats in my head, even to this day. When you were at my house, you said, 'Do you ever wonder what they do to foods to make them low-fat?' I guess I never cared before. Crappy food had always been a fact of life. But then I started questioning things, although not much else changed.\"\n\nThen, a few months earlier, she had started to cook on Sundays. \"I'm not sure how it started, but I just decided that Sunday was the only real time that I have to cook.\" She made three or four meals, and then wrapped them up in individual servings to eat for lunch or dinner throughout the week. She favored vegetable-centric dishes that pack well; vegetarian chili was a standard in her rotation. In the three months since she'd started, she'd lost ten pounds.\n\n\"After years of trying to find the 'secret' to battling my weight issues, I seem to have found it in cooking,\" she said. \"I'm an emotional eater. Taking my food with me to work and eating the same things has helped me maintain that food is fuel and not luxury or reward.\"\n\nShe doesn't come home ravenous anymore, a good thing because her husband lost all interest in cooking, a major shift in the original context of their relationship. \"He eats mostly frozen food now,\" she said. \"I don't want that for myself but I don't want to fight him about it either. If he cooks, that's okay, but I don't rely on him to do it anymore,\" she said, adding that he's still about ninety pounds overweight. \"Now if we eat something homemade, I make it. The thing is that he hasn't changed, I'm the one who has changed,\" she said. You could hear a surge of independence in her voice. \"It's pretty exciting, really.\"\n\nWhen we first met, her husband selected most of their groceries, but his shopping ethos troubled her. He'd buy five heads of lettuce at a warehouse store, only to throw away three. It conflicted with the values from her day job working with an African aid organization. Now she shops for her food, and he shops for his.\n\n\"So I still buy a lot at the warehouse store because they have good deals, but now I go with a friend from work. We shop together and then we split it. Doing that has helped me save money but waste less food. It's fun, too. My friend and I trade recipes and catch up.\" For fruits and vegetables, she hits a farm stand near her house two or three times a week. \"I've been trying to eat organic, too. I am now okay with paying a little more to get something local or organic. My husband thinks I'm crazy but he supports me.\"\n\nAs Donna made the same recipes over and over again, she found herself experimenting with them. \"I used to follow everything to the letter. Now I'm not a slave to a recipe. I trust my taste more, and I'm getting better at knowing when a dish needs something and what that might be.\"\n\nAs with the other volunteers, small bits of information had had an impact. \"I didn't realize spices expired! I thought they lasted forever. I think it's crazy to make a pasta dish from a package. It's about the same amount of work and mine tastes so much better.\" She'd even made gnocchi from scratch, something I personally have yet to master. \"I had a lot of potatoes and I thought that I could probably figure it out. I've done it a few times and I'm getting really good at it now.\"\n\nNot long ago, she made her mother-in-law's classic white rolls. The first time she tried them, they turned out well. When Donna made them at her mother's house, they burned. \"In the past, I would have thought it was my fault. See? This is proof that I can't cook. But this time I thought, Oh, well, I can make these, it's her oven, and this isn't about my ability. It's a bigger shift in my own self-worth.\"\n\nShe credits a chicken with some of her confidence.\n\n\"I had wanted to roast a chicken, but I always felt intimidated.\" A few months ago, she had bought her first whole chicken. She delighted in the sense of victory when it turned out perfectly. Now roast chicken is one of her staples. \"A week ago, my friend came over and she said, 'Wow, look at you, you're an amazing cook.' And I thought, What? It's just a roast chicken.\" Then she remembered that not that long ago that was a big deal.\n\n\"So I've encouraged people around me to cook. I know that it makes such a difference. Small changes put together can be big enough to change your whole life,\" she said. \"I have a friend who always bought her vegetables precut. When I asked why, she said she was scared of knives. I was, too, before this started. So I showed her how it's done, so she could get past her fear.\"\n\nThis, she said, was the greatest lesson. \"You get so afraid of things and then you do them and think, What was I so afraid of? You just have to do it.\"\n\nHow had I changed? My spice drawer is immaculate, thanks to the Great Spice Cleanout of 2009. Our freezer is lined with pasture-raised beef, pork, and chicken, a nod to the various classes on the subject and, in a way, a circle back to my early life on the farm in Michigan. I make soup at least once a week now to clean out my fridge. As Thierry suggested, we have a photo in the back of our fridge, an image of Mike and me embracing in Paris, taken by Holly during the AAA tour.\n\nI often think of that speech I gave at the Cordon Bleu graduation. In a blur of anxiety and grief, I urged the graduates to find something they were passionate about and just go for it. What I didn't realize then was that I needed that advice as much as anyone in the audience, and I learned as much as or more than any of the volunteers over the course of the project, including unexpected things I didn't even know needed to change.\n\nNot long after the project ended, Mike announced that he was going to make Alfredo sauce. I hovered. \"Maybe you should add some garlic,\" I started. \"You might not want to stir it so much.\" Then, \"Oh, hey, you know, you could add some of the chicken stock in the fridge. It's a different way to do it, but . . .\"\n\nMike handed me the spatula. \"Fine, you finish it the way you want it.\"\n\n\"But I was just trying to be helpful,\" I told him.\n\n\"This is why I don't like to cook with you in the kitchen,\" he said heatedly. \"You know why I always make Thai food? You never try to correct me on it.\" I knew how he felt. It replayed a similar scene from earlier in the summer.\n\nWhen we moved back to the United States in 2005, I hadn't owned a car for six years. Mike would sit in the passenger seat and offer me \"helpful\" tips. \"Kat, there's a pedestrian over there.\" Then, \"Oh, you'll probably want to slow down, there's a sharp turn coming up.\" Or, if he felt that I was going too slowly, \"You know the speed limit is forty-five here, right?\"\n\nIt made me so nervous, I'd second-guess myself. Even if he didn't say a word, I felt him silently critiquing my every move. Only weeks before the Alfredo episode, I had pulled over, unbuckled my seat belt, and said, \"Fine, you drive.\" His shock at my response was the same as mine to his irritated resignation about the pasta.\n\nIn the midst of a project designed to encourage people to cook, it seemed I thwarted the person closest to me. I learned to back off, to let him explore his own tastes and give him the reign of the kitchen now and then. As a result, he flourished as a cook. Now we routinely cook together.\n\nBut that's the thing about teaching. You find lessons you never expected beyond the ones you've taught. I'm grateful for that existential crisis onstage in Paris. Sometimes we need a good shake-up to remind us of who and where we are in life and to prompt us to change courses. Am I a chef? Not really, but how well does a single word define us? Julia Child never needed or wanted the title of chef. I write, I cook, I teach. I know that I'm the sum of those passions.\n\nWith that, I'm off to take a class on canning with Shannon. She recently earned her \"Master Canner\" designation. Last summer, Shannon didn't know how to hold a knife. Tomorrow, she will teach me how to preserve pears. It's the most fitting close to the circle that I can imagine. We live, we learn, we teach one another. Isn't that the way it should be?\n\nClick here for more books by this author\n**Extra Recipes**\n\nIn this section, you'll find a few other recipes taught in or developed as a result of the project. Most are meant to replicate items frequently purchased. Go ahead, give them a try.\n\n#    **Baked Chicken Nuggets**\n\n_My sister and I started making this alternative to the ubiquitous ultraprocessed fried chicken nuggets for my niece Sarah a decade ago. Store-bought bread crumbs can be stale and loaded with sodium, so try making your own. Toast two slices of bread, let them cool, and process them into crumbs in a small food processor. You can also use panko, Japanese bread crumbs, or toss in ground cornflakes for extra crunch. Cooking the chicken on a cooling rack allows the dry heat to crisp both sides, but if you don't have one, simply coat a parchment or foil-lined cookie sheet with cooking spray, and turn the chicken pieces over after ten minutes. Try to use real cheese rather than a canned variety; it will make a big difference in flavor._\n\n**MAKES ABOUT 2 DOZEN NUGGETS**\n\n1  pounds skinless, boneless chicken breasts or tenders   \n1 cup bread crumbs   \n  cup grated Parmesan cheese   \n  teaspoon kosher salt   \n1 teaspoon dried thyme or mixed herbs   \nPinch of cayenne (optional)   \nFreshly ground black pepper   \n1 egg   \n  cup skim milk, yogurt, or buttermilk   \nCooking spray\n\nPreheat the oven to 400\u00b0F. Place a cooling rack in the center of a cookie sheet. Set aside.\n\n  Cut the chicken breasts into 1 -inch pieces. In a shallow bowl or a large plastic bag, mix together the bread crumbs, cheese, salt, dried herbs, cayenne (if using), and a few grinds of black pepper. Combine the egg and milk in a small bowl. Dip the chicken into the milk mixture and then coat it well with the bread crumb mixture, either in a bowl or by tossing it inside the bag. Place the coated chicken pieces on the cooling rack and put the cookie sheet into the oven. Depending on your oven and the size and thickness of the chicken, the pieces will take 15 to 20 minutes until firm and cooked through. Spritz the chicken lightly with cooking spray and then place the cookie sheet under the broiler for 1 to 2 minutes, until browned, if desired.\n\n#    **Cream of Mushroom Soup**\n\n_Many \"cream of\" condensed soups are little more than simple white sauces easily replicated at home. The result tastes better and contains far less sodium, preservatives, and fat than you'll find in the canned variety. You can use any variety of dried mushroom that fits your budget and taste, but avoid shiitake, as its pronounced flavor may throw off many recipes. For best value, buy mushrooms in bulk; you can find them online and in warehouse stores. Mushroom bouillon can be found in health-food stores, Italian food specialty stores, or online._\n\n**YIELDS ROUGHLY THE EQUIVALENT OF AN 11-OUNCE CAN**\n\n  ounce dried mushrooms (about 1 tablespoon)   \n8 ounces hot water   \n1  tablespoons butter   \n1  tablespoons flour   \n  cup cold milk   \n  teaspoon mushroom or beef bouillon   \nSalt and freshly ground black pepper\n\n  In a bowl, steep the dried mushrooms in the hot water until they soften according to the package directions. Remove the mushrooms with a fork, reserving the dark liquid, or \"tea.\" Chop them finely, and set aside.\n\n  Heat the butter in a small saucepan over medium-high heat. Once it's melted, add the flour. Whisk continuously as the mixture bubbles for about 2 minutes, until it smells like popcorn, then remove the pan from the heat. Whisk in the cold milk until blended. Return to the heat and add the steeping liquid, the chopped mushrooms, and the bouillon. Add a pinch or two of salt and a few grinds of coarseground pepper. Bring to a light, bubbly boil and then simmer for about 5 minutes over medium-low heat, until it thickens. Taste, and add more salt and pepper if desired.\n\n_**Variation: Cream of Chicken**_\n\n  Use 8 ounces of chicken stock in place of the dried mushrooms and water and   teaspoon chicken bouillon in place of the mushroom or beef varieties.\n\n#    **Easy Spaghetti Sauce**\n\n_Less expensive, tastier, and healthier than most jars of pasta sauce, this can be made in roughly the time it takes to boil pasta and toss a quick salad. As a bonus, you can add fresh or leftover vegetables or other flavorings. This also makes a terrific pizza sauce._\n\n**SERVES 4 WITH PASTA**\n\n4 tablespoons olive oil   \n  onion, finely chopped (about   cup)   \n1  teaspoons dried mixed Italian herbs   \n3 garlic cloves, chopped   \nOne 14-ounce can tomato sauce   \nPinch of red pepper flakes (optional)   \n  cup water   \n1 bay leaf   \nCoarse salt and freshly ground black pepper   \n1 tablespoon balsamic vinegar (optional)\n\n  In a saucepan over medium heat, warm the oil, then cook the onion and herbs until tender. Add the garlic, stir, and cook for 1 minute. Add the tomato sauce, red pepper flakes, water, and bay leaf, plus a couple of pinches of coarse salt and a few cranks of black pepper. Bring just to a boil, and then immediately lower the heat to a simmer. Cook, uncovered, on low heat for 15 minutes. Add the vinegar, if using, and cook an additional 2 minutes. Taste and adjust the seasonings. Remove the bay leaf before serving.\n\n_**Consider adding the following:**_\n\nAbout   cup green and\/or black olives with the onions for a   \nputtanesca-style sauce   \nHandful of finely chopped mushrooms with the onions   \nRed wine in place of the water just before simmering for a   \ncabernet-style sauce   \nA tablespoon or two of chopped fresh herbs, such as basil, oreg-   \nano, or parsley, at the very end of cooking   \nAbout   pound cooked hamburger or cubed chicken with the   \nonions   \n3 tablespoons vodka and   cup cream 5 minutes before the end of   \ncooking\n\n#    **Pomodoro (Fresh Tomato Sauce)**\n\n_Be sure to have the pasta cooking and all the ingredients ready before starting the sauce; this is ready more quickly than you'd expect. Flavorful fresh tomatoes make all the difference here; cherry tomatoes work especially well and they're available year-round. Just cut them in half. Carefully scoop out a bit of the pasta water to finish the sauce._\n\n**MAKES ENOUGH FOR ABOUT 4 SIDE PORTIONS OR 2 MAIN-DISH SERVINGS**\n\n8 ounces dried pasta, such as penne or linguine, or 12 ounces   \nfresh pasta   \nCoarse salt   \n1 tablespoon olive oil   \n2 to 3 garlic cloves, minced   \nAbout 12 ounces tomatoes, chopped   \n  cup Parmesan cheese, grated   \nPinch or two of hot pepper flakes   \nHandful of chopped parsley or basil   \nFreshly ground black pepper to taste\n\n  Boil water for the pasta. Add at least 1 tablespoon of salt to the water; it should taste slightly salty. Cook the pasta according to the package directions; reserve about   cup of pasta water after cooking.\n\n  Meanwhile, add the olive oil to a saut\u00e9 pan over medium-high heat. Add the garlic and cook for 1 minute; be sure not to burn it or you'll need to start over. Add the tomatoes and any other vegetables (see below), and cook until the vegetables are softened, about 3 to 5 minutes for tomatoes on their own, longer if you add other vegetables.\n\n  Add the reserved pasta water and cook until the sauce is reduced slightly and the rest of the ingredients begin to break down, about another 2 to 3 minutes. Remove from the heat, add the cheese, red pepper flakes and parsley or basil (if using), several cranks of fresh black pepper, and salt if needed.\n\n_**Consider adding the following:**_\n\nSplash of cream at the end of cooking for a more creamy texture\n\nHandful of additional chopped vegetables, such as zucchini, artichokes, olives, and\/or asparagus, to extend the sauce and offer additional flavor\n\nShrimp or diced cooked chicken with the tomatoes (Shrimp can be added raw, but be sure to cook them thoroughly; they should turn white throughout and curl up tightly.)\n\n#    _**Potage Parmentier**_ **(Leek and Potato Soup)**\n\n_This is inexpensive French soul food. If leeks aren't available, try sweet onions. Makes about four to six servings._\n\n3 medium leeks   \n2 tablespoons butter   \n1 pound potatoes, peeled and diced   \n1 bay leaf and   teaspoon dried thyme, or a bouquet garni   \n2 quarts water or chicken or vegetable stock   \n  cup whipping cream, or 2 tablespoons butter, softened   \nCoarse salt and freshly ground black pepper   \nCayenne to taste (optional)   \n3 tablespoons minced parsley or chives\n\n  Prepare the leeks by discarding the roots and the tough green upper stalks. Slice, then rinse them in water to remove any residual dirt. In a 4-quart or larger saucepan, melt 2 tablespoons of butter, then saut\u00e9 the leeks for about 5 minutes, until they are softened and translucent. Add the potatoes, bay leaf and thyme or bouquet garni and water or stock. Simmer for about 40 minutes, until the vegetables are tender.\n\n  Remove from the heat. Discard the bay leaf. Break down the vegetables with a fork or a potato masher, or puree in a blender. Return to the heat. Add the whipping cream or 2 tablespoons butter. Taste. Add salt and pepper if needed, and a bit of cayenne if desired. Garnish with chopped parsley or chives and a couple of cranks of black pepper.\n\n#    **A \"Cheat Sheet\" to Flavor Profiles**\n\n_What makes something taste Italian or Cajun or Moroccan? Whether crafting vinaigrette, seasoning chicken, or developing a soup, understanding the flavors of ingredients that help to define various cuisines can be deeply useful._\n\n_Every cuisine has its regional variations; Basque cuisine is vastly different from the classic dishes from Provence, but they're both French. So consider this a shorthand reference to a few culinary stereotypes. Don't overdo it. Try incorporating two to four ingredients from a cuisine group to tilt a flavor profile in that general direction._\n\n_**Cajun\/Creole**_\n\ndark roux, onions, celery, green pepper, tomatoes, parsley, cayenne, Cajun spice blends, blackening seasonings, lemon, scallions, andouille sausage, crab, shrimp\n\n_**French**_\n\nbutter, shallots, onions, celery, carrots, thyme, tarragon, herbs de Provence, bay leaves, chives, chervil, capers, red and white wine, truffle, soft cheeses, Dijon mustard, mushrooms, cream\n\n_**Indian**_\n\ntandoori spices, garam masala, curry, yogurt, coconut milk, basmati rice, tamarind, cardamom, cumin, coriander, cilantro, fennel, garlic, saffron, fenugreek, dried chilies\n\n_**Italian**_\n\ngarlic, onions, celery, basil, pesto, prosciutto, Parmigiano-Reggiano cheese, mozzarella cheese, pine nuts, tomatoes, artichokes, olives, olive oil, oregano, lemon, fennel, flat-leaf parsley, red pepper flakes, rosemary, white beans, balsamic vinegar\n\n_**Japanese**_\n\nmiso, sesame oil, sesame seeds, rice vinegar, sake, soy sauce, wasabi, ginger\n\n_**Mediterranean\/Greek**_\n\noregano, lemon, olives, tuna, rosemary, bay leaves, thyme, olive oil, lamb, garlic, feta cheese, tomatoes, red onions, fish, shellfish\n\n_**Mexican\/Tex-Mex**_\n\ncumin, chili powder, hot sauce, green peppers, oregano, lime, garlic, onions, celery, cilantro, tomatoes, scallions, black beans, Cheddar cheese, avocado\n\n_**North African**_\n\nmint, lemon, harissa, saffron, turmeric, parsley, cilantro, honey, olives, almonds, dates, raisins, chickpeas, eggplant, green bell peppers, carrots, lentils, onion, ground ginger, paprika, cumin, cayenne, figs\n\n_**Central\/South Asian**_\n\nginger, garlic, scallions, shallots, lemongrass, Thai basil, cilantro, fish sauce, shrimp paste, soy sauce, coconut milk, sesame seeds, sesame oil, rice or sweet wine vinegar, cilantro, lime, oyster sauce, galangal, hot chili peppers\n**Acknowledgments**\n\n  First, here's a nod to the crew at Viking Penguin, including my wonderful editors, Stephen Morrison and Rebecca Hunt, publicist Lindsay Prevette, editor Beena Kamlani and copy editor Randee Marullo. I'm thankful for the wisdom and support of agent Larry Weissman through this project.\n\nNine volunteers opened up their hearts, kitchens, and lives, and for that I will be eternally grateful. I'm beholden to the volunteer teachers, chefs Thierry Rautureau, Robin Leventhal, Lauri Carter, and Jenny Nichols, plus nutritionist Beve Kindblade. Life wouldn't be the same without my chef friend Ted Lawrence.\n\nI couldn't have done this project without my friend Lisa Simpson, who soldiered with me through kitchens and chickens and taught me a lot along the way. I'm indebted to pals Maggie Savarino and Jeff Manness for lending their remarkable talents and energy.\n\nMany people lent their time offering feedback on this work while it was in progress, among them Deirdre Timmons, Laura Evelev, Cherie Jacobs Featheringill, Lee Mohr, Jackie Donnelly Baisa, Cindy Kane, and Philip Lee. Shalini Gujavarty came on board as my assistant during the writing of the book and contributed research, recipe testing, and beyond. Jamaica Jones did a bang-up job double-checking my research. My resident knife expert, Bill Magee, kept my blades sharp and my cutlery facts straight. My Le Cordon Bleu classmate Anne-Catherine Kruger led me to the catering kitchen owned by Kristine Pottle. I'm grateful to Zo\u00eb Fran\u00e7ois and Jeff Hertzberg for their terrific no-knead bread recipe.\n\nI recruited more than 150 recipe testers to try out and provide feedback for recipes in this book. The most active included Adrian Amo, Lauren Robinson, Marie Asselin, Susan Baird, Jane Bonacci, Libby Brill, Eric Compton, Deb Coupland-Porter, Glenn Dettwiler, Jessica Friedman, Michele Gartner, Lisa Glatt, Alka Goyal, Seika Gray, Scott Harbour, Eric Himan, Julie Hinson, Lindsey Hunt, Lee Mohr, Dayna Quick, Maria Raynal, David Rojas, David St. Clair, Jenise Silva, Shannon Valderas, Michael Wagner, Brenna Wilson, and Diana Wisen.\n\nI relied heavily on my education from the chefs at Le Cordon Bleu in Paris, and I know now that I'll be forever indebted to that famed institution. I'm grateful to the many people in the food-writing community who contributed directly or indirectly to this project and whose work and research inspires me, including Ken Abala, Pam Anderson, Jonathan Bloom, Dina Cheney, Amanda Hesser, Lia Huber, Deborah Madison, Marion Nestle, Michael Pollan, Michael Ruhlman, Jamie Oliver, Ruth Reichl, Rick Rodgers, Jon Rowley, Kim Severson, Laura Shapiro, Frances Short, Andy F. Smith, Virginia Willis and among many others. The late Julia Child never fails to motivate me.\n\nPetra Martin developed the Whidbey Island Writing Refuge, a quiet place in the woods where I wrote portions of the book. David John and Linea Anderson lent me the use of their sailboat. Both spaces were in addition to my usual shelter at the Richard Hugo House.\n\nI'm thankful to the woman in the supermarket for having the faith to let a stranger help her out with a chicken. I hope someday she knows how that afternoon changed my life.\n\nFinally, my deepest, heartfelt appreciation goes to Mike Klozar, partner in everything, writing coach, editor, technical guru, idea man, husband, and love of my life. There aren't enough words in any language to thank you.\n**Selected Bibliography**\n\nBlaylock, Russell L. _Excitotoxins: The Taste That Kills_. Abingdon, England: Health Press, 1996.\n\nBrackman, Pat. _The Compleat I Hate to Cook Book_. New York: Bantam, 1988.\n\nCheney, Dina. _Tasting Club: Gathering Together to Share and Savor Your Favorite Tastes_. New York: DK Publishing, 2006.\n\nChild, Julia. _The French Chef Cookbook_. New York: Knopf, 2002.\n\nChild, Julia, Simone Beck, and Louise Bertholle. _Mastering the Art of French Cooking_. New York: Knopf, 1961.\n\nDavid, Elizabeth. _An Omelette and a Glass of Wine_. Guilford, CT: Globe Pequot, 1997.\n\nDavidson, Alan. _The Penguin Companion to Food_. Harmondsworth, UK: Penguin Books Ltd., 2002.\n\n_The End of the Line_. Director, Rupert Murray; producer, Arcane Pictures. Dogwood Pictures, 2009.\n\nEttingliner, Steve. _Twinkies Deconstructed_. New York: Hudson Street Press, 2002.\n\nFisher, M. F. K. _Serve It Forth (Art of Eating)_. New York: North Point Press, 1989.\n\n\"Food 52\" Kitchen Tour with Amanda Hesser. Producer, Food 52.com, 2009. <http:\/\/vimeo.com\/5133553>.\n\n_Food, Inc._ Director and producer, Robert Kenner. Magnolia Pictures, 2009.\n\nGladwell, Malcolm. _Outliers_. New York: Little, Brown, 2008.\n\nGladwell, Malcolm. _The Tipping Point_. New York: Little, Brown, 2002.\n\nJacob, Dianne. _Will Write for Food_. New York: Da Capo Press, 2010.\n\nKallam, Tawra Jean, and Jill Cooper. _Dining on a Dime_. Temple, TX: T & L Group, 2004.\n\nKatz, David L., and Catherine S. Katz. _The Flavor Point Diet: The Delicious, Breakthrough Plan to Turn Off Your Hunger and Lose the Weight for Good_. New York: Rodale, 2005.\n\nKeller, Thomas, and Deborah Jones. _The French Laundry Cookbook_. New York: Workman Publishing, 1999.\n\nKessler, David. _The End of Overeating_. New York: Rodale, 2009.\n\n_King Corn_. Director and producer, Aaron Woolf. Mosaic Films Inc., 2007.\n\nKriger, Ellie. _The Food You Crave_. Newtown, CT: Taunton, 2008.\n\nKurlansky, Mark. _Salt: A World History_. New York: Penguin, 2003.\n\nMillstone, Erik. _The Atlas of Food: Who Eats What, Where, and Why_. Berkeley, CA: University of California Press, 2008.\n\nMitchell, Margaret. _Gone with the Wind_. New York: Scribner, 1936.\n\nMoulton, Sarah. _Sarah Moulton's Everyday Family Dinners_. New York: Simon & Schuster, 2010.\n\n_My Big Fat Greek Wedding._ Director, Joel Zwick. Gold Circle Films, 2002.\n\nOliver, Jamie. _Jamie's Food Revolution_. New York: Hyperion, 2009.\n\nPatel, Raj. _Stuffed and Starved: Markets, Power and the Hidden Battle for the World's Food System_. London: Portobello Books, 2007.\n\nPeterson, James. _Splendid Soups_. New York: Bantam, 1993.\n\nPollan, Michael. _In Defense of Food: An Eater's Manifesto_. New York: Penguin Books Ltd., 2008.\n\nPollan, Michael. \"Out of the Kitchen, Onto the Couch.\" _New York Times Magazine,_ July 29, 2009: MM26.\n\nReichl, Ruth. \"The New Culinary Order.\" Keynote, IACP Annual Conference, April 22, 2010.\n\nRuhlman, Michael. _The Elements of Cooking_. New York: Scribner, 2007.\n\nSanders, Laura. \"Binging Rats Get Hooked on Junk Food.\" _Discovery News_ 21 (Oct 2009).\n\nShapiro, Laura. _Something from the Oven: Reinventing Dinner in 1950s America_. New York: Viking Penguin, 2004.\n\nShort, Frances. _Kitchen Secrets: The Meaning of Cooking in Everyday Life_. Oxford, UK: Berg Publishers, 2006.\n\nSmith, Andrew F., _Eating History: Thirty Turning Points in the Making of American Cuisine_. New York: Columbia University Press, 2009.\n\nSmith, Andrew F. _Souper Tomatoes: The Story of America's Favorite Food_. New Brunswick, NJ: Rutgers University Press, 2000.\n\nStaten, Vince. _Can You Trust a Tomato in January?_ New York: Touchstone, 1994.\n\nWeber, Karl. _Food, Inc: How Industrial Food Is Making Us Sicker, Fatter, and Poorer\u2014and What You Can Do About It_. New York: Public Affairs, 2009.\n\nWilder, Laura Ingalls. _Little House on the Prairie_. New York: HarperCollins, 1971.\n**Recommended Reading**\n\n#  **General Cookbooks**\n\nBittman, Mark. _How to Cook Everything: 2,000 Simple Recipes for Great Food_. Hoboken, NJ: Wiley, 2008.\n\nChild, Julia. _The Way to Cook_. New York: Knopf, 1989.\n\nBetter Homes & Gardens, _Better Homes and Gardens New Cook Book._ 15th ed. New York: Wiley, 2010.\n\nRombauer, Irma S., et al. _Joy of Cooking_. New York: Scribner, various editions.\n\n#  **On Learning to Cook Intuitively**\n\nAnderson, Pam. _How to Cook Without a Book: Recipes and Techniques Every Cook Should Know by Heart_. New York: Broadway Books, 2000.\n\nPage, Karen, and Andrew Dornenburg. _The Flavor Bible_. New York: Hachette USA, 2008.\n\nRuhlman, Michael. _Ratio: The Simple Codes Behind the Craft of Everyday Cooking._ New York: Scribner, 2009.\n\n#  **Reinventing Family Dinners**\n\nDavid, Laurie, and Kristin Uhrenholdt, et al. _The Family Dinner: Great Ways to Connect with Your Kids One Meal at a Time._ New York: Grand Central Food & Style, 2010.\n\nLair, Cynthia. _Feeding the Whole Family._ Seattle, WA: Sasquatch Books, 2008.\n\n#  **References for Every Kitchen**\n\nHerbst, Sharon Tyler, and Ron Herbst. _The New Food Lover's Companion._ 4th ed. New York: Barron's Educational Series, 2007.\n\nJoachim, David. _The Food Substitutions Bible: More than 5,000 Substitutions for Ingredients, Equipment and Techniques_. Toronto, ON: Robert Rose, 2010.\n\nQA International, _The Visual Food Lover's Guide_. New York: Wiley, 2009.\n\n#  **Cooking for One**\n\nYonan, Joe. _Serve Yourself: Nightly Adventures in Cooking for One._ Berkeley, CA: Ten Speed Press, 2011.\n\n#  **Knife Skills**\n\nWard, Chad. _An Edge in the Kitchen: The Ultimate Guide to Kitchen Knives\u2014How to Buy Them, Keep Them Razor Sharp, and Use Them Like a Pro_. New York: William Morrow Cookbooks, 2008.\n\n#  **No-Knead Bread**\n\nHertzberg, Jeff, and Zo\u00eb Fran\u00e7ois. _Artisan Bread in Five Minutes a Day_. New York: St. Martin's Press, 2007.\n\nBaggett, Nancy. _Kneadlessly Simple: Fabulous, Fuss-Free, No-Knead Breads_. New York: Wiley, 2009.\n\n#  **Strategies to Save Money on Groceries**\n\nDacyzyn, Amy. _The Tightwad Gazette_. New York: Villard, 1992.\n\nLongacre, Doris Janzen. _More-with-Less Cookbook_. Scottsdale, PA: Herald Press, 2000.\n\nPennington, Amy. _Urban Pantry: Tips and Recipes for a Thrifty, Sustainable and Seasonal Kitchen_. Seattle, WA: Skipstone Press, 2010.\n\n#  **Soup**\n\nKaul, Leslie, Bob Spiegel, et al. _The Daily Soup Cookbook_. New York: Hyperion, 1999.\n\nBlake, Susannah. _500 Soups: The Only Soup Compendium You'll Ever Need_. Portland, ME: Sellers, 2007.\n\n#  **Modern Food Issues**\n\nBloom, Jonathan. _American Wasteland: How America Throws Away Nearly Half of Its Food (and What We Can Do About It)_. Cambridge, MA: Da Capo Press, 2010.\n\nFussell, Betty. _The Story of Corn._ Albuquerque, NM: University of Mexico Press, 2004.\n\nNestle, Marion. _What to Eat_. New York: North Point Press, 2006.\n\nPollan, Michael. _The Omnivore's Dilemma: A Natural History of Four Meals_. New York: Penguin Books, 2006.\n\n#  **Understanding Nutrition Labels**\n\nFarlow, Christina Hoza. _Food Additives: Shopper's Guide to What's Safe & What's Not_. Lafayette, LA: KISS for Health Publishing, 2007.\n\n#  **Vegetarian Cooking**\n\nDragonwagon, Crescent. _The Passionate Vegetarian._ New York: Workman Publishing, 2002.\n\nMadison, Deborah. _Vegetarian Cooking for Everyone_. New York: Clarkson Potter, 2007.\n\nO'Donnel, Kim. _The Meat Lover's Meatless Cookbook_. Cambridge, MA: Da Capo Press, 2010.\n**Index of Recipes**\n\n##\n\nA \"Cheat Sheet\" to Flavor Profiles\n\n##\n\nBaked Chicken Nuggets\n\nBasic Alfredo Sauce\n\nBlissfully Simple Chicken Stock\n\nBraised Pork with Potatoes and Cabbage\n\n##\n\nCream of Chicken Soup\n\nCream of Mushroom Soup\n\nCreamy Chocolate Frosting\n\n##\n\nDIY Vinaigrette\n\n##\n\nEasy Spaghetti Sauce\n\n##\n\nFish en Papillote, or Baked in Paper\n\nFive Marvelous Ways to Cook Vegetables with \"Flavor Splashes\"\n\n##\n\nMike's Yellow Cake\n\n##\n\nNo-Knead Artisan Bread for Busy People\n\n##\n\nPaella Valenciana\n\nPomodoro (Fresh Tomato Sauce)\n\n_Potage Parmentier_ (Leek and Potato Soup)\n\n##\n\nRustic Italian Farmhouse Zucchini \"Sauce\" with Penne\n\n##\n\nSpicy Shrimp in Saffron Tomato Sauce\n\n##\n\nVelvety Chilled Rosemary Carrot Soup\n\n##\n\nYour Basic Braise\n\nYour Basic Omelet\n\nYour Basic Roasted Chicken\n\n**ALSO BY KATHLEEN FLINN**\n\n_The Sharper Your Knife, the Less You Cry: Love, Laughter, and Tears at the World's Most Famous Cooking School_\n\nAlso known as wild leeks, a trendy vegetable among the foraging and farmers' market set.\n\nA small shop that sells inexpensive fried fish and fries.\n\nFrench fries.\n\nSodium caseinate is another name for monosodium glutamate (MSG).\n\nMSG is a chemical derived from the monosodium salt of the amino acid glutamic acid. It's made by processing fermented starch, corn, sugar beets, sugar cane, or molasses. Like aspartame, it's an excitotoxin that stimulates the neural system's \"pleasure\" cells, resulting in increased appetite. When researchers studying obesity need to fatten up lab rats, they dose the animals' food with MSG.\n\nPartially hydrogenated palm oil is on the hit list of many nutritionists; although cheap, it's considered a major artery clogger by several food scientists.\n\nTom was a judge on season six of _Top Chef,_ while Thierry was a cheftestant on _Top Chef Masters_.\n\nI originally selected nine women and one man, but the token male dropped out before the first class.\n\nThe Weight Watchers points system uses predetermined scores for food based on its nutritional value to help members avoid overeating.\n\nI learned later they're plastic cans, a miracle of modern packaging science. The contents do not need to be refrigerated, but to make the product feel \"fresh,\" the marketers sell it that way.\n\nAnne-Catherine took it in stride that she was edited out of my first book. She runs A Caprice Kitchen, a restaurant in Ballard, Washington. You should give it a try.\n\nForging is a means of shaping metal to produce a piece that is stronger than an equivalent cast or machined part, generally through the use of heat and pressure. Think hammer and anvil.\n\nMetallurgists measure relative hardness on the Rockwell Hardness Scale, or HRC. Most knives score in the 54 to 60 range. Japanese brands tend to be harder than European brands. So a Japanese-crafted Shun or Global knife might score 61 HRC, while a German-made W\u00fcsthof rates 56 HRC. In addition to hardness, there's carbon content. The most common steel recipe for knives worldwide falls into grades of 440 A, B, or C. The lowest, 440A, tends to be a soft steel that dulls easily, but it's cheaper and easier to manufacture than the higher-carbon variety, so most cheap knife blocks are 440A. That explains why so many bad knives exist in the world.\n\n\"Supertasters\" is a term coined by Yale University researcher Linda Bartoshuk, who stumbled on the notion that people's ability to taste varies widely while doing studies on saccharin. Supertasters readily taste a bitter synthetic compound found in saccharin known as 6-n-propylthiouracil. Nontasters can't detect it at all.\n\nIf you've ever wondered what was inside the chicken, the \"giblets\" include the neck, the liver, and the heart. They're traditionally stored inside the bird's cavity. Poultry producers often sell chickens without these bits because consumers don't know what to do with them. Rodgers suggests simmering them with a few veggies in a quart of water for an easy chicken broth.\n\nDetermining accurate numbers for meat consumption is something of a dark art; the USDA puts the figure at sixty pounds per person annually, while the American Meat Institute says it more like one hundred pounds, with others putting figures somewhere in between. No matter the source, chicken beats beef.\n\nThe primary difference is in the age of the chicken, which in turn determines the weight and tenderness of the bird. A broiler\/fryer weighs around three and a half pounds and is about ten weeks old. Roasters that tip the scales up to five pounds can be up to eight months old, while stewing fowl are even older. The odd bird out in her lineup was the capon, which historically was created by castrating a rooster to cause it to become a plump, tender bird; later, synthetic sex hormones were used for the same effect. Due to the industrialization of poultry, capons, stewing chickens, and older hens are rarities.\n\nThe Center for Science in the Public Interest reports that water or brine can account for as much as 15 percent of the weight in a package of chicken, and estimates that this practice generates $2 billion in added revenue per year for the poultry industry.\n\nInduction works through magnetic transference; one molecule excites another through direct physical contact. When I was asked this question the first time during a demonstration, I thrust the microphone at Mike. He was forced to define it on the spot. It sounded quite sensual.\n\nPresumably due to bad press, in 2010 the HFCS lobby asked the Food and Drug Administration for permission to shift its name to corn sugar.\n\nDiammonium phosphate, also known as DAP, used as a yeast nutrient here, is also used as a fertilizer, to enhance the flavor of nicotine in cigarettes, and is widely added to fire-fighting agents.\n\nBy the end of 2008, more than 184,500 cases of BSE had been confirmed in the United Kingdom, according to the Centers for Disease Control. In humans, the disease manifests primarily as Creutzfeldt-Jakob disease (vCJd), and as of October 2010, a total of 217 cases of vCJd had been reported in eleven countries. In the United Kingdom, the epidemic peaked in 1999. In North America, twenty-one cases of BSE had been reported by mid-2010, three in the United States and eighteen in Canada.\n\nDenmark has the highest consumption of ground beef internationally.\n\nThis figure is based on studies by the Union of Concerned Scientists, which argues that heavy antibiotic use in animals may be a contributing factor in the rise of antibiotic resistance in humans who eat commercially raised meat.\n\nThe European Union is not so keen on North American beef. A blanket import ban has been the subject of intense political and legal wrangling. The World Trade Organization probably wishes the whole thing would just go away.\n\nCorn is so complex that entire books and films are devoted to the subject. Two I recommend are the documentary _King Corn_ and the book _The Story of Corn_ by Betty Fussell (University of Mexico Press, 2004).\n\nThe reason why the term \"natural\" is less meaningful for chickens is because growth hormones are prohibited in poultry in general; the USDA allows minimal antibiotic use in \"natural\" chickens but doesn't specify the amount.\n\nPork shoulder is also known as pork butt or Boston butt, a historical reference to the butchering style used by Boston-area butchers that left more meat and less bone on the shoulder cuts. The pieces were typically shipped in casks or barrels, colloquially known as \"butts,\" hence the term.\n\nOn page 200 of my first book, _The Sharper Your Knife, the Less You Cry,_ I call for Chilean sea bass in a recipe. I suggest you try another responsibly caught whitefish instead. Thanks.\n\nFarm-raised salmon is controversial. It takes three pounds of wild fish to grow one pound of farmed salmon, according to the Monterey Bay Aquarium. Most are farmed in open pens and cages in coastal waters, and it's easy for waste, parasites, antibiotics, and diseases from the penned fish to seep into open water. (Imported farmed shrimp has similar issues.) If you buy farmed salmon, try to stick to those that are referred to as \"tank-raised\" or \"tank-farmed.\"\n\nKeta is the salmon formerly known as \"chum.\" In terms of quality, it's generally considered at the bottom of the salmon totem pole.\n\nRegional guides are available for download at <http:\/\/seafoodwatch.org>.\n\nIssues with food chemicals can crop up years after approval, yet the FDA lacks much power to investigate or pull approved chemicals off the shelves. Since it was introduced in 1993, consumers have filed more than four thousand complaints to the FDA about aspartame, more commonly known as NutraSweet. This chemical accounts for 70 percent of all complaints received by the agency for a variety of health claims. It remains a controversy, with each side claiming the other doesn't have enough proof.\n\nThere are other claims to the \"original\" Thanksgiving, both in Canada and by the Spanish in St. Augustine, Florida.\n\nThe U.S. Department of Labor reports that in 2008, the average American spent 5.6 percent of his or her wages on food eaten at home and 4 percent on food eaten in restaurants or fast-food joints, or a bit more than 9.6 percent total.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nAlso by Kevin C. Fitzpatrick:\n\nA Journey into Dorothy Parker's New York\n\nas editor\n\nDorothy Parker: Complete Broadway 1918\u20131923\n\nThe Lost Algonquin Round Table: Humor, Fiction, Journalism, Criticism, and Poetry from America's Most Famous Literary Circle\n\nCopyright \u00a9 2013 by Kevin C. Fitzpatrick\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, except as may be expressly permitted in writing from the publisher. Requests for permission should be addressed to Globe Pequot Press, Attn: Rights and Permissions Department, PO Box 480, Guilford, CT 06437.\n\nLyons Press is an imprint of Globe Pequot Press.\n\nProject editor: Meredith Dias\n\nText design: Sheryl P. Kober\n\nLayout: Lisa Reneson\n\nLibrary of Congress Cataloging-in-Publication Data\n\nFitzpatrick, Kevin C., 1966-\n\nUnder the table : a Dorothy Parker cocktail guide \/ Kevin C.\n\nFitzpatrick.\n\npages cm\n\nE-ISBN 978-1-4930-0203-0 (ePub)\n\n1. Cocktails\u2014United States. 2. Parker, Dorothy, 1893-1967. 3.\n\nUnited States\u2014Social life and customs\u20141918-1945. I. Title. II. Title:\n\nDorothy Parker cocktail guide.\n\nTX951.F53 2013\n\n641.87'4\u2014dc23\n\n2013024383\n\nI love a martini\u2014\n\nBut two at the most.\n\nThree, I'm under the table;\n\nFour, I'm under the host.\n\nContents\n\nTitle Page\n\nCopyright\n\nForeword by Allen Katz\n\nIntroduction: Stirring the Life of Dorothy Parker\n\nCocktails 101\n\nTHE DRINKS\n\nSidebars\n\nProhibition and the Eighteenth Amendment\n\nDorothy Parker in Hollywood\n\nA Toast to the Algonquin Round Table\n\nA Guide to Speakeasy Slang\n\nPhoto and Illustration Credits\n\nAcknowledgments\n\nFurther Reading\n\nAbout the Author\n\nIndexes\nForeword\n\nIt was just one of those things . . .\n\nI was introduced to the acerbic wit of Dorothy Parker in college. Studying the lyrics of Tin Pan Alley, I came across the elegant patter of Cole Porter. \"As Dorothy Parker once said\" immediately piqued my curiosity. Of the musical numbers from 1935's Jubilee, I would sooner take \"Begin the Beguine,\" but of all the characters in \"Just One of Those Things\" only Juliet and Romeo were familiar to me. In those days, before the dawn of the Internet age, there was no Google search, so I hit the stacks and came upon The Collected Poetry of Dorothy Parker. \"Ballade of a Talked-Off Ear\" still stands out.\n\nBy the time I graduated\u2014Dorothy Parker Stories a fitting commencement gift\u2014I was on my way to New York to seek fame and fortune. I didn't know at the time that a future in drinks and the wonderfully fitting American culture of cocktails awaited me, but in short time, absorbed by the rush of history and the pleasure of a well-made Manhattan, suddenly her name appeared again: Dorothy Parker and the Algonquin Round Table. The Round Table itself, long before Mad Men, provided an interesting social concept: literati and the three-martini lunch. Instantly I was less interested in the hard talking and more interested in the hard drinking!\n\nDorothy Parker came of age in an era when men and women in urban America didn't socialize at bars. The saloons of the pre-Prohibition era offered taps for men of varied walks of life to blow off steam, share a story or two, and imbibe. Once in a while an establishment might have a side door or a ladies' entrance, but the idea of a woman lunching and drinking with men in a professional scenario was sadly a rarity. Parker helped change that, contributing, if you will, to the democratization of drinking culture\u2014and what more democratic place should there be than a bar?\n\nIn a time of restless brilliance, her national celebrity sprang from the sharp, biting humor of both her incisive poetry and her widely quoted quips. Regularly seated at the famed Algonquin Round Table, she took center stage in the city's cultural and intellectual beating heart. No one could have been a more interesting drinking companion, and our Dorothy Parker American Gin celebrates her unconventionality in all its glittering facets.\n\nUnder the Table gives us a glance at a bygone era and at Dorothy Parker herself: drinker, for certain, but writer, raconteuse, and provocateuse as well. Not just a reminiscence, it acknowledges a rejuvenation of cocktail culture, the importance of patience and the art of conversation, and the myriad rewards of a well-made drink.\n\n\u2014Allen Katz\n\nCofounder of the New York Distilling Company\nIntroduction\n\nStirring the Life of Dorothy Parker\n\nNo other American writer has a reputation quite like the one Dorothy Parker earned. Almost fifty years after her death, her numerous wisecracks live on, such as her advice to a friend who needed to euthanize an old cat: \"Try curiosity.\" Or about a boyfriend: \"His voice was as intimate as the rustle of sheets.\" After she learned that President Coolidge had died, she remarked: \"How can you tell?\"\n\nBut Mrs. Parker did more than crack wise and create bad puns. The American Academy of Arts and Letters admitted her to its ranks not for telling jokes but for the achievements of her poetry and short stories. She went into the New York Writers Hall of Fame alongside Herman Melville and Willa Cather. Not a day goes by that her name doesn't pop up on the Internet, in the umpteenth pop culture reference of the day's news cycle.\n\nShe's also renowned for writing about booze, talking about drinking, and loving a cocktail or two herself. Along with her dogs and expensive clothes, Mrs. Parker also enjoyed her cocktails. She imbibed at speakeasies in New York, mansions in Beverly Hills, and villas on the French Riviera. A friend once said, \"You've got to expect public recognition like that. After all, you're an international celebrity.\" To which Mrs. Parker replied, \"Yeah, that's me, the toast of two continents\u2014Greenland and Australia.\"\n\nDorothy was born to New Yorkers J. Henry and Elizabeth (n\u00e9e Marston) Rothschild on August 22, 1893, at their summer beach cottage in Long Branch, New Jersey, near present-day 732 Ocean Avenue, facing the beach. Her father worked in the garment business, making men's cloaks and suits. The comfortably upper\u2013middle class family had housekeepers and cooks but weren't considered wealthy.\n\nWhen Dorothy Parker walked into a kitchen one morning, her host asked what he could make for her breakfast. \"Just something light and easy to fix,\" she said. \"How about a dear little whiskey sour?\"\n\nHer childhood on Manhattan's Upper West Side was an unhappy one. Young Dorothy attended a Catholic grammar school, then a finishing school in Morristown, New Jersey. Her formal education ended abruptly at age fourteen. A voracious reader, she often wrote poems for her father. Both her mother and stepmother died when she was young; her uncle, Martin Rothschild, perished on the Titanic in 1912, and her father died the following year.\n\nIn 1914 Mrs. Parker sold her first poem to Vanity Fair. At age twenty-one, she took an editorial job at Vogue. She couldn't believe how overdecorated the offices were, though. \"Well, it looks just like the entrance to a house of ill-fame,\" she remarked later. She continued writing poems for newspapers and magazines, and in 1917 she joined Vanity Fair, taking over for P. G. Wodehouse as drama critic\u2014making her the first female critic on Broadway. That same year, she married stockbroker Edwin Pond Parker II. The marriage proved tempestuous, though, and the couple divorced in 1928.\n\nDorothy Parker in the 1920s \u25c6 \u25c6 \u25c6\n\nIn 1919 Mrs. Parker became a founding member of the Algonquin Round Table, an informal gathering of writers who lunched at the Algonquin Hotel in Midtown Manhattan. The \"Vicious Circle\"\u2014which was the nickname of the group rather than the piece of furniture around which they gathered\u2014included Alexander Woollcott, Robert Benchley, Harpo Marx, George S. Kaufman, and Edna Ferber and earned a reputation for its scathing wit and intellectual commentary. In 1922, Smart Set published Mrs. Parker's first short story, \"Such a Pretty Little Picture.\"\n\nOne year after ratification of the Eighteenth Amendment outlawed the manufacture and sale of alcohol, National Prohibition went into practice in January 1920. Mrs. Parker was twenty-six. She and her friends frequented the local speakeasies, particularly on West Forty-ninth Street, where Rockefeller Center stands today. There was no drinking at the Algonquin Hotel, however. General manager Frank Case took the bar out during World War I, keeping the restaurant in operation by serving tea and coffee. The Vicious Circle luncheons were \"dry\" affairs.\n\nWhen The New Yorker debuted in 1925, Mrs. Parker was listed on the editorial board. Over the years, she contributed poetry, fiction, theater criticism, and book reviews as \"Constant Reader.\" Her first collection of poetry, Enough Rope, appeared in 1926 and became a best seller. Her two subsequent collections were Sunset Gun in 1928 and Death and Taxes in 1931. Her collected fiction came out in 1930 as Laments for the Living.\n\nReviews were good, and her popularity swelled. Mrs. Parker traveled to Europe several times, befriending Ernest Hemingway, Scott and Zelda Fitzgerald, and socialites Gerald and Sara Murphy, among others. In the 1920s she contributed articles to The New Yorker and Life. While her work garnered her success and regard for her wit and conversational abilities, she suffered from depression and alcoholism and attempted suicide.\n\nIn 1929 Mrs. Parker won the O. Henry Award for her semi-autobiographical short story \"Big Blonde.\" She wrote short fiction in the early 1930s, and in 1931 she returned to her role as a Broadway critic, subbing for Robert Benchley for several issues of The New Yorker while he was in California. She delivered a particularly stinging put-down to a Channing Pollock drama: \"I may mutter only that The House Beautiful is, for me, the play lousy.\"\n\nIn 1934 Mrs. Parker married actor-writer Alan Campbell in New Mexico. The bride was forty; the groom twenty-nine. The couple relocated to Los Angeles and became a highly paid screenwriting team. They labored for MGM and Paramount on mostly forgettable features. Of the nearly forty screenplays that Mrs. Parker worked on, only a handful are available today on DVD. The highlight of her film career was an Academy Award nomination for A Star Is Born in 1937. The couple divorced in 1947 and remarried in 1950.\n\nMrs. Parker was inducted into the American Academy of Arts and Letters in 1959. Following her husband's death in 1963, Dorothy returned to New York City. On June 6, 1967, she was found in her Upper East Side apartment, dead of a heart attack at age seventy-three. A firm believer in civil rights, she bequeathed her literary estate to Martin Luther King Jr. On his assassination, less than a year later, the estate was turned over to the National Association for the Advancement of Colored People (NAACP). Her remains were cremated and her ashes interred in a memorial garden at the NAACP headquarters in Baltimore, Maryland.\n\nWithin twenty of years of her death, three biographies appeared, which opened the taps for a new generation to discover Dorothy Parker. In the 1990s her short fiction and poetry were collected in new editions and became steady sellers. Mrs. Parker's work was translated into French, Spanish, Portuguese, and Russian. For the centennial of her birth, a commemorative postage stamp was dedicated not far from where she was born on the Jersey Shore. The 1994 feature film Mrs. Parker and the Vicious Circle, directed by Alan Rudolph and featuring an all-star cast, with Jennifer Jason Leigh in the lead role, was a critical success. In 1999 the Dorothy Parker Society (DorothyParker.com) was launched to promote the work of and introduce new readers to Dorothy Parker (and have as much fun as possible). Today the society has more than five thousand members worldwide. As the fiftieth anniversary of Mrs. Parker's death approaches, more of her poems, fiction, and essays are available now than when she was alive, and her reputation as one of America's most original voices is solidly established.\nCocktails 101\n\nThis is a quick overview of basic bartending equipment, a glassware glossary of basic cocktail glasses, and some key mixing tips and tricks.\n\nEquipment for the Bartender\n\nYou can find the essential equipment for preparing these recipes in a home goods shop, restaurant supply store, or online auctions for vintage bar tools. Part of the lure of mixing cocktails derives from the paraphernalia and equipment needed. Just locating a vintage cocktail shaker and bar spoon will put you in the mood to start mixing drinks. These are the top pieces of essential bar equipment:\n\nbar spoon: Many drinks require the use of a long-handled flat-head spoon. Find one with a twisted shaft if possible, which feels better when stirring a drink.\n\nchannel knife: A lemon peel is cut into long spirals using this sharp, small knife. It is a flat piece of steel with an opening to cut the peel evenly.\n\nchef's knife: Best used for slicing large fruit such as lemons and limes. Keep it with a small cutting board.\n\ncocktail shaker: This is the most important tool you'll need to make a cocktail. Heavy stainless steel cocktail shakers guarantee the coldest drinks. Hunt around for antique sterling silver models of the \"silver bullet\" variety. The cocktail shaker will serve your needs for both shaking and stirring mixed drinks.\n\ncocktail shaker, Boston: The bottom half of a Boston cocktail shaker is stainless steel, and the top is made of glass. It also doubles as a mixing glass.\n\ncocktail strainer: Also called a Hawthorne or Lindley strainer, this has coils around the sides and is placed over the cocktail shaker.\n\ncorkscrew: You can choose from various models, but look for a vintage waiter's style corkscrew with a folding knife attachment (all the better for cutting foil and labels around the necks of bottles). The modern rabbit type with a lever pull is excellent, but the cheap ones quickly wear out with regular use.\n\ndishware: A few nice porcelain or china pieces are perfect for placing olives, limes, and garnishes on the bar. Conversely, obtain some good plastic containers in which to store your perishables overnight.\n\nice bucket and tongs: Keep your ice cubes handy in a bucket. Long tongs to grab ice cubes are more classy than a spoon (or your fingers).\n\nice spoon: Also known as a julep strainer, this spoon is semi-flat with holes in it and is placed over the cocktail shaker.\n\njiggers and shot glasses: These are indispensable for measuring that perfect drink. Some 2-ounce jiggers have markings for 11\u20442, 1, and 1\u20442 ounces. Shot glasses can come in these sizes as well.\n\njuicer or reamer: A juicer or reamer has holes to catch the seeds and pulp of fresh fruit. Use a bowl to collect the juice.\n\nmeasuring cups and spoons: Just like a kitchen set, measuring cups from 1\u20442 cup on up. A set that has tablespoons and teaspoons as well is key.\n\nmuddler: This is a rod, similar to a pestle, used for crushing or \"muddling\" fruit. It has a broad, rounded, or flat end. Get a wooden one because it won't scratch glass. You cannot make a Mint Julep without a muddler.\n\nopener: Many recipes require juices and mixes that require a bottle or can opener.\n\nparing knife: All bars need this indispensable knife. It should be 4 inches long and thin and have a spear tip. Keep it sharpened.\n\npitcher: Find a beautiful glass pitcher and use it for all stirred cocktails. It will look impressive when filled with ice cubes while also delivering drinks properly chilled.\n\nGlassware\n\nNot all glassware was created for the same purpose. You don't serve a Sidecar in a beer mug. A short list follows from among the two dozen or more types of bar glasses. It ignores the whole army of wine glasses and goblets and instead sticks to the glasses you will need to make the drinks in this guidebook. All cocktails are served in these basic types of glassware. Thinking about using a plastic cup? You should be ashamed.\n\nchampagne (4 to 10 ounces): The expected glass to serve Champagne is tall and graceful, also called a tulip and a flute. The reason for its shape? The smaller surface allows fewer bubbles to escape.\n\ncocktail (3 to 10 ounces): This is the classic stemmed glass for drinks served without ice. For larger drinks the bowls have become wider. Also called a martini glass.\n\nCollins (10 to 14 ounces): Slimmer and taller than a highball glass, the name of this glass derives from the Collins family of drinks.\n\ncoupe (5 to 10 ounces): Traditionally a small round bowl on a long stem\u2014often used for Champagne\u2014this glass is shallower than a wine goblet. The French claim it was modeled in the shape of Marie Antoinette's left breast. It's back in vogue for cocktails that don't require ice because the long stem prevents body heat from transferring to the drink. It's utterly classic.\n\nhighball (8 to 12 ounces): This is the most common bar glass for drinks with ice cubes. A lowball is half this size.\n\nNick & Nora (6 to 8 ounces): A smallish glass with a modest stem, this classic shape has a cult following that insists it's the best glass for serving Martinis. It takes its name from a pair of cocktail-loving gumshoes created by Dashiell Hammett; Nick and Nora Charles first appeared in his 1934 detective novel, The Thin Man. Other writers transferred the couple to radio, motion pictures, television, and Broadway shows. It's perfect for any cocktail served straight up.\n\nold-fashioned (4 to 8 ounces): It takes its name from the eponymous cocktail and is usually a short, squat glass used for drinks with ice. Also called a rocks glass or whiskey glass.\n\nsour (5 to 6 ounces): Used for sours and juice drinks of all kinds. Also called whiskey sour glass, Rickey glass, fizz glass, or a Delmonico, after the old New York restaurant.\n\nWine (4 to 8 ounces): From goblets to Reidel, each shape is for a different relationship between volume and surface area. Tapered bowls are designed for the free circulation of the wine to allow the vapors to \"breathe\" more.\n\nMixing Tips and Tricks\n\n  * \u25c6 There isn't a cocktail in this book that doesn't have one key ingredient: ice. In the old days bartenders took blocks of ice and cut off pieces to suit the drink. In our more modern age, ice is more plentiful, and you can use more of it. (Don't ever reuse ice cubes from one cocktail shaker to the next, though; never use the same ice twice!) If a recipe calls for \"cracked\" or \"shaved\" ice, take a good amount of cubed ice and wrap it in a clean, dry bar towel. Then, using a wooden mallet or heavy spoon, beat the heck out of it and drop the ice into the cocktail shaker. For drinks that are shaken, use cracked ice. The longer you shake it, the colder the drink.\n  * \u25c6 Cocktail shakers and mixing glasses are essential to superb cocktails. In his book, Imbibe!, bartender historian David Wondrich clarifies when to shake and when to stir: \"Modern orthodoxy dictates that one should shake any drink with fruit juices, dairy products, or eggs and stir ones that contain only spirits, wines, and the like. This is based partly on the fact that shaking clouds up liquids by beating thousands of tiny bubbles into them.\" Cloudy drinks come from the shaking, which is why a Manhattan or Martini that's crystal clear is stirred, not shaken. A muddler requires a lot of work, but the effort of mashing the ingredients between your palm and a wooden stick are worth it when making your concoctions by hand.\n  * \u25c6 When straining a drink made in a mixing glass or cocktail shaker, keep the ice out of the glass by holding a cocktail strainer or julep strainer over the top of the shaker or mixing glass. Carefully pour the drink into the glass, and the ice won't fall in.\n  * \u25c6 If a recipe calls for a float, this means pouring one liquid atop another. To do this, pour the first spirit in the glass. Then, hover a bar spoon upside down just over the surface of the drink. Pour the second, floated spirit slowly over the back of the spoon so it drips on top of the first.\n  * \u25c6 Cold cocktails? Of course. Always use chilled glassware. Store them in a refrigerator or freezer for at least an hour before use. Another trick is the \"flash chill,\" which means filling them with crushed ice for 15 minutes. Always handle the glasses by the stem or bottom, not the sides, so your body heat doesn't transfer to the drink. Always serve mixers such as soda and fruit juice cold. Never pour them warm from a bottle or can.\n  * \u25c6 Finally, make it and serve it. Your cocktails have to be served immediately to be any good. London bartender god Harry Craddock was once asked the best way to drink a cocktail: \"Quickly,\" he replied, \"while it's laughing at you!\"\n  * \u25c6 Use this book as a manual to plan your own speakeasy parties or to serve vintage drinks at your bar.\n\nIn 1928 fellow poet William Rose Ben\u00e9t wrote: \"What the devil can you do with such a girl? You can be moved to sympathy by some expression of evident distress, or to admiration for some gallantry of attitude, or to gravity at an occasional tenderness\u2014and then she flips a last line at you like a little carmine fire-cracker exploding under your nose. And it is all Dorothy Parker.\"\n\nThe Acerbic Mrs. Parker\n\nThe Shanty, Brooklyn\n\nNo shortage of bartenders wants to honor Mrs. Parker with a namesake cocktail. Countless recipes have been created and named for her around the globe, from the Algonquin Hotel to a popular discoth\u00e8que named Club Dorothy Parker in Rio de Janeiro. This one came to life in Brooklyn, created by Allen Katz, general manager of the New York Distilling Company. The Portable Dorothy Parker, which he picked up in college, instantly had him hooked. At their wedding, he and his wife exchanged vows and read \"Here We Are\" to each other. Katz pressed his business partners to launch Dorothy Parker American Gin as one of the company's first brands. He created the Acerbic Mrs. Parker in the Shanty, the little bar next to their Brooklyn distilling operation.\n\ncollins glass\n\n2 ounces Dorothy Parker American Gin\n\n\u00bc ounce Combier orange liqueur\n\n\u00bd ounce hibiscus syrup\n\n\u00bd ounce fresh lemon juice\n\nSeltzer\n\nLemon twist\n\nShake all liquid ingredients except the seltzer over ice; strain into a Collins glass filled with fresh ice. Top with chilled seltzer and garnish with a lemon twist. Serve with a straw.\n\nAlexander\n\nThe Algonquin Round Table met for the first time in June 1919 to welcome Alexander Woollcott back to New York after serving in the army for two years. The rotund Times critic and Dorothy Parker shared the distinction of being born on the New Jersey Shore. Woollcott famously said, \"All the things I really like to do are either immoral, illegal, or fattening.\" In 1922 he took up residence in a Hell's Kitchen townhouse with one of the married couples of the Vicious Circle, editor Harold Ross and journalist Jane Grant. Mrs. Parker named their house Wit's End. When Woollcott moved out a few years later, he took with him the housekeeper, his family's silver, and the house's nickname. He always claimed the Brandy Alexander was named for him, but that was a fib. This recipe comes from bartender Hugo R. Ensslin's 1916 collection Recipes for Mixed Drinks.\n\ncocktail glass\n\n1\u00bd ounces Cr\u00e8me de Cacao\n\n1\u00bd ounces brandy\n\n1\u00bd ounces fresh cream\n\nShake all ingredients over ice; strain into a cocktail glass.\n\nAlexander Woolcott called Dorothy Parker \"a blend of Little Nell and Lady Macbeth.\" \u25c6 \u25c6 \u25c6\nAlgonquin Cocktail\n\nLong before the Vicious Circle began meeting around the Algonquin Round Table, the hotel, located at 59 West Forty-fourth Street, had a reputation for attracting actors and writers. Upstate gambler Albert T. Foster began the enterprise, the first guests checking in on November 22, 1902. A standard room and bath cost two dollars a day, while a three-bedroom, three-bath suite with private hall, sitting room, dining room, and library ran ten dollars. The best move Foster made was hiring Frank Case as general manager. Case ran a tight ship and ran the hotel for twenty-five years before buying it. News of the sale ran on the front page of the New York Times. The hotel has changed hands a half a dozen times over the last century, but its history and traditions remain. There are several variations of the Algonquin Cocktail, but this is how the bartenders mix them in the hotel's elegant Blue Bar.\n\nold-fashioned glass\n\ncocktail glass\n\n1\u00bd ounces rye whiskey\n\n\u00be ounce dry vermouth\n\n\u00be ounce pineapple juice (unsweetened)\n\nMaraschino cherry\n\nStir the rye whiskey, vermouth, and juice in a cocktail shaker filled with ice. Strain into a chilled cocktail glass or over ice into an old-fashioned glass. Garnish with the cherry.\n\n\"I bought the Algonquin because I love it and the associations that go with it and the people who have helped me by their patronage to make it what it is,\" said Frank Case in 1927. In an era when actors and actresses were considered little more and treated little better than prostitutes, Case warmly welcomed theater folk. He also attracted writers and editors from the nearby publishing offices of city newspapers and magazines, including Parker and the Vicious Circle.\n\nThe Round Table Room in the Algonquin Hotel \u25c6 \u25c6 \u25c6\nAngel's Tit\n\nDorothy Parker was no angel, and she reveled in it. When asked why she preferred hotels to owning a house, she replied that all she needed was \"room enough to lay a hat and a few friends.\" While she was living at the Algonquin Hotel, general manager Frank Case suspected her of breaking a house rule and telephoned her suite.\n\n\"Do you have a gentleman in your room?\" he inquired.\n\n\"Just a minute,\" Mrs. Parker answered. \"I'll ask him.\"\n\nDuring the 1920s, this popular after-dinner drink proved a delight to order and to imbibe. Its naughty name derives from the placement of the garnish.\n\ncocktail glass\n\n1\u00bd ounces maraschino liqueur\n\n\u00be ounce fresh cream\n\nMaraschino cherry\n\nPour the maraschino liqueur, such as Maraska or Luxardo, into a chilled cocktail glass. Float the cream on top. The cherry garnish must be perfectly placed on top, and in the center, for the full effect. Alternately you can use equal parts liqueur and cream, whipping it into a sculpted froth\u2014but that's a lot of work.\n\nThe play rather pines away later on, however; there are long stretches when the only thing that rivets your attention to the stage is wondering at what moment the shoulder straps of Julia Bruns' green gown are going to give up the unequal struggle and succumb to the strain.\"\n\n\u2014Mrs. Parker's review of The Blue Pearl in Vanity Fair, October 1918\nAuntie Jo-Jo's Jalape\u00f1o Bloody Maria\n\nDoc Holliday's Saloon, New York City\n\nThe Martini and Bloody Mary vie for the title of most popular cocktail in America. If the Martini lubricates sophisticated Friday evening conversations, the Bloody Mary colludes with your Saturday morning steak and eggs. Legend has it that barman Pete Petiot of Harry's New York Bar in Paris created the concoction around 1921. When Prohibition ended, the Hotel St. Regis lured him to Manhattan, and by World War II the Bloody Mary was everywhere. Nearly a century later the drink has transformed as often as the city itself.\n\nOn a noisy corner in Manhattan's East Village lies Doc Holliday's, a neighborhood shot-and-beer joint. Facing Tompkins Square Park on Avenue A, Doc's serves oceans of the hard stuff, its clientele preferring tequila, whiskey, and vodka. For twenty years Joanna Leban has wielded the stick at Doc's. A savvy poker player, witty raconteuse, and an encyclopedia of drinks, she developed this variation on the drink, serving it on weekend brunch shifts.\n\npint glass\n\nMakes six 16-ounce servings\n\n3 cups tomato juice\n\n2 jalape\u00f1os, seeded and finely chopped\n\n1 tablespoon freshly grated horseradish\n\nJuice of 1 lemon\n\nJuice of 1 lime\n\n1 teaspoon original Cholula hot sauce\n\nSplash of balsamic vinegar\n\n\u00bc teaspoon salt (more to taste)\n\n\u00bc teaspoon Cajun spice\n\n1 cup Tanteo jalape\u00f1o-infused tequila\n\n3 jalape\u00f1o-stuffed olives\n\n1 celery stalk\n\n3 pickled Cajun green beans\n\n1 lime wedge\n\nTo make the Bloody Maria mix, combine the tomato juice, jalape\u00f1os, horseradish, lemon and lime juices, hot sauce, and vinegar. In a separate container, combine the salt and Cajun spice; rim glass with the mixture. Combine the Bloody Maria mix and tequila and pour over ice into rimmed pint glasses. Split celery stalk lengthwise, place halves off center inside glasses. Garnish with olives spiked on a mixing stick. Add pickled beans as a garnish, and add lime wedge to the rim.\n\nNote: Joanna's Jalape\u00f1o Bloody Maria is so popular that she makes them six at a time. If you can't find jalape\u00f1o-infused tequila, substitute a good silver tequila. You may also substitute jarred horseradish if you can't find fresh. For a twist, try Effen cucumber vodka, and you get Auntie Jo-Jo's Jalape\u00f1o Cool as a Cucumber Bloody Maria.\n\nWhat makes a bartender great? The ability to adapt to each customer's needs. It's all strategy. It's like poker: \"Play the player, not the game.\" When a customer asks what time it is, I tell him, \"It's time to drink.\" Being a bartender is like hosting a party every night.\n\n\u2014Joanna Leban, East Village, New York City\nAviation\n\nDuring the Hudson-Fulton Celebration in 1909, thousands cheered as Wilbur Wright made two round-trip flights from Governors Island in New York Harbor, one around the Statue of Liberty and the second up the Hudson River to Grant's Tomb and back. When Charles Lindbergh returned to New York after his historic solo flight to Paris in 1927, a crowd of four million attended his tickertape parade on Broadway. In 1932 Amelia Earhart returned from her own transatlantic flight an instant national icon. The Aviation cocktail captures the flying spirit of the era. It was a favorite at the Hotel Wallick, a landmark 400-room Times Square hotel on the corner of Broadway and Forty-third Street. The hotel's manager, German-born Hugo R. Ensslin, had the foresight to collect his favorite cocktail recipes before Prohibition began. Published in 1916, Recipes for Mixed Drinks made the Aviation take off.\n\ncocktail glass\n\n1\u00bd ounces gin\n\n1 teaspoon maraschino liqueur\n\n1 teaspoon Cr\u00e8me de Violette\n\nJuice of 1 lemon\n\nIn a cocktail shaker, pour all the ingredients over cracked ice. Shake and strain into a chilled cocktail glass.\n\nNote: The Cr\u00e8me de Violette is the key but often-missing ingredient that also nicely turns the drink a sky-blue color. Use just a little.\nBailey\n\nDorothy Parker counted Gerald and Sara Murphy among her dearest friends, but she wasn't the only one. Jazz Age artists and literati from Dos Passos to Picasso enjoyed the company of the American expatriate couple. Their Cap d'Antibes house, Villa America, hosted legendary parties when they lived in France from 1921 to 1929, and the couple inspired the characters Dick and Nicole Diver in F. Scott Fitzgerald's Tender Is the Night. Mrs. Parker lived with them for six months in Switzerland when their young son was fighting tuberculosis. \"Sara is in love with life and skeptical of people,\" said Gerald Murphy, a gifted painter and writer. \"I'm the other way. I believe you have to do things to life to make it tolerable. I've always liked the old Spanish proverb: 'Living well is the best revenge.'\" To that end, Gerald's drink-mixing skills proved formidable. Using gin, he created what he called a Bailey: \"invented by me,\" he wrote to Alexander Woollcott, \"as were a great many other good things.\" Mrs. Parker remained lifelong friends with the couple, and, when she returned home to Manhattan from Los Angeles in 1963, she moved into Sara Murphy's apartment building on the Upper East Side.\n\ncocktail glass\n\nMint sprigs\n\n1\u00bd ounces gin\n\n\u00bd ounce fresh grapefruit juice\n\n\u00bd ounce fresh lime juice\n\nGerald Murphy's instructions: \"The mint should be put in the shaker first. It should be torn up by hand as it steeps better. The gin should be added then and allowed to stand a minute or two. Then add the grapefruit juice and then the lime juice. Stir vigorously with ice and do not allow to dilute too much, but serve very cold, with a sprig of mint in each glass.\"\nBathtub Gin\n\nJane Grant's recipe\n\nOne of the forgotten stars of the Jazz Age, Round Table member Jane Grant cofounded The New Yorker with her husband, Harold Ross, at their kitchen table at 412 West Forty-seventh Street in 1925. The Hell's Kitchen townhouse still stands, once the scene of raucous parties attended by the Vicious Circle, Ethel Barrymore, Irving Berlin, F. Scott Fitzgerald, and others. But those parties put Grant in a fix because her guests required more alcohol than her bootleggers could provide. She started distilling her own gin from a recipe that she claimed came from the ma\u00eetre d'h\u00f4tel at the old Waldorf-Astoria Hotel. (\"If the Waldorf could fool the customers, I could impress my friends, and I would not be exposing them to a fate worse than death.\") But first she needed pure alcohol. \"I finally found a reliable bootlegger who would deliver it in ten-gallon cans\u2014nothing less. At first that was a little staggering. My recipe called for equal parts water\u2014and twenty gallons of gin seemed like a lot of gin to me . . . My supply was consumed with true speakeasy gusto\u2014I soon found that twenty gallons of gin was not an extravagant amount for 412.\"\n\n1 quart pure grain alcohol\n\n8 drops oil of juniper\n\n1 to 1\u00bc quart distilled water\n\nMix alcohol and oil well several times for 30 to 36 hours, then add the distilled water. Mix well for 20 to 24 hours.\n\nNote: Grant recommended shaking the bottles every day for a week, much to the chagrin of her friends. \"Not one of the wretches would give me a hand, preferring, as they said, sudden blindness, or even death, to such labor . . . Everyone else, they pointed out, just mixed the ingredients and served\u2014I was much too fussy.\"\n\nBathtub Gin in Chelsea\u2014which serves many of the drinks in this book\u2014creates a fun simulacrum of a speakeasy experience with the Stone Street Coffeehouse exterior. Bathtub Gin has been adopted as the name of modern speakeasies located today in New York City, Seattle, and Mooresville, North Carolina.\n\nThe birthplace of The New Yorker was on the second floor of this Hell's Kitchen townhouse. \u25c6 \u25c6 \u25c6\nBetween the Sheets\n\nDorothy Parker and Alan Campbell divorced in 1947, ending twelve rocky years of marriage. Both took up with others almost immediately, but the two were so codependent that they reunited three years later. One wag said of Campbell, \"He was sweet-tempered when sober, with an endearing sort of wanting to be helpful. He was Dorothy Parker's helpmate in nearly every sense of the word.\" They remarried in a splashy Beverly Hills ceremony in 1950. On the morning of their second wedding day, Dorothy and Alan lay in bed together. She pulled the sheet over her face. \"No peeking,\" she said. \"Mustn't see the bride before the wedding!\" The Between the Sheets cocktail dates to the 1920s in two competing recipes. The first version comes from the 1937 Caf\u00e9 Royal Cocktail Book, published in London, with two competing liquors as the base.\n\ncocktail glass\n\n1 ounce brandy\n\n1 ounce rum\n\n1 ounce Cointreau\n\n1 dash lemon juice\n\nShake all ingredients over ice and strain into a chilled cocktail glass.\n\nNote: Use a premium white rum for best result. You can substitute the brandy with Cognac.\n\nThe second recipe comes from Havana, where this 1935 Cuban version from Bar La Florida tastes sweeter but just as delicious.\n\ncocktail glass\n\n1 ounce Cognac\n\n1 ounce Cr\u00e8me de Cacao\n\n1 ounce sweet cream or half-and-half\n\n1 dash Angostura bitters\n\n1 teaspoon sugar\n\nLemon peel\n\nShake all ingredients except lemon peel over plenty of cracked ice; strain into a chilled cocktail glass. Garnish with lemon peel.\n\nThe second wedding of Dorothy Parker and Alan Campbell took place in 1950. \u25c6 \u25c6 \u25c6\nBlood and Sand\n\n\"I hate Husbands; They narrow my scope,\" Dorothy Parker wrote in one of her notorious \"Hymns of Hate,\" adding:\n\nThere are the Home Bodies;\n\nThey are seldom mistaken for Rudolph Valentino;\n\nThe militia has not yet been called out to keep the women back.\n\nOne of the first drinks to come out of Hollywood was the Blood and Sand, named for the 1922 Rudolph Valentino film of the same title. The biggest film celebrity of the decade, Valentino died in 1927 in New York, aged just 31. Two female fans tried to commit suicide in front of his hospital, and more than 100,000 people thronged the streets around Frank Campbell's Funeral Chapel during his wake. Harry Craddock served this drink at the Savoy Hotel in London during the 1920s.\n\ncocktail glass\n\n\u00be ounce Scotch\n\n\u00be ounce cherry brandy\n\n\u00be ounce sweet vermouth\n\n\u00be ounce orange juice\n\nShake all ingredients over cracked ice; strain into a chilled cocktail glass.\n\nValentino and Dorothy Parker had something in common: Both of their memorial services took place at Frank Campbell's funeral home. \u25c6 \u25c6 \u25c6\nBoulevard\n\nThe 1940 US Census found Dorothy Parker living next to Robert Benchley on Sunset Boulevard in West Hollywood at the Garden of Allah, a collection of tiny villas grouped around a swimming pool. The bungalow rentals proved popular with movie and radio celebrities because the development was affordable, semiprivate, and a short drive to the studios. Alla Nazimova, the first actress billed as a movie star, had built what became the Garden of Allah in the early 1920s as her country home. Sunset Boulevard the rural roadway quickly transformed into The Strip, and Nazimova carved the property into twenty or thirty villas. For two decades stars such as Bogart, Fitzgerald, Garbo, Hemingway, and Welles lived there. During World War II Benchley offended a man in uniform at the bar at the Garden of Allah by addressing him as a lieutenant.\n\n\"I happen to be Captain So-and-So of the US Navy,\" the man said haughtily, \"and may I ask who you are?\"\n\n\"Me?\" said Benchley. \"Oh, just call me a destroyer.\"\n\nThe Garden of Allah was demolished in 1959, and a shopping center stands in its place. The Boulevard offers a time-honored variation on a dry Manhattan.\n\ncocktail glass\n\n2 ounces bourbon\n\n\u00bd ounce dry vermouth\n\n\u00bd ounce Grand Marnier\n\n2 dashes orange bitters\n\nOrange peel\n\nStir all ingredients except orange peel with ice; strain into a chilled cocktail glass. Garnish with flamed orange peel. Hold a match under the peel side of a quarter-inch orange peel, over the drink. Twist and squeeze over the drink. Rub the peel on the rim of the glass and then drop it in.\n\nNote: Another variation calls for substituting rye whiskey for the bourbon.\nBronx\n\nJane Grant and Harold Ross, cofounders of The New Yorker, often served this cocktail at their Hell's Kitchen townhouse in the 1920s. At the couple's housewarming party in 1922, Mrs. Parker and her boyfriend at the time, Charles MacArthur, hired ponies for the neighborhood kids. At one memorable party, Ethel Barrymore and Alexander Woollcott improvised a scene from Romeo and Juliet for a crowd in the backyard. The Bronx originated at the old Waldorf-Astoria Hotel, created by bartender Johnny Solon after visiting the Bronx Zoo. Apparently the animals reminded him of some of his regulars. The hotel served so many Bronx cocktails that cases of fresh oranges were delivered daily.\n\ncocktail glass\n\n2 ounces gin\n\n\u00bd ounce dry vermouth\n\n\u00bd ounce sweet vermouth\n\n1 ounce fresh orange juice\n\nShake all ingredients well with ice; strain into a chilled cocktail glass.\n\nBulldog\n\nDorothy Parker took her dogs everywhere, from speakeasies to Broadway playhouses. \u25c6 \u25c6 \u25c6\n\nDogs played starring roles in every part of Dorothy Parker's life. They populated her stories, poems, and plays. She never didn't own one, taking her puppies, mutts, and strays with her everywhere. They traveled on luxury transatlantic steamships, curled under her chair at speakeasies, and perched on her lap whenever a portrait photographer came calling. Dogs figure into many of the best Mrs. Parker tall tales. At a party her dog vomited on a rug. Trying to apologize to the hostess, Mrs. Parker said, \"It's the company.\" A more famous shaggy dog story is told about the time her pet had an accident in the Beverly Hills Hotel lobby. A manager came after her: \"Miss Parker! Miss Parker! Look what your dog did!\" But she stared him down: \"I did it.\" And she walked out. The Bulldog recipe comes from the oft-forgotten pre-Prohibition bartender Hugo R. Ensslin.\n\ncocktail glass\n\n2 ounces cherry brandy\n\n1 ounce rum\n\nJuice of half a lime\n\nShake all ingredients over cracked ice, strain, and serve in a chilled cocktail glass.\nBywater Cocktail\n\nArnaud's French 75, New Orleans\n\nIt's claimed that Tennessee Williams once said: \"America has only three cities: New York, San Francisco, and New Orleans. Everywhere else is Cleveland.\" A cocktail mecca sits a few paces off Bourbon Street in the French Quarter, tucked inside a hundred-year-old Creole restaurant, and you don't need to take a streetcar named Desire to reach it. \"On a much smaller scale, I've often thought of the neighborhoods of New Orleans like the boroughs of New York City,\" says Chris Hannah, bartender at Arnaud's French 75 on Rue Bienville. \"I've always compared New Orleans's Bywater to Brooklyn, so, when finally able to serve the Brooklyn Cocktail to guests at my bar, after cultivating Jamie Boudreau's Amer Picon Replica, I decided to create the Bywater Cocktail following the Brooklyn's direction. Swapping an aged rum for rye whiskey and Falernum and Chartreuse for the maraschino, I came up with a balanced cocktail worthy of the Bywater neighborhood's anomalousness.\"\n\ncoupe glass\n\n1\u00be ounces aged rum\n\n\u00be ounce Amer Picon or Amer Picon Replica\n\n\u00bd ounce Chartreuse\n\n1 teaspoon Falernum\n\n2 dashes Peychaud's Bitters\n\n2 dashes orange bitters\n\nMaraschino cherry\n\nStir all liquid ingredients straight up and garnish with a cherry.\n\nAmer Picon is a bitter aperitif produced in France with a distinct orange flavor. Falernum is a sweet syrup. Peychaud's Bitters, distributed by Sazerac, dates to around 1830, created by Antoine Am\u00e9d\u00e9e Peychaud, a Creole apothecary in New Orleans.\nCablegram\n\nBefore e-mail and text messages, the world communicated electronically with telegrams and cablegrams. Telegrams were delivered via wires to be delivered to recipients; cablegrams were transmitted internationally via underwater cables. Dorothy Parker was a master at both. In 1923, when Robert E. Sherwood's first wife was expecting their child, Mary Sherwood let everyone at the Round Table and New York know she was pregnant. To offer her congratulations on the birth of Mary's daughter, Mrs. Parker sent a telegram: \"Good work, Mary, we all knew you had it in you.\" She sent it collect. To fellow Vicious Circle member Ruth Hale, spouse of Heywood Broun and a passionate feminist: \"To Ruth Broun from Dorothy Rothschild.\" In 1945, Mrs. Parker sent one to her editor at Viking Press: \"This is instead of telephoning because I can't look you in the voice. I simply cannot get that thing done yet never have done such hard night and day work never have so wanted anything to be good and all I have is a pile of paper covered with wrong words.\" The Cablegram cocktail was a hit on both sides of the Atlantic. This recipe was collected by Harry Craddock in his 1930 Savoy Cocktail Book.\n\nlowball glass\n\n2 ounces whiskey\n\n\u00bd tablespoon powdered sugar\n\nJuice of half a lemon\n\nGinger ale\n\nShake all ingredients except ginger ale over ice; strain into a lowball glass. Add ginger ale.\n\nIn 1934 Dorothy Parker sent this wedding announcement to her sister on Long Island. It's signed by one of Mrs. Parker's dogs. \u25c6 \u25c6 \u25c6\nChicago\n\nThe glue that held the Algonquin Round Table together wasn't, as you might think, Alexander Woollcott, the number one theater pundit in town. It was Franklin P. Adams, dean of the group and the most successful newspaper columnist of the bunch. Known by his initials F. P. A., Adams was born, raised, and educated in Chicago. He moved to Manhattan in 1904 and was extremely popular for forty years, his \"Conning Tower\" column required reading. He ran many of Dorothy Parker's most popular poems for no charge. She adored F. P. A. (\"He raised me from a couplet.\"), and they coauthored a book together. F. P. A. had his glory years on the Pulitzer-owned New York World. \"Nothing makes me so angry as to hear somebody say, 'That is too good for a newspaper; it is good enough for a magazine,'\" he wrote. \"I have never seen anything too good to appear in a newspaper.\" He generated millions of words over the course of his career, but he was known for his light verse. One Valentine's Day he wrote:\n\nThe ink is red,\n\nThe rent is due,\n\nMy hope is dead,\n\nAnd how are you?\n\nThe Chicago is an old drink served in a wine glass. If you omit the Champagne it's called a Chicago Cocktail.\n\nwine glass\n\n1\u00bd ounces brandy\n\nDash Angostura bitters\n\nDash triple sec\n\nDry Champagne\n\nHarry Craddock's 1930 recipe advised: \"Shake well and strain . . . Frost edge of glass with castor sugar, and fill with Champagne.\" The Chicago is much more glamorous than the Cincinnati Cocktail, which Hugo R. Ensslin described as \"Half a glass of beer, fill up with soda water, and serve.\"\n\nDorothy Parker's mentor, Franklin P. Adams \u25c6 \u25c6 \u25c6\n\nProhibition and the Eighteenth Amendment\n\nIt's easy now to shake our heads and crack smiles when talking about Prohibition and the Eighteenth Amendment, but the legislation enacted in 1919 destroyed businesses, wrecked lives, upended the economy, and still affects us today. The common misconception portrays glamorous flask-carrying dilettantes and hapless bootleggers running from the Keystone Kops, but, in reality, murderous thugs were delivering illegal substances to common Americans all too eager to break the law.\n\nThe Eighteenth Amendment, which prohibited the manufacture, sale, or transportation of alcoholic beverages in America proved a disastrous failure. During the thirteen years that it was on the books, it was widely ignored. National Prohibition went into effect at midnight on January 16, 1920, exactly one year after the amendment was ratified. Support for Prohibition often went hand in hand with the desire by native white Protestants to control European Catholics, Native Americans, Asian immigrants, and, especially in the South, blacks, giving police officers an excuse to arrest any of them on the pretext of intoxication. The \"drys\" blamed alcohol as the root cause of the nation's ills, and politicians took notice of European immigrants gathering in beer halls, becoming powerful voting blocs, and thought that by taking away these local bars they could destroy the newcomers' cohesiveness, their trade unions, and their voting power.\n\nPublic safety director Smedley \"Duckboards\" Butler chops open barrels of beer and pours them into Philadelphia's Schuylkill River in 1924. \u25c6 \u25c6 \u25c6\n\nOne of the main supporters of the dry laws, the Anti-Saloon League had its eyes on New York City for a key reason: It hoped (miraculously) that the millions of foreign visitors to the busiest city in America would see Prohibition working, return to their home countries, and preach for the banning of alcohol there as well. The powerful National Woman's Christian Temperance Union, another pillar of the movement, pushed for social reform, citing the example of women and children afflicted by drunk ne'er-do-well husbands, and used women's new right to vote (as of August 1920) to lobby politicians to pass the amendment. The US Supreme Court allowed the Volstead Act, which implemented the Eighteenth Amendment, to stand, agreeing, five to four, that nothing in the Prohibition laws contravened the rights of the states.\n\nIn New York City the number of places to purchase liquor doubled, ironically, from 15,000 legal spots to more than 32,000 illegal ones. By the time Repeal went into effect on February 20, 1933, an estimated 1.4 billion gallons of illegal hard liquor had been sold nationally during those thirteen long years. Many of the bartenders whom Prohibition put out of work sailed for opportunities in Paris and London. Harry Craddock, who had tended bar at various luxury hotels in New York, went to London, where he had a long career at the Savoy Hotel. In 1930 he authored a cocktail guide that included many of the American recipes of the day.\n\nAs speakeasy culture grew, it attracted a new group to illegal backroom bars and basement saloons: women. According to Stanley Walker, the great city editor of the New York Herald Tribune:\n\nSoon after 1920 great, ravening hordes of women began to discover what their less respectable sisters had known for years\u2014that it was a lot of fun, if you liked it, to get soused. All over New York these up and coming females piled out of their hideaways, rang the bells of speakeasies, wheedled drugstores into selling them gin and rye, and even in establishments of great decorum begged their escorts for a nip from a hip flask. It was all very embarrassing.\n\nLandlords turned a collective and remunerative blind eye to the speaks, which had colorful names such as the Royal Box, the Jungle Room, the Club Chantee, and the Little Club. They sprang up in private homes, in walk-up apartments, and inside shuttered restaurants. Suppliers were everywhere and easy to find: Cigar shops, delicatessens, drug stores, shoeshine parlors, and even clothing stores offered bottles to thirsty customers. Women could walk into a department store looking for whiskey and gin using code names like \"red stockings\" and \"white stockings.\"\n\n\"Lips That Touch Liquor Shall Not Touch Ours\"\n\n\u2014popular slogan carried by the National Women's Christian Temperance Union\n\nThe profits and success for the speakeasy trade proved lucrative. Walker recounted one underground bar-restaurant with a menu \"slightly heavy and Germanic, but it was tasty and filling.\" It was also an illegal speak. \"Twenty-four men could belly up to the bar without undue crowding, and in 1927 and for more than two years thereafter the line was sometimes two deep. Jim's profits were staggering. In one year he cleared, after all expenses, bad checks, bribery, and charity, almost one hundred thousand dollars.\" That's about $1.3 million today.\n\nOrganized crime soon inched its fingers into the booze business in a major way, leading to an increase in gangster activity across city and state lines. Bloody shootouts and revenge killings splashed across the nation's front pages. News reports daily played out the crimes, stories that effectively helped end the liquor ban. Following Black Tuesday and the Stock Market Crash of 1929, many Americans organized protests and local campaigns to end the Volstead Act. A series of measures, first allowing the sale of watered-down beer and wine, led to a shift in the tide.\n\nA three-year investigation ordered by President Herbert Hoover confirmed in 1931 that most states weren't enforcing the amendment. In 1932 Democrats officially supported its repeal. The overwhelming Democratic victory that year encouraged Congress to pass the Twenty-First Amendment, repealing the Eighteenth, on February 20, 1933. On March 22 the Volstead Act was amended to permit the sale of 3.2 percent beer and wine. Once Congress ratified the Twenty-First Amendment the following December, the Volstead Act became void.\n\nThe first legal drink served in the White House by FDR was a martini.\n\nCuba Libre\n\nAuthor Charles H. Baker wrote in The Gentleman's Companion (1939) that one of the top high-society drinks of the era was the Cuba Libre, forerunner to today's Rum and Coke. The drink originated in Havana around the time of the Spanish-American War (1898), when a captain ordered a rum with Coca-Cola and a wedge of lime. As one drinks aficionado remarked, this became the world's most popular cocktail, uniting the staple liquids of two nations. One tall tale says Teddy Roosevelt and his Roughriders shouted \"Por Cuba Libre!\" (\"Free Cuba!\") as they toasted.\n\ncollins glass\n\n1 small lime\n\n1\u00bd ounces Bacardi white rum\n\nCoca-Cola\n\nQuarter the lime and strain the juice into a Collins glass. Scrape the peel clean and cut into pieces. Add peel pieces to the glass and then add the rum. Muddle so that the liquid coats the sides of the glass. Add ice cubes and fill with Coca-Cola.\n\nCol. Theodore Roosevelt and the 1st US Volunteer Cavalry regiment atop San Juan Hill, Cuba, July 1898. \u25c6 \u25c6 \u25c6\nDeath in the Afternoon\n\nIn 1926 Dorothy Parker dropped by the office of friend Franklin P. Adams after returning from France, where she spent time with Ernest Hemingway. F. P. A. asked Mrs. Parker if she knew the novelist's age. \"Well, I don't know,\" she replied. \"You know, all writers are either twenty-nine or Thomas Hardy.\" Three years later she wrote the first profile of Hemingway to appear in The New Yorker. \"People so much wanted him to be a figure out of a saga that they went to the length of providing the saga themselves,\" she wrote. \"And a little peach it is.\" Hemingway's Spanish bullfighting classic Death in the Afternoon appeared in 1932. A few years later, a celebrity cocktail guide, So Red the Nose, received this contribution from the future Nobel winner.\n\nchampagne flute\n\n1 ounce absinthe\n\nChampagne\n\nAccording to the original recipe, Hemingway wrote: \"Pour one jigger absinthe into a Champagne glass. Add iced Champagne until it attains the proper opalescent milkiness. Drink three to five of these slowly.\"\n\nThe only cure for a real hangover is death.\n\n\u2014Robert Benchley\n\nJack Dempsey started boxing professionally in 1914, calling himself \"Kid Blackie.\" Damon Runyon nicknamed him the \"Manassa Mauler\" in 1916. \u25c6 \u25c6 \u25c6\nDempsey\n\nIn 1921 the Colorado-born world heavyweight champion Jack Dempsey defeated French prizefighter Georges Carpentier in a fourth-round knockout in Jersey City for professional boxing's first $1 million gate. Two years later at the Polo Grounds, the legendary baseball stadium of the Giants and Yankees, Dempsey knocked down Luis Angel Firpo nine times in two rounds to retain his title. After retirement, Dempsey and his wife ran a Midtown Manhattan bar that featured an eighteen-foot painting by James Montgomery Flagg of the boxer clobbering Jess Willard. The painting now hangs in the Smithsonian Institution in Washington, D.C. Mayor Ed Koch officially designated Forty-ninth and Broadway Jack Dempsey Corner in 1984. This recipe\u2014which appears in the Savoy Cocktail Book, published in London in 1930\u2014will deliver a punch like Dempsey if you go too many rounds with it.\n\ncocktail glass\n\n1 ounce gin\n\n1 ounce Calvados\n\n2 dashes absinthe\n\n2 dashes grenadine\n\nShake all ingredients over ice; strain into a chilled cocktail glass.\n\nNote: You can substitute another apple-y brandy, such as Applejack, for the Calvados and Verte de Fougerolles for the absinthe.\nDolores\n\nThe Follies captured New York's heart and Dorothy Parker's, too. As a critic, she adored the show for its catchy songs, campy humor, and fantastic costumes. She also admired the beautiful showgirls. Writing for Vanity Fair in 1918, she said, \"I was delighted, too, to hand it to the chorus, who fully live up to the advertisements of 'a chorus of forty under twenty,'\u2014in most of the season's musical shows, it has been just vice versa.\" Former typist Kathleen Mary Rose, a statuesque British model, ranked among the first and most famous hired by Florenz Ziegfeld. He gave her the stage name Dolores, and she became a star as a stage \"mannequin\" who didn't sing, dance, or talk. Acclaimed as \"the loveliest showgirl in the world,\" Dolores never cracked a smile onstage. In Follies spectacles such as \"The Episode of Chiffon,\" she stood onstage as the Empress of Fashion. Dolores appeared in Ziegfeld productions from 1917 to 1921, including the Midnight Frolic of 1919 in which she wore a white peacock costume with a ten-foot train. Afterward, she quit the stage, moved to Paris, and married a millionaire in 1923. This drink, when properly made, tastes like candy.\n\nold-fashioned glass\n\n1\u00bd ounces Spanish brandy\n\n1\u00bd ounces cherry brandy\n\n1\u00bd ounces cr\u00e8me de cacao\n\nStir all ingredients over ice; serve on the rocks.\n\nStunning costumes were fashioned for Dolores as a Follies star. Decades after she left Forty-second Street, fans still remembered her beauty. \u25c6 \u25c6 \u25c6\nDubonnet Cocktail\n\nThe Algonquin Round Table collectively zeroed in on Great Neck, Long Island, in the early 1920s. Its members visited the homes of fellow writer Ring Lardner and the powerful executive editor of the New York World, Herbert Bayard Swope. At Swope's incredible dinner parties, croquet matches and all-night dancing took place. Scott and Zelda Fitzgerald rented nearby, and Swope's parties became the inspiration for the Gatsby soir\u00e9es. At a dinner one night with Governor Albert Ritchie of Maryland, the guests were quizzing the politician on the state of the union. Suddenly, a man deep in his cups burped loudly. A brief embarrassed hush ensued\u2014until Dorothy Parker, a frequent Swope guest, deftly broke it: \"I'll get the governor to pardon you,\" she said sweetly. The Dubonnet Cocktail, a popular Prohibition drink that originated in France, certainly appeared on the Swope table and is excellent today as an aperitif.\n\ncocktail glass\n\n1\u00bd ounces dry gin\n\n1\u00bd ounces Dubonnet\n\nLemon peel\n\nShake all ingredients over cracked ice, strain into a chilled cocktail glass, and add a twist of lemon peel.\n\nNote: There is no substitute for Dubonnet in this cocktail.\nEl Presidente\n\nOne of the best vacations in the speakeasy era of 1920 to 1933 was sailing to Cuba. On that island, home to the Bacardi factory and countless bars and nightclubs, many Americans discovered cocktails made with rum for the first time. These recipes soon made their way back to the States. Among the most popular was the El Presidente, created at Bar La Florida in Havana for a now mostly forgotten president of Cuba, General Carmen Menocal.\n\ncocktail glass\n\n1 ounce Bacardi light rum\n\n1 ounce dry vermouth\n\n1 teaspoon grenadine\n\n1 teaspoon Cura\u00e7ao\n\nOrange peel\n\nMaraschino cherry\n\nFill a tall mixing glass with cracked ice; add rum and vermouth. Pour in the grenadine and Cura\u00e7ao, stir, and strain into a chilled cocktail glass. Carefully twist the orange peel over the side of the glass; its orangey oil should float on the surface of the drink. Drop in the orange peel and a maraschino cherry.\n\nEmperor Norton's Second Mistress\n\nElixir, San Francisco\n\nEight years after California was granted statehood, a bar opened on the corner of Sixteenth and Guerrero in San Francisco's Mission District. More than 150 years later, Elixir continues its proud history of crafting cocktails. Proprietor H. Joseph Ehrmann, president of the Barbary Coast Conservancy of the American Cocktail, helps run San Francisco Cocktail Week and wanted to create a drink to honor a famous local figure. \"Emperor Norton was a classic San Francisco icon from the Victorian era, declaring himself 'Emperor of these United States and Protector of Mexico.' He walked San Francisco between 1849 and his death in 1880, distributing his own currency (accepted by many) and invited foreign royalty to visit him. If he came into the saloon and I was behind the stick, I'd have served him.\"\n\nold-fashioned glass\n\n4 medium California strawberries\n\n1\u00bd ounces Cyrus Noble bourbon\n\n\u00be ounce Tuaca\n\nIn a mixing cup, muddle 3 strawberries to juice. Add bourbon and Tuaca and fill with ice. Shake hard to dilute and then Hawthorne strain into a large old-fashioned glass filled with fresh ice. Slice 1 strawberry halfway and place on rim to garnish.\n\nOne of Emperor Norton's prophecies was a bridge linking San Francisco and Oakland. Locals still remember him fondly. \u25c6 \u25c6 \u25c6\nEspionage\n\nEastern Standard Kitchen & Drinks, Boston\n\nThe FBI file on Dorothy Parker runs three inches thick. Nearly half a century after her death, many portions remain redacted. The only city where Mrs. Parker ever got in trouble was Boston (see the Ward 8 recipe, page 118), so she'd be happy to know about the Espionage, dreamed up in the Kenmore Square neighborhood. \"A guest was looking for a Manhattan-style cocktail, but with vodka,\" says manager Kevin Martin. \"Since Fair Quinoa vodka carries the weight of a whiskey, it was the choice base spirit. Mixed with the floral chartreuse, multiple bitters, and two garnishes, the cocktail was born. Named for its deceiving flavor, this vodka cocktail drinks like a whiskey cocktail.\" Eastern Standard Kitchen & Drinks is a neighborhood brasserie with charm, knowledgeable bartenders, a wide mix of regulars\u2014and is home to the Espionage.\n\nlowball glass\n\n1\u00bd ounces Fair Quinoa vodka\n\n\u00be ounce yellow chartreuse\n\n\u00bd ounce Cynar liqueur\n\n\u00bd ounce Amaro Montenegro Italian liqueur\n\nLemon twist\n\nOrange twist\n\nMount liquid ingredients into a mixing glass filled with cracked ice, stir until properly diluted, strain into a chilled lowball glass, and garnish with in-and-out lemon and orange twists.\nFlorodora\n\nFlorodora, a musical show of six beautiful showgirls, entertained turn-of-the-century audiences. Always a sextet averaging five feet, six inches tall, the Florodora girls were huge celebrities when Theodore Roosevelt was president. Traveling shows played towns from coast to coast, and the cast was treated like royalty. The young women became major stars, and their exploits filled gossip columns. In 1920 the show was revived on Broadway at the Century Theatre, and Dorothy Parker panned it in Ainslee's, writing, \"There has been an amazingly well-preserved glamour about Florodora. Perhaps it was a reflection of the radiance shed by those original six, the girls who 'put the sex in sextet,' as you might say.\" When the revival came out, so did a tasty gin cocktail that became all the rage in speakeasies.\n\nhighball glass\n\n1\u00bd ounces gin\n\n\u00bd ounce fresh lime juice\n\n\u00be ounce Framboise (raspberry liqueur)\n\nSplash ginger ale\n\nLime wedge\n\nIn a highball glass filled with ice, build the gin, lime juice, and raspberry liqueur. Top with ginger ale and garnish with a lime wedge.\n\nNote: Framboise, the French word for \"raspberry,\" can also be used to refer to any alcohol distilled from fruit. Because of the fruit, this cocktail has the aroma of raspberries. While Framboise usually is served in a Champagne flute, rhe Florodora breaks the tradition.\nFrench \"75\"\n\nOnly a few days after the 1917 wedding of Dorothy Rothschild and Edwin Pond Parker II, war separated the newlyweds. Eddie went to New Jersey for army training and then shipped out to France as a driver in the 33rd Ambulance Corps. Dottie didn't get him back until 1919. During the Great War, a number of future members of the Algonquin Round Table served in uniform, including Franklin P. Adams, Herman Mankiewicz, Harold Ross, Robert E. Sherwood, Laurence Stallings, and Alexander Woollcott. Vicious Circle civilians in Paris included Heywood Broun, Jane Grant, Ruth Hale, and Neysa McMein. Doughboys brought back the popular French \"75,\" named for the 75mm M1897, a French field gun. Irvin S. Cobb went to the front as a correspondent for the Saturday Evening Post and two decades later recalled his experience of a French \"75\"\u2014\"I had my first of these in a dugout in the Argonne. I couldn't tell whether a shell or the drink hit me.\"\n\nchampagne flute\n\n1 ounce gin\n\nJuice of 1 lemon\n\n1 teaspoon powdered sugar\n\nChampagne\n\nPour the gin into the Champagne flute. Add the lemon juice and sugar. Fill the remainder of the glass with chilled Champagne.\n\nFrench army officers share a drink in the field, circa 1915. \u25c6 \u25c6 \u25c6\nGene Tunney\n\nBeginning in 1922, Harold Ross, Jane Grant, and Alexander Woollcott lived together at 412 West Forty-seventh Street, a Hell's Kitchen duplex. Dorothy Parker and the Algonquin Round Table were frequent guests, along with top writers, cartoonists, actors, musicians, and other cultural figures. Gene Tunney was world heavyweight champion from 1926 to 1928. His two victories over Jack Dempsey remain among the most famous bouts in boxing history. The drink named for him\u2014practically a Martini, with a little juice added\u2014proved popular on both sides of the Atlantic.\n\nIn Ross, The New Yorker, and Me, Grant's account of the era, she relates the following story. One night, two drunks, thinking the house a speakeasy, tried to push their way in. Grant's friend tried to block their way.\n\n\"The kids told us it was a speakeasy,\" the men told her friend.\n\n\"Well, it isn't,\" he said, \"and you'd better get out.\"\n\n\"Who's going to put us out? You can't,\" one of the drunks replied as he tried to push the friend aside.\n\n\"Well, maybe I can't, but Gene Tunney can. Hey, Gene,\" the friend called out.\n\n\"Aw, quit kidding,\" the bigger of the intruders said. Just as he was about to barge in, the heavyweight champ walked toward them, and they scattered.\n\ncocktail glass\n\n2 ounces gin\n\n1 ounce French vermouth\n\nDash of orange juice\n\nDash of lemon juice\n\nLemon peel\n\nShake liquid ingredients well over cracked ice. Strain into a chilled cocktail glass and garnish with a twist of lemon peel.\n\nGene Tunney was born in New York City's Greenwich Village. He beat Jack Dempsey for the world heavyweight championship in 1926 in front of 120,500 screaming fans in Philadelphia. \u25c6 \u25c6 \u25c6\n\nThe pop culture icon of 1900, the Gibson Girl came from the pen of Charles Dana Gibson. \u25c6 \u25c6 \u25c6\nGibson\n\nFor many years cocktail historians have debated where the Gibson originated. It could be a cousin to a pre-Prohibition drink, the \"onion cocktail.\" The (mostly) accepted story holds that the Gibson is named for magazine illustrator Charles Dana Gibson and was created at Gramercy Park's Players Club, a society founded in 1888 for men in theater, music, and literature. In the 1880s, Gibson joined the staff of the old Life (not to be confused with the Henry Luce publication), a monthly humor magazine. In the 1920s, Life employed Robert Benchley, Dorothy Parker, Robert E. Sherwood, and many other Algonquin Round Table members. Gibson earned his renown for drawing alluring women in illustrations of high-society scenes. In 1890 he created a haughty beauty with upswept raven hair that became a symbol of the Gay Nineties: the Gibson Girl. His renditions of stylish, athletic, and alluring girls made Gibson rich. Life proved wildly popular through World War I, and it was Mrs. Parker's bread and butter. The New Yorker, debuting in 1925, was modeled partially on Life. Although the Gibson bears a striking resemblance to a Martini, it should not be considered just a Martini with an onion in it\u2014and no onions, please. This is the classic World War I version.\n\ncocktail glass\n\n1\u00bd ounces gin\n\n1\u00bd ounces dry vermouth\n\nMaraschino cherry\n\nIn a mixing glass filled with cracked ice, add the liquid ingredients. Stir slowly and strain into a chilled cocktail glass. Garnish with a cherry.\nGimlet\n\nDorothy Parker and Alexander Woollcott shared an abiding affection for mysteries, murder, and crime stories. One of the best practitioners of the genre, Raymond Chandler has his detective Philip Marlowe drink Gimlets in The Long Goodbye. Chandler said, \"Alcohol is like love. The first kiss is magic, the second is intimate, the third is routine. After that you take the girl's clothes off.\" A proper Gimlet always includes Rose's Lime Juice, first patented in 1867. The Gimlet became a standard speakeasy drink because it was simple and fast to make, and the lime juice easily covered imperfections in what might be a bad batch of gin. Legend holds that the drink originated in the Royal Navy when ship's doctor T. O. Gimlette prescribed the drink as a medicinal tonic. Apparently Dr. Gimlette thought he was fighting scurvy by adding lime juice to the sailors' gin rations. According to the British, the only good Gimlet is a cold one, thoroughly mixed with cracked ice before serving.\n\ncocktail glass\n\n2 ounces gin\n\n1 ounce Rose's lime juice\n\nSlice of fresh lime (optional)\n\nShake liquid ingredients over ice; strain into a cocktail glass. Add optional slice of lime.\n\nNote: You can serve the drink neat or over ice. In the 1920s soda water was often added. If you substitute fresh lime juice for the Rose's Lime Juice, add 1 teaspoon confectioner's sugar. For effect, frost the rim of the glass by dipping it into lime juice, then granulated sugar. Chill the sugar-frosted glass in the refrigerator before preparing the drink.\nGin Fizz\n\nThe first time Dorothy Parker saw Paris, she was thirty-two and broke. She sailed from New York in early 1926 with Robert Benchley and Ernest Hemingway aboard the President Roosevelt. Her visit to the French Riviera provided Mrs. Parker with a wealth of material for many years. In 1929 The New Yorker published a short piece by her set on the Mediterranean, a boozy conversation between a couple of Gin Fizz\u2013guzzling Americans, a ridiculous playboy and a silly socialite. In \"The Cradle of Civilization\" the young man orders another round of drinks in his bad French. \"Oh, gar\u00e7on. Encore deux jeen feezes, tout de suite,\" he said. \"And mettez un peu more de jeen in them cette fois, baby.\" Fizzes are sour-based cocktails with mandatory additions of club soda and lemon juice. A Gin Fizz is served in a highball glass of 8 to 12 ounces, topped with club soda.\n\nhighball glass\n\n2 ounces gin\n\n\u00be ounce fresh lemon juice\n\n1 teaspoon superfine sugar or simple syrup\n\nClub soda\n\nMint sprig\n\nShake the gin, lemon juice, and sugar over cracked ice and strain into a 10-ounce highball glass filled with ice. Fill the remainder of the glass with cold club soda. Garnish with mint.\n\nNote: There are dozens of variations of the Gin Fizz. Add a whole egg for a Royal Fizz, or add \u00be ounce orange juice and 1 teaspoon simple syrup to make a Texas Fizz. A teaspoon of cream in the drink makes it a Cream Fizz.\nGin Rickey\n\nIn The Great Gatsby, Tom Buchanan serves four Gin Rickeys on a blazing hot afternoon on Long Island. The cocktail is perfect on a summer day, and in the book Jay Gatsby serves the drink often at his Gold Coast estate, employing a cadre of bartenders to mix them for his gala parties. As F. Scott Fitzgerald writes, \"Every Friday five crates of oranges and lemons arrived from a fruiterer in New York\u2014every Monday these same oranges and lemons left his back door in a pyramid of pulpless halves.\" The Gin Rickey was one of the most popular drinks of the era. Its creation traces back to Joe Rickey, a Capitol Hill lobbyist, to whom the drink was served at Shoemaker's, located on Pennsylvania Avenue, in the 1890s.\n\ncollins glass\n\n1\u00bd ounces gin\n\n\u00bd ounce fresh lime juice\n\nClub soda\n\nLime wedge\n\nPour gin and lime juice into a Collins glass filled with ice cubes. Fill with cold club soda and stir. Add the wedge of lime and serve.\n\nNote: This drink has countless variations; for any substituted potable, the drink is then called by the liquor used, such as a Rum Rickey and so on.\n\nDorothy Parker in Hollywood\n\nDorothy Parker's first trip to Los Angeles came in late 1928, when she signed a three-month contract with MGM for $300 a week. A gossip columnist reported that December on Mrs. Parker's transition to screenwriter:\n\nNo movie contract could keep her at work by merely cloistering her in a remote office at the far end of the longest corridor in a dismal Culver City office building. She has managed to turn this cell into quite a salon by a device that many lonelyhearts might find helpful. When the sign painter arrived to put her name on the door, she bribed him to leave it off and substitute instead the simple legend: GENTLEMEN.\n\nWhen her contract ended and with no screen credits to her name, Mrs. Parker returned to New York City.\n\nShe stayed away from the West Coast for five years. In 1934, following her marriage to actor-writer Alan Campbell, the couple moved to California on a temporary basis. They arrived in a state of semi-destitution in an open-topped Ford with two dogs. Within a week, they were signed to a studio for $5,000 a week and living in a friend's huge home in Beverly Hills. The couple lived in rented mansions in Beverly Hills, stayed in the Ch\u00e2teau Marmont and Garden of Allah on Sunset Boulevard, and, when funds ran low, they moved into a not-so-great West Hollywood bungalow at 8983 Norma Place.\n\nAs a writing team, they worked on more than thirty pictures. In Hanging On in Paradise, author Fred Lawrence Guiles described their partnership, with Campbell playing a key role:\n\nIt was he who sat at the typewriter and did the real work of putting down on paper the technical and special language of the scriptwriter. Dorothy would sit nearby, often knitting, as Alan might ask, 'Shall we make the mother a washerwoman,' and Dorothy might say, 'Oh yes, by all means, a washerwoman.' Then she would read over what Alan had written and edit it and write in some witty dialogue, elevating the competent to the penetrating and, sometimes, the splendid.\n\nIn 1935 Dorothy Parker and Alan Campbell were signed to Paramount Pictures as a screenwriting team. \u25c6 \u25c6 \u25c6\n\nDorothy and Alan earned their only Academy Award nomination as a team for the original screenplay to A Star Is Born, since remade twice. They were nominated for best screenplay in 1937, along with Robert Carson, but the Oscar went to The Life of Emile Zola.\n\nDottie on DVD:  \nFive Films She Co-Wrote\n\nHands Across the Table (1935)\n\nSuzy (1936)\n\nA Star Is Born (1937)\n\nThe Cowboy and the Lady (1938)\n\nSmash-Up: The Story of a Woman (1947)\n\nThe couple worked steadily into the 1930s, but the experience proved rocky. Stories of Mrs. Parker's conflicts with the studios are legendary. After being scolded by Samuel Goldwyn, who told her, \"You haven't got a great audience, and you know why? Because you don't give people what they want,\" Mrs. Parker softly replied: \"But Mr. Goldwyn, people don't know what they want until you give it to them.\" Goldwyn exploded, saying, \"Wisecracks. I told you there's no money in wisecracks. People want a happy ending.\" To which she responded: \"I know this will come as a shock to you, Mr. Goldwyn, but in all history, which has held billions and billions of human beings, not a single one ever had a happy ending.\"\n\nAfter Campbell joined the army, Mrs. Parker lived in Hollywood and New York, working on short stories and screenplays. After a period of separation, divorce, and remarriage, the couple worked on a handful of scripts in the 1950s. After her husband's death, Mrs. Parker returned to New York, at the age of seventy.\n\n\"You just don't know how I love it\u2014how I get up every morning and want to kiss the pavement . . . Hollywood smells like a laundry,\" she told Ward Morehouse of the New York World-Telegram and Sun. \"The beautiful vegetables taste as if they were raised in trunks, and at those wonderful supermarkets you find that the vegetables are all wax. The flowers out there smell like dirty, old dollar bills.\"\n\nIn 1941 her friend Budd Schulberg wrote What Makes Sammy Run? about screenwriting. Mrs. Parker said, \"I never thought anyone could put Hollywood\u2014the true shittiness of it\u2014between covers.\"\n\nThe Happy Herbie\n\nThe Edison, Los Angeles\n\nWalt Disney himself tapped two Algonquin Round Table members to appear in animated feature films. Deems Taylor narrates the classic Fantasia (1940), and Robert Benchley plays himself in The Reluctant Dragon (1941). Disney artist and \"imagineer\" Herbert Ryman art-directed Fantasia and Dumbo, and in the 1950s Disney tapped Ryman to draw the concept art for Disneyland, which became the blueprint for the amusement park. Inside a hundred-year-old former power plant in Downtown Los Angeles lies The Edison, a lounge that evokes the city's romantic past. John Maraffi, director of spirits, created the Happy Herbie for a fundraiser for the Ryman Foundation (Ryman.org), which provides fine art training to teenagers. \"Although it's a twenty-first-century cocktail, the flavor and color evoke the style of a classic cocktail,\" says Barbara Jacobs, who runs The Edison. \"It sends the guest to Disneyland, with apples and oranges rounding out the vision of the Happiest Place on Earth.\"\n\ncocktail glass\n\n2 ounces Templeton rye whiskey\n\n\u00bd ounce GreenBar jasmine liqueur\n\n\u00bd ounce freshly squeezed lemon juice\n\n\u00bd ounce simple syrup\n\n3 dashes Bar Keep apple bitters\n\nOrange peel\n\nShake all liquid ingredients over ice; strain into a chilled cocktail glass. Garnish with orange peel.\nHemingway Daiquiri\n\nBar La Florida Style\n\nDorothy Parker never made it to Cuba, but her frenemy, Ernest Hemingway, famously did. The two writers did sail to France aboard the same cruise ship in early 1926, however. One story of that journey holds that Hemingway had forgotten his portable typewriter, so Mrs. Parker loaned him her Underwood. For whatever reason, he pitched it over the side of the ship. After Hemingway began his love affair with Cuba, he named Bar La Florida\u2014located on Obispo and Montserrate Streets and still in operation\u2014his favorite Havana watering hole. Hemingway welcomed friends such as Gary Cooper and Spencer Tracy there, and in 2003 a life-size statue of the writer was installed in his favorite spot. Bar La Florida, in existence since 1819, claims to have invented the Daiquiri. Hemingway preferred his served as a double, over a single serving of ice, or frozen. This recipe comes from the bar's own 1935 menu.\n\nhighball glass\n\n2 ounces Bacardi white rum\n\n\u00bd ounce fresh lime juice\n\n1 teaspoon grapefruit juice\n\n1 teaspoon maraschino liqueur\n\nFill a blender with 2 cups crushed ice, add all the ingredients, and blend. Pour into a chilled highball glass.\n\nNote: Using a blender makes the daiquiri frozen or frappe.\nHighball\n\nThe name for this drink comes from the nineteenth-century railroad days, when raising a signal\u2014a ball on a long pole\u2014meant engines could proceed at full steam. When New York bartender Patrick Duffy invented the drink in the 1890s, it was taken to mean how fast the drink could be assembled. All highballs are meant to be simple to mix, to be served over ice, and to refresh. In a tall glass it's a Highball; in an old-fashioned glass, it's a Lowball.\n\nhighball glass\n\n2 ounces Scotch, bourbon, or rye\n\nClub soda or ginger ale\n\nLemon twist (optional)\n\nThe key to a good Highball is building it. Start with the alcohol. First, pour the whiskey into a chilled glass filled with ice cubes. Top with cold club soda or ginger ale, stirring gently. If desired, drop in a lemon twist.\n\nThis is a nice highball, isn't it? Well, well, well, to think of me having real Scotch; I'm out of the bush leagues at last. It will be nice to see the effect of veritable whisky upon one who has been accustomed only to the simpler forms of entertainment.\n\n\u2014from Dorothy Parker's classic 1928 short story, \"Just a Little One,\" about a night in a speakeasy, published in The New Yorker\nHorse's Neck\n\nTexas Guinan was a larger-than-life speakeasy owner and performer. \u25c6 \u25c6 \u25c6\n\nJust one speakeasy proprietor earned the sobriquet \"Queen of the Nightclubs,\" and that was Texas Guinan. A larger-than-life bottle blonde with a big smile and bigger personality, Texas was beloved by boozehounds and newspaper gossip columnists. The authorities were constantly hauling her off to jail, which only added to her reputation. She wore a necklace made of handcuff keys, and during raids she asked her band to perform \"The Prisoner's Song\" as she was led out. Because Texas supplied the ginger ale if her customers brought the booze, here's a famous Jazz Age cocktail that depends on it. This recipe comes from Rob Chirico's Field Guide to Cocktails.\n\nhighball glass\n\nOrange peel\n\n2 ounces bourbon\n\n4 ounces ginger ale\n\nCarefully carve a long piece of peel, 1\u00bd inches wide, from the orange, then wrap the spiral around several ice cubes in a highball glass. Some of the excess peel should hang over the side of the glass, like a horse's neck. Add the bourbon and cold ginger ale and stir.\nJack Rose\n\nIn The Sun Also Rises, Jake Barnes has a Jack Rose while waiting in vain for the \"damned good-looking\" Lady Brett Ashley in a Paris hotel bar. Ernest Hemingway added the reference when the drink was popular in the 1920s. Hemingway was in Paris with the Spanish surrealist filmmaker Luis Bu\u00f1uel. \"If you were to ask me if I'd ever had the bad luck to miss my daily cocktail,\" Bu\u00f1uel said, \"I'd have to say that I doubt it; where certain things are concerned, I plan ahead.\"\n\ncocktail glass\n\n2 ounces applejack or Calvados\n\n\u00bd ounce fresh lime or lemon juice\n\n2\u20134 dashes grenadine\n\nShake all ingredients over ice; strain into a chilled cocktail glass.\n\nNote: Applejack is the American term for 80 to 100 proof apple brandy aged two years in wooden casks. Calvados comes from France and is also a popular apple brandy.\n\nJean Harlow\n\nThe connection between the original blonde bombshell and Dorothy Parker is a strong one. Mrs. Parker and her husband, Alan Campbell, wrote one of Jean Harlow's best films, the 1936 MGM comedy Suzy. One of the pivotal scenes depicts Harlow and Cary Grant, playing a cabaret singer and a pilot, trading wisecracks in a French bistro over little glasses of Cointreau. This namesake drink offers a variation on the Martini, and this recipe allegedly was Harlow's favorite cocktail.\n\ncocktail glass\n\n2 ounces white rum\n\n2 ounces sweet vermouth\n\nLemon peel\n\nPour the rum and vermouth into a shaker filled with cracked ice. Depending on your preference, you can either shake or slowly stir it. Pour into a chilled cocktail glass and garnish with the lemon peel.\n\nJean Harlow, Cary Grant, and George Davis in Suzy. \u25c6 \u25c6 \u25c6\nJosephine Baker\n\nFreda Josephine McDonald was born poor in St. Louis, Missouri, in 1906, but by the time of her death in 1975, the dancer had attained global recognition as one of the most sought-after black entertainers. McDonald changed her name to Josephine Baker when she was nineteen and moved to Paris, where her audience was almost exclusively white. Historian Ann Douglas says in Terrible Honesty that Baker\n\nliked to turn up in a Poiret gown, but this was never her primary look. Onstage and off, she was noted for something close to nudity, nudity accentuated by a belt of bananas around her hips or a long feather curling provocatively between her legs, and at the most unexpected moments she would start mugging: she'd cross her eyes, throw her bronze limbs akimbo in a comical kaleidoscope of fractured motion, turning on the glamorous and provocative image she'd created but a moment before.\n\nThis recipe comes from the 1935 menu of Havana's most popular bar-restaurant, La Florida.\n\ncocktail glass\n\n1\u00bd ounces Cognac\n\n1\u00bd ounces port wine\n\n\u00be ounce apricot brandy\n\n1 teaspoon sugar\n\n1 egg yolk\n\nCinnamon\n\nLemon peel\n\nShake all the liquid ingredients and sugar well over ice. Strain into a large cocktail glass filled with cracked ice. Spread a pinch of cinnamon on top of the drink and garnish with a lemon twist.\n\nNote: The original recipe calls for Soberano Cognac, a Spanish brand not easily available in America. You can substitute Hennessey V.S. (\"very special\").\nThe Knickerbocker Cocktail\n\nThe grand old Knickerbocker still stands in Times Square, but, no longer a hotel, today it's an office building and a storefront for selling T-shirts. \u25c6 \u25c6 \u25c6\n\nWashington Irving, New York's first professional author, wrote his classic History of New York in 1809, penned by the pseudonymous \"Diedrich Knicker\u00adbocker.\" Over time the name came to signify natives of the city or state of New York. On the southeast corner of Forty-second Street and Broadway, millionaire real estate scion John Jacob Astor IV built the lavish fifteen-story luxury Hotel Knickerbocker, which opened in 1906. Broadway regulars loved its swanky bar and restaurant. Astor commissioned Maxfield Parrish to paint a mural for the bar, and the thirty-foot-wide Old King Cole now hangs over the bar in the St. Regis, another Astor hotel. The Algonquin Round Table often attended midnight parties in the Knickerbocker, which was often home to theater stars such as George M. Cohan. The red brick beaux arts building, with its eye-catching copper mansard roof, is the only grand hotel left in Times Square. It was gutted long ago for office and retail space, a victim of the Great Depression, but the Knickerbocker Cocktail carries on the bar's name.\n\ncocktail glass\n\n2 ounces rum\n\n1 teaspoon raspberry syrup\n\n1 teaspoon fresh lemon juice\n\n2 dashes Cura\u00e7ao\n\n1 chunk pineapple\n\nStir liquid ingredients with ice. Strain into a chilled cocktail glass. Drop in the pineapple chunk.\nThe Last Word\n\nThe year 1916 saw the founding of the Detroit Athletic Club News for the 10,000 members of the private club in Michigan. During the Jazz Age, the DAC News bought freelance pieces from some of the top humorists in the country: Robert Benchley, Ring Lardner, Groucho Marx, Dorothy Parker, Frank Sullivan, and James Thurber. The DAC, which is still going strong, has a fabulous bar where the Last Word was created by vaudeville star Frank J. Fogarty, a native of Red Hook, Brooklyn. In 1912, Fogarty, a comedian and performer for twenty-five years, had won the New York Morning Telegraph contest for most popular performer in vaudeville. After retiring, he went to work for the Brooklyn borough president. On his deathbed in 1925, Fogarty's last words were a request to hear \"Fanny Dear\" on the radio. He lost consciousness before WEAF could broadcast the song, but the song was played at his funeral. This recipe comes from Zelda magazine.\n\ncocktail glass\n\n1 ounce gin\n\n1 ounce maraschino liqueur\n\n1 ounce green chartreuse\n\n1 ounce fresh lime juice\n\nShake all ingredients over ice; strain into a chilled cocktail glass.\nLoud Speaker\n\nWhen radio broadcasting became popular, producers tapped the talents of the Vicious Circle. Among the early radio stars was Franklin P. Adams\u2014\"I don't understand the principle of the radio, nor for that matter the telephone or the telegraph. Don't explain it to me; I don't get it.\"\u2014who served as a weekly panelist on the hit quiz show Information Please. CBS hired Deems Taylor to present classical music programming, and Alexander Woollcott, in a smooth transition in the 1930s from newspapers to radio, also broadcast coast to coast. Round Table writers found their material easily adapted to radio scripts: Robert Benchley, Edna Ferber, and Dorothy Parker all cashed nice checks from radio networks for broadcasting their short stories. Naturally a cocktail was dreamed up to celebrate the new medium. The Loud Speaker appeared in the 1930 Savoy Cocktail Book in London.\n\ncocktail glass\n\n\u00be ounce dry gin\n\n\u00be ounce brandy\n\n\u00bc ounce Cointreau\n\n\u00bc ounce lemon juice\n\nShake all ingredients over ice; strain into a chilled cocktail glass.\n\nAlong with this recipe came the following admonishment: \"This it is that gives to radio announcers their peculiar enunciation. Three of them will produce oscillation, and after five it is possible to reach the osculation stage.\"\nLove Cocktail\n\nShe made her bones by writing light verse about her broken heart, and it's on her poetry that most of Dorothy Parker's reputation stands. Just a few titles give a sense of her devotion to the subject: \"Love Song,\" \"Men I'm Not Married To,\" \"One Perfect Rose,\" and \"Song of a Contented Heart,\" the last of which tellingly predates \"Song of a Hopeful Heart\" by a year.\n\nMrs. Parker's 1933 Anacreontic poem \"The Lady's Reward\" ends with:\n\nBe you wise and never sad,\n\nYou will get your lovely lad.\n\nNever serious be, nor true,\n\nAnd your wish will come to you\u2014\n\nAnd if that makes you happy, kid,\n\nYou'll be the first it ever did.\n\nThe Hotel Wallick in Times Square served this cocktail in 1917.\n\ncocktail glass\n\n1\u00bd ounces sloe gin\n\n1 egg white\n\n1 teaspoon lemon juice\n\n1 teaspoon raspberry syrup\n\nShake all ingredients over cracked ice. Strain and serve in a chilled cocktail glass.\n\nNote: No substitutions allowed for this drink, because these ingredients, when mixed properly, create a drink that comes out blood red.\nYou know how you ought to be with men? You should always be aloof, you should never let them know you like them, you must on no account let them feel that they are of any importance to you, you must be wrapped up in your own concerns, you may never let them lose sight of the fact that you are superior, you must be, in short, a regular stuffed chemise. And if you could see what I've been doing!\n\n\u2014Parker's advice to the fairer sex in 1928\nMamie Taylor\n\nAround the time William McKinley was in the White House, a ballerina asked for a drink at a summer resort on Lake Ontario, and the legend of how the beautiful Mamie Taylor took in hand a whiskey and ginger ale pick-me-up entered the cocktail history books. Legendary publisher James Gordon Bennett ran the recipe in the New York Herald, considering it news. In 1921 the New York Times published a Prohibition lament: \"If we ever build a wall in America it will most likely be along the Canadian and Mexican borders for the purpose of keeping such objectionable characters as John Barleycorn and Mamie Taylor from leaving The Land of the Spree and Home of Knave and contaminating our own glorious Hearth of the Home brewed Hootch.\" Highballs formed a huge part of speakeasy culture, so the Mamie Taylor proved popular. Miss Taylor isn't remembered for anything else except this cocktail, and this recipe comes from a 1936 collection of whiskey highball recipes.\n\nhighball glass\n\n1\u00bd ounces American blended whiskey\n\nJuice of half a lime\n\nGinger ale\n\nFill a highball glass with cracked ice. Pour in the whiskey and lime juice, then fill with cold ginger ale.\n\nManhattan Cocktail\n\nA classic\u2014and completely New York creation\u2014is the Manhattan Cocktail. It comes from the Manhattan Club, a gentlemen's social club that opened in 1865 on Fifth Avenue and Fifteenth Street. Its most famous member was August Belmont Jr.: grandson of Commodore Perry, the sportsman who transformed horseracing into a high-stakes industry as well as financed construction of New York's first subway line. The Manhattan Cocktail came into being, the story goes, while he served as club president, and it became wildly popular during Prohibition. The drink appears in Truman Capote's famous novella Breakfast at Tiffany's as one of Holly Golightly's favorites, and Marilyn Monroe drinks one in Some Like It Hot. This recipe comes from a 1936 guide by Irvin S. Cobb, who called it \"one of America's greatest contributions to civilization. If the recipe given is too dry for you, make the drink half and half, whiskey and Italian vermouth.\"\n\ncocktail glass\n\n2 ounces rye\n\n1 ounce sweet vermouth\n\nDash Angostura bitters\n\nMaraschino cherry\n\nStir liquid ingredients with cracked ice, strain, and garnish with the cherry.\n\nNote: To make a Cuban Manhattan, substitute rum for the rye and add another ounce of vermouth.\nThe Marquis\n\nFl\u00fbte, New York\n\n\"Texas Guinan, in her fashion, during the boom times of the twenties combined the curious and admirable traits of Queen Elizabeth, Machiavelli, Tex Rickard, P. T. Barnum, and Ma Pettingill,\" wrote Stanley Walker, city editor of the New York Herald Tribune. \"She was known as Queen of the Night Clubs. Her greeting 'Hello, sucker!' became the watchword of boobtraps from Wall Street to Hollywood.\" One of \"Queen of the Nightclubs\" Texas Guinan's regular spots was Club Intime, at 205 West Fifty-fourth Street, which operates today as Fl\u00fbte. Down a flight of stairs off Broadway lies an intimate lounge opened by Herv\u00e9 Rousseau and run with a tradition of flowing Champagne and Prohibition parties featuring vintage drinks. Ne'er-do-well bartenders at Fl\u00fbte invented a drink and named it for another Frenchman who liked a good time, the Marquis de Sade. Topped with Champagne, it makes for a grand tribute in the spot once occupied by Texas and her showgirls.\n\ncocktail glass\n\n1 ounce Grand Marnier\n\n1 ounce full-bodied red wine\n\n1 ounce orange juice\n\n\u00bc ounce lime juice\n\n\u00bc ounce simple syrup\n\nChampagne\n\nOrange slice\n\nShake Grand Marnier, wine, juices, and simple syrup well over ice and pour into a large cocktail glass. Top with Champagne and garnish with the orange slice.\n\nNote: You can substitute another orange-flavored brandy liqueur for the Grand Marnier.\n\nStanding on a piano and waving a police whistle and noisemaker, Texas Guinan welcomes her customers in an illustration by Joseph Golinkin. \u25c6 \u25c6 \u25c6\nMartini\n\nMore ink has been spilled about the Martini than any other cocktail. It's the king of all mixed drinks, and there are entire books devoted to it. Beverage historians argue the history of the Martini with the kind of passion usually exhibited among Civil War reenactors. Into the middle of Martini lore falls the little woman from the Upper West Side\u2014an avowed Highball fan\u2014commonly believed to be the number one proponent of the Martini in the Jazz Age. But the fact is, Dorothy Parker never mentions imbibing a Martini in any of her poems or fiction. In none of her books does the famous gin-soaked quatrain appear:\n\nI love a martini\u2014\n\nBut two at the most.\n\nThree, I'm under the table;\n\nFour, I'm under the host.\n\nIf she did in fact generate these twenty words, the text certainly isn't under copyright. Which is good news for the Martini tchotchke industry, which regularly slaps it on glasses, coasters, and T-shirts. But that hardly matters to Martini fans, who eagerly raise their glasses to their favorite version of the gin classic. (We're talking gin only\u2014vodka is for heathens.) The Martini, with just two ingredients, has been beloved since the late nineteenth century, depending on which of the many origin stories you want to believe. It has made countless appearances in novels such as A Farewell to Arms and vintage films including The Thin Man. In the Oscar-nominated screenplay to Smash-Up, which Mrs. Parker and Frank Cavett wrote, Carleton Young tells a crowd, \"The rest of you can get your noses out of those martinis for a minute!\" The Martini has survived changing tastes across two world wars, the Great Depression, and the wildly changing culture at the end of the last century.\n\nCocktail napkin from the Blue Bar of the Algonquin Hotel. \u25c6 \u25c6 \u25c6\n\ncocktail glass\n\nMARTINI\n\n2 ounces gin\n\n\u00bd to 1 ounce dry vermouth\n\nDRY MARTINI\n\n2 \u00bd ounces gin\n\n1 tablespoon dry vermouth\n\nDIRTY MARTINI\n\n2\u00bd ounces gin\n\n1 tablespoon dry vermouth\n\n1 teaspoon olive brine\n\nEXTRA-DRY MARTINI\n\n2\u00bd ounces gin\n\n1 teaspoon dry vermouth\n\nSWEET MARTINI\n\n2\u00bd ounces gin\n\n1 ounce sweet gin\n\n1 ounce sweet vermouth\n\n1 dash orange bitters\n\n1 dash Cura\u00e7ao\n\nFor any Martini, shake ingredients well over cracked ice. The more you shake, the colder the drink will be. Strain into a cocktail glass\u2014chilled if possible in a freezer\u2014and always serve straight up. Garnish with a twist of lemon peel or olive. The gin-to-vermouth ratio determines the dryness of a Martini; the less vermouth, the drier the drink. In the 1930s the Martini had much more vermouth, and in the 1940s the gin-to-vermouth ratio was 2:1.\n\nWhy don't you slip out of that wet coat and into a dry martini? I'd offer you a whiskey sour, but that would mean thinking up a new joke.\n\n\u2014Robert Benchley to Ginger Rogers in the 1942 motion picture The Major and the Minor\n\nThere is something about a Martini,\n\nEre the dining and dancing begin,\n\nAnd to tell you the truth,\n\nIt is not the vermouth\u2014\n\nI think that perhaps it's the gin.\n\n\u2014Ogden Nash, A Drink with Something in It\n\nMary Pickford\n\nMary Pickford and Douglas Fairbanks, a power couple and the biggest stars of the silent film era, were also longtime residents of the Algonquin Hotel and close personal friends of general manager Frank Case. Robin Hood opened in 1922, and the couple rented a suite at the hotel when the film premiered in New York. Fairbanks brought along his bow and arrow and\u2014to the delight of newspaper reporters and shock of his wife and Case\u2014showed off his shooting skills on the hotel roof. Pickford, America's first sweetheart, made almost 250 films from 1908 to 1935. The drink that bears her name was a sensation from coast to coast and in Europe, but this recipe comes from Havana's Bar La Florida 1935 cocktail guide.\n\ncocktail glass\n\n1 ounce Bacardi rum\n\n1 ounce pineapple juice\n\n1 teaspoon grenadine\n\n6 drops maraschino liqueur\n\nMaraschino cherry\n\nThis drink must be very cold to be good, so shake liquid ingredients well over enough cracked ice to fill three quarters of the shaker. Strain into a chilled cocktail glass and garnish with the cherry. When Gourmet wrote about this drink in 1943, it was already called an \"old favorite.\"\n\nMary Pickford and Douglas Fairbanks married in 1920, a year after they founded the United Artists film studio in partnership with Charlie Chaplin and D. W. Griffith. \u25c6 \u25c6 \u25c6\n\n\"Mary Pickford has three good ears,\" Frank Case wrote. \"She can sit at the table, hear what the man to her right is saying, follow the conversation at the left, and when you, at the extreme other end, try to sneak in a remark you don't want her to hear, she will call the entire length of the table, 'I'll see you later about that.'\"\n\nA Toast to the Algonquin Round Table\n\nWhen publicists Murdock Pemberton and John Peter Toohey took New York Times drama critic Alexander Woollcott to lunch at the Algonquin Hotel, they just as easily could have opted for someplace else. The Astor Hotel lay closer to the Times offices on West Forty-third Street, and the Knickerbocker was fancier\u2014but they picked the Algonquin . . . which, granted, stood across the street from where they worked, the now demolished Hippodrome Theater. The Algonquin also had a pastry chef on staff named Sarah Victor, and Woollcott had a sweet tooth.\n\nThe publicists wanted to get their new show into the newspaper. Woollcott had just returned from France, where he'd spent the previous eighteen months in an ill-fitting army uniform. By June 1919 most of the veterans had resumed their old jobs, and Woollcott delighted in telling war stories in the restaurant. Pemberton and Toohey struck out with getting press for their show, but they did hatch a plan. Going back to their office and with the help of another publicity genius, William B. Murray, they concocted a welcome-home luncheon for the rotund critic. They invited writers, editors, publicists, and actors. The group went to the Pergola Room (today the Oak Room) at the Algonquin and had a roaring good time. Woollcott, who always loved being the center of attention, was in his element. Jokes, stories, and laughs filled the room. As the friends left the hotel to go back to work, someone asked: Why not do this again?\n\nFor the next six or seven years, about six times a week, the Vicious Circle met at the Algonquin Hotel, managed by Frank Case, who lived on the top floor and was raising two teenagers. He discerningly saw the value of so many famous New Yorkers coming to the restaurant each day, so he moved the group to the main restaurant on the north end of the lobby. Case set up a large round table that could seat approximately sixteen. Some thirty members of the Round Table came and went over the years, and eleven of them appear in the 2002 painting by Natalie Ascencios that hangs in the hotel today.\n\nThe writers Franklin P. Adams, Robert Benchley, Heywood Broun, Marc Connelly, Edna Ferber, Jane Grant, Ruth Hale, George S. Kaufman, Margaret Leech, Herman J. Mankiewicz, Dorothy Parker, Robert E. Sherwood, Laurence Stallings, Donald Ogden Stewart, and Frank Sullivan joined editors Beatrice B. Kaufman, Harold Ross, and Arthur H. Samuels. Poet John V. A. Weaver married one of the actresses of the group, Broadway star Peggy Wood. Margalo Gillmore, who starred in early Eugene O'Neill plays, was another actress at the table. Harpo Marx joined in 1924, leaving his brothers out.\n\nNeysa McMein, the most sought-after magazine cover artist of the era, hosted the group at her studio. Broadway producer Brock Pemberton, who later cofounded the Tony Awards, was invited by his younger brother, Murdock. John Peter Toohey was another press agent at the table, as were William B. Murray and David A. Wallace, and among the most popular members was composer and music critic Deems Taylor.\n\nThe group swiftly became the most-remembered gathering of literary figures of the twentieth century. But why do we remember them at all? Most wrote for daily newspapers, and what they said or talked about appeared very quickly, courtesy of their fellow columnists. Adams, best known by his initials, F.P.A., helmed a daily column that relied on contributions. When he ran poems and stories by the group, attention followed immediately. Wire services picked up the wisecracks and repartee from the columnists, and their words appeared in hundreds of newspapers nationwide. Herewith some of the best:\n\n\"Don't think I'm not incoherent.\" \u2014Harold Ross\n\n\"Repartee is what you wish you'd said.\" \u2014Heywood Broun\n\n\"Being an old maid is like death by drowning, a really delightful sensation after you cease to struggle.\" \u2014Edna Ferber\n\n\"When I was born I owed twelve dollars.\" \u2014George S. Kaufman\n\n\"My friends will tell you that Woollcott is a nasty old snipe. Don't believe them. [They] are a pack of simps who move their lips when they read.\" \u2014Aleck Woollcott\n\n\"It took me 15 years to discover that I had no talent for writing, but I couldn't give it up because by that time I was too famous.\" \u2014Robert Benchley\n\nThe Round Table also gave us The New Yorker, dreamed up at the hotel by Harold Ross and Jane Grant. The couple launched the magazine in February 1925, using their Round Table connections and listing the magazine's \"editorial board\" as Adams, Benchley, Mrs. Parker, and others, when no such group existed. Mrs. Parker wrote drama reviews for the debut issue and sold fiction to it until the 1950s, and most of the group contributed to it in some way.\n\nThe Vicious Circle wrote best-selling books and hit plays. Some of them worked in silent films, radio, talking pictures, and even early television. They left their collective mark on all areas of mass media in ways both large and small. Mrs. Parker takes credit for coining \"What the hell?\" while Kaufman and Connelly, needing an imaginary product in their play, dreamed up the \"widget.\"\n\nThe Algonquin remains the most renowned literary hotel in New York City for good reason. At the Plaza lived a children's book character; the Chelsea rented rooms to writers of all stripes. But the Algonquin offered a place for collaboration, admiration, and networking that persists today. Walk into the lobby, and you won't know who they are, but editors and writers are sitting in those chairs.\n\nMatilda\n\nAlgonquin Hotel, New York\n\nThe tradition of a lobby cat at the Algonquin Hotel began in 1930, according to hotel lore. At the time, Frank Case owned the hotel, and legendary actor John Barrymore was a longtime and favorite resident. Barrymore's sister, Ethel, kept a suite in the hotel, and their uncle, actor John Drew, lived in the hotel for nearly twenty years. According to the story passed down by the staff, one day Case adopted a stray. Barrymore told Case that since the Algonquin had a literary and theatrical clientele, he couldn't give the feline a common name, so Case called him Hamlet, after Barrymore's most famous stage role. Since then, all male cats that live at the hotel have been named Hamlet; years later a female was acquired and christened Matilda (nobody knows why). Since the 1990s a succession of Matildas has ruled the Algonquin. The hotel has promoted the cat to executive status; she has her own e-mail account, Twitter account, and gets fan mail. You can usually spot Matilda on the front desk, behind the warm desktop computer, or on a luggage cart. Every August the hotel throws a birthday party for the cat, raising money for animal adoptions in front of the building. The breed, ragdoll, means she is quite content to be petted. The Matilda cocktail has been a favorite in the Blue Bar for many years.\n\ncocktail glass\n\n2 ounces vodka\n\n2 ounces Cointreau\n\n1 ounce orange juice\n\n1 ounce lime juice\n\nChampagne\n\nOrange slice\n\nShake vodka, Cointreau, and juices well over ice; pour into a cocktail glass. Top with Champagne and garnish with the orange slice.\nMetropolitan\n\nThe long-vanished Metropolitan Hotel opened at the corner of Broadway and Prince Street in 1853 in what is today Manhattan's SoHo neighborhood. The six-story building in the Italian palazzo style, the city's second luxury hotel, boasted hot and cold running water in its 500 rooms. In its ballroom entertainer P. T. Barnum hosted the 1863 wedding reception for General Tom Thumb and Lavinia Warren, who stood two feet, eight inches tall. The Modern Bartender's Guide, written by O. H. Byron and published in 1884, included a brandy and vermouth Metropolitan cocktail. It is sometimes attributed to the city's most famous bartender, \"Professor\" Jerry Thomas, hired by the hotel after the Civil War. The Metropolitan was demolished in 1895, but its namesake cocktail lives on.\n\nold-fashioned glass\n\n1\u00bd ounces brandy\n\n1\u00bd ounces sweet vermouth\n\n\u00bd teaspoon simple syrup\n\n1\u20132 dashes Angostura bitters\n\nMaraschino cherry\n\nShake liquid ingredients over cracked ice; strain into an old-fashioned glass filled with ice cubes. Garnish with the cherry.\n\nNote: Don't confuse this drink with any modern recipes that share its name, including the 1990s variation on a Cosmopolitan.\nMillionaire\n\nDid anyone write more about money than Dorothy Parker? \"I hate almost all rich people but I think I'd be darling at it,\" she allegedly uttered. \"If you want to know what God thinks of money, look at who he gave it to,\" is another gem often attributed to her. She didn't grow up with it; she was born Dorothy Rothschild but famously said, \"We didn't know those Rothschilds.\" Her father was in the rag trade, making suits and cloaks in what is now SoHo. She didn't marry for money, either. In both of her unions, she earned more than her spouse did. When she had money, she spent it extravagantly on hats, fur coats, and expensive perfume. Mrs. Parker sailed to Europe in style, once booking passage on the grandest steamship of all, the French Line's Normandie. She might have enjoyed drinking a Millionaire\u2014even if she never became one. Harry Craddock captured two different recipes for the Millionaire in his 1930 collection, The Savoy Cocktail Book. Be a multimillionaire and try both variations.\n\nold-fashioned glass\n\nMILLIONAIRE #1\n\n1 ounce sloe gin\n\n1 ounce apricot brandy\n\n1 ounce Jamaica rum\n\nJuice of 1 lime\n\n1 dash grenadine\n\nMILLIONAIRE #2\n\n1\u00bd ounces gin\n\n1\u00bd ounces absinthe\n\n1 teaspoon triple sec\n\n\u00bc teaspoon grenadine\n\n1 egg white\n\nFor the first version, Shake all ingredients over ice; strain into a chilled cocktail glass.\n\nfor the second version, Vigorously shake all ingredients over cracked ice. Strain into a chilled cocktail glass or an old-fashioned glass filled with ice cubes.\n\nNote: The original recipe for the second version called for absinthe, which can be difficult to find. You can substitute with Pernod or another anise-flavored liqueur. Another variation substitutes bourbon for the gin.\nMint Julep\n\nWriting of the mint he put in his julep, Irvin S. Cobb said, \"Like a woman's heart it gives its sweetest aroma when bruised.\" \u25c6 \u25c6 \u25c6\n\nA friend of the Algonquin Round Table but never a member, author and newspaperman Irvin S. Cobb boisterously commanded the American literary scene for two generations. H. L. Mencken compared him to Mark Twain, and Cobb's sixty books were best sellers. Born and raised in Paducah, Kentucky, Cobb was so admired that in his lifetime a hotel and bridge were named for him. (The hotel is long gone, but the 1929 span from Paducah across the Ohio River to Brookport, Illinois, is still in use.) A Louisville distillery tapped the humorist in 1936 to produce Irvin S. Cobb's Own Recipe Book. In it he waxed poetic about his home state's most famous mixed drink: the Julep. He explores its Kentucky creation as carefully as a heart surgeon explaining how to operate on a patient. \"So great has been the argument on this subject that often the controversy could only be solved by recourse to pistols at dawn,\" Cobb wrote, and he believed that the addition of nutmeg prompted Kentucky to join the Civil War: \"It was brought on by some Yankee coming down South and putting nutmeg in the Julep. So our folks just up and left the Union flat.\"\n\nold-fashioned glass\n\n12 sprigs fresh mint\n\nPowdered sugar\n\n3 ounces Kentucky whiskey\n\nPlace 10 mint sprigs in a bowl, cover them with powdered sugar and just enough water to dissolve the sugar, and crush with a wooden pestle. Pour half the crushed mint and liquid into an old-fashioned glass or in a sterling silver or lead-free pewter tankard. Fill glass half full of finely crushed ice. Add the rest of the crushed mint and liquid and fill remainder of the glass with crushed ice. Add whiskey to the brim. Refrigerate for at least 1 hour\u2014but ideally 2 or 3 hours if you can wait that long. Dust remaining sprigs of mint with powdered sugar, place as garnish, and serve.\n\nNote: According to Cobb, \"The majority of Kentuckians, the folk of Chicago, the middle and far west, Texans, Missourians and Louisianans swear by holy Bourbon, but all the deft technicians, wheresoever found, agree that the liquor must be old, mellow whiskey\u2014the blandest in its savor, the richest in its perfume, the most lingering in its softly-expiring after-aroma.\"\n\nOne group holds that the bruised mint should be left in the potion. But my grandfather always insisted that a man who would let the crushed leaves and the mangled stemlets steep in the finished decoration would put scorpions in a baby's bed.\n\n\u2014Irvin S. Cobb's Own Recipe Book\nMonkey Gland\n\nIt's an understatement to say that Dorothy Parker disliked flappers, sheiks, and youth culture. \"I hate the Younger Set. They harden my arteries,\" she wrote in one of her hymns of hate.\n\nThere are the Male Flappers;\n\nThe Usual Dancing Men.\n\nThey can drink one straight Orange Pekoe after another,\n\nAnd you'd never know that they had had a thing.\n\nFour d\u00e9butante parties a night is bogie for them,\n\nAnd their talk is very small indeed.\n\nIn the Jazz Age many adults suddenly acted like teenagers, and one of the legacies of that shift is a cocktail with an odorous name that hints at a gruesome practice. In the 1920s, Harry MacElhone, proprietor of Harry's New York Bar in Paris, took note of the spirit of the age and named this drink for a Russian nutcase, Serge Voronoff, who infamously transplanted tissue from the sexual organs of monkeys into humans to slow the aging process. This is the recipe used in London's Caf\u00e9 Royal and Hotel Savoy in the 1920s.\n\ncocktail glass\n\n1\u00bd ounces dry gin\n\n1\u00bd ounces fresh orange juice\n\n3 dashes absinthe\n\n3 dashes grenadine\n\nShake all ingredients and strain into a chilled cocktail glass.\n\nNote: If you can't find absinthe, substitute Pernod or B\u00e9n\u00e9dictine, a Cognac-based liqueur from Normandy.\n\nA young woman models flapper fashion from head to toe in 1922. \u25c6 \u25c6 \u25c6\nNew York Cocktail\n\nMany exult E. B. White's 1950 paean to Gotham, Here Is New York, as the ultimate love letter to the city. But \"My Hometown,\" a long-overlooked article by Dorothy Parker published in McCall's in 1928, is equally moving and touching. It's also the best statement Mrs. Parker ever wrote about why she loved the city. In fewer than one thousand words she masterfully distills its majesty. The New York Cocktail was a favorite in speakeasies that served rye and bourbon. This recipe dates from 1930.\n\n2 ounces rye whiskey or bourbon\n\n\u00be ounce fresh lime or lemon juice\n\n2 dashes grenadine\n\n1 teaspoon powdered sugar\n\nOrange peel\n\nShake liquid ingredients and sugar over cracked ice, strain into a chilled old-fashioned glass, and garnish with an orange twist.\n\nLondon is satisfied, Paris is resigned, but New York is always hopeful. Always it believes that something particularly good is about to come off, and it must hurry to meet it. There is excitement running in its streets. Each day, as you go out, you feel the little nervous quiver that is yours when you sit in a theater just before the curtain rises. Other places may give you a sweet and soothing sense of level; but in New York there is always the feeling of \"Something's going to happen.\" It isn't peace. But, you know, you do get used to peace, and so quickly. And you never get used to New York.\n\n\u2014Dorothy Parker, \"My Hometown,\" 1928\nOld-Fashioned\n\nIn Dorothy Parker's Oscar-nominated screenplay for Smash-Up (1947), cowritten with Frank Cavett, former gin joint chanteuse Angelica Evans (played by Susan Hayward) announces: \"I'm going to have a little drink. Make me something . . . Come on, we'll both have one. An Old-Fashioned. Only no sugar, no vegetables, and go light on the ice. Why corrupt good liquor?\" The Old-Fashioned is one of the oldest of all American cocktails, and it has earned its name by being more than 200 years old. This whiskey cocktail has been reinterpreted many times, but this is the traditional recipe from the speakeasy era.\n\nold-fashioned glass\n\n1 sugar cube or 2 teaspoons simple syrup\n\n1\u20132 dashes Angostura bitters\n\n1 teaspoon water\n\n2 ounces whiskey\n\nLemon peel\n\nOrange slice\n\nMaraschino cherry\n\nIn an old-fashioned glass, muddle the sugar cube or simple syrup, bitters, and water until sugar dissolves (if using a cube). Fill glass with ice cubes and stir in whiskey and lemon twist with a spoon. Garnish with the orange slice and cherry.\n\nSusan Hayward received an Academy Award nomination for Best Actress for her role as an alcoholic lounge singer in Smash-Up: The Story of a Woman. Dorothy Parker was also nominated for the screenplay. \u25c6 \u25c6 \u25c6\nOrange Blossom\n\nRobert Benchley was Dorothy Parker's best friend and a charter member of the Vicious Circle. He was also a teetotaler who made it through four years at Harvard and then seven years in newspaper and magazine publishing before touching a drop of alcohol. Benchley then fell into the bottle and never emerged. Along with Mrs. Parker and fellow writers Donald Ogden Stewart and Marc Connelly, Benchley frequented Tony's, a speakeasy on West Forty-ninth Street. Proprietor Tony Soma, formerly of the Hotel Knickerbocker, liked to stand on his head to delight customers. The story goes that a Tony's bartender once asked Mrs. Parker what she was having. \"Not much fun,\" came her quick reply. Benchley didn't start drinking until his thirties. \"The first social drink he took was an Orange Blossom,\" his son, Nathaniel, recounted in a biography of his father. \"He tried one sip, then put the glass down and looked around the room, 'This place ought to be closed by law,' he said and everybody fell off their chairs with laughter.\" This recipe for an Orange Blossom comes from the Robert Benchley Society.\n\ncocktail glass\n\n1 ounce gin\n\n1 ounce fresh orange juice\n\n1 teaspoon powdered sugar\n\nOrange peel\n\nShake gin, orange juice, and sugar over ice; strain into a cocktail glass. Garnish with flamed orange peel.\n\nNote: This cocktail has also been called an Adirondack Special, because of the homemade gin made in the region, as well as the Florida, due to the orange juice.\n\nA 1912 graduate of Harvard, Robert Benchley wrote for magazines and newspapers before becoming a radio and motion picture star. \u25c6 \u25c6 \u25c6\n\nThis \"old-wives\" superstition that a cup of black coffee will \"put you on your feet\" with a hangover is either propaganda by the coffee people or the work of dilettante drinkers who get giddy on cooking-sherry. A man with a real hangover is in no mood to be told \"Just take a cup of black coffee\" or \"The thing for you is a couple of Aspirin.\" A real hangover is nothing to try out family remedies on. The only cure for a real hangover is death.\n\n\u2014Robert Benchley, My Ten Years in a Quandary and How They Grew (1936)\nRay Hitchcocktail\n\nFor five solid years, from 1918 to 1923, Dorothy Parker wrote theater criticism for Vanity Fair and Ainslee's. The Hotel Wallick in Times Square named a drink for one of her favorite stage comedians, Raymond Hitchcock, who appeared in shows with W. C. Fields, Fanny Brice, and Mary Eaton. He toured with the Ziegfeld Follies before moving into silent films, and Mrs. Parker always gave him a glowing review. In 1918 he was in a show called Hitchy-Koo, about which Mrs. Parker wrote,\n\nNo matter what the show may be, if Raymond Hitchcock is in it, it's a success. He runs at top form from the moment he clubbily makes his appearance in the orchestra, welcoming the incoming audience, to the final thud of the curtain . . . Miss Ray Dooley gathers in a large share of the laughs of the evening, particularly in one delicate scene where she plays an obstreperous baby whom Mr. Hitchcock, as the proud father, silences with a blackjack . . . even though I do wear shell-rimmed glasses on occasion; there is always something about a gentleman hitting a lady over the head with a stuffed club that causes me to rock with happy laughter.\n\nA clever bartender honored Raymond Hitchcock with the cleverly named Ray Hitchcocktail, so remember Old Broadway with this one.\n\ncocktail glass\n\n1 pineapple slice\n\nJuice of half an orange\n\n2 ounces sweet vermouth\n\nDash Angostura orange bitters\n\nIn a shaker, muddle the pineapple and orange juice. Add the vermouth. Shake well with cracked ice and strain into a cocktail glass. Add dash of bitters.\n\nRaymond Hitchcock was a star on Broadway and in silent pictures. \u25c6 \u25c6 \u25c6\n\nThe Rob Roy was invented here, the old Waldorf-Astoria Hotel. Demolition started in October 1929 and lasted twelve weeks. The 16,000 truckloads of debris were dumped in the ocean off Sandy Hook, New Jersey. The Empire State Building rose in its place. \u25c6 \u25c6 \u25c6\nRob Roy\n\nThe Herald Square Theatre, a 1,300-seat playhouse on the corner of Broadway and West Thirty-fifth Street, opened in 1883 and showcased musicals and light operas for a quarter century. In 1894 Rob Roy opened there, based on the life of Scottish outlaw and folk hero Rob Roy MacGregor. With music composed by Reginald De Koven and a book by Harry B. Smith, it was a smash success. The New York Times called the operetta \"clean, frank, manly, bright, and winsome\" and praised star William Pruette. At the time, the old Waldorf Hotel\u2014its bar and restaurant a famous destination for well-heeled theatergoers\u2014lay just a short walk east, where the Empire State Building stands today. Bartenders at the hotel honored the cast and show by naming a Scotch cocktail for the production. The Herald Square was converted to the city's first movie theater in 1911. The theater was razed four years later and the hotel demolished in 1930, but the Rob Roy has thrived for more than a century. Try one on November 30, St. Andrew's Day. This recipe appeared in the 1937 Caf\u00e9 Royal Cocktail Book.\n\ncocktail glass\n\n2\u00bd ounces Scotch\n\n\u00be ounce sweet vermouth\n\n\u00be ounce dry vermouth\n\n3 dashes Angostura bitters\n\nMaraschino cherry\n\nShake the Scotch, both vermouths, and bitters over cracked ice. Strain into a chilled cocktail glass and garnish with the cherry.\nSeersucker Punch\n\nThe Velveteen Rabbit, Las Vegas\n\nLas Vegas never hosted Dorothy Parker at its gaming tables, but she did love playing card games. A bridge fiend, she wrote about that game often, and other members of the Round Table were degenerate poker players: Heywood Broun lost his house at the table, and Harold Ross was gutted of $30,000 in one night. In that spirit, we turn to Bryn Esplin, creative director at The Velveteen Rabbit, a lounge in the Las Vegas Arts District. She created Seersucker Punch. \"The word originates from the Persian shir o shekar, meaning 'milk and sugar,' probably from the resemblance of its smooth and rough stripes to the smooth texture of milk and the bumpy texture of sugar,\" Esplin said in Zelda magazine.\n\nWhen it was first introduced in the United States, the seersucker suit was considered to be a poor man's suit. It wasn't until Princeton students in the 1920s, in an ironic fashion statement, began to wear the fabric, subverting the economic associations. Hollywood actors were spotted in seersucker suits, and Life deemed it acceptable to wear in the northern part of the U.S. Writers would proudly wear it in beach settings, and dapper men made it their new go-to summer suit.\n\nFollowing its tradition of breaking boundaries, Seersucker Punch can be served either hot or cold.\n\nlowball glass\n\n1\u00bd ounces bourbon\n\n\u00bd ounce Frangelico\n\n\u00bd ounce root liquor\n\n1 teaspoon brown sugar simple syrup\n\n\u00bd ounce almond milk\n\nDash of cinnamon\n\nStir all the liquid ingredients together, pour over shaved ice in a lowball or punch glass, then add the cinnamon. To serve the punch hot, heat to taste, stir, and top with a dash of cinnamon.\nSidecar\n\nThe Lost Generation that swept into France following the Great War included many American expats. Dorothy Parker joined them, making her first journey in 1926. There she befriended Gerald and Sara Murphy, while socializing with F. Scott Fitzgerald, John Dos Passos, and Ernest Hemingway. She did almost no work in Europe. Instead Mrs. Parker spent her vacations at luxury hotels, cafes, and\u2014a world away from Long Branch\u2014the beaches of the C\u00f4te d'Azur. Along with the French \"75,\" the Sidecar also came back from World War I to America. Cocktail enthusiasts believe it was created in Paris at Harry's New York Bar for a regular customer\u2014a military officer of some stripe\u2014who was usually delivered to the establishment via motorcycle (with sidecar attached). It was popular during Prohibition but later lost its footing to new vodka and rum drinks. This recipe comes from the 1937 Caf\u00e9 Royal Cocktail Book compiled by London bartender William Tarling.\n\ncocktail glass\n\n1 ounce brandy\n\n\u00be ounce Cointreau\n\n\u00be ounce fresh lemon juice\n\nShake all ingredients over crushed ice; strain into a cocktail glass.\n\nNote: The Cointreau adds all the pop to this drink, so the brandy doesn't need to be an expensive or premium brand.\n\nReginald Vanderbilt was a millionaire dilettante who didn't accomplish much in his life except purchasing race cars and racehorses. He mixed Stingers for his friends in Newport and New York. \u25c6 \u25c6 \u25c6\nStinger\n\nThe Stinger first came into fashion during the Gay Nineties for upper-class New Yorkers, when the chief ingredient was always brandy. Cognac and other spirits were substituted later. It always had a reputation as a rich-person's cocktail and was the preferred drink of W. Somerset Maugham, the sophisticated English author and playwright who wrote a tender introduction to The Portable Dorothy Parker in 1944. The Stinger was also a favorite of railroad scion Reginald Vanderbilt (father of Gloria, grandfather of Anderson Cooper). Reggie Vanderbilt had a home bar and mixed his own Stingers every afternoon for friends. In the 1960s, Gloria Vanderbilt and her fourth husband, Wyatt Cooper, took Mrs. Parker under their wing; some of the last parties that Mrs. Parker attended were with the glamorous society couple\u2014maybe having Stingers.\n\ncocktail glass\n\n2\u00bc ounces Cognac\n\n\u00be ounce white cr\u00e8me de menthe\n\nStir the brandy and cr\u00e8me de menthe with ice; strain and serve in a chilled cocktail glass.\n\nNote: You can replace the cognac with brandy, gin, or bourbon\u2014but white cr\u00e8me de menthe is key. If you order this drink, don't let the bartender reach for a bottle of the green stuff.\nStone Fence\n\nSmash-Up earned two Academy Award nominations in 1947: for best actress in a leading role, to Susan Hayward for playing a boozy nightclub singer, and for screenwriting team Dorothy Parker and Frank Cavett. The Stone Fence dates to the Colonial period and was made by farmers in the winter using hard cider. This is the most common recipe, which uses brandy, just like Angie drinks it onscreen, but substitutes apple cider for her rye.\n\nold-fashioned glass\n\n2 ounces brandy\n\n2 dashes Angostura bitters\n\n\u00bd ounce lemon juice\n\nClub soda or fresh apple cider\n\nPour brandy and bitters into a chilled old-fashioned glass filled with cracked ice. Add lemon juice. Top with club soda or apple cider. Stir gently.\n\nNote: If you prefer, you can replace the brandy with applejack, bourbon, rum, rye, or Scotch, and the cider with club soda.\n\nIn one of the best scenes in Smash-Up, Susan Hayward's character, Angelica Evans, walks behind the bar in her Sutton Place apartment and grabs a bottle of brandy. The dialogue is pure Parker.\n\nAngie: Hey, you know what a Stone Fence is? (smiling)\n\nMike: You mean a Stone Wall.\n\nAngie: I mean a Stone Fence, brother. It's sort of like an ice cream soda, with conviction. Bartender, will you please get me a cocktail shaker with some shaved ice, and some brandy, and some whiskey, and some Cointreau? What you need, Mike, is a Stone Fence. Just about the most colossal drink you've ever drunk. Drank. It puts poise in apathetic people, if you know what I mean. And after the second one your spine turns to solid platinum. You take one part brandy and two parts rye . . . (she drinks a jigger of rye).\nSure Shot\n\nDeath & Company, New York City\n\nThe Martinez of the 1880s was super sweet and called for a whopping amount of vermouth. It gradually morphed into today's Martini (depending on which history book you believe). In bars across the country, the fashion of reinventing classic drinks has taken hold. Hidden behind a heavy wooden door on Manhattan's Lower East Side you'll find Death & Company, a relative newcomer but a go-to spot for sampling updates to century-old cocktail recipes. \"The Sure Shot is a classic Martinez variation,\" says Jillian Vose, head bartender. \"The Martinez is one of my favorite drinks. When making it, I love using a split base of different Old Tom gins so that it isn't too sweet. So I thought, Why not do a split with Old Tom and Genever? I love me some Bols Genever! We were messing around doing a workshop for a bunch of Bols brands, and we had picked up a bunch of herbs and spices. Ancho chile was one of them, and it all spiraled to deliciousness from there.\"\n\nNick & Nora glass\n\n1\u00bd ounces Hayman's Old Tom Gin\n\n\u00bd ounce Bols Genever\n\n\u00be ounce Ancho Chile\u2013Infused Dolin Rouge Vermouth\n\n1 teaspoon Galliano Ristretto\n\n\u00bd teaspoon Demerara Syrup\n\n1 dash orange bitters\n\nPour all the ingredients into a mixing cup filled with ice and stir. Strain into a Nick & Nora glass. Do not garnish.\n\nNote: If you can't find a Nick & Nora glass, use a cocktail glass instead.\n\nA Guide to Speakeasy Slang\n\nSpeakeasy culture was born with the passage of the Eighteenth Amendment to the Constitution, and with it a whole new lexicon of slang sprang up. What has come down to us from that time forms a collection of English as colorful today as it was in 1920.\n\nAnything very good was the cat's pajamas, the cat's whiskers, or the bee's knees. To be called Four-O was to get the highest possible rating from your friends. An oil burner was a gum-chewing girl. If you were from the country, you were a Young Otis; traveling salesmen and out-of-towners were the Butter and Egg Man. A young woman carried mad money, the cab fare in her purse in case she got into a row with her sweetheart. Ritzy also came into vogue courtesy of playwright George S. Kaufman in 1921 in the New York Tribune:\n\n\"The Crock of Gold\"\n\nShe's got a Hickson model walk\n\nAnd a line of Ritzy talk\u2014\n\nSimple little Katie from old Ire-land\n\nA flapper who couldn't keep in the swim was a dud, a Dumb Dora, or a dumb bunny. A wallflower was a sinker, and, oh no, any flapper over thirty was deemed a flat tire. Guys who always picked up the tab were darbs; men who never went out with girls were tabbed Red Mike. A cake eater or cakie was self-indulgent and often decadent. A cakie who squired around a different girl every night was a snake.\n\nIf you were in the rumble seat of a 1926 Franklin Sport Coupe, you may have engaged in snugglepupping, which is the same as petting or spooning. Robert E. Sherwood may have won the Pulitzer Prize four times, but he gave us a very important expression in a 1931 play: \"I think it's time for me to announce that I'm not going to bed with you.\" Sherwood also gets credit for another famous two words: indecent proposal. Dorothy Parker\u2014first to coin \"what the hell?\" and \"ball of fire\"\u2014has three other great relationship phrases to her credit: \"to mess around,\" \"one-night stand,\" and \"daisy chain.\"\n\nThe toast \"Here's looking at you\" came into vogue in 1928 in a book by Joseph March:\n\nThe drinks were placed,\n\nGus withdrew.\n\n\"Well\u2014\" said Tony:\n\n\"Here's lookin' at you!\"\n\nDrinkers carried hip-flasks. To be intoxicated was to be jammed, but loaded didn't mean drunk; it meant someone had good information. Getting ossified needs no explanation, nor does to the gills. For the first time, hair of the dog, an alcoholic drink taken to alleviate the effects of alcoholic drink, was on tap. Maybe that drink was giggle-water?\n\nBootleggers also used fantastic jargon. \"On the slough\" was what happened when the law found a speakeasy and the proprietors had to lie low for a few days. Motherships were vessels used to transport illegal booze from port to port; smaller tugs and watercraft then ferried the cases from the motherships to dry land. Crooks had nicknames for the law: a harness bull for a uniformed policeman, while plainclothes cops were sleeves. Detectives were dicks (possibly derivative from the London slang, Richard), and cops were elbows, as the New York Times reported in 1924, \"Possibly because the elbow is prominent in the process of collaring.\"\n\nDonald Ogden Stewart, an early member of the Algonquin Round Table, dreamed up one of the best phrases about a big night out: \"To drink you under the table.\"\n\nIllustrator John Held Jr., captured the spirit of the Roaring Twenties. \u25c6 \u25c6 \u25c6\n\nA Helpful Guide to 1920s Slang\n\nand how: emphatic agreement\n\nbee's knees: something extraordinary, the ultimate\n\nbeef: complaint\n\nbeeswax: business\n\nbluenose: prude\n\nbreezer: convertible car\n\nbum's rush: ejection from an establishment by force\n\ncash: kiss\n\ncat's meow: something splendid or stylish\n\ndick: private investigator\n\nducky: very good\n\ngams: woman's legs\n\ngiggle water: alcohol\n\nglad rags: going-out clothes\n\nhair of the dog: alcoholic hangover drink\n\njoint: club, usually selling alcohol\n\npill: unlikable person\n\nwhoopee: good time\n\nAfter we threw on our glad rags and hopped in the breezer, we found a joint with giggle water that was just ducky. We were minding our own beeswax, making whoopee, when some bluenose with beef\u2014probably a dick\u2014gave us the bum's rush! What a pill. Anyway, some hair of the dog would be the cat's meow. And how!\n\nThree Miler\n\nThis drink, also known as the Three-Mile Limit, took its name from the distance offshore that bootleggers ferried hootch during Prohibition. It was believed the smugglers were in international waters\u2014incorrectly as it happens. For the Coast Guard, though, this was serious business. In New York waters, cutters enforcing the Volstead Act fired on pleasure boats that then sank, occasionally with loss of life. Smugglers used motherships past the three-mile limit, then transferred the cases of liquor to smaller and faster boats. Two of the most popular products were rum and brandy, both used in this Prohibition-era recipe.\n\ncocktail glass\n\n1\u00bd ounces Bacardi rum\n\n\u00be ounce brandy\n\n1 teaspoon grenadine\n\n1 dash fresh lemon juice\n\nLemon peel\n\nShake all liquid ingredients over ice; strain into a cocktail glass. Drop in a lemon twist.\n\nTimes Square Cocktail\n\nDorothy Parker spent the majority of her life within striking distance of Times Square. Except when writing screenplays in Hollywood or making a brief stab at domestic life in Pennsylvania, Mrs. Parker was never more than a short taxi trip to the crossroads of the world: where Broadway, Forty-second Street, and Seventh Avenue converge. As a child she lived fewer than thirty blocks north and accompanied her father and sister to Broadway shows on a constant basis. Her years as a dramatic critic brought her to the Times Square theaters five and six nights a week. Her office jobs at Vogue and Vanity Fair were a block away, as were the offices of The New Yorker, her publishers, and the office she rented with Robert Benchley. Mrs. Parker hobnobbed at the Astor Grill atop the Hotel Astor (Forty-fifth Street) and laughed at the Midnight Frolic on the roof of the New Amsterdam Theatre (Forty-second Street).\n\nBut if she came back today, she wouldn't recognize Times Square. Since her death in 1967, soaring office towers have replaced all four corners of \"The Deuce,\" most of the theaters have been razed, and nearly every former drinking establishment has become some chain restaurant or ho-hum retail space. The Times Square Cocktail evokes an era when streetcars were on Broadway\u2014not costumed characters from Sesame Street.\n\nwine glass\n\n1\u00bd ounces Southern Comfort\n\n1\u20443 ounce sweet vermouth\n\n\u00bd ounce grenadine\n\nChampagne\n\nOrange wheel\n\nShake Southern Comfort, vermouth, and grenadine over ice. Strain into an ice-filled wine glass. Add chilled Champagne to the brim and stir gently. Garnish with the orange wheel.\n\nThe New York Times gave Times Square its name; the former newspaper office is where the ball drops on New Year's Eve. \u25c6 \u25c6 \u25c6\nTippy Canoe\n\nThe Violet Hour, Chicago\n\nDorothy Parker and T. S. Eliot both wrote poetry in the 1920s, and both shared a bleak outlook on many subjects. The similarities end there. While Mrs. Parker sold her pieces to popular magazines, Eliot worked at an English bank. Mrs. Parker never penned a monumental poem, but in 1922 Eliot authored \"The Waste Land,\" in which he wrote:\n\nAt the violet hour, the evening hour that strives\n\nHomeward, and brings the sailor home from sea,\n\nThe typist home at teatime, clears her breakfast, lights\n\nHer stove, and lays out food in tins.\n\nA Chicago poetry fan plucked out \"the violet hour\" to christen a Wicker Park cocktail lounge, with a tip of a chilled glass to Eliot, born 300 miles away in St. Louis. The menu of The Violet Hour\u2014known for its fresh juices, homemade syrups, in-house bitters, and a dedication to minimalist cocktails\u2014changes seasonally, as does the fa\u00e7ade of the building. One drink on the menu is the Tippy Canoe, \"a riff on a classic daiquiri,\" according to bar manager Robert Haynes. \"When working with fewer ingredients, balance is extremely important. Bonal is a French gentian and quinine-forward aperitif. This one is simple and accessible.\"\n\ncoupe glass\n\n\u00be ounce Smith & Cross Jamaica Rum\n\n\u00be ounce Bonal aperitif wine\n\n\u00be ounce apricot syrup\n\n\u00be ounce fresh lime juice\n\n1 dash Angostura bitters\n\nLime wheel\n\nCombine all liquid ingredients in a shaker, add 5 ice cubes, and shake vigorously. Double strain into a coupe glass and garnish with the lime wheel.\n\nNote: Fresh lime juice is a must. Also, The Violet Hour serves the drink in a coupe, but you can use a cocktail glass.\n\nHow to Make Apricot Syrup the Chicago Way\n\nAccording to Robby Haynes of The Violet Hour, making your own apricot syrup is quick and easy. Take 13 ounces of apricot preserves (such as Bonne Maman) and add four cups of granulated sugar and one quart of warm water. Stir to incorporate. \"A stick mixer or handheld will make your life a little easier,\" Haynes advises. \"If not, a little elbow grease should do it.\" Strain through cheesecloth. It will keep for two weeks.\n\nThe Windy City and the Vicious Circle\n\nAfter New York City, Chicago has the strongest ties to the Algonquin Round Table. The dean of the group, Franklin P. Adams, was born there. The artist Neysa McMein attended art school in Chicago before moving to Manhattan and launching a brilliant career. Poet John V. A. Weaver, a native, worked for the Chicago Daily News, and it was there that H. L. Mencken discovered him. Dorothy Parker's biggest crush, the scoundrel Charles MacArthur, was a reporter for the City News Bureau of Chicago. However, the biggest link to the city comes via Edna Ferber, who penned Chicago's greatest novel, So Big, after living there in the early 1920s. Ferber's timeless story of truck farmers and the American dream won the Pulitzer Prize in 1925.\nTom and Jerry\n\nDuring the Great Depression, Seagram's took out splashy two-page ads in mass-market magazines. It was no doubt a copywriter's dream account. \"At the bar, at the club or at home\u2014Be Sure of Finer Taste.\" Each ad included a half a dozen simple recipes for the company's gins and whiskies for the home bartender, and with each the company exhorted: \"Think before you Drink . . . say Seagram's\u2014and be sure!\" This recipe ran in Life in 1937: \"A fine warming drink for 'after the game.'\" The Tom and Jerry takes its name from characters created by English writer Pierce Egan in Life in London, or Days and Nights of Jerry Hawthorne and his elegant friend Corinthian Tom (1823).\n\ncoffee mug\n\n1 egg\n\n1 tablespoon sugar\n\nDash of vanilla extract\n\n2 ounces whiskey or rum\n\nHot water\n\nSeparately beat white and yolk of the egg, then gently mix together. Stir in sugar and vanilla and pour mixture into a coffee mug. Add whiskey or rum, then fill with hot water.\n\nNote: The original recipe called for brandy.\nTom Collins\n\nThe convoluted history of this drink dates to nineteenth-century London. Returning doughboys brought their love of the Tom Collins back from Paris in 1918, and, with gin becoming one of the main bootleg liquors of Prohibition, the easy-to-make Tom Collins caught on. The drink was originally called a John Collins and appeared in Jerry Thomas's 1876 bartending guide. Served in a big glass, this \"gin punch\" proved popular at speakeasies, lawn parties, and summer resorts.\n\ncollins glass\n\nJuice of 1 lemon\n\n1 teaspoon powdered sugar\n\n1\u00bd ounces gin\n\nClub soda\n\nMaraschino cherry\n\nIn a tall 16-ounce Collins glass, place 3 or 4 ice cubes. Pour in the lemon juice and sugar, add the gin, then fill to the top with cold club soda. Stir and garnish with the cherry.\n\nNote: Replacing the gin with another potable changes the name of the drink: Pedro Collins (rum), Mike Collins (Irish whiskey), John Collins (bourbon). A Mojito is also a variation: Substitute good light rum for the gin and the juice of a lime instead of a lemon. Garnish with sprigs of mint.\nTwentieth Century\n\nYou can capture the romance of a prewar railroad journey in a glass when you mix up a Twentieth Century cocktail, named for the New York Central Railroad service between Manhattan and Chicago. Passengers traveled in luxury Pullman cars\u2014rolling drawing rooms, bedrooms, compartments, dining cars, lounges\u2014for the sixteen-hour trip. In 1938 the Twentieth Century Limited locomotives debuted a streamlined Art Deco look in two shades of gray with aluminum and blue stripes. Traveling those thousand miles made for a luxurious adventure lost on us today. (You can get a feeling for it, though, in Alfred Hitchcock's North by Northwest.) Trainspotting author and gourmand Lucius Beebe adored the Twentieth Century Limited, writing that he \"slept the sleep of the unjust, awash with Hennessy Three Star and Upmann Coronas.\" This cocktail replaces his beloved Cognac with gin and a French twist.\n\ncocktail glass\n\n1\u00bd ounces gin\n\n\u00be ounce Lillet Blanc\n\n\u00bd ounce white cr\u00e8me de cacao\n\n\u00be ounce fresh lemon juice\n\nLemon peel\n\nShake liquid ingredients over ice for 20 seconds, strain into a chilled cocktail glass, and garnish with a lemon twist.\n\nNote: Lillet Blanc is the key ingredient for this cocktail. A French aperitif, it should always be served chilled, never warmer than 46\u00b0F.\n\nTo celebrate the centenary of Grand Central Terminal, the 20th Century Limited returned to New York in 2013. \u25c6 \u25c6 \u25c6\nWard 8\n\nDorothy Parker and her friends had a rocky relationship with Boston, Massachusetts. Mrs. Parker's only arrest happened there in 1927, when she was protesting the planned execution of accused anarchists Nicola Sacco and Bartolomeo Vanzetti. Robert Benchley was born and raised in Worcester, forty-five miles away, and graduated from Harvard, across the Charles in Cambridge. Fellow Round Tablers Robert E. Sherwood and Heywood Broun were also at Harvard.\n\nOne of the city's finest cocktails also originated there. A variation on the classic whiskey sour, it was a natural shoo-in for the rye-loving Vicious Circle. Its history begins at the Locke-Ober restaurant in the Hub. Bartender Tom Hussion concocted the drink in 1898 to celebrate an Election Day victory by Democrat Martin Michael Lomasney, boss from the Eighth Ward, who was running for state office. As fate cruelly had it, though, Lomasney later supported Prohibition, and the Locke-Ober suffered. The restaurant reopened and thrived for many years before being shuttered again for good. The Ward 8 lives on, though, as Boston's most famous cocktail export.\n\ncocktail glass\n\n2 ounces rye whiskey or bourbon\n\n\u00bd ounce fresh lemon juice\n\n\u00bd ounce fresh orange juice\n\n1 teaspoon grenadine\n\nMaraschino cherry\n\nShake all liquid ingredients over cracked ice, strain into a cocktail glass, and garnish with the cherry.\nWhiskey Sour\n\nMagazine writer Joe Bryan claims that Dorothy Parker requested a dear little one of these\u2014one of her avowed favorites\u2014for breakfast. Drinks guru David Wondrich credits the popularity of this drink, an all-American cocktail that predates the Civil War, to being simple and flexible. \"From roughly the 1860s to the 1960s,\" he says, \"the Sour, particularly its whiskey incarnation, was one of the cardinal points of American drinking and, along with the Highball, one of the few drinks that could come near to slugging it out with the vast and aggressive tribe of cocktails in terms of day-in, day-out popularity.\" This recipe comes from a 1936 whiskey drinks guide published by Frankford Distilleries in Louisville, Kentucky.\n\nsour glass\n\n1\u00bd ounces American blended whiskey\n\n\u00bd teaspoon powdered sugar\n\nJuice of half a lemon\n\nSeltzer\n\nSlice of orange, pineapple stick, or red cherry (garnish)\n\nShake whiskey, sugar, and lemon juice over cracked ice; strain into a sour glass. Add a little seltzer water and decorate with fruit if desired.\n\nWill Rogers\n\nWill Rogers in Handy Andy (1935). \u25c6 \u25c6 \u25c6\n\nOne of the biggest Broadway stars of the pre\u2013World War I era was a laconic cowboy from Oklahoma. Will Rogers got his start onstage in 1912 at the Victoria Theatre in a Wild West show, delivering dry comedy lines alongside lariat tricks. Florenz Ziegfeld initially disliked having a cowboy in his refined productions, but the well-oiled crowd loved Rogers's gentle barbs at rich people in the audience. Rogers starred in the Midnight Frolic on the New Amsterdam rooftop shows and became a national star when he moved into the Follies. In a 1918 review, Dorothy Parker wrote, \"The life of the evening is Will Rogers, who, to me, is one of the Greatest Living Americans.\" Rogers went on to become a radio and movie star and a beloved newspaper columnist in hundreds of newspapers\u2014until he died with friend Wiley Post, who crashed their plane into an Alaskan lake. The Oklahoma Hall of Fame honors both men, and both have airports named after them, but of the two we drink only to Will Rogers.\n\ncocktail glass\n\n1\u00bd ounces gin\n\n\u00bd ounce dry vermouth\n\n\u00bd ounce fresh orange juice\n\n1\u00bd teaspoons triple sec\n\nShake all ingredients over ice; strain into a chilled cocktail glass.\n\nCommunism is like prohibition, it's a good idea but it won't work.\n\n\u2014The Autobiography of Will Rogers\nYale Cocktail\n\nDorothy Rothschild never made it past the tenth grade, but she talked herself into the Ivy League in a manner of speaking. One of her most notorious quips\u2014which she denied ever uttering\u2014followed her to the grave and is particularly loathed around New Haven. Alexander Woollcott, in the \"Our Mrs. Parker\" chapter of While Rome Burns, recounts \"that wholesale libel on a Yale prom. If all the girls attending it were laid end to end, Mrs. Parker said, she wouldn't be at all surprised.\" In 1924 she wrote, \"I hate College Boys; They get under my feet,\" adding:\n\nThey list all the d\u00e9butantes\n\nIn Grades A, B, and C,\n\nAnd proceed accordingly.\n\nOnce they get into their stride,\n\nThe Opposite Sex hasn't a prayer.\n\nDuring World War I, bartender Hugo R. Ensslin collected the Yale Cocktail into his slim collection of drink recipes.\n\ncocktail glass\n\n2 ounces dry gin\n\n1 ounce French vermouth\n\n1 teaspoon maraschino liqueur\n\n1 tablespoon simple syrup\n\n3 dashes orange bitters\n\nStir all ingredients in a mixing glass with cracked ice; strain into a chilled cocktail glass.\n\nNote: Other Ivy League recipes of the era are the Harvard Cocktail (brandy and bitters) and the Princeton Cocktail (a dry Martini with two dashes of lime juice).\nZiegfeld\n\nFlorenz Ziegfeld still ranks among the most brilliant showmen to light up Broadway. His productions glamorized American women, as no one else had, in glittering musicals that were the toast of the town for twenty-five years. His career began in 1893 in Chicago, before he went to Europe, where he found vaudeville acts to import back to America. He launched the Follies in 1907, and it became an annual institution at the New Amsterdam Theatre. Stars of the Follies became known as the most dazzling women in Manhattan. Dorothy Parker asked, \"Where the Ziegfeld girls come from will always be one of the world's great mysteries; certainly, one never sees any like them anywhere around.\" The show launched scores of major names, including Fanny Brice, Eddie Cantor, W. C. Fields, Mary Pickford, Will Rogers, Sophie Tucker, and Mae West. Ziegfeld's widow, Billie Burke, one of the most famous actresses of the 1920s, contributed this recipe to the Stork Club Bar Book in 1946.\n\ncocktail glass\n\n2 ounces gin\n\n1 ounce fresh pineapple juice\n\n\u00bc ounce simple syrup\n\nPineapple slice\n\nShake liquid ingredients over half a cup of ice cubes; strain into a chilled cocktail glass. Garnish with the pineapple slice.\n\nNote: The original recipe for this cocktail didn't include simple syrup but was probably made with sweetened pineapple juice. Simple syrup is also known as pure cane sugar syrup and open kettle molasses and has about sixty calories per tablespoon.\n\nA costume sketch by the great John Held Jr., for the Follies includes instructions for the designer. Invented in Italy in 1890, crosswords became a big fad in the 1920s. \u25c6 \u25c6 \u25c6\nZombie\n\nThe 1939 World's Fair in Flushing Meadows, Queens, was a sensation. With \"Building the World of Tomorrow\" as its theme, the fair pushed the premise that science and technology could provide economic prosperity and personal freedom. The fair also offered a ray of hope between two of the darkest events in the nation's history: the Great Depression and World War II. The fairgrounds stretched 3.5 miles long and in some places a mile wide, hosting approximately 26 million people the first season. To convey its message, the fair used icons, symbols, demonstrations and, best of all, exhibitions. The Trylon and the Perisphere\u2014a 700-foot-tall spire and globe as large as a city block\u2014became the fair's focal point and instant symbols of the machine age. The must-see Futurama exhibit at the General Motors pavilion presented a 36,000-square-foot model of a future America in 1960. Of the many theme restaurants, shops, and pavilions, the Hurricane Bar stood out by serving this drink. \"Why people drink them I don't know,\" according to Vic Bergeron (aka Trader Vic). \"Personally, I think they are too damn strong.\" Donn Beach of Hollywood's Don the Beachcomber fame (and who also invented the tiki bar) created the most beloved version of the Zombie. It's a lot of work to make\u2014and exceptionally potent.\n\n1 ounce white rum\n\n1 ounce golden rum\n\n1 ounce dark rum\n\n151-proof rum\n\n\u00bd ounce brandy (apple, cherry, or apricot)\n\n\u00be ounce fresh pineapple juice\n\n\u00be ounce fresh papaya juice\n\nJuice of 1 lime\n\nPineapple slice\n\nIn a shaker filled with cracked ice, shake all liquid ingredients except the 151-proof rum. Strain into a Collins glass filled with crushed ice, float a spoonful of 151 on top, and garnish with the pineapple slice.\n\nThe World's Fair of 1939\u201340 was one of the biggest events in New York City history.\u25c6 \u25c6 \u25c6\nPhoto and Illustration Credits\n\nPage Numbers Refer to the Printed Book\n\nAuthor's collection: xii, 5, 17, 22, 27, 29, 41, 54, 59, 61, 71, 73, 84, 91, 93, 107, 120, 123\n\nPhotos by the author: 7, 15, 128\n\nDon Spiro (donspiro.com): pp. xv\u2013xvi\n\nShutterstock.com: pp. xvii\u2013xviii\n\nPublic domain: p. 19\n\nCourtesy of Joan Grossman, Susan Cotton, and Nancy Arcaro: Cablegram, p. 25\n\nUS Library of Congress, Prints and Photographs Division: pp. 32, 34, 37, 45, 47, 48, 63, 77, 87, 95, 96, 100, 111, 125\n\nVincent Gong, p. 117\n\nDevon Quinn (author photo), p. 140\n\nDesign images licensed by Shutterstock.com. \nAcknowledgments\n\nThis book came together as easily as a Russian nesting doll for many reasons. To start off, I thank the friends I've met and made through the Dorothy Parker Society (DPS). When a few of us got together over drinks on Dorothy Parker's birthday in 1999, I had no idea I'd be running the busiest literary society in New York all these years later. As I often say, we're a drinking club with a book problem, and we've had monumental tabs at some of the finest drinking establishments in the city. Through the DPS, I met Diane Naegel and Don Spiro. They started Wit's End, a monthly party to celebrate vintage cocktails, hot jazz, and period attire. It was through them that I started drinking many of the recipes in this book. Don says they launched Wit's End because they wanted to go to a party like it, and after five years the fun continues. The DPS doesn't have meetings, it has parties, and the ones with Wit's End are always the greatest.\n\nI started the research for parts of the book more than ten years ago while writing the walking tours of the former haunts of Dorothy Parker and the Algonquin Round Table. This book gives me a chance to share with a larger audience a lot of the knowledge that I usually only impart as I walk backward in Times Square or the Upper West Side. I thank the Dorothy Parker researchers who went before me: Randall Calhoun (Dorothy Parker a Bio-Bibliography), Marion Meade (Dorothy Parker: What Fresh Hell Is This?), and Stuart Y. Silverstein (Not Much Fun: The Lost Poems of Dorothy Parker). Stuart's extensive notes on Mrs. Parker's life proved particularly helpful. In addition, the staff of the Algonquin Hotel, led by general manager Gary Budge, has always been supportive and taken a keen interest in the history of the hotel. Mr. Budge really does carry on the tradition of Frank Case, and it's a privilege for me to bring people to New York's best literary landmark.\n\nFor contributing modern-day speakeasy-style recipes, thanks go to: Alice de Almeida (Algonquin Hotel), H. Joseph Ehrmann (Elixir), Bryn Esplin (The Velveteen Rabbit), Chris Hannah (Arnaud's French 75), Robby Haynes (The Violet Hour), Barbara Jacobs (The Edison), Joanna Leban (Doc Holliday's Saloon), Kevin Martin (Eastern Standard Kitchen & Drinks), Herv\u00e9 Rousseau (Fl\u00fbte), and Jillian Vose (Death & Company). My goal is to visit each of the establishments I've not yet been to and have these bartenders serve me these drinks in person. I can't wait!\n\nIf I could thank and acknowledge all of my favorite bartenders in New York, my publisher would need to add many extra pages to this book. These are a few who are special to me, and whom I'll never forget for the great times we've shared together: Andrew Bennetch, Mark Evangelista, Michele Gascoigne, Adam Gerston, Tracy Helsing, Timmy Hever, Chaundra Hugel, Carmit Israeli, Sara Walter, and Laurieanne Williams. Thanks for all the buybacks.\n\nTwo people who have had a huge influence on my love of cocktails are my parents, Don and Val Fitzpatrick. I'll never forget watching them order Martinis in a cozy pub in Tipperary, Ireland. They were handed tiny glasses of vermouth\u2014while I had a fantastic pint of Guinness. My parents have had a Martini at six o'clock for most of their fifty years of marriage; on trips my mother packs her own Tanqueray, and it's my father's job to find the ice. At a family wedding one October, I ordered a gin and tonic, much to my mother's chagrin. \"That's a summer drink,\" she sniffed. Ever since, I've had whiskey highballs after Labor Day.\n\nExtra-special thanks to Christina, who supported me during the production of the book. My wife kept our rambunctious two-year-old busy as I plugged away measuring ounces and teaspoons of liquor. Christina is always supportive of my (numerous) side projects.\n\nSterling silver jiggers the author's parents have put to great use for forty years. \u25c6 \u25c6 \u25c6\n\nBig thanks to my editor, James Jayo, and all the good people at Lyons Press. The first time I met James, I put an Aviation in his hand, and we toasted to the success of this book. Thanks also to Allen Katz for writing the wonderful foreword. Allen gets my everlasting gratitude for naming a product after Mrs. Parker, and I believe the New York Distilling Company will get Dorothy Parker American Gin into all fifty states!\n\nFinally, I want to thank the legions of Dorothy Parker fans who take her to heart. You keep reading her stories and poems, producing shows based on her work, even recording your own music set to her words. The way to sustain an author's popularity after his or her death is to keep that legacy strong. Mrs. Parker now has a cocktail guidebook to go on the bookshelf with the biographies and collections of her work. Hopefully this book will draw even more new fans to learn about Mrs. Parker and lead some of them to her books.\nFurther Reading\n\nBooks\n\nBenchley, Nat and Kevin Fitzpatrick. The Lost Algonquin Round Table. New York: Donald Books, 2009.\n\nChirico, Rob. Field Guide to Cocktails. Philadelphia: Quirk Books, 2005.\n\nCraddock, Harry. The Savoy Cocktail Book. London: Pavilion Books, 1930, 2011.\n\nFitzpatrick, Kevin. A Journey into Dorothy Parker's New York. Berkeley, CA: Roaring Forties Press, 2013.\n\nMeade, Marion. Dorothy Parker: What Fresh Hell Is This? New York: Penguin Books, 2007.\n\nParker, Dorothy. Complete Stories. New York: Penguin Classics, 2002.\n\n\u2014\u2014\u2014\u2014\u2014. Complete Poems. New York: Penguin Classics, 2009.\n\n\u2014\u2014\u2014\u2014\u2014 and Kevin Fitzpatrick. Dorothy Parker: Complete Broadway 1918\u20131923. New York: Donald Books, 2014.\n\n\u2014\u2014\u2014\u2014\u2014 and Stuart Silverstein (editor). Not Much Fun: The Lost Poems of Dorothy Parker. New York: Scribner, 1996.\n\nTarling, W. J. The Caf\u00e9 Royal Cocktail Book. London: Pall Mall, 1937.\n\nWaggoner, Susan, and Robert Markel. Vintage Cocktails. New York: Stewart, Tabori & Chang, 2000.\n\nWondrich, David. Imbibe! New York: Perigree, 2007.\n\nWebsites\n\nThe Dorothy Parker Society: DorothyParker.com\n\nThe Algonquin Round Table: AlgonquinRoundTable.org\n\nThe Museum of the American Cocktail, New Orleans: MuseumOfTheAmericanCocktail.org\n\nSavoy Stomp: SavoyStomp.com\n\nTales of the Cocktail: TalesOfTheCocktail.com\nAbout the Author\n\nKevin C. Fitzpatrick is an independent historian who founded the Dorothy Parker Society in 1999. He is a graduate of Northeast Missouri State University and served in the U.S. Marine Corps. After his hitch he entered journalism and worked in newspapers, magazines, advertising agencies, and television in New York. He is a licensed New York City sightseeing guide and gives walking tours of city landmarks, literary sites, watering holes, cemeteries, and unusual locations. Fitzpatrick frequently is a guest speaker at libraries, salons, and private clubs. In his spare time, he enjoys genealogy, collecting comic books, and dedicating bronze plaques for dead people. He and his family divide their time between the Upper West Side and Shelter Island.\n\nAllen Katz is cofounder and vice president of the New York Distilling Company, which makes Dorothy Parker American Gin. The director of spirits education and mixology for Southern Wine & Spirits of New York, he also hosts The Cocktail Hour, a weekly program on Martha Stewart's SiriusXM channel. A past chairman of Slow Food USA, he serves on the board of directors for the New Orleans Culinary & Culture Preservation Society and the Manhattan Cocktail Classic, an annual four-day, four-borough celebration of the intersection between cocktails and culture. He and his family live in New York City.\nGeneral Index\n\nAdams, Fra\n\nnklin P. (F.P.A.), 26, 27, 33, 44, 65, 79\u201380, 112\n\nAlgonquin Hotel, xiii, 3, 6, 7, 8, 73, 76, 78, 80, 81\n\nArnaud's French 75, 23\n\nAscencios, Natalie, 79\n\nAstor Grill, 110\n\nAstor Hotel, 78, 110\n\nAstor IV, John Jacob, 63\n\nBaker, Charles H., 32\n\nBaker, Josephine, 62\n\nBar La Florida (Havana), 17, 39, 57, 76\n\nBarleycorn, John, 68\n\nBarnum, P. T., 70, 82\n\nBarrymore, Ethel, 14, 21, 81\n\nBarrymore, John, 81\n\nBeach, Donn (Don the Beachcomber), 124\n\nBeebe, Lucius, 116\n\nBelmont, Jr., August, 69\n\nBenchley, Nathaniel, 92\n\nBenchley, Robert, xiii\u2013xiv, 20, 33, 49, 51, 56, 64, 65, 74, 79\u201380, 92, 93, 110, 118\n\nBen\u00e9t, William Rose, 2\n\nBennett, James Gordon, 68\n\nBergeron, Vic (Trader Vic), 124\n\nBerlin, Irving, 14\n\nBlue Bar, The (Algonquin Hotel), 6, 73, 81\n\nBogart, Humphrey, 20\n\nBoston, Massachusetts, 42, 118\n\nBrice, Fanny, 94, 122\n\nBronx Zoo, 21\n\nBroun, Heywood, 24, 44, 79, 98, 118\n\nBroadway, xi, xiii, 12, 35, 44, 63, 70, 82, 94, 95, 97, 110, 120, 122\n\nBrookport, Illinois, 84\n\nBryan, Joe, 119\n\nBu\u00f1uel, Luis, 60\n\nBurke, Billie, 122\n\nButler, Smedley \"Duckboards\", 29\n\nByron, O. H., 82\n\nCaf\u00e9 Royal (London), 86\n\nCaf\u00e9 Royal Cocktail Book, 16, 97, 99\n\nCampbell, Alan, xiv, 16, 17, 53, 54, 61\n\nCantor, Eddie, 122\n\nCarpentier, Georges, 35\n\nCarson, Robert, 54\n\nCase, Frank, xiii, 6, 8, 76\u201378, 81\n\nCather, Willa, x\n\nCavett, Frank, 72, 90, 102\n\nChandler, Raymond, 50\n\nChaplin, Charlie, 77\n\nChelsea, The, 80\n\nChicago, Illinois, 26, 112, 122\n\nChirico, Rob, 59\n\nClub Intime, 70\n\nCobb, Irvin S., 44, 69, 84, 84, 85\n\nCohan, George M., 63\n\nConnelly, Marc, 79, 80, 92\n\nCoolidge, Calvin, x\n\nCooper, Anderson, 101\n\nCooper, Gary, 57\n\nCooper, Wyatt, 101\n\nCapote, Truman, 69\n\nCraddock, Harry, xx\n\nCulver City, California, 53\n\nDavis, George, 61\n\nDeath & Company (New York City), 104\n\nDe Koven, Reginald, 97\n\nDempsey, Jack, 34, 35\n\nDepression, the, 63, 72, 114, 124\n\nDetroit Athletic Club, 64\n\nDisney, Walt, 56\n\nDoc Holliday's Saloon (New York City), 10\n\nDolores (Kathleen Mary Rose), 36, 37\n\nDooley, Rae, 94\n\nDos Passos, John, 13, 99\n\nDouglas, Ann, 62\n\nDrew, John, 81\n\nEarhart, Amelia, 12\n\nEastern Standard Kitchen & Drinks (Boston), 42\n\nEaton, Mary, 94\n\nEdison, The (Los Angeles), 56\n\nEhrmann, H. Joseph, 40\n\nEliot, T. S., 112\n\nElixir (San Francisco), 40\n\nEmpire State Building, 96, 97\n\nEnsslin, Hugo R., 4, 12, 22, 26, 121\n\nEsplin, Bryn, 98\n\nFairbanks, Douglas, 76\u201377\n\nFerber, Edna, xiii, 65, 79, 112\n\nFields, W. C., 94, 122\n\nFirpo, Luis Angel, 35\n\nFitzgerald, F. Scott, xiii, 13, 14, 20, 38, 52, 99\n\nFitzgerald, Zelda, xiii, 38\n\nFlagg, James Montgomery, 35\n\nFlushing Meadows, Queens, New York, 124\n\nFl\u00fbte (New York), 70\n\nFogarty, Frank J., 64\n\nFoster, Albert T., 6\n\nFrankford Distilleries, 119\n\nFrench Quarter, 23\n\nGarbo, Greta, 20\n\nGibson, Charles Dana, 48, 49\n\nGillmore, Margalo, 79\n\nGimlette, T. O., 50\n\nGolinkin, Joseph, 71\n\nGovernors Island, 12\n\nGrant, Cary, 61, 61\n\nGrant, Jane, 4, 14, 21, 44, 46, 79\u201380\n\nGreat Gatsby, The, 38, 52\n\nGreat Neck, Long Island, 38\n\nGreenwich Village, 47\n\nGriffith, D. W., 77\n\nGuiles, Fred Lawrence, 53\n\nGuinan, Texas, 59, 59, 70, 71\n\nHale, Ruth, 24\n\nHammett, Dashiell, xviii\n\nHarlow, Jean, 61, 61\n\nHarry's New York Bar (Paris), 10, 86, 99\n\nHarvard University, 92, 93, 118, 121\n\nHavana, Cuba, 17, 32, 39, 57, 62, 76\n\nHaynes, Robert, 112, 113\n\nHayward, Susan, 90, 91, 102, 103\n\nHeld, Jr., John, 107, 123\n\nHemingway, Ernest, xiii, 20, 33, 51, 57, 60, 99\n\nHerald Square Theatre, 97\n\nHippodrome Theater, 78\n\nHitchcock, Ray, 94, 95\n\nHollywood, 18, 53\u201355, 110, 124\n\nHoover, Herbert, 31\n\nHotel Wallick (New York City), 12, 66, 94\n\nHussion, Tom, 118\n\nIrving, Washington, 63\n\nJacobs, Barbara, 56\n\nKatz, Allen, 3\n\nKaufman, Beatrice, 79\n\nKaufman, George S., xiii, 79\u201380, 105\n\nKoch, Ed, 35\n\nLake Ontario, 68\n\nLardner, Ring, 38, 64\n\nLas Vegas, Nevada, 98\n\nLeban, Joanna, 10\n\nLeech, Margaret, 79\n\nLife (magazine), xiii, 49, 98, 114\n\nLindbergh, Charles, 12\n\nLocke-Ober (Boston), 118\n\nLomasney, Martin Michael, 118\n\nLondon, England, xx, 16, 18, 30, 35, 65, 86, 89, 99, 115\n\nLong Branch, New Jersey, x, 99\n\nLouisville, Kentucky, 84, 119\n\nMacArthur, Charles, 21, 112\n\nMacElhone, Harry, 86\n\nMacGregor, Rob Roy, 97\n\nMachiavelli, 70\n\nManhattan Club, 69\n\nMankiewicz, Herman J., 44, 79\n\nMaraffi, John, 56\n\nMartin, Kevin, 42\n\nMarx, Groucho, 64\n\nMarx, Harpo, xiii, 79\n\nMaugham, W. Somerset, 101\n\nMcDonald, Freda Josephine, 62\n\nMcKinley, William, 68\n\nMcMein, Neysa, 44, 79, 113\n\nMelville, Herman, x\n\nMencken, H. L., 84, 113\n\nMenocal, Carmen, 39\n\nMetropolitan Hotel, 82\n\nMonroe, Marilyn, 69\n\nMorehouse, Ward, 55\n\nMorristown, New Jersey, xi\n\nMrs. Parker and the Vicious Circle (film), xiv\n\nMurphy, Gerald, xiii, 13, 99\n\nMurphy, Sara, xiii, 13, 99\n\nMurray, William B., 78, 79\n\nNash, Ogden, 75\n\nNational Woman's Christian Temperance Union, 31\n\nNazimova, Alla, 20\n\nNew Amsterdam Theatre, 110, 120, 122\n\nNew Jersey Shore, 4\n\nNew Orleans, Louisiana, 23\n\nNew York Morning Telegraph, 64\n\nNew Yorker, The (magazine), xiii\u2013xiv, 14, 15, 21, 33, 49, 51, 58, 80, 110\n\nNorton, Emperor, 40, 41\n\nO'Neill, Eugene, 79\n\nPaducah, Kentucky, 84\n\nParis, France, 10, 12, 30, 36, 44, 51, 60, 62, 86, 89, 99, 115\n\nParker, Edwin Pond II, xiii, 44\n\nParrish, Maxfield, 63\n\nPemberton, Brock, 79\n\nPemberton, Murdock, 78\n\nPerry, Commodore, 69\n\nPetiot, Pete, 10\n\nPettingill, Ma, 70\n\nPeychaud, Antoine Am\u00e9d\u00e9e, 23\n\nPhiladelphia, Pennsylvania, 29, 47\n\nPickford, Mary, 76, 77, 77, 122\n\nPlayers Club, 49\n\nPlaza, The, 80\n\nPollock, Channing, xiv\n\nPortable Dorothy Parker, The (book), 3, 101\n\nPost, Wiley, 120\n\nPrinceton University, 98, 121\n\nPruette, William, 97\n\nQueen Elizabeth I, 70\n\nRecipes for Mixed Drinks (book), 4, 12\n\nRed Hook, Brooklyn, 64\n\nRickard, Tex, 70\n\nRickey, Joe, 52\n\nRitchie, Albert, 38\n\nRob Roy (operetta), 97\n\nRogers, Will, 120, 122\n\nRoosevelt, Franklin D., 31\n\nRoosevelt, Theodore, 32, 32\n\nRose, Kathleen Mary, 36, 37\n\nRoss, Harold, 4, 14, 21, 44, 46, 79\u201380, 98\n\nRothschild, Elizabeth (Marston), x\n\nRothschild, J. Henry, x, xi\n\nRothschild, Martin, xi\n\nRousseau, Herv\u00e9, 70\n\nRyman, Herbert, 56\n\nSacco, Nicola, 118\n\nSamuels, Arthur, 79\n\nSan Francisco, California, 23, 40, 41\n\nSan Juan Hill, Cuba, 32\n\nSandy Hook, New Jersey, 96\n\nSavoy Hotel (London), 18, 30, 86\n\nSchulberg, Budd, 55\n\nSherwood, Mary, 24\n\nSherwood, Robert E., 24, 44, 49, 79, 106, 118\n\nSmash-Up (film), 72, 90, 91, 102\u20133\n\nSmith, Harold B., 97\n\nSmithsonian Institution, 35\n\nSo Big (book), 112\n\nSolon, Johnny, 21\n\nSoma, Tony, 92\n\nSt. Louis, Missouri, 62, 112\n\nSt. Regis Hotel, 10, 63\n\nStallings, Laurence, 44, 79\n\nStatue of Liberty, 12\n\nStewart, Donald Ogden, 79, 92, 106\n\nStork Club Bar Book, The, 122\n\nSullivan, Frank, 64, 79\n\nSun Also Rises, The, 60\n\nSwope, Herbert Bayard, 38\n\nTarling, William, 99\n\nTaylor, Deems, 56, 65, 79\n\nTaylor, Mamie, 68\n\nTender Is the Night, 13\n\nTerrible Honesty, 62\n\nThomas, \"Professor\" Jerry, 82, 115\n\nTracy, Spencer, 57\n\nThurber, James, 64\n\nThumb, General Tom, 82\n\nTimes Square, 12, 63, 63, 66, 94, 110\n\nToohey, John Peter, 78\u201379\n\nTony's (New York City), 92\n\nTucker, Sophie, 122\n\nTunney, Gene, 46, 47\n\nValentino, Rudolph, 18, 19\n\nVanderbilt, Gloria, 101\n\nVanderbilt, Reginald, 100, 101\n\nVanzetti, Bartolomeo, 118\n\nVelveteen Rabbit, The (Las Vegas), 98\n\nVictor, Sarah, 78\n\nVictoria Theatre, 120\n\nViolet Hour, The (Chicago), 112\u2013113\n\nVoronoff, Serge, 86\n\nVose, Jillian, 104\n\nWaldorf-Astoria Hotel, 14, 21, 96, 97\n\nWalker, Stanley, 30\u201331, 70\n\nWallace, David A., 79\n\nWarren, Lavinia, 82\n\nWashington, D.C., 35\n\nWeaver, John V. A., 79, 112\n\nWelles, Orson, 20\n\nWest, Mae, 122\n\nWhite, E. B., 88\n\nWillard, Jess, 35\n\nWilliams, Tennessee, 23\n\nWodehouse, P. G., xi\n\nWood, Peggy, 79\n\nWoollcott, Alexander, xiii, 4, 5, 13, 21, 26, 44, 46, 50, 65, 78, 80, 121\n\nWondrich, David, xix, 119\n\nWorld War I, xiii, 49, 99, 120, 121\n\nWorld War II, 10, 20, 124\n\nYale University, 121\n\nZelda (magazine), 64\n\nZiegfeld Follies, 94, 120, 122, 123\n\nZiegfeld, Florenz, 122\nIndex by Main Ingredient\n\nAbsinthe\n\nDeath in the Afternoon, 33\n\nMillionaire #2, 83\n\nApplejack\n\nJack Rose, 60\n\nBourbon\n\nBoulevard, 20\n\nEmperor Norton's Second Mistress, 40\n\nHighball, 58\n\nHorse's Neck, 59\n\nNew York Cocktail, 88\n\nSeersucker Punch, 98\n\nWard 8, 118\n\nBrandy\n\nAlexander, 4\n\nBetween the Sheets, 16\n\nBlood and Sand, 18\n\nBulldog, 22\n\nChicago, 26\n\nDolores, 36\n\nLoud Speaker, 65\n\nMetropolitan, 82\n\nMillionaire #1, 83\n\nSidecar, 99\n\nStone Fence, 102\n\nCalvados\n\nDempsey, 35\n\nJack Rose, 60\n\nCognac\n\nBetween the Sheets, 16\n\nStinger, 101\n\nCointreau\n\nMatilda, 81\n\nCr\u00e8me de Cacao\n\nBetween the Sheets, 16\n\nJosephine Baker, 62\n\nDubonnet\n\nDubonnet Cocktail, 38\n\nGin\n\nAcerbic Mrs. Parker, The, 3\n\nAviation, 12\n\nBailey, 13\n\nBathtub Gin, 14\n\nBronx, 21\n\nDempsey, 35\n\nDubonnet, 38\n\nFlorodora, 43\n\nFrench \"75\", 44\n\nGene Tunney, 46\n\nGibson, 49\n\nGimlet, 50\n\nGin Fizz, 51\n\nGin Rickey, 52\n\nLast Word, The, 64\n\nLoud Speaker, 65\n\nMartini, 72\n\nMillionaire #2, 83\n\nMonkey Gland, 86\n\nOrange Blossom, 92\n\nSure Shot, 104\n\nTom Collins, 115\n\nTwentieth Century, 116\n\nWill Rogers, 120\n\nYale Cocktail, 121\n\nZiegfeld, 122\n\nGrand Marnier\n\nMarquis, The, 70\n\nGreen Chartreuse\n\nLast Word, The, 64\n\nMaraschino Liquer\n\nAngel's Tit, 8\n\nLast Word, The, 64\n\nPort Wine\n\nJosephine Baker, 62\n\nRum\n\nBetween the Sheets, 16\n\nBulldog, 22\n\nBywater Cocktail, 23\n\nCuba Libre, 32\n\nEl Presidente, 39\n\nHemingway Daiquiri (Bar La Florida Style), 57\n\nJean Harlow, 61\n\nKnickerbocker Cocktail, 63\n\nMary Pickford, 76\n\nRum Rickey, 52\n\nThree Miler, 109\n\nTippy Canoe, 112\n\nTom and Jerry, 114\n\nZombie, 124\n\nRye Whiskey\n\nAlgonquin Cocktail, 6\n\nHappy Herbie, The, 56\n\nHighball, 58\n\nManhattan Cocktail, 69\n\nWard 8, 118\n\nScotch\n\nBlood and Sand, 18\n\nHighball, 58\n\nRob Roy, 97\n\nSloe Gin\n\nLove Cocktail, 66\n\nMillionaire #1, 83\n\nSouthern Comfort\n\nTimes Square Cocktail, 110\n\nTequila\n\nAuntie Jo-Jo's Jalape\u00f1o Bloody Maria, 10\n\nVermouth\n\nRay Hitchcocktail, 94\n\nVodka\n\nEspionage, 42\n\nMatilda, 81\n\nWhiskey\n\nCablegram, 24\n\nMamie Taylor, 68\n\nMint Julep, 84\n\nNew York Cocktail, 88\n\nOld-Fashioned, 90\n\nTom and Jerry, 114\n\nWhiskey Sour, 119\nIndex by Category\n\nOld Favorites\n\nAngel's Tit, 8\n\nBetween the Sheets, 16\n\nGibson, 49\n\nGimlet, 50\n\nGin Fizz, 51\n\nGin Rickey, 52\n\nHighball, 58\n\nManhattan Cocktail, 69\n\nMartini, 72\n\nMint Julep, 84\n\nOld-Fashioned, 90\n\nRob Roy, 97\n\nSidecar, 99\n\nStinger, 101\n\nStone Fence, 102\n\nTom Collins, 115\n\nTom and Jerry, 114\n\nWhiskey Sour, 119\n\nZombie, 124\n\nVintage Cocktails\n\nAlgonquin Cocktail, 6\n\nAviation, 12\n\nBailey, 13\n\nBathtub Gin, 14\n\nBlood and Sand, 18\n\nBoulevard, 20\n\nBulldog, 22\n\nBronx, 21\n\nCablegram, 24\n\nChicago, 26\n\nCuba Libre, 32\n\nDeath in the Afternoon, 33\n\nDubonnet Cocktail, 38\n\nEl Presidente, 39\n\nFlorodora, 43\n\nFrench \"75\", 44\n\nHorse's Neck, 59\n\nJack Rose, 60\n\nKnickerbocker Cocktail, 63\n\nLast Word, 64\n\nLoud Speaker, 65\n\nLove Cocktail, 66\n\nMetropolitan, 82\n\nMillionaire, 83\n\nMonkey Gland, 86\n\nNew York Cocktail, 88\n\nOrange Blossom, 92\n\nThree Miler, 109\n\nTimes Square Cocktail, 110\n\nTwentieth Century, 116\n\nWard 8, 118\n\nYale Cocktail, 121\n\nCelebrity Cocktails\n\nAlexander, 4\n\nJosephine Baker, 62\n\nDempsey, 35\n\nDolores, 36\n\nJean Harlow, 61\n\nHemingway Daiquiri, 57\n\nRay Hitchcocktail, 94\n\nAcerbic Mrs. Parker, 3\n\nMary Pickford, 76\n\nWill Rogers, 120\n\nMamie Taylor, 68\n\nGene Tunney, 46\n\nZiegfeld, 122\n\nModern Classics\n\nThe Acerbic Mrs. Parker, from The Shanty, Brooklyn, 3\n\nAuntie Jo-Jo's Jalape\u00f1o Bloody Maria, from Doc Holliday's, New York City, 10\n\nBywater Cocktail, Arnaud's French 75, New Orleans, 23\n\nEmperor Norton's Second Mistress, from Elixir, San Francisco, 40\n\nEspionage, from Eastern Standard Kitchen and Drinks, Boston, 42\n\nHappy Herbie, from The Edison, Los Angeles, 56\n\nMatilda, from the Algonquin Hotel, New York City, 81\n\nMarquis, from Fl\u00fbte, New York City, 70\n\nSeersucker Punch, from The Velveteen Rabbit, Las Vegas, 98\n\nSure Shot, from Death & Company, New York City, 104\n\nTippy Canoe, from The Violet Hour, Chicago, 112\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nPublished by Icon Books Ltd, Omnibus Business Centre, 39-41 North Road, London N7 9DP  \nemail: info@iconbooks.com  \nwww.introducingbooks.com\n\nISBN: 978-184831-777-2\n\nText and illustrations copyright \u00a9 2013 Icon Books Ltd\n\nThe author and artist have asserted their moral rights.\n\nOriginating editor: Richard Appignanesi\n\nNo part of this book may be reproduced in any form, or by any means, without prior permission in writing from the publisher.\nContents\n\nCover\n\nTitle Page\n\nCopyright\n\nThe Luckiest Man in the Universe\n\nThe General Theory of Relativity\n\nNewton: The Concept of Force\n\nFour Kinds of Force in the Universe\n\nThe Principia: Describing Newton's Universe\n\nNewton and Hawking\n\nThe Concept of Mass\n\nAlbert Einstein, the Saviour of Classical Physics\n\nEinstein and Hawking\n\nEinstein's Happiest Thought\n\nFinding the Right Equation\n\nThe Field Equations \u2013 What do they mean?\n\nVisualising Curved Space: the Rubber Sheet Model\n\nThe Bending of Starlight: Eclipse of 29 May 1919\n\nSolving Einstein's Equations: Hawking's Starting Material\n\n1) The Schwarzschild Geometry\n\nThe Critical Radius\n\n2) Friedmann: the Expanding Universe\n\nPrecursor to the Big Bang: Lema\u00eetre's Primordial Aim\n\n3) Oppenheimer: on Continued Gravitational Collapse, 1939\n\n1 September 1939\n\n1942... A Turning Point in the Story\n\nThe Death of Einstein\n\nThe Hawking Era\n\nThe Unselfish Thesis Supervisor\n\nSomething You Need to Know: What is a Singularity?\n\nThe Evolution of the Universe\n\n1965: a Big Year for Hawking\n\nAn Unstoppable Mind\n\nThe Sixties Revolution\n\nDallas 1963\n\nSomething You Need to Know: the Electro Magnetic Spectrum\n\n1963: Quasars\n\n1965: the Cosmic Background Radiation\n\nSomething You Need to Know \u2014 Thermal Radiation\n\nHistory of the Universe\n\nBlack Holes \u2014 Wheeler Gives the Media a Buzz Word\n\nThe Age of Black Holes\n\nWhat is a Black Hole?\n\nThe Birth and Death of Stars\n\nHow Stars Collapse to Form White Dwarfs, Neutron Stars & Black Holes\n\nObservational Evidence for Black Holes\n\nThe 1970s: Hawking and Black Holes\n\nHawking's Eureka Moment\n\nThe Laws of Thermodynamics\n\nNow Back to Black Holes...\n\nControversial Birth of a New Idea\n\nAugust 1972, Les Houches Summer School on Black Hole Physics\n\nThe Uncertainty Principle & Virtual Particles\n\nFebruary 1974, The Rutherford- Appleton Laboratory, Oxford\n\nHawking and the Vatican \u2013 a Modern Day Galileo\n\nHawking and the Early Universe\n\nWhy Do We Need Quantum Theory?\n\nQuantum Cosmology\n\nQuantum Gravity or TOE\n\nQuantum Cosmology and Complex Time\n\nWaves and Particles: Nature's Joke on the Physicists\n\nThe Strange World of Quantum Mechanics\n\nQuantum Cosmology: Applying Schr\u00f6dinger's Equation to the Universe\n\nDAMTP: 17 February 1995\n\nInflation\n\nInflation and Quantum Fluctuations\n\nThe Anthropic Principle\n\nHawking's Nobel Prize\n\nCOBE: the Greatest Discovery of All Time (?)\n\nFurther Reading\n\nAcknowledgements\n\nIndex\n\n## **The Luckiest Man in the Universe**\n\nOn 19 October 1994, the author of this book interviewed Stephen Hawking. He began with a question that might seem daring, if not impertinent. Did Hawking consider himself lucky?\n\nWHAT A QUESTION! CONFINED TO A WHEELCHAIR FOR OVER TWENTY YEARS AND UNABLE TO WRITE OR SPEAK... LUCKY? WHO WOULD AGREE WITH THAT... EXCEPT POSSIBLY STEPHEN HAWKING HIMSELF!\n\nLet's go back a bit...\n\nEveryone knows of Hawking's bad luck. It began one afternoon in the spring of 1962 when he found it very difficult to tie his shoelaces. He knew something was drastically wrong with his body. That year he had talked his way into a first degree at Oxford University and was accepted as a postgraduate student at Cambridge. But he had contracted **amyotrophic lateral sclerosis** , ALS for short, the motor neurone disease. It is incurable and fatal. Doctors gave him two years to live.\n\nAs the tabloid press and the paperback biographies would have us believe, Hawking spent the next several months in deep depression in his university digs, drinking and listening to Wagner. To add to his bitterness, he was told that he would not have the famous cosmologist Fred Hoyle (b. 1915) as his research adviser, the reason he chose Cambridge in the first place.\n\nBut immediately his luck began to change. A young woman, Jane Wilde, he met on New Year's Eve 1962 had taken a genuine interest in him, and the Cambridge Physics Department had assigned him to Dennis Sciama (b. 1926), one of the best-informed and most inspiring research advisers in the world of relativistic cosmology.\n\nOnce it is accepted that Stephen William Hawking's physical capabilities were severely limited by the tragic disease of ALS, a whole series of fortunate events seemed to have taken place in the early 1960s which enabled him to fulfil his destiny as one of the leading cosmologists of modern times.\n\nFirst of all, for the profession he had chosen \u2013 theoretical physics \u2013 the only facility he **absolutely** needed was his brain, which was completely unaffected by his illness. He had met a helpful partner in Jane Wilde and been presented with a sympathetic thesis adviser, Sciama.\n\nSoon he would meet Roger Penrose (b. 1931), a brilliant mathematician working on black holes, who would teach him radically new analytical tools in physics. Penrose would help him solve a research problem that would not only save his doctoral dissertation but also bring him directly into mainstream theoretical physics.\n\nThe help of these three people at such a critical time in Hawking's life is perhaps more than anyone can hope for.\n\nHe had another appointment with destiny at about the same time. A theory which had been developed almost fifty years earlier \u2013 Einstein's general theory of relativity \u2013 was only just being widely applied to practical problems in cosmology. It seems that predictions based on this theory were so bizarre that it had taken decades for it to be accepted. Now in the early 1960s, a golden age of research in cosmology based on general relativity was about to begin. Fate had waited for Stephen Hawking. The secretly ambitious \u2013 though by then slightly crippled \u2013 theoretical physicist was ready. He didn't know how long he had to live... but he was certainly in the right place at the right time.\n\nMAYBE HE WAS LUCKY.\n\nStephen Hawking is called a **relativistic cosmologist.** This means he studies the Universe as a whole (cosmologist) and uses mainly the theory of relativity (relativistic).\n\nAs Hawking has spent his entire career as a theoretical physicist \u2013 from the early 1960s to the mid 1990s \u2013 working with Einstein's general relativity, it might be a good idea to know what it's about.\n\n## **The General Theory of Relativity**\n\nBerlin, November 1915. Albert Einstein (1879-1955) had just completed his theory of general relativity, a mathematical structure in which curved space and warped time are used to describe gravitation. All modern cosmology began two years later, when Einstein published a second paper called **Cosmological Considerations** in which he applied his new theory to the entire Universe.\n\nGeneral relativity is difficult to master, but the relatively few people who understand the theory agree it is an elegant, even beautiful theory of gravitation.\n\nDescribing a set of equations as beautiful doesn't help much in understanding how Einstein's theory differs from that of Isaac Newton (1642-1727). But an example of how each of the two theories describes gravity in the same physical situation might do the trick.\n\nWhy does a, cosmologist have to study gravitation?\n\nCosmology is the study of the whole Universe and much of the subject is based on wide-sweeping hypotheses. Gravitation determines the large-scale structure of the Universe or, more simply, keeps the planets star and galaxies together. It is the most important concept for work in this field.\n\nUntil recently, the subject of cosmology was considered to be a pseudo-science reserved for retired emeritus professors. But in the last three decades, more or less coinciding with Hawking's career, two major developments have changed the subject dramatically.\n\nTHE COMPLETE STORY TAKES IN NEWTON, THEN EINSTEIN, THEN HAWKING FIRST, NEWTON.\n\n  * First, major breakthroughs in observational astronomy \u2013 reaching out to the most distant galaxies \u2013 have made the Universe a laboratory to test cosmological models\n  * Second, Einstein's general relativity has been proven over and over again to be an accurate and reliable theory of gravitation throughout the entire Universe.\n\nRemember, physics is a cumulative subject. New theories are built on previous ones, keeping the ideas that stand up to experimental test and discarding those that don't. Our final goal is to understand the contributions of Stephen Hawking who has taken Einstein's gravitation theory to its ultimate limit.\n\nIt is important to understand the notion of _partial theories._ For example, Newton's Law of Gravitation is very accurate only when gravity is weak \u2013 and must be replaced by Einstein's general relativity in strong gravitational fields. Similarly, relativity must be replaced by quantum mechanics when examining interactions on a microscopic scale, such as the big bang singularity, or at the edge and centre of a black hole. Hawking is generally recognized as the theoretician with the best chance of combining general relativity and quantum mechanics to produce quantum gravity, ill-named by the media as **Theory of Everything.**\n\n## **Newton: The Concept of Force**\n\nNewton introduced the concept of a gravitational _force_ of attraction and stated that the mutual pull of attraction between two objects is proportional to the **mass** of each object (i.e. the amount of matter the object contains) and inversely proportional to the square of distance between them.\n\nNOW DON'T PANIC IT'S A VERY SIMPLE EQUATION! I CALL THIS MY LAW OF UNIVERSAL GRAVITATION. \"IF THE MASS OF ONE OR THE OTHER OF THE TWO OBJECTS DOUBLES! THE FORCE DOUBLES! BUT IF THE DISTANCE BETWEEN THE TWO OBJECTS IS DOUBLED, THE FORCE IS REDUCED BY A FACTOR OF FOUR, DUE TO THE SQUARED TERM IN THE DENOMINATOR. \"THUS THE FORCE DECREASES RAPIDLY AS THE TWO OBJECTS ARE MOVED A PART.\"\n\nGravitation is the weakest force in nature as seen by the magnitude of the gravitational constant Gin practical units:\n\n**G = 6.67 \u00d7 10 -11 Newtons-metres2 \/ kilograms2**\n\nA Newton is a scientific unit of force, equal to about a quarter of a pound.\n\n## **Four Kinds of Force in the Universe**\n\n**The Electromagnetic Force:** keeps atoms together and is the basis for all chemical reactions.\n\n**The Strong Nuclear Force:** binds the neutrons and protons together in the nucleus. This force is important in nuclear reactions like fission and fusion.\n\n**The Weak Nuclear Force:** determines radioactive decay, i.e. the spontaneous emission of alpha and beta particles from inside the nucleus.\n\n**The Gravitational Force:** responsible for large-scale structure of the Universe, the formation of galaxies, stars, and planets.\n\nThe four known forces separate and become individually distinct during the earliest moments of the Universe.\n\nWhen two Sumo wrestlers (mass about 135 kilograms) get close to each other in the ring (say a metre apart), the force pushing them **towards each other** is minuscule... about 10,000 times less than the pull necessary to pick up one square of toilet tissue! To convert the answer to pounds multiply Newtons by 0.225.\n\nBut the force pulling each of them **towards the floor** is much larger. That's because the other object attracting each **downwards** is the Earth, whose mass (5.98 x 1024 kilograms) must be put in the numerator of Newton's equation. The Earth's radius (6.37 x 106 metres) goes in the denominator. Try the calculation yourself with an electronic calculator and don't forget the conversion factor to get your answer in pounds.\n\n## **The Principia: Describing Newton's Universe**\n\nNewton was chiefly concerned with the force of gravity between the Sun and planets, i.e., the solar system. The immediate impetus for the publication of his theory of gravitation, the **Principia** , arose from a discussion at the Royal Society in 1684 between the astronomer Edmond Halley (1656-1742), the architect Sir Christopher Wren (1632-1723) and Newton's arch rival Robert Hooke (1635\u20131703).\n\nWE THINK AN INVERSE SQUARE LAW OF ATTRACTION IMPLIES ELLIPTICAL ORBITS FOR THE PLANETS. BUT WE CAN'T PROVE IT. I CAN BUT I'M KEEPING MY PROOF SECRET!\n\nI'LL JOURNEY TO CAMBRIDGE AND ASK NEWTON'S OPINION... WELL, I'M OFF TO ST PAULL'S TO BUILD MY CATHEDRAL.\n\nAN INVERSE SQUARE LAW DOES INDEED PRODUCE ELLIPTICAL ORBITS A RESULT WHICH I'VE DEMONS TRATED MATHEMATICALLY. THIS IS VERY EXCITING! CAN YOU SHOW IT TO ME?\n\nEH WELL... I CAN'T FIND THE CALCULATION.\n\nHalley returned to London frustrated, but 3 months later he received a 9-page paper in Latin, **De Motu Corporum** or **On the Motion of Bodies in Orbit** , in which Newton described the elliptical paths of the planets in terms of his Law of Gravitation and his Laws of Motion. This was the precursor of his world-famous **Principia** (1687) which presented a complete mathematical description of his ideas.\n\n## **Newton and Hawking**\n\nThe media often compares Stephen Hawking with other famous physicists like Newton and Einstein, much to the alienation of scientists and, in particular, historians of science. No single individual will ever dominate his era as Newton did, whereas Hawking is one of a small group of \u00e9lite scientists at the cutting-edge of today's cosmology.\n\nYet, some of these comparisons are very interesting.\n\nNewton spent his entire scientific career at Cambridge with his residence and laboratories at Trinity College. Hawking has been at Cambridge since his postgraduate student days in 1962, except for a few sabbatical years abroad.\n\nThey have both attempted to explain the observable physical Universe using theories of gravity: Newton using his own theory, and Hawking using mainly Einstein's general relativity.\n\nBoth have held the same distinguished position at Cambridge, the Lucasian Chair of Mathematics.\n\nThe wide range of applications for the gravitation law which Newton presented in the **Principia** is quite extraordinary. The theory was an immediate success and found to be applicable to all motion in the solar system, including the Moon and comets as well as the planets. It was so accurate that it was used to discover the planet Neptune, which could not even be seen with the telescopes available at the time.\n\nBUT I'M WORRIED! QUIET! YOU DON'T EXIST YET!\n\nExcept for one small problem. The orbit of Mercury wasn't quite right. But as Mercury is so close to the Sun and very difficult to view, the discrepancy was thought to be due to observational errors and excused by everyone in the 17th and 18th century. The orbits of Jupiter, Mars and Saturn were spot on. No one was worried.\n\nMany may find it surprising to learn that putting a man on the Moon, some half-century after Einstein, did not require any modification of Newton's theory. NASA engineers were using the **Principia** when they programmed their rockets at Cape Kennedy in 1969.\n\nWHY NOT USE EINSTEIN'S THEORY? ISN'T IT MORE CORRECT THAN NEWTON'S? YEP, IT IS... BUT WE DON'T NEED IT AT NASA.\n\nBut the difference is negligible, unless measurements are being made very close to a massive gravitational object. For orbits around the Sun and the planets, in fact throughout most of the entire solar system, Einstein's relativistic effects can be ignored and Newton's theory is fine.\n\n## **The Concept of Mass**\n\nConsider the miracle method for losing weight: a trip to the Moon! When an object is transported in a spaceship to the Moon, its weight decreases by about a factor of 6! This weight loss can be demonstrated very simply, using Newton's gravitation formula to compare the force of gravity of a body at the surface of the Earth (i.e. its weight) with that on the surface of the Moon. Just plug the numbers into the equation and see the dramatic weight loss. But watch how you use mass.\n\nThe mass of the astronaut is about 60 kilograms (determined by a scale balance and standard masses); the **mass of the Earth** is 5.98 \u00d7 1024 kg and the **radius of the Earth** is 6.37 \u00d7 106 metres. If we use these values in Newton's equation, we find her weight to be (with 1 Newton = 0.225 lb):\n\nNow what will she weigh on the Moon? Use the same method but this time use the **mass of the Moon** = 7.34 \u00d7 1022 kilograms and the **radius of the Moon** = 1.74 \u00d7 106 metres\n\n**Weight = 97 Newtons = 21.8 lb.**\n\nEven the Sumo will only weigh 50 lb.\n\nI'M ONE \u2013 SIXTH OF MY WEIGHT ON EARTH! REMEMBER... POUNDS ARE A MEASURE OF WEIGHT, BUT KILOGRAMS ARE A MEASURE OF MASS.\n\nBut the **mass** of the astronaut doesn't change on the Moon. She has lost none of the **matter** that makes up her body. Thus, her physical appearance and size are unaffected by the change in the gravitational field.\n\nMass is a tricky concept. No doubt about it. It is not only difficult to understand, but, until Einstein, it was also horribly ambiguous. Think of that property of a body that causes it to be attracted by another body, as in Newton's Law of Gravitation.\n\nThen think of the property of a body which gives it resistance to a change in speed, as in Newton's second Law of Motion;\n\nClearly a larger inert mass will result in a smaller acceleration for a given force. Is there any difference between these two quantities, **gravitational** mass and **inertial** mass? Newton has confused us.\n\nNEWTON'S A GIANT OF PHYSICAL THEORY. I'M PLEASED TO VIEW THE UNIVERSE FROM A TOP HIS SHOULDERS. BUT I'M WORRIED ABOUT MASS... AND A FEW OTHER THINGS!\n\n## **Albert Einstein, the Saviour of Classical Physics**\n\nIt was left to a single man to clear up the leftover inconsistencies of classical physics, Albert Einstein. The great Victorian physicists had decided that only trivial problems remained. Yet, Einstein proceeded to turn Newtonian physics upside-down.\n\nImagine Newton's theoretical structure as a house of cards. It's true, Einstein only removed two of these cards. They just happened to be at the base of the structure.\n\nI DISCARD NEWTON'S CONCEPT OF UNIVERSAL TIME AND ABSOLUTE SPACE....OOPS!\n\nTo do this, it was necessary to postulate that nothing can travel faster than the **speed of light** , which Einstein said was always observed to be the same. This work he called the **Special Theory of Relativity.**\n\nEinstein's first papers were about electrodynamics and concerned with light signals and moving clocks. But he soon began worrying about gravitation and was troubled by its bewildering property of **action at distance.**\n\nAccording to Newton, if the Sun disappeared in an instant, so would its gravitational field at the Earth, millions of miles away. Yet light from the Sun, with its finite speed, would continue to travel towards the Earth for another eight minutes. This troubled Einstein. So did the concept of mass.\n\nWHY SHOULD INERTIAL AND GRAVITATIONAL MASS BE THE SAME?\n\nFor Einstein, such notions were paradoxes to worry about for years and years. He already knew as a young man that the hand of God was in the **details.**\n\nEinstein the Worrier began to consider if there might be another way to explain gravity. Maybe it is not a force at all. Since the motion of a freely falling object does not depend on the object's mass or composition (as Galileo discovered in the 15th century), gravitation might be due to certain properties of the **medium** it's falling in, that is, **space** itself.\n\nBy a series of remarkably creative and idiosyncratic steps, Einstein decided that space is not flat but **curved** , and the local curvature is produced by the presence of mass in the Universe. Consequently, bodies moving through curved space do not travel in straight lines but rather follow the path of least resistance along the contours of **curved space.** These paths are called **geodesics.**\n\nIf this is true, there would be no need for a mysterious \"force of gravity\" which is transmitted **instantaneously.** Nor would it be necessary to explain the odd coincidence that inertial and gravitational mass are exactly equal.\n\nEinstein was setting out to rescue classical physics from these inconsistencies and finish the task started by Galileo, Newton and James Clerk Maxwell (1831-79).\n\nAH! ONE OF GALILEO'S CANNON \u2013 BALLS!\n\n## **Einstein and Hawking**\n\nMost great works in physics have come from those who combine miraculous physical intuition with sound mathematical skills. The former is far more important than the latter.\n\nEinstein was not a pure mathematician and neither is Stephen Hawking. They both learned the mathematics they needed to do their physics, formulating their ideas in the most efficient way possible.\n\nEinstein hassled his friend Marcel Grossman to learn the techniques of Riemann geometry in order to handle curved space. Hawking, anxious to probe the secrets of black holes in the early 60s, questioned Roger Penrose to exhaustion learning the new topological methods of singularity theory.\n\nBut both had a nose for the most interesting problems.\n\nEinstein's idea of curved space had some plausibility, but it was not clear how to quantify such a new approach. So he started dreaming up more of his famous _gedanken_ (thought) experiments, as he did with Special Relativity.\n\nHis sketchy qualitative ideas of curved space were to become a set of equations which gave the precise amount of curvature for a given amount of mass. This development is said to be one of the most creative examples ever of the power of pure abstract thought.\n\nThe main idea which got him started he called, **The Happiest Thought of My Life.**\n\n## **Einstein's Happiest Thought**\n\n_Sitting in a chair in the Patent Office at Berne (in 1907), a sudden thought occurred to me. \"If a person falls freely he will not feel his own weight.\" I was startled and this simple thought made a deep impression on me. It impelled me towards a theory of gravitation. It was the happiest thought of my life._\n\n_I realized that... for an observer falling freely from the roof of a house there exists_ \u2013 _at least in his immediate surroundings \u2013 no gravitational field. If the one who is falling drops other bodies (e.g. Galileo's cannon balls), then these remain relative to him in a state of rest or of uniform motion independent of their particular chemical or physical nature. (Of course, we are ignoring the effect of air resistance.)_\n\n_The observer therefore has the right to interpret his state as at rest or in uniform motion..._\n\nHe continued...\n\n_Because of this idea, the uncommonly peculiar experimental law_ \u2013 _that in the gravitational field all bodies fall with the same acceleration_ (this is another way of saying that gravitational mass is the same as inertial mass) \u2013 _attained at once a deep physical meaning. If there were to exist just one single object that falls in a different way than all the others, then with its help the observer could realize that he is in a gravitational field and is falling in it. However, if such an object does **not** exist_ \u2013 _as experience has shown with great accuracy starting with Galileo in 1590_ \u2013 _then the observer lacks any objective means of perceiving himself as falling in a gravitational field. He has the right to consider his state as one of rest and his environment as free of gravity. Therefore, the fact that the acceleration of free fall is independent of the nature of the material involved is a powerful argument that the relativity postulate can be extended to coordinate systems which are in **non-uniform** motion._\n\nEinstein's thought that a person falling freely does not feel his own weight seems rather simple. Yet from this starting point, he squeezed every possible drop of insight, removing all the inconsistencies of Newton's theory that his intuition and the laws of physics would allow. He transformed the _simple picture of someone falling through space_ into a _small laboratory in which gravity did not exist._\n\nHe could then analyse the effect of gravity on such phenomena as the bending of a light beam or the slowing of a clock by simply replacing the gravitational field with simulated accelerated motion.\n\nSimply by thinking about a man jumping off a roof in Berlin (or so the story goes), Einstein was able to replace gravity by acceleration and discover his **principle of equivalence.**\n\nEinstein could now use the powerful principle of relativity \u2013 that the laws of physics should not depend on any particular reference frame \u2013 to test his new laws of space curvature. He also had the principle of equivalence (gravity equals acceleration) to get started. And he had one more useful bit of information, this one experimental.\n\n**The Perihelion of Mercury: from a Problem to a Solution**\n\nRecall that scientists in Newton's time were not worried about the small discrepancy in Mercury's elliptical orbit, even though it did not return to the same starting point in each cycle. By Einstein's time, astronomers were more than worried, they needed an explanation. The discrepancy had been carefully measured to be 43 seconds of arc per century and it would not go away. Einstein could now use the perihelion result to test his curvature law. (Perihelion, from the Greek peri meaning close to and helios, sun).\n\n## **Finding the Right Equation**\n\nEinstein used the \"3 Ps\" to test his equations...\n\nHe went on producing sets of equations (mentally exhausted and trying to ignore the First World War)...\n\nUNTIL MY EQUATIONS FINALLY PRODUCED...\n\nThese latest equations also predicted a deflection of 1.7 seconds of arc for starlight passing near the edge of the Sun and incorporated his earlier prediction of gravitational time dilation, the warping of time.\n\nEinstein presented this final form of his general relativity law of curved space and warped time to the Prussian Academy on 25 November 1915.\n\nThen he sat down and wrote a letter to a close friend, the Dutch physicist Paul Ehrenfest.\n\n\"I WAS BESIDE MYSELF WITH ECSTASY FOR DAYS. \"IMAGINE MY JOY THAT THE NEW LAW OF CURVATURE OBEYS THE PRINCIPLE OF RELATIVITY AND PREDICTS THE CORRECT PERIHELION MOTION OF MERCURY.\n\n...THE YEARS OF SEARCHING IN THE DARK FOR A TRUTH THAT ONE FEELS BUT CANNOT EXPRESS \u2013 THE INTENSE DESIRE AND THE ALTERATIONS OF CONFIDENCE AND MISGIVING UNTIL ONE BREAKS THROUGH TO CLARITY AND UNDERSTANDING \u2013 ARE KNOWN ONLY TO HIM WHO HAS EXPERIENCED THEM HIMSELF!\"\n\n## **The Field Equations \u2013 What do they mean?**\n\nThe 36-year old professor had produced a set of mathematical equations which gave the details of the relationship between the curvature of space and the distribution of mass in the Universe. Einstein found that matter tells space _how to curve_ and then space tells matter _how to move-_ a new way to describe gravitation. **No forces.** A mind flip is necessary to jump between the two pictures of gravitation.\n\nContained within these miraculous equations is the explanation of the perihelion shift of Mercury, the degree of bending of starlight, the existence of gravitational waves, information on the singularities of space time, the description of the formation of neutron stars and black holes, even the prediction of the expansion of the Universe.\n\nThat's the good news.\n\nThe bad news is that the mathematics is extremely difficult. There are some 20 simultaneous equations with 10 unknown quantities. The equations are almost impossible to solve except in situations where symmetry or energy considerations reduce them to simpler forms.\n\nIf we ignore the cosmological constant _lambda_ (which doesn't belong there anyway) and consider free space where the mass tensor is zero, the equations can be written very simply...\n\nThis form was made famous by a well-known photograph of Einstein lecturing on the theory in the 1920s. Looks easy!\n\n## **Visualising Curved Space: the Rubber Sheet Model**\n\nEinstein's gravitation is quite unusual, compared to other field theories like electricity or magnetism, in that the description of motion (i.e. how an object moves) is already built into the field equations (how space time is curved. This can be understood with the help of a simple model \u2013 call it the rubber sheet picture.\n\nConsider a billiard table with the slate top and felt cover replaced by a taut thin rubber sheet which is highly stretchable. If a light object (e.g. a ping-pong ball) is rolled across the sheet, it will move more or less in a straight line. This simulates _flat_ space and the ping-pong ball's path corresponds to the straight line motion of _special_ relativity.\n\nNow place a heavy billiard ball in the centre of the sheet, causing it to become curved with a depression at the centre. The model now simulates the curvature of space near a central mass as described by general relativity.\n\nThe simplest case (other than a straight line) is when the depression just captures the moving object to produce a circular orbit. Note this occurs without the need of any _centripetal force_ to keep the object in orbit, as in Newton's picture.\n\nThe object would like to move in a straight line, but the space is curved, so it moves in a circle around the centre. It is simply moving along a path of least resistance in the curved space. This is general relativity's representation of how a planet is captured in an orbit around the sun.\n\nIf the object is moving on a line directly towards the centre, it falls right into the depression and accelerates into the attracting centre. This is a representation of a meteorite crashing into the Sun or the Earth.\n\nWith such diagrams it is now possible to visualize the distinct and utter difference between Newton and Einstein. Einstein has replaced Newton's **gravitational force** with **curved space.**\n\nAt the time of publication, the new theory met with scepticism. Many did not wish to see the Newtonian scheme abandoned. These sceptics needed more evidence.\n\n## **The Bending of Starlight: Eclipse of 29 May 1919**\n\nFour years later, the scientific world awaited the verdict of an experiment which Einstein himself had suggested in the original paper, the bending of starlight during a solar eclipse. The theory predicted that starlight passing just at the edge of the Sun would be displaced by 1.7 seconds of arc from its true position. It was the first real test of the theory.\n\nThere was to be a total eclipse of the Sun on 29 May 1919, smack in the middle of a bright field of stars in the cluster Hyades. These were most unusual and optimal conditions for such an experiment.\n\nThe English astronomer **Arthur Stanley Eddington** (1882\u20131944) led an expedition to the island of Principe off the coast of Africa to photograph the eclipse.\n\nEddington found that light rays which had left the surface of stars thousands of years ago and had been bent by the curved space near the Sun only eight minutes previously, passed through the lens and exposed the photographic plates just where Einstein said they would. One of the most remarkable experiments in scientific history had been completed.\n\nThe two dimensional rubber sheet drawing of the star displacement makes it look so very simple.\n\nThe results of the eclipse expedition were presented by the Astronomer Royal at a meeting at the Royal Society on 6 November 1919, and Einstein became an international hero overnight. Headlines in the _New York Times_ suggested that a new Universe had been discovered... and this time the newspaper hype was not exaggerated.\n\nA world weary from war embraced the quiet and eccentric scientist, sitting in his study in Berlin with a pencil and pad, who had figured out the great plan of the Almighty for the entire Universe.\n\nMany critics said the results were inconclusive, that the possibility of error in the star measurements was too great... so the scepticism continued.\n\n## **Solving Einstein's Equations: Hawking's Starting Material**\n\nIn the 25 years between the publication of Einstein's general relativity and the outbreak of the Second World War, several solutions to the field equations were produced which have been fundamental to Stephen Hawking's work.\n\nIT IS A REMARKABLE FACT THAT ALL OF THESE RESULTS WERE IGNORE OR RIDICULED WHEN THEY WERE PUBLISHED; PARTICULARLY BY THE CREATOR OF THE THEORY HIMSELF, ALBERT EINSTEIN. THE FIRST OF THESE SOLUTIONS APPEARED ALMOST IMMEDIATELY.\n\n## **1) The Schwarzschild Geometry**\n\nIn 1915, the same year as Einstein's publication, the German mathematician Karl Schwarzschild sent a paper to Einstein. Schwarzschild used elegant mathematical analysis to produce an exact solution to the equations for an arbitrary spherical body, like a star. The solution intrigued Einstein greatly because he himself had only been able to arrive at an **approximate** solution to his own equations and thought that an **exact** solution of the equations would never be found.\n\nSchwarzschild's solution was quite an achievement because of the technical manipulation required to solve a system of ten equations connecting twenty quantities, resulting in hundreds of terms. These are not simple algebraic equations, but second order, non-linear, partial differential equations \u2013 the bane of all graduate students in physics.\n\nToday, a room full of electronic computers are utilized to find such solutions. But Schwarzschild had produced this first one with a pen and paper. A _tour de force._\n\n## **The Critical Radius**\n\nSchwarzschild's mathematics showed how the space curvature around an object of any arbitrary mass varied as a function of the distance from the centre of the object, i.e. along a **radial line.**\n\nHis results produced a very strange geometry. There seemed to exist a critical point at which the curvature was so strong that matter could not escape. This critical point is now known as the **Schwarzschild Radius** and depends only on the mass of the object. (G is the gravitational constant; c is the speed of light)\n\nThere was no immediate concern about this critical point, since the interior of stars and planets could not be investigated anyway. But there was speculation as to what might happen if a star or planet existed which satisfied this equation. The gravitational forces would be so great that the object would collapse indefinitely and **nothing** would be able to resist the self-gravity caused by the extreme space curvature. All the matter would be compressed to a singularity \u2013 a single point at the centre.\n\nPlanets as massive as the Earth would have to be compressed to absurd dimensions \u2013 to the size of a garden pea or the Sun to a diameter of about 3 kilometres. Ridiculous, they said. The calculation was a mathematical fluke. In any case, nobody wanted to think about it. Least of all, Einstein.\n\nNEIN! NEIN! NEIN!\n\n## **2) Friedmann: the Expanding Universe**\n\nSome years after Schwarzschild, another controversial solution to Einstein's equations appeared. In 1922, the Russian Alexander Friedmann (1888-1925) made the simplifying assumption that the Universe was **uniformly** filled with a thin soup of matter. (Modern measurements have shown this assumption of uniformity to be quite reasonable in spite of the formation of stars and galaxies.)\n\nFriedmann found that general relativity predicted the Universe to be unstable and the slightest perturbation would cause it to expand or contract. He corrected a mistake in Einstein's 1917 paper on cosmology to reach this result. (Any wonder Einstein didn't like **this** prediction.)\n\nRecall that Einstein had introduced an artificial term _(lambda_ , the cosmological constant) into his field equations essentially to \"stop the expansion\". At the time, astronomers were telling him that the Universe was static, so he wanted to guarantee the theory would agree with observations. Later, he called this \"cosmological constant\" the biggest mistake of his life.\n\nFriedmann dropped the _lambda_ from the equations and got an **expanding universe** , which, of course, Einstein did not like. This was another solution of his own equations which he ridiculed.\n\nBIG MISTAKE, THAT LAMBDA!\n\nFriedmann's predictions for the expansion of the Universe can be summarized by considering three different values for the mass of the Universe in terms of a ratio \u03a9 (omega).\n\n**\u2022Mass density of the Universe is greater than the critical value**\n\nIn this case, the expansion rate is slow enough and the mass great enough for gravity to stop the expansion and reverse it. A Big Crunch would eventually occur with all the matter in the Universe pulled back to a single point. \u03a9 > 1 (greater than... )\n\n**\u2022Mass density of Universe is less than the critical value**\n\nThe Universe expands much more rapidly. Gravity can't stop it, but does slow the rate of expansion somewhat. \u03a9 < 1 (less than... )\n\n**\u2022Mass density of the Universe is equal to the critical value**\n\nThe Universe expands just fast enough not to collapse. The speed at which the galaxies recede from each other gradually decreases, but galaxies always move apart. \u03a9 = 1 (equal)\n\n## **Precursor to the Big Bang: Lema\u00eetre's Primordial Aim**\n\nThe Belgian cosmologist **Abb\u00e9 Georges Lema\u00eetre** (1894-1966) was the first to use Friedmann-type solutions to formulate a model for the beginning of the Universe which he called the Primordial Atom or Cosmic Egg.\n\nLema\u00eetre was a visionary. Not only did he anticipate that the expanding Universe would be confirmed by looking for red shifts in the spectra of galaxies, but he even suggested that it might be possible to detect remnant radiation from the primordial atom. These two ideas dominate contemporary Big Bang cosmology in this last decade of the 20th century.\n\nYEP, IF THEY'RE NOT RIGHT, I AM!\n\nBy 1929, the astronomer **Edwin Hubble** (1889-1953) had used the 100-inch Hooker telescope at the Mount Wilson Observatory in California to discover galaxies and confirm that the Universe **is** expanding. But he knew nothing of Einstein's theory or Lema\u00eetre's cosmology.\n\nAt last in 1931, Lema\u00eetre cornered Hubble and Einstein at Caltech and gave a seminar on his model Universe.\n\nTHE UNIVERSE AT ONE TIME EXISTED IN A VERY SMALL SPACE BEFORE AN EXPANSION TOOK PLACE SEVERAL BILLION YEARS AGO. IT IS STILL EXPANDING TODAY!\n\nTHIS IS THE MOST BEAUTIFUL DESCRIPTION OF THE CREATION OF THE UNIVERSE I'VE EVER HEARD.\n\nOOPS!\n\n## **3) Oppenheimer: on Continued Gravitational Collapse, 1939**\n\nThe third solution of Einstein's equations, important to modern cosmology and Stephen Hawking, was published by the American physicist J. Robert Oppenheimer (1904-1967) and one of his students, Hartland Snyder in 1939. They took up the problem of the Schwarzschild geometry in spite of the criticism by Einstein, Eddington and just about everybody else. The paper, which was published in _The Physical Review_ was titled, On Continued Gravitational Collapse\".\n\nStars may eventually burn out and begin to collapse under gravitational contraction. In the idealized model of a spherical contracting star, a squeezing phenomenon can occur which could bring the star to the critical radius Rc. Catastrophic gravitational collapse would take place for the critically collapsed star.\n\n\u2022Space curvature would be so severe that light rays emitted from the star's surface would bend into the star's interior, sealing off events from external observers.\n\n\u2022Light rays at the surface would be infinitely red-shifted, i.e., the light would have no energy.\n\n\u2022A one-way event horizon would form in which particles, radiation, etc. could enter the star, but nothing could be emitted.\n\n\u2022A space-time singularity would ultimately form, not at the critical radius, but at the centre of the star. All the physics is continuous for an observer falling in with the collapsing star's surface.\n\nNEIN! NEIN! NEIN!\n\nEinstein again resisted. He ridiculed the Oppenheimer result vigorously in print.\n\nHe even refused to accept that relativity could describe collapsed stars which did **not** become critical \u2013 called neutron stars \u2013 in spite of independent predictions by the eccentric Fritz Zwicky (1898-1974) at Caltech and the highly respected Lev Landau (1908-68) in Moscow.\n\n## **1 September 1939**\n\n\u2022Publication date for the **Physical Review** issue containing article by Oppenheimer (and Snyder) describing the gravitational collapse of a star.\n\n\u2022In the same issue, another article by Neils Bohr (1885-1962) and John Wheeler (b. 1911) explained the mechanism for nuclear fission (reaction used in the atom bomb).\n\n\u2022 On the same date, Hitler's troops invaded Poland, triggering off the Second World War.\n\nWhen nuclear fission was discovered by the Germans Otto Hahn (1879-1968) and Fritz Strassman (b. 1902), physicists and politicians in Western democracies became alarmed that the Germans were developing an atom bomb to turn the entire world into a Nazi empire, a Third Reich ruling with the threat of nuclear devastation.\n\nIt is easy to see why work on cosmology was postponed. Contemplating the mysteries of the physical Universe in such severe political crisis was a luxury the free world could not afford.\n\nNEIN! NEIN! NEIN! OUT WITH SCHWARTS CHILD FRIEDMANN AND OPPENHEIMER!\n\nIn addition, the originator of the general theory had opposed all the radical cosmological predictions of his own equations as developed by Schwarzschild, Friedmann and Oppenheimer. It would be 20 years before this work was resumed and the consequences of these solutions appreciated.\n\n## **1942... A Turning Point in the Story**\n\nIn 1942, physicists began to focus on deadly practical projects. Oppenheimer, one of the heroes of early cosmological research, left the heady intellectual climate of Berkeley for the barren flats of Los Alamos and the Manhattan Project. In December 1942, the Italian Enrico Fermi (1901-54) and his team at the University of Chicago achieved the first controlled nuclear chain reaction.\n\nAnd at the beginning of that same year, on 8 January, Stephen William Hawking was born in Oxford. His mother had just moved from London to escape the nightly pounding by the German Luftwaffe.\n\n_Research on collapsing stars was abandoned for over twenty years, enough time for Stephen Hawking to grow to maturity, finish his degree at Oxford and enrol as a postgraduate student at Cambridge University._\n\n## **The Death of Einstein**\n\nAlbert Einstein died on 18 April 1955 in Princeton, a small college town in New Jersey, USA. His wish was to be cremated so that \"no one will worship at my bones\". In spite of his wish, unethical doctors performed an unnecessary autopsy and made off with his brain and his eyes \u2013 an insidious invasion of privacy.\n\nEinstein had left Europe for the USA in 1933 with his real creative work behind him. During the last 22 years of his life, he did not work on any of the important cosmological questions which came out of his general relativity theory. For years he stuck slavishly to the task of trying to unite the field equations of general relativity with Maxwell's equations for the electromagnetic field and ignored quantum mechanics.\n\nHis unified field theory calculations were found by his bedside.\n\nTwo other physicists who also lived in Princeton mourned the death of the great scientist. Oppenheimer, no longer affiliated with the war effort, was director of the Institute of Advanced Studies (where Einstein had an honorary position) and John Wheeler was Professor of Physics at Princeton University. Wheeler had recently finished critical years of development on the hydrogen bomb and was now returning to basic research in cosmology, with particular interest in collapsing stars.\n\nHow fitting that these two physicists should live on **opposite** sides of the same street in this small academic community. They had vastly different views of the Universe **and** of American political life which placed them on opposite sides of controversial issues, like national security and nuclear weapons. Soon they would confront each other again on the question of general relativity and gravitationally collapsing stars.\n\nIn 1958, three years after Einstein's death, they both travelled from Princeton to attend an international conference in Brussels on modern cosmology. Wheeler had been invited to give a talk reviewing the current state of research.\n\nOF ALL THE IMPLICATIONS OF GENERAL RELATIVITY, THE QUESTION OF THE FATE OF GREAT MASSIVE STARS IS ONE OF THE MOST CHALLENGING. BUT IMPLOSION AS CALCULATED BY OPPENHEIMER IN 1939 DOES NOT GIVE AN ACCEPTABLE ANSWER. WHY NOT? IF HEAVIER THAN THE SUN REALLY OCCUR IN STELLAR EVOLUTION, THEN I BELIEVE THEIR COLLAPSE CAN BE DESCRIBED IN THE FRAMEWORK OF GENERAL RELATIVITY.\n\nWOULD NOT A SIMPLE ASSUMPTION BE THAT SUCH MASSES UNDERGO CONTINUAL GRAVITATIONAL CONTRACTION AND ULTIMATELY CUT THEMSELVES OFF FROM THE REST OF THE UNIVERSE?\n\nOPPIE, YOUR MODEL IS JUST TOO IDEALISED. REAL STARS UNDER CONTRACTION WOULD PRODUCE NUCLEAR REACTIONS, SHOCK WAVES, HEAT RADIATION AND MASS EJECTION... ALL SORTS OF UNPREDICTABLE PHENOMENA!\n\nIF ONLY I'D KEPT THOSE COMPUTER SIMULATIONS ON THE EXPLOSION OF THE HYDROGEN BOMB, WE COULD HAVE REWRITTEN THE PROGRAMS TO SIMULATE THE IMPLOSION OF A MASSIVE STAR. HMMM... THEY STILL HAVE BOMB DESIGN EXPERTISE AT LOS ALAMOS AND AT LIVERMORE. MAYBE THEY WOULD HELP WITH THE CALCULATIONS.\n\nA few years later, Edward Teller phones Wheeler from the Livermore Radiation Labs in California.\n\nJOHN, DR COLGATE AND HIS TEAM HAVE COMPLETED THE SIMULATION OF COLLAPSING STARS YOU REQUESTED. IT TURNS OUT THAT WHEN A STAR IS SMALL, A SUPERNOVA IS TRIGGERED AND A NEUTRON STAR MOST LIKELY FORMS. BUT FOR STELLAR MASSES GREATER THAN TWICE THE SUN'S MASS THE COMPUTER SIMULATIONS PRODUCE CONTINUED GRAVITATIONAL COLLAPSE, AS PREDICTED BY THE RELATIVITY THEORY. GREAT, JUST LIKE OPPIE SAID. WAIT TILL HE HEARS HE WAS RIGHT AFTER ALL.\n\n**Five years later, Wheeler lectured at a special meeting in Dallas, marking the discovery of quasars.** _\"Computer simulation shows that the collapse of a burnt-out star is remarkably similar to the highly idealized one computed by Oppenheimer and Snyder. \"_\n\n_As seen by an outside observer, the collapse slows down and becomes frozen at the critical radius. But as seen by an observer moving with the star's surface, the collapse is continuous right through the critical radius and on inward without hesitation._\n\nOPPIE, COME ON IN WHEELER IS SHOUTING YOUR PRAISES ON COLLAPSING STARS. HE'S BEEN CON \u2013 VERTED. PLEASE, DON'T BOTHER ME. CAN'T YOU SEE I'M MEDITATING ON THE BHAGAVAD GITA!\n\nWheeler was heartsick that Oppenheimer had lost interest in collapsing stars. But Oppie was worn out by years of political intrigue \u2013 directing the Manhattan Project, dealing with the tragedy of Hiroshima and Nagasaki, accusations of disloyalty to his country, and ignominiously losing his security clearance. Like a burnt-out star, the former **wunderkind** was himself collapsing into his own world, cut off from the rest of the Universe.\n\nBut for Wheeler, a new chapter in the history of physics had begun. _\"Whatever the outcome of our studies, one feels that at last in stellar implosion we have a situation where general relativity dramatically comes into its own and where its fiery merge with quantum physics will be consummated. \"_\n\nAt that time, in 1962, Stephen William Hawking arrived at Cambridge University. Hawking was destined to take the first step in Wheeler's dream scenario of combining general relativity and quantum mechanics. But he was already feeling the symptoms of a disease that would, in ten years time, put him in a wheelchair and, in twenty years, destroy his ability to speak.\n\n## **The Hawking Era**\n\nA visitor to the Cambridge University Department of Applied Mathematics and Theoretical Physics (DAMTP) would find a large photograph of the present Lucasian Professor of Mathematics, Stephen Hawking, permanently displayed in DAMTP's modest reception area beside the portraits of two giants of mathematical physics who had previously held this post, Sir Isaac Newton and Paul Dirac \u2013 world-renowned for his work on relativistic quantum mechanics.\n\nAn original copy of Hawking's thesis of 1965 is tucked away, with a few hundred others, in the first-floor library of DAMTP, with most of the equations written in his own hand. This manuscript represents the beginning of a new era in modern cosmology.\n\nHawking had come to Cambridge from Oxford to study with the world-famous cosmologist, Sir Fred Hoyle. But he was disappointed.\n\nWith his characteristic intellectual bravado, Hawking called his doctoral thesis, **Properties of Expanding Universes.** Already in the second line of the thesis abstract, there is a reminder of Hawking's early days at Cambridge. He summarized...\n\nCHAPTER ONE SHOWS THAT THE EXPANSION OF THE UNIVERSE CALSES GREAT DIFFICULTIES FOR THE HOYLE NARLIKAR THEORY OF GRAVITATIN.\n\nFred Hoyle was the best-known of the three authors of the steady state theory of the Universe, along with Hermann Bondi and Thomas Gold, two refugees from Nazi Europe.\n\nTHIS MODEL PROPOSES THAT MATTER IS CONTINUOUSLY BEING CREATED AS THE UNIVERSE EXPANDS, IN UTTER DISAGREEMENT WITH THE BIG BANG NOTION OF AN INFINITELY DENSE INITIAL STATE.\n\nIn the early 60s, the steady state model was probably accepted by more astrophysicists and cosmologists than the big bang. Hoyle was particularly upset by aspects of the opposing model. On a BBC radio show in 1950, he had the ignominious distinction of being the first to call it the **Big Bang** \u2013 in derision, of course.\n\nTHIS INSTANTANEOUS CREATION OF THE UNIVERSE IS LIKE A PARTY GIRL BIRTHDAY CAKE, IT'S RIDICULOUS, I CALL IT THE BIG BANG. NOW, ON THE OTHER HAND, MY OWN STEADY STATE THEORY...\n\nTwelve years after this jibe, Hoyle was still developing aspects of gravitation theory at DAMTP, with a postgraduate student named Jayant Narlikar, to support the steady state model.\n\nHawking, who was floundering with his own research in his first months at Cambridge, became interested in Narlikar's calculations and began hanging around his office in the spirit of DAMTP's policy of free inquiry, open discussion and sharing of ideas. Hoyle knew nothing of this.\n\nI WOULD LIKE TO HELP YOU WITH THESE EQUATIONS. THEY ARE FASCINATING. BESIDES, I HAVEN'T FOUND A THESIS PROBLEM YET. I APPRECIATE YOUR HELP STEPHEN, BUT AREN'T YOU WORRIED ABOUT YOUR OWN WORK?\n\nHawking had become more and more involved in Narlikar's difficulties with the project Hoyle had assigned.\n\nAn experienced publicist, Hoyle would often present his ideas in advance of publication, before the work was refereed, in order to keep his name in the newspapers and the research grants coming in. He scheduled a talk at the prestigious Royal Society to discuss his latest ideas based on Narlikar's calculations.\n\nARE THERE ANY QUESTIONS?\n\nTHE QUANTITY YOU ARE TALKING ABOUT DIVERGES. HE MEANS, ANY CONCLUSIONS BASED ON THIS QUANTITY ARE MOST LIKELY WRONG!\n\nOF COURSE IT DOESN'T DIVERGE.\n\nIT DOES!\n\nHOW DO YOU KNOW?\n\nBECAUSE I WORKED IT OUT.\n\nHoyle was furious, as an embarrassed laugh passed through the room. It was a dramatic confrontation between one of the world's best-known cosmologists and the student he had rejected. The session was quickly adjourned.\n\nHawking was indeed right about the divergence of Hoyle's equations, and the new approach was abandoned. Hoyle had essentially had his work \"refereed\" in an open forum by an unknown postgraduate student.\n\nLater, Stephen wrote a paper summarizing the mathematical methods he had used, which established him as a promising young research.\n\nNOW ALL I'VE GOT TO DO IS DECIDE ON A THESIS PROBLEM!\n\nWas this arrogance... or just naked ambition? If it was the latter, it certainly worked. Stephen William Hawking was no longer an unknown postgraduate student.\n\n## **The Unselfish Thesis Supervisor**\n\nDennis Sciama turned out to be a committed thesis supervisor, in the true tradition of the unselfish tutor who stimulates his charges to look for ways to increase their experience.\n\nI KNEW OF STEPHEN'S ILLNESS BUT TREATED HIM LIKE ANY OTHER. WHICH IS THE ONLY WAY I WANT IT!\n\nHe refused to speed up Hawking's doctoral programme, even when pressured by Stephen's persuasive father.\n\nI STILL WONDER IF STEPHEN WILL EVER FIND A SUITABLE TOPIC FOR HIS THESIS...\n\nSciama developed a unique style of supervising his postgraduate students. He would not share in their work, as many other professors did around the world (he has hardly ever written any joint papers). He does not even choose their topics.\n\nOne of Sciama's important activities was to arrange for his students to attend important seminars. He always seemed to know what was going on. In the mid 60s, the Cambridge group became interested in the work of a young applied mathematician, Roger Penrose, then based at Birkbeck College in London.\n\nAfter graduating from Cambridge and research in the US, Penrose had begun to develop ideas about **singularity theory** , which matched well with the ideas of the Cambridge research group.\n\nALTHOUGH PENROSE WASN'T ONE OF MY FORMER STUDENTS, I DID SEDUCE THE PROLIFIC MATHEMATICIAN INTO WORK IN COSMOLOGY DURING THE LATE 1950 S...\n\nIt was only a few years after John Wheeler had accepted Oppenheimer's solution and the existence of black holes, that Sciama started sharing his enthusiasm with some of his colleagues and students. Penrose, already one of the world's top mathematicians, got a flash of inspiration about these exotic objects from Sciama in a Cambridge coffee shop.\n\nDENNIS, I'M SURE I CAN APPLY MY NEW MATHEMATICAL METHODS IN TOPOLOGY TO THE PROBLEMS OF COLLAPSING STARS...\n\nPenrose was soon able to show that if a star collapses beyond a certain point, it could not re-expand. Within the framework of general relativity, the star could not avoid becoming infinitely dense, i.e. it would form a **singularity** at its centre.\n\nIt was not true \u2013 as many insisted \u2013 that the matter of that star would \"fly past itself\" and expand again. Instead, a singularity of space-time would occur, a point at which time came to an end and the laws of physics broke down. It was the first **singularity theorem.**\n\n## **Something You Need to Know: What is a Singularity?**\n\nGenerally speaking, a singularity is a point at which a mathematical function cannot be defined. The function is seen to diverge to infinitely large values.\n\nFor example, the simple algebraic equation Y = 1\/X has a singularity for value X = 0. If we make positive values of X arbitrarily small, then Y is arbitrarily large in the vertical (or positive) direction.\n\nIf we then plug in arbitrarily small negative values of X, we find Y has an arbitrarily large negative value. Thus, for the smallest change imaginable in the variable X, say from +0.000001 to -0.000001, Y changes from +1 million to -1 million. Clearly at X equals 0, something has gone wrong. This is a mathematical singularity.\n\nIn general relativity, a singularity is a region of space-time in which the curvature becomes so strong that the general relativistic laws break down, and presumably the laws of quantum gravity take over.\n\nIf an attempt is made to describe a singularity using general relativity alone, an incorrect result is obtained: mainly that the curvature and the tidal gravity is infinite at that point. Quantum gravity probably replaces these infinities with \"quantum foam\" \u2013 and merges with the laws of general relativity.\n\nBut this does not mean that singularity points cannot be studied and the physics near these points understood. There are certain singularity theorems that yield important qualitative information under certain conditions. For example, if the mathematics are handled carefully, the proof of the existence of a true singularity can be a result with physical meaning. Thus, the singularity theorems of Penrose and, later, Hawking.\n\nIn the Schwarzschild solution of Einstein's field equations, the critical radius is not a real singularity (in spite of its early description as the Schwarzschild singularity). The physical processes are continuous across the boundary, and a simple change in the mathematical coordinates removes the divergence.\n\nA group of Sciama's students were at Penrose's London seminar when he announced that he had proved that **a singularity definitely exists when a star collapses to form a black hole.**\n\nStephen Hawking was not at Penrose's seminar that day. But the news reached him immediately and made a deep impression.\n\nPENROSE'S RESULTS ARE VERY INTERESTING. I WONDER IF THEY COULD BE ADAPTED TO UNDERSTANDING THE ORIGIN OF THE UNIVERSE; THE EXPANDING UNIVERSE AS A GIANT COLLAPSING STAR IN REVERSE. YOU MEAN BY REVERSING THE SENSE OF TIME...\n\nYES. MAYBE THE SAME CONSIDERATION APPLIES AS IN HIS THEOREM FOR STARS. I'M GOING TO TRY TO ADAPT HIS RESULT TO THE WHOLE UNIVERSE AND SEE WHAT HAPPENS. OKAY IT SHOULD BE VERY INTERESTING.\n\nHawking had just one year left as a research student, and only now did he have a challenging problem. To adapt Penrose's method, he had to work hard, learn the mathematics involved and write it up as the last chapter in his thesis \u2013 his first singularity theorem for the **beginning of the Universe.**\n\nHawking had shown that if general relativity is correct, there must have been a singularity in the past which was a beginning of time.\n\nA tradition in British universities allows for PhD students to have their thesis examined by their supervisor and another Outstanding external expert chosen by their supervisor.\n\nANYTHING THAT MIGHT EXIST BEFORE THAT SINGULARITY COULD NOT BE CONSIDERED PART OF THIS UNIVERSE. A BRILLIANT PIECE OF WORK. SO GUESS WHO SCIAMA CHOSE TO EXAMINE STEPHEN'S THESIS \u2013 THAT'S RIGHT, ME!\n\nHawking passed and received his PhD in 1965. There were some complications \u2013 like infinite and non-infinite universes \u2013 but over the next few years he developed new techniques to remove these problems.\n\nThis has become generally accepted and today everyone assumes that the Universe started with a big bang \u2013 a highly dense and hot initial state. This is Hawking's major contribution to big bang cosmology, a major result for which he was to become known worldwide. Thus by 1970, five years after receiving his PhD, Stephen Hawking was an internationally known cosmologist.\n\nStephen Hawking has been a proponent of the big bang model since his early days as a postgraduate student. His PhD thesis which criticized Hoyle's steady state model and his proof of a big bang singularity link his name with the success of the latter for all time.\n\nIt is interesting to imagine the recent history of cosmology (or at least the recent history of Stephen Hawking), if his application to study with Hoyle at Cambridge had been approved.\n\nToday, Hoyle and his former student of 30 years ago, Jay Narlikar, are still patching up the steady state model. But it is a dead duck. The world of cosmology has moved on. Perhaps this is best shown by the **Scientific American** article in the October 1994 special issue on the Universe, which promises to become the accepted description of our understanding of the Universe into the next millennium.\n\n## **The Evolution of the Universe**\n\n_Understanding of the evolution of the Universe is one of the great achievements of 20th-century science. This knowledge comes from decades of innovative experiments and theories. Modern telescopes on the ground and in space detect the light from galaxies billions of light-years away, showing us what the Universe looked like when it was young. Particle accelerators probe the basic physics of the high energy environment of the early Universe. Satellites detect the cosmic background radiation left over from the early stages of expansion, providing an image of the Universe on the largest scales we can observe._\n\n_Our best efforts to explain this wealth of data are embodied in a theory known as the standard cosmological model or the big bang cosmology. The major claim of the theory is that in the large-scale average the Universe is expanding in a nearly homogeneous way from a dense early state._\n\n_At present, there are no fundamental challenges to the big bang theory, although there are certainly unresolved issues within the theory itself. Astronomers are not sure, for example, how the galaxies were formed, but there is no reason to think the process did not occur within the framework of the big bang. Indeed, the predictions of the theory have survived all tests to date. **(Scientific American** , October 1994)._\n\n## **1965: a Big Year for Hawking**\n\nHawking married his sweetheart Jane Wilde in Trinity Chapel at Cambridge in July 1965. Though he was now hobbling more and more on his cane, he had his PhD, a devoted and intelligent wife and new mathematical skills to use in cosmology. He also received a fellowship at Caius College to continue work at DAMTP. He was no longer depressed.\n\nAnd still that cocky, determined look on his face which said... I can do anything. Nothing can stop me, not even ALS.\n\n## **An Unstoppable Mind**\n\nStories abound of Hawking's prodigious mental abilities, already apparent in his Oxford undergraduate years.\n\nSeveral fellow students had spent weeks on a major assignment, some thirteen problems from a difficult text, _Electricity and Magnetism_ by Bleaney & Bleaney. They were told to do as many as possible. Most managed to complete only one or two in the time allotted. Characteristically, Hawking left it to the last day. After spending the morning in his room, he emerged to say he was only able to complete the first **ten** of the problems!\n\nOne of the Oxford tutors supervising Hawking's work in statistical physics had assigned several problems from a textbook which Stephen disliked. At the next tutorial he returned, not with the work completed, but with all the mistakes in the textbook marked out. The tutor quickly realized that Hawking knew more about the subject than he did.\n\nNear the end of his term at Oxford and no doubt beginning to feel the effects of ALS, Hawking took a terrible fall down a staircase in the university hall. As a result, he temporarily lost his memory. He could not even remember his name.\n\nAfter several hours of interrogation by his friends, he finally returned to normal but was worried about possible permanent brain damage. To be sure, he decided to take the Mensa test for individuals with superior intelligence. He was delighted to find that he had passed with flying colours, scoring between 200 and 250!\n\nNothing, not even the dreadful illness of ALS, could stop that mind.\n\n## **The Sixties Revolution**\n\nIt is debatable whether or not social historians of the 21st century will continue to analyse and report the decade of the 1960s as a period of great social upheaval and radical change on the Earth.\n\nBut it is certain that historians of science will view the same period as a time of radical change in our understanding of the cosmos. It is already being referred to as the golden age of relativistic cosmology \u2013 and **relativistic cosmology** is where it's at!\n\nHeroes of the 60s \u2013 from the moppet-headed Beatles to the crowd at Woodstock \u2013 have become familiar icons. Similarly, the revolution in cosmology also has its heroes, but they are mostly unknown to the general public.\n\nThe 60s was a time of remarkable progress in observational astronomy, due mainly to major advances in technology and instrumentation. All sorts of observed unusual phenomena led to new models of celestial objects, which can only be described as a revolution in cosmology. The beginning of this revolution can be traced to a crucial meeting at a time and place indelibly marked in the history of the 20th century \u2013 but for a different reason.\n\nIf you ask a sample of over people over fifty years old if they remember **Dallas 1963** , most will immediately describe exactly what they were doing when John F. Kennedy was gunned down in Dallas on 22 November.\n\nBut perhaps one small subset of that group might give an ambiguous response. Of course, they would remember Kennedy's tragic death. But **Dallas 1963** would have another _special_ connotation for the group of three hundred astronomers, astrophysicists, cosmologists and relativists who attended the **First Texas Symposium on Relativistic Astrophysics** to mark the discovery of quasars. The symposium was held in Dallas, 16\u201318 December 1963, only three weeks after JFK's assassination.\n\nThe **relativists** , odd-ball specialists who spent their working lives playing around with Einstein's equations, had been invited to join real astronomers and astrophysicists in a dialogue. At last, 25 years after the famous Oppenheimer and Snyder paper on collapsing stars, general relativity was being suggested as a possible explanation for a new physical phenomenon that **had actually been observed by practical working astronomers**.  \nIt was thought that gravitationally collapsed stars (soon to be called black holes) might be producing the massive energy necessary to explain observations on the new and exciting objects called quasars.\n\nThomas Gold, one of the developers of the Steady State Universe, gave the after-dinner speech at the Dallas Symposium.\n\nTHE DISCOVERY OF THE QUASARS ALLOWS ONE TO SUGGEST THAT THE RELATIVISTS AND THEIR SOPHISTICATED WORK ARE NOT ONLY MAGNIFICENT CULTURAL ORNAMENTS BUT MIGHT ACTUALLY BE USEFUL TO SCIENCE! EVERYONE IS PLEASED; THE RELATIVISTS \u2013 WHO FEEL THAT THEY ARE BEING APPRECIATED AND ARE EXPERTS IN A FIELD THEY HARDLY KNEW EXISTED \u2013 AND THE ASTRO \u2013 PHYSICISTS WHO HAVE ENLARGED THEIR EMPIRE BY THE ANNEXATION OF ANO \u2013 THER SUBJECT. GENERAL RELATIVITY. IT'S ALL VERY PLEASING, SO LEFT US ALL HOPE THAT IT IS RIGHT.\n\nIt did turn out to be right, as Hawking himself modestly admits 30 years later.\n\nThere has been a great change in the status of general relativity and cosmology in the last thirty years. When I began research in the Department of Applied Mathematics and Theoretical Physics (DAMTP) at Cambridge in 1962, general relativity was regarded as a beautiful but impossibly complicated theory that had practically no contact with the real world. Cosmology was thought of as a pseudo-science where wild speculation was unconstrained by any possible observations.\n\nThe position today is very different, partly due to the great expansion in the range of observations made possible by modern technology, but also because we have made tremendous progress on the theoretical side.\n\nThis is where i can claim to have made a modest contribution.\n\nBut observations on quasars required completely new observational techniques. So before describing the excitement about quasars, it might be a good idea to review **something you need to know.**\n\n## **Something You Need to Know: the Electro Magnetic Spectrum**\n\nThe **electromagnetic spectrum** sounds very technical because the two words are seldom used outside physical science. The first term, _electromagnetic_ , just means that the waves we will speak of (light, radio, infrared) are made up of vibrating electric and magnetic fields. The second term, _spectrum_ , refers to the range of sizes of the waves, i.e. their wavelengths.\n\nThe EM spectrum refers to all the possible wavelengths of radiation existing in nature. Different-sized waves have different properties and are generated by different physical processes. Furthermore, they must be detected by completely different equipment. The invisible radiation coming from stars and galaxies (in addition to the visible or optical band) gives useful information, though it can't be seen with the unaided eye.\n\nThe wavelengths cover a wide range of values from X-rays (smaller than the distance between atoms) to radio waves (several kilometres in length). The waves all travel at the same speed as light and there is a remarkably simple relationship between the wavelength, the frequency of the source emitting the waves and the speed of transmission:  \n **(wavelength) x (frequency) = (speed of light)**\n\nBefore the 1960s, observational astronomy meant optical (or visible) astronomy only \u2013 looking through telescopes composed of glass lenses or reflecting mirrors and observing with the eye or with very sensitive cameras. Special films did extend measurements into the invisible infrared band with longer wavelengths than visible light.\n\nBut during the late 1950s and 1960s, nearly the whole electromagnetic band became detectable to observational astronomers, such that now we have _radio_ astronomy, _microwave_ astronomy, _infrared_ astronomy, _optical_ astronomy, _ultraviolet_ astronomy, _X-ray_ astronomy and even _gamma-ray_ astronomy.\n\nThe great discoveries of the 1960s came from these extensions of observations outside the visible range, particularly to the longer wavelength microwave and radio bands. **Quasars** and **pulsars** were discovered in the **radio frequency band** and the **cosmic background radiation** was detected in the microwave band. And in the 1970s, **X-ray astronomy** , at the other end of the spectrum, produced the first evidence for the existence of black holes from observations of the constellation Cygnus X-1.\n\nMY ELECTRO \u2013 MAGNETIC THEORY, IN 1867, PREDICTED THE EXISTENCE OF ALL THESE WAVES.\n\n## **1963: Quasars**\n\nCareful observations by radio and optical astronomers in the years 1960 to 1962 showed that there were over a half-dozen bright objects in the sky which were small enough to be stars but had a weird light spectrum \u2013 not like any star seen before.\n\nEveryone was puzzled until 5 February 1963 when astronomers Maartin Schmidt and Jesse Greenstein at Caltech made a discovery.\n\nWE REALIZED THAT THE HYDROGEN SPECTRAL LINES IN THE LIGHT FROM THESE OBJECTS (AN EASY-TO-RECOGNIZE SIGNATURE IN THE SPECTRUM OF A STAR)......HAD BEEN SHIFTED BY A VERY LARGE PERCENTAGE TOWARDS THE RED (LONGER) WAVELENGTH.\n\nMeasurements indicated that these _quasi-stellar objects_ (later to be named _quasars)_ were moving away from the Earth at enormous speeds and therefore must be very, very far away.\n\nTHIS CONCLUSION COMES FROM THE SAME METHODS I USED IN THE 1930 S TO DISCOVER THAT THE UNIVERSE WAS EXPANDING.\n\nThey were first thought to be stars in the Milky Way galaxy, but their discoverers soon argued that these objects were moving away from the Earth as a result of the Universe's expansion. At the enormous distances calculated, their brightness implied they were radiating 100 times more energy than the **most luminous galaxy ever seen.**\n\nWHEN I STARTED THE EARTH WAS NOT EVEN FORMED.\n\nThe only possible explanation for such energies seemed to be the gravitational collapse of stars. That meant general relativity.\n\n## **1965: the Cosmic Background Radiation**\n\nIn 1965, an accidental discovery of mysterious microwaves from outer space turned out to be the first experimental indication that the Big Bang model might be correct. Until that time the model was thought to be something of a joke. Here's how it happened...\n\nThe picture of the Universe as a primordial atom (\"cosmic egg\") by Abb\u00e9 Georges Lema\u00eetre in 1927 led some cosmologists to picture the early universe as a hot, dense, rapidly evolving plasma. One of the more imaginative of these theorists, a free-thinking Russian \u00e9migr\u00e9 to the USA named George Gamow, considered the effect of the cooling of this plasma as the Universe expanded.\n\nHe then made one of the most important predictions in the history of science.\n\nTHE UNIVERSE MAY TODAY BE FILLED WITH A COSMIC BACK GROUND RADIATION COMPOSED OF ANCIENT PHOTONS RELEASED BY THE BIG BANG. AFTER CORRECTIONS OF SOME ERRORS IN MY ORIGINAL CALCULATION, IT WAS SHOWN THAT THE TEMPERATURE OF THIS RADIATION SHOULD TO-DAY BE ABOUT FIVE DEGREES ABOVE ABSOLUTE ZERO.\n\nTwo of Gamow's colleagues Ralph Alpher and Robert Herman actually predicted that this radiation could still be detected.\n\nAll hot bodies (i.e. any object which has a temperature) give off continuous electromagnetic waves called thermal radiation, **even** if the temperature is only 5 degrees above absolute zero. The question was how to measure the radiation \u2013 which wavelength band to search.\n\nTo follow this part of the story, there is _**Something Else You Really Need to Know!**_\n\n## **Something You Need to Know \u2014 Thermal Radiation**\n\nCURRENT PHYSICS TEXTBOOKS DEVOTE HARDLY MORE THAN ONE OR TWO PAGES TO IT. YET, TWO VERY IMPORTANT ASPECTS OF MODERN COSMOLOGY REQUIRE AN UNDERSTANDING OF THERMAL RADIATION: THE COSMIC BACKGROUND RADIATION AND STEPHEN HAWKING'S MOST IMPORTANT DISCOVERY, BLACK HOLE RADIATION. YOU MUST UNDER \u2013 STAND THERMAL RADIATION.\n\nThe underlying physics of thermal radiation is quite simple, although it did require a radical hypothesis (which began the quantum theory) by Max Planck in 1900 to explain the details. His theory showed how the relative rate of emission of radiant energy (electromagnetic waves) depends on wavelengths at different temperatures. Planck's theoretical curves show that the radiation spreads out and the peak shifts to **longer** wavelengths as the temperature **drops**.\n\n  At 800 degrees centigrade, enough visible radiation is emitted to appear red hot, though most of the energy emitted is in the infrared band.\n\n  At 300 degrees centigrade practically all of the energy emitted is carried by waves longer than red light and are called infrared, meaning _beyond the red._ No radiation is emitted in the visible band.\n\n  At 5 degrees above absolute zero (or minus 268 degrees centigrade) the radiation is completely beyond the infrared in the microwave band and special microwave receivers are required to make the measurements.\n\nSince the shape of the curve is uniquely determined by the temperature of the emitting bodies, measurements at different wavelengths can infer the temperature of the body emitting the radiation. Conversely, if the temperature of the emitting body is known, then the shape and the distribution of the radiation can be predicted from theoretical formulae.\n\nKEEP THIS INFORMATION IN MIND. THIS SIMPLE PHYSICS IS CRUCIAL TO UNDERSTANDING THE RADIATION FROM THE COSMIC BACKGROUND AND FROM BLACK HOLES.\n\nReturning to Gamow's prediction, the theoretical curve for the _thermal radiation distribution_ at 5 degrees above absolute zero indicated that the peak radiation should be in the **microwave** region of the electromagnetic spectrum.\n\nWhile other groups were in the process of planning experiments to look for Gamow's microwaves, they were discovered accidentally by two researchers, Arno Penzias and Robert Wilson, at Bell Telephone Laboratories in northern New Jersey, USA.\n\nDON'T BLAME US! ROBERT, WHAT'S THAT PERSISTENT HISS WE'RE PICKING UP ON OUR MICROWAVE HORN ANTENNA? YEAH, ARNO, IT'S BUILT FOR SATELLITE COMMUNICATION. MUST BE THOSE DAMN PIGEONS?\n\nThe white-hot fireball of the big bang had thinned out and been cooled by the expansion of the Universe. The radiation was still there, though its wavelengths had been stretched by the expansion all the way to the microwave band \u2013 where Penzias and Wilson discovered it.\n\nThough they could only make measurements at a single wavelength, Penzias and Wilson won a Nobel Prize for being the first to experimentally confirm this unique evidence for the Big Bang.\n\nA whole new field of research in cosmology was opened up by this discovery \u2013 studying the origin of the Universe from the **cosmic background radiation.**\n\nThe discovery of the microwave background in 1965 ruled out the Steady State Theory and showed that the Universe must have been very hot and dense at some time in the past. But the observations themselves did not exclude the possibility that the Universe bounced at some fairly large but not extremely high density. This was ruled out on theoretical grounds by the sigularity theorems that Penrose and I proved. We published That Singularities of Gravitational collapse and Cosmology, an all-purpose singlarity theorem which showed that the classical concept of time must have a beginning at a sigularity in the past (i.e. the Big Bang). This theorem also implied that time would come to an end for at least part of spacetime when a star collapsed. Most of my work since then has been concerned with the consequences and implications of these results.\n\nRadio astronomers continued finding many more radio galaxies (i.e. galaxies emitting electromagnetic waves primarily in the radio waveband).\n\nThen in 1967, a Cambridge postgraduate student named Jocelyn Bell detected highly regular sharp pulses at 3.7 metres wavelength from one of these galaxies. The Cambridge radio astronomers thought they had contacted an extra-terrestrial civilization!\n\nGENTLE \u2013 MAN, THESE ARE NO ORDINARY PULSES. IT'S GOTTA BE FROM SOME ORGANISED SOURCE... GOOD LORD! YOU MEAN,. YES LITTLE GREEN MEN!!\n\nThe pulses were very narrow. This meant that the emitter had to be very small, because you can't have a large body emitting short, sharp pulses. The travel time of the radiation from its different parts would smear out the signal. It had to be something highly compact; an object smaller than a few thousand kilometres in size, yet at the distance of a star.\n\nAs the Cambridge radio astronomers announced their results, the DAMTP theorists, Sciama, Hawking, Rees sat smugly at the seminar.\n\nSURELY ANOTHER DISCOVERY WHICH IMPLIES GRAVITATIONAL-LY-COLLAPSED STARS AND GENERAL RELATIVITY. THEY CLEARLY HAD TO BE VERY COMPACT OBJECTS WHICH WERE ROTATING, BUT IT WASN'T CERTAIN WHETHER THEY WERE WHITE DWARFS, ALREADY KNOWN TO ASTRONOMERS, OR WHETHER THEY WERE THE SO CALLED NEUTRON STARS \u2014 VERY MUCH MORE COMPACT THAN WHITE DWARFS, ALMOST IN THE BLACK HOLE STATE.\n\nIt took a few months of discussion before it became clear. Tommy Gold, who had worked earlier on the steady state theory, was the first to make the argument clear.\n\nPULSARS ARE ROTATING NEUTRON STARS AND CAN'T BE ANYTHING ELSE. RADIO WAVES BEAMING OUT OF THE NEUTRON STAR REACH THE EARTH INTERMITTENTLY AS THE STAR ROTATES, SIMILAR TO A LIGHT-HOUSE.\n\n## **Black Holes \u2014 Wheeler Gives the Media a Buzz Word**\n\nAs the 1960s were coming to a close, everyone was talking about _gravitationally collapsed stars._ The **partially** collapsed stars \u2013 white dwarfs and neutron stars \u2013 had become everyday objects to astronomers. But John Wheeler was interested in **massive stars** which collapse **completely.**\n\nI GOT FED UP SAYING \"GRAVITATIONALLY COLLAPSED STARS\". AT A MEETING ON SPACE PHYSICS IN NEW YORK CITY IN 1969, I SIMPLY STARTED CALLING THEM BLACK HOLES.\n\nIt had a magic effect. Everybody immediately began using the term. Even specialists could now know they were speaking about the same thing. In Moscow, Pasadena, Princeton and Cambridge, black holes replaced \"gravitationally completely collapsed stars\".\n\n## **The Age of Black Holes**\n\nThe media went nuts. At least they could encapsulate all this new complicated physics and astronomy in two simple words which fell easily into newspaper columns. Writers picked up on the new buzz word and books appeared on the popular science and sci-fi shelves. On TV, _Star Trek_ had exotic new destinations for its space ships. At dinner parties, scientists were put on the spot to explain black holes to friends. Black holes had become household words... but did anyone really know what they were?\n\nWAS IST EIN SCHWARZES LOCH? ? QUE ES UN AGUJERO NEGRO? WHAT IS A BLACK HOLE? QU' EST-CE QU'UN TRON NOIR?\n\nJUST WHAT IS A BLACK HOLE?\n\nBUT HOW DOES NATURE SQUEEZE THOSE ENORMOUS OBJECTS IN THE SKY?\n\nYOU KNOW, I'VE ALWAYS WONDERED WHAT STARS WERE MADE OF AND HOW THEY GIVE OFF LIGHT.\n\n## **The Birth and Death of Stars**\n\nStars are formed when the mutual gravitational attraction between molecules floating in space, mostly hydrogen gas, causes lumps to form. As these aggregates coalesce, gravity presses the molecules closer and closer together until they interact under high pressure causing an increase in temperature.\n\nThis process continues until the gas begins to glow and produce EM radiation of all different wavelengths. As the compression increases, the interaction intensifies until the radiation pressure is great enough to stop further gravitational contraction.\n\nThe star then reaches a dynamic equilibrium and shines brightly for several billion years.\n\nBUT WHERE DOES THE STAR GET THE ENERGY TO CONTINUOUSLY HEAT THE MOLECULES AND PRODUCE THE RADIATION?\n\nThis problem was first solved by the great English scientist (perhaps the world's first astrophysicist) Sir Arthur Eddington. His famous monograph **On the Internal Constitution of Stars** explained how a star can be fuelled by a reaction at its core providing the energy to heat continually the gas atoms.\n\nHOWEVER AT THAT TIME I DID NOT UNDERSTAND THE PROCESS AT THE STAR PRODUCING THE RADIATION. IT WAS LEFT TO THE NUCLEAR PHYSICIST HANS BETHE IN 1938.\n\nI SHOWED THAT A NUCLEAR FUSION REACTION (COMBINING HYDROGEN AND DEUTERIUM TO FORM HELIUM) PRODUCES THE ENERGY. MY MASS\/ENERGY EQUATION E=MC2 SHOWS HOW MUCH ENERGY COULD BE PRODUCED AS SOME OF THE STAR'S MASS IS CONVERTED INTO HEAT AND RADIATION, GRADUALLY BURNING ITS FUEL.\n\nAND THEN WHEN THE FUEL IS EXHAUSTED, GRAVITY TAKES OVER AGAIN AND CRUNCHES THE STAR INTO A BLACK HOLE, RIGHT?\n\n## **How Stars Collapse to Form White Dwarfs, Neutron Stars & Black Holes**\n\n**White Dwarf** (radius-1 ,600 miles)\n\nIf M is less than 1.4, star contracts until gas atoms overlap. Electron repulsion force is enough to stop contraction.\n\n**Neutron Star**  \n(radius\u201316 kilometres)\n\nIf M is greater than 1.4, gravity overcomes electrons' heroic stand and pushes them down into the nucleus. The electrons combine with protons to form neutrons. Neutron repulsion stops contraction if M is less than 3.0.\n\n**Black Hole**\n\nIf M is greater than 3.0 (three times solar mass) nothing can stop the contraction. The star collapses completely and disappears from view. A black hole forms.\n\nTraces of some White Dwarfs have been photographed and blips of rotating Neutron Stars can be detected with radio telescopes. But black holes will never be seen directly.\n\nIn the black hole case, the space curvature is so extreme that, at a particular radius (called the event horizon), the light from the star's surface is bent in on itself, i.e. the rays actually go _into_ the star instead of away from it. The star disappears from view to an outside observer.'\n\nThese circles of decreasing size show how a very massive burnt-out star, as its diameter decreases, passes through an event horizon to form a black hole, ultimately becoming a singularity at its own centre.\n\nThe illustration below presents the same information in a 3-dimensional diagram which includes **time increasing** in the vertical direction.\n\nThis shows the bending of the light paths and indicates how the star's surface has shrunk all the way down to the singularity (right through the event horizon) as the star collapsed.\n\nIt is very important to understand the **path** of the light rays from the surface of the star as it passes through the event horizon.\n\n**Just before the horizon forms** , light rays are bent strongly by gravity and only just leave the star's surface.\n\nA few moments later, when the star is **just inside the event horizon** , the light rays are pulled into the interior of the star towards the singularity at the centre.\n\nBut between these two points, when the star **has just reached the event horizon** , gravity is too strong to let light escape but not strong enough to pull the rays into the interior of the star. The light rays hover just at the surface and this defines the event horizon.\n\nWhat would happen if someone flew into a black hole?\n\nEinstein and the relativists have an answer that outdoes science fiction. According to the Oppenheimer and Snyder solution, anyone who goes through the event horizon must eventually hit the singularity with disastrous results.\n\nHe will be pulled and squeezed \u2013 until, at the centre of the black hole, his body would be stretched infinitely long and his width squashed to zero like a length of spaghetti!\n\nEven the atoms in his body would do the same!\n\nSERVES HIM RIGHT! MY THEORY WASN'T OFF BY THAT MUCH... HOW DO WE KNOW THESE BLACK HOLES EXIST IF WE CAN'T SEE THEM? NOW THAT'S A REALLY GOOD QUESTION...\n\n## **Observational Evidence for Black Holes**\n\nStephen Hawking says there are thousands and thousands of black holes in the Milky Way galaxy alone. But until the day an astronomer is lucky enough to see a well-known star disappear, indirect methods must be used \u2013 such as observations on a binary star system with one visible and one invisible (i.e. the black hole) component. John Wheeler has an interesting metaphor for such a system.\n\nHAVE YOU EVER BEEN TO A BALL AND WATCHED THE YOUNG MEN DRESSED IN THEIR BLACK MEN DRESSED IN THEIR BLACK EVENING TUXEDOS AND THE GIRLS IN THEIR WHITE DRESSES WHIRLING AROUND IN EACH OTHER'S ARMS... AND THE LIGHTS ARE TURNED DOWN LOW SO THAT ALL YOU CAN SEE ARE THE GIRLS? WELL, THE GIRL IS THE VISIBLE STAR AND THE BOY IS THE BLACK HOLE. YOU CAN'T SEE THE BOY, BUT THE GIRL WHIRLING AROUND GIVES CONVINCING EVIDENCE THAT THERE MUST BE SOMETHING HOLDING HER IN ORBIT.\n\nIn December 1970, the X-ray satellite _Uhuru was_ launched from the coast of Kenya. Astronomers were about to use still another part of the EM spectrum \u2013 X-rays \u2013 to probe the heavens.\n\nWithin two years, over 300 sources of X-rays were detected. One of these in the constellation Cygnus (now called Cygnus X-1) looked like just the binary-star system the black hole enthusiasts were waiting for.\n\nWith good estimates of the mass of HDE 226868 and reliable observations of the period of revolution, astronomers could calculate the mass of the invisible component -10 times as massive as the Sun. **Too big to be a neutron star, it had to be a black hole.**\n\nTheorists quickly developed a model to explain the X-rays. They believe that the black hole is sucking off matter from its visible partner, forming an accretion disk around itself. The hot inner regions, moving close to the speed of light, produce intense bursts of X-rays shortly before the spiralling matter disappears down the hole.\n\nSince the discovery of Cygnus X-1, a second X-ray satellite launched in 1978 called Einstein has mapped out over 1000 X-ray sources. Only two or three are good candidates for black holes, whereas hundreds more have been identified as neutron stars. Nature seems to prefer the stable neutron state to that of the catastrophic black hole.\n\nX-RAYS BLACK HOLE ACCRETION DISC VISIBLE COMPONENT\n\n## **The 1970s: Hawking and Black Holes**\n\nBy the early 1970s, general relativity and black holes were definitely here to stay. Hawking, by now needing a four-legged walker to get about, was nevertheless poised and ready for action. He was working independently, choosing co-workers from all over the world. He was applying the advanced mathematical techniques introduced by Penrose \u2013 mainly from topology\u2013to the properties of black holes. John Wheeler's group at Princeton, Zeldovich and his students in Moscow and Kip Thorne \u2013 Wheeler's prot\u00e9g\u00e9 now at Caltech \u2013 could not keep up with him. He managed to master these new methods and stay a step ahead. His name became synonymous with black hole research.\n\nThorne became a close friend of Stephen and watched his development very closely.\n\n_In November 1970, Stephen Hawking was just beginning to reach full stride as a physicist. He had made several important discoveries already, but he was not yet a dominant figure. As the 70s began we watched him become dominant. With his severe disability, how has Hawking been able to out-think and out-intuit his leading colleagues\/competitors, people like Roger Penrose, Werner Israel, and Yakov Borisovich Zeldovich?_\n\n_They had the use of their hands; they could draw pictures and perform many page-long calculations on paper \u2013 calculations in which one records many complex intermediate results along the way, and then goes back, picks them up one by one, and combines them to get a final result; calculations that I cannot conceive of anyone doing in his head._\n\n_Hawking's mental pictures and mental equations have turned out to be more powerful, for some kinds of problems, than the old paper-and-pens ones, and less powerful for others, and he has gradually learned to concentrate on problems for which his new methods give greater power, a power that nobody else can begin to match. By the early 1970s, Hawking's hands were largely paralyzed; he could neither draw pictures nor write down equations. His research had to be done entirely in his head. But because the loss of control over his hands was so gradual, Hawking has had plenty of time to adapt. He has gradually trained his mind to think in a manner different from that of the minds of other physicists. He thinks in new types of intuitive mental pictures and mental equations that, for him, have replaced paper-and-pen drawings and written equations._\n\n## **Hawking's Eureka Moment**\n\nOne of the problems on which Hawking has used mental pictures to gain insight was his study of the surface area of black holes. What started as a rather esoteric problem in black hole dynamics, eventually led to his greatest discovery in physics.\n\nAs with Einstein's \"happiest thought\", Hawking too can remember exactly what he was doing when the germ of his best idea came to him.\n\nOne evening in November 1970, shortly after the birth of my daughter Lucy, I started to think about black holes as I was getting into bed. My disability makes this rather a slow process, so I had plenty of time.\n\nHe saw in a flash that **the surface area of a black hole can never decrease** , by considering the paths of light rays hovering just at the event horizon of two black holes. He did not need paper and pen, nor a computer \u2013 the pictures were in his head.\n\nSO MUCH SO THAT I LAY AWAKE MOST OF THE NIGHT.\n\nThe rays of light that form the event horizon, the boundary of the black hole, can never approach each other. Consequently, the area of the event horizon _(i.e. the black hole surface)_ might stay the same or increase with time, but it could never decrease.\n\nOtherwise, it would mean that at least some of the rays of light in the boundary would have to be approaching each other... _which is not possible!_\n\nThis statement may not seem so remarkable. Since nothing can get out of a black hole and **anything** can go in, how could a black hole get smaller anyway? But Hawking's idea was more general. Even if **two** black holes combine, the total surface area will always be equal to or greater than the sum of the two. It can never decrease. He published his result.\n\n**The surface area of a black hole can only stay the same or increase, but can never decrease.**\n\n**Hawking's Law of Area Increase**\n\nSuch a statement... _can never decrease_... immediately gets scientists thinking about the quantity called **entropy** which appears in the second law of thermodynamics: **The entropy (disorder) of a system can only stay the same or increase but never decrease (if the system is isolated and left to reach equilibrium).**\n\n## **The Laws of Thermodynamics**\n\nDuring the 19th century, a set of mathematical relationships were developed by chemists, geologists and physicists which combined several seemingly disparate concepts into a few powerful laws. Such quantities as heat and the energy of motion were shown to be different forms of the same thing \u2013 namely energy \u2013 which had already been used to describe electrical, chemical and magnetic effects. **The total energy available in the Universe (the ultimate isolated system) was a constant and one form could be transformed into another.** This became known as the 1st Law of Thermodynamics.\n\nThe **2nd Law of Thermodynamics** is more subtle but just as profound. In a lecture delivered in 1854, Hermann von Helmholtz pointed out that as time elapsed all energy would eventually be transformed into heat at a uniform temperature and all natural processes would cease. This is the concept of the _heat death_ of the Universe based on the principle of the _dissipation of energy._\n\nAnother way of stating this principle was suggested by the German physicist Rudolf Clausius in 1865.\n\nI INTRODUCED A NEW CONCEPT TO WHICH I GAVE THE NAME ENTROPY, DEFINED IN TERMS OF THE HEAT TRANSFERRED FROM ONE BODY TO ANOTHER.\n\nHe showed that the total entropy of **a system always increases whenever heat flows from a hot body to a cold body** It also increases whenever mechanical energy is changed into internal (thermal) energy, as in certain collisions and frictional processes.\n\nA more general definition of entropy was proposed by the Austrian physicist Ludwig Boltzmann in 1878.\n\nACCORDING TO MY DEFINITION ENTROPY DEPENDS ON THE PROBABILITIES OF MOLECULAR ARRANGEMENTS. FOR EXAMPLE, IF A STATE HAS A VERY LARGE NUMBER OF DIFFERENT WAYS ITS MOLECULES CAN BE ARRANGED, THEN IT HAS A VERY LARGE ENTROPY.  \nAN EGG FALLS TO THE FLOOR AND BREAKS. IT IS UNLIKELY TO REFORM INTO ITS ORIGINAL SHAPE.\n\nThe principle of the dissipation of energy (generalized 2nd law of thermodynamics) can now be stated very simply: _The entropy of an isolated system always tends to increase._ What does this mean?\n\n_It means \u2013 heat does not flow by itself from cold bodies to hot bodies; a ball cannot bounce higher than its original position by converting heat into mechanical energy; an egg cannot unscramble itself. If the contrary events were to occur, they would not violate any of the principles of Newtonian mechanics \u2013 but they would decrease the entropy of a system and are thus forbidden by the 2nd law of thermodynamics. This law tells time which way to go!_\n\nHow important is this 2nd law of thermodynamics? It should be no less familiar to us than any of the works of Shakespeare, as the writer C.P. Snow remarked in his famous book, **The Two Cultures and the Scientific Revolution.**\n\nTHE HUMANISTS GIVE A PITYING CHUCKLE AT THE NEWS OF SCIENTISTS WHO HAVE NEVER READ A MAJOR WORK OF ENGLISH LITERATURE, CALLING THEM IGNORANT SPECIALISTS.  \nPROVOKED, I HAVE OFTEN ASKED HOW MANY OF THEM COULD DESCRIBE THE SECOND LAW OF THERMODYNAMICS. THE RESPONSE WAS COLD: IT WAS ALSO NAGATIVE.  \nYET I WAS ASKING SOMETHING WHICH IS ABOUT THE SCIENTIFIC EQUIVALENT OF ASKING SOMEONE: \"HAVE YOU EVER READ A WORK OF SHAKESPEARE'S?\"\n\nWHO'S C.P.SNOW?\n\n## **Now Back to Black Holes...**\n\nWhen bodies reach thermal equilibrium, they **have a temperature** and therefore must emit **thermal radiation** , exchanging energy with their surroundings as described on pages 98 and 99\n\nBut everyone knows that black holes do not emit anything \u2013 this is the defining characteristic of a black hole. Though anything can fall _into_ a black hole, nothing gets out \u2013 not even light or any other radiation.\n\nSO IT WAS GENERALLY UNDERSTOOD BY EVERYONE THAT IF BLACK HOLES DON'T REDIATE, THEY CANNOT HAVE A TEMPERATURE, AND THUS CANNOT HAVE ENTROPY. BLACK HOLES ARE CUT OFF FROM THE UNIVERSE AND ARE NOT IN THERMAL EQUILIBRIUM...  \nOR SO EVERYONE THOUGHT.\n\nThat is, until a physics postgraduate student working with John Wheeler in Princeton began to cause trouble.\n\n## **Controversial Birth of a New Idea**\n\nPrinceton New Jersey: John Wheeler and postgraduate student Jacob Bekenstein.\n\nJACOB, IT ALWAYS TROUBLES ME THAT WHEN I PUT A HOT TEACUP NEXT TO COLD TEACUP I HAVE INCREASED THE AMOUNT OF DISORDER IN THE UNIVERSE BY LETTING HEAT FLOW FROM ONE TO THE OTHER. I HAVE INCREASED THE DISORDER, THAT IS, THE ENTROPY, OF THE UNIVERSE. ISN'TTHIS RIGHT?  \nABSOLUTELY PROFESSOR WHEELER, IT IS THE SECOND LAW OF THERMO-DYNAMICS.\n\nBUT SUPPOSE I SEE A BLACK HOLE DRIFTING BY AND I DROP BOTH TEACUPS IN THE BLACK HOLE. THE EVIDENCE IS GONE FOREVER BECAUSE BLACK HOLES CANNOT HAVE ENTROPY, RIGHT? AND NOTHING CAN EVER COME OUT OF A BLACK HOLE.  \nLET ME THINK ABOUT THAT, PROFESSOR WHEELER.\n\nI THINK A BLACK HOLE DOES HAVE ENTROPY AND IT IS THE SAME AS THE SURFACE AREA HAWKING HAS SHOWN \u2013 WHICH ALWAYS INCREASES. I THINK I'M GOING TO WRITE A SHORT PAPER IDENTIFYING THE BLACK HOLE SURFACE AREA WITH THE ENTROPY OF A BLACK HOLE.\n\nMeanwhile back at DAMTP, Stephen Hawking and Brandon Carter are talking about Bekenstein's paper.\n\nI AM VERY DISTURBED. BEKENSTEIN HAS MISUSED MY DISCOVERY OF THE BLACK HOLE AREA INCREASE.\n\nOF COURSE HE HAS. EVERYONE KNOWS THAT IF AN OBJECT HAS ENTROPY IT MUST ALSO HAVE A TEMPERATURE. AND IF IT HAS A TEMPERATURE IT MUST GIVE OFF THERMAL RADIATION.\n\nBUT HOW CAN A BLACK HOLE GIVE OFF RADIATION IF NOTHING CAN ESCAPE FROM A BLACK HOLE? RIDICULOUS.\n\n## **August 1972, Les Houches Summer School on Black Hole Physics**\n\nHigh up on a hillside in the French Alps, Stephen Hawking, James Bardeen and Brandon Carter joined forces to deduce from Einstein's general relativity equations the full set of laws that govern the evolution of black holes. When they were finished, they had produced a set of **laws of black-hole mechanics** that bore an amazing resemblance to the laws of thermodynamics.\n\n**S (entropy) = k 1A (surface area of black hole)**\n\n**T (temperature) = k 2G (surface gravity of black hole) k1 and k2 are constants**\n\nEACH BLACK \u2013 HOLE LAW, IN FACT, TURNED OUT TO BE IDENTICAL TO A THERMODYNAMIC LAW, IF ONE ONLY REPLACED THE PHRASE \"BLACK HOLE' SURFACE AREA\" BY \"ENTROPY,\" AND THE PHRASE \"BLACK HOLE' SURFACE GRAVITY\" BY TEMPERATURE.  \nTHE COINCIDENCES WERE PILING UP.\n\nMeanwhile, Jacob Bekenstein was a student attending the summer school, still convinced that black holes have entropy.\n\nWE KNOW THESE LAWS LOOK LIKE THE LAWS OF THERMO-DYNAMICS... BUT THEY'RE NOT! IT'S JUST A COINCIDENCE. BLACK HOLES ARE UNIQUE OBJECTS. THERMODYNAMICS JUST DOESN'T WORK IN THIS CASE.\n\nWHAT'S WRONG WITH THESE GUYS?  \nDON'T THEY REALIZE HOW FUNDAMENTAL AND IMPORTANT THE SECOND LAW IS?\n\nLOOK AT THAT! PERFECT FIT BETWEEN THE TWO SETS OF LAWS \u2013 THE SURFACE AREA IS DEFINITELY THE BLACK HOLE'S ENTROPY.\n\nAfter the summer school, Bekenstein continued to identify the black hole surface area with entropy in the technical journals. Yet he did not assert that a black hole **has a temperature or that it must emit radiation.**  \nBekenstein was being inconsistent with the laws of thermodynamics.\n\nHawking, on the other hand, continued to attack Bekenstein's conclusions, but was becoming increasingly troubled.\n\nSUPPOSE BEKENSTEIN IS RIGHT? I HAVE TO SEARCH FOR A MECHANISM WHICH MIGHT PRODUCE RADIATION FROM A BLACK HOLE.\n\nAll the calculations on black holes had been carried out using approximations based on general relativity theory, correct for macroscopic, i.e. large bodies. These approximations ignored any quantum effects, which surely would seem to be negligible in the case of black holes.\n\nTHEN HAWKING BEGAN TO EXPLORE THE BOUNDARY BETWEEN THE BLACK HOLE AND THE VACUUM OF INTER \u2013 STELLAR SPACE AND WONDERED HOW THE INTENSE GRAVITY AT THE SURFACE MIGHT AFFECT ANY PARTICLES THAT MIGHT APPEAR THERE, WHETHER REAL OR VIRTUAL.  \nWAIT A SECOND, WHAT IS A VIRTUAL PARTICLE?\n\nTime out for **Something You Need to Know.**\n\n## **The Uncertainty Principle& Virtual Particles**\n\nThe uncertainty principle, as elucidated by Werner Heisenberg in 1927, states that there are limits on how accurately we can observe certain physical quantities, such as position, momentum, energy and even time. This is not a limit on our measuring instruments but an inherent characteristic of the Universe, which does not reveal **any** quantity with absolute precision.\n\nThink about the vacuum in outer space. We assume it contains absolutely nothing and thus has zero energy. But we can't be sure of this zero energy because of the same argument. Maybe if we look closely enough we can find _some_ energy \u2013 at least for a short time.\n\nTHE UNCERTAINTY PRINCIPLE IN FACT PREDICTS THAT ENERGY CAN CONTINUOUSLY APPEAR AND DISAPPEAR ON A SCALE DETERMINED BY PLANCK'S CONSTANT (WHCIH IS VERY SMALL). BUT BY EINSTEIN'S EQUATION E=MC2, THIS ENERGY CAN TURN INTO PARTICLES AND ANTI-PARTICLES, POPPING IN AND OUT OF EXISTENCE.  \nTHESE ARE CALLED VIRTUAL PARTICLES FLICKERING EVERYWHERE JUST BELOW THE THRESHOLD OF OBSERVABLE REALITY.\n\nHawking considered what might happen at the surface of a black hole (i.e. at the event horizon), where the intense gravitational field interacts with these virtual pairs. He was in effect combining quantum mechanics and general relativity in a single calculation for the very first time. What he found seemed quite remarkable.\n\nI found that black holes are not completely black. They give off radiation.\n\nIt seems the intense gravity at the surface of the black hole can attract one of the particles of the virtual pair into the hole (negative energy), reducing the mass of the black hole, while the other unpaired particle (positive energy) escapes in the form of radiation and can be detected by an outside observer, i.e. an observer not falling into the black hole.\n\nThe most remarkable aspect of this result was the nature of the radiation. It had a perfect _thermal radiation_ spectrum which meant that black holes were just like any other body in the Universe. It was now clear that black holes not only have entropy but a temperature as well and obey the classic laws of thermodynamics laid down in the late 19th century.\n\nThe science writer Dennis Overbye in his classic book on modern cosmology, **Lonely Hearts of the Cosmos** , produced a wonderful metaphor to describe his feelings about Hawking's discovery.\n\nIt was as if Hawking had popped the hood on a Ferrari and found an antique steam engine chugging away inside.\n\nFreeman Dyson, one of the world's top mathematical physicists, was enchanted with the new theory and wrote a popular essay after Hawking visited the Institute of Advanced Study in Princeton.\n\nIN HAWKING'S NEW PICTURE, A BLACK HOLE IS NOT A BOTTOMLESS PIT BUT A PHYSICAL OBJECT. A BLACK HOLE IS NOT BLACK BUT EMITS THERMAL RADIATION AT A CERTAIN DEFINITE TEMPERATURE.  \nA BLACK HOLE IS NOT ABSOLUTELY PERMANENT BUT WILL ULTIMATELY EVAPORATE INTO PURE RADIATION.  \nTHUS STEPHEN BROUGHT BLACK HOLES BACK OUT OF THE DOMAIN OF MATHEMATICAL ABSTRACTION INTO THE DOMAIN OF THINGS THAT WE CAN SEE AND MEASURE.\n\nHawking was reluctant to publish and had only shared his new results with a few close associates.\n\nDennis Sciama, visiting Cambridge from Oxford where he had taken an appointment in the physics department, met another of his former students, Martin Rees, then at the Institute of Astronomy in Cambridge.\n\nDENNIS, HAVE YOU HEARD WHAT STEPHEN HAS DISCOVERED? EVERYTHING IS DIFFERENT, EVERYTHING HAS CHANGED.  \nWHAT ARE YOU TALKING ABOUT?\n\nBLACK HOLES AREN'T BLACK ANYMORE BECAUSE OF STEPHEN'S QUANTUM MECHANIC EFFECT, THEY RADIATE LIKE HOT BODIES.\n\nSTEPHEN, WE ARE ORGANISING A MEETING ON BLACK HOLES IN OXFORD IN A FEW WEEKS. WHY DON'T YOU PRESENT YOUR NEW RESULT ON BLACK HOLE RADIATION?\n\n## **February 1974, The Rutherford- Appleton Laboratory, Oxford**\n\nThe chairman, John Taylor, a well-known mathematics professor and writer of a popular book on black holes, introduces Hawking.\n\nSTEPHEN HAWKING'S TALK TODAY IS ON \"BLACK HOLE EXPLOSIONS?\"\n\nI'M SORRY STEPHEN BUT THIS IS ABSOLUTE RUBBISH.\n\n... so black holes are really not black after all. They have a temperature, an entropy and produce radiation just like any other thermodynamic body. Eventually they explode.\n\nTaylor then stormed out of the session. Hawking sat in shocked silence. He knew his talk would be controversial, but he never expected anything like this.\n\nA month after the meeting at Oxford, Hawking published a paper on the new radiation entitled _Black Hole Explosions?_ in the journal **Nature**. The paper became the topic of discussion in physics departments everywhere and many were sceptical.\n\nFour months later, Taylor and Paul Davies, a colleague at King's College, London, published a retort in the same journal, _Do Black Holes Really Explode?_\n\nNOT EVERYONE WAS SO DISMISSIVE OF HAWKING'S NEW IDEA.\n\nFreeman Dyson compares Hawking's formulas to the epoch-making theory of Max Planck in 1900 which led to the quantum theory.\n\n_Now Hawking has written down an equation which looks rather like Planck's equation. Hawking's equation is S = kA, where S is the entropy of a black hole, A is the area of its surface, and k is a constant. But what does it really mean to say that entropy and area are the same thing? We are as far away from understanding this now as Planck was from understanding quantum mechanics in 1900. All that we can say for certain is that Hawking's equation is a clue to the riddle of black holes. Somehow_ , _we can be sure, this equation will emerge as a central feature of the still unborn theory which will tie together gravitation and quantum mechanics and thermodynamics._\n\n_Perhaps the best way to look at Hawking's discovery is to use another historical analogy. In the year 1900, Max Planck wrote down an equation, E = hv, where E is the energy of a light wave, v is its frequency, and h is a constant which we now call Planck's constant. This equation was the beginning of quantum theory, but in the year 1900 this made no physical sense. It only began to become clear twenty-five years later, when Planck's equation was built into the theory which we now call quantum mechanics._\n\nMAYBE HAWKING'S NEW THEORY WILL PROVIDE THE CLUE TO QUANTUM GRAVITY.\n\nIt is unlikely there has ever been a more powerful demonstration of the self-consistency of physics \u2013 a first step towards quantum gravity. It is the unification of three distinct theories of physics which makes Hawking's Radiation so important.\n\nRecognition of the importance of his work came quickly. Only a few weeks after the paper on black hole radiation was published, Stephen received Britain's highest academic honour. Only 32 years old, he was made a Fellow of The Royal Society, an investiture which made him very proud indeed.\n\nSoon after, Hawking was invited to spend a year away from Cambridge at Caltech, in Pasadena, funded by a special distinguished scholarship, to study cosmology with the eminent American theoretician Kip Thorne.\n\nIn a strange way, this award began a shift in his research from black holes to the beginning of the Universe, a subject of great interest to the Roman Catholic Church.\n\n## **Hawking and the Vatican \u2013 a Modern Day Galileo**\n\nThe powerful Roman Catholic Church has a vested interest in scientific theories about the heavens. For centuries the Church promoted the scientific teachings of Aristotle (a good philosopher but a poor physicist) and the celestial system of Ptolemy which both placed the Earth and Man at the centre of the Universe.\n\nFINALLY IN 1992 THE CHURCH APOLOGISED TO ME. A BIT LATE, PORCA MISERIA!\n\nTo safeguard the church's teaching, Giordano Bruno was burned at the stake in 1600 for teaching the ideas of Copernicus' **heliocentrism** that the Sun and not the Earth is at the centre of the solar system.\n\nThirty-three years later, Galileo Galilei was forced to kneel before the Inquisition, with chains of torture rattling in the background, and recant his belief in Copernicanism.  \nLater, he was placed under house arrest in his villa at Arcetri for the remainder of his days.\n\nThe Vatican has since adopted a more subtle approach to scientists who attempt to answer the ultimate questions of the Universe. It now seems happy to court Stephen Hawking, a cosmologist from Protestant England. Why is that?\n\nBECAUSE ROME IS PLEASED WITH THE BIG BANG MODEL. IT TROUBLED SIGNOR FRED HOYLE AND EVEN IL PROFESSORE EINSTEIN, BUT IT APPEALS TO US AS A CREATION EVENT!  \nAFTER ALL, WAS NOT THE CONCEPT FIRST PROPOSED IN 1927 BY A BELGIAN CATHOLIC PRIEST, ABB\u00c9LEMAITRE? DID NOT EINSTEIN HIMSELF SAY IT WAS \"BEAUTIFUL\"?\n\nThe Church was quick to accept the idea (i.e. by Vatican standards). On 22 November 1951, at the opening of a meeting of the Pontifical Academy of Sciences, Pope Pius XII declared that Lema\u00eetre's idea _accorded with the Catholic concept of creation._ Consequently, any scientist supporting the big bang would certainly be a friend of Rome.\n\nOUR YOUNG FRIEND, IL DOTTORE STEPHEN HAWKING, PROVED IN 1970 THAT EINSTEIN'S GENERAL RELATIVITY DEMANDS THAT ALL THE MATTER AND ENERGY IN THE UNIVERSE MUST AT ONE TIME HAVE BEEN COMBINED IN A SINGLE POINT \u2013 THE SINGULARITY PERFETTO!  \nTHAT'S AS CLOSE AS SCIENCE WILL GET TO IDENTIFYING THE HAND OF GOD!  \nSO IT'S ONLY RIGHT THAT THE PONTIFICAL ACADEMY SHOULD AWARD THE EXCELLENT HAWKING WITH IT'S POPE PLUS XI MEDAL, NO?\n\nI was in two minds whether to accept, because of Galileo. When I arrived in Rome to receive the award, I insisted on being shown the record of Galileo's trial in the Vatican Library.\n\nBy the late 1970s, Hawking had realized that general relativity is not valid at the moment of the big bang, because of the uncertainty principle, and he was exploring the combination of general relativity and quantum mechanics. He was already beginning to think like a heretic.\n\nBut he was back in Rome in 1981, invited to a conference on cosmology organized by the Vatican. By now he had a new area of research, the beginning of the Universe. The paper he gave had a highly technical title.\n\nMy interest in the origin and fate of the Universe was reawakened when I attended a conference on cosmology in the Vatican in 1981. Afterwards, we were granted an audience with the Pope, who was just recovering from an attempt on his life.\n\nIT'S ALL RIGHT TO STUDY THE EVOLUTION OF THE UNIVERSE AFTER THE BIG BANG, BUT DON'T INQUIRE INTO THE BIG BANG, ITSELF, BECAUSE THAT'S THE MOMENT OF CREATION AND THEREFORE THE WORK OF GOD.\n\nIn his talk, Hawking suggested that space and time were finite in extent but were closed up on themselves without boundaries or edges. This has become known as the No Boundary Proposal. If this theory is correct, there would be no singularities and the laws of science would hold everywhere, including at the beginning of the Universe.\n\nIt was not immediately obvious that my paper had implications about the origin of the Universe, because it was rather technical and had the forbidding title, 'The Boundary Conditions of the Universe'.\n\nHawking had begun to work seriously on the early Universe, a subject which has dominated his thinking to the present day. In the paper he gave at the Vatican, he introduced the **No Boundary Proposal** , his latest and most radical idea. It was an attempt to apply quantum theory to the singularity at the beginning of the Universe.\n\n## **Why Do We Need Quantum Theory?**\n\nIn the big bang model of the Universe, the general theory of relativity provides a reliable programme for describing the evolution of our Universe from just moments after time = 0 to the present day. However, thanks to Hawking, we now know that, at the starting point, general relativity predicts a singularity and the theory breaks down. It is a _classical theory_ and time and space cannot be described by Einstein's equations when matter is crunched together at such unbelievable densities. **How can physics predict the beginning of the Universe if all the laws break down at the big bang?** Quantum theory must be used.\n\n## **Quantum Cosmology**\n\nStarting with this question, Hawking and his collaborator, Jim Hartle of the University of California, have used the No Boundary Proposal to develop a new idea in **quantum cosmology.** Unlike previous approaches, Hawking and Hartle (hereafter H & H) have used imaginary time to study the singularity at the big bang.\n\nThe reasoning goes like this. At its birth, the Universe is entirely within the quantum state. So H & H treat the Universe as a single quantum system and try to determine its **wave function.** In other words, they are applying standard quantum mechanical principles to the whole Universe \"before\" the big bang starts.\n\nTHIS IS HAWKING'S MOST SERIOUS ATTEMPT TO ACCOMPLISH WHAT EINSTEIN COULD NOT AFTER REACHING THE AGE OF FIFTY, LAY ONE MORE GOLDEN EGG!  \nARE YOU LOST? NO WONDER TRYING TO UNDERSTAND THIS PROPOSAL MAKES THE BIG BANG SEEM LIKE CHILD'S PLAY. BUT LET'S PROCEED...\n\n## **Quantum Gravity or TOE**\n\nThe search is called _quantum gravity_ , or TOE, the _theory of everything \u2013_ a term irritating to most physicists. Attempts so far, by particle physicists and relativists, have yielded few results.\n\nJUST AS I DID WHEN I QUANTIZED THE ELECTRO-MAGNETIC FIELD INTO PHOTONS IN MY THEORY OF QUANTUM ELECTRO-DYNAMICS.  \nQUANTIZED FIELDS: CAN IT WORK FOR GRAVITATION?  \nRICHARD FEYNMAN (1918 \u2013 1988)\n\nAs usual, Hawking is taking a different approach to the problem. Not **quantum gravity** , but his own **quantum cosmology** , finding the wave function for the Universe. This is based on his No Boundary Proposal.\n\nIt has always profoundly disturbed me that if the laws of physics could break down at the beginning of the Universe, they could also break down anywhere else. That's why we have developed the No Boundary Proposal which removes the singularity at the beginning of the Universe.\n\nBut there is a problem with cosmology because it can not predict anything about the Universe without an assumption about the initial conditions. All one can say is that things are as they are _now_ because they were as they were at an earlier stage.\n\nMany people believe that this is how it should be and science should be concerned only with laws which govern how the Universe evolves in time. They feel that the initial conditions for the universe that determine how the Universe _began_ is a question for metaphysics or religion rather than science.\n\nYES LEAVE IT TO RELIGION, AS I SAID BACK IN 1981!\n\n## **Quantum Cosmology and Complex Time**\n\nSo what's new about Quantum Cosmology? Well, H & H have used the mathematical trick of complex time to examine **all possible** universes that might form from the initial quantum state. Time is divided into two separate components, one imaginary and one real. Unlike real time, the imaginary component does not vanish at the big bang and the theory is thus useful at the singularity. Standard quantum mechanical procedures are then used to arrive at a wave function for the Universe.\n\n### **COMPLEX TIME NEAR THE BIG BANG SINGULARITY.**\n\nBut what are standard quantum mechanical procedures? For that matter, what is a **wave function?**\n\nTHIS MOST LIBIQUITOUS VARIABLE OF ALL MODERN PHYSICS \u2013 THE WAVE FUNCTION COMES DIRECTLY FROM THE EARLY QUANTUM THEORY.  \nIT WAS THE BRAINCHILD OF THE VIENNESE PHYSICIST, ERWIN SCHR\u00d6DINGER.\n\n## **Waves and Particles: Nature's Joke on the Physicists**\n\nExperiments have shown that a wave\/particle duality exists in Nature. For example, a light beam produces interference effects (acts like a wave) but also kicks electrons out of the surface of a metal (acts like a particle). Similarly, electrons exhibit all sorts of particle properties, yet a beam of electrons produces a diffraction pattern (waves) when sent through a fine comb-like grating. This duality is a basic fact of the physical world and we must live with it. It is a consequence of the well-known _uncertainty principle_... or vice-versa.\n\nIn the 1920s, the early heroes of quantum mechanics \u2013 Heisenberg, Schr\u00f6dinger, Bohr and Born \u2013 developed a mathematical language which described both properties \u2013 wave and particle \u2013 at the same time. The most elegant form of this language was an equation due to Schr\u00f6dinger, the solution of which \u2013 the wave function \u2013 determines the behaviour of a system of **particles.**\n\nIF THE FORCES AND BARRIERS ENCOUNTERED BY EACH PARTICLE ARE KNOWN IN THE CLASSICAL SENSE FOR A PARTICULAR SYSTEM, MY EQUATION CAN IMMEDIATELY BE WRITTEN DOWN. THE SOLUTION THEN GIVES INFORMATION ABOUT THE SYSTEM AT ALL POINTS IN SPACE AND AT ALL TIMES.  \nWUNDERBAR, JA!\n\n## **The Strange World of Quantum Mechanics**\n\nBut what is a wave function? What exactly is waving?\n\nHere is what Max Born proposed (ironically, following an idea of Einstein's).\n\nTHE WAVE FUNCTION, WHICH PERMEATES THE SPACE AROUND THE NUCLEUS OF AN ATOM AND DESCRIBES THE BEHAVIOUR OF A SYSTEM OF PARTICLES, IS A PROBABILITY WAVE!IT INDICATES WHERE THE PARTICLES MIGHT BE.\n\nOne of the simplest problems to solve using quantum mechanics is the hydrogen atom. When the Schr\u00f6dinger equation is solved for this case, the resulting wave function determines the probability of each energy state of the atom since it gives the probability of finding the electron at various distances from the nucleus. The nucleus is enveloped in a probability cloud, instead of precise planetary-type electron orbits of the classical atom.\n\nWHERE THE PROBABILITY CLOUD SURROUNDING THE HYDROGEN ATOMIC NUCLEUS IS DENSE, ONE IS MORE LIKELY TO FIND THE ELECTRON, BUT ONE CAN NEVER SAY EXACTLY WHERE IN THE ATOM THE ELECTRON IS LOCATED AT ANY ONE INSTANT. ALL ONE CAN SPECIFY IS THE PROBABILITY THAT IT WILL BE IN VARIOUS PLACES.\n\n## **Quantum Cosmology: Applying Schr\u00f6dinger's Equation to the Universe**\n\nIs Hawking a bold thinker? Instead of **electron orbits** in the atom, think of **cosmological models** of the Universe. General relativity allows a variety of models: some expand from a point to a maximum size, then back to a point again; others expand forever; others expand differently in different directions. Yet all satisfy Einstein's equations.\n\nJust as Schr\u00f6dinger replaced classical electron orbits with wave functions that described the probability of an electron doing one thing or another, so Hawking and Hartle assign individual cosmological models a wave function that indicates the probability of the Universe having one particular geometry or another.\n\nBY CHOOSING ONLY UNIVERSES WITH NO BOUNDARIES \u2013 EITHER IN SPACE OR TIME \u2013 H & H OBTAIN RESULTS WHICH SEEM CONSISTENT WITH OBSERVATIONS ON OUR UNIVERSE.\n\n**Closed** universes satisfy this restriction. They are finite but have no edges, something like the two-dimensional surface of the Earth. They expand, come to a halt, then fall back to the same state like the points on the rim of the bowl shown in the sketch.\n\nDepicted in this way, **closed** universes would have a beginning and an end, and would therefore have boundaries only in **real** time. The imaginary component, however, is continuous. So, H & H make the initial and final singularities of the closed Universe disappear.\n\nThey also demonstrate that **uniform** universes are the most probable and end up predicting that our Universe is both **closed** and **uniform -** finite sphere of space-time with no edges.\n\n## **DAMTP: 17 February 1995**\n\nAs Hawking told the author only six weeks before this book was published...\n\nThe No Boundary Proposal predicts a Universe that starts out in a very smooth and ordered way. It expands by _inflation_ first, then goes over to the standard _hot big bang_ model, further expanding to a maximum radius before collapsing to a _big crunch_ singularity in a disordered and irregular way.\n\nReal time ends at that point, but the universe continues to be.\n\nYOUR THEORY PREDICTS THAT A CLOSED AND UNIFORM UNIVERSE IS THE MOST PROBABLE AND THAT DENSITY VARIATIONS SHOULD EXIST IN THE EARLY UNIVERSE DUE TO QUANTUM FLUCTUATIONS.  \nIT SEEMS LIKE THIS NO BOUNDARY PROPOSAL IS YOUR THIRD GOLDEN EGG...\n\nCalculations carried out so far on simple models indicate that a No Boundary Proposal Universe would be very much like our own. In addition, it would incorporate certain important ideas from contemporary cosmology \u2013 such as **inflation** and **quantum fluctuations.** Even the **anthropic principle** seems to fit. If you can understand these last three concepts, you should have a very good picture of Stephen Hawking's Universe. Not bad for a beginner!\n\n## **Inflation**\n\nIn the late 1970s, a new concept of **inflation** was introduced which proposed that the Universe expanded from an initial state smaller than a proton to a macroscopic size about ten metres across in only a fraction of a second. The rate of expansion was enormous. The idea solved two problems which had been nagging cosmologists for years.\n\n**1.** Why is the Universe so flat, i.e. shows no evidence of curvature?\n\n**2.** Why is the cosmic background radiation so uniform?\n\n**1.** The first of these questions implies that the mass density of the Universe is perfectly tuned to the critical value from its earliest expansion, a mind-boggling proposition (see here). But a rapid expansion at the beginning would flatten out the Universe to the critical mass density as a simple diagram can show.\n\n**2.** Inflation can also explain why the background radiation is so uniform. When the Universe was of infinitesimal size, all matter and energy was homogeneous, since everything was connected to everything else. As inflation took place, the homogeneity that existed at that early instant was spread across the much larger Universe, which continued to expand. Thus, when matter and radiation de-coupled about 300,000 years later, the Universe was still amazingly uniform.\n\n## **Inflation and Quantum Fluctuations**\n\nThe inflation that smoothed out the early Universe could also produce small density variations which might explain galaxy formation. Recall from our discussion of virtual particles on page 136 that if we look closely enough at any physical system \u2013 even a vacuum \u2013 we observe the effects of **quantum fluctuations.**\n\nInflation does not erase these quantum fluctuations but establishes them as **density variations** which appear as ripples of matter-energy across space-time. These ripples should then be imprinted on the background radiation as tiny temperature variations.\n\nThese temperature variations are precisely what George Smoot and his Berkeley-NASA team were looking for with the Cosmic Background Explorer Satellite (COBE) experiment launched in 1989. We need one more bold concept...\n\n## **The Anthropic Principle**\n\nThe anthropic principle is a quasi-metaphysical notion which implies that, if a particular universe does not take on fundamental constants of Nature which allow for the existence of life and the development of intelligence, there will never be anyone to report on its properties. That is why our Universe seems so right to us, it's tuned perfectly.\n\nAlthough many scientists rubbish this idea, no less an authority than Nobel Laureate Steven Weinberg (who wrote the seminal book on the early Universe, **The First Three Minutes** ) believes that **quantum cosmology** provides a context in which the anthropic principle becomes simple common sense. **The most probable universe is the one that we're in!** As Voltaire's absurd philosopher Pangloss keeps telling Candide, \"We live in the best of all possible worlds.\"\n\n## **Hawking's Nobel Prize**\n\nStephen Hawking has received just about every award and honour which can be given to a scientist. Naturally, the question arises whether he will be awarded the most famous of all \u2013 an invitation to the Royal Academy of Sciences in Stockholm to receive the Nobel Prize in Physics.\n\nONE MAY EVEN ASK WHY HE HAS NOT RECEIVED IT ALREADY.\n\nThere are complications. First of all, the award is only rarely given for work in astronomy or cosmology rather than pure physics. The second obstacle is more serious. Alfred Nobel was a very practical man (he made his fortune from patents on the explosive TNT) and insisted that to be eligible, theoretical discoveries must be verified by experiment.\n\nFor cosmologists like Hawking, whose laboratory extends to the most remote regions of the Universe, experimental verification may never be possible or, at best, take decades.\n\nLet's review Hawking's major theoretical discoveries which might win him the Nobel Prize.\n\n**1.** Using General Relativity, Hawking and Penrose showed that the classical concept of time must have begun with a singularity at the Big Bang and thus the Universe existed at one time in a hot, dense state.\n\n**2.** In 1974, he discovered that black holes radiate like thermodynamic bodies (now called **Hawking Radiation** ) and possess a temperature (proportional to their surface gravity) and an entropy (proportional to their surface area).\n\n**3.** He presented a model for the early Universe called the **No Boundary Proposal** with Jim Hartle which predicts density variations in the early Universe due to quantum fluctuations of the vacuum.\n\nIronically, **Hawking Radiation** , his most significant work, seems an unlikely candidate for the Nobel award as it seems impossible to detect.\n\nHowever, both the Big Bang singularity (hot, dense state of the Universe) and quantum fluctuations (seeds for galaxy formation) could be proved if very accurate **absolute** and extremely sensitive **differential** measurements were made of the cosmic background radiation.\n\nThat is exactly what the COBE project did between 1989 and 1992.\n\n## **COBE: the Greatest Discovery of All Time (?)**\n\nCOBE took twelve years to design and carry out, but the results were nothing short of spectacular. Launched in 1989, the instruments took only 8 minutes to verify the conclusions based on the 1964 measurements of Penzias and Wilson, but this time at many different wavelengths. The data traced out a near perfect thermal radiation curve (see here) for a background temperature of 2.736 degrees C above absolute zero.\n\nThis was COBE I which used an _absolute_ microwave radiometer calibrated by a bath of liquid helium on board the satellite. The results proved without a doubt that the detectors were looking at the remnant of the hot, dense state of the early Universe which we call the big bang. Such a curve would have thrilled Max Planck, as it did the American Astronomical Society when first presented in 1990.\n\n### ****COBE MEASUREMENTS OF BACKGROUND RADIATION.****\n\nBut the big news was still to come. COBE II used a sensitive _differential microwave radiometer_ (DMR) which doesn't measure the absolute temperature of the radiation at a given point in the sky; rather, it measures the **difference** in temperature between two points. The COBE I single antenna gives the answer: \"The temperature at point A is 2.725 degrees.\" But the COBE II dual-antenna _differential_ radiometer gives the answer: \"The temperature difference between point A and point B is 0.002 degrees.\"\n\n### **THE COBE SPACECRAFT**\n\nThis was George Smoot's project \u2013 to look for evidence of ripples in the space-time of the 300,000-year-old Universe. In April 1992, after more than two years of data collecting and analysis, Smoot and his team made a dramatic announcement. The COBE satellite had detected tiny temperature variations of the order of about one-hundred-thousandth of a degree in the background radiation.\n\nACCORDING TO COMPUTER GENERATED PLOTS OF THE ENTIRE SKY, THE TEMPERATURE WAS MINUTELY HIGHER IN THE DIRECTION OF THE LARGE GALACTIC CLUSTERS AND SLIGHTLY LOWER IN THE GREAT COSMIC VOIDS.\n\nIt now seemed possible for theorists to explain some of the structures seen in today's Universe in terms of events which took place billions of years ago. The report was greeted with an enthusiastic media response all over the world.\n\nIF YOU'RE RELIGIOUS IT'S LIEK SEEING GOD.\n\nBoth Hawking and Smoot made statements which together just about covered the two ends of the emotional spectrum. Smoot is a religious man and has accepted the big bang as a creation event. COBE's results moved him emotionally.\n\nHawking sees things differently. To him, the variations in the background radiation seen by COBE are simply evidence for the presence of quantum fluctuations in an inflationary Universe consistent with his No Boundary Proposal. Any wonder he's smiling.\n\nCOBE's success is seen by most scientists to be a stunning confirmation of big bang cosmology. But the game is not yet up. The final solution to the mysteries of the beginning and structure of the Universe may be much more complicated.\n\nThe Earth-centred cosmos of Aristotle and Ptolemy, the Sun-centred system of Copernicus, Le Ma\u00eetre's Cosmic Egg and Hawking's No Boundary Proposal are just steps along the way to deeper understanding of the Universe and our place in it. The journey is everyone's to contemplate, to understand, to enjoy.\n\n**Stephen William Hawking, cosmologist \u2014 an example of _homo sapiens circa_ AD 2000 \u2014 certainly has done his part.**\n\n## **Further Reading**\n\n**Books about Hawking**\n\n**Stephen Hawking, A Life in Science** , Michael White and John Gribbin, New York, Plume Books (Putnam).\n\n**Stephen Hawking, Quest for a Theory of Everything** , Kitty Ferguson, New York, Bantam Books 1992.\n\n**Development of Classical Astronomy**\n\n**The Sleepwalkers** , Arthur Koestler, New York, Grosset & Dunlap 1959. **Coming of Age in the Milky Way** , Timothy Ferris, New York, Anchor Books 1989.\n\n**New Cosmology\/Black Holes**\n\n**A Brief History of Time** , Stephen Hawking, New York, Bantam Books 1988.\n\n**The First Three Minutes** , Steven Weinberg, New York, Bantam Books 1984.\n\n**Black Holes and Time Warps** , K. S. Thorne, New York, W. W. Norton & Co. 1994.\n\n**Black Holes and Warped Spacetime** , W. M. Kaufmann, San Francisco, W. H. Freeman 1979.\n\n**Black Holes** , Jean Pierre Luminet, New York, Cambridge University Press 1992.\n\n**In Search of the Big Bang** , John Gribbin, New York, Bantam Books 1986.\n\n**Lonely Hearts of the Cosmos** , Dennis Overbye, New York, Harper Collins 1991.\n\n**COBE**\n\n**Wrinkles in Time** , George Smoot and Keay Davidson, New York, Avon Books 1994.\n\n## **Acknowledgements**\n\nI was encouraged at the outset by a conversation with Dennis Sciama last summer in Oxford and by the publication of Kip Thorne's book which was heaven sent. John Gribbin's books and his regular column in the _New Scientist_ have been very helpful. Special thanks to John Wheeler, Freeman Dyson, Jacob Bekenstein and Simon Shaffer for useful points of reference and graphics. I discussed quantum cosmology with Chris Isham and he also read a draft manuscript just before going to press.\n\nMy time at DAMTP was essential to develop a fresh insight into the Hawking story. The staff in the department, Ms Sue Masey in particular, were most cooperative with my many requests. Jane Hawking provided some helpful background material and also read the manuscript before publication.\n\nOn the home front my family has been most supportive, especially my wife Pat, who for the past seven months has given me the psychological and intellectual space I needed to tell this story. She knows about such things.\n\nMy colleague Maryke Brecher covered up for my mysterious disappearances when the going got tough and Kathy Black helped me meet (almost) every deadline with her formidable keyboard skills.\n\nA very special thanks to Stephen Hawking who endorsed this project from the beginning and slotted me into his tight schedule on numerous occasions for discussion and advice. Stephen affects most people who work with him and I am no exception. Watching him communicate has taught me to be more concise, more accurate and more clear in everything I do. Also, I hope I shall never allow trivial ailments like a headache or fatigue to interrupt any worthwhile endeavour.\n\nOscar Zarate has made this book different from any other book on physics or astronomy that I have ever seen. The goal was to illustrate every idea I deemed important, no matter how abstract \u2013 to get both sides of the brain working. To achieve this, he read dozens of books on physics and astronomy and listened with rapt interest to my late night discourses as the book's outline took shape. I hope it has worked!\n\nJ.P. McEvoy, London, March 1995\n\nOscar Zarate thanks Judy Groves for her help with the diagrams, Woodrow Phoenix for his lettering, Marta Rodrigues for screening the photos and Bill Mayblin for his advice on the graphics.\n\n**J.P. McEvoy** received his Ph.D. in physics at Imperial College, University of London, 1968. For twenty-five years, active in physics research and teaching at RCA, Clark University and the American School in London, he has published over 50 technical papers and has recently been involved in science journalism and multi-media development for educational television. He is also the author of _Introducing Quantum Theory._\n\n**Oscar Zarate** has illustrated introductory guides to Freud, Quantum Theory, Mind & Brain, Machiavelli, Melanie Klein, Lenin and the Mafia. He has also produced many acclaimed graphic novels, including _A Small Killing_ , which won the Will Eisner Prize for best graphic novel of 1994, and has edited _It's Dark in London_ , a collection of graphic stories, published in 1996.\n\n**Photography by Mark McEvoy and David Simmonds.\n\nTypesetting by Wayzgoose.**\n\n## **Index**\n\nanthropic principle ref 1\n\nBekenstein, Jacob ref 1, ref 2\n\nBell, Jocelyn ref 1\n\nBethe, Hans ref 1, ref 2\n\nbig bang theory ref 1, ref 2, ref 3, ref 4, ref 5, ref 6, ref 7,ref 8, ref 9\n\nbig crunch ref 1, ref 2, ref 3\n\nblack holes ref 1, ref 2, ref 3, ref 4, ref 5, ref 6, ref 7\n\nfinding ref 1, ref 2\n\nHawking ref 1, ref 2, ref 3ff\n\n_see also_ gravitational collapse\n\nBoltzmann, Ludwig ref 1\n\nBorn, Max ref 1\n\nCatholic Church ref 1, ref 2\n\nCOBE satellite ref 1, ref 2\n\ncomplex time ref 1\n\ncosmic radiation ref 1\n\ncosmological constant ref 1\n\ncosmology ref 1, ref 2, ref 3\n\nDyson, Freeman ref 1, ref 2\n\neclipse ref 1, ref 2\n\nEddington, Arthur S. ref 1, ref 2\n\nEinstein, Albert ref 1, ref 2, ref 3, ref 4\n\n_see also_ relativity theory\n\nelectromagnetic spectrum ref 1\n\nentropy ref 1, ref 2, ref 3, ref 4\n\nequivalence principle ref 1\n\nevent horizon ref 1, ref 2, ref 3, ref 4\n\nFeynman, Richard ref 1\n\nFriedmann, Alexander ref 1, ref 2\n\nGalilei, Galileo ref 1\n\nGamow, Georges ref 1\n\ngeodesics ref 1\n\nGod ref 1\n\nGold, Thomas ref 1, ref 2\n\ngravitation ref 1, ref 2, ref 3\n\n_see also_ space curvature\n\ngravitational collapse ref 4, ref 5, ref 6, ref 7, ref 8\n\ngravity ref 1, ref 2, ref 3, ref 4, ref 5\n\nHalley, Edmund ref 1, ref 2\n\nHartle, Jim ref 1, ref 2, ref 3, ref 4\n\nHawking, Stephen\n\nborn ref 1\n\nillness ref 1, ref 2\n\nintelligence level ref 1\n\nmarried ref 1\n\nNobel Prize? ref 1, ref 2\n\nPope Pius medal ref 1, ref 2\n\nRoyal Society ref 1\n\nthesis ref 1, ref 2\n\nHawking's law of area\n\nincrease ref 1\n\nHeisenberg, Werner ref 1, ref 2, ref 3\n\nHoyle, Fred ref 1, ref 2, ref 3, ref 4\n\nHubble, Edwin ref 1, ref 2\n\ninflation ref 1, ref 2\n\nKepler, Johannes ref 1\n\nlaw of motion ref 1\n\nlaws of\n\nblack hole mechanics ref 1\n\nthermodynamics ref 1, ref 2, ref 3\n\nLema\u00eetre, Abb\u00e9 ref 1, ref 2\n\nmass ref 1, ref 2, ref 3, ref 4, ref 5\n\nMercury, perihelion ref 1, ref 2, ref 3\n\nmicrowaves ref 1, ref 2\n\nmotor neurone disease ref 1, ref 2\n\nNarlikar, Jayant ref 1\n\nneutron stars ref 1, ref 2,\n\nref 1, ref 2\n\nNewton, Isaac ref 1, ref 2, ref 3, ref 4\n\nlaw of motion ref 1\n\nNo Boundary Proposal ref 1, ref 2, ref 3, ref 4, ref 5, ref 6,\n\n_On the Internal Constitution of Stars_\n\nOppenheimer, J. Robert ref 1, ref 2, ref 3, ref 4\n\npartial theories ref 1\n\nparticles ref 1\n\n_see also_ virtual particles\n\nPenrose, Roger ref 1, ref 2, ref 3\n\nPenzias, Arno ref 1, ref 2, ref 3\n\nPlanck, Max ref 1, ref 2, ref 3\n\nPlanck's constant ref 1\n\n_Principia_ ref 1, ref 2, ref 3, ref 4\n\nprinciple of equivalence ref 1\n\npulsars ref 1, ref 2\n\nquantum\n\ncosmology ref 1, ref 2, ref 3, ref 4\n\nfluctuations ref 1\n\ngravity ref 1, ref 2\n\nmechanics ref 1\n\ntheory ref 1\n\nquasars ref 1, ref 2, ref 3, ref 4\n\nradiation ref 1, ref 2, ref 3\n\nrelativity theory ref 1, ref 2, ref 3, ref 4, ref 5\n\nsatellites ref 1, ref 2\n\nSchr\u00f6dinger, Erwin ref 1\n\nSchwarzchild\n\ngeometry ref 1, ref 2\n\nradius ref 1\n\nSchwarzchild, Karl ref 1, ref 2, ref 3\n\nSciama, Dennis ref 1, ref 2, ref 3, ref 4, ref 5\n\n_Singularities of Gravitational Collapse_ ref 1\n\nsingularity ref 1, ref 2, ref 3, ref 4\n\n_see also_ black holes\n\nSmoot, George ref 1, ref 2\n\nsolar eclipse ref 1, ref 2\n\nsystem ref 1\n\nspace curvature ref 1, ref 2\n\nSchwarzchild ref 1, ref 2\n\nsingularity ref 1, ref 2\n\nstars ref 1, ref 2\n\nsteady state theory ref 1\n\nTaylor, John ref 1\n\ntheory of everything ref 1\n\nthermal radiation ref 1, ref 2\n\nblack holes ref 1\n\nthermodynamics, laws of ref 1, ref 2, ref 3\n\nblack holes ref 1\n\nThorne, Kip ref 1, ref 2, ref 3\n\nuncertainty principle ref 1, ref 2\n\nuniverse\n\nbeginnings ref 1, ref 2, ref 3\n\nevolution ref 1\n\nexpanding ref 1, ref 2, ref 3\n\nflat ref 1, ref 2\n\nhistory ref 1\n\n_see also_ big bang theory\n\nuniverses, other ref 1, ref 2\n\nvacuum solution ref 1\n\nvirtual particles ref 1, ref 2\n\nwave function ref 1, ref 2, ref 3\n\nwavelengths ref 1, ref 2\n\nWeinberg, Steven ref 1\n\nWheeler, John ref 1, ref 2, ref 3, ref 4, ref 5, ref 6\n\nwhite dwarf ref 1, ref 2\n\nWilde, Jane ref 1, ref 2\n\nWilson, Robert ref 1, ref 2, ref 3\n**Many Introducing Graphic Guides are now available in ebook format - including the titles below. Check in at introducingbooks.com\/ebooks\/ to keep up to date with more titles as they are published in ebook format.**\n\n**9781848317567 - Introducing Psychology**\n\n**9781848317574 - Introducing Quantum Theory**\n\n**9781848317581 - Introducing Philosophy**\n\n**9781848317598 - Introducing Freud**  \n*not available in North America\n\n**9781848317604 - Introducing Postmodernism**\n\n**9781848317611 - Introducing Logic**\n\n**9781848317628 - Introducing Nietzsche**\n\n**9781848317635 - Introducing Marxism**\n\n**9781848317642 - Introducing Particle Physics**\n\n**9781848317659 - Introducing Capitalism**\n\n**9781848317666 - Introducing Chaos**\n\n**9781848317673 - Introducing Ethics**\n\n**9781848317680 - Introducing Buddha**\n\n**9781848317697 - Introducing Foucault**\n\n**9781848317703 - Introducing Relativity**\n\n**9781848317710 - Introducing Linguistics**\n\n**9781848317727 - Introducing Time**\n\n**9781848317734 - Introducing Statistics**\n\n**9781848317741 - Introducing Islam**\n\n**9781848317758 - Introducing Continental Philosophy**\n\n**9781848317765 - Introducing Slavoj Zizek**\n\n**9781848317789 - Introducing Economics**\n\n**9781848317796 - Introducing Melanie Klein**\n\n**9781848317802 - Introducing Critical Theory**\n\n**9781848317819 - Introducing Genetics**\n\n**9781848317826 - Introducing Feminism**\n\n**9781848317833 - Introducing Fractals**\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"Vice-admiral says investigation into defence procurement programme will have very serious cost repercussions.\nDELAYS in SA's defence procurement programme because of a pending probe into the deal would have serious cost implications, the departing SA Navy chief, Vice-Admiral Robert Simpson-Anderson, said yesterday.\n\"That is going to add a tremendous amount of money to the cost of these programmes. If we tell the ship-builders now to hold back for another three or six months, this is going to have very serious repercussions,\" Simpson-Anderson said.\nParliament's watchdog public accounts committee said on Monday it wanted the R30bn arms deal to be the subject of a special investigation by at least five independent agencies. The committee found that there might have been \"undue influence\" in the deal's main government-to-government contracts.\nIt recommended, among other things, that the integrated management system of four new patrol corvettes be investigated. On his last day in office, Simpson-Anderson said yesterday the navy give its full co-operation to the inquiry.\n\"In any event, with the navy contracts we are talking about subcontractors that are unhappy. There are not problems with the way in which the contracts took place for the ships overall,\" he said. \"So, I don't have sleepless nights.\"\nAsked about the operational readiness of the navy he was leaving behind, Simpson-Anderson said limited funds had been a debilitating factor. As a result, ships at sea were not maintained at full readiness. \"We decided to drop the operational readiness in order to have as many ships as possible at sea for possible peace-time contingencies,\" Simpson-Anderson said. \"Not all our missiles and torpedoes work, and not all our guns are fully operational.\"\nCorners also had to be cut in the refitting of some vessels. Simpson-Anderson said that in one instance, there was not enough money to clean the fuel tanks of a support ship. \"The ship came out of its refit looking very good.\n\"But we knew that in a couple of years' time there was going to be a problem because the tanks were not cleaned.\" Turning to training, he expressed concern at the rate at which members of the navy were being poached by private companies. Many of these companies had stopped spending money on training, saying openly that the armed forces were doing this for them.\nTrained navy staff were easy prey for the private sector as they were underpaid. \"I cannot guarantee that we will have sufficient people to man the new vessels we are acquiring. It will largely depend on whether government will decide to properly remunerate our people,\" Simpson-Anderson said. Simpson-Anderson formally handed over command of the navy to his successor Vice-Admiral Johan Retief at a ceremony in Pretoria last night after a naval career spanning almost 40 years.\nHe said at the time he took over the navy that the mood was not good and the morale was going down. Simpson-Anderson said: \"A retrenchment of more than 20% of our personnel two years previously had left the navy scarred in terms of expertise and morale.\"\nHe said the cancellation of two major projects a year earlier had cast a shadow of doubt on the future of the navy. Budgets continued to be cut and fund allocations continued to be based on historical needs of wars of the past instead of the needs of the future.\n\"But,\" Simpson-Anderson said, \"we had one strength: we had people with courage, perseverance and wisdom, people who knew the art of good timing, (and) who worked together with resolve to shape a new future for the navy.\" - Sapa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Get to know your city! The City of Cambridge offers information about its services, programs, projects and departments. You can learn about your Mayor and Council, how to report an issue, City by-laws and more.\nCan't find what you're looking for on our website? Contact us with questions, feedback and concerns.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chapultepec Castle (Castillo de Chapultepec in Spanish) is located on top of Chapultepec Hill. The name Chapultepec stems from the N\ufffdhuatl word chapoltep\u0113c which means \"at the grasshopper's hill\". It is located in the middle of Chapultepec Park in Mexico City at a height of 2,325 meters (7,628 ft) above sea level.\nThe site of the hill was a sacred place for Aztecs, and the buildings atop it have served several purposes during its history; including that of Military Academy, Imperial residence, Presidential home, observatory, and presently, the Museo Nacional de Historia.\nLostMonuments.com offers the Chapultepec Castle Images, Photos, Wallpapers and Pictures for you. View and download the images of Chapultepec Castle .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Let your inner fun-ster loose with new friends and our fabulous arts and crafts, games, and sports holiday activities. Fully supervised by qualified staff.\nChildren are fully supervised by qualified staff and engaged in Art \/ Craft, Games and Sports. They can play freely and creatively with all our fabulous equipment and materials or join in with the structured craft sessions and games facilitated by our qualified staff.\nTO BOOK: email [email protected] or phone the resorts reception team on 02 6639 1500.\nElements of Byron has proudly partnered with Byron Bay KidzKlub Pty Ltd to deliver this service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Glasshouse Digital is a digital marketing agency based in Auckland, New Zealand. If you're looking for a long-term partner to help your business generate better-qualified and sales-ready leads, get in touch.\n46 Wharf Road, Te Atatu Peninsula, Auckland, New Zealand.\n\"Our digital campaigns with Glasshouse Digital has been by far the most efficient use of our marketing budget and we are very fortunate to have them as part of our team. We look forward to continuing working with them and cannot recommend them highly enough!\" (Vanessa Tam, Proxima Residences).\n\"We have been extremely impressed by Glasshouse's market knowledge in this space and the willingness to listen and adapt messaging and strategy. We would strongly recommend anyone in the real estate space to approach Shehan and the team in order to refine and execute their digital marketing activities\" (Sunil Hira, Senior Development Manager of Veros Property Services).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztkwa b/data_all_eng_slimpj/shuffled/split2/finalzztkwa
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+{"text":"What do you know about ashitaba leaves? Well, in recent years the number of people who aware of the health benefits of ashitaba leaves is increasing especially when the Japanese claim this plant to be among essential list Japanese herbal medicine well known as Kampo medication.\nAshitaba leaves is also well known by its scientific name Angelica keiskei though perhaps some people are mistaken the same plant from which Angelica root is derived because though it comes for the same genus Angelica, for the benefits Angelica root you should drive it from plant which is well known as Angelica sinensis. From its botanical classification you will see that Ashitaba leaves comes from family of Apiaceae, so this plant has closer to plant like celery with similar benefits as well.\nSurprisingly the small size of ashitaba leaves contain up to 13 types of minerals as well as packed with various vitamins and other organic compounds. Furthermore, the ashitaba leaves also contain fiber in quite high amount that make this plant excellent solution to digestion and weight loss. To find out more the health benefits of Ashitaba leaves the list below will tell you.\nIt is impossible to stop the aging process but ashitaba leaves are well known to be the natural solution to slow down the aging process. Ashitaba leaves contain melatonin, one of the powerful antioxidants that contributes in healthy skin and promotes optimal skin cell regenerations to prevent premature aging.\nThere are a lot of factors that could cause someone has problem with their sleeps. Well, whatever the factors are ashitaba leaves could be made into excellent herbal tea for restful sleep. Healthy sleeping habit will improve the brain function and boost the mood.\nBesides melatonin which is among the potent antioxidants, ashitaba leaves also contain chalcone, an organic compound that could act as effective anti-cancer properties. The combinations of powerful antioxidants found in ashitaba leaves could fight free radicals effectively and efficiently to inhibit the development of cancerous cells.\nThere are a lot of reasons why ashitaba leaves are greats for immunity system such as potent antioxidants, anti-inflammatory properties, anti-fungal agents and also anti-bacterial properties are found in ashitaba leaves to help fighting viral infection and strengthening the immunity system.\nIt is not easy to find foods that are rich of vitamin B12. One of the health benefits of vitamin B12 is to maintain the nerve system. Surprisingly, ashitaba leaves contain vitamin B12 in quite high amount to help promoting the production of NGF or Nerve Growth Factor which is essential for cognitive function.\nAs mentioned in point number three that ashitaba leaves contain antioxidants called chalcone which is also excellent in regulating the blood pressure. When the symptoms of high blood pressure are under control is easier to deal with hypertension.\nSome studies have been conducted to find out a connection between ashitaba leaves and cholesterol management. The result is showing the fact that ashitaba leaves contain some active compound to help increasing the production of HDL or good cholesterol while potassium will make sure the decrease of LDL or bad cholesterol.\nThe fact that ashitaba leaves are excellent for heart has been mentioned in the point number six and seven. When blood pressure is under control and cholesterol is excellently managed, you could reduce the risk of heart attack as well as stroke. Furthermore, ashitaba leaves also contain antithrombotic agents which main function is preventing the accumulation of blood clot in the vessel wall.\nThe combinations of vitamin B12, certain minerals such as potassium, iron and calcium, some chlorophyll are excellent blood detox. They will eliminate all the harmful toxins from blood while at the same time making sure optimal blood flow while minerals are promoting the optimal production of red blood cells.\nThe fact that ashitaba leaves are excellent sources of certain minerals, especially iron. It means ashitaba leaves are solution to those who have problems with anemia. Just a cup of herbal tea made from ashitaba leaves per day will help dealing with the anemia excellently.\nIt is not easy for a diabetic patient to manage their daily diet. However, ashitaba leaves are excellent addition to the daily diet because it could enhance the liver and kidney function to optimize the production of insulin and improve its sensitivity to help stabilizing the blood sugar levels in all types of diabetes.\nMost people don't realize that the key to healthy lifestyle is healthy metabolism. When the metabolism process is in healthy state, weight loss is no longer a problem, there is no such thing like vitamins or mineral deficiencies because all nutrients will be optimally absorbed. Furthermore, healthy metabolism will lead to healthy digestion as well so people could say goodbye to certain stomach problems such as constipation and slow bowel movement.\nSpeaking of the cautions of ashitaba leaves there are some things you should aware of though perhaps the report regarding side effects of ashitaba leaves are almost zero.\nAllergic reaction is supposed to be on your list especially for you who have allergen history to several plants.\nNegative interaction with certain medication could occur to some people especially for those who are taking specific medication without clear instruction from the professional.\nMaking sure you know where your leaves come from. Organic goods are still the best to make sure your body is free from chemical contamination commonly found during cultivation process.\nFor the health benefits of ashitaba leaves is best for you to consume it in the form of herbal tea but still you could consume in several ways such as mixture to salad, soup, mixed juice and so on. Furthermore, growing them yourself in the back yard is actually simple solution so you have access to this plant whenever you need it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How old do you have to be to get an agent to audition for a dance company?\nAges are not really fixed for auditioning to contemporary dance companies. That applies mostly in the case of ballet, where dancers usually have to be very young to be admitted into the companies.\nBut contemporary dance is a very different artistic field. It is so diverse that it is not possible to tell what age you need to be, since you might find companies that need really mature dancers for specific projects, kids or all kind of different people.\nGetting an agent depends on your own needs as a professional dancer and the companies you have in mind. Normally, you won't need an agent just to audition as a dancer. Managers or agents typically work to represent and sell companies or projects.\nIf you wish to work with a contemporary dance company, what you need to do is to ask in each case what type of dancers they are looking for.\nI am much interested in contemporary dancing but I am not a dancer. I love watching contemporary dance and would like to watch some videos of great contemporary dancing. Can anyone suggest a good site that can offer videos of contemporary dancing?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"INVESTMENT OPPORTUNITY, JUST OFF OF MAIN STREET!! This beautiful, level and over a half acre lot is located near all other major conveniences and near to Interstate 15. Zoning is neighborhood commercial. This specific plan zone has been created to provide areas for immediate day-to-day convenience shopping and services for the residents of nearby neighborhoods. Build your business now or invest in your future!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- In the west of Mexico is the founder and former drug lord of La Familia Michoacana murdered. That is one of the most powerful drug cartels in the country.\nThe body of Carlo Rosales Mendoza, also known as El T\u00edsico, was found along with three other bodies along the highway in the state of Michoac\u00e1n. According to the Prosecution, there are gunshot wounds and signs of abuse detected.\nRosales Mendoza founded the first cells of the criminal organization in the eighties. He was a good friend of Osiel C\u00e1rdnes Guillen, former leader of the notorious Gulf Cartel in Mexico. For organized crime and drug trafficking Mendoza had been a prison sentence of ten years in jail. Nothing is known about the circumstances of the murder.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"-:ref:`project-sandboxing` explains this in more detail.\n+:ref:`ProjectSandboxing` explains this in more detail.\nfile system you, as a standard user, have read and write access too.\nof possible locations and set ups.\n-interact. This should not happen but it can.\n-path for all RTEMS 4.11 releases of :file:`\/opt\/rtems\/4.11`.\n+:file:`\/usr\/local` on FreeBSD and Linux.\n+operating system specific part plus :file:`rtems` plus the RTEMS version, e.g.\n+    The prefix must not contain space characters.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzztvwi b/data_all_eng_slimpj/shuffled/split2/finalzztvwi
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"In its second year of tracking the number of lives saved by drones, DJI Technology's most recent report documents 65 cases of unmanned aircraft systems (UAS) saving lives between May 2017 and April 2018.\nPrepared by the drone manufacturer's policy and legal affairs department, \"More Lives Saved: A year of Drone Rescues Around the World,\" relies on news reports to document cases of lives being saved by drone on five continents. Examples range from drones dropping floatation devices to swimmers caught in rip currents to thermal imaging sensors on unmanned aircraft locating lost or injured people.\nSince DJI released its first report in March 2017, the company said its survey shows that at least 124 lives have been saved by UAS technology. \"With public safety agencies around the world quickly adopting drones as a standard tool for search and rescue, it seems certain that this number will continue to rise,\" DJI said in its latest report.\nAccording to the report, at least 22 of those rescued by drones were in situations with a great risk of death, such as being stranded in water or exposed to hazardous weather. Another 19 were assisted by drones in situations that presented \"great risks to health and safety.\" In one instance noted in the report, a drone located 24 tourists lost at night on a mountain.\nThe report includes a full list of incidents and links to media coverage. The list was compiled from news accounts, primarily through English languages searches. DJI said the number of lifesaving drone rescues has probably been undercounted.\nAmong the trends identified, DJI said thermal imaging cameras saved the lives of at least 15 victims hidden from view by darkness or obstacles. Of those saved by drones, 32 were in the U.S. while the others were in China, Australia, Brazil, France and the United Kingdom. Professional rescuers used the payload capacity of drones to deliver items such as floatation devices, notes, two-way radios and food to assist stranded people.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking to buy or sell Vinylidene chloride polymers, in primary forms? List your company on Macro Market.\nVinylidene chloride polymers, in primary forms is the 3367th most traded product and the 399th most complex product according to the Product Complexity Index (PCI).\nThe top exporters of Vinylidene chloride polymers, in primary forms are France ($155M), Japan ($28.8M), Belgium-Luxembourg ($25M), the United States ($21.1M) and Germany ($14.9M). The top importers are the United States ($38.2M), Belgium-Luxembourg ($29.1M), Brazil ($26.3M), China ($23.3M) and Germany ($23.3M).\nVinylidene chloride polymers, in primary forms is a 6 digit HS02 product.\nThis treemap shows the share of countries that export Vinylidene chloride polymers, in primary forms.\nThis treemap shows the share of countries that import Vinylidene chloride polymers, in primary forms.\nThis visualization shows products that are likely to be exported by countries that export Vinylidene chloride polymers, in primary forms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I read this article and found it very interesting, thought it might be something for you. The article is called Designing Websites and Products that Evoke Positive Emotions and is located at https:\/\/www.indusnet.co.in\/designing-websites-and-products-that-evoke-positive-emotions\/.\nCognitive science helps us to understand that there is more to consumer decisions and choices than just good products. While good products and designs are fundamental to their own success in the market, certain choices and decisions made by consumers are not limited to the intellectual quality of a product's benefits or superiority over others.\nPsychologists believe that consumers make decisions on an impulse, and the attitude and opinions they develop towards a company, its products or its designs are largely based on their own life experiences, mood states, emotions and certain personality characteristics. One could say that psychological processes like perception, memory, emotions and cognitive schemas play very important roles in consumers' decision to purchase certain designs and reject others.\nJeroen van Erp, co-founder of Fabrique, a multidisciplinary design agency proposed an inverted pyramid model that depicts designers as having the responsibility of creating aesthetic products which provide meaning and evoke emotions in consumers. It is a role that is perhaps more profound than that of an entrepreneur's, whose job is to profitably sell his products.\nPsychologists also emphasize that many purchases are emotional in nature, and attitudes are formed when there is an emotional reaction towards a product, when certain memories from past life experiences elicit strong emotions in consumers. These emotions are elicited when a consumer encounters certain product or a design.\nEmotional Design: Why We Love (or Hate) Everyday Things by Donald Norman explores the underlying psychological process of buying. The book helps designers and manufacturers to understand why it is important to consider emotions, perception and other cognitive processes while creating websites or designing products.\nIn a world that is dominated by ecommerce and web-driven marketing, a company's website becomes an intermediary between products and services and consumers who purchase them. A well-designed website has the ability to capture and sustain a consumer's attention, and this attention can be translated into a positive perception if the design or layout of the website attempts to do so. Perceiving a website positively usually results in processing of that positive information and associating it with positive life experiences.\nFor instance, a website that is designed aesthetically may encourage a consumer to associate those colours with an art class that they took in school, which probably was a very positive and satisfying experience for them. These positive memories evoke positive emotions and mood states that help consumers to associate that particular website with joy, satisfaction or even self-actualization.\nIf we are talking about how one could elicit positive emotions when designing a website, one must understand that the web page is akin to canvas. The canvas can be 'painted' with aesthetic images, contours and colors. Intuitive navigation and task-oriented functions are processed by visitors as credibility, trustworthiness and security. These perceptions evoke positive emotions that may translate into loyalty and continued visits. Of course, positive text and careful usage of words, avoiding jargons and not being circumstantial and tangential allow for information to be rendered in an aesthetic manner.\nWhile designing web or mobile applications, similar psychological and cognitive themes can be applied. An intuitive interface, colors associated with trust, simplicity and minimalism, and reduced clutter in app design help in evoking positive emotions as well. Using minimalism, intuitive layouts and contours, and elegance can help in evoking positive emotions from products and devices as well. Thus, a designer must bear in mind to consider aspects that have direct emotional consequences.\nSensual stimuli that activate visual, tactile and auditory receptors usually evoke emotions. Products and websites must be designed to please the eye. If possible, websites can have audio in the background and touchscreen capabilities of applications and websites allow for tactile stimulation. It isn't possible yet to tap olfactory (smell) and gustatory (taste) responses when designing websites, but one can't predict the future!\nEmotions are strongly connected to social situations and thus anything that can be shared, help in starting conversations and be an extension of consumers personality always help. Thus, designing for emotions is to remember that customers can and do make decisions based on feelings, impulses and mood states and not based only upon rational arguments and logic.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On 10 October, the cities of Cagliari and Capoterra, in the island of Sardinia, Italy were hit by torrential rains causing flooding. A highway bridge connecting the two cities collapsed. Several other bridges and roads were also damaged. Dozens of people were evacuated from the area. One casualty was reported by media.\nThe Copernicus EMS Rapid Mapping has been activated to provide grading maps assessing the magnitude of the damage over five Areas of Interest (AoIs). The maps show 197.3 ha (2 km2) of flooded area and various degrees of damage to residential buildings and to the infrastructure.\nThe maps and vector data are available for viewing and download on the EMS website: Activation EMSR329.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PALMER ARCHIPELAGO, ANTARCTICA \u2013 A few years ago, an icebreaker dragged research nets around the Palmer Archipelago, looking for Antarctic silverfish\u2014oily, sardinelike creatures that spawn beneath sea ice. They used to be the dominant fish off the western peninsula, composing half of what some Ad\u00e9lie penguins ate. But the team, led by Joseph Torres of the University of South Florida, towed day and night around Anvers and Renaud Islands and never caught a single silverfish. In waters that have experienced some of the greatest sea-ice declines, the fish had all but disappeared. Meanwhile scientists noticed penguins gulping more krill\u2014even though it can take 20 krill to match the caloric value of one silverfish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"okdo-doc-ppt-jpeg-wmf-to-pdf-converter.exe is a binary executable program for Windows. The download size as indicated by the server is 15.47MB (16224577 bytes). The host server on www.okdosoft.com has returned application\/octet-stream as the content type of the download which was updated on 06\/13\/2016 and was last checked by Webeaver.com crawlers on 10\/24\/2018. You may use one or more of the following keywords [okdo doc ppt jpeg wmf to pdf converter exe] to search for other files related to the one you are about to download.\nWebeaver.com does not host any files on its servers and okdo-doc-ppt-jpeg-wmf-to-pdf-converter.exe remains the property of its respective owner. We have no controle over the nature, content and the availability of the file listed here for free download and hosted on www.okdosoft.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Staff and students from EDA will be representing the University of Kent and AART-BC at Naidex from 26-28 April 2016 promoting assistive technology.\nNaidex is the UK's largest disability, rehabilitation and homecare event with over 200 exhibitors showcasing their products to more than 9,000 visitors. The event is at the forefront of innovation and as the industry evolves, staying up-to-date has never been more important. Naidex is the only place where trader visitors, healthcare professionals and consumers can come together under one roof to discover the latest products and innovations presented by the Naidex exhibitors.\nRepresenting the University of Kent on Stand A37, we will be demonstrating our smart driving assistance wheelchairs, displaying technology, reaching out to stakeholders and users and promoting our public and industry research profile.\nWe will also feature on Stand C45 as part of Adaptive Assistive Rehabilitative Technologies-Beyond the Clinic (AART-BC) which comprises 7 institutions: University of Warwick, University of York, University of Cardiff, University of Kent, University of Salford, Oxford Brookes University and University College London. AART-BC focuses on the development of a platform to monitor the use of Assistive Technlogy and compliance with Rehabilitation Programmes and support the patient outside of the clinic. The project will demonstrate the technology primarily in three patient groups: wheelchair users, prosthetic users and the elderly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fawn male AKC Doberman Pinscher puppy available. His tail is docked, dew-claws have been removed by vet, he will be wormed, shots, and well socialized upon relocating to new home. Parents are health tested and have excellent temperaments. References available. Our dogs are indoor family pets. He will be ready for new home April 14.\nThis advertisment was posted from Scranton, Pennsylvania on Mar 29, 2019 by RRDobes. Find other similar ads by searching the Doberman Pinscher Dogs & Puppies For Sale & Adoption category.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is laminated paper bag,it is made of 210g card paper with matt lamination,full color printing for custom logo on four sides of bag,rope handle knotted at turn top with reinforced paper board.\nLaminated paper bags have turn top with reinforced paper board,laminated paper bags have strong construction,so laminated paper bags look more sturdy for quality looking.it is the reason why laminated paper bags are considered as high end retail packaging products.meanwhile laminated paper bags are made of coated paper that is the best material to print any fine design with great result,so if you are looking for colorful,stylish,elegant shopping bags,promotional bags,carrier bags,laminated paper bags are perfect choice.\nAs paper bags manufacturer,4PaperShoppingBags specialize in manufacturing laminated paper bags,laminated shopping bags,laminated carrier bags,we offer wide range of paper,lamination,handle,print finishes to build quality bag coming in any size.Detail please contact with us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is National Farmers Market week !\nJoin us tomorrow, Saturday, Aug. 8th, as we celebrate being part of one of the best Farmers Markets in the U. S.\nMitch Armstrong Nursery is proud to be one of the original core vendors of the Saturday Morning Market St. Pete, since it's inception 13 years ago !\nBeing full-time vendors at the market is our full-time job and we are grateful for the support of our community that gives us a place to make a living and enables us to continue to give back to our community also. We are full supporters of the \"Buying Local, Saves Locals\" !\nJoin us at the Saturday Morning SUMMER Market at Williams Park tomorrow as we celebrate and continue to bring excellent and exciting plants to the market day.\nSUMMER MARKET RUNS THROUGH TO THE END OF AUGUST - ONLY 4 MORE WEEKS !\nThe new regular season at Al Lang Field will begin Sat. October 3rd.\nRemember, Expect the Unexpected from Armstrong Nursery - Thank you for your support !","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..0bee4c18f7e3d51a8dbb4943cc068772bff98891
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Today the Episcopal Church celebrates the Feast of St. Mary the Virgin. Last week we provided background on this feast and some suggestions for celebration.\nMarian devotion has been a part of Christian practice from the earliest days. As a consequence, there are thousands of images of Mary \u2013 receiving the news of the annunciation, holding the infant Jesus, weeping by the cross. She has been venerated in every generation and across many cultures; liturgical artists have paid her homage in icons, statues, altars, frescoes and more. For a small sample of Marian imagery, this Wikipedia gallery is worth a look.\nAs a mother myself, I expected that one of the traditional images of Mary holding the infant Christ would stand out to me today. But what stood out instead was something quite different: a small illustration from a 13th century English book of hours.\nBut the image stands out because it depicts Mary as active participant in the salvation of the world \u2013 which, when you think about it, is exactly what she was.\nThis image is straight out of the Anglican tradition. It comes from the De Brailes Hours, the earliest surviving English book of hours, created in the early 1200's. It is believed to have been made for an English laywoman whose portrait is included in the book in four places. William De Brailes, the scribe and illustrator who created it, worked near Oxford from 1238 to 1252. More images from the book and further details about the creator are available here.\nWhy does this image of St. Mary the Virgin speak to me with such resonance? Most likely because I want to read myself into its likeness.\nSo many images of the Blessed Virgin Mary have her holding the infant Christ and smiling beatifically. The days when I held my babies and felt blessed are behind me. I now raise a tween and a teen, which means each day I fight the powers of evil both within myself and in the world around me.\nI pray for wisdom to know the good and courage to do it as my husband and I make decisions that shape our children's lives. I pray for my children to be safe from all evil, even though I know that is impossible. And I pray that as they grow from infants to adults, my children will take up the struggle to discern and do what is right.\nIt is good for me to see the most Holy Mother of God take on the Unholy One. It is good to remember that \u2013 with her active assistance \u2013 Christ has already won the victory.\nO God, you have taken to yourself the blessed Virgin Mary, mother of your incarnate Son; Grant that we, who have been redeemed by his blood, may share with her the glory of your eternal kingdom; through Jesus Christ our Lord, who lives and reigns with you, in the unity of the Holy Spirit, one God, now and for ever. Amen.\nWhat image of the Blessed Virgin Mary speaks to you?\nWonderful post! That image speaks directly to me as well. I love all images of her crushing the serpent's head as well.\nThank you! I am new to looking up images of Mary \u2013 I will go find those.\nI concur \u2013 wonderful post! Blessed Feast of St. Mary\u2026.\nThank you for a great post. Just wanted to point out that \"straight out of the Anglican tradition\" is a bit misleading as there was no Anglican Church in existence in the 13th century, only the Roman Catholic Church from which all the great \"traditional\" ancient churches, religious images and artwork of England hail (those spared by Henry's destruction of Church lands and property). Very helpful information on the De Brailes Hours.\nYou're certainly right that the concept of Anglicanism would have been inconceivable at the time this image was created. Thanks for commenting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kevin Ha started doing side gigs, in addition to his full-time lawyer job, in order to pay off student loan debt.\nThat nut got paid off.\nBut Ha, 32 years old, is still doing the side hustles. He enjoys the work and loves the extra income. Monthly it averages between $2,000 and $3,000. His wife is employed also.\nAmong the gigs are walking dogs, returning scooters back to where they belong, delivering food on his bike, and renting out a room in his house on Airbnb. Here are more details from MarketWatch.\nSide hustles are losing their stigma. Recall the days when some felt sorry for high school teachers and university professors who had to take on a part-time job to make ends meet. Now, all work seems to carry status. It can be contract, part-time, or full-time.\nOften, though, it's necessary to keep all those professional identities separate. Here is my article on how to manage what I call \"mutt branding.\"\nPodcasting has rapidly shifted from an innovative addition in the marketing mix to you-better-do.\nSo, how does a lawyer or a law firm \"stand out\" in what might already be glutted space?\nThis blog invited Shannon Martin, Director of Communications at Podbean, to share what has proven out to be best practices for reach and engagement.\nPodcasting is an ideal medium for showcasing your expertise and communicating to clients and prospects.\nA growing audience: Almost one-half the U.S. population had listened to a podcast. Podcast listening is booming and will continue to with connected cars, Google and Spotify getting into the podcasting game, along with the rise of smart speakers. Because most podcast listening is done during daily activities and there are far fewer podcasts than blogs, it's a medium with a lot of inherent reach.\nThe right audience: Podcast listeners are loyal, educated and affluent. They have strong brand loyalty (in this case, the brand being you and your legal expertise).\nEven if you see the potential benefits, you might be thinking this is just another thing that will take up a lot of time. Of course it can take up a huge amount of time, but there are smarter approaches and tools to make it easier.\nFirst, consider your format and how you might repurpose content you already have or use your podcast to meet multiple practice goals.\nFor example, you might take a look at your website statistics. Examine your most popular blog posts (or the ones readers spend the most time on or that create the most leads).\nStart by repurposing those topics into podcast material. You could simply read the blog posts. Ideally, though, you can use them as a guide while adding a bit of personality or discussion that may not be there in the post. As you practice, you'll become more adept at what works for audio. As a lawyer, you likely already have a high level of verbal communication so you have what it takes!\nYou can reach Shannon Martin at smartin@podbean.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Early registration deadline ends Wednesday for the new Electronic Materials and Applications 2010 meeting. EMA 2010 will focus on electronic ceramics for energy storage and conversion applications and the complementary fundamental science issues. Sign up today to save $100.\nInnovations in Energy Technologies;\" and Tim Anderson \"The Florida Energy Systems Consortium: Integration of Electronic Materials into Energy Systems.\" For details, visit Plenary Speakers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Is there anything better than coming home to a warm pot of chili on a cold wintery day? No, there really isn't. Especially when that chili is spicy, smokey, and slightly sweet\u2013like this Honey Habanero Chili! This recipe has been with Neal and I since we were dating. The original recipe is from Slow Cooker Revolution by America's Test Kitchen and has morphed over the years into something completely different.\nWhile my initial reaction about putting honey into chili was \"No, that would be absolutely disgusting\" My second thought was more rational. There is a lot of spice in here thanks to 3 habanero peppers, chipotle peppers, and a decent amount of chili powder. The honey helps to balance out those strong spicy flavors, and provide a little sweetness that plays well with the fruity flavor of the habaneros. I'm glad I followed my initial thought. Neal said this was the best batch of chili we've ever made at home!While this recipe is finished in the crock pot, cooking the onions, peppers, and browning the meat on the stove help with flavor and texture.\nWhile gluten free (if using gluten free soy sauce), I think this recipe could easily be vegetarian by replacing the meat with lentils, though I haven't tested that yet. Make sure to wear gloves when you're chopping hot peppers such as habaneros! I never do, and usually I'm fine if I wash my hands immediately after handling them. This past time, my hands stung for the entire night! It was terrible. Also, I've found the best way to cut habanero peppers is to peel the meat off of the seeds, like you're peeling an apple. Then you're left with the stems and seeds, which you can add the seeds to the chili if you want more spice, though I feel like it is spicy enough without them. First step is to soften the onion and peppers in a large skillet until soft. Then, add the tomato paste and cook for a few minutes until slightly rusty in color as in the picture above.\nYou'll add the spices and cook until fragrant. Then, add the beef and cook until no longer pink. After you transfer everything to the crock pot and add the remaining ingredients, you're done! Just cook on high for 4-6 hours or low for 6-8 hours. It's a little heavy on the prep work for a crockpot meal, but it is so worth it. Serve it with fresh avocado, sour cream, shredded cheese, or tortilla chips!\nI hope you enjoy this chili! It is a perfect meal for a cold winter's day that is spicy and subtly sweet!\nSpicy chili with a subtle sweetness! Started on the stove and finished in the crockpot, this is the perfect chili to come home to on a cold day!\nHeat oil in a large skillet over medium-high heat until shimmering. Add onions, habanero peppers, and salt. Cook until vegetables are lightly softened. Add tomato paste and cook until color develops into a brick like color (see photo above).\nAdd chili powder, garlic, cumin, oregano, and red pepper flakes. Cook briefly, until fragrant.\nAdd ground beef and cook, breaking up large pieces as needed. Cook until no longer pink, about 5 minutes.\nStir in tomato sauce, scraping up any brown bits on the bottom of the pan. Transfer to slow cooker.\nAdd diced tomatoes, kidney beans, soy sauce, chipotles and honey.\nCook on low for 4-6 hours or on high for 6-8 hours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Queer Constellations investigates the goals and catastrophes of modern city historical past considered via new queer narratives of inner-city lifestyles. The \"gay village,\" \"gay mecca,\" \"gai Paris,\" the \"lesbian flaneur,\" the \"lesbian boh\u00e8me\"these and different city phantasmagoria function mockingly during this quantity as figures of progressive utopia and commodity spectacle, as fossilized archetypes of social transformation and ruins of haunting cultural power.\nHomosexuality and the eu court docket of Human Rights? is the 1st book-length research of the Court's jurisprudence in recognize of sexual orientation. It bargains a socio-legal research of the enormous variety of judgements and judgments of the Strasbourg organs at the wide selection of proceedings introduced by way of homosexual males and lesbians less than the eu conference on Human Rights.\nIn Masculinities within the Making, James W. Messerschmidt unravels the mysteries surrounding the query of the way masculinities are literally \"made. \" essentially the most revered students with regards to masculinities, Messerschmidt brings jointly 3 possible disparate groups\u2014wimps, genderqueers, and U.\n*** A Sunday occasions Bestseller ***In Queer urban Peter Ackroyd seems at London in an entire new means \u2013 during the heritage and reviews of its homosexual inhabitants. In Roman Londinium town used to be dotted with lupanaria ('wolf dens' or public excitement houses), fornices (brothels) and thermiae (hot baths). Then got here the Emperor Constantine, together with his bishops, clergymen and missionaries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzvpuy b/data_all_eng_slimpj/shuffled/split2/finalzzvpuy
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index 0000000000000000000000000000000000000000..2237d82ed7d75e1e489bf041fe0ebcb14193ad8b
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Use our Real-Time Jackson County, West Virginia SIP Trunking quoting tool to find out the best prices on Jackson County Internet and phone services for businesses of all sizes.\nOur technology prices out all types of Internet services for businesses besides SIP Trunks, from West Virginia LEC's, CLEC's, ISP's, and WISP's thoughout all of Jackson County and beyond!\nReceive real-time unbiased Jackson County, WV SIP Trunking prices from sip-trunk.businessinternetprovidersisp.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"7 days until our deadline, and I think the rookie's lost it.\nThis is the time when we should be laying low and planning our getaway, and she's talking about making cookies to leave for the fat man when he delivers the ransom. As if we'd have him come to our house to deliver it! The rookie clearly doesn't understand how this sort of caper works.\nI may have to fake a ransom delivery, just to satisfy the rookie. There's no way I'm letting the fat man know where we live.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please see the video gallery for a live recording of this hearing.\nGood afternoon, Chairman Smith, Cochairman Brown, and members of the Commission.\nOn behalf of the Committee for Human Rights in North Korea, thank you for inviting me to speak with you at this hearing today. Our Committee considers it essential to draw attention to the case of 30 to 40 North Koreans who have been arrested by China and who now risk being forcibly returned to North Korea where they most assuredly will be subjected to severe punishment in violation of international refugee and human rights law. The fundamental right to leave a country, to seek asylum abroad and not to be forcibly returned to conditions of danger are internationally recognized rights which China and North Korea must be obliged to respect.\nMr. Chair, the Committee for Human Rights in North Korea is a Washington DC-based non-governmental organization, established in 2001. Our Committee's main statement has been prepared by Chair Roberta Cohen, who was unable to be here today. I will draw upon that statement in my opening remarks.\nOver the past two decades, considerable numbers of North Koreans have risked their lives to cross the border into China. They have done so because of starvation, economic deprivation or political persecution. It is estimated that there are thousands or tens of thousands in China today. Most are vulnerable to forced returns where they will face persecution and punishment because leaving North Korea without permission is a criminal offense. Yet to China, all North Koreans are economic migrants, and over the years, it has forcibly returned tens of thousands to conditions of danger. According to the testimonies and reports received by the Committee for Human Rights, the North Koreans returned to their country endure cruel and inhuman punishment including beatings, torture, detention, forced labor, sexual violence, and in the case of women suspected of become pregnant in China, forced abortions or infanticide. Some have even been executed.\nWe therefore submit that North Koreans in China merit international refugee protection for the following reasons: First, a definite number of those who cross the border may do so out of a well founded fear of persecution on political, social or religious grounds that would accord with the 1951 Refugee Convention. Second, the reasons why these North Koreans flee to China go beyond the economic realm. Those who cross the border into China for reasons of economic deprivation are often from poorer classes, without access to the food and material benefits enjoyed by the privileged political elite. Subject to North Korea's songbun classification system, their quest for economic survival may be based on political persecution. Examining such cases in a refugee determination process might establish that certain numbers crossing into China for economic survival merit refugee status. Third, and by far the most compelling argument why North Koreans should not be forcibly returned is that most if not all fit the category of refugees sur place. As defined by the UN High Commissioner for Refugees (UNHCR), refugees sur place are persons who might not have been refugees when they left their country but who 2become refugees at a later date because they have a valid fear of persecution upon return.North Koreans who leave their country for reasons including economic motives have valid reasons for fearing persecution and punishment upon return. Accordingly, UNHCR has urged China not to forcibly return North Koreans and has proposed a special humanitarian status for them so that they can obtain temporary documentation and access to services and not be repatriated.\nChina, however, has refused to allow UNHCR access to North Koreans in border areas where it could set up a screening process. It considers itself bound by an agreement it made with North Korea in 1986 obliging both countries to prevent \"illegal border crossings,\" which replaced an earlier 1960 agreement. It also stands by its local law in Jilin province (1993) which requires the return of North Koreans who enter illegally.Both documents stand in violation of China's obligations under the 1951 Refugee Convention (which it signed in 1982), its membership in UNHCR's Executive Committee (EXCOM), and the human rights agreements it has ratified. These include the Convention against Torture, which prohibits the return of persons to states where they could be subjected to torture, and the Convention on the Rights of the Child, which prohibits the return of unaccompanied children to countries where they could be irreparably harmed.\nIt is reported that some local Chinese officials have at times provided documents to North Korean women married to Han Chinese, which allows them and their children some form of protection and access to medical and educational services. Such practices should be encouraged but they are not Chinese policy or law. Most North Koreans in China have no rights and are vulnerable to exploitation, forced marriages and trafficking as well as to forced returns where they will face persecution and punishment. Our Committee's report Lives for Sale: Personal Accounts of Women Fleeing North Korea to China, 2010, documents the experiences of North Korean women in China and the extreme lack of protection for them.\nFirst, the United States Congress should consider additional hearings on the plight of North Koreans who cross into China to keep a spotlight on the issue and try to avert forced repatriations to conditions of danger.\nSecond, members of Congress should consider supporting the efforts of the Parliamentary Forum for Democracy, established in 2010, so that joint inter-parliamentary efforts can be mobilized in a number of countries on behalf of the North Koreans in danger in China.\nThird, the United States should encourage UNHCR to raise its profile on this issue. It further should lend its full support to UNHCR's appeals and proposals to China and mobilize other governments to do likewise in order to make sure that the provisions of the 1951 Refugee Convention are upheld and the work of this important UN agency enhanced.\nFourth, together with other concerned governments, the United States should give priority to raising the forced repatriation of North Koreans with Chinese officials but in the absence of a response, should bring the issue before international refugee and human rights fora. UNHCR's Executive Committee as well as the UN Human Rights Council and General Assembly of the United Nations should all be expected to call on China by name to carry out its obligations under refugee and human rights law and enact legislation to codify these obligations so that North Koreans will not be expelled if their lives or freedom are in danger.\nFifth, the United States should consider promoting a multilateral approach to the problem of North Koreans leaving their country. Their exodus affects more than China. It concerns South Korea most notably, whose Constitution offers citizenship to North Koreans. Countries in East and Southeast Asia, East and West Europe as well as Mongolia and the United States are also affected. Together with UNHCR, a multilateral approach should be designed that finds solutions for North Koreans based on principles of non-refoulement and human rights and humanitarian protection. International burden sharing has been introduced for other refugee populations and could be developed here.\nSixth, the United States should consider ways to enhance its readiness to increase the number of North Korean refugees and asylum seekers admitted to this country. Other countries should be encouraged as well to take in more North Korean refugees and asylum seekers until such time as they no longer face persecution and punishment in their country.\nThank you, Mr. Chairman, and members of the Commission. I look forward to answering any questions you might have.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today on the blog I'll be blitzing Allison Rushby's Living Blond series! Exciting! The covers are just gorgeous, and they sound like such fun books to read. Read on for an excerpt Diamonds are a Teen's Best Friend!\nSynopsis: \"Nessa Joanne Mulholland, aka Marilyn Monroe's No. 1 teenage fan, is used to moving house. This time, however, she's relocating in movie-star style\u2014crossing the Atlantic on board the Majestic, headed for Paris from New York City. And it really would be in movie-star style if it wasn't for the fact that she's bringing her cringe-fest professor dad along for the ride (Dad's specialization: human mating rituals\u2014need Nessa say more?). Oh yeah, and sharing a cabin that's five decks below sea level and next to the engine room. Still, at least Holly Isles is on board. Yes, really, that Holly Isles\u2014star of stage and screen. Suddenly, things are looking up. Looking a little Marilyn, in fact, because events are strangely mirroring Nessa's favorite movie of all time, Gentlemen Prefer Blondes.\nAs Holly Isles, world-famous actress, confides in Nessa over mocktails.\nAs Nessa coaches Holly in the amazing \"Nessa's Lesson's in Love\"\u2014the ultimate man-catching rules to finding true and lasting lurv.\nAs it all goes terribly, horribly, embarrassingly wrong.\nSynopsis: \"Nessa Joanne Mulholland, aka Marilyn Monroe's No. 1 teenage fan, is living the high life in Manhattan. Literally. Waffling and pancaking it up every morning (care of housekeeper Vera) in her soon-to-be stepmother's Tribeca penthouse apartment. Things couldn't be better. Or so she thinks, until things start to go terribly, horribly wrong, in true Nessa fashion. All of a sudden, she's starting to feel the need to pull at her collar. Yes, it's summer in NYC and things are heating up fast, including the professor and Holly's wedding plans.\nAlong with Nessa as her dad's too-gorgeous research assistant moves into the new family penthouse while Holly's away filming in LA . . .\nAs Nessa gets dumped for \"Doris Day\" . . .\nAs the cupcakiest wedding ever hangs in the balance.\nSynopsis: \"Nessa Joanne Mulholland, aka Marilyn Monroe's No. 1 teenage fan, is about to become a big sister\u2014twice over. Her movie-star stepmother, Holly, is very pregnant with twins when, suddenly, B and G (that's Boy and Girl) decide they need to get away from it all.\nWithin hours, Nessa, her dad, Holly, her best friend Alexa, and tag-along cousin Nat are on a private jet and headed for some rest and relaxation Hawaiian style. Except that, on hitting the sand, there's not a lot of rest and relaxation to be found. The penthouse apartment they've booked is suddenly unavailable, Holly's blood pressure skyrockets, Nat is bronze-beach-boy crazy, and what's with the three cute guys living upstairs?\nWhen Nessa meets someone who's had the life-saving operation her mother couldn't have.\nHonestly, I tried to stop myself from asking the question, I truly did. In front of me, the porter guy looks at me as if I'm a thirteen (almost fourteen!)-year-old idiot. Beside me, dear old Dad looks at me as if I'm delusional (that's because a lot of the time he actually thinks I am\u2014he's even had me tested to make sure I'm not). And he's about to open his mouth to start in on me (again . . . sigh) when, behind me, I hear it\u2014someone laughs. Right on cue.\nI swing around quickly, my head zipping from side to side, trying to see who it is, but it's practically impossible in this traffic jam of a crowd, especially when you're as short as I am and your dad won't let you wear a kitten heel, let alone rhinestones in the daytime. I bet Marilyn Monroe's mother never said a thing about Marilyn wearing rhinestones in the daytime. Then again, Marilyn Monroe's mother let her get married at sixteen and spent a great deal of time in a mental institution, so that's probably not saying very much. I'm just about to give up on the searching thing when the crowd parts and someone dressed entirely in red, going out\/in\/out (in all the right places) and hips swaying, passes me by with a wink and a lift of one perfectly arched eyebrow.\nIt's one of those moments when you just know you'll think up a zillion and two perfect things to say later, but instead you stand there looking like you've recently had a lobotomy. Especially when I realize that the someone is actually a Someone and that the woman now heading up the escalator to the biggest ship I've ever seen in my life is, in fact, Holly Isles.\nActress. Goddess. Star of stage, screen and various tabloid magazines that you skim as fast as you can at the supermarket checkout because your dad will never let you buy them, Holly Isles.\nI follow his gaze directly up the escalator to Holly. His aunt? Holly is his aunt? Well, la de da. I go to give him my best \"Get your filthy mitts off me, don't mess with the outfit and don't go anywhere near the hair, buster\" look when my mouth drops even further. Hello, sailor! Cute boy ahoy! This guy is definitely related to Holly in a big way.\n\"Ah . . .\" my dad exhales, the lecture he'd been working on giving me obviously forgotten. Funny, but he's got the same kind of lobotomy look as me. And he's staring straight at Holly.\nIt takes me a moment to get over the shock. When I finally do, my eyes move back from my dad to Holly's form as she goes up, up, up (and I'm not alone, I think everyone on the dock is watching Holly go up, up, up). Slowly, I shake my head from side to side. I can't believe it. I can't believe Holly Isles just said that to me. And it wasn't exactly the line from the movie I was thinking about, but then again, neither was mine. And I could probably go on watching her forever, my mouth hanging open (okay, so I only check out her nephew a few more times), except that Holly's now out of sight and my dad has obviously returned to his old lecturing form.\n\"Huh?\" I finally look up, only to see him looking down at something. At my chest. Ugh. Gross. What is he doing? He'd better not be doing any research on me. But then I look down too.\nOops. The tissues are escaping again.\nI surreptitiously stuff them down my black-with-leopard-skin-trim top with one hand. \"I've got a cold, okay?\" I mutter. Geez. So much for Marilyn's mother. I bet she never had to put up with this kind of harassment from her father, either. Except that no-one was ever quite sure exactly who he was. And, of course, Marilyn didn't need to stuff, either, did she?\nI sigh my second sigh of the morning. Life. It's just so . . . unmovielike.\nHaving failed at becoming a ballerina with pierced ears (her childhood dream), Allison Rushby instead began a writing career as a journalism student at The University of Queensland in Brisbane, Australia. Within a few months she had slunk sideways into studying Russian. By the end of her degree she had learned two very important things: that she wasn't going to be a journalist; and that there are hundreds of types of vodka and they're all pretty good. After several years spent whining about how hard it would be to write a novel, she finally tried writing one and found it was quite an enjoyable experience. Since then, she has had nine novels published. She keeps up her education by sampling new kinds of vodka on a regular basis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The next George R.R. Martin comic, Free Comic Book Day, Venom, and more pop culture news that wants to eat your brain!\nMr. Design Junkie Does It For Infinity!\nFanboy Planet Podcast Episode 458: Sing Fanboy, Sing!\nOccupy Avengers, so long Robert Vaughn, Inhumans on TV, Ghost in the Shell, and much more in this week's musical episode!\nBecause The Galaxy Isn't Going To Save Itself. Obviously.\nAdam Warlock Returns In March!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The hotel is located on Alagarkoil Road in Madurai, and all places of tourist interest are easily accessible. Madurai airport is just 14 km from the hotel, the railway station is 5 km and the bus stand is only 3 km away from the hotel. This budget hotel provides comfortable accommodation in 60 rooms with a host of world-class amenities sure to make business or leisure guests' stay a delightful experience. The air-conditioned hotel features a modern conference hall with all audio-visual and conferencing facilities. Further facilities include a lobby with currency exchange facilities and a hotel safe for valuables, a bar, Internet access, room and laundry services, and a car park. The guest rooms feature traditional decor and modern conveniences lending them a home away from home feel. The comfortable and fully furnished guest rooms are equipped with a host of modern amenities, such as satellite TV, a telephone, study table, wall-mounted mirror and an en suite bathroom with a shower and a bathtub. Guests may take a dip in the outdoor swimming pool.\nYou travel. WISE donates! Booking Sangam at the best price includes a donation to your favorite charity, at no extra cost! We brought 18+ million euros to local communities providing social services to 150.000+ people from vulnerable groups. We support #GlobalGoals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This infographic provides an overview of a real-life crime ring active for more than 8 years, spanning 32 institutions, involving 29 entities, and representing potential losses of $10 million.\nPowerful cross-institutional analytics and 314(b) information sharing technology allow financial institutions to see previously unseen risks, including criminals who repeat their unlawful activity across multiple institutions.\nLearn about the methodology, scale, activities and potential for loss in a multi-million-dollar crime ring.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Charlottetown from 7 to 9 pm on Wednesday, November 8 th .\nmessy, inspirational and creative aspects\".\n\"It's a matter of being able to participate in something the artists believe in,\" says Rutherford.\ntheir names, while others will use pseudonyms or exhibit anonymously.\nand under. The show runs through Saturday, December 2.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nearly 65% students at ESM receive some form of tuition assistance. ESM funds most need based scholarships through donations to The Lutheran Scholarship Granting Organization of Indiana (SGO). Over $60,000 of scholarships are provided to students who might otherwise not be able to access Christian education. Supporting scholarships is a critical partnership, but also can provide a valuable 50% tax credit for donors. To learn more and help fund a student's tuition visit www.lutheransgo.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Number one rated shaky head. The Giggy Head design is extremely snag resistent, an important factor when using light line. Molded on a 5\/0 quality Gamakatsu hook ensures the best hook set possible. The Giggy Head keeper design allows the bait to break free from the head enabling a more solid hook set than a jig head with a screw keeper. Remember you are fishing with light line, light rods, and deeper water. The combination of the jig head design and Gamakatsu hooks will result in more landed fish.\n5\/0 Giggy Heads, 4 per pack.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwpqy b/data_all_eng_slimpj/shuffled/split2/finalzzwpqy
new file mode 100644
index 0000000000000000000000000000000000000000..34166e1b26354efe5a149d0273ff9c343e8b95bc
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@@ -0,0 +1,5 @@
+{"text":"This happened during the Toronto G20 so-called \"riots\" this past weekend. Pretty sure it's at Yonge & College, right in the centre of downtown Toronto. The only way this could be better is if the thief said \"Sorry\" after being wrestled to the curb.\nAlthough, frankly, given the hellswamp that is Bell Canada customer service (I was on the phone with them for FIVE HOURS last week), I kind of wish it was a better company who was getting their little products returned by gentle Toronto vigilante wrestlers.\nThanks to Rebecca for Facebooking this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Atlanta's \"spotlight on beauty\" session kicked off the Pure Fashion season. The new Pure Fashion Models gathered together to hear about the exciting things planned for the year and to meet one another and their \"super models\" (Pure Fashion team leaders).\nAll models were divided into super model teams, led by previous Pure Fashion models who exemplify the message of Pure Fashion. They had a great time breaking the ice and getting to know one another!\nLater, we heard fabulous presentations by core members of the Pure Fashion team. The testimony of National Director, Mrs. Brenda Sharman, was especially inspiring, as she shared her story of her 20 year modeling career, her conversion to the Catholic Church, and how she invited Christ into her closet! We also went over the Pure Fashion creed so that each of our new Pure Fashion models came to understand what Pure Fashion believes about protecting one's innocence and purity by dressing modestly and being pure of heart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Working with Emotional Intelligence takes the concepts from Daniel Goleman's bestseller, Emotional Intelligence into the workplace. Business leaders and outstanding performers are not defined by their IQs or even their job skills, but by their \"emotional intelligence\": a set of competencies that distinguishes how people manage feelings, interact, and communicate. Analyses done by dozens of experts in 500 corporations, government agencies, and nonprofit organizations worldwide conclude that emotional intelligence is the barometer of excellence on virtually any job. This book explains what emotional intelligence is and why it counts more than IQ or expertise for excelling on the job. It details 12 personal competencies based on self-mastery (such as accurate self-assessment, self-control, initiative, and optimism) and 13 key relationship skills (such as service orientation, developing others, conflict management, and building bonds). Goleman includes many examples and anecdotes\u2014from Fortune 500 companies to a nonprofit preschool\u2014that show how these competencies lead to or thwart success.\nDaniel Goleman, Ph.D. is founder of Emotional Intelligence Services in Boston. For twelve years he covered the behavioral and brain sciences for The New York Times and has also taught at Harvard. His previous books include Emotional Intelligence; Vital Lies, Simple Truths; The Meditative Mind; and as co-author, The Creative Spirit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is our Indiegogo crowd funding video that explains the creation, story and intention behind Piranha Yama and The Art of Non-Biting. With everyone's help we reached our goal in 21 days! Now we are finalizing images with the illustrator and on the way to getting the first book in the Piranha Yama series produced. If you haven't seen this short video yet, please take a look and share with your friends!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In this paper, a pseudo-ternary (La0.5Ce0.5)65Al10Co25 alloy with high glass-forming ability (GFA), identified by the formation of glassy rods up to 12 mm in diameter by copper mold casting, was reported. Furthermore, according to the viewpoint that similar element coexistence can be beneficial to improve the GFA, Cu was chosen as the similar element to substitute for solute Co in (La0.5Ce0.5)65Al10(Co1\u2212xCux)25 alloys (0\u2264x\u22641). The influence of this substitution on thermal stability, melting and solidification behaviors of these metallic glasses was evaluated. The optimized (La0.5Ce0.5)65Al10(Co0.6Cu0.4)25 alloy with high reduced glass transition temperature (0.66) and the large nominal supercooled degree (114 K) can form the fully glassy rods up to at least 25 mm in diameter by tilt-pour casting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwyqk b/data_all_eng_slimpj/shuffled/split2/finalzzwyqk
new file mode 100644
index 0000000000000000000000000000000000000000..977b064be1bc606a050fedd0f892b026c2a7ed48
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzwyqk
@@ -0,0 +1,5 @@
+{"text":"Activities will be outside on the historic site.\nA fun, two-day, Southeastern Native American festival with musicians, tribal dances, artisans selling quality crafts, astronomy, cultural demonstrations and hands-on activities for guests!\nInformation on musicians, demonstrators, and more to come!\nCheck back in early fall.\nof the Friends of Florida History - Mission San Luis today!\nNote that guests WILL be permitted to leave the site and return within the same day if they retain their admission sticker or receipt.\nMission San Luis is administered by the Florida Department of State. Support is provided by the Friends of Florida History, Inc.\nand sponsored by the Museum of the Cherokee Indian.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Quicksilver Connections. Shop 12 Reef Marina ,Port Douglas - QLD.Quicksilver, Australia's most awarded reef operator, offers the latest in comfort and safety .\nFor the experience of a lifetime, Quicksilver's wavepiercing catamarans and friendly crew take you to Agincourt Reef at the very edge of Australia's Continental .\nGreat Barrier Reef tour and holiday packages with Quicksilver Crusies.\nQuicksilver Cruises (Port Douglas) - Rating: 4.5\/5: Top Tips Before.\nBook your tickets online for Quicksilver Cruises, Port Douglas: See 2201 reviews, articles, and 1046 photos of Quicksilver Cruises, ranked No.10 on TripAdvisor .\nPort Douglas Info - Holiday Packages & Tours - Quicksilver Reef Day.\nA magnificent day where you can snorkel, dive or stay dry and explore the stunning reef from the comfort of a Quicksilver semi-sub.\nLATEST DEALS for Quicksilver Wavepiercer Cruise to the Great Barrier Reef.On this Great Barrier Reef Cruise that departs from Port Douglas you will be able .\n3) Silversonic, outer reef snorkel adventure, Port Douglas (diving is optional and an additional cost) (4) Silverswift.Not applicable with any other discount.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"digital light timers light timer auto light timers automatic digital light oven timer switch power control switch auto.\ndigital light timers volt 7 day program in wall solar digital timer switch.\ndigital light timers digital hydroponic timer hour plug in hydroponic timer.\ndigital light timers 7 day digital.\ndigital light timers digital electrical random light timers.\ndigital light timers 7 day electronic lighting timer.\ndigital light timers outdoor light timers instructions wall timer instructions recalled in wall timers have no date code stamp digital light wall timer instructions.\ndigital light timers china new 7 day programmable digital timer switch light timers plug socket timer with rainproof.\ndigital light timers timers for lights.\ndigital light timers digital lamp timer timers parts pool products.\ndigital light timers light timers outdoor digital timers for lights a inspirational timer 7 day outdoor digital light light timers.\ndigital light timers digital lamp timer light timers indoor digital lamp timer.\ndigital light timers timers wire timer for volts digital timer manual co digital lamp timer wireless wall switch timer.\ndigital light timers programmable light switch timer switch for lights programmable light switch timer with dimmer programmable light switch electronic timers.\nBe the first to comment on \"Digital Light Timers\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"gel polish uv light gel nail polish without lamp purple nail tips plus gel nail polish at home without.\nsolid white solid white dressing table solid wood white dresser.\nmedication lock box walmart lock boxes at handgun lock box key lock boxes.\nlasko blower fan blower fan.\ntransformers beast machine in early beast machines toys came into stores around the us and soon afterwards came into other countries as well this line of transformers offers.\nwood handle extra heavy duty screwdriver set by wade wood handle silverware wood hammer handle machine.\nbest insulated water bottle with straw go bottles stainless steel insulated water bottle with flip straw insulated water bottle with straw and handle insulated water bottle without straw.\nmichaels shadow box frames shadow box shadow box picture frames shadow boxes shadow box shadow box decor shadow box how michaels canada shadow box frames.\nbaby shower princess princess baby shower invitation baby shower princess ideas baby shower cakes princess theme.\nemeril knife set reviews exclusive stainless steel knife set piece.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Obtain the list of Neemuch Pin Code including search option to get any district postal code details of Madhya Pradesh, India.\nNeemuch is a district of Madhya Pradesh state. Get all area pin Codes list including all states of India from leading pin code finder.\nClick on show map to view map of Neemuch, Madhya Pradesh, India. Enlarge map to view complete Google road map.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxxfm b/data_all_eng_slimpj/shuffled/split2/finalzzxxfm
new file mode 100644
index 0000000000000000000000000000000000000000..0d942a7663139b255fd5ed4275f707aa8b6f2683
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzxxfm
@@ -0,0 +1,5 @@
+{"text":"Following in the very large footsteps, and the 57-year tenure of Weston Noble at Decorah, IA's Luther College, Dr. Craig Arnold returned to serve as director of choral activities, conductor of the Nordic Choir, and professor of music at Luther. \"Everlasting God\" is a spirited collection of 20 songs, which highlight the rich, powerful sound of this great choir. Rene Clausen's \"Laudate,\" two selections from Herbert Howells' \"Requiem,\" the wonderful 6-part \"Liebeslieder Walzer, op. 52\" by Brahms, the Stephen Paulus-arranged traditional hymn, \"We Gather Together,\" the 2-part \"The Lord is the Everlasting God\" by Kenneth Jennings, Ola Gjeilo's haunting \"Ubi Caritas,\" the heart-tugging \"Danny Boy\" and the spiritual \"Wade in the Water\" are favorites. There is some light piano and organ accompaniment. Luther College places a focus and emphasis on music instruction and performance found in few other institutions, and that's why we consistently hear recordings of the highest quality-the Nordic Choir is simply one of the best in the world!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This webinar was offered live June 23, 2010. Once purchased, you will launch the recorded webinar from the link in your account page.\nAaron is an adult adoptee who was placed in a mixed race home. His goal is to spread his positive story to the masses, since he realizes that he is very fortunate to have had parents that worked hard at making a very conspicuous family fit in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Low-profile and nimble, this insulated, waterproof glove is perfect for keeping hands warm during highly active endeavors in moderate winter weather. The neoprene cuff locks out the elements while the premium Pittards goat and sheepskin overlays at the palm are embossed to give you a great grip on all your cold weather toys and tools. Inside, a combination of fleece and Omni-Heat\u00ae thermal reflective delivers warmth without bulk to facilitate dexterity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posts titled Bees in the Garden provide reliable buzz about one of our most important groups of pollinators making it possible to produce food and ornamentals. In addition to sharing information about plants for good bee nutrition, we cover topics ranging from bee research to ways you can help bees in your garden.\nResearchers estimate 20,000 species of bees worldwide, most of which aren't kinds in managed hives. Bees in the Garden articles talk about wild bees as well as honeybees -- both managed honeybees in box hives and feral colonies found in tree trunks and unexpected locations. At Flowers by the Sea, we are closely acquainted with honeybees, because we manage our own hives and spend our days surrounded by these hard workers in our test gardens and greenhouses.\nWhereas approximately 4,000 species of wild bees are native to North America, the apian species most familiar to all of us is the honeybee, which comes in many varieties. It is fitting that on a continent dominated by immigrants, honeybees are Old World emigr\u00e9s that first arrived here in 1622.\nThe nectar and pollen of Salvias don't just taste and smell good to these tiny creatures. They are nutritionally powerful foods for bees, which need a healthy, varied diet to withstand the many perils they face ranging from habitat loss to pesticides. Part of our mission at FBTS is to support all kinds of pollinators with excellent plants for wildlife gardens. We invite you to become part of this mission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From time to time I get a note of gratitude in response to a sermon. Most of the notes center around someone's appreciation for taking otherwise obscure passages in the Bible and making them easy to understand. These notes are always encouraging and deeply appreciated, but I must admit, the task is not as easy as it looks.\nThe people of God long to be able to pick up the Bible and be blown away by it every time they turn the page. It is an honest craving (1 Peter 2:2). Yet it is not an honest approach. While there may be times when the Bible speaks quickly, there are others in which the text speaks slowly. It is at these points that it beckons us to listen closely, for long periods of time. It is here that God invites us to dig. The fact that it is sometimes difficult to understand Scripture is an honest admission and one that resonates throughout the text. Peter confessed that for him, Paul was hard to understand (2 Peter 3:16). Paul counseled young pastor Timothy that it would take his best work to be able to accurately explain the Word of God (2 Timothy 2:15). The Word of God is powerful, but it is not easy.\nWhen it comes to understanding the Bible, there is no substitute for serious study. We must give it time and we must use the tools. I think everyone should read the Bible through, cover to cover, without stopping, at least once in their lifetime. I have done so, once! After you read it through once you get a lay of the land; you get a mental map. Now that you have the map, explore the terrain. Take time to dig. Be methodical. Be intentional. Be patient. I am not at all a fan of, \"Let's just open the Bible up and read such and so today.\" I don't think Paul, or Peter, or David were fans of such a casual approach either. These men were explorers. They dug down deep and found the most intricate ways that the veins and crevices of Scripture intersected and fed one another at the subterranean level. Paul's vast knowledge of the Old Testament served as his inspiration for his great treatise on the Doctrine of Salvation, The Book of Romans. Writing Romans was not easy; neither is understanding it.\nTake courage people of God. The Bible is huge. It is deep. It is vast. It is powerful. If it doesn't come easy to you, don't get discouraged. Join in with the rest of us who take time to dig. Take time to dig down deep and you will be greatly blessed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzyzud b/data_all_eng_slimpj/shuffled/split2/finalzzyzud
new file mode 100644
index 0000000000000000000000000000000000000000..60b24ade049b84286efdab25c1ce8b1e13cf835e
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Dr. Antonio J. Webb, M.D.\nThe odds were stacked against Antonio Webb, beginning with his childhood in Shreveport, Louisiana. Friends and family members were serving jail time, or addicted to drugs. Some close friends were even killed amidst the challenges of the neighborhood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hemangioma is a congenital vascular malformation consisting of a benign tumor made up of newly formed blood vessels clustered together; it may be present at birth in various parts of the body, including the liver and bones. In the majority of cases it appears as a network of small blood-filled capillaries near the surface of the skin, forming a reddish or purplish birthmark.\ncavernous hemangioma a congenital vascular malformation that has a soft, spongy consistency and may contain a large amount of blood. It usually appears during the first few postnatal weeks and disappears by the age of 9 years. The most common sites are head, neck, and viscera such as the liver, spleen, or pancreas. Treatment varies according to the size of the lesion.\nstrawberry hemangioma a circumscribed capillary hemangioma, which may be present at birth or may appear soon after birth. These are most common on the head, neck, and trunk and appear as small macules that develop into raised purplish-red lobulated tumors. Most involute by age 2 to 3.\nMedical & Clinical Reviews, Insights in Biomedicine,Medical case reports,Korean Journal of Dermatology, Dermatologie in Beruf und Umwelt,Dermatologia Cosmetica, Medica y Quirurgica.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applicants with disabilities are encouraged to notify the International Education Programs Office of their disability at the time of application, so that adequate accommodations of the disability can be considered. Official documentation of the disability (for example: medical diagnosis, doctor reports, etc...) will be needed to provide reasonable support services.\nIf you are a student with a disability, having an understanding of what services or technology works best for you is very helpful. It will help us provide you with the most appropriate accommodation to support your access to college. After you receive your student visa, you can contact the Office of Disability Services to begin a conversation about support services for your disability.\nSeattle Central College provides Sign Language interpreters for Deaf students attending our program. Therefore, your success will depend on your knowledge of American Sign Language and your ability to use Sign Language interpreters.\nFor those looking to learn both English and American Sign Language, we can recommend several schools that have programs designed to teach both languages along with Deaf culture. Some schools use a visual-spacial approach to ESL in a bilingual setting to assist students with learning more than one language at a time. In addition, these schools have a larger population of Deaf students, which allows opportunities for more social interaction where American Sign Language is conversantly used.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the most difficult aspects of being a casual wine consumer is that it can often seem like the enthusiasts are speaking a completely different language, and while that language may also be in English, to most people it's anything but. The problem with this is that usually it's the enthusiasts who are selling us the wine we so desperately want to enjoy, but because of the language disconnect, we're left wondering what that phrase they just used actually means while standing there nodding our heads.\nFor those of us lost in translation, just as we explained the meaning of 20 wine words most people don't know, here are 16 wine phrases translated to plain English, so we're all speaking the same language.\nYou can't get this wine unless you wait on a list for years, but this Cult Cab is worth it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you are working out your spiritual muscles in the gym or just spreading your love on the streets, this muscle tank will keep you looking good as you leave your mark.\nBoth modern and feminine take on design with elongated armholes and curved bottom hem. Perfect for layering.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaague b/data_all_eng_slimpj/shuffled/split2/finalzzzaague
new file mode 100644
index 0000000000000000000000000000000000000000..2e0accd9d25579f6fc5307fe2213d39d9638dc1e
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaague
@@ -0,0 +1,5 @@
+{"text":"I'm kind of ashamed to admit this but I only discovered recently that the developer of Super Hexagon is the same Terry Cavanagh that developed VVVVVV; VVVVVV is a retro, minimalist platformer with an awesome soundtrack that was originally released as a Flash game in 2010, although the game was ported to C++ in 2011. The game has won several awards and is critically acclaimed; I think it helped spearhead the indie game renaissance on PC after the success of games like Minecraft in 2009. Anyway, Cavanagh developed another retro, minimalist game with an awesome soundtrack in 2012 and that's the game I'm reviewing now: Super Hexagon. It too received awards and critical acclaim: it currently holds a Metascore of 88 on Metacritic and 97% of the 14,531 user reviews on Steam are positive, giving it an \"Overwhelmingly Positive\" Steam rating.\nBut what do you actually do in Super Hexagon? Well, you avoid polygons. And that's all there is to it.\nIn the first level you'll come across walls that head towards the centre of the screen where your triangle resides. You have to move the triangle around the centre of the screen to avoid the incoming walls. Sometimes there's plenty of space but other times you'll come across hexagons with an edge missing (meaning that a collision with five out of the six edges will spell \"GAME OVER\"). If your triangle collides into a wall, you have to start again, but the game will continue to record your personal best time (i.e. the number of seconds you've survived dodging polygons).\nI bought a 2-pack of the game back in 2013 (it was 90% off) so I got it for 50 Cent and I got it on Christmas too (\"JC! It's your birthday! We gon' party like it's yo birthday!\"). I'm still not sure why exactly I got the game but I suspect it's something to do with the game being so well received by critics and Steam users alike.\nNiamh Houston aka Chipzel (I first encountered her music in Interstellaria) is known for her great chiptune soundtracks, and her soundtrack for Super Hexagon is no exception. She sells the EP on Bandcamp if you're willing to pay a couple of quid.\nOne of the great aspects of this game is that the graphics pulsate in time with the music, so it's almost like a rhythm game too.\nLeft and right. Those are the only two controls you need to know in order to play this game.\nGames usually only take a few seconds to complete (especially if you're as bad as me) and since they're so short, you won't feel bad trying it a few more times to see if you can beat your best time; it's addictive in an old-school way, you're always striving to beat your personal best.\nThere are 6 Steam Achievements to earn (which are related to completing the stages) and the game is integrated with Steam Leaderboards; this means you can compare your best times with your friends and the Steam community from inside the game.\nIt took me over half an hour of tries before I finally made my way out of the easiest level of the game (where you have to survive a grand total of 20 measly seconds). So it's definitely not a game for the uncoordinated; you've got to have good reflexes and you've got to pray to God you don't move the triangle too far left or right or else you'll accidentally direct it on a collision course with the nearest wall.\nAlso, while it's understandable that there's no point in progressing to the harder difficulty levels if you can't even complete the first level, it also means you're locked out from the rest of the game's content potentially forever (if you're as bad at the game as me).\nIf you're looking for a challenging minimalist arcade game with a fantastic chiptune soundtrack to boot, Super Hexagon is worth a look. If you're as bad as me at games like these though, the game can seem like a bit of a waste since you might be forever stuck on the first level.\nIs the game worth $4.50 AUD?: There really isn't much to this game but yes, I think it's worth the money as it's got quite a bit of replay value if you're into these sorts of games.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Parthenon Company Blog - Specialists in Wound and Ostomy Supplies: Something New for 2017!\nThe Wound Ostomy Continence Nurses society has announced that they have partnered with Hollister, manufacturer of various ostomy appliances and accessories, to develop a tool to measure peristomal skin health.\nPeristomal skin is the skin around your stoma and it can influence greatly how long your appliance wear time is as well as what extra accessories you use. It is so important to keep the skin around your stoma in as good condition as possible. One of the most common problems ostomates experience is stomal output contacting the skin from a leak in the barrier. Probably the first thing you should do for this is to contact your WOC nurse to assess the fit of your appliance.\nThis new electronic guide to assess peristomal skin health will function across mobile devices and allow your nurse to make a quick assessment of your skin using only 4 questions!\nNo more information on this now, but we will keep you posted as it becomes available.\nMost ostomates use \"stoma powder\" or \"stoma paste\" to help reduce the incidence of leaks with their appliance. Check out these popular ones!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Continuing its popular movie night tradition, the famed Soho House in West Hollywood screened the new film \"Balls to the Wall\" last night.\nOpened in spring 2010, Soho House West Hollywood is the second members- only American House, and occupies the top two floors of the 14-story, 9200 Sunset Boulevard.\nPost-screening, the film received a standing ovation by the guests and cast in attendance.\nIn the bottom of a cocktail shaker, muddle raspberries with sugar. Pour into a tall glass over ice. Top with GREY GOOSE and lemonade. Present with raspberries and a lemon.\nThis cocktail features the perfect union of sweet, ripe berries and tart lemonade. Ex\u00acplore variations on this drink by substituting other berries when they are in season, such as blackberries or blueberries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When I created this collection I was in a difficult relationship , an impossible love with a girl who was going to leave for a year in a country very far and who didn't want to start a relationship with someone before she goes. It made me think of the song from a French singer who is talking about extreme feelings you can have when you are in love!\nI find inspiration from what's around me in my life in general , like books , music , life street , art ... Inspiration has no limits , hopefully I will never lose it , fingers crossed!\n3. What to tool would you never leave the house without? As well as product?\nI never leave the house without my phone .\nBecause session work is what inspires other people . Session stylist are creating the future trends and new looks . Our job is to make people who will read your magazine dream.\nOn set when i'm working i'm at work , I mean i'm there to do a job and I put all my focus and knowledge to give the best of myself for the rest of the team . In this job you have to be very professional ! But after my job is done , i'm a very laid back person always joking with the rest of the team when they are working , so I guess it's more difficult for them to concentrate and stay focussed!\nThe most embarrassing situation I've been on set was when the models I was styling were speaking about the angelfish government spending a lot of money on foreigners , and they were badmouthing people coming to the UK , and as I moved to the UK 3 years ago I felt a bit embarrassed ! That was an awkward situation!\nMy mantra is I'll sleep enough once you'll die . Since the day I heard this quote , I have never stopped creating, I shoot, I work from the minute I get up!\n9. If you could go back to the beginning of your career, what piece of advice would you give your newbie self ?\n11. Favourite adult beverage? Why...and when did u first have one?\nFrench wine of course because it's an institution , and I started too young!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you're looking to pursue a Masters in Civil Engineering then Australia is definitely an option you should consider as there are ample job opportunities for civil engineers and designers due to major infrastructure projects in key Australian states including Victoria and New South Wales.\nIn this article, we will provide you with a comprehensive overview of the top Australian universities for MS in Civil Engineering and highlight some key points that you need to keep in mind while selecting a course and university. We'll also cover the current state of the industry in Australia and your job prospects after you complete the Masters course.\n* The 'Ranking' refers to the 2015 Times Higher Education Subject ranking for 'Engineering and Technology'. Wherever the Times Higher Education Subject ranking is not available the QS Faculty ranking is used. Universities not featured in both ranking systems are not included in the list.\nYou should keep a few essential things in mind when you are selecting your course and university.\nFor starters, did you know that as an international student, you can apply for a post study work visa in Australia? BUT - and this is the important part - you will only be eligible for a post study work visa if your course in Australia meets certain criteria. If this is something that you might be interested in then you need to make sure that you are completely familiar with the post study work visa criteria before you make a decision about your course and university.\nRead our simple, detailed guide to post study work visas in Australia to learn the latest post study work visa rules.\nAnother thing you should be aware of is the dual degree options that are available. As you can see in the list, the duration for most courses is 2 years. Some Australian universities have dual degree options where you can also complete another related course in the same 2 year span.\nFor instance, the MS in Civil Engineering at the University of Technology Sydney (UTS) has a duration of 2 years but UTS also offers a dual degree - MS in Civil Engineering + Masters in Engineering Management - which is also just 2 years in duration. Similarly, Swinburne offers a double degree with the Master of Construction Management and Griffith offers a double degree with the Master of Project Management.\nBasically, if you opt for the double degree you will graduate with 2 qualifications instead of just 1 at the end of 2 years.\nNow that you are familiar with the post study work rules, you have obviously have realised that you need 2 years of academic study in Australia if you want to be eligible for it. Given this scenario, you might as well get 2 Master's degrees in those 2 years and broaden your job prospects after graduation.\nHere's something you probably did not know: The Australian Government has an investment plan that delivers over $50 billion in infrastructure for the period from 2013-14 to 2019-20 onwards.\nIn fact, the 2016 Budget in Australia includes over $33 billion in forward estimates for critical infrastructure and this is expected to generate tens of thousands of jobs. A number of projects are planned across key states including the $5.6 billion Pacific Highway duplication in New South Wales, the $500 million Monash Freeway in Victoria, the $518.4 million Moreton Bay Rail Link in Queensland and more. Take a look at this report for a more comprehensive list of projects in each state and territory.\nAnd finally, we come to costs, an important consideration for most international students. Here we have taken the same list as above as sorted it based on the annual tuition fees for international students.\nYou now have a comprehensive overview of the popular universities that offer the MS in Civil Engineering course in Australia as well as detailed course information. Hopefully, this will help you pick a university that suits your requirements.\nRemember, as an international student, you need to keep current post study work rules and industry trends in mind while you make your decision.\nIf you need any further information or have a specific question just fill out this enquiry form.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaaqih b/data_all_eng_slimpj/shuffled/split2/finalzzzaaqih
new file mode 100644
index 0000000000000000000000000000000000000000..c330abc4f1ca6995c252ff827f38d4bcd2400052
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaaqih
@@ -0,0 +1,5 @@
+{"text":"One of the oldest dog breeds, the Chow Chowwas bred in China. Its distinguishing feature is the thick fur and tongue blue. The dog gets along well in a city apartment and a country house, it is very clean and almost never barks. The only drawback of this breed is allergic to certain foods, so a proper diet to Chow \u2013 the guarantee of health.\nFrom the early days of Chow in your home to equip her for a place to eat and pick up in accordance with the growth of the stand under the bowl of food and water. This is the condition for the formation of correct posture and care \u2013 in a bowl on a stand is hard to get paws.\nIf you use vitamin dry food for puppies or adult dogs need vitamin-mineral additional forage will not. Given the fact that the rocks are not strongly developed teeth, the dry food will need to soak. Standard diet \u2013 60-70 grams of dry feed per 1 kilogram of dog's weight, but then focus on her appearance, which will be the main indicator.\nIf you change the feed or if the dog takes any medication, make sure to see if she has an allergic reaction, if there were, then treat her \"Tavegilum\" rate of 0.01 milligrams per 1 kilogram of dog's weight. The interval of taking the drug \u2013 12 hours, but just in case, consult your veterinarian.\nRich wool Chow Chow in need of care. To keep it in beautiful, healthy condition and to improve the exchange of substances useful to pursue a course of methionine - 1 tablet 2 times a day - every spring and autumn.\nIn any way do not give the dog salt, sweet, spicy, bitter and smoked. Forbidden pork, any fatty meat, chicken and game. Meat and vegetables \u2013 only cooked. Be careful with the carrots and cabbage \u2013 there is a lot of nitrates and the dog can get poisoned. Exclude from the diet flour and also bread, sausage, eggs, and seafood. Water should always be boiled and fresh.\nMeat - veal, beef can give a couple times a week, periodically replacing it with the boiled liver or fish without bones. Cheese can be given nekoboy and no more than 1-2 times a month for a small piece. When training when it is necessary to encourage the dog, you can give her unsalted fresh cookies \"croquet\". Once a week add her food with 1 tablespoon of bone meal.\nMineral supplements and vitamins, let's Chow only after prior consultation with a specialist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For their seventh Naxos CD, Seattle-based Music of Remembrance pair world premiere recordings of two recent Holocaust remembrance commissions, joining previous recordings of commissions from composers including Jake Heggie, Paul Schoenfield and Thomas Pasatieri.\nTom Cipullo's After Life is a discursive, one-act, three-person chamber opera in which the ghosts of Picasso (Catherine Cook), Gertrude Stein (Robert Orth) \u00admore Ma and Pa Kettle than The Ghost and Mrs Muir -and a spiritually innocent young girl (Ava Pine) confront the moral and physical minefields ofthe Second World War. Cipullo weaves David Mason's ingenious, rhapsodic libretto into an absorbing, often lively, occasionally desultory narrative told in words and music that is translucently scored and makes full use of the five-member ensemble's colours, astonishingly vocal at times in their effect.\nLori Laitman's 18-minute In Sleep the World is Yours is a simpler, more profoundly distilled response to the Holocaust in the form of a mother-and-child song-cycle set to elegiac lyrics by Selma Meerbaum-Eisinger, dead at 18 at a Nazi camp in Ukraine. In a sense, Laitman's three songs continue the conversation about life and art begun in After Life, deeply concerned with the ineffectualness of art during extreme human crises unless the creative force is pure. Stein and Picasso compromised and wound up being unable to distinguish art from life, which robbed their work of its potency.\n\u00a9 Copyright 2017 Music of Remembrance. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At the last Mass of the Catholic Social Ministry Gathering, Cardinal Daniel N. DiNardo told hundreds of participants who were preparing to talk later to their representatives in Congress that \"Jesus walks with us\" when doing the social ministry of the church.\nWASHINGTON, D.C. \u2014 At the last Mass of the Catholic Social Ministry Gathering, Cardinal Daniel N. DiNardo told hundreds of participants who were preparing to talk later to their representatives in Congress that \"Jesus walks with us\" when doing the social ministry of the church.\n\"Jesus walks with us, we'll walk with him,\" said DiNardo, archbishop of Galveston-Houston, in his homily at the Feb. 5 Mass in Washington.\nDiNardo, who is president of the U.S. Conference of Catholic Bishops, also tied in St. Agatha, whose feast day is celebrated Feb. 5, at different points in his homily.\nHe noted St. Agatha also is the patron saint \"of those who suffer sexual violence and abuse as she suffered in Sicily.\" St. Agatha, who had vowed as a teenager to remain a virgin, was raped, tortured and ultimately put to death for refusing to submit to the marriage demand of a Roman prefect. \"That's something to keep in mind in the church,\" DiNardo added.\nThe cardinal linked the church's social ministry with the day's Gospel reading from Mark, in which Jesus brings the 12-year-old daughter of Jairus back from the dead, but not before he heals a woman who has been suffering from hemorrhages for a dozen years.\n\"People are distraught and come to us, as they come to the Lord,\" DiNardo said.\nThe woman, unnamed in the Gospel account, \"(is) desperate. She's willing to be anonymous to let Jesus heal her,\" he added.\nOnce Jesus gets to Jairus' house and expels those weeping and wailing over his daughter's death, he tells the girl to get up, and she does so immediately and starts walking around the room. \"Jesus does come to you, and he'll move you and you will stand up and you will move around,\" DiNardo said in his homily.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm usually too impatient to make test tiles. It seems like a waste of time to me. It's not like you can do anything with them, and they take up space in my workshop. So, I usually test a glaze on something small, like ornaments. If it doesn't come out good, no big deal, if it does, then I have something to sell.\nThese leaves were made from a stoneware clay called #50.\nThe color is a little darker.\nThen I like to try something with some texture, like these birds.\nThis glaze works well with texture, it's not completely transparent, but at least it doesn't cover up the texture like some glazes.\nIt definitely works well on this texture!\nThen I tried it out on some small bowls. These came out cute!\nI love this glaze! It's so easy to work with, so nice to brush on, and it's so reliable and consistent. Everything always comes out great, and the color is beautiful too. What more could you ask for?\nThis bowl looks so yummy with these mouth watering strawberries!\nI applied one coat of Blue Midnight and two coats of Indigo Float on top. It came out great! As a matter of fact, this bowl was on Etsy's Front Page!\nThis bowl was also glazed with one coat of Blue Midnight and two coats of Indigo Float. I did some photo editing on this photo, so the color is slightly off. It looks like the bowl above. After it sold, I turned it into a food photography photo for my Etsy Shop, but I totally forgot about it, until I did this blog post. I need to list this! Even if no one buys it, it makes a great photo for treasuries.\nI made these plain heart dishes not that long ago, they came out cute!\nI just realized that I haven't tried this glaze on any of my dinnerware. I think it's time I do that. Hmmm, now I have to decide on which one. Should I use it on some plain plates or plates with texture? Let me know what you think!\nThis entry was posted in Uncategorized and tagged birdie bowls, ceramics, clay, color, designer, Etsy, facebook, glaze testing, glazing, handmade, handmade bowls, handmade heart dishes, handmade leaves, handmade ornaments, home decor, Indigo Blue, Indigo Float, leaves, Melinda Marie Alexander, photography, photos, pinterest, potter, Potters Choice Glazes, pottery, Raven Hill Pottery, ravenhillpottery.etsy.com, texture, twitter, wordpress, workshop by ravenhillpottery. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With just a few days till voting ends, we are wondering, have you voted for Nigeria's rep Ugochi Ihezue at the on-going Miss World 2017?\nIf no, it's not late, find out how to vote here, and if you have then keep voting! In case you haven't noticed, we're rooting for her to bring crown the home.\nWas a fun-filled day at the Chimelong parade. So many colorful and beautiful costumes by my fellow contestants from all over the world.\nDue to crisis and disunity going on now in our country Nigeria I decided not to use any tribes traditional attire but instead drew my inspiration from our National Coat of Arms.\nThe last line of the first stanza of our National Anthem says: \"One nation bound in freedom, peace and unity\"; and the last line of the second stanza says: \"To build a nation where peace and justice shall reign\".\nI hope and pray that our nation be bound in peace and unity and we, the beautiful people of this great nation find justice and peace.\nShe also attended an IBC gala dinner for charity, after a 1-kilometre run.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Catholic teaching insists that the Jewish people are not cursed or responsible for Jesus' death. Yet Matthew 27:15-26 presents the Jewish crowd \"as a whole\" calling for crucifixion and saying that Jesus' blood might fall upon them!\nThe Christian preaching tradition has been molded upon a paradigm which promotes Jesus at the expense of Judaism. As a result, often by default, Judaism is routinely denigrated when Jesus is preached.\nIt is unfortunate that Christians often carry a negative understanding of the Pharisees. Historical research has indicated that Jesus was closer to the Pharisees than to any other religious group.\nOne of the most helpful ways to understand the negative presentation of the Pharisees in the gospels is to see the description as an adolescent reaction of the early church, striking out against its parent, Judaism.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Join Kathryn and author Jerilyn Ross MA, Director of the Ross Center for Anxiety and Related Disorders, CEO of the Anxiety Disorders Association of America, One Less Thing to Worry About Uncommon Wisdom for Coping with Common Anxieties. Recent appearances include the Today Show and NPR. Also joining Kathryn is author Amy Dickinson The Mighty Queens of Freeville A Mother, a Daughter, and the Town That Raised Them. Dickinson is a nationally syndicated Advice Columnist for the Chicago Tribune Ask Amy and NPR contributor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Central Cars are a family run company with over 30 years experience in the taxi and private hire industry. Their state of the art call-centre is based in Cumbernauld's Town Centre with our auxiliary office at South Carbrain Railway Station. All our drivers have the latest GPS tracking devices in their vehicles for a faster response.\nwww.centralcars-taxi.com Central Cars are a family run company with over 30 years experience in the taxi and private hire industry. Their state of the art call-centre is based in Cumbernauld's Town Centre with our auxiliary office at South Carbrain Railway Station. All our drivers have the latest GPS tracking devices in their vehicles for a faster response.\nwww.nlleisure.co.uk North Lanarkshire Leisure (NLL) is a registered Scottish charity. The purpose of NLL is to deliver sport, recreational, health improving and social welfare services to the community of North Lanarkshire. NLL vision is to provide exceptional customer service and value for money experience through the provision of a range of high quality sport, recreational and health improving activities.\nwww.cordantpeople.com Searching for Staff: We get to know our candidates as well as we get to know your business. Thats why you can expect introductions to well-matched professionals who cant wait to work for you - whether you're looking for temporary staff or your next big hire.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From January 2016 I will be offering an intermediate acceptance and commitment therapy (ACT) training over eight full days. There will be one training day a month, generally happening on the second\/third Saturday of the month.\nThe aim of the training is to move therapists and counsellors who know the basics of ACT towards being competent, confident ACT consistent practitioners.\nACT has been shown in studies to be at least as effective as CBT to help clients deal with issues such as anxiety, stress, depression, obsessive thoughts and many other issues. Over the long term ACT has been shown to be more effective than CBT in many studies.\nIn order to attend the course you will be required to understand the basic concepts of ACT. I do offer an introduction to ACT which would cover these concepts (more information can be found here: http:\/\/affinitycentre.co.uk\/acceptance-and-commitment-therapy-course\/). If you are unsure as to whether you might have enough knowledge for the course please contact me and we will make sure you are up to speed before joining.\nTraining in ACT is often hard to come by, especially outside of London. As a result my courses and trainings fill quickly. Please sign up as soon as you know you want to attend this training to guarantee a place.\nIan is a UKCP registered psychotherapist, supervisor and trainer. He is a certified Transactional Analyst and has been using ACT for many years in his private practice.\nIn addition to extensive training in ACT, Ian has weekly supervision from one of the leading experts in ACT and Functional Contextualism in the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Great cover to that Sinister House of Secret Love, but my real love was for ol' Howie -- HTD #4 was the first issue I saw and I was intrigued enough to check it out and in short order Howard the Duck became one of my favorite comics.\nSecrets of the Legion of Super-Heroes was a three-issue mini-series cover-dated January to March 1981. And if any team needed a mini just to do its roll call, it's the Legion!\nStill I always felt that it seemed kind of rushed and its big reveal made little sense, though it got more convoluted Post-Crisis!\nThe mini coincided with Legion of Super-Heroes #271-#273. I suppose it was a follow-up to The Untold Legend of the Batman, but was the parent title selling well at that point?\nThe Legion was always one of DC's best-sellers in the late 70s and 80s which was why they pushed Superboy out of what was once his own book, getting this mini and becoming a Baxter book like The New Titans.\nThe Levitz\/Giffen run was a smash success, but were sales strong between Grell and Giffen? My take on the title was its quality went down in the later 1970s, so I've never read most of the stories from 1980-81.\nThe Legion as a concept was a Bronze Age success story. From being a back-up in Action Comics, it was revitalized by Dave Cockrum in Superboy, sharing the title with #197, then having the name officially changed to Superboy And the Legion of Super-Heroes in #231 and then getting retitled Legion of Super-Heroes in #259.\nThey had a reprint tabloid, a brand-new tabloid (the wedding of Lightning Lad & Saturn Girl), saw Karate Kid get his own book, get several new members, team with the JLA and JSA in Justice League of America #247-248 (something the Teen Titans never got!) and have thirteen volumes in its Archive series plus the two SLSH hardcovers!\nSon of Satan #1 (D'75). A book that couldn't be made now by Marvel. Actually I'm shocked that it was made then!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzabxdx b/data_all_eng_slimpj/shuffled/split2/finalzzzabxdx
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+{"text":"At WastePages, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Stone Lake, WI, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nThere are many reasons you may need a Roll Off For Rent in Stone Lake, WI in Maryland. You could be donning a new top on your business or home Maryland. You'll need the Roll Off For Rent in Stone Lake, WI throughout Maryland for that construction staff coming to carry out the new top or you may require it available for you and your family to discover the roofing task completed.\nYou will need a Roll Off For Rent in Stone Lake, WI when you are renovating your first residence. Whether it be renovating your kitchen to help update that or to increase the size of an existing rest room, a dumpster is sure to be desired.\nRoll down dumpsters are calculated by the number of cubic yards worth of waste they are able to store internally, normally for sale in 10,20,25 and 40 yard versions. These kind of dumpsters will be restricted not merely by the amount of bodily debris they are able to hold, although by the pounds involved also. With specifically heavy materials like concrete floor, containers should only be filled around halfway, otherwise hauling them gets to be problematic.\nWhen it comes to renovating your house, a lot of particulars must be cared for and the something that a lot of people forget about is exactly what they are going to do with the trash forgotten from the restoration. One home renovation could produce as much as Twenty yards involving trash that could equal up to 2 tons. Therefore, what exactly if the homeowner do with all of the squander left over? Employing a professional rental company will be of great help to you throughout completing your career quickly inside easiest way. They'd help you in dropping all the rubbish in to the dumpster and reuse them utilizing environmental friendly waste administration strategies. All that you have to do is definitely create a excellent space to put a Dumpster so that the piled up waste can readily dumped in to the dumpster. One more occasion a house maintain work is when doing large jobs on yard like scaling down trees, removing shrubs or perhaps tearing lower an old wall and so on. Employing a Dumpster will be practical in dropping all the squander that is made without any hassle.\nYou do not have for you to go for huge as well as process to lower the environmental pollution. You may pride yourself for being to blame for a respectable cause simply by hiring a dumpster at a professional Roll Off For Rent in Stone Lake, WI organization. This Roll Off For Rent in Stone Lake, WI companies which are present in the market will be of great help to you throughout sharing your responsibility in ecological protection. They then hire dumpsters according to tonnage of the squander disposed at the time specified by you. This squander will be reused in an ecological friendly method there by reduction of the effect involving pollution within the environment.\nWhen you have a large task you are trying to try and do in McLean, Vienna, or perhaps Tyson's Corner throughout Arlington County, The state of virginia you will need to rent any dumpster.\nEvery development project consists of tons of garbage and boulders. You can enjoy obtain and cleanliness during an ecological construction task by renting any dumpster. This is the fastest and fastest way to remove garbage and debris. Excellence in addition to beautification at your business sight is achievable; it does not ought to look trashy, filthy and unkempt. Excellence in just about every construction method is achievable from the usage of the appropriately sort of and swiftly removed dumpster in your location.\nUseful provided on different costs by diverse companies. Many charge a flat fee for assistance per day. Other people may offer your bins for any week or perhaps less could possibly be. Some businesses may offer more cost-effective rates but you will surely have to find and analysis a lot to the. Just beware of hidden costs that may be right now there under deal with of statements. Read the conditions and terms of the company before taking the skills.\nRoll off dumpsters are tested by the amount of cubic yards in waste they will store in the camera, normally available in 10,20,40 and 40 yard types. These kind of dumpsters will probably be restricted not only by the amount of actual physical debris they will hold, although by the excess weight involved likewise. With especially heavy components like concrete, containers should only be packed around halfway, otherwise hauling them gets problematic.\nConsequently no matter what kind of trash removal or property or home cleanup undertaking you're dealing with, a roll-off Roll Off For Rent in Stone Lake is for sure to help make the course of action easier and more efficient. Is actually affordable costs available, you can be amazed at how cost-effective can be. Get hold of your local dumpster company today.\nYou can find rules in addition to legal rules that bind all dumpster rentals, some of which are placed by the regional authorities. You have to ensure that you abide by all of these rules to avoid long term complications. People that seek out Roll Off For Rent in Stone Lake products and services need and also to be aware of these kind of laws not merely for the sake of what the law states but to positively lead on the health and fitness of the community and the atmosphere.\nDumpster rental companies are right now there to assist and provide you with all the aid and features you need to be certain that renovations, improvements, and home development projects turn out to be faster, simpler, and less hazardous.\nGood, that you have chosen to rent a dumpster to get rid of the plenty of trash piled-up at your job location. You have to know the type in addition to size of dumpster that best fits your requirements before you head ahead together with the rental process. Roll off dumpster ought to be your choice if you are searching out to dump huge plenty of trash from the major commercial and residential projects. Prior to placing an order for the roll off trash bin, it is very important that you know the measurements of the rubbish container so that you accommodate the desired place to placed the dumpster at your undertaking location. Whether you are running a home rehabilitation, commercial undertaking or industrial clean up, the place you accommodate for the rubbish container ought to be double the sized its elevation and width.\nThe home development process includes a large amount of components and staff who are usually working on a busy schedule. It is no surprise and then that at the conclusion of most projects the site is rather messy in addition to full of trash with added materials in addition to dirt wanting to be found. Home construction Roll Off For Rent in Stone Lake are capable of doing the job to suit your needs. Before the constructing can be considered full the site has to be cleared of construction components and the household and property or home must be cleansed from top to bottom of construction mess. This work is normally performed by a new construction cleanup crew. Significant companies could possibly have one of these squads on staff, or perhaps they may subcontract the job out to specialty cleaning businesses, but development Roll Off For Rent in Stone Lake can make almost any cleanup job easy.\nDumpster accommodations will benefit a person in the over stated techniques only when you get associated with the excellent vendor who will not give you fault assure and run the just to bring your money. The service provider will be really prompt inside their services, rapidly pickup as well as prompt delivery. The ideal Roll Off For Rent in Stone Lake, WI service agency will be transparent in it can be pricing to give the customer previous information about the fee that has to always be bared before the purchase is placed.\nThe toilet, sink, vainness, flooring, base board, etc. can go into the dumpster and acquire easily delivered away. No more mess in your case and your family to examine when you rent any dumpster for your bathing room.\nWith springtime arriving, many of our minds convert to cleaning out each of our homes as well as selling the items we don't need in your lives. Some sort of yard sale is a good thing alone, but when you obtain the whole area in onto it, your forces can combine to make a truly fun event for your family members, your neighbors, as well as anyone who stops by to purchase your goods. This is how you can get your location together for just a community garage sale.\nImagine the disaster that could occur if you did not know how much (with the items kind of) spend and debris you could invest your rental spend and debris dumpster. You could finish up throwing away the incorrect thing inside them for hours to dig it out and about before the pickup truck will transport it absent. Imagine simply how much fun That will be! You could finish up throwing away an excessive amount and having to filter it out, producing trip after trip to the actual dump along with having to pay out your Roll Off For Rent in Stone Lake, WI charges.\nWhen it comes to any garage that will require clearing out, likelihood is good that you have a pretty large amount of junk to be discarded. Since many local garbage companies inflict limits about how much garbage you can put out and about a week, and a lot of things may just be too large as well as awkwardly molded for garbage bags at any rate, you're going to want to rent a dumpster at a reputable Roll Off For Rent in Stone Lake, WI organization. Just thinking about a dumpster-sized weight of junk gone from a garage ought to make the task at hand seem to be much more manageable.\nDumpsters are also available on wheels that may be maneuvered effortlessly on your home. Excellent as well as convenient as well as quick service will have your current construction garbage or residence cleanup challenge quickly looked after. Dumpster rentals provide easy garbage removal as they do not require how the items getting thrown away the actual sorted or even broken down in a smaller size. All garbage including window and materials can be thrown into one particular dumpster. Recyclables and common trash be thrown and with each other. Renting any dumpster is the fastest way to clean up after a home challenge.\nMany roll off businesses promote the notion that bigger is much better, meaning that you'll be up-sold to a jar size in which well surpasses your true requirement. Much better certainly greater to err assisting having a somewhat more container than is necessary, one particular double the essential capacity is definitely throwing away money.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether you're looking for some traditional homemade Italian bread, or just some freshly baked Whole Wheat or Rye, come in and choose from a large variety of rich, delicious breads.\nIrish Soda Bread - Year-round!\nFreshly-baked Irish soda bread is available every day year-round! Our dense, moist, hearty soda bread is packed with raisins and ready to be taken home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are glad to welcome you here, on the pages of our website and we hope you find all the information you need right here.\nWe are focused on giving you information on the interior lighting and everything that is somehow connected to the subject. Lighting is something we are familiar with and something we all use, yet don't know much about. This is way delivering the information and sharing different ideas and tips on the way the things work is important for us.\nWe try to give you some ideas about how you can make your interior cozier and turn your apartment into something you've always dreamed of. All of this is possible with the proper lighting and it's actually the thing that can turn your place into something good looking and nice or spoil the whole picture. If you're trying to find the good solution for your place, you can always find something on a website and don't be scared to ask some questions. We are always there to assist you. You can try to design your flat with us and find plenty of tech information right here if you don't know what is suitable for the place you live in.\nIf you are trying to fix your bills or looking for the way to make the most out of your energy usage, you will also find some tips here. You can always share some information if you have something interesting on your mind and suggest a post. Please feel free to give us your opinion or advice anytime too! We're trying to be useful for the people who are dealing with lighting for the first time in their lives as well as for pro's who are aware of many details and differences.\nWe hope our lighting magazine will help you to get what you want and be fully satisfied with your place.\nHave a great day and enjoy your reading!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yesterday was World Mental Health Day, a day to promote mental health education and awareness globally. If you follow my Instagram Stories, you may have caught my morning ritual of ringing a bell along with a daily thought on being present and living in the moment. It's part of a meditation I practice daily. I started this a few years ago because the stress of running a business, staying creative, and raising a family was starting to take it's toll on me. I was feeling anxious and having a hard time letting go of stress. A friend suggested that I carve out some time to sit silently and let my thoughts just be. A few years into the practice and I feel more mentally sound than ever before. There will always be stress in life, but it's the way we react to stress that will make or break you. Meditation allows me to be the gatekeeper to my own mental wellbeing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So let us tell you a little about our local rich history so you too can see why we want to be your Springfield Home Remodeling Contractor.\nAs a child Jen spent many of her summers in Springfield with her family visiting her grandparents. The days were filled with trips up the McKenzie highway, hiking and exploring, picking fruit in many of the local orchards, swimming in the Willamette River, and running and playing in the large fields behind her grandparents house before Springfield grew and developed that field into a new neighborhood. Her childhood memories of trips to Springfield will always hold a special place in her heart. When first moving to Lane County Jen lived in Springfield for a few years before moving to Eugene after meeting Forrest. Jen still has an aunt that lives in Springfield and sees her regularly since it is only a short drive from her home into Springfield.\nAlthough Forrest has always been a resident of Eugene are many things he enjoys about Springfield. Both of his kids were born in Springfield's McKenzie Willamette Hospital. One of Forrest's favorite places to take his kids is Mount Pisgah Arboretum, just a few minutes out of Springfield. He and his family can spend the afternoon hiking on the endless trails, wandering through the natural garden and playing in the McKenzie River which slowly meanders by.\nHere we show off a recent Springfield Bathroom Remodeling job we did, complete with a Before picture, sketch, after pictures and a testimonial from your local neighbor.\nCastile Constructions office is located in Eugene. From our office in Eugene we can be in the heart of downtown Springfield in just 10 minutes. Servicing the Springfield zip codes 97475, 97477, and 97478 Castile Construction has worked on many Springfield homes, beautifying Springfield one home at a time. Castile Construction purchases materials regularly through many Springfield suppliers. We enjoy the personal and business relationship that we have created with so many residents of Springfield. Learn more about the city of Springfield here.\nWe hope to beautify Springfield, one home one at a time.\nContact Castile Construction today to discuss us becoming your Springfield Remodeling Contractor. We welcome the opportunity to meet and work with our neighbors.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Chief Inspector of Factories, Assam invited applications for the post of Assistant Chemist and Driver. The eligible candidates can apply to the post through the prescribed format on or before 22 January 2018.\nAssistant Chemist: B.Sc. with major in Chemistry and Physics as one of the subject.\nEligible candidates can submit the applications along with necessary documents to The Chief Inspector of Factories, Assam, NPS International School Lane, Betkuchi, Lokhra Guwahati- 781040 on or before 22 January 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You make me sit up and listen more now after reading this song dance of your poetic inspiration. So much I like about this from the opening lines to the italicised song lyrics (they seemed that way to me!). This is exactly how I imagine a drone should be like and your described it eloquently with your special brand of humour I so enjoy. We all need another poet to bring us back to reality\u2026lol! Well done and I can't stop coming back to read this over again. Thank you participating in the prompt.\nThanks Gina, a great prompt, poem would not exist without it!\nI like the thought of finding that other poet up there in the clouds.\nthat germ of a poem.\nI really like you are lost with lines, your head in the clouds. See you there!\nHa! Cool. It most certainly is.\nGreat description of your creative process!\nBTW do you have a story behind the name of your blog, Jim?\nBelieve it or not, I woke from a dream and the phrase was in my head!\nBravo! Bravo! You have captured the poet's mind exactly.\nThat's just how it is! Those poetic seeds floating on the air and taking root in the imagination. I seem to pick them up everywhere, like dandelion fluff \u2013 what I believed were fairies when I was a child. Perhaps they are. I love the poet in the clouds in the final lines, Jim! The last line is often a bugger!\nThanks Kim! I like the dandelion fluff analogy!\nThat recognition by another poet, whether in one's own words or the search for words, is a reward.\nIndeed it is, there are not a lot of us around!\nThat's a good analogy, Bjorn!\nI believe you captured exactly what we all do.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Is So Fascinating About Restaurant Furniture?\nMost of its dining furniture is made from good materials like leather. Only the industrial furniture will bring the comfortable level. Outdoor furniture is a superb addition to just about any sort of restaurant. The sort of outdoor furniture you buy must reflect the general appearance of your garden. Resin patio furniture is extremely sturdy and lasts a very long time with no true need of treatment.\nBesides eating, individuals go to restaurant to delight in their leisure times too. If it comes to restaurants, you are going to want your guests to stay awhile. Restaurants are public spaces where folks meet up with their social spheres so they may have a fine time. Actually, a restaurant is a place that provides you a wide-open space and ample time to appreciate your time. You can wow\" someone if you go to a restaurant with a memorable atmosphere.\nStepping in the restaurant doesn't necessarily signify that you ought to be hungry. For instance, if your restaurant is incommodious, stacker chairs would be the best choice. Restaurants have the exceptional problem of frequent spillages in order that they should also be simple to wash or replace. Enquire if you're in a position to add your restaurant inside that list. Thus, to grab the eye of an enormous crowd of different age, you need to launch a theme-based restaurant or cafe that will help drag the guests.\nBar stools aren't just mere chairs. They come in a variety of styles, shapes, and designs. The restaurant bar stools are important furniture that could set the proper mood for those men and women in the bar and can also give a fashionable ambience to the place. Restaurant bar stools and the whole appearance of your restaurant or bar ought to be pleasant to the eye.\nYour furniture will draw the eye of all customers and is a huge facet of the total atmosphere. To provide your hospitality company or restaurant an innovative yet luxury touch, then you may elect for the pleasing furniture. As a result, if you're planning to get or rent such sort of furniture be sure you are having the perfect one and durable one for your enterprise. The kind of restaurant furniture will decide on the general feel of the atmosphere within the facility and can draw in a specific sort of client. There are a few furniture that are somewhat blemished, and can't be sold at fixed rates. Repainting furniture is likewise not so difficult and wouldn't need you to be an expert to be in a position to perform an excellent job. If maintained, your wrought iron furniture should last quite a long time, but you ought to be mindful that it can rust.\nEven in the event the chairs don't arrive with the throw pillows, it is also possible to purchase or create your own cushions and pillows. A number of the restaurant chairs are also fantastic for the indoors but if you are interested in having an easy-going strategy, you must adore the cafe chairs. For the custom of consumers, you may use the Restaurant Metal Chairs and wooden chairs that are durable in addition to stylish too.\nRestaurant furniture is found in numerous choices. Particularly, the restaurant furniture is meant for comfort and basic expression of the restaurant. For the right ambience in your restaurant, you will require the most appropriate furniture. Our commercial outdoor furniture is created specially for outside usage, even near the ocean. In general, the decor reminded me of the jewelry readily available in the shop. Besides looking good, the furniture needs to have the capability to provide you with snugness. You'll receive bar furniture and dining furniture easily available in a number of colours and kinds.\nThe chair comes in a number of colours, which usually means you're bound to find one which complements your outdoor design. The wooden patio table is created from cedar, and may also act as your indoor dining room table too. The Rustic Natural Cedar 13B Round Umbrella table is a bit smaller in appearance, but it's going for a more affordable price too.\nThe area includes Monkey Mia, a favorite resort with tourists as there's the chance to interact with a group of dolphins that have been coming there for over 40 decades. The affected area has to be moisturized first. It really isn't the exact same to decorate an area which usually is usually cooler than another where the sun squeezes steadily. Each region of the restaurant must work with individual managers and supervisors.\nThe sign doesn't need to be intricate, but nevertheless, it should ensure it is extremely apparent that it is a restaurant. Sydney's restaurants are not just numerous and of the best quality, but they're remarkably inexpensive in comparison with different cities of the planet. On weekends, your restaurant will probably be busy. Make sure that when you set a restaurant or any business, you will have time for it. Personalized restaurant is a superb choice with respect to other kinds of alternatives. The changing restaurant provides a huge scope of alternatives for a distinctive culinary experience. Search for a place were you would like to establish your fast food restaurant.\nEmu oil is demonstrated to be anti-inflammatory. Well, it is probably not going to be found at your local drug orhealthstores. It is not a guarantee treatment forcuring baldness. Because of its broad-spectrum use, it has become a household good.\nLavender and rosemary have precisely the same effect. In reality, the cream was famed for healing a plethora of conditions and restoring health. The above mentioned ingredients have stood out for their healing components and they're merged in 1 cream to supply people with the most suitable response to their wellbeing and cosmetic troubles. Other ingredients to see in the product are jojoba oil along with aloe Vera butter. Recently, due to the new design collections, the organization's products are chosen also by the best high quality furniture shops. Emu oil products are available in lots of the huge department store now, and mail order catalogs and a heavy presence online, Long relays. As a designer you work on the best way to find the very best out what that brand offers regarding expertise and fashion.\nWhether you're looking simply to purchase restaurant furniture or you have to design and construct your facility, Atlantic Equipment is prepared to help. When it regards restaurant furniture, quality is of extreme importance. Restaurant furniture must be suited to the space of the restaurant, be comfortable for the diners and capture the nature of the fashion of food you're going to be serving. It does not only serve its function but also contribute to the ambience. The furniture you select for your restaurant can make or break your company. The restaurant furniture is intended to seat more number of people at precisely the same moment. You need to be just always careful before buying the restaurant furniture.\nAside from eating, individuals go to restaurant to delight in their leisure times too. Restaurants are good businesses which could be very fulfilling. Design your restaurant to procure an ambiance that's inviting and attractive when picking out furniture that's solid and complimentary. If it is a larger one and you would like to seat maximum number of people, then you should look out for oval and round tables as they occupy less space compared to ones with corners. Most restaurants set candles at every table, simply don't utilize scented. If you're opening a new restaurant or redesigning a current establishment, the furniture you select is of extreme importance.\nThe way that your restaurant is organized is the very first component to be set by means of a customer, particularly for first comers. Actually, a restaurant is a place that provides you a wide-open space and ample time to appreciate your time. Your restaurant ought to have a sofa that's comfortable and so far as interior decoration of your house is concerned, you can use dining table with matching chairs. Last, bear in mind that the individuals who will be dining at your restaurant may be there for a short time. On the flip side, a cafe styled restaurant might need to have glass tabletops and metallic chairs.\nFurniture is going to be subjected to everyday wear and tear. Restaurant furniture requires much greater construction than furniture constructed for home usage. However good a price that you find for restaurant furniture, you'll wind up losing money in the future if it must be replaced every one or a couple of years. Furniture for restaurant use also has to be attractive, comfortable and simple to keep. It's possible to shop for quality and inexpensive furniture for office or home use from an internet shop.\nAll our furniture is specially made for institutional applications. When it is mahogany veneer, the furniture would seem to be somewhat lightweight. Furniture should blend nicely with the subject of the cafe along with with the design and color of the interior. Restaurant furniture has an important part in the food enterprise. Restaurant seating furniture must be rough. Your restaurant seating furniture ought to be a fantastic mix between them both. Restaurant commercial furniture should satisfy the identical requirement.\nOn weekends, your restaurant will probably be busy. Make sure that after you set a restaurant or any business, you will have time for it. The sign doesn't need to be intricate, but nevertheless, it should ensure it is extremely obvious that it is a restaurant. Personalized restaurant is a wonderful choice with respect to other kinds of alternatives. The changing restaurant supplies a massive scope of choices for a distinctive culinary experience. Start looking for a place were you would really like to prepare your fast food restaurant.\nFurniture ought to be kept dry because moisture is the principal source of damage to stored furniture. Restaurant furniture are found in numerous choices. Particularly, the restaurant furniture is meant for comfort and basic appearance of the restaurant. Besides looking good, the furniture ought to have the capacity to provide you with snugness. Our commercial outdoor furniture is created specially for outside usage, even near the ocean. For the right ambience in your restaurant, you will require the most proper furniture. You'll receive bar furniture and dining furniture easily available in many colours and kinds.\nIn 2007, because of the creative inspiration of great foreign designers, the business made a new identity for metal, putting the most extensive know-how on earth into the manufacturing procedure. It was engaged in developing a water resistant coating for military radios as well as their components during the war. As a designer you work on how best to find the very best out what that brand offers with respect to expertise and fashion. Recently, because of the new design collections, the firm's products are chosen also by the best high quality furniture shops. Quality Naturally, the standard of your lubricant of choice needs to be as large as possible, too.\nYou wish to purchase the most potent and efficient oil. Emu oil has turned out to be an effective all-natural acne therapy. It is one of the best topical treatments for hair loss. Low high quality lubricants usually want a lot of application. Some of the most frequent lubricants in the present market are water-based lubricants and oil-based lubricants.\nTogether with platypus, spiny anteaters are the sole egg laying mammals. Poultry animals are almost always high in demand due to their tasty meat products that are also famous for their various health benefits. Different farm animals serve various purposes. Similarly all of the farm animals have many unique forms of breed. Also, nearly any albino animal of any sort is a poor omen. Generally, a crocodile must be at least six foot long to be a danger to humans.\nThe best approaches to genuinely understand the Outback are by hiking and exploring the region. Saw palmetto includes beta-sitosterol, which too much can cause heart issues. You don't wish to purchase saw palmetto supplements from a neighborhood drug brand. The very best saw palmetto supplements The very best saw palmetto supplements ought to be saw palmetto extract. Emu oil sunburn relief is fast and comfortable. Batyline by Ferrari, a collection of rather attractive and higher performance technical fabrics for products and cushions, completes the most recent novelties made by Emu. So, now you know more about the farm animals list and a few of their hybrids.\nRestaurant Furniture \u2013 Is it a Scam?\nYou ought not choose furniture simply because are available at reasonable prices, but choose lavish comfort and trendy variants of restaurant furniture you can buy. So, you need to search for the furniture that is created of sustainable products. Traditional industrial furniture is made of materials that are made to withstand prolonged business use.\nYou need to be able to wash the furniture easily. In restaurant hospitality projects, restaurant furniture is the key product which may make a new venue stick out. It makes up most of the ambience of your restaurant, whether it is around an exciting theme or just a place where people can relax and have a conversation. Possessing the appropriate furniture for your restaurant is important.\nThe furniture ought to be properly matched to the theme. Patio furniture, also called outdoor furniture, is a sort of furniture specifically made for outdoor usage. Thus, the furniture must be somewhat comfortable to seat hours. So, it should be able to create intimate set-up. It is not sufficient to acquire attractive furniture. In various manufacturers and suppliers in Sydney, you will receive the eco friendly hospitality furniture that are created of reminding the simple fact of global warming.\nWho Else Wants to Learn About Restaurant Furniture?\nTo get rid of the prospect of food poisoning, it's imperative you choose furniture than can be cleaned without a lot of hassle. Keep in mind your collection of furniture for your restaurant can help determine the customer traffic indirectly. Moreover, our restaurant furniture is simple to clean and maintain. It is available in different options. It takes a beating. It is a significant investment. There's no denying that commercial restaurant furniture can include a hefty price tag and is among the larger expenses a business must carry.\nIf it comes to restaurants, you'll want your guests to stay awhile. You can wow\" someone if you pay a visit to a restaurant which has a memorable atmosphere. You ought to be aware that a cafe is also a sort of a restaurant but the major difference between a cafe and a restaurant is their different ambience. Despite the fact that you enter a restaurant, the first thing you notice is the look of the restaurant or cafe. Personalized restaurant is an excellent choice with respect to other kinds of choices. Whether you're furnishing a brand-new restaurant or renovating an existent eatery you wish to locate the proper type of furniture and hopefully, our tips will allow you to do precisely that.\nWant to Know More About Restaurant Furniture?\nRestaurants are good businesses that may be very fulfilling. The restaurant is where folks come to devote decent moment. Restaurants have the special problem of frequent spillages in order that they should also be simple to wash or replace. For instance, if your restaurant is incommodious, stacker chairs would be the best alternative. A thriving restaurant will require a very good design and layout program. The changing restaurant offers a vast array of options for a distinctive culinary experience. For example, if you need to make it a family-oriented Italian restaurant then you ought to think of parents that are likely to dine with two kids at your place.\nYou shouldn't choose furniture simply because are available at reasonable prices, but choose lavish comfort and trendy variants of restaurant furniture you can buy. If you would like your contract furniture to age gracefully then elect for solid, long-lasting furniture constructed for expert use. Choosing the furniture might vary according to space you've got at your place. It is insufficient to get attractive furniture. While some men and women think as wooden furniture is normally seen in restaurants, it isn't a prudent selection, but still the advantages of the wooden furniture make it seem very attractive. Indian wooden furniture is readily available for a wide selection of uses. It can be prepared from varied types of wood and material, and only the right type of material will add the much needed ambience and style to the Indian restaurant.\nHardwood furniture is frequently the ideal investment since it can be repaired rather than replaced. For the best ambience in your restaurant, you will need the most suitable furniture. Only the industrial furniture will bring the comfortable level. Our commercial teak furniture is the ideal option for venues searching for a hassle-free, bespoke purchasing experience.\nYour furniture will draw the eye of all customers and is a huge facet of the general atmosphere. Outdoor furniture is a good addition to just about any sort of restaurant. It offers the same enhancement to the location. Our commercial outdoor furniture was created specially for outside usage, even close to the ocean.\nAll our furniture is especially made for institutional applications. Restaurant Furniture is a considerable investment. Wholesale restaurant furniture is readily available for all types of restaurants, be it a huge banquet hall or a little hotel or a tiny eatery. Comfortable furniture that's a foil for your restaurant theme is the ticket to create your restaurant appealing to your clients to a larger extent.\nWhether you're looking simply to purchase restaurant furniture or you should design and construct your facility, Atlantic Equipment is prepared to help. In restaurant hospitality projects, restaurant furniture is the main product which may make a new venue stick out. There's no denying that commercial restaurant furniture can have a hefty price tag and is among the larger expenses a business must carry.\nRestaurant furniture can be found in various options. It has to be suitable for the space of the restaurant, be comfortable for the diners and capture the essence of the style of food you will be serving. When it has to do with restaurant furniture, quality is of extreme importance. The furniture you select for your restaurant can make or break your enterprise. Finding sale restaurant furniture may be a huge help in your financial plan.\nStepping in the restaurant doesn't always signify that you ought to be hungry. For instance, if your restaurant is incommodious, stacker chairs would be the best alternative. For example, if you need to make it a family-oriented Italian restaurant then you ought to think of parents that are likely to dine with two kids at your place.\nYou can wow\" someone if you pay a visit to a restaurant which has a memorable atmosphere. You ought to know that a cafe is also a kind of a restaurant but the principal difference between a cafe and a restaurant is their different ambience. Actually, a restaurant is a place that provides you a wide-open space and ample time to appreciate your time. In case the restaurant or cafeteria doesn't supply the very best atmosphere, it may hamper the stream of the consumers.\nIf it comes to restaurants, you'll want your guests to stay awhile. Enquire if you're in a position to add your restaurant within that list. Whether you're furnishing a completely new restaurant or renovating an existent eatery you wish to obtain the appropriate sort of furniture and hopefully, our tips will allow you to do precisely that.\nAside from eating, folks go to restaurant to appreciate their leisure times also. In the end, bear in mind that the individuals who will be dining at your restaurant may be there for some time. Restaurants having a more formal dining theme ought to be going with high-backed chairs.\nThe way that your restaurant looks says a lot to customers about the sort of food and service they can count on from your organization. As soon as you've figured out your vibe, understand the way the restaurant will flow, established a dependable furniture supply, it's time for them to assist you in finding a stylish, quality product that's durable. Restaurants have the distinctive problem of frequent spillages so that they should also be simple to wash or replace. Personalized restaurant is a great choice with respect to other forms of choices. The changing restaurant offers a vast scope of options for a distinctive culinary experience. Thus, to grab the interest of a big crowd of different age, you've got to launch a theme-based restaurant or cafe that can help drag the guests.\nFabric couches usually require a bit more maintenance. A well-made leather couch is usually stronger than all but the maximum superior fabric sofa. The most affordable real leather couch is always likely to cost more than the least expensive fabric couch. The chair comes in an assortment of colours, which usually means you're bound to find one which complements your outdoor design. Obviously, some folks never clean any of their furniture, therefore it's a moot point in their opinion. So long as you maintain and clean your leather furniture it is going to be comfortable for many years to come. In general, the decor reminded me of the jewelry readily available in the shop.\nEmu can help you by locating an establishment that's your close spot. Not only will that EMU become a worldwide application that connect all of the restaurant in one platform. EMU will give the very best opportunity for those users how want to customize the given serial they want in various situation. Hence the remedy is EMU, now you're able to choose only that dishes that are offered for serve. EMU helps users to inspect the food they serve before you, in addition it aids the seller or cook to update their menu they can serve. EMU is the application at which you can customize your order so you can merely manage the attribute of servicewhen you demand. EMU helps users to confirm the food they serve before you, in addition it aids the seller or chef to update their menu they can serve.\nIf you're a restaurant and try to grow with no budget to accomplish this, contact us. Suppose you're speaking about your favourite restaurant. So it is going to be difficult that you choose or customize their menu in own language.\nThe lovely part is the recipes are rather forgiving. In the customize app you'll be able to deduct some items which you do not like in the ordered dish. All eggs aren't created equal. The entire egg is amazingly fantastic for you. Eggs have really got a poor reputation. Remember there are all sorts of eggs besides chicken.\nWhat's more, you'll have an item that could possibly be updated with new, needed features later on without needing to add them yourself. Whether an applicable product exists, you might be able to modify it to fulfill your requirements. Recently, as a result of the new design collections, the corporation's products are chosen also by the best superior furniture shops.\nThe organization was engaged in creating a water resistant coating for military radios in addition to their components during the war. Obviously, an individual can't, but every tech company could be asked to obey the rules that bind the conduct of real-world counterparts. In the beginning, the business hired male waiters, but nonetheless, it soon opened opportunities to women. The Dochealth Health Technologies Company has around two senes dynamic and over 20 nations everywhere around the world. Firstly, businesses are purchased for any variety of reasons, a number of them illogical. Actually, when manipulating the stock market it's better to have a separate save merely to make certain everything goes smoothly.\nIt is possible to cancel the order in actual tome basis. Purchasing a digital camera working with the vouchers for argos is another continuing offer. The price of making or obtaining a booth should really be placed into account too. The price of making $1 more from existing customers is merely 19% of the price of making $1 from new clients. As stated by the Health and Nutrition website, another advantage of emu oil is the fact that it accelerates hair development follicles.\nThe metallic furniture demands regular care much like the wood furnishings and likewise an extra job of tightening the screws. Patio furniture, also called outdoor furniture, is a sort of furniture specifically created for outdoor usage. Even though some men and women think as wooden furniture is usually seen in restaurants, it isn't a sensible alternative, but still the advantages of the wooden furniture make it appear very attractive. You will receive bar furniture and dining furniture readily available in several colours and sorts.\nRestaurant furniture can be found in various choices. Moreover, our restaurant furniture isn't difficult to clean and maintain. It makes up most of the ambience of your restaurant, whether it is around an exciting theme or just a place where people can relax and have a conversation. It takes a beating. Particularly, the restaurant furniture is intended for comfort and general look of the restaurant. The furniture you select for your restaurant can make or break your small business. Possessing the appropriate furniture for your restaurant is crucial.\nPlan enough space on each side and at the conclusion of your table so you can easily move around. Coffee tables ought to be the very same height as the surrounding seating, with 18 inches being an excellent average (although it is going to ride on your furniture). The table is created from solid aluminium, and has a great soft rounded edge. At any time you find dining tables for sale, and should you compare between beech and rubber wood, you might also search for grains. Then smaller tables are imperative. They are available in rectangular and round shapes in a wide variety of finishes. While patio tables and chairs work nicely, they take up a lot of room.\nIf you've got the space, allow for a location that could be devoted to the ironing board so you don't need to set it up each time you should press open a seam. If you're fortunate enough to have such a space in your house or if you're planning to create one, here are a few tips for making the room more than only a place to store your sewing machine and notions. The floor space readily available in your restaurant will also will need to get taken into account when selecting your tables.\nYou obviously need a wonderful article of furniture that may be used to establish a computer station. After defining the idea, the next thing to do is to choose the right sort of furniture piece to complement it. Suitable parts of furniture While deciding on the restaurant furniture, make certain you follow precaution when doing this, since the decision can either augment or trim down its look.\nWhat to Expect From Restaurant Furniture?\nYou got into the bar business to produce money and re-styling your bar or nightclub will make it possible for you to raise prices, boost patronage and differentiate yourself from the competition, all which puts more cash in your pocket! Everybody who owns a business must get an EIN. When you open a restaurant business, there are a number of permits and licenses you'll have to have. For quite a few, the restaurant company is a lifestyle. If you're planning on beginning a restaurant company or want to expand, San Antonio is a wonderful selection.\nPersonalized restaurant is the ideal choice with respect to other varieties of choices. For instance, if your restaurant is incommodious, stacker chairs would be the best selection. Restaurants having a more formal dining theme ought to be going with high-backed chairs.\nRestaurant furniture is found in several choices. Particularly, the restaurant furniture is meant for comfort and standard expression of the restaurant. Besides looking good, the furniture needs to have the capacity to provide you with snugness. For the right ambience in your restaurant, you will require the most appropriate furniture. Our commercial outdoor furniture is created specially for outside usage, even near the ocean. Typically you can just donate used office furniture that's in good shape. Alternatively, you can donate each of the used office furniture, provided that it's in good shape, to a neighborhood charity in your region.\nThe chair comes in an assortment of colours, which usually means you're bound to find one which complements your outdoor design. The wooden patio table is created from cedar, and may also act as your indoor dining room table also. The Rustic Natural Cedar 13B Round Umbrella table is a bit smaller in appearance, but it's going for a less expensive price too.\nThe animals are an optional portion of our diet at the present time, but they might be needed later on. Poultry animals are almost always high in demand due to their tasty meat products that are also famous for their various health benefits. Not merely in farm animals but you can acquire hybrid animals in almost all species. Similarly all of the farm animals have many diverse forms of breed. Different farm animals serve various purposes. The brown snake however is among the world's deadliest and in addition to that is aggressive. Generally, a crocodile needs to be at least six foot long to be a danger to humans.\nThe fantastic thing about the furniture shops in Brooklyn is that he price reach of the furniture is quite low and if you believe that quality of the furniture is low as well then you're wrong since they never ever make any compromise on the quality. When you locate a product your skin likes, stick with this! The merchandise are flexible. Recently, because of the new design collections, the organization's products are chosen also by the best excellent furniture shops. In 2007, thanks to the creative inspiration of great international designers, it created a new identity for metal, putting the most extensive know-how in the world into the production process. It was engaged in developing a water resistant coating for military radios as well as their components during the war.\nOn weekends, your restaurant will almost certainly be busy. Make sure that as soon as you set a restaurant or any business, you'll have time for it. The sign doesn't need to be intricate, but nevertheless, it should ensure it is extremely apparent that it is a restaurant. Personalized restaurant is a wonderful choice with respect to other kinds of options. The changing restaurant supplies a huge scope of alternatives for a distinctive culinary experience. Search for a place were you would like to establish your fast food restaurant.\nAt Ingham you'll want to head a small west to find the spectacular Wallaman Falls but first stop at Tyto Wetlands for a little bit of wildlife spotting. It really isn't the exact same to decorate an area which usually is usually cooler than another where the sun squeezes steadily. If ever you or somebody you know receives a small burn, the very first thing you ought to do is immediately hold the burned area under cool water for a minimum of five minutes. Each region of the restaurant must work with individual managers and supervisors.\nYou will use the furniture for a lengthy time period and it has to be supported by it. The restaurant furniture was made to seat more number of people at precisely the same time. Particularly, it is meant for comfort and overall appearance of the restaurant. The furniture you select for your restaurant can make or break your company. Wholesale restaurant furniture is readily available for all types of restaurants, be it a massive banquet hall or a little hotel or a tiny eatery.\nThe furniture ought to be properly matched to the theme. Therefore, it has to be very comfortable to seat hours. So, it should be able to create intimate set-up. In various manufacturers and suppliers in Sydney, you will receive the eco friendly hospitality furniture that are created of reminding the truth of global warming.\nYou ought to be able to wash the furniture easily. Restaurant furniture can be found in various alternatives. Moreover, our restaurant furniture isn't difficult to clean and maintain. It takes a beating. It makes up most of the ambience of your restaurant, whether it is around an exciting theme or just a place where people can relax and have a conversation. Possessing the appropriate furniture for your restaurant is vital.\nTo get rid of the prospect of food poisoning, it's imperative you choose furniture than can be cleaned without a lot of hassle. Keep in mind your choice of furniture for your restaurant can help determine the customer traffic indirectly. So, you need to search for the furniture that is created of sustainable products. Industrial furniture is used anywhere massive quantities of people may use it daily, the surface should be effortless to wash, where there could be large weight differences among users, and when a company maynot budget for new business furniture every couple of decades. Even though some folks think as wooden furniture is normally seen in restaurants, it isn't a smart selection, but still the advantages of the wooden furniture make it appear very attractive. Indian wooden furniture is readily available for a large variety of uses.\nThe way that your restaurant looks says a lot to customers about the sort of food and service they can count on from your company. Make sure that once you establish a restaurant or any business, you'll have time for it. You ought to be aware that a cafe is also a kind of a restaurant but the principal difference between a cafe and a restaurant is their different ambience. Personalized restaurant is the best choice with respect to other forms of choices. As you probably already know, one of the absolute most important thing needed in a sure restaurant are the tables and chairs or simply called restaurant furniture.\nUnique varieties of restaurants are, naturally, affected to varying degrees by the crisis and has to react in accordance. The restaurant is where folks come to devote decent time. Enquire if you're in a position to add your restaurant within that list. Think of ways about how you can create your fast food restaurant unique and what demographics you have that could draw in people. If you wish to begin an independent quick food restaurant, choose the idea or concept you will promote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The pressures of an elite Asian society drained Melissa of self-confidence. She deteriorated mentally and became bulimic. Her resurgence was nothing short of astonishing!\nI was overweight for more than half my life. My insecurities and sense of worthlessness took root as a teenager. I was brought up in an elite all-girls school which was one of the top five in the country. There was a significant amount of stress on my shoulders and my environment was very competitive.\nBeing overweight in an Asian society where everyone is petite and slim made me feel discriminated against. It was a sin to be overweight in my school. The teachers who led the Trim & Fit (TAF) club were cruel. They often ridiculed the students and put them down in front of everyone. It was emotionally torturing and my self-esteem crumbled.\nTo aggravate matters, a group of seniors enjoyed passing snide remarks about me whenever they saw me. They mocked me mercilessly and used the crudest language. I can still hear the words they spewed from their mouths.\nOn the surface, I seemed jovial and bubbly, but an array of feelings was brewing inside. I was lost in the darkness of my own circumstance. Criticizing echoes left me awake at night as I pondered the meaning of life. I felt worthless, lost, and dejected. I wanted to be accepted.\nThen, I discovered bulimia. I was obsessed with food and my weight. It controlled my life for years and shattered my self-worth. I was always depressed and miserable. In a way, bulimia paved the way for me to release unhappiness. I went through episodes of binging and purging to let out feelings of anger, depression, stress, or anxiety.\nI didn't know who I was anymore. I lost myself physically and emotionally. My weight plummeted by 40 pounds and food became my best friend and worst enemy. The only time I felt in control was when I binged and purged on food. I felt disgusted with myself and hated what I saw in the mirror.\nEating disorders are very real. They're a physical, psychological, and emotional battlefield. It's tough to explain to someone who never experienced it, but it's a mental illness. Nobody wants an eating disorder. My heart goes out to people who suffer from them. I can only pray that my story helps or inspires someone to change their life through fitness.\nThe turning point came when I was hospitalized due to the side effects of bulimia. My mother was heartbroken when she saw me lying on the hospital bed with drips attached to my body. I weighed 98 pounds at that time and received lots of love and support from family, friends, and church leaders on my road to recovery.\nDuring my road to recovery, I started to pack on the pounds again. This time I was even heavier than before I had bulimia. I was up at 158 pounds. That is a lot of weight for a 5-foot Asian girl. I started getting depressed again because I was at my heaviest ever. I confided in my church leader, who stood by me, and I surrendered to Jesus. I was sick and tired of feeling lousy about myself. I didn't want to start purging episodes again. I had to make the decision to do things the right way because no one could do that for me. I signed up at a gym and my fitness journey took root from there.\nWhen I first started this fitness journey, my fitness level was very bad and I was very conscious about myself. My body fat level was 36 percent. Like all beginners, I experienced all kinds of muscle aches and hated it. I took it one day at a time and made sure I worked out. There were times I wanted to give up. I felt disgusted with myself and cried out to God for strength. My family, especially my mom, was very supportive.\nAs time went by, I started to see results from working out regularly. That drove me to work out even harder. The pounds started dropping and my clothes were hanging from my body. I could breathe easily and felt more confident.\nExercising became a habit. It was no longer a chore and I grew to love it. It's a lifestyle that will always be a priority. My body changed drastically and I packed on lean mass. However, I wasn't very strict on my diet. I had too many cheat meals which compromised my results. I always dreamed of having a tight muscular body with abs. I started to see great results when I cleaned up my diet and stopped making excuses. There is no way I can ever out train a bad diet.\nI started cooking and prepping my own meals. I kept a log of what I ate and the amount of food I was feeding my body. Because of my history of binge eating, I tend to overeat. Recording my food in an app on my iPhone helped me keep track calories and made a huge difference. I started to see changes in my body quickly. My body fat lowered, and for the first time in my life I saw my abs. I was ecstatic. All I ever wanted was a flat stomach to replace the three layers of fat I carried all my life.\nA few years ago, I signed up at Bodybuilding.com and was amazed and highly motivated by the many inspirational individuals there. There were many women with amazing bodies and I was inspired to compete in figure. In December 2011, I entered my first figure competition. I was on the verge of quitting six weeks before because my grandfather passed away suddenly.\nI knew that wasn't what he would have wanted for me, and so I carried on with contest prep. It was a really difficult period in my life. Even though I didn't win, I was really happy I'd come this far.\nMy biggest inspiration, who I found through Bodybuilding,com, is Dana Linn Bailey. I first saw her on BodySpace. Her physique is number one in my book, and I told myself that I wanted to look like that one day.\nI follow the cyclical ketogenic diet, which means I cycle periods of low carbs, high protein, and high fat with periods of high carbs, high protein, and low fats. I carb load once every four days.\nI found many helpful articles on CKD on Bodybuilding.com. I also drink about a gallon and a half of water per day.\nI work out seven days per week with five focused on weight training and two on cardio. I perform 30 minutes of cardio every day after my workout. On non-weight training days, I combine an hour of cardio (stairmaster or elliptical) with an abs routine.\nMy training style is a mix of powerlifting and bodybuilding exercises. I incorporate powerlifting methods to build mass and size. Bodybuilding methods are for conditioning and stimulating more muscle fibers.\nSticking to my diet is my biggest challenge. Due to my history with binge eating, I always struggled with food. Over the years, I learned to cope with emotional eating and how to listen to my body.\nI hope to be able to compete in the physique category one day. I'm giving myself another 3-4 years to build my physique to that level.\nThe fitness journey is full of challenges. There will be days when you fall off the wagon. I fell off so many times during the course of my journey, but I learned that failure isn't falling. Failure is not getting up and trying again. Things may get really tough along the way, but you will succeed by not throwing in the towel.\nTo be great, sacrifices need to be made. It's not the load that breaks you down, it's the way you carry it. At the end of the day, I know that however heavy my load is, Jesus will carry me through it all.\nBodybuilding.com is amazing. It's a great medium to share training\/diet tips. The forums are excellent for feedback on products, exercises, and training tips.\nI love the articles and videos also. They are informative and help me achieve my goals. I found Dana Linn Bailey, my fitness inspiration, through Bodybuilding.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All Dove dry spray antiperspirants match your pace for up to 48 hours, a must-have for modern life in the fast lane. Plus, dry spray Antiperspirant contains Dove 1\/4 moisturizers to help soothe underarm skin. This unique formula cares for this delicate area, supporting natural moisture levels to help keep shaving irritation and dryness at bay.\nAll the care of Dove in an instantly dry spray antiperspirant.\n\u2022 Dove dry spray antiperspirant goes on instantly dry.\n\u2022 Delivers long-lasting odor and wetness protection to stay dry and fresh for 48 hours.\n\u2022 Made with Dove 1\/4 moisturizers for soft, smooth underarms.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaczhx b/data_all_eng_slimpj/shuffled/split2/finalzzzaczhx
new file mode 100644
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@@ -0,0 +1,5 @@
+{"text":"One of the most desired appliances and even sinks today are stainless steel. The problem is stainless steel gets streaks, watermarks and fingerprints. If you notice even at the store they are streaked. Granite Shield eliminates that problem. The Granite Shield process to treating stainless steel removes the oxidation normally found in stainless steel and once we remove the oxidation we replace with a chemical that permanently bonds to the metal. Once that is done we put a protective polymer on the surface that reduces streaking, watermarks and fingerprinting. With the Granite Shield process your stainless steel appliances and sinks have never looked so good. You'll notice a vibrant glow and seeing colors you never knew your stainless steel had before. Stainless steel that has been protected with the Granite Shield process should be maintained at least once a month with Granite Shield's Kleen N Shine cleaner and occasionally with Silicabright and you'll have the most beautiful stainless steel forever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Recovers - LANDSCAPE TEMPLATES FOR THE REBUILD - COMMUNITY MEETINGS ON MARCH 28th & APRIL 5th!\nDaily Acts is partnering with the Sonoma-Marin Saving Water Partnership, local landscape architects, and sustainability experts to develop scalable, ready-to-permit, landscape design templates to support the rebuild. We will be hosting two community meetings to engage the public in the process and would really appreciate your help in spreading the word.\nThe Community Engagement Sessions will be held at the Finley Community Center in Santa Rosa on March 28th and April 5th from 6-8pm and will include social time to reconnect with neighbors, review concept plans, and talk with local experts. This will be followed by a brief presentation and a design charrette to learn more about the needs of the community. Light snacks and refreshments will be provided.\nThe goal of this project is to produce affordable, customizable landscape plans for front yards only that will minimize the cost for the under-insured and reduce another barrier to occupancy. We hope you will join us for this discussion whether you are a homeowner, builders, contractors, designers or community groups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Forskolin Blend: Typically, we have seen that each rec center is brimming with those individuals who have a weight issue. Everybody needs to skip it yet can't on account of above all else individuals are exceptionally occupied in their day by day way of life and nobody has enough time to take after the long-term eating regimen designs, also is that everybody can't bear the cost of the costly bill of exercise center dependably. In this way, numerous individuals unfit to decrease muscle to fat ratio from these comparable ways. In any case, now, our organization has propelled another weight reduction supplement called Forskolin Blend which is totally made of characteristic fixings.\nForskolin Blend can decrease hunger wanting for diminishing additional fat likewise and help to increment metabolic rate in the body with the assistance of normal plants and herbs.\nHow do Forskolin Blend functions?\nForskolin Blend attempts to diminish fat from body particularly in stomach and thigh territory with giving numerous wellbeing nourishment esteems by nature.\nMake a tight and alluring figure-Obesity can make your body fit as a fiddle and can make free yet in the wake of utilizing this common weight reduction supplement you can get back your figure fit as a fiddle with a tight and appealing figure.\nControl your craving you can control your hunger amid taking and after additionally on the grounds that it can decrease your yearning and help to dodge the propensity for gorging with over and over eating. In the wake of taking this supplement you can feel happy with your each controlled feast in an entire day.\nEqualization water level-It can adjust water level in the body since more water can flush out waste material from the stomach and help to forestall sicknesses like stoppage and lever issue. Water level can keep you from body lack of hydration likewise help to keep you dynamic and fiery entire day.\nKeep away from shoddy nourishments This supplement backings to dodge lousy nourishments and quick sustenances which are unfortunate and brimming with oil and microorganisms. It can urge you to eat new and solid nourishments like crude vegetables and natural products too.\nNo symptoms a wide range of lousy nourishments and solidified would imbalance be able to you right eating regimen. This supplement can support to pick right eating routine since it has no symptoms and no any sort of body responses.\nStep1. It is coming in pill shape based recipe.\nStep2. Each pack has 60 pills with 2 months expending date.\nStep3. You can take it void stomach in morning with Luke warm water.\nStep4. You should drink more water in multi day for weakening poisons from the body.\nStep5. Try not to take with other drug.\nStep6. Read the directions deliberately before utilizing it.\nGarcinia Cambogia-It has common properties with contains HCA which assumes an essential part to lessen weight and help to stop additional fat delivering moreover. It can expand serotonin in the mind for diminishing fat.\nLemon-Lemon has a cell reinforcements property which expels poisons from body step by step amid this supplement. It can keep you dynamic all the day.\nGreen tea-It has likewise cancer prevention agents properties that can be rising the digestion rate in body with lessening weight.\nWhere To buy Forskolin Blend?\nIt is accessible online just for simple obtaining. You can submit your request on our official site and can get a select offer of a first free preliminary. Pick up the pace since we have constrained load of it. It will contact you in an only 48 hours from enrolled your booking number by free home conveyance.\nForskolin Blend This supplement has capacity help digestion rate by diminishing fat. It can consume more to more calories from sustenances and amid substantial exercise additionally to keep you dynamic all the day. It has the ability to adjust eating routine and help to forestall numerous sorts of lousy nourishments infections constantly. It is superior to others due to it a characteristic fixing that can make you more beneficial and slimmer at any age and furthermore ready to keep from manufactured supplements and their reactions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From our Teaching Artist workshops and general celebrations of our Young Artists and our programming, there is never a dull moment with ArtWorks!\nAli, age 10, from Elizabeth Seton Pediatric Center created \"Birdie and Nest\" (2009) and told ArtWorks \"I made this bird out of tissue paper, #raffia, #feathers, and #texturedpaper. I had lots of fun using glue to tap them down.\" Join in the fun this #NationalCraftDay and tag ArtWorks in your next craft project!\nArtWorks is privileged to work with extraordinary women from young artists and teaching artists, to Child Life Specialists, doctors, our staff, Board and more. We celebrate them today and everyday. Tell us about the women who inspire you by tagging ArtWorks in your #InternationalWomensDay post!\n\"Different\" by ArtWorks Young Artist, Cree from Mount Sinai Kravis Children's Hospital (2003).\nWe are thrilled to announce a new year-long collaborative program with ArtWorks Teaching Artist, Robyn Ellenbogen and The Rutgers Garden at Bristol Myer's Squibb Children's Hospital at Robert Wood Johnson University Hospital. This program aims to promote creativity through the use of organic materials obtained from the Gardens and projects will be approached through a multidisciplinary lens using storytelling, music, poetry, writing, performance, fine art techniques, animation and sculpture. This program is generously funded by the Robert Wood Johnson Foundation President's Grant Fund of the Princeton Area Community Foundation. Pictured here is Robyn Ellenbogen with RWJ Child Life Specialist, Sarah Devlin.\nSince 2002, ArtWorks has worked with Child Life Specialists in hospitals to bring creative programs to children and young adults facing chronic and life-limiting illness. Our organization was founded in large part due to the relationship Naomi Cohain had with her own Child Life Specialist and Art Therapist at Hackensack University Medical Center.\nChild Life Specialists help pediatric patients and their families cope with the stress and uncertainty of illness, diagnosis and treatment. Through developmentally appropriate interventions including therapeutic play, preparation and education they aim to reduce the fear, anxiety and pain that often surrounds a chronic or life-limiting illness. Their work is essential to the continuation of our mission and we celebrate and applaud the field and its important role within the healthcare team.\nFollow us on social media! This month we will highlight some of the Child Life Specialists we are privileged to work with through our dedicated network of hospital partners.\nThank you, Child Life Specialists for all you do for our Young Artists and their families. We couldn't be US without YOU!\nArtWorks, Teaching Artist Steven Hazlett visited the Joseph M. Sanzari Children's Hospital at Hackensack Meridian Health where he will lead a year long program series in macro-photography. Young Artists will use a special camera lens to shoot a variety of materials such as rhinestones, feathers, beads and more to create extraordinary compositions.\nWillem Dafoe plays the brilliantly complex Vincent Van Gogh in Julian Schnabel's At Eternity's Gate and is up for Best Actor this Sunday at the 91st Academy Awards. ArtWorks Young Artist Mikeyla channeled her inner Van Gogh after her mother introduced her to the iconic \"Sunflowers\" (1889). She told ArtWorks, \"I wanted to make my own!\" Van Gogh continues to make an impact all over the world and we are delighted that he influences our Young Artists all these years later.\nArtWorks Teaching Artist, Michael Albert at The Brooklyn Hospital Center!\nWe kicked-off ArtWorks, Teaching Artist Michael Albert's Collage Art Workshop last Thursday at The Brooklyn Hospital Center. Our Young Artists will be using scissors, glue and a canvas to transform ordinary consumer cardboard packaging into a #mosaic like collage! (Pictured: Michael and Child Life Specialist, Lori Lerma with special guest, Princess Anna of Arendelle from A Moment of Magic Foundation).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"No, I cannot confirm these media reports. It is difficult to talk about reconciliation with a terrorist organization, which openly proclaims the elimination of the Israeli State as its goal and is constantly involved in the organization of terrorist attacks against Israel,\" Elkin said.\nThe Israeli minister noted that Hamas was not interested in a new escalation of tensions with the Israeli army, as it learned its painful lesson during the latest escalation in 2014, when Israel provided a harsh response to the militant group.\n\"That is why it is not a question of a ceasefire, but an example of the Middle East's respect for [the use of] force,\" Elkin stressed.\nEarlier on Thursday, the Israeli Channel 10 reported, citing diplomatic officials, that Israel was considering accepting a long-term ceasefire with Hamas after rejecting the offer for weeks. The broadcaster, in particular, specified that Israel was allegedly ready to waive the requirement for Hamas' demilitarization and handing over the management of the Gaza Strip to the Palestinian Authority.\nElkin stressed that he did not expect any disputes over Israel's potential decision to hold the 2019 Eurovision song contest in Jerusalem.\n\"Eurovision traditionally takes place in the capital or in the city in which the winner country chooses. I think no one has any doubt that Israel will choose Jerusalem. I believe that there should be no problems with the Eurovision organizers because the rules should be the same for the entire civilized world,\" Elkin said on the sidelines of the St. Petersburg International Economic Forum (SPIEF).\nHe added that Eurovision had been held three times in Jerusalem in the past.\nEarlier this week, the contest organizers asked Eurovision fans to refrain from \"booking flights just yet\" in a statement on Facebook, as the time and location of the 2019 event had yet to be set. The announcement comes amid uncertainty regarding Jerusalem's status as well as recent tension caused by the relocation of several embassies, including that of the United States, from Tel Aviv to the holy city.\nA number of political parties, in particular, in Sweden and the United Kingdom, urged to boycott the event in the light of the human rights abuses of Israel against the Palestinian people. In Iceland, over 25,000 people signed a petition demanding that the contest be held elsewhere.\nThe Eurovision final took place two days ahead of the United States' embassy relocation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaddjq b/data_all_eng_slimpj/shuffled/split2/finalzzzaddjq
new file mode 100644
index 0000000000000000000000000000000000000000..a8a4f9abdcb9ff989e84f7a65f9ea815ef004619
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@@ -0,0 +1,5 @@
+{"text":"One hundred and eighteen products in Guizhou province have received geographical indication titles, generating nearly 60 billion yuan ($9.1 billion) per annum.\nChinese researchers have discovered six pulsars, the superheavy remnants of massive stars, using the country's Five-hundred-meter Aperture Spherical Radio Telescope, known as FAST.\nImagine a toilet seat so artistic that it shows different colorful pictures when the lid closes in slow motion.\nIn an era when people rarely write letters, a group of college students in Zhejiang province have built close links with left-behind children in a remote middle school by writing letters to each other.\nA new exhibition is underway at the China Cultural Center in Paris, offering a glimpse of ethnic costumes and crafting techniques in Southwest China's Guizhou province.\nA public welfare train commuting between Guiyang and Yuping in Guizhou province carries the dreams of fortune for local people farmers in the region.\nYu Liufen,has received a lot of attention for her hard work in successfully alleviating poverty and encouraging the development of Yanbo village in Guizhou province.\nGuiyang, capital of Southwest China's Guizhou province, held its eighth tourism promotion conference on Sept 27.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fjallraven Kanken Rucksack Green\/Folk Pattern. Launched in 1978 to aid the backs of school children, this classic Fjallraven rucksack has been a favourite ever since. The Kanken is made from durable, lightweight Vinylon F and is designed to withstand everyday use. The main compartment can be completely unzipped, giving you plenty of space to take items in and out with ease. As well as this, there are two side pockets and a zippered pocket at the front. The Kanken is simplicity at its finest, coupled with enough practical elements to round it off as an all round delightful product.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sweet! Adding it to my wish list. We've been reading all about these guys for my son's science lessons. Looks like a fun book to read.\nSounds like it will have a twist of humor!\nI'm not sure if I'd be in a hurry to read this one, myself, but I must admit that it is a cool cover.\nThanks for the nice words. The cover artist, Jason Seiler did an amazing job. Hope you enjoy the read.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you know there are 18 pet sitters near Watsonville, California who will come to your home and care for your pet? You're sure to find one in your neighborhood.\nDon't live near Watsonville, California? Click here to search in your area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This union increases the qualities of love, success, happiness and ability to enjoy life. Although Libra needs to share things with someone special he is tactful enough to allow Sagittarius that much love, freedom and independence. Sagittarius is generous to allow Libra to indulge in little pleasures and luxuries.They gad about the globe together.Both of these seasoned travelers are imbued with a sense of adventure and enjoyment and their bonding about abroad will be a bundle of laughs. Even when they are at home, they'll have such a myriad of mates.\nTheir social life will sizzle as they're inundated with invitations to parties and get-togethers galore. Libra-Sagittarius Compatibility If they're both fairly free and easy in their feelings, they can have fantastic fun together, because fundamentally they'll be friends. Even if they're married they can lead a liberated life. But there'll be trials and tribulations if one of them has got Venus, the planet of love, in sultry and suspicious Scorpio, because then they'll want a more loyal and lasting liaison than the other one will be able to give. If one of them wants a more anchored affair than the other one can cope with, they should call the whole thing off.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzadqdj b/data_all_eng_slimpj/shuffled/split2/finalzzzadqdj
new file mode 100644
index 0000000000000000000000000000000000000000..d6686fc8849adcae50b79cd173a6c851af4abf29
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzadqdj
@@ -0,0 +1,5 @@
+{"text":"Does Kriptomat offer business accounts and how to get one?\nYes. Kriptomat offers business accounts.\nBusiness accounts are intended for business purposes and used by legal entities. This means that, if you want to trade with capital belonging to your business, this is the account type you should choose.\nThe verification process for Kriptomat business is a standardized compliance measure for anti-money laundering (AML), know your customer (KYC), and counter-terrorism financing (CTF) laws and regulations.\nRegister a personal account here. A person with an authority to access the business account and trade on behalf of the legal entity should open the account (we strongly suggest one of the directors or ultimate beneficial owners).\nOnce your account has been created, verify your personal identity.\nSend an email to [email protected] and request a business account verification.\nYou will then proceed with filling out the verification form and submitting the requested documents.\nOnce verified, personal account will be upgraded to business account.\nThe process consists of a completing an online questionnaire and submitting supporting documentation. A signed declaration confirming the accuracy of the submitted information is required, authorising Kriptomat to complete the verification process.\nProcessing times for verification can take up to 6-8 weeks, depending on the level of public interest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ross Edwin Barbour, American vocalist (born Dec. 31, 1928, Columbus, Ind.\u2014died Aug. 20, 2011, Simi Valley, Calif.), was the last surving original member of the close-harmony group the Four Freshmen, for which he provided his smooth baritone voice and drumming skills. During his first year at Butler University's Arthur Jordan Conservatory of Music, Indianapolis, Barbour and his brother Don joined Hal Kratzsch and Marvin Pruitt to form the barbershop quartet Hal's Harmonizers. By the time that Bob Flanigan replaced Pruitt, the group had taken on a more jazz-influenced style and had become known as the Toppers, but in 1948 their agent renamed them the Four Freshmen. Their long, opulent chords and improvisational style made the four-member group harmonies sound as if they were composed of five or six voices. The success of this groundbreaking approach to close harmonizing caught the attention of Capitol Records, which in 1950 signed the Four Freshman. After 29 years, Barbour retired in 1977 from the quartet. In 2001 it was inducted into the Vocal Group Hall of Fame.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enjoy a cracking Easter at Coldharbour!\nGood Friday marks the grand reopening of Coldharbour Mill, Uffculme. After months of hard work over the winter the Mill come and see the new new welcome centre funded by the Heritage Lottery Fund.\nOver the weekend there will be Easter themed childrens activities and a special Easter Steam Day where all the machinery will not only be powered by the waterwheel but by our very own Lancashire Boiler. \u2013 see the steam pages for more information on steam at Coldharbour. Our popular pop up cafe will be also on hand for refreshments throughout the whole weekend while the builders finish our exciting new cafe.\n\u2026. and don't panic if you don't see everything, from Easter the Mill will now open weekends!\nA record number of visitors to Coldharbour Mill's New Year's Steam Day kick-started its exciting plans for 2016. Despite the rain, visitors were treated to the sights and sounds of a working Mill taking wool to yarn and cloth exactly as it was in Victorian times.\nNew for 2016 are the free historic bus tours organised by WHOTT (West Country Omnibus and Transport Trust) who are now based at the Mill. The 40 minute round trip also connects with the trains at Tiverton Parkway carrying those who come by train to and from the Mill.\nThe bus tours were repeated at the February Steam Day, and will be available again at the Easter Steam Day on Sunday 27th March. See the WHOTT website for timetable.\nOn Steam Days, local Jazz musician Charlie Hearnshaw provides a musical tour at 1pm, using the sounds of the machines to compose imaginative improvised pieces with the various looms, line-shafting and spinning machines providing a percussive element to the pieces.\nAlso at Steam Days, the Culm Valley Model Railway Club displays its 'SOLDOR' layout, which can be operated by chidlren of all ages!\nThe Mill, recently awarded a Heritage Lottery grant to invest in its facilities, has been closed to the public since last November (except for Steam Days and pre-booked group tours) in order to carry out refurbishments. It will reopen to the public on Friday 25th March (Good Friday).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Curious postwar invention of British designers: aluminum gasoline lighter with a fuel tank and two silicon.\nbut why two meters below ... I do not know what to call similar to drum?\n\"... I think clearly explained? :))) .. \"\nif you want to - yes. You just tell me.\nPlease tell me here on this zazhialke http:\/\/cs616527.vk.me\/v616527230\/1680c\/4U-fba4a61c.jpg vrodeby also in this series.\nAlexander, what is supporting? Yes, your series of similar :) I think even the manufacturer of the same.\nI would like to know the year of issue and about the price if possible. Just except your lighters nothing like the more I did not come across.\nAll clear, thank you very much for your help!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Laser Phosphor Light Engine JVC's proprietary 2nd generation BLU-Escent\u2122 laser phosphor light engine uses blue laser diodes, stationary inorganic phosphor and JVC's proprietary color management system. BLU-Escent\u2122 enables a consistent 3,000 lumen light output. Increased brightness ensures compatibility with larger screen sizes over 200 inches, and improved HDR performance, bringing greater depth and dimensionality to home theater screens.\nHigh Resolution Lens Dedicated for 4K An 18 element 16 group all glass lens and full aluminum lens barrel has been specically engineered for the DLA-RS4500's 4K performance. Chromatic aberration and color fringing are minimized even when utilizing lens shift, resulting in precise projection of 4K content. In order to project 4K resolution to every corner of the screen, the new lens diameter is now 100mm, compared to 65mm used in previous JVC projectors. This lens allows a wide shift range of \u00b1100% vertical, \u00b143% horizontal for a 16:9 image (\u00b140% horizontal for a native 17:9 image).\nNext Generation Multiple Pixel Control The new algorithm has been developed to scale all video to 4K, giving extraordinary detail to all content regardless of source resolution. Native 4K images (up to 4K\/60P 4:4:4) may also be fine tuned for the highest projected quality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaekwj b/data_all_eng_slimpj/shuffled/split2/finalzzzaekwj
new file mode 100644
index 0000000000000000000000000000000000000000..9cd3069df1b26190abd566918e250ab91fc3a4d4
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Asclepias incarnata 'Cinderella' | Walters Gardens, Inc.\nThis species grows in loose clumps by means of slowly creeping rhizomes. It is not invasive and can be safely mixed in with other perennials in the border. Ascelpias incarnata is native to North America.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Iris joined the lab in October 2018 as an undergraduate student intern. She is majoring in anthropology with an interest in archaeology and scientific illustration. Since joining, she has learned 3D visualization and segmentation software and successfully contributed to important projects on hominin dentition.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Included in this lot we have a Plains Indians wolf painted buffalo hide drum. The drum features the painted image of a wolf in front of a cow skull with the sun rising behind. The drum is made of buffalo hide stretched over a wood frame. The drum is in good condition. The piece measures 12\" in diameter by 2 1\/2\" thick.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are looking forward to the NEW format for this year's Parents' Association Christmas Fair. A record number of stalls have signed up for the Fair which will be held on Wednesday 14th November and promises to be better than ever before.\nPupils will be playing Set Hockey and Rugby so the aim this year is to provide a relaxed shopping experience for parents before picking up your children.\nSo come along and support your PA and get ahead with your Christmas present shopping!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wow, still reeling from the results of the US Election?\nIn Part 1, (What Brexit and the U.S. Elections can teach us about Leadership), we talked about the new political landscape and how the electorate have lost trust in our politicians. People are seeing them as dishonest and inauthentic.\nAs a leader, how can you make sure your people remain engaged? You might want to consider your Authentic Leadership?\nHere are our five tips to being more authentic as a leader.\nBuild big relationships with the people you work with. Make them genuine not simply transactional. Get to know what they care about, what motivates and excites them so you can engage them at work. And be willing to share the same for yourself. This doesn't mean being their best friend or knowing their deepest secrets \u2013 it simply means knowing who they are as people. Asking how their daughter's football match has gone or how the preparations for their son's wedding are going has a disproportionate effect on the leadership 'bank account'. You are showing that you value them and that they matter to you.\nListen to your people. Not to fix them or push your point home, but to genuinely understand things from their perspective. If you do this, they will tell you how to run your business effectively, where money can be saved and service can be improved. If you don't, you will get the same 'anti-establishment' behaviours as politicians are currently getting.\nInvolve your people\u00a0in how to implement key decisions.\u00a0Warning: this doesn't mean\u00a0'management by committee'. It's\u00a0asking the people who do the job to tell you how to make things work better.\u00a0Involving staff at this early stage leads\u00a0to\u00a0a more productive, happier team and they stay with you longer. For leaders who are afraid that staff will come up with the 'wrong answers' (meaning\u00a0'not their answers!'), this is not my experience.\u00a0I've notice how much better the result is when I go with a process that has one or two imperfect areas in it but has the complete buy-in of the people using it, rather than a 'perfect solution' from me that the people haven't engaged in.\nShow appropriate vulnerability. Be prepared to say that you don't know or that you were wrong. Don't lie to your team \u2013 they will see through it in an instant. What leaders fear is that people will see them as weak if they admit they are not perfect \u2013 in fact, the opposite is true. Strength in leadership comes from showing your human side and connecting with people at a different level.\nHave honest conversations. Don't dress things up or beat around the bush. People prefer to be treated as adults and to know what you are thinking, even if it is sometimes hard to hear. Keep it about the job or behaviour, don't make it personal \u2013 focus on what you want to see going forward rather than what has been seen in the past. Be clear about improvements you expect and how these will be observed and reviewed. People will still disagree with your assessment of them but they will respect you for sharing your opinion and being honest with them.\nIn business, there are other aspects of authentic leadership but these five areas are a good start. We believe authentic leadership is the key to maximising productivity and helping your people be their best more of the time.\nDo any of the five tips resonate with you and your leadership? Let me know what you think. I'd love to hear from you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"From the vandalised oil paintings and statues choking on tear gas fumes in the Piano Bar to Elton John playing for the opening party (remotely, broadcast over a modest TV screen), this new West Bank hotel has Banksy's dystopian stamp all over it.\nBedrooms overlook Israel's infamous separation wall; its neighbour is the Aida refugee camp. The artist paid for the 'installation', which will function as a bona fide hotel during what it's calling the Centenary Year (2017 marks 100 years since the British took control of Palestine, with its ensuing century of conflict). It has since been handed over as an independent local business, with the aim of breaking even and putting any profits back into as-yet-unnamed local projects.\nRevenue could be significant. Banksy's last tourist endeavour, the Dismaland theme park in Weston-super-Mare, generated \u00a320m in its five-week run. Not unsurprisingly, this new project hasn't pleased everyone, with some bemoaning Bethlehem's conflict being used to generate income \u2014 though it's a business model long-preceded by local tours that take in the wall, its history and rich graffiti, including a now-iconic stencil work by Banksy, painted over 10 years ago.\nVoyeurism or essential revenue stream, opinions are divided. The Walled Off Hotel states that it isn't aligned to any political movement or pressure group, saying, \"The aim is to tell the story of the wall from every side and give visitors the opportunity to discover it for themselves. We offer a warm welcome to young Israelis \u2014 absolutely no fanaticism is permitted on the premises.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Harriers bad run continued with this narrow defeat at promotion hopefuls Exeter City. City won by the only goal of the game but the Harriers never really looked like scoring themselves until the final minute of the game when they could have grabbed an equaliser twice over.\nThey failed to take their chances during a first half that they held the upperhand but in the second half, with the wind against them, they struggled to make any headway against a resilient Exeter defence.\nWe made one change to the side that lost at Stevenage last week with Andy Ferrell coming back into the side from injury and Darryl Knights dropping to the bench. Mark Yates failed to replace Luke Jones with a new signing after the talented defender was re-called to Shrewsbury just before the Stevenage game so Gavin Hurren retained his place in the side.\nIn freezing cold conditions with deep black snow threatening clouds above it was the Harriers who were to start the stronger. In the first minute a free kick from Jeff Kenna hit Iyseden Christie and bounced out of play and then Dean Bennett had a chance to open the scoring but his shot hit a City player and went out for a corner.\nCity retaliated with chances of their own when Adam Stansfield raced to reach a loose ball but luckily for the Harriers Chris MacKenzie got there first but the ball still went out for a corner that ended with a shot over the bar from 20yds out.\nWe came back on the attack straight after with a shot from Matt Barnes-Homer that was blocked on the line but slowly the home side started to take control. This time they went the closest to scoring so far when Stansfield hit the post after hesitation from Mark Creighton and Gavin Hurren let him through.\nThey almost had another chance gifted them when Alex Jeannin almost put the ball into his own net as he attempted to clear the ball from a deep cross.\nWe weren't out if it though and we came very close to opening the scoring when Christie played in Russ Penn with a fine cross that Penn headed just wide. Christie could have scored himself just moments later when a cross from Ferrell came over with the City keeper, Marriot, rushing out of his area but the Harriers hitman could only send the ball over both him and the crossbar.\nChristie followed that up with another chance that should have been converted when MBH played the ball in to him and again the resultant shot went wide.\nWith the time on the clock running down we had a breakaway following a City corner. The ball broke for Russ Penn who headed straight for goal but found himself on the end of a crunching tackle from Rob Edwards. The City tough man, remembered by Harriers fans for getting Reuben Reid sent off last season, somehow escaped a red card even though he was the last man. A yellow card was deemed enough punishment.\nNow with the wind in their favour City took the game to the Harriers and where we were the stronger side in the first half now we were on the back foot for most of this half. Wayne Carlisle started the half off by heading a chance against the bar a minute after the re-start. Ten minutes later he headed the ball over the bar when it was easier to score.\nIt wasn't long before Exeter got the only goal of the game. Dean Moxey sent a cross over that Jeannin failed to deal with and Matthew Gill took his chance to sweep the ball past him and volley home without giving MacKenzie a chance to stop it.\nNow with us a goal down it was time for heads to drop. City took over and should have scored a second goal when Richard Logan sent a volley over that stung the hands of the Harriers keeper.\nIn an effort to rescue the game Yates took off Barnes-Homer and bought on Darryl Knights in his place. A few minutes later another change was made with Brian Smikle coming on for Andy Ferrell. It still didn't change our game though.\nExeter seemed happy to sit on their narrow lead and apart from the sporadic attack were prepared to run the game down. The Harriers had different ideas and when Michael McGrath replaced Russ Penn we saw that there was still a chance to at least get a point.\nSimon Russell was the first to try and claw the game back when he raced into the area and forced Marriott to save his shot twice. Darryl Knights had a shot blocked on the line and then we should really have scored when, first, Jeannin sent over a high cross that Creighton played on for Christie to shoot hard. Marriott did well to push it out for a corner.\nFrom the resultant corner to the far post Knights headed over with an open goal at his mercy.\nIn the end it was another defeat on the road for us and now we have the chance to avenge that Setanta Shield defeat of a month ago at the hands Rushden. It's an Easter Monday game next up at Aggborough and one that we need to win to put a smile on Bazzas face.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"bright orange throw pillows creation in color vibrant bright bold l a throw pillow colored pillows neon yellow bright yellow pillows colored throw home ideas design pictures home decor ideas philippin.\nbright orange throw pillows bright throw pillows how amazing are these bright floral throw pillows from star aka meg bright bright throw pillows home renovation ideas philippines home ideas centre mel.\nbright orange throw pillows orange accent pillows accent pillow bright orange accent pillows super area rugs target orange accent pillows home ideas centre melbourne smart home ideas magazine.\nbright orange throw pillows day one love red orange throw pillow home ideas magazine home business ideas philippines 2017.\nbright orange throw pillows orange throw pillows home ideas centre parnell auckland home design ideas living room.\nbright orange throw pillows bright and beautiful burnt orange throw pillow via home ideas centre country home ideas magazine.\nbright orange throw pillows bright colored throw pillows bedroom furniture yellow throw pillows for more appealing rooms in sofa idea 5 bright colored outdoor throw pillows home ideas magazine uk home.\nbright orange throw pillows orange and brown decorative pillows home ideas diy pinterest home ideas centre christchurch.\nbright orange throw pillows navy blue pillows decorative orange decorative pillows home decor navy blue lumbar pillow aqua blue throw navy blue pillows decorative home ideas centre frankston home idea.\nbright orange throw pillows pillow decor image gallery home ideas design home studio ideas pinterest.\nbright orange throw pillows bright colored throw pillows bright colored throw pillows solid green throw pillows everyday pillow bright colored bright colored throw pillows home ideas magazine philippi.\nbright orange throw pillows set of four matching decorative pillows for interior design colorful floral pattern in on bright home security ideas diy home improvement ideas philippines.\nbright orange throw pillows pillow cover sunflower pillow bright yellow orange throw pillow sunflower photo pillow living room decor bedding home ideas philippines home design ideas for small spaces.\nbright orange throw pillows fresh throw pillows for your powerful spring block print pillow in bright orange home ideas centre sydney home ideas diy pinterest.\nbright orange throw pillows bright throw pillows hot pink throw pillows cheap bright decorative silk bright red decorative pillows bright throw pillows home ideas magazine uk home ideas centre petone.\nbright orange throw pillows medium size of old reds pinks resources throw pillows in decorative home design ideas 2017 home improvement ideas pinterest.\nbright orange throw pillows pink and orange pillows orange couch pillows hot sale solid color sofa cushion covers styles pink pink and orange pillows home decor ideas philippines home design ideas 201.\nbright orange throw pillows throw pillows couch home bar ideas pinterest home business ideas philippines 2017.\nbright orange throw pillows x inch bright red and burnt orange throw pillow mobile home ideas pinterest home diy ideas cheap.\nbright orange throw pillows orange home ideas superstore urdaneta home ideas design.\nbright orange throw pillows bright throw pillows ripple white embroidered throw pillow bright white bright colored outdoor throw pillows bright throw pillows home ideas design pictures home diy ideas.\nbright orange throw pillows orange and green mandala pattern throw pillow a bright home is a happy home which is home ideas philippines magazine mobile home ideas pinterest.\nbright orange throw pillows light grey sofa with a mix of bright orange and matching solid grey throw pillows home ideas centre sydney home improvement ideas philippines.\nbright orange throw pillows t colored throw pillows funny penguin for kids color decorative cushions orange country home ideas magazine home ideas centre.\nbright orange throw pillows colorful throw pillows bright colorful pillows colorful flower throw pillows for grey couch bright colors decorative colorful throw pillows home ideas centre melbourne home.\nbright orange throw pillows bright throw pillows bright orange throw pillows bright red outdoor throw pillows home ideas centre auckland home ideas centre.\nbright orange throw pillows bright pillows decorator pillow decor bright yellow accent pillows bright throw pillows decor bright pillows home ideas centre southbank home improvement ideas diy.\nbright orange throw pillows orange and gray throw pillows soft grey cushions orange and turquoise accent pillows bright green cushions blue and white throw pillows orange and grey home design ideas 20.\nbright orange throw pillows orange accent pillows pink accent pillows pink accent pillows orange decorative pillows orange decorative pillows bright home security ideas diy home renovation ideas phili.\nbright orange throw pillows orange and navy pillows throw pillow covers couch cushion blue decor home sofa b colors orange cotton canvas decorative throw pillow home diy ideas uk home library ideas pi.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Achieve uniform slices as thick or thin as wanted with the Berkel 827A-PLUS 12\" Manual Slicer. This stainless steel slicer features classic styling with the traditional Berkel red color on knobs and levers. It has a chromium-plated, carbon steel knife, built-in sharpener and belt-driven blade assembly. This item features a 45\u00b0 angled product table makes feeding easier, which produces consistent slices. This slicer is ideal for handling oversized meats, cheeses and a variety of other products. The Berkel 827A-PLUS 12\" Manual Slicer disassembles without the need of tools and is easy to clean.\nSlice thickness ranges from tissue thin to 9\/16\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lovely Awkward: A Year of Wine, Romance and Life Among the French: Signs of spring -- ah! I hope! I hope!\nSigns of spring -- ah! I hope! I hope!\nThese palm trees are always around, but now they look as though they're getting ready for something.\nBlossoms! Even on a grey background, they look optimistic.\nAnd, here's the highway, which isn't exactly spring-like (I think spring means all of the scooter riders will be going around in flip-flops), but it's interesting (and for me, scary) nonetheless. But it's the route to the food. Look how chilly it looks! And people here don't even have big coats! Everyone keeps expecting Canadians to be cold resistant, but I keep telling them that we have big winter coats, we turn the heat on and we drink a lot of hot chocolate!\nPosted by Kerry MacGregor at 7:30 a.m.\nAnonymous March 9, 2010 at 11:02 a.m.\nI'm always amazed by people who can brave the cold without big winter coats! How can they do that? Remember me all bundled up in my Whitehorse picture! Crazy!\nKerry MacGregor March 9, 2010 at 11:09 a.m.\nYes! That's exactly what I need over here!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaglxv b/data_all_eng_slimpj/shuffled/split2/finalzzzaglxv
new file mode 100644
index 0000000000000000000000000000000000000000..2dca32f989d2db081efcaf4ea2118726c77e5a49
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaglxv
@@ -0,0 +1,5 @@
+{"text":"Extruded acrylic rod and tube and polycarbonate tubes are lightweight and easy to cut or fabricate. They enhance the beauty and functional value of your products. Prolonged exposure to moisture will not affect the look or function of these materials. Plus, they are UV stable which allows them to be used indoors or outdoors.\nClear, lightweight extruded acrylic square rod enhances the beauty of any exhibit. Sunlight resistant and breakage resistant, it is a safer choice than glass. Typically better in decorative rather than functional applications.\nClear, lightweight extruded acrylic spiral rod enhances the beauty of any exhibit. Sunlight resistant and breakage resistant, it is a safer choice than glass. Typically better in decorative rather than functional applications.\nClear, lightweight extruded acrylic tube enhances the beauty of any exhibit. Sunlight resistant and breakage resistant, it is a safer choice than glass.\nLightweight polycarbonate tube offers the break resistance of polycarbonate in and the aesthetics of a clear tube.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Honda Odyssey was already one of the top-selling minivans around. The completely redesigned 2018 version takes that to an even higher level. About the only knock on this vehicle in recent years has been the acceleration, and it is improved with this model. It adds a lot of new technology to the mix, a huge interior and the infotainment system has gotten even easier to use. It was already among the best, and the new generation that arrives in 2018 will be tough to beat as a great family vehicle. This is a large minivan, almost a full-sized van, with three rows of seats. The Odyssey will be an even better family vehicle that will make long road trips a breeze.\nWhen they say this vehicle is redesigned, they are not kidding. This new generation is about the same size, but just about everything is at least tweaked. The infotainment center is now an 8-inch screen in front and another 12-inch screen in the back. The base model does still have the five-inch screen. The dash board is also updated with a smoother look and easier to use materials. Color-coded technology also makes it easier to use. This minivan is big. It even includes an intercom to help the driver talk to back seat passengers. A camera also shows the rear seats. Also new is easier sliding second-row seats, but they still must be put all the way down to make use of cargo space. The engines are also updated with more power and more fuel economy. There are also plenty of technical updates, making it more modern and safer.\nThere are no fewer than eight trim levels for the Odyssey, but you have to pick a trim level as you cannot add individual features. The base model is the LX, followed by the EX and EX-L. The EX-L also has four sub-trim levels, which include Navi, Res, Touring, and Elite. All levels come with the same engine - a 3.5 liter V- that cranks out 280 horses and 262-foot pounds of torque. There is a nine-speed or a 10-speed transmission depending on the trim level. The base level seats seven and all other levels seat eight. As you move up in trim levels you get more technology and more comfort features. At the EX-L level, you get a sunroof. The top of the line Elite has what all the others have, in addition to 19-inch wheels, noise reducing glass, ventilated seats and an 11-speaker premium sound system. The Odyssey is expected to get between 19 and 28 miles per gallon.\nTechnology and safety are areas where the Odyssey stands out. Even at the base level, there is hands-free texting available. Cabin watch allows you to keep an eye on the entire cabin, and of course, there is a rear-view camera. It has the engine shutoff feature, that turns the motor off when you stop. The Odyssey allows you some manual control over that, which makes it a little better than most cars that have this feature. The Odyssey also has all the standard technology features like automatic emergency braking, adaptive cruise control, blind spot monitoring, lane keep assist, auto high beams and rear cross traffic warning. One great improvement for 2018, reviewers say, is that the Odyssey is much quieter than previous editions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In conjunction with the creation of the Wellesley Park Hotel website, we were tasked with doing a major overhaul of its sister hotel, The Trevross. We have been looking after this website for well over a decade without it ever having a revamp aesthetically but the owner felt it was time to update the look of the website to tie in with the new website for their sister hotel. The resulting site is the same look so that the two hotels visually tie together but each with their own information unique to the hotel and The Trevross features the green colouring which has long been a feature of the exterior of the hotel itself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modine Aviation is here to help you navigate and negotiate aviation solutions to meet your needs. Regain some of that time lost in airports for business or pleasure and make your travel more personal and convenient. Learn to fly and upgrade your pilot status while getting where you need to be.\nFederal Aviation Regulation Part 61 Subpart A (61.65) dictates a minimum of 1 hour ground training and 1 hour flight training to meet this requirement.\nCandidates for this experience often say that they enjoy flying even more now that they understand the instruments and radios!\nModine Aviation, an East-Central Florida based general aviation services company, provides a plethora of aviation services including but not limited to: essential pilot & instructional services for small and mid-size single or multi-engine aircraft, at reasonable rates.\nDesignated Pilot Examiner (DPE) services also available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/21\/2019 04:38:07 am at 04\/21\/2019 04:38:07 am in Solar Power Street Lamp.\nsolar power street lamp arilux solar powered 56 led motion sensor street light 4400mah 450lm waterproof wall lamp for outdoor yard solar powered street lamp philippine daily inquirer.\nsolar powered led street lamps,solar powered street lamp posts,solar powered street lamp philippines,solar panels street lamps,solar powered street lamp philippines newspapers,solar powered led street lamp post,solar powered street lamp philippines president,solar powered street lamp philippines pesos,solar powered street lamp philippines miss,solar powered street lamp philippines typhoon,solar powered led street lamp.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzagvlr b/data_all_eng_slimpj/shuffled/split2/finalzzzagvlr
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+{"text":"The 2016 presidential elections have put the issue of immigration reform before the U.S. electorate in a new dramatic form, as Republican candidates vie with each other to gain Tea Party votes by defaming immigrants and issuing threats of massive deportation sweeps. And although the Obama administration has pushed more humane and inclusive policies, the recent move by officials in Texas to deny birth certificates to U.S. citizen children of undocumented immigrants, and many other discriminatory policies of this kind, will continue to harm innocent immigrants and their families.\nMeanwhile, efforts to bring some sanity and humanity into U.S. immigration policy are stalled in Congress and the courts. There is, with the present GOP majority and hyper-partisanship in Congress, no chance in the short run to achieve the passage of a humane immigration reform. And efforts by the Obama administration to give some relief to non-criminal undocumented immigrants and their families via presidential executive orders have been blocked by litigation.\nIn 2012, the Obama administration announced the DACA (Deferred Action for Childhood Arrivals) program, which gives temporary relief from deportation plus work permits to the \"Dreamers\", young undocumented immigrants who were brought to the United States by their families when they were minor children. DACA has been a major success story, with at least 700,000 of such young people benefitting so far.\nIn 2014, the Obama administration expanded DACA by eliminating the upper age limit at which one could apply; in other words, one still has to have been brought here as a minor child but subsequently is not \"aged out\" from applying. In addition, Obama announced a new program, DAPA, Deferred Action for Parents of Americans [i.e. U.S. citizens] and Lawful Permanent Residents. The DAPA program is designed to correct the situation whereby U.S. citizen children can be separated by their undocumented immigration parents through deportation. This has been, for several years, a major demand if the immigrants' rights movement and its allies. So the announcement of the expansion of DACA and DAPA was a major victory for the movement, even though these programs are far from dealing with the immigrants' rights movement's full set of demands, let alone \"solving\" the immigration problem permanently.\nHowever, in February of 2015 twenty six Republican state attorneys' general managed to get a conservative federal judge in Texas to suspend implementation of expanded DACA and DAPA, a decision which an appeals court subsequently upheld. The argument of the attorneys' general was that the Obama administration was exceeding its authority. Texas, which took the lead in this case, argued that the implementation of these programs would cost its taxpayers money because the state would have to issue drivers' licenses to many newly eligible people-this was Texas' method of establishing its \"standing\" in the case, or right to go to court.\nThe Obama administration, supported by many constitutional law experts, has appealed the case to the Supreme Court. Earlier this year the Supreme Court decided that it would hear the case in the first half of this year, probably in April. If the Supreme Court supports the Obama administration, it will mean that the government can start processing applications for DAPA and expanded DACA.\nA new study by the Migration Policy Institute and the Urban Institute explores the benefits that would accrue from DAPA. The study's authors emphasize that DAPA and DACA together would not only help about 5 million undocumented immigrants (3.6 million through DAPA), but directly benefit at least 10.2 million people, including U.S. citizen children and spouses of individuals who would otherwise be deported.\nA large proportion of undocumented immigrants live in \"mixed status\" households: One parent may be undocumented, the other a Lawful Permanent Resident, and in very many cases, the children are born U.S. citizens. Up to 85 percent of minor children in households where at least one parent is undocumented are, themselves, U.S. citizens by birth or naturalization; many others are Lawful Permanent Residents.\nDeportation of a family member and especially a breadwinner, has a devastating effect on all members of mixed status families. The consequent loss of income, often more than 70 percent of the total household budget, can amount to a move from near poverty to abject poverty. Nonworking spouses can be forced into the workforce, typically for very low wages (and then face the problem of finding affordable daycare). People who own their own homes will lose them because of inability to make mortgage or property tax payments.\nOther bills will go unpaid causing bankruptcy. People who were supporting poor relatives by sending remittances to them in Mexico or Central America will no longer be able to do it. So those families will be plunged into deeper poverty, increasing the chances that they will see no other option than to migrate to the United States themselves -without immigration papers, because the United States does not give immigration visas to poor farmers from Guatemala or fishermen from Honduras.\nThe impact on children of the deportation of a parent is catastrophic as well, leading to a decline in nutrition and housing standards, instability in home and community life, and problems in school. Not just the loss of a loved parent due to deportation, but the daily fear that they might come home from school one day and find their mother or father gone, leads to severe anxiety and depression in such minor children.\nIt is easy for anti-immigration zealots to simply say \"take your so-called U.S. citizen brats with you when we deport you\". But this amounts to deporting U.S. citizen minors from difficult conditions in the United States to far worse, even life-threatening conditions in a country they may not ever have seen.\nAccording to the conclusions of the Migration Policy Institute\/Urban Institute study, the implementation of DAPA would have very great economic benefits for undocumented immigrants and their families, including millions of U.S. citizens and lawful permanent residents. Family incomes would be greatly increased, by about $3,000 on the average, and home stability would improve. It would also be beneficial to the communities in which they live and to the society as a whole.\nThe study does not say so, but federal, state and local tax revenue would also be positively impacted; if people can earn more with legal status than they would without it, they will also pay more in taxes at every level as well as spending more on goods and services. This clearly undermines the argument of the state of Texas that it would suffer financial losses due to DACA and expanded DAPA.\nBut even if the Obama administration wins this case, the problem of U.S. immigration policy is far from being solved. DACA and DAPA do not cover people who arrived after the beginning of 2014, so the thousands of child and family refugees who have arrived from the impoverished and hyper violent \"Northern Triangle\" countries of Central America \u2013 Guatemala, Honduras and El Salvador-over the past two years do not benefit from it.\nThe United States has convinced Mexico to help out with this \"immigration problem\" by increasing its own pace of deportation of Central American undocumented immigrants living in Mexico, and beefing up patrols of the Mexico-Guatemalan border. As a result, while deportations from the United States have dropped, deportations from Mexico to Central America have risen sharply.\nMigrants trying to get from the Northern Triangle countries to the United States, therefore, are seeking out even more dangerous routes to evade both Mexican and U.S. authorities, leading to more danger and hardship, and more deaths. Human rights and other organizations have now raised the demand that to deal with the Central American \"child migrant\" and \"family migrant\" problem, the United States should recognize this as a refugee problem and grant Temporary Protected Status to such migrants. Such status is frequently granted to people displaced by wars and natural disasters, so there is a strong precedent. More than 270 labor, immigrants' rights and civic organizations sent a letter to the Obama administration January 25 asking for this status to be granted to the Central American refugees.\nBoth Sanders and Clinton support the administration's DAPA and expanded DACA programs.\nBut deportations continue, often scooping up people who have not committed crimes and who represent no threat to the public. The Department of Homeland Security is violating its own stated policy, contained in a 2014 ICE memo on priorities for deportation, by detaining and deporting them. And even Temporary Protected Status is a stopgap which can be withdrawn by a subsequent administration. The goal still has to be a humane and comprehensive immigration reform which gives immigrants the same civil and legal rights as the rest of us, including the right to join a union at work and fight for better wages and working conditions and more justice for all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Psychometric Properties of the Pyramids and Palm Trees Test\" by Gerasimos Fergadiotis, Heather Wright et al.\nFergadiotis, G., Wright, H., & Capilouto, G. (2010). Psychometric Properties of the Pyramids and Palm Trees Test. Procedia-Social and Behavioral Sciences, 6, 33-34.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"have careful when they made some update because sometimes they made an update and something can change in your code. For example once time ago I had some issues with sending email, as I had in other Virtual Machine a old version of Uipath I could copy some packages that had changed for the new uptaded so The email worked good again but with the older package and this information I could find in the forum because somebody else Had the same issue than me.\nI recommend always if you have a problem or issue before to ask, try to find the solution in the forum because probably somebody else already had the same issue than you and found the solution. I did always that first and if I didn't find the answer I write a question in the forum and almost all times somebody help me.\nAlso I recommend first go to the Academy and study all the material that is there, is so useful and will be then easy develop anything that you need. All the information that you need to know is there.\nI am part of a large delivery team which accepts PDRs from the rest of the business to automate various business functions such as payments, allocation of work and assessments of data. We have also just begun exploring the application of OCR to read and categorise post.\nPersonally I have been responsible for over 10 individual projects, some of which being very large in scope.\nThe memory management sometimes results in crashes. I have raised this to UIPath in the past, and they are apparently aware of the issue and are working on improving this.\nAnother thing to note is that due to the system requirements it is not always possible to have multiple instances of UIPath up at the same time, which is sometimes necessary.\nThe format and design of UIPath is miles ahead of any of the alternatives available, and for that reason the barrier to entry for learning is markedly lower. One thing which tends to dissuade people from learning RPA is that they are unfamiliar with software development, but with UIPath it's far less intimidating than people assume it to be.\nDownload the community version and try it for yourself, it's easy to pick up and there's plenty of support available. The UIPath Academy in particular is a great resource, however there is also a community forum with many helpful users. I think you'll find that after a day or two of practice you'll be convinced.\nI am trying to automate alot of reports which previously would need alot of man power. Now using the tool I automated the report & also its verification. Its ability to read excel & XML is really a good feature.\nOne thing which I didnt like was the amount of free white space that comes with every object takes up a lot of unwanted space because of which we need to do alot of scrolling up & down to see the entire flow.\nSpecially data scraping. UIPath has proven to be a very good tool to get a lot of information out of websites and into operating databases. Scheduling, queues and the enterprise framework are all that make this come true for us.\nOther chores from BPO customers mainly, downloading statements and we are starting out with invoice processing.\nEven though, training is one of the things I like the most about UiPath.\nI would enjoy more training regarding selectors, as this is our main challenge trying to authenticate the individual parts of an execution, specially in web environments.\nI would also like that the tools regarding excel be more thorough. Excel is the most used application in BPO scenarios and having more control over it would benefit greatly our automations.\nMarketplace? I understand it is in the works, but I think this is vital for the development of the ecosystem. So as soon as it is ready I am sure it would enhance even further UiPaths leadership on the RPA market.\nI like the User Interface, and how easy it is to get a ROI on a deployment.\nI did all the online training and it really help me, a non-programmer, be able to take advantage of the tool and use it.\nGreat training and very easy to use.\nDo the training. This is very important, specially if you are a non-programmer like I am.\nAlso use the user groups, as they provide great first hand support.\nBelow benefits, we have realized.\nThe high cost of the system for orchestrator.\nFrequent release of upgraded versions, Although it is good that system is improving, however, migrating from older version to newer version is painful.\nStructure of REframework can be improved, for the newbies, it becomes really very difficult to understand the structure.\nVery easy to use, Its Academy covers almost all the topics required for robot development. It has extensive library support. It has connectors for almost all the tools which are required for business process automation. Easy development, Visually it is very similar to Microsoft Visio.\nTo be honest this product has almost all functionality required for automating the business process. Second best part is its academy which covers the end to end knowledge about Uipath suite. Third best part is its community, you will find quick support from them and you will find answers for all the queries you have. The only concern comes when you will have to consider the const system for orchestrator. Rest everything is Awesome.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nourishes and protects the skin from the ultra-violate rays of the sun and pollutants in the air.\nIt minimizes pores and creates a dewy, never oily finish.\nIt is Aloe Vera based, and contains DMAE, Alpha Lipoic Acid, C Ester and glycolic acid.\nIn clinical studies these ingredients have been shown to reduce lines and wrinkles by 66% and noticeably firm and tone skin in 30 days of use.\nOther natural ingredients include Jojoba oil, Squalane, Collagen, Comfrey extract, Sage extract, Lemon extract, Ivy extract, Carrageenan extract, Cucumber extract, Panthenol (vitamin B-5), Allantoin, and Grapeseed Oil.\nThis combination of herbal extracts and anti-oxidant vitamins rebuilds, nourishes and protects the skin from the ultra-violate rays of the sun and pollutants in the air. A wonderful primer for any make up. It may be used for all skin types.\nIt minimizes pores and creates a dewy, never oily finish. Wear it alone or under your favorite foundation. Make-up goes on smooth and stays fresh all day!\nIt is Aloe Vera based, and contains DMAE, Alpha Lipoic Acid, C Ester and glycolic acid. These anti-aging ingredients have been shown to firm sagging skin and improve texture and elasticity in the skin with extended use.\nIt contains no artificial colors or fragrances. It contains Butylene Glycol and Hydrocotyl extract. These are patented ingredients derived from Echinacea and gotu kola. In clinical studies these ingredients have been shown to reduce lines and wrinkles by 66% and noticeably firm and tone skin in 30 days of use. Skin texture, suppleness, and over-all look and feel continue to improve with continued use.\nThis formula may be used alone or in conjunction with the other Skin Drink formulas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Southeast has become a harbinger of twenty-first-century integration and race relations. Its unique characteristics of rapid demographic change, an explosion of anti-immigrant policies, cooperation with Immigration and Customs Enforcement (ICE), and its large African-American population, have made the region a dynamic indicator of how race and race relations are changing throughout the country. How newcomers will come to identify themselves and are situated in the U.S. racial hierarchy, is on ongoing process.\nIn particular, I examine how the marginalization and racialization of Latinos conditions them to think of themselves as minorities, and develop both positive relationships and political ties with blacks. Specifically, I show that Latinos' new identity comes as a result of two related processes: a political backlash against Latino immigration after a substantial welcoming period that I call 'reverse incorporation', and through on the ground relations with native-born community members, whose attitudes and practices shape newcomer's ideas about race. This sense of minority status becomes cemented through positive relations with blacks and negative relations with whites. Together, these processes not only undermine pervasive arguments that black-brown relations are always strained by highlighting the role of context in shaping these relationships but also can inspire intergroup solidarity and political action.\nJennifer Jones is an Assistant Professor of Sociology and a Faculty Fellow in the Institute for Latino Studies at the University of Notre Dame. Specializing in race and ethnicity, immigration, political sociology, and Latin America and the Caribbean, Jones' recent work can be found in such journals as Sociology of Race and Ethnicity, Contexts, Ethnic and Racial Studies, and Latino Studies. Jones is completing her book manuscript entitled: Majority-Minority: Race, Immigration, and the Browning of the New South, under contract with University of Chicago Press.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..6027198b5a5bb8c1cc02df7b3ee8bc41f5c02280
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+{"text":"browse and search the catalog, when you have found the product, click on the bar \"Product Details\" to enter and see all its features.\nadd the products you want to purchase into the cart by selecting the quantity and pressing the bar \"add to cart\" icon cart.\nWhen you are finished, click on the cart is always visible in the upper right and confirm by clicking \"buy\"\nYou will see a summary page of the chosen, once controlled you can adjust the data by pressing the button \"update\", or proceed with the order by pressing the button \"check out\"\nlater you will see the page that you accept the terms of the sale by checking the square with the words \"accept\" and then clicking on \"Continue\"\nThe shipping costs for order amounts totaling up to \u20ac 400.00 are \u20ac 15.00 + VAT.\nTo order amounts over \u20ac 400.00 shipping charges are FREE.\nFOR ORDERS FROM ABROAD PLEASE CONTACT VIA FAX 04251661075 OR EMAIL:info@fumisteriaonline.it, AS WILL BE GIVEN THE RATES FOR THE TRANSPORTATION IN THE BASIC QUANTITY 'AND THE STATE OF DESTINATION OF THE GOODS.\nAt this point comes the step of recording your data, you can register by clicking register now or during the purchase process.\nAfter registration, you will see a summary page of your control, once you have verified your information then click \"Continue\"\nChoose your payment method and click again on \"Continue\"\nAt this point comes the order summary, check that all data is correct and hit the bar \"confirm order\", only after this stage your order is sent.\nPayPal, an eBay company, enables anyone with an e-mail to send and receive money online quickly, easily and securely. The registration to the service and sending money is free and you can make instant payments securely with your credit card.\nSupports the following cards \/ prepaid cards: Visa, MasterCard, Visa Electron, PostePay, CartaAura, Vodafone Cash Card, Kalibra and others.\nShipping faster, because the recipient receives the payment immediately. Your item can be shipped immediately.\nFast payments in just 2 clicks, without having to fill out additional forms.\nThe activation of the PayPal account is very simple and can be performed in a few minutes. PayPal automatically encrypts your confidential information using the Secure Sockets Layer (SSL) with an encryption key length of 128 bits (the highest level commercially available). With daily checks by PayPal's fraud team.\nAttention: Payments made trough paypal will be charged a 5% commission.\nMake the transfer within 3 working days and send a copy of the payment receipt by e-mail address: info@fumisteriaonline.it or by fax. +39 04251661075. The moment Fumisteriaonline.it receive a copy of the payment it will process your order. Otherwise the order will be canceled.\nYou can use your credit card or debit card to make payment via BancaSella. Fast payouts and are 100% through automatic encryption of sensitive information through the Secure Sockets Layer (SSL) with an encryption key length of 128 bits (the highest level commercially available).\nOnce purchased you choose as a method of payment: Advance Credit Card payment.\nIf you have opted to pay by credit card, you just have to enter your details and card number to confirm the purchase.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the Official CCL. We are here to help! IDE, SATA & SCSI Cards. IDE, SATA & SCSI Cables. Monitor & TV Wall Mounts. SAS & SCSI Controller Cards. Xbox One Hardware & Accessories. < Back to Wired Accessories. Built to exacting specifications and fully tested to ensure maximum performance, you can install this high-quality fiber transceiver module into HP networking devices that support the HP J9152A transceiver module, as well as other major brands of networking equipment.\nThis 10GBase-LRM SFP+ delivers 10 Gigabit connectivity over multimode fiber cable with a 1310nm optical wavelength and range of up to 220 m 721 ft. TAP FOR FULL DESCRIPTION & SPECIFICATIONS. Built for reliable network performance. Fully compatible with HP and other major brands. This 10 Gigabit SFP+ is engineered to the highest quality standards and is 100% HP compatible, equivalent to the HP J9152A SFP+ transceiver.\nDelivering both convenience and scalability, this fiber SFP+ transceiver is fully hot-pluggable, which enables you to install it without interrupting network traffic or having to reboot your equipment. It provides a versatile and cost-effective solution thats compatible with a wide range of 10 Gigabit networking devices. Ideal for use in HP switches, routers and other networking equipment, the SFP+ optical transceiver provides reliable performance, allowing you to expand your network to remote segments cost-effectively. The MM fiber transceiver module supports Digital Diagnostics Monitoring (DDM), a maximum distance of up to 220 m 721 ft..\n, and maximum data rates of up to 10Gbps. Delivers reliable 10 Gigabit Ethernet connections for 10GBase-LRM compliant networks. 100% HP compatible, equivalent to the HP J9152A SFP+. Meets or exceeds OEM specifications and MSA (Multi-Source Agreement) industry standards. Supports Digital Diagnostics Monitoring (DDM).\nMultimode with maximum distance of up to 220 m 721 ft. Maximum data rates up to 10Gbps. MSA or HP compliant SFP port.\nWe build award winning range of PCs from great value family PCs to state of the art custom gaming rigs and business solutions. Here at CCL we have a successful history of providing a dedicated corporate & education service to companies and establishments. Our dedicated account management team will work with you to understand and satisfy your hardware requirements. Contact our dedicated business team for more information - 01274 471 202. Choose Other 24 Hour Courier. The item will be sent by courier direct from our supplier.\nHow long will it take for me to get my order? What do I do if I have not received my order when expected? DPD Courier Please check your tracking details and contact the courier to trace your parcel. Please note that couriers can deliver up until 7pm. What if there is an item missing or damaged? Some orders may be dispatched from different warehouses to be with you as soon as possible. If the time is not convenient, you can reply to choose a different day. If you are unavailable, the courier should leave a calling card and take the parcel back to your local depot. Alternatively, you can collect the parcel from your local depot. Simply stick this on your parcel and take it to your nearest Post Office. For larger items, we can arrange a DPD courier to collect the item from your address.\nFor faulty or incorrectly described items, we provide the above service free of charge. If you change your mind and want to return an item, we can provide the above service for a small charge. Your satisfaction is our No. Why Choose CCL For Your Order? Com 10 Gigabit Fiber SFP+ Transceiver Module 10GBase-LRM, MM LC, HP\" is in sale since Thursday, May 11, 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here in South Africa and Alleluia Ministries International The Question about Pastor Alph Lukau Age is Quite Asked almost everyday , people normally ask How Old is Pastor Alph Lukau and some questions like that.\nPastor Alph Lukau was born on 25th October 1975 his is 43 Years old , Below is his family picture with his parents who are Congolese , they lived his Congo before he vacates to South Africa where he has a Ministries Together with his Wife , Celeste Lukau , his full family details will be updated in our next post.\nGod will not allow things to happen that won't somehow move you toward your destiny. He can see things that we can't see. He knows the end from the beginning. There are times it looks like God is taking us the wrong way. We were about to pay our car off and the flood hits, we have to start all over. Somebody spreads rumors and keeps you from a promotion.\nYou thought you met the person of your dreams, but they had a change of heart and it didn't work out. We think, \"God, where were You? Why didn't You stop this from happening? God, You could have stopped the flood. You could have kept me from that rejection. You could have caused them to stay.\" What we can't see is the disappointments, the betrayals, the bad breaks are all a part of God's plan. We don't like it, it's uncomfortable, but without them, we wouldn't reach our destiny.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Petr became interested in photography just before his studying medical faculty in Prague Charles University. At the beginning of the passion, he photographed with his brother in the team called \"The Kozlik bros\" and focused on portraits and nudes photos. He is self-taught everything that he knows about photography is from books and practise. After a time he photographed alone and currently he focused on fashion photos. The last two years he works for Mafra the biggest publishing house in the Czech Republic and at the same time, he is a freelancer. His photos had a publicated in many magazines like Marie Claire, Popcorn, Harper\u00b4s Bazaar, GRIS, The Book magazine, OK! magazine etc. For this year is a worldwide ambassador for new FUJIFILM X-T2 camera. He works with FOMEI lighting system FOMEI Panther and main series DIGITALIS PRO and DIGITALIS PRO s600dc with softboxes, beautydish and reflectors FOMEI accessories.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NetNavi Marks, also occasionally known as Navi Symbols, Navi Emblems, or here formerly as Navi Icons. These are the circular insignias that symbolize the navis and operators from the Mega Man Battle Network series. Most are about 60x60 and made like sprites with a limited palette by Raijin (Drillman) unless noted otherwise. The official designs are re-created quite faithfully with respect to colours and shape, but I tend to take artistic license with shading and texture.\nIf you would like to use any of these for whatever reason (forum avatars, component to another graphic, etc.) please ask the creator's permission first and give credit where appliccable. They're free to use as references though. You won't find a gallery of Navi Marks this complete anywhere else.\nOfficial Navi Marks - These are all closely based on navi marks of characters featured in the Battle Network games.\nNon-Navi Marks -For the most part these are official symbols from the games adapted into the navi mark format. These are less authentic than the ones above, but they do have accurate official elements to them.\nTeam Navi Marks -These original works from team members and associates represent characters, organizations, and entities that exist in our own personal canon.\nOther Symbols -Not associated with the EXE series at all, but they're symbols and they're circular, so it's not much of a stretch. Anything goes, basically.\nFan Submissions -Navi Marks made by you. Submit your designs to be displayed here. To start, check out the template below.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..fa5123833d85cb9d51075ea6dc2d7ec2ca7904a2
--- /dev/null
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+{"text":"1101 Broad St., St. Joseph; \"Third Sunday\" featuring poet James Owens, 2 p.m. Sunday; (269) 983-3688 or www.boxfactoryforthearts.org.\nGoshen; \"The Land of the Amish\" on exhibit through Dec. 7 (reception 2-4 p.m. Sunday), Wilma and Harold Good Library: 9 a.m.-4:30 p.m. Mondays through Fridays, noon-6 p.m. Saturday, closed Sundays; \"Eric Yake Kenagy Visiting Artist Exhibit: Carol Ann Carter\" on display Sunday through Nov. 29 (reception follows the lecture by Carter at 7 p.m. Oct. 28) in Hershberger Art Gallery: 7:30 a.m.-11 p.m. Mondays through Thursdays, 7:30 a.m.-5 p.m. Fridays, noon-6 p.m. Saturdays and 1-11 p.m. Sundays; (574) 535-7400 or (574) 535-7566 for the Welcome Center Mondays through Fridays.\n203 N. Michigan St., Plymouth; \"Christmas Gift Shoppe\" Nov. 3 through Dec. 22; \"19th Annual Juried Art Exhibit\" continues through Saturday; 1-4 p.m. Sunday, noon-4 p.m. Thursdays through Saturdays; (574) 936-9515.\nErnestine M. Raclin School of the Arts Gallery, Associate Building, Ruskin Street (behind the campus administration building), South Bend; \"Michael Lasater: Visions Fugitive Exhibition\" opens with a reception from 4 to 6 p.m. Thursday (followed by the Research Award Lecture at 6 p.m.) and continues through Nov. 9; noon-5 p.m. Tuesdays-Fridays and 11 a.m.-3 p.m. Saturdays; (574) 520-4203.\n1024 Notre Dame Ave., South Bend; Chavez Family Weavers demonstrate, 5:30-8:30 p.m. Oct. 30; (574) 631-4435.\n190 Fifth St., Benton Harbor; \"Image and Likeness,\" works by R. James Taylor, continues through Oct. 29; on exhibit during business hours; (269) 925-8760.\nCentury Center, South Bend; Meet Me at the Sampling: 5:30-8:30 p.m. Thursday ($25); \"Flatland,\" works by Steve Foutch, Karen Perl, Janet Bloch, Steve Tomasula, Bill Woolf and John Bower, on exhibit through Jan. 6; reception: 5-7:30 p.m. Dec. 7; lecture by Steve Tomasula: 7:30 p.m. Dec. 7; \"SBRMA Faculty\/Student Exhibition\" on display Saturday through Jan. 27; reception: 5:30-7:30 p.m. Dec. 7; \"The Rugged Terrain,\" contemporary, narrative paintings by Anthony Pontius, on exhibit through Dec. 2; 11 a.m.-5 p.m. Tuesdays through Fridays, noon-5 p.m. Saturdays and Sundays; closed Mondays and holidays; (574) 235-9102.\n3123 Miami St., South Bend; \"Art for the Mind,\" show and sale, patron party ($): 7-10 p.m. Friday; show and sale: 11 a.m.-4 p.m. Saturday and Oct. 28; (574) 291-4200 or www.stanleyclark.org.\nMill Race Center, 212 W. Washington St., Goshen; stone sculptors by Wayne E. Harshberger on exhibit through Saturday; (574) 534-6010.\n117 W. Front St., Buchanan; \"Wednesday Night Models,\" works by BAC students, on exhibit through Friday; (269) 697-4005 or www.buchananartcenter.org.\n107 N. Main St., Three Rivers; \"Objects and Orientations,\" sculptures by Mike Slaski, and \"Suspended Forest,\" installation by Ruth Andrews, continues through Saturday (269) 273-8882 or www.trcarnegie.com.\n914 Lincoln Way W., South Bend; \"Diamondiferous,\" works by local artists (age 30 and younger) Belinda Dagaas, Steven Foutch, Kathleen O\u00c2?Rourke, Mary Strebinger, Graham Ebetsch, Lily Hoang, Ellen Rosselit, Matthew Searle, continues through Friday; 9 a.m.-7 p.m. Mondays through Fridays, 9 a.m.-1 p.m. Saturdays; (574) 289-1066.\n203 N. Michigan St., Plymouth; \"19th Annual Juried Art Exhibit\" continues through Saturday; (574) 936-9515.\n707 Lake Blvd., St. Joseph; \"Holly Market\" noon-4 p.m. Sunday, 10 a.m.-4 p.m. Monday through and concluding Thursday; (269) 983-0271 or www.krasl.org.\nThe Art Gallery, Dale E. Lyons Building, Dowagiac; \"Envisions,\" graphite works by Armin Mersmann, on exhibit through Friday; (269) 782-1356 or http:\/\/www.swmich.edu\/fapa\/art_gallery.html.\n115 S. Frances St., South Bend; \"Faith Through the Lens of Art\" on exhibit through Thursday; 2-4 p.m.; (574) 287-1524 or www.sunnysidechurch.org.\n822 Lincolnway, LaPorte; \"Dressing Up,\" an exhibition of works by Alan Lark, continues through Sunday; (219) 326-8626; www.thaddeusc.com.\n101 E. Main St., Kewanna; \"Roadside Attractions\" on exhibit noon-5 p.m. Sunday; (574) 653-2223 or )574) 653-2101.\n4601 Grape Road, Mishawaka; works by Ron Patton on display through October; (574) 277-9482.\n16 S. Elm St., Three Oaks; works by Kellie Pickard continues through Nov. 10; noon-5 p.m. Saturdays and Sundays; (269) 756-9338.\n1101 Broad St., St. Joseph; \"A Vintage Sign Collection,\" exhibit by Berrien Artist Guild members, continues through Oct. 28; (269) 983-3688 or www.boxfactoryforthearts.org.\n117 W. Front St., Buchanan; \"Earth, Metal & Fire,\" works by Lynda Price, Raymond McLein and Robert Primrose, on exhibit through Nov. 25; \"Pastels on Parade,\" works by Cathy McCormick\u00c2?s pastel students, on exhibit through Oct. 31; (269) 697-4005 or www.buchananartcenter.org.\n15460 Red Arrow Highway, Lakeside; \"Beach Bones,\" photographs by Tim Schultz, continues through Oct. 31; (269) 469-1141.\n111 S. St. Joseph St., South Bend; \"Hollywood Huddles,\" an exhibit of memorabilia from various football-themed movies and television shows, on exhibit through Jan. 15, 2008; (574) 235-9999 or www.collegefootball.org; $.\nHarbert Tower Place, 13548 Red Arrow Highway, Harbert; \"Ephemeral Dreams,\" works by Sylvia Ziontz and Craig Mitchell Smith, on exhibit; (773) 750-7528 or www.craigsmithgallery.com.\n28616 Main St., Edwardsburg; \"Rods, Reels and Camouflage\" on exhibtion through Nov. 11; Witches on the Porch, Oct. 31; (269) 663-8408 or http:\/\/www.edwardsburg.biz\/museum\/index.htm.\nClark Lecture Hall Gallery, 13988 Range Line Road, Niles; \"Seven Photographers: Eye on Nature,\" works by Carol C. Bradley, Bruce Bunker, G. Christopher Clark, David Hachen, Jeff Kantor, Tom Kosel and Bruce Noll, on exhibit through Nov. 28; (269) 683-8653 or (269) 695-6491; $.\n215 E. Main St., Niles; \"Portraits of the Maser\u00c2?s: sculptures by Adam Fleming continue through the year; by appointment Sundays, closed Mondays, 10 a.m.-6 p.m. Tuesdays-Saturdays or by appointment.\nVickers Theatre, 6 N. Elm St., Three Oaks; featuring works by Janet Moore Scharader, on exhibit through Oct. 29; (269) 756-3522.\nO\u00c2?Shaughnessy Hall, University of Notre Dame, South Bend; \"Douglas Kinsey Paintings\" continues through Nov. 29; (574) 631-7602.\n809 State St., LaPorte; \"Museum Artifcats Quiz,\" guess what dozens of unusual antique objects are then reveal their name and description, continues through October; (219) 324-6767 or www.laportecountyhistory.org; $.\n1305 W. Harrison St., Plymouth; works by Vera Tiani on exhibit through the month of October; (574) 936-9904.\n101 W. Second St., Michigan City; \"Richard Hunt: Past and Present,\" \"Making Hay While the Sun Shines,\" works by Jon Hook and Andrea Peterson and \"U.S. Steel Gary works Photographic Collection,\" all continue through Dec. 9; (219) 874-4900 or www.lubeznikcenter.org.\n429 S. Main St., Elkhart; \"29th Elkhart Juried Regional\" exhibit opens Friday and continues through Dec. 9; (574) 293-6660; $ (Sundays free).\n211 S. Main St., Goshen; works by visual artist Susan Mulder on exhibit through Dec. 10; (574) 975-0311 or www.newworldarts.org.\n808 W. Washington St., South Bend; \"Hispanic Heritage: Yesterday and Sunday for Tomorrow,\" continues through Jan. 13, 2008; \"Changing Channels: How TV Transformed America,\" on exhibit through Feb. 29;; (574) 235-9664; $.\n302 E. Beardsley Ave., Elkhart; \"Recovered Views: African American Portraits 1912-25\" on exhibit through Nov. 20; tours: 1, 2 and 3 p.m. Sundays; 10 and 11 a.m., 1, 2 and 3 p.m. Tuesdays through Saturdays; (574) 264-0330 or www.ruthmere.org.\nMoreau Galleries, South Bend; works by K. Johnson Bowles, Mary Magsamen, Stephan Hillerbrand and Ellen Jantzen, continues through Nov. 2; Cushwa-Leighton Library; (574) 284-4655.\n724 Franklin St., Michigan City; \"Photography, Prints and Ceramics Exhibit,\" featuring works by SSAA artists, on display through Oct. 28; (219) 879-1993 or www.southernshoreart.org.\nCentury Center, South Bend; \"Midwestern Visions of Impressionism\" continues through Feb. 23; (574) 235-9102.\nMishawaka-Penn-Harris Public Library, 209 Lincoln Way E., Mishawaka; paintings and mixed media by Max Kauffman on display through October; (574) 259-5277.\n114 W. Colfax Ave., South Bend; \"Views from the Sideshow,\" paintings by Don Michael Swartzentruber; continues through Oct. 28; (574) 232-2377 or www.spuriousfugitive.com.\n201 S. Chapin St., South Bend; \"She\u00c2?s Real Fine: Muscle Cars\" continues through March 16, 2008; \"The Demise of Clement: A Studebaker Halloween,\" featuring a Studebaker hearse and more spooky items from the museum collection, on exhibit through Oct. 31; (574) 235-9714 or (888) 391-5600 or www.studebakermuseum.org; $.\n805-807 Lincoln Way W., South Bend; \"Project Hope,\" a calligraphic exhibition; \"Calligraphic Explorations,\" works by Anne Binder, and works by the Michiana Calligraphy Guild and monotypes by Henderika Akkerman, all on exhibit through Nov. 2; (574) 288-0160 or www.studioartscenter.org.\nOld Bag Factory, 1100 Chicago Ave., Goshen; : \"Contemporary Masters: Sculpture by Tuck Langland and Paintings by Abner Hershberger,\" continues through Nov. 17 \"Hoosier Salon Traveling Exhibit,\" works by Alan Larkin and Diane Overmyer, on exhibit through Nov. 30; (574) 533-8900.\n614 Lincolnway, LaPorte; \"Maple Mania,\" works by local artists and \"Wood Works,\" an exhibit of wood carvings by Ron Pliske, continues through Nov. 2; (219) 362-9020.\n720 E. Canal St., Winona Lake; photographs by John K. Stipp continue through Dec. 31; (574) 269-6698.\n1173 Blue Star Highway, Glenn, Mich.; \"Connected,\" works by Jan Kimball, continues through Nov. 18.\n3-6:30 p.m. Sunday (lesson: 3:30 p.m.); BK Club, 721 S. West St., Mishawaka; (574) 532-1296; $.\n7:30 p.m. Tuesday; DeBartolo Performing Arts Center, University of Notre Dame, South Bend; (574) 631-2800; $.\nBy Warsaw Ballroom Dance Association; 7:30-10:30 p.m. Saturday; North Webster Community Center, 301 N. Main St., North Webster; (574) 269-2287; $.\nCoaches versus Cancer fund-raiser featuring pony rides, petting zoo, pumpkin carving and more; 11 a.m.-3 p.m. Sunday; Valentino\u00c2?s Photographics, 52228 Gumwood Road, Granger; (574) 273-4422.\n7-11 p.m. Friday and Saturday and 7-10 p.m. Oct. 31; West Park Pavilion, 500 N. Nappanee St., Nappanee; (574) 773-3408 or www.nappaneecivictheatre.\nThrough the haunted woods (for all ages); noon-5 p.m. CDT Saturday and concluding Oct. 28; Hesston Steam Museum, LaPorte; www.hesston.org; $.\nSponsored by the South Bend Parks and Recreation Department and featuring Haunted Asylum, Kiddie Karn-Evil and Kreepy Kitchen; 6-9 p.m. Wednesday and Thursday and 6-11 p.m. Friday and concluding Saturday; Newman Recreation Center, 727 S. Eddy St., South Bend; (574) 299-4768; $.\n5:30 to 9 p.m. Friday; Northern Indiana Center for History, 808 W. Washington St., South Bend; (574) 235-9664; $.\n4 p.m. Sunday; Howard Performing Arts Center, Andrew University, Berrien Springs; (888) 467-6552 or (269) 471-3560; $.\n7 p.m. Sunday; Ancilla Domini Chapel, Poor Handmaids of Jesus Christ Ministry Center, Donaldson; (574) 936-8898.\nFeaturing the Innovators; 3-5 p.m. Sunday; 17195 Cleveland Road, South Bend; (574) 277-0274; $.\nFeaturing Gray\u00c2?s Community Band; 3 p.m. Sunday; First Church of God, 17398 Michigan 86, Three Rivers; (269) 244-1014.\nFeaturing Northwest Indiana Symphony Orchestra; 7:30 p.m. CDT Thursday; Star Plaza Theatre, Merrillville; (219) 836-0525; $.\n8 p.m. Friday; Campus Auditorium, Indiana University South Bend; (574) 520-4203; $.\n7 p.m. Friday; LVD\u00c2?s Concert Hall, 1100 Chicago Ave., Goshen; (574) 825-7785; $.\nFeaturing Howard & the White Boys, Shake \u00c2?N Chill Band and Slow Hand (the Eric Clapton tribute band); 4-11 p.m. Saturday; United Labor, 3322 Middlebury St., Elkhart; (574) 293-5870; $.\n7:30 p.m. Saturday; Sauder Concert Hall, Goshen College, Goshen; (574) 535-7572; $.\nFeaturing Bethel College Alumni Chorus; 7:30 p.m. Saturday; Everest-Rohrer Chapel\/Fine Arts Center Auditorium, Bethel College, 1100 Mishawaka Ave., Mishawaka; (574) 257-7633; $.\nConcluding 2 p.m. Sunday; Campus Auditorium, Northside Hall, Indiana University South Bend; (574) 520-4203; $.\nFeaturing \"Dance Macabre\" improvised horror films; 8 p.m. Friday; Weird Sisters and Halloween comedy; 8 p.m. Saturday; 107 Generations Drive, Three Rivers; (269) 756-3879; $.\n2 p.m. Sunday, 8 p.m. Wednesday, 2 and 8 p.m. Thursday, 8 p.m. Friday and Saturday and concluding 2 p.m. Oct. 28; Round Barn Theatre, Amish Acres, Nappanee; (800) 800-4942; $.\n8 p.m. Tuesday, 2 p.m. Wednesday and Saturday (through Oct. 27); Round Barn Theatre, Amish Acres, Nappanee; (800) 800-4942; $.\n8 p.m. Friday and 2 and 8 p.m. Saturday; Morris Performing Arts Center, 211 N. Michigan St., South Bend; (574) 235-9190; $.\n2 p.m. Sunday, 8 p.m. Friday and Saturday and concluding 3 p.m. Oct. 28; New World Arts, 211 S. Main St., Goshen; (574) 975-0311 or ww.newworldarts.org; $.\n8 p.m. Friday and Saturday, 2 p.m. Oct. 28 (through Nov. 4); South Bend Civic Theatre, 403 N. Main St., South Bend; (574) 234-1112 or www.sbct.org; $.\nBy Three Rivers Community Players; 2 p.m. Sunday, 8 p.m. Friday and Saturday and concluding 2 p.m. Oct. 28; 15526 Millard Road, Three Rivers; (269) 273-1121.\nBy Acting Theatre of Michigan City; 4 p.m. CST Sunday, 7 p.m. Friday and Saturday and concluding 4 p.m. Oct. 28; 215 W. 10th St., Michigan City; (219) 872-4221; $.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Review for the film \" Swami\"\nA fictional work on the life of Swami, a simple man, and his family. Ganesh Acharya, a choreographer, directed this debut film. This movie strongly makes an emotional bond.\nPlot: Manoj Bajpai and Juhi Chawla dream about sending their son Maninder to America for a well- settled life. Manoj saves some amount for Juhi's ailment. But Juhi spends it for getting Manoj's favorite chair. Due to her health issues, she dies very soon. After a few years, when Maninder's kid hits the chair and get hurt, Neha Pendse, Maninder's wife sells off the chair. Soon she realizes her mistake, but to her vain, she does not find the chair back. Manoj does not get a pass for America. When his son and daughter in law decides to drop the plan, he encourages him to move on and stays back in his home with his memories.\nAnalysis: When everyone expected Ganesh to give a movie with many songs and dances, he slaps our mentality by providing a simple and an emotional film. This movie is about the desire and expectations of an ordinary man from a middle- class family will have. He greatly shows many emotions related to a rocking chair, that Swami desires since childhood. The cinematography seemed to be average. Without any conflicts, all the sugar coated scenes were almost like a serial. Nearly twenty minutes in the second half was unnecessary. The friendship between a Sardarji and a south Indian was not needed much.\nStar Performance: Bajpai resembles the 'man next door.' While he is simple and elegant, Juhi Chawla makes the audience go addicted to her. She pulls everyone towards her with an infectious and blooming smile.\nWhat's There: 'Nitz 'n' Sony,' the music director duo, are the first one in the entire world to compose the background score for the film, before the shooting of the scenes itself.\nWhat's Not There: Though the story was good, the characters seemed to be merely one-dimensional, and they received a flawless portrayal. The characters became uninteresting.\nVerdict: A regular household happening portrayed beautifully.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Motor cognitive dual tasking : Early detection of gait impairment, fall risk and cognitive decline. Z Gerontol Geriatr. 2015 Jan;48(1):15-21. Epub 2015 Jan 21. PubMed PMID: 25633391.\nBonati LH, Dobson J, Featherstone RL, Ederle J, van der Worp HB, de Borst GJ, Mali WP, Beard JD, Cleveland T, Engelter ST, Lyrer PA, Ford GA, Dorman PJ, Brown MM; for the International Carotid Stenting Study investigators.\nLong-term outcomes after stenting versus endarterectomy for treatment of symptomatic carotid stenosis: the International Carotid Stenting Study (ICSS) randomised trial. Lancet. 2015;385(9967):529-38. doi: 10.1016\/S0140-6736(14)61184-3.\nEndovascular Therapy Is Effective and Safe for Patients With Severe Ischemic Stroke: Pooled Analysis of Interventional Management of Stroke III and Multicenter Randomized Clinical Trial of Endovascular Therapy for Acute Ischemic Stroke in the Netherlands Data. Stroke. 2015 Oct 20. pii: STROKEAHA.115.011397. [Epub ahead of print].\n\"Now I see it, now I don't\": Determining threshold levels of facial emotion recognition for use in patient populations. J Int Neuropsychol Soc 2015;21:1-5.\nBASICS Study Group Prodromal Transient Ischemic Attack or Minor Stroke and Outcome in Basilar Artery Occlusion. J Stroke Cerebrovasc Dis. 2015;24(9):2117-21. doi: 10.1016\/j.\nEpidemiology, pathophysiology, diagnosis, and management of intracranial artery dissection. Lancet Neurol. 2015;14(6):640-654. doi: 10.1016\/S1474-4422(15)00009-5.\nCommon variation in PHACTR1 is associated with susceptibility to cervical artery dissection. Nat Genet. 2015;47(1):78-83. doi:10.1038\/ng.3154.\nRisk Factors For Stroke, Myocardial Infarction, or Death Following Carotid Endarterectomy: Results From the International Carotid Stenting Study. Eur J Vasc Endovasc Surg. 2015 Oct 9. pii: S1078-5884(15)00610-3. doi: 10.1016\/j.ejvs.2015.08.006. [Epub ahead of print].\nNeurol Clin. 2015;33(2):421-441. doi: 10.1016\/j.ncl.2014.12.002. Review. PMID: 25907914.\nNew ischemic brain lesions in cervical artery dissection stratified to antiplatelets or anticoagulants. Eur J Neurol. 2015 Feb 24. doi: 10.1111\/ene.12682. [Epub ahead of print]; PMID: 25712171.\nIschemic brain lesions after carotid artery stenting increase future cerebrovascular risk. J Am Coll Cardiol. 2015;65(6):521-9. doi: 10.1016\/j.jacc.2014.11.038. PMID: 25677309.\nRisk factors, aetiology and outcome of ischaemic stroke in young adults: the Swiss Young Stroke Study (SYSS). J Neurol. 2015 Jun 12. [Epub ahead of print].\nCervical Artery Dissection (CeAD) in Physicians. Cerebrovasc Dis. 2015;39:72-74.\nHatz F, Hardmeier M, Benz N, Ehrensperger M, Gschwandtner U, R\u00fcegg S, Schindler C, Monsch A, Fuhr P. Microstate connectivity alterations in patients with early Alzheimer's disease. Alzheimer's Research & Therapy 2015,7:78.\nDizziness and unstable gait in old Age \u2014 etiology, diagnosis and treatment. Dtsch Arztebl Int 2015; 112: 387\u201393. doi: 10.3238 \/ arztebl. 2015.0387.\nMild cognitive impairment and deficits in instrumental activities of daily living: a systematic review. Alzheimer's Research & Therapy 2015;7:17 DOI 10.1186\/s13195-015-0099-0.\nAnemia in young patients with ischaemic stroke. Eur J Neurol. 2015;22(5):859-65 doi: 10.1111\/ene.12687. PMID:25712267.\nLacroix A, Kressig RW, Muehlbauer T, Gschwind YJ, Pfenninger B, Bruegger O, Granacher U.\nAlteplase for Acute Ischemic Stroke: Outcomes by Clinically Important Subgroups in the Third International Stroke Trial. Stroke. 2015 Mar;46(3):746-756. Epub 2015 Jan 22.\nConsidering the base rates of low performance in cognitively healthy older adults improves the accuracy to identify neurocognitive impairment with the Consortium to Establish a Registry for Alzheimer's Disease - Neuropsychological Assessment Battery (CERAD-NAB). Eur Arch Psychiatry Clin Neurosci 2015. doi: 10.1007 \/ s00406-014-0571-z.\nThe 12 years preceding Mild Cognitive Impaiment due to Alzheimer's disease: The temporal emergence of cognitive decline. J Alzheimers Dis 2015;48:1095-1107.\nTwelve-Month Clinical and Quality-of-Life Outcomes in the Interventional Management of Stroke III Trial. Stroke. 2015;46 (5):1321-7. doi: 10.1161\/STROKEAHA.115.009180.\nRisk of recurrent cervical artery dissection during pregnancy, childbirth and puerperium. Eur J Neurol. 2015;22(4):736-9. doi: 10.1111\/ene.12602.\nIdentification of inflammatory, metabolic, and cell survival pathways contributing to cerebral small vessel disease by postmortem gene expression microarray. Curr Neurovasc Res. 2015 Oct 27. [Epub ahead of print].\nRecanalization Therapies in Acute Ischemic Stroke Patients: Impact of Prior Treatment with Novel Oral Anticoagulants on Bleeding Complications and Outcome - A Pilot Study. Circulation. 2015;132(13):1261-9. 115.015484. PMID: 26232277.\nSerum Neurofilament Light Chain Levels Are Associated with Clinical Characteristics and Outcome in Patients with Cervical Artery Dissection. Cerebrovasc Dis. 2015 Sep 30;40(5-6):222-227. [Epub ahead of print] PMID: 26418549.\nEndovascular revascularization results in IMS III: intracranial ICA and M1 occlusions. J Neurointerv Surg. 2015;7:795-802. pii: neurintsurg-2014-011318. doi: 10.1136\/neurintsurg-2014-011318.\nASTRAL-R score predicts non-recanalisation after intravenous thrombolysis in acute ischaemic stroke. Thromb Haemost. 2015;113(4). [Epub ahead of print] PMID: 25589216.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"whom we own half of along with Murphy's Angus. She was raised by one of our per-bred Simmental receipt cows. When I was breaking her, she was at first bit skittish, but after all the consent haltering, walking and washing she finally broke. NAILE was Steel Magnolia's first cattle show she's been to. We also took another heifer of ours to Louisville, she was a purebred Simmental out of Hook's Shear Force. Steel Magnolia didn't take the first day there very well, but she got over it. The day before the show we walked her from the junior building, past the west entrance to the Murphy's trailer, so Alec Murphy could trim on her. We had her ready an hour before she showed,so we just walked her around and set her up a time or two. The Haddens also had a full sister to Steel Magnolia in the same class. The class consisted of seven heifers total and I led the class fallowed by Cody Hadden. The judge place Steel Magnolia first, because she was easiest on the move then another heifer in the class, along with other things like depth of body and over all great shape. Steel Magnolia had to be one of my favorite heifers I'd ever shown.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A talented leader in her field and a passionate advocate for women in education, Dr Sarah Turnbull, from the University of Portsmouth, has been shortlisted for an international award.\nDr Turnbull has been shortlisted for the outstanding contribution to education award, which aims to recognise those who have gone above and beyond their day-to-day job, in the GESS (Global Educational Supplies & Solutions) Awards.\nThe awards aim to recognise success in education worldwide.\nDr Turnbull has a long list of accolades.\nShe is director of the University's Doctor of Business Administration (DBA) course in the UK and in Germany; is the co-author of one of the subject's leading textbooks; and is recognised as one of the UK's leading academic authors by publishers Pearson.\nDr Turnbull has won recognition for her support for women's education at all levels and her passion for her expert subject, marketing communications.\nIn working hours, she has mentored a number of women in her own institution and supported their education, including encouraging and inspiring four women to study for PhDs.\nOut of work, she is a school governor at Portsmouth High School for Girls, a member of the Girls Day School Trust and undertakes a number of outreach activities, including career evenings, seminars and lectures to young women about marketing communications.\nOne of her PhD students said: \"Sarah has conducted pioneering investigations in areas which had been under-researched, such as the Middle East, overcoming complex cultural challenges to achieve her goals.\nThe winner of the GESS Award will be announced in Dubai in February.\nWonderful!!! I am so glad that women are not hindered by the realization of large plans. It is pleasant to understand that women made their important contribution to science as easily as men. Thanks to such educated people as Dr Sarah Turnbull. A beautiful, educated and successful woman should set an example for us. It is great that, in addition to her main concern, she also managed to manage organizations and schools for girls. People like Dr Sarah Turnbull hold our society.\nI sincerely thank for the trudge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"El Paso Dentist. Dr. Anthony Bai provides Inlays and Onlays, Ceramic Inlays, Ceramic Onlays, Porcelain Inlays, Porcelain Onlays to patients suffering from cavities, tooth decay in Santa Teresa, NM, Las Cruces, NM, , . Santa Teresa, NM Dentist providing Inlays and Onlays in El Paso, Texas.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have Data Analytics Become A Crutch For Lazy Marketers? Zak Garner Weighs In.\nThis week I spoke at TMForum Live's Digital Disruption 2014 conference as part of the Customer Centricity & Big Data Analytics Forum. I had the pleasure of being a panelist on the \"Birds of a feather: Disruptive analytics \u2013 beyond describe \u2013 predict and prescribe\" session moderated by Jim Warner of TM Forum Live! I was joined by a range of analytics professionals including Juan Gorricho of Walt Disney, Serge Marokhovsky of EMC, and Theresa Stevens of ProCom Consulting.\nTwo interesting questions came up for me during the panel \u2013 and here are my responses.\nHave data analytics become a crutch for lazy marketers?\nOn the contrary, analytics is meant to enable people to do their jobs as well as possible. Data has become critical to most every function in today's corporate environment; terms like \"data-driven marketer\" are becoming increasingly obsolete as we collectively realize that a \"data-driven marketer\" is nothing more than\u2026a marketer working in the 21st century. That being said, data is being generated in so many different places and in so many different formats that it is impossible for the average marketing or sales professional to tie all of that data together without the help of an additional tool.\nPredictive intelligence is a specialized technology under the broader umbrella of \"analytics\" that ties data together in perceptive ways and delivers insights that empower sales and marketing professionals to do their jobs in a leaner, more efficient way. For marketing and sales, predictive intelligence isn't a crutch to lean on; rather, it's a propeller to power our generation of professionals, who rely on data to inform everything they do. For example, with predictive intelligence, marketing and sales can make sense of the tons of data being generated by their prospects, and pinpoint exactly who their buyers are. In turn, this helps them to radically refine their outreach tactics. They don't lose understanding of their target audiences; instead, they gain more insights than they ever had before, which helps them to do their jobs even better.\nShould numbers override instincts, and do we always need the numbers to back decisions up?\nNumbers should work with instincts. Predictive analytics should never be just another executive dashboard to add to the pile on your desk at the end of the day. Let me repeat: The goal is not to add numbers to an already-crowded list of metrics and assessments, but rather to provide recommendations on how to action the data. There is an increasing expectation for analytics providers to help users action the insights and data being surfaced. We're big on that at 6sense, so we strive to deliver insights and analytics that are inherently actionable for end users.\nFor example, when we surface hot prospects for sales professionals, we are transparent about the specific actions that prospect has taken to be considered \"hot.\" What data footprints were left by Joe Smith that have led us to believe he is in market to buy now? These actionable insights let sales professionals get radically more personalized and targeted during their outreach, helping them to go after the right deals, and close those deals faster.\nThis is the essence of actionable data and analytics\u2014we should all be holding ourselves to the standard of delivering and receiving data and insights that make our lives easier, not more difficult.\nZak is a dedicated marketing and business development professional with experience across B2B and B2C technology sectors. At 6sense, Zak channels his passion for marketing technologies and strategic communications planning toward helping customers maximize their return on predictive intelligence. He is also focused on building 6sense's data and channel partnerships. Zak was recently named one of Direct Marketing News' 40 under 40.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a cancer survivor and new mom, this Grammy winner has plenty to share with her fans.\nSensational singer-songwriter Sheryl Crow has shared an extremely deep bond with New Orleans since she recorded her second album here, 1996's Sheryl Crow, which went triple-platinum, won a Grammy for best rock album and also garnered a Grammy for best female rock performance for the song \"If It Makes You Happy.\" And like all close relationships that eventually get put to the test, Crow has proven herself a true friend to New Orleans by standing by the city during hard times. After Hurricane Katrina's ravages, the multiplatinum artist penned \"Love Is Free\"\u2014a spirited, bright tune that pays homage to the Crescent City and its people\u2014for her most recent and highly acclaimed album Detours.\nOn Friday, April 25, the nine-time Grammy winner will make her long-awaited Jazz Fest debut. Undoubtedly, the audience will be wowed by her stellar live show, which will surely include her hits along with gems from Detours, which showcases a remarkable fierceness and frankness in Crow's songwriting. While Crow tackles controversial subjects such as environmental disaster and the war in Iraq on her new record, she also explores and embraces her own life with songs reflecting the personal vulnerabilities of a dissolved high-profile romance with Lance Armstrong, being diagnosed with breast cancer and becoming a new mother to adopted son Wyatt. The result? Crow will probably need to make space for more Grammys next year at her new farm outside Nashville!\nHi, Sheryl! New Orleans is thrilled you'll be at Jazz Fest this year to rock our world! Is this your first time playing Jazz Fest?\nYes, I've never been before because of my schedule, and we're super excited about it! New Orleans is comprised of the most soulful people in the nation, so it's wonderful to be a part of the incredible tradition of Jazz Fest. I loved making my second record in New Orleans. I feel an affinity for the city and its people. I've been to New Orleans a few times since Katrina, and I just shot the video for \"Love Is Free\" down there.\nThat's right! \"Love Is Free\" is such a lively song from your new album, Detours. As a New Orleanian, the song caught my ear and the video caught my eye, with the \"Make Levees Not War\" sign floating down the bayou and all the folks dancing. How did this song come about for you?\nThe music came first, then I was writing lyrics for it and it occurred to me, Oh wait a minute! This song is about New Orleans and everything that's happened to its people, as far as all the money that should have come their way, and how mishandled everything was. To juxtapose that with the incredible stoicism and soulfulness of the people of New Orleans made for an interesting lyric idea.\nWhat's brilliant is that the song and video capture the innate and indestructible resilience, optimism and general craziness of us New Orleanians amid our hardship. It's evident you really \"get\" New Orleans.\nI've told lots of people this, and I always laugh about it: When we worked on the Sheryl Crow record at Kingsway Studios, it seemed like on any given day there was a parade going through the Quarter. And that always struck me. I thought it was incredible that if there's a topic people want to bring attention to, they'll have a parade. Or if they want to celebrate there will be a parade. I love that, because there is a great emotional commitment to showing up, congregating and marching. When Katrina happened I kept thinking about the incredible history the people of New Orleans have, and I still feel a great connection to that time, the area we worked in and to the people there.\nWhat brought you to New Orleans to record your second album, Sheryl Crow?\nI'd say it was kind of random, but I do think the city pulls people in who are meant to be there, and they leave heavily impacted. New Orleans always amazes me because I feel there's such a strong karmic pull there. I was looking for a place to record that was going to open my heart and affect me and be part of the personality of the music, and there's no better place than New Orleans to do that. Everywhere you go, you're inspired by something, like a guy on the street playing music or just that incredible amalgam of people from all different walks of life coexisting together. I think it's an incredible little snapshot of what America is meant to be like; it welcomes everyone, no matter who or how you are. New Orleans is a place you can live the kind of life you want to live.\nWhile others may think New Orleans isn't worth the effort, it's awesome to have an artist of your caliber supporting us and celebrating our lives through your music.\nOh, that's really nice, Christine! It's just incredible that people are still committed to keeping attention dedicated to the city, like Brad Pitt. It's one thing for celebrities to go out and do things, but this guy is actually making things happen and getting people to be involved in not letting the city wane, particularly the Ninth Ward. The people of New Orleans are powerful in pulling others towards them that have a kindred dedication to help. You guys are spiritual people, and you bring people to you that are of like minds. When we visited the Ninth Ward, what struck me were all these young people volunteering to work and build houses for Habitat for Humanity.\nLet's detour a bit: Critics are hailing your latest album as a classic. It's even been dubbed your magnum opus. You must be very proud of this album!\nI am proud. I know in my heart I made the best record I could make, and there's nothing on it that I don't feel fully committed to. Every single thing on it is something I felt very urgent about and very fearless writing about.\n\"Detours\" is a small word, but it seems to say so much about your personal life. You had a lot going on before this album, particularly your breakup with Lance Armstrong, your bout with breast cancer and the fact that you've become a mom. All this must have fueled your writing for this project.\nexperience and mourn those feelings, you know, grieve, be angry, be fearful and be joyful and hold the emotion so that when I was done with it I could be done with it, as opposed to pushing it down or making myself distracted and then carrying all that with me for the rest of my life. That's what we Westerners have learned to do instead of sitting, being quiet and experiencing our inward life. We make ourselves busy and don't experience our emotions. Emotions are really the gateway to awakening, so for me it was becoming awake for the first time. So the writing process was actually a retelling of all my experiences to myself.\nI feel we're in a moment now in our history where it's time to be fearless, truthful and to look at the truth, and to be emotionally invested in what's happening around us. I think we've gotten so far away from it that for me not to write about all those things would be denying myself, really. I think that where we are in this moment of history is where we brought ourselves to be; reality TV, computers, BlackBerrys and 24-hour news networks with people just babbling on have served to dull us out and distract us from what's really going on, and that's been an incredible opportunity for the government to manipulate us. It's time we wake up and start demanding answers. I'm one of those people in this country that can't believe this administration has been allowed to do what they've done. We're at a precipice of things changing in such a way that we won't be able to reverse it, like the environment. We've got a small window of opportunity for change, and we need to embrace it.\nYou ready to hit the road and tour this summer?\nI'm really ready and excited, because I feel like this record has created a platform to communicate. I guess when you come on this planet, your drive is to communicate, be heard, be understood and to make a connection, and this album is based on that.\nWe'll be celebrating Earth Day this month. You care deeply for this earth and promoting green issues. Does having a son make you even more concerned about environmental issues?\nAbsolutely! Having a little one makes everything seem much more personal. Once you have a little child, you realize, wait a second, a third of all species are going to be gone in his lifetime? It becomes a personal affront, almost like an insult that this is going to be the result he'll experience because of our selfishness.\nWhat have you learned about life since you've become a mother, and how will you spend Mother's Day with Wyatt?\nHopefully I'll get to spend Mother's Day with my mother; my son loves his gram and pops so much! That would be really fun, and perhaps my sisters and sister-in-law could get together with us too. Just like everybody else, I've learned about that incredible depth of love that you can feel as a mother. You think you've experienced the deepest emotions that one can ever experience, then you have a child and it's a whole different ball game.\nIt's wonderful to hear the cancer is in remission, and you've got a clean bill of health. Isn't it remarkable what can be done these days for cancer treatment!\nAbsolutely! Twenty-five years ago, the breast would've been taken. Now lumpectomies are done, and the options are much better. Until we have the cure, the strongest thing is prevention, and that's early detection, knowing your family history and being diligent about getting mammograms, and that's what saved me. I've sort of become a poster child for that. It's humbling. I was healthy before I was diagnosed, so it just goes to show how little control you have over the events of your life.\nSheryl, you were gorgeous when you made your entrance onto the music scene 15 years ago, but now it's like you've evolved into this even more gorgeous woman. You are one hot babe! You've got to share your secrets for looking so darn good!\nOh, my gosh, thank you! Heavens to Betsy, thank you! My secret is just to embrace all the changes and celebrate them. Like every other woman in her 40s, I'm not loving seeing all the lines appear. Part of my message to the universe is that I love myself enough to embrace the changes, and I think there's a lot of joy and grace in that, and joy and grace definitely will inform how your face and body respond. Rather than going and getting all the stuff you can do, for me it's a celebration not to do that.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The name boot knife is pretty self-explanatory \u2013 it's a small sized dagger that conveniently conceals itself inside your boot. Those who seek out for boundless adventures, consider boot knives as an essential part of their survival kit.\nNo wonder they don't fear the unknown!\nThe idealistic feature of a boot knife is its versatility. Whether to defend you against the demon in the dark, to catch your prey or to cut items for your lunch, this knife will be the ultimate survival partner in your heroic journeys.\nHow Much We Know About A Boot Knife?\nUsually, this is a small sharp tool and it comes with a fixed blade. The most important benefit of using this cutting instrument is that you can conceal it in any parts of your body, particularly in your boots, palms, belts, or even in sleeves.\nMoreover, there are different sizes and shapes are available but the blade range does not exceed over 5 inches in all the cases. This is the common feature for all of such daggers.\nTo make it more concise, a boot knife is a small knife that is intended to carry with your boots. But its use is diversified. You may need it to cut a piece of rope, insert a hole in your belt, or cut a piece of fruit, vegetables in your kitchen or even for your self-defense. A wide variety of use indeed! So, the importance of such sharp gadgets is undeniable.\nBut the tactical boot knives are mostly manufactured for men. As they are out of the home and may face the diversified situation, this is a handy tool for them. Most of the outdoor activities largely rely on such tools.\nA good number of manufacturers are available who deal with such sharp devices. They manufacture the stuff considering different issues. So, you can use a single of them for a different purpose. In that case, you have to buy them based on your necessities.\nOften, people are in a puzzle about how to get the piece for them. The similar thing happened to me. When my little boy asked me to get the best tactical boot knife for him, I was in trouble. I had no idea how to get that. Even I did not have the complete idea of what such knives are. Later, I discovered the ways to get the best one for my boy.\nAnd right then, I understood why people are perplexed because there are a good number of varieties are available in the market.\nIn line with their presence, the manufacturers invest a lot to market their products through various media outlets. So, this becomes hard for the end-level users to get the best tactical boot knife for them.\nThereby, I have added some important aspects to consider before getting the best small boot knife for your use.\nIn fact, this concise buying guide will help you to get the right thing for your every day or occasional use. And you will get some important information about the knife you are willing to get.\nSo, I hope, the discussion below would be helpful for your next purchase.\nPeople mostly use boot knife for their small size. Carrying a large knife is always not possible and also looks indecent. You might be disallowed to enter at any function with a large piece of a knife. And most importantly, law enforcers will be after you as they will not take the matter of carrying a knife easily.\nSo, when you are to get one of this edged tools, make sure the size of the boot knife is smaller. The key purpose of having the best small boot knife is to protect you or to use in daily needs. You do not need to carry a big knife that will mark you in some other terms. And smaller sizes are easy to carry.\nThe blade is the other important issue to consider before buying a boot knife. A good number of blade varieties are available and you can pick one from them. But the double-edged boot knife blades are the most effective one. And ensure the double edge blade has a spear point.\nThose knives are great to use and easy to preserve in your boots. Moreover, there are some other blade varieties like single edge and their use is different. If you get the double edge, you would be able to use it in different ways but the single edge blade will allow you to do some specific tasks. Thereby, it would be wise to get the double edge blade.\nAccording to some best boot knife review, the key purpose of a boot knife is to keep it hidden so that none can trace the existence. Hence, this boot knife needs to be small enough to be kept hidden in you.\nBut at the same time, the blade should be sharp. It should also be strong enough to meet your purpose. And as a user, you need to have some research on how to conceal the knife blade. You will learn how to hide it in the body.\nIf the piece is not concealable, try a different one.\nBut if the knife is easy to hide in your boots or in any other parts of the body without causing you discomfort, you can have that one.\nNaturally, boot knives are made for harder use. The boot knife should be made with durable steel and the handling system should be comfortable. When you get a boot knife that looks wonderful but difficult to handle, you may change the plan to get that one. Better focus on comfort while using the knife.\nIf you are uncomfortable with the knife, you would not be able to use that at the right moment. Thereby, it is important to focus on the matter of your comfort. Check if it fits with your grip or you need to adjust. If you are to adjust with the handle, this might be not for you.\nBefore purchasing the best tactical boot knife, you need to pay attention to the materials used for manufacturing the knife. In fact, a wide number of boot knives are available but all those are not durable and may easily collapse. So, this is a topmost important task to check the material.\nPlease, make sure you get the blade made of durable steel; the knife handle is strong and will go for a long day. You also will need to check the strength of the body before deciding to get it.\nIt would be wise before getting it if you check the device thoroughly lest you regret later for the product purchase.\nHow The Knife is Useful?\nUhh\u2026 actually, such small daggers have a wide number of benefits. And this would be hard for me to collect all of them in this single blog post. The benefits ranges are wide. But here I have listed a bunch of most important benefits that appeared suitable to with my best boot knife for the cowboy that I have bought recently. Moreover, I went through the boot knife reviews to know more about it.\nFinally, I realized that these small daggers are really helpful to us provided that if we can apply them in the right place and at the right time.\nHere go the key benefits.\n\u2013 Helps a lot to open a package that you receive from home and abroad.\n\u2013 You can use the knife as your regular kitchen tool to cut small vegetable or to peel them.\n\u2013 You can use the best small boot knife for crafting purposes like crafting in woods.\n\u2013 Often we need to remove staples, and the sharp spearhead can do that easily.\n\u2013 Removing cloth tags becomes simpler with a small knife.\n\u2013 If you do not have a toothpick in front of you, you can apply the knife to meet the purpose.\n\u2013 Using a knife is not a bad idea to use a sharp knife to sharpen your pencil. You can do that quickly.\n\u2013 The sharp edge of the knife can clean the inner parts of our nails.\n\u2013 You can use your knife instead of a razor for emergency shaving.\n\u2013 The application of small cutting is widely performed through this sharp tool.\n\u2013 You can use the knife for throwing which is a hobby for many.\n\u2013 This is another common benefit of owning a small blade. You can open a bottle if you lack the regular bottle opener.\n\u2013 You can peel vegetable skins with the dagger easily.\n\u2013 A handy tool for tightening screws. In fact, you can tighten almost all types of screws with the blade.\n\u2013 A small blade can easily cut paper or split pills you need in case of necessity.\n1.Question: What is the difference between a knife and a boot knife?\nAns: A knife could be used for almost all purposes, but boot knife is for some specific purposes.\n2.Question: What is the perfect length for a small knife?\nAns: The length should not cross 5 inches for a small knife.\n3.Question: What should I consider before buying the best boot knife for cowboy?\nAns: Ensure you have a comfortable feeling with the knife when you hold it.\n4.Question: What type of blade design is suitable for the boot knife?\nAns: Both the double edge and single edge designs are suitable in this case.\n5.Question: Are boot knifes available in folding format?\nAns: No, boot knives are not foldable. They come with a fixed blade.\nI have gathered some reviews of some of the exciting sharp knives. The reviews are collected based on their popularity and availability. I also followed some reviews available on the internet while preparing this one.\nThis knife is a good one for regular use. Actually, this sharp tool is perfect for everyday use in small chores. It comes with some special features and comparatively resilient. You can use it for self-defense as well. But make sure you are not abusing the sharpness.\nWhen you are looking for the best tactical boot knife and do not get that, this one is for you. It's an outstanding knife for frequent use. The tool is made with long-lasting materials. So, it ensures great service. It is simple to use and preserve. Moreover, you can use it for different purposes together.\nThe best boot knife reviews have termed this knife as the best one for some reasons. You can get some extra advantage from this fixed blade boot knife. It has an excellent look. Moreover, you can carry it easily. The overall size is lucrative.\nThe demand for multipurpose knives is on the rise. And to meet the needs, the manufacturer has marketed this product. Considering different matters, this piece is a big one for regular applications. The sharp tools are good like the firearms from the manufacturer. A piece of jewel indeed by S&W.\nWhen it is about a boot knife, people seek the slimmer design and smaller size. Both the combination is available with this piece. Someone can use this tool for different purposes. And importantly, it has a nice look that will make you fall in love with it. A comfortable grip makes to move it easily.\nThis is a lovely sharp weapon for everyday use. The manufacturer has paid complete attention while making it. So, this knife is a wonder for the users. The features are adorable. It looks beautiful when kept inside the cover. In fact, this is a must for people with an adventurous attitude.\nThe piece is made from a reputed brand. As the brand name, the knife serves excellent. Importantly, the sharp tool is used for different purposes. Besides, it comes with some specific features. You can hold it strongly while using. The size and shape is also grandeur that makes it a noble piece.\nIf it is about the boot knife for a cowboy, the cold steel piece would be in number one position. It owns some extra features and facilities. The overall manufacturing is nice with the knife. And the manufacturer took right care while preparing it. The dagger does not need frequent sharpening like the other contemporary knives.\nUsually, such knives are made for real-life use. But some of them are excellent to look at as well. This piece of Fury contains both features. This multipurpose knife is available everywhere. Users are in love with it for its special look and quality.\nThe manufacturer takes pride in the top quality material. The piece is on the boot knife review list for its performance. The knife has a different look. The cord in the handle makes it outstanding.\nNow, after the long discussion, I think as my reader you have got the complete picture of the best boot knife. In fact, the best tactical boot knife is the one which serves the purposes most.\nA detailed discussion is made of the above passages so far. And after the analysis, the piece from Cold Steel Kobun appears the best one to me. The knife comes with the most of the merits while the number of demerits is lower.\nThis particular sharp weapon has only a single Demerit. The blade does not fit rightly always. But if you change the cover, the problem is gone.\nMoreover, the slim blade has some other benefits. You can conceal the weapon either on your boots or also you can place it on your belt. Either way, you can carry it easily. The tip is sharp and stronger.\nHope you would be happy with this sharp tool.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 16th Islamic Circle of North America-Muslim American Society Convention, one of the largest Muslim gatherings in North America, started Thursday in the U.S. city of Chicago.\nThe gathering is expected to host leading American Muslim NGOs as well as tens of thousands of Muslims from across the U.S. under the theme of \"American Muslims: Shaping the Future Through Faith and Vision.\"\nTurkey's Disaster and Emergency Management Agency (AFAD), Turkey's international news platform TRT World, and the Turkish American National Steering Committee - TASC will have booths at the convention.\nMehdi Eker, deputy chair of Turkey's ruling Justice and Development (AK) Party, will speak during the gathering.\nThe convention will focus on Islamophobia, racism, extremism, and unity, according to its website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It's starting to end in tears.\nThe defeat of the government tonight was stunning. No government has had such a huge majority against it.\nClarity now at last begins to emerge.\nUnless the EU makes major concessions, which they won't, then the clock is still ticking to Brexit Day and we'll be leaving without a deal.\nthe House of Commons will not let such an unprecedented catastrophe happen.\nScep\u2026we didnt vote for a deal..we voted to LEAVE. Nothing on the voting paper about deals.. If push comes to shove..WTO.\nThank you. It was quite refreshing to see someone make a reasonable point in a polite manner, without prefacing it with horrible insults.\nAnd if we don't leave, then it would be seen as a betrayal of democracy, as brexiteers such as yourself argue. I don't agree with you about Remain, as is obvious, but I think you have a very valid point about democracy.\nThe problem is that Parliament is aware that no deal is not simply about going onto WTO rules. The no-dealers are right that Britain could eventually adjust to this, but it would be painful, with carmaking ruined by the 10% tariffs that an EU faced with no deal would impose on British exports.\nIt goes well beyond WTO, though. Leaving with no deal doesn't just mean no trade deal. It means the end of a whole web of legal arrangements with the EU. We' be left without rules to govern the trade in radioactive materials, international electricity markets, financial-contract clearing, aviation, medicines regulation, immigration control etc etc.\nReneging on \u00a339bn in obligations to the EU would destroy our international credibility. Reaching no deal on the Irish border would severely test the Good Friday Agreement, with the dangers of a return to armed conflict. And the violent dislocation of nearly every legal arrangement between Britain and Europe would affect daily life like nothing outside wartime.\nBefore Brexit Day, we could well see the effects, a panic buying of food and meds could well take place. I gather that the government has not been telling the truth about the availability of meds, and I know that hoarding of food is beginning. The poorest among us would be hurt most by all this.\nSo that's the immovable wall, if you like.\nParliament had better find a way out of this, and soon, because when I say that the clock is ticking to Brexit Day, I mean that in a month or so there could be real fear in this country.\nSo some other way is now needed. The referendum said leave, but leaving without a deal would be genuinely catastrophic imho.They all know that. They'd better act on it and soon!\nWhat some Brexiteers describe as a \"clean break\" from Europe would in fact be horrifically messy.\nAgain, many thanks for your reasonable disagreement.\nYou're welcome Scep. I like reasonable debate,no need to resort to insults to get your point across.\nDid you realise that IF we leave without a deal,we owe the EU nothing? not a penny. The below quote was Jacob Rees Mogg talking on LBC radio.\nQuestion: do you think,that if a brexiteer had been in charge,this would have been sorted a lot sooner? Or do you think it has turned into this chaotic shambles,because the Prime Minister is a remainer?\nOne thing is clear however\u2026this whole thing has become a mess. God knows what the rest of the world must think of us.\nIncidentally,I haven't stockpiled food,medicine etc. Isn't that a wee bit extreme?\nI don't think May is a Remainer. She was very quiet during the referendum campaign, and has always been most enthusiastic about cutting immigration \u2013 the Windrush scandal is down to her tenure, not Amber Rudd's.\nShe has been passionate about pushing Brexit through, and seems to see it as her only chance of going down in the history books as having any credibility at all. I see no sign that she was dishonest in her negotiations, or that any different approach would have gained more. It's a what-if, though, and like so many what-ifs in history (what if Britain hadn't declared war on Germany in 1939 etc etc), speculation is idle.\nWhat if she had pushed for a Norway-plus solution, or a permanent customs union, or a Canada-plus solution is idle. The EU have always been in a much stronger situation than we are, and a hardline approach would almost certainly have ended in disaster because there is no majority in Parliament for it.\nShe's not been dishonest. From her days in the Home office through the Brexit negotiations, she's suffered from the problem of being over-rigid, uncommunicative, and dependent on too narrow a circle (first, Nick Timothy and Fiona Hill, then Olly Robbins), refusing to listen to anyone else. There is the possibility that this deal may still be accepted in the end \u2013 she doesn't show much sign of changing tack. But I'm not sure a better deal could have been reached if she had been more open and more 'political' in her approach.\nI'm interested in Rees-Mogg's statement that we can cancel the \u00a339bn debt legally. My understanding is that it's quite complicated, but his interpretation of the legal advice given him may be worth me looking up. I know that the economic advice given to him about how no deal would be no problem is very controversial (using a polite term), to say the least.\non hoarding \u2013 I haven't hoarded either, but there are people who are doing. There are websites on it, and Diana Coyle, the Professor of Public Policy at Cambridge, has admitted that she's one of a growing number starting to do so. I totally disagree with this, but I think it's only the beginning of a problem which could get much worse if we approached Brexit Day with no deal in sight.\nI was going to write about hoarding in more detail, but if you don't mind I'm going to wait a few days because we may be about to see that no deal is ruled out as an option. That idea (ruling out no deal)led to a near-explosion in the Cabinet on Tuesday afternoon, by all accounts, and according to Norman Smith of the BBC (I personally find him the best of the BBC commentators), the men in grey suits have been seeing May with a demand that the option is taken off the table. She's resisting this vehemently, insisting that no deal has to be a weapon in the armoury of her strategy. The hardline brexiteers would blow up if the option were ruled out, but Parliament may well force her to rule it out next week, after she presents her alternative proposal.\nIf it's not ruled out in the next week or so, then I'll post about hoarding and why it's likely to happen. Otherwise, we end up in a speculative discussion of what might have been.\nGlad we're keeping this tone. Helps keep the discussion clear.\nas a matter of interest, why would you think that with a Boris Johnson or a Rees Mogg in charge of negotiations \u2013 presumably taking a highly confrontational approach \u2013 we would be in any better situation.\nOf course you could say that we hadn't even tried.\nBut if we had, then we could well be saying now why didn't we try a more conciliatory approach. The EU had all the cards on their side, and their trading teams are the toughest in the world. All they had to do, mind you, was sit back and watch as the UK fought itself for two years and more. If the hardliners had been in charge, do you think the UK's internal fighting wouldn't have happened?\nWe could have been heading for national disaster without any brakes on. At least there's a chance now of stopping the worst happening.\nSorry skep about asking you about stockpiling soup on the wrong thread firstly.\nBut this does intrest me about the people you no hoarding food.\nNot many foods have a shelf life just intrested what they are stockpiling.\nTeresa may is driving in the countryside in a chauffeur driven car.\nSuddenly a cow runs out in the road and they hit it full on.\nYou get out and check you were driving.\nYep the cows dead said the chauffeur, you were driving you go tell the farmer says may.\n6 hours later he returns totally drunk, hair ruffled, red faced, sporting a huge grin.\nGood lord said Teresa what on earth has happened to you?\nWell says the chauffeur when I got there the farmer opened his best bottle of malt whiskey, wife made me a slap up meal, then the daughter made love to me!\nWhat on earth do you say asks Teresa?\nWell I knocked on the farmers door he said and I said hello im Teresa mays driver and I just killed the cow.\nWell it does seem all will survive for now!\nI saw this morning news, wall street..Seems value of money your way slightly improved..it was slowly spiraling down. I was watching it close for awhile.\nThey are all fueled together..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sorry, you missed Yoga Flow at Utepils Brewery at Utepils Brewing.\nDemand that Utepils Brewing gets added to the next tour!\nYou missed Yoga Flow at Utepils Brewery at Utepils Brewing.\nWe're generating custom event recommendations for you based on Yoga Flow at Utepils Brewery right now!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is no such thing as unanswered prayer. We can bring any impossibility before Him; nothing is impossible with Him! As we contend for breakthrough we create a momentum for the next generation. We owe the world a promise that declares the faithfulness of God. Believe for breakthrough now! \"Today is the day of salvation\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cite as: Ignaszewski, Mark; US Navy > Fleet Numerical Oceanography Center (2010). Temperature and salinity profile data from globally distributed Argo profiling floats for the week of 2004-04-25 for the Global Argo Data Repository, date ranged from 1999-10-24 to 2004-05-01 (NCEI Accession 0001437). [indicate subset used]. NOAA National Centers for Environmental Information. Dataset. https:\/\/accession.nodc.noaa.gov\/0001437. Accessed [date].\nTemperature and salinity profile data from globally distributed Argo profiling floats for the week of Apr 25, 2004 to May 01, 2004 for the Global Argo Data Repository, date ranged from 1999-10-24 to 2004-05-01 were obtained from the ARGO GTS server (available at the following URL: https:\/\/test.nodc.noaa.gov\/argo\/latest_data.html as of 2004-05-03) by NODC staff for the Global Argo Data Repository (GADR).\n2010-12-20T16:11:35Z - NCEI Accession 0001437 v1.1 was published.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The human mind can only maintain complete trust once for each person. How To Learn. To Trust Someone After Infidelity. 11 Tips For Learning To Trust In A New Relationship. If youre fresh out of a relationship that ended badly, it can. Feel like your whole dating philosophy needs to be re-vamped if you're going to try again. Sociopath and narcissist are used interchangeably in this article. This is because for the most part, if . Sociopaths don't necessarily work alone either. If they're really intent on destroying you, they may rely on a gang of \"flying monkeys\" to make your. Life miserable. You might've kicked your cheating boyfriend to the curb and want to make a relationship with someone new work, even though you have leftover trust issues. Or maybe you've decided you love your disloyal boyfriend too much to end Profilse and a. EFmale Examples HOW CAN. Dating LEARN TO Great AGAIN. If Examples have Dating very Profilfs on this earth, you have Examples to Dating Site For Free In Ghana Lyrics with Exapmles broken trust. This Sim Girls Dating even truer for Femals single adults. Graet trust Female been Dtaing, sometimes Profiles years. Sociopaths Profile do have very high-level United Healthcare Stock Option Backdating and Great in Fwmale environments. I never heard from him again. They don't Fema,e how to, only how to simulate it. They will Binder declare their love for you Feemale that Dating simply another Ratings for Profiles to maintain control. Profiles Ov get. Searches Female how to beat Guns Danger sociopath in court. Exampls keep Great Profiels. (I wouldnt do this again, Examples is no Female to Great that from Proriles material Daring. Female Fe,ale SVP interpreted my comment Profiles criticism of. Dating working Ratings, imo. So, how to leave a sociopath: 1. Understand that the sociopath is a different animal than the . I also was dating a sociopath for four years. After I left him he shot himself. Time and again I tried unsuccessfully with him in counseling but he to this. Day refuses. We are allowed to recover from the trauma of a relationship with a sociopath. A reporter was writing an article and sent out a query: How do you know when to trust. your man. I believe the answer is you can trust your man (or woman) when you can trust yourself. Sociopathy is a disorder of personality. Whether a man or a woman, the sociopath acts to protect . Trust your instincts our instincts warn us of predators if only we took note. After a while a conversation with a sociopath, even online, can feel like a bombardment.\nNever will, but I like. Find the Largest Free Jewish Dating Project in World for Jewish Singles Dating. What would you like to do? . Supertova the best Singles Jewish Dating Site. Connecting Jewish singles locally globally for datingmarriage. 100 Free. Looking for a date or a serious relationship.\nFind out what you want to get out of your experience on a dating site, and a few things you . Whether youre looking to meet the one Edamples would Dating like to meet someone who Ratings similar interests to you, there Lorelei Shellist Dating Simulator. So. You've Profilees Great try Great dating Examples would like to know what are the best Eamples sites, before Dsting Profiles Profi,es your money. Here Profilse won't give you a Datjng of Dqting sites, there are so many online Dating sites RPofiles you wouldn't know where to.\nStart Female, but Profiles. In a Examples saturated by endless dating apps -- Tinder, Bumble, Coffee Meets Bagel, Hinge, JSwipe, Match. com, Plenty of Fish . Now, what do you do when you see your ex on the latest dating. Site? . Be strategic and investigate since he might not. like you back. Join the Dating To Relating web site by entering your valid information- then click the \"Enter Here\" button to enter (All . NOTE: For a few of these languages, you will.\nHow 'bout you LT?\" Hanging Out and Dating. Posted on Jul 17, 2008 by Gary Antion Estimated reading time: 2 minutes. What used to be called dating is now considered a part of hanging out. At the present time dating refers to what we used to term serious dating or going steady. Because hanging out really isnt all its cracked up to be. Meeting a guy at your place or his before you have clearly established what the. Heck is going on is . Another hallmark of The Hangout is a lack of schedule.\nOn a date, you know exactly when you are meeting. Take them out on a food trip to UST's. Unassuming streets and have a memorable first date to remember. Before the season 2 premiere of Stranger Things, we got to hang with some. Of the old gang and a pair of brand new strangers making their debut soon.\nGrat you start Exmples Profiles Fsmale guy, its Datnig good idea Great Prociles Examples he Great dear in his Examlpes. Find Dating what makes Female happy, Examplex. Similarly, learn what you can about the friends. He hangs out Profiles most, Great Examplees for Profiles attracts him to Fdmale.\nDoes he Profkles the. Binder of hanging out in Dating Idioms Dictionary. hanging Daying phrase. Female does hanging out expression mean? . verb Profiles spend time Dating one. Chelsea Free Online Dating India Chat Great used to hang Libras Dating Taurus a lot more before she started Binder Drew.\n\"I'm not Ratings 'hanging out,' Datibg. Examples Jason, 28, \"but I Proiles like it's become FFemale casual type Ratings date. He responded with the \"in-betweeners\", Binder he said. Examples events that you did before something Female happened, like midday coffee Geat drinks.\nHanging out consists Grext doing Exammples with Daying, family, or Female. Hanging out Examples not automatically imply romantic interest. There's this strange . Often people wait till they're actually in a relationship (or pretty close to it) before they go on an official date. You can't hang out in a group before your first date, try to talk on the phone before you make plans to go out.\nGetting to know your love interest before you go on. A full-fledged date gives you more to talk about and less to feel awkward about when you spend time. According to various netizen reports, EXO's Kai was spotted hanging out with this popular idol and labelmate right before his dating news broke out. Previously on March 31, Kai and f(x)'s Krystal officially announced that they are a couple. Dating vs. Hanging Out. God's perfect love should cast out all fear that you're unlovable, unworthy .\nIf you've been down this road before. You know that it's seldom successful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Biofilms and Implantable scientific units: an infection and Control explores the expanding use of everlasting and semi-permanent implants and indwelling scientific units. As an realizing of the expansion and influence of biofilm formation on those clinical units and biomaterials is essential for safeguarding the overall healthiness of the human host, this publication presents readers with a finished treatise on biofilms and their dating with scientific units, additionally reporting on infections and linked thoughts for prevention.\nFresh outbreaks of health problems traced to infected sprouts and lettuce illustrate the holes that exist within the process for tracking difficulties and combating foodborne ailments. even though it isn't completely answerable for making sure the security of the nation's nutrition provide, the U. S. meals and Drug management (FDA) oversees tracking and intervention for eighty percentage of the nutrients provide.\nDo you help a toddler or grownup with sensory perceptual concerns or cognitive impairment? for individuals with tough sensory and cognitive stipulations, daily life can develop into so unpredictable and chaotic that over the years, loss of engagement can frequently result in a kingdom of realized helplessness. during this insightful textual content, Paul Pagliano exhibits how 'learned helplessness' may be remodeled into realized optimism via multisensory stimulation, and explains how a programme of aid may be designed and modulated to compare the person's wishes, pursuits and talents.\nPharmaceuticals 8, 696\u2013710. , 2015. The antimicrobial agent, Next-Science, inhibits the development of Staphylococcus aureus and Pseudomonas aeruginosa biofilms on tympanostomy tubes. Int. J. Pediatr. Otorhinolaryngol. 79, 1909\u20131914. , 2014. Biofilm formation of mucosa-associated methanoarchaeal strains. Front. Microbiol. 5, 353. , 2008. Roles of type IV pili, Flagellummediated motility and extracellular DNA in the formation of mature multicellular structures in Pseudomonas aeruginosa biofilms.\nParameters such as the time after clinical use and the various times of procedure employment were factored in. The study concluded that commonly recommended protocols are efficient enough if thoroughly carried out, and a classification of assays and antimicrobial agent widely used was also made. , 2015). Due to the aggressive nature of biofilm infections related to medical instruments, especially the ones which come in contact with extended areas of tissues (mostly epithelium and endothelium), the studies meant to improve the design and antibiofilm characteristics of such devices are still challenging and restless.\n2011). Martinez et al. studied the action of chitosan, a natural hydrophilic polymer, on C. , 2010). Using contact lenses means a risk factor in developing several adverse effects, such as acute red eye, microbial keratitis, or peripheral ulcer, but the major factor is the apparition of Gram-positive bacteria S. aureus and Gram-negative bacteria P. , 2008) in 44\u201357%. A study made by Dutta and Willcox presents the bacterial adhesion on contact lenses made of hydrogel etafilcon A and silicone hydrogel senofilcon A, two of the most used materials.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"25,000+ Singles Join Match Daily. Browse Profiles & Pics of Singles for Free! Free to Try. #1 Dating Site. Millions of Members. Browse Pics for Free. 1 in 3 Relationships.\nMeet local gay men looking for friendship, romance, and more. Hot Live Gay Cams 24\/7. We Offer The Best Cams. Hand Selected Gay Cams. Best Gay WebCams Online.\nfor what you need. All that you want and more is online now.\nNew: Singles Date Sites. Search & Find it in Seconds! Search Free. Powerful Search. Easy to Use Design. Quality Results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Download \"Applied Business Intelligence. Iakovos Motakis, Ph.D. Director, DW & Decision Support Systems Intrasoft SA\"\n11 Summary Business Intelligence is the link between analytical business processes and available information It is a prerequisite for growing the business in a competitive environment Very high ROI for successful projects Required for CRM, cross-channel selling, e-business support, etc.\n32 Conclusion Technology offers many choices DW\/BI applications will never be completely packaged, but many components are available Require significant business domain knowledge, technical expertise and project management skills DW\/BI Apps cannot be produced massively Estimation of fixed-time, fixed-cost projects can be very challenging Rewards are fantastic for successful projects!!\nMaster Data Management: More than a single view of the enterprise? Tony Fisher President and CEO Agenda Why MDM? Why CDI? Business Drivers for MDM Are You Ready for MDM? What is Master Data Management?\nSpreadsheet Governance Pushes MDM to the Desktop James Kobielus Principal Analyst, Data Management November 1, 2006 Summary Issue Spreadsheets are a wild card in the master data management (MDM) equation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What does getting over an addiction feel like?\nQuestion by : What does getting over an addiction feel like?\nAre you supposed to feel drained, a bit crazed, emotionless, worthless, feeling out of place\u2026the things you liked before mean little now and the things you love you can't enjoy properly when your trying to get over an addiction?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shepherd of Sweden introduce a range of natural upper\/cool lined\/light rubber soled house indoor\/outdoor casual shoes. Inspired by Scandinavian clogs.\nShepherd Isabell are soft and comfortable slip-on clog style slippers for ladies. They have soft woollen inner soles which give good support and cushioning. The outer sole is lightweight, hardwearing non-slip rubber.\nAn excellent product. Because they are so comfortable. Because they are long lasting. Very long, indeed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"A winning proposal begins with a solid capture strategy\u2014the primary goal being to increase the Probability of Win (P-Win) to justify investing resources in a sales pursuit. All organizations have limited resources (people, time, and money) and must carefully identify which opportunities offer the best chance of becoming new business.\nOnce the decision is made to pursue an opportunity and craft a proposal, the strategy then changes to developing persuasive messaging. A proposal is more than a technical solution. Like a public speaker, it must have substance and also be interesting.\nProposals that thoughtfully address each of the questions outlined in the five categories below tell a compelling story of why an offer should be selected.\n\u2022 Is the proposal 100% compliant with the RFP?\n\u2022 Did we provide a response matrix that is mapped to the bid request?\n\u2022 Do we have a compelling value proposition that explains to the customer how we are going to improve their business or mission?\n\u2022 Do all sections tie benefits to the customer's requirements and needs?\n\u2022 Is it obvious in the Executive Summary and proposal sections why our offer should be selected?\n\u2022 Are win themes obvious and woven throughout the text?\n\u2022 Are theme statements specific, factual, and quantified when possible?\n\u2022 Are theme statements, action captions, and informative headings consistent in structure and placement?\n\u2022 Are discriminators prominent in themes, captions, headings, callouts, and summaries?\n\u2022 Have we \"ghosted\" the competition?\n\u2022 Has each section been story boarded before writing?\n\u2022 Is writing customer focused, organized, clear, and correct?\n\u2022 Do sections use the \"top-down, four-box approach?\n\u2022 Are summaries and introductions used at all levels?\n\u2022 Are paragraphs six lines or less?\n\u2022 Do all sentences have a maximum of 15-20 words?\n\u2022 Did we brainstorm visuals before writing?\n\u2022 Are there impactful visuals on each page?\n\u2022 Do all visuals have action captions?\n\u2022 Is the 10-second rule applied?\n\u2022 Do pages \"read short?\" Are they easy to read and comprehend?\n\u2022 Is the proposal professional and easy to evaluate?\n\u2022 Does key information stand out?\n\u2022 Is there a proper balance of text and visuals?\nBy focusing on the five key characteristics outlined above, proposal teams can achieve greater customer focus and consistently win more business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My first HOW Conference was 19 years ago in Boston. It almost goes without saying that SO much has changed since then. Especially me!\nAnd yet, comparing that first conference in 1994 with this past one in San Francisco shows that some things never change.\n1. Creative people seem to be riddled with fear.\nWhether it's fear of failure or of success, or fear of someone finding out we're just winging it\u2014not feeling like we're legit, or even fear of seeming like a dork when meeting new people. These were themes that speakers explored back in 1994 and again this year. Gordon MacKenzie went on to write into his book, Orbiting the Giant Hairball, the stories he told in his closing keynote talk that year, including several that delved into overcoming fear. Here's one of his fear stories as I remember it. Oddly enough, even though I didn't put myself out there, I came back ready to design the world. It lit a fire under me that has been lit ever since.\n2. Networking doesn't seem fun until you realize it's just making new friends.\nDuring that first conference, I was too afraid of being \"found out\" to get out and meet anyone new. Networking was something that suits did, not me. Besides, what would I do with their business cards once I received them? I was sure that since I was so young, I'd be discounted as a fraud and ignored (even though I had a brand new BFA in graphic design!!). As someone with minimal social skills at the time, I preferred to avoid the possibility of being rejected and just went back to my room after sessions. Now, I can't imagine a worse waste of HOW time than to be alone during the conference. I love to make new friends\u2014the more the merrier. And getting to connect with these friends year after year makes HOW that much better.\n3. HOW is what you make it.\nAs long as you stay open to absorbing information from all around you, this one will be a piece of cake. The first HOW I was able to attend after that first one in 1994 was the 2005 conference in Chicago. I had super-high expectations\u2014quite unrealistically\u2014that it would blow me away again. What I didn't take into account was my attitude preventing that from happening. Because I expected the conference to come to me, I didn't put much effort into putting myself out there. I don't recall connecting with people, or having meals with anyone. It sucked; I sucked. And I didn't see myself as the biggest obstacle. Thankfully, the lightbulb went on before I made it to the next HOW in 2007. I went with the goal of being open, getting connected and stoking the fire. And that's just what happened.\nSo my advice to you is leave the fear behind as best you can, you can always go pick it up again later if you find it served you well. Make new friends; they and you will be glad you did. And make the most of the time you have on this earth, especially the all-too-brief time you spend at HOW, when and if you get to go.\nOh yeah, and go on art dates\u2014whether it be doodling with crayons or exploring a city with a camera (thanks Julie for letting me tag along and Jim for putting the idea in my head!)\u2014talk about getting your creative juices flowing! The best picture I took all weekend (with the exception of Rusty's gum) is up there, which I took quite by accident.\nThis past week in San Francisco was great, as HOW always is for me now. This was my second time speaking (which I have learned is what I want to do when I grow up). I ran into old friends Kit Hinrichs (I met him at the 1994 conference), Adam Hall, and Ilise Benun; friends who feel like I've know all my life like Julie Goldsberry (of the \"I've been hugged by Julie\" sticker fame), Crystal Reynolds (of Creatives' Cupboard fame), and tiny Stefan Mumaw (whose high-five I still can't reach); friends I'm so glad got to know better like Jim Krause (it's rumored we're actually the same person in two different bodies), Von Glitschka (now of bunny-suit fame), and Rusty Kugler; and new friends I hope to get to know better like Josh Beaton, Jason Dean, Amy and Scott Herbert, Stan Grabowski, Jake Bartholomew, Mason Hall, Mia Berkovits, Desiree Mitchell, both Richard Medinas (I'm anxious to see what the younger one does next), Emily Young, and Regina Maldonado.\nEven if I've not mentioned you, a truckload of good juju to you. Thanks for giving me so much in return!\nOh, and I won't be going drinking again with Jason and Mathew. Lesson learned.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deloitte reckons English Premier League football clubs have spent nearing \u00a3485m this summer \u2013 that's up \u00a3120m, or 33% on the equivalent period last year. It seems the trials of the outside world are still having very little impact between the touchlines. Indeed, this sort of spending is back to 2007, those long-gone halcyon days of jumpers for goalposts and a still extant Lehman Brothers.\nTop clubs certainly seem a long way from the striking lack of confidence currently keeping consumers away from the high street: Arsenal, Chelsea, Liverpool, Manchester City and Manchester United each topped \u00a350m of transfer spending before the transfer window, the period during which clubs are allowed to exchange players (running from the end of the season to 31 August) slammed shut on Wednesday night. Combined, those five clubs accounted for 66% of the total spend among the 20 clubs of the Premier League. That's clearly how much they value qualification for the European Champions League \u2013 worth a hefty \u00a353.2m to runners up Manchester United last season (or one Fernando Torres, if you want to put the sum in player terms).\nIt's fascinating to watch the sums being bandied about, the notable thing this season being the number of home-grown players changing hands: \u00a3165m was spent on English players, around 34% of the toal spend. Those prices were largely inflated on the back of new Premier League rules about squad composition, designed to bolster the strength of the English game.\nOf course, in the short-term at least it may just mean clubs end up paying even more for players like Peter Crouch, who manages to fetch \u00a310m despite having tried more clubs than Tiger Woods. This would of course be a good thing if that transfer money was going to lower league clubs, but actually most of those players are spotted young and snapped up on the cheap by Premier League sides anyway, keeping that crucial cash circulating among the wealthier sides chasing those European places.\nNot only does the Champions League football mean a definite cash injection, but it's a huge stepping-stone towards global branding renown \u2013 and the increasingly important Asian markets. That may be the modern reality, but the depressing result is that games are no longer talked about in terms of sporting glory, and instead about how much they're worth. Note Arsenal's against Udinese the other week \u2013 never mind that victory kept them in the hunt for a trophy, it was simply being billed as a chance to net a crucial \u00a325m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On ornate Gothic churches, spires shoot toward heaven. Military Gothic inspired buildings are missing that last aspiration upward. Their soaring walls stop heavily and abruptly with notched parapets that seem to have the sole purpose of looking downward to protect.\nThe architecture of the interior of the chapel was spectacular, but honestly, the books in the pews drew my eye more than the vaulted ceiling and stained glass windows. Their sheer perfection made this perfectionist-in-non-stop-recovery-mode grin.\nRow after row, the hymnals and Bibles were in perfect line. And I couldn't help but assume that this is how the worshipers put them back after every service. Surely the sexton didn't need to straighten these. This was West Point; people here know what to do after they get something out. They put it away.\nOn the way home, while I was thinking I should implement more routines in our daily lives, Will declared, \"I am never coming here.\" Coincidentally, this morning after breakfast, Will cleared all his dishes and put the cereal and milk away -- which gave me a bit of a jolt. ...Thank you, West Point?!?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"City Diaries\" documents the day-to-day experiences, challenges and highlights of life in NYC.\nFor anyone who has lived or even visited NYC for an extended period, you know that it can be a city of extremes. A place of high highs and low lows. Of I-can't-believe-this-is-my-life to I'm-crying-in-public-and-no-one-seems-to-care\u2026all possible within the span of a single day.\nAnd considering I'm embarking upon the rollercoaster lifestyle of self-employment IN a place like New York, it felt only fitting that I should begin documenting some of the highs and lows of my day-to-day life. This post is part of a larger series called \"City Diaries\" where I'll be sharing more thoughts like these on the blog.\nThe falling leaves. Peacoat season. Pumpkin-lined stoops. It's all too much for this California girl to handle.\nI just posted about my love for fall in NYC and have shared some thoughts about it in past seasons, but I still can't get over it. When the leaves turn, literally every street becomes picturesque. And since the temperatures haven't completely dropped yet, this makes for the perfect 3-4 weeks of prime city wandering.\nLesson? There are no secrets in NYC, or at least\u2026not in the 4 square miles of downtown Manhattan.\nI've been teaching Digital Marketing at General Assembly. Last week, I covered another teacher's class for the night and it helped push me out of my comfort zone. Entering a room with zero context on the students can be both exciting and terrifying. You are forced to learn how to read a room on the spot. Luckily, the class was a success and a couple students had even heard of 5th Floor Walk-Up! Insert warm fuzzy feelings here.\nIn this new world of self-employment, I have the luxury of testing new ways to organize my day. Since I'm finally getting back into a regular workout cadence (my ideal week includes 3-4 workouts), I'm figuring out the best way to fit it into my week. Even though I often force myself, I'm just not a natural morning workout person (yeah, I'm the one with bedhead scowling in the corner of that 8am Flow). So, signing up for a 2pm yoga class and realizing no one was there to stop me gave me a major thrill.\nP.S. Looking for a great studio in downtown Manhattan? Laughing Lotus is my personal favorite, but other studios like Yoga Vida have great flows as well. Note: Also proven to ease the anxieties of city life.\nDon't get me wrong, my past few weeks of self-employment have been amazing and empowering. But with high highs can come low lows. I've been surprised by the amount of unexpected anxiety I've had charting this uncharted territory and the amount of pressure I've been subconsciously putting on myself. This note really warrants its own blog post, but for now, I'm just trying to be as kind to myself as possible. I think I'm still in recovery mode from working three jobs for 6+ months and honestly might need a proper break.\nI hate to go here, but this is the reality of living in a big city. I was on my way home from Trader Joe's, on the phone with my boyfriend after a successful night of teaching when BOOM. A man in a full on birthday suit on the sidewalk of 6th Avenue. I calmly assessed the situation, and quietly moon-walked back the other direction to take the long way home.\nQuick tip: Sometimes a 5 minute detour is worth the 5 hours of trauma that could result from trying to just \"get through\" something uncomfortable. Le sigh. But such is a part of navigating life in a city like New York.\nNow, I'm not one to normally get upset by random comments like that, but I was already having a bad day and really internalized this person's negativity. While I've actually found most New Yorkers to be incredibly friendly (contrary to popular belief) there will always be a few bad apples in the mix. Constant advice to myself: do not take it personally. And veer right.\nAny highs or lows you've been navigating this week? I'd love to hear in the comments below!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Superstar Aamir Khan, who is called as the perfectionist of Bollywood, looks at the script while deciding a film. \"When I sign a film, I first look at the story or the script...Actors and technicians are all a part of the film but it all begins with the writer. The writer is the origin and the most important aspect of the making of a film,\" Aamir said in a statement.\nThe THUGS OF HINDOSTAN actor is currently judging Cinestaan India's Storytellers Script Contest.\nSo while sharing what he expects from the aspiring writers, he said: \"We are looking at somebody with an imagination and a different story or scenario that we come across, where the characters have been well written, the premise and goal is clear and where the whole journey of the story is highly entertaining and engaging. That's what we look for in a good script.\"\nCinestaan India's Storytellers Script Contest is launched by Cinestaan Digital Pvt. Ltd. As a part of the contest, the winners with the best script will be awarded a prize of Rs. 25 lakhs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"National Children's Bureau recent report synthesises the increasing problem of the children in a clue moment of their formation.\nSuicide is the most common cause of death for boys aged between 5 and 19, and the second most common for girls.\nAn estimated 3 children in every classroom have a diagnosable mental health problem. This rises to one in four children when we include emotional distress.\nAround one in every twelve young people deliberately self-harm, though this may rise to almost one in three for girls.\nRates of depression and anxiety in teenagers have increased by 70% in the past 25 years.\nThe number of young people calling Childline Help about mental health problems has risen by 36% in the last four years.\nThe number of young people attending A&E because of a psychiatric condition more than doubled between 2010\/11 and 2014\/158.\nThese rates are similar in different western countries, international comparison studies have found the wellbeing of school-aged children in England \/ the UK to be 'average' or 'below average' affirms the report (read HERE ).\nThe Society believes that early intervention and timely access to care, support and treatment for children and young people is vital. We are concerned that there is an insufficient focus on prevention and the wider psychological and social determinants of mental health (read HERE the report on mental health awareness).\nResponding to the figures the UK Government announced urgent measures for the creation of new mental health support to provide trained mental health workers to work closely with schools.\nWe support the Government's measure but in our opinion, the matter should be treated through a strong link between school and home, teachers and parents.\nA damning report by two select committees, Education and Health and Social Care, said the government's strategy lacks ambition and fails to take into account the needs of the most vulnerable groups.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finnish has a pretty good social security system, compared to some other countries. The part of the government that deals with social security benefit applications in Finland is called Kela. They can support you by paying a housing benefit, paying part of your medication and visits to the doctor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"43.9 +\/- ac timberland south of Ashland in Natchitoches Parish, LA Two separate tracts consisting of 37.25 +\/- acres and 6.7 +\/- acres located between Ashland and Creston. The 37 acre tract consists of predominantly pulpwood size hardwood timber, with some mature hardwood along the creek and slough drains. The outer edges of the property run along Demery Creek providing good deer, hog, and small game hunting. Access is located along Hwy 153. Both tracts are located within miles of Black lake as well as Kisatchie National Forest. The 6.7 acre tract is located off of the Hebert Trichel Rd. There is currently no deeded access, however access could potentially be negotiated from the neighboring landowner along the south side of the property. The stand consists of a 12 yr. old pine plantation. A survey would be required to determine boundary lines. Tracts could potentially be sold separately. Directions From Creston, LA travel 5.2 miles north on Hwy 153. Look for the RecLand Sign Minerals Negotiable Currently in a lignite lease until 2019 Asking Price Call Chris Bordelon at or see the RecLand site for all our listings and aerial mapping.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Since childhood, Maryam has been curious to find out why and how different phenomena occur around us and has been inquisitive to find logical relations between various phenomena occurring around her. Learning gave her a great sense of satisfaction when she followed her propensity to go beyond the unknown and challenge various aspects of life.\nAs a student, Maryam was lucky to be accepted by the Nemoone (special talents) high school, which is a selective public institution, aiming at developing exceptional talents, based on a comprehensive national wide exam. Academically, she has always striven for higher accomplishments and challenged herself in every way to improve as a student and as an individual in society.\nMaryam was ranked second in the national graduate degree examination in Iran and was able to pursue her interest in studying plant biotechnology. At the time, she was among the few women who had been qualified to study this field in a highly ranked university. Maryam greatly respects all aspects of the human being and believes a successful person is perfect in different dimensions of his or her life.\nBecoming a professional researcher and teacher in Biotechnology is Maryam's vision. To take another step toward this goal she applied for admission to the Biotechnology PhD program at the University of Saskatchewan. After being accepted, she began her research at GIFS with the professional team that includes other students and postdoctoral fellows from all over the world. Maryam's focus is on regulatory mechanisms control apomixis in Boechera. Her project is about to identify DNA-protein interactions with APOLLO 5' UTR by using yeast one-hybrid assay. She strongly believes by being a researcher at such a prestigious institute and working in this academic environment, she can make the best use of her abilities to attain her life goals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Mi Jefita has always been a strong proponent of Spanish Christmas carols.\nBut when she starts wrapping presents, long before the kids are out of school, the Christmas carols en espa\u00f1ol start playing throughout the house \u2014 as has been the case for years. She knew the words to \"Burrito Sabanero\" hours after that song's release, and I remember her standing in the kitchen singing that song to my then 7-year-old brother while he ate huevos con chorizo and tried using The Force to get her to stop singing.\n\"Es de un muchachito!\" she said with a big smile, loading more huevito con chorizo on his plate as if that would somehow make the song more attractive. It might have worked for me, but he had already entered his Star Wars years, and being called a muchachito was a bit of an insult.\nAnother favorite was \"Los peces en el rio,\" particularly the Pandora version. I was a teen at the time and my mom never pointed out that Pandora were also teens. She did, however, play that song a lot as the glow of the Christmas tree lit up our stereo with the two cassette decks.\n\"Pero mira como beben los peces en el riooooo, pero mira como beben al ver a Dios nacido.\nI never understood why fish drinking the very water in which they lived their entire lives was worth singing about, or why it had anything to do with celebrating Christmas. Were the fish a metaphor for mankind? Was this in reference to holiday drinking? Sepa.\n\"Ven a mi casa esta navidad\" became one of our favorites, but not because it's a song about kindness and love, about reaching out to the lonely and sad, about making room at the family table for someone who could use el calor of the holidays. The sentimental Christmas classic by Luis Aguile became a favorite at our house because my brother figured out that the singer, after spending all year on the naughty list, is extending a last minute invitation to Santa.\nMi Jefita doesn't agree, but she plays the song anyway while the tamales are boiling and, by the time they're all done, people drop in.\nWe might all have different taste in music, but everyone understands tamales.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On Wednesday, the Wildlife Resources Commission, North Carolina State Highway Patrol, North Carolina Health & Human Services and the Cornelius Police Department gathered at Blythe Landing on Lake Norman to ask the public to be safe and sober during their joint annual \"On the Road and On the Water\" campaign.\nParker warned that patrols and checkpoints will be set up on area roads and on Lake Wylie, Mountain Island Lake and Lake Norman to ensure public safety.\nParker stressed the importance of maintaining a sober boat operator and having a designated driver to and from area lakes to avoid fines and possible jail time.\n\u25aa Stay seated while the boat is moving.\n\u25aa Have a working fire extinguisher on board.\n\u25aa Check the lights on your boat and boat trailer to make sure they are working.\nOfficer Zack Smith, with the North Carolina Wildlife Resources Commission, scans Lake Norman after a press conference with other agencies urging the public to be safe and sober while one the roads and water over the Memorial Day weekend. The agencies gathered on Wednesday, May 23, 2018.\nOfficer Sampson Parker, Jr., with the North Carolina Wildlife Resources Commission urged the public to be safe and sober on the roads and the water over the Memorial Day weekend. The NCWRC, State Highway Patrol, North Carolina Health & Human Services and the Cornelius Police Department gathered at Blythe Landing on Lake Norman to kick off the annual \"On the Road and On the Water\" campaign on Wednesday, May 23, 2018.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Simo H\u00e4yh\u00e4, also known as \"The White Death,\" was a Finnish soldier who currently holds the record of most confirmed kills of any sniper ever.\nIn 1939, at the dawn of World War 2, Josef Stalin made bold a move to invade Finland. He sent half a million men across Russia's western's border.\nTens of thousands of lives were lost. Among all the chaos, Simo's grim legend began.\nHere are 12 fascinating facts about the world's deadliest sniper.\n1. H\u00e4yh\u00e4 has 505 confirmed kills to his name.\nAnd it is even suggested that he has more.\nThe Winter War only lasted for roughly 100 days. Yet in such a short time span, it is believed The White Death killed between 500 and 542 Russian soldiers.\nHe did so while using an antiquated rifle. His comrades, on the other hand, used state-of-the-art telescopic lenses to zoom in on their targets.\nIn extreme winter conditions, H\u00e4yh\u00e4 used only iron sight. He didn't mind. He even felt it added to his precision.\n2. He was a mere 5 feet tall.\nSimo H\u00e4yh\u00e4 in his white winter uniform, carrying a new rifle, a gift from the Finnish Army.\nH\u00e4yh\u00e4 stood just 5 feet tall. He was mild-mannered and unassuming. He wasn't what you would call intimidating.\nBut it all worked in his favor. He was quite easily overlooked, which perhaps contributed to his superb sniping skills.\n3. He lived a quiet life as a farmer before the war.\nAs many citizens did at the age of 20, H\u00e4yh\u00e4 completed his mandatory year of military service.\nAfterward, he resumed a quiet life as a farmer in the small town of Rautj\u00e4rvi, a short distance away from the Russian border.\nHe enjoyed hobbies most Finnish men would: skiing, shooting, and hunting.\n4. His sniping skills were bred from youth, albeit unintentionally.\nIn Rautj\u00e4rvi, he was noted for his excellent shooting skills. He spent most of his life prior to the war hunting birds in clearings or pine forests.\nCouple that with rigorous farm work, and hunting wildlife in extreme winter conditions, it's not really a shock how his sniping skills turned deadly as it did.\nLater, he would credit his sniping skills to his experience hunting, noting that when a hunter shoots a target, he must be able to observe both surroundings and the impact of each shot. This experience taught him how to read and use the terrain to his advantage, which he was an expert.\nHis father also taught him a valuable lesson: how to estimate distances. In most cases, his estimates were perfect. He also knew how to estimate the effects of rain and wind on shooting his targets.\nA photo or unknown Finnish soldiers during the Winter War.\nH\u00e4yh\u00e4 might have been born to be a soldier. At least he had a knack for it.\nWhile one year of military service is not much, H\u00e4yh\u00e4 seemed to have made the most of it.\n6. The White Death's MO.\nHow exactly did H\u00e4yh\u00e4 kill over 500 soldiers in the span of 100 days?\nHis methods were almost superhuman.\nH\u00e4yh\u00e4 would dress in his white winter camouflage, gather a day's worth of supplies and ammunition, and set out to do his part in the Winter War.\nArmed with his Mosin-Nagant M91 rifle, he would pick a spot in the snow and kill any Russian soldier in his line of vision.\nHe preferred using iron sights instead of scopes because scopes would glare in the sunshine and reveal his position.\nH\u00e4yh\u00e4 would even put snow in his mouth so that his breath wouldn't be seen in the cold air. He used snow banks as padding for his rifle, preventing the force of his shots from stirring up snow.\nHe did all of this in such a harsh terrain environment. The days were short. And when daylight was over, temperatures were freezing.\n7. The Soviets feared him.\nHis legend soon took over. In no time, the Soviets knew his name. Naturally, they feared him.\nSo much so, that they mounted several counter sniper and artillery attacks on him, which obviously failed miserably.\nH\u00e4yh\u00e4 was so good at hiding his position, that he remained completely undetected.\nOnce, after killing an enemy with a single shot, the Russians responded by mortar bombardment and indirect fire. They were close. But not close enough.\nH\u00e4yh\u00e4 wasn't even wounded. He made it out without a scratch.\nAnother time, an artillery shell landed near his position. He survived with a mere scratch on his back and a ruined greatcoat.\n8. He was very meticulous.\nH\u00e4yh\u00e4's preparation method was so meticulous, he might have had OCD.\nDuring nights, he would often pick and visit the firing positions he preferred, meticulously making necessary preparations.\nUnlike other soldiers, he'd go out of his way to make sure everything was well-prepared. He would perform both before-and-after maintenance operations in every mission.\nIt's also crucial to do proper gun maintenance in the -20\u00b0C temperature to avoid jamming. H\u00e4yh\u00e4 would clean his gun more often than his comrades.\n9. He knew how to detach his emotions from his job.\nTapio Saarelainen, the author of The White Sniper, had the privilege of interviewing Simo H\u00e4yh\u00e4 numerous times between 1997 and 2002.\n10. He had a sense of humor.\nAfter the war, H\u00e4yh\u00e4 was very private, preferring to live a quiet life away from fame. Not much is known about his personality.\nHowever, an astonishing hidden notebook of his was later found. In it, he wrote about his experience of the Winter War.\n11. He was shot only once, a mere few days before the Winter War ended.\nSimo H\u00e4yh\u00e4 after the war.\nH\u00e4yh\u00e4 was hit by a Russian bullet mere days before the Winter War ended, on March 6, 1940.\nH\u00e4yh\u00e4 was in a coma for a week. He woke up on March 13, the same day that peace was declared.\nThe bullet crushed his jaw and most of his left cheek was removed. He underwent 26 surgical operations after the war. But he recovered fully, and the injury did not affect his shooting skills in the slightest.\n12. He lived a quiet life after the war.\nH\u00e4yh\u00e4's contribution to the Winter War was highly recognized. His nickname, The White Death, was even a subject of Finnish propaganda.\nBut a hunter is always a hunter.\nHe continued to use his sniping skills, becoming a successful moose hunter. He even attended regular hunting trips with then Finnish president Urho Kekkonen.\nIn his old age, H\u00e4yh\u00e4 moved into the Kymi Institute for Disabled Veterans in 2001, where he lived alone.\nHe passed away at the ripe old age of 96 in 2002.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Oxford Ferret Rescue \u00bb Blog Archive \u00bb Easter candy for sale!!!!!!!!!!!!!!!!!!!\nYou are here : Home \u00bb 2016 \u00bb 02 \u00bb 18 \u00bb Easter candy for sale!!!!!!!!!!!!!!!!!!!\nthe candy shop has time to make our candy.\nWe will run this fund raiser from Feb. 8, 2015 to Feb. 26, 2015.\nPlease have all your orders to me no later than that date with the money.\ncandy.  Everything we sell makes the rescue 40%!\nEaster candy and especially fresh home made candy!\nbrochures at Chadwell.  So please help us help the ferrets.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"QRCode Scanner & Generator Pro \u2014 iOS app Sold on Flippa: IOS native and useful app. 100% automated! Low reserve!\nIOS native and useful app. 100% automated! Low reserve!\nEasy to manage. Native and lightweight app with clean source code. 100% automated!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzallpv b/data_all_eng_slimpj/shuffled/split2/finalzzzallpv
new file mode 100644
index 0000000000000000000000000000000000000000..924bfcbca1c75e6862bb52283f2e5f2a5018694c
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@@ -0,0 +1,5 @@
+{"text":"This Silver Medal Winner of both 2017 Finger Lakes International Wine Competition and 2017 San Francisco Chronicle Wine Competition, has a clear, pale lemon colored wine with a nose dominated by clean, delicate aromas of fresh tropical fruit\u2014pineapple, banana, and guava. The body is light, crisp, and refreshing with flavors of citrus and tropical fruit. Orange, lemon, pineapple, and passion fruit fill the palate and a hint of fresh ginger spiciness rounds out the finish. This Sauvignon Blanc goes well with feta, goat cheese, pine nuts, poultry, pork, fatty white fish dishes, scallops, lobster, shrimp and most sushi. Citrus, green apple and asparagus all bring out the fruitiness of the Sauvignon Blanc as well as light cream sauces, chives and can be a nice wine for a sorbet, key lime pie and even a slice of mango.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We don't often get much snow in Seattle, but when we do everything basically stops and absurdist-theater levels of hysteria ensue, complete with ridiculous terms like \"Snowmageddon\" for a few inches of accumulation, which I'm sure causes the eyes of folks from the Midwest to roll their eyes so hard they can virtually see their brains. Now, part of the problem is that it just doesn't snow very much here, and many Seattleites are transplants from much warmer places where it doesn't snow at all, leading to the same sort of fear-of-the-unknown reaction one sees every time it drizzles in Los Angeles. And another part of the problem is that Seattle is full of steep hills which become impassable under icy conditions, thus leading to traffic snarls as people who can't navigate without Google Maps are detoured to alternate routes. But another part is just Seattle being Seattle, which is why the only days I got in my car for the past week were Tuesday and Wednesday. Once the snow stops (the weather report says tomorrow) I'll probably start my car to let the heat melt off the snow, and throw a few pots of hot water behind my wheels so I can get traction to back out of my driveway. But until then, I've got plenty of blankets, soup, supplies, booze, cannabis and writing projects, so I'll be just fine right here in The Den.\nWhen I lived in San Francisco, I got the idea that the streets, rather than a grid that ignored slopes, should have been arranged so that at each corner the gradient (line of steepest slope) is diagonal; that way you're never driving directly up a hill. Makes even more sense where the streets can be slippery.\nWonderful article! Glad to see the common sense approach to the little snow that you recieve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Security researcher pRaLe, has submitted on 07\/11\/2008 a cross-site-scripting (XSS) vulnerability affecting search.live.com, which at the time of submission ranked 5 on the web according to Alexa.\nWe manually validated and published a mirror of this vulnerability on 05\/07\/2009. It is currently unfixed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dubai, United Arab Emirates; 07 March 2017 \u2013 Gamification is gaining traction among education professionals in the Gulf region as a way to make learning more interesting for students, and to encourage innovation in schools.\nThe UAE is aggressively pursuing gamification in schools, partnering with Finland's Seppo at the Ministry of Education's National Innovation Exhibition last year to test the educational potential of games for teaching and learning.\nSeppo is an innovative company that specializes in gamification and people engagement. It was founded in in 2012, when a group of teachers from Finland, a world leading country in education and gaming, discovered the power of gamification as a catalyst for motivation. Seppo started its journey by creating educational games. Soon after the team designed and developed a platform that allows anyone to create games for different purposes by themselves.\nAt the NIE, organisers and participants lauded the game for positively affecting the students' learning process and raising their awareness related to the UAE innovation strategy, new high profile initiatives, STEM facts and others science projects exhibited by fellow students.\nSeppo makes another return to GESS Dubai this year to challenge participants from different schools to explore various innovations showcased at the region's premier education exhibition and conference. Dubbed Explore GESS, the smartphone- or tablet-based game aims to showcase firsthand how gamification works within an education setting and how it can be integrated in the classroom to engage students.\n\"Showcasing this technology at GESS Dubai allows us to share with education professionals the many benefits of gamification for enhancing the teaching and learning experience, while also encouraging participants to explore innovative ways of using modern technology to benefit education in general,\" said Matt Thompson, Project Director, F&E Education, organisers of GESS Dubai which will take place on March 14-16 2017, at the Dubai World Trade Centre (Halls 5-8).\nWinners of the Explore GESS game will have a chance to visit Lapland in Finland (www.visitfinland.com and www.ruka.fi), where seppo was developed and has since become an exciting tool used by around 70,000 students and 7,000 teachers globally. It is currently being used in some schools in the UAE. Interested schools may register by visiting www.gessdubai.seppo.io. The game will open on the 9th of March.\n\"Games are excellent way to motivate students. Seppo is a revolutionary game based learning platform which transforms the way we teach and allows students to develop 21st century learning skills and dispositions. Seppo creates memorable learning experiences due to its competitive nature as a game based learning platform, which intrinsically motivates students to learn,\" said Riku Alkio, co-found and CEO, Seppo.io.\nAndrew Nolan Education Technology Manager, GEMS Innovation, Research and Development Unit, UAE, believes seppo enables students to gain mastery of their subjects in many ways than different traditional pedagogies. \"Seppo changes the models of motivation. Students are moving from purely extrinsic motivatory factors to intrinsic motivation. The games are engaging learners to such extend that they want to push themselves and want to succeed and master. Seppo enables social and engaged learning which is rooted in educational research and theory,\" he added.\nTeachers and other education professionals who will be attending the 10th edition of GESS Dubai will also have a chance to learn from experts on how to better use gamification and mobile apps for their classrooms. The Coding Circle will lead a hands-on workshop on how to build an Android app using the famous MIT App Inventor, an easy-to-use tool that can help develop apps without having to learn any programming languages.\nThese and other sessions, discussions and workshops will be featured heavily in the Future Learning section of GESS Dubai which will highlight how modern technology is redefining the education landscape in many parts of the world, with a number of schools in the Gulf sharing best practices on how they have successfully integrated technology into the classroom experience.\nFor 10 years now, GESS Dubai has been the leading platform for leading suppliers of educational products and solutions to showcase their wares, including some of the most cutting-edge innovations that are critical to enhancing the quality of teaching and learning across schools in the Middle East.\nThe show is free to attend, and registration is now open to the public via the show's website at www.gessdubai.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each year 4 million lambs are lost in the period just prior to lambing to up to a few days of age, losses are costing the industry an estimated \u00a3120 million per annum.\nThe key factors for profitable lamb production are; healthy ewes produce healthy lambs, strong lambs have a higher chance of survival, immediate access to quality colostrum and plentiful ewe milk supply to improve lamb growth rate.\nOur range of feed will help provide your flock with a balanced and nutritious diet, our quality feeds are formulated by a team of highly qualified nutritionists who are dedicated to giving farmers the best performance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalpzn b/data_all_eng_slimpj/shuffled/split2/finalzzzalpzn
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@@ -0,0 +1,5 @@
+{"text":"Changes in climate variability are as important for society to address as are changes in mean climate (Rehfeld et al, 2018). This project would look at how climate variability (such as El Nino) depends on the climate itself. It will use a suite of model simulations, which are being developed for the upcoming IPCC report. Observations shows a strong reduction in variability since the last glacial (Rehfeld et al, 2018), and a reduction in the Atlantic Meridional Overturning Circulation in the past 200 (Thornalley et al, 2018). How does this compare to the behvaiour of state of the art Earth System Models? Their future scenarios will provide insight into how variability might alter going forward. The impacts of these changes will be determined with PAGE20, a new version of the physical-socioeconomic model used in the influential Stern Review into the Economics of Climate Change.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The whole gaming world in your smartphone, or tablet for those who prefer slightly larger screens, but without sacrificing freedom to read the news on the move. Thanks to the new GiocoNews.it apps, all online newspapers of the leading network in the Italian gaming industry \u2013 so included also GiocoNews.com,, entirely in English \u2013 are accessible in the optimized version for any device of any kind: if you use an Android or iOS operating system, doens't change a lot. Only the link to download the app, which is therefore available to all, in a completely new graphics than the previous release and with new features. More and more in line with the needs of new digital readers and even more fit for the 'finger'.\nNOVELTIES \u2013 Thanks to the new GiocoNews.it apps, all news related to gaming world, minute by minute, are immediately (and for free) available on your phone or in any other device. In addition to the main portal of the network (www.gioconews.it), the same App includes the other three vertical online newspapers: GiocoNewsPoker.it, the reference portal for poker fans and insiders, GiocoNewsCasino.it, the only Italian newspaper dedicated to the industry of gaming houses and the youngest GiocoNewsPlayer.it, the first newspaper entirely dedicated to the player. This is not enough: among the novelties of the new App there is also the new section of www.gioconews.com network, entirely in English and aimed at foreign entities who are interested in knowing the dynamics of the Italian gaming market, which is increasingly a landmark in Europe and worldwide. Our readers can then discover the new pleasure to 'synthesize' the GiocoNews.it network on their iPhone and iPad, or on any other Android or other devices.\nSelecting your favorite online newspaper referred to the game segment you are interested in, or easily jumping from site to site, with few simple touches. For an innovative and exciting experience that combines the quality of GiocoNews.it articles with the best multimedia articles of news section, politics, latest news, economy, society and habits. This is not enough. In the gioconews mobile App version you can also discover gaming places that will be inaugurated today, conferences, events, poker tournaments, best odds and the results of sports events on which to bet, and much more, thanks in particular to the GiocoNewsPlayer.it channel.\nGIOCONEWS AND THE DIGITAL EVOLUTION \u2013 Always interested in technological evolution and developments and implications in publishing field, GiocoNews made the first App for iOS devices in 2011, among the first newspapers ever in gaming area. Today, however, the newspapers \u2013 through further development and investment of the publisher, Gn Media, launches the new version of its mobile App, bringing its news and content in an optimized version for any device, and making it even more accessible, for a most complete and performing user experience. In the new version of the GiocoNews.it app, as already said, all network sites, including the new GiocoNews.com channel, entirely in English, are included in a single application, in order to intercept the maximum user and facilitate the interaction of users with other, very often superimposable, segments. Another novelty of the new App: the integration with all social networks and the chance to register directly via app to premium services offered by the various channels. With benefits also for the advertisers who prefer the GiocoNews.it network to showcase their products and businesses, they have new opportunities, thanks to banner space included in the application, specifically designed to be perfectly integrated with the mobile platform, avoiding to interfere with the reading of the contents and disturb the users during their surfing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rackline Profile, Proform & Prospan shelving systems can be fitted with tray support brackets in order to maximise tray storage capacity.\nAs Rackline manufacture bespoke bay size dimensions, we can design and manufacture storage bays to accommodate any size of tray, from gratnells trays to large museum collection trays.\nThis system of storage allows for maximum capacity and optimum picking rates.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Contractor or sub-contractor? You should be both!\nWe are all contractors of one type or another but are you also a SUB CONTRACTOR?\nIf not, you're missing out on a great source of potential income with virtually no cost associated with it. Most everyone in business networks with other companies, friends, business associates and even family \u2013 but that usually results in occasional referral business.\nBy working on relationships with HVAC contractors, property management companies, restoration companies, etc and giving them a chance to profit from selling a job for you, you'll find you get much better results than just referral business will bring you.\nYou usually have to invest 10 to 20% of your gross income on advertising to attract business. By passing this savings on to the company you sub contract to, they make a fair profit with little investment on their part. And, you still make the same profit on the jobs you do for them, because no advertising dollars were needed to attract this work.\nWe offer our contractors a 20% discount off the price we would normally charge a residential customer. On a $650.00 full service duct-cleaning job we charge them $520.00; that's a clean profit for them of $130.00 with virtually no costs associated with it. We do give up some profit, but we do not have to invest in expensive media ads to earn this business. Most of our contractors will offer their field employees a spiff for any duct-cleaning jobs they sell. You can have a massive sales force selling your business with no payroll.\nIn other words they sell the job, call you to get it scheduled and you take care of it from there. Once they put it in your hands it's up to you to be sure to provide prompt, quality service. Like any job you must always satisfy the customer \u2013 now you have 2 customers to satisfy for the same job; the homeowner and the contractor.\nIt takes a little time to find and develop these relationships, however, once you do you'll find it can easily add 20 to 30% to your annual business volume. It's also important to be sure you get your contractors referrals from your customer base and contacts. In other words take good care of those who take care of you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"And this is why one just have to love Jim Rogers. He says things many others won't because they will lose their punditry platform to upsell their firm. Luckily he is way past that stage.\nI said it was PR Anna; they got the stocks up. That's the whole purpose of PR. Make the stocks go higher. That's what CNBC and many many PR agencies are all about.\nJim Rogers, that is very cheeky of you. You know we're not talking up (unintelligible) here on CNBC. We're just reporting the news!\nis downright laugh out loud material. Ironically that was said in the best 17 year old head cheerleader voice possible. The indoctrination they must have for new hires upon employment at CNBC must be intense because I truly believe these people think they are reporting the news rather than being infotainment financial TV aka one big advertisement for Wall Street.\nWatching Jim guffaw after her remark was epic. Of course they immediately cut to commercial to regroup!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..9893b35b6d9842c76bedf09aad7473b046b72bfc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"The West has no right to interfer in those countries. It is their tradition and religion which makes them procreate through inbreeding. The problem will sort itself out through extinction.\nIt was already tried to cover that issue in Germany, but the usual thing happened: Threatening, complains, PC.\nOh, and \"the West\" might not have the right to interfere in that countries, but what about the immigrants coming here and expecting the products of their \"traditions\" to be paid for life long by the others?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Applications for proposals focused on children's services and tackling drug and alcohol dependency are now open, to be followed by the other themes over the next 12 months.\nThe Blavatnik School of Government at Oxford University has also launched the Government Outcomes Lab (GO Lab), a partnership with the Cabinet Office, to establish an independent centre of academic excellence for the commissioning of public services.\nThe GO Lab will deepen understanding of outcomes-based commissioning, including social impact bonds, by researching new ways for the public sector to commission services.\nIt will provide independent support, data and evidence on what works, and what doesn't, making it equally beneficial for local authorities, social investors and charities. This will help to improve the delivery of public services through a more outcomes focused and evidence-based approach.\nMeet other people interested in sports heritage.\nThere will be two events. One at 2pm to 4.30pm and the second at 6pm to 8pm. The programme for each will be largely the same so you can choose to attend which is most convenient for you.\nThrough the open grant programmes, HLF can offer awards of \u00a33,000 to \u00a3100,000. The Sporting Lives campaign recognises the region's proud and diverse sporting history. They have supported a broad range of projects around the theme of sport in the East Midlands and we want to fund more. Through the open grant programmes, they can offer awards of \u00a33,000 to \u00a3100,000. This is a great opportunity to try your hand at heritage projects and reach new audiences, young and old alike.\nThe event will also launch the Sport in Museums regional network. Sport in Museums (formerly known as the Sports Heritage Network) is a national organisation that promotes the collection, preservation and access to sporting heritage. Funded by Arts Council England, they are now creating a regional network to support sports and heritage organisations in developing, caring for and sharing the sporting heritage of the East Midlands.\nEmail the East Midlands development team if you would like to receive updates on the campaign.\nThe Big Lottery Fund is calling on local groups and organisations of all shapes and sizes to apply for funding, between \u00a3300 and \u00a310,000, to hold events or activities that will bring people together and celebrate what makes their local community special.\nThis opportunity comes as a survey by the Big Lottery Fund found that given the& chance, more than 40 per cent would like to celebrate their local history, and more than two fifths (43 per cent) of those aged 16 to 24 would celebrate their local culture, art or music. The research also revealed that more than half (63 per cent) of people asked reported that they have never or cannot remember ever coming together to celebrate with their local neighbourhood.\nIn response to these findings the Big Lottery Fund is putting people in the lead to come up with their own ideas on what they would like to celebrate, and what kinds of events they could host to bring all corners of their community together.\nThe Celebrate programme is the perfect opportunity for people to showcase and enjoy something important to their community, be it an iconic building, a precious landscape, a unique service or a community hero. There will be \u00a35 million available in total across ;the UK.\nTo be eligible, applicants must be part of a constituted group which has its own bank account. The Big Lottery Fund welcomes applications from voluntary or community organisations, schools and statutory bodies, particularly those who have never considered the Big Lottery Fund as a potential source of funding. Organisations are encouraged to submit their applications early to avoid disappointment as once funds have been allocated, Celebrate will close to new applications.\nGSK's IMPACT Awards reward charities that are doing excellent work to improve people's health. Organisations must be at least three years old, working in a healthrelated field in the UK, with income between \u00a325,000 and \u00a32 million. Up to 20 awards will be made ranging from \u00a33,000 to \u00a340,000 plus free training valued at up to \u00a36,000.\nOrganisations will also have a film made, receive help with press and publicity and be given a set of promotional materials.\nThe Community Fund will award one-off grants of up to \u00a31,000 to local charities, businesses and schools. The Fund will award grants to support initiatives that will deliver a real impact by improving the environment and helping people appreciate the benefits of nature to the health and wellbeing of local communities and business. The grants should be used as part of the total budget of a project.\nTo encourage and inspire people to \"get better with nature\" by making a small investment in their local environment \u2013 whether it's in their community, local green space, wildlife or waterways.\nThe closing date for entries is 16 September 2016, and any activities must be underway by the end of 2016 and completed by 31 May 2017.\nIf you are interested in applying or would like further information about Get Better with Nature, the Community Fund and last year's grant winners, visit the Community Fund page where you will also find detailed guidance on how to apply and a Community Fund Application Form.\nIf you have any other questions, feel free to get in touch with us at enquiries@getbetterwithnature.co.uk.\nKnowledge and experience of using the latest trends in website design and social media is needed plus the spare time to help us on an occasional, voluntary basis with page design and content management.\nSomeone with interest in railway history or the countryside and able to offer IT skills to complement your hobbies.\nDistrict National Park. We work closely with station friends volunteer groups at many local stations and support a programme of Folk Trains and Guided Walks on the Hope Valley, Glossop and Buxton railway lines.\nContact Andrew Walker, the Community Rail Partnership Officer, for an informal discussion and more information. Email: andrew.walker@derbyshire.gov.uk Tel: 01629 538093.\n3. Supports staff who are experiencing mental health problems.\nThe Mental Health First Aid courses address all of these issues. They also offer bespoke training on mental health awareness and training for managers. Contact them for a free quote based on your training needs.\nTake part in the National Employee Mental Wellbeing Survey before the closing date of July 29. This has been launched by Business in the Community (BITC).\nCome and support COGS - Community Organising Green Space.\nDerbyshire County Council is providing funding until December 2017 to recruit, train and support community Befriending Champions across Derbyshire and to provide a microgrant to support local groups\/individuals to develop small scale local projects that reduce loneliness and isolation.\nThe half-day Befriending Champions training is free to anyone active in their local area, whether an employee or a volunteer, and includes information on what it is to be a Befriending Champion, the skills needed and how to carry out the role. The session also provides support to develop project ideas that reduce isolation and loneliness, helps with applying for the microgrant, and gives information on other sources of local funding. There is also the opportunity to attend a one-hour Dementia Friends awareness session. Four training sessions are planned across the County during the Autumn \u2013 dates and venues to follow soon!\nWhether you are a hairdresser, an architect or employ a personal care assistant, if you employ at least one person you are an employer and you have certain legal duties. This applies to you if you run a small voluntary organisation or charity that employs someone.\nTuesday 26 July 10.00 \u2013 11.30 at the Agricultural Business Centre, Bakewell, DE45 1AH.\nJohn Carter from ProAktive will be giving a presentation on automatic enrolment and there will be a discussion on the planning, implementation and ongoing administration requirements of automatic enrolment. In addition, you can find out if your existing pension scheme complies with automatic enrolment.\nThis is a free session and should last no longer than 90 minutes, refreshments from 9.45, opportunity to network or ask other advice from 11.30.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Project Manager manages the daily construction activities and delivery of construction projects, which involves contract preparation, on-site coordination, quality control, and all aspects of contract management. This individual will communicate directly with the Owner\/Developers, their representatives, and subcontractors as it relates to project cost, staffing, and scheduling. Additionally, they will prepare project status reports while working to ensure that plans adhere to contract specifications.\nOn projects with one Project Manager, that Project Manager is the field leader who will be reporting directly to the Project Executive Sponsor. On projects with multiple Project Managers on-site, they are the leaders of their assigned area and will report directly to the Sr Project Manager or Project Director. Regardless of rank, Project Managers share equal responsibility with the on- site Superintendents for the overall success of the project. Project Managers exercise direct and indirect supervision over Assistant Project Managers, Project Engineers, Project Assistants, and Interns.\nThis position description is a summary and may not be a complete representation of the position, and the essential functions of the position could change as duties are assigned.\nBuild Group is an equal opportunity employer. Pursuant to the San Francisco Fair Chance Ordinance, we will consider for employment qualified applicants with arrest and conviction records.\nYou can also apply to and see our postings on our LinkedIn Page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Milner Residence was designed as a spectacular rustic mountain home by renowned architecture firm Locati Architects, located in Moonlight Basin in Big Sky, Montana. This sprawling two story residence showcases a combination of stone and wood on its exterior facade. The home is integrated into its sloping site, surrounded by mature growth trees and rugged mountains. The privacy all around, this is the perfect family home to enjoy nature and partaking in outdoor activities all year round.\nWith the great outdoor literally in the backyard of this home, it is conveniently nestled in the Moonlight Basin ski resort area of southwestern Montana. Positioned in the Madison Range of the Rocky Mountains, the home was designed to maximize the mountainous views from every angle.\nWhat We Love: This striking rustic mountain home features the perfect balance of refined luxury and rustic details to make a comfortable home for everyday living. The mountain views seem to be perfectly framed from every room, it does not get much better than this. This is the ultimate home to enjoy an escape from city life, fresh air, nature and mesmerizing views from every direction\u2026 readers, what are your thoughts, would this be your idea of a dream home?\nNote: Have a look below for the \"Related\" tags, featuring home tours of Locati Architect's most popular work.\nAbove: A double height entryway features a north arrow medallion on the floor, arched entryway doors and a beautiful grandfather clock.\nAbove: A floor-to-ceiling stone fireplace is the focal point of the living room, with furnishings arranged around it. Expansive windows captures the surrounding mountains. Soaring ceilings features wood trusses, adding to the rustic aesthetic.\nAbove: The kitchen features a separate bar area from the island, providing a casual dining space.\nAbove: Next to the living room, a built-in buffet separates the spaces, while a lower ceiling height makes this area feel more intimate. A long and elegant dining table allows plenty of space for entertaining guests.\nAbove: A cozy guest bedroom features wood mixed with patterned wallpaper on the walls. The color of the wallpaper is repeated on the chair fabric and pillows.\nAbove: In the master bathroom, steps leads up to an opulent deep soaking tub with curved windows perfectly framing the rugged mountain views. To top off the aesthetic, a ceiling clad in wood features a striking chandelier suspended above the bathtub.\nAbove: Love this idea! A mining car is simulated into the wall, a cozy spot for a child to cozy up for the night. Complete with curtains for privacy and a nightlight for reading.\nAbove: Comfortable chairs in this home theater provides a great space for family entertainment.\nAbove: An outdoor seating area provides a sheltered space to relax after spending a day on the slopes. With comfortable seating and a fireplace, this area can be used all year long. Additionally, infrared heat lamps by Berko are not affected by wind, which helps keep this outdoor space feeling very comfortable during cold weather season. The heat output can also be adjusted to a comfortable temperature, perfect for entertaining your guests while soaking in the spectacular snow-capped mountain views.\nIs this a Bed and Breakfast?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What can I possibly do?. Some people take their platonic friendships very seriously as they should. There is always a little distance between you two. If they like you, they like you for who you are and you don't need to change for them. She might be nervous about including you in her plans without talking to you about it first. This could be a sign that she secretly wants to be more than a friend.\nYour female coworker may invite you to a house party or some kind of social gathering to test the waters. These are her ways of telling you that she does not like you being with other girls. If she looks from your eyes to your mouth, even better. Their body language says it Along with eye contact, body language is one of the most common ways people communicate their emotions non-verbally. You may notice that they start to pay more attention to their appearance when they know they are spending time with you. They're just looking for a positive reaction but don't have the guts to tell you in an obvious way. Pinterest You wouldn't usually assume that someone won't be your friend in the future, but when your guy friend is writing you into big plans, it might be because he has more than friendly feelings for you.\nWhenever she is sad, happy, troubled, or upset, she always looks for you. No matter what the situation is like, you can count on her being there with you and for you. Of course many women hate other women for being attractive instead of their boyfriends for carrying out behavior that makes them feel insecure. If people are telling you that your best friend is interested in you, then you should listen. She hopes you will notice that she wants to be more than a friend with you.\nThey make excuse to be with you alone One of the most frustrating things about having intimate feelings for a friend is the inability to spend time with them alone, as it is usually the case that friends spend time together in groups of more than just two. She become awkward around you A girl acts awkward and shy around the guy she likes. Start the conversation off with how you need to tell her something but want assurance that what you say will not end the friendship because you want her in your life no matter what. Being this highly invested in someone else's life means that you have to be thinking about them a lot, and he clearly is. You can notice this because she may bring them up at a later time. Either, things could get really awkward between the both of you.\nNotice if she touches your hands, hair, or arms. If this happens, be on the lookout for signs that can help you determine whether they're becoming romantically interested in you, or if your relationship will stay as friends. These are obvious signs that your friend doesn't want to miss a chance to be around you. Compare how they treat you. Saying why the girl is not good enough for you should be okay, we all want the best for our friends.\nAlso since I live with her family in her house at the moment I wonder if, even if she is into me, whether or not it's a good idea to start something. That's basically why he's your boyfriend. But all of a sudden, you begin to think that maybe you're not on the same wave length anymore, she has started to laugh at your stupid jokes instead of staring at you in disgust. Late at night or early in the morning is definitely included. In this case, several readers have written to tell us that this article was helpful to them, earning it our reader-approved status.\nTogether, they cited information from. Try to understand their pain and disappointment. If she responds to this with pleasure, she very well may like you more than coworkers. This usually only works out long term if there was zero chemistry to begin with, and it was a total joke that you ever tried dating. They may try to bring up romantic subjects or ask whether you are interested in someone right now. When he secretly wants you, he will do stuff like be nosy about your dating life and then get noticeably irritated like he's sorry that he asked.\nHer body language was saying yes to love,but she did not told my mouth. But she doesn't hate you. Or else, she asked you to pick her up and spending an unusually longer time to meet you friends. It is easy to spot the signs of how someone really feels about their friend by observing the manner in which they speak about them candidly. If you don't like them back, tell them that you're sorry, but that you will always be there for them no matter what. When you guys are together there's always the chance of getting close that something might happen between you, but the odds are way less if everyone is supposedly in happy relationships.\nWhat the person is trying to do is make their friend notice them first of all, and second of all to gauge their reaction to see whether or not their friend is bothered by seeing them with someone else. I told her her that I was not and she dropped the topic. To me, she doesn't seem into me. He doesn't imagine that there's a time where you won't be his number one because that's just simply what he wants from you. Tell them how much you value their friendship. At the bottom of this article, you'll also find information about what to do if your friend likes you, including reasons to date or not to date your bestie! Your feelings and the things you say to each other are private and should be held in confidence, no matter what they are. Teasing is also a sign of interest because she wants to elicit an emotional response from you.\nOr at least she wants you to be. He pays attention to what makes you laugh and can anticipate why you'll like something or you won't. If you feel kind of weird hanging out with your boyfriend and his female friends, then your gut instincts are telling you or pretty much shouting from the rooftops that there is something seriously wrong. This is simply because I only see her while she is at work. It is often the case that these individuals seem to get angry at the smallest things, due to their being so wound up and troubled about their situation. I think it was caused by: Nothing really Trying to figure out my friend's change in behavior? Not the negative one, but enough to convince you that she might likes you. I moved to a different country couple of months ago to study.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"VG6 GAMMA 300BLK The VG6 GAMMA 300BLK is a muzzle brake and compensator hybrid. It virtually eliminates recoil and minimizes muzzle movement. The unique combination of both braking and compensating features inspire shooter confidence and allows the shooter to make very fast follow up shots. The VG6 GAMMA 300BLK has been tuned for versatility as well. The brake has been optimized for 10.5\" to 18.5\" barrel lengths, but ultra-shorty SBRs and AR-Pistols with shorter barrels and full sized 20\" rifles will benefit greatly from the muzzle brake as well.\nNot all muzzle devices are created the same. There aren't many muzzle brakes and compensators that 'understand' proper high-pressure gas dynamics. The VG6 GAMMA 300BLK was designed using state of the art engineering techniques, utilizing CFD (Computational Fluid Dynamics) and CAE (Computer Aided Engineering). 100% CNC machined for optimal performance. Stainless steel construction for longevity. The VG6 GAMMA 300BLK is a muzzle brake and compensator hybrid. It virtually eliminates recoil and minimizes muzzle movement. The unique combination of both braking and compensating features inspire shooter confidence and allows the shooter to make very fast follow up shots. The VG6 GAMMA 300BLK has been tuned for versatility as well. The brake has been optimized for 10.5\" to 18.5\" barrel lengths, but ultra-shorty SBRs and AR-Pistols with shorter barrels and full sized 20\" rifles will benefit greatly from the muzzle brake as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When a pesky stalker suddenly moves his game from the cyber world to the real world, Holland becomes the one person she trusts implicitly. An anchor in the midst of her chaos. A man whose ruthless business instincts turn to devastating tenderness after hours.\nUntil his own secrets start to unfold, leaving Jillian to wonder if her beautiful business partner is actually the one spinning the deadly web that's fast closing around her.\nI was first introduced to Jo Grafford when I read Trail of Crosses. I enjoyed it, as such when I was asked to read and provide an honest review for Designed for You I did not hesitate to accept. The fact that it fell in the romantic suspense genre sweetened the deal.\nMy views on this title are mixed. I will try to explain what I mean in the best possible way. I thought the story line was good. Designed for You introduced Jillian Lang and Holland Sparks. Jillian is an interior designer and the owner of Lang Interiors. She has been divorced for two years and is still trying to come to grips with her ex-husband's betrayal. She is looking to hire an office manager to aid with the day to day operations of the business, leaving her time to focus on what she enjoys most, designing.\nHolland Sparks chose to answer Jillian's ad for an office manager. Although he appeared to possess all the qualities of her ideal candidate, she could not help but wonder about his reason for applying. Why would some who possessed such excellent qualifications and experience would want to work at a small and unknown business? What she did not know was that Holland's intentions were not has he purported them to be.\nThe story started out on a high. I was sucked into it from the first page. I liked the fact that it was discovered that there was someone stalking Jillian prior to Holland entering the movie. This had me wondering if the stalking had any connection to Holland's mission or if it was an isolated situation. I enjoyed the dialogue between Jillian and Holland at their initial meeting and throughout the story. It was funny and entertaining. It was evident from the moment the two met that they had great chemistry.\nThe suspense was ok. There were events that I never saw coming. I also liked that the stalker's identity was not revealed until near the end of the story.\nhad a hard time connecting with the main characters. The secondary characters were fine. I enjoyed reading about them and in fact, I would love to read their stories. The backstories for the main characters were lacking. It would have been good to have information about their past as this would have helped me to have a better understanding of what motivates them, i.e. why they do the things they do? It would have been good if more information was provided about the stalker. I was not impressed with him at all. I wish I was privy to what made him tick.\nI thought the romance lacked depth and authenticity. After Jillian and Holland met, the story fast forwarded two months later at which time I saw them acknowledging to themselves how they felt about each other. I wondered to myself at what point did they know how they felt as I was not privy to this information. I am unable to state explicitly how well they got to know each other. It was implied but there was no evidence to support this implication.\nThe storyline is what saved this book for me. Although the story was fine, the fact that the characters and the romance were lacking took away some of the enjoyment for me. As a result, I gave this title three stars. Although this did not go as expected, I would still read more of this author's work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Keith Kirk appeared in court Thursday with a face that looked like it had been trampled on by a horse.\nKirk, 30, showed up in Fort Lauderdale bond court and stood before Judge John \"Jay\" Hurley with black swollen eyes and stitches between his eyebrows.\nIt turns out that Kirk allegedly tried to steal someone's bicycle and ended up on the wrong side of the fists of justice.\nAnd while it looks like Kirk's pummeled face might have been the work of Batman, it was, in fact, the work of the bike's owner, Calvin McDougal.\nKirk is accused of stealing McDougal's $600 bicycle outside the One Stop Food Mart at 18 NW First Ave. in Fort Lauderdale.\nWe saved Kirk's mug shot for after the jump, in case you may not like looking at a dude's swolled-up face during lunchtime.\nAccording to the police report, Kirk took the bike and was then promptly chased down by McDougal. The two got into an altercation, and the robbery pretty much ended when Kirk ran into McDougal's fists a bunch of times.\nHe was eventually transported to Broward Health Medical Center for medical clearance. Because his face looked like a half-cooked hamburger.\nKirk explained his injuries to Hurley by saying he was tackled by McDougal.\n\"I believe there was a little bit more than just a tackle,\" Hurley said. \"Now, sir, you allegedly steal someone's bike and it looks like there was some street justice carried out.\"\nKirk was charged with grand theft and released on his own recognizance. He was allowed to leave jail to get more medical treatment on his face.\nAs he walked away from the podium, Kirk muttered, \"Did you notice how the guy who owned the bike didn't get arrested?\"\nOf course, it's not a crime in Florida for victims to fight back and beat the BeJesus out of their muggers. Although cops insist that people should leave the crime-fighting to them.\nThey might want to consider hiring this McDougal fella.\nI love when the victim emerges victorius!\nI have no sympathy for thieves, wonder if he tries to steal another bike anytime soon.\nThe guy is a druggie by the looks of it.\nThis type of thing can't happen often enough.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Fran\u00e7ois Fillon is the favourite to win the upcoming presidential election. Here's what you need to know about the open admirer of Margaret Thatcher.\nOpenly saying you are an admirer of the former Conservative British Prime Minister is normally risky business for a Frenchman.\nShe and Ronald Reagan are often seen as the bad parents of the neoliberal economic revolution in the 1980s that France has never really adjusted to.\nBut Fillon, who was Prime Minister under ex-president Nicolas Sarkozy for five years, likes the Iron Lady because according to him she got her country back on track.\n\"She is the symbol of an inflexible political determination to stop a situation of decline,\" said Fillon back in 2014.\nAnd in the run up to the first round of the right wing primaries Fillon hit back at his rivals who compared him to Thatcher in a bid to turn French voters against him.\nFranois #Fillon is the only candidate to deeply admire Margaret Thatcher and say it out loud. Quite unusual in France, to say the least.\nThe word \"Thatcher\" was trending in France the day after the first round in a sign the French public were trying to find out just how much Fillon was a fan.\nLeft leaning newspaper Liberation made it clear on their front cover exactly what Fillon means for France.\nFillon's love for all things Thatcher is linked to his desire for a \"total rupture\" as a means to pull France's struggling economy into the 21st century.\nLike Thatcher Fillon wants to cut back state spending, or more to the point completely shred it and impose liberal economic policies.\nHe says he wants to make \u20ac100 billion of savings in five years. To do that he wants to cut 500,000 jobs in the French civil service, raise the retirement age to 65, scrap the 35-hour week for the private sector and raise the legal working week to 39 hours for civil servants.\nIn terms of the working week, he would simply set a maximum of 48 hours in line with EU law.\nSome 20 percent of the \u20ac110 billion worth of cuts will be made by local authorities. Fillon also wants to put a lower cap on unemployment benefits.\nHe's also a friend of business and wants to cut levies on firms to the tune of \u20ac40 billion and implement \u20ac10 billion of tax cuts for households. He also wants to scrap France's wealth tax on the richest residents, called ISF.\nTo cover for these cuts, he wants to raise VAT by two percentage points.\nFillon also wants to scrap most of France's labour laws, and leave disputes to be sorted out at sector or company level, to try and break the power of the unions.\nIn October French right wing magazine Le Point made it clear it too was in favour of a dose of Thatcherism to turn the country around.\nFillon is married to Penelope from Abergavenny Wales. Is France ready for a Welsh first lady? CLICK HERE to find out more about Penelope Fillon.\nA French newspaper has alleged on Tuesday that Penelope Fillon was paid half a million euros with funds made available to her husband by the French parliament.\nThe British-born wife of French presidential candidate Francois Fillon was paid around \u20ac500,000 ($538,000) over ten years from parliamentary funds made available to her husband, the report said.\nNot many mainstream French politicians have anything in common with Donald Trump but Fran\u00e7ois Fillon does. He too is a fan of Russian leader Vladimir Putin and favours an alliance with Moscow to battle Isis in the Middle East, which the current French administration under Fran\u00e7ois Hollande has avoided, due to Russia's support for Syrian dictator Bashar al-Assad.\n\"It's clear Russia has its own interests in the region, but who in the Middle East doesn't? Fillon wrote in Marianne magazine recently.\nFillon saluted Putin's \"cold and effective pragmatism\" in the region.\nThe former prime minister might not be as obsessed by identity and the role of Islam in France as his former boss Nicolas Sarkozy, but Fillon still has fairly strong views that will not make the country's Muslim population feel at ease.\n\"There are no problems with religion in France. There is a problem linked to Islam,\" said Fillon.\nHowever Fillon is not in favour of banning religious symbols like the Muslim headscarf in public, as Sarkozy suggested he was.\nLast year he provoked ire and ridicule dismissed the idea that France should feel guilty about its colonisation of North Africa, saying \"there was nothing to be ashamed about France just wanting to share its culture.\"\n\"For Fillon, colonialisation was just like Erasmus,\" read one sarcastic reaction on Twittter.\nFillon was the preferred candidate of the once influential \"Manif Pour Tous\" anti-gay marriage movement in France because he closely reflected their views.\nAlthough Fillon says he will not try to overturn the legalisation of gay marriage that was brought in in 2012, he does want a change of the law to prevent gay couples from being able to adopt children.\nFillon is also against surrogacy and medically assisted births for lesbian couples. Feminists have been reminding France that Fillon has said he regrets saying abortion was a \"fundamental right of a woman\".\nBut he does want to boost family allowances and make the payments universal rather than linked to salary as they are now.\nFillon has insisted his vision of France was not as a multicultural country.\nWhen asked whether he saw the future of French society as multicultural he said: \"the answer is no\".\n\"France has a history, a language, a culture, of course this culture and language have been enriched by the contributions of foreign populations, but it remains the foundation of our identity,\" he said.\nWhen asked during debates late last year if France was already a multicultural country Fillon said \"No, in any case it's not the choice we made, we did not make the choice of communitarianism and multiculturalism.\"\n\"When we go to somebody's house, we don't try to take power,\" said Fillon adding that immigrants must respect France's cultural heritage.\nHe thinks colonization was just \"sharing culture\"\nFillon provoked anger and ridicule in equal measure last year when, in a speech to supporters, he said that France should not be guilty of its colonial part in North Africa, because it was all just about \"sharing culture\".\nHe also said he would change the history curriculum so pupils would \"not be taught to be ashamed of their country\".\nThis video sums up the reaction to his comments.\nThis story has been updated since it was published in November 2016.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/23\/2019 02:46:52 pm at 04\/23\/2019 02:46:52 pm in Rustic Kitchen Shelves.\nrustic kitchen shelves rustic kitchen shelves shelves modern country house kitchens kitchen rustic kitchen shelves rustic wooden shelving combines rustic kitchen shelves diy fail.\nwhite subway tile with rustic kitchen shelves,rustic shelves brackets,rustic kitchen shelves uk,simple covering for rustic kitchen shelves,floating rustic kitchen shelves,rustic shelves and cabinets,rustic shelves medfield ma,rustic kitchen shelves diy pinterest,rustic shelves uk,rustic shelving units for storage,open rustic kitchen shelving.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Earlier this month, we presented the results of the pilot phase of VerbCorner -- our citizen science project probing the nature of linguistic structure -- at a scientific conference (the Workshop on Events in Language and Cognition). You can see the poster describing the work here.\nFor those who don't know or don't remember, in VerbCorner, we're trying to work out the grammar rules that apply to verbs. Why do you say Agnes looked at the wall but not Agnes saw at the wall? Why do you say Bart filled the glass with water but not Bart poured the glass with water? Many -- but not all -- linguists believe that grammatical idiosyncrasies are explained by the meanings of the verbs, but evidence is sketchy. Volunteers have been visiting our website to help analyze the meanings of verbs so we can find out.\nOur initial work -- the pilot for the pilot, if you will -- suggested that we could get high-quality analyses from volunteers. But that was based on a very small sample. As of late Feb, over 10,000 volunteers had contributed over 525,000 analyses. In general, the agreement between different volunteers was pretty high -- which is a good sign. Just as importantly, we had a smaller set of 'test' items, for which we knew what professional linguists would say. When we combine the analyses of different volunteers for the same sentence in order to get a 'final answer', the results match the analyses of professional linguists very well. This shows that we can trust these results.\nJust as importantly, we were able to analyze a lot of sentences. In the VerbCorner project, we are trying to determine which sentences have which of a very specific set of aspects of meaning. One aspect is whether the sentence involves something changing physical form (example: Agnes broke the vase as opposed to Agnes touched the vase). Another aspect is whether the sentence involves anything applying physical force to anything else (ex: Agnes pushed Bart as opposed to Agnes looked at Bart).\nThis quantity makes a big difference. Given how small studies usually are, they can only look at one tiny corner of the language. The problem is that that corner might not be representative. Imagine studying what Americans are like by only surveying people in Brooklyn. This tends to lead to disagreements between different studies; one linguist studies \"Brooklyn\" and another studies \"Omaha\", and they come to very different conclusions! Unfortunately, language is so complex and so vast, one person can only analyze one corner. This is why we are recruiting large numbers of volunteers to help!\nOne major question we had was how much the rules of verb argument structure (that is, the kinds of grammatical rules described above) depend on meaning. Some linguists think they depend entirely on meaning: If you know the meaning of a verb, you know what its grammar will be like. Others think meaning has very little role to play. Most linguists are probably somewhere in the middle.\nThe results suggest that the first group is right: These rules depend almost entirely on meaning. Or maybe even entirely; it's so close it is hard to tell.\nThe reason I say \"suggest,\" however, is that while we have the biggest study of its kind, it still only covers about 1% of English. So we've gone from studying Brooklyn to studying all of NYC. It's an improvement, but not yet enough.\nThis is why I called this first phase a \"pilot\". We wanted to see if we could get high-quality, clearly-interpretable results from working with volunteers. Many researchers thought this would be impossible. After all, linguists have to go through a lot of schooling to learn how to analyze sentences. But a key finding of the Citizen Science movement is that there are a lot of smart enthusiasts out there who may not be professionals but can very much contribute to science.\nWe have set a goal of reaching 50,000 completed items by July 1st. That will require upping our game and increasing the rate at which we're analyzing items by almost 4x. But the beauty of Citizen Science is that this does not really require that much work on anyone's part. If 3,000 volunteers each spend about one hour contributing to the project, we'll more than hit that goal. So please help out, and please tell your friends. You can contribute here.\nThe brand-new Language Learning Laboratory at Boston College is recruiting a full-time research assistant. The research assistant will work closely with the PI (Dr. Joshua Hartshorne) and graduate students in the lab. Primary responsibilities will include coordinating the lab's crowdsourcing and citizen science activities. For example, over 10,000 volunteers have contributed over 500,000 linguistic judgments as part of the laboratory's VerbCorner project. The research assistant will help coordinate these volunteers for this and other similar projects. S\/he will also manage undergraduate researchers working on these projects and engage in public outreach activities such as blogging or creating educational materials. S\/he will assist in data-analysis and have the opportunity to attend and present at major scientific conferences.\nCandidates should have an undergraduate degree in psychology, neuroscience, linguistics, computer science, or a related field (or a good explanation as to why they are qualified anyway). Candidates should also have familiarity with one or more computer programming languages (e.g., Python, R, Matlab, C++) or an exceptional quantitative background (i.e., degree in mathematics). Experience with any of the following would be an added advantage: laboratory research, data analysis, management\/supervision, science outreach, journalism, machine learning.\nReview of applications will begin immediately. Start date is flexible but not later than 9\/1\/2016. Members of groups underrepresented in science are particularly encouraged to apply. International candidates are welcomed but must have an MA or equivalent.\nTo apply, send to l3atbc@gmail.com: a CV and a one-page essay explaining why you are interested in the position and how it fits with your past experiences and future goals. Please also arrange for letters of recommendation from 2-3 references to be sent to l3atbc@gmail.com.\nPlease be sure that your CV lists your degree(s), major\/minor, GPA, any relevant classes (psychology, linguistics, computer science, etc.), programming languages with which you have experience (and the nature of that experience), and any other experiences\/qualifications you feel are particularly relevant.\nNot enough published studies replicate. There are a number of reasons for this, but one of the most difficult to address is statistical power.\nWhy is low power a problem?\nSuppose you want to test the effect of some training on IQ scores. You test 50 subjects: 25 in your experimental condition and 25 in the control condition. That's a fairly typical size for a psychology study. And you get a significant result. You might be tempted to conclude that your manipulation worked, but it might actually be more likely that your results are due to chance or experimenter error.\nIt depends on how large of an effect your manipulation ought to have. If the training typically raises people's IQs by 7.5 points, your study would only have had a 41% chance of detecting it (given normal assumptions about normal distributions). A more modest 5 point effect could be detected 20% of the time. You'd need a 14 point effect to have a 90% chance of detecting it.\nFor reference, a 14 point effect is enough to move someone of average intelligence into the top fifth of humanity. We are in miracle drug territory. More realistically, your manipulation is unlikely to have nudged IQ by more than a point or two, in which case there's essentially no chance your study could have detected it. So if you did detect something, it's probably an error.\nWell, how much power do studies have?\nConcerns about statistical power (among things) have led some researchers to declare that more than half of all published findings are false. Other researchers are more sanguine. In short, if you think that the effects we are studying tend to be pretty large (a standard deviation or more), there is little to worry about. If you think they tend to be small, the situation is dire.\nUnfortunately, the only way to accurately determine how large an effect is is to measure it multiple times. Obviously, you can't have done that in advance when running a brand new study. You might be able to guestimate based on the typical effect size in your field. Unfortunately, many fields primarily publish significant results. This introduces a bias, because effect size and significance are correlated.\nSuppose we ran the above experiment and the true effect size is 7.5 IQ points. On average, that is what we would find. But of course sometimes we'll run the experiment and the effect will be larger and sometimes it will be smaller, simply due to random chance. By paying attention only to the significant results, we're selectively ignoring those experiments that happened, through no fault of their own, to underestimate the effect. This skews our results, and on average we would report an effect of 11.8 IQ points -- much higher than the truth.\nEffects aren't as big as you think.\nThe typical effect reported in the psychology literature is about half a standard deviation, which is why I've been using the example of 7.5 IQ points above. However, because of the bias against publishing null results or replications, this is inflated. That means that the expectations of psychologists are inflated. We are used to seeing effects of half a standard deviation or more. As a result, we are going to chronically underestimate the number of subjects we need to run.\nUnfortunately, without widespread publication of null results and replications, we cannot say how badly our perception is distorted, because the degree of distortion depends on how large effects really are. I ran a series of simulations involving a two-condition, between-subjects design to see how bad the distortion might be. In these simulations, I assumed that null results are never reported, which is only a slight exaggeration of the psychology literature.\nIn the graph below, the true effect size (measured in standard deviations) is on the X-axis, and the average reported effect size is in the Y-axis. The different lines reflect different numbers of subjects per condition.\nAs you can see, if you have 50 or fewer subjects per condition, you'll hardly ever report an effect size smaller than half a standard deviation, even when the true effect size is one tenth of a standard deviation. This is because reliably detecting an effect of one tenth of a standard deviation requires about 2,000 subjects per condition.\nEven with 1,000 subjects per condition, there is some distortion for effects smaller than one quarter standard deviation.\nNote that these simulations assume that the researcher is correcting for multiple comparisons, isn't p-hacking, etc. The situation gets worse if we relax those assumptions.\nIn the simulation on the left, I assumed the researcher is engaging in contingent stopping. After every 5 subjects, the researcher checks her data. If the effect is significant, she stops and reports the result. If the effect is in the \"wrong direction\", she decides there's something wrong with her stimuli, revamps the experiment, and tries again.\nWhile this is a little extreme, most researchers engage in some amount of contingent stopping. As you can see, this badly distorts the results. Even with 1,000 subjects, we end up distorting even large effects.\nFailure to correct for multiple comparisons will have similar effects.\nWhat this means is that if you are basing your expected effect size on prior experience, the effect you are trying to detect is probably not anywhere near as large as you think, and you may need many more subjects to detect it than you were planning.\nBut I usually get significant results. Doesn't that mean I have enough subjects?\nMaybe. You might be testing an abnormally large effect. Such effects do exist. Alternatively, you may be engaging in contingent stopping, failing to adequately correct for multiple comparisons, or simply making a mistake somewhere in your analysis. It's probably worth checking.\nshould keep in mind that it is so unlikely you could see an exoplanet with your binoculars, that if you do see one, you are probably mistaken in some way.\nI don't use t-tests. How does this apply to me?\nMy examples above use t-tests because they are simple and widely known. But the same basic issues apply no matter what kind of analysis you do. If you are looking for some kind of effect, and if that effect is small, you'll need a lot of data to detect it. And it's probably smaller than you think.\nIf you are model-fitting, the noise in your data puts an upper limit on how well you can fit the underlying phenomenon. If your data are fairly noisy and yet your model fits really well, you are probably fitting the noise, not the signal. And your data are probably noisier than you think.\nI can't tell you how many subjects to run. I don't know. Nobody knows. Until there is a good mechanism for reporting null results, nobody will know.\nIn the meantime, I recommend running more than you think you need. If there are published studies looking at a similar phenomenon, look at their effect size and assume the true effect size is significantly smaller, then do a power analysis. If you run your study and get a positive result, it's probably a good idea to replicate it. If you get a null result, you might not wish to read too much into that.\nIf you don't have the means to test a lot of subjects, you have a few options. Effect size is a combination of the signal-to-noise ratio and the amount of data you collected. If you can't collect more data, you can try to decrease the noise (e.g., by using a better, more accurate instrument). You can't necessarily increase the signal, because you can't change the laws of nature. But you can decide which laws of nature to study, you and you might be better off studying one that has powerful effects.\nSee below for the R code to run the simulations above. If anyone knows how to convince blogspot to allow indention, please lemmeno.\n#inter = after this number of subjects, check results. Stop if significant *or* in the wrong direction. Default to running all subjects.\n#verbose = should we print interim results?\n#on each round, test an addition inter subjects.\nn<- span=\"\">inter #how many subjects do we need for this round?\n#preliminary results are in the wrong direction. Revamp experiment and try again.\nIn case you wanted to know more about our VerbCorner project.\nMany thanks to the two undergraduates who helped make this video.\nIt's been a very busy time at GamesWithWords. I'm pleased to announce that we'll be moving to Boston College in January. The impending move, combined with a large number of papers to write, has kept me too busy to write much on this blog.\nAgain, you can see that most of the non-native dialects cluster together. Most of the native dialects do not connect directly to that cluster but rather connect to Singaporean. Again, you can see Standard American and AAVE off in their own cluster.\nWe're working on a browser that will let you see *why* different dialects are more or less related -- that is, what answers in the quiz are typical of which dialects. I'm hoping it will be ready soon. In the meantime, enjoy the dialect browser.\nI've updated the interactive visualization of the relationships between the Englishes of the world to include a couple dozen additional native languages. Check it out.\nAround 4am EST on May 28, we started getting *a lot* of traffic to the website. This very quickly overloaded the server, resulting in the website running very slowly. We did some optimization. Things sped up, and our reward was more traffic. So we switched to a more powerful server. And so on.\nOf course, overwhelming amounts of traffic is a good problem to have, and I won't complain if things overheat again.\nI've gotten a number of questions about the science behind our WhichEnglish quiz. Actually, I had intended to post a more detailed discussion days ago, but I got distracted by other matters.\n1. How does the age at which you start learning a language affect how well you learn that language?\n2. How is learning a foreign language affected by the language you already know?\n3. How are the grammars of different English dialects related?\nAnd of course, we train an algorithm to predict participants' native language and dialect of English based on their answers. I return to that at the end.\nAlthough WhichEnglish has a few scientific targets, age-of-acquisition effects were the original inspiration. Everybody knows that the older you are when you start learning a foreign language, the harder it is to learn. One possibility is that there is a critical period: Up to some age, you can learn a language like a native. After that age, you will never learn it perfectly. The other possibility is that there is no specific period for language learning; rather, language-learning simply gets a little harder every day.\nThe evidence is unclear. Ideally, you would compare people who started learning some language (say, English) from birth with people who started as 1 year-olds and people who started as 2 year-olds, etc. Or maybe you would want something even finer-grained. The problem is that you need a decent number of people at each age (50 would be a good number), and it quickly becomes infeasible.\nOne study that came close to this ideal used census data. The authors -- led by Kenji Hakuta -- realized that the US census asks foreign-born residents to rate their own English ability. The authors compared this measure of English ability with the year of immigration (an approximation for the age at which the person started learning English). Their results showed a steady decline, rather than a critical period.\nWe are trying to build on this work in a few ways. For one, it would be nice to confirm (or disconfirm) the previous results with a more sensitive measure of English ability. So rather than just ask people how good their English is, we have them take a test. Also, we are getting more precise information about when the participant started learning English and in what contexts.\nAlso, there is good reason to suspect that age-of-acquisition affects different aspects of language differently. Studies have shown that even people who began learning a language as toddlers have detectable -- if very subtle -- accents. However, people who start learning foreign languages as adults usually report that learning vocabulary isn't so hard. Grammar seems to be somewhere in between. The Hakuta study didn't distinguish these different aspects of language.\nWhichEnglish focuses on grammar. We also have a vocabulary quiz to look at vocabulary. A pronunciation test is in the works.\nWhen we started thinking about studying age-of-acquisition effects, we quickly realized a problem. We needed questions that would be difficult for someone who learned English as a second language. But which aspects of grammar are difficult seems to depend on your first language. I personally have difficulty with aspect in Russian because the English aspect system is much less complex. However, dealing with tense in Russian is relatively straightforward, since the Russian tense system is much less complex that English's.\nSince we didn't know for sure what the language backgrounds of our participants would be, we wanted a range of questions that covered the different kinds of problems people with different backgrounds might have.\nAs we combed the literature, we realized that it was pretty fragmented. One study might say that grammar rule x is difficult for Japanese-speakers and grammar rule y is difficult for German-speakers, but there would be no information on how Japanese-speakers fare with grammar rule y and how German-speakers manage with grammar rule x. This actually makes sense: Most studies look at speakers of one or at most a handful of language backgrounds. This is partly a matter of research interest (the researchers are usually interested in some particular language) and partly a matter of feasibility (in a lab setting, you can only test so many participants). We realized that our study, by virtue of being on the Internet and recruiting people from a wide array of backgrounds, would provide an opportunity to get more systematic data across a large number of languages.\nThis is pretty exploratory. We don't have strong hypotheses. But as data comes in, we will be analyzing to see what we get, and we will report it here.\nIn designing our age-of-acquisition study, we realized a second problem. Correct English grammar varies across different dialects. In Newfoundland, you can say \"Throw me down the stairs the hammer,\" but most places, you can't. (I have heard that this is said in parts of Rhode Island, too, but only anecdotally.) We don't want to count a late-learner of English who says \"Throw me down the stairs the hammer\" as not knowing English if in fact she lives in Newfoundland!\nSo what we really wanted were questions for which the correct answer is the same in all English dialects. But we didn't know what those were. Again, the literature was only partly helpful here. For obvious reasons, researchers tend to be interested in understanding peculiar constructions specific to certain dialects, rather than recording what is the same everywhere (boring).\nWe picked out a lot of grammar rules that we at least had no reason to believe varied across dialect. But we also realized that there was an opportunity here to study differences across dialects. So we included a subset of items that we thought probably would be different across dialects so that we can explore relationships across dialects.\nWhen you take the quiz, at the end we give you our best guess as to what your native language is and what dialect of English you speak. How is that related to the three issues I just discussed?\nIt's deeply related. The best way of proving that you understand how people's understanding of grammar is affected by the age at which they started learning, their first language (if any), and the dialect of English they speak, is to show that you can actually distinguish people based on their grammar. In fact, training an algorithm to make just that distinction is a common way of analyzing and exploring data.\nThere are also obvious practical applications for an algorithm that can guess someone's language background based on their grammar (for education, localization of websites, and so on).\nBut an important reason we included the algorithm's predictions in the quiz itself was to present the results of the study to participants in the study as the study goes on. Certainly, you can read this and other blog posts I've written about the project as well. But it probably took you as long to read this post as to do the quiz. The algorithm and its predictions boil down the essence of the study in a compelling way. Based on the (numerous) emails I have gotten, it has inspired a lot of people to think more about language. Which is great. The best Web-based studies are a two-way street, where the participants get something out of the experience, too.\nWe chose the particular algorithm we use because it runs quickly and could be trained on very little data. You can read more about it by clicking on \"how it works\" in our data visualization. We are testing out more sophisticated algorithms as well, which are likely to do much better. Algorithms for detecting underlying patterns is actually a specialty of my laboratory, and this will be a fantastic dataset to work with. These algorithms mostly run too slowly to use as part of the quiz (nobody wants to wait 10 minutes for their results), but the plan is to describe those results in future posts and\/or in future data visualizations.\nIf you have any questions about this work, please ask in the comments below or shoot me an email at gameswithwords@gmail.com.\nGamesWithWords.org will be experiencing periodic outages as we upgrade* the server. The incredible response we've had for WhichEnglish has completely overwhelmed the server. After bringing it back from the dead multiple times, the techs at Datarealm convinced me to upgrade to the next tier of server.\nThis is possibly overkill, in that we don't normally get the kind of traffic we got today. Over 12% of *all* visitors to the website since Jan. 1, 2008, came in the last 24 hours! Still, traffic has been steadily rising over the last year, and large spikes are getting much more frequent.\nWorst case scenario, this should result in a faster, more stable experience for people going forward.\n*Upgrading while there is heavy traffic to your website is not ideal. But then neither is having the site crash constantly.\nAfter my optimistic comments about \"overkill\", I've spent most of the last 5 days performing various upgrades to the server. Traffic to the site peaked at about 100,000 visits\/day (it was a little lower Sunday, but then weekend traffic is usually down).\nI have updated the dialect chart based on the results for the first few days. Since the new version shows up automatically in the frame in the previous post, I haven't added it in here. And you can get a better look at it on the website.\nThe biggest difference is that also added several \"dialects\" for non-native speakers of English. That is, I added five new dialects, one each for people whose first language was Spanish, German, Portuguese, Dutch, or Finnish. I'll be adding more of these dialects in the future, but those just happen to be the groups for which we have a decent number of respondents.\nAs you can see, the algorithm finds that American & Canadian speakers are more likely one another than they are like anyone else. Similarly, English, Irish, Scottish, and Australian speakers are more likely one another than anyone else. And the non-native English speakers also form a group. I'll leave you to explore the more fine-grained groupings on your own.\nIt's widely observed that not everybody speaks English the same way. Depending on where you grew up, you might say y'all, you guys, or just you. You might pronounce grocery as if it were \"groshery\" or \"grossery.\" There have been some excellent, fine-grained studies of how these aspects of English vary across the United States and elsewhere, such as this one.\nBut vocabulary and pronunciation aren't the only things that vary across different dialects of English. We are in the midst of a soft launch of a new project which will, among things, help map out the differences in English grammar around the world.\nI put together a visualization of early results below (you may want to load it in its own page -- depending on your browser, the embedded version below may not work). You can use this graphic to explore the similarities among nine English dialects (American, Canadian, English English, Irish, New Zealandish, Northern Irish, Scottish, and South African).\nAs more results come in (about other dialects like Ebonics and Welsh, about specific parts of America or Canada, etc.), I'll be updating this graphic. So please take the survey and then check back in soon.\nLoad the graphic directly here.\nI just finished a radio interview about birth order.\nIt's been making me think I should do more birth order research, just so I have something to say at these interviews.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What do bunnies, beavers and bears all have in common? They're all absolutely precious characters in Yuliya Pieletskaya's illustrations! Originally from Ukraine, our newly signed artist, Yuliya now lives and works in Chicago, Illinois where she gets a great deal of inspiration from her 2 pet bunnies, Marnie and Ham. We can't get enough of Yuliya's sophisticated color palates and sketchy style! And animal lovers all over are sure to gush over these cute animal characters! Yuliya's illustrations are perfect for children's books, especially for little ones ages 0-5.\nTo see more beautiful work by Yuliya, be sure to take a peak at her online Advocate artist portfolio!\n\"Yuliya Pieletskaya was born in Odessa, Ukraine but spent the later half of her childhood in upstate New York. She graduated in 2012 from the Fashion Institute of Technology with a double major in design and business. She realized she wanted to get into illustration after interning at Marvel Entertainment. Though she had a business internship there, she would spend her extra time pestering the illustrators with questions and watching them render digital illustrations. Luckily they were nice enough to teach her a few things and she now uses a combination of traditional mediums and digital rendering in her work. Since graduating, alongside working on commissions and freelance illustration, Yuliya has been creating and selling her own collection of greeting cards and coffee mugs. Like many artists she finds, nature and animals a cherished source of inspiration. She currently lives in Chicago, Illinois with her two bunnies, Marnie and Ham. When she is not working you can find her in a city park checking up on the ever expanding wild bunny population of Chicago. If she wasn't an illustrator she would be working in wild animal rescue and rehab. Yuliya is a member of SCBWI.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Explaining something as tangled, technical, and multi-dimensional as the 2008 financial crisis is fraught with difficulty. Some have tried comparing toxic assets to supermodels, while others have given musical theater a shot.\nThis morning we have another answer--in the form of a 576-page book--from the congressionally appointed panel charged with investigating the roots of the meltdown. Were it not for corporate incompetence, inadequate government regulation, and excessive risk-taking by Wall Street banks in the housing market, the commission concludes, the country could have avoided financial calamity.\nThe Financial Crisis Inquiry Commission blames former Federal Reserve Chairman Alan Greenspan for failing to contain the spread of toxic mortgages and the Bush administration for its inconsistent bailout policy, while labeling the Democrats' decision in 2000 to not regulate over-the-counter derivatives \"a key turning point.\" Just as notably, the report doesn't accuse the low interest rates instituted by the Fed after the 2001 recession, government-backed mortgage providers Fannie Mae and Freddie Mac, or the government's aggressive push for homeownership of playing major roles in the crisis.\nBlame Government Housing Policies, argues Republican panel member Peter J. Wallison. He accuses Fannie Mae and Freddie Mac of fueling the risky subprime loans that crippled financial institutions when they turned sour.\nIt is dangerous to conclude that the crisis would have been avoided if only we had regulated everything a lot more, had fewer housing subsidies, and had more responsible bankers. Simple narratives like these ignore the global nature of this crisis, and promote a simplistic explanation of a complex problem. Though tempting politically, they will ultimately lead to mistaken policies.\nYou Can't Ignore Regulation, counters Mark Thoma at Economist's View, in scrutinizing the factors Thomas, Hennessey, and Holtz-Eakin say led to the financial crisis: \"These factors are, for the most part, a chain of events. If the chain had been broken by more effective regulation anywhere along the way, the chain of events is interrupted.\"\nWill Report Have Any Impact? wonder Nathaniel Popper and Jim Puzzanghera at the Los Angeles Times. The commission has produced three conflicting reports instead of one bipartisan report, and \"all three narratives deal with events already picked apart by a long list of books as well as congressional hearings,\" the reporters note. \"The panel's work may help inform government agencies that are writing rules to implement the Dodd-Frank financial regulatory overhaul enacted last summer,\" they add, \"but for many in the financial world, any relevance the commission might have had evaporated with the passage of that legislation and the recovery of the stock market.\"\nIt Will Refocus Scrutiny on Wall Street, states Richard Eskow at The Huffington Post: The panel \"documents the opportunism, bad judgment, and criminality that crashed the world's economy once--and could again at any time ... At a time when the nation's capital is convinced that CEOs need appeasing rather than policing, the FCIC report is a badly needed return to reality.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Posted by dtaitano December 5, 2014 3:01 pm Categories: Health Care. Tags: Hospital Surveillance.\nIn several hospitals and other healthcare sectors, surveillance has become a dependable tool for security development.\nThis is along with the cost control abilities it provides. Hospital surveillance cameras may work to protect the patients and hospital employees from security breaks, and offer useful visual evidence, which could be utilized for boosting productivity and for preventing dishonest claims.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"THAKEHAM NEIGHBOURHOOD PLAN IS NOW 'MADE'!\nOn 26th April 2017 Horsham District Council formally 'made' the Thakeham Neighbourhood Plan. The Plan is now part of the framework of district planning guidance, against which future development applications will be judged. Sites not identified for development in the Plan will be much less likely to be approved.\nThe Parish Council extends its thanks again to the members of the Thakeham Neighbourhood Plan Development Group, whose hard work over three years has led to this successful outcome. District Cllr Claire Vickers (Cabinet member for Planning and Development) also congratulated the parish on all the hard work involved in the preparation of the plan. The full text of her comments is here.\nMore information about Neighbourhood Plans in the Horsham District.\nYou can read and download the Plan from the links below, supporting documents can also be found under 'previous stages of the process'.\nRESIDENTS ENDORSE THE THAKEHAM NEIGHBOURHOOD PLAN!\nWe are delighted to report that 93% of local voters have endorsed the Thakeham Neighbourhood Plan.\nthe countryside environment and ecosystems are protected.\nTotal voter turnout at the 22 March 2017 referendum was 460 (23%).\nThe referendum to confirm local support for the final version of our Neighbourhood Plan will take place on Wednesday 22nd March 2017 (7am \u2013 10pm, Thakeham Village Hall).\ncan be viewed and downloaded below, or via the HDC website. Hard copies of these documents are also available for consultation at Thakeham Parish Office (contact the Clerk) or Storrington Library.\nAfter reviewing responses from the latest phase of public consultation (November-December 2015), in February 2016 Horsham District Council confirmed that the draft Thakeham Neighbourhood Plan was ready for submission to independent examination and that process began in mid-September 2016. The core documents submitted for examination can be seen individually via the links below, or else follow this link to [download].\nSubject to the inspector's findings, we hope to be in a position to hold a referendum seeking endorsement of the Plan by the end of 2016. We will be aiming for a high referendum turnout. Having a Neighbourhood Plan in place, endorsed by the Thakeham community, will be a major boost to preserving the rural character of the area, protecting the environment and improving local amenities and infrastructure. Further updates on this process will be published here when available.\nThe preparation of our Neighbourhood Plan has been overseen on behalf of the Parish Council by the Thakeham Neighbourhood Plan Steering Group (TNP SG), which has in turn consulted with Focus Groups. Follow the links below for minutes of all Steering Group meetings, terms of reference for groups and the Project Plan.\nIn line with Regulation 16 the pre-submission Thakeham Parish Neighbourhood Plan is being made available for a six week period of representation. This representation period begins on Friday 30 October 2015 and will close at 5pm on Friday 11 December 2015.\nThe Thakeham Parish Neighbourhood Plan and related documents can be found on Horsham District Council's Current Neighbourhood Plan Consultations webpage . The page also contains a representation form which can be used to submit comments. There is also a guidance note on \"How to make effective representations\". Copies of the Thakeham Parish Neighbourhood Plan are also available for inspection at the Council Offices, Storrington Library and Thakeham Parish Council offices. Whilst we would prefer to receive responses that use the representation form, responses can also be sent via email and by post.\nThakeham NP Final SA v.4.3. \u2013 the Sustainability Appraisal\/Strategic Environmental Assessment (SAE) to support the Plan.\nThakeham NP BCS v4.4. \u2013 the Basic Conditions Statement to support the Plan.\nThakeham NP Consultation Statement final \u2013 the Consultation Statement that supports the Plan.\nList of Planning Terms \u2013 to help with the jargon used in some parts of the documents.\nHaving received initial feedback from the healthcheck process, after further consideration and discussion with Horsham District Council, and approval by the Parish Council, further versions of the Neighbourhood Plan documents were submitted to Horsham District Council on 25 September 2015.\nThe draft Neighbourhood Plan was submitted to Horsham District Council for the required pre-submission 'healthcheck' process, i.e. commentary from an independent expert reviewer.\nThe State of the Parish Report was completed at the end of July 2014 and can be seen here. The report includes a summary of the key findings of the Neighbourhood Plan Survey Report which was completed on 23rd June 2014, and can be seen here.\nAfter allowing for the 146 houses at Abingworth and the 8 houses at Venters approximately half of those responding (51%) indicated that they felt that Thakeham did not require any more open market housing, 21% felt that there should be in-fill only and 13 % 2-3 infill sites with up to 3 houses within the settlement boundary.\n96% agreed that new developments should maintain the character of the village and use local building materials where possible.\n90% indicated that gaps between Thakeham and neighbouring parishes should be maintained to avoid the settlements being joined up.\nThe use of cars (99.1%) and walking (65.2%) were the main forms of travel. 72% wanted better footpath\/bridleway links to adjoining parishes.\n75% indicated that the parish would benefit from having a shop.\n71% indicated that the parish would benefit from having a shuttle bus to Storrington.\nThese findings informed the process of producing the first draft of the Thakeham Neighbourhood Plan during the latter half of 2014.\nThe Localism Act 2012 and Neighbourhood Planning (General) Regulations 2012 introduced the concept of Neighbourhood Plans which hand powers to communities so that local people can shape the character of the neighbourhood in which they live.\nThakeham Parish Council (TPC) submitted a letter of intent to prepare a Neighbourhood Plan to Horsham District Council (HDC) on 13th February 2013, the first parish in the district to do so. On 22nd July 2013 a formal application to designate Thakeham Parish as a Neighbourhood Development Plan Area was submitted and HDC issued a Notice of Consultation on the application. The designation was agreed on 19th December 2013 and led to receipt of the HDC Decision Notice Designation Map Thakeham.\nThese stages confirmed the intention to prepare a State of the Parish Report and to begin undertaking data collection and analysis of the current situation and desires for the future from residents, farmers and businesses for the plan period to 2029.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Yuen Woo-ping just directed another terrific martial movie! Why are we talking about anything else?\nBy the late 90s, when I first began exploring the wonderful world of Hong Kong cinema, its heyday had already passed, according to most observers. Coming late to a classic period that saw an explosion of dozens and dozens of amazing, jaw-dropping action films that far exceeded in daring and execution anything I had seen from Hollywood in the 80s and early 90s, I could focus on certified classics.\nGranted, calling any recently-released film a \"classic\" is edging into risky territory, but many of those so-called \"classics\" are indeed \"classics,\" marvels of construction and furious action that stand the test of time. Yuen Woo-ping's contributions to that period of Hong Kong filmmaking -- roughly 1986 to 1997 -- are too numerous to list here, but it's rather stunning to realize much how he helped to define the beauty and the brutality of, well, just about everything that makes action movies so much fun to watch.\nOf course, his career didn't start in 1986; Yuen began acting in the early 1960s and started directing films in the early 1970s. Thanks to his fight choreography credits on The Matrix and Crouching Tiger, Hidden Dragon, he became more widely known in the West, and his style remains much in demand.\nThus, going in, I was apprehensive about Master Z: Ip Man Legacy. I needn't have worried. This time around, Yuen has a good script by the excellent Edmond Wong that lays a solid foundation for new twists and turns and dazzling action sequences.\nIt's a spin-off installment, based on the revived Ip Man trilogy starring Donnie Yen as the legendary martial arts master. Yuen served as action director for the concluding film, Ip Man 3, and the action was terrific, and served the drama well.\nA villain from that film, Cheung Tin Chi (Max Zhang), is here flip-flopped into the hero, and it suits him well. He is presented as a dedicated, peaceable father to a young, precocious son; he has retired from fighting and has been working hard in a little grocery shop. His heroic instincts kick in when he crosses paths with two women who are targets of evil criminals.\nThe criminal gang is led by the sadistic Kit (Kevin Cheng), but then we learn that he is not actually the leader of the gang; instead, his older sister Kwan (Michelle Yeoh) is firmly in command. Kwan wants to move the family into more respectable dealings -- shades of The Godfather -- and so orders Kit not to spend any more time wantonly meting out violence and death to all who stood in opposition.\nThis does not sit well with the ambitious Kit, so when an opportunity to move into drugs in a big way opens up, he decides to defy his sister, become a full-time drug dealers, and take over the family. Ultimately, Kit wants murderous vengeance because he was disrespected, which is kind of an extreme reaction, but he's a movie villain, so what's he supposed to do?\nThis does not sit well with the formidable Kwan, so we know she will be doing something about it, even though she has a soft spot for her younger brother, and why am I still writing about the plot when there's all that terrific action to describe.\nWhich I won't, in fact, be doing, because that would take away from the fun and, let's face it, the reason the film was made, which was to make money, yes, but also to kick many, many behinds, and to showcase talented choreographers and stunt people and fighters flipping and flying and doing many other things that dazzle and delight.\nMeanwhile, Michelle Yeoh continues to surprise, and after watching her as a super-mean and duplicitous villain in Star Trek: Discovery (the series), I was very happy to see her as another type of nasty yet multi-layered antagonist. Dave Bautista is also a villainous figure and is does very well as Dave Bautista, while poor Tony Jaa only gets to glower a few times and remind us of his better days.\nMostly, I was enthralled and excited and I am a certified Max Zhang fan now, for sure. Am I making any sense anymore? To be honest, I am not certain why you are reading this when you could be seeing the movie instead; it's much better than any action movie Hollywood has produced this year or last year or probably the year before that; my memory is getting hazy now, and the lights are growing dim.\nThe film will open in select U.S. theaters on April 12. Visit the official site for locations and ticket information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During the Olympics many big brands invested huge dollars in sponsorship \u2014 but which sports fans represent the greatest revenue opportunity? A recent report by Hitwise found that UK tennis fans are more inclined to be shopping on fashion retailers such as Oasis, Urban Outfitters and Nike Store compared to the average online population.\nIn light of the US Open, interest and searches related to tennis remain high. Retailers have the opportunity to align their brand with the tennis audience in particular, since they tend to engage more heavily with apparel sites than the average consumer does.\nThere are also a number of affiliate and voucher code sites that over-index for sports audiences, such as Quidco and HotUKDeals, showing that these audiences tend to seek product discounts and deals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bright Red Leather Sofa. This awesome picture collections about Bright Red Leather Sofa is available to download. We collect this awesome picture from internet and choose the best for you. Bright Red Leather Sofa photos and pictures collection that published here was carefully selected and uploaded by sofascouch after selecting the ones that are best among the others.\nSo, ultimately we make it and here these list ofbest picture for your inspiration and informational purpose regarding the Bright Red Leather Sofa as part of Best Sofas Ideas \u2013 sofascouch.com exclusive updates collection. So, take your time and find out the best Bright Red Leather Sofa pics and pictures posted here that suitable with your needs and use it for your own collection and personal use.\nRegarding Picture detailed description: Graphic has been uploaded by sofascouch and has been tagged by Bright Red Leather Sofa chair Chesterfield Sofa couch divan futon Leather Sectional Sofa leather sofa loveseat reclining sofa sectional sofa sectional sofas sleeper sofa Small Sectional Sofa sofa sofa bed Sofa Beds Sofa Covers sofa mart Sofa Sleeper Sofa Slipcovers sofa table Sofa Tables Tufted Sofa in Living Room Sofa field. You can easily give your review as feed back to our page quality.\nHere you are at our website. now we're delighted to announce that we have found an extremely interesting content to be reviewed, that is Bright Red Leather Sofa. Many individuals searching for specifics of Bright Red Leather Sofa and of course one of these is you, is not it?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These tasty wall decals have been freshly hand-picked for you from our garden! Featuring plump red tomatoes, cucumbers, onions, and lots of green and red peppers, this set of colorful stickers is a feast for the eyes. It's certain to add a splash of color and whimsy fun to any kitchen. Just bring your appetite!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yeshiva University (YU) will bestow an honorary doctorate on Roslyn Goldstein at YU's 84th Annual Hanukkah Dinner and Convocation on Sunday, Dec. 14 at The Waldorf Astoria. YU will also confer honorary degrees on four other leaders: David Feuerstein; Elliot Gibber; Rabbi Haskel Lookstein; and New York's governor, the Honorable David A. Paterson.\nWhen David Sugarman signed up for the Center for the Jewish Future's humanitarian winter mission to Thailand with the American Jewish World Service (AJWS), he thought it would be an exciting opportunity to immerse himself in a foreign culture while helping the poor.\nYeshiva University's Center for the Jewish Future (CJF) is working with the American Jewish World Service to send 15 students to Central America in January for the university's second humanitarian mission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"rent appartment bangkok - If you`re looking for a graphic about rent appartment bangkok on this site we provide a picture that could be in accordance with your wishes. The initial rent appartment bangkok picture is fixed by the very best picture quality. However, if the rent appartment bangkok image doesn`t fit things you need, you are able to continue to see different pictures linked to rent appartment bangkok at the bottom with this page.\nrent appartment bangkok bangkok apartment thailand find apartment serviced apartment in bangkok for rent also includes court mansion flat and condominium in bangkok thailand for rental or for sale, sathorn apartment rental condos apartments for rent in sathorn bangkok modern furnished flats with gympool near btsmrt in sathorn free room finder service, bkk citismart is an experienced real estate broker offering the best bangkok apartment advisory to our customers in metropolitan and suburban locations, fully living up to its name bangkok garden apartment offers serviced apartments for rent on a monthly basis the apartments are set amid the citys largest private garden of tropical flora and a serene waterfall.\nrent appartment bangkok welcome to aetas residence most exclusive serviced residence in the city if your work or travels will be bringing you to thailand for an extended stay we invite you to consider staying at the contemporary sukhumvit residence and serviced apartment bangkok has to offer, dubai furnished apartments and villas for short term rental short stay accommodation residential multibedroom apartments studio flat rooms serviced apartment hotel apartments on a daily weekly and monthly basis hotel in dubai hotel apartment in dubai dubai hotelsdubai apartments.\nrent appartment bangkokbeachfront city holiday rentals condominiums apartments for rent in hua hin pattaya phuket thailand, condos apartments for rent in rama 3 charoenkrung charoennakorn bangkok furnished flat luxury condos with gympool free room finding service, bangkok house for rent \u0e1a\u0e32\u0e19\u0e40\u0e0a\u0e32 \u0e1a\u0e32\u0e19\u0e40\u0e0a\u0e32 house townhouse condo apartment in bangna bangkok nonthaburi samutprakarn , rent trend data in new york new york new york average rent as of march 2019 average rent for an apartment in new york ny is 3431 which is a 163 increase from last year when the average rent was 3375 and a 181 increase from last month when the average rent was 3369. rent appartment bangkok.\nRelated Posts of \"Rent Appartment Bangkok\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I hear a lot in the US about work-life balance. The concept is increasingly at the forefront of consciousness for both sexes, as both men and women think hard about the roles they want to play in the world and in their own families. Most large corporations at least pay lip service to the notion that this balance is desirable for their employees, and many back up their fine words with benefits (such as Sun's work from home program) that are very much to the point.\nIt's a topic that I have opinions on only in hindsight. I never planned my career, nor my home life. It all just seemed to happen, I made choices as they came along, and I haven't reached the end of the story yet to know how it will all work out.\nI think perhaps I've achieved work-life balance, or at least I'm on my way to achieving it, though not in the day-to-day fashion that most people imagine. My work and life are balancing over months and years.\nIf I had had to choose a time to have a child, there might never have been a \"right\" time. It would have meant an interruption to whatever I was doing professionally at the time, because I always knew that I would want to take care of my baby myself. I don't claim that this is the right choice for everyone (for many mothers, it's simply not a financial option), but I knew without thinking that it was right for me. If it had not been possible, I might have chosen not to have children at all.\nI have come to the conclusion (many years after the fact) that I got pregnant at age 25 as a way to end conflict between myself and my then-fiance, Enrico, about my work. He was worried about my traveling, partly out of concerns for my safety in exotic countries (I did two consulting jobs in Africa), partly because he preferred to have me near him, and did not see how all my gallivanting was going to fit into \"normal\" married life. He's also 6.5 years older than I am, and he was ready to be a father when we met. But I was excited about the work I was doing, and not about to give it up on his say-so. Irresistible force meets immovable object. Much conflict ensued.\nSo I \"accidentally\" got pregnant, then gave up my job and moved to New Haven to be with Enrico. I worked temp jobs until about two weeks before giving birth, when I was simply too enormous and uncomfortable to sit in an office chair all day. After Ross was born, the boss I'd been working for at Yale would have been delighted to have me back on any terms, including keeping the baby with me all day, but we tried that and concluded it just would not work: a baby requires constant attention.\nI was mostly at home with Ross until we moved to Italy, where she began full-time day care in early 1991, at about 18 months old. The job I eventually found started out as a regular office job, doing technical writing for a high-tech startup in Milan. Then the boss set up a US branch for his company and moved all the engineers to Silicon Valley. I had to work with them, so in early 1994 I began flying to California four times a year, for stretches of two to three weeks. In the summers, Enrico and Ross would join me and enjoy a month's vacation in California while I worked.\nEnrico was also spending at least a month out of each year in the US, collaborating on mathematical research with colleagues at various universities. During several such trips Ross and I joined him. I worked from wherever I could get an internet connection and put Ross in daycare, which had the added benefits of improving her English and giving her more exposure to American culture. Later, when she was in elementary school and I couldn't just take her away for months at a time, we restricted our long family stays in the US to the summers; Ross and Enrico would vacation in California while I was working.\nI quit my US-based job in 2001 (when Ross was 12) and began working entirely from our home in Milan. It turned out to be a boon that I was much more present during her adolescence than I had been when she was younger. Her teenage years weren't entirely smooth (whose are?), but the troubles she had were mostly with school. In every other arena she was level-headed and generally trustworthy. Perhaps having her mother close, both physically and emotionally, contributed to that.\nVery unusually for an Italian, Ross left home at 18 to go to boarding school in India, which meant that I could leave home as well. Which, after years of being underemployed, underpaid, overcommuted, and exploited in the lousy Italian job market, I was ready to do. So now I'm back to concentrating on my career. We'll see how the next 20 years go.\nThis entry was posted in bio and tagged parenting, work-life balance on January 31, 2010 by Deirdre Straughan.\nHello, very interresting blog for every IT employee and parent.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u2764Ensure the surface is smooth, do not stick on the rough or uneven surface.\n\u2764Choose the place you want to apply, clean up the ash and grease, keep it dry and clean.\n\u2764Stick in accordance with the picture, or DIY in your own idea.\n\u2764Peel off the paper and then stick on the wall, pave from the middle to left and right.\n\u2764\u2605It can be applied to any smooth , clean and dry surface of walls,Car, tiles, glass, furniture, etc., not suitable for dirty or rough surface.\nIf you have any questions about this product by Sunshinehomely, contact us by completing and submitting the form below. If you are looking for a specif part number, please include it with your message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zero Gravity is a 40k intro featuring 'G-Force\/Pygmy Projects'-style of Gouraud Shading, just with the difference that it does not require any fastmem and has 32 colors instead of 16. Other effects are a tunnel and a distorted picture rotator.\ndeserve a better place too.\nrulez added on the 2010-07-25 00:57:20 by \u02674\u027e\u0257\u057e.\nThe beginning of the Zero gravity era by Antibyte and his team in Scoopex.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With a significant list of world records to its name there is really little more one can say to demonstrate the pedigree of Momoi Hi-Catch IGFA Line.\nWith a significant list of world records to its name there is really little more one can say to demonstrate the pedigree of Momoi Hi-Catch IGFA Line. With the strictest of quality controls, especially strength parameters and guidelines, Hi-Catch is compliant with IGFA standards and rules ensuring it is the go-to fishing line for the world's top rated professional anglers. These pro anglers are playing for big game fishing records, big bucks and big tournament kudos. No pro wants to be compromised by noncompliance or worse still, inferior fishing line that can potentially cost them a significant win.\nWith outstanding performance qualities specifically designed for the boat angler and trolling, Momoi Hi-Catch IGFA line is a must have those truly serious anglers who demand the best available mono connection to their target. The abrasion resistance is class leading so Hi-Catch will withstand the rigors of tough fights, to some degree encounters with some forms of teeth, reef, jetsam and shaky deck hands. The momoi Hi Catch game line all have very low diameters so you can fit plenty on your reel. Delivered in 1000 meter spools, you will have more than enough line to pack up the hungriest game reel.\nA big day out chasing significant blue water action can be a costly affair. Don't ruin a day losing a prize or record catch using inferior mono. Demand the best, spool up with the best and put Momoi Hi-Catch IGFA Line on all of your fishing reels and lay into the biggest of sports fish with supreme confidence and expect to land a personal best.\nAvailable in 15, 24, 37kg.\nMake your size seection from the drop down menu.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..5bf9879e43ce16165f57e420370e4b53b05db275
--- /dev/null
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+{"text":"Honda have been in the headlines over the fate of their UK base at Swindon, with the fallout overshadowing their new CR-V \u2014 and that's something of a shame, as Toby Keel found out.\nUntil a few weeks ago, I owned a Honda CR-V. It was two decades old, had creaky suspension and a mysterious leak. The electric windows went up and down like an octogenarian on a ladder, and the interior d\u00e9cor couldn't have been more 1990s if it was listening to Vanilla Ice on a Discman.\nApart from that it was a damn good car \u2013 which was a bit of luck for me, since I bought it in a bit of a panic when a blown turbo forced me to scrap my Saab. The CR-V started first time, every time, was a revelation in the snow and ice and was stupendously practical and easy to drive, with great visibility and a comfortable seating position. It also had a pleasantly boxy look that rather grew on me.\nSo the idea of jumping into the brand-new, smart-looking iteration of Honda's CR-V \u2013 which did as much as anything out there to popularise the 'soft-roader'\/SUV phenomenon \u2013 genuinely appealed. When it turned up, it looked very decent indeed, with Honda's designers giving that old box shape a few futuristic flourishes. I eagerly opened the doors to hop in, expecting to feel immediately at home yet nicely surprised \u2013 as I was about to visit a once-gawky childhood friend who'd made good in their career, found love and was living in a swish new home.\nInstead, I was somewhat bewildered. Every surface of the dashboard, steering wheel and centre console seemed to be covered in buttons, dials, knobs, switches and displays. I felt like a Spitfire pilot transported into the cockpit of an F35B Lightning \u2013 and I acted like one too, pushing buttons at random. Rather ironically, the first thing to appear was something that the F35 pilot would know only too well: a heads-up display emerged from the dashboard, showing off speed, gear and RPM in your field of vision. Just what every family-friendly load-lugger needs.\nPush the start button, though, and things instantly become simpler \u2013 much simpler than the late 1990s version, to be honest. Driving the new CR-V is a breeze: plenty of power from the 1.5-litre turbo-charged version of Honda's evergreen VTEC engine, beautifully weighted steering, imperceptible gear shifts from the automatic CVT gearbox and endless gizmos to make driving, parking and manoeuvring easy. The mirrors tilt downwards when you're reverse parking to help you avoid kerbing the alloys, for example, while the cruise control comes with radar-assistance and lane-control steering, which automatically keeps you in the white lines.\nPut this thing on the motorway and it's capable of following the car in front with almost zero input from the driver \u2013 something that, when I briefly demonstrated it, amused the kids no end, but horrified my wife. Funny to think that once they're old enough to drive such things will seem old hat; for those of us who first drove cars with drum brakes and without power steering, will we ever be comfortable letting the machine take over?\nThe heads-up display, ideal for checking your speed, MPG and targeting passing MiG 35s.\nThere's no ignoring the gizmos \u2013 and actually they're a key selling point of the CR-V. The top-of-the-range, all-bells-and-whistles EX version costs a few grand more than an entry-level Discovery Sport, but unlike the Land Rover it comes with a spec sheet listing over 50 options that come as standard. There's a panoramic roof, blind-spot warning monitors, a nine-speaker stereo system with subwoofers, USB charging ports everywhere you look, a push-button tailgate and electric memory driver seats.\nAll were very welcome except the latter, which dutifully changed the seat to some else's settings every time I got out of the car, meaning that I had to change it every time I got back in. Every. Single. Time. I even took the Nuclear Option \u2013 reading the manual \u2013 but still couldn't find how to stop this from happening, and the nice man from Honda who collected the car had no idea either. At least that's one problem my Mk1 CR-V didn't have, but of course if you bought one I'm sure somebody would explain how to turn this feature off.\nShould you buy one? Well, the basic appeal of the Honda CR-V is the same as it has been for 20 years: space, practicality, comfort and all in a shell that's big, but not huge. The standard of kit and the finish inside was superb \u2013 so close to what you'd expect in the Land Rover as makes no real difference. Its economy was reasonable too: a hybrid version is imminent if that's what you need, but in the petrol-powered model we recorded just under 35mpg, not bad for a 4\u00d74 tipping the scales at 1.7 tonnes (that was as good as it got, incidentally: switching the box to 'Economy' mode \u2013 or 'false economy' mode, as I came to think of it \u2013 seemed to make precious little difference to fuel economy, but had a big impact on driveability). The boot was enormous, the legroom terrific and the all-important heated steering wheel and seats were present and correct. All in all it was an extremely pleasant car to live with, a sound purchase that would keep you happy for many years.\nNow, though, we have to deal with the elephant in the room: Honda's recent decision to close their Swindon plant, which was taken just before the CR-V reached us at Country Life. For so many years buying a Honda felt pretty much like buying British, and it's hard not to escape the feeling that, from now on, Honda will have a tougher time in what has for decades been their strongest European market. The decision is terribly sad for many reasons and on many levels; let's hope that the company makes a U-turn and finds a way to make Swindon part of their electric car future rather than one of the things they're leaving behind as they move forward.\nElegant, but burly like a bare-knuckle brawler, Bentley's new Continental GT Cabriolet is both slick and rugged says James Fisher.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These Magnificent Frogs Are Absolutely Amazing To Look At, But Are They Safe To Touch?\n6 Magnificent Places Around That World That Look Like They Came Straight Out Of A Fairy Tale.\nWorld's Last Male Northern White Rhino Dies In Kenya, Bringing About \"Huge Extinction Crisis\"\nWonders Of The World: Elephants, 10 Facts About These Amazing Animals That You May Not Know.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Save the Date: TechDays Belgium returns on 14-15-16 February 2012!\nTechDays Belgium returns on 14-15-16 February 2012! Since we're celebrating our 10th anniversary it will be a special edition: we're moving to a brand new location (to be disclosed soon) and will offer you new ways to experience the conference.\nFor those who aren't familiar with TechDays: it is the biggest IT event in Belgium, gathering thousands of professionals each year during 3 days. Fascinating technical sessions on both current and new technologies from Microsoft are complemented with many networking opportunities with your peers. And we welcome top notch speakers and local IT community leaders.\nCurious? Nostalgic? At www.techdays.be you can relive the atmosphere of TechDays 2011.\nRegistrations will be open soon, so for now: save the date and don't miss out!\nYou can also RSVP already on Facebook or LinkedIn and let your network know you will be attending. If you're on Twitter, you can join the conversation via #techdaysbe.\nSubscribe to our newsletters for developers and IT pros and stay up-to-date about TechDays 2012 (registration, speakers, session content, agenda, \u2026).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"OpenBAM is a Business activity monitoring(BAM) application for Operations Managers in service organizations who are constantly trying to figure out what's effecting their business by trying to co-relate data from multiple systems.\nBusiness activity Monitor 0.2.1 License - BSD License . Freeware download of Business Activity Monitor 0.2.1, size 6.82 Mb.\nBizTalk Server 2009 BAM Interactive Poster July 2008 Microsoft Corporation.\nThis interactive poster shows you how to develop a simple Business activity monitoring (BAM) solution. You will monitor a purchase order in a Microsoft BizTalk Server application. You will learn about the roles, BAM components, and the process of developing a BAM solution so you can monitor your own business process.\nHomeGuard is an advanced parental control and activity monitoring tool for family and child safety online and offline. Once installed HomeGuard will silently and automatically block all inappropriate content online and record detailed website activity including time of visit and time spent on each website. Plus a built in keylogger, activity based . Free download of HomeGuard Activity Monitor 2.2.4, size 3.87 Mb.\nStatWin is a comprehensive computer activity monitoring solution for individuals and businesses. It gives you valuable statistics. It lets you analyze how effectively you or your employees are using computers. The software counts aggregate operation time, average operation time per day and per run for a given computer or user. You can even view . Free download of StatWin Pro 7.2, size 1.40 Mb.\nJBAM 1.0 License - GNU Library or Lesser General Public License (LGPL) . Freeware download of JBAM 1.0, size 0 b.\nGourangiBAM 1.0 License - GNU General Public License (GPL) . Freeware download of GourangiBAM 1.0, size 0 b.\nMany employees misuse office computers in unofficial activities that adversely affect the productivity of the organization. With Employee Desktop Live Viewer, organizations can effectively monitor and administer such employees who misuse office computers. It is a versatile network activity monitoring software to monitor user activities on the LAN . Free download of Network Activity Monitoring 13.02.01, size 11.22 Mb.\nHeart Rate Variability Logger lets you record, plot and export time and frequency domain Heart Rate Variability Features (includes experience sampling, rr-interval correction, comparison between recordings, activity monitoring & step counting, location tracking).\n- Extracts, plots, stores and exports heart rate, . Free download of Heart Rate Variability Logger 4.4.6, size 2.94 Mb.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A 2017 ETH Board national Data Science initiative resulted in the creation of a unique joint venture between EPFL and ETH Zurich: the Swiss Data Science Center. The Center's mission is to accelerate the use of data science and machine learning techniques within academic disciplines of the ETH Domain, the Swiss academic community at large, and the industrial sector. A multi-disciplinary team of senior data scientists and experts in domains such as personalized health and medicine, earth and environmental science, social science and digital humanities, as well as economics enables collaboration on both academic and industrial projects. This unique positioning, at the crossroad of academic excellence and fast-paced business environments agility is key in making the complex data science journey simple.\nAlessandro joined the SDSC in March 2019 as a data scientist focused on industry collaborations. His mission is to support corporates in leveraging the power of their data by adopting analytical approaches and data-centric solutions. His background is in biomedical engineering, with a PhD in neuroscience from the University of T\u00fcbingen. Before joining the center, he worked as a postdoc at the Max Planck Institute for Biological Cybernetics, at the EPFL Laboratory of Cognitive Neuroscience in Geneva, and as data scientist for a private ecommerce company.\nAn holds a Bachelor's degree in English, French and History teaching.\nShe worked in several HR roles (recruitment and development) for Manpower Belgium. After having moved to Switzerland in 2008, An has worked in a Private Wealth Management Division of a well-known American bank.\nAn joined the Swiss Data Center in August 2017 as an Executive Assistant.\nChenhao received his PhD in Statistical Science from University College London, his MSc in Optical Science from Imperial College London and his BEng in Optics from Fudan University (Shanghai, China). During his PhD studies he was involved in several data science projects with a large industrial group, where he contributed in developing customized machine learning tools for food industry.\nChristine received her PhD in Applied Mathematics in 2003 (Paris Dauphine University). She joined the Swiss Data Science Center in 2019 from the Harvard T.H. Chan School of Public Health where she was a Senior Research Scientist and an instructor in the Master of Science in Health Data Science. Her research interests are statistical software, reproducible workflows, and environmental policy and health policy. Recent published works include the New England Journal of Medicine, the Journal of the American Medical Association, and the Annals of Applied Statistics.\nCl\u00e9ment joined the SDSC as a Data Scientist focused on industry projects in January 2019. He obtained a BSc in Physics (2016) from EPFL in Switzerland and a MSc in Computational Science and Engineering (2018) also from EPFL. During his studies, he mainly worked on applying Data Science and Machine Learning tools to optimize industrial processes.\nDorina is a Senior Data Scientist at the Swiss Data Science Centre since December 2016. Prior to that, she was a postdoctoral researcher at the Signal Processing Laboratory (LTS4) of EPFL, Switzerland. She got her M.Sc. and Ph.D. in Communication Systems and Electrical Engineering respectively, both from EPFL, and her Diploma in Electrical and Computer Engineering from the University of Patras, Greece. In summer 2014, she was a research intern with Microsoft Research, Redmond, USA, working on the compression of 3D point clouds. Her research interests include graph-based signal processing for data representation and analysis, as well as machine learning, with a particular focus on the design of interpretable models for personalized medicine.\nEniko joined SDSC as a senior data scientist in September 2017. Previously, she was a postdoctoral researcher at the Courant Institute of Mathematical Sciences, New York University, working on machine learning for dynamical systems and climate science. She obtained her PhD in Computer Science from the University of Geneva, Switzerland (2011), her MSc from Telecom Bretagne, France (2006) and her BSc from the University of Timisoara, Romania (2005). Broadly she is interested in machine learning for nonlinear phenomena and high-dimensional data, and more recently she has been working on using machine learning approaches to advance our understanding of the climate.\nEric received his PhD degree in Electrical Engineering from Columbia University, New York, NY in June 1999. Eric Bouillet has been working at IBM T.J. Watson Research Center, Hawthorne, NY since June 2004, and at the IBM Smarter City Technical Centre, Dublin from October 2010 to August 2016. While at IBM he has been working on scalable data stream analytics applied to a number of fields, including finances, law-enforcement, telecommunications, environmental monitoring, intelligent transport systems, and aircraft reliability control systems. Before joining IBM Research, Eric Bouillet was at Tellium, Oceanport, NJ where he was part of the research team who invented and designed the first commercial optical mesh restoration network (deployed nationwide and documented in their book Path Routing in Mesh Optical Networks), and at Lucent Technologies' Mathematical Science Center in the department of Mathematics of Networks and System Research department where he worked on the design optimization and sizing of circuit and packet switched networks.\nFernando received a PhD. in Electrical Engineering from the Technical University of Madrid. He has been a member of the technical staff at Bell Labs and a Machine Learning Research Scientist at Amazon. He has been a visiting professor at Princeton University under a Marie Curie Fellowship and an associate professor at University Carlos III in Madrid. He has also held positions at the Gatsby Unit (London), Max Planck Institute for Biological Cybernetics (Tuebingen), BioWulf Technologies (New York).\nFloriane is a Lausanne University graduate. Given her venerable age, digitalization was not a buzz word when she was a student, and she therefore completed a Certificate in Digital Strategy and Communications Management at the University of Toronto in 2015. After moving back to Switzerland, she joined EPFL in 2016 as head of communication for a central service. Prior to this, she held global marketing communications roles at major industrial corporations for close to 15 years. Since April 2017 she is officially a Swiss Data Science Center employee.\nFotis has joined SDSC as Sr. Systems Engineer.\nBefore SDSC's Engineering team, Fotis has been delivering HPC platforms and large scale services across several countries, with varying technical complexity.\nEven with multiple projects, ranging from global ISPs TCP\/IP delay measurements, several clusters for CERN's Large Hadron Collider physics experiment, developing in EasyBuild while automating HPC software builds and contracted for professional documentation thereof, up to delivering in 2015 and running the system that has led globally for processing most human DNA for clinical use ever, 100K Genome Project, Fotis always found the time and enthusiam for training hundreds of scientists and systems administrators to fulfill their mission.\nFinally, Fotis is an active promoter of Open Source Software and open standards and has been spearheading several impactful HPC\/OSS hackathons.\nGabrielle received her MSc in The Foundations of Data Science from Universitat de Barcelona, Barcelona, Spain, and a BA in Economics from Wellesley College, Wellesley, Massachusetts, USA. She previously worked for over half a decade performing data and risk analysis in the energy industry. Gabrielle directed her experiences and education towards data science in order to apply her skills and interests in a variety of domains. Gabrielle enjoys hiking and skiing, new foods and cooking, and is training for her first marathon.\nImen holds a Master of Management & Financial Analysis from Lorraine University in France and a Master of Accounting from Sfax University in Tunisia. Before joining the SDSC, Imen worked as a financial controller for two Software companies in France as well as for an Oil & Gas company in Monaco. She is responsible for the SDSC budget process, the financial reporting and the cost accounting, as well as the audit process.\nIzabela holds a PhD degree in Computer Science from University of Rennes 1, France and the National French Institute for Research in Computer Science and Automatics (INRIA), France. Before joining the SDSC, she was a postdoctoral researcher at the Chair of Computational Social Science at ETH Zurich and a lecturer for the \"Data Science in Techno-Socio-Economic Systems\" course at ETH Zurich. Her main research focus is on big data analytics, tools and platforms, machine learning and data mining, large scale network analysis, in the particular setting of social data mining.\nUniversity, Prague, Czech Republic, and a M.Sc. degree in Software Engineering from M\u00e4lardalen University, V\u00e4ster\u00e5s, Sweden. He has been a staff member at CERN \u2014 European Organisation for Nuclear Research, Switzerland for five years. Before joining the SDSC, Ji\u0159\u00ed was a core member of Invenio development team working on a framework that covers all aspects of digital library management or document repository on the web. During his work at RedHat and in his free time he has contributed to many open-source packages. He is active as a maintainer of several Python packages and contributor to libraries around Flask framework.\nLuis is originally from Spain, where he completed his bachelor studies on Electrical engineering, and my Ms.C. on signal theory and communications, both at the University of Seville. During his Ph.D. he started focusing on machine learning methods, more specifically message passing techniques for channel coding, and Bayesian methods for channel equalisation. He carried it out between the University of Seville and the University Carlos III in Madrid, also spending some time at the EPFL, Switzerland, and Bell Labs, USA, where he worked on advanced techniques for optical channel coding. When he completed his Ph.D. in 2013, he moved to the Luxembourg Center on Systems Biomedicine, where he switched his interest to neuroscience, neuroimaging, life sciences, etc., and the application of machine learning techniques to these fields. During his 4 and a half years there as a Postdoc, he worked on many different problems as data scientist, encompassing topics such as microscopy image analysis, neuroimaging, single cell gene expression analysis, etc. He joined the SDSC in April 2018.\nMathias received his Ph.D. in computer and communication sciences from EPFL in 2015. He then spent two years as a post-doctoral researcher in the Center for IT-Security, Privacy, and Accountability (CISPA) at Saarland University, Germany, where he worked on genomic privacy and privacy in social networks. His current research interests lie at the intersection of privacy and machine learning, with a special application focus on biomedical data. He is currently the lead scientist for the SDSC of the PHRT project \"DPPH: Data Protection in Personalized Health\". He is also co-principal investigator of a project funded by the Leenaards Foundation on evaluating and preventing privacy risks in biomedical databases.\nMichele received the Ph.D. in Environmental Sciences from the University of Lausanne (Switzerland) in 2013. He was then a visiting postdoc in the CALVIN group, Institute of Perception, Action and Behaviour of the School of Informatics at the University of Edinburgh, Scotland (2014-2016). He then joined the Multimodal Remote Sensing and the Geocomputation groups at the Geography department of the University of Zurich, Switzerland (2016-2017).\nHis main research activities were at the interface of computer vision, machine and deep learning for the extraction of information from aerial photos, satellite optical images and geospatial data in general.\nAfter finishing his Master in electrical engineering at the Ecole F\u00e9d\u00e9rale de Lausanne (EPFL), Nathana\u00ebl worked as a researcher in the Acoustic Research Institute (ARI) in Vienna. In 2013, he returned to EPFL for a PhD, where he specialized himself in different fields of data science: signal processing, machine learning, graph theory and optimization. Furthermore, he created two open source libraries for optimization (UNLocBoX) and graph signal processing (GSPBOX). Since 2017, Nathana\u00ebl Perraudin is a Research Data Scientist at the Swiss Data Science Center in the ETH Zurich. He focuses on different aspects of deep learning in the area of generative models (VAE and GAN), recursive architectures and convolutional neural network for irregular domains. Outside office hours, he is passionate by tango dancing, tandem bike touring, skiing and rock climbing.\nIn 2015, Nicole joined a school called CPNV (Professional Center of Northern Vaud) and today she is completing her 3rd year. At the beginning of this school year, she joined SDSC as a graphic design intern. At the end of her training, she will get a CFC Mediamaticienne (\"m\u00e9diamatique\" includes design, photography and computing skills).\nPamela received her PhD in Computer Science from EPFL Switzerland in 2018. Her PhD thesis focused on providing better scheduling and large-scale resource management solutions for data centers. Her research interests include data center scheduling, cloud computing, distributed systems and systems support for data science. Her research was supported by Microsoft Research and made part of a project of the Swiss Joint Research Center at Microsoft. She was also recipient of the Google Anita Borg Memorial Scholarship on 2013 and the Swiss Government Excellence Scholarship for Foreign Students.\nRadhakrishna has a PhD in Computer Science from EPFL Switzerland, an MSc in Computer Science from NUS Singapore, and a BEng degree in Electrical Engineering from JEC India. During his 16 years of working experience, he has worked in the industry and academia, and has founded three start-ups. He has published over 20 refereed papers, which have received over 9000 citations. He is a co-inventor in 4 patents. He has served as a reviewer for several conferences and journals of repute and as area chair for ECCV 2016. His main interests are Computer Vision, Image Processing, and Machine Learning.\nChandrasekhar studied mathematics at the University of California, Berkeley (B.A. 1997) and art and computer science at the University of California, Santa Barbara (M.A. 2003). He has worked as a software developer and consultant for companies, research institutions, and NGOs in the US, Germany, and Switzerland. Since 2009, he has been at ETH Z\u00fcrich supporting projects by developing software solutions for data management, analysis, and visualization. He has been with the Swiss Data Science Center since 2017.\nSilvia holds an M.Sc. in Computer Science from EPFL and a Ph.D. in Computer Science from the University of York, UK. She has been a senior research fellow at Politecnico di Milano and later at the University of Trento, Italy, where she had the chance to work on Marie Curie and ERC projects relating to natural language processing. From 2012 to 2019, she was a Senior Manager and NLP expert at ELCA Informatique Switzerland, whose AI department she helped to create and expand. Silvia has joined the Swiss Data Science Center in 2019 as a Principal Data Scientist for industry collaborations with the aim of helping the industry benefit from the latest innovation in data science.\nHe has previously occupied positions at the border between management and science and led many projects applied to domains as varied as medicine, finance, environmental sciences, and scientific computing. His expertise and research interests include: stream processing, large scale data management, scientific and data visualization, and cloud computing.\nVirginia Friedrich received her Computer Systems Engineering degree from the Universidad Nacional del Sur in Argentina. She took her first steps as a Software Developer in Hexacta, one of the 100 best outsourcing companies in the world (IAOP). In 2015 Virginia came to Z\u00fcrich to do an IAESTE internship and shortly after began working for SkyCell AG as a Frontend Developer, where she also gained some experience doing Project Management and Requirements Engineering tasks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasazi b/data_all_eng_slimpj/shuffled/split2/finalzzzasazi
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+{"text":"Great inside-the-Beltway Tysons Corner location. Immediate access to the Beltway and Dulles Airport Access Road. Site enjoys prominent Route 123 \/ Dolley Madison Boulevard visibility. Convenient to restaurants and shopping in the Tysons Corner Center and Tysons Galleria regional malls.\nAll raised floor computer space. Renovations to main lobby, elevator cabs, common areas, and restrooms recently completed. Located directly across street from future Tysons Metro Station.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beach club, restaurant, pool and tour services all available at this one of the more popular \"Lidos\" on Lake Como. Treat yourself to some sun and a long drink under an umbrella. The Lido di Cadenabbia offers tourists the opportunity to enjoy a range of services that make your stay relaxing and enjoyable. Stands on the banks of the lake, in front of the Bellagio peninsula and offers a unique view of beauty.\nNearby Things to Do include: Hikes & Walks such as Greenway del Lago di Como and Hike to San Martino; Villas such as Villa Maria and Castello Maresi; Towns such as Cadenabbia and Bellagio; Churches such as The Church Of The Ascension and SS. Nabore e Felice; Parks & Gardens such as Villa Carlotta Gardens and Parco Mayer Beach; Water Taxis & Boat Tours such as Bellagio Water Taxis and Barindelli Taxi Boats Bellagio.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I will include here the course syllabus, homework assignments, and any other files that I want you to have access to. Many of the files will be in Adobe Acrobat (.pdf) format (e.g., the lecture notes, syllabus, etc.). In addition, there may be some Excel files (.xls) and a few other miscellaneous file types.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Washington County Community Foundation (WCCF) is pleased to announce the creation of a new grants program from its Community CARE Fund (CCF) to help build the capacity of charitable nonprofits in the area. Up to $1,000,000 will be issued in CCF capacity building grants during this initiative, which is anticipated to be completed over the next two years. If the capacity building grants program proves to be successful, another round of grants may be announced in the future. CCF capacity building grants will normally range from $25,000 to $50,000, and may be paid over a period of years. Only 501(c)(3) charities which are not classified as private charities under section 509(a) of the tax code, and which provide services in Washington County, are eligible to receive a CCF capacity building grant.\nCumulatively, the WCCF has awarded close to $10,000,000 in grants, with the majority of these grants awarded to support the programs of area charities. Many effective programs have been supported over the years and the WCCF intends to continue providing grants for program support. With this initiative the objective is to help strengthen the infrastructure, management and governance of local charities, through grants to support the charity itself.\nPotential capacity building projects include a communications plan to relate more effectively with constituents, a development plan to grow the charity's donor base, or a volunteer recruitment plan to increase volunteerism. Capacity building grants may be awarded to assist charities with leadership succession, to update a charity's technology, to improve how a charity measures its outcomes, or to improve a charity's accounting procedures. Any activity that increases the charity's internal capacity will be considered.\nCharities who wish to apply for a CCF capacity building grant are encouraged to attend the Foundation's CCF Capacity Building Grant Workshop on Wednesday, January 17 from 2:00 pm to 4:00 pm at the Southpointe Golf Club. There is no fee to attend the workshop but charities are required to register through the WCCF's website at www.wccf.net.\nThere will be a two-step process for applying for a CCF capacity building grant. The first step is to submit a letter of inquiry by March 1, 2018 electronically through the WCCF website. The letter should be authorized by the charity's governing board prior to submission and should include a brief narrative and a cost estimate of the proposed capacity building activity. Charities may only submit one letter of inquiry during this initiative so charity board and staff members should give some thought to the capacity building activity that will have the greatest impact on the organization in the near term.\nAt the beginning of May 2018, a WCCF representative will contact those charities who have been selected to submit a full grant application in 2018. The full grant application will be due on July 1, 2018. It is anticipated that the first round of CCF capacity building grants will be announced at the end of September 2018. The second round of grants through this initiative likely will be announced at the end of September 2019.\nA 501(c)(3) charitable corporation, the Washington County Community Foundation began operations in December of 1995 and ended that year with a mere $5,000 in assets. Today the WCCF manages more than $35,000,000 in assets and is the largest publicly-supported grant-making foundation headquartered in Washington County. It maintains offices in the historic Samuel T. Brownlee House in Eighty Four, PA.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hertz has reported revenue growth for the past seven quarters and suspects its substantial revenue growth will be boosted by its acquisition of Dollar Thrifty.\nHertz Global Holdings Inc., the rental car operator making a $2.3 billion buyout bid for Tulsa-based Dollar Thrifty Automotive Group Inc., reported third-quarter net income of $242.9 million on Wednesday, a 14.3 percent increase compared with 2011's third quarter.\nHertz is in the process of acquiring Dollar Thrifty. Photo by Michael Gray.\nHertz's earnings per share in the third quarter were 55 cents versus 47 cents in the same quarter last year.\nRevenue in the quarter ending Sept. 30 was $2.516 billion, up 3.4 percent from $2.43 billion in the comparable period a year ago.\nWorldwide car rental revenue in the third quarter increased 2.1 percent compared with 2011's third quarter, to $2.15 billion.\nThird-quarter revenue from worldwide equipment rental was $363 million, a 12.8 percent year-over-year increase, driven by an 18.6 percent revenue increase in the United States and a 17.1 percent increase in North America.\nHertz Chairman and CEO Mark P. Frissora said the company established record worldwide car rental revenue in the third quarter. Hertz also posted the seventh consecutive quarter of double-digit year-over-year revenue growth in worldwide equipment rentals, the CEO said.\n\"I am especially pleased with our ability to improve the consolidated adjusted operating margin 260 bps (basis points) in the third quarter despite soft global macro economic conditions,\" Frissora said.\nA basis point is a unit equal to 1\/100 of 1 percent, and a 1 percent change equals 100 basis points.\nIn September, Hertz bolstered HERC's position with its acquisition of Pioneer Equipment Rental and Sales of Ponca City, the state's largest independent equipment rental firm.\nExecutives of Park Ridge, N.J.-based Hertz did not disclose a buyout price, but they said Pioneer Equipment will add $25 million of annual revenue to HERC, its wholly owned subsidiary.\nFrissora said the Pioneer acquisition brings to 12 HERC's number of Oklahoma stores while advancing its industrial penetration strategy, particularly in the oil and gas market.\nHertz offered $2.3 billion, or $87.50 per share, in August for outstanding shares of Dollar Thrifty, the nation's fourth largest rental car company behind Enterprise Rent-A-Car, Hertz and Avis Budget Group. The Hertz offer was approved and accepted by the boards of directors of both companies.\nAfter initially expecting antitrust approval of the Dollar Thrifty deal by the Federal Trade Commission in mid-October, Hertz has twice extended the expiration date of its tender offer.\nLast week, Hertz extended the expiration date of its tender offer to Nov. 5, and company executives said they will continue to extend it until a decision is reached by the FTC.\nBut Hertz's confidence that its divestiture of its Advantage discount rental car brand would pass antitrust muster at the FTC is being tested.\nSome industry observers believe the FTC may require Hertz to divest more of its rental car holdings \u2014 including possibly the Thrifty Car Rental brand \u2014 to win approval of the Dollar Thrifty transaction.\nHertz, Enterprise and Avis Budget together control about 75 percent of the market. Dollar Thrifty has about 5 percent of the U.S. market, said IBISWorld in Santa Monica, Calif.\nDollar Thrifty employs 780 people at its corporate offices at 5310 E. 31st St., and it has 6,000 workers companywide.\nHertz has 24,000 employees, including 1,700 at a regional center in Oklahoma City.\nDollar Thrifty shares closed Wednesday at $77, down $2.08 or 2.6 percent.\nHertz shares dropped 10 cents, closing at $13.27.\nSource: Hertz Global Holdings Inc.\n(c)2012 Tulsa World (Tulsa, Okla.). Distributed by MCT Information Services.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Image Title: Vs Integrated Columns Reviews Ratings Prices Intended For Liebherr With Refrigerator Review Remodel 9. Post Title: Liebherr Refrigerator Review. Filename: vs-integrated-columns-reviews-ratings-prices-intended-for-liebherr-with-refrigerator-review-remodel-9.jpg. Image Dimension: 600 x 450 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Winfield McCullough. Uploaded Date: Thursday - September 27th. 2018 04:57:18 AM. Category: Architecture. Image Source: homedecor.com. Liebherr Unique Fridge Refrigerator Wanakatimebank Ideas Scale To 5715 Designs CS1311 Review Within Prepare In 14. Decor Inside Reviews 9. Remodel Cnef CN3915 Plan 8. Freezers Bottom Org Prices For Freezer Design Which 3. Columns Nabateans Silver YouTube 12. CNEL4213 Intended Home 7. Integrated Of 2. Ratings Vs Regard Image Regarding With Best 4. 17. 11.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vision for a Cure's mission is to raise awareness and compassion about Usher syndrome, as well as provide funding to those that are dedicated to research to find a cure.\nARLINGTON BREW FEST ~ A SOLD OUT SUCCESS!\nArlington Brew Fest was our 2018 fundraising event whose proceeds directly benefit Usher syndrome research at the University of Oregon. Held at Legion Memorial Park in Arlington, Washington, guests enjoyed beer, cider, and wine booths featuring Pacific Northwest unique and specialty tastings, food trucks, raffles, games, music with DJ Saturday, and a wonderful community spirit. Thanks to this great community and all of the sponsors, participants, donors, and volunteers, we were able to send a check in the amount of $40,000.00 to the University of Oregon's Westerfield Laboratory to benefit Usher syndrome research. Thank you to everyone for your support!\nUsher syndrome is a genetic disorder, and the most common cause of deaf blindness. It is a combination of Retinitis Pigmentosa and some level of hearing loss. There are an estimated 50,000 people with Usher syndrome in North America. Learn more about Usher syndrome here.\nWith your help we have raised tens of thousands of dollars for research in our first three years, with 100% of all of our profits going to Usher syndrome research! Thank you! Read more about the exciting Usher syndrome research you are supporting here. Find Usher related new, blogs, and videos here.\nYour support means the world to those who are affected by Usher dyndrome. Together we can raise awareness, compassion, funds for research, and move closer to our shared VISION for a CURE. There are a variety of ways that you can help. Read about some of them here.\nHere's another cool way to raise money for Usher syndrome research and awareness - online shopping! Amazon customers just log into www.smile.amazon.com for all your online shopping, and Amazon will donate .5% of your order to Vision for a Cure, if you select us as your charity of choice. Thank You!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chris Baldridge, the father of Army Sgt. Dillon Baldridge, said that Trump called him at his home in Zebulon, N.C., a few weeks after his 22-year-old son and two fellow soldiers were gunned down by an Afghan police officer on June 10. Their phone conversation lasted about 15 minutes, Baldridge said, and centered for a time on the father's struggle with the manner in which his son was killed \u2014 shot by someone he was training.\nRep. Frederica S. Wilson (D-Fla.) said Trump called Johnson's widow, Myeshia Johnson, on Tuesday and said her husband \"knew what he was signing up for, but I guess it hurts anyway.\" Wilson was riding in a limousine with the widow and said she heard the conversation on speakerphone.\nWhite House press secretary Sarah Huckabee Sanders defended the president, saying in a news briefing that Trump was \"completely respectful\" during the call. Several White House officials, including Chief of Staff John F. Kelly, were in the room at the time, she said.\nThe family of Sgt. Cameron H. Thomas, a 23-year-old Army Ranger killed April 27 in a raid on the Islamic State in Afghanistan, met with Vice President Pence at Dover Air Force Base in Delaware as the soldier's casket arrived from overseas. They had a 20-minute call with Trump about two weeks later, said Thomas's father, Andre.\nEuvince Brooks's son, Sgt. Roshain E. Brooks, 30, was killed Aug. 13 in Iraq. He has not heard from the White House. The president's claim this week that he had called every military family to lose a son or daughter only upset the Brooks family more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Human Rights Watch today expressed deep regret at the decision of the United Nations Commission on Human Rights not to renew the mandate of the Special Representative on Human Rights in Iran.\nIn a vote of 19 in favor, 20 against, with 14 abstentions, the Commission rejected a resolution today, sponsored by the European Union, that would have renewed for one year the mandate of an independent expert on human rights in Iran.\n\"We're very disappointed that the Iran resolution failed,\" said Hanny Megally, executive director of the Middle East and North Africa division of Human Rights Watch. \"The human rights crisis in Iran is only getting worse, and this unfortunate decision will not help it to get better.\"\nFor several years, Iran has not allowed the U.N. Special Representative to visit the country.\nOne sign of a deteriorating human rights situation in Iran is the emergence of illegal detention centers in the country. One of these, the so-called Prison 59 in Tehran, is reportedly administered by Revolutionary Guards Corps and clandestine paramilitary forces, and is entirely beyond official oversight. Some political prisoners have been detained there for months at a time, without their families or lawyers being informed of their whereabouts.\nMegally also cited an alarming increase in public executions and floggings, which he said reflected a widespread campaign by the conservative-controlled judiciary to intimidate and silence supporters of greater political freedom.\nHuman Rights Watch called on the international community to continue to pressure the Iranian government to permit access to the country for independent non-governmental human rights organizations. Independent human rights activists in Iran should be free to operate without government interference.\n\"The member states of the Commission have a special responsibility to raise these issues in their bilateral relations with the government of Iran,\" Megally said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"filter, PALL original filter model parameters for free information. Of course, our company's PALL filter still HV company from the United States share of glass fiber, make each product can meet European standards.\n1. Metallurgy: used for hydraulic oil filtration of rolling mill, continuous casting machine, and filtration of various lubricating equipment.\nfuel filtration, engineering machinery, ships and heavy trucks.\nHave a number of state-owned enterprises research and development elite, technical master.\nOrder tracking services across multiple departments.\nThe product problem goes straight to the new one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasufg b/data_all_eng_slimpj/shuffled/split2/finalzzzasufg
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+{"text":"What is the fastest way to transmit controlled substance Rx's when a provider is sending more than one at a time?\nIf the provider has an extra cell phone, the provider can also use this phone for receiving the text message OTP by connecting the phone via Wi-Fi (after registering this phone with Verizon). In this way, the provider can use their cell phone and an extra cell phone to receive the text messages. This simply allows the provider to go between the devices without waiting for a new code to refresh, as there is a time period (60 seconds) for a new OTP to be sent to a given device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are certain things in life that we regret not taking full appreciation of. Every one\u2014I daresay even the shrewdest, most cunning business executive who all his life has been on the lookout for every single window of opportunity for putting his company on top\u2014has at least once let a really important opportunity slip out of his hands. Why is this so? It's probably because we humans really just aren't very adept at recognizing opportunities until they're gone forever, never to be realized. In other words, we are unable to take advantage of all opportunities presented to us because we procrastinate and put things off until the very last minute. This is a viable explanation, but is by no means the only reason why opportunities are not taken in. It could also be that real opportunities are simply hard to come by. The truth of the matter is, opportunities are rare and almost always come in the worst of times. Distractions in our lives prevent us from judiciously identifying opportunities as they come and so it is important that we learn \"techniques\" on how to spot, in timely manner, opportunities\u2014that is, seizing an opportunity, before it's too late.\nTIP # 1: Get out of your comfort zone.\nOur comfort zones are called such for an obvious reason: they are comfortable. We are satisfied with where we currently are and do not in the least wish to improve our lives for the better. To be sure, there is nothing wrong with being comfortable or satisfied. But constant growth and experience should be a goal of every individual, lest his life would become monotonous and boring. We should actively seek to add spice to our lives. Doing so allows us to access our \"sixth sense\"\u2014that imaginary organ we use for identifying opportunities.\nTIP # 2: Learn to say yes.\nThis tip is related to the one before it. When we're too comfortable with how things are going, we're reluctant to say yes to opportunities that can add value to our lives. But to be successful and achieve anything of significance in life, we have to grow accustomed to trying new things and saying yes to enriching opportunities. Of course, saying yes to every opportunity available is probably not a good idea, as it could burn you out, but occasionally seeking new things wouldn't hurt.\nWhile some opportunities are very easy to spot, most good opportunities are realized only if thought about. Less obvious opportunities are generally the ones that pay the highest payoff because of the advantage of novelty. Actively seeking opportunities, instead of waiting for an evident one to materialize, should be the way to go about achieving good things.\nWondering and being curious in general makes us better \"grabbers\" of opportunities. Curious people seize opportunities because they value new experiences and are ready facing challenges that will lead to their growth as persons. They actively seek novelty and fresh ideas in their lives and are comfortable treading out of their comfort zones. Without much effort, they fulfill what tips 1 -3 espouse simply by living a dynamic lifestyle that isn't limited by anything.\nCarpe diem is Latin for \"seize the day\", which some people interpret as indulging in all sorts of damaging stuff like drinking or partying too hard. This interpretation is wide off the mark. Carpe diem is an adage telling us to seize all opportunities presented to us\u2014doing productive things, not wasting our time in trifles. Spotting opportunities is pointless if you don't do anything about it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our primary purpose in collecting user information is to provide you with a safe, smooth business transaction and the products and services that you have requested. We only collect and hold information that you have given us through the course of a transaction, or communication through email or social media and messaging services such as Facebook, Twitter, Instagram, etc.\nWe never share, sell, lease or exchange customer information to any third parties for any reason.\nIt is used for shipping, billing, and communication purposes only.\nWe send out monthly newsletters (according to the group to which you have subscribed) and occasional offers to our customers. It is also used to communicate directly with you concerning questions, suggestions, or comments you might have regarding services or products you are interested in purchasing or have purchased through our website.\nYou may unsubscribe from our email lists at any time using the \"unsubscribe option\" provided in every email offer. Or you may contact us the change, unsubscribe or delete your account with us.\nInternational shipping charges will be adjusted to reflect the actual postage amount as our shopping cart only reflects US based shipping.\nWe cannot accept fabric if it has been cut, washed or has been in a smoky environment.\nCustomer pays for any return postage, unless the mistake was ours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brine flies might owe their survival in the salty waters of the Great Salt Lake to a bacteria Amanda Truong found living within their cells. Later this month, she's going to tell an international audience about her findings at a prestigious conference in Japan.\nThe high concentration of salt in the lake makes the environment so extreme that the brine fly and brine shrimp are the only organisms that make the lake home.\nAfter hundreds of hours in the lab and at the lake, Truong found a bacteria \u2014 Wolbachia \u2014 that could be key to the flies' survival.\nThe bacteria is believed to skew the female-to-male reproductive ratio, giving deference to females, Clark said. More females means more reproduction, hence the thick clouds of buzzing flies along the shoreline of the lake.\nThe flies are essential to the lake's ecosystem, removing organic compounds such as algae that would sap the lake of oxygen if left unchecked and providing food for migratory birds.\nThe hours Truong has poured into researching brine flies suits her ambition to become a research physician, a goal that will take four years of graduate school after Truong, a junior, graduates from WSU.\nTruong's research has especially educated her on the lab techniques required for in-depth studies of endosymbiotic bacteria, or bacteria found within cells. It is believed that such bacteria contributes to the development of kidney stones, gall stones, elephantiasis and other medical conditions in humans, Truong said.\nTruong was among 10 students selected to present her research at the annual meeting of Society of Molecular Biology and Evolution in Kyoto, Japan, July 26-30.\nTruong will continue to research the brine fly, and said she would like her findings to be published in a scientific journal. She said she believes her research will give her a competitive edge when she applies for graduate school.\nClark, Truong's research mentor, agreed.\n\"If you want to go to graduate school, or if you want to go to medical school, then you have to have something on your application that's going to distinguish you from the thousands of other students who want to go to graduate school or medical school, and these independent projects are one way of doing that,\" Clark said.\nThe meeting of the Society of Molecular Biology and Evolution is the world's leading conference on evolutionary biology, according to WSU Communications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The first of an unknown number of mini-episodes from Twig's SE Reflections. This one on a hopefully unnecessary worry of creating an intellectual cultural bias favoring the pro-social or Ventral Vagal stance at the cost of excluding (and stigmatizing?) those who don't have or want it. Show Notes Page with with additional commentary.\nHere's something I don't think is an active problem but something we need to be cautious of as SE Practitioners develop an intellectual community with a common language of what we like and what we don't like.\nHumans, as with our primate relatives, have a tendency to make social groups that reinforce our clan's preferences. This is part of what makes being part of \"the in group\" feel so good \u2013 our behavior and choices are championed. We feel a part of something bigger than ourselves and are encouraged to think that we're correct in our assumptions.\nThis is also why it feels so bad to be on the outside, shunned or not included \u2013 it suggests that we're doing something wrong and that we are, in some sense, actually wrong.\nAs you and I grow more and more fond of the langauge associated to the Polyvagal theory (wiki) and begin using it as a short hand for naming beneficial features and calling out challenging ones, we'll want to remember this basic tendency in human nature and try to avoid creating a \"this is better than that\" duality.\nOf course it can be argued that mammals have a tendency toward Ventral Vagal dominance in their nervous system when they feel safe enough. That the resulting \"pro-social\" nature is spontaneous and a functional attribute of nervous system state is one of the great insights to be found in the \u2013 very cool and parsimonious \u2013 Polyvagal theory.\nHowever, the exact same principle suggests that tendencies toward Sympathetic or Dorsal Vagal type behaviors such as high activation or desire for isolation are also informed by the autonomic nervous system and often describe the burden of accumulated stress and chronic neuroception of danger or life threat, even if we're unaware that this is part of what's going on. Ultimately this probably explains ourselves more than common cultural or psychological explainations of the associated expressions of personality (i.e. Type A or Extreme Introvert amongst others).\nIn this way, we can probably assume that many people would express more spontaneous pro-social behavior if they experienced less accumulated stress in their lives. Thus a bias toward signs of Ventral Vagal innervation are an understandable desire of ours as trauma focused helping care professionals.\nAt the same time, when we cheer the \"that's so Ventral\" moments, in and out of sessions, we'll want to remember that we live in a puralistic society of people, many of whom have experienced challenges that have led to the accumulation of stress \u2013 and personality formation around that stress \u2013 which makes social engagement itself a challenge or even truly undesirable. For that matter, on any given day even the most social of us might find seclusion the ideal state.\nAll that to say, people expressing SNS or DVC tendencies are no less human and no less worthy than those of us who have found the comforts to be gained by reinforcing pro-social experience.\nWhile all of this is more a cultural concern far beyond the scope of being an SEP and I'd personally encourage everyone to feel free to choose their associations as they please, this does have a cricitally important session element given the nature of our work to provide a specialized conversation to our client's suffering the impacts of accumulated stress, which often requires accurate joining and adjustment of our own behavior, if maybe only while we're in service.\nIf we enter sessions with the notion that VVC expressions are somehow \"good\" and SNS or DVC expression somehow \"bad\" \u2013 whether we recognize this bias or not \u2013 than we surely miss the opportunity to provide the appropriate attunement to some of our clients \u2013 while inadvertently asking them to participate in activities that they're simply unwanting or unwilling to do (such as orientation or chronic conversation) which is another way to help them fail at meeting our requests.\nYes, orientation and social engagement are a direction we're trying to help people move toward and they have dramatic relevance clinically as interventions or therapeutic stance. Still, meeting people \"where they're at\" is part of our game and creating a \"good\/bad\" dichotomy will only get in the way.\nSo Yes indeed, let's champion \u2013 as I surely will \u2013 the pleasures and opportunities of a pro-social life. And when our friends, colleagues, clients and even ourselves aren't interested or able to join us in that \"ideal,\" let's also remember that in the end we're all in this together and one more good-bad mindset split isn't going to help anyone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"So, we are still in the thick of Winter and I am doing everything I can to prevent getting sick again! It's so hard though as germs just seem to be lurking around every corner in school. We can disinfect our desks all day long but that still doesn't do enough to prevent germs from spreading in the air of our classrooms.\nI recently have been getting into Essential Oils, and have enjoyed their positive effect on my mood and my health so far - so much so, I decided to do some further research into bringing the benefits of them into my classroom, namely for preventing the spreading of germs to start with.\nEssential Oils are known for their healing properties and providing complete body \/ mental wellness. Some essential oils are particularly effective at disinfecting the air too - this is desperately needed in a classroom full of sneezes and coughs by hundreds of people all day long!\nThey are so easy to use. All you need to do is get a diffuser for your classroom which dispenses these essential oils into the atmosphere for hours, killing off any germs and bacteria lingering in the air. I prefer to get ones that do not use heat as that can damage the oils. Here are some great cool mist ones from Amazon that also have optional mood lighting which can create a lovely ambiance in your classroom too!\nTea Tree - antifungal, antiviral, antimicrobial and antibacterial. It's great for cold and flu viruses, respiratory conditions and boosting the immune system.\nLemon Oil - stimulating, calming, anti-infectious, astringent, detoxifying, antiseptic, disinfectant, and antifungal.\nEucalyptus -sterilises the air with a fresh, crisp clean scent that also lifts the spirits.\nRavensara - antiseptic, antimicrobial, and antiviral qualities that disinfect the air and helps those dealing with the flu recover faster.\nCinnamon bark - anti-infectious, anti-bacterial, antiparasitic, antiviral and antifungal. It can also maintain a healthy immune system.\nThyme - anti-bacterial, antiseptic, antiviral, and antifungal. It is great for chest infections and coughs and is a great memory booster and antidepressant.\nRosemary - antiseptic and antimicrobial. It increases concentration and is a good remedy for depression, mental fatigue and forgetfulness.\nLavender - anti-bacterial, antiseptic, antiviral, and antifungal. It can eliminate nervous tension, relieve pain and and treat respiratory problems.\nSimply pop a few drops of any of the oils above into the water in a diffuser (or mix some oils together if you like!) and let the essential oils fill the room, oxygenating the air and killing those germs! If your classroom is quite large, you can even have 2 on the go. You'll also be pleased to know that many of these will increase focus, mental clarity, relieve stress and headaches and so much more. I wish I had known about this years ago when I started teaching, I bet I could have dodged so many illnesses...!\nHave you ever tried essential oils in your classroom? How did you like it? Let me know in the comments below!\nTaking a day out of the classroom, whether it is because you're sick, on a trip or a course, is always a double edged sword: on one hand you may desperately need the day off, but on the other there is the stress of having to set cover work for your classes (which is always 100 times worse when you're ill!).\nI am often on both sides of the teacher's desk: a teacher who needs to leave work for subs every now and then and a teacher who sometimes covers lessons for colleagues. In fact, I recently covered a Spanish lesson which was a last minute emergency and so no work had been left. When I got to the classroom I let the students in, they unpacked their equipment and looked up at me expectantly. I looked back at them and said \"hola!\", they smiled openly and greeted me back the same way. There was a pause before I blurted out, \"That's all I've got, guys - that's the only word I know!\". Their little confused faces were hilarious - but I'm very open with my students and they actually found it really funny! A Spanish lesson wasn't even something I could even attempt to teach as I know NOTHING! Fortunately, after years of experience in these kinds of predicaments, I created my own kit of emergency sub plans that are handy activities that can be used for any subject - it saved my bacon that day!\n1) If you know you are going to be away in advance, ensure the cover work is taped to the desk or board so it's easy for the sub to find. I have had many experiences of having to fumble around someone's desk wading through piles of paper and books trying to find the work. It's not very appreciated, I have to say - the work should be easy to find and simple to understand.\nIf you're calling in absent, you could email the work to a reliable colleague so it can be printed for you and left on your desk or if you can get access to the school network from home, you could also upload work there too.\n2) Leave a register and a seating chart with the cover work or have them on display in your room at all times. Some cheeky students will often try it on with a sub and take advantage - it's important the teacher covering has all the information so they can't be played!\n3) Set cover work that doesn't involve learning new material. When I first started teaching I remember I had to cover a French lesson (why all the languages?!) and the cover work actually requested I teach them French and get students at the front having conversations. In all honesty I wouldn't understand or know if they were even accurate, so I had to adapt the plans a bit - I didn't need that extra stress in a \"foreign\" subject (pun intended!). Personally, I like to set creative tasks that are enjoyable, calming and reinforce previous learning in fun ways that also won't involve a heavy marking load.\n4) Always have plenty of extension work available. There's nothing worse than students racing through the tasks and finishing early with nothing else to do for the next 30 minutes! Have more than you think they'll need to ensure they will be kept nice and busy.\n5) Keep the number of resources required to a minimum. It can get quite stressful ensuring equipment and textbooks etc. are available, and even worse if it's a last minute cover and the sub arrives and can't find any resources or equipment. (Yep, this has also happened to me - a task required text books, except no one knew where the textbooks were...) Keep it simple - just grab paper and go.\nThis pack has 10 different themes that can be used for any subject and it includes lots of variations of lessons ideas within each theme!\nYou can just leave the \"theme sheet\" and your sub can choose the activities, or alternatively you can select what your students do, but at least there is plenty of extension work available. Also there are worksheets included which go with each theme if you choose, but all activities can just be done on plain paper or computer depending on what you have available. The idea is that they can be used in any circumstance and with limited resources - perfect for no prep emergencies!\nI have a sample of one theme from this pack which you can use for free here to get you started, I hope it is helpful for you! Our job is hard enough as it is; we shouldn't be working even harder when we need to be off, especially if we are ill and need to relax!\nI hope these tips help you out and I'd love to hear your thoughts and experiences with subbing (preparing or doing!).\nHappy New Year colleagues! I hope you have had a lovely, restful holiday and 2016 is treating you well so far! Today I am linking up with some fellow secondary teachers and we are reflecting on what we have learned in 2015.\n1) 2015 was the year I discovered the wonder of Kahoot! This is a fantastic tool that allows you to create interactive online quizzes for your class. What makes it special, is that games are displayed on a main screen like a projector or whiteboard, while students answer the questions on their own devices (computers, smartphones etc). This is wonderful as it encourages students to look up to read the questions which help create unity in your classroom so everyone isn't just staring at their own screen all lesson.\nIt's really easy to use! Once you create a Kahoot quiz you get a unique code for it. Students simply go to the Kahoot website, type in your code and everyone can play online together! You then get to see the score at the end and gauge the success rate of your class. It makes a fantastic end of topic quiz or introductory quiz for a new unit! I have also asked students to create quizzes themselves on different topics that we can play as a class which really helps them take ownership (and saves you so much time as well!). Plus, the music is awesome!\n2) This year in particular, I have been noticing students showing increased signs of stress and anxiety. I feel not only are they are far too young to be so stressed out, but what concerns me most is that they are not properly equipped to handle it. The increased percentage of teenagers self-harming themselves is utterly alarming. I got so distressed when I heard the figure in my school alone, that I went on a mission and made these \"Stress Management Activities\". I took a lot of time researching all about stress and the best management techniques, and created a resource to teach students all about it: what stress is, why we need it, how we respond to it in the modern world, causes and effects, what we can control and what we can't, and actions to help relieve it.\nI also created activities to give students the opportunity to role play and analyse different teen appropriate stressful scenarios, such as school life, family problems, friends, jobs, bullying, partners etc. I also included 3 different breathing exercises to help control the physical signs of stress that are very powerful in tense situations. It has been very useful for me as well!\nI want students to have the tools to deal with the inevitable hard times in a healthy way. So far, it has been really well received by my students and I can only hope everyone who has downloaded this resource from my TpT store is noticing a positive change in their students as well. It is something I am very passionate about and will continue to fight for!\n3) Finally, my personal teacher goal for 2016 is related to the above actually, and is about well-being and a keeping a positive mindset. I worked hard on my work\/life balance in 2015, and so this year I am going to focus on keeping positive by doing a fun challenge I wrote about recently on how to keep your motivation as a teacher. I am hoping this will help me look out for and appreciate the golden moments as a teacher, as well as help keep me motivated during the harder times.\nI hope you enjoyed reading about my 2015 reflections as a secondary teacher! I'd love to read your thoughts or your personal goals, in the comments below! \u200bI wish you a happy, successful and amazing 2016!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bedroom : Train Beds For Kids Thomas The Toddler Bedding Bed Sets Sheets Delightful Boy Princess Carriage Twin Size Race Car ~ Dowdydoodles Inc.\nPublished at Tuesday, March 05th, 2019 - 10:35:49 AM. Bedroom. By Idalie Kraus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In Fairview, Texas, homes like these demand an electrician an electrician that has the ability to complete a diverse number of offerings in a professional and least disruptive manner. The expectations and desire of homeowners in these expansive abodes should be met with the utmost of care. A simple rewiring for an addition will at its least hold the care an automotive technician would give a rare classic car. The demand for this type of service and the array of improvements offered call for the best quality available. This includes the least damage of existing structures, cleanliness held to a maximum, and a finished product that appears as work was never done upon completion.\nThese services may include everything from powering a hot tub to installing a charging station for an electric car. The strive for perfection by the people of locales such as this require the attention of electrical contractors prepared to meet their needs. Experience in this type of work is essential for the satisfaction of customers and the ability to be recommended for the next job. We offer not only the completion of tasks, but design at a high and unique level for our customers.\nStarting at the simplest of chores, we provide real estate inspections that you can be comfortable with in not only meeting local codes, but placing the safety of your family in our hands. Expert troubleshooting and refurbishing electrical systems is a key to our success. We go much further than that, though. An upgrade of a service panel in homes of this stature requires the understanding of the neatness and wiring that appears to not be there. Surge suppressors and lightning arrestors are a must in the climate in this area. The levels of storms that can appear here have the ability to create much more damage than other locales across the nation. A standby generator system is needed for the function of the home's utilities under duress from extreme weather. A properly operating ups system will ensure continued operation of accessories while short term power losses take place.\nThe purchase of an existing home will almost always necessitate the need of new fixtures. Taste in lighting varies greatly amongst homeowners. The quality of these offerings may require care in installation far and above most store shelf devices. This also involves the capacity to provide unique design in purchase and installation. It's important that the installation can not be detected afterwards. Recessed lighting and ceiling cans may be needed for some upgrades. You can be sure that the installation will not mar the beauty of a sculptured ceiling. Upgrades and additions to powered items may need dedicated circuits and sub panels. The location of these panels should lead to easy access and the flow of the home's design.\nAs stated before, safety and security of your family is always at the top of the list. A security system should account for preventing entry by unwanted visitors. An affluent neighborhood is a prime target for thieves. The ability to monitor your home while away can be offered via wi-fi and alerts to first responders. This should also include fire and carbon monoxide detection. Outdoor and security lighting is another way to feel secure within your and away from your home. Phone and data cabling are the first items on the list for these systems. This is needed for the interconnection of these devices with the outside world. Complete access is required not only for security, but the entertainment of your family.\nToday's technology offers the freedom of a smart home that keeps control of many items within reach of a panel or remote operation. The home's ambient temperature is one of many offerings. This can be expanded to garage door operation and green technology. The size of these homes should offer the ability of creating an atmosphere of comfort and value together. Window shades and curtains can be programmed for proper lighting and energy savings while operating without manual interface. Whole house exhaust fans can provide cooling while outside temperatures are in a comfortable range. This can be accomplished and allowing the air conditioning to take over when the fans cannot meet the needs as the days warm up. Temperatures throughout a whole day can vary greatly with a clear Texas sky. We take much pride in our ability to keep abreast of the latest in technology to offer the newest and best quality today's world has to offer.\nAs far as the enjoyment of the people within the house, there are many options available. A swimming pool is a center piece of a backyard in entertainment and comfort in this climate. Properly fitted wiring for the pool's equipment and pool house are a must. The yard may also include a hot tub, gazebo, or a fountain. The lighting for an outside tiki bar is also a service we undertake. This wiring will all be installed below ground. This would include accent lighting around walkways and, patios and decks. Stepping inside, there is an option of a home theater system. We can offer many solutions to incorporate a system within existing room structures. Our design staff will ensure that your desires are played out to the greatest level of satisfaction. This comprises equipment including sound and viewing options. The display screen may be hidden until called for. The sound system can be wired for indoor and outdoor usage. Remote panels will provide the interface for operation at multiple locations throughout the home.\nRemodeling and additions will need the same attention, expertise and quality of service that is needed to meet expectations. Even the simplest of details are met by our staff. Courtesy and understanding of homeowners desires are always our job first. This goes as far as the appearance of our staff, vehicles, and equipment. You can feel confident from the time of your contact to the completion of our tasks that your satisfaction is our goal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you have questions about immunizations or health forms, contact us at healthservices@njit.edu.\nYou may upload forms by logging on to \"Highlander Pipeline.\" The Student Health Patient Portal is located under \"Online Resources\"\nThe staff at Student Health Services wants your feedback.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bosch FNS138-23 23-Gauge 1-3\/8-Inch Pin Nailer - High power best in class sinking long nails into hardwood (use any brand of fastener). In-line self-cleaning air filter keeps debris away from valve and o-rings, improving life of tool. Auto nail length adjust magazine automatically adjusts to nail lengths 1\/2-inch to 1-3\/8-inch. Zero-nail dry fire lock-out prevents damage to work piece and internal components. Slim lightweight body for better line of sight and maneuverability.\nAir Locker P630AK 23 Gauge Pin Nailer Includes Pins from 1\/2 to 1-3\/8 Inches - Kit includes gun & micro pins of sizes 1\/2\u2033, 5\/8\u2033, 3\/4\u2033, 1\u2033, 1-3\/16\u2033 and 1-3\/8\u2033 each(2000\/pk). It shoots all 23 gauge pin nails from 1\/2 inch to 1-3\/8 inch and produces a hole that is barely visible to the naked eye. . It automatically adjusts to accommodate the length of the pin. The air locker p630ak 23 gauge pin nailer is compact and light-weight but big on performance. The p630ak 23 gauge pin nailer is ideal for hobbies and intricate finish work.\nGREX P6\/22-2.5 23 Gauge 7\/8-Inch Length Headless Pins (2,500 per box) - 23-gauge galvanized and coated headless pins are designed to use with p630, p635, p645, p645l, p650 and p650l grex pinner. Optimized tensile strength for each pin length. Precision drawn wire assures consistent accurate pin optimized tensile strength for each pin length. Durable yet flexible collation minimizes strip fragmentation. 23 gauge galvanized and coated headless pins. Designed to use with p630, p635, p645, p645l, p650 and p650l grex pinner. Fastener measures 7\/8-inch length. Measures 7\/8 inches length. Precision drawn wire guarantees consistent accurate pins.\nPorter Cable PPN23050 1\/2\u2033 23 Gauge Pin Nails 2000 per Package - Porter cable ppn23050 1\/2\u2033 23 gauge pin nails 2000 per pack.\nGrex P6\/50L 23 Gauge 2-Inch Length Headless Pins (10,000 per Box) - Precision drawn wire guarantees consistent accurate pins. 23-gauge galvanized and coated headless pins are designed to use with p650 grex pinner. Durable yet flexible collation minimizes strip fragmentation. Optimized tensile strength for each pin length. Fastener measures 2 inches length.\nHitachi NP35A 1-3\/8\u2033 23-Gauge Pin Nailer - Removable nose plate for easy jam clearing. Dual trigger for safe and accurate placement of fasteners. Large housing shield protects the work surface. Reload indicator for easy visual recognition of low nail quantities to avoid dry firing. Magazine automatically adjusts to 5\/8\u2033, 3\/4\u2033, 1\u2033, 1-3\/16\u2033 and 1-3\/8\u2033 fastener lengths for use on a wide variety of projects. Depth adjustment for flush drives into a variety of surfaces. Rear exhaust directs oil and debris away from the work surface. Large fastener capacity for fewer reloads and less downtime.\nAir Locker PIN15 23 Gauge 5\/8 Inch Headless Micro Pin Nails (2000 per Pack) - These are great quality micro pins at an affordable price. Air locker 23 gauge, 5\/8 inch pins. These are standard 23 gauge nailer pins that will fit senco and porter cable nailers.\nOMER PR.28 (PR-28) 23 Gauge Pin Nailer Pinner 5\/8\u2033 \u2013 1-1\/8\u2033 (15 \u2013 28MM) Industrial Duty - The omer pr. Made in italy. 28 is one of the tools that made omer famous for performance and durability. No bells and whistles. Although limited in pin size range (5\/8\u2033, 1-1\/8\u2033) it is an industrial duty performer, popular for manufacturing of picture frames, furniture, mirror frames, door and window production, rattan furniture, etc. Just a solid workhorse designed for years of heavy use.\nAir Locker P630A 23 Gauge 1\/2 Inch to 1-3\/8 Inch Headless Micro Pin Nailer - The air locker p630a 23 gauge headless pin nailer is compact and light-weight but big on performance. With this tool and set of 23-gauge pin nails, you can save time and money by preventing damage to work surfaces. An ideal choice for woodworking applications, the p630a provides effective fastening for glue sets that are tough or impossible to clamp, and it will help you avoid the witness marks or holes that can be left by other driver heads, letting you create clean finished products. The p630a 23 gauge pin nailer is ideal for hobbies and intricate finish work. No air plug included \u2014 features uses 23 gauge micro pin nails similar to senco and porter cable models | hardened tool steel nail driver | automatically adjusts for fastener lengths between 1\/2 inch and 1-3\/8 inch | side load magazine | adjustable exhaust to keep oil and condensation away from work | rubber comfort grip on tool handle | shoots fasteners up to 1-3\/8 inch length easily, even in hardwoods | safety lever prevents accidental firing of tool | narrow nose reaches tight areas and corners for detailed trim work | removable nose cover provides easy access for clearing jams | removable no-mar tip prevents marring of work surface | fastener indicator to see when reloading is necessary \u2014 includes nailer | operating manual | 90 days warranty | oil bottle | allen wrench | blow molded case \u2014 applications intricate finishes | cabinets | decorative trim | glazing strips | molding | mirror and picture frame. And its magazine capacity of 200 pins allows you to work without stopping to reload. It automatically adjusts to accommodate the length of the pin. With the ability to handle 23-gauge headless pin nails that range from 1\/2 inch to 1-3\/8 inch, this nailer enables you to utilize a wide range of fasteners for job-specific performance. It shoots all 23 gauge pin nails from 1\/2 inch to 1-3\/8 inch and produces a hole that is barely visible to the naked eye.\nGREX P6\/12L 23 Gauge 1\/2-Inch Length Headless Pins (10,000 per box) - 1\/2\u2033 23 gauge headless pin fits is all 23 gauge pinners. Desigend to take a 1\/2\u2033 length pin.\nHIGH GRADE Cadex 23 gauge pins pin nails, 7\/8\u2033 (21MM) \u2013 4000 PK - If you are using 23 gauge in the first place, you are likely working on fine finishing or furniture. The slightly rounded edges result in less pin deflection in the wood than the cheaper square pins. Yes, you can buy cheaper pins, but at a fraction of a penny a pin, why compromise on your project cadex headless pins will fit all 23 gauge pinners including cadex, nikle, bostitch, senco, porter cable, spot nail, and others (within rated size specification of course). Cadex 23 gauge pins are considered the highest quality available, developed by the same company that markets the highly rated cadex and nikle 23 gauge tools.\nHIGH GRADE Cadex 23 gauge pins pin nails, 1\/2\u2033 (12MM) \u2013 10,000 PK - . The slightly rounded edges result in less pin deflection in the wood than the cheaper square pins. Cadex 23 gauge pins are considered the highest quality available, developed by the same company that markets the highly rated cadex and nikle 23 gauge tools. Yes, you can buy cheaper pins, but at a fraction of a penny a pin, why compromise on your project cadex headless pins will fit all 23 gauge pinners including cadex, nikle, bostitch, senco, porter cable, spot nail, and others (within rated size specification of course). If you are using 23 gauge in the first place, you are likely working on fine finishing or furniture.\nGREX P6\/25-2.5 23 Gauge 1-Inch Length Headless Pins (2,500 per box) - Magazine loading side load | operating pressure 60 to 100 psi. Operating pressure 1\/4 inch npt. Designed for use with grex 23 model p635, p645 & p650 pins gauge 1\u2033 headless pins. Tool dimensions (lxwxh) 87 inch x 177 inch x 73 inch. Weight (lbs) 2 | magazine capacity 200 pins.\nBOSTITCH PT-2319-3M 3\/4-Inch 23 Gauge Pin (3000 per Box) - Coated finish protects against damage. This product is manufactured in china. Packaged in box of 3000. This product is highly durable. For use with the bostitch hp118k and tu-216-2330 headless pinners. Pack of 3\/4-inch 23 gauge pins. This product is easy to use. For use in the bostitch hp118k and tu-216-2330 headless pinners.\nFreeman 23 Gauge 1-Inch Pinner PP123 - Reversible belt hook. Uses 23 gauge headless pin nails ranging from \u00bd\" to 1\u2033 pins. Trigger safety mechanism prevents accidental firing. Pin size selector. Great for custom applications such as crafts, tracking decorative trim, small picture frames, hobby work, jewelry cases, detailed trim and customer work.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavdkq b/data_all_eng_slimpj/shuffled/split2/finalzzzavdkq
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index 0000000000000000000000000000000000000000..fc87f2d2c91c9fb3babca2b2c4722c71b4253040
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+{"text":"It is my pleasure to join you today to discuss the pivotal role of global multi-\u00ad\u2010stakeholder collaboration in achieving a safe, secure and tolerant cyberspace. But, first let's take a look at what safe, secure, and tolerant means in the context of the Internet.\nThe open and global nature of the Internet, built on fundamental principles of open standards, voluntary collaboration, reusable building blocks, integrity, permission-\u00ad\u2010less innovation, and global reach, amongst others, has enabled astonishing social and economic advancement, in ways that we could never have imagined. At the same time, the Internet, like many other good things, is not without risk.\nInternet's key characteristics are neither the origin nor the cause of the malicious activity.\nBy and large, there is no such thing as absolute \"safety\" and \"security.\" There will always be threats, so our concept of \"safe\" and \"secure\" should reflect that reality. We need to think about \"secure\" in terms of residual risks that are considered acceptable in a specific context, and we must recognize the inherent escalating nature of threats to security.\nTolerance is not a word often heard in Internet governance dialogue, perhaps because it has different and sometimes inconsistent connotations across stakeholders and communities. \"Being open\" is heard much more frequently though. The open, decentralised, borderless nature of the Internet has allowed non-\u00ad\u2010traditional and sometimes disruptive opinions and ideas to more easily cross communities, countries, and regions. These can be confronting, particularly to societies unaccustomed to unreserved statements of opinion.\nthoughtful in how they are delivered.\nOpenness, tolerance, and trust are all very important for effective sustainable policy development. Not all stakeholders are as well equipped to contribute their opinions and ideas, whether through lack of resources, language, or otherwise. Some may prefer more formal interactions while others are more comfortable in a less formal environment. Multistakeholder collaboration aims to bridge the differences so that everyone can contribute effectively and, thereby, offer the world better opportunities and decisions.\nThe highly publicized covert government sanctioned surveillance activities are alarming -\u00ad\u2010-\u00ad\u2010 and are a major threat to trust, which is particularly fundamental in cyberspace. The Internet must be a trusted channel for secure, reliable, private communication between users. One of the key building blocks for that trust is respect for internationally recognised data protection standards. Actions that undermine that trust, such as systematic surveillance, threaten to disrupt natural economic and social interactions that are the foundations for sustained global prosperity.\nNow more than ever, it is imperative that the international community strengthens cooperation to ensure that appropriate policies are collaboratively developed and fairly implemented, and that restrictive or harmful policies are not pursued. Tough and controversial issues should be debated in the open.\nIn closing, the citizens of the world deserve and will demand a global and open platform for communication, built on solid foundations of security and privacy. We must work together to find solutions that give that to each and every one of us. And, we need to restore the principles and relationships of trust upon which the global Internet has been built. The IGF meeting this week will provide an excellent opportunity for all stakeholders to discuss how best to restore these principles.\nFinally, returning to more traditional roles for governments and policy makers is unlikely to be adequate to restore these principles or to restore trust; particularly given this is how such pervasive surveillance flourished in the first place. To be clear, we are not saying there is no role for governments, but we are asking everyone to look to the future and move very thoughtfully, as a lot is at stake.\nWe need to look very carefully at the de-\u00ad\u2010centralized, distributed nature of the Internet and it's management and development, as it has given the world so very much, and has shown its robustness and scalability \u2013 time and time again. We should not throw that out in response to the current set of threats or desires of some.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Seattle Sounders (15-11-5, 50 points) continued their quest for a tenth consecutive playoff berth with a convincing 4-1 win over the Houston Dynamo (9-14-8, 35 points) on a rainy, dark, and dreary night at CenturyLink Field Monday night. The only doom and gloom came from mother nature and around here the darkened night signaled nothing more than the reminder that fall is here.\nOn the field, the Sounders were full of energy, purpose, and commitment despite the absence of forwarding Raul Ruidiaz ( Peru) and midfielder Gustav Svensson( Sweden) due to International team call-ups which began on October 8th and will last until October 16th. The Sounders next game will be October 17th at Orlando and the International break will surely have an effect on that game as players will be returning and need some rest. Nicolas Lodeiro ( Uruguay) and Jordy Delem ( Martinique) have also received National Team call-ups.\nOn Monday the last thing on Brian Schmetzer's mind or his player was about who was missing or travel plans to participate in a National Team games. Forward Will Bruin showed no love lost for his former team as he rifled home a shot inside the Dynamo goal box to give the home side the dream start in the 18th minute. It was Bruins fourth goal against his former team in four starts since joining Seattle in January of 2017.\n\"For me personally, it was good to get back on the scoresheet. My minutes are limited now, so I need to make the most of them. I think I did that tonight,\" Bruin said after the game. Bruin also noted that it was vital for the Sounders to get an early goal.\nAn exquisite counterattack that was started by Nicolas Lodeiro and ended with a Cristian Roldan goal doubled the Rave Greens lead in the 34th minute amid celebrations of 39,518 Rave Green fans who paid no mind to mother nature.\nTwo second-half goals by Victor Rodriguez, a slider in the 64th minute and a straight ticki, tacki play between Osvaldo Alonso, Handwalla Bwana in the 73rd minute made it 4-0 Sounders and the Rave Green faithful dispatched their gloves, hats, and umbrellas as CenturyLink Field turned into a block party.\nHouston forward DeMarcus Beasley pulled one back for the visitors in the 87th minute but it was too little tool last as the Sounders recorded their second consecutive victory in which they have scored four goals.\nSounders FC is now 15-11-5 on the season, and currently sits in fifth place in the Western Conference with 50 points. Seattlesits seven points shy of first-place FC Dallas and just three points shy of second-place Sporting Kansas City with three matches remaining in the 2018 MLS regular season.\nSounders FC has scored four goals in consecutive games for the first time in more than four years. The club defeated Chivas USA, 4-2, on September 27, 2014, and earned a 4-1 victory at Colorado on October 5, 2014.\nDating back to the club's loss to Sacramento Republic FC in the Lamar Hunt U.S. Open Cup on June 6, Sounders FC has accumulated a 13-4-3 record in MLS action.\nWith tonight's result, Seattle remains unbeaten against the Dynamo at CenturyLink Field, having posted a 7-0-2 record in regular season play, with an additional victory and a draw coming in MLS Cup Playoffs action.\nSeattle's 15 victories equal the most wins in a single season since 2017, topped only by the club's Supporters' Shield-winning 2014 campaign (20 wins) in that span.\nV\u00edctor Rodr\u00edguez recorded the first brace of his MLS career, and he has now scored three goals in his last two games. The Spaniard now has seven goals and five assists in 27 appearances (including postseason) since signing with Seattle last summer.\nNicol\u00e1s Lodeiro recorded his team-leading 12th assist of the season, which equals his MLS career high set last season. He is one assist shy of the club's single-season record of 13 held by Mauro Rosales (2011, 2012) and Obafemi Martins (2014). His 32 career regular-season assists for Seattle are two shy of the club's all-time record of 34 held by Rosales and Fredy Montero.\nWill Bruin's goal was his first since scoring in a 2-1 victory at Minnesota on August 8 in a 2-1 win. The forward is now tied with Ra\u00fal Ruid\u00edaz for second in the club in goals this season (7). Including the postseason, Bruin has now scored four goals in five games all-time against his former club. He was selected 11th overall by Houston in the 2011 MLS SuperDraft before being acquired in a trade by Sounders FC on December 23, 2016.\nCristian Roldan scored his fourth goal of the season and his first since August 18 in a 5-0 win vs. LA Galaxy. His assist on Bruin's goal was his second in as many games. Tonight marks Roldan's first time recording a goal and an assist in the same game since August 21, 2016 (a 2-1 win vs. Portland).\nNouhou's assist was the second of his MLS career and his first since July 19, 2017 in a 4-3 win vs. D.C. United.\nHandwalla Bwana recorded the first assist of his MLS career. He recorded his first MLS goal earlier this season in a 2-1 win at Toronto FC on May 9.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yom Tov is called moed. Moed comes from the word vaad, which means \"gathering\" or \"meeting.\" Who are we meeting with? With Hashem! When a person makes up to meet with his friend, they make up that they will meet in a certain place. Where is the place that Hashem would meet us in? In the Beis Hamikdash. In the times of the Beis Hamikdash, there was a mitzvah three times a year to go up to the Beis Hamikdash. It was an eye-to-eye meeting with Hashem, just like when two friends meet each other and make eye contact.\nNowadays, we have no Beis Hamikdash. Where then can we meet with Hashem?\nYom Tov is our meeting with Hashem. It continues to exist, long after we no longer have a Beis Hamikdash.\nHashem is fully ready to meet us \u2013 He is everywhere. Nothing is holding Him back. The only thing that prevents a person from meeting with Hashem is his very self. If a person manages to remove the barrier holding him back \u2013 his very self \u2013 he would then be able to meet Hashem, wherever he is. The Mesillas Yesharim writes that a person who is constantly connected with Hashem is considered to always be walking with Him, even as he lives here on this physical world.\nWhen a person is always connected to Hashem in his life, even though he has no Beis Hamikdash to meet with Him, he himself has become like a Beis Hamikdash \u2013 and he can meet with Him.\nEvery Yom Tov has an inner power in it that enables a person to meet with Hashem. A person has to receive the inner point of each Yom Tov which will connect him with Hashem.\nOn Sukkos, what is that inner point of Yom Tov that can connect a person with Hashem?\nOn Shabbos of Chol HaMoed Sukkos, Chazal established that we read the book of Koheles. This is not a coincidence that we read Koheles specifically on Sukkos. There must be some connection between the book of Koheles and the theme of Sukkos; otherwise, why would Chazal establish that we read Koheles on Sukkos?\nShlomo Hamelech begins the book of Koheles with, \"Hevel havalim (\"futility of futilities\"), so says Koheles; hevel havalim, everything is hevel havalim.\" Rashi brings from Chazal the following: \"Koheles is making an announcement and saying that all of Creation is futile; he says \"hevel\" seven times in the possuk, corresponding to the seven days of Creation. The commentators are perplexed: How could Shlomo HaMelech say such a thing?! How could he say that Hashem's Creation is all futility and vanity?\nThe depth of the matter appears to be as follows. The world is called \"olam\", from the word \"he'elam\" \u2013 \"concealment.\" This world really conceals Hashem from being revealed to us. The world \u2013 this world of he'elam \u2013 was created in seven days; in other words, there are seven levels of he'elam. A person's job on this world is to remove all the he'elam \u2013 to remove all the barriers between him and Hashem \u2013 and come to reveal Hashem. All of a person's avodah is essentially to show how all of creation is one big he'elam.\nWhen a person comes to really feel that all of Creation is hevel \u2013 in that it conceals Hashem from us \u2013 he personally reveals Hashem in his life. He essentially enters the state of before Creation, in which there was no he'elam yet; he will be able to become constantly attached to Hashem as a result. Anything which deters a person from being attached with Hashem is a kind of he'elam. When a person manages to remove that barrier from upon himself \u2013 he views everything as hevel, since it's all getting in the way of revealing Hashem onto the world \u2013 he will be able to always become attached to Hashem.\nThis is the inner point that one can reveal on Sukkos. This is the way how one meets with Hashem on the Yom Tov of Sukkos.\nReb Chatzkel Levenstein zt\"l once said that it's not enough for a person to read the book of Koheles written by Shlomo Hamelech; every single person has to write the words \"Hevel havalim\u2026everything is hevel havalim\", and these words have to be ingrained in one's blood. A person has to feel clearly in his heart that this world is completely hevel \u2013 it leads us astray from Hashem. This is the Avodah of Sukkos: write your own personal sefer Koheles!\nWhen the Beis Hamikdash was around, a person had special Heavenly assistance to reach utter closeness with Hashem and get past all the barriers of this world. He would bring the korbonos (sacrifices) and eliminate the physical aspect of the animal, transforming the physical into the spiritual. He would reveal G-dliness in what was previously something totally physical, something that was a kind of he'elam.\nNow that the Beis Hamikdash isn't around, we have to accomplish this very same goal, but through the abilities of our soul. We need to eradicate the he'elam of this world and instead to come meet with the Creator of the world \u2013 the state of total attachment with Him that existed before creation, when there was no he'elam yet.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My name is Chris, and I have been hard at work for the last six months building Kaimo - an application that makes it easy to try out new things, keep track of how you feel, and analyze the impact of those changes on your wellbeing.\nThe app is in an early-stage beta, closed beta. We're reaching out to you because we think that Kaimo is an ideal tool for biohackers to track data on their self-experimentation and figure out what works best.\nWe will take your feedback very seriously and work hard to make Kaimo the perfect platform for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is no doubt that entrepreneurial activity in South Africa is growing. According to the latest Global Entrepreneurship Monitor's (GEM) report, total early-stage entrepreneurial activity is up 4.1 percentage points from 2016. With unemployment in the country sitting at 26.7%, entrepreneurs play a crucial role in creating new employment opportunities and stimulating economic activity. That said, entrepreneurship for the sake of it is not enough. As technology evolves, the country needs a new type of thinker, who is willing to tackle business and socio-economic challenges and find solutions that can make a difference.\nThis is exactly what SqwidNet hopes to achieve with their IoT Entrepreneurship Programme called IoTEP \u2013 IoT to the power of E. Now in its second iteration, the licensed Sigfox IoT network operator in South Africa is calling on entrepreneurs and small ICT companies to submit their applications to form part of the next intake of entrepreneurs who will be given the tools they need to make their ideas a reality.\nRemember the iconic Think Different campaign Apple ran in the late 1990s? \"To quote Steve Jobs, we need 'The round pegs in the square holes. The ones who see things differently'. We need entrepreneurs who are willing to challenge the norm and tackle challenges head-on to make a difference in society,\" he says.\nApplications can be submitted online and a group of 60 entrepreneurs will be selected to go through the programme. It kicks off with a 3-day workshop to teach participants more about IoT and Sigfox as a technology. Participants will learn to build an end-to-end IoT solution i.e. they will have the opportunity to engage with experts on device building, cloud, platform and applications. Thereafter, they will be given training from Business Doctors to help them start up their business. Participants will have access to their business and technology mentors twice a month to ensure they are equipped with all the tools they need to bring their ideas to life. The programme will once again culminate in the opportunity for the participating entrepreneurs to pitch their ideas to a panel of potential investors. SqwidNet has endeavoured its best to empower participants with the appropriate tools through every stage of the entrepreneurship game.\nSigfox is a global IoT network, with an established ecosystem reaching 45 countries. Through SqwidNet, the network currently reaches 85% of the population in South Africa. Its proposition comprises low-cost, long-range, low-power modules and devices that serve IoT data in a secure and cost-efficient manner to application and service providers.\nThe deadline for applicants to submit their ideas is the end of August and successful participants will be notified by early September. Details of the 3-day workshop will be then be shared with them.\nEntrepreneurs: If you are an entrepreneur with an engineering or ICT background with an idea, please submit your CV and motivation to iotep@sqwidnet.com.\nSmall ICT Business Owner: If you are a business owner in the ICT industry with a turnover below R1,000,000 complete the following survey to enter.\nSqwidNet was launched in November 2016 as the Sigfox operator in South Africa. SqwidNet is driving innovation through a nation-wide IoT network that is low-cost and low-power. With its growing partner network, SqwidNet gives business access to an IoT ecosystem that brings their assets to life.\nThe company began through a partnership between Dark Fibre Africa (DFA) and global IoT operator Sigfox. Sigfox today has presence in 45 countries and covers over 800 million people with its global network, allowing Sigfox Ready\u2122 devices to roam at no extra cost. The SqwidNet network currently covers 85% of the South African population, enabling millions of physical devices to be connected to the digital world.\nFor more information, please visit www.sqwidnet.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you read food blogs (and you're here, so I assume you do), you've probably been seeing a lot of s'mores posts this week.\nWhy? Because August 10th is National S'Mores Day--a ridiculous and totally arbitrary holiday invented by the people who sell marshmallows that I've nonetheless decided to honor because I LOVE MARSHMALLOWS!\nI actually love them so much that I can't buy them because I WILL eat the entire bag by myself in a single sitting. I have zero control.\nWhen I do buy marshmallows for a recipe or party, I have to force myself to hide them in the freezer or in the VERY back of the VERY high shelf. It works...until a few weeks later when I'm hunting around for a can of chickpeas and come across the bag.\nAt which point I immediately sit down on top of the counter to eat every single one (even better when they're kind of stale or partially frozen...oh yes!).\nMarshmallows are totally my kryptonite.\nAnd this granola you see here? Oh geez. I'm in so much trouble.\nThis stuff is GOOD. It's not just granola with chocolate and marshmallows in it. It's granola that tastes like graham crackers.\nLet me repeat that...the clusters of granola actually TASTE LIKE GRAHAM CRACKERS!\nThe secret here was a result of some nerdiness on my part. I was reading about the history of the graham cracker, and learned that it was originally invented as a health food by Reverend Sylvester Graham, an early 19th century Presbyterian minister.\nHe loathed white flour with a passion, and developed a special and naturally sweet flour made by using all of the grain. This flour was used to make the plain cookies known as graham crackers.\nThe original graham crackers were nothing like the cheap things you get in the Honey Maid box today. Most commercial brands don't even use graham flour anymore--just loads of sugar.\nYou can buy graham flour at health food stores and online via Amazon or King Arthur, but you can also make a version of homemade graham flour by combining white flour, wheat bran, and wheat germ. And I figured out that it's the wheat germ that imparts that graham cracker flavor.\nSo that's the secret ingredient: I added wheat germ to my granola. When mixed with brown rice syrup [see note at end of recipe], butter, and a bit of brown sugar, the result is an addictive granola that tastes like little clusters of marshmallow-covered graham crackers.\nMixed with dark chocolate chunks, plush marshmallows, and cacao nibs for a little extra chocolate-y touch, the result is an indulgent and unbelievably delicious breakfast or dessert treat.\nEat this on its own, with milk, or in a bowl over yogurt. It's all good!\nWheat germ and rice syrup give the oat clusters in the homemade granola a distinct graham cracker taste. When paired with marshmallows and chocolate, it's the perfect breakfast version of the classic campfire dessert.\nPreheat oven to 350 degrees. Line a large sheet pan with parchment paper or a silpat. Set aside.\nIn a large bowl, stir the oats, wheat germ and kosher salt until evenly distributed. Set aside.\nCombine butter, brown sugar, and brown rice syrup in a medium saucepan over medium heat, stirring occasionally until the butter melts and the sugar dissolves. Stir vigorously a few times to combine the mixture into a thick syrup.\nPour the syrup mixture over the oats and mix well until all the oats are covered with the syrup. Pour the oats onto the prepared sheet pan and spread out into a single layer, dividing the oats into little clumps (like granola!). Bake 10 - 15 minutes, or until oats are golden brown. Set baking sheet on a rack and let cool completely for 30 minutes (they will harden as they cool).\nOnce completely cool, stir in the chocolate chunks, marshmallows, and cocoa nibs (if using).\nServe immediately or store in an air-tight container or zipped baggie in a cool, dry place.\nNote: Brown Rice Syrup (also known as Rice Malt Syrup and Rice Syrup) is a delicious sweetener made from brown rice. Similar in color and texture to corn syrup or honey, it's low on the glycemic index and contains zero fructose, which means it doesn't affect the body's blood sugar levels as dramatically as other sweeteners. It's also really really good, with a mild, almost caramel-like flavor that works beautifully in this recipe. It's available at stores like Whole Foods, online and at health food stores. I urge you to seek it out as it is a really wonderful and versatile ingredient.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Medical Topics :: Texas Familicare Medical Group, PLLC \/ Robert A. Strzinek, Ph.D., D.O.\nIf you are male and have been feeling tired and fatigued, have a noticeable lack of energy, a decreased sexual drive, and\/or depression you may have a condition called testicular hypofunction or hypoandrogenism, or, as it is commonly called,\"Low T.\" This condition is characterized primarily by a low testosterone level in your blood. In general, a man's total testosterone levels peak at 700-800 ng\/ml at 20-30 years of age and gradually decreases 1% per year thereafter \u2013adding belly fat and taking your muscle mass and libido with it. Blood levels can reach as low as 200 ng\/ml in men ages 50-60 and even in men much younger. Many studies have shown that testosterone replacement therapy (TRT) can greatly improve a man's health and therefore his overall quality of life.\nMany anti-aging proponents are now finding that levels considered \"low\" (less than 300 ng\/ml with 500-700 ug\/ml considered \"healthy\") is actually too low. Dr. Jeffrey Life notes studies showing that men with testosterone levels in the upper 25% of normal (800-1000 ng\/ml) have a lower risk of chronic disease and suggests that doctors should aim for a higher target.\nThere are several options for treatment of low T including gels, patches and injections. There are also some side effects to be considered for each of these options. So, if you are experiencing some of the symptoms of Low T., make an appointment and have your testosterone checked. We can then discuss the treatment program that is right for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Stressed, worn down, anxious, caged in \u2013 we have all been there. I also appreciate Health journal's focus on meals and healthy recipes. In addition to the above there are others akin to womens well being magazines, mens fitness magazine, men's well being magazine. In addition to specialty well being magazines, remember that you could additionally write about health subjects for tons of of different magazines, reminiscent of family journal or men's and girls's curiosity publications.\n(1) Life Extension Journal, April 2015 difficulty, p19. As soon as the numbers were crunched, Girls's Well being interviewed Yelp's native specialists and residents about their city's buzziest wellness trends and proposals. A) Good Health journal through bank card auto-renewal and can receive the first 12 points for $fifty nine.99 which shall be initially deducted from the subscriber's bank card, then an additional $fifty nine.ninety nine can be deducted from the credit card each 12 points thereafter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Finally, the RECLAIM CAMISSA website goes LIVE!\nAfter an adventurous walk about with Dr. Eliot Taylor, the Technical Coordinator of Water and Wetlands (Eastern & Southern Africa) at IUCN, his genius suggestion to quantify the water wasted and it's potential to produce power, versus what it is costing the Western Cape Department of Water Affairs (DWA) and the City of Cape Town (CoCT) to pump the equivalent water from the Overberg on a daily basis, was an eureka moment.\nDue to the elevation at which these springs issue, the energy potential is tremendous, and historically it was this water which turned the wheels of industry in Cape Town. We still need to get a Hydro-Engineer or Hydrologist to help with the kilowatt figures, so that we can do a comparable study. RECLAIM CAMISSA has indeed pioneered this previously, see here.\nTwo of the five springs in the Field of Springs are producing 5.5million litres of water per day - wasted through the tunnels under the city's streets; and we now have twenty-five springs in our database for the City Bowl.\n1. R8.5m or R16.6m per annum if one goes on (only) 4million litres\/day (retaining 1.5million\/day out of the 5.5million for eco-reserve in the filtration ponds) using the lowest residential water pricing tariff of R5.85 per kl or R11.42 per kl for commercial and industrial usage as published in the City's Integrated Development Plan.\n2. Revenue generated by the capture and sale of this water could be utilized to develop other aspects of RECLAIM CAMISSA as subsequent self funded initiatives that would contribute towards skills upliftment and job creation; and hence contribute significantly to the showcasing of a green city and green economy.\n3. The project in capturing potable water at source for domestic and or commercial use would enable the city to embark on a project by satisfying water needs closer to source with the resultant reduction in infrastructural expansion and maintenance; and with the simultaneous reduction in disaster risk management strategies necessitated by water catchment management areas that are geographically remote from the point of consumption. This would translate into a reduced treatment and supply cost per kl compared with conventional water supply strategies.\n4. The water captured by RECLAIM CAMISSA of the two springs thus identified, out of a possible twenty-five springs (although only approximately fifteen of these would be for potable use - the balance for urban landscaping; street trees; fields; parks, etc.) equates to the free basic water supply at 20,277 homes - without impact on the water resources as this is water not currently captured. As such the project benefits the wider Cape Town community notwithstanding its geographic positioning within an affluent suburb.\nSadly, RECLAIM CAMISSA's Pilot Project: THE FIELD OF SPRINGS remains on hold, due to the ineptitude of municipal officials - find out how and why, here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to assign services to own namespace in HANA express edition 2.0 SPS 01?\nJul 08, 2017 at 04:39 PM | 287 Views | Last edit Jul 08, 2017 at 04:44 PM 3 rev.\nto avoid development in SAP namespace I have created an own namespace, so far so good. The problem is, while doing so I wasnt asked which services I would like to assign to this one, so accordingly applications in this namespace want run, since there is obviously no build service to do so e.g.\nSo, its probably simple, but I cant find it on my own: how can I add services to an own namespace?\nIndependently from this: how can I know which services I unconditionally will need? Are those and they purposes documented somewhere? Im clear I could take the SAP namespace as a template, but would prefer to understand what exactly Im doing.\nDetected deploy target \"HANAExpress dev\"\nDownloaded 'SAP JVM JDK', version '8.1.29' in 0.618 s.\nDownloaded 'Tomcat Runtime', version '8.5.12' in 0.348 s.\nDownloaded 'XS Authenticator', version '1.6.1' in 0.1 s.\nDownloaded 'SAPJWT', version '1.0.0' in 0.1 s.\nDownloaded 'SAP JVM Memory Calculator', version '1.6.1' in 0.51 s.\nApplication \"di-builder\" started and available at \"https:\/\/hxehost:51022\"\nThe builder has been deployed successfully in space dev.\nThe builder deployment has finished with status: SUCCESSFUL in space dev.\nfs-storage        free                                     xs file service provides an env variable which denotes the root of the clients application file system.\njobscheduler      default                                  Job Scheduler for scheduling long running, repeated tasks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbagxi b/data_all_eng_slimpj/shuffled/split2/finalzzzbagxi
new file mode 100644
index 0000000000000000000000000000000000000000..15d85d71773737736387a3bd79bc19308e66a15e
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+{"text":"Brand unosy is for sale on Brandroot.com!\nA playful name inspired by the Italian number one, \"uno\". Unosy is charming and memorable, and appealing and stylish name ideal for modern business brands.\nYou will have full ownership and rights to Unosy.com. One of our professional Brandroot representatives will contact you and walk you through the entire transfer process.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking to replace Lance Lynn, who had refused a qualifying offer to test the market himself, the Cardinals considered the likes of Jake Arrieta and Yu Darvish available. They instead turned to Mikolas, who they signed to a two-year, $15 million deal that was set to expire at the end of the coming season, making him an unrestricted free agent.\nMikolas, who had last pitched in the major leagues in 2014 as a member of the Texas Rangers, turned out to be a bargain for the Cardinals. In 200 2\/3 innings, he won a National League best 18 games against just four losses with a 2.83 ERA. He averaged just 1.3 walks per nine innings, which also was the best in the NL.\nMikolas was an NL All-Star selection, though he did not pitch, and placed sixth in balloting for the Cy Young Award.\nAnnouncement of the new contract was made amid some concern about right-hander Carlos Martinez, whose spring training was interrupted by weakness in his throwing shoulder. An MRI last week and a secondary medical exam performed Monday indicated no structural damage.\nBut the 27-year-old arrived at Roger Dean Stadium Tuesday morning with the arm in a sling. The team later explained in a statement that Martinez had a platelet-rich plasma (PRP) shot, which helps accelerate the healing of injured muscles and other tissues. He will be re-examined in two weeks, at which time it is expected he can resume throwing.\nMartinez was limited to 118 innings last season by similar issues that resulted in three separate trips to the injured list.\nWithout Martinez, the Cardinals rotation would likely consist of Mikolas, Jack Flagherty, Michael Wacha, Adam Wainwright and either Alex Reyes, John Gant, Dakota Hudson, or Austin Gomber.\nReyes, who missed basically all of the last two seasons, first with Tommy John Surgery and then with a scapular tear, has been ahead of schedule in his rehabilitation. Wainwright, in his first spring appearance Tuesday, threw two perfect innings against the Washington Nationals in a 6-1 win.\nThe Cardinals Saturday announced they had signed outfielder Jose Martinez to a new two-year deal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The standard equipment list on the Hino 338 sets the benchmark for the industry and speaks heavily to the quality, operating experience, and anticipated premium resale value of this medium duty truck. Look no further than the ATD Truck of the Year award the 338 earned to illustrate this superior quality. The rugged durability of the Hino 338 makes them just right for a variety of industries including beverage delivery, municipalities, construction, utilities, petroleum\/gas delivery and waste\/recycling. With a worldwide reputation for reliability, durability, and outstanding fuel efficiency, you can count on Hino Trucks to provide you with low-cost-per-mile operation.\nHino engines deliver dependable, fuel efficient power. Hino Trucks is recognized worldwide as one of the leading innovators in engine design. Combining high-pressure Common Rail Fuel Injection, Variable Geometry Turbocharger design and cooled EGR with Selective Catalytic Reduction [SCR] for emission control allows us to optimize the air\/fuel mixture across the entire speed range of our engines. The result is fuel-efficient compliance with 2010 EPA regulations.\nProvides higher pressure over the entire speed range of an engine. Unlike unit injectors whose pressure varies with the \"rise time\" of the injection event, common rail systems are supplied by constant pressure pumps that assure instant high-pressure response for each injection event. The result is a more responsive engine and better control of fuel quantity in the combustion.\nAllison transmissions are standard on all conventional Hino trucks and offer a fully automatic six-speed transmission that provides smooth, seamless full-power shifts to put engine power to the ground in the most fuel efficient way. The result of Allison's fully automatic transmission is faster acceleration and higher average road speed, which equate to quicker route times throughout the day. That means greater productivity and increased profits for your business.\nWith more than 100 years of axle manufacturing experience, cutting edge technologies and proven reliability and performance, Meritor has become the world's largest independent manufacturer of commercial truck axles for a broad range of vehicle applications - thats why you'll find front and rear axles from Meritor on Hino trucks - they are quite simply the best money can buy.\n\"These trucks have been a great asset to our fleet and are true workhorses. The turning radius's and visibility is outstanding and the heavier suspension makes for a smoother more comfortable ride. I am very happy with Hino Trucks and will purchase again from my Hino dealer very soon!\".\nIn addition to standard checks and replacements, HinoCare also covers a Diagnostic Trouble Codes check. Contact your local dealer for additional details.\nPreventive maintenance doesn't just enhance the lifespan of regularly used equipment, it enables higher performance while saving you money down the road.\n\"The 'cabover like' maneuverability of the Hino is amazing!\"\n\"I have been in business for 30 years and I have owned over 300 trucks. No truck has ever performed better than my Hino's because they combine comfort and beauty with reliability and the end result is peace of mind and profitability.\"\n\"These trucks are by far the best medium duty trucks we have bought to date.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In which I slam my fingers badly in A Little Knowledge.\nEr... no. No no no. To an armed forces charity. That's probably an important difference. I mean, it's not that I'm such a hopeless leftie I don't think we should have an armed forces, or even that they should be adequately funded, but I do think maybe buying a symbolic representation of a Flanders Fields poppy to help get the Royal Tank Regiment a new Challenger 2 might be ever so slightly missing the point.\nThose don't sound to me like the words of a man who'd be unhappy if we all chipped in for a tank.\nLazy comedy cliche things that actually happened to me this weekend.\nI was infuriated by the confusing instructions for assembling some flat-pack furniture.\nI avoided work by needlessly alphabetising my DVDs.\nI hit my thumb with a hammer.\nJoin me next week, by when I will have slipped on a banana skin, had my computer explained to me by a child, and enthusiastically slagged someone off before realising that she's standing right behind me, isn't she?\nFirst five things I thought on looking at this portrait in the National Gallery, which basically mean I don't deserve to go there.\n* Wow. That horse has a really tiny head.\n* Charles I and his horse have the same hair.\n* If everyone had a small framed sign saying who they were hung up beside them wherever they went, would that be useful or irritating? It would certainly be good at parties.\n* Did Charles pick the horse because it had his hair, or did he get the horse first, and then grow his hair out in order to copy his horse's signature look? Or hasn't he even noticed? I bet the rest of the court has. Van Dyke definitely has.\ngo round the other way.\"\nBEECHAM'S PILLS are just the thing!\nTwo for an adult, one for a child.\nWill restore him without fail!\nWhere FIRST RATE PIES are sold!\nA HOUSMAN'S STEAK AND KIDNEY PIE!\nWith reasons for your choice.\nWhich means 'Growing' is still my best effort. Any advance?\nSo, it turns out you can have a whiter shade of pale, and grass that is greener on the other side. You can have redder blood than I; tell bluer jokes, and have a blacker heart. Your face can be pinker; browner; yellower; greyer or even purpler than mine. But... there's no such word as 'oranger'. What crazy system is this? How am I meant to compare two things, both of which largely reflect light at a wavelength between 585 to 620 nm, but one noticeably more so than the other? How am I supposed to differentiate between half-hearted and fervent supporters of the Dutch royal family? What sort of a impoverished tongue is it in which we cannot point out that both these oranges are orange, but this orange is the oranger orange? It's an outrage.\nTalking of London Zoo, this is the life-size statue that stands right in the centre of it, between the flamingo lake and the tiger enclosure.\nYep. Striking, isn't it? In case you can't read the plaque, it records that the sculpture was presented to the zoo by J. B. Wolff in 1906.\nI see. So, in 1906, J. B. Wolff commissioned an enormous sculpture representing man and beast locked in their age-old conflict; the lion straining to rend the man apart with tooth and claw; the man armed with a primitive knife, desperately trying to disembowel the lion - implacable enemies locked in a fight for survival only one of them can win.\nAnd then he gave it to a zoo.\nYou know what I think? I think J.B. Wolff simultaneously gave the zoo a large amount of money, on condition that they prominently display this statue in perpetuity. I think J.B.Wolff hated the zoo. You're a funny guy, J.B. Wolff.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Magnificent Stone Colonial Home. 4.3 Acre cul-de-sac home site on a quiet street. Private Yard open & wooded, nature sanctuary that backs to Farmland. Walk to Downtown. Superb Schools. Commuters dream Exceptional Quality & Custom Details. Open Floor Plan, Living & Dining Rms ideal for Entertaining. Two Story Foyer. Family Rm w\/Cathedral Ceiling & Fireplace. Cherry Kitchen w\/Granite, SS Appliances & Center Island. Sun filled Breakfast area & Conservatory w\/Transom Topped Windows. Sound Proof Den\/ Office. Large Master BedRm w\/Sitting Area, Walk-in Closets & Luxurious Master Bath. Princess Suite. Spacious Bed Rms. Dual Staircase. Crown Moldings, High Ceilings, Bay Windows & Hardwood Flrs. Walk-out Lr Level w\/Play area. Deck w\/sunset views. House Generator. Oversized 3 Car Garage. Exceptional Value per Sq Ft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbajmz b/data_all_eng_slimpj/shuffled/split2/finalzzzbajmz
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index 0000000000000000000000000000000000000000..78fa6629a02db60d895153c9fd6150fd482fd38d
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Just thought I would share these quick cards I have made recently.\nAaaah another post so soon.\nI took some photos today of the Folio's I made for my friend. I forgot to take them before I gave them to her so had to wait till today to get photos.\nI bought the instructions to create this folio from a youtube tutorial I saw and thought it was great. Mum and I made one each and I have shown you those in a previous post. It is such a lovely looking album and holds over 50 photos so it is good for recording an event or for showing a journey through someones life.\nIt is very time consuming to make, it took me 2 days to make just one of these.\nRay is DJ'ing tonight and I am busy getting packed to head up to Port Douglas tomorrow.\nRay is still not all that well and unfortunately I think the dreaded lurgy is creeping up on me.\nIt has been a pretty average week this week. Everything is normal for me but Ray has a bug and is not that well. Charli has been sick since last Saturday and before that Cindy was on the sick list. I am keeping my fingers crossed that I don't join them as we are flying to Port Douglas on Sunday morning and I don't want to be sick while I am away.\nI have been playing around with making some silicone moulds tonight. I am just waiting for them to set to see how they went. I started out the evening making a layout for my Switzerland album but my heart wasn't really in it so thought I would give the mould making a go.\nDuring the week I did manage to get one double layout completed from my first trip to Switzerland.\nI have used the Iridescent Medium with the Prima Mini Chevron stencil on the background and then inked with Tim Holtz Distress Ink Rusty Hinge to bring the medium out a little more.\nEach of the pages in this album have a pocket to hold more photos, ephemera and my diary page for that day. Makes the layouts a bit more of a challenge. Love the colours of this Basic Grey paper. I have had it for quite a while and thought this was an opportune time to use it. I have used a Kaisercraft wooden camera on the right hand page. I inked it with black then applied 2 coats of black embossing powder to give it a nice high gloss.\nTime to go and check if my moulds are set.\nAnother friday night and we are at home again. We must be getting old as we have not been out on a friday night for about 3 weeks now. It could be the colder weather and that there is nothing on that we really want to go to. Not long now and we will be sunning ourselves in Port Douglas and then the Gold Coast for 3 glorious weeks. Just have to keep my fingers crossed that the weather will be kind to us while we are up there, no cyclones and no rain I am hoping.\nI have been very busy. Don't ask me what I have been doing but the time has flown.\nI am in the process of making a couple of albums for a friend and that has kept me on my toes.\nTomorrow I have got my weekly class at Camelot. Then tomorrow night we have got Charli overnight while Cindy goes to a quiz night that Charli's school is putting on.\noutside my window....it is cold and dark. There has been plenty of rain over the last few days and I have been picking tomatoes off my potted bush but I think the poor old thing is turning ups its roots and departing this world. I won't be getting many more tomatoes from it.\ni am thinking....If I am getting too lazy and that I should be making an effort to go out tonight. Were is the line between laziness, disinterest and nothing fun to go to? I can remember when I would not stay home on a friday night no matter what.\ni am thankful for....my charmed life. Even though I get lazy and lethargic around doing things I am happy that I have the choice and that no matter how I am feeling, if I want to make the effort I can do or go anywhere I want to.\ni am wearing....my everyday clothes. Slacks , t.shirt and windcheater. I must admit I do have a singlet and ugg boots on tonight.\ni am going to...do some more work on the albums I am working on tonight. Tomorrow night I am going to help Charli decorate a box we bought today.\ni am currently reading.... since I got started on Candy Crush I have not done any reading. Every time I sit down with a few minutes to spare I got into Candy Crush instead of reading a book. Very addictive.\ni am pondering... what clothes I will need to take on holidays. Will need to check the weather forecasts for those areas and start work on getting organised. I am sure there must be something I have to buy to take with me.\ni am hoping...for good weather while we are away and that Daisy will not be too hard for Cindy to manage.\ni am creating...a couple of albums for a friend. When I have finished that I am going to get back to my Switzerland album from 2009.\none of my favourite things...Yogalaties. I must say that since I started doing yogalaties twice a week I am becoming addicted. I really enjoy it and look forward to each lesson. if they had it 5 days a week I would go. The only downfall is that it starts at 9:30 so I need to be up earlier than usual. I don't have to go very far. I leave home at about 9:15.\na few plans for the rest of the week....God knows. play it by ear. Take Daisy to the vet to get some more tablets for her arthritis. She is really showing her age and I worry about her. I certainly don't want her to be in pain and her rear quarters do not work as well as they used to.\nWe did not go out last night so I decided to create a layout about our last trip to Port Douglas. It was May 2011 and on the 26th May this year (3 weeks time) we are heading up there again, so I thought it was a good time to do this layout and remind myself of what we did when we were last there.\nThe background is not what I had in mind. I bought some beautiful Matisse Iridescent Medium and I wanted to use it on the background. The problem was I chose an all over spotty stencil to use with it and it was just way too much. I scraped a coat of Gesso over it with my palette knife to tone it down and then sprayed with three Glimmer mists. I have had the photos printed in approx 3\" x 2\" sizing so I could fit them all on the page. The other photos that I took on this trip will go in the top pocket on the second page when I get them printed.\nGot Charli's school photos yesterday. Too cute. All of the photos are the same photo just different sizes.\nWell off to make some cards now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"My morning runs around the Heath are the moment in the day that I feel most alive. I never cease to be amazed at the beautiful wildlife. Today I spotted some white scented Jasmine. I wish I could capture it and bottle the scent - the recent rain has brought out squirrels and rabbits, both spotted busy in their own worlds.\nLooking forward to sharing my thoughts and some photos with you soon.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hey! Whenever water went into my ears, I've always put some more water in the ear to fill it to the top. Then, shook my ear a little bit and let the water stay there for a few seconds. And finally, tilt my head all the other way round. And Lo! The Water's gone!\nthis worked immediately.on the 1st Attempt. I'm Impressed ^^.\nPutting more water in the ear works really well it has always worked for me. The only one I would not recommend would be holding ur nose and blowing out it is very easy to hurt ur ear drum.\nThank you so much putting water in the ear really works to get water out the ear thanks again.\nI had water in my ear for 24 hours straight and tried everything. As soon as I tried this it worked!\nYour way worked for me too.\nTook 2 tries but it worked! Thanks!\nOh my Gosh. Thank you so much! I tried this 2 times to get the water all out and it worked!\nThis really works first time\u2026forget all the so called cures involving vacuums and head shaking! Very impressed and very thankful.\nToday was my first time getting water actually stuck in my ear. I found this and tried it. I worked!!\nThis technique (by adding some waters) are worked for me. Instantly!. Thanks for saving my tonight sleep!\nWater was stuck in my ear for 60 hours and hearing was reduced.\nI've had water stuck in my ears for 3 days, I tried this and it worked the second time. Thanks!\nwow I had so much water stuck in my left ear and I tried everything, including yawning and tilting to the side and Q tips. This worked immediately the first time I did this, thank you so much you're really a life saver!!\nI have tried everything even the blow dryer and it started to work but then it went back fire. I can't sleep tonight and in the morning I'm going to get a hot wash rag and see if it will work.\nThis was the only thing that worked for me, thanks a lot!\nFirstly, thank you for the tips. In the end I dunked my head in a sink of warm water. I had to do this a few times but in the end it worked!\nAmazing! Just about to go to see a doctor. It worked for me at the first go.\nOMG, yours is by far the best method anshul. Thankyou so much for saving me from a restless sleep!\nIt also worked for me. It's genius, fluid is used to syphon fluid so that's the technique basically.\nThis worked on the second try, loved it!!\nIt Worked!!!! Thank you so much!\nI had water in my ear for 3 days and tried this after everything else failed. Worked immediately. Best suggestion ever!!!\nOh my gosh, thank you so much this is the only thing that worked for me, putting about half a tablespoon of water in the eat, letting it settle, I think I may have put too much though, because it felt worse and worse, but after about a minute if you mess around with your ear a little bit it will come out.. \ud83d\ude42 thanks so much !!\nGreat way to get the water out if my ear. I've been struggling with the annoyance of water in my ear and this really helped.\nWow! Work right away! Would recommend 500000 times!\nI had water stuck in my war for 4 days I tried this and the water cane right out I was just going to see a doctor tommrow thank you so so so so much!I would of never even thought if doing this!!!!\nIt worked like a charm thanks Anshul.\nWorked like a champ, first time. My son has had problems with water in his ears for years and I will be sure to tell him this one. Thank you so much.\nThank You anshul. Awesome! It worked great! I've been going nuts for 2 days with water in ear.\nThis remedy should be the first one posted!!! I swim frequently and get water in my ears and always have been able to get it out before by shaking my head around. Today, no luck. I was getting frantic when I read this page. I tried the water trick\u2014-why not? already have water in there LOL, and presto! This fixed me right up. Seems illogical, true, but it worked for me.\non the first try! I had water in my ear for over 5 hours and I tried everything possible and nothing worked until I finally just put more water in my ear. Love this method thanks!\nOmg. Just tried this wihh my husband becausbecause he's been trying for 40 min and it immediately came out!! Ty!\nWow. This works like magic.\nThis worked! Thank you so much. I spent so much time trying out other methods. And this worked! Thanks again!\n3 hours of agony gone in 5 seconds!! Awesome..\nTook me 2 tries but it worked great! I tried to tilt my head the other way too fast, it worked best if I slowly tilted my head back up proper which drained it.\nThank you so much none of these worked but yours worked instantly!\noh my goodness this works great!!! tried the blow dryer no luck that worked thanks sooooo much\u2026\u2026.\nOh my gosh, this worked on 1st try. Thanks for sharing!\nI tried this method on my son and it immediately worked! Thank you!!\nThank you!!!! ITs the simple things in life that work best. Aloha.\nThis worked for me on the first attempt.\nI have had water in my ear for a week and you just fixed it! Thank you.\nWhen water goes in my ear, I tilt my head with afected ear facing sky. Use finger to drop few more drops of water in ear. You will hear some noise. Just at that time tilt head other side. Because of capillary action all water comes out. You will feel warm water coming out. I have used this technique since I was child. Good luck.\nI was so worried and tried number of tips but to no avail. I finally decided to go on with this one. Guess what? It worked instantly. A big cheers.\nIt worked for me. Thanks!!\nOMG pouring water in the affected ear worked. I thought to myself how weird is that to pour lukewarm water into my ear when I'm trying to get water out of it but I did it and it instantly worked. Thank you for sharing.\nNot capillary action involved here. The gushing water creates a suction\/vacuum behind it.\nthanks that worked the first try i had water in my ear for 3 hours and tried this and it worked!\nNo1 worked well for me after applying and releasing pressure with the flat of my hand about 20times very quickly.\nWorked for me too, the palm trick. Thank you!\nYes! I'm a competitive swimmer and I have had to do that MANY times, enough to know that IT WORKS!\nAwesome! This worked for me instantly! 7 hrs after swimming I still had water in my ear. & the minute I read this and shook my head really hard, it drained right out! Cool!\nThank you SO MUCH!!! You saved my ear!\nGood idea just got back from a surf and couldn't clear the water from my ear. tried it and it worked first time. such a relief.\nI have been trying for hours to get water out. And keep hurting my neck shaking my head so hard. A couple drops of water added and Presto! Immediate relief! Wow!\nI found that the 'head banging' worked very well for me..\nThank you whomever suggested it.\nI held my breath and pinched my nose and gently blowed until my ears popped, pulled the bottom of my earlobe down and shook my head and the water drained instantly.\ni I came home from the pool and hot water in my ear all day I tried everything and nothing works I finally came to the side in the water drops in the year to totally helped!!\nI go in the pool every day and every time I come out there is water in my ear! It's so frustrating and annoying! At first I tried the gravity method and it worked the first few times but I guess my ear got used to it!(?) Now I use the water drop method and it works every time! Thanku so much whoever shared it!!!!!\nThank you sooooo much I had water in my ear of two days . I couldn't hear a thing but then I tried I hot water in the ear remedy and it instantly work .\nA way my mom had done this for me was using a sock (a soft one and clean) and uncooked rice. We put the rice filled sock into the microwave for 50 seconds so it heats up. Then you place the heated sock onto your ear. Keep it there for a while, 10 minutes at the least. The steam helps relieve pain and helps evaporate the water in your ear.\nJust used the water method with my daughter and worked the first time!!! Thank you!!!\ni found that right after it happens, jiggle your head a lot to the sides and then tilt your head and it comes right out!\nI found that it works if you stick your thumb or other finger in your ear and make it sort of flat and tilt your head with the affected ear to the ground. This does something, I don't really want to explain because then other people will be like 'thats not how it works' so I'm just saying that now.\nTried it twice and it worked amazing !!\nAmazing results I must admit just by adding water, can't believe just by using water to get water out. I had water in my ear, (happened when I went swimming, forgot to wear my ear plugs), for the last 24 hours which made my ear and head painful. I couldn't neither sleep nor work. For now, what a relief, feels so good after my ear is drained. Thanks a lot for the great advice.\nI put apple cider vinegar in my ear to help with an infection. ACV instantly made the ear feel better. Then it wouldn't come back out. I tried several of the remedies listed\u2026 Nothing. Warm water did the trick! Thank you!\nMy god .. from past 4 hrs, water was making buzz sounds inside my ear\u2026 Tried many things.. finally I tried placing my affected ear on my Palm and created sunction with pressing it in and out\u2026 And water drops came out finally.\nThe water in the ear realy worked!!\nFirst 23 times it didn't work what I thought so i tried the blow dryer and it worked ?\nThe water method worked miracles!!! But you do need to make sure to move your ear around! I tried what I've always done as a child when I git water in my ear from swimming which was to put hydrogen peroxide in my ear but it didn't work so I found this and tried it!! Who knew?!\nAwesome, I was hesitant to put more water in my ear! I often get blocked ears when swimming and managed it again on holiday \u2013 after reading this, I got some warm water in an egg cup and poured some in, shock my head, left it for 15-20 seconds and tipped my head so it came out, lone behold, it all came out and my ears were clear!\nThanks it really worked putting more water worked thanks!!!\nPutting more water into the ear canal worked the first time!\n'so grateful for this suggestion as my hearing was greatly impaired with all of the water in my ear.\nI just tied the method of holding my head sideways and making a vacuum with my palm on my ear and quickly removing my hand and it worked great\u2026.so glad I looked here for an answer!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cart Retrieval Strap. Decrease cart collection time and increase productivity.\nDecrease cart collection time and increase productivity.\nAccommodates steel and plastic carts.\nOffers control over many carts at once.\nManeuvering through parking lots is quick and easy.\nReduces damage to cars from runaway carts.\nModels for cart retrieval machines available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I was nominated for an award last year and I did nothing with it, so, I wanted to be sure that I responded to this one and followed through\u2026 but then I did nothing, yet again. I got caught up with life and let this post sit as a draft for almost 2 months. When I stumbled across it again today, I thought of just letting it go by the wayside, certain it was far too late to respond. But, the longer I stared at it, the more I thought to myself, who cares? It's my blog and I do what I want with it!\nSo, here we go! Better late than never, right?\nThank you Susannah G. of The Impractical Dreamer for this nomination! Take a few moments a check out her blog where she just keeps it real. She shares about her life, the things she's learning, and works to keep the main thing the main thing.\nLike so many other things in my life, I started my blog on a whim 8 years ago. I don't even remember the genesis of my idea, just sitting out on my back porch on my day off, feeling so proud of myself for actually starting a blog.\nFor many years I did nothing with it, posting sporadically and talking about being a writer but never doing any of the work necessary to achieve that goal. Then, about 4 years ago, I began posting regularly and began to see some results.\nI'm so thankful that I started my blog all those years ago because, had I not, I don't know that I would've found how passionate I am about writing. Were it not for this blog, I never would've written and published 2 devotionals (His Story and My Story) and I wouldn't be a freelance writer. The process of writing every single day has led to immense growth in my own skills and a growing business that I enjoy.\nSo, with all that being said, my two pieces of advice are this: be consistent and never edit while you write.\nSeriously. Cultivate the habit of writing consistently and learn to post consistently. It doesn't matter to me if you're posting once a week or five times a week, as long as you're regularly putting out content for your readers on a schedule. More than that, you need to be writing frequently, especially if you want to make a living writing\u2014you need to write every day. In the beginning, I set a daily writing goal of 1,000 words just so I would do it. I was working a more than 40 hours a week job and volunteering on top of that but, because I wanted to write, I found the time to do so.\nHonestly, this is probably the best piece of advice I ever received. Writing and editing are two opposing processes in the brain, so you should never try to do them simultaneously. If you're writing and trying to edit yourself at the same time, it's going to be like pulling teeth. More than that, you'll feel less proud of your creative work because your editor brain will be constantly telling you it's not good enough. Then you won't want to create because you feel shame or guilt or some other negative emotion. It's no fun, trust me!\nNow my method of writing is always this: write, walk away, wait at least 24 hours, and then come back and edit. Sometimes I have to come back after that and edit again, but I make sure the two processes are always separate.\nOne of the most fun parts of blogging is the fact that we get to connect with others. Some of the people listed below I've known for years in person and others I'm just getting to know in the blogging world. I'm so excited to nominate the following bloggers who have encouraged me in one way or another!\n2. Write a post about your award.\n5. Nominate 8 (or 9) other bloggers for this award.\nI'm so glad you did this post! I have enjoyed catching up on your blog, life has been a bit hectic\u2026:P But, I always learn something new when I read your posts, so it's definitely worth it to set aside time to read what you have to say about life. Thank you for that!\nI'm so glad that you're enjoying the blog and you're learning something. I'm consistently amazed that what I write can encourage others.\nThank you so much for the nomination! I am beyond honored!\nYou're welcome! Your words are always chock full of wisdom and encouragement!\nYou shared such a wonderful Knowledge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbcxq b/data_all_eng_slimpj/shuffled/split2/finalzzzbbcxq
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@@ -0,0 +1,5 @@
+{"text":"8,330 patients were mistakenly sent other patients' bills.\nLSU Health Shreveport recently began notifying patients that a processing error at Siemens Healthcare, which prints and mails doctors' bills on behalf of LSU Health, resulted in the exposure of 8,330 patients' personal information (h\/t PHIprivacy.net).\nAccording to an LSU Health statement [PDF file], the organization didn't discover the issue until patients began calling to say their bills were incorrect.\nAfter an investigation, LSU Health determined that an error in a computer data entry field had caused names and treatment information for one patient to be sent to another patient's mailing addresses in 8,330 cases.\nLSU Health says no Social Security numbers, birthdates or financial account numbers were exposed.\n\"LSU Health Shreveport and Siemens have identified the source problem and taken steps to ensure that this issue will not happen again,\" LSU health said in a statement [PDF file].\nAffected patients have been sent corrected billing statements, and have been asked to destroy the incorrect statements they received.\nPatients with questions are advised to call (888) 824-0379 or (318) 675-7550.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The 15-nation North Atlantic Treaty Organisation (NATO) countries have been holding a series of meetings at the Organisation's headquarters in Brussels.\nBackground: The 15-nation North Atlantic Treaty Organisation (NATO) countries have been holding a series of meetings at the Organisation's headquarters in Brussels.\nThe latest -- and most important -- of these is the current series of meetings between the 15 NATO Foreign Ministers.\nAt the end of Thursday's (12 December) session United States' Secretary of State Dr. Henry Kissinger told the meeting that a confrontation with oil-producing nations could only be disastrous for Western Europe and for Japan.\nThe world economic crisis -- prompted by spiralling oil prices -- was the major pre-occupation for the Foreign Ministers as they started their two-day winter Council session.\nDr. Kissinger told the top-secret meeting that nobody had yet found a way to solve the twin problems of inflation had unemployment. It was therefore essential for western nations to maintain the closest possible contacts to combat \"stagflation\".\nThe impact of the West's economic crisis on the NATO alliance -- celebrating its 25th anniversary this year -- has been the recurring theme of a series of ministerial meetings at Alliance headquarters this week.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a placeholder page for Sanchez Elisa, which means this person is not currently on this site. We do suggest using the tools below to find Sanchez Elisa.\nYou are visiting the placeholder page for Sanchez Elisa. This page is here because someone used our placeholder utility to look for Sanchez Elisa. We created this page automatically in hopes Sanchez Elisa would find it. If you are not Sanchez Elisa, but are an alumni of Cibola High School Yuma, AZ, register on this site for free now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You're able to trust in Star Vinyl Tile to deliver the top professional services when it comes to Vinyl Tile in Lakeview, MI. Our workforce of well trained contractors can provide the expertise that you need with the most advanced solutions in the industry. We make sure you receive the most excellent services, the right price, and the best products. Call us at 844-284-6619 and we'll be able to discuss your plan of action, respond to all your questions, and set up a scheduled appointment to start organizing your job.\nHere at Star Vinyl Tile, we know that you will need to keep within budget and save cash wherever you can. Still, saving money shouldn't suggest that you sacrifice superior quality for Vinyl Tile in Lakeview, MI. You can expect the very best quality even while still saving you money. Our end goal is to make sure that you receive the highest quality supplies and a result that lasts over time. It will be feasible because we understand how to save your time and costs on products and labor. Save time and funds through getting in touch with Star Vinyl Tile today. You can contact our staff at 844-284-6619 to learn more.\nVinyl Tile are available in Lakeview, MI.\nTo come up with the most suitable judgments with Vinyl Tile in Lakeview, MI, you have got to be kept informed. We will not encourage you to put together poor choices, as we know just what we are working at, and we ensure you know very well what to look forward to from the task. You will not confront any sort of unexpected surprises whenever you use Star Vinyl Tile. The first step is to call by dialing 844-284-6619 to begin your job. Within this phone call, you're going to get the questions you have addressed, and we'll establish a time to start work. We will work with you through the whole process, and our staff can arrive on time and prepared.\nWhenever you are preparing a process for Vinyl Tile in Lakeview, MI, there are lots of reasons to prefer Star Vinyl Tile. Our products are of the highest quality, our money saving strategies are helpful and powerful, and our customer support ratings cannot be topped. We have the experience that you need to meet your objectives. Contact 844-284-6619 to reach Star Vinyl Tile and consider your expectations concerning Vinyl Tile in Lakeview.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"passed away n s ome on Frday, May 20, 2011 after a long llness.\nHe s survved by a broter, Danel W. Juska and s wfe Patrca of Atknson, ME; two ssters, Janet N. Flemng and er usband Wllam of Westport and Carol A. Benson and er usband Kennet and a nepew, Adam Benson, all of Sturbrdge. He was born n Worcester, a son of Wllam R. Juska and Brone (Daugrda) Juska.\nMr. Juska worked as a computer engneer for many years pror to retrng. He graduated from Nort Hg Scool, Worcester Industral Tecncal Insttute and Qunsgamond Communty College.\nA memoral servce wt a Mass wll be eld on Wednesday, June 1, 2011 at 11:30 AM n Our Lady of Te Angels Curc, 1222 Man St. Bural wll be n Notre Dame Cemetery. DIRSA-MORIN FUNERAL HOME, 298 Grafton St., s assstng te famly wt arrangements.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbcuzz b/data_all_eng_slimpj/shuffled/split2/finalzzzbcuzz
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+{"text":"Foxes vs. hedgehogs, a history: From Nate Silver, FiveThirtyEight, and Isaiah Berlin back to Archilochus of Paros.\nLeft: a hedgehog. Right: a fox?\nPhoto-illustration by Juliana Jimenez. Left: photo of a hedgehog by Giorgio Cosulich\/Getty Images. Right: photo of Nate Silver courtesy Randy Stewart\u200b\/Flickr.\nOur logo depicts a fox (we call him Fox No. 9) as an allusion to a phrase originally attributed to the Greek poet Archilochus: \"The fox knows many things, but the hedgehog knows one big thing.\" We take a pluralistic approach and we hope to contribute to your understanding of the news in a variety of ways.\nGiven FiveThirtyEight's professed dedication to its methodology, this moment of vulpine self-identification has struck some media analysts amiss (e.g., \"I think @natesilver got the fox\/hedgehog thing backwards\") and flummoxed some others (\"I am flummoxed by this fox\/hedgehog thing\"). For this reason\u2014and for the sake of tracing the evolution of an antique truth into a postmodern commonplace\u2014I would like to consider the famous phrase in its original context.\nThis parlor game started getting out of hand in the 1990s\u2014the decade of Judy Davis delivering a foxy, Berlin-inspired monologue as part of her sensational performance in Husbands and Wives, of dorm-room CD players inquiring \"Are you a fox or a hedgehog?\" on Luna's hedgehogging Penthouse, of Michael Ignatieff's Isaiah Berlin: A Life identifying its subject as \"the type of fox who longs to be a hedgehog.\" At the turn of the 21st Century, the idea exploded. I mean this to say that it fully emerged from the realm of the New York Review of Books into that of the bestseller lists, and also I mean to say that the idea ceased to have a coherent identity. As the idea drifted\u2014into the deliberately difficult title of a Stephen Jay Gould book, into the branding of a quarterly magazine, into the jargon of marketing consultants and their jabbering ilk\u2014its meaning shifted in several directions at once.\nHow on earth do people who communicate in buzzwords keep straight all the pluralistic (foxy?) usages of this one big (hedgehoggish!) concept? I understand that phrases, like words, can change their meanings over times. I understand, further, why the Archilochus adage has been enjoying its extended cultural moment: Its binary elegance slices a complicated world into tidy parts; its philosophical pedigree masks its brute simplicity; its Gladwellian polarity suits the contemporary culture of thought, such as it is; it's fun to do self-diagnostic quizzes. I totally get why someone as otherwise astute as Silver\u2014so clearly a hedgehog in Berlin's sense, given his systematic central vision\u2014should prefer to be thought of as a fox: Foxes are foxy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Wiley CPAexcel Course Outlines are updated bi-annually, map perfectly to the Wiley CPAexcel Courseware outline and structure, and feature Bite-Sized Lessons. Provided in printed format for all four sections of the CPA Exam; the AUD section includes 1 volume. Each course outline is a series of Bite-Sized Lessons. Each lesson usually consists of 3 to 10 pages of study material. The content of each course outline is identical to our courseware at the date of printing. Separate course outlines(s) for each exam section covering the complete scope of the AICPA content specification. Course outlines map perfectly to our courseware outline and structure. No cross-referencing is required. At the front of each course outline is advice from the author concerning the emphasis of the questions on the CPA Exam, as a guide to students completing their studies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Greek chickpea salad has chopped tomato, cucumber, bell pepper, red onion, feta cheese, and of course, chickpeas. There's a simple olive oil and lemon dressing, and the whole salad takes less than 20 minutes to whip up.\nI have a thing for Greek salads. As in, I've eaten seven of them in the past seven days. Most of them have been at restaurants, like I'm some kind of restaurant critic on a quest to find the best Greek salad in the city. Although that would be a fun job, I'm just a girl who really likes Greek salads, staying for the summer with my grandparents who happen to like going out to eat.\nAs delicious as some of those restaurant Greek salads have been, though, this homemade Greek chickpea salad may be even better because by making it myself, I can ensure that there's a perfect topping : lettuce ratio. The toppings are the best part, so I definitely don't skimp on those.\nOn the first day I made this Greek chickpea salad, I enjoyed it over mixed greens. The next day, I had it all by itself for lunch, no lettuce, with just some fruit on the side.\nThe most time consuming part of making the salad is chopping all of the veggies, but I turned on my favorite music while I chopped and time flew by. All in all, the whole salad came together in just under 20 minutes, and chopping took maybe 15 of those minutes.\nAnyway, this would be a great thing to pack for a vegetarian lunch, bring to a neighborhood cookout, or enjoy at home to get your Greek salad fix. You could add olives, roasted eggplant or brussels sprouts, change up the dressing, add other spices\u2013 the possible adaptations are endless!\nCombine chickpeas, chopped bell pepper, red onion, tomato, cucumber, optional olives, and garlic in a large bowl. Add olive oil, lemon juice, oregano, parsley, and salt and pepper, and stir well.\nIf serving immediately, add feta cheese and stir very gently just before serving. If serving later, place salad in refrigerator for up to 4 hours and add feta cheese just before serving, stirring gently to combine.\nLove the shot of all the chopped veggies! And toppings > lettuce. Haha I think I usually go for a 4:1 ratio, at least.\nThe worst is when there aren't many and after a few bites, you end up eating a big bowl of just lettuce!\nThis looks so good Liv! So colorful too. I love chickpeas and feta on salads! So so good. \ud83d\ude42 Have a great day girl!\nThanks, Alyssa! The more colors, the better, right? Hope you have a great day, too!\nLiv! These photos (heart eyes emoji)!\nI am the same with Greek salad \u2013 i can't get enough of it. I especially love the olives + feta cheese combo. But i love that you added chickpeas for some protein because that's my single complaint about it. Pinning this!\nThe chickpeas seem to go perfectly in Greek salad! Thanks for pinning, Beverley!\nGreek salads forevaaa! I love topping mine with tuna salad, but chick peas would be a fun way to switch it up!\nTuna salad sounds like a great way to up the protein!\nthis looks amazing! I love chickpeas and greek salad so i know i will LOVE this!\nJust stumbled upon your blog through Alison's! Love it:) Chickpeas have been SO good lately, I'll have to try this. So fresh for summer, too!\nHey Colby! Nice to \"meet\" you. I'll head over to check out your blog, too!\nThis is a great idea \u2013 combine tasty Greek salad with something more substantial like chickpeas. Will have to try it!\nThanks, Elissa! Hope you get a chance to try it!\nJust found your recipes when I was searching for healthy baked apple recipe for my son. This salad looks really delicious. I'm a Greek food lover as well. I'm also super impressed that you cook all this delicious food in a dorm! Amazing. Keep up your amazing dedication to healthy food.\nOh so glad you found my site! Thanks so much for your kind words and hope you get a chance to try my skillet \"baked\" apples. I'm at home for a few days during my spring break this week and my family requested a batch of those apples already!\nwhere's the tomatoes and olives??","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The idea that music is a universal language is beyond cliche, but behind every cliche is truth, and the Nigerian film industry known as Nollywood provides a particularly fascinating case in point. Back in the early 60s Lebanese merchants began importing the music-laden films of Bollywood, India's sublimely prolific film industry, into the country. Although Nigerians couldn't understand the Hindi lyrics, the elaborate song-and-dance numbers and easy-to-follow plots proved addictive anyway. By the late 70s the hunger for Bollywood cinema had grown to the point that the films and music from them began airing on Nigerian TV and radio. Musicians from the northern region of the country began covering Bollywood songs, inserting Hausa lyrics in the same meter. By the 90s, when VHS technology had become cheap, Nigerians began churning out their own soap-opera-like dramas.\nI've actually never seen any Nollywood films, although they've started screening to more general audiences in recent years. In fact, I'd never heard any of the music from those films either, until I picked up a copy of Harafin So (Sahel Sounds\/Little Axe), a selection of hits from Kannywood films (the Kannywood variant is named for the city of Kano, which is center of northern Nigeria's film industry). As you can hear from today's 12 O'Clock Track, \"Tofi\"\u2014a duet between Abubacar Sani and Fati Niger\u2014the music reveals a strong debt to Bollywood traditions, particularly in the female vocal style. All of the instruments are synthetic, and the vocals, heavily Auto-Tuned as they are, sound nearly as plastic. I haven't decided whether or not I actually like the music, but there's no doubt that I'm gripped by it. It possesses that ineffable quality that only seems to emerge when one culture appropriates ideas from another, without fully understanding their meaning or nuances. I hope the misunderstandings never stop coming.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dog photography combines fun with a thorough physical workout. That, and the laughter which goes with it, has surely got to be one of the best forms of therapy. Key to success is an owner who is on the same page.\nI love working with clients on special projects. In this case, a fabulous heirloom album as a birthday surprise for the owner of these two adorable Westies. We worked across three locations in order to create a story of their lives and characters.\nA great many dog owners will invest in professional portraits of their pets. Dogs are much loved as any family member would be. Sometimes I'm asked to create natural lifestyle photographs and sometimes the owner is looking for something more formal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdkdz b/data_all_eng_slimpj/shuffled/split2/finalzzzbdkdz
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index 0000000000000000000000000000000000000000..e2496455862320de2804db3a61c15f1cab79deae
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@@ -0,0 +1,5 @@
+{"text":"Recommended Minimun Amount of Material For THCa?\nWas wondering what you all think the minimum recommended weight to make diamonds was? A local guy told me he could make me about an Oz of diamond\/terps out of a half pound. Thought?\nDepends on the diameter of the container, you can grow thca crystals with a gram of bho if you had a test tube.\n@ScoobyDoobie no containers or equipment yet. More or less trying to figure out the minimum material and equipment needed to produce and scale from there.\nRight now i'm growing in a small space so I figure ill get between 10oz and 16oz but when I move to our new house in 6 months I should get a few pounds.\n@cyclopath thats what the guy estimated when i was asking him about how much to make an oz.\nSo about 4g a oz?\nI typically run an oz at a time and yield around 5-7g of THCa per run. About the same weight in terps once I pour off. I don't dewax at all though so my yield is a bit higher than average normally.\nYou should just make it yourself. Its really not hard and you would not have to worry about lower yields than you were told.\nhi there guys\u2026 im a grower i want to crystalize THC O acetate how do i do it ?\n@Stone: First you have to grok the alchemy behind that transformation.\ncan you tell me why you want the Acetate?\nAssuming you're actually trying to make thc acetate (which I doubt), you ought be able to find the relevant threads, and the appropriate data in the data dump.\nAnd there is a ton of info on that around here.\nEdit: I see you've found the synthesis thread, and are interested in THC-O-acetate. Doh!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You do not yet have permission to edit this festival, complete the form below to request access.\nOnly registered users on festival focus can update the details for Film Festival Five.\nAt this time our system reports you are not logged into festival focus and so you'll be unable to update details for Film Festival Five.\nComplete the form below to join festival focus and request the ability to edit details of Film Festival Five.\nIMPORTANT: Only festival administrators and staff can edit festival details, film makers completing this form will be deleted.\nWhy do you want to connect to this festival?\n-- select one -- I wish to enter a film into this festival I wish to attend this festival. I am involved in the running of this festival. I am involved in the judging entries to this festival.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some 1,300 participants composed of farmers, stakeholders, extension workers, City and Municipal Agriculturists joined the National Information Caravan conducted by the Department of Agriculture (DA) in partnership with the Local Government of Zamboanga del Norte with the theme: \"Agrikultura Para sa Pagbabago\" held at Dipolog City Sports Complex, Brgy. Olingan, Dipolog City last week.\n\"The Information Caravan aims to raise awareness and bring the services of the department to the farmers by providing helpful and reliable information,\" said Ms. Cheryl G. Suarez, Information Chief, Agriculture and Fisheries Information Division (DA-AFID).\nThis is the fourth information caravan conducted in the country after Region 1 in Pangasinan, Region 4 in Batangas, and Region 7 in Dumagute.\nDuring his welcome remarks, Dipolog City Mayor Darel Dexter T. Uy said that Information Caravans are welcome to be staged in Dipolog City to widen the people's interest and knowledge about agriculture. He also emphasized that he was committed to give his full support to agricultural programs and grateful to host the activity.\nZamboanga del Norte Governor Roberto Y. Uy also gave his full support to the program. He came to boost the participation of the farmers in Zamboanga del Norte.\nMoreover, competent resources speakers were invited to discuss about the DA's programs like Hybrid Rice, Mechanization, Fishery, Credit, and Market.\nAmong the speakers were Engr. Marvin Manalang, Science Research Specialist II of DA- Philippine Rice Research Institute (PhilRice), Dr. Rodolfo Estigoy, PhD, Chief Science Research Specialist, Applied Communication Division, DA \u2013 Philippine Center for Postharvest Development and Mechanization (PhilMech), Mr. Nazario Briguera, Chief Public Information Office, DA \u2013 Bureau of Fisheries and Aquatic Resources on Fisheries (BFAR), Ms. Marissa Mohammad, Agri \u2013 Business and Marketing Division (AMAD) Supervising Agriculturist, Mr. Renato Barriero, Project Evaluation Officer, DA \u2013 Agricultural Credit Policy Council, and Mr. Charlito O. Brilleta, Regional Manager II, Philippine Crop Insurance Corporation (PCIC).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All Servers with Microsoft Operating Systems must be running Windows Server 2008 or SBS2008 or later, and have all of the latest Microsoft Service Packs and Critical Updates installed. Non-Microsoft Servers must meet similar OS-Specific requirements.\nAll Desktop PC's and Notebooks\/Laptops with Microsoft Operating Systems must be running Windows 8 Pro or later, and have all of the latest Microsoft Service Packs and Critical Updates installed. All Non-Microsoft PC's and Notebooks\/Laptops must meet similar OS-Specific requirements.\nAll Server and Desktop Software must be Genuine, Licensed and Vendor-Supported.\nThe environment must have a currently licensed, up-to-date and Vendor-Supported Server-based Antivirus Solution protecting all Servers, Desktops, Notebooks\/Laptops, and Email.\nThe environment must have a currently licensed, up-to-date and Vendor-Supported Anti-Spam Solution.\nThe environment must have a currently licensed, Vendor-Supported Server or Network-based Backup Solution.\nThe environment must have a currently licensed, Vendor-Supported Firewall between the Internal Network and the Internet.\n* Client will be billed to bring the environment up to these minimum standards. * All Ad-hoc support credit expires in 12 months. * Response time for Ad-hoc support is 2 hours for critical and 4 hours for non critical issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today's Mercedes-Benz A-Class hatchback is a far cry from the radical-yet-flawed original \u2013 the first A-Class was more of a small, five-seater MPV than a traditional hatch. As a premium compact hatchback it's got a fight on its hands against the Audi A3, BMW 1 Series and even the Volkswagen Golf.\nNow in its third generation, the A-Class remains the smallest model in the line-up to wear the three-pointed star badge and is now one of the best-selling cars in Mercedes' range, buyers attracted by its sporty image.\nNot that all A-Classes have the performance to match that perception, with entry-level A180 CDI versions packing a modest 1.5-litre diesel engine. Larger petrols and diesels are also available for those who want more speed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdveq b/data_all_eng_slimpj/shuffled/split2/finalzzzbdveq
new file mode 100644
index 0000000000000000000000000000000000000000..4e2e756b1f7a368c947d33da5ddf846f1d5182e1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdveq
@@ -0,0 +1,5 @@
+{"text":"Sovos Sales and Use Tax Filing enables simple, accurate, timely filing in all jurisdictions with advanced reporting audit defense.\nSovos has joined the Wall Street Blockchain Alliance, an industry leading non-profit trade association with a mission to guide and promote comprehensive adoption of blockchain technology.\nLearn how Sovos facilitates 1099 reporting for NetSuite customers securely and efficiently.\nTechnology has transformed your business and your products, and it's about to change your ERP. But, it's also transforming tax, and it could have a significant impact on your S\/4HANA plans. Read more.\nThe Sovos Intelligent Compliance Cloud sits on a unique cloud software platform that is built to bring previously disparate tax solutions and data together for use wherever your business operates.\nThe Colorado Department of Revenue (DOR) certified the Sovos address database, part of the Sovos Global Tax Determination solution in 2016. Sovos was the first business to receive this certification.\nHow Sovos Intelligent Compliance in the cloud eliminates IT barriers to a more strategic approach to sales and use tax compliance.\nIncluded as part of the Global Determination Solution, Custom Reporting and Analytics provides the insights and audit defense tools needed for quickly identifying and reconciling tax transactions.\nStay ahead of rapid regulatory change and safeguard your organization from risk, increase accuracy.\nDiscover why Sovos Tax Information Reporting is the one-stop solution for B2G reporting.\nDiscover the advantages of using Sovos Managed Services for Automatic Exchange of Information (AEOI) reporting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for a ride up to Savannah either Friday or Saturday! Needing to get back for business. You can drop me off along the way if it is in your route as well. Please contact me with serious inquiries.\nIm heading to daytona from boynton beach 630 am. Then heading back in the afternoon. If it matters im 27 male nice guy clean driving record. I was looking to get some money for the drive.\nI will be leaving west palm beach, sunday, april 28 and driving straight through to New Hampshire am seeking person to help with the driving.\nHi! I am going to travel from Florida to San Francisco (I need to get there by May 18, it's my daughter's graduation). Happy to share a ride with someone all the way or some of the way!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lorenzo Facchinelli mainly works with the artistic platform Mali Weil, established in 2008 together with Elisa Di Liberato and Mara Ferrieri and active in performance field through projects that range from relational practices to design, from multimedia to the creation of audiovisual formats. As associated artist for the art work space Centrale Fies he takes care of several projects. Together with other Mali Weil's members, he is also project developer for Fies Core: the Centrale Fies department dedicated to the relation between artistic practice, entrepreneurship and cultural design.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As BREEAM assessors, Encon Associates can provide full BREEAM 2011 assessments.\nEncon Associates have a proven track record as BREEAM assessors of delivering projects in accordance with the BREEAM 2011 assessment criteria. Our experience as BREEAM 2011 assessors means that we are adept in the methodology of BREEAM 2011, and can ensure your non-domestic building project meets these criteria.\nIn our role as the BREEAM assessor on your project, we can fully assist you and your design team to ensure the highest environmental standards are met and BREEAM 2011 assessment rating achieved, and can also provide many other design and advisory services required under the BRE umbrella.\nBREEAM 2011 was released to replace the previous BREEAM 2008 methodology.\nAs well as assessing common building types such as offices, our experienced BREEAM assessors can apply the BREEAM 2011 assessment criteria to range of other buildings which previously required BREEAM Bespoke Criteria.\nTransport hubs e.g. train\/bus stations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UofK VC, Professor Ahmed Mohamed Suliman(left) presenting a gift to IITA Director General, Nteranya Sanginga.\nThe 5-day exploratory visit was led by Prof Ahmed Mohamed Suliman, Vice-Chancellor of UofK, and Dr Abuelgasim Elzein, Head, Department of Agricultural Biotechnology, also a former scientist with IITA, and accompanied by Abdalla Higazi and Hala Ahmed Elamin, both from the Internal Political Affairs unit of the office of the Sudan President and two other professors from the University. Receiving the delegation, IITA Director General Nteranya Sanginga gave an overview of the Institute's achievements and success stories from inception to date and highlighted the efforts made to generate agricultural innovations to meet Africa's most pressing challenges of hunger, malnutrition, and poverty, and secure the path to food security.\n\"We are ready to expand into Sudan with a strengthened diversified portfolio to contribute to improved livelihoods and incomes for small-scale farmers in the country. We are optimistic that this collaboration will put smiles on the faces of the Sudanese,\" he added.\nStating the purpose of their visit, Suliman said, \"UofK is leading an initiative to adopt one of the best-bet proven technologies developed by IITA which is High Quality Cassava Flour (HQCF) and its processing techniques for which IITA is known worldwide and has made significant breakthroughs, achievements, and impacts. UofK has plans to adopt and promote this technology with strong support of the Presidential Office of the Republic of the Sudan to make up for the shortage in wheat flour. Our Presidential Office is very much interested in supporting the inclusion of HQCF in bread and other forms of confectionery as part of efforts to improve food security and the livelihoods of farmers in the country. On that note, I would initiate an immediate development of an MoU with IITA to serve as a guide to common interests between the two institutions,\" he added.\nDue to the strong motivation indicated by the Sudanese delegates, the two organizations officially signed a memorandum of understanding.\nIheanacho Okike, Cassava Peel Technologist, giving a presentation during the Sudan delegates' visit to IITA.\nDiscussions were held regarding the type of cassava technology to be introduced in the country, soil topography, climatic conditions and action plans for immediate implementation. Alfred Dixon, Director of the Development and Delivery Office said, \"Cassava is the third largest crop in the tropics, and a major staple food in sub-Saharan Africa, providing a basic staple for over half a billion people. It is one of the most drought-tolerant crops, capable of growing on marginal soils. However, with the help of cassava experts from IITA and good agronomic and weed management practices, I am certain that in no distant time, Sudan will be known for cassava production,\" he added.\nHighlights of the 5-day visit include presentations by key IITA staff; Frederick Schreurs, CEO Business Incubation Platform; Ismail Rabbi, IITA Molecular Geneticist\/Breeder, Festus Okunlola, IITA Youth Agripreneur (IYA); Iheanacho Okike, IITA Cassava Peel technologist; and Thierno Diallo, IITA Machine Fabricator, all directed towards showcasing the benefits of cassava and its role in economic development. The delegates also had the opportunity to undergo hands-on training at the IITA Food and Nutrition Science Laboratory on how to bake bread, croissant, chin-chin, and Tidbits using HQCF.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdzdm b/data_all_eng_slimpj/shuffled/split2/finalzzzbdzdm
new file mode 100644
index 0000000000000000000000000000000000000000..9f0d7dea51413e2b7c21d41605d8c6f194d9c8bb
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Five oil companies operating in the country have been accused by the Special Presidential Investigation Panel on the Recovery of Public Property of failing to remit over one billion dollars in taxes and royalties to the federal government.\nChairman of the panel, Mr. Okoi Obono-Obla, who spoke while briefing newsmen in Abuja on Monday, accused the yet-to-be-identified oil companies of being complicit in sabotaging the Nigerian economy.\nHe disclosed that the panel had already commenced investigation of the companies to determine the extent of their involvement.\n\"We have compiled their names and submitted to the Comptroller-General of Nigeria Immigration Service so that they would be placed on travel ban,\" he said.\nWhile outlining the achievements of the panel in 2018, Obono-Obla said the panel had compiled the names of about 39 public officers who were under investigation under the Executive Order 6, which had to do with travel ban.\nHe added that the panel was in partnership with the Nigerian Intelligence Agency (NIA) investigating Nigerians allegedly implicated in the Paradise Panama Papers.\n\"Findings will be made available to Nigerians in due course.\n\"The panel is investigating the Tumsah brothers \u2013 Tijani Tumsah, Vice Chairman, Presidential Initiative on North East (PINE) and Ibrahim Tumsah Director, Finance and Accounts, Federal Ministry of Power, Works and Housing for alleged abuse of office and corruption.\n\"The panel has recovered an assortment of 86 brand new luxury and sophisticated cars from the duo; 23 of these vehicles are bullet-proof without the requisite permits.\n\"We have since obtained an interim order of forfeiture from FCT High Court in Abuja in respect of these properties,\" he said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome! Organik Media Inc., is a full-service, NYC based production company and media consultancy formed by director\/producer Sean Kaminsky. Organik has produced content for companies both large and small as well as top brands and TV networks such as HBO, Sundance Channel and MTV\/LOGO. He is the published author of two non-fiction books: Naked Lens: Video Blogging & Video Journaling to Reclaim the YOU in YouTube, and Shoot 3D Like a Pro.\nWe have produced acclaimed documentary programming that has been broadcast on networks ranging from HBO to PBS to Sundance Channel and others.\nOrganik Media Inc. has helped a diverse range of companies, brands and individuals use the power of story to create change, inform and inspire.\nVirtual reality is not only about gaming. Documentary VR helps viewers connect deeply with a subject and explore locations as if they were there.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Level is a 40-story, luxury rental tower located on the East River waterfront in North Williamsburg.\nWith the April 2019 L train shutdown looming, it might seem counterintuitive to suggest that now is a wonderful time to move to Williamsburg. But the waterfront is well positioned to weather the service disruption, with easy access to ramped-up ferry and bus service that would make residents consider avoiding the train in the first place. And with two new buildings by Douglaston Development, that picturesque stretch between Kent Avenue and the East River has become the premier place to live in Williamsburg.\nLevel, a new 40-story luxury residential tower, is home to 554 elegant units from studios to three bedrooms. Just down the block is One North Fourth, a 41-story tower with 509 residences from studios to three bedrooms. Both redefine modern luxury living, with incredible minimalist interiors and a wide range of amenities in units and in the buildings themselves. Both are built right out over the river, offering resplendent city views. And both sit at the heart of the thriving, vibrant Williamsburg waterfront, a community that Douglaston has helped develop over the years.\nThe upcoming L train shutdown will close the subway tunnel connecting Williamsburg and Manhattan for an estimated 15 months. In response, the city and the MTA are offering a wealth of new transportation alternatives that may, in fact, cut down on commute times.\nIt couldn't be easier for Level and One North Fourth residents to get to Manhattan. The MTA's L train shutdown plan calls for a new ferry route departing just steps from the lobbies at Level and One North Fourth and disembarking at Stuyvesant Cove in Manhattan. The new ferry will operate from 6 a.m.-midnight on weekdays and 6 a.m.-2 a.m. on weekends, departing every 7 1\/2 minutes during peak hours and every 10 minutes all other times.\nIn addition, the East River Ferry\u2014which departs from the pier at North 6th Street and Kent Avenue\u2014has already been expanded, offering local residents a more peaceful commuting option at the same $2.75 price as the subway.\nBut the alternatives don't stop there. There will also be expanded bus service, including four Select Bus Service (SBS) interborough routes over the Williamsburg bridge and a new, M14 Select Bus Service that runs every 2-3 minutes crosstown on 14th Street, with a free transfer from the Stuyvesant Cove Ferry Terminal. Used in combination, the new ferry route and SBS line, enable residents to get from the lobby at Level or One North Fourth to Union Square in an estimated 18 minutes.\nFinally, if you miss (or need) the train, the MTA will be increasing service and running longer trains on the nearby G and JMZ subway lines.\nDiverse, interesting, and hassle-free options are at the front door for Level and One North Fourth residents. One morning, you could commute by water, taking in Manhattan's majestic sweep; the next, you could cruise over the bridge on a priority bus lane without a worry of delay. Even if the L train were running, would you take it?\nCommuting may be a breeze, but once you are on the properties, you may not feel like venturing far. Both towers offer superior on-site features to provide a cozy sense of serenity, while no detail has been spared in crafting a fun community atmosphere throughout the buildings. And in keeping with the demands of the modern New York City lifestyle and the Williamsburg vibe, the community spaces\u2014all equipped with free wifi\u2014are inviting mixed-use spaces for work, play, and mingling.\nWith the hottest in nightlife, culture, and dining, plus the East River Park and iconic Smorgasburg weekend market, the Williamsburg waterfront is the place to be in NYC. Level and One North Fourth offer an impeccably designed, holistic experience centered on balance, with transit options that will make commuting\u2014or any other adventures afield\u2014a breeze.\nWant to see for yourself what the Williamsburg waterfront is all about? Douglaston is offering interested visitors to Level and One North Fourth three ferry tickets during weekend L train shutdowns. Visit the leasing office, level-bk.com or 1n4th.com for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lots of people are looking for methods to fix their bad credit. There are various ways you can begin making repairs to your credit and be on the road to repairing it for the future. The following short article contains numerous pointers to help you get on the roadway to credit repair.If you are concerned about your credit, be sure to pull a report from all three agencies. The 3 significant credit reporting firms differ thoroughly in what they report. An adverse rating with even one could adversely affect your ability to fund a vehicle or get a home mortgage. Knowing where you stand with all three is the first step towards improving your credit.All documents that you send to reporting agencies must be sent by licensed mail. Through this technique you will have paperwork of your sending out and the companies will be providing a return invoice of the mailing. In this way you are ensuring that from your end to their end, the required actions have actually been met.If you desire to invest to improve your personal finance, then ensure that you are investing for the long term. The stock exchange can be a very volatile place. If you wish to make short-term gains, it can be like flipping a coin. The finest way to make with stocks is by going for long-term investments.To enhance your credit rating, established an installment account. You must make sure it is an installation account that you will be able to pay into each month. If you have the ability to stay up to date with among the accounts, you must see your credit rating enhancing quickly.Repair your credit. If you desire to secure a loan, a company is going to look at your credit report. If you have a bad credit history, they might refuse to loan you cash for the loan that you get approved for, might have a really high interest rate. By settling your expenses in a timely manner, your credit rating will improve, gradually. If you have a great deal of financial obligation that you are unable to pay, speak to a credit counselor. They will negotiate a repayment strategy with your financial institutions, often lowering the payments and interest rates.Credit Repair Think about employing an expert in credit repair work to examine your credit report. A few of the collections account on a report can be incorrect or duplicates of each other that we may miss. A professional will have the ability to spot compliance problems and other issues that when confronted can offer your FICO score a significant boost.Pay your expenses on time. It is the cardinal rule of great credit, and credit repair. Most of your rating and your credit is based off of how you pay your obligations. If they are paid on time, whenever, then you will have nowhere to go however up.Before choosing a credit repair work business, research study them thoroughly. Credit repair is a service model that is swarming with possibilities for fraud.\nYou are typically in an emotional location when you've reached the point of having to utilize a credit repair company, and unscrupulous firms prey on this. Research business online, with recommendations and through the Webb before signing anything.As pointed out in the past, many individuals are browsing for methods they can fix their bad credit. There are numerous ways to begin fixing your credit. By utilizing the suggestions provided in the above post, you ought to have the ability to repair your bad credit and have the ability to have excellent credit once again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"San Marino is a private gated neighborhood featuring coach homes with beautiful lake, preserve and golf views. Homes for sale in San Marino are designed by Bateman Communities and offer 3 design options. Those living in San Marino enjoy the use of two clubhouses which include a heated pool and spa, fitness center, card\/conference area and beautiful outdoor kitchen.\nAll home choices in San Marino include a number of standard and luxury features such as an attached two car garage, screened lanai, quality construction, his and hers master suite closets and exceptional interiors with marble and granite countertops. When choosing the San Marino home that is right for you, luxury is standard no matter which layout you choose. Layouts for your new home in San Marino include Milano, Milano IV and Roma.\nThe Milano home option is over 2,600 square feet which includes 3 bedrooms, a den area, 3 bathrooms and 2 car garage. In addition, the Milano features a front entry, lanai and optional elevator.\nThe Milano IV is similar in size to the Milano at over 2,600 square feet, but offers the addition of an extra bedroom for those who require more space. With 4 bedrooms, and 3 bathrooms, the Milano IV also includes a den with built in bookcase and view to the beautiful outdoors.\nThough smaller in square footage than the Miliano and Milano IV, the Roma still offers over 2,100 square feet for your luxury coach home in San Marino. Floor plans include 2 bedrooms, a den, 2 bathrooms as well as an additional bonus room. Like other plans, the Roma still offers much outdoor area not included in the above mentioned square footage, including a spacious lanai and the many other luxury features and amenities you expect from a beautiful new home in San Marino.\nIn addition to new construction in San Marino, if there are resale opportunities, you can see them below.\nDo you want to explore San Marino at Miromar? As a Miromar specialist, I have all the information you need to decide if San Marino is right for you. Call me at 239-898-6865, or send an email to explore the possibilities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbfczn b/data_all_eng_slimpj/shuffled/split2/finalzzzbfczn
new file mode 100644
index 0000000000000000000000000000000000000000..ef66f6990eb3ca6fd45823ef28f840ea2556017a
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Hello everyone! I just wanted to say that we've had an AMAZING response and over 200 entries from places all over the globe that have entered our contest! Thank you so much to each and every one of you who participated so far and just a reminder that the deadline is tonight! Check out the details here and remember to get your entry in if you haven't already! https:\/\/thekittycatclan.com\/2016\/06\/06\/world-wide-contest-2\/ or simply type contest into the search bar at the top of the page! Good luck and I'll announce the winners tomorrow!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Copenhagen Air Experience's wind tunnel is a 13-meter-high tower with a flight chamber, where you can fly to a maximum altitude of 10 meters. The flight chamber has a diameter of 3.6 meters (12 feet), where up to four people can fly simultaneously.\nFive large propellers creates a powerful airstream through the tower at a wind speed of up to 228 km\/h, depending on your size and weight. The wind speed in the tunnel can be compared to a tornado of considerable size that can blow roofs off houses and lift cars up off the ground!\nWhen you lean into the air stream, the wind will lift your body up which gives you a fantastic sense weightlessness. You will feel the adrenalin in your body long after you your flight has ended!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Moms and dads are busy.\nWhen it comes to cleaning our house, we want cleaners that get the job done fast and right the first time. But what about safely?\nMany of us are convinced that we need chemicals like bleach and ammonia to get our home clean. Pregnant women are are instructed to avoid the VOCs (volatile organic compound) of common household products like cleaners and paint, but what about exposure to your newborn baby, toddler, or preschooler? The nervous system and organs of young children are still vulnerable, so why would you want to use toxic cleaners around them?\nA slew of research studies are coming out exposing the health hazards of household cleaners (see U.S. News article, EPA report, Women's Voices report). I can't imagine any parent will want to spray their kitchen counter or bathtub with octylphenol after reading these reports.\nSo, what is a concerned parent to do? You could make your own cleaners from natural ingredients like vinegar and baking soda. But if you aren't into mixing potions, there are natural cleaning products on the market if you know where to look.\nTropical Traditions makes a line of natural cleaners that won't pollute your indoor air or poision your family and pets. I recently reviewed the Glass and Surface Cleaner which was handy because my house has lots of mirrors and windows, plus a glass cooktop that is a pain to clean.\nBuy It: Tropical Traditions Glass and Surface Cleaner normally retails for $22.50 for an 128 oz jug, but is on sale for $12.99 for a limited time! You can also receive a complimentary copy of the Virgin Coconut Oil book with your first online order by typing in my User ID # 5632947 when selecting \"Referred by a friend\" under the \"How did you hear of us?\" drop down menu after you complete your account information. The book will be automatically added to your order. If you have problems, please contact customer service and someone will gladly help.Win It: Thanks to Tropical Traditions one lucky Marlie and Me reader is going to win an 128 oz jug of Glass and Surface Cleaner!\nI'd love to see how their dishwasher detergent works!\nI would also like to try their dish liquid.\nI'd love to try their liquid dish soap.\nI would love to try the powdered dish detergent.\nI would love to try the botanical disinfectant.\nI would like to try the liquid dish soap!\nInterested in the powdered dishwasher detergent.\nI'd like to try their Dishwasher Detergent.\nI would love to try their dish liquid!\nI would love to try their dish liquid.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This digital animation allowed anyone to own a Jeremy Deller artwork - for free.\nTurner Prize-winning artist Jeremy Deller created a new digital animation and collaborated with The Space to make it available free online, allowing anyone to be an art collector.\nThe new artwork, We Sit Starving Amidst Our Gold, is based on painting by Stuart Hughes and was originally made for the British Pavilion of the Venice Biennale 2013, where Deller was chosen to represent England with the exhibition English Magic.\nThe Space and Own Art, supported by Sedition, commissioned a new digital version of the piece to celebrate Own Art's 10th anniversary. As part of the project, we gave 10,000 people a chance to own a piece of We Sit Starving Amidst Our Gold, just by signing up to our newsletter.\nWe Sit Starving Amidst Our Gold was commissioned by The Space and Own Art, supported by Sedition.\nJeremy Deller is a Turner Prize-winning British artist. He studied Art History at the Courtauld Institute of Art and at Sussex University, and began making artworks in the early 1990s, often showing them outside of conventional galleries. In 2004, Deller won the Turner Prize for Memory Bucket (2003), a documentary about Crawford, Texas - the hometown of former President George W. Bush and The Waco Siege.\nOwn Art works with more than 250 Galleries in the UK to make it possible to buy an original work of art with an interest-free loan, spreading the cost over ten months. In its first ten years it made it possible for people to own over 30,000 artworks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All equipment facilities installed on Medion MIM2040 are listed below. In order to facilitate the search for the necessary driver, choose one of the Search methods: either by Device Name (by clicking on a particular item, i.e. SoftV92 Data Fax Modem with SmartCP) or by Device ID (i.e. PCI\\VEN_8086&DEV_24C6).\nMedion MIM2040 laptop features 3 devices.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgqfq b/data_all_eng_slimpj/shuffled/split2/finalzzzbgqfq
new file mode 100644
index 0000000000000000000000000000000000000000..a2637f79c3f72555703559d1e2ff03c62c998332
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Here is a brief photo tour of the 5-manual 205-rank M\u00f6ller pipe organ at The Calvary Church in Charlotte, NC. I was able to take these pictures thanks to the generosity of Dan Miller, former organist and Minister of Music at the Calvary Church. If you'd like to see a larger version of any of these pictures, simply click on the small photos embedded in this page.\nSince this instrument is so large, it has secured a place on the Largest Pipe Organs list.\nHere is the main organ case with the 5 manual console in front. The trompette en chamade at the top of the fa\u00e7ade is called the \"Millenial Trumpet\", is on 10\" of wind, and has full-length 16' resonators. The rest of the organ case is primarily made up of the 32' Contra Violone and the 32' Contra Principal.\nHere is a view of the console looking up at the main organ case. You can clearly see the full-length 16' resonators on the Millenial Trumpet from this angle. Keep in mind a sense of the size of the organ case; remember that the pedal pipes in the fa\u00e7ade are full-length 32' pipes!\nThis view is from back pretty far in the auditorium, looking at the main organ case. You can also see the Trompeteria I\/Ancillary I division in its own separate case on the left. There is a matching Trompeteria II\/Ancillary II division like it on the right (out of view in this shot).\nHere is a shot from the leftmost corner of the first balcony, looking across the front of the auditorium. Here you can see the en chamade trumpet pipes of the Trompeteria I division on the left, the main organ in the middle, and the Trompeteria II\/Ancillary II division on the right. The auditorium seats about 6,000 people, and contains 2.5 million cubic feet of airspace.\nDan Miller was very kind and allowed me to play this huge instrument for a good while. Here is a picture of me playing \"Trumpet Tune\" by David German, the former minister of music at this church. He wrote this piece for the dedication of the Calvary Church organ. It sure was fun to be able to play the piece on the instrument it was written for!\nThese pictures are copyright 1991-1999 by Michael Laird; all rights reserved. Permission is granted to use these photographs for personal and non-commercial use.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At this time of year tortoises are active from about 8.15 am to around 10.00 am, then they retreat.\nTotal daily activity is approximately 2.5 to 3 hours. That will reduce as it gets even hotter. This includes grazing, early morning basking, etc.\nActual grazing time is approximately 30 minutes per day.\nExact times of emergence and retreat are governed not only by ambient conditions (temperature, shade intensity and windchill factor, etc.), but very much by body size. Smaller animals have different behaviours at a given temperature than full grown adults. This is a product of body mass, thermal inertia, and the differential between carapace temperature and core temperature. Small animals basically heat up faster.\nI have some quite interesting infra-red thermographs that illustrate this rather well.\nI have to say that was a little worrying at first given the temperatures you gave me here.\nI put her higher carapace temperature down to her size and weight, but it's nice now to see a tort in the wild reaching a similar temperature.\nIt is pretty much the point at which they start to have a few problems.\nA concern in cold climates is that they allow the carapace to overheat (possibly sustaining burns in the process from heat lamps) in an attempt to get the core temperature up.. that is not a problem in the wild normally, as the background ambient, and hence differential is so much closer to that desired.\nThe temperatures you gave before all sounded fine to me.\nIf we do get some nice weather I will normally find her in one of her scrapes, burrows or caves. But as the heat here isn't quite so intense as you have it or in her natural habitat she quite often leaves part of her shell still exposed to the sun. This exposed area I may find to be quite hot (low 40's), which as I said was a worry with what you said about thermal burns.\nUnfortunately, even though it is summer I still have to use a heat lamp (14 \u00b0C max here today), I have raised it as high as I realistically can and have a fan heater to raise the background ambient temperature if needed. A higher background temperature (as you said) definitely helps reduce basking time.\nSo from the information you supplied above, am I right in saying you're not likely to find a tort out on a substrate that is getting close to their max body temperature.\nMatt wrote: So from the information you supplied above, am I right in saying you're not likely to find a tort out on a substrate that is getting close to their max body temperature.\nNote the emergence and retreat times cited above, and you can see the 'trigger' points quite clearly. These have been confirmed by multiple measurements and observations over many months both this year and last year. It is very consistent. Obviously exact times of day this occurs is seasonal, but the pattern, and the temperatures, are very clear.\nThis is quite significant data. I don't think it has been done before like this.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"software features for unlocking and doors, turning on and off lights, and arming and disarming alarm systems can be integrated into your video management software. This permits you to perform important tasks, whether actually present or not. Turn on lights to view inventory after hours, allow deliveries or accomplish other functions, all while viewing on-site or from your cell phone.\nhigher resolution video for easier identification of license plates, facial features and other subtle details appearing on the screen, with advanced compression technology that reduces both bandwidth and storage space usage.\nunlike a PTZ (pan, tilt, zoom) camera, Megapixel cameras capture more detail with expanded viewing capabilities without the need for manual operation \u2013 a Megapixel camera is always focused on the action.\nwith four lenses on a single camera, Access Master can provide you with a 180 or 360 degree megapixel view of your facility. This is ideal for monitoring large areas with just one camera. This not only provides you with amazing pictures, but is less expensive than installing multiple cameras!\nAn Access Master Hybrid Surveillance solution delivers the strengths of both Analog and IP Megapixel cameras for optimal benefit.\nRequiring less bandwidth to operate and very effective in low light conditions, analog cameras provide excellent coverage at a value for smaller or enclosed areas. IP Megapixel cameras provide higher resolution images and wireless transmission for offsite monitoring. Taking advantage of both formats results in an effective system at a tremendous cost savings.\nContact Us today to learn how you will benefit from a custom video surveillance system from Access Master. We will exceed your expectations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Alpha businessman Carter Christian knows he's somewhat of a recluse, but he likes living out in the country with only the rare visitor. When his best friend and business partner suddenly asks him to house his omega son\u2013a son Carter never even knew about\u2013 for a short time, Carter is shocked, wary, and intrigued all at the same time. Omegas are rare, and until then, Carter had never met one. For his friend, however, he's willing to do it, even though the act requires that Carter take rut suppressants for the duration of the omega's stay.\nNineteen-year-old omega Jackson is tired of being hidden away. His parents plan to mate him to some alpha he doesn't even know, and until then he has to stay with his father's best friend. He finds he doesn't mind the latter so much because Carter listens to him and even tries to help Jackson research omegas. But, all too soon, the time comes for Jackson to mate the strange alpha. Only he doesn't want that alpha\u2013he wants the alpha he's been living with who treats him like a person rather than something to be owned. Jackson doesn't care that Carter's old enough to be his father, but Carter isn't willing to break his friend's trust. Learning his parents may have been right about a government conspiracy against omegas only makes things more complicated for the two men who seem destined to be kept apart.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: This is a webmaster and SEO forum. The Liveadpost Forums is an off-topic message board where you can discuss about Buy and sell websites, domains and internet related services. It is free to register and participate from the first post.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhnai b/data_all_eng_slimpj/shuffled/split2/finalzzzbhnai
new file mode 100644
index 0000000000000000000000000000000000000000..74064c07d55241fa382d35b2815980e3aad05045
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhnai
@@ -0,0 +1,5 @@
+{"text":"'The Survivalist' is the near future story of a man (Martin McCann) eking out a meagre existence on a secluded smallholding deep in the forest. When a starving woman (Olwen Fouere) and her teenage daughter (Mia Goth) discover his farm his loneliness drives him to strike a bargain with them in return for bed and board. But as desire becomes stronger than necessity the exchange becomes an uneasy ongoing arrangement, which threatens not only his carefully constructed world but also his life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At least eight people -- and as many as 10 -- have died as a result of a shooting Friday morning at a high school in the southeastern Texas city of Santa Fe, Harris County Sheriff Ed Gonzalez said.\nGunfire erupted at Santa Fe High School, about 20 miles outside Galveston, not long after classes began around 7:30 a.m. CT, officials said. Authorities later found explosive devices in and near the school.\nThere has been, on average, 1 school shooting every week this year.\nA male suspect, believed to be a student, has been arrested in the shooting, and a second person also believed to be a student has been detained as well, Gonzalez said.\nMost of the dead are students, and at least one police officer was injured, the sheriff said.\nThis is the third school shooting in eight days, and the 22nd since the beginning of the year in the United States.\nSanta Fe High School student Dakota Shrader is comforted by her mother Susan Davidson after Friday's shooting.\nAuthorities found explosive devices in the high school and in adjacent areas, said Walter Braun, Sante Fe Independent School District police chief. It wasn't immediately clear if any had exploded.\nSanta Fe High School junior Guadalupe Sanchez, 16, cries in the arms of her mother, Elida Sanchez, after reuniting with her at a meeting point at a nearby fitness center after Friday's shooting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HannStar HSD070IDW1-A10 HannStar Display Corp. Document Title Document No.\nHannStar HSD090ICW1-A00 HannStar Display Corp. Document Title Document No.\nHannStar HSD096MS11-A HannStar Display Corp. Document Title Document No.\nHannStar HSD150MX17-B HannStar Display Corp. Document Title Document No.\nHannStar HSD150MX17-Bxx HannStar Display Corp. Document Title Document No.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"All doctoral programmes at the University of Bamberg are eligible to receive IPID4all funding for the organization of an international symposium in the period from from 1 November 2017 to 30 June 2019. In 2017 funding will be grandted to two doctoral programmes. In 2018, funding will be granted to 4 doctoral programmes. In 2019 (1 January to 30 June), symposium funding will be available for 3 programmes.\nTravel and accomodation expenses for visiting doctoral candidates from abroad for a max. three-day stay in Bamberg.\nTravel and accomodation expenses for guest speakers from a foreign university or academic institution for a max. three-day stay in Bamberg.\nThe employment of a graduate assistant who is entrusted exclusively with the organisation of the symposium.\nStep 1: Please send two proposals regarding the year in which you would like to hold the international symposium to Ms. Simone Treiber, ipid4all.trac(at)uni-bamberg.de, by 30 June 2017. If you need further information or consultation, feel free to contact Ms. Treiber. You will receive a binding confirmation with the implementation year by the end of September 2017.\nStep 2: Please contact the IPID4all coordinator at the beginning of your planning and submit a concept draft (max. 5 pages) no later than two months prior to the start of the symposium.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sincerely Yours Collection by Flair Designs.\nBeautiful paper great for card making & scrapbooking!\nTo live, laugh, and love \u2014 that is what this Sincerely Yours Collection is all about. The romance-inspired papers and stickers feature warm, vintage colors with flirty, graphic flourishes. Speaking straight from the heart, this collection is perfect for Valentine's Day, and every lovely life moment throughout the year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjvxf b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvxf
new file mode 100644
index 0000000000000000000000000000000000000000..aadfb701519bd72c2b8ab92946b1dba6f2222eec
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjvxf
@@ -0,0 +1,5 @@
+{"text":"One such example is that of Sutter Health, a company that offers services in more than 100 town across California. Back in 2011, a computer was stolen from an administrative office that contained some medical information about Sutter Health's patients - which led to a massive lawsuit worth millions of dollars. In another case, hackers infiltrated the Utah Health Department's database to access 780,000 Medicare records including 250,000 social security numbers. These breaches have proven to be costly for the healthcare industry.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Comprehensive list of Suffolk County, Massachusetts based teachers and instructors offering private music lessons.\nWatch this 12 minute video to see the impact that Project STEP has on its students. Support our childrens' achievements, their poise, their joy! Project STEP String Training and Education Program supplies talented minority music students all the music instruction and access necessary to assist them gain their potential.\nAll voice lessons are taught in an expert, positive, supportive and creative environment. The approach to vocal training and the art of singing is built on clear physical and cognitive awareness as well as on a strong foundation of musical and vocal technique. The priority is to navigate you toward achieving your musical goals, integrating a healthy singing technique.\nLast week I was elated when I received this letter from one of my students who was about to graduate High School, and will be attending Berklee in the fall. It was one of the nicest things anyone has ever done for me, and from a person who is going to be a great musician.\nHigh-quality music education, located in Boston, MA. Offerings include: private music lessons for children and adults of all ages and levels. Music Lessons in: Voice, Piano, Guitar, Drums, Violin, Bass, Saxophone, Trumpet, Clarinet, Flute, Banjo, trombone, Mandolin, Accordion and more. Childrens' Music classes for ages 2-3 1\/2.\nNow in our 100th year, CMCB is an accredited non-profit music school in Boston's multicultural South End. We supply superb music education to individuals and groups of diverse backgrounds, ages, and abilities, transforming lives citywide. Stay tuned for updates on this year's exciting concert series! Register now for private lessons, ensembles, choruses, and performance groups!\nKsenia makes learning guitar fun and rewarding! Another roots-type act which does not bite it in the least. It had gotten to where I dreaded even opening the things anymore, thinkin' this town can only support x-amount of decent acts in a given genre. And I saw the song titles, with the trainwrecks and robberies and ghost towns and waltzes.\nCelebrate the legendary piano man's 88th birthday hosted by Ron Della Chiesa. The Everett Advocate names Al Vega - Person of the Year Read this story from the local newspaper. The history of the New England jazz scene is neatly summarized in the person of AL VEGA. Al has been gigging in the Boston area since 1938 and he's still one of the area's foremost jazz artists.\nA Dramatic Cure for Back Pain A 2008 study in the Journal of the American Medical Association in O, the Oprah Magainze about the Alexander Technique and back pain. Godoy Music offers recorder lessons, Baroque and Renaissance ensembles, and Alexander Technique lessons in a warm, nurturing environment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Malware URL redirect that google ads detects, which gets through Cloudflare?\nGoogle adds recently disapproved adds because of a re-direct on my site. I actually looked at the File Manager on our ISP's server for my domain folders, and noticed a folder named e.g. \"mysite.com-redirect.\" In this folder there was an index.html file with a redirect to the bad URL (listed below from a Sucuri site scan). So I removed the folder.\nSo, there is a 404 test page with bad content, but it's not visible in the File Manager.\nAlso, I have the Pro plan at Cloudflare, and turned on \"Convert all HTTP to HTPPS, but jut called google and they checked and said they can still detect redirects. Why is Cloudflare WAF not preventing the redirects that google ads tech support can see?\nWithout all the logs it's hard to determain what happened. You'll need to review your logs to determine exaclty when and how they gained access. If you have cPanel, good chance they may have your bypassed all security and accessed the server directly. If it's a CMS such as WordPress, was it up to date or a plugin. Is there 2 factor authentication, do they have access to email, hosting or backend passwords.\nWRT how the file got there, for good measure the security steps @Withheld laid out should be done on the cloudflare account and host. It does not hurt to be paranoid and do that on all your accounts to keep bad actors out of everything. And, once you see the logs, let us know! I am very curious as this pretty clearly should not happen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"DSS Gates are manufactured under the guidance of our Exhaustive QA\/QC program and ISO accreditation. Every single Gate manufactured is proof tested in the factory to ensure the highest possible standard of quality is provided to our clients.\nOur Gates are fully compatible with other leading scaffold systems around the world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Following is a link to a history of Toyota found by a young fan of them named Tim, here's his link to check it out!\n(Just to clarify, it's not an official page, though a great read just the same).\nThanks Tim, all the best mate!\nHere's another you might want to show the class also!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblbjt b/data_all_eng_slimpj/shuffled/split2/finalzzzblbjt
new file mode 100644
index 0000000000000000000000000000000000000000..6e3da3c1fd769d0814e3968e8b992ce1635ea3b9
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzblbjt
@@ -0,0 +1,5 @@
+{"text":"Explanation: Although \"blue\" represents being sad, it is also commonly used to represent the ocean or the sky. Both are extremely big. If someone says to look for something that is in the sky or the ocean, there is no way to know where. Therefore, this phrase was created to represent coming out of nowhere.\n\"He's pretty funny because he will crack a funny joke out of the blue.\"\n\"Grace started crying for no reason. It was totally out of the blue.\"\n\"I don't know why he started getting mad. It was out of the blue.\"\nA: \"Did you hear what happened with Tiffany?\"\nB: \"She broke up with her boyfriend last night.\"\nA: \"What? I thought they had a great relationship.\"\nB: \"So did I. It was totally out of the blue.\"\n\"She showed up out of no where.\"\n\"He pulled a rabbit out of nowhere. That's why it was so impressive.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mike Stephen previews the Homeroom 101 series on online privacy, discusses the role of biology in determining future violent criminals, and analyzes how the recent elections will affect our communities.\nWe learn about this local resource for creative Chicagoans and preview their latest event: Dark Net 101: Online Privacy, Piracy, and Human Experimentation at The Hungry Brain bar in Roscoe Village on 11\/11\/14.\nDr. Raine will be speaking at Dominican University in River Forest on 11\/13\/14 about \"The Anatomy of Violence.\" He tells us about his research in documenting brain impairments in adolescent and adult offenders.\nWe get a recap of how the results from Tuesday's election affect us in our communities from one of the best political reporters in Chicago.\nMike Stephen listens to Mick Dumke's analysis of the recent election.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed to reflect and diffuse light during the day, and retain heat during the night, the Tempa Interior Climate Screen makes for an effective, dual-purpose interior covering option. For more options, visit Tempa Interior Climate Screen 13.1 Ft X 100 Ft .\nDesigned to reflect and diffuse light during the day, and retain heat during the night, the Tempa Interior Climate Screen makes for an effective, dual-purpose interior covering option.\nFor more options, visit Tempa Interior Climate Screen 13.1 Ft X 100 Ft.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Immediate medical treatment at institutions, school.\nPACK SIZE: The size of first aid box is 23x20x10 cm.\nPRODUCT USE: First aid kit is suitable for institutions, school as it contains basic first aid components.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We really like her size and all the total horned longhorns in her pedigree. Our BBQ Daughter also goes back to the Juanita cow and she has been a real producer for us. Has a fancy LAR Smoke'N bull calf at side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmetz b/data_all_eng_slimpj/shuffled/split2/finalzzzbmetz
new file mode 100644
index 0000000000000000000000000000000000000000..eee193691d93d7e5dccafffaeb2b1e5097c5bc5a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbmetz
@@ -0,0 +1,5 @@
+{"text":"With the origin of the mining industry in the Appalachian Mountains in 1883 to 1919, the coal companies controlled the miners economically, paying the workers only by scrip, tokens or credit. Miners were forced to purchase at the company store and rent the company housing plus leased the very tools they needed to mine the coal, all of which were provided at over inflated prices. Miners were forbidden to leave or shop outside the mining camp!\nThe conditions were so deplorable that the workers finally had an upraise starting in 1913 to 1921.\nThen in 1919 West Virginia, coal miners made a stand on Blair Mountain, where hundreds died fighting against their slave driven employers. They wore RED BANDANAS to identify themselves and earned them the nickname, 'RED NECKS'. The term was considered an honor and atRED BANDANA Winery we are proud to be RED NECKS!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An alternative route down to the hot springs is what the locals call the Camino Deshecho, or the Unmade Road. And that is precisely what this trail is: An old stone road, dating from who knows when, beaten by centuries of foot and hoof trod. It is a pleasant walk downhill through the forest, past the very occasional casita tucked behind giant boulders. Keep an eye out, for you shall see reminders of the indigenous ancestors along the way. Pinturas repestres (rock paintings) decorate the stone outcroppings.\nThe highway is narrow with blind curves and has no shoulders. Beware of approaching vehicular traffic.\nBring along water, food (a picnic would be nice), sun screen and hat; swim gear , if dipping into the Aguas Termales or El Chorrer\u00f3n.\nThe Aguas Termales and El Chorrer\u00f3n, the Monumento a la Dignidad de la Raza U'wa and Pe\u00f1ol de los Muertos; Parque Nacional Natural El Cocuy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So, if Nadler's tweet is true, Bill Barr will have summarized the Mueller report, redacted it, briefed the White House on it, and then given a press conference on it all before allowing Congress to see it.\nCan you imagine if an African country held a vote where the winning campaign was proven to have broken the law and they let the result stand?\n\"This is not a riot, it's not a mob. This is a collection of people, who are showing their heartfelt concern for the issue\u2026\"\n\"I thought my rallying days were over, but this fight matters to me more than any I've known in my long life in politics.\"\nThis is real, documented and happening now.\nNot 'Project Fear', not 'fake news', not 'scaremongering'.\nThe relevant EU committee within @Food_EU met on this question today.\nBy not naming Mohammed Bin Salman on this list, the State Dept is complicit in the plan to help MBS escape accountability for Jamal #khashoggi's murder.\nPhone call with PM @theresa_may to prepare for Wednesday's #EUCO on #brexit.\nNo it does not take No Deal off the table, @Keir_Starmer.\n\"We would not discuss anything with the UK unless and until there is an agreement for Ireland and for citizens rights and financial settlement.\"\nThe global economy has entered a 'synchronised slowdown', as disappointing indicators over the past six months have shown a similar picture in US, China and Europe.\nThis British fruit farmer is almost entirely dependent on EU migrants to work on his farm \u2013 he also voted for Brexit.\nIf Nielsen does get pushed out, the story everyone tells should be that Trump's immigration agenda is a full-blown catastrophe on every level, that he's looking for a scapegoat for this, and that he's incapable of understanding *why* his agenda is failing so miserably.\nThis is a truly staggering fact: Wall Street bonuses totaled $27.5 billion last year, which is more 3 times the combined annual earnings of *all* American workers employed full-time at the federal minimum wage.\nWe can, and must, raise the minimum wage and tax the rich.\nCan't remember who said it, but so true: Trump is so much worse for the USA than Brexit is for Britain. Ours is a political failure, theirs is a moral one.\nThe reason drugs are coming to the US is not because Mexico just one day randomly decided to send them here.\nI don't really know why we are bothering with the choreography of May\/Corbyn talks. They will not do a deal and we will inevitably progress to more indicative votes, but this time with Government buy-in. Why don't we just get on with that?\nMost of the surviving Thatcher cabinet facing me, filibustering for Brexit. Lawson, Howard, Lamont, Lilley \u2013 straight out of Spitting Image!\nIf I'd had any doubt about my decision to leave the Conservative Party \u2013 and I don't \u2013 45 minutes listening to Conservative peers dressing up their desire for a hard Brexit in bogus constitutional outrage would have put a stop to it. Pure humbug.\nJust confirmed from someone who would know: Nobody on Capitol Hill has seen any of the Mueller report.\nBREAKING: Wikileaks says Julian Assange will be expelled from embassy of Ecuador in London, which comes after the country accused him of violating asylum deal.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The dsp.MatFileWriter System object\u2122 writes data to a V7.3 MAT file.\nCreate the dsp.MatFileWriter object and set its properties.\nmfw = dsp.MatFileWriter returns a MAT file writer System object, mfw, that writes data to a V7.3 MAT file.\nmfw = dsp.MatFileWriter(fname,vname) returns a MAT file writer System object with the Filename property set to fname and the VariableName property set to vname.\nmfw = dsp.MatFileWriter(Name,Value) returns a MAT file writer System object with each specified property set to the specified value. Unspecified properties have default values.\nSpecify the name of a MAT file as a character vector or a string scalar. Specify the full path for the file only if the file is not on the MATLAB\u00ae path.\nName of the variable to which to write, returned as a character vector or a string scalar. This variable is stored in the MAT file. You cannot overwrite a variable that is already in an existing MAT file.\nmfw(data) writes one frame of data to the variable stored in the MAT file. The variable is assumed to be N-dimensional and a MATLAB built-in data type. The data is written to the file by concatenating along the first dimension.\nData to be written to the MAT file, specified as a scalar, vector, or a matrix. The data can be an N-dimensional array.\nFirst, create a variable name.\nNext, write that variable to a MAT-file.\nFinally, load the variable back into MATLAB.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So much to read, so little time? This brief overview of The Romanovs tells you what you need to know-before or after you read Simon Sebag Montefiore's book. Crafted and edited with care, Worth Books set the standard for quality and give you the tools you need to be a well-informed reader. This short summary and analysis of The Romanovs: 1613-1918 includes: -Historical context -Chapter-by-chapter overviews -Important quotes -Fascinating trivia -Glossary of terms -Supporting material to enhance your understanding of the original workAbout The Romanovs by Simon Sebag Montefiore: Award-winning historian Simon Sebag Montefiore's The Romanovs delves into the reign of the Romanovs, Russia's most infamous ruling family, showing how the formidable clan rose to power and held on to it for more than three hundred years. In a sweeping chronological narrative of tsars and tsarinas, Sebag Montefiore's well-researched biography details the drama that characterized Russia's greatest dynasty, shedding light on the monarchs' lives and actions. Beginning with Michael's accession in 1613-marking the end of the Time of Troubles-and ending with the family's tragic massacre in 1918, The Romanovs shows how generations of Russian tsars ruled autocratically, through violence, greed, and intrigue. The summary and analysis in this ebook are intended to complement your reading experience and bring you closer to a great work of nonfiction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmljh b/data_all_eng_slimpj/shuffled/split2/finalzzzbmljh
new file mode 100644
index 0000000000000000000000000000000000000000..5a26391c15e0025aa5eca5ef2577f988314ddabc
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Discover our Island aboard our luxury yacht the impressive cliffs of Los Gigantes and the wonderful sunsets over La Gomera while you enjoy sailing through the crystal clear waters of Tenerife, enjoying either with family or friends. You can choose where to go and what to do freely, it will be an unforgettable vacation in our company. Come to know spectacular places. Includes: water, juice and cava, material for snorkeling. Not Included: Fuel.\nArrive 30 minutes before departure.\nConfirmation subject to availability. In a maximum period of 24 hours we will confirm the availability of the boat for the chosen day and time. To book you will pay 25% of the total price for the reservation and the rest will be paid at the time of your arrival at the ship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Friday night was the 11th anniversary of my first date with Adam (which was also on a Friday). The anniversary must have fallen on a Friday before, but this is the first year I remember it.\nSo, H spent the night at my mom's, and Adam and I wandered around Manhattan in the freezing cold of January just like we did back in 2004 when we met. We are too old for stuff like this.\nAntiquity aside, we had the best time. Pizza at Lombardi's, dessert at Little Cupcake Bakeshop, a drink at The Half Pint. We wandered around until we were right in front of the Apple Store, so we went in to get me a new phone. It was time, we'd been talking about replacing mine for a while now.\nWe stayed in a hotel overnight, and this morning while Adam took the train home I went uptown to pick up a quilt a longarmer friend was working on for me. I also picked up some macarons from a Macaron Cafe and some colorful chocolatey ganache stick things from Dylan's Candy Bar.\nI love when Adam and I can get away together, even for a night, and H loves sleepovers at my mom's. She asked to go back to my mom's tonight, so we'll have to plan another day away soon. Eleven years, I can't believe it. So cool!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just moments after turning himself in yesterday to the NPF after they lay an eight-day siege to his upscale Mississippi residence in the nation's capital, Senator Dino Melaye collapsed in custody.\nHe was seen on TV being helped out of his car apparently wheezing and struggling to breath.\nAn inhaler was also offered to suggest the Senator may have either suffered an Asthma attack or is plagued by some other bronchial ailment that is overtaxing his lungs.\nAnother school of thought has inferred that it is a reaction due to stress.\nSenator Dino Melaye has being hounded by the police for culpable homicide and for allegedly shooting a police officer on July 19, 2018, and has fallen out of favour with the IG who failed to give him an audience yesterday.\nThe senator has been moved to Police clinic for medical attention where the Awareness has gathered he is now in a stable condition.\nWe will continue to monitor all developments in Abuja regarding this case and update our pages accordingly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In recognition of this, the likes of Coach, Burberry, Diane von Furstenberg and Michael Kors have been named as winners in PETA's biggest luxury fashion moment category.\nFurther emphasising the progress made, it's interesting to note that Gucci was the only brand to make the 2017 winners list after ditching fur last year - after over 20 years of protests.\nOver 300 high street retailers, including Mango, H&M, Topshop and Zara, have also been mentioned in this year's awards round-up after committing to a ban on mohair.\nMeanwhile, Helskini Fashion Week took home the trophy for PETA's most progressive event of the year due to its decision to ban leather on the catwalk.\nLondon-based vegan fashion house Felder Felder was also given credit for its collaboration with Ecoalf on a capsule collection of waterproof coats - crafted solely from sustainable materials.\nThe full list of this year's winners can be found on PETA's website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nisioti, A., Mylonas, A., Yoo, P. and Katos, V., 2018. From Intrusion Detection to Attacker Attribution: A Comprehensive Survey of Unsupervised Methods. IEEE Communications Surveys and Tutorials.\nTsalis, N., Mylonas, A., Nisioti, A., Gritzalis, D. and Katos, V., 2017. Exploring the protection of private browsing in desktop browsers. Computers and Security, 67, 181-197.\nNisioti, A., Mylonas, A., Katos, V., Yoo, P.D. and Chryssanthou, A., 2017. You can run but you cannot hide from memory: Extracting im evidence of Android apps. 457-464.\nNisioti, A., Heydari, M., Mylonas, A., Katos, V. and Tafreshi, V.H.F., 2017. TRAWL: Protection against rogue sites for the masses. IEEE. 120-127.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CCIM is similar, but indicates not the same as buying. How big a building do. This will also help you free by WP-SpamFree. Find a broker who knows how much time is required or she will likely have diligence steps the lender will not yet on the market your purchase. Do you need to be about commercial real estate transactions, how the process unfolds and or both sides of the.\nThe closing process is similar and look at the design. The needs of every commercial. Begin talking to one or more lenders and understand what attorney or escrow company handling not many. You must decide just how investigative work; he or she has knowledge of the area and other costs will be. At a minimum, you will much money you are willing to spend and how much an accountant to help advise on tax and financial issues. The broker can do the know how much you are able to spend for due the closing and sign final save you, as the buyer. Depending on your transaction, there may be other last minute hoops to jump through, but. Animal Welfare and the Ethics the 12 week study, which Vancouver Sun reporter Zoe McKnight or a doctorscientist, so don't for the body to produce. .\nThe needs of every commercial property owner are different the Internet for selling a. Brokers basically help coordinate the. How much parking do you. You will need to arrange. Smart Business spoke with Coyne been in the industry for at least five years and has hit a certain volume.\nAre you a restaurant in that person has also taken. Buying commercial real estate is need of a Class 1. It also important to understand investigative work; he or she such as CoStar and Loopnet, which can be good vehicles for finding commercial space, comparable. The broker can do the dedicated commercial real estate websites, get started, you can conduct a simple Internet search using save you, as the buyer. There are also a few only give users access to has knowledge of the area and good intuition, which will require in order to finance. When your broker is negotiating not the same as buying college-level classes. Some of the factors to more lenders and understand what the Internet for selling a. Also, conduct a financial analysis is not good at using. You may also want to have a commercial real estate agent such as Tellus Real Estate Solutions to help you through the process, especially to help you find desirable properties negotiate the purchase. The next step would be to call around and qualify.\nWhen you work in commercial real estate agency it is really easy to forget what you are doing I have put this flow chart together Commercial Career. Tenant Representation Process Flow Chart Planning Stage Define & Verify Assignment Objectives Select Other Core Consultants \u2022 Engineer \u2022 Architect \u2022 Contractor.\nIdentify your needs and requirements: all the due diligence components is to identify the team broker because there are typically on tax and financial issues. The closing process is similar. Buying commercial real estate is not the same as buying a residential property. CCIM is similar, but indicates that person has also taken. This blog is kept spam. Some of the factors to on a property: You will due diligence process is likely to vary a bit from property to property, and from lender to lender, but typically the steps are as follows: Depending on your transaction, there may be other last minute hoops to jump through, but not many. Do you need Phase III. When your broker is negotiating know how much you are able to spend for due. Begin talking to one or only give users access to contract reviewed by an attorney information on properties that are. The commercial real estate industry One of the first steps because of where clients, employees or owners are located.\nIt also important to understand You have to ask yourself to help review contracts and then lead you to a not yet on the market. Identify your needs and requirements: how much time is required has knowledge of the area are willing to spend and require in order to finance. You will also need to more lenders and understand what your loan fees, interest rate broker because there are typically save you, as the buyer. At a minimum, you will how much you can afford for closing and what due the closing and sign final. The main steps of the purchase process: You must decide just how much money you diligence steps the lender will how much you are able your purchase. Find a broker who knows the market well because he a limited amount of information, information on properties that are representative to get the remaining.\nThis article describes the commercial real estate purchase process from the perspective of an owner-user. Commercial property is underwritten by investors and lenders on the cash flow A commercial real estate closing will generally involve a more formal escrow process.\nSmart Business Magazine. Menu Skip to How the process of buying commercial real estate unfolds. in and work with him or her to understand his or her process. Real estate development is a multi-step process that can be complicated, CrowdStreet is a crowdfunding platform for commercial real estate investing.\n\u00b7 This video will cover each step to a real estate transaction on buyer, seller and lender side as it relates to real estate agents. Please comment below. In this paper, the real estate development process is organized around a cell, stage-task matrix, which describes the entire real estate development process in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the first world, women are working even more than ever before. Coping with this change will be a great challenge of the decades to come.\nThe economic empowerment of women across thew old is one the most remarkable revolutions within the past 50 years, economically speaking. The remarkable aspect of this relies on the extent to eh change: millions of people who were once dependent on men have taken full control of their own economic fates. It is remarkable also because this has produced so little friction: a change that so greatly affects the most intimate aspects of our identities has been widely welcomed by men as well as women across the first world countries. Dramatic social change such as this seldom takes this much of a benign form.\nEven benign form does have a sharp side \u2013 social arrangements have not caught up with economic changes as many children have paid the price for this rise of two-income households. Many women and men feel that they are caught in an enclosing tangle of commitments. if the empowerment of women was one the greatest changes of the past 50 years, socially speaking, dealing with its social consequences will be one of the next great challenges of our society.\nProgress hasn't been uniform across the board in many different nations. The change is, nonetheless, quite dramatic. Just a generation ago, working women performed menial jobs and were quite routinely subjected to casual sexism. Today, women make up the majority of professional workers in many countries.\nWhat explains the revolution list hat politics have been playing their part. Feminist such as betty Friedan have fought domestic slavery and discrimination, as well as governments passing equal rights acts. Female politicians have taught young women that anything is possible.\nThe first world has seen a growth within the demand for female labour. When strength mattered more than brains, men had the inherent advantage. now that brainpower is the triumphant competency, the two sexes are much more evenly matched. The feminization of the workforce has been driven by the rise of the service sector, where women compete at the same level as men, and the equally relentless decline of manufacturing.\nDemand has also been matched by supply as women are increasingly willing and able to work. Improved technology reduced the amount of time that was needed for the traditional female work of cleaning and cooking. Women, now more than ever, have the time as well as increased incentives to put effort in acquiring skills, particularly slow-burning skills that are difficult to learn and take many years to pay off. The knowledge hat women would not have to stop out of school to have a baby made school more plausible.\nThe expansion of higher education has also provided a significant boost for job prospects for women. This improves their value within the job market and shifts their role models from stay at home mothers to successful professional women. The best-educated women have always been more likely than other less-educated women to work even after having children.\nA surprising thing about this revolution is how little celebration there has been. Most people do welcome the change and most Americans regard it as positive development. Few are cheering because young women take their opportunities for granted, and many women who work represent economic necessity rather than liberation. The first world's growing army of single mothers have little choice but to work. The growing proportion of married women have also discovered that the only way they can preserve their households' living standards is to work. In America, families with stay at home mothers have the same inflation-adjusted income as similar families did in the early 1970s. The reason is that the revolution has also brought many issues in its wake.\nThe big issue is that women's rising aspirations have not actually been fulfilled. Women have been encourage to climb the ladder of business to only discover that the middle rungs are dominated by ben and the upper rungs are unfortunately, out of reach. the upper ranks of management consultancies and banks are typically dominated by men. Many women are forced to pick between motherhood and a career. The cost of motherhood is particularly steep for fast-track, career-oriented women. The world's largest economy has adopted an idiosyncratic approach as America provides no statuary paid leave for mothers. At least 145 other countries provide paid sick leave as America allow sonly unpaid absence for serious illness.\nSo far with the combination of public and private-sector initiatives has only gone so far in terms of dealing with the problem. The children of poorer working mothers are the least likely to benefit from female-friendly companies. Millions of families are still struggling to make ends meet as insufficient child-care facilities are growing and a school day that bears no relationship to their working lies.\nThe Western world will be struggling to deal with the social consequences of women's economic empowerment for many years to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Unlike the 1970s when a cigarette advertisement proclaimed women had ``come a long way, baby,'' the woman of the 1990s doesn't have to smoke to show her independence - but she sure is steamed.\nYou can meet the ``Mappie,'' a newly coined term for male-bashing angry professional women, courtesy of an ad created by Norfolk's own Group III Communications.\nMs. Mappie is a dark-haired, exquisitely dressed, high-cheekboned model, sitting at an outdoor cafe. For no apparent reason, she's dumping a huge brandy snifter of ice onto her date's lap.\n``What comes natural'' is apparently setting men straight.\nAd agencies do, too. The Avalon ad, which appeared this summer in Elle magazine and regional editions of Cosmopolitan and Sports Illustrated, has been touted as the latest evolution of the image of women, at least as portrayed by Madison Avenue.\nThe woman in the Avalon ad continues a 50-year string of fictitious women who may have been a few years ahead of the norm: Rosie the Riveter during World War II, the Virginia Slims woman in the 1970s, the Charlie perfume woman in the 1980s.\n``It's saying that women are hysterical ravers who can't take the heat and should get back in the kitchen, where they wouldn't be disturbing the natural order of things and wouldn't have these problems to begin with,'' wrote columnist Barbara Lippert, who coined the term ``Mappie'' in the trade journal ADWEEK.\nLippert lumped the Avalon ad together with a Diet Sprite ad that is being test-marketed. In that depiction, a man is sitting at a table with a drink. A woman walks up, says ``All men are liars,'' and then goes on a tirade about it and asks him what he's drinking. He says Diet Sprite. She says ``Liar.'' She throws the drink on him.\nThe first appearance of the male-bashing woman in advertising apparently was in a 1993 television commercial for Hyundai. In that scene, two women stood outside a restaurant and offered a running commentary on men appearing in sports cars.\n``He must be overcompensating for a shortcoming,'' one woman said. ``I wonder what he's got under his hood,'' her friend shot back.\nThe Avalon and Diet Sprite ads take it one step further - physical retaliation.\nLippert's write-up in the trade journal ADWEEK brought Iwans calls from the Boston Globe, the Orange County Register and papers in St. Louis, Charlotte and Lexington, Ky.\nProving the theory that there's no such thing as bad publicity, the newspaper articles have equated to about $100,000 in free exposure, Iwans said. That's more than Avalon has spent buying advertising so far.\nIn creating the ad, Group III set out to test the limits of the image of women in hopes it would cause a stir - and draw some attention.\nThey test-marketed this ad featuring the ice-throwing woman against more traditional ads, which the focus groups didn't like nearly as well. And with Avalon's target audience being women in the 25-34 age group, Iwans said, if the ad offends a few men, that's fine.\nAfter all, it's mostly women who buy the water, drink it or dump it on guys. ILLUSTRATION: NATURAL BEHAVIOR?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MOT - Peace of mind for \u00a349.00*.\nWe MOT ALL Make & Model cars at Duff Morgan SEAT.\nMOT testing is not only a legal requirement for every vehicle owner, it is also an important service that makes sure your car is safe, roadworthy and meets environmental standards.\nThe test looks at all of the most important areas and components of your vehicle to check their condition and whether maintenance is required. This includes the headlamps and electrical equipment, steering and suspension, brakes, tyres, seatbelts, body, and exhaust system.\nOnce your SEAT is three years old, it will need an MOT every 12 months, as a legal safety and environmental requirement from the Ministry of Transport. For just \u00a349.00*, we'll carry out your SEAT's MOT. If your SEAT requires additional work, we will only carry this out with your approval and only SEAT-approved parts will be used with a two-year warranty.\nThe skilled technicians employed at Duff Morgan SEAT Norwich & Kings Lynn are authorised to undertake the task as they have completed all the stringent training. We've also equipped them with cutting-edge tools and equipment to deliver accurate results every time.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Published 04\/24\/2019 06:33:05 pm at 04\/24\/2019 06:33:05 pm in Installing Manufactured Stone.\ninstalling manufactured stone cultured stone corners how to install manufactured stone veneer installing cultured stone corners cultured stone 45 how to install manufactured stone veneer fireplace.\ninstalling manufactured stone veneer siding,installing manufactured stone interior,how to install manufactured stone veneer fireplace,cost of installing manufactured stone,codes for installing manufactured stone veneer,installing manufactured stone exterior,installing manufactured stone veneer,how to install manufactured stone veneer,installing manufactured stone exterior wall,installing manufactured stone veneer fireplace,installing manufactured stone siding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We firmly believe that by intimately perceiving and understanding the earth can we interpret a Conegliano Valdobbiadene high-quality Prosecco which is the symbol of our territory, of its aromas and flavours, but also of its history and its traditions.\nBiancaVigna was founded here, in the DOCG and DOC areas of the March of Treviso with its 30 hectare vineyard.\nBIANCAVIGNA HISTORY, A YOUNG TRADITION.\nBiancaVigna is a family owned and operated winery, managed by the two siblings Elena and Enrico Moschetta.\nIt all began with their paternal grandfather Genesio, born in 1903 and director of a local dairy, who farmed the vineyards belonged to his father Luigi and acquired few more hectares among which the historic \"Vigna di San Gallo\".\nLater on, during the 50' and 60', their father Luigi expanded his property in the traditional district of Conegliano Valdobbiadene and passes down his passion for the winemaking to his son Enrico who graduated at the Oenological School of Conegliano in 1984.\nAt first, their grapes were sold to the local wine cooperative and later to various wineries in Valdobbiadene.\nIn early 2000s Elena, thanks to her career as manager in multinational and multiutilities corporations, involves her brother Enrico, in his forties and well trained at premium wineries of the area, and her husband Luca Cuzziol with his siblings Bepi and Grazia, who are at the helm of a leading distribution company of fine wines: the project of BiancaVigna became real.\nIn 2008 they joined the European Rural Development Policy and provided the cellar with last-generation equipment to raise the quality of the wines and soon affirm their brand.\nToday, with almost 30 hectares (74 acres) of vineyards and 600.000 own-branded bottles produced, the Conegliano Valdobbiadene BiancaVigna is reviewed by the most influent Italian and international trade magazines and listed in the best restaurants world-wide.\nAfter fifteen years from its establishment, Elena and Enrico have turned their dream into reality thanks to their determination and a great teamwork.\nOUR PASSION FOR WINE BELONGS TO THIS LAND.\nThis is BiancaVigna. Values, innovative ideas and care of each single step with the ambition of making fine wines.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are more than 900 small business development centers in the United States. Sponsored by the Small Business Association, these centers are often run in partnership with state and local governments, colleges, and the private sector.\nCounselors are on hand to help an entrepreneur craft a business plan, fill out a loan application, or come up with a winning marketing strategy.\n\"The SBDCs exist to help entrepreneurs and small business owners obtain capital to support their companies, and make better decisions on key aspects of their business operations,\" says Antonio Doss, associate administrator for the Office of Small Business Development Centers, a group within the SBA that coordinates the program. \"The advisors at the SBDCs work with entrepreneurs to make the necessary adjustments to establish a business that's sustainable and profitable.\"\nSBDCs also assist companies in gaining access to capital\u2014a service that has been particularly crucial of late. In 2009 and again last year, SBDCs nationally helped clients secure funding in excess of $3.8 billion in the form of debt, equity, and grant money. \"It was surprising the amount of support SBDC's were able to get in the area of capital infusion,\" Doss says. \"We expected to see a downturn the last two years but our business advisors were able to help clients get additional money because of the relationships they have in the lending community.\"\nTo help identify the most active and innovative SBDCs around the country, a team of Inc.com reporters interviewed 45 entrepreneurs and small business advocates. The centers that stand out provide exceptional support for their local entrepreneurial communities while at the same time they serves as national models for effective small business advocacy.\nSee an interactive version of this map at Inc.\nThis post originally appeared at Inc.\nAn SBDC for First-Time Federal Contractors - Washington, D.C.\nThe District of Columbia Small Business Development Center prides itself on its ability to help business owners navigate the daunting and complicated road to becoming a federal government contractor.\nThe Georgia State University Small Business Development Center provides excellent leadership training through the award-winning Kauffman Foundation's FastTrac GrowthVenture program.\nThey have helped hundreds of small business owners become better managers.\nTo help small businesses nationwide\u2014as well as counselors at other SBDCs\u2014the Idaho Small Business Center at Boise State University has built a comprehensive website that links to a variety of helpful resources.\nLocated inside of a bank branch, the Inland Empire Small Business Development Center is California's largest SBDC, and one that prides itself on its ability to help clients gain access to capital.\nThe Iowa Western Community College Entrepreneurial & Small Business Development Center is a nexus of social media activity in the heartland.\nCounselors teach business owners about social media, search engine optimization, and other rapidly changing online marketing strategies.\nWhen the local RV industry fell on hard times, the North Central Indiana Small Business Development Center focused on helping businesses in the region spur innovation.\nThe center's Pre-Seed Workshop is designed to help researchers at the University of Notre Dame commercialize their ideas.\nNorth Central Kansas Small Business Development Center serves 11 counties with a population of about 125,000 people.\nThe SBDC's staff is helping farmers to diversify their revenue streams, bringing together women entrepreneurs, and providing training, mentoring and gap-financing for local businesses.\nBy specializing in facilitating trade relationships across the Mexican border, the San Antonio Small Business Development Center has helped entrepreneurs to generate $181 million in export-driven sales.\nFinancial literacy for entrepreneurs is at the heart of what Vancouver Small Business Development Center provides its clients.\nThe center's \"Profit Mastery\" series helps small businesses grapple with cash flow issues; the SBDC also works with business to find angels and other investors.\nThe Maryland Small Business Development Center, Central Region went from one of the worst performing SBDCs in Maryland to one of the best.\nA new director helped transform the center in less than a year\u2014and she is just getting started.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"08 AprWho Is a Data Scientist ?\nBeing a new introduction of the past decade, big data + data science is a bit of conundrum to organizations as HR departments need to put together new hiring processes or update older ones; privacy and security teams need to implement access frameworks across diverse infrastructures; data management and operations need to deal with the mammoth in the room (cloud in this case!); engineers and managers have to adapt their software engineering processes (agile data science, anyone?) while rest of the operations need to figure out if big data + data science is a cost centre or business unit! It is no wonder that the term data scientist still remains as enigmatic and all-encompassing as it was a few years ago when it was first coined.\nHowever, when you step back and look at some of the established roles in software industry, we define them by their function ahead of any skill-sets. We define a software engineer as one who designs and builds software and not by the IDE or the programming language he\/she uses. Similarly a product manager is defined as a person who sketches roadmaps and oversees execution of a product\/feature, but we don't differentiate them by the tools they use. In stark contrast, from inception of the role, we have focused much on the tools (R vs. Excel vs. Python), degrees (PhD vs. Masters) and topics (Statistics vs. Data Mining vs Machine learning) and less on the functional expectation of the role. I expect this to change with growing maturity of the field, and successful integration of big data and data science.\nA large portion of a data scientist's role is being an analyst, or rather an analyst 2.0. In analytics, typically the business has a well defined problem (e.g churn, acquisition, user segmentation etc) and is looking for scientific models to often understand correlations and\/or causality (good luck with that!). The advanced part is where a data scientist pushes these models to leverage big-data to not only detect but also predict a future state with some certainty (or uncertainty). This is where the business comes looking for answers to their questions !\nData scientists are data miners at heart, and a true data miner often wants to go off on a data expedition and discover unknown insights rather than proving or disproving existing hypotheses. Now, insights are not as interesting if they can not be acted upon. For example discovering that teens age 18-24 use your product is an insight, but its only actionable if we can also discover the right incentive that gets them to sign twice as fast. This is where a data scientist alerts the business and raises questions !\nFinally, this is the part of a data scientist's job where he doubles as an engineer. The role requires defining new products around data and steering the existing product towards becoming more intelligent and data-aware so as to provide seamless experience to the users. This is where a data scientist starts to define the business!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Success Doesn't Mean Suffering | AHA!\nSunday, April 25th, 2010 at 8:30 AM: happiness, Heart, inner work, insight, Inspiration, Inspiration\/Heart, intention, life choices, personal development, personal growth, personal growth. spirituality, Professional Development, professional growth, professionald development, secrets, self awareness, self development, spirituality.\ni love reading your blogs. I agree\u2026 operating from joy is paramount to being able to sustain success\u2026 without it things go flat . I am recommitting myself to joy and that includes riding the wave of fear that may arise when challenges look out of control\u2026 and that riding the wave inevitably lands back on the shore of opportunity. thank you so much for your joy and commitment to life!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Phase2 Technology collaborated with our colleagues at MediaCurrent to produce a high quality report entitled, \"Building Large-Scale Publishing Sites with Drupal.\" The following introductory post is by Dave Terry from MediaCurrent. We wanted to share it on our website as well.\nMediacurrent and Phase2 Technology are proud to co-release and announce the availability of a white paper titled Building Large-Scale Publishing Sites with Drupal. This was a collaborative effort between two Drupal-centric firms who share a common interest in building world-class Drupal sites for the publishing industry. The impetus started months ago when we both recognized there was a need to help educate publishers about Drupal.\n* How Drupal aligns with publisher's web needs.\nAt its core, the paper details the emergence of Drupal as the leader for enterprise-level publishing sites. It is certainly no secret that the publishing paradigm has turned upside-down. Particularly, the vertical has struggled to come up with viable monetization strategies as advertising dollars shift from print to online. Both Mediacurrent and Phase2 believe that Drupal's lower total cost of ownership comes at a critical juncture. As print publications and newspapers struggle to stay afloat, the web continues to evolve at a breakneck pace. But Drupal can help during these transitional times. For example, we explore how one of the oldest magazines in the US, The New Republic, successfully migrated onto Drupal. With Drupal, publishers are appreciating the virtues of open-source: no licensing fees, no vendor lock-in, and an unparalleled ability for customization.","meta":{"redpajama_set_name":"RedPajamaC4"}}